Invariant of a binary form

In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables x and y that remains invariant under the special linear group acting on the variables x and y.

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Ladies and Gentlemen, I´d like to welcome you to our course "Physical Chemistry". My name is Dr. Lauth and today's topic is: How to read binary phase diagrams? How to read phase diagrams of mixtures? e.g. this boiling-point curve of the two components IBA (isobutanol) and IPA (isopropanol). On the abscissa concentration of the mixture is plotted. On the left hand side there´s pure IBA, on the right hand side, there´s pure IPA, in between all possible mixtures. (described with the mole fraction x sub B). On the ordinate temperature is plotted between 80 degrees and 110 degrees. A phase diagram tells us what phases are present at a particular combination of state variables. In order to orient ourselves in this diagram, we look first of all at the homogeneous phases. If we choose a composition x and temperature T, so get to a point in the phase diagram in the above right, we have a homogeneous gaseous mixture. If we choose a point in the lower left, we have a homogeneous liquid mixture. The boundaries of the homogeneous regions are called binodals. Binodals fence off the homogeneous areas to the heterogeneous areas in a phase diagram, and often have specific names: The binodal which marks off the region of the homogeneous fluid to higher temperatures is called bubble point curve (boiling point line); the binodal, which marks off the region of the gas phase to low temperatures is called the dew point curve. Between bubble point curve and dew point curve is a two-phase region. If we plot a point in this area, the system is unstable in an homogeneous state, but will decomposes into two phases - but more on that later. Bubble point curve and dew point curve meet in two points, two so-called invariant points in the phase diagram. At these points the system has no degree of freedom according to Gibbs phase rule (F = C-P +2). Firstly, this is the boiling point of pure IPA. Here liquid and gaseous IPA coexist at exactly 82 degrees Celsius (C = 1, P = 2, F = 1 (p)). On the other side we find the boiling point of pure IBA. Here liquid and gaseous IBA coexist at 108 degrees Celsius. Binodals without any maximum and minimum are typical for phase transitions of ideal mixtures. (a mixture of components A and B being chemically very similar.) We want to discuss heating of a 50:50 mixture using this diagram: If we mix 1 mole of IPA and 1 mole , say , at 80 degrees Celsius, we get a point in the diagram, which is located right here (on the abscissa at 0.5) - So we have a homogeneous liquid phase. We heat the liquid - this means we move up on a vertical. (a so-called isopleth - here marked in green) at about 92 degrees Celsius, the isopleth intersects the bubbling point curve - That is, the mixture begins to boil. The composition of the gas phase above the boiling liquid 50:50 mixture, we may determine using the so called tie line. A tie line is an equilibrium line, an isotherm in a two-phase region. We can draw any number of tie lines in the two-phase region. Each horizontal line between bubbling and dew point curve is a tie line. A tie line connects two phases in equilibrium. In our case it connects the 50 percent liquid phase to the 70 % gas phase. The gas phase is thus enriched in the lower boiling component. A 50:50 mixture of IPA and IBA at 92 ° C begins to boil and the gaseous phase that results from the liquid phase is approximately 70 percent. We want to discuss another issue using the liquid gas phase diagram of IBA / IPA. If we mix 65 percent IPA with 35 percent IBA and heat the mixture to 91.5 degrees Celsius, we´ll end up at a point in the two-phase area of the diagram between bubbling point curve and dew point curve. Here a homogeneous system is not stable but disintegrates along the tie line into a liquid phase and a gas phase. The stoichiometry of the decomposition follows the so-called lever rule. We draw the tie line (in yellow) through the point, we find that it intersects the bubbling point curve x=0.5, and intersects the dew point curve at 0.72. The system is homogeneous unstable and decomposes into a liquid phase with 50 percent IPA (read on the bubbling point curve) and a gas phase with 72 percent IPA (read on the dew point curve). Consider the tie line as a lever, with the initial state point as the fulcrum. The lever arm a(to the liquid side) is approximately twice as long as the lever arm b, which means the amount of gas phase is twice as large as the amount of liquid phase. Using the liquid gas phase diagram of nitrogen and oxygen we may discuss the cooling of air. At 82 Kelvins, the air is still homogeneous gaseous -- with a composition of about 79 percent nitrogen - At 81 Kelvins liquid phase condenses, which is 50% nitrogen. Only at 79 Kelvins, the air is completely condensed. During condensation (or evaporation) of an ideal mixture neither temperature nor composition of the phases are constant. Liquid gas phase diagrams with tie lines and binodals describe the exact course of condensing (or evaporation). In ideal mixtures, the gas phase is always enriched in the lower boiling component. Thus we are able to separate the mixture by distillation. In principle an ideal mixture may be completely separated into the pure components by multiple distillation. This process may be plotted in the phase diagram as a stairs shaped line. Now for non-ideal mixtures. The (non-ideal) mixture of water and formic acid shows a phase diagram with a maximum. The peak is also called an azeotrope. At this point, bubbling point curve and dew point curve intersect for a third time: The azeotrope forms a third invariant point. It boils and condenses exactly as a pure substance. Therefore, it´s not possible to separate a non-ideal mixture into the pure components by distillation. The (non-ideal) mixture of water and ethanol shows a phase diagram with minimum azeotrope. A liquid mixture of 95.6% ethanol will boil at a constant temperature of 78.2 ° C into a gas phase, which is also 95.6% ethanol. Accordingly, you cannot distillate pure ethanol form ethanol / water mixture . Here you can see a phase diagram of the system hexane / nitrobenzene below the boiling point. We may discuss this phase diagram in a similar way as liquid gas phase diagram. We find a binodal, which separates the homogeneous region from the heterogeneous region . above the binodal, the mixture is homogeneous; below the binodal, it consists of two separated liquid phases. In this heterogeneous region, we can arbitrarily draw tie lines. If we prepare a 40:60 nitrobenzene / hexane mixture at 25 ° C ( room temperature) the system is in the state (1). The mixture is homogeneous -- it´s a single phase. If we cool this mixture, the isopleth intersects the binodal at about 20 ° C (state (2)). The mixture then splits up into two liquid phases. At 15 ° C we find state (3). A nitrobenzene-rich liquid phase I and a hexane-rich liquid phase II coexist. The ratio of the two liquid phases can be calculated using the lever rule. Now for solid liquid phase transitions. This is the solid-liquid phase diagram of an ideal mixture (copper / nickel) It looks just like an "ideal" liquid gas phase diagram and we can discuss it in the same way. We find two homogeneous regions, the liquid at high temperatures and the solid at a low temperatures. There are two binodals and these are referred to as solidus and liquidus. The binodals intersect only in two points, namely, the melting points of the pure components. In the two-phase region (between the liquidus and solidus), we can arbitrarily draw tie lines. With the help of tie lines and binodals, we may discuss the solidification of a 60-percent melt. We find, that during solidification of an ideal mixture neither phase composition nor temperature remain constant. This is a typical solid liquid phase diagram of a non-ideal mixture (lithium chloride / potassium chloride). While the two components are perfectly miscible in the liquid phase, however, they form only few mixed crystals (solid solution) in the solid state. Let´s first look for the homogeneous areas in the diagram: we find liquid phase (melt) at high temperature, marked off to lower temperatures by a typical V-shaped liquidus. We find two solid homogeneous regions: potassium chloride mixed crystals (marked in red) and lithium chloride mixed crystals (black featured here) . These homogeneous solid areas may be so narrow that they are almost indistinguishable from the ordinates. All other areas in the diagram are heterogeneous. In the triangular area to the left molten and solid potassium chloride (mixed crystals) coexist. In the triangular area to the right molten and solid lithium chloride mixed crystals coexist. The rectangular area consists of a heterogeneous mixture of solid potassium chloride and solid lithium chloride crystals. As expected, the melting points of the pure components are invariant points (intersections of binodals). And there is another invariant point at the minimum of the liquidus. We call this invariant point eutectic. At this point, molten liquid, solid potassium chloride mixed crystals and solid lithium chloride mixed crystals coexist at a constant temperature of 359 degrees Celsius. Like a pure substance, this mixture melts and solidifies at constant temperature. We want to discuss the solidification of an under-eutectic melt in a schematic solid-liquid phase diagram. We cool the melt (initial state 1), that is we are moving out along an isopleth down to low temperatures. At state (2) we intersect the liquidus. Potassium chloride crystals precipitate from the melt. As we cool down further, the potassium chloride crystals grow, the melt accumulates in lithium chloride and moves along the liquidus to the eutectic. In state (3) large potassium chloride crystals are present in a melt with eutectic composition. Now temperature remains constant until the eutectic mixture has completely solidified. At the final state (4) we find large potassium chloride crystals in a solid eutectic matrix. This matrix is heterogeneous: it consists of an intimate mixture of potassium chloride and lithium chloride crystals. At a eutectic, liquid phase is no longer stable below a certain temperature and decomposes into two solid phases. Conversely at a so called peritectic, a solid phase is no longer stable above a certain temperature and decomposes into another solid phase and a liquid phase with a different composition (this is also called incongruent melting). With congruent melting of a solid compound to a , we find a dystectic, a solid compound formed of two components melts to homogeneous liquid phase with the same composition. We can tell a eutectic by the V-shaped structure of the liquidus, we can tell a peritectic by a kink in the liquidus and the roof-shaped or T-shaped structure of the solidus. If these structures occur and only solid phases are involved, we speak of a eutectoid or by a peritectoid We have here a section of the iron / iron-carbide phase diagram. We want to discuss this diagram. We find a liquid homogeneous area (melt) up here. We find four solid homogeneous regions: Solid alpha-iron bottom left, solid gamma iron (the somewhat broader area here) and delta-iron top left. And we have solid cementite (iron carbide) - that is the right boundary. All other areas in the phase diagram are heterogeneous; We can see several invariant points: - the eutectic at this point (V-shaped liquidus), below this temperature the melt is no longer stable, but decomposes into gamma-iron and cementite. - of the eutectoid at this point (V-shaped solidus): Below this temperature gamma-iron is no longer stable but disintegrates into alpha-iron and cementite. Gamma-iron does not only have a floor temperature, but also a ceiling temperature: That´s where the peritectic is located. Gamma-iron decomposes into delta-iron and melt. Furthermore, there´s a dystectic at this point: cementite melts congruently to the corresponding melt. If you want to discuss a phase diagram, first find the homogeneous areas, then search for then the binodals and designate them as bubbling point curve, dew point curve, liquidus, solidus. Finally determine the invariant points and characterize them as eutectic, peritectic, ect., recording the corresponding phase equilibria. Thank you for watching.

Terminology

A binary form (of degree n) is a homogeneous polynomial Σⁿ
_i=0 (ⁿ
_i)a_n−ixⁿ⁻ⁱyⁱ = a_nxⁿ + (ⁿ
₁)a_n−1xⁿ⁻¹y + ... + a₀yⁿ. The group SL₂(C) acts on these forms by taking x to ax + by and y to cx + dy. This induces an action on the space spanned by a₀, ..., a_n and on the polynomials in these variables. An invariant is a polynomial in these n + 1 variables a₀, ..., a_n that is invariant under this action. More generally a covariant is a polynomial in a₀, ..., a_n, x, y that is invariant, so an invariant is a special case of a covariant where the variables x and y do not occur. More generally still, a simultaneous invariant is a polynomial in the coefficients of several different forms in x and y.

In terms of representation theory, given any representation V of the group SL₂(C) one can ask for the ring of invariant polynomials on V. Invariants of a binary form of degree n correspond to taking V to be the (n + 1)-dimensional irreducible representation, and covariants correspond to taking V to be the sum of the irreducible representations of dimensions 2 and n + 1.

The invariants of a binary form form a graded algebra, and Gordan (1868) proved that this algebra is finitely generated if the base field is the complex numbers.

Forms of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10 are sometimes called quadrics, cubic, quartics, quintics, sextics, septics or septimics, octics or octavics, nonics, and decics or decimics. "Quantic" is an old name for a form of arbitrary degree. Forms in 1, 2, 3, 4, ... variables are called unary, binary, ternary, quaternary, ... forms.

Examples

A form f is itself a covariant of degree 1 and order n.

The discriminant of a form is an invariant.

The resultant of two forms is a simultaneous invariant of them.

The Hessian covariant of a form Hilbert (1993, p.88) is the determinant of the Hessian matrix

H(f)={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x^{2}}}&{\frac {\partial ^{2}f}{\partial x\,\partial y}}\\[10pt]{\frac {\partial ^{2}f}{\partial y\,\partial x}}&{\frac {\partial ^{2}f}{\partial y^{2}}}\end{bmatrix}}.

It is a covariant of order 2n− 4 and degree 2.

The catalecticant is an invariant of degree n/2+1 of a binary form of even degree n.

The canonizant is a covariant of degree and order (n+1)/2 of a binary form of odd degree n.

The Jacobian

\det {\begin{bmatrix}{\frac {\partial f}{\partial x}}&{\frac {\partial f}{\partial y}}\\[10pt]{\frac {\partial g}{\partial x}}&{\frac {\partial g}{\partial y}}\end{bmatrix}}

is a simultaneous covariant of two forms f, g.

The ring of invariants

The structure of the ring of invariants has been worked out for small degrees. Sylvester & Franklin (1879) gave tables of the numbers of generators of invariants and covariants for forms of degree up to 10, though the tables have a few minor errors for large degrees, mostly where a few invariants or covariants are omitted.

Covariants of a binary linear form

For linear forms ax + by the only invariants are constants. The algebra of covariants is generated by the form itself of degree 1 and order 1.

Covariants of a binary quadric

The algebra of invariants of the quadratic form ax² + 2bxy + cy² is a polynomial algebra in 1 variable generated by the discriminant b² − ac of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generated by the discriminant together with the form f itself (of degree 1 and order 2). (Schur 1968, II.8) (Hilbert 1993, XVI, XX)

Covariants of a binary cubic

The algebra of invariants of the cubic form ax³ + 3bx²y + 3cxy² + dy³ is a polynomial algebra in 1 variable generated by the discriminant D = 3b²c² + 6abcd − 4b³d − 4c³a − a²d² of degree 4. The algebra of covariants is generated by the discriminant, the form itself (degree 1, order 3), the Hessian H (degree 2, order 2) and a covariant T of degree 3 and order 3. They are related by the syzygy 4H³=Df²-T² of degree 6 and order 6. (Schur 1968, II.8) (Hilbert 1993, XVII, XX)

Covariants of a binary quartic

The algebra of invariants of a quartic form is generated by invariants i, j of degrees 2, 3. This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series E₄ and E₆. The algebra of covariants is generated by these two invariants together with the form f of degree 1 and order 4, the Hessian H of degree 2 and order 4, and a covariant T of degree 3 and order 6. They are related by a syzygy $jf 3 - Hf 2 i + 4 H 3 + T 2 = 0$ of degree 6 and order 12. (Schur 1968, II.8) (Hilbert 1993, XVIII, XXII)

Covariants of a binary quintic

The algebra of invariants of a quintic form was found by Sylvester and is generated by invariants of degree 4, 8, 12, 18. The generators of degrees 4, 8, 12 generate a polynomial ring, which contains the square of Hermite's skew invariant of degree 18. The invariants are rather complicated to write out explicitly: Sylvester showed that the generators of degrees 4, 8, 12, 18 have 12, 59, 228, and 848 terms often with very large coefficients. (Schur 1968, II.9) (Hilbert 1993, XVIII) The ring of covariants is generated by 23 covariants, one of which is the canonizant of degree 3 and order 3.

Covariants of a binary sextic

The algebra of invariants of a sextic form is generated by invariants of degree 2, 4, 6, 10, 15. The generators of degrees 2, 4, 6, 10 generate a polynomial ring, which contains the square of the generator of degree 15. (Schur 1968, II.9) The ring of covariants is generated by 26 covariants. The ring of invariants is closely related to the moduli space of curves of genus 2, because such a curve can be represented as a double cover of the projective line branched at 6 points, and the 6 points can be taken as the roots of a binary sextic.

Covariants of a binary septic

The ring of invariants of binary septics is anomalous and has caused several published errors. Cayley claimed incorrectly that the ring of invariants is not finitely generated. Sylvester & Franklin (1879) gave lower bounds of 26 and 124 for the number of generators of the ring of invariants and the ring of covariants and observed that an unproved "fundamental postulate" would imply that equality holds. However von Gall (1888) showed that Sylvester's numbers are not equal to the numbers of generators, which are 30 for the ring of invariants and at least 130 for the ring of covariants, so Sylvester's fundamental postulate is wrong. von Gall (1888) and Dixmier & Lazard (1988) showed that the algebra of invariants of a degree 7 form is generated by a set with 1 invariant of degree 4, 3 of degree 8, 6 of degree 12, 4 of degree 14, 2 of degree 16, 9 of degree 18, and one of each of the degrees 20, 22, 26, 30. Cröni (2002) gives 147 generators for the ring of covariants.

Covariants of a binary octavic

Sylvester & Franklin (1879) showed that the ring of invariants of a degree 8 form is generated by 9 invariants of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10, and the ring of covariants is generated by 69 covariants. August von Gall (von Gall (1880)) and Shioda (1967) confirmed the generators for the ring of invariants and showed that the ideal of relations between them is generated by elements of degrees 16, 17, 18, 19, 20.

Covariants of a binary nonic

Brouwer & Popoviciu (2010a) showed that the algebra of invariants of a degree 9 form is generated by 92 invariants. Cröni, Hagedorn, and Brouwer^[1] computed 476 covariants, and Lercier & Olive showed that this list is complete.

Covariants of a binary decimic

Sylvester stated that the ring of invariants of binary decics is generated by 104 invariants the ring of covariants by 475 covariants; his list is to be correct for degrees up to 16 but wrong for higher degrees. Brouwer & Popoviciu (2010b) showed that the algebra of invariants of a degree 10 form is generated by 106 invariants. Hagedorn and Brouwer^[1] computed 510 covariants, and Lercier & Olive showed that this list is complete.

Covariants of a binary undecimic

The ring of invariants of binary forms of degree 11 is complicated and has not yet been described explicitly.

Covariants of a binary duodecimic

For forms of degree 12 Sylvester (1881) found that in degrees up to 14 there are 109 basic invariants. There are at least 4 more in higher degrees. The number of basic covariants is at least 989.

The number of generators for invariants and covariants of binary forms can be found in (sequence A036983 in the OEIS) and (sequence A036984 in the OEIS), respectively.

Invariants of several binary forms

The covariants of a binary form are essentially the same as joint invariants of a binary form and a binary linear form. More generally, on can ask for the joint invariants (and covariants) of any collection of binary forms. Some cases that have been studied are listed below.

Covariants of two linear forms

There are 1 basic invariant and 3 basic covariants.

Covariants of a linear form and a quadratic

There are 2 basic invariants and 5 basic covariants.

Covariants of a linear form and a cubic

There are 4 basic invariants (essentially the covariants of a cubic) and 13 basic covariants.

Covariants of a linear form and a quartic

There are 5 basic invariants (essentially the basic covariants of a quartic) and 20 basic covariants.

Covariants of a linear form and a quintic

There are 23 basic invariants (essentially the basic covariants of a quintic) and 94 basic covariants.

Covariants of a linear form and a quantic

Covariants of several linear forms

The ring of invariants of n linear forms is generated by n(n–1)/2 invariants of degree 2. The ring of covariants of n linear forms is essentially the same as the ring of invariants of n+1 linear forms.

Covariants of two quadratics

There are 3 basic invariants and 6 basic covariants.

Covariants of two quadratics and a linear form

Covariants of several linear and quadratic forms

The ring of invariants of a sum of m linear forms and n quadratic forms is generated by m(m–1)/2 + n(n+1)/2 generators in degree 2, nm(m+1)/2 + n(n–1)(n–2)/6 in degree 3, and m(m+1)n(n–1)/4 in degree 4.

For the number of generators of the ring of covariants, change m to m+1.

Covariants of a quadratic and a cubic

There are 5 basic invariants and 15 basic covariants

Covariants of a quadratic and a quartic

There are 6 basic invariants and 18 basic covariants

Covariants of a quadratic and a quintic

There are 29 basic invariants and 92 basic covariants

Covariants of a cubic and a quartic

There are 20 basic invariants and 63 basic covariants

Covariants of two quartics

There are 8 basic invariants (3 of degree 2, 4 of degree 3, and 1 of degree 4) and 28 basic covariants. (Gordan gave 30 covariants, but Sylvester showed that two of these are reducible.)

Covariants of many cubics or quartics

The numbers of generators of invariants or covariants were given by Young (1898).

References

^ ^a ^b Brouwer, Invariants and covariants of quantics

Brouwer, Andries E.; Popoviciu, Mihaela (2010a), "The invariants of the binary nonic", Journal of Symbolic Computation, 45 (6): 709–720, arXiv:1002.0761, doi:10.1016/j.jsc.2010.03.003, ISSN 0747-7171, MR 2639312, S2CID 30297
Brouwer, Andries E.; Popoviciu, Mihaela (2010b), "The invariants of the binary decimic", Journal of Symbolic Computation, 45 (8): 837–843, arXiv:1002.1008, doi:10.1016/j.jsc.2010.03.002, ISSN 0747-7171, MR 2657667, S2CID 12702092
Cröni, Holger (2002), Zur Berechnung von Kovarianten von Quantiken (Dissertation), Saarbrücken: Univ. des Saarlandes
Dixmier, Jacques; Lazard, D. (1988), "Minimum number of fundamental invariants for the binary form of degree 7", Journal of Symbolic Computation, 6 (1): 113–115, doi:10.1016/S0747-7171(88)80026-9, ISSN 0747-7171, MR 0961375
von Gall, August Freiherr (1880), "Das vollständige Formensystem einer binären Form achter Ordnung", Mathematische Annalen, 17 (1): 31–51, doi:10.1007/BF01444117, ISSN 0025-5831, MR 1510048, S2CID 120828980
von Gall, August Freiherr (1888), "Das vollständige Formensystem der binären Form 7^terOrdnung", Mathematische Annalen, 31 (3): 318–336, doi:10.1007/BF01206218, ISSN 0025-5831, MR 1510486, S2CID 121051862
Gordan, Paul (1868), "Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist", Journal für die reine und angewandte Mathematik, 1868 (69): 323–354, doi:10.1515/crll.1868.69.323, S2CID 120689164
Hilbert, David (1993) [1897], Theory of algebraic invariants, Cambridge University Press, ISBN 978-0-521-44457-6, MR 1266168
Kung, Joseph P. S.; Rota, Gian-Carlo (1984), "The invariant theory of binary forms", Bulletin of the American Mathematical Society, New Series, 10 (1): 27–85, doi:10.1090/S0273-0979-1984-15188-7, ISSN 0002-9904, MR 0722856
Schur, Issai (1968), Grunsky, Helmut (ed.), Vorlesungen über Invariantentheorie, Die Grundlehren der mathematischen Wissenschaften, vol. 143, Berlin, New York: Springer-Verlag, ISBN 978-3-540-04139-9, MR 0229674
Shioda, Tetsuji (1967), "On the graded ring of invariants of binary octavics", American Journal of Mathematics, 89 (4): 1022–1046, doi:10.2307/2373415, ISSN 0002-9327, JSTOR 2373415, MR 0220738
Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-211-77417-5, ISBN 978-3-211-82445-0, MR 1255980
Sylvester, J. J.; Franklin, F. (1879), "Tables of the Generating Functions and Groundforms for the Binary Quantics of the First Ten Orders", American Journal of Mathematics, 2 (3): 223–251, doi:10.2307/2369240, ISSN 0002-9327, JSTOR 2369240, MR 1505222
Sylvester, James Joseph (1881), "Tables of the Generating Functions and Groundforms of the Binary Duodecimic, with Some General Remarks, and Tables of the Irreducible Syzygies of Certain Quantics", American Journal of Mathematics, The Johns Hopkins University Press, 4 (1): 41–61, doi:10.2307/2369149, ISSN 0002-9327, JSTOR 2369149
Young, A. (November 1898). "The Irreducible Concomitants of any Number of Binary Quartics". Proceedings of the London Mathematical Society. s1-30 (1): 290–307. doi:10.1112/plms/s1-30.1.290.

External links

Brouwer, Andries E., Invariants of binary forms

This page was last edited on 20 August 2023, at 12:11

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