text:
User:Physis

Other sites

 * Youtube
 * ungazik

 * Bering Strait School District OpenContent Initiative
 * Physis

 * Extra
 * physis

 * HaskellWiki

On-line examples of Yokuts narratives

 * "The California Indians" by Stephen Powers (1873)
 * "Indian Myths of South Central California" by Alfred L. Kroeber (1907)
 * Myths and Legends of California and the Old Southwest by Katharine Berry Judson (1912)
 * The North American Indian by Edward S. Curtis (1924)

Other Wikimedia projects

 * Wikibooks
 * Physis

 * Wikimedia Commons
 * Physis

Articles saved from stub status

 * Sireniki Eskimo language
 * Fuegians (from dab status, more exactly)
 * Yaranga
 * Intensional logic

Disambiguation page developed into primary topic

 * /Bootstrapping and /Bootstrapping (disambiguation)

Sections

 * Shamanism
 * Shamanism
 * Shamanism
 * Bob the Builder
 * Ket people
 * Mask
 * Eskimo
 * Citing sources/Further considerations
 * Hadza people

Stubs

 * Ayoreo

Other contributions

 * Shamanism
 * Siberian Yupik
 * Orca
 * Tupilaq
 * Soul travel
 * Vision quest
 * Masks among Eskimo peoples
 * Elementary substructure
 * Multimodal logic
 * Silap Inua
 * Libation

Debates, excluding talk pages of the above ones

 * Talk:Lambda calculus
 * Talk:Normal subgroup
 * Talk:Shamanism
 * Talk:Magma (algebra)

Proposals on talk pages

 * Talk:Magma (algebra)
 * Talk:Jan Eskymo Welzl
 * Talk:Integration by substitution, making link to my subpage User:Physis/Substitution theorem for indefinite integral

Discouraging merge

 * Talk:Lambda calculus
 * Shamanistic remnants in Hungarian folklore
 * Transitivity (grammatical category)
 * Talk:Sound mimesis in various cultures

Debts, promises

 * Hermeneutic circle
 * Shamanism (see intended new article /Hunter-gatherer cultures and ecology)
 * /Imre Ruzsa

New redirects

 * Hadzapi pointing to Hadza people

Articles

 * /Substitution theorem for indefinite integral, made inspired from an initiated work on Talk:Integration by substitution
 * /Gödel-Herbrand-Kleene equational calculus
 * /Canonical calculus
 * /Hunter-gatherer cultures and ecology
 * /Warfare among Eskimo peoples

Sections

 * /Shamanhood

Articles yet intact that are planned to work on

 * World tree (Hungarian)
 * Táltos
 * Twin (mythology)

To do
Goals Coordination Eskimos Languages Culture Plans News
 * Diószegi 1960 must be matched with the newer Terebess edition
 * Incorporate sg contribution into Philosophy of mathematics article
 * Soul dualism must be extended using
 * Hungarian coarticle Kettős lélek
 * Shamanism among Eskimo peoples
 * Shamanistic remnants in Hungarian folklore
 * Initialize a "Special language" section in Shamanism (below subsection "Music, songs"), based on Hoppál 2005: 50 and Shamanism among Eskimo peoples
 * Using Fock 1963, extend
 * Waiwai
 * Soul dualism (Fock 1963: 14–19)
 * Shamanism (Fock 1963: 113, 123–132)
 * Imitation of natural sounds in various cultures (Fock 1963: 107)
 * Soul travel (Fock 1963: 16)
 * Using Csonka-Takács & Czövek & Takács 2002,
 * initiate Ethnosemiotics (Darányi 2002: 660)
 * Using Меновщиков 1964,
 * try to extend Shamanism among Eskimo peoples
 * extend Sireniki Eskimo language
 * initiate Sireniki Eskimos
 * Fock 1963: 14 is a good reference for my claims that many seemingly different phenomena of a culture can be understood by grasping the underlying soul concept, add it to
 * Shamanism among Eskimo peoples
 * Shamanism
 * Soul dualism
 * Using Pásztorné & Várterész 2003,
 * extend Hilbert-style deduction system, even rewrite it, see its talk page
 * Add criticism of Johansen 2005 to Soul dualism
 * Arrange consistent link system among
 * Сибирское областничество
 * Siberian separatism
 * wikimedia:Image:SB - Altay shaman with gong.jpg
 * Link together Ágnes Heller and Phronesis
 * Provide links from Yoik to Imitation of natural sounds related to various shamanistic beliefs or practice; and from Katajjaq to Imitation of natural sounds in various cultures
 * Write an article about cut rule (name is redirected, its liberation must be asked from an administrator). Reasons: notion of cut rule crosscuts a plethora of calculi (sequent calculus, natural deduction, maybe even Hilbert-style deduction system), thus, the present solution with being merged into Sequent calculus cannot be justified. It is a facility of "naming", its main function is reuse. The new article must be linked with Structural rule and Cut elimination appropriately.
 * Work Diószegi 1971 into Shamanistic remnants in Hungarian folklore. Work other data in the same volume into the appropriate articles.
 * Mention goog modularizing notion vector area in articles Surface integral and Divergence theorem
 * Check reply to Talk:Vector calculus
 * Watch Talk:Sireniki Eskimo language
 * Do renaming if approved
 * Disambiguation page for Sireniki Eskimo (dispatching Sireniki Eskimo (people) and Sirenik language) must be updated if Sirenik language is moved to Sireniki Eskimo (language)
 * Extend Spider and Cultural depictions of spiders, based on Sireniki Eskimos and Siberian Yupik
 * Mention Siberian Yupik in Sireniki Eskimo (people) at the spider motif
 * Ask image from German Tupilak page to be uploaded to Wikimedia Commons, then insert it into Tupilaq
 * Initiate Earth-diver motif, provide links to it from Ket people, Dualistic cosmology, Loon. Now, it is Ket people that contains good references about the topic.
 * Check whether information stored in page numbers remained intact near 20 November (+- few days)
 * Solve Time dilation (debated in Talk:Time dilation), by possibly initiating a standalone article. The original motivation originates by time dilation myth of Sireniki Eskimos, see Sireniki Eskimos. Peek into Time travel in fiction. For a theoretical background, use, Chronotope, Postmodern literature, . Dia discusses the motif on page 55, analyze also blurring sacred vs profane by.
 * Initiate Shamanic language based on
 * Shamanism among Eskimo peoples
 * and
 * Initiate Shamanhood (possibly merged into Shamanism), based on talk pages of (co)articles Shamanistic remnants in Hungarian folklore = A magyar népi műveltség samanisztikus emlékei
 * Check rename process for Imitation of natural sounds in various cultures and Imitation of natural sounds related to various shamanistic beliefs or practice
 * Check rename process for Shamanistic remnants in Hungarian folklore, and do the same in Hungarian coarticle A magyar népi műveltség samanisztikus emlékei
 * Ask an expert: was Bruno Bettelheim right? Does recent science suppose that archaic mtyhs have a therapeutic value, capable of elaborating work, resolve anxiety, or face flashback? Of course, Bettelheim gives a psychoanalytic background, but the phenomenon itself can be true even if his explanation would be dismissed.
 * Insert "wife exchange for soliciting spirits", "soliciting sea mammals" into Shamanism among Eskimo peoples, e.g. under header “Function”, based on sources Freuchen 1961: 91, 93 and Burch & Forman 1988: 95 and
 * Translate ICC Chukotka pages with google and use it: Index
 * Gulya 1975 (a vízimadarak népe)
 * Paulson 1975: 290 into World tree (Hungarian)
 * Paulson 1975: 291 into Dualistic cosmology
 * Into Imitation of natural sounds in various cultures:
 * Use it as criticism of Onomatopoeia in shamanhood.
 * Judgment
 * Rewrite lead of Hilbert-style deduction system (maybe the whole treatment). The main difference of this approach from the other ones, it is noot the trade-off] between the numbers of axioms vs rules of inference, but the simplicity of judgment concept. The notion of context is dealt with explicitly in [[natural deduction and in sequent calculus, unlike in Hilbert-style deduction system: in the latter approach, judgment consists of the standalone formula itself. See details in Sequent calculus. Possibly write a standalone article about Judgment (mathematical logic), and possibly also one about Context (mathematical logic). Use Judgment (disambiguation), Sequent calculus, Talk:Hilbert-style deduction system. Extend Metatheorem.
 * Consider writing Judgment (type theory). Linkify occurrences of "judgment" in Simply typed lambda calculus using it.
 * Should they have a superarticle? The idea of a Judgment (formal theory) can be supported by.
 * Ethnosemiotics article can use
 * Extend Caribou Inuit per Gabus
 * Voigt 1977:254 into Shamanism standalone, mention also in introductory part of of Shamanism. Voigt 1977: 256… into Shamanism. Voigt attributes it to hunter-gatherer, Hoppál is more critical.
 * Initiate new article Shaman song, possibly by modularizing Shamanism, and link it together with Onomatopoeia, Imitation of natural sounds in various cultures, Onomatopoeia in shamanhood. Use for source:
 * Add Hoppál 1977: 275 to Soul dualism, it explains the main motivation.
 * Extend Kamasins and Kamassian language, and initiate (split off) Kamassian Lament, using a rather detailed source.
 * Write in Hilbert-style deduction system, that closing axioms with all possible generalizations are sometimes approached as a rule of inference, but then the lemma about soundness of rules of inference does not hold any more. See mathematical logic book of Bolyai series.
 * Extend Silap Inua (almost all my Eskimo books can contribute, even the Siberian Yupik ones), and read articles of Études Inuit Studies.
 * Talk:Magma (algebra)
 * Extend Prehistoric medicine
 * Add Hoppál 1977: 275 to Soul dualism, it explains the main motivation.
 * Extend Kamasins and Kamassian language, and initiate (split off) Kamassian Lament, using a rather detailed source.
 * Write in Hilbert-style deduction system, that closing axioms with all possible generalizations are sometimes approached as a rule of inference, but then the lemma about soundness of rules of inference does not hold any more. See mathematical logic book of Bolyai series.
 * Extend Silap Inua (almost all my Eskimo books can contribute, even the Siberian Yupik ones), and read articles of Études Inuit Studies.
 * Talk:Magma (algebra)
 * Extend Prehistoric medicine

Letters

 * /Hilbert-style deduction system
 * /Non-logical symbol

Hadza materials

 * /Spiritual culture of the Hadzabe
 * /Diachronical description of Hadzabe beliefs,
 *
 * /Avaibility of Hadza linguistics (introductory or corpus)

Interested in
/Interested in

Former watchlist

 * /Former watchlist

Off-line work for my contributions under debate

 * /Noaidi
 * /Hermeneutic circle
 * /Certain relatedness of Eskimo cultures, far from homogeneity
 * /Siberian Yupik
 * /Logical axiom
 * /Proportion and ratio

Archive

 * /Archive1

Sandbox
/Sandbox

de:User:Physis fr:User:Physis hu:User:Physis sk:User:Physis sv:Användare:Physis ru:User:Physis

title: User:Physis


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Portal:Humanities/SchoolHeader

W ELCOME TO THE S CHOOL OF        PART OF THE F ACULTY FOR H UMANITIES.

Usage
For use with schools part of the faculty for humanities. Add: To the top of relevant schools pages. Will appear as above except without a link to the school's page it is present on (MediaWiki feature) and with a school filled in for (school_name).

Members
All schools (in fact, all articles) using this template are listed at Category:Global header humanities schools using a template not included in this page.

source: wikimedia

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

title: Portal:Humanities/SchoolHeader


text:
dogmaattinen

Etymology
(see 🇨🇬), ultimately from and adapted with the adjectival suffix.

Adjective

 * 1) dogmatic

title: dogmaattinen


text:
Ali-Naqi Vaziri

Ali-Naqi Vaziri (October 1, 1886 in Tehran – September 9, 1979) was a composer, thinker and a celebrated player of the tar. He is considered a revolutionary icon in the history of 20th-century Persian music. His name was also transcribed as Ali Naghi Vaziri.

Ali-Naqi Vaziri was born in October 1, 1886, in Tehran, Qajar Iran. He was one of the seven children of Musa Khan Vaziri (a prominent official in the Persian Cossack Brigade) and Bibi Khanoom Astarabadi, a notable Iranian writer, satirist and one of the pioneering figures in the women's movement of Iran; her book Ma'ayeb al-Rejal (Failings of Men, also translated as Vices of Men) is considered by some as the first declaration of women's rights in the modern history of Iran. The celebrated painter Hassan Ali Khan Vaziri was his brother.

Ali-Naqi Vaziri was a master of Persian classical music, so he was able to play the tar in a style very reminiscent of that of Mirza Abdollah. He always looked for new dimensions and perspectives in musical expression, and by doing so he revolutionized the style of playing the tar. He was the first to transcribe the classical radif of the Persian music. He developed the sori and koron symbols to annotate Persian quarter-tone notes in standardized musical notation.

Vaziri for years was the director of the Tehran Conservatory of Music and a professor at the University of Tehran.

Innovations
Vaziri was one the first Persian musicians in the 20th century to go to Europe to study music, and after his return to Tehran in 1924. He was for a long time the only traditional instrumentalist familiar with and promoted the theory of Western classical music. He wrote the first transcription of Iranian music using European staff notation in his lute instruction book Dastur-e Tar, published in Iran (1913) and Germany (1923). It contained material from the Iranian oral classical canon (known as the radif), as well as arrangements inspired by composers like Beethoven, Rossini, and Schubert. Vaziri was the first to introduce and promote equal moderation in classical Persian music; in this way, each octave was evenly divided into 24 notes. This method made it possible to use Western system to harmonise traditional Persian melodies. His experiments with musical scale was work in the direction of blending Western polyphony into Persian music. His creation of the 24 step scale was created "with the intention of accommodating the application of Western harmony to musical compositions within Persian modes." He first described this view briefly in The Grammar and then in more detail in Theoretical Music. He invented a new Persian music notation for accidentals, calling the additions sori and koron; the first raises the bottom of a note by a quarter of a step and the second lowers it by a quarter of a step. Vaziri's theory for classical Persian music was heavily rejected since the 1960s.

Vaziri trained students, some of whom became famous in Persian music, including: Abolhassan Saba, Ruhollah Khaleghi, Javad Maroufi, Heshmat Sanjari. He invited some artists of his time (such as Ali Dashti, Ali Akbar Dehkhoda, Gholamreza Rashid-Yasemi, Badiozzaman Forouzanfar and Hassan Taqizadeh) together to establish an "Academy of Fine Arts" with the aim of collecting a culture of musical words and which may have become the basis for the formation of the Academy of Persian Language and Literature.

title: Ali-Naqi Vaziri


text:
UK Health Security Agency

The UK Health Security Agency (UKHSA) is a government agency in the United Kingdom, responsible since April 2021 for England-wide public health protection and infectious disease capability and replacing Public Health England. It is an executive agency of the Department of Health and Social Care (DHSC).

The formation of the UKHSA essentially transferred Public Health England's health improvement functions to DHSC, while its health protection elements form part of the new government agency. Staff and systems were moved into the new organisation in 2021. PHE continued to have a shadow existence until September 2021. UKHSA became fully operational on 1 October 2021.

Formation
A new organisation, initially to be called the Centre for Health Protection, was proposed by Matt Hancock, the Secretary of State for Health and Social Care, in July 2020 to combine NHS Test and Trace, the Joint Biosecurity Centre and the health protection functions of Public Health England (PHE). Under the name of the National Institute for Health Protection (NIHP), the organisation was established by Hancock on 18 August 2020 as a single leadership structure bringing together NHS Test and Trace, the Joint Biosecurity Centre and all of PHE. Due to the ongoing COVID-19 pandemic, the new organisation was not formally established until 1 April 2021, by which time it was called the UK Health Security Agency. It reports directly to the Secretary of State for Health and Social Care.

Baroness Harding was the interim executive chair of the new organisation from 18 August 2020 to 7 May 2021. In 2022, the High Court found that her appointment to the position broke equalities legislation. She had been the chair of NHS Improvement since 2017, and at the time was head of the NHS Test and Trace programme, established in May 2020. She was a former chief executive of the TalkTalk Group who sits in the House of Lords as a member of the Conservative Party and is married to Conservative Party Member of Parliament John Penrose. During questioning by the Science and Technology Committee of the House of Commons, Harding said she held the interim leadership while a full application process was carried out.

In August 2020, Health Secretary Matt Hancock said that the NIHP would learn from South Korea and from Germany's Robert Koch Institute "where their health protection agencies have a huge, primary, focus on pandemic response".

The Telegraph first leaked news of the plans for the new agency on 16 August 2020. They claimed that Public Health England was to be "scrapped" and replaced by a single body combining it with NHS Test and Trace, in response to the COVID-19 pandemic. A leaked memo to staff written by the head of Public Health England, Duncan Selbie, said that the aim of the new body was to boost expertise with "much needed new investment". Selbie apologised to staff that the news of the organisation's demise was briefed to The Telegraph before they were told.

In February 2021, Harding said that the new body would not be "fully staffed and up and running" until October 2021.

On 24 March 2021, Hancock announced that the organisation would be formally established on 1 April 2021 under the new name of the UK Health Security Agency, with Dr Jenny Harries stepping down as England's Deputy Chief Medical Officer to become chief executive, and Ian Peters to be chair. Hancock also confirmed that Harries would take over from Harding as lead of England's test, trace and isolate programme. Harries assumed these roles following a hand-over period which lasted until Harding departed on 7 May 2021.

Role
The responsibilities of the UKHSA include:
 * The health protection functions of Public Health England
 * Planning and executing the response to external health threats such as pandemics
 * The Joint Biosecurity Centre
 * NHS Test and Trace
 * Regulation of Coronavirus diagnostic devices

UKHSA collaborates with Public Health Scotland, Public Health Wales and Northern Ireland's Public Health Agency.

Response
A 2020 BMJ editorial described the creation of the agency as "extremely foolhardy". It characterised the National Institute for Health Protection as "seem[ing] remarkably similar to the Health Protection Agency abolished in 2013." An August 2020 editorial in The Spectator welcomed the return to an organisation similar in remit to that agency; it criticised Public Health England's focus on health improvement topics such as obesity and binge drinking, arguing that these should be tackled by local NHS health teams. In August 2020 The Telegraph welcomed the change, characterising PHE as the quango "responsible for many critical failures over the course of this [COVID-19] pandemic" that had to be scrapped.

On 2 September 2020, more than 70 health organisations wrote to the government to express concern about the future of health improvement work under these changes.

The appointment of Lady Harding as interim executive chair of the new body was criticised by health experts as she did not have a background in health, and because of her political position. The Guardian quoted allies of hers who, in response, said that she had quickly learned after being appointed chair of NHS Improvement in 2017 and that she had a record of "getting things done" while working in business.

The timing of the reorganisation, during the ongoing pandemic response, was criticised by various health experts and other bodies, including the editorial in the BMJ, the Institute for Government, the King's Fund, and Christina Marriott, the chief executive of the Royal Society for Public Health. An editorial in The Guardian compared it to "reorganising a fire brigade as it tries to put out a blaze" and said the decision had been made without proper consultation or scrutiny.

title: UK Health Security Agency


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Mikki Kendall

Mikki Kendall (born October 23, 1976) is an author, activist, and cultural critic. Her work often focuses on current events, media representation, the politics of food, and the history of the feminist movement. Penguin Random House published her graphic novel Amazons, Abolitionists, and Activists in 2019, while her political nonfiction book Hood Feminism was released in early 2020.

Early life and education
Kendall was born in Chicago, Illinois on October 23, 1976 and was raised in the Hyde Park neighborhood of Chicago. Kendall is a 2005 graduate of the University of Illinois at Urbana Champaign. Kendall also has a Masters in Writing and Publishing from DePaul University.

Career
A veteran of the United States Army, Kendall worked in government service until 2013 when she left her job at the Department of Veterans Affairs to pursue her writing career full-time.

Kendall is currently an essayist and cultural critic. She has written for The Guardian, The Boston Globe, NBC News, The Washington Post, Bustle, Essence and Eater (with her essay in Eater "Hot Sauce in Her Bag" named a Best Food Essay by the Association of Food Journalists ). She has appeared as a cultural commentator on NPR, Al Jazeera English, and the BBC.

She is recognizable as a member of Black Twitter. Kendall is also the creator of the viral Twitter hashtags #SolidarityIsForWhiteWomen, which criticized racism in the feminist movement, as well as #FastTailedGirls, a reference to the hypersexualization of Black girls, and #FoodGentrification, about the marginalization of traditional foods by commercial interest.

Kendall edited the science-fiction anthology Hidden Youth for Crossed Genres Press in 2016. Kendall's graphic novel history Amazons, Abolitionists, and Activists: A Graphic History of Women's Fight for Their Rights was released in November 2019 by Ten Speed Press and her political-nonfiction book Hood Feminism: Notes From the Women That a Movement Forgot was released in February 2020 by Viking Books. It criticizes the feminist movement for being largely "the province of the privileged" and ignoring problems that disproportionately impact communities of color, including poverty, housing, medical care, and racism.

Awards and honors

 * 2017: Best Food Essay Award from the Association of Food Journalists for "Hot Sauce in Her Bag: Southern Black identity, Beyoncé, Jim Crow, and the pleasure of well-seasoned food"

title: Mikki Kendall


text:
Wikipedia:Peer review/Bethel Assembly of God Church/archive1

Bethel Assembly of God Church
I've listed this article for peer review because… This page was one deleted. It has been recreated with neutral content

Thanks, CE 09:36, 21 September 2019 (UTC)

I have edited the article a little to improve some of the basics. It needs some more work. Here are a few suggestions for improvements: As it stands, the article is only barely demonstrating notability, though I suspect it is notable. I hope these suggestions help you think about how to expand and solidify the article. Best, The Mirror Cracked (talk) 05:21, 14 November 2019 (UTC)
 * The article needs more reliable, independent sources. I have tagged the languages section as needing a reference, but most of the information in the lede and infobox is unreferenced.
 * If you can find good reliable sources, you could considerably expand the article. It would be good to see the history and development of the church. Apart from it's size, what makes it interesting and unusual?
 * You could add a section about the church's teachings (again, if you have a proper source)
 * What impact/influence has the church had on Bangalore, and the local neighborhood?
 * Expand on what the "outstations" are, and their connection to the main church


text:
Talk:David Durenberger

Untitled
Should really have something about his censure on there- one of only 9 Senators!! — Preceding unsigned comment added by <IP_ADDRESS> (talk) 04:18, 20 February 2012 (UTC)

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title: Talk:David Durenberger


text:
Beevor's axiom

Beevor's Axiom is the idea that the brain does not know muscles, only movements. In other words, the brain registers the movements that muscles combine to make, not the individual muscles that are making the movements. Hence, this is why one can sign their name (albeit poorly) with their foot. Beevor's Axiom was coined by Dr. Charles Edward Beevor, an English neurologist.

Dr. Beevor presented Beevor's Axiom in a series of four lectures from June 3, 1903 to July 4, 1903 before the Royal College of Physicians of London as part of the Croonian Lectures. His experiments showed that when an area of the cortex was stimulated, the body responded with a movement, not just a single muscle. Dr. Beevor concluded that “only co-ordinated movements are represented in the excitable cortex”

In relation to Beevor's Axiom, it has been found that the brain encodes sequences, such as playing the piano, signing our name, wiping off a counter, and chopping vegetables, and once encoded and practiced, it takes less brain activity to perform them. This supports Beevor's Axiom, because the brain can recall movements easier than it can learn them.

Beevor's Axiom is only partially true, however. Most behavior of muscles is encoded in the primary motor cortex (M1) and separated by muscle group. In an effort to understand the encoding in the M1, researchers observed commands of monkeys. Muscle cells changed firing rate according to the direction of the arm movements. Each neuron has one direction that elicits the greatest response. Some M1 neurons encode muscle contractions, while others react to particular movements, regardless of the muscles used to perform them. The key characteristic of the primary motor cortex is its dynamic nature; the M1 changes based on experience. The supplementary motor area (SMA) plays a key role in initiating motion sequences. The premotor cortex (PMA) plays a key role when motor sequences are guided by external events. They map behaviors as opposed to the M1 which maps specific movements. This could cause in issue in brain–computer interface research. If a researcher tries to excite only a muscle, it might be impossible without expecting a full movement.

title: Beevor's axiom


text:
Control structures

Course Navigation

Flow Control Overview
A control structure is like a block of programming that analyses variables and chooses a direction in which to go based on given parameters. The term flow control details the direction the program takes (which way program control "flows"). Hence it is the basic decision-making process in computing; It is a prediction.

Basic Terminologies
Those initial conditions and parameters are called preconditions. Preconditions are the state of variables before entering a control structure. Based on those preconditions, the computer runs an algorithm (the control structure) to determine what to do. The result is called a post condition. Post conditions are the state of variables after the algorithm is run.

An Example
Let us analyse flow control by using traffic flow as a model. A vehicle is arriving at an intersection. Thus, the precondition is the vehicle is in motion. Suppose the traffic light at the intersection is red. The control structure must determine the proper course of action to assign to the vehicle.

 * Precondition: The vehicle is in motion.
 * Treatments of Information through Control Structures
 * Is the traffic light green? If so, then the vehicle may stay in motion.
 * Is the traffic light red? If so, then the vehicle must stop.
 * End of Treatment
 * Post condition: The vehicle comes to a stop.

Thus, upon exiting the control structure, the vehicle is stopped.

can also be form as a structure

IF-THEN Statement
The IF-THEN statement is a simple control that tests whether a condition is true or false. The condition can include a variable, or be a variable. If the variable is an integer 2, it will be true, because any number that is not zero will be true. If the condition is true, then an action specified within the "THEN" statement occurs. If the condition is false, nothing is done. To illustrate:

 * IF variable is true
 * THEN take this course of action.

If the variable indeed holds a value consistent with being true, then the course of action is taken. If the variable is not true, then there is no course of action taken.

IF-THEN-ELSE Statement
IF-THEN statements test for only one action. With an IF-THEN-ELSE statement, the control can "look both ways" so to speak, and take a secondary course of action if the conditional statement is not true. If the condition is true, then an action occurs. If the condition is false, an alternate action specified in the "ELSE" statement occurs. To illustrate:

 * IF variable is true
 * THEN take this course of action
 * ELSE call another routine

In this case, if the variable is true, the program takes a certain course of action and completely skips the ELSE clause. If the variable is false, the control structure calls a routine and completely skips the THEN clause.

Note that you can combine ELSE's with other IF's, allowing several tests to be made. In an IF-THEN-ELSEIF-THEN-ELSEIF-THEN-ELSEIF-THEN structure, tests will stop as soon as the program finds a condition that is true. That's why you'd probably want to put the most "likely" test first, for efficiency (Remembering that ELSE's are skipped if the first condition is true, meaning that the remaining portions of the IF-THEN-ELSEIF... would not be processed). eg:

 * In case your computer doesn't start
 * IF a floppy disk is in the drive
 * THEN remove it and restart
 * ELSE IF you don't have any OS installed
 * THEN install an OS
 * ELSE call the hotline

You can have as many ELSE IF's as you like.

By contrast, you can also place multiple IF-THEN statements in succession to check for every conditional. By this method, the program will not stop evaluating subsequent conditionals when it finds one that is true.

WHILE and DO-WHILE Loops
A WHILE loop is a process in which a loop is initiated until a condition has been met. This structure is useful when performing iterative instructions to satisfy a certain parameter. To illustrate:

 * Precondition: variable X is equal to 1
 * WHILE X is not equal to 9
 * Add 1 to X

This routine will add 1 to X until X is equal to 9, at which point the control structure will quit and move on to the next instruction. Note that when the structure quits, it will not execute the Add function: when X is equal to 9, it will skip over the clause that is attached to the WHILE. This instruction is useful if a parameter needs to be tested repeatedly before acceptance. In the above example, if X is already equal to or greater than 9 when the program reaches the WHILE loop, the program will entirely skip the loop.

DO-WHILE Loops
A DO-WHILE loop is nearly the exact opposite to a WHILE loop. A WHILE loop initially checks to see if the parameters have been satisfied before executing an instruction. A DO-WHILE loop executes the instruction before checking the parameters. To illustrate:

 * DO Add 1 to X
 * WHILE X is not equal 9

As you can see, the example differs from the first illustration, where the DO action is taken before the WHILE. The WHILE is inclusive in the DO. As such, if the WHILE results in a false (X is equal to 9), the control structure will break and will not perform another DO. Note that if X is equal to or greater than 9 prior to entering the DO-WHILE loop, then the loop will never terminate.

FOR Loops
A FOR loop is an extension of a while loop which iterates for a predetermined length. A for loop usually has three commands. The first is used to set a starting point (like x = 0). The second is the end condition (same as in a while loop) and is run every round. The third is also run every round and is usually used to modify a value used in the condition block.

 * FOR X is 0. X is less than 10. add 1 to X.

This loop would be run 10 times (x being 0 to 9) so you won't have to think about the X variable in the loop, you can just put code there. Here is a while loop that does the same:

 * X is 0
 * WHILE X is less than 10
 * add 1 to X

Some programming languages also have a for each loop which will be useful when working with arrays (and collections). A for each loop is an even more automated while loop that cycles through array's and such. Example :

 * FOR EACH student IN students
 * give a good mark to student

 * Next Lesson
 * Previous Lesson
 * Course Home Page

title: Control structures


text:
copybook

Noun

 * 1) A student's exercise book containing samples of good handwriting to be copied.
 * 2)  A notebook containing blank, often lined, pages for writing answers.
 * 3)  A series of instructions or data definitions copied into multiple programs from a shared library; boilerplate.
 * 1)  A series of instructions or data definitions copied into multiple programs from a shared library; boilerplate.

title: copybook


text:
Calculus/Limits/An Introduction to Limits

Limits, the first step into calculus, explain the complex nature of the subject. It is used to define the process of derivation and integration. It is also used in other circumstances to intuitively demonstrate the process of "approaching".

Intuitive Look into Limits
The limit is one of the greatest tools in the hands of any mathematician. We will give the limit an approach. Because mathematics came only due to approaches... remember?!

We designate limit in the form:
 * $$\lim_{x\to a}f(x)$$

This is read as "The limit of $$f$$ of $$x$$ as $$x$$ approaches $$a$$". This is an important thing to remember, it is basic notation which is accepted by the world.

We'll take up later the question of how we can determine whether a limit exists for $$f(x)$$ at $$a$$ and, if so, what it is. For now, we'll look at it from an intuitive standpoint.

Let's say that the function that we're interested in is $$f(x)=x^2$$, and that we're interested in its limit as $$x$$ approaches $$2$$. Using the above notation, we can write the limit that we're interested in as follows:
 * $$\lim_{x\to2}x^2$$

One way to try to evaluate what this limit is would be to choose values near 2, compute $$f(x)$$ for each, and see what happens as they get closer to 2. There are two ways to approach values near 2. One is to approach from below, and the other is to approach from above:

The table above is the case from below.

The table above is the case from above.

We can see from the tables that as $$x$$ grows closer and closer to 2, $$f(x)$$ seems to get closer and closer to 4, regardless of whether $$x$$ approaches 2 from above or from below. For this reason, we feel reasonably confident that the limit of $$x^2$$ as $$x$$ approaches 2 is 4, or, written in limit notation,
 * $$\lim_{x\to2}x^2=4$$

We could have also just substituted 2 into $$x^2$$ and evaluated: $$(2)^2=4$$. However, this will not work with all limits.

Now let's look at another example. Suppose we're interested in the behavior of the function $$f(x)=\frac{1}{x-2}$$ as $$x$$ approaches 2. Here's the limit in limit notation:
 * $$\lim_{x\to 2}\frac{1}{x-2}$$

Just as before, we can compute function values as $$x$$ approaches 2 from below and from above. Here's a table, approaching from below:

And here from above:

In this case, the function doesn't seem to be approaching a single value as $$x$$ approaches 2, but instead becomes an extremely large positive or negative number (depending on the direction of approach). Well, one says such a limit does not exist because no finite number is approached. This arises the concept of infinity: an undefined quantity and the limit is also called infinite limit or limit without a bound.

Note that we cannot just substitute 2 into $$\frac{1}{x-2}$$ and evaluate as we could with the first example, since we would be dividing by 0.

Both of these examples may seem trivial, but consider the following function:
 * $$f(x)=\frac{x^2(x-2)}{x-2}$$

This function is the same as
 * $$f(x)=\left\{\begin{matrix}x^2&\text{if }x\ne2\\ \mbox{undefined}&\text{if }x=2\end{matrix}\right.$$

Note that these functions are really completely identical; not just "almost the same," but actually, in terms of the definition of a function, completely the same; they give exactly the same output for every input.

In elementary algebra, a typical approach is to simply say that we can cancel the term $$(x-2)$$, and then we have the function $$f(x)=x^2$$. However, that would be inaccurate; the function that we have now is not really the same as the one we started with, because it is defined when $$x=2$$, and our original function was specifically not defined when $$x=2$$. This may seem like a minor point, but from making this kind of assumptions we can easily derive absurd results, such that $$0=1$$ (see Mathematical Fallacy § All numbers equal all other numbers in Wikipedia for a complete example). Even without calculus we can avoid this error by stating that:
 * $$f(x)=\frac{x^2(x-2)}{x-2}=x^2\text{ when }x\ne2$$

In calculus, we can introduce a more intuitive and also correct way of looking at this type of function. What we want is to be able to say that, although the function isn't defined when $$x=2$$, it works almost as if it was. It may not get there, but it gets really, really close. For instance, $$f(1.99999)=3.99996$$. The only question that we have is: what do we mean by "close"?

Informal Definition of a Limit
As the precise definition of a limit is a bit technical, it is easier to start with an informal definition; we'll explain the formal definition later.

We suppose that a function $$f$$ is defined for $$x$$ near $$c$$ (but we do not require that it be defined when $$x=c$$).

Notice that the definition of a limit is not concerned with the value of $$f(x)$$ when $$x=c$$ (which may exist or may not). All we care about are the values of $$f(x)$$ when $$x$$ is close to $$c$$, on either the left or the right (i.e. less or greater).

Limit can also be understood as: $$x$$ is infinitely approaching to $$c$$ but never equals to $$c$$, just like the function $$f(x)=\frac{1}{x}$$, which infinitely approaches to $$0$$ but never equals $$0$$.

Rules and Identities
Now that we have defined, informally, what a limit is, we will list some rules that are useful for working with and computing limits. You will be able to prove all these once we formally define the fundamental concept of the limit of a function.

First, the constant rule states that if $$f(x)=b$$ (that is, $$f$$ is constant for all $$x$$) then the limit as $$x$$ approaches $$c$$ must be equal to $$b$$. In other words

 * Example: $$\lim_{x\to6}5=5$$

Second, the identity rule states that if $$f(x)=x$$ (that is, $$f$$ just gives back whatever number you put in) then the limit of $$f$$ as $$x$$ approaches $$c$$ is equal to $$c$$. That is,

 * Example: $$\lim_{x\to6}x=6$$

The next few rules tell us how, given the values of some limits, to compute others.

Notice that in the last rule we need to require that $$M$$ is not equal to 0 (otherwise we would be dividing by zero which is an undefined operation).

These rules are known as identities; they are the scalar product, sum, difference, product, and quotient rules for limits. (A scalar is a constant, and, when you multiply a function by a constant, we say that you are performing scalar multiplication.)

Using these rules we can deduce another. Namely, using the rule for products many times we get that
 * $$\lim_{x\to c}f(x)^n=\left(\lim_{x\to c}f(x)\right)^n=L^n$$ for a positive integer $$n$$.

This is called the power rule.

As a result, we can safely say that all limits for polynomial functions can be deduced into several limits that satisfy the identity rule and thus easier to compute. Find the limit $$\lim_{x\to 2}4x^3$$.
 * Example 1

We need to simplify the problem, since we have no rules about this expression by itself. We know from the identity rule above that $$\lim_{x\to2}x=2$$. By the power rule, $$\lim_{x\to2}x^3=\left(\lim_{x\to2}x\right)^3=2^3=8$$. Lastly, by the scalar multiplication rule, we get $$\lim_{x\to2}4x^3=4\cdot\lim_{x\to2}x^3=4\cdot 8=32$$.

$$\blacksquare$$ Find the limit $$\lim_{x\to2}\Big[4x^3+5x+7\Big]$$.
 * Example 2

To do this informally, we split up the expression, once again, into its components. As above, $$\lim_{x\to2}4x^3=32$$.

Also $$\lim_{x\to2}5x=5\cdot\lim_{x\to2}x=5\cdot 2=10$$ and $$\lim_{x\to2}7=7$$. Adding these together gives
 * $$\lim_{x\to2}\Big[4x^3+5x+7\Big]=\lim_{x\to2}4x^3+\lim_{x\to2}5x+\lim_{x\to2}7=32+10+7=49$$.

Find the limit $$\lim_{x\to2}\frac{4x^3+5x+7}{(x-4)(x+10)}$$.
 * $$\blacksquare$$
 * Example 3

From the previous example the limit of the numerator is $$\lim_{x\to 2}\Big[4x^3+5x+7\Big]=49$$. The limit of the denominator is
 * $$\lim_{x\to2}(x-4)(x+10)=\lim_{x\to2}\big[x-4\big]\cdot\lim_{x\to2}\big[x+10\big]=(2-4)(2+10)=-24$$

As the limit of the denominator is not equal to zero we can divide. This gives
 * $$\lim_{x\to2}\frac{4x^3+5x+7}{(x-4)(x+10)}=-\frac{49}{24}$$.

Find the limit $$\lim_{x\to4}\frac{x^4-16x+7}{4x-5}$$.
 * $$\blacksquare$$
 * Example 4

We apply the same process here as we did in the previous set of examples;
 * $$\lim_{x\to4}\frac{x^4-16x+7}{4x-5}=\frac{\lim\limits_{x\to4}\Big[x^4-16x+7\Big]}{\lim\limits_{x\to4}\Big[4x-5\Big]}=\frac{\lim\limits_{x\to4}x^4-\lim\limits_{x\to4}16x+\lim\limits_{x\to4}7}{\lim\limits_{x\to4}4x-\lim\limits_{x\to4}5}$$.

We can evaluate each of these; $$\lim_{x\to4}(x^4)=256\ ,\ \lim_{x\to4}(16x)=64\ ,\ \lim_{x\to4}(7)=7\ ,\ \lim_{x\to4}(4x)=16\ ,\ \lim_{x\to4}(5)=5.$$ Thus, the answer is $$\frac{199}{11}$$. Find the limit $$\lim_{x\to2}\frac{x^2-3x+2}{x-2}$$.
 * $$\blacksquare$$
 * Example 5

In this example, evaluating the result directly will result in a division by 0. While you can determine the answer experimentally, a mathematical solution is possible as well.

First, the numerator is a polynomial that may be factored: $$\lim_{x\to2}\frac{(x-2)(x-1)}{x-2}$$

Now, you can divide both the numerator and denominator by $$(x-2)$$: $$\lim_{x\to2}(x-1)=(2-1)=1$$

Remember that the limit is a method to determine the approaching value of a function instead of the value of the function itself. So, though the function is undefined at $$x=2$$, the value of the function is approaching to $$1$$ when $$x \rightarrow 2$$
 * $$\blacksquare$$
 * Example 6

Find the limit $$\lim_{x\to0}\left[\frac{1-\cos(x)}{x}\right]$$.

To evaluate this seemingly complex limit, we will need to recall some sine and cosine identities (see Chapter ). We will also have to use two new facts. First, if $$f(x)$$ is a trigonometric function (that is, one of sine, cosine, tangent, cotangent, secant and cosecant functions), and is defined at $$a$$, then $$\lim_{x\to a}f(x)=f(a)$$.

Second, $$\lim_{x\to 0}\frac{\sin(x)}{x}=1$$. This can be proved using squeeze theorem. Note that L'Hospital's rule is not allowed to be used to evaluate this limit because it causes circular reasoning,

in the sense that differentiating $$\sin x$$.requires this limit to equal one, which is exactly the equation that is being proven.

Method 1 (before learning L'Hôpital's rule):

To evaluate the limit, recognize that $$1-\cos(x)$$ can be multiplied by $$1+\cos(x)$$ to obtain $$1-\cos^2(x)$$ which, by our trig identities, is $$\sin^2(x)$$. So, multiply the top and bottom by $$1+\cos(x)$$. (This is allowed because it is identical to multiplying by one.) This is a standard trick for evaluating limits of fractions; multiply the numerator and the denominator by a carefully chosen expression which will make the expression simplify somehow. In this case, we should end up with:

 * $$\lim_{x\to0}\left[\frac{1-\cos(x)}{x}\right]$$
 * $$=\lim_{x\to0}\left[\frac{1-\cos(x)}{x}\cdot\frac{1}{1}\right]$$
 * $$=\lim_{x\to0}\left[\frac{1-\cos(x)}{x}\cdot\frac{1+\cos(x)}{1+\cos(x)}\right]$$
 * $$=\lim_{x\to0}\frac{\big(1-\cos(x)\big)\big(1+\cos(x)\big)}{x\big(1+\cos(x)\big)}$$
 * $$=\lim_{x\to0}\frac{1-\cos^2(x)}{x\big(1+\cos(x)\big)}$$
 * $$=\lim_{x\to0}\frac{\sin^2(x)}{x\big(1+\cos(x)\big)}$$
 * $$=\lim_{x\to0}\left[\frac{\sin(x)}{x}\cdot\frac{\sin(x)}{1+\cos(x)}\right]$$.
 * }
 * $$=\lim_{x\to0}\frac{1-\cos^2(x)}{x\big(1+\cos(x)\big)}$$
 * $$=\lim_{x\to0}\frac{\sin^2(x)}{x\big(1+\cos(x)\big)}$$
 * $$=\lim_{x\to0}\left[\frac{\sin(x)}{x}\cdot\frac{\sin(x)}{1+\cos(x)}\right]$$.
 * }
 * $$=\lim_{x\to0}\frac{\sin^2(x)}{x\big(1+\cos(x)\big)}$$
 * $$=\lim_{x\to0}\left[\frac{\sin(x)}{x}\cdot\frac{\sin(x)}{1+\cos(x)}\right]$$.
 * }
 * $$=\lim_{x\to0}\left[\frac{\sin(x)}{x}\cdot\frac{\sin(x)}{1+\cos(x)}\right]$$.
 * }
 * }

Our next step should be to break this up into $$\lim_{x\to0}\left[\frac{\sin(x)}{x}\right]\cdot\lim_{x\to0}\left[\frac{\sin(x)}{1+\cos(x)}\right]$$ by the product rule. As mentioned above, $$\lim_{x\to0}\left[\frac{\sin(x)}{x}\right]=1$$. Next, $$\lim_{x\to0}\left[\frac{\sin(x)}{1+\cos(x)}\right]=\frac{\lim\limits_{x\to0}\big[\sin(x)\big]}{\lim\limits_{x\to0}\big[1+\cos(x)\big]}=\frac{0}{1+\cos(0)}=0$$.

Thus, by multiplying these two results, we obtain 0.

$$\blacksquare$$

Note that we also cannot apply L'Hospital's rule to evaluate this limit because it causes circular reasoning.

We will now present an amazingly useful result, even though we cannot prove it yet. We can find the limit at $$c$$ of any polynomial or rational function, as long as that rational function is defined at $$c$$ (so we are not dividing by 0). That is, $$c$$ must be in the domain of the function.

We already learned this for trigonometric functions, so we see that it is easy to find limits of polynomial, rational or trigonometric functions wherever they are defined. In fact, this is true even for combinations of these functions; thus, for example, $$\lim_{x\to1}\bigg[\sin(x^2)+4\cos^3(3x-1)\bigg]=\sin(1^2)+4\cos^3(3\cdot 1-1)=\sin 1+4\cos^32$$.

The Squeeze Theorem
The Squeeze Theorem is very important in calculus, where it is typically used to find the limit of a function by comparison with two other functions whose limits are known.

It is called the Squeeze Theorem because it refers to a function $$f$$ whose values are squeezed between the values of two other functions $$g$$ and $$h$$, both of which have the same limit $$L$$. If the value of $$f$$ is trapped between the values of the two functions $$g$$ and $$h$$, the values of $$f$$ must also approach $$L$$.

Expressed more precisely:

Example: Compute $$\lim_{x\to0}x\cdot\sin\left(\tfrac{1}{x}\right)$$.

Since we know that"$-1 \le \sin(\frac{1}{x}) \le 1$"Multiplying $$x$$ into the inequality yields"x"Now we apply the squeeze theorem"x"Since $$\lim_{x\to0}-|x|=\lim_{x\to0}|x|=0$$,

$$\lim_{x\to0}x\cdot\sin\left(\tfrac{1}{x}\right)=0$$

$$\blacksquare$$

Finding Limits
Now, we will discuss how, in practice, to find limits. First, if the function can be built out of rational, trigonometric, logarithmic, or exponential functions, then if a number $$c$$ is in the domain of the function, then the limit at $$c$$ is simply the value of the function at $$c$$:"$\lim_{x \to c} f(x)=f(c)$ when $f(x)$ can be built out of rational, trigonometric, logarithmic, or exponential functions and $c \in$ the Domain of $f(x)$"If $$c$$ is not in the domain of the function, then in many cases (as with rational functions) the domain of the function includes all the points near $$c$$, but not $$c$$ itself. An example would be if we wanted to find $$\lim_{x\to0}\frac{x}{x}$$, where the domain includes all numbers besides 0.

In that case, in order to find $$\lim_{x\to c}f(x)$$ we want to find a function $$g(x)$$ similar to $$f(x)$$, except with the hole at $$c$$ filled in. The limits of $$f$$ and $$g$$ will be the same, as can be seen from the definition of a limit. By definition, the limit depends on $$f(x)$$ only at the points where $$x$$ is close to $$c$$ but not equal to it, so the limit at $$c$$ does not depend on the value of the function at $$c$$. Therefore, if $$\lim_{x\to c}g(x)=L$$, $$\lim_{x\to c}f(x)=L$$ also. And since the domain of our new function $$g$$ includes $$c$$, we can now (assuming $$g$$ is still built out of rational, trigonometric, logarithmic and exponential functions) just evaluate it at $$c$$ as before. Thus we have $$\lim_{x\to c}f(x)=g(c)$$.

In our example, this is easy; canceling the $$x$$'s gives $$g(x)=1$$, which equals $$f(x)=\frac{x}{x}$$ at all points except 0. Thus, we have $$\lim_{x\to0}\frac{x}{x}=\lim_{x\to0}1=1$$. In general, when computing limits of rational functions, it's a good idea to look for common factors in the numerator and denominator.

Specific DNE Situations
Note that the limit might not exist at all (DNE means "does not exist"). There are a number of ways in which this can occur:

"Gap"
 * Gap example.png is a gap (not just a single point) where the function is not defined. As an example, in
 * $$f(x)=\sqrt{x^2-16}$$

 * $$\lim_{x\to c}f(x)$$ does not exist when $$-4\le c\le4$$. There is no way to "approach" the middle of the graph. Note that the function also has no limit at the endpoints of the two curves generated (at $$c=-4$$ and $$c=4$$). For the limit to exist, the point must be approachable from both the left and the right.

 * Note also that there is no limit at a totally isolated point on a graph.

 * "Jump"
 * If the graph suddenly jumps to a different level, there is no limit at the point of the jump. For example, let $$f(x)$$ be the greatest integer $$\le x$$. Then, if $$c$$ is an integer, when $$x$$ approaches $$c$$ from the right $$f(x)=c$$, while when $$x$$ approaches $$c$$ from the left $$f(x)=c-1$$. Thus $$\lim_{x\to c}f(x)$$ will not exist.

 * Vertical asymptote: In $$f(x)=\frac{1}{x^2}$$
 * the graph gets arbitrarily high as it approaches 0, so there is no limit. (In this case we sometimes say the limit is infinite; see the next section.)

]
 * Infinite oscillation: These next two can be tricky to visualize. In this one, we mean that a graph continually rises above and falls below a horizontal line. In fact, it does this infinitely often as you approach a certain $$x$$-value. This often means that there is no limit, as the graph never approaches a particular value. However, if the height (and depth) of each oscillation diminishes as the graph approaches the $$x$$-value, so that the oscillations get arbitrarily smaller, then there might actually be a limit.
 * The use of oscillation naturally calls to mind the trigonometric functions. An example of a trigonometric function that does not have a limit as $$x$$ approaches 0 is
 * $$f(x)=\sin\left(\tfrac{1}{x}\right)$$
 * As $$x$$ gets closer to 0 the function keeps oscillating between $$-1$$ and 1. In fact, $$\sin\left(\tfrac{1}{x}\right)$$ oscillates an infinite number of times on the interval between 0 and any positive value of $$x$$. The sine function is equal to 0 whenever $$x=k\pi$$, where $$k$$ is a positive integer. Between every two integers $$k$$ , $$\sin(x)$$ goes back and forth between 0 and $$-1$$ or 0 and 1. Hence, $$\sin\left(\tfrac{1}{x}\right)=0$$ for every $$x=\tfrac{1}{k\pi}$$. In between consecutive pairs of these values, $$\tfrac{1}{k\pi}$$ and $$\tfrac{1}{(k+1)\pi}$$ , $$\sin\left(\tfrac{1}{x}\right)$$ goes back and forth from 0, to either $$-1$$ or 1 and back to 0. We may also observe that there are an infinite number of such pairs, and they are all between 0 and $$1/\pi$$. There are a finite number of such pairs between any positive value of $$x$$ and $$\tfrac{1}{\pi}$$ , so there must be infinitely many between any positive value of $$x$$ and 0. From our reasoning we may conclude that, as $$x$$ approaches 0 from the right, the function $$\sin\left(\tfrac{1}{x}\right)$$ does not approach any specific value. Thus, $$\lim_{x\to0}\sin\left(\tfrac{1}{x}\right)$$ does not exist.

Determining Limits
The formal way to determine whether a limit exists is to find out whether the value of the limit is the same when approaching from below and above (see at the top of this chapter). The notation for the limit approaching from below (in increasing order) is"$\lim_{x\to c^-}f(x)$, notice the negative sign on the constant $c$"The notation for the limit approaching from above (from decreasing order) is"$\lim_{x\to c^+}f(x)$, notice the positive sign on the constant $c$"For example, let us find the limits of $$f(x)=\frac{1}{x}$$ when $$x$$ is approaching $$0$$ in both directions. In other words, find $$\lim_{x\to 0^-}\frac{1}{x}$$ and $$\lim_{x\to 0^+}\frac{1}{x}$$. Recall the table we made when we are trying to intuitively feel the limit. We can use that to help us. However, if familiar enough with reciprocal functions, we can simply determine the two values by imagining the graph of the function.

The following table is when $$x$$ is approaching from below. Thus, we found that when $$x$$ is approaching from below to $$0$$, the function approaches negative infinity. In mathematical terms:

$$\lim_{x\to 0^-}\frac{1}{x}=-\infty$$

Now let's talk about the approach from above. "We found that $\lim_{x\to 0^+}\frac{1}{x}=\infty$"$$\blacksquare$$

The method of determining if limits exist is relatively intuitive. It only requires some practice to be familiar with the process. Let's use the same example: find $$\lim_{x\to 0}\frac{1}{x}$$. Since we already found that $$\lim_{x\to 0^-}\frac{1}{x}=-\infty$$ and $$\lim_{x\to 0^+}\frac{1}{x}=\infty$$, and obviously,

$$\lim_{x\to 0^-}\frac{1}{x}\ne\lim_{x\to 0^+}\frac{1}{x}$$

We can say that $$\lim_{x\to 0}\frac{1}{x}$$ does not exist. $$\blacksquare$$

Infinity Situations
Now consider the function
 * $$g(x)=\frac{1}{x^2}$$

What is the limit as $$x$$ approaches zero? The value of $$g(0)$$ does not exist; it is not defined.

Notice, also, that we can make $$g(x)$$ as large as we like, by choosing a small $$x$$, as long as $$x\ne0$$. For example, to make $$g(x)$$ equal to $$10^{12}$$, we choose $$x$$ to be $$10^{-6}$$. Thus, $$\lim_{x\to0}\frac{1}{x^2}$$ does not exist.

However, we do know something about what happens to $$g(x)$$ when $$x$$ gets close to 0 without reaching it. We want to say we can make $$g(x)$$ arbitrarily large (as large as we like) by taking $$x$$ to be sufficiently close to 0, but not equal to 0. We express this symbolically as follows:
 * $$\lim_{x\to0}g(x)=\lim_{x\to0}\frac{1}{x^2}=\infty$$

Note that the limit does not exist at $$0$$ ; for a limit, being $$\infty$$ is a special kind of not existing. In general, we make the following definition.

An example of the second half of the definition would be that $$\lim_{x\to0}-\frac{1}{x^2}=-\infty$$.

Applications of Limits
To see the power of the concept of the limit, let's consider a moving car. Suppose we have a car whose position is linear with respect to time (that is, a graph plotting the position with respect to time will show a straight line). We want to find the velocity. This is easy to do from algebra; we just take the slope, and that's our velocity.

But unfortunately, things in the real world don't always travel in nice straight lines. Cars speed up, slow down, and generally behave in ways that make it difficult to calculate their velocities.

Now what we really want to do is to find the velocity at a given moment (the instantaneous velocity). The trouble is that in order to find the velocity we need two points, while at any given time, we only have one point. We can, of course, always find the average speed of the car, given two points in time, but we want to find the speed of the car at one precise moment.

This is the basic trick of differential calculus, the first of the two main subjects of this book. We take the average speed at two moments in time, and then make those two moments in time closer and closer together. We then see what the limit of the slope is as these two moments in time are closer and closer, and say that this limit is the slope at a single instant.

We will study this process in much greater depth later in the book. First, however, we will need to study limits more carefully.


text:
Hellenistic art

Hellenistic art is the art of the Hellenistic period generally taken to begin with the death of Alexander the Great in 323 BC and end with the conquest of the Greek world by the Romans, a process well underway by 146 BC, when the Greek mainland was taken, and essentially ending in 30 BC with the conquest of Ptolemaic Egypt following the Battle of Actium. A number of the best-known works of Greek sculpture belong to this period, including Laocoön and His Sons, Venus de Milo, and the Winged Victory of Samothrace. It follows the period of Classical Greek art, while the succeeding Greco-Roman art was very largely a continuation of Hellenistic trends.

The term Hellenistic refers to the expansion of Greek influence and dissemination of its ideas following the death of Alexander – the "Hellenizing" of the world, with Koine Greek as a common language. The term is a modern invention; the Hellenistic World not only included a huge area covering the whole of the Aegean Sea, rather than the Classical Greece focused on the Poleis of Athens and Sparta, but also a huge time range. In artistic terms this means that there is huge variety which is often put under the heading of "Hellenistic Art" for convenience.

One of the defining characteristics of the Hellenistic period was the division of Alexander's empire into smaller dynastic empires founded by the diadochi (Alexander's generals who became regents of different regions): the Ptolemies in Egypt, the Seleucids in Mesopotamia, Persia, and Syria, the Attalids in Pergamon, etc. Each of these dynasties practiced a royal patronage which differed from those of the city-states. In Alexander's entourage were three artists: Lysippus the sculptor, Apelles the painter, and Pyrgoteles the gem cutter and engraver. The period after his death was one of great prosperity and considerable extravagance for much of the Greek world, at least for the wealthy. Royalty became important patrons of art. Sculpture, painting and architecture thrived, but vase-painting ceased to be of great significance. Metalwork and a wide variety of luxury arts produced much fine art. Some types of popular art were increasingly sophisticated.

There has been a trend in writing history to depict Hellenistic art as a decadent style, following the Golden Age of Classical Greece. The 18th century terms Baroque and Rococo have sometimes been applied to the art of this complex and individual period. A renewed interest in historiography as well as some recent discoveries, such as the tombs of Vergina, may allow a better appreciation of the period.

Architecture
In the architectural field, the dynasties following Hector resulted in vast urban plans and large complexes which had mostly disappeared from city-states by the 5th century BC. The Doric Temple was virtually abandoned. This city planning was quite innovative for the Greek world; rather than manipulating space by correcting its faults, building plans conformed to the natural setting. One notes the appearance of many places of amusement and leisure, notably the multiplication of theatres and parks. The Hellenistic monarchies were advantaged in this regard in that they often had vast spaces where they could build large cities: such as Antioch, Pergamon, and Seleucia on the Tigris.

It was the time of gigantism: thus it was for the second temple of Apollo at Didyma, situated twenty kilometers from Miletus in Ionia. It was designed by Daphnis of Miletus and Paionios of Ephesus at the end of the fourth century BC, but the construction, never completed, was carried out up until the 2nd century AD. The sanctuary is one of the largest ever constructed in the Mediterranean region: inside a vast court (21.7 metres by 53.6 metres), the cella is surrounded by a double colonnade of 108 Ionic columns nearly 20 metres tall, with richly sculpted bases and capitals.

Alexandria
Hellenistic Alexandria originated a distinctive architectural form language that is often referred to as baroque due to its liberal use of ornamentation and its repurposing of structural elements as ornamental elements. Perhaps partly inspired by traditional Egyptian architecture, Alexandrian architects developed new shapes such as segmental, broken, hollow and volute pediments and curved arched, concave and broken entablatures. These baroque shapes seem to have existed in Alexandria at least by the 2nd century BC. Many of them were later adopted across the Roman Empire and also significantly influenced the rock-cut tombs of Petra.

Alexandrian architecture also made heavy use of Corinthian capitals, both on exteriors and interiors. (The earliest known example of a Corinthian exterior in Alexandria is Ptolemy III's temple of Sarapis in the Sarapeion which was constructed sometime between 246-221 BC, though the propylon of the Sanctuary of the Great Gods on Samothrace, which was sponsored by Ptolemy II sometime between 285-246 BC, also already featured Corinthian columns on one of its facades.) Corinthian capitals were sometimes paired with Doric entablatures, a combination that was rarely seen elsewhere in the Hellenistic world. Alexandrian Corinthian capitals exhibited a large variety of shapes and compositions which historians have divided into four distinct types.

Other typical features of Hellenistic Alexandrian architecture were acanthus leaf column bases, which were sometimes inserted between the standard Attic column base and the column shaft of a Corinthian column, and cornices featuring the distinctive Alexandrian flat grooved and/or hollow square modillions.

Roman Second Style frescoes are thought to be inspired by Alexandrian architecture.

Olynthus
The ancient city of Olynthus was one of the architectural and artistic keystones in establishing a connection between the Classical and Hellenistic worlds.

Over 100 homes were found at the Olynthus city site. Interestingly, the homes and other architecture were incredibly well preserved. This allows us to better understand the activities that took place in the homes and how space inside the homes was organized and utilized.

Homes in Olynthus were typically squarer in shape. The desired home was not necessarily large or extravagant, but rather comfortable and practical. This was a mark of civilization that was extremely prominent in Greek culture during the Hellenistic period and beyond. Living a civilized life involved maintaining a sturdy living space, thus many brick-like materials were used in the construction of the homes. Stone, wood, mudbrick, and other materials were commonly used to build these dwellings.

Another element that was increasingly popular during the Hellenistic period was the addition of a courtyard to the home. Courtyards served as a light source for the home as Greek houses were closed off from the outside to maintain a level of privacy. There have been windows found at some home sites, but they are typically high off the ground and small. Because of the issue of privacy, many individuals were forced to compromise on light in the home. Well-lit spaces were used for entertaining or more public activity while the private sectors of the home were dark and closed off which complicated housework.

Courtyards were typically the focus of the home as they provided a space for entertaining and a source of light from the very interior of the home. They were paved with cobblestones or pebbles most often, but there have been discoveries of mosaicked courtyards. Mosaics were a wonderful way for the family to express their interests and beliefs as well as a way to add décor to the home and make it more visually appealing. This artistic touch to homes at Olynthus introduces another element of civilized living to this Hellenistic society.

Pergamon
Pergamon in particular is a characteristic example of Hellenistic architecture. Starting from a simple fortress located on the Acropolis, the various Attalid kings set up a colossal architectural complex. The buildings are fanned out around the Acropolis to take into account the nature of the terrain. The agora, located to the south on the lowest terrace, is bordered by galleries with colonnades (columns) or stoai. It is the beginning of a street which crosses the entire Acropolis: it separates the administrative, political and military buildings on the east and top of the rock from the sanctuaries to the west, at mid-height, among which the most prominent is that which shelters the monumental Pergamon Altar, known as "of the twelve gods" or "of the gods and of the giants", one of the masterpieces of Greek sculpture. A colossal theatre, able to contain nearly 10,000 spectators, has benches embedded in the flanks of the hill.

Sculpture
Pliny the Elder, after having described the sculpture of the classical period notes: Cessavit deinde ars ("then art disappeared"). According to Pliny's assessment, sculpture declined significantly after the 121st Olympiad (296–293 BC). A period of stagnation followed, with a brief revival after the 156th (156–153 BC), but with nothing to the standard of the times preceding it. During this period sculpture became more naturalistic, and also expressive; there is an interest in depicting extremes of emotion. On top of anatomical realism, the Hellenistic artist seeks to represent the character of his subject, including themes such as suffering, sleep or old age. Genre subjects of common people, women, children, animals and domestic scenes became acceptable subjects for sculpture, which was commissioned by wealthy families for the adornment of their homes and gardens; the Boy with Thorn is an example. Realistic portraits of men and women of all ages were produced, and sculptors no longer felt obliged to depict people as ideals of beauty or physical perfection. The world of Dionysus, a pastoral idyll populated by satyrs, maenads, nymphs and sileni, had been often depicted in earlier vase painting and figurines, but rarely in full-size sculpture. The Old Drunkard at Munich portrays without reservation an old woman, thin, haggard, clutching against herself her jar of wine.

Portraiture
The period is therefore notable for its portraits: One such is the Barberini Faun of Munich, which represents a sleeping satyr with relaxed posture and anxious face, perhaps the prey of nightmares. The Belvedere Torso, the Resting Satyr, the Furietti Centaurs and Sleeping Hermaphroditus reflect similar ideas.

Another famous Hellenistic portrait is that of Demosthenes by Polyeuktos, featuring a well-done face and clasped hands.

Privatization
Another phenomenon of the Hellenistic age appears in its sculpture: privatization, seen in the recapture of older public patterns in decorative sculpture. Portraiture is tinged with naturalism, under the influence of Roman art. New Hellenistic cities were springing up all over Egypt, Syria, and Anatolia, which required statues depicting the gods and heroes of Greece for their temples and public places. This made sculpture, like pottery, an industry, with the consequent standardization and some lowering of quality. For these reasons many more Hellenistic statues have survived than is the case with the Classical period.

Second classicism
Hellenistic sculpture repeats the innovations of the so-called "second classicism": nude sculpture-in-the-round, allowing the statue to be admired from all angles; study of draping and effects of transparency of clothing, and the suppleness of poses. Thus, Venus de Milo, even while echoing a classic model, is distinguished by the twist of her hips.

"Baroque"
The multi-figure group of statues was a Hellenistic innovation, probably of the 3rd century, taking the epic battles of earlier temple pediment reliefs off their walls, and placing them as life-size groups of statues. Their style is often called "baroque", with extravagantly contorted body poses, and intense expressions in the faces. The Laocoön Group, detailed below, is considered one of the prototypical examples of the Hellenistic baroque style.

Pergamon
Pergamon did not distinguish itself with its architecture alone: it was also the seat of a brilliant school of sculpture known as Pergamene Baroque. The sculptors, imitating the preceding centuries, portray painful moments rendered expressive with three-dimensional compositions, often V-shaped, and anatomical hyper-realism. The Barberini Faun is one example.

Gauls
Attalus I (269–197 BC), to commemorate his victory at Caicus against the Gauls;— called Galatians by the Greeks – had two series of votive groups sculpted: the first, consecrated on the Acropolis of Pergamon, includes the famous Gaul killing himself and his wife, of which the original is lost; the second group, offered to Athens, is composed of small bronzes of Greeks, Amazons, gods and giants, Persians and Gauls. Artemis Rospigliosi in the Louvre is probably a copy of one of them; as for copies of the Dying Gaul, they were very numerous in the Roman period. The expression of sentiments, the forcefulness of details – bushy hair and moustaches here – and the violence of the movements are characteristic of the Pergamene style.

Great Altar
These characteristics are pushed to their peak in the friezes of the Great Altar of Pergamon, decorated under the order of Eumenes II (197–159 BC) with a gigantomachy stretching 110 metres in length, illustrating in the stone a poem composed especially for the court. The Olympians triumph in it, each on his side, over Giants – most of which are transformed into savage beasts: serpents, birds of prey, lions or bulls. Their mother Gaia comes to their aid, but can do nothing and must watch them twist in pain under the blows of the gods.

Colossus of Rhodes
One of the few city states who managed to maintain full independence from the control of any Hellenistic kingdom was Rhodes. After holding out for one year under siege by Demetrius Poliorcetes (305–304 BC), the Rhodians built the Colossus of Rhodes to commemorate their victory. With a height of 32 meters, it was one of the Seven Wonders of the Ancient World. Progress in bronze casting made it possible for the Greeks to create large works. Many of the large bronze statues were lost – with the majority being melted to recover the material.

Laocoön
Discovered in Rome in 1506 and seen immediately by Michelangelo, beginning its huge influence on Renaissance and Baroque art. Laocoön, strangled by snakes, tries desperately to loosen their grip without affording a glance at his dying sons. The group is one of very few non-architectural ancient sculptures that can be identified with those mentioned by ancient writers. It is attributed by Pliny the Elder to the Rhodian sculptors Agesander, Athenodoros, and Polydorus. Johann Joachim Winckelmann, who first articulated the difference between Greek, Greco-Roman and Roman art, drew inspiration from the Laocoön. Gotthold Ephraim Lessing based many of the ideas in his 'Laocoon' (1766) on Winckelmann's views on harmony and expression in the visual arts.

Sperlonga
The fragmentary Sperlonga sculptures are another series of "baroque" sculptures in the Hellenistic style, perhaps made for the Emperor Tiberius, who was certainly present at the collapse of the seaside grotto in southern Italy that they decorated. The inscriptions suggest the same sculptors made it who made the Laocoön group, or possibly their relations.

"Rococo"
The "Baroque" traits in Hellenistic art, predominately sculpture, have been contrasted with a contemporary trend that has been described as "Rococo". The concept of a Hellenistic "Rococo" was coined by Wilhelm Klein in the early 20th century. Unlike the dramatic "Baroque" sculptures, the "Rococo" trend emphasized playfull motifs, such as satyrs and nymphs. Wilhelm Klein considered the sculpture group "The Invitation to the Dance" to be a prime example of the trend. Also lighthearted depictions of Aphrodite, the goddess of love, and Eros, were seen as typical (as seen, for instance, in the so-called Slipper Slapper Group depicted below). It has later been argued that the preference for the "Rococo" motifs in Hellenistic sculpture can be tied to a changed use of sculpture in general. Private sculpture collecting became more common during the later Hellenistic period, and in such collections there seems to have been a preference for the kinds of motifs characterized as "Rococo".

Neo-Attic
From the 2nd century the Neo-Attic or Neo-Classical style is seen by different scholars as either a reaction to baroque excesses, returning to a version of Classical style, or as a continuation of the traditional style for cult statues. Workshops in the style became mainly producers of copies for the Roman market, which preferred copies of Classical rather than Hellenistic pieces.

Paintings and mosaics
Paintings and mosaics were important mediums in art, but no examples of paintings on panels have survived the fall to the Romans. It is possible to get some idea of what they were like from related media, and what seem to be copies of or loose derivations from paintings in a wider range of materials.

Landscape

Perhaps the most striking element of Hellenistic paintings and mosaics is the increased use of landscape. Landscapes in these works of art are representative of familiar naturalistic figures while also displaying mythological and sacro-idyllic elements. Landscape friezes and mosaics were commonly used to display scenes from Hellenistic poetry such as that by Herondas and Theocritos. These landscapes that expressed the stories of Hellenistic writers were utilized in the home to emphasize that family's education and knowledge about the literary world.

Sacro-idyllic means that the most prominent elements of the artwork are those related to sacred and pastoral themes. This style that emerged most prevalently in Hellenistic art combines sacred and profane elements, creating a dreamlike setting. Sacro-idyllic influences are conveyed in the Roman mosaic "Nile Mosaic of Palestrina" which demonstrates fantastical narratives with a color scheme and commonplace components that illustrate the Nile in its passage from Ethiopia to the Mediterranean. The inclusion of Hellenistic backgrounds can also be seen in works throughout Pompeii, Cyrene, Alexandria. Moreover, specifically in Southern Russia, floral features and branches can be found on walls and ceilings strewn in a disordered yet conventional manner, mirroring a late Greek style. In addition, "Cubiculum" paintings found in Villa Boscoreale include vegetation and a rocky setting in the background of detailed paintings of grand architecture.

Wall paintings
Wall paintings began appearing more prominently in the Pompeian period. These wall paintings were not just displayed in places of worship or in tombs. Often, wall paintings were used to decorate the home. Wall paintings were common in private homes in Delos, Priene, Thera, Pantikapaion, Olbia, and Alexandria. Few examples of Greek wall paintings have survived the centuries. The most impressive, in terms of showing what high-quality Greek painting was like, are those at the Macedonian royal tombs at Vergina. Though Greek painters are given tribute to bringing fundamental ways of representation to the Western World through their art. Three main qualities unique to Hellenistic painting style were three-dimensional perspective, the use of light and shade to render form, and trompe-l'œil realism. Very few forms of Hellenistic Greek painting survive except for wooden pinakes panels and those painted on stone. The most famously known stone paintings are found on the Macedonian Tomb at Agios Athanasios.

Researchers have been limited to studying the Hellenistic influences in Roman frescoes, for example those of Pompeii or Herculaneum. In addition, some of the paintings in Villa Boscoreale clearly echo lost Hellenistic, Macedonian royal paintings.

Mediums and technique
Recent excavations from the Mediterranean have revealed the technology used in Hellenistic painting. Wall art of this period utilized two techniques: secco technique and fresco technique. Fresco technique required layers of lime-rich plaster to then decorate walls and stone supports. On the other hand, no base was necessary for the secco technique, which used gum arabic and egg tempera to paint finalizing details on marble or other stone. This technique is exemplified in the Masonry friezes found in Delos. Both techniques used mediums that were locally accessible, such as terracotta aggregates in the base layers and natural inorganic pigments, synthetic inorganic pigments, and organic substances as colorants.

Recent discoveries
Recent discoveries include those of chamber tombs in Vergina (1987) in the former kingdom of Macedonia, where many friezes have been unearthed. For example, in Tomb II archaeologists found a Hellenistic-style frieze depicting a lion hunt. This frieze found in the tomb supposedly that of Philip II is remarkable by its composition, the arrangement of the figures in space and its realistic representation of nature. Other friezes maintain a realistic narrative, such as a symposium and banquet or a military escort, and possibly retell historical events.

There is also the recently restored 1st-century Nabataean ceiling frescoes in the Painted House at Little Petra in Jordan. As the Nabataeans traded with the Romans, Egyptians, and Greeks, insects and other animals observed in the paintings reflect Hellenism while various types of vines are associated with the Greek god, Dionysus.

Recent archaeological discoveries at the cemetery of Pagasae (close to modern Volos), at the edge of the Pagasetic Gulf have brought to light some original works. The excavations of this site led by Dr. Arvanitopoulos may be connected to various Greek painters in the 3rd and 4th centuries and depict scenes that allude to the reign of Alexander the Great.

In the 1960s, a group of wall paintings was found on Delos. It is evident that the fragments of friezes found were created by a community of painters who lived during the late Hellenistic period. The murals emphasized domestic decoration, conveying the belief these people held that the Delian establishment would remain stable and secure enough for this artwork to be enjoyed by homeowners for many years to come.

Mosaics
Certain mosaics, however, provide a pretty good idea of the "grand painting" of the period: these are copies of frescoes. This art form has been used to decorate primarily walls, floors, and columns.

Mediums and technique
The development of mosaic art during the Hellenistic Period began with Pebble Mosaics, best represented in the site of Olynthos from 5th century BC. The technique of Pebble Mosaics consisted of placing small white and black pebbles of no specific shape, in a circular or rectangular panel to illustrate scenes of mythology. The white pebbles -in slightly different shades- were placed on a black or blue background to create the image. The black pebbles served to outline the image.

In the mosaics from the site of Pella, from the 4th century BC, it is possible to see a more evolved form of the art. Mosaics from this site display the use of pebbles that were shaded in a wider range of colors and tones. They also show early use of terra-cotta and lead wire to create a greater definition of contours and details to the images in the mosaics.

Following this example, more materials were gradually added. Examples of this extended use of materials in mosaics of the 3rd century BC include finely cut stones, chipped pebbles, glass and baked clay, known as tesserae. This improved the technique of mosaics by aiding the artists in creating more definition, greater detail, a better fit, and an even wider range of colors and tones. Despite the chronological order of the appearance of these techniques, there is no actual evidence to suggest that the tessellated necessarily developed from the pebble mosaics.

Opus vermiculatum and opus tessellatum were two different techniques used during this period of mosaic making. Opus tessellatum refers to a redacted tessera (a small block of stone, tile, glass, or other material used in the construction of a mosaic) size followed by an increased variety in shape, color, and material as well as andamento––or the pattern in which the tessera was laid. Opus vermiculatum is oftentimes partnered with this technique but differs in complexity and is known to have the highest visual impact.

The majority of mosaics were produced and laid on site. However, a number of floor mosaics display the use of the emblemata technique, in which panels of the image are created off-site in trays of terra-cotta or stone. These trays were later placed into the setting-bed on the site.

At Delos, colored grouts were used on opus vermiculatum mosaics, but in other regions this is not common. There is one example of colored grout used in Alexandria on the Dog and Askos mosaic. At Samos, the grouts and the tesserae are both colored.

Studying color here is difficult as the grouts are extremely fragile and vulnerable.

Scientifics research has been a source of interesting information with regard to the grouts and tesserae used in Hellenistic Mosaics. Lead strips were discovered on mosaics as a definiting characteristic of the surface technique. Lead strips are absent from the mosaics here. At Delos, lead strips were common on mosaics in the opus tessellatum style. These strips were used to outline decorative borders and geometric decorative motifs. The strips were extremely common on opus vermiculatum mosaics from Alexandria. Because lead strips were present in both styles of surface types, they cannot be the sole characteristic of one type or the other.

Tel Dor mosaic
A rare example of virtuoso Hellenistic style picture mosaic found in the Levantine coast. Through a technical analysis of the mosaic, researchers suggest that this mosaic was created by itinerant craftsman working in situ. Since 2000, over 200 fragments of the mosaic have been discovered at the headline of Tel Dor, however, the destruction of the original mosaic is unknown. Excavators suggest that earthquake or urban renewal is the cause. Original architectural context is unknown, but stylistic and technical comparisons suggest a late Hellenistic period date, estimating around the second half of the second century B.C.E. Analyzing the fragments found at the original site, researchers have found that the original mosaic contained a centralized rectangle with unknown iconography surrounded by a series of decorative borders consisting of a perspective meander followed by a mask-and-garland border. This mosaic consists of two different techniques of mosaic making, opus vermiculatum and opus tessellatum.

Alexander mosaic
An example is the Alexander Mosaic, showing the confrontation of the young conqueror and the Grand King Darius III at the Battle of Issus, a mosaic from a floor in the House of the Faun at Pompeii (now in Naples). It is believed to be a copy of a painting described by Pliny which had been painted by Philoxenus of Eretria for King Cassander of Macedon at the end of the 4th century BC, or even of a painting by Apelles contemporaneous with Alexander himself. The mosaic allows us to admire the choice of colors along with the composition of the ensemble using turning movement and facial expression.

Stag Hunt mosaic
The Stag Hunt Mosaic by Gnosis is a mosaic from a wealthy home of the late 4th century BC, the so-called "House of the Abduction of Helen" (or "House of the Rape of Helen"), in Pella, The signature ("Gnosis epoesen", i.e. Gnosis created) is the first known signature of a mosaicist. The emblema is bordered by an intricate floral pattern, which itself is bordered by stylized depictions of waves. The mosaic is a pebble mosaic with stones collected from beaches and riverbanks which were set into cement. As was perhaps often the case, the mosaic does much to reflect styles of painting. The light figures against a darker background may allude to red figure painting. The mosaic also uses shading, known to the Greeks as skiagraphia, in its depictions of the musculature and cloaks of the figures. This along with its use of overlapping figures to create depth renders the image three dimensional.

Sosos
The Hellenistic period is equally the time of development of the mosaic as such, particularly with the works of Sosos of Pergamon, active in the 2nd century BC and the only mosaic artist cited by Pliny. His taste for trompe-l'œil (optical illusion) and the effects of the medium are found in several works attributed to him such as the "Unswept Floor" in the Vatican museum, representing the leftovers of a repast (fish bones, bones, empty shells, etc.) and the "Dove Basin" (made of small opus vermiculatum tesserae stones) at the Capitoline Museum, known by means of a reproduction discovered in Hadrian's Villa. In it one sees four doves perched on the edge of a gilt bronze basin filled with water. One of them is watering herself while the others seem to be resting, which creates effects of reflections and shadow perfectly studied by the artist. The "Dove Basin" mosaic panel is an emblema, designed to be the central point of an otherwise plain mosaic floor. The emblema was originally an import from the Hellenistic eastern Mediterranean, where, in cities such as Pergamom, Ephesus and Alexandria, there were artists specializing in mosaics. One of them was Sosos of Pergamon, the most celebrated mosaicist of antiquity who worked in the second century BC.

Delos
According to the French archaeologist François Chamoux, the mosaics of Delos in the Cyclades represent the zenith of Hellenistic-period mosaic art employing the use of tesserae to form complex, colorful scenes. This style of mosaic continued until the end of Antiquity and may have had an impact on the widespread use of mosaics in the Western world during the Middle Ages.

Pottery

The Hellenistic Age comes immediately after the great age of painted Ancient Greek pottery, perhaps because increased prosperity led to more use of fine metalware (very little now surviving) and the decline of the fine painted "vase" (the term used for all vessel shapes in pottery). Most vases of the period are black and uniform, with a shiny appearance approaching that of varnish, decorated with simple motifs of flowers or festoons. The shapes of the vessels are often based on metalwork shapes: thus with the lagynos, a wine jar typical of the period. Painted vase types that continued production into the Hellenistic period include Hadra vases and Panathenaic amphora.

Megarian ware
It is also the period of so-called Megarian ware: mold-made vases with decoration in relief appeared, doubtless in imitation of vases made of precious metals. Wreaths in relief were applied to the body of the vase. One finds also more complex relief, based on animals or legendary creatures.

West Slope ware
Red-figure painting had died out in Athens by the end of the 4th century BC to be replaced by what is known as West Slope Ware, so named after the finds on the west slope of the Athenian Acropolis. This consisted of painting in a tan coloured slip and white paint on a fired black slip background with some incised detailing.

Representations of people diminished, replaced with simpler motifs such as wreaths, dolphins, rosettes, etc. Variations of this style spread throughout the Greek world with notable centres in Crete and Apulia, where figural scenes continued to be in demand.

Gnathia vases
Gnathia vases however were still produced not only in Apulian, but also in Campanian, Paestan and Sicilian vase painting.

Canosa ware
In Canosa di Puglia in South Italy, in 3rd century BC burials one might find vases with fully three-dimensional attachments. The distinguishing feature of Canosa vases are the water-soluble paints. Blue, red, yellow, light purple and brown paints were applied to a white ground.

Centuripe ware
The Centuripe ware of Sicily, which has been called "the last gasp of Greek vase painting", had fully coloured tempera painting including groups of figures applied after firing, contrary to the traditional practice. The fragility of the pigments prevented frequent use of these vases; they were reserved for use in funerals, and many were purely for display, for example with lids that did not lift off. The practice perhaps continued into the 2nd century BC, making it possibly the last vase painting with significant figures. A workshop was active until at least the 3rd century BC. These vases are characterized by a base painted pink. The figures, often female, are represented in coloured clothing: blue-violet chiton, yellow himation, white veil. The style is reminiscent of Pompeii and draws more from grand contemporary paintings than on the heritage of the red-figure pottery.

Terracotta figurines
Bricks and tiles were used for architectural and other purposes. Production of Greek terracotta figurines became increasingly important. Terracotta figurines represented divinities as well as subjects from contemporary life. Previously reserved for religious use, in Hellenistic Greece the terracotta was more frequently used for funerary and purely decorative, purposes. The refinement of molding techniques made it possible to create true miniature statues, with a high level of detail, typically painted.

Several Greek styles continued into the Roman period, and Greek influence, partly transmitted via the Ancient Etruscans, on Ancient Roman pottery was considerable, especially in figurines.

Tanagra figurines
Tanagra figurines, from Tanagra in Boeotia and other centers, full of lively colours, most often represent elegant women in scenes full of charm. At Smyrna, in Asia Minor, two major styles occurred side-by-side: first of all, copies of masterpieces of great sculpture, such as the Farnese Hercules in gilt terracotta.

Grotesques
In a completely different genre, there are the "grotesques", which contrast violently with the canons of "Greek beauty": the koroplathos (figurine maker) fashions deformed bodies in tortuous poses – hunchbacks, epileptics, hydrocephalics, obese women, etc. One could therefore wonder whether these were medical models, the town of Smyrna being reputed for its medical school. Or they could simply be caricatures, designed to provoke laughter. The "grotesques" are equally common at Tarsus and also at Alexandria.

Negro
One theme which emerged was the "negro", particularly in Ptolemaic Egypt: these statuettes of Black adolescents were successful up to the Roman period. Sometimes, they were reduced to echoing a form from the great sculptures: thus one finds numerous copies in miniature of the Tyche (Fortune or Chance) of Antioch, of which the original dates to the beginning of the 3rd century BC.

Hellenistic pottery designs can be found in the city of Taxila in modern Pakistan, which was colonized with Greek artisans and potters after Alexander conquered it.

Metallic art
Because of so much bronze statue melting, only the smaller objects still exist. In Hellenistic Greece, the raw materials were plentiful following eastern conquests. The work on metal vases took on a new fullness: the artists competed among themselves with great virtuosity. The Thracian Panagyurishte Treasure (from modern Bulgaria), includes Greek objects such as a gold amphora with two rearing centaurs forming the handles. The Derveni Krater, from near Thessaloniki, is a large bronze volute krater from about 320 BC, weighing 40 kilograms, and finely decorated with a 32-centimetre-tall frieze of figures in relief representing Dionysus surrounded by Ariadne and her procession of satyrs and maenads. The neck is decorated with ornamental motifs while four satyrs in high relief are casually seated on the shoulders of the vase.

The evolution is similar for the art of jewelry. The jewelers of the time excelled at handling details and filigrees: thus, the funeral wreaths present very realistic leaves of trees or stalks of wheat. In this period the insetting of precious stones flourished.

Glass and glyptic art
It was in the Hellenistic period that the Greeks, who until then only knew molded glass, discovered the technique of glass blowing, thus permitting new forms. Beginning in Syria, the art of glass developed especially in Italy. Molded glass continued, notably in the creation of intaglio jewelry.

The art of engraving gems hardly advanced at all, limiting itself to mass-produced items that lacked originality. As compensation, the cameo made its appearance. It concerns cutting in relief on a stone composed of several colored layers, allowing the object to be presented in relief with more than one color. The Hellenistic period produced some masterpieces like the Gonzaga cameo, now in the Hermitage Museum, and spectacular hardstone carvings like the Cup of the Ptolemies in Paris.

Coinage
Coinage in the Hellenistic period increasingly used portraits.

Later Roman copies
Spurred by the Roman acquisition, elite consumption and demand for Greek art, both Greek and Roman artists, particularly after the establishment of Roman Greece, sought to reproduce the marble and bronze artworks of the Classical and Hellenistic periods. They did so by creating molds of original sculptures, producing plaster casts that could be sent to any sculptor's workshop of the Mediterranean where these works of art could be duplicated. These were often faithful reproductions of originals, yet other times they fused several elements of various artworks into one group, or simply added Roman portraiture heads to preexisting athletic Greek bodies.

References and sources

 * References

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title: Hellenistic art


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User:AdamakaCausticLogic

I am Adam Larson aka Caustic Logic. Research at Libyan Civil War: Critical Views blog, A Closer Look On Syria research wiki. --AdamakaCausticLogic (talk) 13:55, 8 February 2015 (UTC)

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Novoye, Jewish Autonomous Oblast

Novoye (Новое) is a rural locality (a selo) in Leninsky District of the Jewish Autonomous Oblast, Russia. According to the 2010 Census, its population was 683.

The village was established in 1869 as the khutor of Novinsky. It later became the settlement of Novy before being renamed as the selo of Novoye.

title: Novoye, Jewish Autonomous Oblast


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Bautista Álvarez

Bautista Goyel Álvarez Domínguez (1933 – 16 September 2017) was a Galician nationalist politician and historian. He was born in San Amaro, Spain. Álvarez, who was a founding member of the Galician People's Union in 1964, also served as the Galician political party's President from 1964 through 1977.

Álvarez died of a heart attack on 16 September 2017 at his sister's home in Beariz, Spain at the age of 84.

title: Bautista Álvarez


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User:Truelyasian

About me

 * Live in metro Atlanta area, Georgia.
 * I am a mechanical engineer, but enjoys casual as well as serious political and scientific discussion on forums and games.
 * I am a moderator for the English Team in Call of War by Bytro Labs.
 * Plays Call of War, Ogame, Politics and War.
 * A distant past adventure of Runescape, Elsword.

title: User:Truelyasian


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User:Jzhou9777

Jzhou9777
I am a student on the Translation Studies MSc at the University of Edinburgh.

title: User:Jzhou9777


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User:Gnowise/sandbox

Gnowise, is a real estate predictive analytics and insight tool headquartered in Kitchener-Waterloo, Ontario. Gnowise provides online real estate predictive and market simulation tools for lenders, investors, appraisers, and real estate professionals.

History
Gnowise was founded by Amir Madadi and Faraz Arbabi. The founders' backgrounds are in data science, machine learning, data analytics, and visualization.

Products

 * Discover & Play is free and limited version for general public
 * Gnowise Pro-Analytics includes market simulation designed for real estate professionals
 * Gnowise Enterprise with unlimited access to Gnowise insights and market simulation options designed for larger organizations

Technology
The Gnowise technology holds millions of diverse data sets, processes and models – then finds patterns and presents them in a personalized way. This gives investors a super-human ability to make smarter decisions.

title: User:Gnowise/sandbox


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MediaWiki:Missinglanguagelinkstext

These pages do not link to their counterpart in $1. Redirects and subpages are not shown.

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mezz

Etymology 1
who introduced Mexican marijuana to the Harlem jazz scene.

Noun

 * 1)  Marijuana.

Adjective

 * 1)  Of or relating to mezzanine financing.

Etymology
From. Cognates include 🇨🇬 and 🇨🇬.

Noun

 * 1) the middle

Etymology
, from.

Noun

 * 1) a means device, method

title: mezz


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Geoffroy d'Ablis

Geoffroy d'Ablis was a Dominican who led the Inquisition in Carcassonne against Cathars such as Peire Autier from 1303 to 1316. He collaborated with Bernard Gui, the inquisitor at Toulouse.

title: Geoffroy d'Ablis