User:Mathsci/sandbox

New-fangled page

When the Jordan curve is smooth (parametrized by arc length) the unit normal vectors give a non-vanishing vector field X0 in the tubular neighbourhood U0 of the curve. Take a polygonal curve in the interior of the curve close to the boundary and transverse to the curve (at the vertices the vector field should be strictly within the angle formed by the edges). By the piecewise linear Jordan–Schoenflies theorem, there is a piecewise linear homeomorphism, affine on an appropriate triangulation of the interior of the polygon, taking the polygon onto a triangle. Take an interior point P in one of the small triangles of the triangulation. It corresponds to a point Q in the image triangle. There is a radial vector field on the image triangle, formed of straight lines pointing towards Q. This gives a series of lines in the small triangles making up the polygon. Each defines a vector field Xi on a neighbourhood Ui of the closure of the triangle. Each vector field is transverse to the sides, provided that Q is chosen in "general position" so that it is not collinear with any of the finitely many edges in the triangulation. Translating if necessary, it can be assumed that P and Q are at the origin 0. On the triangle containing p the vector field can be taken to be the standard radial vector field. Similarly apply the same procedure to the outside, after applying Möbius transformation to map it and ∞ into the finite part of the plane. In this case the neighbourhoods Ui of the triangles have negative indices. Take the vector fields Xi with a negative sign, pointing away from the point at infinity. Together U0 and the Ui's with i ≠ 0 form an open cover of the 2-sphere. Take a smooth partition of unity ψi subordinate to the cover Ui and set


 * $$\displaystyle{X=\sum \psi_i\cdot X_i.}$$

X is a smooth vector field on the two sphere vanishing only at 0 and ∞. The integral curves of this vector field go from the points of the curve to the point P in finite time. Replacing X by f⋅X for an appropriate smooth positive function f, equal to 1 near the curve and near P, the integral curves will all reach P at the same time. The properties of the flow associated to X guarantee that the radial coordinates provided by the integral curves radiating in different directions starting at P give a diffeomorphism between the unit disk and the closure of the interior of the curve. The same procedure can be applied to the outside of the curve, after applying a Möbius transformation to map it and ∞ into the finite part of the plane. The partition of unity Applying a translation if necessary, it can be assumed that P = 0. The two diffeomorphisms with the unit disk patch together to give a smooth diffeomorphism of the Riemann sphere R2 ∪ ∞ carrying the curve onto the unit circle. It carries the inside and outside of the curve onto the regions |z| < 1 and |z| > 1.

Test
suggested otherwise.

Test
The nature of the article means that it is not appropriate to discuss in great detail the tablature discovered in Weimar. That is technical and not the sort of interest for the general readership. It would not have any educational purpose. On the other hand it is reasonable to discuss some of these aspects here on the user talk page.

Rubric, foreword and preface
About the contributor Kirsten Beißwenger.

Brief summary of chapter
The chapter by Beißwenger is primarily interested in Bach's knowledge of other composers, how they influenced Bach's compositions and how he built up his library from other composers. Knowledge of Bach's library and other composers' scores are crucial. Bach's obituary or Nekrolog, recorded by C.P.E. Bach and J.F. Agricola, and their correspondence with J.N. Forkel, shows how Bach respected and valued the works of his contemporaries; it also describes how Bach "loved and studied" other composers in his youth. Just as importantly, Bach's own musical works, either in original works or arrangements, reflect the composers he studied and thus influenced him. Thus the influence of other composers, however, is most directly seen from what remains of Bach's own library. Less indirectly but also of importance were Bach's transcriptions as a pupil, either suggested by his music teachers or discovered on his own.

Bach's music library.

List of Bach's music library.

Recent sources concerning Bach's music library.
 * 1) Dietrich Buxtehude, BuxWV 210, Nun freut euch, lieben Christen g’mein. D-WRa fol. 49/11.
 * 2) Johann Pachelbel, chorale prelude An Wasserflüssen Babylon; Kyrie Gott Vater in Ewigkeit; Fuga. D-WRa fol. 49/11.
 * 3) Johann Adam Reincken, chorale prelude An Wasserflüssen Babylon. D-WRa fol. 49/11.


 * Other sources: Antonio Biffi, François Couperin, Palestrina, Jean Philippe Rameau, Gottfried Heinrich Stölzel, Silvius Leopold Weiss and anon.

Bach's relation with contemporary or earlier composers.

Further directions.

Further test
The first version of the lede was this edit by Gerda on 3 March. There was a copy-edit on 5 March. The new lede was created without consensus by Francis Schonken on 19 March. One minor change was made on 22 March (tune changed to melody). Per WP:BRD and the DYK hook, preferred Gerda's version from 22 March. Gerda added wls to the lede on 23 March. I added Pachelbel to the lede and changed to boy to youth per the DYK. Thoughtfortheday made a minor clarification on 24 March. Franics Schonken then subsituted his preferred lede on 25 March, ignoring the edits of Gerda, Thoughtforthe day and me. I requested a discussion per WP:BRD. That was imediately reverted by Francis Schonken within 2 minutes. On 25 March Francis Schonken disputed with himself about the incoherence of what he had written.

More

 * Dearest God, when will I die?
 * My time runs ever on,
 * And Adam's old heirs,
 * Of whom I am also one,
 * Have this as legacy,
 * They have but a short time
 * Poor and wretched on earth,
 * Then become earth themselves.


 * Ruler over death and life,
 * Let my end one day be good,
 * Teach me to yield the spirit,
 * With courage sure and firm.
 * Help me have an honest grave
 * Near faithful Christian kin,
 * Again at last within the earth,
 * And never more ashamed.

Further
David Yearsley (David Gaynor Yearsley, born April 1965 in Seattle, Washington) is an American organist, harpsicordist and Professor of Musicology. He received his A.B. in American history at Harvard College (1983–1987); and obtained as Ph.D. in musicology from Stanford University in 1994. As an organist he studied with Edward Hansen, Christa Rakich, William Porter, Harald Vogel and Kimberley Marshall. In 1992 he won the first prize in the International Schnitger Organ Competition. In 1994 was won first organ prize for the MAfestival Brugge; he also won first prize there in the category positive-duo with the English organist Annette Richards, to whom they are married. In 1997 Yearsley joined the Music Faculty of Cornell University first as Associate Professor and then as Professor of Musicology. Richards is also a Professor in the Music Faculty.

Lyra Davidica
Here is the English translation of Wie schön leuchtet der Morgenstern from the Lyra Davidica printed in London in 1708 by J. Walsh, J. Hare and P. Randal.

Break
The English metrical translation of the first verse of Philipp Nicolai's German hymn is a composite from the eighteenth and nineteenth century taken from Lyra Davidica (anonymous, 1708), Psalmodia Germanica (Johann Christian Jacobi, 1722) and Christian Chorales for the Chapel and Fireside (Melancthon Woolsey Stryker, 1885). References for all English translations can be found here. In addition, in his 2014 Yale University Press book Tainted Glory in Handel's Messiah, The Unsettling History of the World's Most Beloved Choral Work, Michael Marissen describes the first two English 18th-century translations.

Test1
Property (4) can be checked following Riesz & Sz.-Nagy (1955) and Rubel (1963). For $f$ ≥ 0, monotone and non-decreasing define


 * $$D^+f(x)= \limsup_{s,t\rightarrow x,\,\, s $c$ is contained in an open set of total length < 2ε. Note that an open set $U$ of ($a$,$b$) is canonically the disjoint union of at most countably many open intervals $I$$m$; that allows the total length to be computed |$U$| = ∑ |$I$$m$|. A null set $A$ is one which in which, for every ε' > 0, there is an open $U$ containing $A$ with |$U$| < ε'.

Fix ε, $A$ > 0, and set $c$ = ∑$f$ $n$$f$, $n$ = ∑$g$≤$n$ $N$$f$ and $n$ = ∑$h$>$n$ $N$$f$. Taking $n$ sufficiently large, sup $N$ = ∑$h$>$n$ λ$N$ + μ$n$ < $n$ε. The points $c$ where the slope of $x$ is greater that $h$ are precisely the points $c$ for which


 * $${h(t)-h(s)\over t -s} > c$$

for some $x$, $s$ with $t$ < $s$ < $x$. Every point of ($t$, $s$) shares that property, so that $t$$U$ is an open subset of ($c$, $a$). As such, it can be written as a disjoint union of at most countably many open intervals $b$$I$. Let $k$$J$ be an interval with closure in $k$$I$ and |$k$$J$| = |$k$$I$|/2. By compactness, there are finitely many open intervals of the form ($k$,$s$) covering the closure of $t$$J$; and so their union also has the form $k$$H$ = ($k$$s$,$k$$t$). It follows that


 * $$|J_k| \le |H_k| \le c^{-1}(h(b_k)-h(a_k))$$

so that


 * $$|U_c| = \sum |I_k| =2 \sum |J_k| \le 2\sum |H_k| < 2 \varepsilon.$$

Since $k$ = $f$ + $h$ with $g$ a step function having only finitely many discontinuities at $g$$x$ for $n$ ≤ $n$, it follows that $N$+$D$ = $f$+$D$ + $h$' = $g$+$D$ except at the $h$ points of discontinuity. Taking $N$ = ∪$V$≤$n$ ($N$$x$−δ, $n$$x$+δ) for δ sufficiently small, it follows that the open set $n$$U$ ∪ $c$ contains the points where $V$+$D$ > $f$ while |$c$$U$ ∪ $c$| < 2ε for any $V$ > 0, |$c$$U$| < 2ε. Since ε and $c$ can be taken arbitrarily small, this proves that $c$ ' = 0 almost everywhere.

Test2
Property (4) can be checked following, and. Without loss of generality, it can be assumed that $f$ is a non-negative jump function defined on the compact [$f$,$a$], with discontinuities only in ($b$,$a$).

Note that an open set $b$ of ($U$,$a$) is canonically the disjoint union of at most countably many open intervals $b$$I$; that allows the total length to be computed ℓ($m$) = ∑ ℓ($U$$I$). Recall that a null set $m$ is a subset such that, for any arbitrarily small ε' > 0, there is an open $A$ containing $U$ with ℓ($A$) < ε'. A crucial property of length is that, if $U$ and $U$ are open in ($V$,$a$), then ℓ($b$) + ℓ($U$) = ℓ($V$ ∪ $U$) + ℓ($V$ ∩ $U$). It implies immediately that the union of two null sets is null; and that a finite or countable set is null.

Proposition 1. For $V$ > 0 and a normalised non-negative jump function $c$, let $f$$U$($c$) be the set of points $f$ such that


 * $${h(t)-h(s)\over t -s} > c$$

for some $x$, $s$ with $t$ < $s$ < $x$. Then $t$$U$($c$) is open and has total length ℓ($f$$U$($c$)) ≤ 4 $f$−1 ($c$($f$) – $b$($f$)).

Note that $a$$U$($c$) consists the points $f$ where the slope of $x$ is greater that $h$ near $c$. By definition $x$$U$($c$) is an open subset of ($f$, $a$), so can be written as a disjoint union of at most countably many open intervals $b$$I$ = ($k$$a$, $k$$b$). Let $k$$J$ be an interval with closure in $k$$I$ and ℓ($k$$J$) = ℓ($k$$I$)/2. By compactness, there are finitely many open intervals of the form ($k$,$s$) covering the closure of $t$$J$. On the other hand it is elementary that, if three fixed bounded open intervals have a common point of intersection, then their union contains one of the three intervals: indeed just take the supremum and infimum points to identify the endpoints. As a result the finite cover can be taken as adjacent open intervals ($k$$s$,$k,1$$t$), ($k,1$$s$,$k,2$$t$), ... only intersecting at consecutive intervals. Hence
 * $$\ell(J_k) \le \sum_m (t_{k,m} - s_{k,m}) \le \sum_m c^{-1}(f(t_{k,m})-f(s_{k,m})) \le 2 c^{-1}(f(b_k)-f(a_k)).$$

Finally sum both sides over $k,2$.

Propostion 2. If $k$ is a jump function, then $f$ '($f$) = 0 almost everywhere.

To prove this, define


 * $$Df(x)= \limsup_{s,t\rightarrow x,\,\, s 0, the Dini derivative satisfies $c$$D$($f$) ≤ $x$ almost everywhere, i.e. on a null set.

Choose ε > 0, arbitrarily small. Starting from the definition of the jump function $c$ = ∑ $f$$f$, write $n$ = $f$ + $g$ with $h$ = ∑$g$≤$n$ $N$$f$ and h = ∑$n$>$n$ $N$$f$ where $n$ ≥ 1. Thus $N$ is a step function having only finitely many discontinuities at $g$$x$ for $n$ ≤ $n$ and $N$ is a non-negative jump function. It follows that $h$$D$ = $f$' +$g$$D$ = $h$$D$ except at the $h$ points of discontinuity of $N$. Chossing $g$ sufficiently large so that ∑$N$>$n$ λ$N$ + μ$n$ < ε, it follows that $n$ is a jump function with $h$($h$) − $b$($h$) < ε and satisfies $a$ ≤ $Dh$ off an open set with length less than 4ε/$c$.

By construction is follows that $c$ ≤ $Df$ off an open set with length less than 4ε/$c$. Since ε can be taken arbitrarily small, $c$ and hence $Df$ ' must be 0 almost everywhere.

Lebesgue measure on [0,1]
Let $$U \subset (0,1)$$ be an open set. Thus we can write $$U$$ uniquely as $$U =\bigcup I_k$$, a disjoint union of at most countably many open intervals. As usual the length of an open interval $$(a,b)$$ is given by $$\ell(a,b)=b-a$$. We then wish to give a coherent definition of the length for opens, starting with $$\ell(U) = \sum \ell(I_k).$$ The properties that we establish for the length function should be intuitively obvious if we regard $$\ell(U)$$ as the probability that a random point $$ x\in [0,1]$$ lies in $$U$$.


 * Theorem 1 (easy additivity). If $$U$$ and $$V$$ are finite disjoint unions of open intervals, then $$\ell(U) +\ell(V) = \ell(U\cup V) + \ell(U\cap V).$$
 * Theorem 2 (continuity from below). If $$U_n,\,\,U$$ are open sets with $$U_n \uparrow U,$$ then $$\ell(U_n)\uparrow \ell(U).$$
 * Corollary of proof. If $$U$$ is open, we can find $$K$$ compact in $$U$$ with $$\ell(U\backslash K) \le \varepsilon.$$ (In fact we can take $$K$$ to be a finite union of closed intervals.)
 * Theorem 3 (hard additivity). If $$U$$ and $$V$$ are open sets, then we have $$\ell(U) + \ell(V) =\ell(U\cup V) + \ell(V\cap V).$$
 * Corollary (subadditivity). $$\ell(\bigcup U_i) \le \sum \ell(U_i).$$
 * Covering Lemma. If $$ A \supset B$$ are open sets and $$\varepsilon > 0$$, then we can find $$C$$ open with $$A= B\cup C$$ and $$\ell(C)\le \ell(A) -\ell(B) + \varepsilon.$$
 * Theorem 4 (subadditivity and continuity from above)
 * (1) If $$(U_n)$$ is a sequence of opens such that, for all $$n$$, $$U_n \supseteq U$$ with $$U$$ open, then $$\ell(\bigcup U_n) \le \ell(U) + \sum[\ell(U_n)-\ell(U)].$$
 * (2) If $$(U_n)$$ is a sequence of opens decreasing to $$U$$ open, then $$\ell(U_n)\downarrow \ell(U).$$

YouTube
Johann Sebastian Bach

Test3
It is routine to check from the definitions that


 * if |ℓ($f$) – ℓ($gq$)| = 1 for a simple reflection $g$ and, if $q$ ≠ 1, there is always a simple reflection $g$ such that ℓ($q$) = ℓ($g$) + 1;
 * for $gq$ and $g$ in $h$, ℓ($Γ$) ≤ ℓ($gh$) + ℓ($g$).

Proposition. If $h$ is in $g$ and ℓ($Γ$) = ℓ($gq$) ± 1 for a simple reflection $g$, then $q$e$g$ lies in ±C, and is therefore a positive or negative root, according to the sign.

Replacing $q$ by $g$, only the positive sign needs to be considered. The assertion will be proved by induction on ℓ($gq$) = $g$, it being trivial for m = 0. Assume that ℓ($m$) = ℓ($gs$) + 1. If ℓ($g$) = $g$ > 0, without less of generality it may be assumed that the minimal expression for $m$ ends with ...$g$. Since $t$ and $s$ generate the dihedral group $t$$Γ$, $a$ can be written as a product $g$ = $g$, where $hk$ = ($k$)n or $st$($t$)n and $st$ has a minimal expression that ends with ...$h$, but never with $r$ or $s$. This implies that ℓ($t$) = ℓ($hs$) + 1 and ℓ($h$) = ℓ($ht$) + 1. Since ℓ($h$) < $h$, the induction hypothesis shows that both $m$e$h$ and $s$e$h$ lie in C. It therefore suffices to show that $t$e$k$ has the form $s$e$λ$ + $s$e$μ$ with $t$, $λ$ ≥ 0, not both 0. But that has already been verified in the formulas above.

Poincaré and Klein models
In this section two different models are given for hyperbolic geometry on the unit disk or equivalently the upper half plane.

The group G = SU(1,1) is formed of matrices


 * $$ g = \begin{pmatrix} \alpha & \beta \\ \overline{\beta} & \overline{\alpha}\end{pmatrix}

$$

with


 * $$ |\alpha|^2 -|\beta|^2=1.$$

It is a subgroup of Gc = SL(2,C), the group of complex 2 &times; 2 matrices with determinant 1. The group Gc acts by Möbius transformations on the extended complex plane. The subgroup G acts as automorphisms of the unit disk D and the subgroup G1 = SL(2,R) acts as automorphisms of the upper half plane. If


 * $$C=\begin{pmatrix}1 & i \\ i & 1\end{pmatrix}$$

then


 * $$G=CG_1C^{-1},$$

since the Möbius transformation corresponding M is the Cayley transform carrying the upper half plane onto the unit disk and the real line onto the unit circle.

The Lie algebra $$\mathfrak{g}$$ of SU(1,1) consists of matrices


 * $$ X= \begin{pmatrix} ix & w \\ \overline{w} & -ix \end{pmatrix}$$

with x real. Note that X2 = (|w|2 – x2) I and


 * $$ \det X = x^2 -|w|^2 =-\tfrac12 \operatorname{Tr} X^2.$$

The hyperboloid $$\mathfrak{H}$$ in $$\mathfrak{g}$$ is defined by two conditions. The first is that det X = 1 or equivalently Tr X2 = –2. By definition this condition is preserved under conjugation by G. Since G is connected it leaves the two components with x > 0 and x < 0 invariant. The second condition is that x > 0. For brevity, write X = (x,w).

The group G acts transitively on D and $$\mathfrak{H}$$ and the points 0 and (1,0) have stabiliser K consisting of matrices


 * $$\begin{pmatrix}\zeta & 0 \\ 0 & \overline{\zeta}\end{pmatrix}$$

with |ζ| = 1. Polar decomposition on D implies the Cartan decomposition G = KAK where A is the group of matrices


 * $$a_t = \begin{pmatrix}\cosh t & \sinh t \\ \sinh t & \cosh t\end{pmatrix}.$$

Both spaces can therefore be identified with the homogeneous space G/K and there is a G-equivariant map f of $$\mathfrak{H}$$ onto D sending (1,0) to 0. To work out the formula for this map and its inverse it suffices to compute g(1,0) and g(0) where g is as above. Thus g(0) = β/$μ$ and


 * $$g(1,0)=(|\alpha|^2 +|\beta|^2,-2i\alpha\beta),$$

so that


 * $$\frac\beta\overline{\alpha} = \frac{\alpha\beta} {|\alpha|{}^2} = \frac{2\alpha\beta}{|\alpha|{}^2 + |\beta|{}^2 + 1},$$

recovering the formula


 * $$f(x,w) = \frac{iw}{x + 1}.$$

Conversely if z = iw/(x + 1), then |z|2 = (x – 1)/(x + 1), giving the inverse formula


 * $$ f^{-1}(z) = (x,w) = \left( \frac{1 + |z|{}^2}{1 - |z|{}^2}, \frac{-2iz}{1 - |z|{}^2} \right).$$

This correspondence extends to one between geometric properties of D and $$\mathfrak{H}$$. Without entering into the correspondence of G-invariant Riemannian metrics, each geodesic circle in D corresponds to the intersection of 2-planes through the origin, given by equations Tr XY = 0, with $$\mathfrak{H}$$. Indeed, this is obvious for rays arg z = θ through the origin in D—which correspond to the 2-planes arg w = θ—and follows in general by G-equivariance.

The Beltrami-Klein model is obtained by using the map F(x,w) = w/x as the correspondence between $$\mathfrak{H}$$ and D. Identifying this disk with (1,v) with |v| < 1, intersections of 2-planes with $$\mathfrak{H}$$ correspond to intersections of the same 2-planes with this disk and so give straight lines. The Poincaré-Klein map given by


 * $$K(z) = iF \circ f^{-1}(z) = \frac{2z}{1 + |z|{}^2}$$

thus gives a diffeomorphism from the unit disk onto itself such that Poincaré geodesic circles are carried into straight lines. This diffeomorphism does not preserve angles but preserves orientation and, like all diffeomorphisms, takes smooth curves through a point making an angle less than $\pi$ (measured anticlockwise) into a similar pair of curves. In the limiting case, when the angle is π, the curves are tangent and this again is preserved under a diffeomorphism. The map K yields the Beltrami-Klein model of hyperbolic geometry. The map extends to a homeomorphism of the unit disk onto itself which is the identity on the unit circle. Thus by continuity the map K extends to the endpoints of geodesics, so carries the arc of the circle in the disc cutting the unit circle orthogonally at two given points on to the straight line segment joining those two points. The use of the Beltrami-Klein model corresponds to projective geometry and cross ratios. (Note that on the unit circle the radial derivative of K vanishes, so that the condition on angles no longer applies there.)

The group G1 = SL(2,R) is formed of real matrices


 * $$ g = \begin{pmatrix} a & b \\ c & d\end{pmatrix}

$$

with $$ad -bc =1.$$ The action is by Möbius transformations on the upper half-plane. The Lie algebra of G1 is $$\mathfrak{g}_1$$, the space of 2 x 2 real matrices of trace zero,


 * $$ X = \begin{pmatrix} x & y-t \\ y+t & -x\end{pmatrix}.$$

By transport of construction — conjugating by C — the symmetric bilinear form Tr ad(X)ad(Y) = 4 Tr XY is invariant under conjugation and has signature (2,1). As above X2 = (x2 + y2 – t2) I and


 * $$ \det X = t^2 - x^2 - y^2 =-\tfrac12 \operatorname{Tr} X^2.$$

By Sylvester's law of inertia, the Killing form is, up to equivalence, the unique symmetric bilinear form of signature (2,1) with corresponding quadratic form –x2 –y2 +t2; and G / {±I} or equivalently G1 / {±I} can be identified with SO(2,1).

The group SO(2,1) acts transitively on the two components of the non-zero part of the light cone x2 + y2 = t2. On the interior of the two time-like components, x2 + y2 < t2 with t strictly positive or negative, it is a disjoint union of orbits, namely the hyperboloids x2 + y2 + a2 = t2. For fixed a, say a = 1, the Beltrami-Klein model gives a equivariant homeomorphism f of the hyperboloid x2 + y2 + 1 = t2 onto the open unit disk, f(x,y,t) = (x + iy)/t. Compactifying the hyperboloid by adding a circle at infinity — the rays of the light cone — f extends to a homeomorphism onto the closed unit disk.

Hilbert metric.

Hyperbolic reflection groups
The tessellation of the Schwarz triangles can be viewed as a generalization of the theory of infinite Coxeter groups, following the theory of hyperbolic reflection groups developed algebraically by Jacques Tits and geometrically by Ernest Vinberg. In the case of the Lobachevsky or hyperbolic plane, the ideas originate in the nineteenth-century work of Henri Poincaré and Walther von Dyck. As Joseph Lehner has pointed out in Mathematical Reviews, however, rigorous proofs that reflections of a Schwarz triangle generate a tessellation have often been incomplete, his own 1964 book "Discontinuous Groups and Automorphic Forms", being one example. Carathéodory's elementary treatment in his 1950 textbook "Funktiontheorie", translated into English in 1954, and Siegel's 1954 account using the monodromy principle are rigorous proofs. The approach using Coxeter groups will be summarised here, within the general framework of classification of hyperbolic reflection groups.

Let $\overline{α}$, $r$ and $s$ be symbols and let $t$, $a$, $b$ ≥ 2 be integers, possibly ∞, with


 * $${1 \over a} + {1 \over b} + {1 \over c} < 1.$$

Define $c$ to be the group with presentation having generators $Γ$, $r$ and $s$ that are all involutions and satisfy ($t$)$st$ = 1, ($a$)$tr$ = 1 and ($b$)$rs$ = 1. If one of the integers is infinite, then the product has infinite order. The generators $c$, $r$ and $s$ are called the simple reflections.

Set $t$ = cos π / $A$ if $a$ ≥ 2 is finite and cosh $a$ with $x$ > 0 otherwise; similarly set $x$ = cos π / $B$ or cosh $b$ and $y$ = cos π / $C$ or cosh $c$. Let er, e$z$ and e$s$ be a basis for a 3-dimensional real vector space $t$ with symmetric bilinear form $V$ such that $Λ$(e$Λ$,e$s$) = − $t$, $A$(e$Λ$,e$t$) = − $r$ and $B$(e$Λ$,e$r$) = − $s$, with the three diagonal entries equal to one. The symmetric bilinear form $C$ is non-degenerate with signature (2,1). Define $Λ$(v) = v − 2 $ρ$(v,e$Λ$) e$r$, $r$(v) = v − 2 $σ$(v,e$Λ$) e$s$ and $s$(v) = v − 2 $τ$(v,e$Λ$) e$t$.

Theorem (geometric representation). The operators $t$, $ρ$ and $σ$ are involutions on $τ$, with respective eigenvectors e$V$, e$r$ and e$s$ with simple eigenvalue −1. The products of the operators have orders corresponding to the presentation above (so $t$ has order $στ$, ''etc). The operators $a$, $ρ$ and $σ$ induce a representation of $τ$ on $Γ$ which preserves'' $V$.

The bilinear form $Λ$ for the basis has matrix


 * $$M = \begin{pmatrix}

1 & -C & -B\\ -C & 1 & -A\\ -B & -A & 1\\ \end{pmatrix},$$

so has determinant 1−$Λ$2−$A$2−$B$2−2$C$. If $ABC$ = 2, say, then the eigenvalues of the matrix are 1 and 1 ± ($c$2+$A$2)½. The condition $B$−1 + $a$−1 < ½ immediately forces $b$2+$A$2 > 1, so that Λ must have signature (2,1). So in general $B$, b, $a$ ≥ 3. Clearly the case where all are equal to 3 is impossible. But then the determinant of the matrix is negative while its trace is positive. As a result two eigenvalues are positive and one negative, i.e. Λ has signature (2,1). Manifestly $c$, $ρ$ and $σ$ are involutions, preserving $τ$ with the given −1 eigenvectors.

To check the order of the products like $Λ$, it suffices to note that:


 * 1) the reflections $στ$ and $σ$ generate a finite or infinite dihedral group;
 * 2) the 2-dimensional linear span $τ$ of e$U$ and e$s$ is invariant under $t$ and $σ$, with the restriction of $τ$ positive-definite;
 * 3) $Λ$, the orthogonal complement of $W$, is negative-definite on $U$, and  $Λ$ and $σ$ act trivially on $τ$.

(1) is clear since if $W$ = $γ$ generates a normal subgroup with $στ$−1 = $σγσ$−1. For (2), $γ$ is invariant by definition and the matrix is positive-definite since 0 < cos π / $U$ < 1. Since $a$ has signature (2,1), a non-zero vector $Λ$ in $w$ must satisfy $W$($Λ$,$w$) < 0. By definition, $w$ has eigenvalues 1 and –1 on $σ$, so $U$ must be fixed by $w$. Similarly $σ$ must be fixed by $w$, so that (3) is proved. Finally in (1)


 * $$ \sigma ({\mathbf e}_s) = -{\mathbf e}_s,\,\,\,\,\, \sigma({\mathbf e}_t) = 2 \cos(\pi / a) {\mathbf e}_s + {\mathbf e}_t,\,\,\,\,\, \tau({\mathbf e}_s) = 2 \cos(\pi / a) {\mathbf e}_s + {\mathbf e}_t,\,\,\,\,\, \tau({\mathbf e}_t) = -{\mathbf e}_t,$$

so that, if $τ$ is finite, the eigenvalues of $a$ are -1, $στ$ and $ς$−1, where $ς$ = exp $ς$ / $2πi$; and if $a$ is infinite, the eigenvalues are -1, $a$ and $X$−1, where $X$ = exp $X$. Moreover a straightforward induction argument shows that if $2x$ = $θ$ / $π$ then


 * $$(\sigma\tau)^m({\mathbf e}_s) = [\sin(2m+1)\theta/\sin\theta]{\mathbf e}_s + [\sin 2m\theta/\sin\theta]{\mathbf e}_t, $$
 * $$\tau(\sigma\tau)^m({\mathbf e}_s) = [\sin(2m+1)\theta/\sin\theta]{\mathbf e}_s + [\sin (2m+2)\theta/\sin\theta]{\mathbf e}_t $$

and if $a$ > 0 then


 * $$ (\sigma\tau)^m({\mathbf e}_s) = [\sinh(2m+1)x/\sinh x]{\mathbf e}_s + [\sinh 2mx/\sinh x]{\mathbf e}_t,$$
 * $$\tau(\sigma\tau)^m({\mathbf e}_s) = [\sinh(2m+1)x/\sinh x]{\mathbf e}_s + [\sinh (2m+2)x/\sinh x]{\mathbf e}_t.$$

Let $x$$x$ be the dihedral subgroup of $στ$ generated by $m$ and $s$, with analogous definitions for $2m$$1$ and $s$$2m$. Similarly define $t$$τ$ to be the cyclic subgroup of $στ$ given by the 2-group {1,$m$}, with analogous definitions for $s$$2m$ and $1$$s$. From the properties of the geometric representation, all six of these groups act faithfully on $2m$. In particular $2$$t$ can be identified with the group generated by $Γ$ and $a$; as above it decomposes explicitly as a direct sum of the 2-dimensional irreducible subspace $Γ$ and the 1-dimensional subspace $s$ with a trivial action. Thus there is a unique vector w = e$t$ + $Γ$ e$b$ + $Γ$ e$c$ in $Γ$ satisfying $r$(w) = w and $Γ$(w) = w. Explicitly $r$ = ($Γ$ + $s$)/(1 – $Γ$2) and $t$ = ($V$ + $Γ$)/(1 – $a$2).

Remark on representations of dihedral groups. It is well known that, for finite-dimensional real inner product spaces, two orthogonal involutions $σ$ and $τ$ can be decomposed as an orthogonal direct sum of 2-dimensional or 1-dimensional invariant spaces; for example, this can be deduced from the observation of Paul Halmos and others, that the positive self-adjoint operator ($U$ – $W$)2 commutes with both $S$ and $r$. In the case above, however, where the bilinear form $λ$ is no longer a positive definite inner product, different ad hoc reasoning has to be given.

Theorem (Tits). The geometric representation of the Coxeter group is faithful.

This result was first proved by Tits in the early 1960s and first published in the text of with its numerous exercises. In the text, the fundamental chamber was introduced by an inductive argument; exercise 8 in §4 of Chapter V was expanded by Vinay Deodhar to develop a theory of positive and negative roots and thus shorten the original argument of Tits.

Let X be the convex cone of sums κe$s$ + $μ$e$t$ + $W$e$σ$ with real non-negative coefficients, not all of them zero. For $τ$ in the group $λ$, define ℓ($C$), the word length or length, to be the minimum number of reflections from $AB$, $A$ and $μ$ required to write $B$ as an ordered composition of simple reflections. Define a positive root to be a vector $AC$e$A$, $S$e$T$ or $S$e$T$ lying in X, with $T$ in $Λ$.

It is routine to check from the definitions that


 * if |ℓ($r$) – ℓ($λ$)| = 1 for a simple reflection $s$ and, if $μ$ ≠ 1, there is always a simple reflection $t$ such that ℓ($g$) = ℓ($Γ$) + 1;
 * for $g$ and $r$ in $s$, ℓ($t$) ≤ ℓ($g$) + ℓ($g$).

Proposition. If $r$ is in $g$ and ℓ($s$) = ℓ($g$) ± 1 for a simple reflection $r$, then $g$e$Γ$ lies in ±X, and is therefore a positive or negative root, according to the sign.

Replacing $Γ$ by $V$, only the positive sign needs to be considered. The assertion will be proved by induction on ℓ($gq$) = $g$, it being trivial for m = 0. Assume that ℓ($q$) = ℓ($g$) + 1. If ℓ($q$) = $g$ > 0, without less of generality it may be assumed that the minimal expression for $gq$ ends with ...$g$. Since $h$ and $Γ$ generate the dihedral group $gh$$g$, $h$ can be written as a product $g$ = $Γ$, where $gq$ = ($g$)n or $q$($g$)n and $q$ has a minimal expression that ends with ...$g$, but never with $gq$ or $g$. This implies that ℓ($m$) = ℓ($gs$) + 1 and ℓ($g$) = ℓ($g$) + 1. Since ℓ($m$) < $g$, the induction hypothesis shows that both $t$e$s$ and $t$e$Γ$ lie in X. It therefore suffices to show that $a$e$g$ has the form $g$e$hk$ + $k$e$st$ with $t$, $st$ ≥ 0, not both 0. But that has already been verified in the formulas above.

Corollary (proof of Tits' theorem). The geometric representation is faithful.

It suffices to show that if $h$ fixes e$r$, e$s$ and e$t$, then $hs$ = 1. Considering a minimal expression for $h$ ≠ 1, the conditions ℓ($ht$) = ℓ($h$) + 1 clearly cannot be simultaneously satisfied by the three simple reflections $h$.

Further consequences. ''The roots are the disjoint union of the positive roots and the negative roots. The simple reflection $m$ permutes every positive root other than'' e$h$. For $s$ in $h$, ℓ($t$) is the number of positive roots made negative by $k$.

Fundamental chamber and Tits cone.

Let $s$ be the 3-dimensional closed Lie subgroup of GL($λ$) preserving $s$. As $μ$ can be identified with a 3-dimensional Lorentzian or Minkowski space with signature (2,1), the group $t$ is isomorphic to the Lorentz group O(2,1) and therefore SL±(2,R) / {±$λ$}. Choosing e to be a positive root vector in X, the stabilizer of e is a maximal compact subgroup $μ$ of $g$ isomorphic to O(2). The homogeneous space $r$ = $s$ / $t$ is a symmetric space of constant negative curvature, which can be identified with the 2-dimensional hyperboloid or Lobachevsky plane $$\mathfrak{H}^2$$. The discrete group $g$ acts discontinuously on $g$ / $gq$: the quotient space $g$ \ $q$ / $q$ is compact if $q$, $g$ and $Γ$ are all finite, and of finite area otherwise. Results about the Tits fundamental chamber have a natural interpretation in terms of the corresponding Schwarz triangle, which translate directly into the properties of the tessellation of the geodesic triangle through the hyperbolic reflection group $g$. The passage from Coxeter groups to tessellation can first be found in the exercises of §4 of Chapter V of, due to Tits, and in ; currently numerous other equivalent treatments are available, not always directly phrased in terms of symmetric spaces.

Test7

 * Music and Dialogue, International Project based in the Luhansk Region, Ukraine: Leopold Nikolaus (violin), Anastasia Fedchenko (traverso), Lesya Dermenzhi (violin), Ilona Les (Viola da Gamba), Vitali Alekseenok (harpsichord).
 * (project based in Ukraine): L. Nikolaus, L. Dermenzhi (vns), A. Fedchenko (trav), I. Les (gamba), V. Alekseenok (hpsd)
 * Pachelbel Canon in D on Vimeo (project based in Ukraine): L. Nikolaus, L. Dermenzhi (vns), A. Fedchenko (trav), I. Les (gamba), V. Alekseenok (hpsd)

Test8
This theory is properly explained in physics texts (Bethe) and mathematics texts (Murnaghan) before WW2. It isn't true that any product of double group is a double group, since "double group" refers to a central extension of a group by a cyclic group of order 2. The usual terminology is to refer to spin representations. More generally, for two groups G and H with projective representations σ and τ, σ⊗τ defines a projective representation of G x H; the 2-cocycles with values in $${\mathbb T}$$ = U(1) being multiplied. This is standard terminology on wikipedia: central extension (mathematics), projective representations, spin representations, Schur multipliers, etc. Double groups or spin representations are ubiquitous in mathematics and physics. Likewise wikipedia already has articles like representation theory of SU(2); undergraduate and first year graduate courses in theoretic physics and mathematics cover representations of finite groups, classical groups and their Lie algebras. Cotton's book refers to Bethe and is not self-contained. Murnaghan treats the representations of classical groups and the symmetric group (similarly to the book of Dudley E. Littlewood), but devotes a separate chapter to crystallographic subgroups, which he has attempted to make self-contained where possible.This content should with other wikipedia articles; it's not hard to write about SU(2), SO(3), spin representations of SO(3), double covers, ... Wigner describes in detail the double cover by SU(2) of SO(3) and mentions the double-valued representations, first defined by Weyl. In the same vein, the character theory of SU(2) was dealt with group-theoretically following Weyl, by integrating over SU(2) (Euler angles).

Test10
It is unclear why Pg has written "Mathsci (again)" as he not mentioned me here before. On his user page, he is P G, a retired senior lecturer in the Department of Chemistry at University of Leeds. He states that he retired in 2004. Pg attempted to create a stub Double group which was prodded by and then listed for AfD by D.Lazard at Articles for deletion/Double group. It was also mentioned at Wikipedia talk:WikiProject Mathematics. Pg then campaigned unsuccessfully radically to change the featured article group (mathematics). In real life I've won an LMS prize and have been invited to speak at an ICM. The LMS prize came as the result of two fifty page papers in Inventiones Mathematicae and Annals of Mathematics; it partially concerned projective representations of finite subgroups of SO(3). I know that material extremely well, having given undergraduate and graduate lectures over a long period. My work for the ICM also related to projective unitary representations, but in infinite dimensions, generalising Dirac's "extensor" representations.

During the AfD, I was one of the few people who worked out what was going on in this article. Using the standard resources, I found six WP:RSs which gave crystal clear accounts of the theory. The original idea came from the Physics Nobel laureate Hans Bethe with his 1929 paper. The sources can be described as "Physics for group theory". Pg has stated many times those references in physics are "irrelevant", even the paper of Bethe. Double groups already concern, but in a far deeper way (the ADE corresondence). Looking up "icosahedral group" on nLab gives this nice entry, with references to a former student). During the AfD, I was willing to give the benefit of the doubt, since I could see from the physics references what was going on. Unfortunately, despite having seen references from Hans Bethe, or contemporaries like Weyl, Wigner and von Neumann, Pg has not succeeded in giving any coherent description of double groups (i.e. spin representations of finite subgroups of SO(3)). On the other hand, the article "list of character tables for chemically important 3D point groups" is encyclopedic and links to other wikipedia articles in the normal way.

Requesting that a physicist try to make the stub "double group" coherent is reasonable. I would start with the the words "double-valued (or spin) representations of finite subgroups of SO(3)", and continue from there. Mathsci (talk) 17:00, 17 April 2022 (UTC)
 * Pg has posted twice, but on 18 April has modified his posting without changing date and time stamp. There is a BLP on me on wikipedia. Created without my knowledge, it features five publications that can be found on mathscinet, one of them the invited lecture at a International Congress of Mathematicians; although not giving my university attachments, it shows a Ph.D. (1978–1981, Thouron Award). Pg' edits are extraordinary: he unsuccessfully tried to rewrite group (mathematics) and then was challenged twice by D.Lazard for his sub-stub in "double group", which resulted on Articles for deletion/Double group. Pg has stated his background (lecturer in organic chemistry, retirement at 2003, consistent with statutory retirement age). After the AfD, Petergans created his own preferred version of the article User:Petergans/sandbox. The representation theory of SU(2) gives the correct formula for the special case of the Weyl character formula. As described in the the book "The Classical Groups" by Hermann Weyl, every element of a connected simple compact matrix group is conjugate to a diagonal matrix and the character formula is given as a quotient of alternating sums; for SU(N), these are just determinants. For SU(2), the formula is as given. In representation theory of SU(2), the character of half-integer spin j ≥ 0 is given as $${{\sin \,(j+{1\over 2})\alpha}\over{\sin\, {1\over 2}\alpha }}$$ if the eigenvalues are $$ e^{\pm i\alpha/2}$$. The level = See also = is incorrect per WP:MOS; and so on. As explained below, there is now a systematic way of understanding the character tables of finite subgroups of SU(2) including branching rules from SU(2) and tensor product rules with the 2-dimensional vector representation. Ad hoc presentations of the material have been given in the 1930s by Francis D. Murnaghan and Dudley E. Littlewood.


 * During the AfD, I was one of the few people who worked out what was going on in this article. Using the standard resources, I found six WP:RSs which gave crystal clear accounts of the theory. The original idea came from the Physics Nobel laureate Hans Bethe with his 1929 paper. The sources can be described as "physics for group theory". In Pg's edits, he has stated many times those references in physics are "irrelevant", even the paper of Bethe. Double groups already concern, but in a far deeper way (the ADE correspondence). Looking up "icosahedral group" on nLab gives this nice entry, with references to a former student from my Cambridge College. During the AfD, I was willing to give the benefit of the doubt, since I could see from the physics references what was going on. Unfortunately, despite having seen references from Hans Bethe, or contemporaries like Weyl, Wigner and von Neumann, Pg has not so far provided coherent description of double groups (i.e. "spin representations of finite subgroups of SO(3)"). On the other hand, the article "list of character tables for chemically important 3D point groups" is encyclopedic and links to other wikipedia articles in the normal way.


 * Requesting that a physicist try to make the stub "double group" coherent is reasonable. I would start with the the words "double-valued (or spin) representations of finite subgroups of SO(3)", and continue from there. For mathematicians, this kind of material is well known, but advances and understanding happened in the 1980s, not 1964. In particular, the character tables of finite subgroups of SU(2) appear as eigenvectors of the Smith graph, first introduced by John H. Smith (mathematician) (1969 conference in Calgary). In algebraic combinatorics/representation theory, this has been understood in a systematic way thanks to Ian Macdonald, T. A. Springer, Bertram Kostant, Robert Steinberg, and others. Mathsci (talk) 17:09, 18 April 2022 (UTC)

Test11
Pachelbel's Canon has become hackneyed to the point of becoming muzak. That is reflected in the article itself. On the other hand, for this score, historically informed performances on baroque instruments with appropriate tempo and style do produce dance-like music, where the skillful construction of the canon in three parts can easily be discerned. The audio file that is now in the file is strange, because of its slow tempo, non-existent phrasing and only having two violin parts (the other being a harp). It is hard to recognize any aspects we were taught when studying, for example, Christmas Concerto of Corelli or other baroque string music of the late seventeenth and early eighteenth century. Sometimes audio links are educational—for Frédéric Chopin, the piano works give a good sample of all the different genres; and for Ballade No. 1 (Chopin), all the virtuosos are listed on YouTube as WP:EL. In the case of Pachelbel's Canon, the policy WP:ELYES encourages the provision of properly licensed recordings (possibly of videos). Above I have supplied a standard recording by Christopher Hogwood and the Academy of Ancient Music along with a covid19/Ukraine version which is of good quality and makes me wonder how the lady violinist from Lviv is coping at the moment.

random
Hi. I think know you as an admin active at SPI. During the Tamsin RfA (successful, thank goodness), I followed the cliff-hanging finish. While that was going on, I noticed that you were from Northumberland (i.e. Morpeth), which made me smile. One of the pages I created, W. Gillies Whittaker, shows the part of the world where I came from originally; I have also created content about Sy Cuthbert, e.g. Cuddie's Crag, which was normal for people brought up in Northumberland (and Durham). At the moment I live in the South around Cambridge (I have helped complete 2022 Huntingdonshire District Council election). At the moment CA has started making disruptive edits related to UK politics, which seem counter-productive. He followed my edits; now he has multiple created out-of-date links (WP:Stalker, WP:STALK, WP:STALKER, WP:HARASSMENT, etc) which he uses in a disruptive way to continue the path of following me that started already in 2017. In the case of 2010 Labour Party leadership election (UK), I noticed that CA deleted the section relating to MPs/MEPs. In the UK, we know how that works—we have BBC TV licenses that give us access to news reports, etc. I certainly know who most MPs are (probably enough to remember "money hangers" in Hartlepool). I subsequently suffered stroke while editing wikipedia online (29 December 2017), from which I have gradually recovered. My second language is French (I have worked there for over 14 years); from the IP edits, their first language is French.

goose
The user page "Pg" shows that in 2013 there was a disruptive attempt to edit-war incorrect content about "convolution" on to wikipedia: those definitions conflicted with the standard theory of distributions of Laurent Schwartz and Lars Hörmander. The editor has removed my user name on the user page Pg, as it was being used inappropriately. In 2008, made similarly problematic edits to thier user page – parts had to be removed by  and.

There is no such thing as a "niche gap" - but there is a phenomenon of WP:OWN, displayed by the edits to Pg/sandbox. Constant deletions of reliable sources as "irrelevant" are not helpful. The article "double group" is easy to describe: the material "List of character tables for chemically important 3D point groups" simply has to be generalised to double-valued representations or double covers of the relevant groups. All the mathematical theory was developed and tabulated before 1900. The reluctance to acknowledge mathematical facts about "convolution" is similar to the reluctance to acknowledge facts about double-valued representations: all of these are encyclopedic depending on reliable and verifiable sources. Claims of "irrelevance", "mayhem", the refusals to acknowledge Bethe's contributions are examples of the same "convolution" disruption. There are at least ten text books on "groups and quantum mechanics" devoting several pages to "double group"; some of those WP:RSs are now in the text. Instead of creating coherent content, on the undergraduate level of quantum mechanics (angular momentum) and/or representation theory, the sandbox does not give any context or defintions — a random haphazard list rather than anything encyclopedic. The "List of character tables ...", with companion articles on representation thoery, do a better job, allowing readers to navigate through different aspects.

uk
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a sovereign country in north-western Europe, off the north--western coast of the European mainland. The United Kingdom includes the island of Great Britain, the north--eastern part of the island of Ireland, and many smaller islands within the British Isles. Northern Ireland shares a land border with the Republic of Ireland. Otherwise, the United Kingdom is surrounded by the Atlantic Ocean, with the North Sea to the east, the English Channel to the south and the Celtic Sea to the south-west, giving it the 12th-longest coastline in the world. The Irish Sea separates Great Britain and Ireland. The total area of the United Kingdom is 93,628 sqmi, with an estimated 2020 population of more than 67 million people.

The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a sovereign country in Europe, off the north-­western coast of the continental mainland. It comprises England, Wales, Scotland, and Northern Ireland. The United Kingdom includes the island of Great Britain, the north-­eastern part of the island of Ireland, and many smaller islands within the British Isles. Northern Ireland shares a land border with the Republic of Ireland; otherwise, the United Kingdom is surrounded by the Atlantic Ocean, the North Sea, the English Channel, the Celtic Sea and the Irish Sea. The total area of the United Kingdom is 93,628 square miles (242,500 km2), with an estimated 2020 population of more than 67 million people.

Draft first para UK
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a sovereign country in north-western Europe, off the north--western coast of the continental mainland. It comprises England, Wales, Scotland, and Northern Ireland. The United Kingdom includes the island of Great Britain, the north--eastern part of the island of Ireland, and many smaller islands within the British Isles. Northern Ireland shares a land border with the Republic of Ireland; otherwise, the United Kingdom is surrounded by the Atlantic Ocean, the North Sea, the English Channel, the Celtic Sea and the Irish Sea. The total area of the United Kingdom is 93,628 sqmi, with an estimated 2020 population of more than 67 million people.

more
Template:Did you know nominations/Ich will den Kreuzstab gerne tragen, BWV 56 discography has date/time stamps. The problem with ALT1 was that Dietrich Fischer-Dieskau made (at least) four registered recordings of BWV 56 – in 1951, 1965, 1969 and 1983. On April 15 and 16, I suggested ALT2 and ALT3 where you made some comments, which you responded to. You did not say that you had stopped watchlisting the template; you suggested that another reviewer would comment. On 9 May you asked me for help for translating a German text about Classical Greece; the rest of the book was in English, but I wasted so time time helping you writing material about the hymn connected with the closing chorale of BWV 56. Then on 22 May, NLH5 asked me to pin down the hook, so I improved BWV 56 and it discography. You were pinged at the time and mentioned explicitly (as a courtesy), but refused to respond. Then on 30 May you were alerted to a title change by Ravenpuff, which I objected to. At that stage, you proclaimed that in German "protégé" had connotations. But here this concerned two American baritones (from Texas and Kansas) and the usual mentoring sense applied without any problem: this after all the English-language wikipedia not the German-language wikipedia; and the English-language wiktionary confirms that  "protégé" is the right word. I have no idea why you are attempting to teach native English people how to speak English, when on May 1o you were begging for help from me. In English music journals or books, the term "Kreuzstab cantata" is used. Both singers are baritones, not bass. You had the opportunity to communicate but did not, but in a pointed wait did not collaborate. On the other hand, as User:Fram  has pointed out, your current attempt to write a DYK has been an unmitigated disaster, mostly as a reslt of your own attitude, and the hook has had to be shelved. What you've about other recordings is not reliable, since you just give your own impressions do not match up with what can be found in reliable sources. Being disgruntled by your a poor DYK and then taking that out on me is not reasonable. I have no idea why you have made these misguided comment that Harrell's recording was an easter egg – it means nothing and writing something like that anywhere on wikipedia is unhelpful.

The two singers were baritones; the "Kreuzstab cantata" can be rendered as "cross-staff cantata". I have no idea what you mean about "easter egg" – perhaps you are translating directly into English without realising that it makes no sense. It's like the translations of LouisAlain — unreliable.