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Skepticism

Ancient and Modern Skepticism: an overview
Skepticism has been regarded by Ezequiel de Olaso not only as a way of thinking but also as a way of living which is radically different from others, most of all the philosophical one. However, this was not the meaning that Pyrrho’s legacy adopted in the Early Modern period. This new meaning was not caused by the Early Modern philosophers’ impossibility to read the original texts. It was due to the fact that Skepticism took over a different role, because it was updated in a different context. It is obvious, then, that there is not a direct connection between Ancient Skepticism, whose main thinkers were Pyrrho and Sextus Empiricus, and the one adopted by Early Modern philosophers. Generally speaking, we can point out that both played a confrontative role against dogmatic attitudes, but in the Early Modern period Skepticism did not aim to unperturbedness or quietude as it did in the Ancient times. Philosophers such as Descartes, Leibniz or Hume portrayed the Skeptic as a person who expressed a constant doubt about any topic. James Noxon asserts that both Descartes and Hume considered that they were obliged to test the power of reason as a previous and inevitable step before they could explain their philosophical system. The Pyrrhonic crisis was then a pars destruens that was followed by a pars construens.

Some problems about identity
First of all it is necessary to take into account certain difficulties concerning the different types of Skepticism. Many authors include both Pyrrhonism and Academicism under the label “Skepticism”, despite the fact that Sextus Empiricus himself states what the differences are between the type of Skepticism that he professes, founded by Pyrrho of Ellis (circa 360-275 b.C.), and that of the New Academy, whose main figures were Arcesilas (circa 315 – 241 b.C.) and Carneades (circa 213-129 b.C.). We consider that the authors who adopt this typology usually fall in two mistakes. The first mistake is to consider the New Academy as a kind of Skepticism based on the claim that both Pyrrhonism and Academicism affirm that it is not possible to know the truth. The differences between them consist in the attitude each of them adopts to solve the problem. In order to be faithful to the textual sources, we consider that the above mentioned difference is not exact. The Academics deny the criterion of truth, which entails to assert that we cannot reach the truth because we are incapable of discerning cases of truth from cases of falsity. The Skeptics, on the other side, avoid committing themselves on judgments about the criterion of truth and just accept passively what appears to their senses and follows the laws and customs of their society. Therefore, Pyrrho’s followers do not affirm that they cannot get to know the truth; they just suspend judgment about it and “keep on searching.” Facing the impossibility to reach the truth, Carneades offers a theory of knowledge that consists in asserting that every appearance is potentially deceitful, but despite that fact, we tend to accept some of them and to reject some others. The criterion we use for this distinction is that some appearances persuade us and the rest do not. Thus, he considers that the appearances that are able to persuade us are probable. He also believes that they can acquire different degrees of persuasion based on their coherence. The appearance with the highest level of probability is the one that shows more coherence in its component parts and in doing so allows us to recognize that which it represents. This means that the Academic offers a criterion to discern cases of knowledge –not cases of truth- based on what is more convincing to the knowing subject. The Skeptic, on the contrary, assents unintentionally, without passing through a deliberation process. He does not have recourse to any criterion of eligibility, because that is not a legitimate action to take. The second mistake consists in defining Skepticism as an attitude of “universal doubt.” This interpretation, like the first we have mentioned, belongs not only to contemporary authors but also to many Early Modern philosophers. Concerning this mistake, it may be argued that one of the main features of Skepticism is its opposition to dogmatism. The Skeptic suspends judgment about dogmatic knowledge claims, that is, about assertions that are supposed to be unquestionable, because he considers that it is always possible to oppose to them other claims which are equally trustable. In this way we would arrive at a point of equivalence between them because there would be equally good reasons to accept any of them and it would be impossible to decide which is superior, as the dogmatic pretends. To doubt is not an exclusive feature of the Skeptic because, as De Olaso asserts, dogmatist can also do so. It is an attitude that belongs to anyone who believes that they are able to judge something. That is why doubting is not opposed to asserting something dogmatically. The way to get out of the equivalence of claims to which the dogmatic arguments leads us is not theoretical, but practical. That is why the Skeptical man’s attitude is to abolish doubt through suspense of judgment. This will lead him to unperturbedness about what is being a matter of opinion. As Luis Villoro explains: “Because the main interest of a Pyrrhonist is not to change the world but to reach inner quietude, he chooses to break up with doubt through suspense of judgment. The choice between doubting or not is solved out by means of a practical reason.” As a final comment, we can point out that the fact of not speaking one’s mind out about the truth of falsity of an assessment entails that the Skeptic has a criterion after all, which is to consider that if an assessment cannot be regarded as true or false, it is because it did not reach a certain standard. Therefore, Skepticism is not equal to a lack of criteria. On the contrary, it entails the possession of an unquestionable standard which consists in knowing when it is accurate to suspend judgment or not. This shows once more the dialectical nature of Pyrrhonism. There is not a single Skeptic who doubts about everything. Hence, when we find a Skeptical attitude, we need to enquire what the assessments it is suspending judgment on are and what criterion it is being used. In doing so, we can understand, as Philip Stanley suggests, that “it is possible, without contradiction, to affirm a given proposition under one criterion, suspend judgement on it under a second, and deny it under a third.”

Skepticism in Early Modern times
At the beginning of this article, we stated that Skepticism adopted a new meaning in the Early Modern period. The manuscripts of Sextus Empiricus were re discovered in the mid-fifteenth century, just a few years before the quarrel about “the rule of faith”, which was one of the main topics at issue in the Reformation of 1517. Both Luther and Calvin argued with the Catholic Church about matters of faith and proposed to replace the criterion of authority of the Pope and the councils for the authority of one’s own conscience. One’s own convictions are the only thing that can lead us to justify the acceptance of any religious opinion. This attitude was the starting point of a battle about the need for an accurate justification of the infallible truth of religion. Catholics reacted against reformists showing that the standard that their enemies proposed was not justifiable and that their pretended religious knowledge criterion based on one’s own subjectivity led to complete Skepticism about religious truths. The edition of Sextus’ works in 1562 helped the religious quarrel to turn into an epistemological dispute, because it unfolded the question of justifying the foundations of knowledge in general. As Richard Popkin states, “the problem of finding a criterion of truth, first raised in theological disputes, was then later raised with regard to natural knowledge, leading to la crise pyrrhonienne of the early sixteenth century.” Michel de Montaigne was highly influenced by the works of Sextus. He applied the Skeptical teaching to the intellectual questions of his time, as we can see in his Essays. One of them, the “Apology for Reymond Sebond” reproduces the distinction made by Sextus between dogmaticists, Skeptics and Academicists. Montaigne expresses his preference for the Skeptic attitude, because he believes that it shows a more radical and coherent kind of doubt than the Academic one. He also adheres to the suspense of judgment method that leads to unperturbedness. Gradually, the Aristotelian physics started losing strength against the emerging modern scientific theories, such as Copernicus’, Galileo’s and Newton’s. This opened a new field for the Pyrrhonic doctrines, because this change in the image of the world created a crisis similar to the one that took place in the religious field. But this time the crisis reached all human sciences and philosophy. That is why in the period that goes from seventeenth to eighteenth century, Skepticism played a different role, which departed from the one it assumed in the Renaissance. Popkin explains that “the super-scepticism of Descartes, involved in his demon hypothesis, began a new phase in the history of skepticism that was to be developed by Pascal, Bayle, Huet, and later Hume and Kierkegaard. Descartes refutation of skepticism also made the skeptics turn their attack against his system instead of against their traditional enemies. Hence, the skeptical arguments had to be altered to fit the new opponent, and skepticism in the last half of seventeenth century changed from being anti-Scholastic and anti-Platonic, to being anti-Cartesian.” This new feature, influenced by the philosophy of Descartes, surely contributed to depicting the Pyrrhonist as a man who doubts about everything or who suspends judgment about any matter, and not only about the assessments that involve dogmatism. Academicism also flourished again in the mid-seventeenth century in the works of some of the most relevant anti-Cartesians of that time. They adapted the Academicism that is portrayed by Cicero in Academica in order to attack Cartesianism, as it is evidenced in Simon Foucher’s works.

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User talk:Tclamb/FEniCS

$$\int_\Omega \rho \left( \left(\mathbf{u} \cdot \nabla \mathbf{u} \right) \cdot \mathbf{v} \right) + \mu \left( \left( \nabla \mathbf{u} \right) : \left(\nabla \mathbf{v}\right) \right) d\Omega = \int_\Omega \mathbf{f} \cdot \mathbf{v} - \left( \nabla p \right) \cdot \mathbf{v} \; d\Omega + \oint_\Gamma \left(\nabla \mathbf{u} \cdot \hat \mathbf{n} \right) \cdot \mathbf{v} \; d\Gamma$$

=Function to Get Value at Nearest Point=

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Solar System, technical/Sun

A natural division of astronomical objects, between rocky objects, astronomical objects with solid surfaces, or solids and liquids predominately on the surface, and gas objects, astronomical objects with gases predominately detected and apparently constituting a surface, may be an informative approach toward stellar science. The Earth is an apparent rocky object that has a gaseous envelope. When viewed under certain conditions in radiation astronomy, the Earth appears as a gas object.

Depending primarily upon gas temperature, the presence of gas may be used to determine the composition of the gas object observed, at least the outer layer. Early spectroscopy of the Sun using estimates of "the line intensities of several lines by eye [to derive] the abundances of ... elements ... [concluded] that the Sun [is] largely made of hydrogen."

At right is an image from the GOES 14 Solar X-ray Imager during the most recent quiet period on or above the Sun. Except for X-ray emission that suggests a circular disc with some isolated X-ray sources at specific locations, the Sun is almost invisible. X-rays are primarily emitted from plasmas near 106 K.

Notation
Notation: let the symbol Def. indicate that a definition is following.

Notation: let the symbols between [ and ] be replacement for that portion of a quoted text.

Universals
To help with definitions, their meanings and intents, there is the learning resource theory of definition.

Def. evidence that demonstrates that a concept is possible is called proof of concept.

The proof-of-concept structure consists of
 * 1) background,
 * 2) procedures,
 * 3) findings, and
 * 4) interpretation.

The findings demonstrate a statistically systematic change from the status quo or the control group.

Def. "[t]he star at the center of our solar system" is called the Sun.

Def. "[a] luminous celestial body, made up of plasma (particularly hydrogen and helium) and having a spherical shape" is called a star.

On Wiktionary, after the above definition for a star is the comment, "Depending on context the sun may or may not be included."

Def. "[a]ny small luminous dot appearing in the cloudless portion of the night sky, especially with a fixed location relative to other such dots" is called a star.

Def. "[t]he natural medium emanating from the sun and other very hot sources (now recognised as electromagnetic radiation with a wavelength of 400-750 nm), within which vision is possible" is called light.

Def. "emitting light" is called luminous.

Plasma-object astronomy
"[A] medium-strength flare erupted from the sun on July 19, 2012. The blast also generated the enormous, shimmering plasma loops, which are an example of a phenomenon known as "coronal rain," agency officials said."

"Hot plasma in the corona cooled and condensed along strong magnetic fields in the region" slowly falling back to the solar surface as plasma "rain".

"Many CMEs have also been observed to be unassociated with any obvious solar surface activity".

In the images at right, a CME, or "arcs rise above an active region on the surface of the Sun in this series of images taken by the STEREO (Behind) spacecraft on January 27, 2010. The arcs are plasma, superheated matter made up of moving charged particles (electrons and ions). Just as iron filings arc from one end of a magnet to another, the plasma is sliding in an arc along magnetic field lines. In a movie of STEREO observations made between January 26 and January 29, the dynamic streams were initially just over the Sun’s edge and readily spotted as the Sun rotated them more into view."

"About mid-way through the movie clip, a small coronal mass ejection (a stream of charged particles from the Sun) shoots out and into space at about a million miles per hour, carrying some magnetic field with it. The [first] image shows the beginning of the coronal mass ejection, while the [second] image shows the solar matter leaving the Sun’s corona. Most coronal mass ejections are more bulbous and wide: this one is quite narrow and contained. Nonetheless, NASA solar scientists agree that its speed and characteristics suggest that it was indeed a non-typical coronal mass ejection."

Solar clouds
The February 10, 1956, event "was observed at Sacramento Peak. A bright ball appears above the [Sun's] surface, grows in size and Hα brightness, and explodes upward and outward."

Gaseous-object astronomy
Gaseous objects have been used to approximate a location for the Sun based on Kepler's laws.

No gaseous objects are known to have been emitted, reflected, or deflected by the Sun. No gaseous objects have impacted the surface of the Sun.

"Magnetic clouds represent about one third of ejecta observed by satellites at Earth. Other types of ejecta are multiple-magnetic cloud events (a single structure with multiple subclouds distinguishable) and complex ejecta, which can be the result of the interaction of multiple CMEs."

Mercury
"[H]igh-resolution spectral measurements of Mercury show emission in sodium D lines (Potter and Morgan 1985a). This suggests a substantial sodium population in Mercury's atmosphere ... possibly due to photo-sputtering of the planetary surface". At least in emission yellow astronomy, Mercury is a dwarf gaseous object.

Venus
Venus has been detected as a gaseous object using X-ray through red astronomy.

Earth
"[T]he Earth's atmosphere ... is a relatively bright source of gamma rays produced in interactions of ordinary cosmic ray protons with air atoms". In gamma-ray and X-ray astronomy, Earth is a dwarf gaseous object.

With blue astronomy the Earth appears as a dwarf gaseous or fluid (primarily liquid water) object.

Mars
From the point of view of X-ray astronomy, Mars is a dwarf gaseous object.

Jupiter
Jupiter is a larger gaseous object in X-ray and ultraviolet astronomy.

Jupiter is seen as a gaseous object in orange, red and infrared astronomy.

Green astronomy indicates Io is a dwarf gaseous object.

Saturn
Saturn is a larger gaseous object from X-ray and ultraviolet astronomy.

In orange astronomy, Titan is a gaseous object.

Uranus
Cyan astronomy shows Uranus to be a larger gaseous object than the inner objects of the solar system.

Uranus is a larger gaseous object in green and orange astronomy.

Comets
Cyan and green astronomy demonstrate that many comets are dwarf gaseous objects.

Neptune
The gaseous object Neptune is revealed as such with blue astronomy. Clouds above the blue clouds of Neptune have been observed with infrared astronomy.

Rocky-object astronomy
Rocky objects have been used to approximate a location for the Sun based on Kepler's laws. No rocky objects are known to have been emitted, reflected, or deflected by the Sun. Nor, have any rocky objects impacted the surface of the Sun.

"Sun-grazing comets almost never re-emerge, but their sublimative destruction near the sun has only recently been observed directly, while chromospheric impacts have not yet been seen, nor impact theory developed." "[N]uclei are ... destroyed by ablation or explosion ... in the chromosphere, producing flare-like events with cometary abundance spectra."

"The death of a comet at r ~ Rʘ has been seen directly only very recently (Schrijver et al 2011) using the SDO AIA XUV instrument. This recorded sublimative destruction of Comet C/2011 N3 as it crossed the solar disk very near periheloin q = 1.139Rʘ."

"The phenomenon of flare induced sunquakes - waves in the photosphere - discovered by Kosovichev and Zharkova (1998) and now widely studied (e.g. Kosovichev 2006) should also result from the momentum impulse delivered by a cometary impact."

Earth
The image at right shows what the Earth looks like topographically under the water cover that constitutes more than 50 % of the Earth's actual surface. Based on this sea floor topography, the Earth is a rocky object to depths much less than its radius.

Moon
The Moon has been detected using gamma-ray astronomy. These gamma-rays are produced by cosmic ray bombardment of its rocky surface.

"Four radiometers [aboard Luna 13 ] recorded infrared radiation from the [Moon's] surface".

"Nine out of 10 well-characterized Apollo 17 breccia matrices fall into Group 2, and this includes both the blue-grey breccias which are the dominant rock type at this site".

Meteor astronomy
"Sun-grazing comets almost never re-emerge, but their sublimative destruction near the sun has only recently been observed directly, while chromospheric impacts have not yet been seen, nor impact theory developed." "[N]uclei are ... destroyed by ablation or explosion ... in the chromosphere, producing flare-like events with cometary abundance spectra."

"The death of a comet at r ~ Rʘ has been seen directly only very recently (Schrijver et al 2011) using the SDO AIA XUV instrument. This recorded sublimative destruction of Comet C/2011 N3 as it crossed the solar disk very near periheloin q = 1.139Rʘ."

"The phenomenon of flare induced sunquakes - waves in the photosphere - discovered by Kosovichev and Zharkova (1998) and now widely studied (e.g. Kosovichev 2006) should also result from the momentum impulse delivered by a cometary impact."

"Coronal clouds, type IIIg, form in space above a spot area and rain streamers upon it."

The solar wind originates through the polar coronal holes.

Cosmic-ray astronomy
The surface of the Sun has not been detected with cosmic rays.

"[T]he relative abundances of solar cosmic rays reflect those of the solar photosphere for multicharged nuclei with approximately the same nuclear charge-to-mass ratio."

Neutron astronomy
The surface of the Sun is not a known source of neutrons. No neutron probes have been used to detect the Sun.

Proton astronomy
The Sun and the solar wind, at least that portion that originates through the polar coronal holes apparently from the photosphere, may be major sources of protons within the solar system.

At right is a temporal distribution of solar proton flux in units of particles cm-2 s-1 sr-1 as measured by GOES 11 over the four days from November 2, 2003, to November 4, 2003, in three windows of energy: ≥ 100 MeV (green), ≥ 50 MeV (blue), and ≥ 10 MeV (red). The percentage originating from the surface of the Sun either directly or through the contribution to the solar wind is not indicated.

Electron astronomy
"Energetic photons, ions and electrons from the solar wind, together with galactic and extragalactic cosmic rays, constantly bombard surfaces of planets, planetary satellites, dust particles, comets and asteroids."

Jovian electrons propagate "along the spiral magnetic field of the interplanetary medium [from Jupiter and its magnetosphere to the Sun]".

Positron astronomy
The solar flare at Active Region 10039 on July 23, 2002, exhibits many exceptional high-energy phenomena including the 2.223 MeV neutron capture line and the 511 keV electron-positron (antimatter) annihilation line. In the image at right, the RHESSI low-energy channels (12-25 keV) are represented in red and appear predominantly in coronal loops. The high-energy flux appears as blue at the footpoints of the coronal loops. Violet is used to indicate the location and relative intensity of the 2.2 MeV emission.

During solar flares “[s]everal radioactive nuclei that emit positrons are also produced; [which] slow down and annihilate in flight with the emission of two 511 keV photons or form positronium with the emission of either a three gamma continuum (each photon < 511 keV) or two 511 keV photons." The Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) made the first high-resolution observation of the solar positron-electron annihilation line during the July 23, 2003 solar flare. The observations are somewhat consistent with electron-positron annihilation in a quiet solar atmosphere via positronium as well as during flares. Line-broadening is due to "the velocity of the positronium." "The width of the annihilation line is also consistent ... with thermal broadening (Gaussian width of 8.1 ± 1.1 keV) in a plasma at 4-7 x 105 K. ... The RHESSI and all but two of the SMM measurements are consistent with densities ≤ 1012 H cm-3 [but] <10% of the p and α interactions producing positrons occur at these low densities. ... positrons produced by 3He interactions form higher in the solar atmosphere ... all observations are consistent with densities > 1012 H cm-3. But such densities require formation of a substantial mass of atmosphere at transition region temperatures."

Neutrino astronomy
Neutrinos are hard to detect. The Super-Kamiokande, or "Super-K" is a large-scale experiment constructed in an unused mine in Japan to detect and study neutrinos. The image at right required 500 days worth of data to produce the "neutrino image" of the Sun. The image is centered on the Sun's calculated position. It covers a 90° x 90° octant of the sky (in right ascension and declination). The higher the brightness of the color, the larger is the neutrino flux.

The surface of the Sun is not a known source of neutrinos. Those detected may be from nucleosynthesis within the coronal cloud in the near vicinity of the Sun or perhaps from nucleosynthesis occurring interior to the Sun.

Gamma-ray astronomy
The surface of the Sun has yet to be detected as a gamma ray source, reflector, or in fluorescence.

X-ray astronomy
The image at top shows that the surface of the Sun, its photosphere, apparently neither emits, reflects, nor fluoresces X-rays.

"X-ray photons can be effectively backscattered by photosphere atoms and electrons (Tomblin 1972; Bai & Ramaty 1978). ... [A]t energies not dominated by absorption the backscattered albedo flux must be seen virtually in every solar flare spectrum, the degree of the albedo contribution depending on the directivity of the primary X-ray flux (Kontar et al. 2006). The solar flare photons backscattered by the solar photosphere can contribute significantly (the reflected flux is 50-90 % of the primary in the 30 - 50 keV range for isotropic sources) to the total observed photon spectrum. for the simple case of a power-law-like primary solar flare spectrum (without albedo), the photons reflected by the photosphere produce a broad 'hump' component. Photospheric albedo makes the observed spectrum flatter below ~ 35 keV and slightly steeper above, in comparison with the primary spectrum."

Optical astronomy
"[T]he color index is a simple numerical expression that determines the color of an object, which in the case of a star gives its temperature. To measure the index, one observes the magnitude of an object successively through two different filters, such as U and B, or B and V, where U is sensitive to ultraviolet rays, B is sensitive to blue light, and V is sensitive to visible (green-yellow) light (see also: UBV system). The set of passbands or filters is called a photometric system. The difference in magnitudes found with these filters is called the U-B or B–V color index, respectively. The smaller the color index, the more blue (or hotter) the object is. Conversely, the larger the color index, the more red (or cooler) the object is. This is a consequence of the logarithmic magnitude scale, in which brighter objects have smaller (more negative) magnitudes than dimmer ones. ... The passbands most optical astronomers use are the UBVRI filters, where the U, B, and V filters are as mentioned above, the R filter passes red light, and the I filter passes infrared light. ... These filters were specified as particular combinations of glass filters and photomultiplier tubes."

Ultraviolet astronomy
The Sun's photosphere probably generates little or no ultraviolet rays.

Visual astronomy
"[T]he [visual] color index is a simple numerical expression that determines the color of an object, which in the case of a star gives its temperature. To measure the [visual] index, one observes the magnitude of an object successively through two different filters, such as ... B and V, where ... B is sensitive to blue light, and V is sensitive to visible (green-yellow) light ... The difference in magnitudes found with these filters is called the ... B–V color index ... The smaller the color index, the more blue (or hotter) the object is. Conversely, the larger the color index, the more red (or cooler) the object is. ... For comparison, the yellowish Sun has a B–V index of 0.656 ± 0.005, while the bluish Rigel has a B–V of –0.03 (its B magnitude is 0.09 and its V magnitude is 0.12, B–V = –0.03). "

Violet astronomy
The Sun's photosphere emits over the violet (380-450 nm) band.

Blue astronomy
The surface of the Sun emits blue (450-475 nm) electromagnetic radiation.

Cyan astronomy
The surface of the Sun emits cyan (476-495 nm) electromagnetic radiation.

Green astronomy
The Na I green lines at 568.2 and 568.8 nm arise "in the photospheric layers between log τ5000 ≈ -1 and -2."

Yellow astronomy
Seeing the yellow Sun and feeling the warmth of its rays is probably a student's first encounter with an astronomical yellow radiation source.

The surface of the Sun emits yellow (570-590 nm) radiation.

"Stars of spectral classes F and G, such as our sun Sol, have color temperatures that make them look "yellowish".

Orange astronomy
The surface of the Sun emits in the orange (590 to 620 nm) wavelengths.

The Na I lines at 615.4 and 616.0 nm arise "in the photospheric layers between log τ5000 ≈ -1 and -2."

Red astronomy
The surface of the Sun emits in the red (621 to 750 nm) wavelengths.

"[S]ome observed properties of the Sun still defy explanation, such as the degree of Li depletion" [the "solar Li abundance is roughly a factor of 200 below the meteoritic abundance"].

Infrared astronomy
The surface of the Sun emits in the infrared.

At right is a graph of the expected solar and interstellar infrared emission versus wavelength at various radial distances from the Sun.

The Na I lines at 818.3, 819.4, and 1140 nm in the near infrared arise "in the photospheric layers between log τ5000 ≈ -1 and -2."

Radio astronomy
The surface of the Sun emits radio waves.

Superluminal astronomy
The surface of the Sun is not a known source of superluminal radiation.

Heliognosy
The Sun as a star has a composition. Hydrogen has been detected on the surface of the Sun.

For stars, "the metallicity is often expressed as "[Fe/H]", which represents the logarithm of the ratio of a star's iron abundance compared to that of the Sun (iron is not the most abundant heavy element, but it is among the easiest to measure with spectral data in the visible spectrum). The formula for the logarithm is expressed thus:

$$ [\mathrm{Fe}/\mathrm{H}] = \log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}}\right)_{star}} - \log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}}\right)_{sun}} $$

where $$N_{\mathrm{Fe}}$$ and $$N_{\mathrm{H}}$$ are the number of iron and hydrogen atoms per unit of volume respectively. The unit often used for metallicity is the "dex" which is a (now-deprecated) contraction of decimal exponent. By this formulation, stars with a higher metallicity than the Sun have a positive logarithmic value, while those with a lower metallicity than the Sun have a negative value. The logarithm is based on powers of ten; stars with a value of +1 have ten times the metallicity of the Sun (101). Conversely, those with a value of -1 have one tenth (10 −1), while those with -2 have a hundredth (10−2), and so on. Young Population I stars have significantly higher iron-to-hydrogen ratios than older Population II stars. Primordial Population III stars are estimated to have a metallicity of less than −6.0, that is, less than a millionth of the abundance of iron which is found in the Sun."

The image at right describes graphically the temperature and density of the Sun's atmosphere from the photosphere upwards. "The Sun's photosphere has a temperature between 4500 and 6000 K (with an effective temperature of 5777 K) and a density of about [2 x 10-4kg/m3; other stars may have hotter or cooler photospheres.

Hydrogen
The Balmer series of emission lines from hydrogen occur in the visible spectrum of the Sun at: 397, 410, 434, 486, and 656 nm. Hα is the red line (656 nm).

Helium
Helium "was first detected as an unknown yellow spectral line signature in sunlight during a solar eclipse in 1868 by French astronomer Jules Janssen. Janssen is jointly credited with detecting the element along with Norman Lockyer during the solar eclipse of 1868, and Lockyer was the first to propose that the line was due to a new element, which he named."

Plasma
"[P]lasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms (reduce or increase the number of electrons in them), thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions. "

"For plasma to exist, ionization is necessary. The term "plasma density" by itself usually refers to the "electron density", that is, the number of free electrons per unit volume. The degree of ionization of a plasma is the proportion of atoms that have lost or gained electrons, and is controlled mostly by the temperature. Even a partially ionized gas in which as little as 1% of the particles are ionized can have the characteristics of a plasma (i.e., response to magnetic fields and high electrical conductivity). The degree of ionization, α is defined as α = ni/(ni + na) where ni is the number density of ions and na is the number density of neutral atoms. The electron density is related to this by the average charge state  of the ions through ne =  ni where ne is the number density of electrons."

The solar photosphere is a "weakly ionized [ni/(ni + na)] ~ 10-4, relatively cold and dense plasma".

Heliography
The surface of the Sun is often described by features observed. These are located using heliographic coordinates based on heliographic north and south poles. The surface of the Sun rotates, has a rotational north and south pole, and there is a central meridian.

"[D]ifferent parts of the Sun ... rotate at different rates".

At right is a diagram of the internal rotation in the Sun, showing differential rotation in the outer convective region and almost uniform rotation in the central radiative region. The transition between these regions is called the tachocline.

"Until the advent of helioseismology, the study of wave oscillations in the Sun, very little was known about the internal rotation of the Sun. The differential profile of the surface was thought to extend into the solar interior as rotating cylinders of constant angular momentum. Through helioseismology this is now known not to be the case and the rotation profile of the Sun has been found. On the surface the Sun rotates slowly at the poles and quickly at the equator. This profile extends on roughly radial lines through the solar convection zone to the interior. At the tachocline the rotation abruptly changes to solid body rotation in the solar radiation zone. "

The Stonyhurst Disk is superimposed on an image of the Sun to determine the heliographic coordinates. These are constructed by analogy with the geographical and are characterized by two values, latitude (Φ) and longitude (λ). Latitude is measured from the plane of the solar equator. The first longitude (λ1) is measured from the plane of the "central meridian" as it passes through the rotation axis of the Sun and the line connecting the center of the Sun to the observer. The Carrington longitude (λ2) is measured from the central meridian as it passes through the ascending node of the solar equator at Greenwich noon on January 1, 1854 (JD 2398220.0) and rotating with the sidereal period of 25.38 Earth days.

The two longitudes are associated approximately by the ratio

 * $$\lambda_2=\left\{\frac{\lambda_1}{360^\circ}-\frac{{\rm JD}-2398220}{27.2753}\right\}\cdot360^\circ$$.

The Sun has an equatorial radius of 695,500 km

Surface
Def. "[a] visible surface layer of a star, and especially that of a sun" is called a photosphere.

"When we speak of the surface of the Sun, we normally mean the photosphere." "[T]he photosphere may be thought of as the imaginary surface from which the solar light that we see appears to be emitted. The diameter quoted for the Sun usually refers to the diameter of the photosphere." The photosphere emits visual, or visible, radiation.

Illumination of the Sun's photosphere is in part by gamma rays. "Each gamma ray [that interacts with the photosphere] is converted into several million photons of visible light ... At the visible surface of the Sun, the temperature has dropped to 5,700 K and the density to only 0.2 g/m3 (about 1/6,000th the density of air at sea level). "

"The tremendous power output of the Sun is not due to its high power per volume, but instead due to its large size. ... Above the photosphere visible sunlight is free to propagate into space, and its energy escapes the Sun entirely. The change in opacity is due to the decreasing amount of H− ions, which absorb visible light easily. Conversely, the visible light we see is produced as electrons react with hydrogen atoms to produce H− ions. ... The photosphere has a particle density of ~1023 m−3 (this is about 0.37% of the particle number per volume of Earth's atmosphere at sea level; however, photosphere particles are electrons and protons, so the average particle in air is 58 times as heavy)."

Disc edge
Def. a shape or surface that is round and flat in appearance (middle 17th century usage referring to the seemingly flat round form of the Sun) is called a disc, or disk.

Def. "the radial distance q from the Sun's center such that the following finite Fourier transform is zero:

 * $$F(G; q, a) = \int_{-1/2}^{+1/2} G(q + a \sin \pi s) \cos(2\pi s) ds,$$

where s is a dummy variable, G is the observed solar intensity as a function of the radius, and the parameter a determines the extent of the solar limb used" is called the solar edge.

"When F(G; q, a) = 0, the a dependence of q can be used to choose different points as the edge."

Granulation
There is "turbulence and granulation in the Sun's photosphere." "A typical granule has a diameter on the order of 1,000 kilometers and lasts 8 to 20 minutes before dissipating. At any one time, the Sun's surface is covered by about 4 million granules."

"Supergranulation is a particular pattern of [granules] on the Sun's surface [that] was discovered in the 1950s by A.B.Hart using Doppler velocity measurements showing horizontal flows on the photosphere (flow speed about 300 to 500 m/s, a tenth of that in the smaller granules). Later work (1960s) by Leighton, Noyes and Simon established a typical size of about 30000 km for supergranules with a lifetime of about 24 hours. "

Faculae
"Solar faculae are bright spots that form in the canyons between solar granules".

"The emission of higher than average amounts of radiation later were observed from the solar faculae. "

"Correlations are now known to exist with decreases in luminosity caused by sunspots (generally < - 0.3%) and increases (generally < + 0.05%) caused both by faculae that are associated with active regions as well as the magnetically active 'bright network'. ... Faculae in magnetically active regions are hotter and 'brighter' than the average photosphere and cause temporary increases in [total solar irradiance] TSI. ... Luminosity has also been found to decrease by as much as 0.3% on a 10 day timescale when large groups of sunspots rotate across the Earth's view and increase by as much as 0.05% for up to 6 months due to faculae associated with the large sunspot groups. "

The net effect during periods of enhanced solar magnetic activity is increased radiant output of the sun because faculae are larger and persist longer than sunspots."

"Though sunspots themselves are darker, they form when there are particularly magnetically active regions, which is when larger, brighter, longer-duration Faculae are more common as well".

Holes

Def. "[a] hollow spot in a surface" is called a hole.

Def. "[a]n opening in a solid[, liquid, gas, or plasma]" is called a hole.

Def. "[a] round or irregular patch on [or apparently on] the surface of [an entity] having a different color, texture etc. and generally round in shape" is called a spot.

Def. "[a]n opening through which [entities such as] gases ... can pass" is called a vent.

"Sunspots are temporary phenomena on the photosphere of the Sun that appear visibly as dark spots compared to surrounding regions. They are caused by intense magnetic activity, which inhibits convection by an effect comparable to the eddy current brake, forming areas of reduced surface temperature. Like magnets, they also have two poles. Although they are at temperatures of roughly 3,000–4,500 K (2,727–4,227 °C), the contrast with the surrounding material at about 5,780 K leaves them clearly visible as dark spots, as the luminous intensity of a heated black body (closely approximated by the photosphere) is a function of temperature to the fourth power. If the sunspot were isolated from the surrounding photosphere it would be brighter than an electric arc. Sunspots expand and contract as they move across the surface of the Sun and can be as large as 80,000 kilometers (49,710 mi) in diameter, making the larger ones visible from Earth without the aid of a telescope. They may also travel at relative speeds ("proper motions") of a few hundred m/s when they first emerge onto the solar photosphere."

"For the greater part of the sun-spot period there is practically but one zone of spots in each hemisphere. The departure from this condition of things near or at the time of minimum, when the spots of the dying cycle are approaching the equator, and the forerunners of the new cycle are beginning to appear in high latitudes, is the only case in which the solar spots are distinctly separated into more than a single zone in each hemisphere." "Spöerer's law ... involves that in a minimum year the zone about 15° should be entirely barren".

"First of all there were only fifteen groups seen during the entire year [1901], north and south put together. Of these, seven were in the north, and the mean latitude for the north was 8.6°, exactly the latitude of one spot of the seven, and this very naturally, seeing that it was by far the greatest group of the year, the celebrated "eclipse group."" Bold added. "Greatest group" and "eclipse group" are both relative synonyms for "dominant group".

"[T]he spot-groups have been carefully examined for cases of return, and where it appeared clear that the same group has returned a second time or more frequently, without any temporary disappearance or subsidence, such a long-continued group has been treated as an entity throughout." Bold added. "It has been forgotten that, whatever the cause which produces this variation of rotation rate with latitude, the causes producing difference of rate within any given latitude are more effective still."

"[T]here is a slight retardation of the rotation period from the first cycle to the second, shown by both northern and southern hemispheres."

"Studies of stratigraphic data have suggested that the solar cycles have been active for hundreds of millions of years, if not longer; measuring varves in precambrian sedimentary rock has revealed repeating peaks in layer thickness, with a pattern repeating approximately every eleven years. It is possible that the early atmosphere on Earth was more sensitive to changes in solar radiation than today, so that greater glacial melting (and thicker sediment deposits) could have occurred during years with greater sunspot activity. This would presume annual layering; however, alternate explanations (diurnal) have also been proposed. "

"Analysis of tree rings has revealed a detailed picture of past solar cycles: Dendrochronologically dated radiocarbon concentrations have allowed for a reconstruction of sunspot activity dating back 11,400 years, far beyond the four centuries of available, reliable records from direct solar observation. "

"The earliest surviving record of sunspot observation dates from 364 BC, based on comments by Chinese astronomer Gan De in a star catalogue. By 28 BC, Chinese astronomers were regularly recording sunspot observations in official imperial records. "

The second drawing at right is of a group of sun spots and veiled spots. This group was observed on June 17, 1875, at 07:30 a.m. The interior of these spots appears to have a larger granule size than the outer surface of the photosphere itself.

The spiral sunspot in the third drawing at right is a vortex seen on May 5, 1857, at Rome. "[T]he substance of the photosphere is rushing with an eddying motion [into the spot]."

"Starspots are equivalent to sunspots but located on other stars. Spots the size of sunspots are very hard to detect since they are too small to cause fluctuations in brightness. Observed starspots are in general much larger than those on the Sun, up to about 30 % of the stellar surface may be covered, corresponding to sizes 100 times greater than those on the Sun."

"The distribution of starspots across the stellar surface varies analogous to the solar case, but differs for different types of stars, e.g., depending on whether the star is a binary or not. The same type of activity cycles that are found for the Sun can be seen for other stars, corresponding to the solar (2 times) 11-year cycle. Some stars have longer cycles, possibly analogous to the Maunder minima for the Sun."

"Another activity cycle is the so called flip-flop cycle, which implies that the activity on either hemisphere shifts from one side to the other. The same phenomena can be seen on the Sun, with periods of 3.8 and 3.65 years for the northern and southern hemispheres. Flip-flop phenomena are observed for both binary RS CVn stars and single stars although the extent of the cycles are different between binary and singular stars."

"Manifesting intense magnetic activity, sunspots host secondary phenomena such as coronal loops (prominences) and reconnection events. Most solar flares and coronal mass ejections originate in magnetically active regions around visible sunspot groupings. Similar phenomena indirectly observed on stars are commonly called starspots and both light and dark spots have been measured. "

"The sunspot itself can be divided into two parts:
 * The central umbra, which is the darkest part, where the magnetic field is approximately vertical (normal to the Sun's surface).
 * The surrounding penumbra, which is lighter, where the magnetic field is more inclined."

"[T]he rate of helicity change dH/dt due to horizontal motions is

 * $$\frac{dH}{dt} = -2 \oint ( \mathbf{v} \cdot \mathbf{A_p}) \mathbf{B_n} \mathrm{d}\mathbf{S},$$

where Bn is the vertical component of the magnetic field on the photosphere and v the photospheric horizontal velocity."

"Helicity [(H)] is a quantitative measure of the chiral properties of the structures observed in the solar atmosphere. ... [H]elicity [is] injected to the corona by photospheric horizontal shearing motions (other than differential rotation) or by [vertical] magnetic flux. ... [D]ifferential rotation cannot provide the required helicity to the [CME] field ejected to interplanetary space."

Solar active region AR 10030 contained a group of sunspots including the largest one partially included in the image at the right. It is a planet-sized sunspot showing for the first time the dark cores of the filaments extending into the sunspot. These filaments are thousands of km long by about 100 km wide. The image is recorded on July 15, 2002, using the Swedish Solar Telescope (SST).

"The number of sunspots visible on the Sun is not constant, but varies over an 11-year cycle known as the solar cycle. At a typical solar minimum, few sunspots are visible, and occasionally none at all can be seen. Those that do appear are at high solar latitudes. As the sunspot cycle progresses, the number of sunspots increases and they move closer to the equator of the Sun, a phenomenon described by Spörer's law. Sunspots usually exist as pairs with opposite magnetic polarity. The magnetic polarity of the leading sunspot alternates every solar cycle, so that it will be a north magnetic pole in one solar cycle and a south magnetic pole in the next.

HD 154345b has "a 9.2 year, circular orbit with radius 4.2 AU. ... We also detect a ~ 9 year activity cycle in this star [HD 154345] photometrically and in chromospheric emission. ... We note that the Sun's 11 year activity cycle has a period similar to that of Jupiter's orbit, and that the Mount Wilson survey demonstrated that decadal activity cycles are a common feature of old G stars (Baliunas et al. 1995)."

Naked sunspots
"Naked sunspots are spots seen in Hα to be devoid of associated plage. In magnetograms and K-line little if any opposite polarity field is found, and in soft X-ray images a blank appears in the region of the spot. In almost all cases ... in which naked spots resulted the spot groups had emerged in unipolar regions of the same polarity as the naked spot. At least half of the naked spots are associated with coronal holes."

"[N]aked spots are long-lived and show rotation rates close to the Newton-Nunn curve. Most of the naked spots had bright rims in Hα".

Wolf numbers
"[T]he longest data set, the Wolf sunspot number (R), is determined from visible wavelength observations of the solar surface."

"The Wolf number (also known as the International sunspot number, relative sunspot number, or Zürich number) is a quantity that measures the number of sunspots and groups of sunspots present on the surface of the sun."

"This number has been collected and tabulated by researchers for over 150 years."

"The relative sunspot number $$R$$ is computed using the formula (collected as a daily index of sunspot activity):

 * $$R = k(10g + s) \, $$

where

 * $$s$$ is the number of individual spots,
 * $$g$$ is the number of sunspot groups, and
 * $$k$$ is a factor that varies with location and instrumentation (also known as the observatory factor or the personal reduction coefficient $$K$$). "

Heliometry
"The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as the barycenter. The barycenter is the point between two objects where they balance each other; it is the center of mass where two or more celestial bodies orbit each other. When a moon orbits a planet, or a planet orbits a star, both bodies are actually orbiting around a point that lies away from the center of the primary (larger) body."

"The Sun's motion about the center of mass of the Solar System is complicated by perturbations from the planets. Every few hundred years this motion switches between prograde and retrograde. "

Def. "the mass of the Sun" is called the astronomical unit of mass.

Notation: let the symbol $$M_{\odot}$$ indicate the solar mass.

"The solar mass ($$\begin{smallmatrix}M_\odot\end{smallmatrix}$$) is a standard unit of mass in astronomy, used to indicate the masses of other stars, as well as clusters, nebulanebulae and galaxies. It is equal to the mass of the Sun, about two nonillion kilograms:"

 * $$M_{\odot}=( 1.98855\ \pm\ 0.00025 )\ \times10^{30}\hbox{ kg}$$

"This is about 332,946 times the mass of the Earth or 1,048 times the mass of Jupiter."

"Because the Earth follows an elliptical orbit around the Sun, the solar mass can be computed from the equation for the orbital period of a small body orbiting a central mass. Based upon the length of the year, the distance from the Earth to the Sun (an astronomical unit or AU), and the gravitational constant (G), the mass of the Sun is given by:"

 * $$M_\odot=\frac{4 \pi^2 \times (1\ {\rm AU})^3}{G\times(1\ {\rm year})^2}$$.

"The value of the gravitational constant was derived from 1798 measurements by Henry Cavendish using a torsion balance. The value obtained differed only by about 1% from the modern value. The diurnal parallax of the Sun was accurately measured during the transits of Venus in 1761 and 1769, yielding a value of 9″ (compared to the present 1976 value of 8.794148″). When the value of the diurnal parallax is known, the distance to the Sun can be determined from the geometry of the Earth. "

Notation: let the symbol $$R_{\odot}$$ indicate the solar radius.

The "[s]olar radius is a unit of distance used to express the size of stars in astronomy equal to the current radius of the Sun:

 * $$R_{\odot} = 6.955\times 10^8\,\hbox{m} = 0.0046491\,\hbox{AU} = 2.254\times 10^{-8}\,\hbox{pc}$$

The solar radius is approximately 695,500 kilometres (432,450 miles) or about 110 times the radius of the Earth ($$R_\oplus$$), or 10 times the average radius of Jupiter. It varies slightly from pole to equator due to its rotation, which induces an oblateness of order 10 parts per million."

"The luminosity of stars is measured in two forms: apparent (visible light only) and bolometric (total radiant energy). (A bolometer is an instrument that measures radiant energy over a wide band by absorption and measurement of heating.) When not qualified, "luminosity" means bolometric luminosity, which is measured either in the SI units, watts; or in terms of solar luminosities, $$ L_{\odot} $$, that is, how many times as much energy the object radiates as the Sun".

Notation: let the symbol $$ L_{\odot} $$ represent the solar bolometric luminosity.

"The solar luminosity, [$$ L_{\odot} $$], is a unit of radiant flux (power emitted in the form of photons) conventionally used .. to measure the luminosity of stars. One solar luminosity is equal to the current accepted luminosity of the Sun, which is $3.839 W$, or $3.839 erg/s$. The value is slightly higher, $3.939 W$ (equivalent to $4.382 kg/s$ or $1.9 M_{☉}/d$) if the solar neutrino radiation is included as well as electromagnetic radiation. The Sun is a weakly variable star and its luminosity therefore fluctuates. The major fluctuation is the eleven-year solar cycle (sunspot cycle), which causes a periodic variation of about ±0.1%. Any other variation over the last 200–300 years is thought to be much smaller than this. "

"The solar luminosity is related to the solar irradiance measured at the Earth or by satellites in Earth orbit. The mean irradiance at the top of the Earth's atmosphere is sometimes known as the solar constant, [$$ I_{\odot} $$]. Irradiance is defined as power per unit area, so the solar luminosity (total power emitted by the Sun) is the irradiance received at the Earth (solar constant) multiplied by the area of the sphere whose radius is the mean distance between the Earth and the Sun:
 * $$L_\odot = 4\pi kI_\odot A^2\,$$

where A is the unit distance (the value of the astronomical unit in metres) and k is a constant (whose value is very close to one) that reflects the fact that the mean distance from the Earth to the Sun is not exactly one astronomical unit."

Notation: let the symbol $$ Q_{\odot} $$ represent the net solar charge.

"[A] variety of geophysical and astrophysical phenomena can be explained by a net charge on the Sun of -1.5 x 1028 e.s.u." This figure was later reduced by a factor of five.

 * $$ Q_{\odot} = -0.3\times 10^{28}\,\hbox{esu}.$$

Temperature
"The color of a star, as determined by the peak frequency of the visible light, depends on the temperature of the star's outer layers, including its photosphere. " The effective temperature of the surface of the Sun's photosphere is 5,778 K. The "[t]emperature at [the] bottom of [the Sun's] photosphere [is] 6600 K", while the "[t]emperature at [the] top of [the] photosphere [is] 4400 K". The photosphere is "~400 km" in thickness.

The peak emittance wavelength of 501.5 nm (~0.5 eV) makes the photosphere a primarily green radiation source. The figure at the right shows the extraterrestrial solar spectral irradiance as compared with a blackbody spectrum. The sharper than black-body cutoff at the shorter wavelength end indicates an even lower likelihood that X-rays are emitted from the photosphere.

Irradiance
"Direct irradiance measurements have only been available during the last three cycles and are based on a composite of many different observing satellites. However, the correlation between irradiance measurements and other proxies of solar activity make it reasonable to estimate past solar activity. Most important among these proxies is the record of sunspot observations that has been recorded since ~1610. Since sunspots and associated faculae are directly responsible for small changes in the brightness of the sun, they are closely correlated to changes in solar output. Direct measurements of radio emissions from the Sun at 10.7 cm also provide a proxy of solar activity that can be measured from the ground since the Earth's atmosphere is transparent at this wavelength. Lastly, solar flares are a type of solar activity that can impact human life on Earth by affecting electrical systems, especially satellites. Flares usually occur in the presence of sunspots, and hence the two are correlated, but flares themselves make only tiny perturbations of the solar luminosity."

"Solar irradiance and insolation are measures of the amount of sunlight that reaches the Earth. The equipment used might measure optical brightness, total radiation, or radiation in various frequencies."

Evolution
As stars often occur in binaries or multiple star systems, it is likely that the Sun may have been a member of a binary system or even a multiple star system at some time in the past.

"The solar antapex, the direction opposite of the solar apex, is located near the star Zeta Canis Minoris."

In 1917 the solar antapex has an equatorial location of right ascension (RA) 6h declination (Dec) -34°.

"The solar apex (Apex of the Sun's Way) is the direction that the Sun travels with respect to the Local Standard of Rest ... the "target" within the Milky Way that the Sun appears to be "chasing" as it orbits the galaxy."

"The general direction of the solar apex is southwest of the star Vega near the constellation of Hercules. There are several coordinates for the solar apex. The visual coordinates (as obtained by visual observation of the apparent motion) [are] right ascension (RA) 18:28:0 and declination (dec) of 30° North (in galactic coordinates: 56.24° longitude, 22.54° latitude). The radioastronomical position is RA 18:03:50.2 and dec 30°00′16.8″ (galactic coordinates: 58.87° longitude, 17.72° latitude)."

"The Sun moves towards the apex at about 16.5 km/s. This is relative to the Sun's general orbital speed around the Galactic center, about 220 km/s, already included in the Local Standard of Rest. Thus the Sun gains distance towards the apex at about 1/13 its orbital speed. The sun's motion in the Milky Way is more complex than a simple orbit, it also shifts ("bobs") up and down with respect to the galactic plane. "

Theory
In the model shown at right the Sun and regions around it are labeled.

"The core of the Sun is considered to extend from the center to about 0.2 to 0.25 solar radius. It is the hottest part of the Sun and of the Solar System. It has a density of up to 150 g/cm³ (150 times the density of liquid water) and a temperature of close to 15,000,000 kelvin [15 MK] ... The core is made of hot, dense gas in the plasmic state. The core, inside 0.24 solar radius, generates 99% of the fusion power of the Sun." It is in the core region that solar neutrinos may be produced.

"The radiation zone or radiative zone is a layer of a star's interior where energy is primarily transported toward the exterior by means of radiative diffusion, rather than by convection. Energy travels through the radiation zone in the form of electromagnetic radiation as photons. Within the Sun, the radiation zone is located in the intermediate zone between the solar core at .2 of the Sun's radius and the outer convection zone at .71 of the Sun's radius. "

"Matter in a radiation zone is so dense that photons can travel only a short distance before they are absorbed or scattered by another particle, gradually shifting to longer wavelength as they do so. For this reason, it takes an average of 171,000 years for gamma rays from the core of the Sun to leave the radiation zone. Over this range, the temperature of the plasma drops from 15 million K near the core down to 1.5 million K at the base of the convection zone. "

"Within a radiative zone, the temperature gradient—the change in temperature (T) as a function of radius (r)—is given by:

 * $$\frac{\text{d}T(r)}{\text{d}r}\ =\ -\frac{3 \kappa(r) \rho(r) L(r)}{(4 \pi r^2)(16 \sigma) T^3(r)}$$

where κ(r) is the opacity, ρ(r) is the matter density, L(r) is the luminosity, and σ is the Stefan–Boltzmann constant. Hence the opacity (κ) and radiation flux (L) within a given layer of a star are important factors in determining how effective radiative diffusion is at transporting energy. A high opacity or high luminosity can cause a high temperature gradient, which results from a slow flow of energy. Those layers where convection is more effective than radiative diffusion at transporting energy, thereby creating a lower temperature gradient, will become convection zones. "

"The convection zone of a star is the range of radii in which energy is transported primarily by convection. ... Stellar convection consists of mass movement of plasma within the star which usually forms a circular convection current with the heated plasma ascending and the cooled plasma descending." This is the granular zone in the outer layer of a star.

"The solar dynamo is the physical process that generates the Sun's magnetic field. The Sun is permeated by an overall dipole magnetic field, as are many other celestial bodies such as the Earth. The dipole field is produced by a circular electric current flowing deep within the star, following Ampère's law. The current is produced by shear (stretching of material) between different parts of the Sun that rotate at different rates, and the fact that the Sun itself is a very good electrical conductor (and therefore governed by the laws of magnetohydrodynamics)."

Standard solar model
"The Standard Solar Model (SSM) refers to a mathematical treatment of the Sun as a spherical ball of gas (in varying states of ionisation, with the hydrogen in the deep interior being a completely ionised plasma). This model, technically the spherically symmetric quasi-static model of a star, has stellar structure described by several differential equations derived from basic physical principles. The model is constrained by boundary conditions, namely the luminosity, radius, age and composition of the Sun, which are well determined. The age of the Sun cannot be measured directly; one way to estimate it is from the age of the oldest meteorites, and models of the evolution of the solar system. The composition in the photosphere of the modern-day Sun, by mass, is 74.9% hydrogen and 23.8% helium. All heavier elements, called metals in astronomy, account for less than 2 percent of the mass. The SSM is used to test the validity of stellar evolution theory. In fact, the only way to determine the two free parameters of the stellar evolution model, the helium abundance and the mixing length parameter (used to model convection in the Sun), are to adjust the SSM to "fit" the observed Sun."

"The differential equations of stellar structure, such as the equation of hydrostatic equilibrium, are integrated numerically. The differential equations are approximated by difference equations. The star is imagined to be made up of spherically symmetric shells and the numerical integration carried out in finite steps making use of the equations of state, giving relationships for the pressure, the opacity and the energy generation rate in terms of the density, temperature and composition. "

Evolutionary model
"A star is considered to be at zero age (protostellar) when it is assumed to have a homogeneous composition and to be just beginning to derive most of its luminosity from nuclear reactions (so neglecting the period of contraction from a cloud of gas and dust). To obtain the SSM, a one solar mass stellar model at zero age is evolved numerically to the age of the Sun. The abundance of elements in the zero age solar model is estimated from primordial meteorites. Along with this abundance information, a reasonable guess at the zero-age luminosity (such as the present-day Sun's luminosity) is then converted by an iterative procedure into the correct value for the model, and the temperature, pressure and density throughout the model calculated by solving the equations of stellar structure numerically assuming the star to be in a steady state. The model is then evolved numerically up to the age of the Sun. Any discrepancy from the measured values of the Sun's luminosity, surface abundances, etc. can then be used to refine the model. For example, since the Sun formed, the helium and heavy elements have settled out of the photosphere by diffusion. As a result, the Solar photosphere now contains about 87% as much helium and heavy elements as the protostellar photosphere had; the protostellar Solar photosphere was 71.1% hydrogen, 27.4% helium, and 1.5% metals. A measure of heavy-element settling by diffusion is required for a more accurate model."

"Nuclear reactions in the core of the Sun change its composition, by converting hydrogen nuclei into helium nuclei by the proton-proton chain and (to a lesser extent in the Sun than in more massive stars) the CNO cycle. This decreases the mean molecular weight in the core of the Sun, which should lead to a decrease in pressure. This does not happen as instead the core contracts. By the Virial Theorem half of the gravitational potential energy released by this contraction goes towards raising the temperature of the core, and the other half is radiated away. By the ideal gas law this increase in temperature also increases the pressure and restores the balance of hydrostatic equilibrium. The luminosity of the Sun is increased by the temperature rise, increasing the rate of nuclear reactions. The outer layers expand to compensate for the increased temperature and pressure gradients, so the radius also increases. "

"Most of the neutrinos produced in the sun come from the first step of the pp chain but their energy is so low (<0.425 MeV) they are very difficult to detect. A rare side branch of the pp chain produces the "boron-8" neutrinos with a maximum energy of roughly 15 MeV, and these are the easiest neutrinos to detect.  A very rare interaction in the pp chain produces the "hep" neutrinos, the highest energy neutrinos predicted to be produced by our sun.  They are predicted to have a maximum energy of about 18 MeV."

"All of the interactions described above produce neutrinos with a spectrum of energies. The electron capture of 7Be produces neutrinos at either roughly 0.862 MeV (~90%) or 0.384 MeV (~10%). "

title: Solar System, technical/Sun


text:
User talk:Egm6322.s12.team2.steele.m2

Copyright problems on photos
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title: User talk:Egm6322.s12.team2.steele.m2


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User talk:Egm4313.s12.team11.arrieta

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Inversion (discrete mathematics)

Inversion is a concept in discrete mathematics to measure how much a sequence is out of its natural order.

(An inversion of a permutation is not to be confused with the inverse of a permutation. Also not with point reflection.)

Definitions
A permutation $$\pi$$ has an inversion for each pair of elements that are out of their natural order, i.e. when $$i < j$$ but $$\pi(i) > \pi(j)$$. It may be identified by the pair of places $$(i, j)$$ or by the pair of elements $$\bigl(\pi(i), \pi(j)\bigr)$$. This article favours the former convention (although the latter seems to be more common).

The inversion set of a permutation is the set of all its inversions. For an $$n$$-element permutation the number of potential elements is the triangular number $$\scriptstyle { n \choose 2}$$.

The element-based inversion set is essentially the place-based inversion set of the inverse permutation, just with the elements of the pairs exchanged.

For sequences the element-based inversion set has to be defined as a multiset, as many pairs of places can have the same pair of elements.

The permutation graph has the elements of the permutation as vertices and an edge between elements that are out of order, i.e. an edge is an inversion according to the element-based definition: $$\{\pi(i), \pi(j)\}$$ is an edge iff $$i < j \land  \pi(i) > \pi(j)$$. It is more common to write that $$\{i, j\}$$ is an edge iff $$i < j \land  \pi^{-1}(i) > \pi^{-1}(j)$$. The latter is often written as $$(i - j) \cdot \bigl(\pi^{-1}(i) - \pi^{-1}(j)\bigr) < 0$$.

The permutation graphs of inverse permutations are isomorphic. That of $$\pi^{-1}$$ can be generated from that of $$\pi$$ by changing each vertex $$i$$ to $$\pi(i)$$.

It seems that the permutation graph is only defined with place-based inversions. If it were generalized to apply to sequences, it would thus have to be defined as a multigraph.

The inversion number is the cardinality of the inversion set; thus also the digit sum of the inversion related vectors; and also the number of crossings in the arrow diagram of the permutation.

The parity of a permutation is the parity of its inversion number. The sign of a permutation (the determinant of its matrix) corresponds to the parity: Even permutations have sign 1, odd permutations sign −1.

Inversion related vectors
There are four ways to condense the inversions of a permutation into a vector that uniquely determines it. Three of them are in use. (See sources below).

This article uses the terms left inversion count and right inversion count  like Gnedin (similar to Calude and Deutsch), and the term inversion vector  like Wolfram.

The left and right inversion counts have the feature that interpreted as factorial numbers they determine the permutation's reverse colexicographic and lexicographic index. The inversion vector does not appear to have a similar advantage, but it is widely used anyway.

Left inversion count : With the place-based definition $$l(i)$$ is the number of inversions whose bigger (right) component is $$i$$.
 * $$l(i)$$ is the number of elements in $$\pi$$ greater than $$\pi(i)$$ before $$\pi(i)$$.
 * $$l(i) = \# \left\{ k \mid k < i ~\land~ \pi(k) > \pi(i) \right\}$$

Right inversion count, often called : With the place-based definition $$r(i)$$ is the number of inversions whose smaller (left) component is $$i$$.
 * $$r(i)$$ is the number of elements in $$\pi$$ smaller than $$\pi(i)$$ after $$\pi(i)$$.
 * $$r(i) = \# \{ k \mid k > i ~\land~ \pi(k) < \pi(i) \}$$

Inversion vector : With the element-based definition $$v(i)$$ is the number of inversions whose smaller (right) component is $$i$$.
 * $$v(i)$$ is the number of elements in $$\pi$$ greater than $$i$$ before $$i$$.
 * $$v(i) = \# \{ k \mid k > i ~\land~ \pi^{-1}(k) < \pi^{-1}(i) \}$$

A different way to say the same thing may be more intuitive:
 * $$v\bigl(\pi(i)\bigr)$$ is the number of elements in $$\pi$$ greater than $$\pi(i)$$ before $$\pi(i)$$.
 * $$v\bigl(\pi(i)\bigr) = \# \{ k \mid k < i ~\land~ \pi(k) > \pi(i) \} ~= l(i)$$

The latter definition would also work for a sequence, which does not have an inverse.

Relationship between and :

The first digit of and the last digit of  are always 0, and can be omitted. When these vectors are permuted into each other, the omissible 0 from one does not necessarily become the omissible 0 in the other.

Relationship between and : Both and  can be found with the help of a Rothe diagram, which is a permutation matrix with the 1s represented by dots, and an inversion in every position that has a dot to the right and below it. $$r(i)$$ is the sum of inversions in row $$i$$ of the Rothe diagram, while $$v(i)$$ is the sum of inversions in column $$i$$. The permutation matrix of the inverse is the transpose. Therefore of a permutation is  of its inverse, and vice versa.

Relationship between and :
 * $$\pi(i) = i + r(i) - l(i)$$

This is proven in Gnedin & Olshanski 2012, and mentioned as  in.

Permutations that are equal up to fixed points can be seen as equal. Equivalently, inversion related vectors that are equal up to trailing 0s can be seen as equal. While for and  the omissible 0 on the right is part of the trailing 0s, for  the omissible 0 on the left is separate from them.

Example: Permutations of six elements
The following images show two inverse permutations, their 9 inversions and the corresponding vectors.

Mathematica omits the 0 at the end of the inversion vector. Compare p1 and p2 in Wolfram Alpha. I: p1 = {2,5,4,6,3,1} I: p2 = {6,1,5,3,2,4} I: ToInversionVector[p1] O: {5,0,3,1,0} I: ToInversionVector[p2] O: {1,3,2,2,1} I: ToOrderedPairs[PermutationGraph[p1]] O: {{2,1},{3,1},{4,1},{5,1},{6,1},{4,3},{5,3},{6,3},{5,4}, {1,2},{1,3},{1,4},{1,5},{1,6},{3,4},{3,5},{3,6},{4,5} } I: ToOrderedPairs[PermutationGraph[p2]] O: {{6,1},{3,2},{5,2},{6,2},{5,3},{6,3},{5,4},{6,4},{6,5}, {1,6},{2,3},{2,5},{2,6},{3,5},{3,6},{4,5},{4,6},{5,6} }

Example: All permutations of four elements

The following sortable table shows the 24 permutations of four elements with their place-based inversion sets, inversion related vectors and inversion numbers.

It is shown twice, so different orders can be compared to each other. (The Python script used to create it is shown here.)

The columns with small text are the reflections of the main columns. Sorting by them gives the colexicographic order of the corresponding main column.

Inversion vector and left inversion count  are shown next to each other, because they have the same digits.

Left and right inversion counts are related to the place-based inversion sets shown between them: The nontrivial elements of are the sums of the descending diagonals of the triangle, and those of  are the sums of the of the ascending diagonals.

The default order of the table is the reverse colex order by $$\pi$$, which is the same as the colex order by. (This header cell is highlighted by a darker gray.)

Lex order by $$\pi$$ is the same as lex order by.

Ordering one table by and the other one by  brings inverse permutations next to each other, i.e. those whose matrices are reflected at the main diagonal.

Ordering one table by reflected and the other one by  brings permutations next to each other whose permutation matrices are reflected at the antidiagonal.

Ordering one table by reflected and the other one by  brings permutations next to each other whose permutation matrices are rotated by 180°. (It can be seen, that these permutations' inversion sets are symmetric to each other, which corresponds to and  being symmetric to each other.)

Note that the set of all, the set of all and the set of all reflected  for the same symmetric group are equal. So sorting by two columns that have the omissible 0 at the same end makes these columns equal.

(A more detailled version of this table, including element-based inversion sets and the unused fourth vector, can be found here.)

Weak order of permutations
The Hasse diagram below in the middle shows the 24 inversion sets ordered by the subset relation.

If a permutation is assigned to each inversion set using the place-based definition, the resulting order of permutations is that of the permutohedron, where $$(\pi, \sigma)$$ is an edge iff $$i < j \land  \pi(i)+1 = \pi(j)  \land  (ij) \cdot \pi = \sigma$$, i.e. when only two elements with consecutive values are swapped. (In the central diagram it can be seen on which positions the swapped elements are, by looking which digit changes for the edge.)

The bitwise $$\le$$ ordering of the left and right inversion counts of these permutations gives the same order (because both are diagonal sums of the inversion set triangles).

If a permutation were assigned to each inversion set using the element-based definition, the resulting order of permutations would be that of a Cayley graph, where $$(\pi, \sigma)$$ is an edge iff $$i+1 = j \land  \pi(i) < \pi(j)  \land  (ij) \cdot \pi = \sigma$$, i.e. when only two elements on consecutive places are swapped.

This Cayley graph of the symmetric group is similar to its permutohedron, but with each permutation replaced by its inverse.

Inversions in sequences
For sequences inversion sets and the related vectors are not unique. Their place-based inversion sets are multisets, and their inversion vectors can have bigger digits than would be allowed in a factorial number.

Arrays of permutations
Similar permutations have similar inversion sets and corresponding vectors.

Some can be ordered in arrays, like the one on the right, corresponding to the number triangle.

Cramer 1750
Cramer defines the inversion under the name dérangement to calculate the sign of a permutation.

For the permutation 3 1 2 the dérangements are given as 3 before 1 and 3 before 2.

Cramer, Gabriel: Introduction à l'analyse des lignes courbes algébriques. Genève : chez les Frères Cramer & Cl. Philibert 1750 — Appendice (p. 658)

— 9.1 Cramers rule (p. 286)

Rothe 1800
Rothe essentially defines the right inversion count under the name Stellenexponenten — but with each place bigger by 1.

 * $$s_i$$ is the position of $$\pi_i$$ among the $$\pi_k$$ with $$k \ge i$$ in numerical order. (p. 163)

Strangely he counts from 1 rather than 0, which later leads to avoidable additions and subtractions by 1:
 * $$s_i + 1 = \# \{ \pi_k \mid k>i \land \pi_k<\pi_i \}$$ (p. 165)
 * Each permutation has sign $$+$$ when the sum of the $$s_i - 1$$ is even, and sign $$-$$ when it is odd. (p. 266)

For the permutation $$6~4~3~9~8~10~1~7~2~5$$ the Stellenexponenten are given as $$6~4~3~6~5~5~1~3~1~1$$.

Muir translates Stellenexponent as place-index and calls it “an ill-advised and purposeless modification of Cramers idea of a ‘derangement’”. (The Theory of the Determinant..., p. 55)

"Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. Anwendung der daraus abgeleiteten Satze auf das Eliminationsproblem". In Hindenburg, Carl, ed., Sammlung Combinatorisch-Analytischer Abhandlungen, pp. 263–305, Bey G. Fleischer dem jüngern, 1800.

Laisant 1888
After defining the factorial number system Laisant defines the right inversion count under the name signe figuratif.

For the permutation $$4~3~6~5~1~2$$ the signe figuratif is given as $$(3~2~3~2~0)$$.

Aigner 2007
Now we look at permutations of $$\{1, 2, ..., n\}$$ in word form $$\sigma = a_1, a_2 ... a_n, a_i = \sigma(i)$$. The pair $$\{i, j\}$$ is called an inversion if $$i < j$$ but $$a_i > a_j$$.

A moment's thought shows that $$\{i, j\}$$ is an inversion if and only if the edges $$ia_i$$ and $$ja_j$$ cross in the diagram. Hence the inversion number $$inv(\sigma)$$ equals the number of crossings.

— Word Representation (p. 27 ff)

Comtet 1974
An inversion of a permutation $$\sigma \in \mathfrak{S}[n]$$ is a pair $$(i, j)$$ such that $$1 \le i < j \le n$$ and $$\sigma(i) > \sigma(j)$$. In this case we say that $$\sigma$$ has an inversion in $$(i, j)$$.

He uses the same definition for a non-bijective map in exercise 21 on permutations (p. 266).

— 6.4 Inversions of a permutation of [n] (p. 237)

Cormen et al. 2001
Let $$A[1..n]$$ be an array of n distinct numbers. If $$i < j$$ and $$A[i] > A[j]$$, then the pair $$(i, j)$$ is called an inversion of $$A$$.

— 2-4 Inversions (p. 41) and 5.2-5 (p. 122)

Knuth 1973
If $$ia_j$$, the pair $$(a_i, a_j)$$ is called an inversion of the permutation; for example the permutation 3 1 4 2 has three inversions: (3, 1), (3, 2) and (4, 2).

The inversion table $$b_1 b_2 ... b_n$$ of the permutation $$a_1 a_2 ... a_n$$ is obtained by letting $$b_j$$ be the number of elements to the left of $$j$$ that are greater than $$j$$. In other words, $$b_j$$ is the number of inversions whose second component is $$j$$.

— 5.1.1 Inversions (p. 11)

Pemmaraju & Skiena 2003
A pair of elements $$(\pi(i), \pi(j))$$ in a permutation $$\pi$$ represents an inversion if $$i > j$$ and $$\pi(i) < \pi(j)$$. An inversion is a pair of elements that are out of order

For any $$n$$-permutation $$\pi$$, we can define an inversion vector $$v$$ as follows. For each integer $$i$$, $$1 \le i \le n-1$$, the $$i$$th element of $$v$$ is the number of elements in $$\pi$$ greater than $$i$$ to the left of $$i$$.

— 2.2 Inversions and Inversion Vectors (p. 69)

Vitter & Flajolet 1990
An inversion in permutation $$\sigma$$ is an “out of order” pair $$(\sigma_k, \sigma_j)$$ of elements, in which $$k \sigma_j$$. The number of inversions is thus a measure of the amount of disorder in a permutation.

The inversion table of the permutation $$\sigma = \sigma_1 \sigma_2 \dots \sigma_n$$ is the ordered sequence
 * $$b_1, b_2, \dots, b_n$$, where $$b_k = \Big| \{ 1 \le j <\sigma_k^-1 | \sigma_j > k \} \Big|$$.

— 3.1 Inversions (p. 459)

Grätzer 2016
The authors define the usual strict linear ordering $$1 < 2 < \cdots < i < i+1 < \cdots < n$$ on $$[n]$$ as $$\mathcal{I}_n = \{ (i, j) \in [n] \times [n]  |  i < j  \}$$.

Let $$\sigma$$ be a permutation on $$[n]$$. An inversion of $$\sigma$$ is an ordered pair $$(i, j) \in \mathcal{I}_n$$ satisfying $$\sigma^{-1}(i) > \sigma^{-1}(j)$$. The inversion set of $$\sigma$$ is then defined as $$\mathrm{inv}(\sigma) = \{ (i, j) \in \mathcal{I}_n  |  \sigma^{-1}(i) > \sigma^{-1}(j)  \}$$.

The inversion set of $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 4 & 1 & 2 \end{pmatrix}$$ is given as $$\{(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (4, 5)\}$$.

In the footnote it is mentioned that other authors often use the definition as an ordered pair $$(i, j) \in \mathcal{I}_n$$ such that $$\sigma(i) > \sigma(j)$$. TAoCP is given as an example — despite the fact that Knuth uses essentially the same definition (see above).

— 7-2 Basic objects (p. 221)

Joshi 1989
Let $$x = x_1 x_2 \dots x_n$$ be a sequence of real numbers. By an inversion in $$x$$, we mean a pair $$(x_i, x_j)$$ such that $$ix_j$$. For each $$i$$, let $$d_i$$ = the number of inversions whose first entry is $$x_i$$, i.e.
 * $$d_i = \big| \{ x_j : i < j, x_i > x_j \} \big|$$.

Then the sequence $$(d_1 d_2 \dots d_n)$$ is called the inversion table or inversion vector of $$x$$. The permutation $$2~4~3~5~1$$ has $$(1, 2, 1, 1, 0)$$ as its inversion table.

— Definition 3.12 (p. 188)

Bóna 2012
Bóna first uses the definition with elements: Let $$p = p_1 p_2 \cdots p_n$$ be a permutation. We say that $$(p_i, p_j)$$ is an inversion of $$p$$ if $$i < j$$ but $$p_i > p_j$$. Permutation 31524 has four inversions, namely (3, 1), (3, 2), (5, 2), and (5,4). 2.1 Inversions (p. 43)

But for multisets he uses places instead: An inversion of a permutation $$p = p_1 p_2 \cdots p_n$$ of a multiset is defined just as it was for permutations of sets, that is, $$(i, j)$$ is a inversion if $$i < j$$, but $$p_i > p_j$$· The multiset-permutation 1322 has two inversions, (2, 3), and (2, 4). 2.2 Inversions in Permutations of Multisets (p. 57)

The definition of non-inversions also uses places: look at all non-inversions of a generic permutation $$p = p_1 p_2 \cdots p_n$$; that is, pairs so that $$i < j$$ and $$p_i < p_j$$. 4.4.1.1 An exponential upper bound (p. 141)

Calude et al. 2003
For permutations, an inversion vector is used as an auxiliary array because it fills in the holes made by the elements of the prefix in its defining sequence. For the permutation $$(p[i], \dots, p[n])$$ the right-inversion vector $$(g[i], \dots, g[n-1])$$ is defined by the rule that $$g[i]$$ is the number of $$p[j]$$ to the right of but smaller than $$p[i]$$; in the left-inversion vector, $$g[i]$$ is the number of $$p[j]$$ to the left of but larger than $$p[i]$$.

— Generating Gray codes... (p. 81)

Lothaire 2002
If $$\sigma = \sigma(1) \sigma(2) \dots \sigma(k)$$ is [ a permutation of $$[k]$$ ] an inversion of $$\sigma$$ is a pair $$(i, j)$$ such that $$1 \le i < j \le k$$ and $$\sigma(i) > \sigma(j)$$. 11.1. Prelimiaries (p. 367)

Let $$\sigma$$ be a permutation of $$[n]$$. For $$1 \le i \le n$$, let $$l_i$$ be the number of indices $$j < i$$ such that $$\sigma(j) > \sigma(i)$$. The word $$l = l_1 l_2 \cdots l_n$$ will be called the Lehmer encoding of $$\sigma$$.

To the permutation $$\sigma = 6~8~4~5~9~3~1~2~7$$ corresponds the Lehmer encoding $$l = 0~0~2~2~0~5~6~6~2$$. 11.4. Inversions of permutations with a given shape (p. 372)

Vajnovszki 2011
The pair $$(i, j)$$ is an inversion of $$\pi \in \mathfrak{S}_n$$ if $$i < j$$ but $$\pi_i > \pi_j$$.

An integer sequence $$t_1 t_2 \dots t_n$$ is said to be subexcedent if $$0 \le t_i \le i - 1$$ for $$1 \le i \le n$$, and the set of lenth-$$n$$ subexcedent sequences is denoted by $$S_n$$. The Lehmer code [5] is a bijection code : $$\mathfrak{S}_n \rightarrow S_n$$ which maps each permutation $$\pi = \pi_1 \pi_2 \dots \pi_n$$ to a subexcedent sequence $$t_1 t_2 \dots t_n$$ where, for all $$i, ~ 1 \le i \le n$$, $$~ t_i$$ is the number of inversions $$(j, i)$$ in $$\pi$$ (or equivalently, the number of entries in $$\pi$$ larger than $$\pi_i$$ and on its left). [5]     D. H. Lehmer, Teaching combinatorial tricks to a computer, in Proc. Sympos. Appl. Math., 10 (1960), Amer. Math. Soc., 179-193. Vajnovszki. A new Euler–Mahonian constructive bijection — pp. 1-2

Gnedin & Olshanski 2012
For $$\sigma \in \mathfrak{S}$$, a pair of positions $$(i, j) \in \mathbb{Z} \times \mathbb{Z}$$ is an inversion in $$\sigma$$ if $$i < j$$ and $$\sigma(i) > \sigma(j)$$. If $$(i, j)$$ is an inversion, we say that it is a left inversion for $$j$$, and a right inversion for $$i$$. Introduce the counts of left and right inversions,
 * $$\mathcal{l}_i := \# \{j: j < i, \sigma(j) > \sigma(i) \}, ~ \mathcal{r}_i := \# \{j: j > i, \sigma(j) < \sigma(i) \}, ~ i \in \mathbb{Z}$$,

respectively (of course, for the general $$\sigma \in \mathfrak{S}$$ these quantities may be infinite). 4. A construction from independent geometric variables (p. 624)

The authors proof the following formula for balanced permutations, which include finite permutations:
 * $$\sigma(i) = i + \mathcal{r}_i - \mathcal{l}_i, ~ i \in \mathbb{Z}$$

Lemma 4.6 (p. 627)

Gnedin; Olshanski. The two-sided infinite extension of the Mallows model for random permutations

Deutsch et al. 2008
In a permutation $$\pi = \pi_1 \pi_2 \dots \pi_n$$, an inversion is a pair $$i < j$$ such that $$\pi_i>\pi_j$$. If $$c_i$$ is the number of $$j > i$$ with $$\pi_j < \pi_i$$, then $$(c_1, c_2, \dots, c_n)$$ is called the right inversion vector of $$\pi$$.

Deutsch; Pergola; Pinzani. Six bijections between deco polyominoes and permutations — p. 5

Barth & Mutzel 2004
In a sequence $$\pi = \langle a_0, a_1, \dots, a_{t-1} \rangle$$ of pairwise comparable elements $$a_i ~ (i = 0, 1, \dots , t-1)$$, a pair $$(a_i, a_j)$$ is called an inversion if $$i < j$$ and $$a_i > a_j$$.

A crucial point of the article is illustrated by the images on the right: A bilayer graph that does not describe a map corresponds to a map that has the edges (green) as its domain and one layer of the vertices (blue) as its codomain. The number of crossings in the bilayer graph can be calculated as the number of inversions of the map. In this example the map is $$\pi = \langle 0,1,2,0,3,4,0,2,3,2,4 \rangle$$.

$$\pi_2 = 2$$, $$\pi_3 = 0$$ and $$\pi_6 = 0$$, so with the place-based definition the sequence would have the inversions $$(2, 3)$$ and $$(2, 6)$$. With the element-based definition, chosen by the authors, one could say that the sequence has the inversion $$(2, 0)$$ twice.

— 2 Bilayer Cross Counts and Inversion Numbers (p. 183)

title: Inversion (discrete mathematics)


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 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You do not need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Collingwood (talk) 20:59, 7 July 2012 (UTC) --Collingwood (talk) 20:59, 7 July 2012 (UTC)

title: User talk:Nklrdthy


text:
User talk:Yikrazuul

 Hello and Welcome to Wikiversity Yikrazuul! You can contact us with questions at the colloquium or me personally when you need help. Please remember to sign and date your finished comments when participating in discussions. The signature icon above the edit window makes it simple. All users are expected to abide by our Privacy, Civility, and the Terms of Use policies while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You do not need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Collingwood (talk) 21:00, 7 July 2012 (UTC) --Collingwood (talk) 21:00, 7 July 2012 (UTC)

title: User talk:Yikrazuul


text:
User talk:Egm4313.s12.team1.essenwein

Copyright problems on photos
Hello. There is a problem with one or more files you have uploaded. Every file you upload must have a clear statement of the copyright position. If you took the photos yourself, add PD-self if you are placing it unreservedly in the public domain. If you wish to retain minimal rights, add Cc-by-sa-3.0. If you did not take the photo, please add an explanation of why it can be hosted here. I am afraid that if you do not do this, the file wil be deleted. Let me know if you have any problems.--Collingwood (talk) 21:10, 7 July 2012 (UTC)

title: User talk:Egm4313.s12.team1.essenwein


text:
User talk:Ottava Rima/archives/2012/jul

Account unblocked
This user has been the subject of multiple discussions pertaining to disputes and was previously blocked five times, none of which lasted for an extended length of time. The most recent block was imposed by SB Johnny on 27 November 2011, which is 7 months and 7 days from the present time. I will not repeat the reasoning behind the block, since it is clearly available in the record. However, I will briefly explain my decision to repeal this block.

At this time, the English Wikiversity is a dying project. Unlike the Wikipedia, it never achieved critical mass. Ottava has requested that he be unblocked to work on aspects of the project and for personal reasons. The likelihood of disruption from this user is minimal as there is no community to disrupt. Therefore, based on the lengthy time that Ottava has been barred from the project, I have now unblocked his account to allow him to contribute.

Geoff (talk) 08:20, 6 July 2012 (UTC)

title: User talk:Ottava Rima/archives/2012/jul