LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle l_2}

${\displaystyle l_{2}}$

... is translated to the CAS output ...

Semantic latex: l_2

Confidence: 0

Mathematica

Translation: Subscript[l, 2]

Information

Sub Equations

Subscript[l, 2]

Free variables

Subscript[l, 2]

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('{l}_{2}')

Information

Sub Equations

Symbol('{l}_{2}')

Free variables

Symbol('{l}_{2}')

Tests

Symbolic

Numeric

Maple

Translation: l[2]

Information

Sub Equations

l[2]

Free variables

l[2]

Tests

Symbolic

Numeric

Dependency Graph Information

Is part of

${\begin{aligned}&\int Y_{l_{1}m_{1}}(\theta ,\varphi )Y_{l_{2}m_{2}}(\theta ,\varphi )Y_{l_{3}m_{3}}(\theta ,\varphi )\,\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi \\&\quad ={\sqrt {\frac {(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi }}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\0&0&0\end{pmatrix}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\end{aligned}}$

${\begin{aligned}&{-}{\sqrt {(l_{3}\mp s_{3})(l_{3}\pm s_{3}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}&s_{2}&s_{3}\pm 1\end{pmatrix}}=\\&\quad ={\sqrt {(l_{1}\mp s_{1})(l_{1}\pm s_{1}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}\pm 1&s_{2}&s_{3}\end{pmatrix}}+{\sqrt {(l_{2}\mp s_{2})(l_{2}\pm s_{2}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}&s_{2}\pm 1&s_{3}\end{pmatrix}}.\end{aligned}}$

$l_{1}\ll l_{2},l_{3}$

${\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {2l_{3}+1}}}$

${\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {l_{2}+l_{3}+1}}}$

$\cos(\theta )=(m_{2}-m_{3})/(l_{2}+l_{3}+1)$

${\frac {1}{2}}\int _{-1}^{1}P_{l_{1}}(x)P_{l_{2}}(x)P_{l}(x)\,dx={\begin{pmatrix}l&l_{1}&l_{2}\\0&0&0\end{pmatrix}}^{2}$

Description

integral of the product

spherical harmonic

integer

jm symbol

symbol

Wigner function

better approximation

Legendre polynomial

Regge symmetry

Complete translation information:

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Specify your own input

Formula input
Wikitext context	In [[quantum mechanics]], the '''Wigner 3-j symbols''', also called 3''-jm'' symbols, are an alternative to [[Clebsch–Gordan coefficients]] for the purpose of adding angular momenta.<ref name="Wigner1951">{{cite book \|last=Wigner \|first=E. P. \|editor1-last=Wightman \|editor1-first=Arthur S. \|date=1951 \|chapter=On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups \|title=The Collected Works of Eugene Paul Wigner \|volume=3 \|pages=608–654 \|doi=10.1007/978-3-662-02781-3_42 \|isbn=978-3-642-08154-5 }}</ref> While the two approaches address exactly the same physical problem, the 3-''j'' symbols do so more symmetrically. == Mathematical relation to Clebsch–Gordan coefficients == The 3-''j'' symbols are given in terms of the Clebsch–Gordan coefficients by :<math> \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} \equiv \frac{(-1)^{j_1 - j_2 - m_3}}{\sqrt{2 j_3 + 1}} \langle j_1 \, m_1 \, j_2 \, m_2 \| j_3 \, (-m_3) \rangle. </math> The ''j'' and ''m'' components are angular-momentum quantum numbers, i.e., every {{math\|''j''}} (and every corresponding {{math\|''m''}}) is either a nonnegative integer or [[half-integer\|half-odd-integer]]. The exponent of the sign factor is always an integer, so it remains the same when transposed to the left, and the inverse relation follows upon making the substitution {{math\|''m''<sub>3</sub> → −''m''<sub>3</sub>}}: :<math> \langle j_1 \, m_1 \, j_2 \, m_2 \| j_3 \, m_3 \rangle = (-1)^{-j_1 + j_2 - m_3} \sqrt{2 j_3 + 1} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & -m_3 \end{pmatrix}. </math> == Definitional relation to Clebsch–Gordan coefficients == The CG coefficients are defined so as to express the addition of two angular momenta in terms of a third: :<math> \|j_3\, m_3\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} \langle j_1 \, m_1 \, j_2 \, m_2 \| j_3 \, m_3 \rangle \|j_1 \, m_1 \, j_2 \, m_2 \rangle. </math> The 3-''j'' symbols, on the other hand, are the coefficients with which three angular momenta must be added so that the resultant is zero: :<math> \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} \sum_{m_3=-j_3}^{j_3} \|j_1 m_1\rangle \|j_2 m_2\rangle \|j_3 m_3\rangle \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = \|0 \, 0\rangle. </math> Here <math>\|0 \, 0\rangle</math> is the zero-angular-momentum state (<math>j = m = 0</math>). It is apparent that the 3-''j'' symbol treats all three angular momenta involved in the addition problem on an equal footing and is therefore more symmetrical than the CG coefficient. Since the state <math>\|0 \, 0\rangle</math> is unchanged by rotation, one also says that the contraction of the product of three rotational states with a 3-''j'' symbol is invariant under rotations. == Selection rules == The Wigner 3-''j'' symbol is zero unless all these conditions are satisfied: :<math>\begin{align} & m_i \in \{-j_i, -j_i + 1, -j_i + 2, \ldots, j_i\} \quad (i = 1, 2, 3), \\ & m_1 + m_2 + m_3 = 0, \\ & \|j_1 - j_2\| \le j_3 \le j_1 + j_2, \\ & (j_1 + j_2 + j_3) \text{ is an integer (and, moreover, an even integer if } m_1 = m_2 = m_3 = 0 \text{)}. \\ \end{align}</math> == Symmetry properties == A 3-''j'' symbol is invariant under an even permutation of its columns: :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = \begin{pmatrix} j_2 & j_3 & j_1\\ m_2 & m_3 & m_1 \end{pmatrix} = \begin{pmatrix} j_3 & j_1 & j_2\\ m_3 & m_1 & m_2 \end{pmatrix}. </math> An odd permutation of the columns gives a phase factor: :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_2 & j_1 & j_3\\ m_2 & m_1 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_1 & j_3 & j_2\\ m_1 & m_3 & m_2 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_3 & j_2 & j_1\\ m_3 & m_2 & m_1 \end{pmatrix}. </math> Changing the sign of the <math>m</math> quantum numbers ([[T-symmetry#Microscopic_phenomena:_time_reversal_invariance\|time reversal]]) also gives a phase: :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ -m_1 & -m_2 & -m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}. </math> The 3-''j'' symbols also have so-called Regge symmetries, which are not due to permutations or time reversal.<ref>{{cite journal \|first1=T. \|last1=Regge \|title=Symmetry Properties of Clebsch-Gordan Coefficients \|journal=Nuovo Cimento \|year=1958 \|volume=10 \|issue=3 \|page=544 \|doi=10.1007/BF02859841 \|bibcode=1958NCim...10..544R\|s2cid=122299161 }}</ref> These symmetries are: :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = \begin{pmatrix} j_1 & \frac{j_2+j_3-m_1}{2} & \frac{j_2+j_3+m_1}{2}\\ j_3-j_2 & \frac{j_2-j_3-m_1}{2}-m_3 & \frac{j_2-j_3+m_1}{2}+m_3 \end{pmatrix}, </math> :<math> \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} \frac{j_2+j_3+m_1}{2} & \frac{j_1+j_3+m_2}{2} & \frac{j_1+j_2+m_3}{2}\\ j_1 - \frac{j_2+j_3-m_1}{2} & j_2 - \frac{j_1+j_3-m_2}{2} & j_3-\frac{j_1+j_2-m_3}{2} \end{pmatrix}. </math> With the Regge symmetries, the 3-''j'' symbol has a total of 72 symmetries. These are best displayed by the definition of a Regge symbol, which is a one-to-one correspondence between it and a 3-''j'' symbol and assumes the properties of a semi-magic square:<ref>{{Cite journal \|first1=J. \|last1=Rasch \|first2=A. C. H. \|last2=Yu \|title=Efficient Storage Scheme for Pre-calculated Wigner 3''j'', 6''j'' and Gaunt Coefficients \|journal=SIAM J. Sci. Comput. \|volume=25 \|issue=4 \|year=2003 \|pages=1416–1428 \|doi=10.1137/s1064827503422932 }}</ref> :<math> R= \begin{array}{\|ccc\|} \hline -j_1+j_2+j_3 & j_1-j_2+j_3 & j_1+j_2-j_3\\ j_1-m_1 & j_2-m_2 & j_3-m_3\\ j_1+m_1 & j_2+m_2 & j_3+m_3\\ \hline \end{array}, </math> whereby the 72 symmetries now correspond to 3! row and 3! column interchanges plus a transposition of the matrix. These facts can be used to devise an effective storage scheme.<ref>{{Cite journal \|first1=J. \|last1=Rasch \|first2=A. C. H. \|last2=Yu \|title=Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients \|journal=SIAM J. Sci. Comput. \|volume=25 \|issue=4 \|year=2003 \|pages=1416–1428 \|doi=10.1137/s1064827503422932}}</ref> == Orthogonality relations == A system of two angular momenta with magnitudes {{math\|''j''<sub>1</sub>}} and {{math\|''j''<sub>2</sub>}} can be described either in terms of the uncoupled basis states (labeled by the quantum numbers {{math\|''m''<sub>1</sub>}} and {{math\|''m''<sub>2</sub>}}), or the coupled basis states (labeled by {{math\|''j''<sub>3</sub>}} and {{math\|''m''<sub>3</sub>}}). The 3-''j'' symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations :<math> (2 j_3 + 1)\sum_{m_1 m_2} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j'_3\\ m_1 & m_2 & m'_3 \end{pmatrix} = \delta_{j_3, j'_3} \delta_{m_3, m'_3} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix}, </math> :<math> \sum_{j_3 m_3} (2 j_3 + 1) \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1' & m_2' & m_3 \end{pmatrix} = \delta_{m_1, m_1'} \delta_{m_2, m_2'}. </math> The ''triangular delta'' {{math\|{''j''<sub>1</sub> ''j''<sub>2</sub> ''j''<sub>3</sub>}<!--ignore-->}} is equal to 1 when the triad (''j''<sub>1</sub>, ''j''<sub>2</sub>, ''j''<sub>3</sub>) satisfies the triangle conditions, and is zero otherwise. The triangular delta itself is sometimes confusingly called<ref name="WormerPaldus2006">{{cite journal \|title=Angular Momentum Diagrams \|author1=P. E. S. Wormer \|author2=J. Paldus \|journal=Advances in Quantum Chemistry \|publisher=Elsevier \|volume=51 \|pages=59–124 \|year=2006 \|issn=0065-3276 \|doi=10.1016/S0065-3276(06)51002-0 \|bibcode=2006AdQC...51...59W \|isbn=9780120348510}}</ref> a "3-''j'' symbol" (without the ''m'') in analogy to [[6-j symbol\|6-''j'']] and [[9-j symbol\|9-''j'']] symbols, all of which are irreducible summations of 3-''jm'' symbols where no {{mvar\|m}} variables remain. ==Relation to spherical harmonics== The 3-''jm'' symbols give the integral of the products of three [[spherical harmonics]]{{Citation needed\|date=October 2020\|reason=It is not clear which reference contains this claim.}} :<math> \begin{align} & \int Y_{l_1 m_1}(\theta, \varphi) Y_{l_2 m_2}(\theta, \varphi) Y_{l_3 m_3}(\theta, \varphi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi \\ &\quad = \sqrt{\frac{(2l_1 + 1)(2l_2 + 1)(2l_3 + 1)}{4\pi}} \begin{pmatrix} l_1 & l_2 & l_3 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \end{align} </math> with <math>l_1</math>, <math>l_2</math> and <math>l_3</math> integers. === Relation to integrals of spin-weighted spherical harmonics === Similar relations exist for the [[spin-weighted spherical harmonics]] if <math>s_1 + s_2 + s_3 = 0</math>: :<math> \begin{align} & \int d\mathbf{\hat n} \,_{s_1}\!Y_{j_1 m_1}(\mathbf{\hat n}) \,_{s_2}\!Y_{j_2 m_2}(\mathbf{\hat n}) \,_{s_3}\!Y_{j_3 m_3}(\mathbf{\hat n}) \\ &\quad = \sqrt{\frac{(2j_1 + 1)(2j_2 + 1)(2j_3 + 1)}{4\pi}} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j_3\\ -s_1 & -s_2 & -s_3 \end{pmatrix}. \end{align} </math> == Recursion relations == :<math> \begin{align} & {-}\sqrt{(l_3 \mp s_3)(l_3 \pm s_3 + 1)} \begin{pmatrix} l_1 & l_2 & l_3 \\ s_1 & s_2 & s_3 \pm 1 \end{pmatrix}= \\ &\quad = \sqrt{(l_1 \mp s_1)(l_1 \pm s_1 + 1)} \begin{pmatrix} l_1 & l_2 & l_3 \\ s_1 \pm 1 & s_2 & s_3 \end{pmatrix} + \sqrt{(l_2 \mp s_2)(l_2 \pm s_2 + 1)} \begin{pmatrix} l_1 & l_2 & l_3 \\ s_1 & s_2 \pm 1 & s_3 \end{pmatrix}. \end{align} </math> == Asymptotic expressions == For <math>l_1 \ll l_2, l_3</math> a non-zero 3-''j'' symbol is :<math> \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3+m_3} \frac{d^{l_1}_{m_1, l_3 - l_2}(\theta)}{\sqrt{2l_3 + 1}}, </math> where <math>\cos(\theta) = -2m_3 / (2l_3 + 1)</math>, and <math>d^l_{mn}</math> is a [[Wigner_D-matrix#Wigner_(small)_d-matrix\|Wigner function]]. Generally a better approximation obeying the Regge symmetry is given by :<math> \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{l_2+l_3+1}}, </math> where <math>\cos(\theta) = (m_2 - m_3)/(l_2 + l_3 + 1)</math>. == Metric tensor == The following quantity acts as a [[metric tensor]] in angular-momentum theory and is also known as a ''Wigner 1-jm symbol'':<ref name="Wigner1951"/> :<math>\begin{pmatrix} j \\ m \quad m' \end{pmatrix} := \sqrt{2 j + 1} \begin{pmatrix} j & 0 & j \\ m & 0 & m' \end{pmatrix} = (-1)^{j - m'} \delta_{m, -m'}. </math> It can be used to perform time reversal on angular momenta. == Other properties == :<math>\sum_m (-1)^{j - m} \begin{pmatrix} j & j & J \\ m & -m & 0 \end{pmatrix} = \sqrt{2 j + 1} \, \delta_{J, 0}. </math> :<math> \frac{1}{2} \int_{-1}^1 P_{l_1}(x) P_{l_2}(x) P_{l}(x) \, dx = \begin{pmatrix} l & l_1 & l_2 \\ 0 & 0 & 0 \end{pmatrix}^2, </math> where ''P'' are [[Legendre polynomials]]. == Relation to Racah {{math\|''V''}}-coefficients == Wigner 3-''j'' symbols are related to [[Giulio Racah\|Racah]] {{mvar\|V}}-coefficients<ref>{{Cite journal \|first=G. \|last=Racah \|title=Theory of Complex Spectra II \|journal=[[Physical Review]] \|volume=62 \|issue=9–10 \|pages=438–462 \|year=1942 \|doi=10.1103/PhysRev.62.438 \|bibcode = 1942PhRv...62..438R }}</ref> by a simple phase: :<math> V(j_1 \, j_2 \, j_3; m_1 \, m_2 \, m_3) = (-1)^{j_1 - j_2 - j_3} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix}. </math> ==See also== [[Clebsch–Gordan coefficients]] [[Spherical harmonics]] [[6-j symbol]] [[9-j symbol]] ==References== <references /> <!-- ---------------------------------------------------------- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for a discussion of different citation methods and how to generate footnotes using the<ref>, </ref> and <reference /> tags ----------------------------------------------------------- --> <div class="references"> [[Lawrence Biedenharn\|L. C. Biedenharn]] and J. D. Louck, ''Angular Momentum in Quantum Physics'', volume 8 of Encyclopedia of Mathematics, Addison-Wesley, Reading, 1981. D. M. Brink and G. R. Satchler, ''Angular Momentum'', 3rd edition, Clarendon, Oxford, 1993. * A. R. Edmonds, ''Angular Momentum in Quantum Mechanics'', 2nd edition, Princeton University Press, Princeton, 1960. {{dlmf\|id=34 \|title=3j,6j,9j Symbols\|first=Leonard C.\|last= Maximon}} {{cite book \|first1=D. A. \|last1=Varshalovich \|first2=A. N. \|last2=Moskalev \|first3=V. 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A \|volume=17 \|year=1984 \|page=481 \|doi=10.1088/0305-4470/17/3/011 \|issue=3 \|bibcode=1984JPhA...17..727K }} {{ cite journal \|first1=K. \|last1=Srinivasa Rao \|title=Special topics in the quantum theory of angular momentum \|year=1985 \|volume=24 \| number=1 \| pages=15–26 \|journal=Pramana \|doi=10.1007/BF02894812 \|bibcode = 1985Prama..24...15R \|s2cid=120663002 }} {{ cite journal \|first1=Liqiang \|last1=Wei \|title=Unified approach for exact calculation of angular momentum coupling and recoupling coefficients \|journal=Comp. Phys. Comm. \|year=1999 \|volume=120 \|issue=2–3 \|pages=222–230 \|doi=10.1016/S0010-4655(99)00232-5 \|bibcode=1999CoPhC.120..222W }} {{Cite journal \|first1=J. \|last1=Rasch \|first2=A. C. H. \|last2=Yu \|title=Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients \|journal=SIAM J. Sci. Comput. \|volume=25 \|issue=4 \|year=2003 \|pages=1416–1428 \|doi=10.1137/s1064827503422932 }} </div> ==External links== * {{cite web \|first1=Anthony \|last1=Stone \|url=http://www-stone.ch.cam.ac.uk/wigner.shtml \|title=Wigner coefficient calculator }} * {{cite web \|first1 = A. \|last1 = Volya \|url = http://www.volya.net/vc/vc.php \|archive-url = https://web.archive.org/web/20070929011249/http://www.volya.net/vc/vc.php \|url-status = dead \|archive-date = 2007-09-29 \|title = Clebsch-Gordan, 3-''j'' and 6-''j'' Coefficient Web Calculator }} (Numerical) * {{cite journal \|first1=Paul \|last1=Stevenson \|url=http://personal.ph.surrey.ac.uk/~phs3ps/cleb.html \|title=Clebsch-O-Matic \|journal=Computer Physics Communications \|volume=147 \|issue=3 \|pages=853–858 \|doi=10.1016/S0010-4655(02)00462-9 \|bibcode = 2002CoPhC.147..853S \|year=2002 }} * [http://plasma-gate.weizmann.ac.il/369j.html 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science] (Numerical) * [http://geoweb.princeton.edu/people/simons/software.html Frederik J Simons: Matlab software archive, the code THREEJ.M] * [http://www.sagemath.org/ Sage (mathematics software)] Gives exact answer for any value of j, m * {{cite web \|first1=H. T. \|last1=Johansson \|first2=C. \|last2=Forssén \|title=(WIGXJPF) \|url=http://fy.chalmers.se/subatom/wigxjpf/ }} (accurate; C, fortran, python) * {{cite web \|first1=H. T. \|last1=Johansson \|title=(FASTWIGXJ) \|url=http://fy.chalmers.se/subatom/fastwigxj/ }} (fast lookup, accurate; C, fortran) [[Category:Rotational symmetry]] [[Category:Representation theory of Lie groups]] [[Category:Quantum mechanics]]
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