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Dominant group/Literature

Literature is the art of written works, and is not bound to published sources (although, under circumstances unpublished sources can be exempt). The two major classification of literature are poetry and prose. Others exclude all genres such as romance, crime and mystery, science fiction, horror and fantasy.

The theory of dominant group with respect to literature falls into at least two situations: a dominant group of literature or a dominant group associated with literature.

Dominant group
Examples from primary sources are to be used to prove or disprove each hypothesis. These can be collected per subject or in general.

 * Accident hypothesis: dominant group is an accident of whatever processes are operating.

 * Artifact hypothesis: dominant group may be an artifact of human endeavor or may have preceded humanity.

 * Association hypothesis: dominant group is associated in some way with the original research.

 * Bad group hypothesis: dominant group is the group that engages in discrimination, abuse, punishment, and additional criminal activity against other groups. It often has an unfair advantage and uses it to express monopolistic practices.

 * Control group hypothesis: there is a control group that can be used to study dominant group.
 * Entity hypothesis: dominant group is an entity within each field where a primary author of original research uses the term.
 * Evolution hypothesis: dominant group is a product of evolutionary processes, such groups are the evolutionary process, produce evolutionary processes, or are independent of evolutionary processes.
 * Identifier hypothesis: dominant group is an identifier used by primary source authors of original research to identify an observation in the process of analysis.
 * Importance hypothesis: dominant group signifies original research results that usually need to be explained by theory and interpretation of experiments.
 * Indicator hypothesis: dominant group may be an indicator of something as yet not understood by the primary author of original research.
 * Influence hypothesis: dominant group is included in a primary source article containing original research to indicate influence or an influential phenomenon.
 * Interest hypothesis: dominant group is a theoretical entity used by scholarly authors of primary sources for phenomena of interest.
 * Metadefinition hypothesis: all uses of dominant group by all primary source authors of original research are included in the metadefinition for dominant group.
 * Null hypothesis: there is no significant or special meaning of dominant group in any sentence or figure caption in any refereed journal article.
 * Object hypothesis: dominant group is an object within each field where a primary author of original research uses the term.

 * Obvious hypothesis: the only meaning of dominant group is the one found in Mosby's Medical Dictionary.
 * Original research hypothesis: dominant group is included in a primary source article by the author to indicate that the article contains original research.
 * Primordial hypothesis: dominant group is a primordial concept inherent to humans such that every language or other form of communication no matter how old or whether extinct, on the verge of extinction, or not, has at least a synonym for dominant group.
 * Purpose hypothesis: dominant group is written into articles by authors for a purpose.
 * Regional hypothesis: dominant group, when it occurs, is only a manifestation of the limitations within a region. Variation of those limitations may result in the loss of a dominant group with the eventual appearance of a new one or none at all.
 * Source hypothesis: dominant group is a source within each field where a primary author of original research uses the term.
 * Term hypothesis: dominant group is a significant term that may require a 'rigorous definition' or application and verification of an empirical definition.

Literature
Def. "the body of [all] written work" is called literature.

The theory of literature involves methods of studying and investigating literature, its "nature and function"; "literary theory, criticism, and history; and general, comparative, and national literature."

Def. “[t]he theory or the philosophy of the interpretation of literature and literary criticism” is called literary theory.

Writings
His bringing "writing center practice into line with the authorized knowledge about writing, and his widely followed stricture that tutors are to support the classroom teacher's position completely is clear evidence of how writing centers do not escape domination. Yet one of the benefits of being excluded from the dominant group is that in this position one has less to protect and less to lose."

Abstractions
Def.
 * 1) a "separation from worldly objects",
 * 2) "the withdrawal from one's senses",
 * 3) the "act of focusing on one characteristic of an object rather than the object as a whole group of characteristics; the act of separating said qualities from the object or ideas",
 * 4) the "act of comparing commonality between distinct objects and organizing using those similarities; the act of generalizing characteristics; the product of said generalization",
 * 5) an "idea of an unrealistic or visionary nature", or
 * 6) any "generalization technique that ignores or hides details to capture some kind of commonality between different instances for the purpose of controlling the intellectual complexity of engineered systems, particularly software systems"

is called an abstraction.

The image on the right is an example of the literature of abstraction.

Adventures
Adventure fiction is a genre of fiction in which an adventure, an exciting undertaking involving risk and physical danger, forms the main storyline.

"An adventure is an event or series of events that happens outside the course of the protagonist's ordinary life, usually accompanied by danger, often by physical action. Adventure stories almost always move quickly, and the pace of the plot is at least as important as characterization, setting and other elements of a creative work."

Comedies
"Thus, the use of stereotypes in popular fictional forms such as situation comedies may be rather less unambiguously a reflection of dominant group views than Dyer suggests."

"Mean scores for number of Smile and Laughter responses in telic and paratelic state-dominant groups throughout six successive 100 sec periods during exposure to comedy."

"Yet the situation of women is more complex because of their close involvement with members of the dominant group, which has blurred the boundaries between "us" and "them.""

Comparative literature
"Without doubt, multiculturalism is preferable to the monoculturalist oppression of minorities by the dominant group."

Criticism
Def. “[t]he study, discussion, evaluation, and interpretation of literature” is called literary criticism.

"But my point is that one constant within this struggle remains: that an oppositional culture of non-dominant groups has to define itself against the practices and ideology of the dominant group (or groups), and this inevitably has consequences for form. Indeed, only a very unsophisticated literary criticism could conceive of form and content as distinct entities."

Compositions
A newspaper, or online, feature article is composed of the following:
 * 1) a lede,
 * 2) topic sentence,
 * 3) a body, and an
 * 4) ending.

The ratio of each of these may depend on the audience. In an inverted pyramid style the ratios are about 5:3:2 for lead (including topic sentence), body, and ending.

There is also what's called a "news-peg" or "hook", something that will interest a reader, usually the first sentence or the title.

The following elements should be present: What, When, Where, Why, Who, and How. Nearly all of these elements must appear somewhere in the story.

Crimes
"Accounts of true crime have always been enormously popular among readers. The subgenre would seem to appeal to the highly educated as well as the barely educated, to women and men equally. The most famous chronicler of true crime trials in English history is the amateur criminologist William Roughead, a Scots lawyer who between 1889 and 1949 attended every murder trial of significance held in the High Court of Justiciary in Edinburgh, and wrote of them in essays published first in such journals as The Juridical Review and subsequently collected in best-selling books with such titles as Malice Domestic, The Evil That Men Do, What Is Your Verdict?, In Queer Street, Rogues Walk Here, Knave's Looking Glass, Mainly Murder, Murder and More Murder, Nothing But Murder, and many more…. Roughead's influence was enormous, and since his time "true crime" has become a crowded, flourishing field, though few writers of distinction have been drawn to it."

Debates
Def. “a type of literary composition, taking the form of a discussion or disputation” is called debate.

"On the other hand, it served to maintain the Moluccans' challenge of and resistance to the dominant group's definitions. These examples indicate that it is important to examine the actual use of the notion of 'essentialism' in argument and debate." Debate occurs in public meetings, academic institutions, and legislative assemblies.

Debating is also carried out for educational and recreational purposes, usually associated with educational establishments and debating societies.

Debating topics covered a broad spectrum of topics while the debating societies allowed participants from both genders and all social backgrounds, making them an excellent example of the enlarged public sphere of the Age of Enlightenment. Debating societies were a phenomenon associated with the simultaneous rise of the public sphere, a sphere of discussion separate from traditional authorities and accessible to all people that acted as a platform for criticism and the development of new ideas and philosophy.

John Henley, a clergyman, founded an Oratory in 1726 with the principal purpose of "reforming the manner in which such public presentations should be performed." He made extensive use of the print industry to advertise the events of his Oratory, making it an omnipresent part of the London public sphere. Henley was also instrumental in constructing the space of the debating club: he added two platforms to his room in the Newport district of London to allow for the staging of debates, and structured the entrances to allow for the collection of admission. These changes were further implemented when Henley moved his enterprise to Lincoln's Inn Fields. The public was now willing to pay to be entertained, and Henley exploited this increasing commercialization of British society. By the 1770s, debating societies were firmly established in London society.

Dramas
Def. “[a] composition, normally in prose, telling a story and intended to be represented by actors impersonating the characters and speaking the dialogue” is called a drama.

"Through the sape, there develops what James Scott (1990), in his brilliant essay on resistance strategies in subcultures, calls "hidden transcripts"-a series of disguised messages and attitudes representing a hidden critique of the dominant group's authority."

"Whatever multiplicity of expectations the public may have prior to their first experience with the drama, this system of signs tends to reduce them towards a sameness which conforms with the dominant group's notion of social and artistic "validity" as incorporated into the theater design."

"On the one hand, a theatre seen as part of the unfolding social revolution involves theatre as a catalyst and a pusher in new directions which may not (in this case) represent the interests of an elite or dominant group."

Engineering
Def. the "application of [mathematics and the physical sciences] to the needs of humanity and the development of technology" is called engineering

On the right, as an example of engineering literature, is a cover image of the MuMETAL catalog where engineers, metal suppliers and fabricators have referred to mumetal as the industry standard.

Eroticas
Def. literature "relating to or tending to arouse sexual desire or excitement" is called erotic literature, or erotica.

On the left is a cover image from the new erotica e-book by Elizabeth Black called "PURR a Puss 'n Boots Twisted Tale".

"Here, for a fleeting moment (or occasionally even an entire evening), the existing social/sexual/economic power arrangements are challenged, where the client (who under “normal” circumstances has membership among the hegemonic, socially rewarding, dominant group of sexually conforming elites) temporarily crosses over the dichotomized chasm into the other world, and seeks temporary acceptance among those representing sexually challenging, alternative, erotic communities."

Essays
Def. “[a] written composition of moderate length exploring a particular issue or subject” is called an essay.

An essay is a piece of writing which is often written from an author's personal point of view.

Fantases
Def. a "literary genre generally dealing with themes of magic and [fictive] medieval technology" is called fantasy.

Fictions
Def. a "[l]iterary type using invented or imaginative writing, instead of real facts, usually written as prose” is called fiction.

Geography
At the right is an allegorical painting in the Pinacoteca do Estado de São Paulo, Brazil, 1915.

It is one among many connected to the literature of the time preceded by a long and baroque heritage expressed even in the final years of the nineteenth century in various regions and in art and culture.

On the left is a form of geographic literature consisting of a geographic map showing the locations of Turkic peoples in and around Xinjiang, formerly East Turkestan.

History
Def. “[a] [written] record or narrative description of past events” is called history.

On the right is an illustration of the "Hof van Beselare" in Flanders, Belgium, from the history book, Flandria Illustrata by Antoon Sanders.

Horrors
Def. a "genre of fiction, meant to evoke a feeling of fear and suspense" is called horror.

The Mammoth Book of Best New Horror specifically and exclusively focuses on and publishes current horror fictional literature.

"And like non-disabled women, they have a darker side; they are evil, depraved. They function as symbolic scapegoats for the dominant group, and hence the latter may feel horror and disgust and avert their eyes-or stare."

Humanities
On the right is a graph of Digital Humanities Twitter users with 1,434 nodes and 137,061 directed edges that each symbolize a user following another user.

Humor
Def. the "quality of being amusing, comical, funny" is called humor, or humour.

Intellects
On the right, as an example of intellect literature, is an image of the Issue 2 cover of Intellect Magazine.

Def. "the faculty of thinking, judging, abstract reasoning, and conceptual understanding; the cognitive faculty" is called an intellect.

Laws
Def. “[a] written or understood rule that concerns behaviors and the appropriate consequences thereof” is called law.

Literary history
"To justify the existence and the methods of literary history is entirely superfluous nowadays, and it is no less superfluous to dwell upon the differences and likenesses between it and literary criticism. Our common sense tells us, if we do away with prejudices and futile scholarly discussions, that literary history, working in its own field, is trying neither to replace nor to oppose literary criticism. Literary history thinks that it can help literary criticism; can clear a path for it; can lighten its task of understanding, judging, and classifying literary works and the great movements of human thought. It offers its services as a devoted auxiliary, modest and self-effacing. It has no imperialistic designs: it covers enough territory already to have no need to encroach on that of a neighbor. It prepares the material for the critic but puts no restrictions on the way [she] should use it. If he has faith in impressionistic criticism, if [she] believes that the literary critic should surrender himself to the emotion produced by the book [she] is studying and then should express this emotion with precision and delicacy, he is free to do so. Literary history asks [her] only to base his personal reaction on facts that have been historically verified, to define [her] position clearly, and, when communicating a purely personal reaction to the public, not to believe or to make others believe that he is giving any added information about the work or its writer. "Impressionism", says Lanson, "is the only method that puts us in touch with beauty. Let us, then, use it for this purpose, frankly, but let us limit it to this, rigorously. To distinguish knowing from felling, what we may know from what we should feel; to avoid feeling when we can know, and thinking that we know when we feel: to this, it seems to me, the scientific method of literary history can be reduced."1"

Sweden
The Rök runestone is considered the first piece of written Swedish literature and thus it marks the beginning of the history of Swedish literature.

It was proposed that the inscription has nothing at all to do with the recording of heroic sagas and that it contains riddles which refer only to the making of the stone itself.

The Rök runestone inscription is not connected to heroic deeds in war. Instead it deals with the conflict between light and darkness, warmth and cold, life and death.

"The Rök runestone from central middle Sweden, [is] dated to around 800 CE [...] Combining perspectives and findings from semiotics, philology, archaeology, and history of religion, the study presents a completely new interpretation which follows a unified theme, showing how the monument can be understood in the socio-cultural and religious context of early Viking Age Scandinavia. The inscription consists, according to the proposed interpretation, of nine enigmatic questions. Five of the questions concern the sun, and four of them, it is argued, ask about issues related to the god Odin. A central finding is that there are relevant parallels to the inscription in early Scandinavian poetry, especially in the Eddic poem Vafþrúðnismál."

Mysteries
Def. a "suspenseful, sensational genre of story, book, play or film", such as a "detective story, mystery novel, whodunit, crime fiction", is called a thriller.

"The scripts for the series [I Love a Mystery] were usually themed towards the dark and supernatural, with perhaps the most famous, or infamous (depending on your point of view) scenario being “Temple of the Vampires,” which aroused a great deal of censorial comment when first broadcast as a twenty-episode serial from January 22 through February 16, 1940."

From the Mysteries of the Human Journey, Hegemony: "The persuasion of subordinates to accept the ideology of the dominant group by mutual accommodations that nevertheless preserve the rulers' privileged position."

Narratives
Def. "the systematic recitation of an event or series of events" is called a narrative.

A narrative is a constructive format (as a work of speech, writing, song, film, television, video games, photography or theatre) that describes a sequence of non-fictional or fictional events.

Further, the word "story" may be used as a synonym of "narrative", but can also be used to refer to the sequence of events described in a narrative. A narrative can also be told by a character within a larger narrative. An important part of narration is the narrative mode, the set of methods used to communicate the narrative through a process narration.

Natural sciences
Def. a written work “studying phenomena or laws of the physical world” is called natural science.

An example of a natural science, or sciences, journal is shown on the right with its cover image.

Novels
"Their experience is encoded in the dominant culture as that of exotic Other, "foreigners," as Ralph Connor revealingly titled his novel of immigration in the appropriate(d) discourse. This dominant group has framed the grounds for discussion of a "national literature.""

"This point is crucial to understanding Donald's internalized racism and the novel's resistance to it: the dominant group obtains the consent of the subordinated group not by compulsion but by seduction."

"There is another distinction to be made in considering the Afro-American novel. Baker speaks of experiences in which the dominant group overtly discriminates against the black society and unabashedly allows distinctions that prove its superiority."

Philosophies
Def. a written work “that seeks truth through reasoning rather than empiricism” is called philosophy.

"The Tengrism tangraïsme or sometimes (in Mongolian: Тэнгэр шүтлэг, Tenger shütleg, worship (or religion) of heaven) was the major belief of Xiongnu and Xianbei which consisted of the Turkish population, Mongolian, Hungarian and Bulgarian in antiquity. It focuses around the divinity of the eternal sky Tengri (also transliterated Tangri, Tanrı, Tangra, etc.), and incorporates elements of shamanism, animism, totemism and ancestor worship." Translated using Google Translate.

The Daychin Tengry - Turkic pagan god is shown in the image on the right.

That a text of this philosophy, or religion, probably existed is suggested by the stamp on the left used by Güyük Khan, in a letter of 1246.

Plays
Def. "[a] literary composition, intended to be represented by actors impersonating the characters and speaking the dialogue" or "a theatrical performance featuring actors" is called a play.

Poems
Def. a literary piece written in verse” is called a poem.

Poetrys
"Only those aspects of the minority culture that overlap the dominant culture are recognized by the dominant group."

"Their attitudes toward historical fact are complicated, but not because they are a muted group within a dominant group."

"It began with a rejection of the dominant group and a recognition of acceptance of blackness. In the enumeration and praising of black qualities, it reached its height in an "unfolding" common to both black American and black African poetry."

Romances
The "popularity of old-fashioned romance novels featuring conventional and traditional gender roles seems to defy the stances of the modern-day women's liberation movement."

A romance novel might be characterized as a "hyper-romantic, contrived and extremely unrealistic tales of handsome, manly heroes falling in love with virginal women, enduring a series of adventures, then inexorably ending in a happy resolution."

“Romance novels offer an escape from daily life with the belief that true love really exists.”

"Romance novels [portrayed by the partial cover image on the right] are at once the most scorned and popular form of literature in the world, accounting for as much as 40% of total book sales in much of the world. The average romance reader (and writer) is female, ambitious, leads a very full and busy life, and has an above-average education and intelligence. The livelihood of some of the world’s most critically-acclaimed (mostly male) authors depends on the revenue base generated from the sale of the remarkably diverse genre called ‘romance’, written by and bought overwhelmingly by women."

Science fictions
Def.' "[f]iction in which advanced technology and/or science is a key element" is called science fiction.

"Nerds also collect objects connected with knowledge (atlases and maps; mathematical and scientific equipment such as telescopes and slide rules; etc.), and are avid science fiction fans. ... While they arguably represent a privileged and dominant group, many must reconcile this status with their experience of themselves as relatively powerless, or even subjugated, in their everyday lives."

"All the participants in a dominant culture do not necessarily belong to a dominant group. ... This liquidation is the principal subject of Lovecraft's opus, as I tried to show in "Entre le Fantastique et la Science-Fiction, Lovecraft," Cahiers de l'Herne No. 12 (1969)."

"When groups see themselves as opposed, competing for the same resources, subordinate groups may view the dominant group as cold, exploiting, cruel, and arrogant. ... This is the case in the example at the end of this article, where-it is argued that "Aliens" in recent science fiction films are—among other things—figures for actual historical aliens who enter US borders, legally or illegally."

Sciences
On the right is a cover image for performing health sciences literature reviews.

Hypotheses

 * 1) Usually literature serves the purposes of the dominant group to maintain itself as such.
 * 2) Occasionally literature serves the better good of us all.
 * 3) Literature may also exist from other species or ancestors of hominins.

title: Dominant group/Literature


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User:Rf~enwikiversity/Economics cruft

This is a list of pages which have essentially no useful content and (for the most part) have not been edited in a long time.

 * Capital budgeting
 * Category:Entrepreneurial economics
 * Category:Environmental economics
 * Category:Mathematical economics
 * Category:Measuring the mindedness of the electorate in 2008
 * Debt and money markets
 * Developmental economics
 * Economic indicators
 * Economy of Sierra Leone
 * Engineering at Udayana University
 * Fatih University
 * Free Market Economics
 * Gift economy
 * Hedonic modelling archive
 * International economics
 * Introduction to Macroeconomics
 * Introduction to econometrics
 * Introduction to mathematical economics
 * Land Use and Private Property
 * Measuring the mindedness of the electorate in 2008
 * Personal Finance
 * School:Economics/Game Theory/Games
 * School:Economics/Game theory
 * Statistical economics
 * Topic:Behavioral economics
 * Topic:Economic research
 * Topic:Environmental economics
 * Topic:Free Market Economics
 * Topic:Labour economics
 * Topic:Managerial economics
 * Topic:Microeconomics
 * Topic:Positive economics
 * Topic:Socioeconomics
 * Udayana university
 * Vertical farming
 * Topic:Basics of Economics
 * Eric Von Hippel:Democratizing Innovation
 * Topic:Democratizing innovation
 * The Inevitability of Globalism
 * Wikiworld

Incubate
These pages might have some content, but lack the structure or completeness to be able to stand on their own. They should be expanded or integrated into some other project, or maybe deleted anyway.

 * Neil Postman: Five Things We Need to Know About Technological Change
 * Chemistry and its socioeconomic impact
 * Consumer Theory
 * Game domain model
 * The Tragedy of the Commons

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

title: User:Rf~enwikiversity/Economics cruft


text: Category:Wikiversity official processes

title: Category:Wikiversity official processes


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Linux/mdadm

mdadm is a Linux utility used to manage software RAID devices.

The name is derived from the md (multiple device) device nodes it administers or manages, and it replaced a previous utility mdctl. The original name was "Mirror Disk", but was changed as the functionality increased.

It is free software licensed under version 2 or later of the GNU General Public License - maintained and copyrighted to Neil Brown of Suse.

Types of physical device
mdadm can handle anything which presents to the kernel as a block device. This can encompass whole disks (/dev/sda), partitions (/dev/sda1) and USB flash drives.

RAID Configurations

 * RAID 0 - Block level striping. MD can handle devices of different lengths, the extra space on the larger device is then not striped.
 * RAID 1 - Mirror.
 * RAID 4 - Like RAID 0, but with an extra device for parity.
 * RAID 5 - Like RAID 4, but with the parity distributed across all devices.
 * RAID 6 - Like RAID 5, but with two parity segments per stripe.
 * RAID 10 - Take a number of RAID 1 mirrorsets and stripe across them RAID 0 style.

Non-RAID Configurations

 * LINEAR - Concatenate a number of devices into a single large MD device.
 * MULTIPATH - Provide multiple paths with failover to a single device.
 * FAULTY - A single device which emulates a number of disk fault scenarios for testing and development.
 * CONTAINER - A group of devices managed as one, in which RAID systems can be built.

Types of MD device
The original (standard) form was /dev/mdn where n is a number between 0 and 99. More recent kernels have supported the use of names such as /dev/md/Home. Under kernel 2.4 and earlier these two were the only options. Both of them are non-partitionable.

From kernel 2.6 a new type of MD device was introduced, a partitionable array. The device names were modified by changing md to md_d. The partitions were identified by adding pn; thus /dev/md/md_d2p3 for example.

From kernel 2.6.28 non-partitionable arrays can be partitioned, the partitions being referred to in the same way as for partitionable arrays: /dev/md/md1p2.

Booting
Since support for MD is found in the kernel, there is an issue with using it before the kernel is running. Specifically it will not be present if the boot loader is either (e)LiLo or GRUB legacy. It may not be present for GRUB 2. In order to circumvent this problem a /boot filesystem must be used either without md support, or else with RAID1. In the latter case the system will boot by treating the RAID1 device as a normal filesystem, and once the system is running it can be remounted as md and the second disk added to it. This will result in a catch-up, but /boot filesystems ought to be small.

Create an array
mdadm --create /dev/md0 --level=mirror --raid-devices=2 /dev/sda1 /dev/sdb1

Create a RAID 1 (mirror) array from two partitions. If the partitions differ in size, the array is the size of the smallest partition. You can create a RAID 1 array with more than two devices. This gives you multiple copies. Whilst there is little extra safety in this, it makes sense when you are creating a RAID 5 array for most of your disk space and using RAID 1 only for a small /boot partition. Using the same partitioning for all member drives keeps things simple.

mdadm --create /dev/md1 --level=5 --raid-devices=3 /dev/sda2 /dev/sdb2 /dev/sdc2

Create a RAID 5 volume from three partitions. If the partitions used in your RAID array are not the same size, mdadm will use the size of the smallest from each partition. If you receive an error, such as: "mdadm: RUN_ARRAY failed: Invalid argument", make sure your kernel supports (either via a module or by being directly compiled in) the raid mode you are trying to use. Most modern kernels do, but you never know...

It is possible to create a degraded mirror, with one half missing by replacing a drive name with "missing": mdadm --create /dev/md1 --level=1 --raid-devices=2 /dev/sdb1 missing

The other half mirror is added to the set thus: mdadm --manage /dev/md1 --add /dev/sda1

This is useful when you are adding a disk to a computer which currently isn't mirrored. The new drive is... The computer is then booted off the secondary drive (or a rescue disk), the now idle original disk can be repartitioned if required (no need to format), and then the primary drive submirrors are added.
 * partitioned to match the first (unless you are also repartitioning too)
 * turned into a set of "half-mirrors"
 * formatted with appropriate file system
 * mounted
 * the data is copied over,
 * made bootable
 * its grub config and  mounts changed

Note that the partition types should be changed to 0xFD with fdisk to indicate that they are mirrored devices.

Replace failing disk
First mark the failing disk failed....

mdadm --manage /dev/md127 --fail /dev/sdc1

Then remove it from the array

mdadm --manage /dev/md127 --remove /dev/sdc1

After which it is safe to power down and physically replace the disk.

Then create partition as needed, and add new disk to the array;

mdadm --manage /dev/md127 --add /dev/sdc1

Then check /proc/mdadm for the sync of the device

Recording the array
mdadm --detail /dev/md0

View the status of the multi disk array md0.

mdadm -Es | grep md0 >>/etc/mdadm/mdadm.conf -E examine -s scan

This adds md0 to the configuration file so that it is recognised next time you boot.

You may wish to keep a safe copy of /proc/mdstat. The information will allow you to restart the array manually if mdadm fails to do so.

Growing an array by adding devices
mdadm --add /dev/md1 /dev/sdd1 mdadm --grow /dev/md1 --raid-devices=4

This adds the new device to the array then grows the array to use its space.

In some configurations you may not be able to grow the array until you have removed the internal bitmap. You can add the bitmap back again after the array has been grown. mdadm --grow /dev/md1 -b none mdadm --grow /dev/md1 -b internal

Growing an array by upgrading devices
An array may be upgraded by replacing the devices one by one, either as a planned upgrade or ad hoc as a result of replacing failed devices.

mdadm /dev/md1 --fail /dev/sda1 (replace the first drive with the new, larger one then partition it) mdadm --add /dev/md1 /dev/sda1

Allow the new drive to resync. If replacing all the devices repeat the above for each device, allowing the array to resync between repetitions. Finally, grow the array to use the maximum space available and then grow the filesystem(s) on the RAID array to use the new space.

mdadm --grow /dev/md1 --size=max

Deleting an array
mdadm --stop /dev/md0 # to halt the array mdadm --remove /dev/md0 # to remove the array mdadm --zero-superblock /dev/sd[abc]1 # delete the superblock from all drives in the array (edit /etc/mdadm/mdamd.conf to delete any rows related to deleted array)

Convert an existing partition to RAID 5
Assume that the existing data is on /dev/sda1:

mdadm --create /dev/md1 --level=5 --raid-devices=3 missing /dev/sdb2 /dev/sdc2 mdadm -Es >>/etc/mdadm/mdadm.conf update-initramfs -u dd if=/dev/sda1 of=/dev/md1 (add /dev/md1 to your boot loader's menu) (reboot into /dev/md1) mdadm --add /dev/md1 /dev/sda1 (update your boot loader)

Notes:
 * 1) A partition may be given as missing to act as a placeholder so that it can be added later.
 * 2) The /boot directory should be elsewhere, possibly on /dev/md0 or its own partition.
 * 3) If the reboot fails, do NOT add /dev/sda1 into the array until the problem is corrected!

Known problems
A common error when creating RAID devices is that the dmraid-driver has taken control of all the devices that are to be used in the new RAID device. Error-messages like this will occur:

mdadm: Cannot open /dev/sdb1: Device or resource busy

Typically, the solution to this problem involves adding the "nodmraid" kernel parameter to the boot loader config. Another way this error can present itself is if the device mapper has its way with the drives. Issue 'dmsetup table' see if the drive in question is listed. 'dmsetup remove ' will remove the drive from device mapper and the "Device or resource busy" error will go away as well.

RAID already running
First check if the device isn't in use in another array: cat /proc/mdstat

Probably you will have to stop the array with: mdadm --stop /dev/

Check the /etc/mdadm/mdadm.conf file (and restart system if possible): vi /etc/mdadm/mdadm.conf

Then you should be able to delete the superblock of this device: mdadm --misc --zero-superblock /dev/sdxN

Now the device shouldn't be busy any more.

Sometimes dmraid "owns" the devices and will not let them go. There is a solution.

Tweaking the kernel
To solve this problem, you need to build a new initrd without the dmraid-driver. The following command does this on a system with the "2.6.18-8.1.6.el5"-kernel: mkinitrd --omit-dmraid /boot/NO_DMRAID_initrd-2.6.18-8.1.6.el5.img 2.6.18-8.1.6.el5

After this, the system has to be rebooted with the new initrd. Edit your /boot/grub/grub.conf to achieve this.

Alternatively if you have a self customized and compiled kernel from a distro like Gentoo (the default option in gentoo) which doesn't use initrd then check kernel .conf file in /usr/src/linux for the line
 * 1) CONFIG_BLK_DEV_DM is not configured

If the above line is set as follows: CONFIG_BLK_DEV_DM=yes

then You might have to disable that option, recompile the kernel, put it in /boot and finally edit grub conf file in /boot/grub. PLEASE be careful NOT to disable CONFIG_BLK_DEV_MD=yes

(Note the MD instead of DM) which is essential for raid to work at all!

If both methods have not helped you then booting from live CD probably will (the below example is for starting a degraded raid-1 mirror array and adding a spare hdd to it and syncing. Creating a new one shouldn't be more difficult because the underlying problem was 'Device or resource busy' error): modprobe raid1 mknod /dev/md1 b 9 1 mknod /dev/md3 b 9 3 mdadm --assemble /dev/md1 /dev/hda1 mdadm --assemble /dev/md3 /dev/hda3 mdadm --add /dev/md1 /dev/hdb1 mdadm --add /dev/md3 /dev/hdb3

It might be easier to try and automatically assemble the devices mdadm --assemble --scan

Remember to change the corresponding md* and hd* values with the corresponding ones from your system. You can monitor the sync progress using: cat /proc/mdstat

When the sync is done you can reboot in your Linux normally.

Zeroing the superblock
Another way to prevent the kernel autostarting the raid is to remove all the previous raid-related information from the disks before proceeding with the creation, for example: mdadm --stop /dev/md0 mdadm --zero-superblock /dev/sd[abcd]1

And now the usual create, for example: mdadm --create /dev/md0 --level=5 --raid-devices=4 --spare-devices=0 /dev/sd[abcd]1

Recovering from a loss of raid superblock
There are superblocks on the drives themselves and on the raid (apparently). If you have a power failure, hardware failure, that does not include the drives themselves, and you cannot get the raid to recover in any other way, and wish to recover the data, proceed as follows:

Get a list of the devices in the raid in question: mdadm --detail /dev/md[x]

Result something like this:

/dev/md127: Version : 1.2 Creation Time : Sun Aug 21 23:35:28 2011 Raid Level : raid6 Array Size : 7814047744 (7452.06 GiB 8001.58 GB) Used Dev Size : 1953511936 (1863.01 GiB 2000.40 GB) Raid Devices : 6 Total Devices : 4 Persistence : Superblock is persistent Update Time : Sun Jan 1 11:43:17 2012 State : clean, degraded Active Devices : 4 Working Devices : 4 Failed Devices : 0 Spare Devices : 0 Layout : left-symmetric Chunk Size : 512K Name : clop:1 (local to host clop) UUID : 7ee1e93a:1b011f80:04503b8d:c5dd1e23 Events : 62 Number  Major   Minor   RaidDevice State 0      8       33        0      active sync   /dev/sdc1 1      8       49        1      active sync   /dev/sdd1 2      8       81        2      active sync   /dev/sdf1 3      8       65        3      active sync   /dev/sde1 4      0        0        4      removed 5      0        0        5      removed

RaidDevice order (sdc1,sdd1,sdf1,sde1) and Chunk Size are critical

Record all your raid member parameters: mdadm --examine /dev/sd[abcde...]1 | egrep 'dev|Update|Role|State|Chunk Size'

Look carefully at the Update time. If you have raid members attached to the motherboard and others attached to a raid card, and the card fails, but leaves enough members to keep the raid alive, you want to make a note of that. Look at Array State and Update Time. For example:

/dev/sdc1: Update Time : Wed Jun 15 00:32:35 2011 Array State : AAAA.. ('A' == active, '.' == missing) /dev/sdd1: Update Time : Thu Jun 16 21:49:27 2011 Array State : .AAA.. ('A' == active, '.' == missing) /dev/sde1: Update Time : Thu Jun 16 21:49:27 2011 Array State : .AAA.. ('A' == active, '.' == missing) /dev/sdf1: Update Time : Thu Jun 16 21:49:27 2011 Array State : .AAA.. ('A' == active, '.' == missing) /dev/sdk1: Update Time : Tue Jun 14 07:09:34 2011 Array State : ....AA ('A' == active, '.' == missing) /dev/sdl1: Update Time : Tue Jun 14 07:09:34 2011 Array State : ....AA ('A' == active, '.' == missing)

Devices sdc1, sdd1, sde1 and sdf1 are the last members in the array and will rebuild correctly. sdk1 and sdl1 left the array (in my case due to a raid card failure).

Also note the raid member, starting with 0, the raid needs to be rebuilt in the same order. Chunk size is also important.

Zero the drive superblocks

mdadm --stop /dev/md0 # to halt the array mdadm --remove /dev/md0 # to remove the array mdadm --zero-superblock /dev/sd[cdefkl]1

Reassemble the raid

mdadm --create /dev/md1 --chunk=4096 --level=6 --raid-devices=6 /dev/sdc1 /dev/sdd1 /dev/sdf1 /dev/sde1 missing missing

'missing' tell the create command to rebuild the raid in a degraded state. sdk1 and sdl1 can be added later

Edit /etc/mdadm.conf and add an ARRAY line with a UUID. First get the UUID for your raid:

mdadm -D /dev/md

then:

nano /etc/mdadm.conf

and add something similar to the file (notice there is no # in front of the active line you are adding

ARRAY /dev/md1 UUID=7ee1e93a:1b011f80:04503b8d:c5dd1e23
 * 1) ARRAY /dev/md0 UUID=3aaa0122:29827cfa:5331ad66:ca767371
 * 2) ARRAY /dev/md1 super-minor=1
 * 3) ARRAY /dev/md2 devices=/dev/hda1,/dev/hdb1

Save with

Last, mark the array possilbly dirty with: mdadm --assemble /dev/md1 --update=resync

Monitor the rebuild with

watch -n 1 cat /proc/mdstat

All your data should be recovered!

You can also use Netdata to monitor your mdadm disk arrays.

Increasing RAID ReSync Performance
In order to increase the resync speed, we can use a bitmap, which mdadm will use to mark which areas may be out-of-sync. Add the bitmap with the grow option like below : mdadm -G /dev/md2 --bitmap=internal

 * Note: mdadm - v2.6.9 - 10th March 2009 on Centos 5.5 requires this to be run on a stable "clean" array. If the array is rebuilding the following error will be displayed:

 * mdadm: failed to set internal bitmap.

 * And the following line is added to the log file:
 * md: couldn't update array info. -16

then verify that the bitmap was added to the md2 device using cat /proc/mdstat

you can also adjust Linux kernel limits by editing these files /proc/sys/dev/raid/speed_limit_min and /proc/sys/dev/raid/speed_limit_max

You can also edit this with the sysctl utility sysctl -w dev.raid.speed_limit_min=50000

Increasing RAID5 Performance
To help RAID5 read/write performance, setting the read-ahead & stripe cache size for the array provides noticeable speed improvements.

Note: This tip assumes sufficient RAM availability to the system. Insufficient RAM can lead to data loss/corruption

echo 16384 > /sys/block/md0/md/stripe_cache_size blockdev --setra 16384 /dev/md0 Write Performance: dd if=/dev/zero of=/mnt/family/10gb.16384k.stripe.out bs=1M count=10240 10240+0 records in 10240+0 records out 10737418240 bytes (11 GB) copied, 94.5423 s, 114 MB/s

Read Performance: dd if=/mnt/family/10gb.16384k.stripe.out of=/dev/null bs=1M 10240+0 records in 10240+0 records out 10737418240 bytes (11 GB) copied, 28.5435 s, 376 MB/s

These changes must be done on any reboot (add to an init script to set on start-up)

Monitoring Status

 * , included in  package

cat /proc/mdstat
Review different  outputs :

Output with no devices configured: cat /proc/mdstat Personalities : [linear] [multipath] [raid0] [raid1] [raid6] [raid5] [raid4] [raid10] unused devices: cat /proc/mdstat Personalities : [raid1] [raid6] [raid5] [raid4] md_d0 : active raid5 sde1[0] sdf1[4] sdb1[5] sdd1[2] sdc1[1] 1250241792 blocks super 1.2 level 5, 64k chunk, algorithm 2 [5/5] [UUUUU] bitmap: 0/10 pages [0KB], 16384KB chunk

unused devices:

title: Linux/mdadm


text: Template:Incubate

title: Template:Incubate


text:
User:Musicist.saurav

== 'MUSIC IS GOD... CONVERT TO MUSICISM!' ==

 * Mr. Saurav Sen, a musician of Indian origin believes that musicians all over the world are the followers of the religion named 'Musicism' . To them, music is the utmost resource of inspiration to life and hence if there is something named 'God' then it is bound to be music. As a music teacher and violinist by profession, Mr. Saurav Sen is willing to create/participate a community of musicians for free flowing of musical knowledge throughout the world. He stays at Kolkata, the cultural capital of India.

title: User:Musicist.saurav


text:
User:Egm6322.s12/Report table

 Please keep this table and the wiki code tidy. Use exactly the same format that I created. Remember to log in before you link to your team's  archived  report. Only archived reports are accepted; please do NOT link to current versions. Links to the archived reports should be made within one hour of the deadline. Egm6322.s12 13:16, 3 January 2012 (UTC)

Inspiring video on teamwork and learning: A hole in the wall: How children learn without a teacher, (better video resolution from TED)

Lecture plan, Good model reports

= Comments =

(a) : The deadline for any report is in general on a Wed, unless extended by formal announcement (e.g., to accommodate the travel schedule of students, etc.), or upon a reasonable request.

(b) : Deadline for report R1 extended by one week (from Wed, 25 Jan, to Wed, 1 Feb 2012) for newly formed teams to get organized. Note that the deadline for report R2 remains the same. So don't procrastinate; work on your reports immediately.

(c) : 21:00 UTC = 17:00 EDT = 5:00 pm EDT = 4:00 pm EST. So there are 4 hours difference between EDT and UTC, and 5 hours difference between EST and UTC. From the second Sunday of March until the beginning of November, we use EDT. From the first Sunday of November, the Eastern time changes from EDT back to the EST, and we use 22:00 UTC = 5:00 pm EST.

 Report tables of other courses

title: User:Egm6322.s12/Report table


text:
File talk:Angel of Forgiveness.jpg

Source
Note: The source for this image would seem likely to be http://www.flickr.com/photos/aunto/110997305/. Adambro 13:20, 3 January 2012 (UTC)

title: File talk:Angel of Forgiveness.jpg


text:
Wikiversity:Incubator/Economics and Business Management

Proposed contents:

 * Neil Postman: Five Things We Need to Know About Technological Change
 * Chemistry and its socioeconomic impact
 * Consumer Theory
 * Game domain model
 * The Tragedy of the Commons


text:
Wikiversity:Incubator/Mathematics and Computer Science

Proposed contents:
 * Calculating the square root of a


text:
Wikiversity:Incubator/IT

Proposed contents:
 * Cert III Applications Qualification in Australia

title: Wikiversity:Incubator/IT


text:
Data structures/Applications

"In most applications of data structures, we wish to perform not just a single operation but a sequence of operations, possibly having correlated behavior." Bold added.

Data structures are used in a number of computer applications such as searching, sorting, graph traversing, finding the shortest path from one city to another, resolving traffic problems, in processing online requests or online transactions, alerting users of stock market prices, scheduling of jobs in operating systems, searching details of a student in a university data base ... and so on.

Activities

 * Python Programming/Lists
 * Python Programming/Dictionaries
 * Python Programming/Tuples and Sets
 * Applied Programming/Lists and Tuples
 * Applied Programming/Dictionaries and Sets

title: Data structures/Applications


text: Template:Itsabox

title: Template:Itsabox


text:
User:Egm6936.f10/Orthogonal polynomials

Xiu 2010, Chapter 3, p.25

Polynomials, especially orthogonal polynomials, are widely used in modern approximation theory. It's of great value to review some basis about the orthogonal polynomial. For the purpose of simplicity, we only focus on univariate case here.

=Orthogonal polynomial=

A general $$\displaystyle n$$ - degree polynomial of $$\displaystyle x$$ can be expressed in the form as
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$$\displaystyle P(x)=a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{1}x^{1} + a_{0}x^{0} $$ (ab1) In above expression, the first coefficient $$a_{n}$$ is termed leading coefficient. When $$a_{n} = 1$$, we call the polynomial as monic polynomial.
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Weighted inner product of two real functions
The inner product of two real functions is defined as
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$$\displaystyle \left \langle f(x), g(x) \right \rangle = \int_{S}f(x)g(x)dx $$ (ab2) where $$\displaystyle S$$ is the supporting domain. And the weighted inner product of two real functions is defined as
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$$\displaystyle \left \langle f(x), g(x) \right \rangle_{\omega} = \int_{S}f(x)g(x)\omega(x)dx $$ (ab3) where $$\displaystyle \omega(x)$$ is the weight function.
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Orthogonal polynomial system
A orthogonal system of polynomials is a family of polynomials satisfy that
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$$\displaystyle \int_{S}P_m(x)P_n(x)d\alpha(x) = \gamma_n\delta_{mn}, \;\;\;\; m,n \in {0, 1, 2, ...} $$ (ab4) with respect to some real positive measure $$\displaystyle \alpha $$. Where $$\displaystyle \delta_{mn} $$ is the Kronecker delta function, $$\displaystyle S $$ is the support of $$\displaystyle \alpha $$, $$\displaystyle \gamma_{n} $$ is the positive normalization constants. Apparently,
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$$\displaystyle \gamma_n = \int_{S}P_n(x)^2d\alpha(x), \;\;\;\; n \in {0, 1, 2, ...} $$ (ab5) If $$\displaystyle \gamma_{n} = 1$$, the system is orthonormal.
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Any orthogonal system can be made a orthonormal system by dividing $$\displaystyle P_{n} $$ by $$\displaystyle \gamma_{n} $$, that's $$\displaystyle \widetilde{P_{n}} = P_{n}/\gamma_{n} $$. $$\displaystyle \widetilde{P_{n}}$$ is a orthonormal system.

Usually, the measure $$\displaystyle \alpha $$ can either have a density $$\displaystyle \omega(x) $$ for continuous case, or weight $$\displaystyle \omega_i $$ at points $$\displaystyle x_i $$ for discrete case. The orthogonal relation then becomes
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$$\displaystyle \int_{S}P_m(x)P_n(x)\omega(x)dx = \gamma_n\delta_{mn}, \;\;\;\; m,n \in {0, 1, 2, ...} $$ (ab6) for continuous case, and
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 *
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$$\displaystyle \sum_{i}P_m(x_i)P_n(x_i)\omega_i = \gamma_n\delta_{mn}, \;\;\;\; m,n \in {0, 1, 2, ...} $$ (ab7) for discrete case. Utilizing the definition of weighted inner product of two real functions, we can rewrite the orthogonal relation as
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$$\displaystyle \left \langle P_m(x), P_n(x) \right \rangle_{w} = \gamma_n\delta_{mn}, \;\;\;\; m,n \in {0, 1, 2, ...} $$ (ab8) where
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 *
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$$\displaystyle \gamma_n = \left \langle P_n(x), P_n(x) \right \rangle_{w} = \left \| P_n(x) \right \|_{\omega}^2, \;\;\;\; n \in {0, 1, 2, ...} $$ (ab9)
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Three-term recurrence relation
All orthogonal polynomials satisfy a three-term recurrence relation. And the general recurrence relation is
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$$\displaystyle -xP_{n}(x) = a_{n}P_{n+1}(x) + b_{n}P_{n}(x) + c_{n}P_{n-1}(x), n \geq 1 $$ (ab10)
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Certain restriction is applied to some terms in above equation. These restrictions are $$\displaystyle a_n \neq 0$$, $$\displaystyle c_n \neq 0$$, $$\displaystyle c_n / a_{n-1} > 0$$, $$\displaystyle P_{-1}(x) = 0$$ and $$\displaystyle P_0(x) = 1$$.

Some useful orthogonal polynomials
Before proceeding to orthogonal polynomials, we introduce the Hypergeometric series(Wiki) first. Hypergeometric series is a series that the ratio of two adjacent terms is a simple function which only depends on the index.

 you can look at Battin 1999 Astrodynamics for explanation on hypergeometric series. Egm6322.s12 16:12, 5 January 2012 (UTC)

The Pochhammer symbol(Wiki)$$\displaystyle (a)^n$$ is defined as
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$$\displaystyle (a)^n = \left\{\begin{matrix} a, & n = 0 \\ a(a+1)...(a+n-1), & n = 1,2,... \end{matrix}\right. $$ (ab11) If $$\displaystyle a \in \mathbb N$$ is a positive integer, then
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$$\displaystyle (a)^n = \frac{(a+n-1)!}{(a-1)!}, \;\;\; n > 0 $$ (ab12) For general $$\displaystyle a \in \mathbb R$$
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$$\displaystyle (a)^n = \frac{\Gamma(a+n)}{\Gamma(a)}, \;\;\; n > 0 $$ (ab13) where $$\displaystyle \Gamma (\;)$$ is Gamma function(Wiki), an extension of factorial function. The generalized Hypergeometric series $$\displaystyle {}_{s}F_{r}$$is defined in terms of Pochhammer symbol as
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 *
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$$\displaystyle {}_{s}F_{r}(a_1,a_2,...,a_s;b_1,b_2,...,b_r;z) = \sum_{n=0}^{+\infty}\frac{(a_1)^n...(a_s)^n}{(b_1)^n...(b_r)^n}\frac{z^n}{n!} $$ (ab14) where $$\displaystyle b_i \neq 0,-1,-2,...$$. When $$\displaystyle a_i $$ is a negative integer, the series terminates because $$\displaystyle (-k)^n = 0$$. For example,
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$$\displaystyle {}_{0}F_{0}(\;;\;;z) = \sum_{n=0}^{+\infty}\frac{z^n}{n!} $$ (ab15)
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Legendre polynomials
Historically, Legendre polynomial(Wiki) is derived by Adrien-Marie Legendre(Wiki) when solving Laplace's equation(Wiki) in spherical coordinate system. The domain of the Legendre polynomials presented here is $$\displaystyle I = [-1, 1] $$. Legendre polynomials can be presented either using Hypergeometric series or Rodrigues' formula(Wiki) as In Hypergeometric series:
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$$\displaystyle P_n(x) = {}_{2}F_{n}(-n, n+1; 1; \frac{1-x}{2}) $$ (ab16) In Rodrigues' formular:
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$$\displaystyle P_n(x) = \frac{1}{2^nn!}\frac{dx^n}{d^n}[(x^2 - 1)^n] $$ (ab17)
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And the three-term recurrence relation is
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$$\displaystyle P_{n+1}(x) = \frac{2n+1}{n+1}xP_{n}(x) - \frac{n}{n+1}P_{n-1}(x), \;\;\; n > 0 $$ (ab18)
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The weighted inner product with weight function $$\displaystyle \omega(x) = 1$$ is
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$$\displaystyle \int_{-1}^{+1}P_n(x)P_m(x)dx = \frac{2}{2n+1}\delta_{mn} $$ (ab19) Using above recurrence relation, the first four Legendre polynomials can be derived as $$\displaystyle P_{0}(x) = 1$$ $$\displaystyle P_{1}(x) = x$$ $$\displaystyle P_{2}(x) = \frac{3}{2}x^2 - \frac{1}{2}$$ $$\displaystyle P_{3}(x) = \frac{5}{2}x^3 - \frac{3}{2}x$$ The first five Legendre polynomials
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Hermite polynomials
The Hermite polynomials(Wiki), which arise in Probability(Wiki), is a family of orthogonal polynomials named after Charles Hermite(Wiki). There are two standard version of Hermite polynomials: the probabilists(Wiki)' Hermite polynomials and the physicists(Wiki)' Hermite polynomials. The Hermite polynomials presented here are of the former version and were defined on $$\displaystyle I = \mathbb R $$. The Hermite polynomials expressed in the Hypergeometric series as
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$$\displaystyle H_n(x) = {\sqrt{2}x}^n {}_2F_0 (-\frac{n}{n};-\frac{n-1}{2};-\frac{2}{x^2}) $$ (ab20) The three-term recurrence relation is
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$$\displaystyle H_{n+1}(x) = xH_{n}(x) - nH_{n-1}(x), \;\;\; n > 0 $$ (ab21) The weighted inner product with weight function $$\displaystyle \omega(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ is
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$$\displaystyle \int_{-\infty}^{+\infty}H_n(x)H_m(x)\omega(x)dx = n!\delta_{mn} $$ (ab22) Using above recurrence relation, we can have the first four Hermite polynomials as $$\displaystyle H_{0}(x) = 1$$ $$\displaystyle H_{1}(x) = x$$ $$\displaystyle H_{2}(x) = x^2 - 1$$ $$\displaystyle H_{3}(x) = x^3 - 3x$$
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The first six (probabilists') Hermite polynomials

Laguerre polynomials
The Laguerre polynomials(Wiki) is a family of orthogonal polynomials named after Edmond Laguerre(Wiki). It's the canonical solution of Laguerre's equation:
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$$\displaystyle x\,y'' + (1 - x)\,y' + n\,y = 0\, $$ (ab23) The domain of the Laguerre polynomial presented here is $$\displaystyle I = [0, \infty) $$. Generalizing above second-order linear differential equation as
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$$\displaystyle x\,y'' + (\alpha + 1 - x)\,y' + n\,y = 0\, $$ (ab24) and solving it, we can have the generalized Laguerre polynomial as In Hypergeometric series:
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$$\displaystyle L_n^{\alpha}(x) = \frac{(\alpha + 1)n}{n!} {}_{1}F_{1}(-n; \alpha+1; x ),\;\;\; \alpha > -1 $$ (ab25) In Rodrigues' formular:
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$$\displaystyle L_{n}^{\alpha}(x) = \frac{x^{-\alpha}e^x}{n!}\frac{dx^n}{d^n}(e^{-x}x^{n+\alpha}) $$ (ab26) The three-term recurrence relation is
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$$\displaystyle (n+1)L_{n+1}^{\alpha}(x) = (-x+2n+\alpha+1)L_{n}^{\alpha}(x) - (n+\alpha)L_{n-1}^{\alpha}(x), \;\;\; n > 0 $$ (ab27) The weighted inner product with weight function $$\displaystyle \omega(x) = e^{-x}x^{\alpha}$$ is
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$$\displaystyle \int_{0}^{+\infty}L_n^{\alpha}(x)H_m^{\alpha}(x)\omega(x)dx = \frac{\Gamma(n+\alpha+1)}{n!}\delta_{mn} $$ (ab28) Setting $$\displaystyle \alpha = 0 $$, we obtain the simple Laguerre polynomials.
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$$\displaystyle L_{n}(x) = L_{n}^{0}(x) $$ (ab29) The first four simple Laguerre polynomials are $$\displaystyle L_{0}(x) = 1$$ $$\displaystyle L_{1}(x) = -x + 1$$ $$\displaystyle L_{2}(x) = \frac{1}{2}x^2 - 2x + 1$$ $$\displaystyle L_{3}(x) = -\frac{1}{6}x^3 - \frac{3}{2}x^2 - 3x + 1$$
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The first six Laguerre polynomials

=Polynomial approximation=

We provide the classical approximation theory here. Before giving the theorem, we define $$\displaystyle \mathbb P_n$$ as the linear at most n-degree polynomials space.
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$$\displaystyle \mathbb P_n = span\{x^k, k=0,1,...,n\} $$ (ab30) 【Weierstrass theorem 】: Let $$\displaystyle I $$ be a closed interval and $$\displaystyle f \in C^{0}(I) $$, there will always exists $$\displaystyle n \in \mathbb N $$ and $$\displaystyle p \in \mathbb P_n $$, for any $$\displaystyle \epsilon > 0 $$, such that
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$$\displaystyle \left | f(x)- p(x) \right | < \epsilon, \;\;\; \forall x \in I $$ (ab31)
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=Polynomial projection=

We define the weighted $$\displaystyle L^2$$ space as
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$$\displaystyle L_{\omega}^2(x) := \left \{ v: I \rightarrow \mathbb P | \int_{I}v^2(x)\omega(x)dx < +\infty \right \} $$ (ab32) where $$\displaystyle \omega (x)$$ is the weight function, and $$\displaystyle x \in I$$, $$\displaystyle I$$ is a general interval. And the weighted inner product is
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$$\displaystyle \left \langle f(x),g(x) \right \rangle_{L_{\omega}^2(I)} = \int_{I}f(x)g(x)\omega(x)dx,\;\;\; f(x),g(x) \in L_{\omega}^2(I) $$ (ab33) The norm is
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$$\displaystyle \left \| f(x) \right \|_{L_{\omega}^2(I)} = \left ( \int_{I}f(x)^2\omega(x)dx \right )^{1/2} $$ (ab34)
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We define projection operator$$\displaystyle P_{N}:L_{\omega}^2(I) \rightarrow \mathbb P_{N}$$.

Orthogonal projection
Let $$\displaystyle N$$ is a fixed nonnegative integer and $$\displaystyle \phi_{k}(x), \; k = 0,1,...,N$$ is orthogonal polynomials of degree at most $$\displaystyle N$$ with the weight function $$\displaystyle \omega (x)$$. For any function $$\displaystyle f \in L_{\omega}^2(I)$$
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$$\displaystyle P_{N}f := \sum_{k = 0}^{N}\widetilde{f_k}\phi_k(x) $$ (ab35) where
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$$\displaystyle \widetilde{f_k} := \frac{1}{\left \| \phi_k \right \|_{L_{\omega}^2}^2}\left \langle f,\phi_k \right \rangle_{L_{\omega}^2},\;\;\; 0\leq k \leq N $$ (ab36)
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Convergence rate
The rate of convergence depends on the generality of the interpolated function $$\displaystyle f$$ and the type of interpolating orthogonal bases $$\displaystyle \{ \phi_k \}$$.

When the function $$\displaystyle f$$ is not analytic, the convergence rate is no longer faster than the algebraic rate. For discontinuous functions, the convergence rate deteriorates significantly.

Gibbs phenomenon
Gibbs phenomenon(Wiki) With the increase of the number of orthogonal bases employed, the oscillations will not disappear, which is named the Gibbs phenomenon.

 How can we cure the Gibbs phenomenon? Hylon.Chen 13:37, 19 January 2012 (UTC).

Not easy to cure; just use more and more terms in the series. you may want to add some screenshots from Wolfram Demo Projects. Egm6322.s12 15:35, 26 January 2012 (UTC) Fewer terms used in piecewise approximating discontinuous function using Fourier series

More terms used in piecewise approximating discontinuous function using Fourier series

Courtesy: Above two figures are screenshotted from Wolfram Mathworld.

Discrete projection
For a given function $$\displaystyle f \in L_{\omega}^2 (I)$$, the discrete projection is defined as
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$$\displaystyle I_{N}f := \sum_{k = 0}^{N}\widehat{f_k}\phi_k(x) $$ (ab37) with
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$$\displaystyle \widehat{f_k} = \frac{1}{\left \| \phi_k \right \|_{L_{\omega}^2}^2}\sum_{k=i}^{q}f_k(x^{(i)})\phi_k(x^{(i)})\omega_i,\;\;\; 0\leq k \leq N $$ (ab38) where $$\displaystyle x^{(i)}$$ are integration points and $$\displaystyle w_{i}$$ are the corresponding weights. With certain integration technique(for instance, Gauss quadrature) used, the coefficient $$\displaystyle \widehat f_k$$ approximates the coefficient $$\displaystyle \widetilde{f_k}$$ in the continuous orthogonal projection. Note: The only difference between continuous orthogonal projection and discrete projection is the integration technique approximates the exact continuous integration of $$\displaystyle \widetilde{f_k}$$.
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=Polynomial interpolation=

Polynomial Interpolation(Wiki): Given $$\displaystyle n + 1$$ pairs of $$\displaystyle (x_i,y_i)$$, find a function $$\displaystyle G = G(x) $$ such that $$\displaystyle G(x_i) = y_i$$ for all given $$\displaystyle (x_i,y_i)$$. $$\displaystyle \{x_i\} $$ is the interpolation nodes, $$\displaystyle G $$ is the interpolating polynomial. $$\displaystyle G $$ can be algebraic polynomial, trigonometrical polynomial, piecewise polynomial (local polynomial) and rational polynomial.

Existence
Given a set of points $$\displaystyle (x_i,y_i)$$, we can always find a function $$\displaystyle G(x) $$, such that $$\displaystyle G(x_i) = y_i$$ for all given $$\displaystyle (x_i,y_i)$$.

Uniqueness
For any given set of points $$\displaystyle (x_i,y_i)$$, we can find a unique function $$\displaystyle G(x) $$, such that $$\displaystyle G(x_i) = y_i$$ for all given $$\displaystyle (x_i,y_i)$$.

Interpolation error
Assuming $$\displaystyle f$$ is the given function to be interpolated with the range of $$\displaystyle I$$, where $$\displaystyle x_0,x_1,...,x_n$$ are the distinct nodes within the range, and $$\displaystyle \Pi_nf$$ be the interpolating polynomial of $$\displaystyle nth$$ degree.

【Theorem】: The interpolation error at point $$\displaystyle x$$ with the interval $$\displaystyle I$$ is
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$$\displaystyle E_{N}(x) = f(x)-\Pi_nf(x) = \frac{f^{n+1}(\xi)}{(n+1)!}q_{n+1}(x) $$ (ab39)
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where $$\displaystyle I_x $$ is the smallest interval including $$\displaystyle x,x_0,x_1,...,x_n$$ and $$\displaystyle q_{n+1}(x) = \Pi^{n+1}_{i=0}(x-x_i)$$ is the nodal Lagrange polynomial of degree $$\displaystyle n+1 $$.

For detailed proof of this theorem, one can refer to [[media:nm1.s11.mtg11.djvu|NM1S11 - Mtg 11.2]].

Runge phenomenon
Runge phenomenon (Wiki): When using high-degree polynomials in a interpolation with evenly distributed nodes, $$\displaystyle q_{n+1}(x) = \Pi^{n+1}_{i=0}(x-x_i)$$ will performs rather wildly near the endpoint nodes. This leads to bad convergence at the endpoint nodes. This is called the Runge phenomenon. To cure this phenomenon, one can either use piecewise low-degree polynomials or non-uniformly distributed nodes for high-degree polynomials. Also refer to [[media:nm1.s11.mtg21.djvu|NM1S11 - Mtg 21]].

=Zeros of orthogonal polynomials and quadrature=

【Theorem】: For $$\displaystyle n \geq 1$$, orthogonal polynomials $$\displaystyle P_n, \;\;n \in \mathbb N$$ has exactly $$\displaystyle n $$ real different zeros in $$\displaystyle I $$.

 $$\displaystyle <P_n,1>_w $$ = 0, Why this equality holds? Hylon.Chen 20:45, 24 January 2012 (UTC)

since $$\displaystyle P_0 (x) = 1$$, and by the orthogonality of the family of Legendre polynomials $$\displaystyle \{ P_n \}$$, you have $$\displaystyle <P_n, P_0> = <P_n,1>_w = 0 , \text{ for } n \ge 1$$. Egm6322.s12 15:38, 26 January 2012 (UTC)

For many orthogonal polynomials, their zeros are non-uniformly distributed, while clustered towards the endpoints of the interval $$\displaystyle [-1,1]$$.

Let
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$$\displaystyle I[f] := \int_{I}f(x)w(x)dx $$ (ab40) and define a integration formula with $$\displaystyle n \geq 1$$ points,
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$$\displaystyle U^n[f] := \sum_{i = 1}^{n}f(x^i)w^i $$ (ab41) where $$\displaystyle x^i $$ are a set of nodes and $$\displaystyle w^i $$ are integration weights. We need to find a set of $$\displaystyle \{ x^i,w^i\} $$ such that $$\displaystyle U^n [f] \approx I [f]$$, and hopefully $$\displaystyle \lim_{n\rightarrow \infty}U^n[f] = I [f]$$. Highly accurate integration formulas can be constructed using orthogonal polynomials. Let $$\displaystyle P_n, \;\;n \in \mathbb N$$ be orthogonal polynomials, and let $$\displaystyle z_k^N, \;\;k = 1,2,3,...,N$$ be the zeros of $$\displaystyle P_n $$. Let $$\displaystyle L_k^{N-1}$$ be the $$\displaystyle N-1 $$th-degree Lagrange polynomials through the nodes $$\displaystyle z_k^N$$, and let $$\displaystyle \Pi_{N-1}f(x)=\sum_{k = 1}^{N}f(z_k^N)L_k^{N-1}$$ be the $$\displaystyle N-1$$th-degree interpolation of $$\displaystyle f(x)$$. Then the integration in (ab40) can be approximated by integrating $$\displaystyle \Pi_{N-1}f(x)$$.
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$$\displaystyle \int_{I}f(x)w(x)dx = \sum_{k = 1}^{N}f(z_k^N)w_k $$ (ab42) where
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$$\displaystyle w_k = \int_{I}L_k^{N-1}w(x)dx\,\,\,\, 1\leq k \leq N $$ (ab43) are the weights.
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Formula (ab42) become exact when $$\displaystyle f(x)$$ is any polynomial with degree less than or equal to $$\displaystyle 2N - 1$$ in $$\displaystyle I $$.

See Gaussian quadratureWiki[[media:nm1.s11.mtg7.djvu|NM1S11 - Mtg 7]], Clenshaw-Curtis quadrature(Wiki) and Newton-Cotes quadrature rulesWiki[[media:nm1.s11.mtg8.djvu|NM1S11 - Mtg 8]].

=References=

title: User:Egm6936.f10/Orthogonal polynomials


text:
User:Egm6936.f10/Stochastic systems

The content of this section is mainly from Xiu 2010. For more details, one can refer to Xiu 2010, Chapter 4, p.44

After establishing a deterministic model for a physical system, prior to any simulation being taken, one must set up a proper stochastic model to study the effect of uncertainties in the inputs to the system. This becomes a key step in the stochastic system formulation. More work is devoted here to study some general aspects of this topic.

=Input parameterization=

How to parameterization the input random variables and reduce the dimension of these variables are two important aspects in stochastic processes. we will focus on these two aspects in this section.

Random parameters
When the random inputs to a system are the system parameters, the more important issue is to identify the independent parameters in the set. Such problem can be presented as:
 * $$\displaystyle X = (X_1,X_2,...,X_n), n > 1$$ is the system parameters with a prescribed distribution function $$\displaystyle F_X(x) = P(X \leq x), x \in \mathbb R^n$$, find a set of mutually independent random variables $$\displaystyle Y = (Y_1,Y_2,...,Y_k)\in \mathbb R^k, 1 \leq k \leq n $$, such that $$\displaystyle X = T(Y)$$ for a suitable transformation function $$\displaystyle T$$.

We use a simply example to demonstrate above idea. Suppose the inputs to a system are $$\displaystyle x$$ and $$\displaystyle y$$, and the output of the system is $$\displaystyle U(x,y)$$. Thus, the input random variables are $$\displaystyle X(\omega) = (x, y)$$. Surely, the output $$\displaystyle U(x,y)$$ depends on the input variables $$\displaystyle x$$ and $$\displaystyle y$$.
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$$\displaystyle U(x,y): \Omega \rightarrow \mathbb R $$ (ab1) If $$\displaystyle x$$ and $$\displaystyle y$$ are mutually independent, then we simply let $$\displaystyle Y(\omega) = X(\omega)$$. The output field can be presented as
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$$\displaystyle U(x,y): \mathbb R \times \mathbb R \rightarrow \mathbb R $$ (ab2) If $$\displaystyle x$$ and $$\displaystyle y$$ are not independent of each other, then there exist a function such that $$\displaystyle f(x,y) = 0$$. It is possible to find a random variable $$\displaystyle Y(\omega)$$ to parameterize the relation such that
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 * $$\displaystyle x(\omega) = a(Y(\omega))$$, $$\displaystyle y(\omega) = b(Y(\omega))$$ and $$\displaystyle f(a,b) = 0$$.

Then, the random inputs via $$\displaystyle x$$ and $$\displaystyle y$$ can now be expressed by a single random parameter $$\displaystyle Y$$. The output field becomes
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$$\displaystyle U(x,y): \mathbb R \rightarrow \mathbb R $$ (ab3)
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In practice, there are many input parameters in a system, finding the exact form of the functional dependence among all of them can be a challenging (and unnecessary) task. It is obvious when the only available information for the parameters is the joint probability distributions. Next we will focus on transforming the parameters to a set of independent random parameters by using their distribution functions

Gaussian parameters
When the system parameter has a Gaussian distribution, the problem becomes easy by Theorem 1. Let $$\displaystyle \mathbf X = (X_1,X_2,...,X_n)$$ be a random vector, and $$\displaystyle \mathbf X \doteqdot N(0, \mathbf C)$$, where $$\displaystyle \mathbf C \in \mathbb R^{n \times n}$$ is the covariance matrix and the expectation is assumed to be zero. Let $$\displaystyle \mathbf Z \doteqdot N(0, \mathbf I)$$, where $$\displaystyle \mathbf I $$ is the $$\displaystyle n \times n$$ identity matrix, be a uncorrelated Gaussian vector of size $$\displaystyle n $$. Let $$\displaystyle \mathbf A$$ be an $$\displaystyle n \times n$$ matrix; then according to Theorem 1, $$\displaystyle \mathbf {AZ} \doteqdot N(0, \mathbf {AA}^T)$$. Therefore, if one finds a matrix $$\displaystyle \mathbf A$$ such that $$\displaystyle \mathbf {AA}^T = \mathbf C$$, then $$\displaystyle \mathbf X = \mathbf {AZ}$$ will have the given distribution $$\displaystyle N(0,\mathbf C)$$. Since $$\displaystyle \mathbf C$$ is real and symmetric, solving the problem $$\displaystyle \mathbf {AA}^T = \mathbf C$$ can be readily done via Cholesky's decomposition.

Non-Gaussian parameters
Mathematically, for non-Gaussian parameters, we can utilize the Rosenblatt transformation to accomplish our goal. It's theoretically simply and powerful, but not easy to carry out in practice. Let $$\displaystyle \mathbf X = (X_1,X_2,...,X_n)$$ be a random vector with a (non-Gaussian) pdf $$\displaystyle F_X(x)=P(X \leq x)$$ and let $$\displaystyle \mathbf z = (z_1,z_2,...,z_n)= Tx = T(x_1,x_2,...,x_n)$$ be a transformation defined as
 * $$\displaystyle z_1 = P(X_1 \leq x_1) = F_1(x_1)$$,
 * $$\displaystyle z_2 = P(X_2 \leq x_2 \mid X_1 = x_1) = F_2(x_2 \mid x_1)$$,
 * $$\displaystyle z_n = P(X_n \leq x_n \mid X_{n-1} = x_{n-1},...,X_1 = x_1) = F_n(x_n \mid x_{n-1},...,x_1)$$,
 * $$\displaystyle z_n = P(X_n \leq x_n \mid X_{n-1} = x_{n-1},...,X_1 = x_1) = F_n(x_n \mid x_{n-1},...,x_1)$$,

Then, It can be shown that
 * $$\displaystyle P(Z_i \leq z_i; i = 1,2,...,n)$$
 * $$\displaystyle = \int_...\int d_{x_n}F_n(x_n \mid x_{n-1},...,x_1)...d_{y_1}F_1(x_1)$$
 * $$\displaystyle \int_{0}^{z_n}...\int_{0}^{z_1}d_{z_1}...d_{z_n} = \prod_{i=1}^{n}z_{i}$$,

where $$\displaystyle 0 \leq z_i \leq 1, i = 1,2,...,n)$$. Hence, $$\displaystyle \mathbf Z = (Z_1, Z_2,...,Z_n)$$ are independent and identically distributed random variables with uniform distribution in $$\displaystyle [0,1]^n$$.

In practice, numerical approximations of Rosenblatt transformation are required. And this topic is still understudied.

Dimension reduction
In many cases, the random inputs are stochastic processes. The inputs could be a time-dependent or space-dependent random variables, such as velocity and conductivity. We restate the inputs parameterization problem in stochastic process as follows.
 * Let $$\displaystyle (X_t, t\in \Omega)$$ be a stochastic process that models the random inputs. Find a appropriate transformation function $$\displaystyle T $$ such that $$\displaystyle X_t = T(Z)$$, where $$\displaystyle Z = (Z_1, Z_2,...,Z_d), 1 \leq d$$, are mutually independent.

Note that the index set $$\displaystyle \Omega$$ can be in either the time space or the space domain, and usually is an infinite-dimensional object. Since we require $$\displaystyle d $$ to be a finite integer, the transformation cannot be exact. $$\displaystyle X_t \approx T(Z)$$ in a proper norm or metric, and the accuracy of the approximation will be problem-dependent. A straightforward approach is discretize the index domain $$\displaystyle \Omega$$ into a set of indices and then study the process
 * $$\displaystyle (X_{t_1}, X_{t_2},..., X_{t_n}), t_1, t_2,...,t_n \in \Omega$$.

Above discretization is obviously an approximation. The finer this discretization, the better the approximation. However, this is not desired because a finer discretization leads to a larger dimension of $$\displaystyle n$$ and can significantly increase the computational burden. Some kinds of dimension reduction techniques are required to make the dimension as low as possible while maintaining a satisfactory accuracy.

Karhunen-Loeve expansion
The Karhunen-Loeve is one of the most widely used techniques for reducing dimension in representing random processes.

Let $$\displaystyle \mu_X(t) $$ be the mean of the input process $$\displaystyle X_t $$ and $$\displaystyle C(t,s) = cov(X_t,X_s)$$ be its covariance function. The Karhunen-Loeve expansion of $$\displaystyle X_t$$ is

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$$\displaystyle X_t(\omega) = \mu_X(t)+\sum_{i=1}^{\infty}\sqrt{\lambda_i}\psi_i(t)X_i(\omega) $$ (ab4) where $$\displaystyle \psi_i $$ are the orthogonal eigenfunctions and $$\displaystyle \lambda_i$$ are the corresponding eigenvalues of the eigenvalue problem
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$$\displaystyle \int_{T}C(t,s)\psi_i(s)ds = \lambda_i\psi_i(t), t\in T, $$ (ab5) and $$\displaystyle {X_i(\omega)}$$ are mutually uncorrelated random variables satisfying
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$$\displaystyle \mathbb E[X_i] = 0, \;\; \mathbb E[X_iX_j] = \delta_{ij} $$ (ab6) and defined by
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$$\displaystyle X_i(\omega) = \frac{1}{\sqrt{\lambda_i}}\int_{T}(X_t(\omega)-\mu_X(t))\psi_i(t)dt, \forall i $$ (ab7) In practice, one adopts a finite series expansion, e.g.,
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$$\displaystyle X_t(\omega) = \mu_X(t)+\sum_{i=1}^{d}\sqrt{\lambda_i}\psi_i(t)X_i(\omega), 1\leq d $$ (ab8)
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When to truncate the series is problem-dependent. One important property of KL expansion is Eigenvalues $$\displaystyle \lambda_i$$ decay as index $$\displaystyle i$$ increase. For a given covariance function, the decay rate of the eigenvalues depends inversely on the correlation length. Long correlation length implies that the process is strongly correlated and results in a fast decay of the eigenvalues. The limit of this, infinitely long correlation length, is the fully correlated case where the eignevalues decay to zero immediatedly. Contrary, a weakly correlated process has short correlation length and results in a slow decay of the eignevalues. The limit of this, the uncorrelated process with zero correlation length, has no eigenvalue decay.

$$\displaystyle $$

Gaussian processes
For Gaussian random variables, they are uncorrelated with each other. Hence, we can directly use KL expansion to parameterize a Gaussian process by a finite number of independent Gaussian random variables.

Non-Gaussian processes
When the input random processes are non-Gaussian, their parameterization and dimension reduction are significantly more challenging. The main problem is that, for non-Gaussian distribution, uncorrelation of the random variables $$\displaystyle X_i$$ in (ab4) doesn't imply independence. Hence the KL expansion doesn't provide a way of parameterization with independent variables. Currently, there are not many practical methods for achieving the parameterization procedure. In many practical computations, one often still uses the KL expansion for an input process and then further assumes that the $$\displaystyle X_i$$ are independent. This field is still an active research topic.

 I'm a little confused about the KL expansion and input parameterization. As far as I understand, the KL expansion is to reduce the dimension of the stochastic processes, while the input parameterization is to parameterize the random variables such that they are independent with each other. A general procedure is that we adopt input paramerization techniques, such as Gaussian parameter and Rosenblatt transformation method, to make all the variables independent or uncorrelated. Then we use KL expansion to make the infinite stochastic process domain finite at certain accuracy level. Finally, solve the deterministic PDEs or ODEs. The KL expansion and parameterization are just two different steps. Am i right? Hylon.Chen 21:33, 1 February 2012 (UTC)

yes, these are different steps. you want to do the input parameterization first to define your model. then you use the KL expansion, with a truncation, to obtain a finite-dimensional system. solving the eigenvalue problem (ab5) would result in an infinite number of eigenpairs; only a few selected ones would be used in the finite series of the KL expansion in (ab8). Egm6322.s12 13:30, 2 February 2012 (UTC)

=Formulation of stochastic systems=

We briefly outline the main steps in formulating a stochastic system by taking into account random inputs to a well-established deterministic system. The general procedure is as follow:
 * 1. Parameterize the random input variables;
 * This step is done via some transformation techniques to make all inputs independent with each other.
 * 2. Reduce the dimension of the stochastic process at certain accuracy level;
 * KL expansion can be adopted in this step for uncorrelated random variables. As for correlated random variables, this topic is still active in recent research.
 * 3. Solve the deterministic equations system using traditional methods.

We introduce one partial differential equations (PDEs) as a basic example to illustrate above procedures for formulating a stochastic system. Let's consider a system of PDEs defined in a spatial domain $$\displaystyle D \subset \mathbb R^k,\;\; k = 1,2,3$$, and a time domain $$\displaystyle [0, T], \;\; T > 0 $$,
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$$\displaystyle u_t(x,t,\omega) = \mathcal L (u), \;\; D\times (0,T]\times \Omega$$, $$\displaystyle \mathcal B (u) = 0, \;\; \partial D \times [0, T]\times \Omega$$, $$\displaystyle u = u_0, \;\; D \times {t = 0}\times \Omega, $$ (ab9)
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Where $$\displaystyle \mathcal L $$ is a nonlinear differential operator, $$\displaystyle \mathcal B$$ is the boundary condition operator, $$\displaystyle u_0$$ is the initial condition, and $$\displaystyle \omega \in \Omega$$ denotes the random inputs of the system in a properly defined probability space $$\displaystyle (\Omega, \mathcal F, P) $$. The solution is therefore a random quantity,
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$$\displaystyle u(x,t,\omega): \bar{D}\times[0, T]\times \Omega \rightarrow \mathbb R^{n_u} $$ (ab10) where $$\displaystyle n_u \geq 1$$ is the dimension of $$\displaystyle u$$. The random input to (ab9) take the form of random parameters and random processes. Let us assume that they can all be properly parameterized by a set of independent random variables using the techniques mentioned in the previous sections. Let $$\displaystyle Z = (Z_1, Z_2,...,Z_n) \in \mathbb R^d, d\geq 1$$, be the set of independent random variables characterizing the random inputs. The system (ab9) can be rewritten as
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$$\displaystyle u_t(x,t,Z) = \mathcal L (u), \;\; D\times (0,T]\times \mathbb R^d $$, $$\displaystyle \mathcal B(u) = 0, \;\; \partial D \times [0, T]\times \mathbb R^d $$, $$\displaystyle u = u_0,\;\; D \times {t = 0}\times \mathbb R^d, $$ (ab11)
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The solution is now
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$$\displaystyle u(x,t,Z): \bar{D}\times[0, T]\times \mathbb R^d \rightarrow \mathbb R^{n_u} $$ (ab12) Up to this point, the system (ab12) can be deterministically solved.
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$$\displaystyle $$

=Traditional numerical methods=

We briefly review the traditional methods for solving practical systems with random inputs. Example used in above section will reused in this section for the purpose of illustration.

Monte Carlo Sampling
Monte Carlo sampling(MCS) is a statistical sampling method. The general procedure is as follows.
 * 1. Generate identically and independently distributed random numbers $$\displaystyle Z_i, \;\; i=1, 2,..., N $$, according to the pdf;

 * 2. For each $$\displaystyle i=1, 2,..., N $$, solve the governing equation;

 * 3. Estimate the required solution statistics.

It is obvious that the step 1 and step 3 are preprocessing and postprocessing steps, respectively. Only step 2 requires solution of the original problem, and it involves repetitive simulations of the deterministic counterpart of the problem.

Error estimate of MCS follows immediately from the central limit theorem. The widely adopted concept that the error convergence rate of MCS is inversely proportional to the square root of the number of realizations, which is relatively slow. One advantage of MCS is the convergence rate is independent of the dimension of the random space.

For more information about Monte Carlo Sampling, one can refer to wiki article Monte Carlo Sampling.

Moment equation approach
The objective of the moment equation approach is to compute the moments of the solution directly because in many cases the moments of the solution are what one needs. Consider a ordinary differential equation as
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$$\displaystyle \frac{du}{dt}(t,\omega) = -\alpha(\omega)u,\;\;\; u(0,\omega) = \beta(\omega) $$ (ab13) Taking the expectation of both sides of (ab13) we obtain
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$$\displaystyle \frac{d\mathbb E[u]}{dt}(t) = -\mathbb E[\alpha u],\;\;\; \mu(0) = \mathbb E[\beta] $$ (ab14) This requires a knowledge of $$\displaystyle \mathbb E[\alpha u]$$, which is unknown. We then attempt to derive an equation for this new quantity. Multiplying (ab13) by $$\displaystyle \alpha$$ and taking the expectation,
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$$\displaystyle \frac{d}{dt}\mathbb E[\alpha u](t) = -\mathbb E[\alpha^2 u],\;\;\; \mathbb E[\alpha u](0) = \mathbb E[\alpha\beta] $$ (ab15) A new quantity $$\displaystyle \mathbb E[\alpha^2 u]$$ appears. In general, if we keep multiplying the original system by $$\displaystyle \alpha^k,\;\;\;k=0,1,2,... $$, then we get
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$$\displaystyle \frac{}{dt}\mathbb E[\alpha^k u](t) = -\mathbb E[\alpha^{k+1} u],\;\;\; \mathbb E[\alpha^k u](0) = \mathbb E[\alpha^k\beta] $$ (ab16) The system of equations thus cannot be closed, since new variables keep being introduced into the exist system. This is the well-known closure problem. The typical approach to remedy the difficulty is to assume, for a fixed $$\displaystyle k$$, that the higher-order term is a function of the lower-order ones, that is,
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$$\displaystyle \mathbb E[\alpha^{k+1} u]= g(\mathbb E[u](0), \mathbb E[\alpha u](0), ..., \mathbb E[\alpha^k u](0)) $$ (ab17) where the form of the function $$\displaystyle g$$ is determined on a problem-dependent basis with sufficient justification. There exists, to date, no effective general strategy for solving the closure problem. Also, the error caused by most, if not all, closure techniques is not easy to quantify.
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Perturbation method
In the perturbation method, the random part of the solution is treated as a perturbation. The fundamental assumption is that such a perturbation is small. And this is typically achieved by requiring the standard deviation of the solution to be small.

One can refer to Perturbation method for more details about this method.

$$\displaystyle $$

=References=

title: User:Egm6936.f10/Stochastic systems


text:
User:Lluka

There is no (I) when you all exist...its not about me..it is about us and who we are...

title: User:Lluka


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User talk:Lluka

We are all brilliant!! Beauty at its finniest!!..I have one friend who has a similar understanding of the reality we occupy but we really only two,so for us we don't live in Tibet so its pretty lonely and hard for us to communicate or relay the absolute truth to the masses regarding our collective state of being/mankind +our ages play a factor we 21 & 22 years of age so we fall in the segment of being labeled youths & unfortunately there is a ill global perception off any (youngsters) ability to contribute to important matters due to our alleged lack of experience!! tragic really because in my opinion many of our elders have wasted their experience indulging in senseless thought processes no disrespect meant to anyone by that statement!!.. We think its difficult to occupy a reality that is filled with beings that have allowed their true selves to go dormant.What I mean by this statement is beings today ask themselves who am I? What is the truth? One would think that these two questions have nothing to do with each other but they are are one in the same...If we dive into our minds one starts to realize that ever since birth our subconsciousness as well as our conscious state has been infiltrated by well thought of doctrinal information designed to to create a false you/us,the birth of a social persona/ego/personality.This information is passed down through family generations unknowingly and becomes the basis as to how members of society reflect and express their beliefs and understanding of life to their understanding but there is an obvious fault with that statement. if one is taught everything they know? what does the individual really know? The individual has a mind with someone else's thoughts,ideas and memories through the botch global education system that destroys one's true knowledge and represses the power of the imagination! movies and shows which fall under entertainment..now what people don't know is that entertain means a maze and ment or in Greek mente means mind!! put the pieces together it paints an unpleasant reality the same applies with Government or better yet GOVERN-MENTAL institution am sure you heard that before...That tells us that getting to our minds has always been part of the ill doctrine that has fed the virus that I call the state of society today!! now coming back to the truth or us if we remove all we ever knew /know and view ourselves as a product of our environment we should try to get into contact with the unedited being within ourselves..religion has named it the soul..i call it the absolute truth..it never lies,it cannot actually! Once we find the true self we find absolute truth...that is the key to a world of truth/ a better world not a president or a parliament or parilia mente or parliamentum The truth is self governing !! The answer to everything in order to understand and correct is within ourselves..who wants to argue with the truth? we are the world & the world is us so if you feel your relationship with yourself is broken you should fix it and put a patch on our wound!! you,I,everyone are the saviors of our reality..thank you for your existence as you make me/us who I am today as you give hope!!! may you be peace and may peace be upon you...truth now truth forever!!!

title: User talk:Lluka


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Statistical Analysis/Unit 2 Navigation

This is the main navigation page for Unit 2 of the course Introduction to Statistical Analysis, developed using openly licensed materials from Saylor.org's Introduction to Statistics. Below you will find a full description of Unit 2 in general, as well as for each subunit. Follow the links within each subunit description to access particular topics, or proceed directly to the Unit 2 Content Page.

UNIT 2: PROBABILITY TOPICS
After you've learned to describe and display data, how can you use the sample data to draw conclusions about the populations? To answer this question, you need probability, a subject we will explore over the course of this unit.

If you open a newspaper, you are likely to read headlines like: “Should risky medical procedures be allowed?”; “30% chance of a hurricane this weekend”; “Analysts expect the gas prices to increase this summer”; and “Two brothers meet by accident after being separate for more than 20 years.” All of these topics deal with everyday life and all of them have to do with chance or probability.

The world seems to be full of events that seem unpredictable. Probability theory is a tool that was created to deal with such events more effectively. For example, before getting a surgery, a patient wants to know the chances that the surgery might fail; before taking medication, we want to know the chances that there will be side effects; before leaving our houses, we want to know the chances that it will rain today. Probability deals with the chance of an event occurring. It is a measure of likelihood that takes on values between 0 and 1, inclusive, with 0 representing impossible events and 1 representing certainty and the chances of events occurring fall between these two values. The ability to calculate probability allows us to make better decisions. Probabilities affect our everyday lives.

In this unit, you will learn what probability and its properties are, how probability behaves, and how to calculate and use it. You will study the fundamentals of probability and will work through examples that cover different types of probability problems. These basic probability concepts will provide a foundation for understanding more statistical concepts. You probably already (intuitively) use concepts from probability, but after this unit, you will be able to formally and precisely predict the likelihood of an event occurring given certain constraints. Whether we are evaluating how likely it is it to get more than 50% of the questions correct on a quiz if you guess randomly; predicting the chance that the next storm will arrive by the end of the week; or exploring the relationship between the number of hours students spend at the gym and their performance on an exam, an understanding of the fundamentals of probability is crucial. Make sure you spend time on this unit. Our goal will be to become comfortable with the basic machinery of probability theory and its applications.

Time Advisory
Time Advisory: This unit will take you 6 hours to complete.
 * Subunit 2.1: 1 hours
 * Subunit 2.2: 1 hours
 * Subunit 2.3: 1 hours
 * Subunit 2.4: 3 hours

Learning Outcomes
Upon completion of this this unit, you will be able to:
 * Understand and use the terminology of probability.
 * Determine whether two events are mutually exclusive and whether two events are independent.
 * Calculate probabilities using the Addition Rules and Multiplication Rules.
 * Construct and interpret Contingency Tables.
 * Construct and interpret Venn Diagrams.
 * Construct and interpret Tree Diagrams.

Subunits
Unit two consists of four main topics:

Probability Topics and Terminology
This topic will introduce you the major concepts fundamental to probability and the terminology associated with them.

 * 2.1: Probability Topics and Terminology

Independent and Mutually Exclusive Events

 * 2.2: Independent and Mutually Exclusive Events

Two Basic Rules of Probability

 * 2.3: Two Basic Rules of Probability

Contingency Tables, Venn Diagrams, and Tree Diagrams

 * 2.4: Contingency Tables, Venn Diagrams, and Tree Diagrams

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Kahn Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

title: Statistical Analysis/Unit 2 Navigation