text:
# Methane

Methane
Names
Preferred IUPAC name
Methane[1]
Systematic IUPAC name
Carbane (never recommended[1])
Other names
• Marsh gas
• Natural gas
• Carbon tetrahydride
• Carburetted hydrogen
• Hydrogen carbide
Identifiers
3D model (JSmol)
3DMet
1718732
ChEBI
ChEMBL
ChemSpider
ECHA InfoCard 100.000.739
EC Number
• 200-812-7
59
KEGG
MeSH Methane
RTECS number
• PA1490000
UNII
UN number 1971
• InChI=1S/CH4/h1H4
Key: VNWKTOKETHGBQD-UHFFFAOYSA-N
Properties
CH4
Molar mass 16.043 g·mol−1
Appearance Colorless gas
Odor Odorless
Density
• 0.657 kg·m−3 (gas, 25 °C, 1 atm)
• 0.717 kg·m−3 (gas, 0 °C, 1 atm)[2]
• 422.8 g·L−1 (liquid, −162 °C)[3]
Melting point −182.456 °C (−296.421 °F; 90.694 K)[3]
Boiling point −161.5 °C (−258.7 °F; 111.6 K)[3]
Critical point (T, P) 190.56 K (−82.59 °C; −116.66 °F), 4.5992 megapascals (45.391 atm)
22.7 mg·L−1[4]
Solubility Soluble in ethanol, diethyl ether, benzene, toluene, methanol, acetone and insoluble in water
log P 1.09
14 nmol·Pa−1·kg−1
Conjugate acid Methanium
Conjugate base Methyl anion
−17.4×10−6 cm3·mol−1[5]
Structure
Td
Tetrahedron
0 D
Thermochemistry[6]
35.7 J·(K·mol)−1
186.3 J·(K·mol)−1
−74.6 kJ·mol−1
−50.5 kJ·mol−1
−891 kJ·mol−1
Hazards[7]
GHS labelling:
Danger
H220
P210
NFPA 704 (fire diamond)
Flash point −188 °C (−306.4 °F; 85.1 K)
537 °C (999 °F; 810 K)
Explosive limits 4.4–17%
Related compounds
Related alkanes
Supplementary data page
Methane (data page)
Except where otherwise noted, data are given for materials in their standard state (at 25 °C [77 °F], 100 kPa).

Methane ( MEH-thayn, MEE-thayn) is a chemical compound with the chemical formula CH4 (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Earth makes it an economically attractive fuel, although capturing and storing it poses technical challenges due to its gaseous state under normal conditions for temperature and pressure.

Naturally occurring methane is found both below ground and under the seafloor and is formed by both geological and biological processes. The largest reservoir of methane is under the seafloor in the form of methane clathrates. When methane reaches the surface and the atmosphere, it is known as atmospheric methane.[9] The Earth's atmospheric methane concentration has increased by about 150% since 1750, and it accounts for 20% of the total radiative forcing from all of the long-lived and globally mixed greenhouse gases.[10] It has also been detected on other planets, including Mars, which has implications for astrobiology research.[11]

## Properties and bonding

Methane is a tetrahedral molecule with four equivalent C–H bonds. Its electronic structure is described by four bonding molecular orbitals (MOs) resulting from the overlap of the valence orbitals on C and H. The lowest-energy MO is the result of the overlap of the 2s orbital on carbon with the in-phase combination of the 1s orbitals on the four hydrogen atoms. Above this energy level is a triply degenerate set of MOs that involve overlap of the 2p orbitals on carbon with various linear combinations of the 1s orbitals on hydrogen. The resulting "three-over-one" bonding scheme is consistent with photoelectron spectroscopic measurements.

Methane is an odorless gas and appears to be colorless.[12] It does absorb visible light especially at the red end of the spectrum due to overtone bands, but the effect is only noticeable if the light path is very long. This is what gives Uranus and Neptune their blue or bluish-green colors, as light passes through their atmospheres containing methane and is then scattered back out.[13]

The familiar smell of natural gas as used in homes is achieved by the addition of an odorant, usually blends containing tert-butylthiol, as a safety measure. Methane has a boiling point of −161.5 °C at a pressure of one atmosphere.[3] As a gas, it is flammable over a range of concentrations (5.4–17%) in air at standard pressure.

Solid methane exists in several modifications. Presently nine are known.[14] Cooling methane at normal pressure results in the formation of methane I. This substance crystallizes in the cubic system (space group Fm3m). The positions of the hydrogen atoms are not fixed in methane I, i.e. methane molecules may rotate freely. Therefore, it is a plastic crystal.[15]

## Chemical reactions

The primary chemical reactions of methane are combustion, steam reforming to syngas, and halogenation. In general, methane reactions are difficult to control.

### Selective oxidation

Partial oxidation of methane to methanol, a more convenient, liquid fuel, is challenging because the reaction typically progresses all the way to carbon dioxide and water even with an insufficient supply of oxygen. The enzyme methane monooxygenase produces methanol from methane, but cannot be used for industrial-scale reactions.[16] Some homogeneously catalyzed systems and heterogeneous systems have been developed, but all have significant drawbacks. These generally operate by generating protected products which are shielded from overoxidation. Examples include the Catalytica system, copper zeolites, and iron zeolites stabilizing the alpha-oxygen active site.[17]

One group of bacteria catalyze methane oxidation with nitrite as the oxidant in the absence of oxygen, giving rise to the so-called anaerobic oxidation of methane.[18]

### Acid–base reactions

Like other hydrocarbons, methane is an extremely weak acid. Its pKa in DMSO is estimated to be 56.[19] It cannot be deprotonated in solution, but the conjugate base is known in forms such as methyllithium.

A variety of positive ions derived from methane have been observed, mostly as unstable species in low-pressure gas mixtures. These include methenium or methyl cation CH+
3
, methane cation CH+
4
, and methanium or protonated methane CH+
5
. Some of these have been detected in outer space. Methanium can also be produced as diluted solutions from methane with superacids. Cations with higher charge, such as CH2+
6
and CH3+
7
, have been studied theoretically and conjectured to be stable.[20]

Despite the strength of its C–H bonds, there is intense interest in catalysts that facilitate C–H bond activation in methane (and other lower numbered alkanes).[21]

### Combustion

Methane bubbles can be burned on a wet hand without injury.

Methane's heat of combustion is 55.5 MJ/kg.[22] Combustion of methane is a multiple step reaction summarized as follows:

${\displaystyle {\ce {CH4 + 2 O2 -> CO2 + 2 H2O}}}$
(ΔH = −891 k J/mol, at standard conditions)

Peters four-step chemistry is a systematically reduced four-step chemistry that explains the burning of methane.

Given appropriate conditions, methane reacts with halogen radicals as follows:

{\displaystyle {\begin{aligned}&{\ce {X^{\bullet }{}+CH4->HX{}+CH3^{\bullet }}}\\&{\ce {CH3^{\bullet }{}+X2->CH3X{}+X^{\bullet }}}\end{aligned}}}

where X is a halogen: fluorine (F), chlorine (Cl), bromine (Br), or iodine (I). This mechanism for this process is called free radical halogenation. It is initiated when UV light or some other radical initiator (like peroxides) produces a halogen atom. A two-step chain reaction ensues in which the halogen atom abstracts a hydrogen atom from a methane molecule, resulting in the formation of a hydrogen halide molecule and a methyl radical (CH3). The methyl radical then reacts with a molecule of the halogen to form a molecule of the halomethane, with a new halogen atom as byproduct.[23] Similar reactions can occur on the halogenated product, leading to replacement of additional hydrogen atoms by halogen atoms with dihalomethane, trihalomethane, and ultimately, tetrahalomethane structures, depending upon reaction conditions and the halogen-to-methane ratio.

## Uses

Methane is used in industrial chemical processes and may be transported as a refrigerated liquid (liquefied natural gas, or LNG). While leaks from a refrigerated liquid container are initially heavier than air due to the increased density of the cold gas, the gas at ambient temperature is lighter than air. Gas pipelines distribute large amounts of natural gas, of which methane is the principal component.

### Fuel

Methane is used as a fuel for ovens, homes, water heaters, kilns, automobiles,[24][25] turbines, etc. Activated carbon is used to store methane. Refined liquid methane is used as a rocket fuel,[26] when combined with liquid oxygen, as in the BE-4 and Raptor engines.[27]

As the major constituent of natural gas, methane is important for electricity generation by burning it as a fuel in a gas turbine or steam generator. Compared to other hydrocarbon fuels, methane produces less carbon dioxide for each unit of heat released. At about 891 kJ/mol, methane's heat of combustion is lower than that of any other hydrocarbon, but the ratio of the heat of combustion (891 kJ/mol) to the molecular mass (16.0 g/mol, of which 12.0 g/mol is carbon) shows that methane, being the simplest hydrocarbon, produces more heat per mass unit (55.7 kJ/g) than other complex hydrocarbons. In many cities, methane is piped into homes for domestic heating and cooking. In this context it is usually known as natural gas, which is considered to have an energy content of 39 megajoules per cubic meter, or 1,000 BTU per standard cubic foot. Liquefied natural gas (LNG) is predominantly methane (CH4) converted into liquid form for ease of storage or transport.

As a liquid rocket fuel, methane offers the advantage over kerosene of producing small exhaust molecules. This deposits less soot on the internal parts of rocket motors, reducing the difficulty of booster re-use. The lower molecular weight of the exhaust also increases the fraction of the heat energy which is in the form of kinetic energy available for propulsion, increasing the specific impulse of the rocket. Liquid methane also has a temperature range (91–112 K) nearly compatible with liquid oxygen (54–90 K).

### Chemical feedstock

Natural gas, which is mostly composed of methane, is used to produce hydrogen gas on an industrial scale. Steam methane reforming (SMR), or simply known as steam reforming, is the standard industrial method of producing commercial bulk hydrogen gas. More than 50 million metric tons are produced annually worldwide (2013), principally from the SMR of natural gas.[28] Much of this hydrogen is used in petroleum refineries, in the production of chemicals and in food processing. Very large quantities of hydrogen are used in the industrial synthesis of ammonia.

At high temperatures (700–1100 °C) and in the presence of a metal-based catalyst (nickel), steam reacts with methane to yield a mixture of CO and H2, known as "water gas" or "syngas":

${\displaystyle {\ce {CH4 + H2O <=> CO + 3 H2}}}$

This reaction is strongly endothermic (consumes heat, ΔHr = 206 kJ/mol). Additional hydrogen is obtained by the reaction of CO with water via the water-gas shift reaction:

${\displaystyle {\ce {CO + H2O <=> CO2 + H2}}}$

This reaction is mildly exothermic (produces heat, ΔHr = −41 kJ/mol).

Methane is also subjected to free-radical chlorination in the production of chloromethanes, although methanol is a more typical precursor.[29]

Hydrogen can also be produced via the direct decomposition of methane, also known as methane Pyrolysis. Methane decomposition is a promising route for low-emission hydrogen production since no direct carbon emissions are produced unlike steam methane reforming. Temperatures in excess of 1200 °C are required to break the bonds of methane to produce Hydrogen gas and solid carbon. However, through the use of a suitable catalyst the reaction temperature can be reduced to between 600 °C - 1000 °C depending on the chosen catalyst.[30] The reaction is moderately endothermic as shown in the reaction equation below.[31]

${\displaystyle {\ce {CH4_{(g)}-> C_{(s)}{}+ 2 H2_{(g)}}}}$
(ΔH° = 74.8 kJ/mol)

## Generation

### Geological routes

The two main routes for geological methane generation are (i) organic (thermally generated, or thermogenic) and (ii) inorganic (abiotic).[11] Thermogenic methane occurs due to the breakup of organic matter at elevated temperatures and pressures in deep sedimentary strata. Most methane in sedimentary basins is thermogenic; therefore, thermogenic methane is the most important source of natural gas. Thermogenic methane components are typically considered to be relic (from an earlier time). Generally, formation of thermogenic methane (at depth) can occur through organic matter breakup, or organic synthesis. Both ways can involve microorganisms (methanogenesis), but may also occur inorganically. The processes involved can also consume methane, with and without microorganisms.

The more important source of methane at depth (crystalline bedrock) is abiotic. Abiotic means that methane is created from inorganic compounds, without biological activity, either through magmatic processes or via water-rock reactions that occur at low temperatures and pressures, like serpentinization.[32][33]

### Biological routes

Most of Earth's methane is biogenic and is produced by methanogenesis,[34][35] a form of anaerobic respiration only known to be conducted by some members of the domain Archaea.[36] Methanogens occupy landfills and other soils,[37] ruminants (for example, cattle),[38] the guts of termites, and the anoxic sediments below the seafloor and the bottom of lakes. Rice fields also generate large amounts of methane during plant growth.[39] This multistep process is used by these microorganisms for energy. The net reaction of methanogenesis is:

${\displaystyle {\ce {CO2 + 4 H2 -> CH4 + 2 H2O}}}$

The final step in the process is catalyzed by the enzyme methyl coenzyme M reductase (MCR).[40]

Testing Australian sheep for exhaled methane production (2001), CSIRO
This image represents a ruminant, specifically a sheep, producing methane in the four stages of hydrolysis, acidogenesis, acetogenesis, and methanogenesis.

#### Ruminants

Ruminants, such as cattle, belch methane, accounting for about 22% of the U.S. annual methane emissions to the atmosphere.[41] One study reported that the livestock sector in general (primarily cattle, chickens, and pigs) produces 37% of all human-induced methane.[42] A 2013 study estimated that livestock accounted for 44% of human-induced methane and about 15% of human-induced greenhouse gas emissions.[43] Many efforts are underway to reduce livestock methane production, such as medical treatments and dietary adjustments,[44][45] and to trap the gas to use its combustion energy.[46]

#### Seafloor sediments

Most of the subseafloor is anoxic because oxygen is removed by aerobic microorganisms within the first few centimeters of the sediment. Below the oxygen-replete seafloor, methanogens produce methane that is either used by other organisms or becomes trapped in gas hydrates.[36] These other organisms that utilize methane for energy are known as methanotrophs ('methane-eating'), and are the main reason why little methane generated at depth reaches the sea surface.[36] Consortia of Archaea and Bacteria have been found to oxidize methane via anaerobic oxidation of methane (AOM); the organisms responsible for this are anaerobic methanotrophic Archaea (ANME) and sulfate-reducing bacteria (SRB).[47]

### Industrial routes

This diagram shows a method for producing methane sustainably. See: electrolysis, Sabatier reaction

Given its cheap abundance in natural gas, there is little incentive to produce methane industrially. Methane can be produced by hydrogenating carbon dioxide through the Sabatier process. Methane is also a side product of the hydrogenation of carbon monoxide in the Fischer–Tropsch process, which is practiced on a large scale to produce longer-chain molecules than methane.

An example of large-scale coal-to-methane gasification is the Great Plains Synfuels plant, started in 1984 in Beulah, North Dakota as a way to develop abundant local resources of low-grade lignite, a resource that is otherwise difficult to transport for its weight, ash content, low calorific value and propensity to spontaneous combustion during storage and transport. A number of similar plants exist around the world, although mostly these plants are targeted towards the production of long chain alkanes for use as gasoline, diesel, or feedstock to other processes.

Power to methane is a technology that uses electrical power to produce hydrogen from water by electrolysis and uses the Sabatier reaction to combine hydrogen with carbon dioxide to produce methane. As of 2021, this is mostly under development and not in large-scale use. Theoretically, the process could be used as a buffer for excess and off-peak power generated by highly fluctuating wind turbines and solar arrays. However, as currently very large amounts of natural gas are used in power plants (e.g. CCGT) to produce electric energy, the losses in efficiency are not acceptable.

#### Laboratory synthesis

Methane can be produced by protonation of methyl lithium or a methyl Grignard reagent such as methylmagnesium chloride. It can also be made from anhydrous sodium acetate and dry sodium hydroxide, mixed and heated above 300 °C (with sodium carbonate as byproduct).[citation needed] In practice, a requirement for pure methane can easily be fulfilled by steel gas bottle from standard gas suppliers.

## Occurrence

Methane was discovered and isolated by Alessandro Volta between 1776 and 1778 when studying marsh gas from Lake Maggiore. It is the major component of natural gas, about 87% by volume. The major source of methane is extraction from geological deposits known as natural gas fields, with coal seam gas extraction becoming a major source (see coal bed methane extraction, a method for extracting methane from a coal deposit, while enhanced coal bed methane recovery is a method of recovering methane from non-mineable coal seams). It is associated with other hydrocarbon fuels, and sometimes accompanied by helium and nitrogen. Methane is produced at shallow levels (low pressure) by anaerobic decay of organic matter and reworked methane from deep under the Earth's surface. In general, the sediments that generate natural gas are buried deeper and at higher temperatures than those that contain oil.

Methane is generally transported in bulk by pipeline in its natural gas form, or by LNG carriers in its liquefied form; few countries transport it by truck.

### Atmospheric methane

Methane (CH4) measured by the Advanced Global Atmospheric Gases Experiment (AGAGE) in the lower atmosphere (troposphere) at stations around the world. Abundances are given as pollution free monthly mean mole fractions in parts-per-billion.

In 2010, methane levels in the Arctic were measured at 1850 nmol/mol. This level is over twice as high as at any time in the last 400,000 years. Historic methane concentrations in the world's atmosphere have ranged between 300 and 400 nmol/mol during glacial periods commonly known as ice ages, and between 600 and 700 nmol/mol during the warm interglacial periods. The Earth's oceans are a potential important source of Arctic methane.[48]

Methane is an important greenhouse gas with a global warming potential of 34 compared to CO2 (potential of 1) over a 100-year period, and 72 over a 20-year period.[49][50]

The Earth's atmospheric methane concentration has increased by about 150% since 1750, and it accounts for 20% of the total radiative forcing from all of the long-lived and globally mixed greenhouse gases. The AR6 of the IPCC states: "Observed increases in well-mixed greenhouse gas (GHG) concentrations since around 1750 are unequivocally caused by human activities. Since 2011 (measurements reported in AR5), concentrations have continued to increase in the atmosphere, reaching annual averages of 410 ppm for carbon dioxide (CO2), 1866 ppb for methane (CH4), and 332 ppb for nitrous oxide (N2O) in 2019. (…) In 2019, atmospheric CO2 concentrations were higher than at any time in at least 2 million years (high confidence), and concentrations of CH4 and N2O were higher than at any time in at least 800,000 years (very high confidence). Since 1750, increases in CO2 (47%) and CH4 (156%) concentrations far exceed, and increases in N2O (23%) are similar to, the natural multi-millennial changes between glacial and interglacial periods over at least the past 800,000 years (very high confidence)".[10]

From 2015 to 2019 sharp rises in levels of atmospheric methane have been recorded.[51][52] In February 2020, it was reported that fugitive emissions and gas venting from the fossil fuel industry may have been significantly underestimated.[53]

Climate change can increase atmospheric methane levels by increasing methane production in natural ecosystems, forming a Climate change feedback.[36][54] Another explanation for the rise in methane emissions could be a slowdown of the chemical reaction that removes methane from the atmosphere.[55]

### Clathrates

Methane clathrates (also known as methane hydrates) are solid cages of water molecules that trap single molecules of methane. Significant reservoirs of methane clathrates have been found in arctic permafrost and along continental margins beneath the ocean floor within the gas clathrate stability zone, located at high pressures (1 to 100 MPa; lower end requires lower temperature) and low temperatures (< 15 °C; upper end requires higher pressure).[56] Methane clathrates can form from biogenic methane, thermogenic methane, or a mix of the two. These deposits are both a potential source of methane fuel as well as a potential contributor to global warming.[57][58] The global mass of carbon stored in gas clathrates is still uncertain and has been estimated as high as 12,500 Gt carbon and as low as 500 Gt carbon.[59] The estimate has declined over time with a most recent estimate of ~1800 Gt carbon.[60] A large part of this uncertainty is due to our knowledge gap in sources and sinks of methane and the distribution of methane clathrates at the global scale. For example, a source of methane was discovered relatively recently in an ultraslow spreading ridge in the Arctic.[61] Some climate models suggest that today's methane emission regime from the ocean floor is potentially similar to that during the period of the Paleocene–Eocene Thermal Maximum (PETM) around 55.5 million years ago, although there are no data indicating that methane from clathrate dissociation currently reaches the atmosphere.[60] Arctic methane release from permafrost and seafloor methane clathrates is a potential consequence and further cause of global warming; this is known as the clathrate gun hypothesis.[62][63][64][65] Data from 2016 indicate that Arctic permafrost thaws faster than predicted.[66]

### Extraterrestrial methane

#### Interstellar medium

Methane is abundant in many parts of the Solar System and potentially could be harvested on the surface of another solar-system body (in particular, using methane production from local materials found on Mars[67] or Titan), providing fuel for a return journey.[26][68]

#### Mars

Methane has been detected on all planets of the Solar System and most of the larger moons.[citation needed] With the possible exception of Mars, it is believed to have come from abiotic processes.[69][70]

Methane (CH4) on Mars – potential sources and sinks

The Curiosity rover has documented seasonal fluctuations of atmospheric methane levels on Mars. These fluctuations peaked at the end of the Martian summer at 0.6 parts per billion.[71][72][73][74][75][76][77][78]

Methane has been proposed as a possible rocket propellant on future Mars missions due in part to the possibility of synthesizing it on the planet by in situ resource utilization.[79] An adaptation of the Sabatier methanation reaction may be used with a mixed catalyst bed and a reverse water-gas shift in a single reactor to produce methane from the raw materials available on Mars, utilizing water from the Martian subsoil and carbon dioxide in the Martian atmosphere.[67]

Methane could be produced by a non-biological process called serpentinization[a] involving water, carbon dioxide, and the mineral olivine, which is known to be common on Mars.[80]

## History

In November 1776, methane was first scientifically identified by Italian physicist Alessandro Volta in the marshes of Lake Maggiore straddling Italy and Switzerland. Volta was inspired to search for the substance after reading a paper written by Benjamin Franklin about "flammable air".[81] Volta collected the gas rising from the marsh, and by 1778 had isolated pure methane.[82] He also demonstrated that the gas could be ignited with an electric spark.[82]

Following the Felling mine disaster of 1812 in which 92 men perished, Sir Humphry Davy established that the feared firedamp was in fact largely methane.[83]

The name "methane" was coined in 1866 by the German chemist August Wilhelm von Hofmann.[84][85] The name was derived from methanol.

## Etymology

Etymologically, the word methane is coined from the chemical suffix "-ane", which denotes substances belonging to the alkane family; and the word methyl, which is derived from the German methyl (1840) or directly from the French méthyle, which is a back-formation from the French méthylène (corresponding to English "methylene"), the root of which was coined by Jean-Baptiste Dumas and Eugène Péligot in 1834 from the Greek μέθυ methy (wine) (related to English "mead") and ὕλη hyle (meaning "wood"). The radical is named after this because it was first detected in methanol, an alcohol first isolated by distillation of wood. The chemical suffix -ane is from the coordinating chemical suffix -ine which is from Latin feminine suffix -ina which is applied to represent abstracts. The coordination of "-ane", "-ene", "-one", etc. was proposed in 1866 by German chemist August Wilhelm von Hofmann.[86]

### Abbreviations

The abbreviation CH4-C can mean the mass of carbon contained in a mass of methane, and the mass of methane is always 1.33 times the mass of CH4-C.[87][88] CH4-C can also mean the methane-carbon ratio, which is 1.33 by mass.[89] Methane at scales of the atmosphere is commonly measured in teragrams (Tg CH4) or millions of metric tons (MMT CH4), which mean the same thing.[90] Other standard units are also used, such as nanomole (nmol, one billionth of a mole), mole (mol), kilogram, and gram.

## Safety

Methane is nontoxic, yet it is extremely flammable and may form explosive mixtures with air. Methane is also an asphyxiant if the oxygen concentration is reduced to below about 16% by displacement, as most people can tolerate a reduction from 21% to 16% without ill effects. The concentration of methane at which asphyxiation risk becomes significant is much higher than the 5–15% concentration in a flammable or explosive mixture. Methane off-gas can penetrate the interiors of buildings near landfills and expose occupants to significant levels of methane. Some buildings have specially engineered recovery systems below their basements to actively capture this gas and vent it away from the building.

Methane gas explosions are responsible for many deadly mining disasters.[91] A methane gas explosion was the cause of the Upper Big Branch coal mine disaster in West Virginia on April 5, 2010, killing 29.[92] Natural gas accidental release has also been a major focus in the field of safety engineering, due to past accidental releases that concluded in the formation of jet fire disasters. [93][94]

## Explanatory notes

1. ^ There are many serpentinization reactions. Olivine is a solid solution between forsterite and fayalite whose general formula is (Fe,Mg)2SiO4. The reaction producing methane from olivine can be written as: Forsterite + Fayalite + Water + Carbonic acid → Serpentine + Magnetite + Methane , or (in balanced form): 18 Mg2SiO4 + 6 Fe2SiO4 + 26 H2O + CO2 → 12 Mg3Si2O5(OH)4 + 4 Fe3O4 + CH4

## Citations

1. ^ a b "Front Matter". Nomenclature of Organic Chemistry : IUPAC Recommendations and Preferred Names 2013 (Blue Book). Cambridge: The Royal Society of Chemistry. 2014. pp. 3–4. doi:10.1039/9781849733069-FP001. ISBN 978-0-85404-182-4. Methane is a retained name (see P-12.3) that is preferred to the systematic name 'carbane', a name never recommended to replace methane, but used to derive the names 'carbene' and 'carbyne' for the radicals H2C2• and HC3•, respectively.
2. ^ "Gas Encyclopedia". Archived from the original on December 26, 2018. Retrieved November 7, 2013.
3. ^ a b c d Haynes, p. 3.344
4. ^ Haynes, p. 5.156
5. ^ Haynes, p. 3.578
6. ^ Haynes, pp. 5.26, 5.67
7. ^ "Safety Datasheet, Material Name: Methane" (PDF). USA: Metheson Tri-Gas Incorporated. December 4, 2009. Archived from the original (PDF) on June 4, 2012. Retrieved December 4, 2011.
8. ^ NOAA Office of Response and Restoration, US GOV. "METHANE". noaa.gov. Archived from the original on January 9, 2019. Retrieved March 20, 2015.
9. ^ Khalil, M. A. K. (1999). "Non-Co2 Greenhouse Gases in the Atmosphere". Annual Review of Energy and the Environment. 24: 645–661. doi:10.1146/annurev.energy.24.1.645.
10. ^ a b "Climate Change 2021. The Physical Science Basis. Summary for Policymakers. Working Group I contribution to the WGI Sixth Assessment Report of the Intergovernmental Panel on Climate Change". IPCC. The Intergovernmental Panel on Climate Change. Archived from the original on August 22, 2021. Retrieved August 22, 2021.
11. ^ a b Etiope, Giuseppe; Lollar, Barbara Sherwood (2013). "Abiotic Methane on Earth". Reviews of Geophysics. 51 (2): 276–299. Bibcode:2013RvGeo..51..276E. doi:10.1002/rog.20011. S2CID 56457317.
12. ^ Hensher, David A.; Button, Kenneth J. (2003). Handbook of transport and the environment. Emerald Group Publishing. p. 168. ISBN 978-0-08-044103-0. Archived from the original on March 19, 2015. Retrieved February 22, 2016.
13. ^ P.G.J Irwin; et al. (January 12, 2022). "Hazy Blue Worlds: A Holistic Aerosol Model for Uranus and Neptune, Including Dark Spots". Journal of Geophysical Research: Planets. 127 (6): e2022JE007189. arXiv:2201.04516. doi:10.1029/2022JE007189. PMC 9286428. PMID 35865671. S2CID 245877540.
14. ^ Bini, R.; Pratesi, G. (1997). "High-pressure infrared study of solid methane: Phase diagram up to 30 GPa". Physical Review B. 55 (22): 14800–14809. Bibcode:1997PhRvB..5514800B. doi:10.1103/physrevb.55.14800.
15. ^ Wendelin Himmelheber. "Crystal structures". Archived from the original on February 12, 2020. Retrieved December 10, 2019.
16. ^ Baik, Mu-Hyun; Newcomb, Martin; Friesner, Richard A.; Lippard, Stephen J. (2003). "Mechanistic Studies on the Hydroxylation of Methane by Methane Monooxygenase". Chemical Reviews. 103 (6): 2385–419. doi:10.1021/cr950244f. PMID 12797835.
17. ^ Snyder, Benjamin E. R.; Bols, Max L.; Schoonheydt, Robert A.; Sels, Bert F.; Solomon, Edward I. (December 19, 2017). "Iron and Copper Active Sites in Zeolites and Their Correlation to Metalloenzymes". Chemical Reviews. 118 (5): 2718–2768. doi:10.1021/acs.chemrev.7b00344. PMID 29256242.
18. ^ Reimann, Joachim; Jetten, Mike S.M.; Keltjens, Jan T. (2015). "Chapter 7 Metal Enzymes in "Impossible" Microorganisms Catalyzing the Anaerobic Oxidation of Ammonium and Methane". In Peter M.H. Kroneck and Martha E. Sosa Torres (ed.). Sustaining Life on Planet Earth: Metalloenzymes Mastering Dioxygen and Other Chewy Gases. Metal Ions in Life Sciences. Vol. 15. Springer. pp. 257–313. doi:10.1007/978-3-319-12415-5_7. ISBN 978-3-319-12414-8. PMID 25707470.
19. ^ Bordwell, Frederick G. (1988). "Equilibrium acidities in dimethyl sulfoxide solution". Accounts of Chemical Research. 21 (12): 456–463. doi:10.1021/ar00156a004.
20. ^ Rasul, G.; Surya Prakash, G.K.; Olah, G.A. (2011). "Comparative study of the hypercoordinate carbonium ions and their boron analogs: A challenge for spectroscopists". Chemical Physics Letters. 517 (1): 1–8. Bibcode:2011CPL...517....1R. doi:10.1016/j.cplett.2011.10.020.
21. ^ Bernskoetter, W. H.; Schauer, C. K.; Goldberg, K. I.; Brookhart, M. (2009). "Characterization of a Rhodium(I) σ-Methane Complex in Solution". Science. 326 (5952): 553–556. Bibcode:2009Sci...326..553B. doi:10.1126/science.1177485. PMID 19900892. S2CID 5597392.
22. ^ Energy Content of some Combustibles (in MJ/kg) Archived January 9, 2014, at the Wayback Machine. People.hofstra.edu. Retrieved on March 30, 2014.
23. ^ March, Jerry (1968). Advance Organic Chemistry: Reactions, Mechanisms and Structure. New York: McGraw-Hill Book Company. pp. 533–534.
24. ^ "Lumber Company Locates Kilns at Landfill to Use Methane – Energy Manager Today". Energy Manager Today. September 23, 2015. Archived from the original on July 9, 2019. Retrieved March 11, 2016.
25. ^ Cornell, Clayton B. (April 29, 2008). "Natural Gas Cars: CNG Fuel Almost Free in Some Parts of the Country". Archived from the original on January 20, 2019. Retrieved July 25, 2009. Compressed natural gas is touted as the 'cleanest burning' alternative fuel available, since the simplicity of the methane molecule reduces tailpipe emissions of different pollutants by 35 to 97%. Not quite as dramatic is the reduction in net greenhouse-gas emissions, which is about the same as corn-grain ethanol at about a 20% reduction over gasoline
26. ^ a b Thunnissen, Daniel P.; Guernsey, C. S.; Baker, R. S.; Miyake, R. N. (2004). "Advanced Space Storable Propellants for Outer Planet Exploration" (PDF). American Institute of Aeronautics and Astronautics (4–0799): 28. Archived from the original (PDF) on March 10, 2016.
27. ^ "Blue Origin BE-4 Engine". Archived from the original on October 1, 2021. Retrieved June 14, 2019. We chose LNG because it is highly efficient, low cost and widely available. Unlike kerosene, LNG can be used to self-pressurize its tank. Known as autogenous repressurization, this eliminates the need for costly and complex systems that draw on Earth's scarce helium reserves. LNG also possesses clean combustion characteristics even at low throttle, simplifying engine reuse compared to kerosene fuels.
28. ^ Report of the Hydrogen Production Expert Panel: A Subcommittee of the Hydrogen & Fuel Cell Technical Advisory Committee Archived February 14, 2020, at the Wayback Machine. United States Department of Energy (May 2013).
29. ^ Rossberg, M. et al. (2006) "Chlorinated Hydrocarbons" in Ullmann's Encyclopedia of Industrial Chemistry, Wiley-VCH, Weinheim. doi:10.1002/14356007.a06_233.pub2.
30. ^ Lumbers, Brock (2022). "Mathematical modelling and simulation of the thermo-catalytic decomposition of methane for economically improved hydrogen production". International Journal of Hydrogen Energy. 47 (7): 4265–4283. doi:10.1016/j.ijhydene.2021.11.057. S2CID 244814932. Retrieved June 15, 2022.
31. ^ Lumbers, Brock (2022). "Low-emission hydrogen production via the thermo-catalytic decomposition of methane for the decarbonisation of iron ore mines in Western Australia". International Journal of Hydrogen Energy. 47 (37): 16347–16361. doi:10.1016/j.ijhydene.2022.03.124. S2CID 248018294. Retrieved July 10, 2022.
32. ^ Kietäväinen and Purkamo (2015). "The origin, source, and cycling of methane in deep crystalline rock biosphere". Front. Microbiol. 6: 725. doi:10.3389/fmicb.2015.00725. PMC 4505394. PMID 26236303.
33. ^ Cramer and Franke (2005). "Indications for an active petroleum system in the Laptev Sea, NE Siberia". Journal of Petroleum Geology. 28 (4): 369–384. Bibcode:2005JPetG..28..369C. doi:10.1111/j.1747-5457.2005.tb00088.x. Archived from the original on October 1, 2021. Retrieved May 23, 2017.
34. ^ Lessner, Daniel J. (Dec 2009) Methanogenesis Biochemistry. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net Archived May 13, 2011, at the Wayback Machine
35. ^ Thiel, Volker (2018), "Methane Carbon Cycling in the Past: Insights from Hydrocarbon and Lipid Biomarkers", in Wilkes, Heinz (ed.), Hydrocarbons, Oils and Lipids: Diversity, Origin, Chemistry and Fate, Handbook of Hydrocarbon and Lipid Microbiology, Springer International Publishing, pp. 1–30, doi:10.1007/978-3-319-54529-5_6-1, ISBN 9783319545295, S2CID 105761461
36. ^ a b c d Dean, Joshua F.; Middelburg, Jack J.; Röckmann, Thomas; Aerts, Rien; Blauw, Luke G.; Egger, Matthias; Jetten, Mike S. M.; de Jong, Anniek E. E.; Meisel, Ove H. (2018). "Methane Feedbacks to the Global Climate System in a Warmer World". Reviews of Geophysics. 56 (1): 207–250. Bibcode:2018RvGeo..56..207D. doi:10.1002/2017RG000559. hdl:1874/366386.
37. ^ Serrano-Silva, N.; Sarria-Guzman, Y.; Dendooven, L.; Luna-Guido, M. (2014). "Methanogenesis and methanotrophy in soil: a review". Pedosphere. 24 (3): 291–307. doi:10.1016/s1002-0160(14)60016-3.
38. ^ Sirohi, S. K.; Pandey, Neha; Singh, B.; Puniya, A. K. (September 1, 2010). "Rumen methanogens: a review". Indian Journal of Microbiology. 50 (3): 253–262. doi:10.1007/s12088-010-0061-6. PMC 3450062. PMID 23100838.
39. ^ IPCC. Climate Change 2013: The physical Science Basis Archived October 3, 2018, at the Wayback Machine. United Nations Environment Programme, 2013: Ch. 6, p. 507 IPCC.ch
40. ^ Lyu, Zhe; Shao, Nana; Akinyemi, Taiwo; Whitman, William B. (2018). "Methanogenesis". Current Biology. 28 (13): R727–R732. doi:10.1016/j.cub.2018.05.021. PMID 29990451.
41. ^ "Inventory of U.S. Greenhouse Gas Emissions and Sinks: 1990–2014". 2016. Archived from the original on April 12, 2019. Retrieved April 11, 2019. {{cite journal}}: Cite journal requires |journal= (help)[page needed]
42. ^ FAO (2006). Livestock's Long Shadow–Environmental Issues and Options. Rome, Italy: Food and Agriculture Organization of the United Nations (FAO). Archived from the original on July 26, 2008. Retrieved October 27, 2009.
43. ^ Gerber, P.J.; Steinfeld, H.; Henderson, B.; Mottet, A.; Opio, C.; Dijkman, J.; Falcucci, A. & Tempio, G. (2013). "Tackling Climate Change Through Livestock". Rome: Food and Agriculture Organization of the United Nations (FAO). Archived from the original on July 19, 2016. Retrieved July 15, 2016.
44. ^ Roach, John (May 13, 2002). "New Zealand Tries to Cap Gaseous Sheep Burps". National Geographic. Archived from the original on June 4, 2011. Retrieved March 2, 2011.
45. ^ Roque, Breanna M.; Venegas, Marielena; Kinley, Robert D.; Nys, Rocky de; Duarte, Toni L.; Yang, Xiang; Kebreab, Ermias (March 17, 2021). "Red seaweed (Asparagopsis taxiformis) supplementation reduces enteric methane by over 80 percent in beef steers". PLOS ONE. 16 (3): e0247820. Bibcode:2021PLoSO..1647820R. doi:10.1371/journal.pone.0247820. ISSN 1932-6203. PMC 7968649. PMID 33730064.
46. ^ Silverman, Jacob (July 16, 2007). "Do cows pollute as much as cars?". HowStuffWorks.com. Archived from the original on November 4, 2012. Retrieved November 7, 2012.
47. ^ Knittel, K.; Wegener, G.; Boetius, A. (2019), McGenity, Terry J. (ed.), "Anaerobic Methane Oxidizers", Microbial Communities Utilizing Hydrocarbons and Lipids: Members, Metagenomics and Ecophysiology, Handbook of Hydrocarbon and Lipid Microbiology, Springer International Publishing, pp. 1–21, doi:10.1007/978-3-319-60063-5_7-1, ISBN 9783319600635
48. ^ "Study Finds Surprising Arctic Methane Emission Source". NASA. April 22, 2012. Archived from the original on August 4, 2014. Retrieved March 30, 2014.
49. ^ IPCC Fifth Assessment Report, Table 8.7, Chap. 8, p. 8–58 Archived November 1, 2013, at the Wayback Machine (PDF; 8,0 MB)
50. ^ Shindell, D. T.; Faluvegi, G.; Koch, D. M.; Schmidt, G. A.; Unger, N.; Bauer, S. E. (2009). "Improved Attribution of Climate Forcing to Emissions". Science. 326 (5953): 716–718. Bibcode:2009Sci...326..716S. doi:10.1126/science.1174760. PMID 19900930. S2CID 30881469. Archived from the original on August 7, 2020. Retrieved August 25, 2020.
51. ^ Nisbet, E.G. (February 5, 2019). "Very Strong Atmospheric Methane Growth in the 4 Years 2014–2017: Implications for the Paris Agreement". Global Biogeochemical Cycles. 33 (3): 318–342. Bibcode:2019GBioC..33..318N. doi:10.1029/2018GB006009.
52. ^ McKie, Robin (February 2, 2017). "Sharp rise in methane levels threatens world climate targets". The Observer. ISSN 0029-7712. Archived from the original on July 30, 2019. Retrieved July 14, 2019.
53. ^ Chelsea Harvey Methane Emissions from Oil and Gas May Be Significantly Underestimated; Estimates of methane coming from natural sources have been too high, shifting the burden to human activities Archived February 24, 2020, at the Wayback Machine E&E News via Scientific American February 21, 2020
54. ^ Carrington, Damian (July 21, 2020) First active leak of sea-bed methane discovered in Antarctica Archived July 22, 2020, at the Wayback Machine, The Guardian
55. ^ Ravilious, Kate (July 5, 2022). "Methane much more sensitive to global heating than previously thought – study". The Guardian. Retrieved July 5, 2022.
56. ^ Bohrmann, Gerhard; Torres, Marta E. (2006), Schulz, Horst D.; Zabel, Matthias (eds.), "Gas Hydrates in Marine Sediments", Marine Geochemistry, Springer Berlin Heidelberg, pp. 481–512, doi:10.1007/3-540-32144-6_14, ISBN 9783540321446
57. ^ Miller, G. Tyler (2007). Sustaining the Earth: An Integrated Approach. U.S.A.: Thomson Advantage Books, p. 160. ISBN 0534496725
58. ^ Dean, J. F. (2018). "Methane feedbacks to the global climate system in a warmer world". Reviews of Geophysics. 56 (1): 207–250. Bibcode:2018RvGeo..56..207D. doi:10.1002/2017RG000559. hdl:1874/366386.
59. ^ Boswell, Ray; Collett, Timothy S. (2011). "Current perspectives on gas hydrate resources". Energy Environ. Sci. 4 (4): 1206–1215. doi:10.1039/c0ee00203h.
60. ^ a b Ruppel and Kessler (2017). "The interaction of climate change and methane hydrates". Reviews of Geophysics. 55 (1): 126–168. Bibcode:2017RvGeo..55..126R. doi:10.1002/2016RG000534. Archived from the original on February 7, 2020. Retrieved September 16, 2019.{{cite journal}}: CS1 maint: uses authors parameter (link)
61. ^ "New source of methane discovered in the Arctic Ocean". phys.org. May 1, 2015. Archived from the original on April 10, 2019. Retrieved April 10, 2019.
62. ^ "Methane Releases From Arctic Shelf May Be Much Larger and Faster Than Anticipated" (Press release). National Science Foundation (NSF). March 10, 2010. Archived from the original on August 1, 2018. Retrieved April 6, 2018.
63. ^ Connor, Steve (December 13, 2011). "Vast methane 'plumes' seen in Arctic ocean as sea ice retreats". The Independent. Archived from the original on December 25, 2011. Retrieved September 4, 2017.
64. ^ "Arctic sea ice reaches lowest extent for the year and the satellite record" (Press release). The National Snow and Ice Data Center (NSIDC). September 19, 2012. Archived from the original on October 4, 2012. Retrieved October 7, 2012.
65. ^ "Frontiers 2018/19: Emerging Issues of Environmental Concern". UN Environment. Archived from the original on March 6, 2019. Retrieved March 6, 2019.
66. ^ Reuters (June 18, 2019). "Scientists shocked by Arctic permafrost thawing 70 years sooner than predicted". The Guardian. ISSN 0261-3077. Archived from the original on October 6, 2019. Retrieved July 14, 2019. {{cite news}}: |last= has generic name (help)
67. ^ a b Zubrin, R. M.; Muscatello, A. C.; Berggren, M. (2013). "Integrated Mars in Situ Propellant Production System". Journal of Aerospace Engineering. 26: 43–56. doi:10.1061/(ASCE)AS.1943-5525.0000201.
68. ^ "Methane Blast". NASA. May 4, 2007. Archived from the original on November 16, 2019. Retrieved July 7, 2012.
69. ^ Chang, Kenneth (November 2, 2012). "Hope of Methane on Mars Fades". The New York Times. Archived from the original on June 8, 2019. Retrieved November 3, 2012.
70. ^ Atreya, Sushil K.; Mahaffy, Paul R.; Wong, Ah-San (2007). "Methane and related trace species on Mars: origin, loss, implications for life, and habitability". Planetary and Space Science. 55 (3): 358–369. Bibcode:2007P&SS...55..358A. doi:10.1016/j.pss.2006.02.005. hdl:2027.42/151840.{{cite journal}}: CS1 maint: uses authors parameter (link)
71. ^ Brown, Dwayne; Wendel, JoAnna; Steigerwald, Bill; Jones, Nancy; Good, Andrew (June 7, 2018). "Release 18-050 – NASA Finds Ancient Organic Material, Mysterious Methane on Mars". NASA. Archived from the original on June 7, 2018. Retrieved June 7, 2018.
72. ^ NASA (June 7, 2018). "Ancient Organics Discovered on Mars – video (03:17)". NASA. Archived from the original on June 7, 2018. Retrieved June 7, 2018.
73. ^ Wall, Mike (June 7, 2018). "Curiosity Rover Finds Ancient 'Building Blocks for Life' on Mars". Space.com. Archived from the original on June 7, 2018. Retrieved June 7, 2018.
74. ^ Chang, Kenneth (June 7, 2018). "Life on Mars? Rover's Latest Discovery Puts It 'On the Table' – The identification of organic molecules in rocks on the red planet does not necessarily point to life there, past or present, but does indicate that some of the building blocks were present". The New York Times. Archived from the original on June 8, 2018. Retrieved June 8, 2018.
75. ^ Voosen, Paul (June 7, 2018). "NASA rover hits organic pay dirt on Mars". Science. doi:10.1126/science.aau3992. S2CID 115442477.
76. ^ ten Kate, Inge Loes (June 8, 2018). "Organic molecules on Mars". Science. 360 (6393): 1068–1069. Bibcode:2018Sci...360.1068T. doi:10.1126/science.aat2662. PMID 29880670. S2CID 46952468.
77. ^ Webster, Christopher R.; et al. (June 8, 2018). "Background levels of methane in Mars' atmosphere show strong seasonal variations". Science. 360 (6393): 1093–1096. Bibcode:2018Sci...360.1093W. doi:10.1126/science.aaq0131. PMID 29880682.
78. ^ Eigenbrode, Jennifer L.; et al. (June 8, 2018). "Organic matter preserved in 3-billion-year-old mudstones at Gale crater, Mars". Science. 360 (6393): 1096–1101. Bibcode:2018Sci...360.1096E. doi:10.1126/science.aas9185. PMID 29880683.
79. ^ Richardson, Derek (September 27, 2016). "Elon Musk Shows Off Interplanetary Transport System". Spaceflight Insider. Archived from the original on October 1, 2016. Retrieved October 3, 2016.
80. ^ Oze, C.; Sharma, M. (2005). "Have olivine, will gas: Serpentinization and the abiogenic production of methane on Mars". Geophysical Research Letters. 32 (10): L10203. Bibcode:2005GeoRL..3210203O. doi:10.1029/2005GL022691. S2CID 28981740.
81. ^ Volta, Alessandro (1777) Lettere del Signor Don Alessandro Volta ... Sull' Aria Inflammable Nativa Delle Paludi Archived November 6, 2018, at the Wayback Machine [Letters of Signor Don Alessandro Volta ... on the flammable native air of the marshes], Milan, Italy: Giuseppe Marelli.
82. ^ a b Methane. BookRags. Archived from the original on March 3, 2016. Retrieved January 26, 2012.
83. ^ Holland, John (1841). The history and description of fossil fuel, the collieries, and coal trade of Great Britain. London, Whittaker and Co. pp. 271–272. Retrieved May 16, 2021.
84. ^ Hofmann, A. W. (1866). "On the action of trichloride of phosphorus on the salts of the aromatic monoamines". Proceedings of the Royal Society of London. 15: 55–62. JSTOR 112588. Archived from the original on May 3, 2017. Retrieved June 14, 2016.; see footnote on pp. 57–58
85. ^ McBride, James Michael (1999) "Development of systematic names for the simple alkanes". Chemistry Department, Yale University (New Haven, Connecticut). Archived March 16, 2012, at the Wayback Machine
86. ^ Harper, Douglas. "methane". Online Etymology Dictionary.
87. ^ Jayasundara, Susantha (December 3, 2014). "Is there is any difference in expressing greenhouse gases as CH4Kg/ha and CH4-C Kg/ha?". ResearchGate. Archived from the original on October 1, 2021. Retrieved August 26, 2020.
88. ^ "User's Guide For Estimating Carbon Dioxide, Methane, And Nitrous Oxide Emissions From Agriculture Using The State Inventory Tool" (PDF). US EPA. November 26, 2019. Archived (PDF) from the original on October 1, 2021. Retrieved August 26, 2020.
89. ^ "What does CH4-C mean? – Definition of CH4-C – CH4-C stands for Methane-carbon ratio". acronymsandslang.com. Archived from the original on April 11, 2015. Retrieved August 26, 2020.
90. ^ Office of Air and Radiation, US EPA (October 7, 1999). "U.S. Methane Emissions 1990–2020: Inventories, Projections, and Opportunities for Reductions (EPA 430-R-99-013)" (PDF). ourenergypolicy.org. Archived (PDF) from the original on October 26, 2020. Retrieved August 26, 2020.
91. ^ Dozolme, Philippe. "Common Mining Accidents". About.com. Archived from the original on November 11, 2012. Retrieved November 7, 2012.
92. ^ Messina, Lawrence & Bluestein, Greg (April 8, 2010). "Fed official: Still too soon for W.Va. mine rescue". News.yahoo.com. Archived from the original on April 8, 2010. Retrieved April 8, 2010.
93. ^ OSMAN, Karim; GENIAUT, Baptiste; HERCHIN, Nicolas; BLANCHETIERE, Vincent (2015). "A review of damages observed after catastrophic events experienced in the mid-stream gas industry compared to consequences modelling tools" (PDF). Symposium Series. 160 (25). Retrieved July 1, 2022.
94. ^ Casal, Joaquim; Gómez-Mares, Mercedes; Muñoz, Miguel; Palacios, Adriana (2012). "Jet Fires: a "Minor" Fire Hazard?" (PDF). Chemical Engineering Transactions. 26: 13–20. doi:10.3303/CET1226003. Retrieved July 1, 2022.


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## Set theory seminar – Marion Scheepers: Coding strategies (IV)

October 12, 2010

For the third talk (and a link to the second one), see here. The fourth talk took place on October 12.

We want to show the following version of Theorem 2:

Theorem. Suppose $\kappa$ is a singular strong limit cardinal of uncountable cofinality. Then the following are equivalent:

1. For each ideal $J$ on $\kappa$, player II has a winning coding strategy in $RG(J)$.
2. $2^\kappa<\kappa^{+\omega}$.

Since $2^\kappa$ has uncountable cofinality, option 2 above is equivalent to saying that the instance of ${\sf wSCH}$ corresponding to $\kappa$ holds.

Before we begin the proof, we need to single out some elementary consequences in cardinal arithmetic of the assumptions on $\kappa$. First of all, since $\kappa$ is singular strong limit, then for any cardinal $\lambda<\kappa$, we have that

$\kappa^\lambda=\left\{\begin{array}{cl}\kappa&\mbox{\ if }\lambda<{\rm cf}(\kappa),\\ 2^\kappa&\mbox{\ otherwise.}\end{array}\right.$

Also, since the cofinality of $\kappa$ is uncountable, we have Hausdoff’s result that if $n<\omega$, then $(\kappa^{+n})^{\aleph_0}=\kappa^{+n}$. I have addressed both these computations in my lecture notes for Topics in Set Theory, see here and here.

Proof. $(2.\Rightarrow 1.)$ We use Theorem 1. If option 1. fails, then there is an ideal $J$ on $\kappa$ with ${\rm cf}(\left< J\right>,{\subset})>|J|$.

Note that ${\rm cf}(\left< J\right>,{\subset})\le({\rm cf}(J,{\subset}))^{\aleph_0}$, and $\kappa\le|J|$. Moreover, if $\lambda<\kappa$, then $2^\lambda<{\rm cf}(J,{\subset})$ since, otherwise,

$({\rm cf}(J,{\subset}))^{\aleph_0}\le 2^{\lambda\aleph_0}=2^\lambda<\kappa$.

So ${\rm cf}(J,{\subset})\ge\kappa$ and then, by Hausdorff, in fact ${\rm cf}(J,{\subset})\ge \kappa^{+\omega}$, and option 2. fails.

$(1.\Rightarrow 2.)$ Suppose option 2. fails and let $\lambda=\kappa^{+\omega}$, so $\kappa<\lambda<2^\kappa$ and ${\rm cf}(\lambda)=\omega$. We use $\lambda$ to build an ideal $J$ on $\kappa$ with ${\rm cf}(\left< J\right>,{\subset})>|J|$.

For this, we use that there is a large almost disjoint family of functions from ${\rm cf}(\kappa)$ into $\kappa$. Specifically:

Lemma. If $\kappa$ is singular strong limit, there is a family ${\mathcal F}\subseteq{}^{{\rm cf}(\kappa)}\kappa$ with ${}|{\mathcal F}|=2^\kappa$ and such that for all distinct $f,g\in{\mathcal F}$, we have that ${}|\{\alpha<{\rm cf}(\kappa)\mid f(\alpha)=g(\alpha)|<{\rm cf}(\kappa)$.

In my notes, I have a proof of a general version of this result, due to Galvin and Hajnal, see Lemma 12 here; essentially, we list all functions $f:{\rm cf}(\kappa)\to\kappa$, and then replace them with (appropriate codes for) the branches they determine through the tree $\kappa^{{\rm cf}(\kappa)}$. Distinct branches eventually diverge, and this translates into the corresponding functions being almost disjoint.

Pick a family ${\mathcal F}$ as in the lemma, and let ${\mathcal G}$ be a subfamily of size $\lambda$. Let $S=\bigcup{\mathcal G}\subseteq{\rm cf}(\kappa)\times\kappa$. We proceed to show that $|S|=\kappa$ and use ${\mathcal G}$ to define an ideal $J$ on $S$ as required.

First, obviously $|S|\le\kappa$. Since $\kappa<\lambda=|{\mathcal G}|$ and ${\mathcal G}\subseteq{\mathcal P}(S)$, it follows that ${}|S|\ge\kappa$, or else ${}|{\mathcal P}(S)|<\kappa$, since $\kappa$ is strong limit.

Now define

$J=\{X\subseteq S\mid\exists {\mathcal H}\subseteq{\mathcal G}\,(|{\mathcal H}|<\omega,\bigcup{\mathcal H}\supseteq X)\}.$

Clearly, $J$ is an ideal. We claim that $|J|=\lambda$. First, each singleton $\{f\}$ with $f\in{\mathcal G}$ is in $J$, so ${}|J|\ge\lambda$. Define $\Phi:[{\mathcal G}]^{<\aleph_0}\to J$ by $\Phi({\mathcal H})=\bigcup{\mathcal H})$. Since the functions in ${\mathcal G}$ are almost disjoint, it follows that $\Phi$ is 1-1. Let $G$ be the image of $\Phi$. By construction, $G$ is cofinal in $J$. But then

${}|J|\le|{\mathcal G}|2^{{\rm cf}(\kappa)}=\lambda 2^{{\rm cf}(\kappa)}=\lambda$,

where the first inequality follows from noticing that any $X\in J$ has size at most ${\rm cf}(\kappa)$. It follows that $|J|=\lambda$, as claimed.

Finally, we argue that ${\rm cf}(\left< J\right>,{\subset})>\lambda$, which completes the proof. For this, consider a cofinal ${\mathcal A}\subseteq\left< J\right>$, and a map $f:{\mathcal A}\to[{\mathcal G}]^{\le\aleph_0}$ such that for all $A\in{\mathcal A}$, we have $A\subseteq\bigcup f(A)$.

Since ${\mathcal A}$ is cofinal in $\left< J\right>$, it follows that $f[{\mathcal A}]$ is cofinal in ${}[{\mathcal G}]^{\le\aleph_0}$. But this gives the result, because

${}|{\mathcal A}|\ge{\rm cf}([{\mathcal G}]^{\le \aleph_0},{\subset})={\rm cf}([\lambda]^{\le \aleph_0},{\subset})>\lambda$,

and we are done. $\Box$

## 580 -Cardinal arithmetic (3)

February 9, 2009

It is easy to solve negatively the question immediately following Homework problem 5 on lecture II.1. I asked whether if $X$ is Dedekind-finite but ${\mathcal P}(X)$ is Dedekind-infinite, then it followed that there is an infinite Dedekind-finite set $Y$ such that ${\mathcal P}(Y)\preceq X$.

To exhibit a counterexample, it is enough to know that it is consistent to have an infinite Dedekind finite set $X$ that is the countable union of finite sets (in fact, sets of size 2). Notice that $\omega$ is a surjective image of $X,$ so ${\mathcal P}(X)$ is Dedekind-infinite. Suppose that ${\mathcal P}(Y)\preceq X.$ Then certainly $Y\preceq X,$ so $Y$ is a countable union of finite sets $Y_n.$ If $Y$ is infinite then $Y_n\ne\emptyset$ for infinitely many values of $n.$ But then $\omega$ is also a surjective image of $Y$, so $\omega$ (and in fact $P(\omega)$) injects into ${\mathcal P}(Y)$ and therefore into $X,$ contradiction.

At the end of last lecture we showed Theorem 10, a general result that allows us to compute products $\kappa^\lambda$ for infinite cardinals $\kappa,\lambda,$ namely:

Let $\kappa$ and $\lambda$ be infinite cardinals. Let $\tau=\sup_{\rho<\kappa}|\rho|^\lambda.$ Then

$\displaystyle \kappa^\lambda=\left\{\begin{array}{cl} 2^\lambda & \mbox{if }\kappa\le 2^\lambda,\\ \kappa\cdot\tau & \mbox{if }\lambda<{\rm cf}(\kappa),\\ \tau & \begin{array}{l}\mbox{if }{\rm cf}(\kappa)\le\lambda,2^\lambda<\kappa,\mbox{ and }\\ \rho\mapsto|\rho|^\lambda\mbox{ is eventually constant below }\kappa,\end{array}\\ \kappa^{{\rm cf}(\kappa)} & \mbox{otherwise.}\end{array}\right.$

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# GPT-2

Pretrained model on English language using a causal language modeling (CLM) objective. It was introduced in this paper and first released at this page.

Disclaimer: The team releasing GPT-2 also wrote a model card for their model. Content from this model card has been written by the Hugging Face team to complete the information they provided and give specific examples of bias.

## Model description

GPT-2 is a transformers model pretrained on a very large corpus of English data in a self-supervised fashion. This means it was pretrained on the raw texts only, with no humans labelling them in any way (which is why it can use lots of publicly available data) with an automatic process to generate inputs and labels from those texts. More precisely, it was trained to guess the next word in sentences.

More precisely, inputs are sequences of continuous text of a certain length and the targets are the same sequence, shifted one token (word or piece of word) to the right. The model uses internally a mask-mechanism to make sure the predictions for the token i only uses the inputs from 1 to i but not the future tokens.

This way, the model learns an inner representation of the English language that can then be used to extract features useful for downstream tasks. The model is best at what it was pretrained for however, which is generating texts from a prompt.

## Intended uses & limitations

You can use the raw model for text generation or fine-tune it to a downstream task. See the model hub to look for fine-tuned versions on a task that interests you.

### How to use

You can use this model directly with a pipeline for text generation. Since the generation relies on some randomness, we set a seed for reproducibility:

from tf_transformers.models import GPT2Model
from transformers import GPT2Tokenizer

tokenizer = GPT2Tokenizer.from_pretrained('gpt2-medium')
model = GPT2Model.from_pretrained("gpt2-medium")

text = "Replace me by any text you'd like."
inputs_tf = {}
inputs = tokenizer(text, return_tensors='tf')

inputs_tf["input_ids"] = inputs["input_ids"]
outputs_tf = model(inputs_tf)

### Limitations and bias

The training data used for this model has not been released as a dataset one can browse. We know it contains a lot of unfiltered content from the internet, which is far from neutral. As the openAI team themselves point out in their model card:

Because large-scale language models like GPT-2 do not distinguish fact from fiction, we donâ€™t support use-cases that require the generated text to be true.

Additionally, language models like GPT-2 reflect the biases inherent to the systems they were trained on, so we do not recommend that they be deployed into systems that interact with humans > unless the deployers first carry out a study of biases relevant to the intended use-case. We found no statistically significant difference in gender, race, and religious bias probes between 774M and 1.5B, implying all versions of GPT-2 should be approached with similar levels of caution around use cases that are sensitive to biases around human attributes.

## Training data

The OpenAI team wanted to train this model on a corpus as large as possible. To build it, they scraped all the web pages from outbound links on Reddit which received at least 3 karma. Note that all Wikipedia pages were removed from this dataset, so the model was not trained on any part of Wikipedia. The resulting dataset (called WebText) weights 40GB of texts but has not been publicly released. You can find a list of the top 1,000 domains present in WebText here.

## Training procedure

### Preprocessing

The texts are tokenized using a byte-level version of Byte Pair Encoding (BPE) (for unicode characters) and a vocabulary size of 50,257. The inputs are sequences of 1024 consecutive tokens.

The larger model was trained on 256 cloud TPU v3 cores. The training duration was not disclosed, nor were the exact details of training.

## Evaluation results

The model achieves the following results without any fine-tuning (zero-shot):

(metric) (PPL) (ACC) (ACC) (ACC) (PPL) (PPL) (BPB) (BPC) (PPL) (PPL)
35.13 45.99 87.65 83.4 29.41 65.85 1.16 1,17 37.50 75.20

### BibTeX entry and citation info

@article{radford2019language,
title={Language Models are Unsupervised Multitask Learners},
author={Radford, Alec and Wu, Jeff and Child, Rewon and Luan, David and Amodei, Dario and Sutskever, Ilya},
year={2019}
}


text:
# Invariance of $Z[x]$ under a self-equivalence of the category of commutative rings with 1.

Let $\mbox{Rings}$ be the category of commutative rings with $1$.

Is there an equivalence of categories $F: \mbox{Rings} \to \mbox{Rings}$ s.t.
$$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$

-
$F(\mathbb{Z}[x]) \cong \mathbb{Z}[x]$ already implies $F \cong \text{id}$. Proof: If $|R|$ is the set underlying $R$, then canonically $|R| \cong Hom(\mathbb{Z}[x],R)$. Apply $F$ on the right and get $|R| \cong |F(R)|$. This map is a ring homomorphism: To see this, use naturality to reduce to the case $R = \mathbb{Z}[x]$ or $R = \mathbb{Z}[x,y]$. The first works by definiton, and the second by the description of $\mathbb{Z}[x,y]$ as the coproduct of $\mathbb{Z}[x]$ and $\mathbb{Z}[y]$. –  Martin Brandenburg Feb 19 '11 at 1:35
Well, since mono=injective in rings, you know you can pick out the subcategory of fields in a categorical manner; these are the precisely the rings such that all morphisms out are monomorphisms. Hence, you can make sense of domains (they admit monos into fields), hence the field of fractions of a domain, since this is a universal property. $Q$ is the field of fractions of the initial object, and $Q(x)$ is a universal transcendental extension of $Q$ (finite algebraic extensions are those which have only finitely many morphisms into another field; arbitrary algebraic extensions are colimits... –  Kevin Ventullo Feb 19 '11 at 4:04
of finite extensions). So $F(\mathbb{Z}[x])$ is a domain whose field of fractions is $Q(x)$. Moreover, it admits morphisms into any field, so its intersection with $Q$ is just $Z$. –  Kevin Ventullo Feb 19 '11 at 4:06
Well, it's not quite a solution since I haven't ruled out, say $Z[2x,x^2]$, so I didn't post it as an answer. Universal is probably the wrong word to use, but basically $Q(x)$ is the only transcendental extension of $Q$ such that any other transcendental extension can be factored through it (though not uniquely). –  Kevin Ventullo Feb 19 '11 at 10:09
I just want to remark that I've proven, among several other things of which I don't know if they are useful, that the whole thing works if we replace (rings) to ($k$-algebras) for an algebraically closed field $k$. –  Martin Brandenburg Feb 19 '11 at 15:19

The category $\mbox{Ring}$ is rigid, i.e. every equivalence $\mbox{Ring} \to \mbox{Ring}$ is isomorphic to the identity,

Proof: Let $F : \mbox{Ring} \to \mbox{Ring}$ be an equivalence. The main part is to prove that $A := F(\mathbb{Z}[x])$ is isomorphic to $\mathbb{Z}[x]$. Charles' answer shows that $A$ is a retract of a polynomial ring $\mathbb{Z}[x_1,...,x_n]$, in particular an integral domain. Since $\mathbb{Z}$ is the initial ring, $F$ preserves it and since $\mathbb{Z} \subseteq \mathbb{Z}[x]$ is a split monomorphism, the same is true for $\mathbb{Z} \to A$. Thus we have $\mathbb{Z} \subseteq A \subseteq \mathbb{Z}[x_1,...,x_n]$. Now assume that we already knew that $A$ is a UFD with $\text{tr.deg}(Q(A)/\mathbb{Q}) = 1$. Then [2], Theorem 4.1 would imply that $A \cong \mathbb{Z}[x]$. Note that the only nontrivial input in the lemmas preceding this theorem is Lüroth's theorem applied to $A \otimes \mathbb{Q}$, which is not needed here because be show below directly $Q(A) \cong \mathbb{Q}(x)$.

So let's check the two properties of $A$. Since $A$ is a retract of $\mathbb{Z}[x_1,...,x_n]$, which is UFD, we conclude from [1], Prop. 1.8 that also $A$ is UFD.

Now it remains to prove that $Q(A) \cong \mathbb{Q}(x)$, which Kevin Ventullo has already sketched in the comments. First note that $F$ preserves nontrivial rings since $0$ is the terminal ring. A monomorphism is the same as an injective ring homomorphism (since the forgetul functor from rings to sets is representable), thus $F$ preserves injective ring homomorphisms. Now fields are precisely the nontrivial rings with only one proper ideal, i.e. such that every homomorphism to a nontrivial ring is injective. Thus $F$ preserves fields (and also field extensions).

The universal property of the quotient field and that we already know that $F$ preserves $\mathbb{Z}$, integral domains, fields and injections implies $F(\mathbb{Q}) \cong \mathbb{Q}$ and $Q(A) = F(\mathbb{Q}(x))$. Now the extension $\mathbb{Q}(x)$ of $\mathbb{Q}$ is characterized by the property it is transcendental and every transcendental extension factors over it. Thus it remains to characterize algebraic field extensions $L/K$.

To do this, we call $L/K$ locally finite if for every extension $E/K$ there are only finitely many $K$-homomorphisms $L \to E$. It is clear that every finite extension is locally finite. Besides, every locally finite extension is algebraic: If not, choose a transcendence basis $B$ and extend infinitely many homomorphisms from $K(B)$ to it's algebraic closure to get a contradiction. It follows that algebraic extensions are precisely the filtered colimits of locally finite extensions.

So we have proved that $F(\mathbb{Z}[x]) \cong \mathbb{Z}[x]$. Now let $R$ be a ring. The usual coring structure on $\mathbb{Z}[x]$ induces a ring structure on $\text{Hom}(\mathbb{Z}[x],R)$ which is naturally isomorphic to $R$. Every coring structure on $\mathbb{Z}[x]$ is isomorphic to the usual one ([3], Prop. 3.1). Thus, we get, naturally in $R$,

$R \cong \text{Hom}(\mathbb{Z}[x],R) \cong \text{Hom}(F(\mathbb{Z}[x]),F(R)) \cong \text{Hom}(\mathbb{Z}[x],F(R)) \cong F(R)$.

Thus $F$ is isomorphic to the identity. Actually [3] gives another, similar proof that $\mbox{Ring}$ is rigid, and more generally it classifies the automorphisms of the category of $R$-algebras, where $R$ is an integral domain. I should have checked the literature in the first place.

[1] Douglas L. Costa, Retracts of Polynomial Rings, J. Algebra 44 (1977), pp. 492 - 502

[2] S. Abhyankar, P. Eakin, W. Heinzer, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1970), pp. 310 - 342

[3] W. E. Clark, G. M. Bergman, The Automorphism Class Group of the Category of Rings, J. Algebra 24 (1973), pp. 80 - 99

-
Hi Martin, Can you please explain more why $|R| \cong |F(R)|$ is a ring homomorphism? Also does it depend on the choice of an isomoprhism $\mathbb{Z}[x]\to F(\mathbb{Z}[x])$? –  Nico Bellic Feb 21 '11 at 23:35
I think my proof was incomplete. See the edit. –  Martin Brandenburg Feb 22 '11 at 10:39
Hi Martin, This is probably a stupid question, but looking at Lang didn't yield an answer. How do we know that there is no other purely transcendental, non-isomorphic to $\mathbb{Q}(x)$, extension of $\mathbb{Q}$ that also embeds inside every other transcendental extension of $\mathbb{Q}$? –  Nico Bellic Feb 22 '11 at 22:11
We don't need "purely transcendental", just "transcendental". Assume $K$ is a trancendental extension which embeds into any other. In particular it embeds into $\mathbb{Q}(x)$. In particular, it is purely transcendental and has transcendence degree at most $1$, but it cannot be $0$. Thus $K \cong \mathbb{Q}(x)$. –  Martin Brandenburg Feb 23 '11 at 16:29
I've just realized that I worked with a wrong definition of "purely transcendental". So, the last argument probably has to be replaced by Lüroth's theorem. –  Martin Brandenburg May 20 '11 at 9:21

Here is a new, short proof, that the category of rings is rigid:

Lemma: Let $R,S$ be rings, such that $\text{Alg}(R), \text{Alg}(S)$ are equivalent as categories. Then $R,S$ are isomorphic.

Proof: $R$ is the initial object of $\text{Alg}(R)$ and the comma category $\text{Alg}(R) / R$ is (via unitalization) equivalent the category $\text{Alg}'(R)$ of $R$-algebras which are not assumed to be unital. Now $0$ is a zero object of $\text{Alg}'(R)$, thus null morphisms are defined. A morphism $\alpha : A \times A \to A$ in $\text{Alg}'(R)$ with $\alpha (\text{id}_A,0) = \text{id}_A, \alpha (0,\text{id}_A) = \text{id}_A$ exists if and only if $(x,y) \mapsto x+y$ is multiplicative, i.e. the multiplication on $A$ is trivial, that is $A$ is just a $R$-module. Thus we have reconstructed $\text{Mod}(R)$ categorically from $\text{Alg}(R)$. Finally, we can reconstruct $R$ as the center of $\text{Mod}(R)$. qed

Theorem: Every self-equivalence of $\text{Ring}$ is isomorphic to the identity.

Proof: Let $F : \text{Ring} \to \text{Ring}$ be an equivalence. Now $\text{Alg}(R)$ is just the comma category $R \backslash \text{Ring}$, so we get an equivalence $\text{Alg}(R) \cong \text{Alg}(F(R))$. The lemma implies $F(R) \cong R$, in particular, $F(\mathbb{Z}[x]) \cong \mathbb{Z}[x]$ as rings. But then we get also an isomorphism of corings and then $F \cong \text{id}$, as explained in my previous proof. qed

Remark, however, that this proof is not applicable for automorphisms of $\text{Alg}(R)$.

-
In your lemma it would be more clear to state the assumption that $R,S$ are commutative rings. –  Peter Samuelson May 20 '11 at 18:10
This is nice! I just want to point out that (as you probably know) the category of $R$-modules can be recovered as the category of abelian group objects in $R$-algebras. (This works more generally for algebras over an operad.) –  Akhil Mathew Nov 28 '12 at 13:44
Yes this is essentially what happens in my proof. The group-operation is $\alpha$. –  Martin Brandenburg Nov 28 '12 at 14:34

Here's a proposal for a proof, though I can't finish it.

• A morphism in a category is a regular epimorphism if it coequalizes some pair of morphisms.
• An object $P$ in a category is projective if $\mathrm{Hom}(P,X)\to \mathrm{Hom}(P,Y)$ is surjective for every regular epimorphism $X\to Y$.
• An object $K$ in a category is compact if $\mathrm{Hom}(K,-)$ takes filtered colimits to filtered colimits of sets.
• An object $X$ in a category is irreducible if the only retracts of $X$ are itself and the initial object.

A self-equivalence of a category will preserve the these conditions.

Claim: $\mathbb{Z}[x]$ is (up to isomorphism) the unique irreducible compact projective in commutative rings.

What I do know (I think):

• Regular epis in $\mathrm{Rings}$ are exactly the surjective ring homomorphisms.
• Compact objects in $\mathrm{Rings}$ are exactly the finitely presented rings.

Since polynomial rings are certainly projective, this means that the compact projective objects in $\mathrm{Rings}$ are precisely the retracts of the $\mathbb{Z}[x_1,\dots,x_n]$s.

My intuition is that retracts of a polynomial ring are always isomorphic to a polynomial ring. In which case $\mathbb{Z}[x]$ would be the only irreducible compact projective. But I could be completely wrong about that.

-
Hi Charles. Your plan of attack looks very nice. I think that everything works, possibly except from the last paragraph. I had that intuition too, but I looked it up in MathSciNet, and there's some literature about retracts of polynomial rings... –  Fernando Muro Feb 21 '11 at 7:01


text:
# 32nd M. Smoluchowski Symposium on Statistical Physics

18-21 September 2019
Kraków
Europe/Warsaw timezone

## $q$-Neighbor Majority-Vote Model on Complex Networks

20 Sep 2019, 12:40
20m
Kraków

Talk

### Speaker

Andrzej Krawiecki (Faculty of Physics, Warsaw University of Technology)

### Description

A $q$-neighbor majority-vote model for the opinion formation is introduced in which agents
represented by two-state spins update their opinions on the basis of the opinions of
randomly chosen subsets of $q$ their neighbors ($q$-lobbies). The agents with probability
$(1-2p)$, $0\le p\le1/2$, obey the majority-vote
rule in which the probability of the opinion flip depends only on the sign of the resultant
opinion of the $q$-lobby, and with probability $2p$ act independently and change opinion or
remain in the actual state with equal probabilities. Thus, the parameter $p$ controls the
degree of stochasticity in the model. In the model under study the
agents are located in the nodes of complex networks, e.g., Erd\"os-R\'enyi graphs or
scale-free networks, and the neighborhood of each agent consists of all agents
connected with him/her by edges, out of which the $q$-lobby is chosen randomly at each
step of the Monte Carlo simulation. This model is related to a recently introduced
$q$-neighbor Ising model [A.\ J\c{e}drzejewski et al., Phys.\ Rev.\ E 92, 052105 (2015);
A.\ Chmiel et al., Int.\ J.\ Modern Phys.\ C 29, 1850041 (2018)], with agents obeying
Metropolis opinion update rule, in which, in particular, first-order ferromagnetic transition was
reported, with the width of the hysteresis loop oscillating with $q$. In contrast, in the
$q$-neighbor majority vote model only second-order ferromagnetic transition is observed.
Theory for this transition is presented both in the mean-field approximation, valid for
large mean degrees of nodes and large $q$, and in a more elaborate pair approximation. In the
latter case the predicted location of the critical point $p_{c}$ agrees quantitatively with that
obtained from Monte Carlo simulations for various complex networks with broad range of
mean degrees of nodes and sizes of the $q$-lobby. Finite size scaling analysis shows that
in the vicinity of the critical point the magnetization shows scalin typical for the
mean-field Ising model, with the critical exponent $\beta = 1/2$, but other critical
exponents depend on the topology of the underlying complex network.

### Primary authors

Andrzej Krawiecki (Faculty of Physics, Warsaw University of Technology) Dr. Tomasz Gradowski (Faculty of Physics, Warsaw University of Technology)

### Presentation Materials

 Andrzej Krawiecki.pdf Video presentation
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2006-08-07
2013-03-26

• Orry Verducci
2006-08-07

At the moment I'm getting PHP errors on every page.

For example:
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Any way to fix it? I'm wondering if it's because I'm using PHP 5.1.2.

• lol I have the same problem with all of version of jetbox! in line 73,74,75!

• Bhoff
2009-01-10

Did either of you ever find an answer to this problem?  My client recently changed servers and upgraded to php5, now their cms doesn't work!  I know this is old but if anyone could help that would be great, it's urgent!