formulasearchengine - User contributions [en]

Hydraulic jump

2013-12-02T04:23:09Z

2001:468:C80:4381:AD4A:5011:C2BD:4734: /* Height of the jump */

In mathematics, '''anticommutativity''' is the property of an operation with two or more arguments wherein swapping the position of any two arguments negates the result. Anticommutative [[Operation (mathematics)|operation]]s are widely used in [[algebra]], [[geometry]], [[mathematical analysis]] and, as a consequence, in [[physics]]: they are often called '''antisymmetric [[Operation (mathematics)|operation]]s'''.

== Definition ==
An <math>n</math>-[[Operation_(mathematics)#General_definition|ary operation]] is anticommutative if swapping the order of any two arguments negates the result. For example, a binary operation ∗ is anti-commutative if for all ''x'' and ''y'', {{nowrap|1=''x'' ∗ ''y'' = −(''y'' ∗ ''x'')}}.

More formally, a [[Map (mathematics)|map]] <math> \scriptstyle *:A^n \to \mathfrak{G} </math> from the [[Cartesian product#Cartesian_square_and_n-ary_product|set of all ''n''-tuples]] of elements in a [[Set (mathematics)|set]] ''A'' (where ''n'' is a general integer) to a [[Group (mathematics)|group]] <math> \scriptstyle\mathfrak{G} </math> is anticommutative [[iff|if and only if]]

:<math> {x_1*x_2*\dots*x_n} = \sgn(\sigma) ({x_{\sigma(1)}*x_{\sigma(2)}*\dots* x_{\sigma(n)}}) \qquad \forall\boldsymbol{x} = (x_1,x_2,\dots,x_n) \in A^n</math>

where <math> \scriptstyle\sigma:(n)\to(n) </math> is an arbitrary [[permutation]] of the [[Set (mathematics)|set]] (''n'') of the first ''n'' positive [[integers]] and <math>\mathrm{sgn}(\sigma)</math> is its [[Signature of a permutation|sign]]. This [[equality (mathematics)|equality]] expresses the following concept:
* the value of the operation is unchanged, when applied to all [[Tuple#Formal_definitions|ordered tuple]]s constructed by [[Even and odd permutations|even permutation]] of the elements of a fixed one.
* the value of the operation is the [[Group_(mathematics)#Definition|inverse]] of its value on a fixed [[Tuple#Formal_definitions|tuple]], when applied to all [[Tuple#Formal_definitions|ordered tuple]]s constructed by [[Even and odd permutations|odd permutation]] to the elements of the fixed one. The need for the existence of this [[Group_(mathematics)#Definition|inverse element]] is the main reason for requiring the [[codomain]] <math> \scriptstyle\mathfrak{G} </math> of the [[operation (mathematics)|operation]] to be at least a [[Group (mathematics)|group]].

Note that this is an [[abuse of notation]], since the [[codomain]] of the [[operation (mathematics)|operation]] needs only to be a [[Group (mathematics)|group]]: "−1" does not have a precise meaning since a [[multiplication]] is not necessarily defined on <math> \scriptstyle\mathfrak{G} </math>.

Particularly important is the case {{nowrap|1=''n'' = 2}}. A [[binary operation]] <math> \scriptstyle *:A\times A\to \mathfrak{G} </math> is anticommutative [[iff|if and only if]]

:<math> x_1 * x_2 = -(x_2 * x_1) \qquad\forall(x_1,x_2)\in A\times A</math>

This means that {{nowrap|1=''x''1 ∗ ''x''2}} is the [[Inverse element|inverse]] of the element {{nowrap|1=''x''2 ∗ ''x''1}} in <math> \scriptstyle\mathfrak{G} </math>.

== Properties ==
If the group <math> \scriptstyle\mathfrak{G} </math> is such that

:<math> \mathfrak{-a} = \mathfrak{a} \iff \mathfrak{a} = \mathfrak{0}\qquad \forall \mathfrak{a} \in \mathfrak{G} </math>

i.e. ''the only element equal to its [[Inverse element|inverse]] is the [[neutral element]]'', then for all the [[Tuple#Formal_definitions|ordered tuple]]s such that <math> x_j = x_i </math> for at least two different index <math>i,j</math>

:<math>x_1*x_2*\dots*x_n = \mathfrak{0} </math>

In the case ''<math> n = 2 </math>'' this means

:<math> x_1*x_1 = x_2*x_2 = \mathfrak{0} </math>

== Examples ==

Examples of anticommutative binary operations include:
* [[Subtraction]]
* [[Cross product]]
* Lie bracket of a [[Lie algebra]]
* Lie bracket of a [[Lie ring]]

==See also==
* [[Commutativity]]
* [[Commutator]]
* [[Exterior algebra]]
* [[Operation (mathematics)]]
* [[Symmetry in mathematics]]
* [[Particle statistics]] (for anticommutativity in physics).

== References ==
*{{Citation
| last = Bourbaki
| first = Nicolas
| author-link = Nicolas Bourbaki
| title = Algebra. Chapters 1–3
| place = [[Berlin]]-[[Heidelberg]]-[[New York]]
| publisher = [[Springer-Verlag]]
| chapter = Chapter III. [[Tensor algebra]]s, [[exterior algebra]]s, [[symmetric algebra]]s
| series = Elements of Mathematics
| year = 1989
| pages = xxiii+709
| edition = 2nd printing
| isbn = 3-540-64243-9
| mr = 0979982
| zbl = 0904.00001
}}.

== External links ==
{{Wiktionary}}
*{{springer
| title= Anti-commutative algebra
| id= A/a012580
| last= Gainov
| first= A.T.
| author-link=
}}
*{{MathWorld
|title=Anticommutative
|urlname=Anticommutative}}

[[Category:Abstract algebra]]
[[Category:Binary operations|*Anticommutativity]]

Energy–depth relationship in a rectangular channel

2013-12-02T04:16:18Z

2001:468:C80:4381:AD4A:5011:C2BD:4734: /* Dimensionless momentum function */

{{context|date=November 2010}}
{{noreferences|date=January 2012}}

In [[information theory]], '''Shearer's inequality''' states that if ''X''1, ..., ''X''''d'' are [[random variable]]s and ''S''1, ..., ''S''''n'' are subsets of {1, 2, ..., ''d''} such that every integer between 1 and ''d'' lies in exactly ''r'' of these subsets, then

: <math> H[(X_1,\dots,X_d)] \leq \frac{1}{r}\sum_{i=1}^n H[(X_j)_{j\in S_i}]</math>

where <math> (X_{j})_{j\in S_{i}}</math> is the [[Cartesian product]] of random variables <math>X_{j}</math> with indices ''j'' in <math>S_{i}</math> (so the dimension of this vector is equal to the size of <math>S_{i}</math>).

{{DEFAULTSORT:Shearer's Inequality}}
[[Category:Information theory]]
[[Category:Inequalities]]

Dimensionless Specific Energy Diagrams for Open Channel Flow

2013-12-02T04:14:55Z

2001:468:C80:4381:AD4A:5011:C2BD:4734: /* Dimensionless Diagram */

{{Original research|date=February 2013}}
'''Antimatter comets''' (and '''antimatter meteoroids''') are theoretical [[comet]]s ([[meteoroid]]s) composed solely of [[antimatter]] instead of ordinary [[matter]]. Although never actually observed, and unlikely to exist anywhere within the [[Milky Way]], they have been hypothesized to exist, and their existence, on the presumption that hypothesis is correct, has been put forward as one possible explanations for various observed natural phenomena over the years.

== Hypothesized existence ==
The hypothesis of comets made of anti-matter can be traced back to the 1940s, when physicist [[Vladimir Rojansky]] proposed, in his paper ''The Hypothesis of the Existence of Contraterrene Matter'' the possibility that some comets and meteoroids could be made from "contraterrene" matter (i.e. antimatter).<ref name=NS1974a/> Such objects, Rojanski stated, would (if they existed at all) have their origins outside the solar system.<ref name=Rojansky1940b/> He hypothesized that if there were an antimatter object in orbit in the solar system, it would exhibit the behavior of comets observed in the 1940s: As its atoms annihilated with "terrene" matter from other bodies and [[solar wind]], it would generate volatile compounds and undergo a change of composition to elements with lower [[atomic mass]]es. From this basis he propounded the hypothesis that some objects that had been identified as comets may, in fact, be antimatter objects, suggesting, based upon calculations using the [[Stefan-Boltzmann law]], that it would be possible to determine the existence of such objects within the solar system by observing their temperatures. An antimatter body subjected to normal levels of meteoric bombardment (per 1940s figures), and absorbing half of the energy created by the annihilation of normal matter and antimatter, would have a temperature of {{convert|120|K|C}} for bombardment figures calculated by Wylie or {{convert|1200|K|C}} for calculations by Nininger.<ref name=Rojansky1940c/> In the 1970s, when [[comet Kohoutek]] was observed, Rojanski again suggested hypothesis of anti-matter comets in a letter in ''[[Physical Review Letters]]'', and suggested that [[gamma-ray]] observations be made of the comet to test this hypothesis.<ref name=NS1974a/><ref name=Rojansky1973a/>

Rojansky's original 1940 hypothesis was that perhaps the only bodies within the solar system that could be antimatter were comets and meteoroids, all others being almost certainly normal matter.<ref name=Rojansky1940a/> Experimental evidence gathered since then has not only borne out this restriction but has made the existence of actual antimatter comets and meteoroids themselves seem ever more unlikely. Gary Steigman, assistant professor of Astronomy at [[Yale University]], observed in 1976 that space probes had proven — by the fact that they were not annihilated upon impact — that bodies such as Mars, Venus, and the Moon were not antimatter. He also noted that had any of the planets or similar bodies been antimatter, their interaction with the terrene [[solar wind]] and the sheer strength of the gamma ray emissions that would have resulted{{efn|The formula for the predicted gamma ray flux, resulting from annihilation of solar wind particles (taken to be roughly 2×108 cm−2 sec−1), from a antimatter planet or other solar system body of radius ''r'' at distance ''d'' is <math>F_{gamma} \approx 10^8\left(\frac{r}{d}\right)^2</math> photons cm−2 sec−1. This formula predicts a gamma ray flux for the planet Jupiter that is some six orders of magnitude larger than it is actually observed to be. That is, furthermore, without taking into account the fact that other solar system material in addition to the solar wind infalls to Jupiter.<ref name=Steigman1976c/>}} would have made them readily noticeable long since.<ref name=Steigman1976a/> He noted that not even antimatter [[cosmic rays]] had been found, with all of the nuclei found in studies having been uniformly terrene, the experimental data in several studies made from 1961 onwards by various people excluding the presence of a fractional antimatter composition of cosmic rays any larger than 10−4 of the total. Further, the uniformly terrene nature of the cosmic ray flux indicates that nowhere in the Milky Way are there any sources of heavier antimatter elements (such as carbon), since (although it is not proven) it is a likely assumption that they represent the overall composition of the entire galaxy. They are representative of the galaxy as a whole — goes the logic — and since they ''do'' contain terrene carbon and other atoms, but have not been observed to contain ''any'' antimatter atoms, therefore there is no reasonable source for extrasolar antimatter comets, meteoroids, or any other large scale heavy element objects to originate from, within this galaxy.<ref name=Steigman1976b/>

Martin Beech from the University of Western Ontario (London, Ontario, Canada) referred to the various hypotheses and experimental results that support non existence of antimatter in the Universe. He discussed the Papaelias' formula about the "Velocity-height relation of antimatter meteors"<ref name=Papaelias1987/> and argued that any antimatter comets and meteors that exist must be (at least) extrasolar in origin because the [[nebular hypothesis]] for the [[formation of the solar system]] precludes their being solar. Any antimatter in a pre-formation nebula or planetary accretion disc has a comparatively short lifetime, astronomically speaking, before annihilation with the terrene matter that it is mixed with. This lifetime is measured in the hundreds of years, and so any solar antimatter present at the time that the system was formed will have long since been annihilated. Any antimatter comets and meteors must therefore come from another, antimatter, solar system, and be extrasolar. Furthermore, not only must antimatter meteors be extrasolar in origin, they must have been recently (i.e. within the past 104 ~ 105 years) captured by the solar system. Most meteoroids are broken down to sizes of 10−5g within that timeframe, because of meteoroid-upon-meteoroid collisions. Thus any antimatter meteor must be either extrasolar in origin itself, or broken off from an antimatter comet that is extrasolar in origin. The former are unlikely to exist from observational evidence. Any extrasolar meteoroid would have a [[hyperbolic orbit]], but less than 1% of the observed meteoroids have such, and the process of perturbation of ordinary (terrene) solar objects, by planetary encounters, into hyperbolic trajectories accounts for all of those. Beech concluded that a continued null result, however, does not constitute a proof ('Absence of evidence is not evidence of absence', M. Rees) and a single positive detection negates the arguments presented.<ref name=Beech1988/> Taking into account the work of Alfvẻn, Lehnert and Papaelias, Herbert Shaw detected a seeming necessity of antimatter structures in our vicinity of the Galaxy.<ref name=Shaw1995/>

The Physics that governs an antimatter meteor fall is published by Philip M. Papaelias (assistant professor at the National University of Athens, Greece) who derived a set of formulae that explains how its cosmic velocity and original mass are decreasing and for how long its remaining mass may survive.<ref name=Papaelias1987/><ref name=Papaelias1990/><ref name=Papaelias1991a/><ref name=Papaelias1991b/><ref name=Papaelias1993/><ref name=Papaelias1994/> This framework of Physics helped Ken Bullough from Sheffield (UK), to realize that a small group of comets he was studying were in fact made of antimatter, since they were showing unusual characteristics to distinguishing them from the other ones. This study was radar observations at 73 MHz made in June 1953 at Jodrell Bank as part of the meteor/radio-aurora observational programme. Publication of these data was suppressed because, at that time, no interpretation was possible within the then existing framework of physics and, in addition, the radar echoes were not detected on nominally similar equipment operating at 72 MHz. Decades later, when Bullough was aware about the physics of antimatter meteors and comets, he returned at the Jodrell Bank Observatory (Manchester) to search for the list of those comets. Forty two years after the first observations he published the results which include the short period (6.37 year) [[7P/Pons–Winnecke]] and the [[29P/Schwassmann-Wachmann]] in a group of few dozens of antimatter comets. In addition of this new study, Bullough also analyzed extensively the [[Tunguska event]] and concluded that the explosion was generated by the annihilation of an antimatter meteor in the atmosphere.<ref name=Bullough1995/>

== Hypothesized explanations for observed phenomena ==

=== Tektites ===
In 1947, Mohammad Abdur Rahman Khan, professor at [[Osmania University]] and research associate at the Institute of Meteoretics in the [[University of New Mexico]], put forward the hypothesis that antimatter comets or meteoroids were responsible for [[tektite]]s {{harv|Khan|1947}}. However, this explanation, out of the many proposed explanations for tektites, is considered to be one of the more improbable.<ref name=Bagnall1991/><ref name=Vand1965/>

=== Tunguska event of 1908 ===
By the 1950s, speculating about antimatter comets and meteoroids was a commonplace exercise for astrophysicists. One such, Philip J. Wyatt of [[Florida State University]], suggested that the [[Tunguska event]] may have been a meteor made of antimatter {{harv|Wyatt|1958}}.<ref name=TIME1958a/> [[Willard Libby]] and [[Clyde Cowan]] took Wyatt's idea further {{harv|Cowan|Atluri|Libby|1965}}, having studied worldwide levels of carbon-14 in tree rings and noticing unusually high levels for the year 1909. However, even in 1958 the theoretical flaws in the hypothesis were observed, aside from the evidence that was coming in at the same time from the first gamma ray measurement satellites. For one, the hypothesis did not explain how an antimatter meteor could have managed to survive that low into the Earth's atmosphere, without being annihilated as soon as it encountered terrene matter at the upper levels.<ref name=TIME1958a/><ref name=Steel2008/>
Papaelias suggested a number of mechanisms that prevent the antimatter meteor of being evaporated below the height of 300 kilometers and consequently to reach the ground. Similar mechanisms (e.g. aircap formation) apply to ordinary matter meteorites which survive of evaporation and can be collected from the ground.

=== Ball lightning ===
In 1971, fragments of antimatter comets or meteoroids were hypothesized, by David E. T. F. Ashby of [[Culham Laboratory]] and Colin Whitehead of the U.K. [[Atomic Energy Research Establishment]], as a possible cause for [[ball lightning]] {{harv|Ashby|Whitehead|1971}}. They monitored the sky with gamma-ray detection apparatus, and reported unusually high numbers at 511 keV (kilo-[[electron volt]]s) which is the characteristic gamma ray frequency of a collision between an [[electron]] and a [[positron]]. There were natural explanations for such readings. In particular positrons can be produced indirectly by the action of a thunderstorm, as it creates the unstable isotopes nitrogen-13 and oxygen-15. However, Ashby and Whitehead noted that there were no thunderstorms present at the times that the gamma-ray readings were observed. They instead presented the hypothesis of antimatter meteors as an interesting one that did explain all of what their observations had recorded, and suggested that it merited further investigation.<ref name=NS1971a/><ref name=Charman1972a/>

Ashby and Whitehead's hypothesis, which Dr. Neil Charman (lecturer at the
University of Manchester Institute of Science and Technology) in his 1972
roundup of the several hypothetical explanations of ball lightning characterized
as one of the more bizarre explanations, was based upon the (unproven)
supposition that there was a potential barrier between antimatter and normal
matter. This barrier allowed micrometeoroid and meteoroid fragments that entered
the Earth's atmosphere from space to survive for comparatively lengthy periods,
because the terrene atmospheric molecules would not always possess enough energy
to overcome the barrier and annihilate the antimatter fragments. antimatter
atoms in micrometeors would instead become negatively charged antimatter
ions, as a result of positrons being stripped from them by the photoelectric
effect (and also as a result of secondary effects from annihilation of matter
around them). These negatively charged antimatter ions would be electrically
attracted to the ground in stormy weather, and, gaining enough kinetic energy to
finally overcome the (supposed) repulsive barrier would finally annihilate with
terrene matter to form what is observed as ball lightning.<ref name=Papaelias1990/><ref name=Papaelias1984/> To explain the
survival of an antimatter meteor during its infall flight, Papaelias and Apostolakis introduced
the formula σannihilation=σelastic Π fi, where 0<fi≤1.<ref name=Papaelias1990/> All mechanisms that
may reduce the annihilation cross section σannihilation are involved in the factors fi of
the product Π, including the repulsive potential described above. Several
studies, by using various methods to determine a possible barrier between
ordinary matter and antimatter resulted into controversial discussions, since
the decade of 60s.

Experiments at the LEAR (acronym of Low Energy Accumulating Ring) at CERN have shown a surprising result for a fraction (3%) of antiprotons annihilated by protons of He3 nuclei. The annihilation process is retarded for as much as 100,000,000 times (Eades et al., 1993) than the value derived by theoretical calculations of Enrico Fermi and Edward Teller (1947), the two amongst the greatest figures of modern physics. Thus, the formula described above (σannihilation=σelastic Π fi), which yields in a similar range of magnitude (10−5), for the time duration of the annihilation of antiatoms by atoms, is undoubtedly correct and fairly predicted the high decrease of the rate of annihilation of atoms by antiatoms. This surprising result attracted much attention of the scientific community.{{Citation needed|date=February 2013}}

Antimatter is being produced on the surface of the sun during flares.{{Citation needed|date=February 2013}} The amount of antiprotons that are being produced in only one single flare is so high that could cover the energy consumption of United States of America for a time period of two weeks.{{Citation needed|date=February 2013}} What is important here is that the antiprotons are not being annihilated instantly as expected.{{Citation needed|date=February 2013}} Those antiprotons produced in flares are annihilated far away from the point where they had been produced.{{Citation needed|date=February 2013}} The long distance they travel escaping annihilation is showing a smaller annihilation cross section, which is in accordance with the Papaelias and Apostolakis' formula.<ref name=Papaelias1990/>

In a series of papers, Philip M. Papaelias described how an antimatter meteor
can produce the ball lightning phenomena.<ref name="Papaelias2006b"/><ref name=Papaelias2006b/><ref name=Papaelias2006c/> He also suggested that an
antimatter meteor is continually heated when the annihilation products are
passing through it. Depending on the dimensions of it, those particles depose
part or all of their energy, increasing in that way its temperature. As a
result, the residual mass of the antimatter meteor would reach its melting point
and consequently liquid drops would start to escape from the parental object.
Papaelias calculated the energy absorbed and studied melting processes for 10,
100 or 1000 individual drops, each one glowing separately.<ref name=Papaelias1984/><ref name=Papaelias2010/> He also derived a formula about the mean radious r of the antimatter liquid drops.<ref name=Papaelias1984/><ref name=Papaelias2010/> Twenty two
years later, a ball lightning of about 10–15 m in diameter appeared over
Alexanderplatz, a central location of Berlin, Germany. This bright object, which
was hovering for about 10 minutes at a height which was estimated between 400
and 250 m slowly evolved into a luminous sphere of about 1 m in diameter and was
emitting intense light, maybe equivalent to 10-25 kW of sodium street lighting
lamps. The phenomenon was captured by two web cameras and is showing the ball
lightning slowly changing shape and color. It was observed by Wilfried Heil and
Noemi Zudor, shortly before a thunderstorm on July 29, 2006 at 3:10 AM. The
luminus object was gradually divided into 100 independently glowing smaller
objects that were seen as escaping from the central one.<ref name=Heil2006/>
A remarkable agreement between theoretical prediction and observational confirmation, similar to that of [[Paul Dirac]] who predicted the existence of antimatter in 1928 and four years later [[Carl David Anderson|Carl Anderson]] captured in a photograph the trajectory of a [[positron]] inside a cloud chamber.

=== Phenomena that might be explained by antimatter meteors ===

Additional phenomena that resist explanation for their mysterious properties are puzzling humanity for centuries and can be well clarified under the hypothesis of a slow rate annihilation of an antimatter meteor by atmospheric molecules. One of them is the [[Star of Bethlehem|Christmas star]] described by the [[Evangelist Matthew]] in his [[Gospel]]. Papaelias suggested <ref name=Papaelias2006c/> that an antimatter meteor which had been transformed into a ball lightning after the entrance in the atmosphere can be a reasonable explanation, since hypotheses such as a comet, a straight line configuration of two or three planets, a [[supernova]] explosion or any other known celestial body can not designate any place on Earth and can not stay over the place where Jesus was. The city and the time of the Jesus' birth were known to the three wise men from [[Daniel]] and his [[seventy weeks]] prophecy. What remained to be found was the exact location in the city, which had been revealed to the magi by the movement of the ball lightning.

Another phenomenon that resists explanation for almost half a century is the mystery of the [[Dyatlov Pass incident]]. In 1959, in an excursion at the northern Ural mountains, nine skiers most of which were students or graduates of the Ural Polytechnical University were found dead, with some of them having unusual injuries. The skiers escaped barefoot of their tent, where the outside temperature was -30 degrees of the Celsius scale. A small antimatter meteor that caused their panic was probably the compelling natural force that had been suggested by the Soviet investigators. Its mass can be found by using the formula about the annihilation of an antimatter meteor <ref name=Papaelias1993>{{harvnb|Papaelias|1993|pp=41–46}}</ref> and simple calculations show that it was around a picogram, well within the size of most meteors that may enter in the atmosphere. The high doses of radioactive contamination on clothes of a few victims are strongly supporting the hypothesis of antimatter meteor.

=== Gamma-ray bursts ===
Antimatter comets thought to exist in the [[Oort cloud]] were in the 1990s hypothesized as one possible explanation for [[gamma ray burst]]s.<ref name=Dermer1996/> These bursts can be explained by the [[annihilation]] of matter and antimatter microcomets. The explosion would create powerful gamma ray bursts and accelerate matter to near light speeds.<ref name=Dermer1996/> These antimatter microcomets are thought to reside at distances of more than 1000 [[astronomical units|AU]].<ref name=Dermer1996/> Calculations have shown that comets of around 1 km in radius would shrink by 100 cm if they passed the sun with a perihelion of 1 AU. Microcomets, due to the stresses of solar heating, shatter and burn up much more quickly because the forces are more concentrated within their small masses. Antimatter microcomets would burn up even more rapidly because the annihilation of solar wind with the surface of the microcomet would produce additional heat.<ref name=Dermer1996/> As more gamma-ray bursts were detected in subsequent years, this theory failed to explain the observed distribution of gamma-ray bursts about host galaxies and detections of [[Χ-ray]] lines associated with gamma-ray bursts. The discovery of a [[supernova]] associated with a gamma-ray burst in 2002 provided compelling evidence that massive [[star]]s are the origin of gamma-ray bursts.<ref name=Bloom2002/> Since 2002, more supernovae have been observed to be associated with gamma-ray bursts, and massive stars as the origin of gamma-ray bursts has been firmly established.

== Footnotes ==
{{reflist|group=lower-alpha}}

== References ==
{{reflist|4|refs=

<ref name=Bagnall1991>{{harvnb|Bagnall|1991|pp=124}}</ref>
<ref name=Beech1988>{{harvnb|Beech|1988|pp=215}}</ref>
<ref name=Bloom2002>{{harvnb|Bloom|Kulkarni|Price|Reichart|2002|pp=L45}}</ref>
<ref name=Bullough1995>{{harvnb|Bullough|1995|pp=1533–1551}}</ref>
<ref name=Charman1972a>{{harvnb|Charman|1972|pp=634}}</ref>
<ref name=Dermer1996>{{harvnb|Dermer|1996}}</ref>
<ref name=Heil2006>{{harvnb|Heil|2006}}</ref>
<ref name=NS1971a>{{harvnb|NS|1971a|pp=661}}</ref>
<ref name=NS1974a>{{harvnb|NS|1974a|pp=55}}</ref>
<ref name=Papaelias1984>{{harvnb|Papaelias|1984}}</ref>
<ref name=Papaelias1987>{{harvnb|Papaelias|1987|pp=13}}</ref>
<ref name=Papaelias1990>{{harvnb|Papaelias and Apostolakis|1990|pp=1–13}}</ref>
<ref name=Papaelias1991a>{{harvnb|Papaelias|1991a|pp=105–111}}</ref>
<ref name=Papaelias1991b>{{harvnb|Papaelias|1991b|pp=215–222}}</ref>
<ref name=Papaelias1993>{{harvnb|Papaelias|1993|pp=41–46}}</ref>
<ref name=Papaelias1994>{{harvnb|Papaelias|1994|pp=71–77}}</ref>
<ref name="Papaelias2006b">{{harvnb|Papaelias|2006|}}</ref>
<ref name=Papaelias2006b>{{harvnb|Papaelias|2006|}}</ref>
<ref name=Papaelias2006c>{{harvnb|Papaelias|2006|}}</ref>
<ref name=Papaelias2010>{{harvnb|Papaelias|2010}}</ref>
<ref name=Rojansky1940a>{{harvnb|Rojansky|1940|pp=257}}</ref>
<ref name=Rojansky1940b>{{harvnb|Rojansky|1940|pp=258}}</ref>
<ref name=Rojansky1940c>{{harvnb|Rojansky|1940|pp=259–260}}</ref>
<ref name=Rojansky1973a>{{harvnb|Rojansky|1973|pp=1591}}</ref>
<ref name=Shaw1995>{{harvnb|Shaw|1995|p=436}}</ref>
<ref name=Steel2008>{{harvnb|Steel|2008|}}</ref>
<ref name=Steigman1976a>{{harvnb|Steigman|1976|pp=342}}</ref>
<ref name=Steigman1976b>{{harvnb|Steigman|1976|pp=342–344}}</ref>
<ref name=Steigman1976c>{{harvnb|Steigman|1976|pp=355}}</ref>

<ref name=TIME1958a>{{harvnb|TIME|1958a|}}</ref>
<ref name=Vand1965>{{harvnb|Vand|1965|pp=57}}</ref>

}}

=== Bibliography ===
{{refbegin}}
* {{cite book|ref=harv|title=The Meteorite & tektite collector's handbook: a practical guide to their acquisition, preservation and display|first=Philip M.|last=Bagnall|publisher=Willmann-Bell|year=1991|isbn=9780943396316}}
* {{cite journal|ref=harv|title=A note on antimatter meteors|first=Martin|last=Beech|journal=Earth, Moon, and Planets|issn=0167-9295|volume=40|issue=2|date=February 1988|pages=213–216|doi=10.1007/BF00056024|bibcode=1988EM&P...40..213B}}
* {{cite journal|ref=harv|title=Detection of a Supernova Signature Associated with GRB 011121|first1=J. S.|last1=Bloom|first2=S. R.|last2=Kulkarni|first3=P. A.|last3=Price|first4=D.|last4=Reichart|first5=T. J.|last5=Galama|first6=B. P.|last6=Schmidt|first7=D. A.|last7=Frail|first8=E.|last8=Berger|first9=P. J.|last9=McCarthy| displayauthors = 8|journal=The Astrophysical Journal Letters|volume=572|issue=1|doi=10.1086/341551|date=2002-06-10|bibcode=2002ApJ...572L..45B|arxiv = astro-ph/0203391|pages=L45 }}
* {{cite journal|ref=harv|title=Interactions of antimatter with the atmosphere|first=Ken|last=Bullough|journal=Journal of Atmospheric and Terrestrial Physics|date=November 1995|volume=13|issue=57|pages=1533–1551|doi=10.1016/0021-9169(94)00135-B}}
* {{cite journal|ref=harv|date=1972-12-14|volume=61|issue=880|issn=0262-4079|title=The enigma of ball lightning|first=Neil|last=Charman|magazine=[[New Scientist]]|pages=632–635}}
* {{cite conference|ref=harv|title=Gamma-Ray Bursts from Comet-Antimatter Comet Collisions in the Oort Cloud|url=http://link.aip.org./link/?APCPCS/384/744/1|last=Dermer|first=Charles D.|year=1996|volume=384|issue=1|pages=744–748|doi=10.1063/1.51650|conference=The Third Huntsville Symposium on Gamma-Ray Bursts, Huntsville AL, USA, October 25–27, 1995|editor=C. Kouveliotou, M. S. Briggs, & G. J. Fishman|location=Woodbury|publisher=[[American Institute of Physics]]|isbn=9781563966859}}
* {{cite conference|ref=harv|title=Ball Lightning over Berlin|first=Wilfried|last=Heil|date=August 2006|conference=Proceedings Ninth International Symposium on Ball Lightning, ISBL – 06, 16–19 August 2006, Eindhoven, The Netherlands|editor= Ed. G. C. Dijkhuis}}
* {{cite journal|ref=CITEREFNS1971a|year=1971a|date=1971-03-25|volume=49|issue=744|issn=0262-4079|title=Are antimatter meteorites optical illusions?|magazine=[[New Scientist]]}}
* {{cite journal|ref=CITEREFNS1974a|year=1974a|date=1974-01-10|volume=61|issue=880|issn=0262-4079|title=Is Kohoutek an anti-comet?|magazine=[[New Scientist]]}}
* {{cite book|ref=harv|title=Study of the atom-antiatom annihilation cross section and of the behavior of antimatter meteors in the Earth’s atmosphere|first=Philip M.|last=Papaelias|publisher=Ph. D. Thesis|year=1984|pages=128–130|location=Athens,Greece}}
* {{cite journal|ref=harv|title=Velocity-height relation for antimatter meteors|first=Philip|last=Papaelias|journal=Earth, Moon, and Planets|issn=0167-9295|volume=38|date=May 1987|pages=13–20|doi=10.1007/BF00115933|bibcode=1987EM&P...38...13P}}
* {{cite journal|ref=harv|title=Mass-height relation for antimatter meteors|last1=Papaelias|first1=P. M.|last2=Apostolakis|first2=A.|journal=Earth, Moon, and Planets|issn=0167-9295|volume=49|date=April 1990|pages=1–13|doi=10.1007/BF00053994|bibcode=1990EM&P...49....1P}}
* {{cite journal|ref=harv|title=Momentum loss for antimatter meteors|first=Philip|last=Papaelias|journal=Earth, Moon, and Planets|issn=0167-9295|volume=52|date=February 1991a|pages=105–111|doi=10.1007/BF00054177|bibcode=1991EM&P...52..105P}}
* {{cite journal|ref=harv|title=Energy deposition in antimatter meteors
|first=Philip|last=Papaelias|journal=Earth, Moon, and Planets|issn=0167-9295|volume=55|date=December 1991b|pages=215–222|doi=10.1007/BF00054616|bibcode=1991EM&P...55..215P}}
* {{cite journal|ref=harv|title= Annihilation of antimatter meteors
|first=Philip|last=Papaelias|journal=Earth, Moon, and Planets|issn=0167-9295|volume=60|date=January 1993|pages=41–46|doi=10.1007/BF00612179|bibcode=1993EM&P...60...41P}}
* {{cite journal|ref=harv|title= Lifetime of antimatter meteors|first=Philip|last=Papaelias|journal=Earth, Moon, and Planets|issn=0167-9295|volume=65|date=January 1994|pages=71–77|doi=10.1007/BF00572200|bibcode=1994EM&P...65...71P}}
* {{cite conference|ref=harv|title=Does the annihilation of antimatter meteors produce the phenomena of ball lightning?|first=Philip M.|last=Papaelias|date=August 2006|conference=Proceedings Ninth International Symposium on Ball Lightning, ISBL – 06, 16–19 August 2006, Eindhoven, The Netherlands|editor=Ed. G. C. Dijkhuis}}
* {{cite conference|ref=harv|title=Photographs of ball lightning support evidence of antimatter|first=Philip M.|last=Papaelias|date=August 2006|conference=Proceedings Ninth International Symposium on Ball Lightning, ISBL – 06, 16–19 August 2006, Eindhoven, The Netherlands|editor=Ed. G. C. Dijkhuis}}
* {{cite conference|ref=harv|title=Phenomena of ball lightning in history|first=Philip M.|last=Papaelias|date=August 2006|conference=Proceedings Ninth International Symposium on Ball Lightning, ISBL – 06, 16–19 August 2006, Eindhoven, The Netherlands|editor= Ed. G. C. Dijkhuis}}
* {{cite conference|ref=harv|title=Ball Lightning observations supporting antimatter meteors|first=Philip M.|last=Papaelias|date=October 2010|conference=Actes du XIX Congrès Cosmos et Philosophie|pages=61–65|editor=L' Academie d' Athènes et l' Université d' Athènes|location=Athènes,Grèce}}
* {{cite journal|ref=harv|title=The Hypothesis of the Existence of Contraterrene Matter|first=Vladimir|last=Rojansky|authorlink=Vladimir Rojansky|journal=Astrophysical Journal|date=March 1940|volume=91|pages=257–260|bibcode=1940ApJ....91..257R|doi=10.1086/144161}}
* {{cite journal|ref=harv|title=Comet Kohoutek and Penetrating Rays|first=Vladimir|last=Rojansky|authorlink=Vladimir Rojansky|journal=[[Physical Review Letters]]|volume=31|issue=27|pages=1591|date=December 1973|doi=10.1103/PhysRevLett.31.1591|bibcode=1973PhRvL..31.1591R}}
* {{cite book|ref=harv|title=Craters, Cosmos, and Chronicles: A New Theory of Earth|first=Herbert R.|last=Shaw|publisher=Stanford University Press|year=1994|page=436}}
* {{cite journal|ref=harv|date=2008-06-25|journal=[[Nature (journal)|Nature]]|volume=453|pages=1157–1159|doi=10.1038/4531157a|title=Planetary science: Tunguska at 100|first=Duncan|last=Steel|url=http://www.nature.com./news/2008/080625/full/4531157a.html|issue=7199}}
* {{cite journal|ref=harv|title=Observational Tests of Antimatter Cosmologies|journal=Annual Review of Astronomy and Astrophysics|volume=14|pages=339–372|date=September 1976|first=Gary|last=Steigman|doi=10.1146/annurev.aa.14.090176.002011|bibcode=1976ARA&A..14..339S}}
* {{cite book|ref=harv|title=Ball lightning: an unsolved problem in atmospheric physics|first=Mark|last=Stenhoff|publisher=Springer|year=1999|isbn=9780306461507}}
* {{cite journal|ref=CITEREFTIME1958a|magazine=[[Time (magazine)|Time]]|title=Science: Anti-Meteor?|year=1958a|date=1958-05-05|url=http://time.com./time/magazine/article/0,9171,863347,00.html}}
* {{cite book|ref=harv|first=Vladimir|last=Vand|chapter=Astrogeology: The origin of tektites|title=Advances in Geophysics|volume=1–5|editor=H. E. Landsberg|publisher=Academic Press|year=1965|isbn=9780120188116}}
{{refend}}

== Further reading ==

=== Original publications of the various hypotheses ===
* {{cite journal|ref=harv|last=Khan|first=Mohammad Abdur Rahman|year=1947|title=Contraterrene meteorite impact theory of tektite formation|journal=Contrib. Meteorit. Soc.|volume=4|issue=35}}
* {{cite journal|ref=harv|journal=[[Nature (journal)|Nature]]|volume=230|pages=180–182|date=1971-03-19|doi=10.1038/230180a0|title=Is Ball Lightning caused by Antimatter Meteorites?|first1=David E. T. F.|last=Ashby|first2=Colin|last2=Whitehead|bibcode = 1971Natur.230..180A|issue=5290}}
* {{cite journal|ref=harv|journal=[[Nature (journal)|Nature]]|volume=181|pages=1194|date=1958-04-26|doi=10.1038/1811194b0|title=Possible Existence of Anti-Matter in Bulk|first=Philip J.|last=Wyatt|bibcode = 1958Natur.181.1194W|issue=4617}}
* {{cite journal|ref=harv|journal=[[Nature (journal)|Nature]]|volume=206|pages=861–865|date=1965-05-29|doi=10.1038/206861a0|title=Possible Anti-Matter Content of the Tunguska Meteor of 1908|first1=Clyde|last1=Cowan|first2=C. R.|last2=Atluri|first3=Willard F.|last3=Libby|authorlink1=Clyde Cowan|authorlink3=Willard Libby|bibcode = 1965Natur.206..861C|issue=4987}}

=== Other ===
* {{cite journal|ref=harv|language=Russian|journal=Kosmicheskie Issledovaniya (USSR)|volume=4|issue=1|pages=66–73|date=January/February 1966|last1=Konstantinov|first1=B. P.|last2=Bredov|first2=M. M.|last3=Belyaevskii|first3=A. I.|last4=Sokolov|first4=I. A.|title=O vozmozhnoǐ antiveshchestvennoǐ priode mikrometeorov}}
** ''English translation:'' {{cite journal|ref=harv|last1=Konstantinov|first1=B. P.|last2=Bredov|first2=M. M.|last3=Belyaevskii|first3=A. I.|last4=Sokolov|first4=I. A.|journal=Cosmic Research|volume=4|pages=58|date=January 1966|bibcode=1966CosRe...4...58K|title=The Possible Antimatter Nature of Micrometeorites}}
* {{cite conference|ref=harv|title=The matter-antimatter collision hypothesis for the origin of cosmic gamma-ray bursts|first1=Sabatino|last1=Sofia|first2=H. M.|last2=van Horn|conference=AAS, American Physical Society, and New York Academy of Sciences, Texas Symposium on Relativistic Astrophysics, 7th, Dallas, Tex., Dec. 16–20, 1974|series=Annals of the New York Academy of Sciences|volume=262|date=1975-10-15|pages=197–208|doi=10.1111/j.1749-6632.1975.tb31432.x|bibcode=1975NYASA.262..197S}}
* {{cite journal|ref=harv|journal=Astrophysics and Space Science|year=1976|volume=39|issue=1|pages=L7–L11|doi=10.1007/BF00640527|title=Local gamma ray events as tests of the antimatter theory of gamma ray bursts (Letter to the Editor)|first1=Sabatino|last1=Sofia|first2=Robert E.|last2=Wilson|bibcode=1976Ap&SS..39L...7S}}

{{Comets}}

{{DEFAULTSORT:Antimatter Comet and Meteor}}
[[Category:Antimatter]]
[[Category:Comets]]
[[Category:Meteoroids]]
[[Category:Hypothetical astronomical objects]]