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Ole Nydahl

Ole Nydahl (born 19 March 1941), also known as Lama Ole, is a lama providing Mahamudra teachings in the Karma Kagyu school of Tibetan Buddhism. Since the early 1970s, Nydahl has toured the world giving lectures and meditation courses. With his wife, Hannah Nydahl (1946-2007), he founded Diamond Way Buddhism, a worldwide Karma Kagyu Buddhist organization with over 600 centers for lay practitioners.

Nydahl is the author of more than twenty books (in German and English) about Diamond Way Vajrayana Buddhism, with translations into multiple languages. Titles include The Great Seal: Mahamudra View of Diamond Way Buddhism,The Way Things Are, Entering the Diamond Way, Buddha and Love and Fearless Death.

Early life and education
Ole Nydahl was born north of Copenhagen into an academic family. Growing up in Denmark during the second world war, Nydahl witnessed his parents working in the Danish resistance movement, helping transport Jews to neutral Sweden. In the early 1960s, he served briefly in the Danish Army, then studied philosophy, English, and German at the University of Copenhagen, where he completed the examen philosophicum with the best possible grade. He began but did not finish a doctoral thesis on Aldous Huxley's The Doors of Perception. As a young man, Nydahl was involved in boxing, motorcycles, race car driving and also travelled overland from Denmark to Nepal several times. As described in his autobiography Entering the Diamond Way, his travels were financed through smuggling, for which he was once arrested and detained in Denmark. He used this time in pre-trial detention for meditation and read "Tibetan Yoga and Secret Doctrine" by Walter Evans-Wentz.

Buddhist education
In 1968, Ole Nydahl and his wife Hannah travelled to Nepal on their honeymoon. In Nepal they met their first Buddhist teacher, the Drukpa Kagyu master Lopon Tsechu Rinpoche. In December 1969, the Nydahls met Rangjung Rigpe Dorje, the 16th Karmapa, head of the Karma Kagyu lineage. Nine years after the British woman Freda Bedi became the first Western student of the 16th Karmapa, the Nydahls took refuge and became students of the Karmapa. They studied and meditated in the Himalayas, where they completed the Ngöndro or preliminary practices in six months, and had explanations and empowerments for 8th Karmapa guru yoga meditation practice and other methods. During this time the Nydahls also became students of Mipham Chokyi Lodro, the fourteenth Shamarpa, the 3rd Jamgon Kongtrul, and Kalu Rinpoche in Sonada. From the Karmapa, the Nydahls learned about Vajrayana Buddhism and mahamudra, and received the Kagyu Ngagdzo transmission. From the Shamarpa, they took the Bodhisattva vows and learned about Gampopa's Jewel Ornament of Liberation. They learned phowa from Ayang Rinpoche in 1972. In addition, the Nydahls received teachings and empowerments from various Tibetan lamas, including Dilgo Khyentse, Bokar Rinpoche, Gyaltrul Rinpoche and the Dalai Lama.

Teaching
Upon returning to Europe in 1972, the 16th Karmapa asked Hannah and Ole Nydahl to begin teaching Buddhism and organize meditation centers, first in their native Denmark, then in Germany and other countries. The centers belong to the Karma Kagyu lineage and operate under Ole Nydahl's practical guidance. In the early 90s, Diamond Way Buddhism was founded as a way to protect established centers during the Karmapa controversy.

, there were 635 Diamond Way centers throughout the world. Most are in Europe, Russia, or the United States.

Ole Nydahl regularly travels between them during the year giving lectures and meditation courses. Until 2013, Nydahl taught conscious dying or phowa, as well as other Buddhist meditation practices, but in recent years he has been focused on giving Mahamudra; teachings on the nature of mind. He traveled almost constantly for over 40 years, teaching in a new city nearly every day, until 2017 when his health necessitated reducing his travel schedule.

Together with his students, Nydahl has created Buddhist centers that provide access to Vajrayana meditation methods without requiring an understanding of Tibetan language or culture. In the Diamond Way centers, the meditations and names of the various Buddha forms have been translated into Western languages. Ole Nydahl believes it essential for people to understand and read the meditations in their own language in order for Buddhism to become truly rooted in the West, as noted in the preface to the meditation booklet titled Refuge and the Enlightened Attitude, used in Diamond Way Buddhist Centers, in which Nydahl states, "This practice used to have an exotic edge because the repetition was in Tibetan. It has now been brought into a Western context, enabling us to understand it on a much deeper level."

Diamond Way centers are run entirely by volunteers; the organization does not maintain any paid staff. The organizational structure is intended to be democratic and to function on the basis of idealism and friendship. According to Buddhism Today, the Diamond Way Buddhist magazine, "hierarchical systems will not sell with independent people in the West. Nobody wants a distant teacher on a pedestal or a big organization standing on their shoulders and telling them what to think."

Students in Diamond Way Centers practice the Ngöndro given by Wangchuk Dorje, 9th Karmapa Lama, which are a set of four foundational practices that are intended to prepare the mind for enlightenment, and several forms of guru yoga or meditation on the lama, such as the 16th Karmapa meditation, and the 8th Karmapa meditation (as given by the 16th Karmapa). In a newsletter dated 9 July 2010, Nydahl responded to questions about the types of practices taught in Diamond Way Centers by stating "I never taught anything I was not asked to pass on by the great Sixteenth Karmapa and that its basis was always the Guru Yogas of the Karmapas. Nothing else is practiced in our now 650 Diamond Way centers world-wide where my students meditate side by side." According to Bee Scherer, "This trajectory is a deliberate yet restrictive selection from the vast richness of Kagyu practices", providing an introductory course into Karma Kagyu practices.

Influence
Jørn Borup, a professor of religion at Aarhus University, says that Ole Nydahl is "the most lasting influence on the Buddhist practice scene in Denmark" and "has in many ways been the icon of living Buddhism in Denmark". The total number of Nydahl's adherents is unknown, but can be estimated conservatively to include 15,000 to 70,000 students and casual sympathizers worldwide. In Germany alone, the German Buddhist Union, (Deutsche Buddhistische Union) estimates that about 20,000 persons regularly visit the Diamond Way centers and groups.

The 14th Shamar Rinpoche, Shamarpa, stated in his biography of the 16th Karmapa that "it was Lama Ole who made the Karmapa's name be renowned and through this, he established some 600 dharma centers," noting that Nydahls work "is also the result of Gyalwa Karmapa's activity."

As well as co-founder of Diamond Way with his wife Hannah Nydahl, Ole Nydahl is co-founder and chairman of the board of The Diamond Way Buddhism Foundation. A non-profit international foundation under German law, it supports projects worldwide, such as a library in Karma Guen (Malaga, Spain), which translates and preserves Buddhist texts; organizes cultural events such as Tibetan art exhibitions; and is responsible for building retreat centers and stupas in Europe and Russia.

Lama and lay siddhi-yogi
In 1972, in a letter to Queen Margarethe of Denmark, the 16th Karmapa called Ole and Hannah Nydahl "trusted pupils", entrusting them to establish "a Centre and meditation centre" in Denmark. According to Bee Scherer, a professor of gender studies and religious studies at Canterbury Christ Church University, "Nydahl was not sent back in 1972 already as a "lama" in the sense of a traditionally trained and fully qualified Buddhist teacher." According to Scherer, Nydahl "gained recognition as a Lama (bla ma, traditionally acknowledged teacher)" in the period after the death of the Sixteenth Karmapa in 1981. In August 1983, he was "finally acknowledged as a "Buddhist Master" by the lineage holder Shamar Rinpoche", the 14th Shamarpa, who certified that Nydahl is "appointed Buddhist Master". Scherer notes that since 1995 "the usage of "Lama" by higher Lamas in reference to Nydahl has been documented."

Nydahl's "self-identification and legitimization as a Western Karma bKa' brgyud lay teacher" is an important part of a "continual hagiographical tradition he and his inner circle are writing and rewriting". These hagiographies contribute to the cohesion of the Diamond Way, and "this claim of normative transmission is emphasized in almost every public lecture given by Nydahl himself." According to Scherer, Nydahl presents himself as a lay-siddhi yogi, with a "polarizing style". The yogi-lay element is presented by Nydahl as a legitimization of his position as a teacher within the Karma Kagyu, and "addresses Nydahl's unconventional spiritual formation and education outside the prescribed curriculum of three-year retreats."

Scherer further notes that Nydahls' unconventional role as a siddhi-yogi, and his presentation of Mahamudra-teachings outside "the gradual Tantric trajectory", has historical precedents in the crazy yogis who established Tibetan Buddhism. These precedents are relevant in understanding Nydahl's role as a yogic/lay teacher. Lama Ole's first teacher, Lopon Tsechu Rinpoche, interpreted Nydahl's activity within the context of the Mahāsiddha tradition and the yogi/accomplisher way, stating, "Almost all of the 84 Mahasiddhas followed the lay way, only a few of them were monks and nuns. Nowadays, the lay way is natural and beneficial for many people."

Westernization and neo-orthopraxy
According to Borup, "[Nydahl] and his Diamond Way Buddhism is in no way representative of Buddhism, Tibetan Buddhism or the teachings of the Karma Kagyu lineage." Martin Baumann, a professor of religion at the University of Lucerne, remarked in a newspaper interview "when I listen to his [Nydahl's] alarmingly superficial formulations in his talks I can understand his critics who say that he is presenting a watered-down 'instant Buddhism', a sort of 'Buddhism light' for the West."

In a 2009 article, Bee Scherer, a professor of gender studies and religious studies at Canterbury Christ Church University and a student of Thaye Dorje, as well as a former student of Ole Nydahl, examined if these criticisms apply to the core rituals and practices performed in the Diamond Way. Scherer describes a number of practices, some of which have been partly adapted to the west, while others are fully in line with traditional Karma Kagyu practices, concluding that Baumann's critique applies only partly. According to Scherer, Nydahl's Diamond Way practices can best be described as a "neo-orthopraxy", a new, westernized form of traditional practices. He regrets that Nydahl's ideas are not discussed by Tibet scholars, and opines that they have a duty to counterbalance the prevailing negative criticism by sociologists and students of New Religious Movements.

Role in the Karmapa controversy
When a great Tibetan lama dies, it is tradition in Tibetan Buddhism to find the next reincarnation to continue the work. When Rangjung Rigpe Dorje, the Sixteenth Karmapa (head of the Karma Kagyu) died in 1981, two potential successors were found, Ogyen Trinley Dorje and Trinley Thaye Dorje, causing a major split in the Karma Kagyu. Because Shamar Rinpoche was one of Hannah and Ole Nydahl's main teachers, they supported his recognition of Trinley Thaye Dorje as the 17th Karmapa.

Geoffrey Samuel, an academic expert in the field testified in court, while the recognition of Ogyen Trinley "appears to have been accepted by a majority of Karma Kagyu monasteries and lamas, there remains a substantial minority of monasteries and lamas who have not accepted Ogyen Trinley as Karmapa. In particular, these include the Shamar Rinpoche, who historically has been the person most directly involved in the process of recognition."

It was largely because of the work of Hannah and Ole Nydahl that most European Karma Kagyu centers chose to support Trinley Thaye Dorje. As a result, Trinley Thaye Dorje is the "patron" of the centers of the Diamond Way Buddhism.

Due to his role in the Karmapa controversy, Nydahl has been heavily criticized by the supporters of Ogyen Trinley Dorje, such as the authors Mick Brown and Lea Terhune, a student of Tai Situpa. In connection to this, some blame Nydahl for causing the 1992 split of the Karma Kagyu, and accuse him of breaking the samayas to his teachers, which is deprecated in Vajrayana.

Political views
Some members of the press have criticized Nydahl's version of Diamond Way Buddhism, describing it as featuring "prevalent militaristic appearances, right-wing political views and fierce anti-Islam rhetoric". Others have more positive views of Nydahl's work, describing his dedication to his work and patriotism. Scherer describes Nydahl's representation of himself as "a Buddhist teacher and protector of Western freedom". This addresses his legitimization narrative of himself as emanation of a Buddhist protector and his Kālacakra-linked fierce interpretation of Islam in particular as a key threat to Western freedom and human, especially women's rights. Nydahl has referred to the Islamic religion as "criminal", has called Allah a "terrible god", and has characterized Muslim beliefs as antithetical to freedom of speech and women's rights:

"I seriously hope, you know, that we're not losing the freedom of expression right now, that we're not losing the ability to say what we think even if we step on the toes of some gentlemen from the Near East … who like to beat their wives or stone them or whatever else they do, right?"

Nydahl says that he does not make political comments in his capacity as a lama, but as a "responsible, thinking human being", and that no one can make such statements from a Buddhist perspective because Buddha Shakyamuni did not comment on religious ideas founded centuries after his death.

An online interview with Nydahl also featured the following statement: "Judaism and Christianity are fine. Islam, I warn against. I know the Koran, I know the life story of Mohammad and I think we cannot use that in our society today."

Dispute with German Buddhist Union
Between late 1999 and April 2000 there was a public dispute between the German Buddhist Union and the German branch of Diamond Way, which was a member organisation of the Union. Due to Nydahl's disparaging attitude towards Islam, his political statements, his manner of expressing and presenting himself, and his relationships with women,  there were calls for the expulsion of Nydahl's organisation from the Union. The dispute was resolved at a meeting between the two organizations on 4 October 2000; although differences were clear, they agreed to learn from the past and cooperate in the future. The conversation was described as "a first step" that "should eliminate misunderstandings, and lead to clarity and cooperation". The German branch of Diamond Way (Buddhistischer Dachverband Diamantweg) remained a member of the German Buddhist Union. In 2019 there were discussions and an application to exclude the Diamond Way from the German Buddhist Union (DBU), based on Nydahl's statements about Islam. DBU members were worried about possible damage to reputation and the German section of Diamond Way decided to leave the DBU.

Personal life
Ole Nydahl met his future wife Hannah when he was 10 and she was five. They met again just after Nydhal came out of the army. Hannah Nydahl died of lung cancer in 2007.

In 2014 Nydahl married Alexandra Munoz Barbosa at the Copenhagen Diamond Way Buddhist center. In 2017 they were divorced.

On 31 August 2019 Ole Nydahl and Anne Behrend were married at the Diamond Way Buddhist center in Těnovice, Czech Republic.

In an interview in 2017 Ole Nydahl mentioned he had fathered two children.

On 18 December 2020 Ole Nydahl and Anne Behrend welcomed the birth of their daughter Freya.

title: Ole Nydahl


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Ha-Mim

Ha-Mim (حا میم) is the short form of the name Ha-Mim ibn Mann-Allah ibn Harir ibn Umar ibn Rahfu ibn Azerwal ibn Majkasa (أبا محمد حاميم بن من الله بن حرير بن عمر بن رحفو بن أزروال بن مجكسة), also known as Abu Muhammad; he was a member of the Majkasa sub-tribe of the Ghomara Berbers who proclaimed himself a prophet in 925 near Tetouan in Morocco. He was named after a well-known combination of Qur'anic initial letters.

His claim was widely accepted among the Ghomara of the time, and he established rules for them. He said that he received a revelation in the Berber language, portions of which historian Ibn Khaldun quotes in Arabic: "O You who are beyond sight, who watches the world, release me from my sins! O You who saved Moses from the sea, You believe in Ha-Mim and in his father Abu-Khalaf Mann Allah..."

He died in 927 fighting the Masmuda Berbers near Tangier, and was succeeded politically by his son Isa, who sent an embassy to the Umayyad Caliph Abd-ar-rahman III an-Nasir. His religion's later history is unclear, but it vanished well before even Ibn Khaldun's time.

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

title: Ha-Mim


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Talk:Epistemic theory of truth

What purpose does this page fulfil, apart from identifying this category of truth theories? The theories are better dealt with elsewhere. see fi epistemology Banno 21:20, May 8, 2004 (UTC)

This page was broken off from the main article truth as an editorial convenience. The category ‘epistemic theories of truth’ is not used widely outside the Wikipedia. This article contains only one line about them as a group, before discussing each separately. Finally, only the article on truth has a purposeful link to this page. The three are quite disparate theories. Coherence is dealt with adequately in a separate article. I suggest therefore that we remove the pragmatic and consensus theories to new articles, remove the link from the truth article, and redirect this page to truth. Banno 21:19, May 18, 2004 (UTC)

When I noticed that I had raised this issue months ago, and no one noticed, I decided to just make it so.Banno 21:40, May 18, 2004 (UTC)

title: Talk:Epistemic theory of truth


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Aeolian processes

Aeolian processes, also spelled eolian, pertain to wind activity in the study of geology and weather and specifically to the wind's ability to shape the surface of the Earth (or other planets). Winds may erode, transport, and deposit materials and are effective agents in regions with sparse vegetation, a lack of soil moisture and a large supply of unconsolidated sediments. Although water is a much more powerful eroding force than wind, aeolian processes are important in arid environments such as deserts.

The term is derived from the name of the Greek god Aeolus, the keeper of the winds.

Definition and setting
Aeolian processes are those processes of erosion, transport, and deposition of sediments that are caused by wind at or near the surface of the earth. Sediment deposits produced by the action of wind and the sedimentary structures characteristic of these deposits are also described as aeolian.

Aeolian processes are most important in areas where there is little or no vegetation. However, aeolian deposits are not restricted to arid climates. They are also seen along shorelines; along stream courses in semiarid climates; in areas of ample sand weathered from weakly cemented sandstone outcrops; and in areas of glacial outwash.

Loess, which is silt deposited by wind, is common in humid to subhumid climates. Much of North America and Europe are underlain by sand and loess of Pleistocene age originating from glacial outwash.

The lee (downwind) side of river valleys in semiarid regions are often blanketed with sand and sand dunes. Examples in North America include the Platte, Arkansas, and Missouri Rivers.

Wind erosion
Wind erodes the Earth's surface by deflation (the removal of loose, fine-grained particles by the turbulent action of the wind) and by abrasion (the wearing down of surfaces by the grinding action and sandblasting by windborne particles). Once entrained in the wind, collisions between particles further break them down, a process called attrition.

Worldwide, erosion by water is more important than erosion by wind, but wind erosion is important in semiarid and arid regions. Wind erosion is increased by some human activities, such as the use of 4x4 vehicles.

Deflation
Deflation is the lifting and removal of loose material from the surface by wind turbulence. It takes place by three mechanisms: traction/surface creep, saltation, and suspension. Traction or surface creep is a process of larger grains sliding or rolling across the surface. Saltation refers to particles bouncing across the surface for short distances. Suspended particles are fully entrained in the wind, which carries them for long distances. Saltation likely accounts for 50–70 % of deflation, while suspension accounts for 30–40 % and surface creep accounts for 5–25 %.

Regions which experience intense and sustained erosion are called deflation zones. Most aeolian deflation zones are composed of desert pavement, a sheet-like surface of rock fragments that remains after wind and water have removed the fine particles. The rock mantle in desert pavements protects the underlying material from further deflation. Areas of desert pavement form the regs or stony deserts of the Sahara. These are further divided into rocky areas called hamadas and areas of small rocks and gravel called serirs. Desert pavement is extremely common in desert environments.

Blowouts are hollows formed by wind deflation. Blowouts are generally small, but may be up to several kilometers in diameter. The smallest are mere dimples 1 ft deep and 10 ft in diameter. The largest include the blowout hollows of Mongolia, which can be 5 mi across and 200 to 400 feet deep. Big Hollow in Wyoming, US, extends 9 by 6 mi and is up to 300 ft deep.

Abrasion

Abrasion (also sometimes called corrasion) is the process of wind-driven grains knocking or wearing material off of landforms. It was once considered a major contributor to desert erosion, but by the mid-20th Century, it had come to be considered much less important. Wind can normally lift sand only a short distance, with most windborne sand remaining within 50 cm of the surface and practically none normally being carried above 6 ft. Many desert features once attributed to wind abrasion, including wind caves, mushroom rocks, and the honeycomb weathering called tafoni, are now attributed to differential weathering, rainwash, deflation rather than abrasion, or other processes.

Yardangs are one kind of desert feature that is widely attributed to wind abrasion. These are rock ridges, up to tens of meters high and kilometers long, that have been streamlined by desert winds. Yardangs characteristically show elongated furrows or grooves aligned with the prevailing wind. They form mostly in softer material such as silts.

Abrasion produces polishing and pitting, grooving, shaping, and faceting of exposed surfaces. These are widespread in arid environments but geologically insignificant. Polished or faceted surfaces called ventifacts are rare, requiring abundant sand, powerful winds, and a lack of vegetation for their formation.

In parts of Antarctica wind-blown snowflakes that are technically sediments have also caused abrasion of exposed rocks.

Attrition
Attrition is the wearing down by collisions of particles entrained in a moving fluid. It is effective at rounding sand grains and at giving them a distinctive frosted surface texture.

Collisions between windborne particles is a major source of dust in the size range of 2-5 microns. Most of this is produced by the removal of a weathered clay coating from the grains.

Transport

Wind dominates the transport of sand and finer sediments in arid environments. Wind transport is also important in periglacial areas, on river flood plains, and in coastal areas. Coastal winds transport significant amounts of siliciclastic and carbonate sediments inland, while wind storms and dust storms can carry clay and silt particles great distances. Wind transports much of the sediments deposited in deep ocean basins. In ergs (desert sand seas), wind is very effective at transporting grains of sand size and smaller.

Particles are transported by winds through suspension, saltation (skipping or bouncing) and creeping (rolling or sliding) along the ground. The minimum wind velocity to initiate transport is called the fluid threshold or static threshold and is the wind velocity required to begin dislodging grains from the surface. Once transport is initiated, there is a cascade effect from grains tearing loose other grains, so that transport continues until the wind velocity drops below the dynamic threshold or impact threshold, which is usually less than the fluid threshold. In other words, there is hysteresis in the wind transport system.

Small particles may be held in the atmosphere in suspension. Turbulent air motion supports the weight of suspended particles and allows them to be transported for great distances. Wind is particularly effective at separating sediment grains under 0.05 mm in size from coarser grains as suspended particles.

Saltation is downwind movement of particles in a series of jumps or skips. Saltation is most important for grains of up to 2 mm in size. A saltating grain may hit other grains that jump up to continue the saltation. The grain may also hit larger grains (over 2 mm in size) that are too heavy to hop, but that slowly creep forward as they are pushed by saltating grains. Surface creep accounts for as much as 25 percent of grain movement in a desert.

Vegetation is effective at suppressing aeolian transport. Vegetation cover of as little as 15% is sufficient to eliminate most sand transport. The size of shore dunes is limited mostly by the amount of open space between vegetated areas.

Aeolian transport from deserts plays an important role in ecosystems globally. For example, wind transports minerals from the Sahara to the Amazon basin. Saharan dust is also responsible for forming red clay soils in southern Europe.

Dust storms
Dust storms are wind storms that have entrained enough dust to reduce visibility to less than 1 km. Most occur on the synoptic (regional) scale, due to strong winds along weather fronts, or locally from downbursts from thunderstorms.

Crops, people, and possibly even climates are affected by dust storms. On Earth, dust can cross entire oceans, as occurs with dust from the Sahara that reaches the Amazon Basin. Dust storms on Mars periodically engulf the entire planet. When the Mariner 9 spacecraft entered its orbit around Mars in 1971, a dust storm lasting one month covered the entire planet, thus delaying the task of photo-mapping the planet's surface.

Most of the dust carried by dust storms is in the form of silt-size particles. Deposits of this windblown silt are known as loess. The thickest known deposit of loess, up to 350 m, is on the Loess Plateau in China. This very same Asian dust is blown for thousands of miles, forming deep beds in places as far away as Hawaii. The Peoria Loess of North America is up to 40 m thick in parts of western Iowa. The soils developed on loess are generally highly productive for agriculture.

Small whirlwinds, called dust devils, are common in arid lands and are thought to be related to very intense local heating of the air that results in instabilities of the air mass. Dust devils may be as much as one kilometer high. Dust devils on Mars have been observed as high as 10 km, though this is uncommon.

Deposition

Wind is very effective at separating sand from silt and clay. As a result, there are distinct sandy (erg) and silty (loess) aeolian deposits, with only limited interbedding between the two. Loess deposits are found further from the original source of sediments than ergs. An example of this is the Sand Hills of Nebraska, US. Here vegetation-stabilized sand dunes are found to the west and loess deposits to the east, further from the original sediment source in the Ogallala Formation at the feet of the Rocky Mountains.

Some of the most significant experimental measurements on aeolian landforms were performed by Ralph Alger Bagnold, a British army engineer who worked in Egypt prior to World War II. Bagnold investigated the physics of particles moving through the atmosphere and deposited by wind. He recognized two basic dune types, the crescentic dune, which he called "barchan", and the linear dune, which he called longitudinal or "seif" (Arabic for "sword"). Bagnold developed a classification scheme that included small-scale ripples and sand sheets as well as various types of dunes.

Bagnold's classification is most applicable in areas devoid of vegetation. In 1941, John Tilton Hack added parabolic dunes, which are strongly influenced by vegetation, to the list of dune types. The discovery of dunes on Mars reinvigorated aeolian process research, which increasingly makes use of computer simulation.

Wind-deposited materials hold clues to past as well as to present wind directions and intensities. These features help us understand the present climate and the forces that molded it. For example, vast inactive ergs in much of the modern world attest to late Pleistocene trade wind belts being much expanded during the Last Glacial Maximum. Ice cores show a tenfold increase in non-volcanic dust during glacial maxima. The highest dust peak in the Vostok ice cores dates to 20 to 21 thousand years ago. The abundant dust is attributed to a vigorous low-latitude wind system plus more exposed continental shelf due to low sea levels.

Wind-deposited sand bodies occur as ripples and other small-scale features, sand sheets, and dunes.

Ripples and other small-scale features
Wind blowing on a sand surface ripples the surface into crests and troughs whose long axes are perpendicular to the wind direction. The average length of jumps during saltation corresponds to the wavelength, or distance between adjacent crests, of the ripples. In ripples, the coarsest materials collect at the crests causing inverse grading. This distinguishes small ripples from dunes, where the coarsest materials are generally in the troughs. This is also a distinguishing feature between water laid ripples and aeolian ripples.

A sand shadow is an accumulation of sand on the downwind side of an obstruction, such as a boulder or an isolated patch of vegetation. Here the sand builds up to the angle of repose (the maximum stable slope angle), about 34 degrees, then begins sliding down the slip face of the patch. A sandfall is a sand shadow of a cliff or escarpment.

Closely related to sand shadows are sand drifts. These form downwind of a gap between obstructions, due to the funneling effect of the obstructions on the wind.

Sand sheets
Sand sheets are flat or gently undulating sandy deposits with only small surface ripples. An example is the Selima Sand Sheet in the eastern Sahara Desert, which occupies 60000 km2 in southern Egypt and northern Sudan. This consists of a few feet of sand resting on bedrock. Sand sheets are often remarkably flat and are sometimes described as desert peneplains.

Sand sheets are common in desert environments, particularly on the margins of dune fields, although they also occur within ergs. Conditions that favor the formation of sand sheets, instead of dunes, may include surface cementation, a high water table, the effects of vegetation, periodic flooding, or sediments rich in grains too coarse for effective saltation.

Dunes

A dune is an accumulations of sediment blown by the wind into a mound or ridge. They differ from sand shadows or sand drifts in that they are independent of any topographic obstacle. Dunes have gentle upwind slopes on the windward side. The downwind portion of the dune, the lee slope, is commonly a steep avalanche slope referred to as a slipface. Dunes may have more than one slipface. The minimum height of a slipface is about 30 centimeters.

Wind-blown sand moves up the gentle upwind side of the dune by saltation or creep. Sand accumulates at the brink, the top of the slipface. When the buildup of sand at the brink exceeds the angle of repose, a small avalanche of grains slides down the slipface. Grain by grain, the dune moves downwind.

Dunes take three general forms. Linear dunes, also called longitudinal dunes or seifs, are aligned in the direction of the prevailing winds. Transverse dunes, which include crescent dunes (barchans), are aligned perpendicular to the prevailing winds. More complex dunes, such as star dunes, form where the directions of the winds are highly variable. Additional dune types arise from various kinds of topographic forcing, such as from isolated hills or escarpments.

Transverse dunes
Transverse dunes occur in areas dominated by a single direction of the prevailing wind. In areas where sand is not abundant, transverse dunes take the form of barchans or crescent dunes. These are not common, but they are highly recognizable, with a distinctive crescent shape with the tips of the crescent directed downwind. The dunes are widely separated by areas of bedrock or reg. Barchans migrate up to 30 m per year, with the taller dunes migrating faster. Barchans first form when some minor topographic feature creates a sand patch. This grows into a sand mound, and the converging streamlines of the air flow around the mound build it into the distinctive crescent shape. Growth is ultimately limited by the carrying capacity of the wind, which as the wind becomes saturated with sediments, builds up the slip face of the dune. Because barchans develop in areas of limited sand availability, they are poorly preserved in the geologic record.

Where sand is more abundant, transverse dunes take the form of aklé dunes, such as those of the western Sahara. These form a network of sinuous ridges perpendicular to the wind direction. Aklé dunes are preserved in the geologic record as sandstone with large sets of cross-bedding and many reactivation surfaces.

Draas are very large composite transverse dunes. They can be up to 4000 m across and 400 m high and extend lengthwise for hundreds of kilometers. In form, they resemble a large aklé or barchanoid dune. They form over a prolonged period of time in areas of abundant sand and show a complex internal structure. Careful 3-D mapping is required to determine the morphology of a draa preserved in the geologic record.

Linear dunes
Linear dunes can be traced up to tens of kilometers, with heights sometimes in excess of 70 m. They are typically several hundred meters across and are spaced 1 to 2 kmapart. They sometimes coalesce at a Y-junction with the fork directed upwind. They have a sharp sinuous or en echelon crest. They are thought to form from a bimodal seasonal wind pattern, with a weak wind season characterized by wind directed an at acute angle to the prevailing winds of the strong wind season. The strong wind season produces a barchan form and the weak wind season stretches this into the linear form. Another possibility is that these dunes result from secondary flow, though the precise mechanism remains uncertain.

Complex dunes
Complex dunes (star dunes or rhourd dunes) are characterized by having more than two slip faces. They are typically 500 to 1000 m across and 50 to 300 m high. They consist of a central peak with radiating crests and are thought to form where strong winds can come from any direction. Those in Gran Desierto de Altar of Mexico are thought to have formed from precursor linear dunes due to a change in the wind pattern about 3000 years ago. Complex dunes show Little lateral growth but strong vertical growth and are important sand sinks.

Other dune types
Vegetated parabolic dunes are crescent-shaped, but the ends of the crescent point upwind, not downwind. They form from the interaction of vegetation patches with active sand sources, such as blowouts. The vegetation stabilizes the arms of the dune, and an elongated lake sometimes forms between the arms of the dune.

Clay dunes are uncommon but have been found in Africa, Australia, and along the Gulf Coast of North America. These form on mud flats on the margins of saline bodies of water subject to strong prevailing winds during a dry season. Clay particles are bound into sand-sized pellets by salts and are then deposited in the dunes, where the return of the cool season allows the pellets to absorb moisture and become bound to the dune surface.

Aeolian desert systems
Deserts cover 20 to 25 percent of the modern land surface of the earth, mostly between the latitudes of 10 to 30 degrees north or south. Here the descending part of the tropical atmospheric circulation (the Hadley cell) produces high atmospheric pressure and suppresses precipitation. Large areas of this desert is floored with windblown sand. Such areas are called ergs when they exceed about 125 km2 in area or dune fields when smaller. Ergs and dune fields make up about 20% of modern deserts or about 6% of the Earth's total land surface.

The sandy areas of today's world are somewhat anomalous. Deserts, in both the present day and in the geological record, are usually dominated by alluvial fans rather than dune fields. The present relative abundance of sandy areas may reflect reworking of Tertiary sediments following the Last Glacial Maximum. Most modern deserts have experienced extreme Quaternary climate change, and the sediments that are now being churned by wind systems were generated in upland areas during previous pluvial (moist) periods and transported to depositional basins by stream flow. The sediments, already sorted during their initial fluvial transport, were further sorted by wind, which also sculpted the sediments into eolian landforms.

The state of an aeolian system depends mainly on three things: The amount of sediment supply, the availability of sediments, and the transport capacity of the winds. The sediment supply is largely produced in pluvial periods (periods of greater rainfall) and accumulates by runoff as fan deltas or terminal fans in sedimentary basins. Another important source of sediments is the reworking of carbonate sediments on continental shelves exposed during times of lower sea level. Sediment availability depends on the coarseness of the local sediment supply, the degree of exposure of sediment grains, the amount of soil moisture, and the extent of vegetation coverage. The potential transport rate of wind is usually more than the actual transport, because the sediment supply is usually insufficient to saturate the wind. In other words, most aeolian systems are transport-undersaturated (or sediment-undersaturated).

Aeolian desert systems can be divided into wet, dry, or stabilized systems. Dry systems have the water table well below the surface, where it has no stabilizing effect on sediments. Dune shapes determine whether sediment is deposited, simply moves across surface (a bypass system), or erosion takes place. Wet systems are characterized by a water table near the depositional surface, which exerts a strong control on deposition, bypass, or erosion. Stabilized systems have significant vegetation, surface cement, or mud drapes which dominate the evolution of the system. The Sahara shows the full range of all three types.

The movement of sediments in aeolian systems can be represented by sand-flow maps. These are based on meteorological observations, bedform orientations, and trends of yardangs. They are analogous to drainage maps, but are not as closely tied to topography, since wind can blow sand significant distances uphill.

The Sahara of North Africa is the largest hot desert in the world. Flowlines can be traced from erg to erg, demonstrating very long transport downwind. Satellite observations show yardangs aligned with the sandflow lines. All flowlines arise in the desert itself, and show indications of clockwise circulation roughly like high pressure cells. The greatest deflation occurs in dried lake beds where trade winds form a low-level jet between the Tibesti Mountains and the Ennedi Plateau. The flowlines eventually reach the, sea creating great plume of Saharan dust extending thousands of kilometers into the Atlantic Ocean. This creates a steady rain of silt into the ocean. It is estimated that 260 million tons of sediments are transported through this system each year, but the amount was much greater during the Last Glacial Maximum, based on deep-sea cores. Mineral dust of 0.1–1 microns in size is a good shortwave radiation scatterer and has a cooling effect on climate.

Another example of an aeolian system is the arid interior of Australia. With few topographic barriers to sand movement, an anticlockwise wind system is traced by systems of longitudinal dunes.

The Namib and Oman ergs are fed by coastal sediments. The Namib receives its sediments from the south through narrow deflation corridors from coast that cross more than 100 km of bedrock to the erg. The Oman was created by deflation of marine shelf carbonates during the last Pleistocene lowstand of the sea.

The Loess Plateau of China has been a long-term sink for sediments during the Quaternary ice age. It provides a record of glaciation, in the form of glacial loess layers separated by paleosols (fossil soils). The loess layers were desposited by a strong northwest winter monsoon, while the paleosols record the influence of a moist southeast monsoon.

The African savannah is mostly ergs deposited during the Last Glacial Maximum that are now stabilized by vegetation.

Examples
Major global aeolian systems thought to be linked with weather and climate variation:

 * An average of 132 million tons of dust from the Sahara (primarily the Sahel and Bodélé Depression) across the Atlantic each year.
 * Harmattan winter dust storms in West Africa also occur blowing dust to the ocean.
 * Asian dust originates in the Gobi Desert and reaches Korea, Japan, Taiwan (at times) and even the western US.
 * The 2018 Indian dust storms transported dust from the Thar Desert towards Delhi, Uttar Pradesh, and the Indo-Gangetic Plain.
 * Shamal June–July winds blowing dust in primarily north to south in Saudi Arabia, Iran, Iraq, UAE, and parts of Pakistan.
 * Haboob dust storms in Sudan, Australia, Arizona associated with monsoon.
 * Khamsin dust from Libya, Egypt and Levant in Spring associated with extratropical cyclones.
 * Dust Bowl event in US, carried sand eastward. 5500 tons were deposited in Chicago area.
 * Sirocco sandy winds from Africa/Sahara blowing north into South Europe.
 * Kalahari Desert blowing sand/dust east across southern Africa toward Indian Ocean.
 * Mars in the arid conditions, many aeolian processes have been discovered.

In the geologic record
Aeolian processes can be discerned at work in the geologic record as long ago as the Precambrian. Aeolian formations are prominent in the Paleozoic and Mesozoic of the western US. Other examples include the Permian Rotliegendes of northwestern Europe; the Jurassic–Cretaceous Botucatu Formation of the Parana Basin of Brazil; the Permian Lower Bunter Sandstone of Britain; the Permian-Triassic Corrie Sandstone and Hopeman Sandstone of Scotland; and the Proterozoic sandstones of India and northwest Africa.

Perhaps the best examples of aeolian processes in the geologic record are the Jurassic ergs of the western US. These include the Wingate Sandstone, the Navajo Sandstone, and the Page Sandstone. Individual formations are separated by regional unconformities indicate erg stabilization. The ergs interfingered with adjacent river systems, as with the Wingate Sandstone interfingering with the Moenave Formation and the Navajo Sandstone with the Kayenta Formation.

The Navajo and Nugget Sandstones were part of the largest erg deposit in the geologic record. These formations are up to 700 m thick and are exposed over 265000 km2. Their original extent was likely 2.5 times the present outcrop area. Though once thought to possibly be marine in origin, they are now all but universally regarded as aeolian deposits. They are made up mostly of fine- to medium-sized quartz grains that are well-rounded and frosted, both indications of aeolian transport. The Navajo contains huge tabular crossbed sets with sweeping foresets. Individual crossbed sets dip at an angle of more than 20 degrees and are from 5 to 35 m thick. The formation contains freshwater invertebrate fossils and vertebrate tracks. Slump structures (contorted bedding) are present that resemble those in modern wetted dunes. Successive migrating dunes deposited a vertical stacking of eolian beds between interdune bounding surfaces and regional supersurfaces.

The Permian Rotliegend Group of the North Sea and north Europe contains sediments from adjacent uplands. Erg sand bodies within the group are up to 500 m thick. Study of the crossbedding shows that sediments were deposited by a clockwise atmospheric cell. Drilling core show dry and wet interdune surfaces and regional supersurfaces, and provide evidence of five or more cycles of erg expansion and contraction. A global rise in sea level finally drowned the erg and deposited the beds of the Weissliegend.

The Cedar Mesa Sandstone in Utah was contemporary with the Rogliegend. This formation records at least 12 erg sequences bounded by regional deflation supersurfaces. Aeolian landforms preserved in the formation range from damp sandsheet and lake paleosol (fossil soil) beds to thin, chaotically arranged dune sets to equilibrium erg construction, with dunes 300 to 400 m wide migrating over still larger draas. The draas survived individual climate cycles, and their interdunes were sites of barchan nucleation during arid portions of the climate cycles.

title: Aeolian processes


text:
Nonstandard calculus

In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.

Non-rigorous calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians such as Richard Courant viewed infinitesimals as being naive and vague or meaningless.

Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Howard Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."

History
The history of nonstandard calculus began with the use of infinitely small quantities, called infinitesimals in calculus. The use of infinitesimals can be found in the foundations of calculus independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s. John Wallis refined earlier techniques of indivisibles of Cavalieri and others by exploiting an infinitesimal quantity he denoted $$\tfrac{1}{\infty}$$ in area calculations, preparing the ground for integral calculus. They drew on the work of such mathematicians as Pierre de Fermat, Isaac Barrow and René Descartes.

In early calculus the use of infinitesimal quantities was criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley in his book The Analyst.

Several mathematicians, including Maclaurin and d'Alembert, advocated the use of limits. Augustin Louis Cauchy developed a versatile spectrum of foundational approaches, including a definition of continuity in terms of infinitesimals and a (somewhat imprecise) prototype of an ε, δ argument in working with differentiation. Karl Weierstrass formalized the concept of limit in the context of a (real) number system without infinitesimals. Following the work of Weierstrass, it eventually became common to base calculus on ε, δ arguments instead of infinitesimals.

This approach formalized by Weierstrass came to be known as the standard calculus. After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by Abraham Robinson in the 1960s. Robinson's approach is called nonstandard analysis to distinguish it from the standard use of limits. This approach used technical machinery from mathematical logic to create a theory of hyperreal numbers that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus. An alternative approach, developed by Edward Nelson, finds infinitesimals on the ordinary real line itself, and involves a modification of the foundational setting by extending ZFC through the introduction of a new unary predicate "standard".

Motivation
To calculate the derivative $$f '$$ of the function $$ y =f(x)=x^2$$ at x, both approaches agree on the algebraic manipulations:

 * $$ \frac{\Delta y}{\Delta x} = \frac{(x + \Delta x)^2 - x^2}{\Delta  x} = \frac{2 x \Delta x + (\Delta x)^2}{\Delta x} = 2 x  + \Delta x \approx 2 x$$

This becomes a computation of the derivatives using the hyperreals if $$\Delta x$$ is interpreted as an infinitesimal and the symbol "$$\approx$$" is the relation "is infinitely close to".

In order to make f ' a real-valued function, the final term $$\Delta x$$ is dispensed with. In the standard approach using only real numbers, that is done by taking the limit as $$\Delta x$$ tends to zero. In the hyperreal approach, the quantity $$\Delta x$$ is taken to be an infinitesimal, a nonzero number that is closer to 0 than to any nonzero real. The manipulations displayed above then show that $$\Delta y /\Delta x$$ is infinitely close to 2x, so the derivative of f at x is then 2x.

Discarding the "error term" is accomplished by an application of the standard part function. Dispensing with infinitesimal error terms was historically considered paradoxical by some writers, most notably George Berkeley.

Once the hyperreal number system (an infinitesimal-enriched continuum) is in place, one has successfully incorporated a large part of the technical difficulties at the foundational level. Thus, the epsilon, delta techniques that some believe to be the essence of analysis can be implemented once and for all at the foundational level, and the students needn't be "dressed to perform multiple-quantifier logical stunts on pretense of being taught infinitesimal calculus", to quote a recent study. More specifically, the basic concepts of calculus such as continuity, derivative, and integral can be defined using infinitesimals without reference to epsilon, delta (see next section).

Keisler's textbook
Keisler's Elementary Calculus: An Infinitesimal Approach defines continuity on page 125 in terms of infinitesimals, to the exclusion of epsilon, delta methods. The derivative is defined on page 45 using infinitesimals rather than an epsilon-delta approach. The integral is defined on page 183 in terms of infinitesimals. Epsilon, delta definitions are introduced on page 282.

Definition of derivative
The hyperreals can be constructed in the framework of Zermelo–Fraenkel set theory, the standard axiomatisation of set theory used elsewhere in mathematics. To give an intuitive idea for the hyperreal approach, note that, naively speaking, nonstandard analysis postulates the existence of positive numbers ε which are infinitely small, meaning that ε is smaller than any standard positive real, yet greater than zero. Every real number x is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it. To define the derivative of f at a standard real number x in this approach, one no longer needs an infinite limiting process as in standard calculus. Instead, one sets

 * $$ f'(x) = \mathrm{st} \left( \frac{f^*(x+\varepsilon)-f^*(x)}{\varepsilon} \right),$$

where st is the standard part function, yielding the real number infinitely close to the hyperreal argument of st, and $$f^*$$ is the natural extension of $$f$$ to the hyperreals.

Continuity
A real function f is continuous at a standard real number x if for every hyperreal x'  infinitely close to x, the value f(x' ) is also infinitely close to f(x). This captures Cauchy's definition of continuity as presented in his 1821 textbook Cours d'Analyse, p. 34.

Here to be precise, f would have to be replaced by its natural hyperreal extension usually denoted f* (see discussion of Transfer principle in main article at nonstandard analysis).

Using the notation $$\approx$$ for the relation of being infinitely close as above, the definition can be extended to arbitrary (standard or nonstandard) points as follows:

A function f is microcontinuous at x if whenever $$x'\approx x$$, one has $$f^*(x')\approx f^*(x)$$

Here the point x' is assumed to be in the domain of (the natural extension of) f.

The above requires fewer quantifiers than the (ε, δ)-definition familiar from standard elementary calculus:

f is continuous at x if for every ε > 0, there exists a δ > 0 such that for every x' , whenever |x − x' | < δ, one has |f(x) − f(x' )| < ε.

Uniform continuity
A function f on an interval I is uniformly continuous if its natural extension f* in I* has the following property (see Keisler, Foundations of Infinitesimal Calculus ('07), p. 45):

for every pair of hyperreals x and y in I*, if $$x\approx y$$ then $$f^*(x)\approx f^*(y)$$.

In terms of microcontinuity defined in the previous section, this can be stated as follows: a real function is uniformly continuous if its natural extension f* is microcontinuous at every point of the domain of f*.

This definition has a reduced quantifier complexity when compared with the standard (ε, δ)-definition. Namely, the epsilon-delta definition of uniform continuity requires four quantifiers, while the infinitesimal definition requires only two quantifiers. It has the same quantifier complexity as the definition of uniform continuity in terms of sequences in standard calculus, which however is not expressible in the first-order language of the real numbers.

The hyperreal definition can be illustrated by the following three examples.

Example 1: a function f is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval.

Example 2: a function f is uniformly continuous on the semi-open interval [0,∞) if and only if it is continuous at the standard points of the interval, and in addition, the natural extension f* is microcontinuous at every positive infinite hyperreal point.

Example 3: similarly, the failure of uniform continuity for the squaring function

 * $$x^2$$

is due to the absence of microcontinuity at a single infinite hyperreal point, see below.

Concerning quantifier complexity, the following remarks were made by Kevin Houston:

 * The number of quantifiers in a mathematical statement gives a rough measure of the statement’s complexity. Statements involving three or more quantifiers can be difficult to understand. This is the main reason why it is hard to understand the rigorous definitions of limit, convergence, continuity and differentiability in analysis as they have many quantifiers. In fact, it is the alternation of the $$\forall$$ and $$\exists$$ that causes the complexity.

Andreas Blass wrote as follows:
 * Often ... the nonstandard definition of a concept is simpler than the standard definition (both intuitively simpler and simpler in a technical sense, such as quantifiers over lower types or fewer alternations of quantifiers).

Compactness
A set A is compact if and only if its natural extension A* has the following property: every point in A* is infinitely close to a point of A. Thus, the open interval (0,1) is not compact because its natural extension contains positive infinitesimals which are not infinitely close to any positive real number.

Heine–Cantor theorem
The fact that a continuous function on a compact interval I is necessarily uniformly continuous (the Heine–Cantor theorem) admits a succinct hyperreal proof. Let x, y be hyperreals in the natural extension I* of I. Since I is compact, both st(x) and st(y) belong to I. If x and y were infinitely close, then by the triangle inequality, they would have the same standard part

 * $$c = \operatorname{st}(x) = \operatorname{st}(y).$$

Since the function is assumed continuous at c,

 * $$f(x)\approx f(c)\approx f(y),$$

and therefore f(x) and f(y) are infinitely close, proving uniform continuity of f.

Why is the squaring function not uniformly continuous?
Let f(x) = x2 defined on $$\mathbb{R}$$. Let $$N\in \mathbb{R}^*$$ be an infinite hyperreal. The hyperreal number $$N + \tfrac{1}{N}$$ is infinitely close to N. Meanwhile, the difference

 * $$ f(N+\tfrac{1}{N}) - f(N) = N^2 + 2 + \tfrac{1}{N^2} - N^2 = 2 + \tfrac{1}{N^2}$$

is not infinitesimal. Therefore, f* fails to be microcontinuous at the hyperreal point N. Thus, the squaring function is not uniformly continuous, according to the definition in uniform continuity above.

A similar proof may be given in the standard setting.

Example: Dirichlet function
Consider the Dirichlet function
 * $$I_Q(x):=\begin{cases} 1 & \text{ if }x \text{ is rational}, \\

0 & \text{ if } x \text{ is irrational}. \end{cases}$$

It is well known that, under the standard definition of continuity, the function is discontinuous at every point. Let us check this in terms of the hyperreal definition of continuity above, for instance let us show that the Dirichlet function is not continuous at π. Consider the continued fraction approximation an of π. Now let the index n be an infinite hypernatural number. By the transfer principle, the natural extension of the Dirichlet function takes the value 1 at an. Note that the hyperrational point an is infinitely close to π. Thus the natural extension of the Dirichlet function takes different values (0 and 1) at these two infinitely close points, and therefore the Dirichlet function is not continuous at π.

Limit
While the thrust of Robinson's approach is that one can dispense with the approach using multiple quantifiers, the notion of limit can be easily recaptured in terms of the standard part function st, namely

 * $$\lim_{x\to a} f(x) = L$$

if and only if whenever the difference x − a is infinitesimal, the difference f(x) − L is infinitesimal, as well, or in formulas:

 * if st(x) = a  then st(f(x)) = L,

cf. (ε, δ)-definition of limit.

Limit of sequence
Given a sequence of real numbers $$\{x_n \mid n\in \N\}$$, if $$L\in \R$$ L is the limit of the sequence and

 * $$ L = \lim_{n \to \infty} x_n $$

if for every infinite hypernatural n, st(xn)=L (here the extension principle is used to define xn for every hyperinteger n).

This definition has no quantifier alternations. The standard (ε, δ)-style definition, on the other hand, does have quantifier alternations:

 * $$L = \lim_{n \to \infty} x_n \Longleftrightarrow \forall \varepsilon>0\;, \exists N \in \N\;, \forall n \in \N : n >N \rightarrow |x_n - L| < \varepsilon.$$

Extreme value theorem
To show that a real continuous function f on [0,1] has a maximum, let N be an infinite hyperinteger. The interval [0, 1] has a natural hyperreal extension. The function f is also naturally extended to hyperreals between 0 and 1. Consider the partition of the hyperreal interval [0,1] into N subintervals of equal infinitesimal length 1/N, with partition points xi = i /N as i "runs" from 0 to N. In the standard setting (when N is finite), a point with the maximal value of f can always be chosen among the N+1 points xi, by induction. Hence, by the transfer principle, there is a hyperinteger i0 such that 0 ≤ i0 ≤ N and $$f(x_{i_0})\geq f(x_i)$$ for all i = 0, …, N (an alternative explanation is that every hyperfinite set admits a maximum). Consider the real point

 * $$c= {\rm st}(x_{i_0})$$

where st is the standard part function. An arbitrary real point x lies in a suitable sub-interval of the partition, namely $$x\in [x_i,x_{i+1}]$$, so that st(xi) = x. Applying st to the inequality $$f(x_{i_0})\geq f(x_i)$$, $${\rm st}(f(x_{i_0}))\geq {\rm st}(f(x_i))$$. By continuity of f,
 * $${\rm st}(f(x_{i_0}))= f({\rm st} (x_{i_0}))=f(c)$$.

Hence f(c) ≥ f(x), for all x, proving c to be a maximum of the real function f. See.

Intermediate value theorem
As another illustration of the power of Robinson's approach, a short proof of the intermediate value theorem (Bolzano's theorem) using infinitesimals is done by the following.

Let f be a continuous function on [a,b] such that f(a)<0 while f(b)>0. Then there exists a point c in [a,b] such that f(c)=0.

The proof proceeds as follows. Let N be an infinite hyperinteger. Consider a partition of [a,b] into N intervals of equal length, with partition points xi as i runs from 0 to N. Consider the collection I of indices such that f(xi)>0. Let i0 be the least element in I (such an element exists by the transfer principle, as I is a hyperfinite set). Then the real number $$c=\mathrm{st}(x_{i_0})$$ is the desired zero of f. Such a proof reduces the quantifier complexity of a standard proof of the IVT.

Basic theorems
If f is a real valued function defined on an interval [a, b], then the transfer operator applied to f, denoted by *f, is an internal, hyperreal-valued function defined on the hyperreal interval [*a, *b].

Theorem: Let f be a real-valued function defined on an interval [a, b]. Then f is differentiable at a < x < b if and only if for every non-zero infinitesimal h, the value

 * $$ \Delta_h f := \operatorname{st} \frac{[{}^*\!f](x+h)-[{}^*\!f](x)}{h} $$

is independent of h. In that case, the common value is the derivative of f at x.

This fact follows from the transfer principle of nonstandard analysis and overspill.

Note that a similar result holds for differentiability at the endpoints a, b provided the sign of the infinitesimal h is suitably restricted.

For the second theorem, the Riemann integral is defined as the limit, if it exists, of a directed family of Riemann sums; these are sums of the form

 * $$ \sum_{k=0}^{n-1} f(\xi_k) (x_{k+1} - x_k) $$

where

 * $$a = x_0 \leq \xi_0 \leq x_1 \leq \ldots x_{n-1} \leq \xi_{n-1} \leq x_n = b.$$

Such a sequence of values is called a partition or mesh and

 * $$ \sup_k (x_{k+1} - x_k) $$

the width of the mesh. In the definition of the Riemann integral, the limit of the Riemann sums is taken as the width of the mesh goes to 0.

Theorem: Let f be a real-valued function defined on an interval [a, b]. Then f is Riemann-integrable on [a, b] if and only if for every internal mesh of infinitesimal width, the quantity

 * $$ S_M = \operatorname{st} \sum_{k=0}^{n-1} [*f](\xi_k) (x_{k+1} - x_k) $$

is independent of the mesh. In this case, the common value is the Riemann integral of f over [a, b].

Applications
One immediate application is an extension of the standard definitions of differentiation and integration to internal functions on intervals of hyperreal numbers.

An internal hyperreal-valued function f on [a, b] is S-differentiable at x, provided

 * $$ \Delta_h f = \operatorname{st} \frac{f(x+h)-f(x)}{h} $$

exists and is independent of the infinitesimal h. The value is the S derivative at x.

Theorem: Suppose f is S-differentiable at every point of [a, b] where b − a is a bounded hyperreal. Suppose furthermore that

 * $$ |f'(x)| \leq M \quad a \leq x \leq b. $$

Then for some infinitesimal ε

 * $$ |f(b) - f(a)| \leq M (b-a) + \epsilon.$$

To prove this, let N be a nonstandard natural number. Divide the interval [a, b] into N subintervals by placing N − 1 equally spaced intermediate points:

 * $$a = x_0 < x_1< \cdots < x_{N-1} < x_N = b$$

Then

 * $$ |f(b) - f(a)| \leq \sum_{k=1}^{N-1} |f(x_{k+1}) - f(x_{k})| \leq \sum_{k=1}^{N-1} \left\{|f'(x_k)| + \epsilon_k\right\}|x_{k+1} - x_{k}|.$$

Now the maximum of any internal set of infinitesimals is infinitesimal. Thus all the εk's are dominated by an infinitesimal ε. Therefore,

 * $$ |f(b) - f(a)| \leq \sum_{k=1}^{N-1} (M + \epsilon)(x_{k+1} - x_{k}) = M(b-a) + \epsilon (b-a)$$

from which the result follows.

title: Nonstandard calculus


text:
Rash

A rash is a change of the skin that affects its color, appearance, or texture.

A rash may be localized in one part of the body, or affect all the skin. Rashes may cause the skin to change color, itch, become warm, bumpy, chapped, dry, cracked or blistered, swell, and may be painful. The causes, and therefore treatments for rashes, vary widely. Diagnosis must take into account such things as the appearance of the rash, other symptoms, what the patient may have been exposed to, occupation, and occurrence in family members. The diagnosis may confirm any number of conditions. The presence of a rash may aid diagnosis; associated signs and symptoms are diagnostic of certain diseases. For example, the rash in measles is an erythematous, morbilliform, maculopapular rash that begins a few days after the fever starts. It classically starts at the head, and spreads downwards.

Differential diagnosis
Common causes of rashes include: Uncommon causes:
 * Food allergy
 * Medication side effects
 * Anxiety
 * Allergies, for example to food, dyes, medicines, insect stings, metals such as zinc or nickel; such rashes are often called hives.
 * Skin contact with an irritant
 * Fungal infection, such as ringworm
 * Balsam of Peru
 * Skin diseases such as eczema or acne
 * Exposure to sun (sunburn) or heat
 * Friction due to chafing of the skin
 * Irritation such as caused by abrasives impregnated in clothing rubbing the skin. The cloth itself may be abrasive enough for some people
 * Secondary syphilis
 * Poor personal hygiene
 * Autoimmune disorders such as psoriasis
 * Lead poisoning
 * Pregnancy
 * Repeated scratching on a particular spot
 * Lyme disease
 * Scarlet fever
 * COVID-19 (see )

Diagnostic approach
The causes of a rash are numerous, which may make the evaluation of a rash extremely difficult. An accurate evaluation by a provider may only be made in the context of a thorough history, i.e. medications the patient is taking, the patient's occupation, where the patient has been and complete physical examination.

Points typically noted in the examination include:
 * The appearance: e.g., purpuric (typical of vasculitis and meningococcal disease), fine and like sandpaper (typical of scarlet fever); circular lesions with a central depression are typical of molluscum contagiosum (and in the past, small pox); plaques with silver scales are typical of psoriasis.
 * The distribution: e.g., the rash of scarlet fever becomes confluent and forms bright red lines in the skin creases of the neck, armpits and groins (Pastia's lines); the vesicles of chicken pox seem to follow the hollows of the body (they are more prominent along the depression of the spine on the back and in the hollows of both shoulder blades); very few rashes affect the palms of the hands and soles of the feet (secondary syphilis, rickettsia or spotted fevers, guttate psoriasis, hand, foot and mouth disease, keratoderma blennorrhagicum);
 * Symmetry: e.g., herpes zoster usually only affects one side of the body and does not cross the midline.

A patch test may be ordered, for diagnostic purposes.

Treatment
Treatment differs according to which rash a patient has been diagnosed with. Common rashes can be easily remedied using steroid topical creams (such as hydrocortisone) or non-steroidal treatments. Many of the medications are available over the counter in the United States.

The problem with steroid topical creams i.e. hydrocortisone; is their inability to penetrate the skin through absorption and therefore not be effective in clearing up the affected area, thus rendering the hydrocortisone almost completely ineffective in all except the most mild of cases.

title: Rash


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Autoweek

Autoweek is a car culture publication and magazine based in Detroit, Michigan. It was first published in 1958 and in 1977 the publication was purchased by Crain Communications Inc, its current brand licensor. The magazine was published weekly and focused on motor sports, new car reviews, and old cars, events and DIY. Autoweek now publishes Autoweek.com. Autoweek brand is owned by Crain Communications Inc., publisher of leading industry trade publications Advertising Age and Automotive News, among others, and is based in Detroit, Michigan.

The Autoweek also includes an Autoweek iPhone and iPad app.

As of November 2019 the publication went digital and was no longer available in printed format. Hearst Magazines entered a multi-year licensing deal with Crain Communications to operate the digital and experiential businesses of Autoweek.

History
Autoweek began publication in 1958 as a bi-weekly motorsports newsletter, titled Competition Press. One of the editors involved with the creation of the magazine was professional racer Denise McCluggage. In 1964, distribution was changed to weekly, the title was changed to Competition Press & Autoweek, and vehicle reviews and industry news were included. The name was shortened to Autoweek in 1975.

In 1977, the paper was purchased by Crain Communications, Inc. and eventually changed into a magazine in 1986. In 1988, Leon Mandel was named Publisher, a position he held until November, 2001. Leon's son, Dutch Mandel, joined Autoweek in 1997 holding leadership positions culminating with his time as publisher. In 2016, Mandel left Autoweek and Rory Carroll became interim-publisher and later, publisher. In 2019, Patrick Carone became editorial director.

Editorial direction
At its launch, Autoweek (then titled: Competition Press) aimed to provide information for car and racing enthusiasts that was not readily available through other media outlets. The publication gave thorough coverage of major and minor races, as well as auto shows. Auto racing coverage included full grids, qualifying times, speeds, and tires, for both amateur and professional races. As television and cable began increasing motorsports coverage during the 1970s, 1980s, and 1990s Autoweek began to focus more prominently on automotive enthusiasts. As of 2012, in addition to covering auto racing and auto shows, Autoweek covers vehicle trends, reviews and automotive lifestyle stories.

title: Autoweek


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Wikipedia:Articles for deletion/Thomas-Joseph-Duchesne

Self-added vanity page. Google returns the stupendous sum of zero hits; this article doesn't even come up. Might move to User:Joseph Duchesne and delete the redirect if feeling especially charitable. DopefishJustin 21:24, May 8, 2004 (UTC)

 * Delete. Vanity.  - TB 22:43, 8 May 2004 (UTC)
 * Vanity. Delete -- Cyrius|&#9998 22:43, May 8, 2004 (UTC)
 * Delete, vanity. --Stormie 03:38, May 9, 2004 (UTC)
 * "Joseph grew up to be fine, healthy, and to enjoy typing in 3rd person." Bodnotbod sat at his computer and, with the taste of bile in his throat, voted to delete.
 * But wait! Along came yet another delete vote from out of the clear blue!  Could it be?  Yes!  It's Lucky 6.9 03:08, 10 May 2004 (UTC)
 * Delete. Alcarillo 16:11, 10 May 2004 (UTC)
 * They call it "vanity"? Well, this is very vain. I'm almost sure he's actually looking at himself in the mirror as he's typing it. Either that, or in his armchair. Wiwaxia 00:08, 13 May 2004 (UTC)


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King Peak (Yukon)

King Peak (sometimes called Mount King) is the fourth-highest mountain in Canada and the ninth-highest peak in North America. Situated just west of Mount Logan (highest in Canada), in Yukon, it is considered a satellite peak of the massive mountain.

Climbing history
The first ascent of King Peak was made in 1952 by some students from the University of Alaska. Russell Alston Paige, Keith Hart, Elton Thayer and Bill Atwood walked to the Ogilvie Glacier at the foot of Quartz Ridge where the majority of their supplies had been air dropped. They reached Camp 2 on the west ridge on June 3. After waiting two days for a storm to subside, Hart and Thayer set out for the summit while Atwood remained in camp due to a knee injury. After struggling with rock towers and icy crests, they reached the top on June 6, 1952.

The second and third ascents were also completed in 1952 by an American team who had also just made the first ascent of Mount Augusta. From the south side of King Peak, they made their way up to the east ridge occasionally traversing to the north side to avoid steep slopes. After two failed summit attempts on July 20 and 21, Pete Schoening and Gibson Reynolds succeeded in negotiating the gendarme that had rebuffed the failed attempts and reached the top on July 23. Schoening returned the next day with Dick McGowan and Bill Niendorff to complete the third ascent.

title: King Peak (Yukon)


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Talk:King Peak (Yukon)

Fair use rationale for Image:235 KingPeak.jpg
Image:235 KingPeak.jpg is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in this Wikipedia article constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

Please go to the image description page and edit it to include a fair use rationale. Using one of the templates at Fair use rationale guideline is an easy way to ensure that your image is in compliance with Wikipedia policy, but remember that you must complete the template. Do not simply insert a blank template on an image page.

If there is other fair use media, consider checking that you have specified the fair use rationale on the other images used on this page. Note that any fair use images lacking such an explanation can be deleted one week after being tagged, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.

BetacommandBot (talk) 23:29, 7 March 2008 (UTC)

title: Talk:King Peak (Yukon)