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\chapter*{Introduction}
Let $M$ be a compact smooth $m$-dimensional manifold with boundary
$\partial M \neq \emptyset$.
Assuming that $M$ possesses a $Spin^c$ structure, the fundamental class
in the relative $K$-homology group $K_m (M, \pl M)$ can be realized analytically
in terms of the Dirac operator $\dirac$ (graded if $m$ is even) associated
to a riemannian metric on $M$. More precisely,
according to~\cite[\S 3]{BauDouTay:CRC}, if
$\dirac_e$ is a closed extension of $\dirac$ satisfying the condition
that either $\dirac^*_e \dirac_e$ or $\dirac_e \dirac^*_e$
has compact resolvent ({\em e.g.} both $\dirac_{\rm min}$
and $\dirac_{\rm max}$ are such), then the bounded operator
$F = \dirac_e (\dirac^*_e \dirac_e +1)^{-1/2}$ defines a relative Fredholm module over the
pair of $C^*$-algebras $\bigl(\cC(M), \cC(\pl M) \bigr)$, hence an element
$[\dirac] \in K_m (M, \pl M)$. Moreover, by~\cite[\S 4]{BauDouTay:CRC},
the connecting homomorphism
maps $[\dirac]$ to the fundamental class $[\mathsf{D}_\pl] \in K_{m -1} (\partial M)$
corresponding to the Dirac operator $\mathsf{D}_\pl$
associated to the boundary restriction of the metric and of $Spin^c$ structure.
\nind{Index@$\Index_{[{\mathsf D}]}$}
The map $\Index_{[\dirac]} :K^m (M, \pl M) \rightarrow \Z$,
defined by the pairing of $K$-theory with the $K$-homology class of $[\dirac]$, can be expressed
in cohomological terms by means of Connes'
Chern character with values in
cyclic cohomology~\cite{Con:NDG}. Indeed, the relative $K$-homology group
$ \, K_m (M , \pl M)$, viewed as the Kasparov group
$\, KK_m \bigl(\cC_0 (M\setminus \partial M) ; \C\bigr)$,
can be realized as homotopy classes of Fredholm modules over the Fr\'echet algebra
$ \mathcal J^\infty (\partial M,M) $
of smooth functions on $M$ vanishing to any order on $\partial M$;
$ \mathcal J^\infty (\partial M,M) $ is a
local $C^*$-algebra, $H$-unital and dense in
$\cC_0 (M\setminus \partial M) = \{ f \in \cC (M) \mid f_{|\partial M}=0\}$.
One can therefore define the {\CoChch} of $[\dirac]$
by restricting the operator $F= \dirac_e (\dirac^*_e \dirac_e +1)^{-1/2}$,
or directly $\dirac$, to $\cJ^\infty (\partial M, M)$ and regarding
it as a finitely summable Fredholm module.
The resulting periodic cyclic cocycle corresponds,
via the canonical isomorphism between the periodic cyclic cohomology
$\, HP^{\textrm{ ev/odd}} \bigl(\cJ^\infty (\partial M, M)\bigr)$ and
the \deRham\ homology $\, H_{\rm ev/odd}^{\rm dR}
(M \setminus \partial M ; \C)$ ({\it cf.}~\cite{BraPfl:HAWFSS}),
to the \deRham\ class of the current (with arbitrary support)
associated to the $\hat{A}$-form of the riemannian metric. In fact, one can even recover the
$\hat{A}$-form itself out of local cocycle representatives for the {\CoChch},
as in~\cite[Remark 4, p. 119]{ConMos:TCC} or
\cite[Remark II.1, p. 231]{ConMos:LIF}.
However, the boundary $\pl M$ remains conspicuously absent in such
representations.
\sind{de Rham!homology}
It is the purpose of this paper to provide cocycle representatives for
the {\CoChch} of the fundamental $K$-homology
class $[\dirac] \in K_m (M, \partial M)$
that capture and reflect geometric information about the boundary.
Our point of departure is Getzler's construction~\cite{Get:CHA} of
the {\CoChch} of $[\dirac]$. Cast
in the propitious setting of
\textnm{Melrose}'s \textup{b}-calculus~\cite{Mel:APSIT},
Getzler's cocycle has however the disadvantage,
from the viewpoint of its geometric functionality, of
being realized not in the relative
cyclic cohomology complex proper but in its entire extension. Entire
cyclic cohomology~\cite{Con:ECC}
was devised primarily for handling infinite dimensional geometries and
is less effective as a tool than ordinary cyclic cohomology when dealing with
finite dimensional
$K$-homology cycles. To remedy this drawback we undertook the task of producing
cocycle realizations for the {\CoChch} directly in the
relative cyclic cohomology complex associated to the pair of algebras
$\bigl( \cC^\infty (M), \cC^\infty (\partial M)\bigr)$. This is achieved by adapting and
implementing in the context of relative cyclic cohomology the
retraction procedure of~\cite{ConMos:TCC}, which converts
the entire {\CoChch} into the periodic one. The resulting
cocycles automatically carry information about the boundary and this allows
to derive geometric consequences.
\medskip
It should be mentioned that the relative point of view in the framework
of cyclic cohomology was first exploited in \cite{LesMosPfl:RPC}
to obtain cohomological expressions for $K$-theory invariants associated
to parametric pseudodifferential operators.
It was subsequently employed by Moriyoshi and Piazza to establish a
Godbillon-Vey index pairing for longitudinal
Dirac operators on foliated bundles \cite{MorPia:RPA}.
\medskip
Here is a quick synopsis of the main results of the present paper.
Throughout the paper, we
fix an exact \textup{b}-metric $g$ on $M$, and denote by $\dirac$ the corresponding
\textup{b}-Dirac operator.
We define for each $t > 0$ and any $n \ge m = \dim M$, $n \equiv m$ (mod $2$),
pairs of cochains
\begin{equation}\label{eq:relCochain}
\bigl(\bch^n_t(\dirac), \ch^{n+1}_t (\mathsf{D}_\pl) \bigr)\quad
\text{resp.}\quad \bigl(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^n_t(\dirac), \ch^{n-1}_t(\mathsf{D}_\pl) \bigr)
\end{equation}
over the pair of algebras $ \bigl( \cC^\infty (M), \cC^\infty (\partial M)\bigr)$, given
by the following expressions:
\begin{equation} \label{cht}
\begin{split}
\bch^n_t(\dirac)\, &:=\,
\sum_{j\geq 0} \bCh^{n-2j} (t \dirac) +
B \bTslch^{n+1}_t (\dirac),\\
\ch^{n+1}_t (\mathsf{D}_\pl)\, &:= \,
\sum_{j\geq 0} \Ch^{n-2j+1} (t \mathsf{D}_\pl) +
B \Tslch^{n+2}_t (\mathsf{D}_\pl),\\
\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^n_t(\dirac) \, &:=\,
\bch^n_t(\dirac) +\Tslch^n_t(\mathsf{D}_\pl)\circ i^* .
\end{split}
\end{equation}
In these formul\ae, $\,\Ch^{\bullet} ( \mathsf{D}_\pl)$ stand for the components
of the \textnm{Jaffe-Lesniewski-Osterwalder} cocycle~\cite{JLO:QKT}
representing the {\CoChch} in entire cyclic cohomology,
$\bCh^{\bullet} ( \dirac)$ denote the corresponding \textup{b}-analogue
({\it cf.} \eqref{Eq:DefChern}), while the cochains
$\Tslch^{\bullet}_t (\mathsf{D}_\pl)$, resp. $\bTslch^{\bullet}_t ( \dirac) $
(see \eqref{eq:ML200909263},
are manufactured out of the canonical transgression formula
as in \cite{ConMos:TCC};
$\, i : \partial M \rightarrow M$ denotes the inclusion.
One checks that
\begin{equation} \label{relcohochern}
\begin{split}
(b + B) \bigl(\bch^n_t (\dirac) \bigr) \, &=\,
\ch^{n+1}_t (\mathsf{D}_\pl) \circ i^\ast\\
(b + B) \bigl(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^n_t (\dirac) \bigr) \, &=\,
\ch^{n-1}_t (\mathsf{D}_\pl) \circ i^\ast,
\end{split}
\end{equation}
which shows that the cochains \eqref{eq:relCochain} are cocycles in the
\emph{relative}
total $(b, B)$-complex of
$ (\cC^\infty (M), \cC^\infty (\partial M) ) $. Moreover,
the class of this cocycle in periodic cyclic cohomology is independent
of $t > 0$ and of $n = m + 2k, \, k \in \Z_+$. Furthermore,
its limit as $t \rightarrow 0$ gives the pair of
$\hat A$ currents corresponding to the \textup{b}-manifold M, that is
\begin{equation} \label{eq: 0lim cocycle}
\big(\lim_{t\searrow 0} \bch^n_t (\dirac), \, \lim_{t\searrow 0}
\ch^{n+1}_t (\mathsf{D}_\pl) \big)
= \left(\int_{\tb M} \hat{A}(\bnabla^2_g) \wedge \bullet , \,
\int_{\pl M}\hat{A}(\nabla^2_{g_\partial}) \wedge \bullet\right) \, .
\end{equation}
The notation in the right hand side requires an explanation. With
\begin{equation} \label{eq: A-currents}
\hat{A}(\bnabla^2_g) = \det \left(\frac{\bnabla_g^2/4 \pi i}{\sinh \bnabla_g^2/4 \pi i}\right)^{\frac{1}{2}} ,
\quad \hat{A}(\nabla^2_{g_\partial}) =
\det \left(\frac{\nabla_{g_\partial}^2/4 \pi i}{\sinh \nabla_{g_\partial}^2/4 \pi i}\right)^{\frac{1}{2}} ,
\end{equation}
and $\displaystyle \int_{\tb M} : \bOmega^m (M) \rightarrow \C$ denoting
the \textup{b}-integral of \textup{b}-differential $m$-forms on $M$ associated to the
trivialization of the normal bundle to $\pl M$ underlying the \textup{b}-structure,
both terms in the right hand side of \eqref{eq: 0lim cocycle} are viewed as
$(b, B)$-cochains associated to currents. More precisely, incorporating
the $2 \pi i$ factors which account for the conversion of the Chern character
in cyclic homology into the Chern character in \deRham\ cohomology,
for $M$ even dimensional this identification takes the form
\begin{multline}\label{eq:bA-hat cochain}
\int_{\tb M} \hat{A}(\bnabla^2_g) \wedge \big(f_0, \ldots , f_{2q}\big)\\
= \frac{1}{(2 \pi i)^q (2q)!}
\int_{\tb M} \hat{A}(\bnabla^2_g) \wedge f_0\, df_1\wedge \ldots \wedge f_{2q} \, ,
\end{multline}
respectively
\begin{multline}\label{eq:Abdry-hat cochain}
\int_{\partial M} \hat{A}(\nabla^2_{g_\partial}) \wedge \big(h_0, \ldots , h_{2q-1}\big) \\
= \frac{1}{(2 \pi i)^{q} (2q-1)!}
\int_{\partial M} \hat{A}(\nabla^2_{g_\partial}) \wedge h_0\, dh_1\wedge \ldots \wedge h_{2q-1} .
\end{multline}
Finally (\textit{cf.}~Theorem \ref{t: CC-character} \textit{infra}),
the limit formula \eqref{eq: 0lim cocycle} implies that both
$\big(\bch^n_t (\dirac),\ch^{n+1}_t (\mathsf{D}_\pl) \big)$ and $\big(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^n_t (\dirac),\ch^{n-1}_t (\mathsf{D}_\pl) \big)$
represent the Chern character of the fundamental relative $K$-homology class $[\dirac] \in K_m(M,\partial M)$.
Under the assumption that $\Ker \mathsf{D}_\pl = 0$, we next prove (Theorem
\ref{t:ML20081215} \textit{infra})
that the pair of retracted
cochains $\big(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^n_t (\dirac),\ch^{n-1}_t (\mathsf{D}_\pl) \big)$ has a limit
as $t\to\infty$. For $n$ even, or equivalently $M$ even dimensional,
this limit has the expression
\begin{equation}\label{eq:ML20081215-1-intro}
\begin{split}
\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}_\infty^n(\dirac)&=\sum_{j=0}^{n/2} \kappa^{2j} (\dirac)+
B\bTslch_\infty^{n+1}(\dirac)+\Tslch_\infty^n(\mathsf{D}_\pl)\circ i^* , \\
\ch_\infty^{n-1}(\mathsf{D}_\pl)&= B\Tslch_\infty^{n}(\mathsf{D}_\pl) ,
\end{split}
\end{equation}
with the cochains $\kappa^{\bullet}(\dirac)$, occurring only when $ \Ker\dirac\neq\{0\}$,
given by
\begin{equation} \label{eq: Ker Str}
\kappa^{2j} (\dirac)(a_0, \ldots ,a_{2j})=\,
\Str \bigl(\varrho_H(a_0)\, \go_H(a_1,a_2) \cdots \go_H(a_{2j-1},a_{2j})\bigr) ;
\end{equation}
here $H$ denotes the orthogonal projection onto $ \Ker \dirac$, and
\begin{align*}
\varrho_H(a):=HaH\, , \quad
\go_H(a,b):= \varrho_H(ab)-\varrho_H(a)\varrho_H(b) ,
\text{ for all $a, b \in \cC^\infty (M)$} .
\end{align*}
When $M$ is odd dimensional, the limit cocycle takes the
similar form
\begin{equation}\label{eq:ML20081215-2-intro}
\begin{split}
\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}_\infty^n(\dirac)&=B\bTslch_\infty^{n+1}(\dirac)+ \Tslch_\infty^n(\mathsf{D}_\pl)\circ i^*\, , \\
\ch_\infty^{n-1}(\mathsf{D}_\pl)&= B\Tslch_\infty^n(\mathsf{D}_\pl).
\end{split}
\end{equation}
The absence of cochains of the form $ \kappa^{\bullet} (\mathsf{D}_\pl)$
in the boundary component is due to the assumption that $\Ker \mathsf{D}_\pl = 0$.
\medskip
The geometric implications become apparent when one inspects
the ensuing pairing with
$K$-theory classes. For $M$ even dimensional,
a class in $K^m (M, \pl M)$ can be represented
as a triple $[E,F, h]$, where $E$, $F$ are vector bundles over $M$, which we will
identify with projections $p_E, p_F \in \Mat_N (\cC^\infty (M))$,
and $h :[0, 1] \rightarrow \Mat_N (\cC^\infty (\partial M))$
is a smooth path of projections connecting their restrictions to the boundary
$E_\pl $ and $F_\pl $. For $M$ odd dimensional, a representative of a class in
$K^m (M, \pl M)$ is a triple $(U,V,h)$, where $U,V:M\to U(N)$ are unitaries and $h$
is a homotopy between their restrictions to the boundary $U_{\pl}$ and $V_{\pl}$.
In both cases, the Chern character of $[X, Y , h] \in K^m (M, \partial M)$ is
represented by the relative cyclic homology cycle over the algebras
$\bigl( \cC^\infty (M), \cC^\infty (\partial M)\bigr)$
\begin{equation}
\ch_\bullet \big( [X, Y, h] \big) \, = \,
\Big(
\ch_\bullet(Y) - \ch_\bullet (X) \, , \, - \Tslch_\bullet (h) \Big) \, ,
\end{equation}
where $ \ch_\bullet $, resp. $\Tslch_\bullet $ denote the components of
the standard Chern character in cyclic homology resp.~of its canonical
transgression (see Section \ref{s:TheChernCharacter}).
The pairing $\big\langle [\dirac], \, [X, Y,h] \big\rangle\in\Z$
between the classes $[\dirac] \in K_m(M,\partial M)$ and
$[X, Y , h] \in K^m (M, \partial M)$ acquires the cohomological expression
\begin{equation}\label{eq:relpair even/odd}
\begin{split}
\big\langle [\dirac], \, [X,& Y,h] \big\rangle \, =\,
\big\langle \big( \,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}_t^n (\dirac), \ch_t^{n-1} (\mathsf{D}_\pl) \big), \,
\ch_\bullet [X, Y,h] \big\rangle = \\
= \big\langle& \sum_{j\ge 0} \bCh^{n-2j} (t\dirac) + B\, \bTslch_t^{n+1} (\dirac),
\, \ch_\bullet (Y) -\ch_\bullet (X) \big\rangle \\
&+\big\langle \Tslch_t^n(\mathsf{D}_\pl),\ch_\bullet(Y_\pl)-\ch_\bullet(X_\pl)\big\rangle\\
&- \big\langle \sum_{j\ge 0} \Ch^{n-2j-1} (t\mathsf{D}_\pl) +
B \Tslch_t^{n} (\mathsf{D}_\pl), \,
\Tslch_\bullet (h) \big\rangle ,
\end{split}
\end{equation}
which holds for any $t > 0$.
Letting $t \rightarrow 0$ yields the local form of the pairing formula
\begin{equation} \label{eq: pairing lim0}
\begin{split}
\big\langle& [\dirac], \, [X,Y, h] \big\rangle \, =\,
\\
&= \int_{\tb M} \hat{A}(\bnabla^2_g) \wedge \big( \ch_\bullet (Y) - \ch_\bullet (X) \big)
- \int_{\partial M} \hat{A}(\nabla^2_{g_\partial}) \wedge \Tslch_\bullet (h) .
\end{split}
\end{equation}
It should be pointed out that \eqref{eq: pairing lim0} holds in complete generality, without
requiring the invertibility of $\mathsf{D}_\pl$.
When $M$ is even dimensional
and $\mathsf{D}_\pl$ is invertible, the equality between the above limit and
the limit as $t \rightarrow \infty$ yields, for any $n =2\ell \geq m $, the
identity
\begin{equation} \label{eq:APS-ind}
\begin{split}
\sum_{0\leq k \leq \ell} &
\big\langle \kappa^{2k}(\dirac) , \ch_{2k}(p_F) - \ch_{2k}(p_E)\big\rangle +
\big\langle B \bTslch_\infty^{n+1} (\dirac) , \ch_n(p_F) - \ch_n(p_E) \big\rangle \, +\\
&\quad +
\big\langle \Tslch_\infty^n(\mathsf{D}_\pl), \ch_n (p_{F_\pl})-
\ch_n (p_{E_\pl})\big\rangle \,=\\
&= \int_{\tb M} \hat{A}(\bnabla^2_g) \wedge
\big( \ch_\bullet (p_F) - \ch_\bullet (p_E) \big)
- \int_{\partial M} \hat{A}(\nabla^2_{g_\partial}) \wedge \Tslch_\bullet (h) \\
&\qquad\quad +\big\langle B \Tslch_\infty^{n} (\mathsf{D}_\pl),
\Tslch_{n-1} (h) \big\rangle ,
\end{split}
\end{equation}
where
\begin{displaymath}
\begin{split}
\ch_{2k} (p) = &
\begin{cases}
\tr_0 (p), & \text{for $k=0$}, \\
(-1)^{k} \frac{(2k)!}{k!} \, \tr_{2k}
\big( (p-\frac 12) \otimes p^{\otimes 2k}\big), & \text{for $k>0$} .
\end{cases}
\end{split}
\end{displaymath}
Like the Atiyah-Patodi-Singer index formula~\cite{APS:SARI}, the
equation \eqref{eq:APS-ind}
involves index and eta cochains, only of higher order.
Moreover, the same type of
identity continues to hold in the odd dimensional case.
Explicitly, it takes the form
\begin{equation}
\begin{split}
(-1)^{\frac{n-1}{2}} & \, \textstyle{\big( \frac{n-1}{2} \big)!}\,
\Big(\big\langle B \bTslch_\infty^{n+1} (\dirac) ,
(V^{-1} \otimes V)^{\otimes \frac{n+1}{2}} -
(U^{-1} \otimes U)^{\otimes \frac{n+1}{2}} \big\rangle \, +\\
&\qquad \big\langle \Tslch_\infty^n(\mathsf{D}_\pl),
(V^{-1}_\pl \otimes V_\pl)^{\otimes \frac{n+1}{2}} -
(U^{-1}_\pl \otimes U_\pl)^{\otimes \frac{n+1}{2}} \big\rangle \Big) \,=\\
&= \int_{\tb M}\hat{A}(\bnabla^2_g)\wedge\big( \ch_\bullet (V) - \ch_\bullet (U)\big)
- \int_{\partial M} \hat{A}(\nabla^2_{g_\partial}) \wedge \Tslch_\bullet (h) \\
&\qquad +\big\langle B \Tslch_\infty^{n} (\mathsf{D}_\pl), \Tslch_{n-1} (h) \big\rangle.
\end{split}
\end{equation}
\medskip
The relationship between the relative pairing and the Atiyah-Patodi-Singer index theorem
can actually be made explicit, and leads to interesting geometric consequences. Indeed,
under the necessary assumption that $M$ is even dimensional, we show
({\it cf.} Theorem \ref{t:RelativePairingKTheory}) that
the above pairing can be expressed as follows:
\begin{equation}
\label{Eq:Pairing1}
\begin{split}
\langle [\dirac], [E, F, h] \rangle = \indAPS \dirac^F - \indAPS \dirac^E +
\SF (h , \dirac_\partial) ;
\end{split}
\end{equation}
here $\indAPS$ stands for
the {\APS}-index, and $ \SF (h , \dirac_\partial) $ denotes the spectral
flow along the path of operators $\bigl(\mathsf{D}_\pl^+\bigr)^{h(s)}$;
$\mathsf{D}_\pl^+$ is the restriction of $\sfc(dx)^{-1}\mathsf{D}_\pl$ to the positive
half spinor bundle and $\sfc(dx)$ denotes Clifford multiplication by the inward normal vector.
On applying the {\APS}\ index formula~\cite[Eq. (4.3)]{APS:SARI}, the
pairing takes the explicit form
\begin{equation}
\label{Eq:Pairing2}
\begin{split}
\langle [\dirac], [E, F, h] \rangle \,= &
\int_{\tb M} \hat{A}(\bnabla^2_g) \wedge \big( \ch_\bullet (F)- \ch_\bullet (E) \big) \\
& - \Big( \xi (\mathsf{D}_\pl^{+,F_\partial}) - \xi (\mathsf{D}_\pl^{+,E_\partial}) \Big)
\, + \, \SF (h, \dirac_\partial) ,
\end{split}
\end{equation}
where
\begin{equation} \label{xi}
\xi (\mathsf{D}_\pl^{+,E_\partial}) \, = \,
\frac{1}{2} \Big( \eta (\mathsf{D}_\pl^{+,E_\partial}) \, + \,
\dim \Ker \mathsf{D}_\pl^{+,E_\partial} \Big) .
\end{equation}
Comparing this expression with the local form of the pairing \eqref{eq: pairing lim0}
leads to a generalization
of the {\APS}\
odd-index formula~\cite[Prop. 6.2, Eq. (6.3)]{APS:SARIII}, from trivialized flat bundles
to pairs of equivalent vector bundles in $K$-theory. Precisely ({\it cf.}
Corollary \ref{t: gen APS flat}), if
$E',F'$ are two such bundles on a closed odd dimensional spin manifold $N$,
and $h$ is the homotopy implementing the equivalence of $E'$ with $F'$, then
\begin{equation}
\begin{split}
\xi (\dirac_{g'}^{F'}) - \xi (\dirac_{g'}^{E'})
\, = \, \int_{N} \hat{A} (\nabla_{g'}^2) \wedge
\Tslch_\bullet (h) \, + \, \SF (h , \dirac_{g'}) \, ,
\end{split}
\end{equation}
where $\dirac_{g'}$ denotes the Dirac
operator associated to a riemannian metric $g'$ on $N$; equivalently,
\begin{equation}
\int_0^1 \frac{1}{2} \frac{d}{dt} \big( \eta (p_{h(t)} \, \dirac_{g'} \, p_{h(t)}) \big) dt
\, = \, \int_{N} \hat{A} (\nabla_{g'}^2) \wedge \Tslch_\bullet (h) \, ,
\end{equation}
where $p_{h(t)}$ is the path of projections joining $E'$ and $F'$, and
the left hand side is the natural extension of the
real-valued index in~\cite[Eq. (6.1)]{APS:SARIII}.
\medskip
Let us briefly comment on the main analytical challenges encountered in
the course of proving the results outlined above. In order to compute
the limit as $t\searrow 0$ of the Chern character, one needs
to understand the asymptotic behavior of expressions of the form
\begin{equation}
\begin{split}
\blangle& A_0,A_1,\ldots,A_k\rangle_{\sqrt{t}\dirac}\\
&:=\int_{\Delta_k}
\bTr\bigl( A_0 e^{-\sigma_0 t\dirac^2} A_1 e^{-\sigma_1 t\dirac^2} \ldots A_k e^{-\sigma_k t\dirac^2} \bigr) d\sigma,
\end{split}
\end{equation}
where $\Delta_k$ denotes the standard simplex $\{\sigma_0+\ldots+\sigma_k=1, \sigma_j\ge 0\}$ and $A_0,\ldots,A_k$
are \textup{b}-differential operators of order $d_j, j=0,\ldots,k$;
$d:=\sum_{j=0}^k d_j$ denotes the sum of their orders.
The difficulty here is twofold. Firstly, the \textup{b}-trace is a
\emph{regularized} extension of the trace to \textup{b}-pseudodifferential operators on the
\emph{non-compact} manifold $M\setminus \partial M$ (recall that the \textup{b}-metric
degenerates at $\partial M$).
Secondly, the expression inside the \textup{b}-trace involves a product of
operators. The Schwartz kernel
of the product
$A_0 e^{-\sigma_0 t\dirac^2}A_1 e^{-\sigma_1 t\dirac^2}\cdot \ldots\cdot A_k e^{-\sigma_k t \dirac^2}$
does admit a \emph{pointwise} asymptotic expansion
(see \cite{Wid:STC}, \cite{ConMos:CCN}, \cite{BloFox:APO}),
namely
\begin{equation
\begin{split}
\Bigl(A_0 &e^{-\sigma_0 t\dirac^2} A_1 e^{-\sigma_1 t\dirac^2} \cdot\ldots\cdot A_k e^{-\sigma_k t \dirac^2}\Bigr)(p,p) \\
&=: \sum_{j=0}^{n} a_j(A_0,\ldots,A_k,\dirac)(p)\; t^{\frac{j-\dim M
-d}{2}}+O_p(t^{(n+1-d-\dim M)/2}).
\end{split}
\end{equation}
However, this asymptotic expansion is
only \emph{locally} uniform in $p$; it is not \emph{globally} uniform on the non-compact manifold $M\setminus \partial M$.
A further complication
arises from the fact that the function $a_j(A_0,\ldots,A_k,\dirac)$ is not necessarily integrable.
Nevertheless, a \emph{partie finie}-type regularized integral,
which we denote by $\int_{\tb M} a_j(A_0,\ldots,A_k,\dirac)d\vol $, does exist
and we prove (\emph{cf.} Theorem \ref{t:JLObCommutatorAsymptotic})
that the corresponding \textup{b}-trace admits an asymptotic expansion of the
form
\begin{equation
\begin{split}
\bTr\Bigl(A_0 &e^{-\sigma_0 t\dirac^2} A_1 e^{-\sigma_1 t\dirac^2} \ldots A_k e^{-\sigma_k t \dirac^2}\Bigr) \\
&= \sum_{j=0}^{n} \int_{\tb M} a_j(A_0,\ldots,A_k,\dirac)d\vol \;
t^{\frac{j-\dim M -d}{2}}+\\
&\qquad +O\Bigl(\bigl(\prod_{j=1}^k
\sigma_j^{-d_j/2}\bigr)t^{(n+1-d-\dim M)/2}\Bigr).
\end{split}
\end{equation}
When $\,\mathsf{D}_\pl$ is invertible and hence $\dirac$ is a Fredholm operator, we can also prove
the following estimate (\emph{cf.} \eqref{eq:integrated-multiple-btrace-estimate-large})
\begin{equation
\begin{split}
|\blangle A_0&(I-H),...,A_k(I-H)\rangle_{\sqrt{t}\dirac}|\\
&\le \tilde C_{\delta,\eps}
t^{-d/2-(\dim M)/2-\eps} e^{-t\delta},\quad \text{for \emph{all }} 0<t<\infty ,
\end{split}
\end{equation}
for any $\eps>0$ and any $0<\delta<\inf\specess \dirac^2$. Here, $\specess$
denotes the essential spectrum and $H$ is the orthogonal projection onto $ \Ker\dirac$. This estimate allows us
to compute the limit as $t \nearrow \infty$ and thus
derive the formul\ae\,
\eqref{eq:ML20081215-1-intro} and \eqref{eq:ML20081215-2-intro}.
\medskip
A few words about the organization of the paper are now in order.
We start by recalling, in Chapter \ref{s:preliminaries},
some basic material on relative cyclic cohomology~\cite{LesMosPfl:RPC},
\textup{b}-calculus~\cite{Mel:APSIT} and Dirac operators.
In Section \ref{s:btrace} we discuss in detail the \textup{b}-trace
in the context of a manifold with cylindrical ends.
As a quick illustration of the usefulness of the relative cyclic cohomological
approach in the present context,
we digress in Section \ref{s:McK-S} to establish an
analogue of the well-known McKean--Singer formula for
manifolds with boundary; we
then employ it to recast in these terms
Melrose's proof of the Atiyah-Patodi-Singer index theorem
(\emph{cf.}~\cite[Introduction]{Mel:APSIT}).
In Section
\ref{s: b-trace formula}, refining an observation due to Loya~\cite{Loy:DOB},
we give an effective formula for the \textup{b}-trace,
which will turn out to be a convenient technical device.
After setting up the notation related to \textup{b}-Clifford modules and \textup{b}-Dirac
operators in Section \ref{s:b-Clifford-Dirac}, we revisit in the remainder of the chapter
Getzler's version of the relative entire
{\CoChch} in the setting of \textup{b}-calculus~\cite{Get:CHA}.
In Chapter \ref{s:cylinder-estimates} we prove some crucial estimates for the
heat kernel of a \textup{b}-Dirac operator, which are then applied
in Section \ref{estim Chern} to analyze
the short and long time behavior of the components of the \textup{b}-analogue of
the entire Chern character. As a preparation more standard resolvent and heat kernel estimates are
discussed in Section \ref{sec:basic-estimates}.
The final Chapter \ref{chap:Main} contains our main results:
Section \ref{s: heat expansion} is devoted to asymptotic expansions for the
\textup{b}-analogues of the \textnm{Jaffe-Lesniewski-Osterwalder} components.
The retracted relative cocycle representing
the {\CoChch} in relative cyclic cohomology is
constructed in Section \ref{s:retracted-relative-cocycle}, where we also compute
the expressions of its small and large scale limits.
Finally, Section \ref{s: geom pairing} derives the ensuing pairing formul\ae \,
with the $K$-theory, establishes the connection with the
Atiyah-Patodi-Singer index theorem, and discusses the
geometric consequences.
The paper concludes with an explanatory note (Section \ref{s:conclude})
elucidating the relationship between the results presented here
and the prior work in this direction by
Getzler~\cite{Get:CHA} and Wu~\cite{Wu:CCC}, and clarifying why their
generalized {\APS} pairing is necessarily restricted to almost flat bundles.
\aufm{Matthias Lesch, Henri Moscovici, Markus J. Pflaum}
\newcommand{\comment}[1]{\relax}
\chapter{Preliminaries}
\label{s:preliminaries}
We start by recalling some basic material concerning relative cyclic cohomology,
the Chern character and Dirac operators. Furthermore,
for the convenience of the reader we provide in Sections
\ref{App:bdefbmet}--\ref{s:IndFam} a
quick synopsis of the fundamentals of the \textup{b}-calculus for manifolds with
boundaries due to \textnm{Melrose}.
For further details we refer the reader to the monograph
\cite{Mel:APSIT} and the article \cite{Loy:DOB}.
\section{The general setup}
\label{SubSec:Setup}
Associated to a compact smooth manifold $M$ with boundary $\pl M$, there is
a commutative diagram of Fr\'echet algebras
with exact rows
\nind{J@$\cJ$}
\nind{J@$\cJ^\infty$}
\nind{E@$\cE^\infty(\partial M, M)$}
\begin{equation}
\label{Dia:BasicShExSeq}
\xymatrix{
0 \ar[r] & \cJ^\infty (\partial M, M) \ar[r]\ar[d] &\cC^\infty (M)
\ar[r]^{\!\!\!\!\! \varrho} \ar[d]^\id &\cE^\infty (\partial M , M) \ar[r]\ar[d]^{\mbox{ }_{\|\partial M}} &
0\\ 0 \ar[r] & \cJ (\partial M, M) \ar[r] & \cC^\infty (M) \ar[r] &
\cC^\infty ( \partial M) \ar[r] & 0 .
}
\end{equation}
$\cJ (\partial M, M) \subset \cC^\infty (M)$ is
the closed ideal of smooth functions on $M$ vanishing on $\partial M$,
$\cJ^\infty (\partial M, M) \subset \cJ (\partial M, M)$ denotes the closed
ideal of smooth functions on $M$ vanishing up to infinite order on
$\partial M$, and $\cE^\infty (\partial M , M)$ is the algebra of
Whitney functions\sind{Whitney functions} over the subset $\partial M \subset M$.
More generally, for every closed subset $X\subset M$ the ideal
$\cJ^\infty (X , M) \subset \cC^\infty (M)$ is defined as being
\begin{displaymath}
\cJ^\infty (X , M) :=\big\{ f\in\cC^\infty (M)\mid D f_{|X}=0
\text{ for every differential operator $D$ on $M$} \big\}.
\end{displaymath}
By Whitney's extension theorem (\emph{cf}.~\cite{Mal:IDF,Tou:IFD}), the algebra
$\cE^\infty (X,M)$ of Whitney functions over $X \subset M$ is naturally
isomorphic to the quotient of $\cC^\infty (M)$ by the closed ideal
$\cJ^\infty (X,M)$; we take this as a definition of $\cE^\infty (X,M)$.
The right vertical arrow in diagram
\eqref{Dia:BasicShExSeq} is given by the map
\[
\cE^\infty (X,M) \rightarrow \cC^\infty (X), \quad
F \mapsto F_{\|X} := F + \cJ (X,M) ,
\]
which is a surjection.
Let us check that the Fr\'echet algebra $\cJ^\infty := \cJ^\infty (\partial M ,M)$
is a local $C^*$-algebra. First, by the multivariate
Fa\`a di Bruno formula \cite{ConSav:MFBFA} the unitalization
$\cJ^{\infty,+}$ of $\cJ^\infty$ is seen to be
closed under holomorphic calculus in the unitalization $J^+$ of
the algebra $J := \cC_0 (M\setminus \partial M)$. Since $\cJ^{\infty,+}$ is also dense
in $J^+$, it follows that $\cJ^\infty := \cJ^\infty (\partial M ,M)$ is indeed
a local $C^*$-algebra whose $C^*$-closure is the $C^*$-algebra $J$.
Using this together with
\excision\ in $K$-homology (\emph{cf}.~for example \cite{HigRoe:AKH}),
one can easily check that the space
of equivalence classes of Fredholm modules over $\cJ^\infty$ coincides naturally with the
$K$-homology of the pair of $C^*$-algebras $\big(\cC (M), \cC(\partial M)\big)$.
Moreover, by
\cite[p.~298]{Con:NG} one has the following commutative diagram
\begin{equation}
\label{dia:FredModHP}
\xymatrix{
\hspace{-25mm}\Bigl\{\parbox{5cm}{finitely summable Fredholm\\ modules over $\cJ^\infty$}\Bigr\}\ar[r]^{\ch_\bullet} \ar[d]&
HP^\bullet (\cJ^\infty ) \ar[d] \\
K^\bullet (J) = KK_\bullet (J,\C) \ar[r] & \Hom (K_\bullet (J) ,\C) ,
}
\end{equation}
where the right vertical arrow is given by natural pairing between periodic cyclic cohomology
and $K$-theory, and the lower horizontal arrow by the pairing of $K$-theory
with $K$-homology via the
Fredholm index.
A Dirac, resp. a \textup{b}-{Dirac} operator on $M$ determines
a Fredholm module over $\cJ^\infty$ and therefore a class
in the \Khomology\ of the pair $\big(\cC (M), \cC(\partial M)\big)$.
In this article, we are concerned with geometric representations of
the {\CoChch} of such a class and of the ensuing
pairing with the $K$-theory of the pair $\big(\cC (M), \cC(\partial M)\big)$.
\section{Relative cyclic cohomology}
\label{SubSec:RelCycCoh}
As in \cite{LesMosPfl:RPC}, we associate to a short exact sequence
of Fr\'echet algebras
\begin{equation}
\label{Eq:ShExSeq}
0 \longrightarrow \cJ \longrightarrow \cA \overset{\sigma}{\longrightarrow} \cB \longrightarrow 0,
\end{equation}
with $\cA$ and $\cB$ unital, a short exact sequence of mixed complexes
\begin{equation}
0 \longrightarrow \big( C^\bullet (\cB), b, B\big) \longrightarrow \big( C^\bullet (\cA), b , B\big)
\longrightarrow \big( Q^\bullet , b , B\big) \longrightarrow 0 ,
\end{equation}
where $C^\bullet (\cA)$ denotes the Hochschild cochain complex of a Fr\'echet algebra $\cA$,
b the Hochschild coboundary, and $B$ is the Connes
coboundary (\emph{cf.}~\cite{Con:NDG,Con:NG}).
Recall that the \emph{Hochschild cohomology}\sind{Hochschild (co)homology}
of $\cA$ is computed by the complex
$\big( C^\bullet (\cA), b\big)$,
the \emph{cyclic cohomology}\sind{cyclic (co)homology} of $\cA$ is the
cohomology of the total complex \\
$\big( \tot^\bullet \cB C^{\bullet,\bullet} (\cA), b+B \big)$, where
\nind{T@$\tot^\bullet \cB C^{\bullet,\bullet} (\cA)$}
\nind{C@$ \cB C^{p,q} (\cA)$}
\nind{C@$ \cB C_\text{\rm\tiny per}^{p,q} (\cA)$}
\nind{T@$\tot^\bullet \cB C_\text{\rm\tiny per}^{\bullet,\bullet} (\cA)$}
\begin{displaymath}
\cB C^{p,q} (\cA) =
\begin{cases}
C^{q-p} (\cA) := \big( \cA^{\hatotimes q-p+1}\big)^*, &
\text{for $q\geq p\geq 0$}, \\
0, & \text{otherwise},
\end{cases}
\end{displaymath}
while the \emph{periodic cyclic cohomology}\sind{periodic cyclic (co)homology}
of $\cA$ is the cohomology of the total complex \\
$\big( \tot^\bullet \cB C_\text{\rm\tiny per}^{\bullet,\bullet} (\cA), b+B \big)$,
where
\begin{displaymath}
\cB C_\text{\rm\tiny per}^{p,q} (\cA) =
\begin{cases}
C^{q-p} (\cA) := \big( \cA^{\hatotimes q-p+1}\big)^*, &
\text{for $q\geq p$}, \\
0, & \text{else}.
\end{cases}
\end{displaymath}
In \cite{LesMosPfl:RPC} we noted that the relative cohomology
theories, or in other words the cohomologies of the quotient mixed complex
$\big( Q^\bullet ,b,B \big)$, can be calculated from a particular
mixed complex quasi-isomorphic to $Q^\bullet$, namely from the
direct sum mixed complex
\[
\big( C^\bullet (\cA) \oplus C^{\bullet +1} (\cB) , \widetilde{b},\widetilde{B}\big),
\]
where
\sind{Hochschild (co)homology!relative}\sind{relative!Hochschild (co)homology}
\nind{HH@$HH^\bullet (\cA,\cB)$}
\nind{HC@$HC^\bullet (\cA,\cB)$}
\nind{HP@$HP^\bullet (\cA,\cB)$}
\begin{equation}
\label{Eq:DefCoBdrRelMixDer}
\widetilde b = \left(
\begin{array}{cc}
b & -\sigma^* \\
0 & -b
\end{array}
\right), \quad \text{and} \quad
\widetilde B = \left(
\begin{array}{cc}
B & 0 \\
0 & -B
\end{array}
\right).
\end{equation}
In particular, the \emph{relative Hochschild cohomology}\sind{relative!Hochschild (co)homology}\sind{Hochschild (co)homology!relative} $HH^\bullet (\cA,\cB)$ is
computed by the complex
$\big( C^\bullet (\cA) \oplus C^{\bullet +1} (\cB) , \widetilde{b}\big)$,
the \emph{relative cyclic cohomology}\sind{relative!cyclic (co)homology}\sind{cyclic (co)homology!relative} $HC^\bullet (\cA,\cB)$ by
the complex $ \big( \tot^\bullet \cB C^{\bullet,\bullet} (\cA)\oplus \tot^{\bullet +1}
\cB C^{\bullet,\bullet} (\cB), \widetilde{b} + \widetilde{B} \big) $,
and the \emph{relative periodic cyclic cohomology}\sind{relative!periodic cyclic (co)homology}\sind{periodic cyclic (co)homology!relative} $HP^\bullet (\cA,\cB)$ by
$ \big( \tot^\bullet \cB C_\text{\tiny\rm per}^{\bullet,\bullet} (\cA)\oplus
\tot^{\bullet +1} \cB C_\text{\tiny\rm per}^{\bullet,\bullet} (\cB),
\widetilde{b} + \widetilde{B} \big) $.
Note that of course
\begin{equation}
\begin{split}
\big( \tot^\bullet &\cB C^{\bullet,\bullet} (\cA)\oplus \tot^{\bullet +1}
\cB C^{\bullet,\bullet} (\cB), \widetilde{b} + \widetilde{B} \big) \\
&\simeq \bigl(\tot^\bullet \cB C^{\bullet, \bullet}(\cA,\cB),\widetilde{b}+\widetilde{B}\big),
\end{split}
\end{equation}
where $\cB C^{p,q}(\cA,\cB):=\cB C^{p,q}(\cA)\oplus \cB C^{p,q+1}(\cB)$.
\sind{pairing}
Dually to relative cyclic cohomology, one can define relative cyclic homology
theories. We will use these throughout this article as well, and in particular
their pairing with relative cyclic cohomology. For the convenience of the
reader we recall their definition,
referring to \cite{LesMosPfl:RPC} for more details.
The short exact sequence
\eqref{Eq:ShExSeq} gives rise to the following short exact sequence of
homology mixed complexes
\begin{equation}
\label{Eq:HomMixComSeq}
0 \rightarrow \big( K_\bullet , b, B\big) \rightarrow \big( C_\bullet (\cA), b , B\big)
\rightarrow \big( C_\bullet (\cB), b , B\big) \rightarrow 0 ,
\end{equation}
where here $b$ denotes the Hochschild boundary, and $B$ the Connes boundary.
\sind{Hochschild (co)homology}
The kernel mixed complex $K_\bullet$ is quasi-isomorphic to the direct sum mixed complex
\[
\big( C_\bullet (\cA) \oplus C_{\bullet +1} (\cB) , \widetilde{b},\widetilde{B}\big),
\]
where
\begin{equation}
\label{Eq:DefBdrRelMixDer}
\widetilde b = \left(
\begin{array}{cc}
b & 0 \\
-\sigma_* & -b
\end{array}
\right), \quad \text{and} \quad
\widetilde B = \left(
\begin{array}{cc}
B & 0 \\
0 & -B
\end{array}
\right) .
\end{equation}
This implies that the \emph{relative cyclic homology}\sind{relative!cyclic (co)homology}\sind{cyclic (co)homology!relative} $HC_\bullet (\cA,\cB)$ is the
homology of $\big( \htot_\bullet \mathcal B C_{\bullet,\bullet}
(\cA,\cB) , \widetilde b + \widetilde B\big)$,
where $\mathcal B C_{p,q} (\cA,\cB) = \mathcal BC_{p,q} (\cA) \oplus
\mathcal BC_{p,q+1} (\cB)$. Likewise, the \emph{relative periodic cyclic homology}\sind{relative!periodic cyclic (co)homology}\sind{periodic cyclic (co)homology!relative}
$HP_\bullet (\cA,\cB)$ is the homology of
$\big( \hTot_\bullet \mathcal B C^\text{\tiny per}_{\bullet,\bullet}
(\cA,\cB) , \widetilde b + \widetilde B\big)$, where
$\mathcal B C^\text{\tiny per}_{p,q} (\cA,\cB) =
\mathcal BC^\text{\tiny per}_{p,q} (\cA) \oplus
\mathcal BC^\text{\tiny per}_{p,q+1} (\cB)$.
By \cite[Prop.~1.1]{LesMosPfl:RPC}, the relative cyclic
(co)homology groups inherit a natural pairing
\begin{equation}
\label{Eq:RelCycPair}
\langle - , - \rangle_\bullet : \,
HC^\bullet (\cA,\cB) \times HC_\bullet (\cA,\cB)
\rightarrow \C ,
\sind{pairing}
\nind{$< - , - >$}
\end{equation}
which will be called the {\it relative cyclic pairing}, and which on
chains and cochains is defined by
\begin{equation}
\label{Eq:DualPair}
\begin{split}
\langle - , -\rangle : \; &
\Big( \mathcal B C^{p,q} (\cA) \oplus \mathcal B C^{p,q+1} (\cB) \Big)
\times \Big( \mathcal B C_{p,q} (\cA) \oplus \mathcal B C_{p,q+1} (\cB) \Big)
\rightarrow
\C, \\
& \big( (\varphi,\psi), (\alpha,\beta) \big) \mapsto
\langle \varphi , \alpha \rangle + \langle \psi , \beta \rangle .
\end{split}
\end{equation}
This formula also describes the pairing between the relative periodic cyclic
(co)homology groups.
Returning to diagram \eqref{Dia:BasicShExSeq}, we can now express the
(periodic) cyclic cohomology of the pair
$\big( \cC^\infty (M), \cE^\infty (\partial M,M)\big) $ resp.~of the pair
$\big( \cC^\infty (M) , \cC^\infty (\partial M) \big)$ in terms of
the cyclic cohomology complexes of $\cC^\infty (M)$ and
$\cE^\infty (\partial M,M)$ resp.~$\cC^\infty (\partial M)$. We note that the ideal
$\cJ^\infty (\partial M, M)$ is H-unital, since
$\big(\cJ^\infty (\partial M, M)\big)^2 = \cJ^\infty (\partial M, M)$
(\emph{cf.}~\cite{BraPfl:HAWFSS}).
Hence \excision\ holds true for the ideal $\cJ^\infty (\partial M, M)$, and
any of the above cohomology theories of $\cJ^\infty (\partial M, M)$ coincides
with the corresponding relative cohomology of the pair
$\big( \cC^\infty (M), \cE^\infty (\partial M,M)\big) $. In particular, we
have the following chain of quasi-isomorphisms
\begin{equation}
\label{Eq:DefRelCycCoh}
\begin{split}
\tot^\bullet & \, \cB C^{\bullet,\bullet} \big( \cJ^\infty (\partial M ,M)\big)
\sim_\text{\tiny\rm qism}\tot^\bullet \cB C^{\bullet,\bullet}
\big( \cC^\infty (M), \cE^\infty (\partial M ,M) \big)
\sim_\text{\tiny\rm qism} \\
& \, \sim_\text{\tiny\rm qism}
\tot^\bullet \, \cB C^{\bullet,\bullet} (\cC^\infty (M)) \oplus
\tot^{\bullet+1} \cB C^{\bullet,\bullet} (\cE^\infty (\partial M , M)) .
\end{split}
\end{equation}
Next recall from \cite{BraPfl:HAWFSS} that the map
\[
\begin{split}
\tot^k &\, \cB C_\text{\tiny\rm per}^{\bullet,\bullet} (\cC^\infty (M))
\rightarrow \tot^k \cB C_\text{\tiny\rm per}^{\bullet,\bullet}
(\cE^\infty (\partial M , M)), \\
\psi & \mapsto \Big( \! \big( \cE^\infty (\partial M, M) \big)^{\hatotimes k+1}
\ni F_0 \otimes \ldots \otimes F_k \mapsto
\psi \big( {F_0}_{\|X} \otimes \ldots \otimes {F_k}_{\|X} \big) \! \Big)
\end{split}
\]
between the periodic cyclic cochain complexes is a quasi-isomorphism.
As a consequence of the Five Lemma one obtains quasi-isomorphisms
\begin{equation}
\label{Eq:qismRelCycCoh}
\begin{split}
\tot^\bullet &\,\cB C_\text{\tiny\rm per}^{\bullet,\bullet}
\big( \cJ^\infty (\partial M, M) \big) \sim_\text{\tiny\rm qism}
\tot^\bullet \cB C_\text{\tiny\rm per}^{\bullet,\bullet}
\big( \cC^\infty (M), \cE^\infty (\partial M,M)\big)\sim_\text{\tiny\rm qism} \\
& \, \sim_\text{\tiny\rm qism}
\tot^\bullet \, \cB C_\text{\tiny\rm per}^{\bullet,\bullet} (\cC^\infty (M))
\oplus
\tot^{\bullet+1} \cB C_\text{\tiny\rm per}^{\bullet,\bullet}
(\cC^\infty (\partial M)) .
\end{split}
\end{equation}
In this paper we will mainly work with the relative complexes
over the pair of algebras $\big( \cinf{M},\cinf{\pl M} \big)$, because its cycles
carry geometric information about the boundary, which is lost when considering
only cycles over the ideal $\cJ^\infty (\partial M,M)$. In this respect
we note that periodic cyclic cohomology satisfies \excision\
by \cite{CunQui:EPCC,CunQui:EPCCII},
hence in the notation of \eqref{Eq:ShExSeq},
$HP^\bullet(\cJ)$ is canonically isomorphic to $HP^\bullet(\cA,\cB)$.
\section{The Chern character}
\label{s:TheChernCharacter}\sind{Chern character|(}
For future reference, we recall the Chern character and its transgression in
cyclic homology, both in the even and in the odd case.
\subsection{Even case}
\nind{M@$\Mat_N(\cA),\Mat_\infty(\cA)$}
\nind{C@$\ch_\bullet(e)$}
The Chern character of an idempotent
$e\in\Mat_\infty(\cA) := \lim\limits_{N\to\infty} \Mat_N(\cA)$ is
the class in $ HP_0 (\cA)$ of the cycle given by the formula
\begin{equation}
\ch_\bullet(e) := \tr_0 (e) + \sum_{k=1}^\infty (-1)^k \frac{(2k)!}{k!}
\tr_{2k} \Big( \big(e - \frac{1}{2} \big)\otimes e^{\otimes (2k)}\Big) ,
\end{equation}
where for every $j\in \N$ the symbol $e^{\otimes j}$ is an abbreviation for the
$j$-fold tensor product $e\otimes\dots\otimes e$,
and $\tr_j$ denotes the generalized trace map
$\Mat_N(\cA)^{\otimes j}\longrightarrow \cA^{\otimes j}$.
\nind{e@$e^{\otimes j}$}
\nind{t@$\tr_j$}
\sind{trace!generalized trace map}
\sind{generalized trace map}
If $(e_s)_{0\leq s \leq 1}$ is a smooth path of idempotents, then the
transgression formula reads
\begin{equation}
\label{evenh}
\frac{d}{d s}\ch_\bullet(e_s)
= (b+B)\slch_\bullet(e_s,(2e_s-1)\dot e_s) ;
\end{equation}
here the secondary Chern character $\slch_\bullet$ is given by
\sind{Chern character!secondary}
\begin{equation}
\slch_\bullet(e,h):=\iota (h)\ch_\bullet(e),
\end{equation}
where the map $\iota (h)$ is defined by
\begin{equation}\begin{split}
\iota (h)&(a_0\otimes a_1\otimes \ldots\otimes a_l)\\
&=\sum_{i=0}^l (-1)^i
(a_0\otimes \ldots\otimes a_i\otimes h\otimes a_{i+1}
\otimes \ldots\otimes a_l).
\end{split}
\end{equation}
\sind{Ktheory@$K$-theory!relative}
\sind{relative!Ktheory@$K$-theory}
\nind{K@$K_j(\cA,\cB)$}
\nind{c@$\ch_\bullet(p,q,h)$}
A relative $K$-theory class in $K_0(\cA,\cB)$ can be represented
by a triple $(p,q,h)$ with projections $p,q\in\Mat_N(\cA)$ and $h:[0,1]\to \Mat_N(\cB)$
a smooth path of projections with $h(0)=\sigma(p), h(1)=\sigma(q)$
(\emph{cf.} \cite[Def. 4.3.3]{HigRoe:AKH}, see also \cite[Sec. 1.6]{LesMosPfl:RPC}).
The Chern character of $(p,q,h)$ is represented by the relative cyclic cycle
\begin{equation}\label{eq:ChernCharEven}
\ch_\bullet\bigl( p,q,h\bigr)\, = \,
\Bigl( \ch_\bullet(q)-\ch_\bullet(p) \, ,\, -\Tslch_\bullet(h)\Bigr),
\end{equation}
where
\begin{equation}\label{eq:ChernTransEven}
\Tslch_\bullet(h)=\int_0^1 \slch_\bullet\bigl(h(s),(2h(s)-1)\dot h(s)\bigr) ds.
\end{equation}
That the r.h.s. of Eq.~\eqref{eq:ChernCharEven} is a relative cyclic cycle follows
from the transgression formula Eq.~\eqref{evenh}.
From a secondary transgression formula \cite[(1.43)]{LesMosPfl:RPC}
one deduces that \eqref{eq:ChernCharEven} indeed corresponds
to the standard Chern character on $K_0(\cJ)$ under \excision.
\sind{transgression formula}
\sind{transgression formula!secondary}
\subsection{Odd case}
\nind{G@$\GL_\infty(\cA), \GL_N(\cA)$}
The odd case parallels the even case in many aspects.
Given an element $g\in \GL_\infty (\cA):=\lim\limits_{N\to\infty} \GL_N(\cA),$
the odd Chern character is the following normalized periodic cyclic cycle:
\begin{equation}
\ch_\bullet (g) \,=\, \sum_{k=0}^\infty\, (-1)^k k!
\tr_{2k+1} \big( (g^{-1} \otimes g)^{\otimes (k+1)} \big).
\end{equation}
If $(g_s)_{0\leq s \leq 1}$ is a smooth path in $\GL_\infty(\cA)$, the transgression
formula (\emph{cf}.~\cite[Prop.~3.3]{Get:OCC}) reads
\begin{equation}
\label{Eq:ChTransgress}
\frac{d}{ds} \ch_\bullet (g_s) \, =\, (b+B) \,
\slch_\bullet (g_s, \dot g_s),
\end{equation}
where the secondary Chern character $\slch_\bullet$ is defined by
\begin{align}
\slch_\bullet & (g , h ) = \tr_0 (g^{-1}h) + \\
+ & \sum_{k=0}^\infty (-1)^{k+1} k!
\sum_{j=0}^k \tr_{2k+2} \big( (g^{-1} \otimes g)^{\otimes (j+1)}\otimes
g^{-1} h \otimes (g^{-1} \otimes g)^{ \otimes (k-j)} \big).\nonumber
\end{align}
A relative $K$-theory class in $K_1(\cA,\cB)$ can be represented by
a triple $(U,V,h)$, where $U,V\in\Mat_N(\cA)$ are unitaries and
$h:[0,1]\to \Mat_N(\cB)$ is a path of unitaries joining $\sigma(U)$
and $\sigma(V)$. Putting
\begin{equation}\label{eq:ChernTransOdd}
\Tslch_\bullet(h)=\int_0^1 \slch_\bullet\bigl(h_s,\dot h_s\bigr) ds,
\end{equation}
the Chern character of $(U,V,h)$ is represented by the relative cyclic cycle
\nind{c@$\ch_\bullet( U,V,h)$}
\begin{equation}\label{eq:ChernCharOdd}
\ch_\bullet\bigl( U,V,h\bigr)\, = \,
\Bigl( \ch_\bullet(V)-\ch_\bullet(U) \, ,\, -\Tslch_\bullet(h)\Bigr).
\end{equation}
Again the cycle property follows from the transgression formula
Eq.~\eqref{Eq:ChTransgress} and with the aid of a secondary transgression
formula \cite[(1.15)]{LesMosPfl:RPC} one shows that
\eqref{eq:ChernCharOdd} corresponds to the standard Chern character
on $K_1(\cJ)$ under \excision\ \cite[Thm.~1.7]{LesMosPfl:RPC}.
\sind{Chern character|)}
\section{Dirac operators and $q$-graded Clifford modules}
\label{s:qDirac}
\nind{Clq@$\Cl_q$|(}
To treat both the even and the odd cases simultaneously we make use of the Clifford
supertrace (\emph{cf.~e.g.}~\cite[Appendix]{Get:CHA}).
Denote by $\Cl_q$ the complex Clifford algebra\sind{Clifford algebra!complex}
on $q$ generators, that is $\Cl_q$ is the universal $C^*$-algebra on unitary generators
$e_1,...,e_q$ subject to the relations
\begin{equation}
\label{eq:Clifford-relations}
e_j e_k+e_k e_j= -2 \delta_{jk}.
\end{equation}
Let $\sH=\sH^+\oplus \sH^-$ be a $\Z_2$-graded Hilbert space with grading operator
$\ga$. We assume additionally that $\sH$ is a $\Z_2$-graded right $\Cl_q$-module.
Denote by $\sfcr: \sH \otimes \Cl_q \rightarrow \sH$ the right $\Cl_q$-action
and define operators $E_j:\sH \rightarrow \sH$ for $j=1,\cdots,q$ by
$E_j := \sfcr\big( - \otimes e_j\big)$. Then the $E_j$ are unitary operators
on $\sH$ which anti-commute with $\ga$.
\nind{L@$\sL(\sH)$}
The $C^*$-algebra $\sL(\sH)$ of bounded linear operators on $\sH$ is naturally
$\Z_2$-graded, too. For operators $A,B\in\sL(\sH)$ of pure degree $|A|,|B|$ the
\textit{supercommutator}\sind{supercommutator} is defined by
\begin{equation}\label{eq:def-super-commutator}
[A,B]_\super:= AB-(-1)^{|A||B|}BA.
\end{equation}
Furthermore denote by $\sL_{\Cl_q}(\sH)$ the \textit{supercommutant} of
$\Cl_q$ in $\sH$, that is $\sL_{\Cl_q}(\sH)$ consists of those $A\in \sL(\sH)$ for
which $[A,E_j]_\super=0$, $j=1,...,q$.
For $K\in \sL_{\Cl_q}^1(\sH)=\bigsetdef{A\in \sL_{\Cl_q}(\sH)}{A \text{ trace class }}$ one
defines the \emph{degree $q$ Clifford supertrace}\sind{Clifford supertrace!degree $q$}
\begin{equation}\label{eq:Clifford-super-trace}
\Str_q(K):=(4\pi)^{-q/2}\Tr(\ga E_1\cdot ... \cdot E_q K).
\end{equation}
The following properties of $\Str_q$ are straightforward to verify.
\begin{lemma}\label{l:clifford-trace}
For $K, K_1, K_2\in \sL_{\Cl_q}^1(\sH)$,
one has
\begin{enumerate}
\item $\Str_q K=0$, if $|K|+q$ is odd.
\item $\Str_q$ vanishes on super-commutators: $\Str_q([K_1,K_2]_\super)=0$.
\end{enumerate}
\end{lemma}
Let $(M,g)$ be a smooth riemannian manifold.
Associated to it is the bundle $\Cl (M) := \Cl (T^*M)$ of Clifford algebras. Its fiber over
$p\in M$ is given by the Clifford algebra generated by elements of
$T^*_pM$ subject to the relations
\begin{equation}
\label{eq:cliffordrel}
\xi \cdot \zeta + \zeta \cdot \xi = - 2 g (\xi,\zeta) \quad
\text{for all $\xi,\zeta \in T^*_pM$}.
\end{equation}
\begin{definition}[\emph{cf.}~{\cite[Sec.~5]{Get:CHA}}]
Let $q$ be a natural number. By a
\textit{degree $q$ Clifford module}\sind{Clifford module! degree $q$}
over $M$ one then understands a $\Z_2$-graded complex vector bundle $W \rightarrow M$
together with a hermitian metric $\langle - ,-\rangle$, a Clifford action
$\sfc = \sfcl: \bT^* M \otimes W \rightarrow W$, and an action
$\sfcr: W \otimes \Cl_q \rightarrow W$
such that both actions are graded and unitary and supercommute with each other.
A \textit{Clifford superconnection} on a degree $q$ Clifford module $W$ over
$M$ is a superconnection
\[
\A : \Omega^\bullet (M, W) :=
\Gamma^\infty \big( M;\Lambda^\bullet(T^*M) \otimes W \big) \rightarrow
\Omega^\bullet (M, W)
\]
which supercommutes with the action of $\Cl_q$, satisfies
\[
\big[\A , \sfc (\omega) \big]_\super = \sfc \big(\nabla \omega \big) \quad
\text{for all $\omega \in \Omega^1 (M)$},
\]
and is metric in the sense that
\[
\left\langle \A \xi , \zeta \right\rangle +
\left\langle \xi , \A \zeta \right\rangle =
d \left\langle \xi , \zeta \right\rangle \quad
\text{for all $\xi,\zeta \in \Omega^\bullet (M, W)$}.
\]
Here, and in what follows, $\nabla$ denotes the Levi-Civita connection
belonging to $g$.
The \textit{Dirac operator} associated to a degree $q$ Clifford module
$W$ and a Clifford superconnection $\A$ is defined as the differential
operator
\[
\dirac := \sfcl \circ \A : \Gamma^\infty (M;W) \rightarrow
\Gamma^\infty \big( M;\Lambda^\bullet(T^*M) \otimes W \big) \rightarrow
\Gamma^\infty (M;W).
\]
\end{definition}
\sind{Dirac operator}
\sind{Clifford (super)connection}
In this paper the term ``Dirac operator" will always refer to
the Dirac operator associated to a Clifford (super)connection in the above sense.
Such Dirac operators are automatically formally self-adjoint.
By a \semph{Dirac type operator} we understand a first order differential operator
such that the principal symbol of its square is scalar (\emph{cf.}~\cite{Tay:PDEII}).
\nind{Clq@$\Cl_q$|)}
\subsection{The \textup{JLO} cochain associated to a Dirac operator}\label{ss:JLODO}
Let $M$ be a compact riemannian manifold without boundary
and let $\dirac$ be a Dirac type operator as described above.
Since $M$ is compact and since $\dirac$ is elliptic the
heat operator $e^{-r\dirac^2}, r>0,$ is smoothing and hence
for pseudodifferential operators $A_0,\cdots, A_k \in \pdo^\infty (M,W)$
we put, following \cite[Sec. 2]{Get:CHA},
\begin{equation}
\begin{split}
\langle A_0, \cdots,A_k \rangle_{\dirac_t} \, := &\int_{\Delta_k} \, \Str_q
\big(A_0 \, e^{- \sigma_0 \dirac_t^2} \cdots A_k \, e^{- \sigma_k \dirac_t^2} \big)d\sigma \\
= &\Str_q \big( (A_0,\ldots,A_k )_{\dirac_t} \big),
\end{split}
\end{equation}
where
\begin{equation}
\label{eq:simplex}
\Delta_k:=
\bigsetdef{\sigma=(\sigma_0,...,\sigma_k)\in \R^{k+1}}{\sigma_j\ge 0, \,\sigma_0+\ldots+\sigma_k=1}
\end{equation}
denotes the standard $k$-simplex
and
\begin{equation}\label{eq:ML20090128-3}
(A_0,\ldots,A_k )_{\dirac}
:= \int_{\Delta_k} \, A_0 \, e^{- \sigma_0 \dirac^2} \cdots A_k \, e^{- \sigma_k \dirac^2} d\sigma.
\end{equation}
Furthermore, for smooth functions $a_0,\ldots,a_k\in\cC^\infty(M)$, one puts
\begin{align}
\label{Eq:DefChernClosed}
\Ch^k (\dirac) (a_0,\cdots,a_k) & := \langle a_0, [\dirac,a_1],
\cdots, [\dirac, a_k]\rangle , \\
\label{Eq:DefSlChernClosed}
\begin{split}
\slch^k(\dirac , \mathsf{V} )(a_0,\cdots,a_k) &:= \\
\sum_{0\leq j \leq k} (-1)^{j \, \deg{\mathsf{V}}} \,
\langle a_0, [\dirac,a_1],&
\cdots , [\dirac, a_j], \mathsf{V} , [\dirac, a_{j+1}], \cdots ,
[\dirac, a_k] \rangle.
\end{split}
\end{align}
$\Ch^\bullet(\dirac)$ is, up to a normalization factor depending on $q$,
the \JLO\ cocycle associated to $\dirac$.
For a comparison with the standard non-Clifford covariant \JLO\ cocycle
see also Section \plref{s:CocTransgressNoClifford} below.
Now consider a family of Dirac operators, $\dirac_t$, depending smoothly
on a parameter $t$. The operation $\slch$ will mostly be used with
$V=\dot \dirac_t$ as a second argument. Here $\dirac_t$ is considered of
odd degree regardless of the value of $q$.
$\Ch^\bullet(\dirac_t)$ then satisfies
\begin{equation}\label{eq:cocycle}
b \Ch^{k-1}(\dirac_t) + B \Ch^{k+1}(\dirac_t) =0
\end{equation}
and
\begin{equation}\label{eq:transgression}
\frac{d}{dt} \Ch^k(\dirac_t) +
b\slch^{k-1}(\dirac_t , \dot{\dirac}_t ) +B\slch^{k+1}(\dirac_t,\dot\dirac_t) =0.
\end{equation}
\section[Relative Connes--Chern character]{The relative Connes--Chern character of a
Dirac operator over a manifold with boundary}
\label{Sec:RelConCheDirac}
\sind{ConnesCherncharacter@Connes--Chern character!relative}\sind{relative!ConnesCherncharacter@Connes--Chern character}
In this section, $M$ is a compact manifold with boundary, $g_0$ is a riemannian metric
which is smooth up to the boundary, and $W \rightarrow M$ is a degree $q$ Clifford module.
We choose a hermitian metric $h$ on $W$ together with a
Clifford connection which is unitary with respect to $h$.
Let $\dirac=\dirac(\nabla,g_0)$ be the associated Dirac operator; we suppress the dependence on
$h$ from the notation. Then $\dirac$ is a densely defined operator on the Hilbert space $\sH$
of square-integrable sections of $W$.
\sind{relative!Fredholm module}
According to \cite[Prop.~3.1]{BauDouTay:CRC}, as
outlined in the introduction, $\dirac$ defines a relative Fredholm
module over the pair of $C^*$-algebras $\bigl(\cC(M), \cC(\pl M) \bigr)$. Recall that
the relative Fredholm module is given by\nind{F@$F$} \nind{D@${\mathsf D}_e$}
$F = \dirac_e (\dirac^*_e \dirac_e +1)^{-1/2}$, where $\dirac_e$ is a closed extension of
$\dirac$ such that either $\dirac^*_e \dirac_e$ or $\dirac_e \dirac^*_e$ has compact
resolvent ({\em e.g.}
both the closure $\dirac_{\rm min}=\clos{\dirac}$ and the ``maximal extension" $\dirac_{\rm
max}=(\dirac^t)^*$, which is the adjoint of the formal adjoint, satisfy this condition),
and that the $K$-homology class $[F]$ does not depend on the particular choice
of $\dirac_e$ (see \cite[Prop.~3.1]{BauDouTay:CRC}).
Furthermore, \cite[\S 2]{BauDouTay:CRC} shows that over the $C^*$-algebra
\nindp{$\cC_0 (M\setminus \partial M)$} of continuous functions vanishing at
infinity, whose \Khomology\ is by \excision\ isomorphic to the relative
$K$--homology group $K^\bullet\big(\cC(M),\cC(\partial M)\big)$,
\nind{KCM@$K^\bullet(\cC(M),\cC(\partial M))$}
one has even more freedom to choose closed extensions
of $\dirac$, and in particular the self-adjoint extension $\dirac_{\APS}$ obtained by imposing
{\APS}\ boundary conditions yields the same $K$-homology class as $[F]$\nind{F@$[F]$}
in $K^\bullet\bigl(\cC_0(M\setminus\partial M)\bigr)$.
It is well-known that $\dirac_{\APS}$ has an $m^+$-summable resolvent (\emph{cf. e.g.}
\cite{GruSee:WPP}). Moreover, multiplication by $f\in \cJ^\infty(\partial M,M)$
preserves the {\domain} of $\dirac_{\APS}$ and $[\dirac_{\APS},f]=\sfc(df)$ is bounded.
Thus $\dirac_{\APS}$ defines naturally an $m^+$-summable Fredholm module over
the local $C^*$-algebra
$\cJ^\infty (\partial M,M) \subset \cC_0 (M\setminus \partial M)$. Since by \excision\
in $K$-homology $K^\bullet \big(\cC_0 (M\setminus \partial M)\big)$ is naturally isomorphic to
$K^\bullet \big(\cC(M), \cC(\pl M) \big)$, one concludes that the class $[F]$ of the
relative Fredholm module coincides under this
isomorphism with the class $[\dirac]$ of the $m^+$-summable Fredholm module over
$\cJ^\infty (\partial M,M)$.
Let us now consider the {\CoChch} of $[\dirac]$. According to
\cite{ConMos:TCC}, it can be represented by the truncated \JLO-cocycle
of the operator $\dirac$ (with $n\geq m$ of the same parity as $m$):
\begin{equation}
\ch_t^n (\dirac) = \sum_{k \geq 0} \Ch^{n-2k} (t\dirac) +
B \Tslch_t^{n+1} (\dirac) .
\end{equation}
Recall from \cite{JLO:QKT} that the \JLO-cocycle is given by
\begin{equation}
\label{Eq:JLOdirac}
\begin{split}
\Ch^k (\dirac ) (a_0 , \ldots , a_k ) = &
\int_{\Delta_k} \Str_q\bigl( a_0 e^{-\sigma_0 \dirac^2} [\dirac,a_1]\ldots[\dirac,a_k] e^{-\sigma_k \dirac^2}\bigr)d\sigma, \\
& \text{for } a_0, \ldots , a_k \in \cJ^\infty (\partial M, M) .
\end{split}
\end{equation}
Note that the cyclic cohomology class of $\ch_t^n (\dirac)$ is independent of
$t$, and that $\ch_t^\bullet$ is the {\CoChch} as given in
Diagram \eqref{dia:FredModHP}. To obtain the precise form of the {\CoChch}
$ \ch_t^\bullet (\dirac) \in HP^\bullet \big( \cJ^\infty(\partial M , M) \big)$
one notes first that by \cite{BraPfl:HAWFSS}
$HP^\bullet \big( \cJ^\infty(\partial M , M) \big)$
is isomorphic to the relative \deRham\ cohomology group
$H^\textrm{dR}_\bullet (M,\partial M;\C)$ and then
one has to calculate the limit
$\lim_{t\searrow 0} \ch_t^n (\dirac) (a_0, \ldots , a_k)$.
Since $\Ch^k$ is continuous with respect to the Fr\'echet topology on
$\cJ^\infty (\partial M, M)$, and $\cC^\infty_c (M\setminus \partial M)$
is dense in $\cJ^\infty (\partial M, M)$, it suffices to consider the case where
all $a_j$ in \eqref{Eq:JLOdirac} have compact support in $M\setminus \partial M$.
But in that case one can use standard local heat kernel\sind{heat kernel} analysis
or Getzler's asymptotic calculus as in \cite{ConMos:CCN} or \cite{BloFox:APO} to
show that for $n \geq m$ and same parity as $m$
\begin{equation}
\label{eq:ExpCheConChar}
\lim_{t\searrow 0} \big[\ch_t^n\big]_k (\dirac) (a_0, \ldots , a_k) =
\int_M \go_{\dirac}\wedge a_0 \, da_1 \wedge \ldots da_k .
\end{equation}
Here, $\big[\ch_t^n\big]_k$ denotes the component of $\ch_t^n$ of degree $k$ and
$\go_{\dirac}$ is the local index form of $\dirac$.
By Poincar\'e duality the class of the current \eqref{eq:ExpCheConChar}
in $H^\textrm{dR}_\bullet (M,\partial M)$ depends
only on the absolute \deRham\ cohomology class of $\go_{\dirac}$ in
$H_{\textrm{dR}}^\bullet(M)$.
By the transgression formul\ae\ this cohomology class is independent
of $\nabla$ and $g_0$.
Finally let $g$ be an arbitrary smooth metric on the \emph{interior}
$M^\circ=M\setminus \partial M$ which does not necessarily extend to the boundary.
Then we can still conclude from the transgression formula
that the absolute de Rham cohomology class in $H_\textrm{dR}^\bullet(M)\cong
H_\textrm{dR}(M\setminus\partial M)$ of the index form $\go_{\dirac}$
of $\dirac(\nabla,g)$ represents the {\CoChch} of $[(\dirac(\nabla,g_0)]$.
Summing up and using Diagram \eqref{dia:FredModHP}, we obtain
the following statement.
\begin{proposition}\label{p:ML200909292}
Let $M$ be a compact manifold with boundary and
riemannian metric $g_0$. Let $W$ be a
degree $q$ Clifford module over $M$. For any choice of a hermitian metric $h$ and unitary
Clifford connection $\nabla$ on $W$ the Dirac operator $\dirac = \dirac (\nabla, g_0) $
defines naturally a class $[\dirac] \in K_m(M\setminus\pl M)$.
The {\CoChch} of $[\dirac]$ is independent of
the choice of $\nabla$ and $g_0$. In particular the index map
$\Index_{[\dirac]}:K^\bullet(M,\partial M)\longrightarrow\Z,$
defined by the pairing with $K$-theory, is independent of $\nabla$ and $g_0$.
Furthermore, for \emph{any} smooth riemannian metric $g$ in the \emph{interior}
$M^\circ=M\setminus\partial M$ the de Rham cohomology class $\go_{\dirac(\nabla,g)}$ represents
the {\CoChch} of $\dirac(\nabla,g_0)$.
\end{proposition}
\nind{Index@$\Index_{[{\mathsf D}]}$}
\section{Exact \textup{b}-metrics and \textup{b}-functions on cylinders}
\label{App:bdefbmet}
Let $M$ be a compact manifold with boundary of dimension $m$, let $\partial M$
be its boundary, and denote by $M^\circ$ its interior $M\setminus \partial M$.
Then choose a collar for $M$ which means a diffeomorphism of the form
$(r,\eta) : Y \rightarrow [0,2)\times \partial M$, where
$Y \subset M$ is an open neighborhood of $\partial M = r^{-1} (0)$. The
map $r: Y \rightarrow [0, 2)$ is called the
\textit{boundary defining function}\sind{boundary defining function}
of the collar, the submersion $\eta : Y \rightarrow \partial M$
its \textit{boundary projection}\sind{boundary projection}.
For $s \in \, (0, 2)$ denote by $Y^s$
the open subset $r^{-1} \big( [0,s) \big)$,
put $Y^{\circ s}:= r^{-1}\big( (0,s)\big)$ and finally let
$M^s := M \setminus Y^s$ and $\overline{M^s} := M \setminus Y^{\circ s}$;
likewise $Y^\circ:=Y\setminus \partial M$.
Next, let $x : Y \rightarrow \R$ be the
smooth function $x := \ln \circ r$. Then
$(x,\eta): Y^{\circ 3/2} \rightarrow \, (-\infty, \ln \frac 32) \times \partial M$
is a diffeomorphism of $Y^{3/2}$ onto a cylinder.
After having fixed these data for $M$, we choose the most essential
ingredient for the \textup{b}-calculus, namely an
\emph{exact \textup{b}-metric}\sind{bmetric@\textup{b}-metric}\sind{exactbmetric@exact \textup{b}-metric} for $M$. Following \cite{Mel:APSIT}, one
understands by this a riemannian metric $\bmet$ on $M^\circ$ such that on
$Y^\circ$, the metric can be written in the form
\begin{equation}
{\bmet}_{|Y^\circ}= \frac{1}{r_{|Y^\circ}^2} ( dr \otimes dr)_{|Y^\circ} +
\eta_{|Y^\circ}^*\pmet,
\end{equation}
where $\pmet$ is a riemannian metric on the boundary $\partial M$.
If $M$ is equipped with an exact \textup{b}-metric we will for brevity
call it a \emph{\textup{b}-manifold}. \sind{bmanifold@\textup{b}-manifold}
Clearly, one then has in the cylindrical coordinates $(x,\eta)$
\begin{equation}
{\bmet}_{|Y^\circ}= (dx \otimes dx)_{|Y^\circ} + \eta_{|Y^\circ}^* \pmet .
\end{equation}
\FigbManifold
\FigManifoldCylindricalEnds
This means that the interior $M^\circ$ together with $\bmet$ is a
\emph{complete manifold with cylindrical ends}.
\sind{cylindrical ends}\sind{manifold with cylindrical ends}
Thus although we usually tend to visualize a compact manifold with boundary like
in Figure \ref{fig:M}, a \textup{b}-manifold looks like the one
in Figure \ref{fig:MCylinder}. For calculations it will often be
more convenient to work in cylindrical coordinates and hence next
we are going to show how the smooth functions on $M$ can be described
in terms of their asymptotics on the cylinder.
Consider the cylinder $\R\times \partial M := \R\times \partial M$
together with the product metric
\begin{equation}\label{eq:ML20090219-1}
\cylmet = dx \otimes dx + \operatorname{pr}_2^* \pmet ,
\end{equation}
where here (with a slight abuse of language), $x$ denotes the first coordinate
of the cylinder, and
$\operatorname{pr}_2: \R\times \partial M \rightarrow \partial M$ the
projection onto the second factor.
Next we introduce various algebras of what we choose to call
\emph{\textup{b}-functions}\sind{bfunction@\textup{b}-function} on
$\R\times \partial M$. For $c\in \R$ define
$\bcC \big( (-\infty , c) \times \partial M \big)$
resp.~$\bcC \big( (c,\infty) \times \partial M \big)$ as the algebra
of smooth functions $f$ on $(-\infty , c) \times \partial M$
resp.~on $(c,\infty) \times \partial M $ for which there exist functions
$f^-_0,f^-_1,f^-_2,\ldots \in \cC^\infty (\partial M) $
resp.~$f^+_0,f^+_1,f^+_2,\ldots \in \cC^\infty (\partial M) $ such that the
following asymptotic expansions hold true in $x\in \R$:
\begin{equation}\label{eq:ML20090219-3}
\begin{split}
f (x, -)& \, \sim_{x \rightarrow -\infty}
f^-_0 + f^-_1 e^{x} + f^-_2 e^{2x} + \ldots
\quad \text{resp.}\\
f (x,-)& \sim_{x \rightarrow \infty}
f^+_0 + f^+_1 e^{-x} + f^+_2 e^{-2x} + \ldots \, .
\end{split}
\end{equation}
More precisely, this means that there exists for every $k,l\in \N$ and every
differential operator $D$ on $\partial M$ a constant $C>0$ such that
\begin{equation}\label{eq:ML20090219-4}
\begin{split}
\Big| & \partial^l_x D f (x,p) - 0^l D f^-_0 (p) -
\ldots - k^l Df^-_k (p) e^{kx} \Big|
\leq C e^{(k+1) x} \\ & \hspace{50mm} \text{for all $x\leq c-1$ and
$p\in \partial M$ resp.}\\
\Big| & \partial^l_x D f (x,p) - 0^l D f^+_0 (p) -
\ldots - (-k)^l Df^+_k (p) e^{-kx} \Big|
\leq C e^{-(k + 1) x} \\ & \hspace{50mm} \text{for all $x\geq c+1$ and
$p\in \partial M$}.
\end{split}
\end{equation}
The asymptotic expansion guarantees that
$f \in \bcC \big( (-\infty , c) \times \partial M \big)$ if and only if the
transformed function
$[0,e^c[ \times \partial M \ni (r,p) \mapsto f \big( \ln r , p\big)$
is a smooth function on the collar $[0,e^c[ \times \partial M$.
The concept of \textup{b}-functions has an obvious global meaning on
$M^\circ$. Because of its importance we single it out as
\begin{proposition}\label{p:bsmooth}
A smooth function $f\in \cC^\infty(M^\circ)$ extends to a smooth
function on $M$ if and only if it is a \textup{b}-function.\sind{bfunction@\textup{b}-function}
In other words this means that the restriction map $\cC^\infty(M)\ni f\mapsto f_{|M^\circ}\in
\bcC(M^\circ)$ is an isomorphism of algebras.
\end{proposition}
The claim is clear from the asymptotic expansions \eqref{eq:ML20090219-4}.
The algebra $\bcC \big( \R\times \partial M \big)$ of \textup{b}-functions on the full
cylinder consists of all smooth functions $f $ on $\R \times \partial M$
such that
\[ f_{| (-\infty , 0) \times \partial M} \in
\bcC \big( (-\infty , 0) \times \partial M \big)\quad \text{and}
\quad
f_{| (0,\infty)\times \partial M } \in
\bcC \big( (0,\infty)\times \partial M \big).
\]
Next, we define the algebras of \textup{b}-functions with compact support on the
cylindrical ends by $\bcptC \big( (-\infty , c) \times \partial M \big) := $
\[
\big\{ f \in \bcC \big( (-\infty , c) \times \partial M \big) \mid
f(x,p) = 0 \text{ for $x\geq c-\varepsilon$, $p\in \partial M$ and
some $\varepsilon >0$}\big\}
\]
resp.~by $\bcptC \big( (c,\infty)\times \partial M \big) := $
\[
\big\{ f \in \bcC \big( (c,\infty)\times \partial M \big) \mid
f(x,p) = 0 \text{ for $x\leq c+\varepsilon$, $p\in \partial M$ and
some $\varepsilon >0$}\big\}.
\]
For sections in a vector bundle the notation
$\Gamma^\infty_\textup{cpt}((-\infty,0)\times \partial M;E)$ has the
analogous meaning.
The essential property of the thus defined algebras of
\textup{b}-functions is that the coordinate system
$(x,\eta) : Y^{\circ 3/2} \rightarrow (-\infty , \ln 3/2 ) \times \partial M$
induces an isomorphism
\begin{equation}
(x,\eta)^* : \bcC \big( (-\infty , \ln 3/2 ) \times \partial M \big)
\rightarrow \cC^\infty \big( Y^{\frac 32}\big),
\end{equation}
which is defined by putting
\[
(x,\eta)^* f := \Biggl(
Y^{\frac 32} \ni p \mapsto
\begin{cases}
f \big( x(p), \eta(p)\big) , & \text{if $p \notin \partial M $,} \\
f^-_0 (\eta(p)), & \text{if $p \in \partial M $.}
\end{cases}
\Biggr),
\]
for all $f \in \bcC \big( (-\infty , \ln 3/2 ) \times \partial M \big)$. Under this
isomorphism, $\bcptC \big( (-\infty , 3/2 ) \times \partial M \big)$ is mapped
onto $\cC_\txtcpt^\infty \big( Y^{\frac 32}\big)$.
In this article, we will use the isomorphism $(x,\eta)^*$ to obtain
essential information about solutions of boundary value problems on $M$ by
transforming the problem to the cylinder over the boundary and then performing
computations there with \textup{b}-functions on the cylinder.
The final class of \textup{b}-functions used in this work is the algebra
$\bsS \big( \R\times \partial M \big)$ of
\textit{exponentially fast decreasing functions} or
\textit{\textup{b}-Schwartz test functions} on the cylinder defined as
the space of all smooth functions $f \in \cC^\infty \big( \R\times \partial M \big)$
such that for all $l,n \in \N$ and all differential operators $D \in \Diff (\partial M)$
there exists a $C_{l,D,n} >0$ such that
\begin{equation}
\label{Eq:DefbSchwartz}
\left| \partial^l_x D f(x,p)\right| \leq C_{l,D,N} e^{-n |x|}
\quad \text{for all $x\in \R$ and $p\in \partial M$}.
\end{equation}
Obviously, $(x,y)^*$ maps $\bsS \big( \R\times \partial M \big) \cap
\bcptC \big( (-\infty , \ln 3/2) \times \partial M\big)$ onto
the function space $\cJ^\infty\big(\partial M ,Y^{\frac 32}\big)
\cap \cC_\txtcpt^\infty \big( Y^{\frac 32}\big)$.
\section{Global symbol calculus for pseudodifferential operators}
\label{Sec:GloSymCalPsiOp}
In this section, we briefly recall the global symbol for pseudodifferential
operators which was introduced by Widom in \cite{Wid:CSCPO}
(see also \cite{FulKen:RPG,Pfl:NSRM}).
We assume that $(M_0,g)$ is a riemannian manifold (without boundary),
and that $\pi_E: E \rightarrow M_0$ and $\pi_F: F \rightarrow M_0$ are smooth vector
bundle carrying a hermitian metric $\mu_E$ resp.~$\mu_F$.
In later applications, $M_0$ will be the interior of a given manifold with
boundary $M$.
Recall that there exists an open neighborhood $\Omega_0$ of the diagonal in
$M_0\times M_0$ such that each two points $p,q\in M_0$ can be joined by a
unique geodesic.
Let $\alpha_0$ be a \textit{cut-off function} for $\Omega_0$ which means a smooth map
$M_0\times M_0 \rightarrow [0,1]$ which has support in $\Omega_0$ and is equal to $1$
on a neighborhood of the diagonal. These data give rise to the map
\begin{equation}
\label{eq:DefConIndLin}
\Phi : M \times M \rightarrow TM, \quad
(p,q) \mapsto
\begin{cases}
\alpha_0 (p,q) \exp_p^{-1} (q) & \text{if $(p,q) \in \Omega_0$},\\
0 & \text{else},
\end{cases}
\end{equation}
which is called a \textit{connection-induced linearization}
(\emph{cf.}~\cite{FulKen:RPG}). Next denote for $(p,q)\in \Omega_0$ by
$\tau^E_{p,q} : E_p \rightarrow E_q$ the parallel transport in $E$ along the geodesic
joining $p$ and $q$. This gives rise to the map
\begin{equation}
\label{eq:DefConIndParTrans}
\tau^E : E \times M \rightarrow E, \quad
(e,q) \mapsto
\begin{cases}
\alpha_0 \big( \pi_E(e),q \big) \tau^E_{\pi^E(e),q} (e), &
\text{if $(\pi_E(e),q) \in \Omega_0$},\\
0 & \text{else},
\end{cases}
\end{equation}
which is called a \textit{connection-induced local transport} on $E$.
(\emph{cf.}~\cite{FulKen:RPG}).
Next let us define the symbol spaces $\cS^m \big( T^*M_0 ; \pi_{T^*M}^* E \big) $.
For fixed $m\in \R$ this space consists of all smooth sections
$a : T^*M_0 \rightarrow \pi_{T^*M_0}^* E$
such that in local coordinates $x: U \rightarrow \R^{\dim M_0}$ of
over $U\subset M_0$ open and vector bundle coordinates
$(x,\eta): E_{|U} \rightarrow \R^{\dim M_0 + \dim_\R E}$ the following estimate
holds true for each compact $K\subset U$ and appropriate $C_K >0$ depending on $K$:
\begin{equation}
\big\| \partial_x^\alpha \partial_\xi^\beta ( \eta \circ a ) (\xi) \big\|
\leq C_K \, (1 + \| T^*x (\xi) \| )^{m - |\beta|} \quad
\text{for all $\xi \in T^*_{|K}M_0$}.
\end{equation}
Given a symbol $a \in \cS^m \big( T^*M_0 ; \pi_{T^*M}^* \Hom (E,F) \big) $
one defines now a pseudodifferential operator
$\Op (a) \in \Psi^m \big( M_0; E,F)$ by
\begin{equation}
\begin{split}
\label{Eq:DefOp}
\big( \Op (a) & \, u \big) (p) := \\
:= \, & \frac{1}{(2\pi)^{\dim M}}\hspace{-2mm}
\int\limits_{T_p M_0 \times T^*_p M_0} \hspace{-2mm}
\alpha_0 (p, \exp v) \,
e^{-i \langle v , \xi \rangle} \, a ( p,\xi )
\tau^E (u(\exp v) , p ) \, dv \, d\xi ,
\end{split}
\end{equation}
where $u\in \Gamma^\infty_\text{\rm\tiny cpt} (E)$ and $p\in M_0$.
Moreover, there is a quasi-inverse, the symbol map
$\sigma: \Psi^m \big( M_0 ; E,F \big) \rightarrow
\cS^m \big( T^*M_0 ; \pi_{T^*M}^* \Hom (E,F) \big)$
which is defined by
\begin{equation}
\label{Eq:DefSym}
\sigma (A) (\xi) (e) := A \Big( \alpha_0 (p ,{-}) \, \tau^E(e,{-})
\, e^{i \langle \xi , \Phi (p,{-})\rangle} \Big) \big( \pi(\xi) \big),
\end{equation}
where $p \in M_0, \; \xi \in T^*_pM_0,\; e \in E_p$.
It is a well-known result from global symbol calculus
(\emph{cf.}~\cite{Wid:CSCPO,FulKen:RPG,Pfl:NSRM}) that the map $\Op$
maps $\cS^{-\infty} \big( T^*M_0 ; \pi_{T^*M}^* \Hom (E,F) \big) $ onto
$\Psi^{-\infty} \big( M_0 ; E,F \big)$ and that up to these spaces,
$\Op$ and $\sigma$ are inverse to each other.
\section{Classical \textup{b}-pseudodifferential operators}
\label{Sec:ClassbPseuOp}
Let us explain in the following the basics of the (small) calculus of
\textup{b}-pseudodifferential operators on a manifold with boundary $M$. In our
presentation, we lean on the approach \cite{Loy:DOB}. For more details
on the original approach confer \cite{Mel:APSIT}.
In this section, we assume that $M$ carries a
\textup{b}-metric\sind{bmetric@\textup{b}-metric} denoted by $\bmet$.
Furthermore, let $\pi_E: E\rightarrow M$ and $\pi_F: F\rightarrow M$
be two smooth hermitian vector bundles over $M$, and fix metric connections
$\nabla^E$ and $\nabla^F$.
Then observe that by the Schwartz Kernel Theorem there is an isomorphism
between bounded linear maps
\[
A : \cJ^\infty \big( \partial M, M ; E \big)
\rightarrow \cJ^\infty \big( \partial M, M ; F' \big)'
\]
and the strong dual
$\cJ^\infty \big( \partial (M \times M), M\times M ; E \boxtimes F' \big)'$,
where $\cJ^\infty \big( \partial M, M ; E \big) := \cJ^\infty \big( \partial M, M \big)
\cdot \cC^\infty (M;E)$. This isomorphism is given by
\begin{equation}
A \mapsto K_A := \Big( \cJ^\infty \big( \partial M, M ; E \big) \hatotimes
\cJ^\infty \big( \partial M, M ; F' \big) \ni (u \otimes v) \mapsto
\left\langle Au , v\right \rangle\Big),
\end{equation}
where we have used that
\[
\cJ^\infty \big( \partial (M \times M), M\times M ; E \boxtimes F' \big) =
\cJ^\infty \big( \partial M, M ; E \big) \hatotimes
\cJ^\infty \big( \partial M, M ; F' \big)
\]
with $\hatotimes$ denoting the completed bornological tensor product.
The \textup{b}-volume form $\bvol$ associated to $\bmet$ gives rise to an embedding
\sind{pairing}
\nind{$< - , - >$}
\begin{equation}
\label{Eq:EmbMultAlg}
\begin{split}
& \cM^\infty \big( \partial M, M ; E' \otimes F \big) \hookrightarrow
\cJ^\infty \big( \partial (M \times M), M\times M ; E \boxtimes F' \big)' , \\
& k \mapsto \left( u \otimes v \mapsto \int_{M \times M}
\langle k (p,q) , u(p) \otimes v(q)\rangle \, d\big(\bvol \otimes \bvol \big)(p,q)
\, \right) ,
\end{split}
\end{equation}
which we use implicitly throughout this work.
In the formula for the embedding, $\langle -,-\rangle$ denotes the natural pairing
of an element of a vector bundle with an element of the dual bundle over the same
base point, $u,v$ are elements of $\cJ^\infty \big( \partial M, M ; E \big)$ and
$\cJ^\infty \big( \partial M, M ; F' \big)$ respectively, and
$\cM^\infty \big( X, M ; E)$ denotes for $X\subset M$ closed the space of all
sections $u \in \Gamma^\infty (M\setminus X; E)$ such that in local coordinates
$(y,\eta): \pi_E^{-1}(U) \rightarrow \R^{\dim M +\dim E}$ with $U\subset M$ open
one has for every compact $K\subset U$, $p \in K\setminus X$, and
$\alpha \in \N^{\dim M}$ an estimate of the form
\begin{displaymath}
\left\| \partial^\alpha_y \big(\eta \circ u \big) (p) \right\| \leq
C \frac{1}{\Big(d\big(y(p),y(X\cap U)\big)\Big)^\lambda},
\end{displaymath}
where $C>0$ and $\lambda >0$ depend only on the local coordinate system,
$K$, and $\alpha$. The fundamental property of $\cM^\infty \big( X, M ; E)$
is that
\begin{displaymath}
\cJ^\infty (X,M) \cdot \cM^\infty \big( X, M ; E) \subset \cJ^\infty (X,M; E).
\end{displaymath}
Note that the vector bundle $E$ (and likewise the vector bundle $F$) gives rise
to a pull-back vector bundle
$\operatorname{pr}_{\partial M}^* E_{|\partial M}$ on the cylinder, where
$\operatorname{pr}_{\partial M} :\R\times \partial M
\rightarrow \partial M$ is the canonical projection. This pull-back vector bundle
will be denoted by $E$ (resp.~$F$), too.
As further preparation we introduce two auxiliary functions
$\psi :M\rightarrow [0,1]$ and $\varphi: M \rightarrow [0,1]$ on $M$ which are smooth
and satisfy
$\supp \psi \subset \subset M^1$, $\psi (p) = 1$ for $p\in M^{3/2}$,
$\supp \varphi \subset \subset Y^1 $, and finally $\varphi (p) = 1$ for
$p\in Y^{1/2}$.
Such a pair of functions will be called a {\it pair of auxiliary cut-off functions}.
By a \emph{\textup{b}-pseudodifferential operator}
\sind{bpseudodifferentialoperator@\textup{b}-pseudodifferential operator} of order
$m \in \R$ we now understand a continuous operator
$A: \cJ^\infty (\partial M , M ; E) \rightarrow \cJ^\infty (\partial M, M;F')'$
such that for one (and hence for all) pair(s) of auxiliary cut-off functions the
following is satisfied:
\begin{enumerate}
\item[($\bPsi$1)]
The operator $(1-\varphi)A(1-\varphi)$ is a compactly supported
pseudodifferential operator of order $m$ in the interior $M^\circ$.
\item[($\bPsi$2)]
The operator $\varphi A\psi$ is smoothing. Its integral
kernel $K_{\varphi A\psi}$ has support in $\supp \varphi \times M$
and lies in
$\cJ^\infty \big( \partial (M \times M) , M \times M; E' \boxtimes F\big)$.
\item[($\bPsi$3)]
The operator $\psi A\varphi$ is smoothing. Its integral
kernel $K_{\psi A\varphi}$ has support in $M \times \supp \varphi$ and lies in
$\cJ^\infty \big( \partial (M \times M) , M \times M; E' \boxtimes F\big)$.
\item[($\bPsi$4)]
Consider the induced operator on the cylinder
\[
\begin{split}
\widetilde A : \, & \bsS \big( \R\times \partial M; E \big)
\rightarrow \bsS \big( \R\times \partial M ; F \big)', \\
& u \mapsto \Big( (t,p) \mapsto \big[ (1-\psi)A(1-\psi) \,
\big( (x,\eta)^* u \big) \big] \big( (x,\eta)^{-1} (t,p) \big) \Big) ,
\end{split}
\]
where $\bsS \big( \R\times \partial M; E \big) :=
\bsS ( \R ) \hatotimes \cC^\infty \big( \partial M;E\big)$.
Denote by
$\widetilde a := \sigma (\widetilde A)
\in\cS^m\big( T^*(\R\times \partial M) ;\pi^*\Hom (E, F)\big)$
the complete symbol of $\widetilde A$ defined by Eq.~\eqref{Eq:DefSym}
with respect to the product metric on $\R\times \partial M$.
Then the following conditions hold true:
\begin{enumerate}
\item[(i)]
Let $y$ denote local coordinates of $\partial M$, $(y,\xi)$ the
corresponding local coordinates of $T^* \partial M$, and $\tau$ the
cotangent variable of the cylinder variable $t \in \R$. Then the symbol
$\widetilde a (t,\tau, y, \xi)$
can be (uniquely) extended to an entire function in $\tau \in \C$ such that
uniformly in $t$, uniformly in a strip $|\im \tau | \leq R$ with $R>0$
and locally uniformly in $y$
\begin{displaymath}\hspace{-8mm}
\left\| \partial_t^k\partial_\tau^l\partial_y^\alpha \partial_\xi^\beta
\, \widetilde a \right\| \leq C_{k,l,\alpha,\beta}
\left( 1+ |\tau| + \|\xi\|\right)^{m-l-|\beta|}
\end{displaymath}
for $l\in \N$, $\beta \in \N^{\dim M -1}$.
\item[(ii)]
There exist symbols $\widetilde a_k (\tau,y,\xi) \in
\cS^m \big(\C\times T^*\partial M ;\pi^*\Hom (E, F)\big)$, $k\in \N,$
and
$r_n (t,\tau,y,\xi)\in \cS^m\big( \R\times \C \times T^*\partial M) ;
\pi^*\Hom (E, F)\big)$,
$n\in \N$, which all are entire in $\tau$ and fulfill growth conditions
as in (i) such that for every
$n \in \N$ the following asymptotic expansion holds:
\begin{displaymath}\hspace{4mm}
\widetilde a (t,\tau,y,\xi) =
\sum_{k=0}^n e^{kt} \widetilde a_k (\tau,y,\xi) +
e^{(n+1)t} \, r_n(t,\tau,y,\xi).
\end{displaymath}
\item[(iii)]
The Schwartz kernel $K_{\widetilde B}$ of the operator
$\widetilde B:= \widetilde A - \Op (\widetilde a)$ with
$\Op (\widetilde a)$ defined by Eq.~\eqref{Eq:DefOp}
can be represented in the form
\begin{displaymath}\hspace{-16mm}
K_{\widetilde B} (t,p , t',p')= \int_\R \, e^{i(t-t')\tau} \,
\widetilde b (t,\tau, p,p') \, d\tau
\end{displaymath}
with a symbol
\begin{displaymath}\hspace{5mm}
\widetilde b (t,\tau, p,p') \in \cS^{-\infty}
\big( T^*\R \times \partial M\times \partial M; \pi^* \Hom (E,F) \big)
\end{displaymath}
which is entire in $\tau$ and which for every $\tilde m \in \N$,
$k,l\in \N$ and every pair of differential
operators $D_p$ and $D'_{p'}$ on $\partial M$
(acting on the variable $p$ resp.~$p'$) satisfies the following estimate
uniformly in $t$, $p$, $p'$ and uniformly in a strip
$|\im \tau | \leq R$ with $R>0$
\begin{displaymath}\hspace{-18mm}
\left\| \partial_t^k\partial_\tau^l D_p D'_{p'}
\, \widetilde b \right\| \leq C_{\tilde m, k,l,D,D'}
\left( 1+ |\tau| \right)^{\tilde m}.
\end{displaymath}
\item[(iv)]
There exist symbols
\
\widetilde b_k (\tau,p,p') \in
\cS^{-\infty} \big( \C \times \partial M\times \partial M;
\pi^*\Hom (E, F)\big),
\]
for $k\in \N$ and symbols
\[\hspace{6mm}
r_n (t,\tau,p,p')\in \cS^m\big( \R \times \C \times\partial M\times \partial M;
\pi^*\Hom (E, F)\big),
\]
for $n\in \N$
which all are entire in $\tau$ and fulfill growth conditions
as in (iii) such that for every
$n \in \N$ the following asymptotic expansion holds:
\begin{displaymath}\hspace{4mm}
\widetilde b (t,\tau,p,p') =
\sum_{k=0}^n e^{kt} \, \widetilde b_k (\tau,p,p') +
e^{(n+1)t} \, r_n(t,\tau,p,p').
\end{displaymath}
\end{enumerate}
\end{enumerate}
If in addition to the above conditions the operators $(1-\varphi)A(1-\varphi)$
and $\widetilde A$ are both classical pseudodifferential operators,
then $A$ is a classical \textup{b}-pseudodifferential operator of order $m$.
We denote the space of classical \textup{b}-pseudodifferential operators on $(M,\bmet)$
of order $m$ between $E$ and $F$ by $\bPsi^m (M;E,F)$, and put as usual
$\bPsi^\infty (M;E,F) := \bigcup_{m\in \Z} \bPsi^m (M;E,F)$.
It is straightforward (though somewhat tedious) to check that
$\bPsi^\infty (M;E) := \bPsi^\infty (M;E,E)$ even forms an algebra.
Obviously, $\bPsi^\infty (M;E,F)$ contains as a natural subspace the
space $\bdiff (M;E,F)$ of all \textup{b}-differential operators on $M$ from $E$ to $F$
which means of all local classical \textup{b}-pseudodifferential operators.
The following is immediate to check.
\begin{proposition}
\label{prop:DefEqbDiff}
Let $A \in \bPsi^\infty \big(M ;E,F\big)$. Using the notation from above
the following propositions are then equivalent:
\begin{enumerate}
\item
$A \in \bdiff \big( M; E,F \big)$.
\item
The operators $(1-\varphi)A (1 -\varphi)$ and $\widetilde A$ are
differential operators, and both the operators $\varphi A \psi$ and
$\psi A \varphi$ vanish.
\item
The operator $A$ acts as a differential operator over the interior, i.e.
as a local operator on $\Gamma^\infty \big (E_{|M^\circ} \big)$. In addition,
over the cylinder $(-\infty,0)\times \partial M$
the operator $\widetilde A$ can be written locally in the form
\begin{equation}
\label{eq:DefEqbDiff}
\widetilde A =
\sum_{j + |\alpha| \leq \operatorname{ord} A} a_{j,\alpha} \partial_y^\alpha \partial_x^j ,
\end{equation}
where $a_{j,\alpha} \in \bcC\!\!\big( (-\infty,0) \times U\big)$, $U\subset \partial M$
open and $y:U \rightarrow \R^{\dim M -1}$ are local coordinates of $\partial M$.
\end{enumerate}
\end{proposition}
Over a cylinder $(-\infty,c)\times N$ with $c\in \R$ and $N$ a compact manifold,
we sometimes use the notation $\bcptPsi^\infty \big((-\infty,c)\times N;E,F\big)$
to denote the space of all pseudodifferential operators in
$\bPsi^\infty \big((-\infty,c)\times N;E,F\big)$ having support in some
cylinder $(-\infty,c-\varepsilon ]\times N$ with $\varepsilon >0$.
We also put
\begin{equation}
\label{eq:defbcptdiff}
\begin{split}
\bcptdiff & \big((-\infty,c)\times N;E,F\big) := \\
& \bdiff \big((-\infty,c)\times N;E,F\big) \cap
\bcptPsi^\infty \big((-\infty,c)\times N;E,F\big).
\end{split}
\end{equation}
Note that in condition $(\bPsi 4)$ above, the operator $\widetilde A$
is an element of the space $\bcptPsi^\infty \big((-\infty,3/2)\times \partial M;E,F\big)$.
Throughout this work, we also need the \textup{b}-versions of Sobolev-spaces.
The \textup{b}-Sobolev space $\bSob^m (M;E)$ is defined for $m \in \N$ by
\begin{equation}
\label{Eq:DefbSob}
\bSob^m (M,E) := \big\{ u \in L^2 (M,E) \mid D u \in L^2 (M,E)
\text{ for all $D\in \bdiff^m (M,E)$} \big\}.
\end{equation}
For the definition of $\bSob^m (M,E)$ for arbitrary $m\in \R$
we refer the reader to \cite{Mel:APSIT}. The following result
is straightforward.
\begin{proposition}
\label{Prop:PropbSob}
Let $A \in \bPsi^l (M;E,F)$ be a \textup{b}-pseudodifferential operator.
Then the following holds true:
\begin{enumerate}
\item
$A$ has a natural extension
\begin{equation}
\label{eq:bPsibSob}
A : \bSob^m (M;E) \rightarrow \bSob^{m-l} (M;F),
\end{equation}
which we denote by the same symbol like the original operator.
\item
The \textup{b}-Sobolev-space $\bSob^1 (M,E)$ is the natural {\domain} of any elliptic
first order \textup{b}-pseudodifferential operator acting on sections of $E$.
\item
If $A$ has order $l=0$, then $A$ is bounded.
\end{enumerate}
\end{proposition}
\section{Indicial family}
\label{s:IndFam}
\sind{indicial family}
Assume $A\in \bPsi^m (M;E,F)$. Denote by $\widetilde A$ and $\widetilde a$
the induced operator and its complete symbol on the cylinder
$\R \times \partial M$ as above in condition $(\bPsi 4)$. Consider the
zeroth order term $\widetilde a_0$ in the asymptotic expansion
$(\bPsi 4) (ii)$ and put for $\tau \in \C$, $u \in \Gamma^\infty (\partial M ; E)$
and $p\in \partial M$
\begin{align}
\cI (A) & (\tau) u (p) := \Op \big(\widetilde a_0 (\tau, - )\big) u (p) =\label{eq:DefIndFam} \\
= \, & \frac{1}{(2\pi)^{\dim M -1}} \hspace{-6mm}
\int\limits_{T_p\partial M \times T^*_p\partial M}
\hspace{-2mm} \alpha_0 (p, \exp v) \,
e^{-i \langle v , \xi \rangle} \, \widetilde a_0(\tau,p,\xi ) \,
\tau^E \big( u (\exp v) , p \big) \, dv \, d\xi ,\nonumber
\end{align}
where, as explained in Section \ref{Sec:GloSymCalPsiOp},
$\alpha_0 : M \times M \rightarrow [0,1]$ is a
cut-off function vanishing outside the injectivity radius and $\tau^E$ is
a connection induced parallel transport on $E$. One thus obtains an entire
family $\cI (A)$ of pseudodifferential operators on
the boundary $\partial M$ which is called the \textit{indicial family}\sind{indicial family}
of $A$. The indicial family plays a crucial role in deriving the Atiyah--Patodi--Singer
index formula within the \textup{b}-calculus (\emph{cf.}~\cite{Mel:APSIT}).
\chapter{The \textup{b}-Analogue of the Entire Chern Character}
\label{s:b-Chern-character}
After discussing in Section \ref{s:btrace} the \textup{b}-trace in the
context of a manifold with cylindrical ends, we digress in
Section \ref{s:McK-S} to establish a cohomological analogue of the
well-known McKean--Singer formula in the framework of relative cyclic
cohomology for the pseudodifferential \textup{b}-calculus, and then employ
it to recast Melrose's approach to the proof of the Atiyah--Patodi--Singer
index theorem. In Section \ref{s: b-trace formula} we establish an effective
formula for the \textup{b}-trace, which will be used later in the paper.
The rest of this chapter is devoted to a reformulation of Getzler's version
of the entire Connes--Chern character in the setting of \textup{b}-calculus,
formulated in terms of relative cyclic cohomology.
\section{The \textup{b}-trace}
\label{s:btrace}
From now on we assume that $M$ is a compact manifold with boundary,
that $r: Y \rightarrow [0,2)$ is a boundary defining function,
and that $\bmet $ is an exact \textup{b}-metric on $M$. These are the main
ingredients of the \textup{b}-calculus, which we will use in what follows
(see Sections \ref{App:bdefbmet} to \ref{s:IndFam} for basic definitions
and the monograph \cite{Mel:APSIT} for further material on the
\textup{b}-calculus).
Before we can construct the \textup{b}-analogue of the entire Chern character
we have to recall here however the definition of the \textup{b}-\emph{trace}
(\emph{cf.}~\cite{Mel:APSIT}), since this notion plays an essential role in our work.
It will often be convenient to choose cylindrical coordinates
(see Figure \ref{fig:MCylinder} on page \pageref{fig:MCylinder})
$(x,\eta): Y^\circ \rightarrow \, (-\infty, \ln 2 ) \times \partial M$
over a collar $Y \subset M $ with a boundary defining function
$r : Y \rightarrow [0,2)$ (see Section \ref{App:bdefbmet} for details and notation).
When using these coordinates, we view the interior $M^\circ$
as a manifold with cylindrical ends, and have in this picture
$M^\circ \cong (-\infty,0]\times \pl M\cup_{\pl M} \overline{M^1}$.
All explicit calculations will be done in cylindrical coordinates
as explained in the previous sections.
Let $E$ be a smooth hermitian vector bundle over $M$.
Whenever convenient,
we will tacitly identify elements of $\Gamma^\infty\bigl((-\infty,0]\times \pl M;E\bigr)$,
the sections of $E$ over $(-\infty,0]\times \partial M$, with
$\Gamma^\infty(\pl M;E\rest{\pl M})$--valued smooth functions on $(-\infty,0]$,
\emph{i.e.} elements of $\cC^\infty\bigl((-\infty,0],\Gamma^\infty(\pl M;E\rest{\pl M})\bigr)$,
in the obvious way; \emph{cf.} Section \ref{Sec:ClassbPseuOp}.
Accordingly, we define for $u\in \Gamma^\infty_{\textup{cpt}}\bigl((-\infty,0]\times \pl M;E\bigr)$
the Fourier transform in the cylinder variable, $\hat u(\gl)\in\Gamma^\infty(\pl M;E\rest{\pl E})$, by
\begin{equation}
\hat u(\gl,p):=\int_{-\infty}^\infty e^{-ix\gl} u(x,p) dx.
\end{equation}
Now assume that $A\in \bPsi^{-\infty}(M;E)$
is a smoothing \textup{b}-pseudodifferential operator.
In general, $A$ is not trace class in the usual sense. However, since $A$
has a smooth Schwartz kernel it is locally trace class, in the sense that
$\psi A\varphi$ is trace class for any pair of smooth functions
$\psi,\varphi : M \rightarrow \R$ having compact support in $M^\circ$.
Using the notation from Section \ref{Sec:ClassbPseuOp}, let $\widetilde A$
be the operator induced on the cylinder $(-\infty , 0] \times \partial M$.
We define then an operator valued symbol
$A_\partial (x,\lambda): \Gamma^\infty (E_{|\partial M}) \rightarrow \Gamma^\infty (E_{|\partial M})$
as follows:
for $v\in \Gamma^\infty(E\rest{\pl M})$ put
\begin{equation}\label{eq:DefIndFamAlt}
\bigl(A_\pl(x,\gl)v\bigr)(p):=\Bigl(e^{-ix\gl}\widetilde A\bigl(e^{i\cdot\gl}\otimes v\bigr)\Bigr)(x,p).
\end{equation}
For $v\in\Gamma^\infty(E\rest{\pl M})$ we have $(e^{i\cdot \gl}\otimes v)^\wedge=2\pi \delta_\gl\otimes v$,
hence for
$u \in \Gamma^\infty_{\textup{cpt}} \big( (-\infty , 0] \times \partial M;E \big)$
one then obtains
\begin{equation}
\label{eq:ftbdrop}
\begin{split}
\bigl(\widetilde A u \bigr) (x,p) & =
\frac{1}{2\pi } \int_{-\infty}^{\infty} e^{i x \lambda}
\big( A_\partial (x,\lambda ) \hat u(\lambda, -) \big)(p) \, d\lambda = \\
& = \frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
e^{i (x-\tilde x) \lambda}
\big( A_\partial (x,\lambda ) u(\tilde x, -) \big)(p) \, d\tilde x \, d\lambda.
\end{split}
\end{equation}
We note in passing that $A_\partial (x,\lambda)$ can also be constructed
from the global symbol $\widetilde a$ as
$A_\partial (x,\lambda) = \Op \big( \widetilde a (x,\lambda,-) \big) $
(\emph{cf.}~Sections \ref{Sec:GloSymCalPsiOp} and \ref{Sec:ClassbPseuOp}).
For the properties of $\widetilde a$ see \cite[Sec. 2.2]{Loy:DOB} and Section
\plref{Sec:ClassbPseuOp}. In the small \textup{b}-calculus $\widetilde a(x,\gl,-)$ and hence
$A_\partial (x,\lambda)$ are entire in $\gl$ while in the full \textup{b}-calculus
they are meromorphic \cite{Mel:APSIT}.
In any case one has
\begin{equation}\label{eq:asymptotics-b-symbol}
A_\partial (x,\gl) = \cI(A)(\gl) +O(e^x),\quad x\to - \infty,
\end{equation}
with a family $\cI(A)(\gl), \gl\in\R,$
of classical pseudodifferential operators on $\pl M$ in the parameter dependent
calculus;
\emph{cf. e.g.}~\cite[Sec. 2.1]{LesMosPfl:RPC} for a brief summary of the parameter dependent
calculus.
It turns out that the operator valued function $\, \gl\in\R \mapsto \cI(A)(\gl)$
is exactly the \semph{indicial family} of $A$ as defined in Section \ref{s:IndFam}.
The reason is that in terms of the global symbol
$\widetilde a$ one has $\cI(A)(\gl) = \Op \big( \widetilde a_0 (\lambda,-) \big)$,
where $\widetilde a_0 (\lambda,-)$ is the first term in the asymptotic expansion
of $\widetilde a$ with respect to $x\rightarrow -\infty$.
Denote by $k(x,\tilde x)$, $x,\tilde x \ge 0$, the
$\sL(L^2(\pl M;E\rest{\pl M}))$-valued kernel of $\widetilde A$.
In terms of $A_\partial (x,\gl)$ the kernel $k(x,\tilde x)$ is given by
\begin{equation}\label{eq:KernelOnCylinder}
k(x,\tilde x)=
\frac{1}{2\pi} \int_{-\infty}^\infty e^{i(x-\tilde x )\gl} A_\partial (x,\gl) d\gl.
\end{equation}
Hence, as $R\to \infty$ one has in view of \eqref{eq:asymptotics-b-symbol}
\begin{equation}\label{eq:b-trace-explanation}
\begin{split}
\Tr(A&\rest{\{x\ge -R\}}) = \\
= \, &\Tr(A\rest{M^1})+\int_{-R}^0\Tr_{\pl M}(k(x,x))\;dx\\
= \, &\Tr(A\rest{M^1})+\frac{1}{2\pi}\int_{-\infty}^0\int_{-\infty}^\infty \Tr_{\pl M}\bigl( A_\partial (x,\gl)-\cI(A)(\gl)\bigr) \, d\gl \, dx\\
&+R \frac{1}{2\pi}\int_{-\infty}^\infty
\Tr_{\pl M}\bigl(\cI(A)(\gl)\bigr) \, d\gl+O(e^{-R}),\quad R\to\infty.
\end{split}
\end{equation}
The \emph{finite part} of this expansion is called the {\btrace} of $A$:
\sind{partie finie}
\begin{equation}
\label{eq:b-trace-def}
\bTr(A):=\Tr(A\rest{M^1})+\frac{1}{2\pi}
\int_{-\infty}^0\int_{-\infty}^\infty \Tr_{\pl M}\bigl(A_\partial(x,\gl)-
\cI(A)(\gl)\bigr) \, d\gl \, dx.
\end{equation}
Hence
\begin{equation}\label{eq:b-trace-def-a}
\begin{split}
\Tr(A&\rest{\{x\ge -R\}})\\
&=\bTr(A)+R \frac{1}{2\pi}\int_{-\infty}^\infty
\Tr_{\pl M}\bigl(\cI(A)(\gl)\bigr) d\gl+O(e^{-R}),\quad R\to\infty.
\end{split}
\end{equation}
Its name notwithstanding, the {\btrace} is not a trace. One has, however,
the following crucial formula.
\begin{proposition}[{\cite[Prop.~5.9]{Mel:APSIT}, \cite[Thm.~2.5]{Loy:DOB}}]
\label{p:b-trace-defect}
Assume that $A\in{\bpdo^m (M;E)}$ and $K\in\bpdo^{-\infty}(M;E)$. Then
\begin{equation}
\bTr(AK-KA)=\frac{-1}{2\pi i} \int_{-\infty}^\infty
\Tr_{\pl M}\left( \frac{d\cI(A)(\gl)}{d\gl} \, \cI(K)(\gl) \right)d\gl.
\end{equation}
\end{proposition}
\begin{proof}[Proof in a special case] It is instructive to prove this in the special case that $A$ is a Dirac operator $\dirac$.
We will see in Section \ref{s:b-Clifford-Dirac} below that on the cylinder $\dirac$ takes the form
$\dirac=\Gammabdy \frac{d}{dx}+\mathsf{D}_\pl$ and that $\cI(\dirac)(\gl)=i\Gammabdy \gl+\mathsf{D}_\pl$.
After choosing cut--off functions w.l.o.g. we may assume that $K$ is supported in the interior of
the cylinder and given by
\begin{equation}
\begin{split}
(Ku)(x)&=\frac{1}{2\pi}\int_{-\infty}^\infty e^{ix\gl} k(x,\gl) \hat u(\gl)d\gl\\
&=\frac{1}{2\pi}\int_{-\infty}^\infty\int_{-\infty}^\infty e^{i(x-y)\gl} k(x,\gl) u(y) dyd\gl
\end{split}
\end{equation}
with an operator valued symbol $k(x,\gl)$. Then
\begin{equation}
\begin{split}
(\dirac Ku)(x)&= \frac{1}{2\pi}\int_{-\infty}^\infty e^{ix\gl}(i\gl\Gammabdy+\mathsf{D}_\pl)k(x,\gl) \hat u(\gl)d\gl\\
&\qquad +\frac{1}{2\pi}\int_{-\infty}^\infty e^{ix\gl} \Gammabdy\partial_x k(x,\gl) \hat u(\gl) d\gl.
\end{split}
\end{equation}
Furthermore, since $(\dirac u)^\wedge=\bigl(i\Gammabdy\gl+\mathsf{D}_\pl)\hat u$ we have
\begin{equation}
(K\dirac) u(x)= \frac{1}{2\pi}\int_{-\infty}^\infty e^{ix\gl}k(x,\gl)(i\gl\Gammabdy+\mathsf{D}_\pl) \hat u(\gl)d\gl.
\end{equation}
Consequently
\begin{equation}
\Tr_{\pl M}\bigl((\dirac K-K\dirac)(x,x)\bigr) =\frac{1}{2\pi}\int_{-\infty}^\infty \Tr_{\pl M}\bigl(\Gammabdy \partial_x k(x,\gl)\bigr)d\gl,
\end{equation}
and hence, since by assumption $K$ is supported in $(-\infty,0)\times \pl M$
and taking \eqref{eq:asymptotics-b-symbol} into account, we find
\begin{align}
\int_{-R}^0\Tr_{\pl M}\bigl((&\dirac K-K\dirac)(x,x)\bigr) dx =\frac{-1}{2\pi} \int_{-\infty}^\infty \Tr_{\pl M}\bigl(\Gammabdy k(-R,\gl)\bigr)d\gl\\
&\overset{R\to \infty}{\longrightarrow} \frac{-1}{2\pi i}\int_{-\infty}^\infty \Tr_{\pl M}\Bigl(\frac{d\cI(\dirac)(\gl)}{d\gl} \cI(K,\gl) \Bigr)d\gl.\qedhere
\end{align}
\end{proof}
\noindent
Eq.~\eqref{eq:b-trace-def-a} immediately entails the following result.
\begin{corollary}
\label{prop:smotrcl}
Let $A\in{\bpdo^m} (M;E)$ be a classical \textup{b}-pseudodifferential
operator of order $m < \dim M$.
If the indicial family\sind{indicial family} $\cI(A) $ vanishes, then $A$ is trace class, and
\[
\Tr (A) = \bTr(A ).
\]
\end{corollary}
\section{The relative McKean--Singer formula and the \textup{APS} Index Theorem}
\label{s:McK-S}
\newcommand{\overline\tau}{\overline\tau}
In the introduction to \cite{Mel:APSIT}, the author explained in detail his elegant
approach to the {\APS} index theorem, based on the \textup{b}-calculus.
The relative cohomology point of view allows to make this approach even more
appealing.
Indeed, we will show that the {\APS} index can be obtained as the pairing between a
natural relative cyclic $0$-cocycle, one of whose components is the {\btrace},
and a relative cyclic $0$-cycle constructed out of the heat kernel.\sind{heat kernel}
This pairing leads in fact to a relative version of the McKean--Singer formula.
\sind{pairing}\sind{McKean--Singer formula!relative}\sind{relative!McKean--Singer formula}
We start, a bit more abstractly, by considering an exact sequence of algebras
\begin{equation}\label{eq:ML200909141}
0\longrightarrow \cJ \longrightarrow \cA \overset{\sigma}{\longrightarrow} \cB \longrightarrow 0;
\end{equation}
$\cA, \cB$ are assumed to be unital, $\sigma$ is assumed to be a unital homomorphism.
Let $\tau$ be a \semph{hypertrace} on $\cJ$, \emph{i.e.} $\tau$ satisfies
\begin{equation}\label{eq:ML200909111}
\tau(xa)=\tau(ax) \text{ for } a\in\cA, x\in\cJ .
\end{equation}
Let $\overline\tau:\cA\longrightarrow \C$ be a
linear extension (\semph{regularization}) of $\tau$
to $\cA$, which is not assumed to be tracial. Nevertheless,
$\overline\tau$ induces a cyclic 1-cocycle on $\cB$ as follows:
\begin{equation}\label{eq:ML200911261}
\mu(\sigma(a_0),\sigma(a_1)):= \overline\tau([a_0,a_1]).
\end{equation}
Because of Eq. \eqref{eq:ML200909111},
$\mu$ does indeed depend only on $\sigma(a_0),\sigma(a_1)$. Moreover, the pair $(\overline\tau,\mu)$
is a relative cyclic cocycle. Namely, in the notation of \eqref{Eq:DefCoBdrRelMixDer},
Eq.~\eqref{eq:ML200911261} translates into $\widetilde b (\overline\tau,\mu)=0$.
The relevant example for this paper is the exact sequence
\begin{equation}
0\longrightarrow \bPsi^{-\infty}_{\textrm{tr}}(M;E)\longrightarrow \bPsi^{-\infty}(M;E)^+
\overset{\cI}{\longrightarrow} \sS\longrightarrow 0,
\end{equation}
where $M$ is a compact manifold with boundary equipped with an exact \textup{b}-metric.
The operator trace $\Tr$ on $\bPsi^{-\infty}_{\textrm{tr}}(M;E) = \Ker \cI$ satisfies
\eqref{eq:ML200909111}, and the
{\btrace} $\bTr$ provides its linear extension to $\bPsi^{-\infty}(M;E)^+$.
The indicial map $\cI$ realizes the quotient algebra
\[\sS=\bPsi^{-\infty}(M;E)^+/\bPsi^{-\infty}_{\textrm{tr}}(M;E)\]
as a subalgebra of the unitalized algebra
$\sS(\R,\Psi^{-\infty}(\partial M;E))^+$ of Schwartz functions with
values in the smoothing operators $\Psi^{-\infty}(\partial M;E)$ on the
boundary. That $\sS$ does not equal $\sS(\R,\Psi^{-\infty}(\partial M;E))^+$
(which would be nicer and more intuitive here) has to do with the fine print
of the definition of the $b$--calculus which requires e.g. the analyticity
of the indicial family. For our discussion here these details are not
relevant and hence we will not elaborate further on them.
Going back to the abstract sequence \eqref{eq:ML200909141}
assume now that the algebra $\cA$ is represented as bounded operators on some Hilbert space $\sH$
and that $\cJ\subset \sL^1(\sH)$ consists of trace class operators. Let $D$ be a
self-adjoint unbounded operator affiliated with $\cA$, \emph{i.e.}
bounded continuous functions of $D$ belong to $\cA$.
Furthermore, we assume that we are in a graded (even) situation
and denote the grading operator by $\ga$.
Finally we assume that $D$ is a Fredholm operator and that the orthogonal projection $P_{ \Ker D}\in \cJ$.
We note that in the case of a Dirac operator $\dirac$ on the \textup{b}-manifold $M$ it is well-known that
$\dirac$ is Fredholm if and only if the tangential operator $\mathsf{D}_\pl$ (see Section \plref{s:b-Clifford-Dirac}
and Eq.~\eqref{eq:essspecbd} below) is invertible.
Define
\begin{align}
A_0(t)&:=\alpha D e^{-\frac 12 t D^2},\\
A_1(t)&:= \int_t^\infty D e^{(\frac{t}{2}-s) D^2}\, ds .
\end{align}
Since $D$ is affiliated with $\cA$,
$A_j(t)\in\cA$ for $t>0$ and $j=1,2$.
\subsection{Case 1: $e^{-t D^2}\in\cJ$, for $t>0$}
Under this assumption $A_j(t)\in \cJ$, for $t>0$.
Moreover, $e^{-t D^2}\in C_0^\gl(\cJ)=C_0(\cJ)/((1-\gl)C_0(J))$
defines naturally a class in $H_0^\gl(\cJ)$.
\begin{lemma}\label{l:McKeanSingerHomological} The class of $\ga e^{-t D^2}$ in $H_0^\gl(J)$ equals
that of $\ga P_{ \Ker D}$. In particular, it is independent of $t$.
\end{lemma}
\begin{proof} We calculate
\begin{equation}\label{eq:ML200909143}
\begin{split}
b (A_0\otimes A_1)&= 2 \ga \int_t^\infty D^2 e^{-s D^2}\, ds\\
&= 2\ga \bigl (e^{-t D^2} - P_{ \Ker D} \bigr),
\end{split}
\end{equation}
proving that $\ga e^{-t D^2}$ and $\ga P_{ \Ker D}$ are homologous.
\end{proof}
As an immediate corollary one recovers the classical McKean--Singer formula.
\sind{McKean--Singer formula}
Indeed, since the trace $\tau$ defines a class in $H^0_\gl(\cJ)$ one finds
\begin{equation}
\ind D = \Tr(\ga P_{ \Ker D})=\langle \tau,\ga P_{ \Ker D}\rangle = \langle \tau,\ga e^{-t D^2}\rangle=\Tr(\ga e^{-t D^2}).
\end{equation}
\subsection{Case 2: The general case} The heat operator
$e^{-t D^2}$ gives a class in $H_0^\gl(\cA)$. The pairing of $e^{-t D^2}$
with $\overline\tau$ cannot be expected to be independent of $t$ since $\overline\tau$ is not a trace.
It is however a component of the relative cyclic $0$-cocycle $(\overline\tau,\mu)$.
Therefore, we are led to construct a relative cyclic homology
class from $e^{-t D^2}$. By Eq.~\eqref{Eq:DefBdrRelMixDer}
the relative cyclic complex is given by
\begin{equation}\label{eq:ML200909144}
C_n^\gl(\cA,\cB):=C_n^\gl(\cA)\oplus C_{n+1}^\gl(\cB),\quad \tilde b:=
\begin{pmatrix} b & 0 \\ -\sigma_* & -b \end{pmatrix}.
\end{equation}
From \eqref{eq:ML200909143} we infer
\begin{equation}
\sigma( \ga e^{-t D^2})= \sigma (\ga e^{-t D^2}-\ga P_{ \Ker D})= \frac 12 b \bigl(\sigma(A_0)\otimes \sigma(A_1)\bigr),
\end{equation}
hence
\begin{equation}
\widetilde b { \ga e^{-t D^2} \choose -\frac 12 \sigma(A_0)\otimes \sigma(A_1)}=0,
\end{equation}
\emph{i.e.}~the pair
$\operatorname{EXP}_t(D):=
\bigl(\alpha e^{-t D^2}, -\frac 12\sigma(A_0)\otimes\sigma(A_1)\bigr)$ is a
relative cyclic homology class. Furthermore, since
\begin{equation}
\widetilde b { \frac 12 A_0\otimes A_1 \choose 0}= {\ga (e^{-t D^2}-P_{ \Ker D}) \choose -\frac 12 \sigma(A_0)\otimes \sigma(A_1)},
\end{equation}
the class of $\operatorname{EXP}_t(D)$ in $H_0^\gl(\cA,\cB)$ equals that of
the pair $(\ga P_{ \Ker D},0)$ which, via \excision, corresponds to the class
of $\ga P_{ \Ker D}\in HP_0(\cJ)$. We have thus proved the following result.
\begin{lemma}\label{l:McKeanSingerHomologicalRelative}
The class of the pair $\big(\ga e^{-t D^2}, -\frac 12 \sigma(A_0)\otimes \sigma(A_1)\big)$ in $HC_0^\gl(\cA,\cB)$
equals that of $(\ga P_{ \Ker D},0)$. In particular, it is independent of $t$.
\end{lemma}
Pairing with the relative $0$-cocycle $(\overline\tau,\mu)$ we now obtain the following
relative version of the McKean--Singer
formula:
\begin{equation}
\begin{split}
\ind D&= \Tr(\ga P_{ \Ker D})=\langle (\overline\tau,\mu),
\bigl( \ga e^{-t D^2}, -\frac 12 \sigma(A_0)\otimes \sigma(A_1)\bigr)\rangle\\
&= \overline\tau ( \ga e^{-t D^2}) - \frac 12 \mu(\sigma(A_0),\sigma(A_1))\\
& = \overline\tau (\ga e^{-t D^2}) - \frac 12 \overline\tau ([A_0,A_1]).
\end{split}
\end{equation}
Once known, this identity can also be derived quite directly. Indeed, since
$[A_0,A_1]=2\ga \int_t^\infty D^2 e^{-s D^2}ds$,
\begin{equation}
\begin{split}
\overline\tau (\ga e^{-t D^2}) - \frac 12 \overline\tau ([A_0,A_1])
&=\overline\tau (\ga e^{-t D^2})-\overline\tau\bigl( \ga \int_t^\infty D^2 e^{-s D^2} \, ds \bigr)\\
& =\overline\tau (\ga e^{-t D^2}) +\int_t^\infty\frac{d}{ds}\overline\tau\bigl(\ga e^{-s D^2}\bigr)\, ds\\
& =\lim_{s\to\infty} \overline\tau\bigl(\ga e^{-s D^2}\bigr).
\end{split}
\end{equation}
Let us show that in the case of the Dirac operator on a \textup{b}-manifold
the second summand is nothing but the $\eta$-invariant of the tangential operator.
Indeed, in a collar of the boundary $\dirac$ takes the form
\[
\dirac =\begin{pmatrix} 0 & -\frac{d}{dx}+\Abdy\\ \frac{d}{dx}+\Abdy & 0 \end{pmatrix}=:
\Gammabdy \frac{d}{dx}+\mathsf{D}_\pl,\quad \mathsf{D}_\pl=\begin{pmatrix} 0 & \Abdy \\ \Abdy & 0 \end{pmatrix}.
\]
Hence, one calculates using Proposition \plref{p:b-trace-defect}
\begin{equation}
\begin{split}
\frac 12 \mu(&\cI(A_0,\gl),\cI(A_1,\gl)) \\
&= \frac{-1}{4\pi i} \int_{-\infty}^\infty \Tr_{\pl M} \bigl(\frac{d \cI(A_0,\gl)}{d\gl} \cI(A_1,\gl)\bigr) d\gl\\
&= \frac{-1}{4\pi}\int_{-\infty}^\infty\Tr_{\pl M}\bigl(\ga \Gammabdy \int_t^\infty (i\gl\Gammabdy+\dirac_\pl) e^{-s(\dirac_\pl^2+\gl^2)}\, ds\,\bigr) d\gl\\
&= \frac{-1}{4\sqrt{\pi}} \int_t^\infty \frac{1}{\sqrt{s}}\Tr_{\pl M} \bigl(\begin{pmatrix} -A & 0 \\ 0 & -A\end{pmatrix} e^{-s A^2}\bigr)ds\\
&= \frac{1}{2\sqrt{\pi}}\int_t^\infty \frac{1}{\sqrt{s}} \Tr_{\pl M} \bigl( A e^{-s A^2}\bigr)ds=:
{ \frac 12 \eta_t(A)}.
\end{split}
\end{equation}
Thus if $\dirac$ is Fredholm we have for each $t>0$
\begin{equation}
\ind \dirac = \bTr(\ga e^{-t \dirac^2})- \frac 12 \eta_t(A),
\end{equation}
and taking the limit as $t\searrow 0$ gives the {\APS} index theorem in the \textup{b}-setting.
\section{A formula for the \textup{b}-trace}
\label{s: b-trace formula}
In this section we give an explicit formula for the {\btrace},
based on an observation of \textnm{Loya} \cite{Loy:DOB}, which provides a
convenient tool for subsequent computations.
We first briefly review the Hadamard partie finie integral
in the special case of \textup{b}-functions. Let $f\in\bcC((-\infty,0])$.
From the asymptotic expansion (see~Eq.~\eqref{eq:ML20090219-3})
\begin{equation}\label{eq:ML20090122-1}
f(x)\sim_{x\to -\infty} f_0^-+f_1^- e^x + f_2^- e^{2x}+....
\end{equation}
we infer
\begin{equation}\label{eq:ML20090122-2}
\int_{-R}^0 f(x) dx = f_0^- \; R + c + O(e^{-R}),\quad R\to -\infty.
\end{equation}
The \emph{partie finie integral} of $f$ is then defined to be the constant
term in the asymptotic expansion \eqref{eq:ML20090122-2}, i.e.
\sind{partie finie}
\nind{PartieFinie@Pf$\intop$}
\begin{equation}\label{eq:defPfInt}
\int_{-R}^0 f(x) dx =: f_0^- \; R + \pfint_{-\infty}^0 f(x) dx + O(e^{-R}),\quad
R\to -\infty.
\end{equation}
The definition of the partie finie integral has an obvious extension to
\textup{b}-functions on manifolds with cylindrical ends (see Section
\plref{App:bdefbmet}).\sind{cylindrical ends}
Because of its importance, we single it out as a definition-proposition.
\begin{defprop}\label{defprop15}
Let $M^\circ$ be a riemannian manifold with cylindrical ends
and $M$ the (up to diffeomorphism) unique compact manifold with boundary
having $M^\circ$ as its interior.
For a function $f \in \bcC (M^\circ)$ one has
\begin{displaymath}
\int_{x \geq -R} f \, d\vol =: c \log R + \int_{\tb M} f \, d\vol + O(e^{-R})
\quad \text{as $R\rightarrow \infty$}.
\end{displaymath}
This means that $\int_{\tb M} f \, d\vol$ is the finite part in the asymptotic
expansion of $\int_{x \geq -R} f \, d\vol$ as $R\rightarrow \infty$. More
generally, if $\omega \in \bOmega^m (M)$ is a (top degree) \textup{b}-differential form,
i.e.~a form whose coefficients are in $\bcC (M^\circ)$, then
$\int_{\tb M} \omega$ is defined accordingly as the finite part of
$\int_{x \geq -R} \omega$ as $R\rightarrow \infty$.
\end{defprop}
In local coordinates $y_1,\ldots,y_n$ on $\partial M$, \textup{b}-differential $p$--forms
are sums of terms of the form
\begin{equation}
\go =f(x,y) dx\wedge dy_{i_1}\wedge \ldots\wedge dy_{i_{p-1}}+
g(x,y) dy_{j_1}\wedge \ldots\wedge dy_{j_{p}},
\end{equation}
where $1\le i_1<\ldots<i_{p-1}\le n, \; 1\le j_1<\ldots<j_p\le n$ and $f,g$
are \textup{b}-smooth functions. Putting $\iota^*\go:= g_0^-(y) dy_{j_1}\wedge \ldots\wedge dy_{j_{p}}$
(\emph{cf.}~\eqref{eq:ML20090122-2})
extends to a pullback $\iota^*:\bOmega^p(M)\to \Omega^p(\partial M).$ It is easy to see that
Stokes' Theorem holds for $\int_{\tb M}$ and $\iota$:
\begin{equation}
\int_{\tb M} d\go = \int_{\partial M} \iota^* \go.
\end{equation}
For a \textup{b}-pseudodifferential operator $A\in\bpdo^\bullet(M;E)$ of order $<-\dim M$
the {\btrace} is nothing but the partie finie integral of its kernel over the
diagonal:
\sind{partie finie}
\begin{equation}\label{eq:bTrace-as-Pf}
\bTr(A)=\int_{\tb M} \tr_{p}(K_A(p,p)) d\vol(p),
\end{equation}
where now $K_A(\cdot,\cdot)$ denotes the Schwartz kernel
of $A$ and $\tr_p$ denotes the fiber trace on $E_p$.
Next we mention a useful formula for the partie finie integral in terms of
a convergent integral. By Eq.~\eqref{eq:ML20090219-4},
the asymptotic expansion \eqref{eq:ML20090122-1} may be differentiated,
hence $\partial_x f=O(e^x)$, $x\to -\infty$, is integrable and thus integration by parts
yields
\begin{equation}\label{eq:ML20090122-3}
\begin{split}
\int_{-R}^0 f(x) dx &= Rf(-R)-\int_{-R}^0 x \partial_xf(x) dx\\
&= R f_0^- - \int_{-\infty}^0 x \partial_x f(x)dx+ O(R e^{-R}),\quad
R\to -\infty.
\end{split}
\end{equation}
Hence
\nind{PartieFinie@Pf$\intop$}
\begin{equation}\label{eq:PfIntFormula}
\pfint_{-\infty}^0 f(x) dx = \int_{-\infty}^0 x\partial_xf(x) dx,
\end{equation}
where the integrand on the right hand side is summable in the Lebesgue sense.
Using the tools from the previous paragraphs, we can now prove the following
theorem about the representation of the {\btrace} as a trace of certain trace class operators.
\begin{proposition}\label{t:bTraceAsTrace}
Let $M$ be a compact manifold with boundary and an exact \textup{b}-metric $\bmet$.
Fix a collar $(r,\eta) : Y \rightarrow [0,2)\times \partial M$ of the boundary $\partial M$
as described in Section \ref{App:bdefbmet}, and let
$(x,\eta): \clos{Y^1} \rightarrow \, (-\infty, 0] \times \partial M$
denote the corresponding diffeomorphism onto the cylinder $(-\infty,0]\times\partial M$.
Assume that $A\in{\bpdo^\infty} (M;E)$ is a classical \textup{b}-pseudodifferential operator of
order $<-\dim M$, and that
its kernel is supported within the cylinder $(-\infty,0)\times\partial M$.
Then $x\big[ \frac{d}{dx},A \big]$ is trace class and one has
\begin{equation}\label{eq:bTraceAsTrace}
\begin{split}
\bTr(A)&=-\Tr \left( x \big[ \frac{d}{dx},A \big] \right) =\\
&= -\int_{(-\infty,0)\times\partial M}\, x \frac{d}{dx}
\tr_{x,q}\bigl(K_A(x,q;x,q)\bigr)\, d\vol(x,q),
\end{split}
\end{equation}
where $K_A$ denotes the Schwartz kernel of $A$.
\end{proposition}
\begin{proof} The condition on the support of $A$ is necessary since
the operators $x$ and $\frac{d}{dx}$ are only defined on the cylinder.
However, Proposition \plref{t:bTraceAsTrace} can be extended to arbitrary
$A\in{\bpdo^{<-\dim M}}(M;E)$ in a straightforward way: choose a pair of
cut-off functions $\varphi,\psi\in \cinf{M}$ with support within the cylinder
$(-\infty ,0) \times \partial M$ and such that
$\varphi(x)=1$ for $x\le -2$, $\varphi(x)=0,$ for $x\ge -3/2$,
$\psi(x)=1$ for $x\le -1$ and $\psi(x)=0$ for $x\ge -1/2$.
Finally, choose a cut-off function $\chi\in\cinfz{M\setminus\partial M}$ with
compact support and $\chi (1-\varphi)=1-\varphi$. The definition
of the {\btrace} then immediately shows
that
\[
\bTr(\varphi A)=\bTr(\psi\varphi A)=\bTr(\varphi A \psi)
\]
and hence
\[
\begin{split}
\bTr(A)&=\bTr(\varphi A \psi)+\Tr((1-\varphi) A\chi) =\\
&=-\Tr(x [\frac{d}{dx},\varphi A \psi])+\Tr((1-\varphi) A\chi).
\end{split}
\]
The fact that $x[\frac{d}{dx},A]$ is trace class follows by Prop.~\ref{prop:smotrcl},
since the indicial family of the commutator $[\frac{d}{dx},A]$ vanishes.
We provide two variants of proof for \eqref{eq:bTraceAsTrace}.
\subsection*{1st Variant.}
From equations \eqref{eq:bTrace-as-Pf} and \eqref{eq:PfIntFormula}
we infer
\begin{equation}
\begin{split}
\bTr(A) &= \int_{\tb (-\infty,0)\times\partial M}\tr_{x,q}\bigl(K_A(x,q;x,q)\bigr) d\vol(x,q)\\
&= -\int_{(-\infty,0)\times\partial M}x \frac{d}{dx}\tr_{x,q}\bigl(K_A(x,q;x,q)\bigr) d\vol(x,q).
\end{split}
\end{equation}
This proves the second line of \eqref{eq:bTraceAsTrace}. The first line follows,
since the kernel of $[\frac{d}{dx},A]$ is given
by $[\frac{d}{dx},K_A](x,p;y,q)=\partial_x K_A(x,p;y,q)+\partial_y K_A(x,p;y,q)$
which for $x=y$ equals $\frac{d}{dx} K_A(x,p;x,p)$, \emph{cf.}~Eq.~\eqref{eq:PfIntFormula}.
\subsection*{2nd Variant.}
For $\Re z>0$ the operator $e^{zx}A$ is trace class and the function
\begin{equation}
z\mapsto \Tr(e^{zx}A)
\end{equation}
is holomorphic for $\Re z>0$ and it extends meromorphically to $\Re z>-1$, $0$
is a simple pole and the residue at $0$ equals $\bTr(A)$ (\emph{cf.}~\cite{Loy:DOB}). Hence
\begin{equation}
\begin{split}
\bTr(A) &= \left.\frac{d}{dz} z\Tr(e^{zx}A)\right|_{z=0}\\
&= \left.\frac{d}{dz} \Tr \left( \big[\frac{d}{dx},e^{zx} \big]
A \right)\right|_{z=0}\\
&= \left.\frac{d}{dz} \Tr \left( \big[ \frac{d}{dx},e^{zx}A \big] -e^{zx}
\big[ \frac{d}{dx},A \big] \right)\right|_{z=0}\\
&=-\Tr \left(x \big[\frac{d}{dx},A \big] \right),
\end{split}
\end{equation}
since for $\Re z>0$ the trace of the commutator $\Tr([\frac{d}{dx},e^{zx}A])$,
thanks to the decay of $e^{zx}$, does vanish.
The last claim follows as above.
\end{proof}
\section{\textup{b}-Clifford modules and \textup{b}-Dirac operators}
\label{s:b-Clifford-Dirac}
Let $M$ be a compact manifold with boundary,
$r: Y \rightarrow [0,2)$ a boundary defining function,
and $\bmet $ an exact \textup{b}-metric on $M$, \emph{cf.}~Section \plref{App:bdefbmet}.
If an object is derived from a \textup{b}-metric we indicate
this notationally by giving it a \textup{b}-decoration. This applies
in particular to the various structures derived from the riemannian metric $\bmet$
as described in Section \plref{s:qDirac},
\emph{e.g.}~the (co)tangent bundles $\bT M, \bT^* M$,
the Levi-Civita \textup{b}-connection $\bnabla$ belonging to $\bmet$,
the bundle of Clifford algebras $\bCl (M) := \Cl (\bT^*M)$, and the
\emph{\textup{b}-Clifford superconnection}\sind{bCliffordsuperconnection@\textup{b}-Clifford superconnection}\sind{Clifford superconnection}
$\bA$ on a degree $q$ \textup{b}-Clifford module\sind{bCliffordmodule@\textup{b}-Clifford module}\sind{Clifford module} $W$ over $M$.
For a discussion of $\bcC(M^\circ)$ vs. $\cC^\infty(M)$ we refer
to Section \plref{App:bdefbmet}.
In the remainder of this article, we assume that
a \textup{b}-Clifford superconnection is always of product form near the boundary. This
means that over $Y^s$ for some $s$ with $0<s<2$ the superconnection
has the form
\[
\bA\rest{Y^s} = \eta^* \nabla^\partial + \eta^* \omega^\partial \wedge -,
\]
where $\eta : Y \rightarrow \partial M$ is the boundary projection from
Section \ref{App:bdefbmet}, $\nabla^\partial$ is a metric connection on the
restricted bundle $W\rest{\partial M}$ and
$\omega^\partial\in\Omega^\bullet\bigl(\partial M ;\End\big(W\rest{\partial M}\big)\bigr)$.
Recall that the pull-back covariant derivative $\big(\eta^* \nabla^\partial\big)$
on $W\rest{Y}$ is uniquely defined by requiring for
$\xi\in \Gamma^\infty \big( Y; W\big)$ that
\[
\big(\eta^* \nabla^\partial\big)_V \,\xi =
\begin{cases}
r \frac{\partial \xi}{\partial r},&
\text{if $V= r\frac{\partial}{\partial r} $},\\
\nabla^\partial_{\tilde V}\xi ,& \text{if $V= \tilde V \circ \eta$ for some
$\tilde V \in \Gamma^\infty (\partial M; T\partial M)$}.
\end{cases}
\]
Note that the \textup{b}-metric on $M$ and the metric structure on $W$ give rise to the Hilbert
space $\sH = L^2(M;W)$ of square integrable sections of the \textup{b}-Clifford module.
By assumption, $\Cl_q$ acts on $L^2(M;W)$, hence by Eq.~\eqref{eq:Clifford-super-trace}
one obtains a supertrace
$\Str_q: \sL_{\Cl_q}^1\big( L^2(M;W) \big) \rightarrow \C$.
Similarly the {\btrace} gives rise to a \textup{b}-supertrace
\begin{equation}\label{eq:bSuperTrace}
\begin{split}
&\bStr_q: \bpdosub{\Cl_q}^{<-\dim M}\bigl(M;W\bigr)\longrightarrow \C,\\
& \bStr_q(K):=(4\pi)^{-q/2}\;\;\bTr(\ga E_1\cdot ... \cdot E_q K).
\end{split}
\end{equation}
Here, $\bpdosub{\Cl_q}^\bullet\bigl(M;W\bigr)$ denotes the space of classical
\textup{b}-pseudodifferential operators which lie in the supercommutant of $\Cl_q$ in $\sH$,
\emph{cf.}~\eqref{eq:def-super-commutator} \emph{supra}.
Next consider the natural embedding $T^*\partial M \hookrightarrow \bT^*_{|\partial M}M$.
By the universal property of Clifford algebras one obtains
an embedding of Clifford bundles $\Cl \big(\partial M\big) \hookrightarrow
\bCl \big( \bT^*_{|\partial M}M\big) $. Moreover, the
decomposition $\bT^*\rest{\partial M}M=T^*M\oplus \R \cdot r\frac{\pl}{\pl r} $
induced by $\bmet$ even gives rise to a splitting
$\bCl \big( \bT^*_{|\partial M}M\big) \rightarrow \Cl \big( \partial M \big)$.
Let now $W\rightarrow M$ be a degree $q$ \textup{b}-Clifford module over $M$.
Then $\Cl \big(\partial M\big)$ acts on $W_{|\partial M}$ via the embedding
$\Cl \big(\partial M\big) \hookrightarrow \bCl \big( \bT^*_{|\partial M}M\big)$.
We denote the resulting left action of the boundary Clifford bundle on
$W_{|\partial M}$ again by $\sfc$.
Moreover, the action $W_{|\partial M} \otimes \Cl_q \rightarrow W_{|\partial M}$
extends to a right action
$\sfbcr :W_{|\partial M}\otimes \Cl_{q+1} \rightarrow W_{|\partial M}$ by putting
\begin{equation}
\label{eq:defracbdr}
E_j = \sfbcr (w, e_j ) :=
\begin{cases}
\sfcr \big( w, e_j \big), & \text{for $w \in W_p$, $p\in \partial M$,
$j=1,\cdots , q$}, \\
-\sfcl \big(\frac{dr}{r},w \big), & \text{for $w \in W_p$, $p\in \partial M$,
$j=q+1$},
\end{cases}
\end{equation}
\emph{cf.}~the beginning of Section \ref{s:qDirac}.
It is now easy to check that $W\rest{\partial M}$ together with $\sfc$ and
$\sfbcr$ as Clifford actions becomes a degree $q+1$ Clifford-module over
$\partial M$.
Now we have the ingredients for the \textup{b}-supertrace of a supercommutator:
\begin{proposition}[{\cite[Cor.~5.5]{Get:CHA}}]
\label{p:b-trace-defect-graded}
Let $\dirac$ be a Dirac operator on a $q$-graded \textup{b}-Clifford bundle $W$,
and $K\in\bpdosub{\Cl_q}^{-\infty}\big( M ; W \big)$.
On $\partial M$ put $E_{q+1}:=-\sfcl(\frac{dr}{r})=-\sfcl(dx)$.
Then
\begin{equation}\label{eq:bTraceDefectClifford}
\bStr_q \big([\dirac,K]_\super\big)=
\frac{1}{\sqrt{\pi}} \int_{-\infty}^\infty \Str_{q+1,\partial M}\big( \cI(K,\gl) \big )d\gl.
\end{equation}
\end{proposition}
We do not claim here that $\cI(K,\gl)$ commutes with $\Gammabdy$ in the graded sense.
This is not necessary for the definition of $\Str_{q+1,\partial M}$.
\begin{remark}\label{r:DiracCylinderFormulas}
Another consequence of the previous considerations which we single out for future
reference is the structure of a Dirac operator on a cylinder $\R\times \partial M$
(\emph{cf.}~\eqref{eq:ML20090219-1}). Since all structures are product, $\dirac$ takes the form
\begin{equation}\label{eq:dirac-cylinder}
\dirac=\sfc(dx)\frac{d}{dx}+\mathsf{D}_\pl=:\Gammabdy\Bigl(\frac{d}{dx}+A\Bigr).
\end{equation}
Here $\Gammabdy=\sfc(dx)$ is Clifford multiplication by the normal vector $\frac{d}{dx}$
and $\mathsf{D}_\pl:=\Gammabdy \Abdy$ is the \semph{tangential operator}.
$\mathsf{D}_\pl$ is a Dirac operator on the boundary. Moreover, one has the relations
\begin{equation}
\label{eq:dirac-cylinder-formulas}
\Gammabdy^*=-\Gammabdy, \quad \Gammabdy^2=-I,\quad A^t=
A, \quad \Gammabdy A+A\Gammabdy=\Gammabdy\mathsf{D}_\pl+\mathsf{D}_\pl\Gammabdy=0.
\end{equation}
\end{remark}
Now let $u$ be a section of the Clifford bundle $W$ over the cylinder $\R \times M$.
Then
\begin{equation}
\begin{split}
(\dirac u) (x,p) = \, &
\frac{1}{2\pi} \int_{-\infty}^\infty
\bigl(\sfc(dx)\frac{d}{dx}+\mathsf{D}_\pl\bigr) e^{ix\gl} \hat u (\lambda , p) \, d\gl = \\
= \, &\frac{1}{2\pi} \int_{-\infty}^\infty e^{ix\gl}
\bigl(i\sfc(dx)\gl+\mathsf{D}_\pl\bigr) \hat u (\lambda , p) \, d\gl .
\end{split}
\end{equation}
By Eqs.~\eqref{eq:ftbdrop} and \eqref{eq:asymptotics-b-symbol} this proves the following:
\begin{proposition}\label{p:Dirac-indicial-family}
Let $M$ be a compact manifold with boundary and let $\bmet$ be an
exact \textup{b}-metric on $M$. Furthermore, let $\dirac$ be a Dirac operator on $M$. Then
the indicial family\sind{indicial family} of $\dirac$ is given by $\cI(\dirac)(\gl)=i\gl \sfc(dx)+\mathsf{D}_\pl$.
\end{proposition}
\section{The \textup{b}-JLO cochain}
\label{s:bJLOcc}
The degree $q$ Clifford module approach outlined in Section \plref{s:qDirac}
has advantages when dealing with manifolds with boundary, because the formul\ae\, for
the \JLO-cocycle and its transgression
(\emph{cf.}~\eqref{Eq:cocyclecond}, \eqref{Eq:transgress} below) become simpler.
To make the connection to the standard even and odd Chern character without Clifford action,
from now on
we will also consider \emph{ungraded}\sind{Clifford module!ungraded}
Clifford modules without auxiliary Clifford right action. Therefore we assume that either
\begin{itemize}
\item we are in the \emph{graded}\sind{Clifford module!graded}
case with $q$ Clifford matrices $E_1,\ldots,E_q$ where $\dirac$ is odd,
and $\Str_q$ denotes the Clifford trace defined in Section \plref{s:qDirac},
\end{itemize}
or
\begin{itemize}
\item we are in the \emph{ungraded} case, when there are no Clifford matrices and
no grading operator; this case can be conveniently dealt with by putting
$q=-1$ (which is odd!), $\ga=1$ and $\Str_q:=\Tr=\Tr(\ga \cdot)$.
\end{itemize}
From now on, we assume that $\dirac_t$, $t\in (0,\infty),$ is a family of
self-adjoint differential operators of the form $\dirac_t = f(t) \dirac$ with
$\dirac$ the Dirac operator of a $q$-graded \textup{b}-Clifford module $W$
over the \textup{b}-manifold $M$ ($q\ge -1$ according to the previous explanation)
with \textup{b}-Clifford superconnection $(W,\bA)$ and $f: (0,\infty) \rightarrow \R$ a
smooth function.
$\dirac_t$ are Dirac type operators in the sense of \cite{Tay:PDEII}.
Following Getzler \cite[Sec.~6]{Get:CHA}, we define for
$A_0,\cdots, A_k \in \bpdo^\infty (M,W)$ (\emph{cf.}~Subsection \plref{ss:JLODO})
\begin{equation}
\begin{split}
\blangle A_0, \cdots,A_k \rangle_{\dirac_t} \, := &\int_{\Delta_k} \, \bStr_q
\big(A_0 \, e^{- \sigma_0 \dirac_t^2} \cdots A_k \, e^{- \sigma_k \dirac_t^2} \big)d\sigma \\
= &\bStr_q \big( (A_0,\ldots,A_k )_{\dirac_t} \big).
\end{split}
\end{equation}
Put for $a_0,\ldots, a_k\in \cC^\infty(M)$
\begin{align}
\label{Eq:DefChern}
\bCh^k (\dirac) (a_0,\cdots,a_k) & := \blangle a_0, [\dirac,a_1],
\cdots, [\dirac, a_k]\rangle , \\
\label{Eq:DefSlChern}
\begin{split}
\bslch^k(\dirac , \mathsf{V} )(a_0,\cdots,a_k) &:= \\
\sum_{0\leq j \leq k} (-1)^{j \, \deg{\mathsf{V}}} \,
\blangle a_0, [\dirac,a_1],&
\cdots , [\dirac, a_j], \mathsf{V} , [\dirac, a_{j+1}], \cdots ,
[\dirac, a_k] \rangle.
\end{split}
\end{align}
The operation $\bslch$ will mostly be used with $V=\dot \dirac_t$ as a second argument.
Here $\dirac_t$ is considered of odd degree regardless of the value of $q$.
\begin{remark}\label{rem:ML200908282}
For $q=0$, $k$ even resp. $q=-1$, $k$ odd $\bCh^k(\dirac)$, $\bslch^k(\dirac,\dot\dirac)$
are the \textup{b}-analogues of the even and odd \JLO\ Chern character and its transgression.
\end{remark}
The following result is crucial for this paper. It is essentially due to Getzler \cite[Thm.~6.2]{Get:CHA},
although the following version is not stated explicitly in his paper.
\begin{theorem}\label{P:GETZLER} For $q\ge 0$ we have the following two equations for $\bCh^\bullet(\dirac)$
and $\bslch(\dirac,\dot\dirac)$:
\begin{align}
\label{Eq:cocyclecond}
b\bCh^{k-1}(\dirac_t)+B\bCh^{k+1}(\dirac_t) & =\Ch^k(\mathsf{D}_{\pl,t})\circ i^*,\\
\label{Eq:transgress}
\frac{d}{dt}\bCh^k(\dirac_t) +
b\bslch^{k-1}(\dirac_t , \dot{\dirac}_t ) +B\bslch^{k+1}(\dirac_t,\dot\dirac_t)& =
- \slch^k(\mathsf{D}_{\pl,t} , \dot{\dirac}_{\pl,t} )\circ i^*.
\end{align}
\end{theorem}
These formul\ae\, will be repeatedly used in Section \ref{s:retracted-relative-cocycle} and thereafter.
For notational convenience we will omit the symbol
$\circ i^*$ whenever the context makes clear that
this composition is required.
The theorem can be derived from \cite[Thm.~6.2]{Get:CHA} by introducing the form valued
expression
\begin{equation}\label{eq:ML200911191}
\bllangle A_0, \cdots ,A_k \rrangle \, := \int_{\Delta_k} \, \bStr_q
\big( A_0 \, e^{- \sigma_0 (i \, d \dirac_t + \dirac_t^2)} \cdots
A_k \, e^{- \sigma_k (i \, d \dirac_t + \dirac_t^2)} \big)d\sigma,
\end{equation}
and the combined Chern character $\GCh^\bullet$, defined as
\begin{equation}\label{eq:combined-Chern}
\begin{split}
\GCh^k\, & (\dirac_t) (a_0,\cdots,a_k)\\& :=
\bllangle a_0, [\dirac_t,a_1], \cdots , [\dirac_t,a_k]\rrangle,
\end{split} \quad
a_0, \cdots , a_k \in \cC^\infty ( M ).
\end{equation}
For this, Getzler proves
\begin{equation}\label{eq:Getzler}
(-i\, d+b+B) \GCh^\bullet (\dirac_t) = \GCh^\bullet (\mathsf{D}_{\pl,t})\circ i^* .
\end{equation}
\begin{remark}
Note that in this paper we use self-adjoint Dirac operators while Getzler
uses skew-adjoint ones in \cite{Get:CHA}. Accordingly,
our Dirac operators differ by a factor $-i$ from the Dirac operators in
\cite{Get:CHA}. This explains the appearance of such $i$-factors in our formul\ae\,,
which are not present in \cite{Get:CHA}.
\end{remark}
By carefully tracing all the signs and $i$--factors involved in the graded
form valued Clifford calculus, as well as due to the various conventions, it turns out
that separating \eqref{eq:Getzler} into its scalar and $1$-form parts, using
\cite[Lem.~2.5]{Get:CHA}
\begin{equation}
\label{eq:split-JLO-scalar-form}
\begin{split}
\bllangle& A_0,\cdots,A_k \rrangle =\\
&=\blangle A_0,\cdots,A_k \rangle -i \sum_{j=0}^k \,
\blangle A_0,\cdots,A_{j},dt \wedge \dot \dirac_t,A_{j+1},\cdots,A_k\rangle ,
\end{split}
\end{equation}
one obtains Eqs.~\eqref{Eq:cocyclecond} and~\eqref{Eq:transgress}.
However, for completeness,
we will give a more direct argument in Section \ref{s:CocycleFormula},
without using operator valued forms. The proof below follows the
lines of the standard proof for the \JLO-cocycle representing the
Chern character of a $\theta$-summable Fredholm module (\emph{cf.}~\cite{JLO:QKT},
\cite{GetSze:CCT}).
\section[Cocycle and transgression formul\ae]%
{Cocycle and transgression formul\ae\ for the even/odd \textup{b}-Chern character (without Clifford covariance)}
\label{s:CocTransgressNoClifford}
\newcommand{\warning}[1]{ {\color{red} #1}}
Recall from Remark \plref{rem:ML200908282} that for $q=0$ and $k$ even resp. $q=-1$ and $k$ odd
$\bCh^\bullet(\dirac)$ is the \textup{b}-analogue of the even, resp. odd, \JLO\ Chern character.
We shall relate the ungraded ($q=-1$) case to the graded case with $q=1$.
Starting with an ungraded Dirac operator $\dirac_t$ acting on the Hilbert space $\sH$,
put
\begin{equation}\label{eq:ML200911192}
\widetilde \sH:=\sH\oplus \sH,\quad \ga:=\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix},\quad
\widetilde \dirac_t:=\begin{pmatrix} 0& \dirac_t \\ \dirac_t & 0\end{pmatrix}.
\end{equation}
Then $\widetilde \dirac_t$ is odd with respect to the grading operator $\ga$ and it anti commutes
with
\begin{equation}\label{eq:ML200911193}
E_1:=\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}.
\end{equation}
Note that
\begin{equation}\label{eq:ML200909011}
\widetilde \dirac_t= \ga E_1 \bigl(\dirac_t\otimes I_2\bigr)
\end{equation}
with $I_2$ being the $2\times 2$ identity matrix.
\begin{proposition}\label{p:ML200909032}
Let $\dirac_t$ be ungraded {\rm ($q=-1$)}
and let $\widetilde \dirac_t=\ga E_1 (\dirac_t\otimes I_2)$ be the associated
$1$--graded {\rm ($q=1$)} operator. Then for $k$ odd
\begin{equation}
\begin{split}
\bCh^k(\widetilde \dirac_t) &\, = \frac{1}{\sqrt{\pi}} \bCh^k(\dirac_t),\\
\bslch^{k-1}(\widetilde \dirac,\dot{\widetilde \dirac_t}) &\, =\frac{1}{\sqrt{\pi}} \bslch^{k-1}(\dirac_t,\dot\dirac_t).
\end{split}
\end{equation}
\end{proposition}
Needless to say that these formul\ae\ are valid as well for $\Ch^\bullet$ and $\slch^\bullet$.
\begin{proof}
Using Proposition \plref{p:b-trace-defect-graded} we find for $k$ odd:
\begin{equation}
\label{eq:relOddGradedUngraded}
\begin{split}
\blangle &a_0, [\widetilde \dirac_t,a_1],\ldots,[\widetilde \dirac_t,a_k]\rangle_{\widetilde \dirac_t}\\
&= \blangle (\ga E_1)^{k} a_0, [\dirac_t\otimes I_2,a_1],\ldots,[\dirac_t\otimes I_2,a_k]\rangle_{\dirac_t\otimes I_2}\\
&= \int_{\Delta_k} \frac{1}{\sqrt{4\pi}} \bTr\bigl( \underbrace{(\ga E_1)^{k+1}}_{=1}
a_0 e^{-\sigma_0 \dirac_t^2}[\dirac_t,a_1]\ldots [\dirac_t a_k]e^{-\sigma_k \dirac_t^2}\otimes I_2\bigr)\\
&= \frac{1}{\sqrt{\pi}} \blangle a_0, [\dirac_t,a_1],\ldots,[\dirac_t,a_k]\rangle_{\dirac_t}.
\end{split}
\end{equation}
The calculation for $\bslch^{k-1}(\widetilde \dirac_t,\dot{\widetilde\dirac_t})$ is completely analogous.
\end{proof}
Now we are ready to translate \eqref{Eq:cocyclecond} and \eqref{Eq:transgress} into formul\ae\
for the standard even and odd Chern character without Clifford action.
\subsection{$q=0$} A priori we are in the standard even situation without Clifford right action.
However, $\mathsf{D}_\pl$ is viewed as $\Cl_1$ covariant with respect to the Clifford action
given by $E_1=-\Gammabdy$. On the boundary, $\Gammabdy$ gives a natural identification of the even
and odd half spinor bundle and with respect to the splitting into half spinor bundles $\dirac$ takes the form:
\begin{equation}
\begin{split}
\dirac &=\underbrace{\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}}_{\Gammabdy}
\frac{d}{dx} + \underbrace{\begin{pmatrix} 0 & A \\ A & 0\end{pmatrix}}_{\mathsf{D}_\pl} ;
\end{split}
\end{equation}
$A$ is an ungraded Dirac type operator acting on the positive half spinor bundle (it is the operator whose
positive spectral projection gives the {\APS} boundary condition). In the notation of Eq.~\eqref{eq:ML200911192},
we have $\mathsf{D}_\pl=\widetilde \Abdy$, $E_1=-\Gammabdy$. Thus, Proposition \plref{p:ML200909032} and Theorem \ref{P:GETZLER}
give the following result.
\begin{proposition}\label{p:CocycleTransgressEven}
Let $M$ be an even dimensional compact manifold with boundary with an exact \textup{b}-metric
and let $\dirac_t=f(t)\dirac $ be as before. Writing $\dirac$ in a collar of the boundary (in cylindrical coordinates) in the form
\begin{equation}
\dirac=:\begin{pmatrix} 0 & -\frac{d}{dx}+A\\ \frac{d}{dx} +A\end{pmatrix},
\end{equation}
$A$ is an ungraded Dirac type operator acting on the positive half spinor bundle restricted to the boundary.
Furthermore we have with $A_t=f(t)A$
\begin{equation}
\label{Eq:cocyclecondEven}
b\bCh^{k-1}(\dirac_t)+B\bCh^{k+1}(\dirac_t) =\frac{1}{\sqrt{\pi}}\Ch^k(A_t)\circ i^*,
\end{equation}
\begin{equation}\begin{split}
\label{Eq:transgressEven}
\frac{d}{dt}\bCh^k(&\dirac_t) +
b\bslch^{k-1}(\dirac_t , \dot{\dirac}_t ) +B\bslch^{k+1}(\dirac_t,\dot\dirac_t)\\
& = -\frac{1}{\sqrt{\pi}} \slch^k(A_t, \dot{A}_t)\circ i^*.
\end{split}\end{equation}
\end{proposition}
\subsection{$q=-1$} Now let $\dirac$ be ungraded and put $\widetilde D,\ga, E_1$ as
in Eqs.~\eqref{eq:ML200911192}, \eqref{eq:ML200911193}, \eqref{eq:ML200909011}.
Then by Proposition \plref{p:ML200909032} we have
\begin{align}
\bCh^k(\dirac_t)(a_0,\ldots,a_k)&= \sqrt{\pi}\; \bCh^k(\widetilde \dirac_t)(a_0,\ldots,a_k), \label{eq:ML200909034}\\
\bslch^k(\dirac_t,\dot\dirac_t)(a_0,\ldots,a_k)&= \sqrt{\pi}\; \bslch^k(\widetilde\dirac_t,
\dot{\widetilde\dirac_t})(a_0,\ldots,a_k).
\label{eq:ML200909035}
\end{align}
In the collar of the boundary, we write as usual $\dirac =\Gammabdy \frac{d}{dx}+\mathsf{D}_\pl$, and thus
\begin{equation}
\widetilde\dirac=\underbrace{\begin{pmatrix} 0 & \Gammabdy \\ \Gammabdy & 0 \end{pmatrix}}_{=:\widetilde \Gammabdy}
\frac{d}{dx}+\underbrace{\begin{pmatrix} 0 & \mathsf{D}_\pl\\ \mathsf{D}_\pl &0 \end{pmatrix}}_{=: \widetilde \mathsf{D}_\pl}.\
\end{equation}
$\widetilde\mathsf{D}_\pl$ is $2$--graded with respect to
\begin{equation}
E_1=
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},\quad
E_2 = -\widetilde \Gammabdy =\begin{pmatrix} 0 & -\Gammabdy \\ -\Gammabdy & 0\end{pmatrix}.
\end{equation}
Note that
\begin{equation}
\ga E_1 E_2 = -\Gammabdy \otimes I_2,\quad \widetilde \mathsf{D}_\pl = \ga E_1 (\mathsf{D}_\pl\otimes I_2).
\end{equation}
For even $k$ we have
\begin{equation}
\begin{split}
&\Str_2\bigl( a_0 e^{-\sigma_0 \widetilde\mathsf{D}_{\pl,t}^2} [\widetilde\mathsf{D}_{\pl,t},a_1]
\cdot\ldots\cdot [\widetilde\mathsf{D}_{\pl,t},a_k] e^{-\sigma_k \widetilde\mathsf{D}_{\pl,t}^2}\bigr)\\
=& \frac{1}{4\pi} \Tr\bigl( \ga E_1 E_2 (\ga E_1)^k \bigl( a_0 e^{-\sigma_0 \mathsf{D}_{\pl,t}^2} [\mathsf{D}_{\pl,t},a_1]
\cdot\ldots\cdot [\mathsf{D}_{\pl,t},a_k] e^{-\sigma_k \mathsf{D}_{\pl,t}^2}\bigr) \otimes I_2\bigr)\\
= &- \frac{1}{2\pi} \Tr\bigl(\Gammabdy a_0 e^{-\sigma_0 \mathsf{D}_{\pl,t}^2} [\mathsf{D}_{\pl,t},a_1]
\cdot\ldots\cdot [\mathsf{D}_{\pl,t},a_k] e^{-\sigma_k \mathsf{D}_{\pl,t}^2}\bigr).
\end{split}
\end{equation}
With respect to the grading given by $-i\Gammabdy$ we can now write
\begin{equation}
\frac{-1}{2\pi} \Tr\bigl(\Gammabdy \cdot\bigr)= \frac{1}{2\pi i }\Str_0.
\end{equation}
Together with \eqref{eq:ML200909034} and \eqref{eq:ML200909035} we have thus proved:
\begin{proposition}\label{p:CocycleTransgressOdd}
Let $M$ be an odd dimensional compact manifold with boundary with an exact \textup{b}-metric
and let $\dirac$ be an ungraded Dirac operator. Writing $\dirac$ in a collar of the boundary (in cylindrical coordinates)
in the form
\begin{equation}
\dirac=:\Gammabdy \frac{d}{dx}+\mathsf{D}_\pl,
\end{equation}
$\mathsf{D}_\pl$ is a graded Dirac type operator with respect to the grading operator $-i\Gammabdy$.
Furthermore, we have
\begin{equation}
\label{Eq:cocyclecondOdd}
b\bCh^{k-1}(\dirac_t)+B\bCh^{k+1}(\dirac_t) =\frac{1}{2\sqrt{\pi}i}\Ch^k(\mathsf{D}_\pl)\circ i^*,
\end{equation}
\begin{equation}
\begin{split}\label{Eq:transgressOdd}
\frac{d}{dt}\bCh^k(&\dirac_t) +
b\bslch^{k-1}(\dirac_t , \dot{\dirac}_t ) +B\bslch^{k+1}(\dirac_t,\dot\dirac_t)\\
& = -\frac{1}{2\sqrt{\pi}i} \slch^k(\dirac_t^\pl, \dot{\dirac}_t^\pl)\circ i^*.
\end{split}
\end{equation}
\end{proposition}
\section{Sketch of Proof of Theorem \ref{P:GETZLER}}
\label{s:CocycleFormula}
Recall that Theorem \ref{P:GETZLER} is stated for $q\ge 0$, hence in this section all
Dirac operators will be $q$--graded with $q\ge 0$.
\begin{proposition}\label{p:shuffleRelations} Let $A_0,\ldots,A_k\in \bpdosub{\Cl_q}^{\bullet}\big( M ; W \big)$. Assume that
for all but one index $j_0$ the indicial family is independent of $\gl$ and commutes with
the actions of $E_1,\ldots,E_q$ and $E_{q+1}=-\Gammabdy$ (\emph{cf.} Section
\ref{s:qDirac}). For the possible exception $j_0$ we assume that $A_{j_0}$ is proportional
to $\dot\dirac_t$. Then
\begin{equation}\label{eq:shuffleRelationA}
\blangle A_0,\ldots,A_k\rangle= (-1)^{\eps} \blangle A_k,A_0,\ldots,A_{k-1}\rangle,
\end{equation}
where $\eps=|A_k|(|A_0|+\ldots+|A_{k-1}|)$.
\begin{equation}\label{eq:shuffleRelationB}
\begin{split}
\blangle A_0,\ldots,A_k\rangle &= \sum_{j=0}^k \blangle A_0,\ldots,A_j,1,A_{j+1},\ldots,A_k\rangle\\
&= \sum_{j=0}^k (-1)^{\eps_j} \blangle 1, A_j,\ldots,A_k,A_0,\ldots,A_{j-1}\rangle,
\end{split}
\end{equation}
where $\eps_j= (|A_0|+\ldots+|A_{j-1}|)(|A_j|+\ldots+|A_k|)$.
For $j<k$
\begin{equation}\label{eq:shuffleRelationC}
\begin{split}
\blangle &A_0,\ldots,A_{j-1},[\dirac^2,A_j],A_{j+1},\ldots,A_k\rangle\\
=&\blangle A_0,\ldots,A_{j-2}, A_{j-1}A_j,A_{j+1},\ldots,A_k\rangle\\
&- \blangle A_0,\ldots,A_{j-1}, A_{j}A_{j+1},A_{j+2},\ldots,A_k\rangle.
\end{split}
\end{equation}
Similarly, for $j=k$
\begin{equation}\label{eq:shuffleRelationD}
\begin{split}
\blangle &A_0,\ldots,A_{k-1},[\dirac^2,A_k]\rangle \\
=&\blangle A_0,\ldots,A_{k-2}, A_{k-1}A_k\rangle\\
&- (-1)^ {|A_k|(|A_0|+\ldots+|A_{k-1}|)}\blangle A_k A_0,\ldots,A_{k-1}\rangle.
\end{split}
\end{equation}
\end{proposition}
Note that these formul\ae\ are the same as in Getzler-Szenes \cite[Lemma 2.2]{GetSze:CCT}. In particular there is no
boundary term.
The proof proceeds exactly as the proofs of \cite[Lemma 6.3]{Get:CHA} (1),(2), (4), and we omit the details.
We only note that one has to make heavy use of the following lemma in order to show the vanishing
of certain terms:
\begin{lemma}[Berezin Lemma]\label{l:BerezinLemma}
Let $K\in\sL_{\Cl_q}^1(\sH)$ (\emph{cf.}~Section \ref{s:qDirac}). Then for $j<q$
\[ \Tr(\ga E_1\cdot\ldots\cdot E_j K)=0.\]
\end{lemma}
\begin{proof}
If $j+q$ is odd then moving $\ga$ past $E_1\cdot\ldots\cdot E_j K$ and using the trace
property gives
\begin{equation}
\begin{split}
\Tr(\ga E_1\cdot\ldots\cdot E_j K)&= - \Tr(E_1\cdot\ldots\cdot E_j K \ga)\\
&= - \Tr(\ga E_1\cdot\ldots\cdot E_j K) =0.
\end{split}
\end{equation}
If $j+q$ is even then, since $j<q$, $E_q$ anti commutes with $\ga E_1\cdot \ldots\cdot E_j K$
and hence similarly
\begin{multline*}
\Tr(\ga E_1\cdot\ldots\cdot E_j K)= - \Tr(E_q^2 \ga E_1\cdot\ldots\cdot E_j K)
= \Tr(E_q \ga E_1\cdot\ldots\cdot E_j K E_q)\\ = \Tr(E_q^2 \ga E_1\cdot\ldots\cdot E_j K) = - \Tr(\ga E_1\cdot\ldots\cdot E_jK)=0.\qedhere
\end{multline*}
\end{proof}
We will make repeated use of the equations \eqref{eq:shuffleRelationA}--\eqref{eq:shuffleRelationD}.
Now we can proceed as for a $\theta$--summable Fredholm module. Following
\cite[p. 451]{GraVarFig:ENG} we start with the supercommutator
\begin{equation}\label{eq:SuperCommutatorA}
\int_{\Delta_k} \bStr_q\bigl(\bigl[\dirac_t,a_0 e^{-\sigma_0 \dirac_t^2}[\dirac_t,a_1]\ldots
[\dirac_t,a_k] e^{-\sigma_k \dirac_t^2}\bigr] \bigr)d\sigma,
\end{equation}
with $a_0,\ldots, a_k\in \bcC(M^\circ).$
As in \cite[bottom of p. 37]{Get:CHA} one shows, using Proposition \plref{p:b-trace-defect-graded}
and the fact that $\int_{-\infty}^\infty e^{-\gl^2}d\gl=\sqrt{\pi}$, that this supercommutator equals
\begin{equation}\label{eq:ML200908281}
\langle a_{0,\pl}, [\dirac_t^\partial,a_{1,\pl}],\ldots,[\dirac_t^\partial,a_{k,\pl}]\rangle_{\dirac_t^\partial}.
\end{equation}
It is important to note that here we are in the case $q+1$, where the grading is the induced grading on the boundary
and $E_{q+1}=-\Gammabdy$.
For convenience we will write $\dirac$ instead of $\dirac_t$.
Expanding the supercommutator \eqref{eq:SuperCommutatorA} on the other hand gives
\begin{equation}\label{eq:SuperCommutatorB}
\begin{split}
\blangle &[\dirac,a_0],\ldots,[\dirac,a_k]\rangle\\
&+\sum_{j=1}^k (-1)^{j-1} \blangle a_0,[\dirac,a_1]\ldots,[\dirac,a_{j-1}],[\dirac^2,a_j],\ldots,[\dirac,a_k]\rangle,
\end{split}
\end{equation}
where we have used $[\dirac^2,a_j]=[\dirac,[\dirac,a_j]]_{\Z^2}$.
We can now calculate the effect of $b$ and $B$ on $\bCh$.
\begin{equation}\label{eq:ML200909012}
\begin{split}
B &\bCh^{k+1}(\dirac)(a_0,\ldots,a_k)\\
&= \sum_{j=0}^k (-1)^{kj} \blangle 1,[\dirac,a_j],\ldots,[\dirac,a_k],[\dirac,a_0],\ldots,[\dirac,a_{j-1}]\rangle \\
&= \sum_{j=0}^k \blangle [\dirac,a_0],\ldots,[\dirac,a_{j-1}],1,[\dirac,a_j],\ldots,[\dirac,a_k]\rangle \\
&= \blangle [\dirac,a_0],\ldots,[\dirac,a_k]\rangle,
\end{split}
\end{equation}
where we used \eqref{eq:shuffleRelationB} twice. Thus, the first summand in \eqref{eq:SuperCommutatorB}
equals $B\bCh^{k+1}(\dirac)(a_0,\ldots,a_k).$
Furthermore,
\begin{equation}\label{eq:SuperCommutatorC}
\begin{split}
b &\bCh^{k-1}(\dirac)(a_0,\ldots,a_k)\\
&= \blangle a_0 a_1,[\dirac,a_2],\ldots,[\dirac,a_k]\rangle\\
&\qquad +\sum_{j=1}^{k-1} (-1)^j \blangle a_0,\ldots,[\dirac,a_j a_{j+1}],\ldots,[\dirac,a_k]\rangle\\
&\qquad + (-1)^k \blangle a_k a_0,[\dirac,a_1],\ldots,[\dirac,a_{k-1}]\rangle\\
&= \blangle a_0 a_1,[\dirac,a_2],\ldots,[\dirac,a_k]\rangle\\
&\qquad - \blangle a_0, a_1[\dirac,a_2],\ldots,[\dirac,a_k]\rangle\\
&\qquad + \sum_{j=1}^{k-2} (-1)^{j}\Biggl( \blangle a_0,[\dirac,a_1],\ldots,[\dirac,a_j]a_{j+1},\ldots,[\dirac,a_{k}]\rangle \\
&\qquad \qquad - \blangle a_0,\ldots,[\dirac,a_j],a_{j+1}[\dirac,a_{j+2}],\ldots,[\dirac,a_{k}]\rangle \Biggr)\\
&\qquad + (-1)^{k-1} \blangle a_0,[\dirac,a_1],\ldots,[\dirac,a_{k-1}]a_k\rangle\\
&\qquad +(-1)^{k} \blangle a_k a_0,[\dirac,a_1],\ldots,[\dirac,a_{k-1}]\rangle\\
&=\sum_{j=1}^k (-1)^{j-1} \blangle a_0,[\dirac,a_1],\ldots,[\dirac^2,a_j],\ldots,[\dirac,a_k]\rangle,
\end{split}
\end{equation}
where we have used \eqref{eq:shuffleRelationC} and \eqref{eq:shuffleRelationD}. The right hand side
of \eqref{eq:SuperCommutatorC} equals the sum in the second line of \eqref{eq:SuperCommutatorB}.
Summing up \eqref{eq:SuperCommutatorA}, \eqref{eq:ML200908281}, \eqref{eq:SuperCommutatorB}, \eqref{eq:ML200909012},
and \eqref{eq:SuperCommutatorC} we arrive at Eq.~\eqref{Eq:cocyclecond}.
\medskip
For additional clarity, let us perform two direct checks, for small values of $k$.
\subsection*{Case 1: $k=0$} In this case \eqref{eq:SuperCommutatorB} equals
\begin{equation}
\blangle [\dirac,a_0]\rangle= \blangle 1,[\dirac,a_0]\rangle= B\bCh^1(\dirac)(a_0),
\end{equation}
by \eqref{eq:shuffleRelationB} and we are done in this case.
\subsection*{Case 2: $k=1$} Then \eqref{eq:SuperCommutatorB} equals
\begin{equation}
\blangle [\dirac,a_0],[\dirac,a_1]\rangle + \blangle a_0,[\dirac^2,a_1]\rangle.
\end{equation}
The first summand is $B\bCh^2(a_0,a_1)$ and the second summand equals in view of \eqref{eq:shuffleRelationD}
\begin{equation}
\begin{split}
\blangle a_0 a_1\rangle - \blangle a_1 a_0\rangle = b\bCh^0(\dirac)(a_0,a_1).
\end{split}
\end{equation}
\subsection{The transgression formula}\sind{transgression formula|(}
To prove the transgression formula we proceed analogously
and start with the supercommutator
\begin{multline}\label{eq:TransgressSuperCommutatorA}
\sum_{j=0}^k (-1)^j
\int_{\Delta_{k+1}} \bStr_q\bigl(\bigl[\dirac_t,a_0 e^{-\sigma_0 \dirac_t^2}[\dirac_t,a_1]\ldots\\
\ldots [\dirac_t,a_j] e^{-\sigma_j \dirac_t^2}
\dot D e^{-\sigma_{j+1}\dirac_t^2}\ldots [\dirac_t,a_k] e^{-\sigma_{k+1}\dirac_t^2}
\bigr] \bigr)d\sigma.
\end{multline}
We compute this supercommutator using Proposition \plref{p:b-trace-defect-graded}.
Note that by Proposition \ref{p:Dirac-indicial-family} $\cI(\dot\dirac_t,\gl)$ is proportional to $i\Gammabdy\gl+\dirac^\partial$.
The summand $i\Gammabdy\gl$ contributes a term proportional to $\int_{-\infty}^\infty \gl e^{-\gl^2}d\gl=0$.
The remaining summand gives, since $\int_{-\infty}^\infty e^{-\gl^2}d\gl=\sqrt{\pi}$,
\begin{equation}\label{eq:ML200909021}
\begin{split}
\sum_{j=0}^k (-1)^j \langle &a_{0,\pl},[\dirac_t^\pl,a_{1,\pl}],\ldots,[\dirac_t^\pl,a_{j,\pl}],\dot\dirac_t^\pl,\ldots,[\dirac_t^\pl,a_{k,\pl}]\rangle\\
&= \slch^k(\mathsf{D}_\pl,\dot\mathsf{D}_\pl)(a_{0,\pl},\ldots,a_{k,\pl}).
\end{split}
\end{equation}
Let us again emphasize that here we are in the case $q+1$, where the grading is the induced grading on the boundary
and $E_{q+1}=-\Gammabdy$.
Next we expand the commutator \eqref{eq:TransgressSuperCommutatorA}. However, we will confine ourselves
to small $k$. The calculation is basically the same as in \cite[p. 451]{GraVarFig:ENG}. The only difference is that
on a closed manifold \eqref{eq:TransgressSuperCommutatorA} is a priori $0$ while here it coincides with
the transgressed Chern character on the boundary.
\subsection*{Case 1: $k=0$}
\eqref{eq:TransgressSuperCommutatorA} expands to
\begin{equation}\label{eq:TransgressSuperCommutatorEven}
\blangle [\dirac_t,a_0],\dot\dirac_t]\rangle +\blangle a_0,[\dirac_t,\dot\dirac_t]\rangle.
\end{equation}
On the other hand
\begin{equation}\label{eq:ML200909022}
\begin{split}
B\bslch^1(\dirac_t,\dot\dirac_t)(a_0)&=\bslch^1(\dirac_t,\dot\dirac_t)(1,a_0)\\
&= \blangle 1,\dot\dirac_t,[\dirac_t,a_0]\rangle
- \blangle 1,[\dirac_t,a_0],\dot\dirac_t \rangle\\
&= - \blangle [\dirac_t,a_0],1,\dot\dirac_t\rangle
- \blangle [\dirac_t,a_0],\dot\dirac_t,1 \rangle\\
&= - \blangle [\dirac_t,a_0],\dot\dirac_t]\rangle,
\end{split}
\end{equation}
by \eqref{eq:shuffleRelationB}. Moreover using the well--known formula
\begin{equation}\label{eq:ML200909023}
\frac{d}{dt} e^{-\sigma \dirac_t^2} = - \int_0^\sigma e^{(\sigma - s)\dirac_t^2} [\dirac_t,\dot\dirac_t] e^{-s\dirac_t^2} ds,
\end{equation}
we have
\begin{equation}
\frac{d}{dt} \bCh^0(\dirac_t)(a_0)=-\blangle a_0,[\dirac_t,\dot\dirac_t]\rangle,
\end{equation}
hence altogether
\begin{equation}
\frac{d}{dt} \bCh^0(\dirac_t)+B\bslch^1(\dirac_t,\dot\dirac_t)= -\slch^0(\dirac_t^\pl,\dot\dirac_t^\pl).
\end{equation}
\subsection*{Case 2: $k=1$} To be on the safe side, we also look at an example in the odd case.
We will again make repeated use of the formul\ae\ in Proposition \plref{p:shuffleRelations} without
further mentioning.
Eq. \eqref{eq:TransgressSuperCommutatorA} now expands to
\begin{equation}\label{eq:TransgressSuperCommutatorOdd}
\begin{split}
& \blangle [\dirac_t,a_0],\dot\dirac_t,[\dirac_t,a_1]\rangle -
\blangle [\dirac_t,a_0],[\dirac_t,a_1],\dot\dirac_t\rangle\\
& + \blangle a_0,[\dirac_t,\dot\dirac_t],[\dirac_t,a_1]\rangle
+ \blangle a_0,[\dirac_t,a_1],[\dirac_t,\dot\dirac_t]\rangle\\
& -\blangle a_0,\dot\dirac_t,[\dirac_t^2,a_1]\rangle
- \blangle a_0,[\dirac_t^2,a_1],\dot\dirac_t\rangle.
\end{split}
\end{equation}
On the other hand
\begin{equation}\label{eq:ML200909024}
\begin{split}
B\bslch^2&(\dirac_t,\dot\dirac_t)(a_0)= \bslch^2(\dirac_t,\dot\dirac_t)(1,a_0,a_1)-
\bslch^2(\dirac_t,\dot\dirac_t)(1,a_1,a_0)\\
= &\blangle 1,\dot\dirac_t,[\dirac_t,a_0], [\dirac_t,a_1]\rangle
-\blangle 1,[\dirac_t,a_0],\dot\dirac_t, [\dirac_t,a_1]\rangle\\
& +\blangle 1,[\dirac_t,a_0], [\dirac_t,a_1],\dot\dirac_t\rangle
-\blangle 1,\dot\dirac_t,[\dirac_t,a_1], [\dirac_t,a_0]\rangle\\
& + \blangle 1,[\dirac_t,a_1],\dot\dirac_t, [\dirac_t,a_0]\rangle
-\blangle 1,[\dirac_t,a_1], [\dirac_t,a_0],\dot\dirac_t\rangle\\
=& \blangle [\dirac_t,a_0], [\dirac_t,a_1],1,\dot\dirac_t\rangle
+\blangle [\dirac_t,a_0],1,[\dirac_t,a_1],\dot\dirac_t\rangle\\
&+\blangle [\dirac_t,a_0], [\dirac_t,a_1],\dot\dirac_t,1\rangle
-\blangle [\dirac_t,a_0],\dot\dirac_t, [\dirac_t,a_1],1\rangle\\
& -\blangle [\dirac_t,a_0],\dot\dirac_t,1, [\dirac_t,a_1]\rangle
-\blangle [\dirac_t,a_0],1,\dot\dirac_t, [\dirac_t,a_1]\rangle\\
=& \blangle [\dirac_t,a_0],[\dirac_t,a_1],\dot\dirac_t\rangle
-\blangle [\dirac_t,a_0],\dot\dirac_t,[\dirac_t,a_1]\rangle,
\end{split}
\end{equation}
which equals the negative of the first two summands of \eqref{eq:TransgressSuperCommutatorOdd}.
Furthermore,
\begin{equation}
b \bslch^0(\dirac_t,\dot\dirac_t)(a_0,a_1)=\bslch^0(\dirac_t,\dot\dirac_t)([a_0,a_1])
= \blangle [a_0,a_1],\dot\dirac_t\rangle.
\end{equation}
Applying \eqref{eq:shuffleRelationC} and \eqref{eq:shuffleRelationD} to the last
two summands of \eqref{eq:TransgressSuperCommutatorOdd} we find
\begin{equation}
\begin{split}
\blangle& a_0,\dot\dirac_t,[\dirac_t^2,a_1]\rangle+ \blangle a_0,[\dirac_t^2,a_1],\dot\dirac_t\rangle\\
=& \blangle a_0,\dot\dirac_t a_1\rangle - \blangle a_1a_0,\dot\dirac_t\rangle
+\blangle a_0 a_1,\dot\dirac_t\rangle - \blangle a_0, a_1\dot\dirac_t\rangle \\
=& \blangle a_0,[\dot\dirac_t,a_1]\rangle +b\bslch^0(\dirac_t,\dot\dirac_t)(a_0,a_1),
\end{split}
\end{equation}
hence adding $B\bslch^2(\dirac_t,\dot\dirac_t)(a_0,a_1)$ and $b\bslch^0(\dirac_t,\dot\dirac_t)(a_0,a_1)$
to the right hand side of \eqref{eq:TransgressSuperCommutatorOdd} we obtain
\begin{equation}
\begin{split}
\slch^1(&\dirac_t^\pl,\dot\dirac_t^\pl)(a_{0,\pl},a_{1,\pl})+B\bslch^2(\dirac_t,\dot\dirac_t)(a_0,a_1)
+b\bslch^0(\dirac_t,\dot\dirac_t)(a_0,a_1)\\
=&- \blangle a_0,[\dot\dirac_t,a_1]\rangle + \blangle a_0,[\dirac_t,\dot\dirac_t],[\dirac_t,a_1]\rangle
+ \blangle a_0,[\dirac_t,a_1],[\dirac_t,\dot\dirac_t]\rangle\\
=& -\frac{d}{dt} \bCh^1(\dirac_t)(a_0,a_1)
\end{split}
\end{equation}
in view of \eqref{eq:ML200909023}.
With more effort but in a similar manner, the previous considerations can be extended to arbitrary $k$,
thus proving Eq.~\eqref{Eq:transgress}.
\sind{transgression formula|)}
\chapter{Heat Kernel and Resolvent Estimates}
\label{s:cylinder-estimates}
This is the most technical chapter of the paper. It is devoted to prove
some crucial estimates for the heat kernel\sind{heat kernel} of a \textup{b}-Dirac operator.
These estimates will be used to analyze the short and long time
behavior of the Chern character. Throughout this chapter we will mostly
work in the cylindrical context.
For the convenience of the reader we start by summarizing some
basic estimates for the resolvent and the heat operator associated
to an elliptic operator. These estimates will then be applied
in Section \ref{s:comparison-results} to prove comparison results for
the heat kernel\sind{heat kernel} and {\JLO} integrand of a Dirac operator on a general manifold with
cylindrical ends to those of a corresponding Dirac operator on the model
cylinder. In the remainder of the Chapter we will then prove short and
large time estimates for the \textup{b}-Chern character. This is in
preparation for proving heat kernel asymptotics
in the \textup{b}-setting in Section \ref{s: heat expansion}.
\section{Basic resolvent and heat kernel estimates on general manifolds}
\label{sec:basic-estimates}
During the whole section $M$ will be a riemannian manifold without boundary and
$D_0:\Gamma^\infty(M;W)\longrightarrow \Gamma^\infty(M;W)$
will denote a first order formally self-adjoint elliptic differential operator acting between sections
of the hermitian vector bundle $W$.
We assume that there exists a self-adjoint extension, $D$, of $D_0$.
E.g. if $M$ is complete and $D_0$ is of Dirac type
then $D_0$ is essentially self-adjoint; if $M$ is the interior of a compact manifold
with boundary then $D$ can be obtained by imposing an appropriate boundary condition.
For the following
considerations it is irrelevant which self-adjoint extension is chosen. We just fix one.
\subsection{Resolvent estimates}
\label{ss:ResolventEstimates}
We fix an open sector
$\Lambda:=\bigsetdef{z\in\C\setminus\{0\}}{0< \eps < \arg z < 2\pi-\eps}\subset \C\setminus \R_+$ in the complex plane.
We introduce the following notation: for a function $f:\gL\to \C$ we write
$f(\gl)= O(|\gl|^{\ga+0}), \gl\to\infty, \gl\in\gL$
if for every $\delta>0, \gl_0\in\gL,$ there is a constant $C_{\delta,\gl_0}$ such that
$|f(\gl)|\le C_\delta |\gl|^{\ga+\delta}$ for $\gl\in\gL, |\gl|\ge |\gl_0|$.
We write $f(\gl)=O(|\gl|^{-\infty})$, $\gl\to\infty,\gl\in\gL$ if
$f(\gl)=O(|\gl|^{-N})$ for every $N$; the $O$--constant may depend on $N$.
\sind{Schatten class}
$L^2_s(M;W)$ denotes the Hilbert space of sections of $W$ which are of Sobolev
class $s$. The Sobolev norm of an element $f\in L^2_s(M;W)$ is denoted by
$\|f\|_s$. For a linear operator $T:L^2_s(M;W)\to L^2_t(M;W)$ its operator
norm is denoted by $\|T\|_{s,t}$.
For an operator $T$ in a Hilbert space $\sH$ we denote by $\|T\|_p$
the $p$--th Schatten norm. To avoid confusions the letter $p$ will not
be used for Sobolev orders.
Note that the operator norm of $T$ in $\sH$ coincides with $\|T\|_\infty$.
\begin{proposition}\label{p:ML20080403-A1}
Let $A,B\in\pdo^\bullet(M,W)$ be pseudodifferential operators of order $a,b$
with compact support.\footnote{This means that their
Schwartz kernels are compactly supported in $M\times M$.}
\textup{1.} If $k>(\dim M)/4+a/2$ then $A (D^2-\gl)^{-k}, (D^2-\gl)^{-k}A$ are Hilbert--Schmidt operators for $\gl\not\in\spec D^2$
and we have
\begin{equation}\label{eq:Hilbert-Schmidt-resolvent-estimate}
\| A(D^2-\gl)^{-k}\|_2=O(|\gl|^{a/2+(\dim M)/4-k+0}),\quad \text{as } \gl\to\infty\text{ in }\gL.
\end{equation}
The same estimate holds for $\| (D^2-\gl)^{-k}A\|_2$.
\textup{2.} If $k>(\dim M+a+b)/2$ then $A (D^2-\gl)^{-k}B$ is of trace class for $\gl\not\in\spec D^2$
and
\begin{equation}\label{eq:trace-class-resolvent-estimate}
\| A(D^2-\gl)^{-k}B\|_1= O(|\gl|^{(\dim M+a+b)/2-k+0}),\quad \text{as } \gl\to\infty\text{ in }\gL.
\end{equation}
\textup{3.} Denote by $\pi_1, \pi_2:M\times M\to M$ the projection onto the first resp. second factor
and assume that $\pi_2(\supp A)\cap \pi_1(\supp B)=\emptyset$. Then $A(D^2-\gl)^{-k}B$ is a trace class
operator for any $k\ge 1$ and
\begin{equation}\label{eq:trace-class-smoothing-estimate}
\| A(D^2-\gl)^{-k}B\|_1=O(|\gl|^{-\infty}), \quad \text{as } \gl\to\infty\text{ in }\gL.
\end{equation}
\end{proposition}
\begin{proof}
1. Sobolev embedding and elliptic regularity implies that for $f\in L^2(M;W)$ the section $A(D^2-\gl)^{-k}f$ is
continuous. Moreover, for $r>\dim M/2, |\gl|\ge |\gl_0|,$ and $x$ in the compact set $\supp A=:K$
\begin{equation}
\begin{split}
\|\bigl(A (D^2-\gl)^{-k}f\bigr)(x)\|&\le C \| (D^2-\gl)^{-k}f\|_{a+r,K}\\
&\le C \|(D^2+I)^{(a+r)/2}(D^2-\gl)^{-k} f\|_0\\
&\le C |\gl|^{-k+(a+r)/2} \|f\|.
\end{split}
\end{equation}
For the Schwartz--kernel this implies the estimate
\begin{equation}
\sup_{x\in\supp A} \int_M \|A(D^2-\gl)^{-k}(x,y)\|^2 d\vol(y)\le C |\gl|^{-2k+a+r},
\end{equation}
and since $A$ has compact support, integration over $x$ yields
\begin{equation}
\begin{split}
\| A&(D^2-\gl)^{-k}\|_2^2\\
&\le \int_{\supp A} \int_M \|A(D^2-\gl)^{-k}(x,y)\|^2d\vol(x)d\vol(y)\\
&\le C |\gl|^{-2k+a+r},
\end{split}
\end{equation}
proving the estimate \eqref{eq:Hilbert-Schmidt-resolvent-estimate}.
The estimate for $(D^2-\gl)^{-k}A$ follows by taking the adjoint.
2. The second claim follows from the first one using the H\"older inequality.
3. To prove the third claim we choose cut--off functions $\varphi,\psi\in\cinfz{M}$
with $\varphi=1$ on $\pi_2(\supp A)$, $\psi=1$ on $\pi_1(\supp B)$ and $\supp\varphi\cap\supp \psi=\emptyset$.
Then $A(D^2-\gl)^{-k}B=A\varphi (D^2-\gl)^{-k}\psi B$ and
$\varphi (D^2-\gl)^{-k}\psi$ is a smoothing operator in the
\semph{parameter dependent calculus} (\emph{cf.}~Shubin \cite[Chap. II]{Shu:POS}).
Hence for any real numbers $s,t,N$ we have
\begin{equation}
\|\varphi (D^2-\gl)^{-k}\psi\|_{s,t}\le C(s,t,N)\; |\gl|^{-N}, \quad \text{as } \gl\to\infty\text{ in }\gL.
\end{equation}
Since the Sobolev orders $s,t$ are arbitrary this implies the claim.
\end{proof}
\begin{proposition}\label{p:ML20080403-A2}
Let $A\in\pdo^a(M,W)$ be a pseudodifferential operator with compact support.
\textup{1.} Let $\varphi\in\cinf{M}$ be a smooth function such that $\supp d\varphi$ is compact,
i.e. outside a compact set $\varphi$ is locally constant. Moreover suppose that
$\supp \varphi\cap \pi_1(\supp A)=\emptyset$. Then $\varphi(D^2-\gl)^{-k}A$ is a trace class
operator for any $k\ge 1$ and the estimate \eqref{eq:trace-class-smoothing-estimate}
holds for $\varphi(D^2-\gl)^{-k}A$.
\textup{2.} If $k>(\dim M+a)/2$ then $A (D^2-\gl)^{-k}, (D^2-\gl)^{-k}A$ are trace class operators
and the estimate \eqref{eq:trace-class-resolvent-estimate} holds with $B=I$.
\end{proposition}
\begin{proof}
From
\begin{equation}\label{eq:ML20080403-A7}
(D^2-\gl) \varphi (D^2-\gl)^{-k} A= [D^2,\varphi] (D^2-\gl)^{-k}A+\varphi (D^2-\gl)^{-k+1}A
\end{equation}
we infer since $\varphi A=0$
\begin{equation}\begin{split}
\varphi (&D^2-\gl)^{-k} A\\
&= (D^2-\gl)^{-1}
\begin{cases} [D^2,\varphi] (D^2-\gl)^{-k}A,& k=1,\\
[D^2,\varphi] (D^2-\gl)^{-k}A+\varphi (D^2-\gl)^{-k+1}A,& k>1.
\end{cases}
\end{split}
\end{equation}
Applying Proposition \plref{p:ML20080403-A1}.3 to the right hand side we
inductively obtain the first assertion.
To prove the second assertion we choose a cut--off function $\varphi\in\cinfz{M}$ with $\varphi=1$ on
$\pi_1(\supp A)$. Then we apply Proposition \plref{p:ML20080403-A1}.2 to $\varphi(D^2-\gl)^{-k}A$ and
the proved first assertion to $(1-\varphi)(D^2-\gl)^{-k}A$ to reach the conclusion.
\end{proof}
For the following Proposition it is crucial that we are precise about
{\domain}s of operators:
\begin{definition}\label{d:commutator-compact} By $\Diff^d(M,W)$
we denote the space of differential operators of order $d$
acting on the sections of $W$.
Given a differential operator $A\in\Diff^a(M,W)$ we say
that the commutator $[D^2,A]$ has compact support if
\begin{enumerate}
\item $A$ and $A^t$ map the domain $\dom(D^k)$ into the domain $\dom(D^{k-a})$ for $k\ge a$ and
\item the differential expression $[D^2,A]$ has compact support.
\end{enumerate}
\end{definition}
The main example we have in mind is where $D$ is a Dirac type operator
on a complete manifold and $A$ is multiplication by a smooth function
$\varphi$ such that $d\varphi$ has compact support. Then $[D^2,\varphi]$
has compact support in the above sense.
\begin{proposition}\label{p:ML20081104-A19}
Let $A\in\Diff^a(M,W)$ be a differential operator
such that $[D^2,A]$ has compact support and is of order $\le a+1$.
Then for $k>\dim M+a$ the
commutator $[A,(D^2-\gl)^{-k}]$ is trace class and
\begin{equation}\label{eq:commutator-resolvent-estimate}
\| [A,(D^2-\gl)^{-k}]\|_1= O(|\gl|^{(\dim M+a-1)/2-k+0}),\quad \text{as } \gl\to\infty\text{ in }\gL.
\end{equation}
\end{proposition}
\begin{proof} Note first that since $A$ and $A^t$ map $\dom(D^k)$ into $\dom(D^{k-a})$
the commutator $[A,(D^2-\gl)^{-k}]$ is defined as a linear operator on $L^2(M;W)$
and we have the identity
\begin{equation}\label{eq:ML20081104-A20}
[A,(D^2-\gl)^{-k}] = \sum_{j=1}^k (D^2-\gl)^{-j} [D^2,A](D^2-\gl)^{-k+j-1}.
\end{equation}
Since $k>\dim M+a$ we have in each summand $j>(\dim M+a+1)/2$ or
$k-j+1>(\dim M+1+1)/2$. Say in the first case we apply
Proposition \plref{p:ML20080403-A2}
to $\|(D^2-\gl)^{-j}[D^2,A]\|_1$ and the Spectral Theorem
to estimate $\|(D^2-\gl)^{-k+j-1}\|$ and find
\begin{equation*}
\begin{split}
\|(D^2-\gl)^{-j} &[D^2,A](D^2-\gl)^{-k+j-1}\|_1\\
&\le \|(D^2-\gl)^{-j} [D^2,A]\|_1 \|(D^2-\gl)^{-k+j-1}\|_{\infty}\\
&\le O(|\gl|^{(\dim M+a+1)/2-j+0})\;\cdot\; O(|\gl|^{-k+j-1})\\
&=O(|\gl|^{(\dim M+a-1)/2-k+0}).\qedhere
\end{split}
\end{equation*}
\end{proof}
\subsection{Heat kernel estimates}
\label{ss:HeatKernelEstimates}
From Propositions \plref{p:ML20080403-A1}, \plref{p:ML20080403-A2} we can
derive short and large times estimates for the heat operator $e^{-tD^2}$.
We write
\begin{equation}\label{eq:contour-integral}
\begin{split}
e^{-tD^2}&=\frac{1}{2\pi i}\int_\gamma e^{-t\gl} (D^2-\gl)^{-1} d\gl\\
&=\frac{t^{-k} k!}{2\pi i}\int_{\gamma} e^{-t \gl} (D^2-\gl)^{-k-1} d\gl,
\end{split}
\end{equation}
where integration is over the contour sketched in Figure \ref{fig:ContourNonFredholm}.
The notation $O(t^{\ga-0})$, $O(t^\infty)$ as $t\to 0+$ resp.
$O(t^{\ga-0}), O(t^{-\infty})$ as $t\to\infty$
is defined analogously to the corresponding notation for $\gl\in\gL$ in
the previous Section.
\FigContourNonFredholm
We infer from Propositions
\plref{p:ML20080403-A1}, \plref{p:ML20080403-A2}
\begin{proposition}\label{p:ML20080403-A3}
Let $A,B\in\pdo^\bullet(M,W)$ be pseudodifferential operators of order $a,b$
with compact support.
\textup{1.} For $t>0$ the operators $A e^{-tD^2}, e^{-tD^2} A$
are trace class operators. For $t_0,\eps>0$ there is a constant
$C(t_0,\eps)>0$ such that for all $1\le p\le \infty$
we have the following estimate in the Schatten $p$--norm
\begin{equation}\label{eq:trace-heat-estimate}
\| Ae^{-tD^2}\|_{p}\le C(t_0,\eps)\; t^{-a/2-\frac{\dim M+\eps}{2p}},\quad 0<t\le t_0.
\end{equation}
Note that $C(t_0,\eps)$ is independent of $p$.
The same estimate holds for $\|e^{-tD^2}A\|_p$.
\textup{2.} Denote by $\pi_1, \pi_2:M\times M\to M$ the projection onto the first resp. second factor
and assume that $\pi_2(\supp A)\cap \pi_1(\supp B)=\emptyset$. Then
\begin{equation}\label{eq:heat-smoothing-estimate}
\| Ae^{-tD^2}B\|_1=O(t^\infty), \quad 0<t<t_0,
\end{equation}
with $N$ arbitrarily large.
\textup{3.} Let $\varphi\in\cinf{M}$ be a smooth function such that $\supp d\varphi$ is
compact. Moreover suppose that
$\supp \varphi\cap \pi_1(\supp A)=\emptyset$. Then
the estimate \eqref{eq:heat-smoothing-estimate} also holds
for $\varphi e^{-tD^2}A$.
\end{proposition}
\begin{proof} 1. From Proposition \ref{p:ML20080403-A2} and the contour integral
\eqref{eq:contour-integral} we infer the inequality \eqref{eq:trace-heat-estimate}
for $p=1$. For $p=\infty$ it follows from the Spectral Theorem.
The H\"older inequality implies the following interpolation inequality\sind{interpolation inequality}
for Schatten norms
\begin{equation}\label{eq:Hoelder-interpolation}
\|T\|_p=\Tr(|T|^p)^{1/p}\le \|T\|_\infty^{1-1/p}\|T\|_1^{1/p}, \quad 1\le p\le \infty.
\end{equation}
From this we infer \eqref{eq:trace-heat-estimate}.
The remaining claims follow immediately from the contour integral
\eqref{eq:contour-integral} and the corresponding resolvent estimates.
\end{proof}
For the next result we
assume additionally that $D$ is a Fredholm operator and we denote by $H$ the orthogonal
projection onto $ \Ker D$. $H$ is a finite rank smoothing operator. Let $c:=\min \spec_{\ess} D^2$
be the bottom of the essential spectrum of $D^2$.
Then $e^{-tD^2}(I-H)=e^{-tD^2}-H$ can again be expressed in terms of a contour integral as in
\eqref{eq:contour-integral} where the contour is now depicted in Figure \plref{fig:ContourFredholm}.
\FigContourFredholm
This allows to make large time estimates. The result is as follows:
\begin{proposition}\label{p:ML20080403-A4}
Assume that $D$ is Fredholm and let $A\in \pdo^a(M,W)$ be a pseudodifferential
operator with compact support. Then for any $0<\delta<\inf\spec_{\ess} D^2$
and any $\eps>0$ there is a constant $C(\delta,\eps)$ such that
for $1\le p\le \infty$
\begin{equation}
\| A e^{-tD^2}(I-H)\|_{p} \le C(\delta,\eps)\; t^{-a/2-\frac{\dim M+\eps}{2p}} e^{-t\delta},\quad 0<t<\infty.
\end{equation}
\end{proposition}
\begin{proof} For $t\to 0+$ the estimate follows from Proposition \plref{p:ML20080403-A3}.1.
For $t\to\infty$ and $p=1$ the estimate follows from Proposition \plref{p:ML20080403-A1}
and \eqref{eq:contour-integral} by taking the contour as in Figure \ref{fig:ContourFredholm}.
For $p=\infty$ the estimate is a simple consequence of the Spectral Theorem.
The general case then follows again from
the interpolation inequality \eqref{eq:Hoelder-interpolation}.
\end{proof}
Finally we state the analogue of Proposition \ref{p:ML20081104-A19}
for the heat kernel.\sind{heat kernel}
\begin{proposition}\label{p:ML20081104-A20}
Let $A\in\Diff^a(M,W)$ be a differential operator
such that $[D^2,A]$ has compact support
(in the sense of Definition \textup{\ref{d:commutator-compact}})
and is of order $\le a+1$.
Then for $t>0$ the operator $[A,e^{-tD^2}]$ is of trace class. For $t_0,\eps>0$
there is a constant $C(t_0,\eps)$ such that for all $1\le p\le \infty$
we have the following estimate in the Schatten $p$--norm
\begin{equation}\label{eq:commutator-heat-estimate}
\| [A,e^{-tD^2}]\|_{p}\le C(t_0,\eps)\; t^{-a/2-\frac{\dim M-1+\eps}{2p}},\quad 0<t\le t_0;
\end{equation}
$C(t_0,\eps)$ is independent of $p$.
If $D$ is a Fredholm operator then for any $0<\delta<\inf\spec_{\ess} D^2$
and any $\eps>0$ there is a constant $C(\delta,\eps)$ such that
for $1\le p\le \infty$
\begin{equation}
\| [A, e^{-tD^2}(I-H)]\|_{p} \le C(\delta,\eps)\;
t^{-a/2-\frac{\dim M-1+\eps}{2p}} e^{-t\delta},\quad 0<t<\infty.
\end{equation}
\end{proposition}
\begin{proof} For $p=1$ this follows from Proposition
\ref{p:ML20081104-A19} and the contour integral representation
\eqref{eq:contour-integral} by taking the contours as in Figure \ref{fig:ContourNonFredholm}
for $t\to 0+$ and as in Figure \ref{fig:ContourFredholm} in the Fredholm case
as $t\to\infty$.
For $p=\infty$ the estimates are a simple consequence of the Spectral Theorem.
The general case then follows from
the interpolation inequality \eqref{eq:Hoelder-interpolation}.
\end{proof}
\subsection{Estimates for the \textup{JLO} integrand}
\label{ss:EstimatesJLOIntegrand}
Recall that we denote the standard $k$--simplex by
$\Delta_k:=\bigsetdef{(\sigma_0,...,\sigma_k)\in\R^{k+1}}{\sigma_j\ge 0, \sigma_0+...+\sigma_k=1}$.
Furthermore, recall the notation \eqref{eq:ML20090128-3}.
\begin{proposition}\label{p:multiple-heat-estimate}
Let $A_j\in\Diff^{d_j}(M;W)$, $j=0,...,k$, be $D^{d_j}$--bounded differential operators on
of order $d_j$ on $M$; let $d:=\sum_{j=0}^k d_j$ be the sum of their
orders.
Furthermore, assume that $\supp A_{j_0}$ is compact for at least one index $j_0$.
\textup{1.} For $t_0, \eps>0$ there is a constant $C(t_0,\eps)$ such that
for all $\sigma=(\sigma_0,...,\sigma_k)\in\Delta_k, \sigma_j>0$,
\begin{equation}\label{eq:multiple-heat-estimate-short}
\begin{split}
\| A_0 &e^{-\sigma_0 t D^2}A_1\cdot\ldots\cdot A_k e^{-\sigma_k t D^2}\|_1\\
&\le C(t_0,\eps)\; \Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2} \Biggr)
t^{-d/2-(\dim M)/2-\eps},\quad 0<t\le t_0.
\end{split}
\end{equation}
In particular, if $d_j\le 1, j=0,...,k,$ then
\begin{equation}\label{eq:integrated-multiple-heat-estimate-short}
\|(A_0,...,A_k)_{\sqrt{t}D}\|=O( t^{-d/2-(\dim M)/2-0}),\quad t \to 0+.
\end{equation}
\textup{2.} Assume additionally that $D$ is Fredholm and denote by $H$ the orthogonal
projection onto $ \Ker D$. Then for $\eps>0$ and any $0<\delta<\inf\spec_{\ess} D^2$
there is a constant $C(\delta,\eps)$ such that for all $\sigma\in\Delta_k, \sigma_j>0$
\begin{equation}\label{eq:multiple-heat-estimate-large}
\begin{split}
\| A_0 &e^{-\sigma_0 t D^2}(I-H)A_1\cdot\ldots\cdot A_k e^{-\sigma_k t D^2}(I-H)\|_1\\
&\le C(\delta,\eps)\; \Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2} \Biggr)
t^{-d/2-(\dim M)/2-\eps} e^{-t\delta},\quad \text{for \emph{all} } 0<t<\infty.
\end{split}
\end{equation}
In particular, $d_j\le 1, j=0,...,k,$ then
\begin{equation}\label{eq:integrated-multiple-heat-estimate-large}
\begin{split}
\|(A_0&(I-H),...,A_k(I-H))_{\sqrt{t}D}\|\\
&=O(t^{-d/2-(\dim M)/2-0} e^{-t\delta}),\quad \text{for \emph{all }} 0<t<\infty.
\end{split}
\end{equation}
\end{proposition}
\begin{proof} We first reduce the problem to the case that all $A_j$ are compactly
supported. To this end choose $\varphi_{j_0-1},\varphi_{j_0}\in\cinfz{M}$ such that
$\supp \varphi_{j_0}\cap \supp(1-\varphi_{j_0-1})=\emptyset$ and such that
$\varphi_{j_0}A_{j_0}=A_{j_0}\varphi_{j_0}=A_{j_0}$. Decompose
$A_{j_0-1}=A_{j_0-1}\varphi_{j_0-1}+A_{j_0-1}(1-\varphi_{j_0-1})$.
First we show that the estimates \eqref{eq:multiple-heat-estimate-short}, \eqref{eq:multiple-heat-estimate-large}
hold if we replace $A_{j_0-1}$ by $A_{j_0-1}(1-\varphi_{j_0-1})$:
Case 1. Proposition \ref{p:ML20080403-A3}.3 gives
\begin{equation}\label{eq:ML20080428-A18}
\|A_{j_0-1}(1-\varphi_{j_0-1})e^{-\sigma_{j_0-1} t D^2} \varphi_{j_0}\|_1\le C_{t_0} \sigma_{j_0-1}^Nt^{N},\quad \text{ for } \sigma_{j_0-1}t\le t_0.
\end{equation}
The operator norm of the other factors can be estimated using
the Spectral Theorem, taking into account the $D^{d_j}$-boundedness of $A_j$:
\begin{equation}
\| A_j e^{-\sigma_j tD^2}\|\le C_{t_0} (\sigma_j t)^{-d_j/2},
\quad \text{ for } \sigma_{j}t\le t_0.
\end{equation}
Hence by the H\"older inequality
\begin{equation}
\begin{split}
\| A_0 &e^{-\sigma_0 t D^2}A_1\cdot\ldots\cdot A_{j_0-1}(1-\varphi_{j_0-1})e^{-\sigma_{j_0-1}tD^2}
\cdot A_k e^{-\sigma_k t D^2}\|_1\\
&\le C_{t_0} \Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2} \Biggr)
t^{-\sum\limits_{j=0}^k d_j+N},\quad 0<t\le t_0,
\end{split}
\end{equation}
which is even better than \eqref{eq:multiple-heat-estimate-short}.
Case 2 ($D$ Fredholm).
From \eqref{eq:ML20080428-A18}, Proposition \ref{p:ML20080403-A4}
and the fact that $H$ is a finite rank operator with $e^{-\xi D^2}H=H$ we infer
\begin{equation}\label{eq:ML20080425-1}
\begin{split}
\|A_{j_0-1}(1-\varphi_{j_0-1})&e^{-\sigma_{j_0-1} t D^2} (I-H)\varphi_{j_0}\|_1\\
& \le C_\delta e^{-\sigma_{j_0-1}t\delta},\quad \text{ for all } 0<t<\infty.
\end{split}
\end{equation}
To the other factors we apply
Proposition \ref{p:ML20080403-A4} with $p=\infty$:
\begin{equation}\label{eq:ML20080425-2}
\| A_j e^{-\sigma_j tD^2}(I-H)\|\le C_{\delta}
(\sigma_jt)^{-d_j/2} e^{-\sigma_jt\delta},\quad 0<t<\infty.
\end{equation}
The H\"older inequality combined with
\eqref{eq:ML20080425-1},\eqref{eq:ML20080425-2} gives
\eqref{eq:multiple-heat-estimate-large}.
Altogether we are left to consider $A_0,...,A_{j_0-1}\varphi_{j_0-1},A_{j_0},...A_k$
where now $A_{j_0-1}\varphi_{j_0-1}$ and $A_{j_0}$ are compactly supported.
Continuing this way, also to the right of $j_0$,
it remains to treat the case where \emph{each} $A_j$
has compact support.
Case 1. We apply H\"older's inequality for Schatten norms and Proposition \ref{p:ML20080403-A3}:
\begin{equation}\label{eq:ML20080426}
\begin{split}
\| A_0 e^{-\sigma_0 t D^2} &A_1\cdot\ldots\cdot A_k e^{-\sigma_k t D^2}\|_1\\
&\le \prod_{j=0}^k \|A_j e^{-\sigma_j t D^2}\|_{\sigma_j^{-1}}\\
& \le C(t_0,\eps)\; \prod_{j=0}^k (\sigma_j t)^{-d_j/2-\frac{\dim M+\eps}{2}\sigma_j}\\
& \le C(t_0,\eps)\; \Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2}\Biggr)
t^{-d/2-\frac{\dim M+\eps}{2}}, \quad 0<t\le t_0,
\end{split}
\end{equation}
thanks to the fact that $\sigma\mapsto \sigma^{-\frac{\dim M+\eps}{2}\sigma}$ is bounded as $\sigma\to 0$.
Case 2. If $D$ is Fredholm we estimate
\[\| A_0 e^{-\sigma_0 t D^2}(I-H)A_1\cdot\ldots\cdot A_k e^{-\sigma_k t D^2}(I-H)\|_1\]
using H\"older as in \eqref{eq:ML20080426} and apply Proposition \ref{p:ML20080403-A4}
to the individual factors:
\begin{equation}
\|A_j e^{-\sigma_j t D^2}(I-H)\|_{\sigma_j^{-1}}\le
C_\delta (\sigma_j t)^{-d_j/2-\frac{\dim M+\eps}{2}\sigma_j}e^{-\sigma_jt\delta},
\quad 0<t<\infty,
\end{equation}
to reach the conclusion.
Finally we remark that the inequalities \eqref{eq:integrated-multiple-heat-estimate-short}, \eqref{eq:integrated-multiple-heat-estimate-large}
follow by integrating the inequalities \eqref{eq:multiple-heat-estimate-short}, \eqref{eq:multiple-heat-estimate-large}
over the standard simplex $\Delta_k$. Note that
$\int_{\Delta_k} (\sigma_0\cdot\ldots\cdot\sigma_k)^{-1/2}d\sigma<\infty$.
\end{proof}
\section{Comparison results}\label{s:comparison-results}
Let $M_j$, $j=1,2,$ be complete riemannian manifolds with cylindrical ends,
\cf~Proposition \ref{p:bsmooth}. Assume that $M_1$ and $M_2$ share a common
cylinder component $(-\infty,0]\times Z$.
That is, if $M_j= (-\infty,0]\times Z_j\cup_{Z_j} X_j, j=1,2$,
then $Z$ is (after a suitable identification) a common (union of)
connected component(s) of $Z_1, Z_2$. A typical example will be
$M_2=\R\times Z$.
Suppose that $\dirac_j$ are formally self-adjoint Dirac
operators (\emph{cf.}~Section \plref{s:qDirac})
on $M_j$, $j=1,2$ with
${\dirac_1}\rest{(-\infty,0]\times Z}={\dirac_2}\rest{(-\infty,0]\times Z}
=:\dirac=\sfc(dx)\frac{d}{dx}+\mathsf{D}_\pl$.
The operators $\dirac_j$ are supposed to act on sections of the hermitian
vector bundles $W_j$ such that
${W_1}\rest{(-\infty,0]\times Z}={W_2}\rest{(-\infty,0]\times Z}$.
As explained in Section \ref{App:bdefbmet}, we identify
$\Gamma^\infty \big( (-\infty,c]\times Z;W \big)$, $c\in \R$,
with the completed tensor product
$\cC^\infty \big( (-\infty,c] \big)\hatotimes\Gamma^\infty(Z;W)$
and accordingly
$\bcptGamma \big( (-\infty,c)\times Z;W \big)$ with
$\bcptC \big( (-\infty,c) \big) \hatotimes \Gamma^\infty(Z;W)$.
We want to compare the
resolvents and heat kernels of $\dirac_j$ on the common cylinder
$(-\infty,0]\times Z$.
To this end, we will make repeated use of Remark \ref{r:DiracCylinderFormulas}
without mentioning it every time. There is an intimate relation between
the \emph{spectrum}, $\spec \mathsf{D}_\pl$, of the boundary operator $\mathsf{D}_\pl$
and the \emph{essential spectrum}, $\specess \dirac$, of $D$. We will only
need that
\begin{equation}
\label{eq:essspecbd}
\inf\specess \dirac^2=\inf\spec\mathsf{D}_\pl^2.
\end{equation}
A proof of this can be found in \cite[Sec. 4]{Mul:EIM}.
Concerning notation, $\|T\|_p$ stands for the $p$-th Schatten norm
of an operator $T$ acting on a Hilbert space $\sH$ unless otherwise stated.
\begin{proposition}\label{t:heat-resolvent-comparison}
\textup{1.} Fix an open sector
$\Lambda:=\bigsetdef{z\in\C^*}{\eps < \arg z < 2\pi-\eps}
\subset \C\setminus\R_+$ where $\eps > 0$.
Then on $(-\infty,c]\times Z,c<0,$
the difference of the resolvents $(\dirac_1^2-\gl)^{-1}-(\dirac_2^2-\gl)^{-1}, \gl\in\Lambda,$
is trace class. Moreover, for $N>0$ and $\gl_0\in\Lambda$ there is a constant
$C(c,N,\gl_0)$ such that
\begin{multline}\label{eq:ML20081028-2}
\Bigl\|\bigl( (\dirac_1^2-\gl)^{-1}-(\dirac_2^2-\gl)^{-1}\bigr)
\rest{(-\infty,c]\times Z} \Bigr\|_1\\
\le C(c,N,\gl_0)\; |\gl|^{-N},\quad \text{ for }\gl\in \gL,\,
|\gl|\ge |\gl_0|.
\end{multline}
\textup{2.} On $(-\infty,c]\times Z, c<0$, the difference of the heat kernels
\begin{equation}
\bigl(\dirac_1^l e^{-t\dirac_1^2}-\dirac_2^l e^{-t\dirac_2^2}\bigr)
\rest{(-\infty,c]\times Z}, \quad l\in\Z_+,
\end{equation}
is trace class for $t>0$. Moreover, for $N, t_0>0$ there is a constant
$C(c,l,N,t_0)$ such that
\begin{equation}
\Bigl\|\bigl(\dirac_1^l e^{-t\dirac_1^2}-\dirac_2^l e^{-t\dirac_2^2}\bigr)
\rest{(-\infty,c]\times Z} \Bigr\|_1\\
\le C(c,l,N,t_0)\; t^N,\quad 0<t\le t_0.
\end{equation}
\textup{3. } Assume in addition that $\dirac_1, \dirac_2$ are Fredholm operators and denote by $H_j$
the orthogonal projections onto $ \Ker \dirac_j, j=1,2$.
Then for $0<\delta<\inf\specess \dirac^2$ there is a constant $C(c,\delta)$ such
that
\begin{equation}
\Bigl\|\bigl(\dirac_1^le^{-t\dirac_1^2}(I-H_1)- \dirac_2^le^{-t\dirac_2^2}(I-H_2)\bigr)\rest{(-\infty,c]\times Z} \Bigr\|_1
\le C(c,\delta) e^{-t\delta},
\end{equation}
for $0<t<\infty,\; l\in\Z_+$.
\end{proposition}
\begin{proof}
\textup{1. } We choose cut--off functions $\varphi,\psi\in\cinf{\R}$
such that
\begin{equation}\label{eq:cut-off-functions}
\varphi(x)=\begin{cases} 1, & x\le 4/5 c,\\
0, & x\ge 3/5 c,\\
\end{cases}
\qquad
\psi(x)=\begin{cases} 1, & x\le 2/5 c,\\
0, & x\ge 1/5 c,\\
\end{cases}
\end{equation}
see Figure \ref{fig:CutOff}.
\FigCutOff
We have $\psi\varphi=\varphi$, $\supp d\psi\cap\supp \varphi=\emptyset$. Consider
\begin{equation}
R_{\psi,\varphi}(\gl):= \psi \bigl((\dirac_1^2-\gl)^{-1}-(\dirac_2^2-\gl)^{-1}\bigr) \varphi,
\end{equation}
viewed as an operator acting on sections over $M_1$. Then
\begin{equation}\label{eq:ML20081028-1}
(\dirac_1^2-\gl)R_{\psi,\varphi}(\gl)=[\dirac_1^2,\psi](\dirac_1^2-\gl)^{-1}\varphi-[\dirac_2^2,\psi](\dirac_2^2-\gl)^{-1}\varphi,
\end{equation}
where again $[\dirac_2^2,\psi](\dirac_2^2-\gl)^{-1}\varphi$ is considered as acting on sections over $M_1$.
Since the operators $[D_j^2,\psi], j=1,2$ have compact support which is disjoint from
the support of $\varphi$ we may apply
Proposition \plref{p:ML20080403-A2} to the r.h.s. of \eqref{eq:ML20081028-1}
to infer that $R_{\psi,\varphi}(\gl)$ is trace class and that the estimate \eqref{eq:ML20081028-2}
holds.
\textup{2. }
This follows from 1. and the contour integral representation \eqref{eq:contour-integral}
of the heat kernel. Cf. Proposition \plref{p:ML20080403-A3}.
\textup{3. } This follows from 1. and \eqref{eq:contour-integral} by taking the contour as in Figure \ref{fig:ContourFredholm},
page \pageref{fig:ContourFredholm}.
\end{proof}
We recall from Section \ref{App:bdefbmet} the notation $\bcptdiff^a((-\infty,0)\times Z;W)$
\eqref{eq:defbcptdiff}. In what follows, the subscript \emph{cpt} indicates that
the objects are supported away from $\{0\}\times Z$; it does \emph{not} indicate compact support.
The support of objects in $\bcptdiff^a$, and other spaces having the $\emph{cpt}$ decoration, may
be unbounded towards $\{-\infty\}\times Z$. \emph{Compactly} supported
functions resp. sections are written with a $c$ decoration, e.g. $\cC_c^\infty$ resp. $\Gamma_c^\infty$.
\begin{theorem}\label{t:JLO-comparison}
Let $A_j\in \bcptdiff^{d_j}((-\infty,0)\times Z;W)$ be
$b$--differential operators of order $d_j, j=0,...,k$ which are supported
away from $\{0\}\times Z$. Let $d:=\sum_{j=0}^kd_j$ be the sum of their
orders.
\textup{1. } For $t_0,N>0$ there is a constant $C(t_0,N)$ such that for all
$\sigma\in \Delta_k, \sigma_j>0$
\begin{multline}\label{eq:ML20090128-1}
\Bigl\| A_0 e^{-\sigma_0 t \dirac_1^2}\cdot...\cdot A_l \bigl( e^{-\sigma_l t \dirac_1^2}-e^{-\sigma_l t \dirac_2^2}
\bigr) A_{l+1}\cdot...\cdot A_k e^{-\sigma_k t \dirac_1^2}\Bigr\|_1\\
\le C(t_0,N)\Biggl(\prod_{j=0, j\not=l}^k \sigma_j^{-d_j/2} \Biggr) (\sigma_l t)^N,\quad
0< t<t_0.
\end{multline}
\textup{2. } Assume in addition that $\dirac_1, \dirac_2$ are Fredholm operators and denote by $H_j$
the orthogonal projections onto $ \Ker \dirac_j, j=1,2$.
Then for $0<\delta<\inf \specess \dirac^2$
and all $\sigma\in\Delta_k,\sigma_j>0,$
\begin{multline}\label{eq:ML20090128-2}
\Bigl\| A_0 e^{-\sigma_0 t \dirac_1^2}(I-H_1)\cdot...\cdot A_l \bigl( e^{-\sigma_l t \dirac_1^2}(I-H_1)-e^{-\sigma_l t \dirac_2^2}(I-H_2)\bigr)A_{l+1}\cdot...\\ ...\cdot A_k e^{-\sigma_k t \dirac_1^2}(I-H_1)\Bigr\|_1\\
\le C(\delta)\Biggl(\prod_{j=0, j\not=l}^k \sigma_j^{-d_j/2} \Biggr) e^{-t\delta},\quad
0< t<\infty.
\end{multline}
\end{theorem}
\begin{remark}\label{rem:EstimateOptimality}
With some more efforts one can show that the factors $\Bigl(\prod_{j=0}^k \sigma_j^{-d_j/2}\Bigr)$ on the right
hand sides of the estimates \eqref{eq:ML20090128-1}, \eqref{eq:ML20090128-2},
and also below in \eqref{eq:b-multiple-heat-estimate-short}, \eqref{eq:b-multiple-heat-estimate-large}
are obsolete. But since this is not needed for our purposes we prefer a less
cumbersome presentation.
\end{remark}
\begin{proof}
First note that by Proposition \ref{Prop:PropbSob}
the operator $A(i+\dirac)^{-a}$ is bounded
for $A\in \bcptdiff^a((-\infty,0)\times Z;W)$ and hence
the Spectral Theorem implies that for $t_0>0$ there is a $C(t_0)$ such that
\begin{equation}\label{eq:ML20081028-4}
\|A e^{-t\dirac^2}\|_\infty\le C(t_0) t^{-a/2},\quad 0<t<t_0.
\end{equation}
If $D$ is Fredholm then for $0<\delta<\inf\specess D^2$ and $t_0>0$
there is a $C(t_0,\delta)$ such that
\begin{equation}\label{eq:ML20081028-5}
\|A e^{-t\dirac^2}(I-H)\|_\infty\le C(\delta) e^{-t\delta},\quad t_0<t<\infty.
\end{equation}
\eqref{eq:ML20081028-4} and \eqref{eq:ML20081028-5} together imply that
for $0<\delta<\inf\specess D^2$ there is a $C(\delta)$ such that
\begin{equation}\label{eq:ML20081028-6}
\|A e^{-t\dirac^2}(I-H)\|_\infty\le C(\delta) t^{-a/2} e^{-t\delta},\quad 0<t<\infty.
\end{equation}
The first claim now follows
from the second assertion of Theorem \ref{t:heat-resolvent-comparison}.
Namely, with some $c<0$ such that $\supp A_j\subset (-\infty,c]\times Z$ we find
\begin{equation}\label{eq:ML20081028-7}
\begin{split}
\Bigl\| A_0 &e^{-\sigma_0 t \dirac_1^2}\cdot...\cdot A_l \bigl( e^{-\sigma_l t \dirac_1^2}-e^{-\sigma_l t \dirac_2^2}
\bigr) A_{l+1}\cdot...\cdot A_k e^{-\sigma_k t \dirac_1^2}\Bigr\|_1\\
&\le \Biggl(\prod_{j=0,j\not=l}^k \| A_j e^{-\sigma_j t \dirac_1^2} \|_\infty \Biggr) \;
\bigl\| A_l (i+\dirac)^{-d_l}\bigr\|_\infty \;\cdot\\
& \qquad \cdot \Bigl\|\bigl((i+\dirac_1)^{d_l} e^{-\sigma_l t \dirac_1^2}-(i+\dirac_2)^{d_l}e^{-\sigma_l t \dirac_2^2}\bigr)_{|(-\infty,c]\times Z}\Bigr\|_1\\
&\le C(t_0,N) \Biggl(\prod_{j=0,j\not=l}^k \sigma_j^{-d_j/2} \Biggr) \; \sigma_l^N t^{N-d/2},\quad 0<t<t_0.
\end{split}
\end{equation}
Here we have used \eqref{eq:ML20081028-4}. Since $\sigma_j<1$ an inequality which
is valid for $0<\sigma_j t<t_0$ is certainly also valid for $0<t<t_0$.
Since $N$ is arbitrary \eqref{eq:ML20081028-7} proves the first claim.
2. Similarly, using the third assertion of Theorem \ref{t:heat-resolvent-comparison} and
\eqref{eq:ML20081028-6}
\begin{equation}
\begin{split}
\Bigl\|& A_0 e^{-\sigma_0 t \dirac_1^2}(I-H_1)\cdot...\cdot A_l \bigl( e^{-\sigma_l t \dirac_1^2}(I-H_1)-e^{-\sigma_l t \dirac_2^2}(I-H_2)\bigr)A_{l+1}\cdot...\\
&\qquad ...\cdot A_k e^{-\sigma_k t \dirac_1^2}(I-H_1)\Bigr\|_1\\
&\le \Biggl(\prod_{j=0,j\not=l}^k \| A_j e^{-\sigma_j t \dirac_1^2} (I-H_1)\|_\infty \Biggr)\;
\bigl\| A_l(i+\dirac)^{-d_l}\bigr\|_\infty \cdot\\
&\quad \cdot \Bigl\|\bigl((i+\dirac_1)^{d_l} e^{-\sigma_l t \dirac_1^2}(I-H_1)-
(i+\dirac_2)^{d_l} e^{-\sigma_l t \dirac_2^2}(I-H_2)\bigr)_{|(-\infty,c]\times Z}\Bigr\|_1\\
&\le C(\delta) \Biggl(\prod_{j=0,j\not=l}^k \sigma_j^{-d_j/2} \Biggr) \; t^{-d/2} e^{-t\delta}.
\end{split}
\end{equation}
Together with the proved short time estimate and the fact that the $H_j$ are of finite rank and thus of trace class
we reach the conclusion.
\end{proof}
\section{Trace class estimates for the model heat kernel}
\label{s:estimates-model-heat-kernel}
\nind{Delta@$\Delta_\R$}
We consider the heat kernel\sind{heat kernel} of the Laplacian $\Delta_\R$ on the real
line
\begin{equation}
\label{eq:heat-kernel}
k_t(x,y)=\frac{1}{\sqrt{4\pi t}} e^{-(x-y)^2/4t}.
\end{equation}
By slight abuse of notation we will denote the operator of multiplication
by $\id_\R$ by $X$. We want to estimate the Schatten norms
of $e^{\ga |X|}e^{-t\Delta_\R}e^{\beta |X|}$.
Before we start with this let us note for future reference:
\begin{equation}\label{eq:gaussian-integral}
\begin{split}
\int_\R e^{-\frac{z^2}{\gl t}-\beta z} dz&= \sqrt{\pi \gl t}\; e^{\gl \beta^2 t/4},\\
\int_\R e^{-\frac{z^2}{\gl t}+\beta |z|} dz&\le 2\sqrt{\pi \gl t}\; e^{\gl \beta^2 t/4},
\end{split}
\quad \beta\in\R;\; \gl,t>0.
\end{equation}
Furthermore, we will need the well-known \sindp{Schur's test}:
\begin{lemma}[{\cite[Thm.~5.2]{HalSun:BIO}}]\label{l:Schurs-test}
Let $K$ be an integral operator on a measure space $(\Omega,\mu)$ with kernel
$k:\Omega\times \Omega\to \C$. Assume that there are positive measurable functions
$p,q: \Omega \to (0,\infty)$ such that
\begin{equation}\label{eq:Schurs-test}
\begin{split}
\int_X |k(x,y)| p(y) d\mu(y)&\le C_p\; q(x),\\
\int_X |k(x,y)| q(x) d\mu(x)&\le C_q\; p(y).
\end{split}
\end{equation}
Then $K$ is bounded in $L^2(\Omega,\mu)$ with $\|K\|\le \sqrt{C_pC_q}$.
\end{lemma}
\comment{\begin{proof}
For $f,g\in L^2(\Omega ,\mu)$ we estimate using Cauchy--Schwarz and
\eqref{eq:Schurs-test}
\begin{equation}
\begin{split}
\int_X\int_X &|f(x)| |k(x,y)| |g(y)| d\mu(x)d\mu(y)\\
&= \int_X\int_X \Big( \frac{|f(x)|}{\sqrt{q(x)}} \sqrt{p(y)} |k(x,y)|^{1/2}\Big)\cdot\\
&\qquad
\cdot \Big( \sqrt{q(x)}|k(x,y)|^{1/2} \frac{|g(y)|}{\sqrt{p(y)}}\Big) d\mu(x)d\mu(y)\\
&\le \Big( \int_X\int_X \frac{|f(x)|^2}{q(x)} |k(x,y)| p(y) d\mu(y) d\mu(x)\Big)^{1/2}
\cdot\\
&\qquad \cdot\Big( \int_X\int_X
\frac{|g(y)|^2}{p(y)} |k(x,y)| q(x) d\mu(x) d\mu(y)\Big)^{1/2}\\
&\le \sqrt{C_pC_q} \|f\| \; \|g\|.\qedhere
\end{split}
\end{equation}
\end{proof}}
Now we can prove the following estimate.
\begin{proposition}\label{p:heat-estimate-scalar-laplace}
Let $\Delta_\R=-\frac{d^2}{dx^2}$ be the Laplacian on the real line.
Then for $\ga>\gb>0,t>0, l\in \Z_+$, the integral operator
$e^{-\ga|X|}\bigl(\frac{d}{dx}\bigr)^l e^{-t\Delta_\R} e^{\beta|X|}$
with the (not everywhere smooth) kernel
\begin{equation}\label{eq:heat-estimate-scalar-laplace-1}
\frac{1}{\sqrt{4\pi t}}e^{-\ga |x|} \partial_x^l e^{-(x-y)^2/4t +\gb |y|}
\end{equation}
is $p$-summable for $1\le p\le \infty$ and we have the estimate
\begin{multline}\label{eq:heat-estimate-scalar-laplace-2}
\|e^{-\ga|X|}\bigl(\frac{d}{dx}\bigr)^l e^{-t\Delta_\R} e^{\beta|X|}\|_p\\
\le \bigl( c_1 t^{-\frac{l}{2}-\frac{1}{2p}}+c_2 t^{-\frac{1}{2p}}\bigr)
(\ga-\gb)^{-1/p} e^{(\ga^2+\gb^2)t},\quad 0<t<\infty
\end{multline}
with (computable) constants $c_1,c_2$ independent of $\ga,\gb,p,t$.
\end{proposition}
\begin{remark}\label{rem:EstimateOptimality-a}We are not striving to make these estimates optimal.
We chose to formulate them in such a way that they are sufficient
for our purposes and such that the proofs
do not become too cumbersome.
\end{remark}
\begin{proof} We will prove this estimate for $p=\infty$ using Schur's test
Lemma \plref{l:Schurs-test} and for $p=2$ by estimating the $L^2$-norm
of the kernel. The case $p=1$ will then follow from the semigroup property of
the heat kernel. The result for general $p$ follows from the cases $p=1$ and $p=\infty$
and the interpolation inequality \eqref{eq:Hoelder-interpolation}.
The case $l\ge 2$ can easily be reduced to the case $l\in\{0,1\}$ in view of the
identity
\begin{equation}\label{eq:ML20090127-1}
\partial_x^{2k} e^{-t\Delta_\R} = (-\Delta_\R)^k e^{-t\Delta_\R}=
\partial_t^k e^{-t\Delta_\R}.
\end{equation}
\subsection*{The case $p=\infty$} We apply Schur's test with $p(x)=q(x)=1$. We will make frequent
use of \eqref{eq:gaussian-integral} without explicitly mentioning it all the time.
\begin{equation}\label{eq:heat-estimate-scalar-laplace-3}
\begin{split}
\frac{1}{\sqrt{4\pi t}} &\int_\R e^{-\ga|x| -(x-y)^2/4t +\gb |y|}dy\\
&\le \frac{1}{\sqrt{4\pi t}}e^{-\ga |x|}\int_\R e^{-z^2/4t +\gb |x|+\gb|z|}dz\\
& \le e^{(\gb-\ga)|x|}2 e^{\gb^2 t}\le 2 e^{\gb^2 t}.
\end{split}
\end{equation}
Reversing the roles of $x$ and $y$ one gets similarly
\begin{equation}\label{eq:heat-estimate-scalar-laplace-4}
\begin{split}
\frac{1}{\sqrt{4\pi t}} &\int_\R e^{\gb|y| -(x-y)^2/4t -\ga |x|}dx\\
&\le \frac{1}{\sqrt{4\pi t}}e^{\gb |y|}\int_\R e^{-z^2/4t -\ga |y|+\ga |z|}dz\\
& \le e^{(\gb-\ga)|y|}2 e^{\ga^2 t}\le 2 e^{\ga^2 t}.
\end{split}
\end{equation}
This proves the result for $l=0$ and $p=\infty$.
In the case $l=1$ the integral
\begin{equation}\label{eq:heat-estimate-scalar-laplace-5}
\frac{1}{\sqrt{4\pi t}} \int_\R \frac{|x-y|}{2t}e^{-\ga|x| -(x-y)^2/4t +\gb |y|}dy
\end{equation}
is estimated similarly.
\subsection*{The case $p=2$} We estimate the $L^2$-norm of the kernel on $\R\times\R$ by:
\begin{equation}\label{eq:heat-estimate-scalar-laplace-6}
\begin{split}
\frac{1}{4\pi t} &\int_\R\int_\R e^{-2\ga|x| -\frac{(x-y)^2}{2t}+2\beta|y|} dx dy\\
&\le \frac{1}{4\pi t} \int_\R e^{-2\ga|x|}\int_\R e^{-\frac{z^2}{2t}+2\beta|z|+2\beta|x|} dz dx\\
&\le \frac{1}{\sqrt{2\pi t}}\int_\R e^{-2(\ga-\gb)|x|} e^{2\gb^2 t} dx\\
&= \frac{1}{\sqrt{2\pi t}}\frac{1}{\ga-\gb}e^{2\gb^2 t},
\end{split}
\end{equation}
proving the result for $l=0$ and $p=2$. Again, the case $l=1$ is similar.
\subsection*{The case $p=1$} Put $c=(\ga+\gb)/2$. Then the semigroup property of the heat
kernel gives
\begin{equation}
\begin{split}
\| e^{-\ga|X|} &e^{-t\Delta_\R} e^{\gb|X|}\|_1\\
&\le \| e^{-\ga|X|} e^{-t/2\Delta_\R} e^{c|X|}\|_2
\| e^{- c |X|} e^{-t/2\Delta_\R} e^{\gb |X|}\|_2,
\end{split}
\end{equation}
and using the proved case $p=2$ gives the result for $p=1$.
\end{proof}
The previous Proposition and standard estimates for the heat kernel\sind{heat kernel}
on closed manifolds
(\cf~the Section \ref{sec:basic-estimates}) immediately give the following result
for the heat kernel of the model Dirac operator on the cylinder.
\begin{proposition}\label{p:heat-estimate-model-laplace}
Let $Z$ be a compact closed manifold and $\dirac=\Gammabdy\bigl(\frac{d}{dx}+\Abdy\bigr)$
a Dirac operator on the cylinder $M=\R\times Z$
(\cf~Remark \plref{r:DiracCylinderFormulas}).
Furthermore, let $Q\in \bcptdiff^q((-\infty,0)\times Z;W)$ a \textup{b}-differential operator
of order $q$ with support in some cylindrical end $(-\infty, c)\times Z$.
Then for $\ga>\gb>0,t>0$ the integral operator $e^{-\ga|X|}Q e^{-t\dirac^2} e^{\beta|X|}$
with kernel
\begin{equation}\label{eq:heat-estimate-model-laplace-1}
\frac{1}{\sqrt{4\pi t}}e^{-\ga |x|} Q_x e^{-(x-y)^2/4t +\gb |y|}e^{-tA^2}
\end{equation}
is $p$-summable for $1\le p\le \infty$.
Furthermore, for $\eps>0$, $t_0>0$, there is a constant $C(\eps,t_0)$, such that
for $1\le p\le\infty$, $0<t<t_0$, $0<\beta<\ga$
\begin{equation}
\|e^{-\ga|X|} Q e^{-t \dirac^2} e^{\beta|X|}\|_p
\le C(\eps,t_0) (\ga-\gb)^{-1/p}
t^{-\frac{\dim M+\eps}{2p}-\frac q2}.
\end{equation}
If in addition the operator $A$ is invertible then for $0<\delta<\inf\spec A^2$
and $\eps>0$ there are constants $C_j(\delta,\eps)$, $j=1,2$
such that for $1\le p\le \infty,$
$0<t<\infty$, $0<\gb<\ga$ we have the estimate
\begin{multline}
\label{eq:heat-estimate-model-laplace-2}
\|e^{-\ga|X|} Q e^{-t \dirac^2} e^{\beta|X|}\|_p \\
\le \bigl( C_1(\delta,\eps) t^{-\frac{q}{2}}+C_2(\delta,\eps)\bigr)(\ga-\gb)^{-1/p}
t^{-\frac{\dim M+\eps}{2p}} e^{(\ga^2+\gb^2-\delta)t}.
\end{multline}
\end{proposition}
For the definition of $\bcptdiff^q, \bdiff^q$ see Proposition \plref{prop:DefEqbDiff} and Eq.~\eqref{eq:defbcptdiff}.
\begin{proof}
By Proposition \ref{prop:DefEqbDiff} we may write $Q$ as a sum of operators of the form
\begin{equation}\label{eq:dDiffOp-normalForm}
f(x,p) P \Bigl(\frac{d}{dx}\Bigr)^l
\end{equation}
with
\begin{itemize}
\item $f\in\bcptGamma((-\infty,c)\times Z;\End W)$,
\item $P\in\Diff^{b-l}(Z;W\rest{Z})$ a differential operator of order $b-l$
on $Z$ which is \emph{constant} in $x$-direction.
\end{itemize}
Note that $e^{-\ga|X|}$ commutes with $f$. Furthermore, $f$ is uniformly bounded.
Thus
\begin{equation}\label{eq:ML20090127-2}
\Bigl\| e^{-\ga|X|} f P \partial_x^l e^{-t\dirac^2} e^{\gb |X|}\Bigr\|_p
\le \|f\|_\infty \Bigl\|e^{-\ga|X|} P\partial_x^l e^{-t\dirac^2} e^{\gb |X|}\Bigr\|_p.
\end{equation}
Inside the $p$-norm is now a tensor product of operators
\begin{equation}
e^{-\ga|X|} P \partial_x^l e^{-t\dirac^2} e^{\gb |X|}
= \bigl( e^{-\ga|X|} \partial_x^l e^{-t\Delta_\R} e^{\gb|X|}\bigr)
\otimes \bigl(P e^{-tA^2}\bigr).
\end{equation}
Since the $p$-norm of a tensor product is the product of the $p$-norms
the claim follows from Proposition \ref{p:heat-estimate-scalar-laplace} and
standard elliptic estimates for the closed manifold $Z$
(Propositions \ref{p:ML20080403-A3}, \ref{p:ML20080403-A4}).
\end{proof}
\section[Trace class estimates]{
Trace class estimates for the \textup{JLO} integrand on manifolds with cylindrical ends}
\label{s:estimates-JLO-cylindrical}
The heat kernel estimate for the Dirac operator on the model cylinder from Proposition
\plref{p:heat-estimate-model-laplace}
together with the comparison result in Section \ref{s:comparison-results}
will now be used to obtain trace class estimates for
\textup{b}-differential operators similar to the one in Proposition
\plref{p:multiple-heat-estimate} if the indicial operator of at least
one of the operators $A_0,\ldots,A_k$ vanishes. Let us mention here that
in the following we will use the notation introduced in
Subsection \ref{ss:JLODO}, in particular Eq.~\eqref{eq:ML20090128-3}.
\begin{proposition}\label{p:multiple-heat-estimate-b}
Let $M=(-\infty,0]\times Z\cup_Z X$, where $X$ is a compact manifold with
boundary, be a complete manifold with cylindrical ends
and let $\dirac$ be a Dirac operator on $M$.
Let $A_0,...,A_k\in\bdiff(M;W)$ be \textup{b}-differential operators of order
$d_0,...,d_k; d:=\sum_{j=0}^k d_j$.
Assume that for at least one index
$l\in \{0,...,k\}$ the indicial family of $A_l$ vanishes.
Then for $t>0$, $\sigma\in\Delta_k$ the operator
\begin{equation}
A_0 e^{-\sigma_0 t \dirac^2}A_1\cdot\ldots\cdot A_k e^{-\sigma_k t \dirac^2}
\end{equation}
is trace class. Furthermore, there are the following estimates:
\paragraph*{1} For $t_0>0$, $\eps>0$ there is a constant $C(t_0,\eps)$ such that
for all $\sigma=(\sigma_0,...,\sigma_k)\in\Delta_k$, $\sigma_j>0$,
\begin{equation}\label{eq:b-multiple-heat-estimate-short}
\begin{split}
\| A_0 &e^{-\sigma_0 t \dirac^2}A_1\cdot\ldots\cdot A_k e^{-\sigma_k t \dirac^2}\|_1\\
&\le C(t_0,\eps)\; \Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2} \Biggr)
t^{-d/2-(\dim M)/2-\eps},\quad 0<t\le t_0.
\end{split}
\end{equation}
In particular, if $d_j\le 1, j=0,...,k,$ then
\begin{equation}\label{eq:b-integrated-multiple-heat-estimate-short}
\|( A_0,...,A_k)_{\sqrt{t}\dirac}\|=O( t^{-d/2-(\dim M)/2-0}),\quad t \to 0+.
\end{equation}
\paragraph*{2} Assume additionally that $\dirac$ is Fredholm and denote by $H$ the orthogonal
projection onto $ \Ker \dirac$. Then for $\eps>0$ and any $0<\delta<\inf\spec_{\ess} \dirac^2$
there is a constant $C(\delta,\eps)$ such that for all $\sigma\in\Delta_k, \sigma_j>0$
\begin{equation}\label{eq:b-multiple-heat-estimate-large}
\begin{split}
\| A_0 &e^{-\sigma_0 t \dirac^2}(I-H)A_1\cdot\ldots\cdot A_k e^{-\sigma_k t \dirac^2}(I-H)\|_1\\
&\le C(\delta,\eps)\; \Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2} \Biggr)
t^{-d/2-(\dim M)/2-\eps} e^{-t\delta},\quad \text{for \emph{all} } 0<t<\infty.
\end{split}
\end{equation}
In particular, if $d_j\le 1, j=0,...,k,$ then
\begin{equation}\label{eq:b-integrated-multiple-heat-estimate-large}
\begin{split}
\|( A_0&(I-H),...,A_k(I-H))_{\sqrt{t}\dirac}\|\\
&\le \tilde C(\eps,\delta)\;
t^{-d/2-(\dim M)/2-\eps} e^{-t\delta},\quad \text{for \emph{all }} 0<t<\infty.
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
We first reduce the problem to a problem on the cylinder $(-\infty,0]\times Z$.
Write $A_j=A_j^{(0)}+A_j^{(1)}$ where $A_j^{(0)}$ has compact support and $A_j^{(1)}$
is supported on $(-\infty,c]\times Z$ for some $c<0$.
Then we split $A_0e^{-\sigma_0t \dirac^2}A_1\cdot\ldots\cdot A_ke^{-\sigma_k t\dirac^2}$
(resp. $A_0e^{-\sigma_0t \dirac^2}(I-H)A_1\cdot\ldots\cdot A_ke^{-\sigma_k t\dirac^2}(I-H)$)
into a sum of terms obtained by decomposing $A_j=A_j^{(0)}+A_j^{(1)}$.
To the summands involving at least one term $A_j^{(0)}$ we apply Proposition
\ref{p:multiple-heat-estimate}. To the remaining summand involving only $A_j^{(1)}$
we first apply the comparison Theorem \ref{t:JLO-comparison}
with $\dirac_1=\dirac$ and $\dirac_2=\Gammabdy \frac{d}{dx}+\mathsf{D}_\pl
=\Gammabdy\bigl(\frac{d}{dx}+A\bigr)$,
where $\dirac_2$ acts on sections over the cylinder $\R\times Z$;
\emph{cf.}~Remark \ref{r:DiracCylinderFormulas}.
Hence it remains to prove the claim for the cylinder $M=\R\times Z$ where
each $A_j$ is supported on $(-\infty,c]\times Z$ for some $c<0$.
For definiteness it is not a big loss of generality if
we assume that the indicial family of $A_0$ vanishes (I.e. $l=0$).
Write $A_0=e^{-|X|}\tilde A_0$ with $A_0\in\bcptdiff^{d_0}((-\infty,0)\times Z;W)$.
Let $\gb_0,...,\gb_{k+1}$ be real numbers with $1\ge \gb_0>\gb_1>...>\gb_k>\gb_{k+1}=0$.
Let us assume that $\dirac$ is Fredholm and prove the claim 2. The proof of claim 1.
is similar and left to the reader.
H\"older's inequality yields
\begin{multline}
\bigl \|A_0 e^{-\sigma_0 t \dirac^2}(I-H)A_1\cdot...\cdot A_k e^{-\sigma_k t \dirac^2}(I-H)\bigr\|_1 \\
\le C \bigl\| e^{-\gb_0 |X|} \tilde A_0 e^{-\sigma_0 t\dirac^2}(I-H)e^{\gb_1|X|}\bigr\|_{\sigma_0^{-1}}\cdot \\
\cdot \prod_{j=1}^k \bigl \| e^{-\gb_j|X|} A_j e^{-\sigma_j t \dirac^2}(I-H) e^{\gb_{j+1}|X|}\bigr\|_{\sigma_j^{-1}}.
\end{multline}
The individual factors are estimated by Proposition
\ref{p:heat-estimate-model-laplace} and we obtain
for $0<t<\infty$:
\begin{equation}
\begin{split}
...&\le \prod_{j=0}^k
\bigl( C_{1,j}(\delta,\eps,\gb)
(t\sigma_j)^{-d_j/2}+C_{2,j}(\delta,\eps,\gb)\bigr)\cdot\\
&\qquad \cdot
t^{-\frac{\dim M+\eps}{2}\sigma_j} e^{(\gb_j^2+\gb_{j+1}^2-\delta)\sigma_j t}\\
&\le C(\delta,\eps,\gb,\gamma) \Bigl(\prod_{j=0}^k \sigma_j^{-d_j/2}\Bigr)
t^{-d/2-\frac{\dim M+\eps}{2}} e^{(2\sum \gb_j^2 +\gamma-\delta)t},
\end{split}
\end{equation}
for any $\gamma>0$. The $\gamma>0$ is introduced to compensate $t^{-d/2}$ as $t\to\infty$.
Since we may choose $2\sum \gb_j^2 +\gamma$ as small as we please, the claim is proved.
The remaining cases are treated similarly.
\end{proof}
\section{Estimates for \textup{b}-traces}
\label{s:EstbTraces}
\comment{\marginpar{Due to the commutator formula for the \textup{b}-trace Thm. \eqref{t:bTraceAsTrace}
this section can be presented in a much shorter way. However, since the results
are secured now this steamlining does not have high priority; I wouldn't mind by the
way if somebody else would do it}
Finally we are going to estimate expressions of the form $\bTr(A_0e^{-\sigma_0t \dirac^2}A_1\cdot...\cdot
A_ke^{-\sigma_kt \dirac^2})$.
As a preparation we first discuss the one dimensional case.
\begin{proposition}\label{p:1d-b-estimate}
Let $\Delta_\R=-\frac{d^2}{dx^2}$ on the real line. Furthermore, let
$\varphi_j\in \bcptC(-\infty,0), j=0,...,k$, with $\varphi_j(x)=1$ for $x\le x_0$.
Moreover, given $d_0,...,d_k\in\Z_+$, $d=\sum_{j=0}^k d_j$.
Then for $\eps,\delta>0$ there is a constant $C(\eps,\delta)$
such that for all $\sigma\in\Delta_k, \sigma_j>0,$
\begin{equation}\label{eq:ML20081104-1}
\begin{split}
\bTr(&\varphi_0 \partial_x^{d_0}e^{-\sigma_0t\Delta_\R}\varphi_1\cdot...\cdot
\varphi_k \partial_x^{d_k}e^{-\sigma_kt\Delta_\R})\\
&\le C(\eps,\delta)\Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2-\eps} \Biggr)
t^{-(a+1)/2-\eps}e^{t\delta},\quad 0<t<\infty.
\end{split}
\end{equation}
\end{proposition}
\begin{proof} Fix a $c>0$ and consider the invertible operator $\Delta_\R+c$.
Although $\Delta_\R+c$ is not the square of a first order differential operator
it is clear that the basic estimates of Section \ref{sec:basic-estimates}
hold verbatim for $\Delta_\R+c$ instead of $D^2$. Note that since $\Delta_\R+c$
is invertible, the long time estimates of Section \ref{sec:basic-estimates}
hold with $H=0$. Finally, note that
\begin{equation}
\begin{split}
e^{-t c} \bTr((&\varphi_0 \partial_x^{d_0}e^{-\sigma_0t\Delta_\R}\varphi_1\cdot...\cdot
\varphi_k \partial_x^{d_k}e^{-\sigma_kt\Delta_\R})\\
\bTr((&\varphi_0 \partial_x^{d_0}e^{-\sigma_0t(\Delta_\R+c)}\varphi_1\cdot...\cdot
\varphi_k \partial_x^{d_k}e^{-\sigma_kt(\Delta_\R+c)}),
\end{split}
\end{equation}
and hence we need to show that for $\eps>0$ and $0<\delta<c$ there is
a $C(\eps,\delta)$ such that for all $\sigma\in\Delta_k, \sigma_j>0,$
\begin{equation}\label{eq:ML20081104-2}
\begin{split}
\bTr(&\varphi_0 \partial_x^{d_0}e^{-\sigma_0t(\Delta_\R+c)}\varphi_1\cdot...\cdot
\varphi_k \partial_x^{d_k}e^{-\sigma_kt(\Delta_\R+c)})\\
&\le C(\eps,\delta)\Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2-\eps} \Biggr)
t^{-(a+1)/2-\eps}e^{-t\delta},\quad 0<t<\infty.
\end{split}
\end{equation}
We want to move the $\varphi_j$ to the left. If we commute $\varphi_j$
with a term $e^{-\sigma_{j-1}t(\Delta_\R+c)}$ we apply Proposition
\ref{p:ML20081104-A20} to the term
\begin{equation}
\varphi_0 \partial_x^{d_0}e^{-\sigma_0t(\Delta_\R+c)}\varphi_1...
[\varphi_j,e^{-\sigma_{j-1}t(\Delta_\R+c)}]...
\varphi_k \partial_x^{d_k}e^{-\sigma_kt(\Delta_\R+c)}).
\end{equation}
This expression is of trace class and the trace norm can be estimated by
the r.h.s. of \eqref{eq:ML20081104-2}.
Similarly, if we commute $\varphi_j$ and a derivative $\partial_x$ we need
to estimate the trace norm of a term of the form
\begin{equation}
\varphi_0 \partial_x^{d_0}e^{-\sigma_0t(\Delta_\R+c)}\varphi_1...[\varphi_j,\partial_x]...
\varphi_k \partial_x^{d_k}e^{-\sigma_kt(\Delta_\R+c)}).
\end{equation}
Since $[\varphi,\partial_x]$ is compactly supported the desired estimate
follows from Proposition \plref{p:multiple-heat-estimate}.
Moving all the $\varphi_j$ to the left leaves us to estimate
\begin{equation}
\bTr(\underbrace{\varphi_0...\varphi_k}_{=:\chi(x)} \partial_x^a e^{-t(\Delta_\R+c)}).
\end{equation}
If $a$ is odd this is easily seen to be $0$ since the kernel of $\chi \partial_x^a
e^{-t(\Delta_\R+c)}$ then vanishes on the diagonal. If $a$ is even we find
\begin{equation}
\begin{split}
\bTr(\chi \partial_x^a e^{-t(\Delta_\R+c)})&= e^{-ct}\partial_t^{d/2} \bTr(\chi e^{-t\Delta_\R})\\
&= e^{-ct}\partial_t^{d/2} \textup{p.f.-}\int_{-\infty}^0 \chi(x) \frac{1}{\sqrt{4\pi t}} dx\\
&=e^{-ct}\partial_t^{d/2} (4\pi t)^{-1/2} \int_{-\infty}^0 \chi(x)-1 dx.
\end{split}
\end{equation}
This can certainly be estimated by the r.h.s. of \eqref{eq:ML20081104-2} and
the proof is complete.
\end{proof}
}
Now we come to the main result of this chapter.
\begin{theorem}\label{t:main-b-estimate}
In the notation of Proposition \textup{\ref{p:multiple-heat-estimate-b}}
we now drop the assumption that the indicial family of one of the $A_l$ vanishes.
Then the following estimates hold:
\paragraph*{1} For $\eps>0, t_0>0$ there is a constant $C(\eps,t_0)$ such that
for all $\sigma=(\sigma_0,...,\sigma_k)\in\Delta_k, \sigma_j>0$,
\begin{equation}\label{eq:multiple-btrace-estimate-short}
\begin{split}
\bTr\bigl(A_0 &e^{-\sigma_0 t \dirac^2}A_1\cdot\ldots\cdot A_k e^{-\sigma_k t \dirac^2}\bigr)\\
&\le C(\eps,t_0) \Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2-\eps} \Biggr)
t^{-d/2-(\dim M)/2-\eps},\quad 0<t\le t_0.
\end{split}
\end{equation}
In particular,
\begin{equation}\label{eq:integrated-multiple-btrace-estimate-short}
|\blangle A_0,...,A_k\rangle_{\sqrt{t}\dirac}|=O( t^{-d/2-(\dim M)/2-\eps}),\quad t \to 0+.
\end{equation}
\paragraph*{2} If $\dirac$ is Fredholm then for $\eps>0$ and any $0<\delta<\inf\spec_{\ess} \dirac^2$
there is a constant $C(\eps,\delta)$ such that for all $\sigma\in\Delta_k, \sigma_j>0$
\begin{equation}\label{eq:multiple-btrace-estimate-large}
\begin{split}
\bTr\bigl(A_0 &e^{-\sigma_0 t \dirac^2}(I-H)A_1\cdot\ldots\cdot A_k e^{-\sigma_k t \dirac^2}(I-H)\bigr)\\
&\le C(\eps,\delta) \Biggl(\prod_{j=0}^k \sigma_j^{-d_j/2-\eps} \Biggr)
t^{-d/2-(\dim M)/2-\eps} e^{-t\delta},\quad \text{for \emph{all} } 0<t<\infty.
\end{split}
\end{equation}
In particular,
\begin{equation}\label{eq:integrated-multiple-btrace-estimate-large}
\begin{split}
|\blangle A_0&(I-H),...,A_k(I-H)\rangle_{\sqrt{t}\dirac}|\\
&\le \tilde C_{\delta,\eps}
t^{-d/2-(\dim M)/2-\eps} e^{-t\delta},\quad \text{for \emph{all }} 0<t<\infty.
\end{split}
\end{equation}
\end{theorem}
\begin{proof}
Arguing as in the proof of Proposition \ref{p:multiple-heat-estimate-b}
we may assume that $\dirac$ is the model Dirac operator and that
$A_0,\dots,A_k\in\bcptdiff\bigl((-\infty,0)\times\partial M;W\bigr)$.
By Proposition \plref{t:bTraceAsTrace} we have
\begin{multline}\label{eq:ML200909171}
\bTr\bigl(A_0 e^{-\sigma_0 t \dirac^2} A_1\cdot\ldots\cdot A_k e^{-\sigma_k t \dirac^2}\bigr)\\
= -\sum_{j=0}^k \Tr\bigl(x A_0 e^{-\sigma_0 t \dirac^2}\dots [\frac{d}{dx},A_j]\dots e^{-\sigma_k t \dirac^2}\bigr).
\end{multline}
Although multiplication by $x$ is not a \textup{b}-differential operator it is easy to
see that Proposition \plref{p:multiple-heat-estimate-b} still holds true for the summands on the
right of \eqref{eq:ML200909171}. The reason is that for any fixed $\eps>0$ the function
$x e^{-\eps |x|}$ is bounded. In fact any $0<\eps<1$ will do since the coefficients of
$[\frac{d}{dx},A_j]$ are $O(e^x)$ as $x\to -\infty$.
\end{proof}
\section{Estimates for the components of the entire \textup{b}-Chern character} \label{estim Chern}
We continue to work in the setting of a complete riemannian manifold with
cylindrical ends $M$, which is equivalent to a compact manifold
with boundary with an exact \textup{b}-metric, \cf~Section \ref{App:bdefbmet}.
Furthermore, let $\dirac$ be a Dirac operator on $M$.
\subsection{Short time estimates}
\begin{proposition}
\label{p:ShortTimeEst}
The Chern characters $\bCh^k$ and $\bslch^k$ defined in
\eqref{Eq:DefChern} and \eqref{Eq:DefSlChern} satisfy the following estimates
for $k\in\Z_+$:
\begin{equation}
\begin{split}
\bCh^k (t \dirac)(a_0,\cdots,a_k) & = O ( t^{k-\dim M-0} ), \\
\bslch^k(t \dirac,\dirac)(a_0,\cdots,a_k) & = O ( t^{k-\dim M-0} ),
\end{split}\quad t\to 0+,
\end{equation}
for $a_j\in \bcC(M), j=0,...,k$.
In particular,
\begin{enumerate}
\item $\lim\limits_{t\to 0+} \bCh^k(t\dirac)=0$ for $k>\dim M$.
\item The function $t\mapsto \bslch^k(t\dirac,\dirac)(a_0,...,a_k)$ is integrable on
$[0,T]$ for $T>0$ and $k>\dim M-1$.
\end{enumerate}
\end{proposition}
\begin{proof}
This follows immediately from Theorem \plref{t:main-b-estimate}.
\end{proof}
\subsection{Large time estimates}
Unless otherwise said we assume in this Subsection that
$\mathsf{D}_\pl$ is invertible. Then $\inf\specess \dirac^2=\inf\spec\mathsf{D}_\pl^2>0$
(\cf~Eq.~\eqref{eq:essspecbd}) and hence $\dirac$ is a Fredholm operator.
We denote by $H$ the finite rank orthogonal projection onto the kernel of $\dirac$.
\begin{lemma}\label{l:H1} Let $A_j\in\bdiff(M;W)$
be \textup{b}-differential operators of order $d_j, j=0,\ldots,k;
d:=\sum\limits_{j=0}^k d_j$ the total order.
Furthermore let $H_j=H$ or $H_j=I-H$, $j=0,...,k$ and assume that $H_j=H$ for at least one index $j$.
Then for each $0<\delta<\inf\specess \dirac^2$
\begin{equation}
\begin{split}
\Bigl\|&A_0H_0e^{-\sigma_0 t\dirac^2} A_1 H_1e^{-\sigma_1 t \dirac^2}\ldots A_k H_k e^{-\sigma_k t \dirac^2}\Bigr\|_1\\
&\le C(\delta) \Bigl(\prod_{l\in \{j_1,...,j_q\}} \sigma_{l}^{-d_l/2}\Bigr) t^{-d/2} e^{-(\sigma_{j_1}+...+\sigma_{j_q}) t\delta},\quad 0<t<\infty,
\end{split}
\end{equation}
where $j_1,...,j_q$ are those indices with $H_j=I-H$, $d=\sum_{l\in \{j_1,...,j_q\}} d_l$.
\end{lemma}
\begin{proof} We pick an index $l$ with $H_l=H$. Then H\"older's inequality gives
\begin{equation} \begin{split}
\Bigl\|&A_0H_0e^{-\sigma_0 t\dirac^2} A_1 H_1e^{-\sigma_1 t \dirac^2}\ldots A_k H_k e^{-\sigma_k t \dirac^2}\Bigr\|_1\\
&\le \bigl\| A_l H_l e^{-\sigma_l t \dirac^2}\bigr\|_1 \quad \prod_{j\not=l} \bigl\| A_j H_j e^{-\sigma_j t \dirac^2} \bigr\|_\infty.
\end{split}
\end{equation}
The individual factors are estimated as follows: if $H_j=H$ then
\begin{equation}
\| A_j H e^{-\sigma_j t \dirac^2} \|_p\le \| A_j H\|_p,\quad \text{for } p\in\{1,\infty\}.
\end{equation}
If $H_j=I-H$ then by the Spectral Theorem and Proposition \ref{Prop:PropbSob}
we have for $0<\delta<\inf\spec_\ess \dirac^2$
\[
\| A_j (I-H) e^{-\sigma_j t \dirac^2} \|_\infty \le C(\delta,A_j) (\sigma_j t)^{-d_j/2} e^{-\sigma_jt \delta}, \quad 0<t<\infty.\qedhere
\]
\end{proof}
The next Lemma is extracted from the proof of \cite[Prop. 2]{ConMos:TCC}.
\begin{lemma}\label{l:H2} Let $f:\R_+^q\to \C$ be a (continuous) rapidly decreasing function
of $q\le n$ variables. Then
\begin{equation}
\begin{split}
\lim_{t\to\infty} t^{2q}&\int_{\Delta_n} f(t^2\sigma_1,...,t^2\sigma_q)d\sigma\\
&=\frac{1}{(n-q)!} \int_{\R_+^q} f(u) du
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
Changing variables $u_j=t^2\sigma_j, j=1,...,q; u_j=\sigma_j, j=q+1,...,n$ we find
\begin{equation}
\begin{split}
t^{2q}\int_{\Delta_n}& f(t^2\sigma_1,...,t^2\sigma_j)d\sigma\\
&= \int_{\{t^{-2}(u_1+...+u_q)+u_{q+1}+...+u_n)\le 1\}}
f(u_1,...,u_q)du\\
&= \int_{t^2\Delta_q } f(u_1,...,u_q) \int_{\bigl(1-t^{-2}(u_1+...+u_q)\bigr)\Delta_{n-q}}du\\
&= \frac{1}{(n-q)!}\int_{t^2\Delta_q}\bigl(1-t^{-2}(u_1+...+u_q)\bigr)^{n-q} f(u) du.
\end{split}
\end{equation}
By assumption $f$ is rapidly decreasing, hence we may apply the Dominated Convergence Theorem
to reach the conclusion.
\end{proof}
\subsection{Estimating the transgressed \textup{b}-Chern character}
\begin{proposition}\label{p:estimate-tChern-infty} For $k\ge 1$ and
$a_0,...,a_k\in \bcC(M)$ we have
\begin{equation}
\bslch^k(t\dirac,\dirac)(a_0,...,a_k)=\begin{cases}O(t^{-2}),& k \text{ even },\\
O(t^{-3}), & k \text{ odd },
\end{cases} t\to\infty.
\end{equation}
\end{proposition}
\newcommand{\sqrt{t}}{\sqrt{t}}
\begin{proof} $\slch^k(t\dirac,\dirac)(a_0,...,a_k)$ is a sum of terms
of the form
\begin{equation}
T=\blangle a_0, [t \dirac,a_1],...,[t \dirac,a_{i-1}],\dirac,[t \dirac,a_i],...,[t \dirac,a_k] \rangle_{t \dirac}.
\end{equation}
Writing $A_0=a_0$, $A_j=[\dirac,a_j], j=1,...,i-1,$ $A_j=[\dirac,a_{j-1}], j=i+1,...,k+1$,
$A_i:=\dirac$ we find
\begin{equation}
T=t^k\sum_{H_j\in\{H,I-H\}} \blangle A_0 H_0,...,A_{k+1}H_{k+1}\rangle_{t \dirac},
\end{equation}
where the sum runs over all sequences $H_0,...,H_{k+1}$ with $H_j\in\{H,I-H\}$.
Since $H[\dirac,a_j]H=0, H\dirac=\dirac H=0$ (note $A_i=\dirac$ !) only terms containing no more than $[k/2]+1$ copies of $H$
can give a non-zero contribution.
Consider such a nonzero summand with at least one index $j$ with $H_j=H$ and denote by $q$ the number
of indices $l$ with $H_l=I-H$. Then $q\ge [\frac{k+1}{2}]+1$ and we infer from Lemmas \plref{l:H1},\plref{l:H2}
\begin{equation}
t^k \;\blangle A_0H_0,...,A_{k+1}H_{k+1} \rangle_{t\dirac}
=O(t^{k-2q})=\begin{cases}O(t^{-2}),& k \text{ even, }\\
O(t^{-3}), &k \text{ odd }.
\end{cases}
\end{equation}
We infer from Theorem \ref{t:main-b-estimate} that
the remaining summand with $H_j=I-H$ for all $j$ decays exponentially as $t\to\infty$ and we are done.
\end{proof}
\subsection{The limit as $t\to\infty$ of the \textup{b}-Chern character}
As in \cite{ConMos:TCC} we put
\begin{align}
\varrho_H(A)&:=HAH,\\
\intertext{and}
\go_H(A,B)&:= \varrho_H(AB)-\varrho_H(A)\varrho_H(B).
\end{align}
\begin{proposition}\label{p:ML20090928} Let $a_0,...,a_k\in\bcC(M)$. If $k$ is odd then
\begin{equation}
\lim_{t\to\infty}\bCh^k(t \dirac)(a_0,...,a_k)=0.
\end{equation}
If $k=2q$ is even then
\begin{equation}
\begin{split}
\lim_{t\to\infty}&\bCh^k(t \dirac)(a_0,...,a_{2q})\\
&= \frac{(-1)^q }{q!} \Str\bigl(\varrho_H(a_0)\go_H(a_1,a_2)\dots\go_H(a_{2q-1},a_{2q})\bigr)\\
&=: \kappa^{2q}(\dirac)(a_0,\dots,a_{2q}).
\end{split}
\end{equation}
\end{proposition}
Of course, since $H$ is of finite rank $\varrho_H(a_j)$ and $\go_H(a_j,a_{j+1})$ are of trace class.
\begin{proof}
As in the previous proof we abbreviate $A_0=a_0, A_j=[\dirac,a_j], j\ge 1$ and decompose
\begin{equation}
\blangle A_0,A_1,...,A_k \rangle_{t \dirac}=t^{k}\sum_{H_j\in\{H,I-H\}}
\blangle A_0 H_0,...,A_{k}H_{k} \rangle_{t \dirac}.
\end{equation}
Since $H[\dirac,a_j]H=0$ only terms containing no more than $[k/2]+1$ copies of $H$ can give a nonzero
contribution.
The term containing no copy of $H$ decreases exponentially in view of Theorem
\ref{t:main-b-estimate}.
Consider a term containing $q$ copies of $I-H$.
If the number $k+1-q$ of copies of $H$ is at least one but less than $k/2+1$, which is always the case
if $k$ is odd, then $q>k/2$ and hence in view of Lemmas \plref{l:H1}, \plref{l:H2}
\begin{equation}
t^k \blangle A_0 H_0,...,A_{k}H_{k} \rangle_{t \dirac}=O(t^{k-2q})=O(t^{-1}), \quad t\to \infty.
\end{equation}
If $n=2q$ is even there is exactly one term containing $k/2+1$ copies of $H$, namely
\begin{equation}
\begin{split}
t^{2q}\;\blangle &A_0 H,A_1(I-H),...,A_{2q}H \rangle_{t \dirac}\\
&=t^{2q}\int_{\Delta_{2q}}
\Tr\bigl(\gamma a_0 H e^{-\sigma_0 t^2 \dirac^2}[\dirac,a_1](I-H)\cdot...\cdot e^{-\sigma_{2q}t^2 \dirac^2}H\bigr)d\sigma.
\end{split}
\end{equation}
The integrand depends only on the $q=k/2$ variables $\sigma_1,\sigma_3,...,\sigma_{2q-1}$ and
so we infer from Lemma \plref{l:H2} that the limit as $t\to\infty$ equals
\begin{equation}
\frac{1}{q!}\int_{\R_+^q}\Tr\bigl(\gamma a_0H[\dirac,a_1]e^{-u_1 \dirac^2}(I-H)[\dirac,a_2]H...e^{-u_{2q-1}\dirac^2}(I-H)[\dirac,a_{2q}]H\bigr) du.
\end{equation}
As in \cite[2.2]{ConMos:TCC} one shows that this equals
\[
\frac{(-1)^q }{q!} \Str\bigl(\varrho_H(a_0)\go_H(a_1,a_2)...\go_H(a_{2q-1},a_{2q})\bigr).\qedhere
\]
\end{proof}
\chapter{The Main Results}
\label{chap:Main}
We are now in a position to establish the main results of this paper.
After discussing in Section \ref{s: heat expansion} asymptotic
expansions for the \textup{b}-analogues of the Jaffe-Lesniewski-Osterwalder components,
we construct in Section \ref{s:retracted-relative-cocycle}
the retracted relative cocycle representing the Connes--Chern character
in relative cyclic cohomology and compute its small and large scale limits.
Section \ref{s: geom pairing} derives the ensuing pairing formula
with the $K$-theory, and discusses the geometric consequences.
The final remark (Section \ref{s:conclude}) offers an explanation
for the restrictive eta-pairing which appears in the work of Getzler and Wu.
\section{Asymptotic \textup{b}-heat expansions} \label{s: heat expansion}
\subsection{\textup{b}-Heat expansion}
Let $M$ be a complete riemannian manifold with cylindrical ends and let $\dirac$
be a Dirac operator on $M$ (\cf~Remark \plref{r:DiracCylinderFormulas},
Section \ref{App:bdefbmet}).
Let $Q\in\bdiff^q(M;W)$ be an auxiliary \textup{b}-differential operator of order $q$. It
is well-known (\emph{cf.~e.g.}~\cite{Gil:ITH}) that the Schwartz-kernel of the operator
$Qe^{-t\dirac^2}$ has a \emph{pointwise} asymptotic expansion
\begin{equation}\label{eq:ML20090122-4}
(Qe^{-t\dirac^2})(x,p;x,p)
\sim_{t\to 0+} \sum_{j=0}^\infty a_j(Q,\dirac)(x,p)\; t^{\frac{j-\dim M-q}{2}}.
\end{equation}
The problem is that in general neither $Qe^{-t\dirac^2}$ is of trace class nor
are the local heat invariants $a_j(Q,\dirac)$ integrable over the manifold.
Nevertheless we have the following theorem, which has been used implicitly by
Getzler \cite{Get:CHA}. However, we could not find a reference where the result is
cleanly stated and proved. Therefore, we provide here a proof for the convenience of the
reader.
\begin{theorem}\label{thm:bHeatExpansion}
Under the previously stated assumptions the \textup{b}-heat trace of $Qe^{-t\dirac^2}$ has
the following asymptotic expansion:\sind{bHeatexpansion@\textup{b}-heat expansion}
\begin{equation}
\bTr\bigl(Q e^{-t\dirac^2}\bigr)
\sim_{t\to 0+} \sum_{j=0}^\infty \int_{\tb M} \tr_{p}\bigl(a_j(Q,\dirac)(p)\bigr) d\vol(p) \; t^{\frac{j-\dim M-q}{2}}.
\end{equation}
\end{theorem}
The $\textup{b}$-integral $\int_{\tb M}$ was defined in
Section \plref{s: b-trace formula}, \emph{cf.}~Definition-Proposition
\plref{defprop15}.
\begin{proof}
We first write the operator $Q$ as a sum $Q=Q^{(0)}+Q^{(1)}$ of differential operators with
$Q^{(0)}\in\bcptdiff^q((-\infty,0)\times \pl M;W)$ and $Q^{(1)}$ a differential operator supported in the interior.
By standard elliptic theory (\cite{Gil:ITH}) $Q^{(1)}e^{-t\dirac^2}$ is trace class and
since the asymptotic expansion \eqref{eq:ML20090122-4} is uniform on compact subsets of $M$
the claim follows for $Q^{(1)}$ instead of $Q$.
So it remains to prove the claim for an operator $Q\in \bcptdiff^q((-\infty,0)\times\pl M;W)$; for convenience
we write from now on again $Q$ instead of $Q^{(0)}$.
Next we apply the comparison Theorem \plref{t:heat-resolvent-comparison} which allows us to assume
that $M=\R\times\partial M$ is the model cylinder, $\dirac=\sfc(dx)\frac{d}{dx}+D^\partial$, and $Q$ is supported
on $(-\infty,c)\times\partial M$ for some $c>0$.
Furthermore, we may assume that $Q$ is of the form \eqref{eq:dDiffOp-normalForm}.
Since the heat kernel\sind{heat kernel} of the model operator is explicitly known
(\cf~\eqref{eq:heat-kernel}) we have
\begin{equation}
\begin{split}
\Bigl( f(x,p) &P \partial_x^l e^{-t\dirac^2}\Bigr)(x,p;y,q)\\
&= \frac{1}{\sqrt{4\pi t}} \bigl(\partial_x^l e^{-(x-y)^2/4t}\bigr)
\bigl(P e^{-t A^2}\bigr)(p,q),\qquad \Abdy:=\Gammabdy \mathsf{D}_\pl.
\end{split}
\end{equation}
If $l$ is odd then by induction one easily shows that this kernel vanishes on the diagonal and
hence the {\btrace} $\bTr(Qe^{-t\dirac^2})$ as well as all local heat coefficients
vanish, proving the Theorem in this case. So let $l=2k$ be even. Then
using \eqref{eq:ML20090127-1} and
since on the diagonal $\partial_t^k e^{-t\Delta_\R}(x,x)=\partial_t^k (4\pi t)^{-1/2}=:c_k t^{-1/2-k}$,
we have
\begin{equation}\label{eq:ML20090123-2}
\begin{split}
\Bigl( f(x,p) &P \partial_x^l e^{-t\dirac^2}\Bigr)(x,p;x,p)\\
&= f(x,p)\bigl(P e^{-t A^2}\bigr)(p,p) c_k \; t^{-1/2-k}\\
&\sim_{t\to 0+} \sum_{j=0}^\infty f(x,p) a_j(P,A)(p) c_k \; t^{\frac{j-\dim M-q}{2}}.
\end{split}
\end{equation}
Comparing with \eqref{eq:ML20090122-4} we find for the heat coefficients $a_j(Q,D)$
\begin{equation}\label{eq:ML20090123-3}
a_j(Q,D)(x,p)= f(x,p) a_j(P,A)(p) c_k .
\end{equation}
Furthermore, we have using Theorem \eqref{t:bTraceAsTrace}
\begin{equation}\label{eq:ML20090123-1}
\bTr\bigl(Qe^{-t\dirac^2}\bigr) \\
=\int_{-\infty}^0\int_{\partial M} \tr_{x,p}\Bigl(x\partial_x f(x,p) \bigl(Pe^{-tA^2}\bigr)(p,p)\Bigr)
d\vol_{\partial M}(p) dx.
\end{equation}
$\bigl(Pe^{-tA^2}\bigr)(p,p)$ has an \emph{$x$-independent} asymptotic expansion as $t\to 0+$.
Since $x\partial_x f(x,p)=O(e^{(1-\delta)x}), x\to -\infty$, uniformly in $p$, we can plug
the asymptotic expansion \eqref{eq:ML20090123-2}
into \eqref{eq:ML20090123-1} and use \eqref{eq:ML20090123-3} to find
\begin{equation*}
\begin{split}
\bTr&\bigl(Qe^{-t\dirac^2}\bigr)\\
&\sim_{t\to 0+}\sum_{j=0}^\infty \int_{-\infty}^0 \int_{\partial M} \tr_{x,p}\Bigl(x\partial_x f(x,p) a_j(P,A)(p)\Bigr)
d\vol_{\partial M}(p) dx\; c_k\; t^{\frac{j-\dim M-q}{2}}\\
&\sim_{t\to 0+}\sum_{j=0}^\infty \int_{\tb (-\infty,0)\times \partial M} \tr_{x,p}\bigl( a_j(Q,D)(x,p) \bigr) d\vol(x,p)
\; t^{\frac{j-\dim M-q}{2}}.
\end{split}
\end{equation*}
The claim is proved.
\end{proof}
\subsection{The \textup{b}-trace of the \textup{JLO} integrand}
\newcommand{\nabla_{\mathsf{D}}}{\nabla_{\mathsf{D}}}
To extend Theorem \plref{thm:bHeatExpansion} to expressions of the form
$\bTr\Bigl(A_0 e^{-\sigma_0 t\dirac^2} A_1 e^{-\sigma_1 t\dirac^2} ... A_k e^{-\sigma_k t \dirac^2}\Bigr) $
we use a trick which was already applied successfully in the proof of the local index formula in noncommutative
geometry \cite{ConMos:LIF}. Namely, we successively commute $A_j e^{-\sigma_j t \dirac^2}$ and control
the remainder. We will need the estimates proved in Sections \plref{s:estimates-JLO-cylindrical}
and \plref{s:EstbTraces}.
We first need to introduce some notation (\cf~\cite[Lemma 4.2]{Les:NRP}).
For a \textup{b}-differential operator $B\in\bdiff(M;W)$ we put inductively
\begin{equation}\label{eq:IterCommutator}
\nabla_{\mathsf{D}}^0B:=B,\qquad \nabla_{\mathsf{D}}^{j+1}B:=[\dirac^2,\nabla_D^jB].
\end{equation}
Note that since $\dirac^2$ has scalar leading symbol we have $\ord(\nabla_D^jB)\le j+\ord B$.
The following formula can easily be shown by induction.
\begin{equation}\label{eq:HeatOpCommutator}
\begin{split}
e^{-t\dirac^2}B& = \sum_{j=0}^{n-1} \frac{(-t)^j}{j!} \bigl(\nabla_{\mathsf{D}}^j B \bigr) e^{-t\dirac^2}+\\
&\qquad +\frac{(-t)^n}{(n-1)!}\int_0^1 (1-s)^{n-1} e^{-st\dirac^2}\bigl(\nabla_{\mathsf{D}}^n B\bigr)
e^{-(1-s)t\dirac^2}ds.
\end{split}
\end{equation}
The identity \eqref{eq:HeatOpCommutator} easily allows to prove the following statement about
\semph{local heat invariants}, \cf~\cite{Wid:STC}, \cite{ConMos:CCN}, \cite{BloFox:APO}:
\begin{proposition}\label{p:JLOCommutatorAsymptotic}
Let $A_0,...,A_k\in\bdiff(M;W)$ of order $d_0,...,d_k; d:=\sum_{j=0}^k d_j$. Then the Schwartz kernel
of $A_0 e^{-\sigma_0 t\dirac^2}A_1 e^{-\sigma_1 t\dirac^2}... A_k e^{-\sigma_k t \dirac^2}$
has a pointwise asymptotic expansion
\begin{equation}\label{eq:JLOCommutatorAsymptotic}
\begin{split}
\Bigl(A_0 &e^{-\sigma_0 t\dirac^2} A_1 e^{-\sigma_1 t\dirac^2} ... A_k e^{-\sigma_k t \dirac^2}\Bigr)(p,p) \\
&= \sum_{\ga\in\Z_+^k, |\ga|\le n} \frac{(-t)^{|\ga|}}{\ga!} \sigma_0^{\ga_1}(\sigma_0+\sigma_1)^{\ga_2}...(\sigma_0+...+\sigma_{k-1})^{\ga_k}
\cdot\\
&\qquad\cdot \bigl( A_0 \nabla_{\mathsf{D}}^{\ga_1}A_1 ... \nabla_{\mathsf{D}}^{\ga_k} A_k e^{-t\dirac^2}\bigr)(p,p)+O_p(t^{(n+1-d-\dim M)/2}),\\
&=: \sum_{j=0}^{n} a_j(A_0,...,A_k,\dirac)(p)\; t^{\frac{j-\dim M -d}{2}}+O_p(t^{(n+1-d-\dim M)/2}),
\end{split}
\end{equation}
where $d=\sum\limits_{j=0}^k d_j$.
The asymptotic expansion is locally uniformly in $p$. Furthermore, it is uniform for $\sigma\in \Delta_k$.
\end{proposition}
Again we are facing the problem explained before Theorem \ref{thm:bHeatExpansion}. Still we will be able
to show that one obtains a correct formula by taking the {\btrace} on the left and
partie finie integrals on the right of \eqref{eq:JLOCommutatorAsymptotic}:
\sind{partie finie}
\begin{theorem}\label{t:JLObCommutatorAsymptotic}
Under the assumptions of the previous Proposition \plref{p:JLOCommutatorAsymptotic} we have an asymptotic
expansion
\begin{equation}\label{eq:JLObCommutatorAsymptotic}
\begin{split}
\bTr\Bigl(A_0 &e^{-\sigma_0 t\dirac^2} A_1 e^{-\sigma_1 t\dirac^2} ... A_k e^{-\sigma_k t \dirac^2}\Bigr) \\
&= \sum_{\ga\in\Z_+^k, |\ga|\le n} \frac{(-t)^{|\ga|}}{\ga!} \sigma_0^{\ga_1}(\sigma_0+\sigma_1)^{\ga_2}...(\sigma_0+...+\sigma_{k-1})^{\ga_k}
\cdot\\
&\qquad\cdot \bTr\bigl( A_0 \nabla_{\mathsf{D}}^{\ga_1}A_1 ... \nabla_{\mathsf{D}}^{\ga_k} A_k e^{-t\dirac^2}\bigr)+\\
&\qquad+ O\Bigl(\bigl(\prod_{j=1}^k \sigma_j^{-d_j/2}\bigr)t^{(n+1-d-\dim M)/2}\Bigr),\\
&= \sum_{j=0}^{n} \int_{\tb M} a_j(A_0,...,A_k,\dirac)d\vol \; t^{\frac{j-\dim M -d}{2}}+\\
&\qquad +O\Bigl(\bigl(\prod_{j=1}^k \sigma_j^{-d_j/2}\bigr)t^{(n+1-d-\dim M)/2}\Bigr).
\end{split}
\end{equation}
\end{theorem}
\begin{remark}
The O-constant in \eqref{eq:JLObCommutatorAsymptotic} is independent of $\sigma\in\Delta_k$. However, the
factor $\bigl(\prod_{j=1}^k \sigma_j^{-d_j/2}\bigr)$ inside the $O()$ causes some trouble because it is
integrable over the standard simplex $\Delta_k$ only if $d_1,...,d_k\le 1$.
We do not claim that this factor is necessarily there. It might be an artifact of the inefficiency of
our method. Cf. also Remarks \plref{rem:EstimateOptimality}, \plref{rem:EstimateOptimality-a}.
\end{remark}
\begin{proof}
The strategy of proof we present here can also be used to prove Proposition \plref{p:JLOCommutatorAsymptotic}.
Again by the comparison Theorem \plref{t:JLO-comparison} we may
assume that $\dirac$ is
the model Dirac operator and $A_0,...,A_k\in \bcptdiff((-\infty,0)\times\partial M;W)$.
Using Proposition \plref{t:bTraceAsTrace} we have
\begin{equation}\label{eq:ML20090126-1}
\begin{split}
\bTr\Bigl( A_0 &e^{-\sigma_0t\dirac^2}A_1... A_k e^{-\sigma_k t\dirac^2}\Bigr)\\
&= -\bTr\Bigl(x\bigl[\frac{d}{dx}, A_0 e^{-\sigma_0t\dirac^2}A_1... A_k e^{-\sigma_k t\dirac^2}\bigr]\Bigr)\\
&= -\sum_{j=0}^k \Tr\Bigl(x A_0 e^{-\sigma_0t\dirac^2}A_1...[\frac{d}{dx},A_j]... A_k e^{-\sigma_k t\dirac^2}\Bigr).
\end{split}
\end{equation}
$[\frac{d}{dx},A_j]$ is again in $\bcptdiff((-\infty,0)\times\partial M;W)$ and its indicial family vanishes.
Hence by Proposition \plref{p:multiple-heat-estimate-b}
all summands on the right are of trace class. Cf. also the comment at the end of the proof of
Theorem \plref{t:main-b-estimate}.
It therefore suffices to prove the claim for the summands on the right
of \eqref{eq:ML20090126-1}, i.e. for $\Tr\Bigl(x A_0 e^{-\sigma_0t\dirac^2}A_1... A_k e^{-\sigma_k t\dirac^2}\Bigr)$
where at least one of the $A_j$ has vanishing indicial family.
Applying \eqref{eq:HeatOpCommutator} to $A_1$ we get
\begin{align}\label{eq:HeatOpCommutator-1}
e^{-\sigma_0 t\dirac^2}A_1& = \sum_{j=0}^{n-1} \frac{(-\sigma_0 t)^j}{j!} \bigl(\nabla_{\mathsf{D}}^j A_1 \bigr) e^{-\sigma_0t\dirac^2}+\\
&\qquad +\frac{(-\sigma_0 t)^n}{(n-1)!}\int_0^1 (1-s)^{n-1} e^{-s\sigma_0 t\dirac^2}\bigl(\nabla_{\mathsf{D}}^n A_1\bigr)
e^{-(1-s)\sigma_0 t\dirac^2}ds.\nonumber
\end{align}
Therefore we need to estimate the expression
\begin{equation}\label{eq:ML20090126-2}
x(\sigma_0 t)^n (1-s)^{n-1} A_0 e^{-s\sigma_0 t\dirac^2}\bigl(\nabla_{\mathsf{D}}^n A_1\bigr)
e^{-(1-s)\sigma_0 t\dirac^2} e^{-\sigma_1t\dirac^2}...A_k e^{-\sigma_kt\dirac^2}
\end{equation}
in the trace norm.
If the index $l$ for which the indicial family of $A_l$ vanishes is $0$ we write $A_0$ as $e^{x}\tilde A_0$
with $\tilde A_0\in\bcptdiff((-\infty,0)\times \partial M;W)$ and move $x e^x$ under the trace to the right.
This assures that Proposition \plref{p:multiple-heat-estimate-b} applies to
$\bigl(\nabla_{\mathsf{D}}^n A_1\bigr)e^{-(1-s)\sigma_0 t\dirac^2} e^{-\sigma_1t\dirac^2}...A_k e^{-\sigma_kt\dirac^2}x e^x$.
If $l\ge 1$ we just move $x$ under the trace to the right. After all w.l.o.g. we may assume that $l\ge 1$.
Next we choose an integer $\gb$ such that $A_0(\dirac^2+I)^{-\gb}$ has order $\in \{ 0,1\}$.
Then H\"older's inequality yields
\begin{multline}
(\sigma_0 t)^n (1-s)^{n-1} \Bigl\| A_0 (I+\dirac^2)^{-\gb} e^{-s\sigma_0 t\dirac^2}(I+\dirac^2)^\gb\bigl(\nabla_{\mathsf{D}}^n A_1\bigr)\\
e^{-(1-s)\sigma_0 t\dirac^2} e^{-\sigma_1t\dirac^2}...A_k e^{-\sigma_kt\dirac^2}x\Bigr\|_1\\
\le (\sigma_0 t)^n (1-s)^{n-1} (s\sigma_0t)^{-d_0/2+\gb}\Bigl\|(I+\dirac^2)^\gb\bigl(\nabla_{\mathsf{D}}^n A_1\bigr)\\
e^{-(1-s)\sigma_0 t\dirac^2} e^{-\sigma_1t\dirac^2}...A_k e^{-\sigma_kt\dirac^2}x\Bigr\|_1
\end{multline}
To the remaining trace we apply Proposition \plref{p:multiple-heat-estimate-b} and obtain
\begin{equation}
\begin{split}
...&\le \sigma_0^n t^n (1-s)^{n-1} (s\sigma_0t)^{-d_0/2+\gb} C(t_0,\eps) \bigl((1-s)\sigma_0\bigr)^{-\gb-n/2-d_1/2}\\
&\qquad \bigl(\prod_{j=2}^k \sigma_j^{-d_j/2}\bigr) t^{-d/2+d_0/2-\dim M/2-\eps -n/2-\gb}\\
&\le C(t_0,\eps) s^{-1/2}(1-s)^{n/2-1-\gb-d_1/2} \sigma_0^{\frac{n-d_0-d_1}{2}} \bigl(\prod_{j=2}^k \sigma_j^{-d_j/2}\bigr)
t^{\frac{n-d-\dim M}{2}-\eps}.
\end{split}
\end{equation}
If we choose $n$ large enough the right hand side is integrable in $s$ and we obtain the desired estimate.
In the next step we apply \eqref{eq:HeatOpCommutator} to $e^{-(\sigma_0+\sigma_1)t\dirac^2}$ and $A_2$.
Continuing this way we reach the conclusion after $k$ steps.
\end{proof}
\section{The Connes--Chern character of the relative Dirac class}
\label{s:retracted-relative-cocycle}
\subsection{Retracted Connes--Chern character} \label{ss:retracted-CC}
In this section we assume that $\dirac$ is a Dirac operator on a \textup{b}-Clifford
bundle $W\rightarrow M$ over the \textup{b}-manifold $M$ and
$\dirac_t = t \dirac$ is a family of Dirac type operators.
We now have all tools to apply the method of \cite{ConMos:TCC} to convert
the entire relative {\CoChch}, which was constructed using the {\btrace},
into a finitely supported cocycle.
By integrating Eq.~\eqref{Eq:transgress}, one obtains for
$0 < \varepsilon <t$
\begin{equation}
\begin{split}
\bCh^k & \, (\varepsilon \dirac) - \bCh^k (t \dirac)
= \, b \int_\varepsilon^t \bslch^{k-1} (s \dirac, \dirac) ds \\
& + B \int_\varepsilon^t \bslch^{k+1} (s \dirac, \dirac) ds
+ \int_\varepsilon^t \slch^k (s \mathsf{D}_\pl, \mathsf{D}_\pl)\circ i^* ds .
\end{split}
\end{equation}
$\Ch^\bullet(\mathsf{D}_\pl)$ satisfies the cocycle and transgression formul\ae\
Eq.~\eqref{eq:cocycle}, \eqref{eq:transgression}. Integrating
these we obtain
\begin{equation}\label{eq:ML200911262}
\begin{split}
\Ch^k & \, (\varepsilon \mathsf{D}_\pl) - \Ch^k (t \mathsf{D}_\pl)
= \, b \int_\varepsilon^t \slch^{k-1} (s \mathsf{D}_\pl, \mathsf{D}_\pl) ds \\
& + B \int_\varepsilon^t \slch^{k+1} (s \mathsf{D}_\pl, \mathsf{D}_\pl) ds.
\end{split}
\end{equation}
By Proposition \ref{p:ShortTimeEst} (1),
the limit $\varepsilon \searrow 0$ exists for $k > \dim M$, and
\begin{equation}
\label{Eq:limit}
\begin{split}
&\lim_{\varepsilon \searrow 0} \bCh^k(\varepsilon \dirac)= 0,\\
&\lim_{\varepsilon \searrow 0} \Ch^{k-1}(\varepsilon \mathsf{D}_\pl)= 0,
\end{split}
\qquad \text{for all $k > \dim M $}.
\end{equation}
The second limit statement follows either from an obvious adaption of our calculations to the ordinary
trace or from \cite{ConMos:TCC}.
Hence one gets for $k > \dim M$
\begin{equation}
\begin{split}
\label{Eq:LittlebCapitalBChern}
- \bCh^k (t \dirac) &=
\, b \bTslch_t^{k-1} (\dirac)+
B \bTslch_t^{k+1} (\dirac)
+ \Tslch_t^k (\mathsf{D}_\pl)\circ i^*,\\
- \Ch^{k-1} (t \mathsf{D}_\pl) &= \, b \Tslch_t^{k-2} (\mathsf{D}_\pl)+
B \Tslch_t^{k} (\mathsf{D}_\pl),\\
\end{split}
\end{equation}
where
\begin{equation}\label{eq:ML200909263}
\begin{split}
\bTslch_t^k (\dirac) &:= \int_0^t \slch ^k (s\dirac, \dirac) \, ds,\\
\Tslch_t^{k-1} (\mathsf{D}_\pl) &:= \int_0^t \slch^{k-1} (s\mathsf{D}_\pl, \mathsf{D}_\pl) \, ds.
\end{split}
\end{equation}
The above integrals exist in view of Proposition \plref{p:ShortTimeEst} (2) even for $k\ge \dim M$.
From Eq.~\eqref{eq:ML200911262} and Theorem \ref{P:GETZLER}
we obtain for $k\ge \dim M$:
\begin{equation}
\begin{split}
b\Bigl( &\bCh^k(t\dirac)+B\int_{\eps}^t \bslch^{k+1}(s\dirac,\dirac) ds\Bigr)\\
&=-B \bCh^{k+2}(\eps \dirac) +\Ch^{k+1}(\eps\mathsf{D}_\pl)\circ i^*-b\int_{\eps}^t \slch^k(s\mathsf{D}_\pl,\mathsf{D}_\pl)ds\\
&\longrightarrow -b \Tslch^k_t(\mathsf{D}_\pl)\circ i^*, \quad \eps\to 0+.
\end{split}
\end{equation}
Thus
\begin{equation}
\label{Eq:RellbcB}
\begin{split}
b\Big( \bCh^k (t \dirac) + B\bTslch_t^{k+1} (\dirac) \Big) &= - b \Tslch_t^k (\mathsf{D}_\pl )\circ i^*,\\
b\Big( \Ch^{k-1} (t \mathsf{D}_\pl) + B\bTslch_t^{k} (\mathsf{D}_\pl) \Big) &=0,
\end{split}\quad k\ge \dim M.
\end{equation}
Following \textnm{Connes-Moscovici} \cite{ConMos:TCC}, we define for $k\ge \dim M$
the Chern characters
$\bch^k_t (\dirac)$, $\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k(\dirac)$ and $\ch^{k-1}_t (\mathsf{D}_\pl)$ by
\begin{align}
\bch^k_t (\dirac) & = \sum_{j\geq 0} \bCh^{k-2j} (t \dirac) +
B \bTslch^{k+1}_t (\dirac),\\
\ch^{k-1}_t (\mathsf{D}_\pl) & =
\sum_{j\geq 0} \Ch^{k-2j-1} (t \mathsf{D}_\pl) +
B \Tslch^{k}_t (\mathsf{D}_\pl), \\
\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t(\dirac) \, &=\, \bch^k_t(\dirac) +\Tslch^k_t(\mathsf{D}_\pl)\circ i^*.
\end{align}
Let us now compute $(b+B) \bch^\bullet_t (\dirac)$.
Using Eq.~\eqref{Eq:cocyclecond} and Eq.~\eqref{Eq:RellbcB} above, we write
\begin{equation}
\label{Eq:RelativityCheck}
\begin{split}
b\bch^k_t & (\dirac) + B\bch^k_t (\dirac) = \\
= \, &
\sum_{j \geq 1} \Big( b\bCh^{k-2j} (t\dirac) +
B\bCh^{k-2j + 2} (t \dirac) \Big) \\
& + b \Big( \bCh^k (t\dirac ) + B \bTslch_t^{k+1} (\dirac) \Big) \\
= \, &
\sum_{j \geq 1} \Ch^{k-2j +1} (t \mathsf{D}_\pl)\circ i^* - b \Tslch^k_t (\mathsf{D}_\pl)\circ i^* \\
= \, &
\sum_{j \geq 0} \Ch^{k-2j -1} (t \mathsf{D}_\pl)\circ i^* - b \Tslch^k_t (\mathsf{D}_\pl)\circ i^* \\
= \, &
\ch^{k-1}_t (\mathsf{D}_\pl)\circ i^* - B \Tslch^{k}_t (\mathsf{D}_\pl)\circ i^*
- b \Tslch^k_t (\mathsf{D}_\pl) \circ i^*
\\
= \, & \ch^{k+1}_t (\mathsf{D}_\pl)\circ i^*,
\end{split}
\end{equation}
where the last equality follows from the second line of Eq.~\eqref{Eq:LittlebCapitalBChern}.
In conclusion
\begin{equation}
\begin{split}
(b+B) \bch_t^k(\dirac)&= \ch_t^{k+1}(\mathsf{D}_\pl)\circ i^*\\
(b+B) \,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}_t^k(\dirac)&= \ch_t^{k-1}(\mathsf{D}_\pl)\circ i^*.
\end{split}
\end{equation}
Denoting by $\widetilde b,\widetilde B$ the relative Hochschild resp. Connes'
coboundaries, \emph{cf.}~Eq.~\eqref{Eq:DefCoBdrRelMixDer}, we thus infer
\sind{Hochschild (co)homology}
\begin{equation}
\begin{split}
(\widetilde b+\widetilde B)\bigl(\bch_t^k(\dirac),\ch_t^{k+1}(\mathsf{D}_\pl)\bigr)&=0,\\
(\widetilde b+\widetilde B)\bigl(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}_t^k(\dirac),\ch_t^{k-1}(\mathsf{D}_\pl)\bigr)&=0,
\end{split}
\end{equation}
\emph{i.e.} the pairs $(\bch_t^k(\dirac),\ch_t^{k+1}(\mathsf{D}_\pl))$ and $(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}_t^k(\dirac),\ch_t^{k-1}(\mathsf{D}_\pl))$ are
relative cocycles in the direct sum of total complexes
\begin{equation}\label{eq:ML200909264}
\begin{split}
\tot^k & \, \mathcal B C^{\bullet,\bullet}
(\mathcal C^\infty (M), \mathcal C^\infty (\partial M) ) := \\
& \, :=
\tot^k \, \mathcal B C^{\bullet,\bullet}
(\mathcal C^\infty (M)) \oplus
\tot^{k+1} \mathcal B C^{\bullet,\bullet}
(\mathcal C^\infty (\partial M)).
\end{split}
\end{equation}
By Eq.~\eqref{Eq:RelativityCheck} we have
\begin{equation}
\begin{split}
\Bigl(&\bch_t^k(\dirac)-\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}_t^k(\dirac),\ch_t^{k+1}(\mathsf{D}_\pl)-\ch_t^{k-1}(\mathsf{D}_\pl)\Bigr)\\
&= \Bigl(-\Tslch_t^k(\mathsf{D}_\pl)\circ i^*,-(b+B)\Tslch_t^k(\mathsf{D}_\pl)\Bigr)\\
&=\bigl(\widetilde b+\widetilde B\bigr)\Bigl(0,\Tslch_t^k(\mathsf{D}_\pl)\Bigr),
\end{split}
\end{equation}
hence the two pairs differ only by a coboundary.
Next let us compute
$\big(\bch^{k+2}_t (\dirac),\ch^{k+3}_t (\mathsf{D}_\pl) \big)
- S \big(\bch^k_t (\dirac),\ch^{k+1}_t (\mathsf{D}_\pl) \big)$
in the above relative cochain complex. Using Eq.~\eqref{Eq:LittlebCapitalBChern}
one checks immediately that
\begin{displaymath}
\begin{split}
\bch^{k+2}_t (\dirac) - \bch^k_t (\dirac) = \, &
\bCh^{k+2} (t \dirac) + B \bTslch^{k+3}_t (\dirac)
- B \Tslch^{k+1}_t (\dirac)\\
= \, &
- (b +B ) \Tslch^{k+1}_t (\dirac ) - \Tslch^{k+2}_t (\mathsf{D}_\pl)\circ i^*.
\end{split}
\end{displaymath}
From the second line of Eq.~\eqref{Eq:LittlebCapitalBChern} (or from \cite[Sec.~2.1]{ConMos:TCC})
\begin{displaymath}
\ch^{k+3}_t (\mathsf{D}_\pl) - \ch^{k+1}_t (\mathsf{D}_\pl)
= - (b+B) \Tslch^{k+2}_t (\mathsf{D}_\pl ),
\end{displaymath}
one thus gets
\begin{equation}
\begin{split}
\big(\bch^{k+2}_t &(\dirac),\ch^{k+3}_t (\mathsf{D}_\pl) \big)
- S \big(\bch^k_t (\dirac),\ch^{k+1}_t (\mathsf{D}_\pl) \big)\\
&= (\widetilde b + \widetilde B)
\big(-\bTslch^{k+1}_t (\dirac ),\Tslch^{k+2}_t (\mathsf{D}_\pl ) \big).
\end{split}
\end{equation}
Hence, the relative cocycles
$\big(\bch^{k+2}_t (\dirac),\ch^{k+3}_t (\mathsf{D}_\pl) \big)$
and $S\big(\bch^k_t (\dirac),\ch^{k+1}_t (\mathsf{D}_\pl) \big)$
are cohomologous. Similarly, one gets
\begin{displaymath}
\begin{split}
\bch^k_t & (\dirac) - \ch^k_\tau (\dirac) = \\
= \, & \sum_{j\geq 0}
\big( \bCh^{k-2j} (t \dirac) - \bCh^{k-2j} (\tau \dirac) \big)
+ B \int_\tau^t \bslch^{k+1} (s\dirac,\dirac)\, ds \\
= \, & - (b+B) \sum_{j\geq 0} \int_\tau^t \bslch^{k-2j-1}
(s\dirac,\dirac)\, ds - \sum_{j\geq 0}
\int_\tau^t \slch^{k-2j} (s\mathsf{D}_\pl,\mathsf{D}_\pl)\, ds ,
\end{split}
\end{displaymath}
resp.
\begin{displaymath}
\begin{split}
\ch^{k+1}_t & (\mathsf{D}_\pl) - \ch^{k+1}_\tau (\mathsf{D}_\pl) =
- (b+B) \sum_{j\geq 0} \int_\tau^t \slch^{k-2j}
(s\mathsf{D}_\pl,\mathsf{D}_\pl)\, ds,
\end{split}
\end{displaymath}
hence
$\big(\bch^k_t (\dirac),\ch^{k+1}_t (\mathsf{D}_\pl) \big)$
and $\big(\bch^k_\tau (\dirac),\ch^{k+1}_\tau (\mathsf{D}_\pl) \big)$
are cohomologous in the total relative complex as well.
Thus, we have proved (1)-(3) of the following result.
\begin{theorem} \label{t: CC-character}
\begin{enumerate}
\item The pairs of retracted cochains
$\big(\bch^k_t (\dirac),\ch^{k+1}_t (\mathsf{D}_\pl) \big)$,
$\big(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t (\dirac),\ch^{k-1}_t (\mathsf{D}_\pl) \big)$, $t>0$,
$t>0$,
$k\ge m=\dim M, k-m\in 2\Z$ are cocycles in the relative total complex
$\tot^\bullet \, \mathcal B C^{\bullet,\bullet}
(\mathcal C^\infty (M), \mathcal C^\infty (\partial M) )$.
\item They represent the same class
in $HC^n(\mathcal C^\infty (M), \mathcal C^\infty (\partial M) )$ which
is independent of $t>0$.
\item They represent the same class
in $HP^\bullet(\mathcal C^\infty (M), \mathcal C^\infty (\partial M) )$
which is independent of $k$.
\item Denote by $\bomega_{\dirac}, \go_{\mathsf{D}_\pl}$ the local index forms of
$\dirac$ resp.~$\mathsf{D}_\pl$ \cite[Thm.~4.1]{BerGetVer:HKD}, \emph{cf.} Eq.~\eqref{eq: 0lim cocycle},
\eqref{eq: A-currents} and see Eq.~\eqref{eq:ML200909265} below.
Then one has a pointwise limit
\[
\lim_{t\to 0+}\big(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t (\dirac), \ch^{k-1}_t (\mathsf{D}_\pl) \big)
= \Big(\int_{\tb M} \bomega_{\dirac}\wedge \bullet,
\int_{\pl M} \omega_{\mathsf{D}_\pl} \wedge \bullet\Big).
\]
Moreover,
$\big(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t (\dirac),\ch^{k-1}_t (\mathsf{D}_\pl) \big)$
represents the {\CoChch} of
$[\dirac] \in KK_m(C_0 (M) ; \C)= K_m(M,\partial M)$.
\end{enumerate}
\end{theorem}
The pointwise limit will be explained in the proof below. Up to normalization
constants $\bomega_{\dirac}$ is the $\hat A$ form $\hat A(\bnabla^2_g)$ and
$\omega_{\mathsf{D}_\pl}$ is the $\hat A$ form $\hat A(\nabla^2_{g_\pl})$ on the
boundary. Note also that $\iota^* \go_{\dirac}=\go_{\mathsf{D}_\pl}$ .
\comment{does anyone have the energy
to make $\go_{\dirac}$ more explicit, I guess no}
\begin{proof}
It remains to prove (4). So consider
\[a_0, a_1,\dots, a_j\in \bcC(M^\circ).\]
Using
Getzler's asymptotic calculus (\emph{cf.}~\cite{Get:POS}, \cite[\S 3]{ConMos:CCN}, and
\cite[Thm.~4.1]{BloFox:APO}) one shows that the \emph{local} heat invariants \sind{local heat
invariants}
of
\[ a_0 e^{-\sigma_0 t\dirac^2}[\dirac,a_1] e^{-\sigma_1 t\dirac^2}\ldots
[\dirac,a_j] e^{-\sigma_j t \dirac^2}\]
(\emph{cf.} Proposition \plref{p:JLOCommutatorAsymptotic}) satisfy
\begin{multline
t^j\; \int_{\Delta_j}\str_{q,W_p}\Bigl( a_0 e^{-\sigma_0 t\dirac^2}[\dirac,a_1]
e^{-\sigma_1 t\dirac^2}\ldots [\dirac,a_j] e^{-\sigma_j t \dirac^2}\Bigr)(p,p) \,
d\vol_{\bmet} (p) \\
= \frac{1}{j!}\big(\bomega_{\dirac} \wedge a_0 d a_1\dots\wedge d a_j\big)_{|p}
+ O(t^{1/2}),\quad t\to 0+.\label{eq:ML200909265}
\end{multline}
Here $\str_{q,W_p}$ denotes the fiber supertrace in $W_p$, $q$ indicates the Clifford
degree of $\dirac$, \emph{cf.}~Section \plref{s:qDirac},
the factor $\frac{1}{j!}$ is the volume of the simplex $\Delta_j$.
This statement holds \emph{locally} on any riemannian manifold for any choice
of a self-adjoint extension of a Dirac operator. So it holds for $\dirac$ and
accordingly for $\mathsf{D}_\pl$.
From Theorem \plref{t:JLObCommutatorAsymptotic} and its well-known analogue for
closed manifolds, \emph{cf.}~\cite[Sec.~4]{ConMos:TCC}, we thus infer
\begin{multline}\label{eq:ML200909261}
\lim_{t\to 0+} \bCh^j(t\dirac)(a_0,\dots,a_j)\\
=\frac{1}{j!}\int_{\tb M} \bomega_{\dirac} \wedge a_0 d a_1\dots\wedge d a_j,
\qquad a_0,\ldots, a_j\in\bcC(M^\circ),
\end{multline}
resp.
\begin{multline}\label{eq:ML200909262}
\lim_{t\to 0+} \Ch^{j-1}(t\mathsf{D}_\pl)(a_0,\dots,a_{j-1})\\
=\frac{1}{(j-1)!}\int_{\pl M} \omega_{\mathsf{D}_\pl}
\wedge a_0 d a_1\dots\wedge d a_{j-1},\qquad a_0,\ldots, a_{j-1}\in \cC^\infty(M).
\end{multline}
Furthermore, in view of \eqref{eq:ML200909263} we have for $k\geq \dim M -1$
\begin{equation}
\begin{split}
\lim_{t\to 0+} \bTslch^{k+1}_t(\dirac)(a_0,\dots,a_{k+1})&=0, \qquad
a_0,\ldots,a_{k+1}\in\bcC(M^\circ),\\
\lim_{t\to 0+} \Tslch^{k}_t(\mathsf{D}_\pl)(a_0,\dots,a_k)&=0, \qquad
a_0,\ldots,a_k\in\cC^\infty(M).
\end{split}
\end{equation}
\sind{de Rham!current}
To interpret these limit results we briefly recall the relation between
\deRham\ currents and relative cyclic cohomology classes over
$(\bcC(M),\cC^\infty(M))$, \emph{cf.}~also \cite[Sec.~2.2]{LesMosPfl:RTK}.
Given a \deRham\ current $C$ of degree $j$ then $C$ defines naturally
a cochain $\widetilde C\in C^j(\bcC(M))$ by putting
$\widetilde C(a_0,\dots,a_j):= \frac{1}{j!}\langle C,a_0da_1\wedge\dots\wedge da_j\rangle$.
One has $b\widetilde C=0$ and $B\widetilde C=\widetilde{\pl C}$, where $\pl$
is the codifferential. Because of this identification we will from now on
omit the $\sim$ from the notation if no confusion is possible.
Given a closed \textup{b}-differential form $\go$ on $M$ of even degree.
By $C_\go$ we denote the \deRham\ current $\int_{\tb M} \go\wedge -$.
There is a natural pullback $\iota^*\go$ at $\infty$
(\emph{cf.} Definition and Proposition \plref{defprop15}),
which is a closed even degree form on $\pl M$. We now find
\begin{equation}
\begin{split}
\langle \pl C_{\go} , \alpha \rangle &= \int_{\tb M} \go \wedge d\alpha = \int_{\tb M} d (\go \wedge \alpha)
= \int_{\pl M} \iota^*(\go \wedge \alpha) = \\
&= \int_{\pl M} \iota^*(\go) \wedge \iota^*(\alpha) = \langle C_{\iota^*\go} , \iota^*(\alpha) \rangle .
\end{split}
\end{equation}
In view of Section \plref{SubSec:RelCycCoh} this means that the \emph{pair}
$(C_\go,C_{\iota^* \go})$ is a relative \deRham\ cycle or via the above mentioned
identification between \deRham\ currents and cochains that $(\widetilde{C_\go}, \widetilde{C_{\iota^* \go}})$
is a relative cocycle in the relative total complex Eq.~\eqref{eq:ML200909264}.
If $\go=\sum_{j \ge 0} \go_{2j}$ with closed \textup{b}-differential forms of degree $2j$
then the pair $(\go,\iota^*\go)$ still gives rise to a relative cocycle of degree
$\dim M$ in the relative total complex.
These considerations certainly apply to the even degree forms
$\bomega_\dirac, \omega_{\mathsf{D}_\pl}$ which satisfy
$\iota^*(\bomega)=\omega_{\mathsf{D}_\pl}$.
The limit results \eqref{eq:ML200909261}, \eqref{eq:ML200909262},
\eqref{eq:ML200909263} can then be summarized as
\begin{equation}\label{eq:ML200909265b}
\lim_{t\to 0+}\big(\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t (\dirac), \ch^{k-1}_t (\mathsf{D}_\pl) \big)
= \Big( \int_{\tb M} \bomega_{\dirac} \wedge \bullet ,
\int_{\pl M} \omega_{\mathsf{D}_\pl}\wedge \bullet \Big).
\end{equation}
The limit on left is understood pointwise for each component of pure degree.
Finally we need to relate $\big( \,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t(\dirac),\ch^{k-1}_t(\mathsf{D}_\pl) \big)$ to
the Chern character of $[\dirac]\in K_m(M,\partial M)$. First recall from
Eq.~\eqref{Eq:DefRelCycCoh} that
$HP^\bullet \big( \cC^\infty (M),\cC^\infty (\partial M) \big)$
is naturally isomorphic to $HP^\bullet \big( \cJ^\infty (\partial M, M)\big)$.
Under this isomorphism, the class of the pair
$\big( \,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t(\dirac),\ch^{k-1}_t(\mathsf{D}_\pl) \big)$ is mapped to
${\,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t(\dirac)}_{|\cJ^\infty (\partial M, M)}$, just because elements
of $\cJ^\infty (\partial M, M)$ vanish on $\partial M$.
We note in passing that by \eqref{eq:ML20090219-4}
a smooth function $f$ on $M^\circ$ lies in $ \cJ^\infty (\partial M, M)$
iff in cylindrical coordinates one has for all $l,R$ and every
differential operator $P$ on $\partial M$
\begin{displaymath}
\partial_x^l Df (x,p) = O(e^{Rx}) , \quad x \mapsto -\infty .
\end{displaymath}
In view of \eqref{eq:ML200909265}, the class of
$\big( \,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t(\dirac),\ch^{k-1}_t(\mathsf{D}_\pl) \big)$ in
$HP^\bullet \big( \cJ^\infty (\partial M, M)\big) $ equals that of
$\widetilde{C_{\bomega_\dirac}}$. As explained in Section \ref{Sec:RelConCheDirac},
$HP^\bullet \big( \cJ^\infty (\partial M, M)\big) $ is naturally isomorphic to
$H^\textrm{dR}_\bullet (M,\pl M; \C)$. Under this isomorphism,
$\widetilde{C_{\bomega_\dirac}}$ corresponds to the relative \deRham\ cycle
$\big( C_{\bomega_\dirac} , C_{\omega_{\mathsf{D}_\pl}} \big)$. Finally, note that under
the Poincar\'e duality isomorphism $H^\textrm{dR}_\bullet (M,\pl M; \C)\cong
H^\bullet_\textrm{dR} (M\setminus \pl M;\C) $, the relative \deRham\ cycle
$\big( C_{\bomega_\dirac} , C_{\omega_{\mathsf{D}_\pl}} \big)$ is mapped onto the closed form
$\bomega_\dirac$. This line of argument shows that the class of
$\big( \,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t(\dirac),\ch^{k-1}_t(\mathsf{D}_\pl) \big)$
depends only on the absolute \deRham\ cohomology class of the closed form
$\bomega_\dirac = c \cdot \hat{A} \big( \bnabla^2_g\big)$ on the open manifold
$M\setminus \pl M$. The transgression formula in Chern--Weil theory shows that the
(absolute) \deRham\ cohomology class of $\hat{A}\big( \nabla^2_g\big)$ is independent
of the metric $g$ on $M^\circ$. Thus the \deRham\ class of $\bomega_\dirac$
equals that of $\omega_\dirac = c\cdot \hat{A}\big( \nabla^2_{g_0}\big)$ for any
smooth metric $g_0$. Choosing $g_0$ to be smooth up to the boundary we infer from
Section \ref{Sec:RelConCheDirac} and Proposition \ref{p:ML200909292} that the class of
$\big( \,^\textup{b}\hspace*{-0.2em}\operatorname{\widetilde{ch}}^k_t(\dirac),\ch^{k-1}_t(\mathsf{D}_\pl) \big)$ in
$HP^\bullet \big( \cC^\infty (M),\cC^\infty (\pl M) \big) \cong
HP^\bullet \big( \cJ^\infty (\partial M, M)\big) $ equals that of the
{\CoChch} of $[\dirac]$.
\end{proof}
\subsection{The large time limit and higher $\eta$--invariants}
Let us now assume that the boundary Dirac $\mathsf{D}_\pl$ is invertible.
In view of Proposition \plref{p:estimate-tChern-infty} and
Proposition \plref{p:ShortTimeEst} we can now, for $k\ge\dim M$,
form the transgressed cochain
\begin{equation}
\bTslch_\infty^k(\dirac)(a_0,...,a_k)=\int_0^\infty \bslch^{k} (s\dirac,
\dirac)(a_0,\ldots,a_k) \, ds,
\end{equation}
for $a_0,\ldots,a_k\in \bcC(M^\circ)$.
In view of Eq.~\eqref{eq:ML200909024} we arrive at
\begin{align}
B &\bTslch_\infty^{k+1}(\dirac)(a_0,...,a_k)\nonumber\\
&= \sum_{j=0}^k (-1)^{j+1} \int_0^\infty s^{k+1}\; \blangle [\dirac,a_0],\ldots ,[\dirac,a_j],\dirac,
[\dirac,a_{j+1}],\ldots ,[\dirac,a_k]\rangle\, ds\displaybreak[1]\nonumber \\
&=\sum_{j=0}^k(-1)^{j+1} \int_0^\infty s^{k+1} \int_{\Delta_{k+1}}
\bStr_q\bigl([\dirac,a_0]e^{-\sigma_0 s^2\dirac^2} \ldots \\
&\quad [\dirac,a_j] e^{-\sigma_j s^2\dirac^2}
\dirac e^{-\sigma_{j+1}s^2\dirac^2}
\ldots [\dirac,a_k]e^{-\sigma_{k+1}s^2\dirac^2}\bigr)\, d\sigma \,ds.\nonumber
\end{align}
Together with Proposition \plref{p:ML20090928} we have proved the analogue of \cite[Thm.~1]{ConMos:TCC} in the relative setting:
\begin{theorem}\label{t:ML20081215} Let $k\ge\dim M$ be of
the same parity as $q$ and assume that $\mathsf{D}_\pl$ is invertible.
Then the pair of retracted cochains
$\big(\bch^k_t (\dirac),\ch^{k+1}_t (\mathsf{D}_\pl) \big)$, $t>0$, has a limit as $t\to\infty$.
For $k=2l$ even we have
\begin{equation}\label{eq:ML20081215-1}
\begin{split}
\bch_\infty^k(\dirac)&=\sum_{j=0}^l \kappa^{2j}(\dirac)+B\bTslch_\infty^{k+1}(\dirac)\, ,\\
\ch_\infty^{k+1}(\mathsf{D}_\pl)&= B\Tslch_\infty^{k+2}(\mathsf{D}_\pl)\, .
\end{split}
\end{equation}
If $k=2l+1$ is odd then
\begin{equation}\label{eq:ML20081215-2}
\begin{split}
\bch_\infty^k(\dirac)&=B\bTslch_\infty^{k+1}(\dirac)\, ,\\
\ch_\infty^{k+1}(\mathsf{D}_\pl)&= B\Tslch_\infty^{k+2}(\mathsf{D}_\pl)\, .
\end{split}
\end{equation}
\end{theorem}
\section{Relative pairing formul\ae\, and geometric consequences} \label{s: geom pairing}
\sind{pairing|(}
Let us briefly recall some facts from the theory of
boundary value problems for Dirac operators \cite{BooWoj:EBP}.
Given a Dirac operator $\dirac$ acting on sections
of the bundle $W$ on a compact riemannian manifold with boundary
$(M,g)$. In contrast to the rest of the paper $g$ is
a "true" riemannian metric, smooth and non-degenerate up to the boundary,
and not a \textup{b}-metric. We assume that all structures
are product near the boundary, that is there is a collar
$U=[0,\eps)\times\pl M$ of the boundary such
that $g\rest{U}=dx^2\oplus g\rest{\pl M}$ is a product metric.
In particular the formul\ae\, of Remark \plref{r:DiracCylinderFormulas}
hold.
We assume furthermore that we are in the even situation. That is,
$\dirac$ is odd with respect to a $\Z_2$-grading.
Then in a collar of the boundary $\dirac$ takes the form
\begin{equation}\label{eq:ML20090218-1}
D=\left[\begin{matrix} 0 & D^-\\
D^+ & 0
\end{matrix}\right]
= \left[\begin{matrix} 0 & -\frac{d}{dx}+A^+\\
\frac{d}{dx}+A^+ & 0
\end{matrix}\right]
=\sfc(dx)\frac{d}{dx} +\mathsf{D}_\pl.
\end{equation}
In the matrix notation we have identified $W^+$ and $W^-$ via
$\sfc(dx)$ and put $A^+:=\bigl(\sfc(dx)^{-1}\mathsf{D}_\pl\bigl)\rest{W^+}$.
$A^+$ is a first order self-adjoint elliptic differential operator.
Let $P\in\pdo^0(\partial M;W^+)$
be a pseudodifferential projection with $P-1_{[0,\infty)}(A^+)$
of order $-1$. Then we denote by $\dirac_P$ the operator
$\dirac$ acting on the {\domain}
$\bigsetdef{u\in L^2_1(M;W^+)}{P(u\rest{\partial M})=0}$.
$\dirac_P$ is a Fredholm operator. The Agranovich--Dynin formula
\cite[Prop. 21.4]{BooWoj:EBP}
\begin{equation}\label{eq:AgrDynin}
\ind \dirac^+_P - \ind \dirac^+_Q=\ind (P,Q)=:
\ind\bigl(P:\im Q\longrightarrow \im P\bigr)
\end{equation}
expresses the difference of two such indices
in terms of the relative index of the two projections $P,Q$.
Choosing for $P$ the positive spectral projection
$P_+(A^+)=1_{[0,\infty)}(A^+)$ of $A^+$ we obtain the {\APS} index
\begin{equation}\label{eq:APSindex}
\indAPS \dirac^+= \ind \dirac^+_{P_+(A^+)}.
\end{equation}
We shortly comment on the relative index introduced above
(\emph{cf.}~\cite{AvrSeiSim:IPP}, \cite[Sec.~15]{BooWoj:EBP},
\cite[Sec.~3]{BruLes:BVPI}). Two orthogonal projections $P,Q$ in a Hilbert
space are said to form a Fredholm pair if
$PQ:\im Q\longrightarrow \im P$ is a Fredholm operator. The index of this
Fredholm operator is then called the \emph{relative index},\sind{relative!index} $\ind(P,Q)$,
of $P$ and $Q$. It is easy to see that $P,Q$ form a Fredholm pair if the
difference $P-Q$ is a compact operator. Furthermore, the relative index
is additive
\begin{equation}\label{eq:rel-index-additive}
\ind (P,R)=\ind (P,Q)+\ind (Q,R)
\end{equation}
if $P-Q$ or $P-R$ is compact
\cite[Prop. 15.15]{BooWoj:EBP}, \cite[Thm.~3.4]{AvrSeiSim:IPP}.
In general, just assuming that all three pairs
$(P,Q), (Q,R)$ and $(Q,R)$ are Fredholm is not sufficient for \eqref{eq:rel-index-additive}
to hold.
\sind{spectral flow}
Sometimes the spectral flow of a path of self-adjoint operators can
be expressed as a relative index. The following proposition
is a special case of \cite[Thm.~3.6]{Les:USF}:
\begin{proposition}\label{p:SF-compact-perturbation} Let $T_s=T_0+\widetilde T_s, 0\le s\le 1,$
be a path of self-adjoint Fredholm operators in the Hilbert space $H$. Assume
that $\widetilde T_s$ is a continuous family of bounded $T_0$-compact
operators. Then the spectral flow of $(T_s)_{0\le s\le 1}$ is given by
\[
\SF (T_s)_{0\le s\le 1}=-\ind (P_+(T_1),P_+(T_0)),
\]
where $P_+(T_s):= 1_{[0,\infty)}(T_s)$.
\end{proposition}
\begin{remark} If $T_0$ is bounded then the condition of $T_0$-compactness
just means that the $T_s$ are compact operators. If $T_0$ is unbounded with
compact resolvent then any bounded operator is automatically $T_0$-compact.
The second case is the one of relevance for us.
In \cite[Thm.~3.6]{Les:USF} Proposition \plref{p:SF-compact-perturbation}
is proved for Riesz continuous paths of unbounded Fredholm operators. Since
$s\mapsto \widetilde T_s$ is continuous the map $s\mapsto T_0+\widetilde T_s$
is automatically Riesz continuous \cite[Prop. 3.2]{Les:USF}.
Note that our sign convention for the relative index differs from that of \emph{loc. cit.}
Therefore our formulation of Proposition \ref{p:SF-compact-perturbation} differs from
\cite[Thm.~3.6]{Les:USF} by a sign, too.
\end{remark}
\begin{proposition}\label{p:SFRelAPS}
Let $(\dirac_s)_{0\le s\le 1}$ be a smooth family of self-adjoint
$\Z_2$-graded \emph{(\cf~\eqref{eq:ML20090218-1})} Dirac operators on a compact
riemannian manifold with boundary. We assume that $\dirac_s$ is in product
form near the boundary and that $\dirac_s=\dirac_0+\Phi_s$ with a bundle
endomorphism $\Phi_s\in\Gamma^\infty(M;\End V)$. Then
\begin{equation}
\indAPS \dirac^+_1-\indAPS \dirac^+_0 =-\SF (A_s^+)_{0\le s\le 1}.
\end{equation}
Here, as explained above, $A_s^+=(\sfc(dx)^{-1}\diracbdyvar{s})\rest{W^+}$.
\end{proposition}
\begin{proof} The family $s\mapsto \dirac_{s,\APS}$ is not necessarily
continuous. The reason is that if eigenvalues of $A^+_s$
cross $0$ the family $P_+(A^+_s)$ of {\APS} projections jump.
However, since $A^+_s-A^+_0$ is $0^{\textup{th}}$ order, the corresponding {\APS}
projections $P_+(A^+_s)$ all have the same leading symbol and hence
$P_+(A^+_s)-P_+(A^+_{s'})$ is compact for all $s,s'\in [0,1]$.
Hence we can consider the family $\dirac_{s,P_+(A^+_0)},\; 0\le s\le 1$.
Now the boundary condition is fixed and thus $s\mapsto \dirac_{s,P_+(A^+_0)}$
is a graph continuous family of Fredholm operators
\cite{Nic:MIS,BooFur:MIF,BooLesZhu:CPD}. Therefore, its index is independent of $s$.
Applying the Agranovich--Dynin formula \eqref{eq:AgrDynin} and Proposition
\plref{p:SF-compact-perturbation} we find
\begin{align}
\indAPS(\dirac^+_1)&= \ind \dirac^+_{1,P_+(A^+_0)}+\ind \bigl(P_+(A^+_1),P_+(A^+_0)\bigr)\nonumber \\
&= \ind \dirac^+_{0,P_+(A^+_0)}+\ind \bigl(P_+(A^+_1),P_+(A^+_0)\bigr)\\
&=\indAPS \dirac^+_0 -\SF(A_s^+)_{0\le s\le 1}.\qedhere
\end{align}
\end{proof}
Recall that a smooth idempotent $p:M\longrightarrow \Mat_N(\C)$ corresponds to a smooth
vector bundle $E\simeq \im p$ and using the Grassmann
connection the twisted Dirac operator $\dirac^E$ equals $p(\dirac\otimes \id_N)p$.
To simplify notation we will write $p\dirac p$ for $p(\dirac\otimes\id_N)p$ whenever confusions are unlikely.
We would like to extend Proposition \plref{p:SFRelAPS}
to families of twisted Dirac operators of the form $\dirac_s=p_s \dirac p_s$,
where $p_s:M\longrightarrow \Mat_N(\C)$ is a family of orthogonal projections.
The difficulty is that not
only the leading symbol of $\dirac_s$ but even the Hilbert space of sections, on which the operator
acts, varies.
\begin{proposition}\label{p:SFRelAPSa}
Let $\dirac$ be a self-adjoint $\Z_2$-graded \emph{(\cf~\eqref{eq:ML20090218-1})}
Dirac operator on a compact riemannian manifold with boundary $M$. Let
$p_s:M\longrightarrow \Mat_N(\C)$ be a smooth family of orthogonal projections. Assume
furthermore, that in a collar neighborhood $U=[0,\eps)\times \partial M$ of $\partial M$
we have ${p_s}\rest{U}=p_s^\partial$, i.e. ${p_s}\rest{U}$ is independent of the normal variable. Then
\begin{equation}
\indAPS p_1\dirac^+ p_1-
\indAPS p_0\dirac^+ p_0
=-\SF(p_s A^+ p_s)_{0\le s\le 1}.
\end{equation}
\end{proposition}
\begin{proof}
By a standard trick often used in operator $K$-theory \cite[Prop. 4.3.3]{Bla:KTO} we may
choose a smooth path of unitaries $u:M\longrightarrow \Mat_N(\C)$ such that
$p_s=u_sp_0u_s^*, u_0=\id_N$. Furthermore, we may assume that
$u\rest{[0,\eps)\times \partial M}=u^\partial$
is also independent of the normal variable. Then
$p_s\dirac^+ p_s= u_s\bigl( p_0 u_s^*\dirac^+ u_s p_0\bigr) u_s^*$
and
$\bigl(p_s \dirac^+ p_s\bigr)_{\APS}=
u_s\bigl( p_0 u_s^*\dirac^+ u_s p_0\bigr)_{\APS} u_s^*$.
Since $u_s^*\dirac^+ u_s=\dirac^++u_s^* \sfc(d u_s)$
Proposition \plref{p:SFRelAPS} applies
to the family $p_0 u_s^*\dirac u_s p_0$. Since the spectral flow is invariant
under unitary conjugation we reach the conclusion.
\end{proof}
\begin{definition}\label{def:SF} In the sequel we will write somewhat more suggestively
and for brevity $\SF(p_{\cdot},\mathsf{D}_\pl)$ instead of $\SF(p_s A^+ p_s)_{0\le s\le 1}$.
\end{definition}
\begin{theorem}\label{t:RelativePairingKTheory}
Let $M$ be a compact manifold with boundary and $W$ a degree $q$
Clifford module on $M$. Let $g$ be a smooth riemannian metric
on $M$, $h$ a hermitian metric and $\nabla$ a unitary Clifford connection on $W$. Assume that
all structures are product near the boundary. Let $\dirac=\dirac(\nabla,g)$
be the Dirac operator.
Let
$[p,q,\gamma]\in K^0(M,\partial M)$ be a relative $K$-cycle .
That is $p,q:M\longrightarrow \Mat_N(\C)$
are orthogonal projections and $\gamma:[0,1]\times\partial M\longrightarrow \Mat_N(\C)$
is a homotopy of orthogonal projections with $\gamma(0)=p^\partial, \gamma(1)=q^\partial$. Then
\begin{equation}\label{eq:RelativePairingKTheory}
\begin{split}
\langle [\dirac],\, & [p,q,\gamma]\rangle\\
=&-\indAPS p\dirac^+ p +\indAPS q\dirac^+ q
+\SF(\gamma,\mathsf{D}_\pl),\\
=& \int_{M} \go_{\dirac(\nabla,g)}\wedge \bigl(\ch_\bullet(q)-\ch_\bullet(p)\bigr)
- \int_{\pl M} \go_{\mathsf{D}_\pl(\nabla,g)}\wedge \Tslch_\bullet (h),
\end{split}
\end{equation}
in particular the right hand side of \eqref{eq:RelativePairingKTheory}
depends only on the relative $K$-theory class $[p,q,\gamma]\in K^0(M,\partial M)$
and the degree $q$ Clifford module $W$. It is independent of $\nabla$ and $g$.
In case all structures are \textup{b}-structures, and $\dirac = \dirac (\bnabla, \bmet)$
is the \textup{b}-Dirac operator, then we still have
\begin{equation}
\label{eq:brelpairing}
\langle [\dirac],\, [p,q,\gamma]\rangle =
\int_{\tb M} \bomega_{\dirac}\wedge \bigl(\ch_\bullet(q)-\ch_\bullet(p)\bigr)
- \int_{\pl M} \go_{\mathsf{D}_\pl}\wedge \Tslch_\bullet (h) .
\end{equation}
\end{theorem}
For the fact
that relative $K$-cycles can be represented by triples $[p,q,\gamma]$ as above
we refer to \cite[Thm.~5.4.2]{Bla:KTO}, \cite[Sec.~4.3]{HigRoe:AKH}, see
also \cite[Sec.~1.6]{LesMosPfl:RPC}.
\begin{proof}
Denote the right hand side of \eqref{eq:RelativePairingKTheory} by $I(p,q,\gamma)$.
We first show that $I(p,q,\gamma)$ depends indeed only on the relative $K$-theory class of
$(p,q,\gamma)$. By the stability of the Fredholm index we may assume that in a collar
neighborhood of $\partial M$ the projections $p,q$ do not depend on the normal variable.
After stabilization we need to show the homotopy invariance of $I(p,q,\gamma)$.
Now consider a homotopy $(p_t,q_t,\gamma_t)$ of relative $K$-cycles.
Then by Proposition \plref{p:SFRelAPSa} we have (\emph{cf.}~Figure \ref{fig:SF}, page \pageref{fig:SF})
\begin{align*}
&I(p_1,q_1,\gamma_1)-I(p_0,q_0,\gamma_0)\\
=\, & \hphantom{-} \indAPS(p_1 \dirac^+ p_1)-\indAPS( q_1\dirac^+ q_1)-
\SF\bigl(\gamma_1(\cdot),\mathsf{D}_\pl\bigr)\\
& - \indAPS(p_0 \dirac^+ p_0)+\indAPS( q_0\dirac^+ q_0)
+ \SF\bigl(\gamma_0(\cdot),\mathsf{D}_\pl\bigr)\\
=\, &
- \SF\bigl(\gamma_{\cdot}(0),\mathsf{D}_\pl\bigr)
+ \SF\bigl(\gamma_{\cdot}(1),\mathsf{D}_\pl\bigr)
- \SF\bigl(\gamma_1(\cdot),\mathsf{D}_\pl\bigr) + \SF\bigl(\gamma_0(\cdot),\mathsf{D}_\pl\bigr)\\
=\, &0,
\end{align*}
by the homotopy invariance of the Spectral Flow.
So the l.h.s \emph{and} the r.h.s. of \eqref{eq:RelativePairingKTheory} depend only
on the relative $K$-theory class of $[p,q,\gamma]$. By \excision\ in $K$-theory
(it can of course be shown in an elementary way by exploiting Swan's Theorem) every relative
$K$-theory class can even be represented by a triple $(p,q,p\rest{\pl M})$ such that
$p\rest{[0,\eps)\times \pl M}=q\rest{[0,\eps)\times \pl M}$ and hence
$\gamma(s)=p\rest{\pl M}$ is constant.
\FigSF
Then the twisted version of the {\APS} Index Theorem gives
\begin{equation}
\label{relAPS}
\begin{split}
\indAPS q \dirac^+ q - \indAPS p \dirac^+ p=
\int_M \omega_\dirac \wedge \big( \ch_\bullet (q) - \ch_\bullet (p) \big),
\end{split}
\end{equation}
where $\go_\dirac$ denotes the local index density of $\dirac$. Note that
since the tangential operators of $ p \dirac^+ p$
and of $q \dirac^+ q$ coincide the $\eta$-terms cancel.
As outlined in Section \ref{Sec:RelConCheDirac} (\emph{cf.}~also the proof of Theorem
\plref{t: CC-character}) the {\CoChch} of $[\dirac]$
in $HP^\bullet \bigl(\cJ^\infty (\pl M , M )\bigr) \simeq H_\bullet^{\rm dR}
(M \setminus \partial M ; \C)$ is represented by $\int_M \go_\dirac$.
By construction, the form $\ch_\bullet(q)-\ch_\bullet(p)$ is compactly supported
in $M\setminus \pl M$.
Thus the right hand side of \eqref{relAPS} equals
the pairing $\langle [D], [p,q,p\rest{\pl M}]\rangle$ and the first equality in
\eqref{eq:RelativePairingKTheory} is proved.
\sind{pairing}
To prove the second equality in \eqref{eq:RelativePairingKTheory}
we note that it represents the Poincar{\'e} duality pairing between
the \deRham\ cohomology class of $\go_{\dirac(\nabla,g)}$
(note $\iota^*\go_\dirac=\go_{\mathsf{D}_\pl}$)
and the relative \deRham\ cohomology
class of the pair of forms $(\ch_\bullet(q)-\ch_\bullet(p),\Tslch_\bullet(h))$.
Hence it depends only on the class $[p,q,h]\in K^0 (M,\partial M)$ and on $[\dirac]$.
In the situation above where $p$ and $q$ coincide in a collar of the boundary it
equals $\langle [\dirac],[p,q,\gamma]\rangle$ and hence by homotopy invariance
the claim is proved in general up to Eq.~\eqref{eq:brelpairing}.
For the proof of Eq.~\eqref{eq:brelpairing} note first that for a closed even
\textup{b}-differential form $\omega$ the map (\emph{cf.}~Definition and Proposition \plref{defprop15})
\begin{displaymath}
\begin{split}
\Omega^k (M) \oplus \Omega^{k-1} (\partial M) \rightarrow \C, \quad
(\eta,\tau)\mapsto\int_{\tb M}\omega \wedge\eta -
\int_{\partial M}\iota^*\omega \wedge \tau
\end{split}
\end{displaymath}
descends naturally to a linear form on $H^k_\textrm{dR} (M,\pl M ; \C)$. Hence
the right hand side of Eq.~\eqref{eq:brelpairing} is well-defined and depends only on the
class of $[p,q,h] \in K^0 (M,\pl M)$. As before we may therefore specialize to
$(p,q,p_{|\pl M})$ such that
$p_{|[0,\varepsilon ) \times \pl M}=q_{|[0,\varepsilon ) \times \pl M} $.
Then $\ch_\bullet(q)-\ch_\bullet(p)$
has compact support in $M\setminus \pl M$ and the remaining claim follows from
Theorem \ref{t: CC-character} (4).
\end{proof}
We now proceed to
express the pairing between relative K-theory classes and the fundamental
relative $K$-homology class in cohomological terms. We assume here that
we are in the \textup{b}-setting.
\sind{pairing}
Recall that a relative K-theory class in
$K^0(M,\pl M)$ is represented by a pair of bundles $(E,F)$ over $M$
whose restrictions $E_\pl,F_\pl$ to $\pl M$ are related by a homotopy $h$.
We will explicitly write the formul\ae\, in the even dimensional case and
only point out where the odd dimensional case is different.
The Chern character of $[E,F,h] \in K^0 (M, \partial M)$ is then represented by
the relative cyclic homology class
\begin{equation}
\begin{split}
\ch_\bullet \big( [E,F,h] \big) \,= \,
\Big(
\ch_\bullet(p_F) - \ch_\bullet (p_E) \, , \, - \Tslch_\bullet (h) \Big),
\end{split}
\end{equation}
\emph{cf.}~Eq.~\eqref{eq:ChernCharEven}.
By Theorem \plref{t: CC-character} we have for any $t > 0$
\begin{equation} \label{eq: McKean--Singer formula}
\begin{split}
\big\langle& [\dirac], \, [E,F,h] \big\rangle
\,=\,\big\langle \big( \bch_t^n (\dirac), \ch_t^{n+1} (\mathsf{D}_\pl) \big),\,
\ch_\bullet([E,F,h])\big\rangle \\
=\, & \big\langle \sum_{j \geq 0} \bCh^{n-2j} (t\dirac) + B\, \tb\Tslch_t^{n+1} (\dirac),
\, \ch_\bullet (p_F) -\ch_\bullet (p_E) \big\rangle \\
& - \big\langle \sum_{j \geq 0} \Ch^{n-2j+1} (t\dirac_\partial) +
B \Tslch_t^{n+2} (\dirac_\partial) , \, \Tslch_\bullet (h) \big\rangle.
\end{split}
\end{equation}
Letting $t \searrow 0$ yields, again by Theorem \plref{t: CC-character}, the local form of the pairing:
\begin{equation} \label{eq: local pairing even}
\begin{split}
\big\langle& [\dirac], \, [E,F,h] \big\rangle \\
& = \int_{\tb M} \go_\dirac \wedge \big( \ch_\bullet (p_F) - \ch_\bullet (p_E) \big)
- \int_{\partial M} \go_{\mathsf{D}_\pl}(\partial M) \wedge \Tslch_\bullet (h) .
\end{split}
\end{equation}
If $\mathsf{D}_\pl$ is invertible then, at the opposite end, letting $t \nearrow \infty$ gives in view of Theorem
\plref{t:ML20081215}
\begin{equation} \label{eq: stretched pairing even}
\begin{split}
\big\langle [\dirac], \, [E,F,h] &\big\rangle \, =
\big\langle \sum_{0\leq k \leq \ell} \kappa^{2k} (\dirac)
+ B\tb \Tslch_\infty^{n+1} (\dirac) ,
\, \ch_\bullet(p_F) - \ch_\bullet (p_E) \big\rangle \\
&\qquad- \big\langle B\Tslch_\infty^{n+2} (\dirac_\partial) , \Tslch_\bullet (h) \big\rangle,
\end{split}
\end{equation}
where $2\ell=n$.
By equating the above two limit expressions \eqref{eq: local pairing even}
and \eqref{eq: stretched pairing even}
one obtains the following identity:
\begin{corollary} \label{t: higher APS}
Let $n =2\ell \geq m $ and assume that $\mathsf{D}_\pl$ is invertible.
Then
\begin{equation*
\begin{split}
\big\langle \kappa^0(\dirac) , \, & p_F - p_E \big\rangle + \\
& + \, \sum_{1\leq k \leq \ell} (-1)^k \frac{(2k)!}{k!} \,
\big\langle \kappa^{2k}(\dirac) , \, (p_F-\frac 12)\otimes p_F^{\otimes 2k} -
(p_E-\frac 12)\otimes p_E^{\otimes 2k} \big\rangle \, \\
=\,& \int_{\tb M} \go_\dirac \wedge \big( \ch_\bullet (p_F) - \ch_\bullet (p_E) \big)
- \int_{\partial M} \go_{\mathsf{D}_\pl}\wedge \Tslch_\bullet (h) \\
& - (-1)^{n/2} \frac{n!}{(n/2)!}\,
\big\langle B \tb\Tslch_\infty^{n+1} (\dirac) ,
(p_F-\frac 12)\otimes p_F^{\otimes n} - (p_E-\frac 12)\otimes p_E^{\otimes n} \big\rangle\\
& +\big\langle B \Tslch_\infty^{n+2} (\mathsf{D}_\pl), \Tslch_{n+1} (h) \big\rangle.
\end{split}
\end{equation*}
\end{corollary}
The left hand side plays the role of a `higher' relative index, while
the right hand side contains local geometric terms and
`higher' eta cochains.
The pairing formula acquires a simpler form if one chooses
special representatives for the class $ [E,F,h] $. For example,
one can always assume that
$E_\pl = F_\pl$, in which case one obtains
\begin{equation} \label{eq: stretched pairing even sp}
\begin{split}
\big\langle [\dirac], \, [E,F,h_0] &\big\rangle \, =
\big\langle \sum_{0\leq k \leq \ell} \kappa^{2k} (\dirac)
+ B\tb \Tslch_\infty^{n+1} (\dirac) ,
\, \ch_\bullet(p_F) \big\rangle \\
&\qquad- \big\langle \sum_{0\leq k \leq \ell} \kappa^{2k} (\dirac)
+ B\tb \Tslch_\infty^{n+1} (\dirac) ,
\, \ch_\bullet(p_E) \big\rangle.
\end{split}
\end{equation}
Specializing even more, one can assume $F = \C^N$. Then
the pairing formula becomes
\begin{equation} \label{eq: stretched pairing even xsp}
\begin{split}
\big\langle [\dirac], \, [E,\C^N,h_0] \big\rangle \, = \,
& - \big\langle \sum_{0\leq k \leq \ell} \kappa^{2k} (\dirac)
+ \, B\tb \Tslch_\infty^{n+1} (\dirac) ,
\, \ch_\bullet(p_E) \big\rangle \\
&+\, N ( \dim \Ker \dirac^+ - \dim \Ker \dirac^-) .
\end{split}
\end{equation}
On the other hand, applying Theorem \plref{t:RelativePairingKTheory}
\begin{equation*}
\big\langle [\dirac], \, [E,\C^N,h_0] \big\rangle \, =
\, -\indAPS (p_ED^+ p_E) + N \indAPS D^+ ,
\end{equation*}
one obtains an index formula for the $b$-Dirac operator which is the direct
analogue of Eq.~(3.4) in \cite{ConMos:TCC}:
\begin{corollary} \label{t: b-index formula}
Let $E$ be a vector bundle on $M$
whose restriction to $\pl M$ is trivial and assume $\mathsf{D}_\pl$ to be invertible.
Then for any $n =2\ell \geq m $
\begin{equation} \label{eq: b-index formula}
\begin{split}
\indAPS (p_E D^+ p_E) = \big\langle \sum_{0\leq k \leq \ell} \kappa^{2k}
(\dirac)
+ \, B\tb \Tslch_\infty^{n+1} (\dirac) ,
\, \ch_\bullet(p_E) \big\rangle .
\end{split}
\end{equation}
\end{corollary}
The expression $\indAPS (p_E D^+ p_E)$ is to be understood as follows: if
$p_E^\partial \dirac_\partial p_E^\partial$ is invertible, then it is the Fredholm index of
$p_E\dirac^+p_E $. If $p_E\dirac^+p_E $ is not Fredholm, then chose a metric $\tilde g$
smooth up to the boundary and construct on the Clifford module of $\dirac$ the Dirac
operator $\tilde \dirac$ to the riemannian metric $\tilde g$. Then, by
Theorem.~\ref{t: CC-character} the \CoChch s of $\tilde \dirac$
and $\dirac$ coincide and thus
\begin{displaymath}
\big\langle [\dirac], [E,\C^N,h_0] \big\rangle =
- \indAPS p_E \tilde \dirac^+ p_E + N \indAPS \tilde\dirac^+ .
\end{displaymath}
As a by-product of the above considerations, we can now establish
the following generalization of the
Atiyah-Patodi-Singer odd-index theorem for trivialized flat
bundles (comp.~\cite[Prop.~6.2, Eq.~(6.3)]{APS:SARIII}.
An analogue for even dimensional manifolds has been
subsequently established by Z. Xie \cite{Xie:RIP}.
\begin{corollary} \label{t: gen APS flat}
Let $N$ be a closed odd dimensional spin manifold, and let
$E',F'$ be two vector bundles which are equivalent in
$K$-theory via a homotopy $h$. With $\dirac_{g'}$ denoting the Dirac
operator associated to a riemannian metric $g'$ on $N$, one has
\begin{equation} \label{eq: gen APS flat}
\begin{split}
\xi (\dirac_{g'}^{F'}) - \xi (\dirac_{g'}^{E'})
\, = \, \int_{N} \hat{A} (\nabla_{g'}^2) \wedge
\Tslch_\bullet (h) \, + \, \SF (h , \dirac_{g'}) \, ,
\end{split}
\end{equation}
or equivalently,
\begin{equation} \label{eq: smooth eta}
\begin{split}
\int_0^1 \frac{1}{2} \frac{d}{dt} \big( \eta (p_{h(t)} \, \dirac_{g'} \, p_{h(t)}) \big) dt \, = \,
\int_{N} \hat{A} (\nabla_{g'}^2) \wedge
\Tslch_\bullet (h) \, ,
\end{split}
\end{equation}
where $p_{h(t)})$ is the path of projections joining $E'$ and $F'$.
\end{corollary}
\begin{proof}
This follows from equating the local pairing \eqref{eq:RelativePairingKTheory}
and the relative {\APS} index formula \eqref{Eq:Pairing2}
after the following modifications. We recall that
Eq.~\eqref{eq:RelativePairingKTheory} holds in complete generality without invertibility
hypothesis on $\dirac_\pl$. First by passing to a multiple one can assume
that $N=\pl M$. Then by adding a complement $G'$ we can replace $F'$ by a trivial
bundle. Then both $E'\oplus G'$ and $F'\oplus G'$ extend to $M$.
It remains to notice that both sides of the formula
\eqref{eq: gen APS flat} are additive.
The alternative formulation \eqref{eq: smooth eta} follows immediately
from the known relation (see e.g. \cite[Lemma 3.4]{KirLes:EIM})
\begin{equation*
\xi (\dirac_{g'}^{F'}) - \xi (\dirac_{g'}^{E'}) \, = \, \SF (h , \mathsf{D}_\pl) +
\int_0^1 \frac{1}{2} \frac{d}{dt} \big( \eta (p_{h(t)} \, \dirac_{g'} \, p_{h(t)}) \big) dt .\qedhere
\end{equation*}
\end{proof}
In the odd dimensional case the pairing formul\ae\, are similar, except
that the contribution from the kernel of $\dirac$ does not occur.
Let $(U,V,h)$ be a representative of an odd relative K-theory
class where $U,V:M\to U(N)$ are unitaries and $h$ is a homotopy
between $U_{\pl M}$ and $V_{\pl M}$.
Then
\begin{equation} \label{eq: stretched pairing odd}
\begin{split}
\big\langle [\dirac], \, [U,V,h] &\big\rangle \, =\,
\big\langle B \bTslch_\infty^{n+1} (\dirac) ,
\, \ch_n (U) - \ch_n (V) \big\rangle \\
&\qquad- \big\langle B\Tslch_\infty^{n+2} (\mathsf{D}_\pl) , \Tslch_{n+1} (h) \big\rangle.
\end{split}
\end{equation}
Choosing a representative of the class with $U_{\pl M} = V_{\pl M}$, the above
formula simplifies to
\begin{equation} \label{eq: stretched pairing odd sp}
\begin{split}
\big\langle [\dirac], \, [U,V,h_0] &\big\rangle \, = \,
\big\langle B \bTslch_\infty^{n+1} (\dirac) ,
\, \ch_n (U) - \ch_n (V) \big\rangle ,
\end{split}
\end{equation}
and if moreover one takes $V=\id$, it reduces to
\begin{equation} \label{eq: stretched pairing odd xsp}
\begin{split}
\big\langle [\dirac], \, [U,\id,h_0] &\big\rangle \, = \,
\big\langle B \bTslch_\infty^{n+1} (\dirac) ,
\, \ch_n (U) \big\rangle .
\end{split}
\end{equation}
Finally, the equality between the local form of the pairing \eqref{eq: local pairing even}
and the expression \eqref{eq: stretched pairing odd} gives the following odd
analogue of Corollary \ref{t: higher APS}.
\begin{corollary} \label{t: higher APS odd}
Let $n \geq m $, both odd and assume that $\mathsf{D}_\pl$ is invertible. Then
\begin{equation*
\begin{split}
\big\langle B \bTslch_\infty^{n+1} (\dirac)&,\, \ch_n(U) \big\rangle -
\big\langle B\bTslch_\infty^{n+1} (\dirac),\, \ch_n(V) \big\rangle \, = \\
&= \int_{\tb M} \hat{A} (\bnabla^2_g) \wedge \big( \ch_\bullet (U) - \ch_\bullet (V) \big)
- \int_{\partial M} \hat{A} (\nabla^2_{g'}) \wedge
\Tslch_\bullet (h) \\
& \qquad
- \big\langle B\Tslch_\infty^{n+2} (\mathsf{D}_\pl) , \Tslch_{n+1} (h) \big\rangle.
\end{split}
\end{equation*}
\end{corollary}
\section{Relation with the generalized \textup{APS} pairing}
\label{s:conclude}
Wu~\cite{Wu:CCC} showed that the full cochain $\eta^\bullet (\mathsf{D}_\pl)$ has a finite radius of
convergence, proportional to the lowest eigenvalue of $\vert \mathsf{D}_\pl \vert$
(assumed to be invertible).
Both Wu and Getzler~\cite{Get:CHA} proved, by different
methods, the following generalized Atiyah-Patodi-Singer index formula :
\begin{equation} \label{HAPS}
\indAPS \dirac^E \, = \,
\int_{\tb M} \hat{A} (\bnabla^2_g) \wedge \ch_\bullet(p_E)
\, +\, \big\langle \eta^\bullet (\mathsf{D}_\pl)\circ i^\ast, \, \ch_\bullet(p_E) \big\rangle ,
\end{equation}
for any vector bundle $E = \im p_E$ over $M$ whose restriction to the boundary
satisfies the {\it almost $\partial$-flatness} condition
$ \Vert [\mathsf{D}_\pl , r_\partial (p_E) ] \Vert < \lambda_1 (\vert \mathsf{D}_\pl \vert)$.
Their result does provide a decoupled index pairing,
but only for those classes in $K^m (M, \partial M)$
which can be represented by pairs of almost $\partial$-flat bundles.
Furthermore, if $(E, F , h)$ is such a triple, on applying \eqref{HAPS} and
Theorem \plref{t:RelativePairingKTheory} one obtains
\begin{equation} \label{eq: GWpair}
\begin{split}
\langle [\dirac], [E, F, h] \rangle \,= &
\int_{\tb M} \hat{A}(\bnabla^2_g) \wedge \big(\ch_\bullet(p_F) - \ch_\bullet(p_E) \big)\\
& + \big\langle \eta^\bullet (\mathsf{D}_\pl) , i^\ast\big( \ch_\bullet(p_F) - \ch_\bullet(p_E) \big)\big\rangle
\,+ \, \SF (h , \mathsf{D}_\pl) ,
\end{split}
\end{equation}
where $\SF(h,\mathsf{D}_\pl)$ is an abbreviation for $\SF\bigl( h(s) A^+ h(s)\bigr)_{0\le s\le 1},$
\emph{cf.}~also Eq.~\eqref{eq:ML20090218-1}. By \cite[Proof of Thm.~3.1]{Wu:CCC},
\begin{equation*}
(b+B) \eta^\bullet (\mathsf{D}_\pl) \, =\,- \int_{\pl M} \hat{A}(\nabla^2_{g'}) \wedge -\, .
\end{equation*}
At the formal level
\begin{equation} \label{eq: corrected}
\begin{split}
&\big\langle \eta^\bullet (\mathsf{D}_\pl)\circ i^\ast,
\ch_\bullet(p_F) - \ch_\bullet(p_E) \big\rangle
= \big\langle \eta^\bullet (\mathsf{D}_\pl), (b+B) \Tslch_\bullet (h) \big\rangle \\
& = \big\langle (b+B) \eta^\bullet (\mathsf{D}_\pl), \Tslch_\bullet (h) \big\rangle
\, = \,- \int_{\pl M} \hat{A}(\nabla^2_{g'}) \wedge \Tslch_\bullet (h) .
\end{split}
\end{equation}
However, to ensure that the pairing $ \big\langle \eta^\bullet (\mathsf{D}_\pl), (b+B) \Tslch_\bullet (h) \big\rangle$
makes sense one has to assume that $p_{h(t)}$ satisfy the same almost $\partial$-flatness
condition. Then
$$
\Ker (p_{h(t)} \mathsf{D}_\pl p_{h(t)}) = 0 ,
$$
hence there is no spectral flow along
the path.
Thus, the total eta cochain disappears and
\eqref{eq: corrected} together with \eqref{eq: GWpair} just lead to the known
local pairing formula, \emph{cf.}~Eq.~\eqref{eq:RelativePairingKTheory},
\begin{equation*}
\begin{split}
\langle [\dirac], &[E, F, h] \rangle \\
&= \int_{\tb M} \hat{A}(\bnabla^2_g) \wedge \big(\ch_\bullet(p_F) - \ch_\bullet(p_E) \big)
\, - \,\int_{\pl M} \hat{A}(\nabla^2_{g'}) \wedge \Tslch_\bullet (h) \, .
\end{split}
\sind{pairing|)}
\end{equation*}
The above considerations also show that the total eta cochain
necessarily has a finite radius of
convergence. If it was entire, then $\SF(h, D) = 0$ for any $h(t)$,
which is easy to disprove by a counterexample.
\backmatter
|
{
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}
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\section{Introduction}
The notion of associated variety of a primitive ideal in a universal enveloping algebra is a widely used and very useful one in representation theory, permitting the application of geometric methods, see e.g., \cite{Vog91}. Recently, a similar notion has been introduced for vertex algebras
(\cite{Ara12}) which has turned out to be useful not only in the representation theory of vertex algebras (e.g.,\cite{A2012Dec,AM15,AraKaw18,EkeHel,AEkeren19})
but also in connection with {four dimensional $\mathcal{N}=2$ superconformal field theories} via the 4D/2D correspondence discovered in \cite{BeeLemLie15},
see e.g., \cite{SonXieYan17, Son17, BeeRas,AraMT,BonMenRas19,PanPee20,Ded,XieYan2019lisse}.
Let $\mathfrak{g}$ be a simple complex Lie algebra and $G$ the corresponding adjoint algebraic group. For any choice of level $k$ the associated variety $X_{\L_k(\mathfrak{g})}$ of the simple affine vertex algebra $\L_k(\mathfrak{g})$ is a conic, $G$-invariant
subvariety of $\mathfrak{g}^*$.
While the associated variety of a primitive ideal in $U(\mathfrak{g})$ is always contained in the nilpotent cone $\mathcal{N} \subset \mathfrak{g}^*$, this is not necessarily the case for the variety $X_{\L_k(\mathfrak{g})}$.
Nevertheless,
it was shown in \cite{Ara09b} that
$X_{\L_k(\mathfrak{g})}$ is the closure of a nilpotent orbit $\mathbb{O}_q$,
which depends only on the denominator $q\in \mathbb{Z}_{\geqslant 1}$ of the rational number $k$
whenever $\L_k(\mathfrak{g})$ is an admissible representation \cite{KacWak89} of the affine Kac-Moody algebra
$\widehat{\mathfrak{g}}$.
More explicitly,
we have
\begin{align*}
\overline{\mathbb{O}}_q=
\begin{cases}
\mathcal{N}_q & \text{if }(q,\check{r})=1,\\
{}^L\mathcal{N}_{q/\check{r}} & \text{if }(q,\check{r})\ne 1,
\end{cases}
\end{align*}
where by definition
\begin{align}
\label{eq:Nq}
\begin{split}
\mathcal{N}_q &= \{x\in \mathfrak{g} \colon (\ad x)^{2q}=0\}, \\
\text{and} \quad {}^L\mathcal{N}_q &= \{x\in \mathfrak{g} \colon \pi_{\theta_{\text{s}}}(x)^{2q} = 0\},
\end{split}
\end{align}
and $\pi_{\theta_{\text{s}}}$ is the irreducible finite-dimensional representation of $\mathfrak{g}$
with highest weight $\theta_{\text{s}}$,
$\theta_{\text{s}}$ is the highest short root of $\mathfrak{g}$ and $\check{r}$ is the lacing number of $\mathfrak{g}$, that is, the maximal number of edges in the Dynkin diagram of $\mathfrak{g}$. The irreducibility of the varieties of \eqref{eq:Nq} was checked
in \cite{Ara09b} so that the nilpotent orbit $\mathbb{O}_q$ is indeed well-defined.
Prior to \cite{Ara09b},
the irreducibility of variety $\overline{\mathbb{O}}_q$ had been proved in \cite{Geo04}
in most cases
and
the orbits $\mathbb{O}_q$ were studied in \cite{ElaKacVin08}
as
exceptional nilpotent orbits for those
of principal Levi type and
$(q,\check{r})=1$.
In connection with the 4D/2D correspondence mentioned above,
the variety $\overline{\mathbb{O}}_q$ also appears as the Higgs branch of some
{A}rgyres-{D}ouglas theories \cite{Dan,SonXieYan17,WanXie}.
The purpose of this note is to give a
simple description of the orbit $\mathbb{O}_q$ in terms of primitive ideals. It is well known that the associated variety of the annihilator of an integral highest weight representation of $\mathfrak{g}$ is the closure of a {\emph{special} \cite{Lus79}} nilpotent orbit (\cite{BarVog82,BarVog83}). Since the nilpotent orbit $\mathbb{O}_q$ is not special in general, we need to consider non-integral highest weight representations.
Let $\rho$ be the half sum of positive roots of $\mathfrak{g}$ and
$\check{\rho}$ the half sum of positive coroots of $\mathfrak{g}$. We denote by $J_{\lambda}\subset U(\mathfrak{g})$ the annihilator of the simple highest weight representation $L(\lambda)$
of $\mathfrak{g}$ with highest weight $\lambda$, and by $\operatorname{Var}(I)$ the associated variety of a primitive ideal $I \subset U(\mathfrak{g})$. Our main result is the following.
\begin{Th}\label{MainThm}
Let $q$ be a positive integer, and set
\begin{align*}
\lambda_q=\begin{cases}
\rho/q-\rho&\text{if }(q,\check{r})=1,\\
\check{\rho}/q-\rho&\text{if }(q,\check{r})\ne 1.
\end{cases}
\end{align*}
Then $$\operatorname{Var}(J_{ \lambda_q})=\overline{\mathbb{O}}_q.$$
\end{Th}
The proof of Theorem \ref{MainThm}
goes as follows.
We
use the representation theory of admissible {affine vertex algebras}
to show that $\operatorname{Var}(J_{\lambda_q})$
is contained in $\overline{\mathbb{O}}_q$.
Then we show that
the dimension of $\operatorname{Var}(J_{ \lambda_q})$
is equal to
that of $\overline{\mathbb{O}}_q$ using the formulas obtained in \cite{ElaKacVin08}. Since both varieties are irreducible, the equality follows.
Let $k = -\check{h} + p/q$ be an admissible level and
$\L_k(\lambda)$ the irreducible representation of $\widehat{\mathfrak{g}}$ of highest weight $\lambda + k\Lambda_0$.
By \cite{A12-2}, the irreducible $\widehat{\mathfrak{g}}$-module
$\L_k(\lambda)$ is a module over the vertex algebra $\L_k(\mathfrak{g})$ if and only if
$\lambda+k\Lambda_0$ is an admissible weight whose integral root system is isomorphic to that
of $k\Lambda_0$.
This vertex algebra is quasi-lisse \cite{Ara09b} (see \cite{AraKaw18} for the definition), and it is known that the characters, that is to say graded dimensions, of ordinary modules of lisse, and more generally quasi-lisse, vertex algebras have modular invariance properties \cite{Zhu96,Miyamoto,AraKaw18}. This structure, in turn, permits the application of techniques from the theory of modular forms to the representation theory of such vertex algebras. In our recent work \cite{AvEM} we have applied such techniques to discover and prove isomorphisms between affine vertex algebras and affine $W$-algebras known as collapsing levels. In this context the minimal value among the conformal dimensions of ordinary modules of a quasi-lisse vertex algebra is an important invariant which controls asymptotic behaviour of the characters, and the results of the present paper relate to this circle of ideas through the following result:
\begin{Pro}[{\cite[Proposition 3.12]{AvEM}}]
\label{prop:min_conformal_Lk}
Assume that $k$ is admissible and that
$\lambda+k\Lambda_0$ is an admissible weight whose integral root system is isomorphic to that
of $k\Lambda_0$. Then
\begin{align*}
\operatorname{Var}(J_{\lambda})\subset \overline{\mathbb{O}}_q,\quad
| \lambda+\rho|\geqslant | \lambda_q+\rho|,
\end{align*}
and the equality holds if and only if
$\operatorname{Var}(J_{ \lambda})= \overline{\mathbb{O}}_q$
and $ J_{\lambda}= J_{\lambda_q}$.
In particular, $\L_k(\lambda_q)$ is a simple representation of the vertex algebra $\L_k(\mathfrak{g})$ and
possesses the minimal conformal dimension
among the simple $\L_k(\mathfrak{g})$-modules that belong to the category $\mathcal{O}_k$.
\end{Pro}
We remark that Theorem \ref{MainThm} and Proposition \ref{prop:min_conformal_Lk} together are somewhat similar in spirit to \cite[Proposition 5.10]{BarVog85}.
One sees from the statement of Proposition \ref{prop:min_conformal_Lk} that the weight $\lambda_q+k\Lambda_0$
is not the unique
one that gives the minimal conformal dimension unless $k \in \mathbb{Z}_{\geqslant 0}$. The statement is in some ways more pleasant, however, after Drinfeld-Sokolov reduction.
Let $f\in \mathfrak{g}$ be a nilpotent element, and let $H_{DS,f}^0(?)$ be the Drinfeld-Sokolov reduction
functor associated with $f$.
The following theorem is proven in \cite{AvEM}, too.
\begin{Th}[{\cite[Theorem 4.11]{AvEM}}]
\label{Th:min-conf-dim}
Let $k$ be admissible,
and assume that $f$ admits an even good grading defined by a semisimple element $x^0$.
\begin{enumerate}
\item Let $f\in \overline{\mathbb{O}}_k$, so that $H_{DS,f}^0(\L_k(\mathfrak{g}))\cong \W_k(\mathfrak{g},f)$ (see \cite{AEkeren19}).
Then among the simple positive energy $\W_k(\mathfrak{g},f)$-modules, the minimal conformal dimension is given by $h_{\lambda_q}$ where in general
$$h_\lambda =\frac{|\lambda+\rho|^2-|\rho|^2}{2(k+\check{h})}-\frac{k+\check{h}}{2}|x^0|^2+(x^0,\rho)$$
is the conformal dimension of the $\W_k(\mathfrak{g},f)$-module $H_{DS,f}^0(\L_k(\lambda))$.
\item Suppose further that $f\in \mathbb{O}_k$, so that $\W_k(\mathfrak{g},f)$ is lisse. Then the minimal conformal dimension~$h_{\lambda_q}$ is realised by a unique
simple $\W_k(\mathfrak{g},f)$-module.
\end{enumerate}
\end{Th}
It seems to be widely believed that
a rational, lisse, simple, self-dual conformal vertex algebra $V$
admits a a unique simple module of minimal conformal dimension.
Theorem \ref{Th:min-conf-dim} confirms
this assertion
for exceptional $W$-algebras (\cite{KacWak03,Ara09b})
(in a wider sense, see \cite{AEkeren19})
that are lisse \cite{Ara09b} and rational
(\cite{Ara13, Ara09b, CreLin18,AEkeren19, Fas22,CreLin})\footnote{The rationality of
exceptional $W$-algebras has been recently proved in full generality by McRae \cite{McRae21}.}.
In another direction, let us mention that
one can show that
\begin{align}
\label{eq:small_quantum}
\overline{\mathbb{O}}_q\cong \operatorname{Spec}H^\bullet(\mathfrak{u}_{\zeta}(\mathfrak{g}),\mathbb{C})
\end{align}
based on
the works \cite{GinKum93,BenNakPar14},
where $\mathfrak{u}_{\zeta}(\mathfrak{g})$ is the small quantum group associated with $\mathfrak{g}$ at
the $q$-th root $\zeta$ of unity
provided that $q$ is odd and not a bad prime for $\mathfrak{g}$.
Indeed, for $q > h$, with $h$ the Coxeter number of $\mathfrak{g}$,
$\operatorname{Spec}H^\bullet(\mathfrak{u}_{\zeta}(\mathfrak{g}),\mathbb{C})$ is known to be the nilpotent cone
$\mathcal{N}$ of $\mathfrak{g}$ by \cite{GinKum93}
and $\mathcal{N}=\overline{\mathbb{O}}_q$ by \cite{Ara09b}. For $q \leqslant h$,
the authors of \cite{BenNakPar14} proved that
$\operatorname{Spec}H^\bullet(\mathfrak{u}_{\zeta}(\mathfrak{g}),\mathbb{C})$
is the closure $G.\mathfrak{u}_{I_q}$ of some Richardson nilpotent orbit
described in term of a set
of simple roots $I_q \subset \Pi$ which only depends on $q$.
Here $\Pi$ is a set of simple roots of $\mathfrak{g}$, and $\mathfrak{u}_I$ denotes the nilpotent radical of the standard parabolic subalgebra of $\mathfrak{g}$ corresponding to a subset $I \subset \Pi$. It is straightforward to check that $G.\mathfrak{u}_{I_q}$ coincides with
$\overline{\mathbb{O}_{q}}$ from the tables in \cite{Ara09b}.
The equality \eqref{eq:small_quantum} suggests a strong connection between
the representations of $\mathbb{L}_k(\mathfrak{g})$ and of $\mathfrak{u}_{\zeta}(\mathfrak{g})$,
which will be studied in a forthcoming paper \cite{ACK}.
Finally, we remark that
the statement of Theorem \ref{MainThm} for $(q,\check{r})=1$,
that is,
the equality
$\operatorname{Var}(J_{\rho/q-\rho})
=\mathcal{N}_q$,
seems to be true without the assumption that
$(q,\check{r})=1$.
\subsection*{Acknowledgements.}
The authors thank Anna Lachowska for bringing
the paper \cite{BenNakPar14} to our attention.
TA is partially supported by JSPS KAKENHI Grant Numbers 17H01086, 17K18724,
21H04993.
JvE was supported by the Serrapilheira Institute
(grant number Serra -- 1912-31433) and by CNPq grants 409582/2016-0 and 303806/2017-6.
AM is partially supported by ANR Project GeoLie Grant number ANR-15-CE40-0012.
\section{Admissible affine vertex algebras}
\label{sec:admissible}
In this section we prove the inclusions $\subset$ in Theorem \ref{MainThm}
using
the representation theory of admissible affine vertex algebras.
Fix a triangular decomposition $\mathfrak{g}=\mathfrak{n}_-\+\mathfrak{h}\+\mathfrak{n}_+$,
$\Delta$ the set of roots of $\mathfrak{g}$,
$\Delta_+$ a set of positive roots.
Let $Q$, $P$, $\check{Q}$, $\check{P}$
be the root lattice, the weight lattice, the coroot lattice and the coweight lattice, respectively. Let $\alpha^{\vee}=2\alpha/(\alpha|\alpha)$ denote the coroot corresponding to the root $\alpha$. The Weyl vector and covector are given, respectively, by $\rho = \tfrac{1}{2} \sum_{\alpha \in \Delta} \alpha$ and $\check{\rho} = \tfrac{1}{2} \sum_{\alpha \in \Delta} \alpha^\vee$. Let $h$ and $\check{h}$ be the
Coxeter number
and the dual Coxeter number of $\mathfrak{g}$, respectively. We have
\begin{align}
h=( \check{\rho}|\theta )+1,\quad
\check{h}= ( \rho|\check{\theta})+1,
\label{eq:Coxternum}
\end{align}
where $\theta$ is the highest root of $\mathfrak{g}$. We denote by $\check{\mathfrak{g}}$ the Langlands dual Lie algebra of $\mathfrak{g}$, characterized by its root system $\Delta(\check{\mathfrak{g}}) = \{\alpha^\vee / \check{r} \colon \alpha \in \Delta(\mathfrak{g})\}$.
Let $\widehat{\mathfrak{g}} = \mathfrak{g}[t,t^{-1}] \oplus \mathbb{C} K
$ be the affine Kac-Moody algebra,
with the commutation relations:
$$[x t^m,y t^n] = [x,y] t^{m+n} + m \delta_{m+n,0} (x|y) K,
\quad
[K,\widehat{\mf{g}}]=0,$$
for all $x,y \in\mathfrak{g}$ and all $m,n\in\mathbb{Z}$, where
$(~|~)=\displaystyle{\frac{1}{2 \check{h}}}\times$Killing form
is the normalized invariant inner product of $\mathfrak{g}$ and
and $x t^n$ stands for $x \otimes t^n$, for $x \in \mathfrak{g}$, $n \in \mathbb{Z}$.
Let $\widehat{\mathfrak{g}}=\widehat{\mathfrak{n}}_-\+ \widehat{\mathfrak{h}}\+\widehat{\mathfrak{n}}_+$
be the
standard triangular decomposition,
that is,
$\widehat{\mathfrak{h}} = \mathfrak{h} \oplus \mathbb{C} K$ the Cartan subalgebra of $\widehat{\mathfrak{g}}$,
$\widehat{\mathfrak{n}}_+=\mathfrak{n}_++t\mathfrak{g}[t]$,
$\widehat{\mathfrak{n}}_-=\mathfrak{n}_-+t^{-1}\mathfrak{g}[t^{-1}]$.
Let $\widehat{\mf{h}}^*=\mathfrak{h}^*\+ \mathbb{C}\Lambda_0$ be the dual of $\widehat{\mathfrak{h}}$,
where $(\Lambda_0|K)=1$,
$(\Lambda_0|\mathfrak{h})=0$. The affine Weyl vector is $\hat{\rho}=\rho+\check{h}\Lambda_0$.
Let $\tilde{\mathfrak{h}}=\widehat{\mf{h}}\+ \mathbb{C} D$ be the extended Cartan subalgebra of $\widehat{\mf{g}}$,
$\tilde{\mathfrak{h}}^*=\widehat{\mf{h}}^*\+ \mathbb{C} \delta$ the dual of $\tilde{\mathfrak{h}}$,
with $(\delta|D)=1$, $(\delta|\mathfrak{h})=0$.
Let $\widehat{\Delta}^{\text{re}}\subset \tilde{\mathfrak{h}}^*$ be the real root system,
$\widehat{\Delta}^{\text{re}}_+$ the standard set of positive roots in $\widehat{\Delta}^{\text{re}}$.
We have
\begin{align*}
\widehat{\Delta}^{\text{re}} &= \{\alpha+n\delta \mid n \in \mathbb{Z}, \alpha \in \Delta\}= \widehat{\Delta}^{\text{re}}_+\sqcup \left(-\widehat{\Delta}^{\text{re}}_+\right),\\
&\widehat{\Delta}^{\text{re}}_+
\Delta_+
\sqcup \{\alpha+n\delta\mid \alpha\in \Delta
, \ n> 0\}.
\end{align*}
Let $\widehat{W}=W\ltimes \check{Q}$ be the affine Weyl group of $\widehat{\mf{g}}$ and $\widetilde{W}=W\ltimes \check{P}$ the extended affine Weyl group. Here the action of the element $\beta \in \check{Q}$ or $\check{P}$ is via the translation $t_\beta$ defined as
\begin{align*}
t_\beta(\lambda) = \lambda + (\lambda | \delta) \beta - \left[(\lambda | \beta) + \tfrac{1}{2}|\beta|^2 (\lambda | \delta) \right] \delta.
\end{align*}
For $\lambda\in \widehat\mathfrak{h}^*$, let
$$\widehat{\Delta}(\lambda)=\{\alpha\in \widehat{\Delta}^{\text{re}}\mid {\langle} \lambda+\hat{\rho},\alpha^{\vee}{\rangle}
\in \mathbb{Z}
\},$$
the set of roots integral with respect to $\lambda$. A weight $\lambda \in \widehat\mathfrak{h}^*$
is said to be \emph{admissible} if
\begin{enumerate}
\item $\lambda$ is regular dominant, that is,
${\langle} \lambda+\hat{\rho},\alpha^{\vee}{\rangle} >0$ for all $\alpha\in \widehat{\Delta}_+(\lambda) =\widehat{\Delta}(\lambda)\cap
\widehat{\Delta}^{\text{re}}_+$,
\item $ \mathbb{Q}\widehat{\Delta}^{\text{re}} = \mathbb{Q}\widehat{\Delta}(\lambda)$.
\end{enumerate}
We shall denote by $\Adm^k$ the set of admissible weights of level $k$. The irreducible highest weight representation $\L(\lambda)$ is called
admissible if $\lambda$ is admissible.
Given any $k\in\mathbb{C}$,
let
\begin{align*}
\mathbb{V}^k(\mf{g}) =
U(\widehat{\mf{g}})\otimes_{U(\mathfrak{g}[t]\oplus \mathbb{C} K)}\mathbb{C}_k,
\end{align*}
where $\mathbb{C}_k$ is the one-dimensional representation
of
$\mathfrak{g}[t] \oplus \mathbb{C} K$ on which $\mathfrak{g}[t]$
acts by 0 and $K$ acts as a multiplication by the scalar $k$.
There is a unique vertex algebra structure
on $\mathbb{V}^k(\mathfrak{g})$ such that ${|0\rangle}$ is the image of $1\otimes 1$
in $\mathbb{V}^k(\mathfrak{g})$ and
$$x(z) :=(x_{(-1)}{|0\rangle})(z)= \sum_{n\in\mathbb{Z}} (xt^n) z^{-n-1}$$
for all $x\in \mathfrak{g}$, where we regard $\mathfrak{g}$ as a subspace of $\mathbb{V}^k(\mathfrak{g})$
through the embedding $x \in \mathfrak{g} \hookrightarrow x_{(-1)}{|0\rangle} \in \mathbb{V}^k(\mathfrak{g})$.
The vertex algebra $\mathbb{V}^k(\mathfrak{g})$ is called the {\em universal affine vertex algebra} associated with $\mathfrak{g}$ at level $k$.
Any graded quotient of $\mathbb{V}^k(\mathfrak{g})$ inherits a vertex algebra structure from
$\mathbb{V}^k(\mathfrak{g})$.
In particular,
the unique simple graded quotient $\L_k(\mathfrak{g})$ of $\mathbb{V}^k(\mathfrak{g})$
is a vertex algebra, isomorphic to $\L(k\Lambda_0)$ as a $\widehat{\mathfrak{g}}$-module.
For a graded quotient $V$ of $\mathbb{V}^k(\mathfrak{g})$,
the Zhu $C_2$-algebra \cite{Zhu96} can be defined as
the quotient
$$R_V=V/t^{-2}\mathfrak{g}[t^{-1}] V.$$
There is a surjective linear map
$S(\mathfrak{g})\rightarrow R_V$ that sends the monomial $x_1\dots x_r$, $x_i\in \mathfrak{g}$,
to $(x_1t^{-1})\dots (x_rt^{-1})|0{\rangle}+t^{-2}\mathfrak{g}[t^{-1}] V $,
and the kernel of this map is a Poisson ideal
of $S(\mathfrak{g})=\mathbb{C}[\mathfrak{g}^*]$.
Hence, $R_V$ is a Poisson algebra.
The associated variety of $V$ is by definition
the Poisson variety
\begin{align*}
X_V=\operatorname{Specm}R_V,
\end{align*}
which is a $G$-invariant and
conic subvariety of $\mathfrak{g}^*$.
The simple vertex algebra $\L_k(\mathfrak{g})$ is called the {admissible affine vertex algebra}
if $\L_k(\mathfrak{g})$ is {admissible} as representation over the affine Kac-Moody algebra
$\widehat{\mf{g}}$ associated with $\mathfrak{g}$,
or equivalently,
$k\Lambda_0$ is the admissible weight.
If this is the case,
the level $k$ is called an {admissible number} for $\widehat{\mf{g}}$.
By \cite[Proposition 1.2]{KacWak08}, $k$ is an admissible number if and only if
\begin{align}
k+\check{h}=\frac{p}{q},\quad p,q\in \mathbb{Z}_{\geqslant 1},\
(p,q)=1,\quad p\geqslant \begin{cases}
\check{h}&\text{if }(q,\check{r})=1,\\
h&\text{if }(q,\check{r})\ne 1.
\end{cases}
\label{eq:ad-number}
\end{align}
If this is the case we have
\begin{align*}
\widehat{\Delta}(k\Lambda_0)=
\begin{cases}
\{\alpha+nq\delta\mid \alpha\in \Delta,\ n\in \mathbb{Z}\}
&\text{if }(q,\check{r})=1,\\
\{\alpha+nq\delta\mid \alpha\in \Delta_{\text{long}},\ n\in \mathbb{Z}\}\\
\quad \sqcup
\{\alpha+\frac{nq}{\check{r}}\delta\mid \alpha\in \Delta_{\text{short}},\ n\in \mathbb{Z}\}&\text{if }(q,\check{r})\ne 1,
\end{cases}
\end{align*}
where
$\Delta_{\text{long}}$ and $\Delta_{\text{short}}$ are the sets of long roots and short roots, respectively.
We now recall some results on admissible affine vertex algebras, their associated varieties and modules. To state the results we recall the following subset of admissible weights
\begin{align*}
\Adm^k_{\circ}=\{\lambda\in \Adm^k\mid \widehat{\Delta}(\lambda)=w(\widehat{\Delta}(k\Lambda_0))
\text{ for some }w\in \widetilde{W}
\}.
\end{align*}
\begin{Th}[\cite{Ara09b}]
Let $k$ be an admissible number with denominator $q$ as in \eqref{eq:ad-number}.
Then
\begin{align*}
X_{\L_k(\mathfrak{g})}=\overline{\mathbb{O}}_q.
\end{align*}
\end{Th}
\begin{Th}[\cite{A12-2}]\label{Th:classification-of-simple-modules}
Let $k$ be an admissible number,
$\lambda\in \widehat{\mf{h}}^*$.
Then $\L(\lambda)$ is a $\L_k(\mathfrak{g})$-module if and only if
$\lambda\in \Adm^k_{\circ}$.
\end{Th}
\begin{Pro}\label{Pro:lamq-is-adm}
For a non-negative integer $q$
define $\hat\lambda_q\in \widehat{\mf{h}}^*$ by
\begin{align*}
\hat\lambda_q+\hat{\rho}
=\begin{cases}\frac{1}{q}\left(\rho+p\Lambda_0\right)&\text{ if }(q,\check{r})=1,\\
\frac{1}{q}\left(\check{\rho}+p\Lambda_0\right)&\text{ if }(q,\check{r})\ne 1.
\end{cases}
\end{align*}
If $k$ is an admissible number with denominator $q$, as in \eqref{eq:ad-number}, then $\hat\lambda_q\in \Adm^k_{\circ}$.
\end{Pro}
\begin{proof}
By \eqref{eq:Coxternum},
we have
${\langle} \hat\lambda_q+\hat{\rho},\alpha^{\vee}{\rangle} >0$
for $\alpha\in \widehat{\Delta}^{\text{re}}_+$.
Hence $\hat\lambda_q$ is regular dominant.
It is remain to show that
$\widehat{\Delta}(\hat\lambda_q)=y(\widehat{\Delta}(k\Lambda_0))$ for some $y\in \widetilde{W}$.
Firstly we suppose that $(q,\check{r})=1$,
so that
$(\check{r} p,q)=1$.
Take $c,d\in \mathbb{Z}$ such that
$c\check{r}p+dq=-1$,
and set
$$\mu=c\check{r}\rho\in \check{P}.$$
Then we have
$t_{-\mu}(\alpha+n\delta)=\alpha+\left[n + c\check{r}({\rho} | \alpha)\right]\delta$ and it follows that
\begin{align*}
{\langle} \hat\lambda_q+\hat{\rho},t_{-\mu}(\alpha+n\delta)^{\vee}{\rangle}=
\frac{1}{q}\left(\frac{2np}{(\alpha|\alpha)}+(1+c\check{r}p) (\rho | \alpha^{\vee})\right)\equiv \frac{2}{(\alpha|\alpha)}
\frac{np}{q}\pmod{\mathbb{Z}}.
\end{align*}
Thus
$t_{-\mu}(\alpha+n\delta)\in \widehat{\Delta}(\hat\lambda_q)$ if and only if $n\equiv 0\pmod{q}$, and so we conclude that
\begin{align}
\widehat{\Delta}(\hat\lambda_q)=t_{-\mu}(\widehat{\Delta}(k\Lambda_0)).
\end{align}
Next we suppose that $(q,\check{r})>1$ so that $\check{r}|q$.
Take $c,d\in \mathbb{Z}$ such that $cp-dq=-1$,
and set
\begin{align*}
\mu=c\check{\rho}\in \check{P}.
\end{align*}
Then we have $t_{-\mu}(\alpha+n\delta)=\alpha+\left[n + c(\check{\rho} | \alpha)\right]\delta$ and it follows that
\begin{align*}
{\langle} \hat\lambda_q+\hat{\rho},t_{-\mu}(\alpha+n\delta)^{\vee}{\rangle}=
\frac{1}{q}\left(\frac{2np}{(\alpha|\alpha)}+(1+cp)(\check{\rho} | \alpha^{\vee})\right)\equiv \frac{2}{(\alpha|\alpha)}
\frac{np}{q}\pmod{\mathbb{Z}}.
\end{align*}
Thus,
$t_{-\mu}(\alpha+n\delta)\in \widehat{\Delta}(\hat\lambda_q)$ if and only if $n\equiv \begin{cases}0\pmod{q}&\text{if }
\alpha\in \Delta_{\text{long}}\\
0\pmod{q/\check{r}}&\text{if }\alpha\in \Delta_{\text{short}}\end{cases}$.
We have thus shown that $\widehat{\Delta}(\hat\lambda_q)=t_{-\mu}(\widehat{\Delta}(k\Lambda_0))$.
\end{proof}
Let $\lambda_q$ be the
the restriction of a weight $\hat\lambda_q \in \widehat{\mf{h}}^*$ to $\mathfrak{h}$.
Then
\begin{align}
\lambda_q=\begin{cases}
\rho/q-\rho&\text{if }(q,\check{r})=1,\\
\check{\rho}/q-\rho&\text{if }(q,\check{r})\ne 1
\end{cases}
\end{align}
as in introduction.
\begin{Pro}\label{Pro:included}
We have
$\operatorname{Var}(J_{\lambda_q})\subset \overline{\mathbb{O}}_q$.
\end{Pro}
\begin{proof}
Let $k$ be admissible number with denominator $q$,
and
let $\operatorname{Zhu}(\L_k(\mathfrak{g}))$ be the Zhu algebra of $\L_k(\mathfrak{g})$.
Then $\operatorname{Zhu}(\L_k(\mathfrak{g}))\cong U(\mathfrak{g})/I_k$ for some two-sided ideal $I_k$ of $U(\mathfrak{g})$.
By \cite[Theorem 9.5]{A2012Dec},
we have $$\operatorname{Var}(I_k)=X_{\L_k(\mathfrak{g})}=\overline{\mathbb{O}}_q.$$
By Proposition \ref{Pro:lamq-is-adm} we have $\lambda_q\in \Adm^k_{\circ}$, and so by Theorem \ref{Th:classification-of-simple-modules} in turn,
$\L(\hat\lambda_q)$ is an $\L_k(\mathfrak{g})$-module. Hence,
$L({\lambda}_q)$ is a $\operatorname{Zhu}(\L_k(\mathfrak{g}))$-module.
This immediately implies $I_k\subset J_{ \lambda_q}$,
and therefore
$\operatorname{Var}(J_{ \lambda_q})\subset \operatorname{Var}(I_k)=\overline{\mathbb{O}}_q$.
\end{proof}
\section{Proof of Theorem \ref{MainThm}}
In view of Proposition \ref{Pro:included},
it suffices to show that
\begin{align}
\dim \operatorname{Var}(J_{\lambda_q})=\dim \overline{ \mathbb{O}}_q.
\end{align}
Since $\lambda_q$ is a regular dominant weight, we have
\begin{align*}
\dim \operatorname{Var}(J_{\lambda_q})=\dim \mathcal{N}-|\Delta(\lambda_q)|
\end{align*}
by \cite[Corollary 3.5]{Jos78},
where $$\Delta(\lambda_q)=\{\alpha\in \Delta\mid {\langle} \lambda_q+\rho,\alpha^{\vee}{\rangle} \in \mathbb{Z}\}=\widehat{\Delta}(\hat\lambda_q)\cap \Delta.$$
Therefore it is sufficient to show that
\begin{align}
|\Delta(\lambda_q)|=\dim \mathcal{N}-\dim \overline{\mathbb{O}}_q,
\label{eq:required-equality}
\end{align}
or rather (substituting an equivalent expression for the right hand side), that
$$|\Delta(\lambda_q)| = \dim \mathfrak{g}^f - \operatorname{rk}(\mathfrak{g}),$$
for $f \in \mathbb{O}_q$, where $\mathfrak{g}^f$ is the centralizer of $f$ in $\mathfrak{g}$.
The rest of the section is devoted to a case-by-case proof of \eqref{eq:required-equality}.
\subsection{Proof of \eqref{eq:required-equality} for the simple classical Lie algebras}
\label{sub:classical}
In this paragraph, we show that \eqref{eq:required-equality} holds for all simple Lie
algebra $\mathfrak{g}$ of classical type.
Let $n\in \mathbb{Z}_{>0}$ and assume that $\mathfrak{g}$ is either $\mathfrak{sl}_n$, $\mathfrak{so}_n$ or $\mathfrak{sp}_n$.
First we set up notation. We denote by $\mathscr{P}(n)$ the set
of partitions of $n$ and, unless otherwise specified, we write
an element $\boldsymbol{\lambda}$
of $\mathscr{P}(n)$
as a decreasing sequence $\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_r)$ of positive integers.
Thus,
$$
\lambda_1 \geqslant \cdots \geqslant \lambda_r \geqslant 1\quad
\text{ and }\quad
\lambda_1 + \cdots + \lambda_r = n.
$$
We write $\boldsymbol{\lambda}^t$ for the dual partition of $\boldsymbol{\lambda}$.
$\bullet$ By \cite[Theorem~5.1.1]{CMa}, nilpotent orbits of $\mathfrak{sl}_{n}$ are
parametrized by $\mathscr{P}(n)$. For $\boldsymbol{\lambda}\in\mathscr{P}(n)$,
we shall denote by
$\mathbb{O}_{\boldsymbol{\lambda}}$ the corresponding nilpotent orbit of
$\mathfrak{sl}_n$.
For $f \in \mathbb{O}_{\boldsymbol{\lambda}}$ the dimension of the centralizer of $f$ in $\mathfrak{g}=\mathfrak{sl}_{n}$ is given by
$$\dim \mathfrak{g}^{f}= \sum_{i=1}^t \mu_i^2 - 1,$$
where $(\mu_1,\ldots,\mu_t) = \boldsymbol{\lambda}^t$.
\noindent
$\bullet$ Set
$$\mathscr{P}_{1}(n):=\{\boldsymbol{\lambda} \in \mathscr{P}(n)\; \colon \; \text{multiplicity of each even
part is even}\}.
$$
By \cite[Theorems 5.1.2 and 5.1.4]{CMa},
nilpotent orbits of $\mathfrak{so}_{n}$
are parametrized by $\mathscr{P}_1(n)$, with the exception that each
{\em very even}
partition $\boldsymbol{\lambda} \in\mathscr{P}_{1}(n)$ (i.e., $\boldsymbol{\lambda}$ has only even parts)
corresponds to two nilpotent orbits.
For $\boldsymbol{\lambda}\in \mathscr{P}_1(n)$, not very even, we shall denote by $\mathbb{O}_{\boldsymbol{\lambda}}$
the corresponding nilpotent orbit of $\mathfrak{so}_n$.
(Very even nilpotent orbits do not appear in this work.)
For $f \in \mathbb{O}_{\boldsymbol{\lambda}}$ the dimension of the centralizer of $f$ in $\mathfrak{g}=\mathfrak{so}_{n}$ is given by
$$\dim \mathfrak{g}^{f}= \frac{1}{2}\left(\sum_{i=1}^t \mu_i^2 -
\#\{i \; \colon \; \text{$\lambda_i$ is odd}\}\right),$$
where $(\mu_1,\ldots,\mu_t) = \boldsymbol{\lambda}^t$.
\noindent
$\bullet$
Set
$$
\mathscr{P}_{-1}(n):=\{\boldsymbol{\lambda} \in \mathscr{P}(n)\; \colon \; \text{multiplicity of each odd part is even}\}.
$$
By \cite[Theorem~5.1.3]{CMa}, nilpotent orbits of $\mathfrak{sp}_{n}$
are parametrized by $\mathscr{P}_{-1}(n)$.
For $\boldsymbol{\lambda}=(\lambda_{1},\dots ,\lambda_{r})\in \mathscr{P}_{-1}(n)$,
we shall denote by $\mathbb{O}_{\boldsymbol{\lambda}}$
the corresponding nilpotent orbit of $\mathfrak{sp}_{n}$.
For $f \in \mathbb{O}_{\boldsymbol{\lambda}}$ the dimension of the centralizer of $f$ in $\mathfrak{g}=\mathfrak{sp}_{n}$ is given by
$$\dim \mathfrak{g}^{f}= \frac{1}{2}\left(\sum_{i=1}^t \mu_i^2 + \#\{i \; \colon \; \text{$\lambda_i$ is odd}\}\right),$$
where $(\mu_1,\ldots,\mu_t) = \boldsymbol{\lambda}^t$.
We recall that the \emph{height} of an element $\beta = \sum_{\alpha \in \Delta} k_\alpha \alpha$ of the root lattice $Q$ is by definition the integer $\operatorname{ht}(\beta) = \sum_{\alpha \in \Delta} k_\alpha$. Equivalently $\operatorname{ht}(\beta) = {\langle} \check{\rho}, \beta {\rangle}$. In order to show that \eqref{eq:required-equality} holds in the classical cases,
we exploit a combinatorial formula for
$\# \{\alpha \in \Delta \colon
q | \operatorname{ht}(\alpha) \in \mathbb{Z} \}$ proved in \cite[\S 3.2]{ElaKacVin08}. To state the formula we first write
$$n=q m_0+s_0,$$
with $1 \leqslant s_0 \leqslant q$ if $\mathfrak{g}=\mathfrak{so}_n$ and $n$ even, and $0 \leqslant s_0 \leqslant q-1$
in all other cases. We consider the partition $(q^{m_0},s_0)$ of $n$, and we set
$$
K_n(q) =m_0^2 (q-s_0)+ (m_0+1)^2 s_0.
$$
In fact if $\mathfrak{g} = \mathfrak{sl}_n$ then $K_n(q)-1$ is the dimension of the centralizer of a nilpotent
element $f \in \mathfrak{g}$ associated with the partition $(q^{m_0},s_0)$. By \cite[\S 3.2]{ElaKacVin08},
\begin{align}
\label{eq:EKV_relation}
\# \{\alpha \in \Delta \colon
q | \operatorname{ht}(\alpha) \in \mathbb{Z} \} = d_\mathfrak{g}(q) - \operatorname{rk}(\mathfrak{g}),
\end{align}
where $d_\mathfrak{g}(q)$ is expressed in terms of $n$ and $q$ as follows:
\begin{enumerate}
\item for $\mathfrak{g}=\mathfrak{sl}_n$,
$$d_\mathfrak{g}(q)=K_n(q)-1.$$
\item for $\mathfrak{g}=\mathfrak{sp}_n$,
$$2 d_\mathfrak{g}(q)=K_n(q)+\begin{cases} m_0, & \text{ if } q \text{ is odd, }m_0 \text{ is even}, \\
m_0+1, & \text{ if } q,m_0 \text{ are odd}, \\
0, & \text{ if } q \text{ is even}.\\
\end{cases}$$
\item for $\mathfrak{g}=\mathfrak{so}_n$, $n$ odd,
$$2 d_\mathfrak{g}(q)=K_n(q)- \begin{cases} m_0, & \text{ if } q,m_0 \text{ are odd},\\
m_0+1, & \text{ if } q \text{ is odd, }m_0 \text{ is even}, \\
2m_0+1, & \text{ if } q \text{ is even}.\\
\end{cases}$$
\item for $\mathfrak{g}=\mathfrak{so}_n$, $n$ even,
$$2 d_\mathfrak{g}(q)=K_n(q)- \begin{cases} m_0, & \text{ if } q \text{ is odd, }m_0 \text{ is even}, \\
m_0+1, & \text{ if } q,m_0 \text{ are odd},\\
2m_0, & \text{ if } q,m_0 \text{ are even},\\
2(m_0+1), & \text{ if } q \text{ is even, }m_0 \text{ is odd}.\\
\end{cases}$$
\end{enumerate}
\subsubsection{First case: $(q,\check{r})=1$}
\label{subsub:princ}
In this case we have
$$
|\Delta(\lambda_q)|
= \# \{\alpha \in \Delta \colon {\langle} \rho,\alpha^{\vee}{\rangle} \in q\,\mathbb{Z} \}.
$$
From \eqref{eq:EKV_relation}, using $\operatorname{rk}(\mathfrak{g})= \operatorname{rk}(\check{\mathfrak{g}})$, we have
\begin{align*}
\# \{\alpha \in \Delta(\check{\mathfrak{g}}) \colon {\langle} \rho^\vee,\alpha{\rangle} \in q\,\mathbb{Z} \}
&= \# \{\alpha \in \Delta(\check{\mathfrak{g}}) \colon q | \operatorname{ht}(\alpha) \} \\
&= d_{\check{\mathfrak{g}}}(q) - \operatorname{rk}(\mathfrak{g}),
\end{align*}
(this holds whether or not $(q,\check{r})=1$).
Using Lemma \ref{Lem:verif_d(q)_dual} below, we conclude that it is enough to verify the equality
\begin{align}
\label{eq:verification_d(q)}
\dim \mathfrak{g}^f = d_{\mathfrak{g}}(q)
\end{align}
for $f \in \mathbb{O}_q$.
\begin{lemma}
\label{Lem:verif_d(q)_dual}
For any integer $q$, we have $d_{\check{\mathfrak{g}}}(q)=d_{\mathfrak{g}}(q)$.
\end{lemma}
\begin{proof}
Notice that $\mathfrak{g}$ and $\check{\mathfrak{g}}$
share the same rank $\ell$ and the same exponents $m_1 \leqslant m_2\leqslant \cdots \leqslant m_\ell$.
But it is known that, denoting the number of positive roots of fixed height $i$ by $p_i$,
the sequence $p_1 \geqslant p_2 \geqslant \cdots \geqslant p_{h-1}$, where $h$ is the Coxeter number of $\mathfrak{g}$,
gives a partition of $\Delta_+$ which is the dual partition to the partition $m_{\ell} \geqslant \cdots \geqslant m_2 \geqslant m_1$
given by the exponents (\cite[Theorem 9.1]{Steinberg}).
Then the statement follows from the equality $d_{\check{\mathfrak{g}}}(q)
= \# \{\alpha \in \Delta(\check{\mathfrak{g}}) \colon q | \operatorname{ht}(\alpha) \}+ \operatorname{rk}(\mathfrak{g})$.
\end{proof}
The rest of this paragraph is devoted to checking \eqref{eq:verification_d(q)}
for each classical type.
For $\mathfrak{g}=\mathfrak{sl}_n$ (resp.~$\mathfrak{g}=\mathfrak{so}_n$, $\mathfrak{g}=\mathfrak{sp}_n$), we
let $\boldsymbol{\lambda}$ be the the element of $\mathscr{P}(n)$ (resp.~$\mathscr{P}_{1}(n)$, $\mathscr{P}_{-1}(n)$)
associated with the nilpotent orbit $\mathbb{O}_q$,
and we choose a nilpotent element $f \in \mathbb{O}_q$.
\subsubsection*{Case $\mathfrak{g}=\mathfrak{sl}_n$}
In this case $\mathfrak{g}=\check{\mathfrak{g}}$
and by \cite[Table 2]{Ara09b}, $\boldsymbol{\lambda} = (q^m,s)$, so \eqref{eq:verification_d(q)}
obviously holds.
\subsubsection*{Case $\mathfrak{g}=\mathfrak{so}_n$, $n$ even}
In this case $\mathfrak{g}=\check{\mathfrak{g}}$. Following \cite[Table 2]{Ara09b} there are four sub-cases to analyse.
\begin{enumerate}
\item $\boldsymbol{\lambda} = (q^m,s)$ with $q,m$ odd, $0 \leqslant s\leqslant q$ odd.
Then, clearly,
$$\dim \mathfrak{g}^f=\frac{1}{2}(m^2(q-s) +(m+1)^2 s - m-1) = \dim d_\mathfrak{g}(q),$$
whence \eqref{eq:verification_d(q)}.
\item $\boldsymbol{\lambda} = (q^m,s,1)$ with $q$ odd, $m$ even, $0 \leqslant s\leqslant q-1$ odd.
Then
$$\dim \mathfrak{g}^{f} = \frac{1}{2}(m^2(q-s)+(m+1)^2 (s-1)+(m+2)^2 -m-2).$$
On the other hand, since $n=m q +s+1$ with $0 \leqslant s+1 \leqslant q$,
$$d_{\mathfrak{g}}(q)=\frac{1}{2}(m^2 (q -s-1) +(m+1)^2 (s+1) - m), $$
whence \eqref{eq:verification_d(q)}.
\item $\boldsymbol{\lambda} = (q+1,q^m,s)$ with $q,m$ even, $0 \leqslant s\leqslant q-1$ odd.
Then
$$\dim \mathfrak{g}^{f} = \frac{1}{2}(1+ (m+1)^2(q-s)+(m+2)^2 s -2).$$
On the other hand, since $n=(m+1)q +s+1$ with $0 \leqslant s+1 \leqslant q$,
$$d_{\mathfrak{g}}(q)=\frac{1}{2}((m+1)^2 (q -s-1) +(m+2)^2 (s+1) - 2(m+2)), $$
whence \eqref{eq:verification_d(q)}.
\item $\boldsymbol{\lambda} = (q+1,q^m,q-1,s,1)$ with $q,m$ even, $0 \leqslant s\leqslant q-1$ odd.
Then
$$\dim \mathfrak{g}^{f} = \frac{1}{2}(1+ (m+1)^2 + (m+2)^2 (q-1-s)+(m+3)^2 (s-1) +(m+4)^2 -4).$$
On the other hand, since $n=(m+2)q +s+1$ with $0 \geqslant s+1 \geqslant q$,
$$d_{\mathfrak{g}}(q)=\frac{1}{2}((m+2)^2 (q -s-1) +(m+3)^2 (s+1) - 2(m+2)), $$
whence \eqref{eq:verification_d(q)}.
\end{enumerate}
\subsubsection*{Case $\mathfrak{g}=\mathfrak{so}_n$, $n$ odd}
Following \cite[Table 2]{Ara09b}, there are two sub-cases to analyse.
\begin{enumerate}
\item $\boldsymbol{\lambda} = (q^m,s)$ with $q$ odd, $m$ even, $0 \leqslant s\leqslant q$ odd.
Then
$$\dim \mathfrak{g}^{f} = \frac{1}{2}(m^2(q-s)+(m+1)^2 s -m-1) = d_{\mathfrak{g}}(q),$$
whence \eqref{eq:verification_d(q)}.
\item $\boldsymbol{\lambda} = (q^m,s,1)$ with $q,m$ odd, $0 \leqslant s\leqslant q-1$ odd.
Then
$$\dim \mathfrak{g}^{f} = \frac{1}{2}(m^2(q-s)+(m+1)^2 (s-1)+(m+2)^2 -m-2).$$
On the other hand, $n-1=qm+s$ with $0 \leqslant s \leqslant q-1$.
So
$$d_{\check{\mathfrak{g}}}(q)=\frac{1}{2}(m^2 (q -s) +(m+1)^2 s +m+1) = d_\mathfrak{g}(q) ,$$
by Lemma \ref{Lem:verif_d(q)_dual}, whence \eqref{eq:verification_d(q)}.
\end{enumerate}
\subsubsection*{Case $\mathfrak{g}=\mathfrak{sp}_n$}
Following \cite[Table 2]{Ara09b}, there are two sub-cases to analyse.
\begin{enumerate}
\item $\boldsymbol{\lambda} = (q^m,s)$ with $q$ odd, $m$ even, $0 \leqslant s\leqslant q-1$ even.
Then
$$\dim \mathfrak{g}^{f} = \frac{1}{2}(m^2(q-s)+(m+1)^2 s +m)=d_{\mathfrak{g}}(q),$$
whence \eqref{eq:verification_d(q)}.
\item $\boldsymbol{\lambda} = (q^m,q-1,s)$ with $q$ odd, $m$ even, $0 \leqslant s\leqslant q-1$ even.
Then
$$\dim \mathfrak{g}^{f} = \frac{1}{2}(m^2+(m+1)^2 (q-1-s)+(m+2)^2 s +m).$$
On the other hand, $n+1=q (m+1) +s$ with $0 \leqslant s \leqslant q-1$.
So
$$d_{\check{\mathfrak{g}}}(q)=\frac{1}{2}((m+1)^2 (q -s) +(m+2)^2 s) = d_\mathfrak{g}(q),$$
by Lemma \ref{Lem:verif_d(q)_dual}, whence \eqref{eq:verification_d(q)}.
\end{enumerate}
\subsubsection{Second case: $(q,\check{r}) =\check{r}$}
\label{subsub:coprinc}
In this case $q$ is necessarily even, $\check{r}=2$ and either $\mathfrak{g}=\mathfrak{so}_n$ with $n$ odd or else $\mathfrak{g}=\mathfrak{sp}_n$.
Note that
\begin{align}
\nonumber
|\Delta(\lambda_q)| & = \# \{\alpha \in \Delta \colon
{\langle} \check{\rho},\alpha^{\vee}{\rangle} \in q\,\mathbb{Z} \}&\\
\label{eq:Delta_coprincipal}
& =
\# \{\alpha \in \Delta_{\text{long}} \colon
q | \operatorname{ht}(\alpha) \}+
\# \{\alpha \in \Delta_{\text{short}} \colon
(q/2) | \operatorname{ht}(\alpha) \}.&
\end{align}
\subsubsection*{Case $\mathfrak{g}=\mathfrak{so}_n$, $n$ odd}
Firstly we rewrite \eqref{eq:Delta_coprincipal} as
\begin{align*}
|\Delta(\lambda_q)| & = \# \{\alpha \in \Delta \colon
q | \operatorname{ht}(\alpha) \} & \\
& \qquad + \# \{\alpha \in \Delta_{\text{short}} \colon
(q/2) | \operatorname{ht}(\alpha) \}
- \# \{\alpha \in \Delta_{\text{short}} \colon
q | \operatorname{ht}(\alpha) \}.&
\end{align*}
There are exactly $\operatorname{rk}(\mathfrak{g}) = (n-1)/2$ positive short roots, one of height $i$ for each integer $i=1,2,\ldots,(n-1)/2$. It follows easily that if we write $n=q m_0 +s_0$ where $0 \leqslant s_0 \leqslant q-1$, then we have
\begin{align*}
& \# \{\alpha \in \Delta_{\text{short}} \colon
(q/2) | \operatorname{ht}(\alpha) \}
- \# \{\alpha \in \Delta_{\text{short}} \colon
q | \operatorname{ht}(\alpha) \}
= \begin{cases}
m_0 & \text{ if } m_0 \text{ is even}, \\
m_0 +1 & \text{ if } m_0 \text{ is odd}.
\end{cases}
\end{align*}
We combine equation \eqref{eq:EKV_relation} with the preceding observations to deduce that, in the present case, \eqref{eq:required-equality} is equivalent to
\begin{align}
\label{eq:verification_d(q)-coprincipal-so_n}
\dim \mathfrak{g}^f = d_\mathfrak{g}(q) +\begin{cases}
m_0 & \text{ if } m_0 \text{ is even}, \\
m_0 +1 & \text{ if } m_0 \text{ is odd}.
\end{cases}
\end{align}
We now verify that \eqref{eq:verification_d(q)-coprincipal-so_n} holds.
Following \cite[Table 3]{Ara09b}, there are two sub-cases to analyse.
\begin{enumerate}
\item $\boldsymbol{\lambda}=(q^m,s)$, with $q,m$ even, $0 \leqslant s \leqslant q-1$ odd.
In this case
$$\dim \mathfrak{g}^f = \frac{1}{2}(m^2 (q-s)+(m+1)^2 s-1) $$
On the other hand $m_0=m$ is even, and so
$$d_\mathfrak{g}(q)+m_0 = \frac{1}{2}(m^2 (q-s)+(m+1)^2 s-2m-1) + m $$
whence \eqref{eq:verification_d(q)-coprincipal-so_n}.
\item $\boldsymbol{\lambda}=(q^m,q-1,s,1)$, with $q,m$ even, $0 \leqslant s\leqslant q-1$ odd.
In this case
$$\dim \mathfrak{g}^f = \frac{1}{2}(m^2 +(m+1)^2(q-1-s)+(m+2)^2 (s-1)+(m+3)^2-3) $$
On the other hand $m_0=m+1$ is odd, and so
$$d_\mathfrak{g}(q)+m_0 + 1 = \frac{1}{2}((m+1)^2 (q-s)+(m+2)^2 s-2(m+1)-1) +m+2$$
whence \eqref{eq:verification_d(q)-coprincipal-so_n}.
\end{enumerate}
\subsubsection*{Case $\mathfrak{g}=\mathfrak{sp}_n$}
There are exactly $\operatorname{rk}(\mathfrak{g})=n/2$ positive long roots, one of height $2i+1$ for each integer $i=0,1,\ldots,n/2-1$. In particular
$$ \# \{\alpha \in \Delta_{\text{long}} \colon
q | \operatorname{ht}(\alpha) \}=0$$
since $q$ is even.
Hence \eqref{eq:Delta_coprincipal} becomes
\begin{align*}
|\Delta(\lambda_q)|
& = \# \{\alpha \in \Delta \colon
(q/2) | \operatorname{ht}(\alpha) \}
- \# \{\alpha \in \Delta_{\text{long}} \colon
(q/2) | \operatorname{ht}(\alpha) \}. &
\end{align*}
If $q/2$ is even, then by the above remarks the set $\{\alpha \in \Delta_{\text{long}} \colon
(q/2) | \operatorname{ht}(\alpha) \}$ is empty. If $q/2$ is odd, and we write $n = \left(q/2\right)m_1 +s_1$ where $0 \leqslant s_1 \leqslant \frac{q}{2}-1$, then it follows easily that
$$\{\alpha \in \Delta_{\text{long}} \colon
(q/2) | \operatorname{ht}(\alpha) \} = \begin{cases}
m_1 & \text{ if } m_1 \text{ is even,} \\
m_1+1 & \text{ if } m_1 \text{ is odd.} \\
\end{cases}$$
By combining these facts with \eqref{eq:EKV_relation} we deduce that \eqref{eq:required-equality} is equivalent, in the present case, to
\begin{align}
\label{eq:verification_d(q)-coprincipal-sp_n}
\dim \mathfrak{g}^f = d_\mathfrak{g}(q/2) - \begin{cases}
0 & \text{ if } q/2 \text{ is even,} \\
m_1& \text{ if } q/2 \text{ is odd and } m_1 \text{ is even }, \\
m_1+1 &\text{ if } q/2 \text{ is odd and } m_1 \text{ is odd}. \\
\end{cases}
\end{align}
We now verify that \eqref{eq:verification_d(q)-coprincipal-sp_n} holds.
Following \cite[Table 3]{Ara09b}, there are three sub-cases to analyse.
\begin{enumerate}
\item $\boldsymbol{\lambda}=((q/2)^m,s)$, with $q/2,m$ even, $0 \leqslant s\leqslant q/2-1$ even.
In this case
$$\dim \mathfrak{g}^f = \frac{1}{2}(m^2(q/2-s)+(m+1)^2 s) =d_\mathfrak{g}(q/2)-0,$$
whence \eqref{eq:verification_d(q)-coprincipal-sp_n}.
\item $\boldsymbol{\lambda}=(q/2+1,(q/2)^m,s)$, with $q/2$ odd, $m$ even, $0 \leqslant s\leqslant q/2-1$ even.
In this case
$$\dim \mathfrak{g}^f = \frac{1}{2}(1+(m+1)^2(q/2-s)+(m+2)^2 s+m).$$
On the other hand,
$n=(m+1)(q/2) +s+1$, with $m_1=m+1$ odd, and $0 \leqslant s+1 \leqslant q/2$.
\begin{itemize}
\item If $s< q/2-1$, then $0 \leqslant s+1 <q/2$, and
\begin{align*}
d_\mathfrak{g}(q/2) - m_1& =\frac{1}{2}((m+1)^2(q/2-s-1)+(m+2)^2(s+1)+m+2) - m-2, &
\end{align*}
whence \eqref{eq:verification_d(q)-coprincipal-sp_n}.
\item If $s=q/2-1$, then $n=(m+2)(q/2)$ with $m_1=m+2$ even,
and
\begin{align*}
d_\mathfrak{g}(q/2) - m_1 =\frac{1}{2}((m+1)^2(q/2)+(m+2)^2+m+2) - m-2 ,
\end{align*}
whence \eqref{eq:verification_d(q)-coprincipal-sp_n}.
\end{itemize}
\item $\boldsymbol{\lambda}=(q/2+1,(q/2)^m,q/2-1,s)$, with $q/2$ odd, $m$ even, $0 \leqslant s\leqslant q/2-1$ even.
In this case
$$\dim \mathfrak{g}^f = \frac{1}{2}(1+(m+1)^2 +(m+2)^2(q/2-1-s)+(m+2)^2 s+m).$$
On the other hand,
$n=(m+2)(q/2) + s$, with $m_1=m+2$ even, $0 \leqslant s \leqslant q/2-1$,
and
\begin{align*}
d_\mathfrak{g}(q/2) - m_1& =\frac{1}{2}((m+2)^2 (q/2-s)+(m+3)^2(s+1)-m-2) - (m+2) ,&
\end{align*}
whence \eqref{eq:verification_d(q)-coprincipal-sp_n}.
\end{enumerate}
\subsection{Proof of \eqref{eq:required-equality} for the simple exceptional Lie algebras}
\label{sub:exceptional}
In this paragraph, we show that \eqref{eq:required-equality} holds for all simple Lie
algebra $\mathfrak{g}$ of exceptional type.
As in \S\ref{subsub:princ} above, when $(q,\check{r})=1$ we have
using Lemma \ref{Lem:verif_d(q)_dual},
\begin{align*}
|\Delta(\lambda_q)|
&= \# \{\alpha \in \Delta \colon
{\langle} \rho,\alpha^{\vee}{\rangle} \in q \, \mathbb{Z} \} \\
&=
\# \{\alpha \in \Delta \colon
q | \operatorname{ht}(\alpha^\vee) \} = d_{\check{\mathfrak{g}}}(q) -\operatorname{rk}(\mathfrak{g})
= d_{\mathfrak{g}}(q) -\operatorname{rk}(\mathfrak{g}),
\end{align*}
and hence, by \eqref{eq:EKV_relation},
it is enough to verify that the equality
\eqref{eq:verification_d(q)}
holds for any $f \in \mathbb{O}_q$.
Note that the last three equalities hold even if $(q,\check{r})\not=1$.
Using Tables \ref{Tab:G2-princ}, \ref{Tab:F4-princ}, \ref{Tab:E6-princ}, \ref{Tab:E7-princ} and \ref{Tab:E8-princ} below,
we easily verify that \eqref{eq:verification_d(q)} holds
for $\mathfrak{g}$ any of the simple Lie algebras of type $G_2$, $F_4$,
$E_6$, $E_7$, $E_8$, and for $(q,\check{r})=1$.
In Tables \ref{Tab:G2-princ}, \ref{Tab:F4-princ}, we put in parentheses
the values of $q$ which are not coprime with $\check{r}$.
As we can see, the equality \eqref{eq:verification_d(q)} still holds in these cases.
As in \S\ref{subsub:coprinc} above, when $(q,\check{r})=\check{r}$ we have
\begin{align*}
|\Delta(\lambda_q)|
&=|\Delta(\check{\rho}/{q}-\rho)| \\
&=
\# \{\alpha \in \Delta_{\text{long}} \colon
q | \operatorname{ht}(\alpha) \}+
\# \{\alpha \in \Delta_{\text{short}} \colon (q/\check{r}) | \operatorname{ht}(\alpha) \}.
\end{align*}
Using this, we compute $|\Delta(\lambda_q)|$
for the simple Lie algebras of type $G_2$ and $F_4$ with $(q,\check{r})=\check{r}$.
From Tables \ref{Tab:G2-coprinc} and \ref{Tab:F4-coprinc},
we conclude that \eqref{eq:verification_d(q)} holds.
\bigskip
\begin{minipage}{.5\textwidth}
\centering
{\tiny
\begin{tabular}{|cccc|}
\hline
$q$ & $\mathcal{N}_{q}$ & $\dim \mathcal{N}_{q}$ & $|\Delta(\lambda_q)|$ \\
\hline
$2$ & $\tilde{A}_1$ & $8$& $4$ \\
$(3),4,5$ & $G_2(a_1)$ & $10$ & $2$ \\
$(6), \geqslant 7$ & $G_2$ & $12$ &$0$ \\
\hline
\end{tabular}
\captionof{table}{Data for $G_2$, $(q,\check{r})=1$}
\label{Tab:G2-princ}
}
\end{minipage}
\begin{minipage}{.5\textwidth}
\centering
{\tiny
\begin{tabular}{|cccc|}
\hline
$q$ & ${}^L\mathcal{N}_{q/\check{r}}$ & $\dim {}^L\mathcal{N}_{q/\check{r}}$ & $|\Delta(\lambda_q)|$ \\
\hline
$3$ & ${A}_1$ & $6$& $6$ \\
$6, 9$ & $G_2(a_1)$ & $10$ & $2$ \\
$\geqslant 12$ & $G_2$ & $12$ & $0$ \\
\hline
\end{tabular}
\captionof{table}{Data for $G_2$, $(q,\check{r})=\check{r}$}
\label{Tab:G2-coprinc}
}
\end{minipage}
\bigskip
\begin{minipage}{.5\textwidth}
\centering
{\tiny
\begin{tabular}{|cccc|}
\hline
$q$ & $\mathcal{N}_{q}$ & $\dim \mathcal{N}_{q}$ & $|\Delta(\lambda_q)|$ \\
\hline
$(2)$ & $A_1+\tilde{A}_1$ &$28$& $20$ \\
$3$ & $\tilde{A}_2+{A}_1$ & $36$ & $12$ \\
$(4), 5$ & $F_4(a_3)$ &$40$& $8$ \\
$(6), 7$& $F_4(a_2)$&$44$& $4$ \\
$(8), 9, (10),11$& $F_4(a_1)$& $46$& $2$ \\
$(12), \geqslant 13$ &$F_4$&$48$&$0$\\
\hline
\end{tabular}
\captionof{table}{Data for $F_4$, $(q,\check{r})=1$}
\label{Tab:F4-princ}
}
\end{minipage}
\begin{minipage}{.5\textwidth}
\centering
{\tiny
\begin{tabular}{|cccc|}
\hline
$q$ & ${}^L\mathcal{N}_{q/\check{r}}$ & $\dim {}^L\mathcal{N}_{q/\check{r}}$ & $|\Delta(\lambda_q)|$ \\
\hline
$2$ & $A_1$ &$16$& $32$ \\
$4$ & $A_2+\tilde{A}_1$ & $34$ & $14$ \\
$6$ & $F_4(a_3)$ &$40$& $8$ \\
$8$& $B_3$&$42$& $6$ \\
$10$& $F_4(a_2)$& $44$& $4$ \\
$12,14,16 $&$F_4(a_1)$&$46$& $2$ \\
$\geqslant 18$ &$F_4$ &$48$ &$0$ \\
\hline
\end{tabular}
\captionof{table}{Data for $F_4$, $(q,\check{r})=\check{r}$}
\label{Tab:F4-coprinc}
}
\end{minipage}
\bigskip
\begin{minipage}{.5\textwidth}
\centering
{\tiny
\begin{tabular}{|cccc|}
\hline
$q$ & $\mathcal{N}_{q}$ & $\dim \mathcal{N}_{q}$ & $|\Delta(\lambda_q)|$ \\
\hline
$2$ & $3A_1$ & $40$& $32$ \\
$3$ & $2A_2+A_1$ & $54$ & $18$ \\
$4$ &$D_4(a_1)$ & $58$& $14$ \\
$5$&$A_4+A_1$ & $62$& $10$ \\
$6,7$&$E_6(a_3)$ & $66$ & $6$ \\
$8$&$D_5$& $68$& $4$ \\
$9,10,11$&$E_6(a_1)$&$70$&$2$\\
$\geqslant 12$ & $E_6$ & $72$&$0$\\
\hline
\end{tabular}
\captionof{table}{Data for $E_6$}
\label{Tab:E6-princ}
}
\end{minipage}
\begin{minipage}{.5\textwidth}
\centering
{\tiny
\begin{tabular}{|cccc|}
\hline
$q$ & $\mathcal{N}_{q}$ & $\dim \mathcal{N}_{q}$ & $|\Delta(\lambda_q)|$ \\
\hline
$2$ & $4A_1$ & $70$& $56$ \\
$3$ & $2A_2+A_1$ & $90$ & $36$ \\
$4$ &$A_3+A_2+A_1$ & $100$& $26$ \\
$5$&$A_4+A_2$ & $106$& $20$ \\
$6$&$E_7(a_5)$ & $112$ & $14$ \\
$7$&$A_6$&$114$& $12$ \\
$8$&$E_7(a_4)$&$116$& $10$ \\
$9$&$E_6(a_1)$ &$118$ & $8$ \\
$10,11$&$E_7(a_3)$&$120$& $6$ \\
$12,13$&$E_7(a_2)$&$122$& $4$ \\
$14,\ldots,17$&$E_7(a_1)$&$124$& $2$ \\
$\geqslant 18$&$E_7$& $126$ & $0$\\
\hline
\end{tabular}
\captionof{table}{Data for $E_7$}
\label{Tab:E7-princ}
}
\end{minipage}
\bigskip
\begin{minipage}{.5\textwidth}
\centering
{\tiny
\begin{tabular}{|cccc|}
\hline
$q$ & $\mathcal{N}_{q}$ & $\dim \mathcal{N}_{q}$ & $|\Delta(\lambda_q)|$ \\
\hline
$2$ & $4A_1$ & $128$& $112$ \\
$3$ & $2A_2+A_1$ & $168$ & $72$ \\
$4$ & $2A_3$ & $188$& $52$ \\
$5$ & $A_4+A_3$ & $200$& $40$ \\
$6$ & $E_8(a_7)$ & $208$ &$32$ \\
$7$ & $A_6+A_1$& $212$& $28$ \\
$8$ & $A_7$& $218$& $22$ \\
$9$ & $E_8(b_6)$ & $220$ & $20$ \\
$10,11$ & $E_8(a_6)$ & $224$ &$16$ \\
$12,13$ & $E_8(a_5)$ & $228$ & $12$ \\
$14$ &$E_8(b_4)$ & $230$ & $10$ \\
$15,16,17$&$E_8(a_4)$ & $232$ & $8$ \\
$18,19$&$E_8(a_3)$ & $234$ & $6$ \\
$20,\ldots,23$&$E_8(a_2)$& $236$ & $4$ \\
$24,\ldots,29$&$E_8(a_1)$& $238$ & $2$ \\
$\geqslant 30$ &$E_8$&$240$&$0$\\
\hline
\end{tabular}
\captionof{table}{Data for $E_8$}
\label{Tab:E8-princ}
}
\end{minipage}
\newcommand{\etalchar}[1]{$^{#1}$}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,496
|
package models
// This file was generated by the swagger tool.
// Editing this file might prove futile when you re-run the swagger generate command
import (
"context"
"strconv"
"github.com/go-openapi/errors"
"github.com/go-openapi/strfmt"
"github.com/go-openapi/swag"
"github.com/go-openapi/validate"
)
// Task a structure describing a complete task.
//
// A Task is the main entity in this application. Everything revolves around tasks and managing them.
//
// swagger:model Task
type Task struct {
TaskCard
// The attached files.
//
// An issue can have at most 20 files attached to it.
//
Attachments map[string]TaskAttachmentsAnon `json:"attachments,omitempty"`
// The 5 most recent items for this issue.
//
// The detail view of an issue includes the 5 most recent comments.
// This field is read only, comments are added through a separate process.
//
// Read Only: true
Comments []*Comment `json:"comments"`
// The time at which this issue was last updated.
//
// This field is read only so it's only sent as part of the response.
//
// Read Only: true
// Format: date-time
LastUpdated strfmt.DateTime `json:"lastUpdated,omitempty"`
// last updated by
LastUpdatedBy *UserCard `json:"lastUpdatedBy,omitempty"`
// reported by
ReportedBy *UserCard `json:"reportedBy,omitempty"`
}
// UnmarshalJSON unmarshals this object from a JSON structure
func (m *Task) UnmarshalJSON(raw []byte) error {
// AO0
var aO0 TaskCard
if err := swag.ReadJSON(raw, &aO0); err != nil {
return err
}
m.TaskCard = aO0
// AO1
var dataAO1 struct {
Attachments map[string]TaskAttachmentsAnon `json:"attachments,omitempty"`
Comments []*Comment `json:"comments"`
LastUpdated strfmt.DateTime `json:"lastUpdated,omitempty"`
LastUpdatedBy *UserCard `json:"lastUpdatedBy,omitempty"`
ReportedBy *UserCard `json:"reportedBy,omitempty"`
}
if err := swag.ReadJSON(raw, &dataAO1); err != nil {
return err
}
m.Attachments = dataAO1.Attachments
m.Comments = dataAO1.Comments
m.LastUpdated = dataAO1.LastUpdated
m.LastUpdatedBy = dataAO1.LastUpdatedBy
m.ReportedBy = dataAO1.ReportedBy
return nil
}
// MarshalJSON marshals this object to a JSON structure
func (m Task) MarshalJSON() ([]byte, error) {
_parts := make([][]byte, 0, 2)
aO0, err := swag.WriteJSON(m.TaskCard)
if err != nil {
return nil, err
}
_parts = append(_parts, aO0)
var dataAO1 struct {
Attachments map[string]TaskAttachmentsAnon `json:"attachments,omitempty"`
Comments []*Comment `json:"comments"`
LastUpdated strfmt.DateTime `json:"lastUpdated,omitempty"`
LastUpdatedBy *UserCard `json:"lastUpdatedBy,omitempty"`
ReportedBy *UserCard `json:"reportedBy,omitempty"`
}
dataAO1.Attachments = m.Attachments
dataAO1.Comments = m.Comments
dataAO1.LastUpdated = m.LastUpdated
dataAO1.LastUpdatedBy = m.LastUpdatedBy
dataAO1.ReportedBy = m.ReportedBy
jsonDataAO1, errAO1 := swag.WriteJSON(dataAO1)
if errAO1 != nil {
return nil, errAO1
}
_parts = append(_parts, jsonDataAO1)
return swag.ConcatJSON(_parts...), nil
}
// Validate validates this task
func (m *Task) Validate(formats strfmt.Registry) error {
var res []error
// validation for a type composition with TaskCard
if err := m.TaskCard.Validate(formats); err != nil {
res = append(res, err)
}
if err := m.validateAttachments(formats); err != nil {
res = append(res, err)
}
if err := m.validateComments(formats); err != nil {
res = append(res, err)
}
if err := m.validateLastUpdated(formats); err != nil {
res = append(res, err)
}
if err := m.validateLastUpdatedBy(formats); err != nil {
res = append(res, err)
}
if err := m.validateReportedBy(formats); err != nil {
res = append(res, err)
}
if len(res) > 0 {
return errors.CompositeValidationError(res...)
}
return nil
}
func (m *Task) validateAttachments(formats strfmt.Registry) error {
if swag.IsZero(m.Attachments) { // not required
return nil
}
for k := range m.Attachments {
if swag.IsZero(m.Attachments[k]) { // not required
continue
}
if val, ok := m.Attachments[k]; ok {
if err := val.Validate(formats); err != nil {
if ve, ok := err.(*errors.Validation); ok {
return ve.ValidateName("attachments" + "." + k)
} else if ce, ok := err.(*errors.CompositeError); ok {
return ce.ValidateName("attachments" + "." + k)
}
return err
}
}
}
return nil
}
func (m *Task) validateComments(formats strfmt.Registry) error {
if swag.IsZero(m.Comments) { // not required
return nil
}
for i := 0; i < len(m.Comments); i++ {
if swag.IsZero(m.Comments[i]) { // not required
continue
}
if m.Comments[i] != nil {
if err := m.Comments[i].Validate(formats); err != nil {
if ve, ok := err.(*errors.Validation); ok {
return ve.ValidateName("comments" + "." + strconv.Itoa(i))
} else if ce, ok := err.(*errors.CompositeError); ok {
return ce.ValidateName("comments" + "." + strconv.Itoa(i))
}
return err
}
}
}
return nil
}
func (m *Task) validateLastUpdated(formats strfmt.Registry) error {
if swag.IsZero(m.LastUpdated) { // not required
return nil
}
if err := validate.FormatOf("lastUpdated", "body", "date-time", m.LastUpdated.String(), formats); err != nil {
return err
}
return nil
}
func (m *Task) validateLastUpdatedBy(formats strfmt.Registry) error {
if swag.IsZero(m.LastUpdatedBy) { // not required
return nil
}
if m.LastUpdatedBy != nil {
if err := m.LastUpdatedBy.Validate(formats); err != nil {
if ve, ok := err.(*errors.Validation); ok {
return ve.ValidateName("lastUpdatedBy")
} else if ce, ok := err.(*errors.CompositeError); ok {
return ce.ValidateName("lastUpdatedBy")
}
return err
}
}
return nil
}
func (m *Task) validateReportedBy(formats strfmt.Registry) error {
if swag.IsZero(m.ReportedBy) { // not required
return nil
}
if m.ReportedBy != nil {
if err := m.ReportedBy.Validate(formats); err != nil {
if ve, ok := err.(*errors.Validation); ok {
return ve.ValidateName("reportedBy")
} else if ce, ok := err.(*errors.CompositeError); ok {
return ce.ValidateName("reportedBy")
}
return err
}
}
return nil
}
// ContextValidate validate this task based on the context it is used
func (m *Task) ContextValidate(ctx context.Context, formats strfmt.Registry) error {
var res []error
// validation for a type composition with TaskCard
if err := m.TaskCard.ContextValidate(ctx, formats); err != nil {
res = append(res, err)
}
if err := m.contextValidateAttachments(ctx, formats); err != nil {
res = append(res, err)
}
if err := m.contextValidateComments(ctx, formats); err != nil {
res = append(res, err)
}
if err := m.contextValidateLastUpdated(ctx, formats); err != nil {
res = append(res, err)
}
if err := m.contextValidateLastUpdatedBy(ctx, formats); err != nil {
res = append(res, err)
}
if err := m.contextValidateReportedBy(ctx, formats); err != nil {
res = append(res, err)
}
if len(res) > 0 {
return errors.CompositeValidationError(res...)
}
return nil
}
func (m *Task) contextValidateAttachments(ctx context.Context, formats strfmt.Registry) error {
for k := range m.Attachments {
if val, ok := m.Attachments[k]; ok {
if err := val.ContextValidate(ctx, formats); err != nil {
return err
}
}
}
return nil
}
func (m *Task) contextValidateComments(ctx context.Context, formats strfmt.Registry) error {
if err := validate.ReadOnly(ctx, "comments", "body", []*Comment(m.Comments)); err != nil {
return err
}
for i := 0; i < len(m.Comments); i++ {
if m.Comments[i] != nil {
if err := m.Comments[i].ContextValidate(ctx, formats); err != nil {
if ve, ok := err.(*errors.Validation); ok {
return ve.ValidateName("comments" + "." + strconv.Itoa(i))
} else if ce, ok := err.(*errors.CompositeError); ok {
return ce.ValidateName("comments" + "." + strconv.Itoa(i))
}
return err
}
}
}
return nil
}
func (m *Task) contextValidateLastUpdated(ctx context.Context, formats strfmt.Registry) error {
if err := validate.ReadOnly(ctx, "lastUpdated", "body", strfmt.DateTime(m.LastUpdated)); err != nil {
return err
}
return nil
}
func (m *Task) contextValidateLastUpdatedBy(ctx context.Context, formats strfmt.Registry) error {
if m.LastUpdatedBy != nil {
if err := m.LastUpdatedBy.ContextValidate(ctx, formats); err != nil {
if ve, ok := err.(*errors.Validation); ok {
return ve.ValidateName("lastUpdatedBy")
} else if ce, ok := err.(*errors.CompositeError); ok {
return ce.ValidateName("lastUpdatedBy")
}
return err
}
}
return nil
}
func (m *Task) contextValidateReportedBy(ctx context.Context, formats strfmt.Registry) error {
if m.ReportedBy != nil {
if err := m.ReportedBy.ContextValidate(ctx, formats); err != nil {
if ve, ok := err.(*errors.Validation); ok {
return ve.ValidateName("reportedBy")
} else if ce, ok := err.(*errors.CompositeError); ok {
return ce.ValidateName("reportedBy")
}
return err
}
}
return nil
}
// MarshalBinary interface implementation
func (m *Task) MarshalBinary() ([]byte, error) {
if m == nil {
return nil, nil
}
return swag.WriteJSON(m)
}
// UnmarshalBinary interface implementation
func (m *Task) UnmarshalBinary(b []byte) error {
var res Task
if err := swag.ReadJSON(b, &res); err != nil {
return err
}
*m = res
return nil
}
// TaskAttachmentsAnon task attachments anon
//
// swagger:model TaskAttachmentsAnon
type TaskAttachmentsAnon struct {
// The content type of the file.
//
// The content type of the file is inferred from the upload request.
//
// Read Only: true
ContentType string `json:"contentType,omitempty"`
// Extra information to attach to the file.
//
// This is a free form text field with support for github flavored markdown.
//
// Min Length: 3
Description string `json:"description,omitempty"`
// The name of the file.
//
// This name is inferred from the upload request.
//
// Read Only: true
Name string `json:"name,omitempty"`
// The file size in bytes.
//
// This property was generated during the upload request of the file.
// Read Only: true
Size float64 `json:"size,omitempty"`
// The url to download or view the file.
//
// This URL is generated on the server, based on where it was able to store the file when it was uploaded.
//
// Read Only: true
// Format: uri
URL strfmt.URI `json:"url,omitempty"`
}
// Validate validates this task attachments anon
func (m *TaskAttachmentsAnon) Validate(formats strfmt.Registry) error {
var res []error
if err := m.validateDescription(formats); err != nil {
res = append(res, err)
}
if err := m.validateURL(formats); err != nil {
res = append(res, err)
}
if len(res) > 0 {
return errors.CompositeValidationError(res...)
}
return nil
}
func (m *TaskAttachmentsAnon) validateDescription(formats strfmt.Registry) error {
if swag.IsZero(m.Description) { // not required
return nil
}
if err := validate.MinLength("description", "body", m.Description, 3); err != nil {
return err
}
return nil
}
func (m *TaskAttachmentsAnon) validateURL(formats strfmt.Registry) error {
if swag.IsZero(m.URL) { // not required
return nil
}
if err := validate.FormatOf("url", "body", "uri", m.URL.String(), formats); err != nil {
return err
}
return nil
}
// ContextValidate validate this task attachments anon based on the context it is used
func (m *TaskAttachmentsAnon) ContextValidate(ctx context.Context, formats strfmt.Registry) error {
var res []error
if err := m.contextValidateContentType(ctx, formats); err != nil {
res = append(res, err)
}
if err := m.contextValidateName(ctx, formats); err != nil {
res = append(res, err)
}
if err := m.contextValidateSize(ctx, formats); err != nil {
res = append(res, err)
}
if err := m.contextValidateURL(ctx, formats); err != nil {
res = append(res, err)
}
if len(res) > 0 {
return errors.CompositeValidationError(res...)
}
return nil
}
func (m *TaskAttachmentsAnon) contextValidateContentType(ctx context.Context, formats strfmt.Registry) error {
if err := validate.ReadOnly(ctx, "contentType", "body", string(m.ContentType)); err != nil {
return err
}
return nil
}
func (m *TaskAttachmentsAnon) contextValidateName(ctx context.Context, formats strfmt.Registry) error {
if err := validate.ReadOnly(ctx, "name", "body", string(m.Name)); err != nil {
return err
}
return nil
}
func (m *TaskAttachmentsAnon) contextValidateSize(ctx context.Context, formats strfmt.Registry) error {
if err := validate.ReadOnly(ctx, "size", "body", float64(m.Size)); err != nil {
return err
}
return nil
}
func (m *TaskAttachmentsAnon) contextValidateURL(ctx context.Context, formats strfmt.Registry) error {
if err := validate.ReadOnly(ctx, "url", "body", strfmt.URI(m.URL)); err != nil {
return err
}
return nil
}
// MarshalBinary interface implementation
func (m *TaskAttachmentsAnon) MarshalBinary() ([]byte, error) {
if m == nil {
return nil, nil
}
return swag.WriteJSON(m)
}
// UnmarshalBinary interface implementation
func (m *TaskAttachmentsAnon) UnmarshalBinary(b []byte) error {
var res TaskAttachmentsAnon
if err := swag.ReadJSON(b, &res); err != nil {
return err
}
*m = res
return nil
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,566
|
COO and Co-Founder of TEAMZ Inc
Yanying "Toto" is a Chinese entrepreneur, speaker, and advocate for young women in business and female leadership.
Toto is the co-founder and COO of TEAMZ, Inc, a blockchain solutions company focused on business growth through marketing, development, investment, and network building based in Tokyo. She is a passionate entrepreneur that looks to be a role model for young women entering the business world and advocate for female leadership. She has spoken at various events including TEDx Youth@Tokyo, Girls 20, Musashino International Association, Tokyo University, and Japan Chinese Entrepreneur Association.
Toto started her first business in Beijing at the age of 19. She was previously a Business Consultant at IBM Japan and lead the global expansion team. She has a Masters degree in Intercultural Communication from Tokyo University of Foreign Studies (Japan) and Entrepreneurship at San Diego State University (USA).
Toto is fluent in Chinese, Japanese, and English. She has lived in 4 different countries including China, Japan, USA, and Costa Rica. She is passionate about multiculturalism and working in a world without borders. Toto is an experienced Pipa player and enjoys spreading Chinese culture through music.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,348
|
Q: trouble cloning from branch I am using github and I have a few branches made from a backup branch and I am trying to git clone from the sub branch lets call it "x". I have made a lot of changes and pushed them to the "x" branch but now when I git clone, it pulls that branch + the backup branch. Is there a way to only clone from the "x" branch without git cloning its parent branch? On git the "x" branch looks exactly how I want it and some things are deleted that are in the backup branch however git clone sends those deleted files as well. I am a little confused and I hope this is a simple git clone "url" addition. Thank you anyone who knows what I need to do to only clone the "x" branch.
A: This is what I had to run.
git clone -b branchName --single-branch https://github.com/clone.url
Hope this helps someone
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,880
|
require 'rails_helper'
RSpec.describe BoxesController, type: :controller do
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,522
|
First Light Technologies included for the second year in a row on The Globe and Mail's second-annual ranking of Canada's Top Growing Companies
First Light Technologies is pleased to announce its inclusion on the 2020 Report on Business ranking of Canada's Top Growing Companies.
Canada's Top Growing Companies ranks Canadian companies on three-year revenue growth. First Light Technologies earned its spot with three-year growth of 133%.
First Light's strong growth comes as the company has been at the forefront of solar lighting for the last 11 years. The company is a pioneer in self-contained commercial solar lights . Their approach offers the simplest way of providing outdoor lighting, coupled with overall cost savings of typically 50% or more when compared to wired lighting and no ongoing carbon emissions.
"We're honoured to be included again this year, alongside so many great Canadian success stories. Being one of Canada's top growing companies for a second year in a row is another confirmation that our vision for solar lighting resonates with our customers and we hear that in their success stories," said First Light Technologies CEO Sean Bourquin. "We continue our mission to make First Light solar lighting the clear choice for lighting outdoor community spaces through our team's commitment to our customers and to designing the best products in the market."
About Globe and Mail Canada's Top Growing Companies
Launched in 2019, the Canada's Top Growing Companies editorial ranking aims to celebrate entrepreneurial achievement in Canada by identifying and amplifying the success of growth-minded, independent businesses in Canada. It is a voluntary program; companies had to complete an in-depth application process in order to qualify. In total, 400 companies earned a spot on this year's ranking.
The full list of 2020 winners, and accompanying editorial coverage, is published in the October issue of Report on Business magazine—out now—and online at tgam.ca/TopGrowing.
"The stories of Canada's Top Growing Companies are worth telling at any time, but are especially relevant in the wake of COVID-19 pandemic," says James Cowan, Editor of Report on Business magazine. "As businesses work to rebuild the economy, their resilience and innovation make for essential reading."
"Any business leader seeking inspiration should look no further than the 400 businesses on this year's Report on Business ranking of Canada's Top Growing Companies," says Phillip Crawley, Publisher and CEO of The Globe and Mail. "Their growth helps to make Canada a better place, and we are proud to bring their stories to our readers."
About The Globe and Mail
The Globe and Mail is Canada's foremost news media company, leading the national discussion and causing policy change through brave and independent journalism since 1844. With award-winning coverage of business, politics and national affairs, The Globe and Mail newspaper reaches 5.9 million readers every week in print or digital formats, and Report on Business magazine reaches 2.1 million readers in print and digital every issue. The Globe and Mail's investment in innovative data science means that as the world continues to change, so does The Globe. The Globe and Mail is owned by Woodbridge, the investment arm of the Thomson family.
About First Light Technologies
First Light Technologies designs and manufactures commercial grade, self-contained solar powered lights. The company is dedicated to driving the potential of sustainable, solar lighting by constantly delivering better and simpler solutions.
Kirsten Denham
kdenham@firstlighttechnologies.com
Introducing Extra Warm White Color Temperatures September 28, 2022
Stop Trenching & Wiring to Save Time and Money! July 28, 2022
First Light Goes Hollywood! June 29, 2022
The Lumen Illusion: How System Efficacy Helps Us Look Beyond Lumens When Choosing a Light April 27, 2022
INTERVIEW: HOW THIS SOLAR-POWERED COMPANY CONNECTS WITH CLIENTS TO LIGHT UP THE CONTINENT April 13, 2022
General Lighting
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,983
|
ADD+ON Makes Its Move Back To Basics
By Elizabeth Barnett
Just how ADD+On Software, Inc. had such a great year in 1998--a 75% increase in sales revenue--presents an interesting model for Business Basic developers.
ADD+ON Chief Executive Officer Joe Taylor attributes it to two essential changes: a change in ownership and a change in philosophy. With the change in ownership, ADD+ON has refocused on its one product, rather than being part of a business that was selling multiple products. A more philosophical shift has occurred in that ADD+ON isn't interested so much in volume as in repeat business. But in focusing on repeat business, the company is seeing its volume soar.
Forging VAR Alliances
"Volume for the sake of volume: it's not someplace where we really want to be," says John Lee, ADD+ON's General Manager. ADD+ON strategy lies in hand-picking its value-added resellers (VARs) and Authorized Service Providers (ASPs) and then strongly supporting them.
"We like good, strong VARs and long-term relationships. That's really key to our business," Joe adds. "We are choosey about who we select as partners and we'll continue to be choosey. The people we select are good business people and they're good people."
Nancy Laporte, ADD+ON's Sales Manager says ADD+ON looks for VARs who:
share a similar business philosophy in which they're establishing long-term relationships with their customers
have experience programming in Business Basic and have the capability to customize ADD+ON's product for their applications
are in an area where there are very few other VARs
have three or more employees, indicative of larger, hopefully more stable companies
have recommendations from other VARs and other knowledgeable people
ADD+ON may be selective about its partners, but it's not selective when it comes to giving them information and support. But it wasn't always this way, Nancy says. ADD+ON's VARs have been loyal to the product, but many of them had left in years past, she says. Now, as a result of the renewed focus, she sees them coming back "in droves." The communication is frequent and honest.
"Joe is very realistic," she explains. "People may not like the answers they hear, but at least they know they're getting the whole story, the true story, on what's really happening. And on the sales side, the account management, they're getting called, they're getting paid attention to, and they're getting good information."
Delivering "The Freedom to Grow"
A typical ADD+ON customer is a manufacturing or distribution company, dealing with the day-to-day tasks of transaction processing. The bottom line: they need the ability to enter an order and follow that order through the manufacturing process to the invoicing step to actually shipping the product. Engineered-to-order, made-to-order and made-to-stock manufacturing sectors and companies with multiple locations in the wholesale distribution sector are the core of ADD+ON's customer base.
"Software in general is getting better at it," Joe says, "but ADD+ON is particularly good at processing orders without much labor involved and having that information go where it needs to go fast and accurately." The key, he explains, is to provide the customer service person with all the information needed to process that order quickly and effectively. Then the computer does the rest, relaying the information off to other departments and completely managing inventories. Without effective low-labor automation, a company gets bottlenecked in fulfilling orders and can't grow. And it can't easily or cost-effectively add more people to the job of processing orders because it's just too complicated.
Being able to rely on an automated system that can forward select information to the right people to fulfill and finish the process has changed the scope of business for many ADD+ON customers. Companies' profits climb as ADD+ON solves the processing bottleneck. "That's really what we've done for hundreds and hundreds of customers all across the country over the years: make their processes easier and more effective," Joe says. "That's what the Freedom To Grow is all about. And it really is true."
Industry Hot Buttons
ADD+ON is concentrating these days on developing true three-tier architecture for its product. The ubiquity of PCs and the ease and intuitiveness of graphical user interfaces (GUI) are necessitating GUI front ends for certain people in the industrial manufacturing businesses that comprise a lot of ADD+ON's customers. People on production lines and order entry channels may always need and want character-based UNIX terminals. More and more typically, though, the managers of these enterprises need to manipulate order, inventory and labor data in programs like Microsoft Access and Excel for budgeting and analysis purposes. Right now, ADD+ON is working to allow its customers to choose either GUI or character-based interfaces in the same program. The next phase involves expanding customers' data access capabilities.
"Until a few years ago, access to a client's data wasn't really an issue," Joe says. "In fact, there really wasn't a lot of data, except on tapes. Accounting programs were not geared toward providing users access to data. You had to be a programmer to get access to data."
ADD+ON anticipated the data access need and began in 1994 to normalize its database structure. This huge preparatory leap has poised ADD+ON customers to make full use of any kind of structured query language (SQL) database on the back end of their applications. Then, ADD+ON software, and BASIS PRO/5® upon which it's based, deal with the processing that must take place in between the interface and the data management. "Theoretically, two or three years from now," John says, "you could have ADD+ON Software in a situation where everybody's running a browser front end and an ORACLE back end and using PRO/5 and ADD+ON as the middle piece to move that data around."
Joe adds, "That's really where the whole industry wants to go. We've been evolving and so has BASIS. BASIS has provided us the tools to do that."
Partnering with BASIS
"We depend on BASIS to be forecasting technology for us," John says. But ADD+ON relies on BASIS not only to predict where technology is going but to have PRO/5 and next-generation BASIS products available to meet those technology needs in the future. ADD+ON is excited about the latest forecast, BBjTM, and what it will allow ADD+ON to do. "It moves ADD+ON into a new technology with a minimal effort on our part," Joe explains. "We can continue to focus on features and not have to focus on the technology because Kevin King [BASIS' Director Of Engineering Services] and his group did that for us."
Nancy adds, "People want to be involved in the cutting-edge technology. BBj is giving us the opportunity to stay part of the fast track and still maintain a really mature, solid, accounting and distribution software package."
Joe sees the concept of BBj as perfectly suited to help ADD+ON and its customers move into e-commerce and the Internet. It's his conviction that this is the future of the ADD+ON market. "We're going to find that the Internet is the network," Joe predicts, "with virtual private networks--absolutely--as well as e-commerce with other, unrelated parties." He sees his market, particularly those industrial companies with multiple locations, moving to a model of regional centers, or centralized environments, where software will be available to many, many end users. End users don't care where the software is located, but the people having to go out and administrate and service the system do. Having it all in one place would be so convenient.
"The whole dynamic of how we're distributing software to the marketplace physically as well as operationally is going to change as a result of the Internet," Joe says. "We needed a way to make our software run native to the Internet, and voilà, BASIS announces BBj."
Although the Internet will change how ADD+ON works and how its VARs work, it won't change the application software. "We still need the features, the throughput, the efficiency," Joe stresses. "It'll be interesting to see how it all works out."
Founded in 1981, ADD+ON Software is a business solutions software development company providing full-featured, fully integrated manufacturing, wholesale distribution and accounting applications. ADD+ON's suite of products includes General Accounting, Manufacturing, Distribution and Special Function Applications. Over 5,000 companies use ADD+ON applications to run their businesses. Visit the ADD+ON Web site at www.addonsoftware.com.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,584
|
{"url":"https:\/\/byjus.com\/physics\/ficks-law-of-diffusion\/","text":"# Fick's Law Of Diffusion\n\n## What is Fick\u2019s law of diffusion?\n\nFick\u2019s law of diffusion explains the diffusion process (movement of molecules from higher concentration to lower concentration region). In 1855, Adolf Fick described the Fick\u2019s Law of Diffusion. A diffusion process that obeys Fick\u2019s laws is called normal diffusion or Fickian diffusion. A diffusion process that does NOT obey Fick\u2019s laws is known as Anomalous diffusion or non-Fickian diffusion.\n\nFick\u2019s Law of Diffusion is used to solve the diffusion coefficient D.\n\nThere are two laws that are interrelated ie; Fick\u2019s first law is used to derive Fick\u2019s second law which is similar to the diffusion equation.\n\nAccording to Fick\u2019s law of diffusion,\n\n\u201cThe molar flux due to diffusion is proportional to the concentration gradient\u201d.\n\nThe rate of change of concentration of the solution at a point in space is proportional to the second derivative of concentration with space.\n\n### Fick\u2019s First Law\n\nMovement of solute from higher concentration to lower concentration across a concentration gradient.\n\n$$J = -D\\frac{\\mathrm{d} \\varphi }{\\mathrm{d} x}$$\n\nWhere,\n\nJ: diffusion flux\n\nD: diffusivity\n\n\u03c6: concentration\n\nx: position\n\n### Fick\u2019s Second Law\n\nPrediction of change in concentration along with time due to diffusion.\n\n$$\\frac{\\partial \\varphi }{\\partial t} = D \\frac{\\partial^2 \\varphi }{\\partial x^2}$$\n\nWhere,\n\nD: diffusivity\n\nt: time\n\nx: position\n\n\u03a6: concentration\n\n### Application of Fick\u2019s law\n\n\u2022 Biological application:\n$$flux = -P(c_{2}-c_{1})$$ (from Fick\u2019s first law)\n\nWhere,\n\nP: permeability\n\nc2-c1: difference in concentration\n\n\u2022 Liquids: Fick\u2019s law is applicable for two miscible liquids when they are brought in contact and diffusion takes place at a macroscopic level.\n\u2022 Fabrication of semiconductor: Diffusion equations from Fick\u2019s law are used to fabricate integrated circuits.\n\u2022 Pharmaceutical application\n\u2022 Applications in food industries.\n\nRelated Physics articles:\n\nStay tuned with BYJU\u2019S for more such interesting articles. Also, register to \u201cBYJU\u2019S \u2013 The Learning App\u201d for loads of interactive, engaging Physics-related videos and an unlimited academic assist.\n\nTest Your Knowledge On Ficks Law Of Diffusion!","date":"2021-10-17 22:25:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6991193294525146, \"perplexity\": 2774.6224204002106}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585183.47\/warc\/CC-MAIN-20211017210244-20211018000244-00301.warc.gz\"}"}
| null | null |
layout: post
title: "Jenius Engineer reading list #384"
author: "dedenf"
tags:
- news
- jenius
categories:
- readinglist
- engineer
published: true
canonical: https://jakartadev.org/daily-digest-554/
---
- [Visualizing Neural Networks with the Grand Tour](https://distill.pub/2020/grand-tour/)
- [The Importance of Covariates in Causal Inference: Shown in a Comparison of Two Methods](https://tech.wayfair.com/2020/03/the-importance-of-covariates-in-causal-inference/)
- [The problem with thread^W event loops](https://blog.cloudflare.com/the-problem-with-event-loops/)
- [What the Heck is Backstage Anyway?](https://labs.spotify.com/2020/03/17/what-the-heck-is-backstage-anyway/)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,668
|
» See Pension Plan Information Below
Contract Pilots: Legislation signed by the Governor on 9/11/2002 requires the Department of Forestry and Fire Protection to pay a one-time benefit to eligible survivors of contract pilots who die while performing duties under contract to the Department. The amount is equal to what the survivor would receive if the pilot were covered by the federal Public Safety Officers' Benefits Act. The eligible survivors also receive an amount determined by the Department to commensurate with the death benefit payable to a mid-career firefighter employed by the Department who died in the line of duty.
California Department of Forestry and Fire Protection
Website: calfire.ca.gov
Death benefits are payments to a spouse, children or other dependents if an employee dies from a work-related injury or illness. The amount of the death benefit depends on the number of total and/or partial dependents. Benefits maximums are as follows: 1 total dependent ($250,000), 2 total dependents ($290,000), 3 or more total dependents ($320,000), 1 total plus 1 or more partial dependents ($250,000 plus four times annual support for partial dependents not to exceed $290,000), and 1 or more partial dependents (eight times annual support not to exceed $250,000).
In the case of one or more totally dependent minors, after payment of amounts specified below, death benefits will continue until youngest minor's 18th birthday (disabled minors receive benefits for life). Death benefits are paid at the total temporary disability rate, but not less than $224.00 per week. The period within which to commence proceedings for the collection of death benefits is one year from death where death occurs within one year of date of injury (DOI); or one year from date of last furnishing of any benefits or one year from death where death occurs more than one year from DOI. No such proceedings may be commenced more than 240 weeks from the DOI.
(Reference: Labor Code Section 4702)
Volunteer firefighters: Each member registered as an active firefighting member of any regularly organized volunteer fire department having official recognition and full or partial support of its local government is considered an employee for purposes of workers' compensation. This entitles eligible volunteer firefighters to receive compensation from the local government. If a volunteer firefighter dies while performing duties, then, irrespective of remuneration from this or other employment or from both, the average earnings shall be taken as the maximums fixed for each.
Workers' Compensation Division
Division of Workers' Compensation
1515 Clay Street, 6th Floor
Oakland, CA 94612-1519
E-Mail: dwc@dir.ca.gov
Website: www.dir.ca.gov
See Office Location Directory to identify your closest branch.
California presumptions for firefighters include: Heart, hernia, pneumonia, cancer, tuberculosis, hepatitis, and meningitis. Some bio-chemical, blood-borne pathogens, and methicillin resistant staph.
(Reference: Labor Code Sections 3212, 3212.1, 3212.6, 3212.8, 3212.85, 3212.9)
A maximum burial allowance of $10,000 is available.
There are several retirement systems in the state. Legislation can change benefits within local jurisdictions. It is imperative that you contact specific programs for more details. Benefits for survivors of firefighters who were members of the California Public Employees Retirement System (CalPERS) include:
Group Term Life Insurance provides survivor of State firefighter with a tax-free, lump-sum benefit of $5,000. For members with less than 20 years of State service, benefit is $5,000 plus an amount equal to six months pay.
Special Death Benefit for the surviving spouse, registered domestic partner, unmarried children, or eligible unmarried stepchildren of a State or Local Firefighter who died in performance of duties as a result of an accident or injury. Benefit may provide a monthly allowance equal to 50% of firefighter's final compensation. Allowance may increase to a maximum of 75% of final compensation based on the number of unmarried children under age 22 and if external violence or physical force caused duty-related death. This additional benefit ceases when child marries or reaches age 22.
Spouse or registered domestic partner may instead elect to receive the Alternate Death Benefit. This benefit applies to members under 50 who have 20 or more years of state service. Under this benefit, the eligible spouse or domestic partner may receive a monthly allowance equal to the amount you would have received if the fallen firefighter had retired under a service retirement at age 50 and elected Option 2W. Upon death of spouse or domestic partner, benefit will continue to natural or adopted unmarried children under the age of 18.
Under the 1959 Survivor Benefit Program, survivors of members of the program are eligible for a monthly allowance. If the 1959 Survivor Benefit is greater than the Special Death Benefit, then the difference is paid as the 1959 Survivor Benefit. Monthly allowance levels are as follows: (a) a spouse or registered domestic partner who has care of two or more eligible unmarried children; or three eligible unmarried children only – $1,800; (b) a spouse or registered domestic partner who has care of one eligible unmarried child; or two eligible unmarried children only – $1,500; (c) one eligible unmarried child only; or a spouse or registered domestic partner at age 60 or older – $750; and (d) dependent parents who are at least 60 may be eligible if there are no other eligible survivors – $750 each.
Volunteer firefighters: Length of Service Award System provides $3,000 lump sum payment to the beneficiary if the member was either active or inactive and had accumulated 10 years of service. LOSAP program administered through CalPERS.
Violent Acts: If death is the direct result of a violent act while performing official duties, your beneficiary(ies) may receive a monthly allowance equal to one-half your final compensation.
Lincoln Plaza West
400 Q Street
Website: calpers.ca.gov
(Reference: Your Benefits, Your Future, CALPERS, "State Safety Benefits," 2017)
No mandatory fees or tuition required by the Regents of the University of California, the Board of Directors of the Hastings College of the Law, or the Trustees of the California State University System. This benefit applies to the spouse or children of a public safety officer killed in the performance of active fire suppression and prevention. Law expanded in 2002 to include fee waivers at the California community college level. Educational benefits available to survivors if the firefighter died in the performance of fire suppression and prevention duties, was an employee of a public agency, and was a resident of California. This would include Federal firefighters if they met the criteria listed above. Eligibility must be consistent with the findings of the Worker's Compensation Appeals Board.
The Trustees of California State University may enter into reciprocal agreements with other universities or colleges within the state. This would allow qualified students to attend other universities or colleges without payment of some or all fees or tuition or both. The Trustees may enter into similar reciprocal agreements with public colleges and universities in other states.
Contractors: The current law also covers children and spouses of contractors or employees of contractors performing services for a state, city, county, district or other local public agency. Students must tell the college or university they plan to attend that they are the survivor of a firefighter killed in the line of duty and that they qualify under Sections 68120-68124 of the California Education Code.
The Law Enforcement Personnel Dependents (LEPD) Grant Program: Provides educational grants to each dependent or spouse of a California firefighter killed in the performance of duty or who dies or is totally disabled in a duty-related accident or injury. Grant shall be in an amount equal to that provided a student awarded a Cal Grant Scholarship, ranging from $100 to $12,192 for up to four years. Awards may be used for tuition and fees, books, supplies, and living expenses.
Section 4709 of the Labor Code prohibits proceeds of death benefits received by a dependent of a firefighter killed or disabled in the line of duty from being included when determining financial need for an LEPD grant.
For LEPD information Contact:
California Student Aid Commission
Rancho Cordova, CA 95741-9027
Website: www.csac.ca.gov
(Reference: Provided under California Labor Code 4709 and California Education Code Sections 68120 & 68121)
The Daniel A. Terry Scholarship provides higher education financial assistance to the children of California's fallen firefighters.
Daniel A. Terry served as president of California Professional Firefighters, the largest statewide firefighter organization in California and founder of the California Fire Foundation. For more than three decades, Mr. Terry devoted himself to building a better life for firefighters. His commitment and leadership led to groundbreaking protections for firefighters and their families. Mr. Terry also conceived and led the successful effort to construct a lasting tribute to California's fallen first responders – the California Firefighters Memorial in Sacramento's Capitol Park.
To be eligible for the Daniel A. Terry Scholarship, an applicant must be the natural or legally adopted child of a California firefighter who died in the line of duty and whose name appears or is approved to appear on the California Firefighters Memorial Wall. Applicants must also be under 27 years of age at the application closing date and must possess a high school diploma or equivalent, or be in the final year of high school.
Each scholarship award amount is $2,000.
Daniel A. Terry Scholarship
c/o – California Professional Firefighters
1780 Creekside Oaks Drive
E-mail: cafirefoundation@cpf.org
Website: http://www.cafirefoundation.org/programs/scholarships-grants/daniel-a-terry-scholarships/
(Reference: Click Here to Download the Application)
The Law Enforcement Personnel Dependents (LEPD) Grant Program: Provides educational grants to each dependent or spouse of a California firefighter killed in the performance of duty or who dies or is totally disabled in a duty-related accident or injury. Grant shall be in an amount equal to that provided a student awarded a Cal Grant Scholarship. Awards may be used for tuition and fees, books, supplies, and living expenses.
California Fire Foundation
Website: www.cafirefoundation.org
California Professional Firefighters
Website: http://www.cpf.org
Organizes special funds to provide emergency assistance to families of firefighters killed in the line of duty. The California Fire Foundation publishes a benefits guide, Survivor Benefits on its website.
According to California Labor Code section 4856, when a firefighter has died in the line of duty, then "the employer shall continue providing health benefits to the deceased employee's spouse [and children] under the same terms and conditions provided prior to the death." Qualifying "children" are defined by Government Code section 22822. California Government Code section 22820 provides additional information about health care benefits for fully insured survivors and for partially insured survivors. Government Code section 19849.15 provides information for short-term health benefit coverage for state (Cal Fire) employees only.
Legislation passed in 2002 protects survivors of CalPERS members from an interruption in health benefit coverage. This provision covers family members who were validly enrolled in an approved health benefits plan at the date of the firefighter's death. The firefighter's agency is required to continue to pay the employer's contribution for a specified period for up to 120 days.
For more information regarding the benefit or eligibility, contact CalPERS.
(Reference: California Labor Code section 4856; California Government Code sections 19849.15 and 22820)
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,154
|
#include <fstream>
#include "common/base.h"
#include "common/u/ugnames.h"
#include "common/u/uhdd.h"
#include "common/u/uprocess.h"
#include "common/errdbg/d_printf.h"
#include "common/u/uutils.h"
using namespace std;
FILE* res_os = stdout; //otput stream of transaction results (result of the user's query)
/* global names */
#define POLICY_SINGLETON() NULL,1
#define POLICY_INSTANCE_PER_DB() "DB",(MAX_DBS_NUMBER)
#define POLICY_INSTANCE_PER_SESSION() "SES",(UPPER_SESSIONS_NUM_BOUND)
/* If you want to add a new global IPC object, do it here.
We are going to extract basenames automaticly, so don't
try anything unusual below and preserve markers. */
static UGlobalNamesRegistryItem globalNamesRegistry[] =
{
/* {% GlobalNamesRegistry */
{"SHMEM_GLOBAL", POLICY_SINGLETON()}, /* vmm region info + VMM placeholder when no buffer mapped */
{"SHMEM_GOV", POLICY_SINGLETON()}, /* holding system state and config info (gov_config_struct) */
{"SHMEM_EVENT_LOG", POLICY_SINGLETON()}, /* event logger */
{"SEMAR_EVENT_LOG", POLICY_SINGLETON()}, /* event logger */
{"SHMEM_BUFFERS", POLICY_INSTANCE_PER_DB()}, /* buffer memory */
{"SEMAP_BUFMGR_EXCL_MODE", POLICY_INSTANCE_PER_DB()}, /* regulates the exclusive mode entering by TRN */
{"SHMEM_BUFFERS_LRU", POLICY_INSTANCE_PER_DB()}, /* LRU stats on buffers usage */
// {"SHMEM_SHARED_HEAP", POLICY_INSTANCE_PER_DB()}, /* shared heap */
{"SEMAP_CATALOG_NAMETABLES", POLICY_INSTANCE_PER_DB()}, /* catalog nametables lock */
{"SEMAP_CATALOG_METADATA", POLICY_INSTANCE_PER_DB()}, /* catalog metadata lock */
{"SHMEM_SM_TALK", POLICY_INSTANCE_PER_DB()}, /* used by shared memory-based SM messaging interface */
{"SEMAR_SM_TALK", POLICY_INSTANCE_PER_DB()}, /* used by shared memory-based SM messaging interface */
{"SEMAP_VMM_INIT", POLICY_INSTANCE_PER_DB()}, /* VMM initialisation is serialised with this sem */
{"SHMEM_VMM_CALLBACK_PARAMS", POLICY_INSTANCE_PER_DB()}, /* parameters passed to VMM calback */
{"EVENT_VMM_CALLBACK", POLICY_INSTANCE_PER_SESSION()}, /* VMM callback thread waits on it */
{"EVENT_VMM_CALLBACK_COMPLETED", POLICY_INSTANCE_PER_SESSION()}, /* sem for callback thread to signal call completion */
// {"SEMAP_METADATA", POLICY_INSTANCE_PER_DB()}, /* synchronises access to metadata registry in PH */
// {"SEMAP_INDICES", POLICY_INSTANCE_PER_DB()}, /* synchronises access to indices registry in PH */
// {"SEMAP_FT_INDICES", POLICY_INSTANCE_PER_DB()}, /* synchronises access to full-text indices registry in PH */
// {"SEMAP_TRIGGERS", POLICY_INSTANCE_PER_DB()}, /* synchronises access to triggers registry in PH */
{"SEMAP_LOCKMGR", POLICY_INSTANCE_PER_DB()}, /* serialises requests to lock manager (in SM) */
{"EVENT_LOCK_GRANTED", POLICY_INSTANCE_PER_SESSION()}, /* if transaction request for a lock on DB entity is not satisfied immediately trn waits until the event is signalled (hence if transaction enters the wait state, it can't become a victim for the deadlock-resolution process) */
{"SEMAP_TRN_REGULATION", POLICY_INSTANCE_PER_DB()}, /* currently if checkpoint is active no updater transactions are allowed and vice-versa (earlier mutual exclusion applied to micro-ops but not transactions) */
{"EVENT_NEW_JOB_4_CHECKPOINT_THREAD", POLICY_INSTANCE_PER_DB()}, /* signals that a checkpoint must be activated or snapshots must be advanced */
{"EVENT_READONLY_TRN_COMPLETED", POLICY_INSTANCE_PER_DB()}, /* signals read-only transaction completion */
{"SHMEM_LFS", POLICY_INSTANCE_PER_DB()}, /* lfs state & buffer in shared memory */
{"SEMAP_LFS", POLICY_INSTANCE_PER_DB()}, /* synchronises operation with lfs */
{"SEMAP_CHECKPOINT_FINISHED", POLICY_INSTANCE_PER_DB()}, /* to wait for checkpoint to finish */
{"SHMEM_LOGICAL_LOG", POLICY_INSTANCE_PER_DB()}, /* logical log state & buffer in shared memory */
{"SEMAP_LOGICAL_LOG", POLICY_INSTANCE_PER_DB()}, /* synchronises operation with logical log */
{"EVENT_SM_SHUTDOWN_COMMAND", POLICY_INSTANCE_PER_DB()}, /* signaled by SSMMsg thread in SM when shutdown command arrives via messaging interface */
{"EVENT_RECOVERY_COMPLETED", POLICY_INSTANCE_PER_DB()}, /* signaled when se_rcv completes the recovery */
{"SEMAP_TRNS_TABLE", POLICY_INSTANCE_PER_DB()}, /* synchronises access to transactions table in SM */
{"EVENT_SM_READY", POLICY_INSTANCE_PER_DB()}, /* used to signal initialisation completion when starting SM in the background mode */
{"EVENT_GOV_READY", POLICY_SINGLETON()}, /* used to signal initialisation completion when starting GOV in the background mode */
/* %} */
{NULL}
};
void InitGlobalNames(int rangeBegin, int rangeEnd)
{
UInitGlobalNamesRegistry(globalNamesRegistry, NULL, rangeBegin, rangeEnd);
}
void ReleaseGlobalNames()
{
UReleaseGlobalNamesRegistry();
}
global_name CreateNameOfSmTalk(int databaseId, char *buf, size_t bufSize)
{
const char *namesVec[2];
char bufa[128], bufb[128];
namesVec[0] = UCreateGlobalName("SHMEM_SM_TALK", databaseId, bufa, 128);
namesVec[1] = UCreateGlobalName("SEMAR_SM_TALK", databaseId, bufb, 128);
return UCreateCompoundName(namesVec, 2, buf, bufSize);
}
global_name CreateNameOfEventVmmCalback(int sessionId, char *buf, size_t bufSize)
{
return UCreateGlobalName("EVENT_VMM_CALLBACK", sessionId, buf, bufSize);
}
global_name CreateNameOfEventVmmCalbackCompleted(int sessionId, char *buf, size_t bufSize)
{
return UCreateGlobalName("EVENT_VMM_CALLBACK_COMPLETED", sessionId, buf, bufSize);
}
global_name CreateNameOfEventLockGranted(int sessionId, char *buf, size_t bufSize)
{
return UCreateGlobalName("EVENT_LOCK_GRANTED", sessionId, buf, bufSize);
}
/* empty string is invalid as a global name but NULL is valid, so we use empty string as initializer */
global_name GOVERNOR_SHARED_MEMORY_NAME = "";
global_name CHARISMA_GOVERNOR_IS_READY = "";
global_name SE_EVENT_LOG_SHARED_MEMORY_NAME = "";
global_name SE_EVENT_LOG_SEMAPHORES_NAME = "";
global_name SEDNA_GLOBAL_MEMORY_MAPPING = "";
void SetGlobalNames()
{
static char
GOVERNOR_SHARED_MEMORY_NAME__buf__ [128],
CHARISMA_GOVERNOR_IS_READY__buf__ [128],
SE_EVENT_LOG_SHARED_MEMORY_NAME__buf__ [128],
SE_EVENT_LOG_SEMAPHORES_NAME__buf__ [128],
SEDNA_GLOBAL_MEMORY_MAPPING__buf__ [128];
GOVERNOR_SHARED_MEMORY_NAME =
UCreateGlobalName("SHMEM_GOV", 0, GOVERNOR_SHARED_MEMORY_NAME__buf__, 128);
CHARISMA_GOVERNOR_IS_READY =
UCreateGlobalName("EVENT_GOV_READY", 0, CHARISMA_GOVERNOR_IS_READY__buf__, 128);
SE_EVENT_LOG_SHARED_MEMORY_NAME =
UCreateGlobalName("SHMEM_EVENT_LOG", 0, SE_EVENT_LOG_SHARED_MEMORY_NAME__buf__, 128);
SE_EVENT_LOG_SEMAPHORES_NAME =
UCreateGlobalName("SEMAR_EVENT_LOG", 0, SE_EVENT_LOG_SEMAPHORES_NAME__buf__, 128);
SEDNA_GLOBAL_MEMORY_MAPPING =
UCreateGlobalName("SHMEM_GLOBAL", 0, SEDNA_GLOBAL_MEMORY_MAPPING__buf__, 128);
}
/* empty string is invalid as a global name but NULL is valid, so we use empty string as initializer */
global_name CHARISMA_SM_CALLBACK_SHARED_MEMORY_NAME = "";
global_name CHARISMA_BUFFER_SHARED_MEMORY_NAME = "";
global_name VMM_SM_SEMAPHORE_STR = "";
global_name VMM_SM_EXCLUSIVE_MODE_SEM_STR = "";
global_name SNAPSHOT_CHECKPOINT_EVENT = "";
global_name TRY_ADVANCE_SNAPSHOT_EVENT = "";
global_name CATALOG_NAMETABLE_SEMAPHORE_STR;
global_name CATALOG_MASTER_SEMAPHORE_STR;
//global_name METADATA_SEMAPHORE_STR = "";
//global_name INDEX_SEMAPHORE_STR = "";
//global_name FT_INDEX_SEMAPHORE_STR = "";
//global_name TRIGGER_SEMAPHORE_STR = "";
global_name SEDNA_LFS_SEM_NAME = "";
global_name SEDNA_LFS_SHARED_MEM_NAME = "";
global_name CHARISMA_LOGICAL_LOG_SHARED_MEM_NAME = "";
global_name CHARISMA_LOGICAL_LOG_PROTECTION_SEM_NAME = "";
global_name CHARISMA_CHECKPOINT_SEM = "";
global_name SEDNA_CHECKPOINT_FINISHED_SEM = "";
global_name SEDNA_TRNS_FINISHED = "";
global_name CHARISMA_WAIT_FOR_CHECKPOINT = "";
global_name CHARISMA_DB_RECOVERED_BY_LOGICAL_LOG = "";
global_name CHARISMA_SM_WAIT_FOR_SHUTDOWN = "";
global_name CHARISMA_LRU_STAMP_SHARED_MEMORY_NAME = "";
global_name CHARISMA_SM_SMSD_ID = "";
global_name CHARISMA_SM_IS_READY = "";
void SetGlobalNamesDB(int databaseId)
{
static char
CHARISMA_SM_CALLBACK_SHARED_MEMORY_NAME__buf__ [128],
CHARISMA_BUFFER_SHARED_MEMORY_NAME__buf__ [128],
VMM_SM_SEMAPHORE_STR__buf__ [128],
VMM_SM_EXCLUSIVE_MODE_SEM_STR__buf__ [128],
SNAPSHOT_CHECKPOINT_EVENT__buf__ [128],
SEDNA_CHECKPOINT_FINISHED_SEM__buf__ [128],
// METADATA_SEMAPHORE_STR__buf__ [128],
// INDEX_SEMAPHORE_STR__buf__ [128],
// FT_INDEX_SEMAPHORE_STR__buf__ [128],
// TRIGGER_SEMAPHORE_STR__buf__ [128],
CATALOG_NAMETABLE_SEMAPHORE_STR__buf__ [128],
CATALOG_MASTER_SEMAPHORE_STR__buf__ [128],
SEDNA_LFS_SEM_NAME__buf__ [128],
SEDNA_LFS_SHARED_MEM_NAME__buf__ [128],
CHARISMA_LOGICAL_LOG_SHARED_MEM_NAME__buf__ [128],
CHARISMA_LOGICAL_LOG_PROTECTION_SEM_NAME__buf__ [128],
SEDNA_TRNS_FINISHED__buf__ [128],
TRY_ADVANCE_SNAPSHOT_EVENT__buf__ [128],
CHARISMA_DB_RECOVERED_BY_LOGICAL_LOG__buf__ [128],
CHARISMA_SM_WAIT_FOR_SHUTDOWN__buf__ [128],
CHARISMA_LRU_STAMP_SHARED_MEMORY_NAME__buf__ [128],
CHARISMA_SM_IS_READY__buf__ [128];
CHARISMA_SM_CALLBACK_SHARED_MEMORY_NAME =
UCreateGlobalName("SHMEM_VMM_CALLBACK_PARAMS", databaseId, CHARISMA_SM_CALLBACK_SHARED_MEMORY_NAME__buf__, 128);
CHARISMA_BUFFER_SHARED_MEMORY_NAME =
UCreateGlobalName("SHMEM_BUFFERS", databaseId, CHARISMA_BUFFER_SHARED_MEMORY_NAME__buf__, 128);
VMM_SM_SEMAPHORE_STR =
UCreateGlobalName("SEMAP_VMM_INIT", databaseId, VMM_SM_SEMAPHORE_STR__buf__, 128);
VMM_SM_EXCLUSIVE_MODE_SEM_STR =
UCreateGlobalName("SEMAP_BUFMGR_EXCL_MODE", databaseId, VMM_SM_EXCLUSIVE_MODE_SEM_STR__buf__, 128);
SNAPSHOT_CHECKPOINT_EVENT =
UCreateGlobalName("EVENT_NEW_JOB_4_CHECKPOINT_THREAD", databaseId, SNAPSHOT_CHECKPOINT_EVENT__buf__, 128);
TRY_ADVANCE_SNAPSHOT_EVENT =
UCreateGlobalName("EVENT_READONLY_TRN_COMPLETED", databaseId, TRY_ADVANCE_SNAPSHOT_EVENT__buf__, 128);
SEDNA_CHECKPOINT_FINISHED_SEM =
UCreateGlobalName("SEMAP_CHECKPOINT_FINISHED", databaseId, SEDNA_CHECKPOINT_FINISHED_SEM__buf__, 128);
/*
METADATA_SEMAPHORE_STR =
UCreateGlobalName("SEMAP_METADATA", databaseId, METADATA_SEMAPHORE_STR__buf__, 128);
INDEX_SEMAPHORE_STR =
UCreateGlobalName("SEMAP_INDICES", databaseId, INDEX_SEMAPHORE_STR__buf__, 128);
FT_INDEX_SEMAPHORE_STR =
UCreateGlobalName("SEMAP_FT_INDICES", databaseId, FT_INDEX_SEMAPHORE_STR__buf__, 128);
TRIGGER_SEMAPHORE_STR =
UCreateGlobalName("SEMAP_TRIGGERS", databaseId, TRIGGER_SEMAPHORE_STR__buf__, 128);
*/
CATALOG_NAMETABLE_SEMAPHORE_STR =
UCreateGlobalName("SEMAP_CATALOG_NAMETABLES", databaseId, CATALOG_NAMETABLE_SEMAPHORE_STR__buf__, 128);
CATALOG_MASTER_SEMAPHORE_STR =
UCreateGlobalName("SEMAP_CATALOG_METADATA", databaseId, CATALOG_MASTER_SEMAPHORE_STR__buf__, 128);
SEDNA_LFS_SEM_NAME =
UCreateGlobalName("SEMAP_LFS", databaseId, SEDNA_LFS_SEM_NAME__buf__, 128);
SEDNA_LFS_SHARED_MEM_NAME =
UCreateGlobalName("SHMEM_LFS", databaseId, SEDNA_LFS_SHARED_MEM_NAME__buf__, 128);
CHARISMA_LOGICAL_LOG_SHARED_MEM_NAME =
UCreateGlobalName("SHMEM_LOGICAL_LOG", databaseId, CHARISMA_LOGICAL_LOG_SHARED_MEM_NAME__buf__, 128);
CHARISMA_LOGICAL_LOG_PROTECTION_SEM_NAME =
UCreateGlobalName("SEMAP_LOGICAL_LOG", databaseId, CHARISMA_LOGICAL_LOG_PROTECTION_SEM_NAME__buf__, 128);
SEDNA_TRNS_FINISHED =
UCreateGlobalName("SEMAP_TRN_REGULATION", databaseId, SEDNA_TRNS_FINISHED__buf__, 128);
CHARISMA_DB_RECOVERED_BY_LOGICAL_LOG =
UCreateGlobalName("EVENT_RECOVERY_COMPLETED", databaseId, CHARISMA_DB_RECOVERED_BY_LOGICAL_LOG__buf__, 128);
CHARISMA_SM_WAIT_FOR_SHUTDOWN =
UCreateGlobalName("EVENT_SM_SHUTDOWN_COMMAND", databaseId, CHARISMA_SM_WAIT_FOR_SHUTDOWN__buf__, 128);
CHARISMA_LRU_STAMP_SHARED_MEMORY_NAME =
UCreateGlobalName("SHMEM_BUFFERS_LRU", databaseId, CHARISMA_LRU_STAMP_SHARED_MEMORY_NAME__buf__, 128);
CHARISMA_SM_IS_READY =
UCreateGlobalName("EVENT_SM_READY", databaseId, CHARISMA_SM_IS_READY__buf__, 128);
};
/* The following chars are allowed:
* ! (0x21), # (0x23), % (0x25), & (0x26), ( (0x28), ) (0x29),
* + (0x2B), , (0x2C), - (0x2D), . (0x2E), 0-9 (0x30 - 0x39),
* ; (0x3B), = (0x3D), @ (0x40), A-Z (0x41-0x5A), [ (5B), ] (5D),
* ^ (0x5E), _ (0x5F), ` (0x60), a-z (0x61-0x7A), { (0x7B),
* } (0x7D), ~ (0x7E) */
static const unsigned char
database_name_map[16] = {0x00, 0x00, 0x00, 0x00,
0x56, 0xDE, 0xFF, 0xD4,
0xFF, 0xFF, 0xFF, 0xF7,
0xFF, 0xFF, 0xFF, 0xF6};
/* Is char is allowed within a database name */
#define DATABASE_NAME_ALLOWED_BYTE(byte) \
((byte) & 0x80 ? 0 : (database_name_map[((byte) >> 3)] & (0x80 >> ((byte) & 7))))
void check_db_name_validness(const char* name)
{
if (NULL == name)
throw USER_EXCEPTION2(SE1003,
"database name validation failed (null database name was given)");
size_t len = strlen(name), counter = 0;
/* Name must contain at least one symbol and its length must
* be less or equal than MAX_DATABASE_NAME_LENGTH */
if (len < 1 || len > MAX_DATABASE_NAME_LENGTH)
throw USER_EXCEPTION2(SE4307, "empty or too long database name");
while(counter < len)
{
unsigned char c = name[counter];
if(DATABASE_NAME_ALLOWED_BYTE(c))
counter++;
else
throw USER_EXCEPTION2(SE4307, name);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,935
|
{"url":"http:\/\/tex.stackexchange.com\/questions\/29873\/expanding-variable-inside-picture-environment","text":"# expanding variable inside picture environment\n\n(this version of this question is highly edited based on suggestions)\n\nI thought to quickly create a fax cover sheet last night and, quickly creating one, I ran into a problem I can't figure out so I thought I'd ask around. First, the structure: I have 2 files: one is the style file and the other is the file with the specific fax info. In the style file I've created some variables to hold data that is specified in the info file. Here is the style file:\n\n\n\\newsavebox{\\faxcover}\n\\savebox{\\faxcover}{%\n\\put(0,7){\\makebox{\\bfseries Date:} \\@myfaxdate}\n\\put(0,6){\\makebox{\\bfseries To:} \\@myfaxto}\n\\put(0,5){\\makebox{\\bfseries From:} \\@myfaxfrom}\n\\put(0,4){\\makebox{\\bfseries Re:} \\@myfaxre}\n\\put(0,2){\\makebox{\\bfseries From:} \\@myfaxtelnum}\n\\put(0,1){\\makebox{\\bfseries Re:} \\@myfaxnum}\n}\n\n\\newcommand{\\makefax}{%\n\\begin{picture}(10,10)\n\\put (0,0){\\usebox\\faxcover}\n\\end{picture}\n}\n\n\nHere's the other file (the top-level file)\n\n\\documentclass[10pt]{report}\n\\usepackage{.\/myfax}\n\\begin{document}\n\n\\myfaxdate{09-28-2011}\n\\myfaxto{Ozymandius, King of Kings}\n\\myfaxfrom{Bev}\n\\myfaxre{Two vast and trunkless legs of stone}\n\\myfaxnumpages{3 counting the cover sheet}\n\\myfaxtelnum{(516) 676-4099}\n\\myfaxnum{(800) 123-4567}\n\n\\makefax\n\n\\end{document}\n\n\nWhen I compile, I get an error telling me that \\@myfaxdate is an undefined control sequence. I should add that the purpose of dividing into 2 files is because I want to create my own fax style that I may want others in my org to use (maybe). This will simplify it so all they have to know how to do is to fill in some blanks in the main file.\n\n-\nThere should be a problem either inside or outside: @ is not considered a letter normally by LaTeX. Please add a minimal working example (MWE). \u2013\u00a0 Andrey Vihrov Sep 29 '11 at 7:39\n@AndreyVihrov: except in packages ;-) \u2013\u00a0 \u211daphink Sep 29 '11 at 8:07\n@Andrey - Sorry, I thought that was an MWE. Apparently too much so. I hope this one is clearer. \u2013\u00a0 bev Sep 29 '11 at 20:46\n@Werner - I think that you're correct. I was messing about last night and saw that if I eliminate the boxes, and just use the picture env it compiles. The trouble is that I want to use the boxes as it makes structuring the page so much easier. Do your comments still apply given the edited version of my question? Thx. \u2013\u00a0 bev Sep 29 '11 at 20:49\n\nTry this:\n\n\\documentclass{article}\n\\begin{document}\n\\newcommand{\\myfaxdate}[1]{#1}\n\\begin{picture}(5,5)\n\\put(30,30){Date: \\myfaxdate{9-12-2011}}\n\\end{picture}\n\\end{document}\n\n-\nwell, it works, but it doesn't conform to my wish to have a style file and a user's main file containing the specific info for that specific fax. Thanks tho... \u2013\u00a0 bev Sep 29 '11 at 20:51\n\nThe box in \\savebox is constructed at \"save\" time, not at \"use\" time. By that time the commands have not yet been defined.\n\nTry not using \\savebox and \\usebox at all:\n\n\\newcommand{\\makefax}{%\n\\begin{picture}(10,10)\n\\put(0,7){\\makebox{\\bfseries Date:} \\@myfaxdate}\n\\put(0,6){\\makebox{\\bfseries To:} \\@myfaxto}\n\\put(0,5){\\makebox{\\bfseries From:} \\@myfaxfrom}\n\\put(0,4){\\makebox{\\bfseries Re:} \\@myfaxre}\n\\put(0,2){\\makebox{\\bfseries From:} \\@myfaxtelnum}\n\\put(0,1){\\makebox{\\bfseries Re:} \\@myfaxnum}\n\n\u2022 The \\makebox seems like it could be replaced with just \\textbf.\n\u2022 If you use tabular or \\parbox (or nothing at all) instead of picture, you need not worry about interline space.\n@bev: You seem to misunderstand the purpose of \\savebox. It's not to create a box, it's to typeset text and store the result in a box register. If you say \\usebox{\\mybox} n times, you get n copies of the same, already typeset content. To put content in a box so that you can treat it as a unit use \\parbox or similar. \u2013\u00a0 Andrey Vihrov Sep 30 '11 at 4:58","date":"2015-10-06 16:43:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7334031462669373, \"perplexity\": 2232.780192702906}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-40\/segments\/1443736678861.8\/warc\/CC-MAIN-20151001215758-00080-ip-10-137-6-227.ec2.internal.warc.gz\"}"}
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Q: Cross system process spawning and interaction I've got a piece of software i'm working on that spawns short lived processes on remote systems to run some code (mostly SerialPort IO) that may or may not interact with the spawning application (but it should be assumed it will), and will then terminate on command.
What is the best way to spawn a remote process like this (PSExec? WMI? System.Diagnostics.Process perhaps?) After the processes is spawned how can i tell it to subscribe to it's host? What would the SO community reccommend for a basic event/message based framework for the communication process? Is WCF well suited to the task? would Remoting be easier? What are my options here?
A: I don't think System.Diagnostics.Process allows launching a process on a remote system. So WMI or shelling to something like psexec may be your only options here.
Regarding subscription/interaction options, yes, WCF is the way to go. Avoid Remoting: it offers no benefits for the general case, is no longer being enhanced, and is deprecated in favour of WCF for a variety of reasons (basically distributed objects become a pain in the neck due to versioning and state considerations; messaging is usually easier to set up, understand and maintain. Also remoting had no security if I remember rightly). For setting this up, when the parent process spawns the remote process, it could pass a URL as a command line argument. The parent hosts a WCF service on that URL. Now if the spawned process needs to communicate back to / subscribe to the parent process, it just connects to the URL it was given. If the parent needs to initiate communication, then make the WCF service duplex, or have the spawned process host its own WCF service, and tell the parent the URL via the parent's service.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,853
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For people from Maine, it might seem odd to hear of someone making the trek up to the Pine Tree State from sunny and warm Tennessee on purpose. That's just what Husson University's new Sports Information Director, Kaelyn Angelo, did.
Angelo played two years of soccer at the University of Tennessee before transferring to Manhattan college for her final two years of schooling, graduating in 2015.
Being the director of sports information, you need to have a broad arsenal of abilities to be able to do all the tasks within the job description. Angelo's background gives her just that.
I graduated in May of 2015 and then I had a summer internship that lasted from June of 2015 to August, which was in California," Angelo said. "I worked for an advertising agency that produced the Toyota commercials.
You may be asking, what does the SID do? For Angelo, it's everything from writing press releases to running the social media accounts to even helping committees find new head coaches. Right now, Angelo is apart of a group interviewing for a new men's soccer coach after Jeff Getler stepped down last month after six seasons at the helm.
For the SID, no two days are alike, or lacking in to-do tasks.
"I was expecting the work load to be difficult," Angelo said. "But then again I wasn't, because at Division one schools you have a SID for each sport, and I'm dealing with twenty one sports… I kind of like having a lot of things to do at once because it keeps me busy.
If you've ever been to a sporting event, collegiate or professional, you know that it can be hectic to put, and keep, together. This is also apart of Angelo's job description.
During games, Angelo helps with the statistics side of things. Acting as what she calls a "caller," Angelo will call out a special code to the stat keepers to make sure they don't miss anything that happens.
Part of her crew comes from the Husson-NESCom student body.
This past weekend, Husson hosted the NAC men's basketball conference tournament, with eight teams attending over the weekend.
Student interaction helps Angelo focus on what her work entails on game days. But that's not to say she isn't qualified to take pictures, write game recaps or anything that students help with.
Her internships prior to being hired at Husson also gave her experience in social media as well as advertising, both components to her new SID job. The variety that comes with being the sports information director is what Angelo loves.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 7,429
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Q: Part of terminal line disappears when pressing down arrow I use a custom PS1 to display more relevant information in my terminal, such as if I'm in a git directory and if it's clean or needs to commit changes. However, sometimes when I'm arrowing through commands, part of the terminal line disappears:
@ ~/tests/testing [tests] > grunt
# up arrow, down arrow
@ ~/tests/testing [t
Essentially, the ests] > gets cut off and I'm left with just the [t.
Is there any particular reason why part of the line is getting cut off with this PS1 config?
Here's some additional info:
My TERM env var is xterm-256color. Here's my .bash_profile:
red='\033[0;31m'
yellow='\033[0;32m'
orange='\033[0;33m'
blue='\033[0;34m'
pink='\033[0;35m'
NC='\033[0m'
function is_git {
if git rev-parse --is-inside-work-tree 2>/dev/null; then
return 1
else
return 0
fi
}
function stuff {
if [ $(is_git) ]; then
git_dir="$(git rev-parse --git-dir 2>/dev/null)"
if [ -z "$(ls -A ${git_dir}/refs/heads )" ]; then
echo -en " [${orange}init${NC}]"
return
fi
echo -n " ["
if [ $(git status --porcelain 2>/dev/null| wc -l | tr -d ' ') -ne 0 ]; then
echo -en "${red}"
else
echo -en "${blue}"
fi
echo -en "$(git rev-parse --abbrev-ref HEAD)${NC}]"
fi
}
export PS1="@ \w\[\$(stuff)\]\[\$(tput sgr0)\] > "
A: @i_am_root's suggestion to put the \[ and \] inside the definition of red and the like is a good one. However, per this, bash only processes \[ and \] in PS1, not in text included in PS1 by $(). Therefore, use \001 and \002 (or \x01 and \x02) inside red and the like instead of \[ and \].
Note: Per this answer, only the escape codes should be in the \001 and \002. The text that will be visible to the user should be outside the \001 and \002 so that bash knows it takes up space on the screen and can account for that when redrawing.
A: Bash color codes, escaping characters, assignments, and such get confusing quickly.
Try this code sample which replaces the echo commands by adding to the PS1 variable.
red='\[\033[0;31m\]'
yellow='\[\033[0;32m\]'
orange='\[\033[0;33m\]'
blue='\[\033[0;34m\]'
pink='\[\033[0;35m\]'
NC='\[\033[0m\]'
export PS1="@ \w"
function is_git {
if git rev-parse --is-inside-work-tree 2>/dev/null; then
return 1
else
return 0
fi
}
function stuff {
if [ $(is_git) ]; then
git_dir="$(git rev-parse --git-dir 2>/dev/null)"
if [ -z "$(ls -A ${git_dir}/refs/heads )" ]; then
PS1="${PS1} [${orange}init${NC}]"
return
fi
PS1="$PS1 ["
if [ $(git status --porcelain 2>/dev/null| wc -l | tr -d ' ') -ne 0 ]; then
PS1="${PS1}${red}"
else
PS1="${PS1}${blue}"
fi
PS1="${PS1}$(git rev-parse --abbrev-ref HEAD)${NC}]"
fi
}
stuff
PS1="${PS1}$(tput sgr0) > "
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,723
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\section{Introduction}
The discovery of graphene, the material made of a one-atom-thick carbon layer,
has attracted a lot attention as it provides the realization of a system
where the electrons have conical valence and conduction bands, therefore
behaving at low energies as massless Dirac fermions\cite{geim,kim,rmp}.
This offers the possibility of employing the new material as a test ground
of fundamental concepts in theoretical physics, since the interacting electron
system represents a variant of strongly coupled quantum electrodynamics (QED)
affording quite unusual effects\cite{nil,fog,shy,ter}.
A remarkable feature of such a theory is that a sufficiently strong Coulomb
interaction may open a gap in the electronic spectrum.
This effect was already known from the study of QED \cite{appel}, where it
corresponds to the dynamical breakdown of the chiral $U(2)$ symmetry of the
theory. In the context of graphene, such a mechanism is sometimes alluded
as an exciton instability though, given the absence of a gap between valence
and conduction bands, it becomes more appropriate to describe the effect as
a kind of charge-density-wave instability of the 2D layer. The gap generation
proceeds actually through the development of a non-vanishing average value
of the staggered (sublattice odd) charge density in the underlying honeycomb
lattice, which leads to the generation of a mass and opening of a gap for the
Dirac quasiparticles.
The question of the dynamical gap generation was first
addressed in graphene in the approach to the theory with a large number $N$ of
fermion flavors\cite{khves,gus,ale,son}. The existence of a critical point
for the formation of an excitonic insulator has been also suggested from
second-order calculations of electron self-energy corrections\cite{vafek}.
More recently, Monte Carlo simulations of the field theory have been carried
out in the graphene lattice\cite{drut1,hands}, showing that the chiral symmetry
of the massless theory can be broken above a critical value for the graphene
fine structure constant
$\alpha_c \approx 1.08$ \cite{drut1}. The possibility of
dynamical gap generation has been also studied in the ladder
approximation\cite{gama,fer,me,brey}, leading in the case of static screening of
the interaction to an estimate of the critical coupling
$\alpha_c \approx 1.62$ for $N = 4$ \cite{gama}. Lately, the resolution
of the Schwinger-Dyson formulation of the gap equation has revealed that
the effect of the dynamical polarization can
significantly lower the critical coupling for dynamical gap generation,
down to a value $\alpha_c \approx 0.92$ for $N = 4$\cite{ggg}.
In this paper we take advantage of the renormalization properties of the
Dirac theory in order to assess the effect of the electron self-energy
corrections on the chiral symmetry breaking. In this respect, it has been
found that the renormalization of the quasiparticle properties can have a
significant impact, mainly through the increase of the Fermi velocity at
low energies\cite{khves2,sabio}. Then, we will consider the electron-hole
vertex accounting for the dynamical gap generation in the ladder
approximation, shown schematically in Fig. \ref{one}, and we will supplement
it by self-energy corrections to the electron and hole states. This dressing
of the quasiparticles will have the result of increasing significantly the
critical coupling at which the chiral symmetry breaking takes place. Thus,
under static RPA screening of the interaction potential in the ladder series,
we will find the critical value $\alpha_c \approx 4.9$ at the physical number
of flavors $N = 4$. In agreement
with the trend observed in Ref. \cite{ggg}, we will see however that the more
sensible dynamical screening of the interaction has the effect of lowering
substantially that estimate, down to a value $\alpha_c \approx 1.75$ which is
below the nominal value of the interaction coupling in isolated free-standing
graphene.
\section{Ladder approximation for staggered charge density}
We consider the field theory for Dirac quasiparticles in graphene
interacting through the long-range Coulomb potential, with a Hamiltonian
given by
\begin{eqnarray}
\lefteqn{H = i v_F \int d^2 r \; \overline{\psi}_i({\bf r})
\boldsymbol{\gamma } \cdot \boldsymbol{\nabla} \psi_i ({\bf r}) } \nonumber \\
& & + \frac{e^2}{8 \pi} \int d^2 r_1
\int d^2 r_2 \; \rho ({\bf r}_1)
\frac{1}{|{\bf r}_1 - {\bf r}_2|} \rho ({\bf r}_2) \;\;\;\;\;
\label{ham}
\end{eqnarray}
where $\{ \psi_i \}$ is a collection of $N/2$ four-component Dirac
spinors, $\overline{\psi}_i = \psi_i^{\dagger} \gamma_0 $, and
$\rho ({\bf r}) = \overline{\psi}_i ({\bf r}) \gamma_0 \psi_i ({\bf r})$.
The matrices $\gamma_{\sigma } $ satisfy
$\{ \gamma_\mu, \gamma_\nu \} = 2 \: {\rm diag } (1,-1,-1)$
and can be conveniently represented in terms of Pauli matrices as
$\gamma_{0,1,2} = (\sigma_3, \sigma_3 \sigma_1, \sigma_3 \sigma_2) \otimes
\sigma_3$, where the first factor acts on the two sublattice components of
the graphene lattice.
Our main interest is to study the behavior of the vertex for the staggered
(sublattice odd) charge density
\begin{equation}
\rho_m ({\bf r}) = \overline{\psi}({\bf r}) \psi ({\bf r})
\end{equation}
This operator gives the order parameter for the dynamical gap generation, and
the signal that it gets a nonvanishing expectation value can be obtained from
the divergence of the response function
$\langle T \rho_m ({\bf q}, t) \rho_m (-{\bf q}, 0) \rangle$.
The singular behavior of this susceptibility can be traced back to the
divergence at ${\bf q},\omega_q \rightarrow 0$ of the irreducible vertex
\begin{eqnarray}
\lefteqn{\Gamma ({\bf q},\omega_q;{\bf k},\omega_k) } \nonumber \\
& & = \langle \rho_m ({\bf q},\omega_q) \psi ({\bf k}+{\bf q},\omega_k+\omega_q)
\psi^{\dagger} ({\bf k},\omega_k) \rangle_{\rm 1PI}
\end{eqnarray}
where ${\rm 1PI}$ denotes that $\Gamma$ is made of one-particle irreducible diagrams
without external electron propagators.
In the ladder approximation, the vertex $\Gamma $ is bound to satisfy the
self-consistent equation depicted diagrammatically in Fig. \ref{one}. This
equation can be solved perturbatively by iterating the interaction between
electrons and holes in the vertex, in which case this ends up being represented
by the sum of ladder diagrams. On the other hand, the self-consistent
equation can be written in compact form, specially at momentum transfer
${\bf q} = 0$ and $\omega_q = 0$. We recall at this point the expression of
the free Dirac propagator
\begin{equation}
\langle \psi ({\bf k}, \omega_k ) \psi^{\dagger } ({\bf k}, \omega_k ) \rangle_{\rm free}
= i \frac{-\gamma_0 \omega_k + v_F \boldsymbol{\gamma} \cdot {\bf k} }
{-\omega_k^2 + v_F^2 {\bf k}^2 - i\eta } \gamma_0
\end{equation}
Given that $\Gamma $ must be anyhow proportional to $\gamma_0 $, we get
\begin{align}
& -\frac{-\gamma_0 \omega_p + v_F \boldsymbol{\gamma} \cdot {\bf p} }
{-\omega_p^2 + v_F^2 {\bf p}^2 - i\eta }
\gamma_0 \: \Gamma ({\bf 0},0;{\bf p},\omega_p) \:
\frac{-\gamma_0 \omega_p + v_F \boldsymbol{\gamma} \cdot {\bf p} }
{-\omega_p^2 + v_F^2 {\bf p}^2 - i\eta } \gamma_0 \nonumber \\
& = \frac{\Gamma ({\bf 0},0;{\bf p},\omega_p)}{-\omega_p^2 + v_F^2 {\bf p}^2 - i\eta }
\end{align}
The self-consistent equation for the vertex becomes then
\begin{eqnarray}
\lefteqn{\Gamma ({\bf 0},0;{\bf k},i\omega_k) = \gamma_0 } \nonumber \\
& & + \int \frac{d^2 p}{(2\pi )^2} \frac{d\omega_p}{2\pi }
\frac{\Gamma ({\bf 0},0;{\bf p},i\omega_p)}{\omega_p^2 + v_F^2{\bf p}^2}
V({\bf k}-{\bf p},i\omega_k - i\omega_p) \;\;\;\;\;
\label{self}
\end{eqnarray}
where $V({\bf p},\omega_p)$ stands for the Coulomb interaction. We will deal
in general with the RPA to screen the potential, so that
\begin{equation}
V({\bf p}, \omega_p) = \frac{e^2}{2 |{\bf p}| + e^2 \chi ({\bf p}, \omega_p)}
\end{equation}
in terms of the polarization $\chi $ for $N$ two-component Dirac fermions.
Eq. (\ref{self}) is formally invariant under a
dilatation of frequencies and momenta, which shows that the scale
of $\Gamma ({\bf 0},0;{\bf k},\omega_k)$ is dictated by the high-energy cutoff
$\Lambda $ needed to regularize the integrals. The vertex acquires in general
an anomalous dimension $\gamma_{\psi^2}$, which governs the behavior under
changes in the energy scale\cite{amit}
\begin{equation}
\Gamma ({\bf q},\omega_q;{\bf k},\omega_k) \sim \Lambda^{\gamma_{\psi^2}}
\end{equation}
We recall below how to compute $\gamma_{\psi^2}$, showing that it diverges
at a critical value of the interaction strength $\alpha = e^2/4\pi v_F$.
This translates into a divergence of the own susceptibility
$\langle T \rho_m ({\bf q}, t) \rho_m (-{\bf q}, 0) \rangle$
at momentum transfer ${\bf q} \rightarrow 0$, providing then the signature
of the condensation of
$\rho_m ({\bf r}) = \overline{\psi}({\bf r}) \psi ({\bf r})$
and the consequent development of the gap for the Dirac quasiparticles.
\begin{figure}
\begin{center}
\mbox{\epsfxsize 9.0cm \epsfbox{selfcons.eps}}
\end{center}
\caption{Self-consistent diagrammatic equation for the vertex
$\langle \rho_m ({\bf q},\omega_q) \psi ({\bf k}+{\bf q},\omega_k+\omega_q)
\psi^{\dagger} ({\bf k},\omega_k) \rangle$,
equivalent to the sum of ladder diagrams built from
the iteration of the Coulomb interaction (wavy line) between electron and
hole states (arrow lines).}
\label{one}
\end{figure}
\section{Electron self-energy effects in statically screened ladder approximation}
We deal first with the approach in which electrons and holes are dressed by
self-energy corrections, while the Coulomb interaction in (\ref{self}) is
screened by means of the static RPA with polarization
\begin{equation}
\chi ({\bf p}, 0) = \frac{N}{16}\frac{|{\bf p}|}{v_F}
\end{equation}
The most important self-energy effect comes from the renormalization of the
Fermi velocity at low energies\cite{np2,prbr}, which can be incorporated by
replacing $v_F$ in Eq. (\ref{self}) by the effective Fermi velocity
\begin{equation}
\widetilde{v}_F({\bf p}) = v_F + \Sigma_v ({\bf p})
\end{equation}
dressed with the
self-energy corrections $\Sigma_v ({\bf p})$. The expansion of Eq. (\ref{self})
in powers of $\Sigma_v ({\bf p})$ would amount to the iteration of
self-energy corrections in the electron and hole internal lines
in Fig. \ref{one}, showing that the present approach encodes a systematic
way of improving the sum of ladder diagrams for the vertex $\Gamma $ \cite{note}.
The electron self-energy corrections, as well as the terms of the
ladder series, are given by logarithmically divergent integrals that need to
be cut off at a high-energy scale $\Lambda $. Alternatively, one can also define
the theory at spatial dimension $D = 2 - \epsilon$, what automatically
regularizes all the momentum integrals. After performing the frequency integral,
Eq. (\ref{self}) then becomes
\begin{eqnarray}
\lefteqn{\Gamma ({\bf 0},0;{\bf k},\omega_k) = \gamma_0 } \nonumber \\
& & + \frac{e_0^2}{4\kappa }
\int \frac{d^D p}{(2\pi )^D} \Gamma ({\bf 0},0;{\bf p},\omega_k)
\frac{1}{\widetilde{v}_F({\bf p}) |{\bf p}|} \frac{1}{|{\bf k}-{\bf p}|}
\label{selfcons}
\end{eqnarray}
where $e_0^2$ is related to $e^2$ through an auxiliary
momentum scale $\rho $ such that
\begin{equation}
e_0^2 = \rho^{\epsilon} e^2
\end{equation}
and we have defined the dielectric constant
\begin{equation}
\kappa = 1 + \frac{N e^2}{32 v_F}
\end{equation}
In the ladder approximation, the Fermi velocity gets a divergent correction
only from the ^^ ^^ rainbow'' self-energy diagram with exchange of a single
screened interaction shown in Fig. \ref{rainbow} \cite{np2}. The dressed Fermi
velocity becomes
\begin{equation}
\widetilde{v}_F({\bf p}) = v_F + \frac{e_0^2}{16 \pi^2\kappa}
(4\pi )^{\epsilon /2}
\frac{\Gamma (\tfrac{\epsilon }{2}) \Gamma (\tfrac{1-\epsilon}{2})
\Gamma (\tfrac{3-\epsilon}{2}) }
{ \Gamma (2 - \epsilon) }
\frac{1}{|{\bf p}|^{\epsilon}}
\label{vd}
\end{equation}
The expressions (\ref{selfcons}) and (\ref{vd}) are singular in the limit
$\epsilon \rightarrow 0$. The most convenient way to show that all the poles
in the $\epsilon $ parameter can be renormalized away is to resort at this point
to a perturbative computation of $\Gamma ({\bf 0},0;{\bf k},\omega_k)$.
\begin{figure}
\begin{center}
\mbox{\epsfxsize 3.0cm \epsfbox{rainbow.eps}}
\end{center}
\caption{Electron self-energy correction leading to a divergent renormalization
of the Fermi velocity $v_F$.}
\label{rainbow}
\end{figure}
The solution of (\ref{selfcons}) can be obtained in the form
\begin{equation}
\Gamma ({\bf 0},0;{\bf k},\omega_k) =
\gamma_0
\left(1 + \sum_{n=1}^{\infty} \lambda_0^n
\frac{r_n }{|{\bf k}|^{n\epsilon}} \right)
\end{equation}
with $\lambda_0 = e_0^2/4\pi \kappa v_F $.
Each term in the sum can be obtained from the previous one by expanding
$1/\widetilde{v}_F({\bf p})$ in Eq. (\ref{selfcons}) in powers of $e_0^2$
and noticing that
\begin{eqnarray}
\lefteqn{ \int \frac{d^D p}{(2\pi )^D} \frac{1}{|{\bf p}|^{(m-1)\epsilon} }
\frac{1}{|{\bf p}|} \frac{1}{|{\bf k}-{\bf p}|} }
\nonumber \\
& &
= \frac{(4\pi )^{\epsilon /2}}{4 \pi^{3/2}}
\frac{\Gamma \left(\tfrac{m\epsilon }{2} \right) \Gamma \left(\tfrac{1-m\epsilon}{2} \right)
\Gamma \left(\tfrac{1-\epsilon}{2} \right) }
{ \Gamma \left(\tfrac{1+(m-1)\epsilon}{2} \right) \Gamma \left(1-\tfrac{m + 1}{2}\epsilon \right) }
\frac{1}{|{\bf k}|^{m\epsilon}}
\end{eqnarray}
At each given perturbative level, the vertex displays then a number of poles
at $\epsilon = 0$. The crucial point is that these divergences can be
reabsorbed by passing to physical quantities defined by the multiplicative
renormalization
\begin{eqnarray}
v_F & = & Z_v(v_F)_{\rm ren} \\
\overline{\psi} \psi & = & Z_{\psi^2} (\overline{\psi} \psi )_{\rm ren}
\end{eqnarray}
We observe that the scale of the single Dirac field $\psi $ does not need to be
renormalized in this approach, as self-energy corrections of the
form shown in Fig. \ref{rainbow} with a statically screened interaction do not
modify the frequency dependence of the Dirac propagator.
The renormalized vertex
\begin{equation}
\Gamma_{\rm ren} = Z_{\psi^2} \Gamma
\end{equation}
can be actually made finite at $\epsilon = 0$ with a
suitable choice of momentum-independent factors $Z_v$ and $Z_{\psi^2}$.
$Z_v$ must be chosen to cancel the $1/\epsilon $ pole arising from
$\Gamma (\epsilon /2 )$ in (\ref{vd}), and it
has therefore the simple structure
\begin{equation}
Z_v = 1 + \frac{b_1}{\epsilon }
\end{equation}
with $b_1 = - e^2/16\pi \kappa (v_F)_{\rm ren}$. On the other hand, we have
the general structure
\begin{equation}
Z_{\psi^2} = 1 + \sum_{i=1}^{\infty} \frac{c_i }{\epsilon^i}
\label{poles}
\end{equation}
The position of the different poles must be chosen to enforce the
finiteness of $\Gamma_{\rm ren} = Z_{\psi^2} \Gamma $ in the limit
$\epsilon \rightarrow 0$. The computation of the
first orders of the expansion gives for instance the result
\begin{eqnarray}
c_1 (\lambda ) & = & - \frac{1}{2} \lambda - \tfrac{1}{8} \log(2) \: \lambda^2
- \tfrac{1}{1152} \left( \pi ^2 + 120 \log ^2(2) \right) \lambda^3 \nonumber \\
& & - \tfrac{10 \pi ^2 \log (2)+688 \log ^3(2)+15 \zeta (3)}{6144} \lambda^4
\nonumber \\
& & - \tfrac{13 \pi ^4+2064 \pi ^2 \log ^2(2)+144 \left(716 \log ^4(2)+37 \log (2) \zeta (3)\right)}{737280} \: \lambda^5
\nonumber \\
& & + \ldots \nonumber \\
c_2 (\lambda ) & = & \tfrac{1}{16} \: \lambda^2 +
\tfrac{1}{24} \log(2) \: \lambda^3 \nonumber \\
& & + \tfrac{1}{18432} \left( 5 \pi ^2 + 744 \log ^2(2) \right) \lambda^4 \nonumber \\
& & + \tfrac{110 \pi ^2 \log (2)+8592 \log ^3(2)+135 \zeta (3)}{184320} \: \lambda^5
+ \ldots \nonumber \\
c_3 (\lambda ) & = & - \tfrac{1}{768} \log (2) \: \lambda^4 - \tfrac{1}{184320} \left( \pi ^2+360 \log ^2(2) \right) \lambda^5
\nonumber \\
& & + \ldots \nonumber \\
c_4 (\lambda ) & = & - \tfrac{1}{7680} \log (2) \: \lambda^5 + \ldots
\label{coeff}
\end{eqnarray}
where the series are written in terms of the renormalized coupling
$\lambda$ defined by
\begin{equation}
\lambda \equiv \rho^{-\epsilon} Z_v \lambda_0 = \frac{e^2}{4\pi \kappa (v_F)_{\rm ren}}
\end{equation}
The physical observable in which we are interested is the anomalous
dimension $\gamma_{\psi^2}$. The change in the dimension of $\Gamma_{\rm ren}$
comes from the dependence of $Z_{\psi^2}$ on the only dimensionful scale
$\rho $ in the renormalized theory. Therefore we have\cite{amit}
\begin{equation}
\gamma_{\psi^2} = \frac{\rho }{Z_{\psi^2}} \frac{\partial Z_{\psi^2} }{\partial \rho }
\end{equation}
The original bare theory at $D \neq 2$ does not know about the arbitrary scale
$\rho $, and the
independence of $\lambda_0 = \rho^{\epsilon} \lambda /Z_v $ on that
variable leads to
\begin{equation}
\rho \frac{\partial \lambda }{\partial \rho } =
- \epsilon \lambda - \lambda b_1 (\lambda )
\label{rge}
\end{equation}
At $\epsilon = 0$, this is the well-known expression of the scale
dependence of the effective interaction strength, arising from the
renormalization of the Fermi velocity\cite{np2}.
The anomalous dimension becomes finally\cite{ram}
\begin{equation}
\gamma_{\psi^2} = \frac{\rho }{Z_{\psi^2}}
\frac{\partial \lambda }{\partial \rho }
\frac{\partial Z_{\psi^2} }{\partial \lambda }
= - \lambda \frac{d c_1}{d \lambda }
\label{dreg}
\end{equation}
In the derivation of (\ref{dreg}), it is implicitly assumed that poles
in the $\epsilon $ parameter cannot appear at the right-hand-side of the
equation. For this to be true, the set of equations
\begin{equation}
\frac{d c_{i+1}}{d \lambda } = c_i \frac{d c_1}{d \lambda } - b_1 \frac{d c_i}{d \lambda }
\end{equation}
must be satisfied identically\cite{ram}.
Quite remarkably, we have verified that this is the case, up to the order
$\lambda^{17} $ we have been able to compute numerically the coefficients in
(\ref{poles}). This is the proof of
the renormalizability of the theory, which guarantees that physical quantities
like $\gamma_{\psi^2}$ remain finite in the limit $\epsilon \rightarrow 0$.
From the practical point of view, the important result is the evidence that
the perturbative expansion of $c_1 (\lambda )$
\begin{equation}
c_1 (\lambda ) = \sum_{n} c_1^{(n)} \lambda^n
\end{equation}
approaches a geometric series
in the $\lambda $ variable. The plot of the coefficients $c_1^{(n)}$
computed numerically up to order $\lambda^{17} $ is shown in Fig.
\ref{c1n}. It can be checked that the coefficients
grow exponentially with the order $n$, in such a way that
\begin{equation}
- c_1 (\lambda ) \geq \sum_{n=1}^{\infty} d^n \lambda^n \; + \; {\rm regular} \;\;\; {\rm terms}
\end{equation}
\begin{figure}
\begin{center}
\mbox{\epsfxsize 7.0cm \epsfbox{c1n.eps}}
\end{center}
\caption{Plot of the absolute value of the coefficients $c_1^{(n)}$ in the
expansion of $c_1 (\lambda )$ as a power series of the coupling $\lambda $.}
\label{c1n}
\end{figure}
An estimate of $d$ can be obtained from the coefficients available in the perturbative
series of $c_1 (\lambda )$. The ratio between consecutive $c_1^{(n)}$ increases with
the order $n$, converging towards a limit value.
The best fit of the asymptotic behavior allows us to estimate a radius of convergence
\begin{equation}
\lambda_c \approx 0.56
\end{equation}
This has to be compared with the value
found in the approach neglecting self-energy corrections, which leads to
$\lambda_c \approx 0.45$ \cite{me}, in close agreement with the result of
Ref. \onlinecite{gama}. The critical coupling in the variable
$\lambda $ can be used to draw the boundary for dynamical
gap generation in the space of $N$ and $\alpha = e^2/4\pi (v_F)_{\rm ren} $,
recalling that
\begin{equation}
\lambda = \frac{\alpha}{1 + \frac{N \pi}{8} \alpha }
\end{equation}
The corresponding phase diagram is represented in Fig. \ref{two}.
For $N = 4$, we get in particular the critical coupling $\alpha_c \approx 4.9$,
significantly above the critical value that would be obtained from the radius
of convergence without self-energy corrections ($\alpha_c \approx 1.53$).
\begin{figure}
\begin{center}
\mbox{\epsfxsize 6.0cm \epsfbox{phdbeta4.eps}}
\end{center}
\caption{Phase diagram showing the boundary between the metallic phase and
the phase with dynamical gap generation ($\langle \rho_m \rangle \neq 0$)
in the ladder approximation.
The thin dashed (solid) line represents the phase boundary obtained with static
(dynamic) RPA screening of the interaction potential and no electron self-energy
corrections. The thick dashed (solid) line represents the boundary after including
the effect of the electron self-energy corrections on top of the static (dynamic)
RPA screening of the interaction in the ladder series.
}
\label{two}
\end{figure}
\section{Electron self-energy effects in dynamically screened ladder approximation}
In the framework of the ladder approximation, one can also study the effect
of electron self-energy corrections under dynamical screening of the Coulomb
interaction potential. We can improve
the static RPA by considering the full effect of the
frequency-dependent polarization, which for Dirac fermions
takes the form\cite{np2}
\begin{equation}
\chi ({\bf p}, \omega_p) =
\frac{N}{16} \frac{{\bf p}^2}{\sqrt{v_F^2 {\bf p}^2 - \omega_p^2}}
\label{dyn}
\end{equation}
This expression can be introduced in Eq. (\ref{self}) to look again for
self-consistent solutions for the vertex $\Gamma ({\bf 0},0;{\bf k},\omega_k)$.
Given that in this case we must resort to numerical methods for the
resolution of the integral equation, we can go beyond the self-energy effects
considered before by taking into account the electron self-energy corrections
in the RPA improved with the polarization (\ref{dyn}). In this approach,
the behavior of the dressed Fermi velocity $\widetilde{v}_F({\bf p})$
is given as a function of $g = N e^2/32 \widetilde{v}_F$ by the nonlinear
equation\cite{prbr}
\begin{equation}
\frac{\partial \log \widetilde{v}_F}{\partial \log |{\bf p}|} =
- \frac{8}{N\pi^2}
\left( 1 + \frac{\arccos g}{g \sqrt{1-g^2}} - \frac{\pi}{2} \frac{1}{g}
\right)
\label{vflow}
\end{equation}
We have then used the solution of Eq. (\ref{vflow}) to replace $v_F$ in
Eq. (\ref{self}) by the momentum dependent $\widetilde{v}_F$, which
represents a significant improvement in the sum of self-energy corrections
in the ladder series for the vertex.
In this procedure, we find again that there is a critical coupling in the
variable $\alpha = e^2/4\pi v_F$ at which $\Gamma ({\bf 0},0;{\bf k},\omega_k)$
blows up, marking the boundary between two different regimes where
Eq. (\ref{self}) has respectively positive and negative solutions.
In practice, we have solved the integral equation by defining the
vertex in a discrete set of points in frequency and momentum space. One can
take as independent variables in $\Gamma ({\bf 0},0;{\bf k},\omega_k)$ the modulus of
${\bf k}$ and positive frequencies $\omega_k $. We have adopted accordingly
a grid of dimension $l \times l$ covering those variables, with
$l$ running up to a value of 200 for which it is still viable to invert a
matrix of dimension $l^2$.
As a check of our approach, we have compared the results of the numerical
diagonalization of (\ref{self}), still keeping the undressed Fermi velocity
$v_F$, with the values of the critical coupling in Ref. \cite{ggg}, where the
resolution of the gap equation has been accomplished with the frequency-dependent
polarization. We have relied on the scale invariance of our model to find the
trend of $\alpha_c$ at large $l$, as the critical coupling must obey a
finite-size scaling law
\begin{equation}
\alpha_c (l) = \alpha_c (\infty ) + \frac{c}{l^\nu }
\end{equation}
At $N = 4$, we get $\alpha_c (200) \approx 1.08$ and the estimate
$\alpha_c (\infty ) \approx 0.99$, which turns out to be close to the critical
coupling $\alpha_c \approx 0.92$ found in Ref. \cite{ggg}, providing a nice
check of our computational approach in the case of unrenormalized $v_F$.
The electron self-energy corrections lead anyhow to a substantial increase
in the values of the critical coupling $\alpha_c (l)$. This is a decreasing
function of $l$, as the limit $l \rightarrow \infty$ corresponds to the
large-volume limit of the system. Then, as a result of diagonalizing
Eq. (\ref{self}) with the effective $\widetilde{v}_F$, we have chosen to
represent in Fig. \ref{two} the upper bound $\alpha_c (200)$ to the critical
coupling as a function of $N$.
We observe that, for $N \lesssim 3$, the values of the
critical coupling are larger than those obtained with static screening of
the interaction potential, while the situation is inverted for $N \gtrsim 3$.
In coincidence with the findings of Ref. \cite{ggg}, there is indeed no upper
limit on $N$ for the onset of chiral symmetry breaking in this approach.
At $N = 4$, we get
\begin{equation}
\alpha_c (200) \approx 1.75
\end{equation}
which is substantially
smaller than the value found in Sec. III with the static RPA screening in
the ladder series. These results support the idea that, in the particular
case of graphene ($N = 4$), the nominal coupling of the system in vacuum
($\alpha \approx 2.2$) should be above the critical coupling for dynamical
gap generation. This is reinforced by the fact that other
effects neglected thus far have to do with the electron
self-energy corrections to the own polarization $\chi$. These should lead
to a reduction of the screening and the consequent enhancement of the
effective interaction strength. The values that we find for $\alpha_c$
should be taken in this regard as an upper bound for the critical coupling,
at least when compared with the
result of including the effect of Fermi velocity renormalization in the bare
polarization.
\section{Conclusion}
In this paper we have considered the impact that electron self-energy
corrections may have on the chiral symmetry breaking in the interacting theory
of Dirac fermions. Our starting point has been the ladder approximation for
the electron-hole vertex appearing in the response function for dynamical
gap generation, which we have supplemented by including systematically the
self-energy corrections to electron and hole states in the
ladder series.
In this framework, we have been able to account for the effect of the
Fermi velocity renormalization on the critical coupling for dynamical gap
generation. In this respect, it has been already suggested that
the growth of the Fermi velocity at low energies can have a deep impact to
prevent the chiral symmetry breaking\cite{sabio,see}. The scale dependence
of the Fermi velocity, expressed nonperturbatively in Eq. (\ref{vflow}), has
been already observed in experiments with graphene at very low doping
levels\cite{paco}. Our results show actually that the effect of
renormalization of the Fermi velocity induces a significant reduction in the
strength of the dynamical symmetry breaking in graphene, leading to a critical
coupling $\alpha_c \approx 4.9$ in the case of static RPA screening of the
interaction potential in the ladder series, and to a value
$\alpha_c \approx 1.75$ in the more sensible instance of dynamical screening
of the interaction.
One of the main conclusions of this work is that the screening effects must be
treated accurately in order to make a reliable estimate of the
critical coupling for dynamical gap generation in graphene. This is so
as such an instability depends strongly on the singular behavior of
the Coulomb interaction in the undoped system. In this regard, the situation
is quite different to the case of bilayer graphene, where several low-energy
instabilities have been also predicted\cite{vaf,zhang,nan,lemo}. These can
be traced back to the divergence of objects like the electron-hole
polarization, which results from the particular form of the bandstructure
and does not require a long-range interaction. In monolayer graphene, the
instability towards chiral symmetry breaking appears to be quite sensitive
to many-body corrections to the Coulomb interaction, which makes more delicate
the precise computation of the critical interaction strength.
The other important conclusion is that the value $\alpha_c \approx 1.75$
resulting from the self-energy corrections still remains below the nominal
coupling for graphene in vacuum. This means that an isolated free-standing
layer of the material should be in the phase with dynamical gap generation,
which is apparently at odds with present experimental measurements in suspended
graphene samples. A key observation is however that, if chiral symmetry
breaking is to proceed in graphene according to the present estimates, it is
going to lead to a gap at least three orders of magnitude below the
high-energy scale of the Dirac theory, as found in the resolution of the gap
equation\cite{ggg}. This suggests then that the dynamical gap generation
cannot be discarded in isolated free-standing graphene, though its
experimental signature may be only found in suitable samples, for which the
Fermi level can be tuned within an energy range below the meV scale about
the charge neutrality point.
{\em Acknowledgments.---}
We thank F. Guinea and V. P. Gusynin for very useful discussions.
The financial support from MICINN (Spain) through grant
FIS2008-00124/FIS is also acknowledged.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,358
|
New Boston è un villaggio degli Stati Uniti d'America della contea di Scioto nello Stato dell'Ohio. La popolazione era di 2,272 persone al censimento del 2010. A parte il confine meridionale sul fiume Ohio, New Boston è interamente circondata dalla città di Portsmouth.
Geografia fisica
New Boston è situata a (38.753049, -82.935819).
Secondo lo United States Census Bureau, ha un'area totale di 1,14 miglia quadrate (2,95 km²).
Storia
New Boston fu pianificata il 17 febbraio 1891 da James Skelton, A.T. Holcomb, e M. Stanton. Il villaggio deve il suo nome alla città di Boston, la capitale del Massachusetts, da dove provenivano la maggior parte dei primi coloni.
Società
Evoluzione demografica
Secondo il censimento del 2010, c'erano 2,272 persone.
Etnie e minoranze straniere
Secondo il censimento del 2010, la composizione etnica del villaggio era formata dal 96,0% di bianchi, l'1,1% di afroamericani, lo 0,3% di nativi americani, lo 0,3% di asiatici, lo 0,3% di altre razze, e il 2,1% di due o più etnie. Ispanici o latinos di qualunque razza erano lo 0,9% della popolazione.
Note
Altri progetti
Collegamenti esterni
Villaggi dell'Ohio
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,552
|
Zaraasuchus ("hedgehog crocodile") was a gobiosuchid crocodyliform described in 2004 by Diego Pol and Mark Norell. It was found in the Red Beds of Zos Canyon, in the Gobi Desert of Mongolia, thus making it Late Cretaceous in age.
The type species is Z. shepardi, honouring Dr. Richard Shepard.
Material
The holotype of Z. shepardi is IGM 100/1321, consisting of the posterior region of the skull and lower jaws with articulation with cervical vertebrae, forelimb elements and osteoderms.
Systematics
Pol and Norell (2004) found Zaraasuchus shepardi to be the sister taxon of Gobiosuchus kielanae, united by 14 synapomorphies, primarily from the skull, forming the family Gobiosuchidae.
Sources
Pol, D. & Norell, M. A., (2004). "A new gobiosuchid crocodyliform taxon from the Cretaceous of Mongolia". American Museum Novitates 3458: 1-31.
Late Cretaceous crocodylomorphs of Asia
Terrestrial crocodylomorphs
Late Cretaceous reptiles of Asia
Fossils of Mongolia
Prehistoric pseudosuchian genera
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 487
|
<?php
namespace app\assets;
use yii\web\AssetBundle;
/**
* @author Qiang Xue <qiang.xue@gmail.com>
* @since 2.0
*/
class AppAsset extends AssetBundle
{
public $basePath = '@webroot';
public $baseUrl = '@web';
public $css = [
'css/site.css?v=1',
];
public $js = [
];
public $jsOptions = ['position' => \yii\web\View::POS_HEAD];
public $depends = [
'yii\web\YiiAsset',
'yii\bootstrap\BootstrapAsset',
'yii\bootstrap\BootstrapPluginAsset',
];
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,055
|
(function ( $ ) {
'use strict';
$(document).ready(function() {
$('.variant-table-toggle i.glyphicon').on('click', function(e) {
$(this).toggleClass('glyphicon-chevron-down glyphicon-chevron-up');
$(this).parents('table').find('tr[data-toggle-id="'+$(this).parent().data('id')+'"]').toggle();
});
$('.datepicker').datepicker({format: 'dd.mm.yyyy'});
});
})( jQuery );
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,983
|
DK Gardens welcomes you to our one stop Design Revolution.
A world of fantasy and reality; of transformations, successful conclusions, and spectacular results culminating from innovative, inventive and creative designs.
Why not sit back and let us take you and your company forward into the next millennium with concepts and creativity unparalleled in Landscaping and Interiors.
Our specialisation is in the Creation, Development and Realisation of all types of projects and commissions, which makes us a worldwide leader.
Our mission is to provide all our customers with the complete solution to their dreams, achieved by the genuine spirit of innovation that we have within the DK Gardens.
Today like yesterday, DK Gardens is proud to bring you the joy that comes from a well designed and properly maintained garden.
We can offer total turn key solutions to all our clients for Landscaping and Interiors; and are happy to undertake commissions for the leisure/entertainment industry, municipalities, hotels/ retail, corporate companies and private individuals.
Furthermore we can produce competitive tenders and quotations for your own designs and other landscape schemes.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 637
|
Q: Can't get HDMI audio to work with NVIDIA gpu on Ubuntu 20.04 LTS I've been having an issue where I can't use the speakers on my monitor which is connected to my laptop through HDMI. The display works perfectly and even the sounds does on windows but on Ubuntu it just shows my inbuilt speakers/headphones as the only available source.
Also, a bit out of context but the max volume of my speakers on ubuntu is way less than on Windows. So if there's a solution to that it would be great.
Laptop Specs:
OS: Ubuntu 20.04.2 LTS ×86_64
Host: HP Pavilion Power Laptop 15-cb
Kernel: 5.8.0-55-generic
Uptime: 2 hours, 36 mins
Packages: 2042 (dpkg), 14 (snap)
Shell: bash 5.0.17
Resolution: 1366x768, 1920x1080
DE: GNOME Mutter
Theme: Adwaita
Theme: Yaru-dark [GTK2/3]
Icons: Yaru [GTK2/3]
Terminal: gnome-terminal
GPU: Intel i5-7300HQ (4) @ 3.500GHz
GPU: Intel HD Graphics 630
GPU: NVIDIA GeForce GTX 1050 Mobile
Меmогу: 3464MiB 7846MiB
inxi
Audio:
Device-1: Intel CM238 HD Audio driver: snd_hda_intel
Device-2: NVIDIA GP107GL High Definition Audio driver: snd_hda_intel
Sound Server: ALSA v: k5.8.0-55-generic
aplay -l
**** List of PLAYBACK Hardware Devices ****
card 0: PCH [HDA Intel PCH], device 0: ALC295 Analog [ALC295 Analog]
Subdevices: 0/1
Subdevice #0: subdevice #0
card 0: PCH [HDA Intel PCH], device 3: HDMI 0 [HDMI 0]
Subdevices: 1/1
Subdevice #0: subdevice #0
card 0: PCH [HDA Intel PCH], device 7: HDMI 1 [HDMI 1]
Subdevices: 1/1
Subdevice #0: subdevice #0
card 0: PCH [HDA Intel PCH], device 8: HDMI 2 [HDMI 2]
Subdevices: 1/1
Subdevice #0: subdevice #0
card 0: PCH [HDA Intel PCH], device 9: HDMI 3 [HDMI 3]
Subdevices: 1/1
Subdevice #0: subdevice #0
card 0: PCH [HDA Intel PCH], device 10: HDMI 4 [HDMI 4]
Subdevices: 1/1
Subdevice #0: subdevice #0
Pavucontrol
A: After digging through the internet I was able to find that the error was due to ubuntu automatically applying power control on the sound card and so I was able to get it working by editing the nvidia rules file in ubuntu bios rules directory. In that file you just need to switch the "auto" setting in the last couple lines to "on".
https://forums.developer.nvidia.com/t/no-option-for-audio-over-displayport-hdmi/175889/3
A: I had the same issue but my version and hardware are different. Perhaps it can help...
The command nvidia-smi was returning an error.
I've solved that by applying the procedure describe on the section : "Ubuntu Install Nvidia driver using the CLI method # 2 " of https://www.cyberciti.biz/faq/ubuntu-linux-install-nvidia-driver-latest-proprietary-driver/
And now the sound is ok on ubuntu.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,148
|
\section{Introduction}
Asymmetric dark matter has emerged as a competitive paradigm to thermally produced dark matter. Instead of having a mechanism where the population of Weakly Interacting Massive Particles (WIMPs) is controlled by annihilations, it is possible to have an initial asymmetry between particles and antiparticles. The existence of a conserved quantum number associated with the WIMPs can protect them from decay or co-annihilation. If the particle-antiparticle annihilation is sufficiently strong, antiparticles are eliminated by equal number of particles, and due to the asymmetry, the dark matter (DM) consists of the remaining particles. If annihilations are not strong enough, a mixed case with a substantial number of antiparticles present today is also possible~\cite{Belyaev:2010kp}. Obviously in this mixed scenario DM consists of both particles and antiparticles. The fact that the asymmetry in the DM sector resembles the baryonic asymmetry, makes asymmetric DM easily incorporated in extensions of the Standard Model. It also means that the asymmetries in the baryonic and dark sector might be related. Although the idea of asymmetric DM is not new
\cite{Nussinov:1985xr,Barr:1990ca,Gudnason:2006ug,Gudnason:2006yj}, it has recently attracted a lot of interest~\cite{ Foadi:2008qv,Khlopov:2008ty,Khlopov:2008ty,Dietrich:2006cm,Sannino:2009za,Ryttov:2008xe,Sannino:2008nv,Kaplan:2009ag,Frandsen:2009mi,MarchRussell:2011fi,Frandsen:2011kt,Gao:2011ka,Arina:2011cu,Buckley:2011ye,Lewis:2011zb,Davoudiasl:2011fj,Graesser:2011wi,Bell:2011tn}.
The current status of experimental direct detection of DM is quite intriguing. A signal with annual modulation possibly attributed to DM has been solidly established in DAMA~\cite{Bernabei:2008yi}, and more recently in CoGeNT~\cite{Aalseth:2011wp}. In addition CRESST-II~\cite{Angloher:2011uu} has recently released results compatible with the existence of a light WIMP too. However, experiments such as CDMS ~\cite{Ahmed:2010wy}, and Xenon10/100~\cite{Angle:2011th,Aprile:2011hi} find null evidence for DM, imposing thus severe constraints on WIMP-nucleons cross sections. The fact that some experiments detect DM and some other do not is not the only experimental discrepancy one faces. Upon assuming spin-independent interactions between WIMPs and nuclei, it is clear that DAMA and CoGeNT are at odds, if WIMPs couple to protons and neutrons with equal strength. However if the relative couplings of WIMPs to neutrons and protons satisfy $f_n/f_p\simeq -0.71$ ~\cite{Chang:2010yk,Feng:2011vu}, an agreement of DAMA and CoGeNT is possible, and it indicates a DM-proton cross section $\sigma_p \sim 2 \times 10^{-38} \rm{ cm}^2$ and a DM mass $m_{DM} \approx 8$ GeV (see Fig.~\ref{fig:experiments}). In a recent paper~\cite{DelNobile:2011je} we demonstrated that the isospin violation
needed to produce $f_n/f_p\simeq -0.71$ can be easily accommodated using Standard Model mediators, via interference of two different channels in elastic WIMP-nuclei collisions (see also \cite{Cline:2011zr,Cline:2011uu}). Interfering DM can thus naturally explain the above phenomenological ratio. We showed that if Interfering DM is made of composite asymmetric WIMPs (with electroweak compositeness scale), a simple interference in the WIMP-nucleus collision between a photon exchange (via a dipole type interaction) and a Higgs exchange can produce the required isospin violation. Interestingly this candidate has been proven to arise in strongly interacting models using first principle lattice computations~\cite{Lewis:2011zb}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=.49\textwidt
]{experiments1.pdf
\includegraphics[width=.49\textwidt
]{experiments2.pdf
\caption{\em\label{fig:experiments}Favored regions and exclusion contours in the $(m_{DM}, \sigma_p)$ plane for the standard case $f_n / f_p = 1$ (left panel) and the case $f_n / f_p = - 0.71$ (right panel). The green contour is the $3 \sigma$ favored region by DAMA \cite{Savage:2010tg} assuming no channeling \cite{Bozorgnia:2010xy} and that the signal arises entirely from Na scattering; the blue region is the $90 \%$ CL favored region by CoGeNT; the cyan contour is the $2 \sigma$ favored region by CRESST-II~\cite{Angloher:2011uu}; the dashed line is the exclusion plot by CDMS II Soudan \cite{Ahmed:2010wy}; and the black and blue lines are respectively the exclusion plots from the Xenon10 \cite{Angle:2011th} and Xenon100 \cite{Aprile:2011hi} experiments. The CoGeNT and DAMA overlapping region passing the constraints is shown in red.}
\end{center}
\end{figure}
In this paper we extend the idea of interfering DM by presenting three general interference patterns for fermionic DM that can accommodate the experimental findings. More specifically we show under what conditions interference between $Z$ and a $Z'$; $Z'$ and Higgs, and two Higgs doublets can provide the appropriate isospin violation. In the last case we show that the interference between the two Higgs scalars can also be compatible with Electroweak Baryogenesis~\cite{Cohen:1991iu,Nelson:1991ab,Joyce:1994zn}.
We should also mention that observations of neutron stars put severe constraints on the spin-dependent cross section of fermionic asymmetric WIMPs~\cite{Kouvaris:2010jy}, and bosonic asymmetric WIMPs~\cite{Kouvaris:2011fi}. In our study here we avoid these constraints because our fermionic asymmetric WIMP candidates do not have significant spin-dependent cross section.
\section{$Z$ interfering with $Z'$}
First we will consider a scenario where a fermionic DM particle $\psi$ couples to the $Z$-boson and to a spin-1 state $Z'$.
The $Z$-DM and $Z$-nucleon interaction Lagrangian, including only renormalizable terms, reads
\begin{align}\label{Zint}
\mathscr{L}_Z = \,&\frac{g}{2 \cos \theta_W} Z_{\mu} \bar\psi (v_{\psi}
- a_{\psi} \gamma^5) \gamma^\mu\psi \, + \\ \nonumber
&\frac{g}{2 \cos \theta_W} Z_{\mu} \left[ \bar p\, \gamma^\mu (v_p - a_p \gamma^5) p \, + \bar n\, \gamma^\mu (v_n - a_n \gamma^5) n \right] \ ,
\end{align}
where the $Z$-DM couplings $v_{\psi}$ (vector) and $a_{\psi}$ (axial-vector) are normalized to the usual weak coupling strength. $p$ and $n$ refer respectively to protons and neutrons and the $Z$-nucleon vector and axial-vector couplings are
\begin{align}
v_p = \frac{1}{2} - 2\sin^2 \theta_W \,,\quad v_n = -\frac{1}{2} \,,\quad a_p = 1.36 \,,\quad a_n = -1.18 \,, \nonumber
\end{align}
where we have used the numerical values from \cite{Ellis:2008hf} to estimate $a_p$ and $a_n$.
However, we are not concerned with the axial-vector couplings, since their contribution to the cross section is suppressed with respect to the one given by the vector couplings.
Similarly the $Z'$-DM and $Z'$-nucleon interaction Lagrangian can be written as
\begin{align}\label{Zpint}
\mathscr{L}_{Z'} = \, &\frac{g}{2 \cos \theta_W} Z'_{\mu} \bar\psi (v'_{\psi} - a'_{\psi} \gamma^5) \gamma^\mu\psi \,+ \\ \nonumber
& \frac{g}{2 \cos \theta_W} Z'_{\mu} \left[ \bar p\, \gamma^\mu (v'_p - a'_p \gamma^5) \,p \, + \bar n\, \gamma^\mu (v'_n - a'_n \gamma^5)\ n \right] \ .
\end{align}
As for the $Z$, also in this case the axial-vector couplings contribution to the cross section is negligible.
Possible constraints from colliders on $Z'$ can be safely avoided assuming a leptophobic $Z'$.
As long as the $Z'$ couplings to leptons are small enough, no bounds can be set at present. Using Eqs.~\eqref{Zint} and \eqref{Zpint}, we can write the spin-independent cross section in the zero momentum transfer limit as
\begin{align}\label{sigmaZZp}
\sigma = \frac{ 2 G_F^2\mu_{A}^2}{\pi} \left| f_p \, \mathsf Z + f_n (\mathsf A-\mathsf Z) \right|^2,
\end{align}
where $G_F$ is the Fermi constant, $\mu_A$ is the DM-nucleus reduced mass, and the dimensionless couplings to protons and neutrons are defined as
\begin{eqnarray}
f_p = v_{\psi} v_p + v'_{\psi}v'_p \frac{m^2_{Z}}{m^2_{Z'}} \,,\qquad
f_n = v_{\psi} v_n +v'_{\psi}v'_n \frac{m^2_{Z}}{m^2_{Z'}} \,.
\end{eqnarray}
We already know that in order to alleviate the discrepancy between the different direct detection experimental results we need to
have $m_{DM} \sim 8$ GeV, $f_n / f_p=-0.71$, and the DM-proton cross section $\sigma_p \sim 2 \times 10^{-38}$ cm$^2$.
Thus by fixing these three values we find the following relations for the unknown parameters of the model
\begin{align}
\left| f_p \right| &= \left|v_{\psi} v_p + v'_{\psi}v'_p \frac{m^2_{Z}}{m^2_{Z'}}\right| = \sqrt{\frac{ \sigma_p \pi}{2 G_F^2\mu_{p}^2}} \ ,\\
f_n &= v_{\psi} v_n + v'_{\psi}v'_n \frac{m^2_{Z}}{m^2_{Z'} }= - 0.71f_p \ .
\end{align}
Substituting the numbers and dividing by the known values of the parameters $v_p = 0.055$ and $v_n = -0.5$ we get the two constraints
\begin{align}\label{constZZp}
v_{\psi} + v'_{\psi} \frac{v'_p}{v_p}\frac{m^2_{Z}}{m^2_{Z'}} &= v_{\psi} + 15 \,v'_{\psi} v'_p \left(\frac{m_{Z'}}{100\, \rm{GeV}} \right)^{-2} = \pm 17\,, \\
v_{\psi} + v'_{\psi} \frac{v'_n}{v_n} \frac{m^2_{Z}}{m^2_{Z'}} &= v_{\psi} -1.7 \,v'_{\psi}v'_n \left(\frac{m_{Z'}}{100\, \rm{GeV}} \right)^{-2} = \pm 1.3 \ . \label{fnZZp}
\end{align}
The $Z$-DM coupling $v_{\psi}$ can be constrained using the measurements of the $Z$ decay width into invisible channels. The LEP experiment set strict limits on the number of SM neutrinos, i.e.~$N_{\nu}=2.984\pm0.008$ \cite{Nakamura:2010zzi}. The error in the measurement can be used to constrain non-SM contributions to the $Z$ decay width. Using the uncertainty in the LEP result $\delta_{\rm LEP}=0.008$, this yields
\begin{eqnarray}
v_{\psi}^2 \beta (3-\beta^2) + 2 a_{\psi}^2 \beta^3 < \delta_{\rm LEP} \, ,
\end{eqnarray}
where $\beta = \sqrt{1-4 m^2_{DM}/m^2_Z}$ is the velocity factor.
Assuming a DM mass of $\sim 8$ GeV, with no axial-vector coupling ($a_{\psi} = 0$), the vector coupling $v_{\psi}$ can assume its maximal allowed value $|v_{\psi}| < 0.063$, while for $a_{\psi} = v_{\psi}$ this constraint gives $|v_{\psi}| < 0.045$. Taking into account this strong bound in Eqs.~\eqref{constZZp}, and~\eqref{fnZZp}, it is evident that the bulk contribution is due to the $Z'$ alone. Therefore interference is not relevant for this kind of DM interaction with the SM particles. Similar studies have been performed recently in \cite{Frandsen:2011cg}.
\section{$Z'$ interfering with Higgs}
Before proceeding, let us comment, that the DM signals seen in DAMA/CoGeNT and the null results of the other direct DM experiments cannot be explained simultaneously through a $Z$ and Higgs interference. The reason for this is that a light ($\sim 8$ GeV) Dirac DM particle, with a coupling to the $Z$-boson such that $\sigma_p \sim 2 \times 10^{-38}$ cm$^2$ and $f_n/f_p = -0.71$, is ruled out by the aforementioned LEP constraints. However, as we will demonstrate below, interference between $Z'$ and the Higgs is a viable possibility.
The relevant Higgs ($h$) interaction Lagrangian is
\begin{align}
\mathscr{L}_h = m_{DM}\bar{\psi} \psi -\, h \bar{\psi} (d_h+a_h \gamma^5)\psi
-\frac{m_p}{v_{EW}} f \, h \,( \bar p p +\bar n n ) \ ,
\end{align}
where $d_{h}$ and $a_{h}$ are the dimensionless scalar and pseudo-scalar Higgs-DM couplings respectively. The Higgs field $h$ is here the physical field, i.e.~the oscillation around the vacuum expectation value $v_{EW}$. We have specified a mass term for the DM to point out that it doesn't need to be generated by the vacuum expectation value of the Higgs field
Combining the scalar interaction from this Lagrangian, with the vector one for the $Z'$ as it appears in \eqref{Zpint}, we get the DM-nucleus spin-independent cross section as in Eq.~\eqref{sigmaZZp},
where now the dimensionless couplings to protons and neutrons are defined as
\begin{eqnarray}
f_p = v'_{\psi}v'_p \frac{m^2_{Z}}{m^2_{Z'}} - d_h \frac{f m_p v_{EW}}{ m^2_h} \,,\qquad
f_n = v'_{\psi}v'_n\frac{m^2_{Z}}{m^2_{Z'}} - d_h \frac{f m_p v_{EW}}{ m^2_h} \,.
\end{eqnarray}
As for the $Z$-$Z'$ case, the pseudo-scalar and pseudo-vector couplings of the DM with the Higgs and the $Z'$ respectively lead to negligible contributions to the cross section compared to the scalar and vector ones investigated here.
The constraints are
\begin{align}
|f_p| &= \left| v'_{\psi}v'_p \frac{m^2_{Z}}{m^2_{Z'}} - d_h \frac{f m_p v_{EW}}{ m^2_h} \right| = 0.92 \\
f_n &= v'_{\psi}v'_n\frac{m^2_{Z}}{m^2_{Z'}} - d_h \frac{f m_p v_{EW}}{ m^2_h} = - 0.71f_p = \pm0.65 \,.
\end{align}
These can be rewritten as
\begin{align}
& v'_{\psi}v'_p \left(\frac{m_{Z'}}{100\, \rm{GeV}} \right)^{-2} - 8.3 \times 10^{-3} d_h \left(\frac{m_h}{100\, \rm{GeV}} \right)^{-2} = \pm1.1 \\
& v'_{\psi}v'_n \left(\frac{m_{Z'}}{100\, \rm{GeV}} \right)^{-2} - 8.3 \times 10^{-3} d_h \left(\frac{m_h}{100\, \rm{GeV}} \right)^{-2} = \mp0.78 \ ,
\end{align}
where we have used $f = 0.3$~ \cite{Shifman:1978zn}.
\\
If all the couplings are of order unity and $m_{Z'}, m_h \sim 100$ GeV, the Higgs contribution to the interference is negligible, and the $Z'$ has to directly account for the isospin violation needed to get the desired value of $f_n / f_p$. A substantially lighter Higgs, around $50$~GeV with a coupling $d_h$ in the range $5 - 10$, can lead to a phenomenologically viable interference. Note that such a light Higgs-like state is not immediately ruled out by collider experiments since this state has new decay modes, e.g.~to two DM particles which are not accounted for in the SM (see e.g.~\cite{Kim:2008pp}).
\section{Interference within the Two Higgs doublet model} \label{sec2higgs}
We will now consider a two Higgs doublet model where one of the Higgs fields couples to up-type quarks and the other to down-type quarks. This kind of scenario albeit more general, is similar to the Minimal Supersymmetric Standard Model Higgs sector. We consider Yukawa-type interactions between the two Higgs fields and the fermionic DM $\psi$. We write also effective interactions with the SM proton $p$ and neutron $n$. The interaction Lagrangian is
\begin{align}\label{twohiggs}
\mathscr{L}_{2H} = \, \lambda^{DM}_1 h_1 \bar\psi \psi \,+ \lambda^{p}_1 h_1 \bar p p \, +\, \lambda^{n}_1 h_1 \bar n n
+ \,\lambda^{DM}_2 h_2 \bar\psi \psi \, +
\lambda^{p}_2 h_2 \bar p p \, + \, \lambda^{n}_2 h_2 \bar n n \ .
\end{align}
$ h_1$ and $ h_2$ are here the physical scalars, i.e.~the mass eigenstates after diagonalization, where the original Higgs fields coupled one to the up-type quarks and the other to the down-type. The nucleon couplings are then
\begin{subequations}\label{couplings1}
\begin{align}
\lambda^{p}_1 &= \frac{ \cos\theta}{v_1} \sum_{q_u} \langle p |m_{q_u} \bar{q}_u q_u |p \rangle - \frac{\sin\theta}{v_2} \sum_{q_d} \langle p |m_{q_d} \bar{q}_d q_d |p \rangle \ ,
\\
\lambda^{n}_1 &= \frac{\cos\theta}{v_1} \sum_{q_u} \langle n |m_{q_u} \bar{q}_u q_u |n \rangle - \frac{\sin\theta}{v_2} \sum_{q_d} \langle n |m_{q_d} \bar{q}_d q_d |n \rangle \ ,
\\
\lambda^{p}_2 &=\frac{ \sin\theta }{v_1} \sum_{q_u} \langle p |m_{q_u} \bar{q}_u q_u |p \rangle + \frac{\cos\theta}{v_2} \sum_{q_d} \langle p |m_{q_d} \bar{q}_d q_d |p \rangle \ ,
\\
\lambda^{n}_2 &= \frac{\sin\theta}{v_1} \sum_{q_u} \langle n |m_{q_u} \bar{q}_u q_u |n \rangle + \frac{ \cos\theta}{v_2} \sum_{q_d} \langle n |m_{q_d} \bar{q}_d q_d |n \rangle \ ,
\end{align}
\end{subequations}
where the sums over up-type ($q_{u}$) and down-type ($q_{d}$) quarks account for the scalar quark currents within the nucleons. $v_1$ and $v_2$ are the vacuum expectation values of the two Higgs fields, which obey the relation $v^2_{EW}/2=v^2_1+v^2_2$ ($v_{EW}\simeq 246$~GeV). $\theta$ is the mixing angle needed to diagonalize the Higgs system, and here is a free parameter. We also assume that the DM particle mass is not generated by the vacuum expectation values of the Higgs fields.
The matrix elements $ \langle p,n |m_{q_{u,d}} \bar{q}_{u,d} q_{u,d} |p,n \rangle$ in~\eqref{couplings1} are obtained in chiral perturbation theory, when dealing with light quarks, using the measurements of the pion-nucleon sigma term~\cite{Cheng:1988im}, and in the case of heavy quarks, from the mass of the nucleon via trace anomaly~\cite{Shifman:1978zn, Vainshtein:1980ea}.
The experimental uncertainties, especially in the pion-nucleon sigma term, give rise to differences in the values of these matrix elements. As long as $\lambda^p_i$ and $\lambda^n_i$ are not identical, isospin violation can always be guaranteed.
To evaluate the matrix elements we follow Ref.~\cite{Ellis:2008hf} which makes use of the results found in~\cite{Shifman:1978zn, Vainshtein:1980ea, Cheng:1988im}.
\begin{subequations}\label{couplings}
\begin{align}
\sum_{q_u} \langle p |m_{q_u} \bar{q}_u q_u |p \rangle \approx 105 \, {\rm MeV} \ ,
&\qquad\qquad
\sum_{q_d} \langle p |m_{q_d} \bar{q}_d q_d |p \rangle \approx 417 \, {\rm MeV} \ ,
\\
\sum_{q_u} \langle n |m_{q_u} \bar{q}_u q_u |n \rangle \approx 100 \, {\rm MeV} \ ,
&\qquad\qquad
\sum_{q_d} \langle n |m_{q_d} \bar{q}_d q_d |n \rangle \approx 426 \, {\rm MeV} \ .
\end{align}
\end{subequations}
The spin-independent DM-nucleus cross section can now be calculated using the interaction terms from Eq.~\eqref{twohiggs}
\begin{align}\label{sigma2H}
\sigma = \frac{\mu_{A}^2}{\pi} \left| f_p \, \mathsf Z + f_n (\mathsf A-\mathsf Z) \right|^2,
\end{align}
where the couplings to protons and neutrons are\footnote{The normalization for $f_p$ and $f_n$ here is different than the one used in Eq.~\eqref{sigmaZZp}.}
\begin{eqnarray} \label{fnfp}
f_p = \frac{\lambda^{DM}_1 \lambda^{p}_1}{m^2_{ h_1 }} +\frac{\lambda^{DM}_2 \lambda^{p}_2}{m^2_{ h_2 }} \,,\qquad
f_n = \frac{\lambda^{DM}_1 \lambda^{n}_1}{m^2_{ h_1 }} + \frac{\lambda^{DM}_2 \lambda^{n}_2}{m^2_{ h_2 }} \,.
\end{eqnarray}
Eqs.~\eqref{sigma2H} and~\eqref{fnfp} can be used to study the effects of the interference in a generic two Higgs doublet model.
Substituting the couplings from~\eqref{couplings1}, \eqref{couplings} into~\eqref{fnfp}, and imposing the fitting values for $m_{DM} $, $f_n / f_p$ and $\sigma_p$, we get
the following constraint equations for the unknown parameters:
\begin{align}\label{2Hconst1}
&\lambda^{DM}_1 \left(\frac{v_1}{v_{EW}} \right)^{-1}
\left( \cos\theta - 4.0\, \sin\theta \frac{v_1}{v_2}\right)
\left(\frac{m_{h_1}}{100\, \rm{GeV}} \right)^{-2} + \nonumber \\
& 4.0 \,\lambda^{DM}_2 \left(\frac{v_2}{v_{EW}} \right)^{-1}
\left( \cos\theta + 0.25\, \sin\theta \frac{v_2}{v_1}\right)
\left(\frac{m_{h_2}}{100\, \rm{GeV}} \right)^{-2}
= \pm 3.5 \times 10^2 \ ,
\\
\rule{0cm}{1cm}
\label{2Hconst2}
&\lambda^{DM}_1 \left(\frac{v_1}{v_{EW}} \right)^{-1}
\left( \cos\theta - 4.3 \sin\theta \frac{v_1}{v_2}\right)
\left(\frac{m_{h_1}}{100\, \rm{GeV}} \right)^{-2} + \nonumber \\
& 4.3 \,\lambda^{DM}_2 \left(\frac{v_2}{v_{EW}} \right)^{-1}
\left( \cos\theta + 0.23 \sin\theta \frac{v_2}{v_1}\right)
\left(\frac{m_{h_2}}{100\, \rm{GeV}} \right)^{-2}
= \mp 2.6 \times 10^2 \ .
\end{align}
For natural values of $v_1$ and $v_2 $, i.e.~of the order of $v_{EW}$, and for $ m_{h_1}$ and $m_{h_2}$ of the order of 100-1000 GeV, the DM couplings to the Higgs fields need to be of $ \mathscr{O}(10^3)$ to fit the data. This large couplings are of course unnatural as such. Thus the original DM-Higgs interactions and related couplings, introduced in Eq.~\eqref{twohiggs}, need to be considered as a simple effective description.
We will introduce now a model that will accommodate such large values for the effective couplings, and link it also to Electroweak Baryogenesis \cite{Cohen:1991iu, Nelson:1991ab, Turok:1990in, McLerran:1990zh, Dine:1990fj}. We start by recalling the three Sakharov conditions needed for successful production of a baryon asymmetry
for the model considered here~\cite{Cohen:1991iu}:
the baryon number violation originates from SM sphalerons; out-of-equilibrium conditions are generated by bubble nucleation in a strong first order electroweak (EW) phase transition; a new CP violating phase which we take it to be generated within the two Higgs doublet model.
Thus we will now investigate whether both baryogenesis and the explanation of the direct detection data via interference can be achieved simultaneously using a two Higgs doublet model.
We start by introducing a DM-Higgs effective Lagrangian, which avoids the large DM-Higgs couplings discussed in the end of the last section.
The Lagrangian reads
\begin{align}\label{DMlagr}
\mathscr{L}_{\rm DM} & = (m_{DM} - \frac{ \lambda^{DM}_1 v^2_1 }{ \Lambda } - \frac{ \lambda^{DM}_2 v^2_2 }{ \Lambda } ) \bar{\psi} \psi + \frac{ \lambda^{DM}_1 }{ \Lambda } \phi^{\dagger}_1 \phi_1\bar{\psi} \psi + \frac{ \lambda^{DM}_2 }{ \Lambda } \phi^{\dagger}_2 \phi_2 \bar{\psi} \psi \, ,
\end{align}
where the cut-off $\Lambda$ is assumed to be of the order of 1-10 GeV. We will show that for such a range of $\Lambda$ the required couplings to DM will turn to be between 1-10. Here we indicated the Higgs doublets before EW symmetry breaking by $\phi$. In principle it is not hard to construct a UV complete theory for such a generic effective Lagrangian.
We give one such a model in the Appendix \ref{JV}.
As an underlying two Higgs model we will use the one studied in~\cite{Cohen:1991iu,Nelson:1991ab,Joyce:1994zn}. After implementing the DM part,
the full two Higgs model Lagrangian is
\begin{align}
\mathscr{L}_H = \sum^{2}_{i=1} | D_{\mu} \phi_i|^2 - V(\phi_1, \phi_2) + \mathscr{L}_{\rm DM} + \mathscr{L}_{\rm fermions} + \mathscr{L}_{\rm Yuk} + \mathscr{L}_{\rm gauge} \ ,
\end{align}
where the two Higgs doublets scalar potential is
\begin{align}\label{2Higgspotential}
V(\phi_1, \phi_2) &= \lambda_1(\phi^{\dagger}_1 \phi_1 - v^2_1)^2 + \lambda_2(\phi^{\dagger}_2 \phi_2 - v^2_2)^2 +
\lambda_3[(\phi^{\dagger}_1 \phi_1 - v^2_1)+(\phi^{\dagger}_2 \phi_2 - v^2_2)]^2 \nonumber \\
&+\lambda_4[(\phi^{\dagger}_1 \phi_1)(\phi^{\dagger}_2 \phi_2 ) - (\phi^{\dagger}_1 \phi_2)(\phi^{\dagger}_2 \phi_1)] +
\lambda_5[{\rm{Re}}(\phi^{\dagger}_1 \phi_2) - v_1 v_2 \cos{\xi} ]^2 \nonumber \\
&+\lambda_6[{\rm{Im}}(\phi^{\dagger}_1 \phi_2) - v_1 v_2 \sin{\xi} ]^2 \,,
\end{align}
$\xi$ being the relative CP violating phase between the two Higgs fields, which cannot be entirely rotated away by field redefinitions \cite{Branco:1979pv}. $\mathscr{L}_{\rm fermions}$ and $\mathscr{L}_{\rm gauge}$ account for the fermion covariant derivative terms and the gauge field kinetic terms respectively. Yukawa interactions in $ \mathscr{L}_{\rm Yuk}$ couple the up-type quarks to $ \phi_1$ and the down-type quarks to $ \phi_2$, resulting in identical Higgs-proton and Higgs-neutron couplings as presented in Eqs.~\eqref{twohiggs} and~\eqref{couplings1}. The only relevant SM Yukawa coupling for baryogenesis is the top quark one \cite{Cohen:1991iu}. Due to the specific choice of the interaction between DM and the Higgs fields, baryogenesis is not affected by the presence of the DM sector.
Fitting now the DM direct detection data using the model \eqref{DMlagr}, we get the following constraints
\begin{align}\label{2Hconst1}
&\tilde\lambda^{DM}_1 \left(\frac{\Lambda}{v_{EW}} \right)^{-1}
\left( \cos\theta - 4.0\, \sin\theta \frac{v_1}{v_2}\right)
\left(\frac{m_{h_1}}{100\, \rm{GeV}} \right)^{-2} + \nonumber \\
& 4.0 \, \tilde\lambda^{DM}_2 \left(\frac{\Lambda}{v_{EW}} \right)^{-1}
\left( \cos\theta + 0.25\, \sin\theta \frac{v_2}{v_1}\right)
\left(\frac{m_{h_2}}{100\, \rm{GeV}} \right)^{-2}
= \pm 1.8 \times 10^2 \ ,
\\
\rule{0cm}{1cm}
\label{2Hconst2}
&\tilde\lambda^{DM}_1 \left(\frac{\Lambda}{v_{EW}} \right)^{-1}
\left( \cos\theta - 4.3 \sin\theta \frac{v_1}{v_2}\right)
\left(\frac{m_{h_1}}{100\, \rm{GeV}} \right)^{-2} + \nonumber \\
& 4.3 \, \tilde\lambda^{DM}_2 \left(\frac{\Lambda}{v_{EW}} \right)^{-1}
\left( \cos\theta + 0.23 \sin\theta \frac{v_2}{v_1}\right)
\left(\frac{m_{h_2}}{100\, \rm{GeV}} \right)^{-2}
= \mp 1.3 \times 10^2 \ .
\end{align}
$\tilde\lambda^{DM}_1$ and $\tilde\lambda^{DM}_2$ are here defined so that $2 \tilde\lambda^{DM}_1 v_1 / \Lambda$ and $2 \tilde\lambda^{DM}_2 v_2 / \Lambda$ are the actual couplings of the DM to the physical Higgs fields $h_1$ and $h_2$, respectively,
\begin{equation}
\tilde\lambda^{DM}_1 = \lambda^{DM}_1 (\cos\theta - \sin\theta \frac{\lambda^{DM}_2}{\lambda^{DM}_1} \frac{v_2}{v_1}) \ ,
\qquad\qquad
\tilde\lambda^{DM}_2 = \lambda^{DM}_2 (\cos\theta + \sin\theta \frac{\lambda^{DM}_1}{\lambda^{DM}_2} \frac{v_1}{v_2}) \ .
\end{equation}
Given the potential in Eq.~\eqref{2Higgspotential}, the mixing angle $\theta$ is now given by
\begin{equation}
\tan 2 \theta = \frac{2 v_1 v_2 ( 4 \lambda_3 + g)}{4 v^2_2 (\lambda_2 + \lambda_3) -4 v^2_1 (\lambda_1 + \lambda_3)+ g ( v^2_1- v^2_2)} \ ,
\end{equation}
where $g = \lambda_5 \cos^2 \xi+ \lambda_6 \sin^2 \xi$.
If we assume that $\Lambda \sim 1$ GeV, and that both Higgs fields are light, $\mathscr{O}(100)$ GeV, we find that the DM couplings to the Higgs doublets $\lambda^{DM}_1$ and $\lambda^{DM}_2 $ (or at least one of them) need to be of the order $ \mathscr{O}(10)$ to be able to fulfill the above constraint equations. The size of these couplings is now substantially reduced with respect to the previous model.
Summarizing, since the CP violating phase can be rotated away in the light quark sector \cite{Branco:1979pv} there are no direct implications for the direct detection experiments. A welcome feature is that by including the interaction of the Higgs fields to DM using higher order operators, the energy scale $\Lambda$ can be traded for a more natural value of the dimensionless couplings when fitting their values to direct detection data.
\section{Conclusions}
We have investigated several quantum mechanical interfering patterns for DM scattering off nuclei that can explain the DAMA and CoGeNT results. In particular we considered the case in which DM interacts via $Z$ and $Z'$, $Z'$ and Higgs, and two Higgs fields with or without CP violation. We found that in the first case due to the constraints from the invisible decay width of the $Z$, the dominant contribution should come from the $Z'$ exchange. In the second case, $Z'$ dominates again upon assuming natural values of the Higgs coupling and masses. In the last case we found that an explanation of the DAMA/CoGeNT results based on interference of two Higgs fields besides being phenomenologically viable, is also consistent with the Electroweak Baryogenesis scenarios based on two Higgs doublet models.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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The UN in Uzbekistan
UN Entities in Uzbekistan
Our Team in Uzbekistan
A foundation of justice for building Uzbekistan's businesses
Photo: © Danielle Villasana / UNDP
This year, Uzbekistan's position on the World Bank's Doing Business Index rose seven spots to 69th place, just ahead of Luxembourg, Qatar and South Africa.
What's in a number?
This year, Uzbekistan's position on the World Bank's Doing Business Index rose seven spots to 69th place, just ahead of Luxembourg, Qatar and South Africa. The index also cites Uzbekistan in the top countries showing most progress in improving their national business climate and having jumped from 41st to 22nd in 'enforcing contracts'.
All the while, major international media outlets including the Financial Post, the Economist and CNN have drawn attention to Uzbekistan's new business opportunities, particularly in the tourism sector.
Indeed, when the country is actively working to invite international investment, these rankings greatly strengthen the national image in the business sector. For everyday business owners, Uzbekistan's advancement in the Doing Business Index helps reassure them that conflicts they might face can be legally resolved.
Uzbekistan is presenting itself as a strong and secure place to do business, both for national entrepreneurs and international investors. How it has arrived at this place is a story about where business and justice meet.
Development, business and justice
Business development has been a national priority since Uzbekistan's earliest days of independence, supported by both government and civil society. In fact, by the end of last year, three out of four employed people work in small- and medium- sized enterprises, and those businesses made up almost 60 percent of our economy. However, before 2016, businesspeople were often hamstrung by excessive bureaucracy and unchecked corruption.
Caption: The Imron Textile Group is one of the many medium-sized enterprises that have launched in the last 5 years
Photo: © Karen Cirillo
Setting up a business used to mean jumping through endless hoops, and sometimes facing threats of harassment and extortion. Organized crime and a lack of rule of law stifled new domestic enterprises, while international investors couldn't be confident that agreements made would be respected.
The election of President Shavkat Mirziyoyev and the subsequent national reform process and development strategy) focused on ensuring government and courts worked primarily to meet the needs of the people, with laws to strengthen media autonomy and prioritize citizen interests.
Alongside these reforms was a push to better protect the rights of domestic and international entrepreneurs through the justice system. A notable example was the creation of the nation's first business ombudsman.
If you look at our work at UNDP in Uzbekistan, you will see that business has also been a focus of national development initiatives for more than a decade. We've supported entrepreneurs and initiatives to grow local, regional and national businesses, bring them into international marketplaces and attract global investment.
Equally important, however, has been work to build the trust and confidence of both domestic businesses and potential international investors that transactions and agreements made in Uzbekistan will be protected.
Simplifying national business processes and building investment confidence have been major outcomes of the Rule of Law Partnership project we do together with the Supreme Court of the Republic of Uzbekistan and USAID.
While Uzbekistan's improved business environment has attracted large international brands, we must ensure small businesses also benefit from reform. So far, we have supported that by:
- Creating greater digital transparency and access to justice. The new Supreme Court website, established in partnership with our project, contains easily-accessible information on court activities and judgements, database on legislation, guidance on applicaitons, and a centralized means for paying court fees. When 'time is money', these features let businesses solve conflicts quickly and inexpensively. In addition, the internationally-recognised E-SUD system allows for the online submission of appeals to court, while improving the efficiency and quality of case processing.
- Establishing mediation as a private, faster means for resolving judicial conflicts. UNDP's collaboration with the Supreme Court of Uzbekistan helped establish the "On Mediation" law, which allows citizens to receive justice without going to court, saving unnecessary costs, and preserving business relationships.
- Improving court systems. Simplifying court procedures for hearing claims and introducing pre-trial hearings for settling disputes, measures to reduce stay of proceeding and better court ICT all contribute to higher confidence that the system is working for business owners.
These improvements have now made it now much easier for both citizens and businesses alike to access Uzbekistan's courts, and ensure that their legitimate rights and interests are being adequately protected.
Uzbekistan's meteoric performance in the Doing Business Index is cause for celebration, but also show how much potential there is for Uzbekistan's private sector if the right legal frameworks are in place.
Further improvements are expected for 2020, making setting up and running businesses in the nation easier and improving the speed and ease of accessing justice. Legal entities will be able to submit statements of claim and annexes to defendants and courts electronically through taxpayer accounts. This will not only speed case processing, but also improve Uzbekistan's ranking on the 'Case Studies Performance' indicator of the Doing Business Index.
Last but not least, efforts are being made to make business law more forgiving to first-time offenders. From June 2020, people who have committed their first violation of customs regulations, but who have then voluntarily made payments for goods and performed customs clearance, will be exempt from criminal liability.
As Uzbekistan continues to reform and enhance its business law, its reputation as a destination for business will continue growing, to the benefit of its citizens.
Feruza Nomozova
feruza.nomozova@undp.org +99778 1203460
UN entities involved in this initiative
Goals we are supporting through this initiative
United Nations Uzbekistan
Welcome to the United Nations country team website of Uzbekistan
Resident Coordinator Office
4, Taras Shevchenko Street
Tashkent 100029
Fax:+998 78 120 34 85
Work time: 9.00 - 18.00 (Monday - Friday)
Find out what the UN in Uzbekistan is doing towards the achievement of the Sustainable Development Goals.
Learn about employment opportunities across the UN in Uzbekistan.
© Copyright 2022 United Nations in Uzbekistan
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 3,154
|
{"url":"http:\/\/mathhelpforum.com\/geometry\/122675-geometry-figure-sum-distances-three-points.html","text":"# Math Help - Geometry figure, sum of distances to three points\n\n1. ## Geometry figure, sum of distances to three points\n\nLooking to definition of an ellipse we can see that it is the set of points which the sum of distances to two fixed points is a constant.\nSo i was thinking about a geometry figure that is the set of points which the sum of distances to three fixed points is a constant. Does this geometry figure exist?\nTo check i tried to find a equation to this figure.\n\nTaking the points A,B,C creating an equilateral triangle. With A and C in the X axis and B in the Y axis. The side of the triangle is (2*a).\nSo A(-a,0), C(a,0) and B(0,a*sqrt(3) )\nI used these definitions to simplify the work.\nBeing P(x,y) an generic point, and 2a the constant distance arbitrary choosed.\n\n(Distance PA) + (Distance PB) + (Distance PC) = 2a\n\nsqrt((x+a)\u00b2+(y)\u00b2) + sqrt((x)\u00b2+(y-a*sqrt(3))\u00b2) + sqrt((x-a)\u00b2+y\u00b2) = 2a\n\nDeveloping the polinomial I got this tiny thing:\nx\u00b2+y\u00b2-a\u00b2 +2a(2x+2*sqrt(x\u00b2-2ax+a\u00b2+y\u00b2)-sqrt(3)*y) +2*sqrt(x^4+2x\u00b2y\u00b2+2ax\u00b3+4a\u00b2x\u00b2-2a*sqrt(3)*yx\u00b2+2axy\u00b2-4a\u00b2*sqrt(3)*xy\n+6a\u00b3x-2a\u00b3*sqrt(3)*y+4a\u00b2y\u00b2+3a^4+y^4-2a*sqrt(3)*y^3)) = 0\n\nPutting this equation in a grapher like aGrapher, i got nothing. So i would like to know if this geometry figure i idealize exists or not, and if it exists how does it look like and why my equation doesnt work.\n\nThank you so much, forgive my mistakes in grammar im not native.\n\n2. Hello, Boyrog!\n\nI tried it myself and got into an awful mess . . .\n\nSo i was thinking about a geometry figure that is the set of points\nwhich the sum of distances to three fixed points is a constant.\nDoes this geometry figure exist?\nI didn't bother with an equilateral triangle.\nI used simpler coordinates.\nCode:\n C|\no (0,a)\n* | *\n* | *\n* | *\nB * | * A\n- - o - - - - + - - - - o - -\n(-a,0) | (a,0)\n\nWe want point $P(x,y)$ so that: . $\\overline{PA} + \\overline{PB} + \\overline{PC} \\:=\\:k$\n\nSo: . $\\sqrt{(x-a)^2+y^2} + \\sqrt{(x+a)^2+y^2} + \\sqrt{x^2 + (y-a)^2} \\:=\\:k$\n\nI see no practical way to eliminate the radicals.\n\nMaybe you want to square the equation repeatedly.\n. . I'll wait in the car . . .\n.\n\n3. Thank you for helping Soroban,\nyour definition has helped me. I checked the equation and corrected some mistakes I had done. The correct devoloping is:\n\nPA + PB + PC = K\nCode:\n((x-a)^2+y^2)^0.5+((x+a)^2+y^2)^0.5+(x^2+(y-a)^2)^0.5= K\n\nIsolating the roots and elevating to square till get no square roots:\nCode:\n[(k\u00b2-x\u00b2-a\u00b2-y\u00b2-2ay)\u00b2-4((x\u00b2-a\u00b2)\u00b2+y\u00b2(2x\u00b2+2a\u00b2)+y^4)-4k\u00b2(x\u00b2+(y-a)\u00b2)]\u00b2 - 64k\u00b2[(x\u00b2-a\u00b2)\u00b2+y\u00b2(2x\u00b2+2a\u00b2)+y^4)]*[x\u00b2+(y-a)\u00b2]\nPutting\n\nthis equation in the grapher, we get this kind of flower: (very nice isn't? )","date":"2015-10-04 15:58:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 3, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.811858594417572, \"perplexity\": 1092.9480468265422}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-40\/segments\/1443736675218.1\/warc\/CC-MAIN-20151001215755-00166-ip-10-137-6-227.ec2.internal.warc.gz\"}"}
| null | null |
Q: Backbonejs + MarionetteJD - Converting Backbone list view to MarionetteJS Collection View My BackboneJS "list item view" is defined as follows:
class TagListView extends Backbone.View
el:"#tags"
render: =>
@collection = new TagCollection
@collection.fetch_data
order_by : "name"
, =>
@on_success()
on_success: =>
view_arr = []
@collection.each (tag_model) =>
tag = new TagView {model: tag_model}
view_arr.push tag.render().el
@$el.empty().append view_arr
How would I change this to MarionetteJS collection view? I tried the following, and it doesn't seem to work?
class TagListView extends Marionette.CollectionView
el:"#tags"
itemView:TagView
onBeforeRender: =>
@collection = new TagCollection
@collection.fetch_data
order_by : "name"
, =>
@render()
I'm simply removing the on_sccess() method, since my understanding is that the CollectionView will do the rendering of its items by its render method?
A: You are correct.
The collection view binds to the "add", "remove" and "reset" events of the collection that is specified. Once any of these events happend, collection view will automatically update the view, either part or entier.
Another thing you may want to know is in version 1.0.0, when a collection do fetch, it will not longer trigger the reset event by default. To get the old behavior, pass {reset: true}.
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 2,020
|
{"url":"https:\/\/chemistry.stackexchange.com\/questions\/65141\/most-stable-resonating-structure-of-1-nitro-4-nitrosobenzene","text":"Most stable resonating structure of 1-nitro-4-nitrosobenzene\n\nThe most stable resonating structure of 1-nitro-4-nitrosobenzene is:\n\nIn my view since in each case we have 4 charges and except in case 1 all have complete octets, so 1 is neglected. However, I am facing difficulty in eliminating other two in order to get the desired answer. I noticed that in case 3 it is in conjugation (alternating single and double bonds).\n\nThe answer is not that difficult. You should have a certain number of rules to follow in determining the most stable resonance (not \"resonating\") structure.\n\nIt probably starts with check all for octets (which you correctly did). 1 has ten electrons on the nitroso nitrogen, so that rules it out.\n\nIt probably then goes on to ask you to check for the least number of formal charges (which, judging by your description, you probably did).\n\nThe next step should probably say something about the negative charges being on the most electronegative elements. Electronegative atoms like O, F... (and N to a slightly lesser extent) are quite happy to take on a negative charge whereas carbon will not quite be so pleased.\n\nHere, resonance forms 2 and 4 have the negative charge on carbon, whereas structure 3 has it on oxygen. 3 is therefore the most stable resonance structure.\n\nIncidentally, all of the resonance forms 2 through 4 have complete conjugation throughout the molecule so that's not a factor. The $\\mathrm{sp^2}$-hybridised anionic carbon can take part in conjugation equally well. Conjugation does not require alternating single and double bonds - it only requires an extended, continuous, $\\pi$ system. Generally this means that all atoms along the chain must have unhybridised p orbitals parallel to each other.\n\nSide note: \"Most stable\" resonance structure is a slightly sloppy description. None of these resonance structures actually exist and the true structure is a linear combination of all of these resonance structures. It should more properly be called the \"greatest resonance contributor\" or \"greatest contributor to the resonance hybrid\".","date":"2019-12-15 06:30:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5606901049613953, \"perplexity\": 1103.1724755701973}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575541301598.62\/warc\/CC-MAIN-20191215042926-20191215070926-00523.warc.gz\"}"}
| null | null |
KOFA (1320 AM) is a public radio station airing a wide variety of music programs, along with a few news and talk programs. Licensed to Yuma, Arizona, United States, the station serves the Yuma area. The station is currently owned by Arizona Western College and features programming from National Public Radio and Public Radio International.
History
1320 AM began life as the original KBLU, signing on September 6, 1959. When Eller Telecasting and Combined Communications merged in 1969, the newly formed group had to divest one of KBLU or KYUM at 560, choosing to keep the latter and donate the former to Arizona Western College. On January 1, 1970, the donation took effect, and 1320 AM signed off with the callsign changing to KAWC; 560 AM changed to KBLU.
Arizona Western College immediately relocated the transmitter to its campus and instituted a public radio format, which signed on for the first time on July 11, 1970.
In 2009, the K-Jazz Radio Network with its transmitters in northwestern and northern Arizona entered into agreement with Colorado River Public Media, by which those transmitters simulcast 1320 AM on weekdays.
The station changed its callsign to the current KOFA on July 13, 2017.
References
External links
OFA (AM)
OFA (AM)
NPR member stations
Radio stations established in 1959
1959 establishments in Arizona
Arizona Western College
Full service radio stations in the United States
|
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| 6,388
|
October 30, 2014 By Joseph Zentis
Marstellar, Les & Mary
Les and Mary Marstellar dressed in their colonial costumes at the opening of Fredonia's time capsule in 2001
Ordinary lives?!
Les and Mary Marstellar describe their lives as "ordinary," but that word doesn't feel right when applied to these two Fredonia residents.
So, as I often do when a word seems inappropriate, I picked up my trusty old Webster's New Collegiate Dictionary. It begins by saying that "ordinary" means "of a kind to be expected in the normal order of events." But it continues: "of common quality, rank, or ability; deficient in quality; poor, inferior; lacking in refinement."
No one who knows Les and Mary would ever think of using those words to describe their lives. Not even Les or Mary themselves, and much less those who know them best: their two children, five grandchildren, two great grandchildren, friends, and neighbors.
Les Marstellar in fourth grade at Orr one-room school in Fairview Township, 1925.
On the surface, their lives seem normal for the times in which they live. Both grew up in small communities: Les (born in 1916) in Fredonia, and Mary (1931) in Mercer, and both went to one-room schools.
Les has lived his entire life in Fredonia, except for four years of military service. After graduating from Fredonia Delaware Vocational School, he worked on his father's farm. Les's father, J. R. Marstellar, owned Maple Grove Dairy, which had a retail milk route. After a couple years, Les got a job as salesman at Willis Chevrolet in Mercer, and worked there until he enlisted in the Army Air Corps on September 8, 1941.
Les trained as an airplane mechanic at Sheppard Field in Texas. He was so good that he was retained there as an instructor, then reassigned as an instructor to Amarillo Air Base. In May, 1945, Staff Sergeant Marstellar was transferred the 509 Composite Air Group on Tinian Island in the South Pacific. That was the Atomic Bomb Group. Les's commander was Col. Paul Tibbets, who piloted the Enola Gay when it dropped the first atomic bomb on Japan.
Army Air Corps enlistee Les Marstellar in 1941
"It was all top secret," Les said. "We didn't know anything about it until the bomb was dropped."
Les returned to Fredonia in November, 1945, and worked for a while on the farm before becoming Willis Chevrolet's service manager.
"About three years later, I got the urge to farm again," he said, "so I went back to that. I did some plumbing and heating work on the side, then went into business for myself."
Les also drove school bus for a while, and that turned out to be one of the most important jobs he ever had. One of his passengers was Mary Kelso, whom he married in 1949.
Their son Ed was born in 1950, and daughter Lynn in 1952.
Les and Mary on their wedding day in 1949
"I decided that since I was married, I should have a job that had some benefits," Les said. "So I took the exam for rural carrier here in Fredonia. I didn't get the job, but I served as sub rural carrier for five years. Then I took the exam for postmaster. I was postmaster in Fredonia for 14 years, until I retired in January, 1979."
All that time, Mary served her family as housewife and mother. She was an excellent seamstress who had her own little business sewing for other people"
"We were a very ordinary family," said her daughter, Lynn Rodemoyer. "My father was an extremely hard worker. My mother was the housekeeper; she didn't work outside the home. We didn't have a lot of money, but I wouldn't say we ever really missed out on anything."
Lynn uses glowing terms to describe her parents.
"I've seen my parents do so many things on their own, make something out of nothing, because that's all we had. If we needed something, they made it out of scraps or whatever they had. My mother always made my clothes. It's one of the things I treasure now, even though as a kid I hated that. I can remember one of the greatest thrills of my life was as a junior high school student going to Treasure Island and buying a store-bought dress."
Because they didn't have a lot of money, the Marstellar family traveled very little.
"We went to Florida once, to Niagara Falls once, and to the Grand Canyon once," Mary said.
Lynn's brother Ed remembers one special thing about those trips. Last week, he stopped in to visit his parents as they were preparing to leave for a trip to Missouri. He handed his mother and his father one dollar each.
"You always gave Lynn and me a dollar each for souvenirs on our family trips," he said. "So I'm returning the favor – even though a dollar doesn't buy as much now as it did back then."
"Our Sunday afternoon entertainment," Lynn said, "was to go for a ride in the car and get an ice cream cone later. At the time we probably didn't think it was much of an entertainment, but now we can both see how much that was worth."
Les has been a member of the Fredonia Methodist Church for 85 years, and Mary for as long as they have been married – 61 years. Both have taught Sunday School, and Les has served as church treasurer for 25 years and as a trustee.
"Church has always been the uppermost thing in our lives," Mary said.
Both belonged to the Coolspring Grange until it disbanded, and have served the Fredonia community in countless ways. Les has served on the Fredonia Municipal Authority and the Board of Directors of the Millbank Cemetery. Mary has served as clerk of the Election Board in Fredonia for 30 years.
Les has been a member of the Fredonia Lions Club since 1966. He has an amazing record of perfect attendance for 40 years, has served as president and on various committees, and has received the Melvin Jones Award, the highest honor in the Lions Club. But the Lions Club is more than a service organization for Les.
"I tell everyone the Lions' Den is his second home," Mary said. "One of my nephews said I should be glad it isn't his first home."
Occasionally that seems like a tough call. When it comes to making a choice between Lions Club service and family activities, Les sometimes needs Mary's persuasion to make the right decision.
"Last year," Lynn said, "my oldest son flew into Youngstown for the air show and was part of the static line display. It was also Fredonia's Old Home Week. My father says to my mother, ' I can't go to the air show on Saturday because it's the chicken barbecue, and I've got to be here.' My mother said to him. 'You are 92 years old and you are going to the air show. The Lion's club can get along without you one day.' So he went to the air show."
Les has participated in many activities with the Lions Club, such as the food concession at the Fredonia rodeos and the chicken barbecues. He still is in charge of property and maintenance at the Lions Den, sets up the tables for meals, and keeps the place clean.
Les and a friend building the gazebo on the Fredonia town square in 1988
He has also worked on special Lions Club projects, including the construction of the gazebo on the Fredonia town square in 1988, which was destroyed recently when workers were trimming trees.
"I was sort of the project engineer," Les said. "Three of us from the Lions Club built it. The other two are deceased, so if the town decides to rebuilt it, I'm the one to do it."
Considering that he will be celebrating his 94th birthday on October 17, you might think he isn't serious. "My father is an amazing man," Lynn said. "One time, my husband's brother was roofing their house, so he went out to give them a hand. And my husband's brother has said to this day, 'I will never forget him coming up that ladder, carrying shingles, saying this is the way you do it.'"
Les continues to serve his neighbors any way he can.
Les & Mary in 1993
"My father takes the snow blower and clears the neighborhood sidewalks," Lynn said. "There's a couple down the street. The gentleman must be 25 or 30 years younger than my father, but my father clears the sidewalks for them."
So, how do you decide whether the lives of Mary and Les Marstellar are ordinary or extraordinary? One way would be to observe their relationships with other people, especially the group of friends from the Grange.
"When we were first married, we were the young people of the Grange," Mary said. "The Grange disbanded, but 16 of us met every New Year's Eve for more than 50 years to play cards. There was no smoking and no booze. And there were no deaths and no divorces through all those years. Whenever we sent flowers to someone for an anniversary or something, we always signed the card 'The Young People of the Grange.' My kids have been hysterical over that."
Les and Mary Marstellar with their grandson, Mark Rodemoyer, who is a pilot in the 509th Bomb Wing, a direct descendent of the 509th Composite Group in which Les served during World War II
But the best measure of the quality and success of people's lives is the attitude and gratitude of their own children. Lynn sums it up this way: "If I can give to my children half of what my parents have given to me, I will feel that I have been a good parent."
That, like Ed's dollars for souvenirs, was an extraordinary tribute.
In return, Les and Mary are very proud of their children and grandchildren.
Ed taught math at Fredonia's Lakeview High School for 33 years. His son Bret is serving on a U. S. Navy nuclear submarine, and son Jay is a school teacher. Ed's daughter Kristin Morrison is an attorney in Ohio.
Lynn is a very active homemaker. Her son Mark Rodemoyer is an air force pilot, and son Kurt is a supervisor at Estes Trucking.
Ordinary? Not in this family.
Fredonia Delaware Vocational School grad Les Marstellar
Lynn and Ed Marstellar being towed by their father in the mid-1950s
Les Marstellar with his siblings (l to r): Jean Kuhn, Bob, JoAnn Ulery, Les
Filed Under: Army Air Corps, Biographies, World War II
D'Amore, Dr. Amanto and Florence
Masury, Ohio The Saga of an All-American Family Once upon a time there were three D'Amore brothers – Amanto Primo, Adanto Secundo, and Arcangelo Terzo. From their names, you might think they were … [Read More...]
DeNoi, Tony
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 6,708
|
Q: Randomly Generate a combination of Obstacles on the Z axis I'm attempting to make a 3D Endless Runner-type game. The pathway is 5 blocks wide and 1-4 of the 5 blocks need to generate every 50 units on the Z axis so the player has a way to proceed.
So far, I've been making each combination (31 of them) get chosen from an existing prefab, but making each combination separately is a bit of a pain. Not to mention it takes up more memory then id like it to.
Is there a way to have the blocks generate randomly across the X-Axis every 50 units along the Z-Axis?
Here's the code I've been using so far. (Just figured out how to post it)
using System.Collections;
using System.Collections.Generic;
using UnityEngine;
public class GenerateLevel : MonoBehaviour
{
public GameObject[] section;
public int zPos = 50;
public bool creatingSection = false;
public int secNum;
void Update()
{
if (creatingSection == false)
{
creatingSection = true;
StartCoroutine(GenerateSection());
}
}
IEnumerator GenerateSection()
{
secNum = Random.Range(0, 31);
Instantiate(section[secNum], new Vector3(0, 0, zPos), Quaternion.identity);
zPos += 50;
yield return new WaitForSeconds(2);
creatingSection = false;
}
}
A: I think I get what you're trying to do. So create a single Prefab for a single block and then create unique permutations of a set (like in this answer: here)
public GameObject blockPrefab;
private List<float> possibleXPositions = {10f, 20f, 30f, 40f, 50f};
[...]
IEnumerable<IEnumerable<T>> GetPermutations<T>(IEnumerable<T> items, int count)
{
int i = 0;
foreach(var item in items)
{
if(count == 1)
yield return new T[] { item };
else
{
foreach(var result in GetPermutations(items.Skip(i + 1), count - 1))
yield return new T[] { item }.Concat(result);
}
++i;
}
}
And then use it to generate the permutation you want:
List<List<float>> positions = GetPermutations(possibleXPositions, Random.Range(1,5));
List<float> chosenPositions = positions[Random.Range(0, positions.Count)];
foreach(float x in chosenPositions){
Instantiate(section[secNum], new Vector3(x, 0, zPos), Quaternion.identity);
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,872
|
Illinois IL
Early Childhood Education & Literacy
Equitable Funding
Quality Schools
Bills to Watch
Family Engagement & Organizing
Connection Conversations
Stand As One
Teach Kindness
Dismantling Systemic Racism
Elect Education Champions
Local School Councils in Chicago
Register to Vote, Then Vote
Stand Policy Fellowship
Illinois / Our Blog / Confused About Teacher Pension Funding?
TRS Surcharge
Confused About Teacher Pension Funding?
Legislation, Equitable Funding, Educators, TRS Surcharge | 12/09/2016
Jessica Handy
Government Affairs Director
Jessica advocates at the state capitol for fair school funding and equitable education policies.
There are about 860 school districts in Illinois and just one of them is responsible for paying the employer costs of their teachers' pensions: Chicago Public Schools. The State covers the cost for the rest. Neither the State nor CPS has been a model financial steward for this responsibility—both the State and CPS now have to pay every year to cover the current costs (which we call "normal cost") and debt from past years of skipping payments.
The "normal cost" for Chicago teachers' pensions is $215 million this year. This has become especially relevant at this moment because the General Assembly passed (on a bipartisan vote) an appropriation bill to cover that full amount this year, which the Governor agreed to sign only if there was significant pension reform enacted first. Keep in mind the $215 million doesn't even touch the pension debt that has also amassed; CPS will need to kick in a total of $721 million. The Governor vetoed that bill last week and the Senate overrode the veto. The House has 15 days from then to also override the veto if it is to become law, but assembling the super-majority vote needed to override would be exceedingly difficult and the House is not scheduled to be back in session until January 9.
But let's compare: this year, the State of Illinois will spend about $4 billion for teacher pensions outside of Chicago. By far, most of that is debt. It's about $250 million more than the State of Illinois paid for teacher pensions last year. It is subject to a "continuing appropriation," which means that it gets paid no matter what. No budget? That $4 billion contribution to TRS still gets paid. Here's how state payments to teacher pension systems have gone over the years:
That FY17 $215 million is about 5% of the amount the state will contribute this year to teacher pensions outside of Chicago. It is also the bill that has been termed a "Chicago bailout." But when you consider the normal cost payment in context, it seems pretty clear that this is just one step closer to pension funding parity.
A Big Part of Biden's Plan
Mimi Rodman
Moving Closer to Racial Justice
|
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| 7,492
|
Urbanisation must push development, combat poverty
South Africa should use urbanisation as a tool to push development and combat poverty by providing adequate housing opportunities and ensuring people have access to basic services, says Human Settlements Minister Lindiwe Sisulu.
Minister Sisulu made the remarks when she was addressing delegates at the St George's Hotel in Irene, Tshwane, during a seminar organised by the department in partnership with the University of South Africa.
The seminar, which was attended by practitioners in the human settlements sector, as well as academics was held under the theme "How the Human Settlements Sector is responding to the rapid increase of urbanisation in South Africa".
The seminar, amongst others, provided a platform for sharing information between government officials, academics and various practitioners in the human settlements sector seeking solutions to current challenges such as rapid urbanisation and transformation of cities.
Minister Sisulu told delegates that the country needed proper planning and management of urbanisation and related challenges, such as migration of people seeking better opportunities.
"When not properly designed and managed, cities often pay the high price of negative externalities, such as congestion, contamination and wide inequalities often leading to social unrest and instability," said Minister Sisulu.
She also warned that inability to plan properly would increase poverty in the urban centres, and will see "the concomitant informality continue to be a growing challenge to the future sustainability of our human settlements".
"The real challenges faced by urbanisation, when its positive aspects are recognised, are sustainable in the social, economic and environmental dimensions. Sustainable urban planning and development is necessary to eliminate the causes of segregation and exclusion and support the social, spatial and economic transformation of our cities and towns."
As part of its policies, the Minister said the department places integration and sustainability of residential areas as the core of its projects.
This is also reflected in the Department's Spatial Master Plan, with which it seeks to tackle the problem of racially segregated communities created as part of segregation policies under apartheid.
Most of the key projects of the department, including the N2 Gateway in Western Cape, Cosmo City in Gauteng, Zanemvula in Eastern Cape or Cornubia in Durban are integrated communities.
The developments cater for various segments including Breaking New Ground, mortgage or bonds and social housing.
Migration contributes to shortage of accommodation
Statistician General Dr Pali Lehohla told delegates how the phenomenon of migration also contributed to the shortage of decent accommodation in the cities, prompting the rise in informal settlements.
However, Lehohla commended efforts by the department to tackle the country's shortage of housing, noting that research by StatsSA has indicated notable improvement from eight million formal housing in 2002 to 13 million in 2016.
Other prominent people who attended the seminar included MECs for Human Settlements in KwaZulu-Natal, Ravi Pillay and Mpumalanga, Speedy Mashilo, as well as Select Committee Chairperson Cathy Dlamini and Tshwane Member of Mayoral Committee, Mandla Nkomo. – SAnews.gov.za
Free State leads the pack with 85.7% pass rate
@SAgovnews (@SAgovnews) 21 Jan at 10:57
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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На овој страници налази се списак од 100 Француских департмана поређаних по површини:
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Департмани Француске
Списак департмана Француске по броју становника
Списак департмана Француске по густини насељености
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fr:Départements français classés par superficie
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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\section{Introduction}
Given a real life non-stationary signal $\mathbf{s}(x)$, $x\in{\mathbb{R}}$,
we may be interested in decomposing it into simple components in order to identify features and quasi periodicities hidden in it. We can think, for instance, to a economic index, like the GDP of a nation, or a geophysical signal, like the sea level during a tsunami, as well as to an engineering measure, like the vibrations of a structure or machinery. Standard techniques, like Fourier or wavelet transform, cannot help, in general, to decompose meaningfully non-stationary signals. Whereas, following the idea proposed by Huang et al. in \cite{huang1998empirical}, we can iteratively decompose such signals into a finite sequence of simple components, defined Intrinsic Mode Functions (IMFs), which fulfill two properties:
\textit{i}) the number of extrema and the number of zero crossings must either equal or differ at most by one;
\textit{ii}) considering upper and lower envelopes connecting respectively all the local maxima and minima of the function, their mean has to be zero at any point.
The method originally proposed by Huang et al. in \cite{huang1998empirical}, called Empirical Mode Decomposition (EMD), proved to be unstable to small perturbations. For this reason an alternative algorithm, called Ensemble Empirical Mode Decomposition (EEMD), was proposed in \cite{wu2009ensemble} which is based on the idea of applying EMD to an ensemble of signals produced perturbing hundreds of times the given one with noise. The decomposition is then derived as the average decomposition of the ensemble.
In \cite{lin2009iterative} the authors proposed an alternative technique to the EMD, called Iterative Filtering (IF), which has the very same structure of EMD, but it is stable and convergent both in the continuous setting \cite{cicone2014adaptive,huang2009convergence} and in the discrete one \cite{cicone2017numerical,cicone2017multidimensional}. We point out that IF has been also generalized producing the so called Adaptive Local Iterative Filtering (ALIF) algorithm whose convergence and stability are under investigation \cite{cicone2014adaptive,cicone2017spectral}. For further details on why standard techniques may fail in decomposing a non-stationary signal and the advantages of using EEMD or IF we refer the interested reader to \cite{cicone2017dummies}.
In this work we consider the case of compactly supported and discrete signals. For simplicity we assume that each signal
$\mathbf{s}=\left[\mathbf{s}(x_j)\right]_{j=0}^{n-1}$ (with $n\in{\mathbb{N}}$)
is supported on $[0,1]$ and it is sampled at $n$ points $x_j= \frac{j}{n-1}$, with $j=0,\ldots,n-1$. Without loosing generality we can further assume that $\| \mathbf{s} \|_2 = 1$.
Any signal decomposition method which deals with a compactly supported signal requires assumptions on how the signal extends outside the boundaries, the so called Boundary Conditions (BCs). This aspect has a key role in many applications, for instance in image restoration \cite{pietro2016}, and usually the goal is to employ BCs able to guarantee good accuracy of the approximated solution computed by some numerical algorithm \cite{pietro2017}.
Regarding IF, in \cite{cicone2017numerical} the authors addressed its convergence when it is assumed that the signals extend periodically outside the boundaries. The questions which are still open are: does IF converge also for other kinds of BCs? Given a signal extended artificially outside the boundaries in a certain way, how do the errors introduced outside the boundaries effect the decomposition in the iterations? Given a compactly supported signal what is the best choice in terms of BCs?
The paper is organized as follows.
In Section \ref{sec:IF} we review the IF algorithm applied to discrete and compactly supported signals.
Section \ref{sec:BC} is devoted to a summary of BCs and their properties.
In Section \ref{sec:IFconv} we study the IF convergence when different BCs are chosen for the signal.
In Section \ref{sec:Error} we present an extended version of IF method,
which is useful for addressing the question of how the errors propagate from outside the boundaries to the inside.
This work ends with some numerical examples showing the impact of BCs on decomposition quality and error propagation in Section \ref{sec:Examples},
and concluding remarks in Section \ref{sec:Conclusions}.
\section{Discrete Iterative Filtering}
\label{sec:IF}
In this section we review the IF method focusing on the case of discrete and compactly supported signals $\mathbf{s}$.
We start from the definition of filter
\begin{definition}\label{def:filter}
A non-negative vector $\mathbf{w}=(0,\ldots,0,w_{-l},\ldots,w_{-1},w_{0},w_1,\ldots,w_{l},0,\ldots,0) \in {\mathbb{R}}^n$
such that $w_j>0$ for $j=-l,\ldots,l$ and $\underset{j=-l}{\overset{l}{\sum}} w_j = 1$
is called a \textbf{filter} of length $l$, with $0 < l \leq \lfloor \frac{n-1}{2} \rfloor$.
If a filter is such that $w_{-j}=w_j$ for $j=1,\ldots,l$,
then $\mathbf{w}$ is called \textbf{symmetric}.
If a filter is such that for $0 \leq i < j $, $w_{i} \geq w_{j}$
and for $j < i \leq 0 $, $w_{i} \geq w_{j}$,
then $\mathbf{w}$ is called \textbf{decreasing}.
\end{definition}
In this work we consider only symmetric filters.
When we have a symmetric filter associated with some step $m$, we employ the following notation
\begin{equation}
\mathbf{w}_m=(0,\ldots,0,w_{l_m}^m,\ldots,w_{1}^m,w_{0}^m,w_1^m,\ldots,w_{l_m}^m,0,\ldots,0),
\end{equation}
with $w_{0}^m + 2 \underset{j=1}{\overset{l_m}{\sum}} w_j^m = 1$.
If we assume that some filter shape $h:[-1,1] \rightarrow {\mathbb{R}}$ (symmetric with respect to $y$-axis) has been selected a priori,
like one of the Fokker-Planck filters described in \cite{cicone2014adaptive},
then the elements $w_j^m$ can be computed, for $j=0,1,\ldots,l_m$, by the linear scaling formula
\begin{equation}
w_j^m = h\left(\dfrac{j}{l_m}\right) \dfrac{1}{l_m},
\end{equation}
where $l_m$ is the length that characterizes the filter.
Assuming $\mathbf{s}_1^m=\mathbf{s}$, where the two indices will become clear in the next paragraph, the main step of the IF method,
for $i=0,\ldots,n-1$, is
\begin{eqnarray*}
\mathbf{s}_{k+1}^m(x_i) &=& \mathbf{s}_{k}^m(x_i)-\int_{x_i-\frac{l_m}{n-1}}^{x_i+\frac{l_m}{n-1}} \!\!\!\!\!\!\! \mathbf{s}_k^m(y)
h\left(\dfrac{(x_i-y)(n-1)}{l_m}\right)\dfrac{n-1}{l_m} {\textrm{d}y} \\
&\approx& \mathbf{s}_{k}^m(x_i)-\!\!\!\!\! \sum_{x_j=x_i-\frac{l_m}{n-1}}^{x_i+\frac{l_m}{n-1}}\!\!\!\!\! \mathbf{s}_k^m(x_j)
h\left(\dfrac{(x_i-x_j)(n-1)}{l_m}\right)\dfrac{1}{l_m} \\
&=& \mathbf{s}_{k}^m(x_i)-\!\!\! \sum_{j=i-l_m}^{i+l_m} \!\!\! \mathbf{s}_k^m(x_j)
h\left(\dfrac{i-j}{l_m}\right)\dfrac{1}{l_m} \\
&=& \mathbf{s}_{k}^m(x_i)-\!\!\! \sum_{j=i-l_m}^{i+l_m} \!\!\! \mathbf{s}_k^m(x_j) w_{\vert i-j \vert}^m
\end{eqnarray*}
Algorithm \ref{algo:IF_discrete} provides the pseudocode of the Discrete Iterative Filtering (DIF) Algorithm.
We observe that the first while loop is called Outer Loop, whereas the second one Inner Loop.
In the notation $\mathbf{s}_k^m$, $m$ denotes the step relative to the Outer Loop, while $k$ denotes the step relative to the Inner Loop.
\begin{algorithm}
\caption{\textbf{Discrete Iterative Filtering} IMFs = DIF$(s, h)$}\label{algo:IF_discrete}
\begin{algorithmic}
\STATE $m=1$
\STATE $\mathbf{s}_1^m = \mathbf{s}$
\WHILE{the number of extrema of $\mathbf{s}_1^m$ $\geq 2$}
\STATE $k=1$
\STATE compute the filter length $l_m$ for the signal $\mathbf{s}_k^m$
\STATE compute $\mathbf{w}_m$ (having $h$ and $l_m$)
\WHILE{the stopping criterion is not satisfied}
\STATE $(\mathbf{s}_{k}^m)^{\mathcal{BC}}(x_i)=\mathbf{s}_{k}^m(x_i)$, $i=0,\ldots,n-1$
\STATE apply BCs for computing $(\mathbf{s}_{k}^m)^{\mathcal{BC}}(x_i)$, $i=-l_m,\ldots,-1$ and $i=n,\ldots,n-1+l_m$
\STATE $\mathbf{s}_{k+1}^m(x_i) = \mathbf{s}_{k}^m(x_i) - \underset{j=i-l_m}{\overset{i+l_m}{\sum}} (\mathbf{s}_{k}^m)^{\mathcal{BC}}(x_j) w_{\vert i-j \vert}^m$, $\quad i= 0,\ldots, n-1$
\STATE $k = k+1$
\ENDWHILE
\STATE $\mathbf{f}_m = \mathbf{s}_{k}^m$
\STATE $\mathbf{s}_1^{m+1} = \mathbf{s}_1^{m}-\mathbf{f}_{m}$
\STATE $m = m+1$
\ENDWHILE
\STATE $\mathbf{f}_m = \mathbf{s}_1^m$
\STATE IMFs = $\{ \mathbf{f}_1,\ldots, \mathbf{f}_m \}$
\end{algorithmic}
\end{algorithm}
The idea is to suitably choose the filter length $l_m$ in order to capture the desired frequencies in the IMF $\mathbf{f}_m$,
and this is done by subtracting iteratively from the signal its moving average computed as convolution of the signal itself with the selected filter. In matrix form we have
\begin{equation}\label{eq:MatrixForm}
\mathbf{s}_{k+1}^m=(I-W_m)\mathbf{s}_k^m,
\end{equation}
where $W_m$ is a structured matrix constructed from the filter $\mathbf{w}_m$
and the BCs imposed.
In particular $W_m$ can be written as the sum of two matrices
\begin{equation}
W_{m}^{\mathcal{BC}}=T_{m}+K_{m}^{\mathcal{BC}},
\end{equation}
where the first one is a Toeplitz matrix,
while the second one is a correction matrix which depends on BCs (for details, see Section \ref{sec:BC}).
From (\ref{eq:MatrixForm}), it follows immediately that
\begin{equation}
\mathbf{s}_{k+1}^m = (I-W_m)^k \mathbf{s}_1^m,
\end{equation}
so ideally the first IMF is given by
\begin{equation}\label{eq:First_IMF_fixed_length}
\mathbf{f}_1 = \lim_{k\rightarrow\infty} (I-W_1)^{k} \mathbf{s}.
\end{equation}
However, in the implemented algorithm we do not let $k$ to go to infinity, instead we use a stopping criterion. We can define, for instance, the following quantity
\begin{equation}\label{eq:SD}
\Delta_k^m:=\frac{\|\mathbf{s}_{k+1}^m-\mathbf{s}_{k}^m\|_2}{\|\mathbf{s}_{k}^m\|_2},
\end{equation}
so we can either stop the process when the value $\Delta_k^m$ reaches a certain threshold or we can introduce a limit on the maximal number of iterations for all the Inner Loops. It is also possible to adopt different stopping criteria for different Inner Loops.
\section{Boundary conditions and structured matrices}
\label{sec:BC}
Boundary conditions deal with the problem of extending the signal outside the field of view in which the detection is made.
So let $\mathbf{s}$ be the signal inside the boundaries and let $p$ be the parameter relative to the space outside the boundaries,
we have that,
for $j= 1, \ldots, p$,
Zero BCs are defined as
\begin{equation}
\mathbf{s}(x_{-j}) = 0, \hspace{0.5cm} \mathbf{s}(x_{n-1+j}) = 0,
\end{equation}
Periodic BCs are defined as
\begin{equation}
\mathbf{s}(x_{-j}) = \mathbf{s}(x_{n-j}), \hspace{0.5cm} \mathbf{s}(x_{n-1+j}) = \mathbf{s}(x_{j-1}),
\end{equation}
Reflective BCs are defined as
\begin{equation}
\mathbf{s}(x_{-j}) = \mathbf{s}(x_{j-1}), \hspace{0.5cm} \mathbf{s}(x_{n-1+j}) = \mathbf{s}_{n-j},
\end{equation}
Anti-Reflective BCs are defined as
\begin{equation}
\label{AR1d}
\mathbf{s}(x_{-j})=2\mathbf{s}(x_{0})-\mathbf{s}(x_{j}), \hspace{0.5cm} \mathbf{s}(x_{n-1+j})=2\mathbf{s}(x_{n-1})-\mathbf{s}(x_{n-1-j}).
\end{equation}
According to the BCs imposed, we have a different kind of structured matrix as $W^{\mathcal{BC}}$,
whose elements are defined from values of the filter $\mathbf{w}$ of length $l$.
As said, here we take into account symmetric filters,
since this choice allows to have useful theoretical properties.
In particular,
symmetry is not necessary to get the algebra of Circulant matrices,
associated with Periodic BCs,
while it is necessary to get the algebra of Reflective matrices and
the algebra of Anti-Reflective matrices.
The symmetry property also allows in all these three cases to have a fast transform
that can be employed for computing matrix-vectors products in an efficient way.
In particular, we have Discrete Fourier Transform (DFT) for Circulant matrices,
Discrete Cosine Transform of type III (DCT-III) for Reflective matrices
and Anti-Reflective Transform (ART)
-- strongly linked to Discrete Sine Transform of type I (DST-I) --
for Anti-Reflective matrices.
Unfortunately, this does not hold for Toeplitz matrices, associated with Zero BCs,
i.e. $W^{\mathcal{Z}}=T$, where
\begin{equation}\label{eq:Toeplitz}
T=\left(
\begin{array}{cccccccccc}
w_{0} & w_{1} & w_{2} & \ldots & w_{l} & & & & & \\
w_{1} & w_{0} & w_{1} & w_{2} & \ddots & w_{l} & & & & \\
w_{2} & w_{1} & w_{0} & w_{1} & w_{2} & \ddots & w_{l} & & & \\
\vdots & w_{2} & w_{1} & w_{0} & w_{1} & w_{2} & \ddots & \ddots & & \\
w_{l} & \ddots & w_{2} & w_{1} & \ddots & \ddots & \ddots & \ddots & w_{l} & \\
& w_{l} & \ddots & w_{2} & \ddots & \ddots & w_{1} & w_{2} & \ddots & w_{l} \\
& & w_{l} & \ddots & \ddots & w_{1} & w_{0} & w_{1} & w_{2} & \vdots \\
& & & \ddots & \ddots & w_{2} & w_{1} & w_{0} & w_{1} & w_{2} \\
& & & & w_{l} & \ddots & w_{2} & w_{1} & w_{0} & w_{1} \\
& & & & & w_{l} & \ldots & w_{2} & w_{1} & w_{0}
\end{array}%
\right) _{n\times n}
.
\end{equation}
In case of Periodic BCs, we have Circulant matrices that have this form
\begin{equation}
W^{\mathcal{P}}=\left(
\begin{array}{cccccccccc}
w_{0} & w_{1} & \ldots & w_{l} & & & & w_{l} & \ldots & w_{1} \\
w_{1} & w_{0} & w_{1} & \ddots & w_{l} & & & & \ddots & \vdots \\
\vdots & w_{1} & w_{0} & w_{1} & \ddots & w_{l} & & & & w_{l}\\
w_{l} & \ddots & w_{1} & w_{0} & w_{1} & \ddots & \ddots & & & \\
& w_{l} & \ddots & w_{1} & w_{0} & \ddots & \ddots & w_{l}& & \\
& & w_{l} & \ddots & \ddots & \ddots & w_{1} & \ddots & w_{l} & \\
& & & \ddots & \ddots & w_{1} & w_{0} & w_{1} & \ddots & w_{l} \\
w_{l} & & & & w_{l} & \ddots & w_{1} & w_{0} & w_{1} & \vdots \\
\vdots & \ddots & & & & w_{l} & \ddots & w_{1} & w_{0} & w_{1} \\
w_{1} & \ldots & w_{l} & & & & w_{l} & \ldots & w_{1} & w_{0}%
\end{array}%
\right) _{n\times n}
.
\end{equation}
Denoted by $\mathrm{i}$ the imaginary unit,
$W^{\mathcal{P}}$ can be diagonalized by $Q^{\mathcal{P}}$, defined as
\begin{equation}
[Q^{\mathcal{P}}]_{i,j} = \sqrt{n} F^{-1} =\dfrac{1}{\sqrt{n}} e^{ \frac{2 (i-1) (j-1) \pi \mathrm{i} }{n} }, \hspace{1cm} i,j=1,\ldots,n,
\end{equation}
where $F$ is the $n$-dimensional DFT.
Eigenvalues of $W^{\mathcal{P}}$ can be computed by this formula, for $i=1,\ldots,n$,
\begin{equation}
\label{equ:eigP}
\lambda _{i}^{\mathcal{P}}=w_{0}+2\underset{j=1}{\overset{l}{\sum }}w_{j}\cos\left( \frac{2j(i-1)\pi }{n}\right).
\end{equation}
In the Reflective case, we have $W^{\mathcal{R}}=T+H^{\mathcal{R}}$,
where $H^{\mathcal{R}}$ is a Hankel matrix
\begin{equation}
H^{\mathcal{R}}=\left(
\begin{array}{cccccccccc}
w_{1} & w_{2} & w_{3} & \ldots & w_{l} & & & & & \\
w_{2} & w_{3} & {\reflectbox{$\ddots$}} & {\reflectbox{$\ddots$}} & & & & & & \\
w_{3} & {\reflectbox{$\ddots$}} & {\reflectbox{$\ddots$}} & & & & & & & \\
\vdots & {\reflectbox{$\ddots$}} & & & & & & & & \\
w_{l} & & & & & & & & & \\
& & & & & & & & & w_{l} \\
& & & & & & & & {\reflectbox{$\ddots$}} & \vdots \\
& & & & & & & {\reflectbox{$\ddots$}} & {\reflectbox{$\ddots$}} & w_{3} \\
& & & & & & {\reflectbox{$\ddots$}} & {\reflectbox{$\ddots$}} & w_{3} & w_{2} \\
& & & & & w_{l} & \ldots & w_{3} & w_{2} & w_{1}%
\end{array}%
\right) _{n\times n}
.
\end{equation}
$W^{\mathcal{R}}$ can be diagonalized by $Q^{\mathcal{R}}$,
that is the $n$-dimensional DCT-III,
having entries
\begin{equation}
[Q^{\mathcal{R}}]_{i,j}=\sqrt{\dfrac{2-\delta _{i1}}{n}}\cos \left( \dfrac{%
(i-1)(2j-1)\pi }{2n}\right), \hspace{1cm} i,j=1,\ldots,n,
\end{equation}
where $\delta_{ij}$ denotes the Kronecker delta.
Eigenvalues of $W^{\mathcal{R}}$ can be obtained by the following formula
\begin{equation}
\lambda _{i}^{\mathcal{R}}=\dfrac{[Q^{\mathcal{R}}(W^{\mathcal{R}} \mathbf{e}_{1})]_{i}}{[Q\mathbf{e}_{1}]_{i}},
\end{equation}
where $\mathbf{e}_{1}=(1,0,\ldots ,0)^{T} \in {\mathbb{R}}^n$.
Therefore,
using the variable $t=\frac{(i-1)\pi }{2n}$,
and exploiting
\begin{equation*}
\cos (2jt-t)=\cos (2jt)\cos (t)+\sin (2jt)\sin(t)
\end{equation*}
and
\begin{equation*}
\cos (2jt+t)=\cos (2jt)\cos (t)-\sin (2jt)\sin(t),
\end{equation*}
we get that in the Reflective case eigenvalues can be computed by this formula, for $i=1,\ldots,n$,
\begin{eqnarray}
\label{equ:eigR}
\lambda _{i}^{\mathcal{R}} &=&\underset{j=1}{\overset{n}{\sum }}[W^{\mathcal{R}}]_{1,j}\dfrac{\cos \left( (2j-1)\dfrac{(i-1)\pi }{2n}\right) }{\cos \left( \dfrac{(i-1)\pi }{2n}\right) } \nonumber \\
&=&w_{0}+\underset{j=1}{\overset{l}{\sum }}w_{j}\dfrac{\cos((2j-1)t)+\cos ((2j+1)t)}{\cos \left( t\right) } \nonumber \\
&=&w_{0}+2\underset{j=1}{\overset{l}{\sum }}w_{j}\cos (2jt) \nonumber \\
&=&w_{0}+2\underset{j=1}{\overset{l}{\sum }}w_{j}\cos \left( \frac{j(i-1)\pi }{n}\right)
\end{eqnarray}
In the Anti-Reflective case, the structure of the matrix is more involved, namely
\begin{equation}
W^{\mathcal{AR}}=\left(
\begin{array}{ccccccc}
z_{1}+w_{0} & 0 & \ldots & \ldots & \ldots & 0 & 0 \\
z_{2}+w_{1} & & & & & & \vdots \\
\vdots & & & & & & \vdots \\
z_{l}+w_{l-1} & & & & & & 0 \\
w_{l} & & & \hat{W}^{\mathcal{AR}} & & & w_{l} \\
0 & & & & & & z_{l}+w_{l-1}\\
\vdots & & & & & & \vdots \\
\vdots & & & & & & z_{2}+w_{1} \\
0 & 0 & \ldots & \ldots & \ldots & 0 & z_{1}+w_{0}
\end{array}%
\right) _{n\times n}
,
\end{equation}
where $z_{j}= 2 \underset{k=j}{\overset{l}{\sum }}w_{k}$ and $\hat{W}^{\mathcal{AR}}= P W^{\mathcal{AR}}P^{T}$, with
\begin{equation}
P=\left(
\begin{array}{ccccccc}
0 & 1 & & & & & 0 \\
0 & & 1 & & & & 0 \\
\vdots & & & \ddots & & & \vdots \\
0 & & & & 1 & & 0 \\
0 & & & & & 1 & 0%
\end{array}%
\right) _{(n-2)\times n}
.
\end{equation}
Moreover, $\hat{W}^{\mathcal{AR}}=\hat{T}-\hat{H}^{\mathcal{AR}}$, where $\hat{T}$ is a Toeplitz matrix
\begin{equation}
\hat{T}=\left(
\begin{array}{cccccccc}
w_{0} & w_{1} & \ldots & w_{l} & & & & \\
w_{1} & w_{0} & w_{1} & \ddots & w_{l} & & & \\
\vdots & w_{1} & w_{0} & w_{1} & \ddots & \ddots & & \\
w_{l} & \ddots & w_{1} & w_{0} & \ddots & \ddots & w_{l} & \\
& w_{l} & \ddots & \ddots & \ddots & w_{1} & \ddots & w_{l} \\
& & \ddots & \ddots & w_{1} & w_{0} & w_{1} & \vdots \\
& & & w_{l} & \ddots & w_{1} & w_{0} & w_{1} \\
& & & & w_{l} & \ldots & w_{1} & w_{0}%
\end{array}%
\right) _{(n-2)\times (n-2)}
,
\end{equation}
while $\hat{H}^{\mathcal{AR}}$ is a Hankel matrix
\begin{equation}
\hat{H}^{\mathcal{AR}}=\left(
\begin{array}{cccccccc}
w_{2} & w_{3} & \ldots & w_{l} & & & & \\
w_{3} & {\reflectbox{$\ddots$}} & {\reflectbox{$\ddots$}} & & & & & \\
\vdots & {\reflectbox{$\ddots$}} & & & & & & \\
w_{l} & & & & & & & \\
& & & & & & & w_{l} \\
& & & & & & {\reflectbox{$\ddots$}} & \vdots \\
& & & & & {\reflectbox{$\ddots$}} & {\reflectbox{$\ddots$}} & w_{3} \\
& & & & w_{l} & \ldots & w_{3} & w_{2}
\end{array}%
\right) _{(n-2)\times (n-2)}
.
\end{equation}
$\hat{W}^{\mathcal{AR}}$ can be diagonalized by $\hat{Q}^{\mathcal{AR}}$,
that is the ($n-2$)-dimensional DST-I,
having entries
\begin{equation}
[\hat{Q}^{\mathcal{AR}}]_{i,j}=\sqrt{\dfrac{2}{n-1}}\sin \left( \dfrac{ij\pi }{n-1}\right), \hspace{1cm} i,j=1,\ldots,n-2.
\end{equation}
Eigenvalues of $\hat{W}^{\mathcal{AR}}$ can be obtained by the following formula
\begin{equation}
\hat{\lambda}_{i}^{\mathcal{AR}}=\dfrac{[\hat{Q}^{\mathcal{AR}}(\hat{W}^{\mathcal{AR}}\mathbf{\hat{e}}_{1})]_{i}}{[\hat{Q}^{\mathcal{AR}}\mathbf{\hat{e}}_{1}]_{i}}
\end{equation}
where $\mathbf{\hat{e}}_{1}=(1,0,\ldots ,0)^{T} \in {\mathbb{R}}^{n-2}$.
Therefore,
using the variable $t=\frac{i\pi }{n-1}$,
and exploiting
\begin{equation*}
\sin (2t)=2\sin (t)\cos (t)
\end{equation*}
and
\begin{equation*}
\cos (2t)=1-2\sin ^{2}(t)
\end{equation*}
and
\begin{eqnarray*}
\sin ((j+2)t)-\sin jt) &=& \sin (jt)\cos (2t)+\sin (2t)\cos (jt)-\sin(jt) \\
&=& \sin (jt)(1-2\sin ^{2}(t))+2\sin (t)\cos (t)\cos (jt)-\sin (jt) \\
&=& -2\sin(jt)\sin ^{2}(t)+2\sin (t)\cos (t)\cos (jt)
\end{eqnarray*}
and
\begin{equation*}
2 \sin(jt) \sin(t) = \cos(jt-t) - \cos(jt+t)
\end{equation*}
and\begin{equation*}
2 \cos(t) \cos(jt) = \cos(jt-t) - \cos(jt+t)
\end{equation*}
we get that eigenvalues of $\hat{W}^{\mathcal{AR}}$ can be computed by this formula, for $i=1,\ldots,n-2$,
\begin{eqnarray}
\label{equ:eigAR}
\hat{\lambda}_{i}^{\mathcal{AR}} &=&\underset{j=1}{\overset{n-2}{\sum }}[\hat{W}^{\mathcal{AR}}]_{1,j}\dfrac{\sin
\left( \dfrac{ji\pi }{n-1}\right) }{\sin \left( \dfrac{i\pi }{n-1}\right) } \nonumber \\
&=&w_{0}+w_{1}\dfrac{\sin (2t)}{\sin \left( t\right) }+\underset{j=1}{\overset{l-1}{\sum }}w_{j+1}\dfrac{\sin ((j+2)t)}{\sin \left( t\right) }-w_{j+1}\dfrac{\sin (jt)}{\sin \left( t\right) } \nonumber \\
&=&w_{0}+2w_{1}\cos (t)+\underset{j=1}{\overset{l-1}{\sum }}w_{j+1}\left(2\cos (t)\cos (jt)-2\sin (jt)\sin (t)\right) \nonumber \\
&=&w_{0}+2w_{1}\cos (t)+2\underset{j=1}{\overset{l-1}{\sum }}w_{j+1}\cos((j+1)t) \nonumber \\
&=&w_{0}+2w_{1}\cos (t)+2\underset{j=2}{\overset{l}{\sum }}w_{j}\cos(jt) \nonumber \\
&=&w_{0}+2\underset{j=1}{\overset{l}{\sum }}w_{j}\cos \left(\dfrac{ji\pi }{n-1} \right)
\end{eqnarray}
By Lemma 3.1 in \cite{serra2003antirefl}, we know that the eigenvalues of $W^{\mathcal{AR}}$ are given by 1 with multiplicity
two and by the eigenvalues $\lbrace \hat{\lambda}_{i}^{\mathcal{AR}} \rbrace_{i=1,\ldots,n-2}$ of $\hat{W}^{\mathcal{AR}}$.
Looking at the structure of the matrices presented,
it can be easily verified that
the eigenvector $u_1^{\mathcal{P}}$ associated with $\lambda_1^{\mathcal{P}}=1$ is $(1,\ldots,1)^T$,
the eigenvector $u_1^{\mathcal{R}}$ associated with $\lambda_1^{\mathcal{R}}=1$ is $(1,\ldots,1)^T$,
and the eigenvectors associated with $\lambda_1^{\mathcal{AR}}=\lambda_2^{\mathcal{AR}}=1$ are
$u_1^{\mathcal{AR}}=(0,1,2,\ldots,n-2,n-1)^T$ and $u_2^{\mathcal{AR}}=(n-1,n-2,\ldots,2,1,0)^T$.
We notice that $Q^{\mathcal{P}}$, $Q^{\mathcal{R}}$ and $\hat{Q}^{\mathcal{AR}}$ are all unitary matrices.
However, the last one is linked to $\hat{W}^{\mathcal{AR}}$,
so we also need to know how to diagonalize the original matrix $W^{\mathcal{AR}}$.
In other words, basing on DST-I, we need to introduce ART.
Unfortunately this transform is not unitary, since the matrices in Anti-Reflective algebra are in general not normal.
Recalling theoretical results reported in \cite{arico2011,pietro2016},
we have that $W^{\mathcal{AR}}$ can be diagonalized by $Q^{\mathcal{AR}}$,
that is the $n$-dimensional ART, defined as follows
\begin{equation}
\label{eq:ART}
Q^{\mathcal{AR}}=\left(
\begin{array}{ccccccc}
(n-1) \eta^{-1} & 0 & \ldots & \ldots & \ldots & 0 & 0 \\
(n-2) \eta^{-1} & & & & & & \eta^{-1} \\
(n-3) \eta^{-1} & & & & & & 2 \eta^{-1} \\
\vdots & & & & & & \vdots \\
\vdots & & & \hat{Q}^{\mathcal{AR}} & & & \vdots \\
\vdots & & & & & & \vdots\\
2 \eta^{-1} & & & & & & (n-3) \eta^{-1} \\
\eta^{-1} & & & & & & (n-2) \eta^{-1} \\
0 & 0 & \ldots & \ldots & \ldots & 0 & (n-1) \eta^{-1}
\end{array}%
\right) _{n\times n}
,
\end{equation}
where $\eta = \sqrt{\sum_{j=0}^{n-1} j^2}$ is introduced just to normalize the first and the last column,
which come from eigenvectors $u_1^{\mathcal{AR}}$ and $u_2^{\mathcal{AR}}$.
Summarizing, in this Section we have introduced BCs,
analyzing the properties of matrices associated with their choice,
in particular providing descriptions of matrix structures,
formulas for computation of eigenvalues
and diagonalization results (associated with discrete transforms).
All this will be very useful in Section \ref{sec:IFconv},
in which convergence properties of DIF will be investigated.
\section{Spectral properties and convergence results}
\label{sec:IFconv}
At this stage, we consider some BCs (Periodic or Reflective or Anti-Reflective).
By exploiting results presented in Section \ref{sec:BC},
we are able to prove spectral properties of $W^{\mathcal{BC}}$ and convergence results for DIF algorithm.
\begin{lemma}
\label{lemma:1}
Let $W^{\mathcal{BC}} \in {\mathbb{R}}^{n \times n}$ be the structured matrix constructed from a symmetric decreasing filter $\mathbf{w}$ of length
$0 < l \leq \lfloor \frac{n-1}{2} \rfloor$, imposing some BCs (Periodic or Reflective or Anti-Reflective).
Then $\sigma (W^{\mathcal{BC}})\subseteq [-1,1]$.
\end{lemma}
\begin{proof}
By definition $W^{\mathcal{BC}}$ is a symmetric matrix, so it has a real spectrum.
By considering the following estimate relative to the spectral radius
\begin{equation*}
\rho (W^{\mathcal{BC}})\leq \left\Vert W^{\mathcal{BC}}\right\Vert _{\infty}=
\underset{i}{\max }\underset{j=1}{\overset{n}{\sum }}\left\vert[W^{\mathcal{BC}}]_{i,j}\right\vert=
w_{0}+2\underset{k=1}{\overset{l}{\sum }}w_{k}=1,
\end{equation*}
we can conclude that all eigenvalues lie in the interval $[-1,1]$.
\end{proof}
\begin{theorem}
\label{teo:1}
Let $\mathbf{s} \in {\mathbb{R}}^n$ the signal that has to be decomposed.
Let $\mathbf{v}$ be a symmetric decreasing filter of length $0 < l' \leq \lfloor \frac{n-1}{4} \rfloor$
and $\mathbf{w} = \mathbf{v} \ast \mathbf{v}$ be another symmetric decreasing filter of length $0 < l \leq \lfloor \frac{n-1}{2} \rfloor$
defined by making the convolution of $\mathbf{v}$ with itself.
Then for the matrix $W^{\mathcal{BC}} \in {\mathbb{R}}^{n \times n}$ constructed from $\mathbf{w}$ and some BCs
(Periodic or Reflective or Anti-Reflective),
it holds that $\sigma (W^{\mathcal{BC}})\subseteq [0,1]$.
Moreover,
$\lambda_1^{\mathcal{P}}=1$ and $\lbrace \lambda_{i}^{\mathcal{P}} \rbrace_{i=2,\ldots,n} \subseteq [0,1)$,
$\lambda_1^{\mathcal{R}}=1$ and $\lbrace \lambda_{i}^{\mathcal{R}} \rbrace_{i=2,\ldots,n} \subseteq [0,1)$,
$\lambda_1^{\mathcal{AR}}=\lambda_2^{\mathcal{AR}}=1$ and $\lbrace \lambda_{i}^{\mathcal{AR}} \rbrace_{i=3,\ldots,n} \subseteq [0,1)$.
\end{theorem}
\begin{proof}
As already said,
in case of Periodic or Reflective or Anti-Reflective BCs we are in a matrix algebra;
this means that the matrix product gives rise to a matrix
which can still be interpreted as a matrix constructed from a filter and the same BCs.
In particular,
thanks to the fact that $l' \leq \lfloor \frac{n-1}{4} \rfloor$,
such filter can be computed by means of convolution.
Thus, if we denote by $V^{\mathcal{BC}}$ the structured matrix associated with filter $\mathbf{v}$,
we have that $W^{\mathcal{BC}} = (V^{\mathcal{BC}})^2$.
By Lemma \ref{lemma:1}, $\sigma (V^{\mathcal{BC}})\subseteq [-1,1]$,
therefore $\sigma (W^{\mathcal{BC}})\subseteq [0,1]$.
Moreover,
in case of Periodic BCs,
if we consider (\ref{equ:eigP}) with $i=1$, we get $\lambda_1^{\mathcal{P}}=1$,
while for $1 < i \leq n$ we have a convex combination of cosine values
(each of them strictly less than 1),
except for the first term multiplied by $w_0$ which is equal to 1.
Clearly this sum cannot equal to 1, so it has a value $0 \leq x < 1$.
In case of Reflective BCs,
if we consider (\ref{equ:eigR}) with $i=1$, we get $\lambda_1^{\mathcal{R}}=1$,
while for $1 < i \leq n$ we have a convex combination of cosine values
(each of them strictly less than 1),
except for the first term multiplied by $w_0$ which is equal to 1.
Clearly this sum cannot equal to 1, so it has a value $0 \leq x < 1$.
In case of Anti-Reflective BCs,
if we consider (\ref{equ:eigAR}) for $i=1,\ldots,n-2$
we have a convex combination of cosine values (each of them strictly less than 1),
except for the first term multiplied by $w_0$ which is equal to 1.
Clearly this sum cannot equal to 1, so it has a value $0 \leq x < 1$.
Therefore $n-2$ eigenvalues of $W^{\mathcal{BC}}$ belong to the interval $[0,1)$,
while the remaining two are equal to 1, as we already know.
\end{proof}
In the following Lemma we summarize diagonalization results presented in Section \ref{sec:BC}.
\begin{lemma}
\label{lemma:2}
Let $W^{\mathcal{BC}} \in {\mathbb{R}}^{n \times n}$ be a matrix constructed from
a symmetric filter $\mathbf{w}$ and some BCs (Periodic or Reflective or Anti-Reflective).
Then
\begin{equation}
W^{\mathcal{BC}} = Q^{\mathcal{BC}} D^{\mathcal{BC}} (Q^{\mathcal{BC}} )^{-1},
\end{equation}
i.e. $W^{\mathcal{BC}}$ can be diagonalized by $Q^{\mathcal{BC}}$,
whose columns are eigenvectors of $W^{\mathcal{BC}}$ .
\end{lemma}
Now we are ready to discuss the convergence of DIF algorithm. The method in the limit produces IMFs that are projections of the given signal $\mathbf{s} $ onto the eigenspace of $W^{\mathcal{BC}}$ corresponding to the zero eigenvalue which has algebraic and geometric multiplicity $\zeta\in\{0,\ 1,\ldots,\ n-1\}$. Clearly, if $W^{\mathcal{BC}}$ has only a trivial kernel then the method converges to the zero vector. On the other hand, if we enforce a stopping criterion, the techniques gets approximated IMFs after finitely many steps.
Theorem \ref{teo:2} generalizes theoretical results provided in \cite{cicone2017numerical} only for the case of Periodic BCs.
\begin{theorem}
\label{teo:2}
Let $\mathbf{s} \in {\mathbb{R}}^n$ be the signal that has to be decomposed.
Let $\mathbf{v}$ be a symmetric decreasing filter of length $0 < l' \leq \lfloor \frac{n-1}{4} \rfloor$
and $\mathbf{w} = \mathbf{v} \ast \mathbf{v}$ be another symmetric decreasing filter of length $0 < l \leq \lfloor \frac{n-1}{2} \rfloor$
defined by making the convolution of $\mathbf{v}$ with itself.
Let $W^{\mathcal{BC}} \in {\mathbb{R}}^{n \times n}$ be the matrix constructed from $\mathbf{w}$ and some BCs (Periodic or Reflective or Anti-Reflective),
which can be diagonalized by $Q^{\mathcal{BC}} $,
and $\zeta$ be the number (in the set $\{0,\ 1,\ldots,\ n-1\}$) of its zero eigenvalues.
Let $\alpha^{\mathcal{BC}}$ and $\beta^{\mathcal{BC}}$ be two constants depending on BCs at hand ($\alpha^{\mathcal{P}}=\alpha^{\mathcal{R}}=1$, $\alpha^{\mathcal{AR}}=3$ and $\beta^{\mathcal{P}}=\beta^{\mathcal{R}}=1$, $\beta^{\mathcal{AR}}=2$).
Then, at step $k$ of the inner loop in the DIF method, the first IMF is given by
\begin{equation*}
\mathbf{f}_1 = Q^{\mathcal{BC}} (Z^{\mathcal{BC}} )^k (Q^{\mathcal{BC}})^{-1} \mathbf{s},
\end{equation*}
where $Z^{\mathcal{BC}} = I - D^{\mathcal{BC}}$ can be rewritten in this way
\begin{equation*}
Z^{\mathcal{BC}} = P \left(
\begin{array}{ccccccc}
1-\lambda^{\mathcal{BC}}_1 & & & & & & \\
& 1-\lambda^{\mathcal{BC}}_2 & & & & & \\
& & \ddots & & & & \\
& & & 1-\lambda^{\mathcal{BC}}_{n-\zeta} & & & \\
& & & & 1 & & \\
& & & & & \ddots & \\
& & & & & & 1 \\
\end{array}
\right) P^T
\end{equation*}
by means of a suitable permutation matrix $P$, and the first $\beta^{\mathcal{BC}}$ elements that appear on the diagonal are equal to zero.
Letting $k$ go to infinity, the first outer loop step of the DIF method converges to
\begin{equation*}
\mathbf{f}_1 = Q^{\mathcal{BC}} Z_{\infty} (Q^{\mathcal{BC}})^{-1} \mathbf{s},
\end{equation*}
where $Z_{\infty}$
is a diagonal matrix with entries all zero,
except $\zeta$ diagonal elements equal to one.
Moreover,
fixed $\delta>0$,
for the minimum $k_0\in{\mathbb{N}}$ such that it holds true the inequality
\begin{equation*}
\frac{k_0^{k_0}}{\left(k_0+1\right)^{k_0+1}}<\frac{\delta}{\alpha^{\mathcal{BC}} \|(Q^{\mathcal{BC}})^{-1} \mathbf{s}\|_\infty{\sqrt{n-\beta^{\mathcal{BC}}-\zeta}}},
\end{equation*}
we have that the following stopping criterion is satisfied
\begin{equation*}
\left\| \mathbf{s}_{k+1}^1-\mathbf{s}_k^1\right\|_{2}<\delta, \hspace{0.5cm} \forall k\geq k_0,
\end{equation*}
so the DIF algorithm takes finitely many steps to produce the first IMF.
\end{theorem}
\begin{proof}
The first part of the theorem follows from the definition of DIF method in matrix form and from Lemma \ref{lemma:2}.
In fact,
at step $k$ of the inner loop in the DIF method, the first IMF is given by
\begin{equation*}
\mathbf{f}_1 = (I-W^{\mathcal{BC}})^k \mathbf{s} = Q^{\mathcal{BC}} (Z^{\mathcal{BC}} )^k (Q^{\mathcal{BC}})^{-1} \mathbf{s},
\end{equation*}
Moreover, by Theorem \ref{teo:1} we know that $W^{\mathcal{BC}}$ has $\beta^{\mathcal{BC}}$ eigenvalues equal to 1.
On the other hand, letting $k$ go to infinity, we get
\begin{equation*}
\mathbf{f}_1=\lim_{k\rightarrow \infty} Q^{\mathcal{BC}} (Z^{\mathcal{BC}} )^k (Q^{\mathcal{BC}})^{-1} \mathbf{s}=
Q^{\mathcal{BC}} Z_{\infty} (Q^{\mathcal{BC}})^{-1} \mathbf{s}.
\end{equation*}
The second part of the theorem follows from the next expression
\small
\begin{eqnarray*}
\|\mathbf{s}_{k+1}^1 - \mathbf{s}_k^1\|_2 &=& \|(I-W^{\mathcal{BC}})^{k+1}\mathbf{s} - (I-W^{\mathcal{BC}})^{k} \mathbf{s}\|_2 \\
&=&\| Q^{\mathcal{BC}} (I-D^{\mathcal{BC}})^{k+1} (Q^{\mathcal{BC}})^{-1} \mathbf{s} - Q^{\mathcal{BC}} (I-D^{\mathcal{BC}})^k (Q^{\mathcal{BC}})^{-1} \mathbf{s} \|_2 \\
&=& \| Q^{\mathcal{BC}} (Z^{\mathcal{BC}})^k ((I-D^{\mathcal{BC}})-I) (Q^{\mathcal{BC}})^{-1} \mathbf{s} \|_2 \\
&=& \| Q^{\mathcal{BC}} (Z^{\mathcal{BC}})^k D^{\mathcal{BC}} (Q^{\mathcal{BC}})^{-1} \mathbf{s} \|_2 \\
&\leq& \alpha^{\mathcal{BC}} \| (Z^{\mathcal{BC}})^k D^{\mathcal{BC}} (Q^{\mathcal{BC}})^{-1} \mathbf{s} \|_2 \\
&\leq& \alpha^{\mathcal{BC}} \left\|P \left(
\begin{array}{cccccccc}
(1-\lambda^{\mathcal{BC}}_1 )^k \lambda^{\mathcal{BC}}_1 & & & & & & \\
& \ddots & & & & \\
& & (1-\lambda^{\mathcal{BC}}_{n-\zeta} )^k \lambda^{\mathcal{BC}}_{n-\zeta} & & & \\
& & & 0 & & \\
& & & & \ddots & \\
& & & & & 0 \\
\end{array}
\right) P^T\left(
\begin{array}{c}
\| (Q^{\mathcal{BC}})^{-1} \mathbf{s} \|_\infty \\
\vdots \\
\| (Q^{\mathcal{BC}})^{-1} \mathbf{s} \|_\infty \\
\end{array}
\right)
\right\|_2 \\
&\leq & \alpha^{\mathcal{BC}} {\sqrt{n-\beta^{\mathcal{BC}} -\zeta}} \left(1-\frac{1}{k+1} \right)^k \frac{1}{k+1} \| (Q^{\mathcal{BC}})^{-1} \mathbf{s} \|_\infty
\end{eqnarray*}
\normalsize
where $P$ is a suitable permutation matrix and $\alpha^{\mathcal{BC}}$ depends on the value of $\| Q^{\mathcal{BC}} \|_2$.
So it is equal to $1$ for Periodic and Reflective BCs, since $Q^{\mathcal{P}}$ and $Q^{\mathcal{R}}$ are unitary matrices,
while in the Anti-Reflective case it is equal to $3$ because we can rewrite $Q^{\mathcal{AR}}$
-- see (\ref{eq:ART}) --
as the sum of three matrices,
having non-zero elements only in the first column, in the last column and in the central part
(corresponding to $\hat{Q}^{\mathcal{AR}}$, which is unitary),
then we can apply triangle inequality, exploiting also the fact that the Euclidean norm of each splitting matrix is equal to $1$.
For writing the last inequality, we use the fact that the function $(1-\lambda)^k \lambda$ achieves its maximum at $\lambda=\frac{1}{k+1}$ for $\lambda\in[0,\ 1]$.
Hence the stopping criterion is fulfilled for $k_0$ minimum natural number such that
$\frac{k_0^{k_0}}{\left(k_0+1\right)^{k_0+1}}<\frac{\delta}{\alpha^{\mathcal{BC}} \|(Q^{\mathcal{BC}})^{-1} \mathbf{s}\|_\infty{\sqrt{n-\beta^{\mathcal{BC}}-\zeta}}}$.
\end{proof}
We make here the observation that with the current version of the algorithm we have to reimpose the BCs at every iteration of the inner loop.
In order to reduce the error and its propagation, the idea is to impose the BCs only in the first step of each inner loop, and then let the solution to evolve freely.
This is the reason for introducing the extended approach.
\section{Extended Iterative Filtering}
\label{sec:Error}
Now, once defined the matrix
\begin{equation}
R = \left(\ O_{n\times p}\; I_{n\times n}\; O_{n\times p}\ \right)_{n\times (n+2p)},
\end{equation}
where $O$ is matrix of all zeros and $I$ is the identity matrix, in Algorithm \ref{algo:IF_discrete} we present the pseudocode of Extended Iterative Filtering (EIF) algorithm.
Differently from the DIF method,
it is not based (as DIF algorithm) on a structured matrix $W_{m}^{\mathcal{BC}}\in {\mathbb{R}}^{n\times n}$
(whose structure depends on BCs),
associated with the original signal $\mathbf{s} \in {\mathbb{R}}$,
but on a Circulant matrix $\ddot{W}_m^{\mathcal{P}} \in {\mathbb{R}}^{(n+2p) \times (n+2p)}$
(whose structure does not depend on BCs),
associated with a signal $\mathbf{s}^{\mathcal{BC}} \in {\mathbb{R}}^{(n+2p)}$
extended by means of any BCs.
Hence, since in this framework the BCs do not affect the matrix structure, we have more freedom in their choice.
Any method that extends the original signal in a suitable way outside the boundaries can be employed now.
Moreover, we notice that in Algorithm \ref{algo:EIF} there is only one step involving the use of BCs,
while in Algorithm \ref{algo:IF_discrete} the BCs are employed at every step of the Inner Loop.
Considering that every time we make use of BCs we are adding errors to the decomposition method,
we think that the proposed approach may improve the standard one.
Furthermore, we remark that employing Periodic BCs allows to reduce the computational burden of IF.
In fact the algorithm can be rewritten using Fast Fourier Transform (FFT)
to become what is known as Fast Iterative Filtering (FIF) \cite{cicone2017numerical}.
\begin{algorithm}
\caption{\textbf{Extended Iterative Filtering} IMFs = EIF$(s, h)$}\label{algo:EIF}
\begin{algorithmic}
\STATE $\mathbf{s}^{\mathcal{BC}}(x_i) = \mathbf{s}(x_i)$, $i=0,\ldots,n-1$
\STATE apply BCs to compute $\mathbf{s}^{\mathcal{BC}}(x_i)$, $i=-p,\ldots,-1$ and $i=n,\ldots,n-1+p$
\STATE $m=1$
\STATE $\mathbf{s}_1^m = \mathbf{s}^{\mathcal{BC}}$
\WHILE{the number of extrema of $\mathbf{s}_1^m$ $\geq 2$}
\STATE $k=1$
\STATE compute the filter length $l_m$ for the signal $\mathbf{s}_k^m$
\STATE compute $\mathbf{w}_m$ (having $h$ and $l_m$)
\WHILE{the stopping criterion is not satisfied}
\STATE $\mathbf{s}_{k+1}^m = (I-\ddot{W}_m^{\mathcal{P}}) \mathbf{s}_{k}^m$
\STATE $k = k+1$
\ENDWHILE
\STATE $\mathbf{f}_m = \mathbf{s}_{k}^m$
\STATE $\mathbf{s}_1^{m+1} = \mathbf{s}_1^{m}-\mathbf{f}_{m}$
\STATE $m = m+1$
\ENDWHILE
\STATE $\mathbf{f}_m = \mathbf{s}_1^m$
\STATE IMFs = $\{ R \mathbf{f}_1,\ldots, R \mathbf{f}_m \}$
\end{algorithmic}
\end{algorithm}
We denote by $\mathbf{f}_1\in {\mathbb{R}}^n$ the first approximated IMF computed by the EIF algorithm.
$\mathbf{\bar{f}}_1\in{\mathbb{R}}^n$ represents the first exact IMF, whereas $\mathbf{s}^{\mathcal{BC}}\in{\mathbb{R}}^{n+2p}$ the signal extended according to some a priori chosen boundary conditions and $\ddot{\mathbf{s}}\in{\mathbb{R}}^{n+2p}$ the unknown real signal extended also outside the boundaries.
Then we can split these last two vectors considering the portion of signal inside and outside the field of view identified by the boundaries, and get
\begin{equation}
\mathbf{s}^{\mathcal{BC}} = \mathbf{s}_{\textrm{in}}^{\mathcal{BC}} + \mathbf{s}_{\textrm{out}}^{\mathcal{BC}}
\end{equation}
and
\begin{equation}
\ddot{\mathbf{s}} = \ddot{\mathbf{s}}_{\textrm{in}} + \ddot{\mathbf{s}}_{\textrm{out}}.
\end{equation}
Clearly $\mathbf{s}_{\textrm{in}}^{\mathcal{BC}} = \ddot{\mathbf{s}}_{\textrm{in}}$.
We have that
\begin{eqnarray*}
\mathbf{f}_1 &=& R (I-\ddot{W}_1^{\mathcal{P}})^{k} \mathbf{s}^{\mathcal{BC}} \\
&=& R (I-\ddot{W}_1^{\mathcal{P}})^{k} (\mathbf{s}_{\textrm{in}}^{\mathcal{BC}} + \mathbf{s}_{\textrm{out}}^{\mathcal{BC}} + \ddot{\mathbf{s}}_{\textrm{out}}-\ddot{\mathbf{s}}_{\textrm{out}}) \\
&=& R (I-\ddot{W}_1^{\mathcal{P}})^{k} \ddot{\mathbf{s}} + R (I-\ddot{W}_1^{\mathcal{P}})^{k} (\mathbf{s}_{\textrm{out}}^{\mathcal{BC}} - \ddot{\mathbf{s}}_{\textrm{out}})
\end{eqnarray*}
where the first term tends to $\mathbf{\bar{f}}_1$, while the second term it propagates itself from the boundaries inside the field of view as $k$ grows.
In general it is not possible to estimate the difference $\mathbf{s}_{\textrm{out}}^{\mathcal{BC}} - \ddot{\mathbf{s}}_{\textrm{out}}$.
However it is reasonable to assume that the approximation of the real signal outside the boundaries $\mathbf{s}_{\textrm{out}}^{\mathcal{BC}}$ produces an error $\mathbf{s}_{\textrm{out}}^{\mathcal{BC}} - \ddot{\mathbf{s}}_{\textrm{out}}$ which is a constant function. In order to estimate such error function we can assume that its value equals $\chi=\max\left(\left|\mathbf{s}_{\textrm{in}}^{\mathcal{BC}}\right|\right) = \max\left(\left|\ddot{\mathbf{s}}_{\textrm{in}}\right|\right)$. This is clearly in most cases an overestimation.
Now the question is how the error $R (I-\ddot{W}_1^{\mathcal{P}})^{k} (\mathbf{s}_{\textrm{out}}^{\mathcal{BC}} - \ddot{\mathbf{s}}_{\textrm{out}})$ propagates inside the field of view during the iterations.
We can estimate an upper bound for the error inside the field of view at step $k$ of the iterations using the following formula
\begin{equation}\label{eq:err_UB}
\textrm{err}_k = R (I-\ddot{W}_1^{\mathcal{P}})^{k} \mathbf{u} = R \left(I - {{k}\choose{1}}\ddot{W}_1^{\mathcal{P}} + {{k}\choose{2}}\left(\ddot{W}_1^{\mathcal{P}}\right)^2 +\ldots (-1)^{k} \left(\ddot{W}_1^{\mathcal{P}}\right)^k \right) \mathbf{u},
\end{equation}
where $\mathbf{u}\in{\mathbb{R}}^{n+2p}$ is a vector equal to $\chi$ outside the field of view and $0$ inside.
A potential upper bound of the error for each point $x_i$ inside the field of view can be computed as
\begin{equation}\label{eq:err_UB2}
\textrm{ub}_k (x_i) = \max_{j\leq k}\left(\left|\textrm{err}_j(x_i)\right|\right),
\end{equation}
where $k$ is the number of iterations.
This is just an estimate because in general the actual error introduced by the extension of the signal outside the boundaries is unknown.
If we deal with artificial examples,
and we consider DIF algorithm employing some BCs,
we can compute the actual error using the formula
\begin{equation}\label{eq:err}
\textrm{err}^{\mathcal{BC}}_k (x_i) = \left| \mathbf{f}_1 (x_i) - \mathbf{\bar{f}}_1 (x_i) \right|,
\end{equation}
where $k$ is the number of iterations required to produce the approximated IMF $\mathbf{f}_1$,
while $\mathbf{\bar{f}}_1$ is the first exact IMF.
\FloatBarrier
\section{Numerical results}
\label{sec:Examples}
In order to show how DIF method works when we employ different BCs,
we present four simple examples in which the signal is artificially created through one IMF plus a trend.
In particular, we analyze how the errors introduced outside the boundaries propagate inside the field of view, evaluating the performance of formula \eqref{eq:err_UB} for the a priori estimation of the error when there is no knowledge on the behavior of the signal outside the boundaries.
\subsection{Example 1}
We consider a first signal, shown on the left of Figure \ref{fig:Test_ex_1},
which can safely extended periodically and anti-reflectively outside the boundaries.
On the right of Figure \ref{fig:Test_ex_1}
we can look at decomposition obtained after 3 steps of DIF method with Reflective BCs.
If we apply the DIF algorithm using Periodic, Reflective and Anti-Reflective BCs,
we can extract a first IMF and measure the difference, after a fixed number of steps,
between the exact IMF and the computed IMF measured in absolute.
In Figure \ref{fig:Test_ex_1b} we plot the errors introduced by each extension,
formula \eqref{eq:err},
and we compare them with the a priori estimate of the error computed using formula \eqref{eq:err_UB2}.
Due to a symmetry in the signal, it is sufficient to plot the errors for the left half of the field of view.
As expected,
we produce a minimal error in the decomposition if we extended the signal periodically or anti-reflectively. Furthermore,
the a priori upper bound estimate of the error proves to be correct.
\begin{figure}[h!]
\centering
\includegraphics[width=0.48\textwidth]{ex1fig1}
\includegraphics[width=0.48\textwidth]{ex1fig2}
\caption{(Example 1) Left panel: signal. Right panel: decomposition computed by DIF method with Reflective BCs (black line) compared with exact one (red line).}\label{fig:Test_ex_1}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.48\textwidth]{ex1fig3}
\includegraphics[width=0.48\textwidth]{ex1fig4}
\caption{(Example 1) Upper bound and errors measured in absolute value for Periodic, Reflective and Anti-Reflective BCs relative to 3 iterations (on the left) and 9 iterations (on the right) of DIF method.}\label{fig:Test_ex_1b}
\end{figure}
\FloatBarrier
\subsection{Example 2}
In this second example we consider the signal plotted in the left panel of Figure \ref{fig:Test_ex_2},
while in the right panel
we can look at decomposition obtained after 3 steps of DIF method with Anti-Reflective BCs.
After extending the signal using Periodic, Reflective and Anti-Reflective BCs,
we extract a first IMF using the DIF method.
In Figure \ref{fig:Test_ex_2b} we compare the absolute values of the errors introduced by the different BCs, formula \eqref{eq:err}, and we compute the upper bound by means of \eqref{eq:err_UB2}.
Again, due to a symmetry in the curves, we report only the first half of the error plot.
In this case the reflective extension works best in minimizing the error in the decomposition.
Furthermore, the upper bound on the error allows to correctly estimate the error everywhere inside the field of view.
\begin{figure}[h!]
\centering
\includegraphics[width=0.48\textwidth]{ex2fig1}
\includegraphics[width=0.48\textwidth]{ex2fig2}
\caption{(Example 2) Left panel: signal. Right panel: decomposition computed by DIF method with Anti-Reflective BCs (black line) compared with exact one (red line).}\label{fig:Test_ex_2}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.48\textwidth]{ex2fig3}
\includegraphics[width=0.48\textwidth]{ex2fig4}
\caption{(Example 2) Upper bound and errors measured in absolute value for Periodic, Reflective and Anti-Reflective BCs relative to 3 iterations (on the left) and 9 iterations (on the right) of DIF method.}\label{fig:Test_ex_2b}
\end{figure}
\FloatBarrier
\subsection{Example 3}
In this case the signal, shown in the left panel of Figure \ref{fig:Test_ex_3}, is best extended outside the boundaries using Anti-Reflective BCs.
This fact is confirmed by the error curves \eqref{eq:err} plotted in Figure \ref{fig:Test_ex_3b}.
Due to symmetry properties of the signal under study, we report only the first half of the error curves.
Also in this example the upper bound allows to give a priori estimate of the behavior relative to the error propagation inside the field of view.
\begin{figure}[h!]
\centering
\includegraphics[width=0.48\textwidth]{ex3fig1}
\includegraphics[width=0.48\textwidth]{ex3fig2}
\caption{(Example 3)Left panel: signal. Right panel: decomposition computed by DIF method with Periodic BCs (black line) compared with exact one (red line).}\label{fig:Test_ex_3}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.48\textwidth]{ex3fig3}
\includegraphics[width=0.48\textwidth]{ex3fig4}
\caption{(Example 3)Upper bound and errors measured in absolute value for Periodic, Reflective and Anti-Reflective BCs relative to 3 iterations (on the left) and 9 iterations (on the right) of DIF method.}\label{fig:Test_ex_3b}
\end{figure}
\FloatBarrier
\subsection{Example 4}
In this example we want to study how the errors induced by the extension outside the boundaries depends on the phase of the signal at the boundary. We consider a simple signal, plotted in the left panel of Figure \ref{fig:Test_ex_4}, which is a superposition of a constant trend and a plain sine. The signal support is originally given by the interval $[-333,\ 0]$ which is then extended up to 300 step by step of $\triangle t = 0.01$. Each time we enlarge the support of a $\triangle t$ we redecompose the newly extended signal using DIF with different kind of BCs. For each BCs we compute the relative error
\begin{equation}\label{eq:rel_err}
\textrm{err}^{\mathcal{BC}}_{\textrm{rel}} = \frac{\left\|\mathbf{f}_1 - \mathbf{\bar{f}}_1 \right\|_\infty}{\left\|\mathbf{\bar{f}}_1\right\|_\infty}
\end{equation}
and the relative error upper bound
\begin{equation}\label{eq:rel_err_UB}
\textrm{ub}_{\textrm{rel}} = \frac{\left\|\textrm{ub}_k\right\|_\infty}{\left\|\mathbf{\bar{f}}_1 \right\|_\infty}
\end{equation}
where $\mathbf{\bar{f}}_1$ represents the first exact IMF and $\textrm{ub}_k$ is the error upper bound computed using \eqref{eq:err_UB2} for a fixed number of iterations $k$.
We plot the relative errors in the right panel of Figure \ref{fig:Test_ex_4}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.48\textwidth]{ex4fig1}
\includegraphics[width=0.48\textwidth]{ex4fig2}
\caption{(Example 4) Left panel: signal. Right panel: upper bound, relative errors for different BCs and best extension.}\label{fig:Test_ex_4}
\end{figure}
As expected such curves have the same periodicity as the given signal. Furthermore the relative error upper bound a priori estimate proves to be valid also in this example. Finally this study allows to identify which kind of extension is performing best from a relative error point of view for each value of the phase of the signal at the boundary, dashed curve the right panel of Figure \ref{fig:Test_ex_4}.
\FloatBarrier
\section{Conclusions}\label{sec:Conclusions}
We considered the decomposition of non-stationary signals,
which is a problem of great interest from the theoretical point of view
and has important applications in many different fields.
For instance, it occurs in the identification of hidden periodicities and trends in time series relative to natural phenomena
(like average troposphere temperature)
and economic dynamics
(like financial indices).
Since standard techniques like Fourier or Wavelet Transform are unable to properly capture non-stationary phenomena,
in the last years several ad hoc methods have been proposed in the literature.
Such techniques provide iterative procedures for decomposing a signal into a finite number of simple components.
In this paper we focused on investigating IF algorithm employing different BCs
(Periodic, Reflective and Anti-Reflective BCs),
which give rise to different matrix structures.
We analyzed spectral properties of these matrices and convergence properties of IF method.
We also presented an extended version of IF, in which any BCs can be employed;
this allows to estimate the error propagation from the boundary towards the internal part of the signal.
Numerical experiments show that a suitable choice of BCs is able to improve in a meaningful way the quality
of signal decomposition in IMFs computed by IF method.
We think that this paper can open the way for further interesting developments,
for instance the study of adaptive choice of BCs, basing on the signal at hand,
and the proposal of new accurate BCs,
as done in \cite{pietro2017} for image restoration problem.
\section*{Acknowledgments}
This work was supported by the Istituto Nazionale di Alta Matematica (INdAM) ``INdAM Fellowships in Mathematics and/or Applications cofunded by Marie Curie Actions'', FP7--PEOPLE--2012--COFUND, Grant agreement n. PCOFUND--GA--2012--600198.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,666
|
Irish babies born on January 1st expected to live to 105 years of age – UN
Sorcha Pollak
Irish babies born on January 1st, 2021 are expected to live to 105 years of age, according to Unicef estimates published this morning. The 157 babies(...)
The Offload: No Eddie, rugby is not a safer sport
Gavin Cummiskey
Cautionary tale Junior Seau was the epitome of an NFL linebacker. A 12-time Pro Bowler for the Chargers, the Dolphins and the Patriots, Seau is the c(...)
Young hurler wants to emulate his Waterford heroes
Micheál Ó Muircheartaigh's old line about Seán Óg Ó hAilpín's parentage is a favourite of hurling fans. It still holds true too. Whether coming from (...)
TV Honan: If silence really is golden Waterford will mine nuggets on Sunday
As a giddy mini-bus from McLaughlin's bar crossed the Blackwater Bridge at Youghal, en route to the 2002 Munster final at Páirc Uí Chaoimh, Larry Goga(...)
Autumn Nations Cup: France v Fiji cancelled due to Covid
The opening weekend of the Autumn Nations Cup has been thrown into chaos with Sunday's match between France and Fiji called off after a further four p(...)
Gerry Thornley: Ruddock hasn't enjoyed rub of the green in Ireland career
Gerry Thornley
Back in the summer of 2010, a 19-year-old with a prodigious build and talent was rerouted from the IRB Under-20 World Championship in Argentina, where(...)
Gordon D'Arcy: Losing such a big game shines a very harsh light on certain players
Gordon D'Arcy
September 23, 2020, 05:00
We're putting on the hindsight goggles again. Leinster used the World Cup analogy coming out of lockdown. The theory was sound. Pool matches were the(...)
Dan McFarland needs to shake off Ulster's cobwebs
September 5, 2020, 06:00
There are several ways to interpret Ulster's awful restart. They appear to be mired in pre-season mode. At least the halfbacks cannot mess up the fund(...)
The Irishman who survived the most ruthless adventure race on the planet
"We didn't know what we were letting ourselves in for. We knew it was going to be pretty intense, but nothing prepared us for what unfolded." Ultra en(...)
Gerry Thornley: Players face gruelling 12 months of non-stop rugby
After the longest break comes the longest season, touch wood. Or, put another way, never mind the quality, just feel the quantity, as rugby's various (...)
Gunmen kill two female supreme court judges in Afghanistan 15:17
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,006
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Sascha Schmitz (Soest, 5 de janeiro de 1972), popularmente conhecido como Sasha, é um cantor e compositor alemão. De 2003 até 2005 se apresentou com o alter ego de Dick Brave, líder da banda de rockabilly Dick Brave & The Backbeats. É o mais velho de dois filhos. Cresceu dividido entre os dois pais divorciados.
Carreira
1996—2002
Com o passar dos anos, Sasha juntou dinheiro cantando como voz de fundo de vários artistas.
Em 1998, Schmitz assinou um contrato solo com a gravadora Warner Music. Ele trabalhou com Grant, Di Lorenzo and Pete Smith na gravação do seu álbum Dedicated to... que foi lançado em 1998.
2003—2004
Em 2003, Schmitz começou a se apresentar como o alter ego de "Dick Brave", o vocalista de uma banda de rockabilly chamada Dick Brave & The Backbeats, cujos membros compartilham uma história em uma banda fictícia.
Inspirados em Nick Cave and the Bad Seeds e originalmente concebido como uma brincadeira (Schmitz e seus colegas músicos André "Adriano Batolba" Tolba, Maik "Mike Scott" Schott, Felix "Phil X Hanson" Wiegand e Martell "Matt L. Hanson" realmente "agia" como Dick no palco e durante as entrevistas), o quinteto gravou um álbum inteiro juntos, incluindo versões covers de canções como "Get the Party Started" de Pink, "Freedom" de George Michael, e "Black or White" de Michael Jackson.
Em novembro de 2004, o projeto foi interrompido, após um concerto final em Westfalenhalle Dortmunder em 22 de Novembro de 2004.
Em 7 de janeiro de 2006, a banda novamente formada temporariamente para realizar a festa de casamento de Pink e o piloto de motocross Carey Hart na Costa Rica.
2005—Atualmente
Após um hiato de dois anos, em 2006 lançou o álbum Open Water. nesse álbum, Sasha trabalhou com compositores de vários países Suécia, Inglaterra e Alemanha, entre outros, tais como Fabio Trentini (membro do H-Blockx) e Robin Grubert . Em Dezembro de 2006, lançou o álbum Greatest Hits. Em 2009 lançou seu quinto álbum de estúdio, Good News on a Bad Day obtendo as melhores vendas desde "...You". Em 2011 ele retornou ao seu alter ego de Dick Brave e lançou "Rock'n'Roll Therapy".
Discografia
Álbuns de estúdio
Dedicated to... (1998)
...You (2000)
Surfin' on a Backbeat (2001)
Open Water (2006)
Good News on a Bad Day (2009)
The One (2014)
Compilações
Greatest Hits (2006)
Como Dick Brave and Backbeats
Dick This (2003)
Rock and Roll Therapy (2011)
Filmografia
Goldene Zeiten (2006) — convidado #3
Warum Männer nicht zuhören und Frauen schlecht einparken (2007) — Sven
Ossi's Eleven'' (2008) — Tommy Beck
Ligações externas
de Dick Brave
Cantores da Alemanha
Compositores da Alemanha
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,991
|
Ernst Wilhelm Baader (* 14. Mai 1892 in Berlin; † 1. November 1962 in Hamm) war ein deutscher Arzt und Arbeitsmediziner (Sozialhygieniker bzw. Gewerbehygieniker) sowie Gerichtsmediziner.
Baader studierte Medizin und promovierte 1918 in Berlin über Die Arsentherapie der Syphilis bis zur Salvarsanära. Er wurde 1934 zum Professor ernannt.
Baader begründete in Berlin die erste deutsche arbeitsmedizinische Klinik (klinische Abteilung für Gewerbekrankheiten im Kaiserin-Auguste-Viktoria-Krankenhaus in Berlin-Lichtenberg) und leitete sie von 1925 bis 1945. Er leistete grundlegende Beiträge zur Entwicklung der arbeitsmedizinischen Fachdisziplin in Deutschland.
Baader trat zum 1. Mai 1933 der NSDAP bei (Mitgliedsnummer 2.672.879). Ab 1933 war er als Nachfolger eines aus dem Amt entfernten jüdischen Kollegen ärztlicher Direktor am Städtischen Krankenhaus Berlin-Neukölln.
Nach dem Zweiten Weltkrieg war er von 1945 bis 1955 Leiter des von ihm gegründeten Knappschaftskrankenhauses in Hamm. Zusätzlich wurde er 1951 Honorarprofessor an der Universität Münster. Er war Vorsitzender der Deutschen Gesellschaft für Rheumatologie.
Die Deutsche Gesellschaft für Arbeitsmedizin und Umweltmedizin verleiht den nach ihm benannten E.-W.-Baader-Preis.
Literatur
Gine Elsner: Schattenseiten einer Arztkarriere. Ernst Wilhelm Baader (1892-1962): Gewerbehygieniker und Gerichtsmediziner. VSA-Verlag, Hamburg 2011, ISBN 978-3-89965-466-0.
Philipp Rauh, Karl-Heinz Leven: Ernst Wilhelm Baader (1892-1962) und die Arbeitsmedizin im Nationalsozialismus. Peter Lang, Frankfurt 2013 ISBN 978-3-631-64327-3
Weblinks
Britta Barlage: "'Entlarvung von Simulanten'", in: die tageszeitung vom 7. April 2005
Einzelnachweise
Mediziner (20. Jahrhundert)
Hochschullehrer (Humboldt-Universität zu Berlin)
Hochschullehrer (Westfälische Wilhelms-Universität)
NSDAP-Mitglied
Deutscher
Geboren 1892
Gestorben 1962
Mann
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,912
|
Q: MySQL Row counter in Update statement The following MySQL statement is working fine, and it returns me the rownumber as row, of each result. But now, what I want to do, is setting the column pos with the value of "row", by using an update statement, since I don't want to loop thousands of records with single queries.
Any ideas?
SELECT @row := @row + 1 AS row, u.ID,u.pos
FROM user u, (SELECT @row := 0) r
WHERE u.year<=2010
ORDER BY u.pos ASC LIMIT 0,10000
A: There is a risk using user defined variables
In a SELECT statement, each select expression is evaluated only when sent to the client. This means that in a HAVING, GROUP BY, or ORDER BY clause, referring to a variable that is assigned a value in the select expression list does not work as expected:
A more safe guard method will be
create table tmp_table
(
pos int(10) unsigned not null auto_increment,
user_id int(10) not null default 0,
primary key (pos)
);
insert into tmp_table
select null, u.ID
from user
where u.year<=2010
order by YOUR_ORDERING_DECISION
limit 0, 10000;
alter table tmp_table add index (user_id);
update user, tmp_table
set user.pos=tmp_table.pos
where user.id=tmp_table.user_id;
drop table tmp_table;
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,658
|
"""
Print the Mensaplan of the university ulm in a fancy cli way
"""
import sys
import urllib.request
import json
import datetime
FILES = [
"http://www.uni-ulm.de/mensaplan/data/mensaplan.json",
"http://www.uni-ulm.de/mensaplan/data/mensaplan_static.json"
]
def print_usage():
"""Print the help text"""
print("Usage:")
usage = """
uulm-mensa place
Print the todays menu at the place.
uulm-mensa place [mon, tue, wed, thur, fri]
Print the menu at the place of the given weekday.
Supported places are:
Mensa University: mensa
Bistro: bistro
Burgerbar Southside: burgerbar
CafeteriaB: cafeteriab
Cafeteria West: west
West Side Diner: westside
Mensa Hochschule: hochschule
"""
print(usage)
print("mmk2410 (c) 2015 MIT License")
def get(url):
"""Recieving the JSON file from uulm"""
response = urllib.request.urlopen(url)
data = response.read()
data = data.decode("utf-8")
data = json.loads(data)
return data
def get_day():
"""Function for retrieving the wanted day"""
day = datetime.datetime.today().weekday()
if len(sys.argv) == 3:
if sys.argv[2] == "mon":
day = 0
elif sys.argv[2] == "tue":
day = 1
elif sys.argv[2] == "wed":
day = 2
elif sys.argv[2] == "thur":
day = 3
elif sys.argv[2] == "fri":
day = 4
else:
day = 5
if day > 4:
print("There is no information about the menu today.")
exit(5)
return day
def print_menu(place, static=False):
"""Function for printing the menu
Keyword arguments:
place -- name of the cafeteria / mensa
static -- set true if a static menu exists (default: False)
"""
day = get_day()
if static:
plan = get(FILES[1])
for meal in plan["weeks"][0]["days"][day][place]["meals"]:
if place == "Diner":
print(meal["category"] + " " + meal["meal"])
else:
print(meal["category"] + ": " + meal["meal"])
else:
plan = get(FILES[0])
for meal in plan["weeks"][0]["days"][day][place]["meals"]:
print(meal["category"] + ": " + meal["meal"])
def main():
"""Entry point for the application script"""
if len(sys.argv) >= 2:
cmd = sys.argv[1]
if cmd == "help":
print_usage()
else:
if cmd == "mensa":
print_menu("Mensa")
elif cmd == "bistro":
print_menu("Bistro")
elif cmd == "cafeteriab":
print_menu("CB")
elif cmd == "west":
print_menu("West")
elif cmd == "hochschule":
print_menu("Prittwitzstr")
elif cmd == "westside":
print_menu("Diner", True)
elif cmd == "burgerbar":
print_menu("Burgerbar", True)
else:
print("[ERROR]: No valid place given")
print_usage()
else:
print("[ERROR]: No argument given")
print_usage()
if __name__ == "__main__":
main()
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,199
|
{"url":"http:\/\/bootmath.com\/show-2m-12n11-if-m-is-odd.html","text":"# Show $(2^m-1,2^n+1)=1$ if $m$ is odd\n\nLet $(2^m-1,2^n+1)=1$ and suppose $m$ is even. Also, $m,n,k\\in \\mathbb{N}$. We have\n$$(2^{m}-1)x+(2^n+1)y=1$$\n$$(2^{2k}-1)x+(2^n+1)y=1$$\n$$(2^{2k}-1)y\\equiv 1\\pmod{2^n+1}$$\n$$(2^k+1)(2^k-1)y\\equiv 1\\pmod{2^n+1}$$\nI don\u2019t konw now to procede from here\u2026.\n\n#### Solutions Collecting From Web of \"Show $(2^m-1,2^n+1)=1$ if $m$ is odd\"\n\n$\\overbrace{p\\mid\\color{#0a0}{2^{\\large m}\\!-\\!1}}^{\\color{#08f}{\\Large\\Rightarrow\\,\\ p\\: \\rm\\ odd}},\\color{#c00}{2^{\\large n}\\!+\\!1}\\,$\n$\\Rightarrow\\, {\\rm mod}\\ p\\!:\\ \\color{#0a0}{2^{\\large m}\\! \\equiv}\\!\\!\\!\\!\\!\\!\\!\\!\\overset{\\Large\\ [\\,\\color{#c00}{ -1\\ \\equiv\\ 2^{\\LARGE n}}]^{\\LARGE\\color{#90f}2}\\ \\ \\ \\ }{ 1\\color{#90f}{\\equiv {2^{\\large 2n}}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!$\n$\\Rightarrow\\, {\\rm ord}\\,2\\mid\\overbrace{(\\color{#0a0}m,\\color{#90f}{2n})= (\\color{#0a0}m,\\color{#90f}n)}^{\\Large\\rm\\ \\ by\\ \\color{#0a0}m\\ odd}\\,$\n$\\Rightarrow \\overbrace{\\color{#90f}{\\bf 1}\\equiv\\color{#c00}{2^{\\large\\color{#90f} n}\\!\\equiv -1}}^{\\color{#08f}{\\Large\\Rightarrow\\,\\ p\\ \\mid\\ 2\\quad\\:\\ }}$\n\nRemark $\\$ Above is special case $\\,m$ odd, $\\,k=2\\,$ of the following\n\nLemma $\\,\\ \\ a^{\\large m}\\equiv 1\\equiv a^{\\large kn},\\$ $(m,k) = 1\\,$ $\\Rightarrow\\, a^{\\large (m,n)}\\equiv 1\\,\\Rightarrow\\,a^{\\large n}\\equiv 1$\n\nProof $\\ \\ {\\rm ord}\\ a\\mid m,kn\\, \\Rightarrow\\, {\\rm ord}\\ a\\mid \\underbrace{(m,kn)\\!=\\!(m,n)}_{\\large (m,k)\\,=\\,1}\\mid n\\,\\Rightarrow\\, a^{\\large n}\\equiv 1$","date":"2018-07-22 20:16:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9306312799453735, \"perplexity\": 862.4379214895562}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676593586.54\/warc\/CC-MAIN-20180722194125-20180722214125-00176.warc.gz\"}"}
| null | null |
Florian Hartlieb (* 22. Dezember 1982 in Unna) ist ein deutscher Komponist und Medien-Künstler. Er lebt und arbeitet in Bochum, Deutschland.
Biographie
Florian Hartlieb studierte elektronische Komposition bei Thomas Neuhaus und Roman Pfeifer am ICEM der Folkwang Universität der Künste in Essen sowie bei Karlheinz Essl an der Universität für Musik und darstellende Kunst Wien.
Seine Stücke werden regelmäßig auf international renommierten Festivals und Konferenzen präsentiert, wie beispielsweise der Csound Konferenz (Berklee, 2013), der Linux Audio Conference (Stanford University, 2012), der International Computer Music Conference (Huddersfield, 2011), Tsonami (Buenos Aires, 2011), EMU Festival (Rom, 2010), Musicacoustica Beijing (Peking, 2008) und vielen mehr.
2009 wurde sein Werk "Im vorderen Zimmer des hinteren Raums" vom kanadischen Label CeC auf dem Cache 2009 Sampler veröffentlicht, 2013 erschien sein digitales Mehrkanal-Album Quadrosonic.
Neben seiner künstlerischen Tätigkeit ist Hartlieb Dozent für Klangsynthese, Sampling, Digitale Audiotechnik und Musiktheorie.
Werkverzeichnis (Auswahl)
"Verzweigung" -- 12–Kanal Fixed Media (2013)
"Zersplittert" -- 2–Kanal Fixed Media (2012)
"Caladan" -- 4–Kanal Fixed Media (2011)
"Ist Niemals Jetzt" -- 2 Stimmen, Laptop, Live-Elektronik, 4-Kanal Zuspielung
"Out of the Fridge" -- 2–Kanal Fixed Media (2011)
"Zu Spielen bei offenem Fenster" -- Piano, Live-Elektronik, 4-Kanal Zuspielung (2010)
"Zeitspiel" -- Audiovisuelle Installation (2010, Zusammenarbeit mit Jürgen Heinrich Stutzinger)
"Be here now" -- vier Instrumente, Live Elektronik und Video (2009)
"Im vorderen Zimmer des hinteren Raums" -- 4-Kanal Fixed Media (2009)
"Am I an Image" -- Stoptrickfilm (2009)
"An einem Sonntag" -- 2–Kanal Fixed Media
"Träum ich das ich wach bin oder schlaf ich nur im Stehen?" -- 4–Kanal Fixed Media (2008)
Auszeichnungen
2013 erster Preis des Kompositionswettbewerbs European ERASMUS Composition Competition
2013 Auszeichnung des Kompositionswettbewerbs "Atmosphären"
2009 erster Preis des Kompositionswettbewerbs Jeu de Temps/ Times Play
2006 Exzellenz Stipendium der Folkwang Universität der Künste
Weblinks
Homepage
Monday Sounds
Soundcloud Seite
Youtube-Kanal
Einzelnachweise
Komponist klassischer Musik (21. Jahrhundert)
Deutscher
Geboren 1982
Mann
Medienkünstler (Deutschland)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,701
|
Q: How to implement Client Credential Flow with Azure AD as Identity provider for Cognito We have an application hosted on AWS using Cognito Service, with Azure AD acting as the Identity provider. The solution works great with username and password, authenticating against the Cognito user pool.
We now need the above to work for service-to-service call scenario.
The unattended scheduled service will call another service (all hosted in AWS) but will need to authenticate with access token.
I was thinking of using client credential flow. But I could not find an approach which will work for the above scenario (i.e) client credential flow with Cognito using Azure AD as Identity provider.
When I go Azure AD Application Registration and view the endpoints, the oauth2/token endpoint shows up, but I have not figured out how this will work with Cognito.
Does any one know how to implement this?
A: In cognito if you use client credentials flow, there will not be any federated Identity provider involved. There will be no users so no need to use Azure AD to generate tokens. You will make the access request using Client Id and Client Secret and will be granted an access token that you can use.
https://aws.amazon.com/blogs/mobile/understanding-amazon-cognito-user-pool-oauth-2-0-grants/
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,903
|
FNaF World — компьютерная игра в жанре RPG от разработчика Скотта Коутона, являющиеся первым спин-оффом, к серии игр Five Nights at Freddy's. Игра вышла 21 января 2016 года для Windows и для Android и IOS.
Сюжет
Действие разворачивается в Стране Аниматроников, где живут главные герои. С помощью игрока они должны победить незванных гостей которые стали разрушать их окрестности. На пути у них стоят злобные монстры — Авто-Чиппер, Прыгун, Мальчик с бровями, Поркпатч, Бубба и Охранная система. Сюжет и концовка зависят от действий игрока. При прохождении сюжета, можно найти моменты, в которых игра преобразовывается в загадочную, а иногда и страшную бродилку. Таких моментов в игре много, однако их нужно найти, запустив цепь определённых событий. Игроку придётся навести порядок в стране, найти объекты с глюками и устранить их, зайдя в другое, альтернативное измерение. А также собрать все байты и чипы.
Игровой процесс
Вам предстоит создать собственную команду, используя огромный выбор персонажей из вселенной FNaF. По пути будут попадаться разные враги, с которыми нужно будет сражаться. Игра имеет два режима — Adventure () и Fixed Party ().
В adventure-режиме вы будете иметь возможность менять состав своей команд во время игры. В начале игры игрок может составлять команду только из персонажей из FNaF (кроме Золотого Фредди и Эндо-01) и FNaF 2 (только игрушечные аниматроники без Балун Боя, JJ и Марионетки). В Fixed Party выбора для составления команды больше (помимо вышеупомянутых можно взять в команду Балун Боя, JJ, любого из фантомов и Повреждённого Бонни), но потом не может менять состав команды до самого конца игры. Иногда, сразу после победы над врагами или побега от них, появляется сообщение «A NEW CHALLENGER HAS APPEARED», после чего начинается битва со случайным потенциально играбельным персонажем. В случае победы он присоединится к команде игрока.
Ссылки
Компьютерные ролевые игры
Компьютерные игры, разработанные в США
Компьютерные игры с альтернативными концовками
Компьютерные игры только с однопользовательским режимом игры
Five Nights at Freddy's
Игры для Windows
Компьютерные игры 2016 года
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{"url":"http:\/\/mathhelpforum.com\/algebra\/205420-dividing-both-sides-inequality-one-factor.html","text":"Thread: Dividing both sides of an inequality by one factor\n\n1. Dividing both sides of an inequality by one factor\n\nThis is a system of inequations for which I want to find the set of answers.\n\n$(x-1)(x+2)<(x+2)(x-3)$\n$(x+3)(x+5)>(x+4)(x+3)$\n\nWhat I CAN'T do here is the following:\n\nWe start with the first inequation $(x-1)(x+2)<(x+2)(x-3)$\n\n1) I divide both sides of the inequation by $(x+2)$\nand I get:\n\n$\\frac{(x-1)(x+2)}{(x+2)}<\\frac{(x+2)(x-3)}{(x+2)}$\n\n2) I cancel the $(x+2)$\nin top and bottom of both members of the inequality and I get:\n\n$(x-1)<(x-3)$\n\nIn other words:\n\n$x-1\n\nThen solving this inequality I get:\n\n3) $0<-2$\n\nAnd this is plainly wrong: Then my question is WHY, what's wrong with this procedure.\n\nI know the right one so I don't need the correct procedure, but instead what I'm looking for is an explanation about how come this is incorrect (not just saying because the answer is wrong, I know the correct answer) I thought that perhaps has to do with order of operations...\n\n2. Re: Dividing both sides of an inequality by one factor\n\nOriginally Posted by querti09\nThis is a system of inequations for which I want to find the set of answers.\n\n$(x-1)(x+2)<(x+2)(x-3)$\n$(x+3)(x+5)>(x+4)(x+3)$\n\nWhat I CAN'T do here is the following:\n\nWe start with the first inequation $(x-1)(x+2)<(x+2)(x-3)$\n\n1) I divide both sides of the inequation by $(x+2)$\nand I get:\n\n$\\frac{(x-1)(x+2)}{(x+2)}<\\frac{(x+2)(x-3)}{(x+2)}$\n\n2) I cancel the $(x+2)$\nin top and bottom of both members of the inequality and I get:\n\n$(x-1)<(x-3)$\n\nIn other words:\n\n$x-1\n\nThen solving this inequality I get:\n\n3) $0<-2$\n\nAnd this is plainly wrong: Then my question is WHY, what's wrong with this procedure.\n\nI know the right one so I don't need the correct procedure, but instead what I'm looking for is an explanation about how come this is incorrect (not just saying because the answer is wrong, I know the correct answer) I thought that perhaps has to do with order of operations...\nWhen you divide by the factor $(x+2)$ you don't know if the factor is positive or negative.\n\nIf it were negative you would need to flip over the inequality. I hope this helps.\n\n3. Re: Dividing both sides of an inequality by one factor\n\nOriginally Posted by TheEmptySet\nWhen you divide by the factor $(x+2)$ you don't know if the factor is positive or negative.\nIf it were negative you would need to flip over the inequality. I hope this helps.\nI believe that the coefficient must be positive, since making the \u201cinvisible\u201d coefficients show up, the original inequality\nwould be: $+1(x-1)(x+2)<+1(x+2)(x-3)$\nIf I'm putting well the \u201cinvisible\u201d coefficients then I need to divide by a positive 1 coefficient, and the inequality doesn't change the signs:\n$\\frac{+1(x-1)(x+2)}{+1(x+2)}<\\frac{+1(x+2)(x-3)}{+1(x+2)}$\nWhat I thought as a possibility is to think that this is just something that happens even when the manipulation with the quantities is being done well. I think about it as just a path of solving that doesn't solve the problem, a dead end. Something not to do if I want to get the values for $x$ certainly.\n\n4. Re: Dividing both sides of an inequality by one factor\n\nJust multiply everything and put all terms on the left. Both inequalities are in the form of f(x)<0 and g(x)>0, where f and g are quadratic functions. Graph of quadratic function is called parabola, and the inequality f(x)<0 can be translated to the question \"for which values of x is the parabola of the function f strictly below the abscissa\", and similarly g(x)>0 can be translated to the question \"for which values of x is the parabola of the function g strictly above the abscissa\". So all you have to do is sketch both parabolas (you just have to find the roots of f(x)=0, g(x)=0 and decide on the orientation of the parabola), and then by simple \"visual inspection\" you can get the answers to both of the questions. Both answers will be intervals and all you need to do is find their intersection.\n\n5. Re: Dividing both sides of an inequality by one factor\n\nOriginally Posted by querti09\nThis is a system of inequations for which I want to find the set of answers.\n\n$(x-1)(x+2)<(x+2)(x-3)$\n$(x+3)(x+5)>(x+4)(x+3)$\n\nWhat I CAN'T do here is the following:\n\nWe start with the first inequation $(x-1)(x+2)<(x+2)(x-3)$\n\n1) I divide both sides of the inequation by $(x+2)$\nand I get:\n\n$\\frac{(x-1)(x+2)}{(x+2)}<\\frac{(x+2)(x-3)}{(x+2)}$\n\n2) I cancel the $(x+2)$\nin top and bottom of both members of the inequality and I get:\n\n$(x-1)<(x-3)$\n\nIn other words:\n\n$x-1\n\nThen solving this inequality I get:\n\n3) $0<-2$\n\nAnd this is plainly wrong: Then my question is WHY, what's wrong with this procedure.\n\nI know the right one so I don't need the correct procedure, but instead what I'm looking for is an explanation about how come this is incorrect (not just saying because the answer is wrong, I know the correct answer) I thought that perhaps has to do with order of operations...\nYou can do that if x+ 2> 0 which is, of course, the same as saying x> -2. As long as that is true, then x+ 1< x- 3 which, subtracting x from both sides gives -1< -3. Because that is NOT true, we know that x CANNOT be larger than -2. As long as x< -2, x+2< 0 and dividing both sides by x+2 is dividing by a negative number and so the inequality is reversed: x- 1> x- 3. That reduces to -1> -3 which is TRUE for all x. So the original inequality is true for all x> -2.x^3\n\nYou could also have done this by going ahead and doing the indicated multiplication: $(x- 1)(x+2)= x^3+ x- 2$ and $(x+ 2)(x- 3)= x^2- x- 6$. The inequality is just $x^2+ x- 2< x^2- x- 6$. Now you can just add and subtract from each side to get $2x< -4$ so, because 2 is obviously positive, we can divide by 2 to get x< -2.\n\n6. Re: Dividing both sides of an inequality by one factor\n\nThanks to you all for the help. I have now a better grasp about this.\nWhat I still don't get it Mathoman is:The $x^{2}$ terms, both disappear when I move the second member to the left side of the inequality in order to graph. This makes me really wonder, should this be called a quadratic inequality or a linear inequality? I am confused about this, since when I sketch the inequality I have $2x+4<0$ and $2x+4$ is a polynomial of first grade. What does it make us to label an inequality quadratic or not? The original set of inequalities are quadratic, but later when I made them equal to 0 the sketch I see is a line, not a parabola.\n\n7. Re: Dividing both sides of an inequality by one factor\n\n$-3","date":"2016-12-07 09:00:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 40, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7957943081855774, \"perplexity\": 353.69379472566885}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-50\/segments\/1480698542009.32\/warc\/CC-MAIN-20161202170902-00497-ip-10-31-129-80.ec2.internal.warc.gz\"}"}
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\section{Introduction}
The first extra-Solar X-ray source discovered was the low-mass X-ray
binary Sco~X-1 \citep{Giacconi:1962a}. Its optical counterpart,
V818~Sco, was discovered by \citet{Sandage:1966a}, paving the way for
many subsequent multiwavelength studies. The binary period is widely
accepted to be 18.9\,hr based on the discovery of a photometric
modulation by \citet{Gottlieb:1975a} and spectroscopic confirmation by
\citet{Cowley:1975a}. We now know that Sco~X-1 contains a low-mass
late-type donor transferring mass onto a neutron star at a rather high
rate. The modulation arises from X-ray heating of the donor star,
which also manifests as narrow emission lines of N\,{\sc iii} and
C\,{\sc iii} moving in phase with the donor star
\citep{Steeghs:2002a}.
\citet{Gottlieb:1975a} obtained the period of
$0.787313\pm0.000001$\,days quite remarkably by examining archival
photographic plates from 1889 to 1974. A sinusoidal modulation of
full amplitude around 0.2--0.3\,mag was found in several independent
datasets, with considerable scatter around the mean curve
\citep{Gottlieb:1975a,Wright:1975a}. While the long baseline of
photographic observations defined the period to incredible precision,
the sparse sampling left a plethora of aliases, and
\citet{Gottlieb:1975a} identified strong signals at one-day,
one-month, and one-year aliases of their favored period. Of these,
the one-year alias has been by far the hardest to reject. Several
subsequent photometric studies reproduced the modulation, but none
improved the ephemeris, or resolved the one-year alias issue
\citep{vanGenderen:1977a,Augusteijn:1992a}
Spectroscopic confirmation of this period was suggested by
\citet{Gottlieb:1975a} and \citet{Wright:1975a}, and demonstrated
conclusively by \citet{Cowley:1975a}, who found a
period of $0.787\pm0.006$\,days, and again by
\citet{LaSala:1985a}. Both of these works performed a period search on
the data, but in both cases the frequency resolution was limited by
only observing over a baseline of a week. Other spectroscopic
analyses of these and other data have also found variations at this
period, \citep{Crampton:1976a,Bord:1976a,Steeghs:2002a}, but no other
groups have performed a rigorous independent period search.
Several groups also searched for the orbital period in X-ray data,
with initially no success
\citep{Holt:1976a,Coe:1980a,Priedhorsky:1987a,Priedhorsky:1995a}. The
only positive detection of an orbital period in X-rays came from
\citet{Vanderlinde:2003a} based on a multi-year {\it RXTE}/ASM
dataset. They did not find exactly the \citet{Gottlieb:1975a} period,
but instead the one-year alias (0.78893\,days) with a modulation
around 1\,\%. Given the intensive multi-year coverage of {\it RXTE}
this is surprising, since this dataset should not be susceptible to
the one-year alias problem. \citet{Vanderlinde:2003a} therefore
claimed that their period was the true orbital period and that
\citet{Gottlieb:1975a} had misidentified the alias. While this result
was tantalizing, \citet{Levine:2011a} could not reproduce this period
using a larger {\it RXTE} dataset. They did, however, not use as
sophisticated an analysis as \citet{Vanderlinde:2003a}, leaving open
the possibility that the X-ray period could be real.
Surprisingly, then, fifty years after discovery of the prototypical
LMXB Sco~X-1, there remain doubts about its most fundamental
parameter, the orbital period. While the original optical ephemeris
of \citet{Gottlieb:1975a} has remained the standard reference for the
37\,years since its publication, it remains to be resolved whether
this, or the X-ray period of \citet{Vanderlinde:2003a}, is the true
orbital period. To attempt to resolve these questions, and update the
ephemeris of Sco~X-1 with modern data, we examine here archival
photometry from the All Sky Automated Survey (ASAS). This nine year
dataset has both the long baseline to determine a precise period, and
coverage of a large enough fraction of a year to finally break the
one-year alias problem using optical data.
\section{Observations}
\label{DataSection}
The All Sky Automated Survey (ASAS) monitored Sco~X-1 from 2001 to
2009 \citep{Pojmanski:2002a}. We note that while Sco~X-1 was not
included in the ASAS Catalog of Variable Stars (ACVS) its photometry
is in the ASAS-3 Photometric $V$ Band Catalog in two datasets,
161955--1538.4 and 161955--1538.5. The Sco X-1 datasets include 640
observations from 2001 January 22 to 2009 October 5. With multiyear
coverage spanning typically about 270\,days of the year, it is ideally
suited for obtaining an updated ephemeris and breaking the one-year
alias.
We performed our analysis for a range of choices of data grades and
apertures to optimize our filter criteria. For final analysis, we
retained the 567 grade A or B observations, and used the smallest ASAS
aperture. Inclusion of grade C or worse data, or use of larger
aperture data, significantly reduced the quality of the fits.
\section{Ephemeris}
\label{PeriodSection}
To determine the orbital period we performed a sinuoidal fit to the
data points. Since the scatter around the model is dominated by
intrinsic flickering rather than photometric uncertainties, we
assigned a mean uncertainty of 0.30\,mag to each point to represent
the flickering. This was chosen to produce a minimum $\chi^2$ equal
to the number of degrees of freedom. We then evaluated sinusoidal
fits over a range of trial periods. For each period the best-fitting
mean magnitude, amplitude, and phasing were determined using the
downhill simplex algorithm \citep{Nelder:1965a}. We show the results
in the vicinity of the disputed periods in Fig.~\ref{PeriodFig}.
\begin{figure}
\includegraphics[angle=90,width=3.5in]{fig1.ps}
\caption{$\chi^2$ as a function of trial period for sinusoidal fits to
ASAS data. We show the \citet{Gottlieb:1975a} period of
0.787313\,days and the \citet{Vanderlinde:2003a} period of
0.78893\,days for comparison. We also show calculated one-year
aliases of the preferred period at 0.78562 and 0.78901\,days. The
\citet{Gottlieb:1975a} period is strongly favored by the ASAS data. While
some signal is seen at the alternative periods, all are rejected at
greater than 5-$\sigma$ confidence.}
\label{PeriodFig}
\end{figure}
We see that the \citet{Gottlieb:1975a} period is reproduced exactly to
within the limits of our frequency resolution. Our formal best period
is $0.787313\pm0.000015$\,days. The uncertainty quoted is a formal
1-$\sigma$ error determined from the $\Delta\chi^2=1$ confidence range
in period. We verified the uncertainty using the bootstrap method
with 30 resamplings of the data. This gave a consistent 1-$\sigma$
uncertainty ($1.6\times10^{-5}$). We also show the period of
\citet{Vanderlinde:2003a}, and the one-year aliases with which they
associated it. We find that none of these alternatives are consistent
with the ASAS data, and all can be rejected at better than 5-$\sigma$
confidence. We therefore cannot directly improve on the period of
\citet{Gottlieb:1975a} using the ASAS data, which is not surprising as
that used data drawn from nearly a hundred year baseline. We can,
however, overcome the limitation of that dataset in its vulnerability
to one-year aliases, as the ASAS data has much wider coverage within a
year.
Using the same $\chi^2$ approach, we determine a mean time of minimum
of $2453510.329\pm0.017$. This corresponds to an offset of very close
to 17057 cycles from the time of minimum of \citet{Gottlieb:1975a}.
If we project their time of minimum forwards we predict
$2453510.328\pm0.024$, with equal contributions to the uncertainty
from their time of minimum (quoted as 0.022 cycles) and their period
($10^{-6}$\,days). Our time of minimum is completely consistent with
theirs (a remarkable testament to the accuracy of their historical
ephemeris), but at this point our modern measurement of the time is somewhat
better constrained for use with modern data.
Finally, we show in Figure~\ref{LightcurveFig} the ASAS lightcurve folded on
our derived ephemeris, together with the best fitting sine wave. The
mean $V$ band brightness is 12.63, and the full-amplitude is 0.26\,mag,
comparable to that found by \citet{Gottlieb:1975a} and
\citet{Wright:1975a}.
\begin{figure}
\includegraphics[angle=90,width=3.5in]{fig2.ps}
\caption{Folded and phase-binned ASAS lightcurve of Sco~X-1. The
data have been grouped into 50 phase bins and plotted twice.
Errorbars are empirical and indicate the error on the mean of each
bin. The model plotted is the best fitting sine wave determined
Section~\ref{PeriodSection}.}
\label{LightcurveFig}
\end{figure}
\section{Discussion}
We have established that in optical photometry the 0.787313\,day
period produces a stable modulation over 120\,years of observation.
The ephemeris of \citet{Gottlieb:1975a} reliably and precisely
predicts the time of minimum in the ASAS data, over 17,000
intervening cycles. It is hard to imagine any clock other than the
orbital period providing this stability. This has to be the true
orbital period.
The question then arises as to what, if anything,
\citet{Vanderlinde:2003a} detected. We of course should allow that it
was a spurious detection, until it can be reproduced with data from
the remainder of the {\it RXTE} mission. \citet{Levine:2011a} failed
to reproduce it, but also did not use all the techniques that
\citet{Vanderlinde:2003a} used. Associating it with an alias of the
true orbital period seems unlikely, as {\it RXTE}/ASM data on Sco~X-1
are rather well sampled through the year (just as ASAS data are).
One possible explanation might be if the X-ray signal came at the beat
frequency between the orbital period and a superorbital period of
around a year. Many X-ray binaries have indeed shown super-orbital
periods of tens to hundreds of days \citep[see e.g.][]{Charles:2008a},
although typically all are shorter than a year. The only claim of
such a long period in Sco~X-1, came from early {\it RXTE}/ASM data,
from which \citet{Peele:1996a} suggested a 37\,day period. This
detection has not been sustained in subsequent data, and no
super-orbital period was found by \citet{Farrell:2009a} in {\it
Swift}/BAT data. On longer timescales, \citet{Durant:2010a} and
\citet{Kotze:2010a} both independently suggested a $\sim9$\,year X-ray
modulation is present in {\it RXTE}/ASM data, although this is too
long to account for the \citet{Vanderlinde:2003a} period. This
explanation therefore seems unlikely, and it remains to be seen if the
X-ray period can be reproduced from the full {\it RXTE} mission-long
dataset.
\section{Conclusions}
We have analyzed ASAS data of Sco~X-1 spanning nine years. We can
confirm the period of \citet{Gottlieb:1975a}, while rejecting its
one-year aliases, and also the putative X-ray period of
\citet{Vanderlinde:2003a}. Our updated ephemeris is $T_{\rm min}({\rm
HJD}) = 2453510.329(17)+0.787313(1)E$.
\acknowledgments
This work was supported by the National Science Foundation under Grant
No. AST-0908789. This research has made use of NASA's Astrophysics
Data System.
{\it Facilities:} \facility{ASAS}.
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Goa State Election Commission is an autonomous and statutory body constituted in Indian state of Goa for ensuring that elections are conducted in free, fair and unbiased way. Constitution of India with provisions as per Article 243K and 243 ZA and Article 324 ensures creation and safeguarding of the powers of State Election Commissions. Goa State Election Commission is responsible for conducting elections for Urban Local Bodies like Municipalities, Municipal Corporations, Panchayats and any other specified by Election Commission of India. Goa State Election Commissioner is appointed by Governor of Goa.
History
The State Election Commission of Goa was constituted pursuant to the 73rd and 74th Amendments to the Constitution of India, by amending the Goa Municipalities Act, 1969 and the Goa Panchayat Raj Act, 1993. To ensure the autonomy of the position, the Goa state election commissioner cannot be removed from office except on the grounds and manner specified for judge of High Court. Geeta Sagar, IAS served as the first State Election Commissioner.
List of State Election Commissioners
|}
Powers and Responsibilities
Goa States Election Commissioner is responsible for the following:
Issue notification containing guidelines for conducting elections for Municipal Corporations in State.
Conducting elections for Municipal Corporations in State.
Issue notification containing guidelines for conducting elections for conducting elections for Municipal panchayats in State.
Conducting elections for Municipal panchayats in State.
Laying guidelines for persons eligible to contest in elections for Municipal Corporations in State.
Conducting elections for Municipal panchayats in State.
Model code of conduct are following in elections for local bodies.
Updating Electoral rolls with new additions.
Updating Electoral rolls with removals, if any.
Declaration of results of elections held for Municipal Corporations in State.
Declaration of results of elections held for Municipal panchayats in State.
Ordering repoll if needed.
Composition
Goa State Election Commission is a one-man commission consisting of Chief Election Commissioner. State Election Commissioners are independent persons not holding position or office in any Central or State Government organisations.
Shree W V Ramanamurthy, Retd I.A.S., is the Chief Election Commissioner of Goa. His period of service will be 5 years or attaining an age of 65 years whichever is earlier.
Constitutional Requirements
Goa State Election Commission was formed after amendment of Constitution with 73rd and 74th declaration. State Election Commissions were formed as per Article 243K of the Constitution, similar to setting up of Election commission of India as per Article 324.
See also
Election Commission of India.
References
External links
Official Website
Government of Goa
Elections in Goa
State agencies of Goa
State Election Commissioners of India
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Q: How to listen to incoming events in Android UIThread Having read lots of snippets and tutorials, I'm still (or even more) confused about the road to take. I need a thread/backgroundtask, that listens for incoming events on a socket and report any incomings to the UIThread. What would be the preferred way to choose? Own thread or multitasking? Best way to transfer data to the main thread?
Thanx for any thoughts on the matter.
Regards,
Marcus
Considering your answers below, I've tried the following:
MainActivity:
public class MainActivity extends Activity {
Handler handler = new Handler() {
@Override
public void handleMessage(Message msg) {
toastSomething();
};
};
/** Called when the activity is first created. */
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.main);
threadstarter();
}
protected void threadstarter() {
super.onStart();
Thread backgroundthread = new Thread(new WorkerThread(handler));
backgroundthread.start();
}
public void toastSomething() {
Toast.makeText(this, "hello", Toast.LENGTH_SHORT).show();
} }
An my runnable:
public class WorkerThread implements Runnable {
Handler messageHandler;
WorkerThread(Handler incomingHandler) {
messageHandler = incomingHandler;
}
public void run() {
while (true) {
for (int i = 0; i <= 100000; i++) {
// wait a moment
}
messageHandler.sendEmptyMessage(1);
}
} }
My layout only holds an additional checkbox:
<?xml version="1.0" encoding="utf-8"?>
<TextView
android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:text="@string/hello" />
<CheckBox
android:id="@+id/checkBox1"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:text="CheckBox" />
Good thing is, the toast appears. Bad thing, the cehckbox is unresponsive and the app crashes pretty quick. Isn't that, how it should be done?
Edit: the msg in sendMessage in the WorkerThread seems to be the troublemaker as the exception says, the message is all read in use?
A: only ways to communiacte with UIThread from background are
runOnUIThread(new Runnable {.// your ui stuff goes here.});
handler.post(new Runnable{.// your ui stuff goes here.});
i can't think of anything else.. these 2 are very handy at all situations..
in ru
A: You can communicate like this from background thread to UIThread
Runnable runnable = new Runnable()
{
@Override
public void run() {
for (int i = 0; i <= 10; i++) {
handler.post(new Runnable() {
@Override
public void run() {
progress.setProgress(value);
}
});
}
}
};
A: You can either use the AsyncTask class, and post on the UI thread inside the onPreExecute(...), onPostExecute(...), or onProgressUpdate(...) methods.
Another way is to use a new thread for the background work, and post a Runnable to a Handler.
If you instantiate the Handler on the UI thread, then everything you post to the handler will run on the UI thread. If you instantiate the handler on the background thread then everything you post to the handler will run on the background thread.
A: Here's the code, that's finally working:
UIThread:
public class MainActivity extends Activity {
public static final String LOG_TAG = "UIThread";
Handler handler = new Handler() {
@Override
public void handleMessage(Message msg) {
toastSomething();
//Log.v(LOG_TAG, "main thread");
};
};
/** Called when the activity is first created. */
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.main);
threadstarter();
}
protected void threadstarter() {
super.onStart();
Thread backgroundthread = new Thread(new WorkerThread(handler));
backgroundthread.start();
}
public void toastSomething() {
Toast.makeText(this, "hello", Toast.LENGTH_SHORT).show();
}
}
WorkerThread:
public class MainActivity extends Activity {
public static final String LOG_TAG = "UIThread";
Handler handler = new Handler() {
@Override
public void handleMessage(Message msg) {
toastSomething();
//Log.v(LOG_TAG, "main thread");
};
};
/** Called when the activity is first created. */
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.main);
threadstarter();
}
protected void threadstarter() {
super.onStart();
Thread backgroundthread = new Thread(new WorkerThread(handler));
backgroundthread.start();
}
public void toastSomething() {
Toast.makeText(this, "hello", Toast.LENGTH_SHORT).show();
}
}
HTH someone. Thanx to all for the input.
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"redpajama_set_name": "RedPajamaStackExchange"
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| 6,834
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,914
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Q: How to assign different user roles during registration? I was wondering how can I create different user roles depending on which part of the site user registers? For example, I have three user types. One is admin, another is regular user and third one is moderator.
I have the following table:
create_table "users", force: true do |t|
t.string "username"
t.string "email"
t.string "role"
t.string "password"
end
Inside "role" column I would save one of the following strings: "admin", "regular" or "moderator".
I have already created my authentication system with the help of Michael Hartl's Ruby on Rails Tutorial. Also, I watched railscasts episode #189 Embedded Association where there is a really nice approach of handling multiple user roles.
The problem is that on video user roles are chosen during registration and I don't want user to see user roles. I have two sections of the site. If user registers through one section he will have a role "regular" and if he registers through another section he will have role "moderator". Role "admin' would of course be set differently.
It could be done if I pass parameter "role": "moderator" or "role": "regular" but that doesn't seem to me as the safe option because it would be visible to users and they can easily pass some other parameter there. For example regular users who want to harm the site could pass "role": "moderator".
Thank you for your advices :)
A: I would go with CanCan and Rolify and it's very well documented here:
https://github.com/EppO/rolify/wiki/Tutorial.
There are also some awesome Railscasts about it.
This way you can add all users the same way and simple add a role depending on the page the user chose to sign up from (why is that secure?). I assume that you have some internal sites only accessible by you to add admin and editors?
Now you can use request.original_url and add a switch statement to do something like:
current_user.add_role "admin"
Of course you can also use your approach and just update the user record with:
current_user (or user you are about to create in your user controller) and do a
user.role = "admin"
user.save
Hope it helps!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 311
|
import discord
from discord.ext import commands
from .utils.chat_formatting import escape_mass_mentions, italics, pagify
from random import choice as rndchoice
from random import randint
import os
class Moide:
"""Moide :3."""
def __init__(self, bot):
self.bot = bot
@commands.group(pass_context=True, invoke_without_command=True)
async def moide(self, ctx, *, user: discord.Member=None):
"""Moide alguém."""
botid = self.bot.user.id
if user is None:
user = ctx.message.author
await self.bot.say("Vou te moide " + user.name)
elif user.id == botid:
user = ctx.message.author
botname = self.bot.user.name
await self.bot.say(botname + " Moideu " + user.mention +
" :3 ")
else:
await self.bot.say("Moideu " + user.name + " :3 ")
def setup(bot):
n = Moide(bot)
bot.add_cog(n)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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This work in Level 2 is all about purpose and the transformational power of energy. It is very personal — between you and the Mother of us all, and as with all things true, it is therefore also very universal. When you bring the energy of the Sacred Feminine into and through the vessel of your body there is only the Oneness and Perfection of Life that you see through Her eyes. This experience will change who you "BE" and how you hold yourself in the world — forever.
Gatherings and teachings focus on ceremonies, learning rituals, holding space, Mentoring a Maiden currently enrolled in the Rites of Passage Program, becoming the hollow bone, aligning your intuitive powers, care for the sacred vessel of you, and opening to the Goddess within you and bringing Her wisdom to the world during our infamous Annual Goddess Pageant.
This level of study at the Mystery School, Living the Mysteries, is reserved for women who have completed their Rites of Passage and by invitation only.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,039
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Sally Beauty appoints President and COO
Sally Beauty has appointed Chris Brickman as President and Chief Operating Officer. Brickman has been a member of Sally Beauty's Board of Directors since September 2012 and has served as the company's President and COO since June last year. Prior to joining Sally Holdings, he was President of Kimberly-Clark International.
"The Board and I are very pleased to have an outstanding candidate such as Chris step into the CEO role," said Robert McMaster, Lead Independent Director of the Company's Board. "Chris is a passionate leader with an exceptional track record of success throughout his career. We are confident he will add tremendous value and drive continued growth and development for the organization. The Board is very grateful to Gary Winterhalter for his leadership as CEO and his commitment to the Company's success. We are pleased that he will continue to play a key role with the Company."
Sally Beauty Holdings
Sally Beauty confirms job cuts after "disappointing" Q1 2017
Sally Beauty reports solid full-year but misses estimates
Sally Beauty opens 5,000th store
Sally Beauty CEO 'disappointed' with Q3 2015 results
Sally Beauty update on data
Sally Beauty confirms data breach affecting payment cards
Sally Beauty reports growth
Sally Beauty faces second potential data breach
BCL Naturals hair care introduces eight natural products
Sally Beauty announces Board of Directors retirement
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,453
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Houby (Fungi, Mycetalia) představují velkou skupinu živých organismů dříve řazenou k rostlinám (jako jejich podříše Mycophyta), později Robertem Whittakerem vyčleněnou jako samostatnou říši a v současné době klasifikovanou spolu s např. živočichy jako součást superskupiny Obazoa a kladu Opisthokonta. Její zástupce lze nalézt po celé Zemi a vyskytují se mezi nimi významní rozkladači, parazité či v průmyslu i potravinářství využívané druhy. Mnoho druhů náleží mezi mutualisty žijící v symbióze s cévnatými rostlinami nebo s řasami. K roku 2022 je oficiálně popsáno přes 150 000 druhů hub, ale ve skutečnosti jich existuje řádově více; podle posledních (rok 2021) odhadů publikovaných vědci Mikrobiologického ústavu Akademie věd České republiky je to 6,28 milionu druhů. V Česku je zjištěno asi 10 000 druhů.
V užším pojetí jsou houby (Fungi) stélkaté organismy různého tvaru a velikostí, bez asimilačních barviv (tzn. bez plastidů), s heterotrofní výživou, s buněčnou stěnou chitinózní. Zásobní látkou je glykogen. Houby se rozmnožují buď vegetativně (rozpadem vlákna mycelia), nebo nepohlavními či pohlavními výtrusy.
Věda zabývající se houbami se nazývá mykologie.
Evoluce
Houby se zřejmě vyvinuly z jednoduchých vodních organizmů s bičíkatými sporami; předci hub tedy mohli vypadat podobně jako chytridiomycety. Ztráta bičíku nastala buď jednou, nebo vícekrát. Vznik hub se zřejmě odehrál v pozdních starohorách, v době před 900–570 miliony let. Nové nálezy fosilií vláknité struktury připomínajících podhoubí, u kterých však není dosud možné chemickou analýzou plně vyloučit geologický původ, mohou posunout vznik hub, tehdy jako výhradně mořských organismů, až do doby před 2,4 miliardami let.
V prvohorách došlo k nárůstu rozmanitosti a v pennsylvanu (karbon, před 320–286 miliony lety) se podle fosilních nálezů již vyskytovaly všechny hlavní skupiny hub, jak je známé ze současnosti. V období siluru a devonu představovaly houby hlavní suchozemské organismy. Prototaxity, největší tehdejší organismy (zkameněliny vysoké až 6 m), se totiž podle poměru izotopů uhlíku C13 a C12 v jejich fosilizovaných zbytcích podařilo identifikovat jako houby. K suchozemským organismům tehdy patřili pouze drobní červi, stonožky a bezkřídlý hmyz, žijící hlavně v zemi, výjimečně na povrchu; fotosyntetizující rostliny byly malé, bez kořenů a listů. Živočichové, kteří by se jimi živili, ještě neexistovali.
Stavba těla
Houby jsou jednobuněčné i mnohobuněčné organismy. Z buněčných organel nejsou v cytoplazmě přítomny chloroplasty, proto si nejsou schopny vytvářet organické látky. Základní stavební jednotkou je houbové vlákno (hyfa), které se může rozlišit v podhoubí (mycelium) a v plodnici. Někdy tvoří tzv. nepravá pletiva, jako je plektenchym a pseudoparenchym.
Hyfy
Tělo hub není členěno na jednotlivé orgány, a proto se nazývá stélka. Ta je složena z propletených houbových vláken (hyfa), která vytváří podhoubí. U mnohých hub vyrůstá z podhoubí plodnice.
Systém podhoubí může dosáhnout obrovských rozměrů. Největší houbou a možná vůbec největším žijícím organismem na světě je dřevokazná václavka smrková (Armillaria ostoyae syn. Armillaria solidipes). Vyskytuje se v oregonských Blue Mountains а zabírá téměř 965 hektarů půdy. Objev se podařil díky snahám vysvětlit masivní úhyn stromů ve státních lesích Oregonu v roce 1998. Vzorky odebrané z odumřelých stromů prokázaly nejen napadení václavkou smrkovou, ale také to, že se jedná o geneticky identické houby ze stejného podhoubí a tedy o jeden celistvý organismus (těsné seskupení geneticky identických buněk, které mohou komunikovat a mají společný cíl). Největší vzdálenost mezi nakaženými stromy přitom byla téměř 4 kilometry. Stáří houby bylo odhadnuto na 2400 let, přestože by mohla na Zemi žít již celých 8650 let. To ji řadí i mezi nejstarší organismy obývající planetu. Již dříve, v roce 1992, byly objeveny další dva obří exempláře václavek: Prvním byla Armillaria gallica (václavka hlíznatá) stará 1500 let a zabírající 15 hektarů, vyskytující se v listnatých lesích poblíž Crystal Falls v Michiganu. Druhým byla václavka smrková, jež se rozkládala v jihozápadním Washingtonu na ploše 600 hektarů.
Plodnice
Plodnice je nadzemní "orgán" houby, jehož hlavním úkolem je rozmnožování. Obsahuje totiž (zejména na spodní straně) výtrusy.
Rouško je výtrusorodá vrstva s velkým množstvím kyjovitých výtrusnic s výtrusy. Bývá na spodní ploše klobouku na lupenech nebo v rourkách. Houby s lupeny naspodu klobouku se nazývají lupenaté (např. bedla, muchomůrka, pečárka, ryzec). Někdy mívají i pochvu a plachetku. Houby s rourkami jsou označovány jako rourkaté (např. hřiby, křemenáč, kozák, klouzek).
Výživa hub
Pokud budeme houby dělit dle způsobu, jakým získávají živiny, dostáváme dvě základní skupiny hub – saprofytické (hniložijné) a parazitické (příživné). Saprofytické houby jsou takové, které získávají organické látky pomocí rozkladu odumřelých živočišných či rostlinných těl. Je možno je zařadit mezi rozkladače neboli dekompozitory. Parazitické houby mohou být biotrofní (živí se obsahem buněk ale nezabíjí je) či nekrotrofní (způsobují odumírání tkáně). Dalšími významnými skupinami hub jsou houby formující lišejníky a houby mykorhizní.
Parazitismus
Četné druhy působí škody na rostlinách, zejména na dřevinách (dřevokazné houby), živočiších i člověku tím, že způsobují onemocnění. Tyto se nazývají parazitické (cizopasné) a mohou vyvolávat onemocnění na povrchu těla i ve vnitřních orgánech. K cizopasným druhům na obilí patří padlí, důležitým tropickým parazitoidem je houba rodu Cordyceps.
Existují i dravé houby, například Arthrobotrys dactyloides loví hlístice pomocí specializovaných hyf.
Mykorhiza a lichenismus
Některé druhy hub žijí v mutualistické symbióze (symbióza prospěšná pro oba partnery) s kořeny mnoha rostlin, což nazýváme mykorhiza. Houba přijímá od rostlin různé organické látky, které sama nevytváří, a pomáhá rostlině přijímat vodu s rozpuštěnými minerálními látkami.
Mimoto žijí také v symbióze se řasami nebo sinicemi, zejména složené organismy zvané lišejníky.
Rozkladači
Hniložijné houby jsou (spolu s hniložijnými bakteriemi) nejvýznamnějšími rozkladači odumřelých zbytků různých organismů. Někdy se také nazývají saprofytické houby. Mohou růst i na potravinách.
Využití hub člověkem
Jedlé druhy slouží jako potravina s malou kalorickou hodnotou nebo jako pochutina. Jsou bohaté na vitamíny a minerální látky. Jedovaté druhy nejsou početné, ale pro obsah prudkých jedů nebezpečné.
Kvasinky jsou nezbytné pro mnoho potravinářských technologií (zejména v pekařství a při výrobě alkoholických nápojů). Některé druhy kvasinek se používají k napouštění konzerv a tímto se zamezí kažení potravin. Konzerva vydrží dlouho a nezkazí se. Tento postup je aplikován v oblasti Asie.
Mnohé druhy se rovněž využívají ve farmaceutickém a chemickém průmyslu. U štětičkovce druhu Penicillium notatum byla objevena antibiotika. V potravinářství se vyrábí např. plísňové sýry (camembert, niva, hermelín, …) Jiné druhy hub se využívají k očkování prken a tím se ochrání dřevo před cizími živočichy a houbami. Houby obsadí celý kus dřeva a nepustí jiného parazita na jejich místo, samy však dřevo nepoškodí a nezničí. Dokonce je dřevo díky tomu pevnější. Tento postup objevili vědci v USA na Floridě.
Rozmnožování
U hub známe pohlavní i nepohlavní rozmnožování. V obou případech mohou hrát roli výtrusy čili spory.
Nepohlavní rozmnožování
Při nepohlavním rozmnožování nedochází ke tvorbě pohlavních buněk. Nejjednodušším způsobem mohou dva jedinci vzniknout prostou fragmentací (rozpadem) vláknité stélky vedví. Co se týče jednobuněčných kvasinek, ty se mohou dělit v procesu pučení. Druhou fundamentální možností, jak se nepohlavně rozmnožovat, je u hub také produkce nepohlavních výtrusů, a to buď ve specializovaných sporangiích (takové sporangiospory vznikají např. u rodu Mucor), nebo přímo na houbových vláknech konidioforech (z nichž vznikající spory se označují jako konidie).
Pohlavní rozmnožování
Pohlavní rozmnožování je běžným způsobem rozmnožování u většiny druhů hub. Je spojeno se splýváním buněčných jader a meiotickými děleními, načež jsou touto pohlavní cestou vytvořeny spory (výtrusy). Typicky se útvary, v nichž dozrávají výtrusy pohlavní cestou, označují jako plodnice.
Pohlavní rozmnožování se dá rozdělit na několik fází:
Plazmogamie – splynutí cytoplazmy buněk
Karyogamie – splynutí jader buněk, vzniká zygota s chromozomy jako ostatní buňky
Meióza – redukční dělení; vznik haploidních buněk (haploidní gamety – meiospory)
Příkladem výsledných meiospor jsou askospory vřeckovýtrusných hub anebo basidiospory stopkovýtrusných hub. Gamety splynou a vznikne zygota. U některých hub splývají hyfy (nemají gametangia)
Systematika
Systém hub prošel na začátku 21. století podstatnými úpravami, které ho přibližují fylogeneticky přirozenému členění. Dříve se houby dělily na nižší houby a vyšší houby, ale toto dělení neodpovídalo nárokům na fylogenetickou příbuznost druhů, a tak se od něho upustilo. Podobně se upustilo od samostatného oddělení pro houby nedokonalé (Deuteromycota), což bylo spíše označení pro houby, které se nerozmnožují pohlavně, než přirozený taxon. Jako parafyletický taxon se ve fylogenetických analýzách jeví houby spájivé (Zygomycota), dříve používané oddělení zahrnující houby, v jejichž životním cyklu je odolné zygosporangium. Naopak mykologové nově zařazují mezi houby další skupiny jednobuněčných opisthokontních eukaryot, konkrétně afelidie, kryptomycety a mikrosporidie, vyčleňované v některých systémech eukaryot mimo houby do zvláštní skupiny Opisthosporidia, dříve považované za přirozenou.
Jako aktuální lze uvést systém z r. 2020, zpracovaný v rámci projektu Outline of Fungi and fungus-like taxa (i s účastí mykologů z českých vědeckých pracovišť). Říše Fungi se podle něj dělí na fylogeneticky přirozenou podříši Dikarya, která sdružuje kmeny hub stopkovýtrusých a vřeckovýtrusých, a několik samostatných kmenů, pro přehlednost zpravidla seskupených do dalších 8 podříší (s předpokládanou přirozeností); české názvy dle BioLib, případně dalších uvedených referencí:
Podříše: ROZELLOMYCETA
Kmen: Rozellomycota – mikrosporidie a kryptomycety, zajímavé absencí chitinové buněčné stěny
Podříše: APHELIDIOMYCETA
Kmen: Aphelidiomycota (též Aphelida) – afelidie
Podříše: CHYTRIDIOMYCETA – variabilní, mají bičíkaté pohyblivé spory
Kmen: Caulochytriomycota (dříve součást kmene Chytridiomycota, třídy Spizellomycetes)
Kmen: Chytridiomycota (dříve též Archemycota) – chytridiomycety
Kmen: Monoblepharomycota (dříve součást kmene Chytridiomycota)
Kmen: Neocallimastigomycota
Podříše: BLASTOCLADIOMYCETA – variabilní, bez buněčné stěny, mají bičíkaté pohyblivé spory (dříve součást chytridiomycet)
Kmen: Blastocladiomycota (též Allomycota)
Kmen: Sanchytriomycota
Podříše: BASIDIOBOLOMYCETA
Kmen: Basidiobolomycota
Podříše: OLPIDIOMYCETA – variabilní, mají bičíkaté pohyblivé spory (dříve součást chytridiomycet)
Kmen: Olpidiomycota
Podříše: ZOOPAGOMYCETA – součástí jejich životního cyklu je odolné zygosporangium (dříve součást spájivých hub)
Kmen: Entomophthoromycota
Kmen: Kickxellomycota
Kmen: Zoopagomycota
Podříše: MUCOROMYCETA – součástí jejich životního cyklu je odolné zygosporangium (dříve součást spájivých hub)
Kmen: Calcarisporiellomycota
Kmen: Glomeromycota – účastní se vnitrobuněčné mykorhizy
Kmen: Mortierellomycota
Kmen: Mucoromycota
Podříše: DIKARYA, DIKARYOMYCETA (též NEOMYCOTA)
Kmen: Ascomycota – houby vřeckovýtrusé, houby vřeckaté, askomycety
Podkmen: Pezizomycotina (dříve též Ascomycotina)
Podkmen: Saccharomycotina (dříve též Hemiascomycotina)
Podkmen: Taphrinomycotina (dříve též Archiascomycotina)
Kmen: Basidiomycota – houby stopkovýtrusé, bazidiomycety
Podkmen: Agaricomycotina (obdobné dřívějším Hymenomycetes)
Podkmen: Pucciniomycotina (obdobné dřívějším Urediniomycetes)
Podkmen: Ustilaginomycotina
Podkmen: Wallemiomycotina
Kmen: Entorrhizomycota – parazité rostlinných kořenů
Zjednodušený fylogenetický strom hlavních skupin hub vypadá podle současných představ (r. 2021) následovně:
Odkazy
Poznámky
Reference
Literatura
Související články
Ochrana hub v České republice
Otrava houbami
Houbaření
Externí odkazy
Rok hub aneb kdy rostou (kalendář 1. 1. – 31. 12.) ?
TolWeb – Fungi
Mykologie
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\section{Introduction}
\label{sec:intro}
With the proliferations of Online Social Networks (OSNs) such as Facebook and LinkedIn, the paradigm of viral marketing through the ``word-of-mouth'' effect over OSNs has found numerous applications in modern commercial activities. For example, a company may provide free product samples to a few individuals (i.e, ``seed'' nodes) in an OSN, such that more people can be attracted to buy the company's products through the information cascade starting from the seed nodes.
Kempe~\textit{et.al.}~\cite{Kempe2003} have initiated the studies on the NP-hard $k$-Seed-Selection ($k$SS) problem in OSNs, where the goal is to select $k$ most influential nodes in an OSN under some contagion models such as the independent cascade model and the more general triggering model. After that, extensive studies in this line have appeared to design efficient approximation algorithms for the $k$SS problem and its variations~\cite{LeskovecKGFVG2007,ChenWW2010,Borgs2014,TangXS2014,TangSX2015,NguyenTD2016,CohenDPW2014,Nguyen2017outward,NguyenDT2016,LinCL17}.
However, all the algorithms proposed in these studies can be classified as Influence Maximization (IM) algorithms, because they have the same goal of optimizing the \textit{influence spread} (i.e., the expected number of influenced nodes in the network).
In many applications for viral marketing, one may want to seek a ``best bang for the buck'', i.e., to select a set $S$ of seed nodes with the minimum total cost such that the influence spread (denoted by $f(S)$) is no less than a predefined threshold $\eta$. This problem is called as the Min-Cost Seed Selection (MCSS) problem and has been investigated by some prior work such as~\cite{Chen2014,Kuhnle2017,Goyal2013}. It is indicated by these work that the existing IM algorithms are not appropriate for the MCSS problem, as the IM algorithms require the knowledge on the total cost of the selected seed nodes in advance, while this knowledge is exactly what we pursue in the MCSS problem.
The existing MCSS algorithms~\cite{Chen2014,Kuhnle2017,Goyal2013}, however, suffer from several major deficiencies. First, these algorithms only consider the \textbf{uniform cost (UC)} case where each node has an identical cost 1, so none of their performance ratios holds under the \textbf{general cost (GC)} case where the nodes' costs can be heterogeneous. Nevertheless, the GC case has been indicated by existing work to be ubiquitous in reality~\cite{NguyenDT2016,Goyal2013,NguyenZ2013}. Second, most of them only propose bi-criteria approximation algorithms, which cannot guarantee that the influence spread of the selected nodes is no less than $\eta$. Third, the theoretical bounds proposed by some existing work only hold for OSNs with special influence propagation probabilities, but do not hold for general OSNs. Last but not the least, the existing MCSS algorithms do not scale well to big networks with millions/billions of edges/nodes.
Although the MCSS problem looks like a classical Submodular Set Cover (SSC) problem~\cite{Wolsey1982}, the conventional approximation algorithms for SSC cannot be directly used to find solutions to MCSS in polynomial time, as computing the influence spread of any seed set is essentially a \#P-hard problem~\cite{ChenWW2010}. One possible way for overcoming this hurdle is to apply the existing network sampling methods proposed for the $k$SS problem (e.g., \cite{Borgs2014,TangSX2015,NguyenDT2016}), but it is highly non-trivial to design an efficient sampling algorithm to get a satisfying approximation solution to MCSS, due to the essential discrepancies between MCSS and $k$SS. Indeed, the number of selected nodes in the $k$SS problem is always pre-determined (i.e., $k$), while this number can be uncertain and highly correlated with the generated network samples in MCSS. This requires us to carefully build a quantitative relationship between the generated network samples and the stopping rule for selecting seed nodes, such that a feasible solution with small cost can be found in as short time as possible. Unfortunately, the existing studies have not made a substantial progress towards tackling the above challenges in MCSS, so they suffer from several deficiencies on the theoretical performance ratio, practicability and efficiency, including the ones described in last section.
In this paper, we propose several algorithms for the MCSS problem based on network sampling, and our algorithms advance the prior-art by achieving better approximation ratios and lower time complexity. More specifically, our major contributions include:
\begin{enumerate}
\item {Under the GC case, we propose the first polynomial-time bi-criteria approximation algorithms with provable approximation ratio for MCSS, using a general contagion model (i.e., the triggering model~\cite{Kempe2003}). Given any $\alpha,~\delta\in (0,1)$ and any OSN with $n$ nodes and $m$ edges, our algorithms achieve an $\mathcal{O}(\ln\frac{1}{\alpha})$ approximation ratio and output a seed set $S$ satisfying $f(S)\geq (1-\alpha)\eta$ with the probability of at least $1-\delta$. Our algorithms also achieve an expected time complexity of $\mathcal{O}(\frac{m q}{\alpha^2}\ln\frac{n}{(1-\alpha)\delta\eta})$, where $q$ is the maximum influence spread of any single node in the network.}
\item Under the UC case, our proposed algorithms have $\mathcal{O}(\ln\frac{n\eta}{n-\eta})$ approximation ratios and can output a seed set $S$ satisfying $f(S)\geq \eta$ with probability of at least $1-\delta$. Compared to the existing algorithm~\cite{Chen2014} that has the best-known approximation ratio under the same setting with ours, the running time of our algorithms is at least $\Omega(\frac{n^2}{\ln n})$ times faster, while our approximation ratio can be better than theirs under the same running time.
\item In contrast to some related work such as~\cite{Kuhnle2017}, the performance bounds of all our algorithms do not depend on particularities of the network data (e.g., the influence propagation probabilities of the network), so they are more general.
\item We test the empirical performance of our algorithms using real OSNs with up to 1.5 billion edges. The experimental results demonstrate that our algorithms significantly outperform the existing algorithms both on the running time and on the total cost of the selected seed nodes.
\end{enumerate}
The rest of our paper is organized as follows. We formally formulate our problem in Sec.~\ref{sec:prelim}. We propose bi-criteria approximation algorithms and approximation algorithms for the MCSS problem in Sec.~\ref{sec:bicriteria} and Sec.~\ref{sec:approxforuc}, respectively. The experimental results are presented in Sec.~\ref{sec:pe}. We review the related work in Sec.~\ref{sec:rw} before concluding our paper in Sec.~\ref{sec:conclu}.
\section{Preliminaries}
\label{sec:prelim}
\subsection{Model and Problem Definition}
We model an online social network as a directed graph $G=(V,E)$ where $V$ is the set of nodes and $E$ is the set of edges. Each node $u\in V$ has a cost $C(\{u\})$ which denotes the cost for selecting $u$ as a seed node. For convenience, we define $C(A)=\sum_{u\in A}C(\{u\})$ for any $A\subseteq V$.
When the nodes in a seed set $S\subseteq V$ are influenced, an influence propagation is caused in the network and hence more nodes can be activated. There are many influence propagation/contagion models, among which the Independent Cascade (IC) model and the Linear Threshold (LT) model~\cite{Kempe2003} are the most popular ones. However, it has been proved that both the IC model and the LT model are special cases of the triggering model~\cite{Kempe2003}, so we adopt the triggering model for generality.
In the triggering model, each node $u\in V$ is associated with a \textit{triggering distribution} $\mathcal{I}(u)$ over $2^{N_{in}(u)}$ where $N_{in}(u)=\{v\in V: \langle v,u\rangle\in E\}$. Let $I(u)$ denote a sample taken from $\mathcal{I}(u)$ for any $u\in V$ ($I(u)$ is called a \textit{triggering set} of $u$). The influence propagation with any seed set $A$ under the triggering model can be described as follows. At time $0$, the nodes in $A$ are all activated. Afterwards, any node $u$ will be activated at time $t+1$ iff there exists a node $v\in I(u)$ which has been activated at time $t$. This propagation process terminates when no more nodes can be activated. Let $f(A)(\forall A\subseteq V)$ denote the {Influence Spread} (IS) of $A$, i.e., the expected number of activated nodes under the triggering model. The problem studied in this paper can be formulated as follows:
\begin{definition}[The MCSS problem]
Given an OSN $G=(V,E)$ with $|V|=n$ and $|E|=m$, a cost function $C$, and any $\eta\in (0,n)$, the Min-Cost Seed Selection (MCSS) problem aims to find $S_{opt}\subseteq V$ such that $f(S_{opt})\geq \eta$ and $C(S_{opt})$ is minimized.
\end{definition}
The MCSS problem has been studied by prior work under different settings/assumptions such as the UC setting and the GC setting explained in Sec.~\ref{sec:intro}. Besides, under the \textbf{Exact Value (EV)} setting, it is assumed that the exact value of $f(A)(\forall A\subseteq V)$ can be computed in polynomial time, while this assumption does not hold under the \textbf{Noisy Value (NV)} setting.
The existing MCSS algorithms can also be classified into: (1) \textbf{APproximation (AP) algorithms}~\cite{Chen2014}: these algorithms regard any $A\subseteq V$ satisfying $f(A)\geq \eta$ as a feasible solution; (2) \textbf{Bi-criteria Approximation (BA) algorithms}~\cite{Kuhnle2017,Goyal2013}: these algorithms regard any $A\subseteq V$ satisfying $f(S)\geq (1-\alpha)\eta$ as a feasible solution, where $\alpha$ is any given number in $(0,1)$.
\subsection{The Greedy Algorithm for Submodular Set Cover}
\label{sec:gressc}
It has been shown that $f(\cdot)$ is a \textit{monotone and submodular} function~\cite{Kempe2003}, i.e., for any $S_1\subseteq S_2\subseteq V$ we have $f(S_1)\leq f(S_2)$ and $\forall x\in V\backslash S_2: f(S_1\cup \{x\})-f(S_1)\geq f(S_2\cup \{x\})-f(S_2)$. Therefore, the MCSS problem is an instance of the Submodular Set Cover (SSC) problem~\cite{Wolsey1982,Wan2010}, which can be solved by a greedy algorithm under the EV setting. For clarity, we present a (generalized) version of the greedy algorithm for the SSC problem, shown in Algorithm~\ref{alg:naivegreedy}:
\begin{algorithm}[tp]
$A\leftarrow \emptyset;~\hat{f}(A)\leftarrow 0$\\
\While{$\hat{f}(A) < \Phi $}
{
\ForEach{$u\in V\backslash A$}{
$A'\leftarrow A\cup \{u\}$\\
Compute $\hat{f}(A')$ such that $\mathbb{P}\{(1-\gamma_1){f}(A')\leq \hat{f}(A')\leq (1+\gamma_2){f}(A')\}\geq 1-\theta$ \label{ln:estimateinfluence}
}
$u^*\leftarrow \arg\max_{u\in V\backslash A}\frac{\min\{\hat{f}(A\cup \{u\}),\Phi\}-\hat{f}(A)}{C(\{u\})}$;\\
$A\leftarrow A\cup \{u^*\}$
}
\Return{$A$} \label{ln:returnfinalresult}
\caption{$\mathsf{GreSSC}(\Phi,\gamma_1,\gamma_2,\theta)$} \label{alg:naivegreedy}
\end{algorithm}
Under the EV setting (i.e., $\gamma_1=\gamma_2=\theta=0$), $\mathsf{GreSSC}$ runs in polynomial time and also achieves nice performance bounds for MCSS~\cite{ZhuLZ2016,Goyal2013}. For example, we have:
\begin{fact}[\cite{Goyal2013}]
Let $S'$ be the output of $\mathsf{GreSSC}((1-\alpha)\eta,0,0,0)$ for any $\alpha\in (0,1)$. Then we have $f(S')\geq (1-\alpha)\eta$ and $C(S')\leq (1+\ln\frac{1}{\alpha})C(S_{opt})$. This bound holds even if $f(\cdot)$ is an arbitrary monotone and non-negative submodular function defined on the ground set $V$.
\label{fct:goyalgreedy}
\end{fact}
Unfortunately, it has been shown that calculating $f(A)$ $(\forall A\subseteq V)$ is \#P-hard~\cite{ChenWW2010}. Therefore, implementing $\mathsf{GreSSC}$ under the EV setting is impractical. Indeed, the existing work usually uses multiple monte-carlo simulations to compute $\hat{f}(A')$ in Algorithm~\ref{alg:naivegreedy}~\cite{Kempe2003,LeskovecKGFVG2007}, so $\hat{f}(A')$ is a noisy estimation of $f(A')$. However, such an approach has two drawbacks: (1) The time complexity of $\mathsf{GreSSC}$ is still high (though polynomial); (2) The theoretical performance bounds of $\mathsf{GreSSC}$ under the EV setting (such as Fact~\ref{fct:goyalgreedy}) no longer hold.
\subsection{RR-Set Sampling}
Recently, Borgs. \textit{et. al.}~\cite{Borgs2014} have proposed an elegant network sampling method to estimate the value of $f(A) (\forall A\subseteq V)$, whose key idea can be presented by the following equation:
\begin{eqnarray}
\forall A\subseteq V: f(A)=n\cdot \mathbb{P}\{R\cap A\neq\emptyset\},
\label{eqn:borgsequation}
\end{eqnarray}
where $R$ is a random subset of $V$, called as a Reverse Reachable (RR) set. Under the IC model~\cite{Borgs2014}, an RR-set can be generated by first uniformly sampling $u$ from $V$, then reverse the edges' directions in $G$ and traverse $G$ from $u$ according to the probabilities associated with each edge. According to equation~(\ref{eqn:borgsequation}), the value of $f(A)(\forall A\subseteq V)$ can be estimated unbiasedly by any set $\mathcal{U}$ of RR-sets as follows:
\begin{eqnarray}
\bar{f}(\mathcal{U},A)={n\cdot\sum\nolimits_{R\in \mathcal{U}} X(R, A) }/{|\mathcal{U}|}
\label{eqn:estimation}
\end{eqnarray}
where $X(R,A)\triangleq \min\{1,|R\cap A|\}$.
It is noted that the RR-set sampling method can also be applied to the Triggering model, where equations~(\ref{eqn:borgsequation})-(\ref{eqn:estimation}) still hold~\cite{TangXS2014}.
The work of Borgs. \textit{et. al.}~\cite{Borgs2014} and other proposals~\cite{TangXS2014,TangSX2015,NguyenTD2016} have shown that the influence maximization problem can be efficiently solved by the RR-set sampling method. Nevertheless, how to use this method to solve the MCSS problem still remains largely open.
\begin{algorithm} [tp!]
\KwIn{$ G=(V,E), \delta, \alpha,\sigma,\gamma, \mu, \eta$
\KwOut{A set $S\subseteq V$ satisfying $f(S)\geq (1-\alpha)\eta$ w.h.p.}
%
$W_1\leftarrow \langle (1-\alpha)\eta,\frac{\gamma}{1-\alpha},\frac{\delta}{2\mu} \rangle$; $\Lambda\leftarrow (1-{\alpha}+\gamma)\eta$ \label{ln:setw1}\\
$W_2\leftarrow \langle \eta,\sigma,\frac{\delta}{2} \rangle$;~$T\leftarrow \mathsf{SetT}(W_1,W_2)$ \label{ln:setw2}\\
Generate a set $\mathcal{R}$ of RR-sets satisfying $|\mathcal{R}|=T$ \label{ln:generaterrsets}\\
$S\leftarrow \mathsf{MCA}(\mathcal{R},\Lambda) $;~\Return{$S$} \label{ln:setbiglambda}
\hrule
\textbf{Function} $\mathsf{MCA}(\mathcal{R},\Lambda)$\\
{$A\leftarrow \emptyset$} \\
\lIf{$|V|<\Lambda$}{\Return $\emptyset$}
\While{$\bar{f}(\mathcal{R},A)<\Lambda$}{
$u^*\leftarrow \arg\max_{u\in V\backslash A}\frac{\min\{\bar{f}(\mathcal{R},A\cup \{u\}),\Lambda\}-\bar{f}(\mathcal{R},A)}{C(\{u\})}$;\\
$A\leftarrow A\cup \{u^*\}$
}
\Return{$A$}
\hrule \textbf{Function} $\mathsf{SetT}(W_1,W_2)$\\
$T\leftarrow \lceil \max\{\mathrm{ut}(W_1), \mathrm{lt}(W_2)\} \rceil$;
\Return{$T$}
\caption{Bi-Criteria Approximation Algorithm for General Costs $(\mathsf{BCGC})$}
\label{alg:bcgc}
\end{algorithm}
\section{Bi-Criteria Approximation Algorithms}
\label{sec:bicriteria}
In this section, we propose bi-criteria approximation (BA) algorithms for the MCSS problem {under the GC+NV setting}.
\subsection{The BCGC Algorithm}
Our first bi-criteria approximation algorithm is called $\mathsf{BCGC}$, shown in Algorithm~\ref{alg:bcgc}. $\mathsf{BCGC}$ first generates a set $\mathcal{R}$ of $T$ RR-sets (line~\ref{ln:generaterrsets}) according to the input variables $\delta, \alpha,\sigma,\gamma, \mu$ and $\eta$, and then calls the function $\mathsf{MCA}$ to find a min-cost node set $S$ satisfying $\bar{f}(\mathcal{R},S)\geq \Lambda$, which is returned as the solution to MCSS.
It can be verified that $\bar{f}(\mathcal{R},\cdot)$ is a monotone and non-negative submodular function defined on the ground set $V$, so $\mathsf{MCA}$ is essentially a (deterministic) greedy submodular set cover algorithm similar to $\mathsf{GreSSC}$.
The key issue in $\mathsf{BCGC}$ is how to determine the values of $T$ and $\Lambda$. Intuitively, $T$ and $\Lambda$ should be large enough such that $\mathsf{MCA}$ can output a {feasible solution} $S$ (i.e., $f(S)\geq (1-\alpha)\eta$) with a cost close to $C(S_{opt})$. On the other hand, we also want $T$ and $\Lambda$ to be small such that the time complexity of $\mathsf{BCGC}$ can be reduced. To see how $\mathsf{BCGC}$ achieve these goals, we first introduce the following functions:
\begin{definition}
For any $\Gamma\in (0,n]$ and $\beta,\theta\in (0,1)$, define
\begin{eqnarray}
&&\mathrm{ut}(\langle \Gamma, \beta, \theta\rangle) \triangleq \min\left\{\frac{n^2}{2\beta^2\Gamma^2}\ln\frac{1}{\theta},\frac{2n(\beta+3)}{3\beta^2\Gamma}\ln\frac{1}{\theta}\right\};~~~~\label{eqn:defiofut}\\
&&\mathrm{lt}(\langle\Gamma, \beta, \theta\rangle) \triangleq [{2n }/({\beta^2 \Gamma})]\ln({1}/{\theta}); \label{eqn:defioflt}\\
&&\mathcal{Q}[\Gamma]\triangleq\{A\subseteq V: f(A)<\Gamma\};~D(\Gamma)\triangleq\left({{e}n}/{\lfloor \Gamma\rfloor}\right)^{\lfloor \Gamma\rfloor}, \label{eqn:defiofqgamma}
\end{eqnarray}
where $e$ is the base of natural logarithms.
\end{definition}
\noindent From Eqn.~(\ref{eqn:defiofqgamma}), it can be seen that
\begin{eqnarray}
|\mathcal{Q}[\Gamma]|\leq\sum\nolimits_{i=1}^{\lfloor \Gamma\rfloor} {n \choose i}\leq \sum\nolimits_{i=1}^{\lfloor \Gamma\rfloor}\frac{n^i}{i!}=\sum_{i=1}^{\lfloor \Gamma\rfloor}\frac{{\lfloor \Gamma\rfloor}^i}{i!}\frac{n^i}{\lfloor \Gamma\rfloor^i}\leq D(\Gamma)\nonumber
\end{eqnarray}
Moreover, (\ref{eqn:defiofut})-(\ref{eqn:defiofqgamma}) are useful for Lemmas~\ref{lma:utltbound}-\ref{lma:boundingoursolution}, which can be proved by the concentration bounds in probability theory~\cite{FanL2006}:
\begin{lemma}
Given any $A\subseteq V$ and any set $\mathcal{U}$ of RR-sets, if $f(A)< \Gamma$ and $|\mathcal{U}|\geq \mathrm{ut}(\Gamma,\beta,\theta)$, then we have $\mathbb{P}\{\bar{f}(\mathcal{U},A)\geq (1+\beta)\Gamma\}\leq \theta$; if $f(A)\geq \Gamma$ and $|\mathcal{U}|\geq \mathrm{lt}(\Gamma,\beta, \theta)$, then we have $\mathbb{P}\{\bar{f}(\mathcal{U},A)< (1-\beta) \Gamma \}\leq \theta$
\label{lma:utltbound}
\end{lemma}
\begin{lemma}
Let $\mathcal{U}$ be any set of RR-sets satisfying $|\mathcal{U}|\geq \mathrm{ut}(\Gamma,\beta,\frac{\theta}{D(\Gamma)})$. We have $$\mathbb{P}\{\exists A\in \mathcal{Q}[\Gamma]: \bar{f}(\mathcal{U},A)\geq (1+\beta)\Gamma\}\leq \theta.$$
\label{lma:boundingoursolution}
\end{lemma}
Note that $\mathsf{BCGC}$ set $|\mathcal{R}|=T\geq \max\{\mathrm{ut}(W_1), \mathrm{lt}(W_2)\}$ and $\Lambda=(1-{\alpha}+\gamma)\eta$ (lines~\ref{ln:setw1}-\ref{ln:setw2}). According to Lemma~\ref{lma:boundingoursolution}, when $|\mathcal{R}|\geq \mathrm{ut}(W_1)$ and $\mu\geq D(\eta-\alpha\eta)$, we must have
\begin{eqnarray}
\mathbb{P}\{\exists A\in \mathcal{Q}[(1-\alpha)\eta]: \bar{f}(\mathcal{R},A)\geq (1-\alpha+\gamma)\eta\}\leq {\delta}/{2}\label{eqn:ensurefeasible}
\end{eqnarray}
Moreover, when $|\mathcal{R}|\geq \mathrm{ut}(W_2)$, Lemma~\ref{lma:utltbound} gives us:
\begin{eqnarray}
\mathbb{P}\{\bar{f}(\mathcal{R},S_{opt})< (1-\sigma)\eta\}\leq {\delta}/{2} \label{eqn:ensurear}
\end{eqnarray}
Intuitively, Eqn.~(\ref{eqn:ensurefeasible}) ensures that none of the infeasible solutions in $\mathcal{Q}[(1-\alpha)\eta]$ can be returned by $\mathsf{BCGC}$ with high probability, while Eqn.~(\ref{eqn:ensurear}) ensures that $\bar{f}(\mathcal{R},S_{opt})$ must be close to $\eta$. Using (\ref{eqn:ensurefeasible})-(\ref{eqn:ensurear}), we can get:
\begin{theorem}
When $\sigma,\gamma>0$, $\sigma+\gamma<\alpha<1$ and $\mu\geq D(\eta-\alpha\eta)$, $\mathsf{BCGC}$ returns a set $S\subseteq V$ satisfying $f(S)\geq (1-\alpha)\eta$ and $C(S)\leq (1+\ln\frac{1-\sigma}{\alpha-\gamma-\sigma})C(S_{opt})$ with the probability of at least $1-\delta$.
\label{thm:approximationratio}
\end{theorem}
\begin{proof}
Let $B^*$ denote an optimal solution to the optimization problem: $$\min\{C(A)| \bar{f}(\mathcal{R},A)\geq (1-\sigma)\eta \wedge A\subseteq V\}.$$ As $\mathsf{MCA}$ is essentially a deterministic greedy submodular cover algorithm and $(1-\alpha+\gamma)\eta <(1-\sigma)\eta$, we can use Fact~\ref{fct:goyalgreedy} to get $$C(S)\leq (1+\ln\frac{1-\sigma}{\alpha-\gamma-\sigma})C(B^*).$$ Therefore, if $\bar{f}(\mathcal{R},S_{opt})\geq (1-\sigma)\eta$ holds, then we must have $C(B^*)\leq C(S_{opt})$ and hence $$C(S)\leq (1+\ln\frac{1-\sigma}{\alpha-\gamma-\sigma})C(S_{opt}).$$ According to (\ref{eqn:ensurear}), the probability that $\bar{f}(\mathcal{R},S_{opt})\geq(1-\sigma)\eta$ does not hold is at most $\delta/2$. Besides, as $\bar{f}(\mathcal{R},S)\geq (1-\alpha+\gamma)\eta$, the probability that $\mathsf{BCGC}$ returns an infeasible solution in $\mathcal{Q}[(1-\alpha)\eta]$ is no more than $\mathbb{P}\{\exists A\in \mathcal{Q}[(1-\alpha)\eta]: \bar{f}(\mathcal{R},A)\geq (1-\alpha+\gamma)\eta\}$, which is bounded by $\delta/2$ due to (\ref{eqn:ensurefeasible}). The theorem then follows by using the union bound.
\end{proof}
Note that the approximation ratio of $\mathsf{BCGC}$ nearly matches the approximation ratio (i.e., $1+\ln\frac{1}{\alpha}$) shown in Fact~\ref{fct:goyalgreedy}, which is derived under the EV setting. For example, if we set $\sigma=\gamma=\frac{\alpha}{4}$, then the approximation ratio of $\mathsf{BCGC}$ is at most $1+\ln\frac{2}{\alpha}$, which is larger than $1+\ln\frac{1}{\alpha}$ by only $\ln 2$.
$\mathsf{BCGC}$ spends most of its running time on generating RR-sets. According to the setting of $T$, it can be seen that $\mathsf{BCGC}$ generates at most $\mathcal{O}(\frac{n}{\alpha^2}\ln \frac{n}{\eta\delta})$ RR sets (see the proof of Theorem~\ref{thm:timecomplexityBCGC}). Therefore, we can get:
\begin{theorem}
Let $q=\max\{f(v)|v\in V\}$. $\mathsf{BCGC}$ can achieve the performance bound shown in Theorem~\ref{thm:approximationratio} under the expected time complexity of $\mathcal{O}(\frac{m q}{\alpha^2}\ln\frac{n}{(1-\alpha)\delta\eta})$.
\label{thm:timecomplexityBCGC}
\end{theorem}
\begin{algorithm} [tp!]
\KwIn{\textcolor{black}{$G=(V,E), \delta, \alpha,\sigma,\gamma, \eta$}
\KwOut{A set $S\subseteq V$ satisfying $f(S)\geq (1-\alpha)\eta$ w.h.p.}
$W_1\leftarrow \langle (1-\alpha)\eta,\frac{\gamma}{1-\alpha},\frac{\delta}{6D(\eta-\alpha\eta)} \rangle$;~$W_2\leftarrow \langle \eta,\sigma,\frac{\delta}{6} \rangle$ \label{ln:setttingTtegc1}\\
$T\leftarrow \mathsf{SetT}(W_1,W_2)$;~$\theta\leftarrow {\delta}/{3};~\mathcal{R}\leftarrow \emptyset$ \label{ln:setttingTtegc2}\\%;~$T\leftarrow \lceil \max\{\mathrm{ut}(W_1),
\While{$|\mathcal{R}|\leq T$}{ \label{ln:iterationstart}
Generate some RR-sets and add them into $\mathcal{R}$ until $|\mathcal{R}|=\min\{T,\lceil \mathrm{lt}(\langle \eta,\sigma,\frac{\theta}{3}\rangle)\rceil\}$ \label{ln:iterategenerate2}\\
$S\leftarrow \mathsf{MCA}(\mathcal{R},(1-\alpha+\gamma)\eta)$ \label{ln:findsbigger1minusalphaplus}\\
\lIf{$|\mathcal{R}|=T$}{\Return{$S$} \label{ln:achievesthethreldshold}}
$(\mathcal{U},\mathrm{Pass})\leftarrow \mathsf{TEST}(S,\frac{\gamma}{2(1-\alpha)}, (1-\alpha)\eta,\textcolor{black}{\frac{2\theta}{3}},T-|\mathcal{R}|)$ \label{ln:calltest}\\
\lIf{$\mathrm{Pass}=\mathbf{True}$}{\Return{$S$} \label{ln:returnbypass}}
$\mathcal{R}\leftarrow \mathcal{R}\cup \mathcal{U}$;~$\theta\leftarrow \frac{\theta}{2}$ \label{ln:iterationend}\\
}
\Return{$S$}
\caption{The Trial-and-Error Algorithm for General Costs ($\mathsf{TEGC}$)}
\label{alg:mca}
\end{algorithm}
\subsection{A Trial-and-Error Algorithm}
\label{sec:tae}
It can be seen that $\mathsf{BCGC}$ behaves in an ``once-for-all'' manner, i.e, it generates all the RR-sets in one batch, and then finds a solution using the generated RR-sets. In this section, we propose another ``trial-and-error'' algorithm (called $\mathsf{TEGC}$) for the MCSS problem, which runs in iterations and ``lazily'' generates RR-sets when necessary.
More specifically, $\mathsf{TEGC}$ runs in iterations with the input variables satisfying $\sigma+\gamma<\alpha<1$. In each iteration (lines~\ref{ln:iterationstart}-\ref{ln:iterationend}), it first generates a set $\mathcal{R}$ of RR-sets (line~\ref{ln:iterategenerate2}) according to Lemma~\ref{lma:utltbound} to ensure $$\mathbb{P}\{\bar{f}(\mathcal{R},S_{opt})< (1-\sigma)\eta\}\leq {\theta}/{3},$$
where the parameter $\theta$ will be explained shortly. Then $\mathsf{TEGC}$ calls $\mathsf{MCA}$ in line~\ref{ln:findsbigger1minusalphaplus} to find an approximation solution $S$ satisfying $$\bar{f}(\mathcal{R},S)\geq (1-\alpha+\gamma)\eta.$$ However, it is possible that $S$ is an infeasible solution in $\mathcal{Q}[(1-\alpha)\eta]$. Therefore, $\mathsf{TEGC}$ calls $\mathsf{TEST}$ to judge whether $f(S)\geq (1-\alpha)\eta$ (line~\ref{ln:calltest}). If $\mathsf{TEST}$ returns $\mathrm{Pass}=\mathbf{True}$, it implies that $S$ is a feasible solution w.h.p., so $\mathsf{TEGC}$ terminates and returns $S$. Otherwise, $\mathsf{TEGC}$ enters into another iteration and adds more RR-sets into $\mathcal{R}$. This ``trial and error'' process repeats until $|\mathcal{R}|$ achieves a predefined threshold $T$ (line~\ref{ln:achievesthethreldshold}). As $T=\mathcal{O}(\frac{n}{\alpha^2}\ln \frac{n}{\eta\delta})$ is set similarly with that in $\mathsf{BCGC}$ (lines~\ref{ln:setttingTtegc1}-\ref{ln:setttingTtegc2}), $\mathsf{TEGC}$ is guaranteed to find a feasible solution with high probability.
The parameter $\theta$ in $\mathsf{TEGC}$ is roughly explained as follows. Intuitively, $\theta$ indicates the total probability of the ``bad events'' (e.g., $\{\bar{f}(\mathcal{R},S_{opt})< (1-\sigma)\eta\}$) happen in any iteration of $\mathsf{TEGC}$ when $|\mathcal{R}|<T$. In the first iteration, we set $\theta=\frac{\delta}{3}$ and $\theta$ is decreased by a factor $2$ in every subsequent iteration. We also constrain the probability that the bad events happen when $|\mathcal{R}|=T$ by $\frac{\delta}{3}$ (lines~\ref{ln:setttingTtegc1}-\ref{ln:setttingTtegc2}). Using the union bound, the total probability that $\mathsf{TEGC}$ returns a ``bad'' solution conflicting our performance bounds is upper-bounded by $\frac{\delta}{3}+\sum_{i=0}^\infty \frac{\delta}{3\cdot 2^i}\leq \delta$.
\textcolor{black}{By similar reasoning with that in Theorem~\ref{thm:approximationratio}, we can prove that $\mathsf{TEGC}$ has the same approximation ratio as $\mathsf{BCGC}$:}
\begin{theorem}
When $\sigma,\gamma>0$ and $\sigma+\gamma<\alpha<1$, $\mathsf{TEGC}$ returns a set $S\subseteq V$ satisfying $f(S)\geq (1-\alpha)\eta$ and $C(S)\leq (1+\ln\frac{1-\sigma}{\alpha-\gamma-\sigma})C(S_{opt})$ with the probability of at least $1-\delta$.
\label{thm:artegc}
\end{theorem}
\noindent \textit{\textbf{The Design of function}} $\mathsf{TEST}$: Next, we explain how the function $\mathsf{TEST}(A,\kappa, \Gamma,\beta,L)$ is implemented. $\mathsf{TEST}$ maintains three threshold values $\ell, M, L$. When $L\leq M$, it simply generates $L$ RR-sets and returns them (lines~\ref{ln:returnLrrsetsbegin}-\ref{ln:returnLrrsetsend}). Otherwise, it keeps generating RR-sets $U_1,U_2,\cdots$ until either $\sum_{i=1}^j X(U_i,A)\geq \ell$ or $|\mathcal{U}|= M$, where $\mathcal{U}$ is the set of generated RR-sets. Note that $\mathsf{TEST}$ is called with $L=T-|\mathcal{R}|$, which implies that the total number of generated RR-sets in $\mathsf{TEGC}$ never exceeds $T$.
Intuitively, if $f(A)$ is very large, then $X(U_i,A)$ ($\forall i$) must have a high probability to be $1$, so there is a high probability that $\sum_{i=1}^M X(U_i,A)\geq \ell$ and hence $\mathsf{TEST}$ returns $\mathrm{Pass}=\mathbf{True}$. Conversely, if $f(A)$ is very small, then there is a high probability that $\sum_{i=1}^M X(U_i,A)<\ell$ and hence $\mathsf{TEST}$ returns $\mathrm{Pass}=\mathbf{False}$. By setting the values of $\ell$ and $M$ (line~\ref{ln:settingml}) based on the Chernoff bounds, we get the following theorem:
\begin{theorem}
For any $A\subseteq V$, if $f(A)\geq (1+\kappa)\Gamma$ and $L> M$, then the probability that $\mathsf{TEST}(A,\kappa, \Gamma,\beta,L)$ returns $\mathrm{Pass}=\mathbf{True}$ is at least $1-{\beta}/{2}$; if $f(A)<\Gamma$ and $L> M$, then the probability that $\mathsf{TEST}(A,\kappa, \Gamma,\beta,L)$ returns $\mathrm{Pass}=\mathbf{False}$ is at least $1-{\beta}/{2}$.
\label{thm:betais1}
\end{theorem}
\textcolor{black}{Note that $\mathsf{TEST}$ is called by $\mathsf{TEGC}$ with $A=S$, $\Gamma= (1-\alpha)\eta$ and $\kappa=\frac{\gamma}{2(1-\alpha)}$.} So Theorem~\ref{thm:betais1} implies that $\mathsf{TEGC}$ always returns a feasible solution with high probability. When $(1-\alpha)\eta \leq f(S) < (1-\alpha+\gamma/2)\eta$, it is possible that $\mathsf{TEST}$ returns $\mathrm{Pass}=\mathbf{False}$, but this does not harm the the correctness of $\mathsf{TEGC}$ and only results in more iterations; moreover, the probability for this event to happen can be very small as $\gamma$ is usually small.
\begin{algorithm} [tp!]
$\ell \leftarrow \left\lceil\frac{2(1+\kappa)\Gamma}{(2+\kappa)n}+\frac{8(3+2\kappa)(1+\kappa)}{3\kappa^2}\ln\frac{2}{\beta}\right\rceil$; \label{ln:settingml}
$M\leftarrow \left\lfloor \frac{(2+\kappa)n\ell}{2(1+\kappa)\Gamma}\right\rfloor$\\
$\mathcal{U}\leftarrow \emptyset;~\mathrm{Pass}\leftarrow \mathbf{False};~Z_0\leftarrow 0;$\\
\If{$L\leq M$}{\label{ln:returnLrrsetsbegin}
Generate $L$ RR-sets and add them into $\mathcal{U}$\\
\Return{$(\mathcal{U}, \mathrm{Pass})$ \label{ln:returnLrrsetsend}}
}
\For{$j=1$ \KwTo $M$}{
Generate an RR-set $U_j$ and add it into $\mathcal{U}$\\
$Z_j\leftarrow Z_{j-1}+ \min\{|A\cap U_{j}|,1\}$\\
\If{$Z_j= \ell$}{
$\mathrm{Pass}\leftarrow \mathbf{True}$;~\Return{$(\mathcal{U},\mathrm{Pass})$}
}
}
\Return{$(\mathcal{U}, \mathrm{Pass})$}
\caption{$\mathsf{TEST}(A,\kappa, \Gamma,\beta,L)$}
\label{alg:mca}
\end{algorithm}
\subsection{Theoretical Comparisons for the BA Algorithms}
\label{sec:compareundergc}
We compare the theoretical performance of $\mathsf{BCGC}$ and $\mathsf{TEGC}$ with the state-of-the-art algorithms as follows.
\subsubsection{Comparing with Goyal et.al.'s Results~\cite{Goyal2013}}
To the best of our knowledge, \textbf{the only prior algorithm with a provable performance bound for the MCSS problem under the GC+NV setting is the one proposed in~\cite{Goyal2013}}, which is based on the $\mathsf{GreSSC}$ algorithm. We quote the result of~\cite{Goyal2013} in Fact~\ref{fct:goyalsresult}:
\begin{fact}[\cite{Goyal2013}]
Let $S'=\mathsf{GreSSC}((1-\alpha)\eta,\beta,0,0)$. Then we have $f(S')\geq (1-\alpha)\eta$ and $C(S')\leq (1+\phi)(1+\ln\frac{1}{\alpha})C(S_{opt})$, where $\phi$ and $\beta$ satisfy:
\begin{eqnarray}
\frac{\beta}{1-(1-\beta)(1-{1}/{C(S_{opt})})}=\frac{(1/\alpha)^{\phi}-1}{(1/\alpha)^{\phi+1}-1}
\end{eqnarray}
\label{fct:goyalsresult}
\end{fact}
However, Goyal \textit{et.al.}~\cite{Goyal2013} did not mention how to implement $\mathsf{GreSSC}((1-\alpha)\eta,\beta,0,0)$. A crucial problem is that they require $\hat{f}(A')\leq {f}(A')~(\forall A'\subseteq V)$ in the implementation of $\mathsf{GreSSC}$ (i.e., $\gamma_2=0$); otherwise their proof for the approximation ratio in Fact~\ref{fct:goyalsresult} does not hold. However, as computing $f(A')$ is \#P-hard and $\hat{f}(A')$ in $\mathsf{GreSSC}$ is computed by monte-carlo simulations, $\gamma_2=0$ actually implies that infinite times of monte-carlo simulations should be conducted to compute $\hat{f}(A')$ according to the Chernoff bounds~\cite{FanL2006}. Due to this reason, we think that the approximation ratio shown in Fact~\ref{fct:goyalsresult} is mainly valuable on the theoretical side and is not likely to be implemented in polynomial time.
\subsubsection{Comparing with Kuhnle et.al.'s Results~\cite{Kuhnle2017}}
Very recently, Kuhnle \textit{et.al.}~\cite{Kuhnle2017} have proposed some elegant bi-criteria approximation algorithms for the MCSS problem, but only under the UC setting. We quote their results below:
\begin{fact}[\cite{Kuhnle2017}]
Define the CEA assumption as follows: for any $A\subseteq V$ such that $f(A)<\eta$, there always exists a node $u$ such that $f(A\cup \{u\})-f(A)\geq 1$. Under the UC setting, the STAB algorithms can find a set $S'\subseteq V$ satisfying $|S'|\leq (1+2\rho \eta +\log \eta)|S_{opt}|$ and 1) or 2) listed below with the probability of at least $1-\iota n^3$. The STAB-C1 algorithm has $\mathcal{O}((n+m)|S'|\log\frac{1}{\iota n^2}/\rho^2)$ time complexity. The STAB-C2 algorithm has $\mathcal{O}((n+m)|S'|^2\log |S'|\log\frac{1}{\iota n^2}/\rho^2)$ time complexity.
\begin{enumerate}
\item $f(S')\geq (1-\rho)\eta$ when the CEA assumption holds.
\item $f(S')\geq \eta -(1+\varepsilon)|S_{opt}|$ without the CEA assumption
\end{enumerate}
\label{fct:kuhapproxalg}
\end{fact}
Note that Fact~\ref{fct:kuhapproxalg} depends on the CEA assumption, and there is a large gap between $\eta$ and $f(S')$ (at least $|S_{opt}|$) if without the CEA assumption. In contrast, the performance ratios of $\mathsf{BCGC}$ or $\mathsf{TEGC}$ do not require any special properties of the network and we can guarantee $f(S)\geq (1-\alpha)\eta$ w.h.p. for any $\alpha>0$. Most importantly, Fact~\ref{fct:kuhapproxalg} only holds under the UC setting while our performance bounds shown in Theorems~\ref{thm:approximationratio}-\ref{thm:artegc} hold under the GC setting.
\section{Approximation Algorithms for Uniform Costs}
\label{sec:approxforuc}
In this section, we propose approximation (AP) algorithms for the MCSS problem under the UC+NV setting.
\subsection{The AAUC Algorithm}
We first propose an algorithm called $\mathsf{AAUC}$. $\mathsf{AAUC}$ first generates a set $\mathcal{R}$ of $T$ RR-sets, and then calls $\mathsf{MCA}$ to return a set $S$ satisfying $\bar{f}(\mathcal{R},S)\geq \Lambda=(1+\tau)\eta$, where $\tau$ is a variable in $(0,1)$.
Although $\mathsf{AAUC}$ looks similar to $\mathsf{BCGC}$, its key idea (i.e., how to set the parameters $T$ and $\Lambda$) is very different from that in $\mathsf{BCGC}$, which is explained as follows.
Recall that $\mathsf{MCA}$ is a greedy algorithm. Suppose that $\mathsf{MCA}$ sequentially selects $x_1,x_2,\cdots,x_n$ after it is called by $\mathsf{AAUC}$. Let $B_i(\forall 1\leq i\leq n)$ denote $\{x_1,\cdots,x_i\}$. Let $s=\min\{i|\bar{f}(\mathcal{R},B_i)\geq (1-\tau)\eta\}$ and $t=\min\{i|\bar{f}(\mathcal{R},B_i)\geq (1+\tau)\eta\}$.
As each node has a cost $1$ and $\bar{f}(\mathcal{R},\cdot)$ is submodular, we can use the submodular functions' properties to prove:
\begin{lemma}
For any $\tau\in (0,1)$, we have $|B_{s}|\leq \lceil \ln\frac{n\eta}{n-\eta}\rceil |D^*|$ $+1$, where $D^*=\arg\min_{A\subseteq V\wedge \bar{f}(\mathcal{R},A)\geq (1-\tau)\eta}|A|$.
\label{lma:thearoftminus1}
\end{lemma}
\noindent Clearly, when $\tau$ is sufficiently small, $s$ and $t$ must be very close. Indeed, we can prove:
\begin{lemma}
When $\tau\in (0,\frac{n-\eta}{2n\eta+\eta}]$, we have $|B_{t}|\leq |B_{s}|+1$.
\label{lma:atminus1issmall}
\end{lemma}
\noindent Besides, according to line~\ref{ln:setT} and Lemmas~\ref{lma:utltbound}-\ref{lma:boundingoursolution}, we know that the set $\mathcal{R}$ generated by $\mathsf{AAUC}$ satisfies
\begin{eqnarray}
&&\mathbb{P}\{\exists A\in \mathcal{Q}[\eta]: \bar{f}(\mathcal{R},A)\geq (1+\tau)\eta\}\leq {\delta}/{2} \label{eqn:aaucfeasible}\\
&&\mathbb{P}\{\bar{f}(\mathcal{R},S_{opt})< (1-\tau)\eta\}\leq {\delta}/{2} \label{eqn:aaucoptlarge}
\end{eqnarray}
\textcolor{black}{for any $\tau\in (0,1)$ and $\mu\geq D(\eta)$.} This implies that $S$ is a feasible solution (i.e., $S\notin \mathcal{Q}[\eta]$) and $|D^*|\leq |S_{opt}|$ with high probability. Moreover, according to Lemmas~\ref{lma:thearoftminus1}-\ref{lma:atminus1issmall}, we have $|S|=|B_{t}|\leq |B_{s}|+1$ and $|B_s|\leq \lceil \ln\frac{n\eta}{n-\eta}\rceil |D^*|+1$. Combining all these results gives us:
\begin{algorithm} [tp!]
\KwIn{\textcolor{black}{$ G=(V,E), \eta,\delta, \tau, \mu$}}
\KwOut{A set $S\subseteq V$ satisfying $f(S)\geq \eta$ w.h.p.}
%
$W_1\leftarrow \langle \eta, \tau,\frac{\delta}{2\mu} \rangle;~W_2\leftarrow \langle \eta, \tau,\frac{\delta}{2} \rangle$;~$T\leftarrow \mathsf{SetT}(W_1,W_2)$ \label{ln:setT}\\
Generate a set $\mathcal{R}$ of RR-sets satisfying $|\mathcal{R}|=T$ \\
$S\leftarrow \mathsf{MCA}(\mathcal{R},(1+{\tau})\eta)$;~\Return{$S$} \label{ln:return1plustau}
\caption{Approximation Algorithm for Uniform Costs ($\mathsf{AAUC}$)}
\label{alg:aaucbasic}
\end{algorithm}
\begin{theorem}
When $\tau\in (0,\frac{n-\eta}{2n\eta+\eta}]$ and $\mu\geq D(\eta)$, $\mathsf{AAUC}$ returns a set $S\subseteq V$ such that $f(S)\geq \eta$ and $|S|\leq \lceil \ln\frac{n\eta}{n-\eta}\rceil |{S}_{opt}|+2$ with the probability of at least $1-\delta$.
\label{thm:arofaauc}
\end{theorem}
\begin{proof}
Note that $\bar{f}(\mathcal{R},S)\geq (1+\tau)\eta$ according to line~\ref{ln:return1plustau} of $\mathsf{AAUC}$. Therefore, when $\mu\geq D(\eta)$, the probability that $S$ is an infeasible solution in $\mathcal{Q}[\eta]$ must be no more than $\delta/2$ according to line~\ref{ln:setT} of $\mathsf{AAUC}$ and eqn.~(\ref{eqn:aaucfeasible}).
Let $D^*=\arg\min_{A\subseteq V\wedge \bar{f}(\mathcal{R},A)\geq (1-\tau)\eta}|A|$. When $\tau\in (0,\frac{n-\eta}{2n\eta+\eta}]$, we must have $|S|=|B_t|\leq \lceil \ln\frac{n\eta}{n-\eta}\rceil |D^*|+2$ according to Lemmas~\ref{lma:thearoftminus1}-\ref{lma:atminus1issmall}. Therefore, we have
\begin{eqnarray}
&&\mathbb{P}\{|S|> \lceil \ln\frac{n\eta}{n-\eta}\rceil |{S}_{opt}|+2\}\leq \mathbb{P}\{|D^*|>|{S}_{opt}|\}\nonumber\\
&\leq& \mathbb{P}\{\bar{f}(\mathcal{R},S_{opt})< (1-\tau)\eta\}\leq {\delta}/{2} \label{eqn:lessthandeltadevide2}
\end{eqnarray}
where (\ref{eqn:lessthandeltadevide2}) is due to the definition of $D^*$ and equation~(\ref{eqn:aaucoptlarge}). The theorem then follows by using the union bound.
\end{proof}
By very similar reasoning with that in Theorem~\ref{thm:timecomplexityBCGC}, we can also prove the time complexity of $\mathsf{AAUC}$ as follows:
\begin{theorem}
Let $q=\max\{f(v)|v\in V\}$. $\mathsf{AAUC}$ can achieve the performance bounds shown Theorem~\ref{thm:arofaauc} under the expected time complexity of $\mathcal{O}(\frac{m q}{\varrho^2}\ln\frac{n}{\delta\eta})$ where $\varrho=\frac{n-\eta}{2n\eta+\eta}$.
\label{thm:timeofaauc}
\end{theorem}
\subsection{An Adaptive Trial-and-Error Algorithm}
It can be seen from Theorem~\ref{thm:timeofaauc} that the running time of $\mathsf{AAUC}$ is inversely proportional to $\varrho$, which can be a small number. To address this problem, we propose an adaptive trial-and-error algorithm called $\mathsf{ATEUC}$, shown in Algorithm~\ref{alg:ateuc}. $\mathsf{ATEUC}$ uses a dynamic parameter $\alpha$ for generating RR-sets, and adaptively changes the value of $\alpha$ until a satisfying approximate solution is found.
More specifically, $\mathsf{ATEUC}$ first determines a threshold $T$ using $\varrho$ (lines~\ref{ln:ateucsett1}-\ref{ln:ateucsett2}), then $\mathsf{ATEUC}$ calls a function $\mathsf{Shrink}$ with any $\alpha$ that is larger than $\varrho$. Note that the performance bound of $\mathsf{ATEUC}$ (see Theorem~\ref{thm:arofateuc}) does not depend on the value of $\alpha$, and setting $\alpha>\varrho$ is only for reducing the number of generated RR-sets.
In each iteration, $\mathsf{Shrink}$ first generates a set $\mathcal{R}$ of RR-sets in a similar way with that in $\mathsf{TEGC}$ (line~\ref{ln:taegenrrset2}) to ensure $\mathbb{P}\{\bar{f}(\mathcal{R},S_{opt})< (1-\alpha)\eta\}\leq {\theta}/{3}$, where $\theta$ is set by a similar way with that in $\mathsf{TEGC}$.
After that, $\mathsf{Shrink}$ calls $\mathsf{MCA}$ to find $S_1\subseteq S_2\subseteq V$ such that $\bar{f}(\mathcal{R}, S_1)\geq (1-\alpha)\eta$ and $\bar{f}(\mathcal{R}, S_2)\geq (1+\alpha)\eta$. If $|S_2|> 2|S_1|$, $\mathsf{Shrink}$ decreases $\alpha$ and $\theta$ and enters the next iteration (lines~\ref{ln:decreasethetaandalpha}). If $|S_2|\leq 2|S_1|$, it implies that $\alpha$ is small enough, so $\mathsf{Shrink}$ calls $\mathsf{TEST}$ to judge whether $f(S_2)\geq \eta$ (line~\ref{ln:ateuccalltest}). If $\mathsf{TEST}$ returns $\mathrm{Pass}=\mathbf{True}$, then $\mathsf{Shrink}$ returns $S_2$ as the solution (line~\ref{ln:returns2withrlessT}), otherwise it decreases the value of $\theta$ and enters the next iteration (line~\ref{ln:decreasetheta}).
\begin{algorithm} [tp!]
\KwIn{\textcolor{black}{$G=(V,E), \delta, \eta, \alpha$}}
\KwOut{A set $S\subseteq V$ satisfying $f(S)\geq \eta$ w.h.p.
}
$\varrho\leftarrow \frac{|V|-\eta}{2|V|\eta+\eta}$;~$W_1\leftarrow \langle \eta, \varrho,\frac{\delta}{6D(\eta)} \rangle$; \label{ln:ateucsett1}\\
$W_2\leftarrow \langle \eta, \varrho,\frac{\delta}{6} \rangle$;~$T\leftarrow \mathsf{SetT}(W_1,W_2)$; \label{ln:ateucsett2}\\%~$\theta\leftarrow \frac{\delta}{2}$\\
$(\mathcal{R},S)\leftarrow \mathsf{Shrink}(T,\eta,\alpha,\delta,\varrho)$;~\Return{$S$}
\hrule
\textbf{Function} $\mathsf{Shrink}(T, \eta,\alpha,\delta,\varrho)$\\
$\theta\leftarrow {\delta}/{3};~\mathcal{R}\leftarrow \emptyset$\\
\While{$|\mathcal{R}|\leq T$}{
Generate some RR-sets and add them into $\mathcal{R}$ until $|\mathcal{R}|=\min\{T,\lceil \mathrm{lt}(\langle \eta,\alpha,\frac{\theta}{3}\rangle)\rceil\}$ \label{ln:taegenrrset2} \\
\lIf{$|\mathcal{R}|=T$}{$\alpha\leftarrow \varrho$ \label{ln:settingtau}}
$S_1\leftarrow \mathsf{MCA}(\mathcal{R}, (1-\alpha)\eta)$;~$S_2\leftarrow \mathsf{MCA}(\mathcal{R}, (1+\alpha)\eta)$\\
\If{$S_2\neq \emptyset$}{
\If{$|S_2|\leq 2|S_1|$ \label{ln:s2lessthan2s1}}{
\lIf{$|\mathcal{R}|=T$}{\Return{$(\mathcal{R},S_2)$}\label{ln:returns2withrequalT}}
$(\mathcal{U},\mathrm{Pass})\leftarrow \mathsf{TEST}(S_2,\frac{1}{2}\alpha, \eta,{\frac{2\theta}{3}},T-|\mathcal{R}|)$ \label{ln:ateuccalltest}\\
\lIf{$\mathrm{Pass}= \mathbf{True}$}{\Return{$(\mathcal{R},S_2)$} \label{ln:returns2withrlessT}}
$\mathcal{R}\leftarrow \mathcal{R}\cup \mathcal{U};~\theta\leftarrow {\theta}/{2}$;~\textbf{continue}; \label{ln:decreasetheta}
}
}
$\alpha\leftarrow {\alpha}/{\sqrt{2}};~\theta\leftarrow {\theta}/{2}$ \label{ln:decreasethetaandalpha}\\
}
\Return{$(\mathcal{R},\emptyset)$}
\caption{The Adaptive Trial-and-Error Algorithm for Uniform Costs ($\mathsf{ATEUC}$)}
\label{alg:ateuc}
\end{algorithm}
In the case that the number of RR-sets generated in $\mathsf{Shrink}$ reaches $T$, $\mathsf{Shrink}$ returns $S_2$ by setting $\alpha=\varrho$ (lines~\ref{ln:settingtau},\ref{ln:returns2withrequalT}), so we must have $|S_2|\leq |S_1|+1$ and hence $|S_2|\leq \lceil \ln\frac{n\eta}{n-\eta}\rceil |{S}_{opt}|+2$ according to similar reasoning with that of Lemma~\ref{lma:atminus1issmall} and Theorem~\ref{thm:arofaauc}. However, as $\mathsf{Shrink}$ can return $S_2$ when $|S_2|\leq 2|S_1|$, the value of $\alpha$ is probably much larger than $\varrho$ when $\mathsf{Shrink}$ terminates, so less number of RR-sets can be generated. Moreover, by the setting of $\mathcal{R}$ in line~\ref{ln:taegenrrset2}, we always have $\bar{f}(\mathcal{R},S_{opt})\geq (1-\alpha)\eta$ w.h.p. and hence $|S_1|\leq \lceil \ln\frac{n\eta}{n-\eta}\rceil |S_{opt}|+1$ w.h.p. (using similar reasoning with that in Lemma~\ref{lma:thearoftminus1}). Based on these discussions, we can prove the approximation ratio of $\mathsf{ATEUC}$ as follows:
\begin{theorem}
\textcolor{black}{
With the probability of at least $1-\delta$, $\mathsf{ATEUC}$ returns a set $S\subseteq V$ satisfying $f(S)\geq \eta$ and $|S|\leq 2\lceil \ln\frac{n\eta}{n-\eta}\rceil |{S}_{opt}|+2$. }
\label{thm:arofateuc}
\end{theorem}
\begin{figure*}[htb!]
\centering
\includegraphics[width=0.65\textwidth]{figs/lineTitle_C1_C2_TEGC_BCGC_CELF_alpha02.pdf}
\label{fig:lineTitle_C1_C2_TEGC_BCGC_CELF_alpha02}
\begin{minipage}[htb!]{1.035\textwidth}
\subfigure[wiki-Vote (RT)]{
\includegraphics[width=0.18\textwidth]{figs/wikiVote_generalCost_RunningTime_alpha02.pdf}
\label{fig:wikiVote_generalCost_RunningTime_alpha02}
}
\subfigure[Pokec (RT)]{
\includegraphics[width=0.18\textwidth]{figs/pokec_generalCost_RunningTime_alpha02.pdf}
\label{fig:pokec_generalCost_RunningTime_alpha02}
}
\subfigure[LiveJournal (RT)]{
\includegraphics[width=0.18\textwidth]{figs/LiveJournal_generalCost_RunningTime_alpha02.pdf}
\label{fig:LiveJournal_generalCost_RunningTime_alpha02}
}
\subfigure[Orkut (RT)]{
\includegraphics[width=0.18\textwidth]{figs/orkut_generalCost_RunningTime_alpha02.pdf}
\label{fig:orkut_generalCost_RunningTime_alpha02}
}
\subfigure[Twitter (RT)]{
\includegraphics[width=0.18\textwidth]{figs/twitter_generalCost_RunningTime_alpha02.pdf}
\label{fig:twitter_generalCost_RunningTime_alpha02}
} \\
\subfigure[wiki-Vote (IS)]{
\includegraphics[width=0.18\textwidth]{figs/wikiVote_generalCost_Influence_alpha02.pdf}
\label{fig:wikiVote_generalCost_Influence_alpha02}
}
\subfigure[Pokec (IS)]{
\includegraphics[width=0.18\textwidth]{figs/pokec_generalCost_Influence_alpha02.pdf}
\label{fig:pokec_generalCost_Influence_alpha02}
}
\subfigure[LiveJournal (IS)]{
\includegraphics[width=0.18\textwidth]{figs/LiveJournal_generalCost_Influence_alpha02.pdf}
\label{fig:LiveJournal_generalCost_Influence_alpha02}
}
\subfigure[Orkut (IS)]{
\includegraphics[width=0.18\textwidth]{figs/orkut_generalCost_Influence_alpha02.pdf}
\label{fig:orkut_generalCost_Influence_alpha02}
}
\subfigure[Twitter (IS)]{
\includegraphics[width=0.18\textwidth]{figs/twitter_generalCost_Influence_alpha02.pdf}
\label{fig:twitter_generalCost_Influence_alpha02}
} \\
\subfigure[wiki-Vote (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/wikiVote_generalCost_Cost_alpha02.pdf}
\label{fig:wikiVote_generalCost_Cost_alpha02}
}
\subfigure[Pokec (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/pokec_generalCost_Cost_alpha02.pdf}
\label{fig:pokec_generalCost_Cost_alpha02}
}
\subfigure[LiveJournal (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/LiveJournal_generalCost_Cost_alpha02.pdf}
\label{fig:LiveJournal_generalCost_Cost_alpha02}
}
\subfigure[Orkut (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/orkut_generalCost_Cost_alpha02.pdf}
\label{fig:orkut_generalCost_Cost_alpha02}
}
\subfigure[Twitter (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/twitter_generalCost_Cost_alpha02.pdf}
\label{fig:twitter_generalCost_Cost_alpha02}
}\\
\end{minipage}
\renewcommand{\figurename}{Fig.}
\caption{Comparing the algorithms under the GC setting (IS: influence spread; RT: running time)}
\label{fig:the GC setting_alpha02}
\end{figure*}
\begin{figure*}[htb!]
\centering
\includegraphics[width=0.65\textwidth]{figs/lineTitle_C1_C2_ATEUC_AAUC_CELF_alpha02.pdf}
\label{fig:lineTitle_ATEUC_C1_C2_AAUC_CELF_alpha02}
\begin{minipage}[htb!]{1.035\textwidth}
\subfigure[wiki-Vote (RT)]{
\includegraphics[width=0.18\textwidth]{figs/wikiVote_uniformCost_RunningTime_alpha02.pdf}
\label{fig:wikiVote_uniformCost_RunningTime_alpha02}
}
\subfigure[Pokec (RT)]{
\includegraphics[width=0.18\textwidth]{figs/pokec_uniformCost_RunningTime_alpha02.pdf}
\label{fig:pokec_uniformCost_RunningTime_alpha02}
}
\subfigure[LiveJournal (RT)]{
\includegraphics[width=0.18\textwidth]{figs/LiveJournal_uniformCost_RunningTime_alpha02.pdf}
\label{fig:LiveJournal_uniformCost_RunningTime_alpha02}
}
\subfigure[Orkut (RT)]{
\includegraphics[width=0.18\textwidth]{figs/orkut_uniformCost_RunningTime_alpha02.pdf}
\label{fig:orkut_uniformCost_RunningTime_alpha02}
}
\subfigure[Twitter (RT)]{
\includegraphics[width=0.18\textwidth]{figs/twitter_uniformCost_RunningTime_alpha02.pdf}
\label{fig:twitter_uniformCost_RunningTime_alpha02}
} \\
\subfigure[wiki-Vote (IS)]{
\includegraphics[width=0.18\textwidth]{figs/wikiVote_uniformCost_Influence_alpha02.pdf}
\label{fig:wikiVote_uniformCost_Influence_alpha02}
}
\subfigure[Pokec (IS)]{
\includegraphics[width=0.18\textwidth]{figs/pokec_uniformCost_Influence_alpha02.pdf}
\label{fig:pokec_uniformCost_Influence_alpha02}
}
\subfigure[LiveJournal (IS)]{
\includegraphics[width=0.18\textwidth]{figs/LiveJournal_uniformCost_Influence_alpha02.pdf}
\label{fig:LiveJournal_uniformCost_Influence_alpha02}
}
\subfigure[Orkut (IS)]{
\includegraphics[width=0.18\textwidth]{figs/orkut_uniformCost_Influence_alpha02.pdf}
\label{fig:orkut_uniformCost_Influence_alpha02}
}
\subfigure[Twitter (IS)]{
\includegraphics[width=0.18\textwidth]{figs/twitter_uniformCost_Influence_alpha02.pdf}
\label{fig:twitter_uniformCost_Influence_alpha02}
} \\
\subfigure[wiki-Vote (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/wikiVote_uniformCost_Cost_alpha02.pdf}
\label{fig:wikiVote_uniformCost_Cost_alpha02}
}
\subfigure[Pokec (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/pokec_uniformCost_Cost_alpha02.pdf}
\label{fig:pokec_uniformCost_Cost_alpha02}
}
\subfigure[LiveJournal (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/LiveJournal_uniformCost_Cost_alpha02.pdf}
\label{fig:LiveJournal_uniformCost_Cost_alpha02}
}
\subfigure[Orkut (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/orkut_uniformCost_Cost_alpha02.pdf}
\label{fig:orkut_uniformCost_Cost_alpha02}
}
\subfigure[Twitter (Cost)]{
\includegraphics[width=0.18\textwidth]{figs/twitter_uniformCost_Cost_alpha02.pdf}
\label{fig:twitter_uniformCost_Cost_alpha02}
} \\
\end{minipage}
\renewcommand{\figurename}{Fig.}
\caption{Comparing the algorithms under the UC setting (IS: influence spread; RT: running time)}
\label{fig:the UC setting_alpha02}
\end{figure*}
\subsection{Theoretical Comparisons for the AP Algorithms}
\label{sec:compareunderuc}
To the best of our knowledge, there are no AP algorithms for MCSS with provable performance bounds under the GC+NV setting, and \textbf{only~\cite{Chen2014,Kuhnle2017} have proposed AP algorithms with provable approximation ratios under the UC+NV setting.}
\subsubsection{Comparing with Zhang et.al.'s Results~\cite{Chen2014}}
The work in~\cite{Chen2014} has proved that the $\mathsf{GreSSC}$ algorithm achieves a performance bound stated as follows:
\vspace{-1ex}
\begin{fact}[\cite{Chen2014}]
Let $S'$ be the output of $\mathsf{GreSSC}((1+\xi)\eta,\xi,\xi,0)$. Under the UC setting, if $\xi\leq \frac{\epsilon(n-\eta)}{8n^2(\eta+1)}$ where $\epsilon\in (0,1]$, then we have $f(S')\geq \eta$ and $|S'|\leq \lceil \ln\frac{(1+\epsilon)n\eta}{n-\eta} \rceil |S_{opt}|+1$ for any $\eta\in (0,n)$.
\label{fct:zhangsar}
\end{fact}
However, no previous studies including~\cite{Chen2014} have analyzed the time complexity of $\mathsf{GreSSC}$. So we analyze the time complexity of Zhang \textit{et.al.}'s algorithm by the following lemma:
\vspace{-1ex}
\begin{lemma}
Let $l_i=\min\{f(A)| |A|=i\}$ for any $i\in \{1,\cdots,n\}$ and $\delta$ be any number in $(0,1)$. With probability of at least $1-\delta$, Zhang \textit{et.al.}'s algorithm~\cite{Chen2014} can find a solution $S'$ satisfying the performance bound shown in Fact~\ref{fct:zhangsar} under the time complexity of $\mathcal{O}(\sum_{i=1}^{|S'|} \frac{n^2 m}{l_i\xi^2}\ln\frac{|S'|}{\delta})$.
\label{lma:greedystimecomplexity}
\end{lemma}
Recall that $\mathsf{AAUC}$ outputs $S$ satisfying $f(S)\geq \eta$ and $|S|\leq \lceil \ln\frac{n\eta}{n-\eta}\rceil |{S}_{opt}|+2$ w.h.p. under $\mathcal{O}(\frac{m q}{\varrho^2}\ln\frac{n}{\delta\eta})$ running time, where $q=\max\{f(v)|v\in V\}$ (see Theorem~\ref{thm:timeofaauc}). According to Lemma~\ref{lma:greedystimecomplexity}, even if we set $|S'|=1$, $l_i=n (\forall i)$ and $q=n$ in favor of Zhang \textit{et.al.}'s algorithm, the time complexity of $\mathsf{AAUC}$ is still $\Omega(n^2/\ln n)$ times smaller than that of Zhang \textit{et.al.}'s algorithm, as $\varrho=\Omega(n \xi)$.
Moreover, Theorem~\ref{thm:arofaauc} shows that, although $\mathsf{AAUC}$'s approximation ratio is larger than that in Fact~\ref{fct:zhangsar} by an additive factor 1, its multiplicative factor (i.e., $\lceil \ln\frac{n\eta}{n-\eta}\rceil$) is always smaller than that in Fact.~\ref{fct:zhangsar} (i.e., $\lceil \ln\frac{(1+\epsilon)n\eta}{n-\eta} \rceil$) due to $\epsilon>0$ (note that smaller $\epsilon$ results in higher time complexity and $\epsilon=0$ implies the impractical EV setting). As $\mathsf{AAUC}$ is $\Omega(n^2/\ln n)$ times faster than Zhang \textit{et.al.}'s algorithm even when we set $\epsilon=1$ in Fact~\ref{fct:zhangsar}, the approximation ratio of $\mathsf{AAUC}$ can be better than Zhang \textit{et.al.}'s algorithm~\cite{Chen2014} under the same running time.
\subsubsection{Comparing with Kuhnle et.al.'s Results~\cite{Kuhnle2017}}
The work in~\cite{Kuhnle2017} has also proposed a theoretical bound stated as follows:
\begin{fact}
For any $u\in V$, let $r_u$ be the probability that $u$ remains inactive when all of the neighbors of $u$ in G are activated. Let $r^*=\min_{u\in V} r_u$. There exists an algorithm that can output a set $S'$ satisfying $f(S')\geq\eta$ w.h.p. and $|S'|\leq \left( \frac{1+\varepsilon}{r^*}+\log\frac{\eta}{|S_{opt}|}\right) |S_{opt}|$.
\label{fct:mythaiaa}
\end{fact}
Note that the parameter $r^*$ in Fact.~\ref{fct:mythaiaa} can be very small. For example, in the IC model, we have $r^*=\min_{u\in V} \Pi_{v\in N_{in}(u)}(1-p(v,u))$, where $N_{in}(u)=\{v\in V: \langle v,u\rangle \in E\}$ and $p(v,u)$ is probability that $v$ can activate $u$. Whenever there exists an edge $\langle v,u\rangle\in E$ with $p(v,u)=1$, we will have $r^*=0$ and hence the approximation ratio in Fact~\ref{fct:mythaiaa} becomes infinite. In contrast, $\mathsf{AAUC}$ and $\mathsf{ATEUC}$ always have logarithmic approximation ratios which do not depend on the influence propagation probabilities of the network.
The work in~\cite{Kuhnle2017} has not explained how to implement an algorithm that can achieve the theoretical bounds shown in Fact.~\ref{fct:mythaiaa}, so its time complexity is unclear (note that we cannot follow Fact~\ref{fct:kuhapproxalg} to implement an AP algorithm by setting $\rho=0$, because this would result in infinite time complexity).
\section{Performance Evaluation}
\label{sec:pe}
In this section, we evaluate the performance of our algorithms through extensive experiments on real OSNs.
\subsection{Experimental Settings}
\label{sec:expsetting}
We use public OSN datasets in the experiments, which are shown in Table~\ref{table:datasets}. These data sets are widely used in the literature~\cite{TangSX2015,TangXS2014,NguyenTD2016}, and they can be downloaded from~\cite{snap,TangSX2015}.
\begin{table}[h]
\centering
\small
\begin{tabular}{|c|r|r|c|c|}\hline
Name & \makecell[c]{$n$} & \makecell[c]{$m$} & Type & Average degree\\ \hline
wiki-Vote & 7.1K & 103.7K & directed & 29.1\\ \hline
Pokec & 1.6M & 30.6M & directed & 37.5\\ \hline
LiveJournal & 4.8M & 69.0M & directed & 28.5\\ \hline
Orkut & 3.1M & 117.2M & undirected & 76.3\\ \hline
Twitter & 41.7M & 1.5G & directed & 70.5\\ \hline
\end{tabular}
\setcounter{table}{0}
\caption{Datasets}
\label{table:datasets}
\end{table}
Our experiments are conducted on a linux PC with an Intel(R) Core(TM) i7-6700K 4.0GHz CPU and 64GB memory for the Twitter dataset, while we reduce the memory to 32GB for the other datasets. Following~\cite{Kuhnle2017}, we use the independent cascade model~\cite{Kempe2003} in the experiments. For any $\langle u,v\rangle\in E$, the probability that $u$ can activate $v$ is set to $1/d_{in}(v)$, where $d_{in}(v)$ is the in-degree of $v$. Under the GC setting, the cost of each node is randomly sampled from the uniform distribution with the support $(0,1]$. These settings have been widely adopted in the literature~\cite{TangSX2015,TangXS2014,NguyenTD2016,NguyenZ2013,Goyal2013}.
As in~\cite{TangXS2014,TangSX2015,NguyenTD2016}, our reported data are the average of 10 runs, and we also limit the running time of each algorithm in a run to be within 500 minutes.
The implemented algorithms include:
\subsubsection{CELF} \label{sec:celf}
Following~\cite{Kuhnle2017,Goyal2013}, we implement $\mathsf{GreSSC}$ by adapting the the CELF algorithm~\cite{LeskovecKGFVG2007}, which has used a lazy evaluation technique to reduce the time complexity. CELF uses 10000 monte-carlo simulations to estimate $f(A)$ for any $A\subseteq V$ whenever needed~\cite{LeskovecKGFVG2007}. We notice that Zhang \textit{et.al.}~\cite{Chen2014} have also implemented $\mathsf{GreSSC}$ for Fact~\ref{fct:zhangsar}, but they have not used the parameters in Fact~\ref{fct:zhangsar} and simply used 10000 monte-carlo simulations to estimate $f(A) (\forall A\subseteq V)$.
Therefore, the implementation of $\mathsf{GreSSC}$ in~\cite{Chen2014} does not guarantee the performance bound shown in Fact~\ref{fct:zhangsar} and is essentially the same with CELF.
\subsubsection{STAB-C1 and STAB-C2} These algorithms (from~\cite{Kuhnle2017}) are the state-of-the-art algorithms for the MCSS problem, and they are based on the min-hash sketches proposed by~\cite{CohenDPW2014}. Moreover, it is shown in~\cite{Kuhnle2017} that these algorithms significantly outperform the traditional $k$SS algorithms. However, both STAB-C1 and STAB-C2 are BA algorithms designed only for the UC setting, so we have to adapt them to our case. As they are both greedy algorithms, we adapt them to the GC setting by selecting the node $u^*=\arg\max_{u}\Delta_u/C(\{u\})$ at each step, where $\Delta_u$ is the marginal gain computed by their estimators. To convert them into AP algorithms, we change their stopping condition to $\hat{f}(A)\geq (1+\rho)\eta$ ($\hat{f}(A)$ is their estimated IS of any $A\subseteq V$), because otherwise the solutions output by them are found to be infeasible in our experiments. We also follow the other parameter settings in~\cite{Kuhnle2017} and set $\iota=0.01,\rho=0.2$~\cite{Kuhnle2017}, where $\iota$ and $\rho$ are clarified in Fact~\ref{fct:kuhapproxalg}.
\subsubsection{Our Algorithms} We implement $\mathsf{AAUC}$, $\mathsf{BCGC}$, $\mathsf{TEGC}$ and $\mathsf{ATEUC}$ in our experiments, where we set $\delta=0.01$ in favor of the STAB algorithms\footnote{Note that $\delta$ corresponds to $\iota n^3$ in Fact.~\ref{fct:kuhapproxalg}. However, if we set $\delta=\iota n^3$, then $\iota$ would be very small, and hence the time complexity of the STAB algorithms would be much larger than that in the current setting.}. Note that none of the algorithms compared to us has provable performance bounds in our case. More specifically, the STAB algorithms only provide bi-criteria performance bounds under the UC case; and Zhang~\textit{et.al.}'s implementation~\cite{Chen2014} is essentially the same as CELF, which does not obey the approximation ratio stated in Fact~\ref{fct:zhangsar} but has much less time complexity (see \ref{sec:celf}). Therefore, for fair comparison, we set $\sigma=\gamma={\alpha}/{3}$, $\tau=0.02$ and $\mu=n^8$ to implement our algorithms. Under these parameter settings, the $\mathsf{BCGC}$ and $\mathsf{AAUC}$ algorithms may not obey the theoretical performance bounds shown in Theorem~\ref{thm:approximationratio} and Theorem~\ref{thm:arofaauc}, as these bounds require $\mu={\Theta}(D(\eta))$. However, both $\mathsf{TEGC}$ and $\mathsf{ATEUC}$ obey the theoretical performance bounds shown in Theorem~\ref{thm:artegc} and Theorem~\ref{thm:arofateuc} in all our experiments, and they scale well to billion-scale networks (we will see this shortly).
Finally, we set $\alpha=0.2$ in our algorithms, which corresponds to the setting of $\rho=0.2$ in the STAB algorithms.
\subsection{Experimental Results}
In Fig.~\ref{fig:the GC setting_alpha02}, we compare the algorithms under the GC case.
It can be seen from Fig.~\ref{fig:wikiVote_generalCost_RunningTime_alpha02}-\ref{fig:twitter_generalCost_RunningTime_alpha02} that CELF runs most slowly, and its running time exceeds the time limit for all the datasets except wiki-Vote.
Moreover, both $\mathsf{BCGC}$ and $\mathsf{TEGC}$ significantly outperform the other algorithms on the running time, and they are the only implemented algorithms running (in minutes) within the time limit for the billion-scale network Twitter. This can be explained by the reason that CELF uses the time consuming monte-carlo sampling method, while the STAB algorithms take a long time for building the sketches.
In Figs.~\ref{fig:wikiVote_generalCost_Influence_alpha02}-\ref{fig:twitter_generalCost_Influence_alpha02}, we plot the normalized influence spread (i.e., IS/$(\eta-\alpha\eta)$) of the implemented algorithms, where the IS of any solution is evaluated by $10^4$ monte-carlo simulations. It can be seen that CELF, $\mathsf{BCGC}$, $\mathsf{TEGC}$ and STAB-C2 can output feasible solutions with the influence spread larger than $(1-\alpha)\eta$, but STAB-C1 may output infeasible solutions (especially when $\eta$ is large). This phenomenon has also been reported in~\cite{Kuhnle2017}, which can be explained by the reason that the influence spread estimator used in STAB-C1 is less accurate than that in STAB-C2, so it may output solutions with poorer qualities.
From Figs.~\ref{fig:wikiVote_generalCost_Cost_alpha02}-\ref{fig:twitter_generalCost_Cost_alpha02}, it can be seen that the total costs of the solutions output by our algorithms are lower than those of STAB-C2 and CELF, while STAB-C1 has the best performance on the total cost. This is because that STAB-C1 outputs infeasible solutions with the influence spread less than $(1-\alpha)\eta$, so it selects less nodes than the other algorithms. {On the contrary, STAB-C2 selects many more nodes than what is necessary to achieve the $(1-\alpha)\eta$ threshold on the influence spread, and hence it outputs node sets with larger costs than those of our algorithms.}
In Fig.~\ref{fig:the UC setting_alpha02}, we study the performance of the algorithms under the UC setting, where the results are similar to those in Fig.~\ref{fig:the GC setting_alpha02}. In summary, our algorithms (especially ATEUC) greatly outperform the other baselines on the running time, while they also output feasible solutions with small costs. This can be explained by similar reasons with those for Fig.~\ref{fig:the GC setting_alpha02}.
\section{Related Work}
\label{sec:rw}
Since~\cite{Kempe2003}, a lot of studies have aimed to design efficient $k$SS algorithms. The earlier studies in this line are mostly based on the naive monte-carlo sampling method (e.g.,\cite{LeskovecKGFVG2007,ChenWW2010}), and more recent work~\cite{TangXS2014,TangSX2015,NguyenTD2016,NguyenDT2016} has leveraged more advanced sampling methods~\cite{Borgs2014,DagumKLR1995} to reduce the time complexity, such as the RR-set sampling method. Besides the RR-set sampling method, Cohen~\textit{et.al.}~\cite{CohenDPW2014} have proposed a min-hash sketch based method for $k$SS, which is also used in~\cite{Kuhnle2017}. Some variations of the $k$SS problem have also been studied in \cite{Nguyen2017outward,NguyenDT2016,LinCL17}. However, all these studies belong to the category of influence maximization (IM) algorithms.
Compared with the IM problem, the MCSS problem is less studied in the literature. Recall that we have provided a BA algorithm for MCSS under the GC+NV setting and an AP algorithm for MCSS under the UC+NV setting. To the best of our knowledge, only the work in~\cite{Goyal2013,Kuhnle2017,Chen2014} has provided algorithms with provable performance bounds under the same settings with ours.
Therefore, we have compared our algorithms with~\cite{Goyal2013,Kuhnle2017,Chen2014} in detail in Sec.~\ref{sec:compareundergc} and Sec.~\ref{sec:compareunderuc}. Some variations of the MCSS problem have also been studied in \cite{DinhZNT2014,ZhangNZT2016,ZhuLZ2016,HeJBC2014}, but the models and problem definitions of these proposals are very different from ours, and none of them has considered the MCSS problem under our setting.
\section{Conclusion}
\label{sec:conclu}
We have proposed several algorithms for the Min-Cost Seed Selection (MCSS) problem in OSNs, and compared our algorithms with the state-of-the-art algorithms. The theoretical comparisons reveal that, our algorithms are the first to achieve provable performance bounds and polynomial running time under the case where the nodes have heterogeneous costs, and our algorithms' time complexity outperforms the existing ones in orders of magnitude under the uniform cost case. The experimental comparisons reveal that, our algorithms scale well to big networks, and also significantly outperform the existing algorithms both on the cost and on the running time.
\bibliographystyle{IEEEtran}
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Q: Pipe or Service in Angular? I need to provide simple array of data with years.
What will be right to create: pipe or typically service class?
Iameging, that it is pipe. Then can I iterate this pipe in ngFor like this:
ngFor="item of pipe()"?
A: You should create an array in this case, neither pipe nor a service.
You can create an array by calling a function with the name pipe() though, that would return an array with your specified data.
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"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"http:\/\/www.digplanet.com\/wiki\/Goodstein's_theorem","text":"digplanet beta 1: Athena\nShare digplanet:\n\nAgriculture\n\nApplied sciences\n\nArts\n\nBelief\n\nChronology\n\nCulture\n\nEducation\n\nEnvironment\n\nGeography\n\nHealth\n\nHistory\n\nHumanities\n\nLanguage\n\nLaw\n\nLife\n\nMathematics\n\nNature\n\nPeople\n\nPolitics\n\nScience\n\nSociety\n\nTechnology\n\n\"hydra game\" redirects here. For the game development kit, see HYDRA Game Development Kit.\n\nIn mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Kirby and Paris[1] showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second order arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after G\u00f6del's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of \u03b50-induction in Peano arithmetic. The Paris\u2013Harrington theorem was a later example.\n\nLaurence Kirby and Jeff Paris introduced a graph theoretic hydra game with behavior similar to that of Goodstein sequences: the \"Hydra\" is a rooted tree, and a move consists of cutting off one of its \"heads\" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. Kirby and Paris proved that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very long time.[1]\n\n## Hereditary base-n notation\n\nGoodstein sequences are defined in terms of a concept called \"hereditary base-n notation\". This notation is very similar to usual base-n positional notation, but the usual notation does not suffice for the purposes of Goodstein's theorem.\n\nIn ordinary base-n notation, where n is a natural number greater than 1, an arbitrary natural number m is written as a sum of multiples of powers of n:\n\n$m = a_k n^k + a_{k-1} n^{k-1} + \\cdots + a_0,$\n\nwhere each coefficient ai satisfies 0 \u2264 ai < n, and ak \u2260 0. For example, in base 2,\n\n$35 = 32 + 2 + 1 = 2^5 + 2^1 + 2^0.$\n\nThus the base 2 representation of 35 is 100011, which means 25 + 2 + 1. Similarly, 100 represented in base 3 is 10201:\n\n$100 = 81 + 18 + 1 = 3^4 + 2\\cdot 3^2 + 3^0.$\n\nNote that the exponents themselves are not written in base-n notation. For example, the expressions above include 25 and 34.\n\nTo convert a base-n representation to hereditary base n notation, first rewrite all of the exponents in base-n notation. Then rewrite any exponents inside the exponents, and continue in this way until every digit appearing in the expression is n or less.\n\nFor example, while 35 in ordinary base-2 notation is 25 + 2 + 1, it is written in hereditary base-2 notation as\n\n$35 = 2^{2^2+1}+2+1,$\n\nusing the fact that 5 = 22 + 1. Similarly, 100 in hereditary base 3 notation is\n\n$100 = 3^{3+1} + 2\\cdot 3^2 + 1.$\n\n## Goodstein sequences\n\nThe Goodstein sequence G(m) of a number m is a sequence of natural numbers. The first element in the sequence G(m) is m itself. To get the second, G(m)(2), write m in hereditary base 2 notation, change all the 2s to 3s, and then subtract 1 from the result. In general, the n+1st term G(m)(n+1) of the Goodstein sequence of m is as follows: take the hereditary base n+1 representation of G(m)(n), and replace each occurrence of the base n+1 with n+2 and then subtract one. Note that the next term depends both on the previous term and on the index n. Continue until the result is zero, at which point the sequence terminates.\n\nEarly Goodstein sequences terminate quickly. For example, G(3) terminates at the sixth step:\n\nBase Hereditary notation Value Notes\n2 $2^1 + 1$ 3 Write 3 in base 2 notation\n3 $3^1 + 1 - 1 = 3^1$ 3 Switch the 2 to a 3, then subtract 1\n4 $4^1 - 1 = 3$ 3 Switch the 3 to a 4, then subtract 1. Now there are no more 4s left\n5 $3 - 1 = 2$ 2 No 4s left to switch to 5s. Just subtract 1\n6 $2 - 1 = 1$ 1 No 5s left to switch to 6s. Just subtract 1\n7 $1 - 1 = 0$ 0 No 6s left to switch to 7s. Just subtract 1\n\nLater Goodstein sequences increase for a very large number of steps. For example, G(4) starts as follows:\n\nHereditary notation Value\n$2^2$ 4\n$3^3 - 1 = 2 \\cdot 3^2 + 2 \\cdot 3 + 2$ 26\n$2 \\cdot 4^2 + 2 \\cdot 4 + 1$ 41\n$2 \\cdot 5^2 + 2 \\cdot 5$ 60\n$2 \\cdot 6^2 + 2 \\cdot 6 - 1 = 2 \\cdot 6^2 + 6 + 5$ 83\n$2 \\cdot 7^2 + 7 + 4$ 109\n$\\vdots$ $\\vdots$\n$2 \\cdot 11^2 + 11$ 253\n$2 \\cdot 12^2 + 12 - 1 = 2 \\cdot 12^2 + 11$ 299\n$\\vdots$ $\\vdots$\n\nElements of G(4) continue to increase for a while, but at base $3 \\cdot 2^{402653209}$, they reach the maximum of $3 \\cdot 2^{402653210} - 1$, stay there for the next $3 \\cdot 2^{402653209}$ steps, and then begin their first and final descent.\n\nThe value 0 is reached at base $3 \\cdot 2^{402653211} - 1$. (Curiously, this is a Woodall number: $3 \\cdot 2^{402653211} - 1 = 402653184 \\cdot 2^{402653184} - 1$. This is also the case with all other final bases for starting values greater than 4.[citation needed])\n\nHowever, even G(4) doesn't give a good idea of just how quickly the elements of a Goodstein sequence can increase. G(19) increases much more rapidly, and starts as follows:\n\nHereditary notation Value\n$2^{2^2} + 2 + 1$ 19\n$3^{3^3} + 3$ 7,625,597,484,990\n$4^{4^4} + 3$ $\\approx 1.3 \\times 10^{154}$\n$5^{5^5} + 2$ $\\approx 1.8 \\times 10^{2184}$\n$6^{6^6} + 1$ $\\approx 2.6 \\times 10^{36,305}$\n$7^{7^7}$ $\\approx 3.8 \\times 10^{695,974}$\n\n$8^{8^8} - 1 = 7 \\cdot 8^{(7 \\cdot 8^7 + 7 \\cdot 8^6 + 7 \\cdot 8^5 + 7 \\cdot 8^4 + 7 \\cdot 8^3 + 7 \\cdot 8^2 + 7 \\cdot 8 + 7)}$ $+ 7 \\cdot 8^{(7 \\cdot 8^7 + 7 \\cdot 8^6 + 7 \\cdot 8^5 + 7 \\cdot 8^4 + 7 \\cdot 8^3 + 7 \\cdot 8^2 + 7 \\cdot 8 + 6)} + \\cdots$ $+ 7 \\cdot 8^{(8+2)} + 7 \\cdot 8^{(8+1)} + 7 \\cdot 8^8$ $+ 7 \\cdot 8^7 + 7 \\cdot 8^6 + 7 \\cdot 8^5 + 7 \\cdot 8^4$ $+ 7 \\cdot 8^3 + 7 \\cdot 8^2 + 7 \\cdot 8 + 7$\n\n$\\approx 6 \\times 10^{15,151,335}$\n\n$7 \\cdot 9^{(7 \\cdot 9^7 + 7 \\cdot 9^6 + 7 \\cdot 9^5 + 7 \\cdot 9^4 + 7 \\cdot 9^3 + 7 \\cdot 9^2 + 7 \\cdot 9 + 7)}$ $+ 7 \\cdot 9^{(7 \\cdot 9^7 + 7 \\cdot 9^6 + 7 \\cdot 9^5 + 7 \\cdot 9^4 + 7 \\cdot 9^3 + 7 \\cdot 9^2 + 7 \\cdot 9 + 6)} + \\cdots$ $+ 7 \\cdot 9^{(9+2)} + 7 \\cdot 9^{(9+1)}+ 7 \\cdot 9^9$ $+ 7 \\cdot 9^7 + 7 \\cdot 9^6 + 7 \\cdot 9^5 + 7 \\cdot 9^4$ $+ 7 \\cdot 9^3 + 7 \\cdot 9^2 + 7 \\cdot 9 + 6$\n\n$\\approx 4.3 \\times 10^{369,693,099}$\n$\\vdots$ $\\vdots$\n\nIn spite of this rapid growth, Goodstein's theorem states that every Goodstein sequence eventually terminates at 0, no matter what the starting value is.\n\n## Proof of Goodstein's theorem\n\nGoodstein's theorem can be proved (using techniques outside Peano arithmetic, see below) as follows: Given a Goodstein sequence G(m), we construct a parallel sequence P(m) of ordinal numbers which is strictly decreasing and terminates. Then G(m) must terminate too, and it can terminate only when it goes to 0. A common misunderstanding of this proof is to believe that G(m) goes to 0 because it is dominated by P(m). In fact, the fact that P(m) dominates G(m) plays no role at all. The important point is: G(m)(k) exists if and only if P(m)(k) exists (parallelism). Then if P(m) terminates, so does G(m). And G(m) can terminate only when it comes to 0.\n\nMore precisely, each term P(m)(n) of the sequence P(m) is obtained by applying a function f on the term G(m)(n) of the Goodstein sequence of m as follows: take the hereditary base n+1 representation of G(m)(n), and replace each occurrence of the base n+1 with the first infinite ordinal number \u03c9. For example G(3)(1) = 3 = 21 + 20 and P(3)(1) = f(G(3)(1)) = \u03c91 + \u03c90 = \u03c9 + 1. Addition, multiplication and exponentiation of ordinal numbers are well defined.\n\n\u2022 The base-changing operation of the Goodstein sequence when going from G(m)(n) to G(m)(n+1) does not change the value of f (that's the main point of the construction), thus after the minus 1 operation, P(m)(n+1) will be strictly smaller than P(m)(n). For example, $f(3 \\cdot 4^{4^4} + 4) = 3 \\omega^{\\omega^\\omega} + \\omega= f(3 \\cdot 5^{5^5} + 5)$, hence $f(3 \\cdot 4^{4^4} + 4)$ is strictly greater than $f((3 \\cdot 5^{5^5} + 5 )-1).$\n\nIf the sequence G(m) did not go to 0, it would not terminate and would be infinite (since G(m)(k+1) would always exist). Consequently, P(m) also would be infinite (since in its turn P(m)(k+1) would always exist too). But P(m) is strictly decreasing and the standard order < on ordinals is well-founded, therefore an infinite strictly decreasing sequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals does terminate (and cannot be infinite). This contradiction shows that G(m) terminates, and since it terminates, goes to 0 (by the way, since there exists a natural number k such that G(m)(k) = 0, by construction of P(m) we have that P(m)(k) = 0).\n\nWhile this proof of Goodstein's theorem is fairly easy, the Kirby\u2013Paris theorem,[1] which shows that Goodstein's theorem is not a theorem of Peano arithmetic, is technical and considerably more difficult. It makes use of countable nonstandard models of Peano arithmetic. What Kirby showed is that Goodstein's theorem leads to Gentzen's theorem, i.e. it can substitute for induction up to \u03b50.\n\n## Extended Goodstein's theorem\n\nSuppose the definition Goodstein sequence is changed so that instead of replacing each occurrence of the base b with b+1 it was replaces it with b+2. Would the sequence still terminate? More generally, let b1, b2, b3, \u2026 be any sequences of integers. Then let the n+1st term G(m)(n+1) of the extended Goodstein sequence of m be as follows: take the hereditary base bn representation of G(m)(n), and replace each occurrence of the base bn with bn+1 and then subtract one. The claim is that this sequence still terminates. The extended proof defines P(m)(n) = f(G(m)(n), n) as follows: take the hereditary base bn representation of G(m)(n), and replace each occurrence of the base bn with the first infinite ordinal number \u03c9. The base-changing operation of the Goodstein sequence when going from G(m)(n) to G(m)(n+1) still does not change the value of f. For example, if bn = 4 and if bn+1 = 9, then $f(3 \\cdot 4^{4^4} + 4,4) = 3 \\omega^{\\omega^\\omega} + \\omega= f(3 \\cdot 9^{9^9} + 9,9)$, hence the ordinal $f(3 \\cdot 4^{4^4} + 4,4)$ is strictly greater than the ordinal $f((3 \\cdot 9^{9^9} + 9 )-1,9).$\n\n## Sequence length as a function of the starting value\n\nThe Goodstein function, $\\mathcal{G}: \\mathbb{N} \\to \\mathbb{N} \\,\\!$, is defined such that $\\mathcal{G}(n)$ is the length of the Goodstein sequence that starts with n. (This is a total function since every Goodstein sequence terminates.) The extreme growth-rate of $\\mathcal{G}\\,\\!$ can be calibrated by relating it to various standard ordinal-indexed hierarchies of functions, such as the functions $H_\\alpha\\,\\!$ in the Hardy hierarchy, and the functions $f_\\alpha\\,\\!$ in the fast-growing hierarchy of L\u00f6b and Wainer:\n\n\u2022 Kirby and Paris (1982) proved that\n$\\mathcal{G}\\,\\!$ has approximately the same growth-rate as $H_{\\epsilon_0}\\,\\!$ (which is the same as that of $f_{\\epsilon_0}\\,\\!$); more precisely, $\\mathcal{G}\\,\\!$ dominates $H_\\alpha\\,\\!$ for every $\\alpha < \\epsilon_0\\,\\!$, and $H_{\\epsilon_0}\\,\\!$ dominates $\\mathcal{G}\\,\\!.$\n(For any two functions $f, g: \\mathbb{N} \\to \\mathbb{N} \\,\\!$, $f\\,\\!$ is said to dominate $g\\,\\!$ if $f(n) > g(n)\\,\\!$ for all sufficiently large $n\\,\\!$.)\n\u2022 Cichon (1983) showed that\n$\\mathcal{G}(n) = H_{R_2^\\omega(n+1)}(1) - 1,$\nwhere $R_2^\\omega(n)$ is the result of putting n in hereditary base-2 notation and then replacing all 2s with \u03c9 (as was done in the proof of Goodstein's theorem).\n\u2022 Caicedo (2007) showed that if $n = 2^{m_1} + 2^{m_2} + \\cdots + 2^{m_k}$ with $m_1 > m_2 > \\cdots > m_k,$ then\n$\\mathcal{G}(n) = f_{R_2^\\omega(m_1)}(f_{R_2^\\omega(m_2)}(\\cdots(f_{R_2^\\omega(m_k)}(3))\\cdots)) - 2$.\n\nSome examples:\n\nn $\\mathcal{G}(n)$\n1 $2^0$ $2 - 1$ $H_\\omega(1) - 1$ $f_0(3) - 2$ 2\n2 $2^1$ $2^1 + 1 - 1$ $H_{\\omega + 1}(1) - 1$ $f_1(3) - 2$ 4\n3 $2^1 + 2^0$ $2^2 - 1$ $H_{\\omega^\\omega}(1) - 1$ $f_1(f_0(3)) - 2$ 6\n4 $2^2$ $2^2 + 1 - 1$ $H_{\\omega^\\omega + 1}(1) - 1$ $f_\\omega(3) - 2$ 3\u00b72402653211 \u2212 2\n5 $2^2 + 2^0$ $2^2 + 2 - 1$ $H_{\\omega^\\omega + \\omega}(1) - 1$ $f_\\omega(f_0(3)) - 2$ > A(4,4)\n6 $2^2 + 2^1$ $2^2 + 2 + 1 - 1$ $H_{\\omega^\\omega + \\omega + 1}(1) - 1$ $f_\\omega(f_1(3)) - 2$ > A(6,6)\n7 $2^2 + 2^1 + 2^0$ $2^{2 + 1} - 1$ $H_{\\omega^{\\omega + 1}}(1) - 1$ $f_\\omega(f_1(f_0(3))) - 2$ > A(8,8)\n8 $2^{2 + 1}$ $2^{2 + 1} + 1 - 1$ $H_{\\omega^{\\omega + 1} + 1}(1) - 1$ $f_{\\omega + 1}(3) - 2$ > A3(3,3) = A(A(61, 61), A(61, 61))\n$\\vdots$\n12 $2^{2 + 1} + 2^2$ $2^{2 + 1} + 2^2 + 1 - 1$ $H_{\\omega^{\\omega + 1} + \\omega^\\omega + 1}(1) - 1$ $f_{\\omega + 1}(f_\\omega(3)) - 2$ > f\u03c9+1(64) > Graham's number\n$\\vdots$\n19 $2^{2^2} + 2^1 + 2^0$ $2^{2^2} + 2^2 - 1$ $H_{\\omega^{\\omega^\\omega} + \\omega^\\omega}(1) - 1$ $f_{\\omega^\\omega}(f_1(f_0(3))) - 2$\n\n## Application to computable functions\n\nGoodstein's theorem can be used to construct a total computable function that Peano arithmetic cannot prove to be total. The Goodstein sequence of a number can be effectively enumerated by a Turing machine; thus the function which maps n to the number of steps required for the Goodstein sequence of n to terminate is computable by a particular Turing machine. This machine merely enumerates the Goodstein sequence of n and, when the sequence reaches 0, returns the length of the sequence. Because every Goodstein sequence eventually terminates, this function is total. But because Peano arithmetic does not prove that every Goodstein sequence terminates, Peano arithmetic does not prove that this Turing machine computes a total function.\n\n## References\n\n1. ^ a b c Kirby, L.; Paris, J. (1982). \"Accessible Independence Results for Peano Arithmetic\" (PDF). Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112\/blms\/14.4.285.\n\n## Bibliography\n\nWe're sorry, but there's no news about \"Goodstein's theorem\" right now.\n\n Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }\n\nOops, we seem to be having trouble contacting Twitter","date":"2015-11-28 18:24:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 124, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.919622540473938, \"perplexity\": 621.7970235152318}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-48\/segments\/1448398453656.76\/warc\/CC-MAIN-20151124205413-00070-ip-10-71-132-137.ec2.internal.warc.gz\"}"}
| null | null |
Q: ActiveRecord Collection subtraction Had a question on subtracting queries from similar ActiveRecord collections.
Let say I have one query that is as follows:
all_users = User.all
users_with_adequate_reviews = User.joins(:reviews).select("users.id, count(*) as num_reviews").group(:id).having("num_reviews > 5")
if I do all_users - users_with_adequate_reviews, I get what I would expect from which is users with fewer than review count of 5. How does ActiveRecord relation subtraction know to remove the similar records even though i only select a few attributes from users (primarily the id). Was looking to see documentation on this but couldn't find it anywhere
A: WHERE IS SUBTRACTION METHOD DEFINED ?
Subtraction on ActiveRecord relation is defined on ActiveRecord::Delegation module.
If you're digging that source code, you can see that method is delegated from Array class.
So we need to dig Array's subtraction to understand how ActiveRecord relation's subtraction works.
HOW DOES ARRAY SUBTRACTION WORK ?
This is taken from documentation about Array subtraction / difference.
Array Difference
Returns a new array that is a copy of the original array, removing any
items that also appear in other_ary. The order is preserved from the
original array.
It compares elements using their hash and eql? methods for efficiency.
It means subtraction evaluates two methods : hash && eql? from each object to perform task.
HOW DO THOSE METHODS WORK ON ACTIVE RECORD OBJECT ?
The code below is taken from ActiveRecord::Core module.
def ==(comparison_object)
super ||
comparison_object.instance_of?(self.class) &&
!id.nil? &&
comparison_object.id == id
end
alias :eql? :==
def hash
if id
self.class.hash ^ id.hash
else
super
end
end
You can see both hash & eql? only evaluates class and id.
It means all_users - users_with_adequate_reviews will exclude some objects ONLY IF there are any objects from both elements that have same object's id and object's class.
ANOTHER SAMPLE
irb(main):001:0> users = User.all
User Load (26.4ms) SELECT `users`.* FROM `users` LIMIT 11
=> #<ActiveRecord::Relation [
#<User id: 1, name: "Bob", created_at: "2020-06-09 13:03:45", updated_at: "2020-06-09 13:03:45">,
#<User id: 2, name: "Danny", created_at: "2020-06-09 13:04:14", updated_at: "2020-06-09 13:04:14">,
#<User id: 3, name: "Alan", created_at: "2020-06-09 13:05:30", updated_at: "2020-06-09 13:05:30">,
#<User id: 4, name: "Joe", created_at: "2020-06-09 13:07:00", updated_at: "2020-06-09 13:07:00">]>
irb(main):002:0> users_with_multiple_emails = User.joins(:user_emails).select("users.id, users.name, count(*) as num_emails").group(:id).having("num_emails > 1")
User Load (2.8ms) SELECT users.id, users.name, count(*) as num_emails FROM `users` INNER JOIN `user_emails` ON `user_emails`.`user_id` = `users`.`id` GROUP BY `users`.`id` HAVING (num_emails > 1) LIMIT 11
=> #<ActiveRecord::Relation [#<User id: 1, name: "Bob">]>
irb(main):003:0> users - users_with_multiple_emails
=> [
#<User id: 2, name: "Danny", created_at: "2020-06-09 13:04:14", updated_at: "2020-06-09 13:04:14">,
#<User id: 3, name: "Alan", created_at: "2020-06-09 13:05:30", updated_at: "2020-06-09 13:05:30">,
#<User id: 4, name: "Joe", created_at: "2020-06-09 13:07:00", updated_at: "2020-06-09 13:07:00">]
As you can see all users - users_with_multiple_emails excludes first object (Bob).
Why ? It's because Bob from both elements have same id and class (id: 1, class: User)
Subtraction returns different result if it's like this
irb(main):001:0> users = User.all
User Load (26.4ms) SELECT `users`.* FROM `users` LIMIT 11
=> #<ActiveRecord::Relation [
#<User id: 1, name: "Bob", created_at: "2020-06-09 13:03:45", updated_at: "2020-06-09 13:03:45">,
#<User id: 2, name: "Danny", created_at: "2020-06-09 13:04:14", updated_at: "2020-06-09 13:04:14">,
#<User id: 3, name: "Alan", created_at: "2020-06-09 13:05:30", updated_at: "2020-06-09 13:05:30">,
#<User id: 4, name: "Joe", created_at: "2020-06-09 13:07:00", updated_at: "2020-06-09 13:07:00">]>
irb(main):002:0> users_with_multiple_emails = User.joins(:user_emails).select("users.name, count(*) as num_emails").group(:id).having("num_emails > 1")
User Load (2.3ms) SELECT users.name, count(*) as num_emails FROM `users` INNER JOIN `user_emails` ON `user_emails`.`user_id` = `users`.`id` GROUP BY `users`.`id` HAVING (num_emails > 1) LIMIT 11
=> #<ActiveRecord::Relation [#<User id: nil, name: "Bob">]>
irb(main):003:0> users - users_with_multiple_emails
=> [
#<User id: 1, name: "Bob", created_at: "2020-06-09 13:03:45", updated_at: "2020-06-09 13:03:45">,
#<User id: 2, name: "Danny", created_at: "2020-06-09 13:04:14", updated_at: "2020-06-09 13:04:14">,
#<User id: 3, name: "Alan", created_at: "2020-06-09 13:05:30", updated_at: "2020-06-09 13:05:30">,
#<User id: 4, name: "Joe", created_at: "2020-06-09 13:07:00", updated_at: "2020-06-09 13:07:00">]
This time users_with_multiple_emails only select name & num_emails.
As you can see all users - users_with_multiple_emails doesn't exclude Bob.
Why ? It's because Bob from both elements have different id.
*
*Bob's id from users : 1
*Bob's id from users_with_multiple_emails : nil
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,798
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Thomas Christopher Boyd (14 August 1916 – 15 March 2004) was a British Labour Party politician.
Boyd unsuccessfully fought Isle of Thanet in 1950 and Harborough at the 1951 general election.
At the 1955 general election, he was elected as Member of Parliament (MP) for Bristol North West, defeating the sitting Conservative MP Gurney Braithwaite. He served in the House of Commons for only four years, losing his seat at the 1959 general election to the Conservative Martin McLaren.
After losing his seat at the 1959 general election, Boyd moved to Moffat, Scotland. He then retired to Appledore, Kent.
References
External links
1916 births
2004 deaths
Labour Party (UK) MPs for English constituencies
UK MPs 1955–1959
Politicians from Edmonton
People from Appledore, Kent
People from Moffat
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| 7,844
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Apple removes purchased movies from users' iTunes library
JC Torres - Sep 12, 2018, 11:20pm CDT
Purchasing digital copies of usually physical goods is so easy and convenient these days, most of us don't give a second thought about it. We don't have to carry books around, deal with fragile DVDs, or fumble around for cartridges whenever we want to switch games. But depending on where you bought said digital content, buying might not exactly mean what you think it means. That was the bitter truth that a Canadian iTunes user learned when Apple quietly removed content he already bought because the items were no longer being sold on its digital content store.
Anders G da Silva took to Twitter to reveal his exchange with an iTunes Store customer support representative over the sudden disappearance of his purchased videos. The only explanation he was given was that the content provider of those videos pulled out the movies from the iTunes Store. In exchange, he was being given credits to rent, not buy, other movies.
Me: Hey Apple, three movies I bought disappeared from my iTunes library.
Apple: Oh yes, those are not available anymore. Thank you for buying them. Here are two movie rentals on us!
Me: Wait… WHAT?? @tim_cook when did this become acceptable? pic.twitter.com/dHJ0wMSQH9
— Anders G da Silva (@drandersgs) September 10, 2018
da Silva contested the action and even the "reparation", explaining that he already bought those movies and they should remain, regardless of the content provider. Customer support explained that iTunes is only a storefront and has no control over that aspect of the business. It also increased the offered rental credits.
This situation, however, isn't completely new and has, in fact, been at the heart of the whole DRM situation, especially with music and games. It's the question of who really owns the content you "buy". The words "buy" and "purchase" do imply that you are buying them and they are yours, but digital distribution has turned that definition on its head.
In this particular instance, what you really buy from Apple, Amazon, and even Google are effectively licenses to access the digital content. You don't own the content and neither does Apple. If a publisher decides to pull out their products for one reason or another, you're practically screwed.
Topics DigitalDigital Lifestyledigital mediaiTunesiTunes Store
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{
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Q: how do we know which event fired load event in extjs 4.1 store I am using extjs 4.1 store. Which looks like this:
Ext.define('myStore', {
extend: 'Ext.data.Store',
requires: ['myModel'],
autoLoad: false,
proxy: {
type: 'ajax',
url: '/aaa/bbb',
timeout: '90000',
reader: {
type: 'json',
root: 'data'
}
},
listeners: {
'beforeload': function (store, options) {
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initially I am loading data through proxy. Later based on user interaction, I will extract the data from the store usign store.proxy.reader.rawData and store it in some variable.
Then at later stage, I will load into the store from the variable using loadRawData().
When loadRawData() is called, it also fires the load event.
What I want: I want to diffrentiate between load event fired due to proxy loading the data for the first time vs load event fired due to loadRawData().
A: Upgrading to ExtJS version 5.1 allows you access to store.loadCount which will let you to check whether this is the first load or a subsequent load. http://docs.sencha.com/extjs/5.1/5.1.0-apidocs/#!/api/Ext.data.Store-property-loadCount
If you don't want to upgrade, you could manually implement this feature (e.g. increment a custom variable on your store in the 'load' listener) and check it to see which behaviour you want to perform.
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closeOnClick: true // Closes side-nav on <a> clicks, useful for Angular/Meteor
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# VALUE
# The Representation of Labour in Capitalism
Essays Edited by
Diane Elson
# VALUE
# The Representation of Labour in Capitalism
Essays Edited by
Diane Elson
This edition published by Verso 2015
First published by CSE Books 1979
The Collection © Verso Books 2015
The Contributions © The Contributors 1979, 2015
All rights reserved
The moral rights of the author have been asserted
1 3 5 7 9 10 8 6 4 2
**Verso**
UK: 6 Meard Street, London W1F 0EG
US: 20 Jay Street, Suite 1010, Brooklyn, NY 11201
www.versobooks.com
Verso is the imprint of New Left Books
ISBN-13: 978-1-78478-229-0 (PB)
eISBN-13: 978-1-78478-231-3 (US)
eISBN-13: 978-1-78478-230-6 (UK)
**British Library Cataloguing in Publication Data**
A catalogue record for this book is available from the British Library
**Library of Congress Cataloging-in-Publication Data**
A catalog record for this book is available from the Library of Congress
Printed in the US by Maple Press
## CONTENTS
Introduction
Reading Marx on Value: A Note on the Basic Texts
Aboo Aumeeruddy and Ramon Tortajada
From the Commodity to Capital: Hegel's Dialectic in Marx's _Capital_
Jairus Banaji
Why Labour is the starting point of Capital
Geoffrey Kay
Dialectic of the Value-Form
Chris Arthur
Misreading Marx's Theory of Value: Marx's Marginal Notes on Wagner
Athar Hussain
Marx's Theory of Market-Value
Makoto Itoh, in collaboration with Nobuharu Yokokawa
The Value Theory of Labour
Diane Elson
### **Notes on Contributors**
Chris Arthur teaches philosophy at Sussex University, and has contributed to _Radical Philosophy, Critique_ and _CSE Bulletin_.
Aboo Aumeeruddy is currently working at the Institute of Development Studies, Sussex University. He is a member of the Editorial Committee of _Capital and Class_.
Jairus Banaji lives and works in Bombay, and has contributed to _Capital and Class_.
Diane Elson teaches economics at the University of Manchester. She was formerly a member of the Editorial Committee of _Capital and Class_ and of the Brighton Labour Process Group.
Athar Hussain teaches economics at the University of Keele and is a member of the CSE Money Group. He has contributed to the _CSE Bulletin_ and _Economy and Society_.
Makoto Itoh teaches economics at the University of Tokyo and has contributed to _CSE Bulletin, Capital and Class_ and _Science and Society_.
Geoff Kay teaches economics at the City University, London, and is a member of the Editorial Board of _Critique_. He has contributed to the _CSE Bulletin_.
Ramon Tortajada works in the University of Grenoble, France, and has contributed to _Capital and Class_.
Nobuharu Yokokawa is carrying out research on the work of Ricardo and Marx at the Universities of Tokyo and Cambridge.
## **_INTRODUCTION_**
To some readers, the publication by CSE Books of a collection of essays on Marx's theory of value will simply be an indication of the unwillingness (or inability) of Marxist intellectuals to leave the realms of high theory and produce some politically useful, concrete analysis of the accumulation of capital today. To others it will signify a mystifying refusal to jettison a theory which, it is claimed, has now been shown to be at best redundant, at worst incoherent and without foundations in real social relations. Yet others may expect one more, incomprehensible round in a debate which obsesses 'Marxist economists' but has little significance for those interested in Marx's theory of class or the state or the mode of production. So why has CSE Books chosen to put together this volume? Why is Marx's theory of value important?
It is important because Marx's theory of value is the _foundation_ of his attempt to understand capitalism in a way that is politically useful to socialists. It is not some small and dispensable part of Marx's investigation of capital; it constitutes the basis on which that investigation takes place. If we decide to reject that theory, we are at the same time rejecting precisely those tools of analysis which are Marx's distinctive contribution to socialist thought on the workings of capital. The debate about Marx's value theory is, in fact, a debate about the appropriate method of analysis, about the validity of the concepts which are specific to, and constitute the method of, historical materialism. The outcome of this debate therefore has implications far beyond the way in which we understand prices and profit in the capitalist economy. It has implications for the question of how we should carry out our empirical investigations today of the international restructuring of capital accumulation; of new forms of class struggle, of the capitalist state; and of the possibilities for socialism. It has implications for the fundamental question of whether what is distinctive about Marx's method of analysis is really of any use to socialists today.
Accordingly this collection of essays concentrates on investigating and evaluating the method of analysis instantiated in Marx's theory of value, a method which Marx claimed in his Preface to the French Edition of _Capital_ , I, had not been previously applied to economic subjects. It is not a premise of this book that every word which Marx wrote must inevitably be 'correct', and that the task is simply to propagate them. But it is a premise that much recent debate over Marx's theory of value has been hampered by a mutual incomprehension on matters of method; on the meaning to be attached to terms like 'determination', 'substance', 'measure', 'abstraction', 'form', 'transformation', 'law', 'equivalence', etc.; and on the question of what Marx's theory of value is a theory of, what is its object. Marx himself wrote in the Post-face to the Second German Edition of _Capital_ I,
> 'That the method employed in _Capital_ has been little understood is shown by the various mutually contradictory conceptions that have been formed of it.'
The essays in this book attempt to explore and to resolve some of the differences that exist in current interpretations of Marx's theory of value, recognising that the cause of misunderstandings may lie in inadequacies in Marx's texts, as well as the preconceptions that readers have brought to those texts.
The essays have been written by CSE members from France, India and Japan, as well as from Britain, reflecting the international membership of the CSE. They are not all written from exactly the same perspective; nor do they all reach exactly the same kind of conclusion. But what they have in common is an awareness of the question of method of analysis, raised above. Two of them, the ones by Kay and Hussain, have already appeared in print, in _Critique_ and _Theoretical Practice_ respectively. The rest have been written specially for this collection.
Recognising that there are always readers to whom the topics under discussion are quite new, readers who perhaps are just beginning to read _Capital_ , the first essay in this collection is an annotated guide to reading what Marx wrote on value, prepared by Aboo Aumeeruddy and Ramon Tortajada. They stress the complexity of the relation between Marx's texts and those of classical political economy, in particular of Ricardo, arguing that in Marx's texts there is both a deepening of the analysis begun by classical political economy, and a break with it. Readers with no previous knowledge of Marx's theory of value might find it helpful to begin with this short discussion of Marx's texts, and then turn to the last essay, by Diane Elson, which among other things discusses the various interpretations of Marx's theory of value which have been prominent in CSE debates. This should provide sufficient background for following the more detailed consideration of different aspects of Marx's theory of value presented in the other essays.
The second essay in the collection, by Jairus Banaji, draws our attention to Lenin's well known conclusion that,
> 'It is impossible completely to understand Marx's _Capital_ , and especially its first chapter, without having thoroughly studied and understood the whole of Hegel's _Logic'_ ,
and explores the relation between Hegel's dialectic and Marx's theory of value. In the course of this, Banaji argues that it is quite wrong to suppose that Marx's theory of value is first elaborated for a precapitalist economy of simple commodity producers. Rather, the capitalist mode of production is assumed from the very first sentence of _Capital_. Banaji also shows that Marx was as concerned with appearance as he was with essence, as much with money as with the abstract and reified form of social labour. Most importantly, Banaji shows that the development of the theory of value at the beginning of _Capital_ embodies a dialectical method decomposable into two phases. First is a phase of analysis which begins from an immediate appearance, an historically determinate abstraction, the commodity. By analysing the commodity Marx arrives at the concept of value as the abstract-reified form of social labour. Value then forms the point of departure for the second, synthetic phase, of the investigation which returns to the level of appearance, to the commodity, showing it to be a representation of social relations not immediately apparent.
Geoffrey Kay, in the third essay, replies to some of the criticisms that bourgeois economics makes of Marx's method of analysis in his theory of value. He does so in the form of a discussion of the criticism advanced by Bohm-Bawerk in 1896 which he argues 'remains ahead in its field as the most coherent and systematic challenge to Marxism by any bourgeois economist'. In particular, Kay takes up the challenge that Marx's method of analysis is formalist, a 'purely logical method of deduction' not rooted in a consideration of real social relations. His argument is that this belief arises from a misunderstanding of the method of abstraction that Marx used. The same kind of misunderstanding creates confusion about the form of existence that Marx postulated for value: this was, argues Kay, not a type of labour, but money. Finally Kay considers Bohm-Bawerk's conviction that Marx's theory of value implies that commodities must in practice exchange in ratios proportionate to their relative values, and argues that in fact the possibility of a discrepancy between the two is an essential feature of Marx's analysis, right from the beginning of _Capital_ , I.
A neglected aspect of Marx's theory of value is the subject of the fourth essay by Chris Arthur. This discusses the argument of the third section of _Capital_ , I, Chapter 1, on the value-form, or form of appearance of value. As Arthur comments
> 'From the point of view of formal thinking nothing is going on here except the complication of a tautology-' 'a value is a value is a value!';
and this is perhaps why this section, with its important distinction between the relative and the equivalent forms of value, has been largely ignored. Arthur shows, however, that Marx's method of argument here is not one of formal abstraction, but a dialectical method, which Arthur calls 'the logic of the concrete'. It draws attention to material characteristics of the relation of exchange between commodities which cannot be captured by the methods of formal logic; and its achievement is to lead to an understanding of money as the form of appearance of value, not as a mere numeraire.
The fifth essay by Athar Hussain discusses the way in which Marx's theory of value has been read as a theory of price, in the context of a consideration of Marx's Marginal Notes on Wagner. Wagner overlooked the difference between Marx and Ricardo, and so, suggests Hussain, have many later economists themselves Marxists, or sympathetic to Marxism. This essay originally appeared in 1972 in _Theoretical Practice_ , and since then Hussain has changed his evaluation of Marx's value theory itself. But his comments on the way that other questions have been substituted for Marx's question, in the reading of the theory of value, retain their pertinence. Of particular interest is his treatment of the distinction between concrete and abstract labour, including the controversial view that the latter is not specific to the capitalist mode of production.
The sixth essay, by Makoto Itoh and Nobuharu Yokokawa, discusses the theory of the market process in Marx's theory of value. Its starting point lies in Japanese debates about the status of Marx's discussion of market-value i.e. the determination of value in cases where there are differences in conditions of production for the same kind of commodity. The authors argue that the social value cannot simply be deduced statically and technically as an average of the individual labour-times associated with the different production conditions. In a commodity economy the social value is only made apparent through the process of market competition, which reveals which of the individual conditions of production is the regulative one for that sector. The authors go on to discuss the nature of the additional profit that accrues to producers with conditions of production superior to the regulative condition, and the particular form this takes when land is one of the conditions of production.
The final essay, by Diane Elson, argues that the object of Marx's theory of value is not prices but labour. She suggests it is the traditional interpretation of Marx's theory as a labour theory of value which has been shown to be redundant; but not the value theory of labour which Marx presents in _Capital_. There are various ambiguities and incompletenesses in Marx's presentation, but nevertheless the core of his argument is a coherent and decidedly non-redundant exploration of the contradictions of the capitalist form of the determination of labour. Its political importance lies not in providing the foundation for a proof that capital exploits labour; but rather in providing the foundation for an analysis of the material basis for overcoming that exploitation.
## **Acknowledgements**
I am grateful to the Editorial Board of _Critique_ for permission to reprint the essay by Geoff Kay; and to Dave Smart for translating the essay by Aboo Aumeeruddy and Ramon Tortajada.
Special thanks to Sylvia Worsencroft for helping with the typing; and to John Gaffney of CSE Books for encouraging me to try to meet deadlines, being patient and re-arranging schedules when I didn't, and overseeing the transformation from manuscript to book.
Diane Elson,
Manchester, April 1979.
## _READING MARX ON VALUE: A NOTE ON THE BASIC TEXTS_
## _A Aumeeruddy and R Tortajada_
**Introduction**
Marxist theory (and, in particular, the theory of value) has been, and remains, a source of much controversy. On the one hand, the ruling class ceaselessly and systematically attacks 'marxists', at the same time attempting to recuperate, if not co-opt, Marxist theory by stripping it if its revolutionary content. On the other hand, some Marxists themselves try to turn Marxism into an 'improved political economy'.
If an 'economistic' reading of Marxism has proved possible, this is because there exist certain texts by Marx which admit this reading, while at the same time other of Marx's texts criticise such an approach; Marx's relationship with political economy is consequently very complex, and there can be no question of a key or set of instructions for reading Marx's works.
The reader can, in fact, begin with any text. Nevertheless, two works can be recommended as a starting point — the two series of lectures which Marx prepared for workers' organisations, and which directly address political and social struggles, both against capital and within these organisations themselves.
The first series — subsequently collated by Engels under the title _Wage Labour and Capital_ — dates from August 1847 and was written for the German Workers' Association of Brussels as a contribution to these workers' 'political education'.
The second series of lectures was considered by Marx himself to form a 'course in political economy', although he emphasised 'that it isn't easy to explain all economic questions to the uneducated' (Letter to Engels of 20th May 1865). It dates from June 1865, and was written for the General Council of the International Working-Men's Association. Subsequently published under the title _Wages, Price and Profit_ , it enjoyed massive circulation in pamphlet form, like the first series.
The advantage of starting with these texts lies not only in the fact that they are short, in plentiful supply, cheap, and thus readily available, but also in that Marx himself gave them a pedagogic character: they formed, and continue to form, a model of how to spread Marxist theory within the working class.
There is, however, the apparent paradox that these texts leave relatively obscure — or rather, deal in summary fashion — with certain aspects which are today at the centre of debate on Marx's theory, in particular the notions of value, of value-form, of magnitude of value, of the existence of the commodity, etc.
This is because Marx's principal intention in these lectures is not in fact to make a break with political economy but to call attention to capital itself, or rather to the social relation which appears in machinery as 'accumulated labour'. In doing so, Marx frequently bases himself on political economy, and in particular, as regards the concept of value, on Smith and Ricardo, while at the same time attacking and denouncing the 'Vulgar Economy' that superseded Ricardo's thought in the 1830s. (See 'Afterword' to the Second German Edition of _Capital_ , I.)
In order to understand Marx's relationship with political economy more deeply, it is essential to refer to other texts. The purpose of this note is to present a selection of these. This selection can obviously be neither neutral nor complete, particularly since Marxist theory is itself neither neutral nor finished except, that is, when transformed into its opposite — dogma. The texts which we put forward are thus those which seem to us the most adequate for understanding Marx's relationship with political economy, and are one way or another at the centre of the current discussion on this relationship.
It is common, particularly among 'economists', to start with Volume One of _Capital_ , or in some cases with its forerunners, the _Contribution to the Critique of Political Economy_ , or the _Grundrisse_ or even the _Theories of Surplus Value_ , often neglecting the works of 'the young Marx' which are considered to belong to the field of 'philosophy'. This kind of approach in fact reflects the academic separation of the 'social sciences', and it is consequently not surprising to see an academic like Schumpeter distinguishing a 'sociologist' Marx, a 'philosopher' Marx, an 'economist' Marx, etc. However, Marx's very procedure invites us to reject this separation and not to see _Capital_ as a work of political economy (albeit a 'left-wing' one), nor even as the culmination of a system of thought but as a moment in the development of a theory which sets out to challenge the existing order.
The order in which these texts are presented in no way seeks to define the best order for reading them, nor even to place them chronologically in order of writing or publication. As has been said above there is no 'key' to reading Marx. The _Marginal Notes on Adolph Wagner's 'Lehrbuch der politischen Okonomie'_ have been placed first not because these notes are Marx's least known work on political economy, but simply because this is one of the rare texts in which Marx replies directly to an economist who ventured to criticise his theory of value.
I. Marginal Notes on Adolph Wagner's 'Lehrbuch der politischen Okonomie'
This was written in 1881-82 (or 1879-80) and first published in 1932 as an appendix to the Moscow Marx-Engels-Lenin Institute edition of _Capital_. There are the following translations in English:
1)In _Theoretical Practice_ , No. 5, London, 1972, pp. 40-64;
2)Translated by Terrell Carver under the title _Notes on Adolph Wagner_ in K Marx, _Texts on Method_ , Oxford, Basil Blackwell, 1975, pp. 179-219.
3)Translated by Albert Dragstedt under the title _Marginal Notes to A Wagner's 'Textbook on Political Economy'_ in _Value: studies by Karl Marx_ , London, New Park Publications, 1976, pp. 197-229.
Marx here recalls his analysis of the relations between value, use-value and exchange-value. He emphasises:
a)that exchange-value and value must not be confused, _exchange-value_ merely being the 'phenomenal form' or 'necessary mode of expression' of _value;_
b)that he does not do away with _use-value_ , unlike classical political economy: 'The _value_ of a commodity is expressed in the use-value, that is to say the natural form of the other commodity';
c)that value, use-value and exchange-value are not alternative concepts, in logical opposition to one another, but are the _forms_ in which the commodity presents itself: in other words, three forms which coexist.
It is in this text that Marx repeatedly emphasises that he does not set out from 'value', but from the 'commodity', that is to say, 'the simplest social form in which the product of labour presents itself in contemporary society'. And it will be recalled that the first chapters of both the _Contribution to the Critique of Political Economy_ (1859) and the first volume of _Capital_ (1867) are in fact entitled 'The Commodity'.
II. The different versions of Chapter 1 of _Capital_ , Volume I
The first edition of _Capital_ Volume I, was published in German in 1867 under the supervision of Marx. A French edition prepared by Marx himself, was published in instalments from 1872-75, and a second German edition also prepared by Marx was published in 1873. The first edition in English was published in 1887, a translation by Samuel Moore and Edward Aveling of the third German edition, prepared by Engels, in 1883, with the assistance of notes left by Marx, indicating the passages of the second German edition which were to be replaced by designated passages from the French edition. There are textual differences between all these four editions, but the most important difference is that between Chapter 1 of the first and subsequent editions.
The versions currently most widely available in English are:
1)The Moore-Aveling translation of the Third German edition, incorporating amendments made by Engels for the Fourth German edition, published by Lawrence and Wishart, London, various dates;
2)The new translation of Volume I by Ben Fowkes published by Penguin Books, London, 1976;
3)The text of the Appendix to Chapter 1 of the First German edition, translated by M Roth and W Suchting in _Capital & Class_ No. 4, Spring 1978, pp. 134-150. This deals with the value form.
In reading _Capital_ it is necessary to bear in mind the _Introduction_ sketched out by Marx in August 1857 for _A Contribution to the Critique of Political Economy_. In this 'Introduction' — which was not in fact published with the _Critique_ but can be found in the edition of the _Grundrisse_ prepared by M Nicolaus and published by Penguin Books, 1973, pp. 81-111 — Marx shows that in presenting an exposition of his theory, it was in fact necessary to reverse the order in which it was constituted, a reversal which represents a significant formal aspect of _Capital_. It must also be remembered that the texts published posthumously under the title _Capital_ Volumes II and III and the analyses collected in the _Theories of Surplus Value_ were, as regards essentials, already in existence _before_ Marx wrote the first volume of _Capital_.
Consequently, neither Volumes II or III of _Capital_ nor the _Theories of Surplus Value_ are intended to 'complement' or 'make specific' the 'abstractions' of the first volume of _Capital_. On the contrary, it is on the basis of reading this latter, and more particularly Chapter 1 that certain of the questions raised in Volumes II, III and in _Theories of Surplus Value_ can be analysed. This does not mean that these works are not worthy of attention. On the contrary, study of them is an integral part of research-work aimed at clarifying the relations between Marx and political economy.
The questions which today appear central to numerous debates relating to Marxist theory are posed from the first chapter of _Capital_ , (a chapter which, as Marx himself wrote in his Preface to the First Edition of _Capital_ presents 'the greatest difficulty'.) Among them are: the primacy of the category 'commodity' for the comprehension of capitalist relations; value and its forms; the magnitude and measurement of value; the status of labour and abstraction; the relationship between Marxist theory and Ricardian theory; and the fetishism of commodities.
However, two of the questions raised seem to us to require particular attention insofar as one led Marx to rewrite the beginning of _Capital_ Volume I, and the other defines a certain mode of reading not only this work, but Marxist theory as well.
The first concerns the relation between the study of value and of the forms which it assumes. It is important firstly because the relationship between the _'value'_ of a commodity and its phenomenal form _'exchange-value'_ is at the centre of the debates on Marx's relationship with political economy. Moreover, understanding of the 'general equivalent' and hence of money derives, in Marx's view, from an understanding of the forms of value. Finally, it is this same question which is at the root of the profound reworking of _Capital_ Volume I by Marx between the first and second editions of _Capital_ , a reworking retained in subsequent editions.
In the first edition, Chapter 1 was in fact devoted to the 'commodity' and to 'money', and was divided into three sections: The Commodity, The Process of Commodity-Exchange, Money and Commodity-Circulation. Analysis of the forms of value was consigned to an Appendix at the end of the work, in which Marx analysed these forms systematically and in detail. The three sections became the first three chapters of subsequent editions and the appendix was reintegrated into the first of them.
The second question concerns the process of abstraction. In _Capital_ , and in the first chapter in particular, there is not one single process of abstraction, but two processes of abstraction profoundly _different_ in character. It is therefore worth distinguishing very precisely between them, if only to avoid the very common _confusion_ by which
a) _Capital_ Volume I is seen as an 'abstract' construction (in the sense of being estranged from reality by the adoption of extremely restrictive hypotheses, often cited examples of which are identical organic composition between the various branches of production, and homogeneous labour);
b) Volumes II and III are seen as Marx's attempts to relate his 'abstract' theory to some reality, thus characterising Karl Marx as a 'builder' of economic models to be tested against 'reality'.
On the one hand there is a process of _thought_ or _reasoned abstraction_ , to use the terms Marx himself employs in the _Introduction_ , and on the other hand, on an entirely different level, a process of _real abstraction_.
_Reasoned abstraction_ is to do with the discovery of categories which permit bourgeois social relations to be understood. As Marx emphasised in his pamphlet against Proudhon, 'Economic categories are only the theoretical expression, the abstractions of the social relations of production' ( _The Poverty of Philosophy_ , (1846-47), Lawrence & Wishart, n.d., Chapter II, Second Observation, p. 105). At the same time, he makes it clear that these abstractions are not to be confused with the social relations themselves. Later, in writing the Preface to the first edition of _Capital_ , he returns to this aspect of abstraction, pointing out that 'In the analysis of economic forms moreover, neither microscopes nor chemical reagents are of use. The force of abstraction must replace both.'; and that within bourgeois society this leads one to set out from the simplest and, it would seem, most immediate form, the commodity. Study of the commodity is thus the corner-stone of the analysis of a society characterised by the generalisation of commodity relations to include the labourer himself.
_Real abstraction_ , on the other hand, is not the result of analytical effort, but the consequence of a real process which is at the heart of bourgeois social relations — commodity-exchange.
In bourgeois society, where the private division of social labour prevails, products are the result of private, isolated processes of production operating independently of one another. It is only when production has been completed, that is to say when the labour mobilised has been objectified in a determinate good, that producers' respective acts of production encounter one another on the market, as products offered in exchange for money. And the producers will only know that their products effectively answered a social need if they succeed in exchanging them. Commodity-exchange is the social mode of recognition of the different products, and it is _via_ this exchange that they cease to be products and become commodities, or rather that the commodity which is potential when the product is present on the market becomes real-is realised. It is thus as commodities that the different acts of labour privately carried out in separation from one another become fractions of social labour.
But commodity-exchange is only conceivable if there exists a relation of equivalence between different commodities. From the point of view of their use-values-the physical characteristics of the products-commodities are of course different, hence non-equivalent, and it is precisely this difference which is the motive force of exchange. But in the course of exchange, the use-value of the commodities is abstracted from, and only the social capacity of the commodities to be exchanged is recognised. According to Marx's terms, this 'abstraction' entails abstraction from the specific characteristics of the acts of labour objectified in the commodities. This leaves the commodities as nothing but the result of human labour, without regard to the particular form it takes, in other words of labour 'full stop'. This is abstract labour.
It is because all products participate in this process of abstraction when they become commodities and are therefore recognised as fractions of abstract social labour that one can conceive of establishing a relation of equivalence between them; they belong to the same sphere.
In order to avoid any ambiguity, it is worth emphasising that the process whereby different acts of labour are reduced to abstract labour has nothing whatever to do with the process whereby 'complex labour' is reduced to 'simple labour'. Whereas the first process is involved in the _founding_ of value, the second belongs to a different logic: it relates to the _measurement of magnitudes already constituted_.
III. Results of the Immediate Process of Production
Sometimes known as the 'lost chapter' of _Capital_ , this was written between June 1863 and December 1866 and was first published in 1933, simultaneously in German and Russian. An English translation has recently been made by Rodney Livingstone, and appears as an Appendix to the new Fowkes translation of _Capital_ , I, Penguin Books, 1976, pp. 948-1084.
It was originally planned as a transitional chapter between Volume I and Volume II of _Capital_ , as it is not definitely known why Marx discarded it. It contains both a synthesis of the argument of Volume I, and a further development of the relations between 'value' and 'use-value' in terms of the subsumption of labour to capital. It completes the argument of Volume I by investigating commodities not only as the _premise_ of the formation of capital but also as the _result_ of capitalist production. 'Only on the basis of Capitalist production does the commodity actually become the _universal elementary form of wealth'_. (Op. cit., p. 951).
IV. Introduction drafted for a Contribution to the Critique of Political Economy
Written in 1857 (dated 29th August), it was first published in _Die Neue Zeit_ , 1903 (?); and republished in 1939 by the Moscow Marx-Engels-Lenin Institute. The following English versions are available:
1)Translated by Martin Nicolaus in _Grundrisse_ , Penguin edition, 1973, pp. 81-111;
2)Translated by David McLellan in Marx's _Grundrisse_ , Paladin edition, 1973, pp. 26-57;
3)In _Texts on Method_ , ed Terrell Carver, Oxford, Basil Blackwell, 1975.
Although Marx did not publish this 'Introduction' on the grounds that it anticipated too much the ideas developed in the work itself, it is of fundamental importance both to understanding Marx's 'methods' and to his critique of economic analysis which began from the isolated 'individual' and considered 'production', 'distribution', 'exchange' and 'consumption' only as separate economic 'moments' not as interpenetrating processes.
V. A Contribution to the Critique of Political Economy
Written in 1858-9, it was first published in Berlin in 1859 under the title _Zur Kritik der Politischen Okonomie_. An English translation, edited by Maurice Dobb, was published by Lawrence & Wishart in 1971. The most famous section of it is the Preface, but more relevant to the question of value is the first chapter, entitled _The Commodity_. It is one of the few works published during his lifetime by Marx himself.
VI. Letters on Capital
Marx and Engels carried on a voluminous correspondence. A selection of this was published by Progress Publishers, Moscow, and distributed by Lawrence & Wishart. The following three letters are of particular interest. (Page references to Marx-Engels, _Selected Correspondence_ , Progress Publishers, Moscow, n.d.)
(a)Letters from Marx to Engels, 2nd and 9th August, 1862. (pp. 157-161; 164-165).
It was in connection with Ricardo's theory of ground-rent that Marx, for the first time, came to make the relations between 'value' and 'price' explicit, doing so in terms very close to those used in what would later form _Capital_ , Volume III, Part VI.
From the outset, Marx emphasises that 'Competition does not therefore equate commodities to their value, but to cost prices which are higher than, lower than or equal to their values according to the organic composition of the capitals.' It would seem that Marx is rediscovering the difficulty previously encountered by Ricardo: exchange is not based on the labour-time incorporated if the prices incorporate the general rate of profit.
This difficulty results in two kinds of development. On the one hand, the analysis of the forms of value, which manifests itself in the successive versions of _Capital_ , I, Chapter 1 (Cf. Section II above). On the other hand, the well-known 'transformation of values into prices'.
Besides this first formulation of the analysis of the relationship between value and price, we can see Marx's concern to establish as quickly as possible the connections between the development of theory and the practice or struggle in which he was taking part. In this case, it is analysis of the contradictions of a certain practical solidarity between capitalists and landed proprietors.
It is in fact this concern which leads him: firstly, unlike Ricardo, to distinguish the possibility of 'absolute' rent independently of 'differential' rent; secondly, to base the existence of ground-rent on comparison of values with 'cost prices' (which he confused, at the time, with 'production prices'). This procedure made possible an _identification_ of the spheres of 'value' and 'exchange-value' with one another. Marx himself elsewhere criticised this identification which arises again in the problem of 'transformation'. (Cf. Sections I and VIII in this note).
(b)Letter to Kugelmann, 11th July 1868, (pp. 250-253)
This letter has been quoted and referred to so often that it must, if only because of this, be read in its entirety. May we repeat, however, that like all the other of Marx's texts cited here, it cannot form a key or method for the reading of _Capital_. It certainly raises the question of Marx's complex relationship with political economy — in which Marx simultaneously deepened, and broke with, the latter's analysis.
We would emphasise three points about this letter:
(i) As in his _Marginal Notes on Adolph Wagner's 'Lehrbuch der politischen Okonomie'_ , Marx distances himself from the 'theory' of value, indicating that the _concept_ of value takes second place, in his work, to the analysis of real relations. 'The unfortunate fellow (author of a review of _Capital_ , I) does not see that, even if there were no chapter on 'value' in my book, the analysis of the real relations which I give would contain the proof and demonstration of the real value-relation.' (op. cit. p. 251).
(ii) The second point concerns the relation between the magnitudes of values and exchange-relations. Rebutting the vulgar economists, Marx emphasises that there cannot be _immediate_ identity between 'the real relations of day-to-day exchange' and the 'magnitudes of values' in bourgeois society. But if there is no immediate identity, Marx nevertheless leaves room for a certain ambiguity on the possibility of _mediations_. The 'blindly operating mean' has in fact been interpreted in an 'economistic sense to form the basis for the 'transformation' approach.
(iii) Finally, this letter leaves open the possibility of a naturalist interpretation of the concept of 'law': 'No natural laws can be done away with. What can change in historically different circumstances is only the _form_ in which these laws assert themselves'. (op. cit. p. 251). Consequently; it seems that Marx is here considering 'value' to be an ahistorical concept, and that it is only the form in which it manifests itself, exchange-value, that is historically determined.
VII. The 1844 Manuscripts
Written in 1844, these manuscripts were not published until 1932. The following versions are available in English:
1)'Excerpts from James Mill's Elements of Political Economy', translated by Rodney Livingstone in Marx, _Early Writings_ , Penguin, 1975, pp. 259-278.
2)'Economic and Philosophical Manuscripts', translated by Gregor Benton in _Early Writings_ , Penguin, 1975, pp. 280-400.
3)In _Economic and Philosophical Manuscripts_ , translated by Martin Milligan, Lawrence & Wishart, 1959.
4)In Marx, _Early Writings_ , ed. T Bottomore, London, 1963.
At the end of 1843, Marx began serious study of the works of the principal economists. The first writings explicitly dealing with political economy were subsequently known by the title of the _1844 Manuscripts:_ comprising the _Notes on James Mill_ and, more importantly, the _Economic and Philosophical Manuscripts_.
'Economistic' interpretation of Marxism has neglected the works of Marx's 'youth' and continues to do so. We do not propose to enter into the debate over the 'continuity' (Meszaros, Colletti) or the 'break' (Althusser) between the 'young' and the 'mature' Marx here, but simply to emphasise that, in so far as they constitute a 'turning point' between the critique of philosophy, law and the state (in the writings of 1843, in particular the _Critique of Hegel's Doctrine of the State_ , the _Jewish Question_ , and the _Introduction to the Critique of Hegel's Philosophy of Right_ ) and the critique of political economy, it is essential to _read_ the _1844 Manuscripts_. This is not so as to arrive at the 'correct'(!) interpretation of Marx's thought — there is obviously no _single_ interpretation of the _1844 Manuscripts_ — but in order to consider such questions as that of the relationship between the 'abstraction of labour' elaborated in the _1844 Manuscripts_ and the concept of 'abstract labour'-the substance and measure of value — elaborated in _Capital_. On this the most useful passages are the section on Estranged Labour in the first of the _Economic and Philosophical Manuscripts_ , and in _Excerpts from James Mill's Elements of Political Economy_ , op. cit. pp. 265-278. See, for example, the following passage from the latter:
> 'Thus private property as such is a _surrogate_ , an _equivalent_. Its immediate unity with itself has given way to a relation to _another_. As an _equivalent_ its existence is no longer peculiar to it. It thus becomes a _value_ , in fact an immediate _exchange-value_. Its existence as _value_ is a determination of itself diverging from its immediate nature, external to it, alienated from it, a merely relative existence. The problem of defining this _value_ more precisely, as well as showing how it becomes price, must be dealt with elsewhere. In a situation based on _exchange_ , labour immediately becomes wage-labour.' (Op. cit, p. 268.)
VIII. Theories of Surplus Value, Parts One, Two, Three
Written in 1862-63, this massive examination and critique of the development of economic thought is often referred to as the Fourth Volume of _Capital_. An edition was first published between 1905 and 1910 by Kautsky, but the arrangement of the text differed in various ways from that of the manuscript. A German edition was published in Berlin in 1956-1966 on the basis of the German original kept in the Institute of Marxism-Leninism in Moscow. The English edition published by Lawrence and Wishart, 1969-1972, is a translation of the Berlin edition.
In so far as the manuscripts are devoted to the theory of value and surplus-value in their entirety, it is difficult to pick out particular passages. Nevertheless, in the context of current debates on the theory of value, it seems useful to consider in particular the chapters which Marx devotes to a critical examination of the theories of Smith and Ricardo. In Part One, Chapter III is devoted to Smith. In Part Two, the relevant passages are Chapter X, where Marx compares Smith and Ricardo's theories of cost price; and Chapter XV and XVI, which are devoted to Ricardo's theory of surplus value and of profit. These passages throw light on Marx's analysis of 'classical' theories of value, but only illuminate indirectly his own theory. Consequently, they leave the door open for an 'economistic' reading of Marx's position.
Note also the criticism Marx makes of the 'vulgar economists', in particular that of S Bailey and the (unknown) author of _Observations on Certain Verbal Disputes in Political Economy_ , where Marx is led to tackle the problem of the relations between 'invariable measure' and 'value' (see Part III, Chapter XX, pp. 124-168; cf. also pp. 110-117). We would emphasise three points in this critique:
(i)First of all, there is the question of the relationship between the search for an 'invariable measure' and Marx's theory of value. As far as Marx is concerned, the search for an 'invariable measure' falls outside the problematic of value. The object of the theory of value is not to constitute an 'invariable measure' of the exchange-relations of commodities. A 'measure' of this kind can only be conceived, in Marx's view, if a theory of value has _first_ been constituted. For commodities to be compared with one another in exchange, in terms of exchange-value, it is necessary for the various commodities to be expressions of the same substance. 'The commodities must _already be identical as values'_. (Op. cit. p. 134).
It is in so far as they are fractions of abstract social labour that commodities are expressions of the same substance. The abstraction here has nothing to do with any kind of mental process, but is the social mode whereby men's different acts of labour are recognised in a society in which commodity-exchange prevails.
(ii)Although Marx repeatedly emphasises the difference between the status of value and that of the forms which it is liable to assume, and also the difference between the status of labour in Ricardian theory and in his own theory of value, he nevertheless uses ambiguous formulations on these two points. These ambiguities are, moreover, only very partially resolved when he takes up the question again in _Capital_ , Volume One. The first ambiguity arises from the fact that Marx, on occasion, attributes to Political Economy aims which it was not pursuing, for example, the formulation of a theory of 'value' or of 'surplus-value' whereas its objective, as Marx himself often emphasises, was principally the analysis of exchange-value or the forms of surplus-value. In this particular passage, Marx implies that the search for an 'invariable measure' coincides with the search for the 'value' of commodities:
> 'The problem of an 'invariable measure of value' was simply a spurious name for the quest for the concept, the nature of _value_ itself, the definition of which could not be another value, and consequently could not be subject to variation of value'. (Op. cit. p. 134.)
A second ambiguity, which follows on from the first, concerns the status of labour. Marx's main reproach to Ricardo is not so much that he is oblivious of 'abstract labour', but that he 'continually confuses' it (op. cit. p. 139) with the labour which is represented in use-value.
(iii) Finally, it is in elaborating his criticism with respect to Bailey that Marx emphasises with particular clarity why the determination of the exchange-relation cannot be based solely on the exchange-relation itself.
This critique, which is aimed directly at Bailey, also forms the corner-stone of the criticism directed at the various theories which do base the theory of value solely on the exchange-relation — viz. supply and demand.
IX. Grundrisse: Foundations of the Critique of Political Economy
This is a series of seven notebooks rough-drafted by Marx, chiefly for the purpose of self clarification, during the winter 1857-8. The manuscript became lost under circumstances still unknown and was first effectively published in the German original in Berlin in 1953.
The following versions are available in English:
1)A full version translated by Martin Nicolaus, Penguin, 1973.
2)Extracts in David McLellan, _Marx's Grundrisse_ , MacMillan, 1971, Paladin, 1973.
3)Extracts in Karl Marx, _Precapitalist Economic Formations_ , ed. E Hobsbawm, Lawrence & Wishart, 1964.
In these 'jottings' Marx tackles a number of points which subsequently receive only sketchy treatment; in particular, with respect to value, the relationship of money to value, (in particular Penguin ed. pp. 136-172) and the relationship between exchange-value and private property (previously dealt with in the _1844 Manuscripts_ ). Note also the passages relating to 'Forms preceding capitalist production' (Penguin, pp. 471-514); not only because these passages have been much discussed (and criticised), but also because they contain in germinal form an analysis of the historical genesis of value and of the 'abstraction of labour', a problem which lies at the heart of our preoccupations.
## [_FROM THE COMMODITY TO CAPITAL:
HEGEL'S DIALECTIC IN MARX'S 'CAPITAL'_](04_Contents.xhtml#s3)
## _Jairus Banaji_
THE DIALECTIC IN LENIN AND MARX
Lenin
It is well known that on reading Hegel's _Logic_ late in 1914 Lenin was so profoundly impressed by the impact which he now discovered it had made on Marx that he wrote,
> 'It is impossible completely to understand Marx's _Capital_ , and especially its first chapter, without having thoroughly studied and understood the whole of Hegel's _Logic.'_
Then a startling conclusion: 'Consequently, half a century later none of the Marxists understood Marx!!' (Lenin, 1972, p. 180).
Today, over a century later, this statement has a special significance. In Lenin's day, the Second International had more or less explicitly shifted the philosophical and scientific premises of Marxism in directions quite distant from its Hegelian origins — towards biological evolutionism (Kautsky), varieties of neo-Kantianism (Bernstein among others), and even, through Max Adler, the positivism of Comte (cf. Goldmann, 1959, pp. 280-302). In our own, contemporary period the publication of Della Volpe's _Logica come scienza positiva_ (its original title of the fifties) inaugurated a roughly similar reaction. Conscious repudiation of the dialectic, of the enormous weight of Hegel's method in Marx's development and thus in the formation of Marx's theory, became a fundamental and unifying characteristic of postwar 'Western Marxism'. As in the nascent period of reformism, late in the nineteenth century, so now, in its declining phase, in the sixties, this critique of 'metaphysics' (Della Volpe, 1969) or of 'illusion' (Althusser, 1969) _could only_ take the form of a bizarre philosophical eclecticism ranging across the most divergent and logically incompatible tendencies: from forms of empiricism more (Colletti) or less (Della Volpe) sympathetic to Kant, or simply reflecting the hostility of Positivists to any philosophical connection (Timpanaro), to a revitalised Spinozist rationalism (Althusser). As Althusser's own acknowledgment suggests (1969, pp. 37-38), Della Volpe was the _major_ figure in this belated movement of reaction. The most serious specific consequence of Della Volpe's attack on Hegel was its 'experimentalist' recasting of the dialectic into a conception of science, and of scientific method, strikingly close to Popper's neo-empiricism.1 When he turned to _Capital_ with this underlying conception of the 'reciprocity of fact and theory', or of induction and deduction, or of reason and experience, Della Volpe would not only ascribe to Marx a _labour_ theory of value, but see in the latter a scientific 'hypothesis' which the real development of 'monopoly capitalism' had finally confirmed as true (Della Volpe, 1969, p. 201). Of course, it was with a similar philosophy of science and from similar premises that Popper himself set out to argue the precise opposite.2
Lenin's statement thus takes on a special importance today. The _Logic_ , he says, is fundamental to a correct understanding of Marx's _Capital_ , and 'especially of its first chapter' which contains the theory of value.
On the other hand, how well did Lenin himself understand the _Logic?_ His _Notebooks_ are full of question-marks, of doubts, for Lenin is actually reading Hegel for the first time, with a philosophical past of his own, dominated by a form of empiricism. One group of passages in particular reveals a quite incomplete penetration of the movement of the dialectic. For example, in the concluding sentence of his summary of Hegel's small _Logic_ , Lenin writes,
> 'Cf. concerning the question of Essence versus Appearance
>
> — price and value
>
> — demand and supply versus _Wert (= krystallisierte Arbeit)_
>
> — wages and the price of labour-power' (Lenin, 1972, p. 320).
This says, value = 'essence', price = 'appearance'. Now in the price-form value appears either as pure contingency or as a merely imaginary relationship (cf. _Capital_ , I, p. 197) so that the equations suggested identify appearance with _contingency_. A passage from the essay on 'Dialectics' shows that this is in fact how Lenin understood the matter. In any proposition of language,
> 'already we have the contingent and the necessary, the appearance and the essence; for when we say: John is a man, Fido is a dog... we _disregard_ a number of attributes as _contingent;_ we separate the essence from the appearance, and counterpose the one to the other.' (Lenin, 1972, p. 361.)
In his 'Conspectus' of the _Logic_ , in the summary of Hegel's section on Appearance (in the major _Logic_ this starts with a very short sentence, 'Essence must appear', _Science of Logic_ , p. 479), Lenin criticises Hegel in the following terms:
> 'The shifting of the world in itself further and further _from_ the world of appearances-that is what is so far still not to be seen in Hegel.' (1972, p. 153. All emphases Lenin's.)
And this makes sense. If the world of appearances is a world of pure contingency, if we arrive at the 'essence' of the matter by 'disregarding' its contingent attributes, then it follows that it is the task of scientific cognition to carry through this 'separation', to _'shift_ the world in itself _further and further from_ the world of appearances'. As Lenin correctly notes, Hegel _does not do this_.
The conception that Lenin holds to has two consequences. Firstly, it sees the dialectic as a study of the _opposition_ of essence and appearance. In his own words,
> 'Dialectics is the study of the opposition of the thing-in-itself, of the essence... from the appearance.' (1972, p. 253.)
Secondly, this false conception of the dialectic implies something quite specific for the _method_ of scientific cognition, and this too is quite plain from the _Notebooks_.
> 'Hegel is completely right as against Kant. Thought proceeding from the concrete to the abstract-provided it is correct... does not get away from the truth but comes closer to it. The abstraction of matter, of a law of nature, _the abstraction of value_ , etc., in short _all scientific_ (correct, serious, not absurd) _abstractions reflect nature more deeply, truly and completely_. From living perception to _abstract_ thought, and _from this_ to practice-such is the dialectical path of the cognition of truth...' (1972, p. 171) (Emphasis mine)
But how from _abstract_ thought or from the 'abstraction of value' does one move straight to practice? This is not a question Lenin asks himself at this point. On the other hand, in _Hegel_ , he finds a more subtle, more complex movement, and this puzzles him:
> 'it is strikingly evident that _Hegel sometimes passes from the abstract to the concrete... and sometimes the other way round_... Is not this the inconsistency of an idealist?... Or are there deeper reasons?' (1972, p. 318.)
The 'inconsistency' lies not in Hegel, but in Lenin. For on the one hand, as the _Notebooks_ show, he realises that for Hegel essence 'must appear', that appearance itself is essential (cf. 1972, p. 148, p. 253), but on the other hand, he regards appearances as pure contingency, as that from which we 'abstract' so as to arrive at 'essence', as a world apart from and opposed, or counterposable to essence.
Hegel in Marx
There are countless references to problems of scientific method scattered across the pages of Marx's later work. To draw some of these together here in a form that recapitulates their underlying conceptions: _firstly_ , there is a methodological reference that is basic to any understanding of the architecture of _Capital_ , namely, the distinction Marx repeatedly draws between 'capital in general' and 'many capitals' (cf. Rosdolsky, 1968, Volume I, p. 61 ff.). The former refers to the 'inner nature of capital', to its _'essential_ character' ( _Grundrisse_ , p. 414), and is also called 'the simple concept of capital' (ibid.); by contrast, 'many capitals or competition of capitals, entails a study of capital 'in its _reality_ ' ( _Grundrisse_ , p. 684 note), or in its 'concrete' aspects as they appear reflected on the _surface of society_ , in the _'actual_ movement' of capitals ( _Capital_ , III, p. 25). So in the first place the investigation of capitalist economy is broadly stratified into two levels which contrapose the 'essential character' of capital to its 'concrete' or 'actual' superficial movements. But _secondly_ , the investigation itself is a _movement_ from one level to the other, from essence to concreteness. In the Preface to the first edition (1867) of _Capital_ I, Marx writes that in the analysis of 'economic forms', i.e. of social phenomena as such, the 'power of abstraction' must replace a directly experimental, hence empirical, relation to the object. In what does the power of abstraction consist, however? About this the passage in question leaves no room for doubt. It consists in our ability to identify a _point of departure_ for the movement from one level to the other, and a point of departure which will be simultaneously the _foundation_ of that movement. For Marx here introduces the notion of a 'cell-form' ( _Capital_ , I, p. 90), which he identifies with Hegel's 'in itself ( _An-sich_ , or essence) in the first edition form of Chapter One (cf. Zeleny, 1973, p. 78, n. 8, for the passage). The movement from the cell-form to the concrete is logically continuous, so that, in approaching the concrete forms in which capital appears on the surface of society, we do not abandon the sphere of essential relations as if we were moving across into new territory; rather, we now investigate those very relations in 'their' forms of appearance, i.e. in the forms determined within their own logical movement, as part of this movement.
These intermediate levels of the logical process, which connect abstract and concrete, are as essential as essence in its abstract and simple cell-form. Marx's constant reference to these intermediate levels or 'terms' or 'stages', or to the 'connecting links' ( _Theories_ , 3, p. 453, 2, p. 174, _Capital_ , I, p. 421), implies a _logic of derivation_ (of 'deduction' in the broad sense) which is distinct from the pure deductive method of axiomatic systems (on this see Zeleny, 1973, p. 75 ff., p. 141 ff.). The concrete is derived by stages, from the abstract. Where this process of dialectical-logical derivation collapses, as it does in Classical Economy, Marx refers to 'forced abstractions', to the direct subordination of the concrete to the abstract ( _Theories_ , 1, p. 89, p. 92; 2, p. 164 f., p. 437; 3, p. 87).
Thirdly, in the famous introduction of 1857 ( _Grundrisse_ , p. 100 ff.), the movement of essence from abstract to concrete is also described as a journey from the simple to the combined. The movement of derivation of forms within a framework defined by its logical continuity is thus also a process of 'combination', of the 'concentration' of many 'determinations' into a 'rich totality' which reproduces the concreteness of reality no longer simply as something that impinges confusedly on perception but as something rationally comprehended. These 'determinations' are only the forms derived in the movement of essence as the _form-determinations of essence_ (cf. Rubin, 1972, p. 37 ff.).
Finally, the entire process by which the concrete is reproduced in thought as something rationally comprehended is described in places by Marx as the 'dialectical development' of the 'concept' of capital, and all moments within this movement which are derivable as essential determinations, including, of course, the forms of appearance, no matter how illusory they may be, count as moments (forms, relations) 'corresponding to their concept' (e.g. _Capital_ , III, p. 141). (This is why, despite its illusory and deceptive character, Marx can call the wage-form 'one of the _essential_ mediating forms of capitalist relations of production', _Results_ , p. 1064.)
It is obvious that the methodological references express a consistent and internally unified conception which it is impossible to grasp without reference to the dialectic, that is, to what can now be formally defined as a specific, _non-classical logical type_ of scientific thought, a form of scientific reasoning and proof distinct from generalising inductivism, deductive-axiomatic methods, or any combination of these supposedly characteristic of a 'scientific method in general', e.g. Della Volpe's hypothetico-deductive method.
The point can also be put in these terms: it is impossible to grasp Marx's conception of scientific method outside the framework of Hegel's _Logic_. This is not to claim that, like Lassalle, he simply 'applied an abstract ready-made system of logic' ( _Selected Correspondence_ , p. 123) to the phenomena of capitalist economy. The claim is a different one: the method that Marx followed was a method 'which _Hegel_ discovered' ( _Selected Correspondence_ , p. 121). It was Hegel who first enunciated the conception of a point of departure which is simultaneously the foundation of the movement which it initiates ( _Science of Logic_ , p. 71). For Hegel this was only conceivable because the principle that forms the beginning is not something 'dead', something fixed and static, but something 'self-moving' (Hegel, 1966, p. 104). Hegel's great announcement in the 'Preface' to the _Phenomenology_ is the conception of 'substance' as 'subject', or the conception of a 'self-developing, self-evolving substance', where the term 'substance' can be taken in its classical, Cartesian sense to mean 'that which requires only itself for its existence'. (The importance of this idea for Hegel's work and for its relation to Marx is drawn out by Zeleny, 1973, p. 47 f., p. 98 f.)
As the 'process that engenders its own moments and runs through them' (Hegel, 1966, p. 108) this substance-subject is what Hegel calls _das Wesen_ , essence. Essence cannot be said to be something _'before_ or _in_ its movement', and this movement 'has no substrate on which it runs its course' ( _Science of Logic_ , p. 448). Rather, essence is the movement through which it 'posits itself, 'reflects itself into itself, as the totalising unity of 'essence and form' ( _Science of Logic_ , p. 449). Conversely,
> 'the question cannot therefore be asked, _how form is added to essence_ , for it is only the reflection of essence into essence itself...' ( _Science of Logic_ , p. 449-50)
Moreover, if form is _immanent_ , or
> 'if form is taken as equal to essence, then it is a misunderstanding to suppose that cognition can be satisfied with the 'in itself or with essence, that it can dispense with form, that the basic principle from which we start ( _Grundsatz_ ) renders superfluous the _realisation of essence_ or the _development of form_. Precisely because _form is as essential to essence_ as essence to itself, essence must not be grasped and expressed merely as essence... but as form also, and with the entire wealth of developed form. Only then is it grasped and expressed as something _real.'_ (Hegel, 1966, p. 50)
In this decisive passage Hegel says essence must realise itself, or 'work itself out', and this it can only do through the 'activity of form' ( _Science of Logic_ , p. 453). Only as this self-totalising unity of itself and form does it become something 'real' ( _wirkliches_ ). Otherwise, as immediate substance, substance not mediated through its self-movement, essence remains something abstract and so one-sided and incomplete. It remains something basically _untrue_ , for, as Hegel goes on to say, in the passage cited above, 'the truth is _totality'_ (Hegel, 1966, p. 50), a fusion of essence and form, universal and particular, or the universal drawing out of itself the wealth of particularity.
It is interesting that in a terminology that is almost indistinguishable from Hegel's, Marx articulates an identical conception as early as his _Dissertation_. For Hegel's argument can be summarised in his own words as follows: ' _Appearance is itself essential to essence'_ (Hegel, 1970, p. 21). Now in the _Dissertation_ Marx argues that although they shared the same general principles (Atomist), Democritus and Epicurus evolved diametrically opposed conceptions of knowledge and attitudes towards it. Democritus maintained that 'sensuous appearance does not belong to the Atoms themselves. It is not _objective appearance_ but _subjective semblance (Schein)_. The true principles are the atom and the void...' (Marx, 1975, p. 39). So for Democritus 'the principle _does not enter into the appearance_ , remains without reality and existence', and the real world, the world he perceives, is then _'torn away from the principle_ , left in its own independent reality' (Marx, 1975, p. 40). In Hegel's terms, for Democritus the Atom is devoid of form, has no form of appearance, so that the world of appearances ( _Erscheinungen_ ) necessarily degenerates into a world of pure illusion ( _Schein_ ). 'The Atom remains for Democritus a _pure and abstract category, a hypothesis'_ (Marx, 1975, p. 73. All emphases in this paragraph mine). Or, 'in Democritus there is no _realisation of the principle itself_ (Marx, 1975, p. 56 ff.). On the other hand, if Democritus transforms the world we perceive into pure illusion, Epicurus regards it as 'objective appearance'.
> 'Epicurus was the first to grasp appearance as appearance that is, as alienation of the essence' (Marx, 1975, p. 64).
>
> 'In Epicurus the consequence of the principle itself will be presented' (id. p. 56).
Or for Epicurus the Atom is not a simply abstract and hypothetical determination, it is something 'active', a principle that 'realises itself'.
The Dialectic as Critique of Bourgeois Economy
The conception of Democritus is dominated by the following contradiction: what is true, the principle, remains devoid of any form of appearance, hence something purely abstract and hypothetical; on the other hand, the world of appearances, divorced from any principle, is left as an independent reality. It is not difficult to see that in the critique which Marx developed many years later, classical and vulgar economy emerged as the transfigured expressions of the poles of this contradiction. So Marx would write,
> 'By classical political economy I mean all the economists who... have investigated the real internal relations of bourgeois economy as opposed to the vulgar economists who only flounder around within their forms of appearance' ( _Capital_ , I, p. 174. Translation modified).
>
> 'Vulgar economy feels especially at home in the alienated external appearances of economic relations' ( _Capital_ , III, p. 796. Translation modified),
whereas classical economy, which investigates those relations themselves, seeks to grasp them _'in opposition to their different forms of appearance'_. Classical economy says, the appearances are pure semblance ( _Schein_ ), only the principles are true. So
> 'it is not interested in evolving the different forms through their inner genesis ( _die verschiednen Formen genetisch zu entwickeln_ ) but tries to reduce them to their unity by the analytic method' ( _Theories_ , 3, p. 500. Translation modified).
Again, classical economy 'holds instinctively to the law', 'it tries to rescue the law from the contradictions of appearance', from 'experience based on immediate appearance', while vulgar economy relies here 'as elsewhere on the mere semblance as against the law of appearance ( _gegen das Gesetz der Erscheinung_ )' ( _Capital_ , I, p. 421 f.), that is, as against the notion of appearances as 'essential' ( _Science of Logic_ , p. 500 ff.).
In short, as in the atomism of Democritus, so in bourgeois economy essence and appearance _fall apart_. It follows that classical economy which 'holds to the law', the principle or essence or inner relations, comprehends this only abstractly as a principle that remains 'without reality and existence', as an essence without form, as dead substance or _hypothesis_. In Hegel's terms, its 'principle', the Ricardian labour theory of value, forms an Abstract Identity incapable of passing over into a Concrete Totality, hence into something true. Ricardo
> 'abstracts from what he considers to be accidental',
or the appearances are of no concern to him, his is an essence that can dispose of form.
> 'Another method would be to present the _real process_ in which both what is to Ricardo a merely accidental movement, but what is constant and real, and its law, the average relation, appear as _equally essential'_. (GKP p. 803. Emphasis mine.)
_This_ method is Marx's own, the conception of Epicurus in Antiquity or of Hegel in the modern world.
It follows that the 'abstraction of value' cannot by itself 'reflect nature... truly and completely', as Lenin supposes. As the abstract universal, it is something simple and undeveloped, this form of simplicity is its one-sidedness, it remains a principle that has still to 'realise itself, to become 'active'. And this it can only do by 'entering into appearance', determining itself in appearance or in the whole 'wealth of developed form'. For to trace the movement through which the principle (essence) enters into appearance and acquires reality and existence, is precisely to 'evolve the different forms through their inner genesis', it is to develop conceptually the movement which Marx calls 'the real process of acquiring shape' ( _Theories_ , 3, p. 500, _der wirkliche Gestaltungsprozess_ ).
But finally, it is just as important to bear in mind that this movement through which the forms emerge is only the 'reflection of essence into itself', essence as a movement of reflection or mediation. Henryk Grossmann's example is a good illustration of this. Both in his major study (Grossmann, 1970) and in his critique of Luxemburg's understanding of the Reproduction Schemes (Grossmann, 1971, pp. 45-74), Grossmann saw in the return to the level of appearances or to the concrete, both the chief task of scientific investigation and the main thrust of the method. (Luxemburg was criticised for allegedly confusing different stages in the process of abstraction, that is, for supposing that a _value-schema_ could explain phenomena which presupposed regulation by _prices_.) On the other hand, Grossmann himself proposed an extremely one-sided understanding of Marx's method in _Capital_ , precisely in failing to see in the return to the concrete a process defined by logical continuity. Grossmann writes,
> 'The empirically given world of appearances forms the object of investigation. _This, however, is too complicated to be known directly_. We can approach it only by stages. To this end _numerous simplifying assumptions are made_ , and these enable us to grasp the object of knowledge in its inner structure.' (Grossmann, 1970, p. vi. Emphases mine.)
Grossmann thus sees in the dialectic a 'method of approximation to reality' and in doing so he recasts the relation between abstract and concrete as a progressive 'correction' that again 'takes into account the elements of reality which were initially disregarded' (ibid.). Because this completely ignores the law of motion of the enquiry itself, the conception of substance as self-developing or of the essential movement, Grossmann has no means of explaining on what basis other than pure intuition Marx could select the specific assumptions defining a given level of abstraction. If the 'simplifying assumptions' are the main thing, then, through an opposite route, we are back with Lenin's idea that we arrive at the 'essence' as opposed to the "appearances' by a process of 'abstracting from'. We 'separate' essence from appearance through the series of simplifying assumptions we make, then, reversing the movement, abandon these assumptions step by step so as to arrive at the appearances again. Or, in Hegel's words,
> 'the procedure of the finite cognition of the understanding here is to take up again, equally externally, what it has left out in its creation of the universal by a process of abstraction.' ( _Science of Logic_ , p. 830.)
Indeed, the true logic of this conception is evident in one writer (Himmelmann, 1974, pp. 41-50) who, starting with the notion of appearances as the pure 'other' of essence, drives himself into the conclusion that Marx is concerned with _two different_ objects of analysis, one 'abstract-sociological' and the other 'concrete-economic'.
COMMODITY AND CAPITAL. THE PROBLEM OF THE BEGINNING OF _CAPITAL_
The 'Beginning' in the Literature on _Capital_.
> 'Beginnings are always difficult in all sciences. The understanding of the first chapter, especially the section that contains the analysis of commodities, will therefore present the greatest difficulty.' ( _Capital_ , I, p. 89.)
No section of _Capital_ gave Marx as much trouble as its beginning. Why could he not just begin with Part Two, the transformation of money into capital (as Althusser asks the French readers of _Capital_ to do)? Quite clearly because the whole understanding of _what capital is_ , of its relation to social labour, depends crucially on the exposition of the theory of value. (The sense in which 'value' is used here and throughout this essay will be clarified in the next section.) As Marx says about capital, 'In the concept of value its secret is betrayed' ( _Grundrisse_ , p. 776).
Before turning to an analysis of the structure of the beginning, it would be useful to look briefly at some of the conceptions current in the literature. The most widespread and manifestly incorrect understanding is the one proposed by Meek. Misinterpreting Engels' remarks on the relation of the 'logical' and 'historical' methods _in the critique of political economy_ to be a statement about the relation between the theory and the history _of capitalism as such_ , Meek argues that Marx's logical procedure in _Capital_ reflects the actual historical process of the coming-into-being of capital. The consequence of this mistaken conception is twofold. On the one hand, Marx is supposed to begin _Capital_ with a 'society of simple commodity producers'; however, because the historical existence of a society of this type is problematic, Marx is supposed to start with a _fiction_ , in the fictionalist philosophies of science sense (i.e. a sort of device). So Meek writes,
> 'Marx's postulate of an abstract pre-capitalist society... was not a myth, but rather mythodology' (Meek, 1973, p. 303 f.),
not science fiction but scientific fiction. The second consequence may be stated as follows:
> 'In so far as Marx's logical transition in _Capital_ (from the commodity relation as such to the 'capitalistically modified' form of this relation) is presented by him as the 'mirror image' of a historical transition (from 'simple' to 'capitalist' commodity production), _Marx's procedure becomes formally similar to that of Adam Smith and Ricardo_ , who also believed that the real essence of capitalism could be revealed by analysing the changes which would take place if capitalism suddenly impinged upon _some kind of abstract pre-capitalist society_.' (Meek, 1973, p. xv. Emphasis mine.)
This is a fairly neat way of reabsorbing _Capital_ into the flaccid methodological tradition which Marx himself severely criticised in three whole volumes (the volumes which compose _Theories of Surplus-Value_ ). We shall see later, from Marx's own statements, how in this conception, and in the disguised form of supposedly valid scientific 'fictions', Meek only ascribes to Marx his own totally fictitious conception of science.3
Secondly, passing from the stolid and unshakeable empiricism of the British tradition in philosophy, to the more delicate, but also more hesitant empiricism of the Della Volpe school, there is Colletti who dissolves the dialectic into the 'circle of induction and deduction'. That is, into a twofold process in which the concept is both logically first and empirically or inductively, second. Without explicitly endorsing hypotheticism, Colletti states the conclusion of this conception as follows: 'one must bear in mind that implicit in the logical process is a process of reality which works in the opposite direction' (Colletti, 1973, p. 121) and which makes the concept a _result_ of the _observation_ of reality.
This conception of the twofold process is translated into an interpretation of the beginning of _Capital_ in the following terms-the first moment in the logical process or chain of 'deductive reasoning' forms likewise the last in the real process, or the chain of 'induction'. What is this moment, however? When we turn to the analysis itself, it is a striking fact that no _stable_ identification is evolved.
Passage one:
_'Exchange-value_ presents itself to us in two different respects: on the one hand, as the most comprehensive and broadest generality from which all the other categories are deduced and from which a scientific exposition must begin; on the other hand, as an objective characteristic, as the last (in the inductive chain) and therefore _most superficial_ and abstract characteristic... of the concrete object in question.' (Colletti, 1973. p. 126. Emphasis mine.)
Passage two:
'The work begins its analysis by studying the _'form of value'_ , the _'commodity form'..._ ' (Colletti, 1973, p. 126.)
from which the other forms (money, capital) are derived.
> Passage three:
>
> 'The work develops, together with the deductive process descending from the _commodity_ to money, and from the latter to capital, as an inductive process going back from the generic or secondary features of the object in question to its specific or primary ones... the expository formula commodity-money-capital, shows itself to be also the exposition best-suited to the procedure by which analysis gradually penetrates the object in question, _departing from the non-essential_ or generic aspects and going back to the fundamental or specific ones, _from effects to causes_ and (in short) from the most superficial phenomena to the real basis implicit in them.' (Colletti, 1973, p. 127 f.)
Throughout this exposition, in other words, the commodity = the commodity-form and both (individual commodity and commodity-form) are indifferently and variously characterised as 'the most comprehensive generality', 'the universal', 'most superficial aspect', 'secondary feature', 'non-essential aspect', 'effect'. So in Colletti's understanding, the point of departure in a logical (or logico-deductive) process can actually be something 'non-essential' and yet something which _somehow_ takes us to that which _is_ 'essential' ('fundamental'). As he says, 'departing _from the non-essential'_ we arrive at capital. Or what this entire analysis argues is simply this: the essence of capital does not lie in the _commodity-form_ (that is, in _value_ in the definition to be given later) because the _commodity_ is, after all, only a 'phenomenal form' of capital and as such a merely secondary and subordinate aspect.
Finally, even an obvious sympathy for the dialectic is not a sufficient condition for grasping the structure of the beginning, as the case of Nicolaus shows. According to him,
> 'It is this category, the commodity, which forms the starting point of... _Capital_ I (1867). It is a beginning which is at once concrete, material, almost tangible',
here note the suggested synonyms for 'concrete',
> 'as well as historically specific to capitalist production... Unlike Hegel's _Logic_ , and unlike Marx's own initial attempts earlier, this beginning begins not with a pure, indeterminate, eternal and universal abstraction, but rather with a compound, determinate, delimited and _concrete whole — 'a concentration of many determinations, hence unity of the diverse'.'_ (Nicolaus, 1973, p. 36. Emphasis mine.)
This is really quite strange, for in the very sentence from which Nicolaus quotes Marx, Marx makes it clear that
> 'the concrete is concrete because it is the concentration of many determinations, hence unity of the diverse. It appears in the process of thinking, therefore,... _as a result, not as a point of departure_.' ( _Grundrisse_ , p. 101)
The commodity which forms the starting-point is thus by no stretch of one's dialectical imagination, a 'concrete whole' in the sense suggested by Nicolaus. (The underlying confusion here is the same as Colletti's.) Secondly, because Nicolaus somewhat crudely contraposes 'abstract universal' to 'concrete' (Zeleny, 1973, chapter 4, is also prone to this sort of argument), and discerns in the beginning of _Capital_ only this 'concrete', he is forced into a conclusion which, _if true_ , would render the whole motion of the dialectic something absolutely incomprehensible. This is the conclusion that
> 'the notion that the path of investigation must proceed from simple, general, abstract relations towards complex, particular wholes _no longer appeared to him_ (Marx)... as 'obviously the scientifically correct procedure'.' (Nicolaus, 1973, p. 38. Emphasis mine.)
From the fact that Marx does not begin with a _historically indeterminate_ abstract, production in general, Nicolaus concludes that Marx does not begin with an abstract at all.
It is obvious from these three examples (Meek, Colletti, Nicolaus) that there is a considerable amount of confusion regarding the beginning, even when everyone agrees that the commodity is the starting-point.
How Marx Begins _Capital_.
> 'Compared with your earlier form of presentation, the progress in the sharpness of dialectical exposition is quite striking.' (Engels to Marx, 16.6.67.)
_(a) A Summary of the Argument_.
It would be good to summarise the general argument of the section in advance for the sake of simplicity. The total structure ( _Gesamtaufbau_ ) of _Capital_ is best understood in terms of an image that Marx himself uses at one point. Namely, if it is seen as an 'expanding curve' or spiral-movement composed of specific cycles of abstraction. Each cycle of abstraction, and thus the curve as a whole, begins and ends with the Sphere of _Circulation_ (the realm of appearances), which is finally, at the end of the entire movement, itself determined specifically as the Sphere of the _Competition of Capitals_. The first specific cycle in _Capital_ , the one which initiates the entire movement of the curve, starts with Circulation as the _immediate, abstract_ appearance of the total process of capital, that is, it starts with 'Simple Circulation'. As an immediate appearance of this process, as its _Schein_ , Simple Circulation _presupposes_ this process, which is capital in its totality. The first cycle then moves dialectically from Simple Circulation, or what Marx calls the individual commodity, to capital. This movement will be called the 'dialectical-logical derivation of the concept of capital'. Methodologically, it is itself decomposable into specific phases: an initial phase of Analysis which takes us from the individual commodity to the concept of value, and a subsequent phase of Synthesis which, starting from value, derives the concept of capital through the process Hegel called 'the development of form'. Capital then emerges through this movement as 'nothing else but a value-form of the organisation of productive forces' (Ilyenkov, 1977, p. 85). In the return to the Sphere of Circulation which concludes cycle 1, initiates cycle 2, the individual commodity from which we started is now 'posited', that is, established dialectically, as a form of appearance ( _Erscheinungsform_ ) of capital, and Circulation is posited as both presupposition and result of the Immediate Process of Production. The dialectical status of the Sphere of Circulation thus shifts from being the immediate appearance of a process 'behind it' ( _Schein_ ) to being the posited form of appearance ( _Erscheinung_ ) of this process. (Cf. for example, _Grundrisse_ , p. 358, _Theories_ 3, p. 112, _Results_ p. 949 ff.)
_(b) Capital as Presupposition of the Commodity_.
'In the completed bourgeois system... everything posited is also a presupposition, this is the case with every organic system.' ( _Grundrisse_ , p. 278.)
In the _Grundrisse_ Marx sketches a series of short anticipatory drafts of the plan of his work as a whole. They are, of course more in the nature of notes which he will revise from time to time. In one of these he writes,
> 'In the first section, where exchange-values, money, prices are looked at, commodities always appear as already present... We know that they (commodities) express aspects of social production, but the latter itself is the presupposition. However, they are _not posited_ in this character...' ( _Grundrisse_ , p. 227, Nicolaus' translation slightly modified, Marx's emphasis.)
Here Marx says that at the beginning of the entire movement of investigation the commodity already presupposes social production (capital) of which it is only an 'aspect', or determination, but it is not yet posited as such an aspect. That is, it has still to be _established dialectically_ or dialectico-logically as a determination of the total process of capital. Secondly, this world of commodities that confronts us on the surface of bourgeois society 'points beyond itself towards the economic relations which are posited as relations of production. The internal structure of production therefore forms the second section...' (ibid.). In his original draft of the 1859 _Critique_ , reprinted in the German edition of the _Grundrisse_ , Marx returns to this idea and develops it more explicitly:
> 'An analysis of the specific form of the division of labour, of the conditions of production which are its basis, or of the economic relations into which those conditions resolve, would show that _the whole system of bourgeois production is presupposed_ before exchange-value appears as the simple point of departure on the surface.' (GKP, p. 907. Emphasis mine.)
As the form which confronts us immediately on the surface of society, the commodity _as such_ is our point of departure. But this simple commodity, the point of departure, already presupposes a specific form of the social division of labour, it presupposes the bourgeois mode of production in its totality. On the other hand, at the beginning itself, the commodity has still to be posited as only an 'aspect' or form of appearance of the total process of capital.
Because capital in its totality is the presupposition, when he starts Chapter One of _Capital_ Marx must explicitly _refer to_ this presupposition. And that is exactly what he does. He says,
> 'The wealth of societies _in which the capitalist mode of production prevails_ appears as an immense collection of commodities; the individual commodity appears as its elementary form.' ( _Capital_ , I, p. 125. Emphasis mine.)
So the very first sentence of _Capital_ makes it quite clear that _capital_ is presupposed.
One consequence of this is obvious. The conceptual regime of Part One, Volume One is not some 'abstract pre-capitalist society' of 'simple commodity producers', it is the Sphere of Simple Circulation, or the circulation of commodities as such, and we start with this as the process that is _'immediately present_ on the surface of _bourgeois society'._ ( _Grundrisse_ , p. 255, emphasis mine), we start with it as a reflected sphere of the total process of capital which, however, has still to be determined _as reflected_ , i.e. still to be _posited_. When we examine the simple commodity, or the commodity as such, we only examine _capital_ in its most superficial or immediate aspect. As Marx says,
> 'We proceed from the commodity as _capitalist production in its simplest form_.' ( _Results_ , p. 1060, emphasis mine.)
Indeed, capital _'must_ form _the starting-point_ as well as the finishing point' ( _Grundrisse_ , p. 107, emphasis mine), but as the starting-point capital is taken in its 'immediate being' or as it appears immediately on the surface of society.
It is, therefore, difficult to understand how anything except the most shallow and hasty reading of Marx's _Capital_ could have led to the kind of view proposed by Meek and so many other professional expounders of Marx. The 'abstract pre-capitalist society' that Marx is supposed to have started with is not a fiction that Marx consciously uses in the tradition of certain medieval conceptions of science, but a fiction that 'mythodologists' unconsciously tend to elaborate.
_(c) The Structure of Marx's Concept of Value_.
Also in the remarkably clear original draft of the _Critique_ , Marx writes,
> 'Simple circulation is an abstract sphere of the total process of production of capital which _through its own determinations_ becomes identifiable as a _moment_ , a mere form of appearance of a deeper process that underlies it.' (GKP p. 922 f. emphasis mine.)
This means (i) that the individual commodity contains immediately within itself determinations that can be drawn out of it. 'Our investigation... begins with the analysis of the commodity' ( _Capital_ , I, p. 125). The individual commodity is a 'concrete' in the specific dialectical sense that it comprises a _relation within itself_ ( _Science of Logic_ , p. 75). 'As concrete, it is _differentiated within itself_ ' ( _Science of Logic_ , p. 830), hence something analysable. However, (ii) in drawing out the differentiated determinations that lie within the given object, analysis only initiates, or sets in motion, a process that allows us to _return to_ the commodity and identify it now as a 'moment', a form of appearance, of capital. The immediacy from which we started then becomes what Hegel calls 'a _mediated_ immediacy' ( _Science of Logic_ , p. 99).
Through the movement of analysis Marx draws out the inner determinations of the commodity regarded as a concrete, immediate representation. These determinations are, of course, use-value and exchange-value. _In the first instance_ these differentiated determinations are merely a 'diversity' ( _Science of Logic_ , p. 830), or in Marx's words, (cited Berger, 1974, p. 102, note 37), use-value and exchange-value are simply 'abstract opposites' that split apart in mutual indifference.
Now to say that 'through its own determinations' the simple commodity must become identifiable as a form of appearance of capital is to say that the analysis of the commodity, the drawing out of its inner determinations, must _establish the dialectical-logical basis_ for the derivation of capital, whose own further determination or development will then 'mediate' the immediacy of the commodity as such. _This_ the analysis of the commodity can only do if it takes up _specifically that determination_ which allows us to pass dialectico-logically to a notion like capital. This 'use-value' does not do, because it is the commodity's 'material side which the _most disparate epochs of production..._ have in common' ( _Grundrisse_ , p. 881). Marx begins therefore with 'exchange-value', taking this as the only basis on which he can begin to penetrate the social properties of the commodity.
These properties then appear initially as a sort of 'content' 'hidden within' their 'form of appearance', exchange-value. Insofar as Marx, both in Section 1 and later, calls this 'content' _'value'_ (cf. _Capital_ , I, p. 139: 'We started from exchange-value... in order to track down the value that lay hidden within it'), it is easy to fall into the illusion of supposing that value is something actually contained in the individual commodity. For example, it is easy to suppose that Marx means by value (as quite clearly he did _at one stage_ ) 'the labour objectified in a commodity', and then from there to proceed to the more general identification of labour with value which II Rubin quite correctly polemicised _against_ (Rubin, 1972, p. 111 ff.). But Marx also makes it clear in Chapter One that this is not how he understands the matter. If value appears initially to be a 'content' concealed within its form of appearance, exchange-value, then this false appearance is plainly contradicted when he writes,
'Political economy has indeed analysed value and its magnitude... and has uncovered the content concealed _within these forms_. But it has never once asked the question why this content has assumed that particular form, _that is to say, why labour is represented in value (warum sich also die Arbeit im Wert... darstellt_ ).' ( _Capital_ , I, p. 173-74. Translation slightly modified and emphasis mine.)
In this lucid sentence Marx calls value the _social form as such_..Let us look at this a bit more closely.
Outside of the purely vulgar and quite incorrect Ricardian understanding of Marx's theory of value, which identifies value with the labour objectified in commodities, the usual mode of presentation of the theory in the Marxist literature is the one apparently started by F. Petry and typified in the expository accounts of Rubin, Sweezy and others. In this mode of presentation, the crucial architectural distinction within Marx's value theory is its separation of 'quantitative' and 'qualitative' aspects in the problem of value. For example, Sweezy writes,
> 'The great originality of Marx's value theory lies in its recognition of these two elements of the problem.' (Sweezy in Howard and King, 1976, p. 141 f.)
What _is_ the _'qualitative_ aspect' of the problem of value, however? No sooner do we pose this question, than it becomes evident that the qualitative/quantitative distinction is _not enough_ to render a proper account of Marx's concept of value. Indeed, in the very passages where Marx himself refers to this distinction explicitly, he also says,
> 'Ricardo's mistake is that he is concerned only with the magnitude of value... But the labour embodied in (commodities) _must be represented (dargestellt) as social labour..._ this qualitative aspect of the matter which is contained in the _representation of exchange-value as money (in der Darstellung des Tauschwertes als Geld)_ is not elaborated by Ricardo...' ( _Theories_ , 3, p. 131).
Again, some pages later,
> 'This necessity of representing individual labour as general labour is equivalent to the necessity of representing a commodity as money.' ( _Theories_ , 3, p. 136, translation modified)
In passages such as these Marx isolates two dimensions of the value-process, (a) the representation of the commodity as money, and (b) the representation of (private) individual labour as social labour. The relation between these two dimensions can be described as follows: in the social process of exchange a surface relation, exchange-value, becomes the form of appearance of an inner relation, the relation which connects individual labour to the total social labour. (This connection is, in any case, what we might call a 'material law of society'. Cf. _Selected Correspondence_ , p. 239, p. 251, _Theories_ , 1, p. 44.) The surface-relation is simultaneously a 'relation among things' and the inner relation a 'relation among persons'. Finally, each of these two dimensions of the value-process is susceptible to the earlier-mentioned distinction into qualitative and quantitative aspects, so that the following structure results:
> Dimension | (1) Quantitative | (2) Qualitative
> ---|---|---
> (A) The commodity represented as money... | exchange-value as surface appearance ( _Section One_ ) | the money-form as objective appearance ( _Section Three_ )
> (B) Individual labour represented as social labour... | socially-necessary labour-time ( _Sections One and Two_ ) | abstract labour ( _Sections One and Two_ )
When we look at Marx's final presentation of Chapter One, the 'substance of value' and 'magnitude of value' aspects are taken together, investigated without any specific formal separation, in both of the first two Sections. This is so because, although separable as qualitative and quantitative aspects respectively, they belong to the same dimension of the value-process, the dimension of its inner content as a process within which individual labour is connected to and becomes part of total social labour. On the other hand, this 'content' is logically inseparable from its specific 'form'; or to put the same thing differently, it only becomes something real _through its form_ , which is the representation of the commodity as money. In its 'immediate being' the commodity is only a use-value, a point which Marx repeatedly makes in the _Critique_. Its immediate being is thus the commodity's relation of self-repulsion, or its 'negative' relation to itself as a commodity (cf. _Science of Logic_ , p. 168: 'The negative relation of the one to itself is repulsion'.) The commodity can posit itself as a commodity- _value_ , a product of _social_ labour, only in a form in which it negates itself in its immediate being, hence only in a _mediated form_. This form is _money_. Only through the representation of the commodity _as money_ , or, expressed more concretely, through the individual act of exchange, the transformation of the commodity into money, is individual labour _posited_ as social labour. The concept of value in Marx is constructed as the indestructible unity of these two dimensions, so that _logically it is impossible to understand Marx's theory of value except as his theory of money_ (cf. Backhaus, 1975).4 This is the aspect developed explicitly in Section 3, the _'form_ of value'. In Section 3, moreover, or in this return to the level of appearances, the contradictory determinations of the commodity, which appeared initially as mutually indifferent, become reabsorbed as a _unity_ (money). In Marx's words,
> 'Use-value or the body of the commodity here plays a new role. It becomes... the form of appearance of its own opposite. Instead of splitting apart, the contradictory determinations of the commodity here enter into a relation of mutual reflection' (cited Berger, 1974, p. 102, Zeleny, 1973, p. 78).'
The sequence of Marx's presentation is thus: A(1) → B → A(2).
In Section 1 'exchange-value' figures as pure surface appearance ( _Schein_ ), hence as a quantitative relation of commodities. But already within this section Marx accomplishes a transition to dimension B, whose two aspects (socially-necessary labour, and abstract labour) he then investigates, in this and the following section, without formal separation. Finally, in Section 3, Marx 'returns' to exchange-value, to dimension A, to deal with it no longer as the immediate illusory appearance of the exchange-process but as _objective_ appearance, or _form, Erscheinung_.
_(d) Value (Commodity-form) as the 'Self-evolving Substance'_
In short, value is not labour and 'to develop the concept of capital it is necessary to begin not with labour but with value' ( _Grundrisse_ , p. 259), that is, with the twofold process by which individual labour becomes total labour through the reified appearance-form of the individual act of exchange (transformation of the commodity into money). Regarded as this twofold process of representation, the concept of value can then be formally defined as _the abstract and reified form of social labour_ , and the term 'commodity-form of the product of labour' can be taken as its concrete-historical synonym. It is value, the commodity-form, in this definition, just outlined here, that composes the 'self developing substance' of Marx's entire investigation in _Capital_. As Marx says, value is
> 'the social form as such; its further development is therefore a further development of the social process that brings the commodity out onto the surface of society.' (GKP p. 931)
Or value
> 'contains the whole secret... of all the bourgeois forms of the product of labour'. ( _Selected Correspondence_ , p. 228)
The money-form of value (or money) is 'the first form in which value', social labour in abstract form 'proceeds to the character of capital' ( _Grundrisse_ , p. 259). So as the abstract-reified form of social labour, value 'determines itself first as money, then as capital. In its money-form value obtains its sole form of appearance, and through this the moment of actuality. In its capital-form it _posits_ itself as 'living substance', as a substance become 'dominant subject' ( _Capital_ , I, p. 255 f.), or _posits itself_ as that totalising process which Hegel calls 'essence'. Or, in the concept of value the analysis of the commodity arrives through its own movement at a basis for the dialectical-logical definition of capital.
At the moment of dialectical-logical derivation, this definintion is only the most simple or abstract definition of capital. 'If we speak here of capital, that is still merely a word ( _ein Name_ )', Marx says.
> 'The only aspect in which capital is here posited as distinct from (...) value and from money is that of (...) value which preserves and perpetuates itself in and through circulation.' ( _Grundrisse_ , p. 262)
On the other hand, capital is here _posited_ , or this moment of the determination of capital as a form of value is the initial moment of its positing, so that the 'internal structure of production' which 'forms the second section' (Part Three of Volume I) can now be investigated directly. This is the investigation which, as we know, occupies the major part of Volume I.
In Chapter One, Volume II, Marx _returns_ to the process of circulation, or he comes back to the commodity. But now he can investigate the circulation of commodities directly as a circulation of _capital_. The formal determinations of simple circulation (commodity and money as means of circulation) are now posited as 'aspects', or forms of appearance, of the relations of production which initially they presupposed. Moreover, they are themselves posited as presupposed by capital, as forms essential to the process of realisation. In this spiral return to the point of departure the commodity is treated explicitly as a 'depository of capital' ( _Results_ , p. 975), as one of its 'forms of existence' within the process of circulation. 'The independence of circulation', the aspect in which the commodity initially presented itself to us, 'is here reduced to a mere semblance', that is, an illusory appearance ( _Grundrisse_ , p. 514).
The possibility of returning to the Sphere of Circulation and the necessity of now investigating it directly as a Circulation and, at first implicitly, a Reproduction of Capital signifies the conclusion of the first cycle of abstraction in _Capital_. However, as we have seen, this cycle of abstraction is itself only possible because Chapter One (Volume I) takes us from the individual commodity to value. As we shall see, _both_ are points of departure within the process of thought, but they are nonetheless quite distinct moments of this process. It is now possible to see that Colletti's exposition simply confuses these specific moments. It confuses the individual commodity as _immediate appearance_ of the process of capital with value or the commodity-form as the _essential ground_ of the movement that can finally posit the commodity as a moment, a form of appearance, of its own process. And because of this confusion intrinsic to his analysis, Colletti is forced to ascribe to the _commodity-form_ logical properties that characterise the commodity as such. It is the commodity-form, or value, that Colletti calls 'secondary feature', 'subordinate element', 'non-essential aspect' and so on. He thereby reduces the method of conceptual development (the dialectical development of the concept of capital) to an incomprehensible mystery.
Against Colletti's fabulations, it is important to stress that in the theory of value Marx saw the whole basis for the distinction between himself and the tradition of classical economy. As he told Engels, moreover, 'the matter is too decisive for the whole book' ( _Selected Correspondence_ , p. 228). Which means quite simply that a correct understanding of the chapters that follow, and therefore, of course, our very ability to be able to develop the theory of Marx further, depends on a correct understanding of Chapter One.5 This is something that Lenin came over to seeing _after_ reading the _Science of Logic_. Lenin's flexibility, however, his ability to reassess a problem, his burning restlessness, his capacity to swing from empiricism to Hegel, were only expressions of the fluid, practical mould of his thought. To be able to revise the foundations of your philosophical outlook within the space of three months, you must first be a person who _acts_ , an _actor_. The demolition of the _Logic_ is, however, Colletti's point of departure, and this proceeds not in a world of action, but within the unreal and immobilised world of the university.
Thus in its most simple and essential definition capital is a form of value where value itself is grasped as a form of social labour. From this it follows that when capital seeks to _overcome_ or to _subordinate_ the commodity-form of its own relations of production, to regulate the 'market' according to the combination of its individual wills (cf. Sohn-Rethel, 1975, p. 41 ff.), then it merely seeks to overcome or to subordinate _itself_ as a form of value, or itself in its most essential definition. And this is impossible except as the _contradiction_ which capital becomes.
The Double Structure of the Beginning
> 'If you compare the development from commodity to capital in Marx with development from Being to Essence in Hegel, you will get quite a good parallel...' (Engels to Schmidt, 1.11.91).
In the previous section it was said that the commodity re-emerges as a 'form of _existence'_ of capital at the start of Volume Two. We started with it, however, as capital in its immediate _being_. Translated into the terms of Hegel's _Logic_ , this implies that the movement of Volume One contains a decisive logical step from being to existence which Hegel and, following him, Marx call the 'return into the ground'. The general principle of this retreat into the ground (into essence that _posits itself as_ such),
> 'means in general nothing else but: what _is_ , is not to be taken for a positive immediacy ( _seiendes Unmittelbares_ ) but as something posited.' ( _Science of Logic_ , p. 446)
That is, being is not to be taken as immediate _in the sense of_ un-mediated or lacking all mediation (cf. Henrich, 1971, p. 95 ff.).
> 'Being is the immediate. Since knowledge has for its goal cognition of the true... it does not stop at the immediate and its determinations, but penetrates it on the supposition that at the back of this being there is something else, something other than being itself, that this background constitutes the truth of being.' ( _Science of Logic_ , p. 389. Translation modified in both passages.)
What is this hidden background? As we would expect, it is none other than essence. 'The truth of being is essence' (ibid.).
The terms in which Marx describes the transition from the individual commodity to capital are strikingly reminiscent of these and other passages of the _Logic_. Marx writes, Simple Circulation
> 'does not carry within itself the principle of self-renewal. The moments of the latter (production) are presupposed to it, not posited by it. Commodities constantly have to be thrown into it anew from the outside, like fuel into a fire. Otherwise it flickers out in indifference... Circulation, therefore, which appears as that which is immediately present on the surface of bourgeois society',
which appears as its 'immediate being',
'exists only insofar as it is constantly _mediated_. Looked at in itself, it is the mediation of presupposed extremes (i.e., the two commodities which begin and end the circuit). But it does not posit these extremes. Thus it has to be mediated not only in each of its moments, but as a whole of mediation, as a total process itself. Its immediate being is therefore pure semblance ( _reiner Schein_ ). It is the appearance of a process taking place behind it...' ( _Grundrisse_ , p. 255, translation modified slightly.)
Through our analysis of the simple commodity we arrive at the concept of value and thus at a basis for defining, dialectico-logically, the concept of capital. Now it is capital that produces commodities which form the substance and lifeblood of the process of circulation. Therefore,
> 'Circulation itself returns back into the activity which posits or produces exchange-values. _It returns into it as into its ground.'_ (Ibid. Emphasis mine.)
The 'activity' which forms the 'ground' of the process of circulation and into which it returns (in the sense explained above) is capital in its specific determination as a process of production. The immediate result of this process is the commodity 'impregnated with surplus-value' ( _Results_ , p. 975). So it is possible to say, as Hegel does, that the general course of a dialectical enquiry advances as
> 'a retreat into the ground, into what is primary and true, on which depends and... from which originates, that with which the beginning is made... The ground... is that from which the first proceeds, that which at first appeared as an immediacy.' ( _Science of Logic_ , p. 71.)
The 'first', the commodity, proceeds from capital, whose own development posits it as a mediated immediacy, i.e. a 'moment' of its own process, hence something mediated. This enables us to say that Marx regarded _capital_ as the ground of the movement of his investigation. However, it is through the analysis of the individual commodity that we _arrive at_ this fundamental or ground category (capital). Therefore it is equally true to say that capital, the ground into which we retreat from circulation, from the individual commodity, is a _result_. And 'in this respect the first', the individual commodity, 'is _equally the ground_ ' ( _Science of Logic_ , p. 71, emphasis mine.).
This appears to jeopardise the whole argument, for it appears to ascribe to an immediate appearance (the individual commodity) the character and function of 'ground', that is, of a basis for the movement of the entire investigation. In fact, it is only this statement that finally enables us to reveal the true structure of every dialectical beginning.
About the beginning as such Hegel says at the end of the _Logic_ that
> 'its content is an immediate but an immediate that has the significance and form of _abstract universality_.' ( _Science of Logic_ , p. 827)
Thus a dialectical beginning, such as Marx accomplished, contains two dimensions — a dimension of _immediacy_ (of concreteness) and a dimension of _universality_ (but the universal in its _abstract form_ ). As something universal, however, the latter _presupposes nothing_ — except, Marx will say, the _historical process_ through it has come about (which is why 'the dialectical form of presentation is only correct when it knows _its own limits'_ , GKP p. 945). To say it 'presupposes nothing' except a historical process which lies 'suspended' within it (cf. _Grundrisse_ , p. 460 f.) is to say that the universal which forms the 'significance' of the immediate-concrete that stands at the point of departure is something 'absolute', or related only to itself, 'self-related' ( _Science of Logic_ , p. 70, p. 404, p. 829). _As such_ , as this 'absolute' which relates only to itself, the universal 'counts as the _essence_ of that immediate which forms the starting-point' ( _Science of Logic_ , p. 405, emphasis mine).
Now we know that _capital_ is the _essence_ of the individual commodity (of simple circulation), it is its 'ground' or its 'principle of self-renewal', and the commodity is only a 'moment' of its process. On the other hand, capital itself is only the developed and self-developing form _of value_ , or capital in its own simple definition is value-in-process ( _Grundrisse_ , p. 536), _value_ as the dominant subject, etc. So value is _likewise_ the essence of the simple commodity, and the difference can then be put like this: as the merely _abstract, universal form of capital_ (cf. _Grundrisse_ , p, 776, _Capital_ , I, p. 174, note), value can be called the 'abstract essence' of the simple commodity; the _posited_ form of value, can be called its 'concrete essence'. Or, capital is only the essence of the simple commodity because _value is its own essence_ , the essence of capital, for capital is a _form_ of value. Finally, as the immediate aspect _of value_ , which is its abstract essence, the commodity which forms the starting-point can also be called the ground of the entire movement.
In fact, there is a much clearer form in which this point can be made, as long as it is seen as a statement about the structure of the beginning. Namely, _the beginning is a movement between two points of departure_. (This is what ninety-nine percent of commentators fail to grasp). As the immediate appearance of the total process of capital (this can also be called _value as a totality reflected-into-itself_ for all categories of capital are categories of value), the individual commodity forms the _analytic_ point of departure. From this, however, we do not pass over directly to the concept of capital. By analysing the commodity, drawing out its determinations, we arrive at the concept of _value_ as the abstract-reified form of social labour. This as the ground of all further conceptual determinations (money, capital) forms the _synthetic_ point of departure of _Capital_. (The distinction is clearly understood and explicitly stated by Berger, 1974, p. 86.) The passage from one point to the other forms the structure of the beginning as such. In logical terms this movement is a transition from Immediacy to Mediation, or from Being to Essence. Analysis is simply a prelude, as Marx points out, even if a necessary one, to the process that he calls 'genetic presentation' ( _genetische Darstellung_ ) ( _Theories_ , 3, p. 500). This process is one we have been concerned with through most of this essay. It is the logically continuous movement from the abstract to the concrete, the movement that Hegel calls 'the development of form', the movement that Marx describes as 'the principle entering into appearance', or as the development of the different forms through their inner genesis.
In the dialectical method of development the movement from abstract to concrete is not a straight-line process. One returns to the concrete at expanded levels of the total curve, reconstructing the surface of society by 'stages', as a structure of several dimensions (cf. Hochberger, 1974, p. 155 f., p. 166 f.). And this implies, finally, that in Marx's _Capital_ we shall find a continuous 'oscillation between essence and appearance' (Zeleny, 1973, p. 164 ff.). Yet there is a point at which this movement, the very development of the concept of capital, breaks down in _Capital_ as we have it today. There is a point at which the 'form of enquiry' is no longer reflected back to us in the dialectically perfected shape of a 'method of presentation'. To say this is only to say that Marx's _Capital_ remains incomplete as a reproduction of the concrete in thought, What is remarkable here is not that Marx should have left the book incomplete but that close to four generations of Marxists should have done so. There are, of course, historical reasons why this is so, reasons related to the renovated expansion and qualitative consolidation of capitalism. But one of the most striking manifestations of the underlying crisis in the movement as a whole is the contemporary state of Western Marxism — the ecstatic leap from the uppermost floors of an imposing skyscraper of immobilised dogma to the granite pavements of confused eclecticism.
Footnotes
Wherever possible I have consulted the original of all passages cited from Marx's writings or from Hegel's Preface to the _Phenomenology_ , or from his _Logic_. Thus where existing translations have been modified, it is, of course, only after consultation of the original German text (the _MEW_ edition in Marx's case, and the twenty-volume Frankfurt edition in Hegel's).
1.Della Volpe, like Popper, was fundamentally concerned with the 'demarcation problem', the problem of establishing criteria in terms of which science might be distinguished from metaphysics, e.g. Marx from Hegel (for Della Volpe), or Einstein from Marx (for Popper). Like Popper, Della Volpe saw the hallmark of 'scientific method' in the subordination of our rational constructions to the test of 'experience'. Like Popper, Della Volpe argued that 'experience' could play this role only 'negatively', or as falsification (Della Volpe, 1969, p. 171 f.). Like Popper, finally, he came to see theories or laws as purely tentative, or conjectural or intrinsically 'corrigible' (Della Volpe, 1969, p. 184, p. 186, p. 201). Where, then, can we locate the difference between Marxism and a tradition like Positivism, say? In the fact that Positivism is characterised, for Della Volpe, by its 'worship of facts and repugnance towards hypothesis' (1969, p. 205) — an argument that might have carried conviction in the heyday of Logical Positivism, but certainly does not today.
Perhaps it should be stated here that throughout this essay, a minimal presupposition is the critique of empiricism elaborated in some of the recent philosophy of science, notably, by Wartofsky 1967, Feyerabend 19701, Feyerabend 19702, and Koyre, 1968.
2.It is worth emphasising here that the reaction against Hegel in Western Marxism went _side by side_ with the revival of a systematic, scholarly interest in Hegel and the 'logic of _Capital'_ in post-Stalinist Eastern Europe. Notable representatives of this tendency, whose work has still to be translated into English, are E V Ilyenkov 1960, M Rozental 1957 and J Zeleny 1973 (originally 1962). This body of work and interpretation, together with that done by Rosdolsky 1968, Reichelt 1973, Backhaus 1975 and others _decisively refutes_ the quite shallow nonsense which asserts that 'Hegel's influence on Marx was _largely terminological'_ (Sowell, 1976, p. 54). When Marx refers to his 'flirtation' with Hegel's 'mode of expression' ( _Capital_ , I, p. 103), he is referring only to the _densely Hegelian_ flavour and quality of the _1867 version_ of Chapter One. Many of the explicit references to the _Logic_ which this version contained, and which, from the point of view of the _presentation_ of the argument, were quite irrelevant, were subsequently eliminated by Marx in his reworking of the chapter. For example, the 1867 version contains this sentence: 'This form is rather difficult to analyse because it is simple. It is in a sense the cell form, _or, as Hegel would say, the 'in itself' of money.'_ (cited Zeleny, 1973, p. 78). This was the kind of thing that Marx removed in his reworking.
3.The idea that Marx starts with a 'pre-capitalist' commodity is very widespread among those who read _Capital_ without the faintest conception of its method. It is shared by de Brunhoff 1973, for she tells us that 'exchange value... is first conceived at the level of a commodity production which is _not specific to any particular mode of production'_ (p. 424). This is probably the only _substantive_ , non-obvious point made in the article, which is no accident, because de Brunhoff starts off by asserting that 'the articulation between commodity and capital must be reviewed... _without using a Hegelian method_ , which I _think_ , following _Althusser's_ demonstrations, is _profoundly alien_ to Marx's procedure in _Capital'_ (p. 422, emphases mine). For Althusser see note 5 below.
4.In this sense Zeleny (1973, chapter six) is right in arguing that the 'dialectical-logical derivation of the _money-form_ of value' composes 'the _whole_ of chapter one (not only the section on the individual forms of value)' (p. 91). While this is perfectly true in the specific sense that Marx's theory of value cannot be understood except as a theory of money, it is also the case that Marx's particular presentation, even in its revised form, tends to obscure this fact, so that it is not difficult to suppose, as Rubin (1975 p. 36) wrongly does, that we can actually discuss value without discussing money. For criticisms of Marx see Itoh 1976.
5.This is a point we should make even more strongly against Althusser, who, in attempting to divorce the 'theory of fetishism', the conception of value as a _reified_ form of social labour, from the theory of value, as if these were separate aspects, simply endorses the crude positivism of Joan Robinson and others who find the 'metaphysics' of Chapter One irreconcilable with the naturally superior claims of good common-sense. Joining the camp of positivism, Althusser invitably _finds it quite impossible to understand Chapter One_ , as he confesses, indirectly, to the French readers of Marx. In his characteristically sanctimonious tone, he writes,
'I therefore give the following advice: put the whole of Part One aside for the time being and begin your reading with Part Two... In my opinion, it is impossible to begin (even to begin) to understand Part One until you have read and re-read the whole of Volume I, starting with Part Two.' (Althusser, 1971, p. 79)
Of course, Althusser does not bother to explain to his readers, the readers of _Capital, why_ they should find it any easier to understand Chapter One after reading the rest of the Volume. It seems as if Marx wrote the bulk of the volume to throw light on its introductory chapter, and not precisely the other way round! Or perhaps we are dealing with one of those famous 'inversions'?
Bibliography
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## _WHY LABOUR IS THE STARTING POINT OF CAPITAL_
## _Geoffrey Kay_
Eugen von Bohm-Bawerk (1851-1914) is one of those nineteenth century intellectuals remembered more today as antagonists of Marxism than for the positive contributions they made to their own fields of study. It is true that Bohm-Bawerk's pioneering work on neo-classical economic theory, characterised contemptuously by Bukharin as the economic theory of a leisure class, ensures him a place in the history of economic thought alongside the other founders of that doctrine. But as a critic of Marx from an economic point of view he stands alone. According to Franz X Weiss, the editor of his collected works in the twenties, his book, _Karl Marx and the Close of his System_ ,1 'was rightly regarded as the best criticism of the Marxian theories of value and surplus value' (p. IX). Paul Sweezy endorsed this view in his introduction to the English edition of 1949. 'So far as the United States is concerned', wrote Sweezy, 'all the serious criticisms of Marxian economics... recognise the authority if not the primacy of Bohm-Bawerk in this field; while the similarity of the anti-Marxist arguments in the average textbooks to those of Bohm-Bawerk is too striking not to be considered a coincidence' (pp. IX-X). The recent republication of Bohm-Bawerk's tract underlines this judgment: to this day it remains ahead in its field as the most coherent and systematic challenge to Marxism by any bourgeois economist.
From the time of its first publication in 1896, Bohm-Bawerk's critique has haunted Marxism, and response to it traces a theme in Marxist literature from the very first by Hilferding and Bukharin down to the present day. One reason for this is the close attention that Bohm-Bawerk pays to the text of _Capital_ giving the reader the very definite impression that it is no simple ideological hatchet-job but a determined effort to wrestle with Marx on his own terrain. Another is that he touched a very sensitive area of Marxism when he raised the transformation problem as the Achilles heel of _Capital_. But most importantly, Bohm-Bawerk has haunted Marxism because his is a systematic positivist critique. It challenges not only the content of _Capital_ , but also its method; and in this respect it engages with powerful tendencies within the Marxist movement itself. For the success of positivism as anti-Marxism is not confined to the development of social sciences whose content opposes Marxism directly — in this case neo-classical economics: if anything, its greater victory has been to penetrate Marxism itself and wage war as a fifth columnist. Bohm-Bawerk's polemic has played a most vital part in this counter-offensive and its republication today is no less a challenge to Marxism than it was some eighty years ago.
It was the publication of _Capital_ , III in 1894 that spurred Bohm-Bawerk to pull his various attacks on Marx together. The chapters in this third volume dealing with the formation of the average rate of profit, the famous transformation problem, were, for Bohm-Bawerk, the 'Russian campaign' of Marxism. 'I cannot help myself,' he confesses, 'I see here no explanation and reconciliation of a contradiction but the bare contradiction itself. Marx's third volume contradicts the first. The theory of the average rate of profit and of the prices of production cannot be reconciled with the theory of value' (p. 30). Why? Because 'the "great law of value" which is "immanent in the exchange of commodities"... states and must state that commodities are exchanged according to the socially necessary labour time embodied within them' (p. 12). But although the third volume is the taking-off point of Bohm-Bawerk's critique and his title suggests that his main concern is with the 'close' or 'completion' of the Marxist 'system', the most significant section of his book deals with its 'opening'. 'A firmly rooted system' such as Marxism, he writes, 'can only be effectually overthrown by discovering with absolute precision the point in which the error made its way into this system...' (p. 64). And this point is right at the start of _Capital_ , where Marx 'in the systematic proof of his fundamental doctrine exhibits a logic continuously and palpably wrong' (p. 80). Tracking down and then demonstrating this 'error in the Marxist system' is the actual more fruitful and instructive part of the criticism' (p. 65). It has another dimension. The critique of bourgeois economy runs seam-like through the whole of _Capital_ but in the first few pages of _Capital_ against which Bohm-Bawerk directs the main thrust of his attack, it forms the bedrock of the text. For in establishing his own theoretical ground, Marx simultaneously challenged the foundations of bourgeois economics. Whether Bohm-Bawerk was fully aware of the issues at stake, it is hard to say; even if he were not his instinct took him to the heart of the matter. 'The theory of value', he recognised, 'stands, as it were, in the centre of the entire doctrine of political economy'. And once a 'labour theory of value' is conceded, Marx's conclusions about surplus value, exploitation and the class struggle follow inexorably. 'In the middle part of the Marxian system', Bohm-Bawerk concedes, 'the logical development and connection present a really imposing closeness and intrinsic consistency' (p. 88). This is the first reason why it is so important for him to dismiss once and for all the 'hypothesis' on which it is based — the proposition advanced right at the beginning of _Capital_ that the value of commodities is determined by the amount of (abstract) labour socially necessary for their production. The second is that no other theory of value, and particularly the various forms of neo-classical value theory, can claim legitimacy until Marxism has been thoroughly discredited. Thus when Bohm-Bawerk locks horns with the opening pages of _Capital_ it is not merely Marxism that is thrown into the melting-pot but the whole of bourgeois economy.
THE 'ERROR' IN THE MARXIST SYSTEM
According to Bohm-Bawerk, Marx's method of discovering the nature of value is 'a purely logical proof (p. 63). Starting with an 'old-fashioned' idea derived from Aristotle that the exchange of commodities is a quantitative relationship pre-supposing some property they share in common, 'Marx searches for the 'common factor' which is characteristic of exchange value in the following way: he passes in review the various properties possessed by objects made equal in exchange, and according to the method of exclusion, separates all those that cannot stand the test until at last only one property remains, that of being the product of labour. This therefore must be the sought-for common property' (p. 69). Although this method of 'purely negative proof does not commend itself to Bohm-Bawerk, he accepts it as 'singular but not in itself objectionable' (pp. 68-9). The problem is the way Marx made use of it. And on this point Bohm-Bawerk launches a three-pronged attack. First, he says, Marx rigged the result by leaving certain things out of his 'logical sieve'. Second, letting this by, there are other common facts than labour that Marx has no right to ignore. And third, Marx's entire logic can be reversed and 'value in use could be substituted for labour' (p. 77). These three lines of attack stake out in the clearest possible way the terrain of fundamental theory on which Marxism and neo-classical economics confront each other. The remainder of this section deals with each in turn.
1. Marx's commodity is not coterminous with goods that exchange in the market. The 'gifts of nature', land, natural resources, trees, minerals and so on, exchange in the same way as commodities but unlike them they do not embody labour. Thus Bohm-Bawerk concludes: 'If Marx had not confined his research, at this decisive point, but had sought for the common factor in the exchangeable gifts of nature as well, it would have become obvious that work cannot be the common factor' (p. 73). This is an ingenious gambit. It appears at one stroke to separate Marx's theory from the most obvious and easily verifiable features of reality, while at the same time opening the way for the alternative view that the real common factor of goods, natural as well as man-made, is their utility and scarcity. Furthermore it appears to expose Marx to another criticism that Bohm-Bawerk is quick to press home. Marx was fully aware that non-commodities exchange and their exclusion from the logical sieve can, therefore, be nothing more than a deliberate commission. 'That Marx was truly and honestly convinced of the truth of his thesis I do not doubt', patronises Bohm-Bawerk, 'but the grounds of his conviction are not those which he gives in his system. They are in reality opinions rather than thought out conclusions' (p. 78). And later: 'he knew the result he wished to obtain, and so he twisted and manipulated the long-suffering ideas with admirable skill and subtlety until they yielded the desired result in a respectable syllogistic form' (p. 79). In the language of modern bourgeois theory, Marx's failure to take account of exchangeable goods not produced by labour is evidence of the _ideological_ nature of his theory, the unacceptable subordination of positive economics to value judgement.
Let us get to the heart of the matter — the significance of the undisputed fact that non-commodities such as the free gifts of nature exchange as though they were commodities. Trees grown in a virgin forest, for example, can exchange, that is have a price,2 in much the same way as commodities that embody labour. But this exchangeability does not arise out of their natural properties. It is true, as Marx emphasised continually, that only those items that satisfy some human need will enter into human intercourse; but the nature of this intercourse is not determined by the physical and natural characteristics of the items as such: these make the item a use-value not an exchange value. For virgin trees to be exchanged requires that they not only satisfy some human need — construction material or fuel or whatever-but that they are private property. The natural properties of trees that make them suitable for building or burning merely tells us that some men will make use of them: it does not tell us that these users will also be _buyers_. This requires further specifications: the users of trees must be prevented from appropriating them directly; or, what is to say the same thing, the trees must be the property of some individual whose claim over them is recognised and substantiated socially. For the gifts of nature to enter the market alongside commodities requires the existence of the market and the system of property relations associated with it. The apparent plausibility of Bohm-Bawerk's criticism begins to melt away at this point. Consider land and its 'price', money-rent, which is the most important economic transaction involving gifts of nature. Historically the emergence of money-rent, the exchange of the use of land for money, followed the development of commodity production; that is to say, it happened only after a decisive proportion of agricultural production had taken the form of commodities.3 But what is even more to the point here is that not only is rent historically subsequent to commodity production, but it is also dependent upon it; Ricardo demonstrated this in detail and we have here one of the few parts of classical political economy that was assimilated into neo-classical theory. The magnitude of rent does not determine the prices of commodities; on the contrary, it is determined by these prices. Thus in excluding non-commodities from his sieve Marx was pursuing a line perfectly in keeping not only with classical political economy but also with the school of thought which Bohm-Bawerk represents. This part of Bohm-Bawerk's criticism, so plausible at first sight, collapses completely when confronted with the theory of rent and the logic of Marx's position that we can only analyse the exchange of non-commodities once we have analysed commodities stands its ground with ease.
2. The second prong of Bohm-Bawerk's attack is not dissimilar. He quotes Marx to the effect that 'if the use value of commodities be disregarded there remains only one other property, that of being the products of labour.' And then adds: 'Is it so? I ask today as I asked twelve years ago: is there only one other property? Is not the property of being scarce in relation to demand also common to all exchangeable goods. Or that they are the subjects of supply and demand? Or that they are appropriated? Or that they are natural products?... Or is the property that they cause expense to their producers... common to exhangeable goods?... may not the principle of value reside in any one of these common properties as well as in the property of being the products of labour? For in support of this latter proposition Marx has not adduced a shred of positive evidence.' (p. 75). As this list could be extended indefinitely to include such things as being subject to the law of gravity, in orbit around the sun and so on Bohm-Bawerk surely did not intend that every item on it be taken equally seriously. Some anyway simply do not qualify. To say that all commodities are subject to supply and demand merely says that they are exhange values and therefore is no explanation. Others can be easily rendered consistent with the law of value. What is the expense suffered by producers other than labour? And leaving aside the idea of man's insatiability, for which as a general fact Bohm-Bawerk could not adduce a single shred of positive evidence that could not be countered by other equally positive evidence, is not labour the true scarce factor? The point is not worth pursuing further. Marx' argument is not that commodity exchange arises on the basis of a common factor shared by all commodities. It is more substantial and straight-forward: that in exchange, labour is the common property that regulates the terms of trade. Bohm-Bawerk does not challenge this. He merely misrepresents Marx's argument as formalist — and this brings us to the crux of the matter.
3. The final thrust of Bohm-Bawerk's critique is a confrontation of his own theory with that of Marx in an attempt to show that if Marx had used his 'purely logical method of deduction' correctly he would have reached the conclusion that labour was not the basis of value. 'If Marx had chosed to reverse the order of the examination, the same reasoning that led to the exclusion of value in use would have excluded labour: and then the reasoning that resulted in the crowning of labour might have led him to declare the value in use to be the only property left, and therefore to be the sought-for common property, and value to be the cellular tissue of value in use' (p. 77). At best then, the 'negative' approach cannot dismiss use-value, so that Marx's method fails at one and the same time to establish the premises of its own theory upon an unambiguous base and provide a determinate criticism of the neoclassical alternative. At this point Bohm-Bawerk challenges the dialectical method head on.
We have seen that according to Bohm-Bawerk, Marx's method is 'a purely logical proof, a dialectic deduction from the very nature of exchange' (p. 68). The fact that these two expressions are not equivalent does not strike the Austrian. 'A dialectic deduction from the very nature of exchange' is not 'a purely logical proof: the one refers to a method which is enquiring into a particular phenomenon — i.e. exchange; the other is, as it says, purely logical, it has no particular object. If Marx's method had been of this latter kind it could be represented as follows: imagine a population _P_ consisting of individuals each of which has two characteristics of properties, _A_ and _B_. In every member of the population, these properties _A_ and _B_ , are present; though in specific forms that vary from one individual to the next. Thus the generic property _A_ presents itself as _a_ ', _a_ "... while property _B_ presents itself as _b_ ', _b_ "... And under conditions such as these, neither Marx nor anyone else could claim that property _B_ is the only common property on the grounds it alone can exist in the general form _B_ , while _A_ can only exist in the specific forms _a_ ', _a_ "... For in a purely logical analysis there is no reason why one property should have different characteristics to another. Believing that Marx had in fact employed, and abused a logical method of this kind, Bohm-Bawerk claimed there is no reason why the particular types of labour that exist in commodities can be generalised as the common property of abstract labour, while particular use-value cannot be generalised in exactly the same way. He says this because he sees the problem as a logical one in which labour and utility are merely names given to symbols: the two are formal equivalents in a logical system, therefore they are real equivalents. No logical method can distinguish the one from the other — hence the need for some positive proof. This brings the differences between the two methods into sharp focus. In neo-classical thought, theory is a purely formalist activity with no real content, and its link with the historical process it attempts to confront must be through a leap into observations which are not and cannot be organically related to the theory. The dialectical method makes no such separation. Its theory is never purely formal, but always has a real content. It is, therefore, never separated from the concrete by an unbridgeable gulf. Thus when Marx claims that the specific concrete labour that creates commodities is reduced to abstract labour, while use-value can only exist in particular forms and is not generalised in the same way; he is making a proposition that has to do with the real nature of labour and use-value. The asymmetry between his analysis of labour on the one side and use-value on the other, is not due to a false application of a logic which should treat them as though they were the same: the asymmetry follows from the different natures of labour and use-value. In other words, it is the category under examination that determines the path and the movement of logic in the dialectical method. The methodology of _Capital_ , therefore, and here, particularly its opening section, is inseparably linked to its content, since Marx like Hegel, did not make the separation between logic and category that is characteristic of the 'model-real world' separation that we find in the positivism of the social sciences. The issue at stake is the substantial one of discovering the basis of exchange in capitalist society, and Marx knew better than Bohm-Bawerk that this was not a task of pure logic.
The crux, then, of Bohm-Bawerk's criticism at this point is that the same reasoning that led to the exclusion of value in use could have excluded labour as the basis of value (p. 77). Clearly he pre-supposes a method in which logic stands, so to speak, outside the object of study, and the 'reasoning' he talks of exists solely in the mind and has no organic relation with the world outside. In other words, his formalist logic leads him to conclude that the abstraction from particular use-value to use-value in general is just as acceptable, in principle, as the abstraction from concrete to abstract labour; and that, therefore, on strictly logical grounds there is nothing to choose between the two 'models', one of which takes embodied labour as the common property of commodities and therefore the basis of value, and the other of which casts the function of commodities of satisfying needs in this role. At the centre of this argument lies the idea of general utility which Marx considered and dismissed in a few lines at the very start of _Capital_. 'The usefulness of a thing', he wrote, 'makes it a use-value. But this usefulness does not dangle in mid air. It is conditioned by the physical properties of the commodity and has no existence apart from the latter. It is therefore the physical body of the commodity itself, for instance, iron, corn, a diamond, which is the use-value or useful thing.' ( _Capital_ , I, p. 126). In other words, for Marx, utility means nothing except when it has a particular material form. As something in general, it has no existence and is therefore unreal.
Thus at the very start of _Capital_ , and on a point crucial to the distinction between Marxist and bourgeois theory, we see the close affinity between the methods of Marx and Hegel. In Hegel's objective idealism, the link between the phenomenal world of existence and the reality that stands behind it is indissoluble: in fact, reality is embodied in existence and only becomes real in this way. Thus when Marx says that usefulness has no existence apart from the physical properties of the commodity, he is following Hegel quite closely. For if use-value in general can, by its nature, have no existence, it can also have no reality — except, that is, in the mind of the neo-classical economist. It has the same reality as a dream, for instance, and while a dream might tell us something about the mind of the dreamer it is hardly a reliable guide to the world he inhabits. (See, for instance, Weiskopf, 1949). Bohm-Bawerk does not understand this. He does not understand that Marx dismissed the idea of use-value in general not on logical grounds, but because it has no reality. For Marx it is precisely the fact that use-value can only exist in specific forms that provides the _reason_ for exchange in its most basic form of one commodity for another: i.e. because use-value is specific, commodities differ from each other as use-values and this provides a reason for exchanging them. To insist with Bohm-Bawerk that use-value is not only the reason for exchange in this sense, but also its basis and its measure, posits among other things the category of general utility. But as such a category is incapable by its nature of achieving any form of existence, it is doomed to unreality, and any theory based upon it must be a contentless abstraction.
The obvious rejoinder to this is that the same method can be used to disqualify labour as the common property of commodities, and in a somewhat different context Bohm-Bawerk does in fact take this line.
> 'The plain truth is that... products embody _different kinds_ of labour in _different amounts_ , and every unprejudiced person will admit that this means a state of things exactly contrary to the conditions which Marx demands and must affirm, namely that they embody labour of the same amount and the same kind.' (p. 82)
It is only the qualitative aspect of labour that concerns us here, and Bohm-Bawerk is undoubtedly correct when he states that different commodities embody different kinds of labour. Labour as such must always take the form of concrete labour; it can only exist as this or that type of labour-tailoring or weaving. To posit abstract or general human labour is apparently, therefore, to advance an abstraction which has no more content than the category of general utility that Bohm-Bawerk's criticism implies. In short, because abstract labour can exist only as concrete labour, it is surely impossible to say that labour is the common property of commodities since different commodities are the products of different types of labour. The law of value, no less than the theory of utility, appears to collapse upon its own dialectical foundations.
It is certainly true that at times Marx's method of dealing with this problem is less than satisfactory. The position he appears to adopt at various points in his writings, that all the different types of labour undertaken in capitalist society are nothing more than specific forms of abstract or average labour is, methodologically speaking, no different from Bohm-Bawerk's contention that the multitude of use-values produced and consumed are particular expressions of general use-value. And as such it exposes itself to exactly the same type of criticism. If abstract labour exists only as concrete labour, if it can have no mode of existence apart from concrete labour in all its various forms; then how is it any different from general utility that can only exist as specific use-values? Thus when Marx claims that 'all human labour is an expenditure of human labour-power in a physiological sense and it is in this quality of being equal or abstract that it forms the value of commodities' ( _Capital_ , I, p. 137); Bohm-Bawerk could easily retort that all use-values are use-values in a psychological sense and that it is here that we can discover the secret of value. In which case, the method of the two theories is the same and Bohm-Bawerk's contention that the only way to choose between them is on the grounds of 'positive proof' apparently rests on firm ground.
Let us take the point further. In the _Critique of Political Economy_ Marx writes as follows: 'This abstraction human labour in general exists in the form of average labour, which in any given society the average person can perform, productive expenditure of human muscles, nerves, brains etc.' (op. cit., p. 31). This road to abstract labour is more or less the same as that Bohm-Bawerk would follow to reach general utility and the method employed certainly approximates very closely to that which Bohm-Bawerk characterised as 'purely logical deduction.' Whatever else may be said about it, it is certainly not dialectical; for the abstraction it constructs is a purely mental category that has no existence in its own right. By analogy: to recognise cats and dogs as mammals — specific forms of a genus — may represent a step forward in the biological sciences insofar as we no longer see each species as totally separate and distinct; on the other hand, nobody has ever seen and examined a mammal as such. It is a purely classificatory category and as such has no existence. In the same way, if we constitute abstract labour as the common property of concrete labour — the expenditure of muscles, brains etc. — we are inventing a mental abstraction and not discovering the real abstraction that Marx was after. In analysing exchange value, Marx remarked that 'it cannot be anything other than the mode of expression, the 'form of appearance' of a content distinguishable from it.' ( _Capital_ , I, p. 127) The distinguishable content he was referring to here was, of course, value. Similarly we can talk of concrete labour being the form of appearance of a content distinguishable from it, here meaning abstract labour. But abstract labour defined simply as the common property of concrete labour is not distinguishable at all. It can no more be distinguished from concrete labour than the quality of being a mammal can be distinguished from the feline body of a cat or the canine one of the dog. It cannot be distinguished quite simply because there is nothing to distinguish, because it does not exist.
Moreover, as the few lines cited above show, this method of procedure leads inexorably towards a dehistoricisation of the categories. For if our abstractions are derived merely as common properties — i.e. from specific form to genus — not only can they have no form of existence at all, but equally they have the same status, the same possibility of non-existence and therefore existence, at all periods in history. As regards our immediate subject, labour: if abstract labour is merely the common property of concrete labour, then as concrete labour indisputably exists in every form of society, it must follow that abstract labour has the same universal presence. And if abstract labour is universally present then its product, value, cannot be far from the scene, even where we do not find the definite historical form of commodity production. Thus wherever we depart from dialecticism and employ a mode of abstraction that moves from specific form to genus, we inevitably lose the historical dimension which is such a vital element of Marx's theory.
The plain fact is that despite the odd remarks we might find in _Capital_ and elsewhere in Marx's works to the contrary, abstract labour is not the common property of concrete labour, nor is concrete labour the mode of existence of abstract labour.4 If we take Marx's analysis as a whole and do not focus our attention on individual passages this becomes abundantly clear.
There is no doubt that part of the difficulty in this case arises from the term 'abstract'; the alternative term that Marx sometimes employs, _social labour_ , is preferable in that it is much less prone to ambiguity. While it can easily appear that abstract labour is somehow the interior essence of concrete labour, so that the two cannot exist together on the same plane so to speak, the terms 'individual' or 'specific' when substituted for 'concrete' labour on the one side and 'social' for 'abstract' labour on the other, do not present the same confusion. For in the first place, these terms do not impede us in understanding that any single piece of labour is, at one and the same time, individual in the sense that it is carried out by a particular worker, and social in the sense that it is an organic part of the labour of the whole society, and, moreover, derives a part of its significance from this fact. It is much easier to understand labour as being both individual and social at the same time than it is to understand its being both concrete and abstract simultaneously. And secondly, the terms offer us no temptation to believe that we are dealing with an abstraction to genus from specific forms. When we talk of the common property of individual labour that makes it social, we are not tempted to think in terms of the expenditure of muscles and brains etc., but of the fact that each individual labour shares the character of being part of the labour of society, no matter how different its particular content might be. Whereas the opposition, 'concrete-abstract' labour, can all too easily suggest a tendency for all the different forms of concrete labour to be reduced towards a common content; the opposition, 'individual-social' labour carries no such connotation. In this respect it does not confuse one of the essential features of Marx's theory, that it is variations in the content of individual labour rather than its homogeneity that constitute the real basis for commodity production and the emergence of abstract labour.
Consider an elementary act of exchange where one individual makes a coat and exchanges it for twenty yards of linen made by another individual. In the case of both individuals, the labour is specific and concrete; tailoring in the one case, and weaving in the other. But when they exchange their commodities in order to acquire a different use-value, each individual learns in the most practical way possible that through the expenditure of his own particular type of labour he can acquire the product of another particular type of labour. Through the process of exchange the tailor can by tailoring acquire the product of weaving. When exchange becomes general and all use-values enter the market as commodities, any one type of labour becomes the means to acquire the product of any other type of labour. As Marx puts it: 'one use-value is worth just as much as another provided only that it is present in the appropriate quantity.' ( _Capital_ , I, p. 127) Which means that any one type of labour, when embodied in a commodity, becomes the equivalent in a qualitative sense of every other type of labour that is also embodied in commodities no matter how different they may be. Thus it is not the disappearance of differences among all the various types of concrete labour that provides the form of existence of abstract or social labour; on the contrary it is these differences and their development that provide the necessity for such a form. As concrete labour becomes more varied, that is to say as the division of labour develops and with it commodity production, individual labour ceases to be exclusively individual and increasingly becomes an aspect of social labour. Or to look at it another way, labour remains specific, it is still this or that type of labour, but it does not and cannot operate in isolation. Under a situation of generalised commodity production, even when production is organised on an individual basis, labour is at once individual but also social, at once specific but an organic part of social labour, at once concrete but also abstract. Thus Bohm-Bawerk's criticism that the existence of different forms of labour, particularly skilled and unskilled labour, excludes the possibility of abstract labour, labour as the common factor, is completely superseded, since these differences far from being ignored by Marx or conjured away by some trick of dialectical logic, are posited as the very basis for the existence of abstract labour.
But one problem still remains: if concrete labour is not and cannot be the form of existence of abstract labour what then, is its form? For it must have one otherwise it can achieve no reality. Marx poses and resolves this question in section 3 of the first chapter of _Capital_ , The Value-form or Exchange Value. Labour only becomes abstract once it is embodied in a commodity and constitutes the value of that commodity, so in searching for the form of existence of abstract labour we are merely looking for the value-form. Now in an elementary exchange that involves two commodities, the value of the one is expressed as a definite amount of the other. In Marx's example, where 20 yards of linen exchange for a coat, the coat becomes the form of existence of the linen: 'Use-value becomes the form of appearance of its opposite, value.' ( _Capital_ , I, p. 148) That is to say, when an individual exchanges his commodity for another, the labour he has put into his commodity is now represented directly, and actually, in the use-value of the commodity he acquires.5 When a commodity represents the value of another, Marx calls it the _equivalent form of value_. In a situation of simple exchange, one commodity acts as the equivalent form of value of another; but when exchange becomes more systematic, a single and universal equivalent emerges-i.e. one commodity emerges to act as the form of value of all other commodities: _money_. In its role of universal equivalent, money shows not only that all commodities do in fact have a common property, it acts as this common property. As the medium of circulation, it is the means through which the particular individual concrete labour embodied in any one commodity can become transformed into any, and every, other type of labour. That is to say it is the medium through which concrete labour becomes abstract labour. In a word it is money that is the form of existence of abstract labour.
QUANTITY AND QUALITY
We can now pick up an important matter mentioned in passing at the start: Bohm-Bawerk's firmly held conviction that if the law of value means anything it is that the prices of commodities are proportionate to their values. For him there can be no solution to the transformation problem, for any systematic deviation of prices from (relative) values stands in flat contradiction to the theory of value which he finds in _Capital_ , I. There are two aspects to this issue. The first concerns the quantitative relationship between value and price (of production), and here it can be shown with ease that Bohm-Bawerk defined the question too narrowly. He sees the magnitude of value as the sole determinant of price, whereas it is the magnitude of value in conjunction with its composition that in fact determines prices — and, this, of course, is perfectly consistent with the general proposition that the prices of commodities are determined by their values. Insofar as the problems raised here exist on the same plain of abstraction they are of little fundamental importance, and in this sense Joan Robinson was right to call the transformation problem 'merely an analytical puzzle which like all puzzles ceases to be of interest once it has been solved.' (Robinson and Eatwell, 1973, p. 30) The second aspect of Bohm-Bawerk's critique is much more than a puzzle as it comprises what is perhaps the most important point of separation and opposition of Marxism and bourgeois economics — the relationship between quantity and quality.
We have seen that the contentless abstraction is characteristic of the positivist method with the consequence that theory is separated from the historical process by an unbridgeable gulf. Its concepts cannot progress from the abstract to the concrete through a series of _specifications_ and _mediations_ because they contain nothing that can be specified or mediated. They must therefore attempt to grasp reality directly, _immediately_. So at the very moment that the theory of positivist science is organised around empty abstractions it prizes the operationality of its concepts. Thus Bohm-Bawerk simultaneously proposed a theory of value based upon the utterly intangible notion of generalised utility and advanced the immediate explanation of relative prices as the criterion by which a theory of value is to be judged: a theory must explain the magnitude of value at the same time as it explains its nature. In contradistinction to some modern streams of economic thought, Bohm-Bawerk did not think that 'the whole thing (i.e. value theory) analytically considered was a great fuss about nothing.' (Robinson, 1962, p. 41) He recognised that economic magnitudes could not be explained in terms of each other like a carefully constructed dome that floats of its own accord without any supports. He accepted the need for a theory of value but only in so far as it gave an immediate and direct explanation of relative prices. Thus when he turned to _Capital_ , he took it for granted that the proposition that the value of a commodity is determined by the amount of labour socially necessary for its production meant that commodities would actually exchange for each other, in practice, in a ratio proportionate to their relative values. A cursory reader of the text who has already got a firm idea of value and price from neo-classical economics might find ample evidence to support this view, but in point of fact, it is absolutely inconsistent with Marx's method and entirely unnecessary to the elaboration of his theory.
If the idea that equivalent exchange was in some way or other an essential part of the theoretical structure of _Capital_ was held only by economists such as Bohm-Bawerk, it could be cited simply as a misreading that arose from pre-conceived notions. But in fact the idea recurs continuously within the Marxist camp. In his reply to Bohm-Bawerk, Hilferding toyed with the idea, suggesting that 'under certain specific historical conditions exchange for corresponding values is indispensible.' By adding later that 'Marxist law of value is not cancelled by the data of the third volume but merely modified in a definite way' (p. 157), he gives the impression that the departure from equal exchange is a movement from a norm. In a contemporary contribution Morishima and Catephores deny the idea that conditions of equal exchange have existed historically: that is to say they deny the historical existence of a society of simple commodity production where equivalent exchange took place. But their adherence to the idea as an essential part of Marxist theory is so strong that it surfaces again immediately, only now as a heuristic device, a 'logical simulation': 'the model of simple commodity production (which) is different from the capitalist production model only with respect to the ownership of the means of production.' We are back in a positivist world and soon we learn that not only is the idea of equal exchange merely an element of a model but value itself is an 'analytical device'. Engels' rejection of Sombart's 'interpretation of the concept of value as only a logical tool' is not, so it is suggested, fully consistent with the 'evidence on the total approach of Marx to the question of value that we have tried to present here...' It is interesting that Bohm-Bawerk, though for different reasons, criticised Sombart on the same grounds as Engels. 'For my own part', he wrote, 'I hold it (i.e. value as merely a tool of logic) to be wholly irreconcilable with the letter and spirit of the Marxian teaching.' (p. 103)
These two contributions separated by seventy years testify the on-going fascination of Marxists with the idea of equal exchange. They also specify the only two grounds on which it can be treated: (1) as a real process that existed under definite historical conditions; or (2) as a model. But since the first ground is not historically valid and the second requires a positivism which is utterly inconsistent with Marxism, only one conclusion is possible — equal exchange plays no fundamental part in Marxism. Marx at times might use the idea for simplicity, or exposition, but no substantial part of his theory is dependent upon it. This can be demonstrated through the two most vital points of Marx's theory: first, the establishment of value as a category; and second, the theory of surplus value.
1. The logic Marx uses to track down value at the start of _Capital_ , I seems very definitely to imply an assumption of equal exchange. 'The valid exchange values of a particular commodity,' he says, 'express something equal.' He then gives as an example '1 quarter corn = x cwt. iron. What does this equation signify?' he asks, 'It signifies a common element of identical magnitude in two different things — in 1 quarter of corn and x cwt. of iron.' ( _Capital_ , I, p. 127) Bohm-Bawerk clearly believed that the dimension of 'equal quantities' was so essential to the argument that he did not draw special attention to it when he cited the passage (p. 10). But it is clearly remarks of this nature, several of which can be found in the early sections of _Capital_ , I, that give the impression that Marx was for one reason or another, making substantial use of the notion of equal exchange or simple commodity production. But in this context, which is perhaps the most important, it can be seen that equal exchange is not at all necessary. For the conclusion to which Marx is working, that aside from being use-values 'commodities... have only one common property left, that of being products of labour,' in no way depends upon equal exchange. Suppose double the amount of labour-time is needed to harvest the corn than to smelt the iron, so that the corn has double the value of the iron. An exchange of these commodities still brings a given quantity of the one commodity face to face with a given quantity of another. What does this tell us? It tells us that in two different things, to paraphrase Marx, there exists something common to both, _though not necessarily in equal amounts_. In other words, the position advanced by Marx that the process of exchange reveals the most diverse use-values to share the common property of being the products of labour does not mean, nor does it depend upon, a quite different proposition that only commodities that contain the same amount of labour exchange with each other. Consider the exchange 1 quarter of corn = ¼ cwt. of iron. Here the iron is the equivalent form of value; that is to say its use-value not its value, represents the value of the corn. It is true that only a commodity that is itself a value can get into the position of being an equivalent, but once in this position it is its use-value that represents the value of the relative form. Thus on the side of the equivalent form there exists the possibility of a quantitative "incongruity" between the value the equivalent (use-value) represents, and its own value. This incongruity is an essential feature of Marx's analysis of money and in so far as it presupposes unequal exchange, then, to that extent, it is possible to go beyond arguing that not only does the law of value not rest upon equal exchange, but that in fact the reverse is true: _the law of value presupposes unequal exchange_.
The importance attached to equal exchange by Bohm-Bawerk is understandable. As Hilferding says, his 'mistake is that he confuses value with price, being led to this confusion by his own theory' (p. 156). Marx's position in the opening pages of _Capital_ , I, that commodities are alike in that they are the products of labour, and that this only becomes apparent and real through the process of systematic exchange, is not a theory of prices as understood in neo-classical economics. It is no part of Marx's purpose at this stage of his work to provide a theory of the rates at which commodities exchange for each other; before this can be done it is necessary to discover the nature of the value-and price-forms. In other words, the opening chapters of _Capital_ , I, are an enquiry into the _nature_ of value, value as a _quality;_ the progress from this to the quantitative aspect of the question moves through many mediations. But positivist thought, whether it presents itself in a neo-classical or even a Marxist guise, disregards this progress and leaps directly from quality to quantity. In its modern vulgar form it ignores quality altogether. Bohm-Bawerk's critique collapses on this point which is, one way or another, the most vulnerable of the whole neo-classical edifice.
2. The relationship between economic forms (quality) and their magnitudes (quantity) is just as vital in the theory of surplus value though generally speaking it has received less attention. Bohm-Bawerk deals with the matter only in passing for fairly obvious reasons. 'In the middle part of the Marxian system,' he writes, 'the logical development and connection present a really imposing closeness and logical consistency' (p. 88). This gives an important clue to the manner in which Bohm-Bawerk would have criticised the theory of surplus value had he thought it necessary. In the Marxian system, he would no doubt have argued, labour-power is a commodity which like other commodities exchanges at its value. But remove this false premise and the theory comes apart, for once the wage is no longer tied to the value of labour-power, not only does the rate of surplus value become totally indeterminate, but even its existence as an economic category is called into question. For surplus value is essentially a quantitative phenomenon. It is the difference between two magnitudes, the value produced by a given expenditure of labour-power, and the value of this labour power. Marx took it for granted that the difference between these two magnitudes was positive, but he knows that is not sufficient to prove his point. In order for capital to appropriate this difference as surplus value and profit, it is necessary for him to show that wages will be consistently less than the value labour-power produces, and this is possible only within the framework of his system by assuming what he never establishes positively — namely that wages equal the value of labour-power. 'In this case, as in many others, he manages to glide with admirable dialectical skill over the difficult points of his argument.' He introduces equal exchange but nor for what it is, the only and necessary support for his argument, but as a virtue, as though it were a difficulty for him to overcome. 'Our friend, (Moneybags)... must buy his commodities at their value (and) must sell them at their value,' he writes in _Capital_ , I (p. 269). In _Value, Price and Profit_ he sets himself the same difficult problem. 'To explain the general nature of profits you must start from the theorem that, on an average, commodities are _sold at their real values_ , and _that profits are derived from selling them at their values..._ If you cannot explain profit upon this supposition you cannot explain it at all.' (Marx, 1962, p. 424) Whichever way you look at it, Bohm-Bawerk might well have concluded the existence of surplus value as a consistently positive magnitude depends upon the equal exchange of the commodity labour-power. This is why the proposition that commodities exchange at their value is the pivot of the whole Marxist system. Drop it and you drop what Marx readily admits to be his most important category — surplus value and exploitation. _Hic Rhodus, hic salta_.
It is as well to acknowledge from the outset that the existence of surplus value as a category of capitalist political economy does involve and must involve a tendency for labour-power to exchange at its value. Marx would have been the last person to deny this. But for him the connection between the value of the wage and surplus value was quite the opposite of that attributed to him by the positivist. That is to say for Marx the tendency of the wage to equal the value of labour-power _follows_ from the existence of surplus value as a category and vice versa. Demonstrating this point quickly is virtually impossible, as it involves nothing less than tracing the movement of his theory through the first six chapters of _Capital_ , I, but we can note, at least, the vital point that he has already arrived at the concept of surplus value in _Chapter 4_ well before he has turned to the buying and selling of labour-power in _Chapter 6_. Summarised in the most sketchy detail, the path of his logic passes the following points. In the systematic exchange of commodities, value finds for itself a form of social existence in the use-value or body of one commodity that becomes the universal equivalent — money. The money commodity starts life like any other commodity, but as its role of being universal equivalent becomes socially established, it separates itself off and becomes a commodity on its own. Its own particular use-value drops into the background and it exists more and more exclusively as the form of value of all other commodities. It is at this point, rather than through some artificial assumption of equal exchange, that we get the transformation of quality into quantity. As value, commodities are indistinguishable from one another except in respect to their size, and money, the value-form, confirms this is so far that sums differ from each other only as magnitudes. In the simple circulation of commodities, C — M — C, where money acts merely as a medium of exchange, this property is still latent or secondary. But when we move to the circulation of value, M — C — M, the transformation of money into capital, it becomes manifest and determinant. The reason for the circulation of value is surplus value, the exchange of one sum of money, M, for a larger sum, M1. By the time labour-power comes on to the stage as a commodity, circulation as a whole is firmly subjected to this reason. It must, therefore, like any other commodity act as a medium of circulating value and subject itself to this reason. In the case of labour-power, however, this has special implications. We know that this is a special commodity in that its use-value is to produce all value, surplus value included. Thus if capital is to appropriate the surplus product of society, wages must lie below the value labour produces, i.e. the value of its use-value. At the same time, the purely quantitative nature of value means that the appropriation of surplus value is simultaneously the appropriation of maximum surplus value. Thus capital is driven by its own position not merely to force the wage down below the value produced by labour, but to the lowest possible level — namely the value of labour-power. In a word, the magnitude of the wage, and with it the rate of surplus value, follows from the nature of surplus value as an economic category. Nowhere can the subordination of economic magnitude to economic forms, of quantity to quality, be more vividly apparent.
The conclusion that follows from all this is that Bohm-Bawerk's critique does not have a single point of validity. The Marxist need not concede a single thing to him. But paradoxically, this is what makes his book so valuable, for refuting it, as we have tried to show, involves reaching down to the very fundamentals of Marxism. At the same time, _Karl Marx and the Close of his System_ is indoubtedly the most substantive criticism that any bourgeois economist has ever levelled against _Capital_. It has inspired all other criticisms as Sweezy points out. But more than this, it lays neo-classical theory on the line. For in attempting to attack Marx on what was, if anything his strongest front, the opening of _Capital_ , Bohm-Bawerk revealed the foundations of his own science, and revealed them to be faulty. The Russian retreat of Marxism turns into the Waterloo of neo-classicism, but sadly the latter has not been banished to obscurity. Its flourishing survival, though, is perhaps the least of problems in itself: more important for the working-class movement is the damaging effect its positivistic method has had upon Marxism from within. With the collapse of neo-classicism under the weight of its misconceptions and inconsistencies, this becomes even more important. Within the realm of economics Marxism can no longer be challenged frontally from a neo-classical perspective; it can however be emasculated from within. The republication of Bohm-Bawerk's book will be a timely and valuable addition to the literature only if it is read with this danger in mind.
Notes
1.The Sweezy edition of _Karl Marx and the Close of his System_ , which includes the response by Hilferding, entitled 'Bohm-Bawerk's criticism of Marx', has recently been republished in Britain. All references in the text and notes of this chapter, which simply give a page number, refer to this publication.
2.For example, Marx writes: The price-form... (may)... harbour a qualitative contradiction, with the result that price ceases altogether to express value despite the fact that money is nothing but the price-form of commodities. Things which in and for themselves are not commodities, things such as conscience, honour, etc... can be offered for sale by their holders, and thus acquire the form of commodities through their price. Hence a thing can, formally speaking, have a price without having a value. The expression of price is in this case imaginary. On the other hand, the imaginary price-form may also conceal a real value-relation, or one derived from it, as for instance the price of uncultivated land, which is without value because no human labour is objectified in it'. ( _Capital_ , I, p. 197)
3.In colonies and parts of the world where commodity production was imposed from outside, the process often happened in reverse. Thus in the Highlands of Scotland, for example, the imposition of money rent developed alongside and even preceded the emergence of commodity production and was widely used as an instrument to achieve this latter. (See Prebble, 1969). But this does not invalidate the general point in any way. For the use of money rent as a catalyst of commodity production in one part of the world depended upon the prior development of commodity production upon its own foundations in another.
4.The problem of concrete and abstract labour has become confused with another related question, that of the reduction of skilled to unskilled labour. This issue, which has surfaced again recently-see Rowthorne, 1974 and Kay, 1976-finds its source, like so many other confusions in the work of Bohm-Bawerk (pp. 80-5). Part of the problem is terminological. In the _Critique of Political Economy_ , for example, Marx talks of _simple_ labour, where in context he clearly means abstract labour. The background meaning of 'simple', namely 'uniform' is clearly the one that Marx had in mind, and this lessens the purely terminological possibility of assimilating the notions of abstract and unskilled labour. Whatever ambiguities might arise from language, in theory at least, the relationship between skilled and unskilled labour is not that of concrete and abstract labour. The categories of skill can apply only to concrete labour and even if all concrete labour were unskilled in the sense that it could be performed without any special training or faculties, it would still assume different forms-machine minding and cleaning for example — and therefore would not be uniform like abstract labour. Insisting upon this, that concrete labour can never be the immediate form of abstract labour, does not mean that we dismiss the idea advanced by Marx in the _Grundrisse_ (p. 297) that the process of capitalist development tends towards abstracting labour in the sense of reducing it to a formal activity emptied of content. Only this process of abstraction does not follow immediately from the category of abstract labour, but from the development of the capitalist mode of production as a whole. Braverman is actually sensitive to this issue but he presents it in a facile fashion. (Braverman, 1974, p. 181-2)
5.'The value of the (commodity) can be expressed only as an 'objectivity', a thing which is materially different from the (commodity) itself and yet common to the (commodity) and all other commodities'. ( _Capital_ , I, p. 142). 'Thus the commodity acquires a value-form different from its natural form.' ( _Capital_ , I, p. 143)
Bibliography
von Bohm-Bawerk, E (1975), _Karl Marx and the Close of his System_ , Merlin Press, London. This edition, prepared by Paul Sweezy, also includes R Hilferding's reply, entitled 'Bohm-Bawerk's Criticism of Marx'.
Braverman, H (1974), _Labour and Monopoly Capital_ , Monthly Review Press, London and New York.
Bukharin, N (1972), _The Economic Theory of the Leisure Class_ , Monthly Review Press, London and New York.
Kay, G (1976), 'A Note on Abstract Labour', _Bulletin of the Conference of Socialist Economists_ , March.
Marx, K (1976), _Capital_ , I, Penguin Books, London.
Marx, K (1971), _A Contribution to the Critique of Political Economy_ , Lawrence and Wishart, London.
Marx, K (1973), _Grundrisse_ , Penguin Books, London.
Marx, K (1962), _Selected Works_ , Volume I, Moscow.
Morishima, M and Catephores, G (1975), 'Is there an Historical Transformation Problem?' _Economic Journal_ , June.
Prebble, J (1969), _The Highland Clearances_ , Penguin, London.
Robinson, J (1962), _Economic Philosophy_ , Penguin Books, London.
Robinson, J and Eatwell, J (1973), _An Introduction to Modem Economics_ , McGraw Hill, London.
Rowthorn, B (1974), 'Skilled Labour in the Marxist System', _Bulletin of the Conference of Socialist Economists_ , Spring.
Weisskopf, W A (1949), 'Psychological Aspects of Economic Thought', _Journal of Political Economy_ , Volume LVII, No. 4, August.
## _DIALECTIC OF THE VALUE-FORM_
## _C J Arthur_
Marx admits that the development of the value-form is the most difficult part of his critique of political economy ( _Capital_ , I, p. 90). It is not surprising, therefore, that he continually reworked it. The difficulty in presenting the material in scientific form is indicated by Marx in a letter of 1866 to Kugelmann where he states that _A Contribution to the Critique of Political Economy_ (1859) ought to be summarised at the beginning of _Capital:_ 'not only for completeness, but because even good brains did not comprehend the thing completely correctly; therefore there must be something defective in the first presentation, particularly the analysis of the commodity'. Yet the problem emerged again with _Capital_ (1867) itself. When the first proofs reached Marx he was staying with Kugelmann in Hanover and the latter convinced him that readers needed a supplementary, more didactic, exposition of the form of value, because they lacked dialectics. Marx therefore wrote a special Appendix for the First Volume, for — he explains to Engels — 'the matter is too decisive for the whole book'. ( _Selected Correspondence_ , p. 189). For the second edition Marx rewrote the whole first chapter again — thus making any appendix redundant.
In the Postface to the Second Edition Marx recalls that he 'here and there in the chapter on the theory of value coquetted with the mode of expression peculiar to' Hegel ( _Capital_ , I, p. 102-3). This admission is expressed in the context of a discussion of dialectic — and the development of the value-form is one of the most clearly dialectical passages in _Capital_. I would argue that this section articulates the dialectical relationships of 'value', 'use-value', 'equivalent' and so on, in order to exhibit the concrete structure of commodity exchange and thus correct the one-sided abstractions, and analytical reductions, of the previous sections. It is true to say that the flirtation with Hegel is less evident with the second edition — following the strictures of Kugelmann and Engels, Marx no doubt wanted to give the philistine the least possible excuse for complaining of dialectical paradoxes — so, from the point of view of dialectics, the first edition, and especially its appendix, is of great interest, and we will cite it below.
The Necessity of the Value-Form
By the value-form Marx means the form of appearance of value. Value does not appear as such in the single individual commodity: 'We may twist and turn a single commodity as we wish; it remains impossible to grasp it as a thing possessing value' ( _Capital_ , I, p. 138). Only if a commodity enters into exchange relations with others does it acquire, in the exchange-value it has against these others, a form of appearance of its value.
Classical Political Economy ignored the problem of the value-form in favour of the analysis of the substance, and, even more, the magnitude, of value; in treating commodity relations in an ahistorical fashion, it was led to abstract from the specific forms involved and to concentrate on value as an essential attribute of the product of labour without recognition of the fact that there is a problem about the form of appearance of this content. If the necessity of a material form of appearance of value is not recognised then value theory becomes nothing but metaphysical essentialism founded in abstract thought; an abstraction-value-inheres in the commodity as such. Marx, however, is acutely aware of this:
> 'If we say that, as values, commodities are simply congealed quantities of human labour, our analysis reduces them, it is true, to the level of _abstract value_ , but does not give them a form of value distinct from their natural forms.' ( _Capital_ , I, p. 141).
It is true that the natural form of a coat, for example, bears the imprint of the tailor's labour — one can see a lot of work has gone into it, so to speak. But this has little to do with value for such concrete useful labour is a necessity in all modes of production; it tells us nothing about the specific relations of production concerned. Only if the coat is produced _as a commodity_ does its character as a product of labour that is equatable with all other kinds of labour, that is, taken in abstraction from its specificity as tailoring, give it value. It does not have this character immediately; it can only embody general social labour insofar as immediately private labour is realised as universal social labour, in and through the mediation of exchange, as an emerging result. (See Arthur, 1978).
'However', says Marx, 'it is not enough to express the specific character of the labour which goes to make up value. Human labour-power in its fluid state, or human labour, creates value, but is not itself value. It becomes value only in its coagulated state, in objective form'. ( _Capital_ , I, p. 142). But value can objectify itself concretely only through a form that neglects the specific character of a commodity, as a certain use-value shaped by particular kinds of labour, in favour of its commonality with other products of social labour. The value-form makes this possible insofar as another commodity is posited as representing the value of the first.
Just as in a balance the iron weights represent weight alone for the heavy object being weighed, quite independently of their specific character as iron, so the body of the value-equivalent represents value alone independently of its bodily form, and the concrete useful labour it contains represents only universal human labour in the abstract, from the standpoint of the commodity whose value is expressed in it; in this way the value of a commodity is realised _relative to another_.
Exchange is therefore a crucial presupposition of Marx's investigation; it is the process through which the value-form develops 'from its simplest, almost imperceptible outline to the dazzling money-form'. ( _Capital_ , I, p. 139)
Exchange and Equivalence
Marx assumes that commodity exchange is an exchange of equivalents. Before considering in detail Marx's analysis of the value-form in this light, we deal with two objections that have been raised to this assumption.
Let us first look at the objection raised by Bohm-Bawerk (1975, p. 68):
> 'Where equality and exact equilibrium obtains, no change is likely to occur to disturb the balance. When, therefore, in the case of exchange the matter terminates with a change of ownership of the commodities, it points rather to the existence of some inequality or preponderance which produces the alteration.'
This brilliant observation proves nothing except the profoundly undialectical character of the formalist thinking of the bourgeois critic. For Marx, however, it is precisely through the dialectical unity of use-value and value in the commodity that we understand the basis of the alteration in ownership, on the one hand, and the termination of the transaction resulting in the holding of an equivalent of the original commodity held, on the other. It is precisely the fact that commodities _differ_ as use-values, but are _equivalent_ as values that is the basis of capitalist exchange.
> 'For the owner, his commodity possesses no direct use-value. Otherwise, he would not bring it to market. It has use-value for others; but for himself its only direct use-value is as a bearer of exchange-value, and consequently a means of exchange. He therefore makes up his mind to sell it in return for commodities whose use-value is of service to him. All commodities are non-use-values for their owners, and use-values for their non-owners. Consequently they must all change hands. But this changing of hands constitutes their exchange, and their exchange puts them in relation with each other as values and realises them as values. Hence commodities must be realised as values before they can be realised as use-values.' ( _Capital_ , I, p. 179).
One should note that a purely formal analysis of the relation of exchange cannot sustain the conclusion that exchange _must_ involve exchange of _equivalents_. If Marx's initial presentation of the matter, in _Capital_ , is taken to provide a purely _logical_ argument from exchange, to equivalence, to the substance of this equivalence, then it does not work.
This has been seized on recently by the post-Althusserians:
> 'Marx conceives exchange as an _equation_ , as being effected through the identity of the objects exchanged... But is is by no means inevitable that exchange be conceived as an equation. Exchange may be conceived as being _equivalent_ , in the juridical sense, that is, that both parties to it agree to the equity of the terms of the exchange and receive what they were promised, _but not as an equation_ (there not being any substantive identity between the things exchanged)... Exchange as equation and exchange proportionality as necessity are products of definite theoretical conditions, conditions which give certain questions pertinence... That these questions are theoretical rather than an inevitable part of the nature of things (and for which answers must be sought) is often forgotten. It is possible to argue that prices and exchange-values have no _general_ functions or general determinants, and that there is in general no necessity for the proportions in which commodities exchange.' (Cutler et al., 1977, pp. 12-14)
It is true that scientific questions are not given in the nature of things, but, on the contrary, the nature of things is illuminated by posing, and answering, theoretical questions. What has to be justified is the research programme that embodies a certain theoretical problematic. The entire value-problematic should be junked, it is alleged, since there are no _general_ determinants of exchange. One wonders what these authors would say about Newton's laws of motion; observing the convoluted trajectory of a leaf fluttering to the ground should he not have resigned immediately the search for _general_ determinants of motion? In order to validate Marx's research programme one has to recognise straight away that its pertinence must be limited to exchange in the context of definite historically developed material conditions. Marx says that 'the law of value presupposes for its full development an industrial society in which production is carried on a large scale and free competition prevails, i.e. the modern capitalist society.' ( _Contribution to the Critique of Political Economy_ , p. 69). It follows that his theory of exchange depends not merely on the products of labour being exchanged, but being exchanged under these definite social conditions. Barter of surpluses occuring now and then between self-sufficient communities should be excluded. Likewise if two friends notice that one person has a spare bed while the other has surplus bookshelves they may not bother their heads further than to reassign these use-values; in such a case no qualitative identity or quantative necessity need be posited.
In capitalist economies, however, it is quite different. Here we are dealing with myriads of commodities at once autonomous and interchangeable (and in that sense identical). They are autonomous in that commodities are the product of many individual private production processes, linked only by the market; and identical in the sense that each commodity is interchangeable in definite, known proportions with thousands and thousands of other commodities whether or not it is actually exchanged with them. It is the fact that the interchangeability is independent of any one particular act of exchange, but is nevertheless the unplanned outcome of the sum total of autonomous acts of exchange, which posits capitalist exchange as exchange of equivalents. (See Elson, below.) It is not exchanges as individual acts which posits equivalence, but interchangeability — the fact that we know the exchange-value of one commodity in terms of many other commodities even though it has not actually exchanged with any of them.
In the remainder of this note we presuppose that the form of value expresses a relationship of equivalence and seek to show that formal, undialectical thinking cannot comprehend the analysis of the value-form because it divorces itself from the concrete relationships involved.
The Logic of the Value-Form
Marx proceeds from the simple form of value, to the expanded form, to the general form, to the money form. Schematically, the development can be summarised thus:
1. _Simple Form_ x = y; in this relation the value of x is said to be given in relative form, while y is in the equivalent form.
2. _Expanded Form_ x = y, and x = z, and x = a, and x = n etc.
3. _General Form_ | y = | ! x(x is said to be in universal equivalent form)
---|---|---
z =
a =
b =
n =
4. _Money Form_ As before-but x is an amount of gold or of whatever the historically evolved universal equivalent happens to be in a given society.
From the point of view of formal thinking nothing is going on here except the complication of a tautology — 'a value is a value is a value'. (From a bourgeois point of view no empirical significance attaches to any of it until the money form is mentioned, which can be misrecognised as a reference to familiar phenomena — market prices. Marx, of course, is concerned to show, as against conventionalist theories of money, that it is rooted, in germ, in the simple exchange of values.) This is because, for the formalist, the development of the value-form is the mere elaboration of an abstraction, not the synthesis of the appearance of a real relation.
Stanley Moore (1963) for example, mistakes the development of the value-form as an analysis meant to buttress Marx's earlier derivation of value from exchange-value. He takes Marx to be employing the _principle of abstraction_ throughout this section. He quotes Tarski on this principle as follows:
> 'Every relation which is at the same time reflexive, symmetrical, and transitive is thought of as some kind of equality. Instead of saying therefore that such a relation holds between two things, one can, in this sense, also say that these things are equal in such and such a respect, or-in a more precise mode of speech-that certain properties of the things are identical. Thus, instead of saying that two segments are congruent, or two people equally old, or two words synonomous, it may just as well be stated that the segments are equal in respect of their length, that the people have the same age, or that the meanings of the two words are identical.' (Tarski, 1946, section 30.)
In other words, according to Moore, Marx takes it that the relation between commodities in exchange is _reflexive, symmetrical and transitive;_ and abstracts from this the conclusion that the commodities share an identical property: they have the same value.
According to the canons of formal thought, a relation of equality obtains if and only if it is reflexive, symmetrical and transitive.
1) A relation R is reflexive in the class K when for any x which is a member of K, xRx.
2) A relation R is symmetrical in the class K when for any x and y which are members of K, if xRy and yRx.
3) A relation is transitive in the class when for any x, y and z, which are members of K, if xRy and yRz, then xRz.
I refer to the formal properties of the equality relation (x = x; if x = y then y = x; if x = y, and y = x, then x = z) collectively as 'RST'.
We have already said that Marx takes exchange to be structured as equivalent exchange, but the importance of the _development of the value form_ is precisely that certain contradictions, hidden in the analytical identity of values with each other, emerge insofar as value _appears materially_ as exchange-value, its necessary form of appearance. I will show that Marx's analysis of the value-form, which he characterises as a relation of equivalence, violates RST, or at least that it draws attention to material characteristics of the exchange relation which cannot be expressed in RST forms of analysis. This is because what Marx is tracing in the development of the value-form is not the movement of abstractions but _the logic of the concrete_.
Lack of Reflexivity in the Simple Form of Value
In considering the simple expression of value, 'x = y', Marx argues that, although the relative and equivalent forms are clearly inseparable in it, there is, nonetheless a real polarity here in that these two forms are distributed among _different commodities_ , and he goes on: 'I cannot, for example, express the value of linen in linen: 20 yards of linen = 20 yards of linen is not an expression of value.' ( _Capital_ , I, p. 140) (It simply expresses a definite quantity of an object of use, linen.) Here, then, he denies reflexivity insofar as 'x = x' is said not to be an expression of value. In relating x to itself we cannot do anything except say that it is identical to itself as a determinate body with naturally given properties (relevant to its _use-value_ only). 'x = x' is not an expression of _value_ because value is expressed only in a relation between different commodities and can be assigned to a particular commodity only through the value-form which expresses the value of one commodity _relative_ to another. In 'x = y' _the equivalent 'y'_ stands for the value of x.
To understand this better we must take a step back. Use-value and value are not merely different determinations of the commodity but may be _opposed_ in that, when serving as a use-value, a commodity cannot also be a value, and, when treated as a value, its particular use is ignored; that is, it is either being _exchanged_ , in which relation only value is important and abstraction is made from the particular use of each commodity, or it is being _consumed_ and is no longer viewed as potentially exchangeable.
However, the matter is by no means so simple that we can rest content with this _antithesis_ by assigning use-value to the sphere of consumption and value to the sphere of exchange; because when we say that we abstract from the particular use-values involved in exchanging commodities, it is nonetheless the case that what is exchanged, that is, actually handed over, are use-values which must be _treated as_ values such that a commodity x takes as its value-equivalent the body of y. Marx says in the First Edition of _Capital:_
> 'We stand here at the jumping-off point of all difficulties which hinder the understanding of the _value-form_. It is relatively easy to distinguish the value of a commodity from its use-value, or the labour which forms the use-value from that same labour insofar as it is merely reckoned as the expenditure of human labour power in the commodity-value. If one considers commodity or labour in one form, then one fails to consider it in the other, and vice versa. These abstract opposites fall apart on their own and hence are easy to keep separate. It is different with the _value-form_ which exists only in the relation of commodity to commodity. The use-value or commodity-body is here playing a new role. It is turning into the form of appearance of the commodity- _value_ , thus of its own opposite. Similarly, the _concrete_ , useful labour contained in the use-value turns into its own opposite, to the mere form of realisation of _abstract_ human labour. Instead of falling apart, the opposing determinations of the commodity are reflected against one another. However incomprehensible this seems at first sight, it reveals itself on further consideration to be necessary. The commodity is right from the start a _dual_ thing, use-value _and_ value, product of useful labour _and_ abstract coagulate of labour. In order to manifest itself as what it is, it must therefore _double_ its form.' ( _Value: Studies_ , p. 21.)
Value and use-value enter on a dialectic here in that _value_ , although opposed to _use-value_ in Section 1 of chapter one, cannot, in fact, be separated from its other, because the exchange transaction consists, in actuality, of the handing over of _use-values;_ hence in exchange the value relations have to be mediated in the use-values and the role they play in the exchange relation. A product on its own does not have value; hence a commodity cannot express its value in its _own_ form, as naturally constituted (i.e. as a use-value); but the _double form_ of the commodity (viz. value and use-value) can find expression in the dialectical relation of identity and difference whereby the material use-value y takes on the form of the equivalent of the value of x.
That is to say, the identity of a commodity as a value cannot be expressed through equating it with itself; such a relation to its own self grasps only what it is _immediately_ , namely a use-value. To be a value is to have a status as a _social_ object, which status has to be mediated, therefore, through its equation to another commodity, immediately _different_ from it, yet (in virtue of their common origin in the universal labour of society) of _identical social substance_. A value is identical with itself only in this its other because the substance of value being essentially social 'it can only appear in the social relation between commodity and commodity' ( _Capital_ , I, p. 139)-it has to appear as exchange-value, that is, in _mediated form_.
We see then, that in a purely formal analysis 'abstract opposites fall apart' — a is a and b is b; we are looking at the thing either as a use-value _or_ as a value. Marx on the other hand, is dealing here with the social existence of the commodity as the interpenetration of opposites which are 'reflected against one another' in the value-form in such a way that 'x = x' cannot express a value relation where 'x = y' can-such is _the logic of the concrete_.
Let us turn now to the question of symmetry.
Lack of Symmetry in the Simple Form of Value
While it is clear that the value expression 'y = x' may be derived from that of 'x = y' because this relation has the property of symmetry, Marx stresses the point that these expressions must be taken in a definite direction such that, in the one, x is in relative form, and, in the other, in equivalent form: these are therefore two _different_ expressions of value.
> 'The relative form of the value of linen... presupposes that some other commodity confronts it in the equivalent form. On the other hand, this other commodity, which figures as the equivalent, cannot simultaneously be in the relative form of value. It is not the latter commodity whose value is being expressed. It only provides the material in which the value of the first commodity is expressed. Of course, the expression 20 yards of linen = 1 coat... also includes its converse: 1 coat = 20 yards of linen... But in this case I must reverse the equation, in order to express the value of the coat relatively; and, if I do that, the linen becomes the equivalent instead of the coat. The same commodity cannot, therefore, simultaneously appear in both forms in the same expression of values. These forms rather exclude each other as polar opposites.' ( _Capital_ , I, p. 140)
One essential asymmetry between the commodities in relative form and in equivalent form is that as an equivalent a commodity has the status of _immediate exchangeability_ insofar as it represents _the value of that in the relative form_ , whereas in the relative form a commodity exchanges with its equivalent only through the mediation constituted through this other commodity expressing _its_ value _relative to the first_. (To anticipate our exposition a little) this problem is more obvious if the equivalent is taken to be the money-commodity in that people tend to assume that, unlike the things that express their values in it, money has by nature the special quality of immediate exchangeability. The hypostatisation involved in attributing such a property to the natural form of a commodity was not transcended by those who recognised that gold is not the only use-value that can play the role of money. Only Marx traced the form of immediate exchangeability to its most primitive root in the relationship established in the simplest expression of value, such as 20 yards of linen = 1 coat. (See _Capital_ , I, p. 149-40.)
If we take an exchange, we can consider the matter form the point of view of either party, but Marx insists that there are concretely _two_ such points of view; and this must not be overlooked if we are to stay close to the concrete character of exchange, and avoid getting entrapped in formalisms which omit this vital character of commodity dynamics.
From the point of view of the owner of x the commodity y features merely as its value equivalent. Of course, at the same time, from the point of view of the owner of y it is x that is _its_ equivalent. As Marx puts it in the Appendix to Chapter 1 of First Edition of _Capital_ , I:
> 'Here _both_ , linen and coat, are at the same time in relative value-form and in equivalent form. But, nota bene, for _two different persons_ and in _two different expressions of value_ , which simply occur _at the same time_ '. ( _The Value Form_ , p. 135.)
In the course of explaining this point, Marx makes an interesting reference to _the principle of abstraction_ (see above p. 72): after stressing again that symmetry in exchange actually involves us in _two_ different expressions of value at the same time because each commodity _in turn_ must be taken in relative and equivalent form, he admits that we can draw from either formula, 'x = y' or 'y = x', the conclusion that the values of x and y are identical or equivalent. He says:
> 'We can also express the formula 20 yards of linen = 1 coat... in the following way: 20 yards of linen _and_ 1 coat _are equivalents_ or _both are values of equal magnitude_. Here we do not _express the value_ of either of the two commodities in _the use-value of the other. Neither_ of the two commodities is hence set up _in equivalent form. Equivalent_ means here only _something equal in magnitude_ , both things having been silently reduced in our heads to the abstraction _value.' (The Value Form_ , p. 138.)
Nothing could be clearer: if one loses one's grip on the _concrete character_ of exchange, and the dialectic of the moments value and use-value, by moving into the realm of _abstraction_ then of course _'in our heads'_ everything collapses into a lifeless abstraction — value; and the analysis of the value-form may as well be ignored since all we can discover is the vast tautology 'x = y = x = y = x'. This formalism is, of course, emminently suitable for abstract analysis whereby value precisely becomes nothing more than a standard measure of all things. Moore himself in the paper we cited misses the point precisely in this fashion when he says that in Universal Equivalent Form 'worth is measured by its price in terms of some arbitrarily selected standard commodity, a _numeraire.'_ (Moore, 1963, p. 80). Even those like Ricardo who understand that the search for a standard commodity is misplaced and that it is necessary to analyse the _substance_ of value in terms of labour do not realise that one cannot just assume the substance of value and then see each commodity merely as a given magnitude, a given portion of the total value produced, for these products only become commodities with value insofar as in reality (and not 'in our heads') _exchange imposes this equivalence_ on them through a material process of commensuration (not an ideal comparison) whereby the value of each is externalised in the use-value of the other. The other here is the material equivalent of the first. This irreducible fact about the process of exchange cannot be removed by invoking a formal principle of symmetry leading to a principle of abstraction which reduces each to the same lifeless identity.
Lack of Symmetry in the Universal Equivalent Form
This becomes clear in the general form of value where a universal equivalent expresses the value of all other commodities. As we have said, we need to avoid that formalism which characterises the universal equivalent as a mere _numeraire;_ rather we should grasp it as the concrete exclusion of one commodity from the others, as the incarnation of their _social_ being as values and as products of abstract human labour. We cannot speak about a standard commodity in a way which presupposes that we are concerned with a range of values until we have proved the material validity of the category value, and, since this — value — _attains phenomenal expression_ only through the value-form, it follows that the universal equivalent is by no means a simple convenience of the scientific observer for ordering his data, it is a very concrete necessity for the unification in value of the products of labour-the labour embodied in this universal equivalent represents all human labour (taken as abstracted from its various concrete forms). We have a universal order self-differentiated through the universal equivalent which _unifies_ commodities as _values_ , thus overcoming their _separateness_ as _use-values_.
The opposition between the relative and equivalent forms of value, implicit in the simple form of value, articulates itself concretely in the universal equivalent form:
> 'In _form 1_ the two forms already exclude one another, but _only formally_. According to whether the same equation is read forwards or backwards, each of the two commodities in the extreme positions, like linen and coat, are similarly now in the relative value-form, now in the equivalent-form. At this point it still takes some effort to hold fast to the polar opposition... In _form 3_ the _world of commodities_ possesses general social relative value-form only because and insofar as all the commodities belonging to it _are excluded_ from the _equivalent-form_ or _the form of immediate exchangeability_. Conversely, the commodity which is _in the general equivalent form_ or figures as _general equivalent_ is excluded from the _unified_ and hence _general relative value-form of the world of commodities_.' ( _The Value Form_ , p. 148; compare _Capital_ , I, p. 160-61.)
Here then, _symmetry_ breaks down, as it does in the money form as well, of course. In the _money form_ we have a definite commodity, evolved through historical practice, playing the role of universal equivalent (gold is an obvious example). If other commodities may be formally equated as values with the money commodity it is not the case in fact that each can play the role of money (that is, have the form of universal equivalent — of immediate exchangeability). Marx comments:
> 'Like the relative form of value in general, price expresses the value of a commodity (for instance a ton of iron) by asserting that a given quantity of the equivalent (for instance an ounce of gold) is directly exchangeable with iron. But it by no means asserts the converse, that iron is directly exchangeable with gold... Though a commodity may, alongside its real shape (iron for instance), possess an ideal value-shape or an imagined gold-shape in the form of its price, it cannot simultaneously be both real iron and real gold.' ( _Capital_ , I, p. 197.)
Lack of Transitivity in the Money-Form
We see here, also, that _transitivity_ breaks down in a money economy. Marx says that 'if the owner of the iron were to go to the owner of some other earthly commodity, and were to refer him to the price of iron as proof that it was already money' ( _Capital_ , I, p. 197-8) — he would get a dusty answer, for the owner of this other commodity is not prepared to accept iron even though the iron is worth an amount of gold which is of equivalent value to his own product and for which he would gladly exchange it. Let us say this other commodity is a pound of saffron, then, even if one ton of iron = an ounce of gold = a pound of saffron, it may still be the case that the owner of the saffron will not part with it for the iron, but only for gold, or for some necessity he needs for consumption. One may in imagination take iron to be the universal equivalent, and all the equations will be formally correct, but the exchanges corresponding to them will not occur unless iron is in actuality the universal equivalent, that is, the money commodity. Hegel says:
> 'When the universal is made into a mere form and co-ordinated with the particular, as if it were on the same level, it sinks into a particular itself. Even commonsense in everyday matters is above the absurdity of settling a universal _beside_ the particulars. Would anyone, who wished for fruit, reject cherries, pears and grapes, on the ground that they were cherries, pears or grapes, and not fruit?' (Hegel, 1975, p. 19.)
True-yet in the market-place vendors will reject various other commodities in exchange for their own on the ground that they are not _money_ , their externalised identity as values. Only the money-commodity (e.g. gold) has the _social form_ of universal equivalent (which gives it _immediate exchangeability_ ) in addition to its _formal_ status as a value identical with others. In other words money is _not_ a _'mere_ form' of the abstract universal: value; rather, it _concretely mediates_ the identity of values with each other.
A final point to consider is that, whereas people might concede 1lb of iron does not give its worth in itself, with money (since it has the social form of immediate exchangeability) the illusion arises that — just as it is-it _is_ value: 'the movement through which this process has been mediated vanishes in its own result, leaving no trace behind.' ( _Capital_ , I, p. 187.)
Marx, however, insists that:
> 'the equivalent form of a commodity does not imply that the magnitude of its value can be determined. Therefore, even if we know that gold is money, and consequently directly exchangeable with all other commodities, this still does not tell us how much 10lb of gold is worth, for instance. Money, like every other commodity, cannot express the magnitude of its value except relatively in other commodities.' ( _Capital_ , I, p. 186.)
Hence, even though the value of other commodities is given as a function of the money-commodity, the identity function is a non-starter. This is because value emerges from the dialectical relations of commodity exchange; it is not an abstract essence inhering in a product in a pseudo-natural fashion.
Bibliography
Arthur, C J, 'Marx's Concrete Universal — Labour' in _Inquiry_ 1978.
Bohm-Bawerk, E V, _Karl Marx and the Close of his System_ (Merlin Press, London, 1975).
Cutler, A, Hussain, A, Hindess, B, Hirst P Q, _Marx's 'Capital' and Capitalism Today_ , Volume I, (Routledge, London, 1977).
Hegel, G W F _Hegel's Logic_ trans. W Wallace (Oxford University Press, 3rd ed., 1975).
Marx, K _Contribution to the Critique of Political Economy_ (trans. N I Stone, Chicago, 1904).
Marx, K, 'Chapter One: The Commodity' from the first edition of _Capital-in Value: Studies by Marx_ , trans. A Dragstedt (New Park, London, 1976).
Marx, K, 'The Value Form' (Appendix to first edition of _Capital_ ), trans. M Roth and W Suchting in _Capital and Class_ 4, Spring, 1978.
Marx, K _Capital_ Volume I, (Penguin, Harmondsworth, Middx, 1976).
Marx, K, Letters to Kugelmann, 13 October 1866; and to Engels, 22 June, 1867; in _Selected Correspondence_ (Moscow, 1965).
Moore, S, The Metaphysical Argument in Marx's Labour Theory of Value' in _Cahiers de l'institut de Science Economique appliquee_ , Supplt. No. 140, Aout 1963.
Tarski, A _An Introduction to Modern Logic_ , Second ed., New York, 1946.
## _MISREADING MARX'S THEORY OF VALUE: MARX'S MARGINAL NOTES ON WAGNER_
## _Athar Hussain_
**_Directions for use_**
This article bears the mark of the context in which it appeared. It was published in a review named 'Theoretical Practice' which ceased publication in 1973. The aim of the review was to develop Marxist theory on the assumption that the analyses of Althusser and his associates had removed the obstacles which stood in the way of its further development. This article takes as its point of departure what has been crucial to Althusser's analysis, namely, the assumption that there is a scientific problematic — rules and relation governing discourse — which underlies _Capital_ and that problematic has been masked by layers of the ideological readings of _Capital_ — both by Marxists and non-Marxists. It is this assumption which makes _Marx's Notes on Wagner_ of special importance; what this article does is to gauge the correctness or incorrectness of various readings of the sections of _Capital_ on value by Marx's own comments on a gross misreading of _Capital_ (in particular Section 1 on commodities) by a conservative German economist, Wagner. But once this assumption is removed the arguments of the article become vulnerable. What the assumption of scientificity does is to suspend those questions which cast doubt on the basic concepts of _Capital_ , e.g. value, laws of tendency, etc. Now what we need to take into account is that the barrier to the development of Marxist theory is not simply the misreadings of _Capital_ — of which there are many — but more importantly the concepts of _Capital_ themselves.
Marx's Marginal Notes on Wagner's _Lehrbuch der politischen Okonomie_ constitute one of his last texts. In his introduction to the French paperback edition of _Capital_ , Althusser singles out this text for special mention:
> 'It reveals irrefutably the direction in which Marx's thoughts tended: no longer the shadow of a trace of Feuerbachian humanist or Hegelian influence.' (Althusser, 1971, p. 99.)
Thus, for Althusser, these Notes are important because they specify the epistemological break that detaches science from ideology.
In these Notes, Marx demonstrates that Wagner's reading of _Capital_ takes the form of the suppression of conceptual distinctions and the transformation of concepts into free words, free in the sense that they can be replaced by other words. This transformation, like the rest of Wagner's _Lehrbuch_ , is an effect of a specific problematic, the problematic of Philosophical Anthropology. The theoretical importance of this text derives from the fact that the problematic of Philosophical Anthropology is not confined to Wagner's _Lehrbuch_ but, as will be demonstrated in this introduction, also governs more recent works, including those of certain revisionist economists. In these Notes, Marx not only read Wagner, but also reflects on his own problematic, which thus also makes these Notes nothing less than a reflection of the problematic governing _Capital_. This is the theoretical justification for Althusser's comment that these Notes reveal irrefutably the direction in which Marx's thoughts tended.
Wagner's discussion of _Capital_ centres around the question of the theory of value. Before coming on to the specific effects of his ideological transformation of Marx's discourse, we should make one very general point. Wagner's comment that Marx's theory of value is _'the cornerstone of his socialist system'_ assigns a teleology to the 'theory of value' and thereby denies its autonomy as scientific practice, autonomy in the sense of being governed by the laws specific to that practice. Marx's retort: _'As I have never set up a "socialist system" this is a fantasy of Wagner, Schaffle and tutti quanti'_ , is his affirmation of the autonomy of 'historical materialism'. It is not subjugated to any ideology, not even to a revolutionary ideology. In the Preface to _A Contribution to the Critique of Political Economy_ , Marx affirmed this autonomy of scientific practice in the following words:
> 'At the entrance to science, as at the entrance to hell, the demand must be made,
>
> Qui si convien lasciare ogni sospetto;
>
> ogni vilta convien che qui sia morta.'1 (Op. cit., p. 23.)
Wagner, and as we shall see, he is not the only one, regards labour as the 'common social substance of exchange-value.' Marx points out that exchange-value is the necessary mode of expression ( _Darstellungsweise_ ) of value, and the concept of value is different from the notion of exchange-value, which is invested in the commercial practice of the exchange of commodities. The difference between the two is the difference between 'what is represented' and the 'mode of representation of what is represented'. Marx goes on to specify the order of the discourse in _Capital:_
> 'The progress of our investigation will bring us back to exchange-value as the necessary mode of expression or phenomenal form of value, which, _however_ , we have for the present to consider independently of this form.' ( _Capital_ , I, p. 46, emphasis added.)
The 'order of the discourse', as Marx points out in the _1857 Introduction_ ,2 is distinct from the order of concrete historical events. This difference is a corollary of the fact that discursive practice is a process in thought and the thought-object is different from the real object. This particular difference reveals the error in the historicist reading of _Capital_ according to which the discussion of the concept of value precedes the analysis of the determination of prices of production (i.e., exchange-values denominated in terms of money which in general diverge from values), because prices in the initial stage of capitalism are equal to values while in the later stages they are equal to prices of production.3 The statement by Marx quoted above is based on a theoretically specified relation between value and exchange-value and cannot be construed to specify the order of concrete historical events.
The statement that exchange-value is 'the necessary mode of expression or phenomenal form of value' is crucial to the specification of the difference between Marx and the classical economists. Exchange-value is the necessary mode of expression of value only under a specific mode of production, i.e., one characterised by generalised commodity production. The theoretical connection between value, exchange-value and generalised commodity production is as follows: the generalised commodity production specifies the 'social space', i.e. the capitalist mode of production, in which value is represented in the form of exchange-value (see Ranciere, 1971). Throughout the first chapter of _Capital_ I, the terms 'value-form' and 'exchange-value' are used interchangeably, while the term 'natural form' is used to denote 'use-value'. Wagner overlooks the theoretical connection between the concepts of 'value', 'exchange-value' and 'commodities'. Marx therefore has to remind him that 'for me' (in the first chapter of _Capital_ I), 'Neither "value" nor "exchange-value" are subjects but commodities.'
Nearly all critiques of _Capital_ by bourgeois economists from Bohm-Bawerk to Joan Robinson4 have been based on the assumption that the first chapter of _Capital_ is devoted to the quantitative determination of exchange-value. This particular assumption enables these critics to replace the question asked in the text by another question: what determines the exchange-value of commodities? Marx comments on the effects of the problematic governing bourgeois political economy:
> 'The few economists, among whom is S Bailey, who have occupied themselves with the analysis of form of value' (exchange-value) 'have been unable to arrive at any result, first, because they confuse the form of value with value itself; and second, because, under the coarse influence of the practical bourgeois they exclusively give their attention to the quantitative aspect of the question.' ( _Capital_ , I, p. 56n.)
The coarse influence of the practical bourgeois that Marx is referring to is the object invested in the commercial practice of exchange, i.e., the quantitative magnitude of exchange-value. Marx points out in these Notes that 'apart from this, as every promoter, swindler etc. knows, there is certainly a formation of exchange-value in present day commerce, which has nothing to do with the formation of value.'
The difference between Marx and Ricardo, which Wagner overlooks, is specified by Marx in _Capital_ , I, when he writes:
> 'It is one of the chief failings of classical economy that it has never succeeded, by means of its analysis of commodities, and, in particular, of their value, in discovering that form under which value becomes exchange-value. Even Adam Smith and Ricardo, the best representatives of the school, treat the form of value as a thing of no importance, as having no connection with the inherent nature of commodities. The reason for this is not solely because their attention is entirely absorbed in the analysis of the magnitude of value. It lies deeper. The value-form of the product of labour is not only the most abstract, but is also the most universal form taken by the product in bourgeois production, and stamps that production as a particular species of social production, and thereby gives it its special historical character. If then we treat this mode of production as one eternally fixed by nature for every society, we necessarily overlook that which is the differentia specifica of the value-form, and consequently of the commodity form and capital form, etc. ( _Capital_ , I, p. 85n.)
Ricardo asked the question, what determines the magnitude of value, and provided the answer to it, the value of a commodity is equal to the labour embodied in it. Marx asks a different question, what is the social structure (referred to as 'that form' in the above quotation) in which the value of goods is represented in the form of exchange value? Of course, the statement in the text quoted above is slightly ambiguous, for value does not _become_ exchange-value, but is _represented_ in exchange-value, but this ambiguity is easily removed by referring to other passages from these Notes. Neither Ricardo nor any other bourgeois economist, classical or non-classical, asked the second question. Marx goes on to account for the absence of the second question in the following terms:
> 'If then we treat this mode of production as one eternally fixed by nature for every society, we necessarily overlook that which is the differentia specifica of the value-form'. ( _Capital_ , I, p. 85n.)
What does the statement beginning 'treat this mode of production' (the capitalist mode of production) refer to? Obviously not to the simple fact known from historical chronicles that capitalism has not always existed. In fact, Adam Smith gave a detailed account of the changes in the organisation of production in his discussion of the division of labour. The oversight of Ricardo et al. cannot be corrected by a simple injection of 'time perspective' or by providing that ambiguous 'historical angle' to which Dobb refers.5 The oversight of Ricardo et al. is the oversight of their problematic; the 'treatment of this mode of production as one eternally fixed by nature' is a metaphoric (and hence ambiguous) reference to the problematic governing the discourse of Adam Smith and Ricardo. The main characteristic of that problematic is that it is directly or indirectly determined by the commercial and economic practices specific to the capitalist mode of production. The main effects of that determination, which are specified throughout _Capital_ , are as follows:
> (i) exclusive concentration on the quantitative magnitude of exchange-value;
>
> (ii) the equation of 'surplus labour' with profit — a category which is specific to the capitalist mode of production;
>
> (iii) the failure to distinguish between the value of labour and the value of labour-power.
The reason why Ricardo and bourgeois political economy do not ask the second question can be discovered by determining the theoretical requirements for answering it. The specification of the social structure in which exchange-value is the 'mode of expression of value' requires the concept of the 'mode of production' and the concepts required to specify the pertinent difference of a particular 'mode of production' _vis-a-vis_ others. Marx's counter-question, i.e., the second question above, signifies the change of problematic. The object of the science of history is no longer conceived as a process with a subject, but as a process without a subject. This second question is a question of a specific problematic and it is also a 'non-question' of the problematic governing Ricardo's discourse; the absence of the question is the symptom of the problematic. It is this concept of a process without a subject that Marx owes to Hegel. Althusser points out that in Chapter one of _Capital_ I, Ricardo provides the Generality I, the object of the theoretical labour, while Hegel's 'process without a subject' is used as Generality II, i.e., the means of transformation, to produce Generality III, i.e., historical knowledge.
I have given no demonstration of the assertion that Ricardo's problematic is that of a process with a subject. This demonstration would have to be based on a wider question which I cannot answer here: Is an ideological discourse necessarily governed by the problematic of a process with a subject? Ricardo's exclusive concern with the first question has the necessary consequence, as Marx points out in these Notes, that he can find no connection between his theory of value and the nature of money. Ricardo does not see that money need not be a commodity for generalised commodity production and money (including paper money) as a universal equivalent to be the effects of one and the same social structure, i.e., the capitalist mode of production. Ricardo confined his discussion of money to specie and regarded the value of coin as being equal to the value of the labour embodied in it. In this instance at least, two distinct features of the mode of production are reduced to expressions of labour, the activity of a subject, whereas for Marx the value form and the money form are distinct effects of the mode of generalised commodity production.
Wagner derives exchange-value and use-value from the concept of value. The so-called concept of value is derived by Wagner from 'Man's' natural drive to evaluate ( _schatzen_ ) things of the external world _qua_ goods, i.e., use-values. Wagner goes on to specify the mode of his derivation: 'One starts from the need and the economic nature of man, reaches the concept of the good, and links this to the concept of value.' Marx characterises this mode of derivation as follows: 'Now one can, assuming one feels the "natural drive" of a professor, derive the concept of value in general as follows: endow "the things of the external world" with the attribute "goods" and also "endow them with value" by name.' Marx goes on to point out that, 'But insofar as "attributing value" to the things of the external world is here only another form of words for the expression, endowing them with the attribute "goods", the "goods" themselves are absolutely not attributed "value" as a determination different from their "being goods" as Wagner would like to pretend.'
In other words, Wagner has set himself the task of excluding "use value" from science. He manages this by a play on words. He derives the term value from the notion of goods, i.e., use-values, and then substitutes the term value for use-value. Wagner's reading transforms the two distinct concepts of the scientific discourse of _Capital_ — value and use-value — into two words that are interchangeable with each other. What is the means of this transformation (or alternatively, what is the problematic that governs Wagner's reading of _Capital)?_ Marx specifies it as follows:
> 'What lies in the murky background to the bombastic phrases is simply the immortal discovery that in all conditions man must eat, drink, etc. (one can go no further: clothe himself, have knives and forks or beds and housing, for this is not the case in all conditions); in short, that he must in all conditions either find external things for the satisfaction of his needs pre-existing in nature and take possession of them, or make them for himself from what does pre-exist in nature; in this his actual procedure he thus constatnly relates in fact to certain external things as "use-values", i.e., he constantly treats them as objects for his use; hence use-value is for Rodbertus a "logical" concept; therefore since man must also breathe, 'breath' is a "logical" concept, but for heaven's sake not a "physiological" one'.
In fact, Wagner's problematic is nothing but the problematic of Philosophical Anthropology, i.e. the Feuerbachian-humanist problematic of the early Marx. The characteristic features of this problematic can be schematically enumerated as follows:
> (i) History is a process with the subject 'Man'.
>
> (ii) The subject 'Man', his species-being in the terminology of Feuerbach and the early Marx, is endowed with certain attributes, e.g. he consumes, produces, creates, etc. These attributes, alternatively referred to as the predicates of the subject, constitute the essence of Man. The relation of the subject to its essence can vary within the problematic of Philosophical Anthropology between idealism of the essence and empiricism of the subject on the one hand, and idealism of the subject and empiricism of the essence on the other.
>
> (iii) The banal notion of alienation signifies the relation between the subject, the essence and the alien object.
Alienation signifies the embodiment of the essence into the alien object and the reversal of the relationship between subject and objects, subject and predicates. The following are the immediate effects of the problematic in economic theory:
> (i) Consumption is always the consumption by the species-being 'Man' and not consumption by the supports (Trager) of the relations of production.
>
> (ii) Production is always a relation between Man and nature and not a relation between communal labour (or collective labour) and nature.
The problematic of Philosophical Anthropology, as I have already pointed out, is not, however, restricted to Wagner's _Lehrbuch_. Wagner emphasises the anthropology of consumption, while others focus on the anthropology of production (the _homo faber_ etc.); but in either case, the same subject 'Man' appears under a different mask determined by the variant of the problematic. This same problematic even appears in Maurice Dobb's introduction to the new English translation of _A Contribution to the Critique of Political Economy_ , where it is particularly pernicious because of the trade-mark under which it is marketed — i.e., as an introduction to Marx by a Marxist economist. Dobb specifies Marx's problematic as follows:
> 'It is sometimes said that, whereas for Hegel the dialectic as a principle and structural pattern of development started from abstract Being as Mind or "Spirit", for Marx the dialectic of development started from Nature, and from Man as initially an integral part of Nature. But while part of Nature and subject to the determination of its laws, Man as a conscious being was at the same time capable of struggling with and against Nature — of subordinating it and ultimately transforming it for his own purposes.' (Dobb, 1971, p. 7.)
Further specific effects of the problematic of Philosophical Anthropology need to be pointed out. In the beginning of these Notes, Marx points out that 'Wagner does not distinguish between the concrete character of each kind of labour and the expenditure of labour power common to all these concrete kinds of labour.' If production is treated as the generic activity of 'Man' to satisfy his 'generic needs', then the determinate historical conditions in which labour, i.e., specific kinds of labour, is performed become invisible. The distinction between concrete labour and abstract social labour rests on the following two constituents of the conceptualisation of the process of production:
> (i) production is always production of a specific good;
>
> (ii) production qua production always takes place under determinate historical conditions.
These two aspects of the process of production are aptly specified in the _1857 Introduction:_
> 'Just as there is no production in general' (production always takes place under determinate historical conditions), 'so also there is no general production' (production is always a production of specific products). (Op. cit., p. 196-7.)
The concept of concrete labour refers to the fact (a fact which is not an empirical given but a construct of the general theory of modes of production) that labour is employed in the production of a specific product, while the concept of abstract social labour refers to the fact that labour is performed under specific historical conditions (or as Marx puts it in these Notes, 'the process of making a thing has a social character').
The distinction between abstract social labour and concrete labour is the unseen of the problematic of Philosophical Anthropology, since that problematic, by putting 'Man' in perpetual communion with Nature, suppresses the theoretical preconditions for specifying the determinate historical conditions in which production takes place. Faced with the patent presence of the verbal distinction in _Capital_ , more careful readers than Wagner within this same anthropological problematic reduce it to a relation of 'alienation': labour power being a commodity in the capitalist mode of production, the concrete labour of human beings is 'fetishised' in the labour market into the alien form of abstract social labour. But this interpolation of 'reified' forms between 'Man' and Nature does not alter the misrecognition of the place of the relation between abstract social labour and concrete labours in the theory of the mode of production expounded in _Capital_.6
The problematic of Philosophical Anthropology also enables Wagner to import universal ethical standards into his discourse. On the basis of such standards ('Thou shalt not steal', etc.), Wagner equates the extraction of surplus-value under the capitalist mode of production with robbery. Such importations of ethical standards into political economy are not confined to Wagner. Joan Robinson, in _An Essay on Marxian Economics_ , writes:
> 'Marx's method of treating profit as unpaid labour and the whole apparatus of constant and variable capital and the rate of exploitation keep insistently before the mind of the reader a picture of the capitalist process as a system of piracy, preying upon the very life of the workers. His terminology derives its force from the moral indignation with which it is saturated.' (Op. cit., p. 22.)
Wagner is an apologist for capitalism, Joan Robinson a critic of it, but their respective readings of the concepts of variable and constant capital and the mode of extraction of surplus value in the capitalist mode of production are exactly the same. In these Notes, Marx makes the following comment on Wagner's reading:
> 'Now in my presentation profit on capital is in fact also not "only a deduction or 'theft' from the labourer". On the contrary, I represent the capitalist as the necessary functionary of capitalist production, and indicate at length that he does not only "deduct" or "rob" but enforces the production of surplus-value and thus first helps to create what is to be deducted; I further indicate in detail that even if in commodity exchange only _equivalents_ are exchanged, the capitalist — as soon as he has paid the labourer the real value of his labour power — quite rightfully, i.e., by the right corresponding to this mode of production, obtains surplus-value.'
Note that what is at issue in Marx's comment is not the 'inhuman' effects of the extraction of surplus-value, i.e., of exploitation under the capitalist mode of production (e.g., the lengthening of the working day, disregard for the physical safety of the workers, etc.), but the right of expropriation corresponding to the capitalist mode of production, a right which receives superstructural representation in legal property rights.
While specifying and criticising Wagner's anthropological problematic, Marx also reveals the problematic governing _Capital_ itself. Numerous comments interspersed throughout these Notes are unmistakable symptoms of Marx's problematic. To cite a few examples:
> 'According to Herr Wagner, use-value and exchange-value should be derived _d'abord_ from the concept of value, not as with me from a concrete entity the commodity ( _konkretum der Ware).'_ (As we shall soon see, this _'konkretum der Ware'_ is not the simple empirical presence of the commodity but the historical condition of existence of commodities.) 'Man, if this means the category "Man", then in general he has no needs.'
>
> 'Hence our _vir obscurus_ , who has not even noticed that my analytic method, which does not start from man but from the economically given period of society, has nothing in common with the German professorial concept-linking method.'
>
> 'The labour process, as purposeful activity for the provision of use-values etc. "is equally common to all its" (human life's) "forms of society" and "independent of each of the same". Firstly the individual does not confront the word "use-value", but concrete use-values, and which of these "confront" ( _gegenuberstehen_ ) him (for these people everything "stands" ( _steht_ ), everything pertains to status ( _Stand)_ ), depends completely on the stage of the social process of production, and hence always corresponds to "a social organisation".'
These last three quotations irrefutably point to a complete break with all the variants of Philosophical Anthropology. 'Man in general has no needs', implies the break with the anthropology of consumption; there are no 'generic needs' of the 'species-being' Man. Needs of concrete individuals are always needs in a determinate historical totality. Further on Marx points out that 'an individual's need for the title of Professor or Privy Counsellor, or for a decoration, is possible only in a quite specific "social organisation".'
However, these Notes do not merely give a symptomatic indication of the theoretical terrain of _Capital;_ they go on to specify the order of the discourse and the theoretical function of specific concepts. The starting-point of economic discourse is indicated in a descriptive form at the beginning of _Capital:_
> 'The wealth of those societies in which the capitalist mode of production prevails presents itself as "an immense accumulation of commodities," its unit being a single commodity. Our investigation must therefore begin with the analysis of a commodity.' ( _Capital_ , I, p. 43.)
The Notes on Wagner, however, specify the beginning of economic discourse in the following terms: 'What I start from is the simplest social form in which the labour product is represented in contemporary society, and this is the "commodity".' The descriptive formulation of _Capital_ has been replaced by a formulation based on the fundamental concepts of the general theory of modes of production. To elaborate: the labour-product, i.e., the end-product of economic practice, is represented in a 'social form' because, as I have pointed out above there is no 'production in general' and production always takes place under determinate historical conditions. The representation of the labour-product in a social form is the effect of the determinate historical conditions in which production takes place. The term 'contemporary society' here does not signify society in its immediate 'actuality' but the abstract concept of the existing society or social formation. Elsewhere in these Notes, Marx specifies this:
> 'If one is concerned with analysing the commodity — the simplest concrete entity — all the considerations that have nothing to do with the immediate object of analysis have to be put aside.'
Thus the 'konkretum der Ware' referred to above denotes the determinate historical conditions in which the labour product is represented as a commodity. Marx's statement in these Notes ( _De prime abord_ I do not start from "concepts" and hence do not start from the "concept of value"'), does not counterpose thought constructs or 'concepts' to 'real facts', but counterposes the 'concepts' specific to the problematic of Philosophical Anthropology to the concepts of 'Historical Materialism'. Marx does not start from the concept of 'value', because he had discarded the problematic of Philosophical Anthropology. He starts from the 'concepts' that underlie the statement: 'What I start from is the simplest social form in which the labour-product is represented in contemporary society.'
Later in the same passage, Marx specifies that while analysing the commodity in the form in which it appears he finds that it is on the one hand a 'use-value' and on the other hand a bearer of 'exchange-value'. Marx is not content with the dual representation of the commodity, but goes on to specify that exchange-value is only a 'phenomenal form' ( _Erscheinungsform_ ), an independent mode of representation ( _selbstandige Darstellungsweise_ ) of value. As I pointed out above, it is only under a specific mode of production that exchange is the mode of production of _'value'_. Hence Marx's statement that exchange-value is a historical 'concept', i.e., the concept 'pertinent to' a specific mode of production. The specification of the relation between exchange-value and value leads Marx to modify his representation of the commodity: 'I say specifically...
> "When, at the beginning of this chapter, we said, in common parlance, that a commodity is both a use-value and an exchange-value, we were, accurately speaking, wrong. A commodity is a use-value or object of utility, and a 'value'".'
The commodity is represented as a two-fold thing because the mode of representation of value is distinct from the natural form of the commodity, i.e., the form qua use-value. It should be pointed out that the mode of representation of value (exchange-value) is distinct from value. Hence some of the ambiguous sentences in _Capital_ which bourgeois commentators on Marx rely so heavily on have to be modified accordingly, for example, the following sentence from Chapter 3 of Volume I which is quoted by Robinson: 'Price is the money-name of the labour realised in a commodity.' ( _Capital_ , I, p. 103.)7
The 'value' of a commodity, as Marx points out in these Notes, expresses in a historically developed form something which also exists in every other historical form of society, but in different forms, namely the social character of labour, insofar as the latter exists as the expenditure of 'social' labour power. The substance of value, which, claims Marx in these Notes, Rodbertus, like Ricardo, does not understand, is the 'common character of the labour process'. What is it that gives the labour process a 'common character'? It is the 'relation' between production and consumption, and the concept of that 'relation' in Marx is the 'mode of distribution' of the labour product. If the 'mode of distribution' (which can take different forms, depending on the mode of production) is such that the producer of a good and the consumer of that good are not identical ( _identitas indiscemibilium_ ), then the labour employed in the production of goods has the common character referred to above. In the illustration Marx cites in these Notes, the primitive community is described as the common organism of the labour powers of its members because of the combination of the mode of production with a mode of distribution such that the producer and the consumer of a good are not identical. The capitalist mode of production has a mode of distribution specific to it which is distribution by means of the exchange of equivalents. A substantial part of the much mis-read section of Chapter 1 on 'The Fetishism of Commodities' is concerned with the elaboration of the mode of distribution of commodities, but the discussion there is conducted in terms of 'inter-personal' relations, terms which provide ample scope for the misrecognition of the object of analysis. The Notes on Wagner, however, are completely free of the misleading formulations of the substance of value to be found in Chapter 1 of _Capital_. To give an example, the substance of value is specified in _Capital_ as follows:
> 'Betrachten wir nun das Residuum der Arbeitsprodukte. Es ist nichts von ihnen ubriggeblieben als dieselbe gespenstige Gegenstandlichkeit, eine blosse Gallerte unterschiedloser menschlicher Arbeit, d.h. der Verausgabung menschlicher Arbeitskraft ohne Rucksicht auf die Form ihrer Verausgabung.' ( _Das Kapital_ in Marx-Engels, _Werke_ , Bd. 23, p. 53.) ('Let us now consider the residue of the labour-product. Nothing remains but this phantomnlike objectivity, a mere gelatinous mass of indistinguishable labour, i.e. of human labour power expended regardless of the form of its expenditure.' (Compare with _Capital_ , I, p. 46.)
The substance of value is abstract social labour-abstract because it is labour power expended regardless of the form of its expenditure, social because of the common character of the labour process in the sense referred to above. As Marx argues in _Capital_ ,
> 'Magnitude of value expresses a relation of social production, it expresses the connection that necessarily exists between a certain article and the portion of the total labour-time of society required to produce it.' (Op. cit., Vol I, p. 104.)
The value of a good (not necessarily of a commodity, since the concept of value is not specific to the capitalist mode of production) represents the expenditure of social labour power because the labour-process has the 'common character' we have discussed. The law of value is thus the law of the distribution of the social labour force into different branches of production. In other words, the law of value specifies the relation between abstract social labour and concrete labour; Marx defines concrete labour on the basis of the branch of production in which the labour is employed. He defines the law of value in _Capital_ in the following terms:
> 'The different spheres of production, it is true, constantly tend to an equilibrium: for, on the one hand, while each producer of a commodity is bound to produce a use-value, to satisfy a particular social want, and while the extent of these wants differ quantitatively, still there exists an inner relation which settles their proportion into a regular system, and that system is one of spontaneous growth; and, on the other hand, the law of value of commodities ultimately determines how much of its disposable working time society can expend on each particular class of commodities'. (Vol. I, p. 336.)
The distribution of the social labour force into the various branches of production in the capitalist mode of production is determined by the following:
> (i) the mode of consumption specific to the mode of production;
>
> (ii) the rate of exploitation, i.e., the necessary and surplus portions of social labour time;
>
> (iii) the forces of production, which determine the composition of the means of production in each branch of production — the 'inner relation which settles their proportion into a regular system' referred to by Marx is the detailed matrix of the production of commodities by means of the commodities of Department I, i.e., those that constitute constant capital, and labour;
>
> (iv) and the form of reproduction.
Each of these factors determines the distribution of the social labour force between Departments I and II, and between the branches of production constituting those Departments. The law of value expresses the 'over-determination' of the distribution of the labour force into different branches of production, assuming that labour is paid the full value of its labour power (Marx sees this assumption as a scientifically necessarily procedure, as he remarks in these Notes, whereas Schaffle saw it as 'generous' and others, e.g. Samuelson and Joan Robinson, have believed that Marx subscribed to the so-called 'theory of immiseration'). The factors listed above in a general form determine the distribution of the social labour force in the capitalist mode of production and are specific to that mode. Hence Marx's exclamation in these Notes, 'What a dreadful thing for the "social state"' (i.e., the future socialist society which Schaffle kindly constructed for Marx), 'to violate the _laws of value_ of the capitalist (bourgeois) state.'
Thus it comes as no surprise that Marx affirms in these Notes that 'price formation makes absolutely no difference to the determination of value.' The connection between the law of value and the formation of prices can be formulated as follows. In Volume III, the 'prices of production', i.e. the set of prices that equalise the rate of profit in all branches of production, assuming a given rate of exploitation, are determined on the assumption that the social labour force is distributed such that each branch of production produces no more nor less than the amount demanded of the goods in question, _qua_ means of production or consumption. 'Prices of production' are thus determined by the 'rate of exploitation' and the forces of production, which, as we have seen, define the 'matrix' of the production of the commodities. 'Prices of production' cannot be realised if there is an imbalance between branches of production, i.e. any branch of production producing more or less than the amount demanded of that particular good. The precondition for the realisation of 'prices of production' obtain if and only if the social labour force is distributed in such a way that there is a balance between different branches of production. The relation of interdependence between the distribution of the social labour-force into different branches of production and the quantitative composition of those branches of production is clear once it is taken into account that each product is the product of a series of concrete labours.
Hence there is no inconsistency between the analyses of Volumes I and III of _Capital_ , despite the allegations of Bohm-Bawerk and _tutti quanti_. As these Notes make clear, the analyses of Volume I are based on abstract labour, labour as the expenditure of labour power irrespectively of the useful way in which it is expended. In consequence the analysis of the process of production in Volume I does not refer to any specific branch of production, despite all the concrete illustrations. The problem of the determination of prices, as a theoretical problem, arises only when a distinction is made between different branches of production. This is the justification for the assumption that price is equal to value, an assumption which is removed in Volume III, where the determination of prices is posed as a theoretical problem. This assumption and its subsequent removal do not represent any contradiction but instead 'the order of presentation' of the discourse of _Capital_.
In the Notes on Wagner, Marx suggests the answer to the following important question: Why is value represented in a 'social form' distinct from the natural form of the labour product, i.e., its form _qua_ use-value? _Qua_ product of social labour one good is indistinguishable from another, the distinction between goods being based on their respective attributes _qua_ means of consumption or production, or in short _qua_ their use-values. As Marx points out:
> 'If he (Rodbertus) had further investigated value, he would have found further that in it the thing, the "use-value", counts as a mere _objectification_ of human labour, as an expenditure of equal human labour power, and hence that this content is represented as an objective character of the thing, as a (character) which is materially fitting for itself, although this objectivity does not appear in its natural form (but this makes a special value-form necessary).'
Marx had already answered this question by his use of illustrations in Chapter 1 of _Capital_ I.
> 'In the production of the coat, human labour power, in the shape of tailoring, must have been actually expended. Human labour is therefore accumulated in it. In this aspect the coat is a depository, but though worn to a thread, it does not let this fact show through.') (p. 58.)
The independent value-form or, in other words, the representation ( _Darstellung_ ) of value is not specific to the capitalist mode of production; it is the necessary effect of the 'common character of the labour process'. The specification of the mode of representation ( _Darstellungsweise_ ) proper to each different mode of production (including the socialist mode of production) remains an unfinished theoretical task for historical materialism.
I hope that, notwithstanding the sketchiness of some of these arguments, of which I am well aware, I have succeeded in demonstrating the theoretical importance of the Notes on Wagner. The specific points of importance can be listed schematically as follows:
> (i) an irrefutable proof of the epistemological break with all variants of Philosophical Anthropology;
>
> (ii) an unmistakeable absence of Hegelian modes of expression in discussing the concept of value (this last point is of particular importance, for in _Capital_ itself, as Marx wrote in his Afterword to the Second German Edition (1873), 'I... openly avowed myself to be the pupil of that mighty thinker (Hegel) and even here and there, in the chapter on the theory of value, _coquetted_ with the modes of expression peculiar to him');
>
> (iii) valuable indications as to the 'order of discourse' in _Capital;_ and
>
> (iv) a specification of the theoretical function of the concept of 'value' and of the nature of the relation between 'the formation of value' and 'the formation of prices'.
Notes
1.The verse can be translated as follows:
'Here must all distrust be abandoned, all cowardice must here be dead'. (Dante Alighieri, _The Divine Comedy_ , Inferno, III, 14-15.)
2.'It would be inexpedient and wrong therefore to present the economic categories successively in the order in which they have played the dominant role in history. On the contrary, the relation of succession is determined by their mutual relations in modern bourgeois society and this is quite the reverse of what appears to be natural to them or in accordance with the sequence of historical development.' ( _1857 Introduction_ , in _A Contribution_... , op. cit., p. 213). Note that the emphasis is on the presentation of economic categories in the sequence determined by the mutual relation of those categories in modern bourgeois society. The discussion of value precedes the analysis of the formation of exchange-values or prices of production because of the theoretical relation postulated. Exchange-value is a mode of representation ( _Darstellungsweise_ ) of value. Analysis of the 'order of the discourse' might seem trite or pedantic. So-called 'history of ideas' fails to ask questions about the order of discourse because it implicitly or explicitly subscribes to the empiricist theory of knowledge, according to which the distinction between the order of the discourse and the order of concrete events is not a pertinent one. But once the thought object is distinguished from the real object, this distinction between 'the two sequences' becomes a crucial one.
3.This interpretation was unfortunately lent weight by a remark of Engels in the _Supplement to Capital Volume III_ , that 'the Marxian law of value holds generally... for the whole period of simple commodity production, that is, up to the time when the latter suffers a modification through the appearance of the capitalist form of production' (Vol. III, p. 876). For a more detailed critique of this passage, and of historicist interpretations which rely on it, see Ranciere, 1965. By historicist here, I mean those whose discourse is governed by the problematic of a 'process with a subject'.
The main effects of a historicist problematic are as follows:
(1) History, regardless of its specific forms, is always governed by the same organising principle. For example, history is the history of the struggle of 'Man' with nature, or the history of 'challenges' and responses. (ii) Given the presence of a single organising principle, the historicist problematic suppresses the concepts of the pertinent distinction between one social formation and another, as a necessary effect. The absence of these concepts of pertinent difference in the historicist discourse is represented in the equivalence of 'historical' and 'physical' time. (iii) The historicist problematic is always blended with either empiricism or idealism. The political effects of the historicist problematic take the form of 'reductionism', e.g. economism or ultra-left adventurism. There are many different variants of historicism.
4.Eugen von Bohm-Bawerk was an Austrian economist of the marginalist school. His book _Karl Marx and the Close of his System_ (1896) is based on the alleged contradiction between the analyses of Volumes I and III (see below). Most bourgeois commentators still regard Bohm-Bawerk's critique as a definitive refutation of Marx. See Eugen von Bohm-Bawerk: _Karl Marx and the Close of his System_ (ed. P M Sweezy), Augustus M Kelly, New York 1966 — this translation includes Hilferding's reply to the critique. For Joan Robinson, see her book, _An Essay on Marxian Economics_ , Macmillan, London 1967.
5.In his introduction to _A Contribution_... , Dobb writes, 'The historical perspective from which he (Marx) surveyed the emergent "bourgeois" (capitalist) society of his day at once sets the distinctive focus and emphasis of his economic theory as well as its boundaries (both focus and boundaries which differentiate it sharply from the increasingly narrowed theories of "market equilibria" that were to characterise accepted economic theory at the end of the century and in the present century)' (op. cit., p. 6).
6.I am forced here to link with Adolph Wagner the name of as serious a Marxist scholar and theoretician as Lucio Colletti: 'In the production of commodities,... where social labour is presented as _equal_ or _abstract_ labour, the latter is not merely calculated irrespective of the individual and concrete labours, but also acquires a distinct existence independent of them... This _abstraction_ of labour from the concrete labouring subject, this acquisition of its independence from man, culminates in the form of the modern wage labour... etc.' (Colletti, 1970).
7.Ibid., p. 14. Joan Robinson reads in _Capital_ what she wants to read rather than what is there to be read. On the page following the one from which this quotation is taken, Marx goes on: 'Magnitude of value expresses a relation of social production, it expresses the connection that necessarily exists between a certain article and the portion of the total labour-time of society required to produce it... The possibility, therefore, of quantitative incongruity between price and magnitude of value, or the deviation of the former from the latter, is inherent in the price-form itself' (p. 102). Robinson never asks how on earth the sentence 'price is the money name of the labour realised in a commodity' implies that price is determined by the magnitude of value. (It should be pointed out that these quotations-the one cited by Joan Robinson and the two cited in this footnote-appear in two different paragraphs in the English edition, but in a single one in the German edition: i.e. they constitute part of the same argument. See _Das Kapital_ in Marx-Engels: _Werke_ , Bd. 23, pp. 116-7.)
Bibliography
Althusser, L, 'Preface to _Capital_ Volume One', in _Lenin and Philosophy and Other Essays_ , New Left Books, London, 1971.
von Bohm-Bawerk, E _Karl Marx and the Close of his System_ (ed. P Sweezy), Augustus Kelly, New York, 1966.
Colletti, L _Ideologia e Societa_ , Laterza, Bari, 1970.
Dobb, M, 'Introduction', to _A Contribution to the Critique of Political Economy_ , Lawrence and Wishart, London, 1971.
Marx, K, _A Contribution to the Critique of Political Economy_ , Lawrence and Wishart, London, 1971.
Marx, K _1857 Introduction_ , in _A Contribution_... , op. cit.
Marx, K _Capital_ , Lawrence and Wishart, London, 1974.
Marx, K _Das Kapital_ , in Marx-Engels, _Werke_ , Bd. 23.
Marx, K, 'Marginal Notes on Adolph Wagner's "Lehrbuch der politischen Okonomie" ', translated in _Theoretical Practice_ , No. 5, Spring 1972, pp. 40-64.
Ranciere, J, 'The Concept of Critique and the Critique of Political Economy', _Theoretical Practice_ , No. 2, April 1971, p. 37–47.
Robinson, J _An Essay on Marxian Economics_ , Macmillan, London 1967.
## _MARX'S THEORY OF MARKET VALUE_
## _Makoto Itoh in collaboration with Nobuharu Yokokawa_
The Problems in Marx's Theory of Market-Value
After transforming values of commodities into prices of production ( _Capital_ , III, chapter 9), Marx goes on to discuss market-value in Chapter 10, under the heading: 'Equalisation of the General Rate of Profit Through Competition. Market Prices and Market-Values. Surplus Profit'. Let us first examine the major contents of this complex chapter. It starts off with a review of the logical relation between values and prices of production. According to Marx's transformation procedure,
> 'In the case of capitals of average, or approximately average, composition, the price of production is... the same or almost the same as the value, and the profit the same as the surplus-value produced by them. All other capitals, of whatever composition, tend toward this average under pressure of competition.' ( _Capital_ , III, p. 174.)
Therefore
> 'the sum of the profit... must equal the sum of the surplus value, and the sum of the price of production... equal the sum of its value'. ( _Capital_ , III, p. 173.)
Then Marx suggests:
> 'The really difficult question is this: how is this equalisation of profit into a general rate of profit brought about, since it is obviously a result rather than a point of departure?' ( _Capital_ , III, p. 174.)
Marx seems to answer this question near the end of this chapter, observing that it is capitalist competition which equalises different rates of profit in value terms into a general rate through redistribution of capital. We read:
> 'If the commodities are sold at their values, then, as we have shown, very different rates of profit arise in the various spheres of production, depending on the different organic composition of the masses of capital invested in them. But capital withdraws from a sphere with a low rate of profit and invades others, which yield a higher profit. Through this incessant outflow and influx, or, briefly, through its distribution among the various spheres, which depends on how the rate of profit falls here and rises there, it creates such a ratio of supply to demand that the average profit in the various spheres of production becomes the same, and values are, therefore, converted into prices of production'. ( _Capital_ , III, p. 195.)
Should we understand from this exposition that the allocation of dead and living labour regulated by capital will be changed when values are converted into prices of production? If the equilibrium ratio of supply and demand of each commodity under value relations differs from that under prices of production, can we still regard the former as an actual framework for the analysis of capitalist economy, not as a mere imaginary assumption without any actuality? These problems throw us back to a more fundamental point, i.e. how to prove the real relevance of the equal exchange of abstract labour embodied in commodities. Marx refers to this point just after his question about the formation of a general rate of profit, by asking
> 'how does this exchange of commodity at their real values come about?' ( _Capital_ , III, p. 175.)
To answer this, Marx presents a model of exchange by simple commodity producers where
> 'the labourers themselves are in possession of their respective means of production and exchange their commodities with one another'. ( _Capital_ , III, p. 175.)
and he proceeds to make a famous statement that
> 'the exchange of commodities at their values, or approximately at their values, thus requires a much lower stage than their exchange at their prices of production, which requires a definite level of capitalist development.' ( _Capital_ , III, p. 177.)
Engels in the 'Supplement to Capital, Volume III', and then Rudolf Hilferding in his anti-critique of Bohm-Bawerk (See Sweezy (ed.), 1949), extended this view and asserted a historical-logical transformation theory from values into prices of production. However, the historical-logical transformation theory could not be a final solution. First, simple commodity producers cannot dominate a whole society, unlike capitalist producers, and as a result their exchange relations are not necessarily regulated by _socially_ necessary labour expenditures. Secondly, Marx's _Capital_ from Part III, Volume I onwards, clearly analyses capitalist production, and not a pre-capitalist economy, on the basis of the law of value. Marx's treatment of cost prices in his theory of prices of production also remained incomplete just as Bortkiewicz pointed out. (See Sweezy (ed.), 1949).
In order to overcome these transformation problems, I believe that it is essential to clarify and to utilise Marx's original distinction of the forms and the substance of value. We have to observe the prices of production as a developed form of value, and study how they are determined by the dimensionally different quantity of abstract labour time embodied in commodities, as the social substance of value. The role of prices of production in mediating the social distribution of the labour amounts also has to be clarified. This perspective has been elaborated elsewhere (Itoh, 1976), and will not be repeated here. For our topic is not transformation problems as such. But as we shall see later, it is important to review Marx's theory of market-value in this context, considering its logical relation with the proper theory of prices of production.
After reconsidering the case of the exchange of commodities at their real values, Marx then moves on to investigate how the unique market-value is determined in the case where the individual values of the same kind of commodities are unequal because of the differences in their conditions of production. This investigation of market-value occupies the major portion of this chapter. Marx's attempt to formulate a theory of market-value, however, was not fully completed. In particular, he seems to leave us with two contradictory theories.
One of them defines market-value as determined by the conditions of production. For instance, Marx says in this context:
> 'On the one hand, market-value is to be viewed as the average value of commodities produced in a single sphere, and, on the other, as the individual value of the commodities produced under average conditions of their respective sphere and forming the bulk of the products of that sphere.' ( _Capital_ , III, p. 178.)
In this definition the market-value is regarded as the average of different individual values of commodities produced under different conditions of production, or as the individual value of commodities produced under average and dominant conditions of production. This can be called the 'technical average' theory of market-value. Strictly speaking, the average and the dominant (or most common) conditions of production do not always have the equality which it assumes. In this theory, the situation of demand and supply in the market does not play any role in determining the level of market-value, though it causes fluctuations in market prices around the centre of gravity of market-value.
In contrast, Marx's second theory gives demand an important role in determining the market-value. Marx says for example:
> '... if the demand is so great that it does not contract when the price is regulated by the value of commodities produced under the least favourable conditions, then these determine the market-value. This is not possible unless demand is greater than usual, or if supply drops below the usual level... if the mass of the produced commodities exceeds the quantity disposed of at average market-values, the commodities produced under the most favourable conditions regulate the market-value'. ( _Capital_ , III, p. 179.)
In this context,
> 'it is one of the extremes which determines the market value' ( _Capital_ , III, p. 185).
not the technical average condition of production. We can call this the demand and supply theory of market-value.
Throughout chapter 10 of _Capital_ , III, Marx repeatedly states these two different theories. In which direction should we complete Marx's theory of market-value? Or can we unify Marx's intentions expressed in the two theories? Finally, how should we reconcile the theory of market-value with the theory of prices of production? We shall investigate these points by reviewing Japanese debates on this issue.1 We hope that our investigation will clarify an important aspect of Marx's value theory, and also give an essential theoretical foundation for the theory of ground rent which has just begun to attract the attention of western Marxists.
The Technical Average Theory of Market-Value
If the ratio of demand to supply determines the level of market-value, it may obscure the determination of value by the quantity of abstract labour embodied in the production of the commodity, and it may resemble the marginalist demand and supply theory of price. In order to avoid such a position, the majority of Marxists have traditionally preferred Marx's first definition of market-value, and interpreted the market-value as determined by the average labour time technically necessary to produce a given commodity. This type of interpretation is presented for instance by Itsuro Sakisaka and Masahiko Yokoyama in Japanese debates. According to this technical average theory of market-value, changes in the relation between demand and supply can bring about only deviations of market prices from market-values, so long as the conditions of production remain unchanged. For us, this interpretation raises the question of whether Marx simply made a mistake in presenting the second theory of market-value. Or whether we can make consistent Marx's two theories of market-value? Various attempts have been made to answer these questions. Yokoyama (1955) gives one of the most orthodox interpretations (drawing on Rosenberg (1962–64)). According to Yokoyama, Marx's second type of explanation really concerns a case where the market-value is changed by a shift in the ruling technical conditions (op. cit., p. 147). It is only consistent with the first type of explanation where the increased social demand is satisfied overwhelmingly by the increased supply of commodities produced under the least favourable conditions so that the commodity produced under this condition now forms the bulk of the production of the commodity; or, conversely, where over production excludes commodities produced under worse conditions so that the most favourable conditions become those under which the bulk of the commodity is produced. (op. cit., p. 147-9). This is an attempt to give a consistent interpretation to Marx's second theory of market-value from the viewpoint of the technical average theory. This cannot provide a substantial integration of Marx's different views. First, the least or the most favourable condition of production cannot be a single regulator of market value in the technical average theory, in so far as other conditions do still exist. Secondly, this assertion is easily criticised as an arbitrary interpretation, because Marx himself does not refer to the alteration of the proportional weight of the worse or the better conditions of production in his second version of the theory.
In order to make the explanation entirely consistent with the determination of market-value by the technical average of conditions of production, Fumimaru Yamamoto (1962) came up with the ingenious suggestion that the words 'market-values' in Marx's second theory must all be misprints of 'market prices'. If this were true, there is no 'second theory' and Marx's position is simply reduced to the proposition that the alteration of the relation between demand and supply affects only the market prices but not market-values. However, Yamamoto could give no bibliographical evidence for his misprints theory, and perhaps not surprisingly he could obtain no followers for this interpretation.
Yuichi Ohshima (1974) attempts to be less one-sided. He thinks that Marx's first theory should be regarded as the general theory of market-value, whereas the second theory should not be abandoned but located as a special theory to analyse such cases as monopolistic pricing, some aspects of industrial cycles and the logic of differential rent. However, even in this interpretation, the general and the special theories are not integrated. They are just separated into the different cases. And the technical average theory is regarded as the general theory of market-value without taking the role of the market into consideration. In this way, the importance of the considerations discussed in the second version of the theory is still neglected.
Uno's Theory of Market-Value
The first and the second versions of the theory of market-value originally coexisted without any clear inner relation in Marx's own texts. In the first version, Marx defined market-value entirely on the basis of the static combination of conditions of production, without considering the fluctuation of demand and supply in the market. Whereas in the second, Marx seemed to claim that changes in the ratios between demand and supply immediately determined the regulative condition of production for market-value; As a result, in this version the fluctuation of market-values was not easily distinguishable from that of market prices.2 Attempts to merely add together the two theories, or to maintain the first theory as it stands are both more or less unsatisfactory. We have to try to develop the theory in the direction for which Marx was searching in his dual notion of market-value.
The notion of _market_ -value should not be a merely static and technical definition of value, but it should be related to the dynamic of the market. At the same time, market-value must be presented as the regulator of market prices through the fluctuations of the market. A complete notion of market-value must satisfy these requirements. From such a point of view, Kozo Uno attempted a more substantial reconstruction of Marx's dual theory, by suggesting the notion of market-value as 'social value determined through the mediation of market'. (Uno, 1950-52, Vol. 2, p. 90.) To quote Uno further,
> 'The market-value as the gravitating centre of market price is determined on the basis of an equilibrium of demand and supply. This means that the supply of a commodity increases in relation to the demand for it when the market price raises above the centre, and decreases in the reverse case. Thus, the determination of the market-value of a commodity depends upon the condition of production under which the supply of the commodity is adjusted to the fluctuating demand'. (Uno, 1964, p. 159.)
In this view, the motion of demand and supply in the market, observed in Marx's second theory of market-value, is not related to the fluctuations of market price alone. On the contrary, through the fluctuations of the market price, the commodity economy reveals anarchically under what conditions of production the necessary amount of commodities for the social demand is supplied, showing the level of market-value as the centre of the gravitation of market price. In general, there is no reason to suppose that the regulative condition of production will be one of the extremes, 'on the margin'. Of course, the market-value itself also changes when the regulative condition of production changes. However, such a shift of market-value cannot be directly deduced from observing conditions of production in commodity economy, but must be sought out through the anarchical fluctuations of market price. We see here how the commodity economy actually makes the social value apparent via the motion of market competition while various individual values exist corresponding to the different conditions of production.3 At the same time, the theory of market-value shows the adjustment mechanism of the distribution of socially necessary labour to each sphere of commodity production: the regulative (or standard) condition of production in each sphere is revealed through the motion of market prices. Uno's theory of market-value makes clear these important aspects of value theory as an extension of Marx's dual theory of market-value.
Let us proceed further, to the next problem, namely: what are included in the differences of condition of production which should be discussed here in the theory of market-value? Three sorts of differences in production conditions are conceivable. The first is differences in condition of production which appear in the process of technical improvements of the method of production. Secondly, differences in the scale of capital may result in differences in the cost and the conditions of production of the commodity, even on the same technical basis. The third sort relates to the different and restricted natural conditions represented by land.
Clearly, the first sort of difference of production conditions contains substantially the same problem which is discussed in the theory of temporary extra surplus-value in the first volume of _Capital_ , (p. 300–302). This sort of difference appears and disappears from time to time in the process of technical progress. Hence Uno sometimes suggested that this sort of difference should be regarded as a special case in the theory of market-value. Opposing the orthodox technical average theory of market-value, Uno asserted that differences in natural conditions of production (i.e. land) are directly related to the general theory of market-value. He located the theory of differential rent as an extended development, and not as a revision, of the theory of market-value.
But if technical differences in method of production are strictly regarded as a special case, then, does not Uno's theory of market-value come to depend too much upon persisting differences in conditions of production such as the scale of capital or the grade of land? In our opinion, Uno's notion of market-value shows rather the general formal determination of social value via the market, which is broadly common to _all_ three sorts of differences in production conditions. In this reformulation, the mere average of individual values does not define market value. However, we think that the variant on the 'technical average' theory which defines market value as regulated by the technically dominant or most common condition of production is still substantially relevant to this reformulated theory in the first two cases. For the production condition under which the supply of the commodity is adjusted to the fluctuating demand ordinarily appears as the dominant and most common condition, in the case where it relates either to the technical conditions or to the scale of capital. In contrast, in the case of the restricted natural conditions of production in land, the marginal worst condition which is necessary to satisfy the social demand becomes the regulator of market-value. Therefore, the theory of competition among capitals requires here a specific theory of ground rent, which shows the specific social substance of differential rent.
In order to clarify further the nature of the social substance of value in these different cases, we have to investigate the substance of the extra surplus-value due to uneven technical progress and the substance of the extra profit which is converted into differential rent. The technical average theory interpreted the extra surplus-value acquired by capitalists with superior conditions of production as substantially a transfer of value from the other capitalists in the same sphere operating under production conditions worse than average. However, this interpretation is obviously inapplicable to the case of differential rent, where the worst marginal land regulates the market-value, so that all the commodities produced on better land have a higher market-value than their individual values. In this case, the balance between the market-value and the individual value does not seem to be mutually cancelled by the transfer of the substance of value within the same sphere of production. Hence Marx called this balance which is converted to differential rent 'a false social value'. ( _Capital, III_ , p. 661). It is a difficult problem for the technical average type of market-value theory to explain the social source of this 'false social value'. But before discussing this problem further we shall investigate the logical relation between the theory of prices of production and that of market-value.
Prices of Production and Market-Value
The critical question here is whether to pose the determination of market-value as something quite separate from the determination of price of production. The technical-average theory of market-value makes such a separation because it does not take the role of the anarchic market process into account in the determination of market-value. But in our view there are not two separate mechanisms. Rather there is a single process of competition in which both intra-and inter-sectoral competition play a role and which determines what Marx called 'market prices of production'. ( _Capital_ , III, p. 198). It seems to us clear, both from the title and the structure of Chapter 10, _Capital_ , III, that Marx did intend to relate market-value to price of production.
Uno's theory of market-value helps to clarify this point. According to Uno, the representative condition of production, which determines market-value, is only defined through an anarchic process of intra-sectoral market competition. The pin-pointing of this representative condition of production is necessary for inter-sectoral competition between capitals. Only with reference to such a standard in each sector can profit rates across sectors be compared, and re-allocations of capital tending to equalise those rates take place. At the same time, intra-sectoral competition would be extremely limited and weak without inter-sectoral competition, and thus the latter is a necessary aspect of the definition of market-values.
The theory of prices of production is in a sense more basic than the theory of market-value in developing the law of value as the capitalist law of social reproduction. Nevertheless, the theory of the formation of prices of production through capitalist competition cannot be complete in so far as it lacks a theory of the formation of market-value by competition in each sector. Hence, the theory of market-value should be discussed later than the theory of prices of production, and should be regarded as an integral extension of the theory of prices of production. In this respect, I would like to agree with the suggestion raised by Tsuyoshi Sakurai (1968) and more definitely proposed by Koichiro Suzuki (1962-64) that intra-sectoral capitalist competition should not be discussed as a matter of market-value but as a matter of market price of production from the beginning.
Needless to say, even with an integrated theory of prices of production and market prices of production, we see that capitalist competition to equalise the rate of profit across industrial sectors does not eliminate but necessarily brings about the extra profit to capitalists with better conditions of production than the standard, and therefore representative, conditions in each sector. We can now observe the substance of value obtained in the form of such surplus profit from a new angle. In contrast to the case of the technical average theory of market-value, we need not limit the substantial source of extra profit gained by individual capitals using improved methods of production to the surplus labour extracted in the _same_ sector of industry. The substance of this extra profit can be the transfer of surplus labour extracted in other industrial sectors, just as the substance of some portion of average profit, in the formation of market prices of production consists of transfers of surplus labour from other sectors. At the same time, such a source of extra profit becomes logically conceivable even in the case where capital of worse than the standard condition of production does not exist, and therefore where there is no countervailing transfer of the substance of value within the _same_ industrial sector. This is also true of the substance of the extra profit which is converted into the differential rent. Marx called such a portion of value 'false social value' or 'what society overpays for agricultural products in its capacity as consumer'. ( _Capital_ , III, p. 661). This cannot mean in principle a creation of the substance of value by capitalist competition, nor a deduction from the substance of value of labour power. Therefore, the substance of the differential rent should be regarded as the transfer of a part of social surplus labour to land owners through capitalist competition to determine the market price of production of agricultural products.4
We must certainly clarify the different historical meaning and function of the two kinds of extra profit discussed above. The former, which must be investigated also as the matter of temporary extra surplus value, presented in the first volume of _Capital_ , serves as an incentive to improve methods of production and thus to generate the social production of relative surplus value. As Uno suggests, it may contain the socially necessary labour cost of improving production methods, a cost which is common to more or less all forms of society, and certainly to a socialist society. In contrast, the extra profit which is converted to the differential rent does not have this positive role in increasing productivity, nor does it have a common basis in other forms of society. In that sense, differential rent is simply eliminated under socialism, where the total labour embodied in agricultural products is directly estimated by the actual number of labour hours. The above arguments help to dispel the notions that when labour is combined with improved production methods, it creates temporary extra surplus value as _intensified_ labour; or that 'false social value' in agriculture is _created_ in the process of capitalist competition. By integrating the theory of price of production and the theory of market value into a theory of market price of production, and by distinguishing the form and substance of the latter, we can come to a better understanding of the way the capitalist economy works.
Notes
*The authors would like to thank Sue Himmelweit and Diane Elson for their assistance in clarifying the text and turning it into readable English.
1.It is already known from the English edition of Isaak I Rubin's book, _Essay on Marx's Theory of Value_ , that there was a controversy between Marxian economists about the concept of socially necessary labour in the 1920s in Germany and the USSR. The two versions of the theory of socially necessary labour were summarised by Rubin as follows:
> 'An "economic" concept of necessary labour is that the value of a commodity depends not on the productivity of labour (which expresses that quantity of labour necessary for the production of a commodity under given average technical conditions), but also on the social needs or demand. Opponents of this conception ("technical" version) object that changes in demand which are not accompanied by changes in productivity of labour and in production technique bring about only temporary deviations of market prices from market-values, but not long-run, permanent changes in average prices, i.e., they do not bring about changes in value itself. (Rubin, 1973, p. 185.)
2.Roman Rosdolsky (1977, p. 92), for example, represents a position which is contrary to the orthodox technical average theory, and follows Marx's demand and supply theory of market-value just as it stands. He asserts that market-value is identical with market price within the range of individual values between those of the best and the worst condition of production in the same industry.
3.Though the redefinition of market-value in this way may seem close to Marshallian Marginal theory, it is not in its essence. Unlike the marginalist, we do not take demand for a subjective, individualistic and independent factor which determines the equilibrium price. The fluctuations of demand are to be observed in our view, on the one hand, as a reflection of the anarchical motion of commodity production, and on the other, as the intermediary mechanism revealing the level of social value, which is basically determined from behind by the standard condition of production. In our view, the neo-Ricardians one-sidedly emphasize the technical conditions of production as the determinant of prices, ignoring the role of market competition. The so-called 'indeterminancy of social value' when a commodity is produced under different technical methods with the same cost, which figures in the recent neo-Sraffian critique of Marx, seems at least partly to come from the neglect of such a dynamic role of market competition in revealing the regulative condition of production.
Moreover, our theory of market-value is not a mere formal theory of price like those of the Marginalists or the neo-Ricardians, but also a theory which reveals the relations of labour quantities as the substance of values. Thus, our theory aims at the elucidation of the historically specific form in which, in the (capitalist) commodity economy, differences in labour-time necessary to produce the same sort of good, which arise from differences in production conditions, are related to one another.
4.As Robin Murray (1978) suggests, the surplus labour which is transferred from capitalists to land owners can be within the total surplus labour extracted in the agricultural sector, in so far as the organic composition of capital in agriculture is sufficiently lower than the social average. Such a restrictive condition is not, however, essential for the Marxian principle of differential rent in our view.
Bibliography
Itoh, M (1976), 'A Study of Marx's Theory of Value', _Science and Society_ , 40-43, Fall.
Marx, K (1959), _Capital_ , III, _The Process of Capitalist Production as a Whole_ , Lawrence and Wishart, London.
Murray, R (1978), 'Value and Theory of Rent', Part I, _Capital and Class_ , No. 3.
Ohshima, Y (1974), _Kakaku to Shihon no Riron (Theory of Price and Capital)_ Chapter 7: Shijyo-Kakaku to Shijyo-Kachi (Market Price of Production and Market-Value), Mirai-sha, Tokyo.
Rosdolsky, R (1977), _The Making of Marx's Capital_ , translated by Pete Burgess, Pluto Press, London.
Rozenberg, R I (1962-64), _Comment on K Marx's Capital_ , Moscow 1961, Japanese edition, 5 vols, translated by T Soejima and M Udaka, Aoki-shoten, Tokyo.
Rubin, II (1973), _Essay on Marx's Theory of Value_ , Black Rose Books Ltd., Montreal.
Sakisaka, I (1962), _Marx Keizaigaku no Kihonmondai (The Fundamental Problems in Marxian Economics)_ , Part III, Chapter 3: Shijyo Kachiron to Sotaiteki-Jyoyo-Kachiron (The Theory of Market-Value and the Theory of Relative Surplus Value), Iwanami-shoten, Tokyo.
Sakurai, T (1968), _Seisan Kakaku no Riron (Theories of Price of Production)_ , University of Tokyo Press, Tokyo.
Suzuki, K ed. (1962-64), _Keizaigaku Genriron_ , 2 Vols ( _Principles of Political Economy_ ), University of Tokyo Press, Tokyo.
Sweezy, P ed. (1949) _Karl Marx and the Close of His System_ by Eugen von Bohm-Bawerk and _Bohm-Bawerk's Criticism of Marx_ by Rudolf Hilferding, Kelley, New York.
Uno, K (1950-2), _Keizai Genron_ , 2 vols, ( _Principles of Economics_ ), Iwanami-shoten, Tokyo.
Uno, K (1964), _Keizai Genron (Priciples of Economics)_ , Iwanami-shoten, Tokyo. (We consulted Tomohiko Sekine's translation of this work which is ready for publication in English.)
Yamamoto, F (1962), _Kachiron Kenkyu (Studies on the Theory of Value)_ , Chapter 4: Dai 3-kan dai 10-sho ni okeru "Fumeiryo na Kasho" no Kentou (Studies on the "Ambiguous Parts" of Chapter 10 in the Third Volume), Aoki-shoten, Tokyo.
Yokoyama, M (1955), _Keizaigaku no Kiban (Foundations of Econonomics)_ , Part III: _Marx Kachi-Kakaku-Ron no Kihon Mondai (Fundamental Problems in Marx's Theory of Value and Price)_ , University of Tokyo Press, Tokyo.
## _THE VALUE THEORY OF LABOUR_
## _Diane Elson_
WHAT IS MARX'S THEORY OF VALUE A THEORY OF?
**1. The theory of value: a proof of exploitation?**
Let us first consider the interpretation which is very widespread on the left, particularly among activists, that Marx's theory of value constitutes a proof of exploitation. A good example of this position in CSE debates is that put forward by Armstrong, Glyn and Harrison. Their dogged defence of value rests on the belief that only by employing the category of value can the existence of capitalist exploitation be demonstrated and that to demonstrate this is the point of Marx's value theory:
> Any concept of surplus labour which is not derived from the position that labour is the source of _all_ value is utterly trivial. (Armstrong, Glyn and Harrison, 1978, p. 21.)
Marx does not, however, seem to have shared this view:
> Since the exchange-value of commodities is indeed nothing but a mutual relation between various kinds of labour of individuals regarded as equal and universal labour, i.e. nothing but a material expression of a specific social form of labour, it is a tautology to say that labour is the _only_ source of exchange-value, and accordingly of wealth in so far as this consists of exchange-value... It would be wrong to say that labour which produces use-values is the _only_ source of the wealth produced by it, that is of material wealth. ( _A Contribution to the Critique of Political Economy_ 1, p. 35-36.)
>
> Capital did not invent surplus labour. Wherever a part of society possess the monopoly of the means of production, the worker, free or unfree, must add to the labour-time necessary for his own maintenance an extra quantity of labour-time in order to produce the means of subsistence for the owner of the means of production. ( _Capital_ , I, p. 344.)
Moreover to regard Marx's theory of value as a proof of exploitation tends to dehistoricise value, to make value synonymous with labour-time, and to make redundant Marx's distinction between surplus labour and surplus value. To know whether or not there is exploitation, we must examine the ownership and control of the means of production, and the process whereby the length of the working day is fixed. (See Rowthorn, 1974.) Marx's concern was with the particular _form_ that exploitation took in capitalism (see _Capital_ , I, p. 325), for in capitalism surplus labour could not be appropriated simply in the form of the immediate product of labour. It was necessary for that product to be sold and translated into _money_. As Dobb comments:
The problem for Marx was not to prove the existence of surplus value and exploitation by means of a theory of value; it was, indeed to _reconcile_ the existence of surplus value with the reign of market competition and of exchange of value equivalents. (Dobb, 1971. p. 12.)
The view that Marx's theory of value is intended as a proof of exploitation does, however, have the merit of seeing that theory as a _political_ intervention. The problem is that it poses that politics in a way that is closer to the 'natural right' politics of 'Ricardian socialism' or German Social Democracy, than to the politics of Marx. (See for instance Marx's 'Critique of the Gotha programme', Marx-Engels, _Selected Works_ , Vol.3; also, Dobb, 1973, p. 137-141.) Because of this it has no satisfactory answer to the claim that exploitation in capitalism can perfectly well be understood in terms of the appropriation of _surplus product_ , with no need to bring in value at all. (See for instance Hodgson, 1976; Steedman, 1977.) But in rejecting this interpretation of Marx's value theory we must be careful not to de-politicise that theory. The politics of the theory is a question we shall return to at the end of this paper.
**2. The theory of value: an explanation of prices?**
This approach may be found separately or combined with the one we have just considered. It is the interpretation offered by most Marxist economists in the Anglo-Saxon world, that Marx's theory of value is an explanation of equilibrium or 'natural' prices in a capitalist economy.. As such it is one of a number of theories of equilibrium price, so that, for instance, in Dobb's _Theories of Value and Distribution_ , Marx's theory of value can be examined alongside the theories of Smith, Ricardo, Mill, Jevons, Walras and Marshall, as if it were a theory with the same kind of object. Indeed the main distinction made by Dobb is
> 'between theories that approach the determination of prices, or the relations of exchange, through and by means of conditions of production (costs, input-coefficients and the like) and those that approach it primarily from the side of demand.' (Dobb, 1973, p. 31.)
For Dobb the great divide is between Smith, Ricardo and Marx who are in the first category, and the others, who are in the second. A similar interpretation is offered by Meek:
> 'there is surely little doubt that he (Marx) wanted his theory of value... to do another and more familiar job as well — the same job which theories of value had always been employed to do in economics, that is, to determine prices.' (Meek, 1977, p. 124.)
Of course, it is recognised, within this interpretation, that there are differences between Marx and other economists, even between Marx and Ricardo.
> 'Marx's theory of value was something _more_ than a theory of value as generally conceived: it had the function not only of explaining exchange-value or prices in a quantitative sense, but of exhibiting the historico-social basis in the labour process of an exchange — or commodity — society with labour power itself become a commodity.' (Dobb, 1971, p. 11.)
The way of noting these differences that has become most popular is the distinction between the quantitative-value problem and the qualitative-value problem, introduced by Sweezy. The former is the problem of explaining the quantitative exchange-relation between commodities; the latter is the problem of explaining the social relations which underlie the commodity form. For Sweezy,
> 'The great originality of Marx's value theory lies in its recognition of these two elements of the problem and in its attempt to deal with them simultaneously within a single conceptual framework.' (Sweezy, 1962, p. 25.)
Or as Meek put it,
> 'The qualitative aspect of the solution was directed to the question: why do commodities possess price at all? The quantitative aspect was directed to the question: why do commodities possess the particular prices which they do?' (Meek, 1967, p. 10.)
It is clear that the object of Marx's theory of value is taken, in this tradition, to be the process of exchange or circulation.
> '... the study of commodities is therefore the study of the economic relations of exchange.' (Sweezy, 1962, p. 23.)
Marx is interpreted as explaining this process in terms of a separate, more fundamental process, production. Dobb, for instance, writing an Introduction to Marx's _A Contribution to the Critique of Political Economy_ , suggests that Marx's interest,
> 'is now centred on explaining exchange in _terms of production..._ Exchange relations or market 'appearances' could only be understood... if they were seen as the expression of these more fundamental relations at the basis of society.' (Dobb, 1971, p. 9-10.)
According to Sweezy,
> 'Commodities exchange against each other on the market in certain definite proportions; they also absorb a certain definite quantity (measured in time units) of society's total available labour force. What is the relation between these two facts? As a first approximation Marx assumes that there is an exact correspondence between exchange ratios and labour-time ratios, or, in other words, that commodities which require an equal time to produce will exchange on a one-to-one basis. This is the simplest formula and hence a good starting point. Deviations which occur in practice can be dealt with in subsequent approximations to reality.' (Sweezy, 1962, p. 42.)
It has generally been suggested that this 'first approximation' is maintained throughout the first two volumes of _Capital_ , and relinquished in Volume III, where the category of prices of production is introduced and 'values are transformed into prices.' The adequacy of Marx's 'solution' to the 'transformation problem', and the merits of various alternative solutions have until recently been the chief point of debate in this tradition of interpretation. (No attempt will be made here to review the lengthy literature. For references, see Fine and Harris, 1976.)
In what sense is it held that the labour-time required to produce commodities 'explains' or 'determines' their prices (either as a 'first approximation' or through some 'transformation')? I think two related arguments are deployed in the writings in this tradition. The 'first approximation' of prices to the labour-time required for production is supported by an argument that derives from Adam Smith's example of the principle of equalisation of advantage in a 'deer and beaver' economy. (See, for instance, Sweezy, 1962, p. 45-46.) Suppose we consider two commondities ('deer' and 'beaver'), one of which ('deer') takes one hour to produce, the other of which ('beaver') takes two hours; and suppose that on the market one deer exchanges for one beaver. The argument is that each producer will compare the time it takes him to produce the commodity (in this case by hunting) with its market price, expressed in terms of the other commodity. It is clear that you can get more beavers by producing deer and exchanging than for beaver, than by directly producing beaver. Therefore producers will tend to allocate their time to producing deer rather than beaver. This will increase the supply of deer, reduce the supply of beaver. Other things being equal, this will reduce the market price of deer and increase the market price of beaver. The movement of labour-time from beaver to deer will continue until the market price of deer in terms of beaver is equal to the relative amounts of labour required to produce the two commodities, i.e. until two deer exchange for one beaver. At this point the transfer of labour-time will stop, and the system will be in equilibrium, with prices equal to labour-time ratios.
A more complex argument is deployed to indicate how labour-time determines prices through a 'transformation.' Here labour-time and price of production are related through an equilibrium 'model' of dependent and independent variables. As Meek put it:
> 'In their basic models, all three economists (i.e. Ricardo, Marx and Sraffa) in effect envisage a set of technological and sociological conditions in which a net product or surplus is produced (over and above the subsistence of the worker, which is usually conceived to be determined by physiological and social conditions.) The magnitude of this net product or surplus is assumed to be given independently of prices, and to limit and determine the aggregate level of the profits (and other non-wage incomes) which are paid out of it. The main thing which the models are designed to show is that under the postulated conditions of production the process of distribution of the surplus will result in the simultaneous formation of a determinate average rate of profit and a determinate set of prices for all commodities.' (Meek, 1977, p. 160.)
The magnitude of the net product is measured in terms of the labour time socially required for its production.
The feature of both arguments which it is important to note is that they pose the socially-necessary labour-time embodied in commodities as something quite separate, discretely distinct from, and independent of, price. It is given solely in the process of production, whereas price is given solely in the process of circulation. The two processes are themselves discretely distinct, although they are of course linked. And it is in production that 'the key causal factor', 'the relatively independent 'determining constant" is to be found. (See Meek, 1967, p. 95; Meek, 1977, p. 151.) It follows that we can, in principle, calculate values (i.e. socially-necessary labour-time embodied in commodities) quite independently of prices, and deduce equilibrium prices from those values. The last possibility is often regarded as the indispensable guarantee of the scientific status of Marx's value theory, of its distance from a metaphysical juggling of concepts. (Although, as writers in this tradition generally admit, in practice such a calculation would be impossible to make.)
The reading of Marx as a builder of economic models has been carried to its logical extreme in the recent work of some professional economists, perhaps most notably in the work of Morishima, in which,
> 'the classical labour theory of value is rigorously mathematised in a familiar form parallel to Leontief's inter-sectoral price-cost equations. The hidden assumptions are all revealed and, by the use of the mathematics of the input-output analysis, the comparative statical laws concerning the behaviour of the relative values of commodities (in terms of a standard commodity arbitrarily chosen) are proved. There is a duality between physical outputs and values of commodities, which is similar to the duality between physical outputs and competitive prices. It is seen that the labour theory of value may be compatible with the utility theory of consumers demand or any of its improved variations.' (Morishima, 1973, p. 5.)
All politics is ruthlessly excised in the interests of making Marx a respectable proto-mathematical economist.2
> '(values) are determined only by technological coefficients... they are independent of the market, the class-structure of society, taxes and so on.' (Morishima, 1973, p. 15.)
More important in CSE debates has been the development within this general line of interpretation of an approach which excises not politics as such, but value. Arguing from the same premises as the Sweezy-Meek-Dobb tradition, it has come to the conclusion that,
> 'the project of providing a materialist account of capitalist societies is dependent on Marx's value magnitude analysis _only_ in the negative sense that continued adherence to the latter is a major fetter on the development of the former.' (Steedman, 1977, p. 207; See also Hodgson and Steedman, 1975; Hodgson, 1976; Steedman, 1975a, 1975b.)
The quantity of socially-necessary labour-time embodied in a commodity has been found to be at best redundant to, at most incapable of, the determination of its equilibrium price. The so-called 'Neo-Ricardians' pose instead, as independent variables, the socially-necessary conditions of production and the real wage paid to workers, specified in terms of physical quantities of particular commodities. Unlike Morishima, Steedman does not take such quantities as purely technological: they are assumed to be determined socially and historically and reflect the 'balance of forces' between workers and capitalists in the work place.
There is no doubt that within its own terms this critique of the theory of value, as an explanation of equilibrium prices in terms of labour quantities, is quite correct. Attempts to preserve the traditional Anglo-Saxon version of the theory of value tend to dissolve into positions even more 'Ricardian' than that of the 'Neo-Ricardians' (a point made by Himmelweit and Mohun, 1978). This paper makes no attempt to rescue this traditional 'labour theory of value'. Instead it argues for a quite different reading of Marx's theory of value, in relation to which it is the Sraffa-based critique which is redundant, rather than value.
In some respects even more iconoclastic than the Neo-Ricardians is the work of Cutler, Hindess, Hirst and Hussain. Prefaced by a picture of Christ cleansing the temple, they claim:
> 'It is possible to argue that prices and exchange-values have no _general_ functions or general determinants... Such a change of pertinence of problems would put us not only outside of the Marxist theory of value but also conventional economic theory.' (Cutler et al., 1977, p. 14.)
and declare:
> 'In this book we will challenge the notion that 'value' is such a general determinant' (op. cit., p. 19.)
I too will challenge the notion that value is such a general determinant, in the sense that Cutler et al. understand this, i.e. as a single 'origin' or 'cause' of prices and profits. But my challenge will be directed to the very notion that Marx's theory of value poses value as the origin or cause of anything. Among other things, I shall argue that Marx's concept of a determinant is quite different from those of authors considered in this section.
**3. An abstract labour theory of value?**
It is, of course, by no means original to question whether the 'labour theory of value' discussed in the last section is to be found in the works of Marx, (see for instance Piling, 1972; Banaji, 1976). In recent CSE debates much stress has been placed on abstract labour as a means of differentiating Marx's theory of value from the interpretations so far discussed which are held to apply to Ricardo rather than to Marx. Marx certainly claims that his theory of value differes from that of Ricardo in the attention he pays to the form of labour, and the distinction he introduces between abstract labour and concrete labour. (See for instance, _Theories of Surplus Value_ , Part 2, p. 164, 172.) In _Capital_ we are told that the author,
> 'was the first to point out and examine critically this two-fold nature of labour contained in commodities... this point is crucial to an understanding of political economy.' ( _Capital_ , I, p. 132.)
This point is taken up by Himmelweit and Mohun, 1978, who base their reply to Steedman, 1977, on
> 'a distinction between Ricardian embodied-labour theory of value and a Marxian theory of value based on the category of abstract labour. While the former is intended immediately to be a theory of price, the latter is only so after several mediations.' (op. cit., p. 94.)
They suggest that if we bear this distinction in mind, we shall find that the allegations of redundancy and incoherence, while they apply to Ricardo's theory of value, cannot be sustained for that of Marx.
Their argument is not altogether convincing for two reasons. The first is that Steedman claims to have treated labour as abstract labour and to direct his critique precisely at an abstract labour theory of value (see Steedman, 1977, p. 19), and Himmelweit and Mohun nowhere explicitly confront this claim. Clearly much depends on how the concept of abstract labour is understood. Sweezy, for instance, sees in the concept of abstract labour not an alternative to the concepts of Ricardo and Smith, but a further development and clarification of their work. (Sweezy, 1962, p. 31.) Marx himself did not tend to use 'embodied labour' and 'abstract labour' as if they were opposites, stating for instance that,
> 'The body of the commodity, which serves as the equivalent, always figures as the embodiment of abstract human labour.' ( _Capital_ , I, p. 150.)
The second reason is that their argument becomes circular: they derive the concept of abstract labour from the commodity form, and then wish to use the concept of abstract labour to explain the commodity form (op. cit., p. 73).
In my view the distinction between abstract and concrete labour _is_ an important differentiation between Marx's and Ricardo's theories, but it is not the only differentiation. More fundamental are differences in the object of the theory and the method of analysis. The clarification of these is required before the meaning and significance of the concept of abstract labour becomes apparent.
**4. Labour as the object of Marx's theory of value**
My argument will be, not that Marx's value theory of price is more complex than Ricardo's, but that the object of Marx's theory of value is not price at all. This does not mean that Marx was not concerned with price, nor its relation to the magnitude of value, but that the phenomena of exchange are not the object of the theory. (Again this is not a completely new thought, see Hussain, this volume, p. 84.) My argument is that the _object_ of Marx's theory of value was labour. It is not a matter of seeking an explanation of why prices are what they are and finding it in labour. But rather of seeking an inderstanding of why labour takes the forms it does, and what the political consequences are.
We can see Marx focusing on this question in his first intensive study of Adam Smith ('Economic and Philosophical Manuscripts' in _Early Writings_ , esp. p. 287-9). _The German Ideology_ is a sustained argument for the centrality of this question:
> 'As individuals express their life, so they are. What they are, therefore, coincides with their production, both with _what_ they produce and with _how_ they produce.' (Op. cit., p. 42.)
And in _Capital_ , Marx notes the critical question that separates the direction of his analysis from that of political economy as:
> 'why this content has assumed that particular form, that is to say why labour is expressed in value, and why the measurement of labour by its duration is expressed in the magnitude of the value of the product. These formulas, which bear the unmistakable stamp of belonging to a social formation in which the process of production has mastery over man, instead of the opposite, appear to the political economists' bourgeois consciousness to be as much a self evident and nature-imposed necessity as productive labour itself.' ( _Capital_ , I, p. 174-5.)
Here Marx is signalling, not an 'addition of historical perspective' to political economy, but a difference in the object of the theory, (see also Hussain, this volume, p. 86). It is because labour is the object of the theory that Marx begins his analysis with produced commodities, as being 'the simplest social form in which the labour product is represented in contemporary society.' ( _Marginal Notes on Wagner_ , p. 50); and not, as Bohm-Bawerk claimed, to rig the terms of the explanation of prices (see also Kay, this volume, p. 48-50).
**5. A possible misconception: the social distribution of labour**
The question of why labour takes the forms it does is not simply a _distributional_ question. Here the famous letter to Kugelmann in July 1868 can be very misleading, for Marx writes:
> 'the mass of products corresponding to the different needs require different and quantitatively determined masses of the total labour of society. That this necessity of distributing social labour in definite proportions cannot be done away with by the _particular form_ of social production, but can only change the _form it assumes_ , is self evident. No natural laws can be done away with. What can change in changing historical circumstances, is the _form_ in which these laws operate.' ( _Selected Correspondence_ , p. 251.)
Taken by itself, this letter can lend support to the view that the object of the theory is simply the way in which individuals are distributed and linked together in a pre-given structure of tasks. This view is held by a wide spectrum of writers from the 'Hegelian' I. I. Rubin to the 'anti-Hegelian' Althusser.
For Rubin the theory of value is about the regulation of production in a commodity economy, where 'no one consciously supports or regulates the distribution of social labour among the various industrial branches to correspond with the given state of productive forces.' (Rubin, 1973, p. 77.) From the beginning of his book, Rubin makes it quite clear that the productive forces which constitute the various industrial branches are autonomous products of a material-technical process (Rubin, 1973, p. 1-3). What for him is social is merely the network of links between people in this pre-given structure:
> 'It is also incorrect to view Marx's theory as an analysis of _relations between labour and things_ , things which are the products of labour. The relation of labour to things refers to a given concrete form of labour and a given concrete thing. This is a technical relation which is not, in itself, the subject of the theory of value. The subject matter of the theory of value is the _interrelations of various forms of labour_ in the process of their distribution, which is established through the relation of exchange among things, i.e. products of labour.' (Rubin, 1973, p. 67).
But it is the pre-given structure which has ultimate causal significance:
> 'We can observe that social production relations among people are causally dependent on the material conditions of production and on the distribution of the technical means of production among the different social groups... From the point of view of the theory of historical materialism, this is a general sociological law which holds for all social formations.' (Rubin, 1973, p. 29.)
Clearly there are many differences between Rubin's reading of Marx and that of Althusser, but the latter also invokes the letter to Kugelmann, and writes:
> 'Marx's labour theory of value... is intelligible, but only as a special case of a theory which Marx and Engels called the 'law of value' or the law of the distribution of the available labour power between the various branches of production...' (Althusser, 1977, p. 87.)
or,
> 'the distribution of men into social classes exercising functions in the production process'. (Althusser, 1975, p. 167.)
These 'functions in the production process' are determined by the material and technical conditions of production.
> 'The labour process therefore implies an expenditure of the labour-power of men who, using defined instruments of labour according to adequate (technical) rules, transform the _object_ of labour (either a natural material or an already worked material or raw material) into a useful product... the labour process as a material mechanism is dominated by the physical laws of nature and technology.' (Althusser, 1975, p. 170-1.)
While it is true that such a thesis is 'a denial of every 'humanist' conception of human labour as pure creativity', it is not a denial of, (indeed it positively encourages) a _technicist_ reading of Marx, with potentially disastrous political implications.
What is more immediately important for our consideration of Marx's theory of value is that the technicist reading of the theory, as having as its object the process of distribution of individuals to pre-given places or functions in the production process, tends to lead to a re-introduction of the labour theory of value, albeit in more complex form with reciprocal causality. Not only is labour-time seen as the determinant of exchange-value; exchange-value is also seen as the determinant of labour-time. That is, exchange-values are in equilibrium equal to socially necessary labour-time embodied in commodities; and the distribution of total labour-time between different commodities is regulated by the difference between market price and relative labour-time requirements of different commodities. Rubin in fact presents an exposition of the way in which this works which is practically the same as that of Sweezy. (See Rubin, 1973, chapters 8, 9 and 10; Sweezy, 1962, chapters II and III.)
> 'In a simple commodity economy, the exchange of 10 hours of labour in one branch of production, for example shoe-making, for the product of 8 hours labour in another branch, for example clothing production, necessarily leads (if the shoe-maker and clothes-maker are equally qualified) to different advantages of production in the two branches, and to the transfer of labour from shoe-making to clothing production.' (Rubin, 1973, p. 103.)
The difference is that while Sweezy explicitly acknowledges the provenance of this type of argument in _The Wealth of Nations_ , Rubin claims that he has not repeated 'the mistakes of Adam Smith'. (Rubin, 1973, p. 167.) He claims to differ from Smith in showing that the 'equalisation of advantage' is enforced by an objective social process which compels individuals to behave in this way. But this argument is invalid. There is no social pressure on a simple commodity producer who uses his own or his family's labour (but not hired labour) to compare the different rewards of an hour of labour in different branches of production. (See Banaji, 1977, p. 32 for discussion in the case of peasant agriculture.) It is only capitalists who are _forced_ to account for all labour-time spent in production because they are in competition with other capitalists in the labour market (and all other markets). But capitalists make their calculations in _money_ terms, not by a direct comparison of labour-time with market price, because it is not their own labour-time that they are accounting for.
There is some difference between Rubin's position and Sweezy's position, insofar as the former does not pose value as a category of the production process, whereas the latter does. But this simply means that in Rubin it is the relation between value and exchange-value which is obscured, while in Sweezy (and Meek, Dobb etc.) it is the relation between value and labour-time. What all four authors have in common is a tendency to reduce the categories of the analysis from the three found in Marx's writings (labour-time, value and exchange-value) to two. Rubin identifies value with
> 'that average level around which market prices fluctuate and with which prices would coincide if _social labour_ were proportionately distributed among the various branches of production'. (Rubin, 1973, p. 64);
and thus poses it simply as a category of circulation, and has no systematic distinction between exchange value and value.
Sweezy, Dobb, Meek (and the tradition they represent) identify value with labour-time; for example,
> 'Marx began by defining the 'value' of a commodity as the total quantity of labour which was normally required from first to last to produce it.' (Meek, 1977, p. 95);
and thus pose it simply as a category of production.
Rubin also shares the view that production is a discretely distinct process in which are to be found the 'independent variables' which are of ultimate causal significance.
> '... the moving force which transforms the entire system of value originates in the material-technical process of production. The increase of productivity of labour is expressed in a decrease in the quantity of concrete labour which is factually used up in production, on the average. As a result of this (because of the dual character of labour as concrete and abstract), the quantity of this labour, which is considered 'social' or 'abstract', i.e. as a share of the total, homogeneous labour of the society, decreases. The increase of productivity of labour changes the quantity of abstract labour necessary for production. It causes a change in the value of the products of labour. A change in the value of products in turn affects the distribution of social labour among the various branches of production. _Productivity of labour-abstract labour-value — distribution of social labour:_ this is the scheme of a commodity economy.' (Rubin, 1973, p. 66.)
Thus Rubin is still on the terrain of the labour theory of value. The object of the theory is still located in the process of circulation — it has simply been widened to include the circulation of labour time as well as of the products of labour.
**6. The indeterminateness of human labour**
But if Marx's theory of value does not have as its object the circulation (or distribution) of labour so as to fill the slots in a pre-given structure of production, what is its object? One way of trying to explain would be to say that it is about the determination of the structure of production _as well as_ the distribution of labour in that structure. But that is still far too mechanical, too structural a metaphor. In a vivid passage in the _Grundrisse_ , Marx describes labour thus:
> 'Labour is the living, form-giving fire; it is the transitoriness of things, their temporality, as their formation by living time.' (Op. cit., p. 361.)
It is a fluidity, a potential, which in any society has to be socially 'fixed' or objectified in the production of particular goods, by particular people in particular ways. Human beings are not preprogrammed biologically to perform particular tasks. Unlike ants or bees, there is a potentially vast range in the tasks that any human being can undertake. As Braverman puts it,
> 'Freed from the rigid paths dictated in animals by instinct, human labour becomes indeterminate.' (Braverman, 1974, p. 51).
This fluidity of labour is not simply an attribute of growing industrial economies: human labour is fluid, requiring determination, in all states of society. But it is true that only with industrialisation does the fluidity of labour become immediately _apparent_ , because the jobs that individuals do are obviously _not_ completely determined by 'tradition', religion, family ties etc.,3 and individuals do quite frequently change the job they do. As Marx put it:
> '... We can see at a glance that in our capitalist society a given portion of labour is supplied alternatively in the form of tailoring and in the form of weaving, in accordance with changes in the direction of the demand for labour. This change in the form of labour may well not take place without friction, but it must take place.' ( _Capital_ , I, p. 134.)
Arthur, 1978, recognises that 'in a developed industrial economy social labour, as a productive force, has a fluidity in its forms of appearance' (op. cit. p. 89); but because he fails to distinguish between essence and forms of appearance, he limits this fluidity, this requirement for determination, to capitalist economies. The fact that the essential indeterminateness of human labour is not immediately _apparent_ in pre-capitalist societies does not mean that it does not exist.
So the fundamental question about human labour in all societies is, how is it determined? To speak of 'determination' here does not, of course, mean the denial of _any_ choice on the part of individuals about their work. Rather it is to point to the fact that individuals can't just choose _anything_ , are unable to re-invent the world from scratch, but must choose from among alternatives presented to them.4 As several authors pointed out, Marx's concept of determination is not 'deterministic'. (See for instance, Ollman, 1976, p. 17; Thompson 1978, p. 241-242.) Although Marx stresses that determination can never be simply an exercise of individual wills, he also stresses that it is not independent of and completely exterior to the actions of individuals:
> 'The social structure and the state are continually evolving out of the life process of definite individuals.' ( _German Ideology_ , p. 46.)
But
> 'of individuals, not as they may appear in their own or other people's imagination, but as they _really_ are; i.e. as they operate, produce materially, and hence as they work under definite material limits, pre-suppositions and conditions independent of their will'. ( _German Ideology_ , p. 47.)
Distribution of social labour is not an adequate metaphor for this process of determination, because such distribution always begins from some pre-given, fixed, determinate structure, which is placed outside the process of social determination. What is required is a conceptualisation of a process of social determination that proceeds from the indeterminate to the determinate; from the potential to the actual; from the formless to the formed. _Capital_ is an attempt to provide just that. It uses a method of investigation which is peculiarly Marx's own, a method which he claimed had not previously been applied to economic subjects (Preface to French Edition, _Capital_ , I, p. 104), and which has not been much applied since. I think that it is in large part the difficulties of understanding this method which have lead to mis-readings of Marx's theory of value. The next section considers this method in some detail, and contrasts it with the method of 'the labour theory of value' as traditionally understood.
_Capital_ is, of course, the culmination of work on the social determination of labour that began many years before, and went through various phases. I shall not be discussing the formation of the theory of value presented in _Capital_. I merely note that many of Marx's earlier texts are extremely ambiguous, probably because in investigating the social form that labour takes, Marx _began_ from the problematic of political economy. Part of his transformation of this problematic was carried out by reading into the texts of political economy concerns which were those of Marx, rather than of Ricardo, Smith etc., in particular the concern to locate the substance of value. (See Aumeeruddy and Tortajada, this volume, p. 11-12.) In some texts we may find elements of both a 'labour theory of value' and a 'value theory of labour'. There are symptoms of this even in _Critique of Political Economy_ , published in 1859, eight years before the first volume of _Capital_. In this text there is no clear distinction between value and exchange-value, between the inner relation and its form of appearance, a distinction which plays an important role in the argument of _Capital_ , and which one can see being developed in the commentaries of _Theories of Surplus Value_ , particularly in the critique of Bailey in Part 3. Accordingly, this paper will focus on the theory of value as it appears in _Capital_ , supplementing this where necessary with clarifications deriving from _Theories of Surplus Value;_ and, in a few cases relating to money, from _Critique of Political Economy_.
MISPLACED CONCRETENESS AND MARX'S METHOD OF ABSTRACTION
**1. Rationalist Concepts of Determination**
All of the readings of Marx's value theory so far discussed have in common a misplaced concreteness, in that they understand that theory as a relation between certain already determined, 'given', independent variables located in the process of production, and certain to-be-determined, dependent variables located in the process of circulation. I think this is because it does not occur to the authors we have been considering that there is any other way of understanding the relation of determination. When questions about determination are raised it is usually only to discuss the choice of independent and dependent variables, or whether there _are_ any _general_ determinants. (See for instance Meek, 1977, p. 151-2; Steedman, 1977, p. 25; Cutler, et al. 1977, p. 19). It is simply taken for granted that any theory requires separable determining factors, discretely distinct from what they are supposed to determine. (See Georgescu-Roegen, 1966, p. 42; Oilman, 1976, discusses this in relation to interpretations of Marx's concept of mode of production, p. 5-11.) Althusser's 'structural causality' does not break with that view; it merely puts the independent variables one stage back, behind the 'structure'. Economic phenomena are
> _'determined by a (regional) structure_ of the mode of production, itself determined by _the (global) structure_ of the mode of production.' (Althusser, 1975, p. 185),
but the mode of production itself is constructed of a combination of 'determinate pre-existing elements' which are 'labour power, direct labourers, masters who are not direct labourers, object of production, instruments of production, etc.' (Althusser, 1975, p. 176.)
The abandonment of Althusser's concepts by Cutler et al. does not break with that view either. They dissolve Althusser's self-reproducing 'structures', but only to go back to the 'determinate pre-existing elements' that lie behind them, the 'conditions of existence'. (See for instance Cutler et al., 1977, p. 218-219). Their main distinction is simply to be more agnostic than most other writers in this framework in their choice of independent variables. (See Ohlin Wright, 1979, for a useful classification of different approaches to the 'labour theory of value' in terms of their choice and grouping of variables.)
This approach poses the relation of determination as an effect of some already given, discretely distinct elements or factors on some other, quite separate, element or factors, whose general form is given, but whose position within a possible range is not, using what Georgescu-Roegen calls 'arithmomorphic concepts'. Essentially a _rationalist_ method, it assumes that the phenomena of the material world are like the symbols of arithmetic and formal logic, separate and self-bounded and relate to each other in the same way.5 This is not Marx's method: his theory of value is not constructed on rationalist lines.
**2. Determination in Marx's theory of value: the relation between labour-time, value and exchange-value**
Oilman has pointed out that Marx's concept of the mode of production in the _Preface to A Contribution to the Critique of Political Economy_ is not one of independent variables determining dependent variables. He argues that some of the expressions used to categorise that which determines,
> 'appear to include in their meanings part of the reality which Marx says they 'determine'. Thus, property relations as a system of legal claims came under the heading of superstructure, but they are also a component of the relations of production which 'determine' this superstructure'. (Oilman, 1976, p. 7.)
We can see something similar in the first chapter of _Capital_ I. The first reference to 'determination' is:
> 'It might seem that if the value of a commodity is determined by the quantity of labour expended to produce it, it would be more valuable the more unskilful and lazy the worker who produced it.' ( _Capital_ I, p. 129.)
Marx goes on to explain why this is not so, and concludes:
> 'What exclusively determines the magnitude of value of any article is therefore the amount of labour socially necessary, or the labour-time socially necessary for its production.' ( _Capital_ I, p. 129.)
There is a tendency to misread value as 'exchange-value' or 'price', and to mistake this for a statement of a relation between a dependent and an independent variable — a 'labour theory of value', in short. But just prior to this passage Marx has specifically distinguished value from exchange-value, and stated that for the moment it is value and not exchange-value which is under consideration. Does that mean that Marx is simply giving us a _definition_ of the category value in the above quoted passages, is using 'determine' in the sense of 'logically define'? No, because value is not the _same_ as a quantity of socially necessary labour-time: it is an objectification or materialisation of a certain aspect of that labour-time, its aspect of being simply an expenditure of human labour power in general, i.e. abstract labour. This is a rather peculiar kind of objectification. As Marx says
> 'Not an atom of matter enters into the objectivity of commodities as values; in this it is the direct opposite of the coarsely sensuous objectivity of commodities as physical objects.' ( _Capital_ , I, p. 138.)
Considered simply as physical objects, commodities are objectifications of concrete not abstract labour. The peculiarity of the objectification of abstract labour is in fact signalled by Marx in the reference to 'phantomlike objectivity' in this well known passage:
'Let us look at the residue of the products of labour. There is nothing left of them in each case but the same phantomlike objectivity; they are merely congealed quantities of homogeneous human labour. i.e. of human labour power expended without regard to the form of its expenditure. All these things now tell us is that human labour, i.e. of human labour power expended without regard to the form of its expenditure. All these things now tell us is that human labour-power has been expended to produce them, human labour is accumulated in them. As crystals of this social substance, which is common to them all, they are values — commodity values.' ( _Capital_ , I, p. 128).
We should note the chemical metaphors — 'congealed', 'crystals' — which occur repeatedly in Chapter 1, Vol. 1. For they indicate something of the character of Marx's concept of determination. The quantity of socially necessary labour-time does not determine the magnitude of value in the logical or mathematical sense of an independent variable determining a dependent variable, (or in the sense of defining the meaning of the term 'magnitude of value'), but in the sense that the quantity of a chemical substance in its fluid form determines the magnitude of its crystalline or jellied form. There is a continuity as well as a difference between what determines and what is determined.
But perhaps we have been looking in the wrong direction: what about the relation between value and exchange-value? If value is an objectification of a quantity of socially necessary abstract labour-time and exchange-value is the quantity of one commodity which is exchanged for a given quantity of another, surely these are our two separate variables, the one determining the other? Marx writes of exchange-value as 'the necessary mode of expression, or form of appearance, of value' ( _Capital_ , I, p. 128), but perhaps we could interpret that as meaning that exchange-values are discretely distinct from but _correspond_ to or _approximate_ to values, (as Steedman, 1977, implies in his appendix). After all, Marx writes that the measure of the magnitude of value is labour-time, whereas the magnitude of exchange-value is measured in terms of a quantity of some commodity, or most generally, in terms of money, which would seem to suggest that the two are quite independent.
However, it is extremely difficult to maintain that interpretation if we take into account the much-neglected third section of chapter 1, 'The Value-Form or Exchange Value'. Here Marx suggests that, divorced from its expression as exchange-value, value is simply an abstraction, without practical reality. It cannot stand on its own: it is not a category designating a reality which is independent of exchange-value, but a reality which is manifested through exchange-value. (See Kay, this volume, p. 57-8, and Arthur, ditto, p. 68.)
> 'If we say that, as values, commodities are simply congealed quantities of human labour, our analysis reduces them it is true, to the level of abstract value, but does not give them a form of value, distinct from their natural form.' ( _Capital_ , I, p. 141.)
If a product of labour is a value this must be reflected in some attribute of the product of labour which is immediately apparent, although not immediately recognisable as a reflection of value.6 The simplest form of this reflection is when another commodity stands in a relation of equivalence to the first commodity, and serves as the material in which its value is expressed, as the embodiment of abstract labour. But this is a very limited expression of value, since it only expresses the equivalence of the first commodity with one other commodity. For an adequate expression of value, the first commodity must be able to express its value in terms of a universal equivalent, a commodity directly exchangeable with all other commodities, a commodity whose use value is its interchangeability. As the process of exchange develops one commodity is set apart from the others and comes to play this role, or, as Marx puts it, 'Money necessarily crystallises out of the process of exchange.' ( _Capital_ , I, p. 181.)
Marx thus locates the 'form of value' in the price of a commodity. For Marx, the price of a commodity is not the result of some process quite independent of (discretely distinct from) the formation of its value (the objectification in it of abstract labour). Rather,
> 'the money-form is merely the reflection thrown upon a single commodity by the relations between all other commodities'. ( _Capital_ , I, p. 184.)
This does not mean that money must always be commodity money (i.e. gold); nor that because price is a value-form, price and value are identical.
Marx explicitly recognised that 'money can, in certain functions, be replaced by mere symbols of itself.' ( _Capital_ , I, p. 185), and points out that,
> 'In its form of existence as coin, gold becomes completely divorced from the substance of its value. Relatively valueless objects, therefore, such as paper notes, can serve as coins in place of gold. This purely symbolic character of the currency is still somewhat disguised in the case of metal tokens. In paper money it stands out plainly.' ( _Capital_ , I, p. 244.)
What he is arguing against is the view that money can be completely autonomous, 'a convenient technical device which has been introduced into the sphere of exchange from the outside'. ( _Critique of Political Economy_ , p. 57); the product of a convention rather than of a 'blind' social process. He maintains that there are limits to the extent that paper money can supersede commodity money, in effect rejecting a bifurcation of economic relations into the 'money' and the 'real'. In maintaining that there must be an 'intrinsic connection between money and labour which posits exchange value' ( _Critique of Political Economy_ , p. 57), Marx is denying that value and price are two completely separate variables.
This does not, however, mean that Marx sees value and price as _identical_. Marx expressly criticised Bailey for making this reduction (see _Theories of Surplus Value_ , Part 3, p. 147). There is for Marx both a continuity and a difference between value and price, irrespective of whether price is denominated in gold or in paper.
To summarise: in the argument of _Capital_ , labour-time, value, and exchange-value (price) are not three discretely distinct variables, nor are they identical with one another. There is a continuity as well as a difference between all three. The relation between them (in any combination) is _not_ posed in terms of an independent variable determining a dependent variable.
**3. The measure of value: labour-time and money**
One implication of the above argument is that the analysis of _Capital_ is not predicated on the possibility of calculating values directly in terms of labour-time, quite independently of price, calculated in terms of money (or some numeraire); whereas, as we have already noted, this possibility is central to many readings of the 'labour theory of value' variety. Misconceptions are encouraged here by the fact that in _Capital_ , Marx does not deal with this point explicitly at any length, simply referring us in a footnote to the _Critique of Political Economy_ (see _Capital_ , I, p. 188). Turning to the latter, we find this point discussed in the context of a consideration of Gray's labour-money scheme.7 Gray proposed that a national bank should find out the labour-time expended in the production of various commodities; and in exchange for his commodity the producer would receive an official certificate of its value, consisting of a receipt for as much labour-time as his commodity contained. Marx objects to this on the grounds that it assumes
> 'that commodities could be directly compared with one another as products of social labour. But they are only comparable as the things they are. Commodities are the direct products of isolated independent individual kinds of labour, and through their alienation in the course of individual exchange they must prove that they are general social labour, in other words, on the basis of commodity production, labour becomes social labour only as a result of the universal alienation of individual kinds of labour. But as Gray presupposes that the labour-time contained in commodities is _immediately social_ labour-time, he presupposes that it is communal labour-time of directly associated individuals'. ( _Critique of Political Economy_ , p. 85.)
In other words, the labour-time that can be directly measured in capitalist economies in terms of hours, quite independent of price, is the particular labour-time of particular individuals: labour-time in its private and concrete aspect. This is not the aspect objectified as value, which is its social and abstract aspect. As Marx put it in an earlier passage in _Critique of Political Economy:_
> 'Social labour-time exists in these commodities in a latent state, so to speak, and becomes evident only in the course of their exchange. The point of departure is not the labour of individuals considered as social labour, but on the contrary the particular kinds of labour of private individuals... Universal social labour is consequently not a ready-made pre-requisite but an emerging result.' (Op. cit., p. 45.)
The social necessity of labour in a capitalist economy cannot be determined independent of the price-form: hence values cannot be calculated or observed independently of prices.
But in that case what are we to make of Marx's repeated statements that labour-time is the measure of value? It is not surprising that this leads to misunderstandings, because in _Capital_ Marx does not highlight the conceptual distinction which he makes between an 'immanent' or 'intrinsic' measure, and an 'external' measure, which is the mode of appearance of the 'immanent' measure. This distinction is implicit in the example of the measurement of weight ( _Capital_ , I, p. 148-9), and briefly stated at the beginning of the chapter on Money. Viz:
> 'Money as the measure of value is the necessary form of appearance of the measure of value which is immanent in commodities, namely labour-time.' ( _Capital_ , I, p. 188.)
It is only in the critique of Bailey (in _Theories of Surplus Value_ , Part 3, p. 124-159) that this distinction is explicitly discussed. The 'immanent' measure refers to the characteristics of something that allow it to be measurable as pure quantity; the 'external measure refers to the medium in which the measurements of this quantity are actually made, the scale used, etc. The concept of 'immanent' measure does not mean that the 'external' measure is 'given' by the object being measured. There is room for convention in the choice of a particular medium of measurement, calibration of scale of measurement, etc. It is not, therefore, a matter of counter-posing a realist to a formalist theory of measurement (as Cutler et al., 1977, suggest p. 15). Rather it is a matter of insisting that there are both realist and formalist aspects to cardinal measurability (i.e. measurability as absolute quantity, not simply as bigger or smaller). Things that are cardinally measurable can be added or subtracted to one another, not merely ranked in order of size, (ranking is ordinal measurability).
A useful discussion of this issue is to be found in Georgescu-Roegen, who emphasises that:
> 'Cardinal measurability, therefore, is not a measure just like any other, but it reflects a particular physical property of a category of things.' (Op. cit., p. 49.)
Only things with certain real properties can be cardinally measured. This is the point that Marx is making with his concept of 'immanent' measure, and that he makes in the example, in _Capital_ , I, of the measure of weight (p. 148-9). The external measure of weight is quantities of iron (and there is of course a conventional choice to be made about whether to calibrate them in ounces or grammes, or whether, indeed, to use iron, rather than, say, steel). But unless both the iron and whatever it is being used to weigh (in Marx's example, a sugar loaf) both have weight, iron cannot express the weight of the sugar loaf. Weight is the 'immanent' measure. But it can only be actually measured in terms of a comparison between two objects, both of which have weight and one of which is the 'external' measure, whose weight is pre-supposed.
Thus when Marx says that labour-time is the measure of value, he means that the value of a commodity is measurable as pure quantity because it is an objectification of abstract labour, i.e. of 'indifferent' labour-time, hours of which can be added to or subtracted from one another. As such, as an objectification of pure duration of labour, it has cardinal measurability. This would not be the case if the commodity were simply a product of labour, an objectification of labour in its concrete aspect. For concrete labour is not cardinally measurable as pure time. Hours spent on tailoring and hours spent on weaving are qualitatively different: they can no more be added or subtracted to one another than apples can be added to or subtracted from pears. We can rank concrete labour in terms of hours spent in each task, just as we can rank apples and pears, and say which we have more of. But we can't measure the total quantity of labour in terms of hours, for we have no reason for supposing that one hour of weaving contains as much labour as one hour of tailoring, since they are qualitatively different.
Thus far from entailing that the _medium_ of measurement of value must be labour-time, the argument that labour-time is the (immanent) measure of value entails that labour-time _cannot_ be the medium of measurement. For we cannot, in the actual labour-time we can observe, separate the abstract from the concrete aspect. The only way that labour-time can be posed as the medium of measurement is by making the arbitrary assumption that there is no qualitative difference between different kinds of labour, an assumption that Marx precisely refuses to make with his insistence on the importance of the form of labour.
It is surprising that Cutler et al., 1977, who emphasise their critique of the supposed function of labour-time as a social standard of measurement in _Capital_ , do not refer to Marx's distinction between 'immanent' and 'external' measure. Had they done so, they might have realised that it is _money_ , and not labour-time, which functions as the social standard of measurement, in Marx's _Capital_ , as in capitalist society itself. The reason that labour-time is stressed as the measure of value, is to argue that money in itself does not make the products of labour commensurable. They are only commensurable insofar as they are objectifications of the abstract aspect of labour.
None of these confusions are new. Unfortunately the following comment that Marx made on Boisguillebert8 remains of relevance today:
> 'Boisguillebert's work proves that it is possible to regard labour-time as the measure of the value of commodities, while confusing the labour which is materialised in the exchange value of commodities and measured in time units with the direct physical activity of individuals.' ( _Critique of Political Economy_ , p. 55.)
One implication of this discussion of the measure of value which we should note is that the value-magnitude equations which Marx uses in _Capital_ , do not refer to directly observable labour-time magnitudes (the direct physical activity of individuals), but are a way of indicating the intrinsic character, or substance, of the directly observable money magnitudes. Marx generally introduces these equations in their general form e.g. the value of a commodity = (C + V) + S; and then gives a specific example. These specific examples are always couched in money terms, _never_ in terms of hours of labour-time. For example, the value of a commodity = (£410 constant + £90 variable) + £90 surplus (cf. _Capital_ , I, p. 320). This does not mean that Marx is identifying values and prices; rather that he is indicating the inner value character of monetary magnitudes. The reason why Marx does not simply work at the level of money is that he wants to uncover social relations, such as the rate of surplus-value, which do not directly appear in money form.
Perhaps we can summarise this argument by saying that what Marx proposes is that in a capitalist economy (labour)-time becomes money in a more than purely metaphorical sense. Labour-time and money are not posed as discretely distinct variables which have to be brought into correspondence. Rather the relation between them is posed as one of both continuity and difference. Significantly the metaphors used to characterise this relation are not mechanical ('articulation'), nor mathematical/logical ('correspondence', 'approximation') but chemical and biological terms ('crystallisation', 'incarnation', 'embodiment', 'metabolism', 'metamorphosis'). The idea they carry is that of 'change of form'.
**4. The analysis of form determination: the method of historical materialism.**
Some may feel we have proved too much. They will suggest that in demonstrating that Marx's value theory has been misread, we have also demonstrated that it is incoherent; that it must fail to provide a proper explanation of labour, or prices, or anything else, because it does not pose determinants completely independent of what is determined. Surely, it will be said, this must inevitably make the argument 'circular'. This would be the case if Marx were seeking to provide explanations _ab initiò_ , were seeking to explain the 'origins' of phenomena in factors external to them; to set out their necessary and sufficient conditions of existence in terms of combinations of other factors, in the manner of an economic or sociological model. _But this was not Marx's project_.
Marx saw the determination of social forms as an historical process; a process eventuating through time in which every precipitated form becomes in turn dissolved, changes into a new form, a process whose dynamic is internal to it, which has no external 'cause', existing outside of history, of which it is an effect. This entails a view of the world as a qualitatively changing continuum, not an assembly of discretely distinct forms (see Oilman, 1976, especially Chapters 2 and 3). There is no methodological preface to _Capital_ which systematically expounds this view, but there are indications of it in the Post-face to the Second Edition of _Capital_ , I, where we are told that Marx's main concern with phenomena is
> 'the law of their variation, of their development, i.e. of their transition from one form into another, from one series of connections into a different one.' (Op. cit., p. 100.)
and that,
> 'economic life offers us a phenomenon analogous to the history of evolution in other branches of biology'. (Op. cit., p. 101.)
This view of the determination of social forms is expounded more systematically by Engels in _Anti-Duhring_ , an exposition read in manuscript by Marx and issued with his knowledge (see _Anti-Duhring_ , p. 14). In it Engels writes that
> 'Political economy is therefore essentially a _historical_ science. It deals with material that is historical, that is, constantly changing. (Op. cit. p. 204.)
This view of form determination as an historical process is not simply a matter of noting that the social forms of a particular epoch have not always existed (see Banaji, 1976, p. 37-8). It is a matter of analysing them as determinate and yet transient: as the Marxist historian Edward Thompson puts it,
> 'In investigating history we are not flicking through a series of 'stills', each of which shows us a moment of social time transfixed into a single eternal pose: for each of these stills is not only a moment of being but also a moment of becoming... Any historical moment is both the result of prior process and an index towards the direction of its future flow.' (Thompson, 1978, p. 239.)
The method of analysis appropriate for analysing historical process is not the mathematico-logical method of specifying independent and dependent variables, and their relation. Such a method can only identify static structures, and is forced to pose a qualitative change as a sudden discontinuity, a quantum leap between structures; and not as a process, a qualitatively changing continuum. (See Georgescu-Roegen, 1966, p. 29-41 for a useful discussion of this issue.) The point is that to analyse historical process we need 'a different kind of logic, appropriate to phenomena which are always in movement'. (Thompson, 1978, p. 230.)
But what kind of logic? Trying to explain the determination of a form by describing the succession of previous forms will not do. This only tells us what came after what; not how forms are crystallised and might re-dissolve. And in any case, where are we to start such a sequence, and how can we avoid posing the starting point as an 'origin', itself outside the historical process? Marx rejects this approach as early as 1844; in the _Economic and Philosophical Manuscripts_ we find:
> 'We must avoid repeating the mistake of the political economist who bases his explanations on some imaginary primordial condition. Such a primordial condition explains nothing.' ( _Early Writings_ , p. 323.)
Such a sequential approach also finds it difficult to avoid posing the earlier forms as inevitably leading to the later, a problem discussed by Marx in the _1857 Introduction;_ this discussion concludes:
> 'It would therefore be unfeasible and wrong to let the economic categories follow one another in the same sequence as that in which they were historically decisive. Their sequence is determined, rather, by their relation to one another in modern bourgeois society, which is precisely the opposite of that which seems to be their natural order or which corresponds to historical development. The point is not the historic position of the economic relations in the succession of different forms of society... Rather, their order within modern bourgeois society.' (Op. cit., p. 107-8.)
This is an elaboration of the conclusion of 1844:
> 'We shall start out from a _present day_ economic fact.' ( _Early Writings_ , p. 323).
In other words, we start from the form that we want to understand, and we do not go backwards in time; rather we consider how to treat it as the precipitate of an on-going process without detaching it from that process.
Marx's solution was not to go _outside_ the form looking for factors to explain it, but to go _inside_ the form, to probe beneath its immediately apparent appearance. (See Banaji, this volume, pp. 17-21 for a detailed discussion of this point.) Going inside the form is achieved by treating it as the temporary precipitate of opposed _potentia;_ what Thompson calls a moment of becoming, a moment of co-existent opposed possibilities, 'double-edged and double-tongued' (Thompson, 1978, p. 305-6). But these opposed _potentia_ are not discretely distinct building blocks; rather they are different aspects of the continuum of forms in process, they share a continuity as well as a difference. It is in this sense that Marx treats determinate forms as embodiments of contradiction. In the same way elliptical motion can be treated as the resultant of two opposing _potentia:_ a tendency of one body to continually move away from another, and an opposing tendency to move towards it. (Cf. _Capital_ , I, p. 198.)
These different, counter-posed aspects are often referred to by Marx as 'determinants' or 'determinations' (just as the opposed movements whose resultant is the ellipse are referred to as 'determinants'). But that does not mean that the form is produced or caused by the 'determination' or 'determinants' acting in some autonomous way. For instance, Marx writes that the case of Robinson Crusoe contains 'all the essential determinants of value'. ( _Capital_ , I, p. 170), but he quite clearly does not mean that Robinson Crusoe's labour is objectified as value. In fact, Marx goes further and claims that the determinants of value 'necessarily concern mankind' 'in all situations' ( _Capital_ , I, p. 164); but he quite clearly does not mean that value is eternally present. The point is that the determinants are not independent variables, but are simply aspects, one-sided abstractions singled out as a way of analysing the form.
The analysis of a form into its determinants is, however, only the first phase of the investigation. After this phase of individuation of a moment from the historical process, and dissection of the tendencies or aspects counterposed in it, comes the phase of synthesis, of reconstitution of the appearance of the form, and of re-immersing it in process (see Banaji, this volume, p. 28). This second phase does not simply take us back to where we began, but beyond it, because it enables us to understand our starting point in a different light, as predicated on other aspects of a continuous material process. It suggests new abstractions which need to be made from a different angle, in order to capture more of the process. The phase of synthesis brings us back to continuities which the phase of analysis has deliberately severed. The whole method moves in an ever-widening spiral, taking account of more and more aspects of the historical process from which the starting point was individuated and detached.
What kind of knowledge does this method give? It cannot give a Cartesian Absolute Knowledge of the world, its status as true knowledge validated by some epistemological principle. Rather it is based upon a rejection of that aspiration as a form of idealism (see Ruben, 1977, especially p. 99). It is taken for granted, in this method, that the world has a material existence outside our attempts to understand it; and that any category we use to cut up the continuum of the material world can only capture a partial knowledge, a particular aspect seen from a certain vantage point. This is explicitly recognised in the discussion of method in the _1857 Introduction_ :
> 'for example, the simplest economic category, say e.g. exchange value, pre-supposes population, moreover a population producing in specific relations; as well as a certain kind of family, or commune, or state, etc. It can never exist other than as _an abstract, one-sided relation within an already given, concrete, living whole.' (1875 Introduction_ , p. 101, emphasis added.)
The second phase of the investigation, the phase of synthesis helps to correct the one-sidedness intrinsic to the first phase of analysis, by suggesting other perspectives which must be investigated; new, interrelated ways of cutting up the continuum. These in turn are necessarily one-sided, but the phase of synthesis based on them again helps to correct their one-sidedness. So by following this procedure a more and more complete understanding of the material world can be gained 'in which thought appropriates the concrete'. But there remains always a necessary distance between our understanding of the world, and the world itself:
> 'The totality as it appears in the head, as a totality of thoughts, is a product of a thinking head, which appropriates the world in the only way it can, a way different from artistic, religious, practical and mental appropriation of this world. The real subject retains its autonomous existence outside the head just as before.' ( _1857 Introduction_ , p. 101.)
The appropriation of the world can never be completed in thought; it requires practical action.
We must now examine this method at work in Marx's theory of value. None of the argument so far entails that there are no ambiguities and inconsistencies in that theory, for we have not yet subjected Marx's theory of value to critical scrutiny. There is certainly a danger in using this method of analysis, a danger which Marx explicitly recognised, that 'the movement of the categories appears as the real act of production' ( _1857 Introduction_ , p. 101). That is, a category of analysis, which as such is a one-sided abstraction, becomes transformed into a self-developing entity; and the historical process becomes transformed into the expression of this entity. The categories of analysis produce our knowledge of the world: but they do not produce the world itself. Marx argues that Hegel
> 'fell into the illusion of conceiving the real as the product of thought concentrating itself, probing its own depths, and unfolding itself out of itself, by itself.' ( _1857 Introduction_ , p. 101.)
In my view the 'capital-logic' approach9 falls into the same illusion, taking capital not as a one-sided abstraction, a category of analysis, but as an entity; and understanding the historical process of form determination as the product of the self-development of this entity. One of the key questions considered in the next Section is how far Marx himself succumbed to this illusion.
MARX'S VALUE THEORY OF LABOUR
**1. Aspects of labour: social and private, abstract and concrete**
In analysing the form of labour in capitalist society, Marx made use of four categories of labour, the opposing pair, abstract and concrete; and the opposing pair, social and private. He did not begin the argument of _Capital_ (or _Critique of Political Economy_ ) from these categories, but I think it is easier to evaluate his argument if we first consider what these categories mean; and what Marx claimed to have established about the relation between these aspects of labour in capitalist society, as determinants of the form of labour.
The first point I want to make is that these are _not_ concepts of different _types_ of labour. It is not that some labour is private and some social; some concrete and some abstract. Or that labour is at some stage private, and becomes at another stage social, or at some stage concrete, and becomes at another stage abstract. They are concepts of different _aspects_ of labour (cf. the 'two-fold nature' or 'dual character' of labour embodied in commodities); and as such they are all one-sided abstractions.
The second point is that they are concepts pertaining to all epochs of history. They are concepts of some of those 'determinations which belong to all epochs... No production will be thinkable without them'. ( _1857 Introduction_ , p. 85.) Where historical epochs differ is the way that these aspects are represented, i.e. the way they appear. Here we need to distinguish between 'formless' appearance as scattered, seemingly unconnected symptoms, and crystallisation into a distinct form of appearance, a representation which enables the aspect to be grasped as a unity: and which gives what Marx calls 'a practical truth' to the abstraction (cf. _1857 Introduction_ , p. 105). Marx did _not_ regard abstractions which do not have such a 'practical truth' as invalid (cf. _1857 Introduction_ , p. 85, p. 105). The criterion Marx put forward for a valid abstraction was that it should be 'a rational abstraction in so far as it really brings out and fixes the common element and thus saves us repetition'. ( _1857 Introduction_ , p. 85.) What he suggested was that such valid abstractions do not have the same status for all historical epochs: they have a different significance in epochs in which they have a 'practical truth'. In such circumstances, the process bringing to light the common aspect is not only a mental process. The mental process has its correlate in a real social process which gives the common aspect a distinct form of appearance, albeit quite possibly a fetishised form of appearance, so that the common aspect represented may be misrecognised if we go only by appearances.
The third point is that the two pairs of abstractions (abstract/concrete; social/private) must not be collapsed into one. There is a tendency to suggest that 'abstract' means the same as 'social', and 'concrete' the same as 'private'. (See Kay, this volume, p. 56; and Hussain, this volume, p. 95). There _is_ some overlapping in meaning, but the two pairs, as concepts, are nevertheless distinct. What Marx argues is that in the specific conditions of capitalism, the distinctions between the two pairs tend, as a practical reality, to be obliterated: the concrete aspect of labour is 'privatised', and the social aspect of labour is 'abstracted'. These points are not uncontroversial, so it is necessary to deal with them in more depth.
There is a tendency to suppose that Marx analysed capitalism as a form of production in which labour starts off as 'concrete' and 'private'; in the process of exchange this labour, by now embodied in products, is then transformed into a different type of labour 'abstract' and 'social' (cf. in particular Rubin, 1973, p. 70; also Arthur, 1978, p. 93-5). Certainly Marx does refer to commodities as 'the products of private individuals who work independently of each other'. ( _Capital_ , I, p. 165); and claims that 'Universal social labour is consequently not a ready made pre-requisite but an emerging result.' ( _Critique of Political Economy_ , p. 45.) And he does discuss the labour process 'independently of any specific social formation'. ( _Capital_ , I, p. 283.) But this does not signify a departure from his position that
> 'Individuals producing in society — hence socially determined individual production — is, of course, the point of departure'. ( _1857 Introduction_ , p. 83.)
Rather, it signifies his analysis of the problem that
> 'in this society of free competition, the individual appears detached from the natural bonds etc. which in earlier historical periods make him the accessory of a definite and limited human conglomerate'. ( _1857 Introduction_ , p. 83.)
In _Capital_ , Marx continued to begin from the position that
> 'as soon as men start to work for each other in any way, their labour also assumes a social form'. ( _Capital_ , I, p. 164.)
The problem was to locate this social form in a capitalist society, where it appeared that men as producers are private individuals free from social forms; or rather that social forms have no independent effectivity and are simply the result of private decisions, individual choices; the cash nexus simply a way of aggregating and reconciling these choices. Marx contrasted this form of appearance with examples of pre-capitalist societies, in which the social forms constraining individuals in production were immediately apparent: the patriarchal family, feudal rights and duties etc. (See _Critique of Political Economy_ , p. 32-4; _Capital_ , I, p. 169-171.) When Marx refers to 'private individuals' in _Capital_ , he is referring precisely to this appearance:
> 'Since the producers do not come into social contact until they exchange the products of their labour, the specific social characteristics of their private labour appear only within this exchange.' ( _Capital_ , I, p. 165.)
(We should perhaps note that Marx is at this stage of the discussion abstracting from the _internal_ organisation of each producing unit.) What Marx means is that capitalist production is private in the sense that the social relation of each producing unit to all others is _latent_ , hidden; not in the sense that labour as an activity has no social character, and only acquires one _after_ its embodiment in commodities. Marx's argument is not that the process of exchange _confers_ a social form on hitherto private labour — but that it brings out the social character which is already latent, albeit bringing it out in a fetishised form, as a 'social relation between things'.
The concept of 'concrete labour' overlaps with the concept of 'private labour', since it is a concept of subjective human activity 'determined by its aim, mode of operation, object, means and result' ( _Capital_ , I, p. 132). What it adds to the notion of the individual, subjective aspects of human labour is the notion of labour as 'a process between man and nature' ( _Capital_ , I, p. 283) in which labour takes many different, specific forms: tailoring, weaving, spinning etc. etc. It is the concept of diversity and heterogeneity of labour. The 'private' and 'concrete' aspects of labour are in fact coincidental in capitalist societies, where the different kinds of labour appear to be undertaken as a result of the choices made by the individuals doing them (even if very constrained choices). This is not the case in pre-capitalist forms of production, where,
> 'The natural form of labour, its particularity... is here its immediate social form.' ( _Capital_ , I, p. 170);
and individuals appear to have little choice about the kind of work they do.
The term 'concrete labour' is rather unfortunate, in that it is a hindrance to our recognition that 'concrete labour' is a one-sided abstraction, the concept _not_ of labour as 'the concentration of many determinations', as a living whole, a determinate form, (which is how Marx uses the term 'concrete' in the _1857 Introduction_ ); but rather the concept of certain aspects of labour ( _one_ of the 'many determinations'). The concept of concrete labour abstracts from labour as a living whole its subjective, qualitative, diverse aspects, which are in all epochs reflected as characteristics of the product in terms of its use-value.
Marx's discussion of the labour process 'independently of any specific social formation' ( _Capital_ , I, p. 283) does not license us to take concrete labour as the concept of a 'given', determinate reality upon which social relations are superimposed. He specifically mentions that this is a presentation of 'simple and abstract elements' ( _Capital_ , I, p. 290); and that 'labour process' and 'valorisation process' are 'two _aspects_ of the production process' of commodities. ( _Capital_ I, p. 304, emphasis added). Failure to take note of this tends to lead either in the direction of technological determinism (cf. Rubin, 1973), or to posing the socialist project in terms of the impossible task of removing _any_ social mediation between the individual and her work (cf. Colletti, 1976, especially p. 66).
It is generally accepted that concrete labour is a category pertinent to all epochs; but the same is not accepted of abstract labour. (Hussain, this volume, and Itoh, 1976, are exceptions).10 Generally writers who stress the importance of abstract labour insist that it is a category pertinent only to commodity production (cf. Rubin, 1973; Arthur, 1976 and 1978). To argue otherwise, it is suggested, makes the theory of value an 'eternal' theory, true of all societies, and not specific to capitalism. But this implication does not seem to me to follow from the proposition that abstract labour is a category pertinent to all epochs. The belief that it does possibly stems from a misreading of Marx's claim that it is abstract labour which forms the substance of value as a definition of abstract labour, or the assumption that abstract labour is the concept of a type of labour, and must therefore produce _something_ , a something which Marx calls 'value'. But as we shall presently see, the categories of value and abstract labour are arrived at independently, not derived from one another.
At this point we simply note that abstract labour, like concrete labour, is not the concept of a _type_ of labour, but of certain aspects of human labour. This is certainly indicated by the phrase 'the dual character of the labour embodied in commodities'. In _Capital_ these aspects are at first defined only negatively, as aspects which remain when we disregard the particular, useful, aspect of labour ( _Capital_ , I, p. 128), and this has perhaps contributed to the confusion. But these aspects are subsequently characterised as those of 'quantities of homogeneous human labour' (op. cit., p. 128), and of 'human labour pure and simple, the expenditure of human labour in general' (op. cit., p. 135). In other words, it is the concept of the unity or similarity of human labour, differentiated simply in terms of quantity, duration. It is _not_ an assumption that all work is physiologically identical. Rather, it draws attention to the fact that all work takes time and effort, irrespective of what kind of work it is. Marx specifically claims that this aspect of labour 'in all situations... must necessarily concern mankind, although not to the same degree at different stages of development' ( _Capital_ , I, p. 164), and offers a brief discussion of the way it is of concern in the case of Robinson Crusoe, European feudalism, peasant family production and communal production ( _Capital_ , I, p. 169-72).
The concept of abstract labour overlaps somewhat with the concept of social labour, in that both view the activity of labour 'objectively' in detachment from particular individuals. Both investigate labour from the point of view of the collectivity, looking at any particular expenditure of labour-power not as an isolated self-generating activity, but as part of a collective effort. What the concept of abstract labour adds to the concept of social labour is the idea of _quantity_ , labour is viewed not simply as part of a collective effort, but as a definite fraction of a quantitatively specified total.
The four categories that we have been discussing are thus concepts of four _potentia_ , which can never exist on their own as determinate forms of labour. Labour always has its abstract and concrete, its social and private aspects. Marx poses any particular determinate form of labour as a precipitate of these four different aspects of labour. What is specific to a particular kind of society is the relation of these aspects to one another and the way in which they are represented in the precipitated forms. Marx concludes that in capitalist society the abstract aspect is _dominant_. The social character of labour is established precisely through the representation of the abstract aspect of labour:
> 'Only because the labour-time of the spinner and the labour-time of the weaver represent universal labour-time... is the social aspect of the labour of the two individuals represented for each of them...' ( _Critique of Political Economy_ , p. 33).
>
> 'the specific social character of private labours carried on independently of each other consists in their equality as human labour.' ( _Capital_ , I, p. 167).
The concrete and private aspects of labour are mediated by the abstract aspect. The labour of the individual producer
> 'can satisfy the manifold needs of the individual producer himself only in so far as every particular kind of useful private labour can be exchanged with i.e. counts as the equal of, every other kind of useful private labour. Equality in the full sense between different kinds of labour can be arrived at only if we abstract from their real inequality, if we reduce them to the characteristic they have in common, that of being the expenditure of human labour-power, of human labour in the abstract.' ( _Capital_ , I, p. 166).
A useful summary of Marx's conclusion, that in the determination of the form of labour in capitalist society it is the abstract aspect which is dominant, can be found in _Results of the Immediate Process of Production_ , originally planned as Part Seven of Volume I of _Capital_ , and serving both as a summary of Volume I and a bridge to Volume II. Here Marx writes that, in the social terms of a capitalist society,
> 'labour does not count as a productive activity with specific utility, but simply as a value-creating substance, as social labour in general which is in the act of objectifying itself, and whose sole feature of interest is its _quantity.' (Results..._ , p. 1012).
This does not mean that the particular useful qualities of labour, its concrete aspect, do not matter, but rather thay they matter only in so far as they affect the quantity of human labour expended in production. The domination of abstract labour signifies 'a social formation in which the process of production has mastery over man, instead of the opposite' ( _Capital_ , I, p. 175). For Marx, money and capital are both forms of this domination. The theory of value is the foundation for this conclusion.
Thus, Marx's argument is not that the abstract aspect of labour is the product of capitalist social relations, but that the latter are characterised by the dominance of the abstract aspect over other aspects of labour. In these conditions, abstract labour comes to have a 'practical truth' because the unity of human labour, its differentiation simply in terms of quantity of labour, is not simply recognised in a mental process, but has a correlate in a real social process, that goes on quite independently of how we reason about it. Marx argues, not that some particular type of labour can in capitalist society be identified as purely abstract labour, but that the abstract aspect of labour is 'objectified' or 'crystallised', that 'the equality of the kinds of human labour takes on a physical form' ( _Capital_ , I, p. 164). The objectification of the concrete aspect of labour is universal, but the objectification of the abstract aspect of labour is not: it is specific to capitalist social relations. This objectification at some stages in accumulation of capital may take the form of 'Labour in the form of standardised motion patterns', labour as 'an interchangeable part' and 'in this form come ever closer to corresponding, in life, to the abstraction employed by Marx in analysis of the capitalist mode of production' (Braverman, 1974, p. 182). It may take the form of mobility of labour: ( _cf. Results_... , p. 10134; this is also stressed by Arthur, 1978). But its most basic and simplest form is the objectification of abstract labour as a characteristic of the product of labour, reflected in its exchange value. And for this reason Marx begins the exposition of _Capital_ with
> 'the simplest social form in which the labour product is represented in contemporary society, and this is the ' _commodity_ '' ( _Marginal Notes on Wagner_ , p. 50).
We must now consider the argument by which he tries to establish that the abstract aspect of labour is objectified, and the way in which this establishes the domination of abstract labour.
**2. The phase of analysis: from the commodity to value**
The first phase of Marx's theory of value begins from the commodity11 and proceeds to value, the substance of which is argued to be objectified abstract labour. The commodity is analysed dialectically as a moment of co-existence of two opposed aspects, use value and exchange value; and then exchange value, as the aspect specific to capitalism is subject to further scrutiny. The movement of the argument from exchange value to value and its substance does present some problems, and has provoked charges from Bohm-Bawerk12 to Cutler et al., that the conclusions Marx draws cannot legitimately be drawn.
The problem is two-fold: the status of the argument that in exchange commodities are made equivalent to one another, signifying that 'a common element of identical magnitude exists in two different things.' ( _Capital_ , I, p. 127), and the argument that this common element is an objectification of abstract labour. We might note that the questionable status of Marx's arguments here has largely been overlooked by the 'labour theory of value' tradition of interpretation, because it has ignored the structure of Marx's own argument, and argued from pre-given quantities of labour to prices. I think there is undoubtedly a problem in the way Marx presents his argument, so that some results quite easily appear to be deductions from a formalist and ahistorical concept of exchange. But in my view the analysis is not inherently formalist, and formalist elements in its presentation can be replaced with more satisfactory arguments, some of which Marx develops elsewhere, particularly in _Theories of Surplus Value_.
Let us first consider the argument about exchange, equivalence and the 'common element'.
> 'Let us now take two commodities, for example corn and iron. Whatever their exchange relation may be, it can always be represented by an equation in which a given quantity of corn is equated to some quantity of iron, for instance 1 quarter of corn = x cwt. of iron. What does this equation signify? It signifies that a common element of identical magnitude exists in two different things, in 1 quarter of corn and similarly in x cwt. of iron. Both are therefore equal to a third thing, which in itself is neither the one nor the other. Each of them, so far as it is exchange value, must therefore be reducible to this third thing.' ( _Capital_ , I, p. 127).
The above passage does tend to suggest that (as Cutler et al., 1977, claim) Marx regards exchange _per se_ as an act which reduces the goods exchanged to instantiations of a common element, _equates_ them, and deduces his results from this formal concept of exchange. This impression is reinforced by a later passage where Marx approvingly quotes Aristotle's dictum:
'There can be no exchange without equality, and no equality without commensurability.' ( _Capital_ , I, p. 151.)
The objection that Cutler et al. raise is that while for a transaction to be an exchange, it is necessary that both parties to it agree to the terms of the exchange, there is no necessity for this to entail the reduction of the goods exchanged to a common element (op. cit., p. 14). It is not hard to find examples of exchange where such a reduction is absent, even in developed capitalist societies — for example, the exchange of gifts at Christmas; the exchange of the products of domestic labour in the household. (See also Arthur, this volume, p. 71.) The exchanges here depend very specifically on the kind of goods exchanged, and upon particular relations of personal obligation and reciprocity. In such exchanges the goods exchanged are not reduced to a common element, are not made equivalents; they are not commensurated, though they may be compared. Such exchanges are not, however, accomplished by buying and selling. Clearly, in considering the exchange of commodities, Marx _is_ considering a process of sale and purchase, even if he does not emphasise this at this particular point in the argument. Moreover, the example of exchange of corn and iron, cited above, is simply _one_ instance of exchange abstracted from a very large number of ex changes, as Marx's preceding paragraph makes clear ( _Capital_ , I, p. 127). The characteristics of the exchange of corn and iron are not held to depend simply on that one exchange, considered in isolation, but on the whole process of exchange from which this one example has been abstracted. Although Marx does not make the point very clearly, I think we can conclude that he is not considering exchange _per se_ , but a particular form of exchange, capitalist commodity exchange. His argument that such exchange is a process of equation, of reduction of the goods exchanged to equivalence is not an argument from a formal, a-historical concept of exchange, but from a _specific_ social relation, capitalist commodity exchange.
This reading is supported by Marx's much more explicit discussion of this point in the course of his critique of Bailey, in _Theories of Surplus Value_ , Part 3. Here Marx specifically argues _against_ the idea that a single act of exchange in itself reduces the goods exchanged to equivalence (see _Theories of Surplus Value_ , Part 3, p. 132; p. 142; p. 144). Rather he argues that reduction to equivalence depends upon the general exchangeability, through the market, of every commodity with every other commodity:
> 'the commodity has a thousand different kinds of value... as many kinds of value as there are commodities in existence, all these thousand expressions always express _the same value_. The best proof of this is that all these different expressions are _equivalents_ which not only can replace one another in this expression, but do replace one another in exchange itself.' ( _Theories of Surplus Value_ , Part 3, p. 147).
The same point is made in _Critique of Political Economy:_
> 'A commodity functions as an exchange value if it can freely take the place of a definite quantity of any other commodity, irrespective of whether or not it constitutes a use-value for the owner of the other commodity.' (Op. cit., p. 44).
This general exchangeability does not simply depend on the individual characteristics of the owners of the goods, or of the goods themselves, for the rates at which the goods in any particular exchange are exchanged depend not only on the parties to that transaction, but upon all the other exchanges simultaneously taking place. This kind of exchange is a social, not an individual process. The abstraction of _a_ commodity with _an_ exchange value can only be made on the presupposition that this commodity is simply one of a very large number of interchangeable commodities, a presupposition that Marx has made clear in the opening sentence of _Capital_.
> 'The wealth of societies in which the capitalist mode of production prevails appears as an 'immense collection of commodities'.' ( _Capital_ , I, p. 125).
In fact, as will later emerge, this kind of general interchangeability of goods can only become the dominant form of exchange on the basis of capitalist relations of production, in which labour is separated from the means of production. (See Brenner, 1977, especially p. 51). But in Chapter 1, the categories for analysing capitalist relations of production have not been elaborated, so this point is not explicitly made. (Although it is clear from _Grundrisse_ , p. 509, that Marx was well aware of it.) To summarise: Marx's claim that exchange of commodities entails their equivalence does not derive from an ahistorical and formal _concept_ of exchange, but from observation of a specific, capitalist process of exchange, in which goods actually _are_ socially commensurated, the visible expression of which is their prices.
Marx is not alone in describing this kind of exchange in terms of equivalence: it is a general feature of the work of economists of all kinds.13 Where Marx differs is in arguing that such equivalence needs a separate concept, 'value'. Why, for instance, can we not treat this equivalence simply by selecting one commodity as the numeraire in terms of which the exchange values of all other commodities are presented? Does not this correspond to the capitalist economy in which the money commodity serves as numeraire? And if so, surely we must agree with Bailey that value is a 'scholastic invention' ( _Theories of Surplus Value_ , Part 3, p. 137).
The argument about the 'common element' that Marx gives in _Capital_ is quite inadequate to deal with the above point. In the first section of Chapter 1 he gives the famous 'simple geometrical example':
> 'In order to determine and compare the areas of all rectilinear figures we split them up into triangles. Then the triangle itself is reduced to an expression totally different from its visible shape: half the product of the base and the altitude. In the same way the exchange values of commodities must be reduced to a common element, of which they represent a greater or lesser quantity.' (Op. cit., p. 127).
But this fails to indicate why we should not follow the numeraire approach. Indeed it even encourages the latter, because it poses the question in terms of a process of reasoning and measurement that takes place in our heads. But, as Marx stresses in Section 4 of Chapter 1, the equivalence of commodities is not established in the same way as the equivalence of triangles, but as the result of a social process. The agents in this process do not seek to establish the interchangeability of all products, but simply to exchange their own products. The exchange ratios are formed as a result of an iterative, competitive process, not on the basis of rationally deduced formulae. Money emerges as universal equivalent, not as the result of a rational social convention, but from an unplanned historical process.
The critical point is that if we treat the equivalence of commodities in terms of a numeraire commodity, we must presuppose the equivalence of commodities, but we have still not answered the question 'As what do they become exchangeable?' In what relation do they stand in the social process that enables one commodity to become the numeraire? This point emerges much more clearly from Marx's discussion of Bailey in _Theories of Surplus Value_ , Part 3, (p. 133-47) than it does in _Capital_. Much of the most sophisticated modern economics, whether of the Sraffian or neo-classical variety, prefers to sidestep this question by not treating the formation of exchange-values as a social process at all. It assumes exchangeability and focuses almost exclusively on the question of consistency. The central question it asks is whether a set of exchange-values (prices) can be deduced from given premises which will be consistent with some criterion set by the economist, such as the reproduction of the structure of production, or the attainment by each consumer of his 'preferred' consumption bundle, given the assumptions about how economic agents react to prices. Finding such a consistent set of exchange-values is called proving the 'existence' of an equilibrium set of exchange-values. But it is a very attenuated concept of existence, referring to the formal solution of an arithmomorphic model, not to the real world process of exchange.
An earlier generation of neo-classical economists were more robust; and so are many policy-orientated neo-classical economists today, who must eschew the theoretical rigour and purity of general equilibrium models if they are to be able to make policy prescriptions. They give the same answer to the question 'As what do commodities become exchangeable?' as was given by Bohm-Bawerk: commodities become equivalents as yielders of utility, of satisfaction. The exchange process is explained in terms of commodity owners commensurating different commodities in terms of the satisfaction they bring. Marx rejects this view, but does not set out very clearly the reasons why, quite possibly because although this has come to be the dominant view among economists, it was not so in Marx's day.
Some of the argument of the first chapter of _Capital_ , I, may give the impression that Marx denied any role to use-value in the process of exchange (cf. 'the exchange relation of commodities is characterised precisely by its abstraction from their use-values', _Capital_ , I, p. 127). But as his later argument makes clear, Marx is far from denying that use-value plays an important role in the process of exchange: what he is rejecting is the idea that the _equivalence_ of commodities can be explained in terms of use-value. There are, I think, two aspects to this rejection. One is that Marx argued that it is in terms of _difference_ that use-value is important, not in terms of _equivalence_ (cf. _Capital_ , I, p. 259). The other is that Marx argued that a _purely_ subjective approach to the exchange process could not capture certain crucial features of it (cf. _Theories of Surplus Value_ , Part 3, p. 163).
To argue that commodities are equated as use-values entails the view that commodities are wanted for the utility (or satisfaction) they bring; their characteristics as particular use-values are simply a means to the end of getting satisfaction. Utility or satisfaction represents 'the common essence of all wants, the unique want into which all wants can be merged' (Georgescu-Roegen, 1966, p. 195).14 Marx, however, rejected this idea of the reducibility of wants to a common want.
> 'As use-values, commodities differ above all in quality, while as exchange values they can only differ in quantity, and therefore do not contain any atom of use value.' ( _Capital_ , I, p. 128).
And certainly everyday experience yields much support for the irreducibility of wants — bread cannot save someone dying of thirst.
The reducibility of wants remains inherent in most varieties of neo-classical price theory,15 even though the nineteenth century idea that the satisfaction yielded by a commodity could, in principle, be measured and the satisfactions yielded by different commodities added and subtracted, has been abandoned (see Georgescu-Roegen, 1966, chapter 3).
Marx's rejection of use-value as a basis for the equivalence of commodities does not mean, contrary to what is sometimes claimed, that Marx rejects _any_ subjective element as a determinant of the exchange process. Marx was prefectly well aware that
> 'Commodities cannot themselves go to market and perform exchanges in their own right. We must, therefore, have recourse to their guardians, who are the possessors of commodities.' ( _Capital_ , I, p. 178),
and he recognised that the occasion for exchange is the desire of commodity owners (for whatever reasons) for use values other than the ones they possess. But he also recognised another aspect of the exchange process, which is that while the formation of exchange-values is necessarily the result of the actions of commodity owners, to each commodity owner entering the market it appears that the exchange ratios are already given.16
> 'These magnitudes vary continually, independently of the will, fore-knowledge and actions of the exchangers. Their own movement within society has for them the form of a movement made by things, and these things, far from being under their control, in fact control them.' ( _Capital_ , I, p. 168).
In so far as each commodity owner wants to exchange his own use-value for some other use-value, the process of exchange is composed of individual, subjective acts. But in so far as the exchange-values appear to be 'given' to each commodity owner it is a general social process which takes place 'behind the backs' of the commodity owners (cf. _Capital_ , I, p. 180). Marx wishes to capture in his categories _both_ the subjective, individual and the social, general aspects of the process, to encompass
> 'the crucial ambivalence of our human presence in our history, part-subjects, part-objects, the voluntary agents of our own involuntary determinations.' (Thompson, 1978, p. 280).
It is, I think, for this reason that he treats the equivalence of commodities in a way that is often found extremely puzzling,17 as a _substantial_ equivalence. That is, Marx does not treat this equivalence as a matter of some common characteristic in terms of which commodities are commensurated by their owners; but in terms of a unifying 'common element' or 'substance' which the commodities themselves embody, and which is designated by the separate category 'value'. The equivalence of commodities is explained in terms of the nature of this substance, not in terms of subjective commensuration by commodity owners (cf. _Capital_ , I, p. 166).
Unfortunately, Marx does not explicitly discuss the implications of treating the equivalence of commodities as 'substantial', and the considerations which underlie his treatment are not introduced until Section 4 of Chapter 1, 'The Fetishism of the Commodity and its Secret.' This encourages two kinds of misconception: the misconception that Marx's method is formalist, his 'common element' simply a common characteristic in terms of which we can (subjectively) commensurate commodities; and the misconception that Marx's method is idealist, his value substance an idealist reification of the equivalence or continuity between commodities. It was on the basis of the first misconception that Bohm-Bawerk attacked Marx's argument. (See Kay, this volume, pp.50-54). And certainly if Marx's procedure had been formalist in the manner postulated by Bohm-Bawerk, it would have been totally arbitrary to locate abstract labour as the common characteristic. But Bohm-Bawerk ignores the force of the term 'substance'.
The notion that Marx's use of the term 'substance' signals an idealist, metaphysical approach has more plausibility, for 'substance' is a term with a certain philosophical history. It has frequently been used to designate an absolute entity which underlies and produces all particular forms. Thus in the work of Spinoza, there is a single substance, labelled 'God', and all material things or thoughts are conceived of as the modes of being of this entity. (See Oilman, 1976, p. 30.) Marx himself criticised Hegel for 'comprehending _substance_ as _subject'_ in _The Holy Family_ (1845) (see Arthur, 1978, p. 88); but perhaps his own method in _Capital_ is vulnerable to the same criticism, as is argued by Moore, 1971? Marx claims in _Theories of Surplus Value_ that value 'is not an absolute, is not conceived as an entity' (op. cit., Part 3, p. 130) but how far is this true?
In my view, Marx poses commodities as substantially equivalent in the same way that in natural science, light, heat and mechanical motion are posed as substantially equivalent, as forms which are interchangeable as embodiments of a common substance, which is self-activating, in the sense of not requiring some outside intervention, some 'prime mover' to sustain it and transform it, i.e. as forms of energy. Similarly different chemicals are posed as substantially equivalent as forms of self-activating matter.18 Only with such a concept is a materialist account of the process of transformation and conservation of energy and matter possible, an account of this process as one of _natural_ history, proceeding with a dynamic internal to it, and requiring no extra-natural 'cause', no _deus ex machina_ to sustain it.
There is a danger that 'energy' or 'matter' will be reified into absolute entities; but properly understood, they are not discretely distinct from particular forms of energy or matter, rather they are concepts of the continuity between these different forms. Their self-activity is not posed teleologically, as goal-directed or by design. The concept of the equivalence of forms of energy or matter in terms of the substance of energy or matter is thus a materialist, not an idealist concept.
The transformation of one commodity into another, insofar as the rates of transformation are determined 'behind the backs' of the commodity owners, is akin to a process of natural hstory, a process that seems to have objective 'laws' of its own which operate over and above the volitions of the individuals carrying it out. Hence Marx poses this process in terms of substantial equivalence, but with 'substance' understood in materialist terms — as an abstraction with a practical reality insofar as one form of the substance is actually transformed into another form, and not in idealist terms, as an absolute entity realising its goals.
There is an important difference between the interchangeability of forms of energy, and of commodities, the substance of the equivalence in the latter case must be human. Though value appears as a relation of objects to one another, we know that it cannot be so. As Marx tartly observes:
> 'No scientist to date has yet discovered what natural qualities make definite proportions of snuff, tobacco and paintings 'equivalents' for one another.' ( _Theories of Surplus Value_ , Part 3, p. 130).
Marx implicitly rejects the procedure of treating the process of capitalist exchange 'as if' agency could stem from some non-human source, a 'structure' or an 'invisible hand'. Though it does not appear to be so, the equivalence of commodities must essentially be a relation between people, not between the commodities as physical objects. Therefore, though the form of the relation must be posed in terms that capture its naturalistic appearance, the content of the relation must be posed in terms that capture its human essence. Hence the substance of value must be the human self-activity, the human energy, embodied in the commodities; the commodities under consideration are,
> 'products of social activity, the result of expended human energy, _materialised labour_. As objectification of social labour, all commodities are crystallisations of the same substance.' ( _Critique of Political Economy_ , p. 29).
This all seems to have been so obvious to Marx that he took it for granted without discussion.19 The underlying consideration, that the equivalence of commodities is
> 'only a representation in objects, an objective expression, of a relation between men, a social relation, the relationship of men to their reciprocal productive activity.' ( _Theories of Surplus Value_ , Part 3, p. 147)
is not made explicit until Section 4 of Chapter 1 on the fetishism of the commodity, but is, I think, present in the argument from the outset. What Marx was concerned with making explicit was 'the particular form which labour assumes as the substance of value', and he often writes as if this is the major question separating him from Ricardo, rather than more fundamental questions of the object of the theory and the method of investigation (cf. _Theories of Surplus Value_ , Part 2, p. 172). The social substance of commodities as values cannot be labour as such, for this has a two-fold nature, a qualitative aspect as concrete labour, as well as a quantitave aspect as abstract labour. As values, commodities differ only quantitatively, they are all interchangeable: their substance must be homogeneous, uniform. Thus we are led to the conclusion that the substance of value must be the abstract aspect of labour. As values, substantial equivalents, commodities must be objectifications of abstract labour.
> 'The product of labour is an object of utility in all states of society; but it is only a historically specific epoch of development which presents the labour expended in the production of a useful article as an 'objective' property of that article, i.e. as its value.' ( _Capital_ , I, p. 154).
This conclusion has been reached by starting from the simplest form of the product of labour, the commodity; splitting it into two aspects, use value and exchange-value; further examining exchange-value, as a historically specific form of exchange relation, and establishing what this form of appearance must presuppose as a product of a socio-historical process. The methodological premises required to establish this result are those of historical materialism; the 'real' premises those of capitalist commodity exchange. If they are rejected, then the result cannot be established.
The argument in this phase of analysis concludes that the equivalence of commodities presupposes the objectification of the abstract aspect of labour, but it does not show how such objectification can take place. In fact it is a rather puzzling conclusion, as Marx signals with his use of the phrase 'phantom-like objectivity' ( _Capital_ , I, p. 128). The next stage of the argument, the phase of synthesis, attempts to show how objectification of abstract labour does take place, and how the abstract aspect of labour becomes dominant. At the same time it shows the problematical character of this domination, its tenuous and transient character, the fact that once achieved it is not immutably fixed, but liable to disintegration as a result of its own internal oppositions.
It has been argued by Itoh, 1976, that there is an inconsistency in the first chapter of _Capital_ , I, between Sections I and II (the phase of analysis) and Section III (the phase of synthesis) because the first two sections rest on the assumption of the interchangeability of commodities, and the third points to the difficulties of this interchange, to the fact that the equivalence can break down. For Itoh this implies that there is a Ricardian residual in Marx's argument in the first two sections. I disagree with this conclusion. In my view, there is no inconsistency. It is rather that Marx begins the analysis from the most immediate appearance of the commodity, as a product of labour interchangeable with, in a relation of equivalence to, a multitude of other products; in effect, from a set of equilibrium exchange relations. This appearance does not directly signal the problematical character of the equivalence of commodities, and hence among other things lends plausibility to the idea that aggregate supply is always equal to aggregate demand (Say's Law). Marx was, I think, well aware that this appearance of equilibrium is a one-sided abstraction from a process which is fundamentally one of disequilibrium. The second phase of the argument shows the contradictions of exchange equivalence, and makes apparent the necessity of revising the impressions that stem from the immediate appearance of exchange-value.
**3. The phase of synthesis: from value to price**
The phase of synthesis encompasses the whole of the rest of Part One of _Capital_ , I. In it Marx discusses the way that the objectification of abstract labour occurs and how this entails the dominance of abstract labour; and also shows the precarious nature of this objectification. It is about the operation of the 'law of value' which fundamentally means the 'law' of the process by which abstract labour is objectified. The term 'law' and the explicit comparison of the law of value with 'a regulative law of nature' (cf. _Capital_ , I, p. 168) is once more a reference to the naturalistic aspect of this process, the fact that it takes place 'behind the backs' of the commodity owners. But it is important to note that Marx does not have a rigid, 'deterministic' concept of a 'regulative law'. He criticised such a concept in one of his earliest writings on political economy:
> '... Mill succumbs to the error, made by the entire Ricardo School, of defining _abstract law_ without mentioning the fluctuations or the continual suspension by which it comes into being... the monetary co-incidence (of cost of production and price) is succeeded by the same fluctuations and the same disparity. This is the _real_ movement, then, and the above-mentioned law is no more than an abstract, contingent and one-sided moment in it.' ('Excerpts from James Mill's Elements of Political Economy', _Early Writings_ , p. 260.)20
And he was careful to avoid such an 'abstract law' in the argument of _Capital:_
> 'Under capitalist production, the general law acts as the prevailing tendency only in a very complicated and approximate manner, as a never ascertainable average of ceaseless fluctuations.' ( _Capital_ , III, p. 161).
The 'law of value' is often posed as a relation between value and price, but this is because price is the form through which the objectification of abstract labour is achieved. Establishing this result is the first step of the phase of syntheses.
The problem is to explain the process by which abstract labour, an aspect of labour, becomes 'objectified' as the value of a commodity. Marx's argument is that this requires the abstract labour embodied in a commodity (e.g. linen) to be expressed 'objectively', as a 'thing which is materially different from the linen itself and yet common to the linen and all other commodities' ( _Capital_ , I, p. 142). This can be done if one commodity functions as the bearer of value (or value-form), and reflects the value of the commodities exchanged with it. Section III of chapter 1, _Capital_ , I, is devoted to exploring the implications of this 'determination of reflection' (cf. _Capital_ , I, p. 149). The simplest implication is that,
> '... the natural form of commodity B becomes the value form of commodity A, in other words the physical body of commodity B becomes a mirror for the value of commodity A'. ( _Capital_ , I, p. 144).
Marx calls the commodity which serves as the bearer of value the equivalent form; and the commodity whose value is being reflected, the relative form. The next implication that Marx draws, is that in order to function as a bearer or representation of value, the equivalent form must be 'directly exchangeable' ( _Capital_ , I, p. 147). That is, its exchangeability (the possibility of exchanging it) must not depend upon its own use-value, nor on the character of the actual, individual labour embodied in it. In this it must differ from all other commodities, where, as we have already seen, their use-value and the private characteristics of their owners play a role in their exchangeability. In the case of the equivalent form, its exchangeability must instead depend upon its _social_ position as equivalent. But this social position 'can only arise as the joint contribution of the whole world of commodities' ( _Capital_ , I, p. 159). That is, no individual commodity owner can decide to make his commodity an equivalent form: this can only come about as the byproduct of the actions of each commodity owner trying to exchange his own commodity for others he would rather have (see also _Capital_ , I, p. 180).
Direct exchangeability will remain in only an embryonic form unless the equivalent form is a _universal_ equivalent, in which _all_ other commodities have their abstract labour objectified, their value reflected. The physical form of such a universal equivalent 'counts as the visible incarnation, the social chrysalis state, of all human labour' ( _Capital_ , I, p. 159). The full establishment of direct exchangeability requires a further condition that there should be a _unique_ universal equivalent, a commodity whose 'specific social function, and consequently its social monopoly (is) to play the part of universal equivalent in the world of commodities' ( _Capital_ , I, p. 162).
And at this point we can make an empirical check on the line of argument. The argument has implied that in capitalist societies there should be a tendency for one commodity to be excluded from the ranks of all other commodities, to have conferred upon it the social monopoly of direct exchangeability with all other commodities. Can such a commodity be found? If not, then something must be wrong with Marx's argument. On inspection we do find such a commodity: gold-money. The implication is not, of course that the universal equivalent must _always_ be gold money. As we have already seen, Marx goes on to note that gold, for some purposes, can be replaced as universal equivalent by symbols of itself, by paper money. The implication is rather, that gold-money as the universal equivalent is a necessary precursor to paper money. At the root of the argument here is Marx's rejection of the view that the universal equivalent can be established 'by a convention', i.e. by a conscious and simultaneous decision of all commodity owners to invest some material form with the properties of universal equivalent. Rather he takes the view that 'Money necessarily crystallises out of the process of exchange' ( _Capital_ , I, p. 181), and that it certainly cannot be treated 'as if' established 'by a convention'.
The fact that we do find a commodity with the social monopoly of direct exchangeability with all other commodities does not prove the correctness of Marx's argument that such a commodity is the visible expression of objectified abstract labour. Rather it has the negative effect of not disproving it, of not halting the line of argument, but allowing it to proceed. This is all an empirical check on the argument can ever do. The question of when we have sufficiently grasped the real relations under investigation, when we know enough about them to proceed to practical action, is not one that can ever be finally decided by an empirical test. It must always be a matter of judgement.
There is a problem with Marx's exposition of the role of gold-money as universal equivalent, 'direct incarnation of all human labour', in that he does not distinguish sufficiently clearly between money as a medium of exchange and the money form of value (money as universal equivalent). Money in itself is not specific to the capitalist mode of production (see Brenner, 1977), and the fact that money is functioning as a medium of exchange does not mean that it is functioning as an expression of value, the 'direct incarnation of all human labour'. This distinction is ellided in many of the statements made in Chapter 2, 'The Process of Exchange', creating the impression that where there is money, there is also value. Money as medium of exchange is certainly a necessary precursor to the money form of value, but in Chapter 2 Marx overstresses the continuity at the expense of the difference. To recapitulate the argument: beginning from an economy in which the capitalist mode of production is dominant and in which there are capitalist relations of exchange (i.e. the general exchangeability of products of labour through a process of sale and purchase), we arrived through analysis at the conclusion that this presupposes value (i.e. the objectification of abstract labour); we then considered the conditions for the objectification of abstract labour and concluded that this implies a universal equivalent that reflects and is the expression of value. Gold-money in capitalist economies does have the characteristics necessary for being a universal equivalent. But being a universal equivalent is itself predicated upon the social relations of the capitalist mode of production.
Marx's line of argument is not formalist but begins from real premises in the specific social relations of capitalism; and it does survive empirical checks, in that a social phenomenon can be found corresponding to what is posited by the argument of the phase of synthesis. Nevertheless, it leads us to an extraordinary conclusion, the extraordinariness of which Marx notes quite explicitly in the last section of Chapter 1, _Capital_ , I:
> 'If I state that work or boots stand in a relation to linen because the latter is the universal incarnation of abstract human labour, the absurdity of the statement is self evident. Nevertheless, when the producers of coats and boots bring these commodities into a relation with linen, or with gold or silver (and this makes no difference here), as the universal equivalent, the relation between their own private labour and the collective labour of society appears to them in exactly this absurd form'. ( _Capital_ , I, p. 169).
The point is made even more vividly in a passage included in the First Edition of _Capital_ , but not in subsequent editions, and recently brought to our attention by Arthur, 1978. The objectification of abstract labour through its incarnation in the universal equivalent
> '... is as if alongside and external to lions, tigers, rabbits and all other actual animals, which form grouped together the various kinds, species, sub-species, families etc. of the animal kingdom, there existed also in addition the _animal_ , the individual incarnation of the entire animal kingdom.' (quoted by Arthur, 1978, p. 98).
The objectification of abstract labour entails its dependent expression in a determinate form, the form of the money commodity. But does not this conclusion, that objectified abstract labour (value) has an independent expression, undermine Marx's claim that value is not conceived as an absolute entity? Here it is helpful to bear in mind another little-noticed distinction drawn by Marx, that between 'internal independence' and 'external independence' (cf. _Capital_ , I, p. 209). Value lacks the 'internal independence' necessary for it to be an entity because it is always one side of a unity of value and use-value, i.e. the commodity. But the value side of the commodity can be given 'external independence' if the commodity is bought into a relation with another commodity which serves only to reflect value. This produces the illusory appearance that value in its money form _is_ an independent entity; but the autonomy it confers on value is only relative. It is this externally independent expression, in objectified form, of a one-sided abstraction, the abstract aspect of labour, which is the fetishism of commodities. Unlike the fetishism of 'the misty realm of religion' it is not an ideological form, a product of our way of looking at things; but a product of the particular form of the determination of labour, of particular relations of production.
In the form of the universal equivalent, abstract labour is not only objectified: it is established as the dominant aspect of labour. The concrete aspect serves only to express the abstract aspect of human labour; for the usefulness of the labour embodied in the universal equivalent consists in 'making a physical object which we at once recognise as value' ( _Capital_ , I, p. 150). The private aspect of the labour embodied in it serves only to express the social aspect: individual producers cannot decide to produce the universal equivalent until it has already been established as universal equivalent by a 'blind' social process. The social aspect of the labour embodied in it, its social necessity, consists in producing a commodity which functions simply as the incarnation of abstract labour. This does not mean that the private, concrete and social aspects of labour are being extinguished, obliterated; that the labour embodied in the universal equivalent is simply abstract labour. What it means is that other aspects of labour are subsumed as expressions of abstract labour. The form of the universal equivalent reflects only abstract labour.
The argument of _Capital_ , I, goes on to show the dominance of the universal equivalent, the money form of value, over other commodities, and how this domination is expressed in the self-expansion of the money form of value i.e. in the capital form of value. Further it shows that the domination of the capital form of value is not confined to labour 'fixed' in products, it extends to the immediate process of production itself, and to the reproduction of that process. The real sub-sumption of labour as a form of capital (see _Results of Immediate Process of Production_ , p. 1019-1038) is a developed form of the real subsumption of the other aspects of labour as expressions of abstract labour in the universal equivalent, the money form of value.
In discussing the domination of objectified abstract aspect of labour, through the capital form of value, Marx refers to value as 'the subject of a process', valorising itself 'independently' ( _Capital_ , I, p. 255). Here again it seems as if value is being posed as an absolutely independent entity. It is indeed these references which form the point of departure of the capital-logic approach. It does seem as if here is a case where Marx is mistaking 'the movement of the categories' for the 'real act of production'. But we need to recall the distinction, made earlier, between external and internal independence; and the fact that these references occur in a discussion of the circulation of capital, i.e. of the form of appearance of valorisation in money terms. At this level it certainly appears that value is 'the subject of a process, endowed with a life of its own. But there is more to it than immediately meets the eye; which Marx signals in these ironic words:
> 'By virtue of being value, it has acquired the occult ability to add value to itself. It brings forth living offspring, or at least lays golden eggs.' ( _Capital_ , I, p. 255.)
We are reminded of the ironical references to the mysterious abilities of the commodity, its 'metaphysical subtleties and theological niceties', at the beginning of the section on the fetishism of commodities ( _Capital_ , I, p. 162). In my view, value appearing as the subject of a process', valorising itself 'independently' is posed by Marx as one more aspect of the fact that,
> 'the commodity reflects the social characteristics of men's own labour as objective characteristics of the products of labour themselves, as the socio-natural properties of these things.' ( _Capital_ , I, p. 165.)
The 'determination of reflection' whereby the abstract labour of one commodity is objectified by its expression in the money form of value is what underlies Marx's statements about the relation of value to price (exchange-value expressed in the money form). It should be clear from earlier sections of this paper that the references in the first two sections of Chapter 1, Volume I of _Capital_ (i.e. the phase of analysis) to the determination of the magnitude of value by labour-time do not constitute an argument about the relation of value and price, but about the relation of value and its internal measure. It is in Sections 3 and 4 of Chapter 1 that we find the first references to value as a regulator of exchange ratios, most notably:
> 'It becomes plain that it is not the exchange of commodities which regulates the magnitude of their value, but rather the reverse, the magnitude of the value of commodities which regulates the proportion in which they exchange.' ( _Capital_ , I, p. 156.)
and,
> '... in the midst of the accidental and ever-fluctuating exchange relations between the products, the labour-time socially necessary to produce them asserts itself as a regulative law of nature'. ( _Capital, I, p_. 156.)
It will be apparent from my earlier argument that it would be a mistake to interpret 'regulate' in terms of a relation between a dependent and an independent variable. Rather we should understand it in terms of the way in which the inner character of some form regulates its representation at the level of appearance, its reflection. Thus the molecular structure of a chemical substance regulates the representation of the substance in the form of a crystal, and the cell-structure of a living organism regulates the form of the organism's body.
We should note that in the passages quoted above, Marx confines himself to saying that values 'regulate' exchange ratios. He says nothing specific about the form of this regulation; in particular, he does not commit himself to the view that the exchange ratios expressed in the equivalent form, directly represent the magnitude of values (i.e. that prices are equal to values). There is a passage in the discussion of the General Form of Value which is rather more ambiguous:
> 'In this form, when they are all counted as comparable with linen, all commodities appear not only as qualitatively equal, as values in general, but also as values of quantitatively comparable magnitude. Because the magnitudes of their values are expressed in one and the same material, the linen, these magnitudes are now reflected in each other. For instance, 10 lb of tea = 20 yards of linen, and 40 lb of coffee = 20 yards of linen. Therefore, 10 lb of tea = 40 lb of coffee, in other words, 1 lb of coffee contains only a quarter as much of the substance of value, that is, labour, as 1 lb of tea.' ( _Capital_ , I, p. 159.)
The last sentence certainly suggests an equality of magnitude of value and price. (Marx argues that here linen is playing the role of money). But I think we have to pay particular attention to the unstressed reference to 'appearance'. Marx in this stage of the argument is returning from consideration of the inner substance of the relations between commodities to their appearance. The point is that on the basis of the investigation so far, it appears that commodities exchange in ratios which reflect directly the magnitude of their values, and there is as yet no basis for challenging that appearance. In writing _Capital_ , I, Marx was however well aware that at a later stage of the investigation conclusions based on this appearance would have to be challenged. He signals this in his footnote reference to 'the insufficiency of Ricardo's analysis of the magnitude of value' which 'will appear from the third and fourth books of this work' ( _Capital_ , I, p. 173).
Such an awareness is not to be found in _Critique of Political Economy_ published in 1859, eight years before _Capital_ , I, and which does not contain the same careful distinction between substance (or inner structure) and appearance, failing, for instance, to make a systematic distinction between value and exchange-value.
In _Capital_ , I, Marx takes no steps to dispel the appearance that prices directly represent values as magnitudes. But this is not quite the same as making the assumption that prices are approximately equal to values, and subsequently relaxing it. Rather, in _Capital_ , I, the argument abstracts from consideration of the social relations that imply that prices cannot directly represent the magnitude of values. This is often explained in terms of _Capital_ , I, dealing with 'capital in general' and _Capital_ , III, where the form of representation of the magnitude of value is explicitly considered, dealing with 'many capitals' (cf. Rosdolsky, 1977, p. 41-50). The trouble with this explanation is that it often leads to confusion about competition: to the view, for instance, that _Capital_ , I, abstracts from competition. This is clearly not the case: competition is an essential feature of capitalism; capital can only exist in the form of many capitals. It is not competition that Marx abstracts from in Volume I, but the question of the distribution of value between capitals.
More helpful is the distinction that Marx himself makes at the beginning of _Results of the Immediate Process of Production_ , a distinction between considering the commodity simply as the product of labour, and considering it as the product of capital (i.e. of self-valorising labour). Marx indicates that his procedure in Volume I is to begin from the commodity viewed simply as the product of labour, because this is its immediate form of appearance. The investigations of Volume I show precisely the superficiality of this immediate appearance of the commodity, revealing that the commodity, as the 'immediate result of the capitalist process of production', embodies not only value, but also surplus value; is represented not only in the price but in the profit form.
This forces a reconsideration of the representation of magnitudes of value by prices, which is undertaken in _Capital_ , III, where the concept of price of production is elaborated. A discussion of the adequacy of the conclusions reached is beyond the scope of this paper. Here we need merely note that the analysis of the relation between prices and values presented in Volume III does not rest on different premises from that offered in Volume I, but is a further development of the same analysis, attempting to encompass features of the capitalist mode of production from which Volume I abstracts.
Marx not only claims that values regulate, in the sense explained, prices. He also points to the possibility of breakdown of this regulation. In order for the abstract aspect of the labour embodied in a commodity to be objectified, the commodity must have a price. But this price
> 'may express both the magnitude of value of the commodity and the greater or lesser quantity of money for which it can be sold under given circumstances. The possibility, therefore, of a quantitative incongruity between price and magnitude of value, i.e. the possibility that the price may diverge from the magnitude of value, is inherent in the price-form itself.' ( _Capital_ , I, p. 196).
Money as universal equivalent is a necessary condition for the objectification of abstract labour, but not a sufficient condition for its objectification in a quantitatively determinate, socially necessary form. The realisation of the magnitude of value in the price form is precarious because of the relative autonomy of the circulation of money from the production of commodities. In the relation between the two processes,
> 'commodities as use-values confront money as exchange values. On the other hand, both sides of this opposition are commodities, hence themselves unities of use-value and value. But this unity of differences is expressed at two opposite poles, and at each pole in an opposite way.' ( _Capital_ , I, p. 199).
There is no necessary relation between relinquishing one's own use-value in the commodity form and acquiring someone else's use-value; for one can choose to hold money, a commodity which, unlike any other, is normally exchangeable at any time for any commodity. But the magnitude of value of money is necessarily indeterminate, for there is no universal equivalent uniquely reflecting its value, but a whole series of reflections in the quantities of all other commodities that a given amount of money will purchase (see _Capital_ , I, p. 147). The timing and sequence of purchases and sales of different goods can thus have an independent effect upon prices, and at any moment in time there is no necessary identity of aggregate sales and aggregate purchases.
But if the assertion of the relative autonomy of the circulation of money from the production of commodities
> 'proceeds to a certain critical point, their unity violently makes itself felt by producing–a crisis. There is an antithesis, immanent in the commodity, between use-value and value, between private labour, which must simultaneously manifest itself as directly social labour, and a particular concrete kind of labour, which simultaneously counts as merely abstract universal labour, between the conversion of things into persons and the conversion of persons into things; the antithetical phases of the metamorphosis of the commodity are the developed forms of motion of this immanent contradiction. These forms therefore imply the possibility of crises, though no more than the possibility. For the development of this possibility into a reality a whole series of conditions is required, which do not yet exist from the standpoint of the simple circulation of commodities.' ( _Capital_ , I, p. 209.)
Our observations of capitalist economies tell us that not only is this possibility of crisis realised, it is also — temporarily — resolved, in the sense that restructuring takes place and there is recovery from the crisis. Thus there are clearly limits to the extent to which the circulation of money departs from the production of commodities; or, in other words, to the extent to which price departs from the magnitude of value. What sets these limits can only be established after a good deal more investigation. Given the categories of analysis established so far, all that we can say is that these limits must take the form of some pressure on commodity producers to represent labour-time expended in production in money terms, to account in money terms for every moment.21 To establish how such pressure is brought to bear requires an analysis of capitalist production. It is quite illegitimate to argue that the pressure must come from capital's 'need' to reproduce itself. Here I am in agreement with Cutler et al. who reject such reasoning as functionalist and economistic (op. cit. 1977, p. 71). But I would also stress that nowhere does Marx present an argument of this type.
It is true that the investigations of _Capital_ , I, proceed for the most part on the assumption of equilibrium — the reflection of the magnitude of value in the price of commodities (exchange of equivalents) — rather than on the assumption of disequilibrium — the failure of this reflection to be quantitatively determinate (exchange of non-equivalents). But this is because the assertion of the relative autonomy of the circulation of money from the production of commodities shows up in terms of the distribution of profit between capitals (see _Capital_ , I, p. 262-6 for a preliminary indication of this), precisely the question from which Marx abstracts in Volume I. The major concern of Volume I is to establish how it is that labour comes to count 'simply as a value-creating substance', how this entails the subsumption of labour as a form of capital. In doing this Marx follows the procedure of first examining the equilibrium aspect of the process he is considering, its 'law', but he also indicates that this is merely one side of the process, and that the forms of the process of the determination of labour in capitalist economies imply disequilibrium and crisis, just as much as equilibrium and 'law'
**4. The political implications of Marx's value analysis**
We began by rejecting the view that Marx's value analysis constitutes a proof of exploitation, but argued that such a rejection did not necessarily lead to a de-politicisation of that analysis. We must now briefly return to the question of politics; briefly, because any attempt to treat this question in depth would require at least another essay. In my view the political merit of Marx's theory of value, the reason why it is helpful for socialists, is that it gives us a tool for analysing how capitalist exploitation works, and changes and develops; for understanding capitalist exploitation in process. And as such, it gives us a way of exploring where there might be openings for a materialist political practice, a practice which in Colletti's words 'subverts and subordinates to itself the conditions from which it stems' (Colletti, 1976, p. 69).
In support of this view I will make just three short points: firstly, the theory of value enables us to analyse capitalist exploitation in a way that overcomes the fragmentation of the experience of that exploitation; secondly, it enables us to grasp capitalist exploitation as a contradictory, crisis-ridden process, subject to continual change; thirdly, it builds into our understanding of how the process of exploitation works, the possibility of action to end it.
The first point stems from the premise that those who experience capitalist exploitation do not need a theory to tell that something is wrong. The problem is that the experience of capitalist exploitation is fragmentary and disconnected, so that it is difficult to tell exactly what is wrong, and what can be done to change it. In particular, there is a problem of a bifurcation of money relations and labour process relations, so that exploitation appears to take two separate forms: 'unfair' money wages or prices, and/or arduous work with long hours and poor conditions. The politics that tend to arise spontaneously from this fragmented experience is in turn bifurcated: it is a politics of circulation and/or a politics of production. By a politics of circulation I mean a politics that concentrates on trying to change money relations in a way thought to be advantageous to the working class. Examples are struggles to raise money wages, control money prices; control and remove the malign influence of the operation of the financial system, direct flows of investment funds; make transfers of money income through a welfare state, etc. By a politics of production, 1 mean a politics that concentrates on trying to improve conditions of production; shorten the working day, organise worker resistance on the shop-floor; build up workers' co-operatives, produce an 'alternative plan' (cf. Lucas Aerospace Workers Plan), etc. Both these kinds of politics have been pursued by the labour movement in both Marx's day and ours. The point is not that these kinds of politics are in themselves wrong, but that they have been pursued in isolation from one another (even when pursued at the same time by the same organisation), as if there were two separate arenas of struggle, circulation and production; money relations and labour process relations.
What Marx's theory of value does is provide a basis for showing the link between money relations and labour process relations in the process of exploitation. The process of exploitation is actually a unity; and the money relations and labour process relations which are experienced as two discretely distinct kinds of relation, are in fact onesided reflections of particular aspects of this unity. Neither money relations nor labour process relations in themselves constitute capitalist exploitation; and neither one can be changed very much without accompanying changes in the other. (For examples of Marx's argument on this point, see 'Wages, Price and Profit' in Marx-Engels, _Selected Works_ , Vol. 2; and _Critique of Political Economy_ , p. 83-6). Marx's theory of value is able to show this unity of money and labour process because it does not pose production and circulation as two separate, discretely distinct spheres, does not pose value and price as discretely distinct variables.
The importance of the second point, that capitalist exploitation is analysed as a contradictory process, not a static 'fact', is that it enables us to grasp both how exploitation survives, despite the many changes in its form, changes which the politics of circulation and the politics of production have helped to bring about; and also how it has an inbuilt tendency to disintegrate in the form in which it exists at any moment, and to be constituted in another form. The key to understanding this contradictory process is that although money relations and labour process relations are aspects of the same unity, internally dependent on other, they are nevertheless relatively autonomous from one another. In that relative autonomy lie the seeds of potential crisis. This is important politically, not because such a crisis in itself constitutes the breakdown of capitalism — it clearly does not — but because it indicates a potential space for political action; for the self conscious collective regulation of the processes of production and distribution, rather than their regulation through 'blind' market forces.
But Marx's theory of value does not simply analyse the determination of labour in capitalist society in a way that indicates potential space for political action. Its third virtue is that it also builds into the analysis, not only potential space for political action, but the possibility of taking political action. Now the possibility of taking political action against the capitalist form of the determination of labour, against capitalist exploitation, is taken for granted by all socialists. But the strange thing is that this possibility has all too often not been built into the concepts with which socialists have analysed the process of exploitation. Instead exploitation has been analysed as a closed system, and political action against it — class struggle — has been introduced, to impinge upon this system, from the outside. It may impinge as 'the motor of history' pushing the system on over time, at a slower or more rapid pace; or as the independent variable determining the level of wages, or the length of the working day, or the particular form or tempo of the restructuring of capital after crisis. Whatever formula is used, the same drawback is there: class struggle only enters the analysis as a _deus ex machina_. This leaves us unable to think of the transition from capitalism to socialism as an historical process, a metamorphosis consciously brought about by collective action; rather than as a leap between two fixed, pre-given structures, or as a simple extension of socialist forms considered as already co-existing with capitalist ones (for a longer discussion of this point, see Elson, 1979).
Edward Thompson has recently presented an impassioned critique of Althusserian Marxism on this very point (Thompson, 1978), and it seems to me that his critique is equally applicable to the model-building of most Marxist economics; and to the relentlessly unfolding dialectic of the capital-logic school. All of them analyse capitalist exploitation without using concepts which contain _within them_ the recognition of the possibility of conscious collective action against that exploitation. There is a bifurcation between their analysis of what capitalist exploitation is, and their analysis of the politics of ending it. If the 'structure' really is 'in dominance'; if the independent variables are simply 'given', and the dependent variables uniquely determined by them; of capital really is 'dominant subject'; then we are left without a material basis for political action.
In my view, and here I differ from Thompson, the same bifurcation does not occur in Marx's _Capital_. This offers us neither a structure in dominance, nor a model of political economy, nor a self-developing, all-enveloping entity. Rather it analyses, for societies in which the capitalist mode of production prevails, the determination of labour as an historical process of forming what is intrinsically unformed; arguing that what is specific to capitalism is the domination of one aspect of labour, abstract labour, objectified as value. On this basis it is possible to understand why capital can appear to be the dominant subject, and individuals simply bearers of capitalist relations of production; but it is also possible to establish why this is only half the truth. For Marx's analysis also recognises the _limits_ to the tendency to reduce individuals to bearers of value-forms. It does this by incorporating into the analysis the subjective, conscious, particular aspects of labour in the concepts of private and concrete labour; and the collective aspect of labour in the concept of social labour. The domination of the abstract aspect of labour, in the forms of value, is analysed, not in terms of the obliteration of other aspects of labour, but in terms of the subsumption of these other aspects to the abstract aspect. That subsumption is understood in terms of the mediation of the other aspects by the abstract aspect, the translation of the other aspects of labour into money form. But the subjective, conscious and collective aspects of labour are accorded, in the analysis, a relative autonomy. In this way the argument of _Capital_ does incorporate a material basis for political action. Subjective, conscious and collective aspects of human activity are accorded recognition. The political problem is to bring together these private, concrete and social aspects of labour without the mediation of the value forms, so as to create particular, conscious collective activity directed against exploitation. Marx's theory of value has, built into it, this possibility.
Its realisation, in my view, would be helped if socialists were to use the tools which Marx's theory of value provides to analyse the particular forms of determination of labour which prevail in capitalist countries today. This essay is offered as a contribution to the restoration to working condition of those tools.
Notes
I should like to thank the many comrades in Brighton and Manchester with whom I have discussed value theory over the last few years; and in particular Ian Steedman for reading and commenting on the manuscript of this essay. The responsibility for its idiosyncracies remains mine alone. I would welcome comments from readers via CSE Books
1.Hereafter referred to as _Critique of Political Economy_.
2.As Steedman, 1976, has pointed out, Morishima's 'Generalised Fundamental Marxian Theorem' in fact incorporates a concept of value rather different from that of Marx.
3.A notable exception is the sexual division of labour. The impression that this is determined by 'natural' biological factors is not completely undermined.
4.In the technical analysis of choice theory, an individual chooses from within the choice set, but does not choose the choice set itself. The question of who chooses the choice set, or more strictly speaking, of how the choice set comes to be delineated, is a serious problem generally assumed away by exponents of choice-logic.
5.As Georgescu-Roegen puts it, _'discrete_ distinction constitutes the very essence of logic.' (Op. cit., 1966, p. 21). This interesting writer, who may be unfamiliar to CSE members, is an unconventional economist, who is well acquainted with the works of Hegel and Marx; and who critises the arithmomorphism of neo-classical economics from the stronghold of a wide knowledge of mathematics and philosophy.
6.'Value, therefore, does not have its description branded on its forehead; rather it transforms every product into a social hieroglyphic.' ( _Capital_ , I, p. 167).
7.John Gray (1799-1850) was an economic pamphleteer and utopian socialist. His scheme has many similarities to the one later put forward by Proudhon.
8.Boisguillebert (1646-1714) was a Frenchman, one of the first writers in the tradition of classical political economy.
9.By the 'capital-logic' approach, I mean the approach which one-sidedly emphasises capital (or value in process, self-expanding value) as the 'dominant subject' (cf. _Capital_ , I, p. 255). Rosdolsky, 1977, is a prominant example, and the point of departure for much other 'capital-logic' writing.
10.Although I agree with Hussain and Itoh that abstract labour is a concept pertinent to all epochs, I differ in my interpretation of what it means.
11.But not from 'simple commodity production'. As may already be apparent from my remarks on Marx's rejection of the sequential method of investigation I do not think that Marx followed Adam Smith and postulated some pre-capitalist mode of simple commodity production as the starting point for his theory of value. For a detailed treatment of this point, see Banaji, this volume, p. 14-45.
12.See Kay, this volume, for a discussion of Bohm-Bawerk's critique of Marx.
13.Cutler et al., 1977, are wrong to argue (p. 14) that marginal utility theories of commodity exchange do not explain exchange in terms of equivalence. It is perfectly true that the act of exchange is explained in terms of a _difference_ in _total_ utility, each commodity owner would get greater utility from some different combination of goods than the one he possesses, and hence enters into exchange. But the quantities exchanged and hence the rate of exchange, are explained precisely in terms of _equivalence_ of _marginal_ utility. (See for instance, Dobb, 1973, p. 183-4; Georgescu-Roegen, 1966, p. 191.)
14.Georgescu-Roegen, 1966, Chapter 3, has a useful discussion of the fundamental issue at stake here: that of the commensurability of commodities as use-values. Unfortunately, most of his argument is probably inaccessible to the non-economist.
15.Such a reduction can be avoided by postulating a lexicographic preference ordering of commodities (i.e. an ordering made on the same basis as the ordering of words in a dictionary). This gives an order of priority in which wants are to be satisfied, and entails comparability, but not commensurability, of commodities as use-values. This postulate is not the one normally adopted in proving the existence theorems of neo-classical general equilibrium theory, but I am assured that these theorems could be proved, even for lexicographic preference orderings, and hence do not depend on the reducibility of wants. I find it harder to see how the process of formation of exchange values can be explained on this basis, where the process of comparing quantities of commodities in terms of quantities of a common satisfaction is ruled out. The postulate of lexicographic preference ordering seems to me much more suited to a different task: that of explaining the choices of an individual faced with a given set of prices.
16.This 'givenness' of prices is recognised in the general equilibrium theorems of neo-classical economics. But the question of _how_ the prices are given seems no longer to be raised. An earlier generation of neo-classical economists did try to tackle this problem. For instance Walras offered an explanation in terms of _cries au hasard_ , and Edgeworth in terms of 'recontracting'. (See Schumpeter, 1963, p. 1002). Both of these are subjective explanations, in which prices are determined directly by producers, and not 'behind their backs'.
17.Cf. 'the "substance of value" — a phrase that has puzzled many modern readers', Dobb, 1971, p. 10.
18.Marx uses the term 'substance' in a chemical context in his example of the relation between butyric acid and propyl formate. ( _Capital_ , I, p. 141.) Both are forms of the same underlying chemical substance, C4 H8 O2. They are equivalent substances in their chemical composition as C4 H8 O2 but different arrangements of the atoms in the molecule give them different physical properties; but that does not mean that C4 H8 O2 is discretely distinct from either butyric acid or propyl formate — it is their essence, as opposed to their form of appearance.
19.At least I have not yet come across any explicit discussion by Marx of what he means by 'substance'; nor have I found any helpful secondary literature on this point. Perhaps any reader who has found such material would let me know.
20.Mill and Ricardo did, of course, recognise that prices in the market fluctuate considerably. But this was regarded as surface 'noise' which masked rather than manifested the underlying relations. (See Banaji, this volume, p. 14-45 for a further discussion of the relation between underlying relations and appearances in classical political economy).
21.This does not mean that every hour of labour is objectified as the same quantity of value and represented by the same quantity of money. Hours of different kinds of labour may be objectified as different quantities of value, and represented by different quantities of money. Marx deals with this question in terms of the relation between skilled and unskilled labour. It is beyond the scope of this paper to discuss the adequacy of Marx's treatment of this point, but we may note that the literature commenting specifically upon it is as full of misconceptions as the more general writings on Marx's theory of value
Nor does this mean that the purpose of value theory is to generate pricing rules by which the representation of labour-time in money must be governed to secure the reproduction of a particular pattern of labour-time expenditure. The fact that no consistent rules can be generated, in the case of joint production to link the labour-time socially necessary for the production of an individual commodity and the price of that commodity does not, therefore, invalidate Marx's value theory (for an amplification of this point see Himmelweit and Mohun, 1978, Sections 4 and 5).
Rather, Marx's value theory provides us with a tool for analysing why the elaboration of pricing rules becomes necessary in the development of capitalism, giving rise to the whole modern panoply of accountants, capital budgeting experts and value analysts (sic), and also to the concern of modern economists with finding the 'optimum' pricing rules. It also provides us with the tools to investigate a phenomenon with which Marx was little concerned, perhaps because in his day it was of little practical relevance, the _contradictions_ inherent in such pricing rules, of which the contradictions of attempts to account for the labour-time spent in joint production are a good example.
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|
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| 8,013
|
\section{Introduction}
The nonlinear transport of electrons in two-dimensional (2D)
electron systems placed in a perpendicular magnetic field has been
extensively studied in the past in connection with the breakdown of
the quantum Hall effect at high current densities.$^1$ More
recently, it was realized that the current causes substantial
modifications of the resistance even in the region of weak magnetic
fields and relatively high temperatures, when the Landau levels are
thermally mixed so the Shubnikov-de Haas oscillations (SdHO) are
suppressed.
The present interest to the static (dc) nonlinear transport in 2D
systems is stimulated by observation of two important phenomena.
First, in high-mobility systems there appears a special kind of
magnetotransport oscillations, when the resistance oscillates as a
function of either magnetic field or electric current.$^{2-4}$
Second, it is found that the current substantially decreases the
resistance even at moderate applied voltages.$^{3,5}$ The observed
phenomena are of quantum origin, they are caused by the Landau
quantization of electron states and reflect the influence of the
current on the quantum contribution to resistivity. The oscillating
behavior is explained by modification of the electron spectrum in
the presence of high Hall field,$^{2,3,6}$ while the decrease of the
resistance is most possibly governed by modification of electron
diffusion in the energy space, which leads to the oscillating
non-equilibrium contribution to the distribution function of
electrons.$^{7}$ A theory describing both these phenomena in a
unified way has been recently presented.$^8$
In contrast to the Hall field-induced resistance oscillations, the
phenomenon of decreasing resistance has not been studied extensively
in experiment. Though the available data$^5$ support the
theory$^{7,8}$ predicting nontrivial changes in the distribution
function as a result of dc excitation under magnetic fields, they
are not sufficient for definite interpretation of the observed
phenomenon in terms of this theory. For better understanding of the
physical mechanisms of nonlinear behavior, further investigations
are necessary.
In this paper, we undertake the studies of nonlinear
magnetotransport in double quantum wells (DQWs), which are
representative for the systems with two closely separated occupied
2D subbands. In contrast to the quantum wells with a single occupied
subband, the positive magnetoresistance,$^9$ which originates from
the Landau quantization, is modulated in DQWs by the
magneto-intersubband (MIS) oscillations.$^{10}$ These oscillations,
whose maxima correspond to integer ratios of the subband splitting
energy $\Delta_{12}$ to the cyclotron energy $\hbar \omega_c$, are
caused by periodic variation of the probability of elastic
intersubband scattering of electrons by the magnetic field as the
density of electron states becomes an oscillating function of
energy. As a result, the changes in the quantum contribution to the
conductivity are directly seen from the corresponding changes of the
MIS oscillation amplitudes. In particular, we observe a remarkable
manifestation of nonlinearity in DQWs, the inversion of the MIS
oscillation picture, which appears when the quantum
magnetoresistance changes from positive to negative as a result of
increased current (Fig. 1). By adopting the ideas of the theory of
Ref. 7, we explain basic features of our experimental data and
determine the inelastic relaxation time of electrons in our samples.
The paper is organized as follows. In Sec. II we describe the
experimental details and present the results of our measurements. In
Sec. III we generalize the theory of Ref. 7 to the case of
two-subband occupation. A discussion, including comparison of
experimental results with the results of our calculations, is given
in Sec. IV. The last section contains the concluding remarks.
\section{Experiment}
The samples are symmetrically doped GaAs double quantum wells with
equal widths $d_{W}=14$ nm separated by Al$_{x}$Ga$_{1-x}$As
barriers with width $d_{b}$=1.4, 2, and 3.1 nm. Both layers are
shunted by ohmic contacts. Over a dozen specimens of both the Hall
bars and van der Pauw geometries from three wafers have been
studied. We have studied the dependence of the resistance of
symmetric balanced GaAs DQWs on the magnetic field $B$ at different
applied voltages and temperatures. While similar results has been
obtained in all samples with different configuration and barrier
width, we focus on measurements performed on two samples with
barrier width $d_{b}$=1.4 nm. The samples have mobilities of $9.75
\times 10^{5}$ cm$^{2}$/V s (sample A) and $4.0 \times 10^{5}$
cm$^{2}$/V s (sample B) and total electron density $n_s =1.01 \times
10^{12}$ cm$^{-2}$. The samples are Hall bars of width 200 $\mu$m
and length 500 $\mu$m between the voltage probes. The resistance
$R=R_{xx}$ was measured by using the standard low-frequency lock-in
technique for low value of the current. We also use DC current,
especially for high-current measurements. The results obtained with
AC and DC techniques are similar. The subband separation
$\Delta_{12}$, found from the MIS oscillation periodicity at low
$B$, is 3.7 meV for sample A and 5.1 meV for sample B.
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf1.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) Magnetoresistance of the
sample A for three different currents $I$ at $T=1.4$ K. The
oscillations are inverted with the increase of the current. The
inset shows the linear and non-linear (at $I=200$ $\mu$A)
magnetoresistance in the low-field region.}
\end{figure}
The resistance of the sample A as a function of magnetic field at
different currents is presented in Figs. 1 and 2. At small currents,
the magnetoresistance is positive and modulated by the large-period
MIS oscillations clearly visible above $B=0.1$ T. The small-period
SdHO, superimposed on the MIS oscillation pattern, appear at higher
fields in the low-temperature measurements (Fig. 1). With increasing
current $I$, the amplitudes of the MIS oscillations decrease, until
a flip of the MIS oscillation picture occurs. This flip, which we
associate with inversion of the quantum component of the
magnetoresistance from positive to negative, starts from the region
of lower fields and extends to higher fields as the current
increases. Therefore, one can introduce a characteristic,
current-dependent inversion field $B_{inv}$. The inset to Fig. 2
shows the behavior of the magnetoresistance near the point of
inversion. In this point, apart from the transition from negative to
positive quantum magnetoresistance, we observe an additional feature
that looks like splitting of the MIS oscillation peaks or appearance
of the next harmonic of the MIS oscillations. This feature persists
in higher magnetic fields. In contrast to the MIS oscillations, the
SdHO are not inverted by the current, as seen in Fig. 1. However,
the SdHO amplitudes decrease as the current increases until the SdHO
completely disappear in the low-field region. We attribute this
suppression of the SdHO to electron heating at high current
densities.
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf2.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) Magnetoresistance of the
sample A for different currents at $T=4.2$ K. The inset shows
inversion of the quantum magnetoresistance around $B=0.2$ T.}
\end{figure}
The amplitudes of inverted MIS oscillations increase with
increasing current and become larger than the MIS oscillation
amplitudes in the linear regime. At low temperatures the ratio
of the corresponding amplitudes varies between 2 and 3; see Fig. 1.
However, when the current increases further, the amplitudes of
inverted peaks slowly decrease, this decrease goes faster in the
region of lower magnetic fields. This property is seen in Figs. 3
and 4, where the magnetoresistance data for the sample B is
presented. The typical current dependence of the inverted peak
amplitudes at $T=1.4$ K is shown in the inset to Fig. 3. The
behavior of magnetoresistance at 4.2 K, shown in Fig. 4, is similar.
In the chosen interval of magnetic fields, the SdHO at 4.2 K are
suppressed even in the linear regime. The splitting of the MIS
oscillation peaks is clearly visible in Fig. 4 at $I=80$ $\mu$A. For
$I = 100$ $\mu$A this splitting apparently develops in the frequency
doubling of the MIS oscillations. Further increase of the current
suppresses this feature, leading to a more simple picture of
inverted MIS oscillations.
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf3.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) Magnetoresistance of the
sample B at $T=1.4$ K. The values of the current are 10 (bold), 30,
50 (dash), 100 (bold dash), 150, 200 (short dash), and 300 (bold)
$\mu$A. The inset shows amplitudes of the inverted peaks at $B=0.34$
T.}
\end{figure}
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf4.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) Magnetoresistance of the
sample B at $T=4.2$ K. The values of the current ($\mu$A) are 1, 50,
80, 100, 120, and 150 for the curves marked by the numbers from 1 to
6, respectively. The other curves corresponds to the currents of 200
(short dash), 250 (bold dash) 300 (solid), 350 (dash), and 400
(bold) $\mu$A.}
\end{figure}
\section{Theory}
The theoretical interpretation of our data is based on the physical
model of Dmitriev {\em et al.},$^{7}$ generalized to the two-subband
case. The elastic scattering of electrons is assumed to be much
stronger than the inelastic one. This scattering maintains nearly
isotropic carrier distribution at moderate currents, when the
momentum gained by an electron moving in the electric field between
the scattering events is much smaller than the Fermi momentum. Since
the intersubband elastic scattering is also much stronger than the
inelastic scattering, the isotropic part of electron distribution
function, $f_{\varepsilon}$, is common for both subbands and depends
only on the electron energy $\varepsilon$. When the current of
density $j$ flows through the sample, the kinetic equation for this
function is written as
\begin{equation}
\frac{P}{D_{\varepsilon} \sigma_d} \frac{\partial}{\partial
\varepsilon} \sigma_d (\varepsilon) \frac{\partial}{\partial
\varepsilon} f_{\varepsilon} =- J_{\varepsilon}(f),
\end{equation}
where $P=j^2 \rho_d$ is the power of Joule heating (the energy
absorbed per unit time over a unit square of electron system)
expressed through the diagonal resistivity $\rho_d$, and
$D_{\varepsilon}$ is the density of states. The function $\sigma_d
(\varepsilon)$ can be written through the electron Green's
functions, which are determined by the interaction of electrons with
static disorder potential in the presence of magnetic field. The
free-electron states in the magnetic field described by the vector
potential $(0,Bx,0)$ are characterized by the quantum numbers $j$,
$n$, and $p_y$, where $j=1,2$ numbers the electron subband of the
quantum well, $n$ is the Landau level number, and $p_y$ is the
continuous momentum. Using the free-electron basis, one obtains
\begin{eqnarray}
\sigma_{d}(\varepsilon)= \frac{e^2}{2 \pi m} {\rm Re} \left[
Q_{\varepsilon}^{AR} - Q_{\varepsilon}^{AA} \right], \\
Q_{\varepsilon}^{ss'} = \frac{2 \omega_c}{L^2}
\sum_{nn'} \sum_{jj'} \sqrt{(n+1)(n'+1)} \sum_{p_y p_y'} \nonumber \\
\times \left< \left< G^{jj',s}_{\varepsilon}(n+1 p_y, n'+1 p'_y)
G^{j'j,s'}_{\varepsilon}(n' p'_y, n p_y) \right> \right>,
\end{eqnarray}
where $e$ is the electron charge, $m$ is the effective mass of
electron, $G^{jj',s}_{\varepsilon}$ are the retarded ($s=R$) and
advanced ($s=A$) Green's functions, and $L^2$ is the normalization
square. The Zeeman splitting is neglected, so the electrons are
assumed to be spin-degenerate. The double angular brackets in Eq.
(3) denote averaging over the random potential. In terms of the
Green's functions, the density of states is given by
\begin{equation}
D_{\varepsilon}=\frac{2}{\pi L^2} \sum_{j n p_y} {\rm Im} \left<
\left< G^{jj,A}_{\varepsilon}(n p_y,n p_y) \right> \right>= \frac{2
m}{\pi \hbar^2} \sum_j {\rm Im} S_{j\varepsilon}.
\end{equation}
The dimensionless function $S_{j\varepsilon}$ is found from the
implicit equation
\begin{eqnarray}
S_{j\varepsilon}=\frac{\hbar \omega_c}{2 \pi} \sum_{n}
\frac{1}{\varepsilon-\hbar \omega_c(n+1/2)-\varepsilon_j-\Sigma_{j
\varepsilon} }, \\
\Sigma_{j \varepsilon}=\sum_{j'} \frac{\hbar}{\tau_{jj'}}
S_{j'\varepsilon},~~~~~~~~~ \nonumber
\end{eqnarray}
where $\omega_c$ is the cyclotron energy, $\varepsilon_j$ is the
subband energy, and $\tau_{jj'}$ are the quantum lifetimes of
electrons with respect to intrasubband ($j'=j$) and intersubband
($j' \neq j$) scattering. Equation (5) is valid when the correlation
length of the disorder potential is smaller than the magnetic
length, and the disorder-induced energy broadening of the subbands
is smaller than the subband separation $\Delta_{12}=\varepsilon_2-
\varepsilon_1$. It corresponds to the the self-consistent Born
approximation (SCBA).
According to the definition (2), the diagonal conductivity is
\begin{equation}
\sigma_d=\int d \varepsilon \left(-\frac{\partial
f_{\varepsilon}}{\partial \varepsilon} \right)
\sigma_{d}(\varepsilon).
\end{equation}
Therefore, multiplying the kinetic equation (1) by the density of
states $D_{\varepsilon}$ and energy $\varepsilon$, and integrating
it over $\varepsilon$, one obtains the balance equation $P=P_{ph}$,
where $P_{ph}=-\int d \varepsilon ~\! \varepsilon D_{\varepsilon}
J_{\varepsilon}(f)$ is the power lost to the lattice vibrations
(phonons).
Below we consider the case of classically strong magnetic field,
$\omega_c \tau_{tr} \gg 1$, when $\sigma_{d}(\varepsilon)$ is
written in terms of $S_{j\varepsilon}$ as
\begin{equation}
\sigma_{d}(\varepsilon)=\frac{4e^2}{m \omega_c^2}
\left[\frac{n_1}{\tau^{tr}_{11}} ({\rm Im} S_{1 \varepsilon})^2 +
\frac{n_2}{\tau^{tr}_{22}} ({\rm Im} S_{2 \varepsilon})^2
+\frac{n_s}{\tau^{tr}_{12}} {\rm Im} S_{1 \varepsilon} {\rm Im} S_{2
\varepsilon} \right],
\end{equation}
where $n_1$ and $n_2$ are the electron densities in the subbands,
$n_s=n_1+n_2$, and $\tau^{tr}_{jj'}$ are the transport times of
electrons. Both $\tau_{jj'}$ and $\tau^{tr}_{jj'}$ are determined by
the expressions
\begin{equation}
\begin{array}{c} 1/\tau_{jj'} \\
1/\tau^{tr}_{jj'} \end{array}
\left\} = \frac{m}{\hbar^3} \int_0^{2 \pi}
\frac{d \theta}{2 \pi} w_{jj'} \left( \sqrt{ (k^2_{j} +
k^2_{j'})F_{jj'}(\theta)} \right) \times \right\{\begin{array}{c} 1 \\
F_{jj'}(\theta) \end{array} ,
\end{equation}
where $w_{jj'}(q)$ are the Fourier transforms of the correlators of
the scattering potential, $F_{jj'}(\theta)=1 - 2 k_j k_{j'} \cos
\theta/(k_j^2 + k_{j'}^2)$, and $k_j$ is the Fermi wavenumber for
the subband $j$. The electron densities in the subbands are
expressed as $n_{j}=k^2_{j}/2 \pi$.
In DQWs, where the energy separation between the subbands is usually
small compared to the Fermi energy, the difference $k^2_{1}-k^2_{2}$
is small in comparison with $k^2_{1}+k^2_{2}$ so that $n_1 \simeq
n_2 \simeq n_s/2$. Furthermore, in the symmetric (balanced) DQWs,
where the electron wave functions are delocalized over the layers
and represent themselves symmetric and antisymmetric combinations of
single-layer orbitals, one has nearly equal probabilities for
intrasubband and intersubband scattering owing to $w_{11}(q) \simeq
w_{22}(q) \simeq w_{12}(q)$, provided that interlayer correlation of
the scattering potentials is weak. Therefore, $\tau_{jj} \simeq
\tau_{12} \simeq 2 \tau$, and $\tau^{tr}_{jj} \simeq \tau^{tr}_{12}
\simeq 2 \tau_{tr}$, where $\tau$ and $\tau_{tr}$ are the averaged
quantum lifetime and transport time, respectively.
In these approximations, Eq. (7) is written in the most simple way:
\begin{equation}
\sigma_d (\varepsilon) \simeq \sigma^{(0)}_d {\cal
D}^2_{\varepsilon},~~~~ {\cal D}_{\varepsilon}=\frac{1}{2}({\cal
D}_{1\varepsilon}+{\cal D}_{2\varepsilon}), ~~~~{\cal D}_{j
\varepsilon} = 2 {\rm Im} S_{j \varepsilon}
\end{equation}
where $\sigma^{(0)}_d=\sigma^2_{\bot} \rho_0$,
$\sigma_{\bot}=e^2n_s/ m \omega_c$ is the Hall conductivity, and
$\rho_0=m/e^2 \tau_{tr} n_s$ is the classical resistivity. The
function ${\cal D}_{\varepsilon}= 1 + \gamma_{\varepsilon}$ is the
dimensionless density of states, containing oscillating (periodic in
$\hbar \omega_c$) part $\gamma_{\varepsilon}$. Therefore, it is
convenient to solve the kinetic equation by representing the
distribution function as a sum $f^{0}_{\varepsilon}+ \delta
f_{\varepsilon}$, where the first term slowly varies on the scale of
cyclotron energy, while the second one rapidly oscillates.$^{7}$ The
first term satisfies the equation
\begin{equation}
\kappa \frac{\partial^2}{\partial \varepsilon^2} f^{0}_{\varepsilon}
=- J_{\varepsilon}(f^{0}),~~~\kappa=\frac{\pi \hbar^2 j^2
\rho_0}{2m}.
\end{equation}
Solution of this equation can be satisfactory approximated by a
heated Fermi distribution. This is always true if the
electron-electron scattering dominates over the electron-phonon
scattering and over the electric-field effect described by the
left-hand side of Eq. (10). In this case, the Fermi distribution of
electrons is maintained against the field-induced diffusion in the
energy space, while the electron-phonon scattering determines the
effective electron temperature $T_e$. In the general case, a
numerical solution of Eq. (10) involving electron-phonon scattering
in the collision integral$^{11}$ confirms that $f^{0}_{\varepsilon}$
is very close to the heated Fermi distribution.
The equation for the oscillating part, $\delta f_{\varepsilon}$, is
then written in the following form:
\begin{equation}
{\cal D}_{\varepsilon} \frac{\partial^2}{\partial \varepsilon^2}
\delta f_{\varepsilon} + 2 \frac{\partial {\cal
D}_{\varepsilon}}{\partial \varepsilon} \frac{\partial}{\partial
\varepsilon} \delta f_{\varepsilon} + \kappa^{-1}
J_{\varepsilon}(\delta f) = -2 \frac{\partial {\cal
D}_{\varepsilon}}{\partial \varepsilon} \frac{\partial
f^{0}_{\varepsilon}}{\partial \varepsilon} .
\end{equation}
Below we search for the function $\delta f_{\varepsilon}$ in the
form $\delta f_{\varepsilon}=(\partial f^{0}_{\varepsilon}/\partial
\varepsilon) \varphi_{\varepsilon}$, where $\varphi_{\varepsilon}$
is a periodic function of energy. Taking into account that the main
mechanism of relaxation of the distribution $\delta f_{\varepsilon}$
is the electron-electron scattering, one may represent the
linearized collision integral $J_{\varepsilon}(\delta f)$ as
\begin{eqnarray}
J_{\varepsilon}(\delta f) = - \frac{1}{\tau_{in}} \frac{\partial
f^{0}_{\varepsilon}}{\partial \varepsilon} \frac{1}{ {\cal N} {\cal
D}_{ \varepsilon}} \sum_{j j' j_1 j'_1} M_{jj',j_1 j'_1} \left<
{\cal D}_{j \varepsilon} {\cal D}_{j_1 \varepsilon+ \delta
\varepsilon} {\cal D}_{j' \varepsilon'} {\cal D}_{j'_1 \varepsilon'
- \delta \varepsilon} \right. \nonumber \\
\left. \times [\varphi_{\varepsilon} + \varphi_{\varepsilon'}-
\varphi_{\varepsilon + \delta \varepsilon} -\varphi_{\varepsilon' -
\delta \varepsilon} ] \right>_{\varepsilon', \delta
\varepsilon},~~~~{\cal N}=\sum_{j j' j_1 j'_1} M_{jj',j_1 j'_1},
\end{eqnarray}
where $\delta \varepsilon$ is the energy transferred in
electron-electron collisions, $M_{jj',j_1 j'_1}$ is the probability
of scattering (when electrons from the states $j$ and $j'$ come to
the states $j_1$ and $j'_1$), ${\cal N}$ is the normalization
constant, and the angular brackets $\left< \ldots
\right>_{\varepsilon', \delta \varepsilon}$ denote averaging over
the energies $\varepsilon'$ and $\delta \varepsilon$. Expression
(12) is a straightforward generalization of the result of Ref. 7.
The characteristic inelastic scattering time $\tau_{in}$ describes
the relaxation at low magnetic fields, when ${\cal D}_{j
\varepsilon}$ are close to unity. In this case the collision
integral acquires the most simple form $J_{\varepsilon}(\delta f) =
-\delta f_{\varepsilon}/\tau_{in}$, i.e. the relaxation time
approximation is justified.
The resistivity $\rho_d = \sigma^{(0)}_d/\sigma^2_{\bot}$ is
written, according to Eq. (6), in the form
\begin{equation}
\rho_{d}= \rho_0 \int d \varepsilon {\cal D}^2_{\varepsilon} \left(
-\frac{\partial f^{0}_{\varepsilon}}{\partial \varepsilon} \right)
\left(1+ \frac{\partial \varphi_{\varepsilon}}{\partial \varepsilon}
\right),
\end{equation}
where we have taken into account that $\partial
f_{\varepsilon}/\partial \varepsilon \simeq (\partial
f^{0}_{\varepsilon}/\partial \varepsilon ) \left[1+ \partial
\varphi_{\varepsilon}/\partial \varepsilon \right]$. Therefore, in
order to calculate the resistivity, one should find
$\varphi_{\varepsilon}$ by using Eqs. (11) and (12). In general, Eq.
(12) is an integro-differential equation that cannot be solved
analytically. However, the property of periodicity allows one to
expand $\varphi_{\varepsilon}$ in series of harmonics,
$\varphi_{\varepsilon} =\sum_{k} \varphi_k \exp(2 \pi i k
\varepsilon/\hbar \omega_c)$, and represent Eq. (11) as a system of
linear equations:
\begin{equation}
(Q^{-1}+k^2) \varphi_k + \sum_{k'=-\infty}^{\infty}\left[
(2kk'-k'^2) \gamma_{k-k'}+Q^{-1} C_{kk'} \right] \varphi_{k'} = 2 i
k \frac{\hbar \omega_c}{2 \pi} \gamma_k,
\end{equation}
where
\begin{equation}
Q= \frac{2 \pi^3 j^2}{e^2 n_s \omega^2_c}\frac{\tau_{in}}{\tau_{tr}}
\end{equation}
is a dimensionless parameter characterizing the nonlinear effect of
the current on the transport. The matrix $C_{kk'}$, whose explicit
form is not shown here, describes the effects of electron-electron
scattering beyond the relaxation time approximation.
The harmonics of the density of states, $\gamma_k$, as well as the
coefficients $C_{kk'}$, which are expressed in terms of products of
these harmonics, are proportional to the Dingle factors $\exp(-k
\pi/\omega_c \tau)$. Therefore, searching for the coefficients
$\varphi_k$ at weak enough magnetic fields, when $e^{-\pi/\omega_c
\tau}$ is small, one can take into account only a single ($k=\pm 1$)
harmonic. Within this accuracy, one should also neglect the sum in
Eq. (14). This leads to a simple solution $\varphi_{\pm 1}=\pm i
\gamma_{\pm 1}(\hbar \omega_c/\pi) Q/(1+Q)$. Since $\gamma_{+1} +
\gamma_{-1}=-2 e^{-\pi/\omega_c \tau} \cos(\pi \Delta_{12}/\hbar
\omega_c)$, Eq. (13) is reduced to a simple analytical expression
for the resistivity:
\begin{eqnarray}
\frac{\rho_{d}}{\rho_0}= 1 + e^{-2 \pi/\omega_c \tau}
\frac{1-3Q}{1+Q}\left(1+\cos \frac{2 \pi \Delta_{12}}{\hbar
\omega_c}\right) \nonumber \\
-4 e^{-\pi/\omega_c \tau} {\cal T} \cos \left( \frac{2 \pi
\varepsilon_F}{\hbar \omega_c} \right) \cos \left( \frac{\pi
\Delta_{12}}{\hbar \omega_c} \right).
\end{eqnarray}
The second term in this expression, proportional to $e^{-2
\pi/\omega_c \tau}$, differs from a similar term of the
single-subband theory$^7$ by the modulation factor $\left[1+\cos (2
\pi \Delta_{12}/\hbar \omega_c) \right]/2$ describing the MIS
oscillations. The last term in Eq. (16) describes the SdHO, which
are thermally suppressed because of the factor ${\cal T}=(2 \pi^2
T_e/\hbar \omega_c)/\sinh(2 \pi^2 T_e/\hbar \omega_c)$. The Fermi
energy $\varepsilon_F$ is counted from the middle point between the
subbands, $(\varepsilon_1+\varepsilon_2)/2$, and, therefore, is
directly proportional to the total electron density, $\varepsilon_F=
\hbar^2 \pi n_s/2m$.
\section{Results and discussion}
The basic features of our experimental findings can be understood
within Eqs. (16) and (15). In the linear regime, when the parameter
$Q$ is small, this equation gives a good description of the MIS
oscillations experimentally investigated in Ref. 10. As the current
increases, the amplitudes of these oscillations decrease, and then
the flip occurs, when the MIS peaks become inverted. In contrast,
the SdHO peaks are not affected by the the current directly, and
their decrease is caused by the effect of heating. The flip of the
MIS oscillations corresponds to $Q=1/3$. Since $Q$ is inversely
proportional to the square of the magnetic field, there exists the
inversion field, $B_{inv}$, determined from the equation $Q=1/3$,
where $Q$ is given by Eq. (15). This feature is observed in our
experiment, see the inset to Fig. 2. For the sample B, we have
extracted $B_{inv}$ for several values of the current. The results
are shown in Fig. 5. At 4.2 K the experimental points follow the
linear $B_{inv}(I)$ dependence predicted by Eq. (15). Since the
ratio $B_{inv}/I$ is proportional to the square root of the
inelastic relaxation time $\tau_{in}$, we are able to estimate this
time from experimental data as $\tau_{in} \simeq 64$ ps at $T=4.2$
K. Assuming the $T^{-2}$ scaling of this time,$^{7}$ one obtains
$\hbar/\tau_{in}=6.6$ mK at $T=1$ K, which is not far than
the theoretical estimate $\hbar/\tau_{in} =4$ mK at $T=1$ K based
on the consideration of electron-electron scattering.$^{7}$
The positions of experimental points at $T=1.4$ K also fit within
this picture if the electron heating is taken into account. The
increase of electron temperature with increasing current (heating
effect) leads to deviation of the $B_{inv}(I)$ dependence from
linearity because of temperature dependence of $\tau_{in}$, and this
deviation is essential at $T=1.4$ K; see Fig. 5. The same consideration,
applied to the high-mobility sample A, gives the inelastic scattering
time $\tau_{in} \simeq 108$ ps at $T=4.2$ K, which is very close to
the theoretical estimate.
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf5new.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) Dependence of the
inversion field on the current for the sample B at $T=4.2$ K and
$T=1.4$ K. (points) The dashed lines correspond to a linear
$B_{inv}(I)$ dependence assuming $\tau_{in}= 64$ ps at 4.2 K (580
ps at 1.4 K). The solid lines represent the calculated $B_{inv}(I)$
dependence taking into account electron heating by the current.}
\end{figure}
When the current becomes high enough ($Q \gg 1$), Eq. (16) predicts
saturation of the resistance, when the amplitudes of inverted MIS
peaks are three times larger than the amplitudes of the MIS peaks in
the linear regime ($Q \ll 1$). We indeed observe the regime resembling
a saturation, with almost three times increase in the amplitudes of
inverted peaks for both samples at $T=1.4$ K (see Figs. 1 and 3).
For higher temperatures the behavior is similar, though the maximum
amplitudes of inverted peaks are only slightly larger than the amplitudes
in the linear regime. We explain this by the effect of heating on the
characteristic times. Though the resistivity in the high-current regime
($Q \gg 1$) no longer depends on $\tau_{in}$, there is a sizeable
decrease in the quantum lifetime $\tau$ with increasing temperature,$^{10}$
which takes place because the electron-electron scattering contributes
into $\tau$. As a result, the Dingle factor decreases, and the quantum
contribution to the resistance becomes smaller as the electrons are heated.
At higher initial temperature, when $\tau_{in}$ is smaller, the regime
$Q \gg 1$ requires higher currents. The corresponding increase in
heating reduces the quantum contribution, so the maximum amplitudes of
inverted peaks never reach the theoretical limit and are expected to
decrease with increasing initial temperature. The slow suppression
of the inverted peaks with further increase in the current (see the inset to
Fig. 3) is explained by the same mechanism. This conclusion is supported by
the experimental observation that the suppression is more efficient at lower
magnetic fields, when the Dingle factor $\exp(-\pi/\omega_c \tau)$ is more
sensitive to the temperature dependence of quantum lifetime $\tau$.
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf6new.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) Calculated
magnetoresistance of the sample B at $T=4.2$ K and different
currents: 1, 50, 80, 100, 120, and 150 $\mu$A for the curves marked
by the numbers from 1 to 6; the other curves corresponds to $I=200$
(short dash), 250 (bold dash), 300 (solid), and 400 (bold) $\mu$A.
The additional (dashed) line 1 shows the linear magnetoresistance
determined by the SCBA calculation of the density of states in Eq.
(13).}
\end{figure}
To illustrate the above-discussed relation of the basic theoretical
predictions to our experiment, we present the results of theoretical
calculations according to Eqs. (15) and (16) in Fig. 6. The
calculations are done for the sample B at 4.2 K, so the theoretical
curves show the expected behavior of the measured magnetoresistance
from Fig. 4. We take into account the effect of heating, described
by using the collision integral for interaction of electrons with
acoustic phonons$^{11}$ and temperature dependence of the quantum
lifetime $\tau$ of electrons determined empirically from the studies
of the MIS oscillations in the linear regime.$^{10}$ The theoretical
plots demonstrate a reasonable qualitative agreement with the
experiment. However, the theory predicts a slower suppression of the
inverted peak amplitudes with increasing current at weak magnetic fields.
This may be a consequence of underestimated heating,$^{12}$ because the
screening effect on the electron-phonon interaction$^{13}$ has not been
taken into account in the calculation of the power loss to acoustic
phonons. Similar calculations carried out for different samples at
different temperatures are also in agreement with experimental data.
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf7new.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) (a) Calculated
magnetoresistance of the sample B at $T=4.2$ K and $I=120$ $\mu$A.
The plot 1 correspond to simple theory [Eq. (16)], while the others
represent the results of numerical solution of Eq. (14) for the
cases of subband-independent electron-electron scattering (2) and
only intrasubband electron-electron scattering (3). (b) The same
plots, where the SdHO contribution is excluded. The density of
states is found within the SCBA.}
\end{figure}
The simple theory fails do describe the interesting and unexpected
feature observed in our experiment, the current-induced splitting of
the MIS oscillation peaks. This kind of nonlinear behavior is
well-reproducible, we see it in different samples. We have found
that a possible explanation of this feature can be based on the
theory presented in Sec. III, if higher harmonics of the
distribution function $\delta f_{\varepsilon}$ are taken into
account. We have carried out a numerical solution of the system of
equations (14) under some simplifying assumptions about the
collision integral. In the first case, we have assumed equal
probabilities for all possible electron-electron scattering
processes, so the matrix $M_{jj',j_1 j'_1}$ in Eq. (12) is replaced
by a constant. Another limiting case we consider is the complete
neglect of intersubband transitions in electron-electron collisions,
when $M_{jj',j_1 j'_1} \propto \delta_{jj_1} \delta_{j'j'_1}$. This
case is also reasonable, since electron-electron scattering at low
temperatures assumes a small momentum transfer, so the intersubband
scattering contribution should be suppressed owing to reduction of
the overlap integrals of envelope wave functions of electrons. Then,
the coefficients $\gamma_k$ and $C_{kk'}$ have been determined by
using the density of states numerically calculated within the SCBA;
see Eq. (5). The results, corresponding to $I=120$ $\mu$A for the
sample B are presented in Fig. 7. In the low-field region, where the
MIS peaks are inverted, the calculation shows a considerable
increase in their amplitudes above 0.2 T, where contribution of
higher harmonics of the density of states becomes essential. This
enhancement occurs because of the current-induced mixing between
different harmonics of the distribution function, formally coming
from the term with $\gamma_{k-k'}$ in the sum in Eq. (14). In
contrast, in the linear regime, the SCBA magnetoresistance is close
to the magnetoresistance calculated within the single-harmonic
approximation [Eq. (16)]; see Fig. 6. Above 0.27 T, where the Landau
levels become separated, one can see features associated with the
specific semi-elliptic shape of the SCBA density of states. In the
vicinity of the inversion field ($B_{inv} \simeq 0.4$ T), where the
contribution of the first harmonic of the distribution function is
suppressed ($Q \simeq 1/3$) while the higher harmonics are still
active, two sets of MIS peaks are seen. It is not surprising,
because higher harmonics of the density of states contain the
factors $\cos(k \pi \Delta_{12}/\hbar \omega_c)$ describing higher
harmonics of the MIS oscillations. Above the inversion field, the
resistance is considerably smaller than the resistance predicted by
the single-harmonic approximation, and a splitting of the MIS peaks
occurs. The splitting increases with the increase of the magnetic
field. These effects are caused by the contribution of higher
harmonics of the density of states in the collision integral.
Indeed, in the single-harmonic approximation the collision integral
contains only the outcoming term proportional to
$\varphi_{\varepsilon}$. This approximation becomes insufficient in
higher magnetic fields, when incoming terms in the collision
integral (12) are also important, so the relaxation of the
distribution function, which counteracts the diffusion of electrons
in the energy space, becomes less efficient. This means that the
effect of the current on the distribution function increases, and
the resistance is lowered. The described suppression of the
collision-integral term is more significant in the regions of the
MIS resonances, when $\Delta_{12}/\hbar \omega$ is integer, because
the peaks of the density of states are the narrowest in these
conditions, and the energies transferred in the electron-electron
collisions, $\delta \varepsilon$, are small. Away from the MIS
resonances, the energy space for electron-electron scattering
increases, especially when the intersubband transitions are allowed
(see curve 2 in Fig. 7). Therefore, the relaxation is less
suppressed as compared to the center of the MIS peak, and the effect
of the current is weaker. The above consideration explains why the
centers of the MIS peaks drop down, so the peak splitting takes
place.
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf8new.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) The same as in Fig. 7
for the Gaussian model of the density of states.}
\end{figure}
The SCBA has a limited applicability for description of the density
of electron states in the magnetic field. In particular, it leads to
non-physically sharp edges of the density of states, which generate
the harmonics $\gamma_k$ with large $k$ in Eq. (14). This apparently
leads to an overestimate of the effect of the current on the
resistance in the region where the MIS peaks are inverted, see Fig.
7. To avoid such singularities, and to have a further insight into
the problem of nonlinear magnetoresistance, we have considered the
expression
\begin{equation}
{\cal D}^{(G)}_{1,2 \varepsilon}=\frac{\hbar \omega_c}{ \sqrt{\pi}
\Gamma(\omega_c)} \sum_{n=-\infty}^{\infty} \exp \frac{[\varepsilon
\pm \Delta_{12}/2 -\hbar \omega_c(n+1/2)]^2}{\Gamma^2(\omega_c)}.
\end{equation}
which corresponds to the Gaussian model for the density of states
and describes two independent sets of Landau-level peaks from each
subband (strictly speaking, the Landau-level peaks are not
independent because of elastic intersubband scattering, as follows
from Eq. (5), see more details in Ref. 14). The magnetic-field
dependence of the broadening energy $\Gamma$ has been set to make
the first [proportional to $\cos(2 \pi \varepsilon/\hbar\omega_c)$]
harmonics of ${\cal D}^{(G)}_{j \varepsilon}$ and ${\cal D}_{j
\varepsilon}$ equal. The results of the calculations using ${\cal
D}^{(G)}_{j \varepsilon}$ instead of the SCBA density of states are
shown in Fig. 8. The magnetoresistance in the region of inversion
appears to be nearly the same as predicted by the simple
single-harmonic theory. In the region above the inversion field, the
splitting of the MIS peaks does not take place if the intersubband
electron-electron scattering is forbidden. This is understandable
from the discussion given above: if different subbands contribute
into the density of states independently, the efficiency of
electron-electron collisions does not depend on the ratio
$\Delta_{12}/\hbar \omega$ and the reduction of the collision
integral owing to incoming terms causes just a uniform suppression
of the whole MIS peak. In the SCBA, when the shape of ${\cal D}_{j
\varepsilon}$ depends on this ratio, the splitting of the MIS peaks
does not necessarily require the intersubband electron-electron
scattering.
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf9new.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) Evolution of the
nonlinear magnetoresistance calculated using the parameters of the
sample B when the current varies from 100 to 150 $\mu$A with the step
of 10 $\mu$A. The Gaussian model of the density of states and the
assumption of subband-independent electron-electron scattering are
used.}
\end{figure}
If the intersubband electron-electron scattering is allowed, the
magnetoresistance pictures obtained within the Gaussian model, as
well as within the SCBA model above the inversion point,
qualitatively reproduce the features we observe experimentally.
The results of calculations presented in Fig. 9 demonstrate
that varying the current in a relatively narrow range leads
to a dramatic reconstruction of the magnetoresistance oscillation
pattern.
\begin{figure}[ht]
\begin{center}\leavevmode
\includegraphics[width=8cm]{nonf0.eps}
\end{center}
\addvspace{-0.8 cm} \caption{(Color online) Comparison of the
measured and calculated nonlinear magnetoresistance in the sample A
at $T=4.2$ K and $I=75$ $\mu$A. The Gaussian model of the density of
states and the assumption of subband-independent electron-electron
scattering are used in the calculations.}
\end{figure}
Numerical calculation of magnetoresistance in the high-mobility
sample A also gives the results very similar to what we see
experimentally. To demonstrate this, we have put experimental and
calculated curves together in Fig. 10. Apart from a weak negative
magnetoresistance at low fields and a slight decrease in the MIS
oscillations frequency with increasing $B$ (the features we see in
all our samples$^{10,15}$ both in linear and nonlinear regimes), the
agreement between experiment and theory is good.
\section{Conclusions}
Investigation of nonlinear transport of 2D electrons in magnetic
fields enriches the knowledge of the quantum kinetic properties of
electron systems and of the microscopic processes responsible for
the observed modifications of the resistivity. In our work, we have
demonstrated that using double quantum well systems opens wide
possibilities for studying the nonlinear behavior. The presence of
the MIS oscillations, which modulate the quantum component of the
resistivity, allows us to investigate the current dependence of the
quantum magnetoresistance. In particular, we are able to determine
the magnetic fields $B_{inv}$ corresponding to the current-induced
inversion of the magnetoresistance. This inversion manifests itself
in a spectacular way, as a flip of the MIS oscillation pattern. We
point out that this behavior resembles recently observed$^{15}$
inversion of the MIS oscillations by the low-frequency (35 GHz)
microwave radiation. This is not surprising, because the physical
mechanism in both cases is similar. Apart from the flip of the MIS
oscillations, we have observed a wholly unexpected quantum
phenomenon, the splitting of the MIS oscillation peaks in the region
of fields above the inversion point $B_{inv}$.
We have shown that the theoretical explanation of all the observed
phenomena can be based on the kinetic equation for the isotropic
non-equilibrium part of electron distribution function. This
function oscillates with energy owing to oscillations of the density
of electron states in the magnetic field. The effect of electric
current on this function, the increase of electron diffusion in the
energy space, is equilibrated by the inelastic electron-electron
scattering. Theoretical explanation of the most of observed
phenomena is done in a simple single-harmonic approach, which
allowed us to determine the inelastic relaxation time $\tau_{in}$ by
comparison of experimental data with theory. The values of
$\tau_{in}$ for different samples are close to the theoretical
estimates of this time, and confirm the predicted$^7$ temperature
dependence $\tau_{in} \propto T^{-2}$. Thus, our data on the
inelastic relaxation time in double quantum well samples are in
agreement with the data obtained in single quantum well samples.$^5$
The description of the splitting of MIS oscillations requires a more
detailed numerical analysis including consideration of higher
harmonics of both the density of states and the distribution
function. Apart from the verification of the basic principles of the
theory of Ref. 7, this analysis demonstrates sensitivity of the
nonlinear behavior to the shape of the density of electron states
and to the details in description of inelastic scattering.
Therefore, investigation of nonlinear magnetoresistance in
relatively weak magnetic fields offers a tool for studying the
electron states and scattering mechanisms both in single and double
quantum wells.\\
This work was supported by CNPq and FAPESP (Brazilian agencies).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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{"url":"https:\/\/proofwiki.org\/wiki\/Polynomial_Long_Division","text":"# Polynomial Long Division\n\n## Technique\n\nLet $\\map {P_n} x$ be a polynomial in $x$ of degree $n$.\n\nLet $\\map {Q_m} x$ be a polynomial in $x$ of degree $m$ where $m \\le n$.\n\nThen $\\map {P_n} x$ can be expressed in the form:\n\n$\\map {P_n} x \\equiv \\map {Q_m} x \\map {D_{n - m} } x + \\map {R_k} x$\n\nwhere:\n\n$\\map {D_{n - m} } x$ is a polynomial in $x$ of degree $n - m$\n$\\map {R_k} x$ is a polynomial in $x$ of degree $k$, where $k < m$, or may be null.\n\nHence we can define $\\dfrac {\\map {P_n} x} {\\map {Q_m} x}$:\n\n$\\dfrac {\\map {P_n} x} {\\map {Q_m} x} = \\map {D_{n - m} } x + \\dfrac {\\map {R_k} x} {\\map {Q_m} x}$\n\nThe polynomial $\\map {R_k} x$ is called the remainder.\n\nThe procedure for working out what $\\map {D_{n - m} } x$ and $\\map {R_k} x$ are is called (polynomial) long division.\n\n## Proof\n\nLet $\\ds \\map {P_n} x = \\sum_{j \\mathop = 0}^n p_j x^j$.\n\nLet $\\ds \\map {Q_m} x = \\sum_{j \\mathop = 0}^m q_j x^j$.\n\nFirst calculate $\\map {Q'_m} x = \\map {Q_m} x \\times \\dfrac {p_n} {q_m} x^{n - m}$.\n\nThis gives:\n\n $\\ds \\map {Q'_m} x$ $=$ $\\ds \\sum_{j \\mathop = 0}^m \\frac {p_n q_j} {q_m} x^{n - m + j}$ $\\ds$ $=$ $\\ds \\sum_{j \\mathop = n - m}^n \\frac {p_n q_{j - n + m} } {q_m} x^j$ $\\ds$ $=$ $\\ds p_n x^n + \\sum_{j \\mathop = n - m}^{n - 1} \\frac {p_n q_{j - n + m} } {q_m} x^j$\n\nThen evaluate:\n\n$\\map {P'_{n - 1} } x = \\map {P_n} x - \\map {Q'_m} x$\n\nwhich (after some algebra) works out as:\n\n$\\ds \\map {P_n} x - \\map {Q'_m} x = \\sum_{j \\mathop = n - m}^{n - 1} \\frac {p_n q_{j - n + m} } {q_m} x^j + \\sum_{j \\mathop = 0}^{n - m - 1} p_j x^j$\n\nSo we see that $\\map {P_n} x - \\map {Q'_m} x$ is a polynomial in $x$ of degree $n - 1$.\n\nLet $\\dfrac {p_n} {q_m} = d_{n - m}$.\n\nHence we have:\n\n$\\map {P_n} x = d_{n - m} x^{n - m} \\map {Q_m} x + \\map {P'_{n - 1} } x$\n\nWe can express $\\map {P'_{n - 1} } x$ as:\n\n$\\ds \\map {P'_{n - 1} } x = \\sum_{j \\mathop = 0}^{n - 1} p'_j x^j$\n\nRepeat the above by subtracting $\\ds \\frac {p'_{n - 1} } {q_m} x^{n - m - 1} \\map {Q_m} x$ from $\\map {P'_{n - 1} } x$, and letting $\\dfrac {p'_{n - 1} } {q_m} = d_{n - m - 1}$.\n\nHence:\n\n$\\map {P'_{n - 1} } x = d_{n - m - 1} x^{n - m - 1} \\map {Q_m} x + \\map {P''_{n - 2} } x$\n\nThe process can be repeated $n - m$ times.\n\nIt can be seen that after the last stage, we have:\n\n$\\map {P_n} x = \\map {D_{n - m} } x \\map {Q_m} x + \\map {R_k} x$\n\nwhere:\n\n$\\ds \\map {D_{n - m} } x = \\sum_{j \\mathop = 0}^{n - m} d_j x^j$\n$\\map {R_k} x$ is a polynomial of degree at most $m - 1$.\n\n$\\blacksquare$","date":"2022-05-28 02:00:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9663230776786804, \"perplexity\": 157.62230501302426}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652663011588.83\/warc\/CC-MAIN-20220528000300-20220528030300-00495.warc.gz\"}"}
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Q: Substitution of a random variable into a stochastic integral I am faced with the following fact while reading a paper:
Let $\{ \mathcal{F}_t \}_{t \in [0,T]} $ be the filtration generated by a one-dimensional Brownian motion $\{ B_t \}_{t \in [0,T]} $ defined on some probability space. Let $\{X_t (y) \}_{t \in [s,T]}$ be an adapted process, for each $y \in \mathbb{R}.$ Then, for any random variable $\eta$ that is $ \mathcal{F}_s$-measurable, the author claims that for every $t \in [s,T],$
$$ \bigg\{ \int_s^t X_r (y) \, dB_r \bigg\} \bigg|_{y= \eta} = \int_s^t X_r (\eta) \, dB_r.$$
The argument is that the stochastic integral $\int_s^t X_r (y) \, dB_r$ is $\sigma \big\{ B_r - B_s, r \in [s,T] \big\}$-adapted and is therefore independent of $ \mathcal{F}_s$, and in particular, independent of $\eta$. Therefore, direct substitution is allowed.
I am wondering if there is any result in the literature that guarantees direct substitution of a random variable into a stochastic integral, given its independence? I cannot show it directly from its definition.
A: I think that the assertion is, in general, wrong. The problem is, essentially, the following: If $F(y)$ is for each $y \in \mathbb{R}$ a random variable which is only defined up to a null set, then the expression
$$F(y) \bigg|_{y=\eta}$$
is ill-defined. There are null sets building up; depending on which representation we choose for $F(y)$ we get totally different results.
The same problem pops up when one tries to generalize the pull out property of the conditional expectation (see this question).
Here is an illustrating example for the problem your are considering:
Fix $0 < s \leq T <\infty$ and consider
$$X_r(y) := 1_{\{y\}}(B_s) \qquad \eta := B_s.$$
Since $\mathbb{P}(B_s=y)=0$ for any $y \in \mathbb{R}$, we have by Itô's isometry
$$\mathbb{E} \left( \left| \int_s^t X_r(y) \, dB_r \right|^2 \right) ,$$
and therefore
$$F(y) := \int_s^t X_r(y) \, dB_r = 0.$$
(The stochastic integrals is only defined up to null set and therefore we can choose $F(y,\omega)=0$, $\omega \in \Omega$ as a representative.) Hence, $F(\eta) =F(B_s)=0$ almost surely. On the other hand, we have
$$\int_s^t X_r(\eta) \, dB_r = \int_s^t 1_{\{B_s\}}(B_s) \, dB_r = B_t-B_s.$$
Thus,
$$0 = F(\eta) \neq \int_s^t X_r(\eta) \, dB_r = B_t-B_s.$$
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Many time you need in your laravel application integration multiple file uploading functionality for upload any file or some specific file. you can done this type of task easily helping of dropzone js. dropzon js give to easy interface for uploading mulriple file uploading with awesome front-end design. how to integration dropzone js in your laravel application. it is so easy task. we are here provide all tutorial step by step so you can integrate or make multiple file uploading in your laravel application.
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Looking my following code sample i left empty style and jquery section. so, i write this code after blade file code. so when you can integrate it then must be write in one file depend on you.
Please also check our demo for Multiple file uploading using DropzoneJS.
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Q: How can I call a base class's method via reflection?
Possible Duplicate:
Use reflection to invoke an overridden base method
Normally I can call my base class from within an overridden method like this:
public override void Foo(Bar b)
{
base.Foo(b);
}
How can I make this same call with reflection?
Edit: to explain a bit, I'm trying to use AOP to guard my library's entry points from uninitialized operation (in my case there is no "initialize" call prior to the library's usage). So the relevant calls will technically end up inside the class (by virtue of the AOP), but the pre-compiled code will be written in a separate class. In other words, I want the following advice applied to all of my entry points:
if (!initialized)
return base.<method>(<arguments>);
I suppose the IL trick shown in Use reflection to invoke an overridden base method will work for me - I was just hoping there was something cleaner in my case since it feels more legitimate.
A: EDIT: OK, so a bit of clarification; you're not trying to call the base class's implementation from outside either class. Instead, from within the overridden class, you want to reflectively call the parent method's implementation.
... Why? Your object statically knows what it is, and therefore statically knows what its base class is. You can't have two base classes, and you can't dynamically "assign" a base class to a pre-existing child class.
To answer your question, you can't. Any attempt to reflectively call "Foo" using the current instance will infinitely recurse, because the invocation of a MethodInfo makes the method call as if it were coming from outside ('cause it is), and so the runtime will obey inheritance/overriding behaviors. You would have to use the IL hack from the related question.
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← "'Jesus is the Christ.' (Acts 9.22) Can Jesus be called Shiva?"
Audrey Hughes, Funeral Oration →
Posted on July 17, 2016 by interfaithashram
The following article John Dupuche was published in Theology@ McAuley, E-Journal, Australian Catholic University, 2004
"Renewing Christian Anthropology in Terms of Kashmir Shaivism".
Rev. Dr. John Dupuche was Pastor of Nazareth Parish, Ricketts Point, Melbourne. He is senior lecturer at the University of Divinity, and Honorary Fellow at the Australian Catholic University, and chair of the Catholic Interfaith Committee of the Archdiocese. He has a doctorate in Sanskrit, specialising in Kashmir Shaivism and is particularly interested in its interface with Christianity. His book Abhinavagupta: the Kula Ritual as Elaborated in Chapter 29 of the Tantrāloka was published in 2003; Jesus, the Mantra of God in 2005, Towards a Christian Tantra in 2009. He has written many articles in these fields.
Email: jeandupuche@gmail.com
website: johndupuche.com
My good friend Bettina Bäumer[1] relates the following story:
"It was a seminar in Vienna University where [Karl Rahner] also spoke and I gave my first ever paper on KS [Kashmir Shaivism] on anupāya. After listening very attentively, he took me aside after the discussion and said [Wir sind nur Waisenkinder] [which she glosses as] "we are orphans compared to what these Indians have discovered!" (Waisenkinder means we are far behind or more primitive, spiritually)."[2]
The first generations of Christians moved out of the Jewish framework into the thought-world of the Greeks and reinterpreted their faith in a new way. Now with the end of the colonial era, where the East was interesting only if it was exotic, we are witnessing a massive new shift. Rahner's comment to Bettina Bäumer reflects his awareness that the Hindu thought must profoundly affect Christian theology, making Christians qualify categories and images that are so familiar as to be unquestioned.
Christian anthropology, as presently understood, is profoundly dualistic: God and man, heaven and earth, nature and grace, faith and reason, Church and State, sin and grace, good and evil etc. But St Paul says: "all are one in Christ Jesus".[3] New anthropologies are needed.[4]
The method of this paper is to present some aspects of Indian and Christian thought. I will weave between Christianity and Kashmir Shaivism ending not with syncretism but reinterpretation. I will speak of consciousness in place of the word 'God', of emanation in place of creation, of ignorance in place of sin, recognition in place of redemption, of identity instead of faith, of universal bliss instead of eternal life.
These pairs of terms – consciousness / God etc. – are not deemed to be equivalent. Neither are they being compared but only connected. What light can one throw on the other? What questions are posed? Can the Christian experience be expounded – not falsely – in these terms, given, as we know, that Christian vocabulary cannot adequately express Christian experience? Can these Sanskrit terms become the vehicle for a theology which leads to the knowledge of the Christ who exceeds all that can be said of him?
This attempt will be the beginnings of a Shaiva Christianity or a Christian Shaivism.
It is part of the future task of theology. In the opinion of David Tracy "the inter-religious dialogue will become an integral part of all Christian theological thought."[5]
God and consciousness:
In the Shaivism of Kashmir, consciousness, also called 'Śiva', is pure awareness without any object of awareness. However, consciousness is not ignorant of itself. Awareness is self-aware not dividedly but identically. This auto-illumination of consciousness is the Supreme Word (paravāc) and is expressed as "I am" (aham). This consciousness is not the impersonal Brahma as in the famous phrase "Thou art That" (tat-tvam-asi) which is found in the reflections of Raimon Panikkar. Rather, in Kashmir Shaivism the ultimate reality is supremely personal but not individual, always Subject and never object.
The divine Subject cannot, therefore, ultimately be the object of fitting discourse but transcends all that can be said. Discourse about God gives way to silence and union, not as subject to subject but as identity, one Subject, "God who is all in all."[6]
Creation and emanation
a. Like the mirror which can reflect any object precisely because it does not necessarily portray any particular object, so too the Supreme Word contains every expression and is limited to none. Out of freedom (svātantrya), indeed out of a sense of play (līlā) the Word is expressed in the multiplicity of the universe. This universe is therefore the expression of Consciousness who both transcends the expression and is the expression; just as the dancer is the stance he adopts and is not confined to that stance. Śiva is his work, yet at the same time transcends his work. The universe is the dance of Śiva Naṭarāja ('Lord of the Dance').
This dance is at the same time emanation (sṛṣṭi), maintenance (stithi) and dissolution (saṁhāra) since all is flux and change in this vibrating universe.
b. A few words now, on the Judeo-Christian idea of creation, which may at first seem totally different from the Hindu view.
The Hebrew word ōlām first meant both heaven and earth. It is only in later Hebrew that it came to mean the 'world'. The Greek word kosmos, for its part, refers to the order of the universe formed out of pre-existent chaos.[7] The Septuagint, therefore, in choosing the word kosmos to translate the Hebrew ōlām colours the meaning of this latter term.
The term kosmos occurs most frequently in the Johannine writings, some 105 times, which is two and a half times more frequently than in the rest of the New Testament.[8] It can have a quite neutral meaning in itself[9] although it is full of possibility because the kosmos proceeds from the logos and is essentially linked to it.[10] The word kosmos can also have a positive meaning because God loves the world.[11] Later in the Gospel it acquires a negative meaning when the world is seen as hostile to Jesus.[12]
c. It is against this Greek view of kosmos formed out of chaos that Athanasius teaches the doctrine of creatio ex nihilo.
"Prior to the debates of Athanasius with Arius, the theory of creatio ex nihilo was propounded, if at all, with uncertainty … [but] with this assertion of creatio ex nihilo came a recognition by Athanasius of a clear and substantial distinction between God and the created order, between the uncreated, non-contingent and asomatic Creator and the contingent and somatic creation, called into being from nothing by the will of God."[13]
This Athanasian view has become dominant even though an emanationist interpretation of creation is available in the neo-Platonic Christian tradition.[14]
The seeming opposition between Hindu emanation and Athanasian creation may not, however, be insuperable. In Hindu thought there is a distinction between the expresser and the expression but not a separation. The term 'mantra' can refer both to the deity and to the phonic expression of that deity, to the reciter and to the mantra she recites. The speaker both transcends her word and is her word. When the speaker fully communicates herself, she and her word are not dual but identical, distinct but not divided. The one leads to the other; the one is the other and, even if our minds construct a separation, in reality there is none. The analogue for understanding the formation of the world, therefore, can be the dancer or the poet or prophet rather than the architect. Indeed, the first account in Genesis sees creation as a prophetic act. God is his word and transcends his word. But word is work and work is word. The work of creation is God and is not God. This is all the more true in the Indian philosophical system, which is based on the word rather than on objective reality, on revelation rather than on being (esse).
Sin and ignorance:
Similarly, in the Shaivism of Kashmir the human being is the expression of Śiva and in that sense is Śiva. The human being is, therefore, essentially Śiva who in the inmost depth of human consciousness speaks the primordial Word and proclaims, "I am" (aham). To quote Jacques Dupuis,
"God has been reached from both ends, as the "Father in heaven" and as more intimate to myself than I am" (interior intimo meo) (St Augustine, Confessions III.6.11)."[15]
However, the expression is also a limitation. The emanation of the world is both an expression of the divine Light (prakāśa) and a concealment (tirodhāna) of that Light which continues to diminish until it reaches the state of inertia (jaḍatā), just as the ripples in the pond eventually peter out. Thus Śiva delights to be his opposite, consciousness being reduced to ignorance, light being completely obscured.
The human being who does not understand these things sees herself as merely human. The individual says: 'I am this person and not that person. I am such and not otherwise.' This divisive attitude is an error, an ignorance (avidyā), which is not a lack of information but an absence of wisdom. It is even a lie, since in the depths of one's being the truth is always known. This failure to understand is the primary fault or stain (mala) confusing the individual self (ahaṁkāra) with the universal Self (aham), either to inflate the importance of the individual self or to reduce the universal self to the human level.
It is said, in classical Catholic moral theology, that for a sin, either of commission or omission, to be perpetrated there must be sinful matter, knowledge and consent. The sin is grave if all three elements are grave and full; the sin is venial if one of the elements is partial. Knowledge would seem, therefore, to be a constituent part of the sinful act. However, there are many texts in the Gospel which also describe sin as ignorance. Not only the famous 'Father, forgive them; they do not know not what they are doing' (Lk 23.34 ), but also: 'Blind? If you were, you would not be guilty, but since you say, "We see", your guilt remains.'(Jn 9.41) Or again: 'The [servant] who did not known [what his master wants], but deserves to be beaten for what he has done, will receive fewer strokes.' (Lk 12.48) The 'strokes of the lash' are given, even though there is no conscious act of disobedience.
Ignorance (avidyā) in Kashmir Shaivism is a failure to know the truth; an absence of enlightenment which means that the individual cannot but perform acts which are disastrous both personally and for others. Revelation is not only concerning the good but also concerning the true nature of evil.
The acknowledgment of the Self (aham) does not involve the elimination of the individual self (ahaṁkāra). Absorption (saṁhāra) does not mean annihilation but reinterpretation: understanding that the limited self is an expression of the true self and that one is really "I am". It is extremely difficult to cease identifying with the individual self. Indeed, in Scriptural terms it is a 'dying to oneself'.[16] This is more than the elimination of unrighteous thoughts and actions, the abandonment of selfishness. It is a fundamental change of perception, a rebirth, and regeneration.
Even if the soul is declared to be immortal[17] it is not absolute and does not necessarily exist. It could, if God so willed, simply cease to exist. No ultimate reliance can be placed upon the soul or the will. The individual self is indeed real and not imaginary, but is essentially contingent and in this sense profoundly unreal. Only God is truly real.
This ignorance leads to acts that are absurd and divisive, bearing a harvest of unfortunate consequences (karma), which may take lifetimes to redress.
Where the Western mind distinguishes in order to understand, the Hindu mind absorbs in order to perceive the essential nature of things. The Western mind says 'one is not the other'; the Hindu mind says that one is essentially the other: sarvaṃ-sarvātmakam
Redemption and recognition:
The purpose of the teachings of Kashmir Shaivism is to lead the disciple to the act of recognition (pratyabhijñā) where he recognises his essential truth and concomitantly understands the relative nature of his individual self. He comes to see that his individual self is essentially an expression of the divine self and that his essential reality is divine. St Paul puts it perfectly: "I live now not with my own life but with the life of Christ who lives in me."[18] According to Kashmir Shaivism, the saving moment is essentially a change of perception. The practitioner turns away from idolising all limited things and recognises the essential nature of reality. This dying to oneself is not just a moral attitude, but also a profound change of perception, a new ontology. The individual self ceases to be the centre of focus and is reabsorbed into its origin.
Faith and identity
If faith implies devotion, and if devotion is understood to mean separation, there is no place for that sort of faith in Kashmir Shaivism. If, however, faith implies identity (tādātmya) then Kashmir Shaivism is profoundly concerned with faith, for its aim is to acquire identity with Śiva, indeed to attain the very state of Śiva (śivatā). It is a resting; not in a separate self but in one's own true self (sva-ātma-viśrantī) identified with the divine Self.
Panikkar puts it well:
Eternal life and universal bliss:
The act of recognition leads to the divine state which is not self-absorption but universal bliss (jagad-ānanda); a state beyond action (kalpa) and thought (vikalpa), a state transcending thought (nirvikalpa) and which all thoughts and actions only partially express. The practitioner is not aloof from the world but fully present. The panoply is not something apart from her but is indeed her very self, the expression of her own being, and is therefore welcomed as she welcomes her own self.
This is the 'attitude of Bhairava' (bhairava-mudrā), where, if the meditator looks within, into his own heart, he sees the whole world; if he opens his eyes and looks upon the world, he sees himself, for the world and he are one. Whether the eyes are open or shut he sees the same. His eyes are both open and shut, for he is in the world as in his own body but not defined by it.
It is not a state available only after death but can be achieved in time. The practitioner is liberated while alive (jīvan-mukta), so that his every word is mantra and his every act is ritual.
St Augustine, on seeing a drunken man, said in all humility and against the Pelagians, "There but for the grace of God go I". The outlook proposed by Kashmir Shaivism would add: 'He is not apart from me, someone other than me. He is my very self.'
Indeed, true knowledge of an object is possible only by identification with that object. I can truly know the mountain only if I am the mountain. Only God can truly know God, only God can fully worship God.[20] That is why Jesus, the true High Priest, must be "God from God, Light from Light". Furthermore, if God wishes to speak to humans it is only by means of the divine Word being also human. Again, if God is to be worshipped by humans it is only by humans being divine. The Christian can truly know God only be being God in a profound sense, by means of theosis,
The means of coming to recognition:
In order to achieve that result, Kashmir Shaivism proposes four means (upāya), which are based on four forms of knowledge.
a. The forms of knowledge:
The simple statement 'I see the mountain' distinguishes clearly between three forms of knowledge: firstly the object of knowledge, the mountain; next, the means of knowledge, the seeing; and lastly the knowing subject, the viewer, 'I'. Thus there is object (prameya), means, (pramāṇa) and subject (pramātṛ). However this division into three is transcended by a fourth: where all are unified as one; where the object known, the means of knowledge and the subject are one and the same, namely the knower (pramiti).[21] In other words, the Self sees the Self by means of the Self. Indeed, all is simply the Self, "I am". All is light. [22]
b. The four means (upāya), which was the topic of Bettina Bäumer's paper that so impressed Karl Rahner, are based on those four forms of knowledge and each can lead to the 'attitude of Bhairava'.
The least exalted method is based on the object, i.e. on practices that are varied according to the character of the practitioner. The next is based on the means of knowledge where, by reasoning and reflection, he comes to the act of recognition. The next focuses on the subject where the subject more directly and immediately perceives his own true nature. But that method is still imperfect because the practitioner sees himself as distinct from the means and the object of knowledge.[23]
c. The most exalted means is really a non-means (anupāya) because in fact there is no path to follow: the goal is reached suddenly and totally, due to an intense descent of energy (śakti-pāta), an immense outpouring of grace (anugraha). Nothing more is to be done; there is no need for repeated practice or deeper understanding.
"The revelation [of this Light] is given once and for all, after which there is no means."[24]
"The reality of Consciousness shines forth by its own radiance. What is the value, therefore of those [means to make him known]?"[25]
The anupāya is described largely in negative terms since the light of consciousness cannot be described by what is less than the fullness of that light:
"The supreme state is neither 'being' nor 'non-being', neither duality [nor non-duality], for it is beyond the realm of words. It is located on the apophatic (akathya) level. It is with energy, it is without energy."[26]
"[The Light of consciousness] is not a mantra, not a divinity whose mantra is recited, nor a reciter of mantras. [The Light] is neither initiation nor initiator nor initiated: It is the supreme Lord."[27]
Therefore the usual acts of religion are unnecessary:
"For them there is no mantra, no meditation, no cultic worship, nor visualisation, nor the commotion involved in ordinary initiation, consecration of the master etc."[28]
Conflicting emotions also lose their significance:
"[For those who have attained this highest state], notions of pleasure and pain, fear and anguish, disappear completely: the knower has arrived at supremely non-differentiated thought."[29]
The practitioner who has achieved this state is not introverted. Rather, universal bliss confers universal bliss.
"They have no other work to accomplish but to confer grace".[30]
"The worldly person works assiduously for himself, and does nothing in favour of others, but the one who, having overcome all impurities, has achieved the divine state works solely for the benefit of others."[31]
c. However, according to the thirteenth century commentator Jayaratha, the term 'non-means' (an-upāya) can also be understood as 'a very reduced means' (alpopaāya)[32] or a 'subsidiary means' (parikaratvam).[33] He lists a certain number of the reduced means.
"The sight of the Perfected Beings and yoginīs, the eating of the 'oblation', a teaching, a transition (?), spiritual practice, service of the Teacher."[34]
Any one of these is sufficient to bring a person to full realisation, suddenly and without any need to engage in practices to deepen the realisation.
Yet, the ones who receive such an immense outpouring of grace are few in number. The vast majority of beings need to follow one or other of the three lesser paths, according to the measure of grace given to them:
"However, those whose consciousness is not utterly pure receive grace only by following one of the paths."[35]
What sort of Kashmir Shaiva Christology emerges form all this?
On seeing (darśana) Jesus or hearing a teaching (kathanam), the disciple experiences his own consciousness expanding. He then knows both Jesus and his own self, and indeed realises that Jesus is his own very self, for only like can see like, only the same can see the same. In fact, not only is the self of Jesus the very self of the disciple but the whole world too is an expression of the one Self. In short, the sight and teaching of Jesus are examples of the "very reduced means" (alpopāya) noted above.
But more; in contemplating Jesus and so arriving at consciousness, the disciple penetrates to the utterly Transcendent (anuttara) so that it becomes clear to him that Jesus of Nazareth is essentially the "I am", the Supreme Word (paravāc), the self-revelation of Consciousness.
Since from that Expression all other expressions derive, Jesus looks upon the world and sees it as the expression of his self. Jesus is the Lord of the Dance.
"He is the image of the unseen God and the first-born of all creation… for in him were created all things in heaven and on earth … all things were created through him and for him…. [36]
In the events of the Sacred Triduum Jesus knows both the depths and the height; knowing good and evil, able to descend lower than any because he knows the height. The Paschal Mystery is the moment of supreme revelation. Although the Word of God has been revealed in various ways since the dawn of time, the Word incarnate is best able to reveal to flesh, since flesh needs flesh. Flesh best reveals flesh to itself. In the fullness of his living and dying he is the perfect expression of heaven and earth. Jesus, therefore, is able to provide the knowledge, which leads to the utterly Transcendent (anuttara). He is the Light that brings all to Light. The Word made flesh makes all flesh Word.
God wanted … all things to be reconciled through him and for him, everything in heaven and everything on earth."[37]
All is non-dual (a-dvaita). All is one.
Abhinavagupta Tantrāloka with the Commentary of Jayaratha. Re-edited by R.C. Dwivedi and Navjivan Rastogi, enlarged with an introduction by Navjivan Rastogi and reprinted in 8 volumes. Delhi, Motilal Banarsidass, 1987.
Bettina Bäumer 'The Four Spiritual Ways (upāya) in the Kashmir Saiva Tradition' in Regional Spiritualities, pp. 3-22.
Brown, Raymond The Gospel according to John. New York: Doubleday and Company Inc. 1966, Vol.1; 1970, Vol.2.
Cassem, N.H. "A Grammatical and Contextual Inventory of the use of kosmos in the Johannine Corpus with some Implications for a Johannine Cosmic Theology," in NTS 19 (1972-1973) 81-91.
Denzinger, Heinrich and Schönmetzer, Adolf. Enchiridion Symbolorum. Freiburg im
Breisgau: Herder, 1967.
Dupuis, Jacques Christanitiy and the Religions. Maryknoll, New York: Orbis Books, 2002. [First published as Il cristianismo e le religioni: Dallo scontro all'incontro. Brescia, Edizioni Queriniana, 2001.]
Palamas, Gregory Triads, Edited with an introduction by John Meyendorff,
translated by Nicholas Gendle, preface by Jaroslav Pelikan. London: SPCK, 1983.
Panikkar, Raimon 'On Christian identity' in Cornille, Catherine (ed.) Many
Mansions., Maryknoll, New York: Orbis Books, 2002. pp.121-144.
Pettersen, Alvyn Athanasius and the Human Body, Bristol: The Bristol Press, 1990.
Tracy, David Dialogue with the Other; The inter-religious Dialogue. Louvain: Peeters Press, 1990.
[1] Prof. Dr. Bettina Bäumer, Institute of Religious Studies, University of Vienna.
[2] Personal communication, 9 April 2004.
[3] Gal. 3.28.
[4] Cardinal Ratzinger, in the recent ad limina visit of the Australian Catholic Bishops, "spoke of the need for the Church to present a Christian anthropology which opens out to the world a deeper understanding of the human condition …. A positive vision of what it means to be a human being…'Letter of Archbishop Hart, dated 1 April 2004, to all priests of the Archdiocese.
[5] David Tracy, Dialogue with the Other; The inter-religious Dialogue. Louvain: Peeters Press, 1990. p.94.
[6] I Cor.15.28.
[7] Raymond Brown, The Gospel according to John. New York: Doubleday and Company Inc. 1966, Vol.1, p. 508.
[8] N.H. Cassem, "A Grammatical and Contextual Inventory of the use of kosmos in the Johannine Corpus with some Implications for a Johannine Cosmic Theology," NTS 19 (1972-1973) 81.
[9] Jn. 3.16. See also Jn 11.9 ; 17.5, 24; 21.25.
[10] Brown, The Gospel according to John. Vol.1, p.25.
[11] Jn 1.29; 3.16; 4.42; 6.51; 8.12; 9.5.
[12] See 12.31; 14.17, 22, 27, 30; 15.18-19; 16. 8, 11, 20, 33; 17.6, 9, 14-16.
[13] Alvyn Pettersen, Athanasius and the Human Body, Bristol: The Bristol Press, 1990. p. 5.
[14] Tracy, Dialogue with the Other. p.86.
[15] Jacques Dupuis, Christanity and the Religions. Maryknoll: New York, Orbis Books, 2002. p.123.
[16] Cf. Mk 8.35.
[17] Ecumenical Council Lateran V, Bull "Apostolici regiminis". In Denzinger H and Schönmetzer A. Enchiridion Symbolorum, no.1440. Freiburg im Breisgau: Herder, 1967. p.353.
[18] Gal.2.19.
[19] Raimon Panikkar, "On Christian identity" in Cornille, Catherine (ed.) Many Mansions, Maryknoll, New York: Orbis Books, 2002. p.139.
[20] This is a commonplace of Hindu thought.
[21] This is fourth in the listing but in fact underlies all three separate forms.
[22] This notion of light seeing its light by means of its own light is found in the theology of Gregory Palamas, the last of the Greek Doctors of the Church. He makes a very striking analogy with the eye. After referring to St Paul (II Cor.12.2) he pictures a sun of infinite radiance and size – at the centre of which all stands but now transformed into an eye. Paul, like that eye, is in light and seeing light. There are no limits. "If [the visual faculty] looks at itself it sees light; if it looks at the object of its sight that is also light; and if it looks at the means it uses to see, that too is light; that is what union is: let all that be one." Triads, II.3.36. London: SPCK, 1983. p. 66.
[23] "(In this method śāmbhavupāya there is still) a conception of a difference between method and goal (upāya-upeya-kalpanā), whereas (in the case of anupāya) there is not even a trace of any difference. For in the non-way, who is to be liberated, how and from what?" TĀ 3.272-273. Bettina Bäumer, 'The Four Spiritual Ways (upāya) in the Kashmir Śaiva Tradition' in Regional Spiritualities, pp.17-18.
[24] TĀ 2.2b.
[25] TĀ 2.10a.
[26] TĀ 2.33.
[28] TĀ 2. 37.
[30] TĀ 2.38b.
[32] Tantrāloka vol.2. p.312, line 13.
[33] Tantrāloka vol.7. p.3420, line12.
[34] Tantrāloka vol.2, p.312 lines 13-14.
[36] Col.1.15-16.
This entry was posted in Christian tantra, Hindu Christian relations, Interreligious dialogue, Interreligious dialogue, Melbourne, Kashmir Shaivism. Bookmark the permalink.
2 Responses to "Renewing Christian Anthropology in Terms of Kashmir Shaivism",
ananthrajeev says:
Its evident what ur attempting.. ur struck by hindu philosophy but ur ego doesnt allow to shove christianity. In turn, u want to appropriate hindu thoughts to satisfy ur identity crisis. Jesus was a great yogi, no doubt about that. But ur attempts at preserving ur identity is less effective in spiritual path. There is no need to bring out a justifiable theology out of syncretism and place jesus at the top. Its just ur mind playing little games with u.. The message of jesus to serve humanity like u would serve urself is enough for a believer to reach the truth. But the more u try to bring in separation, bias and favouritism, ur still in the loop. Loose ur affliations , identity and kill ur ego in christ( since ur a beleiver) to discover urself. U can try fooling urself into believing that grace alone is enough..! But ask urself again..Have u shred away the last drop of ur ego..??
33Dee says:
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/precalculus\/precalculus-6th-edition-blitzer\/chapter-10-section-10-7-probability-exercise-set-page-1120\/47","text":"## Precalculus (6th Edition) Blitzer\n\n$\\frac{33}{40}$\nStep 1. The total number is: $8+11+14+7=40$ Step 2. The probability of choosing a professor is $P(prof)=\\frac{8+11}{40}=\\frac{19}{40}$ Step 3. The probability of choosing a male is $P(male)=\\frac{8+14}{40}=\\frac{22}{40}$ Step 4. The probability of choosing a male professor is $P(male\\ prof)=\\frac{8}{40}$ Step 5. The probability of choosing a professor or a male is $P(prof\\ or\\ male)=P(prof)+P(male)-P(male\\ prof)=\\frac{19}{40}+\\frac{22}{40}-\\frac{8}{40}=\\frac{33}{40}$","date":"2020-05-27 12:42:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9621340036392212, \"perplexity\": 325.5640907505125}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347394074.44\/warc\/CC-MAIN-20200527110649-20200527140649-00553.warc.gz\"}"}
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{"url":"https:\/\/www.toolinux.it\/ewma-covariance-matrix-python.html","text":"Ewma Covariance Matrix Pythonp6a This motivated Zangari ( 1994 ) to propose a modification of UWMA called exponentially weighted moving average (EWMA) estimation. By guiding you to the right analysis and giving you clear results, Minitab helps you find meaningful solutions to your toughest business problems Feature List * New or Improved Assistant * Measurement Sy. v9 The EW functions support two variants of exponential weights. Released documentation is hosted on read the docs. This paper proposes a general multivariate exponentially weighted moving average chart, in which the smoothing matrix is full, instead of one having only diagonal elements. Motor failure in multi-leaf collimators (MLC) is a common reason for unscheduled accelerator maintenance, disrupting the workflow of a radiotherapy treatment centre. There is also no problem with having duplicate periods in the result rng = pd. Specifically, it's a measure of the degree to which two variables are linearly associated. Covariance matrix from samples vectors. We adopted the Python DISPY distributed computation platform for computation assignment and let the. O'Reilly members get unlimited access to live online training experiences, plus books, videos, and digital. 1 Languages: Multilingual File Size: 287. During some periods, a particular volatility or correlation may be. are considered for monitoring of variance-covariance matrix when the\u00a0. Simply import the NumPy library and use the np. More concisely, we can define the whole correlation matrix by:\u0393t\u2254D-1t\u2211tD-1t. Compare this with the fitted equation for the ordinary least squares model: Progeny = 0. By default, method = \"unbiased\", The covariance matrix is divided by one minus the sum of squares of the weights, so if the weights are the default (1\/n) the conventional unbiased estimate of the covariance matrix with divisor (n - 1) is obtained. Standard Deviation is the square root of the Variance. mr e5z It is suitable for the simulation of very large portfolios. (the correlation matrix is the covariance matrix normalized with individual standard deviations; it has ones on its diagonal), along with a list of nominal values and standard deviations: >>> (u3, v3, sum3) = uncertainties. S: is the sample covariance matrix. The sample estimators for the mean and covariance matrix are, respectively, the sample ARCH and GARCH models (to be studied later in more detail for the variance modeling). Calculate the efficient frontier with the new mu and Sigma. Udemy Importing Finance Data with Python from Free Web Sources. Python is a programming language that provides toolkits for machine learning and analysis, such as scikit-learn, numpy, scipy, pandas, and related data visualization using matplotlib. The estimated covariance rate between variables X and Y on day n \u2212 1 can be calculated as: covn = \u03c1A,B \u00d7\u03c3A\u03c3B = 0. dx Search: Portfolio Volatility Python. - Estimate Rating Transition Matrix with Cohort and Hazard Rate Approach - Credit Scores with Logistic Regression - Compute Operational Value at Risk (VaR) and Expected Shortfall (ES) using Monte Carlo Simulation based on Poisson and Log-Normal distribution - Run R Scripts for online statistical data analysis - Live Currency Rates & Gold Price. Also known as the auto-covariance matrix, dispersion matrix, variance matrix, or variance-covariance matrix. An Individual moving range (I-MR ) chart is used when data is continuous and not collected in subgroups. It is calculated using numpy\u2018s corrcoeff() method. 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A review on outlier\/anomaly detection in time series data. Click card to see definition \ud83d\udc46. and higher-way analysis of variance (ANOVA), analysis of covariance (ANOCOVA), multiple linear regression, stepwise regression, response surface prediction, ridge regression, and one-way multivariate analysis of variance (MANOVA). o5u Nonetheless, a winner in a kaggle competition is required only to attach a code for the replication of the winning result. 104 operators in Python, 393 linear algebra, 105-106 or keyword, 401 matrix operations in, 377-379 order method, 375 ndarray arrays, 80 OS X, setting up Python on, 9-10 outer method, 368, 369 Boolean indexing, 89-92 outliers, filtering, 201-202 creating, 81-82 output variables, 58-59 data types for, 83-85 fancy indexing, 92-93 P. (EWMA)=\u03bb\u03c3 n\u221212 +(1\u2212\u03bb)u n\u221212 where:EWMA=Exponentially weighted moving average\u03c3 n2 =Variance today\u03bb=Degree of weighting\u03c3 n\u221212 =Variance yesterdayu n\u221212 =Squared return yesterday \ufeff Recursive means. Uniwersytet Ekonomiczny w Krakowie Coding in Python Wynik: 30\/30 lis 2021 Coding in R Wynik: 27\/30 lis 2021. The EWMA model is a special case of the IGARCH(1,1) model where volatility innovations have infinite persistence. These are pdist (distribution), ddist (density), qdist (quantile) and rdist (random number generation), in addition to dskewness and dkurtosis to return the conditional density skewness and kurtosis values. Estimate a covariance matrix, given data and weights. n9 Financial Risk Forecasting is a complete introduction to practical quantitative risk management, with a focus on market risk. Options, Futures, and Other Derivatives, 10th Edition. \u2019BLFM\u2019: use estimates of expected return vector and covariance matrix based on Black Litterman applied to a Risk Factor model specified by the user. Using these links is the quickest way of finding all of the relevant EViews commands and functions associated with a general topic such as equations, strings, or statistical distributions. Note that the calculations are different for data in subgroups. The normalized version of the covariance, the correlation coefficient, however, shows by its magnitude the strength of the linear relation. Viewed 1k times 0 I have weekly return data in ascending order. RiskMetrics\u00ae is actually a special case of the GARCH approach. \u2022 Audit Lead for new, more responsive internal Exponentially Weighted Moving Average (EWMA) Value-at-Risk methodology in R and MATLAB. When adjust=True (default), the EW function is calculated using weights $$w_i = (1 - \\alpha)^i$$. The following links provide quick access to summaries of the help command reference material. \u2019ewma1\u2019\u2019: use ewma with adjust=True, see EWM for more details. When this covariance matrix becomes too small, recursive least squares algorithms respond slow to changes in model parameters. In addition to availability of regression coefficients computed recursively, the recursively computed residuals the construction of statistics to investigate parameter instability. This systematic review aims to provide an introduction and guide for researchers who are interested in quality-related issues of physical sensor data. ami py \u2022 Extend the example from Multi-Variate modeling \u2013 Two stock portfolio, MSFT and IBM, 50% weight each \u2013 Daily data from Jan 2000 to two different end dates: Dec. Minitab is the unmatched, all-in-one data analysis and statistics software for everyone that lets data be used for what it is worth. More specifically, we say that rt - \u03bc~EWMA(\u03bb) if: \u2211t + 1 = (1 - \u03bb)(rt - \u03bc)(rt - \u03bc) '. The reason for saying that, even though there are two sets of scores, T and U, for each of X and Y respectively, is that they have maximal covariance. So it is highly unlikely, a chance of 1 in 370, that a data point, $$\\overline{x}$$, calculated from a subgroup of $$n$$ raw $$x$$-values, will lie outside these bounds. The question you have to ask yourself is whether you consider:. j6a The well-known MEWMA is directed at changes in. But when I calculate the eigenvalues (with np. In pandas, the std () function is used to find the standard Deviation of the series. Worked example: There are several ways to extend the EWMA model to generate predictions. More specifically, we say that r t-\u03bc ~ EWMA \u03bb if: \u2211 t + 1 = 1-\u03bb r t-\u03bc r t-\u03bc ' + \u03bb \u2211 t V-Lab uses \u03bb = 0. How to Perform an ANCOVA in Python - Statology 111000. pdf - Free download as PDF File (. 3 where s2 ewma = (\u03bb\/(2\u2212\u03bb)s2) and s is the standard. In this blog, we shall discuss on Gaussian Process Regression, the basic concepts, how it can be implemented with python from scratch and also using the GPy library. rm (str, optional) \u2013 The risk measure used to optimze the portfolio. var(a) method to calculate the. 9h 92 to update correlation and covariance rates. matrix or array of the quality characteristics. Published on September 17, 2020 by Pritha Bhandari. Standard deviation in statistics, typically denoted by \u03c3, is a measure of variation or dispersion (refers to a distribution's extent of stretching or squeezing) between values in a set of data. by the way your link only shows univariate EWMA. Multivariate exponentially weighted moving a verage (MEWMA) charts are among the best control charts. Covariance indicates the level to which two variables vary together. Basic knowledge of molecular biology and genetics is preferred but not required. Use classical methods in Minitab Statistical Software, integrate with open-source languages R or Python, or boost your capabilities further with machine learning algorithms like Classification and Regression Trees (CART\u00ae) or TreeNet\u00ae and Random Forests\u00ae, now available in Minitab's Predictive Analytics Module. Naval Research Logistics, 2013, 60(8), 625-636. A high standard deviation means that values are generally far from the mean, while a low standard deviation. yr DataFrame data: Input time series data:return: The success flag, model date and a trained lad filtering object:rtype: tuple[bool, str, LADFilteringModel object] >>> data raw interpolated 2020-01-01 1326. This chapter explains why, ultimately 2. Multivariate DCC-GARCH covariance matrix. Distance; GH-124: Fixing the Envelop filter as missing loop variables were not. Journal of Empirical Finance, 10:603-621. Intuitively, the historical correlation (or equivalently variance-covariance) matrix needs to be adjusted to the new information environment. A simulation using the SMC approach is not capable of predicting scenarios during times of crisis if the covariance matrix was. , 1992) that the $$(k,l)$$th element of the covariance matrix of the $$i$$th EWMA, $$\\Sigma_{Z_i}$$, is $$\\Sigma_{Z_i}(k,l) = \\lambda_k \\lambda_l \\, \\frac{\\left[ 1-(1-\\lambda_k)^i (1-\\lambda_l)^i \\right]}{(\\lambda_k + \\lambda_l - \\lambda_k \\lambda_l )} \\, \\sigma_{k,l} \\, ,$$ where $$\\sigma_{k,l}$$ is the $$(k,l)$$th element of $$\\Sigma$$, the covariance matrix of the $$X$$'s. This algorithm computes a harmonic model for the 'training' portion of the input data and subtracts that from the original results. 547 About Excel Weighted Covariance. Exponentially weighted moving average (EWMA) is a popular IIR filter. First, let's create dummy time series data and try implementing SMA using just Python. 34615789769413313] Python: Calculate Sharpe Ratio adjustments (optionally) p=Portfolio(returns) # by default Sharpe Ratio adjustments are on unless we turn them off. 8 is the final version that supported Python 2. Now that we have had a look at the main characteristics of the variance and covariance. h3 bi3 1 Inverse and Adjoint of a Square Matrix A. [2] Standard Errors assume that the covariance matrix of the errors is correctly specified. It tells you, on average, how far each value lies from the mean. The Seventh Edition of Introduction to Statistical Quality Control provides a comprehensive treatment of the major aspects of using statistical methodology for quality control and improvement. fmq Covariance is a measure of relationship between the variability of 2 variables - covariance is scale dependent because it is not standardized. EWMA has a higher reversion rate than GARCH (1,1). The difference between the EWMA & SMA methods to the VCV approach lies in the calculation of the underlying volatility of returns. Python and R use exponential weighted average (EWMA), Arima autoregressive moving average model to predict time series. WAX-ML makes JAX-based programs easy to use for end-users working with pandas and xarray for data manipulation. Calculations for GARCH(1,1) and EWMA are to be done on separate sheets of the same Excel File. Variance Covariance Approach \u2013 Exponentially weighted moving average (EWMA) We will now look at how to calculate the exponentially weighted moving average (EWMA) VCV VaR. Introducing Time Series with pandas\u00b6. Added a parameter rescale to arch_model that allows the estimator to rescale data if it may help parameter estimation. gaussian_kde\u73b0\u5b9ePython\u793a\u4f8b\u3002\u60a8\u53ef\u4ee5\u8bc4\u4ef7\u793a\u4f8b. Bayesian regressions (part 1) October 6, 2011 Cathy O'Neil, mathbabe. The standard context for PCA as an exploratory data analysis tool involves a dataset with observations on pnumerical variables, for each of n entities or individuals. 27 us4 function in SAS\/IML to create the sample covariance matrix for a given matrix [Ref. It differs from the Python list data type in the following ways: N-dimensional. Posible values are: \u2019hist\u2019: use historical estimates. Complete 2-in-1 Python for Business and Finance Bootcamp. It was released on October 19, 2017 - over 4 years ago. cov() function only supports weights given to individual measurements (i. \u2019FM\u2019: use estimates of expected return vector and covariance matrix based on a Risk Factor model specified by the user. Collaborate with chetanrg05 on pandas-self-practice notebook. >>> import num py as np>>> python \u534f\u65b9\u5dee\u77e9\u9635 _num py \u534f\u65b9\u5dee\u77e9\u9635 num py. Multivariate exponentially weighted moving average (MEWMA) charts are among the best control charts for detecting small changes in any direction. These algorithms include: Minimum Covariance Determinant; Empirical Covariance; Covariance Estimator with Shrinkage; Semi-Covariance Matrix; Exponentially-\u00a0. R\u8bed\u8a00arima\uff0c\u5411\u91cf\u81ea\u56de\u5f52\uff08VAR\uff09\uff0c\u5468\u671f\u81ea\u56de\u5f52(PAR)\u6a21\u578b\u5206\u6790\u6e29\u5ea6\u65f6\u95f4\u5e8f\u5217. 1bf For example, REGION is a higher level summary of STATE. To be specific, we present the correlation matrix in the format of a heatmap in Figure 1. You can rate examples to help us improve the quality of examples. bbw and improved its Sharpe ratio by 34% by estimating the covariance matrix with EWMA and shrinkage methods Conducted performance attribution analysis on 1,400+ fixed income mutual funds over the past 14 quarters using the Campisi model, analyzed attribution results statistically, and composed a research report summarizing findings. Anomaly Detection This will take a dive into common methods of doing time series analysis, introduce a new algorithm for online ARIMA, and a number of variations of Kalman filters with barebone implementations in Python. The correlation estimate for two variables A and B on day n \u2212 1 is 0. Its weighting scheme replaces the quandary of how much data to use with a similar quandary as to how aggressive a decay factor \u03bb to use. The basic object is a timestamp. Let us define Ct as the volatility of a market variable on day t as estimated from day t - 1 Exponentially weighted moving average estimation is widely used, but. The exponentially-weighted moving average (EWMA) model calculates covariances by placing more emphasis on recent observations via a decay factor, \u03bb. iloc[-1] def ewma_cov_pd(rets, alpha=0. Triangular Arbitrage Strategies for Forex & Commodities. The relative weight is determined by setting the half-life of the rate of decay, and it differs between the short. Build your own projects and share them online!. Garch Model Python Github Every day, TRB and thousands of other voices read, write, and share important stories on Medium. Standard Deviation | A Step by Step Guide with Formulas. Instructions 100 XP Use the exponential weighted covariance matrix from risk_models and exponential weighted historical returns function from expected_returns to calculate Sigma and mu. to the EWMA model, \u201c\u2026 it is often found to generate short-run forecasts of the variance-covariance matrix that are as good as those of more sophisticated volatility models \u2026\u201d (page 805). becomes large, the covariance matrix may be expressed as: \\Sigma_{Z_i} = \\frac{\\lambda}{2 - \\lambda} \\Sigma \\,. While I prefer R for the majority of my analyses, I recommend Minitab over JMP because of Minitab's excellent technical support and its ease of use. bob said: Congratulations, you have just identified problem #1 with MC VaR. Simulation of 3 stocks (AMZN, GOOG, and AAPL) available for download from GitHub. is the covariance matrix of the input data. as well academic utilization of R and Python. Write a NumPy program to compute the covariance matrix of two given arrays. 1 \u5f15\u8a00\u6211\u4eec\u5728 \u300a\u6b63\u786e\u7406\u89e3 Barra \u7684\u7eaf\u56e0\u5b50\u6a21\u578b\u300b\u4ecb\u7ecd\u4e86 Barra \u7684\u591a\u56e0\u5b50\u6a21\u578b\u3002\u8be5\u6587\u8ba8\u8bba\u7684\u91cd\u70b9\u5728\u4e8e\u4ece\u4e1a\u52a1\u4e0a\u8bf4\u660e\u56fd\u5bb6\u3001\u884c\u4e1a\u3001\u98ce\u683c\u7eaf\u56e0\u5b50\u6295\u8d44\u7ec4\u5408\u7684\u542b\u4e49\uff0c\u800c\u975e\u5177\u4f53\u7684\u6570\u5b66\u8ba1\u7b97\u3002\u4e0d\u8fc7\uff0c\u540e\u6765\u6211\u610f\u8bc6\u5230\u6211\u7ed9\u81ea\u5df1\u6316\u4e86\u4e00\u4e2a\u5751\u3002. Answer (1 of 2): The paper says > an exponentially-weighted moving average on the [data], with more recent observations having a higher weight than those from the more distant past. CUSUM and EWMA charts only monitored the selected quantitative response variable's changes without considering the risk factors nor their corresponding model parameters. Interpreting the scores in PLS \u2014 Process Improvement using Data. veu Basket Strategy (Index-Index, Index-Stocks). We adopted the Python DISPY distributed. Documentation Documentation from the main branch is hosted on my github pages. The well-known MEWMA is directed at changes in the mean vector. While the frequency of the new PeriodIndex is inferred from the timestamps by default, you can specify any frequency you want. 26 Full PDFs related to this paper. 4 - \u0646\u0631\u0645 \u0627\u0641\u0632\u0627\u0631 \u0632\u0628\u0627\u0646 \u0628\u0631\u0646\u0627\u0645\u0647 \u0646\u0648\u06cc\u0633\u06cc \u067e\u0627\u06cc\u062a\u0648\u0646 [286,121] Microsoft Edge v99. To see my original article on the basics of using the BarChart OnDemand API click here. 73% of the area (in R: pnorm(+3)-pnorm(-3) gives 0. It does not attempt to model market conditional heteroskedasticity any more than UWMA does. EWMA data point can be calculated as: EWMA t = \u03bbp(I) t +(1 \u2212\u03bb)EWMA t\u22121 (2) where \u03bb de\ufb01nes the impact of older data compared to new data. Baca Dan Streaming Artikel Var In Python Value At Risk In Python Varcovariance Var Stock Var Single Var Part 1 Semoga Bermanfaat. The Standard Deviation denoted by sigma is a measure of the spread of numbers. The Exponentially Weighted Moving Average (EWMA) was used to estimate the current variance in a setting where it might have a changing over time. Hands-On Data Analysis with Pandas: A Python data science handbook for data collection, wrangling, analysis, and visualization [2 ed. WAX-ML is a research-oriented Python library providing tools to design powerful machine learning algorithms and feedback loops working on streaming data. 6: Histogram of price increments of DAX and Dow Jones stock indices between. EWMA is a particular case of GARCH (1,1) where the reversion rate is zero. Use the exponential weighted covariance matrix from risk_models and exponential weighted historical returns function from expected_returns to calculate Sigma\u00a0. The rugarch package contains a set of functions to work with the standardized conditional distributions implemented. WAX-ML makes JAX-based programs easy to use for end-users working with. When there are active constraints, that is, , the variance-covariance matrix is given by where and. Note that the cumulative statistics is also a windowed with n = k. EWMA covariance matrix using pandas. About Filter Kalman Sklearn Python. \u2022 Fotran90 to Python \u2022 SQLite with Python \u2022 EWMA smoothing length Indeed, a covariance matrix is supposed to be symmetric and positive-definite. mgo (1) m k ( n) = 1 n \u2211 i = k \u2212 n + 1 k x i = 1 n S k ( n) Below for k = n we use the notation X k ( k) = X k. python - covariance isn't positive definite - Stack Overflow. Attended by more than 6,000 people, meeting activities include oral presentations, panel sessions, poster presentations, continuing education courses, an exhibit hall (with state-of-the-art statistical products and opportunities), career placement services, society and section business. PCA starts with computing the covariance matrix Whitening We have used PCA to reduce the dimension of the data. fj3 Sklavounos, Edoh, and Plytas applied EWMA and CUSUM control charts for Root to Local (R2L) intrusion and ${\\rm{\\Sigma }}$ was the covariance matrix with diagonal as 1 and data transformation, and communications among Fog nodes. A positive value for the covariance indicates the variables have a linear relationship. \u2022 Wrote Python code to forecast covariance matrix based on the in-sample data with both MA and EWMA method and implement optimization algorithm on in-sample data to construct the ETF using no. President Kissell Research Group and Adjunct Faculty Member Gabelli School of Business, Fordham University Manhasset, NY, United States. As the persistence parameter under EWMA is lowered, which of the following would be true: A. jgb The resulting fitted equation from Minitab for this model is: Progeny = 0. cases of individual observations the covariance matrix is estimated according to Holmes and Mer- gen(1993). It is a matrix in which i-j position defines the correlation between the i th and j th parameter of the given data-set. Titus 2 is a Portable Format for Analytics (PFA) implementation for Python 3. (i) the exponentially weighted moving average (EWMA) model; for an N \u00d7 N variance-covariance matrix \u03a9 to be internally consistent is. covariance (str, optional) \u2013 The method used to estimate the covariance matrix: The default is \u2018hist\u2019. n_components: int: Number of states in the model. fm In other words we should use weighted least squares with weights equal to 1 \/ S D 2. 1pa com (python\/data-science news) Python Musings #7: Simulating FSAs in lieu of real postal code data. SAS topics include data management, manipulation, cleaning, macros, and matrix computations. (This is a change from versions prior to 0. More details on these plans will be discussed in later editions of the RiskMetrics Monitor. The difference between variance, covariance, and correlation is: Variance is a measure of variability from the mean. Python\u4e2d\u7684ARIMA\u6a21\u578b\u3001SARIMA\u6a21\u578b\u548cSARIMAX\u6a21\u578b\u5bf9\u65f6\u95f4\u5e8f\u5217\u9884\u6d4b. As a part of a statistical analysis engine, I need to figure out a way to identify the presence or absence of trends and seasonality patterns in a given set of time series data. Exponentially Weighted Moving Average Change Detection. date_range ('1\/29\/2000', periods=6, freq='D') ts2 = Series (randn (6), index=rng) ts2. fit_transform(data) Though a simple Google search for python ZCA Whitening gives an answer LW is the Ledoit and Wolf method, ROB is the robust method from the MASS package and EWMA an. The most straightforward method is to choose some historical data for your n assets, generate the covariance matrix on the excess returns (perhaps by using. Mar 17, 2020 Expected portfolio volatility= SQRT (WT * (Covariance Matrix) * W). (1991) as well as Shamma and Shamma (1992) proposed the double EWMA (DEWMA) scheme which is the extended version of Roberts (1959)\u2019s EWMA scheme where the smoothing parameter is applied twice to further improve the sensitivity of the EWMA scheme towards very small shifts. Tracking the tracker: Time Series Analysis in Python from First Principles. To account for this, an exponentially weighted moving average (EWMA) is taken for each asset. rand (2, 2) print data cov = calcCov (data) eigvals, eigvec = np. Optional: To show the process mean and sigma. A columnar udf object is defined by ts. Python Pandas - Descriptive Statistics. Clustering based on similarity\u00a0. Sensor data quality plays a vital role in Internet of Things (IoT) applications as they are rendered useless if the data quality is bad. This window shifts forward for each new data point. Here is an example of Matrix-based calculation of portfolio mean and variance: When $$w$$ is the column-matrix of portfolio weights, $$\\mu$$ the column-matrix of expected returns, and $$\\Sigma$$ the return covariance matrix We talk a lot about the importance of diversification, asset allocation, and portfolio construction at Listen Money Matters. 20, you'll get a MultiIndex DataFrame because Panel is deprecated. 1 Idempotent and Nilpotent Matrices. The exponential covariance matrix: gives more weight to recent data. The setting of the lines and characters is demonstrated in the example programs below. o7b Covariance matrices: The inter-class covariance matrix (equal to the unbiased covariance matrix for the means of the various classes), the intra-class covariance matrix for each of the classes (unbiased), the total intra-class covariance matrix, which is a weighted sum of the preceding ones, and the total covariance matrix calculated for all. 53 qq 3 So EWMA (1) = 40 EWMA for time 2 is as follows EWMA (2) = 0. Long-run Covariance Estimation; Python 3. Abstract Accurate calculation of the Average Run Length (ARL) for exponentially weighted moving average (EWMA) charts might be a tedious task. Python gaussian_kde - \u5df2\u627e\u523030\u4e2a\u793a\u4f8b\u3002\u8fd9\u4e9b\u662f\u4ece\u5f00\u6e90\u9879\u76ee\u4e2d\u63d0\u53d6\u7684\u6700\u53d7\u597d\u8bc4\u7684scipystatskde. The covariance is normalized by N-ddof. exponentially weighted covariance. String describing the type of covariance parameters used by the model. n_features: int: Dimensionality of the Gaussian emissions. The result is shown in Figure 1. 'naive' is used to compute the naive (standard) covariance matrix. Full PDF Package Download Full PDF Package. You can specify the smoothing factor in terms of halflife, span, or center of mass. The key is to notice that it depends on what the weights. This means that, instead of using both risk and return information as in the Markowitz portfolio selection, the portfolio is constructed using only measures of risk. Mean of all the elements in a NumPy Array. generalizing further to normalized weights. This is accomplished, loosely speaking, by \"multiplying\" the historic returns by the revised correlation matrix to yield updated correlation-adjusted returns. Python Data Analysis; exponentially weighted moving average (EWMA) model] and GARCH approaches are both exponential smoothing weighting methods. Create a CSV or tab-delimited file similar to your Amazon file, but add columns for the closing prices of Google and Apple. The covariance of two portfolio returns, each denoted by their own set of weights, say w a, w b can also be found using matrix algebra. cov () can be used to compute covariance between series (excluding missing values). More specifically, we say that rt - \u03bc~EWMA(\u03bb) if: \u2211t + 1 = (1 - \u03bb)(rt - \u03bc)(rt - \u03bc) + \u03bb\u2211t V-Lab uses \u03bb = 0. The Variance-Covariance VaR method makes a number of assumptions. Different methodologies will be tested to obtain the variance-covariance matrix, in particular we will test the historical moving average model, the EWMA model and the DCC-GARCH(1,1) model. 93 and Volatility of XYZ using share prices 67. Learn more about bidirectional Unicode characters. the number of features like height, width, weight, \u2026). A test of covariance-matrix forecasting methods. {\\Sigma }} \\) was the covariance matrix with diagonal as 1 and data transformation, and communications among Fog nodes. Probability and probability distribution plots. The Covariance Matrix is also known as dispersion matrix and variance-covariance matrix. Binned scatterplots, boxplots, bubble plots, bar charts, correlograms, dotplots, heatmaps, histograms, matrix plots, parallel plots, scatterplots, time series plots, etc. The EWMA chart (Exponentially Weighted Moving Average) is a variable data control chart that blends the current data point with an average of the previous data points. The Exponentially Weighted Moving Average ( EWMA) covariance model assumes a specific parametric form for this conditional covariance. for detecting small changes in any direction. \ud83d\udd16 Version updates and fixes: GH-76\/GC-24: Add easier creating and handling of factors for categorical variables; GH-123: Bug in the Euclidean on Accord. INTRODUCTION TO PORTFOLIO ANALYSIS IN PYTHON. The three-dimensional covariance matrix is shown as To create the 3\u00d73 square covariance matrix, we need to have three-dimensional data. Therefore, the GARCH variance-covariance matrix lacks of robustness, thus, the variance-covariance matrix obtained through the EWMA was the chosen to input into the model. y t = \u2211 i = 0 t w i x t \u2212 i \u2211 i = 0 t w i, where x t is the input and y t is the result. Kevin Sheppard's MFE toolbox for Matlab and Arch package for Python have EWMA and GARCH. outer ( v, v) correlation = covariance \/ outer_v. Must be one of \u2018spherical\u2019, \u2018tied\u2019, \u2018diag\u2019, \u2018full\u2019. Numerical integration of Marchenko-Pastur distribution. For calculating the EWMA Volatility, I implemented the following functions: after exhausting my options, I end up converting a MatLab matrix calculation to Python code and it does the vol with decay calculation perfectly in matrix form. Let's see an example of using pd. Mean Reverting Strategies like Pair Trading using Z score Model. 69960 Name: tas, dtype: float64. cov(min_periods=None, ddof=1) [source] \u00b6 Compute pairwise covariance of columns, excluding NA\/null values. exponentially weighted moving average covariance matrix yanyachen\/arimaMisc documentation built on May 4, 2019, 2:30 p. Time series prediction is all about forecasting the future. x2q From Figures 3 (a)-(c), it is revealed that the smaller the smoothing parameter (i. Jon Danielsson \"Financial risk forecasting\" has EWMA and GARCH for R and Matlab and looks like Python now too. Predicting MLC replacement needs ahead of time would allow for proactive maintenance scheduling, reducing the impact MLC replacement has on treatment workflow. This code was written by Michael Rabba. 5sw 94, the parameter suggested by RiskMetrics for daily returns, and \u03bc is the sample average of the returns numpy. \u5bfc\u8bfb1\u3001 \u4f5c\u4e3a\u897f\u5b66\u4e1c\u6e10--\u6d77\u5916\u6587\u732e\u63a8\u8350\u7cfb\u5217\u62a5\u544a\u7b2c\u4e94\u5341\u4e09\u7bc7\uff0c\u672c\u6587\u63a8\u8350\u4e86Valeriy Zakamulin\u4e8e2015\u5e74\u53d1\u8868\u7684\u8bba\u6587\u300aA Test of Covariance-Matrix Forecasting Methods\u300b\u30022\u3001 \u91d1\u878d\u8d44\u4ea7\u6536\u76ca\u7387\u534f\u65b9\u5dee\u77e9\u9635\u7684\u4f30\u8ba1\u548c\u9884\u6d4b\u5728\u91d1\u878d\u4f17\u591a\u9886\u57df\u5982\u8d44\u4ea7\u914d\u7f6e\u3001\u98ce\u9669\u7ba1\u7406\u7b49\u4e2d\u5177\u6709\u6838\u5fc3\u5730\u4f4d\u3002\u76ee\u524d\u5173\u4e8e\u4e0d\u540c\u534f\u65b9\u5dee\u77e9\u9635\u9884\u6d4b\u65b9\u6cd5\u7684\u6bd4\u8f83\u7814\u7a76\u8fd8. In order to detect outliers I use the percentile 99. Convert to correlaon matrix W, and twist this matrix in order to construct \ufb01nal correlaon C, with the correlaon of the individual models in the diagonal blocks. Kevin Sheppard used to have an implementation of the EWMA 2006 covariance matrix but I don't see it anymore. Pandas Basics and GroupBy: Intro to Python Data Science. Udemy Python for Excel: Use xlwings for Data Science and Finance. V is the covariance matrix, and W T is the transpose of the matrix W. vector \u03bc0 and variance-covariance matrix 0, has an upper control limit of Lu =\u03c72 p,1\u2212\u03b1. Recent landslide detection studies have focused on pixel-based deep learning (DL) approaches. I like the flexibility of using Pandas objects and functions but when the set of assets grows the function is becomes very slow: import pandas as pd import numpy as np. As evident in the chart above, large moves in the S&P tend to cluster around major events\u2014Black Monday in 1987, the global financial crisis, and the covid-19 pandemic, most. cov () covs [3] # covariance matrix as of period 4; could be DatetimeIndex Out [7]: 0 1 2 0 0. 2 Kevin Sheppard August 04, 2016 Contents 1 Contents 3 2 Indices and tables 125 Bibliography 127 i ii arch Documentation, Release 3. Kalman-filter is just an algorithm that tune this unknown parameters in a smart way. The parameter \u03bb in the exponential weighted moving average (EWMA) \u03c3n 2 = \u03bb \u03c3 n-1 2 + (1-\u03bb) U 2 n-1- model is 0. s2w The EWMA is widely used in finance, the main applications being technical analysis and volatility modeling. One of the simplest is something like this: Compute the EWMA of the time series and use the last point as an intercept, inter. Asymptotic covariance matrix of $\\bar{\\pmb x}$ Hot Network Questions Do arrows count as trinkets for Prestidigitation? Was the Saturn V assembly carried out on the crawler-transporter or on the VAB's ground floor? How to convert std::vector to a vector of pairs std::vector> using STL. 4so @abstractmethod def compute_variance (self, parameters: NDArray, resids: NDArray, sigma2: NDArray, backcast: Union [float, NDArray], var_bounds: NDArray,)-> NDArray: \"\"\" Compute the variance for the ARCH model Parameters-----parameters : ndarray Model parameters resids : ndarray Vector of mean zero residuals sigma2 : ndarray Array with same size as resids to store the conditional variance. 7gt In Python, create a PriceSeries class. First we convert it into a time Series. D students in BME, Math, Physics, and other quantitative sciences. We can define a population in which a regression equation describes the relations between Y and some predictors, e. The reason for $$c_n = \\pm 3$$ is that the total area between that lower and upper bound spans 99. For example, we'll require volatility for sharpe ratio, sortino ratio and etc. Documentation from the main branch is hosted on my github pages. me7 I want to compute the covariance C of n measurements of p quantities, where each individual quantity measurement is given its own weight. It mainly targets six-sigma professionals. Contents 1 Introduction 2 2 Stationarity 4 3 A central limit theorem 9 4 Parameter estimation 18 5 Tests 22 6 Variants of the GARCH(1,1) model 26 7 GARCH(1,1) in continuous time 27. 5i Last Updated: December 2, 2020. First, we calculate s1, s2, s3, s4, where c = 4, as shown in range F4:F7. Example #1 Let's consider 5 data points as per below table: And parameter a = 30% or 0. cov Nipper 2019-05-19 13:16:54 114 0 python \/ pandas \/ covariance \/ weighted-average \/ covariance-matrix. orc The L-shaped matrix helps display relationships among any two different groups of people, processes, materials, machines, or environmental factors. The (FCM) predicts the volatilities and correlations of the factors, thus. We show that the chart is competitive,. Learning Python Programming A-Z with Real World Simulations. The estimated standard deviations on day n \u2212 1 for variables A and B are 2% and 2. udf () with a python function, a return type and a list of input columns. RiskMetrics 2006 EWMA for Python is here. 7i9 Wax is what you put on a surfboard to avoid slipping. But it is too simple, we already know\u00a0.","date":"2022-06-28 22:03:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 2, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.48137789964675903, \"perplexity\": 2465.908053073707}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103617931.31\/warc\/CC-MAIN-20220628203615-20220628233615-00439.warc.gz\"}"}
| null | null |
{"url":"http:\/\/databasefaq.com\/index.php\/answer\/262107\/ruby-dictionary-why-is-reject-not-rejecting","text":"ruby,dictionary , Why is reject not rejecting?\n\nQuestion:\n\nTag: ruby,dictionary\n\nThis code is my initial stab at the Hamming Distance problem on Exercism.io, but it fails the case when string a is longer than string b, and I'm trying to understand why.\n\ndef self.compute(a, b)\na.split('').reject.with_index { |c, i| c == b[i] }.size\nend\n\n\nI got around the problem by trimming the first string...\n\ndef self.compute(a, b)\na[0...b.size].split('').reject.with_index { |c, i| c == b[i] }.size\nend\n\n\n...but I don't understand why reject is including the extra characters. When I check the comparisons, they seem to be coming up false, as I would expect, yet are still included in the result.\n\nCan anyone tell me why?\n\nI don't understand why reject is including the extra characters. When I check the comparisons, they seem to be coming up false\n\nCorrect. And when you're rejecting, false means \"accept\" - the opposite of reject.\n\nThe problem is merely that you're not grasping what \"reject\" means. When you're up against a question like this, debug. In this case, the way to do that is to eliminate the superfluous material and focus on the thing that's confusing you. Remove the size call and just look at the results of the reject call:\n\ndef compute(a, b)\na.split('').reject.with_index { |c, i| c == b[i] }\nend\nresult = compute(\"hey\", \"ha\")\nputs result\n\n\nThe output is \"e\" and \"y\". And this makes sense:\n\n\u2022 On the first pass, \"h\" == \"h\" and is rejected.\n\n\u2022 On the second pass, \"e\" != \"a\" and is accepted.\n\n\u2022 On the third pass, \"y\" has nothing to compare it with, so it cannot succeed; thus we fail to reject \u2014 and so the \"y\" is accepted. That's what you're asking about.\n\nRelated:\n\nRuby access words in string\n\nruby\nI don't understand the best method to access a certain word by it's number in a string. I tried using [] to access a word but instead it returns letter. puts s # => I went for a walk puts s[3] # => w ...\n\nregex to pull in number with decimal or comma\n\nruby,regex\nThis is my line of code: col_value = line_item[column].scan(\/\\d+.\/).join().to_i When I enter 30,000 into the textfield, col_value is 30. I want it to bring in any number: 30,000 30.5 30.55 30000 Any of these are valid... Is there a problem with the scan and or join which would cause it...\n\nError when trying to install app with mysql2 gem\n\nmysql,ruby-on-rails,ruby,ruby-on-rails-4\nIm trying to install an open source rails 3.2.21 application that uses the mysql2 gem, but when i try and run the bundle commant I get the following error: Fetching: mysql2-0.3.18.gem (100%) Building native extensions. 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I have an Array of Objects, along this line: [ { name: \"foo1\", location: \"new york\" }, { name: \"foo2\", location: \"new york\" }, { name: \"foo3\", location: \"new york\"...","date":"2019-09-15 05:54:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.17427517473697662, \"perplexity\": 5197.747108660859}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514570740.10\/warc\/CC-MAIN-20190915052433-20190915074433-00556.warc.gz\"}"}
| null | null |
module SpreeUnifiedPayment
module Generators
class InstallGenerator < Rails::Generators::Base
def add_javascripts
append_file 'vendor/assets/javascripts/spree/backend/all.js', "//= require admin/spree_unified_payment\n"
end
def add_stylesheets
inject_into_file 'vendor/assets/stylesheets/spree/frontend/all.css', " *= require store/spree_unified_payment\n", :before => /\*\//, :verbose => true
end
def add_migrations
run 'bundle exec rake railties:install:migrations FROM=spree_unified_payment'
end
def run_migrations
run 'bundle exec rake db:migrate'
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,383
|
\section{\label{introduction}Introduction}
Total correlations present in a system can be separated in a purely quantum part and a classical part. Quantum correlations can be classified as those associated with non-separability (entanglement) and other quantum correlations, together quantified by the quantum discord (QD) \cite{Zurek2001PRL,Henderson2001JPA}. For quantum information processing it is relevant to know the dynamics of these different kinds of correlations.
Entanglement dynamics can be considered well understood, in its general lines, for bipartite quantum systems interacting with quantum environments (independent or common), presenting phenomena like sudden death \cite{yu2004PRL}, revivals \cite{bellomo2007PRL,bellomo2008PRA,mazzola2009PRA} or trapping \cite{bellomo2008trapping}, depending on the Markovian or non-Markovian nature of the environments. Dynamics of QD has also been investigated for two-qubit systems in the presence of both Markovian \cite{maziero2010PRA,ferraro2010PRA} and non-Markovian \cite{fanchini2010PRA,wang2010PRA} quantum environments.
In this paper we analyze the dynamics of correlations in a two-qubit system where each qubit is subject to a phase noisy laser modeled as a classical field. This is a commonly used model for the interaction between light and matter, and thus the dynamics of correlations for this situation merits investigation. The broader question underlying our work is whether all kinds of behavior of correlations that can be described using a quantum environment also can be well described using a classical approximation of the environment. One might for example expect that a classical environment should not be able to store quantum correlations on its own, and that therefore revivals of quantum correlations in the system may be affected. For the phase-noisy laser, are there qualitative features that will be lost by treating the field classically as opposed to quantum-mechanically? Among other things, we find that this model can describe both decay and oscillatory behavior of quantum correlations.
\section{Model: a qubit subject to a phase-noisy laser}
Our system consists of a pair of qubits (two-level atoms), $A$ and $B$, each driven by a local phase noisy laser. Each atom under the action of its own laser is described, in a rotating frame and for resonant atom-field interaction, by the Hamiltonian \cite{Cresser2009}
\begin{equation}\label{Hamiltonian}
\hat{H}=\lambda \left[\sigma_-\mathrm{e}^{\mathrm{i}\Phi(t)}+\sigma_+\mathrm{e}^{-\mathrm{i}\Phi(t)}\right],
\end{equation}
where the laser is described as a classical field with a randomly fluctuating phase $\Phi(t)$. The interaction between each qubit and its local field mode is assumed to be strong enough so that, for sufficiently long times, the dissipation effects of the vacuum radiation modes on the qubit dynamics can be neglected. This could be feasible by considering, as a qubit, an atom in a cavity subject to a resonant interaction with the phase noisy laser but out of resonance with cavity mode frequencies in order to inhibit effects like spontaneous emission. In this phase-noisy model the phase undergoes a Wiener process, i.e. $\Phi(t)$ is white noise with a correlation function $\langle\dot{\Phi}(t)\dot{\Phi}(t+\tau)\rangle=2d\delta(\tau)$ where $d$ is a diffusion rate. In Eq.~(\ref{Hamiltonian}), $\lambda$ is the atom-field coupling strength, $\sigma_+=\ket{1}\bra{0}$ and $\sigma_-=\ket{0}\bra{1}$ are the atomic raising and lowering operators, where $\ket{0}$ and $\ket{1}$ are the ground and excited state of the atom, respectively. The field correlation function, corresponding to the above phase correlation function, is a complex colored noise $e^{i\Phi(t)}$ described by $\langle\mathrm{e}^{i \Phi}(t)\mathrm{e}^{-i \Phi}(t+\tau)\rangle=\mathrm{e}^{-d\tau}$.
The Hamiltonian of Eq.~(\ref{Hamiltonian}) leads to a local-in-time non-Markovian master equation of the form \cite{Cresser2009}
\begin{equation}\label{unitalmastereq}
\dot{\rho}(t)=\sum_{i=1}^3(\gamma_i\sigma_i\rho\sigma_i-\gamma_i\rho),
\end{equation}
where $\sigma_i$ ($i=1,2,3$) are the atomic pseudo-spin Pauli matrices and
\begin{equation}\label{gammaandGamma}
\gamma_1=\gamma_2=-\frac{\dot{\Gamma}_1}{4\Gamma_1},\ \gamma_3=-\frac{1}{2}\left(\frac{\dot{\Gamma}_2}{\Gamma_2}
-\frac{\dot{\Gamma}_1}{2\Gamma_1}\right).
\end{equation}
This master equation is derived by the Nakajima-Zwanzig projection operator method and the functions $\Gamma_1,\Gamma_2$ can be expressed in terms of the system parameters. In particular $\Gamma_1(t)=\mathrm{e}^{-\frac{dt}{2}}\left[\cosh\left(\frac{1}{2}t\sqrt{d^2-16\lambda^2}\right)
+d\sinh\left(\frac{1}{2}t\sqrt{d^2-16\lambda^2}\right)/\sqrt{d^2-16 \lambda ^2}\right]$, while the expression for $\Gamma_2(t)$ is cumbersome and is not given here. Because the $\Gamma_i$ are time-dependent, the master equation of Eqs.~(\ref{unitalmastereq}) and (\ref{gammaandGamma}) has a quasi-Lindblad form, that is a master equation that resembles the Lindblad form but has time-dependent decay rates $\gamma_i$ (which also can be negative) \cite{Cresser2009}. Moreover, the above single-qubit master equation is a unital master equation \cite{andersson2007JMO}. A master equation $\partial\rho/\partial t=\mathcal{L}_t(\rho)$ is defined to be \emph{unital} if the maximally mixed state $\frac{1}{2}\hat{I}$ is a fixed point, that is if $\mathcal{L}_t(\hat{I})\equiv0$. Unital master equations can be solved by Kraus-type decomposition methods giving the single-atom reduced density matrix elements at time $t$ as \cite{andersson2007JMO}
\begin{eqnarray}\label{singleatomreduceddensitymatrix}
\rho_{11}(t)&=&[(1+\Lambda_3)\rho_{11}(0)+(1-\Lambda_3)\rho_{00}(0)]/2,\nonumber\\
\rho_{10}(t)&=&[(\Lambda_1+\Lambda_2)\rho_{10}(0)+(\Lambda_1-\Lambda_2)\rho_{01}(0)]/2,
\end{eqnarray}
with $\rho_{11}=1-\rho_{00}$, $\rho_{10}=\rho^*_{01}$. All the $\Lambda_i(t)\equiv\Lambda_i$ are time-dependent and related to the rates $\gamma_i$ by
\begin{equation}\label{Lambdaandgamma}
\Lambda_i(t)=\mathrm{e}^{-2 \int_0^t \mathrm{d}t'[\gamma_j(t')+\gamma_k(t')]},
\end{equation}
with the conditions $\Lambda_i+\Lambda_j\leq1+\Lambda_k$, where $\{i,j,k\}$ run over the cyclic permutations of $\{1,2,3\}$. Note also that $\Lambda_j(0)=1$ and $\Lambda_j(t)\geq0$. Using Eq.~(\ref{gammaandGamma}) in Eq.~(\ref{Lambdaandgamma}) we have $\Lambda_1=\Lambda_2=\Gamma_2$, $\Lambda_3=\Gamma_1$ and the resulting reduced density matrix elements for the single atom driven by a phase noisy laser reduces, from Eq.~(\ref{singleatomreduceddensitymatrix}), to $\rho_{11}(t)=\frac{1}{2}[(1+\Gamma_1)\rho_{11}(0)+(1-\Gamma_1)\rho_{00}(0)]$ and $\rho_{10}=2\Gamma_2\rho_{10}(0)$.
It has been shown that for $d/\lambda<4$, $\Gamma_1$ oscillates while $\Gamma_2$ shows oscillatory behavior only for $d/\lambda <2.606$ \cite{Cresser2009}; for $d/\lambda>4$, neither $\Gamma_1$ nor $\Gamma_2$ shows oscillatory behavior. Finally, in the limit $d/\lambda\rightarrow0$, the two decay rates $\Gamma_1,\Gamma_2$ become periodic functions of $\lambda t$, more specifically $\Gamma_1\rightarrow\cos(2\lambda t)$ and $\Gamma_2\rightarrow\cos^2(\lambda t)$. In this limit, the probability distribution of the phase is a steady state distribution, so that the effective dynamics is the statistical average, equally weighted, of all the evolutions with phase between $0$ and $2\pi$. An analogous but simpler setup, described by a uniform phase distribution with only two values ($0,\pi$), has been investigated in Ref.~\cite{LoFranco2010arxive}.
\section{Quantifiers of two-qubit correlations\label{sec:quantifiers of correlations}}
To obtain the expressions of the correlation quantifiers we need the two-qubit density matrix elements. We construct the two-qubit reduced density matrix at time $t$ by the knowledge of the evolution of single-qubit reduced density matrices, according to a standard procedure \cite{bellomo2007PRL}, with the single-qubit reduced density matrix evolution given by Eqs.~(\ref{singleatomreduceddensitymatrix}) and (\ref{Lambdaandgamma}). Thus, we obtain the explicit expressions of the two-qubit density matrix elements at time $t$ in terms of the functions $\Lambda_i$ for any initial two-qubit state. These expressions are however quite cumbersome and are not reported here.
We take as two-qubit initial states the extended Werner-like (EWL) states \cite{bellomo2008PRA}
\begin{equation}\label{EWLstates}
\hat{\rho}^\Phi=r \ket{\Phi}\bra{\Phi}+\frac{1-r}{4}I_4,\quad
\hat{\rho}^\Psi=r \ket{\Psi}\bra{\Psi}+\frac{1-r}{4}I_4,
\end{equation}
where $r$ indicates the purity of the initial states, $I_4$ is the $4\times4$ identity matrix,
$\ket{\Phi}=\alpha\ket{01}+\beta\mathrm{e}^{i\delta}\ket{10}$ and $\ket{\Psi}=\alpha\ket{00}+\beta\mathrm{e}^{i\delta}\ket{11}$ are the Bell-like states where $\alpha,\beta$ \blue{are} non-negative real numbers and $\alpha^2+\beta^2=1$. These are mixed states reducing to Werner states for $\alpha=\beta=\pm1/\sqrt{2}$ or to Bell-like states for $r=1$. The density matrix elements of the EWL states are such that the resulting density matrix has an ``X'' structure with nonzero elements only along the main diagonal and anti-diagonal.
In the standard basis $\mathcal{B}=\{\ket{1}\equiv\ket{11},\ket{2}\equiv\ket{10},\ket{3}\equiv\ket{01},\ket{4}\equiv\ket{00}\}$ and for general different environments characterized by different values of $\Lambda_i^S$ $(i=1,2,3;\ S=A,B)$, the time-dependent two-qubit density matrix elements for $\hat{\rho}^\Phi$ are
\begin{eqnarray}\label{twoqubitelements}
&\rho^\Phi_{jj}(t)=\frac{1}{4}\left\{1-r\left[\Lambda^A_3\Lambda^B_3
+(-1)^j(1-2\alpha^2)(\Lambda^A_3-\Lambda^B_3)\right]\right\},&\nonumber\\
&\rho^\Phi_{ll}(t)=\frac{1}{4}\left\{1+r\left[\Lambda^A_3\Lambda^B_3
+(-1)^l(1-2\alpha^2)(\Lambda^A_3+\Lambda^B_3)\right]\right\},&\nonumber\\
&\rho^\Phi_{1+k4-k}(t)=\frac{\alpha\beta r}{2}\left[f(\Lambda)\cos\delta
+\mathrm{i}(-1)^{k+1}g(\Lambda)\sin\delta\right],&
\end{eqnarray}
where $j=1,4$, $l=2,3$, $k=0,1$, $f(\Lambda)=\Lambda^A_1\Lambda^B_1+\Lambda^A_2\Lambda^B_2$ and $g(\Lambda)=\Lambda^A_1\Lambda^B_2+\Lambda^A_2\Lambda^B_1$. The density matrix elements for the initial state $\hat{\rho}^\Psi$ are obtained from Eq.~(\ref{twoqubitelements}) by changing $1\leftrightarrow2$ and $3\leftrightarrow4$. Note that under the dynamical conditions here considered, the two-qubit state maintains an X structure during the evolution.
In order to describe the entanglement dynamics we use the concurrence, which for an X state is given by \cite{yu2007QIC}
$C_\rho^X(t)=2\mathrm{max}\{0,K_1(t),K_2(t)\}$ where $K_1(t)=|\rho_{23}(t)|-\sqrt{\rho_{11}(t)\rho_{44}(t)}$ and $K_2(t)=|\rho_{14}(t)|-\sqrt{\rho_{22}(t)\rho_{33}(t)}$. The EWL states of Eq.~(\ref{EWLstates}) present the same initial value of the concurrences, $C(0)=2\mathrm{max}\{0,(\alpha\beta+1/4)r-1/4\}$, from which one finds that there is initial entanglement when $r>r^\ast=(1+4\alpha\beta)^{-1}$. Using the time-dependent density matrix elements of Eq.~(\ref{twoqubitelements}), it is readily seen that in our system the concurrence at time $t$ is the same for both the EWL states of Eq.~(\ref{EWLstates}), that is $C_\rho^\Phi(t)=C_\rho^\Psi(t)=C(t)$. For example, for initial Bell states ($r=1, \alpha=1/\sqrt{2},\delta=0,\pi$) and different local conditions, concurrence is given by $C(t)=\frac{1}{2}\left(\Lambda^A_1 \Lambda^B_1+\Lambda^A_2\Lambda^B_2+\Lambda^A_3 \Lambda^B_3-1\right)$.
In order to quantify total correlations, $\mathcal{T}$, present in the two-qubit system and to distinguish a quantum $\mathcal{D}$ and a classical part $\mathcal{J}$ of them, we use the notion of quantum discord \cite{Zurek2001PRL,Henderson2001JPA}. The calculation of discord and classical correlations requires a maximization procedure and this has been analytically solved only for certain class of quantum states (Bell-diagonal and X states) \cite{Luo2008PRA,ali2010PRA}. In particular, here we give the explicit expressions of these quantifiers for the initial condition $\alpha=\beta=1/\sqrt{2}, \delta=0,\pi$ for which the two-qubit density matrix always has a Bell-diagonal form $\rho_\mathrm{B}=[\mathbb{I}\otimes\mathbb{I}+\sum_{j=1}^3c_{j}\sigma_j\otimes\sigma_j]/4$. For this class of states, the following expressions for $\mathcal{T}$ and $\mathcal{J}$,
with $\mathcal{D}=\mathcal{T}-\mathcal{J}$, hold \cite{Luo2008PRA}:
\begin{eqnarray}\label{total, discord and classical}
\mathcal{T}=2+\sum_{i,s}\lambda_i^s\mathrm{log}\lambda_i^s \quad
\mathcal{J}=\sum_{i}^2 \frac{1+(-1)^ic}{2}\mathrm{log}[1+(-1)^ic],
\end{eqnarray}
where $i=1,2; s=\pm$, $c\equiv\mathrm{max}\{|c_1|,|c_2|,|c_3|\}$, $\lambda_1^\pm=(1\pm c_1\pm c_2-c_3)/4$, $\lambda_2^\pm=(1\pm c_1\mp c_2+c_3)/4$. If the initial state is $\hat{\rho}^\Phi$, the $c_i$ coefficients are $c_1=r\mathrm{max}\{\Lambda^A_1 \Lambda^B_1,\Lambda^A_2 \Lambda^B_2 \}$, $c_2=r\mathrm{min}\{\Lambda^A_1\Lambda^B_1,\Lambda^A_2\Lambda^B_2 \}$ and $c_3=-r\Lambda^A_3\Lambda^B_3$. If the initial state is $\hat{\rho}^\Psi$, previous coefficients change as $c_1\rightarrow c_1$, $c_2\rightarrow -c_2$ and $c_3\rightarrow-c_3$, that is a relabeling of the $\lambda$ eigenvalues: the quantifiers $\mathcal{T}$, $\mathcal{D}$ and $\mathcal{J}$ thus coincide for both initial states.
\section{Dynamics of correlations due to a phase-noisy laser}
We use the general results of the previous section to analyze the dynamics of correlations. In particular, for initial EWL states in identical environments ($\Gamma_j^A=\Gamma_j^B=\Gamma_j,\, j=1,2$) we obtain $C_\rho^\Phi(t)=C_\rho^\Psi(t)=C=\frac{1}{2}\left(4 r \alpha\beta\Gamma_2^2+r \Gamma^2_1-1\right)$. In Fig.~\ref{Concurrence and entropy}(a) we plot concurrence evolution for initial Bell states, for two different values of the ratio $d/\lambda=5,0.1$. The behavior for more general initial mixed states can be shown to be qualitatively similar. One sees that while concurrence presents Markovian-like decay for $d/\lambda= 5$, it is subject to non-Markovian revival for $d/\lambda=0.1$, as already found in other systems \cite{bellomo2007PRL,LoFranco2010arxive}. In order to evidence the fact that for large times the two-qubit system goes toward a maximally mixed state as a consequence of the unital evolution of the single qubits, in Fig.~\ref{Concurrence and entropy}(b) we plot the evolution of the von Neumann entropy, $S=-\mathrm{Tr}\{\rho(t)\log\rho(t)\}$, for the same values of $d/\lambda$ (von Neumann entropy is the same for both initial EWL states $S_\rho^\Phi(t)=S_\rho^\Psi(t)=S$). It is seen that, for any value of $d/\lambda$, $S$ tends to its maximum value 2, corresponding to the two-qubit maximally mixed state, with oscillating behavior when memory effects are present ($d/\lambda=0.1$).
\begin{figure}
\begin{center}
{\includegraphics[width=0.45 \textwidth]{fig1a}
\hspace{0.5 cm}
\includegraphics[width=0.45 \textwidth]{fig1b}}
\caption{\label{Concurrence and entropy}\footnotesize Time evolution of concurrence $C$ (panel (a)) and von Neumann entropy $S$ (panel (b)) starting from initial Bell states ($r=1,\alpha=1/\sqrt{2},\delta=0, \pi$) for the ratios $d/\lambda=5$ (black solid curve) and $d/\lambda=0.1 $ (blue dashed curve).}
\end{center}
\end{figure}
We can also separately investigate the evolution of quantum and classical correlations. In Fig.~\ref{quantum vs classical} we plot the evolution of total, quantum and classical correlations of Eq.~(\ref{total, discord and classical}) for the value of the ratio $d/\lambda=0.1$ for initial Bell states.
\begin{figure}
\begin{center}
\includegraphics[width=0.45 \textwidth]{fig2}
\caption{\label{quantum vs classical}\footnotesize Time evolution of quantum discord $\mathcal{D}$ (black solid curve), classical correlations $\mathcal{J}$ (red dashed curve) and total correlations $\mathcal{T}$ (blue dot-dashed curve), starting from initial Bell states ($r=1, \alpha=1/\sqrt{2}, \delta=0, \pi$) for the $d/\lambda=0.1$. In the inset $\mathcal{D}$ (black solid curve) and concurrence $C$ (blue dashed curve) are plotted.}
\end{center}
\end{figure}
From the plot one sees that, in this non-Markovian case, total and classical correlations show an oscillating decay while quantum discord has a non-purely oscillating decay (i.e., there are points of discontinuity for the derivative during the time regions where discord is very close to zero) and it decays much faster than classical correlations. On the other hand, as seen from the inset of Fig.~\ref{quantum vs classical}, discord definitively disappears later than entanglement, with time regions when it is different from zero while entanglement is absent.
\section{Conclusions \label{par:Conclusion}}
In this paper we have analyzed the dynamics of correlations between two initially entangled independent qubits each locally subject to a phase noisy laser. We found explicit expressions for various quantifiers of correlations, such as concurrence and quantum discord, in terms of the decay rates present in the single-qubit unital master equation which here describes the action of phase noisy laser. We discussed how the dynamics of correlations depends on the ratio between the phase diffusion rate $d$ and the atom-laser coupling strength $\lambda$.
Although the light field is treated as classical, this model can describe revivals of quantum correlations.
For large values of $d/\lambda$ Markovian-like decay occurs, while non-Markovian effects become relevant for small values of this ratio, giving place for example to revival of entanglement and of discord. Moreover, in the non-Markovian regime, quantum discord presents an oscillatory decay faster than that of classical correlations, with time regions where it is nonzero and entanglement is zero. It is therefore evident that the phase diffusion rate and coupling strength of each atom-laser affect the qualitative time-behavior of quantum correlations in a crucial way.
\section*{References}
\providecommand{\newblock}{}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,123
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package org.springframework.cloud.config.monitor;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertNotNull;
import static org.junit.Assert.assertNull;
import java.util.Map;
import org.junit.Test;
import org.springframework.core.io.ClassPathResource;
import org.springframework.http.HttpHeaders;
import com.fasterxml.jackson.core.type.TypeReference;
import com.fasterxml.jackson.databind.ObjectMapper;
/**
* @author Dave Syer
*
*/
public class GitlabPropertyPathNotificationExtractorTests {
private GitlabPropertyPathNotificationExtractor extractor = new GitlabPropertyPathNotificationExtractor();
private HttpHeaders headers = new HttpHeaders();
@Test
public void pushEvent() throws Exception {
// See http://doc.gitlab.com/ee/web_hooks/web_hooks.html#push-events
Map<String, Object> value = new ObjectMapper().readValue(
new ClassPathResource("gitlab.json").getInputStream(),
new TypeReference<Map<String, Object>>() {
});
this.headers.set("X-Gitlab-Event", "Push Event");
PropertyPathNotification extracted = this.extractor.extract(this.headers, value);
assertNotNull(extracted);
assertEquals("application.yml", extracted.getPaths()[0]);
}
@Test
public void nonPushEventNotDetected() throws Exception {
// See http://doc.gitlab.com/ee/web_hooks/web_hooks.html#push-events
Map<String, Object> value = new ObjectMapper().readValue(
new ClassPathResource("gitlab.json").getInputStream(),
new TypeReference<Map<String, Object>>() {
});
this.headers.set("X-Gitlab-Event", "Issue Event");
PropertyPathNotification extracted = this.extractor.extract(this.headers, value);
assertNull(extracted);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,613
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Q: Explaining the GTAV Stock Market As a little experiment, I invested all of Franklin's money on the LCN in one company: Maze Bank. I then sent him to sleep over and over for a week (game time), checking his portfolio every time he woke up to see how the stock was doing. What I saw baffled me.
The stock fluctuated 1 or 2% for the whole week. I bought it at around $14.60 and it stayed within roughly a dollar of that figure the entire week.
The Graph
Firstly, the graph apparently reflects this past week, but it seems to say that the stock spent time at $70 a share, and was now resting at $56. In fact, according to the graph, it had only once dipped below $26 the entire week... Nowhere near the $14.60 figure I usually saw.
How am I reading the graph wrong?
Stock Information
On the right-hand side, the "Stock Information" seemed equally confusing. The "High" and "Low" seem to refer to the highest and lowest the stock had EVER been. (It certainly didn't reflect the week's fluctuations.)
"Last" (and thus "Change"/"% Change") didn't seem to reflect any price I saw over the period of a week, either. (Does anyone know what time frame the "Last" figure comes from?)
And that's only the LCN. Does the BAWSAQ have more accurate graphs/figures?
On the whole, all these graphs and figures seemed very confusing. Can anyone help make sense of them?
A: Django, I agree with you - the chart on each individual stock is at best difficult to decipher, and at worst meaningless. I own several LCN stocks on all 3 characters and, like you, I noticed that so far (over several in-game days - well over a week) they have moved very little, but their "details" page shows wild swings up and down each day. It's like there's some 100ms-long peak each day that if I just happen to be looking at the market in that instant, I'd double or triple my money.
Similarly, the Last and %Change don't make much sense much of the time.
So I can't answer your question, but I've decided to only pay attention to the High and Low numbers, and buy when a stock is within about 10% of the swing to the Low number. For example, if the low is $10 and the high is $100, then the swing is $90. 10% of the swing is $9, so I'll buy if the stock is between $10-19 (or if it goes below $10 of course).
I think the general consensus is that stocks WILL climb back towards their High value, so I have faith that eventually this will pay off. For the last couple of in-game weeks, though, all of my bought-low stocks are just staying low, with one exception. I bought VAG (insert your jokes about VAG here [insert your jokes about "insert" here]) at about $6 and I got 70,000 shares of it. It went up about $0.70 shortly thereafter which made me $40K or so.
Good luck.
A: LCN is not a very lucrative market outside of the assassination missions. BAWSAQ on the other hand can make you rich relatively quickly. I made $1.8 million with one of my characters in three days time. The graphs are very useful but alone they will only scare you or give you false hope. It's important to check in at the Rockstar social club. The in game charts for BAWSAQ only show the last five hours while the social club charts show the last week. Watch the social club charts for a stock that is crashing and compare it with the in game charts. When the stock reaches it's low point, and the in game chart show an uptick in stock value buy. From here on out its best to watch the social club charts but keep an eye on the in game charts for any sudden major drops. Also keep an eye on Pißwasser. It's is consistently lucrative. I bought in at $1.23 and sold at $24. Good luck.
Also in response to your initial questions "last" is the price of the stock before it last changed. You can't really say how often that is (some stocks linger at the same price for longer than others) but the information is useful and I will explain why in a minute. The "%change" is the % of the last recorded price that the difference of the new price and the original price is. This sounds confusing but it's really very simple. If a stock is $10 and jumps to $15 the %change is 50% because $10-$15 is $5 and $5 is 50% of $10. This is also useful and I will also explain why. Pay no attention to the high, it is the price of the stock when the game launched and most stocks don't get anywhere near their initial worth. I wish they would give us the average high price, but what are you gonna do? The low is very useful. When the price is at the low point BUY. It may drop a little but stay confident, don't sell, and wait for it to rise. Now for selling, and this is when those figures come into play. The "last" will not always be lower than the current price when a stock is on the rise. As long as the %change stays above -5% to -10% the stock will still likely rise. When comes time that you think you want to sell watch the I'm game graphs for a major down tick. If you follow these tips along with my other post you stand to make up 2500% increase on your initial investment, and that's just the most I've made so far. I think with the right stock I could make at least 5000% ROI (that's Return On Investment). The BAWSAQ graphs do reflect real market movement though, but both graphs are needed to make informed investments.
A: Part of the problem here is that the LCN is not a true market. It's semi-random with scripted elements driven by events in the game. General consensus is that the graphs for LCN stock are just there to be pretty and don't ever change. I've heard, but haven't confirmed for myself, that the scripted events will change what the historical graph looks like temporarily to match the events the game describes.
I did an experiment to confirm this. I looked at the LCN page for Maze Bank. Then, without saving I jumped into online mode. After a few seconds, I next switched back to story mode. When I looked at the Maze Bank page again the history graph was completely different, but the current value was the same(well only a few cents off).
First Maze Bank Graph:
Second Maze Bank Graph:
So, yeah, it's just random noise meant to sell the "realism" of the LCN exchange.
If the graph did make sense the values would mean:
*
*High: The highest historical valuation of the stock.
*Low: The lowest historical valuation of the stock.
*Current: The current value of the stock.
*Last: The value at the close of the previous day of trading
*Change and %Change The change from the current price and the last value.
BAWSAQ works differently. It's semi-modeled to behave like a real market. All the players who buy stock effect the price so long as they have online access. How much they effect it is contentious. There have been a few organized pump and dump schemes, but the results are not quite obvious. This is why BAWSAQ has more sensible graphs that maintain an accurate history of the stock value.
A: When it comes to LCN don't pay any attention to the graph. It's there to give you a general idea how the market is moving. So for instance it may be around the low and go up a dollar or two, on the graph it would look like it has made a huge jump, and in context it the rest of the market it has. Stocks rarely go up by that much at any given time. BAWSAQ models the real world stock market a little bit more closely. It is generally meant to be watched closely (so your entire play time consists of watching the market) because stocks only go up or down a small amount, but they change very often. Again the chart doesn't mean much. Your best bet when playing the market is to watch stocks and buy them at their lowest and sell whenever there is an increase. Never sell them when they are at a decrease at what you bought them at because they WILL go back up again eventually. One last tip: buy into cheap stocks, so after you do the missions LifeInvader trades at a low around 1.79 and at most goes up to 3.0 on a good upswing. If you buy into this you can purchase a lot of stocks and can potentially double your money, whereas most stocks will only give you a small amount of a return by comparison because you have fewer stocks overall.
Edit:one thing I forgot to mention, in the main menu of the stock market there are stories about things happening with businesses such as food poisoning at Burger Shot. Those are pretty good indicators on what stock is going to make a big change fairly soon, or has just made a big change.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,053
|
Court strikes down abortion clinic "buffer zone": In Plain English
By Amy Howe
on Jun 27, 2014 at 5:22 pm
In 2007, Massachusetts passed a law that makes it a crime to stand on a public road or sidewalk within thirty-five feet of any abortion clinic in the state. Yesterday the Supreme Court struck down the Massachusetts "buffer zone," siding with a group of abortion opponents who argued that the law was unconstitutional because it prevented them from being able to counsel and offer assistance to women entering the clinics. But (much like yesterday's decision in the recess appointments case, which I discussed in Plain English here), although all nine Justices agreed that the Massachusetts law cannot stand, there was no consensus on the reasoning that they used to reach that result. Let's talk about the decision in McCullen v. Coakley in Plain English.
Although we often think of Justice Anthony Kennedy as the pivotal vote on the Court in high-profile cases, yesterday it was Chief Justice John Roberts who played that role, writing an opinion that had the support of the four more liberal Justices — Ginsburg, Breyer, Sotomayor, and Kagan. Roberts began by emphasizing that public streets and sidewalks have historically had a special significance, for purposes of the First Amendment, as a place for public discussion and debate. They are, the Court observed, one of the few places left where you can run into speech that you might not otherwise hear, and might not want to hear. Given that special significance, the Court continued, the government's power to regulate the content of speech on streets and sidewalks is very limited.
But with the "buffer zone" law, the Court reasoned, Massachusetts isn't regulating content – that is, the law doesn't target what people are saying. This is so even if the law only applies at abortion clinics – which, the plaintiffs had argued, meant that all of the speech that it regulated was speech about abortion. A violation of the law doesn't hinge on what you say, the Court explained; all that matters is where you say it. If you are in the buffer zone, you can violate the law without saying a word. The Court acknowledged that Massachusetts couldn't pass this kind of law just because anti-abortion protesters or counselors made other people uncomfortable, but it underscored that the concerns that prompted the legislature to pass this law – trying to keep access to clinics open and keep the area around clinics safe – aren't aimed at what people are saying. The Court also found that, although the law does not apply to clinic employees, that doesn't mean that it discriminates based on who is speaking. In the Court's view, the exemption simply allows employees – not only the escorts who walk women into the clinic but also the guy who shovels the sidewalk in front of the clinic – to do their jobs.
Although the Supreme Court's caselaw puts fairly tight limits on the government's ability to regulate the content of speech (limits that, the Court concluded, do not apply to the Massachusetts law), the government has more leeway to regulate other aspects of speech: where and when you can speak and how. For these kinds of regulations, courts look at whether the law is "narrowly tailored" – that is, does it restrict more speech than necessary to advance the government's goals? The Massachusetts "buffer zone" law, the Court determined, fails even under this more lenient test. On the one hand, the Court suggested, the burden that the law imposes on the plaintiffs in this case is significant: talking to someone on the sidewalk and handing her a pamphlet in the hope of convincing her not to have an abortion is exactly the kind of speech that the First Amendment protects. And it doesn't help the state's cause that the plaintiffs can still stand outside the buffer zones and chant or hold signs, because they are instead trying to counsel people.
On the other hand, the Court noted, the law does restrict more speech than it needs to. No other state uses a buffer zone like this one, which suggests that there are alternatives that Massachusetts has overlooked. First and foremost, if the goal is to protect patients and more broadly maintain order outside clinics (which, the Court agreed, is a legitimate interest), there is a separate provision of the law that specifically addresses misconduct outside clinics and imposes criminal penalties for violations. Or the state could enact other laws to deal with it. Either option would allow the state to target particular individuals who block access to clinics or harass women, without penalizing people like the plaintiffs who say that they are just trying to talk to women. The Court also expressed skepticism that access to and public safety around clinics are actually problems anywhere other than one specific clinic: "For a problem shown to arise only once a week in one city at one clinic, creating 35-foot buffer zones at every clinic across" the state "is hardly a narrowly tailored solution."
Nor, the Court continued, can the state justify the restrictions by saying that it tried other options but they didn't work; the five Justices pointedly observed that they saw no sign that the state had tried to rely on those other options to prosecute anyone in the last seventeen years. And even if the fixed buffer zone was easier for police than some of the other options, the Court was unmoved: "[T]he prime objective of the First Amendment is not efficiency."
Justices Scalia, Kennedy, Thomas, and Alito all agreed with their five colleagues that the Massachusetts law violates the First Amendment. But unlike the majority, Scalia (writing for himself and Justices Kennedy and Thomas) did regard the law as targeting speech on abortion, so he would subject it and similar laws to a more stringent test. And even more importantly, Scalia would overrule a 2000 decision by the Court that upheld a different buffer zone in Colorado. "Protecting people from speech they do not want to hear," he made clear, "is not a function that the First Amendment allows the government to undertake in public streets and sidewalks."
The end result in this case was not much of a surprise; many people (including me) had predicted after the oral argument that the law was likely to fall. But the vote was perhaps more unexpected; many abortion rights supporters probably would not have anticipated that some of the more reliably liberal Justices like Ginsburg and Sotomayor would join the Chief Justice and vote to strike down the law. One possibility may be – as Kevin Russell suggested in his post yesterday – that the more liberal Justices may have recognized that there were not enough votes for the Massachusetts law. And so they may have been willing to sacrifice this law to ensure that, if challenged, other buffer zones are subjected to (and may be able to survive) the same, less stringent test that the Court used in the case.
The case is also interesting because of what it may signal for the challenge to the Affordable Care Act's contraception mandate, in which we are still waiting on the Court's decision. Although the precise issues before the Court are different, both involve the intersection of the First Amendment (the McCullen plaintiffs' desire to counsel women who might be seeking abortion and the Hobby Lobby families' firm opposition to providing their female employees with birth control) and women's reproductive rights. If the oral argument in Hobby Lobby is any indicator, don't look for the same kind of unanimity, even with regard to just the result, if (as we expect) the Court issues its decision on Monday. But whether the vote is nine-zero, five-four, or something in between, we will be back to cover it in Plain English.
Posted in Merits Cases, Plain English / Cases Made Simple, Everything Else
Recommended Citation: Amy Howe, Court strikes down abortion clinic "buffer zone": In Plain English, SCOTUSblog (Jun. 27, 2014, 5:22 PM), https://www.scotusblog.com/2014/06/court-strikes-down-abortion-clinic-buffer-zone-in-plain-english/
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,876
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Barnard's Star b (also designated GJ 699 b) was a proposed super-Earth-mass exoplanet orbiting Barnard's Star in the constellation of Ophiuchus, six light-years away from Earth. The exoplanet's discovery by an international team of astronomers – including the European Southern Observatory and Carnegie Institution for Science – was officially announced on 14 November 2018. More recent studies, in 2021 and 2022, have concluded that the radial velocity signal corresponding to Barnard's Star b is most likely an artifact of stellar activity, and thus the planet does not exist.
Characteristics
Barnard's Star b technically remains a planet candidate as it has been proposed with a confidence figure of 99%. The research team that made the announcement will continue observations to ensure that no improbable variations in brightness and motion in the star might account for the discovery. Direct imaging opportunities of the planet from large ground-based telescopes, or potentially the Nancy Grace Roman Space Telescope, are expected within ten years of 2018. There is an outside chance that a transit of the star might also allow for imaging.
The planet was proposed through the radial velocity method, the most common planet-hunting technique. A "wobble" observed in Barnard's Star's motion was confirmed to have a period of about 233 days, corresponding to a semi-major axis of 0.4 AU for a proposed companion. The mass of the likely planetary body was then deduced to be about 3.2 Earth mass. Lead astronomer Ignasi Ribas notes: "We used observations from seven different instruments, spanning 20 years of measurements, making this one of the largest and most extensive datasets ever used for precise radial-velocity studies."
Barnard's Star b is expected to be frigid, with an equilibrium temperature of around . Its orbital distance, though close to the star by solar system standards, is around the snow line for a dim red dwarf like Barnard's Star. This is the point where volatile compounds such as water condense to form ice and thus outside the assumed habitable zone where temperatures are right for surficial liquid water. However, new research suggests that heat generated by geothermal processes could warm pockets of water beneath the surface of the planet, potentially providing havens for life to evolve.
Astronomers expect to find more such "snow line" planets as proto-planetary accretion is favorable in this temperature range. A second planetary companion for Barnard's Star has been suggested based on unconfirmed "wobbles" in the current system.
References
Disproven exoplanets
Ophiuchus (constellation)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,213
|
Q: How to add a Netbeans project as a dependency to a Gradle project? I am new to gradle and most of my existing projects are in ant (netbeans projects).
Do I have to create gradle project for each of those projects that I would want to reuse?
Can I straightaway declare existing netbeans projects as dependencies in my gradle project? If yes, how?
Thanks.
A: The simplest approach is to add a dependency on files produced by Ant build (they are usually in build/dist). This would be similar to Gradle dependencies with file directories
Better solution is to start using repository manager: Ivy, Artifactory, Nexus. Then update your NetBeans projects to publish built artifacts into this repository and your Gradle projects can easily refer to them. Check for more details in http://gradle.org/docs/current/userguide/artifact_dependencies_tutorial.html
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 970
|
{"url":"https:\/\/www.clutchprep.com\/physics\/practice-problems\/144995\/find-the-ratio-of-speeds-of-an-electron-and-a-negative-hydrogen-ion-one-having-a-1","text":"Relationships Between Force, Field, Energy, Potential Video Lessons\n\nConcept\n\n# Problem: Find the ratio of speeds of an electron and a negative hydrogen ion (one having an extra electron) accelerated through the same voltage, assuming non-relativistic final speeds. Take the mass of the hydrogen ion to be 1.67x10-27 kg.\n\n###### FREE Expert Solution\n\nKinetic energy:\n\n$\\overline{){\\mathbf{K}}{\\mathbf{=}}\\frac{\\mathbf{1}}{\\mathbf{2}}{\\mathbf{m}}{{\\mathbf{v}}}^{{\\mathbf{2}}}}$\n\nWhen an electron and the hydrogen ion are accelerated through the same potential, the acquire equal kinetic energy.\n\n91% (481 ratings)\n###### Problem Details\n\nFind the ratio of speeds of an electron and a negative hydrogen ion (one having an extra electron) accelerated through the same voltage, assuming non-relativistic final speeds. Take the mass of the hydrogen ion to be 1.67x10-27 kg.","date":"2021-06-21 09:51:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5786385536193848, \"perplexity\": 2178.719900762048}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488269939.53\/warc\/CC-MAIN-20210621085922-20210621115922-00515.warc.gz\"}"}
| null | null |
package org.unitime.timetable.model.base;
import java.io.Serializable;
import java.util.Date;
import java.util.HashSet;
import java.util.Set;
import org.unitime.timetable.model.CourseType;
import org.unitime.timetable.model.RefTableEntry;
import org.unitime.timetable.model.StudentSectioningStatus;
/**
* Do not change this class. It has been automatically generated using ant create-model.
* @see org.unitime.commons.ant.CreateBaseModelFromXml
*/
public abstract class BaseStudentSectioningStatus extends RefTableEntry implements Serializable {
private static final long serialVersionUID = 1L;
private Integer iStatus;
private String iMessage;
private Date iEffectiveStartDate;
private Date iEffectiveStopDate;
private Integer iEffectiveStartPeriod;
private Integer iEffectiveStopPeriod;
private StudentSectioningStatus iFallBackStatus;
private Set<CourseType> iTypes;
public static String PROP_STATUS = "status";
public static String PROP_MESSAGE = "message";
public static String PROP_START_DATE = "effectiveStartDate";
public static String PROP_STOP_DATE = "effectiveStopDate";
public static String PROP_START_SLOT = "effectiveStartPeriod";
public static String PROP_STOP_SLOT = "effectiveStopPeriod";
public BaseStudentSectioningStatus() {
initialize();
}
public BaseStudentSectioningStatus(Long uniqueId) {
setUniqueId(uniqueId);
initialize();
}
protected void initialize() {}
public Integer getStatus() { return iStatus; }
public void setStatus(Integer status) { iStatus = status; }
public String getMessage() { return iMessage; }
public void setMessage(String message) { iMessage = message; }
public Date getEffectiveStartDate() { return iEffectiveStartDate; }
public void setEffectiveStartDate(Date effectiveStartDate) { iEffectiveStartDate = effectiveStartDate; }
public Date getEffectiveStopDate() { return iEffectiveStopDate; }
public void setEffectiveStopDate(Date effectiveStopDate) { iEffectiveStopDate = effectiveStopDate; }
public Integer getEffectiveStartPeriod() { return iEffectiveStartPeriod; }
public void setEffectiveStartPeriod(Integer effectiveStartPeriod) { iEffectiveStartPeriod = effectiveStartPeriod; }
public Integer getEffectiveStopPeriod() { return iEffectiveStopPeriod; }
public void setEffectiveStopPeriod(Integer effectiveStopPeriod) { iEffectiveStopPeriod = effectiveStopPeriod; }
public StudentSectioningStatus getFallBackStatus() { return iFallBackStatus; }
public void setFallBackStatus(StudentSectioningStatus fallBackStatus) { iFallBackStatus = fallBackStatus; }
public Set<CourseType> getTypes() { return iTypes; }
public void setTypes(Set<CourseType> types) { iTypes = types; }
public void addTotypes(CourseType courseType) {
if (iTypes == null) iTypes = new HashSet<CourseType>();
iTypes.add(courseType);
}
public boolean equals(Object o) {
if (o == null || !(o instanceof StudentSectioningStatus)) return false;
if (getUniqueId() == null || ((StudentSectioningStatus)o).getUniqueId() == null) return false;
return getUniqueId().equals(((StudentSectioningStatus)o).getUniqueId());
}
public int hashCode() {
if (getUniqueId() == null) return super.hashCode();
return getUniqueId().hashCode();
}
public String toString() {
return "StudentSectioningStatus["+getUniqueId()+" "+getLabel()+"]";
}
public String toDebugString() {
return "StudentSectioningStatus[" +
"\n EffectiveStartDate: " + getEffectiveStartDate() +
"\n EffectiveStartPeriod: " + getEffectiveStartPeriod() +
"\n EffectiveStopDate: " + getEffectiveStopDate() +
"\n EffectiveStopPeriod: " + getEffectiveStopPeriod() +
"\n FallBackStatus: " + getFallBackStatus() +
"\n Label: " + getLabel() +
"\n Message: " + getMessage() +
"\n Reference: " + getReference() +
"\n Status: " + getStatus() +
"\n UniqueId: " + getUniqueId() +
"]";
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,248
|
Q: How can I tell if my SQL Server DB performance is hardware-limited? Testing an app currently under single-user load - as the test data has increased to production sizes (400k-2M rows per table), some SELECT sp's are not quite fast enough anymore (with limited test data, used to be <30ms each, now it's 100-200ms, but there are several, so the delay is becoming apparent in the UI).
A: You can use DBCC SQLPERF("waitstats"). This will return the wait times of what tasks your SQL server was waiting on. Detailed explanations of each counter can be found online. You can use this information to find out your bottlenecks.
Also, turn on the client statistics in query analyzer to see the wait times on the client side.
I am assuming you hardware has not changed since your initial test, so since they are constant, I wouldn't doubt them.
A: Thoughts:
*
*hardware is almost never an issue: it's poor design and code
*always test with near-production data quality and quantity
Some solutions:
Run the missing index DMV to see, well, missing indexes:
SELECT
migs.avg_total_user_cost * (migs.avg_user_impact / 100.0) * (migs.user_seeks + migs.user_scans) AS improvement_measure,
'CREATE INDEX [missing_index_' + CONVERT (varchar, mig.index_group_handle) + '_' + CONVERT (varchar, mid.index_handle)
+ '_' + LEFT (PARSENAME(mid.statement, 1), 32) + ']'
+ ' ON ' + mid.statement
+ ' (' + ISNULL (mid.equality_columns,'')
+ CASE WHEN mid.equality_columns IS NOT NULL AND mid.inequality_columns IS NOT NULL THEN ',' ELSE '' END
+ ISNULL (mid.inequality_columns, '')
+ ')'
+ ISNULL (' INCLUDE (' + mid.included_columns + ')', '') AS create_index_statement,
migs.*, mid.database_id, mid.[object_id]
FROM sys.dm_db_missing_index_groups mig
INNER JOIN sys.dm_db_missing_index_group_stats migs ON migs.group_handle = mig.index_group_handle
INNER JOIN sys.dm_db_missing_index_details mid ON mig.index_handle = mid.index_handle
WHERE migs.avg_total_user_cost * (migs.avg_user_impact / 100.0) * (migs.user_seeks + migs.user_scans) > 10
ORDER BY migs.avg_total_user_cost * migs.avg_user_impact * (migs.user_seeks + migs.user_scans) DESC
...and most expensive DMV queries
SELECT TOP 20
qs.sql_handle,
qs.execution_count,
qs.total_worker_time AS Total_CPU,
total_CPU_inSeconds = --Converted from microseconds
qs.total_worker_time/1000000,
average_CPU_inSeconds = --Converted from microseconds
(qs.total_worker_time/1000000) / qs.execution_count,
qs.total_elapsed_time,
total_elapsed_time_inSeconds = --Converted from microseconds
qs.total_elapsed_time/1000000,
st.text,
qp.query_plan
FROM
sys.dm_exec_query_stats AS qs
CROSS APPLY sys.dm_exec_sql_text(qs.sql_handle) AS st
CROSS apply sys.dm_exec_query_plan (qs.plan_handle) AS qp
ORDER BY qs.total_worker_time DESC
Otherwise, this SO question has good tips from me and other SQL high rep types: https://stackoverflow.com/q/4118156/27535 (I won't copy/paste all 3 longish answers)
A: Log system resources or look at task manager to see how many system resources are used by the processes.
A: Interesting thing that should be considered is the version of MSSQL 2000 running
There are four versions of the binaries
*
*Express
*Standard
*Professional
*Enterprise
Each of those versions have limits in terms of RAM and CPU.
It is worth exploring the possibility that the amount of data currently stored has simply outgrown the capabilities of the version of MSSQL 2000 due to queries needing more RAM to fulfuill queries/subqueries or inadequate CPU utilization. You may require upgrading the binary version to the MSSQL 2000 Entrprise version (probably a long shot becasue of how old your version of MSSQL is) or the best version your budget can afford.
You may even want to get out of MSSQL 2000 since 2008 is the latest and has current support available. Again, this could be a budget issue. If you are already using Enterprise, or your budget cannot allow for any major upgrade, now you can explore DB Statistics or DB Design.
Disclaimer : I'm not a SQL Server DBA
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,843
|
require 'sauce/capybara'
require 'pathname'
require 'json'
require 'bundler/setup'
Bundler.require(:default, :test)
$USE_BROWSER = ARGV.last || :windows7chrome
class CapybaraTestCase < Test::Unit::TestCase
include Capybara::DSL
include DriverHelpers
def setup
use($USE_BROWSER)
Capybara.app_host = environment
Capybara.default_wait_time = 10
Capybara.run_server = false
end
end
class ConnectionCheckTest < CapybaraTestCase
def test_connection
@host = HostSession.new(:host, self)
@host.visit "http://www.google.com/"
@host.action do
assert page.has_css?("span", text: "Google Search")
end
end
end
module DriverHelpers
def use_configed(chosen_browser)
supportedBrowsers = JSON.parse(Pathname.new(File.dirname(__FILE__) + '/BrowserList.json').read)
supportedBrowsers.each do |os, browserHash|
browserHash.each do |browserName, browserArray|
browserArray.each do |browser|
ident = "#{os}#{browserName}#{browser['short_version']}".downcase.gsub(' ', '').to_sym
if ident == chosen_browser
Sauce.config do |c|
c[:browsers] = [[os, browserName, browser['short_version']]]
c[:name] = "#{os} - #{browserName} - #{browser['short_version']}"
end
Capybara.register_driver ident do |app|
Sauce::Capybara::Driver.new(app)
end
break
end
end
end
end
Capybara.current_driver = chosen_browser
Capybara.javascript_driver = chosen_browser
end
def use(browser)
if respond_to? "use_#{browser}"
public_send "use_#{browser}"
else
use_configed(browser.to_sym)
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,140
|
Q: What is the exact meaning of "he was assumed my rapture"? In this this sentence "he was assumed my rapture", does the author mean that the character is in rapture over something ? Is there a subtle nuance ? I can't find anywhere the expression "assumed by rapture".
Thanks!
A: "He was assumed my rapture," makes no sense in English. "He was consumed by rapture," though, means he was so enraptured or enchanted, so overcome by emotion, that his intellect was "consumed"; i.e. he was unable to think, just to perceive.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,864
|
url link Demonstrate all of your current customers and followers that you simply do really appreciate their assist and value their time, make this happen by delivering email messages that are informative, contrary to spam. Prevent delivering them e-mails about income only. Attempt to consist of a strategy to a standard dilemma, a clever new way to utilize a product or service, or perhaps a specific advertising in every single electronic mail.
The value of e mail marketing and advertising is lively and very well, as you have witnessed. Heed the recommendations in this article and continue looking for ways to found an revolutionary and powerful electronic mail advertising campaign for your customers.
Focus on your e-newsletter for a every week plan that makes perception for people who study it. Choose a weekday for a newsletter that goals enterprise-connected troubles so your buyer can make full use of it during the typical workweek. Family members or leisure relevant newsletters, however, needs to be mailed for the weekend break.
Your issue collection ought to package a punch and make the reader pay attention. Whenever a discounted or discount coupon is presented in the topic, the likelihood of it staying wide open raises drastically. Try out to get their focus via something unique, similar to a new item, anything free, or a discount. Using a excellent issue collection will considerably increase the chance that the electronic mail is opened up.
marketing strategy has an exit sign for all those that like to opt out or unsubscribe. It does cost funds to receive and send email messages, even though it can't be considered a considerable amount. If individuals perceive you as a spammer, it can hurt your business. This might also make folks block your email address internet,and that's not good if you're trying to get your item or service on the market.
Remember that your clients are increasingly more very likely to use smartphones or another mobile devices to view your emails. As the display screen image resolution is lower on these products, you may have a more compact space with which to function. Knowing the limitations of the mobile phone monitors will assist you to make e-mails that your particular customers can read.
Have you seen your e-mail right now? If you're much like many people, you almost certainly examine your e-mail around 2 times everyday. E mail advertising is a terrific way to reach out to your potential clients and present versions. Follow this advice regarding how to successfully use e-mail advertising and marketing to your benefit.
E mail marketing and advertising can be produced more potent by offering your visitors possibilities. Permit the subscriber stipulate how many times they would like to get communications through your organization, exactly how much private information they relinquish to you personally, and the way numerous email messages they would like on your part within a presented time frame. They will likely be significantly secure once you provide them with much more control over the complete method. If you are planning on beginning a advertising campaign by way of email, it is vital that you receive authorization from every single particular person you intend on contacting. In case you don't get authorization initial, you will get many spam grievances and in many cases get rid of or else faithful consumers. In case your business has branded previous client communications, make certain that your electronic mail marketing plan is steady with this branding. Ensure each e-mail involves exactly the same coloration system along with your business emblem. Having a solid brand recognized along with your customers may help change more visitors to consumers simply because they have confidence in the organization currently.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,045
|
Jean Joseph Delarge (ur. 6 kwietnia 1906 w Liège; zm. 7 lipca 1977 tamże) – belgijski bokser.
Delarge brał udział na Igrzyskach Olimpijskich w 1924 roku, gdzie uczestniczył w zawodach wagi półśredniej. Zdobył wówczas złoty medal.
W pierwszej rundzie zawodów pokonał Louisa Sauthiera. W kolejnej rundzie wygrał z Patrickiem O'Hanrahanem. W ćwierćfinałach Delarge pokonał Roya Ingrama, w półfinałach Douglasa Lewisa. W walce o złoty medal wygrał z Héctorem Méndeze.
Delarge w latach 1925–1930 stoczył 12 walk zawodowych – 3 wygrał, 6 przegrał i 3 zremisował.
Przypisy
Linki zewnętrzne
Belgijscy bokserzy
Belgijscy medaliści olimpijscy
Ludzie urodzeni w Liège
Medaliści Letnich Igrzysk Olimpijskich 1924
Urodzeni w 1906
Zmarli w 1977
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,657
|
{"url":"https:\/\/leanprover-community.github.io\/archive\/stream\/113489-new-members\/topic\/How.20to.20use.20dvd_of_mul_left_dvd.20and.20friends.3F.html","text":"Stream: new members\n\nTopic: How to use dvd_of_mul_left_dvd and friends?\n\nJordan Scales (Oct 11 2020 at 20:15):\n\nHey :) I'm toying around with some modular arithmetic, and am struggling to prove a basic premise: 3 \u2223 n \u2192 \u00ac 5 \u2223 n \u2192 \u00ac 15 \u2223 n\n\nI'd like to prove this by assuming divisibility by 15, and finding a contradiction with divisibility by 5. However, dvd_of_mul_left_dvd is giving me pause.\n\nexample (n : \u2115) : 3 \u2223 n -> \u00ac 5 \u2223 n -> \u00ac 15 \u2223 n := begin\nintros h\u2083 h\u2085 h,\nhave q : 15 = 3 * 5, { refl },\nrw q at h,\nhave contra := dvd_of_mul_left_dvd h,\nend\n\n\nspecifically:\n\ntype mismatch at application\ndvd_of_mul_left_dvd h\nterm\nh\nhas type\n3 * 5 \u2223 n\nbut is expected to have type\n?m_3 * ?m_4 \u2223 ?m_5\n\n\nI am unsure why the pattern matching isn't working here. Any pointers?\n\nKevin Buzzard (Oct 11 2020 at 20:16):\n\nWhat do I need to import\/open to get this working? Can you post a #mwe?\n\nJordan Scales (Oct 11 2020 at 20:17):\n\nmy apologies! I have the following imported:\n\nimport tactic.suggest\nimport algebra.divisibility\n\n\nso the entire (not working :( ) file is as follows:\n\nimport tactic.suggest\nimport algebra.divisibility\n\nexample (n : \u2115) : 3 \u2223 n -> \u00ac 5 \u2223 n -> \u00ac 15 \u2223 n := begin\nintros h\u2083 h\u2085 h,\nhave q : 15 = 3 * 5, { refl },\nrw q at h,\nhave contra := dvd_of_mul_left_dvd h,\nend\n\n\nKevin Buzzard (Oct 11 2020 at 20:21):\n\nYou don't have enough imports, apparently: you seem to have managed to avoid importing the fact that the naturals are a commutative monoid.\n\nimport tactic\n\nexample (n : \u2115) : 3 \u2223 n -> \u00ac 5 \u2223 n -> \u00ac 15 \u2223 n := begin\nintros h\u2083 h\u2085 h,\nhave q : 15 = 3 * 5, { refl },\nrw q at h,\nhave contra := dvd_of_mul_left_dvd h,\nsorry\nend\n\n\nworks fine.\n\nJordan Scales (Oct 11 2020 at 20:22):\n\nfascinating! works for me as well :) thank you.\n\nKevin Buzzard (Oct 11 2020 at 20:22):\n\nThese facts used to be in core Lean 3 but in the community fork we ripped them out and moved them to mathlib, so now you have to explicitly import them.\n\nKevin Buzzard (Oct 11 2020 at 20:22):\n\nI didn't know this wasn't there by default any more because every Lean file I write starts with import tactic.\n\nJordan Scales (Oct 11 2020 at 20:23):\n\nstarting every file with import tactic is something I can definitely accept\n\nLast updated: May 14 2021 at 06:16 UTC","date":"2021-05-14 06:36:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.36162373423576355, \"perplexity\": 5078.344283456876}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991648.40\/warc\/CC-MAIN-20210514060536-20210514090536-00306.warc.gz\"}"}
| null | null |
Q: error LGHT0103 : The system cannot find the file Running a WixSharp Managed Setup up project I'm getting this error.
Using WixUI_Common
C:\agent\_work\66\s\src\ext\UIExtension\wixlib\Common.wxs(7) : error LGHT0103 : The system cannot find the file
This location C:\agent\_work\66\s\src\ext\UIExtension\wixlib\Common.wxs(7) doesn't locate in my PC.
I have only C:\ in the above path
I have no clue from where I get it.
What can be the problem?
A: what that resolved my issue:
Uninstall all WixSharp libraries from the solution and reinstall them.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,254
|
@interface RegisterViewController ()<UITableViewDataSource, UITableViewDelegate, TTTAttributedLabelDelegate>
@property (nonatomic, assign) RegisterMethodType medthodType;
@property (nonatomic, strong) Register *myRegister;
@property (strong, nonatomic) TPKeyboardAvoidingTableView *myTableView;
@property (strong, nonatomic) UIButton *footerBtn;
@property (strong, nonatomic) EaseInputTipsView *inputTipsView;
@property (assign, nonatomic) BOOL captchaNeeded;
@property (strong, nonatomic) NSString *phoneCodeCellIdentifier;
@property (strong, nonatomic) NSDictionary *countryCodeDict;
@property (assign, nonatomic) NSInteger step;
@end
@implementation RegisterViewController
+ (instancetype)vcWithMethodType:(RegisterMethodType)methodType registerObj:(Register *)obj{
RegisterViewController *vc = [self new];
vc.medthodType = methodType;
vc.myRegister = obj;
vc.step = 0;
return vc;
}
- (void)viewDidLoad{
[super viewDidLoad];
self.view.backgroundColor = kColorWhite;
[self.navigationController.navigationBar setupClearBGStyle];
self.phoneCodeCellIdentifier = [Input_OnlyText_Cell randomCellIdentifierOfPhoneCodeType];
_captchaNeeded = NO;
// self.title = @"注册";
if (!_myRegister) {
self.myRegister = [Register new];
}
if (!_countryCodeDict) {
_countryCodeDict = @{@"country": @"China",
@"country_code": @"86",
@"iso_code": @"cn"};
}
// 添加myTableView
_myTableView = ({
TPKeyboardAvoidingTableView *tableView = [[TPKeyboardAvoidingTableView alloc] initWithFrame:self.view.bounds style:UITableViewStylePlain];
[tableView registerClass:[Input_OnlyText_Cell class] forCellReuseIdentifier:kCellIdentifier_Input_OnlyText_Cell_Text];
[tableView registerClass:[Input_OnlyText_Cell class] forCellReuseIdentifier:kCellIdentifier_Input_OnlyText_Cell_Password];
[tableView registerClass:[Input_OnlyText_Cell class] forCellReuseIdentifier:kCellIdentifier_Input_OnlyText_Cell_Captcha];
[tableView registerClass:[Input_OnlyText_Cell class] forCellReuseIdentifier:kCellIdentifier_Input_OnlyText_Cell_Phone];
[tableView registerClass:[Input_OnlyText_Cell class] forCellReuseIdentifier:self.phoneCodeCellIdentifier];
// tableView.backgroundColor = kColorTableSectionBg;
tableView.dataSource = self;
tableView.delegate = self;
tableView.separatorStyle = UITableViewCellSeparatorStyleNone;
[self.view addSubview:tableView];
[tableView mas_makeConstraints:^(MASConstraintMaker *make) {
make.edges.equalTo(self.view);
}];
tableView.estimatedRowHeight = 0;
tableView.estimatedSectionHeaderHeight = 0;
tableView.estimatedSectionFooterHeight = 0;
tableView;
});
[self setupNav];
self.myTableView.tableHeaderView = [self customHeaderView];
self.myTableView.tableFooterView=[self customFooterView];
[self configBottomView];
}
- (void)refreshCaptchaNeeded{
if (_medthodType == RegisterMethodPhone && _step <= 0) {
self.captchaNeeded = NO;
[self.myTableView reloadData];
}else{
//写死,APP 不需要
self.captchaNeeded = NO;
[self.myTableView reloadData];
// __weak typeof(self) weakSelf = self;
// [[Coding_NetAPIManager sharedManager] request_CaptchaNeededWithPath:@"api/captcha/register" andBlock:^(id data, NSError *error) {
// if (data) {
// NSNumber *captchaNeededResult = (NSNumber *)data;
// if (captchaNeededResult) {
// weakSelf.captchaNeeded = captchaNeededResult.boolValue;
// }
// [weakSelf.myTableView reloadData];
// }
// }];
}
}
- (void)viewWillAppear:(BOOL)animated{
[super viewWillAppear:animated];
[self.navigationController setNavigationBarHidden:NO animated:YES];
[self refreshCaptchaNeeded];
}
- (void)viewWillDisappear:(BOOL)animated{
[super viewWillDisappear:animated];
[self.view endEditing:YES];
}
- (EaseInputTipsView *)inputTipsView{
if (!_inputTipsView) {
_inputTipsView = ({
EaseInputTipsView *tipsView = [EaseInputTipsView tipsViewWithType:EaseInputTipsViewTypeRegister];
tipsView.valueStr = nil;
__weak typeof(self) weakSelf = self;
tipsView.selectedStringBlock = ^(NSString *valueStr){
[weakSelf.view endEditing:YES];
weakSelf.myRegister.email = valueStr;
[weakSelf.myTableView reloadData];
};
UITableViewCell *cell = [_myTableView cellForRowAtIndexPath:[NSIndexPath indexPathForRow:1 inSection:0]];
[tipsView setY:CGRectGetMaxY(cell.frame)];
[_myTableView addSubview:tipsView];
tipsView;
});
}
return _inputTipsView;
}
#pragma mark - Nav
- (void)setupNav{
if (self.navigationController.childViewControllers.count <= 1) {
self.navigationItem.leftBarButtonItem = [UIBarButtonItem itemWithBtnTitle:@"取消" target:self action:@selector(dismissSelf)];
}
}
- (void)dismissSelf{
[self dismissViewControllerAnimated:YES completion:nil];
}
#pragma mark - Top Bottom Header Footer
- (void)configBottomView{
UIView *bottomView = [UIView new];
UIButton *bottomBtn = ({
UIButton *button = [UIButton new];
button.titleLabel.font = [UIFont systemFontOfSize:15];
[button setTitleColor:kColorDark2 forState:UIControlStateNormal];
[button setTitle:@"已有 Coding 账号?" forState:UIControlStateNormal];
__weak typeof(self) weakSelf = self;
[button bk_addEventHandler:^(id sender) {
if (weakSelf.navigationController.viewControllers.count > 1) {
[weakSelf.navigationController popToRootViewControllerAnimated:YES];
}else{
LoginViewController *vc = [[LoginViewController alloc] init];
vc.showDismissButton = YES;
[weakSelf.navigationController pushViewController:vc animated:YES];
}
} forControlEvents:UIControlEventTouchUpInside];
button;
});
[bottomView addSubview:bottomBtn];
[bottomBtn mas_makeConstraints:^(MASConstraintMaker *make) {
make.left.top.right.equalTo(bottomView);
make.height.mas_equalTo(25);
}];
[self.view addSubview:bottomView];
[bottomView mas_makeConstraints:^(MASConstraintMaker *make) {
make.left.right.bottom.equalTo(self.view);
make.height.mas_equalTo(50 + kSafeArea_Bottom);
}];
}
- (void)changeMethodType{
if (_medthodType == RegisterMethodPhone) {
RegisterViewController *vc = [RegisterViewController vcWithMethodType:RegisterMethodEamil registerObj:_myRegister];
[self.navigationController pushViewController:vc animated:YES];
}else{
[self.navigationController popViewControllerAnimated:YES];
}
}
- (UIView *)customHeaderView{
UIView *headerV = [[UIView alloc] initWithFrame:CGRectMake(0, 0, kScreen_Width, 60)];
UILabel *headerL = [UILabel labelWithFont:[UIFont systemFontOfSize:30] textColor:kColorDark2];
headerL.text = self.step > 0? @"设置密码": @"注册";
[headerV addSubview:headerL];
[headerL mas_makeConstraints:^(MASConstraintMaker *make) {
make.left.offset(kPaddingLeftWidth);
make.bottom.offset(0);
make.height.mas_equalTo(42);
}];
return headerV;
}
- (UIView *)customFooterView{
UIView *footerV = [[UIView alloc] initWithFrame:CGRectMake(0, 0, kScreen_Width, 150)];
//button
_footerBtn = [UIButton buttonWithStyle:StrapSuccessStyle andTitle:self.step > 0? @"注册": @"下一步" andFrame:CGRectMake(kLoginPaddingLeftWidth, 20, kScreen_Width-kLoginPaddingLeftWidth*2, 50) target:self action:@selector(sendRegister)];
[footerV addSubview:_footerBtn];
__weak typeof(self) weakSelf = self;
RAC(self, footerBtn.enabled) = [RACSignal combineLatest:@[RACObserve(self, myRegister.global_key),
RACObserve(self, myRegister.phone),
RACObserve(self, myRegister.email),
RACObserve(self, myRegister.password),
RACObserve(self, myRegister.confirm_password),
RACObserve(self, myRegister.code),
RACObserve(self, myRegister.j_captcha),
RACObserve(self, captchaNeeded)]
reduce:^id(NSString *global_key,
NSString *phone,
NSString *email,
NSString *password,
NSString *confirm_password,
NSString *code,
NSString *j_captcha,
NSNumber *captchaNeeded){
BOOL enabled;
if (weakSelf.medthodType == RegisterMethodEamil) {
enabled = (global_key.length > 0 &&
password.length > 0 &&
(!captchaNeeded.boolValue || j_captcha.length > 0) &&
email.length > 0);
}else if (weakSelf.step > 0){
enabled = (global_key.length > 0 &&
password.length > 0 &&
confirm_password.length > 0 &&
// [confirm_password isEqualToString:password] &&
(!captchaNeeded.boolValue || j_captcha.length > 0) &&
(phone.length > 0 && code.length > 0));
}else{
enabled = (global_key.length > 0 &&
(!captchaNeeded.boolValue || j_captcha.length > 0) &&
(phone.length > 0 && code.length > 0));
}
return @(enabled);
}];
//label
UITTTAttributedLabel *lineLabel = ({
UITTTAttributedLabel *label = [[UITTTAttributedLabel alloc] initWithFrame:CGRectZero];
label.textAlignment = NSTextAlignmentCenter;
label.font = [UIFont systemFontOfSize:14];
label.textColor = kColorDark2;
label.numberOfLines = 0;
label.linkAttributes = kLinkAttributes;
label.activeLinkAttributes = kLinkAttributesActive;
label.delegate = self;
label;
});
NSString *tipStr = @"点击注册,即同意《Coding 服务条款》";
lineLabel.text = tipStr;
[lineLabel addLinkToTransitInformation:@{@"actionStr" : @"gotoServiceTermsVC"} withRange:[tipStr rangeOfString:@"《Coding 服务条款》"]];
CGRect footerBtnFrame = _footerBtn.frame;
lineLabel.frame = CGRectMake(CGRectGetMinX(footerBtnFrame), CGRectGetMaxY(footerBtnFrame) +15, CGRectGetWidth(footerBtnFrame), 15);
[footerV addSubview:lineLabel];
return footerV;
}
#pragma mark - Table view data source
- (NSInteger)tableView:(UITableView *)tableView numberOfRowsInSection:(NSInteger)section{
NSInteger num = _medthodType == RegisterMethodEamil? 3: _step > 0? 2: 3;
return _captchaNeeded? num +1 : num;
}
- (UITableViewCell *)tableView:(UITableView *)tableView cellForRowAtIndexPath:(NSIndexPath *)indexPath{
NSString *cellIdentifier;
if (_medthodType == RegisterMethodEamil) {
cellIdentifier = (indexPath.row == 3? kCellIdentifier_Input_OnlyText_Cell_Captcha:
indexPath.row == 2? kCellIdentifier_Input_OnlyText_Cell_Password:
kCellIdentifier_Input_OnlyText_Cell_Text);
}else{
if (_step > 0) {
cellIdentifier = (indexPath.row == 2? kCellIdentifier_Input_OnlyText_Cell_Captcha:
kCellIdentifier_Input_OnlyText_Cell_Text);
}else{
cellIdentifier = (indexPath.row == 2? self.phoneCodeCellIdentifier:
indexPath.row == 1? kCellIdentifier_Input_OnlyText_Cell_Phone:
kCellIdentifier_Input_OnlyText_Cell_Text);
}
}
Input_OnlyText_Cell *cell = [tableView dequeueReusableCellWithIdentifier:cellIdentifier forIndexPath:indexPath];
cell.isBottomLineShow = YES;
__weak typeof(self) weakSelf = self;
if (_medthodType == RegisterMethodEamil) {
if (indexPath.row == 0) {
[cell setPlaceholder:@" 用户名" value:self.myRegister.global_key];
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.myRegister.global_key = [valueStr trimWhitespace];
};
}else if (indexPath.row == 1){
cell.textField.keyboardType = UIKeyboardTypeEmailAddress;
[cell setPlaceholder:@" 邮箱" value:self.myRegister.email];
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.inputTipsView.valueStr = valueStr;
weakSelf.inputTipsView.active = YES;
weakSelf.myRegister.email = valueStr;
};
cell.editDidEndBlock = ^(NSString *textStr){
weakSelf.inputTipsView.active = NO;
};
}else if (indexPath.row == 2){
[cell setPlaceholder:@" 设置密码" value:self.myRegister.password];
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.myRegister.password = valueStr;
};
}else{
[cell setPlaceholder:@" 验证码" value:self.myRegister.j_captcha];
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.myRegister.j_captcha = valueStr;
};
}
}else{
if (_step > 0) {
if (indexPath.row == 0){
[cell setPlaceholder:@" 设置密码" value:self.myRegister.password];
cell.textField.secureTextEntry = YES;
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.myRegister.password = valueStr;
};
}else if (indexPath.row == 1){
[cell setPlaceholder:@" 重复密码" value:self.myRegister.password];
cell.textField.secureTextEntry = YES;
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.myRegister.confirm_password = valueStr;
};
}else{
[cell setPlaceholder:@" 验证码" value:self.myRegister.j_captcha];
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.myRegister.j_captcha = valueStr;
};
}
}else{
if (indexPath.row == 0) {
[cell setPlaceholder:@" 用户名" value:self.myRegister.global_key];
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.myRegister.global_key = [valueStr trimWhitespace];
};
}else if (indexPath.row == 1){
cell.textField.keyboardType = UIKeyboardTypeNumberPad;
[cell setPlaceholder:@" 手机号码" value:self.myRegister.phone];
cell.countryCodeL.text = [NSString stringWithFormat:@"+%@", _countryCodeDict[@"country_code"]];
cell.countryCodeBtnClickedBlock = ^(){
[weakSelf goToCountryCodeVC];
};
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.myRegister.phone = valueStr;
};
}else if (indexPath.row == 2){
cell.textField.keyboardType = UIKeyboardTypeNumberPad;
[cell setPlaceholder:@" 手机验证码" value:self.myRegister.code];
cell.textValueChangedBlock = ^(NSString *valueStr){
weakSelf.myRegister.code = valueStr;
};
cell.phoneCodeBtnClckedBlock = ^(PhoneCodeButton *btn){
[weakSelf phoneCodeBtnClicked:btn withCaptcha:nil];
};
}
}
}
return cell;
}
- (CGFloat)tableView:(UITableView *)tableView heightForRowAtIndexPath:(NSIndexPath *)indexPath{
return 65.0;
}
#pragma mark TTTAttributedLabelDelegate
- (void)attributedLabel:(TTTAttributedLabel *)label didSelectLinkWithTransitInformation:(NSDictionary *)components{
[self gotoServiceTermsVC];
}
#pragma mark Btn Clicked
- (void)phoneCodeBtnClicked:(PhoneCodeButton *)sender withCaptcha:(NSString *)captcha{
if (![_myRegister.phone isPhoneNo]) {
[NSObject showHudTipStr:@"手机号码格式有误"];
return;
}
sender.enabled = NO;
NSMutableDictionary *params = @{@"phone": _myRegister.phone,
@"phoneCountryCode": [NSString stringWithFormat:@"+%@", _countryCodeDict[@"country_code"]]}.mutableCopy;
if (captcha.length > 0) {
params[@"j_captcha"] = captcha;
}
__weak typeof(self) weakSelf = self;
[[CodingNetAPIClient sharedJsonClient] requestJsonDataWithPath:@"api/account/register/generate_phone_code" withParams:params withMethodType:Post autoShowError:captcha.length > 0 andBlock:^(id data, NSError *error) {
if (data) {
[NSObject showHudTipStr:@"验证码发送成功"];
[sender startUpTimer];
}else{
[sender invalidateTimer];
if (error && error.userInfo[@"msg"] && [[error.userInfo[@"msg"] allKeys] containsObject:@"j_captcha_error"]) {
[weakSelf p_showCaptchaAlert:sender];
}else if (captcha.length <= 0){
[NSObject showError:error];
}
}
}];
}
- (void)p_showCaptchaAlert:(PhoneCodeButton *)sender{
SDCAlertController *alertV = [SDCAlertController alertControllerWithTitle:@"提示" message:@"请输入图片验证码" preferredStyle:SDCAlertControllerStyleAlert];
UITextField *textF = [UITextField new];
textF.layer.sublayerTransform = CATransform3DMakeTranslation(5, 0, 0);
textF.backgroundColor = [UIColor whiteColor];
[textF doBorderWidth:0.5 color:nil cornerRadius:2.0];
UIImageView *imageV = [YLImageView new];
imageV.backgroundColor = [UIColor lightGrayColor];
imageV.contentMode = UIViewContentModeScaleAspectFit;
imageV.clipsToBounds = YES;
imageV.userInteractionEnabled = YES;
[textF doBorderWidth:0.5 color:nil cornerRadius:2.0];
NSURL *imageURL = [NSURL URLWithString:[NSString stringWithFormat:@"%@api/getCaptcha", [NSObject baseURLStr]]];
[imageV sd_setImageWithURL:imageURL placeholderImage:nil options:(SDWebImageRetryFailed | SDWebImageRefreshCached | SDWebImageHandleCookies)];
[alertV.contentView addSubview:textF];
[alertV.contentView addSubview:imageV];
[textF mas_makeConstraints:^(MASConstraintMaker *make) {
make.left.equalTo(alertV.contentView).offset(15);
make.height.mas_equalTo(25);
make.bottom.equalTo(alertV.contentView).offset(-10);
}];
[imageV mas_makeConstraints:^(MASConstraintMaker *make) {
make.right.equalTo(alertV.contentView).offset(-15);
make.left.equalTo(textF.mas_right).offset(10);
make.width.mas_equalTo(60);
make.height.mas_equalTo(25);
make.centerY.equalTo(textF);
}];
//Action
__weak typeof(imageV) weakImageV = imageV;
[imageV bk_whenTapped:^{
[weakImageV sd_setImageWithURL:imageURL placeholderImage:nil options:(SDWebImageRetryFailed | SDWebImageRefreshCached | SDWebImageHandleCookies)];
}];
__weak typeof(self) weakSelf = self;
[alertV addAction:[SDCAlertAction actionWithTitle:@"取消" style:SDCAlertActionStyleCancel handler:nil]];
[alertV addAction:[SDCAlertAction actionWithTitle:@"确定" style:SDCAlertActionStyleDefault handler:nil]];
alertV.shouldDismissBlock = ^BOOL (SDCAlertAction *action){
if (![action.title isEqualToString:@"取消"]) {
[weakSelf phoneCodeBtnClicked:sender withCaptcha:textF.text];
}
return YES;
};
[alertV presentWithCompletion:^{
[textF becomeFirstResponder];
}];
}
- (void)sendRegister{
NSString *tipStr = nil;
if (![_myRegister.global_key isGK]) {
tipStr = @"用户名仅支持英文字母、数字、横线(-)以及下划线(_)";
}else if (_step > 0 && ![_myRegister.confirm_password isEqualToString:_myRegister.password]){
tipStr = @"密码输入不一致";
}
if (tipStr) {
[NSObject showHudTipStr:tipStr];
return;
}
__weak typeof(self) weakSelf = self;
if (_medthodType == RegisterMethodPhone && _step <= 0) {
[self.footerBtn startQueryAnimate];
NSDictionary *gkP = @{@"key": _myRegister.global_key};
[[CodingNetAPIClient sharedJsonClient] requestJsonDataWithPath:@"api/user/check" withParams:gkP withMethodType:Get andBlock:^(id data, NSError *error) {
if (!error && [data[@"data"] boolValue]) {//用户名还未被注册
NSDictionary *phoneCodeP = @{@"phone": _myRegister.phone,
@"verifyCode": _myRegister.code,
@"phoneCountryCode": [NSString stringWithFormat:@"+%@", _countryCodeDict[@"country_code"]],
};
[[CodingNetAPIClient sharedJsonClient] requestJsonDataWithPath:@"api/account/register/check-verify-code" withParams:phoneCodeP withMethodType:Post andBlock:^(id data, NSError *error) {
[weakSelf.footerBtn stopQueryAnimate];
if (!error) {
//手机验证码通过校验
RegisterViewController *vc = [RegisterViewController new];
vc.medthodType = RegisterMethodPhone;
vc.myRegister = weakSelf.myRegister;
vc.step = 1;
[weakSelf.navigationController pushViewController:vc animated:YES];
}
}];
}else{
[weakSelf.footerBtn stopQueryAnimate];
if (!error) {
[NSObject showHudTipStr:@"用户名已存在"];
}
}
}];
}else{
NSMutableDictionary *params = @{@"channel": [Register channel],
@"global_key": _myRegister.global_key,
@"password": [_myRegister.password sha1Str],
@"confirm": [_myRegister.password sha1Str]}.mutableCopy;
if (_medthodType == RegisterMethodEamil) {
params[@"email"] = _myRegister.email;
}else{
params[@"phone"] = _myRegister.phone;
params[@"code"] = _myRegister.code;
params[@"country"] = _countryCodeDict[@"iso_code"];
params[@"phoneCountryCode"] = [NSString stringWithFormat:@"+%@", _countryCodeDict[@"country_code"]];
}
if (_captchaNeeded) {
params[@"j_captcha"] = _myRegister.j_captcha;
}
[self.footerBtn startQueryAnimate];
[[Coding_NetAPIManager sharedManager] request_Register_V2_WithParams:params andBlock:^(id data, NSError *error) {
[weakSelf.footerBtn stopQueryAnimate];
if (data) {
[self.view endEditing:YES];
[Login setPreUserEmail:self.myRegister.global_key];//记住登录账号
[((AppDelegate *)[UIApplication sharedApplication].delegate) setupTabViewController];
if (weakSelf.medthodType == RegisterMethodEamil) {
kTipAlert(@"欢迎注册 Coding,请尽快去邮箱查收邮件并激活账号。如若在收件箱中未看到激活邮件,请留意一下垃圾邮件箱(T_T)。");
}
}else{
[weakSelf refreshCaptchaNeeded];
}
}];
}
}
#pragma mark VC
- (void)gotoServiceTermsVC{
NSString *pathForServiceterms = [[NSBundle mainBundle] pathForResource:@"service_terms" ofType:@"html"];
WebViewController *vc = [WebViewController webVCWithUrlStr:pathForServiceterms];
[self.navigationController pushViewController:vc animated:YES];
}
- (void)goToCountryCodeVC{
__weak typeof(self) weakSelf = self;
CountryCodeListViewController *vc = [CountryCodeListViewController new];
vc.selectedBlock = ^(NSDictionary *countryCodeDict){
weakSelf.countryCodeDict = countryCodeDict;
};
[self.navigationController pushViewController:vc animated:YES];
}
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,478
|
Heart Bedford
Biggleswade Attack Leaves Victim Shocked
8 April 2014, 12:25 | Updated: 8 April 2014, 12:27
Five people have been arrested after a 40 year old man was assaulted and robbed whilst waiting for a lift in Potton Road, Biggleswade last week.
Heart's learned the victim was waiting at the side the road last Tuesday morning (April 1) at about 7.20am, when he was approached by a man who tried to take his phone out of his hand. He pulled the phone away, but the offender punched him in the face. The victim was then set upon by the first man who was joined by two others.
The three men, all described as white and in their twenties, hit and kicked him while he was on the floor and demanded money from him. Luckily, a passer-by stopped his car and shouted at them, at which point they took the victim's bag and ran off in the direction of Devon Drive, shortly afterwards getting into a red Vauxhall Astra.
The victim was left with cuts and bruises on his face and was extremely shocked by the whole ordeal.
Later that day, five people were arrested in Bedford in connection with the offence and these are currently on police bail while further enquiries are carried out.
Det Con Surinder Ram would like to hear from anyone who saw the incident, which happened across from the new housing estate.
"While some people have been arrested in connection with this robbery, we do still need to hear from anyone who saw it or passed it. By all accounts there were a couple of very kind people who helped the victim and we know that others would have seen it. If you were there, please do make contact with us as soon as you can," said Det Con Ram.
If you can help, please contact Det Con Ram on 01234 275346, or call the 24 hour non-emergency police number 101.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,369
|
{"url":"http:\/\/www.physicsforums.com\/showthread.php?p=4204223","text":"# Segment of a circle calculation\n\nby bergie7isu\nTags: calculation, circle, segment\n P: 2 I'm working to calculate the cross-sectional area of a lathe turned feature machined with a radiused insert. My calculations have essentially led me to the equation for the area of a segment of a circle. Area=r^2\/2*(\u220f\/180*C-sin(C)) where r is the circle's radius and C is the central angle of the associated sector. In this particular case, I have a predetermined area and need to determine the central angle. How do I solve for C? In it's most basic form, I have: y=x+sin(x) Solve for x. Any help is appreciated. Thanks!\n Sci Advisor HW Helper PF Gold P: 11,968 Basically, you can't find a finite expression for the inverse here. However, various appoximative techniques might be used. IF, for example, y is \"sufficiently close to 0\", a power series expansion about x=0 might zoom onto the solution fairly quickly. To show how this might be done: We have: $$y=x+x-\\frac{x^{3}}{3!}+\\frac{x^{5}}{5!}+++$$ when expanding the sine function in its power series (a finite segment of that highly accurate when x is close to zero) We now invert the power series, by assuming: $$x=a_{1}y+a_{2}y^{2}+a_{3}y^{3}+++$$ where the solution x(y) of the original equation boils down to determining the a's. Inserting the latter in the former, we get: $$y=2*(a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}+a_{5}y^{5}+++)-\\frac{1}{6}(a_{1}^{3}y^{3}+3a_{1}^{2}a_{2}y^{4}+3a_{1}^{2}a_{3}y^{5}+3a _{2}^{2}a_{1}y^{5})+\\frac{1}{30}a_{1}^{5}y^{5}++$$ where terms of higher orders in y than 5 are dropped. Now, we simply compare coefficients in each power to \"y\", to determine the a's. We get: $$a_{1}=\\frac{1}{2}, a_{2}=0, a_{3}=-\\frac{1}{48}, a_{4}=0, a_{5}=\\frac{1}{640}$$ Thus, to fifth order accuracy, you have: $$x=\\frac{1}{2}y-\\frac{1}{48}y^{3}+\\frac{1}{640}y^{5}$$\n Sci Advisor HW Helper PF Gold P: 11,968 Hmm, my coefficients were wrong. We have: $$a_{3}=\\frac{1}{96}$$ $$a_{5}=-\\frac{1}{4800}$$ or, I hope.. You'd better check for yourself.\nP: 2\n\n## Segment of a circle calculation\n\narildno, I appreciate the response! Unfortunately, I can't assume y is close to zero in this situation. I agree with your guidance that a finite solution for x isn't possible. It seems that only ordered pairs are achievable (set a value for y, guess a value for x, and iterate until x converges on a solution). I generated a spreadsheet that does the iteration automatically. Thanks again for the help!\n\nThis situation is still a little tough to wrap my head around. It would seem that, with a finite area of a segment, I should be able to calculate the central angle. I've yet to intuitively explain why I can't calculate this feature when I have a real, finite, pre-defined segment area that I can measure.\n\n#continuouseducation\nHomework\nHW Helper\nThanks\nP: 8,880\n Quote by bergie7isu This situation is still a little tough to wrap my head around. It would seem that, with a finite area of a segment, I should be able to calculate the central angle. I've yet to intuitively explain why I can't calculate this feature when I have a real, finite, pre-defined segment area that I can measure.\nIt is determined, but calculability is another matter. We are accustomed to treating strandard functions as exact answers. E.g. if the answer to a problem is \"x = sin(\u03c0\/7)\" you'd accept that. But in practice, since it is irrational, you can only turn that into a number by iterative approximation. If the problem's answer is \"that x for which x+sin(x) = 2\", we find that less satisfying, yet in reality it is no different. If answers of that form were to crop up reguarly in some contexts we might define a new function for it, and if we want a numerical answer we iterate.\n Sci Advisor HW Helper PF Gold P: 11,968 To generalize this, once you have a desired approximation X* for some \"y\" of yours (the closer X* is \"y\", the faster the convergence will be!), we may write: $$x=X*+\\epsilon$$ where epsilon is some \"small function\" of \"y\" and X* Inserting this into your equation yields: $$y=X*+\\epsilon+\\sin(X*+\\epsilon)=X*+\\sin(X*)(1-\\frac{\\epsilon^{2}}{2}+-+)+\\cos(X*)(\\epsilon-\\frac{\\epsilon^{3}}{3!}+-+)$$ Now, we arrange this to: $$\\frac{y-X*-\\sin(X*)}{1+\\cos(X*)}=\\epsilon+\\frac{\\sin(X*)}{1+\\cos(X*)}(-\\frac{\\epsilon^{2}}{2}+-+)+\\frac{\\cos(X*)}{1+\\cos(X*)}(-\\frac{\\epsilon^{2}}{3!}+-+)$$ ----------------------------- Remember now that the numerator of LHS is \"close to zero\", by assumption that X* is an approximate solution!. Breezing past, for now, the potential trouble of a \"too small\" denominator on LHS, we term LHS for \"kappa\", and expand \"epsilon\" in the power series. $$\\epsilon=a_{1}\\kappa+a_{2}\\kappa^{2}+++$$ Thus, we get, trivially $a_{1}=1$, and to second order: $$0=a_{2}\\kappa^{2}-\\frac{3\\sin(X*)+\\cos(X*)}{3!(1+\\cos(X*))}\\kappa^{2}$$ that is: $$a_{2}=\\frac{3\\sin(X*)+\\cos(X*)}{3!(1+\\cos(X*))}$$ and so on. Agreed so far?\n P: 583 Is this the part where I ask if anyone's thought of using a different formula? The basic formula for the area of a segment is $A_{segment} = A_{sector} - A_{triangle}$. If we know the length of the chord between the outer endpoints of the radii and the distance from the center to that chord, the area of the segment can be found by $A_{segment} = \\frac{x r^2}{2} - \\frac{hc}{2} = \\frac{x r^2 - hc}{2}$, where c is the length of the chord, h is the distance of the chord from the center, and x is in radians. In this case, $x = \\frac{2A_{segment}+hc}{r^2}$. If we can't find c and h, would it not be reasonable to say that $\\displaystyle y = x + sin(x) = x + \\lim_{r\u2192\\infty}\\sum_{n=0}^{r} \\frac{(-1)^n x^{n+1}}{(2n+1)!}$? Thus, our approximation for x gets increasingly better as r approaches infinity. Either way, remember that your answer for x will be in radians.","date":"2014-03-08 07:32:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.767117440700531, \"perplexity\": 485.96579862257585}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-10\/segments\/1393999653836\/warc\/CC-MAIN-20140305060733-00097-ip-10-183-142-35.ec2.internal.warc.gz\"}"}
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\section{Introduction}
Major depressive disorders (MDDs) are a constantly growing economic and societal problem~\cite{OECD2018}. The harsh collateral social side-effects of COVID-19 (social isolation, employment loss, bereavement, grief) have further exacerbated this already substantial problem~\cite{moreno2020mental, ons2021corona}. MDDs are also one of the leading causes of disability worldwide, accompanied by high socio-economic costs~\cite{deloitte2020cost}. As a result, accurately identifying, treating and thereby reducing the prevalence of MDDs is a major public health goal and challenge. However, globally, demand for mental health support greatly outstrips supply.Therefore, advances in digital health tools and phenotyping technologies that can support clinicians in this effort are crucial to ensure better and more widespread access to high-quality mental health support services and treatment.
Speech has great potential to be a source of such phenotypes and to provide unique preventative and predictive information about depression~\cite{cummins2015review,low2020review, yamamoto2020using, abbas2021remote}. However, current research in this space is not without its limitations. These become apparent when we consider a) the datasets used, b) core assumptions regarding effects of depression severity, and c) additional modalities collected alongside speech.
If we look at machine learning approaches that have arguably dominated speech-depression research over the last ten years, the majority of these works use the Audio/Visual Emotion Challenge (AVEC) datasets~\cite{valstar2013avec,Ringeval2017avec}. Whilst these investigations have undoubtedly advanced our knowledge in modelling depressed speech, their continuous use in pseudo-competition settings raises concerns relating to Goodhart's Law~\cite{Miller2020} and overfitting~\cite{hawkins2004problem}. Therefore, in order for the speech-depression community to continue to be driven forward, it is imperative that new databases are (additionally) used henceforth.
Another critical issue in speech-depression research is that the majority of works assume depression severity is in, and of itself, the most dominant and important measurable effect in speech. However, depression has a highly heterogeneous clinical profile; it may be the case therefore, that speech alterations observed within depression could be more strongly (and more explainably) associated with subsets of core depressive symptoms, rather than severity as an absolute measure. Preliminary works exploring this conjecture have indeed demonstrated that speaking rate measures are more strongly correlated with mood and psychomotor retardation measurements than with overall depression severity~\cite{Horwitz2013Importance, trevino2011phonologically}. These findings, however, are from small sample sizes and should be regarded as preliminary only.
Finally, no research to date has considered placing depression assessment via speech - whether with respect to depressive severity or depressive symptoms - alongside assessments of depression from other modalities; other than vision~\cite{cohn2018multimodal, girard2015automated, pampouchidou2017automatic}. For example, to the best of the authors' knowledge, there is no research examining how speech alterations may be complemented with reaction times and error rates in more classic neuropsychology protocols~\cite{nikolin2021investigation, owen2005n}. If we are to attempt to understand how the speech signal may be related to depressive symptomatology, it makes sense to examine it alongside and within the context of such classic protocols, leveraging complementary information to gain a better understanding of this relationship.
In response to the limitations described above, this paper presents a set of preliminary analyses conducted on a novel, large, online multimodal dataset collected by Thymia Limited (henceforth Thymia). In the following sections, we detail how for the first time, we combined speech elicitation tasks with the n-Back Task, an experimental psychology protocol targeting Working Memory~\cite{nikolin2021investigation, 10.3389/fpsyg.2019.00004, ROSE2006149}. Presented analyses include a set of machine learning experiments that highlight the complementary information contained within the speech and n-Back features when predicting the presence or absence of depression. Importantly, we additionally present a set of experiments that highlight the association between different speech and n-Back markers at the symptom level of depression, allowing us an initial insight into how speech and the n-Back Task may be used in tandem to better target different core depressive symptoms.
\section{Data Collection}
As part of its core mission Thymia \cite{thymiawebsite} (a London based mental health tech startup) is actively collecting
multimodal -- video, speech and behavioural -- data to develop models targeting remote assessment and monitoring of depression. In order to maximise access to the studies, the data collection needs to run on a browser, operating system and device agnostic platform.
\subsection{Thymia Research Platform}
In the past few years several platforms for online studies have gained popularity \cite{research-internet}, but to the best of our knowledge none offer the level of flexibility and security required for our intended task. Thymia, therefore, developed and implemented our own research platform.
The Thymia Research platform allows the hosting of complex, remote, one-off or longitudinal multimodal studies where detailed, informed consent and demographic data can be gathered, questionnaires can be completed, and gamified activities can be assigned to participants on a schedule. During these activities, data from the device's camera, keyboard, mouse/trackpad and/or touch screen can be streamed to a secure backend\footnote{The Thymia platform is fully compliant with the 2018 EU General Data Protection Regulations, is ISO27001-certified and NHS Toolkit-compliant. All of our research is reviewed by independent and/or University research ethics committees. The Thymia platform has been successfully used not only by Thymia, but also by several UK university research groups.}. Throughout the experiment protocols, when launching activities that require media recording, participants are reminded that their camera and/or microphone will be switched on for recording and they are free to opt out.
\subsection{Online Study Setup}
The dataset used in this work is part of a larger online study\footnote{"Does mood affect speech patterns and reactions? A Proof of Concept study." This study, including all Information Sheets and Consent Forms, has been reviewed by an independent research ethics expert working under the auspices of the Association of Research Managers and Administrators. All subjects read a detailed Information Sheet prior to beginning the experiment, could remotely ask questions and expressly consented to participate knowing they were free to withdraw at any point. All data collected were handled according to GDPR and participants were compensated for their time.} running on the Thymia Research platform. It consists of demographic and psychiatric questionnaires, speech eliciting tasks and gamified experimental protocols targeting visual processing, attention, psychomotor response and working memory.
Participants were pre-screened and split into two groups: a patient and a healthy age- and gender-matched control group. Both groups consisted of adult, native English speakers, aged 18 to 75 (evenly split across age groups), with normal or corrected-to-normal vision, no hearing, language or speech impairments and - for the control group - no prior history of psychiatric illness. The patient group participants must have had a formal MDD diagnosis by a GP, clinical psychologist or psychiatrist at least two months prior to participating.
A study session is completed via the participant's laptop or smart device without any researcher supervision. The session includes standardised questionnaires to gather information about demographics and mood, including the \textit{Patient Health Questionnaire - 8} (PHQ-8)~\cite{kroenke2009phq}, a well-established depression scale used commonly in research which aims to assess a number of core depressive symptoms, including fatigue, working memory impairment, anhedonia and low mood. Participants also completed speech eliciting tasks, including an Image Description Task, and short point-and-click (or screen tapping) tasks measuring reaction times, accuracy and error rates including an implementation of the classical n-Back task~\cite{owen2005n}.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{image_description.png}
\caption
The Image Description Task; the image represents a cafe' scene with several people. The image has many elements that are animated: the bird is moving, people are talking.}
\label{fig:image_description}
\vspace{-5mm}
\end{figure}
\subsubsection{Image Description Task}
In the Image Description Task, participants are encouraged to describe what they see in the image whilst keeping their camera and microphone on; while performing the task participants can see their own camera feed as feedback and reminder of the fact that they are being recorded. The image itself is a rich, animated illustration depicting a caf\'{e} environment filled with people at different tables (Figure~\ref{fig:image_description}).
\subsubsection{n-Back Task}
Performance in the n-Back Task offers insights into working memory dysfunction in depression~\cite{nikolin2021investigation, ROSE2006149}. The implementation used as part of this study is presented as a card memory game. A rounded edged card (as in a deck of cards) with a single digit number or letter in the middle of it appears in the centre of a blank screen for a short interval; this is then followed by a blank screen; followed by another card containing another number or letter; followed by a blank screen etc. The participant must tap/click the screen when the current number or letter matches the number or letter that appeared \textit{n} cards back.
There are two difficulty levels based on the value of \textit{n}. The different difficulty levels are based on progressive cognitive loading. In the first block, \textit{n} equals one (1) (i.e. the current target card must match the card one card back); in the second block \textit{n} increases to two (2) (i.e. the current target card matches the card appearing two cards before it). Each participant saw 3 practice blocks for each n-Back load, followed by 6 experiment blocks for each n-Back load. Match number and position were pseudo-randomised across blocks and block order was counterbalanced across participants.
\begin{table*}[t!]
\caption{Sociodemographic, Depression (Low: PHQ-8 $<$ 10, High: PHQ-8 $\geq$ 10) and Activity distributions in the experimental data.
}
\vspace{-2mm}
\label{tab:demographics}
\centering
\begin{tabular}{c c c c c c c c c}
\toprule
\multicolumn{1}{c}{\textbf{Partition}} &
\multicolumn{2}{c}{\textbf{\#Participants}} &
\multicolumn{2}{c}{\textbf{Age}} &
\multicolumn{2}{c}{\textbf{Binary PHQ-8}} &
\multicolumn{2}{c}{\textbf{Total Activity Time}} \\
\multicolumn{1}{c}{\textbf{}} &
\multicolumn{1}{c}{\textbf{Male}} &
\multicolumn{1}{c}{\textbf{Female}} &
\multicolumn{1}{c}{\textbf{Mean}} &
\multicolumn{1}{c}{\textbf{SD}} &
\multicolumn{1}{c}{\textbf{\#Low}} &
\multicolumn{1}{c}{\textbf{\#High}} &
\multicolumn{1}{c}{\textbf{Speech}} &
\multicolumn{1}{c}{\textbf{n-Back}} \\
\midrule
Training & 387 & 388 & 34.94 & 12.64 & 501 & 274 & 10:19:36 & 4 days, 17:13:01 \\
Test & 97 & 97 & 35.39 & 13.24 & 125 & 69 & 2:21:33 & 1 day, 2:12:18 \\
\bottomrule
\end{tabular}
\vspace{-2mm}
\end{table*}
\section{Dataset}
Our experimental dataset consists of 969 participants who performed a range of activities within a single session on the Thymia Research Platform using their own personal devices. The presented analysis focuses on two specific data modalities gathered through the platform: audio data from the Image Description Task, and behavioural data from the n-Back Task.
We performed a stratified split of the dataset into training and test sets (80/20 \%), keeping the same proportions of genders, age groups and PHQ-8 distribution (Table~\ref{tab:demographics}).
\subsection{Quality Controls}
Given the real-life nature of the dataset, each participant using their own personal device, we performed a number of quality controls on the different data modalities.
For the audio recordings, we implemented an automated audio quality pipeline to flag the audio files as having good or bad quality based on two criteria: (i) the quality of extracted audio features, and (ii) the presence of speech activity. Out of 917 files, the audio quality pipeline detected 51 bad quality audio files that were rejected from subsequent analysis. Through manual inspection of these 51 files, we confirmed that they were all cases of either high environmental noise, microphone turned-off or malfunctioning, or participant not speaking. A principal component projection of the extracted features of each audio file in the dataset, confirms the difference between the distributions of good and bad audio files (Figure~\ref{fig:audio_quality}).
For the
n-Back Task, we confirmed that the various metrics of performance are within expected ranges and vary with n-Back load as well as known covariates such as age; noting a general decrease in n-Back performance with age~\cite{10.3389/fpsyg.2018.02208}.
\section{Experimental Settings}
We investigated the impact of combining multiple data modalities, namely audio data from the Image Description Task and behavioural data from the n-Back Task, in a binary PHQ-8 classification paradigm (PHQ-8 $<$ 10 vs PHQ-8 $\geq$ 10). Details on the features and the models used are provided in the following.
\begin{figure}[t!]
\centering
\includegraphics[width=0.925\linewidth]{interspeech_2022_audio_quality_pca.pdf}
\caption
First and second principal components of the audio features from the Image Description Task highlighting the difference in distribution of good and bad audio files.}
\label{fig:audio_quality}
\vspace{-5mm}
\end{figure}
\subsection{Speech Features}
The audio recordings of the Image Description Task were processed to extract a range of acoustic and linguistic features. We extracted 88 acoustic features as defined in the \textit{extended Geneva Minimalistic Acoustic Parameter Set} (eGeMAPS) \cite{eyben2015geneva} using
\textsc{openSMILE}~\cite{eyben2010opensmile}. In addition, we extracted a curated set of 28 features describing speech rate, pitch, voice quality and formant properties, using the Parselmouth package \cite{parselmouth} as a Python interface to Praat~\cite{boersma2001praat}. To quantify the linguistic content,
we first transcribed the
files using Amazon Transcribe~\cite{aws-transcribe}. We then used the spaCy library~\cite{spacy2} to extract
25 linguistic features describing speech-rate, pause-rate and part-of-speech usage.
\subsection{n-Back Features}
For each session, the data collected from the n-Back Task consists of a sequence of clicks and a corresponding sequence of targets and non-targets. From these sequences, we calculated standard features that quantify the performance in the task, namely precision, recall, false-positive rate, as well as reaction time. These features were calculated separately for the two n-Back loads (1-Back and 2-Back), yielding
8 n-Back features.
\subsection{Models}
All models, training and calibration procedures are implemented using the scikit-learn package~\cite{scikit-learn}.
\begin{figure}[b!]
\centering
\includegraphics[width=\linewidth]{interspeech_2022_test_roc.pdf}
\caption{ROC curves calculated on the hold-out test set for the Speech, n-Back and Multimodal models.}
\label{fig:test_roc}
\end{figure}
\subsubsection{Speech and n-Back Models}
First, we investigated five unimodal models: four speech models and an n-Back model. The speech models are; (i) an eGeMAPS model; (ii) a Praat model; (iii) a Linguistic model; and (iv) a Speech model, which is the early fusion model of the 141 acoustic and linguistic features. All models consist of a binary Random Forest classifier with input features from their specific feature representations. All models receive 3 additional features as input, namely age, gender, and device setup. This last categorical feature encodes the personal device setup on which the session was performed (i.e. Laptop+Trackpad, Laptop+Mouse, Mobile/Tablet, Desktop). Standard rescaling is applied to all numerical features, while gender is encoded as binary and device setup is one-hot encoded. The unimodal models are fitted and calibrated on the training set using a cross-validated random parameter search with 100 iterations and 10 cross-validation folds. The hyperparameters tuned were \textit{\#Trees}, \textit{Max. Depth}, \textit{Min. \#Samples/Split} and \textit{Max. Rel. \#Features}.
\subsubsection{Multimodal Model}
We combined the Speech and n-Back unimodal models into a voting ensemble to create a multimodal model. The predictions of this model are given by a soft-voting rule, whereby the multimodal prediction is given by the most likely class label after averaging the predicted class probabilities across the unimodal models.
\begin{table}[]
\caption{Cross-validated model performances when classifying low (PHQ-8 $<$ 10) versus high (PHQ-8 $\geq$ 10) depression on the training set}
\label{tab:cv_performance}
\centering
\begin{tabular}{@{}l r r@{}}
\toprule
\multicolumn{1}{l}{\textbf{Model}} &
\multicolumn{2}{c}{\textbf{ROC-AUC}} \\
\multicolumn{1}{c}{\textbf{}} &
\multicolumn{1}{c}{\textbf{Mean}} &
\multicolumn{1}{c}{\textbf{SD}} \\
\midrule
eGeMAPS & 0.620 & 0.032 \\
Praat & 0.607 & 0.056 \\
Linguistic & 0.625 & 0.045 \\
n-Back & 0.619 & 0.074 \\
Speech & 0.631 & 0.024 \\ \midrule
Multimodal & \textbf{0.652} & 0.037 \\
\bottomrule
\end{tabular}
\end{table}
\subsubsection{Feature Analysis}
To gain further insights into the performance of our multimodal system, we ran a set of linear regression analyses on our training set. The aim of this testing was to establish the importance of different eGeMAPs, Praat, Linguistic and n-Back features when predicting either individual items (questions) within the PHQ-8 scale, or predicting overall depression severity as given by the overall PHQ-8 score. We modelled each feature separately, and included age, gender and personal device setup (encoded as dummy variables) as covariates. We ranked feature importance in predicting each item, or the overall score using the R$^{2}$ value of each model.
\begin{table}[t!]
\caption{Top features for predicting either a single PHQ-8 item or total PHQ-8 score}
\label{tab:feature-ranks}
\begin{tabular}{@{}lrrr@{}}
\toprule
\textbf{Item} & \textbf{Feature} & \textbf{R$^{2}$} & \textbf{Beta}\\ \midrule
Item \#1 & Loudness Peaks Per Sec. (eGeMAPS) & .051 & -.054\\
Item \#2 & Loudness Peaks Per Sec. (eGeMAPS) & .039 & -.049\\ \midrule
Item \#3 & 1-Back Reaction Time & .034 & .089\\
Item \#4 & 1-Back False Positive & .049 & .096\\ \midrule
Item \#5 & CV Spectral Flux Voiced (eGeMAPS) & .052 & .001\\
Item \#6 & Syllable Rate (Praat) & .042 & .001\\ \midrule
Item \#7 & Loudness Peaks Per Sec. & .034 & -.073\\
Item \#8 & overall n-Back Precision & .033 & .055\\ \midrule
Total & Loudness Peaks Per Sec. & .049 & -.322\\ \bottomrule
\end{tabular}
\end{table}
\section{Results and Discussion}
We use the area under the ROC curve (ROC-AUC) as the performance metric to compare the models. We see that the performance of the unimodal models on the training set is well-above chance level on average (Table~\ref{tab:cv_performance}). This performance demonstrates that both speech and n-Back features contain predictive information about the corresponding PHQ-8 score. In addition, the average performance of the Multimodal model is higher than all unimodal models (Table~\ref{tab:cv_performance}), suggesting that the two modalities contain complementary PHQ-8 information.
To validate these results, we tested the Speech, n-Back and Multimodal models on the test set, which was not touched during model training and hyperparameter calibration. The performance of all models on the test set is qualitatively similar to the training set, with the Multimodal model outperforming the unimodal ones (Figure~\ref{fig:test_roc}).
Our linear regression analysis offers insights into why the fusion of our Speech and n-Back features improves results. When comparing the top-ranking features for each PHQ-8 item (Table~\ref{tab:feature-ranks}), we can see that the items relating to anhedonia (Item \#1), depressed mood (Item \#2), change in appetite (Item \#5), feelings of worthlessness (Item \#6) and problems concentrating (Item \#7) all returned a speech feature as the top-ranked feature. While the top feature for the somatic (Item \#3 and \#4) and psychomotor (Item \#8) items were from the n-Back. Interestingly, n-Back features are ranked in the top three positions for the somatic and psychomotor items highlighting the strength of this task in capturing these changes.
\section{Conclusion}
Digital phenotyping offers the chance to aid depression diagnosis and management by providing objective information based on cognitive, physiological and behavioural cues. The results presented in this paper demonstrate that speech features, and metrics derived from n-Back Task performance offer complementary information when predicting depression. This is shown in two ways: by the increase in predictive performance when fusing both modalities, and feature space analysis, indicating that features from the different modalities strongly align to different items within the PHQ-8 domain. To the best of the authors' knowledge, this is the first time such a result has been shown. Future work will focus on repeating the fusion results in more complex models and exploring how the addition of facial information changes the feature space dynamics.
\newpage
\bibliographystyle{IEEEtran}
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"redpajama_set_name": "RedPajamaArXiv"
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Q: SystemC running inside a virtual machine, timing issues or corrupt results? Is it ok to run a SystemC based simulation on a guest OS inside a virtual machine? Can simulation time be affected by this? I know that SystemC time is simulated and not actually tied to hardware timers.
And will running dozens of instances of SystemC simulations in the virtual machine configured with 4 cores (physical machine has 8) affect the results?
A: There are no problems running SystemC on a virtual machine. I do it regularly with VirtualBox. I run SystemC on Linux and Windows virtual machines, both 32-bit and 64-bit.
Unless there is a bug in the virtual machine software, a software application should behave identically when run on a physical or virtual machine.
Running multiple SystemC simulations concurrently on a virtual machine is also fine. The limit to how many simulations you can run concurrently, will be based on how much RAM you have available.
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{"url":"http:\/\/michaelkorshandbagsoutlet.com.co\/pendleton-eco-iiuibe\/does-xef4-have-a-dipole-moment-a00371","text":"# does xef4 have a dipole moment\n\nDipole moment is equal to the product of the partial charge and the distance. (a) H3O+ (b) PCl4 (c) SnCl3 (d) BrCl4 (e) ICl3 (f) XeF4 (g) SF2. Paper by Super 30 Aakash Institute, powered by embibe analysis.Improve your score by 22% minimum while there is still time. The lone pair leads to the creation of a strong dipole moment. does xef4 have a dipole moment - question answered here at HaveYourSay.org - leading question and answers website. A molecule is said to be polar when it has a dipole moment, creating partial positive and negative charges and forming unsymmetrical bonds. Polar molecules sometimes, but not necessarily, have a net charge equivalent to zero. 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For example, each of the two carbon-oxygen bonds in CO 2 has a dipole moment, but the CO 2 molecule has no dipole moment because the dipole moments of the two carbon-oxygen bonds are identical in magnitude and opposite in direction, resulting in a vector sum of zero. Hence it does not have a permanent dipole moment. 0 0. 98. We have step-by-step solutions for your textbooks written by Bartleby experts! Which one of the following molecules has a dipole moment greater than zero? not TeCl and XeF2. A property of polar molecules that are permanent dipoles is the possession of a permanent dipole moment, but dipole moments are not guaranteed solely by uneven distribution of charge. The SO2 molecule has a dipole moment. This geometry has a dipole moment since the electronegative difference between F and S does not cancel. D. X e F 4 -has a planar structure, so it is non-polar Which one of the following molecules does NOT have a dipole moment? Only a polar compound has a dipole moment. Which of the following is a non-polar molecule having one or more polar bonds? 1 year ago. For each of the following, does the molecule have a permanent dipole moment? (a) ClF5 (b) 2ClO (c) 24TeCl (d) PCl3 (e) SeF4 (f) 2PH (g) XeF2. Or if you need more Dipole Moment practice, you can also practice Dipole Moment practice problems. XeF4. If the molecule XeF2Br2 is nonpolar does it have a dipole moment? 3. bond will be polarized. SO2. Add your answer and earn points. Dipole Moments: There is a measure of how strongly an atom in a covalent bond will attract the shared electrons. The Questions and Answers of Which of the following would have a permanent dipole moment? H--Br H--Cl H--I H--F NH3 6. XeF4 and XeF6.XeF4 is pyramidal with one lone pair and XeF6 is distorted octahedral having one lone pair. 10 - Explain how the dipole moment could be used to... Ch. A molecule may not have a dipole moment despite containing bonds that do. The equation for dipole moment is as follows. 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\section{Introduction}
Nearly forty years ago, Bloom and Gilman found \cite{Bloom:1970xb} that in electron
scattering on protons the inclusive structure function $F_2$ in the
resonance region oscillates around the DIS scaling curve and, after
averaging, closely resembles it. This phenomenon is one of the ways
quark--hadron duality reveals itself in physical processes. Generally
quark--hadron duality establishes a relationship between the
quark--gluon description of a certain phenomenon, which is
theoretically justified in the DIS region, and the hadronic
description, which is more convenient at medium and low energies.
Understanding duality is also essential when establishing
relationships between exclusive and inclusive processes. For a recent
and detailed review of duality we refer the reader to
Ref.~\cite{Melnitchouk:2005zr}.
So far, most theoretical studies of quark--hadron duality in lepton
scattering were dealing with nucleon targets. The topic becomes of
great practical interest when turning to nuclear targets and neutrino
sources. The current precision measurements of the oscillation
parameters require an efficient and accurate description of the
neutrino--nucleus cross sections. Of particular interest is the
resonance region and the possibility of linking it with the DIS
region. A hadronic description of the neutrino-nucleus cross sections
at low $Q^2$ requires the vector and axial transition form factors for
each resonance. For the majority of the resonances, these transition
form factors are not well constrained. Provided that one can establish
that quark-hadron duality holds with a reasonable accuracy, one could
think of using the DIS results for estimating the neutrino-nucleus
cross sections in the resonance region. In that respect it is worth
mentioning that in nuclei, the Fermi motion of the nucleons smears the
observables, so that the averaging in the resonance region required
for duality, proceeds to a certain extent automatically. The issue
whether quark--hadron duality holds with sufficient accuracy in
lepton-nucleus scattering, requires further theoretical and
experimental investigation. The present paper addresses this issue
from the theoretical point of view.
Recent electron scattering measurements at Jefferson Laboratory (JLab)
have confirmed the validity of Bloom--Gilman duality for the proton,
deuterium \cite{Niculescu:2000tk} and iron \cite{Arrington:2003nt}
structure functions. Further experimental efforts are required for
neutrino scattering. Among the upcoming neutrino experiments,
Miner$\nu$a\cite{Boehnlein:2007zz,SolanoSalinas:2007zza,minerva} and
SciBooNE\cite{AlcarazAunion:2007zz,Hiraide:2007zz,sciboone} aim at
measurements with carbon, iron and lead nuclei as targets. From the
theoretical side, recent investigations of the phenomenon of duality
for electron and neutrino scattering on nucleons include the works
reported in Refs.~
\cite{Matsui:2005ns,Graczyk:2005uv,Lalakulich:2006yn}. These studies
differ in the way they treat the resonant contributions and the way
they parameterize the DIS structure functions. This paper extends the
study of Ref.~\cite{Lalakulich:2006yn} about the duality phenomenon in
the nucleon to nuclei.
For a free nucleon target, the structure functions generally depend on
the transferred energy $\nu=E-E'$ and four-momentum $Q^2=-q_\mu
q^\mu$. At low $Q^2$ the $\nu$--distributions reveal several peaks,
which correspond to various baryon resonances. We briefly sketch our
theoretical approach to resonance production in nuclei in
Section~\ref{nucleus}. The nuclear structure functions are defined in
Section~\ref{defSF}. At high $Q^2$ the structure functions exhibit
scaling behavior, which is discussed in Section~\ref{DIS}. Comparing
the structure functions in these two regions allows one to check the
basic features of duality and compare its validity for different
targets and incoming leptons. Our results are presented in
Section~\ref{electrons} for electrons and Section~\ref{neutrinos} for
neutrinos. Conclusions are given in Section~\ref{summary}.
\section{Formalism}
We consider inclusive charged-current (CC) neutrino scattering from nuclei and its electromagnetic counterpart
\begin{equation}
\nu_l(k^\mu) + A \to l^- (k^{\prime \mu}) + X \ ,
\qquad
l^- (k^\mu) + A \to l^- (k^{\prime \mu}) + X \ ,
\end{equation}
where $l$ is the lepton flavor, $A$ represents a nucleus with mass
number $A$, and $k^\mu=(E,\vec k)$ and $k^{\prime\mu}=(E',\vec k')$ are
the four--momenta of the incoming and outgoing lepton respectively. We work in the
laboratory frame of reference. The coordinate system is chosen such
that the $z$-axis lies along the direction of the virtual photon, so
that the transferred momentum is given by
$q^\mu=k^\mu-k^{\prime\mu}=(\nu, 0,0, q^z)$. The lepton scattering
proceeds in the $xz$--plane. In this section, we investigate the structure functions
$F_2$, $2xF_1$ and $xF_3$, the latter being nonzero for neutrino
reactions only. To this end, CP-violation effects are neglected for
the case of electron scattering.
\subsection{Resonance production on a nucleus \label{nucleus}}
For lepton--nucleus scattering we describe the struck nucleus as a
collection of bound nucleons. Assuming an independent--particle shell
model, each nucleon occupies a nuclear shell $\alpha$ with a
characteristic binding energy $e_{\alpha}$ and is described by the
bound--state spinor $u_\alpha$. In the impulse approximation, an
impinging lepton interacts with a single bound nucleon. Hence, the
nuclear cross section can be expressed as an incoherent sum over all
nucleons of one--nucleon cross sections weighted with the
corresponding nucleon momentum distributions $n_\alpha$. For example,
for a carbon nucleus, one has
\[
\begin{array}{l} \displaystyle
\frac{ d\sigma^{ {}_{\; 6}^{12}C } }{dQ^2 d\nu} = \int d^3 p \biggl[
2 \frac{d\sigma_{\nu p} \left|_{1s^{1/2}} \right. }{dQ^2 d\nu} n^{(p)}_{1s^{1/2}}(|\vec p|)
+ 4 \frac{d\sigma_{\nu p} \left|_{1p^{3/2}} \right. }{dQ^2 d\nu} n^{(p)}_{1p^{3/2}}(|\vec p|)
\\[3mm] \displaystyle \hspace*{27mm} \displaystyle
+ 2 \frac{d\sigma_{\nu n} \left|_{1s^{1/2}} \right.}{dQ^2 d\nu} n^{(n)}_{1s^{1/2}}(|\vec p|)
+ 4 \frac{d\sigma_{\nu n} \left|_{1p^{3/2}} \right.}{dQ^2 d\nu} n^{(n)}_{1p^{3/2}}(|\vec p|)
\biggr].
\end{array}
\]
This allows us to employ the one--body lepton-nucleon vertex that can
be well constrained in experiments with a proton and deuteron target.
The four--momentum of the bound nucleon
can be written as $p^\mu=(m_N-e_\alpha, \vec{p})$. Both the
bound--state spinor $u_\alpha(\vec p)$ and the corresponding binding
energies are computed in the Hartree approximation to the
$\sigma-\omega$ Walecka--Serot
model~\cite{Serot:1984ey,Furnstahl:1996wv}. Binding energies for
carbon and iron are summarized in Table~\ref{tab:e0}. For each shell,
the nucleon momentum distribution $n_\alpha(|\vec p|)$ is constructed
from the bound--state spinors, the normalization convention being
\[
\int d^3p \; n_\alpha(|\vec p|) = 1 \; .
\]
These $n_\alpha(|\vec p|)$ are shown in Fig.~\ref{fig:n_alpha}, for
the case of a carbon nucleus. Clearly, for a specific shell, the proton and
neutron distributions are almost identical.
\begin{table}
\caption{Binding energies (MeV) for carbon and iron nuclei}
\[
\begin{array}{ccc}
& proton & neutron
\\
{}^{12}C: & &
\\
1s^{1/2} & 47.76 & 51.17
\\
1p^{3/2} & 16.76 & 19.87
\\[2mm]
{}^{56}Fe: & &
\\
1s^{1/2} & 57.19 & 63.66
\\
1p^{3/2} & 43.11 & 50.12
\\
1p^{1/2} & 39.32 & 46.00
\\
1d^{5/2} & 27.64 & 34.84
\\
2s^{1/2} & 17.77 & 24.41
\\
1d^{3/2} & 16.55 & 23.01
\\
1f^{7/2} & 12.11 & 19.17
\\
2p^{3/2} & - & 5.99
\end{array}
\]
\label{tab:e0}
\end{table}
\begin{figure}[hbt]
\epsfig{figure=DisplaySF.ps,angle=-90,width=0.5\textwidth}
\caption{Momentum distributions for proton and neutron shells in carbon.}
\label{fig:n_alpha}
\end{figure}
After the interaction takes place inside the nucleus, the reaction
products can escape the nucleus without interactions or they can
undergo elastic and/or inelastic rescatterings with the other
nucleons. Thus, the reaction strength is redistributed between
different channels. All these processes are called the final state
interactions (FSI). The effect of FSI can be large for a specific
exclusive process, for example for quasi-elastic nucleon knockout
\cite{Martinez:2005xe}, where the cross section can be suppressed by a
factor of 2. In one--pion production, the outgoing pion can be
absorbed in the nucleus and thus mimic a quasi--elastic event.
For a duality study, however, it suffices to consider inclusive
reactions. Consequently, since the outgoing hadrons and the residual
nucleus are not detected, we can make the assumption, following
Ref.~\cite{Benhar:2006nr}, that FSI can be disregarded.
Recently, duality in lepton--nucleon scattering was investigated
theoretically within the Sato--Lee \cite{Sato:2003rq}, Rein--Sehgal
\cite{Rein:1980wg} and Dortmund--group \cite{Lalakulich:2006sw}
models for resonance production. In this paper, we follow the approach
used in \cite{Lalakulich:2006sw} and extend it to calculate the
nuclear structure functions. In particular, in the resonance region we
take into account the first four low--mass baryon resonances
$P_{33}(1232)$, $P_{11}(1440)$, $D_{13}(1520)$, $S_{11}(1535)$ and
describe the vertices of their leptoproduction within a
phenomenological form-factor approach.
The nucleon structure functions ${\cal W}_i$ are defined by the standard expansion of the hadronic tensor
\begin{equation}
W_{\mu\nu}=
-g_{\mu\nu}{\cal W}_1
+ \frac{p_\mu p_\nu}{m_N^2} {\cal W}_2
- i\varepsilon_{\mu\nu\lambda\sigma} \frac{p^\lambda q^\sigma}{2 m_N^2} {\cal W}_3
+ \frac{q_\mu q_\nu}{m_N^2} {\cal W}_4
+ \frac{p_\mu q_\nu + p_\nu q_\mu}{m_N^2} {\cal W}_5 \ .
\label{Wmunu-nucleon}
\end{equation}
Each ${\cal W}_i$ depends on two independent kinematic variables, for
example $Q^2$ and $\nu$, which are determined exclusively by the
lepton kinematics. Another set of variables, namely $Q^2$ and $W$, is
also possible, since the invariant mass $W$, defined as $W^2=(p+q)^2$,
for a free target nucleon can be uniquely related to $Q^2$ and $\nu$:
$W^2=m_N^2 +2m_N\nu -Q^2$. The analytical expressions for the
one--nucleon structure functions $F_1=m_N {\cal W}_1$, $F_2=\nu {\cal
W}_2$, $F_3=\nu {\cal W}_3$ in terms of form factors for a free
nucleon as well as the form factors themselves are given in
\cite{Lalakulich:2006sw}. The Fermi motion of the bound nucleon
modifies the expression for the scalar product $({q\cdot p})$, so that the
invariant mass $W^2=(p+q)^2$ will now depend on the nucleon momentum
and binding energy. The variables $Q^2$ and $\nu$, being determined by
lepton kinematics only, remain unaffected. Strictly speaking, the
expansion in Eq.~(\ref{Wmunu-nucleon}) is only valid for a free
(on--mass shell) target nucleon. For a bound nucleon, all inclusive
observables depend not only on $\nu$ and $Q^2$, but also on an
additional independent kinematical variable, which can be chosen to be
$p_\mu p^\mu=p^2$. Here, we make the assumption (see
\cite{Ferree:1995fb} for a detailed discussion) that expression
(\ref{Wmunu-nucleon}) can still be used to define the bound--nucleon
structure functions, and recalculate them keeping the kinematical
variable $p^2$ as an independent one. The results are given below for
the $W_2$ and $W_3$ structure functions. Equating $p^2=m_N^2$, the
free-nucleon results of \cite{Lalakulich:2006sw} are easily
reproduced. For the spin-3/2 resonances ($P_{33}(1232)$ and
$D_{13}(1520)$ in our case) one has
\begin{equation}
{\cal W}_i(Q^2,\nu, p^2)=\frac{2}{3m_N} V_i(Q^2,\nu,p^2) R(W,M_R),
\end{equation}
where $R(W,M_R)$ is the finite representation of the $\delta-$function
$\delta(W^2 - M_R^2)$, which gives the relativistic Breit--Wigner distribution:
\[
R(W,M_R)=\frac{M_R \Gamma_R}{\pi} \frac{1}{(W^2-M^2_R)^2+M_R^2 \Gamma_R^2} \ ,
\]
and the $V_i$ are given below. The upper and lower signs are for the positive ($P_{33}(1232)$) and negative ($D_{13}(1520)$) parity resonances, respectively.
\begin{eqnarray}
V_2 &=&
\frac{(C_3^V)^2 + (C_3^A)^2}{M_R^2} Q^2 \left[ {q\cdot p} +p^2 +M_R^2 \right]
+\left( \frac{(C_4^V)^2}{m_N^2}
+ \frac{(C_5^V)^2(Q^2+M_R^2)}{m_N^2 M_R^2}
+\frac{2 C_4^V C_5^V}{m_N^2} \right)Q^2
\left[ {q\cdot p} +p^2 \mp m_N M_R \right]
\nonumber
\\[3mm]
&+&\frac{C_3^V C_4^V}{m_N M_R} Q^2 \left[ {q\cdot p} + p^2 + M_R^2 \mp 2 m_N M_R \right]
+\frac{C_3^A C_4^A}{m_N M_R} Q^2 \left[ {q\cdot p} + p^2 + M_R^2 \pm 2 m_N M_R \right]
+ C_3^A C_5^A \frac{m_N}{M_R} Q^2
\nonumber
\\[3mm]
&+&\frac{C_3^V C_5^V}{m_N M_R } Q^2 \left[ {q\cdot p} + p^2 + M_R^2 \mp 2 m_N M_R +Q^2\right]
+\left[ ({C_5^A})^2 \frac{m_N^2}{M_R^2}
+ \frac{(C_4^A)^2}{m_N^2} Q^2 \right]
\left[ {q\cdot p}+ p^2 \pm m_N M_R \right],
\label{calW2}
\end{eqnarray}
\begin{eqnarray}
V_3 &=&
2\frac{C_3^V C_3^A}{M_R^2} \left[ 2(Q^2-{q\cdot p})^2 +M_R^2(3Q^2-4{q\cdot p}) \right]
+2\left[ \frac{C_4^V C_4^A}{m_N^2}(Q^2 -{q\cdot p}) -C_4^V C_5^A \right] (Q^2-{q\cdot p})
\nonumber
\\[3mm]
&+& 2\frac{C_5^V C_3^A {q\cdot p} - C_4^V C_3^A(Q^2-{q\cdot p})}{M_R m_N}
\left[2M_R^2 \mp 2 m_N M_R +Q^2 -{q\cdot p} \right]
+ 2\left[ C_5^V C_5^A -\frac{C_5^V C_4^A}{m_N^2}(Q^2-{q\cdot p}) \right] {q\cdot p}
\nonumber
\\[3mm]
& + & 2 \left[C_3^V C_5^A \frac{m_N}{M_R} -
\frac{C_3^V\, C_4^A}{M_R m_N} (Q^2-{q\cdot p})
\right]
\left(2 M_R^2 \pm 2 m_N M_R + Q^2 - {q\cdot p} \right)
\label{calW3}
\end{eqnarray}
For spin-1/2 resonances we have
\[
{\cal W}_i(Q^2,\nu, p^2)=\frac{1}{m_N} V_i(Q^2,\nu, p^2) R(W,M_R) \; ,
\]
where
\begin{equation}
V_2=2 m_N^2 \left[\frac{(g_1^V)^2}{\mu^4}Q^4 + \frac{(g_2^V)^2}{\mu^2}Q^2 + (g_1^A)^2 \right] \ ,
\end{equation}
\begin{equation}
V_3= 4 m_N^2 \left[ \frac{g_1^V g_1^A}{\mu^2}Q^2 + \frac{g_2^V g_1^A}{\mu}(M_R\pm m_N) \right] \ ,
\end{equation}
and $\mu=m_N+M_R$. The upper and lower signs again correspond to
positive ($P_{11}(1440)$) and negative ($S_{11}(1535)$) parity
resonances, respectively. In the case of electroproduction, all axial
form factors should be put equal to zero and the weak vector form
factors should be replaced by the electromagnetic ones for proton or
neutron, depending on the target nucleon.
To make the article self-contained, we present the transition form
factors for each resonance. Electromagnetic and weak vector form factors were determined in
\cite{Lalakulich:2006sw} by fitting the electroproduction data on
helicity amplitudes in the region $Q^2<3\; \mathrm{GeV}^2$. Recently, it was
shown \cite{Vereshkov:2007xy} that in order to satisfy the asymptotics
for helicity amplitudes at $Q^2\to \infty$, as prescribed by
perturbative QCD, the vector form factors should also exhibit a
certain asymptotic $Q^2$ behavior. Therefore, we refitted the form
factors according to this prescription. In the region $Q^2 \le 4 \; \mathrm{GeV}^2$,
however, the difference between our new fit and the one performed in
\cite{Lalakulich:2006sw} falls within the accuracy of the
experimentally extracted helicity amplitudes. To be on the safe side
for higher $Q^2$ values, further attempts to improve the fits of the
form factors (for example, in accordance to upcoming data on helicity
amplitudes) will be done within the framework of the arguments
presented in \cite{Vereshkov:2007xy}. The axial form factors are the
ones used in \cite{Lalakulich:2006yn} for the ``fast'' fall--off
case. Thus, we use the following form factors
\begin{equation}
\begin{array}{ll}
P_{33}(1232): & C_3^{(p)}=\frac{2.14/D_V}{1+Q^2/4 M_V^2}, \quad
C_4^{(p)}=\frac{-1.56/D_V}{(1+Q^2/7.3 M_V^2)^2}, \quad
C_5^{(p)}=\frac{0.83/D_V}{(1+Q^2/0.95 M_V^2)^2},
\\[3mm]
& C_i^{(n)}=C_i^{(p)}, \qquad C_i^{V}=C_i^{(p)},
\\[3mm]
& C_3^A=0, \quad C_4^A=-C_5^A/4, \quad C_5^A=\frac{1.2/D_A}{1+Q^2/3M_A^2},
\quad C_6^A=m_N^2 \frac{C_5^A}{m_\pi^2+Q^2},
\end{array}
\end{equation}
\begin{equation}
\begin{array}{ll}
P_{11}(1440): & g_1^{(p)} = \frac{2.2/D_V}{1+Q^2/1.2 M_V^2}\left[ 1.+0.97\ln\left(1.+\frac{Q^2}{1\; \mathrm{GeV}^2}\right)\right], \quad
g_2^{(p)} = \frac{-0.76/D_V}{(1+Q^2/43 M_V^2)^2} \left[1 - 2.08 \ln\left(1+\frac{Q^2}{1\; \mathrm{GeV}^2}\right) \right],
\\[3mm]
& g_{i}^{(n)}=-g_{i}^{(p)}, \qquad g_{i}^{V}=g_{i}^{(n)}-g_{i}^{(p)},
\\[3mm]
& g_1^A=\frac{-0.51/D_A}{1+Q^2/3M_A^2}, \qquad g_3^A=\frac{(M_R+m_N)m_N}{Q^2+m_\pi^2} g_1^A{}^{(P)},
\end{array}
\end{equation}
\begin{equation}
\begin{array}{ll}
D_{13}(1520): & C_3^{(p)}=\frac{2.95/D_V}{1+Q^2/8.0 M_V^2}, \quad
C_4^{(p)}=\frac{-1.05/D_V}{(1+Q^2/17 M_V^2)^2}, \quad
C_5^{(p)}=\frac{-0.48/D_V}{(1+Q^2/37 M_V^2)^2} .
\\[3mm]
& C_3^{(n)}=\frac{-1.13/D_V}{1+Q^2/8.0 M_V^2}, \quad
C_4^{(n)}=\frac{0.46/D_V}{(1+Q^2/17 M_V^2)^2}, \quad
C_5^{(n)}=\frac{-0.17/D_V}{(1+Q^2/37 M_V^2)^2} ,
\\[3mm]
& C_{i}^{V}=C_{i}^{(n)}-C_{i}^{(p)},
\\[3mm]
& C_3^A=0, \quad C_4^A=0, \quad C_5^A=\frac{-2.1/D_A}{1+Q^2/3M_A^2}, \quad C_6^A=m_N^2 \frac{C_5^A}{m_\pi^2+Q^2},
\end{array}
\end{equation}
\begin{equation}
\begin{array}{ll}
S_{11}(1535): & g_1^{(p)}=\frac{1.87/D_V}{1+Q^2/1.2 M_V^2} \left[1+7.07\ln\left(1+ \frac{Q^2}{1\; \mathrm{GeV}^2}\right) \right] \ , \quad
g_2^{(p)}=\frac{0.64/D_V}{(1+Q^2/17 M_V^2)^2} \left[1 + 1.0 \ln\left(1+\frac{Q^2}{1\; \mathrm{GeV}^2}\right) \right] \ ,
\\[3mm]
& g_{i}^{(n)}=-g_{i}^{(p)}, \qquad g_{i}^{V}=g_{i}^{(n)}-g_{i}^{(p)},
\\[3mm]
& g_1^A=\frac{-0.21/D_A}{1+Q^2/3M_A^2}, \quad g_3^A=\frac{(M_R-m_N)m_N}{Q^2+m_\pi^2} g_1^A \ .
\end{array}
\end{equation}
Here, $D_V=(1+Q^2/M_V^2)^2$ with $M_V=0.84\; \mathrm{GeV}$ and
$D_A=(1+Q^2/M_A^2)^2$ with $M_A=1.05\; \mathrm{GeV}$. The weak form factors
presented here are determined for the excitation of the $R^+$
resonance state, i.e. for neutrino scattering on a neutron. For the
excitation of the double charged states, which is possible for
isospin-3/2 resonances in neutrino--proton scattering, the isospin
relation gives an additional factor $\sqrt{3}$ for each form factor.
For the resonance widths we use the so called running widths $\Gamma_R(W)$, as they were presented in Ref.~\cite{Lalakulich:2006sw}:
\begin{equation}
\Gamma_R(W)=\Gamma_R^0\left( \frac{p_\pi(W)}{p_\pi(M_R)} \right)^{2s_R},
\label{widths}
\end{equation}
where $s_R$ is the spin of the resonance, on--shell widths are $\Gamma_\Delta^0=0.12\; \mathrm{GeV}$,
$\Gamma_{P1440}^0=0.350\; \mathrm{GeV}$, $\Gamma_{D1520}^0=0.125\; \mathrm{GeV}$, $\Gamma_{S1535}^0=0.150\; \mathrm{GeV}$, and
\[
p_{\pi}(W)=\frac1{2W}\sqrt{(W^2-m_N^2-m_\pi^2)^2-4m_N^2 m_\pi^2} \ .
\]
\subsection{Definition of the nuclear structure functions \label{defSF}}
For nuclear targets, the nuclear structure functions ${\cal W}_i^A$
can be defined in the standard manner by means of the expansion of the nuclear hadronic
tensor
\begin{equation}
W^A_{\mu\nu}=
-g_{\mu\nu}{\cal W}_1^A
+ \frac{p_\mu^A p_\nu^A}{M_A^2} {\cal W}_2^A
- i\varepsilon_{\mu\nu\lambda\sigma} \frac{p_A^\lambda q^\sigma}{2 M_A^2} {\cal W}_3^A
+ \frac{q_\mu q_\nu}{M_A^2} {\cal W}_4^A
+ \frac{p_\mu^A q_\nu + p_\nu^A q_\mu}{M_A^2} {\cal W}_5^A \; ,
\label{Wmunu-nucleus}
\end{equation}
where $p^{A, \, \mu}=(M_A, \vec 0)$ is the four--momentum of the
target nucleus with mass $M_A$ in the laboratory frame of reference.
In the impulse approximation we are dealing with the bound nucleon as
a target, so we must relate the one--bound--nucleon structure
functions introduced in the previous section to nuclear ones. We
follow the prescription of Ref.~\cite{Atwood:1972zp} and express the
nuclear structure functions in terms of the nucleon ones in terms of a
convolution of the type
\begin{equation}
W^A_{\mu\nu}= \sum\limits_{\alpha} \int d^3 p \; (2j_\alpha+1) n_\alpha(p) (W_{\mu\nu (\alpha)}^p + W_{\mu\nu (\alpha)}^n) \ ,
\label{WAW}
\end{equation}
where $\alpha$ extends over single--particle shells in the target
nucleus and $2j_\alpha+1$ specifies their occupancies.
It is worth stressing that in the original paper \cite{Atwood:1972zp}
as well as in \cite{Ferree:1995fb} an additional phase--space
correction factor $E_p/m_N$ is introduced in the
expression~(\ref{WAW}) to preserve the space volume under Lorentz
transformation. Since we construct a momentum distribution from wave
functions normalized as $u_\alpha ^\dagger u_\alpha = 1$ for each
shell $\alpha$, our correction factor must be equal to~$1$.
Substituting (\ref{Wmunu-nucleon}) and (\ref{Wmunu-nucleus}) in (\ref{WAW}), one arrives at
\begin{equation}
\begin{array}{l} \displaystyle
{\cal W}_1^A(Q^2,\nu)=\sum\limits_{\alpha} {\cal W}_1^{(\alpha)}(Q^2,\nu)
=\sum\limits_{\alpha} \int d^3 p \; (2j_\alpha+1) n_\alpha(p)
\biggl[ {\cal W}_1(Q^2,\nu, p^2)
+ {\cal W}_2(Q^2,\nu, p^2)\frac{|\vec p|^2 -p_z^2}{m_N^2} \biggr]
\ ,
\\[4mm] \displaystyle
{\cal W}_2^A(Q^2,\nu) =\sum\limits_{\alpha} {\cal W}_2^{(\alpha)}(Q^2,\nu)
=\sum\limits_{\alpha} \int d^3 p \; (2j_\alpha+1) n_\alpha(p)
{\cal W}_2(Q^2,\nu, p^2)
\left[\frac{|\vec p|^2 -p_z^2}{m_N^2}\frac{Q^2}{q_z^2}
+ \left( \frac{(p\cdot q)}{m_N\nu} \right)^2 \left( 1+ \frac{p_z}{q_z}\frac{Q^2}{(p\cdot q)} \right)^2
\right].
\end{array}
\label{calWA}
\end{equation}
This prescription guarantees, that as $Q^2$ tends to zero, the
longitudinal structure function ${\cal W}_L$ also tends to zero as
expected for the real photon:
\begin{equation}
\lim\limits_{Q^2\to 0} \left[ \frac{\nu^2}{Q^2} {\cal W}_2^A (Q^2,
\nu) - {\cal W}_1^A (Q^2, \nu) \right] =0 \; .
\end{equation}
In neutrino experiments one can also measure the ${\cal W}_3$
structure function, for which our definition gives:
\begin{equation}
{\cal W}_3^A(Q^2,\nu)=\sum\limits_{\alpha} \int d^3 p \; (2j_\alpha+1)
n_\alpha(p) {\cal W}_3(Q^2,\nu, p^2) \frac{M_A}{m_N^2} \frac{p^0 q^z -
\nu p^z }{q_z} \; .
\end{equation}
Note that ${\cal W}_3$ depends on the nucleus mass $M_A$. Realizing
that the Bjorken variable for a nucleus ($x_A=Q^2/2 M_A \nu$) differs
from the one for a nucleon ($x=Q^2/2m_N\nu$), the function that is
independent of $M_A$ is $x_A F_3^A$:
\begin{equation}
x_A F_3^A =
\sum\limits_{\alpha} \int d^3 p \; (2j_\alpha+1) n_\alpha(p)
xF_3(Q^2,\nu, p) \frac{1}{m_N} \frac{p^0 q^z - \nu p^z }{q_z} \; .
\end{equation}
According to the definition (\ref{calWA}), $F_2^A=\nu {\cal W}_2^A$ and
$x_A F_1^A = x_A M_A {\cal W}_1^A$ are also independent on $M_A$.
Within the adopted approach there is no unambiguous recipe
for deciding whether one should keep $m_N$ in the denominators
of~(\ref{calWA}) or replace it with some effective mass, that corrects
for the binding energy. For the numerical calculations presented here,
we have opted to use the expression ~(\ref{calWA}) and interpret the
$m_N$ as the free nucleon mass.
The integration over $d^3p=|\vec{p}|^2 \, d|\vec{p}| \; d\cos\gamma_p
\; d\varphi_p$ in Eq.~(\ref{calWA}) is performed in the following
way. Integration over the azimuthal angle $d\varphi_p$ gives $2\pi$,
since no structure function depends on it. The phase space in the
plane determined by the absolute momentum value $|\vec{p}|$ and polar
angle $ \gamma _p$ is restricted by the condition $W^2>W_{min}^2$. For one--pion
production one has that $W_{min}=m_N+m_\pi$. For a a bound nucleon
this condition translates into
\begin{equation}
p_0^2-|\vec{p}|^2+2p_0\nu
-2|\vec{p}|\sqrt{Q^2+\nu^2} \cos\gamma_p - Q^2 > W_{min}^2 \; .
\label{phasespace}
\end{equation}
When performing the $d^3 {p}$ integrations, the above condition
determines the boundaries of the absolute bound-nucleon momentum
$| \vec{p} | $ for a given $\cos \gamma _p$, $Q^2$ and $\nu$:
\begin{equation}
|\vec{p}|_{\pm} = -\sqrt{Q^2+\nu^2}\cos \gamma_p
\pm \sqrt{(Q^2+\nu^2)\cos^2\gamma_p +p_0^2 +2 p_0\nu -Q^2 -W_{min}^2 } \ .
\label{p_pm}
\end{equation}
The sign of the quantity $W_{min}^2 + Q^2 - p_0^2 - 2p_0\nu$
discriminates between two classes of kinematic conditions. In what
follows we provide a discussion of the values of $p_{min}$ and
$p_{max}$ in the phase-space integration $ \int _{p_{min}} ^{p_{max}}
dp $ for a positive and negative sign of $W_{min}^2 + Q^2 - p_0^2 -
2p_0\nu$. For
\begin{equation}
W_{min}^2 + Q^2 - p_0^2 - 2p_0\nu<0 \ ,
\label{also-free}
\end{equation}
the $|\vec{p}|_{-}$ calculated according to ({\ref{p_pm}) is negative, so one
should take $p_{min}(Q^2,\nu,\cos\gamma_p)=0$. This means that the
phase space (\ref{also-free}) is accessible for a nucleon with
arbitrarily small three-momentum, including $|\vec{p}|=0$, as is the
case for a free nucleon. When the condition (\ref{also-free}) is
fullfilled, $p_{max}(Q^2,\nu,\cos\gamma_p) =
|\vec{p}|_{+}(Q^2,\nu,\cos\gamma_p)$ for all polar angles $\gamma_p$.
Increasing the phase space for the bound nucleon does not necessarily
imply that the cross section grows, because each point in the phase
space gets weighted with a momentum distribution of the type shown in
Fig.~\ref{fig:n_alpha}. Cross sections and structure functions for
high $|\vec{p}|$ are strongly suppressed and the major contributions
stem from the momenta inside the Fermi sphere.
For
\begin{equation}
W_{min}^2 + Q^2 - p_0^2 - 2p_0\nu>0 \ ,
\label{bound-only}
\end{equation}
the $|\vec{p}|_{\pm}$ are only defined for backward moving target
nucleons. The restrictions on $\cos\gamma_p$ for given $Q^2$ and $\nu$
come from the condition
\begin{equation}
(Q^2+\nu^2)\cos^2\gamma_p +p_0^2 +2 p_0\nu -Q^2 -W_{min}^2 > 0 \ ,
\end{equation}
which gives
\begin{equation}
-1 < \cos\gamma_p(Q^2,\nu) < -\sqrt{\frac{W_{min}^2 + Q^2 - p_0^2 - 2p_0\nu}{Q^2+\nu^2}} \ .
\label{bound-cosgamma}
\end{equation}
Since the minimal value of the three-momentum $|\vec{p}|_{-}$ is
positive in this case, the accessibility to this $(Q^2,\, \nu)$ region
crucially depends on a nucleon already moving, which is only possible
for a bound nucleon. This region of phase space grows in importance
with increasing $Q^2$.
For $Q^2=0.1\; \mathrm{GeV}^2$ and different $\nu$, the typical phase spaces
available for a $1s^{1/2}$ proton in carbon are shown in
Fig.~\ref{fig:phasespace}. We use polar coordinates for the variables
$|\vec{p}| $ and $ \gamma_p$. The left (right) panel corresponds with
the condition (\ref{bound-only}) ((\ref{also-free})). For each $Q^2$
and $\nu$, thick lines represent $|\vec{p}|_{+}$ and thin lines
$|\vec{p}|_{-}$. The points where the $|\vec{p}|_+$ and $|\vec{p}|_- $
lines coincide correspond to the upper boundary on $\cos\gamma_p$, as
calculated in Eq.(\ref{bound-cosgamma}). Remark that the available
phase space in ($|\vec{p}|, \cos \gamma_p)$ is contained within a circle. At
$\nu=0.4\; \mathrm{GeV}$ the left part of the circle is not shown because it
corresponds to values of $|\vec{p}|$ larger than 1 GeV. The momentum
distribution of nucleons in nuclei will reduce those contributions to
negligible proportions.
\begin{figure}[hbt]
\epsfig{figure=pmax-Q201-cosgamma-bound.ps,angle=-90,width=0.49\textwidth}
\hfill
\epsfig{figure=pmax-Q201-cosgamma-free.ps,angle=-90,width=0.49\textwidth}
\caption{Sketch of the available $(|\vec{p}| \, ,\gamma_p)$ phase space
in polar coordinates. We consider a $1s^{1/2}$ proton in carbon for
$Q^2=0.1 \; \mathrm{GeV}^2$ and different $\nu$. The $ |\vec{p} | $ is
expressed in GeV. The left (right) panel corresponds with kinematics
conditions obeying the condition of Eq.~(\ref{bound-only}) (of Eq.~(\ref{also-free})).}
\label{fig:phasespace}
\end{figure}
The phase space collapses to one point $\cos\gamma_p=-1$,
$|\vec{p|}=\sqrt{Q^2+\nu_{min}^2}$ for
$\nu_{min}=-p_0+W_{min}$. Remark that for a bound nucleon the minimal
value of $\nu$ does not depend on $Q^2$. Physically this means that for any $Q^2$
there is a bound nucleon moving backward fast enough to fulfill the
requirement $(p+q)^2>W_{min}^2$. Thus, contrary to the free nucleon
case, for the off-shell nucleon the pion production threshold is
defined in terms of $\nu$ rather than invariant mass and, strictly
speaking, is independent of $Q^2$.
At high $Q^2$, however, using $\nu_{min}=-p_0+W_{min}$ is not
convenient for calculations, because all observables are strongly
suppressed for large $\vec{p}$ inspite of the fact that the phase
space is available. In our numerical calculations we have not
considered nucleon momenta beyond three times the Fermi momentum.
We stress that the phase space boundaries derived here depend
on our assumption about the form of the four--momentum for the bound
nucleon, which was taken as $p^\mu=(m_N-e_{\alpha}, \vec{p})$.
\subsection{DIS region and scaling variable \label{DIS}}
In the kinematical regime of high $Q^2$ and $\nu$, the so-called
Bjorken limit, the
structure functions depend only on the Bjorken variable
$x=Q^2/2m_N\nu$ when one neglects higher-twist effects. This phenomenon of no observed $Q^2$ dependence for a
fixed $x$ value is called Bjorken scaling. At these energies, the
lepton scattering on nucleons and nuclei is dominated by deep
inelastic scattering with a multiple--particle hadronic final state.
Deep inelastic scattering on nuclei was intensively studied
experimentally since the sixties. This experimental information will be
used as DIS input for our investigation.
For electron--carbon scattering, $F_2$ was measured by the BCDMS
Collaboration~\cite{Bollini:1981cr,Benvenuti:1987zj} for
$30\; \mathrm{GeV}^2<Q^2<200\; \mathrm{GeV}^2$. We choose several sets of data at different
$Q^2=Q^2_{DIS}$: $30, \, 45$ and $50 \; \mathrm{GeV}^2$. As expected from Bjorken
scaling, for most of the $x$ region the data coincide with an accuracy
better than $5\%$. For iron, the neutrino scattering results are
available from the CCFR~\cite{Seligman:1997mc} and
NuTeV~\cite{Tzanov:2005kr} collaborations.
Scaling structure functions are conventionally plotted against the
Bjorken variable $x$. Violation of Bjorken scaling comes from
target-mass corrections and higher-twist effects. In the scaling
region, the Nachtmann variable $\xi= 2x/(1+\sqrt{1 + 4 m_N^2
x^2/Q^2})$ was shown \cite{Nachtmann:1973mr,Georgi:1976ve} to be a
better alternative, because it implicitly includes the kinematical
part of the target-mass correction, which can be important at large
$x$ and low $Q^2$. Expanding the inverse of this variable in a power
series of $1/Q^2$, we recover the variable $1/\xi\approx \omega'=(2m_N
\nu + m_N^2)/Q^2$, used by Bloom and Gilman in their pioneering work
on duality. For large $\nu$, one has $\omega'\approx 1/x$.
\section{Duality in electroproduction \label{electrons}}
In the case of an isoscalar target nucleon, and for $Q^2>0.5\; \mathrm{GeV}^2$,
it was shown \cite{Lalakulich:2006yn} that Bloom--Gilman duality holds
at the level of $20\%$.
Here, we compute the nuclear structure functions $F_2^A$ and $x_AF_1^A$
along the model outlined in Section II. The results of our
description of the resonance region in terms of hadronic degrees of
freedom are then compared to DIS data.
In Figure~\ref{fig:C12-em}, the bound--nucleon structure functions
$F_2$ and $2x F_1$ for a proton in the $1s^{1/2}$ and $1p^{3/2}$
carbon shells are contrasted with the structure functions for a free
proton. They are plotted versus the Nachtmann variable $\xi$ for
$Q^2=0.2, \, 0.85$, and $2.4 \; \mathrm{GeV}^2$, with the largest $Q^2$ curves
covering the largest $\xi$ values. Similar to the free--nucleon case,
for a fixed $Q^2$, the peak at larger $\xi$ corresponds to the
$\Delta$ resonance and the peak at smaller $\xi$ corresponds to the
second resonance region. One can easily notice the effect of smearing:
the two resonance regions are distinguishable only at low $Q^2$. Fermi
smearing proceeds differently for different shells, which in turn
introduces an additional averaging when summing over shells.
One can also observe that the bound--nucleon curves extend to higher
$\xi$ values than the free nucleon ones. This additional contribution
comes from the phase space at low $\nu$ values, which is shown in the
left panel of Fig.~\ref{fig:phasespace} and discussed in
Section~\ref{defSF}. At high $\xi$ the $F_2$ and $2 x F_1$ for the $1s^{1/2}$
shell are significantly lower than for the $1p^{3/2}$ shell. At high $\xi$
the phase space extends to relatively large bound-nucleon momenta.
For those momenta the momentum
distribution for a shell close to the Fermi surface (like $1p^{3/2}$
in carbon) is larger than for a deep-lying shell (like $1s^{1/2}$ in
carbon).
\begin{figure}[htb]
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-Na-C12-em-sp.ps,angle=-90,width=\textwidth}
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=cal2xF1-Na-C12-em-sp.ps,angle=-90,width=\textwidth}
\end{minipage}
\caption{(color online) Structure functions $F_2$ (left) and $2xF_1$ (right) for a free proton (solid curve), for $1s^{1/2}$ (dashed curve) and $1p^{3/2}$ (dash--dotted curve) protons in ${}^{12}$C. The three sets of curves correspond to $Q^2=0.2$, $0.85$, and $2.4\; \mathrm{GeV}^2$}
\label{fig:C12-em}
\end{figure}
Fig.~\ref{fig:F2-C12-em} shows the carbon structure function per
nucleon $F_2^{e{}^{12} C}/A$ in the resonance region for several $Q^2$
values, from $0.45$ to $3.3 \; {\mathrm{GeV}}^2$. When investigating
duality for a free nucleon, we took the average over free proton and
neutron targets, thus considering the isoscalar structure
function. Since the carbon nucleus contains an equal number of protons
and neutrons, averaging over isospin is performed automatically. At
$Q^2=0.45 \; \mathrm{GeV}^2$, the $\Delta$ peak is pronounced and can still be
distinguished from the second resonance peak, which is also
visible. At higher $Q^2$ one cannot distinguish the resonance
structure anymore and the first and second resonance region merge into
one broad peak.
\begin{widetext}
\begin{figure}[h!bt]
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-Na-C12-em.ps,angle=-90,width=\textwidth}
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-int-C12-em.ps,angle=-90,width=\textwidth}
\end{minipage}
\caption{(color online) Duality for the $F_2^{e {}^{12}C}$
structure function. (Left) Resonance curves $F_2^{e {}^{12}C}/12$ as
a function of $\xi$, for $Q^2 = 0.45, 0.85, 1.4, 2.4$ and $3.3\;
{\mathrm{GeV}}^2$ (indicated on the spectra), compared with the
experimental data \cite{Bollini:1981cr,Benvenuti:1987zj} in the DIS
region at $Q^2_{DIS}=30$, $45$ and $50\; \mathrm{GeV}^2$. (Right) Ratio $I_2$
defined in Eq.(\ref{eq:Int}) for the free nucleon (dash-dotted line),
and $^{12}$c. We consider the under limits determined by
$\tilde{W}=1.1 \; \mathrm{GeV}$ (solid line) and by the threshold value (dotted
line).}
\label{fig:F2-C12-em}
\end{figure}
\end{widetext}
In the left panel of Fig.~\ref{fig:F2-C12-em}, the resonance structure
functions are compared with data obtained by the BCDMS
Collaboration~\cite{Bollini:1981cr,Benvenuti:1987zj} in muon--carbon
scattering in the DIS region ($Q^2\sim 30-50 \;
{\mathrm{GeV}}^2$). They are shown as experimental points connected by
smooth curves. For different $Q^2$ values, the curves agree within
$5\%$ in most of the $\xi$ region, as expected from Bjorken
scaling. One observes that, as $Q^2$ increases, the resonance peaks
decrease in height and slide along the DIS curve. This means that
global duality holds for electron scattering on nuclei. To
characterize local duality, we consider the ratio of the integrals of
the resonance (res) and DIS structure functions
\begin{equation}
I_i(Q^2) =
\frac{ \int_{\xi_{\rm min}}^{\xi_{\rm max}} d\xi\
{\cal F}_i^{(\rm res)}(\xi,Q^2) }
{ \int_{\xi_{\rm min}}^{\xi_{\rm max}} d\xi\
{\cal F}_i^{(\rm DIS)}(\xi,Q^2_{DIS}) }\ ,
\label{eq:Int}
\end{equation}
where ${\cal F}_i$ denotes $F_2^A$ or $x_A F_3^A$ (used later for neutrino scattering). The value $Q^2_{DIS}$ is
taken as the actual $Q^2$ value for a given experimental data set. For
electron--carbon scattering we choose the data set
\cite{Benvenuti:1987zj} at $Q^2_{DIS}=50 \; \mathrm{GeV}^2$, because it covers
most of the $\xi$ region.
For a proton target \cite{Niculescu:2000tk}, the integration limits for
$\xi$ are conventionally chosen equal for both integrals and are
defined in such a way as to cover the first and second resonance
regions for each $Q^2$. For a free nucleon, this requirement is
written as \cite{Lalakulich:2006yn}
\begin{equation}
\xi_{\rm min}^{N} = \xi(W=1.6~{\rm GeV},\, Q^2), \qquad
\xi_{\rm max}^{N} = \xi(W=1.1~{\rm GeV},\, Q^2),
\label{11-16}
\end{equation}
where the invariant mass for a free nucleon can be expressed in terms
of $\nu$ and $Q^2$ as $W^2=(p+q)^2=m_N^2 +2m_N\nu -Q^2$. The upper
value $W=1.6 \; \mathrm{GeV}$ is chosen in such a way as to cover the mass range
of the four resonances taken into account, the heaviest one with the
mass $M_R=1.535 \; \mathrm{GeV}$. The lower value $W=1.1 \; \mathrm{GeV}$ is chosen close to
the pion--production threshold $W_{thr}=1.08\; \mathrm{GeV}$.
In a nuclear target, the invariant mass of the struck nucleon
depends on the initial momentum of the target nucleon. On the other
hand, the structure functions, as well as other observables, are
defined as integrals over the initial nucleon momentum. This prevents
one from using $W$ in defining the integration limits. One needs an
alternative variable, which can be easily determined from the lepton
kinematics.
Experimentally one often (see, for example, \cite{Sealock:1989nx})
``defines'' the effective variable $\tilde{W}$ by the relation
$\tilde{W}^2=m_N^2+2m_N\nu-Q^2$. Notice that $\tilde{W}$ is only an
invariant for $\vec{p}=0$. However, it gives a reasonable
feeling of the invariant mass region involved in the problem. In
particular, the resonance curves presented in all figures are plotted in the
region from the pion--production threshold up to $\tilde{W}=2\; \mathrm{GeV}$.
As was illustrated in Fig.~\ref{fig:phasespace}, bound backward-moving
nucleons allow lower $\nu$ values beyond the free--nucleon limits.
Thus, as discussed at the end of Secton~\ref{defSF}, the
threshold for the structure functions is now defined in terms of $\nu$
or $\tilde{W}$, rather than $W$.
Hence, we consider two different cases in choosing the
$\xi$ integration limits for the ratio (\ref{eq:Int}). First, for a
given $Q^2$, we choose the $\xi$ limits as in
Eq.~(\ref{11-16}). That amounts to defining them by the condition
\begin{equation}
\xi_{\rm min} = \xi(\tilde{W}=1.6~{\rm GeV},\, Q^2), \qquad
\xi_{\rm max} = \xi(\tilde{W}=1.1~{\rm GeV},\, Q^2) \ .
\label{11-16-}
\end{equation}
We refer to this choice as integrating ``from 1.1 GeV''. The integration
limits for the DIS curve always correspond to this choice. As a
second choice, for
each $Q^2$ we integrate the resonance curve from the threshold, that
is from as low $\tilde{W}$ as achievable for the nucleus under
consideration. This corresponds to the threshold value at higher $\xi$
and is referred to as integrating ``from threshold''. With this choice we
guarantee that the extended kinematical regions typical for resonance
production from nuclei are taken into account.
Since there is no natural threshold for the $\xi_{min}$, for
both choices it is estimated from $\tilde{W}=1.6\; \mathrm{GeV}$, as defined in
Eq.~(\ref{11-16-}).
The results for the ratio in Eq.~(\ref{eq:Int}) are shown in the right
panel of Fig.~\ref{fig:F2-C12-em}. The curve for the isoscalar
free-nucleon case is the same as in Ref.~\cite{Lalakulich:2006yn} with
the ``GRV'' parameterization for the DIS structure function. One can
see that the carbon curve obtained by integrating ``from threshold''
lies above the one obtained by integrating ``from 1.1 GeV'', the
difference increasing with $Q^2$. This indicates that the threshold
region becomes more and more significant, as one can see from
Fig.~\ref{fig:C12-em}.
The closer the ratio (\ref{eq:Int}) gets to 1, the higher the accuracy
of local duality is. Our calculations for a carbon target show that:
1) the ratio grows with $Q^2$, just like in the isoscalar free-nucleon
case; 2) the ratio is lower than the free-nucleon value for both
choices of the integration limits. This means that the integrated
resonance contribution is always smaller than the integrated DIS one.
\begin{figure}[h!b]
\epsfig{figure=calF2-Na-C12-em-DIS.ps,angle=-90,width=0.55\textwidth}
\caption{Electromagnetic structure functions $F_2$ in DIS
region for a free isoscalar nucleon as obtained via the GRV
parameterization at $Q^2=10\; \mathrm{GeV}^2$ (solid curve) and for a
carbon nucleus as measured experimentally at
$Q^2=30, \, 45$ and $50\; \mathrm{GeV}^2$.}
\label{fig:F2-C-em-DIS}
\end{figure}
In search for an explanation for this discrepancy, we compare how
nuclear effects influence the resonance and DIS curves. As it was
illlustrated in Fig.~\ref{fig:C12-em}, the nuclear effects suppress
the resonance peaks by 40\%-50\%, broaden them and shift them to lower
$\xi$ values. The experimental DIS values for the carbon nucleus, on
the other hand, are only $5\%-10\%$ lower than the DIS curve for the
free isoscalar nucleon. This is illustrated in
Fig.~\ref{fig:F2-C-em-DIS}, where the DIS structure function $F_2$ for
a carbon target is compared to the GRV parameterization for the free
isoscalar nucleon at $Q^2=10\; \mathrm{GeV}^2$. In conclusion, one can say that
nuclear effects have a much more dramatic effect in the resonance
region than in the DIS regime.
Similar calculations can be done for other nuclei. First of all, it
would be interesting to compare an isoscalar nucleus with a nucleus
with neutron excess. We show the structure functions $(F_2^{e\,
{}^{56}\! Fe}/56)$ and $(F_2^{e\, {}^{52}\! Fe}/52)$ versus
$\tilde{W}$ for several $Q^2$ values in the left panel of
Fig.~\ref{fig:F2-FeC-em}. The structure functions for $^{52}$Fe are
only marginally higher than those for $^{56}$Fe. This can be explained
by the fact that the electromagnetic $\Delta$-production cross section
is equal for proton and neutron targets. In the second resonance
region, the cross sections on the proton are typically $5\%-30\%$
higher than those for the neutron. The overall effect, however, is
hardly visible for an excess of 4 neutrons out of 56 nucleons.
\begin{figure}[h!b]
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-ratio-W-em-Fe2Fe-.ps,angle=-90,width=\textwidth}
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-x-ratio-em.ps,angle=-90,width=\textwidth}
\end{minipage}
\caption{(color online)
(Left) Structure functions $F_2$ for electroproduction on iron-52 and iron-56 versus $\tilde{W}$. Curves are for $Q^2 = 0.2, 0.45, 0.85, 1.4$, and $2.4 \; \mathrm{GeV}^2$ (indicated on the spectra).
(Left) Ratio $(F_2^{e {}^{56}Fe}/56)/(F_2^{e {}^{12}C}/12)$ versus Bjorken variable $x$ compared to DIS data \cite{Arneodo:1996rv}.
}
\label{fig:F2-FeC-em}
\end{figure}
From an experimental point of view, it is also interesting to compare
carbon with iron target nuclei. In the right panel of
Fig.~\ref{fig:F2-FeC-em}, we plot the ratio of structure functions
$(F_2^{e {}^{56}Fe}/56)/(F_2^{e {}^{12}C}/12)$ versus $x$ for several
values of $Q^2$ ranging from $0.2\; \mathrm{GeV}^2$ to $3.3\; \mathrm{GeV}^2$. All curves
are shown in the $\xi$ region corresponding to $1.1\; \mathrm{GeV}<\tilde{W}<2.0
\; \mathrm{GeV}$.
We stress that there is little
physical meaning in the fine structure of the curves in the right
panel of Fig.~\ref{fig:F2-FeC-em}. The peaks in the curves,
for example, do not coincide with the resonance peaks. As one can see,
the iron structure functions appear to be very close to the carbon
ones: for each $Q^2$ the ratio of the iron to carbon structure
functions does not deviate more than $5\%$ from the value of $1$. When
averaged, this ratio slightly decreases with increasing $Q^2$, a
behavior which is also exhibited by the DIS data presented in the same
figure. The latter were measured by the NMC Collaboration
\cite{Arneodo:1996rv}, the mean $Q^2$ in the experiment varying from
$20\; \mathrm{GeV}^2$ for $x \sim 0.1$ to $60 \; \mathrm{GeV}^2$ for $x \to 1$.
\section{Duality in neutrinoproduction \label{neutrinos}}
In a previous paper \cite{Lalakulich:2006yn} it was demonstrated that
in neutrino reactions quark--hadron duality does not hold for proton
and neutron targets separately. This is a principle feature of
neutrino interactions, stemming from fundamental isospin arguments.
For the charged current reaction $\nu_\mu \, p \to \mu^- \, R^{++}$,
only isospin-3/2 $R^{++}$ resonances are excited, in particular the
$P_{33}(1232)$ resonance. Because of isospin symmetry constraints, the
neutrino-proton structure functions for these resonances are three
times larger than the neutrino-neutron ones. In neutrino-neutron
scattering, both the isospin-3/2 resonances and the isospin-1/2
resonances contribute to the structure functions. The interplay
between the resonances of different isospins allows for duality to
hold with reasonable accuracy for the average over the proton and
neutron targets. It appears reasonable that one may expect a similar
picture to emerge in neutrino reactions with nuclei.
\begin{figure}[htb]
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-Na-Fe56-weak.ps,angle=-90,width=\textwidth}
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calxF3-Na-Fe56-weak.ps,angle=-90,width=\textwidth}
\end{minipage}
\caption{ (color online) The computed resonance curves $F_2^{\nu \,
{}^{56}\! Fe}/56$ and $x_{Fe} F_3^{\nu \, {}^{56}\! Fe}/56$ as a function of $\xi$,
for $Q^2 = 0.2, 0.45, 0.85$, $1.4$, and $2.4\; {\mathrm{GeV}}^2$. The
calculations are compared with the DIS data from
Refs.~\cite{Seligman:1997mc,Tzanov:2005kr}. The DIS data refer to
measurements at $Q^2_{DIS}=7.94$, $12.6$ and $19.95 \; \mathrm{GeV}^2$.
}
\label{fig:Fe56-nu}
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=F2-int-Fe56.ps,angle=-90,width=\textwidth}
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=xF3-int-Fe56.ps,angle=-90,width=\textwidth}
\end{minipage}
\caption{ (color online) Ratios $I_2^{\nu \, {}^{56}Fe}$ and $I_3^{\nu
\, {}^{56}Fe}$ defined in Eq.~(\ref{eq:Int}) for the free nucleon
(dash-dotted line) and $^{56}$Fe. For $^{56}$Fe the results are
displayed for two choices of the underlimit in the integral:
$\tilde{W}=1.1 \; \mathrm{GeV}$ (solid line) and threshold (dotted line). For
each of these two choices we have used two sets of DIS data in
determining the denominator of Eq.~(\ref{eq:Int}). These sets of
DIS data are obtained at $Q^2_{DIS}=12.59$ and 19.95 GeV$^2$. }
\label{fig:I-Fe56-nu}
\end{figure}
The structure functions $F_2^A$ and $x_A F_3^A$ for neutrino--iron
scattering are shown in Fig.~\ref{fig:Fe56-nu}.
The curves for the
isoscalar free nucleon case is identical to the one presented in
Ref.~\cite{Lalakulich:2006yn} with the ``fast'' fall--off
of the axial form factors for the isospin-1/2 resonances.
Like for the electron-carbon results of
Fig.~\ref{fig:C12-em}, the resonance structure is hardly
visible. Indeed for each $Q^2$ the computed resonance curves display
one broad peak. The resonance structure functions are compared with
the experimental data in DIS region obtained by the CCFR
\cite{Seligman:1997mc} and NuTeV \cite{Tzanov:2005kr}
collaborations. It appears that the resonance curves slide along the
DIS curve, which indicates global duality. Like for the electron
results discussed in previous section, however, the resonance $F_2^A$
and $x_A F_3^A$ predictions are noticeably lower than the DIS
measurements.
The ratios $I_2^{\nu \, {}^{56}\! Fe}$ and $I_3^{\nu \, {}^{56}\! Fe}$
defined in Eq.(\ref{eq:Int}) are shown in
Fig.~\ref{fig:I-Fe56-nu}. Our results show, that 1) these ratios are
significantly smaller than 1; 2) they are significantly smaller than
the one for the free nucleon ; 3) $I_2$ is lower than the
corresponding ratio for electroproduction; 4) $I_2$ and $I_3$
slightly decrease with $Q^2$ which is the opposite behavior of what
was observed for electrons.
\begin{figure}[htb]
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-Na-Fe56-weak-sp.ps,angle=-90,width=\textwidth}
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-Na-Fe56-DIS.ps,angle=-90,width=\textwidth}
\end{minipage}
\caption{ (color online)
(Left) Weak structure functions $F_2$ in the resonance region for a free neutron (solid curve), for $1s^{1/2}$ (dashed curve), $1p^{3/2}$ (dash--dotted curve), $1d^{5/2}$ (long-dashed curve) and $1f^{7/2}$ (short-dashed curve) neutrons in ${}^{56}Fe$. The three sets of curves correspond to $Q^2=0.2$, $0.85$, and $2.4\; \mathrm{GeV}^2$.
(Right) Weak structure functions $F_2$ in DIS region for a free isoscalar nucleon as obtained via GRV parameterization at $Q^2=10\; \mathrm{GeV}^2$ (solid curve) and for iron-56 nucleus as measured experimentally \cite{Seligman:1997mc,Tzanov:2005kr} at $Q^2=7.94$, $12.59$ and $19.95 \; \mathrm{GeV}^2$.
}
\label{fig:Fe56-free}
\end{figure}
In an attempt to explain the above observations, we compare the free
isoscalar structure functions with the $^{56}$Fe ones. In the left
panel of Fig.~\ref{fig:Fe56-free} the structure function $F_2^{A}$ for
a neutron in the $1s^{1/2}$, $1p^{3/2}$, $1d^{5/2}$ and $1f^{7/2}$
iron shells are contrasted with the structure function for a free
neutron. For a fixed $Q^2$, the peak at the larger value of the
Nachtmann variable corresponds to the $\Delta$ resonance. The peak at
smaller $\xi$ corresponds to the second resonance region. It is clear
that the nuclear effects reduce the peaks by about $30-50\%$ and shift
them to lower $\xi$ values in comparison with the free nucleon
case. The suppression is most significant for the single-particle
shells close to the Fermi surface.
In
close resemblance to what was observed in the discussion of the
electron-nucleus cross sections of previous section, the peculiar
Fermi smearing pattern for each shell introduces additional averaging
when summing over shells. For a bound proton, the effect of
suppression is nearly the same. In the right panel of
Fig.~\ref{fig:Fe56-free}, measured DIS structure functions for
$^{56}$Fe at various $Q^2$ are compared to the GRV parameterization
($Q^2=10\; \mathrm{GeV}^2$) for a free isoscalar nucleon
\cite{Lalakulich:2006yn}. It is obvious that the measured nuclear DIS
structure functions are very similar to the free-nucleon ones. Thus, we
predict a substantial nuclear reduction of the resonance strength,
whereas the data in the DIS region do not point to such a reduction.
This explains the computed low values of the ratios in
Fig.~\ref{fig:I-Fe56-nu}.
We wish to stress that the low values of $I_2^{\nu \, {}^{56}Fe}$ and
$I_3^{\nu \, {}^{56}Fe}$ are not related to the neutron excess. We
remind that the neutron structure functions for the $\Delta$ resonance
are 3 times smaller than the proton ones. The structure
function for isoscalar $^{52}$Fe is only about 5\% larger than for
$^{56}$Fe. This is shown in Fig.~\ref{fig:F23-Fe2Fe-nu} for $F_2$ and
$x_A F_3$. The effect can be easily estimated from $(26\cdot 3\cdot
f+30\cdot f)/(26\cdot 3\cdot f+26\cdot f)\approx 1.04$, where $f$ is
the neutron structure function in the $\Delta$ region. In the second
resonance region the difference is even smaller.
\begin{figure}[htb]
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-ratio-W-nu-Fe2Fe-.ps,angle=-90,width=\textwidth}
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calxF3-ratio-W-nu-Fe2Fe-.ps,angle=-90,width=\textwidth}
\end{minipage}
\caption{(color online) Structure functions $F_2^A$ (left) and
$x_A F_3^A/A$ (right) for neutrinoproduction on iron-52 and
iron-56 versus $\tilde{W}$. Curves in the resonance region are
for $Q^2 = 0.2, 0.45, 0.85, 1.4$, and $2.4 \; \mathrm{GeV}^2$ (indicated
on the spectra).
}
\label{fig:F23-Fe2Fe-nu}
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calF2-x-ratio-nu.ps,angle=-90,width=\textwidth}
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\textwidth}
\epsfig{figure=calxF3-x-ratio-nu.ps,angle=-90,width=\textwidth}
\end{minipage}
\caption{(color online) Ratios $(F_2^{\nu \, {}^{56}\! Fe}/56)/(F_2^{\nu \, {}^{12}\! C}/12)$ (left)
and $(x_{{}^{56}Fe} F_3^{\nu\, {}^{56}\! Fe/56)}/(x_{{}^{52}\! Fe}F_3^{\nu \, {}^{12}C}/12)$ (right)
versus Bjorken variable $x$
for $Q^2 =0.2, 0.45, 0.85, 1.4, 2.4$ and $3.3\; {\mathrm{GeV}}^2$
}
\label{fig:F23-FeC-nu}
\end{figure}
It is also interesting to make a comparison with the carbon
nucleus. The ratios of iron to carbon structure functions $F_2^A$ and
$x_A F_3^A$ versus $x$ are shown in Fig.~\ref{fig:F23-FeC-nu}. For each
$Q^2$ the $\xi$ range corresponds to $1.1<\tilde{W}<2.0 \; \mathrm{GeV}$. Like
in the case of electromagnetic reaction, the ratios are close to $1$,
but a bit lower in general and the average is slightly increasing with
$Q^2$. Remark that the peaks in Fig.~\ref{fig:F23-FeC-nu} are not
related to resonances and that the fluctuations which are of the
order of 5\% can be attributed to subtleties in the shell structure of
the various target nuclei.
\section{Summary \label{summary} }
In view of the current experimental activities, there is great need
for an efficient framework for reliably predicting neutrino--nucleus
cross sections and for a deeper understanding of quark--hadron duality
in nuclei. We performed a phenomenological study of duality in
electron-nucleus and neutrino-nucleus structure functions.
Using the Dortmund-group model for the production of the {f}{i}rst four
lowest-lying nucleon resonances and using single-particle
wavefunctions from the Hartree approximation to the relativistic
$\sigma \omega$ model, we computed the structure functions $x_A
F_1^A$, $F_2^A$ and
$x_A F_3^A$ in the resonance region for carbon and iron targets and
compared them with the measured DIS ones. At the same time we
compared the computed resonance structure functions for nuclei with
those for a free nucleon. For quantitative comparisons, we de{f}{i}ned
the ratios $I_{i}(Q^2)$ of integrated resonance to DIS structure
functions. Perfect quark-hadron duality is reached for $I_{i}(Q^2)$
values of unity.
Summarizing our results, we observe that the computed resonance contribution to
the lepton--nucleus structure functions is qualitatively consistent with the
measured DIS structure functions. This means that global quark--hadron
duality holds for nuclei. The
computed integrated resonance strength, however, is about half of the
measured DIS one. Contrary to the free nucleon case, where the
ratios $I_{i}(Q^2)$ are at the level of $0.8$, we find for nuclei
$0.6$ for electroproduction and $0.4$ for neutrinoproduction. This
points towards a scale dependence in the role of the nuclear
effects. It is obvious that nuclear effects act differently at lower
$Q^2$ (resonance regime) than at higher $Q^2$ (DIS regime).
In our presented analysis we include the resonance contributions and
ignored the role of the background terms. Further investigations
require a theoretical or phenomenological model for the background
contributions in the first and second resonance region. One could for
example estimate the role of the background contribution to the
$\Delta$--resonance region within the context of the non-linear sigma
model \cite{Hernandez:2007qq}. Extending these or similar models to
higher $W$ values and incorporating them in a model for lepton
reactions with nuclei could be the next step in exploring
quark--hadron duality.
\acknowledgements
The authors acknowledge financial support from the Research Foundation - Flanders (FWO),
and the Research Council of Ghent University.
\bibliographystyle{apsrev}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,173
|
{"url":"https:\/\/q-and-answers.com\/mathematics\/question17163820","text":"# The bearing of Mr Fosu from Mr Woode is 349\u00b0. Find the bearing of Mr Woode from Mr Fosu.\n\nrscott400 \u00a0\u00a0\u00b7\u00a0\u00a0 12.08.2020 04:01\n01.07.2019 00:20\n\n$$y=85(\\frac{3}{4})^x$$\n\nexplanation:\n\nsince we have given that\n\ncost of a new textbook = \\$85\n\naccording to question, the resale value of a textbook decreases by 25% with each previous owner,\n\nso, our function becomes,\n\n$$y=85(1-\\frac{25}{100})^x\\\\\\\\y=85(\\frac{100-25}{100})^x\\\\\\\\y=85(\\frac{75}{100})^x\\\\\\\\y=85(\\frac{3}{4})^x$$\n\nhere, y is the total value of the text book after x owners.\n\nhence, function will be\n\n$$y=85(\\frac{3}{4})^x$$\n\n29.06.2019 21:00\n14 estimated and 13.62 exacti\u2019m pretty sure..\n28.06.2019 08:30\n\nanswer: the resulting cross section is a rectangle. solution: a plane parallel to the base of a triangular prism will intersect a cross section that is the same shape as its bases. so the cross section is a triangle.\n\nstep-by-step explanation:\n\n28.06.2019 08:00\n\noof\n\nstep-by-step explanation:\n\n### Other questions on the subject: Mathematics\n\nWhat is the range of this function? all real numbers such that y \u2264 40 all real numbers such that y \u2265 0 all real numbers such that 0 \u2264 y \u2264 40 all real numbers such that 37.75 \u2264 y \u2264...\nMathematics\n21.06.2019 12:30\nFormulate the indicated conclusion in nontechnical terms. be sure to address the original claim. the foundation chair for a hospital claims that the mean number of filled overnight...\nMathematics\n21.06.2019 20:40\nAcarton of juice contains 64 ounces miss wilson bought six cartons of juice how many ounces of juice did she buy...\nMathematics\n21.06.2019 23:00\nPlzz ! if you were constructing a triangular frame, and you had wood in the length of 4 inches, 4 inches, and 7 inches, would it make a triangle? would you be able to create a f...\nMathematics\n22.06.2019 01:00\nWhat is the solution to the system of equations? y=1.5-3 y=-x...\nMathematics\n22.06.2019 01:30\n5cakes cost 3.50 how much do 7 cakes cost...\nMathematics\n22.06.2019 01:30\nIf point a= (10,4) and b= (2,19) what is the length of ab 17 units 15 units 23 units 12 units...\nMathematics\n22.06.2019 05:00\nThe figure shows triangle abc with medians a f, bd, and ce. segment a f is extended to h in such a way that segment gh is congruent to segment ag. triangle abc with medians ce, a f...\nMathematics\n22.06.2019 05:40\nGraph a sine function whose amplitude is 3, period is 4\u03c0 , midline is y = 2, and y-intercept is (0,\u20092) . the graph is not a reflection of the parent function over the x-axis. use t...\nMathematics\n22.06.2019 06:00","date":"2023-03-21 05:36:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4919028580188751, \"perplexity\": 1903.364901147502}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296943625.81\/warc\/CC-MAIN-20230321033306-20230321063306-00310.warc.gz\"}"}
| null | null |
package org.basex.query.func.fn;
import org.basex.query.*;
import org.basex.query.expr.*;
import org.basex.query.util.*;
import org.basex.query.util.list.*;
import org.basex.query.value.item.*;
import org.basex.query.value.node.*;
import org.basex.query.value.type.*;
import org.basex.query.var.*;
import org.basex.util.*;
import org.basex.util.hash.*;
/**
* Runtime expression, created by non-deterministic functions.
*
* @author BaseX Team 2005-15, BSD License
* @author Christian Gruen
*/
abstract class RuntimeExpr extends ParseExpr {
/** Arguments. */
Var[] params;
/**
* Creates a new function item containing this expression as body.
* @param expr expression
* @param args number of arguments
* @param sc static context
* @param qc query context
* @return function item
*/
static FuncItem funcItem(final RuntimeExpr expr, final int args,
final StaticContext sc, final QueryContext qc) {
final VarScope vsc = new VarScope(sc);
final Var[] params = new Var[args];
for(int p = 0; p < args; p++) params[p] = vsc.newLocal(qc, null, null, true);
expr.params = params;
return new FuncItem(sc, new AnnList(), null, expr.params, FuncType.ANY_FUN, expr,
qc.value, qc.pos, qc.size, args);
}
/**
* Constructor.
* @param info input info
*/
protected RuntimeExpr(final InputInfo info) {
super(info);
}
@Override
public void checkUp() throws QueryException {
throw Util.notExpected();
}
@Override
public Expr compile(final QueryContext qc, final VarScope scp) throws QueryException {
throw Util.notExpected();
}
@Override
public boolean has(final Flag flag) {
throw Util.notExpected();
}
@Override
public boolean removable(final Var var) {
throw Util.notExpected();
}
@Override
public VarUsage count(final Var var) {
throw Util.notExpected();
}
@Override
public Expr inline(final QueryContext qc, final VarScope scp, final Var var, final Expr ex)
throws QueryException {
throw Util.notExpected();
}
@Override
public Expr copy(final QueryContext qc, final VarScope scp, final IntObjMap<Var> vs) {
throw Util.notExpected();
}
@Override
public boolean accept(final ASTVisitor visitor) {
throw Util.notExpected();
}
@Override
public int exprSize() {
throw Util.notExpected();
}
@Override
public void plan(final FElem e) {
throw Util.notExpected();
}
@Override
public String toString() {
return "Runtime function";
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,418
|
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