formulasearchengine - User contributions [en]

Charlier polynomials

2014-01-05T16:56:21Z

217.184.64.130: /* References */ putting in the umlauts.

In [[mathematics]], '''Racah polynomials''' are [[orthogonal polynomials]] named after [[Giulio Racah]], as their orthogonality relations are equivalent to his orthogonality relations for [[Racah coefficient]]s.

The Racah polynomials were first defined by {{harvtxt|Wilson|1978}} and are given by
:<math>p_n(x(x+\gamma+\delta+1)) = {}_4F_3\left[\begin{matrix} -n &n+\alpha+\beta+1&-x&x+\gamma+\delta+1\\
\alpha+1&\gamma+1&\beta+\delta+1\\ \end{matrix};1\right].</math>

{{harvtxt|Askey|Wilson|1979}} introduced the ''q''-Racah polynomials defined in terms of [[basic hypergeometric function]]s by
:<math>p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\
aq&bdq&cq\\ \end{matrix};q;q\right].</math>
They are sometimes given with changes of variables as
:<math>W_n(x;a,b,c,N;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&cq^{x-n}\\
aq&bcq&q^{-N}\\ \end{matrix};q;q\right].</math>

==References==
*{{Citation | last1=Askey | first1=Richard | last2=Wilson | first2=James | title=A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols | doi=10.1137/0510092 | id={{MathSciNet | id = 541097}} | year=1979 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=10 | issue=5 | pages=1008–1016}}
*{{citation|first=J.|last= Wilson|title= Hypergeometric series recurrence relations and some new orthogonal functions|series= Ph.D. thesis|publisher= Univ. Wisconsin, Madison|year= 1978}}
[[Category:Orthogonal polynomials]]

{{algebra-stub}}

Fox H-function

2013-04-01T06:32:24Z

217.184.226.204: ***.

In mathematics, a '''±1–sequence''' is a [[sequence]] of numbers, each of which is either 1 or −1. One example is the sequence {{math|(''x''1, ''x''2, ''x''3, ...)}}, where {{math|1=''x''''i'' = (−1)''i''+1}}.

Such sequences are commonly studied in [[discrepancy theory]].

==Erdős discrepancy problem==


Let ''S''=(''x''1, ''x''2, ''x''3,...) be a ±1–sequence, where ''x''''j'' denotes the ''j''th term. The ''Erdős discrepancy problem'' asks whether there exists a sequence ''S'' and an integer ''C''''S'', such that for any two positive integers ''d'' and ''k'',
: <math> \left| \sum_{i=1}^k x_{id} \right| \leq C_S.</math>
{{Asof|2010|October}}, this problem is currently being studied by the [[Polymath Project]].

==Barker codes==
{{main|Barker code}}

A '''Barker code''' is a sequence of ''N'' values of +1 and −1,
:<math>a_j</math> for j = 1, 2, …, N
such that
:<math>\left|\sum_{j=1}^{N-v} a_j a_{j+v}\right| \le 1\,</math>
for all <math>1 \le v < N</math>.<ref>{{cite book |author=Barker, R. H. |chapter=Group Synchronizing of Binary Digital Sequences |title=Communication Theory |place=London |publisher=Butterworth |pages=273–287 |year=1953}}</ref>

Barker codes of length 11 and 13 are used in [[direct-sequence spread spectrum]] and [[pulse compression|pulse compression radar]] systems because of their low [[autocorrelation]] properties.

==See also==
*[[Binary sequence]]
*[[Discrepancy of hypergraphs]]

==Notes==
{{reflist}}

==References==
*{{cite book|title=The Discrepancy Method: Randomness and Complexity|last=Chazelle|first=Bernard|author-link=Bernard Chazelle|publisher=Cambridge University Press|ISBN=0-521-77093-9}}

==External links==
*[http://michaelnielsen.org/polymath1/index.php?title=The_Erd%C5%91s_discrepancy_problem The Erdős discrepancy problem] – Polymath Project

{{DEFAULTSORT:±1-Sequence}}
[[Category:Binary sequences]]

Eulerian number

2012-06-12T22:54:22Z

217.184.105.77: /* Eulerian numbers of the second kind */ *** hyphen to minus.

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Appell series

2012-06-10T15:01:48Z

217.184.241.248: /* References */ *** correction.

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Humbert series

2012-06-10T13:39:24Z

217.184.241.156: /* Related series */ *** formatting.

[[File:Dust on the a solar panel.jpg|right|thumb|Dust often accumulates on the glass of solar panels - seen here as black dots - which reduces the amount of light available to the panel.]]
'''Solar cell efficiency''' is the ratio of the electrical output of a solar cell to the incident energy in the form of sunlight. The [[energy conversion efficiency]] (η) of a [[solar cell]] is the percentage of the [[solar energy]] to which the cell is exposed that is converted into [[Electricity|electrical energy]].<ref name="Energy Efficiency and Renewable Energy">{{cite web|title=Photovoltaic Cell Conversion Efficiency|url=http://www.eere.energy.gov/basics/renewable_energy/pv_cell_conversion_efficiency.html|publisher=U.S. Department of Energy|accessdate=19 May 2012}}</ref> This is calculated by dividing a cell's power output (in [[watt]]s) at its maximum power point (Pm) by the [[irradiance|input light]] (''E'', in W/m2) and the [[surface area]] of the solar cell (''Ac'' in m2).

:<math>\eta = \frac{P_{m}}{E \times A_c}</math>

By convention, solar cell efficiencies are measured under standard test conditions (STC) unless stated otherwise. STC specifies a temperature of 25 °C and an irradiance of 1000 W/m2 with an air mass 1.5 (AM1.5) spectrum. These conditions correspond to a clear day with sunlight incident upon a sun-facing 37°-tilted surface with the sun at an angle of 41.81° above the horizon.<ref>ASTM G 173-03, "Standard Tables for Reference Solar Spectral Irradiances: Direct Normal and Hemispherical on 37° Tilted Surface," ASTM International, 2003.</ref><ref>{{Cite web|title = Solar Spectral Irradiance: Air Mass 1.5|publisher = National Renewable Energy Laboratory|url = http://rredc.nrel.gov/solar/spectra/am1.5/|accessdate=2007-12-12}}</ref> This represents solar noon near the spring and autumn equinoxes in the continental United States with surface of the cell aimed directly at the sun. Under these test conditions a solar cell of 20% efficiency with a 100 cm2 (0.01 m2) surface area would produce 2.0 watts of power.

The efficiency of the solar cells used in a [[photovoltaic system]], in combination with latitude and climate, determines the annual energy output of the system. For example, a solar panel with 20% efficiency and an area of 1 m² will produce 200 watts of power at STC, but it can produce more when the sun is high in the sky and will produce less in cloudy conditions and when the sun is low in the sky. In central Colorado, which receives daily insolation of 2200 Wh/m²,<ref name="nrel.gov">Photovoltaic Solar Resource of the United States, http://www.nrel.gov/gis/images/map_pv_national_lo-res.jpg</ref> such a panel can be expected to produce 440 kWh of energy per year. However, in Michigan, which receives only 1400 kWh/m²/yr,<ref name="nrel.gov"/> annual energy yield will drop to 280 kWh for the same panel. At more northerly European latitudes, yields are significantly lower: 175kWh annual energy yield in southern England.<ref>Solar photovoltaics: data from a 25-m2 array in Cambridgeshire in 2006, http://www.inference.phy.cam.ac.uk/withouthotair/c6/page_40.shtml</ref>

Several factors affect a cell's conversion efficiency value, including its [[Reflectivity#Reflectance|reflectance]] efficiency, [[thermodynamic efficiency limit|thermodynamic efficiency]], [[Theory of solar cells#Charge carrier separation|charge carrier separation]] efficiency, and [[Thermal conduction|conduction]] efficiency values.<ref name="Energy Efficiency and Renewable Energy"/> Because these parameters can be difficult to measure directly, other parameters are measured instead, including [[quantum efficiency]], [[Open-circuit voltage|VOC]] ratio, and [[Solar cell#Efficiency|fill factor]]. Reflectance losses are accounted for by the quantum efficiency value, as they affect "external quantum efficiency." Recombination losses are accounted for by the quantum efficiency, VOC ratio, and fill factor values. Resistive losses are predominantly accounted for by the fill factor value, but also contribute to the quantum efficiency and VOC ratio values.

In {{currentyear}}, the highest efficiencies have been achieved by using multiple junction cells at high solar concentrations (44.7% by The Fraunhofer Institute for Solar Energy Systems ISE, Soitec, CEA-Leti and the Helmholtz Center Berlin <ref>http://www.sciencedaily.com/releases/2013/09/130923204214.htm</ref> )

==Factors affecting energy conversion efficiency==

===Thermodynamic efficiency limit===
{{Main|Thermodynamic efficiency limit}}
[[File:ShockleyQueisserFullCurve.svg|thumb|The [[Shockley-Queisser limit]] for the efficiency of a single-junction solar cell under unconcentrated sunlight. This calculated curve uses actual solar spectrum data, and therefore the curve is wiggly from IR absorption bands in the atmosphere. This efficiency limit of ~34% can be exceeded by [[multijunction solar cell]]s.]]
The maximum theoretically possible conversion efficiency for sunlight is 86% due to the entropy of the photons emitted by the sun's surface.<ref>{{Cite journal|title=On the Thermodynamic Limit of Photovoltaic Energy Conversion| author=A. De Vos and H. Pauwels|doi=10.1007/BF00901283|journal=Appl. Phys.|volume=25|page=119|year=1981 }}</ref>

However, solar cells operate as quantum energy conversion devices, and are therefore subject to the "thermodynamic efficiency limit". Photons with an energy below the band gap of the absorber material cannot generate a hole-electron pair, and so their energy is not converted to useful output and only generates heat if absorbed. For photons with an energy above the band gap energy, only a fraction of the energy above the band gap can be converted to useful output. When a photon of greater energy is absorbed, the excess energy above the band gap is converted to kinetic energy of the carrier combination. The excess kinetic energy is converted to heat through phonon interactions as the kinetic energy of the carriers slows to equilibrium velocity.

Solar cells with multiple band gap absorber materials improve efficiency by dividing the solar spectrum into smaller bins where the thermodynamic efficiency limit is higher for each bin.<ref>{{Cite journal|title=Limiting efficiencies for multiple energy-gap quantum devices| author=Cheng-Hsiao Wu and Richard Williams|doi=10.1063/1.331859|journal=J. Appl. Phys.|volume=54|page=6721 |year=1983|bibcode = 1983JAP....54.6721W }}</ref>

===Quantum efficiency===
{{Main|Quantum efficiency}}
As described above, when a photon is absorbed by a solar cell it can produce an electron-hole pair. One of the carriers may reach the p-n junction and contribute to the current produced by the solar cell; such a carrier is said to be ''collected''. Or, the carriers [[Carrier generation and recombination|recombine]] with no net contribution to cell current.

Quantum efficiency refers to the percentage of photons that are converted to electric current (i.e., collected carriers) when the cell is operated under short circuit conditions. The "external" quantum efficiency of a silicon solar cell includes the effect of optical losses such as transmission and reflection. If some of these losses can be recaptured by other portions of the solar cell array (for example via oblique angles of incidence) the aggregate external quantum
efficiency of the system may be increased despite a lower internal quantum efficiency. However, it is often useful to look at the quantum efficiency of the light left after the reflected and transmitted light has been lost. "Internal" quantum efficiency refers to the efficiency with which photons that are not reflected or transmitted out of the cell can generate collectable carriers

Quantum efficiency is most usefully expressed as a ''spectral'' measurement (that is, as a function of photon wavelength or energy). Since some wavelengths are absorbed more effectively than others, spectral measurements of quantum efficiency can yield valuable information about the quality of the semiconductor bulk and surfaces. Quantum efficiency alone is not the same as overall energy conversion efficiency, as it does not convey information about the fraction of power that is converted by the solar cell.

===Maximum power point===
A solar cell may operate over a wide range of [[voltage]]s (V) and [[current (electricity)|currents]] (I). By increasing the resistive load on an irradiated cell continuously from zero (a ''[[short circuit]]'') to a very high value (an ''open circuit'') one can determine the [[maximum power theorem|maximum power]] point, the point that maximizes V×I; that is, the load for which the cell can deliver maximum electrical power at that level of irradiation. (The output power is zero in both the short circuit and open circuit extremes).

A high quality, monocrystalline silicon solar cell, at 25 °C cell temperature, may produce 0.60 volts open-circuit (''VOC''). The cell temperature in full sunlight, even with 25 °C air temperature, will probably be close to 45 °C, reducing the open-circuit voltage to 0.55 volts per cell. The voltage drops modestly, with this type of cell, until the short-circuit current is approached (''ISC''). Maximum power (with 45 °C cell temperature) is typically produced with 75% to 80% of the open-circuit voltage (0.43 volts in this case) and 90% of the short-circuit current. This output can be up to 70% of the ''VOC x ISC'' product. The short-circuit current (''ISC'') from a cell is nearly proportional to the illumination, while the open-circuit voltage (''VOC'') may drop only 10% with an 80% drop in illumination. Lower-quality cells have a more rapid drop in voltage with increasing current and could produce only 1/2 ''VOC'' at 1/2 ''ISC''. The usable power output could thus drop from 70% of the ''VOC x ISC'' product to 50% or even as little as 25%. Vendors who rate their solar cell "power" only as ''VOC x ISC'', without giving load curves, can be seriously distorting their actual performance.

The maximum power point of a [[photovoltaic]] varies with incident illumination. For example, accumulation of dust on photovoltaic panels reduces the maximum power point.<ref>{{Cite journal|author=A. Molki|doi=10.1088/0031-9120/45/5/F03|title=Dust affects solar-cell efficiency|journal=Physics Education|volume=45|pages=456–458|year=2010|bibcode = 2010PhyEd..45..456M }}</ref> For systems large enough to justify the extra expense, a [[maximum power point tracker]] tracks the instantaneous power by continually measuring the voltage and [[Electric current|current]] (and hence, power transfer), and uses this information to dynamically adjust the load so the maximum power is ''always'' transferred, regardless of the variation in lighting.

===Fill factor===
Another defining term in the overall behavior of a solar cell is the ''[[fill factor]]'' (''FF''). This is the available ''power'' at the ''maximum power point'' (''Pm'') divided by the ''open circuit voltage'' (''VOC'') and the ''short circuit current'' (''ISC''):

:<math>FF = \frac{P_{m}}{V_{OC} \times I_{SC}} = \frac{\eta \times A_c \times E}{V_{OC} \times I_{SC}}.</math>

The fill factor is directly affected by the values of the cell's series and shunt resistances. Increasing the shunt resistance (Rsh) and decreasing the series resistance (Rs) lead to a higher fill factor, thus resulting in greater efficiency, and bringing the cell's output power closer to its theoretical maximum.<ref name=nelson>{{Cite book|title = The Physics of Solar Cells|author = Jenny Nelson|publisher = Imperial College Press|year = 2003|isbn = 978-1-86094-340-9|url=http://www.solarfreaks.com/download/file.php?id=281}}</ref>

==Comparison of energy conversion efficiencies==
{{Main|Photovoltaics}}

Energy conversion efficiency is measured by dividing the electrical power produced by the cell by the light power falling on the cell. Many factors influence the electrical power output, including spectral distribution, spatial distribution of power, temperature, and resistive load applied to the cell. [[International Electrotechnical Commission|IEC]] standard 61215 is used to compare the performance of cells and is designed around terrestrial, temperate conditions, using its standard temperature and conditions (STC): [[irradiance]] of 1 kW/m2, a spectral distribution close to solar radiation through AM ([[airmass]]) of 1.5 and a cell temperature 25 °C. The resistive load is varied until the peak or maximum power point (MPP) is achieved. The power at this point is recorded as [[Watt-peak]] (Wp). The same standard is used for measuring the power and efficiency of PV modules,

Air mass has an effect on power output. In space, where there is no atmosphere, the spectrum of the sun is relatively unfiltered. However, on earth, with air filtering the incoming light, the solar spectrum changes. To account for the spectral differences, a system was devised to calculate this filtering effect. Simply, the filtering effect ranges from [[air mass coefficient|Air Mass]] 0 (AM0) in space, to approximately Air Mass 1.5 on Earth. Multiplying the spectral differences by the quantum efficiency of the solar cell in question will yield the efficiency of the device. For example, a silicon solar cell in space might have an efficiency of 14% at AM0, but have an efficiency of 16% on earth at AM 1.5. Terrestrial efficiencies typically are greater than space efficiencies. Note, however, that the
incident photons in space have considerably
more energy, so the solar cell might produce
considerably more power in space, despite the lower efficiency as indicated by reduced percentage of the total incident energy captured.

[[File:PVeff(rev131204)a.jpg|400 px|right|Reported timeline of solar cell energy conversion efficiencies (from National Renewable Energy Laboratory (USA)]]
Solar cell efficiencies vary from 6% for amorphous silicon-based solar cells to 44.0% with multiple-junction production cells and 44.4% with multiple dies assembled into a hybrid package.<ref>{{Cite web
|url = http://cleantechnica.com/2012/10/15/solar-junction-powers-up-cpv-with-new-conversion-efficiency-record/|title = Solar Junction Breaks Its Own CPV Conversion Efficiency Record Read more at http://cleantechnica.com/2012/10/15/solar-junction-powers-up-cpv-with-new-conversion-efficiency-record/#jfRH5o5FL1W8hZC8.99 | accessdate = 2013-12-18 | date = 2013-12-18}}</ref><ref>{{Cite web
| url = http://cleantechnica.com/2013/06/23/solar-cell-efficiency-world-record-set-by-sharp-44-4/
| title = Solar Cell Efficiency World Record Set By Sharp — 44.4% Read more at http://cleantechnica.com/2013/06/23/solar-cell-efficiency-world-record-set-by-sharp-44-4/#Xpme8KCkfHcDZu4E.99
| accessdate = 2013-07-28
| date = 2013-07-28
}}</ref> Solar cell energy conversion efficiencies for commercially available ''multicrystalline Si'' solar cells are around 14-19%.<ref>{{Cite web
|url = http://www.ise.fraunhofer.de/veroeffentlichungen/konferenzbeitraege/2007/22nd-european-photovoltaic-solar-energy-conference-and-exhibition-milano-italy-2007/silicon-solar-cells-with-screen-printed-front-side-metallization-exceeding-19-efficiency
| title = Silicon Solar Cells with Screen-Printed Front Side Metallization Exceeding 19% Efficiency
}}</ref> The highest efficiency cells have not always been the most economical — for example a 30% efficient multijunction cell based on exotic materials such as gallium arsenide or indium selenide and produced in low volume might well cost one hundred times as much as an 8% efficient amorphous silicon cell in mass production, while only delivering about four times the electrical power.

However, there is a way to "boost" solar power. By increasing the light intensity, typically photogenerated carriers are increased, resulting in increased efficiency by up to 15%. These so-called "[[Concentrated photovoltaics|concentrator systems]]" have only begun to become cost-competitive as a result of the development of high efficiency GaAs cells. The increase in intensity is typically accomplished by using concentrating optics. A typical concentrator system may use a light intensity 6-400 times the sun, and increase the efficiency of a one sun GaAs cell from 31% at AM 1.5 to 35%.

A common method used to express economic costs of electricity-generating systems is to calculate a price per delivered [[kilowatt-hour]] (kWh). The solar cell efficiency in combination with the available irradiation has a major influence on the costs, but generally speaking the overall system efficiency is important. Using the commercially available solar cells (as of 2006) and system technology leads to system efficiencies between 5 and 19%. As of 2005, photovoltaic electricity generation costs ranged from ~0.60 US$/kWh (0.50 €/kWh) (central Europe) down to ~0.30 US$/kWh (0.25 €/kWh) in regions of high solar irradiation. Note that installation costs and cost of the installation space may alter US$/kWH. (A solar installation in [[Palm Springs]] California will cost
more per kWH than in the surrounding desert. Additionally, vertical installations may increase economic efficiency by reducing real estate costs even if they decrease the energy efficiency of the underlying solar cells by suboptimal placement and angle to incident light). This electricity is generally fed into the electrical grid on the customer's side of the meter. The cost can be compared to prevailing retail electric pricing (as of 2005), which varied from between 0.04 and 0.50 US$/kWh worldwide. These cost/kWh calculations will vary depending on the assumed useful life of the system. Most c-Si panels are warranted for 25 years and should see 35+ years of useful life.

===Solar cells and energy payback===

The energy payback time is defined as the recovery time required for generating the energy spent for manufacturing a modern photovoltaic module. By some estimates, it is typically from 1 to 4 years<ref name="NRELPV">{{Cite web|format=PDF|url=http://www.nrel.gov/docs/fy05osti/37322.pdf|title=What is the Energy Payback for PV?|accessdate=2008-12-30}}</ref><ref>{{Cite journal|author=M. Ito, K. Kato, K. Komoto, et al.|title=A comparative study on cost and life-cycle analysis for 100 MW very large-scale PV (VLS-PV) systems in deserts using m-Si, a-Si, CdTe, and CIS modules|journal=Progress in Photovoltaics: Research and Applications|volume=16|pages=17–30|year=2008|doi=10.1002/pip.770}}</ref> depending on the module type and location. With a typical lifetime of 20 to 30 years, this means that, if these estimates of manufacturing energy are reliable, modern solar cells would be net energy producers, i.e. they would generate more energy over their lifetime than the energy expended in producing them.<ref name="NRELPV"/><ref>{{Cite web|format=PDF|url=http://alpha.chem.umb.edu/chemistry/ch471/evans%20files/Net_Energy%20solar%20cells.pdf|title=Net Energy Analysis For Sustainable Energy Production From Silicon Based Solar Cells|accessdate=2011-09-13}}</ref><ref>{{Cite journal| title=Can Solar Cells Ever Recapture the Energy Invested in their Manufacture?| first=Richard| last= Corkish| journal=Solar Progress| volume=18| issue= 2| pages= 16–17| year=1997| url=http://www.csudh.edu/oliver/smt310-handouts/solarpan/pvpayback.htm}}</ref> Generally, thin-film technologies - despite having comparatively low conversion efficiencies - achieve significantly shorter energy payback times than conventional systems (often < 1 year).<ref>{{Cite journal|author=K. L. Chopra, P. D. Paulson, and V. Dutta|title=Thin-film solar cells: An overview Progress in Photovoltaics|journal=Research and Applications|volume=12|pages=69–92|year=2004}}</ref>

Crystalline silicon devices are approaching the theoretical limiting efficiency of 29.4%<ref name="Richter2013">{{Cite journal| author=A. Richter, M. Hermle, S.W. Glunz|title = Reassessment of the limiting efficiency for crystalline silicon solar cells|journal = IEEE Journal of Photovoltaics|volume = 3|issue = 4|pages = 1184–1191|date = Oct 2013|doi = 10.1109/JPHOTOV.2013.2270351}}</ref> and achieve an energy payback period of 1–2 years.<ref name="NRELPV"/><ref name="sciencedaily-highest-efficiency">{{Cite web|url=http://www.sciencedaily.com/releases/2008/10/081023100536.htm |title=Highest silicon solar cell efficiency ever reached |date=24 October 2008 |publisher=ScienceDaily |accessdate=9 December 2009}}</ref>

However, solar cell manufacture is dependent on and presupposes the existence of a complex global industrial manufacturing system. This comprises not only the fabrication systems typically accounted for in estimates of manufacturing energy, but the contingent mining, refining and global transportation systems, as well as other energy intensive critical support systems including finance, information, and security systems. The uncertainty of that energy component confers uncertainty on any estimate of payback times derived from that estimate, considered by some to be significant.<ref>Trainer, FE (2007) "Renewable Energy Cannot Sustain a Consumer Society"</ref>

==See also==
{{Portal|Renewable energy|Energy}}
*[[Energy and the environment]]
*[[Efficient energy use|Energy efficiency]]

==References==
{{Reflist|2}}

==External links==
{{Commons category|Photovoltaics}}
{{Commons|solar cell}}
*{{dmoz|Science/Technology/Energy/Renewable/Solar/Solar_Electric/|Solar electric}}

{{Solar energy}}
{{Use dmy dates|date=April 2011}}

[[Category:Solar cells]]
[[Category:Photovoltaics]]

Barnes integral

2012-06-09T04:14:16Z

217.184.105.112: /* References */ *** adding JFM numbers.

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