text:
\section{\bf 1. Introduction.}

In a recent work, [\putref{Dow40}], I have looked at $p$--forms on tesselations
of the three--sphere. In this follow--up, I expand on the higher--dimensional
aspects of the formalism initiated there. The generating functions are
presented for any $p$--form although I concentrate, for actual $\zeta$--function\
computations, on (coexact) forms of the middle rank, $p=(d-1)/2$, on the odd
$d$--dimensional factored sphere, S$^d/\Gamma$. The reason for this is that the
eigenvalues are perfect squares and the expressions for the spectral objects
can be taken a long way in terms of known quantities.

The deck group, $\Gamma$, is the complete symmetry group of a regular polytope in
$n$, $=d+1$, dimensions. In another terminology, it is a real reflection
group. These have all been classified.

This paper should be looked upon as a direct continuation of  [\putref{Dow40}]
and, to avoid repetition, I will use, without derivation, any necessary
equations and results of this reference As there, the analysis is presented
as an example of spectral theory in bounded domains that is easily, and
explicitly, managed {\it via} images. It largely consists of bolting together
already existing pieces of knowledge and taking the expressions a little
further than seems to exist in the literature.

The quantum field theory of anti--symmetric fields has a certain importance,
{\it e.g.}\ [\putref{SandTy,Obukhov,CandT,Reuter}]. I will consider a selection of
spectral objects, such as heat--kernel expansion coefficients, the Casimir
energy and the eta invariant, as examples.

\section{\bf 2. Spectrum and generating functions}
The coexact eigenvalues of the Hodge de--Rham Laplacian, $d\delta+\delta d$, on the
$d$--sphere are standard,
  $$
  \lambda^{CE}(p,l)=\big(l+(d+1)/2\big)^2-\big((d-1)/2-p\big)^2\,,
  \quad l=0,1,\ldots\,\,,
  \eql{ceig2}
  $$
which specialises to
  $$
  \mu^{CE}(p,l)=\big(l+p+1\big)^2\,,
  \quad l=0,1,\ldots\,\,,
  \eql{ceigmo}
  $$
for middle rank forms, if $d$ is odd.

The corresponding degeneracies, $g^{CE}_b(p,l)$, are best encoded in the
generating function defined by,
  $$
  g^{CE}_b(p,\sigma)\equiv\sum_{l=0}^\infty g^{CE}_b(p,l)\,\sigma^l\,,
  \eql{gf}
  $$
where the label, $b=r$ or $a$, indicates the conditions satisfied by the
$p$--form on the boundary, $\partial{\cal M}$, of the fundamental domain, ${\cal M}$, for
the action of $\Gamma$ on S$^d$. Absolute (`$a$') conditions arise when the form
is symmetric under this action while relative (`$r$') ones originate from
anti--symmetric behaviour. The relation is the duality one,
  $$
  g^{CE}_b\big(p,\sigma\big)=g^{CE}_{*b}\big(d-1-p,\sigma\big)\,,
  \eql{du}
  $$
which is a consequence of the $\mbox{\open\char82}^n$ duality,
  $$
  h^{CCC}_{*b}\big(n-p,\sigma\big)=h^{CCC}_b\big(p,\sigma\big)\,,
  \eql{cccd}
  $$
for the closed--coclosed functions, $h^{CCC}$, and the relation between coexact and closed,
  $$
  g^{CE}_b\big(p,\sigma\big)=h^{CCC}_b\big(p+1,\sigma\big)\,.
  \eql{identi}
  $$

An important fact is that forms of the middle rank are self--dual in the
sense that $g^{CE}_b\big((d-1)/2,\sigma\big)=g^{CE}_{*b}\big((d-1)/2,\sigma\big)$.

Equations are developed in [\putref{Dow40}] that allow the generating function
to be found in closed form in terms of the degrees, ${\bf
d}=(d_1,d_2,\ldots,d_d)$, that define the polytope (reflection) group, $\Gamma$.
The case of $p=1$, $d=3$ was treated in detail there. Now I expose the
general result for {\it any} coexact $p$--form in any dimension $d$,
  $$
  g^{CE}_a\big(p,\sigma\big)=(-1)^{d+1+p}{\sum_{q=0}^{d-1-p}
  \,(-1)^qe_{d-q}\big(\sigma^{d_1},\ldots,
  \sigma^{d_d}\big)\over\sigma^{p+1}\prod_{i=1}^d\big(1-\sigma^{d_i}\big)}\,,
  \eql{ggen}
  $$
where the $e_q$ are the elementary symmetric functions.

It is useful to write out the relative generating function from the duality
relation (\puteqn{du}),
  $$\eqalign{
  g^{CE}_r\big(p,\sigma\big)&=(-1)^p{\sum_{q=0}^p
  \,(-1)^qe_{d-q}\big(\sigma^{d_1},\ldots,
  \sigma^{d_d}\big)\over\sigma^{d-p}\prod_{i=1}^d\big(1-\sigma^{d_i}\big)}\,,\cr
  &=(-1)^{p+d}{\sum_{q=d-p}^d
  \,(-1)^qe_q\big(\sigma^{d_1},\ldots,
  \sigma^{d_d}\big)\over\sigma^{d-p}\prod_{i=1}^d\big(1-\sigma^{d_i}\big)}\,.
  }
  \eql{ggen2}
  $$
I derive these expressions later, while developing the formalism.

As in [\putref{Dow40}], the behaviour under the inversion $\sigma\to1/\sigma$ is
important. From (\puteqn{ggen}),
  $$
  g^{CE}_a\big(p,1/\sigma\big)=(-1)^{p+1}\sigma^{p+1}{\sum_{q=0}^{d-1-p}
  \,(-1)^qe_q\big(\sigma^{d_1},\ldots,
  \sigma^{d_d}\big)\over\prod_{i=1}^d\big(1-\sigma^{d_i}\big)}\,,
  \eql{ggen3}
  $$
and combined with (\puteqn{ggen2}), this gives,
  $$
   T^{CE}_a(p,1/\sigma)-(-1)^dT^{CE}_r(p,\sigma)=\sigma^{p-(d-1)/2}\,(-1)^{p+1}\,,
   \eql{tdu}
  $$
after defining,
  $$
  T^{CE}_b(p,\sigma)=\sigma^{(d+1)/2}\,g^{CE}_b(p,\sigma)\,.
  \eql{teepees}
  $$

For self--dual forms, $d$ is odd and (\puteqn{tdu}) gives the {\it symmetrical}
part of the `cylinder kernel',
  $$
  T^{CE}_b(p,1/\sigma)+T^{CE}_b(p,\sigma)=(-1)^{p+1}\,,\quad (b=a,\,r)\,,
  \eql{sdu}
  $$
which can be employed to advantage when evaluating the $\zeta$--function\ in the next
section.

\section{\bf 3. Zeta functions and heat--kernels}

A useful spectral organising quantity is the coexact $\zeta$--function,
  $$
  \zeta^{CE}_b(p,s)=\sum_{l=0}^\infty {g^{CE}_b(p,l)\over\lambda^{CE}(p,l)^s}\,,
  $$
because on a manifold, with or without a boundary, there is the general
decomposition,
  $$
  \zeta_b(p,s)=\zeta_b^{CE}(p,s)+\zeta_b^{CE}(p-1,s)\,,
  \eql{zetot}
  $$
of the total $\zeta$--function\ for the Hodge--de Rham Laplacian.

Only for middle rank forms can $\zeta^{CE}$ be related, using (\puteqn{ceigmo}),
to the generating functions and, for the remainder of this section, I will
make this simplifying restriction so that $d=2p+1$. The $\zeta$--function\ is then,
  $$\eqalign{
  \zeta^{CE}_a(p,s)&={i\Gamma(1-2s)\over2\pi}\int_{C_0}d\tau\,(-\tau)^{2s-1}\,
  T_a^{CE}(p,\tau)\cr
  &={i\Gamma(1-2s)\over2\pi}\int_{C_0}d\tau\,(-\tau)^{2s-1}\,
  e^{-\tau(d+1)/2}\,g^{CE}_a(p,\tau)
  }
  \eql{zet0}
  $$
where I have introduced $\tau=-\log \sigma$ and understand, notationally,
$T^{CE}_a(p,\tau)\equiv T^{CE}_a(p,\sigma)$ and $g^{CE}_a(p,\tau)\equiv
g^{CE}_a(p,\sigma)$. (I have written the absolute quantity, but this equals the
relative one.)

Therefore, from (\puteqn{ggen}),
  $$
  \zeta^{CE}_a(p,s)=(-1)^p
  {i\Gamma(1-2s)\over2\pi}\int_{C_0}\!\!d\tau\,(-\tau)^{2s-1}
  \sum_{q=0}^{d-1-p}\!\! (-1)^q
 {e_{d-q}\big(e^{-d_1\tau},\ldots,
  e^{-d_d\tau}\big)\over\prod_{i=1}^d\big(1-e^{-d_i\tau}\big)}\,.
  \eql{zet1}
  $$

I now recall the integral representation of the Barnes $\zeta$--function,
  $$\eqalign{ \zeta_d(s,a|{\bf d})=&{i\Gamma(1-s)\over2\pi}\int_{C_0}
  d\tau {e^{-a\tau}
  (-\tau)^{s-1}\over\prod_{i=1}^d\big(1-e^{-d_i\tau}\big)}\,,\cr
  }
  \eql{barn}
  $$
so that (\puteqn{zet1}) becomes,
  $$\eqalign{
    \zeta^{CE}_a(p,s)&=(-1)^p\,\sum_{q=0}^{d-1-p}
    (-1)^q\sum_{{i_1<i_2<\ldots<i_q}\atop=1}^d
    \zeta_d\big(2s,\Sigma d-d_{i_1}-\ldots-d_{i_q}\,|\,{\bf d}\big)\cr
    &=(-1)^p\,\sum_{q=0}^{d-1-p}
    (-1)^q\sum_{{i_1<i_2<\ldots<i_{d-q}}\atop=1}^d
    \zeta_d\big(2s,d_{i_1}+\ldots+d_{i_{d-q}}\,|\,{\bf d}\big)\,\cr
     &=(-1)^{p+d}\,\sum_{q=d-p}^d
    (-1)^q
    \sum_{{i_1<i_2<\ldots<i_q}\atop=1}^d
    \zeta_d\big(2s,d_{i_1}+\ldots+d_{i_q}\,|\,{\bf d}\big)\,,\cr
    }
    \eql{zeta1}
  $$
by a simple reordering.

As an example, I calculate the values $\zeta_a^{CE}(p,-k/2)$, $k\in\mbox{\open\char90}$.
Averaging the first and third lines of (\puteqn{zeta1}),
  $$\eqalign{
   \zeta_a^{CE}(p,-k/2)=(-1)^p{k!\over2(d+k)!\prod d_i}\bigg(&
   \sum_{q=0}^{d-1-p}
    (-1)^{q+k}+\sum_{q=d-p}^{d}
    (-1)^q\bigg)\cr
&\times\,\sum_{{i_1<i_2<\ldots<i_q}\atop=1}^d
   B^{(d)}_{d+k}(d_{i_1}+\ldots+d_{i_q}\,|\,{\bf d})\,.\cr
}
  $$

If $k$ is even, the two sums combine,
  $$\eqalign{
   \zeta_a^{CE}(p,-k)&={(-1)^p(2k)!\over2(d+2k)!\prod d_i}
   \sum_{q=0}^d
    (-1)^q\!\!\sum_{{i_1<i_2<\ldots<i_q}\atop=1}^d
   B^{(d)}_{d+2k}(d_{i_1}+\ldots+d_{i_q}\,|\,{\bf d})\,.\cr
}
  \eql{rec2}
  $$
Special interest is attached to the value $k=0$,
  $$\eqalign{
   \zeta_a^{CE}(p,0)&={(-1)^p\over 2d!\prod d_i}
   \sum_{q=0}^d(-1)^q\sum_{{i_1<i_2<\ldots<i_q}\atop=1}^d
   B^{(d)}_d(d_{i_1}+\ldots+d_{i_q}\,|\,{\bf d})\,.\cr
}
  \eql{rec3}
  $$

To evaluate these, I note that the Barnes $\zeta$--function\ satisfies the recursion,
[\putref{Barnesa}],
  $$
  \zeta_d(s,a+d_i\,|\,{\bf d})=\zeta_d(s,a\,|\,{\bf d})
  -\zeta_{d-1}(s,a\,|\,\hat{\bf d}_i)\,,
  $$
whose limiting iteration is, [\putref{Dow9}],
  $$\eqalign{
  \zeta_d(s,a+d_1+\ldots+d_d\,|\,{\bf d})&
  -\sum_{*=1}^d\zeta_d(s,a+d_1+\ldots+*+\ldots+d_d\,|\,{\bf d})\cr
  &+\sum_{*=1}^d\sum_{*=1}^d\zeta_d(s,a+d_1+\ldots
  +*+\ldots+*+\ldots+d_d\,|\,{\bf d})\cr
  &\vdots\cr
  &+(-1)^{d-1}\sum_{i=1}^d\zeta_d(s,a+d_i\,|\,{\bf d})
  +(-1)^d\zeta_d(s,a\,|\,{\bf d})\cr
  &=a^{-s}\,.
  }
  \eql{urec}
  $$

In the first summation, the star denotes that one of the $d$'s is to be
omitted, in turn. In the second summation, every two different pairs of $d$'s
must be successively omitted, and so on. This is the notation of Barnes
[\putref{Barnesa}]. The star summations and omissions correspond to the more
conventional  ordered summations in {\it e.g.}\ (\puteqn{zeta1}).

The iteration, (\puteqn{urec}), on setting $s$ to specific values or by
extracting the poles of the $\zeta$--function, leads to identities involving the
generalised Bernoulli polynomials. For example, setting $s$ equal to $-2k$,
it is possible to safely equate $a$ to zero and (\puteqn{urec}) gives the values
of the expressions (\puteqn{rec2}) and (\puteqn{rec3}). I find,
  $$\eqalign{
   \zeta_a^{CE}(p,-k)&=0\cr
  \zeta_a^{CE}(p,0)&={1\over2}\,(-1)^{p+1}\,,
  }
  \eql{rell}
  $$
which show that the coexact middle form heat--kernel coefficients,
$C^{CE}_{k+d/2}$, $k=1,2,\ldots$, vanish and that the constant term,
$C^{CE}_{d/2}$, equals $(-1)^{p+1}/2$. This derivation is related to, but
independent of, that presented in [\putref{Dow40}]. It does not depend on the
fact that the $C_{d/2}(p)$ coefficient for a general $p$--form vanishes on
the {\it doubled} fundamental domain,
  $$
  C^b_{d/2}(p)+C_{d/2}^{*b}(p)=0\,,\quad\forall p\,.
  \eql{ceeid}
  $$
and could be taken as a proof of this fact.

Furthermore, as mentioned at the end of the previous section, the symmetrical
part of the integrand, (\puteqn{sdu}), produces (\puteqn{rell}) immediately,
bypassing explicit use of the recursion formulae which I have given, though,
for completeness.

The heat--kernel expansion terminates with the constant term, which
generalises the result in [\putref{CandH}] on the full sphere. It is almost
obvious that factoring the sphere will not alter this fact. The only question
would be the effect of the fixed points.

Adding the two (equal) constants for absolute and relative conditions gives
the constant for the doubled fundamental domain. In particular, the value,
$(-1)^{p+1}$, holds for the full sphere. This agrees with the known value,
[\putref{CandH,ELV,Dow8}].

The remaining heat--kernel coefficients, $C_{k/2}^{CE}$, follow from the
`positive' poles of the coexact $\zeta$--function\ at $s=(d-k)/2$, for $k=0,1,\ldots,d-1$,
which themselves result from the known poles of the Barnes $\zeta$--function\ according to
(\puteqn{zeta1}),
  $$\eqalign{
   C^{CE}_{k/2}=(-1)^p{\Gamma\big((d-k)/2\big)
   \over2k!(d-k-1)!\prod d_i}\bigg(&
   \sum_{q=0}^{d-1-p}
    (-1)^q-\sum_{q=d-p}^{d}
    (-1)^{q+k}\bigg)\cr
&\times\,\sum_{{i_1<i_2<\ldots<i_q}\atop=1}^d
   B^{(d)}_k(d_{i_1}+\ldots+d_{i_q}\,|\,{\bf d})\,.\cr
}
  $$
If $k$ is odd, the summations combine to allow the pole part of (\puteqn{urec})
to come into use showing that the boundary coefficients, {\it i.e. } $C_{k/2}^{CE}$
for $k$ odd, are zero, again generalising the result in [\putref{Dow40}]. As
before, this conclusion follows more easily from (\puteqn{sdu}).

There are no such `topological' simplifications or cancellations for the
other values of the $\zeta$--function, for example for the coexact Casimir energy,
  $$\eqalign{
  E&={1\over2}\,\zeta^{CE}_a(p,-1/2)\cr
  &={(-1)^p\over2(d+1)!\prod d_i}
   \sum_{q=0}^p
    (-1)^q\!\!\sum_{{i_1<i_2<\ldots<i_q}\atop=1}^d
   B^{(d)}_{d+1}\big(d_{i_1}+\ldots+d_{i_q}\,|\,{\bf d}\big)\,,
  }
  $$
 and one is reduced to actual computation.
\section{\bf4. Extensions and elaborations}
Although the expression, (\puteqn{zet0}), for $\zeta_b^{CE}(p,s)$ is valid just
for the middle rank forms on S$^d/\Gamma$, it has significance for {\it all} $p$
when the factored sphere is realised as the base of a generalised cone in
$\mbox{\open\char82}^{d+1}$, [\putref{DandKi}], [\putref{BKD}]. I have also referred to this
construction as a bounded M\"obius corner, [\putref{Dow40,Dowl}]. The
separation of variables into radial and angular introduces a term that
effectively cancels the second part of (\puteqn{ceig2}). Gilkey, [\putref{Gilkey}]
\S4.7.5, refers to the resulting operator as the {\it normalised} spherical
Laplacian.

In this case the $T$ quantities defined in (\puteqn{teepees}) are {\it bone
fide} cylinder kernels (without propagation significance) and the
corresponding absolute $\zeta$--function\ is,
   $$\eqalign{
  \zeta^{CE}_a(p,s)&={i\Gamma(1-2s)\over2\pi}\int_{C_0}d\tau\,(-\tau)^{2s-1}\,
  T^{CE}_a(p,\tau)\cr
  &=(-1)^p{i\Gamma(1-2s)\over2\pi}\int_{C_0}\!\!d\tau\,(-\tau)^{2s-1}
  e^{-\big((d-1)/2-p\big)\,\tau}\,\,\times\cr
  &\hspace{****************}\sum_{q=0}^{d-1-p}\!\!
  (-1)^q {e_{d-q}\big(e^{-d_1\tau},\ldots,
  e^{-d_d\tau}\big)\over\prod_{i=1}^d\big(1-e^{-d_i\tau}\big)}\cr
&=(-1)^d\!\!\sum_{q=p+1}^d(-1)^q \!\! \sum_{{i_1<i_2<\ldots<i_q}\atop=1}^d
\!\!\!
    \zeta_d\big(2s,{d-1\over2}-p+d_{i_1}+\ldots+d_{i_q}\,|\,{\bf d}\big)\,,
  }
  \eql{zet4}
  $$
with duality giving the relative $\zeta_r(p,s)=\zeta_a(d-1-p,s)$,
  $$
   \zeta^{CE}_r(p,s)=(-1)^d\!\!\sum_{q=d-p}^d(-1)^q \!\!
   \sum_{{i_1<i_2<\ldots<i_q}\atop=1}^d
\!\!\!
    \zeta_d\big(2s,p-{d-1\over2}+d_{i_1}+\ldots+d_{i_q}\,|\,{\bf d}\big)\,.
    \eql{zet5}
  $$

These $\zeta$--functions\ appear as useful intermediate quantities  but have no independent
dynamical significance. In our work on the ball, Dowker and Kirsten
[\putref{DandKi}], they were referred to as `modified' $\zeta$--functions. The present
results would allow us to extend the ball calculations to factored bases in a
systematic fashion. For example the computations of the scalar functional
determinants reported in [\putref{Dowl}] could be generalised to $p$--forms.

An expression for the modified $\zeta$--function\ on the full sphere is  given in equn.
(42) in [\putref{DandKi}]. A related formula can be obtained from our present
results by adding the absolute and relative expressions on the hemisphere,
for which all the degrees are one. Hence from (\puteqn{zet4}) and (\puteqn{zet5}),
  $$\eqalign{
  \zeta^{CE}_{sphere}(p,s)=(-1)^d\bigg(\sum_{q=p+1}^d(-1)^q&\comb
  dq\,\zeta_d\big(2s,{d-1\over2}-p+q\,|\,{\bf d}\big)+\cr
  &\sum_{q=d-p}^d(-1)^q\comb dq\,\zeta_d\big(2s,p-{d-1\over2}+q\,|\,{\bf
  d}\big)\bigg)\,,
  }
  $$
which, after some manipulation, is equivalent to the form given in
[\putref{DandKi}].\marginnote{CHECK SIGN!}

More generally, adding relative and absolute produces the results for a
`doubled' fundamental domain. Rather than give the full expressions, I will
only look at the consequences of the duality relation, (\puteqn{tdu}) which
yields,
  $$
  T_{a+r}^{CE}(p,\tau)-(-1)^dT_{a+r}^{CE}(p,-\tau)=2(-1)^{p+d}
  \cosh\big(p-(d-1)/2\big)\tau\,,
  $$
allowing $\zeta$--function\ values to be easily found as powers of $p-(d-1)/2$. I will not
do this in detail and only remark that these values are independent of the
factoring, $\Gamma$.
\section{\bf5. The eta invariant}

An important spectral quantity is the eta invariant, $\eta(0)$, which gives a
measure of the asymmetry of the spectrum. Originally introduced by Atiyah,
Patodi and Singer, [\putref{APS}], as a boundary `correction' to an index, it
has achieved an independent life, and its computation has become a standard
challenge. A number of approaches, simple and sophisticated, are available.
The original one, [\putref{APS}], employs the $G$--index theorem. According to
Donnelly, [\putref{Donnelly}], Millson was the first to evaluate $\eta(0)$ on
lens spaces by direct calculation. A direct spectral computation, in the
particular case of spherical space forms, is also mentioned in [\putref{APS}]
and attributed to Ray. Such a calculation was given, later, by Katase,
[\putref{Katase}], on quotients of the 3--sphere. The analysis is somewhat
involved and the result is just the general angle form given previously. Even
so, I outline my own version below.

The actual numbers for the various homogeneous (fixed point free) quotients
of S$^3$ were computed by Gibbons {\it et al}, [\putref{GPR}], who performed
the group average by summing over the angles that define the elements.
Something similar was done by Seade [\putref{Seade}]. In [\putref{Dow11}], I
offered an algebraic alternative to this rather cumbersome geometric
technique. The eta invariant on spherical space forms has been systematically
investigated by Gilkey [\putref{Gilkey,Gilkey3}].

The eta function, $\eta(s)$, measures the asymmetry of the boundary part of
an operator and, as such, is computed on a closed manifold. As an exercise, I
wish to find it for fundamental domains associated with the quotient
S$^d/\Gamma$ and these have a boundary. In fact, things are not quite so bad
because I can work on the doubled fundamental domains resulting from the
restriction to the direct rotational polytope group. However, the domain does
have edges and vertices.

The fundamental domain, ${\cal M}$, of the spherical tesselation is the base of
the generalised (metric) cone formed, on $\mbox{\open\char82}^n$, by the set of reflecting
hyperplanes that define the extended group $\Gamma$. As such, it is part of the
boundary of this cone, or M\"obius corner (kaleidoscope), the other part
being the union of the flat sides. Restricting to the rotational subgroup of
$\Gamma$ turns the cone into a periodic one whose boundary is just its base, the
doubled fundamental domain, $2{\cal M}$. There are, however, singularities of
codimension two, corresponding to the edges of the fundamental domain, and of
codimension three from the vertices, {\it cf } [\putref{DandCh}].

The signature eta function on a $d$--manifold, $N$, (typically a boundary) is
neatly expressed in terms of the middle rank coexact eigenforms, $\phi_l$,
by, [\putref{Cheeger}],
 $$
  \eta(s)=\sum_l \int_N {\phi^*_l\wedge d\phi_l\over \mu^{s+1/2}_l}\,,
 $$
where $\mu_l=\mu^{CE}(p,l)$ of (\puteqn{ceigmo}), and $p=(d-1)/2$ is odd,
($d=4D-1$). For each label, $l$, there are, possibly, two coexact eigenforms
(`positive' and `negative') that can be chosen\footnote{ The restriction
$d=4D-1$ is necessary for this. I have also allowed for complex eigenforms,
although they can be arranged to be real. I have not distinguished
notationally between the positive and negative types. } to be eigenforms of
$*d$,
  $$
  *d\,\phi_l=\pm \,\omega_l\, \phi_l\,,\quad \omega_l=\sqrt\mu_l\,,
  $$
and the sum is over both types. The $\omega$ spectrum is not generally
symmetric.

Despite my preference for the algebraic method, since all the hard work has
been done on the 3--sphere, I initially use angle summation. Only the
signature eta function will be considered and I now derive, again, the
expression obtained long ago in [\putref{APS}].

In physicist's language, the signature eta function in four dimensions is
just the transverse spin--one spectral asymmetry function on the
three--dimensional boundary, [\putref{Dowqg,Dowit}]. For spherical factors, the
necessary spectral information has been given a number of times before in
various connections, {\it e.g.}\ [\putref{Dow10,DandJ}], and repeated in
[\putref{Dow11,Dow12}].

In terms of the positive and negative, spin--1 `Hamiltonian' $\zeta$--functions, on
S$^3/\Gamma$,
  $$\eqalign{
  \zeta^+(s)&=\sum_{\overline L=1}^\infty {d^+(\overline L)\over(\overline
  L+1) ^s}\cr
  \zeta^-(s)&=\sum_{\overline L=3}^\infty {d^-(\overline L)\over(\overline
  L-1) ^s}=\sum_{\overline L=1}^\infty {d^-(\overline L+2)\over(\overline
  L+1) ^s}\,,\cr
  }
  \eql{zetaf}
  $$
the eta function is defined to be,
  $$
  \eta(s)= \zeta^+(s)- \zeta^-(s)\,.
  $$

The degeneracies, $d^{\pm}$, follow from character theory and are given in
the just cited references. The eta function can be written as the group
average,
  $$
  \eta(s)={1\over|\Gamma|}\sum_\gamma \eta(\gamma,s)\,,
  \eql{ga}
  $$
where, by the algebra detailed in [\putref{Dow12}], the partial eta function,
$\eta(\gamma,s)$, is,
  $$
  \eta(\gamma,s)=2\sum_{n=1}^\infty{1\over n^s}{\sin\alpha\sin n\beta-\sin\beta\sin
  n\alpha\over\cos\alpha-\cos\beta}\,.
  \eql{ga2}
  $$
The sum over $\gamma$ in (\puteqn{ga}) is a sum over the angles, $\alpha$ and $\beta$.

The important value is $\eta(0)$ and substitution into (\puteqn{ga2}) shows that
there are no problems with the fixed points. As usual, the identity element
gives zero as do the other special values, $\alpha=0$, $\beta\ne0$, which
correspond to the fixing of a 2--flat in the ambient $\mbox{\open\char82}^4$. The summation
over $n$ is then trivially performed using $2\sum_{n=1}^\infty \sin n\theta=\cot
\theta/2$ giving,
  $$
  \eta(0)=-{1\over|\Gamma|}\sum_{\alpha\ne0;\,\beta} \cot\alpha/2\,\cot\beta/2\,,
  \eql{etav}
  $$
which is, apart from the summation restriction, the standard formula
\footnote{ This derivation is, no doubt, the same as those of Millson and Ray
mentioned earlier.}.

The values of $\alpha$ and $\beta$ corresponding to the elements of the several
polytope groups can now be inserted and the group average performed using the
class decompositions given in [\putref{Dow40}] which were taken from Hurley
[\putref{Hurley}] and Chang [\putref{Chang}]. I find the values for the doubled
fundamental domain,
  $$\eqalign{
  \eta(0)&=-{2\over5\sqrt5}\,\,,\qquad\qquad\qquad \{3^3\} \cr
  &=-{5\over16}\,,\qquad\qquad\qquad\quad \{3^24\}\cr
  &=-{29\over48}\,,\qquad\qquad\qquad\quad\{343\}\cr
  &=-{2341\over5400}-{118\over75\sqrt5}\,, \qquad\,\{3^25\}\,.
  }
  $$
The novelty is the presence of the surd in two cases and the simple fractions
in the others. By contrast, in the evaluation of the Casimir energy all
irrationalities cancel.

The eta function on a lune is, almost trivially, zero because all group
elements fix a 2--flat `axis' of rotation, and these contribute nothing.

Without going through the mode analysis, it is reasonable that the Dirac eta
invariant will be given by the standard, basic expression as given, in Hanson
and R\"omer, [\putref{HandR}], {\it e.g.}, see [\putref{EGH}],
  $$
  \eta_S(0)=-{1\over2|\Gamma|}\sum_{\alpha\ne0;\,\beta} {\rm cosec\,}\alpha/2\,\,{\rm cosec\,}\beta/2\,.
  \eql{etas}
  $$
Numerical evaluation yields the following values,
  $$\eqalign{
  \eta_S(0)&=-{1\over5\sqrt5}\,\,,\,\,\qquad\qquad\qquad\quad \{3^3\} \cr
  &=-{89\over768}-{9\over32\sqrt2}\,,\qquad\qquad \{3^24\}\cr
  &=-{1867\over1728}-{9\over8 \sqrt2}\,,\qquad\qquad\{343\}\cr
  &=-{37291\over7200}+{277\over75}{1\over\sqrt5}\,, \qquad\,\{3^25\}\,.
  }
  $$

The presence of the surds implies that it is not possible to find alternative
expressions for the eta invariant purely in terms of the degrees, $d_i$, as
it is for {\it homogeneous}, fixed point free quotients, [\putref{Dow11}], or
for the Casimir energy.

Incidentally, this conclusion seems not in agreement with the work of
Degeratu, [\putref{Degeratu}], which relates the coefficients in the Laurent
expansion of the Molien (Poincar\'e) series directly to the (Dirac) eta
invariants associated with the boundaries, S$^3/\Gamma_i$, of the orbifolds,
$\mbox{\open\char67}^2/\Gamma_i$ where the $\Gamma_i$ are subgroups of $\Gamma$.

Cheeger, [\putref{Cheeger}], discusses the eta invariant on a generalised cone.
For the standard situation of a smooth manifold, $M$, a generalised cone is
attached to the boundary, $N$, converting $M$ into $X$, a compact space, with
a conical singularity, on which the index can be calculated. In this way,
Cheeger shows that the standard Atiyah--Patodi--Singer formula for Sig($M$)
follows from spectral analysis on the cone, the boundary $\eta(0)$ arising
now from the effect of the cone apex. Also mentioned is the non--standard eta
function on manifolds with boundaries or with conical points and the
possibility that it might be irrational is raised. My computation seems to
confirm this and I leave it at this point.
\section{\bf6. Developing the formalism -- the double Poincar\'e series}

In this section I present a derivation of my basic formulae, (\puteqn{ggen}) and
(\puteqn{ggen2}), from the recursions for the various generating functions given
in [\putref{Dow40}] which are defined as sums over the mode label, as in
(\puteqn{gf}). Although not necessary, I will do this using double generating
functions obtained from the previous ones by summing also over the form rank,
$p$. Such double series are used by Ray on spheres, [\putref{Ray}], but my
approach is different in detail and, in fact, refers to the expressions {\it
after} the group average. They allow for a compressed treatment.

I start with the degeneracy of harmonic polynomial forms on $\mbox{\open\char82}^n$ and
define the double Poincar\'e series, a finite, `fermionic' polynomial in $z$,
  $$
  h_b(z,\sigma)=\sum_{p=0}^n h(p,\sigma)\,z^p={1-\sigma^2\over|\Gamma|}
  \sum_A{{\rm det\,}(1+zA)\over{\rm det\,}(1-\sigma A)}\,\chi^*(A)\,.
  \eql{dhpf}
  $$
It is possible to think of $z$ as a fugacity.

Using standard identities, explicit forms are,
  $$
  h_a(z,\sigma)=(1+z\sigma)\prod_{i=1}^d{1+z\sigma^{m_i}
  \over1-\sigma^{d_i}}\,,
  \eql{dhpf2}
  $$
and
  $$\eqalign{
  h_r(z,\sigma)&=(z+\sigma)\prod_{i=1}^d{z+\sigma^{m_i}
  \over1-\sigma^{d_i}}\cr
  &=z^n\,h_a\big({1\over z},\sigma\big)\,,
  }
  \eql{dhpf3}
  $$
where the final factor in the numerator has been extracted using $m_n=1$.
Equation (\puteqn{dhpf}), without the harmonic factor $(1-\sigma^2)$, is Solomon's
theorem, [\putref{Solomon}], Bourbaki, [\putref{Bourbaki}] p.136, Kane,
[\putref{Kane}] \S22.4,. See also, relatedly, Sturmfels, [\putref{Sturmfels}],
p.37, Exercise 5. The explicit forms are algebraic while the group average is
geometric.

I introduce the notion of polynomial dual, or reciprocal, by the definition,
  $$
  ^*\!f(z)\equiv z^n\,f(1/z)\,,\quad{\it i.e. }\,\,\, ^*{}^*f(z)=f(z)\,,
  $$
on a polynomial, $f$, of (unwritten) degree $n$, so that, {\it e.g.}, the relation
(\puteqn{dhpf3}) reads,
  $$
  h_{*b}(z,\sigma)=\,^*h_b(z,\sigma)\,.
  \eql{hd}
  $$
This helps notationally when dealing with the maximum form rank, $p=n$, on
$\mbox{\open\char82}^n$.

The recursions given in [\putref{Dow40}] transcribe into formulae that can be
solved algebraically. I give the basic steps. For example, the harmonic and
closed harmonic generating functions are related by the recursion,
  $$
  h_b(p,\sigma)=h_b^C(p,\sigma)+\sigma h^C_b(p+1,\sigma)
  \eql{rec}
  $$
which, on account of $h^C_b(p,\sigma)=0$ $(p>n)$, becomes, using the same basic
symbols,
  $$
  h_b(z,\sigma)=\overline h_b^{\,C}(z,\sigma)+{\sigma\over z}\, \big(\overline h^{\,C}_b(z,\sigma)-
  \overline h^{\,C}_b(0,\sigma)\big)\,,
   \eql{rec1}
  $$
where $\overline h$ is defined by,
  $$
  \overline h_b^{\,C}(z,\sigma)=h_b^{\,C}(z,\sigma)-\sigma^2\delta_{ba}\,.
  \eql{hbar}
  $$

In going from (\puteqn{rec}) to (\puteqn{rec1}), the term involving $\sigma^2$ has
been inserted by hand, {\it via} (\puteqn{hbar}), for the reason mentioned in
[\putref{Dow40}] for adding a term, $\delta_{ba}\delta_{p0}\,\delta_{l2}$, to the
solution of the recursion for $h^C_b(p,l)$. It is needed to ensure the
required zero mode end point value, $h_b^C(0,\sigma)=\delta_{ba}$, which is not
covered by the exact sequence that provides the recursion.

It is useful at this point to list some of the end point values,
  $$\eqalign{
  &h_b^C(0,\sigma)=\delta_{ba},\,\quad
 ^*\!h^{C}_b(0,\sigma)=^*\!\!h_b(0,\sigma),\,\quad h^{CCC}_b(0,\sigma)=\delta_{ba}\,,\cr
 &^*\!h^{CCC}_b(0,\sigma)=\delta_{br},\,\quad ^*\!h^{CC}_b(0,\sigma)=\delta_{br},\,
 \quad h^{CC}_b(0,\sigma)=h_b(0,\sigma)\,.
  }
  \eql{epvs}
  $$
The identities established in [\putref{Dow40}] also take on an elegant
appearance. For example the supertraces,
  $$
  h_b(-\sigma,\sigma)=\delta_{ba}(1-\sigma^2)\,,\quad h^{CC}_b(-\sigma,\sigma)=\delta_{ba}\,.
  \eql{sids}
  $$
The first equation is actually an easy consequence of (\puteqn{dhpf}).

Thus, setting $z=-\sigma$ in (\puteqn{rec1}), gives,
  $$
  h_b(-\sigma,\sigma)=\overline h_b^C(0,\sigma)=(1-\sigma^2)\,\delta_{ba}\,,
  $$
as an algebraic check.

The solution of (\puteqn{rec1}) is,
  $$
  h_b^C(z,\sigma)={z\over z+\sigma}\,h_b(z,\sigma)
  +{1+z\sigma\over1+z/\sigma}\,\delta_{ba}\,.
  \eql{hch}
  $$

For the closed and the closed--coclosed functions there are no `correction
terms' and the recursion formula, which has the same form as (\puteqn{rec1}),
gives,
  $$
  h_b^{CCC}(z,\sigma)={z\over z+\sigma}\,
  \,h^{CC}_b(z,\sigma)+{\sigma\over z+\sigma}\,\delta_{ba}\,,
  \eql{clsph}
  $$
using $h^{CCC}_b(0,\sigma)=\delta_{ba}$.

The left--hand side of (\puteqn{clsph}) is the quantity required as it encodes
the closed degeneracies on the $d$--sphere, [\putref{IandT}]. The inputs are
the explicit forms (\puteqn{dhpf2}) and (\puteqn{dhpf3}) which can be used after
relating $h^{CC}$ and $h^C$ by duality on $\mbox{\open\char82}^n$. This yields the various
equivalent forms,
  $$\eqalign{
  h^{CC}_b(z,\sigma)&=z^n\,h^C_{*b}\big({1\over z},
  \sigma\big)=\,^*\!h^{C}_{*b}(z,\sigma)\cr
  &={z^n\over 1+z\sigma}\,h_{*b}\big({1\over
  z},\sigma\big)+{z^n\sigma( z+\sigma)\over 1+z\sigma}\,\delta_{*ba}\cr
  &={1\over 1+z\sigma}\,h_{b}\big(z,\sigma\big)+{z^n \sigma(z+\sigma)
  \over 1+z\sigma}\,\delta_{br}\,.
  }
  \eql{hcc}
  $$
and
  $$\eqalign{
  ^*\!h^{CC}_b(z,\sigma)&=h^{C}_{*b}(z,\sigma)
  ={z\over z+\sigma}\,h_{*b}\big(z,\sigma\big)+{1+z\sigma\over 1+z/\sigma}\,\delta_{br}\,,\cr
  &
  }
  \eql{hcc*}
  $$
using (\puteqn{hch}). These expressions exhibit immediately the end values in
(\puteqn{epvs}).

Substitution into (\puteqn{clsph}) produces,
  $$
  h^{CCC}_b(z,\sigma)={1\over (1+\sigma/z)(1+\sigma z)}\,h_b(z,\sigma)
  +{z^n z\sigma\over 1+z\sigma}\,\delta_{br}
  +{\sigma\over z+\sigma}\,\delta_{ba}\,.
  \eql{hcccz}
  $$

For consistency, duality on $\mbox{\open\char82}^n$ in the form,
  $$
  h^{CCC}_b(z,\sigma)=\,^{*\!}h^{CCC}_{*b}(z,\sigma)\,,
  \eql{hdu}
  $$
can easily be checked. Hence it is sufficient to restrict to absolute
conditions and the explicit form (\puteqn{dhpf2}) results in the final
expression,
  $$
   h^{CCC}_a(z,\sigma)={z\over z+\sigma}\prod_{i=1}^d{1+z\sigma^{m_i}
  \over1-\sigma^{d_i}}+{\sigma\over z+\sigma}\,,
  \eql{hcccz2}
  $$
which is equivalent to (\puteqn{ggen}), taking into account the relation
(\puteqn{identi}).

One way of showing this, is to go backwards and derive the recursion
satisfied by the double Poincar\'e series constructed directly from
(\puteqn{ggen}), virtually paralleling the previous analysis. For simplicity, I
rename $h^{CCC}_a(*,\sigma)$ as $H_a(*)$.

From (\puteqn{ggen}) it is straightforward to derive the form rank recursion,
  $$
   \sigma^{p+1}H_a(p+1)+\sigma^p\, H_a(p)={e_p(\sigma^{d_1},\ldots
   ,\sigma^{d_d})\over\prod_{i=1}^d(1-\sigma^{d_i})}\,.
   \eql{frec}
  $$

I next note that, because of the end point value $^{*\!}H(0)=0$, the top
limit in the sum defining $H(z)$ can be put at $p=d$ (appropriate for working
on S$^d$) and the construction of the generating function of $z$ allows the
recursion (\puteqn{frec}) to be rewritten and then solved, much as before, to
give,
  $$
  H_a(\sigma z)={z\over 1+z}\prod_{i=1}^d{1+z\sigma^{d_i}
  \over1-\sigma^{d_i}}+{1\over1+z}\,,
  $$
which is the same as (\puteqn{hcccz2}) after setting $z\to z/\sigma$, moreover, one
can see how the exponents, $m_i$, turn into the degrees, $d_i$. In this,
slightly synthetic way, I have justified the forms (\puteqn{ggen}) and
(\puteqn{ggen2}). A direct demonstration is possible.

For convenience I give the expression for the absolute coexact double
generating function that follows from (\puteqn{hcccz2}) and the relation
(\puteqn{identi}),
  $$
   g^{CE}_a(z,\sigma)={1\over z+\sigma}\bigg[\prod_{i=1}^d{1+z\sigma^{m_i}
  \over1-\sigma^{d_i}}-1\bigg]\,,
  \eql{gcea}
  $$
a neat encapsulation of my results. For completeness, the relative version
reads \footnote{ When checking the various relations it is necessary to note
that, algebraically, $g^{CE}_a(p,\sigma)=1$ for $p=-1$.},
  $$\eqalign{
   g^{CE}_r(z,\sigma)&={1\over 1+z\sigma}\bigg[\prod_{i=1}^d{z+\sigma^{m_i}
  \over1-\sigma^{d_i}}+z^n\sigma\bigg]\cr
  &=z^{d-1}\bigg(g^{CE}_a\big({1\over z},\sigma\big)+z\bigg)\,.
  }
  \eql{gcer}
  $$
I remark again that the $g^{CE}$ are polynomials in $z$.

  On the $d$--hemisphere,
  $$
   g^{CE}_a(z,\sigma)\bigg|_{hemisphere}={1\over
   z+\sigma}\bigg[\bigg({1+z\over1-\sigma}\bigg)^d-1\bigg]\,,
   \eql{cehsa}
  $$
and
  $$
   z^d g^{CE}_r\big({1\over z},\sigma\big)\bigg|_{hemisphere}={z\over
   z+\sigma}\bigg[\bigg({1+z\over1-\sigma}\bigg)^d+{\sigma\over z}\bigg]\,,
   \eql{cehsr}
  $$
while a supertrace result is,
  $$
   \sigma \,g^{CE}_a(-\sigma,\sigma)+d=\sum_{i=1}^d {1\over1-\sigma^{d_i}}\,.
  $$

\section{\bf7. The 0--form case}

When $z=0$, {\it i.e. } $p=0$, the above expressions for $g^{CE}(z,\sigma)$ do not give
the degeneracies for the {\it general} $\,0$--form. These are, of course,
known, but the most convenient way of including them in the present formalism
is to extend the range of $l$ down to $-1$, which corresponds to a {\it zero
mode}, as is apparent from (\puteqn{ceig2}). This mode exists only for absolute
({\it i.e. } Neumann) conditions and is a uniform function, a polynomial of zero
order. Hence the {\it complete} $0$--form generating function is,
  $$
  h_b(\sigma)=\delta_{ba}+\sigma\, g^{CE}_b(0,\sigma)\,,
  $$
which, from (\puteqn{gcea}) and (\puteqn{gcer}), yield the known forms,
  $$\eqalign{
  h_a(\sigma)&=h_N(\sigma)={1\over\prod_{i=1}^d (1-\sigma^{d_i})}\cr
  h_r(\sigma)&=h_D(\sigma)={\sigma^{d_0}\over\prod_{i=1}^d (1-\sigma^{d_i})}\,,
  \quad d_0=\Sigma_i\, m_i\,.\cr
  }
  $$

\section{\bf8. The double fundamental domain and a check}
Formulae (\puteqn{gcea}) and (\puteqn{gcer}) give the generating functions on the
fundamental domain, ${\cal M}$. That on the doubled domain, $2{\cal M}$, is obtained
by adding these two expressions. This can be checked explicitly for the
hemisphere since the full sphere degeneracies are standard, [\putref{IandT}].
To this end, I split these as in [\putref{DandKi}],
  $$
  g^{CE}(p,l)={(l+d)!\over p!\,(d-p-1)!\,l!}
  \bigg({1\over l+1+p}+{1\over l+d-p}\bigg)\,,\quad l=0,1,\ldots\,\,,
  \eql{stdeg}
  $$
the two parts of which are related by $p\to d-1-p$. Each part corresponds to
a hemisphere contribution. Assuming, for a moment, that this is true, the
combinatorial identity used in [\putref{DandKi}], equn.(41), allows the
eigenvalue generating functions on the hemisphere to be found from
(\puteqn{stdeg}). Then,
  $$\eqalign{
  g^{CE}_a(p,\sigma)\bigg|_{hemisphere}&=\sum_{m=p+1}^d
  \comb{m-1}p{1\over(1-\sigma)^m}\cr
  g^{CE}_r(p,\sigma)\bigg|_{hemisphere}&=\sum_{m=d-p}^d
  \comb{m-1}{d-p-1}{1\over(1-\sigma)^m}\,.
  }
  $$
Construction of the double Poincar\'e series turns these into (\puteqn{cehsa})
and (\puteqn{cehsr}) by simple algebra, (replace the lower limits by zero),
confirming that the two parts do give the two hemisphere contributions. That
is,
  $$
  g^{CE}(z,\sigma)\bigg|_{sphere}=g^{CE}_a(z,\sigma)\bigg|_{hemisphere}
  +g^{CE}_r(z,\sigma)\bigg|_{hemisphere}\,,
  $$
which is a special case of,
  $$
  g^{CE}(z,\sigma)\bigg|_{2{\cal M}}=g^{CE}_a(z,\sigma)\bigg|_{{\cal M}}
  +g^{CE}_r(z,\sigma)\bigg|_{{\cal M}}\,,
  $$
a $p$--form generalisation of the known scalar result.

As well as the sum of relative and absolute expressions, the difference is
also of interest (see the next section). It is preferable to remove the $z^d$
term in (\puteqn{gcer}) and define the combination,
 $$
  w(z,\sigma)=g_r(z,\sigma)-g_a(z,\sigma)-z^d
  \eql{diffg}
 $$
which is a polynomial of degree $d-1$ and is anti--reciprocal,
 $$
  {}^*w(z,\sigma)=-w(z,\sigma)\,.
  \eql{sr}
 $$
\section{\bf9. The counting function}

The counting function, $N(\lambda)$, could be considered the basic global
spectral object\footnote{Probably it first appears in Sturm's 1829 treatment
of the roots of the secular equation}. In general terms, let the eigenvalues,
$\lambda_i$, be ordered linearly $\lambda_1\le\lambda_2\le\ldots\le \lambda_i\le\ldots$ with
$i$ a counting label and degeneracies accounted for by equality. Then a
definition of $N(\lambda)$ is
  $$
  N(\lambda)=\sum_{\lambda_i<\lambda}1+\sum_{\lambda_i=\lambda}{1\over2}\,.
  \eql{cfun1}
  $$
Alternatively, if the spectrum is described by the distinct {\it eigenlevels}
$\lambda(n)$, $n=0,1,2,\ldots$, with explicit degeneracies,
$g(n)=g\big(\lambda(n)\big)$, {\it cf } Baltes and Hilf [\putref{BaandH}],
  $$
  N(\lambda)=\sum_{\lambda(n)<\lambda}g(n)+{1\over2}\,g(n)\,\delta_{\lambda,\lambda(n)}\,.
\eql{cfun2}
  $$
In the case under consideration in this paper, $n$ is $l$, the polynomial
order, and has a dynamical significance. The eigenvalues are functions of
$l$, hence the notation.

The value of $N(\lambda)$ depends on the particular function $\lambda(n)$ in the
sense that, for a fixed argument, $\lambda$, it will vary if the form of $\lambda(n)$
is changed. One could, for example, add a variable constant to the
eigenvalues. On the other hand, the evaluation of $N(\lambda)$ at an eigenvalue,
$G(n)\equiv N\big(\lambda(n)\big)$, is the accumulated degeneracy and does not
depend on the form of $\lambda(n)$,
  $$
  G(n)=\sum_{n'=0}^n g(n')+{1\over2}g(n)\,.
  $$
$G(n)$ satisfies the recursion,
  $$
  G(n)-G(n-1)={1\over2}\big(g(n)+g(n-1)\big)
  $$
which translates into the relation,
  $$
  G(\sigma)={1\over2}\bigg({1+\sigma\over1-\sigma}\bigg)\,g(\sigma)\,.
  \eql{cfun3}
  $$
between generating functions, $G(\sigma)=\sum_{n=0}^\infty G(n)\sigma^n$ {\it etc. }

This equation can be applied to the situation in this paper and, for example,
from (\puteqn{gcea}) I find,
  $$
  G^{CE}_a(\sigma)={1\over2}\bigg({1+\sigma\over1-\sigma}\bigg)
  {1\over z+\sigma}\bigg[\prod_{i=1}^d{1+z\sigma^{m_i}
  \over1-\sigma^{d_i}}-1\bigg]\,.
  \eql{cfun4}
  $$

It is possible to extract the individual accumulated degeneracies\footnote{
The degeneracies themselves can be extracted but will not be exhibited here.}
by writing,
  $$
  G(l)={1\over2\pi i}\oint_C d\sigma\,{1\over\sigma^{l+1}}\,G(\sigma)
  $$
where the contour $C$ circles the origin. The singularities of $N(\sigma)$ lie
only at $\sigma=1$, and it converges as $|\sigma|\to\infty$, so the contour can be
deformed into one, $C'$, around $\sigma=1$,
  $$
  G(l)=-{1\over2\pi i}\oint_{C'} d\sigma\,{1\over\sigma^{l+1}}\,G(\sigma)\,,
  $$
and the calculation is one of residues. In principle this gives an explicit
expression for $N(l)$. As an example, I treat the hemisphere using
(\puteqn{cehsa}) and (\puteqn{cfun3}),
  $$\eqalign{
  G^{CE}_a(l)&=-{1\over4\pi i}\oint_{C'} d\sigma\,{1\over\sigma^{l+1}}
  {1+\sigma\over(1-\sigma)(z+\sigma)}
  \bigg[\bigg({1+z\over1-\sigma}\bigg)^d-1 \bigg]\cr
  &={(-1)^d\over2d!}\,(1+z)^d{d^d\over d\sigma^d}
  {1+\sigma\over(z+\sigma)\sigma^{l+1}}\bigg|_{\sigma=1}-{1\over1+z}\cr
  &=\sum_{r=0}^dA^d_r(z)(l+1)(l+2)
    \ldots(l+r)-{1\over1+z}\,,
  }
  \eql{enel}
  $$
where,
  $$\eqalign{
  A^d_r(z)&={(-1)^{d-r}\over2\, d!}\comb dr(1+z)^d{d^{d-r}\over d\sigma^{d-r}}{1+\sigma\over z+\sigma}
  \bigg|_{\sigma=1}\cr
  &=(-1)^{d-r}{(1+z)^d\over2(d-r)!r!} {d^{d-r}\over
  d\sigma^{d-r}}\bigg(1+{1-z\over
  z+\sigma}\bigg)\bigg|_{\sigma=1}\cr
  &={1\over d!}(1+z)^{d-1}\,,\quad r=d\cr
  &= {1\over 2r!}(1-z)(1+z)^{r-1}\,,\quad r<d\,.}
  \eql{adr}
  $$

Separating the $r=d$ and $r=0$ terms, (\puteqn{enel}) reads,
  $$\eqalign{
  G^{CE}_a(l)={(l+1)\ldots(l+d)\over d!}\,&(1+z)^{d-1}-{1\over2}\,+\cr
  &{1-z\over2}\sum_{r=1}^{d-1}{(1+z)^{r-1}\over r!}\,(l+1)(l+2)\ldots(l+r)\,,}
  \eql{geel}
  $$
which is exact and is, correctly, a finite polynomial in $z$.

It should be noted that, because of the final term in the particular
definition of the counting function, (\puteqn{cfun1}), $G(l)$ is half an
integer.

For a given eigenvalue form, such as (\puteqn{ceig2}), the counting function,
$N(\lambda)$ can be determined by the condition that, if $\lambda(l)\le\lambda<\lambda(l+1)$
then $N(\lambda)=G(l)$.

Particular interest lies in the asymptotic behaviour of $N(\lambda)$ as
$\lambda\to\infty$, in relation to Weyl's conjecture. It is tolerably clear that
the leading term will follow from (\puteqn{geel}) as $l\to \infty$ since, from
(\puteqn{ceig2}) $\lambda\sim l^2$ for very large $\lambda$. The highest power of $l$ is
  $$
  G^{CE}_a(l)\sim {l^d\over d!}\,(1+z)^{d-1}
  $$
which corresponds, after slight manipulation, to Weyl's term
  $$
  N(\lambda)\sim \comb {d-1}p {|{\cal M}|\over(4\pi)^{d/2}}
  {1\over\Gamma(1+d/2)}\,\lambda^{d/2}
  $$
for coexact $p$--forms when ${\cal M}$ is the hemisphere. With more algebraic
work, these calculations can be extended to the general tessellation
(\puteqn{cfun4}).

The point here is that this leading term depends only on the degeneracies and
not on the specific eigenvalues, so long as $\lambda=l^2+o(l)$. It is possible to
go further and derive a form generalisation of the Weyl--Polya conjecture
which says that, in the scalar case, $N_D(\lambda)\le N_N(\lambda)$. Proceeding as
for the scalar case in B\'erard and Besson, [\putref{BandB}], by analogy to the
definition, (\puteqn{diffg}), I define, the modified difference of accumulated
degeneracies,
  $$
  W(z,\sigma)={1\over2}{1+\sigma\over1-\sigma}\,w(z,\sigma)
  =G^{CE}_r(z,\sigma)-G^{CE}_a(z,\sigma)-{1\over2}{1+\sigma\over1-\sigma}\,z^d\,.
  $$
This is an anti--reciprocal polynomial of degree $d-1$ in $z$,
  $$
  {}^*W(z,\sigma)=-W(z,\sigma)\,,
  $$
a statement of duality.

Hence, writing the polynomial as,
  $$
  W(z,\sigma)=w_0+w_1z+\ldots+w_{d-1}z^{d-1}\,,
  $$
one has, $w_i=-w_{d-1-i}$ and, if $d$ is odd, the middle coefficient
$w_{(d-1)/2}$ is zero corresponding to the self--duality of the middle rank
form discussed earlier.

It is then easy to show that $W(z,\sigma)$ vanishes at $z=1$ and also, for odd
$d$, at $z=-1$.

It is also a fact that the lower half coefficients are all negative and the
upper half all positive \footnote{ Unfortunately I have not yet been able to
prove this, but symbolic manipulation verifies it.},
  $$\eqalign{
  w_i&<0\,,\quad i=0,1,\ldots,d/2-1\cr
  w_i&>0\,,\quad i=d/2,1,\ldots,d-1\,,\cr
  }
  $$
which constitutes our generalisation of the Weyl--Polya conjecture.

\section{\bf10. Conclusion}

The main formal results are the Poincar\'e series (\puteqn{gcea}), (\puteqn{gcer}),
for the coexact Laplacian degeneracies on $d$--dimensional fundamental
domains. Some of the specific consequences, such as the termination of the
heat--kernel expansion, are not unexpected. Results involving S$^3$ generally
extend, in some way, to higher odd dimensions.

I note that the heat--kernel expansions are all of the conventional form.
Because of the fixed points, logarithmic terms might have been expected but
it seems that these are hard to generate, [\putref{BandHe}], [\putref{Seeley}].
The $\zeta$--function\ has the standard meromorphic structure and the implication is that
the image method (the group average) automatically yields the Friedrichs
extension.

The consequences of an irrational eta invariant for the signature have yet to
be determined.
\newpage

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\end{putreferences}

\bye

redpajama_set_name: RedPajamaArXiv


text:
Q: APK Crashes immediately on android while clicking on the app icon, Kivymd I'm making an android app project using the Kivy framework. I have created 50% of my project. It works on Windows perfectly then I think to first try and check is it easily converting to APK file  and runs on android? I have created 4 screens SignIn, Signup, HomeScreen and > CameraUI.
Using an opencv library feeding from the webcam. I don't know what should we put in the cv2.VideoCapture() paraenthesis that it should open the android phone camera for now running on the windows i put 0 in the paranthesis
Now I'm stuck in APK it crashes immediately while clicking on its icon. As there is no error showing while creating an APK using > Buildozer. I search a lot to solve the problem but didn't make it. I found a lot of questions related to the same problem but I can't find the solution.
The main.py code is below
from kivymd.app import MDApp
from kivy.lang import Builder
from kivymd.uix.dialog import MDDialog
from kivymd.uix.button import MDFlatButton
from kivy.uix.screenmanager import ScreenManager, Screen
from kivy.core.window import Window
from kivy.graphics.texture import Texture
from kivy.clock import Clock
import cv2

Window.size = (350, 600)

class LoginScreen(Screen):
    pass

class RegistrationScreen(Screen):
    pass

class HomeScreen(Screen):
    pass

class WebCamScreen(Screen):
    flag = None

    def do_start(self):
        flag = False
        self.capture = cv2.VideoCapture(0)
        Clock.schedule_interval(self.load_video, 1.0 / 24.0)

    def load_video(self, *args):
        ret, frame = self.capture.read()
        self.image_frame = frame
        # frame = frame[220:220+250, 400:400+250, :]
        buffer = cv2.flip(frame, 0).tostring()
        image_texture = Texture.create(size=(frame.shape[1], frame.shape[0]), colorfmt="bgr")
        image_texture.blit_buffer(buffer, colorfmt="bgr", bufferfmt="ubyte")
        self.ids.img.texture = image_texture

    def capture_image(self):
        try:
            image_name = "first_pic.jpg"
            cv2.imwrite(image_name, self.image_frame)
        except AttributeError:
            self.flag = MDDialog(
                title="WARNING!",
                text="Please Start the Camera First...!",
                buttons=[
                    MDFlatButton(
                        text="Okay",
                        theme_text_color="Custom",
                        text_color="orange",
                        on_release=self.close_dialog
                    ),
                ],
            )
            self.flag.open()

    def close_dialog(self, obj):
        self.flag.dismiss()

class MainApp(MDApp):

    def build(self):
        screen_manger = ScreenManager()
        screen_manger.add_widget(LoginScreen(name="login"))
        screen_manger.add_widget(RegistrationScreen(name="registration"))
        screen_manger.add_widget(HomeScreen(name="home"))
        screen_manger.add_widget(WebCamScreen(name="camera"))

        return screen_manger

if __name__ == "__main__":
    MainApp().run()

KV code
<LoginScreen>
    MDFloatLayout:
        MDLabel:
            text: "Sign In"
            pos_hint: {"center_x": 0.35, "center_y": 0.7}
            size_hint: (0.5, 0.5)
            font_style: "H4"

        MDTextField
            hint_text: "Username"
            mode: "rectangle"
            pos_hint: {"center_x": 0.5, "center_y": 0.6}
            size_hint_x: 0.8
            icon_left: "account-circle"

        MDTextField
            hint_text: "Password"
            mode: "rectangle"
            size_hint_x: 0.8
            password: True
            icon_left: "key-variant"
            pos_hint: {"center_x": 0.5, "center_y": 0.47}

        MDFlatButton:
            text: "Forget Password?"
            theme_text_color: "Custom"
            text_color: "orange"
            pos_hint: {"center_x": 0.74, "center_y": 0.39}

        MDRaisedButton:
            text: "Sign In"
            md_bg_color: "orange"
            pos_hint: {"center_x": 0.5, "center_y": 0.3}
            on_release:
                root.manager.transition.direction = "left"
                root.manager.current = "home"

        MDLabel:
            text: "Don't have an account?"
            size_hint: (0.5, 0.5)
            font_style: "Caption"
            pos_hint: {"center_x": 0.49, "center_y": 0.15}

        MDFlatButton:
            text: "Register"
            theme_text_color: "Custom"
            text_color: "orange"
            pos_hint: {"center_x": 0.7, "center_y": 0.15}
            on_release:
                root.manager.transition.direction = "left"
                root.manager.current = "registration"

<RegistrationScreen>
    MDFloatLayout:
        MDLabel:
            text: "Sign Up"
            pos_hint: {"center_x": 0.35, "center_y": 0.75}
            size_hint: (0.5, 0.5)
            font_style: "H4"

        MDTextField
            hint_text: "Username"
            mode: "rectangle"
            pos_hint: {"center_x": 0.5, "center_y": 0.65}
            size_hint_x: 0.8
            icon_left: "account-circle"

        MDTextField
            hint_text: "Phone Number"
            mode: "rectangle"
            size_hint_x: 0.8
            password: True
            icon_left: "dialpad"
            pos_hint: {"center_x": 0.5, "center_y": 0.53}

        MDTextField
            hint_text: "Password"
            mode: "rectangle"
            size_hint_x: 0.8
            password: True
            icon_left: "key-variant"
            pos_hint: {"center_x": 0.5, "center_y": 0.41}

        MDRaisedButton:
            text: "Sign Up"
            md_bg_color: "orange"
            pos_hint: {"center_x": 0.5, "center_y": 0.3}

        MDLabel:
            text: "Already have an account?"
            size_hint: (0.5, 0.5)
            font_style: "Caption"
            pos_hint: {"center_x": 0.49, "center_y": 0.15}

        MDFlatButton:
            text: "Sign In"
            theme_text_color: "Custom"
            text_color: "orange"
            pos_hint: {"center_x": 0.75, "center_y": 0.15}
            on_release:
                root.manager.transition.direction = "right"
                root.manager.current = "login"

<HomeScreen>
    MDFloatLayout:
        MDRaisedButton:
            text: "Register Your Face"
            size_hint: (0.7, 0.05)
            md_bg_color: "orange"
            pos_hint: {"center_x": 0.5, "center_y": 0.85}
            on_release:
                root.manager.transition.direction = "left"
                root.manager.current = "camera"

        MDRaisedButton:
            text: "Compare Faces"
            size_hint: (0.7, 0.05)
            md_bg_color: "orange"
            pos_hint: {"center_x": 0.5, "center_y": 0.7}

        MDRaisedButton:
            text: "Remove Face Data"
            size_hint: (0.7, 0.05)
            md_bg_color: "orange"
            pos_hint: {"center_x": 0.5, "center_y": 0.55}

<WebCamScreen>
    MDFloatLayout:
        MDRaisedButton:
            text: "Start Camera"
            size_hint_x: None
            size_hint_y: None
            md_bg_color: "orange"
            pos_hint: {"center_x": 0.5, "center_y": 0.95}
            on_release:
                root.do_start()

        Image:
            id: img
            size_hint_x: 0.85
            size_hint_y: 0.5
            pos_hint: {"center_x": 0.5, "center_y": 0.6}

        MDIconButton:
            icon: "camera"
            md_bg_color: "orange"
            pos_hint: {"center_x": .5, "center_y": .2}
            on_release:
                root.capture_image()

I'm doing it on Google Colab following this blog 2. The Google Colab Way!. I'm running this on Widnows 11.
and my buildozer.spec code is
# (str) Title of your application
title = Face Recognizer

# (str) Package name
package.name = myapp

# (str) Package domain (needed for android/ios packaging)
package.domain = org.test

# (str) Source code where the main.py live
source.dir = .

# (str) Application versioning (method 1)
version = 0.1

# (list) Application requirements
# comma separated e.g. requirements = sqlite3,kivy
requirements = python3,kivy,kivymd,opencv,Pillow,urllib3,Kivy-Garden,numpy

# (str) Supported orientation (one of landscape, sensorLandscape, portrait or all)
orientation = portrait

# change the major version of python used by the app
osx.python_version = 3

# Kivy version to use
osx.kivy_version = 1.9.1

# (bool) Indicate if the application should be fullscreen or not
fullscreen = 0

# (list) Permissions
android.permissions = CAMERA

# (str) Path to a custom kivy-ios folder
#ios.kivy_ios_dir = ../kivy-ios
# Alternately, specify the URL and branch of a git checkout:
ios.kivy_ios_url = https://github.com/kivy/kivy-ios
ios.kivy_ios_branch = master

# Another platform dependency: ios-deploy
# Uncomment to use a custom checkout
#ios.ios_deploy_dir = ../ios_deploy
# Or specify URL and branch
ios.ios_deploy_url = https://github.com/phonegap/ios-deploy
ios.ios_deploy_branch = 1.10.0

# (bool) Whether or not to sign the code
ios.codesign.allowed = false

# (int) Log level (0 = error only, 1 = info, 2 = debug (with command output))
log_level = 2

# (int) Display warning if buildozer is run as root (0 = False, 1 = True)
warn_on_root = 1

And is it mandatory to put the version of the library/package in the buildozer.spec file like (kivymd==this_version).
And thanks in advance :)
And i will realy appreciate the answers to solve my this problem <3.

A: You can use logcat to debug your app, find in gooogle how to use it!

A: This issue is solved by adding the kivy and kivymd versions which I/you used in mine/your project. For my case it was kivy==2.1.0 and kivymd==1.1.1

You can use logcat to debug your app, find in gooogle how to use it!

While the answer of @Сервер Чауш is also useful. the logcat is better for debugging.

redpajama_set_name: RedPajamaStackExchange


text:
The KGB and Me
By Andrei Malgin
I was lucky as a young man. Unlike my fellow college-age friends who were stuck behind the Iron Curtain, in 1977 I was given the opportunity at age 19 to study journalism for three years at the University of Warsaw as an exchange student.
I would have stayed on for a full five years and earned my degree there, but in June 1980, on the eve of the Summer Olympics in Moscow, the KGB decided to purge the ranks of Soviet citizens living abroad and I was sent home. For the next 10 years I was denied an international passport and the right to travel abroad.
I wondered what I had done to deserve such treatment. After all, there had to be some reason for it. Following the events of 1991, when the KGB was having troubles, I used some help from acquaintances and the fact that I was a Moscow city councilor to steal a look at my KGB file.
Of course, I could never pull that off now, but there was a brief period of a few months back in those chaotic days when it was possible.
My first surprise was the size of the file concerning my student years. It turns out that while I innocently led the life of a typical student — meeting women, carousing with friends, hanging out in the library and working odd jobs to get by — people were watching me and reporting my activities to the political police.
Of course, the names of the informants did not appear in writing, but it was possible to guess their identities based on the facts included in the KGB file.
All of the information fell into three categories: material gathered from "friends" in the "bezpieka," the Polish version of the KGB; information obtained from Russian Embassy employees in Warsaw; and reports from Russian informants that were sent directly to Moscow.
Not knowing anything at all about my life, the embassy staff simply did the minimum by submitting meaningless formal replies, or created "facts" out of thin air — such as the allegation that I was planning to flee Poland and escape to Sweden.
The Polish "friends" of the KGB submitted far more substantive information, sometimes entrapping their victims to secure it. One young man invited me to buy a copy of Alexander Solzhenitsyn's book "The Gulag Archipelago," which was banned in the Soviet Union, then described the transaction in his damning report.
My file even noted that I was friends with a girl whose sister had immigrated to England. The bezpieka agents suggested to their Moscow colleagues that this was no chance acquaintance and deserved particular attention.
But the biggest surprise for me was the fact that the KGB obtained most of its information from my closest friend and dormitory roommate, a Soviet student just like me who had traveled with me to Poland.
I never would have expected it from him and hoped I might be mistaken: After all, no surnames were included in the paperwork. Unfortunately, I later learned it was true.
In 1997, Poland, unlike Russia, adopted a law on lustration and even formed the Institute of National Remembrance, tasked with exploring possible collusion between the bezpieka and political and public figures.
Ironically, two Poles who regularly informed on me did not sink into obscurity but enjoyed good careers: one as a prominent journalist, the other as the Polish ambassador to an Asian country.
Even my informant roommate went on to achieve what we could have only dreamed about in our student days, attaining the pinnacle of journalistic success as the host of his own show on NTV television.
It turns out that with or without lustration, the result is the same — but with one major difference: Poland no longer has political police, whereas Russia does. And that is more important than exposing yesterday's informants.
Andrei Malgin is a journalist, literary critic and blogger.
The views expressed in opinion pieces do not necessarily reflect the position of The Moscow Times.
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redpajama_set_name: RedPajamaCommonCrawl


The Government has announced an increase to the Ethnic Communities Development Fund following the 15 March terror attacks.
Minister for Ethnic Communities Jenny Salesa announced on 13 April 2019 that she will co-host a series of meetings across the country with key Muslim leaders.
We recently spoke with Shirish Paranjape, a leader in the Christchurch community, to find out more about his governance board experience.
The following organisations are available to provide those who need support in wake of the Christchurch terror attacks.
The Office of Ethnic Communities Nominations Service supported Pengbo Jiang's appointment to the Department of Internal Affairs' Lottery Significant Projects and Community Facilities Committee.
New Zealand youth are inspiring action on race relations in New Zealand.
More of Us, released 21 March 2019, is a compilation of poetry written by 46 former refugees and migrants from 29 countries, all now living in New Zealand.
The book was published by Landing Press, a small Wellington-based company that aims to publish good poetry for people to enjoy.
The book release was held at the National Library in Wellington by Michael Wood, Under-Secretary to the Minister for Ethnic Communities.
New interactive, online tool will give government agencies, employers and media an understanding of superdiversity!


text:
Alopecurus aquaticus  es una especie no aceptada y puede referirse a:
Alopecurus aquaticus descrita por (Dumort.) Tinant es el Alopecurus arundinaceus de Poir.
Alopecurus aquaticus descrita por Dumort. es el Alopecurus pratensis var. aquaticus de (Dumort.) Mathieu

Referencias

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October means Canzine, art, bikes, zombies and Halloween!
October has been a busy month for dandyhorse and all things bike related. We participated in Nuit Blanche, attended the Ontario Youth Bike Summit and continued with our series on the proposed Harbord bike lane makeover. And this month's fun is not over, we are still looking forward to the upcoming Canzine festival, and of course Halloween!
Toronto's all night city-wide art festival, Nuit Blanche, seems increasingly popular with, well, everyone. This year's insane crowds made it difficult to tour the art-filled city by bike, while at the same time much of the city's art was bike-filled. Chinese dissident artist Ai WeiWei wowed the throngs of spectators/art-zombies who ventured to Nathan Philip Square to see what a 3000-bike art sculpture looks like up close. And dandyhorse helped out with the Kim Adam's exhibit of his intriguing two-headed bikes, which force users to question the meaning of relationships and cooperation in our postmodern, late-capitalist, urban-individualist society (or something like that).
Maybe not as exciting as an all night art party, but definitely as inspiring, was the inaugural Ontario Youth Bike Summit which took place over October 6-7 at North York Civic centre. The event, hosted by Share The Road, had youth leaders from across the province come together to discuss bike advocacy and planning strategy with bike policy experts, and ended with a youth presentation to MPPs at a closing reception at Queen's Park.
Speaking of bike policy and planning, this month we continued our ongoing coverage of The Mysterious Case of the Harbord bike lane. Our latest installment follows the trail of bread crumbs found between the Harbord Bakery, The Harbord BIA offices and City Hall. While details continue to surface, the mystery of the dissappearing bike lane between Spadina and Borden remains unsolved.
dandyhorse is also super-bram-stoked to be participating in this years Canzine festival. Canada's largest festival of zine and underground culture is coming to Toronto this Sunday, at the 918 Bathurst Centre. dandyhorse will be there with our friends at Broken Pencil, and we'll have our latest issue available. We recently talked with Ian Sullivan Cant of Broken Pencil about independent media, zine culture and the Canzine festival.
When the evil anti-bike forces rear their ugly heads, a great way to be reminded of the terrific cycling community we have here in Toronto is to go on a group bike ride. And October means we can express our own, much more creative take on being ugly monsters with this month's costumed Halloween rides and celebrations. The critical mass Halloween ride critical mask will take place on Friday, October 25. And Cycle Toronto (which suspects that the bike lanes approved for Landsdowne must themselves be celebrating Halloween by dressing up as invisible ghosts) will be hosting a Phantom Ride along Landsdowne on Tuesday, October 29.
If you are left wondering about the connection between zombies and bikes, dandy contributor Albert Koehl solves that mystery in his new short essay. We all know the zombie apocalypse is just around the corner, and as cyclists we already have the jump on survival. Find out why in Avoiding the Zombie Apocalypse by Bike. Illustrated beautifully(and seen above) by new dandy contributor Ness Lee. Ness Lee will be at Canzine this Sunday Oct. 20, too.
So many great events still to check out this month!
This Year's Canzine festival of zines and underground culture is this Sunday, October 20! dandyhorse will be there and we'll have our latest edition available. You can find us with our friends at the Broken Pencil table. Hope to see you there! At 918 Bathurst centre, Sunday 1-7 p.m.
Clay and Paper Theatre's 14th annual Night of Dread on October 26 is an evening of costumed wonder not to be missed!
Copyright © *2013 *Dandyhorse Media Inc.*, All rights reserved.


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