Tachyons in relativity theory.

In special relativity, a faster-than-light particle would have space-like four-momentum, in contrast to ordinary particles that have time-like four-momentum. Although in some theories the mass of tachyons is regarded as imaginary, in some modern formulations the mass is considered real, the formulas for the momentum and energy being redefined to this end. Moreover, since tachyons are constrained to the spacelike portion of the energy–momentum graph, they could not slow down to subluminal speeds.

Mass.

In a Lorentz invariant theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "bradyons" in discussions of tachyons) must also apply to tachyons. In particular the energy–momentum relation: (where p is the relativistic momentum of the bradyon and m is its rest mass) should still apply, along with the formula for the total energy of a particle: This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy. When "v" is larger than "c", the denominator in the equation for the energy is imaginary, as the value under the radical is negative. Because the total energy must be real, the numerator must "also" be imaginary: i.e. the rest mass m must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.  In some modern formulations of the theory, the mass of tachyons is regarded as real.

Speed.

One curious effect is that, unlike ordinary particles, the speed of a tachyon "increases" as its energy decreases. In particular, formula_3 approaches zero when formula_4 approaches infinity. (For ordinary bradyonic matter, "E" increases with increasing speed, becoming arbitrarily large as "v" approaches "c", the speed of light). Therefore, just as bradyons are forbidden to break the light-speed barrier, so too are tachyons forbidden from slowing down to below "c", because infinite energy is required to reach the barrier from either above or below.  As noted by Albert Einstein, Tolman, and others, special relativity implies that faster-than-light particles, if they existed, could be used to communicate backwards in time.

Neutrinos.

In 1985, Chodos proposed that neutrinos can have a tachyonic nature. The possibility of standard model particles moving at superluminal speeds can be modeled using Lorentz invariance violating terms, for example in the Standard-Model Extension. In this framework, neutrinos experience Lorentz-violating oscillations and can travel faster than light at high energies. This proposal was strongly criticized.

Cherenkov radiation.

A tachyon with an electric charge would lose energy as Cherenkov radiation—just as ordinary charged particles do when they exceed the local speed of light in a medium (other than a hard vacuum). A charged tachyon traveling in a vacuum, therefore, undergoes a constant proper time acceleration and, by necessity, its world line forms a hyperbola in space-time. However reducing a tachyon's energy "increases" its speed, so that the single hyperbola formed is of "two" oppositely charged tachyons with opposite momenta (same magnitude, opposite sign) which annihilate each other when they simultaneously reach infinite speed at the same place in space. (At infinite speed, the two tachyons have no energy each and finite momentum of opposite direction, so no conservation laws are violated in their mutual annihilation. The time of annihilation is frame dependent.)  Even an electrically neutral tachyon would be expected to lose energy via gravitational Cherenkov radiation (unless gravitons are themselves tachyons), because it has a gravitational mass, and therefore increases in speed as it travels, as described above. If the tachyon interacts with any other particles, it can also radiate Cherenkov energy into those particles. Neutrinos interact with the other particles of the Standard Model, and Andrew Cohen and Sheldon Glashow used this to argue that the faster-than-light neutrino anomaly cannot be explained by making neutrinos propagate faster than light, and must instead be due to an error in the experiment. Further investigation of the experiment showed that the results were indeed erroneous.

Causality.

Causality is a fundamental principle of physics. If tachyons can transmit information faster than light, then according to relativity they violate causality, leading to logical paradoxes of the "kill your own grandfather" type. This is often illustrated with thought experiments such as the "tachyon telephone paradox" or "logically pernicious self-inhibitor."  The problem can be understood in terms of the relativity of simultaneity in special relativity, which says that different inertial reference frames will disagree on whether two events at different locations happened "at the same time" or not, and they can also disagree on the order of the two events (technically, these disagreements occur when the spacetime interval between the events is'space-like', meaning that neither event lies in the future light cone of the other).  If one of the two events represents the sending of a signal from one location and the second event represents the reception of the same signal at another location, then as long as the signal is moving at the speed of light or slower, the mathematics of simultaneity ensures that all reference frames agree that the transmission-event happened before the reception-event. However, in the case of a hypothetical signal moving faster than light, there would always be some frames in which the signal was received before it was sent so that the signal could be said to have moved backward in time. Because one of the two fundamental postulates of special relativity says that the laws of physics should work the same way in every inertial frame, if it is possible for signals to move backward in time in any one frame, it must be possible in all frames. This means that if observer A sends a signal to observer B which moves faster than light in A's frame but backwards in time in B's frame, and then B sends a reply which moves faster than light in B's frame but backwards in time in A's frame, it could work out that A receives the reply before sending the original signal, challenging causality in "every" frame and opening the door to severe logical paradoxes. Mathematical details can be found in the tachyonic antitelephone article, and an illustration of such a scenario using spacetime diagrams can be found in "Baker, R. (2003)"

Reinterpretation principle.

The reinterpretation principle asserts that a tachyon sent "back" in time can always be "reinterpreted" as a tachyon traveling "forward" in time, because observers cannot distinguish between the emission and absorption of tachyons. The attempt to "detect" a tachyon "from" the future (and violate causality) would actually "create" the same tachyon and send it "forward" in time (which is causal).  However, this principle is not widely accepted as resolving the paradoxes. Instead, what would be required to avoid paradoxes is that unlike any known particle, tachyons do not interact in any way and can never be detected or observed, because otherwise a tachyon beam could be modulated and used to create an anti-telephone or a "logically pernicious self-inhibitor". All forms of energy are believed to interact at least gravitationally, and many authors state that superluminal propagation in Lorentz invariant theories always leads to causal paradoxes.

Fundamental models.

In modern physics, all fundamental particles are regarded as excitations of quantum fields. There are several distinct ways in which tachyonic particles could be embedded into a field theory.

Fields with imaginary mass.

In the paper that coined the term "tachyon", Gerald Feinberg studied Lorentz invariant quantum fields with imaginary mass. Because the group velocity for such a field is superluminal, naively it appears that its excitations propagate faster than light. However, it was quickly understood that the superluminal group velocity does not correspond to the speed of propagation of any localized excitation (like a particle). Instead, the negative mass represents an instability to tachyon condensation, and all excitations of the field propagate subluminally and are consistent with causality. Despite having no faster-than-light propagation, such fields are referred to simply as "tachyons" in many sources.  Tachyonic fields play an important role in modern physics. Perhaps the most famous is the Higgs boson of the Standard Model of particle physics, which has an imaginary mass in its uncondensed phase. In general, the phenomenon of spontaneous symmetry breaking, which is closely related to tachyon condensation, plays an important role in many aspects of theoretical physics, including the Ginzburg–Landau and BCS theories of superconductivity. Another example of a tachyonic field is the tachyon of bosonic string theory.  Tachyons are predicted by bosonic string theory and also the Neveu-Schwarz (NS) and NS-NS sectors, which are respectively the open bosonic sector and closed bosonic sector, of RNS Superstring theory prior to the GSO projection. However such tachyons are not possible due to the Sen conjecture, also known as tachyon condensation. This resulted in the necessity for the GSO projection.

Lorentz-violating theories.

In theories that do not respect Lorentz invariance, the speed of light is not (necessarily) a barrier, and particles can travel faster than the speed of light without infinite energy or causal paradoxes. A class of field theories of that type is the so-called Standard Model extensions. However, the experimental evidence for Lorentz invariance is extremely good, so such theories are very tightly constrained.

Fields with non-canonical kinetic term.

By modifying the kinetic energy of the field, it is possible to produce Lorentz invariant field theories with excitations that propagate superluminally. However, such theories, in general, do not have a well-defined Cauchy problem (for reasons related to the issues of causality discussed above), and are probably inconsistent quantum mechanically.

History.

The term was coined by Gerald Feinberg in a 1967 paper titled "Possibility of Faster-Than-Light Particles". He had been inspired by the science-fiction story "Beep" by James Blish. Feinberg studied the kinematics of such particles according to special relativity. In his paper he also introduced fields with imaginary mass (now also referred to as tachyons) in an attempt to understand the microphysical origin such particles might have.  The first hypothesis regarding faster-than-light particles is sometimes attributed to German physicist Arnold Sommerfeld in 1904, and more recent discussions happened in 1962 and 1969.  In September 2011, it was reported that a tau neutrino had traveled faster than the speed of light in a major release by CERN; however, later updates from CERN on the OPERA project indicate that the faster-than-light readings were due to a faulty element of the experiment's fibre optic timing system.

In fiction.

Tachyons have appeared in many works of fiction. They have been used as a standby mechanism upon which many science fiction authors rely to establish faster-than-light communication, with or without reference to causality issues. The word "tachyon" has become widely recognized to such an extent that it can impart a science-fictional connotation even if the subject in question has no particular relation to superluminal travel (a form of technobabble, akin to "positronic brain").

A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Most physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such particles did exist, they could be used to build a tachyonic antitelephone and send signals faster than light, which (according to special relativity) would lead to violations of causality. No experimental evidence for the existence of such particles has been found.

-{zh-tw:迅子;zh-cn:快子}-(英语:tachyon)也称为-{zh-tw:快子;zh-cn:迅子}-、速子,是一种理论上预测的超光速次原子粒子。这种由相对论衍生出的假想粒子,总是以超过光速的速度在运动。快子与一般物质(相应称为慢子(tardyon))的相互作用可能不明显,所以即使其存在也不一定能侦测得到。在狭义相对论中,快子具有类空的四维动量和虚的原时,并被限定在能量-动量图中的类空区间部分。因此,它无法降低速度至亚光速状态。


History.

Originally, the top-level domain space was organized into three main groups: "Countries", "Categories", and "Multiorganizations". An additional "temporary" group consisted of only the initial DNS domain, arpa, and was intended for transitional purposes toward the stabilization of the domain name system.

Types.

As of 2015, IANA distinguishes the following groups of top-level domains:  Countries are designated in the Domain Name System by their two-letter ISO country code; there are exceptions, however (e.g.,.uk). This group of domains is therefore commonly known as country-code top-level domains (ccTLD). Since 2009, countries with non–Latin-based scripts may apply for internationalized country code top-level domain names, which are displayed in end-user applications in their language-native script or alphabet, but use a Punycode-translated ASCII domain name in the Domain Name System.  Generic top-level domains (formerly Categories) initially consisted of gov, edu, com, mil, org, and net. More generic TLDs have been added, such as info.  The authoritative list of currently existing TLDs in the root zone is published at the IANA website at https://www.iana.org/domains/root/db/.

Internationalized country code TLDs.

An internationalized country code top-level domain (IDN ccTLD) is a top-level domain with a specially encoded domain name that is displayed in an end user application, such as a web browser, in its language-native script or alphabet (such as the Arabic alphabet), or a non-alphabetic writing system (such as Chinese characters). IDN ccTLDs are an application of the internationalized domain name (IDN) system to top-level Internet domains assigned to countries, or independent geographic regions.  ICANN started to accept applications for IDN ccTLDs in November 2009, and installed the first set into the Domain Names System in May 2010. The first set was a group of Arabic names for the countries of Egypt, Saudi Arabia, and the United Arab Emirates. By May 2010, 21 countries had submitted applications to ICANN, representing 11 scripts.

Infrastructure domain.

The domain arpa was the first Internet top-level domain. It was intended to be used only temporarily, aiding in the transition of traditional ARPANET host names to the domain name system. However, after it had been used for reverse DNS lookup, it was found impractical to retire it, and is used today exclusively for Internet infrastructure purposes such as in-addr.arpa for IPv4 and ip6.arpa for IPv6 reverse DNS resolution, uri.arpa and urn.arpa for the Dynamic Delegation Discovery System, and e164.arpa for telephone number mapping based on NAPTR DNS records. For historical reasons, arpa is sometimes considered to be a generic top-level domain.

Reserved domains.

A set of domain names is reserved by the Internet Engineering Task Force as special-use domain names per authority of Request for Comments (RFC) 6761. The practice originated in RFC 1597 for reserved address allocations in 1994, and reserved top-level domains in RFC 2606 of 1999. RFC 6761 reserves the following four top-level domain names to avoid confusion and conflict. Any such reserved usage of those TLDs should not occur in production networks that utilize the global domain name system:  RFC 6762 reserves the use of.local for link-local host names that can be resolved via the Multicast DNS name resolution protocol.  RFC 7686 reserves the use of.onion for the self-authenticating names of Tor hidden services. These names can only be resolved by a Tor client because of the use of onion routing to protect the anonymity of users.  Internet-Draft draft-wkumari-dnsop-internal-00 proposes reserving the use of.internal for "names which do not have meaning in the global context but do have meaning in a context internal to their network", and for which the RFC 6761 reserved names are semantically inappropriate.

Historical domains.

In the late 1980s, InterNIC created the nato domain for use by NATO. NATO considered none of the then existing TLDs as adequately reflecting their status as an international organization. Soon after this addition, however, InterNIC also created the int TLD for the use by international organizations in general, and persuaded NATO to use the second level domain "nato.int" instead. The "nato" TLD, no longer used, was finally removed in July 1996.  Other historical TLDs are cs for Czechoslovakia (now using cz for Czech Republic and sk for Slovakia), dd for East Germany (using de after reunification of Germany), yu for SFR Yugoslavia and Serbia and Montenegro (now using ba for Bosnia and Herzegovina, hr for Croatia, me for Montenegro, mk for North Macedonia, rs for Serbia and si for Slovenia), and zr for Zaire (now cd for the Democratic Republic of the Congo). In contrast to these, the TLD su has remained active despite the demise of the Soviet Union that it represents. Under the chairmanship of Nigel Roberts, ICANN's ccNSO is working on a policy for retirement of ccTLDs that have been removed from ISO 3166.

Proposed domains.

Around late 2000, ICANN discussed and finally introduced aero, biz, coop, info, museum, name, and pro TLDs. Site owners argued that a similar TLD should be made available for adult and pornographic websites to settle the dispute of obscene content on the Internet, to address the responsibility of US service providers under the US Communications Decency Act of 1996. Several options were proposed including xxx, "sex" and "adult". The.xxx domain went live in 2011.  An older proposal consisted of seven new gTLDs: arts, firm, info, nom, rec, shop, and web. Later biz, info, museum, and name covered most of these old proposals.  During the 32nd International Public ICANN Meeting in Paris in 2008, ICANN started a new process of TLD naming policy to take a "significant step forward on the introduction of new generic top-level domains". This program envisioned the availability of many new or already proposed domains, as well as a new application and implementation process. Observers believed that the new rules could result in hundreds of new gTLDs being registered.  On 13 June 2012, ICANN announced nearly 2,000 applications for top-level domains, which began installation throughout 2013. Donuts Inc. invested $57 million in more than 300 applications while Famous Four Media applied for 61 new domains. The first seven – "bike", "clothing", "guru", "holdings", "plumbing", "singles", and "ventures" – were released in 2014.

Alternative DNS roots.

ICANN's slow progress in creating new generic top-level domains, and the high application costs associated with TLDs, contributed to the creation of alternate DNS roots with different sets of top-level domains. Such domains may be accessed by configuration of a computer with alternate or additional (forwarder) DNS servers or plugin modules for web browsers. Browser plugins detect alternate root domain requests and access an alternate domain name server for such requests.

Pseudo-domains.

Several networks, such as BITNET, CSNET, UUCP, existed that were in widespread use among computer professionals and academic users, but were not interoperable directly with the Internet and exchanged mail with the Internet via special email gateways. For relaying purposes on the gateways, messages associated with these networks were labeled with suffixes such as bitnet, oz, csnet, or uucp, but these domains did not exist as top-level domains in the public Domain Name System of the Internet.  Most of these networks have long since ceased to exist, and although UUCP still gets significant use in parts of the world where Internet infrastructure has not yet become well established, it subsequently transitioned to using Internet domain names, and pseudo-domains now largely survive as historical relics. One notable exception is the 2007 emergence of SWIFTNet Mail, which uses the swift pseudo-domain.  The anonymity network Tor formerly used the top-level pseudo-domain onion for Tor hidden services, which can only be reached with a Tor client because it uses the Tor onion routing protocol to reach the hidden service to protect the anonymity of users. However, the pseudo-domain became officially reserved in October 2015. i2p provides a similar hidden pseudo-domain,.i2p.  BT hubs use the top-level pseudo-domain home for local DNS resolution of routers, modems and gateways.

A top-level domain (TLD) is one of the domains at the highest level in the hierarchical Domain Name System of the Internet. The top-level domain names are installed in the root zone of the name space. For all domains in lower levels, it is the last part of the domain name, that is, the last label of a fully qualified domain name. For example, in the domain name www.example.com, the top-level domain is com. Responsibility for management of most top-level domains is delegated to specific organizations by the Internet Corporation for Assigned Names and Numbers (ICANN), which operates the Internet Assigned Numbers Authority (IANA), and is in charge of maintaining the DNS root zone.

顶级域(或顶级域名;英语:Top-level Domain;英文缩写:英语:TLD)是互联网DNS等级之中的最高级的域,它保存于DNS根域的名字空间中。顶级域名是域名的最后一个部分,即是域名最后一点之后的字母,例如在codice_1这个域名中,顶级域是codice_2(或codice_3),大小写视为相同。


Toponymy.

The Tibesti Mountains are named for the Toubou people, also written "Tibu" or "Tubu", that inhabit the area. In the Kanuri language, "tu" means "rock" or "mountain" and "bu" means "a person" or "dweller," and thus "Toubou" roughly translates to "people of the mountains" and "Tibesti" to "place where the mountain people live."  Most of the mountain names are derived from Arabic as well as the Tedaga and Dazaga languages. The term

Geography.

Location.

The mountains lie on the border between Chad and Libya, straddling the Chadian regions of Borkou and Tibesti and the Libyan districts of Murzuq and Kufra, around north of N'djamena and south-southeast of Tripoli. The range is adjacent to Niger and located approximately halfway between the Mediterranean Sea and Lake Chad, just south

Topography.

The highest peak in the Tibesti Mountains, as well as the highest point in Chad and the Sahara Desert, is the Emi Koussi, located at the southern end of the range. Other prominent peaks include Pic Toussidé at and the Ehi Timi on its western side, the Tarso Yega, the Tarso Tieroko, the Ehi Mousgou, the Tarso Voon, the Ehi Sunni, and the Ehi Yéy near the center of the range. The Mouskorbe and the Kegueur Terbi are two peaks notable for their height in the northeastern part of the mountain range.

Hydrology.

Five rivers in the northern half of the Tibesti Mountains flow to Libya and are part of the Mediterranean Basin, while the southern half belongs to the endorheic basin of Lake Chad. However, none of the rivers travel long distances, as the water evaporates in the desert heat or seeps into the ground, although the latter may be carried great distances by subterranean aquifers.  The wadis in the Tibesti are called "enneris". The water originates from the storms that periodically rage over the

Geology.

The Tibesti Mountains are a large area of tectonic uplift that, according to contemporary theory, resulted from a mantle plume in the craton of the African lithosphere, which is about thick. This tectonic uplift may have been accompanied by the opening, and subsequent closure via subduction, of a rift zone. The heart of the Tibesti is composed of schist, basalt and diorite of the Ediacaran period, one of six exposures of Precambrian crystalline rock in North Africa. These are overlaid by sandstone of the Paleozoic era, while the peaks consist of volcanic rock.  The continental hotspot activity began as early as the Oligocene, although the basalt in the area dates primarily from the Miocene to the Lower Pleistocene and, in places, to the Holocene. Due to the comparatively slow movement of the African plate—between per year since the Oligocene—there is

Climate.

The Tibesti climate is substantially less dry than that of the surrounding Sahara Desert. Rainfall events are more intense and more frequent, but they are also highly variable from year to year. Indeed, the Tibesti can see seven to eight years pass between storms. In the south of the range, this variation is largely due to oscillations of the Intertropical Convergence Zone (ITCZ), which steadily moves northward toward Northern Chad from November until August, accompanied by the humid monsoonal air. Normally, the ITCZ repels the Harmattan, a dry trade wind that blows south-west from the Sahara Desert, and brings rainfall to southern Tibesti. However, sometimes the front retires early, before reaching the Tibesti, resulting one or

Flora and fauna.

The Tibesti Mountains are part of the Tibesti-Jebel Uweinat montane xeric woodlands ecoregion, which covers.

Flora.

The flora in the Tibesti is Saharomontane, mixing Mediterranean, Sahara, Sahel and Afromontane vegetation. Biodiversity and endemism levels are much higher in the Tibesti than in the Aïr Mountains or the Ennedi Plateau, although the vegetation's coverage is highly dependent on rainfall. A number of oases lie along the courses of the "enneris" such as Enneri Yebige, which is virtually unexplored. These oases, which are more numerous to the north and west of the range, have vegetation of the genera "Acacia", "Ficus" (fig trees), "Hyphaene" and "Tamarix". Most gueltas are lined with macrophytes including "Cyperus laevigatus", "Equisetum ramosissimum", "Juncus fontanesii" and "Scirpus holoschoenus". "Acacia nilotica" grow near these water basins. "Myrtus nivellei" and "Nerium oleander" grow between elevations of in the western part of the range, while "Tamarix nilotica" grow at similar elevations in its northern part. Downstream, where the current of the "enneris" is slower and the riverbed is deeper, there are dense thickets of "Tamarix aphylla" and "Salvadora persica" (locally known as "yii").  Around the edge of the Tibesti, where the canyons exit the range, are doum palms ("Hyphaene thebaica",

Fauna.

The large mammals present in the ecoregion are the addax ("Addax nasomaculatus"), Barbary sheep ("Ammotragus lervia"), dorcas gazelle ("Gazella dorcas"), and rhim gazelle ("Gazella leptoceros"). However, rodents are the most represented order of mammals in the Tibesti, and include the Agag gerbil ("Gerbillus agag"), Balochistan gerbil ("Gerbillus nanus"), Cairo spiny mouse ("Acomys cahirinus"), lesser Egyptian jerboa ("Jaculus jaculus"), Libyan jird ("Meriones libycus"), Mzab gundi ("Massoutiera Mzabi") and the North African gerbil ("Gerbillus campestris"). Also present are cats such as the African wildcat ("Felis silvestris lybica") and, more rarely, the caracal ("Caracal caracal") and the Sudan cheetah ("Acinonyx jubatus soemmeringii"), as well as several canine species including the Egyptian wolf ("Canis anthus lupaster"), fennec fox ("Vulpes zerda") and Rüppell's fox ("Vulpes rueppellii"). The striped hyena ("Hyaena hyaena") may also occupy the range. There may be as many as 50 endangered painted hunting dogs ("Lycaon pictus") in the Tibesti, although some regard these relict populations as extirpated, partially due to the Darfur refugee turmoil and other Sudan generated conflicts. Bats are heavily represented in the Tibesti, including the desert long-eared bat ("Otonycteris hemprichii"), greater mouse-tailed bat ("Rhinopoma microphyllum"), Hamilton's tomb bat ("Taphozous hamiltoni"), Mauritian tomb bat ("Taphozous mauritianus") and the trident bat ("Asellia tridens"). The Cape hare ("Lepus capensis"), desert hedgehog ("Paraechinus aethiopicus"), olive baboon ("Papio anubis"), rock hyrax ("Procavia capensis") and the Saharan striped polecat ("Ictonyx libyca") also populate the area.  Reptile and amphibian fauna is poor in the Tibesti range. There are gecko populations, including the Algerian sand gecko ("Tropiocolotes steudneri"), northern sand gecko ("Tropiocolotes

Population.

The principal settlements in the Tibesti are Bardaï, Aozou and Zouar, respectively located in the center, north and west of the range, and each situated on oases along the "enneris". Bardaï, at an elevation of, is the capital of Chad's Tibesti Region with around 1,500 inhabitants. It is connected to Zouar by a track that crosses Tarso Toussidé. The village of Omchi is accessible from Bardaï via Aderké, or from Auzou via Irbi. These rough tracks extend southward towards Yebbi Souma and

History.

Human settlement.

There is evidence of human occupation of the Tibesti dating back to the Stone Age, when richer paleovegetation facilitated human habitation. The Toubou settled in the region in the 5th century BC and established trade relations with the Carthaginian civilization. Around this time, Herodotus portrayed the Toubou, which he labeled "Ethiopians" because of their skin color, and described as having a language akin to the "cry of bats."  According to historian Raffael Joorde, a Roman traveler and diplomat named Julius Maternus explored with the king of the Garamantes the territory of the Tibesti mountains, while this king did a military campaign (or a "razzia") against rebellious subjects in the region an Agisymba  In the 12th century, the geographer Muhammad al-Idrisi spoke of a "country of Zaghawa negroes", or camel herders,

Regional relations and colonization.

The Ottoman Empire came into contact with the Toubou in 1560. Their relationship broke out into conflict in the late 17th century, with the Turks favoring the authority of local Maïna at the expense of the authority of the Derdé. In 1780, the Derdé initiated an attack against the Ottomans. The Ottoman retaliation killed 60 percent the Toubou population. The Teda then began launching attacks on Turkish caravans in an effort to disrupt Ottoman trade. In 1890, Maï Getty Tchénimémi established diplomatic relations with the Ottoman Empire and began receiving firearms. Meanwhile, Derdé Chachaï allied with the Senussi Arabs and agreed that the southern half of the Tibesti could serve as a fallback base for the Senussi in their struggle against the French Colonial Army. The Tibesti was then effectively split in two, with the "pro-Maï" in

Modern history.

Chadian Civil War.

Chad gained independence from France in 1960 and in 1965 the Chadian government led by François Tombalbaye imposed its administrative and judicial authority in the Tibesti. Mere days after the withdrawal of French troops from the region rebellion erupted in Bardaï, followed by numerous small-scale clashes over subsequent months and a more famous clash again in Bardaï in September. In response, the Tombalbaye government imposed travel and trade restrictions on the Toubou and voided the traditional power of the then Derdé, Oueddei Kichidemi. Kichidemi went into exile in Libya the following year and became a national symbol in Chad for opposition to the government. These events were a direct trigger of the First Chadian Civil War, which lasted from 1965 to 1979.  In 1968, the French Army, at the request of Tombalbaye, intervened in an attempt to put an end to the rebellion, killing 25 rebels and 121 civilians in various towns, including Zouar. However, the French leader of the intervention force, General Edouard Cortadellas, admitted that the Tibesti area was basically ungovernable, remarking, "I believe we should draw a line below [the Tibetsi region] and leave them to their stones.

Chadian–Libyan conflict.

The Aouzou strip in the Tibesti region was defined for the first time in the discussions between France and Italy after World War I, in relation to an award to Italy for the victory in that war. At the Paris Peace Conference, 1919, France agreed to give some Saharan territories to Italian Libya: the area around Aouzou in the Tibesti region.  After many discussions during the 1920s, in 1935 the Franco-Italian Agreement was signed between Benito Mussolini and Pierre Laval, which included a provision under which Italy would receive the Aouzou strip, which was to be added to Libya. But the "instruments of ratification" of the Mussolini-Laval Treaty never were exchanged with France by the Italian Parliament (that wanted more territories). Despite this

Tibesti War.

By 1986, following a series of military defeats, the GUNT had begun to disintegrate along with relations between Goukouni and Gaddafi. In August, Goukouni was arrested by the Libyans, which spurred his troops to attack Libyan positions in the Tibesti, forcing their withdrawal from the mountains. In October the remnant GUNT forces pledged themselves to Habré.  Libya sought to retake Bardaï and Zouar, and sent a task-force of 2,000 troops with T-62 tanks and heavy support by the Libyan Air Force into the

Scientific exploration and research.

Due to its isolation and geopolitical situation, the Tibesti Mountains were long unexplored by scientists. The German Gustav Nachtigal, dispatched by Otto von Bismarck, was the first European to explore, albeit with great difficulty, the Tibesti in 1869. Prior to Nachtigal's expedition, an American along with two missionaries had attempted to enter the Tibesti range, but were killed by a Toubou warrior. While Nachtigal provided an accurate description of the population, his account discouraged any new adventure into the Tibesti for over 40 years, as he was convicted by the Toubou of spying for Bismarck and only released after the intervention of Maï Arami Tetimi. Later expeditions carried out between 1920 and 1970 yielded valuable information on the geology and petrology of

Climbing history.

The first mountain to be climbed by a westerner in the Tibesti was Pic Toussidé in 1869 by Gustav Nachtigal during his exploration of the area, although the peak is not technically difficult. In 1938, Emi Koussi was summited by Wilfred Thesiger. The first sporting climb was probably that of the peak and needle of Botoum, at and, respectively, by a Swiss expedition led by Edouard Wyss-Dunant in 1948.  In 1957, P.R. Steele, R.F. Marks and W.W. Tuck of the University of Cambridge embarked on an expedition to Tarso Tieroko, which Thesiger had described as "probably the most beautiful peak in Tibesti," and which offered an interesting climbing opportunity. They started at

Economy.

Natural resources.

Although gold was long known to exist in small quantities, substantial deposits were discovered in 2012. Diamonds and emeralds have also been found. The mountains and their surroundings could contain significant quantities of uranium, tin, tungsten, niobium, beryllium, lead, zinc, copper, platinum, and nickel. The natron extracted from Trou au Natron on Tarso Toussidé is used as a food supplement for animals. The Soborom

Agriculture.

The oases to the west and north of the range have the easiest access. Where mosquitoes do not abound, they support several villages, such as Zouar, where indigenous plant species have largely been replaced by some 56,000 date palms ("Phoenix dactylifera") for the production of dates. These are harvested between late July and early August. In winter, when reserves are depleted, it is not uncommon for the cores of the dates and the fiber of the palms to be ground into a paste and consumed. These palm groves can also grow millet and corn, but the crop is spotty and sometimes swept away by early flooding. The banks of the "enneris" grow desert gourd ("Citrullus colocynthis"), which are collected in October to extract the bitter seeds which, after being

Tourism.

Tibesti has tourism potential, with its primary attractions being the palm groves, rock and parietal art, canyons, basalt spires, volcanoes and hot springs. However, the Zouar airfield cannot accommodate aircraft with more than a 20-seat capacity. This pushes the cost of air tickets from N'djamena up to 60 percent of the cost to fly from N'djamena to major European cities. Furthermore, tourist accommodations are virtually absent.

Conservation.

A protected area in the Tibesti range has been proposed to preserve the area's rhim gazelle ("Gazella leptoceros") and Barbary sheep ("Ammotragus lervia")

Culture.

The Tibesti Mountains are renowned for their rock and parietal art. Around 200 engraving sites—including some 1800 different representations—and 100 painting sites have been identified. Most date from between the 5th and 3rd millennia BC, although some date from the 6th millennium BC, centuries before the arrival of the Toubou. The art has suffered the effects of time, including weathering from sand blown by the wind. The earliest works often portray animals that have since died out due to climate change in the Tibesti, including elephants, rhinoceros, hippopotamus and giraffes. More recent art includes ostriches and antelopes, which have an exceptional presence on the edge of the range, as well as gazelles and sheep, which are more common.  Later works, dated less than 2000 years old, portray domesticated animals, such as oxen and camels. Other engravings portray warriors, known as "owoza", dressed in feathers or spiked ornaments and armed with bows, shields, assegai, or traditional knives. Some portray dance scenes. The walls of a canyon near Bardaï have engravings that measure over in height, including that of the "man of Gonoa," Gonoa being the name of the "enneri" that runs through the valley. These primarily show hunting scenes. The paintings, often drawn with a red "Faidherbia albida" base, signify a turn towards domestication in the Tibesti, because the scenes are more pastoral. The Tibesti art is unique in the Sahara because of the absence of inscriptions, the relative lack of chariots, and the low representation of camels and horses until comparatively recently. The art has remained important to the Toubou; around 1200 AD, a man named Yerbou engraved a palm leaf into rock, symbolizing his love for a married woman.  In 1989, French painter and sculptor Jean Vérame used the natural surroundings of the Ehi Kourné, a few kilometers south of Bardaï, to create multidimensional land art works by painting rocks. The project was supported by the Chadian president and the United Nations Development Programme, as well as private corporations such as the French oil company Total S.A. Over time, the Klein Blue paint turned patina white and red, and today is pinkish in color.  The volcanic spires of the Tibesti, along with a stylized sheep's head, were displayed on a 20 CFA franc postage stamp issued by Republic of Chad in 1961.  The Tibesti range was featured in the 1966 short story "Le Mura di Anagoor" ("The Walls of Anagoor") by the Italian novelist Dino Buzzati. In the story, a local guide offers to show a traveller the walls of a great city that is absent from the maps. The city is exceedingly opulent, yet exists in total autarky and does not submit to local authority. The traveler waits many years, in vain, to enter the Tibesti city.

The Tibesti Mountains are a mountain range in the central Sahara, primarily located in the extreme north of Chad, with a small extension into southern Libya. The highest peak in the range, Emi Koussi, lies to the south at a height of and is the highest point in both Chad and the Sahara. Bikku Bitti, the highest peak in Libya, is located in the north of the range. The central third of the Tibesti is of volcanic origin and consists of five shield volcanoes topped by large craters: Emi Koussi, Tarso Toon, Tarso Voon, Tarso Yega and Toussidé. Major lava flows have formed vast plateaus that overlie Paleozoic sandstone. The volcanic activity was the result of a continental hotspot that arose during the Oligocene and continued in some places until the Holocene, creating fumaroles, hot springs, mud pools and deposits of natron and sulfur. Erosion has shaped volcanic spires and carved an extensive network of canyons through which run rivers subject to highly irregular flows that are rapidly lost to the desert sands.

提贝斯提高原,又称提贝斯提山脉或提贝斯提山地,是撒哈拉沙漠中部、乍得北部由一系列火山组成的熔岩高地,平均海拔超过2000米,其最高峰库西山是一座死火山,海拔3415米。


Explanation.

A liquid crossing its standard freezing point will crystalize in the presence of a seed crystal or nucleus around which a crystal structure can form creating a solid. Lacking any such nuclei, the liquid phase can be maintained all the way down to the temperature at which crystal homogeneous nucleation occurs.  Homogeneous nucleation can occur above the glass transition temperature, but if homogeneous nucleation has not occurred above that temperature, an amorphous (non-crystalline) solid will form.  Water normally freezes at 273.15 K (0 °C or 32 °F), but it can be "supercooled" at standard pressure down to its crystal homogeneous nucleation at almost 224.8 K (−48.3 °C/−55 °F). The process of supercooling requires that water be pure and free of nucleation sites, which can be achieved by processes like reverse osmosis or chemical demineralization, but the cooling itself does not require any specialised technique. If water is cooled at a rate on the order of 10 K/s, the crystal nucleation can be avoided and water becomes a glass—that is, an amorphous (non-crystalline) solid. Its glass transition temperature is much colder and harder to determine, but studies estimate it at about 136 K (−137 °C/−215 °F). Glassy water can be heated up to approximately 150 K (−123 °C/−189.4 °F) without nucleation occurring. In the range of temperatures between 231 K (−42 °C/−43.6 °F) and 150 K (−123 °C/−189.4 °F), experiments find only crystal ice.  Droplets of supercooled water often exist in stratus and cumulus clouds. An aircraft flying through such a cloud sees an abrupt crystallization of these droplets, which can result in the formation of ice on the aircraft's wings or blockage of its instruments and probes, unless the aircraft is equipped with an appropriate de-icing system. Freezing rain is also caused by supercooled droplets.  The process opposite to supercooling, the melting of a solid above the freezing point, is much more difficult, and a solid will almost always melt at the same temperature for a given pressure. For this reason, it is the melting point which is usually identified, using melting point apparatus; even when the subject of a paper is "freezing-point determination", the actual methodology is "the principle of observing the disappearance rather than the formation of ice". It is possible, at a given pressure, to superheat a liquid above its boiling point without it becoming gaseous.  Supercooling is often confused with freezing-point depression. Supercooling is the cooling of a liquid below its freezing point without it becoming solid. Freezing point depression is when a solution can be cooled below the freezing point of the corresponding pure liquid due to the presence of the solute; an example of this is the freezing point depression that occurs when salt is added to pure water.

Constitutional supercooling.

Constitutional supercooling, which occurs during solidification, is due to compositional solid changes, and results in cooling a liquid below the freezing point ahead of the solid–liquid interface. When solidifying a liquid, the interface is often unstable, and the velocity of the solid–liquid interface must be small in order to avoid constitutional supercooling.  Supercooled zones are observed when the liquidus temperature gradient at the interface is larger than the temperature gradient.  or  The slope of the liquidus phase boundary on the phase diagram is formula_3  The concentration gradient is related to points, formula_4 and formula_5, on the phase diagram:  For steady-state growth formula_7 and the partition function formula_8 can be assumed to be constant. Therefore, the minimum thermal gradient necessary to create a stable solid front is as expressed below.  For more information, see the equation (3) of

In animals.

In order to survive extreme low temperatures in certain environments, some animals use the phenomenon of supercooling that allow them to remain unfrozen and avoid cell damage and death. There are many techniques that aid in maintaining a liquid state, such as the production of antifreeze proteins, or AFPs, which bind to ice crystals to prevent water molecules from binding and spreading the growth of ice. The winter flounder is one such fish that utilizes these proteins to survive in its frigid environment. Noncolligative proteins are secreted by the liver into the bloodstream. Other animals use colligative antifreezes, which increases the concentration of solutes in their bodily fluids, thus lowering their freezing point. Fish that rely on supercooling for survival must also live well below the water surface, because if they came into contact with ice nuclei they would freeze immediately. Animals that undergo supercooling to survive must also remove ice-nucleating agents from their bodies because they act as a starting point for freezing. Supercooling is also a common feature in some insect, reptile, and other ectotherm species. The potato cyst nematode larva ("Globodera rostochiensis") could survive inside their cysts in a supercooled state to temperatures as low as, even with the cyst encased in ice.  Supercooling is a last resort for animals. The best option is to move to a warmer environment if possible. As an animal gets farther and farther below its original freezing point the chance of spontaneous freezing increases dramatically for its internal fluids, as this is a thermodynamically unstable state. The fluids eventually reach the supercooling point, which is the temperature at which the supercooled solution freezes spontaneously due to being so far below its normal freezing point. Animals unintentionally undergo supercooling and are only able to decrease the odds of freezing once supercooled. Even though supercooling is essential for survival, there are many risks associated with it.

In plants.

As demonstrated by animals, plants can also survive extreme cold conditions brought forth during the winter months. Many plant species located in northern climates can acclimate under these cold conditions by supercooling, thus these plants survive temperatures as low as −40 °C. Although this supercooling phenomenon is poorly understood, it has been recognized through infrared thermography. Ice nucleation occurs in certain plant organs and tissues, debatably beginning in the xylem tissue and spreading throughout the rest of the plant. Infrared thermography allows for droplets of water to be visualized as they crystalize in extracellular spaces.  Supercooling inhibits the formation of ice within the tissue by ice nucleation and allows the cells to maintain water in a liquid state and further allows the water within the cell to stay separate from extracellular ice. Cellular barriers such as lignin, suberin and the cuticle inhibit ice nucleators and force water into the supercooled tissue. The xylem and primary tissue of plants are very susceptible to cold temperatures because of the large proportion of water in the cell. Many boreal hardwood species in northern climates have the ability to prevent ice spreading into the shoots allowing the plant to tolerate the cold. Supercooling has been identified in the evergreen shrubs "Rhododendron ferrugineum" and "Vaccinium vitis-idaea" as well as "Abies", "Picea" and "Larix" species. Freezing outside of the cell and within the cell wall does not affect the survival of the plant. However, the extracellular ice may lead to plant dehydration.

Applications.

One commercial application of supercooling is in refrigeration. Freezers can cool drinks to a supercooled level so that when they are opened, they form a slush. Another example is a product that can supercool the beverage in a conventional freezer. The Coca-Cola Company briefly marketed special vending machines containing Sprite in the UK, and Coke in Singapore, which stored the bottles in a supercooled state so that their content would turn to slush upon opening.  Supercooling was successfully applied to organ preservation at Massachusetts General Hospital/Harvard Medical School. Livers that were later transplanted into recipient animals were preserved by supercooling for up to 96 hours (4 days), quadrupling the limits of what could be achieved by conventional liver preservation methods. The livers were supercooled to a temperature of –6 °C in a specialized solution that protected against freezing and injury from the cold temperature.  Another potential application is drug delivery. In 2015, researchers crystallized membranes at a specific time. Liquid-encapsulated drugs could be delivered to the site and, with a slight environmental change, the liquid rapidly changes into a crystalline form that releases the drug.  In 2016, a team at Iowa State University proposed a method for "soldering without heat" by using encapsulated droplets of supercooled liquid metal to repair heat sensitive electronic devices. In 2019, the same team demonstrated the use of undercooled metal to print solid metallic interconnects on surfaces ranging from polar (paper and Jello) to superhydrophobic (rose petals), with all the surfaces being lower modulus than the metal.  Eftekhari et al. proposed an empirical theory explaining that supercooling of ionic liquid crystals can build ordered channels for diffusion for energy storage applications. In this case, the electrolyte has a rigid structure comparable with that of a solid electrolyte, but the diffusion coefficient can be as large as in liquid electrolytes. Supercooling increases the medium viscosity but keeps the directional channels open for diffusion.

Supercooling, also known as undercooling, is the process of lowering the temperature of a liquid or a gas below its freezing point without it becoming a solid. It achieves this in the absence of a seed crystal or nucleus around which a crystal structure can form. The supercooling of water can be achieved without any special techniques other than chemical demineralization, down to −48.3 °C (−55 °F). Droplets of supercooled water often exist in stratus and cumulus clouds. An aircraft flying through such a cloud sees an abrupt crystallization of these droplets, which can result in the formation of ice on the aircraft's wings or blockage of its instruments and probes.

过冷(Supercooling,又译超冷冻)是一种物理现象,透过降低液体或气体的温度,但不使其凝固的过程,能做到让水瞬间凝冰的效果。


Animal vivisection.

Research requiring vivisection techniques that cannot be met through other means is often subject to an external ethics review in conception and implementation, and in many jurisdictions use of anesthesia is legally mandated for any surgery likely to cause pain to any vertebrate.  In the United States, the Animal Welfare Act explicitly requires that any procedure that may cause pain use "tranquilizers, analgesics, and anesthetics", with exceptions when "scientifically necessary". The act does not define "scientific necessity" or regulate specific scientific procedures, but approval or rejection of individual techniques in each federally funded lab is determined on a case-by-case basis by the Institutional Animal Care and Use Committee, which contains at least one veterinarian, one scientist, one non-scientist, and one other individual from outside the university.  In the United Kingdom, any experiment involving vivisection must be licensed by the Home Secretary. The Animals (Scientific Procedures) Act 1986 "expressly directs that, in determining whether to grant a licence for an experimental project, 'the Secretary of State shall weigh the likely adverse effects on the animals concerned against the benefit likely to accrue.  In Australia, the Code of Practice "requires that all experiments must be approved by an Animal Experimentation Ethics Committee" that includes a "person with an interest in animal welfare who is not employed by the institution conducting the experiment, and an additional independent person not involved in animal experimentation."  Anti-vivisectionists have played roles in the emergence of the animal welfare and animal rights movements, arguing that animals and humans have the same natural rights as living creatures, and that it is inherently immoral to inflict pain or injury on another living creature, regardless of the purpose or potential benefit to mankind.

Vivisection and anti-vivisection in the 19th century.

At the turn of the 19th century, medicine was undergoing a transformation. The emergence of hospitals and the development of more advanced medical tools such as the stethoscope are but a few of the changes in the medical field. There was also an increased recognition that medical practices needed to be improved, as many of the current therapeutics were based on unproven, traditional theories that may or may not have helped the patient recover. The demand for more effective treatment shifted emphasis to research with the goal of understanding disease mechanisms and anatomy. This shift had a few effects, one of which was the rise in patient experimentation, leading to some moral questions about what was acceptable in clinical trials and what was not. An easy solution to the moral problem was to use animals in vivisection experiments, so as not to endanger human patients. This, however, had its own set of moral obstacles, leading to the anti-vivisection movement.

François Magendie (1783–1855).

One polarizing figure in the anti-vivisection movement was François Magendie. Magendie was a physiologist at the Académie Royale de Médecine in France, established in the first half of the 19th century. Magendie made several groundbreaking medical discoveries, but was far more aggressive than some of his other contemporaries with his use of animal experimentation. For example, the discovery of the different functionalities of dorsal and ventral spinal nerve roots was achieved by both Magendie, as well as a Scottish anatomist named Charles Bell. Bell used an unconscious rabbit because of "the protracted cruelty of the dissection", which caused him to miss that the dorsal roots were also responsible for sensory information. Magendie, on the other hand, used conscious, six-week-old puppies for his own experiments. While Magendie's approach was more of an infringement on what we would today call animal rights, both Bell and Magendie used the same justification for vivisection: the cost of animal lives and experimentation was well worth it for the benefit of humanity.  Many viewed Magendie's work as cruel and unnecessarily torturous. One note is that Magendie carried out many of his experiments before the advent of anesthesia, but even after ether's discovery it was not used in any of his experiments or classes. Even during the period before anesthesia, other physiologists expressed their disgust with how he conducted his work. One such visiting American physiologist describes the animals as "victims" and the apparent sadism that Magendie displayed when teaching his classes. The cruelty in such experiments actually even led to Magendie's role as an important figure in animal rights legislation. He was so despised in Britain that his experiments were cited in the drafting of the Cruel Treatment of Cattle Act 1822 and the Cruelty to Animals Act 1876.

David Ferrier and the Cruelty to Animals Act 1876.

The Cruelty to Animals Act, 1876 in Britain determined that one could only conduct vivisection on animals with the appropriate license from the state, and that the work the physiologist was doing had to be original and absolutely necessary. The stage was set for such legislation by physiologist David Ferrier. Ferrier was a pioneer in understanding the brain and used animals to show that certain locales of the brain corresponded to bodily movement elsewhere in the body in 1873. He put these animals to sleep, and caused them to move unconsciously with a probe. Ferrier was successful, but many decried his use of animals in his experiments. Some of these arguments came from a religious standpoint. Some were concerned that Ferrier's experiments would separate God from the mind of man in the name of science. Some of the anti-vivisection movement in England had its roots in Evangelicalism and Quakerism. These religions already had a distrust for science, only intensified by the recent publishing of Darwin's Theory of Evolution in 1859.  Neither side was pleased with how the Cruelty to Animals Act 1876 was passed. The scientific community felt as though the government was restricting their ability to compete with the quickly advancing France and Germany with new regulations. The anti-vivisection movement was also unhappy, but because they believed that it was a concession to scientists for allowing vivisection to continue at all. Ferrier would continue to vex the anti-vivisection movement in Britain with his experiments when he had a debate with his German opponent, Friedrich Goltz. They would effectively enter the vivisection arena, with Ferrier presenting a monkey, and Goltz presenting a dog, both of which had already been operated on. Ferrier won the debate, but did not have a license, leading the anti-vivisection movement to sue him in 1881. Ferrier was not found guilty, as his assistant was the one operating, and his assistant did have a license. Ferrier and his practices gained public support, leaving the anti-vivisection movement scrambling. They made the moral argument that given recent developments, scientists would venture into more extreme practices to operating on "the cripple, the mute, the idiot, the convict, the pauper, to enhance the “interest” of [the physiologist's] experiments".

Human vivisection.

It is possible that human vivisection was practiced by some Greek anatomists in Alexandria in the 3rd century BC. Celsus in "De Medicina" and the early-Christian writer Tertullian state that Herophilos of Alexandria vivisected at least 600 live prisoners.  Unit 731, a biological and chemical warfare research and development unit of the Imperial Japanese Army, undertook lethal human experimentation during the period that comprised both the Second Sino-Japanese War, and the Second World War (1937–1945). In Mindanao, Moro Muslim prisoners of war were subjected to various forms of vivisection by the Japanese, in many cases without anesthesia.  Nazi human experimentation involved many medical experiments on live subjects, such as vivisections by Josef Mengele, usually without anesthesia.  Vivisection without anesthesia was an execution method employed by the Khmer Rouge at the Tuol Sleng prison. Only seven people survived the four-year run of the prison before its liberation by the Vietnamese army in January 1979.

Vivisection (), also known as V-section, is surgery conducted for experimental purposes on a living organism, typically animals with a central nervous system, to view living internal structure. The word is, more broadly, used as a pejorative catch-all term for experimentation on live animals by organizations opposed to animal experimentation, but the term is rarely used by practising scientists. Human vivisection, such as live organ harvesting, has been perpetrated as a form of torture. However, as "vivisection" etymologically means a surgery on a living being, all forms of open surgery on living people are literally "human vivisection".

从词源上来说,活体解剖是指为为生理学或病理学科学研究而对活着的动物进行的解剖。最近,在人们,尤其是动物权利分子的观念中,活体解剖这一条目被广泛地应用到任何形式的有动物受到伤害的实验中,无论这些实验是否有明确的书面表明是活体解剖。


Units.

Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre (cm) is the volume of a cube whose sides are one centimetre (1 cm) in length.  In the International System of Units (SI), the standard unit of volume is the cubic metre (m). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus  so Small amounts of liquid are often measured in millilitres, where In the same way, large amounts can be measured in megalitres, where  Various other traditional units of volume are also in use, including the cubic inch, the cubic foot, the cubic yard, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, the hogshead, the acre-foot and the board foot.

Related terms.

"Capacity" is defined by the Oxford English Dictionary as "the measure applied to the content of a vessel, and to liquids, grain, or the like, which take the shape of that which holds them". (The word "capacity" has other unrelated meanings, as in e.g. capacity management.) Capacity is not identical in meaning to volume, though closely related; the capacity of a container is always the volume in its interior. Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are closely related: one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the capacity of a vehicle's fuel tank is rarely stated in cubic feet, for example, but in gallons (an imperial gallon fills a volume of 0.1605 cu ft).  The "density" of an object is defined as the ratio of the mass to the volume. The inverse of density is "specific volume" which is defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied.  The volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m s]).

Volume in calculus.

In calculus, a branch of mathematics, the volume of a region "D" in R is given by a triple integral of the constant function formula_1 over the region and is usually written as:  In cylindrical coordinates, the volume integral is  In spherical coordinates (using the convention for angles with formula_4 as the azimuth and formula_5 measured from the polar axis; see more on conventions), the volume integral is

Volume formulas.

Volume ratios for a cone, sphere and cylinder of the same radius and height.

The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.  Let the radius be "r" and the height be "h" (which is 2"r" for the sphere), then the volume of cone is  the volume of the sphere is  while the volume of the cylinder is  The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.

Volume formula derivations.

Sphere.

The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness "dx". The calculation for the volume of a sphere with center 0 and radius "r" is as follows.  The surface area of the circular disk is formula_10.  The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is  or  where y or z can be taken to represent the radius of a disk at a particular x value.  Using y as the disk radius, the volume of the sphere can be calculated as  Now  Combining yields formula_15  This formula can be derived more quickly using the formula for the sphere's surface area, which is formula_16. The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to

Cone.

The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well.  However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally thin circular disks of thickness "dx". The calculation for the volume of a cone of height "h", whose base is centered at (0, 0, 0) with radius "r", is as follows.  The radius of each circular disk is "r" if "x" = 0 and 0 if "x" = "h", and varying linearly in between—that is,  The surface area of the circular disk is then  The volume of the cone can then be calculated as  and after extraction of the constants  Integrating gives us

Volume in differential geometry.

In differential geometry, a branch of mathematics, a volume form on a differentiable manifold is a differential form of top degree (i.e., whose degree is equal to the dimension of the manifold) that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. Integrating the volume form gives the volume of the manifold according to that form.  An oriented pseudo-Riemannian manifold has a natural volume form. In local coordinates, it can be expressed as  where the formula_24 are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold, and formula_25 is the determinant of the matrix representation of the metric tensor on the manifold in terms of the same basis.

Volume in thermodynamics.

In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's volume per unit of mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, volume is related to the pressure and temperature of an ideal gas by the ideal gas law.

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i. e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

体积(英语:Volume)是物件占有多少空间的量。体积的国际单位制是立方米。一件固体物件的体积是一个数值用以形容该物件在空间所占有的空间。一维空间物件(如线)及二维空间物件(如正方形)在三维空间中均为零体积。


Introduction and definition.

The concept of vector space will first be explained by describing two particular examples:

First example: arrows in the plane.

The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, and, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the "sum" of the two arrows and is denoted. In the special case of two arrows on the same line, their sum is the arrow on this line whose

Second example: ordered pairs of numbers.

A second key example of a vector space is provided by pairs of real numbers and. (The order of the components and is significant, so such a pair is also called an ordered

Definition.

In this article, vectors are represented in boldface to distinguish them from scalars.  A vector space over a field is a set together with two operations that satisfy the eight axioms listed below. In the following, denotes the Cartesian product of with itself, and denotes a mapping from one set to another.  Elements of are commonly called "vectors". Elements of are commonly called "scalars".  In the two examples above, the field is the field of the real numbers and the set of the vectors consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively.  To qualify as a vector space, the set and the operations of addition and multiplication must adhere to a number of requirements called axioms. In the list below, let, and be arbitrary vectors in, and and scalars in.  These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition

Alternative formulations and elementary consequences.

Vector addition and scalar multiplication are operations, satisfying the closure property: and are in for all in, and, in. Some older sources mention these properties as separate axioms.  In the parlance of abstract algebra, the first four axioms are equivalent to requiring the set of vectors to be an abelian group under addition. The remaining axioms give this group an

History.

Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space. Around 1636, French mathematicians René Descartes and Pierre de Fermat founded analytic geometry by identifying solutions to an equation of two variables with points on a plane curve. To achieve geometric solutions without using coordinates, Bolzano introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors. This work was made use of in the conception of barycentric coordinates by Möbius in 1827. The foundation of the definition of vectors was Bellavitis' notion of the bipoint, an oriented segment one of whose ends is the origin and the other one a target. Vectors were reconsidered with the presentation of complex numbers by Argand and Hamilton and the inception of

Examples.

Coordinate space.

The simplest example of a vector space over a field is the field itself, equipped with its standard addition and multiplication. More generally, all -tuples (sequences of length ) of

Complex numbers and other field extensions.

The set of complex numbers, that is, numbers that can be written in the form for real numbers and where is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: and for real numbers,,, and. The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic.  In fact, the example of complex numbers is essentially

Function spaces.

Functions from any fixed set to a field also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions and is the function given by and similarly for multiplication. Such function spaces occur in many geometric situations, when is the real line or

Linear equations.

Systems of homogeneous linear equations are closely tied to vector spaces. For example, the solutions of are given by triples with arbitrary,, and. They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they

Basis and dimension.

"Bases" allow one to represent vectors by a sequence of scalars called "coordinates" or "components". A basis is a (finite or infinite) set of vectors, for convenience often indexed by some index set, that spans the whole space and is linearly independent. "Spanning the whole space" means that any vector can be expressed as a finite sum (called a "linear combination") of the basis elements:  where the are scalars, called the coordinates (or the components) of the vector with respect to the basis, and elements of. Linear independence means that the coordinates are uniquely determined for any vector in the vector space.  For example, the coordinate vectors,, to, form a basis of, called the standard basis, since any vector can be uniquely expressed as a linear combination of these vectors: The corresponding coordinates,,, are just the Cartesian coordinates of the vector.  Every vector space has a basis. This follows from Zorn's lemma, an equivalent formulation of the Axiom of Choice. Given the other axioms of Zermelo–Fraenkel set theory, the existence of bases is equivalent to the axiom of choice. The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or

Linear maps and matrices.

The relation of two vector spaces can be expressed by "linear map" or "linear transformation". They are functions that reflect the vector space structure, that is, they preserve sums and scalar multiplication:  An "isomorphism" is a linear map such that there exists an inverse map, which is a map such that the two possible compositions and are identity maps. Equivalently, is both one-to-one (injective) and onto (surjective). If there exists an isomorphism between and, the two spaces are said to be "isomorphic"; they are then essentially identical as vector spaces, since all identities holding in are, via, transported to similar ones in, and vice versa via. For example, the "arrows in the plane" and "ordered pairs of numbers" vector spaces in the introduction are isomorphic: a planar arrow departing at the origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the - and -component of the arrow, as shown in the image at the right. Conversely, given a pair, the arrow going by to the right (or to the left, if is negative), and up (down, if is negative) turns back the arrow.  Linear maps between two vector spaces form a vector space, also denoted. The space of linear maps from to is called the "dual vector space", denoted. Via the injective natural map, any vector space can be embedded into its "bidual"; the map is an isomorphism if and only if the space is finite-dimensional.  Once a basis of is chosen, linear maps are completely determined by specifying the images of the basis vectors, because any element of is expressed uniquely as a linear combination of them. If, a 1-to-1 correspondence between fixed bases of and gives rise to a linear map that maps any basis element of to the corresponding basis element of. It is an isomorphism, by its very definition. Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is "completely classified" (up to isomorphism) by its dimension, a single number. In particular, any "n"-dimensional -vector space is isomorphic to. There is, however, no "canonical" or preferred isomorphism; actually an isomorphism is equivalent to the choice of a basis of, by mapping the standard basis of to, via. The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context; see below.

Basic constructions.

In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the definitions given below, they are also characterized by universal properties, which determine an object by specifying the linear maps from to any other vector space.

Subspaces and quotient spaces.

A nonempty subset "W" of a vector space "V" that is closed under addition and scalar multiplication (and therefore contains the 0-vector of "V") is called a "linear subspace" of "V", or simply a "subspace" of "V", when the ambient space is unambiguously a vector space. Subspaces of "V" are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set "S" of vectors is called its span, and it is the smallest subspace of "V" containing the set "S". Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of "S".  A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane. A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane. In a vector space of finite dimension, a vector hyperplane is thus a subspace of dimension.  The counterpart to subspaces are "quotient vector spaces". Given any subspace, the quotient space "V"/"W" (""V" modulo "W"") is defined as follows: as a set, it

Direct product and direct sum.

The "direct product" of vector spaces and the "direct sum" of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.  The "direct product" formula_11 of a family of vector spaces "V" consists of the set of

Tensor product.

The "tensor product", or simply, of two vector spaces "V" and "W" is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map is called bilinear if "g" is linear in both variables v and w. That is to say, for fixed w the map is linear in the sense above and likewise for fixed v.  The tensor product is a particular vector

Vector spaces with additional structure.

From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces "per se" do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures.  A vector space may be given a partial order ≤, under which some vectors can be compared. For example, "n"-dimensional real space R can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions where "f" denotes the positive part of "f" and "f" the negative part.

Normed vector spaces and inner product spaces.

"Measuring" vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted formula_14 and formula_15, respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm formula_16. Vector spaces endowed with such data are known as "normed vector spaces" and "inner product

Topological vector spaces.

Convergence questions are treated by considering vector spaces "V" carrying a compatible topology, a structure that allows one to talk about elements being close to each other. Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x and y in "V", and "a" in "F" vary by a bounded amount, then so do and. To make sense of specifying the amount a scalar changes, the field "F" also has to carry a topology in this context; a common choice are the reals or the complex numbers.  In such "topological vector spaces" one can consider series of vectors. The infinite sum denotes the limit of the corresponding finite partial sums of the sequence ("f") of elements of "V". For example, the "f" could be (real or complex) functions belonging to some function space "V", in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples. A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces

Algebras over fields.

General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an "algebra over a field". Many algebras stem from functions on some geometrical object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone–Weierstrass theorem, for example, relies on Banach algebras which are both Banach spaces and algebras.  Commutative algebra makes great use of rings of polynomials in one or several variables, introduced above. Their multiplication is both commutative and associative. These rings and their quotients form the basis of algebraic geometry, because they are rings of functions

Applications.

Vector spaces have many applications as they occur frequently in common circumstances, namely wherever functions with values in some field are involved. They provide a framework to deal with analytical and geometrical problems, or are used in the Fourier transform. This list is not exhaustive: many more applications exist, for example in optimization. The minimax theorem of game theory stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector spaces methods. Representation theory fruitfully transfers the good understanding of linear algebra and vector spaces to other mathematical domains such as group theory.

Distributions.

A "distribution" (or "generalized function") is a linear map assigning a number to each "test" function, typically a smooth function with compact support, in a continuous way: in the above terminology the space of distributions is the (continuous) dual of the test function space. The latter space is endowed with a topology that takes into account not only "f" itself, but also all its higher derivatives. A standard example is the result of integrating a test function "f" over some domain Ω: When the set consisting of a single point, this reduces to the

Fourier analysis.

Resolving a periodic function into a sum of trigonometric functions forms a "Fourier series", a technique much used in physics and engineering. The underlying vector space is usually the Hilbert space "L"(0, 2π), for which the functions sin "mx" and cos "mx" ("m" an integer) form an orthogonal basis. The Fourier expansion of an "L" function "f" is  The coefficients "a" and "b" are called Fourier coefficients of "f", and are calculated by the formulas  In physical terms the function is represented as a superposition of sine waves and the coefficients give information about the function's frequency spectrum. A complex-number form of Fourier series is also commonly used. The concrete formulae above are consequences of a more general mathematical duality called Pontryagin duality. Applied to the group R, it yields the classical Fourier transform; an application in physics are reciprocal

Differential geometry.

The tangent plane to a surface at a point is naturally a vector space whose origin is identified with the point of contact. The tangent plane is the best linear approximation, or linearization, of a surface at a point. Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceived of

Generalizations.

Vector bundles.

A "vector bundle" is a family of vector spaces parametrized continuously by a topological space "X". More precisely, a vector bundle over "X" is a topological space "E" equipped with a continuous map such that for every "x" in "X", the fiber π("x") is a vector space. The case dim is called a line bundle. For any vector space "V", the projection makes the product into a "trivial" vector bundle. Vector bundles over "X" are required to be locally a product of "X" and some (fixed) vector space "V": for every "x" in "X", there is a neighborhood "U" of "x" such that the restriction of π to π("U") is isomorphic to the trivial bundle. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space "X") be "twisted" in the large (that

Modules.

"Modules" are to rings what vector spaces are to fields: the same axioms, applied to a ring "R" instead of a field "F", yield modules. The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases,

Affine and projective spaces.

Roughly, "affine spaces" are vector spaces whose origins are not specified. More precisely, an affine space is a set with a free transitive vector space action. In particular, a vector space is an affine space over itself, by the map If "W" is a vector space, then an affine subspace is a subset of "W" obtained by translating a linear subspace "V" by a fixed vector ; this space is denoted by (it is a coset

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called "scalars". Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called "axioms", listed below, in. For specifying that the scalars are real or complex numbers, the terms real vector space and complex vector space are often used.

向量空间是现代数学中的一个基本概念。是线性代数研究的基本对象。