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# Dirichlet problem associated with Dunkl Laplacian on $W$-invariant open sets
Mohamed Ben Chrouda
Department of Mathematics, High Institute of Informatics and Mathematics
5000 Monastir, Tunisia
E-mail: [email protected]
and
Khalifa El Mabrouk
Department of Mathematics, High School of Sciences and Technology
4011 Hammam Sousse, Tunisia
E-mail: [email protected]
###### Abstract
Combining probabilistic and analytic tools from potential theory, we
investigate Dirichlet problems associated with the Dunkl Laplacian
$\Delta_{k}$. We establish, under some conditions on the open set
$D\subset\mathbb{R}^{d}$, the existence of a unique continuous function $h$ in
the closure of $D$, twice differentiable in $D$, such that
$\Delta_{k}h=0\quad\textrm{in}\;D\quad\textrm{and}\quad
h=f\quad\textrm{on}\;\partial D.$
We also give a probabilistic formula characterizing the solution $h$. The
function $f$ is assumed to be continuous on the Euclidean boundary $\partial
D$ of $D$.
## 1 Introduction
In their monograph [2], J. Bliedtner and W. Hansen developed four descriptions
of potential theory using balayage spaces, families of harmonic kernels, sub-
Markov semigroups and Markov processes. They proved that all these
descriptions are equivalent and gave a straight presentation of balayage
theory which is, in particular, applied to the generalized Dirichlet problem
associated with a large class of differential and pseudo-differential
operators.
Let $W$ be a finite reflection group on $\mathbb{R}^{d}$, $d\geq 1$, with root
system $R$ and we fix a positive subsystem $R_{+}$ of $R$ and a nonnegative
multiplicity function $k:R\to\mathbb{R}_{+}$. For every $\alpha\in R$, let
$H_{\alpha}$ be the hyperplane orthogonal to $\alpha$ and $\sigma_{\alpha}$ be
the reflection with respect to $H_{\alpha}$, that is, for every
$x\in\mathbb{R}^{d}$,
$\sigma_{\alpha}x=x-2\frac{\langle x,\alpha\rangle}{|\alpha|^{2}}\alpha$
where $\langle\cdot,\cdot\rangle$ denotes the Euclidean inner product of
$\mathbb{R}^{d}$. C. F. Dunkl introduced in [4] the operator
$\Delta_{k}=\sum_{i=1}^{d}T_{i}^{2},$
which will be called later _Dunkl Laplacian_ , where, for $1\leq i\leq d$,
$T_{i}$ is the differential-difference operator defined for $f\in
C^{1}(\mathbb{R}^{d})$ by
$T_{i}f(x)=\frac{\partial f}{\partial x_{i}}(x)+\sum_{\alpha\in
R_{+}}k(\alpha)\alpha_{i}\frac{f(x)-f(\sigma_{\alpha}x)}{\langle\alpha,x\rangle}.$
Our main goal in this paper is to investigate the _Dirichlet problem_
associated with the Dunkl Laplacian. More precisely, given a bounded open set
$D\subset\mathbb{R}^{d}$ and a continuous real-valued function $f$ on
$D^{c}:=\mathbb{R}^{d}\setminus D$, we are concerned with the following
problem:
$\displaystyle\left\\{\begin{array}[]{rcll}\Delta_{k}h&=&0&\mbox{in }D,\\\
h&=&f&\mbox{on }D^{c}.\end{array}\right.$ (1)
We mean by a solution of (1) every function $h:\mathbb{R}^{d}\to\mathbb{R}$
which is continuous in $\mathbb{R}^{d}$, twice differentiable in $D$ and such
that both equations in (1) are pointwise fulfilled. In the particular case
where $D$ is the unit ball of $\mathbb{R}^{d}$, M. Maslouhi and E. H. Youssfi
[11] solved problem (1) by methods from harmonic analysis using the Poisson
kernel for $\Delta_{k}$ which is introduced by C. F. Dunkl and Y. Xu [5]. It
should be noted that, for balls with center $a\not=0$, the Poisson kernel for
$\Delta_{k}$ is not known up to now.
Let us briefly introduce our approach. It is well known (see [6] and
references therein) that there exists a càdlàg $\mathbb{R}^{d}$-valued Markov
process
$X=(\Omega,{\mathcal{F}},{\mathcal{F}}_{t},X_{t},P^{x}),$
which is called _Dunkl process_ , with infinitesimal generator
$\frac{1}{2}\Delta_{k}$. For a given bounded Borel function
$h:\mathbb{R}^{d}\to\mathbb{R}$, we define
$H_{U}h(x)=E^{x}[h(X_{\tau_{U}})]$
for every $x\in\mathbb{R}^{d}$ and every bounded open subset $U$ of
$\mathbb{R}^{d}$, where
$\tau_{U}=\inf\\{t>0;X_{t}\notin U\\}$
denotes the first exit time from $U$ by $X$. We first show that if $h$ is
continuous in $\mathbb{R}^{d}$ and twice differentiable in $D$ then
$\Delta_{k}h=0$ in $D$ if and only if $h$ is $X$-harmonic in $D$, i.e.,
$H_{U}h(x)=h(x)$ for every open set $U$ such that $\overline{U}\subset D$ (we
shall write $U\Subset D$) and for every $x\in U$. We then conclude, using the
general framework of balayage spaces [2], that problem (1) admits at most one
solution. Moreover, if the open set $D$ is regular for the Dunkl process, then
$H_{D}f$ will be the solution of (1) provided it is of class $C^{2}$ in $D$.
For some examples of Markov processes, namely Brownian motion or
$\alpha$-stable process, some additional geometric assumptions on the
Euclidean boundary $\partial D$ of $D$ permit a decision on the regularity of
$D$. In fact, it is well known that $D$ is regular, with respect to Brownian
motion or $\alpha$-stable process, whenever each boundary point of $D$
satisfies the ”cone condition”. For a particular choice of the root system
$R$, we shall prove in Section 3 that the cone condition is still sufficient
for the regularity of $D$ with respect to the Dunkl process. However, we could
not know whether this result holds true for arbitrary root systems. In this
setting, we only show that balls of center $0$ are regular.
Finally, assuming that $D$ is regular, the study of problem (1) is equivalent
to the study of smoothness of $H_{D}f$. Indeed, as was mentioned above, (1)
has a solution if and only if
$H_{D}f\in C^{2}(D).$
To that end, we need to assume that $D$ is _$W$ -invariant_ which means that
$\sigma_{\alpha}(D)\subset D$ for every $\alpha\in R$. Hence, using the fact
that the operator $\Delta_{k}$ is hypoelliptic in $D$ (see [7, 10]) we prove
that $H_{D}f$ is infinitely differentiable in $D$. Thus, we not only deduce
the existence and uniqueness of the solution to
$\displaystyle\left\\{\begin{array}[]{rcll}\Delta_{k}h&=&0&\mbox{in }D,\\\
h&=&f&\mbox{on }\partial D,\end{array}\right.$ (2)
but we also prove that $h$ is given by the formula
$h(x)=E^{x}[f(X_{\tau_{D}})]$.
Throughout this paper, let $\lambda=\gamma+\frac{d}{2}-1$ and assume that
$\lambda>0$.
## 2 Harmonic Kernels
For the sake of simplicity, we assume in all the following that
$|\alpha|^{2}=2$ for every $\alpha\in R$. It follows from [4] that, for $f\in
C^{2}(\mathbb{R}^{d})$,
$\Delta_{k}f(x)=\Delta f(x)+2\sum_{\alpha\in
R_{+}}k(\alpha)\left(\frac{\langle\nabla
f(x),\alpha\rangle}{\langle\alpha,x\rangle}-\frac{f(x)-f(\sigma_{\alpha}(x))}{\langle\alpha,x\rangle^{2}}\right),$
(3)
where $\Delta$ denotes the usual Laplacian on $\mathbb{R}^{d}$. M. Rösler has
shown in [13] that $\frac{1}{2}\Delta_{k}$ generates a Feller semigroup
$P_{t}^{k}(x,dy)=p_{t}^{k}(x,y)w_{k}(y)dy$ which has the expression
$p_{t}^{k}(x,y)=\frac{1}{c_{k}t^{\gamma+\frac{d}{2}}}\exp\left(-\frac{|x|^{2}+|y|^{2}}{2t}\right)E_{k}\left(\frac{x}{\sqrt{t}},\frac{y}{\sqrt{t}}\right),$
(4)
where $E_{k}(\cdot,\cdot)$ is the Dunkl kernel associated with $W$ and $k$
(see [5]), the constant $c_{k}$ is taken such that $P_{1}^{k}1\equiv 1$,
$\;\gamma=\sum_{\alpha\in R_{+}}k(\alpha)$ and $w_{k}$ is the $W$-invariant
weight function defined on $\mathbb{R}^{d}$ by
$w_{k}(y)=\prod_{\alpha\in R_{+}}|\langle y,\alpha\rangle|^{2k(\alpha)}.$
Let $X=(\Omega,{\mathcal{F}},{\mathcal{F}}_{t},X_{t},P^{x})$ be the Dunkl
process in $\mathbb{R}^{d}$ with transition kernel $P_{t}^{k}(x,dy)$. For
every bounded open subset $D$ of $\mathbb{R}^{d}$, let $\tau_{D}$ be the first
exit time from $D$ by $X$. A point $z\in\partial D$ is said to be _regular_
(for $D$) if $P^{z}[\tau_{D}=0]=1$ and _irregular_ if $P^{z}[\tau_{D}=0]=0$.
Notice that by Blumenthal’s zero-one law, each boundary point of $D$ is either
regular or irregular. It is also easy verified that the fact that Dunkl
process has right continuous paths yields that $P^{x}[\tau_{D}=0]=0$ if $x\in
D$ and $P^{x}[\tau_{D}=0]=1$ if $x\in\mathbb{R}^{d}\setminus\overline{D}$.
###### Proposition 1.
$E^{x}[\tau_{D}]<\infty$ for every $x\in\mathbb{R}^{d}$ and every bounded open
subset $D$ of $\mathbb{R}^{d}$.
###### Proof.
Let $D$ be a bounded open subset of $\mathbb{R}^{d}$, $x\in\mathbb{R}^{d}$ and
choose $r>0$ such that the ball $B=B(0,r)$ contains $x$ and $D$. Then,
applying Fubini’s theorem and using spherical coordinates,
$\displaystyle E^{x}[\tau_{B}]$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}E^{x}[\mathbf{1}_{B}(X_{s})]ds$
$\displaystyle=$
$\displaystyle\int_{0}^{r}t^{2\lambda+1}\int_{0}^{\infty}\int_{S^{d-1}}p_{s}^{k}(x,tz)w_{k}(z)\sigma(dz)ds\,dt.$
Here and in all the following, $\sigma$ denotes the surface area measure on
the unit sphere $S^{d-1}$ of $\mathbb{R}^{d}$. It is well known (see [13, 14])
that for every $x,y\in\mathbb{R}^{d}$ and $s>0$,
$p_{s}^{k}(x,y)=\frac{1}{c_{k}^{2}}\int_{\mathbb{R}^{d}}e^{-\frac{s}{2}|\xi|^{2}}E_{k}(-ix,\xi)E_{k}(iy,\xi)w_{k}(\xi)d\xi$
and
$\int_{S^{d-1}}E_{k}(ix,\xi)w_{k}(\xi)\sigma(d\xi)=\frac{c_{k}}{2^{\lambda}\Gamma(\lambda+1)}j_{\lambda}(|x|),$
where
$j_{\lambda}(z):=\Gamma(\lambda+1)\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n}}{4^{n}n!\Gamma(n+\lambda+1)}$
is the Bessel normalized function. Hence
$\displaystyle E^{x}[\tau_{D}]$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}E^{x}[\mathbf{1}_{B}(X_{s})]ds$ (5)
$\displaystyle=$
$\displaystyle\frac{1}{2^{2\lambda-1}(\Gamma(\lambda+1))^{2}}\int_{0}^{r}t^{2\lambda+1}\int_{0}^{\infty}j_{\lambda}(ut)j_{\lambda}(u|x|)u^{2\lambda-1}dudt$
$\displaystyle=$
$\displaystyle\frac{2^{2\lambda-1}\Gamma(\lambda+1)\Gamma(\lambda)}{2^{2\lambda-1}(\Gamma(\lambda+1))^{2}}\int_{0}^{r}t^{2\lambda+1}(\max(t,|x|))^{-2\lambda}dt$
$\displaystyle=$
$\displaystyle\frac{r^{2}}{2\lambda}-\frac{|x|^{2}}{2\lambda+2}<\infty.$ (6)
In order to get (5) one should think about formula (11.4.33) in [1]. ∎
Let $D$ be a bounded open subset of $\mathbb{R}^{d}$. For every
$x\in\mathbb{R}^{d}$, the exit distribution $H_{D}(x,\cdot)$ from $D$ by the
Dunkl process starting at $x$ will be called harmonic measure relative to $x$
and $D$. That is, for every Borel subset $A$ of $\mathbb{R}^{d}$,
$H_{D}(x,A)=P^{x}(X_{\tau_{D}}\in A).$
It is clear that $H_{D}(x,\cdot)=\delta_{x}$ the Dirac measure at $x$ whenever
$x\in\partial D$ is regular or $x\not\in\overline{D}$. We define
${}^{W}\\!\\!D:=\cup_{w\in
W}w(D)\quad\mbox{and}\quad\Gamma_{D}:=\overline{{}^{W}\\!\\!D}\setminus D.$
In other words, ${}^{W}\\!\\!D$ is the smallest open set containing $D$ which
is invariant under the reflection group $W$. The following theorem ensures
that $H_{D}(x,\cdot)$ is supported by $\Gamma_{D}$ for every
$x\in\overline{D}$.
###### Theorem 2.
Let $D$ be a bounded open subset of $\mathbb{R}^{d}$. Then for every
$x\in\overline{D}$,
$P^{x}\left(X_{\tau_{D}}\in\Gamma_{D}\right)=1.$ (7)
###### Proof.
It is easily seen that for every regular boundary point $x$,
$P^{x}(X_{\tau_{D}}\in\Gamma_{D})=\delta_{x}(\Gamma_{D})=1$. Now, assume that
$x\in D$ or $x\in\partial D$ is irregular and consider the function $\digamma$
defined for every $y,z\in\mathbb{R}^{d}$ by $\digamma(y,z)=0$ if
$z\in\\{\sigma_{\alpha}y;\alpha\in R_{+}\\}$ and $\digamma(y,z)=1$ otherwise.
Let
$Y_{t}:=\sum_{s<t}\mathbf{1}_{\\{X_{s^{-}}\neq
X_{s}\\}}\digamma(X_{s^{-}},X_{s}),\quad t>0.$
It follows from [6, Proposition 3.2] that for every $t>0$, $P^{x}(Y_{t}=0)=1$
and consequently
$P^{x}\left(\mathbf{1}_{\\{X_{s^{-}}\neq
X_{s}\\}}\digamma(X_{s^{-}},X_{s})=0;\forall s>0\right)=1.$
Then, since $P^{x}(0<\tau_{D}<\infty)=1$ we deduce that
$P^{x}\left(\mathbf{1}_{\\{X_{\tau_{D}^{-}}\neq
X_{\tau_{D}}\\}}\digamma(X_{\tau_{D}^{-}},X_{\tau_{D}})=0\right)=1.$
On the other hand, seeing that $X_{\tau_{D}^{-}}\in\overline{D}$ on
$\\{0<\tau_{D}<\infty\\}$ we have
$\left\\{X_{\tau_{D}}\not\in\Gamma_{D},0<\tau_{D}<\infty\right\\}\subset\left\\{\mathbf{1}_{\\{X_{\tau_{D}^{-}}\neq
X_{\tau_{D}}\\}}\digamma(X_{\tau_{D}^{-}},X_{\tau_{D}})=1\right\\}.$
This finishes the proof. ∎
Let $\mathcal{O}$ be the set of all bounded open subsets of $\mathbb{R}^{d}$.
In the following, we denote by $\mathcal{B}_{b}(\mathbb{R}^{d})$ the set of
all bounded Borel measurable functions on $\mathbb{R}^{d}$. For every
$D\in\mathcal{O}$ and $f\in\mathcal{B}_{b}(\mathbb{R}^{d})$, let $H_{D}f$ be
the function defined on $\mathbb{R}^{d}$ by
$H_{D}f(x)=E^{x}\left[f(X_{\tau_{D}})\right]=\int f(y)H_{D}(x,dy).$
Since $X$ is a Hunt process, it follows from the general framework of balayage
spaces studied by J. Bliedtner and W. Hansen in [2] that, for every
$D\in\mathcal{O}$ and $f\in\mathcal{B}_{b}(\mathbb{R}^{d})$ with compact
support, $H_{D}f$ is continuous in $D$ and for every $V\Subset D$,
$H_{V}H_{D}=H_{D}\quad\textrm{in}\;\;V.$ (8)
Since $\textrm{supp}\,H_{D}(x,\cdot)\subset\Gamma_{D}$ for every
$x\in\overline{{}^{W}\\!\\!D}$, it is trivial that
$H_{D}f(x)=H_{D}\left(1_{\Gamma_{D}}f\right)(x),\quad
x\in\overline{{}^{W}\\!\\!D}.$
Hence, we immediately conclude that $H_{D}f$ is continuous in $D$. For every
$D\in\mathcal{O}$ and every $f\in\mathcal{B}_{b}(\Gamma_{D})$, it will be
convenient to denote again
$H_{D}f(x)=\int f(y)H_{D}(x,dy),\quad x\in\overline{{}^{W}\\!\\!D}.$ (9)
Let $U$ be an open subset of $\mathbb{R}^{d}$. A locally bounded function
$h:^{W}\\!\\!\\!\\!U\rightarrow\mathbb{R}$ is said to be _$X$ -harmonic_ in
$U$ if $H_{D}h(x)=h(x)$ for every open set $D\Subset U$ and every $x\in D$. If
$U$ is bounded and $h$ is continuous in $\overline{{}^{W}\\!U}$ then $h$ is
$X$-harmonic in $U$ if and only if for every $x\in U$,
$h(x)=H_{U}h(x).$ (10)
In fact, let $x\in U$ and let $(U_{n})_{n\geq 1}$ be a sequence of nonempty
bounded open subsets of $\mathbb{R}^{d}$ such that $x\in U_{n}\Subset U_{n+1}$
and $U=\cup_{n}U_{n}$. Then $(\tau_{U_{n}})_{n}$ converges to $\tau_{U}$
almost surely. Hence, the continuity of $h$ on $\overline{{}^{W}\\!U}$
together with the quasi-left-continuity of the Dunkl process yield that
$H_{U}h(x)=\lim_{n}H_{U_{n}}h(x)$. The following proposition follows
immediately from (10).
###### Proposition 3.
Let $U\in\mathcal{O}$ and let $h$ be a continuous function on
$\overline{{}^{W}\\!U}$. If $h$ is $X$-harmonic in $U$, then
$\max_{x\in\overline{{}^{W}\\!\\!U}}h(x)=\max_{x\in\Gamma_{U}}h(x)\quad\textrm{and}\quad\min_{x\in\overline{{}^{W}\\!\\!U}}h(x)=\min_{x\in\Gamma_{U}}h(x).$
We shall denote by $G^{k}$ the Green function of $\Delta_{k}$ which is defined
for every $x,y\in\mathbb{R}^{d}$ by
$G^{k}(x,y)=\int_{0}^{\infty}p_{t}^{k}(x,y)dt.$
Since $p_{t}^{k}$ is symmetric in $\mathbb{R}^{d}\times\mathbb{R}^{d}$, we
obviously see that the Green function $G^{k}$ is also symmetric in
$\mathbb{R}^{d}\times\mathbb{R}^{d}$. Therefore, it follows from [3, Theorem
VI.1.16] that for every $D\in\mathcal{O}$ and for every
$x,y\in\mathbb{R}^{d}$,
$\int G^{k}(x,z)H_{D}(y,dz)=\int G^{k}(y,z)H_{D}(x,dz).$ (11)
Furthermore, for every $y\in\mathbb{R}^{d}$, the function $G^{k}(\cdot,y)$ is
excessive, that is, $G^{k}(\cdot,y)$ is lower semi-continuous in
$\mathbb{R}^{d}$ and $\int p_{t}^{k}(x,z)G^{k}(z,y)w_{k}(z)dz\leq G^{k}(x,y)$
for every $t>0$ and $x\in\mathbb{R}^{d}$. Consequently, it follows from [2,
Theorem IV.8.1] that $G^{k}(\cdot,y)$ is hyperharmonic on $\mathbb{R}^{d}$,
i.e., for every $D\in\mathcal{O}$ and for every $x\in\mathbb{R}^{d}$,
$\int G^{k}(z,y)H_{D}(x,dz)\leq G^{k}(x,y).$ (12)
###### Lemma 4.
Let $f\in C^{2}_{c}(\mathbb{R}^{d})$ and $D\in\mathcal{O}$. For every
$x\in\mathbb{R}^{d}$,
$\int G^{k}(x,y)\Delta_{k}f(y)w_{k}(y)dy=-2f(x).$ (13)
In particular,
$H_{D}f(x)-f(x)=\frac{1}{2}E^{x}\left[\int_{0}^{\tau_{D}}\Delta_{k}f(X_{s})ds\right].$
(14)
###### Proof.
To get (13) it suffices to recall that
$\frac{\partial}{\partial t}P_{t}^{k}=\frac{1}{2}P_{t}^{k}\Delta_{k},\quad
t>0.$
Then, we integrate over $t$ and use the fact that $\lim_{t\rightarrow
0}P_{t}^{k}f(x)=f(x)$ and $\lim_{t\rightarrow\infty}P_{t}^{k}f(x)=0$ for every
$x\in\mathbb{R}^{d}$. Formula (14) follows from (13) and the strong Markov
property. ∎
Let $U$ be an open subset of $\mathbb{R}^{d}$. A function
$h:^{W}\\!\\!\\!\\!U\rightarrow\mathbb{R}$ is said to be _$\Delta_{k}$
-harmonic_ in $U$ if $h\in C^{2}(U)$ and $\Delta_{k}h(x)=0$ for every $x\in
U$.
###### Theorem 5.
Let $U$ be an open subset of $\mathbb{R}^{d}$ and let $h\in C(^{W}\\!\\!U)$.
If $h\in C^{2}(U)$ then $h$ is $\Delta_{k}$-harmonic in $U$ if and only if $h$
is $X$-harmonic in $U$.
###### Proof.
Let $D\Subset U$ and let $x\in D$. Then
$H_{D}h(x)-h(x)=\frac{1}{2}E^{x}\left[\int_{0}^{\tau_{D}}\Delta_{k}h(X_{s})ds\right].$
(15)
In fact, choose an open set $V$ such that $D\Subset V\Subset U$, $f\in
C^{2}_{c}(\mathbb{R}^{d})$ which coincides with $h$ in $V$ and let $\psi=h-f$.
Then using (14) we obtain
$H_{D}h(x)-h(x)=\frac{1}{2}E^{x}\left[\int_{0}^{\tau_{D}}\Delta_{k}f(X_{s})ds\right]+H_{D}\psi(x).$
(16)
For every $y\in\mathbb{R}^{d}$, let $N(y,dz)$ be the Lévy kernel of the Dunkl
process $X$ which is given by the following formula [6]
$N(y,dz)=\sum_{\alpha\in R_{+},\langle y,\alpha\rangle\neq
0}\frac{k(\alpha)}{\langle\alpha,y\rangle^{2}}\delta_{\sigma_{\alpha}y}(dz).$
(17)
Since $\psi=0$ on $V$, it follows from [8, Theorem 1] that
$H_{D}\psi(x)=E^{x}\left[\int_{0}^{\tau_{D}}\int\psi(z)N(X_{s},dz)ds\right].$
(18)
On the other hand, by (3) and (17) we easily see that for every $y\in D$,
$\Delta_{k}f(y)=\Delta_{k}h(y)-2\int\psi(z)N(y,dz).$ (19)
Thus formula (15) is obtained by combing (16), (18) and (19) above. Now, $h$
is obviously $X$-harmonic in $U$ whenever it is $\Delta_{k}$-harmonic in $U$.
Conversely, assume that $h$ is $X$-harmonic in $U$ and let $x\in U$. Since
$h\in C(^{W}\\!\\!U)\cap C^{2}(U)$ then $\Delta_{k}h$ is continuous in $U$ and
consequently for every $\varepsilon>0$ there exists an open neighborhood
$D\Subset U$ of $x$ such that $|\Delta_{k}h(y)-\Delta_{k}h(x)|\leq\varepsilon$
for every $y\in D$. Using formula (15), we obtain
$|\Delta_{k}h(x)|=\frac{1}{E^{x}[\tau_{D}]}\left|E^{x}\left[\int_{0}^{\tau_{D}}\left(\Delta_{k}h(X_{s})-\Delta_{k}h(x)\right)ds\right]\right|\leq\varepsilon.$
Hence $\Delta_{k}h(x)=0$ as desired. ∎
## 3 Regular Sets
A bounded open subset $D$ of $\mathbb{R}^{d}$ is said to be _regular_ if each
$z\in\partial D$ is regular for $D$. A complete study of regularity is
developed by J. Bliedtner and W. Hansen in [2]. It follows that a point
$z\in\partial D$ is regular for $D$ if and only if for every $f\in
C(\Gamma_{D})$,
$\lim_{x\in D,x\rightarrow z}H_{D}f(x)=f(z).$
Consequently, $H_{D}f$ is continuous on ${}^{W}\\!\\!\overline{D}$ whenever
$D$ is regular and $f\in C(\Gamma_{D})$.
###### Example 6.
For all $R>r>0$, the ball $B(0,R)$ and the annulus
$C(r,R)=\\{x\in\mathbb{R}^{d};\;r<\|x\|<R\\}$ are regular.
In fact, by [2, Proposition VII.3.3], it is sufficient to find a neighborhood
$V$ of $z\in\partial D$ and a real function $u$ such that
* i)
$u$ is positive in $V\cap D$,
* ii)
$u$ is $X$-harmonic in $V\cap D$,
* iii)
$\lim_{x\in V\cap D,x\rightarrow z}u(x)=0$.
Consider $V=\mathbb{R}^{d}\backslash\\{0\\}$ and $g$ the function defined on
$V$ by
$g(x)=\frac{1}{|x|^{2\lambda}}.$
Using formula (3), simple computation shows that $g$ is $\Delta_{k}$-harmonic
in $V$ which yields, by theorem 5, that $g$ is $X$-harmonic in $V$. Let
$z\in\mathbb{R}^{d}$ such that $|z|=R$ and consider
$u(x)=g(x)-\frac{1}{R^{2\lambda}},\quad x\in V.$
It is clear that $u$ satisfy (i), (ii) and (iii) above with $D=B(0,R)$ or
$D=C(r,R)$. Hence $z$ is regular for $D$. Similarly, taking
$u(x)=\frac{1}{r^{2\lambda}}-g(x),\quad x\in V,$
we conclude that all points $z\in\mathbb{R}^{d}$ such that $|z|=r$ are regular
for $C(r,R)$.
A sufficient condition for regularity, known as the cone condition, is given
in the following theorem for a particular root system $R$.
###### Theorem 7.
Let $(e_{1},...,e_{d})$ be the canonical basis of $\mathbb{R}^{d}$ and
consider the root system $R=\\{\pm e_{i},\;1\leq i\leq d\\}$. Let $D$ be a
bounded open subset of $\mathbb{R}^{d}$ and let $z\in\partial D$. Assume that
there exists a cone $C$ of vertex $z$ such that $C\cap B(z,r)\subset D^{c}$
for some $r>0$. Then $z$ is regular for $D$.
###### Proof.
It is trivial that $P^{z}[\tau_{D}\leq t]\geq P^{z}[X_{t}\in C\cap B(z,r)]$
for all $t>0$. Therefore, in virtue of Blumenthal’s zero-one low, it is
sufficient to show that $\liminf_{t\rightarrow 0}P^{z}[X_{t}\in C\cap B(z,r)]$
is positive. Denote $C_{0}=C-z$, then
$\displaystyle P^{z}[X_{t}\in C\cap B(z,r)]$ $\displaystyle=$
$\displaystyle\int_{C\cap B(z,r)}p_{t}^{k}(z,y)w_{k}(y)dy$ (20)
$\displaystyle=$ $\displaystyle\frac{1}{t^{\gamma}}\int_{C_{0}\cap
B(0,\frac{r}{\sqrt{t}})}p_{1}^{k}(\frac{z}{\sqrt{t}},\frac{z}{\sqrt{t}}-y)w_{k}(z-\sqrt{t}y)dy.$
It is trivial to see, from (4), that
$p_{1}^{k}(\frac{z}{\sqrt{t}},\frac{z}{\sqrt{t}}-y)=e^{-\frac{|y|^{2}}{2}}e^{-\langle\frac{z}{t},z-\sqrt{t}y\rangle}E_{k}\left(\frac{z}{t},z-\sqrt{t}y\right).$
Let $k_{i}=k(e_{i})$ and $y_{i}=\langle y,e_{i}\rangle$ for every
$y\in\mathbb{R}^{d}$ and $i\in\\{1,...,d\\}$. It is known [16] that for all
$x,y\in\mathbb{R}^{d}$,
$e^{-\langle
x,y\rangle}E_{k}(x,y)=\prod_{i=1}^{d}M(k_{i},2k_{i}+1,-2x_{i}y_{i}).$
$M(k_{i},2k_{i}+1,\cdot)$ denotes the Kummer’s function defined on
$\mathbb{R}$ by
$M(k_{i},2k_{i}+1,s)=\sum_{n\geq
0}\frac{(k_{i})_{n}}{(2k_{i}+1)_{n}}\frac{s^{n}}{n!}=1+\frac{k_{i}}{2k_{i}+1}s+\frac{k_{i}(k_{i}+1)}{(2k_{i}+1)(2k_{i}+2)}\frac{s^{2}}{2!}+\cdots\quad.$
Therefore, for any $y\in\mathbb{R}^{d}$ and $t>0$, we have
$\begin{array}[]{lll}\displaystyle\frac{1}{t^{\gamma}}e^{-\langle\frac{z}{t},z-\sqrt{t}y\rangle}E_{k}\left(\frac{z}{t},z-\sqrt{t}y\right)w_{k}\left(z-\sqrt{t}y\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle=\prod_{i=1}^{d}\frac{M\left(k_{i},2k_{i}+1,-2\frac{z_{i}}{t}(z_{i}-\sqrt{t}y_{i})\right)(z_{i}-\sqrt{t}y_{i})^{2k_{i}}}{t^{k_{i}}}.\end{array}$
First, it is clear that
$\frac{1}{t^{k_{i}}}M\left(k_{i},2k_{i}+1,-2\frac{z_{i}}{t}(z_{i}-\sqrt{t}y_{i})\right)(z_{i}-\sqrt{t}y_{i})^{2k_{i}}=\left\\{\begin{array}[]{ll}1&\textrm{if}\;k_{i}=0\\\
y_{i}^{2k_{i}}&\textrm{if}\;z_{i}=0.\end{array}\right.$
Next, assume that $k_{i}>0$ and $z_{i}\neq 0$ for some $i\in\\{1,\cdots,d\\}$.
Then, it follows from the integral representation of $M(k_{i},2k_{i}+1,\cdot)$
that
$\displaystyle\frac{1}{t^{k_{i}}}M\left(k_{i},2k_{i}+1,-2\frac{z_{i}}{t}(z_{i}-\sqrt{t}y_{i})\right)$
$\displaystyle=$
$\displaystyle\frac{\Gamma(2k_{i}+1)}{\Gamma(k_{i})\Gamma(k_{i}+1)}\int_{0}^{1}\frac{1}{t^{k_{i}}}e^{-2\frac{z_{i}}{t}(z_{i}-\sqrt{t}y_{i})u}u^{k_{i}-1}(1-u)^{k_{i}}du$
$\displaystyle=$
$\displaystyle\frac{\Gamma(2k_{i}+1)}{\Gamma(k_{i})\Gamma(k_{i}+1)}\int_{0}^{\frac{1}{t}}e^{-2z_{i}(z_{i}-\sqrt{t}y_{i})v}v^{k_{i}-1}(1-tv)^{k_{i}}dv.$
Now, applying the Lebesgue dominated convergence theorem, we obtain
$\lim_{t\rightarrow
0}\frac{1}{t^{k_{i}}}M\left(k_{i},2k_{i}+1,-2\frac{z_{i}}{t}(z_{i}-\sqrt{t}y_{i})\right)=\frac{\Gamma(k_{i}+1)}{\sqrt{\pi}z_{i}^{2k_{i}}}.$
Thus
$\begin{array}[]{lll}\displaystyle\lim_{t\rightarrow
0}\frac{1}{t^{\gamma}}e^{-\langle\frac{z}{t},z-\sqrt{t}y\rangle}E_{k}\left(\frac{z}{t},z-\sqrt{t}y\right)w_{k}\left(z-\sqrt{t}y\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\geq\prod_{i=1}^{d}\min\left(1,y_{i}^{2k_{i}},\frac{\Gamma(k_{i}+1)}{\sqrt{\pi}}\right)=:\theta(y).\end{array}$
Hence, Fatou’s lemma applied to (20) yields that
$\liminf_{t\rightarrow 0}P^{z}[X_{t}\in C\cap
B(z,r)]\geq\int_{C_{0}}e^{-\frac{|y|^{2}}{2}}\theta(y)dy>0.$
∎
## 4 Dirichlet Problem
This section is devoted to study the following Dirichlet problem : Giving a
regular open subset $D$ of $\mathbb{R}^{d}$ and a function $f\in
C(\Gamma_{D})$, we shall investigate existance and uniqueness of function
$h\in C(\overline{{}^{W}\\!\\!D})\cap C^{2}(D)$ satisfying the boundary value
problem
$\left\\{\begin{array}[]{rcll}\Delta_{k}h&=&0&\;\textrm{in}\;D,\\\
h&=&f&\;\textrm{in}\;\Gamma_{D}.\end{array}\right.$ (21)
For every square integrable functions $\varphi$ and $\psi$ on $\mathbb{R}^{d}$
with respect to the measure $w_{k}(x)dx$, we define
$\langle\varphi,\psi\rangle_{k}=\int\varphi(x)\psi(x)w_{k}(x)dx.$
###### Lemma 8.
For every bounded open set $D$ and for every $\varphi,\psi\in
C^{2}_{c}(\mathbb{R}^{d})$,
$\langle
H_{D}\psi,\Delta_{k}\varphi\rangle_{k}=\langle\Delta_{k}\psi,H_{D}\varphi\rangle_{k}.$
(22)
###### Proof.
Applying formula (13) to $\psi$, we have
$\langle H_{D}\psi,\Delta_{k}\varphi\rangle_{k}=-\frac{1}{2}\int
G^{k}(z,y)\Delta_{k}\psi(y)w_{k}(y)dyH_{D}(x,dz)\Delta_{k}\varphi(x)w_{k}(x)dx.$
(23)
Then (22) is obtained by Fubini’s theorem and formulas (11) and (13). Here,
since $\varphi$ and $\psi$ are with compact supports, formulas (12) and (6)
justify the transformation of the integrals in (23) by Fubini’s theorem. ∎
A set $D$ is called _$W$ -invariant_ if ${}^{W}\\!\\!D=D$ which, in turn, is
equivalent to $\Gamma_{D}=\partial D$. We finally have the necessary tools at
our disposal for solving the following Dirichlet problem.
###### Theorem 9.
Let $D$ be a $W$-invariant regular open subset of $\mathbb{R}^{d}$. For every
function $f\in C(\partial D)$, there exists one and only one function $h\in
C(\overline{D})\cap C^{2}(D)$ such that
$\left\\{\begin{array}[]{rcll}\Delta_{k}h&=&0&\;\textrm{in}\;D,\\\
h&=&f&\;\textrm{in}\;\partial D.\end{array}\right.$ (24)
Moreover, $h$ is given by
$h(x)=\int_{\partial D}f(y)H_{D}(x,dy),\quad x\in\overline{D}.$
###### Proof.
In virtue of Theorem 5, we observe that for $f\in C(\partial D)$, every
solution $h$ of (21) satisfies necessarily :
$\left\\{\begin{array}[]{ll}h\textrm{ is X-harmonic in}\;D,\\\
h=f\;\textrm{in}\;\partial D.\end{array}\right.$ (25)
Then, by Proposition 3, (24) admits at most one solution. The function
$H_{D}f$ is $X$-harmonic in $D$ by (8). Moreover, the regularity of $D$ yields
that $H_{D}f$ is a continuous extension of $f$ to $\overline{D}$. Therefore,
according to Theorem 5, $H_{D}f$ will be the unique solution of (24) provided
it is twice differentiable in $D$. On the other hand, it has been shown in [7]
that $\Delta_{k}$ is hypoelliptic in $D$ (see also [10]), i.e., a continuous
function $g$ in $D$ which satisfies
$\langle g,\Delta_{k}\varphi\rangle_{k}=0\quad\textrm{for all}\;\;\varphi\in
C^{\infty}_{c}(D)$ (26)
is necessary infinitely differentiable in $D$. Thus to complete the proof we
only need to show that (26) holds true for $g=H_{D}f$. To this end let
$\varphi\in C^{\infty}_{c}(D)$ and let $(f_{n})_{n\geq
1}\subset\leavevmode\nobreak\ C^{2}_{c}(\mathbb{R}^{d})$ be a sequence which
converges uniformly to $f$ in $\partial D$. Since $H_{D}\varphi(y)=0$ for all
$y\in\mathbb{R}^{d}$, applying (22) we obtain
$\langle H_{D}f_{n},\Delta_{k}\varphi\rangle_{k}=0,\quad n\geq 1.$ (27)
On the other hand,
$\displaystyle\sup_{x\in\overline{D}}|H_{D}f_{n}(x)-H_{D}f(x)|\leq\sup_{y\in\partial
D}|f_{n}(y)-f(y)|\longrightarrow 0\quad\textrm{as}\quad
n\longrightarrow\infty.$
Hence $H_{D}f$ satisfies (26) by letting $n$ tend to $\infty$ in (27). ∎
It should be noted that the hypothesis ”$D$ is $W$-invariant” is only needed
to get the hypoellipticity of $\Delta_{k}$. For open set $D$ which is not
$W$-invariant, the question whether $\Delta_{k}$ is hypoelliptic in $D$ or not
remained open. In the case of positive answer, analogous arguments as in the
proof of Theorem 9 will immediately imply that $H_{D}f$ is the unique solution
of problem (21).
Let us notice that, using methods from harmonic analysis, M. Maslouhi and E.
H. Youssfi [11] studied problem (24) in the special case where $D=B$ is the
unit ball of $\mathbb{R}^{d}$. They proved that, for any $f\in C(\partial B)$,
the function $h$ given by
$h(x)=\int_{\partial B}P_{\kappa}(x,y)f(y)w_{k}(y)\sigma(dy),\;x\in B$
is the unique solution of (24), where $P_{\kappa}$ denotes the Poisson kernel
introduced by C. F. Dunkl and Y. Xu [5]. Hence, our above theorem immediately
yields that for every $x\in B$,
$H_{B}(x,dy)=P_{\kappa}(x,y)w_{k}(y)\sigma(dy).$
## References
* [1] Abramowitz, M. and Stegun, I. A. (1984). Handbook mathematical functions. Verlag Harri Deutsch. Frankfurt-Main.
* [2] Bliedtner, J. and Hansen, W. (1986). Potentiel theory. An analytic and probabilistic approach to balayage. Springer-Verlag.
* [3] Blumenthal, R. M. and Getoor, R. K. (1968). Markov processes and potential theory. Academic Press.
* [4] Dunkl, C. F. (1989). Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311 167–183.
* [5] Dunkl, C. F. and Xu, Y. (2001). Othogonal polynomials of sevaral variables. Cambridge University Press.
* [6] Gallardo, L. and Yor, M. (2006). A chaotic representation property of the multidimensional Dunkl processes. Ann. Proba. 34 1530–1549.
* [7] Hassine, K. (2014). Mean value propoerty associated with the Dunkl Laplace opertor. Preprint. arXiv:1401.1949v1.
* [8] Ikeda, N. and Watanabe, S. (1962). On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2 79–95.
* [9] Mejjaoli, H. and Trimèche, K. (2001). On a mean value property associated with the dunkl Laplacian operator and applications. Integral Transforms Spec. Funct. 12 279–302.
* [10] Mejjaoli, H. and Trimèche, K. (2004). Hypoellipticity and hypoanalyticity of the Dunkl Laplacian operator. Integral Transforms Spec. Funct. 15 523–548.
* [11] Maslouhi, M. and Youssfi, E. H. (2007). Harmonic functions associated to Dunkl operators. Monatsh. Math. 152 337–345.
* [12] Rösler, M. (1999). Positivity of Dunkl’s intertwining operator. Duke Math. J. 98 445–463.
* [13] Rösler, M. (1998). Generalized Hermite polynomials and heat equation for Dunkl operators. Commun. Math. Phys. 192 519–542.
* [14] Rösler, M. (2003). A positive radial product formula for Dunkl kernel. Trans. Am. Math. Soc. 355 2413–2438.
* [15] Trimèche, K. (2001). The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual. Integral Transforms Spec. Funct. 12 349–374.
* [16] Xu, Y. (1997). Orthogonal polynomials for a family of product weight functions on the spheres. Can. J. Math. 49 175–192.
|
arxiv-papers
| 2014-02-07T10:43:48 |
2024-09-04T02:49:57.939305
|
{
"license": "Public Domain",
"authors": "Mohamed Ben Chrouda and Khalifa El Mabrouk",
"submitter": "Khalifa El Mabrouk",
"url": "https://arxiv.org/abs/1402.1597"
}
|
1402.1644
|
Surface patterns in drying films of silica colloidal dispersions
F. Boulognea†∗, F. Giorgiutti-Dauphinéa, L. Paucharda
We report an experimental study on the drying of silica colloidal dispersions.
Here we focus on a surface instability occurring in a drying paste phase
before crack formation which affects the final film quality. Observations at
macroscopic and microscopic scales reveal the occurrence of the instability,
and the morphology of the film surface. Furthermore, we show that the addition
of adsorbing polymers on silica particles can be used to suppress the
instability under particular conditions of molecular weight and concentration.
We relate this suppression to the increase of the paste elastic modulus.
00footnotetext: a UPMC Univ Paris 06, Univ Paris-Sud, CNRS, F-91405. Lab
FAST, Bat 502, Campus Univ, Orsay, F-91405, France. Fax: +33 1 69 15 80 60;
Tel: +33 1 69 15 80 46; E-mail: [email protected]
† Now at: Department of Mechanical and Aerospace Engineering, Princeton
University, Princeton, NJ 08544, USA.
## 1 Introduction
Patterns arising in soft materials such as elastomers, gels and biological
tissues receive a growing attention 1. The understanding and the control of
the underlying instabilities are crucial for technological applications
(microelectronics, microfluidics), for biological systems (wrinkling of human
skin or drying fruit 2) or for medical applications where gels are used as
biological scaffolds for tissues or organs. In addition, various patterns
which affects the surface of films are reported in the literature. One of the
most studied concerns wrinkles observed when a hard skin, sitting on a soft
layer, is compressed. Beyond a critical strain, it results in a periodic
sinusoidal deformation of the interface 3, 4, 5, 6 for which the periodicity
has been derived theoretically 2, 7. When the strain is further increased, a
secondary instability occurs leading to wrinkle-to-fold transition 8, 9; this
results in a localization of the deformation 10. Under a biaxial compressive
stress, it has been shown that a repetitive wrinkle-to-fold transition
produces a hierarchical network of folds 11, 12. These patterns are observed
in various systems such as glassy polymers 13, 3, polyethylene sheets 2 or
foams 14. Another group of patterns arises in soft layers supported by rigid
substrates 1. Under compressive stresses, an instability often called creasing
is manifested by localized and sharp structures at the free surface. These are
commonly observed in elastomers 15, 16, and hydrogels 17, or in the situation
of a rising dough in a bowl 18. Despite similarities in their morphologies,
origins of folds and creases are strongly different. Whereas folds are a
secondary instability of wrinkles, creases are a direct deformation of a flat
film (creases are also known as sulci 19, 20). Moreover drying films can
produce characteristic crack patterns as they are dried21, 22. The drying-
induced cracks can invade the surface and propagate simultaneously into the
volume of the medium with evaporation of solvent, resulting in the division of
the plane into polygonal domains.
In this paper, we experimentally investigate patterns displayed at the surface
of drying films of silica nanoparticles in an aqueous solvent. We discuss the
occurrence of these patterns, and we examine their main features using
different imaging techniques and rheological measurements. Moreover, the
polymer/silica interaction is usually used to tune the mechanical properties
of composite films. Consequently, we propose here a method to suppress these
structures by adding adsorbing polymers to the silica nanoparticles.
## 2 Experimental
Fig. 1: Experimental setup. Dry or moist air is produced by an air flow from
the ambient atmosphere through desiccant or water respectively to the box.
Depending on the humidity measured by the humidity sensor inside the box, a
solenoid valve is actioned to converge the humidity to the desired value. A
honeycomb grid (represented in its actual size) is used to improve the
contrast of the surface corrugations (Schlieren technique).
### 2.1 Controlled drying conditions
Experiments consist in drying a colloidal suspension in a circular glass Petri
dish (inner diameter $2R=5.6$ cm) placed in a chamber which has controlled
temperature ($22\pm 2$ ∘C) and relative humidity ($R_{H}=50\pm 2$%) 23. The
setup is sketched in figure 1. Top and bottom walls of the chamber are
transparent for visualization by transmitted light. The bottom wall is
carefully adjusted horizontally prior each experiment. Since the sample
contrast is very low, the Schlieren technique24 is used: a honeycomb grid is
positioned between the bottom wall and an extended light source. A camera
(Nikon D300), located at the top of the chamber, records a photograph each
minute.
In the following, $m_{i}$ varies in the range $1$ to $9$ g, resulting in a
initial thickness $h_{i}$ varying in the range from $\simeq 0.3$ to $2.5$ mm.
### 2.2 Colloidal dispersions
Table 1: Main properties of silica dispersions used in this paper. The volume fraction is noted $\phi_{0}$. Values of the particle diameters come from reference 25 (p. 324). Silica | diameter $2a$ ($\mathrm{nm}$) | $\phi_{0}$ | $\rho$ ($\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$) | pH
---|---|---|---|---
SM | 10 | 0.15 | 1180 | 9.9
HS | 16 | 0.19 | 1250 | 9.8
TM | 26 | 0.20 | 1260 | 9.2
We use three aqueous dispersions of silica colloidal particles: Ludox SM-30,
HS-40 and TM-50, commercially available from Sigma-Aldrich. The pH is in the
range of 9-10, and so the particle surface bears a high negative charge
density 26. SM-30 is used without treatments, while HS-40, and TM-50 are
diluted using pure water (milliQ quality, resistivity: $18$ M$\Omega$.cm) at
pH 9.5 by addition of NaOH. A weight ratio of 90/10 (HS-40/water) and 75/25
(TM-50/water) are chosen to obtain similar initial volume fractions. In the
following, SM (unmodified), HS and TM designate these dispersions; their main
properties are reported in Table 1.
The effect of polymer chains on TM particles is investigated using
polyethylene oxyde (PEO) or alternatively polyvinylpyrrolidone (PVP). Indeed,
a high affinity for silica surfaces is known for PEO and PVP resulting in an
adsorption on silica particles. To simplify the notations, polymers are noted
$P_{i}$ as follow. Different molecular weights are studied ($P_{0}$:
$300\text{\,}\mathrm{Da}$, $P_{1}$: $600\text{\,}\mathrm{Da}$, $P_{2}$:
$3350\text{\,}\mathrm{Da}$, $P_{3}$: $6000\text{\,}\mathrm{Da}$, $P_{4}$:
$35\text{\,}\mathrm{kDa}$, $P_{5}$: $600\text{\,}\mathrm{kDa}$ and $P_{6}$:
$8\text{\,}\mathrm{MDa}$) for PEO and $P_{7}$: $40\text{\,}\mathrm{kDa}$ for
PVP. Polymers are purchased from Sigma-Aldrich and are dissolved in pure water
at pH 9.5. This solution, with a weight concentration of polymers noted
$C_{p}$, and is used for the dilution of TM-50. As a result, the final polymer
concentration in TM dispersion is $C_{p}/4$.
The adsorption of polymers on silica particles has been largely studied in the
literature 27, 28, 29, 30. In particular, it has been shown, from adsorption
isotherms, that silica particles are totally covered with polymers for
concentrations above $1$ $\mathrm{mg}\text{\,}{\mathrm{m}}^{-2}$ for PEO 27
and PVP 31. In our experiments, the polymer concentrations do not exceed
$\lesssim 0.01$ $\mathrm{mg}\text{\,}{\mathrm{m}}^{-2}$, which is much lower
than the covering concentration. In this concentration range, polymers adsorb
on particles to form necklaces29; the dispersed state is stable since the
interactions between necklaces are repulsive32.
### 2.3 Visualisation techniques
Observation using an optical microscopy (DM2500 Leica microscope) by
transmitted light is used at a macroscopic scale.
At a microscopic scale, the surface profile is investigated using an Atomic
Force Microscope (AFM, Vicco) in tapping mode on dried TM samples (scanning
area are $80\times 80$ ${\mathrm{\SIUnitSymbolMicro m}}^{2}$) prepared with an
initial weight $m_{i}=6$ g. To complete these measurements at the onset of the
propagation, an optical profiler is used (Taylor Hobson, $10\times$, working
distance $7$ mm). This contact-less method allows us to measure the surface
profile of consolidating materials with a typical surface area $1.6\times 1.6$
mm2.
## 3 Results
### 3.1 Macroscopic observations
#### 3.1.1 Temporal evolution
Fig. 2: Time evolution of a flat film of a TM sample (initial weight is
$m_{i}=6$ g). Each image is focused on the film surface (in the center, far
from the petri dish edge) and is taken at the same region of the film. The
arrow in image (4) gives the mean direction of propagation of the structures.
Image (6) shows the final pattern made of sinuous structures superposed to a
network of channeling cracks. Scale bar: $2$ mm.
During water removal, particles concentrate and the film thickness decreases.
At a given time, a network of fine dark lines progressively invades the flat
region of the film from the center to the edges. The time evolution of the
film surface is shown in figure 2. Starting from an homogeneous surface (image
1 in figure 2), surface corrugation can be observed after a few minutes of
evaporation (image 2 in figure 2). The corrugation patterns probably result
from Rayleigh-Bénard or Bénard-Marangoni convective instabilities33. However,
the convective cells disappear after a period of time; the surface recovers
its visual homogeneity (image 3 in figure 2). Then, at time $t_{onset}$,
structures progressively invade the surface of the flat film (image 4 in
figure 2) and results in the pattern shown in image 5 in figure 2. Finally,
the classical crack pattern forms in the film (image 6 in figure 2).
#### 3.1.2 Onset time
Starting from an homogeneous surface, the structures appear where the film is
thinner, i.e. in the center, far from the meniscus. For TM dispersions, we
observed that below a critical initial weight $m_{i}^{c}=1.4\pm 0.3$ g, i.e.,
$h_{i}^{c}=0.4\pm 0.1$ mm, no structure forms even if cracks do.
Fig. 3: Onset time $t_{onset}$ of structures as a function of the initial
weight and corresponding initial film thickness, for three samples: TM, TM +
$P_{5}$ (PEO, $600\text{\,}\mathrm{kDa}$) at $C_{p}=0.1$% and TM + $P_{6}$
(PVP, $40\text{\,}\mathrm{kDa}$) at $C_{p}=0.2$%. Lines are guides for the
eye. Error bars reported for TM samples are similar for other samples.
In addition, above this critical initial weight, the onset time $t_{onset}$
needed to observe the structures growth, is measured for dispersions of
different initial weights deposited in the container, ranging from $1$ to $9$
g. Results are reported in figure 3 and show that the onset time increases
linearly with the initial weight.
Table 2: Final state of sample surfaces for the different samples. Sample | SM | HS | TM | TM + ($P_{0}$, $P_{1}$, $P_{2}$, $P_{3}$) | TM + ($P_{4}$, $P_{5}$, $P_{6}$, $P_{7}$)
---|---|---|---|---|---
| | | | $\forall C_{p}$ | $C_{p}<C_{p}^{c}$ | $C_{p}>C_{p}^{c}$
Observation | $\forall h$, no pattern | $\forall h$, no pattern | patterns if $h>h_{c}$ | patterns if $h>h_{c}$ | patterns if $h>h_{c}$ | $\forall h$, no pattern
Similar experiments have been carried out with the other samples and results
are summarized in table 2. For SM and HS samples, no structure appear at the
film surface, independently of the investigated range of film thicknesses.
Consequently, we studied the addition of adsorbing polymers only on TM
samples. For a sample of initial weight $m_{i}=3$ g deposited in the container
($h_{i}=1$ mm), using PEO with molecular weights larger than $6$ kDa, the
formation of the structures is suppressed above a critical concentration
$C_{p}^{c}=0.13\%$ (within an uncertainty of $0.03\%$ for $P_{3}$ and $0.01\%$
for $P_{4}$, $P_{5}$ and $P_{6}$). Thus, the weight amount of polymers
necessary to suppress the structures is found to be independent of the
molecular weight (for $M_{w}\geq 6\textrm{kDa}$). However, for shorter polymer
chains ($M_{w}<6\textrm{kDa}$), and for concentrations up to $C_{p}=0.6$%, the
structures still form.
Moreover, addition of PVP ($40$ kDa) to TM particles results in suppression of
the structures for $C_{p}^{c}(PVP)=0.33\pm 0.03$%. We notice that the
molecular weight of a polymer unit for PEO $M_{w}^{PEO}=44$ g/mol and for PVP
$M_{w}^{PVP}=111$ g/mol:
$\frac{M_{w}^{PEO}}{M_{w}^{PVP}}\simeq\frac{C_{p}^{c}(PVP)}{C_{p}^{c}(PEO)}$
(1)
As a result, structures are suppressed for a critical number of polymer units
($160\pm 10$ units per colloidal particle, i.e. $7.6\pm 0.6$
$\mathrm{\SIUnitSymbolMicro g}\text{\,}{\mathrm{m}}^{-2}$). We checked that
drying kinetics are not affected by the addition of polymers by weight
measurements 23.
#### 3.1.3 State of the material
The drying of colloidal dispersion is a balance between two opposite fluxes:
(a) the solvent flux tends to accumulate the particles to the free surface,
whereas (b) the diffusion process smooths concentration gradients. The
competition between both processes usually states either the formation of a
skin at the surface of a drying film or the consolidation in the bulk phase.
In that way, the relevant dimensionless number is a Péclet number defined as
the ratio of a diffusion timescale $h^{2}/D$ and an advection timescale
$h/V_{e}$: $\textrm{Pe}(h)=\frac{V_{e}h}{D}$ 34, where $h$ is the film
thickness, $V_{e}$ is the evaporation rate and $D$ a diffusion coefficient.
The evaporation rate is found to be equal to $V_{e}\simeq 1\times 10^{-8}$m/s
under the same drying conditions23. The diffusion coefficient is deduced from
the Stokes-Einstein relation $D=k_{B}T/(6\pi\eta_{s}a)$, where $\eta_{s}=$
$1\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$ is the solvent viscosity. From
thicknesses lying in the range $[0.4,2.9]$ mm, the Péclet number is between
$0.2$ and $1.8$. The value close to $1$ does not allow us to distinguish
between skin formation or consolidation in the bulk.
However, to determine the state of the material, we gently collected a
fraction of the material located at the surface of our samples (e.g. TM
sample, $m_{i}=9$ g, typically $1/4$ of the paste thickness); the sample was
operated during the propagation of structures and before the apparition of
cracks. We measured a volume fraction of $\phi=38\pm 2\%$ which is compatible
with a elastic material ($\phi>0.35$) of non-aggregated particles
($\phi<0.61$) 35; this material, deposited in a test tube filled thereafter
with deionized water, can be redispersed after shaking.
### 3.2 Visualisations
#### 3.2.1 Optical microscopy
Fig. 4: Dynamics of the structure propagation in the center of a drying TM
film. The time laps between pictures (a) and (b) is $21$ seconds. The
situation indicated by arrows is detailed in the inserts. Picture (c) shows
the final pattern. (d) The series of photographs show a morphogenetic sequence
(time laps between each picture: $1$ s) leading to developed branching
process. Scale bars represent $200\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
The propagation of structures was firstly observed by optical microscopy. The
sequence of images in figure 4(a, b, c) reveals the dynamics of formation of
the structures in the center of the sample. A propagating branch (2)
approaches a prior one (1), then crosses. Moreover, in figure 4(c), the final
pattern in the sequence shows an increase of the structures contrast, and the
formation of a new generation of branches, dividing domains. The series of
photographs in figure 4(d) show a typical detailed process of the splitting of
one branch into two branches.
Fig. 5: Final pattern (a) in the center of the petri dish and (b) at $1$ cm
from the edge located at the bottom of the picture (The vector $\vec{r}$
designates the radial direction.). Scale bars represent $100$
$\mathrm{\SIUnitSymbolMicro m}$. Fig. 6: (a) Surface profile (AFM) of a dried
TM sample ($m_{i}=6$ g). The 3D profile on the right corresponds to the region
inside the yellow square in image on the left. (b) Surface profile obtained
using an optical profiler: a drying region with propagating structures from
the background to the foreground are shown in the profiles along the lines 1
and 2.
As shown in previous pictures, the structures exhibit an isotropic pattern in
the center of the sample. However, while structures propagate in a thickness
gradient naturally imposed by the meniscus at the edge of the container, they
become preferentially oriented radially and ortho-radially (figure 5(b)).
A meniscus at the edge of the container extends over few times the capillary
length scale $\kappa^{-1}=\sqrt{\frac{\gamma}{\rho g}}\simeq 2$ mm ($\gamma$
is the surface tension of water). At the final stage, we obtain isotropic
patterns (figure 5(a)) in a region covering $40$% of the total surface area
which corresponds to the flat film area.
#### 3.2.2 AFM profilometry
The surface topography of a TM sample is reported in figure 6(a). The typical
structure height ranges between $1$ $\mathrm{\SIUnitSymbolMicro m}$ and $4$
$\mathrm{\SIUnitSymbolMicro m}$. In particular, we notice the asymmetric shape
of the surface profile, as an evidence of two inclined planar surfaces
connected by a transition region of about $10$ $\mathrm{\SIUnitSymbolMicro
m}$. As a result, such measurements during the propagation of structures are
shown in figure 6(b). The red region of the surface corresponds to a thicker
layer; this can be related to the presence of the microscope objective
limiting the evaporation flux 36. In front of the structures, the film is
flat, and an arch-shaped profile is shown at the onset of the instability.
During the drying process, this shape evolves to an asymmetric shape. The
kinks present near the junction of arches (bottom graph) is due to the absence
of interference fringes with the optical profiler at the singularities
resulting in an artifact. Thus, we only consider the parts of the profile
highlighted in red.
Moreover, in the case of experiments carried out with SM and HS samples for
$m_{i}\in[2,12]$ g, no structure can be observed. This statement is also
confirmed using AFM, and optical profiler images.
Finally, measurements using AFM and optical profiler techniques reveal the
absence of structures in TM with polymers above $C_{p}^{c}$ as in the case of
observations using optical microscopy. Note that polymer addition below
$C_{p}^{c}$ leads to a lower structure height. For instance, addition of PEO
($600\text{\,}\mathrm{kDa}$, $C_{p}=0.1$%) decreases the maximum height to
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ ($m_{i}=6$ g).
### 3.3 Rheology
Fig. 7: Rheological measurements of the elastic modulus $G_{0}$ obtained from
small oscillatory shear flow tests (plate-plate geometry) for different
samples of pure particles (SM, HS and TM) and with addition of PEO ($600$ kDa)
in TM. Error bars indicate extreme values.
In the following the rheological properties of pastes are investigated at the
onset of structures formation. In order to compare the elastic moduli of
different silica pastes, measurements are performed at a well-defined
consolidation time: the onset time for TM films and the same consolidation
time for other systems. Samples are prepared using the same protocol: the
initial weights range from $7$ to $9$ g. We collect surface samples
corresponding to the 4/5th of the film thickness in the center of the
container. A part is used for a dried extract in order to deduce the volume
fraction $\phi$. The paste elastic modulus is measured using a rheometer
(Anton Paar MCR501) in the plate-plate geometry (diameter: $24.93$ mm) with a
solvent trap. The gap is adjusted between $0.3$ and $0.7$ mm accordingly to
the amount of paste used for the measurement. Small oscillatory shear flow
tests are performed with an amplitude of $0.2$%, and a frequency sweep from
$100$ Hz to $1$ Hz for $200$ s per point. Under these conditions, the elastic
modulus $G^{\prime}(\omega)$ does not vary significantly with the frequency
(relative variations lower than $10$%) and the deformation amplitude up to few
percent above which the elastic modulus decreases. The average value is
denoted by $G_{0}$ in the following.
In this way, five samples are considered: SM, HS and TM dispersions and binary
mixture of TM and PEO ($600\text{\,}\mathrm{kDa}$) at $C_{p}=0.2$% and
$C_{p}=0.4$%. For TM samples (without polymers), since structures appear at
$\phi=38\pm 2$%, we select other samples (where no structures are developed)
with similar volume fraction. Results are shown in figure 7. Our results on
the HS sample are consistent with values available in the literature37. Note
that, for all studied samples, the loss modulus ($G^{\prime\prime}(\omega)$)
appears to be at least one order of magnitude lower than $G_{0}$. For pure
dispersions, the elastic modulus decreases of nearly one order of magnitude
when the mean particle size varies from $10$ to $26$ nm. On the contrary, the
addition of a small amount of PEO adsorbing on TM particles results in the
increase of the paste elastic modulus.
## 4 Discussion
Our observations concerning the dynamics of formation and the characteristics
of the pattern show that the structures observed here can be scarcely related
to winkles. Indeed, wrinkles are periodic deformations of the interface5
whereas our patterns are clearly localized.
Folds, described as a secondary instability arising from wrinkles8, present
strong similarities such as the localization of the deformation, the
subdivision of domains and the crossing patterns11. However, the observation
of wrinkles is expected but it has never been observed in our systems.
Moreover, wrinkles and folds result from the deformation of a skin in contact
with a softer material whereas at the onset of the structures formation, the
film seems to behave like an homogeneous paste phase.
Observations at the onset of patterning (top graph of figure 6(b)) suggest
some morphological similarities with creases patterns of swelling elastomers
reported in the literature 16 while the mechanism is different. The observed
arches evolve to a“saw-tooth roof shape” as shown by AFM measurements (Fig.
6(a)). This transition is illustrated in the bottom graph of figure 6(b) where
the onset of the lost of symmetry can be seen. Indeed, this transition may be
attributed to the fact that colloidal pastes are able to change their
microstructures. A possible explanation of the symmetry breakage from the
onset of the instability to the final state can be that any further strain
energy is dissipated in a shear of the material (creep flow).
The second question addressed in this paper concerns the absence of structures
for SM and HS particles, as well as the suppression by the addition of
adsorbing polymers.
It results that mechanical properties of pastes depend on the colloidal
dispersion or the concentration of additional polymers. The strain in the
paste phase at the onset time can be deduced its elastic modulus. Indeed,
assuming that the internal stress during the drying process comes from the
flux of water through the porous matrix made by silica particles, the greatest
stress occurs at the drying surface where the pore pressure is the larger38.
There, the stress $\sigma$ scales as follows:
$\sigma\sim\frac{h\eta_{s}}{3k}V_{e}$ (2)
where $k$ is the permeability of the porous material. This expression
justifies that below a critical thickness $h_{i}^{c}$, no structure are
visible because the tensile stress is not large enough to trigger the
mechanical instability. Estimating the permeability from Carman-Kozeny
relation at $\phi=38$ % ($k=6\times 10^{-18}$ ${\mathrm{m}}^{2}$), we deduce a
critical strain $\epsilon_{c}$ at the onset of the structures formation:
$\epsilon_{c}\sim\frac{h_{i}^{c}\eta_{s}}{3kG_{0}}V_{e}\simeq 0.2$ (3)
where $G_{0}=11$ kPa for TM. It is noteworthy that for this deformation, the
elastic modulus of the material is still comparable to the one measured at low
deformation amplitudes. Moreover, the elastic modulus is measured just prior
the propagation of the structures (that occurs over a timescale of 10 min) at
a controlled volume fraction. Thus, this elastic modulus must be considered as
an order of magnitude that could be underestimated, which means that this
critical strain could beoverestimated.
Measurements show that above a critical elastic modulus, structures are
suppressed. This critical value is estimated by $G_{0}^{c}=15\pm 5$ kPa from
figure 7. Indeed, for pastes exhibiting larger elastic moduli, shrinkage
induced by the drying is withstood by the paste network leading to a low
strain. Further shrinkage is frustrated by adhesion of the paste onto the
substrate and results in tensile stress and so cracks formation. However, for
pastes exhibiting lower elastic moduli, the strain is larger leading to
possible surface corrugations. Equation 3 also suggest that $G_{0}^{c}$ should
depend on the film thickness if $\epsilon_{c}$ does not. From our
observations, we cannot conclude on a dependence of $G_{0}^{c}$ with the film
thickness because near the threshold, the presence or absence of structures
are difficult to be stated. Indeed, the structures density decreases with the
film thickness as shown in the inset of figure 3.
In the case of pure colloidal dispersions (SM, HS, TM), the particle size
affects the elastic modulus (Fig. 7), consequently the permeability. Indeed,
fitting the elastic moduli as a function of the particle size with a power
law, $G_{0}\propto a^{p}$, leads to values $p\in[-1,-3]$. The inaccuracy comes
from the large error bars and the little number of available samples. Taking
into account the particle size dependency in the permeability with Carman-
Kozeny model, the deformation varies as $\epsilon\propto a^{-(p+2)}$. A power
coefficient $p$ lower than $-2$ would be consistent with our observations on
the suppression effect. We can also note that our description assumes that
changing Ludox only results in varying particle size. It is worth to note that
other parameters may have effects, such as polydispersity (see reference 25
(p. 324) for measurements) which can modify the permeability 39.
Finally, we also observed that patterns are hierarchical (Fig. 2) which
introduces different lengthscales. The final pattern settles after $250$
minutes in the given example. This suggests that the refinement is related to
the consolidation of the material that can be analyzed by the increase of the
elastic modulus in time which increases the stress in the material.
## 5 Conclusion
In this paper, we report experimental observations on surface instabilities
during the drying of silica colloidal dispersions. These patterns grow if
films are thicker than a critical thickness and that the onset time increases
linearly with the initial film thickness and it is concomitant with a paste
phase sitting on the substrate. Measurements using microscopy techniques
highlight the surface corrugations resulting first in arches that evolves to a
saw tooth roof shape. We also provide a method to prevent these patterns
consisting in the addition of a small amount of polymers which increases the
material elastic modulus. From this technique, we estimated a critical elastic
modulus for structures formation.
As a perspective, it would be interesting to refine the description of the
evolution from arches to the saw-tooth roof shape which is attributed to a
creep flow caused by the nature of the material. Thus, a competition between
the creasing instability and the creep flow might may exist in the selection
of the final characteristic distance separating structures. In particular, in
a future work, the evolution of structures could be related to the
characteristics (onset time, final size) of the different generations in order
to establish a criterion for the distance between structures and to precise
the influence of loss of symmetry in the development of the instability.
Further theoretical developments might focus on similarities and differences
of surface instabilities in visco-elasto-plastic materials compared to visco-
elastic gels.
## References
* Li _et al._ 2012 X. Li, D. Ballerini and W. Shen, _Biomicrofluidics_ , 2012, 6, 011301.
* Cerda and Mahadevan 2003 E. Cerda and L. Mahadevan, _Phys. Rev. Lett._ , 2003, 90, 074302.
* Huraux _et al._ 2012 K. Huraux, T. Narita, B. Bresson, C. Fretigny and F. Lequeux, _Soft Matter_ , 2012, 8, 8075–8081.
* Xuan _et al._ 2012 Y. Xuan, X. Guo, Y. Cui, C. Yuan, H. Ge, B. Cui and Y. Chen, _Soft Matter_ , 2012, 8, 9603–9609.
* Chen and Crosby 2013 Y.-C. Chen and A. Crosby, _Soft Matter_ , 2013, 9, 43–47.
* Genzer and Groenewold 2006 J. Genzer and J. Groenewold, _Soft Matter_ , 2006, 2, 310–323.
* Groenewold 2001 J. Groenewold, _Physica A: Statistical Mechanics and its Applications_ , 2001, 298, 32–45.
* Brau _et al._ 2013 F. Brau, P. Damman, H. Diamant and T. Witten, _Soft Matter_ , 2013, 9, 8177–8186.
* Brau _et al._ 2011 F. Brau, H. Vandeparre, A. Sabbah, C. Poulard, A. Boudaoud and P. Damman, _Nat Phys_ , 2011, 7, 56–60.
* Pocivavsek _et al._ 2008 L. Pocivavsek, R. Dellsy, A. Kern, S. Johnson, B. Lin, K. Lee and E. Cerda, _Science_ , 2008, 320, 912–916.
* Kim _et al._ 2011 P. Kim, M. Abkarian and H. Stone, _Nat Mater_ , 2011, 10, 952–957.
* Reis 2011 P. Reis, _Nat Mater_ , 2011, 10, 907–909.
* Ebata _et al._ 2012 Y. Ebata, A. Croll and A. Crosby, _Soft Matter_ , 2012, 8, 9086–9091.
* Reis _et al._ 2009 P. Reis, F. Corson, A. Boudaoud and B. Roman, _Phys. Rev. Lett._ , 2009, 103, 045501.
* Hong _et al._ 2009 W. Hong, X. Zhao and Z. Suo, _Applied Physics Letters_ , 2009, 95, 111901–3.
* Cai _et al._ 2012 S. Cai, D. Chen, Z. Suo and R. Hayward, _Soft Matter_ , 2012, 8, 1301–1304.
* Trujillo _et al._ 2008 V. Trujillo, J. Kim and R. Hayward, _Soft Matter_ , 2008, 4, 564–569.
* Cai _et al._ 2010 S. Cai, K. Bertoldi, H. Wang and Z. Suo, _Soft Matter_ , 2010, 6, 5770–5777.
* Hohlfeld and Mahadevan 2011 E. Hohlfeld and L. Mahadevan, _Phys. Rev. Lett._ , 2011, 106, 105702.
* Tallinen _et al._ 2013 T. Tallinen, J. Biggins and L. Mahadevan, _Phys. Rev. Lett._ , 2013, 110, 024302.
* Xu _et al._ 2009 P. Xu, A. Mujumdar and B. Yu, _Drying Technology_ , 2009, 27, 636 – 652.
* Routh 2013 A. Routh, _Rep. Prog. Phys._ , 2013, 76, 046603.
* Boulogne _et al._ 2013 F. Boulogne, F. Giorgiutti-Dauphiné and L. Pauchard, Oil Gas Sci. Technol. \- Rev. IFP Energies nouvelles, 2013.
* Weinstein 2010 L. Weinstein, _The European Physical Journal Special Topics_ , 2010, 182, 65–95.
* Bergna 1994 H. Bergna, _The Colloid Chemistry of Silica_ , American Chemical Society, 1994.
* Iler 1979 R. Iler, _The Chemistry of Silica: Solubility, Polymerization, Colloid and Surface Properties and Biochemistry of Silica_ , Wiley-Interscience, 1979.
* Lafuma _et al._ 1991 F. Lafuma, K. Wong and B. Cabane, _Journal of Colloid and Interface Science_ , 1991, 143, 9–21.
* Otsubo 1990 Y. Otsubo, _Langmuir_ , 1990, 6, 114–118.
* Wong _et al._ 1992 K. Wong, P. Lixon, F. Lafuma, P. Lindner, O. Charriol and B. Cabane, _Journal of Colloid and Interface Science_ , 1992, 153, 55–72.
* Parida _et al._ 2006 S. Parida, S. Dash, S. Patel and B. Mishra, _Advances in Colloid and Interface Science_ , 2006, 121, 77–110.
* Ilekti 2000 P. Ilekti, _Ph.D. thesis_ , Université Pierre et Marie Curie, 2000.
* Cabane _et al._ 1997 B. Cabane, K. Wong, P. Lindner and F. Lafuma, _Journal of Rheology_ , 1997, 41, 531–547.
* Bassou and Rharbi 2009 N. Bassou and Y. Rharbi, _Langmuir_ , 2009, 25, 624–632.
* Routh and Russel 1998 A. Routh and W. Russel, _AIChE J._ , 1998, 44, 2088–2098.
* Boulogne _et al._ 2014 F. Boulogne, L. Pauchard, F. Giorgiutti-Dauphiné, R. Botet, R. Schweins, M. Sztucki, J. Li, B. Cabane and L. Goehring, _EPL_ , 2014, 105, 38005.
* Parneix _et al._ 2010 C. Parneix, P. Vandoolaeghe, V. Nikolayev, D. Quéré, J. Li and B. Cabane, _Phys. Rev. Lett._ , 2010, 105, 266103.
* Di Giuseppe _et al._ 2012 E. Di Giuseppe, A. Davaille, E. Mittelstaedt and M. François, _Rheologica Acta_ , 2012, 51, 451–465.
* Brinker and Scherer 1990 C. Brinker and G. Scherer, _Sol-Gel Science: The Physics and Chemistry of Sol-Gel Processing_ , Academic Press, 1990.
* Scherer and Swiatek 1989 G. Scherer and R. Swiatek, _Journal of Non-Crystalline Solids_ , 1989, 113, 119–129.
## 6 acknowledgments
The authors thank Triangle de la Physique for the rheometer (Anton Paar, MCR
501) and A. Aubertin for designing the experimental setup. We are grateful to
B. Cabane, J.-P. Hulin, G. Gauthier and C. Quilliet for fruitful discussions,
A. Chennevière, C. Poulard and F. Restagno for guidance and sharing their AFM
and optical profiler.
|
arxiv-papers
| 2014-02-07T14:05:52 |
2024-09-04T02:49:57.947270
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fran\\c{c}ois Boulogne, Fr\\'ed\\'erique Giorgiutti-Dauphin\\'e, Ludovic\n Pauchard",
"submitter": "Fran\\c{c}ois Boulogne",
"url": "https://arxiv.org/abs/1402.1644"
}
|
1402.1837
|
# Pion masses in 2-flavor QCD with $\eta$ condensation
Sinya Aoki1,2 and Michael Creutz3 1Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan
3Center for Computational Sciences, University of Tsukuba, Ibaraki 305-8571,
Japan
3Physics Department 510A, Brookhaven National Laboratory, Upton, NY
11973,USA111This manuscript has been authored under contract number DE-
AC02-98CH10886 with the U.S. Department of Energy. Accordingly, the U.S.
Government retains a non-exclusive, royalty-free license to publish or
reproduce the published form of this contribution, or allow others to do so,
for U.S. Government purposes.
###### Abstract
We investigate some aspects of 2-flavor QCD with $m_{u}\not=m_{d}$ at low-
energy, using the leading order chiral perturbation theory including anomaly
effects. While nothing special happens at $m_{u}=0$ for the fixed
$m_{d}\not=0$, the neutral pion mass becomes zero at two critical values of
$m_{u}$, between which the neutral pion field condenses, leading to a
spontaneously CP broken phase, the so-called Dashen phase. We also show that
the ”topological susceptibility” in the ChPT diverges at these two critical
points. We briefly discuss a possibility that $m_{u}=0$ can be defined by the
vanishing the ”topological susceptibility. We finally analyze the case of
$m_{u}=m_{d}=m$ with $\theta=\pi$, which is equivalent to $m_{u}=-m_{d}=-m$
with $\theta=0$ by the chiral rotation. In this case, the $\eta$ condensation
occurs at small $m$, violating the CP symmetry spontaneously. Deep in the
$\eta$ condensation phase, three pions become Nambu-Goldstone bosons, but they
show unorthodox behavior at small $m$ that $m_{\pi}^{2}=O(m^{2})$, which,
however, is shown to be consistent with the chiral Ward-Takahashi identities.
###### pacs:
12.38.Gc, 13.75.Cs, 21.65.Mn, 26.60.Kp
††preprint: YITP-14-12
## I Introduction
One of possible solutions to the strong CP problem is ”massless up quark”,
where the $\theta$ term in QCD can be rotated away by the chiral rotation of
up quark without affecting other part of the QCD action. This solution,
unfortunately, seems to be ruled out by results from lattice QCD
simulationsNelson:2003tb .
In a series of papersCreutz:1995wf ; Creutz:2003xu ; Creutz:2003xc ;
Creutz:2005gb ; Creutz:2013xfa , however, one of the present authors has
argued that a concept of ”massless up quark” is ill-defined if other quarks
such as a down quark are all massive, since no symmetry can guarantee
masslessness of up quark in this situation due to the chiral anomaly. In
addition, it has been also argued that a neutral pion becomes massless at some
negative value of up quark mass for the positive down quark mass fixed, and
beyond that point, the neutral pion field condenses, forming a spontaneous CP
breaking phase, so-called a Dashen phaseDashen:1971aa . Furthermore, at the
phase boundary, the topological susceptibility is claimed to diverge due to
the massless neutral pion, while it may become zero at the would-be “massless
up quark” point.
The purpose of this letter is to investigate above properties of QCD with non-
degenerate quarks in more detail, using the chiral perturbation theory (ChPT)
with the effect of anomaly included as the determinant term. For simplicity,
we consider the $N_{f}=2$ case with $m_{u}\not=m_{d}$, but a generalization to
an arbitrary number of $N_{f}$ is straightforward with a small modification.
Our analysis explicitly demonstrates the above-mentioned properties such as an
absence of any singularity at $m_{u}=0$ and the existence of the Dashen phase
with the appearance of a massless pion at the phase boundaries. We further
apply our analysis to the case of $m_{u}=m_{d}=m$ with $\theta=\pi$, which is
equivalent to $m_{u}=-m_{d}=-m$ with $\theta=0$ by the chiral rotation. We
show that, while $\eta$ condensation occurs, violating the CP symmetry
spontaneously, three pions become Nambu-Glodstone (NG) bosons at $m=0$ deep in
the $\eta$ condensation phase. We also show a unorthodox behavior at small $m$
that $m_{\pi}^{2}=O(m^{2})$, which is indeed shown to be consistent with the
chiral Ward-Takahashi identities (WTI).
## II Phase structure, masses and topological susceptibility
The theory we consider in this letter is given by
$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle\frac{f^{2}}{2}{\rm
tr}\,\left(\partial_{\mu}U\partial^{\mu}U^{\dagger}\right)-\frac{1}{2}{\rm
tr}\,\left(M^{\dagger}U+U^{\dagger}M\right)$ (1) $\displaystyle-$
$\displaystyle\frac{\Delta}{2}\left(\det U+\det U^{\dagger}\right),$
where $f$ is the pion decay constant, $M$ is a quark mass matrix, and $\Delta$
is a positive constant giving an additional mass to an eta meson. Differences
between an ordinary ChPT and the above theory we consider are the presence of
the determinant term222Based on the large $N$ behavior, it is standard to use
$(\log\det U)^{2}$ term to incorporate anomaly effects into ChPTWitten:1980sp
; Rosenzweig:1979ay ; Kawarabayashi:1980dp ; Arnowitt:1980ne . Since we can
determine the phase structure only numerically in this case, we instead employ
$\det U$ term in our analysis, which leads to he phase structure determined
analytically. We have checked that the two cases lead to a qualitatively
similar phase structure except at large quark masses., which breaks $U(1)$
axial symmetry, thus representing the anomaly effect, and field $U\in
U(N_{f})$ instead of $U\in SU(N_{f})$. We here ignore $\det U$ terms with
derivatives for simplicity, since they do not change our conclusions. For
$N_{f}=2$, without a loss of generality, the mass term is taken as
$\displaystyle M$ $\displaystyle=$ $\displaystyle
e^{i\theta}\left(\begin{array}[]{cc}m_{u}&0\\\ 0&m_{d}\\\
\end{array}\right)\equiv e^{i\theta}2B\left(\begin{array}[]{cc}m_{0u}&0\\\
0&m_{0d}\\\ \end{array}\right),$ (6)
where $B$ is related to the magnitude of the chiral condensate, $m_{0u,0d}$
are bare quark masses, and $\theta$ represents the $\theta$ parameter in QCD.
We consider that any explicit $F\tilde{F}$ term in the action has been rotated
into the mass matrix. In the most of our analysis, we take $\theta=0$, but an
extension of our analysis to $\theta\not=0$ is straight-forward.
Let us determine the vacuum structure of the theory at $m_{u}\not=m_{d}$.
Minimizing the action with
$\displaystyle U(x)$ $\displaystyle=$ $\displaystyle
U_{0}=e^{i\varphi_{0}}e^{i\sum_{a=1}^{3}\tau^{a}\varphi_{a}},$ (7)
we obtain the phase structure given in Fig. 1, which is symmetric with respect
to $m_{+}\equiv m_{u}+m_{d}=0$ axis and $m_{-}\equiv m_{d}-m_{u}=0$ axis,
separately. The former symmetry is implied by the chiral rotation that
$U\rightarrow e^{i\pi\tau^{1}/2}Ue^{i\pi\tau^{1}/2}$ ($\psi\rightarrow
e^{i\pi\gamma_{5}\tau^{1}/2}\psi$ for the quark), while the latter by the
vector rotation that $U\rightarrow e^{i\pi\tau^{1}/2}Ue^{-i\pi\tau^{1}/2}$
($\psi\rightarrow e^{i\pi\tau^{1}/2}\psi$ )333This phase structure seems
incompatible with the 1+1 flavor QCD with rooted staggered quarks, which is
symmetric individually under $m_{u}\rightarrow-m_{u}$ or $m_{d}\rightarrow-
m_{d}$. This suggests that the rooted trick for the staggered quarks can not
be used in a region at $m_{d}m_{u}<0$..
Figure 1: Phase structure in $m_{u}$-$m_{d}$ plain, where the CP breaking
Dashen phase are shaded in blue, while the CP preserving phase with
$U_{0}=\tau^{3}$ (lower right) or $U_{0}=-\tau^{3}$ (upper left) are shaded in
red.
In the phase A (white), $U_{0}={\bf 1}_{2\times 2}$ (upper right) or
$U_{0}=-{\bf 1}_{2\times 2}$(lower left), while $U_{0}=\tau^{3}$ (lower right)
or $U_{0}=-\tau^{3}$ (upper left) in the phase C (shaded in red). In the phase
B (shaded in blue), we have nontrivial minimum with
$\displaystyle\sin^{2}(\varphi_{3})$ $\displaystyle=$
$\displaystyle\frac{(m_{d}-m_{u})^{2}\\{(m_{u}+m_{d})^{2}\Delta^{2}-m_{u}^{2}m_{d}^{2}\\}}{4m_{u}^{3}m_{d}^{3}}$
(8) $\displaystyle\sin^{2}(\varphi_{0})$ $\displaystyle=$
$\displaystyle\frac{(m_{u}+m_{d})^{2}\Delta^{2}-m_{u}^{2}m_{d}^{2}}{4m_{u}m_{d}\Delta^{2}},$
(9)
which breaks CP symmetry spontaneously, since $\langle\pi^{0}\rangle={\rm
tr}\,\tau^{3}(U_{0}-U_{0}^{\dagger})/(2i)=2\cos(\varphi_{0})\sin(\varphi_{3})$
and $\langle\eta\rangle={\rm
tr}\,(U_{0}-U_{0}^{\dagger})/(2i)=2\sin(\varphi_{0})\cos(\varphi_{3})$. This
phase, where the neutral pion and the eta fields condense, corresponds to the
Dashen phase. The spontaneous CP breaking 2nd-order phase transition occurs at
the boundaries of the Dashen phase: Lines between phase A and phase B, on
which $\sin^{2}\varphi_{3}=\sin^{2}\varphi_{0}=0$, are defined by
$(m_{d}+m_{u})\Delta+m_{d}m_{u}=0$ (a line $\overline{aa^{\prime}}$) and
$(m_{d}+m_{u})\Delta-m_{d}m_{u}=0$ (a line $\overline{bb^{\prime}}$), while
those between $B$ and $C$, on which
$\sin^{2}\varphi_{3}=\sin^{2}\varphi_{0}=1$, are given by
$(m_{d}-m_{u})\Delta+m_{d}m_{u}=0$ (a line $\overline{ab}$) and
$(m_{d}-m_{u})\Delta-m_{d}m_{u}=0$ (a line $\overline{a^{\prime}b^{\prime}}$).
Note that $\sin^{2}\varphi_{3}=1$ also on a $m_{+}=0$ line.
We next calculate pseudo-scalar meson masses in each phase. Expanding $U(x)$
around $U_{0}$ as $U(x)=U_{0}e^{i\Pi(x)/f}$ with
$\displaystyle\Pi(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\displaystyle\frac{\eta(x)+\pi_{0}(x)}{\sqrt{2}}&\pi_{-}(x)\\\
\pi_{+}(x)&\displaystyle\frac{\eta(x)+\pi_{0}(x)}{\sqrt{2}}\\\
\end{array}\right),$ (12)
the mass term is given by
$\displaystyle{\cal L}^{M}$ $\displaystyle=$
$\displaystyle\frac{m_{+}(\vec{\varphi})}{4f^{2}}\left\\{\eta^{2}(x)+\pi_{0}^{2}(x)+2\pi_{+}(x)\pi_{-}(x)\right\\}$
(13) $\displaystyle+$ $\displaystyle\frac{\delta
m}{2f^{2}}\eta^{2}(x)-\frac{m_{-}(\vec{\varphi})}{2f^{2}}\eta(x)\pi_{0}(x),$
where
$m_{\pm}(\vec{\varphi})=m_{\pm}\cos(\varphi_{0})\cos(\varphi_{3})+m_{\mp}\sin(\varphi_{0})\sin(\varphi_{3})$
with $\delta m=2\Delta\cos(2\varphi_{0})$. While the charged meson mass
$m_{\pi_{\pm}}$ is simply given by
$m_{\pi_{\pm}}^{2}=m_{+}(\vec{\varphi})/(2f^{2})$, mass eigenstates,
$\displaystyle\left(\begin{array}[]{c}\tilde{\pi}_{0}(x)\\\ \tilde{\eta}(x)\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2X}}\left(\begin{array}[]{c}X_{+}^{1/2}\pi_{0}(x)+X_{-}^{1/2}\eta(x)\\\
X_{-}^{1/2}\pi_{0}(x)-X_{+}^{1/2}\eta(x)\\\ \end{array}\right),$ (18)
have
$\displaystyle m_{\tilde{\pi}_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2f^{2}}\left[m_{+}(\vec{\varphi})+\delta m-X\right],$
(19) $\displaystyle m_{\tilde{\eta}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2f^{2}}\left[m_{+}(\vec{\varphi})+\delta m+X\right],$
(20)
where $X=\sqrt{m_{-}(\vec{\varphi})^{2}+\delta m^{2}}$ and $X_{\pm}=X\pm\delta
m$. We here choose $\tilde{\pi}_{0}$ and $\tilde{\eta}$ such that
$m_{\tilde{\pi}_{0}}^{2}\leq m_{\tilde{\eta}}^{2}$. It is then easy to see
$m_{\tilde{\pi}_{0}}^{2}\leq m_{\pi_{\pm}}^{2}\leq m_{\tilde{\eta}}^{2}$.
By plugging $\varphi_{0}$ and $\varphi_{3}$ into the above formula, we obtain
meson masses in each phase. Here we show that $m_{\tilde{\pi}_{0}}^{2}=0$ at
all phase boundaries, to demonstrate that the phase transition is indeed of
second order. In the phase A, we have
$\displaystyle m_{\tilde{\pi}_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2f^{2}}\left[|m_{+}|+2\Delta-\sqrt{m_{-}^{2}+4\Delta}\right],$
(21)
which becomes zero at $(m_{d}+m_{u})\Delta+m_{d}m_{u}=0$ (on
$\overline{aa^{\prime}}$) and at $(m_{d}+m_{u})\Delta-m_{d}m_{u}=0$ (on
$\overline{bb^{\prime}}$). Note that nothing special happens at $m_{u}=0$ (a
massless up quark) at $m_{d}\not=0$ as
$m_{\tilde{\pi}_{0}}^{2}=(|m_{d}|+2\Delta-\sqrt{m_{d}^{2}+4\Delta})/(2f^{2})$.
In the phase C, we obtain
$\displaystyle m_{\tilde{\pi}_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2f^{2}}\left[|m_{-}|-2\Delta-\sqrt{m_{+}^{2}+4\Delta}\right],$
(22)
$m_{\tilde{\pi}_{0}}^{2}=0$ at $(m_{d}-m_{u})\Delta+m_{d}m_{u}=0$ (on
$\overline{ab}$) and at $(m_{d}-m_{u})\Delta-m_{d}m_{u}=0$ (on
$\overline{a^{\prime}b^{\prime}}$). In addition, it is easy to check that the
massless condition for $\tilde{\pi}_{0}$ that $m_{+}(\vec{\varphi})+\delta
m=\sqrt{m_{-}(\vec{\varphi})^{2}+\delta m^{2}}$ in the phase B can be
satisfied only on all boundaries of the phase B.
So far, we have shown three claims in Creutz:1995wf ; Creutz:2003xu ;
Creutz:2003xc ; Creutz:2013xfa that (1) the Dashen phase with spontaneous CP
breaking by the pion condensate exists in non-degenerate 2-flavor QCD, (2) the
massless neutral pion appears at the boundaries of the Dashen phase, and (3)
nothing special happens at $m_{u}=0$ except at $m_{d}=0$.
We now consider the relation between the topological susceptibility and
$m_{u}$ in the ChPT. To define the topological susceptibility in ChPT, let us
consider the chiral U(1) WTI given by
$\displaystyle\langle\left\\{\partial^{\mu}A_{\mu}^{0}(x)+{\rm
tr}\,M(U^{\dagger}(x)-U(x))-2N_{f}q(x)\right\\}{\cal O}(y)\rangle$ (23)
$\displaystyle=$ $\displaystyle\delta^{(4)}(x-y)\langle\delta^{0}{\cal
O}(y)\rangle$
where $A_{\mu}=f^{2}{\rm tr}\\{U^{\dagger}(x)\partial_{\mu}U(x)-U(x)\partial
U^{\dagger}(x)\\}$ is the U(1) axial current, ${\cal O}$ and $\delta^{0}{\cal
O}$ are an arbitrary operator and its infinitesimal local axial U(1) rotation,
respectively, and $2N_{f}q(x)\equiv\Delta\\{\det U(x)-\det U^{\dagger}(x)\\}$
corresponds to the topological charge density. Taking ${\cal O}(y)=q(y)$ and
integrating over $x$, we define the topological susceptibility in the ChPT
through WTI as
$\displaystyle 2N_{f}\chi\equiv\int
d^{4}x\langle\\{\partial^{\mu}A_{\mu}^{0}(x)+{\rm
tr}\,M(U^{\dagger}(x)-U(x))\\}q(y)\rangle,$
$\displaystyle=\frac{\Delta^{2}}{4}\int d^{4}x\langle
q(x)q(y)\rangle+\frac{\Delta}{2}\langle\det U(x)+\det U^{\dagger}(x)\rangle,$
(24)
where the second term comes from $\delta^{0}q(x)$ in ChPT, which is absent in
QCD, but represent an effect of the contact term of $q(x)q(y)$ in ChPT. The
leading order in ChPT gives
$\displaystyle 2N_{f}\chi$ $\displaystyle=$
$\displaystyle-\frac{4\Delta^{2}m_{+}(\vec{\varphi})}{m_{+}(\vec{\varphi})^{2}-m_{-}(\vec{\varphi})^{2}+2m_{+}(\vec{\varphi})\delta
m}+\Delta.$ (25)
At $m_{u}=0$, we have $m_{+}(\vec{\varphi})=m_{-}(\vec{\varphi})=|m_{d}|$ and
$\delta m=2\Delta$, so that
$\displaystyle 2N_{f}\chi=-4\Delta^{2}|m_{d}|/(4|m_{d}|\Delta)+\Delta=0,$ (26)
which confirms the statement that (4) $\chi=0$ at $m_{u}=0$. Since the
denominator of $\chi$ is proportional to $m_{\tilde{\pi}_{0}}^{2}\times
m_{\tilde{\eta}}^{2}$, $\chi\rightarrow-\infty$ on all phase boundaries since
$m_{\tilde{\pi}_{0}}^{2}=0$ and $m_{+}(\vec{\varphi})>0$, which again confirms
the statement that (5) $\chi$ negatively diverges at the phase boundaries
where the neutral pion becomes massless.
We have confirmed the five statements in Ref. Creutz:1995wf ; Creutz:2003xu ;
Creutz:2003xc ; Creutz:2013xfa , (1) – (5) in the above, by the ChPT analysis.
In addition, we have found a new CP preserving phase, the phase C, which has
$U_{0}=\pm\tau^{3}$ instead of $U_{0}=\pm{\bf 1}_{2\times 2}$ of the phase A.
Since the phase C occurs at rather heavy quark masses such that
$m_{u,d}=2Bm^{0}_{u,d}=O(\Delta)$, however, the leading order ChPT analysis
may not be reliable for the phase C. Indeed, the phase C seems to disappear if
$(\log\det U)^{2}$ is employed instead of $\det U$. Other properties, (1) –
(5), on the other hand, are robust, since they already occur near the origin
($m_{u}=m_{d}=0$) in the $m_{u}-m_{d}$ plain and they survive even if
$(\log\det U)^{2}$ is used.
The property (4) suggests an interesting possibility that one can define
$m_{u}=0$ at $m_{d}\not=0$ in 2-flavor QCD from a condition that $\chi=0$.
This is different from the standard statement that the effect of $\theta$ term
is rotated away at $m_{u}=0$. We instead define $m_{u}=0$ from $\chi=0$, which
is equivalent to an absence of the $\theta$ dependence if higher order
cumulants of topological charge fluctuations are all absent. A question we may
have is whether $\chi=0$ is a well defined condition or not. As already
discussed in Ref. Creutz:1995wf ; Creutz:2003xu ; Creutz:2003xc ;
Creutz:2013xfa , a value of $\chi$, and thus the $\chi=0$ condition, depend on
its definition at finite lattice spacing (cut-off). Although one naively
expect such ambiguity of $\chi$ disappears in the continuum limit, we must
check a uniqueness of $\chi$ explicitly in lattice QCD calculations by
demonstrating that $\chi$ from two different definitions but at same physical
parameters agree in the continuum limit. If the uniqueness of $\chi$ can be
established, one should calculate $\chi$ at the physical point of 1+ 1+1
flavor QCD in the continuum limit. If $\chi\not=0$ in the continuum limit, the
solution to the U(1) problem by the massless up quark ( $\chi=0$ in our
definition) is ruled out.
## III Degenerate 2-flavor QCD at $\theta=\pi$
In the remainder of this letter, as an application of our analysis, we
consider the 2-flavor QCD with $m_{u}=m_{d}=m$ and $\theta=0$, which is
equivalent to the 2-flavor QCD with $m_{u}=-m_{d}$ but $\theta=0$. In both
systems, we have a SU(2) symmetry generated by
$\\{\tau^{1},\tau^{2},\tau^{3}\\}$ for the former or
$\\{\tau^{1}\gamma_{5},\tau^{2}\gamma_{5},\tau^{3}\\}$ for the latter. We here
give results for the former case, but a reinterpretation of results in the
latter case is straightforward.
The vacuum is given by $\varphi_{3}=0$ and
$\displaystyle\cos\varphi_{0}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lc}1,&2\Delta\leq m\\\
\displaystyle\frac{m}{2\Delta},&-2\Delta<m<2\Delta\\\ -1,&m\leq-2\Delta\\\
\end{array}\right.,$ (30)
which leads to
$\displaystyle\langle\bar{\psi}i\gamma_{5}\psi\rangle$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lc}0,&m^{2}\geq 4\Delta^{2}\\\ \pm
2\sqrt{1-\displaystyle\frac{m^{2}}{4\Delta^{2}}},&m^{2}<4\Delta^{2}\\\
\end{array}\right.,$ (33) $\displaystyle\langle\bar{\psi}\psi\rangle$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{lc}2,&2\Delta\leq m\\\
\displaystyle\frac{m}{\Delta},&-2\Delta<m<2\Delta\\\ -2,&m\leq-2\Delta\\\
\end{array}\right.,$ (37)
showing the spontaneous CP symmetry breaking at $m^{2}<4\Delta^{2}$. Note that
$\langle\bar{\psi}\psi\rangle^{2}+\langle\bar{\psi}i\gamma_{5}\psi\rangle^{2}=4$
at all $m$.
PS meson masses are calculated as
$\displaystyle m_{\pi}^{2}$ $\displaystyle=$ $\displaystyle
m_{\pi_{\pm}}^{2}=m_{\pi_{0}}^{2}=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2f^{2}}2|m|,&m^{2}\geq
4\Delta^{2}\\\
\displaystyle\frac{1}{2f^{2}}\frac{m^{2}}{\Delta},&m^{2}<4\Delta^{2}\\\
\end{array}\right.,$ (40) $\displaystyle m_{\eta}^{2}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2f^{2}}\left[2|m|-4\Delta\right],&m^{2}\geq
4\Delta^{2}\\\
\displaystyle\frac{1}{2f^{2}}\frac{4\Delta^{2}-m^{2}}{\Delta},&m^{2}<4\Delta^{2}\\\
\end{array}\right.,$ (43)
where $\eta$ becomes massless at the phase boundaries at $m^{2}=4\Delta^{2}$,
showing that $\eta$ is the massless mode associated with the spontaneous CP
symmetry breaking phase transition, while three pion become massless Nambu-
Goldstone modes at $m=0$. Fig. 2 represents these behaviors.
Figure 2: $m_{\pi}^{2}$ (blue) and $m_{\eta}^{2}$ (red) in unit of
$\frac{\Delta}{f^{2}}$ as a function of $m$.
As mentioned before, although ChPT analysis around the phase transition points
at $m^{2}=4\Delta^{2}$ may not be reliable444Indeed, both CP preserving phase,
which correspond to the phase C, and the phase transition points, disappear if
$(\log\det U)^{2}$ term is employed instead of $\det U$ term. In this case CP
is broken at all $m$., we can trust the results near $m=0$ that the CP
symmetry is spontaneously broken by the $\eta$ condensation in the degenerate
2-flavor QCD with $\theta=\pi$ and three pions become massless NG bosons at
$m=0$. Pion masses, however, behaves as $m_{\pi}^{2}=m^{2}/(2f^{2}\Delta)$
near $m=0$, contrary to the orthodox PCAC relation that
$m_{\pi}^{2}=|m|/(2f^{2})$555A similar behavior has been predicted in a
different contextKnecht:1994zb ; Stern:1997ri ; Stern:1998dy .. Let us show
that this unorthodox relation can be explained by the WTI. The integrated WTI
for the non-singlet chiral rotation with $\tau^{3}$ and ${\cal O}={\rm
tr}\,\tau^{3}(U^{\dagger}-U)$ reads
$\displaystyle m\int d^{4}x\,{\rm tr}\,\tau^{3}(U^{\dagger}-U)(x){\rm
tr}\,\tau^{3}(U^{\dagger}-U)(y)\rangle$ (44) $\displaystyle=$
$\displaystyle-2\langle{\rm tr}\,(U+U^{\dagger})(y)\rangle,$
which leads to
$\displaystyle m_{\pi_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{m}{f^{2}}\cos\varphi_{0}=\frac{m^{2}}{f^{2}}\frac{m}{2\Delta}.$
(45)
This tells us that one $m$ explicitly comes from the WTI, the other $m$ from
the VEV of $\bar{\psi}\psi$, giving the unorthodox relation that
$\displaystyle m_{\pi}^{2}$ $\displaystyle=$
$\displaystyle\frac{m^{2}}{2f^{2}\Delta}.$ (46)
It is interesting and challenging, because of sign problems, to confirm this
prediction by lattice QCD simulations with $\theta=\pi$, and to consider
possible applications of this to particle physics Dashen:1971aa .
###### Acknowledgements.
S.A thanks Dr. T. Hatsuda for useful comments. S.A is partially supported by
Grant-in-Aid for Scientific Research on Innovative Areas(No.2004:20105001) and
for Scientific Research (B) 25287046 and SPIRE (Strategic Program for
Innovative REsearch).
## References
* (1) D. R. Nelson, G. T. Fleming and G. W. Kilcup, Phys. Rev. Lett. 90, 021601 (2003) [hep-lat/0112029].
* (2) M. Creutz, Phys. Rev. D 52, 2951 (1995) [hep-th/9505112].
* (3) M. Creutz, Phys. Rev. Lett. 92, 201601 (2004) [hep-lat/0312018].
* (4) M. Creutz, Phys. Rev. Lett. 92, 162003 (2004) [hep-ph/0312225].
* (5) M. Creutz, PoS, LAT2005:119 (2006).
* (6) M. Creutz, Annals Phys. 339, 560 (2013) [arXiv:1306.1245 [hep-lat]].
* (7) R. F. Dashen, Phys. Rev. D 3, 1879 (1971).
* (8) E. Witten, Annals Phys. 128, 363 (1980).
* (9) C. Rosenzweig, J. Schechter, and C.G. Trahern, Phys. Rev. D21 (1980)3388.
* (10) Ken Kawarabayashi and Nobuyoshi Ohta. Nucl. Phys. B175 (1980) 477.
* (11) Richard L. Arnowitt and Pran Nath. Nucl. Phys. B209 (1982) 234.
* (12) M. Knecht and J. Stern, hep-ph/9411253.
* (13) J. Stern, In *Mainz 1997, Chiral dynamics: Theory and experiment* 26-45 [hep-ph/9712438].
* (14) J. Stern, [hep-ph/9801282].
|
arxiv-papers
| 2014-02-08T09:52:02 |
2024-09-04T02:49:57.967692
|
{
"license": "Public Domain",
"authors": "Sinya Aoki and Michael Creutz",
"submitter": "Sinya Aoki",
"url": "https://arxiv.org/abs/1402.1837"
}
|
1402.1922
|
1em1em
# Amortised Resource Analysis and Typed Polynomial Interpretations
(extended version)††thanks: This research is partly supported by FWF (Austrian
Science Fund) project P25781.
Martin Hofmann
Institute of Computer Science
LMU Munich Germany
email [email protected] Georg Moser
Institute of Computer Science
University of Innsbruck Austria
email [email protected]
###### Abstract
We introduce a novel resource analysis for typed term rewrite systems based on
a potential-based type system. This type system gives rise to polynomial
bounds on the innermost runtime complexity. We relate the thus obtained
amortised resource analysis to polynomial interpretations and obtain the
perhaps surprising result that whenever a rewrite system $\mathcal{R}$ can be
well-typed, then there exists a polynomial interpretation that orients
$\mathcal{R}$. For this we adequately adapt the standard notion of polynomial
interpretations to the typed setting.
_Key words_ : Term Rewriting, Types, Amortised Resource Analysis, Complexity
of Rewriting, Polynomial Interpretations
## 1 Introduction
In recent years there have been several approaches to the automated analysis
of the complexity of programs. Mostly these approaches have been developed
independently in different communities and use a variety of different, not
easily comparable techniques. Without hope for completeness, we mention work
by Albert et al. AAGP:2011 that underlies COSTA, an automated tool for the
resource analysis of Java programs. Related work, targeting C programs, has
been reported by Alias et al. ADFG:2010 . In Zuleger et al. ZGSV:2011 further
approaches for the runtime complexity analysis of C programs is reported,
incorporated into LOOPUS. Noschinski et al. NoschinskiEG13 study runtime
complexity analysis of rewrite systems, which has been incorporated in AProVE.
Finally, the RaML prototype HAH:2012 provides an automated potential-based
resource analysis for various resource bounds of functional programs and TCT
AvanziniM13a is one of the most powerful tools for complexity analysis of
rewrite systems.
Despite the abundance in the literature almost no comparison results are known
that relate the sophisticated methods developed. Indeed a precise comparison
often proves difficult. For example, on the surface there is an obvious
connection between the decomposition techniques established by Gulwani and
Zuleger in GulwaniZuleger:2010 and recent advances on this topic in the
complexity analysis of rewrite systems, cf. AvanziniM13 . However, when
investigated in detail, precise comparison results are difficult to obtain. We
exemplify the situation with a simple example that will also serve as running
example throughout the paper.
###### Example 1.1.
Consider the following term rewrite system (TRS for short)
$\mathcal{R}_{\mathsf{que}}$, encoding a variant of an example by Okasaki
(Okasaki:1999, , Section 5.2).
$\displaystyle 1\colon$
$\displaystyle\mathsf{chk}(\mathsf{que}(\mathsf{nil},r))$
$\displaystyle\to\mathsf{que}(\mathsf{rev}(r),\mathsf{nil})$
$\displaystyle\hskip 25.83325pt7\colon$
$\displaystyle\mathsf{enq}(\mathsf{0})$
$\displaystyle\to\mathsf{que}(\mathsf{nil},\mathsf{nil})$ $\displaystyle
2\colon$
$\displaystyle\mathsf{chk}(\mathsf{que}(x\mathrel{\mathsf{\sharp}}xs,r))$
$\displaystyle\to\mathsf{que}(x\mathrel{\mathsf{\sharp}}xs,r)$
$\displaystyle\hskip 25.83325pt8\colon$
$\displaystyle\mathsf{rev^{\prime}}(\mathsf{nil},ys)$ $\displaystyle\to ys$
$\displaystyle 3\colon$
$\displaystyle\mathsf{tl}(\mathsf{que}(x\mathrel{\mathsf{\sharp}}f,r))$
$\displaystyle\to\mathsf{chk}(\mathsf{que}(f,r))$ $\displaystyle\hskip
25.83325pt9\colon$ $\displaystyle\mathsf{rev}(xs)$
$\displaystyle\to\mathsf{rev^{\prime}}(xs,\mathsf{nil})$ $\displaystyle
4\colon$ $\displaystyle\mathsf{snoc}(\mathsf{que}(f,r),x)$
$\displaystyle\to\mathsf{chk}(\mathsf{que}(f,x\mathrel{\mathsf{\sharp}}r))$
$\displaystyle\hskip 25.83325pt10\colon$
$\displaystyle\mathsf{hd}(\mathsf{que}(x\mathrel{\mathsf{\sharp}}f,r))$
$\displaystyle\to x$ $\displaystyle 5\colon$
$\displaystyle\mathsf{rev^{\prime}}(x\mathrel{\mathsf{\sharp}}xs,ys)$
$\displaystyle\to\mathsf{rev^{\prime}}(xs,x\mathrel{\mathsf{\sharp}}ys)$
$\displaystyle\hskip 25.83325pt11\colon$
$\displaystyle\mathsf{hd}(\mathsf{que}(\mathsf{nil},r))$
$\displaystyle\to\mathsf{err\\_head}$ $\displaystyle 6\colon$
$\displaystyle\mathsf{enq}(\mathsf{s}(n))$
$\displaystyle\to\mathsf{snoc}(\mathsf{enq}(n),n)$ $\displaystyle\hskip
25.83325pt12\colon$ $\displaystyle\mathsf{tl}(\mathsf{que}(\mathsf{nil},r))$
$\displaystyle\to\mathsf{err\\_tail}$
$\mathcal{R}_{\mathsf{que}}$ encodes an efficient implementation of a queue in
functional programming. A queue is represented as a pair of two lists
$\mathsf{que}(f,r)$, encoding the initial part $f$ and the reversal of the
remainder $r$. Invariant of the algorithm is that the first list never becomes
empty, which is achieved by reversing $r$ if necessary. Should the invariant
ever be violated, an exception ($\mathsf{err\\_head}$ or
$\mathsf{err\\_tail}$) is raised.
We exemplify the physicist’s method of amortised analysis Tarjan:1985 . We
assign to every queue $\mathsf{que}(f,r)$ the length of $r$ as _potential_.
Then the amortised cost for each operation is constant, as the costly reversal
operation is only executed if the potential can pay for the operation, compare
Okasaki:1999 . Thus, based on an amortised analysis, we deduce the optimal
linear runtime complexity for $\mathcal{R}$.
On the other hand let us attempt an application of the interpretation method
to this example. Termination proofs by interpretations are well-established
and can be traced back to work by Turing Turing:49 . We note that
$\mathcal{R}_{\mathsf{que}}$ is polynomially terminating. Moreover, it is
rather straightforward to restrict so-called _polynomial interpretations_ BN98
suitably so that compatibility of a TRS $\mathcal{R}$ induces polynomial
runtime complexity, cf. BonfanteCMT01 . Such polynomial interpretations are
called _restricted_. However, it turns out that no restricted polynomial
interpretation can exist that is compatible with $\mathcal{R}_{\mathsf{que}}$.
The reasoning is simple. The constraints induced by
$\mathcal{R}_{\mathsf{que}}$ imply that the function $\mathsf{snoc}$ has to be
interpreted by a linear polynomial. Thus an exponential interpretation is
required for enqueuing ($\mathsf{enq}$). Looking more closely at the different
proofs, we observe the following. While in the amortised analysis the
potential of a queue $\mathsf{que}(f,r)$ depends only on the remainder $r$,
the interpretation of $\mathsf{que}$ has to be monotone in both arguments by
definition. This difference induces that $\mathsf{snoc}$ is assigned a
strongly linear potential in the amortised analysis, while only a linear
interpretation is possible for $\mathsf{snoc}$.
Still it is possible to precisely relate amortised analysis to polynomial
interpretations if we base our investigation on many-sorted (or typed) TRSs
and make suitable use of the concept of _annotated types_ originally
introduced in HofmannJ03 . More precisely, we establish the following results.
We establish a novel runtime complexity analysis for typed constructor rewrite
systems. This complexity analysis is based on a potential-based amortised
analysis incorporated into a type system. From the annotated type of a term
its derivation height with respect to innermost rewriting can be read off (see
Theorem 3.1). The correctness proof of the obtained bound rests on a suitable
big-step semantics for rewrite systems, decorated with counters for the
derivation height of the evaluated terms. We complement this big-step
semantics with a similar decorated small-step semantics and prove equivalence
between these semantics. Furthermore we strengthen our first result by a
similar soundness result based on the small-step semantics (see Theorem 4.1).
Exploiting the small-step semantics we prove our main result that from the
well-typing of a TRS $\mathcal{R}$ we can read off a typed polynomial
interpretation that orients $\mathcal{R}$ (see Theorem 5.1).
While the type system exhibited is inspired by Hoffmann et al. HoffmannH10a
we generalise their use of annotated types to arbitrary (data) types.
Furthermore the introduced small-step semantics (and our main result) directly
establish that any well-typed TRS is terminating, thus circumventing the
notion of partial big-step semantics introduced in HoffmannH10b . Our main
result can be condensed into the following observations. The physicist’s
method of amortised analysis conceptually amounts to the interpretation method
if we allow for the following changes:
* •
Every term bears a potential, not only constructor terms.
* •
Polynomial interpretations are defined over annotated types.
* •
The standard compatibility constraint is weakened to orientability, that is,
all ground instances of a rule strictly decrease.
Our study is purely theoretic, and we have not (yet) attempted an
implementation of the provided techniques. However, automation appears
straightforward. Furthermore we have restricted our study to typed
(constructor) TRSs. In the conclusion we sketch application of the established
results to innermost runtime complexity analysis of untyped TRSs.
This paper is structured as follows. In the next section we cover some basics
and introduce a big-step operational semantics for typed TRSs. In Section 3 we
clarify our definition of annotated types and provide the mentioned type
system. We also present our first soundness result. In Section 4 we introduce
a small-step operational semantics and prove our second soundness result. Our
main result will be stated and proved in Section 5. Finally, we conclude in
Section 6, where we also mention future work.
## 2 Typed Term Rewrite Systems
Let $\mathcal{C}$ denote a finite, non-empty set of _constructor symbols_ and
$\mathcal{D}$ a finite set of _defined function symbols_. Let $S$ be a finite
set of (data) types. A family $(X_{A})_{A\in S}$ of sets is called _$S$
-typed_ and denotes as $X$. Let $\mathcal{V}$ denote an $S$-typed set of
_variables_ , such that the $\mathcal{V}_{s}$ are pairwise disjoint. In the
following, variables will be denoted by $x$, $y$, $z$, …, possibly extended by
subscripts.
Following JR99 , a _type declaration_ is of form $[{A_{1}\times\cdots\times
A_{n}}]\to{C}$, where $A_{i}$ and $C$ are types. Type declarations serve as
input-output specifications for function symbols. We write $A$ instead of
$[{}]\to{A}$. A _signature_ $\mathcal{F}$ (with respect to the set of types
$S$) is a mapping from $\mathcal{C}\cup\mathcal{D}$ to type declarations. We
often write ${f}{:}\,{[{A_{1}\times\cdots\times A_{n}}]\to{C}}$ if
$\mathcal{F}(f)=[{A_{1}\times\cdots\times A_{n}}]\to{C}$ and refer to a type
_declaration_ as a type, if no confusion can arise. We define the $S$-typed
set of terms
$\operatorname{\mathcal{T}}(\mathcal{D}\cup\mathcal{C},\mathcal{V})$ (or
$\operatorname{\mathcal{T}}$ for short): (i) for each $A\in S$:
$\mathcal{V}_{A}\subseteq\operatorname{\mathcal{T}}_{A}$, (ii) for
$f\in\mathcal{C}\cup\mathcal{D}$ such that
$\mathcal{F}(f)=[{A_{1},\dots,A_{n}}]\to{A}$ and
$t_{i}\in\operatorname{\mathcal{T}}_{A_{i}}$, we have
$f(t_{1},\dots,t_{n})\in\operatorname{\mathcal{T}}_{A}$. Type assertions are
denoted ${t}{:}\,{C}$. Terms of type $A$ will sometimes be referred to as
instances of $A$: a term of list type, is simply called a list. If
$t\in\operatorname{\mathcal{T}}(\mathcal{C},\varnothing)$ then $t$ is called a
_ground constructor term_ or a _value_. The set of values is denoted
$\operatorname{\mathcal{T}}(\mathcal{C})$. The ($S$-typed) set of variables of
a term $t$ is denoted $\operatorname{\mathcal{V}\mathsf{ar}}(t)$. The root of
$t$ is denoted $\operatorname{\mathsf{rt}}(t)$ and the size of $t$, that is
the number of symbols in $t$, is denoted as $\lvert{t}\rvert$. In the
following, terms are denoted by $s$, $t$, $u$, …, possibly extended by
subscripts. Furthermore, we use $v$ (possibly indexed) to denote values.
A _substitution_ $\sigma$ is a mapping from variables to terms that respects
types. Substitutions are denoted as sets of assignments:
$\sigma=\\{x_{1}\mapsto t_{1},\dots,x_{n}\mapsto t_{n}\\}$. We write
$\operatorname{\mathsf{dom}}(\sigma)$ ($\operatorname{\mathsf{rg}}(\sigma)$)
to denote the domain (range) of $\sigma$;
$\operatorname{\mathcal{V}\mathsf{rg}}(\sigma)\mathrel{:=}\operatorname{\mathcal{V}\mathsf{ar}}(\operatorname{\mathsf{rg}}(\sigma))$.
Let $\sigma$ be a substitution and $V$ be a set of variables;
${\sigma}\\!\restriction\\!{V}$ denotes the restriction of the domain of
$\sigma$ to $V$. The substitution $\sigma$ is called a _restriction_ of a
substitution $\tau$ if
${\tau}\\!\restriction\\!{\operatorname{\mathsf{dom}}(\sigma)}=\sigma$. Vice
versa, $\tau$ is called _extension_ of $\sigma$. Let $\sigma$, $\tau$ be
substitutions such that
$\operatorname{\mathsf{dom}}(\sigma)\cap\operatorname{\mathsf{dom}}(\tau)=\varnothing$.
Then we denote the (disjoint) union of $\sigma$ and $\tau$ as
$\sigma\mathrel{\uplus}\tau$. We call a substitution $\sigma$ _normalised_ if
all terms in the range of $\sigma$ are values. In the following, all
considered substitutions will be normalised.
A _typing context_ is a mapping from variables $\mathcal{V}$ to types. Type
contexts are denoted by upper-case Greek letters. Let $\Gamma$ be a context
and let $t$ be a term. The typing relation
${\Gamma}\sststile{}{}{}{{t}{:}\,{A}}$ expresses that based on context
$\Gamma$, $t$ has type $A$ (with respect to the signature $\mathcal{F}$). The
typing rules that define the typing relation are given in Figure 2, where we
forget the annotations. In the sequel we sometimes make use of an abbreviated
notation for sequences of types $\vec{A}=A_{1},\dots,A_{n}$ and terms
$\vec{t}\mathrel{:=}t_{1},\ldots,t_{n}$.
A typed rewrite rule is a pair $l\to r$ of terms, such that (i) the type of
$l$ and $r$ coincides, (ii) $\operatorname{\mathsf{rt}}(l)\in\mathcal{D}$, and
(iii)
$\operatorname{\mathcal{V}\mathsf{ar}}(l)\supseteq\operatorname{\mathcal{V}\mathsf{ar}}(r)$.
An $S$-typed _term rewrite system_ (_TRS_ for short) over the signature
$\mathcal{F}$ is a finite set of typed rewrite rules. We define the _innermost
rewrite relation_
$\mathrel{\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}}_{\mathcal{R}}}$
for typed TRSs $\mathcal{R}$. For terms $s$ and $t$,
$s\mathrel{\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}}_{\mathcal{R}}}t$
holds, if there exists a context $C$, a normalised substitution $\sigma$ and a
rewrite rule ${l\to r}\in\mathcal{R}$ such that $s=C[l\sigma]$, $t=C[r\sigma]$
and $s$, $t$ are well-typed. In the sequel we are only concerned with
_innermost_ rewriting. A TRS is _orthogonal_ if it is left-linear and non-
overlapping BN98 ; TeReSe . A TRS is _completely defined_ if all ground
normal-forms are values. These notions naturally extend to typed TRS. In
particular, note that an orthogonal typed TRS is confluent.
###### Definition 2.1.
We define the _runtime complexity_ (with respect to $\mathcal{R}$) as follows:
$\operatorname{\mathsf{rc}}(n)\mathrel{:=}\max\\{\operatorname{\mathsf{dh}}(t,\to)\mid\text{$t$
is basic and $\lvert{t}\rvert\leqslant n$}\\}\hbox to0.0pt{$\;$,\hss}$
where a term $t=f(t_{1},\dots,t_{k})$ is called _basic_ if $f$ is defined, and
the terms $t_{i}$ are only built over constructors and variables.
${\sigma}\sststile{}{0}{{x}\Rightarrow{v}}x\sigma=v$
${\sigma}\sststile{}{0}{{c(x_{1},\dots,x_{n})}\Rightarrow{c(v_{1},\dots,v_{n})}}\lx@proof@logical@and
c\in\mathcal{C}x_{1}\sigma=v_{1}\cdots x_{n}\sigma=v_{n}$
---
${\sigma}\sststile{}{m+1}{{f(x_{1},\ldots,x_{n})}\Rightarrow{v}}\lx@proof@logical@and
f(l_{1},\ldots,l_{n})\to r\in\mathcal{R}\exists\tau\ \forall i\colon
x_{i}\sigma=l_{i}\tau{\sigma\mathrel{\uplus}\tau}\sststile{}{m}{{r}\Rightarrow{v}}$
${\sigma}\sststile{}{m}{{f(t_{1},\ldots,t_{n})}\Rightarrow{v}}\lx@proof@logical@and\begin{minipage}[b]{129.16626pt}
all $x_{i}$ are fresh \hfil\\\
${\sigma\mathrel{\uplus}\rho}\sststile{}{m_{0}}{{f(x_{1},\ldots,x_{n})}\Rightarrow{v}}$
\hfill\end{minipage}{\sigma}\sststile{}{m_{1}}{{t_{1}}\Rightarrow{v_{1}}}\cdots{\sigma}\sststile{}{m_{n}}{{t_{n}}\Rightarrow{v_{n}}}m=\sum_{i=0}^{n}m_{i}$
Here $\rho\mathrel{:=}\\{x_{1}\mapsto v_{1},\ldots,x_{n}\mapsto v_{n}\\}$.
Recall that $\sigma$, $\tau$, and $\rho$ are normalised.
Figure 1: Operational Big-Step Semantics
We study _typed_ _constructor_ TRSs $\mathcal{R}$, that is, for each rule
$f(l_{1},\dots,l_{n})\to r$, the $l_{i}$ are constructor terms. Furthermore,
we restrict to _completely defined_ and _orthogonal_ systems. These
restrictions are natural in the context of functional programming. If no
confusion can arise from this, we simply call $\mathcal{R}$ a TRS.
$\mathcal{F}$ denotes the signature underlying $\mathcal{R}$. In the sequel,
$\mathcal{R}$ and $\mathcal{F}$ are kept fixed.
###### Example 2.1 (continued from Example 1.1).
Consider the TRS $\mathcal{R}_{\mathsf{que}}$ and let
$S=\\{\mathsf{Nat},\mathsf{List},\mathsf{Q}\\}$, where $\mathsf{Nat}$,
$\mathsf{List}$, and $\mathsf{Q}$ represent the type of natural numbers, lists
over over natural number, and queues respectively. Then
$\mathcal{R}_{\mathsf{que}}$ is an $S$-typed TRSs over signature
$\mathcal{F}$, where the signature of some constructors is as follows:
$\displaystyle\mathsf{0}\colon$ $\displaystyle\mathsf{Nat}$
$\displaystyle\hskip 25.83325pt\mathsf{s}\colon$
$\displaystyle[{\mathsf{Nat}}]\to{\mathsf{Nat}}$
$\displaystyle\mathsf{que}\colon$
$\displaystyle[{\mathsf{List}\times\mathsf{List}}]\to{\mathsf{Q}}$
$\displaystyle\mathsf{nil}\colon$ $\displaystyle\mathsf{List}$
$\displaystyle\hskip 25.83325pt\mathrel{\mathsf{\sharp}}\colon$
$\displaystyle[{\mathsf{Nat}\times\mathsf{List}}]\to{\mathsf{List}}\hbox
to0.0pt{$\;$.\hss}\hskip 21.52771pt$
In order to exemplify the type declaration of defined function symbols,
consider
${\mathsf{snoc}}{:}\,{[{\mathsf{Q}\times\mathsf{Nat}}]\to{\mathsf{Q}}}\hbox
to0.0pt{$\;$.\hss}$
As $\mathcal{R}$ is completely defined any derivation ends in a value. On the
other hand, as $\mathcal{R}$ is non-overlapping any innermost derivation is
determined modulo the order in which parallel redexes are contracted. This
allows us to recast innermost rewriting into an operational big-step semantics
instrumented with resource counters, cf. Figure 1. The semantics closely
resembles similar definitions given in the literature on amortised resource
analysis (see for example JLHSH09 ; HoffmannH10a ; HAH12b ).
Let $\sigma$ be a (normalised) substitution and let $f(x_{1},\dots,x_{n})$ be
a term. It follows from the definitions that
$f(x_{1}\sigma,\dots,x_{n}\sigma)\mathrel{\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}}^{\ast}_{\mathcal{R}}}v$
iff ${\sigma}\sststile{}{}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}$. More,
precisely we have the following proposition.
###### Proposition 2.1.
Let $f$ be a defined function symbol of arity $n$ and $\sigma$ a substitution.
Then ${\sigma}\sststile{}{m}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}$ holds iff
$\operatorname{\mathsf{dh}}(f(x_{1}\sigma,\dots,x_{n}\sigma),\mathrel{\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}}_{\mathcal{R}}})=m$
holds.
###### Proof.
In proof of the direction from left to right, we show the stronger statement
that ${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$ implies
$\operatorname{\mathsf{dh}}(t\sigma,\mathrel{\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}}_{\mathcal{R}}})=m$
by induction on the size of the proof of the judgement
${\sigma}\sststile{}{m}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}$. For the
opposite direction, we show that if
$\operatorname{\mathsf{dh}}(t\sigma,\mathrel{\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}}_{\mathcal{R}}})=m$,
then ${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$ by induction on the length of
the derivation $D\colon
t\sigma\mathrel{\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}}^{\ast}_{\mathcal{R}}}v$.
∎
The next (technical) lemma follows by a straightforward inductive argument.
###### Lemma 2.1.
Let $t$ be a term, let $v$ be a value and let $\sigma$ be a substitution. If
${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$ and if $\sigma^{\prime}$ is an
extension of $\sigma$, then
${\sigma^{\prime}}\sststile{}{m}{{t}\Rightarrow{v}}$. Furthermore the sizes of
the derivations of the corresponding judgements are the same.
## 3 Annotated Types
Let $S$ be a set of types. We call a type $A\in S$ _annotated_ , if $A$ is
decorated with resource annotation. These annotations will allow us to read
off the potential of a well-typed term $t$ from the annotations.
###### Definition 3.1.
Let $S$ be a set of types. An _annotated type_ ${A}^{\vec{p}}$, is a pair
consisting of a type $A\in S$ and a vector $\vec{p}=(p_{1},\dots,p_{k})$ over
non-negative rational numbers, typically natural numbers. The vector $\vec{p}$
is called _resource annotation_.
Resource annotations are denoted by $\vec{p}$, $\vec{q}$, $\vec{u}$,
$\vec{v}$, …, possibly extended by subscripts and we write $\mathcal{A}$ for
the set of such annotations. For resource annotations $(p)$ of length $1$ we
write $p$. The empty annotation $()$ is written $0$. We will see that a
resource annotation does not change its meaning if zeroes are appended at the
end, so, conceptually, we can identify $()$ with $(0)$. If
$\vec{p}=(p_{1},\dots,p_{k})$ we write $k=\lvert\vec{p}\rvert$ and
$\max\vec{p}=\max_{i}p_{i}$. We define the notations $\vec{p}\leqslant\vec{q}$
and $\vec{p}+\vec{q}$ and $\lambda\vec{p}$ for $\lambda\geqslant 0$ component-
wise, filling up with $0$s if needed. So, for example $(1,2)\leqslant(3,4,5)$
and $(1,2)+(3,4,5)=(4,6,5)$. Furthermore, we recall the additive shift
HoffmannH10a given by
$\operatorname{\triangleleft}(\vec{p})\mathrel{:=}(p_{1}+p_{2},p_{2}+p_{3},\dots,p_{k-1}+p_{k},p_{k})\hbox
to0.0pt{$\;$.\hss}$
We also define the interleaving $\vec{p}\interleave\vec{q}$ by
$(p_{1},q_{1},p_{2},q_{2},$ $\dots,p_{k},q_{k})$ where, as before the shorter
of the two vectors is padded with $0$s. Finally, we use the notation
$\Diamond\vec{p}=p_{1}$ for the first entry of an annotation vector.
If no confusion can arise, we refer to annotated types simply as types. In
contrast to Hoffmann et al. HoffmannH10a ; Hoffmann:2011 , we generalise the
concept of annotated types to arbitrary (data) types. In HoffmannH10a only
list types, in Hoffmann:2011 list and tree types have been annotated.
###### Definition 3.2.
Let $\mathcal{F}$ be a signature. Suppose
$\mathcal{F}(f)=[{A_{1}\times\cdots\times A_{n}}]\to{C}$, such that the
$A_{i}$ ($i=1,\dots,n$) and $C$ are types. Consider the annotated types
$A_{i}^{\vec{u_{i}}}$ and ${A}^{\vec{v}}$. Then an _annotated type
declaration_ for $f$ is a type declaration over annotated types, decorated
with a number $p$:
$[{A_{1}^{\vec{u_{1}}}\times\cdots\times
A_{n}^{\vec{u_{n}}}}]\xrightarrow{p}{{C}^{\vec{v}}}\hbox to0.0pt{$\;$.\hss}$
The set of annotated type declarations is denoted as
$\mathcal{F}_{\mathsf{pol}}$.
We write $A^{0}$ instead of $[{}]\xrightarrow{0}{{A}^{0}}$. We lift signatures
to _annotated signatures_
$\mathcal{F}\colon\mathcal{C}\cup\mathcal{D}\to(\operatorname{\mathcal{P}}(\mathcal{F}_{\mathsf{pol}})\setminus\varnothing)$
by mapping a function symbol to a non-empty set of annotated type
declarations. Hence for any $f\in\mathcal{C}\cup\mathcal{D}$ we allow multiple
types. If $f$ has result type $C$, then for each annotation $C^{\vec{q}}$
there should exist exactly one declaration of the form
$[{A_{1}^{\vec{p_{1}}}\times\cdots\times
A_{n}^{\vec{p_{n}}}}]\xrightarrow{p}{C^{\vec{q}}}$ in $\mathcal{F}(f)$.
Moreover, constructor annotations are to satisfy the _superposition principle_
: If a constructor $c$ admits the annotations
$[{A_{1}^{\vec{p_{1}}}\times\cdots\times
A_{n}^{\vec{p_{n}}}}]\xrightarrow{p}{C^{\vec{q}}}$ and
$[{A_{1}^{\vec{p^{\prime}_{1}}}\times\cdots\times
A_{n}^{\vec{p^{\prime}_{n}}}}]\xrightarrow{p^{\prime}}{C^{\vec{q^{\prime}}}}$
then it also has the annotations
$[{A_{1}^{\lambda\vec{p_{1}}}\times\cdots\times
A_{n}^{\lambda\vec{p_{n}}}}]\xrightarrow{\lambda p}{C^{\lambda\vec{q}}}$
($\lambda\geqslant 0$) and
$[{A_{1}^{\vec{p_{1}}+\vec{p^{\prime}_{1}}}\times\cdots\times
A_{n}^{\vec{p_{n}}+\vec{p^{\prime}_{n}}}}]\xrightarrow{p+p^{\prime}}{C^{\vec{q}+\vec{q^{\prime}}}}$.
Note that, in view of superposition and uniqueness, the annotations of a given
constructor are uniquely determined once we fix the annotated types for result
annotations of the form $(0,\dots,0,1)$ (remember the implicit filling up with
$0$s). An annotated signature $\mathcal{F}$ is simply called signature, where
we sometimes write ${f}{:}\,{[{A_{1}\times\cdots\times
A_{n}}]\xrightarrow{p}{C}}$ instead of $[{A_{1}\times\cdots\times
A_{n}}]\xrightarrow{p}{C}\in\mathcal{F}(f)$.
###### Example 3.1 (continued from Example 2.1).
In order to extend $\mathcal{F}$ to an annotated signature we can set
$\displaystyle\mathcal{F}(\mathsf{0})$
$\displaystyle\mathrel{:=}\\{\mathsf{Nat}^{\vec{p}}\mid\vec{p}\in\mathcal{A}\\}$
$\displaystyle\hskip 2.15277pt\mathcal{F}(\mathsf{s})$
$\displaystyle\mathrel{:=}\\{[{\mathsf{Nat}^{\operatorname{\triangleleft}(\vec{p})}}]\xrightarrow{\Diamond\vec{p}}{\mathsf{Nat}^{\vec{p}}}\mid\vec{p}\in\mathcal{A}\\}$
$\displaystyle\mathcal{F}(\mathsf{nil})$
$\displaystyle\mathrel{:=}\\{\mathsf{List}^{\vec{p}}\mid\vec{p}\in\mathcal{A}\\}$
$\displaystyle\hskip 2.15277pt\mathcal{F}(\mathrel{\mathsf{\sharp}})$
$\displaystyle\mathrel{:=}\\{[{\mathsf{Nat}^{0}\times\mathsf{List}^{\operatorname{\triangleleft}(\vec{p})}}]\xrightarrow{\Diamond\vec{p}}{\mathsf{List}^{\vec{p}}}\mid\vec{p}\in\mathcal{A}\\}$
$\displaystyle\mathcal{F}(\mathsf{que})$
$\displaystyle\mathrel{:=}\\{[{\mathsf{List}^{\vec{p}}\times\mathsf{List}^{\vec{q}}}]\xrightarrow{0}{\mathsf{Q}^{{\vec{p}\interleave\vec{q}}}}\mid\vec{p},\vec{q}\in\mathcal{A}\\}$
In particular, we have the typings
$\mathrel{\mathsf{\sharp}}:[{\mathsf{Nat}^{0}\times\mathsf{List}^{7}}]\xrightarrow{7}{\mathsf{List}^{7}}$
and
$\mathrel{\mathsf{\sharp}}:[{\mathsf{Nat}^{0}\times\mathsf{List}^{(10,7)}}]\xrightarrow{3}{\mathsf{List}^{(3,7)}}$
and
$\mathsf{que}:[{\mathsf{List}^{1}\times\mathsf{List}^{3}}]\xrightarrow{0}{\mathsf{Q}^{{(1,3)}}}$.
We omit annotations for the defined symbols and refer to Example 3.3 for a
complete signature with different constructor annotations.
The next definition introduces the notion of the potential of a value.
###### Definition 3.3.
Let $v=c(v_{1},\dots,v_{n})\in\operatorname{\mathcal{T}}(\mathcal{C})$ and let
$[{A_{1}\times\cdots\times A_{n}}]\xrightarrow{p}{C}\in\mathcal{F}(c)$. Then
the _potential_ of $v$ is defined inductively as
$\Phi({v}{:}\,{C})\mathrel{:=}p+\Phi({v_{1}}{:}\,{A_{1}})+\cdots+\Phi({v_{n}}{:}\,{A_{n}})\hbox
to0.0pt{$\;$.\hss}$
Note that by assumption the declaration in $\mathcal{F}(c)$ is unique.
###### Example 3.2 (continued from Example 3.1).
It is easy to see that for any term $t$ of type $\mathsf{Nat}^{0}$, we have
$\Phi({t}{:}\,{\mathsf{Nat}^{0}})=0$ and
$\Phi({t}{:}\,{\mathsf{Nat}^{\lambda}})=\lambda t$.
If $l$ is a list then $\Phi({l}{:}\,{\mathsf{List}^{(p,q)}})=p\cdot\lvert
l\rvert+q\cdot\binom{\lvert l\rvert}{2}$. where $\lvert l\rvert$ denotes the
length of $l$, that is the number of $\mathrel{\mathsf{\sharp}}$ in $l$. Let
$\lvert l\rvert=\ell$. We proceed by induction on $\ell$. Let $\ell=0$. Then
$\Phi({\mathsf{nil}}{:}\,{\mathsf{List}^{(p,q)}})=0$ as required. Suppose
$\ell=\ell^{\prime}+1$:
$\displaystyle\Phi({n\mathrel{\mathsf{\sharp}}l^{\prime}}{:}\,{\mathsf{List}^{(p,q)}})$
$\displaystyle=p+\Phi({n}{:}\,{\mathsf{Nat}^{0}})+\Phi({l^{\prime}}{:}\,{\mathsf{List}^{(p+q,q)}})$
$\displaystyle=p+(p+q)\cdot\ell^{\prime}+q\cdot\binom{\ell^{\prime}}{2}$
$\displaystyle=p\cdot\ell+q\cdot\left[\binom{\ell^{\prime}}{1}+\binom{\ell^{\prime}}{2}\right]=p\cdot\ell+q\cdot\binom{\ell}{2}\hbox
to0.0pt{$\;$.\hss}$
More generally, we have
$\Phi({l}{:}\,{\mathsf{List}^{\vec{p}}})=\sum_{i}p_{i}\binom{\lvert
l\rvert}{i}$. Finally, if $\mathsf{que}(l,k)$ has type $\mathsf{Q}$ then
$\Phi({\mathsf{que}(l,k)}{:}\,{\mathsf{Q}^{{\vec{p}\interleave\vec{q}}}})=\Phi({l}{:}\,{\mathsf{List}^{\vec{p}}})+\Phi({k}{:}\,{\mathsf{List}^{\vec{q}}})$.
The _sharing relation_
$\curlyvee\\!({A^{\vec{p}}}\\!\mid\\!{A_{1}^{\vec{p_{1}}},A_{2}^{\vec{p_{2}}}})$
holds if $A=A_{1}=A_{2}$ and $\vec{p_{1}}+\vec{p_{2}}=\vec{p}$. The subtype
relation is defined as follows: ${A}^{\vec{p}}\mathrel{<:}{B}^{\vec{q}}$, if
$A=B$ and $\vec{p}\geqslant\vec{q}$.
###### Lemma 3.1.
If
$\curlyvee\\!({A^{\vec{p}}}\\!\mid\\!{A_{1}^{\vec{p_{1}}},A_{2}^{\vec{p_{2}}}})$
then
$\Phi({v}{:}\,{A^{\vec{p}}})=\Phi({v}{:}\,{A_{1}^{\vec{p_{1}}}})+\Phi({v}{:}\,{A_{2}^{\vec{p_{2}}}})$
holds for any value of type $A$. If ${A}^{\vec{p}}\mathrel{<:}{B}^{\vec{q}}$
then $\Phi({v}{:}\,{{A}^{\vec{p}}})\geqslant\Phi({v}{:}\,{{B}^{\vec{q}}})$
again for any $v:A$.
###### Proof.
The proof of the first claim is by induction on the structure of $v$. We note
that by superposition together with uniqueness the additivity property
propagates to the argument types. For example, if we have the annotations
$\mathsf{s}:[{\mathsf{Nat}^{2}}]\xrightarrow{4}{\mathsf{Nat}^{3}}$ and
$\mathsf{s}:[{\mathsf{Nat}^{4}}]\xrightarrow{6}{\mathsf{Nat}^{5}}$ and
$\mathsf{s}:[{\mathsf{Nat}^{x}}]\xrightarrow{10}{\mathsf{Nat}^{y}}$ then we
can conclude $x=6$, $y=8$, for this annotation must be present by
superposition and there can only be one by uniqueness.
The second claim follows from the first one and nonnegativity of potentials. ∎
${{x_{1}}{:}\,{A_{1}^{\vec{u_{1}}}},\dots,{x_{n}}{:}\,{A_{n}^{\vec{u_{n}}}}}\sststile{}{p}{{f(x_{1},\dots,x_{n})}{:}\,{{C}^{\vec{v}}}}\lx@proof@logical@and
f\in\mathcal{C}\cup\mathcal{D}[{A_{1}^{\vec{u_{1}}}\times\cdots\times
A_{n}^{\vec{u_{n}}}}]\xrightarrow{p}{{C}^{\vec{v}}}\in\mathcal{F}(f)$
${\Gamma}\sststile{}{p^{\prime}}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma}\sststile{}{p}{{t}{:}\,{C}}p^{\prime}\geqslant
p$
---
${\Gamma_{1},\dots,\Gamma_{n}}\sststile{}{p}{{f(t_{1},\dots,t_{n})}{:}\,{C}}\lx@proof@logical@and\begin{minipage}[b]{172.22168pt}
all $x_{i}$ are fresh\\\
${{x_{1}}{:}\,{A_{1}},\dots,{x_{n}}{:}\,{A_{n}}}\sststile{}{p_{0}}{{f(x_{1},\dots,x_{n})}{:}\,{C}}$
\end{minipage}\begin{minipage}[b]{137.77734pt} $p=\sum_{i=0}^{n}p_{i}$\\\
${\Gamma_{1}}\sststile{}{p_{1}}{{t_{1}}{:}\,{A_{1}}}\ \cdots\
{\Gamma_{n}}\sststile{}{p_{n}}{{t_{n}}{:}\,{A_{n}}}$ \end{minipage}$
${\Gamma,{x}{:}\,{A}}\sststile{}{p}{{t}{:}\,{C}}{\Gamma}\sststile{}{p}{{t}{:}\,{C}}$
${\Gamma,{z}{:}\,{A}}\sststile{}{p}{{t[z,z]}{:}\,{C}}\lx@proof@logical@and{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}}\sststile{}{p}{{t[x,y]}{:}\,{C}}\curlyvee\\!({A}\\!\mid\\!{A_{1},A_{2}})\text{$x$,
$y$ are fresh}$
${\Gamma,{x}{:}\,{A}}\sststile{}{p}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma,{x}{:}\,{B}}\sststile{}{p}{{t}{:}\,{C}}A\mathrel{<:}B$
${{x}{:}\,{A}}\sststile{}{0}{{x}{:}\,{A}}$
${\Gamma}\sststile{}{p}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma}\sststile{}{p}{{t}{:}\,{D}}D\mathrel{<:}C$
Figure 2: Type System for Rewrite Systems
The set of typing rules for TRSs are given in Figure 2. Observe that the type
system employs the assumption that $\mathcal{R}$ is left-linear. In a
nutshell, the method works as follows: Let $\Gamma$ be a typing context and
let us consider the typing judgement ${\Gamma}\sststile{}{p}{{t}{:}\,{A}}$
derivable from the type rules. Then $p$ is an upper-bound to the amortised
cost required for reducing $t$ to a value. The derivation height of $t\sigma$
(with respect to innermost rewriting) is bound by the difference in the
potential before and after the evaluation plus $p$. Thus if the sum of the
potential of the arguments of $t\sigma$ is in
$\operatorname{\mathsf{O}}(n^{k})$, where $n$ is the size of the arguments,
then the runtime complexity of $\mathcal{R}$ lies in
$\operatorname{\mathsf{O}}(n^{k})$.
Recall that any rewrite rule $l\to r\in\mathcal{R}$ can be written as
$f(l_{1},\dots,l_{n})\to r$ with
$l_{i}\in\operatorname{\mathcal{T}}(\mathcal{C},\mathcal{V})$. We introduce
_well-typed_ TRSs.
###### Definition 3.4.
Let $f(l_{1},\dots,l_{n})\to r$ be a rewrite rule in $\mathcal{R}$ and let
$\operatorname{\mathcal{V}\mathsf{ar}}(f(\vec{l}))=\\{y_{1},\dots,y_{\ell}\\}$.
Then $f\in\mathcal{D}$ is _well-typed_ wrt. $\mathcal{F}$, if we obtain
${{y_{1}}{:}\,{B_{1}},\dots,{y_{\ell}}{:}\,{B_{\ell}}}\sststile{}{p-1+\sum_{i=1}^{n}k_{i}}{{r}{:}\,{C}}\hbox
to0.0pt{$\;$,\hss}$ (1)
for all $[{A_{1}\times\cdots\times
A_{n}}]\xrightarrow{p}{C}\in\mathcal{F}(f)$, for all types $B_{j}$
($j\in\\{1,\dots,\ell\\}$), and all costs $k_{i}$, such that
${{y_{1}}{:}\,{B_{1}},\dots,{y_{\ell}}{:}\,{B_{\ell}}}\sststile{}{k_{i}}{{l_{i}}{:}\,{A_{i}}}$
is derivable. A TRS $\mathcal{R}$ over $\mathcal{F}$ is _well-typed_ if any
defined $f$ is well-typed.
Contrary to analogous definitions in the literature on amortised resource
analysis the definition recurs to the type system in order to specify the
available resources in the type judgement (1). This is necessary to adapt
amortised analysis to rewrite systems.
Let $\Gamma$ be a typing context and let $\sigma$ be a substitution. We call
$\sigma$ _well-typed (with respect to $\Gamma$)_ if for all
$x\in\operatorname{\mathsf{dom}}(\Gamma)$ $x\sigma$ is of type $\Gamma(x)$. We
extend Definition 3.3 to substitutions $\sigma$ and typing contexts $\Gamma$.
Suppose $\sigma$ is well-typed with respect to $\Gamma$. Then
$\Phi({\sigma}{:}\,{\Gamma})\mathrel{:=}\sum_{x\in\operatorname{\mathsf{dom}}(\Gamma)}\Phi({x\sigma}{:}\,{\Gamma(x)})$.
We state and prove our first soundness result.
###### Theorem 3.1.
Let $\mathcal{R}$ and $\sigma$ be well-typed. Suppose
${\Gamma}\sststile{}{p}{{t}{:}\,{A}}$ and
${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$. Then
$\Phi({\sigma}{:}\,{\Gamma})-\Phi({v}{:}\,{A})+p\geqslant m$.
###### Proof.
Let $\Pi$ be the proof deriving ${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$
and let $\Xi$ be the proof of ${\Gamma}\sststile{}{p}{{t}{:}\,{A}}$. The proof
of the theorem proceeds by main-induction on the length of $\Pi$ and by side-
induction on the length of $\Xi$.
1. 1.
Suppose $\Pi$ has the form
${\sigma}\sststile{}{m}{{x}\Rightarrow{v}}x\sigma=v\hbox to0.0pt{$\;$,\hss}$
such that $t=x$ and $v=x\sigma$. Wlog. $\Xi$ is of form
${{x}{:}\,{A}}\sststile{}{0}{{x}{:}\,{A}}$. Then
$\Phi({\sigma}{:}\,{\Gamma})=\Phi({x\sigma}{:}\,{A})$ and the theorem follows.
2. 2.
Suppose $\Pi$ has the form
${\sigma}\sststile{}{m}{{c(x_{1},\dots,x_{n})}\Rightarrow{c(v_{1},\dots,v_{n})}}\lx@proof@logical@and
c\in\mathcal{C}x_{1}\sigma=v_{1}\cdots x_{n}\sigma=v_{n}\hbox
to0.0pt{$\;$,\hss}$
such that $t=c(x_{1},\dots,x_{n})$ and $v=c(v_{1},\dots,v_{n})$. Further wlog.
we suppose that $\Xi$ ends in the following judgement:
${{x_{1}}{:}\,{A_{1}^{\vec{u_{1}}}},\dots,{x_{n}}{:}\,{A_{n}^{\vec{u_{n}}}}}\sststile{}{p}{{c(x_{1},\dots,x_{n})}{:}\,{{C}^{\vec{w}}}}\hbox
to0.0pt{$\;$.\hss}$
Then we have $[{A_{1}^{\vec{u_{1}}}\times\cdots\times
A_{n}^{\vec{u_{n}}}}]\xrightarrow{p}{{C}^{\vec{w}}}\in\mathcal{F}(c)$ and
thus:
$\Phi({\sigma}{:}\,{\Gamma})+p=p+\sum_{i=1}^{n}\Phi({x_{i}\sigma}{:}\,{A_{i}^{\vec{u_{i}}}})=p+\sum_{i=1}^{n}\Phi({v_{i}}{:}\,{A_{i}^{\vec{u_{i}}}})=\Phi({c(v_{1},\dots,v_{n})}{:}\,{C^{\vec{w}}})\hbox
to0.0pt{$\;$,\hss}$
from which the theorem follows.
3. 3.
Suppose $\Pi$ ends in the following rule:
${\sigma}\sststile{}{m+1}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}\lx@proof@logical@and\exists\
f(l_{1},\dots,l_{n})\to r\in\mathcal{R}\exists\tau\ \forall i\colon
x_{i}\sigma=l_{i}\tau{\sigma\mathrel{\uplus}\tau}\sststile{}{m}{{r}\Rightarrow{v}}\hbox
to0.0pt{$\;$.\hss}$
Then $t=f(x_{1},\dots,x_{n})$ and
$f(x_{1},\dots,x_{n})\sigma=f(l_{1},\dots,l_{n})\tau$. Suppose
$\operatorname{\mathcal{V}\mathsf{ar}}(f(\vec{l}))=\\{y_{1},\dots,y_{\ell}\\}$
and let
$\operatorname{\mathcal{V}\mathsf{ar}}(l_{i})=\\{y_{i1},\dots,y_{il_{i}}\\}$
for $i\in\\{1,\dots,n\\}$. As $\mathcal{R}$ is left-linear we have
$\operatorname{\mathcal{V}\mathsf{ar}}(f(l_{1},\dots,l_{n}))=\biguplus_{i=1}^{n}\operatorname{\mathcal{V}\mathsf{ar}}(l_{i})$.
We set $\Gamma={x_{1}}{:}\,{A_{1}},\dots,{x_{n}}{:}\,{A_{n}}$. By the
assumption ${\Gamma}\sststile{}{p}{{t}{:}\,{A}}$ and well-typedness of
$\mathcal{R}$ we obtain
${\overbrace{{y_{1}}{:}\,{B_{1}},\dots,{y_{\ell}}{:}\,{B_{\ell}}}^{{}=:\Delta}}\sststile{}{p-1+\sum_{i=1}^{n}k_{i}}{{r}{:}\,{C}}\hbox
to0.0pt{$\;$,\hss}$
as in (1). By main induction hypothesis together with the above equation, we
have
$\Phi({\sigma\mathrel{\uplus}\tau}{:}\,{\Delta})-\Phi({v}{:}\,{C})+p-1+\sum_{i=1}^{n}k_{i}\geqslant
m$. Furthermore, we have
$\displaystyle\Phi({\sigma}{:}\,{\Gamma})$
$\displaystyle=\sum_{i=1}^{n}\Phi({x_{i}\sigma}{:}\,{A_{i}})=\sum_{i=1}^{n}\left(k_{i}+\Phi({y_{i1}\tau}{:}\,{B_{i1}})+\cdots+\Phi({y_{il_{i}}\tau}{:}\,{B_{il_{i}}})\right)$
$\displaystyle=\Phi({\sigma\mathrel{\uplus}\tau}{:}\,{\Delta})+\sum_{i=1}^{n}k_{i}\hbox
to0.0pt{$\;$.\hss}$
Here the first equality follows by an inspection on the case for the
constructors. In sum, we obtain
$\Phi({\sigma}{:}\,{\Gamma})-\Phi({v}{:}\,{C})+p=\Phi({\sigma\mathrel{\uplus}\tau}{:}\,{\Delta})+\sum_{i=1}^{n}k_{i}-\Phi({v}{:}\,{C})+p\geqslant
m+1\hbox to0.0pt{$\;$,\hss}$
from which the theorem follows.
4. 4.
Suppose the last rule in $\Pi$ has the form
${\sigma}\sststile{}{m}{{f(t_{1},\dots,t_{n})}\Rightarrow{v}}\lx@proof@logical@and{\sigma\mathrel{\uplus}\rho}\sststile{}{m_{0}}{{f(x_{1},\ldots,x_{n})}\Rightarrow{v}}{\sigma}\sststile{}{m_{1}}{{t_{1}}\Rightarrow{v_{1}}}\cdots{\sigma}\sststile{}{m_{n}}{{t_{n}}\Rightarrow{v_{n}}}m=\sum_{i=0}^{n}m_{i}\hbox
to0.0pt{$\;$.\hss}$
We can assume that $t$ is linear, compare the case employing the share
operator. Hence the last rule in the type inference $\Xi$ is of the following
form.
${\Gamma_{1},\dots,\Gamma_{n}}\sststile{}{p}{{f(t_{1},\dots,t_{n})}{:}\,{C}}\lx@proof@logical@and{\overbrace{{y_{1}}{:}\,{A_{1}},\dots,{y_{n}}{:}\,{A_{n}}}^{{}=:\Delta}}\sststile{}{p_{0}}{{f(\vec{y})}{:}\,{C}}{\Gamma_{1}}\sststile{}{p_{1}}{{t_{1}}{:}\,{A_{1}}}\cdots{\Gamma_{n}}\sststile{}{p_{n}}{{t_{n}}{:}\,{A_{n}}}p=\sum_{i=0}^{n}p_{i}\hbox
to0.0pt{$\;$.\hss}$
By induction hypothesis:
$\Phi({\sigma}{:}\,{\Gamma_{i}})-\Phi({v_{i}}{:}\,{A_{i}})+p_{i}\geqslant
m_{i}$ for all $i=1,\dots,n$. Hence
$\sum_{i=1}^{n}\Phi({\sigma}{:}\,{\Gamma_{i}})-\sum_{i=1}^{n}\Phi({v_{i}}{:}\,{A_{i}})+\sum_{i=1}^{n}p_{i}\geqslant\sum_{i=1}^{n}m_{i}\hbox
to0.0pt{$\;$.\hss}$ (2)
Again by induction hypothesis we obtain:
$\Phi({\sigma\mathrel{\uplus}\rho}{:}\,{\Delta})-\Phi({v}{:}\,{C})+p_{0}\geqslant
m_{0}\hbox to0.0pt{$\;$.\hss}$ (3)
Now
$\Phi({\sigma}{:}\,{\Gamma})=\sum_{i=1}^{n}\Phi({\sigma}{:}\,{\Gamma_{i}})$
and
$\Phi({\sigma\mathrel{\uplus}\rho}{:}\,{\Delta})=\Phi({\rho}{:}\,{\Delta})=\sum_{i=1}^{n}\Phi({v_{i}}{:}\,{A_{i}})$.
Due to (2) and (3), we obtain
$\displaystyle\Phi({\sigma}{:}\,{\Gamma})+\sum_{i=0}^{n}p_{i}$
$\displaystyle=\sum_{i=1}^{n}\Phi({\sigma}{:}\,{\Gamma_{i}})+\sum_{i=1}^{n}p_{i}+p_{0}$
$\displaystyle\geqslant\sum_{i=1}^{n}\Phi({v_{i}}{:}\,{A_{i}})+\sum_{i=1}^{n}m_{i}+p_{0}\geqslant\Phi({v}{:}\,{C})+\sum_{i=0}^{n}m_{i}\hbox
to0.0pt{$\;$,\hss}$
and thus $\Phi({\sigma}{:}\,{\Gamma}-\Phi({v}{:}\,{C})+p\geqslant m$.
5. 5.
Suppose $\Xi$ is of form
${\Gamma}\sststile{}{p^{\prime}}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma}\sststile{}{p}{{t}{:}\,{C}}p^{\prime}\geqslant
p\hbox to0.0pt{$\;$.\hss}$
By side-induction on ${\Gamma}\sststile{}{p}{{t}{:}\,{C}}$ together with
${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$ we conclude
$\Phi({\sigma}{:}\,{\Gamma})-\Phi({v}{:}\,{A})+p\geqslant m$. Then the theorem
follows from the assumption $p^{\prime}\geqslant p$.
6. 6.
Suppose $\Xi$ is of form
${\Gamma,{x}{:}\,{A}}\sststile{}{p}{{t}{:}\,{C}}{\Gamma}\sststile{}{p}{{t}{:}\,{C}}\hbox
to0.0pt{$\;$.\hss}$
We conclude by side-induction together with
${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$ we conclude
$\Phi({\sigma}{:}\,{\Gamma})-\Phi({v}{:}\,{A})+p\geqslant m$. Clearly
$\Phi({\sigma}{:}\,{\Gamma,{x}{:}\,{A}})\geqslant\Phi({\sigma}{:}\,{\Gamma})$
and the theorem follows.
7. 7.
Suppose $\Xi$ is of form
${\Gamma,{z}{:}\,{A}}\sststile{}{p}{{t[z,z]}{:}\,{C}}\lx@proof@logical@and{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}}\sststile{}{p}{{t[x,y]}{:}\,{C}}\curlyvee\\!({A}\\!\mid\\!{A_{1},A_{2}})$
By assumption ${\sigma}\sststile{}{m}{{t[z,z]}\Rightarrow{v}}$; let
$\rho\mathrel{:=}\sigma\mathrel{\uplus}\\{x\mapsto z\sigma,y\mapsto
z\sigma\\}$. As ${\sigma}\sststile{}{m}{{t[z,z]}\Rightarrow{v}}$, we obtain
${\rho}\sststile{}{m}{{t[x,y]}\Rightarrow{v}}$ by definition. From the side-
induction on
${\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}}\sststile{}{p}{{t[x,y]}{:}\,{C}}$ and
${\rho}\sststile{}{m}{{t[x,y]}\Rightarrow{v}}$ we conclude that
$\Phi({\rho}{:}\,{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}})-\Phi({v}{:}\,{C}+p\geqslant
m\hbox to0.0pt{$\;$.\hss}$
The theorem follows as by definition of $\rho$ and Lemma 3.1, we obtain
$\Phi({\sigma}{:}\,{\Gamma,{z}{:}\,{A}})=\Phi({\rho}{:}\,{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}})\hbox
to0.0pt{$\;$.\hss}$
8. 8.
Suppose $\Xi$ is of form
${\Gamma,{x}{:}\,{A}}\sststile{}{p}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma,{x}{:}\,{B}}\sststile{}{p}{{t}{:}\,{C}}A\mathrel{<:}B$
By assumption ${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$ and by induction
hypothesis
$\Phi({\sigma}{:}\,{\Gamma,{x}{:}\,{B}})-\Phi({v}{:}\,{A})+p\geqslant m$. By
definition of the subtype relation
$\Phi({x\sigma}{:}\,{A})\geqslant\Phi({x\sigma}{:}\,{B})$. Hence the theorem
follows.
9. 9.
Suppose $\Xi$ is of form
${\Gamma}\sststile{}{p}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma}\sststile{}{p}{{t}{:}\,{D}}D\mathrel{<:}C$
The case follows similarly to the sub-case before by induction hypothesis.
From this the theorem follows.
The second assertion of the theorem follows from the first together with the
assumption that every defined symbol in $\mathcal{F}$ is well-typed and
Proposition 2.1. ∎
###### Example 3.3 (continued from Example 1.1).
Consider the TRS $\mathcal{R}_{\mathsf{que}}$ from Example 1.1. We detail the
signature $\mathcal{F}$, starting with the constructor symbols.
$\displaystyle\mathsf{0}\colon\\!$ $\displaystyle\mathsf{Nat}^{p}$
$\displaystyle\hskip 8.61108pt\mathsf{s}\colon\\!$
$\displaystyle[{\mathsf{Nat}^{p}}]\xrightarrow{p}{\mathsf{Nat}^{p}}$
$\displaystyle\hskip 8.61108pt\mathsf{err\\_head}\colon\\!$
$\displaystyle\mathsf{Nat}^{p}$ $\displaystyle\hskip
8.61108pt\mathsf{que}\colon\\!$
$\displaystyle[{\mathsf{List}^{p}\times\mathsf{List}^{q}}]\xrightarrow{0}{\mathsf{Q}^{{(p,q)}}}$
$\displaystyle\mathsf{nil}\colon\\!$ $\displaystyle\mathsf{List}^{q}$
$\displaystyle\hskip 8.61108pt\mathrel{\mathsf{\sharp}}\colon\\!$
$\displaystyle[{\mathsf{Nat}^{0}\times\mathsf{List}^{q}}]\xrightarrow{q}{\mathsf{List}^{q}}$
$\displaystyle\hskip 8.61108pt\mathsf{err\\_tail}\colon\\!$
$\displaystyle\mathsf{Q}^{{(0,1)}}\hbox to0.0pt{$\;$,\hss}$
where $p,q\in{\mathbb{N}}$. Furthermore we make use of the following types for
defined symbols.
$\displaystyle\mathsf{chk}\colon$
$\displaystyle[{\mathsf{Q}^{{(0,1)}}}]\xrightarrow{3}{\mathsf{Q}^{{(0,1)}}}$
$\displaystyle\hskip 10.76385pt\mathsf{tl}\colon$
$\displaystyle[{\mathsf{Q}^{{(0,1)}}}]\xrightarrow{4}{\mathsf{Q}^{{(0,1)}}}$
$\displaystyle\hskip 10.76385pt\mathsf{hd}\colon$
$\displaystyle[{\mathsf{Q}^{{(0,1)}}}]\xrightarrow{1}{\mathsf{Nat}^{0}}$
$\displaystyle\mathsf{rev^{\prime}}\colon$
$\displaystyle[{\mathsf{List}^{1}\times\mathsf{List}^{0}}]\xrightarrow{1}{\mathsf{List}^{0}}$
$\displaystyle\hskip 10.76385pt\mathsf{rev}\colon$
$\displaystyle[{\mathsf{List}^{1}\times\mathsf{List}^{0}}]\xrightarrow{2}{\mathsf{List}^{0}}$
$\displaystyle\mathsf{snoc}\colon$
$\displaystyle[{\mathsf{Q}^{{(0,1)}}\times\mathsf{Nat}^{0}}]\xrightarrow{5}{\mathsf{Q}^{{(0,1)}}}$
$\displaystyle\hskip 10.76385pt\mathsf{enq}\colon$
$\displaystyle[{\mathsf{Nat}^{6}}]\xrightarrow{1}{\mathsf{Q}^{{(0,1)}}}\hbox
to0.0pt{$\;$,\hss}\hskip 10.76385pt$
Let $\mathcal{F}$ denote the induced signature. Based on the above definitions
it is not difficult to verify that $\mathcal{R}_{\mathsf{que}}$ is well-typed
wrt. $\mathcal{F}$. We show that $\mathsf{enq}$ is well-typed. Consider rule
6. First, we observe that $6$ resource units become available for the
recursive call, as
${{n}{:}\,{\mathsf{Nat}^{6}}}\sststile{}{6}{{\mathsf{s}(n)}{:}\,{\mathsf{Nat}^{6}}}$
is derivable. Second, we have the following partial type derivation; missing
parts are easy to fill in.
${{n}{:}\,{\mathsf{Nat}^{6}}}\sststile{}{6}{{\mathsf{snoc}(\mathsf{enq}(n),n)}{:}\,{\mathsf{Q}^{{(0,1)}}}}{{n_{1}}{:}\,{\mathsf{Nat}^{6}},{n_{2}}{:}\,{\mathsf{Nat}^{0}}}\sststile{}{6}{{\mathsf{snoc}(\mathsf{enq}(n_{1}),n_{2})}{:}\,{\mathsf{Q}^{{(0,1)}}}}\lx@proof@logical@and{{q}{:}\,{\mathsf{Q}^{{(0,1)}}},{m}{:}\,{\mathsf{Nat}^{0}}}\sststile{}{5}{{\mathsf{snoc}(q,m)}{:}\,{\mathsf{Q}^{{(0,1)}}}}\begin{minipage}[b]{150.69397pt}
\mbox{}
\hfill${{n_{2}}{:}\,{\mathsf{Nat}^{0}}}\sststile{}{0}{{n_{2}}{:}\,{\mathsf{Nat}^{0}}}$\\\
${{n_{1}}{:}\,{\mathsf{Nat}^{6}}}\sststile{}{1}{{\mathsf{enq}(n_{1})}{:}\,{\mathsf{Q}^{{(0,1)}}}}$
\end{minipage}$
Considering rule 7, it is easy to see that
${{n}{:}\,{\mathsf{Nat}^{6}}}\sststile{}{0}{{\mathsf{que}(\mathsf{nil},\mathsf{nil})}{:}\,{\mathsf{Q}^{{(0,1)}}}}$
is derivable. Thus $\mathsf{enq}$ is well-typed and we conclude optimal linear
runtime complexity of $\mathcal{R}_{\mathsf{que}}$.
### Polynomial bounds
Note that if the type annotations are chosen such that for each type $A$ we
have $\Phi({v}{:}\,{A})\in\operatorname{\mathsf{O}}(n^{k})$ for $n=\lvert
v\rvert$ then
$\operatorname{\mathsf{rc}}_{\mathcal{R}}(n)\in\operatorname{\mathsf{O}}(n^{k})$
as well. The following proposition gives a sufficient condition as to when
this is the case and in particular subsumes the type system in HoffmannH10a .
###### Theorem 3.2.
Suppose that for each constructor $c$ with
$[{A_{1}^{\vec{u_{1}}}\times\cdots\times
A_{n}^{\vec{u_{n}}}}]\xrightarrow{p}{C^{\vec{w}}}\in\mathcal{F}(c)$, there
exists $\vec{r}_{i}\in\mathcal{A}$ such that
$\vec{u_{i}}\leqslant\vec{w}+\vec{r}_{i}$ where
$\max{\vec{r}}_{i}\leqslant\max\vec{w}=:r$ and $p\leqslant r$ with
$\lvert\vec{r}_{i}\rvert<\lvert\vec{w}\rvert=:k$. Then
$\Phi({v}{:}\,{C^{\vec{w}}})\leqslant r\lvert v\rvert^{k}$.
###### Proof.
The proof is by induction on the size of $v$. Note that, if $k=0$ then
$\Phi({v}{:}\,{C^{\vec{w}}})=0$. This follows by superposition and uniqueness.
Otherwise, we have
$\displaystyle\Phi({c(v_{1},\dots,v_{n})}{:}\,{C^{\vec{w}}})$
$\displaystyle\leqslant
r+\Phi({v_{1}}{:}\,{A_{1}^{\vec{w}+\vec{r}_{1}}})+\dots+\Phi({v_{n}}{:}\,{A_{n}^{\vec{w}+\vec{r}_{n}}})$
$\displaystyle\leqslant r(1+\lvert v_{1}\rvert^{k}+\lvert
v_{1}\rvert^{k-1}+\dots+\lvert v_{n}\rvert^{k}+\lvert v_{n}\rvert^{k-1})$
$\displaystyle\leqslant r(1+\lvert v_{1}\rvert+\dots+\lvert
v_{n}\rvert)^{k}=r\lvert v\rvert^{k}\hbox to0.0pt{$\;$.\hss}$
Here we employ Lemma 3.1 to conclude for all $i=1,\dots,n$:
$\Phi({v_{i}}{:}\,{A_{i}^{\vec{w}+\vec{r}_{i}}})=\Phi({v_{i}}{:}\,{A_{i}^{\vec{w}}})+\Phi({v_{i}}{:}\,{A_{i}^{\vec{r}_{i}}})\hbox
to0.0pt{$\;$.\hss}$
Based on this observation we apply induction hypothesis to obtain the second
line. Furthermore in the last line we employ the multinomial theorem. ∎
We note that our running example satisfies the premise to the proposition. In
concrete cases more precise bounds than those given by Theorem 3.2 can be
computed as has been done in Example 3.2. The next example clarifies that
potentials are not restricted to polynomials.
###### Example 3.4.
Consider that we annotate the constructors for natural numbers as
${\mathsf{0}}{:}\,{\mathsf{Nat}^{\vec{p}}}$ and
${\mathsf{s}}{:}\,{[{\mathsf{Nat}^{2\vec{p}}}]\xrightarrow{\Diamond\vec{p}}{\mathsf{Nat}^{\vec{p}}}}$.
We then have, for example, $\Phi({t}{:}\,{\mathsf{Nat}^{1}})=2^{t+1}-1$.
As mentioned in the introduction, foundational issues are our main concern.
However, the potential-based method detailed above seems susceptible to
automation. One conceives the resource annotations as variables and encodes
the constraints of the typing rules in Figure 2 over these resource variables.
## 4 Small-Step Semantics
${}\sststile{}{0}{\langle{x},{\sigma}\rangle\to\langle{v},{\sigma}\rangle}x\sigma=v$
${}\sststile{}{0}{\langle{c(x_{1},\dots,x_{n})},{\sigma}\rangle\to\langle{c(v_{1},\dots,v_{n})},{\sigma}\rangle}\lx@proof@logical@and
c\in\mathcal{C}x_{1}\sigma=v_{1}\cdots x_{n}\sigma=v_{n}$
---
${}\sststile{}{0}{\langle{f(v_{1},\dots,v_{n})},{\sigma}\rangle\to\langle{f(x_{1},\dots,x_{n})},{\sigma\mathrel{\uplus}\rho}\rangle}\lx@proof@logical@and\forall
i\colon\text{$v_{i}$ is a value}\rho=\\{x_{1}\mapsto v_{1},\dots,x_{n}\mapsto
v_{n}\\}\text{$f$ is defined and all $x_{i}$ are fresh}$
${}\sststile{}{1}{\langle{f(x_{1},\dots,x_{n})},{\sigma}\rangle\to\langle{r},{\sigma\mathrel{\uplus}\tau}\rangle}\lx@proof@logical@and
f(l_{1},\dots,l_{n})\to r\in\mathcal{R}\forall i\colon x_{i}\sigma=l_{i}\tau$
${}\sststile{}{1}{\langle{f(\dots,t_{i},\dots)},{\sigma}\rangle\to\langle{f(\dots,u,\dots)},{\sigma^{\prime}}\rangle}{}\sststile{}{1}{\langle{t_{i}},{\sigma}\rangle\to\langle{u},{\sigma^{\prime}}\rangle}$
Note that the substitutions $\sigma$, $\sigma^{\prime}$, $\tau$, and $\rho$
are normalised.
Figure 3: Operational Small-Step Semantics
The big-step semantics, the type system, and Theorem 3.1 provide an amortised
resource analysis for typed TRSs that yields polynomial bounds. However,
Theorem 3.1 is not directly applicable, if we want to link this analysis to
the interpretation method. We recast the method and present a small-step
semantics, which will be used in our second soundness results (Theorem 4.1
below), cf. Figure 3. As the big-step semantics, the small-step semantics is
decorated with counters for the derivation height of the evaluated terms.
Suppose
${}\sststile{}{}{\langle{s},{\sigma}\rangle\to\langle{t},{\sigma^{\prime}}\rangle}$
holds for terms $s,t$ and substitutions $\sigma,\sigma^{\prime}$. An
inspection of the rules shows that $\sigma^{\prime}$ is an extension of
$\sigma$. Moreover we have the following fact.
###### Lemma 4.1.
Let $s,t$ be terms, let $\sigma$ be a normalised substitution such that
$\operatorname{\mathcal{V}\mathsf{ar}}(s)\subseteq\operatorname{\mathsf{dom}}(\sigma)$
and suppose
${}\sststile{}{}{\langle{s},{\sigma}\rangle\to\langle{t},{\sigma^{\prime}}\rangle}$.
Then $\sigma^{\prime}$ extends $\sigma$ and $s\sigma=s\sigma^{\prime}$.
###### Proof.
The first assertion follows by induction on the relation
${}\sststile{}{}{\langle{s},{\sigma}\rangle\to\langle{t},{\sigma^{\prime}}\rangle}$.
Now suppose
$\sigma={\sigma^{\prime}}\\!\restriction\\!{\operatorname{\mathsf{dom}}(\sigma)}$.
Then
$s\sigma=s({\sigma^{\prime}}\\!\restriction\\!{\operatorname{\mathsf{dom}}(\sigma)})=s\sigma^{\prime}$.
∎
The transitive closure of the judgement
${}\sststile{}{m}{\langle{s},{\sigma}\rangle\to\langle{t},{\tau}\rangle}$ is
defined as follows:
1. 1.
${}\sststile{}{m}{\langle{s},{\sigma}\rangle\twoheadrightarrow\langle{t},{\tau}\rangle}$
if ${}\sststile{}{m}{\langle{s},{\sigma}\rangle\to\langle{t},{\tau}\rangle}$
2. 2.
${}\sststile{}{m_{1}+m_{2}}{\langle{s},{\sigma}\rangle\twoheadrightarrow\langle{u},{\rho}\rangle}$
if
${}\sststile{}{m_{1}}{\langle{s},{\sigma}\rangle\to\langle{t},{\tau}\rangle}$
and
${}\sststile{}{m_{2}}{\langle{t},{\tau}\rangle\twoheadrightarrow\langle{u},{\rho}\rangle}$.
The next lemma proves the equivalence of big-step and small-step semantics.
###### Lemma 4.2.
Let $\sigma$ be a normalised substitution, let $t$ be a term,
$\operatorname{\mathcal{V}\mathsf{ar}}(t)\subseteq\operatorname{\mathsf{dom}}(\sigma)$,
and let $v$ be a value. Then ${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$ if
and only if
${}\sststile{}{m}{\langle{t},{\sigma}\rangle\twoheadrightarrow\langle{v},{\sigma^{\prime}}\rangle}$,
where $\sigma^{\prime}$ is an extension of $\sigma$.
###### Proof.
First we prove the direction from left to right.
1. 1.
Suppose $\Pi$ has the form:
${\sigma}\sststile{}{0}{{x}\Rightarrow{v}}x\sigma=v\hbox to0.0pt{$\;$,\hss}$
such that $t=x$ and $v=x\sigma$. Hence we obtain
${}\sststile{}{0}{\langle{x},{\sigma}\rangle\twoheadrightarrow\langle{v},{\sigma}\rangle}$.
2. 2.
Suppose $\Pi$ has the form:
${\sigma}\sststile{}{0}{{c(x_{1},\dots,x_{n})}\Rightarrow{c(v_{1},\dots,v_{n})}}\lx@proof@logical@and
c\in\mathcal{C}x_{1}\sigma=v_{1}\cdots x_{n}\sigma=v_{n}\hbox
to0.0pt{$\;$,\hss}$
such that $t=c(x_{1},\dots,x_{n})$ and $v=c(v_{1},\dots,v_{n})$. Again, we
directly obtain
${}\sststile{}{0}{\langle{t},{\sigma}\rangle\twoheadrightarrow\langle{v},{\sigma}\rangle}$.
3. 3.
Suppose the last rule in $\Pi$ if of form:
${\sigma}\sststile{}{m+1}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}\lx@proof@logical@and
f(l_{1},\dots,l_{n})\to r\in\mathcal{R}\forall i\colon
x_{i}\sigma=l_{i}\tau{\sigma\mathrel{\uplus}\tau}\sststile{}{m}{{r}\Rightarrow{v}}\hbox
to0.0pt{$\;$,\hss}$
where $t=f(x_{1},\dots,x_{n})$. By hypothesis there exists an extension
$\sigma^{\prime}$ of $\sigma\mathrel{\uplus}\tau$ such that
${}\sststile{}{m}{\langle{r},{\sigma\mathrel{\uplus}\tau}\rangle\twoheadrightarrow\langle{v},{\sigma^{\prime}}\rangle}$.
Furthermore, we have
${}\sststile{}{1}{\langle{t},{\sigma}\rangle\to\langle{r},{\sigma\mathrel{\uplus}\tau}\rangle}$.
Thus
${}\sststile{}{m+1}{\langle{t},{\sigma}\rangle\twoheadrightarrow\langle{v},{\sigma^{\prime}}\rangle}$.
By definition
$\operatorname{\mathsf{dom}}(\sigma)\cap\operatorname{\mathsf{dom}}(\tau)=\varnothing$.
Hence $\sigma^{\prime}$ is an extension of $\sigma$.
4. 4.
Finally, suppose the last rule in $\Pi$ has the form
${\sigma}\sststile{}{m}{{f(t_{1},\dots,t_{n})}\Rightarrow{v}}\lx@proof@logical@and{\sigma\mathrel{\uplus}\rho}\sststile{}{m_{0}}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}{\sigma}\sststile{}{m_{1}}{{t_{1}}\Rightarrow{v_{1}}}\cdots{\sigma}\sststile{}{m_{n}}{{t_{n}}\Rightarrow{v_{n}}}m=\sum_{i=0}^{n}m_{i}\hbox
to0.0pt{$\;$,\hss}$
where $t=f(t_{1},\dots,t_{n})$. By induction hypothesis (and repeated use of
Lemma 2.1), we have for all $i=1,\dots,n$:
${}\sststile{}{m_{i}}{\langle{t_{1}},{\sigma_{i-1}}\rangle\twoheadrightarrow\langle{v_{1}},{\sigma_{i}}\rangle}$,
where we set $\sigma_{0}=\sigma$ and note that all $\sigma_{i}$ are extensions
of $\sigma$. As
${}\sststile{}{0}{\langle{f(v_{1},\dots,v_{n})},{\sigma_{n}}\rangle\to\langle{f(x_{1},\dots,x_{n})},{\sigma_{n}\mathrel{\uplus}\rho}\rangle}$
we obtain:
${}\sststile{}{\sum_{i=1}^{n}m_{i}}{\langle{f(t_{1},\dots,t_{n})},{\sigma}\rangle\twoheadrightarrow\langle{f(x_{1},\dots,x_{n})},{\sigma_{n}\mathrel{\uplus}\rho}\rangle}\hbox
to0.0pt{$\;$.\hss}$ (4)
Furthermore, by Lemma 2.1 and the induction hypothesis there exists a
substitution $\sigma^{\prime}$ such that
${}\sststile{}{m_{0}}{\langle{f(x_{1},\dots,x_{n})},{\sigma_{n}\mathrel{\uplus}\rho}\rangle\twoheadrightarrow\langle{v},{\sigma^{\prime}}\rangle}\hbox
to0.0pt{$\;$,\hss}$ (5)
where $\sigma^{\prime}$ extends $\sigma_{n}\mathrel{\uplus}\rho$ (and thus
also $\sigma$ as
$\operatorname{\mathsf{dom}}(\sigma_{n})\cap\operatorname{\mathsf{dom}}(\rho)=\varnothing$).
From (4) and (5) we obtain
${}\sststile{}{m}{\langle{t},{\sigma}\rangle\twoheadrightarrow\langle{v},{\sigma^{\prime}}\rangle}$.
This establishes the direction from left to right. Now we consider the
direction form right to left. The proof of the first reduction
${}\sststile{}{m}{\langle{t},{\sigma}\rangle\to\langle{u},{\sigma^{\prime\prime}}\rangle}$
in $D$ is denoted as $\Xi$.
1. 1.
Suppose $\Xi$ has either of the following forms
${}\sststile{}{0}{\langle{x},{\sigma}\rangle\to\langle{v},{\sigma}\rangle}x\sigma=v\qquad{}\sststile{}{0}{\langle{c(x_{1},\dots,x_{n})},{\sigma}\rangle\to\langle{c(v_{1},\dots,v_{n})},{\sigma}\rangle}\lx@proof@logical@and
x_{1}\sigma=v_{1}\cdots x_{n}\sigma=v_{n}$
Then the lemma follows trivially.
2. 2.
Suppose $\Xi$ has the form
${}\sststile{}{0}{\langle{f(v_{1},\dots,v_{n})},{\sigma}\rangle\to\langle{f(x_{1},\dots,x_{n})},{\sigma\mathrel{\uplus}\rho}\rangle}\lx@proof@logical@and\forall
i\colon\text{$v_{i}$ is a value}\rho=\\{x_{1}\mapsto v_{1},\dots,x_{n}\mapsto
v_{n}\\}\text{$f$ is defined and all $x_{i}$ are fresh}\hbox
to0.0pt{$\;$.\hss}$
We apply the induction hypothesis to conclude
${\sigma\mathrel{\uplus}\rho}\sststile{}{m}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}$.
Moreover, we observe that ${\sigma}\sststile{}{0}{{v_{i}}\Rightarrow{v_{i}}}$
holds for all $i=1,\dots,n$. (This follows by a straightforward inductive
argument.) From this we derive
${\sigma}\sststile{}{0}{{f(v_{1},\dots,v_{n})}\Rightarrow{v}}$ as follows:
${\sigma}\sststile{}{0}{{f(v_{1},\dots,v_{n})}\Rightarrow{v}}\lx@proof@logical@and{\sigma\mathrel{\uplus}\rho}\sststile{}{m}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}{\sigma}\sststile{}{0}{{v_{1}}\Rightarrow{v_{1}}}\cdots{\sigma}\sststile{}{0}{{v_{n}}\Rightarrow{v_{n}}}\hbox
to0.0pt{$\;$.\hss}$
3. 3.
Suppose $\Xi$ has the form
${}\sststile{}{1}{\langle{f(x_{1},\dots,x_{n})},{\sigma}\rangle\to\langle{r},{\sigma\mathrel{\uplus}\tau}\rangle}\lx@proof@logical@and
f(l_{1},\dots,l_{n})\to r\in\mathcal{R}\forall i\colon
x_{i}\sigma=l_{i}\tau\hbox to0.0pt{$\;$,\hss}$
such that $\sigma^{\prime}$ is an extension of $\sigma\mathrel{\uplus}\tau$.
By induction hypothesis we conclude
${\sigma\mathrel{\uplus}\tau}\sststile{}{m^{\prime}}{{r}\Rightarrow{v}}$. In
conjunction with an application of the rule
${\sigma}\sststile{}{m+1}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}\lx@proof@logical@and
f(l_{1},\dots,l_{n})\to r\in\mathcal{R}\forall i\colon
x_{i}\sigma=l_{i}\tau{\sigma\mathrel{\uplus}\tau}\sststile{}{m^{\prime}}{{r}\Rightarrow{v}}\hbox
to0.0pt{$\;$,\hss}$
we derive
${\sigma}\sststile{}{m^{\prime}+1}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}$ as
required.
4. 4.
Suppose $\Xi$ has the form
${}\sststile{}{1}{\langle{f(\dots,t_{i},\dots)},{\sigma}\rangle\to\langle{f(\dots,u,\dots)},{\sigma^{\prime\prime}}\rangle}{}\sststile{}{1}{\langle{t_{i}},{\sigma}\rangle\to\langle{u},{\sigma^{\prime\prime}}\rangle}\hbox
to0.0pt{$\;$,\hss}$
such that $\sigma^{\prime}$ is an extension of $\sigma^{\prime\prime}$. Then
by induction hypothesis we obtain:
${\sigma^{\prime\prime}}\sststile{}{m^{\prime}}{{f(\dots,u,\dots)}\Rightarrow{v}}$.
Furthermore by induction hypothesis we have
${\sigma}\sststile{}{1}{{t_{i}}\Rightarrow{v_{1}}}$
5. 5.
Suppose the initial sequence of $D$ is based on the following reductions,
where $m=\sum_{i=1}^{n}m_{i}+m^{\prime}$.
$\displaystyle{}\sststile{}{m_{1}}{\langle{f(t_{1},\dots,t_{n})},{\sigma}\rangle\twoheadrightarrow\langle{f(v_{1},\dots,t_{n})},{\sigma_{1}}\rangle}$
$\displaystyle\qquad\vdots$
$\displaystyle{}\sststile{}{m_{n}}{\langle{f(v_{1},\dots,t_{n})},{\sigma}\rangle\twoheadrightarrow\langle{f(v_{1},\dots,v_{n})},{\sigma_{n}}\rangle}$
$\displaystyle{}\sststile{}{0}{\langle{f(v_{1},t_{2},\dots,t_{n})},{\sigma_{n}}\rangle\to\langle{f(x_{1},\dots,x_{n})},{\sigma_{n}\mathrel{\uplus}\rho}\rangle}\hbox
to0.0pt{$\;$.\hss}$
We apply induction hypothesis on
${}\sststile{}{m^{\prime}}{\langle{f(x_{1},\dots,x_{n})},{\sigma^{\prime}\mathrel{\uplus}\rho}\rangle\to\langle{v},{\sigma^{\prime}}\rangle}$
and conclude:
${\sigma^{\prime}\mathrel{\uplus}\rho}\sststile{}{m^{\prime}}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}$.
Again by induction hypothesis and inspection of the corresponding proofs, we
obtain ${\sigma_{i-1}}\sststile{}{m_{i}}{{t_{i}}\Rightarrow{v_{i}}}$ for all
$i=1,\dots,n$. (We set $\sigma_{0}\mathrel{:=}\sigma$.) Due to Lemma 4.1 we
have $t_{i}\sigma_{i}=t_{i}\sigma$. Thus, for all $i$,
${\sigma}\sststile{}{m_{i}}{{t_{i}}\Rightarrow{v_{i}}}$. Note that
$\operatorname{\mathsf{dom}}(\sigma_{n})\cap\operatorname{\mathsf{dom}}(\rho)=\varnothing$.
Hence, from
${\sigma_{n}\mathrel{\uplus}\rho}\sststile{}{m^{\prime}}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}$
we obtain
${\sigma\mathrel{\uplus}\rho}\sststile{}{m^{\prime}}{{f(x_{1},\dots,x_{n})}\Rightarrow{v}}$.
Thus ${\sigma}\sststile{}{m}{{t}\Rightarrow{v}}$ follows.
∎
We extend the notion of potential (cf. Definition 3.3) to ground terms.
###### Definition 4.1.
Let
$t=f(t_{1},\dots,t_{n})\in\operatorname{\mathcal{T}}(\mathcal{D}\cup\mathcal{C})$
and let $[{A_{1}\times\cdots\times
A_{n}}]\xrightarrow{p}{C}{q}\in\mathcal{F}(f)$. Then the _potential_ of $t$ is
defined as follows:
$\Phi({t}{:}\,{C})\mathrel{:=}(p-q)+\Phi({t_{1}}{:}\,{A_{1}})+\cdots+\Phi({t_{n}}{:}\,{A_{n}})\hbox
to0.0pt{$\;$.\hss}$
Note that by assumption the declaration in $\mathcal{F}(f)$ is unique.
###### Example 4.1 (continued from Example 3.3).
Recall the types of $\mathsf{que}$ and $\mathsf{chk}$. Let
$q=\mathsf{que}(f,r)$ be a queue. We obtain
$\Phi({\mathsf{chk}(q)}{:}\,{\mathsf{Q}^{{(0,1)}}})=3+\Phi({q}{:}\,{\mathsf{Q}^{{(0,1)}}})=3+\Phi({f}{:}\,{\mathsf{List}^{0}})+\Phi({r}{:}\,{\mathsf{List}^{1}})=3+\lvert
r\rvert$.
###### Lemma 4.3.
Let $\mathcal{R}$ and $\sigma$ be well-typed. Suppose
${\Gamma}\sststile{}{p}{{t}{:}\,{A}}$. Then we have
$\Phi({\sigma}{:}\,{\Gamma})+p\geqslant\Phi({t\sigma}{:}\,{A})$.
###### Proof.
Let $\Xi$ denote the proof of ${\Gamma}\sststile{}{p}{{t}{:}\,{A}}$.
1. 1.
Let $t=x$ and thus wlog. $\Xi$ is of form
${{x}{:}\,{A}}\sststile{}{0}{{x}{:}\,{A}}\hbox to0.0pt{$\;$.\hss}$
Then
$\Phi({\sigma}{:}\,{\Gamma})=\Phi({x\sigma}{:}\,{A})=\Phi({t\sigma}{:}\,{A})$,
from which the lemma follows.
2. 2.
Let $t=f(x_{1},\dots,x_{n})$ where $f\in\mathcal{C}\cup\mathcal{D}$. Thus
wlog. $\Xi$ is of form
${{x_{1}}{:}\,{A_{1}^{\vec{u_{1}}}},\dots,{x_{n}}{:}\,{A_{n}^{\vec{u_{n}}}}}\sststile{}{p}{{f(x_{1},\dots,x_{n})}{:}\,{{C}^{\vec{v}}}}\lx@proof@logical@and
f\in\mathcal{C}\cup\mathcal{D}[{A_{1}^{\vec{u_{1}}}\times\cdots\times
A_{n}^{\vec{u_{n}}}}]\xrightarrow{p}{{C}^{\vec{v}}}\in\mathcal{F}(f)\hbox
to0.0pt{$\;$.\hss}$
Hence we obtain
$\Phi({\sigma}{:}\,{\Gamma})+p=\sum_{i=1}^{n}\Phi({x_{i}\sigma}{:}\,{A_{i}^{\vec{u_{i}}}})+p=\Phi({t\sigma}{:}\,{C^{\vec{v}}})\hbox
to0.0pt{$\;$,\hss}$
and the lemma follows.
3. 3.
Suppose $t=f(t_{1},\dots,t_{n})$, such that $\vec{t}\not\in\mathcal{V}$ and
$f\in\mathcal{C}\cup\mathcal{D}$. Thus $\Xi$ is of form
${\Gamma_{1},\dots,\Gamma_{n}}\sststile{}{p}{{f(t_{1},\dots,t_{n})}{:}\,{A}}\lx@proof@logical@and{\overbrace{{x_{1}}{:}\,{A_{1}},\dots,{x_{n}}{:}\,{A_{n}}}^{{}=:\Delta}}\sststile{}{p_{0}}{{f(x_{1},\dots,x_{n})}{:}\,{A}}{\Gamma_{1}}\sststile{}{p_{1}}{{t_{1}}{:}\,{A_{1}}}\cdots{\Gamma_{n}}\sststile{}{p_{n}}{{t_{n}}{:}\,{A_{n}}}\hbox
to0.0pt{$\;$,\hss}$
where $p=\sum_{i=0}^{n}p_{i}$. Then by induction hypothesis we have
$\Phi({\sigma}{:}\,{\Gamma_{i}})+p_{i}\geqslant\Phi({t_{i}\sigma}{:}\,{A_{i}})$
for all $i=1,\dots,n$. Hence
$\sum_{i=1}^{n}\Phi({\sigma}{:}\,{\Gamma_{i}})+\sum_{i=1}^{n}p_{i}\geqslant\sum_{i=1}^{n}\Phi({t_{i}\sigma}{:}\,{A_{i}})$.
Let $\rho\mathrel{:=}\\{x_{1}\mapsto t_{1}\sigma,\dots,x_{n}\mapsto
t_{n}\sigma\\}$. Again by induction hypothesis we have
$\Phi({\rho}{:}\,{\Delta})+p_{0}\geqslant\Phi({f(x_{1},\dots,x_{n})\rho}{:}\,{A})$.
Note that $f(x_{1},\dots,x_{n})\rho=t\sigma$ and $x_{i}\rho=t_{i}\sigma$ by
construction. We obtain
$\displaystyle\Phi({\sigma}{:}\,{\Gamma})+\sum_{i=0}^{n}p_{i}$
$\displaystyle=\sum_{i=1}^{n}\Phi({\sigma}{:}\,{\Gamma_{i}})+p_{0}\geqslant\sum_{i=1}^{n}\Phi({t_{i}\sigma}{:}\,{A_{i}})+p_{0}$
$\displaystyle=\sum_{i=1}^{n}\Phi({x_{i}\rho}{:}\,{A_{i}})+p_{0}=\Phi({\rho}{:}\,{\Delta})+p_{0}$
$\displaystyle\geqslant\Phi({t\sigma}{:}\,{A})\hbox to0.0pt{$\;$.\hss}$
4. 4.
Suppose $\Xi$ is of form:
${\Gamma}\sststile{}{p^{\prime}}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma}\sststile{}{p}{{t}{:}\,{C}}p^{\prime}\geqslant
p$
By induction hypothesis, we have
$\Phi({\sigma}{:}\,{\Gamma})+p\geqslant\Phi({t\sigma}{:}\,{A})$. Then the
lemma follows from the assumption $p^{\prime}\geqslant p$.
5. 5.
Suppose $\Xi$ ends with one of the following structural rules
${\Gamma,{x}{:}\,{A}}\sststile{}{p}{{t}{:}\,{C}}{\Gamma}\sststile{}{p}{{t}{:}\,{C}}\hskip
43.05542pt{\Gamma,{z}{:}\,{A}}\sststile{}{p}{{t[z,z]}{:}\,{C}}\lx@proof@logical@and{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}}\sststile{}{p}{{t[x,y]}{:}\,{C}}\curlyvee\\!({A}\\!\mid\\!{A_{1},A_{2}})$
We only consider the second rule, as the first alternatives follows trivially.
Let $\rho\mathrel{:=}\sigma\mathrel{\uplus}\\{x\mapsto z\sigma,y\mapsto
z\sigma\\}$; by induction hypothesis, we have
$\Phi({\rho}{:}\,{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}})+p\geqslant\Phi({t[x,y]\rho}{:}\,{A})$.
By definition of $\rho$ and Lemma 3.1, we obtain
$\Phi({\sigma}{:}\,{\Gamma,{z}{:}\,{A}})=\Phi({\rho}{:}\,{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}})\hbox
to0.0pt{$\;$.\hss}$
Hence
$\Phi({\sigma}{:}\,{\Gamma,{z}{:}\,{A}})+p\geqslant\Phi({t[z,z]\sigma}{:}\,{A})$
follows from $t[x,y]\rho=t[z,z]\sigma$.
6. 6.
Suppose $\Xi$ ends either in a sub- or in a supertyping rule:
${\Gamma,{x}{:}\,{A}}\sststile{}{p}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma,{x}{:}\,{B}}\sststile{}{p}{{t}{:}\,{C}}A\mathrel{<:}B\hskip
43.05542pt{\Gamma}\sststile{}{p}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma}\sststile{}{p}{{t}{:}\,{D}}D\mathrel{<:}C$
Consider the second rule. We have to show that
$\Phi({\sigma}{:}\,{\Gamma})+p\geqslant\Phi({t\sigma}{:}\,{C})$. This follows
from induction hypothesis, which yields
$\Phi({\sigma}{:}\,{\Gamma})+p\geqslant\Phi({t\sigma}{:}\,{D})$ as
$\Phi({t\sigma}{:}\,{D})\geqslant\Phi({t\sigma}{:}\,{C})$ by definition of the
subtyping relation. The argument for the first rule is similar. This concludes
the inductive argument.
∎
We obtain our second soundness result.
###### Theorem 4.1.
Let $\mathcal{R}$ and $\sigma$ be well-typed. Suppose
${\Gamma}\sststile{}{p}{{t}{:}\,{A}}$ and
${}\sststile{}{m}{\langle{t},{\sigma}\rangle\to\langle{u},{\sigma^{\prime}}\rangle}$.
Then $\Phi({\sigma}{:}\,{\Gamma})-\Phi({u\sigma^{\prime}}{:}\,{A})+p\geqslant
m$. Thus if for all ground basic terms $t$ and types $A$:
$\Phi({t}{:}\,{A})\in\operatorname{\mathsf{O}}(n^{k})$, where
$n=\lvert{t}\rvert$, then
$\operatorname{\mathsf{rc}}_{\mathcal{R}}(n)\in\operatorname{\mathsf{O}}(n^{k})$.
###### Proof.
Let $\Pi$ be the proof of the judgement
${}\sststile{}{m}{\langle{t},{\sigma}\rangle\to\langle{u},{\sigma^{\prime}}\rangle}$
and let $\Xi$ denote the proof of ${\Gamma}\sststile{}{p}{{t}{:}\,{A}}$. The
proof proceeds by main-induction on the length of $\Pi$ and by side-induction
on the length of $\Xi$. We focus on some interesting cases.
1. 1.
Suppose $\Pi$ has the form
${}\sststile{}{0}{\langle{x},{\sigma}\rangle\to\langle{u},{\sigma}\rangle}x\sigma=u\hbox
to0.0pt{$\;$,\hss}$
such that $t=x$ and $u=x\sigma$. As $\sigma$ is normalised $u$ is a value.
Wlog. we can assume that $\Xi$ is of form
${{x}{:}\,{A}}\sststile{}{0}{{x}{:}\,{A}}$. It suffices to show
$\Phi({\sigma}{:}\,{\Gamma})\geqslant\Phi({u\sigma}{:}\,{A})$, which follows
from Lemma 4.3 as $x\sigma=u=u\sigma$.
2. 2.
Suppose $\Pi$ has the form
${}\sststile{}{0}{\langle{c(x_{1},\dots,x_{n})},{\sigma}\rangle\to\langle{c(u_{1},\dots,u_{n})},{\sigma}\rangle}\lx@proof@logical@and
x_{1}\sigma=u_{1}\cdots x_{n}\sigma=u_{n}\hbox to0.0pt{$\;$,\hss}$
such that $t=c(x_{1},\dots,x_{n})$ and $u=c(x_{1}\sigma,\dots,x_{n}\sigma)$,
which again is a value. Further let $\Xi$ end in the judgement:
${{x_{1}}{:}\,{A_{1}^{\vec{u_{1}}}},\dots,{x_{n}}{:}\,{A_{n}^{\vec{u_{n}}}}}\sststile{}{p}{{c(x_{1},\dots,x_{n})}{:}\,{{C}^{\vec{v}}}}\hbox
to0.0pt{$\;$.\hss}$
Let
$\Gamma={x_{1}}{:}\,{A_{1}^{\vec{u_{1}}}},\dots,{x_{n}}{:}\,{A_{n}^{\vec{u_{n}}}}$;
by Lemma 4.3 we have
$\Phi({\sigma}{:}\,{\Gamma})+p\geqslant\Phi({t\sigma}{:}\,{A})=\Phi({u\sigma}{:}\,{A})$
as $t\sigma=u=u\sigma$.
3. 3.
Suppose $\Pi$ has the form
${}\sststile{}{0}{\langle{f(v_{1},\dots,v_{n})},{\sigma}\rangle\to\langle{f(x_{1},\dots,x_{n})},{\sigma\mathrel{\uplus}\rho}\rangle}\lx@proof@logical@and\forall
i\colon\text{$v_{i}$ is a value}\rho=\\{x_{1}\mapsto v_{1},\dots,x_{n}\mapsto
v_{n}\\}\text{$f$ is defined and all $x_{i}$ are fresh}$
Then $t=f(v_{1},\dots,v_{n})$ is ground, as all $v_{i}$ are values. Hence, we
have
$t\sigma=t=f(x_{1},\dots,x_{n})\rho=f(x_{1},\dots,x_{n})(\sigma\mathrel{\uplus}\rho)\hbox
to0.0pt{$\;$.\hss}$
The last equality follows as
$\operatorname{\mathsf{dom}}(\sigma)\cap\operatorname{\mathsf{dom}}(\rho)=\varnothing$.
By Lemma 4.3 we have
$\Phi({\sigma}{:}\,{\Gamma})+p\geqslant\Phi({t\sigma}{:}\,{A})$. Then the
theorem follows as $t\sigma=f(x_{1},\dots,x_{n})(\sigma\mathrel{\uplus}\rho)$
from above.
4. 4.
Suppose $\Pi$ has the form
${}\sststile{}{1}{\langle{f(x_{1},\dots,x_{n})},{\sigma}\rangle\to\langle{r},{\sigma\mathrel{\uplus}\tau}\rangle}\lx@proof@logical@and
f(l_{1},\dots,l_{n})\to r\in\mathcal{R}\forall i\colon
x_{i}\sigma=l_{i}\tau\hbox to0.0pt{$\;$.\hss}$
Then $t=f(x_{1},\dots,x_{n})$ and
$f(x_{1},\dots,x_{n})\sigma=f(l_{1},\dots,l_{n})\tau$. Suppose
$\operatorname{\mathcal{V}\mathsf{ar}}(f(\vec{l}))=\\{y_{1},\dots,y_{\ell}\\}$
and let
$\operatorname{\mathcal{V}\mathsf{ar}}(l_{i})=\\{y_{i1},\dots,y_{il_{i}}\\}$
for $i\in\\{1,\dots,n\\}$. As $\mathcal{R}$ is left-linear we have
$\operatorname{\mathcal{V}\mathsf{ar}}(f(l_{1},\dots,l_{n}))=\biguplus_{i=1}^{n}\operatorname{\mathcal{V}\mathsf{ar}}(l_{i})$.
We set $\Gamma={x_{1}}{:}\,{A_{1}},\dots,{x_{n}}{:}\,{A_{n}}$. By the
assumption ${\Gamma}\sststile{}{p}{{t}{:}\,{A}}$ and well-typedness of
$\mathcal{R}$ we obtain
${\overbrace{{y_{1}}{:}\,{B_{1}},\dots,{y_{\ell}}{:}\,{B_{\ell}}}^{{}=:\Delta}}\sststile{}{p-1+\sum_{i=1}^{n}k_{i}}{{r}{:}\,{C}}\hbox
to0.0pt{$\;$,\hss}$ (6)
as in (1). We have
$\displaystyle\Phi({\sigma}{:}\,{\Gamma})+p$
$\displaystyle=\sum_{i=1}^{n}\Phi({x_{i}\sigma}{:}\,{A_{i}})+p$
$\displaystyle=\sum_{i=1}^{n}\left(k_{i}+\Phi({y_{i1}\tau}{:}\,{B_{i1}})+\cdots+\Phi({y_{il_{i}}\tau}{:}\,{B_{il_{i}}})\right)+p$
$\displaystyle=\Phi({\tau}{:}\,{\Delta})+\sum_{i=1}^{n}k_{i}+(p-1)+1$
$\displaystyle\geqslant\Phi({r\tau}{:}\,{C})+1\geqslant\Phi({r(\sigma\mathrel{\uplus}\tau)}{:}\,{C})+1\hbox
to0.0pt{$\;$.\hss}$
Here the first equality follows by an inspection on the cases for the
constructors and
$\Phi({\tau}{:}\,{\Delta})+\sum_{i=1}^{n}k_{i}+(p-1)\geqslant\Phi({r\tau}{:}\,{C})$
follows due to Lemma 4.3 and (6). Furthermore note that
$r\tau=r(\sigma\mathrel{\uplus}\tau)$, as
$\operatorname{\mathsf{dom}}(\sigma)\cap\operatorname{\mathsf{dom}}(\tau)=\varnothing$.
5. 5.
Suppose the last rule in $\Pi$ has the form
${}\sststile{}{1}{\langle{f(t_{1},\dots,t_{n})},{\sigma}\rangle\to\langle{f(u,\dots,t_{n})},{\sigma^{\prime}}\rangle}{}\sststile{}{1}{\langle{t_{1}},{\sigma}\rangle\to\langle{u},{\sigma^{\prime}}\rangle}\hbox
to0.0pt{$\;$.\hss}$
Wlog. the last rule in the type inference $\Xi$ is of the following form,
where we can assume that every variable occurs at most once in
$f(t_{1},\dots,t_{n})$.
${\underbrace{\Gamma_{1},\dots,\Gamma_{n}}_{{}=:\Gamma}}\sststile{}{p}{{f(t_{1},\dots,t_{n})}{:}\,{C}}\lx@proof@logical@and{\overbrace{{x_{1}}{:}\,{A_{1}},\dots,{x_{n}}{:}\,{A_{n}}}^{{}=:\Delta}}\sststile{}{p_{0}}{{f(\vec{x})}{:}\,{C}}{\Gamma_{1}}\sststile{}{p_{1}}{{t_{1}}{:}\,{A_{1}}}\cdots{\Gamma_{n}}\sststile{}{p_{n}}{{t_{n}}{:}\,{A_{n}}}p=\sum_{i=0}^{n}p_{i}\hbox
to0.0pt{$\;$.\hss}$
By induction hypothesis on
${}\sststile{}{1}{\langle{t_{1}},{\sigma}\rangle\to\langle{u},{\sigma^{\prime}}\rangle}$
and ${\Gamma_{1}}\sststile{}{p_{1}}{{t_{1}}{:}\,{A_{1}}}$ we obtain (i)
$\Phi({\sigma}{:}\,{\Gamma_{1}})-\Phi({u\sigma^{\prime}}{:}\,{A_{1}})+p_{1}\geqslant
1$ and $n-1$ applications of Lemma 4.3 yield (ii)
$\Phi({\sigma}{:}\,{\Gamma_{i}})+p_{i}\geqslant\Phi({t_{i}\sigma}{:}\,{A_{i}})$
for all $i=2,\dots,n$. We set $\rho\mathrel{:=}\\{x_{1}\to
u\sigma^{\prime},x_{2}\to t_{2}\sigma,\dots,x_{n}\to t_{n}\sigma\\}$. Another
application of Lemma 4.3 on
${\Delta}\sststile{}{p_{0}}{{f(x_{1},\dots,x_{n})}{:}\,{C}}$ yields (iii)
$\Phi({\rho}{:}\,{\Delta})+p_{0}\geqslant\Phi({f(x_{1}\rho,x_{2}\rho,\dots,x_{n}\rho)}{:}\,{C})$.
Finally, we observe
$\Phi({\sigma}{:}\,{\Gamma})=\sum_{i=1}^{n}\Phi({\sigma}{:}\,{\Gamma_{i}}$.
The theorem follows by combining the equations in (i)–(iii).
6. 6.
Suppose $\Xi$ is of form:
${\Gamma}\sststile{}{p^{\prime}}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma}\sststile{}{p}{{t}{:}\,{C}}p^{\prime}\geqslant
p$
By side-induction on ${\Gamma}\sststile{}{p}{{t}{:}\,{C}}$ and
${}\sststile{}{m}{\langle{t},{\sigma}\rangle\to\langle{u},{\sigma^{\prime}}\rangle}$
we conclude that
$\Phi({\sigma}{:}\,{\Gamma})-\Phi({u\sigma^{\prime}}{:}\,{A})+p\geqslant m$.
Then the theorem follows from the assumption $p^{\prime}\geqslant p$.
7. 7.
Suppose $\Xi$ is of form:
${\Gamma,{x}{:}\,{A}}\sststile{}{p}{{t}{:}\,{C}}{\Gamma}\sststile{}{p}{{t}{:}\,{C}}$
We conclude by side-induction that
$\Phi({\sigma}{:}\,{\Gamma})-\Phi({u\sigma^{\prime}}{:}\,{A}+p\geqslant m$. As
$\Phi({\sigma}{:}\,{\Gamma,{x}{:}\,{A}})\geqslant\Phi({\sigma}{:}\,{\Gamma})$
the theorem follows.
8. 8.
Suppose $\Xi$ is of form:
${\Gamma,{z}{:}\,{A}}\sststile{}{p}{{t[z,z]}{:}\,{C}}\lx@proof@logical@and{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}}\sststile{}{p}{{t[x,y]}{:}\,{C}}\curlyvee\\!({A}\\!\mid\\!{A_{1},A_{2}})$
By assumption
${}\sststile{}{m}{\langle{t[z,z]},{\sigma}\rangle\to\langle{u},{\sigma^{\prime}}\rangle}$;
let $\rho\mathrel{:=}\sigma\mathrel{\uplus}\\{x\mapsto z\sigma,y\mapsto
z\sigma\\}$. By side-induction on
${\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}}\sststile{}{p}{{t[x,y]}{:}\,{C}}$ and
${}\sststile{}{m}{\langle{t[x,y]},{\rho}\rangle\to\langle{u},{\sigma^{\prime}}\rangle}$
we conclude that for all
$\Phi({\rho}{:}\,{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}})-\Phi({u\sigma^{\prime}}{:}\,{A})+p\geqslant
m$. By definition of $\rho$ and Lemma 3.1, we obtain
$\Phi({\sigma}{:}\,{\Gamma,{z}{:}\,{A}})=\Phi({\rho}{:}\,{\Gamma,{x}{:}\,{A_{1}},{y}{:}\,{A_{2}}})$,
from which the theorem follows.
9. 9.
Suppose $\Xi$ ends either in a sub- or in a supertyping rule:
${\Gamma,{x}{:}\,{A}}\sststile{}{p}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma,{x}{:}\,{B}}\sststile{}{p}{{t}{:}\,{C}}A\mathrel{<:}B\hskip
43.05542pt{\Gamma}\sststile{}{p}{{t}{:}\,{C}}\lx@proof@logical@and{\Gamma}\sststile{}{p}{{t}{:}\,{D}}D\mathrel{<:}C$
Consider the first rule. By assumption
${}\sststile{}{m}{\langle{t},{\sigma}\rangle\to\langle{u},{\sigma^{\prime}}\rangle}$
and by definition
$\Phi({\sigma}{:}\,{\Gamma,{x}{:}\,{A}})\geqslant\Phi({\sigma}{:}\,{\Gamma,{x}{:}\,{B}})$.
Thus the theorem follows by side-induction hypothesis.
∎
## 5 Typed Polynomial Interpretations
We adapt the concept of polynomial interpretation to typed TRSs. For that we
suppose a mapping $\llbracket{\cdot}\rrbracket$ that assigns to every
_annotated_ type $C$ a subset of the natural numbers, whose elements are
ordered with $>$ in the standard way. The set $\llbracket{C}\rrbracket$ is
called the _interpretation_ of $C$.
###### Definition 5.1.
An _interpretation $\gamma$ of function symbols_ is a mapping from function
symbols and types to functions over ${\mathbb{N}}$. Consider a function symbol
$f$ and an annotated type $C$ such that
$\mathcal{F}(f)\owns[{A_{1}\times\cdots\times A_{n}}]\xrightarrow{p}{C}$. Then
the interpretation
$\gamma(f,C)\colon\llbracket{A_{1}}\rrbracket\times\cdots\times\llbracket{A_{n}}\rrbracket\to\llbracket{C}\rrbracket$
of $f$ is defined as follows:
$\gamma(f,C)(x_{1},\dots,x_{n})\mathrel{:=}x_{1}+\cdots+x_{n}+p\hbox
to0.0pt{$\;$.\hss}$
Note that by assumption the declaration in $\mathcal{F}(f)$ is unique and thus
$\gamma(f,C)$ is unique. Interpretations of function symbols naturally extend
to interpretation on ground terms.
$\llbracket{{f(t_{1},\dots,t_{n})}{:}\,{C}}\rrbracket^{\gamma}\mathrel{:=}\gamma(f,C)(\llbracket{{t_{1}}{:}\,{A_{1}}}\rrbracket^{\gamma},\dots,\llbracket{{t_{n}}{:}\,{A_{n}}}\rrbracket^{\gamma})\hbox
to0.0pt{$\;$.\hss}$
Let $\mathcal{R}$ be a well-typed and let the interpretation $\gamma$ of
function symbols in $\mathcal{F}$ be induced by the well-typing of
$\mathcal{R}$. Then by construction
$\llbracket{{t}{:}\,{A}}\rrbracket^{\gamma}=\Phi({t}{:}\,{A})$.
###### Example 5.1 (continued from Example 3.3).
Based on Definition 5.1 we obtain the following definitions of the
interpretation of function symbols $\gamma$. We start with the constructor
symbols.
$\displaystyle\gamma(\mathsf{0},\mathsf{Nat}^{p})$ $\displaystyle=0$
$\displaystyle\hskip 8.61108pt\gamma(\mathsf{s},\mathsf{Nat}^{p})(x)$
$\displaystyle=x+p$ $\displaystyle\hskip
8.61108pt\gamma(\mathsf{err\\_head},\mathsf{Nat}^{p})$ $\displaystyle=0$
$\displaystyle\gamma(\mathsf{nil},\mathsf{List}^{q})$ $\displaystyle=0$
$\displaystyle\hskip
8.61108pt\gamma(\mathrel{\mathsf{\sharp}},\mathsf{List}^{q})(x,y)$
$\displaystyle=x+y+q$ $\displaystyle\hskip
8.61108pt\gamma(\mathsf{err\\_tail},\mathsf{Q}^{{(0,1)}})$ $\displaystyle=0$
$\displaystyle\gamma(\mathsf{que},\mathsf{Q}^{{(0,1)}})(x,y)$
$\displaystyle=x+y\hbox to0.0pt{$\;$,\hss}$
where $p,q\in{\mathbb{N}}$. Similarly the definition of $\gamma$ for defined
symbols follows from the signature detailed in Example 3.3. It is not
difficult to see that for any rule $l\to r\in\mathcal{R}_{\mathsf{que}}$ and
any substitution $\sigma$, we obtain
$\llbracket{l\sigma}\rrbracket^{\gamma}>\llbracket{r\sigma}\rrbracket^{\gamma}$.
We show this for rule 1.
$\displaystyle\llbracket{{\mathsf{chk}(\mathsf{que}(\mathsf{nil},r\sigma))}{:}\,{\mathsf{Q}^{{(0,1)}}}}\rrbracket^{\gamma}$
$\displaystyle=\llbracket{{r\sigma}{:}\,{\mathsf{List}^{1}}}\rrbracket^{\gamma}+3>0$
$\displaystyle=\llbracket{{\mathsf{rev}(r\sigma)}{:}\,{\mathsf{List}^{0}}}\rrbracket^{\gamma}+\llbracket{{\mathsf{nil}}{:}\,{\mathsf{List}^{1}}}\rrbracket^{\gamma}$
$\displaystyle=\llbracket{{\mathsf{que}(\mathsf{rev}(r\sigma),\mathsf{nil})}{:}\,{\mathsf{Q}^{{(0,1)}}}}\rrbracket^{\gamma}\hbox
to0.0pt{$\;$.\hss}$
Orientability of $\mathcal{R}_{\mathsf{que}}$ with the above given
interpretation implies the optimal linear innermost runtime complexity.
We lift the standard order $>$ on the interpretation domain ${\mathbb{N}}$ to
an order on terms as follows. Let $s$ and $t$ be terms of type $A$. Then $s>t$
if for all well-typed substitutions $\sigma$ we have
$\llbracket{{s\sigma}{:}\,{A}}\rrbracket^{\gamma}>\llbracket{{t\sigma}{:}\,{A}}\rrbracket^{\gamma}$.
###### Theorem 5.1.
Let $\mathcal{R}$ be well-typed, constructor TRS over signature $\mathcal{F}$
and let the interpretation of function symbols $\gamma$ be induced by the type
system. Then $l>r$ for any rule ${l\to r}\in\mathcal{R}$. Thus if for all
ground basic terms $t$ and types $A$:
$\llbracket{{t}{:}\,{A}}\rrbracket^{\gamma}\in\operatorname{\mathsf{O}}(n^{k})$,
where $n=\lvert{t}\rvert$, then
$\operatorname{\mathsf{rc}}_{\mathcal{R}}(n)\in\operatorname{\mathsf{O}}(n^{k})$.
###### Proof.
Let $l=f(l_{1},\dots,l_{n})$ and let $x_{1},\dots,x_{n}$ be fresh variables.
Suppose further $\mathcal{F}(f)\owns[{A_{1}\times\cdots\times
A_{n}}]\xrightarrow{p}{C}$. As $\mathcal{R}$ is well-typed we have
${\overbrace{{x_{1}}{:}\,{A_{1}},\dots,{x_{n}}{:}\,{A_{n}}}^{{}=:\Gamma}}\sststile{}{p}{{f(x_{1},\dots,x_{n})}{:}\,{C}}\hbox
to0.0pt{$\;$,\hss}$
for $p\in{\mathbb{N}}$.
Now suppose that $\tau$ denotes any well-typed substitution for the rule $l\to
r$. It is standard way, we extend $\tau$ to a well-typed substitution $\sigma$
such that $l\tau=f(x_{1},\dots,x_{n})\sigma$. By definition of the small-step
semantics, we obtain
${}\sststile{}{1}{\langle{f(x_{1},\dots,x_{n})},{\sigma}\rangle\to\langle{r},{\sigma\mathrel{\uplus}\tau}\rangle}\hbox
to0.0pt{$\;$.\hss}$
Then by Lemma 4.1,
$\Phi({\sigma}{:}\,{\Gamma})+p>\Phi({r(\sigma\mathrel{\uplus}\tau)}{:}\,{C})$
and by definitions, we have:
$\Phi({l\tau}{:}\,{C})=\Phi({f(x_{1}\sigma,\dots,x_{n}\sigma)}{:}\,{C})=\sum_{i=1}^{n}\Phi({x_{i}\sigma}{:}\,{A_{i}})+p=\Phi({\sigma}{:}\,{\Gamma})+p\hbox
to0.0pt{$\;$.\hss}$
Furthermore, observe that $r(\sigma\mathrel{\uplus}\tau)=r\tau$ as
$\operatorname{\mathsf{dom}}(\sigma)\cap\operatorname{\mathsf{dom}}(\tau)=\varnothing$.
In sum, we obtain $\Phi({l\tau}{:}\,{C})>\Phi({r\tau}{:}\,{C})$, from which we
conclude
$\llbracket{{l\tau}{:}\,{C}}\rrbracket^{\gamma}{\gamma}>\llbracket{{r\tau}{:}\,{C}}\rrbracket^{\gamma}$.
As $\tau$ was chosen arbitrarily, we obtain ${\mathcal{R}}\subseteq{>}$. ∎
We say that an interpretation _orients_ a typed TRS $\mathcal{R}$, if
${\mathcal{R}}\subseteq{>}$. As an immediate consequence of the theorem, we
obtain the following corollary.
###### Corollary 5.1.
Let $\mathcal{R}$ be a well-typed and constructor TRS. Then there exists a
typed polynomial interpretation over ${\mathbb{N}}$ that orients
$\mathcal{R}$.
At the end of Section 3 we have remarked on the automatabilty of the obtained
amortised analysis. Observe that Theorem 5.1 gives rise to a conceptually
quite different implementation. Instead of encoding the constraints of the
typing rules in Figure 2 one directly encode the orientability constraints for
each rule, cf. contejean:2005 .
## 6 Conclusion
This paper is concerned with the connection between amortised resource
analysis, originally introduced for functional programs, and polynomial
interpretations, which are frequently used in complexity and termination
analysis of rewrite systems.
In order to study this connection we established a novel resource analysis for
typed term rewrite systems based on a potential-based type system. This type
system gives rise to polynomial bounds for innermost runtime complexity. A key
observation is that the classical notion of potential can be altered so that
not only values but any term can be assigned a potential. Ie. the potential
function $\Phi$ is conceivable as an interpretation. Based on this observation
we have shown that well-typedness of a TRSs $\mathcal{R}$ induces a typed
polynomial interpretation orienting $\mathcal{R}$.
Apart from clarifying the connection between amortised resource analysis and
polynomial interpretation our results seems to induce two new methods for the
innermost runtime complexity of typed TRSs as indicated above.
We emphasise that these methods are not restricted to typed TRSs, as our cost
model gives rise to a _persistent_ property. Here a property is persistent if,
for any typed TRS $\mathcal{R}$ the property holds iff it holds for the
corresponding untyped TRS $\mathcal{R}^{\prime}$. While termination is in
general not persistent TeReSe , it is not difficult to see that the runtime
complexity is a persistent property. This is due to the restricted set of
starting terms. Thus it seems that the proposed techniques directly give rise
to novel methods of automated innermost runtime complexity analysis.
In future work we will clarify whether the established results extend to the
multivariate amortised resource analysis presented in HAH12b . Furthermore, we
will strive for automation to assess the viability of the established methods.
## References
* (1) E. Albert, P. Arenas, S. Genaim, and G. Puebla. Closed-form upper bounds in static cost analysis. JAR, 46(2), 2011.
* (2) C. Alias, A. Darte, P. Feautrier, and L. Gonnord. Multi-dimensional rankings, program termination, and complexity bounds of flowchart programs. In Proc. 17th SAS, volume 6337 of LNCS, pages 117–133, 2010\.
* (3) M. Avanzini and G. Moser. A combination framework for complexity. In Proc. 24th RTA, volume 21 of LIPIcs, pages 55–70, 2013\.
* (4) M. Avanzini and G. Moser. Tyrolean complexity tool: Features and usage. In Proc. 24th RTA, volume 21 of LIPIcs, pages 71–80, 2013\.
* (5) F. Baader and T. Nipkow. Term Rewriting and All That. Cambridge University Press, 1998.
* (6) G. Bonfante, A. Cichon, J.-Y. Marion, and H. Touzet. Algorithms with polynomial interpretation termination proof. JFP, 11(1):33–53, 2001.
* (7) E. Contejean, C. Marché, A.-P. Tomás, and X. Urbain. Mechanically proving termination using polynomial interpretations. Journal of Automated Reasoning, 34(4):325–363, 2005.
* (8) S. Gulwani and F. Zuleger. The reachability-bound problem. In Proc. PLDI’10, pages 292–304. ACM, 2010.
* (9) J. Hoffmann. Types with Potential: Polynomial Resource Bounds via Automatic Amortized Analysis. PhD thesis, Ludwig-Maximilians-Universiät München, 2011.
* (10) J. Hoffmann, K. Aehlig, and M. Hofmann. Multivariate amortized resource analysis. ACM Trans. Program. Lang. Syst., 34(3):14, 2012.
* (11) J. Hoffmann, K. Aehlig, and M. Hofmann. Resource aware ML. In Proc. 24th CAV, volume 7358 of LNCS, pages 781–786, 2012\.
* (12) J. Hoffmann and M. Hofmann. Amortized resource analysis with polymorphic recursion and partial big-step operational semantics. In Proc. 8th APLAS, volume 6461 of LNCS, pages 172–187, 2010\.
* (13) J. Hoffmann and M. Hofmann. Amortized resource analysis with polynomial potential. In Proc. 19th ESOP, volume 6012 of LNCS, pages 287–306, 2010\.
* (14) M. Hofmann and S. Jost. Static prediction of heap space usage for first-order functional programs. In Proc. 30th POPL, pages 185–197. ACM, 2003.
* (15) S. Jost, H.-W. Loidl, K. Hammond, N. Scaife, and M. Hofmann. “Carbon Credits” for resource-bounded computations using amortised analysis. In Proc. 2nd FM, volume 5850 of LNCS, pages 354–369. Springer Verlag, 2009.
* (16) J.-P. Jouannaud and A. Rubio. The higher-order recursive path ordering. In Proc. 14th LICS, pages 402–411. IEEE Computer Society, 1999\.
* (17) L. Noschinski, F. Emmes, and J. Giesl. Analyzing innermost runtime complexity of term rewriting by dependency pairs. JAR, 51(1):27–56, 2013.
* (18) C. Okasaki. Purely functional data structures. Cambridge University Press, 1999.
* (19) R. Tarjan. Amortized computational complexity. SIAM J. Alg. Disc. Meth, 6(2):306–318, 1985.
* (20) TeReSe. Term Rewriting Systems, volume 55 of Cambridge Tracks in Theoretical Computer Science. Cambridge University Press, 2003.
* (21) A. Turing. Checking a large routine. In In Report of a Conference on High Speed Automatic Calculating Machines, pages 67–69. University Mathematics Lab, Cambridge University, 1949\.
* (22) F. Zuleger, S. Gulwani, M. Sinn, and H. Veith. Bound analysis of imperative programs with the size-change abstraction. In Proc. of 18th International Symposium on Static Analysis, volume 6887 of LNCS, pages 280–297. Springer Verlag, 2011.
|
arxiv-papers
| 2014-02-09T07:08:04 |
2024-09-04T02:49:57.977087
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Martin Hofmann and Georg Moser",
"submitter": "Georg Moser",
"url": "https://arxiv.org/abs/1402.1922"
}
|
1402.2031
|
# Deeply Coupled Auto-encoder Networks for
Cross-view Classification
Wen Wang, Zhen Cui, Hong Chang, Shiguang Shan, Xilin Chen
Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
{wen.wang, zhen.cui, hong.chang, shiguang.shan, xilin.chen}@vipl.ict.ac.cn
(November 2013)
###### Abstract
The comparison of heterogeneous samples extensively exists in many
applications, especially in the task of image classification. In this paper,
we propose a simple but effective coupled neural network, called Deeply
Coupled Autoencoder Networks (DCAN), which seeks to build two deep neural
networks, coupled with each other in every corresponding layers. In DCAN, each
deep structure is developed via stacking multiple discriminative coupled auto-
encoders, a denoising auto-encoder trained with maximum margin criterion
consisting of intra-class compactness and inter-class penalty. This single
layer component makes our model simultaneously preserve the local consistency
and enhance its discriminative capability. With increasing number of layers,
the coupled networks can gradually narrow the gap between the two views.
Extensive experiments on cross-view image classification tasks demonstrate the
superiority of our method over state-of-the-art methods.
## 1 Introduction
Real-world objects often have different views, which might be endowed with the
same semantic. For example, face images can be captured in different poses,
which reveal the identity of the same object; images of one face can also be
in different modalities, such as pictures under different lighting condition,
pose, or even sketches from artists. In many computer vision applications,
such as image retrieval, interests are taken in comparing two types of
heterogeneous images, which may come from different views or even different
sensors. Since the spanned feature spaces are quite different, it is very
difficult to classify these images across views directly. To decrease the
discrepancy across views, most of previous works endeavored to learn view-
specific linear transforms and to project cross-view samples into a common
latent space, and then employed these newly generated features for
classification.
Though there are lots of approaches used to learn view-specific projections,
they can be divided roughly based on whether the supervised information is
used. Unsupervised methods such as Canonical Correlation Analysis (CCA)[14]
and Partial Least Square (PLS) [26] are employed to the task of cross-view
recognition. Both of them attempt to use two linear mappings to project
samples into a common space where the correlation is maximized, while PLS
considers the variations rather than only the correlation in the target space.
Besides, with use of the mutual information, a Coupled Information-Theoretic
Encoding (CITE) method is developed to narrow the inter-view gap for the
specific photo-sketch recognition task. And in [30], a semi-coupled dictionary
is used to bridge two views. All the methods above consider to reduce the
discrepancy between two views, however, the label information is not
explicitly taken into account. With label information available, many methods
were further developed to learn a discriminant common space For instance,
Discriminative Canonical Correlation Analysis (DCCA) [16] is proposed as an
extension of CCA. And In [22], with an additional local smoothness
constraints, two linear projections are simultaneously learnt for Common
Discriminant Feature Extraction (CDFE). There are also other such methods as
the large margin approach [8] and the Coupled Spectral Regression (CSR) [20].
Recently, multi-view analysis [27, 15] is further developed to jointly learn
multiple specific-view transforms when multiple views (usually more than 2
views) can be available.
Although the above methods have been extensively applied in the cross-view
problem, and have got encouraging performances, they all employed linear
transforms to capture the shared features of samples from two views. However,
these linear discriminant analysis methods usually depend on the assumption
that the data of each class agrees with a Gaussian distribution, while data in
real world usually has a much more complex distribution [33]. It indicates
that linear transforms are insufficient to extract the common features of
cross-view images. So it’s natural to consider about learning nonlinear
features.
A recent topic of interest in nonlinear learning is the research in deep
learning. Deep learning attempts to learn nonlinear representations
hierarchically via deep structures, and has been applied successfully in many
computer vision problems. Classical deep learning methods often stack or
compose multiple basic building blocks to yield a deeper structure. See [5]
for a recent review of Deep Learning algorithms. Lots of such basic building
blocks have been proposed, including sparse coding [19], restricted Boltzmann
machine (RBM) [12], auto-encoder [13, 6], etc. Specifically, the (stacked)
auto-encoder has shown its effectiveness in image denoising [32], domain
adaptation [7], audio-visual speech classification [23], etc.
As we all known, the kernel method, such as Kernel Canonical Correlation
Analysis(Kernel CCA) [1], is also a widely used approach to learn nonlinear
representations. Compared with the kernel method, deep learning is much more
flexible and time-saving because the transform is learned rather than fixed
and the time needed for training and inference process is beyond the limit of
the size of training set.
Inspired by the deep learning works above, we intend to solve the cross-view
classification task via deep networks. It’s natural to build one single deep
neural network with samples from both views, but this kind of network can’t
handle complex data from totally different modalities and may suffer from
inadequate representation capacity. Another way is to learn two different deep
neural networks with samples of the different views. However, the two
independent networks project samples from different views into different
spaces, which makes comparison infeasible. Hence, building two neural networks
coupled with each other seems to be a better solution.
In this work, we propose a Deeply Coupled Auto-encoder Networks(DCAN) method
that learns the common representations to conduct cross-view classification by
building two neural networks deeply coupled respectively, each for one view.
We build the DCAN by stacking multiple discriminative coupled auto-encoders, a
denoising auto-encoder with maximum margin criterion. The discriminative
coupled auto-encoder has a similar input corrupted and reconstructive error
minimized mechanism with the denoising auto-encoder proposed in [28], but is
modified by adding a maximum margin criterion. This kind of criterion has been
used in previous works, like [21, 29, 35], etc. Note that the counterparts
from two views are added into the maximum margin criterion simultaneously
since they both come from the same class, which naturally couples the
corresponding layer in two deep networks. A schematic illustration can be seen
in Fig.1.
The proposed DCAN is related to Multimodal Auto-encoders [23], Multimodal
Restricted Boltzmann Machines and Deep Canonical Correlation Analysis [3]. The
first two methods tend to learn a single network with one or more layers
connected to both views and to predict one view from the other view, and the
Deep Canonical Correlation Analysis build two deep networks, each for one
view, and only representations of the highest layer are constrained to be
correlated. Therefore, the key difference is that we learn two deep networks
coupled with each other in representations in each layer, which is of great
benefits because the DCAN not only learn two separate deep encodings but also
makes better use of data from the both two views. What’s more, these
differences allow for our model to handle the recognition task even when data
is impure and insufficient.
The rest of this paper is organized as follows. Section 2 details the
formulation and solution to the proposed Deeply Coupled Auto-encoder Networks.
Experimental results in Section 3 demonstrate the efficacy of the DCAN. In
section 4 a conclusion is given.
## 2 Deeply Coupled Auto-encoder Networks
In this section, we first present the basic idea. The second part gives a
detailed description of the discriminative coupled auto-encoder. Then, we
describe how to stack multiple layers to build a deep network. Finally, we
briefly describe the optimization of the model.
### 2.1 Basic Idea
Figure 1: An illustration of our proposed DCAN. The left-most and right-most
schematic show the structure of the two coupled network respectively. And the
schematic in the middle illustrates how the whole network gradually enhances
the separability with increasing layers, where pictures with solid line border
denote samples from view 1, those with dotted line border denote samples from
view 2, and different colors imply different subjects.
As shown in Fig.1, the Deeply Coupled Auto-encoder Networks(DCAN) consists of
two deep networks coupled with each other, and each one is for one view. The
network structures of the two deep networks are just like the left-most and
the right-most parts in Fig.1, where circles means the units in each layers
(pixels in a input image for the input layer and hidden representation in
higher layers), and arrows denote the full connections between adjacent
layers. And the middle part of Fig.1 illustrates how the whole network
projects samples in different views into a common space and gradually enhances
the separability with increasing layers.
The two deep networks are both built through stacking multiple similar coupled
single layer blocks because a single coupled layer might be insufficient, and
the method of stacking multiple layers and training each layer greedily has be
proved efficient in lots of previous works, such as those in [13, 6]. With the
number of layers increased, the whole network can compactly represent a
significantly larger set of transforms than shallow networks , and gradually
narrow the gap with the discriminative capacity enhanced.
We use a discriminative coupled auto-encoders trained with maximum margin
criterion as a single layer component. Concretely, we incorporate the
additional noises in the training process while maximizing the margin
criterion, which makes the learnt mapping more stable as well as discriminant.
Note that the maximum margin criterion also works in coupling two
corresponding layers. Formally, the discriminative coupled auto-encoder can be
written as follows:
$\displaystyle\quad\min_{f_{x},f_{y}}\quad L(X,f_{x})+L(Y,f_{y})$ (1)
$\displaystyle s.t.\quad
G_{1}(H_{x},H_{y})-G_{2}(H_{x},H_{y})\leq\varepsilon,$ (2)
where $X,Y$ denote inputs from the two views, and $H_{x},H_{y}$ denote hidden
representations of the two views respectively. $f_{x}:X\longrightarrow
H_{x},f_{y}:Y\longrightarrow H_{y}$ are the transforms we intend to learn, and
we denote the reconstructive error as $L(\cdot)$, and maximum margin criterion
as $G_{1}(\cdot)-G_{2}(\cdot)$, which are described detailedly in the next
subsection.$\varepsilon$ is the threshold of the maximum margin criterion.
### 2.2 Discriminative coupled auto-encoder
In the problem of cross-view, there are two types of heterogenous samples.
Without loss of generality, we denote samples from one view as
$X=[x_{1},\cdots,x_{n}]$ , and those from the other view as
$Y=[y_{1},\cdots,y_{n}]$, in which $n$ is the sample sizes. Noted that the
corresponding labels are known, and $H_{x},H_{y}$ denote hidden
representations of the two views we want to learn.
The DCAN attempts to learn two nonlinear transforms $f_{x}:X\longrightarrow
H_{x}$ and $f_{y}:Y\longrightarrow H_{y}$ that can project the samples from
two views to one discriminant common space respectively, in which the local
neighborhood relationship as well as class separability should be well
preserved for each view. The auto-encoder like structure stands out in
preserving the local consistency, and the denoising form enhances the
robustness of learnt representations. However, the discrimination isn’t taken
into consideration. Therefore, we modify the denoising auto-encoder by adding
a maximum margin criterion consisting of intra-class compactness and inter-
class penalty. And the best nonlinear transformation is a trade-off between
local consistency preserving and separability enhancing.
Just like the one in denoising auto-encoder, the reconstructive error
$L(\cdot)$ in Eq.(1) is formulated as follows:
$\displaystyle
L(X,\Theta)=\sum_{x\in{X^{p}}}{\mathbb{E}_{\tilde{x}\sim{P(\tilde{x}|x)}}}\|\hat{x}-x\|$
(3) $\displaystyle
L(Y,\Theta)=\sum_{y\in{Y^{p}}}{\mathbb{E}_{\tilde{y}\sim{P(\tilde{y}|y)}}}\|\hat{y}-y\|$
(4)
where $\mathbb{E}$ calculates the expectation over corrupted versions
$\tilde{X},\tilde{Y}$ of examples $X,Y$ obtained from a corruption process
$P(\tilde{x}|x),P(\tilde{y}|y)$.
$\Theta=\\{W_{x},W_{y},b_{x},b_{y},c_{x},c_{y}\\}$ specifies the two nonlinear
transforms $f_{x},f_{y}$ , where $W_{x},W_{y}$ is the weight matrix, and
$b_{x},b_{y},c_{x},c_{y}$ are the bias of encoder and decoder respectively,
and $\hat{X},\hat{Y}$ are calculated through the decoder process :
$\begin{split}\hat{X}=s(W_{x}^{T}H_{x}+c_{x})\\\
\hat{Y}=s(W_{y}^{T}H_{y}+c_{y})\end{split}$ (5)
And hidden representations $H_{x},H_{y}$ are obtained from the encoder that is
a similar mapping with the decoder,
$\begin{split}H_{x}=s(W_{x}\tilde{X}+b_{x})\\\
H_{y}=s(W_{y}\tilde{Y}+b_{y})\end{split}$ (6)
where $s$ is the nonlinear activation function, such as the point-wise
hyperbolic tangent operation on linear projected features, i.e.,
$s(x)=\frac{e^{ax}-e^{-ax}}{e^{ax}+e^{-ax}}$ (7)
in which $a$ is the gain parameter.
Moreover, for the maximum margin criterion consisting of intra-class
compactness and inter-class penalty, the constraint term
$G_{1}(\cdot)-G_{2}(\cdot)$ in Eq.(1) is used to realize coupling since
samples of the same class are treated similarly no matter which view they are
from.
Assuming $S$ is the set of sample pairs from the same class, and $D$ is the
set of sample pairs from different classes. Note that the counterparts from
two views are naturally added into $S,D$ since it’s the class rather than the
view that are considered.
Then, we characterize the compactness as follows,
$\displaystyle
G_{1}(H)=\frac{1}{2N_{1}}\sum\limits_{I_{i},I_{j}\in{S}}\|h_{i}-h_{j}\|^{2},$
(8)
where $h_{i}$ denotes the corresponding hidden representation of an input
$I_{i}\in{X\bigcap{Y}}$ and is a sample from either view 1 or view 2, and
$N_{1}$ is the size of $S$.
Meanwhile, the goal of the inter-class separability is to push the adjacent
samples from different classes far away, which can be formulated as follows,
$\displaystyle
G_{2}(H)=\frac{1}{2N_{2}}\sum\limits_{\tiny\begin{subarray}{c}I_{i},I_{j}\in{D}\\\
I_{j}\in{KNN(I_{i})}\end{subarray}}\|h_{i}-h_{j}\|^{2},$ (9)
where $I_{j}$ belongs to the $k$ nearest neighbors of $I_{i}$ with different
class labels, and $N_{2}$ is the number of all pairs satisfying the condition.
And the function of $G_{1}(H),G_{2}(H)$ is illustrated in the middel part of
Fig.1. In the projected common space denoted by $S$, the compactness term
$G_{1}(\cdot)$ shown by red ellipse works by pulling intra-class samples
together while the penalty term $G_{2}(\cdot)$ shown by black ellipse tend to
push adjacent inter-class samples away.
Finally, by solving the optimization problem Eq.(1), we can learn a couple of
nonlinear transforms $f_{x},f_{y}$ to transform the original samples from both
views into a common space.
### 2.3 Stacking coupled auto-encoder
Through the training process above, we model the map between original sample
space and a preliminary discriminant subspace with gap eliminated, and build a
hidden representation $H$ which is a trade-off between approximate
preservation on local consistency and the distinction of the projected data.
But since real-world data is highly complicated, using a single coupled layer
to model the vast and complex real scenes might be insufficient. So we choose
to stack multiple such coupled network layers described in subsection 2.2.
With the number of layers increased, the whole network can compactly represent
a significantly larger set of transforms than shallow networks, and gradually
narrow the gap with the discriminative ability enhanced.
Training a deep network with coupled nonlinear transforms can be achieved by
the canonical greedy layer-wise approach [12, 6]. Or to be more precise, after
training a single layer coupled network, one can compute a new feature $H$ by
the encoder in Eq.(6) and then feed it into the next layer network as the
input feature. In practice, we find that stacking multiple such layers can
gradually reduce the gap and improve the recognition performance (see Fig.1
and Section 3).
### 2.4 Optimization
We adopt the Lagrangian multiplier method to solve the objective function
Eq.(1) with the constraints Eq.(2) as follows:
$\begin{split}\min_{\Theta}\quad&\lambda(L(X,\Theta)+L(Y,\Theta))+(G_{1}(H)-G_{2}(H))+\\\
&\gamma(\frac{1}{2}\|W_{x}\|_{F}^{2}+\frac{1}{2}\|W_{y}\|_{F}^{2})\end{split}$
(10)
where the first term is the the reconstruction error, the second term is the
maximum margin criterion, and the last term is the shrinkage constraints
called the Tikhonov regularizers in [11], which is utilized to decrease the
magnitude of the weights and further to help prevent over-fitting. $\lambda$
is the balance parameter between the local consistency and empirical
separability. And $\gamma$ is called the weight decay parameter and is usually
set to a small value, e.g., 1.0e-4.
To optimize the objective function (10), we use back-propagation to calculate
the gradient and then employ the limited-memory BFGS (L-BFGS) method [24, 17],
which is often used to solve nonlinear optimization problems without any
constraints. L-BFGS is particularly suitable for problems with a large amount
of variables under the moderate memory requirement. To utilize L-BFGS, we need
to calculate the gradients of the object function. Obviously, the object
function in (10) is differential to these parameters $\Theta$, and we use
Back-propagation [18] method to derive the derivative of the overall cost
function. In our setting, we find the objective function can achieve as fast
convergence as described in [17].
## 3 Experiments
In this section, the proposed DCAN is evaluated on two datasets, Multi-PIE [9]
and CUHK Face Sketch FERET (CUFSF) [34, 31].
### 3.1 Databases
Multi-PIE dataset [9] is employed to evaluate face recognition across pose.
Here a subset from the 337 subjects in 7 poses
($-45^{\circ},-30^{\circ},-15^{\circ},0^{\circ},15^{\circ},30^{\circ},45^{\circ}$),
3 expression (Neutral,Smile, Disgust), no flush illumination from 4 sessions
are selected to validate our method. We randomly choose 4 images for each pose
of each subject, then randomly partition the data into two parts: the training
set with 231 subjects (i.e., $231\times 7\times 4=6468$ images) and the
testing set with the rest subjects.
CUHK Face Sketch FERET (CUFSF) dataset [34, 31] contains two types of face
images: photo and sketch. Total 1,194 images (one image per subject) were
collected with lighting variations from FERET dataset [25]. For each subject,
a sketch is drawn with shape exaggeration. According to the configuration of
[15], we use the first 700 subjects as the training data and the rest subjects
as the testing data.
### 3.2 Settings
All images from Multi-PIE and CUFSF are cropped into 64$\times$80 pixels
without any preprocess. We compare the proposed DCAN method with several
baselines and state-of-the-art methods, including CCA [14], Kernel CCA [1],
Deep CCA [3], FDA [4], CDFE [22], CSR [20], PLS [26] and MvDA [15]. The first
seven methods are pairwise methods for cross-view classification. MvDA jointly
learns all transforms when multiple views can be utilized, and has achieved
the state-of-the-art results in their reports [15].
The Principal Component Analysis (PCA) [4] is used for dimension reduction. In
our experiments, we set the default dimensionality as 100 with preservation of
most energy except Deep CCA, PLS, CSR and CDFE, where the dimensionality are
tuned in [50,1000] for the best performance. For all these methods, we report
the best performance by tuning the related parameters according to their
papers. Firstly, for Kernel CCA, we experiment with Gaussian kernel and
polynomial kernel and adjust the parameters to get the best performance. Then
for Deep CCA [3], we strictly follow their algorithms and tune all possible
parameters, but the performance is inferior to CCA. One possible reason is
that Deep CCA only considers the correlations on training data (as reported in
their paper) so that the learnt mode overly fits the training data, which thus
leads to the poor generality on the testing set. Besides, the parameter
$\alpha$ and $\beta$ are respectively traversed in [0.2,2] and [0.0001,1] for
CDFE, the parameter $\lambda$ and $\eta$ are searched in [0.001,1] for CSR,
and the reduced dimensionality is tuned for CCA, PLS, FDA and MvDA.
As for our proposed DCAN, the performance on CUFSF database of varied
parameters, $\lambda,k$, is shown in Fig.3. In following experiments, we set
$\lambda=0.2,\gamma=1.0e-4$, $k=10$ and $a=1$. With increasing layers, the
number of hidden neurons are gradually reduced by $10$, _i.e.,_ $90,80,70,60$
if four layers.
Method | Accuracy
---|---
CCA[14] | 0.698
KernelCCA[10] | 0.840
DeepCCA[3] | 0.599
FDA[4] | 0.814
CDFE[22] | 0.773
CSR[20] | 0.580
PLS[26] | 0.574
MvDA[15] | 0.867
DCAN-1 | 0.830
DCAN-2 | 0.877
DCAN-3 | 0.884
DCAN-4 | 0.879
Table 1: Evaluation on Multi-PIE database in terms of mean accuracy. DCAN-k
means a stacked k-layer network.
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.816 | 0.588 | 0.473 | 0.473 | 0.515 | 0.511
$-30^{\circ}$ | 0.816 | 1.000 | 0.858 | 0.611 | 0.664 | 0.553 | 0.553
$-15^{\circ}$ | 0.588 | 0.858 | 1.000 | 0.894 | 0.807 | 0.602 | 0.447
$0^{\circ}$ | 0.473 | 0.611 | 0.894 | 1.000 | 0.909 | 0.604 | 0.484
$15^{\circ}$ | 0.473 | 0.664 | 0.807 | 0.909 | 1.000 | 0.874 | 0.602
$30^{\circ}$ | 0.515 | 0.553 | 0.602 | 0.604 | 0.874 | 1.000 | 0.768
$45^{\circ}$ | 0.511 | 0.553 | 0.447 | 0.484 | 0.602 | 0.768 | 1.000
(a) CCA, $Ave=0.698$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.878 | 0.810 | 0.756 | 0.706 | 0.726 | 0.737
$-30^{\circ}$ | 0.878 | 1.000 | 0.892 | 0.858 | 0.808 | 0.801 | 0.757
$-15^{\circ}$ | 0.810 | 0.892 | 1.000 | 0.911 | 0.880 | 0.861 | 0.765
$0^{\circ}$ | 0.756 | 0.858 | 0.911 | 1.000 | 0.938 | 0.759 | 0.759
$15^{\circ}$ | 0.706 | 0.808 | 0.880 | 0.938 | 1.000 | 0.922 | 0.845
$30^{\circ}$ | 0.726 | 0.801 | 0.861 | 0.759 | 0.922 | 1.000 | 0.912
$45^{\circ}$ | 0.737 | 0.757 | 0.765 | 0.759 | 0.845 | 0.912 | 1.000
(b) KernelCCA, $Ave=0.840$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.854 | 0.598 | 0.425 | 0.473 | 0.522 | 0.523
$-30^{\circ}$ | 0.854 | 1.000 | 0.844 | 0.578 | 0.676 | 0.576 | 0.566
$-15^{\circ}$ | 0.598 | 0.844 | 1.000 | 0.806 | 0.807 | 0.602 | 0.424
$0^{\circ}$ | 0.425 | 0.578 | 0.806 | 1.000 | 0.911 | 0.599 | 0.444
$15^{\circ}$ | 0.473 | 0.676 | 0.807 | 0.911 | 1.000 | 0.866 | 0.624
$30^{\circ}$ | 0.522 | 0.576 | 0.602 | 0.599 | 0.866 | 1.000 | 0.756
$45^{\circ}$ | 0.523 | 0.566 | 0.424 | 0.444 | 0.624 | 0.756 | 1.000
(c) DeepCCA, $Ave=0.599$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.847 | 0.754 | 0.686 | 0.573 | 0.610 | 0.664
$-30^{\circ}$ | 0.847 | 1.000 | 0.911 | 0.847 | 0.807 | 0.766 | 0.635
$-15^{\circ}$ | 0.754 | 0.911 | 1.000 | 0.925 | 0.896 | 0.821 | 0.602
$0^{\circ}$ | 0.686 | 0.847 | 0.925 | 1.000 | 0.964 | 0.872 | 0.684
$15^{\circ}$ | 0.573 | 0.807 | 0.896 | 0.964 | 1.000 | 0.929 | 0.768
$30^{\circ}$ | 0.610 | 0.766 | 0.821 | 0.872 | 0.929 | 1.000 | 0.878
$45^{\circ}$ | 0.664 | 0.635 | 0.602 | 0.684 | 0.768 | 0.878 | 1.000
(d) FDA, $Ave=0.814$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.854 | 0.714 | 0.595 | 0.557 | 0.633 | 0.608
$-30^{\circ}$ | 0.854 | 1.000 | 0.867 | 0.746 | 0.688 | 0.697 | 0.606
$-15^{\circ}$ | 0.714 | 0.867 | 1.000 | 0.887 | 0.808 | 0.704 | 0.579
$0^{\circ}$ | 0.595 | 0.746 | 0.887 | 1.000 | 0.916 | 0.819 | 0.651
$15^{\circ}$ | 0.557 | 0.688 | 0.808 | 0.916 | 1.000 | 0.912 | 0.754
$30^{\circ}$ | 0.633 | 0.697 | 0.704 | 0.819 | 0.912 | 1.000 | 0.850
$45^{\circ}$ | 0.608 | 0.606 | 0.579 | 0.651 | 0.754 | 0.850 | 1.000
(e) CDFE, $Ave=0.773$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.914 | 0.854 | 0.763 | 0.710 | 0.770 | 0.759
$-30^{\circ}$ | 0.914 | 1.000 | 0.947 | 0.858 | 0.812 | 0.861 | 0.766
$-15^{\circ}$ | 0.854 | 0.947 | 1.000 | 0.923 | 0.880 | 0.894 | 0.775
$0^{\circ}$ | 0.763 | 0.858 | 0.923 | 1.000 | 0.938 | 0.900 | 0.750
$15^{\circ}$ | 0.710 | 0.812 | 0.880 | 0.938 | 1.000 | 0.923 | 0.807
$30^{\circ}$ | 0.770 | 0.861 | 0.894 | 0.900 | 0.923 | 1.000 | 0.934
$45^{\circ}$ | 0.759 | 0.766 | 0.775 | 0.750 | 0.807 | 0.934 | 1.000
(f) MvDA, $Ave=0.867$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.872 | 0.819 | 0.730 | 0.655 | 0.708 | 0.686
$-30^{\circ}$ | 0.856 | 1.000 | 0.881 | 0.825 | 0.754 | 0.737 | 0.650
$-15^{\circ}$ | 0.807 | 0.874 | 1.000 | 0.869 | 0.865 | 0.781 | 0.681
$0^{\circ}$ | 0.757 | 0.854 | 0.896 | 1.000 | 0.938 | 0.858 | 0.790
$15^{\circ}$ | 0.688 | 0.777 | 0.854 | 0.916 | 1.000 | 0.900 | 0.823
$30^{\circ}$ | 0.708 | 0.735 | 0.788 | 0.834 | 0.918 | 1.000 | 0.916
$45^{\circ}$ | 0.719 | 0.715 | 0.697 | 0.752 | 0.832 | 0.909 | 1.000
(g) DCAN-1, $Ave=0.830$
| $-45^{\circ}$ | $-30^{\circ}$ | $-15^{\circ}$ | $0^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---
$-45^{\circ}$ | 1.000 | 0.905 | 0.876 | 0.783 | 0.714 | 0.779 | 0.796
$-30^{\circ}$ | 0.927 | 1.000 | 0.954 | 0.896 | 0.850 | 0.825 | 0.730
$-15^{\circ}$ | 0.867 | 0.929 | 1.000 | 0.905 | 0.905 | 0.867 | 0.757
$0^{\circ}$ | 0.832 | 0.876 | 0.925 | 1.000 | 0.958 | 0.896 | 0.808
$15^{\circ}$ | 0.765 | 0.865 | 0.907 | 0.951 | 1.000 | 0.929 | 0.874
$30^{\circ}$ | 0.779 | 0.832 | 0.870 | 0.916 | 0.945 | 1.000 | 0.949
$45^{\circ}$ | 0.794 | 0.777 | 0.785 | 0.812 | 0.876 | 0.938 | 1.000
(h) DCAN-3, $Ave=0.884$
Table 2: Results of CCA, FDA [4], CDFE [22], MvDA [15] and DCAN on MultiPIE
dataset in terms of rank-1 recognition rate. DCAN-k means a stacked k-layer
network. Due to space limitation, the results of other methods cannot be
reported here, but their mean accuracies are shown in Table 1.
### 3.3 Face Recognition across Pose
First, to explicitly illustrate the learnt mapping, we conduct an experiment
on Multi-PIE dataset by projecting the learnt common features into a 2-D space
with Principal Component Analysis (PCA). As shown in Fig.2. The classical
method CCA can only roughly align the data in the principal directions and the
state-of-the-art method MvDA [15] attempts to merge two types of data but
seems to fail. Thus, we argue that linear transforms are a little stiff to
convert data from two views into an ideal common space. The three diagrams
below shows that DCAN can gradually separate samples from different classes
with the increase of layers, which is just as we described in the above
analysis.
Figure 2: After learning common features by the cross-view methods, we project
the features into 2-D space by using the principal two components in PCA. The
depicted samples are randomly chosen form Multi-PIE [9] dataset. The “$\circ$”
and “$+$” points come from two views respectively. Different color points
belong to different classes. DCAN-k is our proposed method with a stacked
k-layer neural network.
Next, we compare our methods with several state-of-the-art methods for the
cross-view face recognition task on Multi-PIE data set. Since the images are
acquired over seven poses on Multi-PIE data set, in total $7\times 6=42$
comparison experiments need to be conducted. The detailed results are shown in
Table 2,where two poses are used as the gallery and probe set to each other
and the rank-1 recognition rate is reported. Further, the mean accuracy of all
pairwise results for each methods is also reported in Table 1.
From Table 1, we can find the supervised methods except CSR are significantly
superior to CCA due to the use of the label information. And nonlinear methods
except Deep CCA are significantly superior to the nonlinear methods due to the
use of nonlinear transforms. Compared with FDA, the proposed DCAN with only
one layer network can perform better with 1.6% improvement. With increasing
layers, the accuracy of DCAN reaches a climax via stacking three layer
networks. The reason of the degradation in DCAN with four layers is mainly the
effect of reduced dimensionality, where 10 dimensions are cut out from the
above layer network. Obviously, compared with two-view based methods, the
proposed DCAN with three layers improves the performance greatly (88.4% vs.
81.4%). Besides, MvDA also achieves a considerably good performance by using
all samples from all poses. It is unfair to compare these two-view based
methods (containing DCAN) with MvDA, because the latter implicitly uses
additional five views information except current compared two views. But our
method performs better than MvDA, 88.4% vs. 86.7%. As observed in Table 2,
three-layer DCAN achieves a largely improvement compared with CCA,FDA,CDFE for
all cross-view cases and MvDA for most of cross-view cases. The results are
shown in Table 2 and Table 1.
### 3.4 Photo-Sketch Recognition
Method | Photo-Sketch | Sketch-Photo
---|---|---
CCA[14] | 0.387 | 0.475
KernelCCA[10] | 0.466 | 0.570
DeepCCA[3] | 0.364 | 0.434
CDFE[22] | 0.456 | 0.476
CSR[20] | 0.502 | 0.590
PLS[26] | 0.486 | 0.510
FDA[4] | 0.468 | 0.534
MvDA[15] | 0.534 | 0.555
DCAN-1 | 0.535 | 0.555
DCAN-2 | 0.603 | 0.613
DCAN-3 | 0.601 | 0.652
Table 3: Evluation on CUFSF database in terms of mean accuracy. DCAN-k means a
stacked k-layer network.
(a)
(b)
Figure 3: The performance with varied parameter values for our proposed DCAN.
The sketch and photo images in CUFSF [34, 31] are respectively used for the
gallery and probe set. (a) Varied $\lambda$ with fixed $k=10$. (b) Varied $k$
with fixed $\lambda=0.2$.
Photo-Sketch recognition is conducted on CUFSF dataset. The samples come from
only two views, photo and sketch. The comparison results are provided in Table
3. As shown in this table, since only two views can be utilized in this case,
MvDA degrades to a comparable performance with those previous two-view based
methods. Our proposed DCAN with three layer networks can achieve even better
with more than 6% improvement, which further indicates DCAN benefits from the
nonlinear and multi-layer structure.
Discussion and analysis: The above experiments demonstrate that our methods
can work very well even on a small sample size. The reasons lie in three
folds:
1. (1)
The maximum margin criterion makes the learnt mapping more discriminative,
which is a straightforward strategy in the supervised classification task.
2. (2)
Auto-encoder approximately preserves the local neighborhood structures.
For this, Alain et al. [2] theoretically prove that the learnt representation
by auto-encoder can recover local properties from the view of manifold. To
further validate that, we employ the first 700 photo images from CUFSF
database to perform the nonlinear self-reconstruction with auto-encoder. With
the hidden presentations, we find the local neighbors with 1,2,3,4,5 neighbors
can be preserved with the probability of 99.43%, 99.00%, 98.57%, 98.00% and
97.42% respectively. Thus, the use of auto-encoder intrinsically reduces the
complexity of the discriminant model, which further makes the learnt model
better generality on the testing set.
3. (3)
The deep structure generates a gradual model, which makes the learnt transform
more robust. With only one layer, the model can’t represent the complex data
very well. But with layers goes deeper, the coupled networks can learn
transforms much more flexible and hence can be allowed to handle more complex
data.
## 4 Conclusion
In this paper, we propose a deep learning method, the Deeply Coupled Auto-
encoder Networks(DCAN), which can gradually generate a coupled discriminant
common representation for cross-view object classification. In each layer we
take both local consistency and discrimination of projected data into
consideration. By stacking multiple such coupled network layers, DCAN can
gradually improve the learnt shared features in the common space. Moreover,
experiments in the cross-view classification tasks demonstrate the superior of
our method over other state-of-the-art methods.
## References
* [1] S. Akaho. A kernel method for canonical correlation analysis, 2006.
* [2] G. Alain and Y. Bengio. What regularized auto-encoders learn from the data generating distribution. arXiv preprint arXiv:1211.4246, 2012.
* [3] G. Andrew, R. Arora, J. Bilmes, and K. Livescu. Deep canonical correlation analysis.
* [4] P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman. Eigenfaces vs. fisherfaces: Recognition using class specific linear projection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(7):711–720, 1997.
* [5] Y. Bengio, A. Courville, and P. Vincent. Representation learning: A review and new perspectives. 2013\.
* [6] Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle. Greedy layer-wise training of deep networks, 2007.
* [7] M. Chen, Z. Xu, K. Weinberger, and F. Sha. Marginalized denoising autoencoders for domain adaptation, 2012.
* [8] N. Chen, J. Zhu, and E. P. Xing. Predictive subspace learning for multi-view data: a large margin approach, 2010.
* [9] R. Gross, I. Matthews, J. Cohn, T. Kanade, and S. Baker. The cmu multi-pose, illumination, and expression (multi-pie) face database, 2007.
* [10] D. R. Hardoon, S. Szedmak, and J. Shawe-Taylor. Canonical correlation analysis: An overview with application to learning methods. Neural Computation, 16(12):2639–2664, 2004.
* [11] T. Hastie, R. Tibshirani, and J. J. H. Friedman. The elements of statistical learning, 2001.
* [12] G. E. Hinton, S. Osindero, and Y.-W. Teh. A fast learning algorithm for deep belief nets. Neural computation, 18(7):1527–1554, 2006.
* [13] G. E. Hinton and R. R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507, 2006.
* [14] H. Hotelling. Relations between two sets of variates. Biometrika, 28(3/4):321–377, 1936.
* [15] M. Kan, S. Shan, H. Zhang, S. Lao, and X. Chen. Multi-view discriminant analysis. pages 808–821, 2012.
* [16] T.-K. Kim, J. Kittler, and R. Cipolla. Discriminative learning and recognition of image set classes using canonical correlations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(6):1005–1018, 2007.
* [17] Q. V. Le, J. Ngiam, A. Coates, A. Lahiri, B. Prochnow, and A. Y. Ng. On optimization methods for deep learning, 2011.
* [18] Y. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller. Efficient backprop. In Neural networks: Tricks of the trade, pages 9–50. Springer, 1998\.
* [19] H. Lee, A. Battle, R. Raina, and A. Y. Ng. Efficient sparse coding algorithms, 2007.
* [20] Z. Lei and S. Z. Li. Coupled spectral regression for matching heterogeneous faces, 2009.
* [21] H. Li, T. Jiang, and K. Zhang. Efficient and robust feature extraction by maximum margin criterion. Neural Networks, IEEE Transactions on, 17(1):157–165, 2006.
* [22] D. Lin and X. Tang. Inter-modality face recognition. pages 13–26, 2006.
* [23] J. Ngiam, A. Khosla, M. Kim, J. Nam, H. Lee, and A. Y. Ng. Multimodal deep learning, 2011.
* [24] J. Nocedal and S. J. Wright. Numerical optimization, 2006.
* [25] P. J. Phillips, H. Wechsler, J. Huang, and P. J. Rauss. The feret database and evaluation procedure for face-recognition algorithms. Image and vision computing, 16(5):295–306, 1998.
* [26] A. Sharma and D. W. Jacobs. Bypassing synthesis: Pls for face recognition with pose, low-resolution and sketch, 2011.
* [27] A. Sharma, A. Kumar, H. Daume, and D. W. Jacobs. Generalized multiview analysis: A discriminative latent space, 2012.
* [28] P. Vincent, H. Larochelle, Y. Bengio, and P.-A. Manzagol. Extracting and composing robust features with denoising autoencoders, 2008\.
* [29] F. Wang and C. Zhang. Feature extraction by maximizing the average neighborhood margin. In Computer Vision and Pattern Recognition, 2007. CVPR’07. IEEE Conference on, pages 1–8. IEEE, 2007.
* [30] S. Wang, L. Zhang, Y. Liang, and Q. Pan. Semi-coupled dictionary learning with applications to image super-resolution and photo-sketch synthesis, 2012.
* [31] X. Wang and X. Tang. Face photo-sketch synthesis and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(11):1955–1967, 2009.
* [32] J. Xie, L. Xu, and E. Chen. Image denoising and inpainting with deep neural networks, 2012.
* [33] S. Yan, D. Xu, B. Zhang, H.-J. Zhang, Q. Yang, and S. Lin. Graph embedding and extensions: a general framework for dimensionality reduction. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(1):40–51, 2007.
* [34] W. Zhang, X. Wang, and X. Tang. Coupled information-theoretic encoding for face photo-sketch recognition, 2011.
* [35] B. Zhao, F. Wang, and C. Zhang. Maximum margin embedding. In Data Mining, 2008. ICDM’08. Eighth IEEE International Conference on, pages 1127–1132. IEEE, 2008.
|
arxiv-papers
| 2014-02-10T04:15:23 |
2024-09-04T02:49:57.991463
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wen Wang, Zhen Cui, Hong Chang, Shiguang Shan, Xilin Chen",
"submitter": "Hong Chang",
"url": "https://arxiv.org/abs/1402.2031"
}
|
1402.2073
|
# Mining Images in Biomedical Publications: Detection and Analysis of Gel
Diagrams
Tobias Kuhn 1 Mate Levente Nagy Tobias Kuhn – [email protected] 3
ThaiBinh Luong Mate Levente Nagy — [email protected] 2 and Michael
Krauthammer2,3 ThaiBinh Luong – [email protected] Michael Krauthammer –
[email protected] (1)Department of Humanities, Social and Political
Sciences, ETH Zurich, Switzerland. (2)Department of Pathology, Yale University
School of Medicine, New Haven, CT, USA. (3)Program for Computational Biology
and Bioinformatics, Yale University, New Haven, CT, USA.
###### Abstract
Authors of biomedical publications use gel images to report experimental
results such as protein-protein interactions or protein expressions under
different conditions. Gel images offer a concise way to communicate such
findings, not all of which need to be explicitly discussed in the article
text. This fact together with the abundance of gel images and their shared
common patterns makes them prime candidates for automated image mining and
parsing. We introduce an approach for the detection of gel images, and present
a workflow to analyze them. We are able to detect gel segments and panels at
high accuracy, and present preliminary results for the identification of gene
names in these images. While we cannot provide a complete solution at this
point, we present evidence that this kind of image mining is feasible.
## Introduction
A recent trend in the area of literature mining is the inclusion of images in
the form of figures from biomedical publications [1, 2, 3]. This development
benefits from the fact that an increasing number of scientific articles are
published as open access publications. This means that not just the abstracts
but the complete texts including images are available for data analysis. Among
other things, this enabled the development of query engines for biomedical
images like the Yale Image Finder [4] and the BioText Search Engine [5].
Below, we present our approach to detect and access gel diagrams. This is an
extended version of a previous workshop paper [6].
As a preparatory evaluation to decide which image type to focus on, we built a
corpus of 3 000 figures that allows us to reliably estimate the numbers and
types of images in biomedical articles. These figures were drawn randomly from
the open access subset of PubMed Central and then manually annotated. They
were split into subfigures when the figure consisted of several components.
Figure 1 shows the resulting categories and subcategories. This classification
scheme is based on five basic image categories: Experimental/Microscopy,
Graph, Diagram, Clinical and Picture, each divided into multiple
subcategories. It shows that bar graphs (12.4%), black-on-white gels (12.0%),
fluorescence microscopy images (9.4%), and line graphs (8.1%) are the most
frequent subfigure types (all percentages are relative to the entire set of
images).
Figure 1: Categorization of images from open access articles of PubMed
Central.
We targeted different kinds of graphs (i.e., diagrams with axes) in previous
work [7], and we decided to focus this work on the second most common type of
images: gel diagrams. They are the result of gel electrophoresis, which is a
common method to analyze DNA, RNA and proteins. Southern, Western and Northern
blotting [8, 9, 10] are among the most common applications of gel
electrophoresis. The resulting experimental artifacts are often shown in
biomedical publications in the form of gel images as evidence for the
discussed findings such as protein-protein interactions or protein expressions
under different conditions. Often, not all details of the results shown in
these images are explicitly stated in the caption or the article text. For
these reasons, it would be of high value to be able to reliably mine the
relations encoded in these images.
A closer look at gel images reveals that they follow regular patterns to
encode their semantic relations. Figure 2 shows two typical examples of gel
images together with a table representation of the involved relations. The
ultimate objective of our approach (for which we can only present a partial
solution here) is to automatically extract at least some of these relations
from the respective images, possibly in conjunction with classical text mining
techniques. The first example shows a Western blot for detecting two proteins
(14-3-3$\sigma$ and $\beta$-actin as a control) in four different cell lines
(MDA-MB-231, NHEM, C8161.9, and LOX, the first of which is used as a control).
There are two rectangular gel segments arranged in a way to form a $2\times 4$
grid for the individual eight measurements combining each protein with each
cell line. A gel diagram can be considered a kind of matrix with pictures of
experimental artifacts as content. The tables to the right illustrate the
semantic relations encoded in the gel diagrams. Each relation instance
consists of a condition, a measurement and a result. The proteins are the
entities being measured under the conditions of the different cell lines. The
result is a certain degree of expression indicated by the darkness of the
spots (or brightness in the case of white-on-black gels). The second example
is a slightly more complex one. Several proteins are tested against each other
in a way that involves more than two dimensions. In this case, the use of “+”
and “–” labels is a frequent technique to denote the different possible
combinations of a number of conditions. Apart from that, the principles are
the same. In this case, however, the number of relations is much larger. Only
the first eight of a total of 32 relation instances are shown in the table to
the right. In such cases, the text rarely mentions all these relations in an
explicit way, and the image is therefore the only accessible source.
|
---
| Condition | Measurement | Result
---|---|---
MDA-MB-231 | 14-3-3$\sigma$ | high expression
NHEM | 14-3-3$\sigma$ | no expression
C8161.9 | 14-3-3$\sigma$ | high expression
LOX | 14-3-3$\sigma$ | low expression
MDA-MB-231 | $\beta$-actin | high expression
NHEM | $\beta$-actin | high expression
C8161.9 | $\beta$-actin | high expression
LOX | $\beta$-actin | high expression
|
---
| Condition | Measurement | Result
---|---|---
IL-1$\beta$ (–) | DEX (–) | RU486 (–) | p-p38 | low expression
IL-1$\beta$ (+) | DEX (–) | RU486 (–) | p-p38 | high expression
IL-1$\beta$ (–) | DEX (+) | RU486 (–) | p-p38 | no expression
IL-1$\beta$ (+) | DEX (+) | RU486 (–) | p-p38 | low expression
IL-1$\beta$ (–) | DEX (–) | RU486 (+) | p-p38 | no expression
IL-1$\beta$ (+) | DEX (–) | RU486 (+) | p-p38 | high expression
IL-1$\beta$ (–) | DEX (+) | RU486 (+) | p-p38 | low expression
IL-1$\beta$ (+) | DEX (+) | RU486 (+) | p-p38 | high expression
… | | | … | …
Figure 2: Two examples of gel images from biomedical publications (PMID
19473536 and 15125785) with tables showing the relations that could be
extracted from them
## Background
In principle, image mining involves the same processes as classical literature
mining [11]: document categorization, named entity tagging, fact extraction,
and collection-wide analysis. However, there are some subtle differences.
Document categorization corresponds to image categorization, which is
different in the sense that it has to deal with features based on the two-
dimensional space of pixels, but otherwise the same principles of automatic
categorization apply. Named entity tagging is different in two ways:
pinpointing the mention of an entity is more difficult with images (a large
number of pixels versus a couple of characters), and OCR errors have to be
considered. Fact extraction in classical literature mining involves the
analysis of the syntactic structure of the sentences. In images, in contrast,
there are rarely complete sentences, but the semantics is rather encoded by
graphical means. Thus, instead of parsing sentences, one has to analyze
graphical elements and their relation to each other. The last process,
collection-wide analysis, is a higher-level problem, and therefore no
fundamental differences can be expected. Thus, image mining builds upon the
same general stages as classical text mining, but with some subtle yet
important differences.
Image mining on biomedical publications is not a new idea. It has been applied
for the extraction of subcellular location information [12], the detection of
panels of fluorescence microscopy images [13], the extraction of pathway
information from diagrams [14], and the detection of axis diagrams [7]. Also,
there is a large amount of existing work on how to process gel images [15, 16,
17, 18, 19] and databases have been proposed to store the results of gel
analyses [20]. These techniques, however, take as input plain gel images,
which are not readily accessible from biomedical papers, because they make up
just parts of the figures. Furthermore, these tools are designed for
researchers who want to analyze their gel images and not to read gel diagrams
that have already been analyzed and annotated by a researcher. Therefore,
these approaches do not tackle the problem of recognizing and analyzing the
labels of gel images. Some attempts to classify biomedical images include gel
figures [21], which is, however, just the first step in locating them and
analyzing their labels and their structure. To our knowledge, nobody has yet
tried to perform image mining on gel diagrams.
## Approach and Methods
Figure 3 shows the procedure of our approach to image mining from gel
diagrams. It consists of seven steps: figure extraction, segmentation, text
recognition, gel detection, gel panel detection, named entity recognition and
relation extraction.111Due to the fact that many figures consist of multiple
panels of different types, we go straight to gel segment detection without
first classifying entire images. Most gel panels share their figure with other
panels, which makes automated classification difficult at the image level.
Figure 3: The procedure of our approach: (1) figure extraction, (2)
segmentation, (3) text recognition, (4) gel detection, (5) gel panel
detection, (6) named entity recognition, and (7) relation extraction.
Using structured article representations, the first step is trivial. For steps
two and three, we rely on existing work. The main focus of this paper lies on
steps four and five: the detection of gels and gel panels. In the discussion
section, we present some preliminary results on step six of recognizing named
entities, and sketch how step seven could be implemented, for which we cannot
provide a concrete solution at this point.
To practically evaluate our approach, we ran our pipeline on the entire open
access subset of PubMed Central (though not all figures made it through the
whole pipeline due to technical difficulties).
### Figure Extraction
A large portion of the articles of the open access subset of the PubMed
Central database are available as structured XML files with additional image
files for the figures. We only use these articles so far, which makes the
figure extraction task very easy. It would be more difficult, though
definitely feasible, to extract the figures from PDF files or even bitmaps of
scanned articles (see [22] and http://pdfjailbreak.com for approaches on
extracting the structure of articles in PDF format).
### Segmentation and Text Recognition
For the next two steps — segment detection and subsequent text recognition —,
we rely on our previous work [23, 24]. This method includes the detection of
layout elements, edge detection, and text recognition with a novel pivoting
approach. For optical character recognition (OCR), the Microsoft Document
Imaging package is used, which is available as part of Microsoft Office 2003.
Overall, this approach has been shown to perform better than other existing
approaches for the images found in biomedical publications [23]. We do not go
into the details here, as this paper focuses on the subsequent steps.
Due to some limitations of the segmentation algorithm when it comes to
rectangles with low internal contrast (like gels), we applied a complementary
very simple rectangle detection algorithm.
### Gel Segment Detection
Based on the results of the above-mentioned steps, we try to identify gel
segments. Such gel segments typically have rectangular shapes with darker
spots on a light gray background, or — less commonly — white spots on a dark
background. We decided to use machine learning techniques to generate
classifiers to detect such gel segments. To do so, we defined 39 numerical
features for image segments: the coordinates of the relative position (within
the image), the relative and absolute width and height, 16 grayscale histogram
features, three color features (for red, green and blue), 13 texture features
(coarseness, presence of ripples, etc.) based on [25], and the number of
recognized characters.
To train the classifiers, we took a random sample of 500 figures, for which we
manually annotated the gel segments. In the same way, we obtained a second
sample of another 500 figures for testing the classifiers.222We double-checked
these manual annotations to check their quality, which revealed only four
misclassified segments in total for the training and test samples (0.016% of
all segments). We used the Weka toolkit and opted for random forest
classifiers based on 75 random trees. Using different thresholds to adjust the
trade-off between precision and recall, we generated a classifier with good
precision and another one with good recall. Both of them are used in the next
step. We tried other types of classifiers including naive Bayes, Bayesian
networks [26], PART decision lists [27], and convolutional networks [28], but
we achieved the best results with random forests.
### Gel Panel Detection
A gel panel typically consists of several gel segments and comes with labels
describing the involved genes, proteins, and conditions. For our goal, it is
not sufficient to just detect the figures that contain gel panels, but we also
have to extract their positions within the figures and to access their labels.
This is not a simple classification task, and therefore machine learning
techniques do not apply that easily. For that reason, we used a detection
procedure based on hand-coded rules.
In a first step, we group gel segments to find contiguous gel regions that
form the center part of gel panels. To do so, we start with looking for
segments that our high-precision classifier detects as gel segments. Then, we
repeatedly look for adjacent gel segments, this time applying the high-recall
classifier, and merge them. Two segments are considered neighbors if they are
at most 50 pixels apart333We are using absolute distance values at this point.
A more refined algorithm could apply some sort of relative measure. However,
the resolution of the images does not vary that much, which is why absolute
values worked out well so far. and do not have any text segment between them.
Thus, segments which could be gel segments according to the high-recall
classifier make it into a gel panel only if there is at least one high-
precision segment in their group. The goal is to detect panels with high
precision, but also to detect the complete panels and not just parts of them.
We focus here on precision because low recall can be leveraged by the large
number of available gel images. Furthermore, as the open access part of PubMed
Central only makes up a small subset of all biomedical publications, recall in
a more general sense is anyway limited by the proportion of open access
publications.
As a next step, we collect the labels in the form of text segments located
around the detected gel regions. For a text segment to be attributed to a
certain gel panel, its nearest edge must be at most 30 pixels away from the
border of the gel region and its farthest edge must not be more than 150
pixels away. We end up with a representation of a gel panel consisting of two
parts: a center region containing a number of gel segments and a set of labels
in the form of text segments located around the center region.
To evaluate this algorithm, we collected yet another sample of 500 figures, in
which 106 gel panels in 61 different figures were revealed by manual
annotation.444Again, these manual annotations were double-checked to ensure
their quality. Five errors were found and fixed in this process. Based on this
sample, we manually checked whether our algorithm is able to detect the
presence and the (approximate) position of the gel panels.
## Results
The top part of Table 1 shows the result of the gel detection classifier. We
generated three different classifiers from the training data, one for each of
the threshold values 0.15, 0.3 and 0.6. Lower threshold values lead to higher
recall at the cost of precision, and vice versa. In the balanced case, we
achieved an F-score of 75%. To get classifiers with precision or recall over
90%, F-score goes down significantly, but stays in a sensible range. These two
classifiers (thresholds 0.15 and 0.6) are used in the next step. To interpret
these values, one has to consider that gel segments are greatly outnumbered by
non-gel segments. Concretely, only about 3% are gel segments. More
sophisticated accuracy measures for classifier performance, such as the area
under the ROC curve [29], take this into account. For the presented
classifiers, the area under the ROC curve is 98.0% (on a scale from 50% for a
trivial, worthless classifier to 100% for a perfect one).
| Method | Threshold | Precision | Recall | F-score | ROC area
---|---|---|---|---|---|---
Segments | Random forests | 0.15 | 0.439 | 0.909 | 0.592 | $\left.\begin{matrix}~{}\\\ ~{}\\\ \end{matrix}\right\\}$ 0.980
0.30 | 0.765 | 0.739 | 0.752
0.60 | 0.926 | 0.301 | 0.455
Naive Bayes | | 0.172 | 0.739 | 0.279 | 0.883
Bayesian network | | 0.394 | 0.531 | 0.452 | 0.914
PART decision list | | 0.631 | 0.496 | 0.555 | 0.777
Convolutional networks | | 0.142 | 0.949 | 0.248 |
Panels | Hand-coded rules | | 0.951 | 0.368 | 0.530 |
Table 1: The results of the gel segment detection classifiers (top) and the
gel panel detection algorithm (bottom)
The results of the gel panel detection algorithm are shown in the bottom part
of Table 1. The precision is 95% at a recall of 37%, leading to an F-score of
53%. The comparatively low recall is mainly due to the general problem of
pipeline-based approaches that the various errors from the earlier steps
accumulate and are hard to correct at a later stage in the pipeline.
Table 2 shows the results of running the pipeline on PubMed Central. We
started with about 410 000 articles, the entire open access subset of PubMed
Central at the time we downloaded them (February 2012). We successfully parsed
the XML files of 94% of these articles (for the remaining articles, the XML
file was missing or not well-formed, or other unexpected errors occurred). The
successful articles contained around 1 100 000 figures, for some of which our
segment detection step encountered image formatting errors or other internal
errors, or was just not able to detect any segments. We ended up with more
than 880 000 figures, in which we detected about 86 000 gel panels, i.e.
roughly ten out of 100 figures. For each of them, we found on average 3.6
labels with recognized text. After tokenization, we identified about 76 000
gene names in these gel labels, which corresponds to 6.8% of the tokens.
Considering all text segments (including but not restricted to gel labels),
only 3.3% of the tokens are detected as gene names.555The low numbers are
partially due to the fact that a considerable part of the tokens are “junk
tokens” produced by the OCR step when trying to recognize characters in
segments that do not contain text.
Total articles | 410 950
---|---
Processed articles | 386 428
Total figures from processed articles | 1 110 643
Processed figures | 884 152
Detected gel panels | 85 942
Detected gel panels per figure | 0.097
Detected gel labels | 309 340
Detected gel labels per panel | 3.599
Detected gene tokens | 1 854 609
Detected gene tokens in gel labels | 75 610
Gene token ratio | 0.033
Gene token ratio in gel labels | 0.068
Table 2: The results of running the pipeline on the open access subset of
PubMed Central
## Discussion
The presented results show that we are able to detect gel segments with high
accuracy, which allows us to subsequently detect whole gel panels at a high
precision. The recall of the panel detection step is relatively low, but with
about 37% still in a reasonable range. As mentioned above, we can leverage the
high number of available figures, which makes precision more important than
recall. Running our pipeline on the whole set of open access articles from
PubMed Central, we were able to retrieve 85 942 potential gel panels (around
95% of which we can expect to be correctly detected).
The next step would be to recognize the named entities mentioned in the gel
labels. To this aim, we did a preliminary study to investigate whether we are
able to extract the names of genes and proteins from gel diagrams. To do so,
we tokenized the label texts and looked for entries in the Entrez Gene
database to match the tokens. This look-up was done in a case-sensitive way,
because many names in gel labels are acronyms, where the specific
capitalization pattern can be critical to identify the respective entity. We
excluded tokens that have less than three characters, are numbers (Arabic or
Latin), or correspond to common short words (retrieved from a list of the 100
most frequent words in biomedical articles). In addition, we extended this
exclusion list with 22 general words that are frequently used in the context
of gel diagrams, some of which coincide with gene names according to
Entrez.666These words are: _min_ , _hrs_ , _line_ , _type_ , _protein_ , _DNA_
, _RNA_ , _mRNA_ , _membrane_ , _gel_ , _fold_ , _fragment_ , _antigen_ ,
_enzyme_ , _kinase_ , _cleavage_ , _factor_ , _blot_ , _pro_ , _pre_ ,
_peptide_ , and _cell_. Since gel electrophoresis is a method to analyze genes
and proteins, we would expect to find more such mentions in gel labels than in
other text segments of a figure. By measuring this, we get an idea of whether
the approach works out or not. In addition, we manually checked the gene and
protein names extracted from gel labels after running our pipeline on 2 000
random figures. In 124 of these figures, at least one gel panel was detected.
Table 3 shows the results of this preliminary evaluation. Almost two-thirds of
the detected gene/protein tokens (65.3%) were correctly identified. 9% thereof
were correct but could be more specific, e.g. when only “actin” was recognized
for “$\beta$-actin” (which is not incorrect but of course much harder to map
to a meaningful identifier). The incorrect cases (34.6%) can be split into two
classes of roughly the same size: some recognized tokens were actually not
mentioned in the figure but emerged from OCR errors; other tokens were
correctly recognized but incorrectly classified as gene or protein references.
| absolute | relative
---|---|---
Total | 156 | 100.0%
Incorrect | 54 | 34.6%
– Not mentioned (OCR errors) | 28 | 17.9%
– Not references to genes or proteins | 26 | 16.7%
Correct | 102 | 65.3%
– Partially correct (could be more specific) | 14 | 9.0%
– Fully correct | 88 | 56.4%
Table 3: Numbers of recognized gene/protein tokens in 2 000 random figures
Although there is certainly much room for improvement, this simple gene
detection step seems to perform reasonably well.
For the last step, relation extraction, we cannot present any concrete results
at this point. After recognizing the named entities, we would have to
disambiguate them, identify their semantic roles (condition, measurement or
something else), align the gel images with the labels, and ultimately quantify
the degree of expression. To improve the quality of the results, combinations
with classical text mining techniques should be considered. This is all future
work. We expect to be able to profit to a large extent from existing work to
disambiguate protein and gene names [30, 31] and to detect and analyze gel
spots (see the existing work mentioned above).
It seems reasonable to assume that these results can be combined with existing
techniques of term disambiguation and gel spot detection at a satisfactory
level of accuracy. We plan to investigate this in future work.
As mentioned above, we have started to investigate how the gel segment
detection step could be improved by the use of the image recognition technique
of convolutional networks (ConvNet) [28]. We started with a simplified
approach to the one presented in [32]. In this approach, images are tiled into
small quadratic pieces. We used a single network (and not several parallel
networks), based on $48\times 48$ input tile images with three layers of
convolutions. The first layer takes eight $5\times 5$ convolutions and is
followed by a $2\times 2$ sub-sampling. The second layer takes twenty four
$5\times 5$ convolutions and is followed by a $3\times 3$ sub-sampling. The
last layer takes seventy two $6\times 6$ convolutions, which leads to a fully
connected layer. We trained our ConvNet on the 500 images of the training set,
where we manually annotated the tiles as _gel_ and _non-gel_. With the use of
EBLearn [33], this trained ConvNet classified the tiles of the 500 images of
our testing set. The classified tiles can then be reconstructed into a mask
image, as shown in Figure 4. A manual check of the clusters of recognized gel
tiles led to the results shown in Table 1. Recall is very good (95%) but
precision is very poor (14%), leading to an F-score of 25%. This is much worse
than the results we got with our random forest approach, which is why ConvNet
is currently not part of our pipeline. We hope, however, that we can further
optimize this ConvNet approach and combine it with random forests to exploit
their (hopefully) complementary benefits. Using ConvNet to classify complete
images as _gel-image_ or _non-gel-image_ and adjusting the classification to
account for unbalanced classes, we were able to obtain an F-score of 74%,
which makes us confident that a combination of the two approaches could lead
to a significant improvement of our gel segment detection step. As an
alternative approach, we will try to run ConvNet on down-scaled entire panels
rather than small tiles, as described in [34]. Furthermore, we will experiment
with parallel networks instead of single ones to improve accuracy.
Figure 4: Original and mask image after ConvNet classification for an
exemplary image from PMID 14993249. Green means _gel_ ; brown means _other_ ;
and white means _not enough gradient information_.
The results obtained from our gel recognition pipeline indicate that it is
feasible to extract relations from gel images, but it is clear that this
procedure is far from perfect. The automatic analysis of bitmap images seems
to be the only efficient way to extract such relations from existing
publications, but other publishing techniques should be considered for the
future. The use of vector graphics instead of bitmaps would already greatly
improve any subsequent attempts of automatic analysis. A further improvement
would be to establish accepted standards for different types of biomedical
diagrams in the spirit of the Unified Modeling Language, a graphical language
widely applied in software engineering since the 1990s. Ideally, the resulting
images could directly include semantic relations in a formal notation, which
would make relation mining a trivial procedure. If authors are supported by
good tools to draw diagrams like gel images, this approach could turn out to
be feasible even in the near future.
Concretely, we would like to take the opportunity to postulate the following
actions, which we think should be carried out to make the content of images in
biomedical articles more accessible:
* •
Stop pressing diagrams into bitmaps! Unless the image only consists of one
single photograph, screenshot, or another kind of picture that only has bitmap
representation, vector graphics should be used for article figures.
* •
Let data and metadata travel from the tools that generate diagrams to the
final articles! Whenever the specific tool that is used to generate the
diagram “knows” that a certain graphical element refers to an organism, a
gene, an interaction, a point in time, or another kind of entity, then this
information should be stored in the image file, passed on, and finally
published with the article.
* •
Use RDF vocabularies to embed semantic annotations in diagrams! Tools for
creating scientific diagrams should use RDF notation and stick to existing
standardized schemas (or define new ones if required) to annotate the diagram
files they create.
* •
Define standards for scientific diagrams! In the spirit of the Unified
Modeling Language, the biomedical community should come up with standards that
define the appearance and meaning of different types of diagrams.
Obviously, different groups of people need to be involved in these actions,
namely article authors, journal editors, and tool developers. It is relatively
inexpensive to follow these postulates (though it might require some time),
which in turn would greatly improve data sharing, image mining, and scientific
communication in general. Standardized diagrams could be the long sought
solution to the problem of how to let authors publish computer-processable
formal representations for (part of) their results. This can build upon the
efforts of establishing an open annotation model [35, 36].
## Conclusions
Successful image mining from gel diagrams in biomedical publications would
unlock a large amount of valuable data. Our results show that gel panels and
their labels can be detected with high accuracy, applying machine learning
techniques and hand-coded rules. We also showed that genes and proteins can be
detected in the gel labels with satisfactory precision.
Based on these results, we believe that this kind of image mining is a
promising and viable approach to provide more powerful query interfaces for
researchers, to gather relations such as protein-protein interactions, and to
generally complement existing text mining approaches. At the same time, we
believe that an effort towards standardization of scientific diagrams such as
gel images would greatly improve the efficiency and precision of image mining
at relatively low additional costs at the time of publication.
## Competing Interests
The authors declare that they have no competing interests.
## Authors’ Contributions
TK was the main author and main contributor of the presented work. He was
responsible for designing and implementing the pipeline, gathering the data,
performing the evaluation, and analyzing the results. MLN applied, trained,
and evaluated the ConvNet classifier, and contributed to the annotation of the
test sets. TL built and analyzed the corpus for the preparatory evaluation. MK
contributed to the conception and the design of the approach and to the
analysis of the results. All authors have been involved in drafting or
revising the manuscript, and all authors read and approved the final
manuscript.
## Acknowledgments
This study has been supported by the National Library of Medicine grant
5R01LM009956.
## References
* [1] Yu H, Lee M: Accessing bioscience images from abstract sentences. _Bioinformatics_ 2006, 22(14):e547–e556.
* [2] Zweigenbaum P, Demner-Fushman D, Yu H, Cohen KB: Frontiers of biomedical text mining: current progress. _Briefings in Bioinformatics_ 2007, 8(5):358–375.
* [3] Peng H: Bioimage informatics: a new area of engineering biology. _Bioinformatics_ 2008, 24(17):1827–1836, [http://dx.doi.org/10.1093/bioinformatics/btn346].
* [4] Xu S, McCusker J, Krauthammer M: Yale Image Finder (YIF): a new search engine for retrieving biomedical images. _Bioinformatics_ 2008, 24(17):1968–1970, [http://dx.doi.org/10.1093/bioinformatics/btn340].
* [5] Hearst MA, Divoli A, Guturu H, Ksikes A, Nakov P, Wooldridge MA, Ye J: BioText Search Engine. _Bioinformatics_ 2007, 23(16):2196–2197, [http://dx.doi.org/10.1093/bioinformatics/btm301].
* [6] Kuhn T, Krauthammer M: Image Mining from Gel Diagrams in Biomedical Publications. In _Proceedings of the 5th International Symposium on Semantic Mining in Biomedicine (SMBM 2012)_ , University of Zurich (Zurich, Switzerland) 2012:26–33, [http://www.zora.uzh.ch/64476/].
* [7] Kuhn T, Luong T, Krauthammer M: Finding and Accessing Diagrams in Biomedical Publications. In _Proceedings of the American Medical Informatics Association (AMIA) 2012 Annual Symposium_ , American Medical Informatics Association (Bethesda, MD, USA) 2012.
* [8] Southern E: Detection of specific sequences among DNA fragments separated by gel electrophoresis. _J. Mol. Biol._ 1975, 98(3):503–517.
* [9] Alwine JC, Kemp DJ, Stark GR: Method for detection of specific RNAs in agarose gels by transfer to diazobenzyloxymethyl-paper and hybridization with DNA probes. _Proceedings of the National Academy of Sciences_ 1977, 74(12):5350.
* [10] Burnette WN: Western Blotting: Electrophoretic transfer of proteins from sodium dodecly sulfate-polyacrylamide gels to unmodified nitrocellulose and radiographic detection with antibody and radioiodinated protein A. _Anal. Biochem._ 1981, 112:195–203.
* [11] De Bruijn B, Martin J: Getting to the (c)ore of knowledge: mining biomedical literature. _International Journal of Medical Informatics_ 2002, 67(1-3):7–18.
* [12] Murphy RF, Kou Z, Hua J, Joffe M, Cohen WW: Extracting and Structuring Subcellular Location Information from On-line Journal Articles: The Subcellular Location Image Finder. In _Proceedings of the IASTED International Conference on Knowledge Sharing and Collaborative Engineering (KSCE-2004)_ , ACTA Press (Calgary, AB, Canada) 2004:109–114, [http://www.actapress.com/Abstract.aspx?paperId=17244].
* [13] Qian Y, Murphy RF: Improved recognition of figures containing fluorescence microscope images in online journal articles using graphical models. _Bioinformatics_ 2008, 24(4):569–576, [http://dx.doi.org/10.1093/bioinformatics/btm561].
* [14] Kozhenkov S, Baitaluk M: Mining and integration of pathway diagrams from imaging data. _Bioinformatics_ 2012, 28(5):739–742, [http://dx.doi.org/10.1093/bioinformatics/bts018].
* [15] Lemkin PF: Comparing two-dimensional electrophoretic gel images across the Internet. _Electrophoresis_ 1997, 18(3-4):461–470.
* [16] Luhn S, Berth M, Hecker M, Bernhardt J: Using standard positions and image fusion to create proteome maps from collections of two-dimensional gel electrophoresis images. _Proteomics_ 2003, 3(7):1117–1127.
* [17] Cutler P, Heald G, White IR, Ruan J: A novel approach to spot detection for two-dimensional gel electrophoresis images using pixel value collection. _Proteomics_ 2003, 3(4):392–401.
* [18] Rogers M, Graham J, Tonge RP: Statistical models of shape for the analysis of protein spots in two-dimensional electrophoresis gel images. _Proteomics_ 2003, 3(6):887–896.
* [19] Zerr T, Henikoff S: Automated band mapping in electrophoretic gel images using background information. _Nucleic Acids Research_ 2005, 33(9):2806–2812.
* [20] Schlamp K, Weinmann A, Krupp M, Maass T, Galle P, Teufel A: BlotBase: a northern blot database. _Gene_ 2008, 427(1-2):47–50.
* [21] Rodriguez-Esteban R, Iossifov I: Figure mining for biomedical research. _Bioinformatics_ 2009, 25(16):2082–2084, [http://dx.doi.org/10.1093/bioinformatics/btp318].
* [22] Ramakrishnan C, Patnia A, Hovy EH, Burns GA, et al.: Layout-aware text extraction from full-text PDF of scientific articles. _Source code for biology and medicine_ 2012, 7:255–258.
* [23] Xu S, Krauthammer M: A new pivoting and iterative text detection algorithm for biomedical images. _J. of Biomedical Informatics_ 2010, 43(6):924–931, [http://dx.doi.org/10.1016/j.jbi.2010.09.006].
* [24] Xu S, Krauthammer M: Boosting text extraction from biomedical images using text region detection. In _Biomedical Sciences and Engineering Conference (BSEC), 2011_ , IEEE (New York City, NY, USA) 2011:1–4.
* [25] Haralick RM, Shanmugam K, Dinstein I: Textural Features for Image Classification. _IEEE Transactions on Systems, Man, and Cybernetics_ 1973, 3(6):610–621, [http://dx.doi.org/10.1109/TSMC.1973.4309314].
* [26] Cooper GF, Herskovits E: A Bayesian method for the induction of probabilistic networks from data. _Machine learning_ 1992, 9(4):309–347.
* [27] Frank E, Witten IH: Generating Accurate Rule Sets Without Global Optimization. In _Proceedings of the Fifteenth International Conference on Machine Learning_ , Morgan Kaufmann Publishers (Burlington, MA, USA) 1998:144–151.
* [28] LeCun Y, Bengio Y: Convolutional networks for images, speech, and time series. _The handbook of brain theory and neural networks_ 1995, 3361.
* [29] Bradley AP: The use of the area under the ROC curve in the evaluation of machine learning algorithms. _Pattern recognition_ 1997, 30(7):1145–1159.
* [30] Tanabe L, Wilbur WJ: Tagging gene and protein names in biomedical text. _Bioinformatics_ 2002, 18(8):1124–1132.
* [31] Lu Z, Kao HY, Wei CH, Huang M, Liu J, Kuo CJ, Hsu CN, Tsai R, Dai HJ, Okazaki N, et al.: The gene normalization task in BioCreative III. _BMC bioinformatics_ 2011, 12(Suppl 8):S2.
* [32] Barbano PE, Nagy ML, Krauthammer M: Energy-Based Architecture for Classification of Publication Figures. In _Proceedings of the Biomedical Science and Engineering Center Conference (BSEC 2013)_ , IEEE (New York City, NY, USA) 2013.
* [33] Sermanet P, Kavukcuoglu K, LeCun Y: EBlearn: Open-source energy-based learning in C++. In _Proceedings of the 21st International Conference on Tools with Artificial Intelligence (ICTAI’09)_ , IEEE (New York City, NY, USA) 2009:693–697.
* [34] Krizhevsky A, Sutskever I, Hinton G: ImageNet classification with deep convolutional neural networks. _Advances in Neural Information Processing Systems_ 2012, 25:1106–1114.
* [35] Ciccarese P, Ocana M, Garcia Castro LJ, Das S, Clark T: An open annotation ontology for science on web 3.0. _J Biomed Semantics_ 2011, 2(Suppl 2):S4.
* [36] Sanderson R, Ciccarese P, Van de Sompel H: Open Annotation Data Model. Community draft, W3C 2013, [http://www.openannotation.org/spec/core/20130208/index.html].
|
arxiv-papers
| 2014-02-10T09:16:13 |
2024-09-04T02:49:58.006800
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tobias Kuhn, Mate Levente Nagy, ThaiBinh Luong, Michael Krauthammer",
"submitter": "Tobias Kuhn",
"url": "https://arxiv.org/abs/1402.2073"
}
|
1402.2185
|
# Relativity, the Special Theory,
explained to Children
(from 7 to 107 years old)
Charles-Michel Marle
Institut de Mathématiques de Jussieu
Université Pierre et Marie Curie
Paris, France
(15 July 2005)
###### Abstract
The author thinks that the main ideas or Relativity Theory can be explained to
children (around the age of 15 or 16) without complicated calculations, by
using very simple arguments of affine geometry. The proposed approach is
presented as a conversation between the author and one of his grand-children.
Limited here to the Special Theory, it will be extended to the General Theory
elsewhere, as sketched in conclusion.
For Agathe, Florent, Basile, Mathis, Gabrielle,
Morgane, Quitterie and my future other grand-children
## 1 Prologue
Maybe one day, one of my grand-children, at the age of 15 or 16, will ask me:
— Grand-father, could you explain what is Relativity Theory? My Physics
teacher lectured about it, talking of rolling trains and of lightnings hitting
the railroad, and I understood almost nothing!
This is the discussion I would like to have with her (or him).
— Do you know the theorem: the diagonals of a parallelogram meet at their
middle point?
— Yes, I do! I even know that the converse is true: if the diagonals of a
plane quadrilateral meet at their middle point, that quadrilateral is a
parallelogram. And I believe that I know a proof!
— Good! You know all the stuff needed to understand the basic idea of
Relativity theory! However, we must first think about Time and Space.
— Time and space seem to me very intuitive, and yet difficult to understand in
deep!
— Many people feel the same. The true nature of Time and Space is mysterious.
Let us say that together, Time and Space make the frame in which all physical
phenomena take place, in which all material objects evolve, including our
bodies. We should keep a modest mind profile on such a subject. We cannot hope
to understand all the mysteries of Time and Space. We should only try to
understand some of their properties and to use them to describe physical
phenomena. We should be ready to change the way we think about Time and Space,
if some experimental evidence shows that we were wrong.
— But if we do not know what are Time and Space, how can we hope to understand
some of their properties, and to be able to use them?
— By building mental pictures of Time and Space. Unfortunately we, poor
limited human beings, cannot do better: we know the surrounding world only
through our senses (enhanced by the measurement and observation instruments we
have built) and our ability of reasoning. Our reasoning always apply to the
mental pictures we have built of reality, not to reality itself.
Let me now indicate how the mental pictures of Time and Space used by
scientists have evolved, mainly from Newton to Einstein.
## 2 Newton’s and Leibniz’s views about Time and Space
### 2.1 Newtonian Time
The great scientist Isaac Newton [2] (1642–1727) used, as mental picture of
Time, a straight line $\cal T$, going to infinity on both sides, hence without
beginning nor end, with no privileged origin. Each particular time, for
example “now”, or ”three days ago at the sunset at Paris”, corresponds to a
particular element of that straight line.
Observe that Newton considered, without any discussion, that for each event
happening in the universe, there was a corresponding well defined time
(element of the straight line $\cal T$), the time at which that event happens.
— Where is that straight line $\cal T$? Is it drawn in some plane or in space?
— Nowhere! You should not think about the straight line of Time $\cal T$ as
drawn in something of larger dimension. Newton considered Time as an abstract
straight line, because successive events are linearly ordered, like points on
a straight line. Don’t forget that $\cal T$ is a mental picture of Time, not
Time itself! However, that mental picture is much more than a confuse idea: it
has very well defined mathematical properties. In modern language, we say that
$\cal T$ is endowed with an _affine structure_ and with an _orientation_.
— What is an affine structure? and what is its use?
— An affine structure allows us to compare two time intervals and to take
their ratio, for example to say that one of these intervals is two times the
other one. Newton considered the comparison of two time intervals as possible,
even when they were many centuries or millenaries apart, and to take their
ratio. In modern mathematical language, that property determines an _affine
structure_.
For the mathematician, that property means that we can apply transforms to
$\cal T$ by sliding it along itself, without contraction nor dilation, and
that these transforms (called _translations_) do not change its properties.
For the physicist, it means that the physical laws remain the same at all
times.
Another important property of Time: it always flows from past to future. To
take it into account, we endow $\cal T$ with an _orientation_ ; it means that
we consider the two directions (from past to future and from future to past)
as different, not equivalent, for example by choosing the direction from past
to future as preferred. We then say that $\cal T$ is _oriented_.
### 2.2 Newton’s absolute space
— OK, I roughly agree with that mental picture, although it does not account
for the main property of Time: it flows continuously and we cannot stop it!
And what about Space?
— Newton identified Space with the three dimensional space of geometers,
denoted by $\cal E$ : the space in which there are various figures made of
planes, straight lines, spheres, polyhedra, which obey the theorems developed
in Euclidean geometry: Thales and Pythagoras theorems, the theorem which says
that the diagonals of a parallelogram meet at their middle point, $\ldots$
### 2.3 The concept of Space-Time
Newton used Time and Space to describe the motion of every object $A$ of the
physical world as follows. That object occupies, at each time $t$ (element of
$\cal T$) for which it exists, a position $A_{t}$ in Space $\cal E$. The
motion of of $A$ is described by its successive positions $A_{t}$ when $t$
varies in $\cal T$.
Let me introduce now a new concept, that of Space-Time [1], due to the German
mathematician Hermann Minkowski (1864–1909). That concept was not used in
Mechanics before the discovery of Special Relativity. That is very
unfortunate, since its use makes much easier the understanding of the
foundations of Classical Mechanics, as well as those of Relativistic
Mechanics. Therefore I use it now, with the absolute Time and Space of Newton,
although Newton himself did not use that concept.
Newton Space-Time is simply the product set ${\cal E}\times{\cal T}$, whose
elements are pairs (called _events_) $(x,t)$, made by a point $x$ of $\cal E$
and a time $t$ of $\cal T$.
— What is the use of that Space-Time?
— It is very convenient to describe motions. For example, the motion of a
material particle $a$ (a very small object whose position, at each time
$t\in{\cal T}$, is considered as a point $a_{t}\in{\cal E}$), is described by
a line in ${\cal E}\times{\cal T}$, made by the events $(a_{t},t)$, for all
$t$ in the interval of time during which $a$ exists. That line is called the
_world line_ of $a$.
You will see on Figure 1 (where, for simplicity, Space is represented as a
straight line, as if it were one-dimensional) the world lines of three
particles, $a$, $b$ and $c$.
Figure 1: World lines in Newton Space-Time.
* •
The world line of $b$ est parallel to the Time axis $\cal T$: that particle is
at rest, il occupies a fixed position in the absolute Space $\cal E$.
* •
The world line of $c$ is a slanting straight line. The trajectory of that
particle in absolute Space $\cal E$ is a straight line and its velocity is
constant.
* •
The world line of $a$ is a curve, not a straight line. It means that the
velocity of $a$ changes with time.
### 2.4 Absolute rest and motion
For Newton, _rest_ and _motion_ were absolute concepts: a physical object is
at rest if its position in Space does not change with time; otherwise, it is
in motion.
— It seems very natural. Why should we change this view?
— Because nothing is at rest in the Universe! The Earth rotates around its
axis and around the Sun, which rotates around the center of our Galaxy. And
there are billions of galaxies in the Universe, all moving with respect to the
others! For these reasons, Newton’s concept of an absolute Space was
criticized very early, notably by his contemporary, the great mathematician
and philosopher Gottfried Wilhelm Leibniz (1647–1716).
### 2.5 Reference frames
— But without knowing what is at rest in the Universe, how Newton managed to
study the motions of the planets?
— To study the motion of a body $A$, Newton, and after him almost all
scientists up to now, used a _reference frame_. It means that he used another
body $R$ which remained approximately rigid during the motion he wanted to
study, and he made as if that body was at rest. Then he could study the
_relative motion_ of $A$ with respect to $R$.
Assuming that Newton’s absolute Space $\cal E$ exists, we recover the
description of absolute motion of $A$ by choosing, for $R$, a body at rest in
$\cal E$. The corresponding reference frame is called the _absolute fixed
frame_.
The body $R$ used to determine a reference frame can be, for example,
* •
the Earth (if we want to study the motion of a falling apple),
* •
the trihedron made by the straight lines which join the center of the Sun to
three distant stars (if we want to study the motions of the planets in the
solar system).
### 2.6 Galilean frames and Leibniz Space-Time
All reference frames are not equivalent. A _Galilean frame_ 111 In memory of
Galileo Galilei, (1564–1642), the founder of modern Physics., also called an
_inertial frame_ , is a reference frame in which the _principe of inertia_
holds true. That principle, first formulated for absolute motions in Newton’s
absolute space $\cal E$, says that the (absolute) motion of a free particle
takes place on a straight line, at a constant speed. But, as shown by Newton
himself, that principe remains true for the _relative motion_ of a free
particle with respect to some particular reference frames, the _Galilean
frames_.
More exactly, let us assume that the principle of inertia holds true for the
relative motion of free particles with respect to the reference frame defined
by the rigid body $R_{1}$. What happens for the relative motion of these free
particles with respect to another reference frame, defined by another rigid
body $R_{2}$? It is easy to see that the principle of inertia still holds true
_if and only if_ the relative motion of $R_{2}$ with respect to $R_{1}$ is a
motion by translation at a constant speed.
The absolute frame, if it exists, therefore appears as a Galilean frame among
an infinite number of other Gallilean frames, that no measurement founded on
mechanical properties can distinguish from the others. For this reason,
several scientists, following Leibniz, doubted about its existence.
Leibniz accepted Newton’s concept of an absolute Time, but not that of an
absolute Space. His views were not successful during his life, probably
because at that time nobody saw how to cast them in a mathematically rigorous
setting. Now we can do that; let me explain how.
We will consider that at each time $t\in{\cal T}$, there exists a _Space at
time $t$_, denoted by ${\cal E}_{t}$, whose properties are those of the three-
dimensional Euclidean space of geometers. We must consider that the Spaces
${\cal E}_{t_{1}}$ and ${\cal E}_{t_{2}}$, at two different times $t_{1}$ and
$t_{2}$, $t_{1}\neq t_{2}$, have no common element. Leibniz Space-Time, which
will be denoted by $\cal U$ (for Universe), is the disjoint union of all the
Spaces ${\cal E}_{t}$ for all times $t\in{\cal T}$. So, according to Leibniz
views, we still have a Space-Time, but no more an absolute space ! The next
picture shows,
* •
on the left side, Newton Space-Time ${\cal E}\times{\cal T}$, with the two
projections $p_{1}:{\cal E}\times{\cal T}\to{\cal E}$ and $p_{2}:{\cal
E}\times{\cal T}\to{\cal T}$;
* •
on the right side, Leibniz Space-Time $\cal U$, endowed with only one natural
projection onto absolute Time $\cal T$, still denoted by $p_{2}:{\cal
U}\to{\cal T}$; the horizontal lines represent the Spaces ${\cal
E}_{t}=p_{2}^{-1}(t)$, for various values of $t\in{\cal T}$.
Figure 2: Newton and Leibniz Space-Time.
— But how do you put together the Spaces at various times ${\cal E}_{t}$ to
make Leibniz Space-Time $\cal U$? Are they stacked in an arbitrary way?
— Of course no! Leibniz Space-Time $\cal U$ is a $4$-dimensional affine space,
fibered (via an affine map) over Time $\cal T$, which is itself a
$1$-dimensional affine space. Its fibres, the Spaces ${\cal E}_{t}$ at various
times $t\in{\cal T}$, are $3$-dimensional Euclidean spaces. The affine
structure of $\cal U$ is determined by the _principle of inertia_ of which we
have already spoken. That principle can be formulated in a way which does not
use reference frames, by saying:
_The world line of any free particle is a straight line._
So formulated, the principle of inertia can be applied to Newton Space-Time
${\cal E}\times{\cal T}$ and to Leibniz Space-Time $\cal U$ as well. More, it
_determines_ the affine structure of $\cal U$, since one can easily show that
the affine structure for which it holds true, if any, is unique. A physical
law, the _principle of inertia_ , is so embedded in the geometry of Leibniz
Space-Time $\cal U$.
By using a reference frame $R$, one can split Leibniz Space-Time into a
product of two factors: a space ${\cal E}_{R}$, fixed with respect to that
frame, and the absolute Time $\cal T$. But of course, the space ${\cal E}_{R}$
depends on the choice of the reference frame $R$. For that reason, it seems
that before 1905, not many scientists were aware of the fact that by dropping
Newton’s absolute Space $\cal E$, they already had completely changed the
conceptual setting in which motions are described:
* •
according to Newton, absolute Space ${\cal E}$ and absolute Time ${\cal T}$
were directly related to reality, while Space-Time ${\cal E}\times{\cal T}$
was no more than a mathematical object, not very interesting (he did not use
it) and not directly related to reality;
* •
but according to Leibniz’s views, when expressed as done above, it is Space-
Time $\cal U$ which is directly related to reality, as well as absolute Time
$\cal T$; absolute Space $\cal E$ no more exists.
## 3 Relativity
Einstein [1] was led to drop Leibniz Space-Time when trying to reconcile the
theories used in two different parts of Physics: Mechanics on one hand,
Electromagnetism and Optics on the other hand.
According to the theory built by the great Scotch physicist James Clerk
Maxwell (1831–1879), electromagnetic phenomena propagate in vacuum as waves,
with the same velocity in all directions, independently of the motion of the
source of these phenomena. Maxwell soon understood that light was an
elecromagnetic wave, and lots of experimental results confirmed his views.
### 3.1 The luminiferous ether, a short lived hypothesis
In Leibniz Space-Time (as well as in Newton Space-Time) _relative velocities
behave additively_. In that setting, it is with respect to _at most one
particular reference frame_ that light can propagate with the same velocity in
all directions. Physicists introduced a new hypothesis: electromagnetic waves
were considered as vibrations of an hypothetic, very subtle, but highly rigid
medium called the _luminiferous ether_ , everywhere present in space, even
inside solid bodies. They thought that it was with respect to the ether’s
reference frame that light propagates at the same velocity in all directions.
This new hypothesis amounts to come back to Newton’s absolute Space identified
with the ether. There were even physicists who introduced additional
complications, by assuming that the ether, partially drawn by the motion of
moving bodies, could deform with time!
— But if the luminiferous ether really exists, accurate measurements of the
velocity of light in all directions should allow the determination of the
Earth’s relative velocity with respect to the ether!
— Good remark! These measurements were made several times, notably by Albert
Abraham Michelson (1852–1931) and Edward Williams Morley (1838–1923), between
1880 et 1887. No relative velocity of the Earth with respect to the
luminiferous ether could be detected.
These results remained not understood until 1905, despite many attempts. The
most interesting of these attempts was that due to Hendrik Anton Lorentz
(1853–1928) and George Francis FitzGerald (1851–1901). Independently, they
proposed the following hypothesis: when a rigid body, for example a rule or
the arm of an interferometer, is moving with respect to the luminiferous
ether, that body contracts slightly in the direction of its relative
displacement.
— So that is the famous relativistic contraction my teacher spoke about!
— No! Not at all! Lorentz and FitzGerald considered that contraction as a true
physical effect of the relative motion of a body with respect to the ether.
This assumption is now completely abandoned, together with the luminiferous
ether! The relativistic contraction of lengths and dilation of times has
nothing to do with it: rather than a real phenomenon, it is only an
appearance, like the following effect of perspective. Imagine that you look at
a 20 centimeters rule, from a distance of, say two meters from its center.
That rule looks shorter when it is not perpendicular to the straight line
which joins your eye to its center than when it is. It may even seem to be
reduced to a point when it lies along that straight line. As we will soon see,
the relativistic contraction of lengths and dilation of times has a similar
origin.
### 3.2 Minkowski Space-Time
Einstein was the first 222 The great French mathematician Jules Henri Poincaré
(1854–1912) has, almost simultaneously and independently, presented very
similar ideas [3], without explicitly recommending to drop the concept of an
absolute Time. to understand (in 1905) that the results of Michelson and
Morley experiments could be explained by a deep change of the properties
ascribed to Space and Time. At that time, his idea appeared as truly
revolutionary. But now it may appear as rather natural, if we think along the
following lines:
_When we dropped Newton Space-Time in favour of Leibniz Space-Time, we
recognized that there is no absolute Space, but that Space depends on the
choice of a reference frame. Maybe Time too is no more absolute than Space,
and depends on the choice of a reference frame!_
— But if we drop absolute Time, which properties are left to our Space-Time?
— In 1905, Einstein implicitly considered that Space-Time still was a
$4$-dimensional affine space, which will be called _Minkowski Space-Time_ and
will be denoted by $\cal M$. He implicitly considered too that _translations_
of $\cal M$ leave its properties unchanged, and he assumed that the _principe
of inertia_ still holds true in $\cal M$ when expressed without the use of
reference frames:
_The world line of any free particle is a straight line._
He also kept the notion of a _Galilean frame_. In $\cal M$, a Galilean frame
is determined by a direction of straight line (not any straight line, a _time-
like_ straight line, as we will see below). Given a Galilean frame $R$,
Minkowski Space-Time $\cal M$ can be split into a product ${\cal
E}_{R}\times{\cal T}_{R}$ of a three-dimensional Space ${\cal E}_{R}$ and a
one-dimensional Time ${\cal T}_{R}$, which both depend on $R$. Let me recall
that in Leibniz Space-Time $\cal U$, a Galilean frame $R$ allowed us to split
$\cal U$ into a product ${\cal E}_{R}\times{\cal T}$ of a three-dimensional
Space ${\cal E}_{R}$, which depended on $R$, and the one-dimensional absolute
Time $\cal T$, which did not depend on $R$. That is the main difference
between Leibniz’s and Einstein’s views about Space and Time.
Under these hypotheses, the properties of Space-Time follow from two
principles:
* •
the _Principle of Relativity:_ all physical laws have the same expression in
all Galilean frames;
* •
the _Principle of Constancy of the velocity of light:_ the modulus of the
velocity of light is an universal constant, which depends neither on the
Galilean frame with respect to which it is calculated, nor on the motion of
the source of that light.
— You said that a direction of straight line was enough to determine a
Galilean frame. But how is that possible, since we no more have an absolute
Time?
— That determination will follow from the pinciple of constancy of the
velocity of light. Let us call _light lines_ the straight lines in $\cal M$
which are possible world lines of light signals. Given an event $A\in{\cal
M}$, the light lines through $A$ make a $3$-dimensional cone, the _light cone
with apex $A$_; the two layers of that cone are called _the past half-cone_
and _the future half-cone_ with apex $A$. Since it is assumed that
translations leave unchanged the properties of Space-Time, the light cone with
another event $B$ as apex is deduced from the light cone with apex $A$ by the
translation which maps $A$ onto $B$.
Apart from light lines, there are two other kinds of straight lines in $\cal
M$:
* •
_time-like straight lines_ , which lie _inside_ the light cone with any one of
their elements as apex;
* •
and _space-like straight lines_ , which lie _outside_ the light cone with any
of their element as apex.
I can now explain how the direction of a time-like straight line $\cal A$
determines a Galilean frame $R$. That frame is such that the rigid bodies at
rest in it are those whose all material points have, as world lines, straight
lines parallel to $\cal A$. The straight lines parallel to $\cal A$ will be
called the _isochorous lines_ 333 The word _isochorous_ , already used in
Thermodynamics, refers here to a set of events which all occur at the same
spatial location at various times, in similarity with the word _isochronous_
which refers to a set of events which all occur simultaneously in time at
various spatial locations. of the reference frame $R$; each of these lines is
a set of events which all happen at the same place in the Space ${\cal E}_{R}$
of our frame $R$. For each event $M\in{\cal M}$, the set of all other events
which occur at the same time as $M$, for the Time ${\cal T}_{R}$ of our
Galilean frame $R$, will be called the _isochronous subspace_ through $M$, for
the Galilean frame $R$. It is a $3$-dimensional affine subspace ${\cal
E}_{R,\,M}$ of $\cal M$ containing the event $M$, and the other isochronous
subspaces for $R$ are all the $3$-dimensional subspaces of $\cal M$ parallel
to ${\cal E}_{R,M}$. They are determined by the property: the length covered
by a light signal, calculated in the reference frame $R$, during a given time
interval, also evaluated in that reference frame, _is the same in any two
opposite directions._
In a schematic $2$-dimensional Space-Time (or in a plane section containing
$\cal A$ of the “true” $4$-dimensional Space-Time), the direction of
isochronous subspaces is easily obtained as shown on the left part of Figure
3: we take the two light lines ${\cal L}^{g}$ and ${\cal L}^{d}$ through an
event $A\in{\cal A}$ (the red lines on that figure); we take another event
$A_{1}\in A$, for example in the future of $A$, and we build the parallelogram
$A\,A_{1}^{g}\,A_{2}\,A_{1}^{d}$ with two sides supported by ${\cal L}^{g}$
and ${\cal L}^{d}$, with $A$ as one of its apices and $A_{1}$ as center. The
isochronous subspaces are all the straight lines parallel to the space-like
diagonal $A_{1}^{g}\,A_{1}^{d}$ of that parallelogram. Three of these lines
are drawn (in blue) on Figure 3, ${\cal E}_{R,\,A}$, ${\cal E}_{R,\,A_{1}}$
and ${\cal E}_{R,\,A_{2}}$.
— Why?
— A light signal starting from $A$ covers, during the time interval between
events $A$ and $A_{1}$, the lengths $A_{1}\,A_{1}^{g}$ towards the left and
$A_{1}\,A_{1}^{d}$ towards the right. These lengths are equal because
$A_{1}^{g}\,A_{1}^{d}$ is the diagonal of a parallelogram whose center is
$A_{1}$.
Figure 3: Construction of Space and Time relative to a Galilean frame
— What for the “true” $4$-dimensional Minkowski Space-Time $\cal M$ ? And what
are the Space ${\cal E}_{R}$ and the Time ${\cal T}_{R}$ of our reference
frame $R$?
— It is the same, as shown on the right side of Figure 3. Take the event
$A_{2}$ on the light line $\cal A$ such that $A_{1}$ is the middle point of
$A\,A_{2}$. Consider the future light half-cone with apex $A$ and the past
light half-cone with apex $A_{2}$. Their intersection is a $2$-dimensional
sphere $S$. The unique affine hyperplane ${\cal E}_{R,\,A_{1}}$ which contains
$S$ is an isochronous subspace for the Galilean frame determined by the
direction of $\cal A$ (in blue on Figure 3). The other isochronous subspaces
for that Galilean frame are all the hyperplanes parallel to ${\cal
E}_{R,\,A_{1}}$. The Space ${\cal E}_{R}$ is the set of all the isochorous
lines, _i.e_ the set of all straight lines parallel to $\cal A$, and the Time
${\cal T}_{R}$ the set of all isochronous subspaces. Minkowski Space-Time
$\cal M$ splits into the product ${\cal E}_{R}\times{\cal T}_{R}$, or in other
words can be identified with that product, because a pair made by an
isochorous line and an isochronous subspace determine a unique element of
$\cal M$, the event at which they meet.
— What happens if you change your Galilean frame?
— Of course, as for Galilean frames in Leibniz Space-Time, the direction of
isochorous lines (the straight world lines of points at rest with respect to
the chosen Galilean frame) is changed. Moreover, contrary to what happened in
Leibniz Space-Time, the direction of isochronous subspaces is also changed!
Therefore, the chronological order of two events can be different when it is
appreciated in two different Galilean frames!
### 3.3 Metric properties of Minkowski Space-Time
Up to now, we have compared the lengths of two straight line segments in $\cal
M$ only when they were supported by parallel straight lines. That was allowed
by the _affine structure_ of $\cal M$. We need more, because the spectral
lines of atoms allow us to build clocks and to compare time intervals measured
in two different Galilean frames.
Let $A\,A_{1}$ and $A\,B_{1}$ be two straight line segments supported by two
different time-like straight lines $\cal A$ and $\cal B$, which meet at the
event $A$. Let $R_{\cal A}$ and $R_{\cal B}$ be the Galilean frames determined
by the directions of $\cal A$ and $\cal B$, respectively. We assume that the
time intervals corresponding to $A\,A_{1}$ measured in $R_{\cal A}$, and to
$A\,B_{1}$ measured in $R_{\cal B}$, are the same. Let $B^{\prime}$ be the
event at which the time-like straight line $\cal B$ meets the isochronous
subspace ${\cal E}_{R_{\cal A},A_{1}}$ containing $A_{1}$ of the Galilean
frame $R_{\cal A}$ (figure 4). Since the events $A_{1}$ and $B^{\prime}$ are
synchronous for $R_{\cal A}$, the time interval corresponding to $A\,B_{1}$
appears longer than the time interval corresponding to $A\,A_{1}$ when both
are observed in the reference frame $R_{\cal A}$, by the ratio
$\displaystyle\frac{A\,B_{1}}{A\,B^{\prime}}$. That ratio is the _ratio of
dilation of times_ of the Galilean frame of $R_{\cal B}$, when observed in the
Galilean frame $R_{\cal A}$. Similarly,
$\displaystyle\frac{A\,A_{1}}{A\,A^{\prime}}$ is the ratio of dilation of
times of the Galilean frame $R_{\cal A}$ when observed in the Galilean frame
$R_{\cal B}$. According to the Principle of Relativity, these two Galilean
frames must play the same role with respect to the other, which implies the
equality
$\displaystyle\frac{A\,A_{1}}{A\,A^{\prime}}=\frac{A\,B_{1}}{A\,B^{\prime}}$.
By a well known property of hyperbolae, that equality holds _if and only if
$A_{1}$ and $B_{1}$ lie on the same arc of hyperbola which has the light lines
${\cal L}^{d}$ and ${\cal L}^{g}$ (which meet at $A$ and are contained in the
two-dimensional plane which contains $\cal A$ and $\cal B$) as asymptotes_. Or
more generally, on the same hyperboloid with the light cone of $A$ as
asymptotic cone.
Figure 4: Comparison of times.
The comparison of lengths on two non-parallel space-like straight lines is
similar to the comparison of time intervals. Let $A\,A^{d}$ and $A\,B^{d}$ be
two segments supported by two space-like straight lines which meet at the
event $A$. They are of equal length _if and only if $A^{d}$ and $B^{d}$ lie on
the same hyperboloid with the light cone of $A$ as asymptotic cone_.
## 4 Conclusion
The comparison of time intervals and lengths presented above allows a very
natural introduction of the pseudo-Euclidean metric of Minkowski Space-Time.
The construction of isochronous subspaces in two different Galilean frames, as
presented above, leads to the formulas for Lorentz transformations with a
minimum of calculations. The pictures we have presented allow a very easy
explanation of the apparent contraction of lengths and dilation of times
associated to a change of Galilean frames and a very simple explanation,
without complicated calculations, of the (improperly called) paradox of
Langevin’s twins.
By explaining that the affine structure of Space-Time should be questioned, a
smooth transition towards General Relativity, suitable from children from 8 to
108 years old, seems possible.
Acknowledgements. The author thanks the team “Analyse algébrique” of the
“Institut de Mathématiques de Jussieu” and his University for taking in charge
his registration fee at this International Conference.
## References
* [1] Einstein, A., Lorentz, H.A., Weyl, H., Minkowski, H., The Principles of Relativity, a collection of original papers on the special and general theory of relativity, with notes by A. Sommerfeld. Methuen and Company, 1923. Reprinted by Dover Publications, Inc., New York.
* [2] Newton, Isaac, Principes mathématiques de la Philosophie naturelle, tomes I et II, translated by Madame la Marquise du Chastellet, chez Desaint et Saillant, Paris, 1759. Reprinted by the Éditions Jacques Gabay, Paris, 1990.
* [3] Poincaré, Henri, La Mécanique nouvelle, book containing the text of a lecture presented at the congress of the “Association française pour l’avancement des sciences” (Lille, 1909), the paper dated 23 July 1905 Sur la dynamique de l’électron, Rendiconti del Circolo matematico di Palermo XXI (1906), and a “Note aux Comptes Rendus de l’Académie des Sciences” with the same title dated 15 June 1905; Gauthier-Villars, Paris, 1924; reprinted by the Éditions Jacques Gabay, Paris, 1989.
|
arxiv-papers
| 2014-02-01T20:05:14 |
2024-09-04T02:49:58.020653
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Charles-Michel Marle (IMJ)",
"submitter": "Charles-Michel Marle",
"url": "https://arxiv.org/abs/1402.2185"
}
|
1402.2300
|
# Feature and Variable Selection in Classification
Aaron Karper
###### Abstract
The amount of information in the form of features and variables available to
machine learning algorithms is ever increasing. This can lead to classifiers
that are prone to overfitting in high dimensions, high dimensional models do
not lend themselves to interpretable results, and the CPU and memory resources
necessary to run on high-dimensional datasets severly limit the applications
of the approaches.
Variable and feature selection aim to remedy this by finding a subset of
features that in some way captures the information provided best.
In this paper we present the general methodology and highlight some specific
approaches.
## 1 Introduction
As machine learning as a field develops, it becomes clear that the issue of
finding good features is often more difficult than the task of using the
features to create a classification model. Often more features are available
than can reasonably be expected to be used, because using too many features
can lead to overfitting, hinders the interpretability, and is computationally
expensive.
### 1.1 Overfitting
One of the reasons why more features can actually hinder accuracy is that the
more features we have, the less can we depend on measures of distance that
many classifiers (e.g. SVM, linear regression, k-means, gaussian mixture
models, …) require. This is known as the curse of dimensionality.
Accuracy might also be lost, because we are prone to overfit the model if it
incorporates all the features.
###### Example.
In a study of genetic cause of cancer, we might end up with 15 participants
with cancer and 15 without. Each participant has 21’000 gene expressions. If
we assume that any number of genes in combination can cause cancer, even if we
underestimate the number of possible genomes by assuming the expressions to be
binary, we end with $2^{21^{\prime}000}$ possible models.
In this huge number of possible models, there is bound to be one arbitrarily
complex that fits the observation perfectly, but has little to no predictive
power [Russell et al., 1995, Chapter 18, Noise and Overfitting]. Would we in
some way limit the complexity of the model we fit, for example by discarding
nearly all possible variables, we would attain better generalisation.
### 1.2 Interpretability
If we take a classification task and want to gain some information from the
trained model, model complexity can hinder any insights. If we take up the
gene example, a small model might actually show what proteins (produced by the
culprit genes) cause the cancer and this might lead to a treatment.
### 1.3 Computational complexity
Often the solution to a problem needs to fulfil certain time constraints. If a
robot takes more than a second to classify a ball flying at it, it will not be
able to catch it. If the problem is of a lower dimensionality, the
computational complexity goes dows as well.
Sometimes this is only relevant for the prediciton phase of the learner, but
if the training is too complex, it might become infeasible.
### 1.4 Previous work
This article is based on the work of [Guyon and Elisseeff, 2003], which gives
a broad introduction to feature selection and creation, but as ten years
passed, the state-of-the-art moved on.
The relevance of feature selection can be seen in [Zhou et al., 2005], where
gene mutations of cancer patients are analysed and feature selection is used
to conclude the mutations responsible.
In [Torresani et al., 2008], the manifold of human poses is modelled using a
dimensionality reduction technique, which will presented here in short.
Kevin Murphy gives an overview of modern techniques and their justification in
[Murphy, 2012, p. 86ff]
### 1.5 Structure
In this paper we will first discuss the conclusions of Guyon and Elisseeff
about the general approaches taken in feature selection in section 2, discuss
the creation of new features in section 3, and the ways to validate the model
in section 4. Then we will continue by showing some more recent developments
in the field in section 5.
## 2 Classes of methods
In [Guyon and Elisseeff, 2003], the authors identify four approaches to
feature selection, each of which with its own strengths and weaknesses:
Ranking
orders the features according to some score.
Filters
build a feature set according to some heuristic.
Wrappers
build a feature set according to the predictive power of the classifier
Embedded methods
learn the classification model and the feature selection at the same time.
If the task is to predict as accurately as possible, an algorithm that has a
safeguard against overfitting might be better than ranking. If a pipeline
scenario is considered, something that treats the following phases as blackbox
would be more useful. If even the time to reduce the dimensionality is
valuable, a ranking would help.
### 2.1 Ranking
Variables get a score in the ranking approach and the top $n$ variables are
selected. This has the advantage that $n$ is simple to control and that the
selection runs in linear time.
###### Example.
In [Zhou et al., 2005], the authors try to find a discriminative subset of
genes to find out whether a tumor is malignant or benign111Acutally they
classify the tumors into 3 to 5 classes.. In order to prune the feature base,
they rank the variables according to the correlation to the classes and make a
preliminary selection, which discards most of the genes in order to speed up
the more sophisticated procedures to select the top 10 features.
There is an inherrent problem with this approach however, called the xor
problem[Russell et al., 1995]:
Figure 1: A ranking procedure would find that both features are equally
useless to separate the data and would discard them. If taken together however
the feature would separate the data very well.
It implicitly assumes that the features are uncorrelated and gives poor
results if they are not. On figure 1, we have two variables $X$ and $Y$, with
the ground truth roughly being $Z=X>5\;\mathtt{xor}\;Y>5$. Each variable taken
separately gives absolutely no information, if both variables were selected
however, it would be a perfectly discriminant feature. Since each on its own
is useless, they would not rank high and would probably be discarded by the
ranking procedure, as seen in figure 1.
###### Example.
Take as an example two genes $X$ and $Y$, so that if one is mutated the tumor
is malignant, which we denote by $M$, but if both mutate, the changes cancel
each other out, so that no tumor grows. Each variable separately would be
useless, because $P(M=true|X=true)=P(Y=false)$, but
$P(M=true|X=true,Y=false)=1$ –
### 2.2 Filters
While ranking approaches ignore the value that a variable can have in
connection with another, filters select a subset of features according to some
determined criterion. This criterion is independent of the classifier that is
used after the filtering step. On one hand this allows to only train the
following classifier once, which again might be more cost-effective. On the
other hand it also means that only some heuristics are available of how well
the classifier will do afterwards.
Filtering methods typically try to reduce in-class variance and to boost
inter-class distance. An example of this approach is a filter that would
maximize the correlation between the variable set and the classification, but
minimize the correlation between the variables themselves. This is under the
heuristic, that variables, that correlate with each other don’t provide much
additional information compared to just taking one of them, which is not
necessarily the case, as can be seen on figure 2: If the variable is noisy, a
second, correlated variable can be used to get a better signal, as can be seen
in figure 2.
Figure 2: Features might be identically distributed, but using both can reduce
variance and thus confusion by a factor of $\sqrt{n}$
A problem with the filtering approach is that the performance of the
classifier might not depend as much as we would hope on the proxy measure that
we used to find the subset. In this scenario it might be better to assess the
accuracy of the classifier itself.
### 2.3 Wrappers
Wrappers allow to look at the classifier as a blackbox and therefore break the
pipeline metaphor. They optimize some performance measure of the classifier as
the objective function. While this gives superior results to the heuristics of
filters, it also costs in computation time, since a classifier needs to be
trained each time – though shortcuts might be available depending on the
classifier trained.
Wrappers are in large search procedures through feature subset space – the
atomic _movements_ are to add or to remove a certain feature. This means that
many combinatorical optimization procedures can be applied, such as simulated
annealing, branch-and-bound, etc. Since the subset space is $2^{N}$, for $N$
the number of features, it is not feasible to perform an exhaustive search,
therefore greedy methods are applied: The start can either be the full feature
set, where we try to reduce the number of features in an optimal way
(_backward elimination_) or we can start with no features and add them in a
smart way (_forward selection_). It is also possible to replace the least
predictive feature from the set and replace it with the most predictive
feature from the features that were not chosen in this iteration.
### 2.4 Embedded
Wrappers treated classifiers as a black box, therefore a combinatorical
optimization was necessary with a training in each step of the search. If the
classifier allows feature selection as a part of the learning step, the
learning needs to be done only once and often more efficiently.
A simple way that allows this is to optimize in the classifier not only for
the likelihood of the data, but instead for the posterior probability (MAP)
for some prior on the model, that makes less complex models more probable. An
example for this can be found in section 5.2.
Somewhat similar is SVM with a $\ell_{1}$ weigth constraint
222$\ell_{p}(\mathbf{w})=\|\mathbf{w}\|_{p}=\sqrt[p]{\sum|w_{i}|^{p}}$ . The
1:1 exchange means that non-discriminative variables will end up with a 0
weight. It is also possible to take this a step further by optimizing for the
number of variables directly, since $l_{0}(w)=\lim_{p\rightarrow 0}l_{p}(w)$
is exactly the number of non-zero variables in the vector.
## 3 Feature creation
In the previous chapter the distinction between variables and features was not
necessary, since both could be used as input to the classifier after feature
selection. In this section _features_ is the vector offered to the classifier
and _variables_ is the vector handed to the feature creation step, i.e. the
_raw inputs_ collected. For much the same reasons that motivated feature
selection, feature creation for a smaller number of features compared to the
number of variables provided.
Essentially the information needs to be compressed in some way to be stored in
fewer variables. Formally this can be expressed by mapping the high-
dimensional space through the bottleneck, which we hope results in recovering
the low dimensional concepts that created the high-dimensional representation
in the first place. In any case it means that typical features are created,
with a similar intuition to efficient codes in compression: If a simple
feature occurs often, giving it a representation will reduce the loss more
than representing a less common feature. In fact, compression algorithms can
be seen as a kind of feature creation[Argyriou et al., 2008].
This is also related to the idea of manifold learning: While the variable
space is big, the actual space in that the variables vary is much smaller – a
manifold333A manifold is the mathematical generalization of a surface or a
curve in 3D space: Something smooth that can be mapped from a lower
dimensional space. of hidden variables embedded in the variable space.
###### Example.
In [Torresani et al., 2008] the human body is modelled as a low dimensional
model by _probabilistic principal component analyis_ : It is assumed that the
hidden variables are distributed as Gaussians in a low dimensional space that
are then linearly mapped to the high dimensional space of positions of pixels
in an image. This allows them to learn _typical_ positions that a human body
can be in and with that track body shapes in 3d even if neither the camera,
nor the human are fixed.
## 4 Validation methods
The goal up to this point was to find a simple model, that performs well on
our training set, but we hope that our model will perform well in data, it has
never seen before: minimizing the _generalization error_. This section is
concerned with estimating this error.
A typical approach is _cross-validation_ : If we have independent and
identically distributed datapoint, we can split the data and train the model
on one part and measure its performance on the rest. But even if we assume
that the data is identically distributed, it requires very careful curation of
the data to achieve independence:
###### Example.
Assume that we take a corpus of historical books and segment them. We could
now cross-validate over all pixels, but this would be anything but
independent. If we are able to train our model on half the pixels of a page
and check against the other half, we would naturally perform quite well, since
we are actually able to learn the style of the page. If we split page-wise, we
can learn the specific characteristics of the author. Only if we split author-
wise, we might hope to have a resemblence of independence.
Another approach is _probing_ : instead of modifying the data set and
comparing to other data, we can modify the _feature space_. We add random
variables, that have no predictive power to the feature set. Now we can
measure how well models fare against pure chance444 This can take the form of
a significance test.. Our performance measure is then the signal-to-noise
ratio of our model.
## 5 Current examples
### 5.1 Nested subset methods
In the nested subset methods the feature subset space is greedily examined by
estimating the expected gain of adding one feature in forward selection or the
expected loss of removing one feature in backward selection. This estimation
is called the _objective function_. If it is possible to examine the objective
function for a classifier directly, a better performance is gained by
embedding the search procedure with it. If that is not possible, training and
evaluating the classifier is necessary in each step.
###### Example.
Consider a model of a linear predictor $p(\mathbf{y}|\mathbf{x})$ with $M$
input variables needing to be pruned to $N$ input variables. This can be
modeled by asserting that the real variables $x_{i}^{\star}$ are taken from
$\mathbb{R}^{N}$, but a linear transformation $A\in\mathbb{R}^{N\times M}$ and
a noise term $n_{i}=\mathcal{N}(0,\sigma_{x}^{2})$ is added:
$\mathbf{x_{i}}=A\mathbf{x^{\star}_{i}}+n_{i}$
In a classification task555The optimization a free interpretation of [Guo,
2008], we can model
$y=\mathop{\mathrm{Ber}}\nolimits(\mathop{\mathrm{sigm}}\nolimits(\mathbf{w}\cdot\mathbf{x}^{\star}))$.
This can be seen as a generalisation of PCA666Principal component analysis
reduces the dimensions of the input variables by taking only the directions of
the largest eigenvalues. to the case where the output variable is taken into
account ([West, 2003] and [Bair et al., 2006] develop the idea). Standard
supervised PCA assumes that the output is distributed as a gaussian
distribution, which is a dangerous simplification in the classification
setting[Guo, 2008].
The procedure iterates over the eigenvectors of the natural parameters of the
joint distribution of the input and the output and adds them if they show an
improvement to the current model in order to capture the influence of the
input to the output optimally. If more than $N$ variables are in the set, the
one with the least favorable score is dropped. The algorithms iterates some
fixed number of times over all features, so that hopefully the globally
optimal feature subset is found.
### 5.2 Logistic regression using model complexity regularisation
In the paper _Gene selection using logistic regressions based on AIC, BIC and
MDL criteria_[Zhou et al., 2005] by Zhou, Wang, and Dougherty, the authors
describe the problem of classifying the gene expressions that determine
whether a tumor is part of a certain class (think malign versus benign). Since
the feature vectors are huge ($\approx 21^{\prime}000$ genes/dimensions in
many expressions) and therefore the chance of overfitting is high and the
domain requires an interpretable result, they discuss feature selection.
For this, they choose an embedded method, namely a normalized form of logistic
regression, which we will describe in detail here:
Logistic regression can be understood as fitting
$p_{\mathbf{w}}(\mathbf{x})=\frac{1}{1+e^{\mathbf{w}\cdot\mathbf{x}}}=\mathop{\mathrm{sigm}}\nolimits(\mathbf{w}\cdot\mathbf{x})$
with regard to the separation direction $\mathbf{w}$, so that the confidence
or in other words the probability $p_{\mathbf{w}^{\star}}(\mathbf{x}_{data})$
is maximal.
This corresponds to the assumption, that the probability of each class is
$p(c|\mathbf{w})=\mathop{\mathrm{Ber}}\nolimits(c|\mathop{\mathrm{sigm}}\nolimits(\mathbf{w}\cdot\mathbf{x}))$
and can easily be extended to incorporate some prior on $\mathbf{w}$,
$p(c|\mathbf{w})=\mathop{\mathrm{Ber}}\nolimits(c|\mathop{\mathrm{sigm}}\nolimits(\mathbf{w}\cdot\mathbf{x}))\,p(w)$
[Murphy, 2012, p. 245].
The paper discusses the priors of the Akaike information criterion (AIC), the
Bayesian information criterion (BIC) and the minimum descriptor length (MDL):
#### AIC
The Akaike information criterion penaltizes degrees of freedom, the $\ell_{0}$
norm, so that the function optimized in the model is $\log
L(\mathbf{w})-\ell_{0}(\mathbf{w})$. This corresponds to an exponential
distribution for $p(\mathbf{w})\propto\exp(-\ell_{0}(\mathbf{w}))$. This can
be interpreted as minimizing the variance of the models, since the variance
grows exponentially in the number of parameters.
#### BIC
The Bayesian information criterion is similar, but takes the number of
datapoints $N$ into account: $p(\mathbf{w})\propto
N^{-\frac{\ell_{0}(\mathbf{w})}{2}}$. This has an intuitive interpretation if
we assume that a variable is either ignored, in which case the specific value
does not matter, or taken into account, in which case the value influences the
model. If we assume a uniform distribution on all such models, the ones that
ignore become more probable, because they accumulate the probability weight of
all possible values.
#### MDL
The minimum descriptor length is related to the algorithmic probability and
states that the space necessary to store the descriptor gives the best
heuristic on how complex the model is. This only implicitly causes variable
selection. The approximation for this value can be seen in the paper itself.
Since the fitting is computationally expensive, the authors start with a
simple ranking on the variables to discard all but the best 5’000. They then
repeatedly fit the respective models and collect the number of appearances of
the variables to rank the best 5, 10, or 15 genes. This step can be seen as an
additional ranking step, but this seems unnecessary, since the fitted model by
construction would already have selected the best model. Even so they still
manage to avoid overfitting and finding a viable subset of discriminative
variables.
### 5.3 Autoencoders as feature creation
Autoencoders are deep neural networks777Deep means multiple hidden layers.
that find a fitting information bottleneck (see 3) by optimizing for the
reconstruction of the signal using the _inverse_ transformation888A truely
inverse transformation is of course not possible..
Figure 3: An autoencoder network
Deep networks are difficult to train, since they show many local minima, many
of which show poor performance [Murphy, 2012, p. 1000]. To get around this,
Hinton and Salakhutdinov [Hinton and Salakhutdinov, 2006] propose pretraining
the model as stacked restricted Bolzmann machines before devising a global
optimisation like stochastical gradient descent.
Restricted Bolzmann machines are easy to train and can be understood as
learning a probability distribution of the layer below. Stacking them means
extracting probable distributions of features, somewhat similar to a
distribution of histograms as for example HoG or SIFT being representative to
the visual form of an object.
It has long been speculated that only low-level features could be captured by
such a setup, but [Le et al., 2011] show that, given enough resources, an
autoencoder can learn high level concepts like recognizing a cat face without
any supervision on a 1 billion image training set.
The impressive result beats the state of the art in supervised learning by
adding a simple logistic regression on top of the bottleneck layer. This
implies that the features learned by the network capture the concepts present
in the image better than SIFT visual bag of words or other human created
features and that it can learn a variety of concepts in parallel. Further
since the result of the single best neuron is already very discriminative, it
gives evidence for the possibility of a _grandmother neuron_ in the human
brain – a neuron that recognizes exactly one object, in this case the
grandmother. Using this single feature would also take feature selection to
the extreme, but without the benefit of being more computationally
advantageous.
### 5.4 Segmentation in Computer Vision
A domain that necessarily deals with a huge number of dimensions is computer
vision. Even only considering VGA images, in which only the actual pixel
values are taken into account gives $480\times 640=307^{\prime}200$ datapoints
per image.
For a segmentation task in document analysis, where pixels need to be
classified into regions like border, image, and text, there is more to be
taken into account than just the raw pixel values in order to incorporate
spartial information, edges, etc. With up to 200 heterogeneous features to
consider for each pixel, the evaluation would take too long to be useful.
This section differs from the previous two in that instead of reviewing a
ready made solution to a problem, it shows the process of producing such a
solution.
The first thing to consider is whether or not we have a strong prior of how
many features are useful. In the example of cancer detection, it was known
that only a small number of mutation caused the tumor, so a model with a
hundred genes could easily be discarded. Unfortunately this is not the case
for segmentation, because our features don’t have a causal connection to the
true segmentation. Finding good features for segmentation requires finding a
good proxy feature set for the true segmentation.
Next we might consider the loss of missclassification: In a computer vision
task, pixel missclassifications are to be expected and can be smoothed over.
Computational complexity however can severely limit the possible applications
of an algorithm. As [Russell et al., 1995] note, using a bigger dataset can be
more advantageous than using the best algorithm, so we would favour an
efficient procedure over a very accurate one, because it would allow us to
train on a bigger training set. Since the variables are likely to be
correlated, ranking will give bad results.
Taking this into account, we would consider $L_{1}$ normalized linear
classifiers, because of the fast classification and training (the latter due
to [Yuan et al., 2010], in which linear time training methods are compared).
Taking linear regression could additionally be advanageous, since its _soft_
classification would allow for better joining of continuous areas of the
document.
## 6 Discussion and outlook
Many of the concepts presented in [Guyon and Elisseeff, 2003] still apply,
however the examples fall short on statistical justification. Since then
applications for variable and feature selection and feature creation were
developed, some of which were driven by advances in computing power, such as
high-level feature extraction with autoencoders, others were motivated by
integrating prior assumptions about the sparcity of the model, such as the
usage of probabilistic principal component analysis for shape reconstruction.
The goals of variable and feature selection – avoiding overfitting,
interpretability, and computational efficiency – are in our opinion problems
best tackled by integrating them into the models learned by the classifier and
we expect the embedded approach to be best fit to ensure an optimal treatment
of them. Since many popular and efficient classifiers, such as support vector
machines, linear regression, and neural networks, can be extended to
incorporate such constraints with relative ease, we expect the usage of
ranking, filtering, and wrapping to be more of a pragmatic first step, before
sophisticated learners for sparse models are employed. Advances in embedded
approaches will make the performance and accuracy advantages stand out even
more.
Feature creation too has seen advances, especially in efficient
generalisations of the principal component analysis algorithm, such as kernel
PCA (1998) and supervised extensions. They predominantly rely on the bayesian
formulation of the PCA problem and we expect this to drive more innovation in
the field, as can be seen by the spin-off of reconstructing a shape from 2D
images using a bayesian network as discussed in [Torresani et al., 2008].
## References
* [Argyriou et al., 2008] Argyriou, A., Evgeniou, T., and Pontil, M. (2008). Convex multi-task feature learning. Machine Learning, 73(3):243–272.
* [Bair et al., 2006] Bair, E., Hastie, T., Paul, D., and Tibshirani, R. (2006). Prediction by supervised principal components. Journal of the American Statistical Association, 101(473).
* [Guo, 2008] Guo, Y. (2008). Supervised exponential family principal component analysis via convex optimization. In Advances in Neural Information Processing Systems, pages 569–576.
* [Guyon and Elisseeff, 2003] Guyon, I. and Elisseeff, A. (2003). An introduction to variable and feature selection. The Journal of Machine Learning Research, 3:1157–1182.
* [Hinton and Salakhutdinov, 2006] Hinton, G. E. and Salakhutdinov, R. R. (2006). Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507.
* [Le et al., 2011] Le, Q. V., Ranzato, M., Monga, R., Devin, M., Chen, K., Corrado, G. S., Dean, J., and Ng, A. Y. (2011). Building high-level features using large scale unsupervised learning. arXiv preprint arXiv:1112.6209.
* [Murphy, 2012] Murphy, K. P. (2012). Machine learning: a probabilistic perspective. The MIT Press.
* [Russell et al., 1995] Russell, S. J., Norvig, P., Canny, J. F., Malik, J. M., and Edwards, D. D. (1995). Artificial intelligence: a modern approach, volume 74. Prentice hall Englewood Cliffs.
* [Torresani et al., 2008] Torresani, L., Hertzmann, A., and Bregler, C. (2008). Nonrigid structure-from-motion: Estimating shape and motion with hierarchical priors. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 30(5):878–892.
* [West, 2003] West, M. (2003). Bayesian factor regression models in the ”large p, small n” paradigm. Bayesian statistics, 7(2003):723–732.
* [Yuan et al., 2010] Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., and Lin, C.-J. (2010). A comparison of optimization methods and software for large-scale l1-regularized linear classification. The Journal of Machine Learning Research, 9999:3183–3234.
* [Zhang et al., 2011] Zhang, X., Yu, Y., White, M., Huang, R., and Schuurmans, D. (2011). Convex sparse coding, subspace learning, and semi-supervised extensions. In AAAI.
* [Zhou et al., 2005] Zhou, X., Wang, X., and Dougherty, E. R. (2005). Gene selection using logistic regressions based on aic, bic and mdl criteria. New Mathematics and Natural Computation, 1(01):129–145.
|
arxiv-papers
| 2014-02-10T21:05:58 |
2024-09-04T02:49:58.031650
|
{
"license": "Public Domain",
"authors": "Aaron Karper",
"submitter": "Aaron Karper",
"url": "https://arxiv.org/abs/1402.2300"
}
|
1402.2415
|
arxiv-papers
| 2014-02-11T10:01:15 |
2024-09-04T02:49:58.041228
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pradeep Singla, Devraj Gautam",
"submitter": "Pradeep Singla",
"url": "https://arxiv.org/abs/1402.2415"
}
|
|
1402.2431
|
ITEP-TH- 02/14
RG limit cycles
K.Bulycheva$\,{}^{a}$, A.Gorsky$\,{}^{a,b}$
a Institute of Theoretical and Experimental Physics, Moscow 117218, Russia
b Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia
[email protected]
[email protected]
Contribution to the ”Pomeranchuk-100” Volume
Abstract
In this review we consider the concept of limit cycles in the renormalization
group flows. The examples of this phenomena in the quantum mechanics and field
theory will be presented.
## 1 Generalities
It is usually assumed that the RG flow connects fixed points, starting at a UV
repelling point and terminating at a IR attracting point. However it turned
out that this open RG trajectory does not exhaust all possibilities and the
clear-cut quantum mechanical example of the nontrivial RG limit cycle has been
found in [1] confirming the earlier expectations. This example triggered the
search for patterns of this phenomena which was quite successful. They have
been identified both in the systems with finite number of degrees of freedom
[2, 3, 4, 5] and in the field theory framework [6, 7, 8]. Now the cyclic RG
takes its prominent place in the world of RG phenomena however the subject
certainly deserves much more study.
The appearance of critical points corresponds to phase transitions of the
second kind, hence there exists a natural question concerning the connection
of RG cycles with phase transitions. The very phenomenon of the cyclic RG flow
has been interpreted in the important paper [7] as a kind of generalization of
the BKT phase transitions in two dimensions. One can start from a usual
example of an RG flow connecting UV and IR fixed points and then consider a
motion in a parameter space which results in a merging of the fixed points.In
[7] it was argued that when the parameter goes into the complex region the
cyclic behaviour of the RG flow gets manifested and a gap in the spectrum
arises. This happens similar to the BKT transition case when a deconfinement
of vortices occurs at the critical temperature and the conformal symmetry is
restored at lower temperatures. The appearance of the RG cycles can be also
interpreted as the peculiar anomaly in the classical conformal group [9]. This
anomaly has the origin in some ”falling to the center” UV phenomena which
could have quite different reincarnations. We would like to emphasize one more
generic feature of the phenomena — the cyclic RG usually occurs in the system
with at least two couplings. One of them undergoes the RG cyclic flow while
the second determines the period of the cycle.
The collision of the UV and IR fixed points can be illustrated in a quite
general manner as follows. Assume that there are two couplings $(\alpha,g)$ in
the theory and we focus at the renormalization of the coupling which enjoys
the following $\beta$-function
$\beta_{g}=(\alpha-\alpha_{0})-(g-g_{0})^{2},$ (1)
which vanishes at the hypersurface in the parameter space
$g=g_{0}\pm\sqrt{\alpha-\alpha_{0}}.$ (2)
It was argued in [7] that the collision of two roots at $\alpha=\alpha_{0}$
can be interpreted as the collision of UV and IR fixed points. Upon the
collision the points move into the complex $g$ plane and an RG cycle emerges.
The period of the cycle can be immediately estimated as
$T\propto\int_{g_{UV}}^{g_{IR}}\frac{dg}{\beta(\alpha;g)}\propto\frac{1}{\sqrt{\alpha-\alpha_{0}}}.$
(3)
The phenomena is believed to be generic once the beta–function has the form
(1). Note that is was shown that the RG cycles are consistent with the
c-theorem [10].
Breaking of the conformal symmetry results in the generation of the mass scale
which has non-perturbative nature. Due to the RG cycles the scale is not
unique and the whole tower with the Efimov-like scaling gets manifested
$E_{n+1}=\lambda E_{n},$ (4)
where $\lambda$ is fixed by the period of RG cycle.
In the examples available we could attempt to trace the physical picture
behind. It turns out that the origin of two couplings is quite general. One
coupling does not break the conformal symmetry which is exact in some subspace
of the parameter space. The second coupling plays the role of UV
regularization which can be imposed in one or another manner. It breaks the
conformal symmetry however some discrete version of the scale symmetry
survives which is manifested in the cycle structure. The UV regularization
will have different reincarnations in the examples considered: the account of
the finite size of the nuclei, contact interaction in the model of
superconductivity or the brane splitting in the supersymmetric models.
Historically the first example of this phenomena has been found long time ago
by Efimov [11] in the context of nuclear physics. He considered the three-body
system when two particles are near threshold and have attractive potential
with the third particle. It was shown that two–particle bound states are
absent in the spectrum, but there is a tower of the three–particle bound
states with the geometrical scaling corresponding to $\lambda\approx 22,7$.
The review of the RG interpretation of the Efimov phenomena can be found in
[12].
When considering the system with finite number of degrees of freedom the
meaning of the RG flows has to be clarified. To this aim some UV cutoff should
be introduced. In the first example in [1] the step of the RG corresponds to
the integrating out the highest energy level taking into account its
correlation with the rest of the spectrum. This approach has a lot in common
with the renormalization procedure in the matrix models considered in [13].
The same UV cutoff for formulation of RG procedure has been used for the
Russian Doll (RD) model describing the restricted BCS model of
superconductivity [14]. In that case the coupling providing the Cooper pairing
undergoes the RG cycle while the CP-violating parameter defines the period.
In the second class of examples the UV cutoff is introduced not at high energy
scale but at small distances. The RG cycles have been found in the non-
relativistic Calogero-like models with $\frac{1}{r^{2}}$ potential which
enjoys naive conformal symmetry [3, 4, 5]. The RG flow is formulated in terms
of the short distance regularization of the model. It is assumed that the wave
function with $E=0$ at large $r$ does not depend on the UV cutoff at small
$r$. This condition yields the equation for the parameter of a cutoff in the
regularization potential. This equation has multiple solutions which can be
interpreted as the manifestation of the tower of shallow bound states with the
Efimov scaling in the regularized Calogero model with attraction. The scaling
factor in the tower is determined by the Calogero coupling constant which
reflects the remnant of the conformal group upon the regularization.
The list of the field theory examples in different dimensions with the cyclic
RG flows is short but quite representative. In two dimensions the explicit
example with the RG cycle has been found in some range of parameters in the
sin-Gordon model. The cycle manifests itself in the pole structure of the
$S$-matrix. Efimov-like tower of states corresponds to the specific poles with
the Regge-like behavior of the resonance masses [6]
$m_{n}=m_{s}e^{\frac{n\pi}{h}},$ (5)
where $h$ is a certain parameter of the model. Moreover it was argued that the
$S$-matrix behaves universally under the cyclic RG flows. The tower of Efimov
states scales in the same manner as in the quantum mechanical case.
The origination of the cyclic RG behavior in the sine–Gordon model is not
surprising. Indeed it was argued in [7] that the famous
Berezinsky–Kosterlitz–Thouless (BKT) phase transition in XY system belongs to
this universality class. On the other hand one can map the XY system at the
$T$ temperature into the sin-Gordon theory with the parameters:
$L_{SG}=T(\partial\phi)^{2}-4z\cos\phi,$ (6)
and look at the renormalization of the interaction coupling. The
$\beta$–functions read as
$\beta_{u}=-2v^{2},\qquad\beta_{v}=-2uv,$ (7)
where
$u=1-\frac{1}{8\pi T},\qquad v=\frac{2z}{T\Lambda^{2}},$ (8)
and $\Lambda$ is the UV cutoff introduced to regularize the vortex core. The
form of $\beta$ functions implies the existence of the limit cycle with the
following expression for the correlation length:
$\xi_{BKT}\Lambda\propto\exp\left(\frac{c}{\sqrt{|T-T_{c}|}}\right),$ (9)
above the phase transition. This RG behavior gets mapped into the RG cycle in
the sine–Gordon model.
The example of the Efimov tower in 2+1 dimensions has been found in [15] in
the holographic representation. The model is based on the $D3-D5$ brane
configuration and corresponds to the large $N$ 3d gauge theory with
fundamentals enjoying $\mathcal{N}=4$ supersymmetry. In addition the magnetic
field and the finite density of conserved charge are present. At strong
coupling the gauge theory is described in terms of the probe $N_{f}$ flavor
branes in the nontrivial $AdS_{5}\times S^{5}$ geometry when the $U(1)$ bulk
gauge field is added providing the magnetic field in the boundary theory.
The generation of the tower of the Efimov states happens at some value of the
“filling fraction” $\nu$ in external magnetic field. The phase transition
corresponds to the change of the minimal embedding of the probe $D5$ branes in
the bulk geometry with the BKT critical behavior of the order parameter. In
that case the order parameter gets identified with the condensate $\sigma$
which behaves as:
$\sigma\propto\exp\left(-\frac{1}{\nu}\right).$ (10)
Above the phase transition the embedding gets changed and the brane becomes
extended in one more coordinate. The scale associated with this extension into
new dimension is nothing but the nonperturbative scale amounting to the mass
gap. The phenomena of the cyclic RG flow in this case has the Breitenlohner-
Freedman instability as the gravitational counterpart.
In four dimensions the most famous example of the Efimov tower is the so-
called Miransky scaling for the condensate in the magnetic field. In [16] was
argued that the chiral condensate is generated in the external magnetic field
in the abelian theory with the following behavior:
$\langle\bar{\Psi}\Psi\rangle\propto\Lambda^{3}\exp(-\frac{c}{\sqrt{\alpha-\alpha_{crit}}}),$
(11)
where $\alpha$ is the fine–structure constant, and $c$ is some parameter of
the model.
More recent example [17] of the Efimov tower in four dimensions concerns the
Veneziano limit of QCD when $N_{f},N_{c}\rightarrow\infty$ while the ratio
$x=\frac{N_{f}}{N_{c}}$ is fixed. It turns out that this parameter can be
considered as the variable in the RG flow which reminds the finite-dimensional
examples. At some value of RG scale the tower of condensates gets generated
with geometrical Efimov scaling. The period of the RG cycle reads as:
$T\propto\frac{\kappa}{\sqrt{x_{c}-x}},$ (12)
where $x_{c}$ is the critical value of the $x$ parameter. Finally the 4d
example with the RG cycle has been found in the $\mathcal{N}=2$ SUSY gauge
theory in the $\Omega$-background [8]. In that case the gauge coupling
undergoes the RG cycle whose period is determined by the parameter of the
$\Omega$-background,
$T\propto\epsilon^{-1}.$ (13)
The appearance of the RG cycle in this model can be traced from its relation
with the quantum integrable systems of the spin chain type.
In this review we provide the reader with the examples of this phenomenon. The
list of the systems with finite number of degrees of freedom involves the
Calogero model and the relativistic model with the classical conformal
symmetry describing the external charge in graphene. Another finite-
dimensional example concerns the RD model of the restricted BCS
superconductivity. The field theory examples concern the $3d$ and $4d$
theories in external fields. We shall focus on their brane representations and
use their relations to the finite dimensional integrable systems.
## 2 RG cycles in non-relativistic quantum mechanics
In this Section we consider the example of the limit cycle in RG in the non-
relativistic system with the inverse-square potential, or the Calogero system:
$H=\frac{\partial^{2}}{\partial r^{2}}-\frac{\mu(\mu-1)}{r^{2}}.$ (14)
The distinctive feature of the system described by the Hamiltonian (14) is its
conformality. Namely, the operators $(H,D,K)$, where $D$ is the dilation
generator and $K$ is the conformal boost, generate the conformal
$\mathfrak{sl}_{2}$ algebra (see Section 4).
The eigenfunctions of (14) having finite energy immediately break this
symmetry; more non-trivial is the fact that even the ground state breaks
conformality. Namely, the solution to the $H\psi=0$ equation is the following:
$\psi_{0}=c_{+}r^{\mu}+c_{-}r^{1-\mu}.$ (15)
This solution is scale-invariant only if one of the coefficients $c_{\pm}$ is
zero. If both the coefficients are present, they define an intrinsic length
scale $L=\left(c_{+}/c_{-}\right)^{1/(-2\mu+1)}$. Requiring that the quantity
$c_{+}/c_{-}$ which describes the ground-state solution be invariant under the
change of scale,
$\frac{c_{+}}{c_{-}}=-r_{0}^{-2\mu+1}\frac{\gamma-\mu+1}{\gamma+\mu},$ (16)
we arrive at the beta-function for the $\gamma$ parameter,
$\beta_{\gamma}=\frac{\partial\gamma}{\partial\log
r_{0}}=-\left(\gamma+\mu\right)\left(\gamma-\mu+1\right)=\left(\mu-\frac{1}{2}\right)^{2}-\left(\gamma-\frac{1}{2}\right)^{2},$
(17)
where $r_{0}$ is the RG scale. We can identify $\gamma=\mu-1,\gamma=-\mu$
points, i.e. solutions with $c_{+}=0$, $c_{-}=0$, with UV and IR attractive
points of the renormalization group flow [7].
If $\mu=i\nu$ is imaginary, i.e. the potential is attractive, then the
equation (17) allows us to determine the period of the renormalization group:
$T=-\int_{-\nu+1}^{\nu}\frac{d\gamma}{\beta_{\gamma}}=\frac{\pi}{\nu-\frac{1}{2}}.$
(18)
This means that an infinite number of scales is generated, differing by a
factor of $\exp\left(-\frac{\pi}{\nu-\frac{1}{2}}\right)$. To see this
explicitly, we find the solutions to the Schrödinger equation at finite
energies. In the attractive potential the solution (15) can be written as:
$\psi_{0}\propto\sqrt{r}\sin\left(\left(\nu-\frac{1}{2}\right)\log\left(\frac{r}{r_{0}}\right)+\alpha\right).$
(19)
We observe that this solution oscillates indeterminately in the vicinity of
the origin and there is no way to fix the $\alpha$ constant. To regularize
this behaviour, we can break the scale invariance at the level of the
Hamiltonian and introduce a regularizing potential. Two most popular
regularizations involve the square-well potential [4, 5] or the delta-shell
potential [3]. One more choice is to introduce a $\delta$-function at the
origin [7].
Choosing the square-well regularization,
$V(r)=\left\\{\begin{array}[]{l}-\frac{\nu(\nu-1)}{r^{2}},r>R,\\\
-\frac{\lambda}{R^{2}},r\leq R,\end{array}\right.$ (20)
we require that the action of the dilatation operator on the wavefunction
inside the well and outside it be equal at $r=R$. This condition amounts to
the equation on $\lambda$,
$\sqrt{\lambda}\cot\sqrt{\lambda}=\frac{1}{2}+\nu\cot\left(\nu\log\left(\frac{R}{r_{0}}\right)\right).$
(21)
The multivalued function $\lambda(R)$ can be chosen to be continuous [5].
The wavefunction regular at infinity is given as a combination of the Bessel
functions [5],
$\psi\left(r,\kappa_{m}\right)=\sqrt{r}(-1)^{m}\left(ie^{-i\nu\frac{\pi}{2}}J_{i\nu}\left(\kappa_{m}r\right)-ie^{i\nu\frac{\pi}{2}}J_{-i\nu}\left(\kappa_{m}r\right),\right),$
(22)
where $\kappa$ is the energy of the state. The spectrum consists of infinitely
many shallow bound states with adjacent energies differing by an exponential
factor,
$\frac{\kappa_{m+1}}{\kappa_{m}}=e^{-\frac{\pi}{\nu}}.$ (23)
Note that the coordinate enters the wavefunction (22) only in combination with
energy, and the spectrum is generated by the dilation operator:
$\psi_{m+1}=\exp\left(-\frac{\pi}{\nu}r\partial_{r}\right)\psi_{m}.$ (24)
One can think of that relation as that the action of the dilatation operator
shifts zeroes of the wave function from the area of $r<R$ to the area with the
inverse square potential, and one step of (24) evolution corresponds to
elimination of a single zero in the area with the square-well potential. Since
the wave function oscillates infinitely at the origin, the elimination of all
the zeroes would require an infinite number of steps, and in this way a whole
tower of states gets generated.
## 3 RG cycle in graphene
In this Section we shall consider the similar problem in 2+1 dimensions which
physically corresponds to the external charge in the planar graphene layer.
The problem has the classical conformal symmetry and is the relativistic
analogue of the conformal non-relativistic Calogero-like system. Due to
conformal symmetry we could expect the RG cycles and Efimov-like states in
this problem upon imposing the short distance cutoff. The issue of the charge
in the graphene plane has been discussed theoretically [19, 20, 21] and
experimentally [22, 23]. It was argued that indeed there is the tower of
”quasi-Rydberg” states with the exponential scaling [24]. The situation can be
interpreted as an atomic collapse phenomena similar to the instability of
$Z>137$ superheavy atoms in QED [25].
Turn now to the consideration of an electron in graphene which interacts with
an external charge. The two-dimensional Hamiltonian reads as,
$H_{D}=v_{F}\sigma_{i}p^{i}+V(r),\qquad i=1,2.$ (25)
The external charge creates a Coulomb potential,
$V(r)=-\frac{\alpha}{r},\qquad r\geq R.$ (26)
As we shall see, the solution in presence of the potential (26) oscillates
indefinitely at the origin and needs to be regularized by some cutoff $R$.
Hence close enough to the origin $r\leq R$ the potential (26) gets replaced by
some constant potential $V_{reg}(r,\lambda(R))$. The renormalization condition
for the $\lambda$ parameter is that the zero-energy wave function is not
dependent on the short-distance regularization. This condition is chosen
similarly to that of the renormalization of the Calogero system (see Section
2). Hence our primary task is to find the zero-energy solution to the Dirac
equation,
$H_{D}\psi_{0}=0.$ (27)
Since the Hamiltonian commutes with the $J_{3}$ operator,
$J_{3}=i\frac{\partial}{\partial\varphi}+\sigma_{3},\qquad\left[H_{D},J_{3}\right]=0,$
(28)
we can look for the solutions of (27) in the form:
$\psi_{0}=\begin{pmatrix}\chi_{0}(r)\\\
\xi_{0}(r)e^{i\varphi}\end{pmatrix},\qquad J_{3}\psi_{0}=\psi_{0}.$ (29)
Then in polar coordinates the equation (27) reads as:
$\left\\{\begin{array}[]{l}-i\hbar
v_{F}\left(\partial_{r}+\frac{1}{r}\right)\xi_{0}=-V(r)\chi_{0},\\\ -i\hbar
v_{F}\partial_{r}\chi_{0}=-V(r)\xi_{0},\end{array}\right.$ (30)
which is equivalent to:
$\left\\{\begin{array}[]{l}\xi_{0}(r)=i\hbar
v_{F}(V(r))^{-1}{\partial_{r}\chi_{0}},\\\
\partial_{r}^{2}\chi_{0}+\left(\frac{1}{r}-\frac{V^{\prime}(r)}{V(r)}\right)\partial_{r}\chi_{0}+\frac{V^{2}(r)}{\hbar^{2}v_{F}^{2}}\chi_{0}=0.\end{array}\right.$
(31)
For the potential $V=-\frac{\alpha}{r}$ we get the following equation on
$\chi_{0}(r)$:
$\partial_{r}^{2}\chi_{0}+\frac{2}{r}\partial_{r}\chi_{0}+\frac{\beta^{2}}{r^{2}}\chi_{0}=0,\qquad\beta=\frac{\alpha}{\hbar
v_{F}}.$ (32)
Supposing that $\beta^{2}=\frac{1}{4}+\nu^{2}$ we write the solution as:
$\chi_{0}=\sqrt{r}\left(c_{-}\left(\frac{r}{r_{0}}\right)^{-i\nu}+c_{+}\left(\frac{r}{r_{0}}\right)^{i\nu}\right)\propto\sqrt{r}\sin\left(\nu\log\frac{r}{r_{0}}+\varphi\right).$
(33)
We see that this solution shares the properties of the ground-state Calogero
wavefunction (15), namely at nonzero $c_{\pm}$ it generates its own intrinsic
length scale and it oscillates indeterminately at the origin. In order to fix
the $\varphi$ constant we need to introduce a cut-off potential. Hence we
consider the solution in the potential:
$V(r)=\left\\{\begin{array}[]{l}-\frac{\alpha}{r},\qquad r>R,\\\
V_{reg}=-\hbar v_{F}\frac{\lambda}{R},\qquad r\leq R.\end{array}\right.$ (34)
The dilatation operator acts on $\chi$ as following:
$r\partial_{r}\chi_{0}=\left(\frac{1}{2}+\nu\cot\left(\nu\log\frac{r}{r_{0}}\right)\right)\chi_{0}.$
(35)
For the constant potential $V_{reg}$ we get from $(\ref{Dechi})$:
$\partial^{2}_{r}\chi_{0}^{reg}+\frac{1}{r}\partial_{r}\chi_{0}^{reg}+\frac{\lambda^{2}}{R^{2}}\chi_{0}^{reg}=0.$
(36)
Choosing the solution of (36) which is regular at the origin we obtain,
$\chi_{0}^{reg}\propto J_{0}\left(\lambda\frac{r}{R}\right).$ (37)
Computing the action of the dilation operator on the solution in the area of
constant potential and equating it to the action of the dilation operator (35)
we get the equation on the $\lambda$ regulator parameter:
$\frac{1}{2}+\nu\cot\left(\nu\log\left(\frac{R}{r_{0}}\right)\right)=-\lambda\frac{J_{1}(\lambda)}{J_{0}(\lambda)}.$
(38)
The equation (38) defines $\lambda$ as a multi-valued function of $R$. The
period of the RG flow corresponds to jump from one branch of the $\lambda(R)$
function to another.
Now we proceed to find the bound states in the (26) potential. We consider
again the Dirac equation,
$H_{D}\psi_{\kappa}=-\hbar v_{F}\kappa\psi_{\kappa}.$ (39)
Then the equation on $\chi$ analogous to (31) is as following:
$\partial^{2}_{r}\chi_{\kappa}+\frac{2\beta-\kappa r}{\beta-\kappa
r}\frac{1}{r}\partial_{r}\chi_{\kappa}+\left(\frac{\beta}{r}-\kappa\right)^{2}\chi_{\kappa}=0.$
(40)
Asymptotically when $r\gg\frac{\beta}{\kappa}$ the solution of (40) regular at
infinity is given by the Hankel function,
$\chi_{\kappa}\propto H_{0}^{(1)}(i\kappa r).$ (41)
At small $r\ll\frac{\beta}{\kappa}$ the solution is not regular at the origin,
$\chi_{\kappa}\propto\sqrt{r}\sin\left(\nu\log\frac{r}{r_{0}}\right),$ (42)
and we are again in need for the regulator potential. Solving again the Dirac
equation (39) in presence of the constant potential $V_{reg}$ and computing
the action of the dilatation operator,
$r\partial_{r}\chi_{k}^{reg}=-\left(\lambda-\kappa
R\right)\frac{J_{1}\left(\lambda-\kappa R\right)}{J_{0}\left(\lambda-\kappa
R\right)}\chi_{\kappa}^{reg},$ (43)
we can equate (43) to the action of the dilatation operator on (42) and get
the equation on the spectrum of the bound states,
$\frac{1}{2}+\nu\cot\left(\nu\log\left(\kappa
R\right)\right)=-\left(\lambda-\kappa R\right)\frac{J_{1}\left(\lambda-\kappa
R\right)}{J_{0}\left(\lambda-\kappa R\right)}.$ (44)
This condition gives the spectrum of infinitely many shallow bound states,
$\kappa_{n}=\kappa_{*}\exp\left(-\frac{\pi
n}{\nu}\right),\qquad\kappa\to\infty.$ (45)
## 4 Anomaly in the ${\bf so}(2,1)$ algebra
Let us make some comments on the algebraic counterpart of the phenomena
considered following [9]. As we have mentioned the conformal symmetry is the
main player since Hamiltonians under consideration are scale invariant before
regularization. Actually this group can be thought of as the example of
spectrum generating algebra when the Hamiltonian is one of the generators or
is expressed in terms of the generators in a simple manner. This is familiar
from the exactly or quasi-exactly solvable problems when the dimension of the
representation selects the size of the algebraic part of the spectrum.
Let us introduce the generators of the ${\bf so}(2,1)$ conformal algebra
$J_{1},J_{2},J_{3}$: the Calogero Hamiltonian,
$J_{1}=H=p^{2}+V(r),$ (46)
the dilatation generator,
$J_{2}=D=tH-\frac{1}{4}(pr+rp),$ (47)
and the generator of special conformal transformation,
$J_{3}=K=t^{2}H-\frac{t}{2}(pr+rp)+\frac{1}{2}r^{2}.$ (48)
They satisfy the relations of the ${\bf so}(2,1)$ algebra:
$[J_{2},J_{1}]=-iJ_{1},\qquad[J_{3},J_{1}]=-2iJ_{2},\qquad[J_{2},J_{3}]=iJ_{3}.$
(49)
The singular behavior of the potential at the origin amounts to a kind of
anomaly in the ${\bf so}(2,1)$ algebra,
$A(r)=-i[D,H]+H,$ (50)
which in $d$ space dimensions can be presented in the following form:
$A(r)=-\frac{d-2}{2}V(r)+(r^{i}\nabla_{i})V(r).$ (51)
The simple arguments imply the following relation
$\frac{d}{dt}\langle D\rangle_{\mathrm{ground}}=E_{\mathrm{ground}},$ (52)
where the matrix element is taken over the ground state.
It turns out that (52) is fulfilled for the singular potentials in Calogero-
like model or in models with contact potential, $V(r)=g\delta(r)$. The
expression for anomaly does not depend on the regularization chosen. Moreover
more detailed analysis demonstrates that the anomaly is proportional to the
$\beta$-function of the coupling providing the UV regularization as can be
expected.
A similar calculation of the anomaly for the graphene case can be performed
for arbitrary state,
$\left\langle{\frac{dD}{dt}}\right\rangle_{\psi}=\braket{\Xi}_{\psi}=-\int
d^{2}x\psi^{*}(V(x)+x_{i}\partial_{i}V(x))\psi,$ (53)
which yields using square-well regularization:
$\braket{\Xi}_{\psi}=\hbar
v_{F}\frac{\lambda(R)}{R}\frac{\int\limits^{R}_{0}r|\psi|^{2}dr}{\int\limits^{\infty}_{0}r|\psi|^{2}dr}.$
(54)
It is convenient to use the two-dimensional identity in (51),
$\nabla\frac{\vec{r}}{r}=2\pi\delta(\vec{r}),$ (55)
which simplifies the calculation of the anomaly for any normalized bound
state,
$\frac{d}{dt}\langle D\rangle_{\Psi}=-g\pi\int d^{2}r\delta(r)|\Psi(r)|^{2}.$
(56)
## 5 RG cycles in models of superconductivity
In this Section we explain how the cyclic RG flows emerge in truncated models
of superconductivity. To this aim we shall first describe the Richardson model
and then consider its generalization to the RD model which enjoys the cyclic
RG flow. These models are distinguished by the finiteness of the number of
fermionic levels. The relation with the integrable many-body systems proves to
be quite useful.
### 5.1 Richardson model versus Gaudin model
Let us recall the truncated BCS-like Richardson model of superconductivity
[26] with some number of doubly degenerated fermionic levels with the energies
$\epsilon_{j\sigma},j=1,\dots,N$. It describes the system of a fixed number of
the Cooper pairs. It is assumed that several energy levels are populated by
Cooper pairs while levels with the single fermions are blocked. The
Hamiltonian reads as
$H_{BCS}=\sum_{j,\sigma=\pm}^{N}\epsilon_{j\sigma}c^{+}_{j\sigma}c_{j\sigma}-G\sum_{jk}c^{\dagger}_{j+}c^{\dagger}_{j-}c_{k-}c_{k+},$
(57)
where $c_{j\sigma}$ are the fermion operators and $G$ is the coupling constant
providing the attraction leading to the formation of the Cooper pairs. In
terms of the hard-core boson operators it reads as
$H_{BCS}=\sum_{j}\epsilon_{j}b^{\dagger}_{j}b_{j}-G\sum_{jk}b^{\dagger}_{j}b_{k},$
(58)
where
$[b^{\dagger}_{j},b_{k}]=\delta_{jk}(2N_{j}-1),\qquad
b_{j}=c_{j-}c_{j+},\qquad N_{j}=b^{\dagger}_{j}b_{j}.$ (59)
The eigenfunctions of the Hamiltonian can be written as,
$|M\rangle=\prod_{i}^{M}B_{i}(E_{i})|\mathrm{vac}\rangle,\qquad
B_{i}=\sum_{j}^{N}\frac{1}{\epsilon_{j}-E_{i}}b^{\dagger}_{j},$ (60)
provided the Bethe ansatz equations are fulfilled,
$G^{-1}=-\sum_{j}^{N}\frac{1}{\epsilon_{j}-E_{i}}+\sum_{j}^{M}\frac{2}{E_{j}-E_{i}}.$
(61)
The energy of the corresponding states reads as:
$E(M)=\sum_{i}E_{i}.$ (62)
It was shown in [27] that the Richardson model is exactly solvable and closely
related to the particular generalization of the Gaudin model. To describe this
relation it is convenient to introduce the so-called pseudospin ${\bf sl}(2)$
algebra in terms of the creation-annihilation operators for the Cooper pairs,
$t_{j}^{-}=b_{j},\qquad t_{j}^{+}=b^{\dagger}_{j},\qquad t^{0}_{j}=N_{j}-1/2.$
(63)
The Richardson Hamiltonian commutes with the set of operators $R_{i}$,
$R_{i}=-t^{0}_{i}-2G\sum^{N}_{j\neq
i}\frac{t_{i}t_{j}}{\epsilon_{i}-\epsilon_{j}},$ (64)
which are identified as the Gaudin Hamiltonians,
$[H_{BCS},R_{j}]=[R_{i},R_{j}]=0.$ (65)
Moreover the Richardson Hamiltonian itself can be expressed in terms of the
operators $R_{i}$ as:
$H_{BCS}=\sum_{i}\epsilon_{i}R_{i}+G\left(\sum
R_{i}\right)^{2}+\mathrm{const}.$ (66)
The number $N$ of the fermionic levels coincides with the number of sites in
the Gaudin model and the coupling constant in the Richardson Hamiltonian
corresponds to the ”twisted boundary condition” in the Gaudin model. The Bethe
ansatz equations for the Richardson model (61) exactly coincide with the ones
for the generalized Gaudin model. It was argued in [2] that the Bethe roots
correspond to the excited Cooper pairs that is natural to think about the
solution to the Baxter equation as the wave function of the condensate. In
terms of the conformal field theory Cooper pairs correspond to the screening
operators [28].
For the nontrivial degeneracies of the energy levels $d_{j}$ the BA equations
read as:
$G^{-1}=-\sum_{j}^{N}\frac{d_{j}}{\epsilon_{j}-E_{i}}+\sum_{j\neq
i}^{M}\frac{2}{E_{j}-E_{i}}.$ (67)
### 5.2 Russian Doll model of superconductivity and twisted XXX spin chains
The important generalization of the Richardson model describing
superconductivity is the so-called RD model [2]. It involves the additional
dimensionless parameter $\alpha$ and the RD Hamiltonian reads as:
$H_{RD}=2\sum_{i}^{N}(\epsilon_{i}-G)N_{i}-\bar{G}\sum_{j<k}(e^{i\alpha}b^{+}_{k}b_{j}+e^{-i\alpha}b^{+}_{j}b_{k}),$
(68)
with two dimensionful parameters $G,\eta$ and $\bar{G}=\sqrt{G^{2}+\eta^{2}}$.
In terms of these variables the dimensionless parameter $\alpha$ has the
following form:
$\alpha=\arctan\left(\frac{\eta}{G}\right).$ (69)
It is also useful to consider two dimensionless parameters $g,\theta$ defined
as $G=gd$ and $\eta=\theta d$ where $d$ is the level spacing. The RD model
reduces to the Richardson model in the limit $\eta\rightarrow 0$.
The RD model turns out to be integrable as well. Now instead of the Gaudin
model the proper counterpart is the generic quantum twisted XXX spin chain
[29]. The transfer matrix of such spin chain model $t(u)$ commutes with the
$H_{RD}$ which itself can be expressed in terms of the spin chain
Hamiltonians.
The equation defining the spectrum of the RD model reads as:
$e^{2i\alpha}\prod_{l=1}^{N}\frac{E_{i}-\varepsilon_{l}+i\eta}{E_{i}-\varepsilon_{l}-i\eta}=\prod_{j\neq
i}^{M}\frac{E_{i}-E_{j}+2i\eta}{E_{i}-E_{j}-2i\eta},$ (70)
and coincides with the BA equations for the spin chain.
Taking the logarithm of the both sides of the equation (70) we obtain:
$\alpha+\pi
Q_{i}+\sum_{l=1}^{N}\arctan\left(\frac{\eta}{E_{i}-\varepsilon_{l}}\right)-\sum_{j=1}^{M}\arctan\left(\frac{2\eta}{E_{i}-E_{j}}\right)=0.$
(71)
Note that here we have added an arbitrary integer term to account for
generically multivalued arctangent function.
The RG step amounts to integrating out the $N$-th degree of freedom in the RD
model, or equivalently to integrating out the $N$-th inhomogeneity in the XXX
chain. This results into renormalization of the twist. From (71) it is easy to
see that:
$\arctan\left(\frac{\eta}{G_{N}}\right)-\arctan\left(\frac{\eta}{G_{N-1}}\right)=\sum_{i=1}^{M}\arctan\left(\frac{2\eta}{E_{i}-\varepsilon_{N}}\right).$
(72)
When $M=1$ it implies that:
$G_{N-1}-G_{N}=\frac{G_{N}^{2}+\eta^{2}}{\varepsilon_{N}-G_{N}-E},$ (73)
which is a discrete version of the (1) equation. Of course the same relation
can be derived from the RD Hamiltonian (68). If we consider the wavefunction
$\psi=\sum_{i}^{N}\psi_{i}b_{i}^{\dagger}|0\rangle$, the Schrödinger equation
for a state with one Cooper pair amounts to:
$\left(\varepsilon_{i}-G-E\right)\psi_{i}=(G+i\eta)\sum_{j=1}^{i-1}\psi_{j}+(G-i\eta)\sum_{j=i+1}^{N}\psi_{j}.$
(74)
Integration out the $N$-th degree of freedom amounts to expressing $\psi_{M}$
in terms of the other modes,
$\psi_{N}=\frac{G+i\eta}{\varepsilon_{N}-G-E}\sum_{j=1}^{N-1}\psi_{j},$ (75)
and substituting it back into the Schrödinger equation (74). The $G_{N-1}$
constant in the resulting equation will differ from the initial $G_{N}$ value
as in (73).
The key feature of the RD model is the multiple solutions to the gap equation.
The gaps are parameterized as follows:
$\Delta_{n}=\frac{\omega}{\sinh t_{n}},\qquad t_{n}=t_{0}+\frac{\pi
n}{\theta},\qquad n=0,1,\dots,$ (76)
where $t_{0}$ is the solution to the following equation:
$\tan(\theta t_{0})=\frac{\theta}{g},\qquad 0<t_{0}<\frac{\pi}{\theta}.$ (77)
and $\omega=dN$ for equal level spacing. Here
$E^{2}=\varepsilon^{2}+|\Delta|^{2}$. This behavior can be derived via the
mean field approximation [14]. The gap with minimal energy defines the ground
state, and the other values of the gap describe excitations. In the limit
$\theta\rightarrow 0$ the gaps $\Delta_{n>0}\rightarrow 0$ and
$t_{0}=\frac{1}{g},\qquad\Delta_{0}=2\omega\exp\left(-\frac{1}{g}\right),$
(78)
therefore the standard BCS expression for the gap is recovered. At the weak
coupling limit the gaps behave as:
$\Delta_{n}\propto\Delta_{0}e^{-\frac{n\pi}{\theta}}.$ (79)
In terms of the solutions to the BA equations the multiple gaps correspond to
the choices of the different branches of the logarithms, i.e. to different
choices of the integer $Q$ parameter in (70).
If the degeneracy of the levels is $d_{n}$ then the RD model gets modified a
little bit and is related to the higher spin XXX spin chain. The local spins
$s_{i}$ are determined by the corresponding higher pair degeneracy $d_{i}$ of
the $i$-th level,
$s_{i}=d_{i}/2,$ (80)
and the corresponding BA equations read as:
$e^{2i\alpha}\prod_{l=1}^{N}\frac{E_{i}-\varepsilon_{l}+id_{l}+i\eta}{E_{i}-\varepsilon_{l}-id_{l}-i\eta}=\prod_{j\neq
i}^{M}\frac{E_{i}-E_{j}+2i\eta}{E_{i}-E_{j}-2i\eta}.$ (81)
### 5.3 Cyclic RG flows in the RD model
The RD model of truncated superconductivity enjoys the cyclic RG behavior [2].
The RG flows can be treated as the integrating out the highest fermionic level
with appropriate scaling of the parameters using the procedure developed in
[1]. The RG equations read as (73):
$g_{N-1}=g_{N}+\frac{1}{N}(g_{N}^{2}+\theta^{2}),\qquad\theta_{N-1}=\theta_{N}.$
(82)
At large $N$ limit the natural RG variable is identified with
$s=\log(N/N_{0})$ and the solution to the RG equation is:
$g(s)=\theta\tan\left(\theta
s+\tan^{-1}\left(\frac{g_{0}}{\theta}\right)\right).$ (83)
Hence the running coupling is cyclic,
$g(s+\lambda)=g(s),\qquad g(e^{-\lambda}N)=g(N),$ (84)
with the RG period,
$\lambda=\frac{\pi}{\theta},$ (85)
and the total number of the independent gaps in the model is:
$N_{cond}\propto\frac{\theta}{\pi}\log N.$ (86)
The multiple gaps are the manifestations of the Efimov-like states. The sizes
of the Cooper pairs in the $N$-th condensates also have the RD scaling. The
cyclic RG can be derived even for the single Cooper pair.
What is going on with the spectrum of the model during the period? It was
shown in [14] that it gets reorganized. The RG flow experiences
discontinuities from $g=+\infty$ to $g=-\infty$ when a new cycle gets started.
At each jump the lowest condensate disappears from the spectrum,
$\Delta_{N+1}(g=+\infty)=\Delta_{N}(g=-\infty),$ (87)
indicating that the $(N+1)$-th state wave function plays the role of $N$-th
state wave function at the next cycle (see (75)).
The same behavior can be derived from the BA equation [14]. To identify the
multiple gaps it is necessary to remind that the solutions to the BA equations
are classified by the integers $Q_{i},i=1,\dots,M$ parameterizing the branches
of the logarithms. If one assumes that $Q_{i}=Q$ for all Bethe roots then this
quantum number gets shifted by one at each RG cycle and was identified with
the integer parameterizing the solution to the gap equations,
$\Delta_{Q}\propto\Delta_{0}\exp^{-\lambda Q}.$ (88)
At the large $N$ limit the BA equations of the RD model reduce to the BA
equation of the Richardson-Gaudin model with the rescaled coupling,
$G_{Q}^{-1}=\eta^{-1}({\alpha}+\pi Q),$ (89)
which can be treated as the shifted boundary condition in the generalized
Gaudin model parameterized by an integer. Let us emphasize that the unusual
cyclic RG behavior is due to the presence of two couplings in the RD model.
## 6 Triality in the integrable models and RG cycles
| | |
---|---|---|---
RS modelQC duality$\scriptstyle{\begin{matrix}\text{non-relativistic}\\\
\text{limit}\end{matrix}}$XXX
chain$\scriptstyle{\begin{matrix}\text{semiclassical}\\\
\text{limit}\end{matrix}}$RD modelCalogero systemQC dualityGaudin
modelRichardson model Figure 1: Besides the triality shown on the picture, a
bispectral duality acts on RS/Calogero and XXX/Gaudin sides of the
correspondence. Being originated from three-dimensional mirror-symmetry [30],
this duality interchanges coordinates with Lax eigenvalues in the classical
systems, and inhomogeneities with twists in quantum ones.
In this Section we summarize several dualities between the integrable models
and consider the realizations of the cyclic RG flows in these systems. The
question is motivated by the close relationship between the restricted BCS
models and spin chains. Actually there are three different families of models
related with each other by the particular identifications of phase spaces and
parameters. The first family concerns the system of fermions (Richardson-
Russian Doll) which develop superconducting gap. The second family involves
the spin systems of twisted inhomogeneous Gaudin-XXX-XXZ type and their
generalizations. The third family involves the Calogero-Ruijsenaars (CR) chain
of the integrable many body systems.
We look for the answers on the following questions
* •
What is the condition yielding the RG equation for some coupling in each
family?
* •
What is the RG variable?
* •
What determines the period of the cycle?
In the superconducting system at RG step one decouples the highest energy
level and looks at the renormalization of the interaction coupling constant.
The RG time is identified with the number of energy levels $t=\log N$. The
period of RG is defined by the T-asymmetric parameter of RD model.
In the spin chain model the RG step corresponds to the ”integrating out” one
”highest” inhomogeneity with the corresponding renormalization of the twist.
The period of the RG flow is fixed by the Planck constant in the quantum spin
chain. In the bispectral dual spin chain [33] one now ”integrates out” one of
the twists and ”renormalizes” the inhomogeneity. Since the Planck constant
gets inverted upon bispectrality $\hbar_{spin}\rightarrow\hbar_{spin}^{-1}$
the period of the RG cycle gets inverted as well. Note that the RG equation in
the superconducting model can be mapped into BAE in the spin chain [14]. The
condition yealding the RG equation corresponds to the independence of the
Bethe root on the RG step.
For two-body system with attractive rational potential one can define the RG
condition as the continuity of the zero-energy wave function under the
changing the cutoff scale at small $r$. This condition imposes the RG equation
at the cutoff UV coupling constant. This RG equation has the cycle with the
period
$T_{Cal}=\frac{\pi}{\nu-\frac{1}{2}}.$ (90)
as was shown in Section 2.
The Quantum-Classical (QC) duality [30, 31] relates the quantum spin chain
systems and the classical Calogero–type systems. Through the QC
correspondence, the rational Gaudin model can be linked with the rational
Calogero system spin chain inhomogeneities being the Calogero coordinates, and
the twist in the spin chain (which is a single variable in our case) being the
Lax matrix eigenvalue. It is also possible to make a bispectrality
transformation of rational Calogero model, which interchanges Lax eigenvalues
with coordinates. This means that now the Calogero coordinates correspond to
the twists at the spin chain side. In this case the Calogero coupling gets
inverted which means that the period of the RG cycle gets inverted as well.
To consider the mapping of RG cycles in the Calogero system and the spin chain
we need the generalization of QC duality to the quantum-quantum case. The
spectral problem in Calogero model has been identified with the KZ equation
involving the Gaudin Hamiltonian,
$\frac{d}{dz_{i}}\Psi=H_{gaud}\Psi+\lambda\Psi.$ (91)
Since we formulate RG condition on the Calogero side for the $E=0$ state, the
inhomogeneous term in the KZ equation is absent. The simplest test of the
mapping of the RG cycles under QQ duality concerns the identifications of the
periods. On the spin chain side it is identified with the Planck constant
while at the Calogero side the period is defined by the coupling constant. The
following identification holds for QC duality [31]:
$\hbar_{spin}=\nu,$ (92)
which implies that the periods of the cycles at the Calogero and spin chain
sides match.
The Efimov-like tower in these families have the following interpretations. In
the superconducting system it corresponds to the family of the gaps
$\Delta_{n}$ with the Efimov scaling responsible of the scale symmetry broken
down to the discrete subgroup. In the spin chain it corresponds to the
different branches of the solutions to the BAE which can be also interpreted
in terms of the allowed set of twists. Finally in the CR family it corresponds
to the family of the shallow bound states near the continuum threshold.
## 7 RG cycles in $\Omega$-deformed SUSY gauge theories
In this Section we shall explain how the RG flows in $\Omega$\- deformed SUSY
gauge theories can be reformulated in terms of the brane moves. Why the very
RG cycles could be expected in the deformed gauge theories? The answer is
based on the identification of the quantum spin chains in one or another
context in the SUSY gauge theory. Once such quantum spin chain has been found
we can apply the results of the previous sections where the place of the RG
cycles in the spin chain framework has been clarified.
First, we shall briefly review the $\Omega$-deformation of the SUSY gauge
theories. Then we make some general comments concerning the realization of the
gauge theories as the worldvolume theories on $D$-branes to explain how the
parameters of the gauge theory are identified with the brane coordinates.
### 7.1 Four-dimensional $\Omega$-deformed gauge theory
The Bethe ansatz equations can be encountered not only in the models of
superconductivity, but also in gauge theories. The quantum XXX spin chain
governs the moduli space of vacua of an $\Omega$-deformed four-dimensional
theory in the Nekrasov–Shatashvili limit, i.e. when one of the deformation is
chosen to be zero: $\epsilon_{2}=0,\epsilon_{1}=\epsilon$ [35]. Since the
quantum XXX spin chain displays a cyclic RG behaviour, as we have seen in the
Section 5, it is interesting to identify this phenomenon in the four-
dimensional gauge theory.
Consider a four-dimensional $\mathcal{N}=2$ theory with matter hypermultiplet,
which has a vanishing $\beta$-function, i.e. when $N_{f}=2N_{c}$. This theory
is dual to a classical inhomogeneous twisted XXX chain, in a sense that the
Seiberg-Witten curve for the gauge theory coincides with the spectral curve
for the spin chain. The twist of the spin chain is identified with the modular
parameter of the curve and with the complexified coupling of the gauge theory,
the inhomogeneities of the spin chain get mapped into masses of the
hypermultiplets. For more information on the correspondence between classical
integrable systems and gauge theories the reader can consult [36].
The $\Omega$-deformation is introduced to regularize the instanton divergence
in the partition function of the gauge theory [37]. We can consider the four-
dimensional theory as a reduction of the six-dimensional $\mathcal{N}=1$
theory with metric:
$ds^{2}=2dzd\bar{z}+\left(dx^{m}+\Omega^{mn}x_{n}d\bar{z}+\bar{\Omega}^{mn}x_{n}dz\right)^{2},\qquad
m=1,\ldots,4,$ (93)
i.e. we can consider the theory on a four-dimensional space, fibered over a
two-dimensional torus. One can imagine the $\epsilon_{1,2}$ deformation
parameters as chemical potentials for the rotations in two orthogonal planes
in four-dimensional Euclidean space. One can also think that the Euclidean
$\mathbb{R}^{4}$ space gets substituted by a sphere $S^{4}$ with finite
volume.
The non-trivial $\Omega$-deformation modifies the correspondence between gauge
theories and integrable systems. Namely, in the Nekrasov-Shatashvili limit the
$\Omega$-deformed gauge theory corresponds to a quantum XXX spin chain with
$\epsilon$ playing the role of the Planck constant [35]. This deformed gauge
theory also appears to be dual to the two-dimensional effective theory on a
worldsheet of a non-abelian string [39].
Consider $\Omega$-deformed $\mathcal{N}=2$ SQCD with $SU(L)$ gauge group, $L$
fundamental hypermultiplets with masses $m^{f}_{i}$ and $L$ antifundamental
hypermultiplets with masses $m^{af}_{i}$. Let us denote the set of the
eigenvalues of the adjoint scalar in the vector multiplet by $\vec{a}$. We can
expand the deformed partition function around $\epsilon=0$ to identify the
prepotential and effective twisted superpotential,
$\log\mathcal{Z}\left(\vec{a},\epsilon_{1},\epsilon_{2}\right)\sim\frac{1}{\epsilon_{1}\epsilon_{2}}\mathcal{F}(\vec{a},\epsilon)+\frac{1}{\epsilon_{2}}\mathcal{W}(\vec{a},\epsilon).$
(94)
The effective twisted superpotential is a multivalued function, with the
branch fixed by the set of integers $\vec{k}$:
$\mathcal{W}(\vec{a},\epsilon)=\frac{1}{\epsilon}\mathcal{F}(\vec{a},\epsilon)-2\pi
i\vec{k}\cdot\vec{a},\qquad\vec{k}\in\mathbb{Z}^{L}.$ (95)
The equation on vacua,
$\frac{\partial\mathcal{W}(\vec{a},\epsilon)}{\partial\vec{a}}=\vec{n},\qquad\vec{n}\in\mathbb{Z}^{L},$
(96)
provides the condition on $\vec{a}$,
$\vec{a}=\vec{m}^{f}-\epsilon\vec{n}.$ (97)
This theory admits the existence of non-abelian strings probing the four-
dimensional space-time. The two-dimensional worldsheet theory of the non-
abelian string involves $L$ fundamental chiral multiplets with twisted masses
$M^{F}_{i}$ and $L$ antifundamental multiplets with twisted masses
$M^{AF}_{i}$, which are identified as:
$M^{F}_{i}=m^{f}_{i}-\frac{3}{2}\epsilon,\qquad
M^{AF}_{i}=m^{af}_{i}+\frac{1}{2}\epsilon.$ (98)
The two-dimensional theory also contains an adjoint chiral multiplet with mass
$\epsilon$. The rank of the gauge group $N$ (or equivalently the number of
non-abelian strings) is given in terms of $\vec{n}$ vector by the relation:
$N+L=\sum_{l=1}^{L}n_{l}.$ (99)
The modular parameters of the four-dimensional and the two-dimensional
theories are related as:
$\tau_{2d}=\tau_{4d}+\frac{1}{2}(N+1).$ (100)
The effective twisted worldsheet superpotential is given in terms of the four-
dimensional superpotential:
$\mathcal{W}_{4d}\left(a_{i}=m^{f}_{i}-\epsilon
n_{i},\epsilon\right)-\mathcal{W}_{4d}\left(a_{i}=m^{f}_{i}-\epsilon,\epsilon\right)=\mathcal{W}_{2d}\left(\\{n_{i}\\}\right).$
(101)
The two-dimensional superpotential depends on the set of eigenvalues of the
adjoint scalar in vector representation $\lambda_{i}$, $i=1,\ldots,N$. The set
of equations $\partial\mathcal{W}_{2d}/\partial\lambda=0$ appears to be
equivalent to the Bethe ansatz equations for the XXX spin chain:
$\prod_{l=1}^{L}\left(\frac{\lambda_{j}-M^{F}_{l}}{\lambda_{j}-M^{AF}_{l}}\right)=\exp\left(2\pi
i\tau_{4d}\right)\prod_{k\neq
j}^{N}\left(\frac{\lambda_{j}-\lambda_{k}-\epsilon}{\lambda_{j}-\lambda_{k}+\epsilon}\right).$
(102)
The Planck constant in the spin chain is identified with the $\epsilon$
deformation parameter. The complexified coupling parameter plays the role of
twist in the spin chain. The renormalization of the spin chain amounts to
decoupling of one fundamental and one anti-fundamental chiral multiplet. In
the four-dimensional theory it corresponds to the decrease of the number of
flavors $N_{f}\to N_{f}-2$ simultaneously with reducing the rank of the gauge
group $N_{c}\to N_{c}-1$. Therefore the theory remains conformal. The
renormalization of the coupling constant analogous to (73) derived from the
relation (102) for $N=1$ is:
$\exp\left(2\pi
i(\tau_{L}-\tau_{L-1})\right)=\frac{\lambda-M^{F}_{L}}{\lambda-M^{AF}_{L}}.$
(103)
If we choose the masses to be equidistant with spacing $\delta m$, the change
in the coupling constant during one step of RG flow is:
$\exp\left(2\pi i(\tau_{L}-\tau_{L-1})\right)\propto\frac{\epsilon}{\delta
m}.$ (104)
Hence a number of nonperturbative scales emerges in a theory, analogously to
the generation of the Efimov scaling in the Calogero model. These scales
correspond to multiple gaps in the superconducting model:
$\Delta_{n}\propto\Delta_{0}\exp\left(-\frac{\pi n\delta m}{\epsilon}\right).$
(105)
Note that the emergence of cyclic RG evolution is a feature caused by the
$\Omega$-deformation, since in a non-deformed theory a decoupling of the heavy
flavor does not lead to any cyclic dynamics.
### 7.2 $3d$ gauge theories and theories on the brane worldvolumes
Let us briefly explain the main points concerning the geometrical engineering
of the gauge theories on the $D$-branes suggesting the reader to consult the
details in the review paper [40]. The $Dp$ brane is the $(p+1)$-dimensional
hypersurface in the ten-dimensional space-time which supports the $U(1)$ gauge
field. This feature provides the possibility to built up the gauge theories
with the desired properties. Let us summarize the key elements of the
”building procedure”.
* •
A stack of coinciding $N$ $D$-branes supports $U(N)$ gauge theory with the
maximal supersymmetry.
* •
Displacing some branes from the stack in the transverse direction corresponds
to the Higgs mechanism in the $U(N)$ gauge theory and the distance between
branes corresponds to the Higgs vev.
* •
To reduce the SUSY one imposes some boundary conditions at some coordinates
using other types of branes or rotates some branes.
* •
All geometrical characteristics of the brane configurations have the meaning
of parameters of the gauge theory like couplings or vevs of some operators in
the gauge theory on their worldvolumes.
* •
If we move some brane through another one the brane of smaller dimension could
be created. The Hanany-Witten move is the simplest example (see fig. 2).
* •
Since generically we have branes of different dimensions in the configuration,
for example, $N$ $D2$ branes and $M$ $D4$ branes we have simultaneously $U(N)$
2+1 dimensional gauge theory and $U(M)$ dimensional 4+1 dimensional theory on
the brane worldvolumes. These theories coexist simultaneously hence there is
highly nontrivial interplay between two gauge theories.
Figure 2: Hanany-Witten move. Here vertical lines are $NS5$ branes, horizontal
lines are $D3$ branes, and circles are $D5$ branes. When a $D5$ brane is moved
through a sequence of $NS5$ branes the linking number between them is
conserved hence additional $D3$ branes appear.
Let us explain now how these brane rules can be used to engineer the gauge
theories which are related with the quantum spin chains. Our main example is a
$3d$ $\mathcal{N}=2$ quiver gauge theory.
The brane configuration relevant for this theory is built as follows. We have
$M$ parallel $NS5$ branes extended in $(012456)$, $N_{i}$ $D3$ branes extended
in $(0123)$ between $i$-th and $(i+1)$-th $NS5$ branes, and $M_{i}$ $D5$
branes extended in $(012789)$ between $i$-th and $(i+1)$-th $NS5$ branes (see
table 3). From this brane configuration we obtain the $\prod_{i}^{M}U(N_{i})$
gauge group on the $D3$ branes worldvolume with $M_{i}$ fundamentals for the
$i$-th gauge group. The distances between the $i$-th and $(i+1)$-th $NS5$
branes yield the complexified gauge coupling for $U(N_{i})$ gauge group while
the coordinates of the $D5$ branes in the $(45)$ plane correspond to the
masses of fundamentals. The positions of the $D3$ branes on $(45)$ plane
correspond to the coordinates on the Coulomb branch in the quiver theory. The
additional $\Omega$ deformation reduces the theory with $\mathcal{N}=4$ SUSY
to the $\mathcal{N}=2^{*}$ theory, i.e. an $\mathcal{N}=2$ theory with massive
adjoint. It is identified as $3d$ gauge theory when the distance between $NS5$
is assumed to be small enough. We assume that one coordinate is compact that
is the theory lives on $\mathbb{R}^{2}\times S^{1}$.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
D3 | $\times$ | $\times$ | $\times$ | $\times$ | | | | | |
NS5 | $\times$ | $\times$ | $\times$ | | $\times$ | $\times$ | $\times$ | | |
D5 | $\times$ | $\times$ | $\times$ | | | | | $\times$ | $\times$ | $\times$
Figure 3: Brane construction of the $3d$ quiver theory.
The mapping of the gauge theory data into the integrability framework goes as
follows. In the NS limit of the $\Omega$-deformation the twisted
superpotential in $3d$ gauge theory on the $D3$ branes gets mapped into the
Yang-Yang function for the $XXZ$ chain [35]. The minimization of the
superpotential yields the equations describing the supersymmetric vacua and in
the same time they are the Bethe ansatz equations for the $XXZ$ spin chain,
generally speaking the nested Bethe ansatz equations. That is $D3$ branes are
identified with the Bethe roots which are distributed according to the ranks
of the gauge groups at each of $M$ steps of nesting $\prod_{i}^{M}U(N_{i})$.
The positions of the $D5$ branes in the $(45)$ plane correspond to the
inhomogeneities in the $XXZ$ spin chain. The anisotropy of the $XXZ$ chain is
defined by the radius of the compact dimensions while the parameter of the
$\Omega$ deformation plays the role of the Planck constant in the $XXZ$ spin
chain. At small radius the XXZ spin chain turns to the XXX spin chain. The
twists in the spin chain correspond to the coordinates of the $NS5$ branes in
the (78) plane, and the Fayet–Iliopoulos parameters in the three-dimensional
theory [30].
One step of the RG flow corresponds to elimination of one inhomogeneity in the
spin chain resulting in renormalization of the twists. In the three-
dimensional theory this means that the integration of one massive flavor leads
to the renormalization of the FI parameters. In terms of the transformations
of the brane configurations this process receives transparent geometrical
interpretation:
* •
The RG step is the removing of one $D5$ brane which amounts to the
renormalization of the position of $NS5$ branes or twists.
* •
The period of the RG cycle is fixed by the number of $NS5$ branes [34], since
it was identified with the Planck constant in the spin chain.
* •
At some scale the twists flow from $+\infty$ to $-\infty$.
## 8 Conclusion
Are there any general lessons which we could learn for the quantum field
theory from the very existence of the cyclic RG flows? The most important
point is that there is some fine structure at the UV scale which is reflected
in the Efimov tower with the BKT scaling behavior. Moreover the cyclic flows
imply the interplay between the UV and IR cutoffs in the theory which usually
was attributed to the noncommutative theory. This mixing presumably could shed
the additional light on the dimensional transmutation phenomena in the field
theory and provide the examples for the simultaneous generation of the
multiple scales.
The presence of two parameters in RG is quite common however probably some
additional properties of these parameters are required. In particular in many
(although not all) examples the period of the cycle is fixed by the “filling
fraction” in some external field which could be magnetic field or parameter of
$\Omega$ background. The latter has the meaning of the Planck constant in the
auxiliary finite dimensional integrable model. This could suggest that the
very issue can be formulated purely in terms of the quantum phase space since
the Planck constant can be interpreted as the external field applied to the
classical phase space.
Actually we could expect the relation of RG cycles with some refinement of the
path integral in quantum mechanics. As an aside remark note that the attempt
to get the rigorous mathematical formulation of the renormalization of the QFT
leaded to the motivic generalization of the path integral. It corresponds to
some fine structure at the regulator scale which has some similarities with
the discussion above. The RG cycle in the quantum rational Calogero model
implies the intimate relation with the knot theory since the knot invariants
at the rational Calogero coupling are the characteristics of the Calogero
spectrum (cf. [34]).
As we already mentioned, cyclic renormalization dynamics is connected with
BKT–pairing of partons in two-dimensional model. One could wonder whether this
connection is universal. One four-dimensional example of such pairing has to
be mentioned. It is bion condensation in 3+1 dimensions. The RG analysis of
the model involving the gas of bions and electrically charged W-bosons has
been considered in [42] where the RG flows involves the fugacities for
electric and magnetic components and the coupling constant. The coupled set of
the RG equations has been solved explicitly in the self-dual case and the
solution to the RG equations for the fugacities obtained in [42] is identical
to the solution for the coupling in the RD model upon the analytic
continuation. The period of the RG in the solution above is fixed by the RG
invariant which has been identified with the product of the UV values of the
electric and magnetic fugacities $y_{e}\times y_{m}$. The similarity between
the RG behavior is not accidental since the mapping of the gauge theory and
the perturbed XY model has been found in [42].
We would like to emphasize that the investigation of various aspects of limit
cycles in RG dynamics still remains on its early stage and there is a
considerable number of open questions. The RG cycles can have numerous
applications to different aspects of mathematical physics. In this case the RG
dynamics is considered as an example of non-trivial dynamical system.
The work of A.G. and K.B. was supported in part by grants RFBR-12-02-00284 and
PICS-12-02-91052. The work of K.B. was also supported by the “Dynasty”
fellowship. A.G. thanks FTPI at University of Minnesota where the part of this
work has been done for the hospitality and support. We would like to thank N.
Nekrasov and F. Popov for useful discussions and comments.
## References
* [1] S. Glazek and K. Wilson, “Limit cycles in quantum theories,” Phys.Rev.Lett. 89 (2002) 23401, arXiv:hep-th/0203088.
* [2] A. LeClair, J. M. Roman and G. Sierra, “Russian doll renormalization group and superconductivity,” Phys. Rev. B 69, 20505 (2004) arXiv:cond-mat/0211338.
* [3] E. Braaten, H. -W. Hammer, “Universality in few-body systems with large scattering length,” Phys. Rept. 428, 259-390 (2006), arXiv:cond-mat/0410417.
* [4] M. Bawin and S. A. Coon, “The Singular inverse square potential, limit cycles and selfadjoint extensions,” Phys. Rev. A 67, 042712 (2003), arXiv:quant-ph/0302199.
E. Braaten and D. Phillips, “The Renormalization group limit cycle for the
1/r**2 potential,” Phys. Rev. A 70, 052111 (2004), arXiv:hep-th/0403168.
* [5] S. R. Beane, P. F. Bedaque, L. Childress, A. Kryjevski, J. McGuire and U. van Kolck, “Singular potentials and limit cycles,” Phys. Rev. A 64, 042103 (2001), arXiv:quant-ph/0010073.
* [6] A. Leclair, J. M. Roman and G. Sierra, “Russian doll renormalization group, Kosterlitz-Thouless flows, and the cyclic sine-Gordon model,” Nucl. Phys. B 675, 584 (2003), arXiv:hep-th/0301042.
* [7] D. B. Kaplan, J. -W. Lee, D. T. Son and M. A. Stephanov, “Conformality Lost,” Phys. Rev. D 80, 125005 (2009), arXiv:0905.4752 [hep-th].
* [8] A. Gorsky, “SQCD, Superconducting Gaps and Cyclic RG Flows,” arXiv:1202.4306 [hep-th].
* [9] G. N. J. Ananos, H. E. Camblong, C. Gorrichategui, E. Hernadez and C. R. Ordonez, “Anomalous commutator algebra for conformal quantum mechanics,” Phys. Rev. D 67, 045018 (2003), arXiv:hep-th/0205191.
H. E. Camblong and C. R. Ordonez, “Renormalization in conformal quantum
mechanics,” Phys. Lett. A 345, 22 (2005), arXiv:hep-th/0305035.
S. Moroz and R. Schmidt, “Nonrelativistic inverse square potential, scale
anomaly, and complex extension,” Annals Phys. 325, 491 (2010), arXiv:0909.3477
[hep-th].
G. N. J. Ananos, H. E. Camblong and C. R. Ordonez, “SO(2,1) conformal anomaly:
Beyond contact interactions,” Phys. Rev. D 68, 025006 (2003), arXiv:hep-
th/0302197.
* [10] T. L. Curtright, X. Jin and C. K. Zachos, “RG flows, cycles, and c-theorem folklore,” Phys. Rev. Lett. 108, 131601 (2012) [arXiv:1111.2649 [hep-th]].
* [11] V. Efimov, “Energy levels arising from resonant two-body forces in a three-body system,” Phys. Lett. B33, 563 (1970).
V. Efimov, “Energy levels of three resonantly interacting particles,” Nucl.
Phys. A210, 157 (1973).
* [12] H. -W. Hammer and L. Platter, “Efimov physics from a renormalization group perspective,” Phil. Trans. Roy. Soc. Lond. A 369, 2679 (2011), arXiv:1102.3789 [nucl-th].
* [13] E. Brezin, J. Zinn-Justin, “Renormalization group approach to matrix models,” Phys. Lett. B288, 54-58 (1992), arXiv:hep-th/9206035.
* [14] A. Anfossi, A. Leclair and G. Sierra, “The elementary excitations of the exactly solvable Russian doll BCS model of superconductivity,” Journal of Statistical Mechanics: 05011 (2005), arXiv:cond-mat/0503014 [cond-mat.supr-con].
* [15] K. Jensen, A. Karch, D. T. Son and E. G. Thompson, “Holographic Berezinskii-Kosterlitz-Thouless Transitions,” Phys. Rev. Lett. 105, 041601 (2010), arXiv:1002.3159 [hep-th].
* [16] V. A. Miransky, “Dynamics of Spontaneous Chiral Symmetry Breaking and Continuum Limit in Quantum Electrodynamics,” Nuovo Cim. A 90, 149 (1985).
V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, “Catalysis of dynamical
flavor symmetry breaking by a magnetic field in (2+1)-dimensions,” Phys. Rev.
Lett. 73, 3499 (1994) [Erratum-ibid. 76, 1005 (1996)] [hep-ph/9405262].
* [17] D. Arean, I. Iatrakis, M. Jarvinen and E. Kiritsis, “The discontinuities of conformal transitions and mass spectra of V-QCD,” JHEP 1311, 068 (2013), arXiv:1309.2286 [hep-ph].
* [18] A. Gorsky and F. Popov, ”Atomic collapse in graphene and cyclic RG flow”, arxiv:1312.7399.
* [19] A. Shytov, M. Katsnelson and L. Levitov, “Vacuum Polarization and Screening of Supercritical Impurities in Graphene,” Phys. Rev. Lett. 99, 236801 (2007), arXiv:0705.4663 [cond-mat.mes-hall].
* [20] V. Pereira, V. Kotov and A. Castro Neto, “Supercritical Coulomb Impurities in Gapped Graphene,”, Phys.Rev. B78, 085101 (2008), arXiv:0803.4195 [cond-mat.mes-hall].
* [21] M. Fogler, D. Novikov and B. Shklovskii, “Screening of a hypercritical charge in graphene,” Phys. Rev. B 76, 233402 (2007), arXiv:0707.1023 [cond-mat.mes-hall].
* [22] Y. Wang et al., Nat. Phys 8, 653 (2012).
* [23] Y. Wang et al., Science 340, 734 (2013).
* [24] A. Shytov, M. Katsnelson and L. Levitov, “Atomic Collapse and Quasi-Rydberg States in Graphene,” Phys. Rev. Lett. 99, 246802 (2007), arXiv:0708.0837 [cond-mat.mes-hall].
* [25] Y.Pomeranchuk and Y. Smorodinsky, J.Phys. USSR, 9,97 (1945).
Y. Zeldovich and V. Popov, Sov.Phys.Usp. 14, 673 (1972).
* [26] R.Richardson, ”A restricted class of exact eigenstates of the pairing-force Hamiltonian”, Phys. Lett 3, (1963) 277.
M. C. Cambiaggio, A. M. F. Rivas and M. Saraceno, “Integrability of the
pairing hamiltonian,” Nucl.Phys. A 624, 157 (1997), arXiv:nucl-th/9708031.
* [27] M. C. Cambiaggio, A. M. F. Rivas and M. Saraceno, “Integrability of the pairing hamiltonian,” Nucl.Phys. A 624, 157 (1997), arXiv:nucl-th/9708031.
* [28] G. Sierra, “Conformal field theory and the exact solution of the BCS Hamiltonian,” Nucl. Phys. B 572, 517 (2000), arXiv:hep-th/9911078.
M. Asorey, F. Falceto and G. Sierra, “Chern-Simons theory and BCS
superconductivity,” Nucl. Phys. B 622, 593 (2002), arXiv:hep-th/0110266.
* [29] C. Dunning and J. Links, ”Integrability of the Russian doll BCS model”, Nucl.Phys. B702 (2004) 481, arXiv:cond-mat/0406234 [cond-mat.stat-mech].
* [30] D. Gaiotto and P. Koroteev, On Three Dimensional Quiver Gauge Theories and Integrability, JHEP 1305, 126 (2013) [arXiv:1304.0779 [hep-th]].
* [31] A. Gorsky, A. Zabrodin and A. Zotov, “Spectrum of Quantum Transfer Matrices via Classical Many-Body Systems,” arXiv:1310.6958 [hep-th].
* [32] A. Veselov, Calogero quantum problem, Knizhnik-Zamolodchikov equation, and Huygens principle, Theor.Math.Phys. 98, i.3 (1994) 368-376.
* [33] M. R. Adams, J. Harnad, J. Hurtubise, Lett. Math. Phys., Vol. 20, Num. 4, 299-308 (1990).
E. Mukhin, V. Tarasov, A Varchenko, ”Bispectral and (glN, glM) dualities,
discrete versus differential”, Advances in Mathematics, Volume 218, 2008
216-265;
* [34] K. Bulycheva and A. Gorsky, “BPS states in the Omega-background and torus knots,” arXiv:1310.7361 [hep-th].
* [35] N. Nekrasov and S. Shatashvili, “Supersymmetric vacua and Bethe ansatz,” Nucl.Phys.B, Proc.Suppl.192–193 2009:91–112 arXiv:0901.4744 [hep-th].
N. A. Nekrasov and S. L. Shatashvili, “Quantization of Integrable Systems and
Four Dimensional Gauge Theories,” arXiv:0908.4052 [hep-th].
* [36] A. Gorsky, A. Mironov, “Integrable Many-Body Systems and Gauge Theories,” arXiv:hep-th/0011197.
* [37] N. Nekrasov, “Seiberg-Witten Prepotential From Instanton Counting,” Adv.Theor.Math.Phys.7:831-864 (2004), arXiv:hep-th/0206161.
* [38] M. Shifman, A. Yung, “Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories,” Rev. Mod. Phys. 79, 1139 (2007), hep-th/0703267.
* [39] N. Dorey, T. Hollowood, S. Lee, “Quantization of Integrable Systems and a 2d/4d Duality,” arXiv:1103.5726 [hep-th].
N. Dorey, “The BPS spectra of two-dimensional supersymmetric gauge theories
with twisted mass terms,” JHEP 9811, 005 (1998) [hep-th/9806056].
* [40] A. Giveon and D. Kutasov, “Brane dynamics and gauge theory,” Rev. Mod. Phys. 71, 983 (1999) [hep-th/9802067].
* [41] P. Bedaque, H. Hammer and U. van Kolck, “Renormalization of the three-body system with short-range interaction”, Phys.Rev.Lett 82 (1999) 463, arXiv:nucl-th/9809025.
* [42] E. Poppitz, M. Unsal, ”‘Seiberg-Witten and ’Polyakov-like’ magnetic bion confinements are continuously connected,” JHEP 1107, 082 (2011). [arXiv:1105.3969 [hep-th]].
|
arxiv-papers
| 2014-02-11T10:30:24 |
2024-09-04T02:49:58.046620
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K. Bulycheva, A. Gorsky",
"submitter": "Ksenia Bulycheva",
"url": "https://arxiv.org/abs/1402.2431"
}
|
1402.2497
|
# Generalized Monge-Ampère Capacities
E. Di Nezza Institut Mathématiques de Toulouse
Université Paul Sabatier
31062 Toulouse
France [email protected] and Chinh H. Lu Chalmers
University of Technology
Mathematical Sciences
412 96 Gothenburg
Sweden [email protected]
(Date:
The authors are partially supported by the french ANR project MACK)
###### Abstract.
We study various capacities on compact Kähler manifolds which generalize the
Bedford-Taylor Monge-Ampère capacity. We then use these capacities to study
the existence and the regularity of solutions of complex Monge-Ampère
equations.
###### Contents
1. 1 Introduction
2. 2 Generalized Monge-Ampère Capacities
1. 2.1 Energy classes
2. 2.2 The $(\varphi,\psi)$-Capacity
3. 2.3 Proof of Theorem A
3. 3 Another proof of the Domination Principle
4. 4 Applications to Complex Monge-Ampère equations
1. 4.1 Proof of Theorem B
2. 4.2 (Non) Existence of solutions
3. 4.3 Proof of Theorem C
4. 4.4 Non Integrable densities
5. 4.5 The case of semipositive and big classes
6. 4.6 Critical Integrability
## 1\. Introduction
Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$ and let
$D$ be an arbitrary divisor on $X$. Consider the complex Monge-Ampère equation
(1.1) $(\omega+dd^{c}\varphi)^{n}=f\omega^{n},$
where $0\leq f\in L^{1}(X)$ is such that
$\int_{X}f\omega^{n}=\int_{X}\omega^{n}$. It follows from [20] and [16] that
equation (1.1) has a unique normalized solution in the finite energy class
$\mathcal{E}(X,\omega)$. We say that the solution $\varphi$ is normalized if
$\sup_{X}\varphi=0$.
If $f$ is strictly positive and smooth on $X$, we know from the seminal paper
of Yau [24] that the solution is also smooth on $X$. Recall that this solves
in particular the Calabi conjecture and allows to construct Ricci flat metrics
on $X$ whenever $c_{1}(X)=0$.
Given $f$ positive and smooth on $X\setminus D$, it is natural to investigate
the regularity of the solution. In [15] we have proved in many cases that the
solution $\varphi$ is smooth in $X\setminus D$.
As in the classical case of Yau [24], the most difficult step is to establish
an a priori $\mathcal{C}^{0}$-estimate. This estimate is much more difficult
in our situation since in general the solution is not globally bounded. A
natural idea is to bound the normalized solution from below by a singular
quasi plurisubharmonic function (qpsh for short). This is where generalized
Monge-Ampère capacities play a crucial role.
We recall the notion of the classical capacity ${\rm Cap}_{\omega}$ introduced
and studied in [22] and [19]:
${\rm Cap}_{\omega}(E)=\sup\left\\{\int_{E}(\omega+dd^{c}u)^{n}\ \ \big{|}\ \
u\in{\rm PSH}(X,\omega),\ -1\leq u\leq 0\right\\},\ E\subset X.$
A strong comparison between the Lebesgue measure and ${\rm Cap}_{\omega}$, as
is needed in a celebrated method due to Kołodziej [21], does not hold in our
setting. We therefore study other capacities to provide an a priori
$\mathcal{C}^{0}$-estimate. In dealing with complex Monge-Ampère equations in
quasiprojective varieties we were naturally lead to work with generalized
capacities of type ${\rm Cap}_{\psi-1,\psi}$ in [15] (see below for their
definition).
In this paper, we make a systematic study of these capacities as well as the
more general ${\rm Cap}_{\varphi,\psi}$ capacities: let $\varphi,\psi$ be two
$\omega$-plurisubharmonic functions on $X$ such that $\varphi<\psi$ on $X$
modulo possibly a pluripolar set. The $(\varphi,\psi)$-Capacity of a Borel
subset $E\subset X$ is defined by
${\rm Cap}_{\varphi,\psi}(E):=\sup\left\\{\int_{E}(\omega+dd^{c}u)^{n}\ \
\big{|}\ \ u\in{\rm PSH}(X,\omega),\ \varphi\leq u\leq\psi\right\\}.$
Here, for a $\omega$-psh function $u$, $(\omega+dd^{c}u)^{n}$ is the non-
pluripolar Monge-Ampère measure of $u$. See Section 2 for the definition. When
$\varphi\equiv\psi-1$, we drop the index $\varphi$ and denote the
$(\psi-1,\psi)$-Capacity by ${\rm Cap}_{\psi}$,
${\rm Cap}_{\psi}:={\rm Cap}_{\psi-1,\psi}.$
This is exactly the generalized capacity used in our previous paper [15]. If
moreover $\psi$ is constant, $\psi\equiv C$, we recover the Monge-Ampère
capacity defined above
${\rm Cap}_{C}={\rm Cap}_{\omega}.$
Given any subset $E\subset X,$ we define the outer $(\varphi,\psi)$-capacity
of $E$ by
${\rm Cap}_{\varphi,\psi}^{*}(E):=\inf\left\\{{\rm Cap}_{\varphi,\psi}(U)\
\big{|}\ U\ \text{is an open subset of }X,\ E\subset U\right\\}.$
We say that the $(\varphi,\psi)$-capacity characterizes pluripolar sets on $X$
if for any subset $E\subset X$, the following holds
${\rm Cap}_{\varphi,\psi}^{*}(E)=0\Longleftrightarrow\ \text{E is a pluripolar
subset of}\ X.$
If $E\subset X$ is a Borel subset we set
$h_{\varphi,\psi,E}(x):=\sup\left\\{u(x)\ \big{|}\ u\in{\rm
PSH}(X,\omega),u\leq\psi\ {\rm on}\ X,\ u\leq\varphi\;\text{q.e.}\
E\right\\}.$
Here, quasi everywhere (q.e. for short) means outside a pluripolar set. Let
$h_{\varphi,\psi,E}^{*}$ be its upper semicontinuous regularization which we
call the $(\varphi,\psi)$-extremal function of $E$. We establish a useful
characterization of the $(\varphi,\psi)$-capacity in terms of the relative
extremal function for any subset.
When $\varphi$ belong to the finite energy class $\mathcal{E}(X,\omega)$ we
can bound ${\rm Cap}_{\varphi,\psi}$ by $F({\rm Cap}_{\omega})$ for some
positive function $F$ which vanishes at $0$. This uniform control turns out to
be very useful in studying convergence of the complex Monge-Ampère operator
since it allows us to replace quasi-continuous functions by continuous ones
without affecting the final result. We also prove that the generalized Monge-
Ampère capacity ${\rm Cap}_{\varphi,\psi}$ characterizes pluripolar sets when
the lower weight is in $\mathcal{E}(X,\omega)$:
Theorem A. Assume that $\varphi\in\mathcal{E}(X,\omega)$ and $\psi\in{\rm
PSH}(X,\omega)$ such that $\varphi<\psi$ modulo a pluripolar subset.
* (i)
Let $E\subset X$ be a Borel subset of $X$, and denote by $h_{E}$ the
$(\varphi,\psi)$-extremal function of $E$. The outer $(\varphi,\psi)$-capacity
of $E$ is given by
${\rm
Cap}_{\varphi,\psi}^{*}(E)=\int_{\\{h_{E}<\varphi\\}}\mathrm{MA}\,(h_{E})=\int_{X}\left(\frac{-h_{E}+\psi}{-\varphi+\psi}\right)\mathrm{MA}\,(h_{E}),$
where $h_{E}:=h^{*}_{\varphi,\psi,E}$ is the $(\varphi,\psi)$-extremal
function of $E$.
* (ii)
There exists a function $F:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
$\lim_{t\to 0^{+}}F(t)=0$ and such that for all Borel subset $E$,
${\rm Cap}_{\varphi,\psi}(E)\leq F({\rm Cap}_{\omega}(E)).$
* (iii)
${\rm Cap}_{\varphi,\psi}$ characterizes pluripolar sets.
We stress that the function $F$ in $(ii)$ is quite explicit (see Theorem 2.9).
As we have underlined, these generalized capacities play an important role in
studying complex Monge-Ampère equations on quasi-projective varieties (see
[15]). We give in the second part of this paper several other applications.
We consider the following complex Monge-Ampère equation
(1.2) $(\omega+dd^{c}\varphi)^{n}=e^{\lambda\varphi}f\omega^{n},\
\lambda\in\mathbb{R}.$
Assume that $0<f\in\mathcal{C}^{\infty}(X\setminus D)$ satisfies Condition
$\mathcal{H}_{f}$, i.e. $f$ can be written as
$f=e^{\psi^{+}-\psi^{-}},\ \ \psi^{\pm}\ {\rm are\ quasi\ psh\ functions\ on}\
X\ ,\ \psi^{-}\in L^{\infty}_{\rm loc}(X\setminus D).$
When $\lambda=0$ and $f$ satisfies $\int_{X}f\omega^{n}=\int_{X}\omega^{n}$,
we proved in [15] that there is a unique normalized solution in
$\mathcal{E}(X,\omega)$ which is smooth on $X\setminus D$. When $\lambda>0$
and $\int_{X}f\omega^{n}<+\infty$ the same result holds since the
$\mathcal{C}^{0}$ estimate follows easily from the comparison principle.
Consider now the case when $\lambda<0$. In this case solutions do not always
exist and when they do, there may be many of them. Our result here says that
any solution in $\mathcal{E}(X,\omega)$ (if exists) is smooth on $X\setminus
D$.
Theorem B. Let $0<f\in\mathcal{C}^{\infty}(X\setminus D)\cap L^{1}(X)$. Assume
that $f$ satisfies Condition $\mathcal{H}_{f}$ and
$\varphi\in\mathcal{E}(X,\omega)$ is a solution of
$(\omega+dd^{c}\varphi)^{n}=e^{\lambda\varphi}f\omega^{n},\ \lambda<0.$
Then $\varphi$ is smooth on $X\setminus D$.
Note that when $\lambda<0$ and equation (1.2) has a solution in
$\mathcal{E}(X,\omega)$, the measure $\mu=f\omega^{n}$ is dominated by
$\mathrm{MA}\,(u)$ for some $u\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$. In
particular, $f\in L^{1}(X)$.
We next investigate the case when $\lambda>0$ and $f$ is not integrable on
$X$. Of course solutions do not always exist. But observe that when $\varphi$
is singular enough $e^{\varphi}f$ will be integrable on $X$ and it is then
reasonable to find a solution. For example, one can look at densities of the
type
$f\simeq\frac{1}{|s|^{2}},$
which is not integrable. Here $s$ is a holomorphic section of the line bundle
associated to $D$. Such densities have been considered by Berman and Guenancia
in their study of the compactification of the moduli space of canonically
polarized manifolds [5]. They have shown that there exists a unique solution
$\varphi\in\mathcal{E}(X,\omega)$ which is smooth in $X\setminus D$. As
another application of the generalized Monge-Ampère capacities we show in the
following result that in a general context whenever a solution in
$\mathcal{E}(X,\omega)$ exists it is smooth outside $D$.
Theorem C. Assume $0<f\in\mathcal{C}^{\infty}(X\setminus D)$ satisfies
Condition $\mathcal{H}_{f}$. If the equation
$(\omega+dd^{c}\varphi)^{n}=e^{\lambda\varphi}f\omega^{n},\ \lambda>0$
admits a solution $\varphi\in\mathcal{E}(X,\omega)$ then $\varphi$ is smooth
on $X\setminus D$.
Let us stress that in Theorem C we do not assume that
$\int_{X}f\omega^{n}<+\infty$. It turns out that the existence of solutions in
$\mathcal{E}(X,\omega)$ is equivalent to the existence of subsolutions in this
class, these are easy to construct in concrete situations (see Example 4.7).
We also obtain a similar result in the case of semipositive and big classes
(see Theorem 4.8 and Example 4.9).
Finally we use generalized capacitites to study the critical integrability of
a given $\phi\in{\rm PSH}(X,\omega)$.
Theorem D. Let $\phi\in{\rm PSH}(X,\omega)$ and
$\alpha=\alpha(\phi)\in(0,+\infty)$ be the canonical threshold of $\phi$, i.e.
$\alpha=\alpha(\phi):=\sup\\{t>0\ \ \big{|}\ \ e^{-t\phi}\in L^{1}(X)\\}.$
Then there exists $u\in{\rm PSH}(X,\omega)$ with zero Lelong number at all
points such that $e^{u-\alpha\phi}$ is integrable. Moreover, there exists a
unique $\varphi\in\mathcal{E}(X,\omega)$ such that
$(\omega+dd^{c}\varphi)^{n}=e^{\varphi-\alpha\phi}\omega^{n}.$
It turns out that one can even chose $u=\chi\circ\phi$ in
$\mathcal{E}(X,\omega)$, as an explicit function of $\phi$ with attenuated
singularities (see Theorem 4.10).
The paper is organized as follows. In section 2 we recall some known facts on
energy classes, we introduce generalized capacities on compact Kähler
manifolds and prove Theorem A. As an application of the generalized capacities
we give another proof of the domination principle in $\mathcal{E}(X,\omega)$
in Section 3. In Section 4 we use generalized capacities to study complex
Monge-Ampère equations as (1.2). The proof of Theorem D will be given in
Section 4 as well.
Acknowledgements. We would like to thank Vincent Guedj and Ahmed Zeriahi for
constant help, many suggestions and encouragements. We also thank Robert
Berman and Bo Berndtsson for useful discussions. We are indebted to Henri
Guenancia for a careful reading and very useful comments on a previous draft
version of this paper.
## 2\. Generalized Monge-Ampère Capacities
Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$. In
this section we prove some basic properties of the $(\varphi,\psi)$-capacity
and of the relative $(\varphi,\psi)$-extremal functions.
### 2.1. Energy classes
###### Definition 2.1.
We let ${\rm PSH}(X,\omega)$ denote the class of $\omega$-plurisubharmonic
functions ($\omega$-psh for short) on $X$, i.e. the class of functions
$\varphi$ such that locally $\varphi=\rho+u$, where $\rho$ is a local
potential of $\omega$ and $u$ is a plurisubharmonic function.
Let $\varphi$ be some unbounded $\omega$-psh function on $X$ and consider
$\varphi_{j}:=\max(\varphi,-j)$ the ”canonical approximants”. It has been
shown in [20] that
${\bf 1}_{\\{\varphi_{j}>-j\\}}(\omega+dd^{c}\varphi_{j})^{n}$
is a non-decreasing sequence of Borel measures. We denote its limit by
$\mathrm{MA}\,(\varphi)=(\omega+dd^{c}\varphi)^{n}:=\lim_{j\to+\infty}{\bf
1}_{\\{\varphi_{j}>-j\\}}(\omega+dd^{c}\varphi_{j})^{n}.$
###### Definition 2.2.
We denote by $\mathcal{E}(X,\omega)$ the set of $\omega$-psh functions having
full Monge-Ampère mass:
$\mathcal{E}(X,\omega):=\left\\{\varphi\in{\rm PSH}(X,\omega)\ \ \big{|}\ \
\int_{X}\mathrm{MA}\,(\varphi)=\int_{X}\omega^{n}\right\\}.$
Let us stress that $\omega$-psh functions with full Monge-Ampère mass have
mild singularities. In particular, any $\varphi\in\mathcal{E}(X,\omega)$ has
zero Lelong numbers $\nu(\varphi,\cdot)=0$ (see [20, Corollary 1.8]). We also
recall that, for every $\varphi\in\mathcal{E}(X,\omega)$ and any $\psi\in{\rm
PSH}(X,\omega)$, the _generalized comparison principle_ is valid, namely
$\int_{\\{\varphi<\psi\\}}(\omega+dd^{c}\psi)^{n}\leq\int_{\\{\varphi<\psi\\}}(\omega+dd^{c}\varphi)^{n}.$
###### Definition 2.3.
Let $\chi:\mathbb{R}^{-}\to\mathbb{R}^{-}$ be an increasing function such that
$\chi(0)=0$ and $\chi(-\infty)=-\infty$. We denote by
$\mathcal{E}_{\chi}(X,\omega)$ the class of $\omega$-psh functions having
finite $\chi$-energy:
$\mathcal{E}_{\chi}(X,\omega):=\left\\{\varphi\in\mathcal{E}(X,\omega)\;\,|\;\,\chi(-|\varphi|)\in
L^{1}(\mathrm{MA}\,(\varphi))\right\\}.$
For $p>0$, we use the notation
${\mathcal{E}}^{p}(X,\omega):=\mathcal{E}_{\chi}(X,\omega),\text{ when
}\chi(t)=-(-t)^{p}.$
### 2.2. The $(\varphi,\psi)$-Capacity
In this subsection we always assume that $\varphi,\psi\in{\rm PSH}(X,\omega)$
are such that $\varphi<\psi$ quasi everywhere on $X$. The
$(\varphi,\psi)$-capacity of a Borel subset $E\subset X$ is defined by
${\rm Cap}_{\varphi,\psi}(E):=\sup\left\\{\int_{E}\mathrm{MA}\,(u)\ \ \big{|}\
\ u\in{\rm PSH}(X,\omega),\ \varphi\leq u\leq\psi\right\\}.$
When $\varphi\equiv\psi-1$, to simplify the notation we simply denote
${\rm Cap}_{\psi}:={\rm Cap}_{\psi-1,\psi}.$
If moreover $\psi\equiv C$ is constant we recover the Monge-Ampère capacity
introduced in [2], [22], [19]. The following properties of the
$(\varphi,\psi)$-Capacity follow straightforward from the definition.
###### Proposition 2.4.
(i) If $E_{1}\subset E_{2}\subset X$ then ${\rm
Cap}_{\varphi,\psi}(E_{1})\leq{\rm Cap}_{\varphi,\psi}(E_{2})$ .
(ii) If $E_{1},E_{2},\cdots$ are Borel subsets of $X$ then
${\rm
Cap}_{\varphi,\psi}\left(\bigcup_{j=1}^{\infty}E_{j}\right)\leq\sum_{j=1}^{+\infty}{\rm
Cap}_{\varphi,\psi}(E_{j}).$
(iii) If $E_{1}\subset E_{2}\subset\cdots$ are Borel subsets of $X$ then
${\rm
Cap}_{\varphi,\psi}\left(\bigcup_{j=1}^{\infty}E_{j}\right)=\lim_{j\to+\infty}{\rm
Cap}_{\varphi,\psi}(E_{j}).$
The outer $(\varphi,\psi)$-capacity of $E$ is defined by
${\rm Cap}_{\varphi,\psi}^{*}(E):=\inf\left\\{{\rm Cap}_{\varphi,\psi}(U)\
\big{|}\ U\ \text{is an open subset of }X,\ E\subset U\right\\}.$
We say that the $(\varphi,\psi)$-capacity characterizes pluripolar sets on $X$
if for any subset $E\subset X$, the following holds
${\rm Cap}_{\varphi,\psi}^{*}(E)=0\Longleftrightarrow\ \text{E is a pluripolar
subset of}\ X.$
###### Definition 2.5.
If $E\subset X$ is a Borel subset we set
$h_{\varphi,\psi,E}:=\sup\left\\{u\in{\rm PSH}(X,\omega),\ u\leq\varphi\
\text{quasi\ everywhere\ on}\ E,u\leq\psi\ \text{on}\ X\right\\},$
where ”quasi everywhere” means outside a pluripolar set. The upper
semicontinuous regularization of $h_{\varphi,\psi,E}$ is called the relative
$(\varphi,\psi)$-extremal function of $E$.
###### Proposition 2.6.
Let $E\subset X$.
* (i)
The function $h_{\varphi,\psi,E}^{*}$ is $\omega$-psh. It satisfies
$\varphi\leq h_{\varphi,\psi,E}^{*}\leq\psi$ on $X$ and
$h_{\varphi,\psi,E}^{*}=\varphi$ quasi everywhere on $E.$
* (ii)
If $P\subset E$ is pluripolar, then $h_{\varphi,\psi,E\setminus P}^{*}\equiv
h_{\varphi,\psi,E}^{*}$; in particular $h_{\varphi,\psi,P}^{*}\equiv\psi$.
* (iii)
If $(E_{j})$ are subsets of $X$ increasing towards $E\subset X$, then
$h_{\varphi,\psi,E_{j}}^{*}$ decreases towards $h_{\varphi,\psi,E}^{*}$.
* (iv)
If $h^{*}_{\varphi,\psi,E}\equiv\psi$ then $E$ is pluripolar.
###### Proof.
The statement $(i)$ is a standard consequence of Bedford-Taylor’s work [2].
Set $E_{1}:=E\setminus P$, and denote by $h=h^{*}_{\varphi,\psi,E},\
h_{1}=h^{*}_{\varphi,\psi,E_{1}}$ the corresponding $(\varphi,\psi)$-extremal
functions of $E,E_{1}$. Since $E_{1}\subset E$ it is clear that $h_{1}\geq h$.
On the other hand $h_{1}=\varphi$ quasi everywhere on $E_{1}$ hence on $E$.
This yields $h_{1}\leq h$ whence equality.
Let us prove $(iii)$. Since $(E_{j})$ is increasing,
$h_{j}:=h^{*}_{\varphi,\psi,E_{j}}$ is decreasing toward $h\in{\rm
PSH}(X,\omega).$ It is clear that $h\geq h^{*}_{\varphi,\psi,E}$. By
definition, for each $j\in\mathbb{N}$, $h_{j}=\varphi$ quasi everywhere on
$E_{j}$. It then follows that $h=\varphi$ quasi everywhere on $E$. We then
infer that $h\leq h^{*}_{\varphi,\psi,E}$, hence the equality.
To prove $(iv)$ assume that $h^{*}_{\varphi,\psi,E}\equiv\psi$. By definition
of $h:=h^{*}_{\varphi,\psi,E}$ and by Choquet’s lemma we can find an
increasing sequence $(u_{j})$ such that $u_{j}=\varphi$ on $E$ and
$\left(\lim_{j\rightarrow+\infty}u_{j}\right)^{*}=h$. Note that
$E\subset\left\\{\left(\limsup_{j\rightarrow+\infty}u_{j}\right)<\left(\limsup_{j\rightarrow+\infty}u_{j}\right)^{*}\right\\},$
modulo a pluripolar set. The latter is also pluripolar, hence $E$ is
pluripolar. ∎
###### Theorem 2.7.
If $\varphi\in\mathcal{E}(X,\omega)$ and $E\subset X$ is pluripolar then ${\rm
Cap}_{\varphi,\psi}^{*}(E)=0$.
###### Proof.
Assume that $\varphi\in\mathcal{E}(X,\omega)$ and fix a pluripolar set
$E\subset X$. By translating $\psi$ and $\varphi$ by a constant we can assume
that $\psi\leq 0$. It follows from [20, Proposition 2.2] that
$\varphi\in\mathcal{E}_{\chi}(X,\omega)$ for some convex increasing function
$\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$. We can find
$u\in\mathcal{E}_{\chi}(X,\omega),u\leq 0$ such that
$E\subset\\{u=-\infty\\}$. We claim that
${\rm
Cap}_{\varphi,\psi}(\\{u<-t\\})\leq\frac{-2}{\chi(-t)}\left(E_{\chi}(u)+2^{n}E_{\chi}(\varphi)\right),\
\forall t>0.$
Indeed, let $v\in PSH(X,\omega)$ such that $\varphi\leq v\leq\psi.$ We obtain
immediately that
$\int_{\\{u<-t\\}}\mathrm{MA}\,(v)\leq\frac{1}{-\chi(-t)}\int_{\\{u<-t\\}}(-\chi\circ
u)\mathrm{MA}\,(v).$
From this and [20, Proposition 2.5] we get
$\int_{\\{u<-t\\}}MA(v)\leq\frac{-2}{\chi(-t)}\left(E_{\chi}(u)+E_{\chi}(v)\right).$
This coupled with the fundamental inequality in [20, Lemma 2.3] yield the
claim. Since for any $t>0$, $E\subset\\{u<-t\\}$ we obtain
${\rm Cap}_{\varphi,\psi}^{*}(E)\leq{\rm Cap}_{\varphi,\psi}(u<-t)\to 0\ \
\text{as}\ \ t\to+\infty.$
∎
From now on we fix $\varphi,\psi$ two functions in $\mathcal{E}(X,\omega)$
such that $\varphi<\psi$ quasi everywhere on $X$.
Given any $u\in{\rm PSH}(X,\omega)$ such that $u\leq 0$, it follows from [20,
Example 2.14] (see also the Main Theorem in [12]) that $u_{p}:=-(-u)^{p}$
belongs to $\mathcal{E}(X,\omega)$ for any $0<p<1$. The same arguments can be
applied to get the following result:
###### Lemma 2.8.
Let $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ be any measurable function.
Assume that there exists $q>0$ such that
$\sup_{t\leq-1}|\chi(t)|(-t)^{-q}=C<+\infty.$
Then for any $u\in{\rm PSH}(X,\omega)$ such that $u\leq-1$ and any
$0<p<\frac{1}{q+1}$ we have
$\int_{X}|\chi\circ u_{p}|\mathrm{MA}\,(u_{p})\leq A,$
where $u_{p}:=-(-u)^{p}$ and $A$ is a positive constant depending only on
$C,p,q$.
###### Proof.
In the proof we use $A$ to denote various positive constants which are under
control. By considering $u^{j}:=\max(u,-j)$, the canonical approximants of
$u$, and letting $j\to+\infty$ it suffices to treat the case when $u$ is
bounded. We compute
$\omega+dd^{c}u_{p}=\omega+p(1-p)(-u)^{p-2}du\wedge
d^{c}u+p(-u)^{p-1}dd^{c}u.$
We thus get
$0\leq\omega+dd^{c}u_{p}\leq(-u)^{p-1}(\omega+dd^{c}u)+\omega+(-u)^{p-2}du\wedge
d^{c}u.$
We need to verify the following bounds:
$\int_{X}|\chi\circ u_{p}|(-u)^{p-1}(\omega+dd^{c}u)^{k}\wedge\omega^{n-k}\leq
A$
and
$\int_{X}|\chi\circ u_{p}|(-u)^{p-2}du\wedge
d^{c}u\wedge(\omega+dd^{c}u)^{k}\wedge\omega^{n-k-1}\leq A,$
where $k=0,1,...,n$. Let us consider the first one. By assumption we have
$|\chi\circ u_{p}|(-u_{p})^{-q}\leq C.$
To bound the first term, it thus suffices to get a bound for
$\int_{X}(-u)^{p-1+pq}(\omega+dd^{c}u)^{k}\wedge\omega^{n-k},$
which is easy since $p+pq-1<0$. For the second one it suffices get a bound for
$\int_{X}(-u)^{p-2+pq}du\wedge
d^{c}u\wedge(\omega+dd^{c}u)^{k}\wedge\omega^{n-k-1},$
which follows easily by the same reason and by integration by parts.
∎
We know from Theorem 2.7 that ${\rm Cap}_{\varphi,\psi}$ vanishes on
pluripolar subsets of $X$. This suggests that ${\rm Cap}_{\varphi,\psi}$ is
dominated by $F({\rm Cap}_{\omega})$, where $F$ is some positive function
vanishing at $0$. The following result gives an explicit formula of $F$.
###### Theorem 2.9.
Let $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ be a convex increasing
function and $\varphi\in\mathcal{E}_{\chi}(X,\omega)$. Let $q>0$ be a positive
real number such that
(2.1) $\sup_{t\leq-1}|\chi(t)|(-t)^{-q}<+\infty.$
Then for any $p<\frac{1}{1+q}$ there exists $C>0$ depending on
$p,q,\chi,\varphi$ such that
${\rm Cap}_{\varphi,0}(K)\leq\frac{C}{\left|\chi\left(-{\rm
Cap}_{\omega}(K)^{\frac{-p}{n}}\right)\right|}\ ,\ \forall K\subset X.$
As a concrete example, when $\varphi\in\mathcal{E}^{q}(X,\omega)$ for some
$q>0$ and $p<1/(1+q)$, then we can take $F(s):=s^{\frac{pq}{n}}$ for $s>0$,
getting
${\rm Cap}_{\varphi,0}(K)\leq C\,{\rm Cap}_{\omega}(K)^{\frac{pq}{n}}.$
###### Proof.
Fix $p>0$ such that $p(q+1)<1$. Let $V_{K}$ be the extremal
$\omega$-plurisubharmonic function of $K$:
$V_{K}:=\sup\\{u\ \ \big{|}\ \,u\in{\rm PSH}(X,\omega),\,u\leq 0\,\
\rm{on}\,\,K\\},$
and $M_{K}:=\sup_{X}V_{K}^{*}$. It follows from (2.1) and Lemma 2.8 that the
function
$u=-(-V_{K}^{*}+M_{K}+1)^{p}$
belongs to $\mathcal{E}_{\chi}(X,\omega)$. Fix $h\in{\rm PSH}(X,\omega)$ be
such that $\varphi\leq h\leq 0$. It follows from Lemma 2.10 below that
$\int_{X}|\chi\circ u|\mathrm{MA}\,(h)\leq C_{1},$
where $C_{1}>0$ only depends on $\chi$, $p,q$ and $\varphi$. Therefore, using
the fact that $V_{K}^{*}\equiv 0$ quasi everywhere on $K$ we get
$\displaystyle\int_{K}\mathrm{MA}\,(h)\leq\int_{X}\frac{|\chi\circ
u|}{|\chi(-M_{K}^{p})|}\omega_{h}^{n}\leq\frac{C_{1}}{|\chi(-M_{K}^{p})|}.$
It follows from [19] that $M_{K}\geq C_{2}{\rm Cap}(K)^{-1/n}.$ This coupled
with the above yield the result. ∎
###### Lemma 2.10.
Assume that $\chi$, $p,q$ and $\varphi$ are as in Theorem 2.9. Then there
exists $C>0$ depending on $\chi,p,q,\varphi$ such that
$\int_{X}|\chi(-(-u)^{p})|\mathrm{MA}\,(v)\leq C,\ \forall u,v\in{\rm
PSH}(X,\omega),\ \sup_{X}u=-1,\ \varphi\leq v\leq 0.$
###### Proof.
We argue by contradiction, assuming that there are two sequences $(u_{j})$,
$(v_{j})$ of functions in ${\rm PSH}(X,\omega)$ such that $\sup_{X}u_{j}=-1$,
$\varphi\leq v_{j}\leq 0$, and
$\int_{X}|\chi(-(-u_{j})^{p})|\mathrm{MA}\,(v_{j})\geq 2^{(n+2)j},\ \forall
j\in\mathbb{N}.$
Set
$u:=\sum_{j=1}^{+\infty}2^{-j}u_{j},\ v=\sum_{j=1}^{+\infty}2^{-j}v_{j}.$
Then $u\in{\rm PSH}(X,\omega)$, $u\leq-1$. Moreover, it follows from Lemma 2.8
that
$u_{p}:=-(-u)^{p}\in\mathcal{E}_{\chi}(X,\omega).$
We also have $\varphi\leq v\leq 0$, in particular
$v\in\mathcal{E}_{\chi}(X,\omega)$. But
$\int_{X}|\chi\circ
u_{p}|\mathrm{MA}\,(v)\geq\sum_{j=1}^{+\infty}2^{j}=+\infty,$
which contradicts [20, Proposition 2.5]. ∎
###### Proposition 2.11.
Let $E$ be a Borel subset of $X$ and set $h_{E}:=h^{*}_{\varphi,\psi,E}$ the
relative $(\varphi,\psi)$-extremal function of $E$. Then
$\mathrm{MA}\,(h_{E})\equiv 0\ {\rm on}\ \\{h_{E}<\psi\\}\setminus\bar{E}.$
###### Proof.
We first assume that $\psi$ is continuous on $X$. Set $h:=h_{E}$ and let
$x_{0}\in X\setminus\bar{E}$ be such that $(h-\psi)(x_{0})<0.$ Let
$B:=B(x_{0},r)\subset X\setminus\bar{E}$ be a small ball around $x_{0}$ such
that $\sup_{\bar{B}}(h-\psi)(x)=-2\delta<0.$ Let $\rho$ be a local potential
of $\omega$ in $B.$ Shrinking $B$ a little bit we can assume that
$\sup_{\bar{B}}|\rho|<\delta$ and ${\rm osc}_{\bar{B}}\psi<\delta/2$. By
definition of $h$ and by Choquet’s lemma we can find an increasing sequence
$(u_{j})_{j}\subset\mathcal{E}(X,\omega)$ such that $u_{j}=\varphi$ quasi
everywhere on $E$, $u_{j}\leq\psi$ on $X$, and $(\lim_{j}u_{j})^{*}=h$. For
each $j,k\in\mathbb{N}$, we solve the Dirichlet problem to find
$v_{j}^{k}\in{\rm PSH}(X,\omega)\cap L^{\infty}(X)$ such that
$\mathrm{MA}\,(v_{j}^{k})=0$ in $B$ and $v_{j}^{k}\equiv\max(u_{j},-k)$ on
$X\setminus B$. Since
$\rho+v_{j}^{k}\leq\rho+h\leq-\delta+\psi\leq\sup_{\bar{B}}\psi-\delta$
on $\partial B$, we deduce from the maximum principle that
$v_{j}^{k}\leq\inf_{\bar{B}}\psi-\delta/2-\rho\leq\psi$ on $B$. Furthermore,
taking $k$ big enough such that $\psi\geq-k$, we can conclude that
$v_{j}^{k}\leq\psi$ on $X$. For $j\in\mathbb{N}$ fixed, by the comparison
principle $(v_{j}^{k})_{k}$ decreases to $v_{j}\in\mathcal{E}(X,\omega)$. Then
$u_{j}\leq v_{j}\leq h$ since $v_{j}=u_{j}=\varphi$ on $E$ and $v_{j}\leq\psi$
on $X$. It follows from [20] that the sequence of Monge-Ampère measures
$MA(v_{j}^{k})$ converges weakly to $MA(v_{j})$. Thus $MA(v_{j})(B)=0.$ On the
other hand, $v_{j}$ increases almost everywhere to $h$ and these functions
belong to $\mathcal{E}(X,\omega).$ The same arguments as in [20, Theorem 2.6]
show that $MA(v_{j})$ converges weakly to $MA(h)$. We infer that $MA(h)(B)=0$.
It remains to remove the continuity hypothesis on $\psi$. Let $(\psi_{j})$ be
a sequence of continuous functions in ${\rm PSH}(X,\omega)$ decreasing to
$\psi$ on $X$. Let $h_{j}:=h_{\varphi,\psi_{j},E}^{*}$ be the relative
$(\varphi,\psi_{j})$-extremal function of $K$. Then $h_{j}$ decreases to $h$,
hence $\mathrm{MA}\,(h_{j})$ converges weakly to $\mathrm{MA}\,(h)$. Denote by
$V:=\\{h<\psi\\}\setminus\bar{E}$. Now, fix $\varepsilon>0$ and $U$ an open
subset of $X$ such that
${\rm Cap}_{\omega}\left[(U\setminus V)\cup(V\setminus
U)\right]\leq\varepsilon.$
From the first step we know that $\mathrm{MA}\,(h_{j})$ vanishes on $V$. Thus
$\displaystyle\int_{V}\mathrm{MA}\,(h)$ $\displaystyle\leq$
$\displaystyle\int_{U}\mathrm{MA}\,(h)+F(\varepsilon)$ $\displaystyle\leq$
$\displaystyle\liminf_{j\to+\infty}\int_{U}\mathrm{MA}\,(h_{j})+F(\varepsilon)$
$\displaystyle\leq$
$\displaystyle\liminf_{j\to+\infty}\int_{V}\mathrm{MA}\,(h_{j})+2F(\varepsilon)$
$\displaystyle=$ $\displaystyle 2F(\varepsilon),$
It suffices now to let $\varepsilon\to 0$ since $\lim_{\varepsilon\to
0}F(\varepsilon)=0$ thanks to Theorem 2.9. ∎
###### Lemma 2.12.
Let $E\subset X$ be a Borel subset and $h_{E}:=h^{*}_{\varphi,\psi,E}$ be its
relative $(\varphi,\psi)$-extremal function. Then we have
${\rm Cap}_{\varphi,\psi}(E)\leq\int_{\\{h_{E}<\psi\\}}\mathrm{MA}\,(h_{E}).$
###### Proof.
Observe first that the $(\varphi,\psi)$-capacity can be equivalently defined
by
${\rm Cap}_{\varphi,\psi}(E):=\sup\left\\{\int_{E}\mathrm{MA}\,(u)\ |\
u\in{\rm PSH}(X,\omega),\ \varphi<u\leq\psi\right\\}.$
For simplicity, set $h:=h_{E}$. Now take any $u\in{\rm PSH}(X,\omega)$ such
that $\varphi<u\leq\psi$. Then
$E\subset\\{h<u\\}\subset\\{h<\psi\\},$
where the first inclusion holds modulo a pluripolar set. The comparison
principle for functions in $\mathcal{E}(X,\omega)$ (see [20]) yields
$\int_{E}MA(u)\leq\int_{\\{h<u\\}}MA(u)\leq\int_{\\{h<u\\}}MA(h)\leq\int_{\\{h<\psi\\}}MA(h).$
By taking the supremum over all candidates $u$, we get the result. ∎
The following result says that the inequality in Lemma 2.12 is an equality if
$E$ is a compact or open subset of $X$.
###### Theorem 2.13.
Let $E$ be an open (or compact) subset of $X$ and let
$h_{E}:=h^{*}_{\varphi,\psi,E}$ be the $(\varphi,\psi)$-extremal function of
$E$. The $(\varphi,\psi)$-capacity of $E$ is given by
${\rm Cap}_{\varphi,\psi}(E)=\int_{\\{h_{E}<\psi\\}}MA(h_{E}).$
###### Proof.
From Lemma 2.12 above we get one inequality. We now prove the opposite one.
Set $h:=h_{E}$. Assume first that $E$ is a compact subset of $X.$ Let
$(\psi_{j})$ be a sequence of continuous $\omega$-psh functions decreasing to
$\psi$. Denote by $h_{j}:=h_{\varphi,\psi_{j},E}^{*}$. It is easy to check
that $h_{j}$ decreases to $h$ and that ${\rm Cap}_{\varphi,\psi_{j}}(E)$
decreases to ${\rm Cap}_{\varphi,\psi}(E)$. Since $h_{j}$ is a candidate
defining the $(\varphi,\psi_{j})$-capacity of $E$, it follows from Proposition
2.11 and Lemma 2.12 that
(2.2) ${\rm
Cap}_{\varphi,\psi_{j}}(E)=\int_{\\{h_{j}<\psi_{j}\\}}MA(h_{j})=\int_{E}MA(h_{j}).$
Fix $j_{0}\in\mathbb{N}.$ Since $h_{j}\leq h_{j_{0}}$ and $\psi\leq\psi_{j}$,
for any $j>j_{0}$
$\int_{\\{h_{j}<\psi_{j}\\}}MA(h_{j})\geq\int_{\\{h_{j_{0}}<\psi\\}}MA(h_{j}).$
Fix $\varepsilon>0$ and replacing $\psi$ by a continuous function
$\tilde{\psi}$ such that ${\rm
Cap}_{\omega}(\\{\tilde{\psi}\neq\psi\\})<\varepsilon$. Arguing as in the
proof of Proposition 2.11 we get
$\displaystyle\liminf_{j\rightarrow+\infty}\int_{\\{h_{j_{0}}<\psi\\}}MA(h_{j})\geq\int_{\\{h_{j_{0}}<\psi\\}}MA(h).$
Taking the limit for $j\rightarrow+\infty$ in (2.2) we get
${\rm Cap}_{\varphi,\psi}(E)\geq\int_{\\{h<\psi\\}}MA(h).$
We now assume that $E\subset X$ is an open set. Let $(K_{j})$ be a sequence of
compact subsets increasing to $E$. Then clearly
$h_{j}:=h^{*}_{\varphi,\psi,K_{j}}\searrow h$ and ${\rm
Cap}_{\varphi,\psi}(K_{j})\nearrow{\rm Cap}_{\varphi,\psi}(E)$. We have
already proved that ${\rm
Cap}_{\varphi,\psi}(K_{j})\geq\int_{\\{h_{j}<\psi\\}}MA(h_{j})$. For each
fixed $k\in\mathbb{N}$, we have
$\liminf_{j\to+\infty}\int_{\\{h_{j}<\psi\\}}MA(h_{j})\geq\liminf_{j\to+\infty}\int_{\\{h_{k}<\psi\\}}MA(h_{j})\geq\int_{\\{h_{k}<\psi\\}}MA(h).$
Then letting $k\to+\infty$ and using the first part of the proof we get
$\liminf_{j\to+\infty}{\rm
Cap}_{\varphi,\psi}(K_{j})\geq\int_{\\{h<\psi\\}}MA(h).$
On the other hand, it is clear that $\lim_{j\to+\infty}{\rm
Cap}_{\varphi,\psi}(K_{j})={\rm Cap}_{\varphi,\psi}(E)$, and hence
${\rm Cap}_{\varphi,\psi}(E)\geq\int_{\\{h<\psi\\}}MA(h).$
∎
Now we want to give a formula for the outer $(\varphi,\psi)$-capacity. Assume
that $E$ is a Borel subset of $X$. We introduce an auxiliary function
(2.3)
$\phi:=\phi_{\varphi,\psi,E}=\begin{cases}\frac{-h_{\varphi,\psi,E}^{*}+\psi}{-\varphi+\psi}\quad{\rm
if}\ \varphi>-\infty\\\ \;0\quad\quad\quad{\rm if}\
\varphi=-\infty\end{cases}.$
Observe that $\phi$ is a quasicontinuous function, $0\leq\phi\leq 1$ and
$\phi=1$ quasi everywhere on $E$.
###### Theorem 2.14.
Let $E\subset X$ be a Borel subset and denote by
$h_{E}:=h^{*}_{\varphi,\psi,E}$ the $(\varphi,\psi)$-extremal function of $E$.
Then
${\rm
Cap}_{\varphi,\psi}^{*}(E)=\int_{\\{h_{E}<\psi\\}}\mathrm{MA}\,(h_{E})=\int_{X}\left(\frac{-h_{E}+\psi}{-\varphi+\psi}\right)\,\mathrm{MA}\,(h_{E}).$
To prove Theorem 2.14 we need the following results.
###### Lemma 2.15.
Let $(u_{j})$ be a bounded monotone sequence of quasi-continuous functions
converging to $u$. Let $\chi$ be a convex weight and
$\\{\varphi_{j}\\}\subset\mathcal{E}_{\chi}(X,\omega)$ be a monotone sequence
converging to $\varphi\in\mathcal{E}_{\chi}(X,\omega)$. Then
$\int_{X}u_{j}\,\mathrm{MA}\,(\varphi_{j})\xrightarrow[j\rightarrow+\infty]{}\int_{X}u\,\mathrm{MA}\,(\varphi).$
###### Proof.
Fix $\varepsilon>0.$ Let $U$ be an open subset of $X$ with ${\rm
Cap}_{\omega}(U)<\varepsilon$ and $v_{j},v$ be continuous functions on $X$
such that $v_{j}\equiv u_{j}$ and $v\equiv u$ on $K:=X\setminus U.$ By Theorem
2.9 (and by letting $\varepsilon\to 0$) it suffices to prove that
$\int_{X}v_{j}\,\mathrm{MA}\,(\varphi_{j})\xrightarrow[j\rightarrow+\infty]{}\int_{X}v\,\mathrm{MA}\,(\varphi)\,.$
From Dini’s theorem $v_{j}$ converges uniformly to $v$ on $K$. Thus, using
again Theorem 2.9, the problem reduces to proving that
$\int_{X}v\,\mathrm{MA}\,(\varphi_{j})\xrightarrow[j\rightarrow+\infty]{}\int_{X}v\,\mathrm{MA}\,(\varphi)\,.$
But the latter obviously follows since $v$ is continuous on $X$. The proof is
thus complete. ∎
###### Proposition 2.16.
Let $E$ be a compact or open subset of $X$ and let
$h_{E}:=h^{*}_{\varphi,\psi,E}$ denote the $(\varphi,\psi)$-extremal function
of $E$. Then
${\rm
Cap}_{\varphi,\psi}(E)=\int_{\\{h_{E}<\psi\\}}\mathrm{MA}\,(h_{E})=\int_{X}\left(\frac{-h_{E}+\psi}{-\varphi+\psi}\right)\,\mathrm{MA}\,(h_{E}).$
###### Proof.
The first equality has been proved in Theorem 2.13. Set $h:=h_{E}$ and
$\phi:=\phi_{\varphi,\psi,E}=\frac{-h_{E}+\psi}{-\varphi+\psi}$. Observe that
$\\{h<\psi\\}=\\{\phi>0\\}$ modulo a pluripolar set and $\phi\leq 1.$ Thus
$\int_{\\{h<\psi\\}}\mathrm{MA}\,(h)\geq\int_{X}\phi\,\mathrm{MA}\,(h).$
Assume that $E$ is compact. By Proposition 2.11 and Theorem 2.13 we have
${\rm Cap}_{\varphi,\psi}(E)=\int_{E}\mathrm{MA}\,(h).$
Since $\phi=1$ quasi everywhere on $E$ we obtain
$\int_{E}\mathrm{MA}\,(h)\leq\int_{X}\phi\,\mathrm{MA}\,(h).$
We assume now that $E\subset X$ is an open subset. Let $(K_{j})$ be a sequence
of compact subsets increasing to $E$. Then
${\rm Cap}_{\varphi,\psi}(E)=\lim_{j\rightarrow+\infty}{\rm
Cap}_{\varphi,\psi}(K_{j})=\lim_{j\rightarrow+\infty}\int_{X}\phi_{j}\,\mathrm{MA}\,(h_{j}),$
where $h_{j}:=h_{\varphi,\psi,K_{j}}^{*}$ and
$\phi_{j}:=\phi_{\varphi,\psi,K_{j}}$ is defined by (2.3). Since $\phi_{j}$ is
quasicontinuous for any $j$ and $\phi_{j}\searrow\phi$, the conclusion follows
from Lemma 2.15. ∎
###### Lemma 2.17.
Let $u,v$ be $\omega$-plurisubharmonic functions. Let $G\subset X$ be an open
subset. Set $E=\\{u<v\\}\cap G$ and $h_{E}:=h_{\varphi,\psi,E}^{*}$. Then
${\rm Cap}_{\varphi,\psi}^{*}(E)={\rm
Cap}_{\varphi,\psi}(E)=\int_{\\{h_{E}<\psi\\}}\mathrm{MA}\,(h_{E})=\int_{X}\left(\frac{-h_{E}+\psi}{-\varphi+\psi}\right)\,\mathrm{MA}\,(h_{E}).$
###### Proof.
We start showing the first identity. First, just by definition ${\rm
Cap}_{\varphi,\psi}^{*}(E)\geq{\rm Cap}_{\varphi,\psi}(E)$. Fix
$\varepsilon>0$. There exists a function $\tilde{v}\in\mathcal{C}(X)$ such
that
${\rm Cap}_{\omega}(\\{\tilde{v}\neq v\\})<\varepsilon.$
Clearly $E\subset\left(\\{u<\tilde{v}\\}\cap G\right)\cup\\{\tilde{v}\neq
v\\}$ and so, applying Theorem 2.9 we get
$\displaystyle{\rm Cap}_{\varphi,\psi}^{*}(E)$ $\displaystyle\leq$
$\displaystyle{\rm Cap}_{\varphi,\psi}(\\{u<\tilde{v}\\}\cap
G)+F(\varepsilon)$ $\displaystyle\leq$ $\displaystyle{\rm
Cap}_{\varphi,\psi}(E)+2F(\varepsilon),$
where $F(\varepsilon)\to 0$ as $\varepsilon\to 0.$ Taking the limit as
$\varepsilon\rightarrow 0$ we arrive at the first conclusion.
Let now $\\{K_{j}\\}$ be a sequence of compact sets increasing to $G$ and
$\\{u_{j}\\}$ be a sequence of continuous functions decreasing to $u$. Then
$E_{j}=\\{u_{j}+1/j\leq v\\}\cap K_{j}$ is compact and $E_{j}\nearrow E$. Set
$h:=h_{\varphi,\psi,E},\,\phi:=\frac{-h_{E}+\psi}{-\varphi+\psi},\,h_{j}:=h_{\varphi,\psi,E_{j}}^{*},\,\phi_{j}:=\frac{-h_{E_{j}}+\psi}{-\varphi+\psi}.$
Observe that $h_{j}\searrow h$ and $\phi_{j}\searrow\phi$. By Proposition 2.16
and Lemma 2.15 we have
$\displaystyle{\rm Cap}_{\varphi,\psi}(E)$ $\displaystyle=$
$\displaystyle\lim_{j\rightarrow+\infty}{\rm Cap}_{\varphi,\psi}(E_{j})$
$\displaystyle=$
$\displaystyle\lim_{j\rightarrow+\infty}\int_{X}\phi_{j}\,\mathrm{MA}\,(h_{j})$
$\displaystyle=$
$\displaystyle\int_{X}\phi\,\mathrm{MA}\,(h)\leq\int_{\\{h<\psi\\}}\mathrm{MA}\,(h).$
Furthermore, for each fixed $k\in\mathbb{N}$, using Theorem 2.9 we can argue
as above to get
$\liminf_{j\to+\infty}\int_{\\{h_{j}<\psi\\}}\mathrm{MA}\,(h_{j})\geq\liminf_{j\to+\infty}\int_{\\{h_{k}<\psi\\}}\mathrm{MA}\,(h_{j})\geq\int_{\\{h_{k}<\psi\\}}\mathrm{MA}\,(h).$
Letting $k\to+\infty$ and using Proposition 2.16 again we get
${\rm Cap}_{\varphi,\psi}(E)\geq\int_{\\{h<\psi\\}}\mathrm{MA}\,(h),$
which completes the proof. ∎
We are now ready to prove Theorem 2.14.
###### Proof.
As usual, for simplicity, set $h:=h_{E}$. By definition of the outer capacity
there is a sequence $(O_{j})$ of open sets decreasing to $E$ such that ${\rm
Cap}_{\varphi,\psi}^{*}(E)=\lim_{j\rightarrow+\infty}{\rm
Cap}_{\varphi,\psi}(O_{j})$. Furthermore by Choquet’s lemma there exists a
sequence $(u_{j})$ of $\omega$-psh functions such that $u_{j}\equiv\varphi$
quasi everywhere on $E$, $u_{j}\leq\psi$ on $X$ and $u_{j}\nearrow h$. Since
${\rm Cap}_{\varphi,\psi}^{*}$ vanishes on pluripolar sets (see Theorem 2.7)
we can assume that $u_{j}\equiv\varphi$ on $E$. For any $j$, we set
$E_{j}=O_{j}\cap\\{u_{j}<\varphi+1/j\\}$ and
$h_{j}:=h_{\varphi,\psi,E_{j}}^{*}$. Then $(E_{j})$ is a decreasing sequence
of open subsets such that $E\subset E_{j}\subset O_{j}$ and $u_{j}-1/j\leq
h_{j}\leq h$, thus $h_{j}\nearrow h$. Clearly ${\rm
Cap}_{\varphi,\psi}^{*}(E)=\lim_{j\rightarrow+\infty}{\rm
Cap}_{\varphi,\psi}(E_{j})$. By Lemma 2.17 and Lemma 2.15 we get
$\lim_{j\rightarrow+\infty}{\rm
Cap}_{\varphi,\psi}^{*}(E_{j})=\lim_{j\rightarrow+\infty}{\rm
Cap}_{\varphi,\psi}(E_{j})=\lim_{j\rightarrow+\infty}\int_{X}\phi_{j}\,\mathrm{MA}\,(h_{j})=\int_{X}\phi\,\mathrm{MA}\,(h),$
where $\phi_{j}:=\phi_{\varphi,\psi,E_{j}}$ is defined by (2.3). ∎
###### Corollary 2.18.
Let $K\subset X$ be a compact set and $(K_{j})$ a sequence of compact subsets
decreasing to $K$. Then
* (i)
${\rm Cap}_{\varphi,\psi}^{*}(K)={\rm
Cap}_{\varphi,\psi}(K)=\lim_{j\to+\infty}{\rm Cap}_{\varphi,\psi}(K_{j})$,
* (ii)
$h_{\varphi,\psi,K_{j}}^{*}\nearrow h_{\varphi,\psi,K}^{*}$.
###### Proof.
The first equality in statement (i) comes straightforward from Theorem 2.13
and Theorem 2.14. The second one follows from (ii) and Theorem 2.14. It
remains to prove (ii). Since $(K_{j})$ decreases to $K$,
$h_{j}:=h_{\varphi,\psi,K_{j}}^{*}$ increases to some
$h_{\infty}\in\mathcal{E}(X,\omega)$. Clearly $h_{\infty}\leq h$. Thus we need
to prove that $h_{\infty}\geq h$. Since
$\\{h_{\infty}<h\\}\subset\\{h_{\infty}<\psi\\}\setminus K$ modulo a
pluripolar set,
$\int_{\\{h_{\infty}<h\\}}\mathrm{MA}\,(h_{\infty})\leq\int_{\\{h_{\infty}<\psi\\}\setminus
K}\mathrm{MA}\,(h_{\infty}).$
From Proposition 2.11 we know that
$\int_{\\{h_{j}<\psi\\}\setminus K_{j}}\mathrm{MA}\,(h_{j})=0,\,\forall
j\in\mathbb{N}.$
Fix $\varepsilon>0$ and let $\psi_{\varepsilon}\in\mathcal{C}(X)$ such that
${\rm Cap}_{\omega}(\\{\psi_{\varepsilon}\neq\psi\\})<\varepsilon$. Then for
each fixed $k\in\mathbb{N}$, we have
$\displaystyle\int_{\\{h_{\infty}<\psi\\}\setminus
K_{k}}\mathrm{MA}\,(h_{\infty})$ $\displaystyle\leq$
$\displaystyle\int_{\\{h_{\infty}<\psi_{\varepsilon}\\}\setminus
K_{k}}\mathrm{MA}\,(h_{\infty})+F(\varepsilon)$ $\displaystyle\leq$
$\displaystyle\liminf_{j\to+\infty}\int_{\\{h_{\infty}<\psi_{\varepsilon}\\}\setminus
K_{k}}\mathrm{MA}\,(h_{j})+F(\varepsilon)$ $\displaystyle\leq$
$\displaystyle\liminf_{j\to+\infty}\int_{\\{h_{\infty}<\psi\\}\setminus
K_{k}}\mathrm{MA}\,(h_{j})+2F(\varepsilon)$ $\displaystyle\leq$
$\displaystyle\liminf_{j\to+\infty}\int_{\\{h_{j}<\psi\\}\setminus
K_{k}}\mathrm{MA}\,(h_{j})+2F(\varepsilon)$ $\displaystyle=$ $\displaystyle
2F(\varepsilon),$
where $F(\varepsilon)\to 0$ as $\varepsilon\to 0$ thanks to Theorem 2.9. Thus,
letting $\varepsilon\to 0$ then $k\to+\infty$ and using the domination
principle below (Proposition 3.1) we can conclude that $h_{\infty}\geq h$. ∎
### 2.3. Proof of Theorem A
Let us briefly resume the proof of Theorem A. Statements (i) and (ii) have
been proved in Theorem 2.14 and Theorem 2.9 respectively. One direction of the
last staement has been proved in Theorem 2.7. Now, if $E$ is a Borel subset of
$X$ such that ${\rm Cap}_{\varphi,\psi}^{*}(E)=0$ then it follows from Theorem
2.14 that
$\int_{\\{h_{\varphi,\psi,E}^{*}<\psi\\}}\mathrm{MA}\,(h_{\varphi,\psi,E}^{*})=0.$
We then can apply the domination principle (see [7] or Proposition 3.1 below
for a proof) to conclude.
## 3\. Another proof of the Domination Principle
The following domination principle was proved by Dinew using his uniqueness
result [16], [7]. As an application of the $(\varphi,\psi)$-Capacity we
propose here an alternative proof.
###### Proposition 3.1.
If $u,v\in\mathcal{E}(X,\omega)$ such that $u\leq v$ $MA(v)$-almost everywhere
then $u\leq v$ on $X.$
###### Proof.
We first claim that for every $\varphi\in\mathcal{E}(X,\omega)$ such that
$0\leq\varphi-u\leq C$ for some constant $C>0$ and for any $s>0$ one has
$\int_{\\{v<u-s\\}}MA(\varphi)=0.$
Indeed, fix $s>0$ and let $\varphi$ be such a function. Let $C>0$ be a
constant such that $\varphi-u\leq C$ on $X.$ Choose $\delta\in(0,1)$ such that
$\delta C<s.$ Now, by using the comparison principle and the fact that
$0\leq\varphi-u\leq C$ we get
$\displaystyle\delta^{n}\int_{\\{v<u-s\\}}MA(\varphi)$ $\displaystyle=$
$\displaystyle\int_{\\{v<u-s\\}}(\delta\omega+dd^{c}\delta\varphi)^{n}$
$\displaystyle\leq$
$\displaystyle\int_{\\{v<\delta\varphi+(1-\delta)u-s\\}}MA\left(\delta\varphi+(1-\delta)u\right)$
$\displaystyle\leq$
$\displaystyle\int_{\\{v<\delta\varphi+(1-\delta)u-s\\}}MA(v)$
$\displaystyle\leq$ $\displaystyle\int_{\\{v<u\\}}MA(v)=0.$
Thus, the claim is proved. Now for each $t>0$ let $h_{t}$ denote the
$(u,0)$-extremal function of the open set $G_{t}:=\\{u<-t\\}.$ It is clear
that for every $t>0,$ $h_{t}\in\mathcal{E}(X,\omega)$ and
$\sup_{X}(h_{t}-u)<+\infty.$ The previous step yields
$\int_{\\{v<u-s\\}}MA(h_{t})=0,\ \forall s>0.$
Fix $\varepsilon>0$. Let $\tilde{u}$ be a continuous function on $X$ such that
${\rm Cap}_{\omega}(\\{u\neq\tilde{u}\\})<\varepsilon$. Since $h_{t}$
increases to $0$ (see Lemma 3.2 below), we infer that
$\int_{\\{v<\tilde{u}-s\\}}\omega^{n}\leq\liminf_{t\to+\infty}\int_{\\{v<u-s\\}}\mathrm{MA}\,(h_{t})+{\rm
Cap}_{u,0}(\\{u\neq\tilde{u}\\}).$
Repeating this argument we get
$\int_{\\{v<u-s\\}}\omega^{n}\leq\varepsilon+{\rm
Cap}_{u,0}(\\{u\neq\tilde{u}\\}).$
Letting $\varepsilon\to 0$ and using Theorem 2.9 we get ${\rm
Vol}(\\{v<u-s\\})=0$, for any $s>0$ which implies that $u\leq v$ on $X$ as
desired. ∎
###### Lemma 3.2.
Let $v\in{\rm PSH}(X,\omega).$ For each $t>0$, set $G_{t}:=\\{v<-t\\}$. Denote
by $h_{t}$ the $(\varphi,0)$-extremal function of $G_{t}$. Then $h_{t}$
increases quasi everywhere on $X$ to $0$ when $t$ increases to $+\infty$.
###### Proof.
We know that $h_{t}$ increases quasi everywhere to $h\in\mathcal{E}(X,\omega)$
and that $h\leq 0$. By Theorem 2.7 (up to consider $-(-v)^{p}$ with
$p\in(0,1)$ instead of $v$), we get
$\lim_{t\to+\infty}{\rm Cap}_{\varphi,0}(G_{t})=0.$
It follows from Theorem 2.13 that for each $t>0$,
$\int_{\\{h<0\\}}MA(h_{t})\leq\int_{\\{h_{t}<0\\}}MA(h_{t})={\rm
Cap}_{\varphi,0}(G_{t}).$
We thus get
$\int_{\\{h<0\\}}MA(h)\leq\liminf_{t\to+\infty}\int_{\\{h<0\\}}MA(h_{t})=0.$
The comparison principle yields ${\rm Vol}(\\{h<0\\})=0$ which completes the
proof. ∎
###### Remark 3.3.
Lemma 3.2 is stated and proved in the case $\psi\equiv 0$. Observe that it
also holds for any $\psi\in\mathcal{E}(X,\omega)$ such that $\varphi<\psi$. To
see this we can follow the same arguments of above but for the final step
where we get $\psi\leq h\;\,\mathrm{MA}\,(h)$-almost everywhere. We then
conclude using the domination principle.
## 4\. Applications to Complex Monge-Ampère equations
In this section (in the same spirit of [15]) we prove Theorem B by using ${\rm
Cap}_{\psi}:={\rm Cap}_{\psi-1,\psi}$. Let us recall the setting. Let $X$ be a
compact Kähler manifold of dimension $n$ and let $\omega$ be a Kähler form on
$X$. Let $D$ be an arbitrary divisor on $X$. Consider the complex Monge-Ampère
equations
(4.1) $(\omega+dd^{c}\varphi)^{n}=e^{\lambda\varphi}f\omega^{n},\
\lambda\in\mathbb{R}.$
We say that $f$ satisfies Condition $\mathcal{H}_{f}$ if
$f=e^{\psi^{+}-\psi^{-}},\ \ \psi^{\pm}\ {\rm are\ quasi\ psh\ functions\ on}\
X\ ,\ \psi^{-}\in L^{\infty}_{\rm loc}(X\setminus D).$
We have already treated the case when $\lambda=0$ in [15]. If $\lambda>0$ and
$f$ is integrable then the same arguments can be applied. More precisely,
$\mathcal{C}^{0}$-estimates follow from comparison principle while the
$\mathcal{C}^{2}$ estimate follows exactly the same way as in [15].
The case when $\lambda<0$ is known to be much more difficult. We need the
following observation where we make use of the generalized capacity ${\rm
Cap}_{\psi}$:
###### Lemma 4.1.
Let $\varphi\in\mathcal{E}(X,\omega)$ be normalized by $\sup_{X}\varphi=0$.
Assume that there exist a positive constant $A$ and $\psi\in{\rm
PSH}(X,\omega/2)$ such that $\mathrm{MA}\,(\varphi)\leq e^{-A\psi}\omega^{n}$.
Then there exists $C>0$ depending only on $\int_{X}e^{-2A\varphi}\omega^{n}$
such that
$\varphi\geq\psi-C.$
Observe that for all $A>0$ and $\varphi\in\mathcal{E}(X,\omega)$,
$e^{-A\varphi}\omega^{n}\in L^{1}(X)$ as follows from Skoda integrability
theorem [23], [25], since functions in $\mathcal{E}(X,\omega)$ have zero
Lelong number at all points [20].
###### Proof.
Set
$H(t)=\left[{\rm Cap}_{\psi}(\\{\varphi<\psi-t\\})\right]^{1/n},\ t>0.$
Observe that $H(t)$ is right-continuous and $H(+\infty)=0$ (see [15, Lemma
2.6]). It follows from [15, Lemma 2.7] that ${\rm Cap}_{\omega}\leq 2^{n}{\rm
Cap}_{\psi}$. By a strong volume-capacity domination in [19] we also have
${\rm Vol}_{\omega}\leq\exp{\left(\frac{-C_{1}}{{\rm
Cap}_{\omega}^{1/n}}\right)},$
where $C_{1}$ depends only on $(X,\omega)$. Thus using [15, Proposition 2.8]
and the assumption on the measure $\mathrm{MA}\,(\varphi)$, we get
$\displaystyle s^{n}{\rm Cap}_{\psi}(\\{\varphi<\psi-t-s\\})$
$\displaystyle\leq$
$\displaystyle\int_{\\{\varphi<\psi-t\\}}\mathrm{MA}\,(\varphi)$
$\displaystyle\leq$
$\displaystyle\int_{\\{\varphi<\psi-t\\}}e^{-A\varphi}e^{A\psi}\mathrm{MA}\,(\varphi)$
$\displaystyle\leq$
$\displaystyle\left[\int_{X}e^{-2A\varphi}\omega^{n}\right]^{1/2}\left[\int_{\\{\varphi<\psi-t\\}}\omega^{n}\right]^{1/2}$
$\displaystyle\leq$ $\displaystyle C_{2}\left[{\rm
Cap}_{\psi}(\\{\varphi<\psi-t\\})\right]^{2},$
where $C_{2}$ depends on $\int_{X}e^{-2A\varphi}\omega^{n}$. We then get
$sH(t+s)\leq C_{2}^{1/n}H(t)^{2},\ \forall t>0,\forall s\in[0,1].$
Then by [17, Lemma 2.4] we get $\varphi\geq\psi-C_{3}$, where $C_{3}$ only
depends on $\int_{X}e^{-2A\varphi}\omega^{n}$. ∎
Now, we are ready to prove Theorem B.
### 4.1. Proof of Theorem B
It suffices to treat the case when $\lambda=-1$. Since $f$ satisfies Condition
$\mathcal{H}_{f}$ we can write $\log f=\psi^{+}-\psi^{-}$, where $\psi^{\pm}$
are qpsh functions on $X$, $\psi^{-}$ is locally bounded on $X\setminus D$ and
there exists a uniform constant $C>0$ such that
$dd^{c}\psi^{\pm}\geq-C\omega,\;\sup_{X}\psi^{+}\leq C.$
We apply the smoothing kernel $\rho_{\varepsilon}$ in Demailly regularization
theorem [13] to the functions $\varphi$ and $\psi^{\pm}$. For $\varepsilon$
small enough, we get
$dd^{c}\rho_{\varepsilon}(\varphi+\psi^{-})\geq-
C_{1}\omega,\;\;dd^{c}\rho_{\varepsilon}(\psi^{+})\geq-
C_{1}\omega,\;\;\sup_{X}\rho_{\varepsilon}(\psi^{+})\leq C_{1},$
where $C_{1}$ depends on $C$ and the Lelong numbers of the currents
$C\omega+dd^{c}\psi^{\pm}$. By the classical result of Yau [24], for each
$\varepsilon$, there exists a unique smooth $\omega$-psh function
$\phi_{\varepsilon}$ satisfying
$\mathrm{MA}\,(\phi_{\varepsilon})=e^{c_{\varepsilon}+\rho_{\varepsilon}(\psi^{+})-\rho_{\varepsilon}(\varphi+\psi^{-})}\omega^{n}=g_{\varepsilon}\omega^{n},\
\ \sup_{X}\phi_{\varepsilon}=0,$
where $c_{\varepsilon}$ is a normalization constant such that
$\int_{X}g_{\varepsilon}\omega^{n}=\int_{X}e^{-\varphi}f\omega^{n}=\int_{X}\omega^{n}.$
Since by Jensen’s inequality $e^{\rho_{\varepsilon}(-\varphi+\log
f)}\leq\rho_{\varepsilon}(e^{-\varphi+\log f})$ and
$e^{\rho_{\varepsilon}(-\varphi+\log f)}$ converges point-wise to
$e^{-\varphi}f$ on $X$, it follows from the general Lebesgue dominated
convergence theorem that $e^{\rho_{\varepsilon}(-\varphi+\log f)}$ converges
to $e^{-\varphi}f$ in $L^{1}(X)$ when $\varepsilon\downarrow 0$. This means
that $c_{\varepsilon}$ converges to zero when $\varepsilon\rightarrow 0$. It
then follows from [15, Lemma 3.4] that $\phi_{\varepsilon}$ converges in
$L^{1}(X)$ to $\varphi-\sup_{X}\varphi$. We now apply the $\mathcal{C}^{2}$
estimate in [15, Theorem 3.2] to get
$n+\Delta\phi_{\varepsilon}\leq
C_{3}e^{-2\rho_{\varepsilon}(\varphi+\psi^{-})}\leq
C_{4}e^{-2(\varphi+\psi^{-})},$
where $C_{3},C_{4}$ are uniform constants (do not depend on $\varepsilon$).
Now, we need to bound $\varphi$ from below. By the assumption on $f$ we have
$\mathrm{MA}\,(\varphi)=e^{\psi^{+}-(\varphi+\psi^{-})}\omega^{n}\leq
e^{-(\varphi+\psi^{-}-C)}\omega^{n}.$
Consider $\psi:=\frac{1}{2C+2}(\varphi+\psi^{-})$. Since this function belongs
to ${\rm PSH}(X,\omega/2)$ we can apply Lemma 4.1 to get
$\varphi-\sup_{X}\varphi\geq\psi-C_{5}.$
This gives $\varphi\geq C_{6}\psi^{-}-C_{7}$. Applying again this argument to
$\phi_{\varepsilon}$ and noting that $c_{\varepsilon}$ converges to $0$, and
hence under control, we get
$\phi_{\varepsilon}\geq\rho_{\varepsilon}(\varphi+\psi^{-})-C_{8}\geq
C_{9}\psi^{-}-C_{10}.$
We can now conclude using the same arguments in [15, Section 3.3].
### 4.2. (Non) Existence of solutions
In the previous subsection, no regularity assumption on $D$ has been done. We
now discuss about the existence of solutions in concrete examples, assuming
more information on $D,f$.
Let $D=\sum_{j=1}^{N}D_{j}$ be a simple normal crossing divisor on $X$.
Reacall that ”simple normal crossing” means that around each intersection
point of $k$ components $D_{j_{1}},...,D_{j_{k}}$ ($k\leq N$), we can find
complex coordinates $z_{1},...,z_{n}$ such that for each $l=1,...,k$ the
hypersurface $D_{j_{l}}$ is locally given by $z_{l}=0$.
For each $j$, let $L_{j}$ be the holomorphic line bundle defined by $D_{j}$.
Let $s_{j}$ be a holomorphic section of $L_{j}$ defining $D_{j}$, i.e
$D_{j}=\\{s_{j}=0\\}$. We fix a hermitian metric $h_{j}$ on $L_{j}$ such that
$|s_{j}|:=|s_{j}|_{h_{j}}\leq 1/e$.
We assume that $f$ has the following particular form:
(4.2) $f=\frac{h}{\prod_{j=1}^{N}|s_{j}|^{2}(-\log|s_{j}|)^{1+\alpha}},\
\alpha>0,$
where $h$ is a bounded function: $0<1/B\leq h\leq B,\;B>0$.
In this subsection we always assume that $\lambda<0$.
###### Proposition 4.2.
Assume that $f$ satisfies (4.2) with $0<\alpha\leq 1$. Then there is no
solution in $\mathcal{E}(X,\omega)$ to equation
$(\omega+dd^{c}\varphi)^{n}=e^{\lambda\varphi}f\omega^{n}.$
###### Proof.
We can assume (up to normalization) that $\lambda=-1$. Then observe that if
there exists $\varphi\in\mathcal{E}(X,\omega)$ such that
$(\omega+dd^{c}\varphi)^{n}=e^{-\varphi}\mu,$ where $\mu$ is a positive
measure, then we can find $A>0$ such that
$\mu\leq A\left(\omega+dd^{c}u\right)^{n},$
where $u:=e^{(\varphi-\sup_{X}\varphi)/n}$ is a bounded $\omega$-psh function.
Indeed, $u$ is a $\omega$-psh function and
$\omega+dd^{c}u\geq\omega+\frac{u}{n}dd^{c}\varphi\geq\frac{u}{n}(\omega+dd^{c}\varphi)\geq
0.$
This coupled with [15, Proposition 4.4 and 4.5] yields the conclusion. ∎
The above analysis shows that there is no solution if the density has
singularities of Poincaré type or worse. We next show that when $f$ is less
singular than the Poincaré type density (i.e. $\alpha>1$), equation (4.1) has
a bounded solution provided $\lambda=-\varepsilon$ with $\varepsilon>0$ very
small. We say that $f$ satisfies Condition $\mathcal{S}(B,\alpha)$ for some
$B>0$, $\alpha>0$ if
$f\leq\frac{B}{\prod_{j=1}^{N}|s_{j}|^{2}(-\log|s_{j}|)^{1+\alpha}}.$
###### Theorem 4.3.
Assume that $f$ satisfies Condition $\mathcal{S}(B,\alpha)$ with $\alpha>1$.
We also normalize $f$ so that $\int_{X}f\omega^{n}=\int_{X}\omega^{n}$. Then
for $\lambda=-\varepsilon$ with $\varepsilon>0$ small enough depending only on
$C,\alpha,\omega$, there exists a bounded solution $\varphi$ to (4.1).
The solution is automatically continuous on $X$. In particular, it is also
smooth on $X\setminus D$ if $f$ is smooth there.
###### Proof.
The last statement follows easily from our previous analysis. Let us prove the
existence. We use the Schauder Fixed Point Theorem. Let $C=C(2B,\alpha)$ be
the constant in Lemma 4.4 below. Choose $\varepsilon>0$ very small such that
$e^{\varepsilon C}\leq 2$. Consider the following compact convex set in
$L^{1}(X)$:
$\mathcal{C}:=\\{u\in{\rm PSH}(X,\omega)\ \ \big{|}\ \ -C\leq u\leq 0\\}.$
Let $\psi\in\mathcal{C}$ and $c_{\psi}$ be a constant such that
$\int_{X}e^{-\varepsilon\psi+c_{\psi}}f\omega^{n}=\int_{X}\omega^{n}.$
Since $-C\leq\psi\leq 0$, it is clear that $-C\varepsilon\leq c_{\psi}\leq 0$.
Let $\varphi$ be the unique bounded $\omega$-psh function such that
$\sup_{X}\varphi=0$ and
$(\omega+dd^{c}\varphi)^{n}=e^{-\varepsilon\psi+c_{\psi}}f\omega^{n}.$
The density on the right-hand side satisfies Condition $\mathcal{S}(B,\alpha)$
since $c_{\psi}\leq 0$ and since $e^{\varepsilon C}\leq 2$. We thus get from
Lemma 4.4 below that $\varphi\geq-C$. Thus we have defined a mapping from
$\mathcal{C}$ to itseft
$\Phi:\mathcal{C}\rightarrow\mathcal{C},\ \ \ \Phi(\psi):=\varphi.$
Let us prove that $\Phi$ is continuous on $\mathcal{C}$. Let $\psi_{j}$ be a
sequence in $\mathcal{C}$ which converges to $\psi$ in $L^{1}(X)$. Denote by
$c_{j}:=c_{\psi_{j}},\ \ c:=c_{\psi},\ \Phi(\psi_{j})=\varphi_{j},\
\Phi(\psi)=\varphi.$
It is enough to prove that any cluster point of the sequence $(\varphi_{j})$
is equal to $\varphi$. Therefore, we can assume that $\varphi_{j}$ converges
to $\varphi_{0}$ in $L^{1}(X)$ and up to extracting a subsequence that
$\psi_{j}$ converges almost everywhere to $\psi$ on $X$ and also that $c_{j}$
converges to $c_{0}\in[-C\varepsilon,0]$. Since
$e^{-\varepsilon\psi_{j}+c_{j}}f$ converges in $L^{1}(X)$ to
$e^{-\varepsilon\psi+c_{0}}f$ in $L^{1}(X)$ and almost everywhere, it follows
from [15, Lemma 3.4] that
$(\omega+dd^{c}\varphi_{0})^{n}=e^{-\varepsilon\psi+c_{0}}f\omega^{n}.$
It is clear that $c_{0}=c$ and it follows from Hartogs’ lemma that
$\sup_{X}\varphi_{0}=0$. Thus $\varphi_{0}=\varphi$. This concludes the
continuity of $\Phi$.
Now, since $\mathcal{C}$ is compact and convex in $L^{1}(X)$ and since $\Phi$
is continuous on $\mathcal{C}$, by Schauder Fixed Point Theorem there exists a
fixed point of $\Phi$, say $\varphi$. Then $\varphi-c_{\varphi}/{\varepsilon}$
is the desired solution. ∎
We refer the reader to [15, Section 4.2] for the proof of the following lemma.
###### Lemma 4.4.
Assume that $f$ satisfies Condition $\mathcal{S}(B,\alpha)$ with $\alpha>1$,
$B>0$. Let $\varphi\in\mathcal{E}(X,\omega)$ be the unique function such that
$(\omega+dd^{c}\varphi)^{n}=f\omega^{n},\ \sup_{X}\varphi=0.$
Then $\varphi\geq-C$ with $C=C(B,\alpha)>0$.
### 4.3. Proof of Theorem C
Assume that $\varphi\in\mathcal{E}(X,\omega)$ satisfies
$(\omega+dd^{c}\varphi)^{n}=e^{\lambda\varphi}f\omega^{n},\lambda>0.$
Up to rescaling $\omega$ it suffices to treat the case when $\lambda=1$. The
proof of Theorem C is quite similar to that of Theorem B. The difference here
is that $f$ is not integrable. For convenience of the reader we rewrite the
arguments here. Since $f$ satisfies Condition $\mathcal{H}_{f}$ we can write
$\log f=\psi^{+}-\psi^{-}$, where $\psi^{\pm}$ are qpsh functions on $X$,
$\psi^{-}$ is locally bounded on $X\setminus D$ and there exists a uniform
constant $C>0$ such that
$dd^{c}\psi^{\pm}\geq-C\omega,\;\sup_{X}\psi^{+}\leq C.$
We apply the smoothing kernel $\rho_{\varepsilon}$ in Demailly regularization
theorem [13] to the functions $\varphi$ and $\psi^{\pm}$. For $\varepsilon$
small enough, we get
$dd^{c}\rho_{\varepsilon}(\psi^{-})\geq-
C_{1}\omega,\;\;dd^{c}\rho_{\varepsilon}(\varphi+\psi^{+})\geq-
C_{1}\omega,\;\;\sup_{X}\rho_{\varepsilon}(\varphi+\psi^{+})\leq C_{1},$
where $C_{1}$ depends on $C$, the Lelong numbers of the currents
$C\omega+dd^{c}\psi^{\pm}$ and $\sup_{X}\varphi$. By the classical result of
Yau [24], for each $\varepsilon$, there exists a unique smooth $\omega$-psh
function $\phi_{\varepsilon}$ satisfying
$\mathrm{MA}\,(\phi_{\varepsilon})=e^{c_{\varepsilon}+\rho_{\varepsilon}(\varphi+\psi^{+})-\rho_{\varepsilon}(\psi^{-})}\omega^{n}=g_{\varepsilon}\omega^{n},\
\ \sup_{X}\phi_{\varepsilon}=0,$
where $c_{\varepsilon}$ is a normalization constant such that
$\int_{X}g_{\varepsilon}\omega^{n}=\int_{X}e^{\varphi}f\omega^{n}=\int_{X}\omega^{n}.$
Since by Jensen’s inequality $e^{\rho_{\varepsilon}(\varphi+\log
f)}\leq\rho_{\varepsilon}(e^{\varphi+\log f})$ and
$e^{\rho_{\varepsilon}(\varphi+\log f)}$ converges point-wise to
$e^{\varphi}f$ on $X$, it follows from the general Lebesgue dominated
convergence theorem that $e^{\rho_{\varepsilon}(\varphi+\log f)}$ converges to
$e^{\varphi}f$ in $L^{1}(X)$ when $\varepsilon\downarrow 0$. This means that
$c_{\varepsilon}$ converges to zero when $\varepsilon\rightarrow 0$. It then
follows from Lemma 3.4 in [15] that $\phi_{\varepsilon}$ converges in
$L^{1}(X)$ to $\varphi-\sup_{X}\varphi$. We now apply the $\mathcal{C}^{2}$
estimate in Theorem 3.2 in [15] to get
$n+\Delta\phi_{\varepsilon}\leq C_{3}e^{-2\rho_{\varepsilon}(\psi^{-})}\leq
C_{4}e^{-2\psi^{-}},$
where $C_{3},C_{4}$ are uniform constants (do not depend on $\varepsilon$).
Now, we need to bound $\varphi$ from below. By the assumption on $f$ we have
$\mathrm{MA}\,(\varphi)=e^{\varphi+\psi^{+}-\psi^{-}}\omega^{n}\leq
e^{-(\psi^{-}-C_{1})}\omega^{n}.$
Consider $\psi:=\frac{1}{2C}\psi^{-}$. Since this function belongs to ${\rm
PSH}(X,\omega/2)$ we can apply Lemma 4.1 to get
$\varphi-\sup_{X}\varphi\geq\psi-C_{5}.$
Now the remaining part of the proof follows by exactly the same way as we have
done in [15, Section 3.3].
### 4.4. Non Integrable densities
When $0\leq f\notin L^{1}(X)$ it is not clear that we can find a solution
$\varphi\in\mathcal{E}(X,\omega)$ of equation
$(\omega+dd^{c}\varphi)^{n}=e^{\varphi}f\omega^{n}.$
We show in the following that it suffices to find a subsolution. Another
similar result has been proved by Berman and Guenancia in [5] using the
variational approach. We provide here a simple proof using our generalized
Monge-Ampère capacities.
###### Theorem 4.5.
Let $0\leq f$ be a measurable function such that
$\int_{X}f\omega^{n}=+\infty$. If there exists $u\in\mathcal{E}(X,\omega)$
such that $\mathrm{MA}\,(u)\geq e^{u}f\omega^{n}$ then there is a unique
$\varphi\in\mathcal{E}(X,\omega)$ such that
$\mathrm{MA}\,(\varphi)=e^{\varphi}f\omega^{n}.$
###### Proof.
The uniqueness follows easily from the comparison principle. Indeed, one can
find a proof in [5, Proposition 3.1]. We now establish the existence. For each
$j\in\mathbb{N}$ we can find $\varphi_{j}\in{\rm PSH}(X,\omega)\cap
L^{\infty}(X)$ such that
$(\omega+dd^{c}\varphi_{j})^{n}=e^{\varphi_{j}}\min(f,j)\omega^{n}.$
It follows from the comparison principle that $\varphi_{j}$ is non-increasing
and $\varphi_{j}\geq u$. Then
$\varphi_{j}\downarrow\varphi\in\mathcal{E}(X,\omega)$ and by continuity of
the complex Monge-Ampère operator along decreasing sequence in
$\mathcal{E}(X,\omega)$ we get
$\mathrm{MA}\,(\varphi)=e^{\varphi}f\omega^{n}.$
Indeed, since $\mathrm{MA}\,(\varphi_{j})$ converges weakly to
$\mathrm{MA}\,(\varphi)$, from Fatou’s lemma we get
$\mathrm{MA}\,(\varphi)\geq e^{\varphi}f\omega^{n}$
in the sense of positive Borel measures. To get the reverse inequality we need
to show that the right-hand side has full mass, i.e.
$\int_{X}e^{\varphi}f\omega^{n}=\int_{X}\omega^{n}.$
Fix $\varepsilon>0$. Since $\varphi$ is $\omega$-psh, in particular quasi-
continuous, we find $U$ an open subset of $X$ such that ${\rm
Cap}_{\omega}(U)<\varepsilon$ and $\varphi$ is continuous on $K:=X\setminus
U$. Then $\varphi$ is bounded on $K$ and hence $f$ must be integrable on $K$.
We thus can apply the Lebesgue Dominated Convergence Theorem on $K$ to get
$\lim_{j\to+\infty}\int_{K}\mathrm{MA}\,(\varphi_{j})=\lim_{j\to+\infty}\int_{K}e^{\varphi_{j}}\min(f,j)\omega^{n}=\int_{K}e^{\varphi}f\omega^{n}.$
We can assume that $\varphi_{j}\leq 0$. It follows from Theorem 2.9 that
$\int_{U}\mathrm{MA}\,(\varphi_{j})\leq{\rm Cap}_{u,0}(U)\leq
F(\varepsilon)\rightarrow 0\ \ {\rm as}\ \varepsilon\downarrow 0.$
This implies that
$\displaystyle\int_{X}e^{\varphi}f\omega^{n}$ $\displaystyle\geq$
$\displaystyle\int_{K}e^{\varphi}f\omega^{n}=\lim_{j\to+\infty}\int_{K}\mathrm{MA}\,(\varphi_{j})$
$\displaystyle=$
$\displaystyle\int_{X}\mathrm{MA}\,(\varphi_{j})-\lim_{j\to+\infty}\int_{U}\mathrm{MA}\,(\varphi_{j})$
$\displaystyle\geq$ $\displaystyle\int_{X}\omega^{n}-F(\varepsilon).$
By letting $\varepsilon\to 0$ we get
$\int_{X}e^{\varphi}f\omega^{n}=\int_{X}\omega^{n}$, which completes the
proof. ∎
###### Remark 4.6.
Theorem 4.5 also holds if $\omega$ is merely semipositive and big.
###### Example 4.7.
Let $D=\sum_{j=1}^{N}D_{j}$ be a simple normal crossing divisor on $X$. Assume
that the $D_{j}$ are defined by $s_{j}=0$, where $s_{j}$ are holomorphic
sections such that $|s_{j}|<1/e$. Consider the following density
$f=\frac{1}{\prod_{j=1}^{N}|s_{j}|^{2}}.$
Then for suitable positive constants $C_{1},C_{2}$ the following function
$\varphi:=-2\sum_{j=1}^{N}\log(-\log|s_{j}|+C_{1})-C_{2}$
is a subsolution of $\mathrm{MA}\,(\varphi)=e^{\varphi}f\omega^{n}$. In fact,
it suffices to find a function $u\in\mathcal{E}(X,\omega/2)$ such that
$e^{u}f$ is integrable (see Example 4.9).
### 4.5. The case of semipositive and big classes
In this section we try to extend our result in Theorem C to the case of
semipositive and big classes. Let $\theta$ be a smooth closed semipostive
$(1,1)$-form on $X$ such that $\int_{X}\theta^{n}>0$. Assume that
$E=\sum_{j=1}^{M}a_{j}E_{j}$ is an effective simple normal crossing divisor on
$X$ such that $\\{\theta\\}-c_{1}(E)$ is ample. Let $0\leq f$ is a non-
negative measurable function on $X$. Consider the following degenerate complex
Monge-Ampère equation
(4.3) $(\theta+dd^{c}\varphi)^{n}=e^{\varphi}f\omega^{n}.$
As in Theorem C we obtain here a similar regularity for solutions in
$\mathcal{E}(X,\omega)$:
###### Theorem 4.8.
Assume that $0<f\in\mathcal{C}^{\infty}(X\setminus D)$ satisfies Condition
$\mathcal{H}_{f}$. Let $\theta$ and $E$ be as above. If there is a solution in
$\mathcal{E}(X,\omega)$ of equation (4.3) then this solution is also smooth on
$X\setminus(D\cup E)$.
Note that in Theorem 4.8 we do not assume that $f$ is integrable on $X$. We
also stress that there is at most one solution in $\mathcal{E}(X,\theta)$ (see
[5]).
###### Proof.
We adapt the proof of Theorem 3 in [15] where we followed essentially the
ideas in [8]. Assume that $\varphi\in\mathcal{E}(X,\theta)$ is a solution to
equation (4.3). By assumption on $f$ we can find a uniform constant $C>0$ such
that
$f=e^{\psi^{+}-\psi^{-}},\ \ dd^{c}\psi^{\pm}\geq-C\omega^{n},\ \
\sup_{X}\psi^{+}\leq C,\ \sup_{X}\varphi\leq C,\ \ \psi^{-}\in L^{\infty}_{\rm
loc}(X\setminus D).$
We regularize $\varphi$ and $\psi^{\pm}$ by using the smoothing kernel
$\rho_{\varepsilon}$ in Demailly’s work [13]. Then for $\varepsilon>0$ small
enough we have
$dd^{c}\rho_{\varepsilon}(\psi^{-})\geq-
C_{1}\omega,\;\;dd^{c}\rho_{\varepsilon}(\varphi+\psi^{+})\geq-
C_{1}\omega,\;\;\sup_{X}\rho_{\varepsilon}(\varphi+\psi^{+})\leq C_{1},$
where $C_{1}$ depends on $C$ and the Lelong numbers of the currents
$C\omega+dd^{c}\psi^{\pm}$. For each $\varepsilon>0$ by the famous result of
Yau [24] there exits a unique smooth $\phi_{\varepsilon}\in{\rm
PSH}(X,\theta+\varepsilon\omega)$ normalized by $\sup_{X}\phi_{\varepsilon}=0$
such that
$(\theta+\varepsilon\omega+dd^{c}\phi_{\varepsilon})^{n}=e^{c_{\varepsilon}+\varphi_{\varepsilon}+\psi^{+}_{\varepsilon}-\psi^{-}_{\varepsilon}}\omega^{n}=g_{\varepsilon}\omega^{n},$
where $c_{\varepsilon}$ is a normalized constant. As in the proof of Theorem 3
in [15] we can prove that $c_{\varepsilon}$ converges to $0$ as
$\varepsilon\downarrow 0$. We then can show that $\phi_{\varepsilon}$
converges in $L^{1}$ to $\varphi-\sup_{X}\varphi$. Now, we can apply Theorem
5.1 and Theorem 5.2 in [15] to get uniform bound on $\phi_{\varepsilon}$ and
$\Delta_{\omega}\phi_{\varepsilon}$ on every compact subset of
$X\setminus(D\cup E)$. From this we can get the smoothness of $\varphi$ on
$X\setminus(D\cup E)$ as in [15]. ∎
It follows from Theorem 4.5 (which is also valid in the case of semipositive
and big classes) that to solve the equation it suffices to find a subsolution
in $\mathcal{E}(X,\theta)$. We show in the following example that in some
cases it is easy to find a subsolution in $\mathcal{E}(X,\theta)$.
###### Example 4.9.
We consider the density given in Example 4.7. Assume that $\theta$ satisfies
$\\{\theta\\}-c_{1}(E)>0$, where $E=\sum_{j=1}^{M}{a_{j}E_{j}}$ is an
effective simple normal crossing divisor on $X$. Assume that $E_{j}$ is
defined by the zero locus of a holomorphic section $\sigma_{j}$ such that
$|\sigma_{j}|<1/e$. Then for some constants $p\in(0,1)$ and $a>0$,
$A\in\mathbb{R}$ the following function
$u:=-\left(-a\sum_{j=1}^{N}\log|s_{j}|-\frac{1}{2}\sum_{j=1}^{M}a_{j}\log|\sigma_{j}|\right)^{p}-A$
belongs to $\mathcal{E}(X,\theta/2)$ and verifies
$\int_{X}e^{u}f\omega^{n}=2^{-n}\int_{X}\theta^{n}$. It follows from [4] that
there exists $v\in\mathcal{E}(X,\theta/2)$ such that $v\leq 0$ and
$(\theta/2+dd^{c}v)^{n}=e^{u}f\omega^{n}.$
It is easy to see that $\varphi:=u+v\in\mathcal{E}(X,\theta)$ is a subsolution
of (4.3).
### 4.6. Critical Integrability
Recently, Berndtsson [6] solved the openness conjecture of Demailly and Kollár
[14] which says that given $\phi\in{\rm PSH}(X,\omega)$ and
$\alpha(\phi)=\sup\\{t>0\ \ \big{|}\ \ e^{-t\phi}\in L^{1}(X)\\}<+\infty,$
then one has $e^{-\alpha\phi}\notin L^{1}(X)$ (a stronger version of the
openness conjecture has been quite recently obtained by Guan and Zhou [18]).
In the following result, we use the generalized capacity to show that
$e^{-\alpha\phi}$ is however not far to be integrable in the following sense:
###### Theorem 4.10.
Let $\phi\in{\rm PSH}(X,\omega)$ and $\alpha=\alpha(\phi)\in(0,+\infty)$ be
the canonical threshold of $\phi$. Then we can find $\varphi\in{\rm
PSH}(X,\omega)$ having zero Lelong number at all points of $X$ such that
$\int_{X}e^{\varphi-\alpha\phi}\omega^{n}<+\infty.$
One can moreover chose $\varphi=\chi\circ\phi\in\mathcal{E}(X,\omega)$ for
some $\chi$ increasing convex function. We thank S. Boucksom and H. Guenancia
for indicating this.
###### Proof.
Let $\alpha_{j}$ be an increasing sequence of positive numbers which converges
to $\alpha$. By assumption we have $e^{-\alpha_{j}\phi}$ is integrable on $X$.
We can assume that $\phi\leq 0$. We solve the complex Monge-Ampère equation
$(\omega+dd^{c}\varphi_{j})^{n}=e^{\varphi_{j}-\alpha_{j}\phi}\omega^{n}.$
For each $j$, since $e^{-\alpha_{j}\phi}$ belongs to $L^{p_{j}}$ for some
$p_{j}>1$, it follows from the classical result of Kołodziej [21] that
$\varphi_{j}$ is bounded. Moreover, the comparison principle reveals that
$\varphi_{j}$ is non-increasing. Now, we need to bound $\varphi_{j}$ uniformly
from below by some singular quasi-psh function.
Let $1/2>\varepsilon>0$ be a very small positive number. By assumption we know
that
$e^{(\varepsilon-\alpha)\phi}\in L^{p}(X),\ \
p=p_{\varepsilon}:=\frac{\alpha-\varepsilon/2}{\alpha-\varepsilon}>1.$
Set $\psi:=\varepsilon\phi\in{\rm PSH}(X,\omega/2)$ and consider the function
$H(t):=\left[{\rm Cap}_{\psi}(\varphi_{j}<\psi-t)\right]^{1/n},\ \ t>0.$
It follows from [15, Lemma 2.7] that ${\rm Cap}_{\omega}\leq 2^{n}{\rm
Cap}_{\psi}$. By a strong volume-capacity domination in [19, Remark 5.10] we
also have
${\rm Vol}_{\omega}\leq\exp{\left(\frac{-C_{1}}{{\rm
Cap}_{\omega}^{1/n}}\right)},$
where $C_{1}$ depends only on $(X,\omega)$. Fix $t>0,s\in[0,1]$. Using [15,
Proposition 2.8] and Hölder inequality we get
$\displaystyle s^{n}{\rm Cap}_{\psi}(\\{\varphi_{j}<\psi-t-s\\})$
$\displaystyle\leq$
$\displaystyle\int_{\\{\varphi_{j}<\psi-t\\}}\mathrm{MA}\,(\varphi_{j})$
$\displaystyle\leq$
$\displaystyle\int_{\\{\varphi_{j}<\psi-t\\}}e^{-\varphi_{j}}e^{\psi}\mathrm{MA}\,(\varphi_{j})$
$\displaystyle\leq$
$\displaystyle\int_{\\{\varphi_{j}<\psi-t\\}}e^{(\varepsilon-\alpha)\phi}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\left[\int_{X}e^{(\varepsilon/2-\alpha)\phi}\omega^{n}\right]^{1/p}\left[\int_{\\{\varphi_{j}<\psi-t\\}}\omega^{n}\right]^{1/q}$
$\displaystyle\leq$ $\displaystyle C_{2}\left[{\rm
Cap}_{\psi}(\\{\varphi_{j}<\psi-t\\})\right]^{2},$
where $p=p_{\varepsilon}:=\frac{\alpha-\varepsilon/2}{\alpha-\varepsilon}>1$
and $q>1$ is the exponent conjugate of $p$. The constant $C_{2}>0$ depends on
$\varepsilon$ and also on $\int_{X}e^{(\varepsilon/2-\alpha)\phi}\omega^{n}$.
We then get
$sH(t+s)\leq C_{2}^{1/n}H(t)^{2},\ \forall t>0,\forall s\in[0,1].$
Then by [17, Lemma 2.4] we get
$\varphi_{j}\geq\varepsilon\phi-C_{\varepsilon},$
where $C_{\varepsilon}$ only depends on $\varepsilon$ and
$\int_{X}e^{(\varepsilon/2-\alpha)\phi}\omega^{n}$. Then we see that
$\varphi_{j}$ decreases to $\varphi\in{\rm PSH}(X,\omega)$ and $\varphi$
satisfies
$\varphi\geq\varepsilon\phi-C_{\varepsilon}.$
Since $\varepsilon$ is arbitrarily small we conclude that $\varphi$ has zero
Lelong number everywhere on $X$. Finally, it follows from Fatou’s lemma that
$e^{\varphi-\alpha\phi}$ is integrable on $X$.
We now show that $\varphi$ can be chosen to be in $\mathcal{E}(X,\omega)$,
more precisely $\varphi=\chi\circ\phi$,
$\int_{X}e^{\chi\circ\phi-\alpha\phi}\omega^{n}<+\infty,$
for some $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ increasing convex
function such that $\chi(-\infty)=-\infty$ and $\chi^{\prime}(-\infty)=0$.
Note that $\chi\circ\phi\in\mathcal{E}(X,\omega)$ thanks to [12]. We are
grateful to H. Guenancia for the following constructive proof.
We can always assume that $\phi\leq-1$. For each $k\in\mathbb{N}$ let
(4.4) $a_{k}:=\log\int_{X}e^{-(\alpha-2^{-k-1})\phi}\omega^{n}<+\infty.$
Define the sequence $(c_{k})$ inductively by
(4.5) $c_{1}=a_{1},\ c_{k+1}:=\max(c_{k}+4k,a_{k+1}),\ \forall k\geq 1.$
Define another sequence $(t_{k})$ by
(4.6) $t_{1}:=1,\ t_{k+1}:=2^{k+1}(c_{k+1}-c_{k}),\ \forall k\geq 1.$
Define $\chi:(-\infty,-1]\rightarrow\mathbb{R}^{-}$ by
$\chi(-t):=-2^{-k}t-c_{k}\ \ {\rm if}\ \ t\in[t_{k},t_{k+1}],\ \forall k\geq
1.$
It follows from (4.4) that
$e^{(\alpha-2^{-k-1})t}\ {\rm
Vol}(\phi<-t)\leq\int_{X}e^{-(\alpha-2^{-k-1})\phi}\omega^{n}\leq e^{c_{k}}.$
Thus using (4.5), (4.6) and the above inequality we get
$\displaystyle\int_{X}e^{\chi(\phi)-\alpha\phi}\omega^{n}$ $\displaystyle\leq$
$\displaystyle e^{\chi(-1)+\alpha}+\alpha\int_{1}^{+\infty}e^{\alpha
t+\chi(-t)}{\rm Vol}(\phi<-t)dt$ $\displaystyle\leq$ $\displaystyle
C+\alpha\sum_{k=1}^{+\infty}\int_{t_{k}}^{t_{k+1}}e^{\alpha t+\chi(-t)}\ {\rm
Vol}(\phi<-t)dt$ $\displaystyle\leq$ $\displaystyle
C+\alpha\sum_{k=1}^{+\infty}\int_{t_{k}}^{t_{k+1}}e^{c_{k}+2^{-k-1}t-2^{-k}t-c_{k}}dt$
$\displaystyle\leq$ $\displaystyle
C+\alpha\sum_{k=1}^{+\infty}\int_{t_{k}}^{t_{k+1}}e^{-2^{-k-1}t}dt$
$\displaystyle\leq$ $\displaystyle
C+\alpha\sum_{k=1}^{+\infty}2^{k+1}e^{-2^{-k-1}t_{k}}$ $\displaystyle\leq$
$\displaystyle C+\alpha\sum_{k=1}^{+\infty}2^{k+1}e^{-2^{-1}(c_{k}-c_{k-1})}$
$\displaystyle\leq$ $\displaystyle
C+\alpha\sum_{k=1}^{+\infty}2^{k+1}e^{-2(k-1)}$ $\displaystyle\leq$
$\displaystyle C+4\alpha.$
∎
The above result is quite optimal as the following example shows:
###### Example 4.11.
Let $(X,\omega)$ be a compact Kähler manifold and $D$ be a smooth complex
hypersurface on $X$ defined by a holomorphic section $s$ such that $|s|\leq
1/e$. Consider
(4.7) $\phi=2\log|s|-(-\log|s|)^{p},\ \ p\in(0,1).$
By rescaling $\omega$ we can assume that $\phi\in{\rm PSH}(X,\omega)$. Then
for any $q>0$
$\int_{X}\frac{e^{-\phi}}{(-\phi)^{q}}\omega^{n}=+\infty.$
The example above has been given in [1] in the case of one complex variable
which is locally similar to our setting. Assume now that $\phi$ is given by
(4.7). It follows from Theorem 4.10 that we can find $\varphi\in{\rm
PSH}(X,\omega)$ having zero Lelong number everywhere such that
$\int_{X}e^{\varphi-\phi}\omega^{n}<+\infty.$
In this concrete example one such function $\varphi$ can be given explicitly
by
$\varphi=-(\log|s|)^{p}-(1+\varepsilon)\log(\log|s|),\ \varepsilon>0.$
###### Proof of Theorem D.
It follows from the above proof of Theorem 4.10 that there exists
$u\in\mathcal{E}(X,\omega/2)$ such that $e^{u-\alpha\phi}$ is integrable. We
then can argue as in Example 4.9 to find a subsolution which also yields a
solution thanks to Theorem 4.5. The uniqueness follows from the comparison
principle (see [5]). ∎
## References
* [1] P. Ahag, U. Cegrell, S. Kolodziej, , H. H. Pham, A. Zeriahi, Partial pluricomplex energy and integrability exponents of plurisubharmonic functions, Advances Math. 222 (2009), 2036–2058.
* [2] E. Bedford, B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1-40.
* [3] S. Benelkourchi, V. Guedj, A. Zeriahi, Plurisubharmonic functions with weak singularities, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist. Vol. 86 (2009), 57-74.
* [4] R. J. Berman, S. Boucksom, V. Guedj, A. Zeriahi, A variational approach to complex Monge-Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179-245.
* [5] R. J. Berman, H. Guenancia, Kähler-Einstein metrics on stable varieties and log canonical pairs, arXiv:1304.2087.
* [6] B. Berndtsson, The openness conjecture for plurisubharmonic functions, arXiv:1305.5781.
* [7] T. Bloom, N. Levenberg, Pluripotential energy, Potential Analysis, Volume 36, Issue 1 (2012), 155-176.
* [8] S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Monge-Ampère equations in big cohomology classes, Acta Math. 205 (2010), no. 2, 199-262.
* [9] U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998), no. 2, 187-217.
* [10] U. Cegrell, The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 1, 159-179.
* [11] U. Cegrell, S. Kołodziej, A. Zeriahi, Subextension of plurisubharmonic functions with weak singularities, Math. Z. 250 (2005), no. 1, 7-22.
* [12] D. Coman, V. Guedj, A. Zeriahi, Domains of definitions of Monge-Ampère operators on compact Kähler manifolds, Math. Z. 259 (2008), no. 2, 393-418.
* [13] J. P. Demailly, Regularization of closed positive currents and intersection theory, J. Alg. Geom. 1 (1992), no. 3, 361-409.
* [14] J. P. Demailly, J. Kollár, Semicontinuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 4, 525-556.
* [15] E. Di Nezza, H. C. Lu, Complex Monge-Ampère equations on quasi-projective varieties, arXiv:1401.6398.
* [16] S. Dinew, Uniqueness in $\mathcal{E}(X,\omega)$, J. Funct. Anal. 256 (2009), no. 7, 2113-2122.
* [17] P. Eyssidieux, V. Guedj, A. Zeriahi, Singular Kähler Einstein metrics, Journal of the American Mathematical Society, Volume 22, Number 3, (2009), 607-639.
* [18] Q. Guan, X. Zhou, Strong openness conjecture for plurisubharmonic functions, arXiv: 1311.3781.
* [19] V. Guedj, A. Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), no. 4, 607-639.
* [20] V. Guedj, A. Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), no. 2, 442-482.
* [21] S. Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998) 69-117.
* [22] S. Kołodziej, The complex Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), no. 3, 667-686.
* [23] H. Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans $\mathbb{C}^{n}$, Bull. Soc. Math. de France 100 (1972), 353-408. J. Amer. Math. Soc. 3 (1990), no. 3, 579-609.
* [24] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411.
* [25] A. Zeriahi,Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions, Indiana Univ. Math. J., 50 (2001), 671-703.
|
arxiv-papers
| 2014-02-11T14:21:10 |
2024-09-04T02:49:58.061108
|
{
"license": "Public Domain",
"authors": "Eleonora Di Nezza and Chinh H. Lu",
"submitter": "Chinh Lu Hoang",
"url": "https://arxiv.org/abs/1402.2497"
}
|
1402.2539
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-016 LHCb-PAPER-2013-066 25 March 2014
Measurement of $\Upsilon$ production in $\mathrm{p}\mathrm{p}$ collisions at
$\sqrt{s}=2.76\mathrm{\,Te\kern-2.07413ptV}$
The LHCb collaboration111Authors are listed on the following pages.
The production of $\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and
$\Upsilon(3\mathrm{S})$ mesons decaying into the dimuon final state is studied
with the LHCb detector using a data sample corresponding to an integrated
luminosity of $3.3\mbox{\,pb}^{-1}$ collected in proton-proton collisions at a
centre-of-mass energy of $\sqrt{s}=2.76$ TeV. The differential production
cross-sections times dimuon branching fractions are measured as functions of
the $\Upsilon$ transverse momentum and rapidity, over the ranges $p_{\rm
T}<15$ GeV/$c$ and $2.0<y<4.5$. The total cross-sections in this kinematic
region, assuming unpolarised production, are measured to be
$\displaystyle\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon(1\mathrm{S})\mathrm{X}\right)\times{\cal
B}\left(\Upsilon(1\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}\right)$
$\displaystyle=$ $\displaystyle 1.111\pm 0.043\pm 0.044\rm\,nb,$
$\displaystyle\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon(2\mathrm{S})\mathrm{X}\right)\times{\cal
B}\left(\Upsilon(2\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}\right)$
$\displaystyle=$ $\displaystyle 0.264\pm 0.023\pm 0.011\rm\,nb,$
$\displaystyle\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon(3\mathrm{S})\mathrm{X}\right)\times{\cal
B}\left(\Upsilon(3\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}\right)$
$\displaystyle=$ $\displaystyle 0.159\pm 0.020\pm 0.007\rm\,nb,$
where the first uncertainty is statistical and the second systematic.
Submitted to Eur. Phys. J. C
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25, S. Amato2, S. Amerio22, Y. Amhis7, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel39, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson50, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30, W.
Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,r, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez36,p, A. Camboni36, P. Campana18,38, D. Campora
Perez38, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,j, A. Cardini15, H.
Carranza-Mejia50, L. Carson50, K. Carvalho Akiba2, G. Casse52, L. Castillo
Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph.
Charpentier38, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38,
X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47,
J. Closier38, C. Coca29, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A.
Comerma-Montells36, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G.
Corti38, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz
Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J. Dalseno46, P.
David8, P.N.Y. David41, A. Davis57, I. De Bonis4, K. De Bruyn41, S. De
Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
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Donleavy52, F. Dordei11, M. Dorigo39, P. Dorosz26,o, A. Dosil Suárez37, D.
Dossett48, A. Dovbnya43, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49, U. Egede53, V. Egorychev31, S. Eidelman34,
S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5,
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Farry52, D. Ferguson50, V. Fernandez Albor37, F. Ferreira Rodrigues1, M.
Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M. Fiorini16,f, C. Fitzpatrick38,
M. Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C.
Frei38, M. Frosini17,38,g, J. Fu21, E. Furfaro24,l, A. Gallas Torreira37, D.
Galli14,d, M. Gandelman2, P. Gandini59, Y. Gao3, J. Garofoli59, J. Garra
Tico47, L. Garrido36, C. Gaspar38, R. Gauld55, E. Gersabeck11, M. Gersabeck54,
T. Gershon48, Ph. Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47, L.
Giubega29, V.V. Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38, A.
Gomes1,a, H. Gordon38, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado
Cardoso38, E. Graugés36, G. Graziani17, A. Grecu29, E. Greening55, S.
Gregson47, P. Griffith45, L. Grillo11, O. Grünberg61, B. Gui59, E. Gushchin33,
Yu. Guz35,38, T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, T.W.
Hafkenscheid64, S.C. Haines47, S. Hall53, B. Hamilton58, T. Hampson46, S.
Hansmann-Menzemer11, N. Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann61,
J. He38, T. Head38, V. Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A.
Hernando Morata37, E. van Herwijnen38, M. Heß61, A. Hicheur1, D. Hill55, M.
Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55, N. Hussain55, D.
Hutchcroft52, D. Hynds51, V. Iakovenko44, M. Idzik27, P. Ilten56, R.
Jacobsson38, A. Jaeger11, E. Jans41, P. Jaton39, A. Jawahery58, F. Jing3, M.
John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N. Jurik59, M.
Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, M.
Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C. Khurewathanakul39, S.
Klaver54, O. Kochebina7, I. Komarov39, R.F. Koopman42, P. Koppenburg41, M.
Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11, M. Kreps48, G.
Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k, V.
Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31,38, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, T. Latham48, C. Lazzeroni45,
R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32, J.
Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11, Y. Li3, M.
Liles52, R. Lindner38, C. Linn11, F. Lionetto40, B. Liu15, G. Liu38, S.
Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40, H. Lu3, D.
Lucchesi22,r, J. Luisier39, H. Luo50, E. Luppi16,f, O. Lupton55, F.
Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O. Maev30,38, S. Malde55, G.
Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5, U. Marconi14, P.
Marino23,t, R. Märki39, J. Marks11, G. Martellotti25, A. Martens8, A. Martín
Sánchez7, M. Martinelli41, D. Martinez Santos42, F. Martinez Vidal63, D.
Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A.
Mazurov16,38,f, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
McSkelly52, B. Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,t, R. Mountain59, F.
Muheim50, K. Müller40, R. Muresan29, B. Muryn27, B. Muster39, P. Naik46, T.
Nakada39, R. Nandakumar49, I. Nasteva1, M. Needham50, N. Neri21, S. Neubert38,
N. Neufeld38, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,q, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-
Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R.
Oldeman15,e, G. Onderwater64, M. Orlandea29, J.M. Otalora Goicochea2, P.
Owen53, A. Oyanguren36, B.K. Pal59, A. Palano13,c, F. Palombo21,u, M.
Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51, L. Pappalardo16,
C. Parkes54, C.J. Parkinson9, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, C. Pavel-Nicorescu29, A. Pazos Alvarez37, A. Pearce54, A.
Pellegrino41, G. Penso25,m, M. Pepe Altarelli38, S. Perazzini14,d, E. Perez
Trigo37, P. Perret5, M. Perrin-Terrin6, L. Pescatore45, E. Pesen65, G.
Pessina20, K. Petridis53, A. Petrolini19,j, E. Picatoste Olloqui36, B.
Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50, M. Plo
Casasus37, F. Polci8, G. Polok26, A. Poluektov48,34, E. Polycarpo2, A.
Popov35, D. Popov10, B. Popovici29, C. Potterat36, A. Powell55, J.
Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39,
G. Punzi23,s, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Redford55, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S.
Ricciardi49, A. Richards53, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts58, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez
Perez37, S. Roiser38, V. Romanovsky35, A. Romero Vidal37, M. Rotondo22, J.
Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P. Ruiz Valls36, G.
Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B. Saitta15,e,
V. Salustino Guimaraes2, B. Sanmartin Sedes37, R. Santacesaria25, C.
Santamarina Rios37, E. Santovetti24,l, M. Sapunov6, A. Sarti18, C.
Satriano25,n, A. Satta24, M. Savrie16,f, D. Savrina31,32, M. Schiller42, H.
Schindler38, M. Schlupp9, M. Schmelling10, B. Schmidt38, O. Schneider39, A.
Schopper38, M.-H. Schune7, R. Schwemmer38, B. Sciascia18, A. Sciubba25, M.
Seco37, A. Semennikov31, K. Senderowska27, I. Sepp53, N. Serra40, J. Serrano6,
P. Seyfert11, M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52,
L. Shekhtman34, O. Shevchenko43, V. Shevchenko62, A. Shires9, R. Silva
Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, T. Skwarnicki59, N.A.
Smith52, E. Smith55,49, E. Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D.
Sokoloff57, F.J.P. Soler51, F. Soomro39, D. Souza46, B. Souza De Paula2, B.
Spaan9, A. Sparkes50, F. Spinella23, P. Spradlin51, F. Stagni38, S. Stahl11,
O. Steinkamp40, S. Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S.
Stracka23,38, M. Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38,
L. Sun57, W. Sutcliffe53, S. Swientek9, V. Syropoulos42, M. Szczekowski28, P.
Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, E. Teodorescu29, F. Teubert38, C. Thomas55, E. Thomas38, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D.
Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4,53, S. Tourneur39,
M.T. Tran39, M. Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M.
Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin62, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez
Sierra37, S. Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M.
Vesterinen11, B. Viaud7, D. Vieira2, X. Vilasis-Cardona36,p, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß61, H.
Voss10, J.A. de Vries41, R. Waldi61, C. Wallace48, R. Wallace12, S.
Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, A.D. Webber54, D.
Websdale53, M. Whitehead48, J. Wicht38, J. Wiechczynski26, D. Wiedner11, L.
Wiggers41, G. Wilkinson55, M.P. Williams48,49, M. Williams56, F.F. Wilson49,
J. Wimberley58, J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A.
Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y. Xie50,38, Z. Xing59, Z. Yang3,
X. Yuan3, O. Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, F. Zhang3, L.
Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3, A.
Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
62National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
63Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
64KVI - University of Groningen, Groningen, The Netherlands, associated to 41
65Celal Bayar University, Manisa, Turkey, associated to 38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
## 1 Introduction
Studies of the production of heavy quark-antiquark bound systems, such as the
$\mathrm{b}\overline{}\mathrm{b}$ states $\Upsilon(1\mathrm{S})$,
$\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ (indicated generically as
$\Upsilon$ in the following) in hadron-hadron interactions probe the dynamics
of the colliding partons and provide a unique insight into quantum
chromodynamics (QCD). The total production cross-sections and spin
configurations of these heavy quarkonium states are currently not reproduced
by the theoretical models. These include the colour singlet model [1, 2, 3, 4,
5], recently improved by adding higher-order contributions [6, 7], the colour-
evaporation model [8], and the non-perturbative colour octet mechanism [9, 10,
11], which is investigated in the framework of non-relativistic QCD. The first
complete next-to-leading order calculation of $\Upsilon$ production properties
[12], based on the non-relativistic QCD factorisation scheme, provides a good
description of the measured differential cross-sections at large transverse
momentum, $p_{\rm T}$, but overestimates the data at low $p_{\rm T}$.
The production of $\Upsilon$ mesons in proton-proton ($\mathrm{p}\mathrm{p}$)
collisions occurs either directly in parton scattering or via feed-down from
the decay of heavier prompt bottomonium states, like $\upchi_{\mathrm{b}}$
[13, 14, 15, 16], or higher-mass $\Upsilon$ states. The latter source
complicates the theoretical description of bottomonium production [17, 18].
The Large Hadron Collider provides a unique possibility to study bottomonium
and charmonium hadroproduction in $\mathrm{p}\mathrm{p}$ interactions at
different collision energies and discriminate between various theoretical
approaches. This study presents the first measurement of the inclusive
production cross-sections of the three considered $\Upsilon$ mesons in
$\mathrm{p}\mathrm{p}$ collisions at a centre-of-mass energy of
$\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$. The measurements are performed
as functions of the $\Upsilon$ transverse momentum and rapidity, $y$,
separately in six bins of $p_{\rm T}$ in the range $\mbox{$p_{\rm
T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and five bins of $y$ in the range
$2.0<y<4.5$. The results are reported as products of the cross-sections and
the branching fractions of $\Upsilon$ mesons into the dimuon final state. This
analysis is complementary to those performed by the ATLAS [19], CMS [20] and
LHCb [21, 22] collaborations and allows studies of the $\Upsilon$ production
cross-section at forward rapidities as a function of the centre-of-mass
energy.
## 2 Detector and data sample
The LHCb detector [23] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $\mathrm{b}$ or $\mathrm{c}$ quarks. The detector includes a high-
precision tracking system consisting of a silicon-strip vertex detector
surrounding the $\mathrm{p}\mathrm{p}$ interaction region, a large-area
silicon-strip detector located upstream of a dipole magnet with a bending
power of about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and
straw drift tubes placed downstream. The combined tracking system provides a
momentum measurement with relative uncertainty that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with large transverse momentum. Different types of
charged hadrons are distinguished by information from two ring-imaging
Cherenkov detectors [24]. Photon, electron and hadron candidates are
identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers [25].
The analysis is carried out using a sample of data corresponding to an
integrated luminosity of $3.3\mbox{\,pb}^{-1}$ collected in
$\mathrm{p}\mathrm{p}$ collisions at
$\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$. Events of interest are
preselected by a trigger consisting of a hardware stage, based on information
from the calorimeter and muon systems, followed by a software stage, which
applies a full event reconstruction. The presence of two muon candidates with
the product of their $p_{\rm T}$ larger than 1.68 $($GeV$/c$$)^{2}$ is
required in the hardware trigger. At the software stage, the events are
required to contain two well reconstructed tracks with hits in the muon
system, having total and transverse momenta greater than
$6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, respectively. The selected muon
candidates are further required to originate from a common vertex and have an
invariant mass larger than $4.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
To determine the acceptance, reconstruction and trigger efficiencies, fully
simulated signal samples are reweighted to reproduce the multiplicity
distributions for reconstructed primary vertices, tracks and hits in the
detector observed in the data. The simulation is performed using the LHCb
configuration [26] of the Pythia 6.4 event generator [27]. Here, decays of
hadronic particles are described by EvtGen [28] in which final-state photons
are generated using Photos [29]. The interaction of the generated particles
with the detector and its response are implemented using the Geant4 toolkit
[30, 31] as described in Ref. [32].
## 3 Signal selection and cross-section determination
The selection strategy used in the previous LHCb studies on $\Upsilon$
production [21, 22] is applied here. It includes selection criteria that
ensure good quality track and vertex reconstruction. In addition, the muon
candidates are required to have $p>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$\mbox{$p_{\rm T}$}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. To further reduce
background contamination, a set of additional requirements is employed in this
analysis. It consists of tightened criteria on track quality [33], muon
identification [34] and a good quality of a global fit of the dimuon vertex
with a primary vertex constraint [35].
The invariant mass distribution of the selected
$\Upsilon\\!\rightarrow\upmu^{+}\upmu^{-}$ candidates is shown in Fig. 1 for
the full kinematic range. The distribution is described by a function similar
to the one used in the previous studies on $\Upsilon$ production [21, 22]. It
models the signal component using the sum of three Crystal Ball functions
[36], one for each of the $\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and
$\Upsilon(3\mathrm{S})$ signals, and includes an exponential component to
account for combinatorial background. The position and width of the Crystal
Ball function describing the $\Upsilon(1\mathrm{S})$ meson are allowed to
vary, while the mass differences between $\Upsilon$ states are fixed to their
known values [37] along with parameters describing the radiative tail, as
determined from simulation studies. The widths of the $\Upsilon(2\mathrm{S})$
and $\Upsilon(3\mathrm{S})$ peaks are constrained to the value of the width of
the $\Upsilon(1\mathrm{S})$ signal scaled by the ratio of their masses to the
$\Upsilon(1\mathrm{S})$ mass. In total, five parameters are extracted from the
fit for the signal component: the yields of $\Upsilon(1\mathrm{S})$,
$\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ states, the
$\Upsilon(1\mathrm{S})$ mass resolution and its peak position. The latter is
found to be consistent with the known mass of the $\Upsilon(1\mathrm{S})$
meson [37], while reasonable agreement is observed between the data and
simulation for the $\Upsilon(1\mathrm{S})$ mass resolution.
LHCb$\sqrt{s}$=2.76$\mathrm{\,Te\kern-1.00006ptV}$$m_{\upmu^{+}\upmu^{-}}$$\left[\\!{\mathrm{\,Ge\kern-1.20007ptV\\!/}c^{2}}\right]$
Candidates/(50${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
Figure 1: Invariant mass distribution of selected
$\Upsilon\\!\rightarrow\upmu^{+}\upmu^{-}$ candidates with $\mbox{$p_{\rm
T}$}<15{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and $2.0<y<4.5$. The result of
the fit described in the text is illustrated with a red solid line, while the
signal and background components are shown with magenta dotted and blue dashed
lines, respectively. The three peaks correspond to the
$\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$
mesons (from left to right).
The $\Upsilon$ production cross-sections are measured separately in six bins
of $p_{\rm T}$ and five bins of $y$ since the limited amount of data does not
allow a measurement of double differential cross-sections. For a given $p_{\rm
T}$ or $y$ bin, the differential cross-section for the inclusive $\Upsilon$
production of the three different states decaying into the dimuon final state
is determined as
$\dfrac{{\mathrm{d}}\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon{\mathrm{X}}\right)}{\mathrm{d}\mbox{$p_{\rm
T}$}}\times\mathcal{B}\left(\Upsilon\\!\rightarrow\upmu^{+}\upmu^{-}\right)=\dfrac{N^{\mathrm{corr}}_{\Upsilon}}{\mathcal{L}\times\Delta\mbox{$p_{\rm
T}$}}\;,$ (1a)
$\dfrac{{\mathrm{d}}\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon{\mathrm{X}}\right)}{\mathrm{d}y}\times\mathcal{B}\left(\Upsilon\\!\rightarrow\upmu^{+}\upmu^{-}\right)=\dfrac{N^{\mathrm{corr}}_{\Upsilon}}{\mathcal{L}\times\Delta
y}\;,$ (1b)
where $N^{\mathrm{corr}}_{\Upsilon}$ is the efficiency-corrected yield of
$\Upsilon\\!\rightarrow\upmu^{+}\upmu^{-}$ decays, $\mathcal{L}$ stands for
the integrated luminosity and $\Delta\mbox{$p_{\rm T}$}\,(\Delta y)$ denotes
the $\mbox{$p_{\rm T}$}\,(y)$ bin size. For the mass fits in individual
$p_{\rm T}$ and $y$ bins, the $\Upsilon(1\mathrm{S})$ peak position is fixed
to the value obtained from the fit for the full kinematic range, while the
$\Upsilon(1\mathrm{S})$ mass resolution is parameterised with a function of
$p_{\rm T}$ and $y$ using simulation. The total observed signal yields and
their statistical uncertainties for $\Upsilon(1\mathrm{S})$,
$\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ mesons obtained by
summation over $\mbox{$p_{\rm T}$}\,(y)$ bins are $1139\pm 37\,(1145\pm 37)$,
$271\pm 20\,(270\pm 20)$ and $158\pm 16\,(156\pm 16)$, respectively. These
results are in good agreement with the total signal yields obtained from the
fit to the reconstructed dimuon invariant mass for the full kinematic range.
Based on the mass fit results in individual bins, the efficiency-corrected
yield for each kinematic region is determined as
$N^{\mathrm{corr}}_{\Upsilon}=\sum_{i}\dfrac{w^{\Upsilon}_{i}}{\varepsilon^{\mathrm{tot}}_{i}}\;,$
(2)
where $w^{\Upsilon}_{i}$ is a signal weight factor,
$\varepsilon^{\mathrm{tot}}_{i}$ is the total signal event efficiency and the
sum runs over all candidates $i$. The $w^{\Upsilon}_{i}$ factor accounts for
the background subtraction and is obtained from the fit using the sPlot
technique [38]. The total signal event efficiency is calculated for each
$\Upsilon\\!\rightarrow\upmu^{+}\upmu^{-}$ candidate as
$\varepsilon^{\mathrm{tot}}=\varepsilon^{\mathrm{acc}}\times\varepsilon^{\mathrm{rec}}\times\varepsilon^{\mathrm{trg}}\times\varepsilon^{\upmu\mathrm{ID}}\;,$
(3)
where $\varepsilon^{\mathrm{acc}}$ is the detector acceptance,
$\varepsilon^{\mathrm{rec}}$ is the reconstruction and selection efficiency,
$\varepsilon^{\mathrm{trg}}$ is the trigger efficiency and
$\varepsilon^{\upmu\mathrm{ID}}$ is the efficiency of muon identification. The
efficiencies $\varepsilon^{\mathrm{acc}}$, $\varepsilon^{\mathrm{rec}}$ and
$\varepsilon^{\mathrm{trg}}$ are determined using simulation and further
corrected using data-driven techniques to account for small differences in
muon reconstruction efficiency between data and simulation [34, 33, 39]. The
efficiency $\varepsilon^{\upmu\mathrm{ID}}$ is measured directly from data
using a tag-and-probe method on a large sample of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ decays. The total efficiency-corrected
signal yields obtained by summation over $\mbox{$p_{\rm T}$}\,(y)$ bins for
$\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$
mesons are $3678\pm 144\,(3684\pm 143)$, $875\pm 76\,(869\pm 75)$ and $527\pm
65\,(515\pm 64)$, respectively.
The integrated luminosity of the data sample is estimated with the beam-gas
imaging method [40, 41, 42, 43, 44]. It is based on the beam currents and the
measurements of the angles, offsets and transverse profiles of the two
colliding bunches, which is achieved by reconstructing beam-gas interaction
vertices.
## 4 Systematic uncertainties
Previous LHCb studies of $\Upsilon$ production [21, 22] showed that the signal
efficiency depends on the initial polarisation of $\Upsilon$ mesons. This
property was measured in $\mathrm{p}\mathrm{p}$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ by the CMS collaboration at central
rapidities and large $p_{\rm T}$ and was found to be small [45]. Polarisation
of other vector quarkonium states, such as
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\uppsi\mathrm{(2S)}$ mesons was studied in $\mathrm{p}\mathrm{p}$ collisions
at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ by the LHCb [46, 47] and ALICE
[48] collaborations and was also found to be small. This analysis is performed
assuming zero polarisation of $\Upsilon$ mesons and no corresponding
systematic uncertainty is assigned.
Table 1: Relative systematic uncertainties (in $\%$) affecting the $\Upsilon$ production cross-section measurements in the full kinematic region. The total uncertainties are obtained by adding the individual effects in quadrature. Source | $\Upsilon(1\mathrm{S})$ | $\Upsilon(2\mathrm{S})$ | $\Upsilon(3\mathrm{S})$
---|---|---|---
Luminosity | 2.3 | 2.3 | 2.3
Fit model and range | 0.5 | 1.0 | 2.3
Data-simulation agreement | | |
Radiative tails | 1.0 | 1.0 | 1.0
Multiplicity reweighting | 0.6 | 0.4 | 2.0
Efficiency corrections | 0.7 | 1.0 | 1.0
Track reconstruction | $2\times 0.4$ | $2\times 0.4$ | $2\times 0.4$
Selection variables | 1.0 | 1.0 | 1.0
Trigger | 2.0 | 2.0 | 2.0
Total | 3.6 | 3.7 | 4.7
The systematic uncertainties affecting the $\Upsilon$ cross-section
measurements presented in this paper are summarised in Table 1. These
uncertainties are strongly correlated between bins. The largest contribution
arises from the absolute luminosity scale, which is determined with a 2.3%
uncertainty. It is dominated by the vertex resolution of beam-gas interactions
and detector alignment [44].
The influence of the signal extraction technique is studied by varying the fit
range and the signal and background parameterisations used in the fit model.
The fits are also performed with floating mass and resolution of the
$\Upsilon(1\mathrm{S})$ peak and without constraints for the
$\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ masses. The spread of the
extracted signal yields between these scenarios is taken as the corresponding
systematic uncertainty. It ranges from 0.4 to 33% for different $\mbox{$p_{\rm
T}$}\,(y)$ bins and amounts to 0.5%, 1.0% and 2.3% for the
$\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$
cross-section measurements in the full kinematic region, respectively.
The possible mismodeling of bremsstrahlung simulation for the radiative tail
and its effect on the signal shape was addressed in the previous LHCb analysis
[22]. It leads to an additional uncertainty of 1.0%.
Several systematic uncertainties are related to the determination of the total
efficiency components in Eq. (3). The detector acceptance, reconstruction and
selection efficiencies are determined using simulated samples. These are
corrected using an iterative procedure to match the multiplicity distributions
for reconstructed primary vertices, tracks and hits in the detector with those
observed in data. The systematic uncertainty associated with this reweighting
procedure is assessed by varying the number of iterative steps. It ranges from
0.4 to 4.8% for different $\mbox{$p_{\rm T}$}\,(y)$ bins and is found to be
0.6%, 0.4% and 2.0% for the $\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$
and $\Upsilon(3\mathrm{S})$ cross-section measurements in the full kinematic
region, respectively.
The $\varepsilon^{\mathrm{rec}}$ efficiency is corrected using data-driven
techniques for a small difference in the muon reconstruction efficiency
between data and simulation [34, 33]. The $\varepsilon^{\upmu\mathrm{ID}}$
efficiency is determined from data using alternative methods, based on a tag-
and-probe approach on a large sample of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ decays. The difference between these
methods is taken as the corresponding systematic uncertainty. It is combined
with the uncertainties associated with the correction factors discussed above
and propagated to the $\Upsilon$ cross-section measurements using 400 pseudo-
experiments. The resulting uncertainty ranges from 1.0 to 13% for different
$\mbox{$p_{\rm T}$}\,(y)$ bins and amounts to 0.7%, 1.0% and 1.0% for the
$\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$
cross-section measurements in the full kinematic region, respectively.
To account for differences between the actual tracking efficiency and that
estimated with simulation using data-driven techniques [33, 39], a systematic
uncertainty of 0.4% is assigned per track.
Good agreement between the data and reweighted simulation is observed for all
selection variables used in this analysis, in particular for the $\chi^{2}$ of
the dimuon vertex fit and the $\chi^{2}$ of the global fit [35]. The
discrepancies do not exceed 1.0%, which is conservatively taken as a
systematic uncertainty to account for the disagreement between the data and
simulation.
The systematic uncertainty associated with the trigger requirements is
assessed by studying the performance of the dimuon trigger, described in Sect.
2, for events selected using the single muon high-$p_{\rm T}$ trigger [49].
The fractions of signal $\Upsilon(1\mathrm{S})$ events selected using both
trigger requirements are compared for the data and simulation in bins of
dimuon $p_{\rm T}$, and a systematic uncertainty of 2.0% is assigned.
## 5 Results
The integrated $\Upsilon$ production cross-sections times dimuon branching
fractions in the kinematic region $\mbox{$p_{\rm
T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$ are measured to
be
$\displaystyle\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon(1\mathrm{S})\mathrm{X}\right)\times{\cal
B}\left(\Upsilon(1\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}\right)$
$\displaystyle=$ $\displaystyle 1.111\pm 0.043\pm 0.044\rm\,nb,$
$\displaystyle\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon(2\mathrm{S})\mathrm{X}\right)\times{\cal
B}\left(\Upsilon(2\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}\right)$
$\displaystyle=$ $\displaystyle 0.264\pm 0.023\pm 0.011\rm\,nb,$
$\displaystyle\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon(3\mathrm{S})\mathrm{X}\right)\times{\cal
B}\left(\Upsilon(3\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}\right)$
$\displaystyle=$ $\displaystyle 0.159\pm 0.020\pm 0.007\rm\,nb,$
where the first uncertainty is statistical and the second systematic.
The single differential cross-sections times dimuon branching fractions are
shown as functions of $p_{\rm T}$ and $y$ in Fig. 2 and summarised in Table 2.
The total uncertainties of the results are dominated by statistical effects in
all $p_{\rm T}$ and $y$ bins. In addition to the data, Fig. 2 reports
theoretical predictions, based on the next-to-leading order non-relativistic
QCD calculation [18], for the $\Upsilon$ differential cross-sections in the
kinematic region $6<\mbox{$p_{\rm T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and $2.0<y<4.5$. The long-distance matrix elements used in the calculations
are fitted to CDF [50] and D0 [51] results for $\Upsilon(1\mathrm{S})$
production in $\mathrm{p}\overline{}\mathrm{p}$ collisions at $\sqrt{s}=1.8$
and 1.96$\mathrm{\,Te\kern-1.00006ptV}$. The predictions include the feed-down
contributions from higher excited S-wave and P-wave
$\mathrm{b}\overline{}\mathrm{b}$ states. Good agreement between the data and
predictions is found for all three $\Upsilon$ states. The dependence of the
$\Upsilon$ cross-sections on $y$ is found to be more pronounced than at higher
collision energies [21, 22], which is in line with theoretical expectations
presented for example in Ref. [52].
LHCb$\Upsilon(1\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$LHCb$\Upsilon(1\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$LHCb$\Upsilon(2\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$LHCb$\Upsilon(2\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$LHCb$\Upsilon(3\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$LHCb$\Upsilon(3\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$$p_{\rm
T}$$\left[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\right]$$y$$p_{\rm
T}$$\left[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\right]$$y$$p_{\rm
T}$$\left[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\right]$$y$
$\mathcal{B}_{\Upsilon(1\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}}\times\tfrac{\mathrm{d}\upsigma}{\mathrm{d}\mbox{$p_{\rm
T}$}}~{}\left[\tfrac{\mathrm{nb}}{{\mathrm{\,Ge\kern-0.63004ptV\\!/}c}}\right]$
$\mathcal{B}_{\Upsilon(1\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}}\times\tfrac{\mathrm{d}\upsigma}{\mathrm{d}y}~{}\left[\mathrm{nb}\right]$
$\mathcal{B}_{\Upsilon(2\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}}\times\tfrac{\mathrm{d}\upsigma}{\mathrm{d}\mbox{$p_{\rm
T}$}}~{}\left[\tfrac{\mathrm{nb}}{{\mathrm{\,Ge\kern-0.63004ptV\\!/}c}}\right]$
$\mathcal{B}_{\Upsilon(2\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}}\times\tfrac{\mathrm{d}\upsigma}{\mathrm{d}y}~{}\left[\mathrm{nb}\right]$
$\mathcal{B}_{\Upsilon(3\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}}\times\tfrac{\mathrm{d}\upsigma}{\mathrm{d}\mbox{$p_{\rm
T}$}}~{}\left[\tfrac{\mathrm{nb}}{{\mathrm{\,Ge\kern-0.63004ptV\\!/}c}}\right]$
$\mathcal{B}_{\Upsilon(3\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}}\times\tfrac{\mathrm{d}\upsigma}{\mathrm{d}y}~{}\left[\mathrm{nb}\right]$
Figure 2: Differential cross-sections for $\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ mesons times dimuon branching fractions as functions of $p_{\rm T}$ (left) and $y$ (right). The inner error bars indicate the statistical uncertainty, while the outer error bars indicate the sum of statistical and systematic uncertainties in quadrature. The next-to-leading order non-relativistic QCD predictions [18] are shown by the solid yellow band. Table 2: Cross-sections for $\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ mesons times dimuon branching fractions (in $\rm\,nb$) in bins of $p_{\rm T}$ and $y$ without normalisation to the bin sizes. The first uncertainty is statistical and the second is systematic. $\mbox{$p_{\rm T}$}~{}\left[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\right]$ | $\Upsilon(1\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$ | $\Upsilon(2\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$ | $\Upsilon(3\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$
---|---|---|---
0–2 | $0.257\pm 0.021\pm 0.011$ | $0.066\pm 0.012\pm 0.007$ | $0.023\pm 0.007\pm 0.002$
2–3 | $0.167\pm 0.014\pm 0.007$ | $0.028\pm 0.007\pm 0.002$ | $0.024\pm 0.008\pm 0.002$
3–4 | $0.154\pm 0.016\pm 0.009$ | $0.038\pm 0.008\pm 0.002$ | $0.023\pm 0.008\pm 0.001$
4–6 | $0.277\pm 0.023\pm 0.013$ | $0.065\pm 0.011\pm 0.003$ | $0.038\pm 0.010\pm 0.002$
6–10 | $0.212\pm 0.019\pm 0.008$ | $0.048\pm 0.010\pm 0.002$ | $0.033\pm 0.008\pm 0.001$
10–15 | $0.043\pm 0.008\pm 0.003$ | $0.020\pm 0.007\pm 0.001$ | $0.018\pm 0.006\pm 0.002$
$y$ | $\Upsilon(1\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$ | $\Upsilon(2\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$ | $\Upsilon(3\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}$
2.0–2.5 | $0.404\pm 0.034\pm 0.022$ | $0.101\pm 0.019\pm 0.005$ | $0.061\pm 0.016\pm 0.003$
2.5–3.0 | $0.321\pm 0.018\pm 0.012$ | $0.086\pm 0.010\pm 0.004$ | $0.053\pm 0.008\pm 0.003$
3.0–3.5 | $0.227\pm 0.013\pm 0.008$ | $0.050\pm 0.007\pm 0.002$ | $0.029\pm 0.005\pm 0.001$
3.5–4.0 | $0.124\pm 0.011\pm 0.005$ | $0.025\pm 0.005\pm 0.001$ | $0.007\pm 0.003\pm 0.001$
4.0–4.5 | $0.035\pm 0.008\pm 0.002$ | $0.001\pm 0.003\pm 0.001$ | $0.005\pm 0.004\pm 0.001$
Table 3: Ratios of the $\Upsilon(2\mathrm{S})$ to $\Upsilon(1\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ to $\Upsilon(1\mathrm{S})$ cross-sections times dimuon branching fractions as functions of $p_{\rm T}$ and $y$. The first uncertainty is statistical and the second is systematic. $\mbox{$p_{\rm T}$}~{}\left[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\right]$ | $\mathcal{R}^{\mathrm{2}S/1S}$ | $\mathcal{R}^{\mathrm{3}S/1S}$
---|---|---
0–2 | $0.257\pm 0.053\pm 0.009$ | $0.090\pm 0.030\pm 0.006$
2–3 | $0.165\pm 0.044\pm 0.007$ | $0.141\pm 0.050\pm 0.010$
3–4 | $0.244\pm 0.056\pm 0.007$ | $0.148\pm 0.055\pm 0.006$
4–6 | $0.233\pm 0.043\pm 0.007$ | $0.138\pm 0.037\pm 0.005$
6–10 | $0.227\pm 0.051\pm 0.006$ | $0.157\pm 0.041\pm 0.004$
10–15 | $0.474\pm 0.179\pm 0.031$ | $0.413\pm 0.155\pm 0.029$
$y$ | |
2.0–2.5 | $0.249\pm 0.051\pm 0.007$ | $0.152\pm 0.042\pm 0.006$
2.5–3.0 | $0.266\pm 0.033\pm 0.007$ | $0.164\pm 0.026\pm 0.007$
3.0–3.5 | $0.219\pm 0.032\pm 0.004$ | $0.129\pm 0.025\pm 0.003$
3.5–4.0 | $0.204\pm 0.046\pm 0.004$ | $0.060\pm 0.026\pm 0.003$
Figure 3 illustrates the ratios of the $\Upsilon(2\mathrm{S})$ to
$\Upsilon(1\mathrm{S})$, $\mathcal{R}^{\mathrm{2}S/1S}$, and
$\Upsilon(3\mathrm{S})$ to $\Upsilon(1\mathrm{S})$,
$\mathcal{R}^{\mathrm{3}S/1S}$, cross-sections times dimuon branching
fractions as functions of $p_{\rm T}$ and $y$. Here, most of the systematic
uncertainties on the cross-sections cancel, while the statistical
uncertainties remain significant. The ratios are found to be in good agreement
with the corresponding results obtained in the previous analyses on $\Upsilon$
production at $\sqrt{s}=7$ and $8\mathrm{\,Te\kern-1.00006ptV}$ [21, 22]. The
measured $\mathcal{R}^{\mathrm{2}S/1S}$ and $\mathcal{R}^{\mathrm{3}S/1S}$ are
also consistent with theoretical predictions presented in Refs. [53, 52, 54],
where the $\Upsilon(3\mathrm{S})$ meson is considered as a mixture of normal
$\mathrm{b}\overline{}\mathrm{b}$ and hybrid
$\mathrm{b}\overline{}\mathrm{b}\mathrm{g}$ states. Table 3 lists
$\mathcal{R}^{\mathrm{2}S/1S}$ and $\mathcal{R}^{\mathrm{3}S/1S}$ for each
$p_{\rm T}$ and $y$ bin.
LHCb$\Upsilon(2\mathrm{S})/\Upsilon(1\mathrm{S})$LHCb$\Upsilon(2\mathrm{S})/\Upsilon(1\mathrm{S})$LHCb$\Upsilon(3\mathrm{S})/\Upsilon(1\mathrm{S})$LHCb$\Upsilon(3\mathrm{S})/\Upsilon(1\mathrm{S})$$p_{\rm
T}$$\left[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\right]$$y$$p_{\rm
T}$$\left[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\right]$$y$
$\mathcal{R}^{\rm 2S/1S}$
$\mathcal{R}^{\rm 2S/1S}$
$\mathcal{R}^{\rm 3S/1S}$
$\mathcal{R}^{\rm 3S/1S}$
Figure 3: Ratios of the $\Upsilon(2\mathrm{S})$ to $\Upsilon(1\mathrm{S})$
and $\Upsilon(3\mathrm{S})$ to $\Upsilon(1\mathrm{S})$ cross-sections times
dimuon branching fractions as functions of $p_{\rm T}$ and $y$. The error bars
indicate the total uncertainties of the results obtained by adding statistical
and systematic uncertainties in quadrature.
To provide a reference for a future LHCb measurement of $\Upsilon$ production
with $\mathrm{p}\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=5$ TeV, the
$\Upsilon$ cross-sections are measured in the reduced kinematic region
$\mbox{$p_{\rm T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.5<y<4.0$.
The corresponding integrated cross-sections times dimuon branching fractions
in this kinematic region are
$\displaystyle\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon(1\mathrm{S})\mathrm{X}\right)\times{\cal
B}\left(\Upsilon(1\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}\right)$
$\displaystyle=$ $\displaystyle 0.670\pm 0.025\pm 0.026\rm\,nb,$
$\displaystyle\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon(2\mathrm{S})\mathrm{X}\right)\times{\cal
B}\left(\Upsilon(2\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}\right)$
$\displaystyle=$ $\displaystyle 0.159\pm 0.013\pm 0.007\rm\,nb,$
$\displaystyle\upsigma\left(\mathrm{p}\mathrm{p}\rightarrow\Upsilon(3\mathrm{S})\mathrm{X}\right)\times{\cal
B}\left(\Upsilon(3\mathrm{S})\\!\rightarrow\upmu^{+}\upmu^{-}\right)$
$\displaystyle=$ $\displaystyle 0.089\pm 0.010\pm 0.004\rm\,nb.$
## 6 Conclusions
The production of $\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and
$\Upsilon(3\mathrm{S})$ mesons is observed for the first time in
$\mathrm{p}\mathrm{p}$ collisions at a centre-of-mass energy of
$\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$ at forward rapidities with a data
sample corresponding to an integrated luminosity of 3.3$\mbox{\,pb}^{-1}$. The
$\Upsilon$ differential production cross-sections times dimuon branching
fractions are measured separately as functions of the $\Upsilon$ transverse
momentum and rapidity for $\mbox{$p_{\rm
T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$. The theoretical
predictions, based on the next-to-leading order non-relativistic QCD
calculation, provide a good description of the data at large $p_{\rm T}$. The
ratios of the $\Upsilon(2\mathrm{S})$ to $\Upsilon(1\mathrm{S})$ and
$\Upsilon(3\mathrm{S})$ to $\Upsilon(1\mathrm{S})$ cross-sections times dimuon
branching fractions as functions of $p_{\rm T}$ and $y$ are found to be in
agreement with the corresponding results obtained at higher collision
energies.
## Acknowledgements
We thank G. Bodwin, L. S. Kisslinger, A. K. Likhoded and A. V. Luchinsky for
fruitful discussions about bottomonium production. In addition, we are
grateful to K.-T. Chao, H. Han and H.-S. Shao for the next-to-leading order
non-relativistic QCD predictions for prompt $\Upsilon$ production at
$\sqrt{s}=2.76\mathrm{\,Te\kern-1.00006ptV}$. We also express our gratitude to
our colleagues in the CERN accelerator departments for the excellent
performance of the LHC. We thank the technical and administrative staff at the
LHCb institutes. We acknowledge support from CERN and from the national
agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and
Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland);
INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania);
MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal
and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC
(United Kingdom); NSF (USA). We also acknowledge the support received from the
ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France),
KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC
(Spain), GridPP (United Kingdom). We are indebted to the communities behind
the multiple open source software packages we depend on. We are also thankful
for the computing resources and the access to software R&D tools provided by
Yandex LLC (Russia).
## References
* [1] V. G. Kartvelishvili, A. K. Likhoded, and S. R. Slabospitsky, $\mathrm{D}$ meson and $\uppsi$ meson production in hadronic interactions, Sov. J. Nucl. Phys. 28 (1978) 678
* [2] C.-H. Chang, Hadronic production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ associated with a gluon, Nucl. Phys. B172 (1980) 425
* [3] E. L. Berger and D. Jones, Inelastic photoproduction of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\Upsilon$ by gluons, Phys. Rev. D23 (1981) 1521
* [4] R. Baier and R. Rückl, Hadronic collisions: a quarkonium factory, Z. Phys. C19 (1983) 251
* [5] R. Baier and R. Rückl, Hadronic production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\Upsilon$: transverse momentum distributions, Phys. Lett. B102 (1981) 364
* [6] J. Campbell, F. Maltoni, and F. Tramontano, QCD corrections to ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\Upsilon$ production at hadron colliders, Phys. Rev. Lett. 98 (2007) 252002, arXiv:hep-ph/0703113
* [7] P. Artoisenet et al., $\Upsilon$ production at Fermilab Tevatron and LHC energies, Phys. Rev. Lett. 101 (2008) 152001, arXiv:0806.3282
* [8] A. D. Frawley, T. Ullrich, and R. Vogt, Heavy flavor in heavy-ion collisions at RHIC and RHIC II, Phys. Rept. 462 (2008) 125, arXiv:0806.1013
* [9] G. T. Bodwin, E. Braaten, and G. P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D51 (1995) 1125, arXiv:hep-ph/9407339
* [10] P. Cho and A. K. Leibovich, Color-octet quarkonia production, Phys. Rev. D53 (1996) 150, arXiv:hep-ph/9505329
* [11] P. Cho and A. K. Leibovich, Color-octet quarkonia production II, Phys. Rev. D53 (1996) 6203, arXiv:hep-ph/9511315
* [12] B. Gong, L.-P. Wan, J.-X. Wang, and H.-F. Zhang, Complete next-to-leading-order study on the yield and polarization of $\Upsilon\mathrm{(1S,2S,3S)}$ at the Tevatron and LHC, arXiv:1305.0748
* [13] ATLAS collaboration, G. Aad et al., Observation of a new $\upchi_{\mathrm{b}}$ state in radiative transitions to $\Upsilon(1\mathrm{S})$ and $\Upsilon(2\mathrm{S})$ at ATLAS, Phys. Rev. Lett. 108 (2012) 152001, arXiv:1112.5154
* [14] D0 collaboration, V. M. Abazov et al., Observation of a narrow mass state decaying into $\Upsilon(1\mathrm{S})+\upgamma$ in $\mathrm{p}\overline{}\mathrm{p}$ collisions at $\sqrt{s}=1.96$ TeV, Phys. Rev. D86 (2012) 031103, arXiv:1203.6034
* [15] LHCb collaboration, R. Aaij et al., Measurement of the fraction of $\Upsilon(1\mathrm{S})$ originating from $\upchi_{\mathrm{b}}\mathrm{(1P)}$ decays in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, JHEP 11 (2012) 31, arXiv:1209.0282
* [16] LHCb collaboration, Observation of $\upchi_{\mathrm{b}}\mathrm{(3P)}$ state at LHCb in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, LHCb-CONF-2012-020
* [17] A. Likhoded, A. Luchinsky, and S. Poslavsky, Production of $\upchi_{\mathrm{b}}$-mesons at LHC, Phys. Rev. D86 (2012) 074027, arXiv:1203.4893
* [18] K. Wang, Y.-Q. Ma, and K.-T. Chao, $\Upsilon(1\mathrm{S})$ prompt production at the Tevatron and LHC in nonrelativistic QCD, Phys. Rev. D85 (2012) 114003, arXiv:1202.6012
* [19] ATLAS Collaboration, G. Aad et al., Measurement of Upsilon production in 7 TeV pp collisions at ATLAS, Phys. Rev. D87 (2013) 052004, arXiv:1211.7255
* [20] CMS Collaboration, S. Chatrchyan et al., Measurement of the $\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ cross sections in pp collisions at $\sqrt{s}$ = 7 TeV, Phys. Lett. B727 (2013) 101, arXiv:1303.5900
* [21] LHCb collaboration, R. Aaij et al., Measurement of $\Upsilon$ production in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, Eur. Phys. J. C72 (2012) 2025, arXiv:1202.6579
* [22] LHCb collaboration, R. Aaij et al., Production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\Upsilon$ mesons in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$, JHEP 06 (2013) 64, arXiv:1304.6977
* [23] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [24] M. Adinolfi et al., Performance of the LHCb RICH detector at the LHC, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
* [25] A. A. Alves Jr. et al., Performance of the LHCb muon system, JINST 8 (2013) P02022, arXiv:1211.1346
* [26] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [27] T. Sjöstrand, S. Mrenna, and P. Skands, Pythia 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [28] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [29] P. Golonka and Z. Was, Photos Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [30] Geant4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [31] Geant4 collaboration, S. Agostinelli et al., Geant4: a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [32] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. Phys. Conf. Ser. 331 (2011) 032023
* [33] R. Aaij et al., Measurement of the track reconstruction efficiency at LHCb, LHCb-DP-2013-002, in preparation
* [34] F. Archilli et al., Performance of the muon identification at LHCb, JINST 8 (2013) P10020, arXiv:1306.0249
* [35] W. D. Hulsbergen, Decay chain fitting with a Kalman filter, Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
* [36] T. Skwarnicki, A study of the radiative cascade transitions between the $\Upsilon^{\prime}$ and $\Upsilon$ resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
* [37] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001, and 2013 partial update for the 2014 edition
* [38] M. Pivk and F. R. Le Diberder, sPlot: a statistical tool to unfold data distributions, Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
* [39] LHCb collaboration, R. Aaij et al., Prompt $\mathrm{K}^{0}_{\rm\scriptscriptstyle S}$ production in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=0.9\mathrm{\,Te\kern-1.00006ptV}$, Phys. Lett. B693 (2010) 69, arXiv:1008.3105
* [40] M. Ferro-Luzzi, Proposal for an absolute luminosity determination in colliding beam experiments using vertex detection of beam-gas interactions, Nucl. Instrum. Meth. A553 (2005) 388
* [41] LHCb collaboration, R. Aaij et al., Absolute luminosity measurements with the LHCb detector at the LHC, JINST 7 (2012) P01010, arXiv:1110.2866
* [42] P. Hopchev, LHCb beam-gas imaging results, arXiv:1107.1492
* [43] P. Hopchev, Absolute luminosity measurement at LHCb, PhD thesis, Université de Grenoble, Grenoble, 2011, CERN-THESIS-2011-210
* [44] C. Barschel, Precision luminosity measurement at LHCb with beam-gas imaging, PhD thesis, Fakultät für Mathematik, Informatik and Naturwissenschaften der RWTH Aachen University, Aachen, 2013
* [45] CMS collaboration, S. Chatrchyan et al., Measurement of the $\Upsilon(1\mathrm{S})$, $\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ polarizations in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, Phys. Rev. Lett. 110 (2013) 081802, arXiv:1209.2922
* [46] LHCb collaboration, R. Aaij et al., Measurement of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ polarization in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, Eur. Phys. J. C73 (2013) 2631, arXiv:1307.6379
* [47] LHCb collaboration, R. Aaij et al., Measurement of $\uppsi{\mathrm{(2S)}}$ polarisation in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, LHCb-PAPER-2013-067, in preparation
* [48] ALICE collaboration, B. Abelev et al., ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ polarization in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, Phys. Rev. Lett. 108 (2012) 082001, arXiv:1111.1630
* [49] R. Aaij et al., The LHCb trigger and its performance in 2011, JINST 8 (2013) P04022, arXiv:1211.3055
* [50] CDF collaboration, D. Acosta et al., $\Upsilon$ production and polarization in $\mathrm{p}\overline{}\mathrm{p}$ collisions at $\sqrt{s}=1.8\mathrm{\,Te\kern-1.00006ptV}$, Phys. Rev. Lett. 88 (2002) 161802
* [51] D0 collaboration, V. M. Abazov et al., Measurement of inclusive differential cross sections for $\Upsilon(1\mathrm{S})$ production in $\mathrm{p}\overline{}\mathrm{p}$ collisions at $\sqrt{s}=1.96\mathrm{\,Te\kern-1.00006ptV}$, Phys. Rev. Lett. 94 (2005) 232001, arXiv:hep-ex/0502030
* [52] L. S. Kisslinger and D. Das, $\Upsilon$ production in $\mathrm{p}\mathrm{p}$ collisions for forward rapidities at LHC, Mod. Phys. Lett. A28 (2013) 1350067, arXiv:1207.3296
* [53] L. S. Kisslinger, M. X. Liu, and P. McGaughey, Heavy quark qtate production in $\mathrm{p}\mathrm{p}$ collisions, Phys. Rev. D84 (2011) 114020, arXiv:1108.4049
* [54] L. S. Kisslinger, $\Upsilon$ production in $\mathrm{p}\mathrm{p}$ collisions at LHC, Mod. Phys. Lett. A27 (2012) 1250074, arXiv:1201.1033
|
arxiv-papers
| 2014-02-11T15:54:58 |
2024-09-04T02:49:58.077849
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. Anderlini,\n J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, M. Borsato, T.J.V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto,\n J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M. Calvi, M. Calvo Gomez,\n A. Camboni, P. Campana, D. Campora Perez, A. Carbone, G. Carboni, R.\n Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G.\n Casse, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph.\n Charpentier, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid\n Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C.\n Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A.\n Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, I. Counts, B. Couturier,\n G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie, C.\n D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, S. Donleavy, F.\n Dordei, M. Dorigo, P. Dorosz, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger, R. Ekelhof,\n L. Eklund, I. El Rifai, Ch. Elsasser, S. Esen, A. Falabella, C. F\\\"arber, C.\n Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues,\n M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, J. Fu, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P.\n Gandini, Y. Gao, J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld,\n E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, A. Gianelle, S. Giani', V.\n Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, P.\n Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, T.W. Hafkenscheid, S.C. Haines, S. Hall,\n B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, L.\n Henry, J.A. Hernando Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, D. Hill,\n M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, N. Hussain, D. Hutchcroft,\n D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans,\n P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B.\n Jost, N. Jurik, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach,\n M. Kelsey, I.R. Kenyon, T. Ketel, B. Khanji, C. Khurewathanakul, S. Klaver,\n O. Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S.\n Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, M. Liles, R. Lindner, C.\n Linn, F. Lionetto, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N.\n Lopez-March, P. Lowdon, H. Lu, D. Lucchesi, J. Luisier, H. Luo, E. Luppi, O.\n Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G.\n Manca, G. Mancinelli, M. Manzali, J. Maratas, U. Marconi, P. Marino, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, F. Martinez Vidal, D. Martins Tostes, A.\n Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov, M. McCann, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, M. Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, R.\n Mountain, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik,\n T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neri, S. Neubert, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M.\n Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano,\n F. Palombo, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, L.\n Pappalardo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pearce, A. Pellegrino,\n G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, P. Perret, M.\n Perrin-Terrin, L. Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini,\n E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, A. Pistone, S.\n Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A.\n Popov, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A.\n Pritchard, C. Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B.\n Rachwal, J.H. Rademacker, B. Rakotomiaramanana, M. Rama, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, S. Reichert, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, A.B. Rodrigues, E. Rodrigues, P. Rodriguez\n Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T.\n Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, M. Schiller, H. Schindler, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, G. Simi, M. Sirendi, N. Skidmore,\n T. Skwarnicki, N.A. Smith, E. Smith, E. Smith, J. Smith, M. Smith, H. Snoek,\n M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B.\n Spaan, A. Sparkes, F. Spinella, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S. Stracka, M.\n Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah, L. Sun, W. Sutcliffe, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, D. Szilard, T. Szumlak,\n S. T'Jampens, M. Teklishyn, G. Tellarini, E. Teodorescu, F. Teubert, C.\n Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, L.\n Tomassetti, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti, A.\n Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez Sierra, S.\n Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D.\n Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, J.A. de Vries, R. Waldi, C.\n Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley,\n J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Dmytro Volyanskyy",
"url": "https://arxiv.org/abs/1402.2539"
}
|
1402.2554
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-018 LHCb-PAPER-2013-065 March 19, 2014
Measurements of the $B^{+}$, $B^{0}$, $B^{0}_{s}$ meson and $\mathchar
28931\relax^{0}_{b}$ baryon lifetimes
The LHCb collaboration†††Authors are listed on the following pages.
Measurements of $b$-hadron lifetimes are reported using $pp$ collision data,
corresponding to an integrated luminosity of 1.0 fb-1, collected by the LHCb
detector at a centre-of-mass energy of $7$$\mathrm{\,Te\kern-1.00006ptV}$.
Using the exclusive decays
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*}(892)^{0}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$, $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ and
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ the
average decay times in these modes are measured to be
$\tau_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ | = $1.637\ \pm$ 0.004 $\pm$ 0.003 ${\rm\,ps}$,
---|---
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}}$ | = $1.524\ \pm$ 0.006 $\pm$ 0.004 ${\rm\,ps}$,
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ | = $1.499\ \pm$ 0.013 $\pm$ 0.005 ${\rm\,ps}$,
$\tau_{\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax}$ | = $1.415\ \pm$ 0.027 $\pm$ 0.006 ${\rm\,ps}$,
$\tau_{B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi}$ | = $1.480\ \pm$ 0.011 $\pm$ 0.005 ${\rm\,ps}$,
where the first uncertainty is statistical and the second is systematic. These
represent the most precise lifetime measurements in these decay modes. In
addition, ratios of these lifetimes, and the ratio of the decay-width
difference, $\Delta\Gamma_{d}$, to the average width, $\Gamma_{d}$, in the
$B^{0}$ system, $\Delta\Gamma_{d}/\Gamma_{d}=-0.044\pm 0.025\pm 0.011$, are
reported. All quantities are found to be consistent with Standard Model
expectations.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, A. Affolder51, Z. Ajaltouni5, J.
Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G. Alkhazov29, P. Alvarez
Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y. Amhis7, L.
Anderlini17,g, J. Anderson39, R. Andreassen56, M. Andreotti16,f, J.E.
Andrews57, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli37, A.
Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,n, M. Baalouch5, S.
Bachmann11, J.J. Back47, A. Badalov35, V. Balagura30, W. Baldini16, R.J.
Barlow53, C. Barschel38, S. Barsuk7, W. Barter46, V. Batozskaya27, Th.
Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S. Belogurov30, K.
Belous34, I. Belyaev30, E. Ben-Haim8, G. Bencivenni18, S. Benson49, J.
Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van Beuzekom40, A.
Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,i, P.M. Bjørnstad53, T. Blake47,
F. Blanc38, J. Blouw10, S. Blusk58, V. Bocci24, A. Bondar33, N. Bondar29, W.
Bonivento15,37, S. Borghi53, A. Borgia58, M. Borsato7, T.J.V. Bowcock51, E.
Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J. Bressieux38, D.
Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51, A. Bursche39,
G. Busetto21,r, J. Buytaert37, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez35,p, A. Camboni35, P. Campana18,37, D. Campora
Perez37, A. Carbone14,d, G. Carboni23,l, R. Cardinale19,j, A. Cardini15, H.
Carranza-Mejia49, L. Carson49, K. Carvalho Akiba2, G. Casse51, L. Castillo
Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles8, Ph.
Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37,
X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46,
J. Closier37, C. Coca28, V. Coco37, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, I. Counts55, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz
Torres59, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, J. Dalseno45, P.
David8, P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De
Capua53, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De
Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D.
Derkach54, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, S.
Donleavy51, F. Dordei11, M. Dorigo38, P. Dorosz25,o, A. Dosil Suárez36, D.
Dossett47, A. Dovbnya42, F. Dupertuis38, P. Durante37, R. Dzhelyadin34, A.
Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52, V. Egorychev30, S. Eidelman33,
S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5,
Ch. Elsasser39, S. Esen11, A. Falabella16,f, C. Färber11, C. Farinelli40, S.
Farry51, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M.
Ferro-Luzzi37, S. Filippov32, M. Fiore16,f, M. Fiorini16,f, C. Fitzpatrick37,
M. Fontana10, F. Fontanelli19,j, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,37,g, E. Furfaro23,l, A. Gallas Torreira36, D. Galli14,d,
M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, J. Garra Tico46, L.
Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T.
Gershon47, Ph. Ghez4, A. Gianelle21, S. Giani’38, V. Gibson46, L. Giubega28,
V.V. Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes1,a,
H. Gordon37, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado
Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S.
Gregson46, P. Griffith44, L. Grillo11, O. Grünberg60, B. Gui58, E. Gushchin32,
Yu. Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C. Haen37, T.W.
Hafkenscheid62, S.C. Haines46, S. Hall52, B. Hamilton57, T. Hampson45, S.
Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60,
J. He37, T. Head37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, M. Heß60, A. Hicheur1, D. Hill54, M. Hoballah5,
C. Hombach53, W. Hulsbergen40, P. Hunt54, N. Hussain54, D. Hutchcroft51, D.
Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten55, R. Jacobsson37, A. Jaeger11,
E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, N. Jurik58, M. Kaballo9, S. Kandybei42, W.
Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, B. Khanji20,
C. Khurewathanakul38, S. Klaver53, O. Kochebina7, I. Komarov38, R.F.
Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K.
Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M.
Kucharczyk20,25,37,k, V. Kudryavtsev33, K. Kurek27, T. Kvaratskheliya30,37,
V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W.
Lambert41, E. Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11,
Y. Li3, M. Liles51, R. Lindner37, C. Linn11, F. Lionetto39, B. Liu15, G.
Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-March38, P. Lowdon39,
H. Lu3, D. Lucchesi21,r, J. Luisier38, H. Luo49, E. Luppi16,f, O. Lupton54, F.
Machefert7, I.V. Machikhiliyan30, F. Maciuc28, O. Maev29,37, S. Malde54, G.
Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5, U. Marconi14, P.
Marino22,t, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, A. Martín
Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2, A.
Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, A. Mazurov16,37,f, M.
McCann52, J. McCarthy44, A. McNab53, R. McNulty12, B. McSkelly51, B.
Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N.
Minard4, J. Molina Rodriguez59, S. Monteil5, D. Moran53, M. Morandin21, P.
Morawski25, A. Mordà6, M.J. Morello22,t, R. Mountain58, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N. Neufeld37, A.D.
Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,q, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin31, T. Nikodem11, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,e, G.
Onderwater62, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A.
Oyanguren35, B.K. Pal58, A. Palano13,c, M. Palutan18, J. Panman37, A.
Papanestis48,37, M. Pappagallo50, L. Pappalardo16, C. Parkes53, C.J.
Parkinson9, G. Passaleva17, G.D. Patel51, M. Patel52, C. Patrignani19,j, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pearce53, A. Pellegrino40, G.
Penso24,m, M. Pepe Altarelli37, S. Perazzini14,d, E. Perez Trigo36, P.
Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen63, G. Pessina20, K.
Petridis52, A. Petrolini19,j, E. Picatoste Olloqui35, B. Pietrzyk4, T.
Pilař47, D. Pinci24, A. Pistone19, S. Playfer49, M. Plo Casasus36, F. Polci8,
G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C.
Prouve45, V. Pugatch43, A. Puig Navarro38, G. Punzi22,s, W. Qian4, B.
Rachwal25, J.H. Rademacker45, B. Rakotomiaramanana38, M. Rama18, M.S. Rangel2,
I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, S. Reichert53, M.M.
Reid47, A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, D.A. Roberts57, A.B. Rodrigues1, E.
Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero
Vidal36, M. Rotondo21, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P.
Ruiz Valls35, G. Sabatino24,l, J.J. Saborido Silva36, N. Sagidova29, P.
Sail50, B. Saitta15,e, V. Salustino Guimaraes2, B. Sanmartin Sedes36, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,l, M. Sapunov6, A.
Sarti18, C. Satriano24,n, A. Satta23, M. Savrie16,f, D. Savrina30,31, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42,f, Y.
Shcheglov29, T. Shears51, L. Shekhtman33, O. Shevchenko42, V. Shevchenko61, A.
Shires9, R. Silva Coutinho47, G. Simi21, M. Sirendi46, N. Skidmore45, T.
Skwarnicki58, N.A. Smith51, E. Smith54,48, E. Smith52, J. Smith46, M. Smith53,
H. Snoek40, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro38, D. Souza45, B. Souza
De Paula2, B. Spaan9, A. Sparkes49, F. Spinella22, P. Spradlin50, F. Stagni37,
S. Stahl11, O. Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58, B.
Storaci39, S. Stracka22,37, M. Straticiuc28, U. Straumann39, R. Stroili21,
V.K. Subbiah37, L. Sun56, W. Sutcliffe52, S. Swientek9, V. Syropoulos41, M.
Szczekowski27, P. Szczypka38,37, D. Szilard2, T. Szumlak26, S. T’Jampens4, M.
Teklishyn7, G. Tellarini16,f, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, L.
Tomassetti16,f, D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E.
Tournefier4,52, S. Tourneur38, M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6,
P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37, A. Ukleja27, A. Ustyuzhanin61,
U. Uwer11, V. Vagnoni14, G. Valenti14, A. Vallier7, R. Vazquez Gomez18, P.
Vazquez Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M.
Veltri17,h, G. Veneziano38, M. Vesterinen11, B. Viaud7, D. Vieira2, X.
Vilasis-Cardona35,p, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A.
Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, J.A. de Vries40, R. Waldi60,
C. Wallace47, R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K.
Watson44, A.D. Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J.
Wiechczynski25, D. Wiedner11, L. Wiggers40, G. Wilkinson54, M.P.
Williams47,48, M. Williams55, F.F. Wilson48, J. Wimberley57, J. Wishahi9, W.
Wislicki27, M. Witek25, G. Wormser7, S.A. Wotton46, S. Wright46, S. Wu3, K.
Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,b, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61National Research Centre Kurchatov Institute, Moscow, Russia, associated to
30
62KVI - University of Groningen, Groningen, The Netherlands, associated to 40
63Celal Bayar University, Manisa, Turkey, associated to 37
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
## 1 Introduction
Within the framework of heavy quark expansion (HQE) theory [1, 2, 3, 4, 5, 6,
7], $b$-hadron observables are calculated as a perturbative expansion in
inverse powers of the $b$-quark mass, $m_{b}$. At zeroth order the lifetimes
of all weakly decaying $b$ hadrons are equal, with corrections appearing at
order $1/m_{b}^{2}$. Ratios of $b$-hadron lifetimes can be theoretically
predicted with higher accuracy than absolute lifetimes since many terms in the
HQE cancel. The latest theoretical predictions and world-average values for
the $b$-hadron lifetimes and lifetime ratios are reported in Table 1. A
measurement of the ratio of the $\mathchar 28931\relax^{0}_{b}$ baryon
lifetime, using the $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ decay mode111Charge conjugation is implied throughout this
paper, unless otherwise stated., to that of the $B^{0}$ meson lifetime has
recently been made by the LHCb collaboration [8] and is not yet included in
the world average.
In this paper, a measurement of the lifetimes of the $B^{+}$, $B^{0}$ and
$B^{0}_{s}$ mesons and $\mathchar 28931\relax^{0}_{b}$ baryon is reported
using $pp$ collision data, corresponding to an integrated luminosity of 1.0
fb-1, collected in 2011 with the LHCb detector at a centre-of-mass energy of
$7$$\mathrm{\,Te\kern-1.00006ptV}$. The lifetimes are measured from the
reconstructed $b$-hadron decay time distributions of the exclusive decay modes
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*}(892)^{0}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
$\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$. Collectively, these are referred to as
$H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ decays. In
addition, measurements of lifetime ratios are reported.
As a result of neutral meson mixing the decay time distribution of neutral
$B^{0}_{q}$ mesons ($q\in\\{s,d\\}$) is characterised by two parameters,
namely the average decay width $\Gamma_{q}$ and the decay width difference
$\Delta\Gamma_{q}$ between the light (L) and heavy (H) $B^{0}_{q}$ mass
eigenstates. The summed decay rate of $B^{0}_{q}$ and $\overline{B}^{0}_{q}$
mesons to a final state $f$ is given by [9, 10, 11]
$\langle\Gamma(B^{0}_{q}(t)\rightarrow
f)\rangle\equiv\Gamma(B^{0}_{q}(t)\rightarrow
f)+\Gamma(\overline{B}^{0}_{q}(t)\rightarrow f)=R^{f}_{q,\rm
L}e^{-\Gamma_{q,\rm L}t}+R^{f}_{q,\rm H}e^{-\Gamma_{q,\rm H}t},$ (1)
where terms proportional to the small flavour specific asymmetry, $a_{\rm
fs}^{q}$, are ignored [12]. Therefore, for non-zero $\Delta\Gamma_{q}$ the
decay time distribution of neutral $B^{0}_{q}$ decays is not purely
exponential. In the case of an equal admixture of $B^{0}_{q}$ and
$\overline{B}^{0}_{q}$ at $t=0$, the observed average decay time is given by
[11]
$\tau_{B_{q}^{0}\rightarrow
f}=\frac{1}{\Gamma_{q}}\frac{1}{1-y_{q}^{2}}\left(\frac{1+2\mathcal{A}^{f}_{\Delta\Gamma_{q}}y_{q}+y_{q}^{2}}{1+\mathcal{A}^{f}_{\Delta\Gamma_{q}}y_{q}}\right),$
(2)
where $y_{q}\equiv\Delta\Gamma_{q}/(2\Gamma_{q})$ and
$\mathcal{A}^{f}_{\Delta\Gamma_{q}}\equiv(R_{q,\rm H}^{f}-R_{q,\rm
L}^{f})/(R_{q,\rm H}^{f}+R_{q,\rm L}^{f})$ is an observable that depends on
the final state, $f$. As such, the lifetimes measured are usually referred to
as effective lifetimes. In the $B^{0}_{s}$ system, where
$\Delta\Gamma_{s}/\Gamma_{s}=0.159\pm 0.023$ [13], the deviation from an
exponential decay time distribution is non-negligible. In contrast, in the
$B^{0}$ system this effect is expected to be small as
$\Delta\Gamma_{d}/\Gamma_{d}$ is predicted to be $(42\pm 8)\times 10^{-4}$ in
the Standard Model (SM) [14, 15]. Both the BaBar [16, 17] and Belle [18]
collaborations have measured $|\Delta\Gamma_{d}/\Gamma_{d}|$ and the current
world average is $|\Delta\Gamma_{d}/\Gamma_{d}|=0.015\pm 0.018$ [13]. A
deviation in the value of $\Delta\Gamma_{d}$ from the SM prediction has
recently been proposed [19] as a potential explanation for the anomalous like-
sign dimuon charge asymmetry measured by the D0 collaboration [20]. In this
paper, $\Delta\Gamma_{d}/\Gamma_{d}$ is measured from the effective lifetimes
of $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*}(892)^{0}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays, as proposed in Ref. [21].
The main challenge in the measurements reported is understanding and
controlling the detector acceptance, reconstruction and selection efficiencies
that depend upon the $b$-hadron decay time. This paper is organised as
follows. Section 2 describes the LHCb detector and software. The selection
criteria for the $b$-hadron candidates are described in Sec. 3. Section 4
describes the reconstruction efficiencies and the techniques used to correct
the decay time distributions. Section 5 describes how the efficiency
corrections are incorporated into the maximum likelihood fit that is used to
measure the signal yields and lifetimes. The systematic uncertainties on the
measurements are described in Sec. 6. The final results and conclusions are
presented in Sec. 7.
Table 1: Theoretical predictions and current world-average values [13] for
$b$-hadron lifetimes and lifetime ratios.
Observable Prediction World average $\tau_{B^{+}}$[${\rm\,ps}$ ] – $1.641\pm
0.008$ $\tau_{B^{0}}$[${\rm\,ps}$ ] – $1.519\pm 0.007$
$\tau_{B^{0}_{s}}$[${\rm\,ps}$ ] – $1.516\pm 0.011$ $\tau_{\mathchar
28931\relax^{0}_{b}}$[${\rm\,ps}$ ] – $1.429\pm 0.024$
$\tau_{B^{+}}/\tau_{B^{0}}$ $1.063\pm 0.027$ [22, 23, 15] $1.079\pm 0.007$
$\tau_{B^{0}_{s}}/\tau_{B^{0}}$ $1.00\phantom{0}\pm 0.01\phantom{0}$ [24, 25,
23, 15] $0.998\pm 0.009$ $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{B^{0}}$
0.86–0.950 [3, 26, 27, 28, 29, 30, 23, 31, 32] $0.941\pm 0.016$
## 2 Detector and software
The LHCb detector [33] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector (TT) located
upstream of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and
three stations of silicon-strip detectors and straw drift tubes placed
downstream. The combined tracking system provides a momentum, $p$, measurement
with relative uncertainty that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for charged particles with high transverse momentum, $p_{\rm
T}$. Charged hadrons are identified using two ring-imaging Cherenkov detectors
[34]. Photon, electron and hadron candidates are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [35]. The right-handed coordinate system adopted has the $z$-axis
along the beam line and the $y$-axis along the vertical. The trigger [36]
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage, which applies a full event
reconstruction.
Two distinct classes of tracks are reconstructed using hits in the tracking
stations on both sides of the magnet, either with hits in the VELO (long
track) or without (downstream track). The vertex resolution of $b$-hadron
candidates reconstructed using long tracks is better than that for candidates
reconstructed using downstream tracks. However, the use of long tracks
introduces a dependence of the reconstruction efficiency on the $b$-hadron
decay time.
In the simulation, $pp$ collisions are generated using Pythia 6.4 [37] with a
specific LHCb configuration [38]. Decays of hadronic particles are described
by EvtGen [39], in which final state radiation is generated using Photos [40].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [41, *Agostinelli:2002hh] as
described in Ref. [43].
## 3 Candidate selection
The reconstruction of each of the
$H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ decays is
similar and commences by selecting ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays. Events passing the hardware trigger
contain dimuon candidates with high transverse momentum. The subsequent
software trigger is composed of two stages. The first stage performs a partial
event reconstruction and requires events to have two well-identified
oppositely charged muons with an invariant mass larger than
$2.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The selection at this stage has
a uniform efficiency as a function of decay time. The second stage performs a
full event reconstruction, calculating the position of each $pp$ interaction
vertex (PV) using all available charged particles in the event. The average
number of PVs in each event is approximately $2.0$. Their longitudinal ($z$)
position is known to a precision of approximately $0.05\rm\,mm$. If multiple
PVs are reconstructed in the event, the one with the minimum value of
$\chi^{2}_{\rm IP}$ is associated with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate, where $\chi^{2}_{\rm
IP}$ is the increase in the $\chi^{2}$ of the PV fit if the candidate
trajectory is included. Events are retained for further processing if they
contain a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ pair that forms a vertex that is
significantly displaced from the PV. This introduces a non-uniform efficiency
as function of decay time.
The offline sample of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson
candidates is selected by requiring each muon to have $p_{\rm T}$ larger than
500${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate to be displaced from
the PV by more than three times its decay length uncertainty. The invariant
mass of the two muons, $m(\mu^{+}\mu^{-})$, must be in the range
$[3030,3150]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The $b$-hadron candidate selection is performed by applying kinematic and
particle identification criteria to the final-state tracks, the details of
which are reported in Sec. 3.1 to 3.5. No requirements are placed on variables
that are highly correlated to the $b$-hadron decay time, thereby avoiding the
introduction of additional biases. All final-state particles are required to
have a pseudorapidity in the range $2.0<\eta<4.5$. In addition, the
$z$-position of the PV ($z_{\rm PV}$) is required to be within $100\rm\,mm$ of
the nominal interaction point, where the standard deviation of the $z_{\rm
PV}$ distribution is approximately $47\rm\,mm$. These criteria cause a
reduction of approximately $10\%$ in signal yield but define a fiducial region
where the reconstruction efficiency is largely uniform.
The maximum likelihood fit uses the invariant mass,
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$, and proper decay time,
$t$, of each $b$-hadron candidate. The decay time of the $b$-hadron candidate
in its rest frame is derived from the relation $t=m\,l/q$, where $m$ is its
invariant mass and the decay length, $l$, and the momentum, $q$, are measured
in the experimental frame. In this paper, $t$ is computed using a kinematic
decay-tree fit (DTF) [44] involving all final-state tracks from the $b$-hadron
candidate with a constraint on the position of the associated PV. Unlike in
the trigger, the position of each PV is calculated using all available charged
particles in the event after the removal of the $b$-hadron candidate final-
state tracks. This is necessary to prevent the final-state tracks from biasing
the PV position towards the $b$-hadron decay vertex and helps to reduce the
tails of the decay-time resolution function. This prescription does not bias
the measured lifetime using simulated events. The $\chi^{2}$ of the fit,
$\chi^{2}_{\rm DTF}$, is useful to discriminate between signal and background.
In cases where there are multiple $b$-hadron candidates per event, the
candidate with the smallest $\chi^{2}_{\rm DTF}$ is chosen. The $z$-position
of the displaced $b$-hadron vertices are known to a precision of approximately
$0.15\rm\,mm$.
Studies of simulated events show that in the case of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
($B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$)
decays, imposing requirements on $\chi^{2}_{\rm DTF}$ introduces a dependence
of the selection efficiency on the decay time if the $K^{+}$ and $\pi^{-}$
($K^{+}$ and $K^{-}$) tracks are included in the DTF. If no correction is
applied to the decay time distribution, the measured lifetime would be biased
by approximately $-2\rm\,fs$ relative to the generated value. Using simulated
events it is found that this effect is correlated to the opening angle between
the $K^{+}$ and $\pi^{-}$ ($K^{+}$ and $K^{-}$) from the $K^{*0}$ ($\phi$)
decay. No effect is observed for the muons coming from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay due to the larger opening
angle in this case. To remove the effect, the calculation of $\chi^{2}_{\rm
DTF}$ for the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ and
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
channels is performed with an alternative DTF in which the assigned track
parameter uncertainties of the kaon and pion are increased in such a way that
their contribution to the $b$-hadron vertex position is negligible.
Candidates are required to have $t$ in the range $[0.3,14.0]{\rm\,ps}$. The
lower bound on the decay time suppresses a large fraction of the prompt
combinatorial background that is composed of tracks from the same PV, while
the upper bound is introduced to reduce the sensitivity to long-lived
background candidates. In the case of the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays, the lower bound is increased to
0.45${\rm\,ps}$ to compensate for the worse decay time resolution in these
modes.
In events with multiple PVs, $b$-hadron candidates are removed if they have a
$\chi^{2}_{\rm IP}$ with respect to the next best PV smaller than $50$. This
requirement is found to distort the decay time distribution, but reduces a
source of background due to the incorrect association of the $b$ hadron to its
production PV.
The invariant mass is computed using another kinematic fit without any
constraint on the PV position but with the invariant mass of the
$\mu^{+}\mu^{-}$ pair, $m(\mu^{+}\mu^{-})$, constrained to the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [45]. Figures 1 and 2 show
the $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ distributions for the
selected candidates in each final state and Table 2 gives the corresponding
signal yields.
\begin{overpic}[scale={0.75}]{figs/BuJpsiKFlat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ }
\put(81.0,31.0){\scriptsize$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ }
\end{overpic}\begin{overpic}[scale={0.75}]{figs/BdJpsiKstarFlat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ }
\put(81.0,31.0){\scriptsize$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ }
\end{overpic}\begin{overpic}[scale={0.75}]{figs/Bd2JpsiKSFlat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ }
\put(81.0,31.0){\scriptsize$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ } \end{overpic}
Figure 1: Distributions of the (left) mass and (right) decay time of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ candidates and their associated
residual uncertainties (pulls). The data are shown by the black points; the
total fit function by the black solid line; the signal contribution by the red
dashed line and the background contribution by the blue dotted line.
\begin{overpic}[scale={0.75}]{figs/BsJpsiPhiFlat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ }
\put(81.0,31.0){\scriptsize$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ }
\end{overpic}\begin{overpic}[scale={0.75}]{figs/Lambdab0Flat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$} \put(81.0,31.0){\scriptsize$\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$} \end{overpic}
Figure 2: Distributions of the (left) mass and (right) decay time of
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
$\mathchar 28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ candidates and their associated residual
uncertainties (pulls). The data are shown by the black points; the total fit
function by the black solid line; the signal contribution by the red dashed
line and the background contribution by the blue dotted line.
Table 2: Estimated event yields for the five $b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ channels selected using the criteria described in Sec. 3.1 to 3.5. Channel | Yield
---|---
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ | $229\,434\pm 503$
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ | $70\,534\pm 312$
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ | $17\,045\pm 175$
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ | $18\,662\pm 152$
$\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax$ | $3\,960\pm\ \,89$
### 3.1 Selection of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
The $B^{+}$ candidates are reconstructed by combining the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates with a charged
particle that is identified as a kaon with $p_{\rm T}$ larger than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $p$ larger than
$10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The invariant mass,
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})$, must be in the range
$[5170,5400]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the lower bound
is chosen to remove feed-down from incompletely reconstructed
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
decays. The $\chi^{2}_{\rm DTF}$ of the fit, which has 5 degrees of freedom,
is required to be less than 25. Multiple $B^{+}$ candidates are found in less
than $0.02\%$ of selected events.
### 3.2 Selection of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decays
The $K^{*0}$ candidates are reconstructed by combining two oppositely charged
particles that are identified as a kaon and a pion. The pion and $K^{*0}$ must
have $p_{\rm T}$ greater than $0.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, respectively. The invariant mass,
$m(K^{+}\pi^{-})$, must be in the range
$[826,966]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The $B^{0}$ candidates are reconstructed by combining the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $K^{*0}$ candidates. The
invariant mass, $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\pi^{-})$,
must be in the range $[5150,5340]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
where the upper bound is chosen to remove the contribution from
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}$ decays. The $\chi^{2}_{\rm DTF}$ of the fit, which
has 3 degrees of freedom, is required to be less than 15. Multiple $B^{0}$
candidates are found in $2.2\%$ of selected events.
### 3.3 Selection of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays
The $K^{0}_{\rm\scriptscriptstyle S}$ candidates are formed from the
combination of two oppositely charged particles that are identified as pions
and reconstructed as downstream tracks. This is necessary since studies of
simulated signal decays demonstrate that an inefficiency depending on the
$b$-hadron decay time is introduced by the reconstruction of the long-lived
$K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar 28931\relax$ particles using
long tracks. Even so, it is found that the acceptance of the TT still depends
on the origin of the tracks. This effect is removed by further tightening of
the requirement on the position of the PV to $z_{\rm PV}>-50\rm\,mm$.
For particles produced close to the interaction region, this effect is
suppressed by the requirements on the fiducial region for the PV, which is
further tightened by requiring that , to account for the additional acceptance
introduced by the TT.
The downstream pions are required to have $p_{\rm T}$ greater than
$0.1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $p$ greater than
$2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $K^{0}_{\rm\scriptscriptstyle S}$
candidate must have $p_{\rm T}$ greater than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be well separated from the $B^{0}$
decay vertex, to suppress potential background from
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decays
where the kaon has been misidentified as a pion. The $\chi^{2}$ of the
$K^{0}_{\rm\scriptscriptstyle S}$ vertex fit must be less than 25 and the
invariant mass of the dipion system, $m(\pi^{+}\pi^{-})$, must be within
$15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
$K^{0}_{\rm\scriptscriptstyle S}$ mass [45]. For subsequent stages of the
selection, $m(\pi^{+}\pi^{-})$ is constrained to the known
$K^{0}_{\rm\scriptscriptstyle S}$ mass.
The invariant mass, $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})$, of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ candidate combination must be in the range
$[5150,5340]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the upper bound
is chosen to remove the contribution from
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays. The $\chi^{2}_{\rm DTF}$ of the
fit, which has 6 degrees of freedom, is required to be less than 30. Multiple
$B^{0}$ candidates are found in less than $0.4\%$ of selected events.
### 3.4 Selection of
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
decays
The $\phi$ candidates are formed from two oppositely charged particles that
have been identified as kaons and originate from a common vertex. The
$K^{+}K^{-}$ pair is required to have $p_{\rm T}$ larger than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The invariant mass of the
$K^{+}K^{-}$ pair, $m(K^{+}K^{-})$, must be in the range
$[990,1050]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The $B^{0}_{s}$ candidates are reconstructed by combining the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate with the $K^{+}K^{-}$
pair, requiring the invariant mass, $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-})$, to be in the range
$[5200,5550]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The $\chi^{2}_{\rm
DTF}$ of the fit, which has 3 degrees of freedom, is required to be less than
15. Multiple $B^{0}_{s}$ candidates are found in less than $2.0\%$ of selected
events.
### 3.5 Selection of $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays
The selection of $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ candidates follows a similar approach to that
adopted for $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays. Only downstream protons and
pions are used to reconstruct the $\mathchar 28931\relax$ candidates. The
pions are required to have $p_{\rm T}$ larger than
$0.1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, while pions and protons must have
$p$ larger than $2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $\mathchar
28931\relax$ candidate must be well separated from the $\mathchar
28931\relax^{0}_{b}$ decay vertex and have $p_{\rm T}$ larger than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $\chi^{2}$ of the $\mathchar
28931\relax$ vertex fit must be less than 25 and $m(p\pi^{-})$ must be within
$6{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $\mathchar
28931\relax$ mass [45]. For subsequent stages of the selection, $m(p\pi^{-})$
is constrained to the known $\mathchar 28931\relax$ mass.
The invariant mass, $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar
28931\relax)$, of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\mathchar 28931\relax$ candidate combination must be in the range
$[5470,5770]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The $\chi^{2}_{\rm
DTF}$ of the fit, which has 6 degrees of freedom, is required to be less than
30. Multiple $\mathchar 28931\relax^{0}_{b}$ candidates are found in less than
$0.5\%$ of selected events.
## 4 Dependence of efficiencies on decay time
Section 3 described the reconstruction and selection criteria of the
$H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ decays and
various techniques that have been used to minimise the dependence of selection
efficiencies upon the decay time. After these steps, there remain two effects
that distort the $b$-hadron decay time distribution. These are caused by the
VELO-track reconstruction efficiency, $\varepsilon_{\rm VELO}$, and the
combination of the trigger efficiency, $\varepsilon_{\rm trigger}$, and
offline selection efficiency, $\varepsilon_{\rm selection|trigger}$. This
section will describe these effects and the techniques that are used to
evaluate the efficiencies from data control samples.
### 4.1 VELO-track reconstruction efficiency
The largest variation of the efficiency with the decay time is introduced by
the track reconstruction in the VELO. The track finding procedure in the VELO
assumes that tracks originate approximately from the interaction region [33,
46]. In the case of long-lived $b$-hadron candidates this assumption is not
well justified, leading to a loss of reconstruction efficiency for charged
particle tracks from the $b$-hadron decay.
\begin{overpic}[scale={0.39}]{figs/data_online.pdf} \put(79.0,59.7){(a)}
\end{overpic}\begin{overpic}[scale={0.39}]{figs/data_offline.pdf}
\put(79.0,59.7){(b)} \end{overpic}
Figure 3: VELO-track reconstruction efficiency for kaon tracks reconstructed
using the (a) online and (b) offline algorithms as a function of the kaon
$\rho$, as defined in Eq. (3). The red solid lines show the result of an
unbinned maximum likelihood fit using the parameterisation in Eq. (4) to the
background subtracted data (black points).
The distance of closest approach of the track to the $z$-axis is defined as
$\rho$ $\displaystyle\equiv$
$\displaystyle\frac{\left|(\boldsymbol{d}-\boldsymbol{v})\cdot(\boldsymbol{p}\times\boldsymbol{\hat{z}})\right|}{\left|\boldsymbol{p}\times{\boldsymbol{\hat{z}}}\right|},$
(3)
where $\boldsymbol{p}$ is the momentum of the final-state track from a
$b$-hadron candidate decaying at point $\boldsymbol{d}$,
$\boldsymbol{\hat{z}}$ is a unit vector along the $z$-axis and
$\boldsymbol{v}$ is the origin of the VELO coordinate system. During data
taking the position of the LHCb VELO is monitored as a function of time and is
centred around the LHC beam line. Using a control sample of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates where the $K^{+}$ is reconstructed as a downstream track, the VELO-
track reconstruction efficiency, $\varepsilon_{\rm VELO}(\mbox{$\rho$})$, is
computed as the fraction of these tracks that are also reconstructed as long
tracks. From samples of simulated $b$-hadron decays, it is observed that
$\varepsilon_{\rm VELO}(\mbox{$\rho$})$ can be empirically parameterised by
$\varepsilon_{\rm VELO}(\mbox{$\rho$})=a(1+c\mbox{$\rho$}^{2}),$ (4)
where the parameters $a$ and $c$ are determined from a fit to the unbinned
efficiency distribution.
Table 3: VELO reconstruction efficiency in data for kaon tracks reconstructed with the online and offline algorithms. In both cases, the correlation coefficient between $a$ and $c$ is 0.2. | $a$ | $c$ [$\rm\,mm^{-2}$]
---|---|---
Online | $0.9759\pm 0.0005$ | $-0.0093\pm 0.0007$
Offline | $0.9831\pm 0.0004$ | $-0.0041\pm 0.0005$
Figure 3 shows the VELO-track reconstruction efficiency obtained using this
method and Table 3 shows the corresponding fit results. Since different
configurations of the VELO reconstruction algorithms are used within the LHCb
software trigger (online) and during the subsequent processing (offline), it
is necessary to evaluate two different efficiencies. The stronger dependence
of the online efficiency as a function of $\rho$ is due to the additional
requirements used in the first stage of the software trigger such that it
satisfies the required processing time.
Applying the same technique to a simulated sample of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
yields qualitatively similar behaviour for $\varepsilon_{\rm
VELO}(\mbox{$\rho$})$. Studies on simulated data show that the efficiency for
kaons and pions from the decay of $\phi$ and $K^{*0}$ mesons is smaller than
for the kaon in $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ decays, due to the small opening between the particles in the
$\phi$ and $K^{*0}$ decays, as discussed in Sec. 3. In addition, there are
kinematic differences between the calibration $B^{+}$ sample and the signal
samples. Scaling factors on the efficiency parameters are derived from
simulation to account for these effects, and have typical sizes in the range
$[1.04,1.65]$, depending on the decay mode and final-state particle being
considered.
The distortion to the $b$-hadron candidate decay time distribution caused by
the VELO-track reconstruction is corrected for by weighting each $b$-hadron
candidate by the inverse of the product of the per-track efficiencies. The
systematic effect introduced by this weighting is tested using simulated
samples of each channel. The chosen efficiency depends on whether the particle
is reconstructed with the online or offline variant of the algorithm. Studies
on simulated data show that tracks found by the online tracking algorithm are
also found by the offline tracking efficiency. For example, the efficiency
weight for each $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ candidate takes the form
$\displaystyle w_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}$ $\displaystyle=$ $\displaystyle
1/\left(\varepsilon^{\mu^{+}}_{\rm VELO,online}\ \varepsilon^{\mu^{-}}_{\rm
VELO,online}\ \varepsilon^{K^{+}}_{\rm VELO,offline}\
\varepsilon^{\pi^{-}}_{\rm VELO,offline}\right),$ (5)
since the two muons are required to be reconstructed online, while the kaons
and the pions are reconstructed offline.
In the case of the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ channels, since no VELO information is used when
reconstructing the $K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax$ particles, the candidate weighting functions take the form
$w=1/\left(\varepsilon^{\mu^{+}}_{\rm VELO,online}\ \varepsilon^{\mu^{-}}_{\rm
VELO,online}\right)$.
### 4.2 Trigger and selection efficiency
The efficiency of the second stage of the software trigger depends on the
$b$-hadron decay time as it requires that the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson is significantly
displaced from the PV. A parameterisation of this efficiency,
$\varepsilon_{\rm trigger}(t)$, is obtained for each
$b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ decay mode by
exploiting a corresponding sample of
$b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ candidates that
are selected without any displacement requirement. For each channel, the
control sample corresponds to approximately $40\%$ of the total number of
signal candidates. A maximum likelihood fit to the unbinned invariant mass
distribution $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ is performed
to determine the fraction of signal decays that survive the decay-time biasing
trigger requirements as a function of decay time.
The same technique is used to determine the decay time efficiency of the
triggered candidates caused by the offline selection, $\varepsilon_{\rm
selection|trigger}(t)$, which is introduced by the requirement on the
detachment of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons in the
sample used to reconstruct the $b$-hadron decays. The combined selection
efficiency, $\varepsilon_{\rm selection}(t)$, is given by the product of
$\varepsilon_{\rm trigger}(t)$ and $\varepsilon_{\rm selection|trigger}(t)$.
Figure 4 shows $\varepsilon_{\rm selection}(t)$ obtained for the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ channel
as a function of decay time. The efficiencies obtained for the other
$H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ channels
are qualitatively similar. Studies using simulated events show that the
efficiency drop below $0.5{\rm\,ps}$ is caused by the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ displacement requirement. The
dip near $1.5{\rm\,ps}$ appears because the PV reconstruction in the software
trigger is such that some final-state tracks of short-lived $b$-hadron decays
may be used to reconstruct an additional fake PV close to the true $b$-hadron
decay vertex. As a result the reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson does not satisfy the
displacement requirement, leading to a decrease in efficiency.
The efficiency parameterisation for each channel is used in the fit to measure
the corresponding $b$-hadron lifetime. An exception is made for the $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ channel where, owing to its smaller event yield,
$\varepsilon_{\rm selection}(t)$ measured with
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays is used instead. The validity of
this approach is checked using large samples of simulated events.
Figure 4: Combined trigger and selection efficiency, $\varepsilon_{\rm
selection}(t)$, for $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates.
## 5 Maximum likelihood fit
For each channel, the lifetime is determined from a two-dimensional maximum
likelihood fit to the unbinned $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X)$ and $t$ distributions. The full probability density function (PDF)
is constructed as ${\cal P}=f_{s}({\cal S}_{m}\times{\cal
S}_{t})+(1-f_{s})({\cal B}_{m}\times{\cal B}_{t})$, where $f_{s}$ is the
signal fraction, determined in the fit, and ${\cal S}_{m}\times{\cal S}_{t}$
and ${\cal B}_{m}\times{\cal B}_{t}$ are the
($m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$, $t$) PDFs for the signal
and background components, respectively. A systematic uncertainty is assigned
to the assumption that the PDFs factorise.
The signal mass PDF, ${\cal S}_{m}$, is modelled by the sum of two Gaussian
functions. The free parameters in the fit are the common mean, the width of
the narrower Gaussian function, the ratio of the second to the first Gaussian
width and the fraction of the first Gaussian function. The background mass
distribution, ${\cal B}_{m}$, is modelled by an exponential function with a
single free parameter.
The signal $b$-hadron decay time distribution is described by an exponential
function with decay constant given by the $b$-hadron lifetime,
$\tau_{H_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X}$. The
signal decay time PDF, ${\cal S}_{t}$, is obtained by multiplying the
exponential function by the combined $t$-dependent trigger and selection
efficiency described in Sec. 4.2. From inspection of events in the sidebands
of the $b$-hadron signal peak, the background decay time PDF, ${\cal B}_{t}$,
is well modelled by a sum of three exponential functions with different decay
constants that are free in the fit. These components originate from a
combination of prompt candidates, where all tracks originate from the same PV,
and long-lived candidates where tracks from the associated PV are combined
with other tracks of long-lived particles. For each channel the exponential
functions are convolved with a Gaussian resolution function with width
$\sigma$ and mean $\Delta$, an offset of the order of a few femtoseconds that
is fixed in the fit. Using a sample of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ background events, the decay
time resolution for $H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X$ channels reconstructed using long tracks has been measured to be
approximately $45\rm\,fs$ [47]. For
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays, which use downstream tracks to
reconstruct the $K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar 28931\relax$
particles, a similar study of an event sample composed of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons combined with two
downstream tracks, reconstructed as either a $K^{0}_{\rm\scriptscriptstyle S}$
or $\mathchar 28931\relax$, has determined the resolution to be $65\rm\,fs$.
The systematic uncertainties related to the choice of resolution model are
discussed in Sec. 6.
The negative log-likelihood, constructed as
$-\ln{\cal L}=-\alpha\sum_{\mathrm{events}\;i}{w_{i}\ln{{\cal P}}},$ (6)
is minimised in the fit, where the weights $w_{i}$ correspond to the per-
candidate correction for the VELO reconstruction efficiency described in Sec.
4.1. The factor $\alpha=\sum_{i}w_{i}/\sum_{i}w_{i}^{2}$ is used to include
the effect of the weights in the determination of the uncertainties [48].
Figures 1 and 2 show the result of fitting this model to the selected
candidates for each channel, projected onto the corresponding
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ and $t$ distributions.
As a consistency check, an alternative fit procedure is developed where each
event is given a signal weight, $W_{i}$, determined using the sPlot [49]
method with $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ as the
discriminating variable and using the mass model described above. A weighted
fit to the decay time distribution using the signal PDF is then used to
measure the $b$-hadron lifetime. In this case, the negative log-likelihood is
given by Eq. (6) where $w_{i}$ is replaced with $W_{i}w_{i}$ and
$\alpha=\sum_{i}(W_{i}w_{i})/\sum_{i}(W_{i}w_{i})^{2}$. The difference between
the results of the two fitting procedures is used to estimate the systematic
uncertainty on the background description.
## 6 Systematic uncertainties
The systematic effects affecting the measurements reported here are discussed
in the following and summarised in Tables 4 and 5.
The systematic uncertainty related to the VELO-track reconstruction efficiency
can be split into two components. The first uncertainty is due to the finite
size of the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ sample, reconstructed using downstream kaon tracks, which is used
to determine the per-candidate efficiency weights and leads to a statistical
uncertainty on the $\varepsilon_{\rm VELO}(\mbox{$\rho$})$ parameterisation.
The lifetime fits are repeated after varying the parameters by $\pm 1\sigma$
and the largest difference between the lifetimes is assigned as the
uncertainty. The second uncertainty is due to the scaling factors, which are
used to correct the efficiency for phase-space effects, obtained from
simulated events. The fit is repeated using the unscaled efficiency and half
of the variation in fit results is assigned as a systematic uncertainty. These
contributions, of roughly the same size, are added in quadrature in Table 4.
A number of additional consistency checks are performed to investigate
possible mismodelling of the VELO-track reconstruction efficiency. First,
$\varepsilon_{\rm VELO}(\mbox{$\rho$})$ is evaluated in two track momentum and
two track multiplicity bins and the event weights recalculated. Using both
data and simulated events, no significant change in the lifetimes is observed
after repeating the fit with the updated weights and, therefore, no systematic
uncertainty is assigned. Secondly, to assess the sensitivity to the choice of
parameterisation for $\varepsilon_{\rm VELO}(\mbox{$\rho$})$ (Eq. 4), the
results are compared to those with linear model for the efficiency. The effect
is found to be negligible and no systematic uncertainty is applied. Thirdly,
the dependence of the VELO-track reconstruction efficiency on the azimuthal
angle, $\phi$, of each track is studied by independently evaluating the
efficiency in four $\phi$ quadrants for both data and simulation. No
dependence is observed. Finally, the efficiency is determined separately for
both positive and negative kaons and found to be compatible.
The techniques described in Sec. 4 to correct the efficiency as a function of
the decay time are validated on simulated data. The lifetime is fit in each
simulated signal mode and the departure from the generated lifetime,
$\Delta\tau$, is found to be compatible with zero within the statistical
precision of each simulated sample. The measured lifetimes in the data sample
are corrected by each $\Delta\tau$ and a corresponding systematic uncertainty
is assigned, given by the size of the statistical uncertainty on the fitted
lifetime for each simulated signal mode.
The assumption that $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ is
independent of the decay time is central to the validity of the likelihood
fits used in this study. It is tested by re-evaluating the signal weights of
the alternative fit in bins of decay time and then refitting the entire sample
using the modified weights. The systematic uncertainty for each decay mode is
evaluated as the sum in quadrature of the lifetime variations, each weighted
by the fraction of signal events in the corresponding bin.
For each signal decay mode, the effect of the limited size of the control
sample used to estimate the combined trigger and selection efficiency is
evaluated by repeating the fits with $\varepsilon_{\rm selection}(t)$ randomly
fluctuated within its statistical uncertainty. The standard deviation of the
distribution of lifetimes obtained is assigned as the systematic uncertainty.
The alternative likelihood fit does not assume any model for the decay time
distribution associated with the combinatorial background. Therefore, the
systematic uncertainty associated to the modelling of this background is
evaluated by taking the difference in lifetimes measured by the nominal and
alternative fit methods.
The fit uses a double Gaussian function to describe the
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ distribution of signal
candidates. This assumption is tested by repeating the fit using a double-
sided Apollonios function [50] where the mean and width parameters are varied
in the fit and the remaining parameters are fixed to those from simulation.
The differences in lifetime with respect to the default results are taken as
systematic uncertainties.
As described in Sec. 5 the dominant background in each channel is
combinatorial in nature. It is also possible for backgrounds to arise due to
misreconstruction of $b$-hadron decays where the particle identification has
failed. The presence of such backgrounds is checked by inspecting events in
the sidebands of the signal and re-assigning the mass hypotheses of at least
one of the final-state hadrons. The only contributions that have an impact are
$\mathchar 28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ decays in the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ channel
where a proton is misidentified as a kaon and the cross-feed component between
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays where pion and protons are misidentified.
In the case of $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays, the fit is repeated including a contribution of $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ decays. The two-dimensional PDF is determined from simulation,
while the yield is found to be $6\%$ from the sidebands of the $B^{0}_{s}$
sample. This leads to the effective lifetime changing by $0.4\rm\,fs$, which
is assigned as a systematic uncertainty. A similar procedure is used to
determine the invariant mass shape of the cross-feed background between
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays, while the decay time is modelled with the
exponential distribution of the corresponding signal mode. A simultaneous fit
to both data samples is performed in order to constrain the yield of the
cross-feed and the resulting change in lifetime of $-0.3\rm\,fs$ and
$+1.1\rm\,fs$ for $B^{0}$ and $\mathchar 28931\relax^{0}_{b}$, respectively,
is assigned as a systematic uncertainty.
Another potential source of background is the incorrect association of signal
$b$ hadrons to their PV, which results in an erroneous reconstruction of the
decay time. Since the fitting procedure ignores this contribution, a
systematic uncertainty is evaluated by repeating the fit after including in
the background model a component describing the incorrectly associated
candidates. The background distribution is determined by calculating the decay
time for each $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ decay with respect to a randomly chosen PV from the previous
selected event. In studies of simulated events the fraction of this background
is less than $0.1\%$. Repeating the fit with a $1\%$ contribution results in
the lifetime changing by $0.1\rm\,fs$ and, therefore, no systematic
uncertainty is assigned.
The measurement of the effective lifetime in the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
channel is integrated over the angular distributions of the final-state
particles and is, in the case of uniform angular efficiency, insensitive to
the different polarisations of the final state [47]. To check if the angular
acceptance introduced by the detector geometry and event selection can affect
the measured lifetime, the events are weighted by the inverse of the angular
efficiency determined in Ref. [47]. Repeating the fit with the weighted
dataset leads to a shift of the lifetime of $-1.0\rm\,fs$, the same as is
observed in simulation. The final result is corrected by this shift, which is
also assigned as a systematic uncertainty. The $B^{0}_{s}$ effective lifetime
could also be biased due to a small $C\\!P$-odd S-wave component from
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
decays that is ignored in the fit. For the $m(K^{+}K^{-})$ mass range used
here (Sec. 3), Ref. [51] indicates that the S-wave contribution is $1.1\%$.
The effect of ignoring such a component is evaluated by repeating the fit on
simulated experiments with an additional $1\%$ $C\\!P$-odd component. A change
in the lifetime of $-1.2\rm\,fs$ is observed, which is used to correct the
final lifetime and is also taken as a systematic uncertainty. Finally, as
described in Sec. 3, only events with a decay time larger than $0.3{\rm\,ps}$
are considered in the nominal fit. This offset leads to a different relative
contribution of the heavy and light mass eigenstates such that the lifetime
extracted from the exponential fit does not correspond to the effective
lifetime defined in Eq. (2). A correction of $-0.3\rm\,fs$ is applied to
account for this effect.
The presence of a production asymmetry between $B^{0}$ and $\overline{B}^{0}$
mesons could bias the measured
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ effective lifetime, and therefore
$\Delta\Gamma_{d}/\Gamma_{d}$, by adding additional terms in Eq. (2). The
production asymmetry is measured to be $A_{\rm P}(B^{0})=(0.6\pm 0.9)\%$ [52],
the uncertainty of which is used to estimate a corresponding systematic
uncertainty on the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ lifetime of $1.1\rm\,fs$. No
uncertainty is assigned to the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
lifetime since this decay mode is flavour-specific222Flavour-specific means
that the final state is only accessible via the decay of a $B^{0}_{(s)}$ meson
and accessible by a meson originally produced as a $\overline{B}^{0}_{(s)}$
only via oscillation. and the production asymmetry cancels in the untagged
decay rate. For the $B^{0}_{s}$ system, the rapid oscillations, due to the
large value of $\Delta m_{s}=17.768\pm 0.024{\rm\,ps^{-1}}$ [53], reduce the
effect of a production asymmetry, reported as $A_{\rm P}(B^{0}_{s})=(7\pm
5)\%$ in Ref. [52], to a negligible level. Hence, no corresponding systematic
uncertainty is assigned.
There is a $0.02\%$ relative uncertainty on the lifetime measurements due to
the uncertainty on the length scale of LHCb [53], which is mainly determined
by the VELO modules $z$ positions. These are evaluated by a survey, having an
accuracy of $0.1\rm\,mm$ over the full length of the VELO ($1000\rm\,mm$), and
refined through a track-based alignment. The alignment procedure is more
precise for the first track hits, that are less affected by multiple
scattering and whose distribution of $z$ positions have an RMS of
$100\rm\,mm$. In this region, the differences between the module positions
obtained from the survey and track-based alignment are within $0.02\rm\,mm$,
which is taken as systematic uncertainty. The systematic uncertainty related
to the momentum scale calibration affects both the $b$ hadron candidate mass
and momentum and, therefore, cancels when computing the decay time.
The systematic uncertainty related to the choice of $45\rm\,fs$ for the width
of the decay-time resolution function ($65\rm\,fs$ in the case of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$) is evaluated by changing the width by $\pm
15\rm\,fs$ and repeating the fit. This change in width is larger than the
estimated uncertainty on the resolution and leads to a negligible change in
the fit results. Consequently, no systematic uncertainty is assigned.
Furthermore, to test the sensitivity of the lifetimes to potential
mismodelling of the long tails in the resolution, the resolution model is
changed from a single Gaussian function to a sum of two or three Gaussian
functions with parameters fixed from simulation. Repeating the fit with the
new resolution model causes no significant change to the lifetimes and no
systematic uncertainty is assigned. The lifetimes are insensitive to the
offset, $\Delta$, in the resolution model.
Several consistency checks are performed to study the stability of the
lifetimes, by comparing the results obtained using different subsets of the
data in terms of magnet polarity, data taking period, $b$-hadron and track
kinematic variables, number of PVs in the event and track multiplicity. In all
cases, no trend is observed and all lifetimes are compatible with the nominal
results.
Table 4: Statistical and systematic uncertainties (in femtoseconds) for the values of the $b$-hadron lifetimes. The total systematic uncertainty is obtained by combining the individual contributions in quadrature. Source | $\tau_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ | $\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}}$ | $\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ | $\tau_{\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax}$ | $\tau_{B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi}$
---|---|---|---|---|---
Statistical uncertainty | 3.5 | 6.1 | 12.8 | 26.5 | 11.4
VELO reconstruction | 2.0 | 2.3 | 0.9 | 0.5 | 2.3
Simulation sample size | 1.7 | 2.3 | 2.9 | 3.7 | 2.4
Mass-time correlation | 1.4 | 1.8 | 2.1 | 3.0 | 0.7
Trigger and selection eff. | 1.1 | 1.2 | 2.0 | 2.0 | 2.5
Background modelling | 0.1 | 0.2 | 2.2 | 2.1 | 0.4
Mass modelling | 0.1 | 0.2 | 0.4 | 0.2 | 0.5
Peaking background | – | – | 0.3 | 1.1 | 0.4
Effective lifetime bias | – | – | – | – | 1.6
$B^{0}$ production asym. | – | – | 1.1 | – | –
LHCb length scale | 0.4 | 0.3 | 0.3 | 0.3 | 0.3
Total systematic | 3.2 | 3.9 | 4.9 | 5.7 | 4.6
Table 5: Statistical and systematic uncertainties (in units of $10^{-3}$) for the lifetime ratios and $\Delta\Gamma_{d}/\Gamma_{d}$. For brevity, $\tau_{B^{0}}$ $(\tau_{\overline{B}^{0}})$ corresponds to the measurement of $\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}}$ $(\tau_{\overline{B}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\overline{K}^{*0}})$. The total systematic uncertainty is obtained by combining the individual contributions in quadrature. Source | $\tau_{B^{+}}/\tau_{B^{0}}$ | $\tau_{B^{0}_{s}}/\tau_{B^{0}}$ | $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{B^{0}}$ | $\tau_{B^{+}}/\tau_{B^{-}}$ | $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern 0.63004pt\overline{\kern-0.63004pt\mathchar 28931\relax}^{0}_{b}}$ | $\tau_{B^{0}}/\tau_{\overline{B}^{0}}$ | $\Delta\Gamma_{d}/\Gamma_{d}$
---|---|---|---|---|---|---|---
Statistical uncertainty | 5.0 | 8.5 | 18.0 | 4.0 | 35.0 | 8.0 | 25.0
VELO reconstruction | 1.6 | 1.7 | 1.1 | – | – | – | 4.1
Simulation sample size | 2.0 | 2.2 | 2.8 | 2.1 | 5.3 | 3.0 | 6.3
Mass-time correlation | 1.6 | 1.2 | 2.3 | – | – | – | 4.7
Trigger and selection eff. | 1.1 | 1.8 | 1.5 | – | – | – | 4.0
Background modelling | 0.3 | 0.1 | 1.5 | 0.2 | 3.0 | 1.4 | 3.8
Mass modelling | 0.2 | 0.4 | 0.2 | 0.1 | 0.2 | 0.2 | 0.8
Peaking background | – | 0.3 | 0.7 | – | – | – | 0.5
Effective lifetime bias | – | 1.0 | – | – | – | – | –
$B^{0}$ production asym. | – | – | – | – | – | 8.5 | 1.9
Total systematic | 3.2 | 3.7 | 4.4 | 2.1 | 6.1 | 9.1 | 10.7
The majority of the systematic uncertainties described above can be propagated
to the lifetime ratio measurements in Table 7. However, some of the
uncertainties are correlated between the individual lifetimes and cancel in
the ratio. For the first set of ratios and for $\Delta\Gamma_{d}/\Gamma_{d}$,
the systematic uncertainty from the VELO-reconstruction efficiency weights and
the LHCb length scale are considered as fully correlated. For the second set
of ratios, other systematic uncertainties, as indicated in Table 5, cancel,
since the ratio is formed from lifetimes measured using the same decay mode.
In contrast to the situation for the measurement of the $B^{0}$ lifetime in
the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
mode, the $B^{0}$ production asymmetry does lead to a systematic uncertainty
on the measurement of
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}/\tau_{\overline{B}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}}$ since terms like $A_{\rm P}\cos(\Delta m_{d}t)$ do
not cancel in the decay rates describing the decays of $B^{0}$ and
$\overline{B}^{0}$ mesons to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}$ final states. The effect of candidates where the
flavour, via the particle identification of the decay products, has not been
correctly assigned is investigated and found to be negligible.
## 7 Results and conclusions
The measured $b$-hadron lifetimes are reported in Table 6. All results are
compatible with existing world averages [13]. The reported $\tau_{\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax}$ is smaller by approximately $2\sigma$ than a
previous measurements from LHCb [8]. With the exception of the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ channel, these are the single most precise
measurements of the $b$-hadron lifetimes. The $B^{0}_{s}$ meson effective
lifetime is measured using the same data set as used in Ref. [47] for the
measurement of the $B^{0}_{s}$ meson mixing parameters and polarisation
amplitudes in $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays. The $B^{0}_{s}$ meson effective lifetime computed from
these quantities is compatible with the lifetime reported in this paper and a
combination of the two results is, therefore, inappropriate.
Table 6: Fit results for the $B^{+}$, $B^{0}$, $B^{0}_{s}$ mesons and
$\mathchar 28931\relax^{0}_{b}$ baryon lifetimes. The first uncertainty is
statistical and the second is systematic.
Lifetime Value [${\rm\,ps}$ ]
$\tau_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$
1.637 $\pm$ 0.004 $\pm$ 0.003
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}}$
1.524 $\pm$ 0.006 $\pm$ 0.004
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ 1.499 $\pm$ 0.013 $\pm$ 0.005
$\tau_{\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax}$ 1.415 $\pm$ 0.027 $\pm$ 0.006
$\tau_{B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}$ 1.480 $\pm$ 0.011 $\pm$ 0.005
Table 7 reports the ratios of the $B^{+}$, $B^{0}_{s}$ and $\mathchar
28931\relax^{0}_{b}$ lifetimes to the $B^{0}$ lifetime measured in the
flavour-specific $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ channel. This decay mode provides a better normalisation than
the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ channel due to the lower statistical
uncertainty on the $B^{0}$ meson lifetime and due to the fact that the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
lifetime only depends quadratically on $\Delta\Gamma_{d}/\Gamma_{d}$, as shown
in Eq. (7). The statistical and systematic uncertainties from the absolute
lifetime measurements are propagated to the ratios, taking into account the
correlations between the systematic uncertainties. All ratios are consistent
with SM predictions [24, 25, 22, 30, 23, 31, 32, 15] and with previous
measurements [13]. Furthermore, the ratios $\tau_{B^{+}}/\tau_{B^{-}}$,
$\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern
0.70004pt\overline{\kern-0.70004pt\mathchar 28931\relax}^{0}_{b}}$ and
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}/\tau_{\overline{B}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}}$ are reported. Measuring any of these different from
unity would indicate a violation of $C\\!PT$ invariance or, for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
decays, could also indicate that $\Delta\Gamma_{d}$ is non-zero and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ is not
$100\%$ flavour-specific. No deviation from unity of these ratios is observed.
Table 7: Lifetime ratios for the $B^{+}$, $B^{0}$, $B^{0}_{s}$ mesons and
$\mathchar 28931\relax^{0}_{b}$ baryon. The first uncertainty is statistical
and the second is systematic.
Ratio Value
$\tau_{B^{+}}/\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}$ 1.074 $\pm$ 0.005 $\pm$ 0.003
$\tau_{B^{0}_{s}}/\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}$ 0.971 $\pm$ 0.009 $\pm$ 0.004 $\tau_{\mathchar
28931\relax^{0}_{b}}/\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}$ 0.929 $\pm$ 0.018 $\pm$ 0.004 $\tau_{B^{+}}/\tau_{B^{-}}$ 1.002
$\pm$ 0.004 $\pm$ 0.002 $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern
0.70004pt\overline{\kern-0.70004pt\mathchar 28931\relax}^{0}_{b}}$ 0.940 $\pm$
0.035 $\pm$ 0.006
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}/\tau_{\overline{B}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}}$ 1.000 $\pm$ 0.008 $\pm$ 0.009
The effective lifetimes of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays are used to measure
$\Delta\Gamma_{d}/\Gamma_{d}$. Flavour-specific final states such as
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ have
$\mathcal{A}_{\Delta\Gamma_{d}}^{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}=0$, while
$\mathcal{A}_{\Delta\Gamma_{d}}^{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}=\cos(2\beta)$ to a good approximation
in the SM, where
$\beta\equiv\arg\left[-(V_{cd}V^{*}_{cb})/(V_{td}V^{*}_{tb})\right]$ is one of
the CKM unitarity triangle angles. Hence, the two effective lifetimes can be
expressed as
$\displaystyle\tau_{B^{0}\rightarrow J/\psi K^{*0}}$
$\displaystyle=\frac{1}{\Gamma_{d}}\frac{1}{1-y_{d}^{2}}\left(1+y_{d}^{2}\right),$
(7) $\displaystyle\tau_{B^{0}\rightarrow J/\psi K^{0}_{S}}$
$\displaystyle=\frac{1}{\Gamma_{d}}\frac{1}{1-y_{d}^{2}}\left(\frac{1+2\cos(2\beta)y_{d}+y_{d}^{2}}{1+\cos(2\beta)y_{d}}\right).$
(8)
Using the effective lifetimes reported in Table 6 and
$\beta=(21.5^{+0.8}_{-0.7})^{\circ}$ [13], a fit of $\Delta\Gamma_{d}$ and
$\Gamma_{d}$ to the expressions in Eq. (7) and Eq. (8) leads to
$\displaystyle\Gamma_{d}$ $\displaystyle=\phantom{+}0.656\pm 0.003\pm
0.002{\rm\,ps^{-1}},$ (9) $\displaystyle\Delta\Gamma_{d}$
$\displaystyle=-0.029\pm 0.016\pm 0.007{\rm\,ps^{-1}},$ (10)
where the first uncertainty is statistical and the second is systematic. The
correlation coefficient between $\Delta\Gamma_{d}$ and $\Gamma_{d}$ is $0.43$
when including statistical and systematic uncertainties. The combination gives
$\frac{\Delta\Gamma_{d}}{\Gamma_{d}}=-0.044\pm 0.025\pm 0.011,$ (11)
consistent with the SM expectation [14, 15] and the current world-average
value [13].
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages we depend on. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia).
## References
* [1] V. A. Khoze and M. A. Shifman, Heavy quarks, Sov. Phys. Usp. 26 (1983) 387
* [2] M. A. Shifman and M. Voloshin, Preasymptotic effects in inclusive weak decays of charmed particles, Sov. J. Nucl. Phys. 41 (1985) 120
* [3] M. A. Shifman and M. Voloshin, Hierarchy of lifetimes of charmed and beautiful hadrons, Sov. Phys. JETP 64 (1986) 698
* [4] I. I. Bigi, N. Uraltsev, and A. Vainshtein, Nonperturbative corrections to inclusive beauty and charm decays: QCD versus phenomenological models, Phys. Lett. B293 (1992) 430, arXiv:hep-ph/9207214
* [5] I. I. Bigi, The QCD perspective on lifetimes of heavy-flavor hadrons, arXiv:hep-ph/9508408
* [6] N. Uraltsev, Heavy quark expansion in beauty and its decays, arXiv:hep-ph/9804275
* [7] M. Neubert, B decays and the heavy-quark expansion, Adv. Ser. Direct. High Energy Phys. 15 (1998) 239, arXiv:hep-ph/9702375
* [8] LHCb collaboration, R. Aaij et al., Precision measurement of the $\mathchar 28931\relax^{0}_{b}$ baryon lifetime, Phys. Rev. Lett. 111 (2013) 102003, arXiv:1307.2476
* [9] K. Hartkorn and H. Moser, A new method of measuring $\Delta\Gamma/\Gamma$ in the $B_{s}^{0}$ anti-$B_{s}^{0}$ system, Eur. Phys. J. C8 (1999) 381
* [10] I. Dunietz, R. Fleischer, and U. Nierste, In pursuit of new physics with $B_{s}$ decays, Phys. Rev. D63 (2001) 114015, arXiv:hep-ph/0012219
* [11] R. Fleischer and R. Knegjens, Effective lifetimes of $B_{s}$ decays and their constraints on the $B_{s}^{0}$-$\bar{B}_{s}^{0}$ mixing parameters, Eur. Phys. J. C71 (2011) 1789, arXiv:1109.5115
* [12] U. Nierste, CP asymmetry in flavor-specific B decays, arXiv:hep-ph/0406300
* [13] Heavy Flavor Averaging Group, Y. Amhis et al., Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of early 2012, arXiv:1207.1158, updated results and plots available at: http://www.slac.stanford.edu/xorg/hfag/
* [14] A. Lenz and U. Nierste, Theoretical update of $B_{s}-\bar{B}_{s}$ mixing, JHEP 06 (2007) 072, arXiv:hep-ph/0612167
* [15] A. Lenz and U. Nierste, Numerical updates of lifetimes and mixing parameters of B mesons, arXiv:1102.4274
* [16] BaBar collaboration, B. Aubert et al., Limits on the decay-rate difference of neutral $B$ mesons and on CP, T, and CPT violation in $B^{0}\overline{B}^{0}$ oscillations, Phys. Rev. Lett. 92 (2004) 181801, arXiv:hep-ex/0311037
* [17] BaBar collaboration, B. Aubert et al., Limits on the decay-rate difference of neutral $B$ mesons and on CP, T, and CPT violation in $B^{0}\overline{B}^{0}$ oscillations, Phys. Rev. D70 (2004) 012007, arXiv:hep-ex/0403002
* [18] Belle collaboration, T. Higuchi et al., Search for time-dependent CPT violation in hadronic and semileptonic B decays, Phys. Rev. D85 (2012) 071105, arXiv:1203.0930
* [19] G. Borissov and B. Hoeneisen, Understanding the like-sign dimuon charge asymmetry in $p\bar{p}$ collisions, Phys. Rev. D87 (2013) 074020, arXiv:1303.0175
* [20] D0 collaboration, V. M. Abazov et al., Study of CP-violating charge asymmetries of single muons and like-sign dimuons in $p\overline{p}$ collisions, Phys. Rev. D89 (2014) 012002, arXiv:1310.0447
* [21] T. Gershon, $\Delta\Gamma_{d}$: a forgotten null test of the standard model, J. Phys. G38 (2011) 015007, arXiv:1007.5135
* [22] M. Beneke et al., The $B^{+}-B^{0}_{d}$ lifetime difference beyond leading logarithms, Nucl. Phys. B639 (2002) 389, arXiv:hep-ph/0202106
* [23] E. Franco, V. Lubicz, F. Mescia, and C. Tarantino, Lifetime ratios of beauty hadrons at the next-to-leading order in QCD, Nucl. Phys. B633 (2002) 212, arXiv:hep-ph/0203089
* [24] M. Beneke, G. Buchalla, and I. Dunietz, Width Difference in the $B_{s}-\bar{B_{s}}$ System, Phys. Rev. D54 (1996) 4419, arXiv:hep-ph/9605259
* [25] Y.-Y. Keum and U. Nierste, Probing penguin coefficients with the lifetime ratio $\tau(B_{s})/\tau(B_{d})$, Phys. Rev. D57 (1998) 4282, arXiv:hep-ph/9710512
* [26] N. Uraltsev, On the problem of boosting nonleptonic b baryon decays, Phys. Lett. B376 (1996) 303, arXiv:hep-ph/9602324
* [27] I. I. Bigi, M. A. Shifman, and N. Uraltsev, Aspects of heavy quark theory, Ann. Rev. Nucl. Part. Sci. 47 (1997) 591, arXiv:hep-ph/9703290
* [28] D. Pirjol and N. Uraltsev, Four fermion heavy quark operators and light current amplitudes in heavy flavor hadrons, Phys. Rev. D59 (1999) 034012, arXiv:hep-ph/9805488
* [29] M. Voloshin, Reducing model dependence of spectator effects in inclusive decays of heavy baryons, Phys. Rev. D61 (2000) 074026, arXiv:hep-ph/9908455
* [30] C. Tarantino, Beauty hadron lifetimes and B meson CP violation parameters from lattice QCD, Eur. Phys. J. C33 (2004) S895, arXiv:hep-ph/0310241
* [31] F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, $\mathchar 28931\relax^{0}_{b}$ lifetime puzzle in heavy quark expansion, Phys. Rev. D68 (2003) 114006, arXiv:hep-ph/0303235
* [32] F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, Spectator effects and lifetimes of heavy hadrons, Phys. Rev. D70 (2004) 094031, arXiv:hep-ph/0407004
* [33] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [34] M. Adinolfi et al., Performance of the LHCb RICH detector at the LHC, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
* [35] A. A. Alves Jr et al., Performance of the LHCb muon system, JINST 8 (2013) P02022, arXiv:1211.1346
* [36] R. Aaij et al., The LHCb trigger and its performance in 2011, JINST 8 (2013) P04022, arXiv:1211.3055
* [37] T. Sjöstrand, S. Mrenna, and P. Skands, Pythia 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [38] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [39] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [40] P. Golonka and Z. Was, Photos Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [41] Geant4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [42] Geant4 collaboration, S. Agostinelli et al., Geant4: a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [43] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. Phys. : Conf. Ser. 331 (2011) 032023
* [44] W. D. Hulsbergen, Decay chain fitting with a Kalman filter, Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
* [45] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001, and 2013 partial update for the 2014 edition
* [46] O. Callot, FastVelo, a fast and efficient pattern recognition package for the Velo, LHCb-PUB-2011-001
* [47] LHCb collaboration, R. Aaij et al., Measurement of $C\\!P$-violation and the $B^{0}_{s}$-meson decay width difference with $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ and $B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays, Phys. Rev. D87 (2013) 112010, arXiv:1304.2600
* [48] Y. Xie, sFit: a method for background subtraction in maximum likelihood fit, arXiv:0905.0724
* [49] M. Pivk and F. R. Le Diberder, sPlot: a statistical tool to unfold data distributions, Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
* [50] D. M. Santos and F. Dupertuis, Mass distributions marginalized over per-event errors, submitted to Nucl. Instrum. Meth. A (2013) arXiv:1312.5000
* [51] LHCb collaboration, R. Aaij et al., Amplitude analysis and branching fraction measurement of $\overline{B}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$, Phys. Rev. D87 (2013) 072004, arXiv:1302.1213
* [52] LHCb collaboration, R. Aaij et al., First measurement of time-dependent $CP$ violation in $B_{s}^{0}\rightarrow K^{+}K^{-}$ decays, JHEP 10 (2013) 183, arXiv:1308.1428
* [53] LHCb collaboration, R. Aaij et al., Precision measurement of the $B^{0}_{s}-\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ oscillation frequency $\Delta m_{s}$ in the decay $B^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$, New J. Phys. 15 (2013) 053021, arXiv:1304.4741
|
arxiv-papers
| 2014-02-11T16:40:32 |
2024-09-04T02:49:58.090027
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. Anderlini,\n J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, M. Borsato, T.J.V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto,\n J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M. Calvi, M. Calvo Gomez,\n A. Camboni, P. Campana, D. Campora Perez, A. Carbone, G. Carboni, R.\n Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G.\n Casse, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph.\n Charpentier, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid\n Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C.\n Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A.\n Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, I. Counts, B. Couturier,\n G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie, C.\n D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, S. Donleavy, F.\n Dordei, M. Dorigo, P. Dorosz, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger, R. Ekelhof,\n L. Eklund, I. El Rifai, Ch. Elsasser, S. Esen, A. Falabella, C. F\\\"arber, C.\n Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues,\n M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, A. Gianelle, S. Giani', V.\n Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, P.\n Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, T.W. Hafkenscheid, S.C. Haines, S. Hall,\n B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, D. Hill, M.\n Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, N. Hussain, D. Hutchcroft, D.\n Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P.\n Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B.\n Jost, N. Jurik, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach,\n I.R. Kenyon, T. Ketel, B. Khanji, C. Khurewathanakul, S. Klaver, O.\n Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S.\n Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, M. Liles, R. Lindner, C.\n Linn, F. Lionetto, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N.\n Lopez-March, P. Lowdon, H. Lu, D. Lucchesi, J. Luisier, H. Luo, E. Luppi, O.\n Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G.\n Manca, G. Mancinelli, M. Manzali, J. Maratas, U. Marconi, P. Marino, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev,\n Z. Mathe, C. Matteuzzi, A. Mazurov, M. McCann, J. McCarthy, A. McNab, R.\n McNulty, B. McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A.\n Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, M.\n Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, F. Muheim, K.\n M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar,\n I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C.\n Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, A.\n Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, G. Onderwater, M. Orlandea, J.M. Otalora Goicochea,\n P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, L. Pappalardo, C. Parkes, C.J. Parkinson, G.\n Passaleva, G.D. Patel, M. Patel, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pearce, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini,\n E. Perez Trigo, P. Perret, M. Perrin-Terrin, L. Pescatore, E. Pesen, G.\n Pessina, K. Petridis, A. Petrolini, E. Picatoste Olloqui, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, A. Pistone, S. Playfer, M. Plo Casasus, F. Polci, G.\n Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A.\n Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H. Rademacker, B.\n Rakotomiaramanana, M. Rama, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven,\n S. Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards,\n K. Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, A.B.\n Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A.\n Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz\n Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V.\n Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, G.\n Simi, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E. Smith,\n J. Smith, M. Smith, H. Snoek, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, F. Spinella, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S. Stone, B.\n Storaci, S. Stracka, M. Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah,\n L. Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer,\n V. Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, J.A. de\n Vries, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, G.\n Wormser, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang,\n X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Greig Cowan Dr",
"url": "https://arxiv.org/abs/1402.2554"
}
|
1402.2633
|
Identification and correction of sample mix-ups
in expression genetic data: A case study
Karl W. Broman∗,1, Mark P. Keller†, Aimee Teo Broman∗,
Christina Kendziorski∗, Brian S. Yandell‡,§, Śaunak Sen∗∗,2, Alan D. Attie†
∗Department of Biostatistics and Medical Informatics, †Department of
Biochemistry, ‡Department of Statistics, and §Department of Horticulture,
University of Wisconsin–Madison, Madison, Wisconsin 53706, and ∗∗Department of
Epidemiology and Biostatistics, University of California, San Francisco,
California 94107
Running head: Correcting sample mix-ups in eQTL data
Key words: quality control, microarrays, genetical genomics, mislabeling
errors, eQTL
1Corresponding author:
| Karl W Broman
---|---
| Department of Biostatistics and Medical Informatics
| University of Wisconsin–Madison
| 2126 Genetics-Biotechnology Center
| 425 Henry Mall
| Madison, WI 53706
| Phone: | 608–262–4633
| Email: | `[email protected]`
2Present address: Division of Biostatistics, Department of Preventive
Medicine, University of Tennessee Health Science Center, Memphis, TN 38163
Abstract
In a mouse intercross with more than 500 animals and genome-wide gene
expression data on six tissues, we identified a high proportion (18%) of
sample mix-ups in the genotype data. Local expression quantitative trait loci
(eQTL; genetic loci influencing gene expression) with extremely large effect
were used to form a classifier to predict an individual’s eQTL genotype based
on expression data alone. By considering multiple eQTL and their related
transcripts, we identified numerous individuals whose predicted eQTL genotypes
(based on their expression data) did not match their observed genotypes, and
then went on to identify other individuals whose genotypes did match the
predicted eQTL genotypes. The concordance of predictions across six tissues
indicated that the problem was due to mix-ups in the genotypes (though we
further identified a small number of sample mix-ups in each of the six panels
of gene expression microarrays). Consideration of the plate positions of the
DNA samples indicated a number of off-by-one and off-by-two errors, likely the
result of pipetting errors. Such sample mix-ups can be a problem in any
genetic study, but eQTL data allow us to identify, and even correct, such
problems. Our methods have been implemented in an R package, R/lineup.
Introduction
To map the genetic loci influencing a complex phenotype, one seeks to
establish an association between genotype and phenotype. In such an effort,
the maintenance of the concordance between genotyped and phenotyped samples
and data is critical. Sample mislabelings and other sample mix-ups will weaken
associations, resulting in reduced power and biased estimates of locus
effects. In traditional genetic studies, one has limited ability to detect
sample mix-ups and almost no ability to correct such problems. Inconsistencies
between subjects’ sex and X chromosome genotypes may reveal some problems, and
in family studies, some errors may be revealed through Mendelian
inconsistencies at markers, but we will generally be blind to most errors.
In expression genetics studies, in which genome-wide gene expression is
assayed along with genotypes at genetic markers, the presence of expression
quantitative trait loci (eQTL) with profound effect on gene expression
(particularly local-eQTL, in which a polymorphism near a gene affects the
expression of that gene) provides an opportunity to not just identify but also
correct sample mix-ups.
In a mouse intercross with more than 500 animals and genome-wide gene
expression data on six tissues, we identified a high proportion (18%) of
sample mix-ups in the genotype data. We further identified a small number of
mix-ups among the expression arrays in each tissue.
A number of investigators have developed methods for identifying such sample
mix-ups (Westra et al. 2011; Schadt et al. 2012; Lynch et al. 2012; Ekstrøm
and Feenstra 2012), and a similar approach was applied by Baggerly and Coombes
(2008, 2009) in their forensic bioinformatics analyses of the Duke debacle. We
have developed a further approach that is simple but effective. We illustrate
its use through a particularly dramatic example.
Methods
Mice and genotyping
C57BL/6J (abbreviated B6 or B) and BTBR _T_ + _tf_ /J (abbreviated BTBR or R)
mice were purchased from the Jackson Laboratory (Bar Harbor, ME) and bred at
the University of Wisconsin–Madison. The _Lep ob_ mutation was introgressed
into all strains using heterozygous parents to generate homozygous _Lep ob/ob_
offspring. F2 mice, all _Lep ob/ob_, were the offspring of F1 parents derived
from a cross between BTBR females and B6 males (Figure S1). F2 mice and a
small number of parental and F1 controls were genotyped with the 5K GeneChip
(Affymetrix).
Gene expression microarrays
Gene expression was assayed with custom two-color ink-jet microarrays
manufactured by Agilent Technologies (Palo Alto, CA). RNA preparations were
performed at Rosetta Inpharmatics (Merck & Co.). Six tissues were considered:
adipose, gastrocnemius muscle (abbreviated gastroc), hypothalamus (abbreviated
hypo), pancreatic islets (abbreviated islet), kidney, and liver. Tissue-
specific mRNA pools were used for the second channel, and gene expression was
quantified as the ratio of the mean log10 intensity (mlratio). For further
details, see Keller et al. (2008).
Sample mix-ups in the gene expression arrays
Let $x^{s}_{ip}$ denote the gene expression measure for sample $i$ at array
probe $p$ in tissue $s$. We first considered each probe and each pair of
tissues and calculated the between-tissue correlation across samples, omitting
any samples with missing data for that probe in either tissue. We identified
the subset of probes, for each tissue pair, with correlation $>$ 0.75. With
this subset of probes, we then calculated the correlation between sample $i$
in tissue $s$ and sample $j$ in tissue $t$; call it $r^{st}_{ij}$. As an
illustration, consider the schematic in Figure 1: for each pair of tissues, we
identified the subset of probes with high between-tissue correlation (the
shaded region) and then evaluated the correlation between a sample in one
tissue and another sample in the other tissue, across that subset of probes.
Figure 1: Scheme for evaluating the similarity between expression arrays for
different tissues. We first consider the expression of each array probe for
samples assayed for both tissues (A) and calculate the between-tissue
correlation in expression (B). We identify the subset of array probes with
correlation $>$ 0.75 (shaded region in C) and calculate the correlation in
gene expression for one sample in the first tissue and another sample in the
second tissue, across these selected probes. This forms a similarity matrix
(D), for which darker squares indicate greater similarity. Orange squares
indicate missing values (samples assayed in one tissue but not the other).
We then summarized the similarity between sample $i$ in tissue $s$ and sample
$j$ in the other tissues by the median correlation across tissue pairs that
include tissue $s$, $r^{s}_{ij}=\text{median}\\{r^{st}_{ij}:t\neq s\\}$. Of
course, we considered only pairs of tissues $(s,t)$ for which sample $i$ was
measured in tissue $s$ and sample $j$ was measured in tissue $t$.
Sample mix-ups in tissue $s$ were identified as samples $i$ for which the self
similarity, $r^{s}_{ii}$, was small, but for which there existed some array
with high similarity: $\max_{j\neq i}r^{s}_{ij}$ is large. We then inferred
the correct label for sample $i$ in tissue $s$ to be $\arg\max_{j\neq
i}r^{s}_{ij}$. In other words, viewing $r^{s}_{ij}$ as a similarity matrix, we
were looking for rows with a small value on the diagonal, but with some large
off-diagonal element in that row. In order to ensure confidence in the
relabeling of such samples, we compared the maximum value in the row to the
second-highest value.
To further investigate possible sample duplicates within a tissue, we
considered the subset of probes with correlation $>$ 0.75 with at least one
other tissue, and then calculated between-sample correlations, across the
chosen subset of probes, within that tissue.
Sample mix-ups in the DNA samples
In our investigation of potential sample mix-ups in the DNA samples, we first
calculated multipoint genotype probabilities at all markers and at
pseudomarker positions between markers. The pseudomarker positions were placed
at evenly spaced locations between markers, with a maximum spacing of 0.5 cM
between adjacent markers or pseudomarkers. The multipoint genotype
probabilities calculations were performed via a hidden Markov model (HMM),
with an assumed genotyping error rate of 0.2% and with the Carter-Falconer map
function (Carter and Falconer 1951).
We first considered each tissue, individually, and identified the subset of
probes with a strong local-eQTL. We considered all array probes with known
genomic location and on an autosome, identified the nearest marker or
pseudomarker to the location of the probe, and calculated a LOD score (log10
likelihood ratio) assessing the association between genotype at that location
and the gene expression of that probe. The LOD score was calculated by Haley-
Knott regression (Haley and Knott 1992), a quick approximation to standard
interval mapping (Lander and Botstein 1989). Calculations were performed at a
single location for each array probe, rather than with a scan of the genome.
We chose the subset of probes with LOD $>$ 100.
Continuing to focus on one tissue at a time, we considered the set of local-
eQTL locations and the corresponding probe or probes. (Generally there was a
single probe corresponding to a given eQTL location, but in a small number of
instances for each tissue, there were a pair of probes at the same eQTL
location; for islet, there were three eQTL with three corresponding probes,
and for adipose there was one such trio.) For each eQTL position and for each
mouse, we took the genotypes with maximal multipoint probability to be the
observed eQTL genotype, provided that this exceeded 0.99; if no genotype had
probability $>$ 0.99, the observed eQTL genotype was treated as missing.
Considering each eQTL in a tissue individually, we then formed a $k$-nearest
neighbor classifier, with $k=40$, for predicting eQTL genotype from the
expression values for the corresponding probe or probes. For a given mouse, if
more than 80% of the 40 nearest neighbors, by Euclidean distance, shared the
same observed eQTL genotype, this was taken to be the inferred eQTL genotype
for that mouse. If no more than 80% of the 40 nearest neighbors shared a
common genotype, the inferred eQTL genotype was treated as missing.
In order to filter out samples that were clearly incorrect and improve our
classifiers, we then calculated the proportion of matches, for each sample,
between the observed eQTL genotypes and the corresponding inferred eQTL
genotypes, omitted samples for which the proportion of matches was $<$ 0.7,
and rederived the $k$-nearest neighbor classifiers with the subset of samples
deemed likely correct.
As an illustration, consider the schematic in Figure 2: for each tissue, we
identified a subset of array probes with strong local-eQTL, we derived
classifiers for predicting eQTL genotype from the corresponding expression
phenotypes, and then constructed a matrix of inferred eQTL genotypes. As a
measure of similarity between a DNA sample and an mRNA sample, we calculated
the proportion of matches between the observed eQTL genotypes for the DNA
sample and the inferred eQTL genotypes for the mRNA sample.
Figure 2: Scheme for evaluating the similarity between genotypes and
expression arrays. We first identify a set of probes with strong local eQTL.
For each such eQTL, we use the samples with both genotype and expression data
(A) to form a classifier for predicting eQTL genotype from the expression
value (B). We then compare the observed eQTL genotypes for one sample to the
inferred eQTL genotypes, from the classifiers, for another sample (C). The
proportion of matches, between the observed and inferred genotypes, forms a
similarity matrix (D), for which darker squares indicate greater similarity.
Orange squares indicate missing values (for example, samples with genotype
data but no expression data).
To combine the tissue-specific similarity measures across the six tissues, we
simply took the overall proportion of matching genotypes, across all eQTL and
across all tissues.
As in the investigation of sample mix-ups within the expression arrays, we
treated the proportions of matches between observed and inferred eQTL
genotypes as a similarity matrix. Problem DNA samples were identified as rows
for which the value on the diagonal (the self similarity) was small. In such
rows, we inferred the correct label to be that of the maximal off-diagonal
value, provided that this maximum was large and was well above the second-
largest value.
QTL analysis
To characterize the improvement in results following correction of sample mix-
ups, we performed QTL analysis with several traits of interest, including the
expression traits in each tissue, with the original data and with the
corrected data. In the corrected data, we omitted the DNA samples that could
not be verified to be correct (that is, those with no corresponding gene
expression data.)
_Insulin:_ We first considered a clinical phenotype of considerable interest:
10 week plasma insulin. QTL analysis was performed by Haley-Knott regression
(Haley and Knott 1992), with log insulin, and with sex included as an
interactive covariate (that is, allowing the effects of QTL to be different in
the two sexes).
_Agouti and tufted coat:_ We considered two simple Mendelian traits: agouti
coat color (due to a single gene on chromosome 2) and tufted coat (due to a
single gene on chromosome 17). QTL analysis was performed treating each
phenotype as a binary trait (Xu and Atchley 1996; Broman 2003). To handle
possible marker genotyping errors at the causal loci, we took the observed
genotypes to be those with maximal multipoint probability, provided that this
exceeded 0.99; if no genotype had probability $>$ 0.99, the observed genotype
was treated as missing.
_eQTL analyses:_ We considered each of the six tissues individually, and
focused on the subset of probes with known genomic location on an autosome or
the X chromosome. For hypothalamus tissue, we omitted a batch of 119 poorly
behaved arrays, though these had been included in our efforts to identify
sample mix-ups.
Expression measures were transformed to normal quantiles. That is, the
expression measures were converted to ranks $R_{i}\in\\{1,\dots,n\\}$ and then
transformed to $y_{i}=\Phi^{-1}[(R_{i}-0.5)/n]$, where $\Phi^{-1}$ is the
inverse of the normal cumulative distribution function.
QTL analysis was performed by Haley-Knott regressions with sex included as an
interactive covariate. We considered the maximal peak for each array probe on
each chromosome, and inferred the presence of a QTL if the LOD score exceeded
5, a 5% genome-wide significance level established by computer simulation.
An inferred eQTL was considered a local-eQTL if the 2-LOD support interval
contained the genomic location of the corresponding array probe; otherwise, it
was considered a _trans_ -eQTL.
Software
All analyses were conducted with R (R Development Core Team 2013). QTL
analyses were performed with the R package, R/qtl (Broman et al. 2003). Our
methods for identifying sample mix-ups have been assembled as an R package,
R/lineup, available at http://github.com/kbroman/lineup as well as The
Comprehensive R Archive Network (CRAN; http://cran.r-project.org).
Data availability
The genotype and gene expression microarray data are available at the QTL
Archive, now part of the Mouse Phenome Database:
http://phenome.jax.org/db/q?rtn=projects/projdet&reqprojid=532
Results
We first became aware of potential problems in the samples through the
identification of six duplicate DNA samples and 32 mice whose X chromosome
genotypes were incompatible with their sex. We genotyped 554 F2 mice at 2060
informative SNPs, including 20 on the X chromosome. Three samples were
assigned “no call” at all markers and not considered further. Six pairs were
seen to be duplicates, with over 98% genotype identity across typed markers
(Table S1).
The F2 mice were the offspring of F1 siblings derived by crossing BTBR females
to B6 males (Figure S1). F2 females should be homozygous BTBR (RR) or
heterozygous (BR) on the X chromosome; F2 males should be hemizygous B or R.
(Note that homozygous and hemizygous genotypes could not be distinguished.)
However, 19 females exhibited some homozygous B6 genotypes on the X, and 17
males exhibited some heterozygous genotypes (Figure S2). While four of these
males had a single heterozygous genotype that was likely a genotyping error,
the 19 females and the other 13 males were clearly indicated to have swapped
sex. There were an additional 53 females and 50 males with homozygous RR or
hemizygous R genotypes for all markers on the X chromosome, compatible with
either sex.
In cleaning the genotype data, we omitted a set of seven samples, including
one pair of the sample duplicates, with poorly behaved data. (They showed a
high rate of apparent genotyping errors, an unusually large proportion of
homozygous genotypes, and an unusually large number of apparent crossovers.)
For the other five pairs of duplicates, we omitted one sample from each pair.
Sample mix-ups in the gene expression arrays
For each of six tissues (adipose, gastroc, hypo, islet, kidney, liver),
approximately 500 F2 mice were assayed for gene expression with two-color
Agilent arrays with tissue-specific pools (Table S2). A small number of poorly
behaved arrays were omitted. We later discovered a batch of 119 poorly behaved
arrays for hypo, but these were included in the analyses described here. There
were 527 mice assayed for at least one of the six tissues, but not all mice
were assayed for all tissues. In particular, there were 27 mice assayed only
for gene expression in kidney, and 43 mice assayed for all tissues except
kidney. Further, 27 mice were genotyped but were not subject to gene
expression analysis.
To identify potential sample mix-ups among gene expression arrays, we first
identified, for each pair of tissues, a subset of array probes with high
between-tissue correlations. Consideration of all probes would greatly reduce
the apparent correlation between arrays, due to the abundance of unexpressed
genes. For example, for Mouse3567, the correlation between gene expression in
kidney and in liver, across all 40,572 probes, is 0.32, while for the subset
of 155 probes with correlation $>$ 0.75 between kidney and liver, the
correlation is 0.78. (See Figure S3.)
Figure S4 contains density estimates of the between-tissue correlations for
all array probes. The densities are organized by tissue, with the panel for
each tissue containing the five tissue pairs involving that tissue. There are
some small differences among tissue pairs, but the vast majority of between-
tissue correlations are between -0.25 and 0.50. Table S3 contains the numbers
of probes for each pair of tissues with correlations exceeding 0.70, 0.75,
0.80, and 0.90, respectively. We focused on probes with correlations $>$ 0.75,
of which there were between 46 and 200 probes per tissue pair.
For each pair of tissues, we calculated the correlations among samples across
the subset of correlated probes. For each tissue, we then summarized the
similarity between each sample in that tissue and each sample in other tissues
by the median correlations, across the tissue pairs that included the target
tissue.
Figure S5 contains histograms of the similarity measures for each tissue,
separating the self-self similarities (the diagonal elements) and the self-
nonself similarities (the off-diagonal elements). There are a number of clear
outliers: small self-self similarities and large self-nonself similarities.
The self-nonself similarities follow a bimodal distribution, with the lower
mode corresponding to opposite-sex pairs and the upper mode corresponding to
same-sex pairs. The chosen probes included a probe in _Xist_ (involved in X
chromosome inactivation) and probes on the Y chromosome.
To identify problem samples in each tissue, we considered for each sample, the
self similarity vs. the maximum similarity (that is, the values on the
diagonal of the similarity matrix and the maximum values in each row). These
are displayed in Figure 3.
Figure 3: Self similarity (median correlation across tissue pairs) versus
maximum similarity for the expression arrays for each tissue. The diagonal
gray line corresponds to equality. Green points are inferred to be sample mix-
ups. Gray points correspond to arrays for which the self similarity is
maximal. Red points correspond to special cases (see the text). There were 27
samples assayed only for kidney; these have missing self similarity values.
The vast majority of samples in each tissue were indicated to be correctly
labeled: the self similarity was the maximum similarity. But for each tissue,
there were at least a few samples which were more like some other sample in
the other tissues. In each case, we infer the correct label to be that with
the maximal similarity. In Figure S6, we display the second-highest similarity
vs. the maximum similarity for each sample in each tissue. The problem samples
(colored green) are generally well away from the diagonal, indicating good
support for our ability to infer the correct label.
The red points in Figure 3 and Figure S6 are special cases: The Mouse3188
sample is highlighted as a potential problem in both islet and gastroc (being
slightly off the diagonal line), but this is because that sample was involved
in array swaps in two different tissues (adipose and hypo). This is the only
sample indicated to be mislabeled in multiple tissues. We also highlight
Mouse3484 in gastroc, which appeared to be a mixture (more on this below).
The inferred errors are displayed in Figure 4. For adipose, we identified two
problems. The samples for Mouse3583 and Mouse3584 were swapped, and there was
a three-way swap among Mouse3187, Mouse3188, and Mouse3200, with the sample
labeled Mouse3187 really being Mouse3188, that labeled Mouse3188 really being
Mouse3200, and that labeled Mouse3200 really being Mouse3187.
Figure 4: The mRNA sample mix-ups for the six tissues. Double-headed arrows
indicate a sample swap. The trio of points in adipose corresponds to a three-
way swap. The pink circles with a single-headed arrow, in islet and liver, are
sample duplicates. The questionable case in kidney indicates a potential
sample mixture arrayed twice.
For gastroc, there was a single sample swap, between Mouse3655 and Mouse3659.
For hypo, there were 9 pairs of sample swaps. For islet, the samples Mouse3598
and Mouse3599 were swapped, and the sample labeled Mouse3296 was really a
duplicate (or _unintended technical replicate_) of the Mouse3295 sample. For
liver, the sample labeled Mouse3142 really corresponded to Mouse3143
(Mouse3142 was not assayed for gene expression in liver), and the sample
labeled Mouse3141 was really a duplicate of the Mouse3136 sample.
For kidney, the samples for Mouse3510 and Mouse3523 were swapped, and
Mouse3484 was also seen to be a problem. We believe that the samples for
Mouse3484 and Mouse3503 may have been mixed and assayed twice in duplicate
(more below). There were 27 samples that were assayed for gene expression only
in kidney; for these, the self similarity cannot be calculated. We have
limited ability to detect mix-ups for these samples, but none were very close
to any sample in other tissues, and so they can, at least provisionally, be
assumed to be correctly labeled.
To further illustrate the sample swaps, Figure S7 contains scatter plots of
the gastroc arrays labeled Mouse3655 and Mouse3659 against the arrays in the
other tissues with those labels. For each pair of tissues, we plot the array
probes with between-tissue correlation $>$ 0.75. Mouse3655 in gastroc is
correlated with Mouse3659 in other tissues, while Mouse3659 in gastroc is
correlated with Mouse3655 in other tissues, indicating a clear swap between
these samples within gastroc.
Figure S8 contains similar scatter plots for a pair of inferred duplicates,
with the sample labeled Mouse3141 in liver really being a duplicate of the
Mouse3136 liver sample. Mouse3136 liver and Mouse3141 liver are each
correlated with Mouse3136 in other tissues and not with Mouse3141, and the two
samples are extremely highly correlated with each other (see the two central
panels in the bottom row). In Figure S9, we display the between-sample
correlations for samples with these two labels, for all pairs of tissues, with
the pairs including liver highlighted in red. The Mouse3136 samples are
correlated for all tissue pairs; the Mouse3141 samples are correlated for all
tissue pairs not involving liver, and the Mouse3141 liver sample is correlated
with all Mouse3136 samples in other tissues.
The Mouse3484 and Mouse3503 samples in kidney appear to be sample duplicates,
but these samples are correlated with each of Mouse3484 and Mouse3503 in the
other tissues. We’re inclined to believe that the two kidney samples were
mixed and arrayed in duplicate, but we are not able to prove this point.
Figure S10 contains scatter plots for the two samples in kidney vs. all
tissues; the central panels in the second row from the bottom indicate that
the two samples are highly correlated and so likely replicates, but all
scatter plots here show strong correlation. Figure S11 contains the between-
sample correlations for both sample labels in all tissue pairs; contrast this
with Figure S9, for the simple duplicate in liver. Mouse3484 kidney and
Mouse3503 kidney are strongly correlated with both samples in the other
tissues, but not so strongly as Mouse3484 and Mouse3503 are with themselves in
the non-kidney pairs. And for tissue pairs not including kidney, Mouse3484 and
Mouse3503 are much more weakly correlated.
As we were unable to resolve the problems with Mouse3484 and Mouse3503 in
kidney, these two arrays were omitted from later analyses. The two simple
sample duplicates, one in islet and one in liver, were combined and assigned
the correct label. The other sample mix-ups were relabeled as inferred in
Figure 4.
Expression of the _Xist_ gene (involved in X chromosome inactivation and so
highly expressed in females but not males) and of genes on the Y chromosome is
a useful diagnostic for the sex of an mRNA sample. In Figure S12, we display,
for each tissue, the average expression across for Y chromosome genes vs. the
expression of _Xist_ , with the original data and after correction of the
sample mix-ups in the expression arrays. Just three of the sample-swaps (one
in gastroc and two in hypo) involved opposite-sex pairs. These show up clearly
in the left column, with the original data, and are resolved after correction
of the sample mix-ups. The unusual pattern of expression in hypo, with a
bimodal distribution for the Y chromosome genes in males and a large number of
females with relatively low _Xist_ expression, was due to a set of 119 poorly
behaved arrays.
Sample mix-ups in the genotypes
Having corrected the sample mix-ups among the gene expression arrays, we
turned to potential problems in the genotypes. For each tissue, we considered
the 36,364 autosomal array probes with known genomic location and identified
those with a strong local-eQTL, having LOD score $>$ 100 for the association
between the probe expression measures and genotype at the corresponding
location.
For each such probe, we created a $k$-nearest neighbor classifier (with k=40),
for predicting eQTL genotype from the expression phenotype. For example, in
Figure 5, we display the expression, in islet, of probe 499541 (on chromosome
1) vs. genotype at the nearest marker. At this probe, there are three clear
groups of mice, with B6 homozygotes B6 (BB) having high expression, BTBR
homozygotes (RR) having low expression, and heterozygotes (BR) intermediate.
There are a number of mice whose expression does not match their observed eQTL
genotype; the classifier infers a different eQTL genotype. The points
highlighted in pink have expression at the boundary between the BB and BR
groups and are left unassigned. (To assign an inferred eQTL genotype to a
point, we required that 80% of the nearest neighbors had a common eQTL
genotype.)
Figure 5: Plot of islet expression vs observed genotype for an example probe.
Points are colored by the inferred genotype, based on a k-nearest neighbor
classifier, with yellow, green, and blue corresponding to BB, BR, and RR,
respectively, where B = B6 and R = BTBR. Salmon-colored points lie at the
boundary between two clusters and were not assigned.
For sets of probes mapping to approximately the same genomic location, we
considered the probes’ expression jointly. Examples of pairs of probes mapping
to the same location are shown in Figure S13, with points colored by observed
eQTL genotype.
We considered 45–115 eQTL per tissue; their locations on the genetic map of
markers is shown in Figure S14. The majority of eQTL had a single
corresponding probe. There were 3–14 eQTL per tissue with a pair of
corresponding probes. For islet, there were three eQTL with three
corresponding probes, and for adipose there was one such trio.
For each tissue, we calculated the proportion of matches between the observed
eQTL genotypes for each DNA sample and the inferred eQTL genotypes from each
mRNA sample, as a measure of similarity between the DNA and mRNA samples. We
further calculated a combined measure of similarity as the overall proportion
of mismatches, pooling all six tissues.
Figure S15 contains histograms of the similarity measures for each tissue,
separating the self-self similarities (the diagonal elements) and the self-
nonself similarities (the off-diagonal elements). There are a number of clear
outliers: small self-self similarities and large self-nonself similarities.
To identify problem DNA samples, we again considered the self similarity vs.
the maximum similarity (that is, the values on the diagonal of the similarity
matrix vs. the maximum values in each row). Figure 6 contains a scatterplot of
these values. Gray points, with maximum similarity equal to the self
similarity, are inferred to be corrected labeled. Green points, with small
self similarity but large maximum similarity, are inferred to be incorrect,
but are fixable. Red points concern DNA samples for which no corresponding
mRNA sample can be found.
Figure 6: Self similarity (proportion matches between observed and inferred
eQTL genotypes, combined across tissues) versus maximum similarity for the DNA
samples. The diagonal gray line corresponds to equality. Samples with missing
self similarity (at bottom) were not intended to have expression assays
performed. Gray points correspond to DNA samples that were correctly labeled.
Green points correspond to sample mix-ups that are fixable (the correct label
can be determined). Red points comprise both samples mix-ups that cannot be
corrected as well as samples that may be correct but cannot be verified as no
expression data is available.
Detailed results for the six tissues, with tissue-specific similarity values,
are shown in Figure S16. The points are colored as in Figure 6, based on the
combined similarity measure. The points with missing self similarity (at the
bottom of each panel) were not intended to be assayed for gene expression in
that tissue. The tissue-specific results are concordant with the overall
conclusions, with two caveats. First, there are a number of green points
(corresponding to mislabeled, but fixable, DNA samples), with low maximum
similarity in each tissue. These correspond to samples for which gene
expression assays were not performed for that tissue, the bulk of which are
for the 27 samples that were assayed only for gene expression in kidney and
the 43 samples that were assayed for all tissues except kidney. Second, for
hypo, the strength of eQTL associations were weaker, and fewer eQTL were
considered, than for the other tissues, and so there is less separation
between the green and pink points.
In Figure S17, we display the second-highest similarity vs. the maximum
similarity, for the combined similarity measures accounting for all tissues.
The fixable mis-labeled samples (in green) are all well away from the
diagonal, indicating good support for our ability to infer the correct label.
The inferred mix-ups among the DNA samples are displayed in Figure 7 according
to the arrangement of the samples on the 96-well genotyping plates. Black dots
indicate that the correct DNA sample was placed in the correct well. The blue
arrows point from the well in which a DNA sample was supposed to be placed, to
the well where it was actually placed. For example, on plate 1631, the sample
in well D02 was placed in the correct well but was also placed in well B03.
The sample that belonged in B03 was placed in B04, the sample that belonged in
B04 was placed in E03, and the sample belonging in E03 was not found (but, as
indicated by the green arrowhead, there was no corresponding gene expression
data).
Figure 7: The DNA sample mix-ups on the seven 96-well plates used for
genotyping. Black dots indicate that the correct DNA was put in the well. Blue
arrows point from where a sample should have been placed to where it was
actually placed; the different shades of blue convey no meaning. Red X’s
indicate DNA samples that were omitted. Orange arrowheads indicate wells with
incorrect samples, but the sample placed there is of unknown origin. Purple
and green arrow-heads indicate cases where the sample placed in the well was
incorrect, but the DNA that was supposed to be there was not found; with the
purple cases, there was corresponding gene expression data, while for the
green cases, there was no corresponding gene expression data. Pink circles
(e.g., well D02 on plate 1631) indicate sample duplicates. Gray dots indicate
that the sample placed in the well cannot be verified, as there was no
corresponding gene expression data. Gray circles indicate controls or unused
wells.
While there were many long-range sample swaps, particularly for samples
belonging in the eleventh column of plate 1629, the bulk of the errors
occurred on plates 1632 and 1630, with a long series of off-by-one and off-by-
two errors indicative of single-channel pipetting mistakes.
Let us describe a small portion of the further errors. On plate 1632, the
sample belonging in well E07 was placed in the correct well but was also
placed in the well below, F07. The sample belonging in well F07 was not found
but had no corresponding gene expression data. The sample placed in well G07
was incorrect but had no corresponding gene expression data, and so presumably
corresponds to that which should have been in the well above, F07. The sample
belonging in well G07 was placed one below, H07. There are then a series of
off-by-one errors, except that the sample belonging in well C09 was actually
placed in well G01, while the sample belonging in well D09 was placed in both
well E01 and on plate 1629 (well C11).
Of the 554 DNA samples that were genotyped, 10 were omitted due to poorly
behaved genotypes (including a pair of replicates), 435 were found to be
correctly labeled, and 8 were possibly correct but could not be verified due
to lack of gene expression assays. However, 5 samples were duplicates of other
samples, 84 were incorrectly labeled but the correct label could be assigned,
and 12 were incorrectly labeled and the correct label could not be identified.
Thus, at least 18% of the samples were involved in sample mix-ups.
We had initially become suspicious of possible sample mix-ups through the
identification of 36 mice whose X chromosome genotypes were inconsistent with
their sex. After correction of the sample mix-ups, there were no such
discrepancies. Only a small portion of the problems were identified through
such sex/genotype incompatibilities, because the majority of sample mix-ups
were off-by-one errors in the genotype plates, and the samples were arranged
on the plates so that adjacent samples were often the same sex.
The large discrepancies between expression and eQTL genotype seen in Figure 5
and Figure S13 are largely eliminated following correction of the inferred
sample mix-ups. Figure S18 shows the same examples, but with the corrected
data. Panels A-D of Figure S18 correspond to the panels in Figure S13; the
genotypes are now more clearly separated, though some overlap remains and
there are a few outliers (most notably, in Figure S18B). Panel E of Figure S18
corresponds to Figure 5; following correction of the sample mix-ups, there is
no overlap between the three genotype groups.
QTL mapping results
It should come as no surprise that the correction of the sample mix-ups,
particularly the 18% mix-ups in the DNA samples, leads to great improvement in
QTL mapping results. Figure 8 contains LOD curves for 10 week insulin level
with the original and corrected datasets. With the original data, four
chromosomes had LOD score $>$ 4; after correction of the sample mix-ups, nine
chromosomes have LOD score $>$ 4.
Figure 8: LOD curves for 10 week insulin level, before (salmon color) and
after (blue) correction of the sample mix-ups.
Two coat-related traits were recorded for the F2 mice: agouti and tufted
coats. Concerning agouti coat: BTBR mice have tan coats, while B6 mice are
black; this is due to a gene on chromosome 2, and the BTBR allele is dominant.
Mapping the agouti coat color as a binary phenotype, the LOD score on
chromosome 2 increased from 64 to 110 after correction of the sample mix-ups
(Figure S19A). While the corrected data still contained inconsistencies
between genotype and coat color, the number of inconsistencies decreased from
47 to 7 (Table S4).
Tufted coat is due to a single gene on chromosome 17, with the BTBR allele
(with the tufted phenotype) being recessive to the B6 allele (not tufted).
Mapping this phenotype as a binary trait, the LOD score on chromosome 17
increased from 64 to 107 after correction of the sample mix-ups (Figure S19B).
While, as with agouti, the corrected data still contained inconsistencies
between genotype and phenotype, the number of inconsistencies decreased from
37 to 4 (Table S5).
Finally the corrected data resulted in a great increase in the numbers of
inferred eQTL in the six tissues (Figure 9). For each array probe with know
genomic position, we performed a genome scan, including sex as an interactive
covariate (that is, allowing the QTL effect to be different in the two sexes).
For each array probe, we counted the number of chromosomes have a peak LOD
score above 5. Such a peak, on the chromosome containing the probe, was
considered a local-eQTL if the 2-LOD support interval contained the probe
location; other peaks were called _trans_ -eQTL. The inferred number of local-
eQTL increased by 7% across tissues (with a somewhat smaller increase in
hypo). The inferred number of _trans_ -eQTL increased by 37% across tissues
(though only by 8% in hypo). The modest increases in hypo were due in part to
the omission of 119 poorly behaved arrays. The increased numbers of inferred
eQTL is also seen with more stringent thresholds; the numbers of eQTL with LOD
$\geq$ 10 are shown in Figure S20.
Figure 9: Numbers of identified local- and _trans_ -eQTL with LOD $\geq$ 5,
with the original data (red) and after correction of the sample mix-ups
(blue), across 37,797 array probes with known genomic location. An eQTL was
considered local if the 2-LOD support interval contained the corresponding
probe; otherwise it was considered _trans_.
Discussion
In a mouse intercross with over 500 animals and gene expression microarray
data on six tissues, we identified and corrected sample mix-ups involving 18%
of the DNAs, along with a small number of mix-ups in each batch of expression
arrays. The QTL mapping results improved markedly following the correction of
mix-ups, but it was perhaps most surprising just how strong the results were
prior to the corrections.
To align the expression arrays, we first identified subsets of genes with
strong between-tissue correlation in expression, and then considered the
correlations between samples across these subsets of genes. To align genotypes
and expression arrays, we identified transcripts with strong local-eQTL,
formed predictors of eQTL genotype from expression values, and calculated the
proportion of matches between the observed eQTL genotypes for a DNA sample and
the predicted eQTL genotypes for an mRNA sample.
This approach applies quite generally: Whenever one has two data matrices, $X$
and $Y$, whose rows should correspond, one should check that the rows do in
fact correspond. The simplest approach is to first identify subsets of
associated columns (in which a column of $X$ is associated with a column of
$Y$) and then calculate some measure of similarity between rows of $X$ and
rows of $Y$, across that subset of columns.
Similar approaches have been described by a number of groups. Westra et al.
(2011) considered a number of public datasets and found an overall rate of 3%
sample mix-ups, with one dataset (Choy et al. 2008) having 23% mix-ups. Schadt
et al. (2012) showed that, with the tight connection between genotypes and
gene expression phenotypes, external eQTL information can, in principle, be
used to identify individuals participating in a gene expression study: Genome-
wide gene expression is just as revealing of individual identities as genome-
wide genotype data. Lynch et al. (2012) highlighted issues arising in large
tumor studies and focused particularly on a number of experimental design
issues, such as plate layout. Ekstrøm and Feenstra (2012) considered the
identification of sample mix-ups in genome-wide association studies, focusing
on a small number of phenotypes, such as blood group data, with strong
genotype-phenotype associations. Also relevant is the forensic bioinformatics
work of Baggerly and Coombes (2008, 2009), particularly their efforts to
correct mix-ups in data files. Finally, Jun et al. (2012) recently described
methods for detecting mixtures in DNA samples based on genotype or sequencing
data, and there is considerable work on detecting mislabeled microarrays
(e.g., Zhang et al. 2009; Bootkrajang and Kabán 2013).
There are a number of opportunities for improvement in our approach. In
particular, a number of critical parameters (such as the LOD score for
choosing eQTL, and the number of nearest neighbors and the minimum vote in the
$k$-nearest neighbor classifier) were chosen in an _ad hoc_ way. The choice of
such parameters influences the variation within and the separation between the
self-self and self-nonself distributions of similarity measures, and thus our
ability to identify errors. In addition, other classification methods might be
used, though the $k$-nearest neighbor classifier has an important advantage:
It works well even in the presence of mis-classification error in the
“training” data.
Perhaps the most important lesson from this work is the value of investigating
aberrations. One should follow up any observed inconsistencies in data, to
identify the source. In particular, one should not rely solely on LOD scores
or other summary statistics, but also inspect plots of genotype versus
phenotype, such as that in Figure 5.
Of course, there are many possible errors that we couldn’t see by these
approaches. For example, all of the tissues (including the DNA) for a pair of
animals might be swapped, or there may be mix-ups within the clinical
phenotypes (such as plasma insulin levels). And some mix-ups are detectable
but not correctable.
We have not identified any between-tissue mix-ups in the expression data, but
such errors are possible. For that type of error, it may be useful to consider
the gene expression bar code developed by Zilliox and Irizarry (2007).
The correction of inferred sample mix-ups, as we have done, may introduce bias
towards larger estimated eQTL effects. We believe that, in the current study,
there is little risk of such bias, as the data provide strong evidence for
specific sample labels. If the correction of sample mix-ups were accompanied
by a higher level of uncertainty, one might consider omitting samples rather
than assigning the inferred labels, though such an approach could also incur
some bias.
Finally, one might ask, following these findings: What is an acceptable error
rate in a research study? And what laboratory procedures should be instituted
to avoid such errors? There exist procedures to help protect against errors,
both for genotypes (e.g., Huijsmans et al. 2007a, b) and for microarrays
(Grant et al. 2003; Imbeaud and Auffray 2005; Walter et al. 2010), but they
are not always put into practice. However, as the current study indicates,
with expression genetic data, one can accommodate a high rate of errors
provided that one applies appropriate procedures to detect and correct such
errors.
Acknowledgments
The authors thank Angie Oler, Mary Rabaglia, Kathryn Schueler, and Donald
Stapleton for their work on the underlying project, and Amit Kulkarni for
providing annotation information for the expression microarrays. This work was
supported in part by National Institutes of Health grants GM074244 (to K.W.B.)
and DK066369 (to A.D.A).
## Literature Cited
* Baggerly and Coombes (2008) Baggerly, K. A., and K. R. Coombes, 2008 Run batch effects potentially compromise the usefulness of genomic signatures for ovarian cancer. J. Clin. Oncol. 26: 1186–1187.
* Baggerly and Coombes (2009) Baggerly, K. A., and K. R. Coombes, 2009 Deriving chemosensitivity from cell lines: Forensic bioinformatics and reproducible research in high-throughput biology. Ann. Appl. Stat. 3: 1309–1334.
* Bootkrajang and Kabán (2013) Bootkrajang, J., and A. Kabán, 2013 Classification of mislabelled microarrays using robust sparse logistic regression. Bioinformatics 29: 870–877.
* Broman (2003) Broman, K. W., 2003 Mapping quantitative trait loci in the case of a spike in the phenotype distribution. Genetics 163: 1169–1175.
* Broman et al. (2003) Broman, K. W., H. Wu, S. Sen, and G. A. Churchill, 2003 R/qtl: QTL mapping in experimental crosses. Bioinformatics 19: 889–890.
* Carter and Falconer (1951) Carter, T. C., and D. S. Falconer, 1951 Stocks for detecting linkage in the mouse, and the theory of their design. J. Genet. 50: 307–323.
* Choy et al. (2008) Choy, E., R. Yelensky, S. Bonakdar, R. M. Plenge, R. Saxena et al., 2008 Genetic analysis of human traits in vitro: drug response and gene expression in lymphoblastoid cell lines. PLoS Genet. 4: e1000287.
* Ekstrøm and Feenstra (2012) Ekstrøm, C. T., and B. Feenstra, 2012 Detecting sample misidentifications in genetic association studies. Stat. Appl. Genet. Mol. Biol. 11: Article 13.
* Grant et al. (2003) Grant, G. R., E. Manduchi, A. Pizarro, and C. J. Stoeckert, 2003 Maintaining data integrity in microarray data management. Biotechnol. Bioeng. 84: 795–800.
* Haley and Knott (1992) Haley, C. S., and S. A. Knott, 1992 A simple regression method for mapping quantitative trait loci in line crosses using flanking markers. Heredity 69: 315–324.
* Huijsmans et al. (2007a) Huijsmans, C. J. J., F. G. C. Heilmann, A. G. M. van der Zanden, P. M. Schneeberger, and M. H. A. Hermans, 2007a Single nucleotide polymorphism profiling assay to exclude serum sample mix-up. Vox Sang. 92: 148–153.
* Huijsmans et al. (2007b) Huijsmans, R., J. Damen, H. van der Linden, and M. Hermans, 2007b Single nucleotide polymorphism profiling assay to confirm the identity of human tissues. J. Mol. Diagn. 9: 205–213.
* Imbeaud and Auffray (2005) Imbeaud, S., and C. Auffray, 2005 ‘The 39 steps’ in gene expression profiling: critical issues and proposed best practices for microarray experiments. Drug Discov. Today 10: 1175–1182.
* Jun et al. (2012) Jun, G., M. Flickinger, K. N. Hetrick, J. M. Romm, K. F. Doheny et al., 2012 Detecting and estimating contamination of human DNA samples in sequencing and array-based genotype data. Am. J. Hum. Genet. 91: 839–848.
* Keller et al. (2008) Keller, M. P., Y. Choi, P. Wang, D. B. Davis, M. E. Rabaglia et al., 2008 A gene expression network model of type 2 diabetes links cell cycle regulation in islets with diabetes susceptibility. Genome Res. 18: 706–716.
* Lander and Botstein (1989) Lander, E. S., and D. Botstein, 1989 Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps. Genetics 121: 185–199.
* Lynch et al. (2012) Lynch, A. G., S.-F. Chin, M. J. Dunning, C. Caldas, S. Tavaré et al., 2012 Calling sample mix-ups in cancer population studies. PLoS ONE 7: e41815.
* R Development Core Team (2013) R Development Core Team, 2013 R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
* Schadt et al. (2012) Schadt, E. E., S. Woo, and K. Hao, 2012 Bayesian method to predict individual SNP genotypes from gene expression data. Nat. Genet. 44: 603–608.
* Walter et al. (2010) Walter, M., A. Honegger, R. Schweizer, S. Poths, and M. Bonin, 2010 Utilization of AFFX spike-in control probes to monitor sample identity throughout Affymetrix GeneChip Array processing. BioTechniques 48: 371–378.
* Westra et al. (2011) Westra, H.-J., R. C. Jansen, R. S. N. Fehrmann, G. J. te Meerman, D. van Heel et al., 2011 MixupMapper: correcting sample mix-ups in genome-wide datasets increases power to detect small genetic effects. Bioinformatics 27: 2104–2111.
* Xu and Atchley (1996) Xu, S., and W. R. Atchley, 1996 Mapping quantitative trait loci for complex binary diseases using line crosses. Genetics 143: 1417–1424.
* Zhang et al. (2009) Zhang, C., C. Wu, E. Blanzieri, Y. Zhou, Y. Wang et al., 2009 Methods for labeling error detection in microarrays based on the effect of data perturbation on the regression model. Bioinformatics 25: 2708–2714.
* Zilliox and Irizarry (2007) Zilliox, M. J., and R. A. Irizarry, 2007 A gene expression bar code for microarray data. Nat. Methods 4: 911–913.
Identification and correction of sample mix-ups
in expression genetic data: A case study
SUPPLEMENT
Karl W. Broman∗, Mark P. Keller†, Aimee Teo Broman∗,
Christina Kendziorski∗, Brian S Yandell‡,§, Śaunak Sen∗∗, Alan D. Attie†
∗Department of Biostatistics and Medical Informatics, †Department of
Biochemistry, ‡Department of Statistics, and §Department of Horticulture,
University of Wisconsin–Madison, Madison, Wisconsin 53706, and ∗∗Department of
Epidemiology and Biostatistics, University of California, San Francisco,
California 94107
Figure S1: The behavior of the X chromosome in the intercross (BTBR $\times$
B6) $\times$ (BTBR $\times$ B6). In the F2 generation, females are homozygous
BTBR or heterozygous, while males are hemizygous BTBR or B6. The small bar is
the Y chromosome.
Figure S2: X chromosome genotypes for 19 female mice and 17 male mice with
genotypes that are incompatible with their sex. Females should be homozygous
BTBR (RR, blue) or heterozygous (green). Males should be hemizygous B6 (BY,
yellow) or hemizygous BTBR (RY, blue). The top four males have a single
incompatibility that could reasonably be a genotyping error.
Figure S3: Example scatterplot of gene expression in liver versus kidney for a
single individual (Mouse3567). Gray points are all probes on the array; red
points are the 155 probes with correlation across mice $>$ 0.75 between liver
and kidney.
Figure S4: Density estimates of the between-tissue correlations for all probes
on the expression arrays. In each panel, the distributions for the five pairs
of tissues, including a given tissue, are displayed.
Figure S5: Histograms of similarity measures for the expression arrays for
each tissue, versus all other tissues combined. The panels on the left include
self-self similarities (along the diagonal of the similarity matrices); the
panels on the right include all self-nonself similarities (the off-diagonal
elements of the similarity matrices). Self-self values $<$ 0.8 and self-
nonself values $>$ 0.8 are highlighted with red tick marks. The two modes in
the self-nonself distributions are for opposite-sex and same-sex pairs.
Figure S6: Second highest similarity (median correlation across tissue pairs)
versus maximum similarity for the expression arrays for each tissue. The
diagonal gray line corresponds to equality. Green points correspond to arrays
inferred to be sample mix-ups. Gray points correspond to arrays for which the
self similarity is maximal. Red points correspond to special cases, as in
Figure 1 (see the text).
Figure S7: Scatterplots for expression in pairs of tissues for an inferred
sample swap, between Mouse3655 and Mouse3659 in gastroc.
Figure S8: Scatterplots for expression in pairs of tissues for an inferred
sample duplicate, with Mouse3136 in liver also arrayed as Mouse3141 liver. In
the bottom row, the panels with gray points are identical data, and the panels
with red points are the unintended duplicates.
Figure S9: Between-tissue correlations for pairs of tissues for an inferred
sample duplicate, with Mouse3141 in liver really being a duplicate of
Mouse3136 in liver. Correlations are calculated using tissue-pair-specific
probes that show between-tissue correlation, across all mice, of $>$ 0.75.
Tissue pairs are abbreviated by the first letter of the tissues’ names. Red
points involve Mouse3136 liver, green points involve Mouse3141 liver, and the
purple point involves both.
Figure S10: Scatterplots for expression in pairs of tissues for a potential
sample mixture, of Mouse3484 and Mouse3503 in kidney. In the second from the
bottom row, the panels with gray points are identical data, and the panels
with red points are the unintended duplicates.
Figure S11: Between-tissue correlations for pairs of tissues for a potential
sample mixture, of Mouse3484 and Mouse3503 in kidney. Correlations are
calculated using tissue-pair-specific probes that show between-tissue
correlation, across all mice, of $>$ 0.75. Tissue pairs are abbreviated by the
first letter of the tissues’ names. Red points involve Mouse3484 kidney, green
points involve Mouse3503 kidney, and the purple point involves both.
Figure S12: Scatterplots of the average expression for four Y chromosome genes
versus expression of the _Xist_ gene in each tissue, before and after
correction of sample mix-ups. Females are in red; males are in blue. The
unusual pattern in hypothalamus is due to a batch of 120 poorly behaved
arrays.
Figure S13: Example scatterplots of islet expression for pairs of probes at
the same genomic location.
Figure S14: Positions of local eQTL used for the aligning the expression
arrays and genotype data. Marker locations are indicated by horizontal line
segments on the genetic map. The points to the right of each chromosome
indicate the eQTL locations, with different colors for different tissues.
Figure S15: Histograms of similarities between the genotypes and the
expression arrays (the proportion of matches between observed and inferred
eQTL genotypes) for each tissue. The panels on the left include self-self
similarities (along the diagonal of the similarity matrices); the panels on
the right include all self-nonself similarities (the off-diagonal elements of
the similarity matrices). Self-self values $<$ 0.8 and self-nonself values $>$
0.8 are highlighted with red tick marks.
Figure S16: Self similarity (proportion matches between observed and inferred
eQTL genotypes, considering each tissue separately) versus maximum similarity
for the DNA samples. The diagonal gray line corresponds to equality. Samples
with missing self similarity (at top) did not have an expression assay
performed for that tissue. Points are colored based on the inferred status of
the corresponding samples based on the combined information from all tissues.
Gray points correspond to DNA samples that were correctly labeled. Green
points correspond to sample mix-ups that are fixable (the correct label can be
determined). Red points comprise both samples mix-ups that cannot be corrected
as well as samples that may be correct but cannot be verified as no expression
data is available.
Figure S17: Second highest similarity (proportion matches between observed and
inferred eQTL genotypes, combined across tissues) versus maximum similarity
for the DNA samples. The diagonal gray line corresponds to equality. Gray
points correspond to DNA samples that were correctly labeled. Green points
correspond to sample mix-ups that are fixable (the correct label can be
determined). Red points comprise both samples mix-ups that cannot be corrected
as well as samples that may be correct but cannot be verified as no expression
data is available.
Figure S18: Panels A-D contain the example scatterplots of islet expression
for pairs of probes at the same genomic location, as in Figure S13, following
correction of the sample mix-ups. Panel E contains the plot of islet
expression vs observed genotype for an example probe, as in Figure 5,
following correction of the sample mix-ups.
Figure S19: LOD curves for agouti (A) and tufted (B) coat traits with the
original data (red) and after correction of the sample mix-ups (blue).
Figure S20: Numbers of identified local- and _trans_ -eQTL with LOD $\geq$
10, with the original data (red) and after correction of the sample mix-ups
(blue), across 37,797 array probes with known genomic location. An eQTL was
considered local if the 2-LOD support interval contained the corresponding
probe; otherwise it was considered _trans_.
Table S1: Duplicate DNA samples Mouse 1 | Mouse 2 | No. matches | No. typed markers | % mismatches
---|---|---|---|---
Mouse3259 | Mouse3269 | 2017 | 2022 | 0.2
Mouse3267 | Mouse3362 | 1933 | 1966 | 1.7
Mouse3287 | Mouse3290 | 2012 | 2016 | 0.2
Mouse3317 | Mouse3318 | 1964 | 1996 | 1.6
Mouse3353 | Mouse3354 | 2026 | 2031 | 0.2
Mouse3553 | Mouse3559 | 1998 | 2008 | 0.5
Table S2: Numbers of gene expression arrays Tissue | # arrays | # omitted | # kept
---|---|---|---
adipose | 497 | 4 | 493
gastroc | 498 | 2 | 496
hypo | 494 | 1 | 493
islet | 499 | 1 | 498
kidney | 482 | 1 | 481
liver | 491 | 1 | 490
Table S3: Numbers of probes, for each tissue pair, with large between-tissue correlation Tissue 1 | Tissue 2 | corr $>$ 0.70 | corr $>$ 0.75 | corr $>$ 0.80 | corr $>$ 0.90
---|---|---|---|---|---
adipose | gastroc | 199 | 143 | 99 | 30
adipose | hypo | 110 | 72 | 50 | 7
adipose | islet | 216 | 159 | 106 | 38
adipose | kidney | 255 | 186 | 135 | 51
adipose | liver | 159 | 113 | 79 | 19
gastroc | hypo | 79 | 55 | 43 | 10
gastroc | islet | 180 | 132 | 92 | 33
gastroc | kidney | 219 | 164 | 109 | 43
gastroc | liver | 149 | 102 | 71 | 23
hypo | islet | 127 | 82 | 57 | 10
hypo | kidney | 131 | 92 | 60 | 17
hypo | liver | 63 | 46 | 33 | 6
islet | kidney | 269 | 200 | 146 | 42
islet | liver | 152 | 97 | 64 | 24
kidney | liver | 245 | 155 | 106 | 30
Table S4: Genotype versus phenotype at the agouti locus
| | Original | | Corrected
---|---|---|---|---
| | Coat color | | Coat color
Chr 2 genotype | | Tan | Black | | Tan | Black
BB | | 26 | 114 | | 5 | 126
BR | | 249 | 15 | | 255 | 2
RR | | 88 | 6 | | 92 | 0
B = B6 allele; R = BTBR allele
Table S5: Genotype versus phenotype at the tufted locus
| | Original | | Corrected
---|---|---|---|---
| | Tufted coat | | Tufted coat
Chr 17 genotype | | No | Yes | | No | Yes
BB | | 151 | 7 | | 153 | 0
BR | | 258 | 9 | | 256 | 0
RR | | 21 | 92 | | 4 | 106
B = B6 allele; R = BTBR allele
|
arxiv-papers
| 2014-02-11T20:25:43 |
2024-09-04T02:49:58.105336
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Karl W. Broman, Mark P. Keller, Aimee Teo Broman, Christina\n Kendziorski, Brian S. Yandell, Saunak Sen, Alan D. Attie",
"submitter": "Karl Broman",
"url": "https://arxiv.org/abs/1402.2633"
}
|
1402.2660
|
# An isometrically universal Banach space with a monotone Schauder basis
Joanna Garbulińska-Wȩgrzyn
Institute of Mathematics, Jan Kochanowski University (POLAND)
Faculty of Mathematics and Computer Science, Jagiellonian University (POLAND)
[email protected]
###### Abstract
We present an isometric version of the complementably universal Banach space
$\mathcal{B}$ with a monotone Schauder basis. The space $\mathcal{B}$ is
isomorphic to Pełczyński’s space with a universal basis as well as to Kadec’
complementably universal space with the bounded approximation property.
MSC (2010) Primary: 46B04. Secondary: 46M15, 46M40.
Keywords: Monotone Schauder basis, linear isometry.
## 1 Introduction
A Banach space $X$ is _complementably universal_ for a given class of spaces
if every space from the class is isomorphic to a complemented subspace of $X$.
In 1969 Pełczyński [10] constructed a complementably universal Banach space
with a Schauder basis. In 1971 Kadec [4] constructed a complementably
universal Banach space for the class of spaces with the _bounded approximation
property_ (BAP). In the same year Pełczyński [8] showed that every Banach
space with BAP is complemented in a space with a basis. Pełczyński &
Wojtaszczyk [11] constructed in 1971 a universal Banach space for the class of
spaces with a finite-dimensional decomposition. Applying Pełczyński’s
decomposition argument [9], one immediately concludes that all three spaces
are isomorphic. It is worth mentioning a negative result of Johnson &
Szankowski [3] saying that no separable Banach space can be complementably
universal for the class of all separable spaces. The author in [2] presented a
natural extension property that describes an isometric version of the Kadec-
Pełczyński-Wojtaszczyk space. The constructed space is unique, up to isometry,
for the class of Banach spaces with finite-dimensional decomposition and
isomorphic to the Kadec-Pełczyński-Wojtaszczyk space.
In this note we present an isometric version of the complementably universal
Banach space with a monotone Schauder basis. Most of the arguments are
inspired by the recent works [2], [6] and [7].
## 2 Preliminaries
A _projectional resolution of the identity_ (briefly: _PRI_) on a Banach space
$X$ is a sequence of norm-one projections $\\{P_{n}\\}_{n\in\omega}$ of $X$
satysfying following conditions:
1. (1)
$P_{n}\circ P_{m}=P_{min\\{n,m\\}}=P_{m}\circ P_{n}$ for every $n,m\in\omega$;
2. (2)
$\operatorname{dim}(P_{n}[X])=n$;
3. (3)
$X=\operatorname{cl}\bigcup_{n\in\omega}P_{n}[X]$.
A _Schauder basis_ is a sequence $\\{e_{n}\\}_{n\in\omega}$ of vectors in a
Banach space $X$ such that for every $x\in X$ there are uniquely determined
scalars $\\{a_{n}\\}_{n\in\omega}$ such that
$x=\sum_{n=0}^{\infty}a_{n}e_{n},$
where the convergence of the series is taken with respect to the norm. Once
this happen, for each $n\in\omega$ there is a cannonical projection $P_{n}$
defined by
$P_{n}(\sum_{i\in\omega}a_{i}e_{i})=\sum_{i<n}a_{i}e_{i}.$
A Banach space $X$ has a _monotone Schauder basis_ if and only if it has a PRI
$\\{P_{n}\\}_{n\in\omega}$.
On the other hand, the basis is _monotone_ if $\|P_{n}\|\leq 1$ for every
$n\in\omega$.
Recall that a Banach space $X$ is _1-complemented_ in $Y$ if there exists a
projection $P:Y\to X$ such that $\|P\|\leq 1$ and $P[Y]=X$.
Given Banach spaces $Y\subseteq X$, we say that $Y$ is an _initial subspace_
of $X$ if there is a sequence of norm-one projections
$\\{P_{n}\\}_{n\in\omega}$ satysfying conditions (1), (3) and
1. $1^{\circ}$
for each $n\in\omega$ the image $P_{n+1}-P_{n}$ is 1-dimensional,
2. $2^{\circ}$
$X=P_{0}[Y]$.
Typical examples of initial subspaces are linear spans of initial parts of a
Schauder basis. Note that, an initial subspace is 1-complemented and the
trivial space is initial in $Y$ if and only if $Y$ has a monotone Schauder
basis.
Given a Schauder basis $\\{e_{n}\\}_{n\in\omega}$ in $X$, given a subset
$S\subset\omega$, we say that $\\{e_{n}\\}_{n\in S}$ is a _cannonically
1-complemented subbasis_ if the linear operator $P_{S}:X\to X$ defined by
conditions $P_{S}e_{n}=e_{n}$ for $n\in S$, $P_{S}e_{n}=0$ for $n\notin S$,
has norm $\leq$ 1.
Finally, we say that a basis $\\{v_{n}\\}_{n\in\omega}$ is _isometric_ to a
subbasis of $\\{e_{m}\\}_{m\in\omega}$ if there is an increasing function
$\varphi:S\to\omega$ such that the linear operator $f$ defined by equations
$f(v_{n})=e_{\varphi(n)}$ ($S\subseteq\omega$) is a linear isometric
embedding.
Every finite-dimensional Banach space $E$ is isometric to $\mathbb{R}^{n}$
with some norm $\|\cdot\|$. We shall say that $E$ is _rational_ if
$E=\mathbb{R}^{n}$ with a norm such that its unit ball is a polyhedron spanned
by finitely many vectors whose every coordinate is a rational number.
Equivalently, $X$ is rational if, up to isometry, $X=\mathbb{R}^{n}$ with a
“maximum norm" $\|\cdot\|$ induced by finitely many functionals
$\varphi_{0},\dots,\varphi_{m-1}$ such that
$\varphi_{i}[\mathbb{Q}^{n}]\subseteq\mathbb{Q}$ for every $i<m$. More
precisely,
$\|x\|=\max_{i<m}{\mid\varphi_{i}(x)\mid}$
for $x\in\mathbb{R}^{n}$. Typical examples of rational Banach spaces are
$\ell_{1}(n)$ and $\ell_{\infty}(n)$, the $n$-dimensional variants of
$\ell_{1}$ and $\ell_{\infty}$, respectively. On the other hand, for
$1<p<\infty$, $n>1$, the spaces $\ell_{p}(n)$ are not rational. Of course,
every rational Banach space is polyhedral. An operator
$T:{\mathbb{R}^{n}\to\mathbb{R}^{m}}$ is _rational_ if
$T[\mathbb{Q}^{n}]\subseteq\mathbb{Q}^{m}$.
It is clear that there are (up to isometry) only countably many rational
Banach spaces and for every $\varepsilon>0$, every finite-dimensional space
has an $\varepsilon$-isometry onto some rational Banach space.
Let $X$ is a set. A convex hull is the minimal convex set containing $X$.
Let $\mathfrak{K}$ be a category and $A,B\in\mathfrak{K}$. By
$\mathfrak{K}(A,B)$ we will denote the set of all $\mathfrak{K}$-morphisms
from $A$ to $B$. A _subcategory_ of $\mathfrak{K}$ is a category
$\mathfrak{L}$ such that each object of $\mathfrak{L}$ is an object of
$\mathfrak{K}$ and each arrow of $\mathfrak{L}$ is an arrow of $\mathfrak{K}$.
Category $\mathfrak{L}$ is _cofinal_ in $\mathfrak{K}$ if for every
$A\in\mathfrak{K}$ there exist an object $B\in\mathfrak{L}$ such that the set
$\mathfrak{K}(A,B)$ is nonempty. Let $\mathfrak{K}$ be a category.
$\mathfrak{K}$ has the amalgamation property if for every objects
$A,B,C\in\mathfrak{K}$ and for every morphisms $f\in\mathfrak{K}(A,B)$,
$g\in\mathfrak{K}(A,C)$ we can find object $D\in\mathfrak{K}$ and morphisms
$f^{\prime}\in\mathfrak{K}(B,D)$, $g^{\prime}\in\mathfrak{K}(C,D)$ such that
$f^{\prime}\circ f=g^{\prime}\circ g$.
Category $\mathfrak{K}$ has the _joint embedding property_ if for every
objects $A,B\in\mathfrak{K}$ we can find some object $C\in\mathfrak{K}$ such
that there exist morphisms $f\in\mathfrak{K}(A,C)$, $g\in\mathfrak{K}(B,C)$.
## 3 The Amalgamation
###### Lemma 1.
(Amalgamation Lemma) Let $Z$, $X$, $Y$ be finite-dimensional Banach spaces,
such that $i:Z\to X$, $j:Z\to Y$ are isometric embeddings and $\\{0\\}$,
$i[Z]$, $j[Z]$ are initial subspaces of $Z$, $X$, $Y$, respectively. Then
there exists a finite-dimensional Banach space $W$, isometric embeddings
$i^{\prime}:X\to W$, $j^{\prime}:Y\to W$ such that $i^{\prime}[X]$,
$j^{\prime}[Y]$ are initaial subspaces of $W$ and the following commutes:
$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime}}$$\textstyle{W}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{j}$$\textstyle{X.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$
###### Proof.
Let $Z$, $X$, $Y$ be finite-dimensional Banach spaces
($\operatorname{dim}(Z)=N$, $\operatorname{dim}(X)=M$,
$\operatorname{dim}(Y)=K$ and $N\leq M,K$), where $\\{0\\}$ is an initial
subspace $Z$, $i[Z]$ is an initial subspace of $X$ and $j[Z]$ is an initial
subspace of $Y$, with sequences of norm-one projections $\\{P_{n}\\}_{n\leq
N}$, $\\{Q_{n}\\}_{n\leq M-N}$, $\\{R_{n}\\}_{n\leq K-N}$, respectively.
Observe that $Q_{0}[X]:=i(Z)=Z$ and $R_{0}[Y]:=j(Z)=Z$ (we assume that $i,j$
are inclusions).
We define $W$ as a $(X\oplus Y)/\Delta$, where $\Delta=\\{(z,-z):z\in Z\\}$.
Given $(x,y)\in X\oplus Y$, define a norm $\|(x,y)\|=\|x\|_{X}+\|y\|_{Y}$. Let
$i^{\prime}_{X}(x)=(x,0)+\Delta$ and $j^{\prime}_{Y}(y)=(0,y)+\Delta.$ Then
$i^{\prime},j^{\prime}$ are isometric embeddings (see [2] or [6]).
We have to show that $i^{\prime}[X],j^{\prime}[Y]$ are initial subspaces of
$W$ (observe that $\operatorname{dim}(W)=M+K-N$). We will definie sequences of
projections $\\{S_{n}\\}_{n\leq M}$, $\\{T_{n}\\}_{n\leq K}$ such that
$S_{0}:=i^{\prime}\circ Q_{M-N}=(Q_{M-N},0)+\Delta$ and
$T_{0}:=j^{\prime}\circ R_{K-N}=(0,R_{K-N})+\Delta$. Note that
$Q_{0}[X]=R_{0}[Y]$. Define $S_{n}(x,y):=(Q_{M-N}(x),R_{n}(y))+\Delta$ for
$n\geq 1$. Similarly, $T_{n}(x,y):=(Q_{n}(x),R_{K-N}(y))+\Delta$ for $n\geq
1$.
It is easy to show that $\|S_{n}\|\leq 1$ and $\|T_{n}\|\leq 1$. Observe that:
1. 1.
$\operatorname{dist}((Q_{M-N}(x),R_{n}(y))+\Delta)\leq\|(Q_{M-N}(x),R_{n}(y))\|=\|Q_{M-N}(x)\|_{X}+\|R_{n}(y)\|_{Y}\leq\|x\|_{X}+\|y\|_{Y}$;
2. 2.
$\operatorname{dist}((Q_{n}(x),R_{K-N}(y))+\Delta)\leq\|(Q_{n}(x),R_{K-N}(y))\|=\|Q_{n}(x)\|_{X}+\|R_{K-N}(y)\|_{Y}\leq\|x\|_{X}+\|y\|_{Y}$.
This completes the proof ∎
Fix $\varepsilon>0$ and fix a linear operator $f:X\to Y$ such that
$(1+\varepsilon)^{-1}\cdot\|x\|\leq\|f(x)\|\leq\|x\|$
for $x\in X$ and $f[X]$ is an initial subspace of $Y$. Consider the following
category $\mathfrak{K}_{f}^{\varepsilon}$. The objects of
$\mathfrak{K}_{f}^{\varepsilon}$ are pairs $(i,j)$ of linear operators $i:X\to
Z$, $j:Y\to Z$ between Banach spaces with initial subspaces such that
1. 1.
$i[X]$, $j[Y]$ are initial subspaces of $Z$;
2. 2.
$\|i\|\leq 1$ and $\|j\|\leq 1$;
3. 3.
$\|i(x)-j(f(x))\|\leq\varepsilon\cdot\|x\|$ for $x\in X$.
Given $a_{0}=(i_{0},j_{0})$ and $b_{0}=(i_{1},j_{1})$ in
$\mathfrak{K}_{f}^{\varepsilon}$, where $i_{k}:X\to{Z_{k}}$ and
$j_{k}:Y\to{Z_{k}}$ for $k<2$, an arrow from $a_{0}$ to $b_{0}$ is defined to
be a linear operator $T:{Z_{0}}\to{Z_{1}}$ such that $\|T\|\leq 1$, $T\circ
i_{0}=i_{1}$, and $T\circ j_{0}=j_{1}$.
###### Lemma 2.
The category $\mathfrak{K}_{f}^{\varepsilon}$ has an initial object
$(i_{0},j_{0})$ such that both $i_{0}$, $j_{0}$ are canonical isometric
embeddings into $X\oplus Y$ with a suitable norm $\|\cdot\|$ and $X\oplus Y$
has initial subspaces $X$, $Y$.
###### Proof.
Define
$G=\\{(x,-f(x))\in X\times Y:x\in\varepsilon^{-1}B_{X}\\}.$
Recall that $B_{X}$ and $B_{Y}$ are the unit balls of $X$ and $Y$
respectively. Finally, let $K$ be the convex hull of
$G\cup(B_{X}\times\\{0\\})\cup(\\{0\\}\times B_{Y})$ and let
$\|(x,y)\|_{K}=\inf\\{\|u\|_{X}+\|v\|_{Y}+\varepsilon\|w\|_{X}:(x,y)=(u,0)+(0,v)+(w,-f(w)),(x,y)\in
K\\}$
on $K$. We claim that $\|\cdot\|_{K}$ is as required.
Define linear operators $i_{0}(x)=(x,0)$ and $j_{0}(y)=(0,y)$. We check that
$\|i_{0}(x)-j_{0}(f(x))\|_{K}\leq~{}\varepsilon\|x\|_{X}$.
We have that $\|(x,-f(x))\|_{K}\leq\varepsilon\|x\|_{X}$. This implies that
$\|i_{0}(x)-j_{0}(f(x))\|_{K}=\|(x,-f(x))\|_{K}\leq\varepsilon\|x\|_{X}.$
This proves that $(i_{0},j_{0})$ is an object of the category
$\mathfrak{K}_{f}^{\varepsilon}$.
Let $(w,-f(w))\in G$, $(u,0)\in B_{X}\times\\{0\\}$, $(0,v)\in\\{0\\}\times
B_{Y}$ and let $\|u\|\leq 1$, $\|v\|\leq 1$, $\|w\|\leq\varepsilon^{-1}$. Then
$(x,y)\in K$ is a linear combination:
$\displaystyle(x,y)$ $\displaystyle=(u,0)+(0,v)+(w,-f(w))=(u+w,v-f(w)).$
Now we prove that $\|(x,0)\|_{K}=\|x\|_{X}$ and $\|(0,y)\|_{K}=\|y\|_{Y}$.
Suppose that $\|x\|=1$ and $y=0$; then $v=f(w)$.
It is easy to show that $\|(x,0)\|_{K}\leq\|x\|_{X}$ (we take $u=x$, then
$w=0$ and $v=f(w)=0$). Observe that
$\displaystyle\|u\|_{X}+\|v\|_{Y}+\varepsilon\|w\|_{X}=\|u\|_{X}+\|f(w)\|_{Y}+\varepsilon\|w\|_{X}\geq$
$\displaystyle\geq\|u\|_{X}+(1-\varepsilon)\|w\|_{X}+\varepsilon\|w\|_{X}=\|u\|_{X}+\|w\|_{X}\geq\|x\|_{X}.$
Suppose that $x=0$ and $\|y\|=1$, then $u=-w$. It is obvious that
$\|(0,y)\|_{K}\leq\|y\|_{Y}$ (we take $v=y$, then $f(w)=0$ and $u=-w=0$). On
the other hand
$\displaystyle\|u\|_{X}+\|v\|_{Y}+\varepsilon\|w\|_{X}=\|w\|_{X}+\|v\|_{Y}+\varepsilon\|w\|_{X}=$
$\displaystyle=(1+\varepsilon)\|w\|_{X}+\|v\|_{Y}\geq(1+\varepsilon)\|w\|_{X}+\|v\|_{Y}\geq\|f(w)\|_{Y}+\|v\|_{Y}\geq\|y\|_{Y}.$
This proves that $i_{0}$, $j_{0}$ are isometric embeddings.
We have to check that the convex hull $K$ is a unit ball of the norm
$\|\cdot\|_{K}$. Let
$B_{K}=\\{(x,y):\|(x,y)\|_{K}\leq 1\\}.$
The inclusion $K\subseteq B_{K}$ is obvious. To prove that $B_{K}\subseteq K$,
fix $(x,y)$ such that $\|(x,y)\|_{K}<~{}1$. Then
$\|u\|_{X}+\|v\|_{Y}+\varepsilon\|w\|_{X}<1$
for some $u,v,w$ such that $x=u+w$ and $y=v-f(w)$. Observe that $(u,0)\in
B_{X}\times\\{0\\}$, $(0,v)\in\\{0\\}\times B_{Y}$, $(w,-f(w))\in G$ and
$(u,0)+(0,v)+(w,-f(w))=(x,y)$.
Denote $\alpha=\|u\|_{X}+\|v\|_{Y}+\varepsilon\|w\|_{X}$, then $\alpha<1$. Let
$\lambda_{1}=\frac{\|u\|_{X}}{\alpha}$, $\lambda_{2}=\frac{\|v\|_{Y}}{\alpha}$
and $\lambda_{3}=\frac{\varepsilon\|w\|_{X}}{\alpha}$. Then
$(u,0)=\lambda_{1}(\frac{\alpha}{\|u\|_{X}}(u,0))\in B_{X}\times\\{0\\}$,
$(0,v)=\lambda_{2}(\frac{\alpha}{\|v\|_{Y}}(0,v))\in\\{0\\}\times B_{Y}$ and
$(w,-f(w))=\lambda_{3}(\frac{\alpha}{\varepsilon\|w\|_{X}}(w,-f(w)))\in G$.
We have to check that the pair $(i_{0},j_{0})$ is an initial object. This
means that for every $(i,j)\in Obj(\mathfrak{K}_{f}^{\varepsilon})$ there
exists a unique linear operator $T:X\oplus Y\to Z$ such that $T\circ i_{0}=i$,
$T\circ j_{0}=j$ and the norm of $T$ is less or equal to 1.
Fix $(i,j)\in Obj(\mathfrak{K}_{f}^{\varepsilon})$ and define
$T(x,y)=i(x)+j(y)$. It is clear that this is the only possibility for $T$. We
will check that $\|T\|_{Z}\leq 1$.
Let $(x,y)\in K$, then
1. $1^{\circ}$
if $(x,y)\in B_{X}\times\\{0\\}$ then $\|T(x,0)\|_{Z}=\|i(x)\|_{Z}\leq 1$,
2. $2^{\circ}$
if $(x,y)\in\\{0\\}\times B_{Y}$ then $\|T(0,y)\|_{Z}=\|j(y)\|_{Z}\leq 1$,
3. $3^{\circ}$
if $(x,y)\in G$ then
$\|T(x,-f(x))\|_{Z}=\|i(x)-j(f(x))\|_{Z}\leq\varepsilon\cdot\|x\|_{Z}\leq\varepsilon\cdot\varepsilon^{-1}=1$.
Let $X$ and $Y$ be finite-dimensional Banach spaces
($\operatorname{dim}(X)=M$, $\operatorname{dim}(Y)=K$ and $M\leq K$), where
$\\{0\\}$ is an initial subspace $X$ and $f[X]$ is an initial subspace of $Y$,
with sequences of norm-one projections $\\{P_{n}\\}_{n\leq M}$ and
$\\{Q_{n}\\}_{n\leq K-M}$, respectively. Observe that $Q_{0}[Y]:=f(X)$.
We prove that $X,Y$ are initial subspaces of $X\oplus Y$
($\operatorname{dim}(X\oplus Y)=M+K$). We define sequences of projecions
$\\{R_{n}\\}_{n\leq K}$, $\\{S_{n}\\}_{n\leq M}$ such that $R_{0}:=i\circ
P_{M}$ and $S_{0}:=j\circ Q_{K-M}$.
Let $R_{n}:=i\circ P_{M}+j\circ f\circ P_{n}=(P_{M},f\circ P_{n})$ for $n\leq
M$ and $R_{M+1+n}:=i\circ P_{M}+j\circ Q_{n+1}=(P_{M},Q_{n+1})$ for ${n\leq
K-M-1}$. Similarly $S_{n}:=i\circ P_{n}+j\circ Q_{K-M}=(P_{n},Q_{K-M})$ for
$n\leq M$.
It is easy to show that $\|R_{n}\|_{K}\leq 1$ and $\|S_{n}\|_{K}\leq 1$.
Observe that if
1. $1^{\circ}$
$\|(P_{M},f\circ P_{n})\|\leq 1$ for $n\leq M$, we take $u=P_{M}$, $v=0$,
$w=0$, $f(w)=f\circ P_{n}$.
2. $2^{\circ}$
$\|(P_{M},Q_{n+1})\|\leq 1$ for ${n\leq K-M-1}$, we take $u=P_{M}$, $v=0$,
$w=0$, $f(w)=Q_{n+1}$.
3. $3^{\circ}$
$\|(P_{n},Q_{K-M})\|\leq 1$ for $n\leq M$, we take $u=P_{n}$, $v=0$, $w=0$,
$f(w)=Q_{K-M}$.
∎
First version of the proof of the lemma about extending $\varepsilon$-isometry
between Banach spaces can be found in [2] and [7].
## 4 A construction
In order to make some statements shorter, we shall consider $1$-bounded
operators, this means linear operators of norm at most 1 only.
We shall now prepare the setup for our construction.
We now define the relevant category $\mathfrak{K}$. The objects of
$\mathfrak{K}$ are finite-dimensional Banach spaces. Given finite-dimensional
spaces $X$, $Y$, an $\mathfrak{K}$-arrow is an isometric embedding $f:X\to Y$
such that $f[X]$ is an initial object of $Y$ and $\\{0\\}$ is an initial
object of $X$ (both have a monotone Schauder basis).
Denote by $\mathfrak{L}$ the subcategory of $\mathfrak{K}$ consisting of all
rational $\mathfrak{K}$-arrows. Obviously, $\mathfrak{L}$ is countable.
Looking at the proof of Lemma 1, we can see that $\mathfrak{L}$ has the
amalgamation property. We now use the concepts from [6] for constructing a
“generic" sequence in $\mathfrak{L}$. First of all, a sequence in a fixed
category $\mathfrak{C}$ is formally a covariant functor from the set of
natural numbers $\omega$ into $\mathfrak{C}$.
Up to isomorphism, every sequence in $\mathfrak{L}$ corresponds to a chain
$\\{X_{n}\\}_{n\in\omega}$ of finite-dimensional subspaces with initial
subspaces. By this way, the monotone Schauder basis of a Banach space $X$ is
translated into the existence of a sequence in $\mathfrak{K}$ whose co-limit
is $X$. For the sake of convenience, we shall denote a sequence by $\vec{U}$,
having in mind a chain $\\{U_{n}\\}_{n\in\omega}$ of finite-dimensional spaces
with the initial subspaces. Given $U_{n}\subseteq U_{m}$, the
$\mathfrak{K}$-arrow $f^{m}_{n}:U_{n}\to U_{m}$ is such that the image
$f^{m}_{n}[U_{n}]$ is an initial subspace of $U_{m}$.
Following [6], we shall say that a sequence $\vec{Y}$ in $\mathfrak{L}$ is
Fraïssé if it satisfies the following condition:
1. (A)
Given $n\in\omega$, and an $\mathfrak{L}$-arrow $f:{Y_{n}}\to Z$, there exist
$m>n$ and an $\mathfrak{L}$-arrow $g:Z\to{Y_{m}}$ such that $g\circ f$ is the
arrow from $Y_{n}$ to $Y_{m}$.
It is clear that this definition is purely category-theoretic. The name
“Fraïssé sequence", as in [6], is motivated by the model-theoretic theory of
Fraïssé limits explored by Roland Fraïssé [1]. One of the results in [6] is
that every countable category with amalgamations has a Fraïssé sequence.
###### Theorem 1 ([6]).
The category $\mathfrak{L}$ has a Fraïssé sequence.
From now on, we fix a Fraïssé sequence $\\{Y_{n}\\}_{n\in\omega}$ in
$\mathfrak{L}$. As usual, we assume that the embeddings are inclusions. Let
$\mathcal{B}$ be the completion of the union $\bigcup_{n\in\omega}Y_{n}$.
## 5 Universality
###### Theorem 2.
Let $X$ be a Banach space with a monotone Schauder basis. Then there exists an
isometric embedding $e:X\to\mathcal{B}$ such that $e[X]$ is an initial
subspace of $\mathcal{B}$.
###### Proof.
Fix a Banach space $X$ with a monotone Schauder basis and let this be
witnessed by a chain $\\{X_{n}\\}_{n\in\omega}$ together with suitable
projections $\\{Q_{n}\\}_{n\in\omega}$.
We construct inductively $1$-bounded operator $e_{n}:X_{n}\to Y_{k_{n}}$ such
that
1. (1)
$e_{n}[X_{n}]$ is an initial subspace of $Y_{k_{n}}$,
2. (2)
$\|e_{n+1}\restriction X_{n}-e_{n}\|<2^{-n}$.
Recall that, according to our previous agreement, we consider only $1$-bounded
operators. We may assume that $X_{0}=Y_{0}=\\{0\\}$, therefore it is clear how
to start the induction. Suppose $e_{n}$ (and $k_{n}\in\omega$) have already
been defined. By Lemma 2, there exist $i:X_{n}\to~{}W$ and
$j:Y_{k_{n}}\to~{}W$, where $W=X_{n}\oplus_{e_{n}}Y_{k_{n}}$, and the
following conditions are satisfied:
1. (3)
$\|j\circ e_{n}-i\|<2^{-n}$.
Using Lemma 1, we may further extend $W$ so that there exists also a
$\ell:X_{n+1}\to W$ satisfying
1. (4)
$\ell\restriction X_{n}=i$.
Applying Lemma 2 and Lemma 1 we preserve condition (1). Recall that $Y_{n}$ is
a rational Banach space. Thus, we can extend $W$ further, so that the extended
arrow from $Y_{n}$ to $W$ will become rational. Doing this, we make some
“error" of course, although we can still preserve (3) and (4), because all the
inequalities appearing there are strict. Now we use the fact that
$\\{Y_{n}\\}_{n\in\omega}$ is a Fraïssé sequence. Specifically, we find
$k_{n+1}>k_{n}$, rational operators $g:W\to{Y_{k}}$ and $H:Y_{k_{n+1}}\to W$
such that $H\circ g=\operatorname{id}_{W}$ and $g\circ j$ is the inclusion
$Y_{k_{n}}\subseteq Y_{k}$.
Define $e_{n+1}=g\circ\ell$. This finishes the inductive construction.
Passing to the limits, we obtain $1$-bounded operator $e:X\to\mathcal{B}$ .
Condition (2) imply that $e$ is an isometric embedding. In particular, $e[X]$
is an initial subspace of $\mathcal{B}$.
∎
###### Corollary 3.
The space $\mathcal{B}$ is isomorphic to Pełczyńki’s complementably universal
space for Schauder bases, as well as to Kadec’s complementably universal space
for the bounded approximation property.
###### Proof.
See the proof in [2, Corollary 5.2]. ∎
## 6 Isometric uniqueness
Proofs are done analogous like in [2]. We present only the theorems and
references, when we can find the proofs.
Let us consider the following extension property of a Banach space $E$:
1. (B)
Given a pair $X\subseteq Y$ of finite-dimensional Banach spaces with monotone
Schauder basis such that $\\{0\\}$, $X$ are initial subspaces of those spaces,
respectively, given an isometric embedding $f:X\to E$ such that $f[X]$ is an
initial subspace of $E$, for every $\varepsilon>0$ there exists an
$\varepsilon$-isometric embedding $g:Y\to E$ such that $\|g\restriction
X-f\|<\varepsilon$ and $g[Y]$ is an initial subspace of $E$.
###### Theorem 4.
$\mathcal{B}$ satisfies condition (B).
###### Proof.
See the proof in [2, Theorem 6.1]. ∎
###### Lemma 3.
Assume $X$ satisfies condition (B). Then, given $\varepsilon,\delta>0$, given
finite-dimensional spaces $E\subseteq F$, given an $\varepsilon$-isometric
embedding $f:E\to X$ such that $f[E]$ is an initial subspace of $X$, there
exists a $\delta$-isometric embedding $g:F\to X$ such that $\|g\restriction
E-f\|<\varepsilon$ and $g[F]$ is an initial subspace of $X$.
###### Proof.
See the proof in [2, Lemma 6.2]. ∎
###### Theorem 5.
Let $\mathcal{B}$ and $\mathcal{K}$ be Banach spaces with monotone Schauder
bases satisfying condition (B) and let $h:A\to B$ be a bijective linear
isometry between finite-dimensonal subspaces, where $A$ and $B$ are initial
subspaces of $\mathcal{B}$ and $\mathcal{K}$, respectively. Then for every
$\varepsilon>0$ there exists a bijective linear isometry
$H:\mathcal{B}\to\mathcal{K}$ that is $\varepsilon$-close to $h$. In
particular, $\mathcal{B}$ and $\mathcal{K}$ are linearly isometric.
###### Proof.
See the proof in [2, Theorem 6.3]. We use standard back-and-forth argument.
Instead of codition (4) from proof of [2, Theorem 6.3], we have that
$f_{n}[A_{n}]$ and $g_{n}[B_{n}]$ are initial subspaces of $\mathcal{K}$ and
$\mathcal{B}$, respectively. ∎
## References
* [1] R. Fraïssé, Sur quelques classifications des systèmes de relations, Publ. Sci. Univ. Alger. Sér. A. 1 (1954) 35–182
* [2] J. Garbulińska, Isometric uniqueness of a complementably universal Banach space for Schauder decompositions, Banach J. Math. Anal. 8 (2014), no. 1, 211–220
* [3] W.B. Johnson, A. Szankowski, Complementably universal Banach spaces, Studia Math. 58 (1976) 91–97
* [4] M. I. Kadec, On complementably universal Banach spaces, Studia Math. 40 (1971) 85–89
* [5] N. Kalton, Universal spaces and universal bases in metric linear spaces, Studia Math. 61 (1977), 161–191
* [6] W. Kubiś, Fraïssé sequences: category-theoretic approch to universal homogeneus structures, preprint, arxiv.org/abs/0711.1683
* [7] W. Kubiś, S. Solecki, A proof of uniqueness of the Gurariĭ space, Israel J. Math. 195 (2013), 449–456
* [8] A. Pełczyński, Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis, Studia Math. 40 (1971) 239–243
* [9] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228
* [10] A. Pełczyński, Universal bases, Studia Math. 32 (1969), 247-268
* [11] A. Pełczyński, P. Wojtaszczyk, Banach spaces with finite-dimensional expansions of identity and universal bases of finite-dimensional subspaces, Studia Math. 40 (1971) 91–108
|
arxiv-papers
| 2014-02-11T21:04:36 |
2024-09-04T02:49:58.116093
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joanna Garbuli\\'nska-W\\c{e}grzyn",
"submitter": "Joanna Garbuli\\'nska - W{\\ke}grzyn",
"url": "https://arxiv.org/abs/1402.2660"
}
|
1402.2665
|
# Large eddy simulation requirements for the Richtmyer-Meshkov Instability
Britton J. Olson [email protected] Lawrence Livermore National Laboratory
Livermore, CA 94550 Jeff Greenough Lawrence Livermore National Laboratory
Livermore, CA 94550
###### Abstract
The shock induced mixing of two gases separated by a perturbed interface is
investigated through Large Eddy Simulation (LES) and Direct Numerical
Simulation (DNS). In a simulation, physical dissipation of the velocity field
and species mass fraction often compete with numerical dissipation arising
from the errors of the numerical method. In a DNS the computational mesh
resolves all physical gradients of the flow and the relative effect of
numerical dissipation is small. In LES, unresolved scales are present and
numerical dissipation can have a large impact on the flow, depending on the
computational mesh. A suite of simulations explores the space between these
two extremes by studying the effects of grid resolution, Reynolds number and
numerical method on the mixing process. Results from a DNS are shown using two
different codes, which use a high- and low-order numerical method and show
convergence in the temporal and spectral dependent quantities associated with
mixing. Data from an unresolved, high Reynolds number LES are also presented
and include a grid convergence study. A model for an effective viscosity is
proposed which allows for an _a posteriori_ analysis of the simulation
database that is agnostic to the LES model, numerics, and the physical
Reynolds number of the simulation. An analogous approximation for an effective
species diffusivity is also presented. This framework is then used to estimate
the effective Reynolds number and Schmidt number of future simulations,
elucidate the impact of numerical dissipation on the mixing process for an
arbitrary numerical method, and provide guidance for resolution requirements
of future calculations.
## I Introduction
The mixing of fluids is enhanced in the presence of fully developed turbulent
flow. The turbulent cascade transports entrained fluid from the large scale
eddies to the small scale eddies, increasing the net interfacial surface area
and the speed at which the fluids molecularly diffuse. This fluid dynamics
process is of importance in numerous applications in engineering and nature.
For example, Inertial Confinement Fusion (ICF) capsules are known to be
Rayleigh-Taylor unstable during the late compression phase of the ignition
process. If this instability transitions to turbulence, the rate at which the
ablator mixes with the fuel rapidly increases, potentially impacting capsule
performance. Given the extreme conditions of ICF, physicists must rely heavily
on computational and theoretical models to elucidate the actual state of the
mixing.
Large eddy simulation (LES) is a powerful simulation tool for capturing the
large scale dynamics of unsteady fluid flow. Of LES, W.C. Reynolds Reynolds
(1990) wrote, “The objective of large eddy simulations is to compute the
three-dimensional time-dependent details of the largest scales of motion
(those responsible for the primary transport) using a simple model for the
smaller scales. LES is intended to be useful in the study of turbulence
physics at high Reynolds number, in the development of turbulence models, and
for predicting flows of technical interest in demanding complex situations
where simpler model approaches (e.g. Reynolds stress transport) are
inadequate”.
Traditional LES approaches use high-order numerics and explicit sub-grid scale
(SGS) models to account for the unresolved scales of motion at or below the
grid cut-off frequency. Although a complete overview of existing SGS models is
not given here, a review of general SGS model development and scale invariance
is given by Meneveau and Katz Meneveau & Katz (2000) with select analysis of
popular SGS models. The low numerical dissipation associated with high-order
schemes is a desired attribute in LES as it allows for a broader range of
length scales to be captured on the computational mesh. Indeed, Kravchenko and
Moin Kravchenko & Moin (1997) found that errors in the SGS model and numerical
truncation were reduced when high-order methods, with lower numerical
dissipation and a broader range of resolved scales, were used. Since the
fidelity of an LES calculation is proportional to the percentage of energy
explicitly captured on the mesh, using a scheme with less numerical
dissipation will generally produce more accurate results.
In all LES approaches, dissipation works to inhibit and damp out energy in the
fine scales. Dissipation is introduced into the simulation by the numerical
method, the physical viscosity or the SGS model viscosity, if one is used. In
the absence of an explicit SGS model viscosity, the method must rely on the
underlying numerical discretization scheme to supply the “implied” SGS
viscosity through numerical dissipation. Schemes which have no SGS model and
no physical transport properties are classified as Implicit Large Eddy
Simulation (ILES) methods. A complete development of various ILES methods is
given by Grinstein, Margolin, and Rider Grinstein et al. (2007) with
additional development of the general ILES approach given by Boris Boris et
al. (1992). In the present work, we have considered methods that include the
Navier-Stokes properties with and with out explicit SGS terms.
For a given LES calculation with unresolved scales of motion, the effects of
the three sources of dissipation are difficult to segregate. Although the
physical transport coefficients are directly known, or not included in the
case of ILES, their relative effect on the solution will depend on the amount
of model and numerical dissipation. SGS dissipation and numerical dissipation
will effectively vanish relative to the physical dissipation as the grid
resolution increases and the DNS limit is approached. However, DNS solutions
are often not computationally feasible for high Reynolds number flows.
Furthermore, if physical viscous terms are not included, as is typical with
ILES, the notion of a DNS limit is nonexistent.
Efforts to quantify the dissipation of LES methods through an effective
viscosity have been previously made. Aspden et al. Aspden et al. (2008)
derived an effective viscosity model for an incompressible fluid which was
verified by a suite of viscous calculations for sustained isotropic
turbulence. Aspden’s model was particularly instructive in that it was applied
_a posteriori_ to data and allowed for an effective Reynolds number to be
computed for viscous and inviscid simulations. Grinstein et al. Grinstein et
al. (2011) briefly showed for a Taylor-Green vortex that there exists a
connection between under-resolved LES calculations at high Reynolds number and
resolved DNS calculations at a much lower Reynolds number, implying an
effective Reynolds number for the LES calculations. Thornber et al. Thornber
et al. (2007) examined the numerical viscosity of decaying isotropic
turbulence using high-order methods in ILES calculations. Zhou et al. Zhou et
al. (2014) provided a method for estimating effective viscosities for ILES
calculations where approximations of the dissipation rate are made to derive
an effective viscosity.
LES studies of the Richtmyer-Meshkov instability have led to significant
scientific insight and have been done using the gamut of LES methodologies.
Hill et al.Hill et al. (2006) used the stretched vortex SGS method and a
hybrid WENO scheme to simulate the effect of shock Mach number on RM growth
with reshock. Thornber et al. Thornber et al. (2010) used ILES and a high-
order Gudonov-type finite volume method to investigate the dependence of
initial conditions on RM induced mixing. Shankar and Lele Shankar & Lele
(2012) used a 6${}^{\text{th}}$-order compact finite difference scheme and an
explicit hyper-viscosity model Kawai & Lele (2008); Cook (2007) to perform LES
studies of recent experiments Balakumar et al. (2008) of the RM instability.
Although a full review of all the LES studies of the RM instability is outside
the scope of this paper, the aforementioned examples serve to illustrate the
diversity of numerical methods and models used in LES of the RM instability.
Results from independent studies which employ different LES methodologies and
different numerical methods will be subject to some degree of variability.
With the exception of DNS, results from different methods should not be
expected to be identical.
To elucidate uncertainties in the LES approach, a comparison study of two
Large Eddy Simulation methodologies is made by simulating the Richtmyer-
Meshkov instability. The range between the viscous (DNS) and inviscid (Euler)
limits is explored by variation of the physical viscosity. A grid convergence
study in conducted at each Reynolds number for both LES approaches. The
resulting database of simulation data allows the various sources of
dissipation to be explored and is unprecedented for three-dimensional RM
instability. A new a posteriori analysis is proposed which treats all methods,
resolutions and Reynolds numbers in one common framework, which includes a
formulation for both an effective viscosity and an effective species
diffusivity.
The paper is divided into five subsequent sections. Section II gives an
overview of the equations of motion of multi-component flow which are being
solved in the LES/DNS calculations. The numerical methods of the two codes and
LES models are outlined as well. Section III includes results for the high-
and low-Reynolds number cases, showing diagnostics for mixing and scale
dependent energy. Dependence of the results on grid resolution and numerical
method are discussed. In Section IV a framework for comparing results of LES
calculations at different Reynolds numbers, grid resolutions and using
different numerics is given. An effective viscosity and diffusivity are
proposed which collapse the data and provide an estimate for an effective
Reynolds number, Péclet number, and Schmidt number for the flow. Additional
discussion and suggestions for predicting the requirements for LES/DNS
calculations is given in Section V and a summary of the present work is given
in the Conclusion in Section VI.
## II LES Methodology
Variation of the numerical method is achieved by using two different LES codes
for simulating the RM instability. Both codes, Ares and Miranda, are developed
at Lawrence Livermore National Laboratory and are capable of solving the
compressible Navier-Stokes equations in three spatial dimensions. In this
section, an overview of the equations of motion is given. A brief summary of
each LES solver is provided including discussion of the numerical method and
the LES model, if any.
### II.1 Equations of Motion
The compressible multi-component Navier-Stokes equations for $N$ fluids can be
written in strong conservation form as:
$\displaystyle\frac{\partial\rho Y_{i}}{\partial t}$
$\displaystyle+\nabla\cdot\left(\rho Y_{i}{\bf u}+{\bf J}_{i}\right)=0,\ \ \
\text{for i=1,2,..,N}$ (1) $\displaystyle\frac{\partial\rho{\bf u}}{\partial
t}$ $\displaystyle+\nabla\cdot\left(\rho{\bf
uu^{T}}+\underline{\bf\delta}p-\underline{\bf\tau}\right)=0$ (2)
$\displaystyle\frac{\partial E}{\partial t}$
$\displaystyle+\nabla\cdot\left({\bf u}\left(E+p\right)+{\bf q}-{\bf
u\cdot\underline{\tau}}\right)=0$ (3)
where $\rho$ is the density, $Y_{i}$ is the mass fraction of species $i$, $\bf
u$ is the velocity vector, $E=\rho\left(e+{\bf u}^{2}/2\right)$ is the total
energy of the mixture, $T$ is the temperature of the gas, $e$ is the internal
energy, $p$ is the pressure, and $\underline{\bf\delta}$ is the Kronecker
delta tensor. The diffusive mass flux ${\bf J}_{i}$, viscous stress tensor
$\underline{\tau}$, and energy flux $\bf q$ are given by
$\displaystyle{\bf J_{i}}$ $\displaystyle=-\rho\left(D_{i}\nabla
Y_{i}-Y_{i}\sum_{j=1}^{N}D_{j}\nabla Y_{j}\right),$ (4)
$\displaystyle\underline{\tau}$
$\displaystyle=2\mu\underline{S}+\left(\beta-\frac{2}{3}\mu\right)\left(\nabla\cdot{\bf
u}\right)\underline{\delta},$ (5) $\displaystyle{\bf q}$ $\displaystyle={\bf
q}_{T}+{\bf q}_{E}$ (6)
where the strain rate tensor $\underline{S}$, the conductive heat flux ${\bf
q}_{T}$ and the interdiffusional enthalpy flux ${\bf q}_{E}$ are written as
$\displaystyle\underline{S}$ $\displaystyle=\frac{1}{2}\left(\nabla{\bf
u}+\left(\nabla{\bf u}\right)^{T}\right)$ (7) $\displaystyle{\bf q}_{T}$
$\displaystyle=-\kappa\nabla T$ (8) $\displaystyle{\bf q}_{E}$
$\displaystyle=\sum_{i}^{N}h_{i}{\bf J}_{i}$ (9)
and where $h_{i}$ is the individual species enthalpy Cook (2009).
#### II.1.1 Mixture equation of state
The Navier-Stokes terms in eqs. 1, 2 and 3 contain the physical transport
coefficients $\mu$, $\beta$, $\kappa$ and $D_{i}$; which are the shear
viscosity, bulk viscosity, thermal conductivity, and species diffusivity,
respectively. For low-Mach number flow, the temperature dependence of the
species diffusivities is small. Once the shock wave has passed, the mean
turbulent Mach number of the mixing layer remains below 0.05 for all time.
Therefore, for problem simplification, a constant physical viscosity is
prescribed through a Reynolds number, species diffusivity through a constant
Schmidt number and thermal conductivity through a constant Prandtl number as
follows;
$\mu_{i}=\frac{\rho_{0,i}V_{0}\lambda_{0}}{Re_{\lambda_{0},i}},$ (10)
$D_{f}=\frac{\mu_{i}}{\rho_{0,i}Sc_{i}}$ (11)
$\kappa_{f}=\frac{c_{p}\mu_{f}}{Pr}$ (12)
where $V_{0}$ is the post-shock velocity, $\lambda_{0}$ is the fastest growing
perturbed wave length (eq. 20) and $c_{p}$ is the specific heat capacity at
constant pressure. For the present calculations, the initial Reynolds numbers
($Re_{\lambda_{0}}$) in the pre-shocked air and SF6 are 30,000 and 180,000,
respectively. The Schmidt numbers ($Sc$) are 1.11 and 0.18 in the the Air and
SF6, respectively, and give a constant diffusivity, $D_{f}$. The Prandtl
number ($Pr$) is 1.0 and is based on the mixture viscosity, $\mu_{f}$, which
is given as,
$\mu_{f}=\left(\sum_{i=1}^{2}\frac{Y_{i}}{\mu_{i}}\right)^{-1},$ (13)
where the species index $i$ refers to Air ($i=1$) and SF6 ($i=2$). The
constant thermodynamic and species transport properties are summarized in
table 1. The ideal gas law is assumed, giving temperature and pressure as
$\displaystyle T$ $\displaystyle=\frac{(\gamma_{f}-1)e}{R_{f}},$ (14)
$\displaystyle p$ $\displaystyle=(\gamma_{f}-1)\rho e.$ (15)
The mixture ratio of specific heats ($\gamma_{f}$) and mixture specific gas
constant ($R_{f}$) are given as
$\displaystyle\gamma_{f}$ $\displaystyle=\frac{c_{p}}{c_{v}},$ (16)
$\displaystyle R_{f}$
$\displaystyle=R_{\text{univ}}\sum_{i=1}^{2}\frac{Y_{i}}{M_{w,i}}$ (17)
with
$\displaystyle c_{p}$
$\displaystyle=R_{\text{univ}}\sum_{i=1}^{2}\frac{Y_{i}\gamma_{i}}{M_{w,i}\left(\gamma_{i}-1\right)},$
(18) $\displaystyle c_{v}$
$\displaystyle=R_{\text{univ}}\sum_{i=1}^{2}\frac{Y_{i}}{M_{w,i}\left(\gamma_{i}-1\right)}$
(19)
and where $R_{\text{univ}}=8.314\times 10^{7}$ [erg/K/mol] is the universal
gas constant.
Gasi | $\gamma_{i}$ | $\mu_{i}$ [g/cm$\cdot$s] | $D_{f}$ [cm2/s] | $M_{w,i}$ [g/mol] | Re${}_{\lambda_{0},i}$ | Sc0,i
---|---|---|---|---|---|---
Air ($i=1$) | 1.4 | $18.26\times 10^{-5}$ | $15.05\times 10^{-2}$ | 28.8 | $30\times 10^{3}$ | 1.11
SF6 ($i=2$) | 1.1 | $14.75\times 10^{-5}$ | $15.05\times 10^{-2}$ | 146.0544 | $180\times 10^{3}$ | 0.18
Table 1: Constant thermodynamics and molecular transport properties for the
present study.
### II.2 The Miranda code
The Miranda code has been used extensively for simulating turbulent flows with
high Reynolds numbers and multi-component mixing Cook et al. (2004); Cabot &
Cook (2006); Olson & Cook (2007); Olson et al. (2011). Miranda uses a
10${}^{\text{th}}$-order compact differencing scheme for spatial
differentiation and a 5-stage, 4${}^{\text{th}}$-order Runge-Kutta scheme for
temporal integration. Full details of the numerical method are given by
CookCook (2007). For numerical regularization of the sharp, unresolved
gradients in the flow, artificial fluid properties are used to locally damp
structures which exist on the length scales of the computational mesh.
In this approach, artificial diffusion terms are added to the physical ones
which appear in Eqs. 4, 5 and 8. This method of AFLES was originally proposed
by Cook Cook (2007) and has been altered by computing the artificial bulk
viscosity term using $\nabla\cdot\mbox{\boldmath$u$}$ rather than $S$
(magnitude of the strain rate tensor). Mani et al.Mani et al. (2009) showed
that this modification substantially decreased the dissipation error of the
method. The artificial transport coefficients are computed by taking higher
derivatives of the resolved fields. The explicit form for the terms and
various test problems for validating the method are given in the references
Cook (2007); Johnsen et al. (2010); Olson & Lele (2013).
### II.3 ARES
ARES is an Arbitrary Lagrange Eulerian (ALE) code developed at Lawrence
Livermore National Laboratory (LLNL). The Lagrange time step uses a second
order predictor-corrector method. The Gauss divergence theorem is applied to
solve the discrete finite difference equations Wilkins (1963) of the
compressible multi-component Navier-Stokes equations (eqs. 1-3). The spatial
derivatives are approximated using a second-order finite difference method.
Artificial viscosity Kolev & Rieben (2009) is applied to damp out spurious,
high frequency oscillations which are generated near shocks and contact
discontinuities.
Velocities are defined as nodal quantities, while density and internal energy
are defined at zone centers using piecewise constant profiles. After each
Lagrangian time step, a second order remap is applied to all variables (nodal
and cell centered) to a new mesh, in keeping with the general ALE methodology.
For the simulations of this study, a fixed Eulerian mesh is used.
Although the ARES code includes an adaptive mesh refinement (AMR) capability
Berger & Oliger (1984); Berger & Colella (1989), it was not exercised in this
study to facilitate a direct comparison with Miranda. No explicit sub-grid
scale model is applied to the equations of motion for all simulations
presented in this study.
## III Richtmyer-Meshkov Instability
### III.1 Problem Setup
To focus the scope of the present study, only the single shock RM problem is
considered. In this case, dependence on the initial conditions is strong,
therefore a particular realization of initial conditions is used for both
codes at all resolutions and Reynolds numbers. The problem is solved in the
post shock interface frame of reference, such that after the shock passes
through the interface, it remains motionless in one dimension (1D). This
motion was analytically prescribed and verified numerically in 1D.
Figure 1: Schematic setup of the Richtmyer-Meshkov instability showing the
initial conditions (left) and the 1D evolution of the shock waves and
interface locations on the $x-t$ diagram (right). The “sponge” boundary
conditions are used to absorb the outward moving shock wave with minimal
spurious reflection. The states from region 1-5 are given in Table 2.
A Mach 1.18 shock wave is initialized in air, ahead of a perturbed interface
of sulfur-hexaflouride (SF6). The shock wave is initialized at $x_{s}$ such
that the two discontinuities intersect at $x=0$ (see Fig. 1). The shock wave
satisfies the Rankine-Hugoniot jump conditions, which are used to prescribe
the conditions ahead of and behind the shock wave. The states in regions 1-5
of Figure 1 are explicitly given in Table 2 below. The three dimensional
domain extents are 16cm$\times$8cm$\times$8cm in the $x$,$y$ and $z$
directions, respectively.
Region | $p~{}\mathrm{[g/(cm\cdot s^{2})]}$ | $\rho~{}\mathrm{[g/cm^{3}]}$ | $u_{x}~{}\mathrm{[cm/s]}$ | Species
---|---|---|---|---
1 | 1.36e6 | 1.42e-3 | 3.33e3 | Air
2 | 0.931e6 | 1.08e-3 | -6.33e3 | Air
3 | 0.931e6 | 5.50e-3 | -6.33e3 | SF6
4 | 1.53e6 | 8.66e-3 | 0.0 | SF6
5 | 1.53e6 | 1.55e-3 | 0.0 | Air
Table 2: Values for initial flow field in the post shock air (region 1), pre
shock air (region 2) and pre shock SF6 (region 3). The final states after the
transmission and reflection of the shock wave are given for the SF6 and Air,
regions 4 and 5, respectively.
#### III.1.1 Perturbed initial interface
The perturbation of the initial interface is necessary to generate baroclinic
vorticity, instability growth and eventual transition to turbulence. The
perturbation is defined in Fourier space as a power spectrum which is a
function of the two-dimensional wave number. In this study, the general form
for the power spectrum suggested by Thornber et al. Thornber et al. (2010) is
assumed as
$P(k)=\begin{cases}Ck^{m},&k_{\text{min}}<k<k_{\text{max}},\\\
0,&\text{otherwise},\end{cases}$ (20)
where $k=\sqrt{k_{y}^{2}+k_{z}^{2}}$ is the two-dimensional wave number. For
the present study, $C=\lambda_{0}/10$, $\lambda_{0}=1/k_{\text{max}}$ , $m$ is
set to -2, ($k_{\text{min}}$,$k_{\text{max}}$) is set to (4,16) and the random
phase shifts used to construct the Fourier modes were determined _a priori_
and used to initialize all calculations. Since $k_{\text{max}}$ is less than
the Nyquist wave number on the coarsest mesh, all initial fields are
spectrally exact.
The interface height is therefore given as
$\eta(y,z)=\sum_{j}^{k_{y}}\sum_{k}^{k_{z}}P(k)\cos(k_{y}y+\phi_{y,j})\sin(k_{z}z+\phi_{z,k}),$
(21)
where $\phi$ are the set of random numbers used for all initializations. The
mass fraction fields are diffusely initialized over a finite width using a
hyperbolic tangent function as
$\displaystyle Y_{SF_{6}}(x,y,z,\tau=0)$
$\displaystyle=\frac{1}{2}\left(1+\tanh\left(\frac{x-\eta(y,z)}{\delta_{p}}\right)\right),$
(22) $\displaystyle Y_{Air}(x,y,z,\tau=0)$
$\displaystyle=1-Y_{SF_{6}}(x,y,z,\tau=0)$ (23)
where $\delta_{p}$ is the initial interface thickness and is set to
$\lambda_{0}/4$ for all calculations and where the non-dimensional time is
given as $\tau=tV_{0}/\lambda_{0}$. Prior to first shock the two fluids have a
constant ambient temperature of 297 K, which is implicitly given the values of
Table 1 and 2 and the ideal gas equation of state.
Figure 2: Iso-volume of the mass fraction of $SF_{6}$ between .1 and .9 for
cases B, C, and D (top to bottom) from Miranda (left) and Ares (right)
calculations at the nominal Reynolds number at $tV_{0}/\lambda_{0}=35$. Data
from mesh D show the existence of a broad range of length scales in the mixing
layer.
### III.2 Low Reynolds number DNS
To establish a baseline for convergence to the resolved scales, a grid
refinement study was conducted at a Reynolds number 1/25${}^{\text{th}}$ the
nominal value. This reduction in Reynolds number (and subsequent reductions)
was achieved by multiplying the species diffusivity and viscosities by the
relevant factor, thereby maintaining a constant Schmidt number. At this
Reynolds number it was possible to approach the DNS limit for the given high-
resolution grid spacing selected. Table 3 shows the various resolutions
selected and the resulting number of total grid points.
Mesh | $N_{x}$ | $N_{y}$ | $N_{z}$ | Total Pts.
---|---|---|---|---
A | 128 | 64 | 64 | 0.5 M
B | 256 | 128 | 128 | 4.2 M
C | 512 | 256 | 256 | 33.5 M
D | 1024 | 512 | 512 | 268.4 M
Table 3: Computational mesh parameters for various levels of refinement and
the resulting number of total grid points. Grids were uniformly spaced in all
three coordinate directions.
#### III.2.1 Mixing region growth
Several integral measures of the mixing region are compared here. These global
measures show the time dependent mixing state and are typically used for
experimental comparison where only gross mixing measures are available. The
mixing width is defined as
$W=4\int_{-\infty}^{\infty}\langle Y_{\text{SF6}}\rangle\langle
Y_{\text{Air}}\rangle dx,$ (24)
where the $\langle\cdot\rangle$ operator denotes planar averages taken in the
$y$-$z$ plane and is defined as
$\displaystyle\langle f\rangle(x,t)$ $\displaystyle=\frac{1}{A}\int
f(x,y,z,t)dydz\text{ , where }$ (25) $\displaystyle A$ $\displaystyle=\int
dydz.$ (26)
Another measure of mixing is the “mixedness”, which is the ratio of mixed
fluid to entrained fluid defined as
$\Theta=\frac{\int_{-\infty}^{\infty}\langle
Y_{\text{SF6}}Y_{\text{Air}}\rangle dx}{\int_{-\infty}^{\infty}\langle
Y_{\text{SF6}}\rangle\langle Y_{\text{Air}}\rangle dx}.$ (27)
For fully developed three-dimensional mixing, this quantity approaches Cabot &
Cook (2006) $\approx 0.8$. $\Theta$ represents the $2^{\text{nd}}$ statistical
moment of mixing and can be identically related to the variance of the mass
fractions as
$\Theta=1+4\frac{\int_{-\infty}^{\infty}\langle
Y_{\text{SF6}}^{\prime}Y_{\text{Air}}^{\prime}\rangle dx}{W}$ (28)
where the primed values are defined as $F^{\prime}=F-\langle F\rangle$.
Therefore, where $W$ is an integral measure of the mean of species mass
fraction, $\Theta$ is an integral measure of its variance or fluctuations.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 3: Non-dimensional mixing width vs. time for meshes A-D at the reduced
Reynolds number from Miranda (a) and Ares (b). Data between codes at the
finest resolution are plotted in (c) and show that agreement worsens with time
and is at most 2% different at $\tau=40$.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 4: Mixedness ($\Theta$) vs. time for meshes A-D at the reduced
Reynolds number from Miranda (a) and Ares (b). Data between codes at the
finest resolution are plotted in (c) and show that differences arise early on
and remain constant over the observed time.
Figures 3 and 4 show the time history of $W/\lambda_{0}$ and $\Theta$ at the
various resolutions for the two codes. Curves for resolutions C and D are
nearly indistinguishable for $W/\lambda_{0}$ through $\tau=40$ in Miranda
(Fig. 3a) and up to $\tau=15$ in Ares (Fig. 3b) . The comparison of the fine
mesh calculation between codes (Fig. 3c) shows the solutions differ more as
time progresses, reaching approximately 2% difference at $\tau=40$.
Figure 4a shows values for $\Theta$ are converged for $\tau>10$ in Miranda.
For Ares in Figure 4b, the opposite occurs, where convergence is most
pronounced for $\tau<10$. At $\tau=5$, $W$ is constant and $\Theta$ differs
between the two codes (Figure 4c) which (by enforcing equation 28) implicates
a larger mass fraction variance in Miranda at $\tau=5$ and indeed, for all
time. This statement is confirmed by differences in the power spectra of the
mass fraction given below.
#### III.2.2 Spectra
Evaluating the wave number dependence of the fluctuating turbulent quantities
can elucidate characteristics of the flow physics as well as the numerical
errors associated with the particular LES approach. In LES comparisons,
directly measuring the energy of high wave numbers will indicate the range of
scales which are resolved on the LES mesh. Spectra are computed at each
$y$-$z$ plane where $4\langle Y_{\text{SF6}}\rangle\langle
Y_{\text{Air}}\rangle>0.7$. The two-dimensional Fourier transforms from these
$N$ planes are then averaged, binned into annuli and plotted as a function of
two-dimensional wave number, $k$. This procedure is applied to the fluctuating
mass fraction field as well as the velocity field, which are plotted in Figure
5 and 6, which shows the spectra for both Ares and Miranda at all resolutions
at $\tau=35$.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 5: Power spectra of the fluctuating velocity at $\tau=35$ for Miranda
(a) and Ares (b) for meshes A-D at the reduced Reynolds number. Convergence
for wave numbers less than 70 is observed in both codes. The difference of the
spectra for mesh D between the codes (c) is negligible up to wave number 100.
The $k^{-5/3}$ fiducial is plotted (dashed) and shows a lack of an inertial
subrange.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 6: Power spectra of the mass fraction of SF6 at $\tau=35$ for Miranda
(a) and Ares (b) for meshes A-D at the reduced Reynolds number. Convergence
for wave numbers less than 100 is observed in both codes. The difference of
the spectra for mesh D between the codes (c) is noticeable for wave numbers
larger than 15. The $k^{-5/3}$ fiducial is plotted (dashed) and shows a lack
of an inertial subrange.
The spectra show excellent convergence at the low wave numbers and a strong
trend toward convergence at the higher wave numbers. For the velocity spectra,
data from the finest resolution grid of Ares and Miranda (Figure 5c) are
nearly indistinguishable for all wave numbers below $k=100$. The spectra of
the mass fraction indicate that nearly all scales are captured for both
Miranda (Figure 6a) and Ares (Figure 6b), as no differences are observed
between spectra from meshes C and D. However, Ares and Miranda data are
converging to different solutions in the high wave numbers, indicating a
dependence on the numerical method. Since the integral of the power spectral
density is proportional to the variance, Figure 6c supports the previous
assertion made in Section III.2.1, that Miranda has a larger mass fraction
variance.
#### III.2.3 Dissipation Measures
Numerical dissipation is most active on the fine scales which are unresolved
on the computational grid. Quantities which are more dependent on the small
scales, therefore, will exhibit larger sensitivity to both grid resolution and
numerical method. To explore these sensitivities, the domain integrated
enstrophy and normalized scalar dissipation rate are computed to explore the
high wave number behavior of dissipation of the velocity field as well as the
scalar field. Enstrophy is given by
$\Omega(t)=\int_{V}\rho\|\mathbf{\omega}\|^{2}\ \ {dxdydz}$ (29)
where $\mathbf{\omega}=\nabla\times\mathbf{u}$. The scalar (mass fraction)
dissipation rate is defined as
$\chi(t)=\int_{V}D_{SF_{6}}\nabla Y_{SF_{6}}\cdot\nabla Y_{SF_{6}}\ {dxdydz}.$
(30)
Given that the simplified equation of state produces a constant value for
diffusivity in the mixing layer, the diffusivity may be pulled outside the
integral. Therefore, comparing $\chi/D_{SF_{6}}$ allows data from LES
calculation of various Schmidt and Reynolds numbers to be compared directly.
Differences between the codes and resolutions are largest at the temporal
maxima of the enstrophy and scalar dissipation rate curves in Figures 7 and 8.
This maximum occurs at around $\tau=12$ for enstrophy and at $\tau=8$ for
scalar dissipation and is indicative of the time when the flow is becoming
damped by the dissipation scales. Therefore, energy coupling to higher modes
has taken place and the flow is beginning to transition to broadband
turbulence. Note here that the “turbulence” referred to is in the
diffusive/dissipative regime as the flow is relaxing and decaying and is not
being driven. For the scalar field, this dissipation threshold occurs slightly
before that of the velocity field.
Convergence is significantly slower for these global measures of dissipation
as compared to the mean mixing measures. In both Miranda and Ares (Figure
7a-b), enstrophy values are getting closer under grid refinement but have not
fully converged by mesh D. The difference between mesh C and D in Miranda is
smaller than that in Ares. The difference between the enstrophy at the finest
mesh (Figure 7c) between the two codes is large, at nearly 10%.
The measure of scalar dissipation also exhibits slow convergence, with the
mesh D solution differing from that of mesh C by approximately 20% for both
Miranda and Ares (Figure 8a-b). The difference between codes in
$\chi/D_{SF_{6}}$ at the mesh D resolution (Figure 8c) is even larger than the
differences in the enstrophy as might be expected, given that the power
spectra of the species mass fraction differ more than the power spectra of the
velocity.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 7: Time history of the domain integrated enstrophy ($\Omega$, eq. 29)
for meshes A-D at the reduced Reynolds number from Miranda (a) and Ares (b).
Data between codes at the finest resolution are plotted in (c).
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 8: Time history of the domain integrated scalar dissipation rate
($\chi/D_{SF_{6}}$, eq. 30) for meshes A-D at the reduced Reynolds number from
Miranda (a) and Ares (b). Data between codes at the finest resolution are
plotted in (c).
As will be shown later, the behavior of the mean flow at this reduced Reynolds
number is largely dependent on the Reynolds number. With no inertial subrange,
the smallest viscous scales will directly impact the large scales and alter
the energy containing scales. As the Reynolds number gets sufficiently large
and the inertial range forms and broadens, this dependence will gradually
subside. Indeed, Grinstein et al. Grinstein et al. (2011) have suggested that
DNS at low Reynolds number can resemble poorly resolved LES calculations at
infinite Reynolds numbers, loosely linking the notion of grid dependence with
Reynolds number dependence. This will be explored further in Section IV.
### III.3 High Reynolds number LES
The second set of calculations were conducted at the Reynolds number given in
Table 1, which is close to the experimental conditions of previous studies
Jacobs & Krivets (2005); Jacobs & Sheely (1996); Vetter & Sturtervant (1995)
of RMI. The required number of grid points needed for a DNS at this high
Reynolds number is approximately $\sim 4\times 10^{12}$, which exceeds the
capability of today’s computational resources. Simulations using the grids of
Table 3 are therefore under resolved with respect to the viscous length
scales. Therefore, the actual diffusion length scales of the simulation will
be dependent on the dissipation from the numerics and the model. Both of which
should vanish under grid refinement but will depend heavily on the numerical
method and grid spacing.
The large energy containing scales will become increasingly independent of the
fine scales associated with the grid as the inertial subrange between the two
broadens. It is this scale separation and independence of the solution on the
fine scale which is probed in a requisite grid convergence study (Figure 2) of
an LES calculation. Therefore, the energy containing portions of the flow
field and global/integral observables such as the mixing width and mixedness
will exhibit converging behavior. However, metrics which are biased to the
small scales such as the scalar dissipation rate and enstrophy diverge under
grid refinement and show stronger dependencies on the numerical dissipation.
To explore this grid convergence at high Reynolds numbers, a grid resolution
study was conducted for both numerical methods on meshes given in table 3. As
in the DNS study, the temporal mixing widths and mixedness are plotted for
both codes and all resolutions in Figure 9 and Figure 10. Convergence is less
pronounced (as compared to the DNS convergence study) and curves diverge with
time. However, for early time ($\tau<25$) the solutions are nearly
indistinguishable at the fine mesh resolution.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 9: Non-dimensional mixing width vs. time for meshes A-D at the nominal
Reynolds number from Miranda (a) and Ares (b). Data between codes at the
finest resolution are plotted in (c) and show that agreement worsens with time
and is at most 5% different at $\tau=40$.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 10: Mixedness ($\Theta$) vs. time for meshes A-D at the nominal
Reynolds number from Miranda (a) and Ares (b). Data between codes at the
finest resolution are plotted in (c) and show that the differences grow with
time and are approximately 5% at $\tau=40$.
The range of resolved scales can be readily examined by looking at the spectra
of velocity and mass fraction fluctuations. Figure 11 shows the power spectra
of velocity as a function of the two-dimensional wave number, $k$, computed as
was described in the previous section. The power spectra between codes at the
fine resolution are in good agreement for wave number less than 30, after
which, they diverge. The inertial range following the $\sim k^{-5/3}$ spans
over a wider range of wave numbers in Miranda than in Ares by approximately a
factor of two on mesh D. At the coarsest calculation (mesh A) the spectra for
both Ares and Miranda do not exhibit inertial ranges.
The mass fraction spectra for the two codes (Figures 12a and 12b) show
converged behavior up through wave number 80. Furthermore, on mesh D,
solutions from Miranda and Ares (Figure 12), have equally wide inertial ranges
and agree quite well for all wave numbers plotted.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 11: Power spectra of the fluctuating velocity at $\tau=35$ for Miranda
(a) and Ares (b) for meshes A-D at the nominal Reynolds number. The difference
of the spectra for mesh D between the codes (c) number 40. The $k^{-5/3}$
fiducial is plotted (dashed) and shows a broader inertial subrange as compared
to the DNS spectra.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 12: Power spectra of the mass fraction at $\tau=35$ for Miranda (a)
and Ares (b) for meshes A-D at the nominal Reynolds number. The difference of
the spectra for mesh D between the codes (c) is small over the entire range of
plotted wave numbers. The $k^{-5/3}$ fiducial is plotted (dashed) and shows a
that the LES maintains an inertial range before the numerical dissipation
effects begin to dominate.
Quantitative measures of dissipation exhibit the largest differences in the
under resolved LES calculations. Since these measures are biased towards
gradients of the finest scales (where numerical dissipation is most active),
grid and scheme dependence will be most apparent. The time histories of
enstrophy and normalized scalar dissipation are plotted in Figure 13 and 14,
respectively. The local maxima of the curves increase in value as the grid is
refined. Values of enstrophy from the mesh C resolution in Miranda are close
to those in Ares from mesh D, suggesting that Miranda is capturing finer
length scales by roughly a factor of two. For the scalar dissipation in Figure
14, the disparity is not as large and Ares mesh D data lie somewhere between
Miranda mesh C and D data.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 13: Time history of the domain integrated enstrophy ($\Omega$, eq. 29)
for the nominal Reynolds number LES. The divergent behavior of the data in (a)
and (b) suggest that the velocity length scales are proportional to grid
spacing. The comparison of Miranda and Ares (c) on mesh D show the peak
enstrophy values of the Miranda calculation on mesh C are approximately
equivalent to those of the Ares calculation on mesh D.
(a) Miranda
(b) Ares
(c) Mesh D for Miranda & Ares
Figure 14: Time history of the domain integrated scalar dissipation rate
($\chi/D_{SF_{6}}$, eq. 30) for meshes A-D at the nominal Reynolds number from
Miranda (a) and Ares (b). Data between codes at the finest resolution are
plotted in (c).
For the high Reynolds numbers calculations, the data clearly suggest that the
flow is under resolved. Although the mean flow field still exhibits dependence
on the fine grid scales (Figure 9 and 10), the effect is decreasing under grid
refinement. Indeed, as the range of resolved scales grows larger with
increased resolution, the effect of the new small scales on the large scales
decreases. This effect can be directly seen in the convergence of the power
spectra (Figures 11 and 12). As higher wave number energy is introduced
through grid refinement, the effect on the lower wave numbers decreases. In
the limit of infinite scale separation, the large scale solution will approach
the Reynolds number independent solution. Thus, there is a notional connection
between grid convergence and Reynolds number independence.
Conversely, there also exists a connection between grid dependence and
Reynolds number dependence. Grinstein et al. Grinstein et al. (2011) showed
(for the Taylor-Green vortex) comparisons of low Reynolds number calculations
and under-resolved high Reynolds number calculations. They found close
correlations between the data, suggesting that poor numerical resolution has a
similar effect as large amounts of physical viscosity on a well resolved grid.
Both mechanisms act like a viscosity, damping the fine scales and reducing the
length of the inertial range. In the following section, we seek a general way
of comparing arbitrary simulation data which considers grid resolution,
Reynolds number and numerical method through formulation of an effective
viscosity.
## IV An effective viscosity for assessing the numerical dissipation in LES
schemes
The data presented in the previous sections demonstrate a dependence on
Reynolds number, grid resolution and LES method. The differences arise from
the small length scales associated with dissipation. In this section, an
effective viscosity is proposed as an _a posteriori_ diagnostic to determine
an effective Reynolds number and an effective Kolmogorov length scale of the
flow for a given grid size, numerical method and physical Reynolds number. An
analogous effective diffusivity is also proposed, which suggests an effective
Batchelor scale and an effective Schmidt number.
Given the strong grid dependence in the high wave numbers on the spectra and
on profiles of the gradient based quantities, the previous LES in Section
III.3 were poorly resolved with respect to the viscous and diffusion length
scales. For under resolved calculations, the dissipation provided by the
Navier-Stokes terms can be small compared to the dissipation of the SGS model
or the numerical discretization. This has motivated the exclusion of the
Navier-Stokes terms entirely in previous ILES studies Grinstein et al. (2011);
Thornber et al. (2010); Latini et al. (2007) of RMI. Doing so can reduce the
computational cost of the simulation but, used as a general approach, has
certain disadvantages. DNS solutions will be impossible to generate or to
approach under grid convergence. The fine scales of turbulence in an Euler
calculation will always scale with those of the grid. Enstrophy, scalar
dissipation rate and other high-order measures of turbulent mixing will never
converge. Furthermore, having never approached the transition between DNS and
LES regime, LES schemes which neglect physical transport terms will have less
confidence in the assumption of the Reynolds number independence for modeling
realistic flows.
A general LES scheme can use any arbitrary set of numerical methods with any
arbitrary set of SGS models. Typically, one selects numerics which balances
the overall cost of the flux approximation with adequate resolving power and
low numerical dissipation. SGS models are often selected or developed
independently of the numerical scheme and motivated by physical properties of
the turbulence. Some LES approaches combine the two and rely on the natural
dissipation of the numerics to act as the SGS model of the scheme. In all such
cases, there exists a non-neglible amount of numerical dissipation which often
cannot be directly quantified. Careful post-processing of the data can reveal
the artifacts of the dissipative nature of the scheme when comparisons are
made. Quantities such as enstrophy and scalar dissipation rate are biased
toward the high wave numbers and will show greater sensitivity to dissipation
compared to conventional measures, such as turbulent kinetic energy (TKE).
Computing an effective viscosity for LES calculations is instructive in that
it allows the net effect of all diffusive processes to be compared on equal
terms. In the absence of an explicit SGS model (as in ILES) previous efforts
have shown the utility of an effective viscosity. Grinstein and Guirguis
Grinstein & Guirguis (1992) compared viscous theory and simulation of two-
dimensional shear layer to relate modified equations to an implicit sub-grid
scale model. More recently, Aspden et al. provided a method for computing an
effective viscosity for incompressible sustained isotropic turbulence. This
viscosity was computed for the entire domain as,
$\nu_{e}=\epsilon/D,$ (31)
where
$D=\frac{1}{V}\int_{V}\mathbf{u}\cdot\nabla^{2}\mathbf{u}\ \mathrm{d}V$ (32)
and where $\epsilon$ is the kinetic energy dissipation rate, evaluated
directly from the domain time rate of change of kinetic energy. Aspden showed
that $\nu_{e}$ continuously transitioned between the two extremes; from fully
resolved (DNS) where $\nu_{f}/\nu_{e}\to 1$, to under resolved, quasi-inviscid
calculations where $\nu_{f}/\nu_{e}\to 0$, where subscript $f$ denotes the
physical viscosity.
For compressible turbulence and RMI in particular, we found this form to be
insufficient for providing an _a posteriori_ approximation of the effective
viscosity of the flow. Firstly, $D$ is not Galilean invariant and will change
in magnitude for arbitrary frames of reference as is the case for shock
induced mixing. Secondly, in compressible flow, $\nu$ has thermodynamic
dependence and may not be moved outside of the Laplacian of $\bf u$ and
therefore the relationship between $\epsilon$ and $D$ will not hold, in
general, for a compressible fluid. Like Aspden, however, we do seek an
identical behavior at the limits of DNS and Euler calculations.
The motion of viscous fluids converts kinetic energy irreversibly to internal
energy. The rate of this conversion due to viscous effects is the dissipation
rate ($\epsilon$) and is given Landau & Lifshitz (2004) by
$\rho\epsilon=\underline{\tau}:\nabla{\bf u}\ .$ (33)
Substituting for the stress tensor ($\underline{\tau}$) of a compressible
Netwonian fluid, we have
$\rho\epsilon=2\mu{\bf
S}^{2}+\left(\beta-\frac{2}{3}\mu\right)\left(\nabla\cdot{\bf u}\right)^{2}\
.$ (34)
SGS models seek to account for sub-grid scale turbulent motion associated
primarily with the rotational portion of $S$ and solenoidal portion of the
velocity field. Therefore, if we neglect the purely dilatational term we have
$\rho\epsilon=2\mu{\bf S}^{2}\ .$ (35)
This is starting point for many SGS models used in the LES community. Perhaps
the most ubiquitous of which is the Smagorinsky model, which approximates
viscous dissipation as
$\epsilon=2(C_{s}\Delta x)^{2}{\bf S}^{3}\ \ $ (36)
and therefore the SGS viscosity can be written as
$\mu_{Smag}=(C_{s}\Delta x)^{2}\rho{\bf S}\ \ .$ (37)
Explicit model viscosity will therefore only be dynamically active in regions
of the flow where high wave number turbulent energy exists. In resolved
regions the dynamic model will vanish at a rate of $(\Delta x)^{2}$.
The above overview and description of this particular LES model is not
intended to defend nor refute its usage as an LES model. Rather, its
attributes and characteristics are highlighted here only to give context and a
starting point for the proposed diagnostic of the present work. To measure
more precisely when the smallest scales of turbulent motion become resolved,
an effective viscosity based on the Smagorinsky model and the SGS model of
Cook is written as,
$\mu^{*}=C_{\mu}\rho|\nabla^{2}\mathbf{S}|\Delta x^{4}$ (38)
which is equivalent to Cook’s model, with $r=2$ and to Smagorinski where $\bf
S$ is replaced with $(\Delta x)^{2}\nabla^{2}{\bf S}$. The effect of the
Laplacian operator is to amplify the localization of the artificial terms in
unresolved regions and to give a convergence rate of $(\Delta x)^{4}$ in
regions of resolved flow. Therefore if we write the effective viscosity as
$\mu_{\text{eff}}=C_{\mu}\rho|\nabla^{2}\mathbf{S}|\Delta x^{4}+\mu_{f}$ (39)
we have $\mu_{f}/\mu_{\text{eff}}\to 1$ for DNS flows and
$\mu_{f}/\mu_{\text{eff}}\to 0$ for inviscid or highly under revolved
calculations. This form is Galilean invariant, general for compressible flow
and can be computed either locally or integrated over some domain. The
coefficient, $C_{\mu}$ requires closure (which will be discussed below) but is
constant for a given numerical method.
#### IV.0.1 An A Posteriori Analysis of Numerical Dissipation
The effective viscosity can be computed at every point in the domain on an
existing data set. For comparison purposes, the derivative operator involved
in computing $\bf S$ and in taking the Laplacian should be identical between
the two codes. For the present study, a simple $2^{\text{nd}}$ order central
finite difference method is used for both Miranda and Ares data. For ease in
comparison, a single value for the effective viscosity,
$\overline{\mu_{\text{eff}}}$, is approximated by taking the peak value of the
span average of $\mu_{e}$, written as
$\overline{\mu_{\text{eff}}(t)}=\max\left({\langle\mu_{\text{eff}}({\bf
x},t\rangle}\right).$ (40)
Data for the Laplacian non-dimensionalized by the post shock velocity
($V_{0}$) and the smallest characteristic wave length of the initial
perturbation spectrum ($\lambda_{0}$) are plotted in Figure 15a for $\tau=35$
versus the non-dimensional inverse grid spacing or the number of points per
initial wave length. Data from the two Reynolds numbers at all resolutions are
plotted for both Miranda (blue) and Ares (red). An additional case which used
a Reynolds number 100 times larger than the nominal value of Table 1 was also
run and represents the inviscid limit of the flow.
(a) Inviscid scaling
(b) Viscous scaling
Figure 15: Left: Non-dimensional Laplacian of the strain-rate tensor, ${\bf
S}$, for all the cases in table 3 as a function of inverse grid spacing.
Right: Viscous scaling of the non-physical viscosity as a function of the grid
Reynolds number expression. Blue symbols are data from Miranda and red are
from Ares. The triangle, square and circle symbols correspond to a Reynolds
number of $100Re_{\lambda_{0}}$, $Re_{\lambda_{0}}$ and $Re_{\lambda_{0}}/25$,
respectively. The plus symbols reference additional cases described in Table
4.
When $\mathbf{S}$ becomes resolved, $\nabla^{2}{\bf S}$ will converge and the
whole expression in Eq. 38 will vanish as $(\Delta x)^{4}$. This rapid
convergence can be seen in Figure 15 in the circle symbols, which is data from
the DNS calculation. The slope of convergence is clearly steeper than that of
the LES calculations (triangles and squares) and indicates that $\bf S$ is
nearly converged.
For cases where the flow is clearly under-resolved, the magnitude of the
effective viscosity (see Figure 15a) will be proportional to $\Delta x^{-m}$
where $|m|<4$. For single shock RMI, both data sets suggest that the value of
$m$ is approximately -1.4. It will be shown later that for LES of high
Reynolds number turbulent flows, the value of $m$ is predicted by turbulence
theory to be $-4/3$, which is approximately $5\%$ of the measured value. These
convergence slopes are then used to non-dimensionalize the data over all
Reynolds numbers. At the point where the slope becomes $(-4+m)/2$, the
approximation is made that the artificial viscosity and the physical viscosity
are equivalent or that $\mu^{*}/\mu_{f}=1$. The degree of freedom used to
enforce this constraint gives an explicit value for $C_{\mu}$, which is
dependent on the numerical method of the scheme, but independent of grid
spacing and the physical viscosity of the problem. The values of $C_{\mu}$
were 8.11 and 63.13 in Miranda and Ares, respectively.
With $C_{\mu}$ in hand, the entire expression for $\mu^{*}$ is known and can
be non-dimensionalized by physical viscosity. The x-axis is also modified to
include the effects of both physical viscosity and the grid spacing by
computing the quotient $Re_{\lambda_{0}}^{R}/Re_{\Delta x}$. Here,
$Re_{\lambda_{0}}$ is the large scale Reynolds number given by $\rho
V_{0}\lambda_{0}/\mu_{f}$ and $Re_{\Delta x}$ is the grid Reynolds number
given by $\rho V_{0}\Delta x/\mu_{f}$. The exponent $R$ is given exactly as
$R=1+1/m$, which ensures that there is collapse of the data at different
physical viscosities. Note, that if the convergence of $\mu^{*}$ in the Euler
regime gives $m=-1$, then $R=0$ and the data collapse with $1/Re_{\Delta x}$.
The non-dimensionalization is performed and the data from all the cases are
plotted in Figure 15 along with the fiducial slopes for the different
convergence rates in each regime. The x-axis is shifted by a constant such
that $\alpha Re_{\lambda_{0}}^{R}/Re_{\Delta x}=1$ when $\mu^{*}/\mu_{f}=1$
where $\alpha$ is a constant for each code. For Miranda, $\alpha=10^{n}$ with
$n=-1.46$ and in Ares, $n=-1.16$. To the left of this line, the flow is under-
resolved and mostly dominated by non-physical dissipation. To the right,
physical viscosity has a large effect on the smallest of length scales and the
fourth order convergence indicates DNS levels of resolution.
With this form of the artificial viscosity and after having made the
aforementioned non-dimensionalization, one can readily answer two pertinent
questions for LES: given the numerics and SGS model of an LES approach, 1)
what resolution is needed for a DNS level calculation? 2) what is the
effective Reynolds number of an under-resolved LES calculation? The first asks
at which point the viscous scales become numerically resolved. The critical
point at which this transition occurred (when $\mu^{*}/\mu_{f}=1)$ is given as
$Re_{\lambda_{0}}^{R}/Re_{\Delta x}=1/\alpha$, where $\alpha$ was 28.84 in
Miranda and 14.46 in Ares. The ratio between the two ($\alpha_{A}/\alpha_{M}$)
can be used to compare DNS requirements. For example, for a given
$Re_{\lambda_{0}}$, if Miranda is predicted to reach a DNS regime at $\Delta
x_{0}$, Ares will reach a DNS regime at $\alpha_{A}/\alpha_{M}\Delta x_{0}$ or
$\Delta x_{0}/2.0$. Additionally, for a constant $\Delta x_{0}$ for both
codes, if Miranda can compute a DNS at $Re_{\lambda_{0}}$, Ares can compute a
DNS at $\left(\alpha_{A}/\alpha_{M}\right)^{-m}Re_{\lambda_{0}}$ or
$Re_{\lambda_{0}}/2.64$ using the same grid spacing.
The second question is relevant to under-resolved LES flows where the effect
of physical viscosity may be small and therefore, any Reynolds number which
uses that viscosity will have arbitrary significance. Instead, the effective
viscosity (Eq. 39) can be used to give a more realistic approximation of an
effective Reynolds number of the flow. This Reynolds number will be more
indicative of the resolved length scale separation between large production
scales and small dissipation scales. Reynolds number independence and
convergence of the large scale flow features will be highly dependent on this
Reynolds number. This effective Reynolds number can be written as
$Re_{\text{eff}}=Re_{0}\cdot\left(\frac{1}{1+\frac{\mu^{*}}{\mu_{f}}}\right).$
(41)
As $\mu^{*}/\mu_{f}$ vanishes with convergence of the DNS solution, the
effective Reynolds number will simply be the physical Reynolds number. For
under-resolved LES flows, $\mu^{*}/\mu_{f}$ will be arbitrarily large and lead
to a substantially lower effective Reynolds number of the flow.
Figure 16: Non-dimensional Kolmogorov length scale vs. effective Reynolds
number at $\tau=30$. The symbol references are the same as in Figure 15. The
dashed line is from Kolmogorov theory (Eq. 43) and agrees quite well the
measured values.
To verify that a smaller effective Reynolds number does indeed lead to a
smaller range of length scales in the flow, the Kolmogorov length scale is
evaluated within the mixing layer. The Kolmogorov length scale is computed as
$\eta_{\text{eff}}=\left(\frac{\nu^{3}}{\epsilon}\right)^{1/4}$ (42)
where $\nu=\mu_{\text{eff}}/\rho$ is the effective viscosity and where
$\epsilon=2\nu{\mathbf{S}}^{2}$ is being used to approximate the effective
dissipation rate. The effective Kolmogorov length scale plotted in Figure 16
shows a clear relationship with the effective Reynolds number and a small
dependence on the physical Reynolds number of the flow. Indeed, for
sufficiently high Reynolds number, one may assume a balance between the mean
turbulent kinetic energy and dissipation rate, $\lambda_{0}\sim
k^{3/2}/\epsilon$, as suggested from Kolmogorov theory. Therefore, using the
definition of $\eta_{\text{eff}}$, one may write an approximate scaling of
$\eta_{\text{eff}}$ in terms of the effective Reynolds number as
$\frac{\eta_{\text{eff}}}{\lambda_{0}}\sim Re_{\text{eff}}^{-3/4}.$ (43)
This approximate scaling is plotted in Figure 16 which shows good agreement
with the actual data. The relationship in Equation 43 implies a scaling of the
effective viscosity with grid spacing. Earlier, it was reported that
$\mu^{*}\sim\left(1/\Delta x\right)^{m}$ where $m$ was measured to be
$\approx-1.4$. One can derive an exact value for $m$ using Eq. 43, the
definition of $Re_{\text{eff}}$, and the approximation that $\eta\sim\Delta x$
and show that $m=-4/3$. As the data have indicated, this value and the
assumptions needed to derive it, are valid for small vales of $\alpha
Re_{\lambda_{0}}^{R}/Re_{\Delta x}$, away from the DNS regime.
#### IV.0.2 Effective Species diffusivity
In problems of turbulent multi-component mixing, numerical dissipation will
directly affect the diffusive flux of differing materials. Therefore, the
resolved gradients of species mass fraction will largely depend on the
numerical scheme, grid resolution and the Reynolds and Schmidt numbers of the
flow. By similar arguments as the effective viscosity, construction of an
effective diffusivity can elucidate the differences between methods,
resolutions and physical parameters used in LES. Using the form of the
effective viscosity as a template and using $\nabla Y\cdot\nabla Y$ as an
indicator for scalar dissipation, the numerical portion is written as
$D^{*}=C_{D}c_{s}\left|\nabla^{2}\left(\sqrt{\nabla Y\cdot\nabla
Y}\right)\right|\Delta x^{4}$ (44)
where $c_{s}$ is the sound speed and $C_{D}$ is a code dependent coefficient.
The form of $D^{*}$ follows that of $\mu^{*}$ where the magnitude of $S$ has
been replaced with the magnitude of $\nabla Y$ and where $C_{\mu}\rho$ has
been replaced with $C_{D}c_{s}$. For two component flow the $Y$ can be the
mass fraction from either gas. The effective diffusivity is the sum of the
numerical and physical portion, written as
$D_{\text{eff}}=D^{*}+D_{f}\ .$ (45)
Similar to the $\mathbf{S}$ in the effective viscosity expression, as $\nabla
Y$ becomes resolved in the DNS limit $D_{f}/D_{\text{eff}}\to 1$. For under
resolved simulations where the numerical diffusivity dominates,
$D_{f}/D_{\text{eff}}\to 0$. Figure 17a shows the Laplacian of $|\nabla Y|$
non-dimensionalized by inviscid mean flow variables and plotted as a function
of the number of grid points per $\lambda_{0}$. The data show two convergence
rates and can be non-dimensionalized in an analogous fashion to
$\mu_{\text{eff}}$, where the Péclet number
($Pe_{\lambda_{0}}=Sc_{0}Re_{\lambda_{0}}$) is the relevant non-dimensional
number. Data indicate that $m=-1.4$ (the same value as $\mu_{\text{eff}}$)
which is the slope of the data from the under resolved calculation. The
coefficient $\alpha$ used to scale the x-axis such that $D^{*}/D_{f}=1$ when
$\alpha Pe_{\Delta x}^{R}/Pe_{\lambda_{0}}=1$ is $10^{1.77}$ in Miranda and
$10^{1.71}$ in Ares. Again, by construction, $R=1+1/m$, which is constant for
all cases and codes. This gives coefficients $C_{D}$ of .039 and .097 for
Miranda and Ares, respectively. Figure 17 shows the non-dimensional numerical
diffusion in the under resolved and resolved regions. The fiducial slopes
indicate where the flow is becoming resolved on the grid. Ares (red) data are
shifted slightly to the left of the Miranda data, indicating that Miranda
solutions reach DNS levels of convergence at a slightly coarser resolution
than Ares. Therefore, for a given grid resolution and physical Reynolds
number, one would expect higher values of $Pe_{\lambda_{0}}$ and smaller
scalar length scales in Miranda.
(a) Inviscid scaling
(b) Diffusive scaling
Figure 17: Left: Non-dimensional Laplacian of the magnitude of the scalar
gradient, $\sqrt{\nabla Y\cdot\nabla Y}$, for all the cases in table 3 as a
function of inverse grid spacing. Right: Viscous scaling of the non-physical
diffusivity as a function of the grid Péclet number expression. Blue symbols
are data from Miranda and red are from Ares. The triangle, square and circle
symbols correspond to a Reynolds number of $100Re_{\lambda_{0}}$,
$Re_{\lambda_{0}}$ and $Re_{\lambda_{0}}/25$, respectively. The plus symbols
reference additional cases described in Table 4. Figure 18: Non-dimensional
Batchelor length scale vs. effective Péclet number at $\tau=30$. The symbol
references are the same as in Figure 17. The dashed line is the scaling for
$\lambda_{bch}$ as predicted by Kolmogorov theory (Eq. 48) which shows good
agreement with the data.
Similar to the Kolmogorov length scale, the Batchelor scale describes the
smallest length scales in the scalar gradient that can exist before diffusion
dominates. This length scale can be related to the Kolmogorov scale as
$\lambda_{bch}=\frac{\eta}{Sc_{\text{eff}}^{1/2}}\ $ (46)
where the effective Schmidt number is defined as
$Sc_{\text{eff}}=\frac{\mu_{\text{eff}}}{\rho D_{\text{eff}}}\ .$ (47)
The non-dimensional Batchelor scale is plotted in Figure 18 and shows an
exponential relationship with $Pe_{\text{eff}}$. Indeed, Kolmogorov theory
also suggest a scaling of the Batchelor scale and Péclet number as
$\frac{\lambda_{bch}}{\lambda_{0}}\sim\left(Pe_{\text{eff}}\right)^{-3/4},$
(48)
which is plotted as dashed line in Figure 18 and shows good agreement with the
actual data. Similar to the artificial viscosity, it can be shown that the
artificial species diffusivity scales as $D^{*}\sim\left(1/\Delta
x\right)^{m}$. Measured data and theory predict the value of $m$ to be,
respectively, $-1.4$ and $-4/3$, identical to the values associated with
$\mu^{*}$ in the LES regime. The effective Schmidt number is also plotted vs.
$Pe_{\text{eff}}$ in Figure 19 and shows that Miranda data have a slightly
higher Schmidt number than Ares.
Figure 19: Effective Schmidt number vs. effective Péclet number at $\tau=30$
in the mixing layer. The physical Schmidt number was between .18 and 1.11 (see
Table 1) on the heavy and light side, respectively. The symbol references are
the same as in Figure 17.
## V Discussion of LES requirements
LES results and the effective viscosity/diffusivity suggest that dissipation
from the numerical method, grid resolution, and physical properties affect the
small scales of motion. The above framework enables all three sources of
dissipation to be assessed directly by examination of the large data set. As
one might expect, the low order code produced larger amounts of effective
viscosity than the higher order code. The difference between the two can be
quantified as the equivalent $\Delta x_{lo}$ needed in the low order code
(Ares), to have an equal amount of effective viscosity as the high order code
(Miranda) at grid spacing $\Delta x_{ho}$. The ratio between mesh spacing when
$\mu^{*}_{lo}/\mu^{*}_{ho}=1$ is defined as $N\equiv\Delta x_{ho}/\Delta
x_{lo}$. It was observed that the value of $N$ was dependent on the level of
resolution of the physical viscous scales. Near the DNS limit, it was
previously shown that $N=(\alpha_{ho}/\alpha_{lo})$. Away from the DNS regime
$N$ was larger as evidenced by the poor collapse between codes in Figure 15b
for small values of $\alpha Re^{R}_{\lambda}/Re_{\Delta x}$. The upper bound
for $N$ can be approximated as $N=\left(C_{\mu,ho}/C_{\mu,lo}\right)^{1/m}$
which assumes that the $\nabla^{2}{\bf S}$ is the same between codes for a
given case. Therefore, the equivalent grid spacing can be expressed as
$\frac{\alpha_{ho}}{\alpha_{lo}}\leq
N\leq\left(\frac{C_{\mu,ho}}{C_{\mu,lo}}\right)^{1/m}\ .$ (49)
It is also important to note that for three-dimensional time dependent
simulations, the additional cost of running a calculation at a finer grid
spacing scales approximately with $N^{4}$ and will be less if using AMR. The
predicted bound of $N$ for the viscous terms at $\tau=30$ was $2.0\leq N\leq
4.3$ which is consistent with the time histories of enstrophy and in the
spectra of the velocity field. For example, in the velocity spectra and
enstrophy plots (Figures 11 and 13), the data from the mesh D Ares calculation
lies in between data from mesh B and C from the Miranda calculation. These
Miranda data are 2 and 4 times as coarse which is consistent with the
predicted bounds on $N$ evaluated from Equation 49.
The resolution difference of the scalar field was less pronounced than in the
velocity field and equation 49 (where $\mu$ is replaced with $D$) gives
$1.15\leq N\leq 1.92$. Here, the mass fraction spectra and scalar dissipation
(Figures 12 and 14) show that the data from the scalar dissipation rate and
the spectra of the density are slightly less than a factor of two different
between Ares and Miranda, which again is consistent with the $N$ from Equation
49.
The value of $N$ for the viscous and diffusive scales are supported by the
measured effective Kolmogorov and Batchelor length scales. Both the Kolmogorov
and Batchelor length scales represent the smallest length scales of turbulent
motion, where fluctuations are dissipated by the viscosity of diffusivity. The
lower numerical dissipation in Miranda leads to smaller values of these inner
diffusive scales and therefore, a broader inertial range of turbulent
fluctuations (see Figure 11). As stated earlier, it is this range which must
be sufficiently large as to produce large scale LES results which are grid
independent and which approximate a real flow in the Reynolds number
independent regime. From the present data, it was observed that sufficient
scale separation occurred at and above $Re_{\text{eff}}=2500$, which was
represented on grid C and D in Miranda and grid D in Ares. Such information
could be used in approximating the resolution requirements for a given scheme
and Reynolds number if one wanted to compute either a DNS solution or grid
independent LES solution. We note that a grid independent LES calculation of a
Reynolds number dependent flow must be a DNS if the flow is truly grid
independent. Furthermore, as was implicit in the analysis of Aspden et al.,
the maximum Reynolds number that a given mesh can capture at DNS resolution
must always be less than the effective Reynolds number of an Euler calculation
on that same mesh.
As an _a posteriori_ test of the above analysis to approximate the level of
resolution of the simulated flow, two additional simulations were run. One in
Miranda using $Re_{\lambda_{0}}/10$ at mesh C resolution and one in Ares using
$Re_{\lambda_{0}}/50$ at mesh C resolution. For the viscous terms, the
relative resolution metric, $\alpha Re^{R}_{\lambda_{0}}/Re_{\Delta x}$, was
0.87 in Miranda and 1.34 in Ares. The Miranda case falls in the under-resolved
regime (since $\alpha Re^{R}_{\lambda_{0}}/Re_{\Delta x}<1$) and the predicted
value by a fit from data in Figure 15b of $\mu^{*}/\mu_{f}$ is 1.59. The
actual measured value for $\mu^{*}/\mu_{f}$ was 1.57 which is strikingly close
to the predicted value considering the analysis is in logarithmic space. These
data are also plotted in Figure 15b as cross symbols and show good agreement
with the other non-dimensionalized data. This same assessment is made for the
Ares data and for both viscous and diffusive terms. The results are summarized
in Table 4 and plotted as cross symbols in Figures 15b and 17b.
Case | $\alpha Re^{R}_{\lambda_{0}}/Re_{\Delta x}$ | $\mu^{*}/\mu_{f}$ | $\alpha Pe^{R}_{\lambda}/Pe_{\Delta x}$ | $D^{*}/D_{f}$
---|---|---|---|---
| | measured | predicted | | measured | predicted
$Re_{\lambda_{0}}/10$ (Miranda) | 0.87 | 1.57 | 1.59 | 1.30 | 0.64 | 0.65
$Re_{\lambda_{0}}/25$ (Ares) | 1.34 | 0.18 | 0.45 | 3.74 | 0.016 | 0.022
Table 4: Summary of an _a posteriori_ test of the analysis in Section IV.0.1
using two independent calculations in Ares and Miranda. The analysis predicts
the observed dissipation measures ($\mu^{*}/\mu_{f}$ and $D^{*}/D_{f}$) quite
well, when compared to the collapsed data in Figures 15b and 17b.
It is certainly not feasible to conduct the full analysis contained in this
study for every LES problem one encounters. However, once the coefficients are
determined, one can expect Cook (2007) them to be fairly universal across a
broad range of turbulent flows. Therefore, on new problems where little
resolution requirement information is known, one can quite easily compute
$\mu_{f}/\mu_{\text{eff}}$ as a scalar quantity in the flow field and
determine which regions are under resolved or where
$\mu_{f}/\mu_{\text{eff}}<<1$. Such an indicator can be useful for flagging
regions which are to undergo adaptive mesh refinement (AMR) or indicate where
numerical errors arising from non-physical dissipation are expected to be most
pronounced. Furthermore, since it has been shown that
$\mu^{*}/\mu_{f}\sim\left(1/\Delta x\right)^{m}$ in the under-resolved LES
regime, a reasonable prediction in the grid spacing needed to reach the
minimal DNS requirement ($\mu^{*}=\mu_{f}$) can be provided by the expression,
$\Delta x_{DNS}\approx\frac{\Delta
x_{LES}}{\left(\mu^{*}/\mu_{f}\right)_{LES}^{1/m}}$ (50)
where the “LES” subscripts make reference to value from an under-resolved LES
calculation.
## VI Conclusion
We have investigated the effects of numerical method, grid resolution and
Reynolds number on the Richtmyer-Meshkov instability through a suite of LES
and DNS calculation in the Ares and Miranda codes. Four mesh resolutions were
used between the two codes in the simulation of the RMI using five different
Reynolds numbers. Large scale integral quantities such as mixing layer width
and integral mixedness were compared and showed close agreement under
refinement. Frequency dependent terms demonstrated dependence on the mesh,
scheme and Reynolds number of the flow. Gradient based terms which were
related to dissipation rates also showed large dependence on the difference
sources of dissipation. The results confirm the expected behavior, that the
high-order method captures and a broader range of length scales and has better
convergence than the low-order method. Although this finding is not
particularly novel, the fidelity of the simulation database is novel, and
therefore, has been interrogated to establish a new framework for LES
comparisons.
A simple form for an effective viscosity and diffusivity were proposed and
applied _a posteriori_ to the data and which indicate the cumulative amount of
dissipation in the flow field. The effective viscosity and diffusivity
scalings collapse all the data between codes, resolutions and physical
Reynolds numbers in one common framework which indicates the breadth of the
dynamic range of scales supported in a particular LES calculation. An
effective Reynolds number was also constructed which indicated that grid
independence occurs at $Re_{\text{eff}}>2500$ and that the smallest viscous
and diffusive scales supported on the grid are proportional to, respectively,
the effective Reynolds number and Péclet number to the -3/4 power. The
effective viscosity and diffusivity can be used to determine regions of under
resolved flow and make predictions of the level of resolution needed to either
produce a DNS result or an LES solution which is grid independent. The
predictive capability of the framework was assessed for two additional,
independent calculations which showed excellent collapsed onto the original
data.
## Acknowledgements
This work was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under contract number DE-
AC52-07NA27344. The authors wish to thank A. Cook, W. Cabot, O. Schilling and
B. Morgan for many valuable discussions and for help in running the codes.
## References
* (1)
* Aspden et al. (2008) Aspden, A., Nikiforakis, N., Dalziel, S. & Bell, J. B. (2008), ‘Analysis of implicit LES methods’, Comm. App. Math. and Comp. Sci. 3-1.
* Balakumar et al. (2008) Balakumar, B. J., Orlicz, G. C., Tomkins, C. D. & Prestridge, K. P. (2008), ‘Simultaneous particle-image velocimetry-planar laser-induced fluorescence measurements of Richtmyer-Meshkov instability growth in a gas curtain with and without reshock’, Physics of Fluids 20, 124103.
* Berger & Colella (1989) Berger, M. & Colella, P. (1989), ‘Local adaptive mesh refinement for shock hydrodynamics’, J. Comp. Phys. 82, 64–84.
* Berger & Oliger (1984) Berger, M. & Oliger, J. (1984), ‘Adaptive mesh refinement for hyperbolic partial differential equations’, J. Comp. Phys. 53, 484–512.
* Boris et al. (1992) Boris, J. P., Grinstein, F. F., Oran, E. S. & Kolbe, R. L. (1992), ‘New insights into large eddy simulation’, Fluid Dynamics Research 10, 199–229.
* Cabot & Cook (2006) Cabot, W. H. & Cook, A. W. (2006), ‘Reynolds number effects on Rayleigh-Taylor instability with possible implications for type-1a supernovae’, Nature Phys. 2, 562.
* Cook (2007) Cook, A. W. (2007), ‘Artificial fluid properties for large-eddy simulation of compressible turbulent mixing’, Physics of Fluids 19, 055103.
* Cook (2009) Cook, A. W. (2009), ‘Enthalpy diffusion in multicomponent flows’, Physics of Fluids 21, 055109.
* Cook et al. (2004) Cook, A. W., Cabot, W. & Miller, P. L. (2004), ‘The mixing transition in Rayleigh-Taylor instability’, J. Fluid Mech. 511, 333.
* Grinstein et al. (2007) Grinstein, F. F., Margolin, L. G. & Rider, W. J. (2007), Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics, Cambridge University Press.
* Grinstein et al. (2011) Grinstein, F., Gowardhan, A. & Wachtor, A. (2011), ‘Simulations of Richtmyer-Meshkov instabilities in planar shock-tube experiments’, Physics of Fluids 23, 034106.
* Grinstein & Guirguis (1992) Grinstein, F. & Guirguis, R. (1992), ‘Effective viscosity in the simulation of spatially evolving shear flows with monotonic FCT models’, J. Comp. Phys. 101, 165–175.
* Hill et al. (2006) Hill, D. J., Pantano, C. & Pullin, D. I. (2006), ‘Large-eddy simulation and multi scale modeling of a Richtmyer-Meshkov instability with reshock’, J. Fluid Mech. 557, 29–61.
* Jacobs & Krivets (2005) Jacobs, J. & Krivets, V. (2005), ‘Experiments on the late-time development of single-mode Richtmyer-Meshkov instability’, Physics of Fluids 17 (3), 034105.
* Jacobs & Sheely (1996) Jacobs, J. & Sheely, J. (1996), ‘Experimental study of incompressible Richtmyer-Meshkov instability’, Physics of Fluids 8 (2), 405–415.
* Johnsen et al. (2010) Johnsen, E., Larsson, J., Bhagatwala, A. V., Cabot, W. H., Moin, P., Olson, B. J., Rawat, P. S., Shankar, S. K., Sjögreen, B., Yee, H., Zhong, X. & Lele, S. K. (2010), ‘Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves’, J. Comp. Phys. 229, 1213–1237.
* Kawai & Lele (2008) Kawai, S. & Lele, S. K. (2008), ‘Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes’, J. Comp. Phys. 227, 9498–9526.
* Kolev & Rieben (2009) Kolev, T. & Rieben, R. (2009), ‘A tensor artificial viscosity using a finite element approach’, J. Comp. Phys. 228, 8836–66.
* Kravchenko & Moin (1997) Kravchenko, A. G. & Moin, P. (1997), ‘On the Effect of Numerical Errors in Large Eddy Simulations of Turbulent Flows’, J. Comp. Phys. 131, 310–322.
* Landau & Lifshitz (2004) Landau, L. & Lifshitz, E. (2004), Fluid Mechanics, Pergamon Press.
* Latini et al. (2007) Latini, M., Schilling, O. & Don, W. S. (2007), ‘Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer-Meshkov instability’, J. Comp. Phys. 221, 805–836.
* Mani et al. (2009) Mani, A., Larsson, J. & Moin, P. (2009), ‘Suitability for artificial bulk viscosity for large-eddy simulation of turbulent flows with shocks’, J. Comp. Phys. 228, 7368–74.
* Meneveau & Katz (2000) Meneveau, C. & Katz, J. (2000), ‘Scale Invariance and Turbulence Models for Large-Eddy Simulation’, Annu. Rev. Fluid Mech. 32, 1–32.
* Olson & Cook (2007) Olson, B. J. & Cook, A. W. (2007), ‘Rayleigh-Taylor shock waves’, Physics of Fluids 19, 128108.
* Olson et al. (2011) Olson, B. J., Larsson, J., Lele, S. K. & Cook, A. W. (2011), ‘Non-linear effects of the combined Rayleigh-Taylor/Kelvin-Helmholtz instability’, Physics of Fluids 23, 114107.
* Olson & Lele (2013) Olson, B. J. & Lele, S. K. (2013), ‘Directional artificial fluid properties for compressible large-eddy simulation’, J. Comp. Phys. 246, 207–220.
* Reynolds (1990) Reynolds, W. C. (1990), ‘Whither Turbulence? Turbulence at the Crossroads: The potential and limitations of direct and large-eddy simulation’, Lecture Notes in Physics 357, 313–343.
* Shankar & Lele (2012) Shankar, S. K. & Lele, S. K. (2012), ‘Numerical Investigation of Turbulence in Re-shocked Richtmyer-Meshkov Unstable Curtain of Dense Gas’, Int. Sym. of Shock Waves pp. 329–334.
* Thornber et al. (2010) Thornber, B., Drikakis, D. & Youngs, D. (2010), ‘The influence of initial conditions on turbulent mixing due to Richtmyer-Meshkov instability’, J. Fluid Mech. 654, 99–139.
* Thornber et al. (2007) Thornber, B., Mosedale, A. & Drikakis, D. (2007), ‘On the implicit large eddy simulations of homogeneous decaying turbulence’, J. Comp. Phys. 226, 1902–1929.
* Vetter & Sturtervant (1995) Vetter, M. & Sturtervant, B. (1995), ‘Experiments on the Richtmyer-Meshkov instability of an air/SF6 interface’, Shock Waves 4, 247.
* Wilkins (1963) Wilkins, M. (1963), ‘Calculation of elastic plastic flow’, Report UCRL, LLNL, CA 7322.
* Zhou et al. (2014) Zhou, Y., Grinstein, F. F., Wachtor, A. J. & Haines, B. M. (2014), ‘Estimating the effective Reynolds number in implicit large-eddy simulation’, Phys. Rev. E 89, 013303.
|
arxiv-papers
| 2014-02-11T21:13:09 |
2024-09-04T02:49:58.124157
|
{
"license": "Public Domain",
"authors": "Britton J. Olson and Jeffrey A. Greenough",
"submitter": "Britton Olson",
"url": "https://arxiv.org/abs/1402.2665"
}
|
1402.2712
|
11institutetext: School of Computer and Mathematical Sciences
Auckland University of Technology, New Zealand
11email: [email protected], 11email: [email protected]
# Dynamic Partial Sorting
Jiamou Liu Kostya Ross
###### Abstract
The dynamic partial sorting problem asks for an algorithm that maintains lists
of numbers under the link, cut and change value operations, and queries the
sorted sequence of the $k$ least numbers in one of the lists. We first solve
the problem in $O(k\log(n))$ time for queries and $O(\log(n))$ time for
updates using the tournament tree data structure, where $n$ is the number of
elements in the lists. We then introduce a layered tournament tree data
structure and solve the same problem in $O(\log_{\varphi}^{*}(n)k\log(k))$
time for queries and $O\left(\log(n)\cdot\log^{2}(\log(n))\right)$ for
updates, where $\varphi$ is the golden ratio and $\log_{\varphi}^{*}(n)$ is
the iterated logarithmic function with base $\varphi$.
## 1 Introduction
The problem setup. Many practical applications store data in a collection of
key-value pairs where the keys are drawn from an ordered domain. In such
applications, queries would be made on the order statistics of values within a
subset of keys. Consider as an example the loan data of a library. This can be
represented by an ordered map whose keys are structured indices indicating the
category of the book, and whose values are the number of times the book was
borrowed since acquisition. One possible type of queries involves retrieving
the most popular (or least popular) books from various categories or
subcategories. Facilitating such queries is an inherently dynamic problem;
firstly, the subsets of the ordered map for whose values order statistics are
desired can vary, and secondly, ordered maps typically represent mutable data,
which requires values and keys to change.
Existing algorithms and data structures cannot effectively solve this problem.
If we want to represent an mutable ordered map, the standard solution (a self-
balancing binary search tree) cannot efficiently extract order statistics
about its values. On the other hand, existing selection algorithms for data
with structured keys rely on the data being static, which in a dynamic
context, would force a re-run of the algorithm on every change. Neither of
these are desirable, especially when the data being represented by the ordered
map is large. This leads to a need for a solution that can effectively extract
order statistics about values, while being amenable to data mutation.
Abstractly, we may view an ordered map as a list of numbers, where elements
are arranged in the list by their keys. The above queries then amount to
obtaining order statistics of numbers within intervals of the list. Formally,
we propose the dynamic partial sorting problem, which is stated as follows:
Maintain a collection of lists $\ell_{1},\ell_{2},\cdots,\ell_{m}$ of numbers,
while allowing the following partial sorting operation:
* •
$\mathsf{psort}(\ell_{i},k)$: Return the $k$ smallest numbers in $\ell_{i}$ if
$k$ is at most the size of $\ell_{i}$, and all elements in $\ell_{i}$
otherwise. The output should be in increasing order.
We also support the following update operations:
* •
$\mathsf{changeval}(\ell_{i},x,x^{\prime})$: Suppose $x$ is a number in
$\ell_{i}$; change $x$ to $x^{\prime}$.
* •
$\mathsf{link}(\ell_{i},\ell_{j})$: Link the lists $\ell_{i}$ and $\ell_{j}$
by attaching the tail of $\ell_{i}$ to the head of $\ell_{j}$.
* •
$\mathsf{cut}(\ell_{i},x)$: Suppose $x$ is a number in $\ell_{i}$; separate
$\ell_{i}$ into two lists, such that the first list contains all elements from
the head of $\ell_{i}$ to $x$ inclusive, and the second list contains all
other elements of $\ell_{i}$.
We assume the parameter $x$ in the $\mathsf{cut}(\ell_{i},x)$ and
$\mathsf{changeval}(\ell_{i},x,x^{\prime})$ operations points directly to the
element $x$ in $\ell_{i}$, and therefore no searching is necessary. In this
paper, we are only going to focus on the $\mathsf{link}$, $\mathsf{cut}$,
$\mathsf{changeval}$ and $\mathsf{psort}$ operations as defined above. We
observe that these operations also permit partial sorting on arbitrary
intervals in a list.
Dynamically maintaining a sorted list of numbers is a well-explored topic.
Existing solutions include utilizing various self-balancing binary search
trees [1]. These data structures are not suitable for the dynamic partial
sorting problem, as here we require elements in the lists to preserve their
orders while extracting order statistics from the lists. To the authors’
knowledge, there has not been work formally addressing the dynamic partial
sorting problem. Here we describe some naive algorithms for solving the
problem:
The first naive solution to the dynamic partial sorting problem is to simply
put numbers in a linked list. Thus $\mathsf{link}(\ell,\ell^{\prime})$,
$\mathsf{cut}(\ell,x)$ and $\mathsf{changeval}(\ell,x,x^{\prime})$ are solved
in constant time, but to perform $\mathsf{psort}(\ell,k)$, we run an optimal
static algorithm such as quick select [7] and partial quicksort [12], which
take time $O(n+k\log(k))$.
The second naive solution to the dynamic partial sorting problem is to store
the numbers in each list in a priority queue. This allows us to perform
$\mathsf{psort}(\ell,k)$ by repeatedly removing and returning the minimum
item, and then re-inserting those items afterwards. The running time of
$\mathsf{psort}(\ell,k)$ is $O(k\log(n))$, where $n$ is the number of elements
in $\ell$. We can perform $\mathsf{link}(\ell,\ell^{\prime})$ and
$\mathsf{cut}(\ell,x)$ by successively inserting or deleting elements from the
priority queues of the lists. Hence each of these operations takes
$O(n\log(n))$.
Related work. Bordim et al have employed a partial sorting algorithm to solve
problems in common-channel communication over single-hop wireless sensor
networks [2]. Additionally, the problem has been generalized to sorting
intervals [9]. The asymptotic time complexity of partial sorting has been
thoroughly studied [11, 8, 5].
Several data structures for partial sorting have been described. Navarro and
Paredes proposed one such structure in [13], but it is optimized for use of
memory, rather than time, and is both amortized and online. Duch et al
presented another structure in [3] for the selection problem which can be used
for partial sorting. However the structure is not dynamic, and depends heavily
on the length of the input data.
Contribution of the paper. The goal of the paper is to design a solution to
the dynamic partial sorting problem where the query and update operations have
better time complexity. We first describe a solution that is based on the
tournament tree data structure. A tournament tree of a list of numbers is a
full balanced binary tree whose leaves are the elements of the list and the
value of every internal node is the minimum of the values of its two children.
Hence any node in the tournament tree stores the minimum number in the subtree
rooted at this node. Based on this observation, we perform the
$\mathsf{psort}(\ell,k)$ operation in time $O(k\log(n))$. We perform
$\mathsf{changeval}(\ell,x,x^{\prime})$ in $O(\log(n))$ time by updating the
path from $x$ to the root. The link and cut operations are handled in a
similar way as linking and cutting balanced binary trees, and thus take time
$O(\log(n))$.
The tournament tree solution to the partial sorting problem allows efficient
query and update operations. However, the time complexity of the
$\mathsf{psort}(\ell,k)$ operation depends both on $k$ and the size $n$ of the
list $\ell$. In practical applications where $n$ could be much larger than
$k$, it is desirable to make the running time of the query operation
independent from $n$. Therefore we develop another dynamic algorithm that
solves the dynamic partial sorting problem with the following properties:
* •
We handle $\mathsf{psort}(\ell,k)$ in such a way that the size $n$ of $\ell$
has minimal influence on the time complexity of the operation.
* •
The time complexity of the update operations is not much worse than the
tournament-tree-based algorithm above. More precisely, the update operations
run in $o(\log^{2}(n))$.
To this end, we introduce a recursive data type called the layered tournament
tree data structure. The main idea is that, instead of using one tournament
tree to store the items in a list, we use multiple layers of tournament trees.
The layers extend downwards. The top layer consists of the tournament tree of
the list. This tournament tree is partitioned into teams where each team can
be viewed as a path segment of the tree. Each of these teams is then
represented by a tournament tree in the layer below, where elements of the
team correspond to leaves in the tree. The tournament tree of a team is again
partitioned into teams which are represented by tournament trees in the
subsequent layer. This process continues until the team consists of only one
node. Since we maintain the tournament trees as balanced trees, we can
guarantee that a tree in a particular layer has logarithmic size compared to
the corresponding tree in the layer above.
We define the partial sort operations for tournament trees on every layer of
the data structure. Using an iterative algorithm that recursively calls the
partial sort operation in lower layers, we perform the
$\mathsf{psort}(\ell,k)$ operation on the original list $\ell$. The time
complexity of the operation is $O\left(\log_{\varphi}^{*}(n)k\log(k)\right)$
where $n$ is the number of items in $\ell$, $\varphi=\frac{\sqrt{5}+1}{2}$ is
the golden ratio and $\log_{\varphi}^{*}(n)$ is the iterated logarithmic
function with base $\varphi$ (See Section 5 for a definition). Since the
function $\log_{\varphi}^{*}(n)$ is almost constant even for very large values
of $n$, the running time of $\mathsf{psort}(\ell,k)$ is almost independent
from $n$. The time complexity of the $\mathsf{link}(\ell,\ell^{\prime})$,
$\mathsf{cut}(\ell,x)$ and $\mathsf{changeval}(\ell,x,x^{\prime})$ operations
is $O\left(\log n\cdot\log^{2}(\log n)\right)$.
Organization. Section 2 introduces the tournament tree data structure. Section
3 describes the solution to the dynamic partial sorting problem using
tournament trees. Section 4 introduces the layered tournament tree data
structure. Section 5 and Section 6 discuss the algorithms for the
$\mathsf{psort}(\ell,k)$ operation and the update operations using layered
tournament trees, respectively. Section 7 concludes the paper and discusses
future work.
## 2 Tournament Trees
A list is an ordered tuple of numbers. We write a list $\ell$ as
$a_{1},a_{2},a_{3},\ldots,a_{k}$ where $k$ and each element $a_{i}$ is a
natural number. Throughout the paper we assume that the elements in a list are
pairwise distinct.
Trees. We assume a pointer-based computation model for our tree data
structure. This means that every node in the tree has a reference that points
to its parent. We normally use $T$ for a tree and $V$ for the set of nodes in
$T$. The size of a tree $T$ is $|V|$. For every node $v\in V$, we use $p(v)$
to denote the parent of $v$ if $v$ is not the root, and set
$p(v)=\mathsf{null}$ otherwise.
We will use binary trees to represent lists of numbers. The fields of any node
$v\in V$ in a binary tree consist of a tuple
$(p(v),\mathsf{le}(v),\mathsf{ri}(v),\mathsf{val}(v))$
where $\mathsf{le}(v),\mathsf{ri}(v)$ are respectively the left child and
right child of $v$. The field $\mathsf{val}(v)$ is a integer value associated
with the node $v$.
We use $T(v)$ to denote the subtree rooted at $v$. A path is a set of nodes
$\\{u_{0},u_{1},\ldots,u_{m}\\}$ where $m\in\mathbb{N}$, $u_{0}$ is a leaf and
$u_{i+1}=p(u_{i})$ for $0\leq i<m$. We call $m$ the length of the path.The
height $h(T)$ of a tree $T$ is the maximum length of any path in $T$. A binary
tree $T$ is balanced if for every node $v\in V$,
$|h(T(\mathsf{le}(v))-h(T(\mathsf{ri}(v))|\leq 1$. A binary tree is full if
every internal node has exactly two children, i.e., the $\mathsf{le}(v)$ and
$\mathsf{ri}(v)$ fields are both non-null.
Tournament trees. The tournament tree data structure is inspired by the
tournament sort algorithm, which uses the idea of a single-elimination
tournament in selecting the next element [10]. Formally, the data structure is
defined as follows:
###### Definition 1
A tournament tree of a list $\ell$ of numbers $a_{1},a_{2},a_{3},\ldots,a_{n}$
is a balanced full binary tree $T$ that satisfies the following properties:
1. 1.
The tree has exactly $n$ leaves whose values are $a_{1},a_{2},\ldots,a_{n}$
respectively.
2. 2.
For every internal node $v\in V$, if $\mathsf{val}(\mathsf{le}(v))=a_{i}$ and
$\mathsf{val}(\mathsf{ri}(v))=a_{j}$, then $i<j$ and
$\mathsf{val}(v)=\mathsf{min}\\{a_{i},a_{j}\\}$.
See Figure 1 for an example of a tournament tree. Intuitively, one can view a
tournament tree of a list of numbers as a binary search tree, where the
numbers are stored in the leaves. The key of each leaf in the binary search
tree is the index of the number it stores in the list, and the value is the
number itself.
22336292444783$\ell:$692478
Figure 1: A tournament tree of a list $\ell=3,6,9,2,4,7,8$. Edges in principal
paths are bolded.
As a tournament tree is balanced, its height is logarithmic with respect to
the number of leaves. More specifically we prove the following lemma.
###### Lemma 1
If $T$ is a tournament tree with $n$ leaves where $n>0$, then the height of
$T$ is not more than $\log_{\varphi}(n)$ where $\varphi$ is the golden ratio.
###### Proof
It suffices to show that the least number of leaves $f(h)$ in any tournament
tree with height $h\geq 0$ is $\varphi^{h}$, where
$\varphi=\frac{\sqrt{5}+1}{2}$ is the golden ratio. The lemma can be easily
proved using the following observation. Note that here we use the fact that a
tournament tree is balanced and full.
$f(h)\geq\begin{cases}1&\text{ if $h=0$,}\\\ 2&\text{ if $h=1$,}\\\
f(h-1)+f(h-2)&\text{ if $h\geq 2$.}\end{cases}$
∎
## 3 Dynamic Partial Sorting With Tournament Trees
We now describe an algorithm for solving the dynamic partial sorting problem
based on tournament trees. The algorithm assumes that any list $\ell$ of
numbers is represented as a tournament tree $T$, whose leaves are the elements
of $\ell$. Therefore, we will refer to a list and its tournament tree
interchangeably. Furthermore, when we refer to an element $x$ of $\ell$, we
also mean the leaf $u$ in $T$ with value $x$ and vice versa. All terms that
relate to a tournament tree $T$ carry forward to the corresponding list
$\ell$. Hence the nodes, root, leaves, and internal nodes of $\ell$ refer to
the equivalent concepts in $T$.
Let $\ell$ be a list of numbers. We list the elements of $\ell$ from small to
large as $x_{1},x_{2},\ldots,x_{n}$. By definition, the root of $\ell$ has the
smallest value. Therefore to find the minimum element $x_{1}$, we simply
return the root. For finding the subsequent $x_{i}$’s, we make the following
definitions.
###### Definition 2
Let $T$ be a tournament tree. For any nodes $u,v$ in $T$, we write $u\sim v$
if $\mathsf{val}(u)=\mathsf{val}(v)$.
As we assume that any list $\ell$ contains pairwise distinct numbers, the
equivalence relation $\sim$ partitions the nodes in a tournament tree into
disjoint paths.
###### Definition 3
The principal path $\mathsf{Path}(u)$ of a node $u$ is the equivalence class
$\\{v\mid u\sim v\\}$. The value of $\mathsf{Path}(u)$ is $\mathsf{val}(u)$.
Intuitively we view $\mathsf{Path}(u)$ as a path that originates from a leaf
in $T$ and extends upwards, and every node in $\mathsf{Path}(u)$ “gains” its
value from this leaf. Hence we single out this leaf and define the following.
###### Definition 4
The origin of a principal path $P$ is the leaf in $P$.
Later when referring to “a principal path” in the tree $T$, we mean
$\mathsf{Path}(u)$ for some node $u$ in $T$. Note that the second least number
in $T$ is the value of a sibling of some node in the principal path of $T$’s
root. In general, for any $1\leq i<n$, let $P_{i}$ denote the principal path
in $T$ with value $x_{i}$. The number $x_{i+1}$ is $\mathsf{val}(u)$ where $u$
is a sibling of some node in
$P_{1}\cup P_{2}\cup\cdots\cup P_{i}.$
Hence in computing the $(i+1)$th smallest number in $\ell$ one would need to
examine all principal paths whose origins are $x_{1},x_{2},\ldots,x_{i}$, and
the values of the siblings of nodes on these paths. Formally, we make the
following definition.
###### Definition 5
Let $u$ be an internal node in a tournament tree $T$. The subordinate
$\mathsf{sub}(u)$ of $u$ is a child of $u$ that does not belong to the same
principal path as $u$.
Based on the above observation, to perform $\mathsf{psort}(\ell,k)$, we first
output the root of $\ell$ (along with its value), and then apply the
following: Whenever a node $u$ is returned, we continue to examine the
subordinates of all nodes in the principal path of $u$. This process is
continued until we return $\mathsf{min}\\{k,n\\}$ nodes in $\ell$. During this
process we use a priority queue to store the nodes examined so far. Formally
we describe the operation in Algorithm 1.
Algorithm 1 $\mathsf{\mathsf{psort}}(\ell,k)$
1:$u\leftarrow$ the root of $\ell$
2:Make a new priority queue $Q$
3:for $k$ iterations do
4: Output $\mathsf{val}(u)$
5: while $u\neq\mathsf{null}$ do
6: $y\leftarrow\mathsf{sub}(u)$
7: $\mathsf{insert}$($Q,y$)
8: $u\leftarrow$ the child of $u$ with the same value as $u$, or
$\mathsf{null}$ if no such child exists
9: $u\leftarrow$ $\mathsf{deletemin}$($Q$), or $\mathsf{null}$ if $Q$ is empty
To perform $\mathsf{changeval}(\ell,x,x^{\prime})$, we first change the value
of the leaf $x$ to $x^{\prime}$. This can make the values of every ancestor of
$x$ incorrect; thus, we walk the path from $x$ to the root of $\ell$, and set
the value of every ancestor of $x$ to be the minimum value of its children.
For an exact description, see Algorithm 2.
Algorithm 2 $\mathsf{changeval}(\ell,x,x^{\prime})$
$\mathsf{val}(x)\leftarrow x^{\prime}$; $v\leftarrow p(x)$
while $v\neq\mathsf{null}$ do
if
$\mathsf{val}(v)\neq\mathsf{min}\\{\mathsf{val}(\mathsf{le}(v)),\mathsf{val}(\mathsf{ri}(v))\\}$
then
$\mathsf{val}(v)\leftarrow\mathsf{min}\\{\mathsf{val}(\mathsf{le}(v)),\mathsf{val}(\mathsf{ri}(v))\\}$
$v\leftarrow p(v)$
The link and cut operations are handled in a similar way as linking and
cutting self-balancing binary search trees as described in [14].
* •
Link. For the $\mathsf{link}(\ell,\ell^{\prime})$ operation, we let $T_{1}$
and $T_{2}$ denote the tournament trees of $\ell$ and $\ell^{\prime}$
respectively. Without loss of generality, we assume that $h(T_{1})>h(T_{2})$;
the other case is symmetric. We would like to join $T_{1}$ and $T_{2}$ so that
all leaves in $T_{1}$ are to the left of the leaves in $T_{2}$ in the
resulting tree. For this operation, we follow right child pointers from the
root of $T_{1}$ until we reach a node $x$ such that $h(T_{1}(x))\leq
h(T_{2})$. We then cut the subtree $T_{1}(x)$ away from $T_{1}$, and replace
it with a new node $u$; we set $\mathsf{le}(u)$ to be $x$, $\mathsf{ri}(u)$ to
be the root of $T_{2}$, and $\mathsf{val}(u)$ as the minimum of the values of
$u$’s two children. This change can cause the new tree to become unbalanced,
and may also require us to modify the values of the nodes on the path from $u$
to the root. To solve these problems, we walk the path from $u$ to the root;
at each node $v$ on the path, we must perform two tasks. Firstly, we check
whether $T(v)$ is unbalanced; if it is, we perform a left tree rotation on its
right child $v^{\prime}$ and then we set $\mathsf{val}(v^{\prime})$ to be the
minimum of the values of its children. Secondly, we correct $\mathsf{val}(v)$
to be the minimum of the values of its children. We only need to perform a
rotation once for any join, as the height of any subtree of $T_{1}$ has
increased by at most 1 as part of this process. Note that the resulting tree
is a balanced full binary tree. See Algorithm 3. In this description, we use
$\mathsf{rotateleft}(u)$ to refer to a left tree rotation of the node $u$.
Algorithm 3 $\mathsf{\mathsf{link}}(T_{1},T_{2})$ (For the $h(T_{1})>h(T_{2})$
case)
1:$x\leftarrow$ the root of $T_{1}$, $x^{\prime}\leftarrow$ the root of
$T_{2}$
2:while $h(T_{1}(x))>h(T_{2})$ do
3: $x\leftarrow\mathsf{ri}(x)$
4:Create a new node $u$
5:$\mathsf{ri}(p(x))\leftarrow u$, $\mathsf{le}(u)\leftarrow x$,
$\mathsf{ri}(u)\leftarrow x^{\prime}$ $\triangleright$ Form a new tree with
left subtree $T_{1}(x)$ and right subtree $T_{2}$
6:$\mathsf{val}(u)\leftarrow\mathsf{min}\\{\mathsf{val}(u),\mathsf{val}(x^{\prime})\\}$
7:$y\leftarrow u$
8:while $y\neq\mathsf{null}$ do
9: $z\leftarrow\mathsf{le}(p(u))$
10: if $h(y)>h(z)+1$ then
11: $\mathsf{rotateleft}$($y$)
12:
$\mathsf{val}(p(z))\leftarrow\mathsf{min}\\{\mathsf{val}(z),\mathsf{val}(\mathsf{ri}(p(z))\\}$
13:
$\mathsf{val}(y)\leftarrow\mathsf{min}\\{\mathsf{val}(\mathsf{le}(y)),\mathsf{val}(\mathsf{ri}(y))\\}$
14: $y\leftarrow p(y)$
* •
Cut. To perform the $\mathsf{cut}(\ell,x)$ operation, we need to split the
tournament tree $T$ of $\ell$ at the leaf $u$ where $\mathsf{val}(u)=x$, such
that $u$ and all leaves to its left belong to one tournament tree, and all
leaves to its right belong to another. For this operation, we first walk the
path from $u$ to the root, deleting every edge on the path and incident to it.
We also remove any internal nodes which have no children as part of this
process. This breaks the tree into a collection of subtrees, the root of each
of which was a child of a node on the path from $u$ to the root. We then link
the subtrees containing nodes to the left of $u$ (and $u$ itself) to form a
tournament tree $T_{1}$, and the subtrees containing the other nodes to form
another tournament tree $T_{2}$. See Algorithm 4.
Algorithm 4 $\mathsf{\mathsf{cut}}(T,u)$
1:$x\leftarrow p(u)$; $y\leftarrow u$
2:Create two empty tournament trees $T_{1},T_{2}$
3:$T_{1}\leftarrow$ $T(y)$
4:while $x\neq\mathsf{null}$ do
5: if $y=\mathsf{le}(x)$ then
6: $T_{2}\leftarrow$ $\mathsf{link}$($T_{2},T(\mathsf{ri}(x))$)
7: else
8: $T_{1}\leftarrow$ $\mathsf{link}$($T(\mathsf{le}(x)),T_{1}$)
9: $y\leftarrow x$; $x\leftarrow p(x)$
###### Theorem 3.1
There is an algorithm that solves the dynamic partial sorting problem which
performs the $\mathsf{psort}(\ell,k)$ operation in time $O(k\log(n))$, and
performs the $\mathsf{link}(\ell,\ell^{\prime})$, $\mathsf{cut}(\ell,x)$ and
$\mathsf{changeval}(\ell,x,x^{\prime})$ operations in time $O(\log(n))$, where
$n$ is the size of the list $\ell$.
###### Proof
We analyze the time complexity of the above operations.
* (a)
$\mathsf{psort}(\ell,k)$. By Lemma 1, every path of the tournament tree is
bounded by $\log_{\varphi}(n)$. This means that when the
$\mathsf{psort}(\ell,k)$ operation outputs an element $x$, it inserts at most
$\log_{\varphi}n$ nodes into the priority queue. Hence the priority queue has
size bounded by $k\log_{\varphi}n$. If we use an efficient priority queue
implementation, the time complexity of the operation is $O(k\log(n))$.
* (b)
$\mathsf{changeval}(\ell,x,x^{\prime})$. By Lemma 1 we must modify at most
$\lceil\log_{\varphi}(n)\rceil+1$ nodes, and each modification consists of an
assignment and a two-way comparison, each of which takes constant time. Thus,
we have at most $2(\lceil\log_{\varphi}(n)\rceil+1)$ constant-time operations,
which makes $\mathsf{changeval}(\ell,x,x^{\prime})$ an $O(\log(n))$ operation.
* (c)
$\mathsf{link}(\ell,\ell^{\prime})$. Let $T_{1},T_{2}$ be the tournament trees
of $\ell$ and $\ell^{\prime}$ respectively. Let $m=|h(T_{1})-h(T_{2})|$. As
discussed above, the $\mathsf{link}(T_{1},T_{2})$ operation performs at most
one rotation and up to $m$ many changes to the values of nodes while walking
the path from $u$ to the root. Therefore the $\mathsf{link}(T_{1},T_{2})$
operation takes time $O(m)$, which is $O(\log(n))$.
* (d)
$\mathsf{cut}(\ell,x)$. Let $T$ be the tournament tree of $\ell$ and $u$ be
the leaf with value $x$. Let $P=\\{u_{0},u_{1},u_{2},\ldots,u_{k}\\}$ be the
path in $T$ from $u_{0}=u$ to the root of $T$ where $u_{i+1}=p(u_{i})$ for all
$0\leq i<k$. By Algorithm 4, the $\mathsf{cut}(T,u)$ operation separates $T$
into a collection of tournament trees
$\widehat{T}_{1},\widehat{T}_{2},\ldots,\widehat{T}_{k}$
where each $\widehat{T}_{i}$ is either the left or the right subtree of
$u_{i}$. Since $T$ is balanced, one could easily prove by induction on $i$
that
$h(\widehat{T}_{i})\leq 2i-1.$
The $\mathsf{cut}(T,u)$ operation then iteratively joins the trees
$\widehat{T}_{1},\ldots,\widehat{T}_{k}$ to form two trees $T_{1}$ and
$T_{2}$, where $T_{1}$ contains all leaves to the left of and including $u$,
and $T_{2}$ contains all leaves to the right of $u$. We note from (c) that the
time required for any $\mathsf{link}$ operation is linear on the height
difference between the two trees being joined. The total running time of the
sequence of $\mathsf{link}$ operations performed is therefore at most
$\displaystyle 2\sum_{i\geq
1}^{k-1}\left(h\left(\widehat{T}_{i+1}\right)-h\left(\widehat{T}_{i}\right)\right)$
$\displaystyle=2\left(h\left(\widehat{T}_{k}\right)-h\left(\widehat{T}_{1}\right)\right)$
$\displaystyle\leq 2(2k-1).$
The value of $k$ is at most $h(T)$ which is bounded by $\log_{\varphi}(n)$.
Thus, the total time required for $\mathsf{cut}(T,u)$ is $O(\log(n))$.
∎
## 4 Layered Tournament Trees
In this section we present an alternative solution to the dynamic partial
sorting problem, where the running time of $\mathsf{psort}(\ell,k)$ is
(almost) independent from $n$. The algorithm uses a data structure that
consists of layers of tournament trees, which we call the layered tournament
tree (LTT) data structure. Intuitively, the LTT data structure maintains a
number of layers that extend downwards, where each layer consists of a number
of tournament trees. The tree in the top layer is the tournament tree of
$\ell$. A tree in any lower layer stores a principal path in a tree in the
layer above. Formally, we make the following definitions. Throughout, let
$\ell$ be a list of distinct numbers.
###### Definition 6
Let $T$ be the tournament tree of $\ell$. Let
$P=\\{u_{0},u_{1},\ldots,u_{k}\\}$ be a principal path in $T$ where $u_{0}$ is
the origin of $P$ and $u_{i+1}=p(u_{i})$ for $0\leq i<k$. We define the team
of $P$ as the list of numbers
$t=\mathsf{val}(\mathsf{sub}(u_{k})),\mathsf{val}(\mathsf{sub}(u_{k-1})),\ldots,\mathsf{val}(\mathsf{sub}(u_{1})).$
A team in the tournament tree $T$ is a team of some principal path $P$ in
$\ell$.
Note that only a principal path with more than one element has a team. We
generally use the small case letter $t$ to denote a team.
###### Definition 7
We define a layered tournament tree (LTT) of $\ell$ as a set $\Gamma_{\ell}$
of tournament trees that satisfies the following:
* •
If $\ell$ consists of a single number $x$, then $\Gamma_{\ell}=\\{S\\}$ where
$S$ consists of a single node whose value is $x$.
* •
Otherwise, $\Gamma_{\ell}$ contains a tournament tree $T$ of $\ell$ as well as
an LTT $\Gamma_{t}$ for each team $t$in $T$. In other words,
$\Gamma_{\ell}=\\{T\\}\cup\bigcup\left\\{\Gamma_{t}\mid t\text{ is a team in
}T\right\\}.$
When the list $\ell$ is clear from the context, we drop the subscript writing
$\Gamma_{\ell}$ simply as $\Gamma$. We next define layers in a layered
tournament tree $\Gamma$ of $\ell$.
###### Definition 8
Let $T$ be a tournament tree in $\Gamma$. We say that
* •
$T$ is in layer $0$ of $\Gamma$ if $T$ is a tournament tree of $\ell$; and
* •
$T$ is in layer $i$ of $\Gamma$, where $i>0$, if $T$ is a tournament tree of a
team $t$ in a layer-$(i-1)$ tree in $\Gamma$.
We call $\ell$ the layer-$0$ team, and the team $t$ mentioned above a
layer-$i$ team in $\Gamma$. If a tree $T$ is in layer $i$ of $\Gamma$, we call
it a layer-$i$ tree in $\Gamma$. The layer number of $\Gamma$ is the maximum
$i\geq 0$ such that a tree is in layer $i$ of $\Gamma$.
Let $P$ be a principal path in a layer-$i$ tree of $\Gamma$, where $i\geq 0$
and the length of $P$ is at least 1. By Def. 6 and Def. 8, $\Gamma$ contains a
tournament tree $T$ of the team of $P$ in layer-$(i+1)$. We call $T$ the team
tree of $P$. The team tree $\mathsf{Team}(u)$ of any node $u$ is the team tree
of the principal path containing $u$.
Recall that the origin of a principal path $P$ is the leaf in $P$. We
introduce the following notions:
* •
Suppose $u$ is an internal node in a layer-$i$ tree $T\in\Gamma$. We define
$\mathsf{down}(u)$ as the origin $v$ of the principal path in the team tree
$\mathsf{Team}(u)$ such that $\mathsf{val}(v)=\mathsf{val}(\mathsf{sub}(u))$.
* •
Suppose $u$ is a leaf in a layer-$i$ tree $T\in\Gamma$ where $i>0$. We define
$\mathsf{up}(u)$ as the internal node $v$ in a layer-$(i-1)$ tree such that
$\mathsf{down}(v)=u$.
This finishes the description of the LTT data structure; see Figure 2 for an
example of an LTT.
333395574484644459Layer 07Layer 1668559Layer 289Layer 3
Figure 2: The LTT of the list $\ell=3,9,5,7,8,4,6$. The $\mathsf{up}$ and
$\mathsf{down}$ nodes are indicated by a dashed grey line. The layer number is
3. The team of 3 is a list 4,5,9. The team of 5 is a list with a single
element 7. The team of 4 is 6,8. These teams form their own team trees at
layer 1.
Remark. Intuitively the layered tournament tree is similar in concept to a
dynamic tree as described by Tarjan and Sleator [14]. However by Def. 8 a
dynamic tree has only two layers while a layered tournament tree can have
arbitrarily-many.
In subsequent sections, we describe the $\mathsf{psort}$, $\mathsf{link}$,
$\mathsf{cut}$ and $\mathsf{changeval}$ operations for the LTT data structure.
The factors that determine the time complexity of these operations are 1) the
height of a layer-$i$ tree in an LTT $\Gamma$ for $i\geq 0$; and 2) the layer
number in the LTT $\Gamma$.
To analyze the height of a layer-$i$ tree in a LTT $\Gamma$ for any $i\geq 0$,
we recall the following function.
###### Definition 9
Let $b>1$ be a real number. The iterated logarithm with base $b$
$\log^{*}_{b}(n)$ of a number $n>b$ is the smallest $i\geq 0$ such that
$\underbrace{\log_{b}\cdots\log_{b}}_{i}(n)\leq 1.$
It is known that the iterated logarithm function is defined for all $b\leq
e^{1/e}$. The function $\log_{b}^{n}$ is known to be extremely slow-growing;
for example, when $b$ is the golden ratio $\varphi$, $\log^{*}_{b}(10^{6})=6$
and $\log^{*}_{b}(10^{10000})=7$. More precisely, $\log^{*}_{b}(n)$ is the
inverse of the power tower function with base $b$ defined as
$b\uparrow\uparrow n=\underbrace{b^{b^{\iddots^{b}}}}_{n}$
Hence we have the following lemma, which we state without a proof.
###### Lemma 2
For any $b\geq e^{1/e}$, for all $i\geq 0$ we have
$\exists n^{\prime}>0\forall n>n^{\prime}:\
\log^{*}_{b}(n)\leq\underbrace{\log_{b}\cdots\log_{b}}_{i}(n).$
###### Lemma 3
For any $i\geq 1$, the size of any layer-$i$ team is at most
$\underbrace{\log_{\varphi}\cdots\log_{\varphi}}_{i}(n)$, where $n$ is the
size of the list $\ell$. Furthermore layer number of the LTT data structure is
at most $\log^{*}_{\varphi}(n)$.
###### Proof
By Lemma 1 the height of any tournament tree is at most $\log_{\varphi}(m)$
where $m$ is the number of leaves in the tree. The first statement of the
lemma follows directly from the fact that the number of leaves in a layer-$i$
tree is at most the height of a layer-$(i-1)$ tree in $\Gamma$. The second
statement follows directly from the first statement.∎
As an example, suppose the list $\ell$ contains a million numbers. The layer
number in the LTT of $\ell$ is at most $\log^{*}_{\varphi}(10^{6})\leq 6$.
## 5 The $\mathsf{psort}(\ell,k)$ Operation With LTT
We now describe the algorithm for solving the dynamic partial sorting problem
using the LTT data structure. Similarly to Section 3, we assume that a list
$\ell$ is represented by an LTT $\Gamma$. More specifically, we assume that
the elements of $\ell$ are the leaves of the layer-0 tree in $\Gamma$. In this
section we will refer to a list and its LTT interchangeably. All terms that
relate to a team tree $T$ carry forward to the corresponding list $\ell$.
Hence the nodes, root, leaves, and internal nodes of $\ell$ refer to the
equivalent concepts in $T$.
We describe the partial sorting operation $\mathsf{psort}(\ell,k)$ on an LTT
$\Gamma$ of the list $\ell$. We use
$x_{1}<x_{2}<\ldots<x_{n}$
to denote the numbers in $\ell$ in ascending order. Intuitively the algorithm
is similar to the $\mathsf{psort}(\ell,k)$ operation described in Section 3.
The algorithm searches for and outputs each $x_{i}$ iteratively by exploring
the layer-0 tournament tree $T$. The smallest number $x_{1}$ is the value of
the root of $T$. If $k=1$ or $\ell$ contains only one element, then the
algorithm stops after outputting $x_{1}$. Otherwise, to find the second-
smallest number $x_{2}$ in $\ell$, let $P$ be the principal path of the root
of $T$. The number $x_{2}$ is the least number in the team of $P$. Unlike
Algorithm 1, where we check through the subordinates of all nodes in $P$, here
we recursively apply the partial sort operation on the tournament tree of the
layer-1 team of $P$. In this way, the search continues in a lower layer.
To formally describe the $\mathsf{psort}(\ell,k)$ operation, we use an
iterator, which is defined as follows.
###### Definition 10
Let $\ell$ be a list of numbers. An iterator of $\ell$ is a data structure
$\mathrm{It}(\ell)$ that supports an operation $\mathsf{next}(\ell)$ with the
following property: In the $i$th call to $\mathsf{next}(\ell)$, the operation
outputs $x_{i}$ if $i\leq n$; and outputs $\mathsf{null}$ otherwise.
An iterator $\mathrm{It}(\ell)$ maintains a priority queue $Q$, which is going
to contain nodes in $T$. The $\mathsf{psort}(\ell,k)$ operation amounts to
creating an iterator $\mathrm{It}(\ell)$ and calling $\mathsf{next}(\ell)$ $k$
times to obtain the list $x_{1},x_{2},\ldots,x_{k}$. We use $u_{i}$ to denote
the leaf with value $x_{i}$ in the layer-0 tree of $\ell$ for $1\leq i\leq n$.
For convenience, we consider the output of $\mathsf{next}(\ell)$ to be the
leaf $u_{i}$, rather than its value $x_{i}$.
To create an iterator for $T$, the algorithm simply creates an empty priority
queue $Q$. We describe the $\mathsf{next}(\ell)$ operation by induction on the
number of elements in $\ell$. When the operation $\mathsf{next}(\ell)$ is
called the first time, we return the origin of $\mathsf{Path}(r)$, where $r$
is the root of $\ell$. In subsequent calls to $\mathsf{next}(\ell)$, if $\ell$
contains only one element, then the algorithm returns $\mathsf{null}$. Suppose
$\ell$ contains more than one element, and assume that we have defined
iterators of lists with fewer elements than $\ell$.
Suppose $i\geq 1$ and we have made $i$ calls to $\mathsf{next}(\ell)$ which
outputs the nodes
$u_{1},u_{2},\ldots,u_{i}$
. Algorithm 5 implements the $\mathsf{next}(\ell)$ operation for the $(i+1)$th
call.
Algorithm 5 $\mathsf{next}(\ell)$ (The $(i+1)$th call)
1:if $\mathsf{Team}(u_{i})$ is not empty then
2: Create an iterator $\mathrm{It}(\mathsf{Team}(u_{i}))$
3: $a\leftarrow\mathsf{next}(\mathsf{Team}(u_{i}))$
4: Insert $\mathsf{up}(a)$ to $Q$ with value $\mathsf{val}(\mathsf{sub}(a))$
5:if $Q$ is not empty then
6: $x\leftarrow\mathsf{deletemin}(Q)$
7: $u_{i+1}\leftarrow$ the origin of $\mathsf{Team}(\mathsf{sub}(x))$
8: $b\leftarrow\mathsf{next}(\mathsf{Team}(x))$
9: if $\mathsf{up}(b)\neq\mathsf{null}$ then
10: Insert $\mathsf{up}(b)$ to $Q$ with value $\mathsf{val}(\mathsf{sub}(b))$
11: Output $u_{i+1}$
12:else
13: Output $\mathsf{null}$
To show the correctness of the algorithm above, we make the following
definition:
###### Definition 11
Let $v$ be a node in a tournament tree $T$. The superordinate of $v$ is a node
$\mathsf{sup}(v)$ in $T$ whose subordinate belongs to the principal path
$\mathsf{Path}(v)$. The superordinate set of a set $U$ of nodes is
$\mathsf{sup}(U)=\\{\mathsf{sup}(v)\mid v\in U\\}.$
For the next definition, we take a set $U$ of nodes in $T$.
###### Definition 12
A node $v$ is an $U$-candidate if there is some $u\in U$ such that
$v\in\mathsf{Path}(u)$ and for any $w\in\mathsf{Path}(u)$,
$\mathsf{val}(\mathsf{sub}(w))<\mathsf{val}(\mathsf{sub}(v))$ if and only if
$w\in\mathsf{sup}(U)$. We denote the set of $U$-candidates as $\mathsf{C}(U)$.
###### Lemma 4
For every $1\leq i<n$,
$\mathsf{sup}(u_{i+1})\in\mathsf{C}(\\{u_{1},\ldots,u_{i}\\})$.
###### Proof
We prove this lemma by induction on $i$. By definition of the tournament tree
$T$, $u_{2}$ is the subordinate of a node $v\in\mathsf{Path}(u_{1})$.
Furthermore, $\mathsf{val}(u_{2})$ is the smallest number in the team of
$\mathsf{val}(u_{1})$. Hence $\mathsf{sup}(u_{2})\in\mathsf{C}(\\{u_{1}\\})$.
Suppose the statement holds for $i\geq 1$. Let $x$ be the superordinate of the
node $u_{i+1}$. Our goal is to show that
$x\in\mathsf{C}(\\{u_{1},\ldots,u_{i}\\})$. For any node
$v\in\mathsf{Path}(x)$, we have $\mathsf{val}(v)<\mathsf{val}(u_{i+1})$ as
otherwise $v$ would not be in the same principal path as $x$. Hence the head
of the principal path $\mathsf{Path}(x)$ is $u_{j}$ for some $1\leq j\leq i$.
Let $w$ be a node in $\mathsf{Path}(x)$. Suppose
$\mathsf{val}(\mathsf{sub}(w))<\mathsf{val}(\mathsf{sub}(x))$. Since
$\mathsf{val}(\mathsf{sub}(x))=\mathsf{val}(u_{i+1})$, the team of
$\mathsf{Path}(\mathsf{sub}(w))$ would contain a number that has smaller value
than $u_{i+1}$. Therefore $w$ must be $\mathsf{sup}(u_{j})$ for some $1\leq
j\leq i$. This means that $w\in\mathsf{sup}(\\{u_{1},\ldots,u_{i}\\})$.
Conversely, suppose $w\in\mathsf{sup}(\\{u_{1},\ldots,u_{i}\\})$. Then by
choice of $u_{i+1}$ we have
$\mathsf{val}(\mathsf{sub}(w))<\mathsf{val}(u_{i+1})=\mathsf{val}(\mathsf{sub}(x))$.
Thus $x$ is in $\mathsf{C}(\\{u_{1},\ldots,u_{i}\\})$. ∎
The next lemma implies the correctness of Alg. 5.
###### Lemma 5
For any $i\geq 1$, the $i$th call to $\mathsf{next}(\ell)$ returns the node
$u_{i}$ if $i\leq n$, and $\mathsf{null}$ otherwise.
###### Proof
We prove the lemma by induction on the number of times $\mathsf{next}(\ell)$
is called. It is clear that in the first call to $\mathsf{next}(\ell)$, the
algorithm returns the node $u_{1}$ which is the origin of the principal path
that contains the root of $\ell$. Consider the second call to
$\mathsf{next}(\ell)$. If $\ell$ contains only one node $u_{1}$, then
$\mathsf{Team}(u_{1})$ does not exist and the priority queue $Q$ is empty at
line 5. If $\ell$ contains more than one element, then $\mathsf{Team}(u_{1})$
is defined. At line 5, $Q$ will store the element $x=\mathsf{up}(a)$, where
$a=\mathsf{next}(\mathsf{Team}(u_{1}))$ is the node with the smallest value in
$\mathsf{Team}(u_{1})$. By definition $\mathsf{C}(\\{v_{1}\\})=\\{x\\}$.
For the inductive step, suppose we are running $\mathsf{next}(\ell)$ the
$(i+1)$th time, where $i\geq 1$. We assume the following inductive assumption:
When the algorithm reaches line 5,
1. (I1)
if $\ell$ contains no more than $i$ elements, then the priority queue $Q$ is
empty;
2. (I2)
if $\ell$ contains at least $i+1$ elements, then the priority queue $Q$
contains exactly those nodes in $\mathsf{C}(\\{u_{1},\ldots,u_{i}\\})$.
If $\ell$ contains no more than $i$ elements, then by (I1) the algorithm
returns $\mathsf{null}$ and $Q$ remains empty. Now suppose $\ell$ contains at
least $i+1$ elements. By (I2), when the algorithm reaches line 5, the priority
queue $Q$ contains exactly those nodes in
$\mathsf{C}(\\{u-1,\ldots,u_{i}\\})$. Let $x$ be the least element in $Q$. By
Lemma 4, $x$ is the superordinate $\mathsf{sup}(u_{i+1})$ of $u_{i+1}$. Thus
the algorithm would locate and return the node $u_{i+1}$.
We then need to verify that the $\mathsf{next}(\ell)$ operation preserves the
inductive invariants (I1) and (I2). It is clear that $(I1)$ holds at line 5 of
the $(i+2)$th call to $\mathsf{next}(\ell)$.
To verify (I2), let $S$ and $S^{\prime}$ denote the sets of nodes stored in
the priority queue $Q$ at line 5 in the $(i+1)$th and the $(i+2)$th call to
$\mathsf{next}(\ell)$, respectively. Let $b$ be the leaf that has the next
smallest value in $\mathsf{Team}(x)$ after $x$. After we finish the $(i+1)$th
call to $\mathsf{next}(\ell)$, $Q$ would store the set
$S\setminus\\{x\\}\cup\\{\mathsf{up}(b)\\}$. In the $(i+2)$th call to
$\mathsf{next}(\ell)$, before reaching Line 5, the algorithm would add the
node $\mathsf{up}(a)$ to $Q$ where $a$ has the least value in
$\mathsf{Team}(u_{i+1})$. Therefore we have
$S^{\prime}=S\setminus\\{x\\}\cup\\{\mathsf{up}(a),\mathsf{up}(b)\\}=\mathsf{C}(\\{u_{1},\ldots,u_{i},u_{i+1}\\}).$
Hence (I2) is preserved. ∎
As described above, the $\mathsf{psort}(\ell,k)$ operation amounts to creating
an iterator of $\ell$ and calling the $\mathsf{next}(\ell)$ operation $k$
times. By Lemma 5, the operation outputs the desired numbers
$x_{1},x_{2},\ldots,x_{k}$ in increasing order.
Time complexity. We now analyze the time complexity of the
$\mathsf{psort}(\ell,k)$ operation. Suppose $t$ is a layer-$i$ team in
$\Gamma$. Any call to the $\mathsf{next}(t)$ operation may in turn trigger a
sequence of calls to the $\mathsf{next}(t^{\prime})$ operations on teams in
lower layers. The algorithm maintains a priority queue for every team for
which an iterator is created.
Each call to $\mathsf{next}(t)$ performs a fixed number of priority queue
operations (such as insert and $\mathsf{deletemin}$), at most two calls to the
$\mathsf{next}(t^{\prime})$ operation on some layer-$(i+1)$ team $t^{\prime}$,
and a fixed number of other elementary operations. Among these operations, the
first call to $\mathsf{next}(t^{\prime})$ occurs immediately after the
$(i+1)$-iterator of $t^{\prime}$ is created. This call to
$\mathsf{next}(t^{\prime})$ simply involves a pointer lookup and thus takes
constant time. Furthermore, by Lemma 1, the number of leaves of the team tree
of $t^{\prime}$ is at most $\log_{\varphi}(m)$ where $m$ is the number of
elements in $t$.
Suppose we perform $k$ calls to $\mathsf{next}(t)$ where $k\geq 1$. Note that
for any team $t^{\prime}$ in layer $j>i$, the algorithm would make at most
$k-1$ calls to $\mathsf{next}(t^{\prime})$. With every call to
$\mathsf{next}(t^{\prime})$, the number of elements stored in the priority
queue increases by at most 2. Thus the number of elements stored in any
priority queue is at most than $2k$. Therefore, using a heap implementation of
priority queues, the time for inserting an element to or deleting the minimum
element from the priority queue takes $O(\log(k))$.
Summing up the above costs over all $k$ calls, the operations perform $O(k)$
number of priority queue operations, $k-1$ calls to $\mathsf{next}$ on trees
in a layer down, and other operations that take a total of $O(k)$ time. We use
$\mu(k,m)$ to denote the time taken by $k$ calls to $\mathsf{next}(t)$ where
the team tree of $t$ has $m$ leaves. Assuming an efficient priority queue
implementation, there is a constant $d>0$ such that.
$\mu(k,m)\leq\begin{cases}dk\log k+\mu(k-1,\log_{\varphi}(m))&\text{if
$m>1$;}\\\ d&\text{otherwise.}\end{cases}$ (1)
###### Lemma 6
The $\mathsf{psort}(\ell,k)$ operation runs in time
$O(\log^{*}_{\varphi}(n)k\log(k))$ where $n$ is the size of the list $\ell$.
###### Proof
The $\mathsf{psort}(\ell,k)$ operation makes $k$ calls to the
$\mathsf{next}(\ell)$ operation. Therefore the running time of
$\mathsf{psort}(\ell,k)$ is $\mu(k,n)$ where $n$ is the size of $\ell$. By (1)
we get
$\mu(k,n)\leq dk\log(k)+d(k-1)\log(k)+d(k-2)\log(k)+\cdots+d(k-s+1)\log(k)+d,$
where $n$ is the number of elements in $\ell$ and $s$ is layer number of
$\Gamma$. By Lemma 3, $s\leq\log^{*}_{\varphi}(n)$. Therefore the total time
taken by $\mathsf{psort}(\ell,k)$ is $O(\log^{*}_{\varphi}(n)k\log(k))$.∎
## 6 The Update Operations With LTT
We describe the update operations assuming that all lists are represented by
the LTT data structure. Unless stated otherwise, all occurrences of
$\mathsf{link},\mathsf{cut}$ and $\mathsf{changeval}$ refer to the update
operations defined in this section, but not to the operations with the same
names in Section 3. As explained in Section 5, the arguments of the
$\mathsf{link}$, $\mathsf{cut}$ and $\mathsf{changeval}$ operations consist of
LTTs (representing lists) and leaves in the layer-0 tree of the corresponding
LTTs (representing elements in the lists).
In the following we define the $\mathsf{link}(\ell,\ell^{\prime})$,
$\mathsf{cut}(\ell,x)$ and $\mathsf{changeval}(\ell,x,x^{\prime})$ operations
by induction on the maximum layer number in the argument LTTs
$\ell,\ell^{\prime}$. If an LTT consists of only one layer, it contains only
one node. Therefore the $\mathsf{cut}$ and $\mathsf{changeval}$ operations
performed on such an LTT are trivial. To perform the
$\mathsf{link}(\ell,\ell^{\prime})$ operation where both $\ell,\ell^{\prime}$
consist of one layer, we create a new node $v$ and set $\mathsf{le}(v)$ and
$\mathsf{ri}(v)$ as $\ell$ and $\ell^{\prime}$ respectively in the layer-0
tree, and create a layer-1 tree with a single node whose value is the larger
of the values of the nodes in $\ell$ and $\ell^{\prime}$. In subsequent
sections we define the $\mathsf{changeval}(\ell,x,x^{\prime})$,
$\mathsf{link}(\ell,\ell^{\prime})$ and $\mathsf{cut}(\ell,x)$ operations
where $\ell$ and $\ell^{\prime}$ have more than one layer. The inductive
hypothesis assumes correct implementation of $\mathsf{link}$ and
$\mathsf{cut}$ on LTTs with fewer layers.
### 6.1 The $\mathsf{expose}(\ell,u)$ and
$\mathsf{changeval}(\ell,u,x^{\prime})$ Operation
The $\mathsf{changeval}(\ell,x,x^{\prime})$ operation assumes that $x$ is a
leaf in the layer-0 tree of the LTT representing $\ell$ and changes its value
to $x^{\prime}$. Note that after changing the value of $x$ to $x^{\prime}$,
the LTT structure may be broken. Thus we should apply other procedures to
preserve the LTT. This is achieved using an $\mathsf{expose}(\ell,u)$
operation where $u=p(x)$.
Intuitively, the $\mathsf{expose}(\ell,u)$ operation is a “fix up” operation
that maintains the LTT structure on the path from $u$ to the root of the tree,
once a change has occurred on a child. It walks the path from $u$ to the root,
and performs the following procedures in each step: It first separates $u$
from its principal path from below, so that both its left child
$\mathsf{le}(u)$ and right child $\mathsf{ri}(u)$ are detached from the
principal path of $u$. It then links the smaller of $\mathsf{le}(u)$ and
$\mathsf{ri}(u)$ with the principal path of $u$ and sets $\mathsf{val}(u)$ as
the smaller value of its children. Finally, it repeats the same process to set
$p(u)$ as the new $u$.
To separate and link the principal paths mentioned above, we use the
$\mathsf{cut}$ and $\mathsf{link}$ operations on the team trees of the
corresponding principal paths. Note that in the above operation, we may change
the subordinate of $u$. This requires us to change the value of
$\mathsf{down}(u)$ in the team tree $\mathsf{Team}(u)$, which can be performed
by calling
$\mathsf{changeval}(\mathsf{Team}(u),\mathsf{down}(u),\mathsf{max}\\{\mathsf{le}(u),\mathsf{ri}(u)\\})$
recursively. Note that the team trees used as arguments of the
$\mathsf{cut},\mathsf{link}$ operations and the recursive call to
$\mathsf{changeval}$ have strictly fewer layers than $\ell$. Thus, by the
inductive hypothesis, these operations have been defined. For an exact
description, see Algorithm 6.
Algorithm 6 $\mathsf{expose}(\ell,u)$
1:$x\leftarrow u$
2:while $x\neq\mathsf{null}$ do
3: $(z,z^{\prime})\leftarrow(\mathsf{le}(x),\mathsf{ri}(x))$ if
$\mathsf{val}(\mathsf{le}(x))<\mathsf{val}(\mathsf{ri}(x))$; otherwise
$(z,z^{\prime})\leftarrow(\mathsf{ri}(x),\mathsf{le}(x))$
4: $\mathsf{val}(x)\leftarrow\mathsf{val}(z)$
5: $T_{1},T_{2}\leftarrow$ $\mathsf{cut}$($\mathsf{Team}(x),\mathsf{down}(x)$)
6: $T_{1}\leftarrow$ $\mathsf{link}$($T_{1},\mathsf{Team}(z)$)
7: $\mathsf{changeval}$($T_{1},\mathsf{down}(x),\mathsf{val}(z^{\prime})$)
$\triangleright$ Change the value of $\mathsf{down}(x)$ in the layer below
8: $x\leftarrow p(x)$
We now analyze the correctness of the $\mathsf{expose}(\ell,u)$ operation.
More specifically, let $v$ be an internal node in the LTT data structure. We
use the following invariants:
1. (J1)
$\mathsf{val}(v)=\mathsf{min}\\{\mathsf{val}(\mathsf{le}(v)),\mathsf{val}(\mathsf{ri}(v))\\}$
2. (J2)
$\mathsf{val}(\mathsf{down}(v))=\mathsf{val}(\mathsf{sub}(v))$
3. (J3)
If $v$ has a child $v^{\prime}$ that is an internal node and
$\mathsf{val}(v)=\mathsf{val}(v^{\prime})$, then
$\mathsf{down}(v),\mathsf{down}(v^{\prime})$ belong to the same team tree
$\mathsf{Team}(v)$ and $\mathsf{down}(v)$ is to the left of
$\mathsf{down}(v^{\prime})$ in $\mathsf{Team}(v)$.
Intuitively, the three invariants state that the LTT structure is maintained.
Indeed, (J1) states that the value of $v$ is assigned according to the
tournament tree property, (J2) states that $\mathsf{down}(v)$ has the correct
value, and (J3) states that the team tree of $\mathsf{down}(v)$ is correctly
maintained.
###### Definition 13
Let $v$ be a node in the LTT data structure of $\ell$. The parent-down closure
of $v$ is the minimal set $\mathsf{Pd}(v)$ of nodes in the LTT that contains
$v$ and for any node $w\in\mathsf{Pd}(v)$,
1. 1.
$p(w)\in\mathsf{Pd}(v)$ if $w$ is not the root of a tree; and
2. 2.
$\mathsf{down}(w)\in\mathsf{Pd}(v)$ if $w$ is not a leaf in a tree.
Note that the $\mathsf{expose}(\ell,u)$ operation may only update the values,
as well as split and join team trees, for nodes in the set $\mathsf{Pd}(u)$.
Hence intuitively, $\mathsf{Pd}(u)$ denotes the “region of operation” in the
LTT $\ell$ of $\mathsf{expose}(\ell,u)$. For the next lemma, recall that we
assume by the inductive hypothesis that a correct implementation of
$\mathsf{link}$ and $\mathsf{cut}$ can be called on LTTs with fewer layers
than $\ell$.
###### Lemma 7
After running $\mathsf{expose}(\ell,u)$, (J1)–(J3) hold for every node
$v\in\mathsf{Pd}(u)$.
###### Proof
The proof proceeds by induction on the number of layers in $\ell$. The
statement is clear for $\ell$ with a single layer (which consists of only one
node). Now suppose $\ell$ contains $m$ layers where $m>1$. Take a node
$v\in\mathsf{Pd}(u)$ that is in layer-0 of the LTT $\ell$. Then $v$ is set as
$x$ by some iteration of the $\mathsf{while}$-loop. During this iteration,
(J1) holds after running Line 4, (J2) holds after running Line 7 and (J3)
holds after running Line 6 for the node $v$.
Suppose that (J1)–(J3) hold for all nodes in $\mathsf{Pd}(u)$ on some
layer-$i$ and $v\in\mathsf{Pd}(u)$ is an internal node in a layer-$(i+1)$ tree
of the LTT data structure. Then by definition of $\mathsf{Pd}(u)$, there is
some leaf $w$ in the subtree rooted at $v$ such that
$w=\mathsf{down}(w^{\prime})$ for some $w^{\prime}\in\mathsf{Pd}(u)$. Let $w$
be the rightmost leaf with this property. The algorithm must have made a call
$\mathsf{changeval}(T_{1},w,\mathsf{val}(z^{\prime}))$ during its execution.
In this call to $\mathsf{changeval}$, the $\mathsf{while}$-loop visits $v$ and
make (J1)–(J3) hold for $v$ using Line 4, Line 7 and Line 6 respectively. ∎
### 6.2 The $\mathsf{link}(\ell,\ell^{\prime})$ and $\mathsf{cut}(\ell,x)$
Operations
The $\mathsf{link}(\ell,\ell^{\prime})$ operation is performed similarly to
linking two balanced binary search trees. The operation compares the layer-0
trees of $\ell$ and $\ell^{\prime}$ and links the tree with a smaller height
as a subtree of the other.
Before we describe the $\mathsf{link}(\ell,\ell^{\prime})$ operation, we
describe the tree rotation operation for LTTs, which is an important
subroutine. Here, we describe the left rotation $\mathsf{rotateleft}(\ell,u)$,
where $u$ is a right child in an LTT $\ell$; the right rotation operation is
symmetric. To perform $\mathsf{rotateleft}(\ell,u)$, we first separate both
$u$ and its parent $p(u)$ from the rest of their principal paths from above
and below. We then perform the left rotation on $u$ as if for a normal binary
tree. Lastly, we restore the principal paths of $p(u)$ by calling the
$\mathsf{expose}(\ell,p(u))$ operation. This will fix the principal paths we
separated in this operation and preserve the structure of the LTT. See
Algorithm 7.
Algorithm 7 $\mathsf{rotateleft}(\ell,u)$
1:$y\leftarrow p(u)$;
2:if $y$ is not the root then
3: $\mathsf{cut}$($\mathsf{Team}(p(y)),\mathsf{down}(p(y))$) $\triangleright$
Separate $y$ from above
4:$\mathsf{cut}$($\mathsf{Team}(y),\mathsf{down}(y)$) $\triangleright$
Separate $y$ from below
5:$\mathsf{cut}$($\mathsf{Team}(u),\mathsf{down}(u)$) $\triangleright$
Separate $u$ from below
6:$\mathsf{ri}(y)\leftarrow\mathsf{le}(u)$; $\mathsf{le}(u)\leftarrow y$
$\triangleright$ Perform the left rotation on $u$
7:$\mathsf{expose}$($\ell,y$)
The following lemma is implied from Lemma 7 and the proof is straightforward.
###### Lemma 8
Let $y$ be the parent of $u$. After running $\mathsf{rotateleft}(\ell,u)$,
(J1)–(J3) hold for every node $v\in\mathsf{Pd}(y)$.
We now describe the $\mathsf{link}(\ell,\ell^{\prime})$ operation. For
simplicity in this section we only describe the case when the layer-0 tree of
$\ell$ has a greater or equal height than the layer-0 tree of $\ell^{\prime}$;
the other case is symmetric. We first find a node $u$ on the rightmost path in
the layer-0 tree of $\ell$ such that $T(u)$ has the same height as the layer-0
tree $T^{\prime}$ of $\ell^{\prime}$. We then create a new node $v$, making it
a child of $p(u)$ if $u$ is not the root, and set $T(u)$ as $v$’s left subtree
and $T^{\prime}$ as $v$’s right subtree. We then fix the principal paths by
calling $\mathsf{expose}$ on $v$. This operation may leave the resulting
layer-0 tree unbalanced. Hence we walk the path from $v$ to the root and find
a node $y$ on this path such that the subtree $T(p(y))$ is unbalanced, and we
call $\mathsf{rotateleft}$ on $y$. See Algorithm 8. This finishes the
description of the $\mathsf{link}(\ell,\ell^{\prime})$ operations. Note that
inside this operation, all recursive subroutine calls to $\mathsf{link}$ and
$\mathsf{cut}$ are made on argument LTTs with fewer layers than $\ell$, and
are thus defined by the inductive hypothesis.
Algorithm 8 $\mathsf{link}(\ell,\ell^{\prime})$
1:$T,T^{\prime}\leftarrow$ the layer-$0$ tournament trees of
$\ell,\ell^{\prime}$ respectively
2:$r_{1},r_{2}\leftarrow$ the roots of $T,T^{\prime}$ respectively
3:Follow $\mathsf{ri}$ pointers from $r_{1}$ to find $u$ such that $T(u)$ and
$T^{\prime}$ have the same height
4:Create a new node $v$ and the corresponding node $\mathsf{down}(v)$ in the
layer below
5:$p(v)\leftarrow p(u)$
6:$\mathsf{le}(v)\leftarrow u$; $\mathsf{ri}(v)\leftarrow r_{2}$
7:$\mathsf{expose}$($\ell,v$)
8:Following $p$ pointers from $v$ until we reach $y$ such that $T(p(y))$ is
unbalanced
9:If such $y$ exists, $\mathsf{rotateleft}$($\ell,y$)
We perform the $\mathsf{cut}(\ell,u)$ operation in a similar way as Alg. 4 in
Section 3. The operation first calls $\mathsf{changeval}$ on $u$ to assign it
a value smaller than all numbers in $\ell$ (we call it $-\infty$ for
convenience). In this way, all nodes on the path from $u$ to the root form a
principal path. The operation then walks the path from $u$ to the root,
joining all subtrees to its left into a new tree and all subtrees to its right
into another new tree. Finally it restores the value of $u$ and joins $u$ to
the first new tree. We perform all the joining of trees using the
$\mathsf{link}$ operation; see Alg. 9.
Algorithm 9 $\mathsf{\mathsf{cut}}(\ell,u)$
1:$a\leftarrow\mathsf{val}(u)$; $\mathsf{changeval}$($\ell,u,-\infty$)
2:$x\leftarrow p(u)$; $y\leftarrow u$
3:Create two empty tournament trees $T_{1},T_{2}$
4:while $x\neq\mathsf{null}$ do
5: if $y=\mathsf{le}(x)$ then
6: $T_{2}\leftarrow$ $\mathsf{link}$($T_{2},T(\mathsf{ri}(x))$)
7: else
8: $T_{1}\leftarrow$ $\mathsf{link}$($T(\mathsf{le}(x)),T_{1}$)
9: $y\leftarrow x$; $x\leftarrow p(x)$
10:$\mathsf{val}(u)\leftarrow a$; $\mathsf{link}$($T_{1},u$) $\triangleright$
Link $T_{1}$ with the restored $u$
### 6.3 Time Complexity of the Update Operations
We now analyze the time complexity of the update operations. For any list
$\ell$ with $n$ elements, we define $s_{i}(n)$ as the maximum number of
elements of a layer-$i$ team in the LTT of $\ell$. It is clear that
$s_{0}(n)=n$. By Lemma 3, for all $n>0$ we have
$\displaystyle s_{\log^{*}_{\varphi}(n)}(n)=1,\text{ and }$
$\displaystyle\forall i\geq 0:\ s_{i+1}(n)\leq\log_{\varphi}(s_{i}(n))$ (2)
For convenience, we set $s_{i}(n)=1$ for all $i>\log^{*}_{\varphi}(n)$. We
will express the complexity of the update operations using the variables
$s_{i}(n)$.
###### Lemma 9
For any $i\geq 0$, there is a constant $n_{0}>0$ such that for all $n>n_{0}$
we have
$\prod_{j\geq i+1}s_{j}(n)\leq s_{i}(n)$
###### Proof
As $s_{j}(n)=1$ for all $n>0$ and $j\geq\log_{\varphi}^{*}(n)$, the statement
is clear for $i\geq\log_{\varphi}^{*}(n)-1$. The proof proceeds by induction
on $i$. Fix $0<i<\log_{\varphi}^{*}(n)$ and suppose there is $n_{0}$ such that
the statement holds for all $n>n_{0}$. Then for all $n\geq n_{0}$ we have
$\displaystyle\prod_{j\geq i}s_{j}(n)$
$\displaystyle=s_{i}(n)\cdot\prod_{j\geq i}s_{j}(n)$ $\displaystyle\leq
s^{2}_{i}(n)$ (by the ind. hyp.)
$\displaystyle\leq\log^{2}_{\varphi}(s_{i-1}(n))$ (by (2))
Take $n^{\prime}$ such that
$\log^{2}_{\varphi}(s_{i-1}(n^{\prime}))\leq s_{i-1}(n^{\prime}).$
Then for all $n\geq\mathsf{max}\\{n^{\prime},n_{0}\\}$
$\prod_{j\geq i}s_{j}(n)\leq\log^{2}_{\varphi}(s_{i-1}(n))\leq s_{i}(n).$
∎
Recall that the $\mathsf{expose}(\ell,u)$ operation performs a number of
iterations. We analyze the running time of each iteration separately. Without
loss of generality, we assume in the next lemma that the list $\ell$ contains
no fewer elements than $\ell^{\prime}$.
###### Lemma 10
Let $n$ be the number of elements in the list $\ell$. The following hold for
the update operations:
1. (a)
Each iteration of $\mathsf{expose}(\ell,u)$ runs in time
$O\left(s_{2}^{2}(n)\right)$.
2. (b)
The $\mathsf{expose}(\ell,u)$ and $\mathsf{changeval}(\ell,u,x^{\prime})$
operations run in time $O\left(s_{1}(n)s_{2}^{2}(n)\right)$.
3. (c)
The $\mathsf{join}(\ell,\ell^{\prime})$ operation runs in time
$O\left(d(\ell,\ell^{\prime})\cdot s_{2}^{2}(n)\right)$ where
$d(\ell,\ell^{\prime})$ is the height difference between the layer-0 trees of
$\ell$ and $\ell^{\prime}$.
4. (d)
The $\mathsf{cut}(\ell,u)$ operation runs in time
$O\left(s_{1}(n)s_{2}^{2}(n)\right)$.
###### Proof
We prove the lemma by induction on the layer number of $\ell$. The statements
are clear if $\ell$ consists of a single layer. For the case when $\ell$ has
more than one layer, we prove each statement as follows:
1. (a)
We use $\mathsf{T}_{\mathsf{exp}}(n,0)$ to denote the maximal running time of
each iteration of $\mathsf{expose}(\ell,u)$. It is clear that the number of
iterations is bounded by the length of the path from $u$ to the root, which is
at most $s_{1}(n)$. Hence the total running time of $\mathsf{expose}(\ell,u)$
is $s_{1}(n)\mathsf{T}_{\mathsf{exp}}(n,0)$.
Note also that each iteration of $\mathsf{expose}(\ell,u)$ may make a
recursive call to $\mathsf{expose}$ on a team in the layer below, and this
recursive call may trigger further recursive calls to $\mathsf{expose}$ on
lower layers of the LTT. Thus for $0\leq i\leq\log^{*}_{\varphi}(n)$ and any
layer-$i$ team $t$, we define $\mathsf{T}_{\mathsf{exp}}(n,i)$ as the maximal
running time of an iteration in a recursive call $\mathsf{expose}(t,v)$ that
is made within $\mathsf{expose}(\ell,u)$. Since the recursive call
$\mathsf{expose}(t,v)$ consists of at most $s_{i+1}(n)$ iterations, the total
running time of $\mathsf{expose}(t,v)$ is at most
$s_{i+1}(n)\mathsf{T}_{\mathsf{exp}}(n,i)$.
To prove (a), we prove by induction on $i$ that
$\mathsf{T}_{\mathsf{exp}}(n,i)$ is $O(s_{i+2}^{2}(n))$ for all $0\leq
i\leq\log_{\varphi}^{*}(n)$.
It is clear that
$\mathsf{T}_{\mathsf{exp}}\left(n,\log^{*}_{\varphi}(n)\right)=1$. Now suppose
$t$ is a layer-$i$ team where $i<\log^{*}_{\varphi}(n)$. Each iteration in a
recursive call $\mathsf{expose}(t,v)$ makes one call to $\mathsf{cut}$ and one
call to $\mathsf{link}$. Both of these subroutine calls are made on teams in
the next layer down, which by the inductive hypothesis takes
$O(s_{i+2}(n)s^{2}_{i+3}(n))$. The iteration also recursively calls
$\mathsf{expose}$ on a team in the next layer down. By the above argument this
takes $s_{i+2}(n)\mathsf{T}_{\mathsf{exp}}(n,i+1)$. Lastly the iteration also
performs a fixed number of other elementary operations. Therefore we obtain
the following expression for $0\leq i<\log^{*}_{\varphi}(n)$:
$\mathsf{T}_{\mathsf{exp}}(n,i)\leq
c_{1}s_{i+2}(n)s^{2}_{i+3}(n)+s_{i+2}(n)\mathsf{T}_{\mathsf{exp}}(n,i+1)+c_{2}$
where $c_{1},c_{2}>0$ are constants. For convenience we drop the parameter $n$
in the above expression to get
$\mathsf{T}_{\mathsf{exp}}(i)\leq
c_{1}s_{i+2}s^{2}_{i+3}+s_{i+2}\mathsf{T}_{\mathsf{exp}}(i+1)+c_{2}$ (3)
Applying telescoping on (3), we obtain
$\displaystyle\mathsf{T}_{\mathsf{exp}}(0)\leq\ $ $\displaystyle
c_{1}s_{2}s_{3}^{2}+c_{1}s_{2}s_{3}s^{2}_{4}+\cdots+c_{1}s_{2}\cdots
s_{\log^{*}_{\varphi}(n)}s_{\log^{*}_{\varphi}(n)+1}s^{2}_{\log^{*}_{\varphi}(n)+2}$
$\displaystyle+c_{2}+c_{2}s_{2}+\cdots+c_{2}s_{2}\ldots
s_{\log_{\varphi}^{*}(n)}$ $\displaystyle\leq\ $ $\displaystyle
c_{1}\sum_{i=1}^{\log^{*}_{\varphi}(n)}\left(s_{i+2}\prod_{j=2}^{i+2}s_{j}\right)+c_{2}\sum_{i=2}^{\log^{*}_{\varphi}(n)}\prod_{j=2}^{i}s_{j}$
$\displaystyle\leq\ $ $\displaystyle
c_{1}\sum_{i=1}^{\log^{*}_{\varphi}(n)}s_{2}s_{3}^{2}s_{i+2}+c_{2}\log^{*}_{\varphi}(n)s_{2}s^{2}_{3}$
(by Lemma 9) $\displaystyle\leq\ $ $\displaystyle
c_{1}\log^{*}_{\varphi}(n)s_{2}s_{3}^{3}+c_{2}\log^{*}_{\varphi}(n)s_{2}s^{3}_{3}$
Hence the running time of a single iteration in $\mathsf{expose}(\ell,u)$ is
$O(\log^{*}_{\varphi}(n)s_{2}(n)s_{3}^{3}(n))$. By Lemma 2,
$\log^{*}_{\varphi}(n)$ is $O(s_{3}(n))$ and thus
$\mathsf{T}_{\mathsf{exp}}(n,0)$ is $O(s_{2}(n)s_{3}^{4}(n))$, which by (2),
is $O(s_{2}^{2}(n))$.
2. (b)
This statement follows directly from (a) and the fact that the maximum number
of iterations performed by the $\mathsf{expose}(\ell,u)$ operation is
$s_{1}(n)$.
3. (c)
For the $\mathsf{link}(\ell,\ell^{\prime})$ operation we use the following
inductive hypothesis: Any calls to $\mathsf{cut}$ and $\mathsf{expose}$ on
teams at layer-1 of the LTT $\ell$ takes time $cs_{2}(n)s_{3}^{2}(n)$ for some
constant $c>0$.
Let $T$ and $T^{\prime}$ be the top layer trees of $\ell$ and $\ell^{\prime}$
respectively and $d(\ell,\ell^{\prime})$ be the height difference between $T$
and $T^{\prime}$. Recall that the $\mathsf{link}(\ell,\ell^{\prime})$
operation finds a node $u$ on the rightmost path of $T$ such that $T(u)$ and
$T^{\prime}$ have the same height and links $T(u)$ and $T^{\prime}$ to a new
node below this node. Hence the $\mathsf{expose}(\ell,v)$ operation in
$\mathsf{link}(\ell,\ell^{\prime})$ consists of $d(\ell,\ell^{\prime})$
iterations. By (a), this call to $\mathsf{expose}(\ell,v)$ takes time
$c_{1}\cdot d(\ell,\ell^{\prime})\cdot s_{2}^{2}(n)$, where $c_{1}$ is a
constant.
The $\mathsf{link}(\ell,\ell^{\prime})$ operation also makes a call to
$\mathsf{rotateleft}(\ell,y)$ which consists of three calls to $\mathsf{cut}$
and one call to $\mathsf{expose}$ on teams at a lower layer. By the inductive
hypothesis, these subroutine calls to takes time $c_{2}s_{2}(n)s_{3}^{2}(n)$
for some constant $c_{2}>c$. The $\mathsf{link}(\ell,\ell^{\prime})$ operation
also performs $O(d(\ell,\ell^{\prime}))$ many other elementary operations.
Therefore the running time of $\mathsf{link}(\ell,\ell^{\prime})$ is at most
$c_{1}d(\ell,\ell^{\prime})s_{2}^{2}(n)+c_{2}s_{2}(n)s_{3}^{2}(n)+c_{3}d(\ell,\ell^{\prime}).$
Note that we may pick $c$ to be bigger than $c_{1}+c_{3}$ and therefore the
above expression is at most
$(c_{1}+c_{3})d(\ell,\ell^{\prime})s_{2}^{2}(n)+c_{2}s_{2}(n)s_{3}^{2}(n)$
which is at most $c\cdot d(\ell,\ell^{\prime})\cdot s_{2}^{2}(n)$ when $n$ is
sufficiently large. Therefore the running time for
$\mathsf{link}(\ell,\ell^{\prime})$ is $O(d(\ell,\ell^{\prime})s_{2}^{2}(n))$.
4. (d)
Let $T$ be the top-layer tree of $\ell$. The cut operation first makes a call
to $\mathsf{changeval}(T,u,-\infty)$, which by (b) takes time
$O\left(s_{1}(n)s^{2}_{2}(n)\right)$. It then walks the path from $u$ to the
root. Let $P=\\{u_{0},u_{1},\ldots,u_{m}\\}$ be the path in $T$ from $u_{0}=u$
to the root of $T$ where $u_{i+1}=p(u_{i})$ for all $0\leq i<m$. It is clear
that $m\leq s_{1}(n)$ and thus the traversal itself takes time $O(s_{1}(n))$.
By Alg. 9, the $\mathsf{cut}(\ell,u)$ operation separates $T$ into a
collection of trees
$\widehat{T}_{1},\widehat{T}_{2},\ldots,\widehat{T}_{k}$
where each $\widehat{T}_{i}$ is either the left or the right subtree of
$u_{i}$. As $T$ is balanced, one could easily prove by induction on $i$ that
$h\left(\widehat{T}_{i}\right)\leq 2i-1.$
The $\mathsf{cut}(\ell,u)$ operation then iteratively joins the trees
$\widehat{T}_{1},\widehat{T}_{2},\ldots,\widehat{T}_{k}$ to form two trees
$T_{1}$ and $T_{2}$. Let $n_{i}$ be the number of leaves in the tree
$\widehat{T}_{i}$. By (c) the total running time of the sequence of
$\mathsf{link}$ operations performed is at most
$\displaystyle\ 2\sum_{i\geq
1}^{m-1}\left(h\left(\widehat{T}_{i+1}\right)-h\left(\widehat{T}_{i}\right)\right)\cdot
s^{2}_{2}\left(n_{i+1}\right)$ $\displaystyle\leq$ $\displaystyle\
2\sum_{i\geq
1}^{m-1}\left(h\left(\widehat{T}_{i+1}\right)-h\left(\widehat{T}_{i}\right)\right)\cdot
s^{2}_{2}(n)$ $\displaystyle\leq$ $\displaystyle\
2\left(h\left(\widehat{T}_{m}\right)-h\left(\widehat{T}_{1}\right)\right)\cdot
s^{2}_{2}(n)$ $\displaystyle\leq$ $\displaystyle\ 2s_{1}(n)s^{2}_{2}(n).$
Therefore the overall running time of the $\mathsf{cut}(\ell,u)$ operation is
$O\left(s_{1}(n)s^{2}_{2}(n)\right)$. ∎
###### Theorem 6.1
There is an algorithm that solves the dynamic partial sorting problem which
performs the $\mathsf{psort}(\ell,k)$ operation in time
$O(\log^{*}_{\varphi}(n)k\log k)$, and performs the
$\mathsf{link}(\ell,\ell^{\prime})$, $\mathsf{cut}(\ell,x)$ and
$\mathsf{changeval}(\ell,x,x^{\prime})$ operations in time
$O\left(\log(n)\cdot\log^{2}(\log(n))\right)$, where $n$ is the size of the
list $\ell$.
###### Proof
The correctness of the $\mathsf{psort}(\ell,k)$ operation follows from Lemma
5. For correctness of the update operation, assume that (J1)–(J3) hold for
every node in the LTT data structure. Suppose we perform the
$\mathsf{changeval}(\ell,u,x^{\prime})$ operation. Since $u$ is a leaf in
$\ell$, by Lemma 7, (J1)–(J3) still hold for every node in the LTT. Suppose we
perform the $\mathsf{link}(\ell,\ell^{\prime})$ operation. The
$\mathsf{expose}(\ell,v)$ operation in Line 7 in Alg. 8 preserves (J1)–(J3)
for every node. If the operation performs $\mathsf{rotateleft}(\ell,y)$ in
Line 9, then by Lemma 8 (J1)–(J3) also hold for every node and thus
$\mathsf{link}(\ell,\ell^{\prime})$ is correct. Lastly, suppose we perform the
$\mathsf{cut}(\ell,u)$ operation. Then (J1)–(J3) still hold by the correctness
of $\mathsf{changeval}$ and $\mathsf{join}$.
The complexity of the $\mathsf{psort}(\ell,k)$ operations follows directly
from Lemma 6. The complexity of the update operations follows from Lemma 10
and Lemma 3.∎
## 7 Conclusion and Future Work
This paper presents data structures for solving the dynamic partial sorting
problem.We propose here two possible directions of optimizing the layered
tournament trees: on query size and on intervals. In both cases, we seek to
perform optimizations by determining an optimal query size or interval, and
then creating a data structure which performs this query optimally. This is
similar in principle to optimized BSTs as presented in [4].
We can perform these optimizations either statically or dynamically. In the
case of optimizing for query size, in the static case, we have a table of
queries and the probability that a query will have that length (similarly to
the optimal BST). We then determine an expected query length, and make a
structure to perform queries of that length optimally. In the dynamic case,
the structure keeps track of query probabilities, and dynamically rebuilds
itself when the expected query length changes. When optimizing for intervals,
one would take a similar approach, except to optimize access to a particular
interval or set of intervals that are frequently queried.
As the layered tournament tree structure is designed for very large data sets,
other optimizations to consider for the structure are parallelism, external
memory use optimization, and persistence (as described in [6]). In particular,
the first two of these are suitable for extremely large data sets, and require
different analysis of the structure, and likely a different implementation as
well.
## References
* [1] Andersson, A., Fagerberg, R., Larsen, K.: Balanced Binary Search Trees. In: Mehta, D., Sahni, S., eds: Handbook of Data Structures and Applications, 182–205, 2002
* [2] Bordim, J., Nakano, K., Shen, H.: Sorting on Single-Channel Wireless Sensor Networks. In: Hsu, F., Ibarra, H., Saldaña, R., eds, Proc. of the International Symposium on Parallel Architectures, Algorithms and Networks (I-SPAN’02), 133–138, 2002
* [3] Duch, A., Jiménez, R., Martínez, C.: Selection by rank in $k$-dimensional binary search trees. In: Random Structures and Algorithms, appeared online 2012
* [4] Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, the MIT Press. 356–369, 2002
* [5] Floyd, R., Rivest, R.: Expected time bounds for selection. In: Communications of the ACM 18(3). 165–172, 1975
* [6] Haim, K.: Persistent Data Structures. In: Mehta, D., Sahni, S., eds: Handbook of Data Structures and Applications, 182–205, 2002
* [7] Hoare, C.: Quicksort. Computer Journal, 5:10–15, 1962.
* [8] Huang, H., Tsai, T., Quickselect and the Dickman function. In: Combinatorics, Probability and Computing 11(4), 353–371, 2000
* [9] Jiménez, R., Martínez, C.: Interval Sorting. In: Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP 2010), Part I, 238–248, 2010
* [10] Knuth, D.: The Art of Computer Programming, Sorting and Searching, Volume 3, 141–142, 1998
* [11] Kuba, M.: On Quickselect, partial sorting and Multiple Quickselect. In: Information Processing Letters 99(5), 181–186, 2006
* [12] Martínez, C.: Partial quicksort. In: Arge, L., Italiano, G., Sedgewick, R., eds. Proc. of the 6th ACM-SIAM Workshop on Algorithm Engineering and Experiments (ALENEX) and the 1st ACM-SIAM Workshop on Analytic Algorithmics and Combinatorics (ANALCO), 224–228, 2004
* [13] Navarro, G., Paredes, R.: On Sorting, Heaps and Minimum Spanning Trees. In: Algorithmica 57(4), 585–620, 2010
* [14] Sleator, D., Tarjan, R.: Self-Adjusting Binary Search Trees. In: Journal of the ACM 32(3), 625–686, 1985
|
arxiv-papers
| 2014-02-12T01:36:36 |
2024-09-04T02:49:58.137717
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiamou Liu and Kostya Ross",
"submitter": "Jiamou Liu",
"url": "https://arxiv.org/abs/1402.2712"
}
|
1402.2878
|
# Number of unique Edge-magic total labelings on Path $P_{n}$
Mukkai S. Krishnamoorthy
Rensselaer Polytechnic Institute, Troy, NY
Allen Lavoie
Washington University, St. Louis, MO
Ali Dasdan,
Hypergrowth, San Fransico Bay Area, CA
Bharath Santosh
Rennselaer Polytechnic Institute, Troy, NY
###### Abstract
Edge-magic total labeling was introduced by [2]. The number of edge-magic
solutions for cycles have been explored in [1]. This sequence is mentioned in
On Line Encyclopedia of Integer Sequences (OEIS) [3]. In this short note, we
enumerate the number of unique edge-magic total labelings on Path $P_{n}$
## 1 Introduction
Edge-magic labeling (EMTL) has been studied in the past with an application
towards communication networks. Given a simple undirected graph $G=(V,E)$, let
$\lambda$ be a mapping from the numbers $1,2,\cdots,|V|+|E|$ to the vertices
and edges of graph G, such that each element has an unique label. The weight
of an edge is obtained as the sum of the labels of that edge and its two end
vertices. An edge-magic total labeling is labeling in which the weight of
every edge is the same. The weight of the each edge is said to be a magic
constant. Figure 1 illustrates an example for a path of length 5 with a magic
constant of 16.
Figure 1: Path of length 5
Paper by Wallis et al [4] describes existence of edge-magic total labeling of
many types of graphs including $P_{n}$. The aim of this note is to enumerate
unique edge-magic total labeling of Path $P_{n}$.
## 2 Main Results
Our results are summarized in the following Table.
Path Length | Number of Solutions
---|---
0 | 1
1 | 3
2 | 12
3 | 28
4 | 48
5 | 240
6 | 944
7 | 5344
8 | 23408
9 | 133808
10 | 751008
11 | 5222768
12 | 37898776
13 | 292271304
Table 1: Number of Edge-Magic Total Labels.
As far as we have seen this series 1,3,12,28,48,240,944,5344,23408 does not
appear in OEIS.
### 2.1 Method and Program
We started with a simple python program to obtain all edge-magic solutions of
paths of lengths 2 to 17. Paths of length 2 means that there will be three
vertices and 2 edges, a total of 5 graph elements.
import itertools
for j in range(5,27,2):
x = range(1,j+1)
sum2 = 0
for a in itertools.permutations(x):
x = list(a)
sum1 = x[0]+x[1]+x[2]
d = 1
for i in range(2,j-2,2):
if (sum1== x[i]+x[i+1]+x[i+2]):
d = 1
else:
d = 0
break
if (d==1):
if (a[0]<a[j-1]):
#print x, sum1
sum2 = sum2+1
print j, "\t", sum2
This program generated one permutation at a time, and checked for whether the
assignment leads to an edge-magic labeling.
However it is too slow and we could not compute past the path length of 7. We
further optimized our python code and utilized the bounds on magic sum, $k$,
similar to the one used in the paper[1]. Let $f(r)=\frac{r\times(r+1)}{2}$
$\frac{f(2\times n+1)+f(n-1)}{n}\leavevmode\nobreak\ \leq\leavevmode\nobreak\
k\leavevmode\nobreak\ \leq\frac{2\times f(2\times n+1)-f(n+2)}{n}$
With this improvement and a shortcircuit optimization, we are able to get up
to a path length of 13. All our code is located in the following github
location https://github.com/allenlavoie/path-counting .
We will like to point the total number of edge-magic solutions for paths form
a strict (albeit weak) upper bound for total number of edge-magic solutions
for cycles of the same length.
## References
* [1] A. Baker and J. Sawada, ”Magic Labelings on Cycles and Wheels,” Lecture Notes in Computer Science 5165 Combinatorial Optimization and Applications. Second International Conference, COCOA 2008. pp. 361-373
* [2] R. D.Godbold and P.J.. Slater, ”All cycles are edge-magic,” Bulletin of the ICA vol. 22 (1998) pp. 93-97.
* [3] _On Line Encyclopedia of Integer Sequences, Seq A 145692_ http://oeis.org/A145692
* [4] W. D. Wallis et. al, ”Edge-magic total labelings,” Australasian Journal of Combinatorics vol. 22, 2000. pp.177-190
|
arxiv-papers
| 2014-02-10T18:17:07 |
2024-09-04T02:49:58.153141
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mukkai S. Krishnamoorthy, Allen Lavoie, Ali Dasdan and Bharath Santosh",
"submitter": "Mukkai Krishnamoorthy",
"url": "https://arxiv.org/abs/1402.2878"
}
|
1402.2918
|
# Confidence Bands for Distribution Functions:
A New Look at the Law of the Iterated Logarithm
Lutz Dümbgen∗ and Jon A. Wellner∗∗
(March 2014)
###### Abstract
We present a general law of the iterated logarithm for stochastic processes on
the open unit interval having subexponential tails in a locally uniform
fashion. It applies to standard Brownian bridge but also to suitably
standardized empirical distribution functions. This leads to new goodness-of-
fit tests and confidence bands which refine the procedures of Berk and Jones
(1979) and Owen (1995). Roughly speaking, the high power and accuracy of the
latter procedures in the tail regions of distributions are essentially
preserved while gaining considerably in the central region.
∗Work supported by Swiss National Science Foundation.
∗∗Work supported in part by NSF DMS 11-04832 and NI-AID Grant 2R01
AI291968-04.
#### AMS subject classifications.
62E20, 62G15, 62G30.
#### Key words.
Confidence band, limit distribution, sub-exponential tails, submartingale,
tail regions.
## 1 Introduction
Let $\widehat{F}_{n}$ be the empirical distribution function of independent
random variables $X_{1},X_{2},\ldots,X_{n}$ with unknown distribution function
$F$ on the real line. Let us recall some well-known facts about
$\widehat{F}_{n}$ (cf. Shorack and Wellner 1986): The stochastic process
$\bigl{(}\widehat{F}_{n}(x)\bigr{)}_{x\in\mathbb{R}}$ has the same
distribution as $\bigl{(}\widehat{G}_{n}(F(x))\bigr{)}_{x\in\mathbb{R}}$,
where $\widehat{G}_{n}$ is the empirical distribution of independent random
variables $U_{1},U_{2},\ldots,U_{n}$ with uniform distribution on $[0,1]$.
This enables us to construct confidence bands for the distribution function
$F$. A well-known classical method are Kolmogorov-Smirnov confidence bands:
Let
$\mathbb{U}_{n}(t)\ :=\ n^{1/2}(\widehat{G}_{n}(t)-t),$
and let $\kappa_{n,\alpha}^{\rm KS}$ be the $(1-\alpha)$-quantile of
$\|\mathbb{U}_{n}\|_{\infty}\ :=\ \sup_{t\in[0,1]}|\mathbb{U}_{n}(t)|.$
Then with probability at least $1-\alpha$,
$F(x)\ \in\ [\widehat{F}_{n}(x)\pm n^{-1/2}\kappa_{n,\alpha}^{\rm
KS}]\quad\text{for all}\ x\in\mathbb{R}.$ (1)
Equality holds if $F$ is continuous. Since $\mathbb{U}_{n}$ converges in
distribution in $\ell_{\infty}([0,1])$ to standard Brownian bridge
$\mathbb{U}$, $\kappa_{n,\alpha}^{\rm KS}$ converges to the
$(1-\alpha)$-quantile $\kappa_{\alpha}^{\rm KS}$ of $\|\mathbb{U}\|_{\infty}$.
In particular, the simultaneous confidence intervals in (1) have width
$O(n^{-1/2})$ uniformly in $x\in\mathbb{R}$.
Another method, based on a goodness-of-fit test by Berk and Jones (1979), was
introduced by Owen (1995): Let $\kappa_{n,\alpha}^{\rm BJ}$ be the
$(1-\alpha)$-quantile of
$T_{n}^{\rm BJ}\ :=\ n\sup_{t\in(0,1)}K(\widehat{G}_{n}(t),t),$
where
$K(s,t)\ :=\ s\log\frac{s}{t}+(1-s)\log\frac{1-s}{1-t}$
for $s\in[0,1]$ and $t\in(0,1)$. This leads to an alternative confidence band
for $F$: With probability at least $1-\alpha$,
$nK(\widehat{F}_{n}(x),F(x))\ \leq\ \kappa_{n,\alpha}^{\rm BJ}\quad\text{for
all}\ x\in\mathbb{R}.$ (2)
As shown by Jager and Wellner (2007), the asymptotic distribution of
$T_{n}^{\rm BJ}$ remains the same if one replaces $K(s,t)$ by a more general
function; in particular, one may interchange its two arguments. Moreover,
$\kappa_{n,\alpha}^{\rm BJ}\ =\ \log\log(n)+2^{-1}\log\log\log(n)+O(1).$
From this one can deduce that (2) leads to confidence intervals with length at
most
$2\bigl{(}2\gamma_{n}F(x)(1-F(x))\bigr{)}^{1/2}+2\gamma_{n}\quad\text{where}\quad\gamma_{n}:=\frac{\kappa_{n,\alpha}^{\rm
BJ}}{n}=(1+o(1))\frac{\log\log n}{n},$
uniformly in $x\in\mathbb{R}$; see (K.5) in Section 6.2. Hence they are
substantially shorter than the Kolmogorov-Smirnov intervals for $F(x)$ close
to $0$ or $1$. But in the central region, i.e. when $F(x)$ is bounded away
from $0$ and $1$, they are of width $O(n^{-1/2}(\log\log n)^{1/2})$ rather
than $O(n^{-1/2})$. An obvious goal is to refine these methods and combine the
benefits of the Kolmogorov-Smirnov and Berk-Jones confidence bands. Methods of
this type have been proposed by various authors, see Mason and Schuenemeyer
(1983) and the references cited therein.
A key for understanding the asymptotics of $T_{n}^{\rm BJ}$ but also the new
methods presented later are suitable variants of the law of the iterated
logarithm (LIL). For Brownian bridge $\mathbb{U}$ the LIL states that
$\limsup_{t\downarrow 0}\frac{\mathbb{U}(t)}{\sqrt{2t\log\log(1/t)}}\ =\
\limsup_{t\uparrow 1}\frac{\mathbb{U}(t)}{\sqrt{2(1-t)\log\log(1/(1-t))}}\ =\
1$ (3)
almost surely. Various refinements of this result have been obtained. One
particular consequence of Kolmogorov’s upper class test (cf. Erdös 1942, or
Ito and McKean 1974, Chapter 1.8) is the following result: For $t\in(0,1)$
define
$\displaystyle C(t)\ $ $\displaystyle:=\ \log\log\frac{e}{4t(1-t)}\ =\
\log\bigl{(}1-\log(1-(2t-1)^{2})\bigr{)}\ \geq\ 0,$ $\displaystyle D(t)\ $
$\displaystyle:=\ \log(1+C(t)^{2})\ \geq\ 0.$
Then for any fixed $\nu>3/4$,
$\sup_{t\in(0,1)}\Bigl{(}\frac{\mathbb{U}(t)^{2}}{2t(1-t)}-C(t)-\nu
D(t)\Bigr{)}\ <\ \infty$ (4)
almost surely. Note that $C(t)=C(1-t)$, $D(t)=D(1-t)$, and, as $t\downarrow
0$,
$\displaystyle C(t)\ $ $\displaystyle=\
\log\log(1/t)+O\bigl{(}\log(1/t)^{-1}\bigr{)},$ $\displaystyle D(t)\ $
$\displaystyle=\ 2\log\log\log(1/t)+O\bigl{(}(\log\log(1/t))^{-1}\bigr{)}.$
This explains why (4) follows from Kolmogorov’s test and shows the connection
between (4) and (3). Note also that
$\lim_{t\to 1/2}\frac{C(t)}{(2t-1)^{2}}\ =\ \lim_{t\to
1/2}\frac{D(t)}{(2t-1)^{4}}\ =\ 1.$
In the present paper we prove statements similar to (4) for general stochastic
processes on $(0,1)$. In Section 2 we state a general condition on a
stochastic process $X=(X(t))_{t\in(0,1)}$ such that for any fixed $\nu>1$,
$\sup_{t\in(0,1)}\bigl{(}X(t)-C(t)-\nu D(t)\bigr{)}\ <\ \infty$
almost surely. In particular, the stochastic process
$X(t)\ :=\ \frac{\mathbb{U}(t)^{2}}{2t(1-t)}$
satisfies this condition. Then in Section 3 these general results are applied
to
$X_{n}(t)\ :=\ nK(\widehat{G}_{n}(t),t).$
It turns out that for any fixed $\nu>1$,
$T_{n,\nu}\ :=\ \sup_{t\in(0,1)}\bigl{(}nK(\widehat{G}_{n}(t),t)-C(t)-\nu
D(t)\bigr{)}$ (5)
converges in distribution to
$T_{\nu}\ :=\
\sup_{t\in(0,1)}\Bigl{(}\frac{\mathbb{U}(t)^{2}}{2t(1-t)}-C(t)-\nu
D(t)\Bigr{)}.$
Asymptotic statements like this refer to $n\to\infty$, unless stated
otherwise. Moreover, if $U_{n:1}<U_{n:2}<\cdots<U_{n:n}$ are the order
statistics of $U_{1},U_{2},\ldots,U_{n}$, then for fixed $\nu>1$,
$\tilde{T}_{n,\nu}\ :=\
\max_{j=1,2,\ldots,n}\bigl{(}(n+1)K(t_{nj},U_{n:j})-C(t_{nj})-\nu
D(t_{nj})\bigr{)}\ \to_{\mathcal{L}}\ T_{\nu},$
where
$t_{nj}\ :=\ \frac{j}{n+1}\quad\text{for}\ j=1,2,\ldots,n.$
To test the null hypothesis that $F$ is equal to a given continuous
distribution function $F_{o}$, consider the test statistic
$T_{n,\nu}(F_{o})\ :=\
\sup_{x\in\mathbb{R}}\bigl{(}nK(\widehat{F}_{n}(x),F_{o}(x))-C(F_{o}(x))-\nu
D(F_{o}(x))\bigr{)}.$ (6)
Under the null hypothesis, $T_{n,\nu}(F_{o})$ has the same distribution as
$T_{n,\nu}$. Hence if $\kappa_{n,\nu,\alpha}$ denotes the
$(1-\alpha)$-quantile of $T_{n,\nu}$, one may reject the null hypothesis at
level $\alpha\in(0,1)$ if $T_{n,\nu}(F_{o})$ exceeds $\kappa_{n,\nu,\alpha}$.
In Section 4 we investigate the power of this new test in more detail. In
particular we show that it attains the detection boundary for Gaussian mixture
models as specified by Donoho and Jin (2004).
The statistic $\tilde{T}_{n,\nu}$ leads to a new confidence band for $F$: Let
$-\infty=X_{n:0}<X_{n:1}\leq X_{n:2}\leq\cdots\leq X_{n:n}<X_{n:n+1}=\infty$
be the order statistics of $X_{1},X_{2},\ldots,X_{n}$, and let
$\tilde{\kappa}_{n,\nu,\alpha}$ and $\kappa_{\nu,\alpha}$ be the
$(1-\alpha)$-quantile of $\tilde{T}_{n,\nu}$ and $T_{\nu}$, respectively. Then
$\tilde{\kappa}_{n,\nu,\alpha}\to\kappa_{\nu,\alpha}$, and with probability at
least $1-\alpha$, the following is true: For $0\leq j\leq n$ and $X_{n:j}\leq
x<X_{n:j+1}$,
$F(x)\ \in\ [a_{nj},b_{nj}],$
where $a_{n0}:=0$, $b_{nn}:=1$ and
$\displaystyle a_{nj}$
$\displaystyle:=\min\bigl{\\{}u\in[0,1]:nK(t_{nj},u)\leq C(t_{nj})+\nu
D(t_{nj})+\tilde{\kappa}_{n,\nu,\alpha}\bigr{\\}}\ \ \text{if}\ j>0,$
$\displaystyle b_{nj}$
$\displaystyle:=\max\bigl{\\{}u\in[0,1]:nK(t_{n,j+1},u)\leq C(t_{n,j+1})+\nu
D(t_{n,j+1})+\tilde{\kappa}_{n,\nu,\alpha}\bigr{\\}}\ \ \text{if}\ j<n.$
Since $C(t_{nj})+\nu D(t_{nj})+\tilde{\kappa}_{n,\nu,\alpha}$ is no larger
than
$C(t_{n1})+\nu D(t_{n1})+\tilde{\kappa}_{n,\nu,\alpha}\ =\ (1+o(1))\log\log n$
for $1\leq j\leq n$, our confidence bands have similar accuracy as those of
Owen (1995) in the tail regions while achieving the usual root-$n$ consistency
everywhere. A more precise comparison is provided in Section 5. Thereafter we
relate our methods to a negative result of Bahadur and Savage (1956) about the
nonexistence of confidence bands with vanishing width in the tails. Finally we
discuss briefly an interesting alternative approach to goodness-of-fit tests
and confidence bands by Aldor-Noiman et al. (2013) and Eiger et al. (2013).
All proofs and technical arguments are deferred to Section 6. Section 7
contains supplementary material including a quantitative version of Bahadur
and Savage (1956, Theorem 2) and decision theoretic considerations about the
Gaussian mixture model of Donoho and Jin (2004).
## 2 A general non-Gaussian LIL
Our conditions and results involve the function
$\mathop{\mathrm{logit}}\nolimits:(0,1)\to\mathbb{R}$ with
$\mathop{\mathrm{logit}}\nolimits(t)\ :=\ \log\Bigl{(}\frac{t}{1-t}\Bigr{)}.$
Its inverse is the logistic function $\ell:\mathbb{R}\to(0,1)$ with
$\ell(x)\ :=\ \frac{e^{x}}{1+e^{x}}\ =\ \frac{1}{e^{-x}+1},$
and
$\ell^{\prime}(x)\ =\ \ell(x)(1-\ell(x))\ =\ \frac{1}{e^{x}+e^{-x}+2}.$
We consider stochastic processes $X=(X(t))_{t\in\mathcal{T}}$ on subsets
$\mathcal{T}$ of $(0,1)$ which have locally uniformly sub-exponential tails in
the following sense:
###### Condition 2.1.
There exist a real constant $M\geq 1$ and a non-increasing funtion
$L:[0,\infty)\to[0,1]$ such that $L(c)=1-O(c)$ as $c\downarrow 0$, and
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{t\in[\ell(a),\ell(a+c)]\cap\mathcal{T}}X(t)>\eta\Bigr{)}\
\leq\ M\exp(-L(c)\eta)$ (7)
for arbitrary $a\in\mathbb{R}$, $c\geq 0$ and $\eta\in\mathbb{R}$.
###### Theorem 2.2.
Suppose that $X$ satisfies Condition 2.1. For arbitrary $\nu>1$ and
$L_{o}\in(0,1)$ there exists a real constant $M_{o}\geq 1$ depending only on
$M$, $L(\cdot)$, $\nu$ and $L_{o}$ such that
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{t\in\mathcal{T}}\bigl{(}X(t)-C(t)-\nu
D(t)\bigr{)}>\eta\Bigr{)}\ \leq\ M_{o}\exp(-L_{o}\eta)\quad\text{for
arbitrary}\ \eta\geq 0.$
###### Remark 2.3.
Suppose that $X$ satisfies Condition 2.1, where $\inf(\mathcal{T})=0$ and
$\sup(\mathcal{T})=1$. For any $\nu>1$, the supremum $T_{\nu}(X)$ of $X-C-\nu
D$ over $\mathcal{T}$ is finite almost surely. But this implies that
$\lim_{t\to\\{0,1\\}}\bigl{(}X(t)-C(t)-\nu D(t)\bigr{)}\ =\ -\infty$
almost surely. For if $1<\nu^{\prime}<\nu$, then
$X(t)-C(t)-\nu D(t)\ \leq\ T_{\nu^{\prime}}(X)-(\nu-\nu^{\prime})D(t),$
so the claim follows from $T_{\nu^{\prime}}(X)<\infty$ almost surely and
$D(t)\to\infty$ as $t\to\\{0,1\\}$.
###### Remark 2.4.
Our definition of the function $D=\log(1+C^{2})$ may look somewhat arbitrary.
Indeed, we tried various choices, e.g. $D=2\log(1+C)$. Theorem 2.2 is valid
for any nonnegative function $D$ on $(0,1)$ such that $D(1-\cdot)=D(\cdot)$
and $D(t)/\log\log\log(1/t)\to 2$ as $t\downarrow 0$. The special choice
$D=\log(1+C^{2})$ yielded a rather uniform distribution of
$\mathop{\mathrm{arg\,max}}_{(0,1)}(X-C-\nu D)$ when
$X(t)=\mathbb{U}(t)^{2}/(2t(1-t))$ and $\nu$ close to one.
Our first example for a process $X$ satisfying Condition 2.1 is squared and
standardized Brownian bridge:
###### Lemma 2.5.
Let $\mathcal{T}=(0,1)$ and $X(t)=\mathbb{U}(t)^{2}/(2t(1-t))$ with standard
Brownian bridge $\mathbb{U}$. Then Condition 2.1 is satisfied with $M=2$ and
$L(c)=e^{-c}$.
In particular, Lemma 2.5 and Theorem 2.2 yield (4) for any $\nu>1$.
## 3 Implications for the uniform empirical process
As indicated in the introduction, Theorem 2.2 may be applied to the uniform
empirical process $\widehat{G}_{n}$ in two ways. A first version concerns
$\mathcal{T}=(0,1)$ and
$X_{n}(t)\ :=\ nK(\widehat{G}_{n}(t),t).$
###### Lemma 3.1.
The stochastic process $X_{n}$ satisfies Condition 2.1 with $M=2$ and
$L(c)=e^{-c}$.
Combining this lemma, Theorem 2.2 and Donsker’s Theorem for the uniform
empirical process yields the following result:
###### Theorem 3.2.
For any fixed $\nu>1$,
$T_{n,\nu}\ =\ \sup_{t\in(0,1)}\bigl{(}X_{n}(t)-C(t)-\nu D(t)\bigr{)}$
converges in distribution to the random variable
$T_{\nu}\ :=\
\sup_{t\in(0,1)}\Bigl{(}\frac{\mathbb{U}(t)^{2}}{2t(1-t)}-C(t)-\nu
D(t)\Bigr{)}.$
For the computation of confidence bands it is more convenient to work with the
following stochastic process on
$\mathcal{T}_{n}:=\\{t_{nj}:j=1,2,\ldots,n\\}$:
$\tilde{X}_{n}(t_{nj})\ :=\ (n+1)K(t_{nj},U_{n:j}).$
###### Lemma 3.3.
The stochastic process $\tilde{X}_{n}$ satisfies Condition 2.1 with $M=2$ and
$L(c)=e^{-c}$.
Again we may combine this with Theorem 2.2 and Donsker’s theorem for partial
sum processes to obtain a new limit theorem:
###### Theorem 3.4.
For any fixed $\nu>1$,
$\tilde{T}_{n,\nu}\ =\
\sup_{t\in\mathcal{T}_{n}}\bigl{(}\tilde{X}_{n}(t)-C(t)-\nu D(t)\bigr{)}$
converges in distribution to the random variable $T_{\nu}$ defined in Theorem
3.2.
## 4 Goodness-of-fit tests
As explained in the introduction, we may reject the null hypothesis that $F$
is a given continuous distribution function $F_{o}$ at level $\alpha$ if
$T_{n,\nu}(F_{o})\ =\
\sup_{x\in\mathbb{R}}\bigl{(}nK(\widehat{F}_{n}(x),F_{o}(x))-C(F_{o}(x))-\nu
D(F_{o}(x))\bigr{)},$
exceeds $\kappa_{n,\nu,\alpha}$. Note also that the latter supremum may be
expressed as the maximum of $2n+1$ terms, replacing the argument $(x)$ with
$(X_{n:i})$ and $(X_{n:i}\,-)$ for $1\leq i\leq n$ or with
$(F_{o}^{-1}(1/2))$.
As shown in the next lemma, for any fixed citical value $\kappa>0$, the
probability that $T_{n,\nu}(F_{o})\leq\kappa$ is small if the quantity
$\Delta_{n}(F,F_{o})\ :=\
\sup_{\mathbb{R}}\frac{\sqrt{n}|F-F_{o}|}{\sqrt{\Gamma(F_{o})F_{o}(1-F_{o})}+\Gamma(F_{o})/\sqrt{n}}$
is large, where $\Gamma(t):=C(t)+1$ for $t\in[0,1]$ with $C(0):=C(1):=\infty$.
Note that $\Gamma(t)t(1-t)\to 0$ as $(0,1)\ni t\to\\{0,1\\}$.
###### Lemma 4.1.
For any critical value $\kappa>0$ there exists a constant $B_{\nu,\kappa}$
such that
$\mathop{\mathrm{I\\!P}}\nolimits_{F}\bigl{(}T_{\nu,n}(F_{o})\leq\kappa\bigr{)}\
\leq\ B_{\nu,\kappa}\,\Delta_{n}(F,F_{o})^{-4/5}.$ (8)
Here and subsequently, the subscript $F$ in
$\mathop{\mathrm{I\\!P}}\nolimits_{F}(\cdot)$ or
$\mathop{\mathrm{I\\!E}}\nolimits_{F}(\cdot)$ specifies the true distribution
function of the random variables $X_{1},X_{2},\ldots,X_{n}$. Now consider an
arbitrary sequence $(F_{n})_{n}$ of distribution functions. Then for any fixed
level $\alpha\in(0,1)$, Lemma 4.1 and the fact that
$\kappa_{n,\nu,\alpha}\to\kappa_{\nu,\alpha}<\infty$ imply that
$\mathop{\mathrm{I\\!P}}\nolimits_{F_{n}}\bigl{(}T_{n,\nu}(F_{o})>\kappa_{n,\nu,\alpha}\bigr{)}\
\to\ 1$
provided that
$\Delta_{n}(F_{n},F_{o})\ \to\ \infty.$ (9)
In particular, (9) is satisfied if $F_{n}\equiv F_{*}\not\equiv F_{o}$ for all
sample sizes $n$. Thus our test has asymptotic power one for any fixed
distribution function different from $F_{o}$.
#### Detecting Gaussian mixtures.
We consider a testing problem studied in detail by Donoho and Jin (2004). The
null hypothesis is given by $F_{o}=\Phi$, the standard Gaussian distribution
function, whereas
$F_{n}(x)\ :=\ (1-\varepsilon_{n})\Phi(x)+\varepsilon_{n}\Phi(x-\mu_{n})$
for certain numbers $\varepsilon_{n}\in(0,1)$ and $\mu_{n}>0$. By means of
Lemma 4.1 one can derive the following result:
###### Lemma 4.2.
(a) Suppose that $\varepsilon_{n}=n^{-\beta+o(1)}$ for some fixed
$\beta\in(1/2,1)$. Further let $\mu_{n}=\sqrt{2r\log(n)}$ for some
$r\in(0,1)$. Then $\Delta_{n}(F_{n},\Phi)\to\infty$ if
$r\ >\ r_{*}(\beta):=\begin{cases}\beta-1/2&\text{if}\ \beta\in(1/2,3/4],\\\
\bigl{(}1-\sqrt{1-\beta}\bigr{)}^{2}&\text{if}\ \beta\in[3/4,1).\end{cases}$
(b) Suppose that $\varepsilon_{n}=n^{-1/2+o(1)}$ such that
$\pi_{n}:=\sqrt{n}\varepsilon_{n}\to 0$. Then
$\Delta_{n}(F_{n},\Phi)\to\infty$ if $\mu_{n}=\sqrt{2s\log(1/\pi_{n})}$ for
some $s>1$.
As explained by Donoho and Jin (2004), any goodness-of-fit test at fixed level
$\alpha\in(0,1)$ has trivial asymptotic power $\alpha$ whenever
$\varepsilon_{n}=n^{-\beta}$ for some $\beta\in(1/2,1)$ and
$\mu_{n}=\sqrt{2r\log(n)}$ with $r<r_{*}(\beta)$. Thus our new test provides
another example of an asymptotically optimal procedure in this particular
setting. Other procedures with asymptotic power one whenever $r>r_{*}(\beta)$
are Tukey’s higher criticism test (Donoho and Jin 2004) or the generalized
Berk–Jones tests (Jager and Wellner 2007).
In the setting of part (b), the latter two classes of tests can fail to have
asymptotic power one if $\mu_{n}=\sqrt{2s\log(1/\pi_{n})}$ for fixed $s>1$ but
$\pi_{n}\to 0$ sufficiently slow. On the other hand, one can show that any
level-$\alpha$ test of $F_{o}$ versus $F_{n}$ has trivial asymptotic power
whenever $\mu_{n}\leq\sqrt{2s\log(1/\pi_{n})}$ for an arbitrary fixed $s<1$. A
rigorous proof is provided with the supplementary material.
Parts (a) and (b) of Lemma 4.2 are well connected. For let
$\varepsilon_{n}=n^{-\beta+o(1)}$ for some $\beta\in(1/2,3/4]$, and
$\mu_{n}=\sqrt{2r\log(n)}$ for some $r>\beta-1/2$. Then $s:=r/(\beta-1/2)>1$
and with $\pi_{n}:=\sqrt{n}\varepsilon_{n}=n^{1/2-\beta+o(1)}$ we may rewrite
$\mu_{n}$ as
$\mu_{n}\ =\ \sqrt{2s(\beta-1/2)\log(n)}\ =\ \sqrt{(2s+o(1))\log(1/\pi_{n})}.$
## 5 Confidence bands
The confidence bands of Owen (1995) may be described as follows: For $0\leq
j\leq n$ let $s_{nj}:=j/n$. With confidence $1-\alpha$ we may claim that for
$0\leq j\leq n$ and $X_{n:j}\leq x<X_{n:j+1}$,
$F(x)\ \in\ [a_{nj}^{\rm BJO},b_{nj}^{\rm BJO}],$
where
$\displaystyle b_{nj}^{\rm BJO}\ $ $\displaystyle:=\
\begin{cases}\max\bigl{\\{}b\in(s_{nj},1):K(s_{nj},b)\leq\gamma_{n}^{\rm
BJ}\bigr{\\}}&\text{for}\ 0\leq j<n,\\\ 1&\text{for}\ j=n,\end{cases}$
$\displaystyle a_{nj}^{\rm BJO}\ $ $\displaystyle:=\ 1-b_{n,n-j}^{\rm BJO},$
and
$\gamma_{n}^{\rm BJ}\ =\ \frac{\kappa_{n,\alpha}^{\rm BJ}}{n}\ =\
\frac{\log\log n}{n}(1+o(1)).$
Our new method is analogous: With confidence $1-\alpha$, for $0\leq j\leq n$
and $X_{n:j}\leq x<X_{n:j+1}$, the value $F(x)$ is contained in
$[a_{nj},b_{nj}]$, where
$\displaystyle b_{nj}\ $ $\displaystyle:=\
\begin{cases}\max\bigl{\\{}u\in(t_{n,j+1},1):K(t_{n,j+1},u)\leq\gamma_{n}(t_{n,j+1})\bigr{\\}}&\text{for}\
0\leq j<n,\\\ 1&\text{for}\ j=n,\end{cases}$ $\displaystyle a_{nj}\ $
$\displaystyle:=\ 1-b_{n,n-j},$
and
$\gamma_{n}(t)\ :=\ \frac{C(t)+\nu D(t)+\tilde{\kappa}_{n,\nu,\alpha}}{n+1}$
for $t\in\mathcal{T}_{n}$. Asymptotically the new confidence band is
everywhere at least as good as Owen’s (1995) band, and in the central region
it is infinitely more accurate:
###### Theorem 5.1.
For any fixed $\alpha\in(0,1)$,
$\max_{j=0,1,\ldots,n-1}\frac{b_{nj}-s_{nj}}{b_{nj}^{\rm BJO}-s_{nj}}\ =\
\max_{j=1,2,\ldots,n}\frac{s_{nj}-a_{nj}}{s_{nj}-a_{nj}^{\rm BJO}}\ \to\ 1,\\\
$
while
$\displaystyle\max_{j=0,1,\ldots,n}(b_{nj}^{\rm BJO}-s_{nj})\ =\
\max_{j=0,1,\ldots,n}(s_{nj}-a_{nj}^{\rm BJO})\ $ $\displaystyle=\
(1+o(1))\sqrt{\frac{\log\log n}{2n}},$
$\displaystyle\max_{j=0,1,\ldots,n}(b_{nj}-s_{nj})\ =\
\max_{j=0,1,\ldots,n}(s_{nj}-a_{nj})\ $ $\displaystyle=\ O(n^{-1/2}).$
To be honest, the asymptotic statement in the first part of Theorem 5.1
requires huge sample sizes to materialize. In our numerical experiments it
turned out that for sample sizes $n$ up to $10000$ and very small indices $j$,
the ratio $(b_{nj}-s_{nj})/(b_{nj}^{\rm BJO}-s_{nj})$ is between $1.5$ and $2$
but drops off quickly as $j$ gets larger.
#### Numerical example.
The left panel in Figure 1 depicts for $n=500$, $\nu=1.1$ and $\alpha=5\%$ the
confidence limits $a_{nj}$ and $b_{nj}$ as functions of
$j\in\\{0,1,\ldots,n\\}$. The dotted (yellow) line in the middle represents
the values $s_{nj}$. The corresponding quantile
$\tilde{\kappa}_{n,\nu,\alpha}$ was estimated in $40000$ Monte-Carlo
simulations as $4.2471$, and this leads to the maximal value
$\gamma_{n}(t_{n1})=0.0151$. In the right panel one sees the centered
boundaries $a_{nj}-s_{nj}$ and $b_{nj}-s_{nj}$. In addition the centered
boundaries $a_{nj}^{\rm BJO}-s_{nj}$ and $a_{nj}^{\rm BJO}-s_{nj}$ are shown
as dashed (and cyan) lines, based on the estimated quantile
$\kappa_{n,\alpha}^{\rm BJ}=5.6615$ and $\gamma_{n}^{\rm BJO}=0.0113$. The
additional horizontal lines are the values $\pm n^{-1/2}\kappa_{n,\alpha}^{\rm
KS}=\pm 0.0604$ for the Kolmogorov-Smirnov bands.
Figure 2 shows the same as the right panel in Figure 1, but with sample sizes
$n=2000$ and $n=8000$ in the left and right panel, respectively.
Figure 1: The confidence limits $a_{nj},b_{nj}$ (left panel) and the centered
confidence limits $a_{nj}-s_{nj},b_{nj}-s_{nj}$ (right panel) for $n=500$,
$\nu=1.1$ and $\alpha=5\%$.
Figure 2: Centered confidence limits for $n=2000,8000$ and $\nu=1.1$,
$\alpha=5\%$.
#### Accuracy in the tails.
The confidence bands described here yield an upper bound for $F$ with limit
$b_{n0}^{\rm BJO}$ or $b_{n0}$ at $-\infty$ and a lower bound for $F$ with
limit $a_{nn}^{\rm BJO}$ or $a_{nn}$ at $+\infty$. The proof of Theorem 5.1
reveals that
$b_{n0}^{\rm BJO},b_{nn}\ =\ \frac{\log\log n}{n}(1+o(1))\quad\text{and}\quad
a_{nn}^{\rm BJO},a_{nn}\ =\ 1-\frac{\log\log n}{n}(1+o(1)).$
On the other hand, the proof of Theorem 2 of Bahadur and Savage (1956) shows
that we cannot expect substantially more accuracy in the tails. Their
arguments can be adpated to show that for any $(1-\alpha)$-confidence band and
any $c>0$, the limit of the upper band at $-\infty$ is smaller than $c/n$ with
probability at most $(1-c/n)^{-n}\alpha$. The same bound holds true for the
probability that the limit of the lower bound at $\infty$ is greater than
$1-c/n$. For a proof we refer to the supplementary material.
#### An alternative approach via the union-intersection principle.
Aldor-Noiman et al. (2013) and Eiger et al. (2013) propose to use a union-
intersection type goodness-of-fit test and related confidence bands. Under the
null hypothesis that $F\equiv F_{o}$, the test statistic $F_{o}(X_{n:i})$ and
$U_{n:i}$ follow a beta distribution with parameters $i$ and $n+1-i$. Denoting
its distribution function with $B_{ni}$, two resulting p-values would be
$B_{ni}(F_{o}(X_{n:i}))$ and $1-B_{ni}(F_{o}(X_{n:i}))$. Thus one can reject
the null hypothesis at level $\alpha$ if the test statistic
$\min_{i=1,2,\ldots,n}\
\min\bigl{\\{}B_{ni}(F_{o}(X_{n:i})),1-B_{ni}(F_{o}(X_{n:i}))\bigr{\\}}$
is lower or equal to the $\alpha$-quantile $\kappa_{n,\alpha}^{\rm UI}$ of
$\min_{i=1,2,\ldots,n}\
\min\bigl{\\{}B_{ni}(U_{n:i}),1-B_{ni}(U_{n:i})\bigr{\\}}.$ (10)
A corresponding $(1-\alpha)$-confidence band for $F$ may be constructed as
follows: With confidence $1-\alpha$ one may claim that for $0\leq j\leq n$ and
$X_{n:j}\leq x<X_{n:j+1}$,
$F(x)\ \in\ [a_{nj}^{\rm UI},b_{nj}^{\rm UI}],$
where $a_{n0}^{\rm UI}:=0$, $b_{nn}^{\rm UI}:=1$, and
$\displaystyle a_{nj}^{\rm UI}\ $ $\displaystyle:=\
B_{nj}^{-1}(\kappa_{n,\alpha}^{\rm UI})\quad\text{for}\ j>1,$ $\displaystyle
b_{nj}^{\rm UI}\ $ $\displaystyle:=\ B_{nj}^{-1}(1-\kappa_{n,\alpha}^{\rm
UI})\quad\text{for}\ j<n.$
The results of Eiger et al. (2013) indicate that this goodness-of-fit test has
similar properties as the one of Berk and Jones (1979). Indeed, if one
considers the closely related test statistic
$\max_{i=1,2,\ldots,n}\,(n+1)K(t_{ni},F_{o}(X_{n:i}))$
one may consider $\exp\bigl{(}-(n+1)K(t_{ni},U_{ni})\bigr{)}$ as a simple
surrogate for the minimum of the two p-values $B_{ni}(F_{o}(X_{n:i}))$ and
$1-B_{ni}(F_{o}(X_{n:i}))$.
A possible weakness of the union-intersection approach is that it ignores
correlations between the random variables $U_{n:i}$. Elementary calculations
reveal that for $1\leq i<j\leq n$,
$\mathrm{Corr}(U_{n:i},U_{n:j})\ =\
\exp\bigl{(}-(\ell(t_{nj})-\ell(t_{ni}))/2\bigr{)}.$
Thus the correlation of two neighbors $U_{n:i}$ and $U_{n:i+1}$ is rather
large if $t_{ni}$ is close to $1/2$ but much smaller if $t_{ni}$ is close to
$0$ or $1$. As a result, the minimum in (10) tends to be attained for indices
$i$ such that $t_{ni}$ is close to $0$ or $1$. With our additive correction
term $-C(t_{ni})-\nu D(t_{ni})$ we try to account for such effects.
## 6 Proofs
### 6.1 Proofs for Section 2
###### Proof of Theorem 2.2.
For symmetry reasons it suffices to prove upper bounds for
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{\mathcal{T}\cap[1/2,1)}\bigl{(}X-C-\nu
D\bigr{)}>\eta\Bigr{)}.$
Note first that for $t,t^{\prime}\in(0,1)$,
$\Bigl{|}\log\frac{t^{\prime}(1-t^{\prime})}{t(1-t)}\Bigr{|}\ \leq\
\Bigl{|}\log\frac{t^{\prime}}{t}\Bigr{|}+\Bigl{|}\log\frac{1-t^{\prime}}{1-t}\Bigr{|}\
=\
\bigl{|}\mathop{\mathrm{logit}}\nolimits(t^{\prime})-\mathop{\mathrm{logit}}\nolimits(t)\bigr{|}.$
(11)
Consequently,
$\displaystyle C(t^{\prime})\ $ $\displaystyle=\
\log\log\Bigl{(}\frac{e}{4t(1-t)}\frac{t(1-t)}{t^{\prime}(1-t^{\prime})}\Bigr{)}$
$\displaystyle\leq\
\log\bigl{(}\exp(C(t))+\bigl{|}\mathop{\mathrm{logit}}\nolimits(t^{\prime})-\mathop{\mathrm{logit}}\nolimits(t)\bigr{|}\bigr{)}$
$\displaystyle=\
C(t)+\log\bigl{(}1+\exp(-C(t))\bigl{|}\mathop{\mathrm{logit}}\nolimits(t^{\prime})-\mathop{\mathrm{logit}}\nolimits(t)\bigr{|}\bigr{)}$
$\displaystyle\leq\
C(t)+\bigl{|}\mathop{\mathrm{logit}}\nolimits(t^{\prime})-\mathop{\mathrm{logit}}\nolimits(t)\bigr{|}$
and since $x\mapsto\log(1+x^{2})$ has derivative $2x/(1+x^{2})\leq 1$,
$\displaystyle D(t^{\prime})\ \leq\
D(t)+\bigl{|}\mathop{\mathrm{logit}}\nolimits(t^{\prime})-\mathop{\mathrm{logit}}\nolimits(t)\bigr{|}.$
Now let $(a_{k})_{k\geq 0}$ be sequence of real numbers with $a_{0}=0$ such
that
$a_{k}\ \to\ \infty\quad\text{and}\quad 0<\delta_{k}:=a_{k+1}-a_{k}\ \to\
0\quad\text{as}\ k\to\infty.$ (12)
Then it follows from
$0\leq\mathop{\mathrm{logit}}\nolimits(t)-\mathop{\mathrm{logit}}\nolimits(\ell(a_{k}))\leq\delta_{k}$
for $t\in[\ell(a_{k}),\ell(a_{k+1})]$ that
$\displaystyle\sup_{\mathcal{T}\cap[\ell(a_{k}),\ell(a_{k+1})]}$
$\displaystyle\bigl{(}X-C-\nu D)$ $\displaystyle\leq\
\sup_{\mathcal{T}\cap[\ell(a_{k}),\ell(a_{k+1})]}X\,-\,C(\ell(a_{k}))-\nu
D(\ell(a_{k}))\,+\,(1+\nu)\delta_{k}$ $\displaystyle\leq\
\sup_{\mathcal{T}\cap[\ell(a_{k}),\ell(a_{k+1})]}X\,-\,C(\ell(a_{k}))-\nu
D(\ell(a_{k}))\,+\,(1+\nu)\delta_{*}$
with $\delta_{*}:=\max_{k\geq 0}\delta_{k}$. Thus Condition 2.1 implies that
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}$
$\displaystyle\sup_{\mathcal{T}\cap[1/2,1)}(X-C-\nu D)>\eta\Bigr{)}$
$\displaystyle\leq\ \sum_{k\geq
0}\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{\mathcal{T}\cap[\ell(a_{k}),\ell(a_{k+1})]}(X-C-\nu
D)>\eta\Bigr{)}$ $\displaystyle\leq\ \sum_{k\geq
0}\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{\mathcal{T}\cap[\ell(a_{k}),\ell(a_{k+1})]}X\,>\eta-(1+\nu)\delta_{*}+C(\ell(a_{k}))+\nu
D(\ell(a_{k}))\Bigr{)}$ $\displaystyle\leq\ M\exp((1+\nu)\delta_{*})\exp(-\eta
L(\delta_{*}))\cdot G,$
where
$\displaystyle G\ $ $\displaystyle:=\ \sum_{k\geq
0}\exp\bigl{(}-L(\delta_{k})C(\ell(a_{k}))-L(\delta_{k})\nu
D(\ell(a_{k}))\bigr{)}$ $\displaystyle=\ \sum_{k\geq
0}\Bigl{(}\log\frac{e}{4\ell^{\prime}(a_{k})}\Bigr{)}^{-L(\delta_{k})}\Bigl{(}1+\Bigl{(}\log\log\frac{e}{4\ell^{\prime}(a_{k})}\Bigr{)}^{2}\Bigr{)}^{-\nu
L(\delta_{k})}.$
For any number $a\geq 0$,
$1\ \leq\ \log\frac{e}{4\ell^{\prime}(a)}\ =\ \log\frac{e(e^{a}+e^{-a}+2)}{4}\
\in\ \bigl{(}a+\log(e/4),a+1\bigr{]}.$
Now we define
$a_{k}\ :=\ \delta_{*}A(k)\quad\text{with}\quad A(s)\ :=\ \frac{s}{\log(e+s)}$
for some $\delta_{*}>0$ such that $L(\delta_{*})\geq L_{o}\in(0,1)$. Note that
$A(\cdot)$ is a continuously differentiable function on $[0,\infty)$ with
$A(0)=0$, limit $A(\infty)=\infty$ and derivative
$A^{\prime}(s)\ =\
\frac{1}{\log(e+s)}\Bigl{(}1-\frac{s}{(e+s)\log(e+s)}\Bigr{)}\ \in\
\Bigl{(}0,\frac{1}{\log(e+s)}\Bigr{)}.$
This implies that (12) is indeed satisfied with
$\delta_{k}\ \leq\ \frac{\delta_{*}}{\log(e+k)}\ =\ O\bigl{(}(\log
k)^{-1}\bigr{)}\quad\text{as}\ k\to\infty.$
Moreover, as $k\to\infty$,
$\displaystyle\Bigl{(}\log$
$\displaystyle\frac{e}{4\ell^{\prime}(a_{k})}\Bigr{)}^{-L(\delta_{k})}\Bigl{(}1+\Bigl{(}\log\log\frac{e}{4\ell^{\prime}(a_{k})}\Bigr{)}^{2}\Bigr{)}^{-\nu
L(\delta_{k})}$ $\displaystyle=\
O\bigl{(}a_{k}^{-L(\delta_{k})}\log(a_{k})^{-2\nu L(\delta_{k})}\bigr{)}$
$\displaystyle=\ O\bigl{(}k^{-L(\delta_{k})}(\log k)^{L(\delta_{k})}(\log
k)^{-2\nu L(\delta_{k})}\bigr{)}$ $\displaystyle=\ O\bigl{(}k^{-1+O(1/\log
k)}(\log k)^{-(2\nu-1)L(\delta_{k})}\bigr{)}$ $\displaystyle=\
O\bigl{(}k^{-1}(\log k)^{-(2\nu-1+o(1))}\bigr{)}.$
Since $2\nu-1>1$, this implies that $G<\infty$. Hence the asserted inequality
is true with $M_{o}=2M\exp((1+\nu)\delta_{*})\cdot G$. ∎
###### Proof of Lemma 2.5.
To verify Condition 2.1 here, recall that if
$\mathbb{W}=(\mathbb{W}(t))_{t\geq 0}$ is standard Brownian motion, then
$(\mathbb{U}(t))_{t\in(0,1)}$ has the same distribution as
$\bigl{(}(1-t)\mathbb{W}(s(t))\bigr{)}_{t\in(0,1)}$ with
$s(t):=t/(1-t)=\exp(\mathop{\mathrm{logit}}\nolimits(t))$. Hence for
$a\in\mathbb{R}$ and $c\geq 0$,
$\displaystyle\sup_{t\in[\ell(a),\ell(a+c)]}X(t)\ $
$\displaystyle=_{\mathcal{L}}\
\sup_{t\in[\ell(a),\ell(a+c)]}\frac{(1-t)^{2}\mathbb{W}(s(t))^{2}}{2t(1-t)}$
$\displaystyle=\
\sup_{t\in[\ell(a),\ell(a+c)]}\frac{\mathbb{W}(s(t))^{2}}{2s(t)}$
$\displaystyle=\ \sup_{s\in[e^{a},e^{a+c}]}\frac{\mathbb{W}(s)^{2}}{2s}$
$\displaystyle=_{\mathcal{L}}\
\sup_{u\in[1,e^{c}]}\frac{\mathbb{W}(u)^{2}}{2u}.$
But it is well-known that $(\mathbb{W}(u)/u)_{u\geq 1}$ is a reverse
martingale. Thus $\bigl{(}\exp(\lambda W(u)/u)\bigr{)}_{u\geq 1}$ is a
nonnegative reverse submartingale for arbitrary real numbers $\lambda$. Hence
it follows from Doob’s inequality for nonnegative submartingales that for any
$\eta>0$,
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{u\in[1,e^{c}]}\frac{\mathbb{W}(u)^{2}}{2u}\geq\eta\Bigr{)}\
$ $\displaystyle\leq\
\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{u\in[1,e^{c}]}\frac{\mathbb{W}(u)^{2}}{u^{2}}\geq\frac{2\eta}{e^{c}}\Bigr{)}$
$\displaystyle\leq\
2\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{u\in[1,e^{c}]}\mathbb{W}(u)/u\geq\sqrt{2e^{-c}\eta}\Bigr{)}$
$\displaystyle=\
2\inf_{\lambda>0}\,\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{u\in[1,e^{c}]}\exp\bigl{(}\lambda\mathbb{W}(u)/u\bigr{)}\geq\exp\bigl{(}\lambda\sqrt{2e^{-c}\eta}\bigr{)}\Bigr{)}$
$\displaystyle\leq\
2\inf_{\lambda>0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp\bigl{(}\lambda\mathbb{W}(1)/1\bigr{)}\exp\bigl{(}-\lambda\sqrt{2e^{-c}\eta}\bigr{)}$
$\displaystyle=\
2\inf_{\lambda>0}\,\exp\bigl{(}\lambda^{2}/2-\lambda\sqrt{2e^{-c}\eta}\bigr{)}$
$\displaystyle=\ 2\exp(-e^{-c}\eta).$
∎
### 6.2 Various properties of the function $K(\cdot,\cdot)$
Before starting with a function $K(\cdot,\cdot)$ itself, let us introduce two
auxiliary functions:
$\displaystyle H(x)\ $ $\displaystyle:=\ x-\log(1+x),\quad x\in(-1,\infty),$
$\displaystyle\tilde{H}(z)\ $ $\displaystyle:=\ -\log(1-z)-z\ =\ H(-z),\quad
z\in(-\infty,1).$
Elementary algebra shows that for $s,t\in(0,1)$,
$K(s,t)\ =\
sH\Bigl{(}\frac{t-s}{s}\Bigr{)}+(1-s)\tilde{H}\Bigl{(}\frac{t-s}{1-s}\Bigr{)}.$
This representation will be useful for $s$ close to $0$ or $1$.
###### Lemma 6.1.
Both functions $H:[0,\infty)\to[0,\infty)$ and $\tilde{H}:[0,1)\to[0,\infty)$
are bijective, strictly increasing and strictly convex. Moreover,
$\displaystyle H(x)\ $ $\displaystyle\in\
\Bigl{[}1+x-\sqrt{1+2x},\frac{x^{2}}{2+x}\Bigr{]}\quad\text{for}\
x\in[0,\infty),$ $\displaystyle\tilde{H}(z)\ $ $\displaystyle\in\
\bigl{[}-\log(1-z^{2})/2,-\log(1-z)\bigr{]}\quad\text{for}\ z\in[0,1).$
The inverse functions $H^{-1}:[0,\infty)\to[0,\infty)$ and
$\tilde{H}^{-1}:[0,\infty)\to[0,1)$ are strictly increasing and strictly
concave with
$\displaystyle H^{-1}(y)\ $ $\displaystyle\in\
\bigl{[}\sqrt{2y+y^{2}/4}+y/2,\sqrt{2y}+y\bigr{]},$
$\displaystyle\tilde{H}^{-1}(y)\ $ $\displaystyle\in\
\bigl{[}1-e^{-y},\sqrt{1-e^{-2y}}\bigr{]}.$
The proof of this lemma is elementary and thus omitted. Now we are ready to
state essential properties of $K(\cdot,\cdot)$:
#### (K.0)
With the convention that $0\log 0:=0$ one can easily verify that the function
$K:[0,1]\times(0,1)\to\mathbb{R}$ is continuous. In particular,
$K(0,t)=-\log(1-t)$ and $K(1,t)=-\log t$. Moreover, $K(1-s,1-t)=K(s,t)$ for
arbitrary $s\in[0,1]$ and $t\in(0,1)$.
#### (K.1)
For $s,t\in(0,1)$,
$\frac{\partial K(s,t)}{\partial s}\ =\
\mathop{\mathrm{logit}}\nolimits(s)-\mathop{\mathrm{logit}}\nolimits(t)\quad\text{and}\quad\frac{\partial
K(s,t)}{\partial t}\ =\ -\frac{s}{t}+\frac{1-s}{1-t}\ =\ \frac{t-s}{t(1-t)}.$
(The latter formula is true even for $s\in[0,1]$.) In particular, $K(s,t)\geq
0$ with equality if, and only if, $s=t$.
#### (K.2)
For $s,t\in(0,1)$,
$\displaystyle\frac{\partial^{2}K(s,t)}{\partial s^{2}}\ $ $\displaystyle=\
\frac{1}{s(1-s)},\quad\frac{\partial^{2}K(s,t)}{\partial s\partial t}\ =\
-\frac{1}{t(1-t)}\quad\text{and}$
$\displaystyle\frac{\partial^{2}K(s,t)}{\partial t^{2}}\ $ $\displaystyle=\
\frac{s}{t^{2}}+\frac{1-s}{(1-t)^{2}}\ =\
\frac{(t-s)^{2}+s(1-s)}{t^{2}(1-t)^{2}}.$
In particular, the Hessian matrix of $K$ at $(s,t)$ has positive diagonal
elements and non-negative determinant $(t-s)^{2}/(s(1-s)t^{2}(1-t)^{2})$. This
implies that $K$ is convex on $[0,1]\times(0,1)$.
#### (K.3)
For fixed $u\in(0,1)$ and arbitrary $0<t<t^{\prime}<1$,
$\frac{K(0,t^{\prime})}{K(0,t)},\frac{K(t^{\prime}u,t^{\prime})}{K(tu,t)},\frac{K(t^{\prime},t^{\prime}u)}{K(t,tu)}\
\in\
\Bigl{(}\frac{t^{\prime}}{t},\frac{t^{\prime}(1-t)}{(1-t^{\prime})t}\Bigr{)}.$
###### Proof.
Since $K(tu,tu)=0$, it follows from (K.1) that
$K(tu,t)\ =\ \int_{tu}^{t}\frac{\partial K(tu,x)}{\partial x}\,dx\ =\
\int_{tu}^{t}\frac{(x-tu)}{x(1-x)}\,dx\ =\
\int_{u}^{1}\frac{t(v-u)}{v(1-tv)}\,dv.$
These formulae remain true if we replace $u$ with $0$. On the other hand,
since $K(tu,tu)=0=\partial K(s,tu)/\partial s$ for $s=tu$, a suitable version
of Taylor’s formula and (K.2) imply that
$K(t,tu)\ =\ \int_{tu}^{t}(t-x)\frac{\partial^{2}}{\partial x^{2}}K(x,tu)\,dx\
=\ \int_{tu}^{t}\frac{(t-x)}{x(1-x)}\,dx\ =\
\int_{u}^{1}\frac{t(1-v)}{v(1-tv)}\,dv,$
But for any $v\in(0,1)$,
$\frac{\partial}{\partial t}\log\frac{t}{1-tv}\ =\ \frac{1}{t(1-tv)}\ \in\
\Bigl{(}\frac{1}{t},\frac{1}{t(1-t)}\Bigr{)}\ =\
\bigl{(}\log^{\prime}(t),\mathop{\mathrm{logit}}\nolimits^{\prime}(t)\bigr{)}.$
Thus for $0<t<t^{\prime}<1$,
$\frac{t^{\prime}}{1-t^{\prime}v}\Big{/}\frac{t}{1-tv}\ \in\
\Bigl{(}\frac{t^{\prime}}{t},\frac{t^{\prime}(1-t)}{(1-t^{\prime})t}\Bigr{)},$
and this entails the asserted inequalities for the three ratios
$K(0,t^{\prime})/K(0,t)$, $K(t^{\prime}u,t^{\prime})/K(tu,t)$ and
$K(t^{\prime},t^{\prime}u)/K(t,tu)$. ∎
#### (K.4)
To verify Theorems 3.2, 3.4 and 5.1 we have to approximate $K$ by a simpler
function $\tilde{K}$ given by
$\tilde{K}(s,t)\ :=\ \frac{(s-t)^{2}}{2t(1-t)}.$
Indeed, for arbitrary $s,t\in(0,1)$ and
$c:=\bigl{|}\mathop{\mathrm{logit}}\nolimits(s)-\mathop{\mathrm{logit}}\nolimits(t)\bigr{|}$,
$\frac{K(s,t)}{\tilde{K}(s,t)},\frac{K(s,t)}{\tilde{K}(t,s)}\ \in\
[e^{-c},e^{c}].$
###### Proof.
It follows from (K.1-2) and Taylor’s formula that
$K(s,t)\ =\ \frac{(s-t)^{2}}{2\xi(1-\xi)}$
for some $\xi$ between $\min\\{s,t\\}$ and $\max\\{s,t\\}$. Hence
$\frac{K(s,t)}{\tilde{K}(s,t)}\ =\
\frac{t(1-t)}{\xi(1-\xi)}\quad\text{and}\quad\frac{K(s,t)}{\tilde{K}(t,s)}\ =\
\frac{s(1-s)}{\xi(1-\xi)}$
are both contained in $[e^{-c},e^{c}]$, according to (11). ∎
#### (K.5)
For arbitrary $\gamma>0$ and $s\in[0,1]$, $t\in(0,1)$, the inequality
$K(s,t)\leq\gamma$ implies that
$(t-s)^{\pm}\ \leq\ \begin{cases}\sqrt{2\gamma s(1-s)}+(1-2s)^{\pm}\gamma,\\\
\sqrt{2\gamma t(1-t)}+(2t-1)^{\pm}\gamma.\end{cases}$
In particular,
$|s-t|\ \leq\
\min\bigl{\\{}\sqrt{2s(1-s)\gamma},\sqrt{2t(1-t)\gamma}\bigr{\\}}+\gamma.$
###### Proof.
The first inequality has been proved by Dümbgen (1998), but for the reader’s
convenience and the proof of the new part, a complete derivation is given
here: For symmetry reasons, it suffices to consider the case $0\leq s<t<1$ and
derive the upper bounds for $\delta:=t-s=(t-s)^{+}$.
Let us first treat the case $s=0$: Here $K(s,t)=-\log(1-t)\geq t$. Thus
$K(0,t)\leq\gamma$ implies that $\delta=t\leq\gamma=\sqrt{2\gamma
s(1-s)}+(1-2s)\gamma$. Moreover, $\sqrt{2\gamma t(1-t)}+(2t-1)^{+}\gamma\geq
t\bigl{(}\sqrt{2(1-t)}+(2t-1)^{+}\bigr{)}$, and elementary considerations show
that $\sqrt{2(1-t)}+(2t-1)^{+}\geq 1$.
Now let $0<s<t<1$ and $\delta:=t-s$. It follows from $K(s,s)=0$ and (K.1) that
$K(s,t)\ =\ \int_{s}^{t}\frac{\partial K(s,y)}{\partial y}\,dy\ =\
\int_{0}^{\delta}\frac{x}{(s+x)(1-s-x)}\,dx\ \geq\
\int_{0}^{\delta}\frac{x}{s(1-s)+(1-2s)x}\,dx.$
In case of $s\geq 1/2$, the latter integral is not smaller than
$\delta^{2}/(2s(1-s))$, and $K(s,t)\leq\gamma$ implies the upper bound
$\delta\leq\sqrt{2\gamma s(1-s)}$. In case of $s<1/2$, we obtain the bound
$K(s,t)\ \geq\ \int_{0}^{\delta}\frac{x}{\alpha+\beta x}\,dx\ =\
\frac{\delta}{\beta}-\frac{\alpha}{\beta^{2}}\log\Bigl{(}1+\frac{\beta\delta}{\alpha}\Bigr{)}\
=\ \frac{\alpha}{\beta^{2}}H\Bigl{(}\frac{\beta\delta}{\alpha}\Bigr{)}$
with $\alpha:=s(1-s)>0$, $\beta:=1-2s>0$ and the auxiliary function $H$ from
Lemma 6.1. Consequently, the inequality $K(s,t)\leq\gamma$ entails that
$H(\beta\delta/\alpha)\leq\beta^{2}\gamma/\alpha$, so
$\delta\ \leq\ (\alpha/\beta)H^{-1}(\beta^{2}\gamma/\alpha)\ \leq\
\sqrt{2\gamma\alpha}+\beta\gamma\ =\ \sqrt{2\gamma s(1-s)}+(1-2s)\gamma.$
On the other hand,
$K(s,t)\ =\ \int_{s}^{t}\frac{y-s}{y(1-y)}\,dy\ =\
\int_{0}^{\delta}\frac{\delta-x}{(t-x)(1-t+x)}\,dx\ \geq\
\int_{0}^{\delta}\frac{\delta-x}{t(1-t)+(2t-1)x}\,dx.$
In case of $t\leq 1/2$, the latter integral is at least
$\delta^{2}/(2t(1-t))$, and we may conclude from $K(s,t)\leq\gamma$ that
$\delta$ is bounded by $\sqrt{2\gamma t(1-t)}$. In case of $t>1/2$, we define
$a:=t(1-t)>0$, $b:=2t-1>0$ and may write
$K(s,t)\ \geq\ \int_{0}^{\delta}\frac{\delta-x}{a+bx}\,dx\ >\
\int_{0}^{\delta}\frac{x}{a+bx}\,dx\ =\
\frac{a}{b^{2}}H\Bigl{(}\frac{b\delta}{a}\Bigr{)}.$
The second inequality in the previous display follows from the fact that
$f(x):=1/(a+bx)$ is strictly decreasing on $[0,\delta]$. Thus
$\int_{0}^{\delta}(\delta-x)f(x)\,dx-\int_{0}^{\delta}xf(x)\,dx$ equals
$\int_{0}^{\delta}(\delta-2x)f(x)\,dx\ =\
\int_{0}^{\delta}(\delta-2x)(f(x)-f(\delta/2))\,dx$
and is strictly positive. Hence the preceding considerations yield the upper
bound $\sqrt{2\gamma a}+b\gamma=\sqrt{2\gamma t(1-t)}+(2t-1)\gamma$ for
$\delta$. ∎
#### (K.6)
For $s\in(0,1)$ and $\gamma>0$ let $b=b(s,\gamma)\in(s,1)$ solve the equation
$K(s,b)\ =\ \gamma.$
Then
$\frac{b-s}{sH^{-1}(\gamma/s)}\ \begin{cases}\leq\ 1,\\\ \to\ 1&\text{as}\
s,\gamma\to 0,\end{cases}$ (13) $\frac{b-s}{\sqrt{2\gamma s(1-s)}}\ \to\
1\quad\text{as}\ \frac{\gamma}{s(1-s)}\to 0,$ (14)
$\frac{b-s}{(1-s)\tilde{H}^{-1}(\gamma/(1-s))}\ \in\ [s,1].$ (15)
###### Proof.
With $\delta:=(b-s)/s>0$ we may write
$\frac{\gamma}{s}\ =\ \frac{K(s,s+s\delta)}{s}\ =\
H(\delta)+\frac{1-s}{s}\tilde{H}\Bigl{(}\frac{s\delta}{1-s}\Bigr{)}.$
Since $\tilde{H}\geq 0$, this implies that $H(\delta)\leq\gamma/s$, which is
equivalent to $b-s\leq sH^{-1}(\gamma/s)$. On the other hand, it follows from
the expansion $-\log(1-z)=\sum_{k=1}^{\infty}z^{k}/k=z+\tilde{H}(z)$ that
$\frac{\gamma}{s}\ =\
H(\delta)+\frac{1-s}{s}\sum_{k=2}^{\infty}\Bigl{(}\frac{s\delta}{1-s}\Bigr{)}^{k}/k\
\leq\ H(\delta)+\frac{s\delta^{2}}{2(1-s-s\delta)}.$
As $c:=\max\\{s,\gamma\\}\to 0$, it follows from $\delta\leq
H^{-1}(\gamma/s)\leq\sqrt{2\gamma/s}+\gamma/s$ that
$\displaystyle 1-s-s\delta\ $ $\displaystyle\geq\ 1-s-\sqrt{2s\gamma}-\gamma\
=\ 1-O(c),$ $\displaystyle s\delta^{2}\ $ $\displaystyle\leq\
s\bigl{(}\sqrt{2\gamma/s}+\gamma/s\bigr{)}^{2}\ =\ O(c)\gamma/s,$
whence
$\frac{\gamma}{s}\ \leq\ H(\delta)+O(c)\frac{\gamma}{s}.$
Consequently,
$b-s\ \geq\ sH^{-1}\bigl{(}(1-O(c))\gamma/s\bigr{)}\ \geq\
(1-O(c))\,sH^{-1}(\gamma/s),$
the latter inequality following from concavity of $H^{-1}$. This proves (13).
As to (14), let $c:=\sqrt{\gamma/(s(1-s))}<1/2$, and define the points
$t(x)=t(s,\gamma,x):=s+\sqrt{2\gamma s(1-s)x}=s+cs(1-s)\sqrt{2x}$ for
$x\in[0,2]$. Then
$0\ <\
\mathop{\mathrm{logit}}\nolimits(t(x))-\mathop{\mathrm{logit}}\nolimits(s)\ <\
\log\frac{1+2c}{1-2c}\ \to\ 0\quad\text{as}\ c\to 0.$
Consequently, by (K.4),
$K(s,t(x))\ =\ (1+o(1))\tilde{K}(t(x),s)\ =\ (1+o(1))\gamma x$
uniformly in $x\in[0,2]$. This shows that
$b(s,\gamma)=t(1+o(1))=s+\sqrt{2\gamma s(1-s)}(1+o(1))$ as $c\to 0$.
Finally, let $\delta:=(b-s)/(1-s)$. Then it follows from $\tilde{H}(z)\geq
z^{2}/2\geq H(z)$ that
$\frac{\gamma}{1-s}\ =\
\frac{s}{1-s}H\Bigl{(}\frac{1-s}{s}\delta\Bigr{)}+\tilde{H}(\delta)\
\begin{cases}\geq\ \tilde{H}(\delta),\\\ \leq\
(1-s)\delta^{2}/(2s)+\tilde{H}(\delta)\ \leq\ \tilde{H}(\delta)/s.\end{cases}$
Consequently, by concavity of $\tilde{H}^{-1}(\cdot)$,
$s\tilde{H}^{-1}(\gamma/(1-s))\ \leq\ \tilde{H}^{-1}(s\gamma/(1-s))\ \leq\
\delta\ \leq\ \tilde{H}^{-1}(\gamma/(1-s)),$
which yields (15). ∎
### 6.3 Proofs for Section 3
Before proving Lemma 3.1 let us recall that for $s\in\mathbb{R}$ and
$t\in(0,1)$,
$K(s,t)\ :=\ \sup_{\lambda\in\mathbb{R}}\,\bigl{(}\lambda
s-\log(1-t+te^{\lambda})\bigr{)}\ =\ \begin{cases}\displaystyle
s\log\frac{s}{t}+(1-s)\log\frac{1-s}{1-t}&\text{if}\ s\in[0,1],\\\
\infty&\text{else}.\end{cases}$
Indeed, Hoeffding (1963) showed that for a random variable
$Y\sim\mathrm{Bin}(n,t)$ and $s\in\mathbb{R}$,
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits(Y\geq ns)\ $
$\displaystyle\leq\ \exp\Bigl{(}-n\sup_{\lambda\geq 0}\,\bigl{(}\lambda
s-\log(1-t+te^{\lambda})\bigr{)}\Bigr{)}\ =\ \exp(-nK(s,t))\quad\text{if}\
s\geq t,$ $\displaystyle\mathop{\mathrm{I\\!P}}\nolimits(Y\leq ns)\ $
$\displaystyle\leq\ \exp\Bigl{(}-n\sup_{\lambda\leq 0}\,\bigl{(}\lambda
s-\log(1-t+te^{\lambda})\bigr{)}\Bigr{)}\ =\ \exp(-nK(s,t))\quad\text{if}\
s\leq t.$
###### Proof of Lemma 3.1.
We imitate and modify a martingale argument of Berk and Jones (1979, Lemma
4.3) which goes back to Kiefer (1973). Note first that $\widehat{G}_{n}(t)/t$
is a reverse martingale in $t\in(0,1)$, that means,
$\mathop{\mathrm{I\\!E}}\nolimits\bigl{(}\widehat{G}_{n}(s)/s\,\big{|}\,(\widehat{G}_{n}(t^{\prime}))_{t^{\prime}\geq
t}\bigr{)}\ =\ \widehat{G}_{n}(t)/t\quad\text{for}\ 0<s<t<1.$
Consequently, for $0<t<t^{\prime}<1$ and $0\leq u\leq 1$,
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\inf_{s\in[t,t^{\prime}]}\widehat{G}_{n}(s)/s\leq
u\Bigr{)}\ $ $\displaystyle=\ \inf_{\lambda\leq
0}\,\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{s\in[t,t^{\prime}]}\exp(\lambda\widehat{G}_{n}(s)/s-\lambda
u)\geq 1\Bigr{)}$ $\displaystyle\leq\ \inf_{\lambda\leq
0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda\widehat{G}_{n}(t)/t-\lambda
u)$
by Doob’s inequality for non-negative submartingales. But
$n\widehat{G}_{n}(t)\sim\mathrm{Bin}(n,t)$, so
$\displaystyle\inf_{\lambda\leq
0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda\widehat{G}_{n}(t)/t-\lambda
u)\ $ $\displaystyle=\ \inf_{\lambda\leq
0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp\bigl{(}\lambda
n\widehat{G}(t)-n\lambda tu\bigr{)}$ $\displaystyle=\
\exp\Bigl{(}-n\sup_{\lambda\leq 0}\bigl{(}\lambda
tu-\log(1-t+te^{\lambda})\bigr{)}\Bigr{)}$ $\displaystyle=\ \exp(-nK(tu,t)).$
Thus
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\inf_{s\in[t,t^{\prime}]}\widehat{G}_{n}(s)/s\leq
u\Bigr{)}\ \leq\ \exp(-nK(tu,t))\quad\text{for all}\ u\in[0,1].$
One may rewrite this inequality as
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{s\in[t,t^{\prime}]}nK\bigl{(}t\min\\{\widehat{G}_{n}(s)/s,1\\},t\bigr{)}\geq\eta\Bigr{)}\
\leq\ \exp(-\eta)\quad\text{for all}\ \eta\geq 0.$
For if $\eta>-n\log(1-t)$, the probability on the left hand side equals $0$.
Otherwise there exists a unique $u=u(t,\eta)\in[0,1]$ such that
$nK(tu,t)=\eta$. But then
$nK\bigl{(}t\min\\{\widehat{G}_{n}(s)/s,1\\},t\bigr{)}\geq\eta\quad\text{if,
and only if,}\quad\widehat{G}_{n}(s)/s\leq u.$
Finally, it follows from property (K.3) of $K(\cdot,\cdot)$ that for $t\leq
s\leq t^{\prime}$,
$K\bigl{(}\min\\{\widehat{G}_{n}(s),s\\},s\bigr{)}\ =\
K\bigl{(}s\min\\{\widehat{G}_{n}(s)/s,1\\},s\bigr{)}\ \leq\
e^{c}K\bigl{(}t\min\\{\widehat{G}_{n}(s)/s,1\\},t\bigr{)}$
with
$c:=\mathop{\mathrm{logit}}\nolimits(t^{\prime})-\mathop{\mathrm{logit}}\nolimits(t)$.
Hence
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{s\in[t,t^{\prime}]}nK\bigl{(}\min\\{\widehat{G}_{n}(s),s\\},s\bigr{)}\geq\eta\Bigr{)}\
\leq\ \exp(-e^{-c}\eta)\quad\text{for all}\ \eta\geq 0.$
Since $\bigl{(}\widehat{G}_{n}(t)\bigr{)}_{t\in(0,1)}$ has the same
distribution as $\bigl{(}1-\widehat{G}_{n}((1-t)\,-)\bigr{)}_{t\in(0,1)}$, and
because of the symmetry relations $K(s,t)=K(1-s,1-t)$ and
$\mathop{\mathrm{logit}}\nolimits(1-t)=-\mathop{\mathrm{logit}}\nolimits(t)$,
the previous inequality implies further that
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}$
$\displaystyle\sup_{s\in[t,t^{\prime}]}nK\bigl{(}\max\\{\widehat{G}_{n}(s),s\\},s\bigr{)}\geq\eta\Bigr{)}$
$\displaystyle=\
\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{s\in[t,t^{\prime}]}nK\bigl{(}\min\\{1-\widehat{G}_{n}(s),1-s\\},1-s\bigr{)}\geq\eta\Bigr{)}$
$\displaystyle=\
\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{s\in[1-t^{\prime},1-t]}nK\bigl{(}\min\\{\widehat{G}_{n}(s),s\\},s\bigr{)}\geq\eta\Bigr{)}$
$\displaystyle\leq\ \exp(-e^{-c}\eta)\quad\text{for all}\ \eta\geq 0.$
Consequently, since
$K(\cdot,s)=\max\bigl{\\{}K(\min\\{\cdot,s\\},s),K(\max\\{\cdot,s\\},s)\bigr{\\}}$,
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\sup_{s\in[t,t^{\prime}]}nK(\widehat{G}_{n}(s),s)\geq\eta\Bigr{)}\
\leq\ 2\exp(-e^{-c}\eta)\quad\text{for all}\ \eta\geq 0.$
∎
###### Proof of Theorem 3.2.
For any fixed $\delta\in(0,1/2)$, it follows from Donsker’s invariance
principle for the uniform empirical process and the continuous mapping theorem
that
$\sup_{t\in[-\delta,\delta]}\Bigl{(}\frac{\mathbb{U}_{n}(t)^{2}}{2t(1-t)}-C(t)-\nu
D(t)\Bigr{)}\ \to_{\mathcal{L}}\ \sup_{[-\delta,\delta]}\bigl{(}X-C-\nu
D\bigr{)},$
where $X(t)=\mathbb{U}(t)^{2}/(2t(1-t))$. With
$X_{n}(t)=nK(\widehat{G}_{n}(t),t)$ it follows from property (K.4) of
$K(\cdot,\cdot)$ that
$\frac{\mathbb{U}_{n}(t)^{2}}{2t(1-t)}\ =\ n\tilde{K}(\widehat{G}_{n}(t),t)\
=\ X_{n}(t)(1+r_{n}(t))$
with
$\sup_{t\in[\delta,1-\delta]}|r_{n}(t)|\ \leq\
\bigl{(}1-n^{-1/2}\delta^{-1}\|\mathbb{U}_{n}\|_{\infty}\bigr{)}^{-2}-1\ =\
O_{p}(n^{-1/2}).$
Thus
$\sup_{[-\delta,\delta]}\bigl{(}X_{n}-C-\nu D\bigr{)}\ \to_{\mathcal{L}}\
\sup_{[-\delta,\delta]}\bigl{(}X-C-\nu D\bigr{)}.$
But Theorem 2.2 implies that for any $1<\nu^{\prime}<\nu$, the random
variables $T_{n,\nu^{\prime}}$ and $T_{\nu^{\prime}}$ satisfy the inequalities
$\mathop{\mathrm{I\\!P}}\nolimits(T_{n,\nu^{\prime}}>\eta)\leq
M_{o}\exp(-L_{o}\eta)$ and
$\mathop{\mathrm{I\\!P}}\nolimits(T_{\nu^{\prime}}>\eta)\leq
M_{o}\exp(-L_{o}\eta)$ for arbitrary $\eta\in\mathbb{R}$ and some constants
$L_{o}\in(0,1)$, $M_{o}\geq 1$. Consequently for any $\rho>0$,
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}$
$\displaystyle\sup_{[\delta,1-\delta]}(X_{n}-C-\nu D)<\sup_{(0,1)}(X_{n}-C-\nu
D)\Bigr{)}$ $\displaystyle\leq\
\mathop{\mathrm{I\\!P}}\nolimits\bigl{(}T_{n,\nu^{\prime}}-(\nu-\nu^{\prime})D(\delta)>-\rho\bigr{)}+\mathop{\mathrm{I\\!P}}\nolimits\bigl{(}X_{n}(1/2)\leq-\rho\bigr{)}$
$\displaystyle\leq\ M_{o}\exp\bigl{(}L_{o}\rho-
L_{o}(\nu-\nu^{\prime})D(\delta)\bigr{)}+\mathop{\mathrm{I\\!P}}\nolimits(X(1/2)\leq-\rho)+o(1)$
because $X_{n}(1/2)\to_{\mathcal{L}}X(1/2)$, and
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}$
$\displaystyle\sup_{[\delta,1-\delta]}(X-C-\nu D)<\sup_{(0,1)}(X-C-\nu
D)\Bigr{)}$ $\displaystyle\leq\ M_{o}\exp\bigl{(}L_{o}\rho-
L_{o}(\nu-\nu^{\prime})D(\delta)\bigr{)}+\mathop{\mathrm{I\\!P}}\nolimits(X(1/2)\leq-\rho).$
Setting $\rho=(\nu-\nu^{\prime})D(\delta)/2$, the limits of the right hand
sides become arbitrarily small for sufficiently small $\delta$. This shows
that $T_{n,\nu}=\sup_{(0,1)}\bigl{(}X_{n}-C-\nu D\bigr{)}$ converges in
distribution to $T_{\nu}$. ∎
Our proof of Lemma 3.3 involves an exponential inequality for Beta
distributions from Dümbgen (1998). For the reader’s convenience, its proof is
included in the supplementary material.
###### Lemma 6.2.
Let $s,t\in(0,1)$, and let $Y\sim\mathrm{Beta}(mt,m(1-t))$ for some $m>0$.
Then
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits(Y\leq s)\ $ $\displaystyle\leq\
\inf_{\lambda\leq 0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda Y-\lambda
s)\ \leq\ \exp(-mK(t,s))\quad\text{if}\ s\leq t,$
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits(Y\geq s)\ $ $\displaystyle\leq\
\inf_{\lambda\geq 0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda Y-\lambda
s)\ \leq\ \exp(-mK(t,s))\quad\text{if}\ s\geq t.$
###### Proof of Lemma 3.3.
We utilize a well-known representation of uniform order statistics: Let
$E_{1},E_{2},\ldots,E_{n+1}$ be independent random variables with standard
exponential distribution, i.e. $\mathrm{Gamma}(1)$, and let
$S_{j}:=\sum_{i=1}^{j}E_{i}$. Then
$(U_{ni})_{i=1}^{n}\ =_{\mathcal{L}}\ (S_{i}/S_{n+1})_{i=1}^{n}.$
In particular,
$U_{n:i}\sim\mathrm{Beta}(i,n+1-i)=\mathrm{Beta}\bigl{(}(n+1)t_{ni},(n+1)(1-t_{ni})\bigr{)}$
and $\mathop{\mathrm{I\\!E}}\nolimits U_{n:i}=t_{ni}$. Furthermore, for $2\leq
k\leq n+1$, the random vectors $(S_{i}/S_{k})_{i=1}^{k-1}$ and
$(S_{i})_{i=k}^{n+1}$ are stochastically independent. This implies that
$(U_{n:i}/t_{ni})_{i=1}^{n}$ is a reverse martingale, because for $1\leq
j<k\leq n$,
$\mathop{\mathrm{I\\!E}}\nolimits\Bigl{(}\frac{U_{n:j}}{t_{nj}}\,\Big{|}\,(S_{i})_{i=k}^{n+1}\Bigr{)}\
=\
\mathop{\mathrm{I\\!E}}\nolimits\Bigl{(}\frac{S_{j}}{t_{nj}S_{k}}\cdot\frac{S_{k}}{S_{n+1}}\,\Big{|}\,(S_{i})_{i=k}^{n+1}\Bigr{)}\
=\ \frac{j}{t_{nj}k}\cdot\frac{S_{k}}{S_{n+1}}\ =\ \frac{U_{n:k}}{t_{nk}}.$
Consequently, for $1\leq j\leq k\leq n$ and $0<u<1$, it follows from Doob’s
inequality and Lemma 6.2 that
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\min_{j\leq i\leq
k}\frac{U_{n:i}}{t_{ni}}\leq u\Bigr{)}\ $ $\displaystyle=\
\inf_{\lambda<0}\,\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\min_{j\leq i\leq
k}\exp\Bigl{(}\lambda\frac{U_{n:i}}{t_{ni}}-\lambda u\Bigr{)}\geq 1\Bigr{)}$
$\displaystyle\leq\
\inf_{\lambda<0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp\bigl{(}\lambda
U_{n:j}-\lambda ut_{nj}\bigr{)}$ $\displaystyle\leq\
\exp\bigl{(}-(n+1)K(t_{nj},t_{nj}u)\bigr{)}.$
Again one may reformulate the previous inequalities as follows: For any
$\eta>0$,
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\max_{j\leq i\leq
k}(n+1)K\Bigl{(}t_{nj},t_{nj}\min\Bigl{\\{}\frac{U_{n:i}}{t_{ni}},1\Bigr{\\}}\Bigr{)}\geq\eta\Bigr{)}\
\leq\ \exp(-\eta).$
But property (K.3) of $K(\cdot,\cdot)$ implies that for $j\leq i\leq k$,
$K\bigl{(}t_{ni},\min\\{U_{n:i},t_{ni}\\}\bigr{)}\ \leq\
e^{c}K\Bigl{(}t_{nj},t_{nj}\min\Bigl{\\{}\frac{U_{n:i}}{t_{ni}},1\Bigr{\\}}\Bigr{)}$
with
$c:=\mathop{\mathrm{logit}}\nolimits(t_{nk})-\mathop{\mathrm{logit}}\nolimits(t_{nj})$.
Consequently,
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\max_{j\leq i\leq
k}(n+1)K\bigl{(}t_{ni},\min\\{U_{n:i},t_{ni}\\}\bigr{)}\geq\eta\Bigr{)}\ \leq\
\exp(-e^{-c}\eta)\quad\text{for all}\ \eta>0.$
Since $(1-U_{n:n+1-i})_{i=1}^{n}$ has the same distribution as
$(U_{n:i})_{i=1}^{n}$, a symmetry argument as in the proof of Lemma 3.1
reveals that
$\mathop{\mathrm{I\\!P}}\nolimits\Bigl{(}\max_{j\leq i\leq
k}(n+1)K(t_{ni},U_{n:i})\geq\eta\Bigr{)}\ \leq\
2\exp(-e^{-c}\eta)\quad\text{for all}\ \eta>0.$
∎
###### Proof of Theorem 3.4.
One can use essentially the same arguments as in the proof of Theorem 3.2.
This time one has to utilize the well-known fact that
$(U_{n:i})_{i=1}^{n}\ =\
\bigl{(}t_{ni}+n^{-1/2}\mathbb{V}_{n}(t_{ni})\bigr{)}_{i=1}^{n}$
where the uniform quantile process $\mathbb{V}_{n}$ with
$\mathbb{V}_{n}(t):=\sqrt{n}(\widehat{G}_{n}^{-1}(t)-t)$ converges in
distribution in $\ell_{\infty}([0,1])$ to a Brownian bridge $\mathbb{V}$; see
e.g. Shorack and Wellner (1986), pages 86, 93, and 637-644. ∎
### 6.4 Proofs for Sections 4 and 5
###### Proof of Lemma 4.1.
Suppose that $T_{n,\nu}(F_{o})\leq\kappa$. Then the inequalities in (K.5)
imply that
$|\widehat{F}_{n}-F_{o}|\ \leq\
\sqrt{2\tilde{\Gamma}(F_{o})F_{o}(1-F_{o})/n}+\tilde{\Gamma}(F_{o})/n,$
where $\tilde{\Gamma}(t):=C(t)+\nu D(t)+\kappa$. Multiplying this inequality
with $n$ and utilizing the triangle inequality
$|\widehat{F}_{n}-F_{o}|\geq|F-F_{o}|-|\widehat{F}_{n}-F|$ leads to
$n|F-F_{o}|\ \leq\
\sqrt{2n\tilde{\Gamma}(F_{o})F_{o}(1-F_{o})}+\tilde{\Gamma}(F_{o})+n|\widehat{F}_{n}-F|.$
(16)
Now our goal is to get rid of the term $n|\widehat{F}_{n}-F|$ on the right
hand side. Defining the auxiliary stochastic process
$W_{n}\ :=\ \frac{n(\widehat{F}_{n}-F)^{2}}{F(1-F)}$
with the convention $0/0:=0$, we may rewrite (16) as
$\displaystyle n|F-F_{o}|\ $ $\displaystyle\leq\
\sqrt{2n\tilde{\Gamma}(F_{o})F_{o}(1-F_{o})}+\tilde{\Gamma}(F_{o})+\sqrt{W_{n}nF(1-F)}$
$\displaystyle\leq\
\sqrt{2n\tilde{\Gamma}(F_{o})F_{o}(1-F_{o})}+\tilde{\Gamma}(F_{o})+\sqrt{W_{n}nF_{o}(1-F_{o})}+\sqrt{W_{n}n|F-F_{o}|}$
$\displaystyle\leq\
\sqrt{n(4\tilde{\Gamma}(F_{o})+2W_{n})F_{o}(1-F_{o})}+\tilde{\Gamma}(F_{o})+\sqrt{W_{n}n|F-F_{o}|},$
(17)
where we utilized the inequalities $|a(1-a)-b(1-b)|\leq|a-b|$ for
$a,b\in[0,1]$ and $\sqrt{c+d}\leq\sqrt{c}+\sqrt{d}\leq\sqrt{2c+2d}$ for
$c,d\geq 0$. Note that inequality (17) is of the form $Y_{n}\leq
V_{n}+\sqrt{W_{n}Y_{n}}$ with the nonnegative processes $Y_{n}=n|F-F_{o}|$ and
$V_{n}=\sqrt{n(4\tilde{\Gamma}(F_{o})+2W_{n})F_{o}(1-F_{o})}+\tilde{\Gamma}(F_{o})$.
But $Y_{n}\leq V_{n}+\sqrt{W_{n}Y_{n}}$ is equivalent to $Y_{n}/V_{n}\leq
1+\sqrt{W_{n}/V_{n}}\sqrt{Y_{n}/V_{n}}$, and this may be rewritten as
$\bigl{(}\sqrt{Y_{n}/V_{n}}-\sqrt{W_{n}/V_{n}}/2\bigr{)}^{2}\leq
1+(W_{n}/V_{n})/4$, so
$\sqrt{Y_{n}/V_{n}}\ \leq\ \sqrt{W_{n}/V_{n}}/2+\sqrt{1+(W_{n}/V_{n})/4}\
\leq\ 1+\sqrt{W_{n}/V_{n}}.$
Consequently,
$\displaystyle n|F-F_{o}|\ $ $\displaystyle\leq\
\bigl{(}1+\sqrt{W_{n}/V_{n}}\bigr{)}^{2}\Bigl{(}\sqrt{n(4\tilde{\Gamma}(F_{o})+2W_{n})F_{o}(1-F_{o})}+\tilde{\Gamma}(F_{o})\Bigr{)}$
$\displaystyle\leq\
\bigl{(}1+\sqrt{W_{n}/\kappa}\bigr{)}^{2}\sqrt{4+2W_{n}/\kappa}\Bigl{(}\sqrt{n\tilde{\Gamma}(F_{o})F_{o}(1-F_{o})}+\tilde{\Gamma}(F_{o})\Bigr{)}$
$\displaystyle\leq\
2\bigl{(}1+\sqrt{W_{n}/\kappa}\bigr{)}^{5/2}\Bigl{(}\sqrt{n\tilde{\Gamma}(F_{o})F_{o}(1-F_{o})}+\tilde{\Gamma}(F_{o})\Bigr{)},$
because $V_{n}\geq\tilde{\Gamma}(F_{o})\geq\kappa$. Finally, since
$B^{\prime}=B^{\prime}_{\nu,\kappa}:=\max\bigl{\\{}\sup_{(0,1)}\tilde{\Gamma}/\Gamma,1\bigr{\\}}<\infty$,
we obtain the inequality
$\frac{n|F-F_{o}|}{\sqrt{n\Gamma(F_{o})F_{o}(1-F_{o})}+\Gamma(F_{o})}\ \leq\
2B^{\prime}\bigl{(}1+\sqrt{W_{n}/\kappa}\bigr{)}^{5/2}\quad\text{if}\
T_{n,\nu}(F_{o})\leq\kappa.$ (18)
On the left hand side stands a function
$\Delta_{n}=\Delta_{n}(\cdot,F,F_{o})$, and its supremum over $\mathbb{R}$
equals $\Delta_{n}(F,F_{o})$. Thus it suffices to show that for a suitable
constant $B_{\nu,\kappa}$,
$\mathop{\mathrm{I\\!P}}\nolimits_{F}\Bigl{(}2B^{\prime}\bigl{(}1+\sqrt{W_{n}(x)/\kappa}\bigr{)}^{5/2}\geq\Delta_{n}(x)\Bigr{)}\
\leq\ B_{\nu,\kappa}\,\Delta_{n}(x)^{-4/5}$
for any $x\in\mathbb{R}$. Indeed,
$2B^{\prime}\bigl{(}1+\sqrt{W_{n}(x)/\kappa}\bigr{)}^{5/2}\geq\Delta_{n}(x)$
is equivalent to
$W_{n}(x)\ \geq\
\kappa\max\bigl{\\{}0,\Delta_{n}(x)^{2/5}(2B^{\prime})^{-2/5}-1\bigr{\\}}^{2}.$
Since $\mathop{\mathrm{I\\!E}}\nolimits W_{n}(x)\leq 1$, it follows from
Markov’s inequality that the latter inequality occurs with probability at most
$\kappa^{-1}\max\bigl{\\{}0,\Delta_{n}(x)^{2/5}(2B^{\prime})^{-2/5}-1\bigr{\\}}^{-2}\
=\
\max\bigl{\\{}0,B^{\prime\prime}\Delta_{n}(x)^{2/5}-\kappa^{1/2}\bigr{\\}}^{-2}$
with a certain constant $B^{\prime\prime}=B^{\prime\prime}_{\nu,\kappa}$. This
bound is trivial if $B^{\prime\prime}\Delta_{n}(x)^{2/5}<1+\kappa^{1/2}$,
which is equivalent to
$\Delta_{n}(x)^{4/5}<B_{\nu,\kappa}:=(1+\kappa^{1/2})^{2}/(B^{\prime\prime})^{2}$.
Otherwise,
$\displaystyle\max\bigl{\\{}$ $\displaystyle
0,B^{\prime\prime}\Delta_{n}(x)^{2/5}-\kappa^{1/2}\bigr{\\}}^{-2}$
$\displaystyle=\
\bigl{(}B^{\prime\prime}-\kappa^{1/2}\Delta_{n}(x)^{-2/5}\bigr{)}^{-2}\Delta_{n}(x)^{-4/5}\
\leq\ B\Delta_{n}^{-4/5}.$
∎
###### Proof of Lemma 4.2.
In what follows we use frequently the elementary inequalities
$\frac{\phi(x)}{x+1}\leq\ \Phi(-x)\ \leq\ \frac{\phi(x)}{x}\quad\text{for}\
x>0,$ (19)
where $\phi(x):=\Phi^{\prime}(x)=\exp(-x^{2}/2)/\sqrt{2\pi}$. In particular,
as $x\to\infty$,
$\displaystyle\Phi(-x)\ $ $\displaystyle=\ \exp(-x^{2}/2+O(\log
x))\quad\text{and}$ $\displaystyle C(\Phi(x))\ $ $\displaystyle=\
\log\bigl{(}O(1)+\log(1/\Phi(-x))\bigr{)}\ =\ 2\log(x)-\log(2)+o(1).$
Now consider two sequences $(x_{n})_{n}$ and $(\mu_{n})_{n}$ tending to
$\infty$ and $F_{o}=\Phi$,
$F_{n}=(1-\varepsilon_{n})\Phi+\varepsilon_{n}\Phi(\cdot-\mu_{n})$. Then the
inequalities (19) imply that
$\displaystyle\Gamma(F_{o}(x_{n}))F_{o}(x_{n})(1-F_{o}(x_{n}))\ $
$\displaystyle=\ (2\log(x_{n})+O(1))\Phi(-x_{n})(1+o(1))$ $\displaystyle=\
\exp\bigl{(}-x_{n}^{2}/2+O(\log(x_{n}))\bigr{)}.$
Moreover,
$F_{o}(x_{n})-F_{n}(x_{n})\ =\
\varepsilon_{n}\bigl{(}\Phi(\mu_{n}-x_{n})-\Phi(-x_{n})\bigr{)}\ =\
\varepsilon_{n}\Phi(\mu_{n}-x_{n})(1+o(1)),$
because $\Phi(-x_{n})\leq\phi(x_{n})/x_{n}$ while
$\Phi(\mu_{n}-x_{n})\ \geq\ \begin{cases}1/2&\text{if}\ \mu_{n}\geq x_{n},\\\
\displaystyle\frac{\phi(x_{n}-\mu_{n})}{x_{n}-\mu_{n}+1}\ \geq\
\frac{\phi(x_{n})\exp(\mu_{n}^{2}/2)}{x_{n}+1}&\text{if}\
\mu_{n}<x_{n}.\end{cases}$
Consequently, $\Delta_{n}(F_{n},\Phi)\to\infty$ if
$\frac{n\varepsilon_{n}\Phi(\mu_{n}-x_{n})}{n^{1/2}\exp\bigl{(}-x_{n}^{2}/4+O(\log(x_{n}))\bigr{)}+O(\log(x_{n}))}\
\to\ \infty.$ (20)
In part (a) with $\varepsilon_{n}=n^{-\beta+o(1)}$ and $\beta\in(1/2,1)$ we
imitate the arguments of Donoho and Jin (2004) and consider
$\mu_{n}\ =\ \sqrt{2r\log(n)}\quad\text{and}\quad x_{n}\ =\ \sqrt{2q\log(n)}$
with $0<r<q\leq 1$. Then by (19),
$\displaystyle n\varepsilon_{n}\Phi(\mu_{n}-x_{n})\ $ $\displaystyle=\
n^{1-\beta-(\sqrt{q}-\sqrt{r})^{2}+o(1)},$ $\displaystyle
n^{1/2}\exp\bigl{(}-x_{n}^{2}/4+O(\log(x_{n}))\bigr{)}\ $ $\displaystyle=\
n^{1/2-q/2+o(1)},$ $\displaystyle O(\log(x_{n}))\ $ $\displaystyle=\
n^{o(1)},$
so the left hand side of (20) equals
$\frac{n^{1-\beta-(\sqrt{q}-\sqrt{r})^{2}+o(1)}}{n^{1/2-q/2+o(1)}+n^{o(1)}}\
=\ \frac{n^{1/2-\beta+q/2-(\sqrt{q}-\sqrt{r})^{2}+o(1)}}{1+n^{(q-1)/2+o(1)}}\
=\
\frac{n^{1/2-\beta+2\sqrt{r}\sqrt{q}-\sqrt{q}^{2}/2-r+o(1)}}{1+n^{(q-1)/2+o(1)}}.$
The exponent in the enumerator is maximal in $q\in(r,1]$ if
$\sqrt{q}=\min\\{2\sqrt{r},1\\}$, i.e. $q=\min\\{4r,1\\}$, and this leads to
$\begin{cases}1/2-\beta+r&\text{if}\ r\leq 1/4,\\\
1-\beta-(1-\sqrt{r})^{2}&\text{if}\ r\geq 1/4.\end{cases}$
Thus when $\beta\in(1/2,3/4)$ we should choose $r\in(\beta-1/2,1/4)$ and
$q=4r$. When $\beta\in[3/4,1)$ we should choose
$r\in\Bigl{(}\bigl{(}1-\sqrt{1-\beta}\bigr{)}^{2},1\Bigr{)}$ and $q=1$.
As to part (b), we consider the more general setting that
$\varepsilon_{n}=n^{-\beta+o(1)}$ for some $\beta\in[1/2,3/4)$, where
$\pi_{n}=\sqrt{n}\varepsilon_{n}\to 0$. The latter constraint is trivial when
$\beta>1/2$ but relevant when $\beta=1/2$. Now we consider
$\mu_{n}\ :=\ \sqrt{2s\log(1/\pi_{n})}\quad\text{and}\quad x_{n}\ :=\
\sqrt{2q\log(1/\pi_{n})}$
with arbitrary constants $0<s<q$. Now
$\displaystyle n\varepsilon_{n}\Phi(\mu_{n}-x_{n})\ $ $\displaystyle=\
n^{1/2}\pi_{n}\Phi(\mu_{n}-x_{n})$ $\displaystyle=\
n^{1/2}\pi_{n}^{1+(\sqrt{q}-\sqrt{s})^{2}+o(1)},$ $\displaystyle
n^{1/2}\exp\bigl{(}-x_{n}^{2}/4+O(\log(x_{n}))\bigr{)}\ $ $\displaystyle=\
n^{1/2}\pi_{n}^{q/2+o(1)},$ $\displaystyle O(\log(x_{n}))\ $ $\displaystyle=\
\pi_{n}^{o(1)},$
so the left hand side of (20) equals
$\frac{n^{1/2}\pi_{n}^{1+(\sqrt{q}-\sqrt{s})^{2}+o(1)}}{n^{1/2}\pi_{n}^{q/2+o(1)}+\pi_{n}^{o(1)}}\
=\
\frac{\pi_{n}^{1+\sqrt{q}^{2}/2-2\sqrt{q}\sqrt{s}+s+o(1)}}{1+n^{-1/2}\pi_{n}^{-q/2+o(1)}}\
=\
\frac{\pi_{n}^{1+\sqrt{q}^{2}/2-2\sqrt{q}\sqrt{s}+s+o(1)}}{1+n^{-1/2+(\beta-1/2)q/2+o(1)}}.$
The exponent of $\pi_{n}$ becomes minimal in $q\in(s,\infty)$ if
$\sqrt{q}=2\sqrt{s}$, i.e. $q=4s$. Then we obtain
$\frac{\pi_{n}^{1-s+o(1)}}{1+n^{-1/2+(2\beta-1)s+o(1)}}\ =\
\frac{\pi_{n}^{1-s+o(1)}}{1+\sqrt{n}^{(4\beta-2)s-1+o(1)}},$
and this converges to $\infty$ if the exponents of $\pi_{n}$ and $\sqrt{n}$
are negative and non-positive, respectively. This is the case if $1<s\leq
1/(4\beta-2)$. (Note that $4\beta-2<1$ because $\beta<3/4$.) ∎
###### Proof of Theorem 5.1.
By symmetry it suffices to analyze the differences $b_{nj}-s_{nj}$ and
$b_{nj}^{\rm BJO}-s_{nj}$ for $0\leq j<n$.
Recall the notation $b(s,\gamma)$ for the unique number $b\in(s,1)$ such that
$K(s,b)=\gamma$, introduced in (K.6). There we considered only $s\in(0,1)$,
but it follows from $K(0,b)=-\log(1-b)$ that
$b(0,\gamma)=1-\exp(-\gamma)=\gamma+o(1)$ as $\gamma\to 0$. For $0\leq j<n$,
we may write
$b_{nj}^{\rm BJO}\ =\ b(s_{nj},\gamma_{n}^{\rm BJO})\quad\text{and}\quad
b_{nj}\ \leq\ b(t_{n,j+1},\gamma_{n}(t_{n1}))$
Recall that
$\gamma_{n}^{\rm BJ}\ =\ \frac{\log\log
n}{n}(1+o(1))\quad\text{and}\quad\gamma_{n}(t_{n1})\ =\ \frac{\log\log
n}{n}(1+o(1)).$
Moreover, since $K(s_{nj},\cdot)$ is convex on $[s_{nj},1)$, the numbers
$b(s_{nj},\gamma)$ are concave in $\gamma\geq 0$. In particular, with
$\tilde{\gamma}_{n}$ denoting the maximum of $\gamma_{n}^{\rm BJO}$ and
$\gamma_{n}(t_{n1})$,
$\frac{b_{nj}^{\rm BJO}-s_{nj}}{b(s_{nj},\tilde{\gamma}_{n})-s_{nj}}\ \geq\
\frac{\gamma_{n}^{\rm BJ}}{\tilde{\gamma}_{n}}\ \to\ 1$
uniformly in $0\leq j<n$. Hence it suffices to show that
$\limsup_{n\to\infty}\,\max_{0\leq
j<n}\frac{b(t_{n,j+1},\tilde{\gamma}_{n})-s_{nj}}{b(s_{nj},\tilde{\gamma}_{n})-s_{nj}}\
\leq\ 1.$ (21)
First we consider indices $j\leq j(n,1):=\lceil(\log\log n)^{1/2}\rceil$. Note
that for $j=0$,
$b(s_{nj},\tilde{\gamma}_{n})-s_{nj}\ =\ b(0,\tilde{\gamma}_{n})\ =\
\tilde{\gamma}_{n}(1+o(1)),$
and we may deduce from (13) and $\lim_{y\to\infty}H^{-1}(y)/y=1$ that
uniformly in $1\leq j\leq j(n,1)$,
$\displaystyle b(s_{nj},\tilde{\gamma}_{n})-s_{nj}\ $ $\displaystyle\geq\
(1+o(1))s_{nj}H^{-1}(\tilde{\gamma}_{n}/s_{nj})$ $\displaystyle=\
(1+o(1))\tilde{\gamma}_{n}\frac{H^{-1}(n\tilde{\gamma}_{n}/j)}{n\tilde{\gamma}_{n}/j}$
$\displaystyle\geq\ \tilde{\gamma}_{n}(1+o(1)).$
On the other hand, since
$t_{n,j+1}-s_{nj}\ =\ \frac{1-s_{nj}}{n+1}\ <\ n^{-1}\ =\
o(\tilde{\gamma}_{n}),$
we may conclude that uniformly in $0\leq j\leq j(n,1)$,
$\displaystyle b(t_{n,j+1},\tilde{\gamma}_{n})-s_{nj}\ $ $\displaystyle\leq\
b(t_{n,j+1},\tilde{\gamma}_{n})-t_{n,j+1}+n^{-1}$ $\displaystyle\leq\
t_{n,j+1}H^{-1}(\tilde{\gamma}_{n}/t_{n,j+1})+n^{-1}$ $\displaystyle=\
\tilde{\gamma}_{n}\frac{H^{-1}(\tilde{\gamma}_{n}/t_{n,j+1})}{\tilde{\gamma}_{n}/t_{n,j+1}}+n^{-1}$
$\displaystyle\leq\ \tilde{\gamma}_{n}(1+o(1)).$
Hence (21) holds true if we restrict $j$ to the interval
$\\{0,\ldots,j(n,1)\\}$.
Next we consider indices $j$ between $j(n,1)$ and $j(n,2):=\lceil
n\tilde{\gamma}_{n}^{1/3}\rceil$, i.e. $j(n,2)/n\to 0$ and
$t_{n,j+1}/s_{nj}\to 1$ uniformly in $j(n,1)\leq j\leq j(n,2)$. Then it
follows from (13), together with $H^{-1}(y)\geq y$ and monotonicity of
$H^{-1}(\cdot)$, that uniformly in $j_{n1}\leq j\leq j_{n2}$,
$\displaystyle\frac{b(t_{n,j+1},\tilde{\gamma}_{n})-s_{nj}}{b(s_{nj},\tilde{\gamma}_{n})-s_{nj}}\
$ $\displaystyle=\
(1+o(1))\frac{t_{n,j+1}H^{-1}(\tilde{\gamma}_{n}/t_{n,j+1})+n^{-1}}{s_{nj}H^{-1}(\tilde{\gamma}_{n}/s_{nj})}$
$\displaystyle=\
(1+o(1))\frac{t_{n,j+1}H^{-1}(\tilde{\gamma}_{n}/t_{n,j+1})}{s_{nj}H^{-1}(\tilde{\gamma}_{n}/s_{nj})}$
$\displaystyle\leq\ (1+o(1))\frac{t_{n,j+1}}{s_{nj}}$ $\displaystyle=\
1+o(1).$
Hence (21) is satisfied with $\\{j(n,1),\ldots,j(n,2)\\}$ in place of
$\\{0,1,\ldots,n-1\\}$.
Now consider $j(n,3):=n-j(n,2)$. Uniformly in $j(n,2)\leq j\leq j(n,3)$, the
product $s_{nj}(1-s_{nj})$ is larger than $\tilde{\gamma}_{n}^{1/3}(1+o(1))$,
so $\tilde{\gamma}_{n}/(s_{nj}(1-s_{nj}))\to 0$. Moreover,
$\mathop{\mathrm{logit}}\nolimits(t_{n,j+1})-\mathop{\mathrm{logit}}\nolimits(s_{nj})\to
0$, and it follows from (14) that
$\displaystyle\frac{b(t_{n,j+1},\tilde{\gamma}_{n})-s_{nj}}{b(j/n,\tilde{\gamma}_{n})-s_{nj}}\
$ $\displaystyle\leq\
(1+o(1))\frac{\sqrt{2\tilde{\gamma}_{n}t_{n,j+1}(1-t_{n,j+1})}+n^{-1}}{\sqrt{2\tilde{\gamma}_{n}s_{nj}(1-s_{nj})}}$
$\displaystyle=\ 1+o(1)+O(\tilde{\gamma}_{n}^{-1/3}n^{-1})$ $\displaystyle=\
1+o(1)$
uniformly in $j(n,2)\leq j\leq j(n,3)$.
Finally, we may conclude from (15), concavity of $\tilde{H}^{-1}(\cdot)$ and
the inequality $H^{-1}(y)\geq 1-e^{-y}$ that that uniformly for $j(n,3)\leq
j\leq n-1$,
$\displaystyle\frac{b(t_{n,j+1},\tilde{\gamma}_{n})-s_{nj}}{b(j/n,\tilde{\gamma}_{n})-s_{nj}}\
$ $\displaystyle\leq\
(1+o(1))\frac{(1-t_{n,j+1})\tilde{H}^{-1}(\tilde{\gamma}_{n}/(1-t_{n,j+1}))+(1-s_{nj})/n}{(1-s_{nj})\tilde{H}^{-1}(\tilde{\gamma}_{n}/(1-s_{nj}))}$
$\displaystyle\leq\
(1+o(1))\Bigl{(}1+\frac{(1-s_{nj})/n}{(1-t_{n,j+1})\tilde{H}^{-1}(\tilde{\gamma}_{n}/(1-t_{n,j+1}))}\Bigr{)}$
$\displaystyle=\ (1+o(1))\bigl{(}1+O(n^{-1}\tilde{\gamma}_{n}^{-2/3})\bigr{)}$
$\displaystyle=\ 1+o(1).$
These considerations prove (21).
It remains to analyze the maximum of $b_{nj}^{\rm BJO}-s_{nj}$ and
$b_{nj}-s_{nj}$, respectively, over $j=0,1,\ldots,n$. Note first that by
(K.5),
$b_{nj}^{\rm BJO}-s_{nj}\ \leq\
\sqrt{2\tilde{\gamma}_{n}s_{nj}(1-s_{nj})}+\tilde{\gamma}_{n}\ \leq\
\sqrt{\tilde{\gamma}_{n}/2}+\tilde{\gamma}_{n}\ =\
(1+o(1))\sqrt{\frac{\log\log n}{2n}}.$
On the other hand, for $j(n):=\lfloor(n+1)/2\rfloor$, (14) implies that
$b_{n,j(n)}^{\rm BJO}-s_{n,j(n)}\ \geq\ (1+o(1))\sqrt{2\gamma_{n}^{\rm
BJ}s_{n,j(n)}(1-s_{n,j(n)})}\ =\ (1+o(1))\sqrt{\frac{\log\log n}{2n}}.$
This proves the assertion about $\max_{j}(b_{nj}^{\rm BJO}-s_{nj})$. As to the
new confidence bounds, note first that by (K.5),
$\displaystyle b_{nj}-s_{nj}\ $ $\displaystyle\leq\ b_{nj}-t_{nj}+n^{-1}$
$\displaystyle\leq\
\sqrt{2\gamma_{n}(t_{nj})t_{nj}(1-t_{nj})}+\gamma_{n}(t_{n1})+n^{-1}$
$\displaystyle\leq\
n^{-1/2}\sqrt{h(t_{nj})+2t_{nj}(1-t_{nj})\tilde{\kappa}_{n,\nu,\alpha}}+O(n^{-1}\log\log
n),$
where $h(t):=2t(1-t)\bigl{(}C(t)+\nu D(t)\bigr{)}$ is a continuous function on
$(0,1)$ with limit $0$ as $t\to\\{0,1\\}$. Consequently, $\sup_{(0,1)}h$ is
finite and
$\max_{j=0,1,\ldots,n}\,(b_{nj}-s_{nj})\ \leq\
n^{-1/2}\sqrt{\sup_{(0,1)}h+\tilde{\kappa}_{n,\nu,\alpha}/2}+O(n^{-1}\log\log
n)\ =\ O(n^{-1/2}).$
∎
#### Acknowledgements.
We are grateful to Guenther Walther for stimulating talks about likelihood
ratio tests in nonparametric settings. In particular, Rivera and Walther
(2013) inspired us to reformulate the Law of the Iterated Logarithm in terms
of processes with subexponential tails. Many thanks go to Rudy Beran for
drawing our attention to Bahadur and Savage (1956).
## References
* [1] S. Aldor-Noiman, L. D. Brown, A. Buja, W. Rolke, and R. A. Stine, The power to see: A new graphical test of normality, Amer. Statist., 67 (2013), pp. 249–260.
* [2] R. R. Bahadur and L. J. Savage, The nonexistence of certain statistical procedures in nonparametric problems, Ann. Math. Statist., 27 (1956), pp. 1115–1122.
* [3] R. H. Berk and D. H. Jones, Goodness-of-fit test statistics that dominate the Kolmogorov statistics, Z. Wahrsch. Verw. Gebiete, 47 (1979), pp. 47–59.
* [4] D. Donoho and J. Jin, Higher criticism for detecting sparse heterogeneous mixtures, Ann. Statist., 32 (2004), pp. 962–994.
* [5] L. Dümbgen, New goodness-of-fit tests and their application to nonparametric confidence sets, Ann. Statist., 26 (1998), pp. 288–314.
* [6] A. M. Eiger, B. Nadler, and C. Spiegelman, The calibrated kolmogorov-smirnov test, tech. rep., Department of Computer Science, Weizmann Institute of Science; Department of Statistics, Texas A&M University, 2013. (http://arxiv.org/abs/1311.3190)
* [7] P. Erdös, On the law of the iterated logarithm, Ann. of Math. (2), 43 (1942), pp. 419–436.
* [8] W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58 (1963), pp. 13–30.
* [9] K. Itô and H. P. McKean, Jr., Diffusion processes and their sample paths, Springer-Verlag, Berlin, 1974. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125.
* [10] L. Jager and J. A. Wellner, Goodness-of-fit tests via phi-divergences, Ann. Statist., 35 (2007), pp. 2018–2053.
* [11] J. Kiefer, Iterated logarithm analogues for sample quantiles when $p_{n}\downarrow 0$, in Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California, 1973, pp. 227–244.
* [12] L. Le Cam and G. L. Yang, Asymptotics in statistics – Some basic concepts, Springer Series in Statistics, Springer-Verlag, New York, second ed., 2000.
* [13] D. M. Mason and J. H. Schuenemeyer, A modified Kolmogorov-Smirnov test sensitive to tail alternatives, Ann. Statist., 11 (1983), pp. 933–946.
* [14] A. B. Owen, Nonparametric likelihood confidence bands for a distribution function, J. Amer. Statist. Assoc., 90 (1995), pp. 516–521.
* [15] C. Rivera and G. Walther, Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics, Scand. J. Statist., 40 (2013), pp. 752–769.
* [16] G. R. Shorack and J. A. Wellner, Empirical processes with applications to statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986.
## 7 Supplementary material
### 7.1 A remark on moment-generating functions
Somewhat hidden in our proofs of Lemmas 3.1 and 3.3 is a basic fact about
moment generating functions which is stated in a slightly different form by
Rivera and Walther (2013) and possibly of independent interest: Suppose that
$X$ is a real-valued random variable with mean $\mu$ and moment-generating
function $m_{X}$,
$m_{X}(t)\ :=\ \mathop{\mathrm{I\\!E}}\nolimits\exp(tX).$
We assume that $m_{X}<\infty$ in a neighborhood of zero. In particular, all
moments of $X$ are finite. A standard application of Markov’s inequality
yields
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits(X\geq x)\ $ $\displaystyle\leq\
\exp\bigl{(}-K(x)\bigr{)}\quad\text{for all}\ x\geq\mu,$
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits(X\leq x)\ $ $\displaystyle\leq\
\exp\bigl{(}-K(x)\bigr{)}\quad\text{for all}\ x\leq\mu,$
where
$K(x)\ :=\ \sup_{t\in\mathbb{R}}\bigl{(}tx-\log m_{X}(t)\bigr{)}\
\begin{cases}\displaystyle=\ \sup_{t\geq 0}\bigl{(}tx-\log
m_{X}(t)\bigr{)}&\text{if}\ x\geq\mu,\\\ \displaystyle=\ \sup_{t\leq
0}\bigl{(}tx-\log m_{X}(t)\bigr{)}&\text{if}\ x\leq\mu.\end{cases}$
The latter facts follow from the fact that $\log m_{X}$ is a convex function
with derivative $\mu$ at $0$. Note also that $K:\mathbb{R}\to[0,\infty]$ is a
convex, lower semi-continuous function with $K(\mu)=0$ and
$\lim_{|x|\to\infty}K(x)=\infty$. From this one can derive the following
inequalities:
###### Lemma 7.1.
For arbitrary $\eta>0$,
$\left.\begin{array}[]{c}\mathop{\mathrm{I\\!P}}\nolimits(K(X)\geq\eta\
\text{and}\ X\geq\mu)\\\ \mathop{\mathrm{I\\!P}}\nolimits(K(X)\geq\eta\
\text{and}\ X\leq\mu)\end{array}\\!\\!\right\\}\ \leq\ \exp(-\eta),$
and thus
$\mathop{\mathrm{I\\!P}}\nolimits(K(X)\geq\eta)\ \leq\ 2\exp(-\eta).$
###### Proof of Lemma 7.1.
By symmetry, it suffices to show that
$\mathop{\mathrm{I\\!P}}\nolimits(K(X)\geq\eta\ \text{and}\ X\geq\mu)$ is not
greater than $\exp(-\eta)$. Since $K:[\mu,\infty)\to[0,\infty]$ is convex and
lower semi-continuous with $K(\mu)=0$ and $\lim_{x\to\infty}K(x)=\infty$, the
point
$x_{\eta}\ :=\ \max\bigl{\\{}x\geq\mu:K(x)\leq\eta\bigr{\\}}$
is well-defined. When $K(x_{\eta})=\eta$, convexity of $K$ and $K(\mu)=0$
imply that $K(x)<\eta$ for all $x\in[\mu,x_{\eta})$. Hence
$\mathop{\mathrm{I\\!P}}\nolimits(K(X)\geq\eta\ \text{and}\ X\geq\mu)\ =\
\mathop{\mathrm{I\\!P}}\nolimits(X\geq x_{\eta})\ \leq\ \exp(-K(x_{\eta}))\ =\
\exp(-\eta).$
When $K(x_{\eta})<\eta$, we may conclude from monotonicity and lower
semicontinuity of $K$ that $K(x)=\infty$ for all $x>x_{\eta}$. But this
implies that
$\mathop{\mathrm{I\\!P}}\nolimits(K(X)\geq\eta\ \text{and}\ X\geq\mu)\ =\
\mathop{\mathrm{I\\!P}}\nolimits(X>x_{\eta})\ =\
\sup_{x>x_{\eta}}\mathop{\mathrm{I\\!P}}\nolimits(X\geq x)\ =\ 0.$
∎
### 7.2 Exponential inequalities for beta distributions
Let $s,t\in(0,1)$, and let $Y\sim\mathrm{Beta}(mt,m(1-t))$ for some $m>0$.
Then
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits(Y\leq s)\ $ $\displaystyle\leq\
\inf_{\lambda\leq 0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda Y-\lambda
s)\ \leq\ \exp(-mK(t,s))\quad\text{if}\ s\leq t,$
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits(Y\geq s)\ $ $\displaystyle\leq\
\inf_{\lambda\geq 0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda Y-\lambda
s)\ \leq\ \exp(-mK(t,s))\quad\text{if}\ s\geq t.$
###### Proof.
In case of $s\geq t$, Markov’s inequality yields that
$\mathop{\mathrm{I\\!P}}\nolimits(Y\geq s)\ =\ \inf_{\lambda\geq
0}\,\mathop{\mathrm{I\\!P}}\nolimits(\lambda Y-\lambda s\geq 0)\ \leq\
\inf_{\lambda\geq 0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda Y-\lambda
s)\ =\ \inf_{\lambda\geq 0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda
mY-\lambda ms).$
The latter step is trivial but convenient for the next consideration: We may
write $Y=G/(G+G^{\prime})$ with independent random variables
$G\sim\mathrm{Gamma}(mt)$ and $G^{\prime}\sim\mathrm{Gamma}(m(1-t))$.
Moreover, it is well-known that $Y$ and $G+G^{\prime}$ are stochastically
independent with $\mathop{\mathrm{I\\!E}}\nolimits(G+G^{\prime})=m$.
Consequently, by Jensen’s inequality and Fubini’s theorem,
$\displaystyle\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda mY-\lambda ms)\ $
$\displaystyle=\
\mathop{\mathrm{I\\!E}}\nolimits\exp\bigl{(}\lambda\mathop{\mathrm{I\\!E}}\nolimits\bigl{(}G-s(G+G^{\prime})\,\big{|}\,Y\bigr{)}\bigr{)}$
$\displaystyle=\
\mathop{\mathrm{I\\!E}}\nolimits\exp\bigl{(}\lambda\mathop{\mathrm{I\\!E}}\nolimits\bigl{(}(1-s)G-\lambda
sG^{\prime}\,\big{|}\,Y\bigr{)}\bigr{)}$ $\displaystyle\leq\
\mathop{\mathrm{I\\!E}}\nolimits\mathop{\mathrm{I\\!E}}\nolimits\bigl{(}\exp(\lambda(1-s)G-\lambda
sG^{\prime})\,\big{|}\,Y\bigr{)}$ $\displaystyle=\
\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda(1-s)G-\lambda sG^{\prime})$
$\displaystyle=\
\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda(1-s)G)\mathop{\mathrm{I\\!E}}\nolimits\exp(-\lambda
sG^{\prime})$ $\displaystyle=\ (1-\lambda(1-s))^{-mt}(1+st)^{-m(1-t)}$
$\displaystyle=\
\exp\Bigl{(}-m\bigl{(}t\log(1-\lambda(1-s))+(1-t)\log(1+\lambda
s)\bigr{)}\Bigr{)}$
for $0\leq\lambda<1/(1-s)$. (For $\lambda\geq 1/(1-s)$ the expectation of
$\exp(\lambda(1-s)G)$ would be infinite.) Elementary calculations show that
$t\log(1-\lambda(1-s))+(1-t)\log(1+\lambda s)$ is maximal for
$\lambda=(s-t)/(s(1-s))\in[0,1/(1-s))$, and this yields the bound
$\inf_{\lambda\geq 0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda Y-\lambda
s)\ \leq\ \exp(-mK(t,s)).$
In case of $s\leq t$, the previous result may be applied to
$1-Y\sim\mathrm{Beta}(m(1-t),mt)$:
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits(Y\leq s)\ =\
\mathop{\mathrm{I\\!P}}\nolimits(1-Y\geq 1-s$ $\displaystyle)\ \leq\
\inf_{\lambda\geq
0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda(1-Y)-\lambda(1-s)\bigr{)}$
$\displaystyle\begin{cases}\displaystyle=\ \inf_{\lambda\leq
0}\,\mathop{\mathrm{I\\!E}}\nolimits\exp(\lambda Y-\lambda s),\\\ \leq\
\exp(-mK(1-t,1-s))\ =\ \exp(-mK(t,s)).\end{cases}$
∎
### 7.3 Further details about Gaussian mixtures
As in Section 4 we consider the standard Gaussian distribution function $\Phi$
and the alternative distribution functions
$F_{n}\ :=\ (1-\varepsilon_{n})\Phi+\varepsilon_{n}\Phi(\cdot-\mu_{n}),$
where $\varepsilon_{n}\downarrow 0$ and $\mu_{n}\to\infty$. Optimal tests of
$H_{0}:F\equiv\Phi$ versus $H_{1}:F\equiv F_{n}$ reject for large values of
the log-likelihood ratio statistic
$\sum_{i=1}^{n}\log\frac{dF_{n}}{d\Phi}(X_{i})\ =\
\sum_{i=1}^{n}\log(1+V_{n}(X_{i}))$
with
$V_{n}(x)\ =\ \varepsilon_{n}\bigl{(}\exp(\mu_{n}x-\mu_{n}^{2}/2)-1\bigr{)}.$
If $(\mu_{n})_{n}$ is chosen such that
$\sum_{i=1}^{n}\log(1+V_{n}(X_{i}))\ \to_{p}\ 0\quad\text{when}\ F\equiv\Phi,$
(22)
then for any sequence of tests $\phi_{n}:\mathbb{R}^{n}\to[0,1]$,
$\limsup_{n\to\infty}\bigl{|}\mathop{\mathrm{I\\!E}}\nolimits_{F_{n}}\phi_{n}(X_{1},\ldots,X_{n})-\mathop{\mathrm{I\\!E}}\nolimits_{\Phi}\phi_{n}(X_{1},\ldots,X_{n})\bigr{|}\
=\ 0;$
see LeCam and Yang (2000).
###### Lemma 7.2.
Suppose that $\varepsilon_{n}=n^{-\beta+o(1)}$ for some $\beta\in[1/2,3/4)$
and $\pi_{n}=n^{1/2}\varepsilon_{n}\to 0$. Then (22) is satisfied if
$\mu_{n}=\sqrt{2s\log(1/\pi_{n})}$ for some fixed $s\in(0,1)$.
###### Proof of Lemma 7.2.
Note that for $v>-1$,
$\log(1+v)\ =\ v-\frac{v^{2}}{2(1+\xi(v))}$
with $\xi(v)\geq\min\\{0,v\\}$. Consequently, since $V_{n}>-\varepsilon_{n}$,
$\sum_{i=1}^{n}V_{n}(X_{i})-\frac{1}{2(1-\varepsilon_{n})}\sum_{i=1}^{n}V_{n}(X_{i})^{2}\
\leq\ \sum_{i=1}^{n}\log(1+V_{n}(X_{i}))\ \leq\ \sum_{i=1}^{n}V_{n}(X_{i}).$
But it follows from $\mathop{\mathrm{I\\!E}}\nolimits_{\Phi}(V_{n}(X_{1}))=0$
that
$\displaystyle\mathop{\mathrm{I\\!E}}\nolimits_{\Phi}\biggl{(}\Bigl{(}\sum_{i=1}^{n}V_{n}(X_{i})\Bigr{)}^{2}\biggr{)}\
$ $\displaystyle=\ n\mathop{\mathrm{Var}}\nolimits_{\Phi}(V_{n}(X_{1}))$
$\displaystyle=\
n\varepsilon_{n}^{2}\bigl{(}\mathop{\mathrm{I\\!E}}\nolimits_{\Phi}\exp(2\mu_{n}X_{1}-\mu_{n}^{2})-1\bigr{)}$
$\displaystyle=\ \pi_{n}^{2}(\exp(\mu_{n}^{2})-1)$ $\displaystyle=\
\pi_{n}^{2(1-s)}-\pi_{n}^{2}$ $\displaystyle\to\ 0$
because $s<1$, and
$\mathop{\mathrm{I\\!E}}\nolimits_{\Phi}\Bigl{(}\frac{1}{2(1-\varepsilon_{n})}\sum_{i=1}^{n}V_{n}(X_{i})^{2}\Bigr{)}\
=\
\frac{n\mathop{\mathrm{Var}}\nolimits_{\Phi}(V_{n}(X_{1}))}{2(1-\varepsilon_{n})}\
\to\ 0.$
∎
### 7.4 Bahadur and Savage (1956) revisited
Let $(L_{n},U_{n})$ be a $(1-\alpha)$-confidence band for $F\in\mathcal{F}$
with a given class $\mathcal{F}$ of distribution functions. That means
$L_{n}=L_{n}(\cdot,\boldsymbol{X}_{n})$ and
$U_{n}=U_{n}(\cdot,\boldsymbol{X}_{n})$ are non-decreasing functions on the
real line depending on the data vector $\boldsymbol{X}_{n}=(X_{i})_{i=1}^{n}$
such that
$\mathop{\mathrm{I\\!P}}\nolimits_{F}\bigl{(}L_{n}\leq F\leq U_{n}\ \text{on}\
\mathbb{R}\bigr{)}\ \geq\ 1-\alpha\quad\text{for any}\ F\in\mathcal{F}.$
We assume that $\mathcal{F}$ is convex and satisfies
$F(\cdot-\mu)\in\mathcal{F}$ for any $F\in\mathcal{F}$ and $\mu\in\mathbb{R}$.
This is true if, for instance, $\mathcal{F}$ corresponds to all mixtures of
Gaussian distributions with variance one. Then Theorem 2 of Bahadur and Savage
(1956) may be modified as follows:
###### Theorem 7.3.
Let $(L_{n},U_{n})$ be a $(1-\alpha)$-confidence band for $F\in\mathcal{F}$.
For any $\varepsilon\in(0,1)$,
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits_{F}\Bigl{(}\inf_{x\in\mathbb{R}}U_{n}(x)<\varepsilon\Bigr{)}\
$ $\displaystyle\leq\ (1-\varepsilon)^{-n}\alpha,$
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits_{F}\Bigl{(}\sup_{x\in\mathbb{R}}L_{n}(x)>1-\varepsilon\Bigr{)}\
$ $\displaystyle\leq\ (1-\varepsilon)^{-n}\alpha.$
Setting $\varepsilon=c/n$ for some fixed $c>0$ reveals that
$\inf_{x\in\mathbb{R}}U_{n}(x)<c/n$ or $\sup_{x\in\mathbb{R}}L_{n}(x)\leq
1-c/n$ with probability at most $(1-c/n)^{-n}\alpha=e^{c}\alpha+o(1)$,
respectively.
###### Proof of Theorem 7.3.
By symmetry, it suffices to prove the claim about $U_{n}$. By monotonicity of
$U_{n}$,
$\mathop{\mathrm{I\\!P}}\nolimits_{F}\Bigl{(}\inf_{x\in\mathbb{R}}U_{n}(x)<\varepsilon\Bigr{)}\
=\
\sup_{x\in\mathbb{R},\delta\in(0,\varepsilon)}\mathop{\mathrm{I\\!P}}\nolimits_{F}(U_{n}(x)<\delta).$
Hence it suffices to show that
$\mathop{\mathrm{I\\!P}}\nolimits_{F}(U_{n}(x)<\delta)\leq(1-\varepsilon)^{-n}\alpha$
for any single point $x\in\mathbb{R}$ and $\delta\in(0,\varepsilon)$. To this
end consider $F_{\varepsilon,\mu}:=(1-\varepsilon)F+\varepsilon F(\cdot-\mu)$
for our given $\varepsilon$ and some $\mu\in\mathbb{R}$. Note that
$\mathcal{L}_{F_{\varepsilon,\mu}}(\boldsymbol{X}_{n})$ describes the
distribution of
$\tilde{\boldsymbol{X}}_{n}\ :=\ \bigl{(}Y_{i}+\xi_{i}\mu\bigr{)}_{i=1}^{n}$
with $2n$ independent random variables
$\xi_{1},\xi_{2},\ldots,\xi_{n}\sim\mathrm{Bin}(1,\varepsilon)$ and
$Y_{1},Y_{2},\ldots,Y_{n}\sim F$. In particular, for any event
$A_{n}\subset\mathbb{R}^{n}$,
$\displaystyle\mathop{\mathrm{I\\!P}}\nolimits_{F_{\varepsilon,\mu}}(\boldsymbol{X}_{n}\in
A_{n})\ $ $\displaystyle=\
\mathop{\mathrm{I\\!P}}\nolimits(\tilde{\boldsymbol{X}}_{n}\in A_{n})$
$\displaystyle\geq\
\mathop{\mathrm{I\\!P}}\nolimits\bigl{(}\tilde{\boldsymbol{X}}_{n}\in
A_{n},\xi_{1}=\xi_{2}=\cdots=\xi_{n}=0\bigr{)}$ $\displaystyle=\
(1-\varepsilon)^{n}\mathop{\mathrm{I\\!P}}\nolimits_{F}(\boldsymbol{X}_{n}\in
A_{n}).$
Consequently, since $F_{\varepsilon,\mu}\in\mathcal{F}$, too, we may conclude
from
$\mathop{\mathrm{I\\!P}}\nolimits_{F_{\varepsilon,\mu}}\bigl{(}L_{n}\leq
F_{\varepsilon,\mu}\leq U_{n}\ \text{on}\ \mathbb{R}\bigr{)}\ \geq\ 1-\alpha$
that
$\displaystyle\alpha\ $ $\displaystyle\geq\
\mathop{\mathrm{I\\!P}}\nolimits_{F_{\varepsilon,\mu}}\bigl{(}U_{n}(x)<F_{\varepsilon,\mu}(x)\bigr{)}$
$\displaystyle\geq\
(1-\varepsilon)^{n}\mathop{\mathrm{I\\!P}}\nolimits_{F}\bigl{(}U_{n}(x)<(1-\varepsilon)F(x)+\varepsilon
F(x-\mu)\bigr{)}$ $\displaystyle\geq\
(1-\varepsilon)^{n}\mathop{\mathrm{I\\!P}}\nolimits_{F}\bigl{(}U_{n}(x)<\varepsilon
F(x-\mu)\bigr{)}.$
But for sufficiently small (negative) $\mu$, the value $\varepsilon F(x-\mu)$
is greater than or equal to $\delta$. Then we may conclude that
$\alpha\geq(1-\varepsilon)^{n}\mathop{\mathrm{I\\!P}}\nolimits_{F}(U_{n}(x)<\delta)$.
∎
|
arxiv-papers
| 2014-02-12T18:17:56 |
2024-09-04T02:49:58.159894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Lutz Duembgen and Jon A. Wellner",
"submitter": "Lutz Duembgen",
"url": "https://arxiv.org/abs/1402.2918"
}
|
1402.2949
|
der Philosophisch-naturwissenschaftlichen
Fakultät der Universität Bern
vorgelegt von
Aaron Karper
Leiter der Arbeit:
Professor Dr. Thomas Strahm
Institut für Informatik und angewandte Mathematik
CHAPTER: INTRODUCTION
CHAPTER: COMPUTABILITY
CHAPTER: COMPLEXITY
[1]
A.V. Aho, M.S. Lam, R. Sethi, and J.D. Ullman.
Compilers: principles, techniques, and tools, volume 1009.
Pearson/Addison Wesley, 2007.
[2]
R.G. Downey and M.R. Fellows.
Parameterized complexity, volume 3.
springer New York, 1999.
[3]
Gábor Etesi and István Németi.
Non-turing computations via malament–hogarth space-times.
International Journal of Theoretical Physics, 41:341–370,
[4]
Y. Futamura.
Partial evaluation of computation process–an approach to a
Higher-Order and Symbolic Computation, 12(4):381–391, 1999.
[5]
J.Y. Girard, P. Taylor, and Y. Lafont.
Proofs and types, volume 7.
Cambridge University Press Cambridge, 1989.
[6]
L.K. Grover.
A fast quantum mechanical algorithm for database search.
In Proceedings of the twenty-eighth annual ACM symposium on
Theory of computing, pages 212–219. ACM, 1996.
[7]
N.D. Jones, C.K. Gomard, and P. Sestoft.
Partial evaluation and automatic program generation.
Peter Sestoft, 1993.
[8]
Neil D. Jones.
Computability and complexity: from a programming perspective.
MIT Press, Cambridge, MA, USA, 1997.
[9]
H.P. Nilsson.
Porting gcc for dunces.
Master Thesis, 5:43–54, 2000.
[10]
Armin Rigo.
Representation-based just-in-time specialization and the psyco
prototype for python.
In Proceedings of the 2004 ACM SIGPLAN symposium on Partial
evaluation and semantics-based program manipulation, PEPM '04, pages 15–26,
New York, NY, USA, 2004. ACM.
[11]
S.J. Russell and P. Norvig.
Artificial intelligence: a modern approach, volume 3.
Prentice hall Englewood Cliffs, NJ, 2009.
[12]
M. Sipser.
Introduction to the Theory of Computation, volume 27.
Thomson Course Technology Boston, MA, 2006.
[13]
D. van Dalen.
Algorithms and decision problems: A crash course in recursion theory.
G. Gabbay and F. Guenthner (eds.), Handbook of Philosophical
Logic, 1:409–478, 1983.
[14]
T.L. Veldhuizen.
C++ templates are turing complete.
Available at citeseer. ist. psu. edu/581150. html, 2003.
Aaron Karper
der Philosophisch-naturwissenschaftlichen
Fakultät der Universität Bern
vorgelegt von
Aaron Karper
Leiter der Arbeit:
Professor Dr. Thomas Strahm
Institut für Informatik und angewandte Mathematik
CHAPTER: INTRODUCTION
CHAPTER: COMPUTABILITY
CHAPTER: COMPLEXITY
[1]
A.V. Aho, M.S. Lam, R. Sethi, and J.D. Ullman.
Compilers: principles, techniques, and tools, volume 1009.
Pearson/Addison Wesley, 2007.
[2]
R.G. Downey and M.R. Fellows.
Parameterized complexity, volume 3.
springer New York, 1999.
[3]
Gábor Etesi and István Németi.
Non-turing computations via malament–hogarth space-times.
International Journal of Theoretical Physics, 41:341–370,
[4]
Y. Futamura.
Partial evaluation of computation process–an approach to a
Higher-Order and Symbolic Computation, 12(4):381–391, 1999.
[5]
J.Y. Girard, P. Taylor, and Y. Lafont.
Proofs and types, volume 7.
Cambridge University Press Cambridge, 1989.
[6]
L.K. Grover.
A fast quantum mechanical algorithm for database search.
In Proceedings of the twenty-eighth annual ACM symposium on
Theory of computing, pages 212–219. ACM, 1996.
[7]
N.D. Jones, C.K. Gomard, and P. Sestoft.
Partial evaluation and automatic program generation.
Peter Sestoft, 1993.
[8]
Neil D. Jones.
Computability and complexity: from a programming perspective.
MIT Press, Cambridge, MA, USA, 1997.
[9]
H.P. Nilsson.
Porting gcc for dunces.
Master Thesis, 5:43–54, 2000.
[10]
Armin Rigo.
Representation-based just-in-time specialization and the psyco
prototype for python.
In Proceedings of the 2004 ACM SIGPLAN symposium on Partial
evaluation and semantics-based program manipulation, PEPM '04, pages 15–26,
New York, NY, USA, 2004. ACM.
[11]
S.J. Russell and P. Norvig.
Artificial intelligence: a modern approach, volume 3.
Prentice hall Englewood Cliffs, NJ, 2009.
[12]
M. Sipser.
Introduction to the Theory of Computation, volume 27.
Thomson Course Technology Boston, MA, 2006.
[13]
D. van Dalen.
Algorithms and decision problems: A crash course in recursion theory.
G. Gabbay and F. Guenthner (eds.), Handbook of Philosophical
Logic, 1:409–478, 1983.
[14]
T.L. Veldhuizen.
C++ templates are turing complete.
Available at citeseer. ist. psu. edu/581150. html, 2003.
|
arxiv-papers
| 2014-02-10T21:35:37 |
2024-09-04T02:49:58.173515
|
{
"license": "Public Domain",
"authors": "Aaron Karper",
"submitter": "Aaron Karper",
"url": "https://arxiv.org/abs/1402.2949"
}
|
1402.2982
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-017 LHCb-PAPER-2013-068 23 May 2014
A study of $C\\!P$ violation in $B^{\pm}\rightarrow DK^{\pm}$ and
$B^{\pm}\rightarrow D\pi^{\pm}$ decays with $D\rightarrow K^{0}_{\rm
S}K^{\pm}\pi^{\mp}$ final states
The LHCb collaboration†††Authors are listed on the following pages.
A first study of $C\\!P$ violation in the decay modes
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}]_{D}h^{\pm}$ and
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{\mp}\pi^{\pm}]_{D}h^{\pm}$, where $h$ labels a $K$ or $\pi$ meson and $D$
labels a $D^{0}$ or $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ meson,
is performed. The analysis uses the LHCb data set collected in $pp$
collisions, corresponding to an integrated luminosity of 3$\mbox{\,fb}^{-1}$.
The analysis is sensitive to the $C\\!P$-violating CKM phase $\gamma$ through
seven observables: one charge asymmetry in each of the four modes and three
ratios of the charge-integrated yields. The results are consistent with
measurements of $\gamma$ using other decay modes.
Published in Phys. Lett. B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25, S. Amato2, S. Amerio22, Y. Amhis7, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel39, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson50, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30, W.
Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,r, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez36,p, A. Camboni36, P. Campana18,38, D. Campora
Perez38, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,j, A. Cardini15, H.
Carranza-Mejia50, L. Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20,
L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph.
Charpentier38, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38,
X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47,
J. Closier38, C. Coca29, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A.
Comerma-Montells36, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G.
Corti38, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz
Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J. Dalseno46, P.
David8, P.N.Y. David41, A. Davis57, I. De Bonis4, K. De Bruyn41, S. De
Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D.
Derkach55, O. Deschamps5, F. Dettori42, A. Di Canto11, H. Dijkstra38, S.
Donleavy52, F. Dordei11, M. Dorigo39, P. Dorosz26,o, A. Dosil Suárez37, D.
Dossett48, A. Dovbnya43, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49, U. Egede53, V. Egorychev31, S. Eidelman34,
S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5,
Ch. Elsasser40, S. Esen11, A. Falabella16,f, C. Färber11, C. Farinelli41, S.
Farry52, D. Ferguson50, V. Fernandez Albor37, F. Ferreira Rodrigues1, M.
Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M. Fiorini16,f, C. Fitzpatrick38,
M. Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C.
Frei38, M. Frosini17,38,g, J. Fu21, E. Furfaro24,l, A. Gallas Torreira37, D.
Galli14,d, M. Gandelman2, P. Gandini59, Y. Gao3, J. Garofoli59, J. Garra
Tico47, L. Garrido36, C. Gaspar38, R. Gauld55, L. Gavardi9, E. Gersabeck11, M.
Gersabeck54, T. Gershon48, Ph. Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47,
L. Giubega29, V.V. Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38,
A. Gomes1,a, H. Gordon38, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A.
Granado Cardoso38, E. Graugés36, G. Graziani17, A. Grecu29, E. Greening55, S.
Gregson47, P. Griffith45, L. Grillo11, O. Grünberg61, B. Gui59, E. Gushchin33,
Yu. Guz35,38, T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, T.W.
Hafkenscheid64, S.C. Haines47, S. Hall53, B. Hamilton58, T. Hampson46, S.
Hansmann-Menzemer11, N. Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann61,
J. He38, T. Head38, V. Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A.
Hernando Morata37, E. van Herwijnen38, M. Heß61, A. Hicheur1, D. Hill55, M.
Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55, N. Hussain55, D.
Hutchcroft52, D. Hynds51, V. Iakovenko44, M. Idzik27, P. Ilten56, R.
Jacobsson38, A. Jaeger11, E. Jans41, P. Jaton39, A. Jawahery58, F. Jing3, M.
John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N. Jurik59, M.
Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, M.
Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C. Khurewathanakul39, S.
Klaver54, O. Kochebina7, I. Komarov39, R.F. Koopman42, P. Koppenburg41, M.
Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11, M. Kreps48, G.
Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k, V.
Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31,38, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, T. Latham48, C. Lazzeroni45,
R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32, J.
Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11, Y. Li3, M.
Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G. Liu38, S.
Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40, H. Lu3, D.
Lucchesi22,r, J. Luisier39, H. Luo50, E. Luppi16,f, O. Lupton55, F.
Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O. Maev30,38, S. Malde55, G.
Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5, U. Marconi14, P.
Marino23,t, R. Märki39, J. Marks11, G. Martellotti25, A. Martens8, A. Martín
Sánchez7, M. Martinelli41, D. Martinez Santos42, F. Martinez Vidal63, D.
Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A.
Mazurov16,38,f, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
McSkelly52, B. Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,t, R. Mountain59, F.
Muheim50, K. Müller40, R. Muresan29, B. Muryn27, B. Muster39, P. Naik46, T.
Nakada39, R. Nandakumar49, I. Nasteva1, M. Needham50, N. Neri21, S. Neubert38,
N. Neufeld38, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,q, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-
Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R.
Oldeman15,e, G. Onderwater64, M. Orlandea29, J.M. Otalora Goicochea2, P.
Owen53, A. Oyanguren36, B.K. Pal59, A. Palano13,c, F. Palombo21,u, M.
Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51, L. Pappalardo16,
C. Parkes54, C.J. Parkinson9, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, C. Pavel-Nicorescu29, A. Pazos Alvarez37, A. Pearce54, A.
Pellegrino41, G. Penso25,m, M. Pepe Altarelli38, S. Perazzini14,d, E. Perez
Trigo37, P. Perret5, M. Perrin-Terrin6, L. Pescatore45, E. Pesen65, G.
Pessina20, K. Petridis53, A. Petrolini19,j, E. Picatoste Olloqui36, B.
Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50, M. Plo
Casasus37, F. Polci8, G. Polok26, A. Poluektov48,34, E. Polycarpo2, A.
Popov35, D. Popov10, B. Popovici29, C. Potterat36, A. Powell55, J.
Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39,
G. Punzi23,s, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Redford55, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S.
Ricciardi49, A. Richards53, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts58, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez
Perez37, S. Roiser38, V. Romanovsky35, A. Romero Vidal37, M. Rotondo22, J.
Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P. Ruiz Valls36, G.
Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B. Saitta15,e,
V. Salustino Guimaraes2, B. Sanmartin Sedes37, R. Santacesaria25, C.
Santamarina Rios37, E. Santovetti24,l, M. Sapunov6, A. Sarti18, C.
Satriano25,n, A. Satta24, M. Savrie16,f, D. Savrina31,32, M. Schiller42, H.
Schindler38, M. Schlupp9, M. Schmelling10, B. Schmidt38, O. Schneider39, A.
Schopper38, M.-H. Schune7, R. Schwemmer38, B. Sciascia18, A. Sciubba25, M.
Seco37, A. Semennikov31, K. Senderowska27, I. Sepp53, N. Serra40, J. Serrano6,
P. Seyfert11, M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52,
L. Shekhtman34, O. Shevchenko43, V. Shevchenko62, A. Shires9, R. Silva
Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, T. Skwarnicki59, N.A.
Smith52, E. Smith55,49, E. Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D.
Sokoloff57, F.J.P. Soler51, F. Soomro39, D. Souza46, B. Souza De Paula2, B.
Spaan9, A. Sparkes50, F. Spinella23, P. Spradlin51, F. Stagni38, S. Stahl11,
O. Steinkamp40, S. Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S.
Stracka23,38, M. Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38,
L. Sun57, W. Sutcliffe53, S. Swientek9, V. Syropoulos42, M. Szczekowski28, P.
Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, E. Teodorescu29, F. Teubert38, C. Thomas55, E. Thomas38, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D.
Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4,53, S. Tourneur39,
M.T. Tran39, M. Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M.
Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin62, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez
Sierra37, S. Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M.
Vesterinen11, B. Viaud7, D. Vieira2, X. Vilasis-Cardona36,p, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß61, H.
Voss10, J.A. de Vries41, R. Waldi61, C. Wallace48, R. Wallace12, S.
Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, A.D. Webber54, D.
Websdale53, M. Whitehead48, J. Wicht38, J. Wiechczynski26, D. Wiedner11, L.
Wiggers41, G. Wilkinson55, M.P. Williams48,49, M. Williams56, F.F. Wilson49,
J. Wimberley58, J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A.
Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y. Xie50,38, Z. Xing59, Z. Yang3,
X. Yuan3, O. Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, F. Zhang3, L.
Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3, A.
Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
62National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
63Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
64KVI - University of Groningen, Groningen, The Netherlands, associated to 41
65Celal Bayar University, Manisa, Turkey, associated to 38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
## 1 Introduction
A precise measurement of the unitarity triangle angle
$\gamma=\arg{\left(-\frac{V_{ud}V_{ub}^{*}}{V_{cd}V_{cb}^{*}}\right)}$ is one
of the most important tests of the Cabibbo Kobayashi Maskawa (CKM) mechanism.
This parameter can be accessed through measurements of observables in decays
of charged $B$ mesons to a neutral $D$ meson and a kaon or pion, where $D$
labels a $D^{0}$ or $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ meson
decaying to a particular final state accessible to $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$. Such decays are sensitive to
$\gamma$ through the interference between $b\rightarrow c\bar{u}s$ and
$b\rightarrow u\bar{c}s$ amplitudes. They offer an attractive means to measure
$\gamma$ because the effect of physics beyond the Standard Model is expected
to be negligible, thus allowing interesting comparisons with other
measurements where such effects could be larger.
The determination of $\gamma$ using $B^{\pm}\rightarrow DK^{\pm}$ decays was
first proposed for $D$ decays to the $C\\!P$-eigenstates $K^{+}K^{-}$ and
$\pi^{+}\pi^{-}$ (so-called “GLW” analysis) [1, 2]. Subsequently, the analysis
of the $K^{\pm}\pi^{\mp}$ final state was proposed (named “ADS”) [3, 4], where
the suppression between the colour favoured $B^{-}\rightarrow D^{0}K^{-}$ and
suppressed $B^{-}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}$ decays is compensated by the
CKM suppression of the $D^{0}\rightarrow K^{+}\pi^{-}$ decay relative to
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$,
resulting in large interference. The LHCb collaboration has published the two-
body ADS and GLW analyses [5], the Dalitz analysis of the decay
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}h^{\pm}h^{\mp}]_{D}K^{\pm},~{}(h=\pi,K)$ [6] and the ADS-like analysis of
the decay mode
$B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}]_{D}K^{\pm}$ [7], where
$[X]_{D}$ indicates a given final state $X$ produced by the decay of the $D$
meson. These measurements have recently been combined to yield the result
$\gamma=(72.0^{+14.7}_{-15.6})^{\circ}$ [8], which is in agreement with the
results obtained by the BaBar and Belle collaborations of
$\gamma=(69^{+17}_{-16})^{\circ}$ [9] and $\gamma=(68^{+15}_{-14})^{\circ}$
[10], respectively. In analogy to studies in charged $B$ meson decays,
sensitivity to $\gamma$ can also be gained from decays of neutral $B$ mesons,
as has been demonstrated in the LHCb analysis of
$B^{0}\rightarrow[K^{+}K^{-}]_{D}K^{*0}$ decays [11].
The inclusion of additional $B^{\pm}\rightarrow DK^{\pm}$ modes can provide
further constraints on $\gamma$. In this Letter, an analysis of the
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ final states is
performed, the first ADS-like analysis to use singly Cabibbo-suppressed (SCS)
decays, as proposed in [12]. The two decays,
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}]_{D}h^{\pm}$ and
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{\mp}\pi^{\pm}]_{D}h^{\pm}$, are distinguished by the charge of the
$K^{\pm}$ from the decay of the $D$ meson relative to the charge of the $B$
meson, so that the former is labelled “same sign” (SS) and the latter
“opposite sign” (OS).
In order to interpret $C\\!P$-violating effects using these three-body decays
it is necessary to account for the variation of the $D$ decay strong phase
over its Dalitz plot due to the presence of resonances between the particles
in the final state. Instead of employing an amplitude model to describe this
phase variation, direct measurements of the phase made by the CLEO
collaboration are used, which are averaged over large regions of the Dalitz
plot [13]. The same CLEO study indicates that this averaging can be employed
without a large loss of sensitivity. The use of the CLEO results avoids the
need to introduce a systematic uncertainty resulting from an amplitude model
description.
The analysis uses the full 2011 and 2012 LHCb $pp$ collision data sets,
corresponding to integrated luminosities of 1 and 2$\mbox{\,fb}^{-1}$ and
centre-of-mass energies of $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and
$8\mathrm{\,Te\kern-1.00006ptV}$, respectively. The results are measurements
of $C\\!P$-violating observables that can be interpreted in terms of $\gamma$
and other hadronic parameters of the $B^{\pm}$ meson decay.
## 2 Formalism
The SS decay $B^{+}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{+}\pi^{-}]_{D}K^{+}$ can proceed via a $D^{0}$ or $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ meson, so that the decay amplitude
is the sum of two amplitudes that interfere,
$\displaystyle A(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle S}\pi})=A_{\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}}(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}\pi})+r_{B}e^{i(\delta_{B}+\gamma)}A_{D^{0}}(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle S}\pi}),$ (1)
where $A_{\\{D^{0},\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}\\}}(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle S}\pi})$ are the $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decay amplitudes to a specific
point in the $K^{0}_{\rm\scriptscriptstyle S}K^{+}\pi^{-}$ Dalitz plot. The
amplitude ratio $r_{B}$ is $\frac{|A(B^{+}\rightarrow
D^{0}K^{+})|}{|A(B^{+}\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}K^{+})|}=0.089\pm 0.009$ [8] and
$\delta_{B}$ is the strong phase difference between the $B^{+}\rightarrow
D^{0}K^{+}$ and $B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}$ decays. The calculation of
the decay rate in a region of the Dalitz plot requires the evaluation of the
integral of the interference term between the two $D$ decay amplitudes over
that region. Measurements have been made by the CLEO collaboration [13], where
quantum-correlated $D$ decays are used to determine the integral of the
interference term directly in the form of a “coherence factor”,
$\kappa_{K^{0}_{\rm\scriptscriptstyle S}K\pi}$, and an average strong phase
difference, $\delta_{K^{0}_{\rm\scriptscriptstyle S}K\pi}$, as first proposed
in Ref. [14]. The coherence factor can take a value between 0 and 1 and is
defined through the expression
$\kappa_{K^{0}_{\rm\scriptscriptstyle
S}K\pi}e^{-i\delta_{K^{0}_{\rm\scriptscriptstyle S}K\pi}}\equiv\frac{\int
A^{*}_{K^{0}_{\rm\scriptscriptstyle
S}K^{-}\pi^{+}}(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K\pi})A_{K^{0}_{\rm\scriptscriptstyle
S}K^{+}\pi^{-}}(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K\pi})dm^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K}dm^{2}_{K\pi}}{A^{\textrm{int.}}_{K^{0}_{\rm\scriptscriptstyle
S}K^{-}\pi^{+}}A^{\textrm{int.}}_{K^{0}_{\rm\scriptscriptstyle
S}K^{+}\pi^{-}}},$ (2)
where $A^{\textrm{int.}}_{K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}}=\int|A_{K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}}(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K\pi})|^{2}dm^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K}dm^{2}_{K\pi}$. This avoids the significant modelling uncertainty incurred
by the determination of the strong phase difference between the $D^{0}$ and
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ amplitudes at each point in
the Dalitz region from an amplitude model. The decay rates, $\Gamma$, to the
four final states can therefore be expressed as
$\displaystyle\Gamma_{\textrm{SS, }DK}^{\pm}$ $\displaystyle=z[\quad$
$\displaystyle\hskip 2.84544pt1$ $\displaystyle+\hskip
7.11317ptr_{B}^{2}r_{D}^{2}$
$\displaystyle+2r_{B}r_{D}\kappa_{K^{0}_{\rm\scriptscriptstyle
S}K\pi}\cos(\delta_{B}\pm\gamma-\delta_{K^{0}_{\rm\scriptscriptstyle
S}K\pi})\quad]$ $\displaystyle\Gamma_{\textrm{OS, }DK}^{\pm}$
$\displaystyle=z[$ $\displaystyle r_{B}^{2}$ $\displaystyle+\hskip
14.22636ptr_{D}^{2}$
$\displaystyle+2r_{B}r_{D}\kappa_{K^{0}_{\rm\scriptscriptstyle
S}K\pi}\cos(\delta_{B}\pm\gamma+\delta_{K^{0}_{\rm\scriptscriptstyle
S}K\pi})\quad]$ (3)
where $r_{D}$ is the amplitude ratio for $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}\pi^{-}$ with respect to $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{-}\pi^{+}$ and $z$ is the normalisation
factor. Analogous equations can be written for the $B^{\pm}\rightarrow
D\pi^{\pm}$ system, with $r_{B}^{\pi}$ and $\delta_{B}^{\pi}$ replacing
$r_{B}$ and $\delta_{B}$, respectively. Less interference is expected in the
$B^{\pm}\rightarrow D\pi^{\pm}$ system where the value of $r_{B}^{\pi}$ is
much lower, approximately $0.015$ [8]. These expressions receive small
corrections from mixing in the charm system which, though accounted for in
Sect. 7, are not explicitly written here. At the current level of precision
these corrections have a negligible effect on the final results.
Observables constructed using the decay rates in Eq. (3) have a sensitivity to
$\gamma$ that depends upon the value of the coherence factor, with a higher
coherence corresponding to greater sensitivity. The CLEO collaboration
measured the coherence factor and average strong phase difference in two
regions of the Dalitz plot: firstly across the whole Dalitz plot, and secondly
within a region $\pm 100{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the
$K^{*}(892)^{\pm}$ resonance, which decays to $K^{0}_{\rm\scriptscriptstyle
S}\pi^{\pm}$, where, though the sample size is diminished, the coherence is
higher. The measured values are $\kappa_{K^{0}_{\rm\scriptscriptstyle
S}K\pi}=0.73\pm 0.08$ and $\delta_{K^{0}_{\rm\scriptscriptstyle S}K\pi}=8.3\pm
15.2^{\circ}$ for the whole Dalitz plot, and
$\kappa_{K^{0}_{\rm\scriptscriptstyle S}K\pi}=1.00\pm 0.16$ and
$\delta_{K^{0}_{\rm\scriptscriptstyle S}K\pi}=26.5\pm 15.8^{\circ}$ in the
restricted region. The branching fraction ratio of $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}\pi^{-}$ to $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{-}\pi^{+}$ decays was found to be $0.592\pm
0.044$ in the whole Dalitz plot and $0.356\pm 0.034$ in the restricted region
[13].
Eight yields are measured in this analysis, from which seven observables are
constructed. The charge asymmetry is measured in each of the four decay modes,
defined as $\mathcal{A}_{\textrm{SS,
}DK}\equiv\frac{N^{DK^{-}}_{\textrm{SS}}-N^{DK^{+}}_{\textrm{SS}}}{N^{DK^{-}}_{\textrm{SS}}+N^{DK^{+}}_{\textrm{SS}}}$
for the $B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}]_{D}K^{\pm}$ mode and analogously for the other modes. The
ratios of $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$
yields are determined for the SS and OS decays, $\mathcal{R}_{DK/D\pi\textrm{,
SS}}$ and $\mathcal{R}_{DK/D\pi\textrm{, OS}}$ respectively, and the ratio of
SS to OS $B^{\pm}\rightarrow D\pi^{\pm}$ yields,
$\mathcal{R}_{\textrm{SS/OS}}$, is measured. The measurements are performed
both for the whole $D$ Dalitz plot and in the restricted region around the
$K^{*}(892)^{\pm}$ resonance.
## 3 The LHCb detector and data set
The LHCb detector [15] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4 % at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6 % at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
(IP) resolution of 20${\,\upmu\rm m}$ for tracks with large transverse
momentum. Different types of charged hadrons are distinguished by particle
identification (PID) information from two ring-imaging Cherenkov (RICH)
detectors [16]. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. The software trigger searches for a track with
large $p_{\rm T}$ and large IP with respect to any $pp$ interaction point,
also called a primary vertex (PV). This track is then required to be part of a
two-, three- or four-track secondary vertex with a high $p_{\rm T}$ sum,
significantly displaced from any PV. A multivariate algorithm [17] is used for
the identification of secondary vertices consistent with the decay of a $b$
hadron.
Samples of around two million $B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{\mp}\pi^{\pm}]_{D}\pi^{\pm}$ and two million
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{\mp}\pi^{\pm}]_{D}K^{\pm}$ decays are simulated to be used in the
analysis, along with similarly-sized samples of
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$,
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}]_{D}\pi^{\pm}$
and $B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$ decays
that are used to study potential backgrounds. In the simulation, $pp$
collisions are generated using Pythia [18, *Sjostrand:2007gs] with a specific
LHCb configuration [20]. Decays of hadronic particles are described by EvtGen
[21], in which final state radiation is generated using Photos [22]. The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit [23, *Agostinelli:2002hh] as described in
Ref. [25].
## 4 Candidate selection
Candidate $B\rightarrow[K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}]_{D}K$
and $B\rightarrow[K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}]_{D}\pi$
decays are reconstructed in events selected by the trigger and then the
candidate momenta are refit, constraining the masses of the neutral $D$ and
$K^{0}_{\rm\scriptscriptstyle S}$ mesons to their known values [26] and the
$B^{\pm}$ meson to originate from the primary vertex [27]. Candidates where
the $K^{0}_{\rm\scriptscriptstyle S}$ decay is reconstructed using “long” pion
tracks, which leave hits in the VELO and downstream tracking stations, are
analysed separately from those reconstructed using “downstream” pion tracks,
which only leave hits in tracking stations beyond the VELO. The signal
candidates in the former category are reconstructed with a better invariant
mass resolution.
The reconstructed masses of the $D$ and $K^{0}_{\rm\scriptscriptstyle S}$
mesons are required to be within 25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and 15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively, of their known
values. Candidate $B^{\pm}\rightarrow DK^{\pm}$ decays are separated from
$B^{\pm}\rightarrow D\pi^{\pm}$ decays by using PID information from the RICH
detectors. A boosted decision tree (BDT) [28, 29] that has been developed for
the analysis of the topologically similar decay mode
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{-}]_{D}h^{\prime\pm}$ is applied to the reconstructed candidates.
The BDT was trained using simulated signal decays, generated uniformly over
the $D^{0}$ Dalitz plot, and background candidates taken from the $B^{\pm}$
invariant mass region in data between 5700 and
7000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. It exploits the displacement
of tracks from the decays of long-lived particles with respect to the PV
through the use of $\chi^{2}_{\textrm{IP}}$ variables, where
$\chi^{2}_{\textrm{IP}}$ is defined as the difference in $\chi^{2}$ of a given
PV fit with and without the considered track. The BDT also employs the
$B^{\pm}$ and $D$ candidate momenta, an isolation variable sensitive to the
separation of the tracks used to construct the $B^{\pm}$ candidate from other
tracks in the event, and the $\chi^{2}$ per degree of freedom of the decay
refit. In addition to the requirement placed on the BDT response variable,
each composite candidate is required to have a vector displacement of
production and decay vertices that aligns closely to its reconstructed
momenta. The cosine of the angle between the displacement and momentum vectors
is required to be less than 0.142$\rm\,rad$ for the
$K^{0}_{\rm\scriptscriptstyle S}$ and $D^{0}$ candidates, and less than
0.0141$\rm\,rad$ (0.0100$\rm\,rad$) for long (downstream) $B^{\pm}$
candidates.
Additional requirements are used to suppress backgrounds from specific
processes. Contamination from $B$ decays that do not contain an intermediate
$D$ meson is minimised by placing a minimum threshold of 0.2${\rm\,ps}$ on the
decay time of the $D$ candidate. A potential background could arise from
processes where a pion is misidentified as a kaon or vice versa. One example
is the relatively abundant mode
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}]_{D}h^{\pm}$, which has a branching fraction around ten times
larger than the signal. These are suppressed by placing requirements on both
the $D$ daughter pion and kaon, making use of PID information. For
$K^{0}_{\rm\scriptscriptstyle S}$ candidates formed from long tracks, the
flight distance $\chi^{2}$ of the candidate is used to suppress background
from $B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}]_{D}h^{\pm}$ decays.
Where multiple candidates are found belonging to the same event, the candidate
with the lowest value of the refit $\chi^{2}$ per degree of freedom is
retained and any others are discarded, leading to a reduction in the sample
size of approximately 0.3 %.
The $B^{\pm}$ invariant mass spectra are shown in Fig. 1 for candidates
selected in the whole $D$ Dalitz plot, overlaid with a parametric fit
described in Sect. 5. The $D$ Dalitz plots are shown in Fig. 2 for the
$B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$ candidates
that fall within a nominal $B^{\pm}$ signal region in $B^{\pm}$ invariant mass
(5247–5317${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$). The dominant
$K^{*}(892)^{\pm}$ resonance is clearly visible within a horizontal band, and
the window around this resonance used in the analysis is indicated.
SS candidates
OS candidates
Figure 1: Distributions of $B^{\pm}$ invariant mass of the SS and OS samples
for the (a, c, e, g) $B^{\pm}\rightarrow DK^{\pm}$ and (b, d, f, h)
$B^{\pm}\rightarrow D\pi^{\pm}$ candidates in the full data sample. The fits
are shown for (a, b, e, f) $B^{+}$ and (c, d, g, h) $B^{-}$ candidates. Fit
PDFs are superimposed.
Figure 2: Dalitz plot distribution of candidates selected in (a) the
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle S}K\pi]_{D}K^{\pm}$ and (b)
the $B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle S}K\pi]_{D}\pi^{\pm}$
decay modes, where the data in the SS and OS modes, and the two
$K^{0}_{\rm\scriptscriptstyle S}$ categories, are combined. Candidates
included are required to have a refitted $B^{\pm}$ mass in a nominal signal
window between 5247${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
5317${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The kinematic boundary is
added in blue, and the restricted region around the $K^{*}(892)^{\pm}$
resonance is indicated by horizontal red lines.
## 5 Invariant mass fit
In order to determine the signal yields in each decay mode, simultaneous fits
are performed to the $B^{\pm}$ invariant mass spectra in the range
5110${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to
5800${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in the different modes, both
for candidates in the whole $D$ Dalitz plot, and for only those inside the
restricted region around the $K^{*}(892)^{\pm}$ resonance. The data samples
are split according to the year in which the data were taken, the decay mode,
the $K^{0}_{\rm\scriptscriptstyle S}$ type and the charge of the $B$
candidate. The fit is parameterised in terms of the observables described in
Sect. 2, rather than varying each signal yield in each category independently.
The probability density function (PDF) used to model the signal component is a
modified Gaussian function with asymmetric tails, where the unnormalised form
is given by
$f(m;m_{0},\alpha_{L},\alpha_{R},\sigma)\equiv\left\\{{\exp[-(m-m_{0})^{2}/(2\sigma^{2}+\alpha_{L}(m-m_{0})^{2})]\textrm{
for
}m<m_{0},\atop\exp[-(m-m_{0})^{2}/(2\sigma^{2}+\alpha_{R}(m-m_{0})^{2})]\textrm{
for }m>m_{0},}\right.$ (4)
where $m$ is the reconstructed mass, $m_{0}$ is the mean $B$ mass and $\sigma$
determines the width of the function. The $\alpha_{L,R}$ parameters govern the
shape of the tail. The mean $B$ mass is shared among all categories but is
allowed to differ according to the year in which the data were collected. The
$\alpha_{L}$ parameters are fixed to the values determined in the earlier
analysis of $B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}]_{D}h^{\pm}$ [6]. The $\alpha_{R}$ parameters are common to
the $B^{\pm}\rightarrow D\pi^{\pm}$ and $B^{\pm}\rightarrow DK^{\pm}$, SS and
OS categories, and are allowed to vary in the fit. Only the width parameters
$\sigma(B^{\pm}\rightarrow DK^{\pm})$ are allowed to vary in the fit. The
ratios $\sigma(B^{\pm}\rightarrow D\pi^{\pm})/\sigma(B^{\pm}\rightarrow
DK^{\pm})$ are fixed according to studies of the similar mode
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}]_{D}h^{\pm}$. The fitted values for
$\sigma(B^{\pm}\rightarrow DK^{\pm})$ vary by less than 10% around
14${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The total yield of
$B^{\pm}\rightarrow D\pi^{\pm}$ decays is allowed to vary between the
different $K^{0}_{\rm\scriptscriptstyle S}$ type and year categories. The
yields in the various $D$ decay modes and different charges, and all the
$B^{\pm}\rightarrow DK^{\pm}$ yields, are determined using the observables
described in Sect. 2, rather than being fitted directly.
In addition to the signal PDF, two background PDFs are required. The first
background PDF models candidates formed from random combinations of tracks and
is represented by a linear function. In the fit within the restricted Dalitz
region, where the sample size is significantly smaller, the slope of the
linear function fitting the $B^{\pm}\rightarrow D\pi^{\pm}$ data is fixed to
the value determined in the fit to the whole Dalitz plot. The second
background PDF accounts for contamination from partially reconstructed
processes. Given that the contamination is dominated by those processes that
involve a real $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}$ decay, the PDF is fixed to the shape determined from the
more abundant mode $B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}h^{\pm}$. The
yields of both these background components are free to vary in each data
category.
A further significant background is present in the $B^{\pm}\rightarrow
DK^{\pm}$ samples due to $\pi\rightarrow K$ misidentification of the much more
abundant $B^{\pm}\rightarrow D\pi^{\pm}$ mode. This background is modelled in
the $B^{\pm}\rightarrow DK^{\pm}$ spectrum using a Crystal Ball function [30],
where the parameters of the function are common to all data categories in the
fit and are allowed to vary. The yield of the background in the
$B^{\pm}\rightarrow DK^{\pm}$ samples is fixed with respect to the fitted
$B^{\pm}\rightarrow D\pi^{\pm}$ signal yield using knowledge of the RICH
particle identification efficiencies that is obtained from data using samples
of $D^{*\pm}\rightarrow[K\pi]_{D}\pi^{\pm}$ decays. The efficiency for kaons
to be selected is found to be around 84 % and the misidentification rate for
pions is around 4 %.
Production and detection asymmetries are accounted for, following the same
procedure as in Refs. [5, 7]. Values for the $B^{\pm}$ production and $K$
detection asymmetries are assigned such that the combination of production and
detection asymmetries corresponds to the raw asymmetry observed in
$B^{\pm}\rightarrow J/\psi K^{\pm}$ decays [31]. The detection asymmetry
assigned is $-0.5\pm 0.7\,\%$ for each unit of strangeness in the final state
to account for the differing interactions of $K^{+}$ and $K^{-}$ mesons with
the detector material. An analogous asymmetry is present for pions, though it
is expected to be much smaller, and the detection asymmetry assigned is
$0.0\pm 0.7\,\%$. Any potential asymmetry arising from a difference between
the responses of the left and right sides of the detector is minimised by
combining approximately equal data sets taken with opposite magnet polarity.
A further correction is included to account for non-uniformities in the
acceptance over the Dalitz plot. This efficiency correction primarily affects
the $\mathcal{R}_{\textrm{SS/OS}}$ observable, given the difference in the
Dalitz distributions for the two $D$ meson decay modes. A correction factor,
$\zeta$, is found by combining the LHCb acceptance, extracted from the
simulated signal sample, and amplitude models, $A_{\textrm{SS,
OS}}(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle S}\pi})$, for the Dalitz
distributions of the SS or OS decays,
$\zeta\equiv\frac{\int_{\mathcal{D}}\textrm{d}m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K}\textrm{d}m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}\pi}[\epsilon(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}\pi})\times|A_{\textrm{OS}}(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}\pi})|^{2}]}{\int_{\mathcal{D}}\textrm{d}m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K}\textrm{d}m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}\pi}[\epsilon(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}\pi})\times|A_{\textrm{SS}}(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle S}\pi})|^{2}]},$ (5)
where $\epsilon(m^{2}_{K^{0}_{\rm\scriptscriptstyle
S}K},m^{2}_{K^{0}_{\rm\scriptscriptstyle S}\pi})$ is the efficiency at a point
in the Dalitz plot. The typical deviation of $\zeta$ from unity is found to be
around 5 %. The acceptance is illustrated in Fig. 3, where bins of variable
size are used to ensure that statistical fluctuations due to the finite size
of the simulated sample are negligible. The Dalitz distributions are
determined using the fact that little interference is expected in
$B^{\pm}\rightarrow D\pi^{\pm}$ decays and, therefore, the flavour of the $D$
meson is effectively tagged by the charge of the pion. In this case, the
Dalitz distributions are given by considering the relevant $D^{0}$ decay
($D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{-}\pi^{+}$ for SS and
$D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}\pi^{-}$ for OS). These
$D^{0}$ decay Dalitz distributions are known and amplitude models from CLEO
are available [13] from which the Dalitz distributions can be extracted.
Figure 3: Dalitz acceptance determined using simulated events and normalised
relative to the maximum efficiency.
Due to the restricted sample size under study, small biases exist in the
determination of the observables. The biases are determined by generating and
fitting a large number of simulated samples using input values obtained from
the fit to data, and are typically found to be around 2 %. The fit results are
corrected accordingly.
The fit projections, with long and downstream $K^{0}_{\rm\scriptscriptstyle
S}$-type categories merged and 2011 and 2012 data combined, are given for the
fit to the whole Dalitz plot in Fig. 1. The signal purity in a nominal mass
range from $5247{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to
5317${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is around 85 % for the
$B^{\pm}\rightarrow DK^{\pm}$ samples and 96 % for the $B^{\pm}\rightarrow
D\pi^{\pm}$ samples. The signal yields derived from the fits to both the whole
and restricted region of the Dalitz plot are given in Table 1. The fitted
values of the observables are given in Table 2, including their systematic
uncertainties as discussed in Sect. 6. The only significant difference between
the observables fitted in the two regions is for the value of
$\mathcal{R}_{\textrm{SS/OS}}$. This ratio is expected to differ
significantly, given that the fraction of $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{-}\pi^{+}$ decays that are expected to lie
inside the restricted portion of the Dalitz plot is around 75 %, whereas for
$D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}\pi^{-}$ the fraction is
around 44 % [13]. This accounts for the higher value of
$\mathcal{R}_{\textrm{SS/OS}}$ in the restricted region. The ratios between
the $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$ yields
are consistent with that measured in the LHCb analysis of
$B^{\pm}\rightarrow[K\pi]_{D}h^{\pm}$, $0.0774\pm 0.0012\pm 0.0018$ [5]. The
$C\\!P$ asymmetries are consistent with zero in the $B^{\pm}\rightarrow
D\pi^{\pm}$ system, where the effect of interference is expected to be small.
The asymmetries in the $B^{\pm}\rightarrow DK^{\pm}$ system,
$\mathcal{A}_{\textrm{SS, }DK}$ and $\mathcal{A}_{\textrm{OS, }DK}$, which
have the highest sensitivity to $\gamma$ are all compatible with zero at the
$2\sigma$ level. The correlations between $\mathcal{R}_{\textrm{SS/OS}}$ ratio
and the ratios $\mathcal{R}_{DK/D\pi\textrm{, SS}}$ and
$\mathcal{R}_{DK/D\pi\textrm{, OS}}$ are $-16\,\%$ ($-13\,\%$) and $+16\,\%$
($+16\,\%$), respectively, for the fit to the whole Dalitz plot
($K^{*}(892)^{\pm}$ region). The correlation between the
$\mathcal{R}_{DK/D\pi\textrm{, SS}}$ and $\mathcal{R}_{DK/D\pi\textrm{, OS}}$
ratios is $+11\,\%$ ($+15\,\%$). Correlations between the asymmetry
observables are all less than 1 % and are neglected.
Table 1: Signal yields and their statistical uncertainties derived from the fit to the whole Dalitz plot region, and in the restricted region of phase space around the $K^{*}(892)^{\pm}$ resonance. | Whole Dalitz plot | $K^{*}(892)^{\pm}$ region
---|---|---
Mode | $DK^{\pm}$ | $D\pi^{\pm}$ | $DK^{\pm}$ | $D\pi^{\pm}$
SS | 145 $\pm$ | 15 | 1841 $\pm$ | 47 | 97 $\pm$ | 12 | 1365 $\pm$ | 38
OS | 71 $\pm$ | 10 | 1267 $\pm$ | 37 | 26 $\pm$ | 6 | 553 $\pm$ | 24
Table 2: Results for the observables measured in the whole Dalitz plot region, and in the restricted region of phase space around the $K^{*}(892)^{\pm}$ resonance. The first uncertainty is statistical and the second is systematic. The corrections for production and detection asymmetries are applied, as is the efficiency correction defined in Eq. (5). Observable | Whole Dalitz plot | $K^{*}(892)^{\pm}$ region
---|---|---
$\mathcal{R}_{\textrm{SS/OS}}$ | 1.528 $\pm$ | 0.058 $\pm$ | 0.025 | 2.57 $\pm$ | 0.13 $\pm$ | 0.06
$\mathcal{R}_{DK/D\pi\textrm{, SS}}$ | 0.092 $\pm$ | 0.009 $\pm$ | 0.004 | 0.084 $\pm$ | 0.011 $\pm$ | 0.003
$\mathcal{R}_{DK/D\pi\textrm{, OS}}$ | 0.066 $\pm$ | 0.009 $\pm$ | 0.002 | 0.056 $\pm$ | 0.013 $\pm$ | 0.002
$\mathcal{A}_{\textrm{SS, }DK}$ | 0.040 $\pm$ | 0.091 $\pm$ | 0.018 | 0.026 $\pm$ | 0.109 $\pm$ | 0.029
$\mathcal{A}_{\textrm{OS, }DK}$ | 0.233 $\pm$ | 0.129 $\pm$ | 0.024 | 0.336 $\pm$ | 0.208 $\pm$ | 0.026
$\mathcal{A}_{\textrm{SS, }D\pi}$ | $-0.025$ $\pm$ | 0.024 $\pm$ | 0.010 | $-0.012$ $\pm$ | 0.028 $\pm$ | 0.010
$\mathcal{A}_{\textrm{OS, }D\pi}$ | $-0.052$ $\pm$ | 0.029 $\pm$ | 0.017 | $-0.054$ $\pm$ | 0.043 $\pm$ | 0.017
## 6 Systematic uncertainties
The largest single source of systematic uncertainty is the knowledge of the
efficiency correction factor that multiplies the
$\mathcal{R}_{\textrm{SS/OS}}$ observable. This uncertainty has three sources:
the uncertainties on the CLEO amplitude models, the granularity of the Dalitz
divisions in which the acceptance is determined, and the limited size of the
simulated sample available to determine the LHCb acceptance. Of these, it is
the modelling uncertainty that is dominant. In addition, an uncertainty is
assigned to account for the fact that interference is neglected in the
computation of the efficiency correction factor, which is shared between the
$D\pi^{\pm}$ and $DK^{\pm}$ systems.
Uncertainties on the parameters that are fixed in the PDF are propagated to
the observables by repeating the fit to data whilst varying each fixed
parameter according to its uncertainty. An additional systematic uncertainty
is calculated for the fit to the restricted $K^{*}(892)^{\pm}$ region, where
the $D\pi^{\pm}$ combinatorial background slopes are fixed to the values
determined in the fit to the whole Dalitz plot.
Uncertainties are assigned to account for the errors on the $B^{\pm}$
production asymmetry and the $K^{\pm}$ and $\pi^{\pm}$ detection asymmetries.
The effect of the detection asymmetry depends on the pion and kaon content of
the final state, and the resulting systematic uncertainty is largest for the
$\mathcal{A}_{\textrm{SS, }DK}$ and $\mathcal{A}_{\textrm{OS, }D\pi}$
observables.
The absolute uncertainties on the particle identification efficiencies are
small, typically around 0.3 % for kaon efficiencies and 0.03 % for pion
efficiencies. Of the four main sources of systematic error, these result in
the smallest uncertainties on the experimental observables.
In Table 3, the sources of systematic uncertainty are given for each
observable in the fit to the whole Dalitz plot. Similarly those for the fit in
the restricted region are given in Table 4.
Table 3: Absolute values of systematic uncertainties, in units of $10^{-2}$, for the fit to the whole Dalitz plot. Observable | Eff. correction | Fit PDFs | Prod. and det. asymms. | PID | Total
---|---|---|---|---|---
$\mathcal{R}_{\textrm{SS/OS}}$ | 2.40 | 0.50 | $-$ | 0.01 | 2.45
$\mathcal{R}_{DK/D\pi\textrm{, SS}}$ | 0.01 | 0.38 | $-$ | 0.02 | 0.38
$\mathcal{R}_{DK/D\pi\textrm{, OS}}$ | 0.01 | 0.19 | $-$ | 0.01 | 0.19
$\mathcal{A}_{\textrm{SS, }DK}$ | 0.14 | 0.44 | 1.71 | 0.01 | 1.78
$\mathcal{A}_{\textrm{OS, }DK}$ | 0.36 | 2.13 | 0.99 | 0.01 | 2.37
$\mathcal{A}_{\textrm{SS, }D\pi}$ | 0.02 | 0.05 | 0.99 | $<0.01$ | 0.99
$\mathcal{A}_{\textrm{OS, }D\pi}$ | 0.03 | 0.10 | 1.71 | $<0.01$ | 1.72
Table 4: Absolute values of systematic uncertainties, in units of $10^{-2}$, for the fit in the restricted region. Observable | Eff. correction | Fit PDFs | Prod. and det. asymms. | PID | Total
---|---|---|---|---|---
$\mathcal{R}_{\textrm{SS/OS}}$ | 6.08 | 0.53 | $-$ | 0.01 | 6.10
$\mathcal{R}_{DK/D\pi\textrm{, SS}}$ | 0.01 | 0.25 | $-$ | 0.02 | 0.25
$\mathcal{R}_{DK/D\pi\textrm{, OS}}$ | 0.01 | 0.21 | $-$ | 0.01 | 0.21
$\mathcal{A}_{\textrm{SS, }DK}$ | 0.13 | 2.27 | 1.71 | 0.01 | 2.85
$\mathcal{A}_{\textrm{OS, }DK}$ | 0.04 | 2.38 | 0.99 | 0.01 | 2.57
$\mathcal{A}_{\textrm{SS, }D\pi}$ | 0.04 | 0.17 | 0.99 | $<0.01$ | 1.00
$\mathcal{A}_{\textrm{OS, }D\pi}$ | 0.06 | 0.09 | 1.71 | $<0.01$ | 1.72
## 7 Interpretation and conclusions
The sensitivity of this result to the CKM angle $\gamma$ is investigated by
employing a frequentist method to scan the $\gamma-r_{B}$ parameter space and
calculate the $\chi^{2}$ probability at each point, given the measurements of
the observables with their statistical and systematic uncertainties combined
in quadrature, accounting for correlations between the statistical
uncertainties. The effects of charm mixing are accounted for, but $C\\!P$
violation in the decays of $D$ mesons is neglected. Regions of $1\sigma$,
$2\sigma$ and $3\sigma$ compatibility with the measurements made are indicated
by the dark, medium and light blue regions, respectively, in Fig. 4. The small
sample size in the current data set results in a bound on $\gamma$ that is
only closed for the $1\sigma$ contour.
Figure 4: Scans of the $\chi^{2}$ probabilities over the $\gamma-r_{B}$
parameter space for (a) the whole Dalitz fit and (b) the fit inside the
$K^{*}$ region (b). The contours are the usual $n\sigma$ profile likelihood
contours, where $\Delta\chi^{2}=n^{2}$ with $n=1\textrm{ (dark blue),
}2\textrm{ (medium blue), and }3\textrm{ (light blue)}$. The $2\sigma$ contour
encloses almost all of the parameter space shown, so a central value of
$\gamma$ and relevant bounds are not extracted. The result is seen to be
compatible with the current LHCb measurement of $\gamma$, indicated by the
point at ($\gamma=72.0^{\circ}$ and $r_{B}=0.089$), at a level between 1 and
$2\sigma$.
Although it is not possible to measure $\gamma$ directly using these results
alone, it is of interest to consider how this result relates to the previous
LHCb $\gamma$ determination, obtained from other $B^{\pm}\rightarrow DK^{\pm}$
modes [8], since it will be included in future combinations. In order to aid
this comparison, the scans of the $\gamma-r_{B}$ space plots are shown in Fig.
4(a) for the measurement made using the whole $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K\pi$ Dalitz plot and in Fig. 4(b) for that
made in the restricted region. The current LHCb average, extracted from a
combination of $B^{\pm}\rightarrow DK^{\pm}$ analyses [8], is shown as a point
with error bars at $\gamma=72.0^{\circ}$ and $r_{B}=0.089$. The LHCb average
lies within the $2\sigma$ region allowed by the measurements presented in this
Letter. It is interesting to note that the bound determined in the $\gamma-
r_{B}$ space indicates a more stringent constraint when using the restricted
region, where the coherence is higher. This, and the fact that the
measurements in this Letter are limited by their statistical precision,
motivates the use of this region in future analyses of these decays in a
larger data sample. Combination with analyses in other, more abundant channels
with sensitivity to the same parameters will yield more stringent constraints
upon $\gamma$.
In summary, for the first time a measurement of charge asymmetries and
associated observables is presented in the decay modes
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}]_{D}h^{\pm}$ and
$B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle
S}K^{\mp}\pi^{\pm}]_{D}h^{\pm}$, and no significant $C\\!P$ violation is
observed. The results of the analysis are consistent with other measurements
of observables in related $B^{\pm}\rightarrow DK^{\pm}$ modes, and will be
valuable in future global fits of the CKM parameter $\gamma$.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages we depend on. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia).
## References
* [1] M. Gronau and D. London, How to determine all the angles of the unitarity triangle from $B_{d}^{0}\rightarrow DK^{0}_{\rm\scriptscriptstyle S}$ and $B^{0}_{s}\rightarrow D\phi$, Phys. Lett. B253 (1991) 483
* [2] M. Gronau and D. Wyler, On determining a weak phase from charged $B$ decay asymmetries, Phys. Lett. B265 (1991) 172
* [3] D. Atwood, I. Dunietz, and A. Soni, Enhanced $C\\!P$ violation with $B\rightarrow KD^{0}(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0})$ modes and extraction of the CKM angle $\gamma$, Phys. Rev. Lett. 78 (1997) 3257, arXiv:hep-ph/9612433
* [4] D. Atwood, I. Dunietz, and A. Soni, Improved methods for observing $C\\!P$ violation in $B^{\pm}\rightarrow K^{\pm}D$ and measuring the CKM phase $\gamma$, Phys. Rev. D63 (2001) 036005, arXiv:hep-ph/0008090
* [5] LHCb collaboration, R. Aaij et al., Observation of $C\\!P$ violation in $B^{\pm}\rightarrow DK^{\pm}$ decays, Phys. Lett. B712 (2012) 203, arXiv:1203.3662
* [6] LHCb collaboration, R. Aaij et al., A model-independent Dalitz plot analysis of $B^{\pm}\rightarrow DK^{\pm}$ with $D\rightarrow K^{0}_{\rm S}h^{+}h^{-}$ ($h=\pi,K$) decays and constraints on the CKM angle $\gamma$, Phys. Lett. B718 (2012) 43, arXiv:1209.5869
* [7] LHCb collaboration, R. Aaij et al., Observation of the suppressed ADS modes $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}K^{\pm}$ and $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$, Phys. Lett. B723 (2013) 44, arXiv:1303.4646
* [8] LHCb collaboration, R. Aaij et al., A measurement of $\gamma$ from a combination of $B^{\pm}\rightarrow Dh^{\pm}$ analyses, Phys. Lett. B726 (2013) 151, arXiv:1305.2050
* [9] BaBar collaboration, J. Lees et al., Observation of direct CP violation in the measurement of the Cabibbo-Kobayashi-Maskawa angle $\gamma$ with $B^{\pm}\rightarrow D^{(*)}K^{(*)\pm}$ decays, Phys. Rev. D87 (2013) 052015, arXiv:1301.1029
* [10] Belle collaboration, K. Trabelsi, Study of direct CP in charmed B decays and measurement of the CKM angle gamma at Belle, arXiv:1301.2033, presented at CKM2012, Cincinnati, USA, 28\. Sep.–2. Oct. 2012
* [11] LHCb collaboration, R. Aaij et al., Measurement of $C\\!P$ observables in $B^{0}\rightarrow DK^{*0}$ with $D\rightarrow K^{+}K^{-}$, JHEP 03 (2013) 67, arXiv:1212.5205
* [12] Y. Grossman, Z. Ligeti, and A. Soffer, Measuring $\gamma$ in $B^{\pm}\rightarrow K^{\pm}(KK^{*})_{D}$ decays, Phys. Rev. D67 (2003) 071301(R), arXiv:hep-ph/0210433
* [13] CLEO collaboration, J. Insler et al., Studies of the decays $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$ and $D^{0}\rightarrow K_{S}^{0}K^{+}\pi^{-}$, Phys. Rev. D85 (2012) 092016, arXiv:1203.3804
* [14] D. Atwood and A. Soni, Role of a charm factory in extracting CKM-phase information via $B\rightarrow DK$, Phys. Rev. D68 (2003) 033003, arXiv:hep-ph/0304085
* [15] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [16] M. Adinolfi et al., Performance of the LHCb RICH detector at the LHC, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
* [17] V. V. Gligorov and M. Williams, Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree, JINST 8 (2013) P02013, arXiv:1210.6861
* [18] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [19] T. Sjöstrand, S. Mrenna, and P. Skands, A brief introduction to PYTHIA 8.1, Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
* [20] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [21] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [22] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [23] Geant4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [24] Geant4 collaboration, S. Agostinelli et al., Geant4: a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [25] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. Phys. Conf. Ser. 331 (2011) 032023
* [26] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001, and 2013 partial update for the 2014 edition
* [27] W. D. Hulsbergen, Decay chain fitting with a Kalman filter, Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
* [28] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
* [29] R. E. Schapire and Y. Freund, A decision-theoretic generalization of on-line learning and an application to boosting, Jour. Comp. and Syst. Sc. 55 (1997) 119
* [30] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
* [31] LHCb collaboration, R. Aaij et al., Measurements of the branching fractions and $C\\!P$ asymmetries of $B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}$ and $B^{\pm}\rightarrow\psi{(2S)}\pi^{\pm}$ decays, Phys. Rev. D85 (2012) 091105(R), arXiv:1203.3592
|
arxiv-papers
| 2014-02-12T21:00:31 |
2024-09-04T02:49:58.179935
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. Anderlini,\n J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, M. Borsato, T.J.V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto,\n J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M. Calvi, M. Calvo Gomez,\n A. Camboni, P. Campana, D. Campora Perez, A. Carbone, G. Carboni, R.\n Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G.\n Casse, L. Cassina, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M.\n Charles, Ph. Charpentier, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba,\n X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, I.\n Counts, B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R.\n Currie, C. D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n S. Donleavy, F. Dordei, M. Dorigo, P. Dorosz, A. Dosil Su\\'arez, D. Dossett,\n A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger,\n R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, S. Esen, A. Falabella, C.\n F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, J. Fu, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, J. Garra Tico, L. Garrido, C.\n Gaspar, R. Gauld, L. Gavardi, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, A. Gianelle, S. Giani', V. Gibson, L. Giubega, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, L. Grillo, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen,\n T.W. Hafkenscheid, S.C. Haines, S. Hall, B. Hamilton, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He,\n T. Head, V. Heijne, K. Hennessy, P. Henrard, L. Henry, J.A. Hernando Morata,\n E. van Herwijnen, M. He\\ss, A. Hicheur, D. Hill, M. Hoballah, C. Hombach, W.\n Hulsbergen, P. Hunt, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M.\n Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F.\n Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, N. Jurik, M.\n Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, M. Kelsey, I.R.\n Kenyon, T. Ketel, B. Khanji, C. Khurewathanakul, S. Klaver, O. Kochebina, I.\n Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, M. Liles, R. Lindner, C. Linn, F. Lionetto, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, P. Lowdon, H. Lu, D.\n Lucchesi, J. Luisier, H. Luo, E. Luppi, O. Lupton, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, M.\n Manzali, J. Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, F. Martinez Vidal, D. Martins Tostes, A. Massafferri, R. Matev, Z.\n Mathe, C. Matteuzzi, A. Mazurov, M. McCann, J. McCarthy, A. McNab, R.\n McNulty, B. McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A.\n Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, M.\n Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, F. Muheim, K.\n M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar,\n I. Nasteva, M. Needham, N. Neri, S. Neubert, N. Neufeld, A.D. Nguyen, T.D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem,\n A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, G. Onderwater, M. Orlandea, J.M. Otalora Goicochea,\n P. Owen, A. Oyanguren, B.K. Pal, A. Palano, F. Palombo, M. Palutan, J.\n Panman, A. Papanestis, M. Pappagallo, L. Pappalardo, C. Parkes, C.J.\n Parkinson, G. Passaleva, G.D. Patel, M. Patel, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pearce, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, E. Perez Trigo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, A. Pistone, S. Playfer, M. Plo\n Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov,\n B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C.\n Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H.\n Rademacker, B. Rakotomiaramanana, M. Rama, M.S. Rangel, I. Raniuk, N.\n Rauschmayr, G. Raven, S. Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P.\n Robbe, D.A. Roberts, A.B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S.\n Roiser, V. Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F.\n Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, M. Schiller, H. Schindler, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, G. Simi, M. Sirendi, N. Skidmore,\n T. Skwarnicki, N.A. Smith, E. Smith, E. Smith, J. Smith, M. Smith, H. Snoek,\n M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B.\n Spaan, A. Sparkes, F. Spinella, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S. Stracka, M.\n Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah, L. Sun, W. Sutcliffe, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, D. Szilard, T. Szumlak,\n S. T'Jampens, M. Teklishyn, G. Tellarini, E. Teodorescu, F. Teubert, C.\n Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, L.\n Tomassetti, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti, A.\n Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez Sierra, S.\n Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D.\n Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, J.A. de Vries, R. Waldi, C.\n Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley,\n J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Daniel Johnson",
"url": "https://arxiv.org/abs/1402.2982"
}
|
1402.3156
|
# Second main theorems for meromorphic mappings intersecting moving
hyperplanes with truncated counting functions and unicity problem
Si Duc Quang Department of Mathematics, Hanoi National University of
Education
136-Xuan Thuy, Cau Giay, Hanoi, Vietnam Email: [email protected]
###### Abstract.
In this article, we show some new second main theorems for the mappings and
moving hyperplanes of ${\mathbf{P}}^{n}({\mathbf{C}})$ with truncated counting
functions. Our results are improvements of recent previous second main
theorems for moving hyperplanes with the truncated (to level $n$) counting
functions. As their application, we prove a unicity theorem for meromorphic
mappings sharing moving hyperplanes.
††footnotetext: 2010 Mathematics Subject Classification: Primary 32H30,
32A22; Secondary 30D35.
Key words and phrases: Nevanlinna, second main theorem, meromorphic mapping,
moving hyperplane.
## 1\. Introduction
The theory of the Nevanlinna’s second main theorem for meromorphic mappings of
${\mathbf{C}}^{m}$ into the complex projective space
${\mathbf{P}}^{n}({\mathbf{C}})$ intersecting a finite set of fixed
hyperplanes or moving hyperplanes in ${\mathbf{P}}^{n}({\mathbf{C}})$ was
started about 70 years ago and has grown into a huge theory. For the case of
fixed hyperplanes, maybe, the second main theorem given by Cartan-Nochka is
the best possible. Unfortunately, so far there has been a few second main
theorems with truncated counting functions for moving hyperplanes. Moreover,
almost of them are not sharp.
We state here some recent results on the second main theorems for moving
hyperplanes with truncated counting functions.
Let $\\{a_{i}\\}_{i=1}^{q}$ be meromorphic mappings of ${\mathbf{C}}^{m}$ into
the dual space ${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ in general position. For
the case of nondegenerate meromorphic mappings, the second main theorem with
truncated (to level $n$) counting functions states that.
Theorem A (see [4, Theorem 2.3] and [6, Theorem 3.1]). Let
$f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a meromorphic
mapping. Let $\\{a_{i}\\}_{i=1}^{q}\ (q\geq n+2)$ be meromorphic mappings of
${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ in general
position such that $f$ is linearly nondegenerate over
$\mathcal{R}(\\{a_{i}\\}_{i=1}^{q}).$ Then
$||\
\dfrac{q}{n+2}T_{f}(r)\leq\sum_{i=1}^{q}N_{(f,a_{i})}^{[n]}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)).$
We note that, Theorem A is still the best second main theorem with truncated
counting functions for nondegenerate meromorphic mappings and moving
hyperplanes available at present. In the case of degenerate meromorphic
mappings, the second main theorem for moving hyperplanes with counting
function truncated to level $n$ was first given by M. Ru-J. Wang [5] in 2004.
After that in 2008, D. D. Thai-S. D. Quang [7] improved the result of M. Ru-J.
Wang by proved the following second main theorem.
Theorem B (see [7, Corollary 1]). Let
$f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a meromorphic
mapping. Let $\\{a_{i}\\}_{i=1}^{q}$ $(q\geq 2n+1)$ be $q$ meromorphic
mappings of ${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ in
general position such that $(f,a_{i})\not\equiv 0\ (1\leq i\leq{q}).$ Then
$\bigl{|}\bigl{|}\quad\dfrac{q}{2n+1}\cdot
T_{f}(r)\leq\sum_{i=1}^{q}N^{[n]}_{(f,a_{i})}(r)+O\bigl{(}\max_{1\leq
i\leq{q}}T_{a_{i}}(r)\bigl{)}+O\bigl{(}\log^{+}T_{f}(r)\bigl{)}.$
These results play very essential roles in almost all researches on truncated
multiplicity problems of meromorphic mappings with moving hyperplanes.
Hovewer, in our opinion, the above mentioned results of these authors are
still weak.
Our main purpose of the present paper is to show a stronger second main
theorem of meromorphic mappings from ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$ for moving targets. Namely, we will prove the
following.
###### Theorem 1.1.
Let $f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a meromorphic
mapping. Let $\\{a_{i}\\}_{i=1}^{q}\ (q\geq 2n-k+2)$ be meromorphic mappings
of ${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ in general
position such that $(f,a_{i})\not\equiv 0\ (1\leq i\leq q),$ where
$k+1=\mathrm{rank}_{\mathcal{R}\\{a_{i}\\}}(f)$. Then the following assertions
hold:
$\displaystyle\mathrm{(a)}\ ||\
\dfrac{q}{2n-k+2}T_{f}(r)\leq\sum_{i=1}^{q}N_{(f,a_{i})}^{[k]}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)),$ $\displaystyle\mathrm{(b)}\ ||\
\dfrac{q-n+2k-1}{n+k+1}T_{f}(r)\leq\sum_{i=1}^{q}N_{(f,a_{i})}^{[k]}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)).$
We may see that Theorem 1.1(a) is a generalization of Theorem A and also is an
improvement of Theorem B. Theorem 1.1(b) is really stronger than Theorem B.
Remark.
1) If $k\geq\dfrac{n+1}{2}$ then Theorem 1.1(a) is stronger than Theorem
1.1(b). Otherwise, if $k<\dfrac{n+1}{2}$ then Theorem 1.1(b) is stronger than
Theorem 1.1(a).
2) If $k=0$ then $f$ is constant map, and hence $T_{f}(r)=0.$
3) Setting $t=\frac{2n-k+2}{3n+3}$ and $\lambda=\frac{n+k+1}{3n+3},$ we have
$t+\lambda=1$. Thus, for all $1\leq k\leq n$ we have
$\displaystyle\max\biggl{\\{}\dfrac{q}{2n-k+2},\dfrac{q-n+2k-1}{n+k+1}\biggl{\\}}$
$\displaystyle\geq\dfrac{q}{2n-k+2}\cdot
t+\dfrac{q-n+2k-1}{n+k+1}\cdot\lambda$
$\displaystyle=\dfrac{2q-n+2k-1}{3n+3}\geq\dfrac{2q-n+1}{3n+3}.$
4) If $k\geq 1$, we have the following estimates:
* •
$\min_{\frac{n+1}{2}\leq k\leq
n,(k\in\mathbf{Z})}\left(\dfrac{q}{2n-k+2}\right)\geq\dfrac{q}{2n-\frac{n+1}{2}+2}=\dfrac{2q}{3(n+1)}$.
* •
$\min_{1\leq
k\leq\frac{n+1}{2},(k\in\mathbf{Z})}\left(\dfrac{q-n+2k-1}{n+k+1}\right)=\min_{1\leq
k\leq\frac{n+1}{2},(k\in\mathbf{Z})}\left(\dfrac{q-3n-3}{n+k+1}+2\right)$
$\geq\begin{cases}\dfrac{2q}{3(n+1)}&\text{ if }q\geq 3n+3\\\
\dfrac{q-n+1}{n+2}&\text{ if }q<3n+3\end{cases}$
Thus
$\min_{1\leq k\leq
n}\biggl{\\{}\max\bigl{\\{}\dfrac{q}{2n-k+2},\dfrac{q-n+2k-1}{n+k+1}\bigl{\\}}\biggl{\\}}\geq\begin{cases}\dfrac{2q}{3(n+1)}&\text{
if }q\geq 3n+3\\\ \dfrac{q-n+1}{n+2}&\text{ if }q<3n+3.\end{cases}$
Therefore, from Theorem 1.1 and Remark (1-4) we have the following corollary.
###### Corollary 1.2.
Let $f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a meromorphic
mapping. Let $\\{a_{i}\\}_{i=1}^{q}\ (q\geq 2n+1)$ be meromorphic mappings of
${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ in general
position such that $(f,a_{i})\not\equiv 0\ (1\leq i\leq q).$
$\mathrm{(a)}$ Then we have
$||\dfrac{2q-n+1}{3(n+1)}T_{f}(r)\leq\sum_{i=1}^{q}N_{(f,a_{i})}^{[n]}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)).$
$\mathrm{(b)}$ If $q\geq 3n+3$ then
$||\dfrac{2q}{3(n+1)}T_{f}(r)\leq\sum_{i=1}^{q}N_{(f,a_{i})}^{[n]}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)).$
$\mathrm{(c)}$ If $q<3n+3$ then
$||\dfrac{q-n+1}{n+2}T_{f}(r)\leq\sum_{i=1}^{q}N_{(f,a_{i})}^{[n]}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)).$
As applications of these second main theorems, in the last section we will
prove a unicity theorem for meromorphic mappings sharing moving hyperplanes
regardless of multiplicities. To state our main result, we give the following
definition.
Let $f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a meromorphic
mapping. Let $k$ be a positive integer or maybe $+\infty$. Let
$\\{a_{i}\\}_{i=1}^{q}$ be “slowly” (with respect to $f$) moving hyperplanes
in ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position such that
$\dim\ \\{z\in{\mathbf{C}}^{m}:(f,a_{i})(z)\cdot(f,a_{j})(z)=0\\}\leq
m-2\quad(1\leq i<j\leq q).$
Consider the set $\mathcal{F}(f,\\{a_{i}\\}_{i=1}^{q},k)$ of all meromorphic
maps $g:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ satisfying the
following two conditions:
(a) $\min\\{\nu_{(f,a_{i})}(z),k\\}=\min\\{\nu_{(g,a_{i})}(z),k\\}\quad(1\leq
i\leq q),$ for all $z\in{\mathbf{C}}^{m}$,
(b) $f(z)=g(z)$ for all $z\in\bigcup_{i=1}^{q}\mathrm{Zero}(f,a_{i})$.
We wil prove the following
###### Theorem 1.3.
Let $f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a meromorphic
mapping. Let $\\{a_{i}\\}_{i=1}^{q}$ be slowly (with respect to $f$) moving
hyperplanes in ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position such that
$\dim\ \\{z\in{\mathbf{C}}^{m}:(f,a_{i})(z)\cdot(f,a_{j})(z)=0\\}\leq
m-2\quad(1\leq i<j\leq q).$
Then the following assertions hold:
a) If $q>\frac{9n^{2}+9n+4}{4}$ then $\sharp\
\mathcal{F}(f,\\{a_{i}\\}_{i=1}^{q},1)\leq 2,$
b) If $q>3n^{2}+n+2$ then $\sharp\ \mathcal{F}(f,\\{a_{i}\\}_{i=1}^{q},1)=1.$
Acknowledgements. This work was done during a stay of the author at Vietnam
Institute for Advanced Study in Mathematics. He would like to thank the
institute for their support.
## 2\. Basic notions and auxiliary results from Nevanlinna theory
(a) Counting function of divisor.
For $z=(z_{1},\dots,z_{m})\in{\mathbf{C}}^{m}$, we set
$\|z\|=\Big{(}\sum\limits_{j=1}^{m}|z_{j}|^{2}\Big{)}^{1/2}$ and define
$\displaystyle B(r)$ $\displaystyle=\\{z\in{\mathbf{C}}^{m};\|z\|<r\\},\quad
S(r)=\\{z\in{\mathbf{C}}^{m};\|z\|=r\\},$ $\displaystyle d^{c}$
$\displaystyle=\dfrac{\sqrt{-1}}{4\pi}(\overline{\partial}-\partial),\quad\sigma=\big{(}dd^{c}\|z\|^{2}\big{)}^{m-1},$
$\displaystyle\eta$
$\displaystyle=d^{c}\text{log}\|z\|^{2}\land\big{(}dd^{c}\text{log}\|z\|\big{)}^{m-1}.$
Thoughout this paper, we denote by $\mathcal{M}$ the set of all meromorphic
functions on ${\mathbf{C}}^{m}$. A divisor $E$ on ${\mathbf{C}}^{m}$ is given
by a formal sum $E=\sum\mu_{\nu}X_{\nu}$, where $\\{X_{\nu}\\}$ is a locally
family of distinct irreducible analytic hypersurfaces in ${\mathbf{C}}^{m}$
and $\mu_{\nu}\in\mathbf{Z}$. We define the support of the divisor $E$ by
setting $\mathrm{Supp}\,(E)=\cup_{\nu\neq 0}X_{\nu}$. Sometimes, we identify
the divisor $E$ with a function $E(z)$ from ${\mathbf{C}}^{m}$ into
$\mathbf{Z}$ defined by $E(z):=\sum_{X_{\nu}\ni z}\mu_{\nu}$.
Let $k$ be a positive integer or $+\infty$. We define the truncated divisor
$E^{[k]}$ by
$E^{[k]}:=\sum_{\nu}\min\\{\mu_{\nu},k\\}X_{\nu},$
and the truncated counting function to level $k$ of $E$ by
$\displaystyle
N^{[k]}(r,E):=\int\limits_{1}^{r}\frac{n^{[k]}(t,E)}{t^{2m-1}}dt\quad(1<r<+\infty),$
where
$\displaystyle n^{[k]}(t,E):=\begin{cases}\int\limits_{\mathrm{Supp}\,(E)\cap
B(t)}E^{[k]}\sigma&\text{ if }m\geq 2,\\\ \sum_{|z|\leq t}E^{[k]}(z)&\text{ if
}m=1.\end{cases}$
We omit the character [k] if $k=+\infty$.
For an analytic hypersurface $E$ of ${\mathbf{C}}^{m}$, we may consider it as
a reduced divisor and denote by $N(r,E)$ its counting function.
Let $\varphi$ be a nonzero meromorphic function on ${\mathbf{C}}^{m}$. We
denote by $\nu^{0}_{\varphi}$ (resp. $\nu^{\infty}_{\varphi}$) the divisor of
zeros (resp. divisor of poles) of $\varphi$. The divisor of $\varphi$ is
defined by
$\nu_{\varphi}=\nu^{0}_{\varphi}-\nu^{\infty}_{\varphi}.$
We have the following Jensen’s formula:
$\displaystyle
N(r,\nu^{0}_{\varphi})-N(r,\nu^{\infty}_{\varphi})=\int\limits_{S(r)}\text{log}|\varphi|\eta-\int\limits_{S(1)}\text{log}|\varphi|\eta.$
For convenience, we will write $N_{\varphi}(r)$ and $N^{[k]}_{\varphi}(r)$ for
$N(r,\nu^{0}_{\varphi})$ and $N^{[k]}(r,\nu^{0}_{\varphi})$, respectively.
(b) The first main theorem.
Let $f$ be a meromorphic mapping of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$. For arbitrary fixed homogeneous coordinates
$(w_{0}:\cdots:w_{n})$ of ${\mathbf{P}}^{n}({\mathbf{C}})$, we take a reduced
representation $f=(f_{0}:\cdots:f_{n})$, which means that each $f_{i}$ is
holomorphic function on ${\mathbf{C}}^{m}$ and
$f(z)=(f_{0}(z):\cdots:f_{n}(z))$ outside the analytic set
$I(f):=\\{z;f_{0}(z)=\cdots=f_{n}(z)=0\\}$ of codimension at least $2$.
Denote by $\Omega$ the Fubini Study form of ${\mathbf{P}}^{n}({\mathbf{C}})$.
The characteristic function of $f$ (with respect to $\Omega$) is defined by
$\displaystyle
T_{f}(r):=\int_{1}^{r}\dfrac{dt}{t^{2m-1}}\int_{B(t)}f^{*}\Omega\wedge\sigma,\quad\quad
1<r<+\infty.$
By Jensen’s formula we have
$\displaystyle T_{f}(r)=\int_{S(r)}\log||f||\eta+O(1),$
where $\|f\|=\max\\{|f_{0}|,\dots,|f_{n}|\\}$.
Let $a$ be a meromorphic mapping of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ with reduced representation
$a=(a_{0}:\dots:a_{n})$. We define
$m_{f,a}(r)=\int\limits_{S(r)}\text{log}\dfrac{||f||\cdot||a||}{|(f,a)|}\eta-\int\limits_{S(1)}\text{log}\dfrac{||f||\cdot||a||}{|(f,a)|}\eta,$
where $\|a\|=\big{(}|a_{0}|^{2}+\dots+|a_{n}|^{2}\big{)}^{1/2}$ and
$(f,a)=\sum_{i=0}^{n}f_{i}\cdot a_{i}.$
Let $f$ and $a$ be as above. If $(f,a)\not\equiv 0$, then the first main
theorem for moving hyperplaness in value distribution theory states
$T_{f}(r)+T_{a}(r)=m_{f,a}(r)+N_{(f,a)}(r)+O(1)\ (r>1).$
For a meromorphic function $\varphi$ on ${\mathbf{C}}^{m}$, the proximity
function $m(r,\varphi)$ is defined by
$m(r,\varphi)=\int\limits_{S(r)}\log^{+}|\varphi|\eta,$
where $\log^{+}x=\max\big{\\{}\log x,0\big{\\}}$ for $x\geqslant 0$. The
Nevanlinna’s characteristic function is defined by
$T(r,\varphi)=N(r,\nu^{\infty}_{\varphi})+m(r,\varphi).$
We regard $\varphi$ as a meromorphic mapping of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{1}({\mathbf{C}})^{*}$, there is a fact that
$T_{\varphi}(r)=T(r,\varphi)+O(1).$
(c) Lemma on logarithmic derivative.
As usual, by the notation $``||\ P"$ we mean the assertion $P$ holds for all
$r\in[0,\infty)$ excluding a Borel subset $E$ of the interval $[0,\infty)$
with $\int_{E}dr<\infty$. Denote by $\mathbf{Z}_{+}$ the set of all
nonnegative integers. The lemma on logarithmic derivative in Nevanlinna
theorey is stated as follows.
###### Lemma 2.1 (see [8, Lemma 3.11]).
Let $f$ be a nonzero meromorphic function on ${\mathbf{C}}^{m}.$ Then
$\biggl{|}\biggl{|}\quad
m\biggl{(}r,\dfrac{\mathcal{D}^{\alpha}(f)}{f}\biggl{)}=O(\log^{+}T_{f}(r))\
(\alpha\in\mathbf{Z}^{m}_{+}).$
(d) Family of moving hyperplanes.
We assume that thoughout this paper, the homogeneous coordinates of
${\mathbf{P}}^{n}({\mathbf{C}})$ is chosen so that for each given meromorphic
mapping $a=(a_{0}:\cdots:a_{n})$ of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ then $a_{0}\not\equiv 0$. We set
$\tilde{a}_{i}=\dfrac{a_{i}}{a_{0}}\text{ and
}\tilde{a}=(\tilde{a}_{0}:\tilde{a}_{1}:\cdots:\tilde{a}_{n}).$
Let $f:{\mathbf{C}}^{m}\rightarrow{\mathbf{P}}^{n}({\mathbf{C}})$ be a
meromorphic mapping with the reduced representation $f=(f_{0}:\cdots:f_{n}).$
We put $(f,a):=\sum_{i=0}^{n}f_{i}a_{i}$ and
$(f,\tilde{a}):=\sum_{i=0}^{n}f_{i}\tilde{a}_{i}.$
Let $\\{a_{i}\\}_{i=1}^{q}$ be $q$ meromorphic mappings of ${\mathbf{C}}^{m}$
into ${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ with reduced representations
$a_{i}=(a_{i0}:\cdots:a_{in})\ (1\leq i\leq q).$ We denote by
$\mathcal{R}(\\{a_{i}\\})$ (for brevity we will write $\mathcal{R}$ if there
is no confusion) the smallest subfield of $\mathcal{M}$ which contains
${\mathbf{C}}$ and all ${a_{i_{j}}}/{a_{i_{k}}}$ with $a_{i_{k}}\not\equiv 0.$
###### Definition 2.2.
The family $\\{a_{i}\\}_{i=1}^{q}$ is said to be in general position if
$\dim(\\{a_{i_{0}},\ldots,a_{i_{n}}\\})_{\mathcal{M}}=n+1$ for any $1\leq
i_{0}\leq\cdots\leq i_{n}\leq q$, where
$(\\{a_{i_{0}},\ldots,a_{i_{n}}\\})_{\mathcal{M}}$ is the linear span of
$\\{a_{i_{0}},\ldots,a_{i_{N}}\\}$ over the field $\mathcal{M}.$
###### Definition 2.3.
A subset $\mathcal{L}$ of $\mathcal{M}$ (or $\mathcal{M}^{n+1}$) is said to be
minimal over the field $\mathcal{R}$ if it is linearly dependent over
$\mathcal{R}$ and each proper subset of $\mathcal{L}$ is linearly independent
over $\mathcal{R}.$
Repeating the argument in ([1, Proposition 4.5]), we have the following:
###### Proposition 2.4 (see [1, Proposition 4.5]).
Let $\Phi_{0},\ldots,\Phi_{k}$ be meromorphic functions on ${\mathbf{C}}^{m}$
such that $\\{\Phi_{0},\ldots,\Phi_{k}\\}$ are linearly independent over
${\mathbf{C}}.$ Then there exists an admissible set
$\\{\alpha_{i}=(\alpha_{i1},\ldots,\alpha_{im})\\}_{i=0}^{k}\subset\mathbf{Z}^{m}_{+}$
with $|\alpha_{i}|=\sum_{j=1}^{n}|\alpha_{ij}|\leq k\ (0\leq i\leq k)$ such
that the following are satisfied:
(i)
$\\{{\mathcal{D}}^{\alpha_{i}}\Phi_{0},\ldots,{\mathcal{D}}^{\alpha_{i}}\Phi_{k}\\}_{i=0}^{k}$
is linearly independent over $\mathcal{M},$ i.e,
$\det{({\mathcal{D}}^{\alpha_{i}}\Phi_{j})}\not\equiv 0.$
(ii)
$\det\bigl{(}{\mathcal{D}}^{\alpha_{i}}(h\Phi_{j})\bigl{)}=h^{k+1}\det\bigl{(}{\mathcal{D}}^{\alpha_{i}}\Phi_{j}\bigl{)}$
for any nonzero meromorphic function $h$ on ${\mathbf{C}}^{m}.$
## 3\. Proof of Theorem 1.1
In order to prove Theorem 1.1 we need the following.
###### Lemma 3.1.
Let $f:{\mathbf{C}}^{m}\rightarrow{\mathbf{P}}^{n}({\mathbf{C}})$ be a
meromorphic mapping. Let $\\{a_{i}\\}_{i=1}^{q}$ $(q\geq n+1)$ be $q$
meromorphic mappings of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ in general position. Assume that there
exists a partition $\\{1,\ldots,q\\}=I_{1}\cup I_{2}\cdots\cup I_{l}$
satisfying:
$\mathrm{(i)}$ $\\{(f,\tilde{a}_{i})\\}_{i\in I_{1}}$ is minimal over
$\mathcal{R}$, and $\\{(f,\tilde{a}_{i})\\}_{i\in I_{t}}$ is linearly
independent over $\mathcal{R}\ (2\leq t\leq l),$
$\mathrm{(ii)}$ For any $2\leq t\leq l,i\in I_{t},$ there exist meromorphic
functions $c_{i}\in\mathcal{R}\setminus\\{0\\}$ such that
$\sum_{i\in
I_{t}}c_{i}(f,\tilde{a}_{i})\in\biggl{(}\bigcup_{j=1}^{t-1}\bigcup_{i\in
I_{j}}(f,\tilde{a}_{i})\biggl{)}_{\mathcal{R}}.$
Then we have
$T_{f}(r)\leq\sum_{i=1}^{q}N^{[k]}_{(f,a_{i})}+o(T_{f}(r))+O(\max_{1\leq i\leq
q}T_{a_{i}}(r)),$
where $k+1=\mathrm{rank}_{\mathcal{R}}(f)$.
Proof. Let $f=(f_{0}:\cdots:f_{n})$ be a reduced representation of $f$. By
changing the homogeneous coordinate system of ${\mathbf{P}}^{n}({\mathbf{C}})$
if necessary, we may assume that $f_{0}\not\equiv 0.$ Without loss of
generality, we may assume that $I_{1}=\\{1,\ldots.,k_{1}\\}$ and
$I_{t}=\\{k_{t-1}+1,\ldots,k_{t}\\}\ (2\leq t\leq l),\text{ where
}1=k_{0}<\cdots<k_{l}=q.$
Since $\\{(f,\tilde{a}_{i})\\}_{i\in I_{1}}$ is minimal over $\mathcal{R}$,
there exist $c_{1i}\in\mathcal{R}\setminus\\{0\\}$ such that
$\sum_{i=1}^{k_{1}}c_{1i}\cdot(f,\tilde{a}_{i})=0.$
Define $c_{1i}=0$ for all $i>k_{1}.$ Then
$\sum_{i=1}^{k_{l}}c_{1i}\cdot(f,\tilde{a}_{i})=0.$
Because ${\\{c_{1i}(f,\tilde{a}_{i})\\}}_{i=k_{0}+1}^{k_{1}}$ is linearly
independent over $\mathcal{R},$ Lemma 2.4 yields that there exists an
admissible set
$\\{\alpha_{1(k_{0}+1)},\ldots,\alpha_{1k_{1}}\\}\subset\mathbf{Z}^{m}_{+}$
$(|\alpha_{1i}|\leq k_{1}-k_{0}-1\leq\mathrm{rank}_{\mathcal{R}}f-1=k)$ such
that the matrix
$\ A_{1}=\left(\mathcal{D}^{\alpha_{1i}}(c_{1j}(f,\tilde{a}_{j}));k_{0}+1\leq
i,j\leq k_{1}\right)$
has nonzero determinant.
Now consider $t\geq 2.$ By constructing the set $I_{t}$, there exist
meromorphic mappings $c_{ti}\not\equiv 0\ (k_{t-1}+1\leq i\leq k_{t})$ such
that
$\sum_{i=k_{t-1}+1}^{k_{t}}c_{ti}\cdot(f,\tilde{a}_{i})\in\biggl{(}\bigcup_{j=1}^{t-1}\bigcup_{i\in
I_{t}}{(f,\tilde{a}_{i})}\biggl{)}_{\mathcal{R}}.$
Therefore, there exist meromorphic mappings $c_{ti}\in\mathcal{R}\ (1\leq
i\leq k_{t-1})$ such that
$\sum_{i=1}^{k_{t}}c_{ti}\cdot(f,\tilde{a}_{i})=0.$
Define $c_{ti}=0$ for all $i>k_{t}.$ Then
$\sum_{i=1}^{k_{l}}c_{ti}\cdot(f,\tilde{a}_{i})=0.$
Since $\\{c_{ti}(f,\tilde{a}_{i})\\}_{i=k_{t-1}+1}^{k_{t}}$ is
$\mathcal{R}$-linearly independent, by again Lemma 2.4 there exists an
admissible set
$\\{\alpha_{t(k_{t-1}+1)},\ldots,\alpha_{tk_{t}}\\}\subset\mathbf{Z}^{m}_{+}$
$(|\alpha_{ti}|\leq k_{t}-k_{t-1}-1\leq\mathrm{rank}_{\mathcal{R}}f-1=k)$ such
that the matrix
$\
A_{t}=\left(\mathcal{D}^{\alpha_{ti}}(c_{1j}(f,\tilde{a}_{j}));k_{t-1}+1\leq
i,j\leq k_{t}\right)$
has nonzero determinant.
Consider the following $(k_{l}-1)\times k_{l}$ matrix
$\displaystyle T$
$\displaystyle=\left(\mathcal{D}^{\alpha_{ti}}(c_{1j}(f,\tilde{a}_{j}));k_{0}+1\leq
i\leq k_{t},1\leq j\leq k_{t}\right)$
$\displaystyle=\left[\begin{array}[]{cccc}\mathcal{D}^{\alpha_{12}}(c_{11}(f,\tilde{a}_{1}))&\cdots&\mathcal{D}^{\alpha_{12}}(c_{1k_{l}}(f,\tilde{a}_{k_{l}}))\\\
\mathcal{D}^{\alpha_{13}}(c_{11}(f,\tilde{a}_{1}))&\cdots&\mathcal{D}^{\alpha_{13}}(c_{1k_{l}}(f,\tilde{a}_{k_{l}}))\\\
\vdots&\vdots&\vdots\\\
\mathcal{D}^{\alpha_{1k_{1}}}(c_{11}(f,\tilde{a}_{1}))&\cdots&\mathcal{D}^{\alpha_{1k_{1}}}(c_{1k_{l}}(f,\tilde{a}_{k_{l}}))\\\
\mathcal{D}^{\alpha_{2k_{1}+1}}(c_{21}(f,\tilde{a}_{1}))&\cdots&\mathcal{D}^{\alpha_{2k_{1}+1}}(c_{2k_{l}}(f,\tilde{a}_{k_{l}}))\\\
\mathcal{D}^{\alpha_{2k_{1}+2}}(c_{21}(f,\tilde{a}_{1}))&\cdots&\mathcal{D}^{\alpha_{2k_{1}+2}}(c_{2k_{l}}(f,\tilde{a}_{k_{l}}))\\\
\vdots&\vdots&\vdots\\\
\mathcal{D}^{\alpha_{2k_{2}}}(c_{21}(f,\tilde{a}_{1}))&\cdots&\mathcal{D}^{\alpha_{2k_{2}}}(c_{2k_{t}}(f,\tilde{a}_{k_{l}}))\\\
\vdots&\vdots&\vdots\\\
\mathcal{D}^{\alpha_{lk_{l-1}+1}}(c_{l1}(f,\tilde{a}_{1}))&\cdots&\mathcal{D}^{\alpha_{lk_{l-1}+1}}(c_{lk_{l}}(f,\tilde{a}_{k_{l}}))\\\
\mathcal{D}^{\alpha_{lk_{l-1}+2}}(c_{l1}(f,\tilde{a}_{1}))&\cdots&\mathcal{D}^{\alpha_{lk_{l-1}+2}}(c_{lk_{l}}(f,\tilde{a}_{k_{l}}))\\\
\vdots&\vdots&\vdots\\\
\mathcal{D}^{\alpha_{lk_{l}}}(c_{lk}(f,\tilde{a}_{1}))&\cdots&\mathcal{D}^{\alpha_{lk_{l}}}(c_{lk_{l}}(f,\tilde{a}_{k_{l}}))\\\
\end{array}\right].$
Denote by $D_{i}$ the subsquare matrix obtained by deleting the $(i+1)$-th
column of the minor matrix $T$. Since the sum of each row of $T$ is zero, we
have
$\det D_{i}={(-1)}^{i-1}\det D_{1}={(-1)}^{i-1}\prod_{j=1}^{l}\det A_{j}.$
Since $\\{a_{i}\\}_{i=1}^{q}$ is in general position, we have
$\det(\tilde{a}_{ij},\ 1\leq i\leq n+1,0\leq j\leq n)\not\equiv 0.$
By solving the linear equation system $(f,\tilde{a}_{i})=\tilde{a}_{i0}\cdot
f_{0}+\ldots+\tilde{a}_{in}\cdot f_{n}\ (1\leq i\leq n+1),$ we obtain
(3.2) $\displaystyle f_{v}=\sum_{i=1}^{n+1}A_{vi}(f,\tilde{a}_{i})\
(A_{vi}\in\mathcal{R})\text{ for each }0\leq v\leq n.$
Put $\Psi(z)=\sum_{i=1}^{n+1}\sum_{v=0}^{n}|A_{vi}(z)|\
(z\in{\mathbf{C}}^{m}).$ Then
$\ \ ||f(z)||\leq\Psi(z)\cdot\max_{1\leq i\leq
n+1}\bigl{(}|(f,\tilde{a}_{i})(z)|\bigl{)}\leq\Psi(z)\cdot\max_{1\leq i\leq
q}\bigl{(}|(f,\tilde{a}_{i})(z)|\bigl{)}\ (z\in{\mathbf{C}}^{m}),$
and
$\displaystyle\int\limits_{S(r)}\log^{+}\Psi(z)\eta$
$\displaystyle\leq\sum_{i=1}^{n+1}\sum_{v=0}^{n}\int\limits_{S(r)}\log^{+}|A_{vi}(z)|\eta+O(1)$
$\displaystyle\leq\sum_{i=1}^{n+1}\sum_{v=0}^{n}T(r,A_{vi})+O(1)$
$\displaystyle=O(\max_{1\leq i\leq q}T_{a_{i}}(r))+O(1).$
Fix
$z_{0}\in{\mathbf{C}}^{m}\setminus\bigcup_{j=1}^{q}\biggl{(}\mathrm{Supp}\,(\nu^{0}_{(f,\tilde{a}_{j})})\cup\mathrm{Supp}\,(\nu^{\infty}_{(f,\tilde{a}_{j})})\biggl{)}.$
Take $i\ (1\leq i\leq q)$ such that
$|(f,\tilde{a}_{i})(z_{0})|=\max_{1\leq j\leq q}(|f,\tilde{a}_{j})(z_{0})|.$
Then
$\displaystyle\dfrac{|\det
D_{1}(z_{0})|\cdot||f(z_{0})||}{\prod_{j=1}^{q}|(f,\tilde{a}_{i})(z_{0})|}$
$\displaystyle=\dfrac{|\det
D_{i}(z_{0})|}{\prod_{{\mathrel{\mathop{{j=0}}\limits_{{j\neq
i}}}}}^{q}|(f,\tilde{a}_{j})(z_{0})|}\cdot\biggl{(}\dfrac{||f(z_{0})||}{|(f,\tilde{a}_{i})(z_{0})|}\biggl{)}$
$\displaystyle\leq\Psi(z_{0})\cdot\dfrac{|\det
D_{i}(z_{0})|}{\prod_{{\mathrel{\mathop{{j=1}}\limits_{{j\neq
i}}}}}^{q}|(f,\tilde{a}_{j})(z_{0})|}.$
This implies that
$\displaystyle\log\dfrac{|\det
D_{1}(z_{0})|.||f(z_{0})||}{\prod_{j=1}^{q}|(f,\tilde{a}_{j})(z_{0})|}$
$\displaystyle\leq\log^{+}\biggl{(}\Psi(z_{0})\cdot\biggl{(}\dfrac{|\det
D_{i}(z_{0})|}{\prod_{j=1,j\neq
i}^{q}|(f,\tilde{a}_{j})(z_{0})|}\biggl{)}\biggl{)}$
$\displaystyle\leq\log^{+}\biggl{(}\dfrac{|\det
D_{i}(z_{0})|}{\prod_{j=1,j\neq
i}^{q}|(f,\tilde{a}_{j})(z_{0})|}\biggl{)}+\log^{+}\Psi(z_{0}).$
Thus, for each
$z\in{\mathbf{C}}^{m}\setminus\bigcup_{j=1}^{q}\biggl{(}\mathrm{Supp}\,(\nu^{0}_{(f,\tilde{a}_{j})})\cup\mathrm{Supp}\,(\nu^{\infty}_{(f,\tilde{a}_{j})})\biggl{)},$
we have
$\displaystyle\log\dfrac{|\det
D_{1}(z)|.||f(z)||}{\prod_{i=1}^{q}|(f,\tilde{a}_{i})(z)|}\leq\sum_{i=1}^{q}\log^{+}\biggl{(}\dfrac{|\det
D_{i}(z)|}{\prod_{j=1,j\neq
i}^{q}|(f,\tilde{a}_{j})(z)|}\biggl{)}+\log^{+}\Psi(z)$
Hence
(3.3) $\displaystyle\log||f(z)||+\log\dfrac{|\det
D_{1}(z)|}{\prod_{i=1}^{q}|(f,\tilde{a}_{i})(z)|}\leq\sum_{i=1}^{q}\log^{+}\biggl{(}\dfrac{|\det
D_{i}(z)|}{\prod_{j=1,j\neq
i}^{q}|(f,\tilde{a}_{j})(z)|}\biggl{)}+\log^{+}\Psi(z).$
Note that
$\displaystyle\dfrac{\det D_{i}}{\prod_{j=1,j\neq i}^{q}(f,\tilde{a}_{j})}$
$\displaystyle=\dfrac{\det D_{i}/f_{0}^{q-1}}{\prod_{j=1,j\neq
i}^{q}\biggl{(}(f,\tilde{a}_{j})/f_{0}\biggl{)}}$
$\displaystyle=\left[\begin{array}[]{cccc}\dfrac{\mathcal{D}^{\alpha_{12}}\biggl{(}\dfrac{c_{11}(f,\tilde{a}_{1})}{f_{0}}\biggl{)}}{\dfrac{(f,\tilde{a}_{1})}{f_{0}}}&\cdots&\dfrac{\mathcal{D}^{\alpha_{12}}\biggl{(}\dfrac{c_{1k_{l}}(f,\tilde{a}_{k_{l}})}{f_{0}}\biggl{)}}{\dfrac{(f,\tilde{a}_{k_{l}})}{f_{0}}}\\\
\vdots&\vdots&\vdots\\\
\dfrac{\mathcal{D}^{\alpha_{lk_{l}}}\biggl{(}\dfrac{c_{l1}(f,\tilde{a}_{1})}{f_{0}}\biggl{)}}{\dfrac{(f,\tilde{a}_{1})}{f_{0}}}&\cdots&\dfrac{\mathcal{D}^{\alpha_{lk_{l}}}\biggl{(}\dfrac{c_{lk_{l}}(f,\tilde{a}_{k_{l}})}{f_{0}}\biggl{)}}{\dfrac{(f,\tilde{a}_{k_{l}})}{f_{0}}}\end{array}\right]$
(The determinant is counted after deleting the $i$-th column in the above
matrix).
Each element of the above matrix has a form
$\dfrac{\mathcal{D}^{\alpha}\biggl{(}\dfrac{c(f,\tilde{a}_{j})}{f_{0}}\biggl{)}}{\dfrac{(f,\tilde{a}_{j})}{f_{0}}}=\dfrac{\mathcal{D}^{\alpha}\biggl{(}\dfrac{c(f,\tilde{a}_{j})}{f_{0}}\biggl{)}}{\dfrac{c(f,\tilde{a}_{j})}{f_{0}}}\cdot
c\ (c\in\mathcal{R}).$
By lemma on logarithmic derivative lemma, we have
$\displaystyle\biggl{|}\biggl{|}\quad\quad
m\biggl{(}r,\dfrac{\mathcal{D}^{\alpha}\biggl{(}\dfrac{c(f,\tilde{a}_{j})}{f_{0}}\biggl{)}}{\dfrac{(f,\tilde{a}_{j})}{f_{0}}}\biggl{)}$
$\displaystyle\leq
m\biggl{(}r,\dfrac{\mathcal{D}^{\alpha}\biggl{(}\dfrac{c(f,\tilde{a}_{j})}{f_{0}}\biggl{)}}{\dfrac{c(f,\tilde{a}_{j})}{f_{0}}}\biggl{)}+m(r,c)$
$\displaystyle=O\biggl{(}\log^{+}T\biggl{(}r,\dfrac{c(f,\tilde{a}_{j})}{f_{0}}\biggl{)}\biggl{)}+O(\max_{1\leq
i\leq q}T(r,a_{i}))$ $\displaystyle=O(\log^{+}T_{f}(r))+O(\max_{1\leq i\leq
q}T(r,a_{i})).$
This yields that
$\biggl{|}\biggl{|}\quad m\left(r,\dfrac{\det D_{i}}{\prod_{j=1,j\neq
i}^{q}(f,\tilde{a}_{j})}\right)=O(\log^{+}T_{f}(r))+O(\max_{1\leq j\leq
q}T_{a_{j}}(r))\ (1\leq i\leq q).$
Hence
$\biggl{|}\biggl{|}\quad\quad\sum_{i=1}^{q}m\left(r,\dfrac{\det
D_{i}}{\prod_{j=1,j\neq
i}^{q}(f,\tilde{a}_{j})}\right)=O(\log^{+}T_{f}(r))+O(\max_{1\leq j\leq
q}T_{a_{j}}(r)).$
Integrating both sides of the inequality (3.3), we have
$\displaystyle\biggl{|}\biggl{|}\ \int_{S(r)}\log||f||\eta$
$\displaystyle+\int_{S(r)}\log\biggl{(}\dfrac{|\det{D}_{0}|}{\prod_{i=1}^{q}|(f,\tilde{a}_{i})|}\biggl{)}\eta$
$\displaystyle\leq\sum_{i=1}^{q}\int_{S(r)}\log^{+}\biggl{(}\dfrac{|\det
D_{i}|}{\prod_{j=1,j\neq
i}^{q}|(f,\tilde{a}_{j})|}\biggl{)}\eta+\int_{S(r)}\log^{+}\Psi(z)\eta$
$\displaystyle=\sum_{i=1}^{q}m\biggl{(}r,\dfrac{\det D_{i}}{\prod_{j=1,j\neq
i}^{q}(f,\tilde{a}_{j})}\biggl{)}+O(\max_{1\leq i\leq q}T_{a_{i}}(r))$
$\displaystyle=O(\log^{+}T_{f}(r))+O(\max_{0\leq i\leq q-1}T_{a_{i}}(r)).$
Hence
$||\ \ T_{f}(r)+\int\limits_{S(r)}\text{log}\dfrac{|\det
D_{1}|}{\prod_{i=1}^{q}|(f,\tilde{a}_{i})|}\eta=O(\log^{+}T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)),\ \text{i.e, }$ $\displaystyle||\ T_{f}(r)$
$\displaystyle=\int\limits_{S(r)}\text{log}\dfrac{\prod_{i=1}^{q}|(f,\tilde{a}_{i})|}{|\det
D_{1}|}\eta+O(\log^{+}T_{f}(r))+O(\max_{1\leq i\leq q}T_{a_{i}}(r))$
$\displaystyle=\int\limits_{S(r)}\text{log}\prod_{i=1}^{q}|(f,\tilde{a}_{i})|\eta-\int\limits_{S(r)}\text{log}|\det
D_{1}|\eta+O(\log^{+}T_{f}(r))+O(\max_{1\leq i\leq q}T_{a_{i}}(r))$ (3.4)
$\displaystyle\leq N_{\prod_{i=1}^{q}(f,\tilde{a}_{i})}(r)-N(r,\nu_{\det
D_{1}})+O(\log^{+}T_{f}(r))+O(\max_{1\leq i\leq q}T_{a_{i}}(r)).$
###### Claim 3.5.
$||\ N_{\prod_{i=1}^{q}(f,\tilde{a}_{i})}(r)-N(r,\nu_{\det
D_{1}})\leq\sum_{i=1}^{q}N^{[k]}_{(f,a_{i})}(r)+O(\max_{1\leq i\leq
q}T_{a_{i}}(r)).$
Indeed, fix $z\in{\mathbf{C}}^{m}\setminus I(f)$, where $I(f)=\\{f_{0}=\cdots
f_{n}=0\\}$. We call $i_{0}$ the index satisfying
$\nu^{0}_{(f,\tilde{a}_{i_{0}})}(z)=\min_{1\leq i\leq
n+1}\nu^{0}_{(f,\tilde{a}_{i})}(z).$
For each $i\neq i_{0},i\in I_{s}$, we have
$\displaystyle\nu^{0}_{\mathcal{D}^{\alpha_{sk_{s-1}+j}}(c_{si}(f,\tilde{a}_{i}))}(z)$
$\displaystyle\geq\min_{\beta\in\mathbf{Z}_{+}^{m}\text{ with
}\alpha_{sk_{s-1}+j}-\beta\in\mathbf{Z}_{+}^{m}}\\{\nu^{0}_{\mathcal{D}^{\beta}c_{si}\mathcal{D}^{\alpha_{st_{s-1}+j}-\beta}(f,\tilde{a}_{i})}(z)\\}$
$\displaystyle\geq\min_{\beta\in\mathbf{Z}_{+}^{n}\text{ with
}\alpha_{sk_{s-1}+j}-\beta\in\mathbf{Z}_{+}^{n}}\bigl{\\{}\max\\{0,\nu^{0}_{(f,\tilde{a}_{i})}(z)-|\alpha_{sk_{s-1}+j}-\beta|\\}$
$\displaystyle\hskip 90.0pt-(\beta+1)\nu^{\infty}_{c_{si}}(z)\bigl{\\}}$
$\displaystyle\geq\max\\{0,\nu_{(f\tilde{a}_{i})}^{0}(z)-k\\}-(k+1)\nu_{c_{si}}^{\infty}(z)$
On the other hand, we also have
$\displaystyle\nu^{\infty}_{\mathcal{D}^{\alpha_{sk_{s-1}+j}}(c_{si}(f,\tilde{a}_{i}))}(z)\leq(|\alpha_{sk_{s-1}+j}|+1)\nu^{\infty}_{c_{si}(f,\tilde{a}_{i})}(z)\leq(k+1)(\nu^{\infty}_{c_{si}}(z)+\nu^{0}_{a_{i0}}(z)).$
Thus
$\nu_{\mathcal{D}^{\alpha_{sk_{s-1}+j}}(c_{si}(f,\tilde{a}_{i}))}(z)\geq\max\\{0,\nu_{(f\tilde{a}_{i})}^{0}(z)-k\\}-(k+1)\bigl{(}2\nu_{c_{si}}^{\infty}(z)+\nu^{0}_{a_{i0}}(z)\bigl{)}$
Since each element of the matrix $D_{i_{0}}$ has a form
$\mathcal{D}^{\alpha_{sk_{s-1}+j}}(c_{si}(f,\tilde{a}_{i}))\ (i\neq i_{0})$,
one estimates
(3.6) $\displaystyle\nu_{D_{1}}(z)=\nu_{D_{i_{0}}}(z)\geq\sum_{i\neq
i_{0}}\left(\max\\{0,\nu_{(f\tilde{a}_{i})}^{0}(z)-k\\}-(k+1)\bigl{(}2\nu_{c_{si}}^{\infty}(z)+\nu^{0}_{a_{i0}}(z)\bigl{)}\right).$
We see that there exists $v_{0}\in\\{0,\ldots,n\\}$ with $f_{v_{0}}(z)\neq 0$.
Then by (3.2), there exists $i_{1}\in\\{1,\ldots,n+1\\}$ such that
$A_{v_{0}i_{1}}(z)\cdot(f,\tilde{a}_{i_{1}})(z)\neq 0$. Thus
(3.7)
$\displaystyle\nu^{0}_{(f,\tilde{a}_{i_{0}})}(z)\leq\nu^{0}_{(f,\tilde{a}_{i_{1}})}(z)\leq\nu^{\infty}_{A_{v_{0}i_{1}}}(z)\leq\sum_{A_{vi}\not\equiv
0}\nu^{\infty}_{A_{vi}}(z).$
Combining the inequalities (3.6) and (3.7), we have
$\displaystyle\nu^{0}_{\prod_{i=1}^{q}(f,\tilde{a}_{i})}(z)$
$\displaystyle-\nu_{\det D_{1}}(z)$ $\displaystyle\leq\sum_{i\neq
i_{0}}\left(\min\\{\nu_{(f,\tilde{a}_{i})}^{0}(z),k\\}+(k+1)\bigl{(}2\nu_{c_{si}}^{\infty}(z)+\nu^{0}_{a_{i0}}(z)\bigl{)}\right)+\sum_{A_{vi}\not\equiv
0}\nu^{\infty}_{A_{vi}}(z)$
$\displaystyle\leq\sum_{i=1}^{q}\left(\min\\{\nu_{(f,\tilde{a}_{i})}^{0}(z),k\\}+(k+1)\bigl{(}2\nu_{c_{si}}^{\infty}(z)+\nu^{0}_{a_{i0}}(z)\bigl{)}\right)+\sum_{A_{vi}\not\equiv
0}\nu^{\infty}_{A_{vi}}(z),$
where the index $s$ of $c_{si}$ is taken so that $i\in I_{s}$. Integrating
both sides of this inequality, we obtain
$\displaystyle||\ \ N_{\prod_{i=1}^{q}(f,\tilde{a}_{i})}(r)$
$\displaystyle-N(r,\nu_{\det D_{1}})$
$\displaystyle\leq\sum_{i=1}^{q}\left(N^{[k]}_{(f,\tilde{a}_{i})}(r)+(k+1)\biggl{(}2N_{\frac{1}{c_{si}}}(r)+N_{a_{i0}}(r)\biggl{)}\right)+\sum_{A_{vi}\not\equiv
0}N_{{1}/{A_{vi}}}(r)$ (3.8)
$\displaystyle=\sum_{i=1}^{q}N^{[k]}_{(f,a_{i})}(r)+O(\max_{1\leq i\leq
q}T_{a_{i}}(r)).$
The claim is proved.
From the inequalities (3.4) and the claim, we get
$||\ \
T_{f}(r)\leq\sum_{i=1}^{q}N^{[k]}_{(f,a_{i})}(r)+O(\log^{+}T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)).$
The lemma is proved. $\square$
Proof of Theorem 1.1.
(a). We denote by $\mathcal{I}$ the set of all permutations of $q-$tuple
$(1,\ldots,q)$. For each element $I=(i_{1},\ldots,i_{q})\in\mathcal{I}$, we
set
$N_{I}=\\{r\in{\mathbf{R}}^{+};N^{[k]}_{(f,a_{i_{1}})}(r)\leq\cdots\leq
N^{[k]}_{(f,a_{i_{q}})}(r)\\}.$
We now consider an element $I=(i_{1},\ldots,i_{q})$ of $\mathcal{I}$. We will
construct subsets $I_{t}$ of the set $A_{1}=\\{1,\ldots,{2n-k+2}\\}$ as
follows.
We choose a subset $I_{1}$ of $A$ which is the minimal subset of $A$
satisfying that $\\{(f,\tilde{a}_{i_{j}})\\}_{j\in I_{1}}$ is minimal over
$\mathcal{R}$. If $\sharp I_{1}\geq n+1$ then we stop the process.
Otherwise, set $A_{2}=A_{1}\setminus I_{1}$. We consider the following two
cases:
* •
Case 1. Suppose that $\sharp A_{2}\geq n+1$. Since
$\\{\tilde{a}_{i_{j}}\\}_{j\in A_{2}}$ is in general position, we have
$\left((f,\tilde{a}_{i_{j}});j\in
A_{2}\right)_{\mathcal{R}}=\left(f_{0},\ldots,f_{n}\right)_{\mathcal{R}}\supset\left((f,\tilde{a}_{i_{j}});j\in
I_{1}\right)_{\mathcal{R}}\not\equiv 0.$
* •
Case 2. Suppose that $\sharp A_{2}<n+1$. Then we have the following:
$\displaystyle\dim_{\mathcal{R}}\left((f,\tilde{a}_{i_{j}});j\in
I_{1}\right)_{\mathcal{R}}\geq k+1-(n+1-\sharp I_{1})=k-n+\sharp I_{1},$
$\displaystyle\dim_{\mathcal{R}}\left((f,\tilde{a}_{i_{j}});j\in
A_{2}\right)_{\mathcal{R}}\geq k+1-(n+1-\sharp A_{2})=k-n+\sharp A_{2}.$
We note that $\sharp I_{1}+\sharp A_{2}=2n-k+2$. Hence the above inequalities
imply that
$\displaystyle\dim_{\mathcal{R}}$
$\displaystyle\biggl{(}\bigl{(}(f,\tilde{a}_{i_{j}});j\in
I_{1}\bigl{)}_{\mathcal{R}}\cap\bigl{(}(f,\tilde{a}_{i_{j}});j\in
A_{2}\bigl{)}_{\mathcal{R}}\biggl{)}$
$\displaystyle\geq\dim_{\mathcal{R}}\left((f,\tilde{a}_{i_{j}});j\in
I_{1}\right)_{\mathcal{R}}+\dim_{\mathcal{R}}\left((f,\tilde{a}_{i_{j}});j\in
A_{2}\right)_{\mathcal{R}}-(k+1)$ $\displaystyle=k-n+\sharp I_{1}+k-n+\sharp
A_{2}-(k+1)=1.$
Therefore, from the above two case, we see that
$\bigl{(}(f,\tilde{a}_{i_{j}});j\in
I_{1}\bigl{)}_{\mathcal{R}}\cap\bigl{(}(f,\tilde{a}_{i_{j}});j\in
A_{2}\bigl{)}_{\mathcal{R}}\neq\\{0\\}.$
Therefore, we may chose a subset $I_{2}\subset A_{2}$ which is the minimal
subset of $A_{2}$ satisfying that there exist nonzero meromorphic functions
$c_{i}\in\mathcal{R}\ (i\in I_{2})$,
$\sum_{i\in I_{2}}c_{i}(f,\tilde{a}_{i})\in\biggl{(}\bigcup_{i\in
I_{1}}(f,\tilde{a}_{i})\biggl{)}_{\mathcal{R}}.$
By the minimality of the set $I_{2}$, the family
$\\{(f,\tilde{a}_{i_{j}})\\}_{j\in I_{2}}$ is linearly independent over
$\mathcal{R}$, and hence $\sharp I_{2}\leq k+1$ and
$\sharp(I_{2}\cup I_{2})\leq\min\\{2n-k+2,n+k+1\\}.$
If $\sharp(I_{2}\cup I_{2})\geq n+1$ then we stop the process.
Otherwise, by repeating the above argument, we have a subset $I_{3}$ of
$A_{3}=A_{1}\setminus(I_{1}\cup I_{2})$, which satisfies the following:
* •
there exist nonzero meromorphic functions $c_{i}\in\mathcal{R}\ (i\in I_{3})$
so that
$\sum_{i\in I_{3}}c_{i}(f,\tilde{a}_{i})\in\biggl{(}\bigcup_{i\in I_{1}\cup
I_{2}}(f,\tilde{a}_{i})\biggl{)}_{\mathcal{R}},$
* •
$\\{(f,\tilde{a}_{i_{j}})\\}_{j\in I_{3}}$ is linearly independent over
$\mathcal{R}$,
* •
$\sharp I_{3}\leq k+1$ and $\sharp(I_{1}\cup\cdots\cup
I_{3})\leq\min\\{2n-k+2,n+k+1\\}$.
Continuing this process, we get the subsets $I_{1},\ldots,I_{l}$, which
satisfy:
* •
$\\{(f,\tilde{a}_{i_{j}})\\}_{j\in I_{1}}$ is minimal over $\mathcal{R}$,
$\\{(f,\tilde{a}_{i_{j}})\\}_{j\in I_{t}}$ is linearly independent over
$\mathcal{R}\ (2\leq t\leq l),$
* •
for any $2\leq t\leq l,j\in I_{t},$ there exist meromorphic functions
$c_{j}\in\mathcal{R}\setminus\\{0\\}$ such that
$\sum_{j\in
I_{t}}c_{j}(f,\tilde{a}_{i_{j}})\in\biggl{(}\bigcup_{s=1}^{t-1}\bigcup_{j\in
I_{s}}(f,\tilde{a}_{i_{j}})\biggl{)}_{\mathcal{R}},$
* •
$n+1\leq\sharp(I_{1}\cup\cdots\cup I_{l})\leq\min\\{2n-k+2,n+k+1\\}$.
Then the family of subsets $I_{1},\ldots,I_{t}$ satisfies the assumptions of
the Lemma 3.1. Therefore, we have
$\displaystyle||\ T_{f}(r)\leq\sum_{j\in
J}N^{[k]}_{(f,a_{i_{j}})}+o(T_{f}(r))+O(\max_{1\leq i\leq q}T_{a_{i}}(r)),$
where $J=I_{1}\cup\cdots\cup I_{l}$. Then for all $r\in N_{I}$ (may be outside
a finite Borel measure subset of ${\mathbf{R}}^{+}$) we have
$\displaystyle||\ T_{f}(r)$ $\displaystyle\leq\dfrac{\sharp
J}{q-(2n-k+2)+\sharp J}\biggl{(}\sum_{j\in
J}N^{[k]}_{(f,a_{i_{j}})}(r)+\sum_{j=2n-k+3}^{q}N^{[k]}_{(f,a_{i_{j}})}(r)\biggl{)}$
(3.9) $\displaystyle+o(T_{f}(r))+O(\max_{1\leq i\leq q}T_{a_{i}}(r)).$
Since $\sharp J\leq 2n-k+2$, the above inequality implies that
(3.10) $\displaystyle||\
T_{f}(r)\leq\dfrac{2n-k+2}{q}\sum_{i=1}^{q}N^{[k]}_{(f,a_{i})}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)),\quad r\in N_{I}.$
We see that $\bigcup_{I\in\mathcal{I}}N_{I}={\mathbf{R}}^{+}$ and the
inequality (3.10) holds for every $r\in N_{I},I\in\mathcal{I}$. This yields
that
$T_{f}(r)\leq\dfrac{2n-k+2}{q}\sum_{i=1}^{q}N^{[k]}_{(f,a_{i})}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r))$
for all $r$ outside a finite Borel measure subset of ${\mathbf{R}}^{+}$. Thus
$||\
\dfrac{q}{2n-k+2}T_{f}(r)\leq\sum_{i=1}^{q}N^{[k]}_{(f,a_{i})}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)).$
The assertion (a) is proved.
(b) We repeat the same argument as in the proof of the assertion (a). If
$n+k+1>2n-k+1$ then the assertion (b) is a consequence of the assertion (a).
Then we now only consider the case where $n+k+1\leq 2n-k+1$.
From (3.9) with a note that $\sharp J\leq n+k+2$, we have
$\displaystyle||\ T_{f}(r)$
$\displaystyle\leq\dfrac{n+k+1}{q-(2n-k+2)+n+k+1)}\sum_{i=1}^{q}N^{[k]}_{(f,a_{i})}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r))$
$\displaystyle=\dfrac{n+k+1}{q-n+2k-1}\sum_{i=1}^{q}N^{[k]}_{(f,a_{i})}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r))\ r\in N_{I}.$
Repeating again the argument in the proof of assertion (a), we see that the
above inequality holds for all $r\in{\mathbf{R}}^{+}$ outside a finite Borel
measure set. Then the assertion (b) is proved. $\square$
## 4\. Proof of Theorem 1.3
In order to prove Theorem 1.3, we need the following.
4.1. Let $f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a
meromorphic mapping with a reduced representation $f=(f_{0}:\ldots:f_{n})$.
Let $\\{a_{i}\\}_{i=1}^{q}$ be “slowly” (with respect to $f$) moving
hyperplanes of ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position such that
$\dim\\{z\in{\mathbf{C}}^{m}:(f,a_{i})(z)=(f,a_{j})(z)=0\\}\leq m-2\quad(1\leq
i<j\leq q).$
For $M+1$ elements
$f^{0},\ldots,f^{M}\in\mathcal{F}(f,\\{a_{j}\\}_{j=1}^{q},1)$, we put
$T(r)=\sum_{k=0}^{M}T(r,f^{k}).$
Assume that $a_{i}$ has a reduced representation
$a_{i}=(a_{i0}:\cdots:a_{in}).$ By changing the homogeneous coordinate system
of ${\mathbf{P}}^{n}({\mathbf{C}}),$ we may assume that $a_{i0}\not\equiv 0\
(1\leq i\leq q).$
We set $\ F^{jk}_{i}:=\dfrac{(f^{k},a_{j})}{(f^{k},a_{i})}\quad(1\leq i,j\leq
q,\ 0\leq k\leq M).$
###### Lemma 4.1.
Suppose that $q\geq 2n+1$. Then
$||\ T_{g}(r)=O(T_{f}(r))\text{ for each
}g\in\mathcal{F}(f,\\{a_{i}\\}_{i=1}^{q},1).$
Proof. By Corollary 1.2(a), we have
$\displaystyle\parallel\ \dfrac{2q-n+1}{3(n+1)}T_{g}(r)$
$\displaystyle\leq\sum_{i=1}^{q}N_{(g,a_{i})}^{[n]}(r)+o(T_{g}(r)+T_{f}(r))$
$\displaystyle\leq n\sum_{i=1}^{q}N_{(g,a_{i})}^{[1]}(r)+o(T_{g}(r)+T_{f}(r))$
$\displaystyle=\sum_{i=1}^{q}nN_{(f,a_{i})}^{[1]}(r)+o(T_{g}(r)+T_{f}(r))$
$\displaystyle\leq qnT_{f}(r)+o(T_{g}(r)+T_{f}(r)).$
Hence $||\quad T_{g}(r)=O(T_{f}(r)).$ $\square$
###### Definition 4.2 (see [2, p. 138]).
Let $F_{0},\ldots,F_{M}$ be nonzero meromorphic functions on
${\mathbf{C}}^{m}$, where $M\geq 1$. Take a set
$\alpha:=(\alpha^{0},\ldots,\alpha^{M-1})$ whose components $\alpha^{k}$ are
composed of $m$ nonnegative integers, and set
$|\alpha|=|\alpha^{0}|+\ldots+|\alpha^{M-1}|.$ We define Cartan’s auxiliary
function by
$\Phi^{\alpha}\equiv\Phi^{\alpha}(F_{0},\ldots,F_{M}):=F_{0}F_{1}\cdots
F_{M}\left|\begin{array}[]{cccc}1&1&\cdots&1\\\
\mathcal{D}^{\alpha^{0}}(\frac{1}{F_{0}})&\mathcal{D}^{\alpha^{0}}(\frac{1}{F_{1}})&\cdots&\mathcal{D}^{\alpha^{0}}(\frac{1}{F_{M}})\\\
\vdots&\vdots&\vdots&\vdots\\\
\mathcal{D}^{\alpha^{M-1}}(\frac{1}{F_{0}})&\mathcal{D}^{\alpha^{M-1}}(\frac{1}{F_{1}})&\cdots&\mathcal{D}^{\alpha^{M-1}}(\frac{1}{F_{M}})\\\
\end{array}\right|$
###### Lemma 4.3 (see [2, Proposition 3.4]).
If $\Phi^{\alpha}(F,G,H)=0$ and
$\Phi^{\alpha}(\frac{1}{F},\frac{1}{G},\frac{1}{H})=0$ for all $\alpha$ with
$|\alpha|\leq 1$, then one of the following assertions holds :
(i) $F=G,G=H$ or $H=F$
(ii) $\frac{F}{G},\frac{G}{H}$ and $\frac{H}{F}$ are all constant.
###### Lemma 4.4 (see [6, Lemma 4.7]).
Suppose that there exists
$\Phi^{\alpha}=\Phi^{\alpha}(F_{i_{0}}^{j_{0}0},\ldots,F_{i_{0}}^{j_{0}M})\not\equiv
0$ with $1\leq i_{0},j_{0}\leq q,\ |\alpha|\leq\dfrac{M(M-1)}{2},\
d\geq|\alpha|.$ Assume that $\alpha$ is a minimal element such that
$\Phi^{\alpha}(F_{i_{0}}^{j_{0}0},\ldots,F_{i_{0}}^{j_{0}M})\not\equiv 0$.
Then, for each $0\leq k\leq M$, the following holds:
$\parallel
N_{(f^{k},a_{j_{0}})}^{[d-|\alpha|]}(r)+M\sum_{j\neq{j_{0},i_{0}}}N_{(f^{k},a_{j})}^{[1]}(r)\leq
N_{\Phi^{\alpha}}(r)\leq T(r)-M\cdot N^{[1]}_{(f^{k},a_{i_{0}})}(r)+o(T(r)).$
And hence
$||\quad
N_{(f^{k},a_{j_{0}})}^{[d-|\alpha|]}(r)+M\sum_{j\neq{j_{0}}}N_{(f^{k},a_{j})}^{[1]}(r)\leq
T(r)+o(T(r)).$
4.2. Proof of Theorem 1.3
a) Assume that $q>\frac{9n^{2}+9n+2}{2}$. Suppose that there exist three
distinct elements
$f^{0},f^{1},f^{2}\in\mathcal{F}(f,\\{a_{j}\\}_{j=1}^{q},1).$
Suppose that there exist two indices $i,j\in\\{1,\ldots,q\\}$ and
$\alpha=(\alpha_{0},\alpha_{1})\in(\mathbf{Z}_{+}^{n})^{2}$ with $|\alpha|\leq
1$ such that $\Phi^{\alpha}(F_{j}^{i0},F_{j}^{i1},F_{j}^{i2})\not\equiv 0$. By
Lemma 4.4, we have
$2\sum_{t\neq i}N_{(f^{0},a_{t})}^{[1]}(r)\leq T(r)+o(T_{f}(r)).$
Hence, by Corollary 1.2(b) we have
$\displaystyle\parallel T(r)$
$\displaystyle\geq\dfrac{2}{3}\sum_{k=1}^{3}\sum_{t\neq
i}N_{(f^{k},a_{t})}^{[1]}(r)+o(T_{f}(r))\geq\dfrac{2}{3n}\sum_{k=1}^{3}\sum_{t\neq
i}N_{(f^{k},a_{t})}^{[n]}(r)+o(T_{f}(r))$
$\displaystyle\geq\dfrac{4(q-1)}{9n(n+1)}T(r)+o(T_{f}(r)).$
Letting $r\longrightarrow+\infty$, we get $1\geq\frac{4(q-1)}{9n(n+1)}$, i.e.,
$q\leq\frac{9n^{2}+9n+4}{4}$. This is a contradiction.
Then for two indices $i,j$ $(1\leq i<j\leq q)$, we have
$\Phi^{\alpha}(F_{j}^{i0},F_{j}^{i1},F_{j}^{i2})\equiv 0\text{ and
}\Phi^{\alpha}(F_{i}^{j0},F_{i}^{j1},F_{i}^{j2})\equiv 0$
for all $\alpha=(\alpha_{0},\alpha_{1})\ \text{ with }|\alpha|\leq 1.$ By
Lemma 4.3, there exists a constant $\lambda$ such that
$F_{j}^{i0}=\lambda F_{j}^{i1},F_{j}^{i1}=\lambda F_{j}^{i2},\text{ or
}F_{j}^{i2}=\lambda F_{j}^{i0}.$
For instance, we assume that $F_{j}^{i0}=\lambda F_{j}^{i1}$. We will show
that $\lambda=1.$
Indeed, assume that $\lambda\neq 1$. Since $F_{j}^{i0}=F_{j}^{i1}$ on the set
$\bigcup_{k\neq j}\\{z:(f,a_{k})(z)=0\\},$ we have that
$F_{j}^{i0}=F_{j}^{i1}=0$ on the set $\bigcup_{k\neq
j}\\{z:(f,a_{k})(z)=0\\}.$ Hence $\bigcup_{k\neq
j}\\{z:(f,a_{k})(z)=0\\}\subset\\{z:(f,a_{i})(z)=0\\}.$ It follows that
$\\{z:(f,a_{k})(z)=0\\}=\emptyset\ (k\neq i,j).$ We obtain that
$\parallel\dfrac{2(q-2)}{3(n+1)}T_{f}(r)\leq\sum_{k\neq i,k\neq
j}N_{(f,a_{k})}^{[n]}(r)+o(T_{f}(r))=o(T_{f}(r)).$
This is a contradiction. Thus $\lambda=1\ (1\leq i<j\leq q).$
Define
$I_{1}=\\{i\in\\{1,\ldots,q-1\\}:F_{q}^{i0}=F_{q}^{i1}\\},$
$I_{2}=\\{i\in\\{1,\ldots,q-1\\}:F_{q}^{i1}=F_{q}^{i2}\\},$
$I_{3}=\\{i\in\\{1,\ldots,q-1\\}:F_{q}^{i2}=F_{q}^{i0}\\}.$
Since $\sharp(I_{1}\cup I_{2}\cup I_{3})=\sharp\\{1,\ldots,q-1\\}=q-1\geq
3n-2$, there exists $1\leq k\leq 3$ such that $\sharp\ I_{k}\geq n$. Without
loss of generality, we may assume that $\sharp\ I_{1}\geq n$. This implies
that $f^{0}=f^{1}$. This is a contradiction.
Thus, we have $\sharp\ \mathcal{F}(f,\\{a_{i}\\}_{i=1}^{q},1)\leq 2.$
b) Assume that $q>3n^{2}+n+2$.
Take $g\in\mathcal{F}(f,\\{a_{i}\\}_{i=1}^{q},1).$ Suppose that $f\neq g.$ By
changing indices if necessary, we may assume that
$\underbrace{\dfrac{(f,a_{1})}{(g,a_{1})}\equiv\dfrac{(f,a_{2})}{(g,a_{2})}\equiv\cdots\equiv\dfrac{(f,a_{k_{1}})}{(g,a_{k_{1}})}}_{\text{
group
}1}\not\equiv\underbrace{\dfrac{(f,a_{k_{1}+1})}{(g,a_{k_{1}+1})}\equiv\cdots\equiv\dfrac{(f,a_{k_{2}})}{(g,a_{k_{2}})}}_{\text{
group }2}$
$\not\equiv\underbrace{\dfrac{(f,a_{k_{2}+1})}{(g,a_{k_{2}+1})}\equiv\cdots\equiv\dfrac{(f,a_{k_{3}})}{(g,a_{k_{3}})}}_{\text{
group
}3}\not\equiv\cdots\not\equiv\underbrace{\dfrac{(f,a_{k_{s-1}+1})}{(g,a_{k_{s-1}+1})}\equiv\cdots\equiv\dfrac{(f,a_{k_{s}})}{(g,a_{k_{s}})}}_{\text{
group }s},$
where $k_{s}=q.$
For each $1\leq i\leq q,$ we set
$\sigma(i)=\begin{cases}i+n&\text{ if $i+n\leq q$},\\\ i+n-q&\text{ if
$i+n>q$}\end{cases}$
and
$P_{i}=(f,a_{i})(g,a_{\sigma(i)})-(g,a_{i})(f,a_{\sigma(i)}).$
By supposition that $f\neq g$, the number of elements of each group is at most
$n$. Hence $\dfrac{(f,a_{i})}{(g,a_{i})}$ and
$\dfrac{(f,a_{\sigma(i)})}{(g,a_{\sigma(i)})}$ belong to distinct groups. This
means that $P_{i}\not\equiv 0\ (1\leq i\leq q)$.
Fix an index $i$ with $1\leq i\leq q.$ It is easy to see that
$\displaystyle\nu_{P_{i}}(z)\geq\min\\{\nu_{(f,a_{i})},\nu_{(g,a_{i})}\\}+\min\\{\nu_{(f,a_{\sigma(i)})},\nu_{(g,a_{\sigma(i)})}\\}+\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,\sigma(i)}}}}}^{q}\nu_{(f,a_{v})}^{[1]}(z)$
outside a finite union of analytic sets of dimension $\leq m-2.$ Since
$\min\\{a,b\\}+n\geq\min\\{a,n\\}+\min\\{b,n\\}$ for all positive integers $a$
and $b$, the above inequality implies that
$\displaystyle
N_{P_{i}}(r)\geq\sum_{v=i,\sigma(i)}\left(N^{[n]}_{(f,a_{v})}(r)+N^{[n]}_{(g,a_{v})}(r)-nN^{[1]}_{(f,a_{v})}(r)\right)+\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,\sigma(i)}}}}}^{q}N^{[1]}_{(f,a_{v})}(r).$
On the other hand, by the Jensen formula, we have
$\displaystyle N_{P_{i}}(r)=$ $\displaystyle\int_{S(r)}\log|P_{i}|\eta+O(1)$
$\displaystyle\leq$
$\displaystyle\int_{S(r)}\log(|(f,a_{i})|^{2}+|(f,a_{\sigma(i)}|^{2})^{\frac{1}{2}}\eta+\int_{S(r)}\log(|(g,a_{i})|^{2}+|(g,a_{\sigma(i)}|^{2})^{\frac{1}{2}}\eta+O(1)$
$\displaystyle\leq$ $\displaystyle T_{f}(r)+T_{g}(r)+o(T_{f}(r)).$
This implies that
$\displaystyle T_{f}(r)+T_{g}(r)\geq$
$\displaystyle\sum_{v=i,\sigma(i)}\left(N^{[n]}_{(f,a_{v})}(r)+N^{[n]}_{(g,a_{v})}(r)-nN^{[1]}_{(f,a_{v})}(r)\right)$
$\displaystyle+$ $\displaystyle\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,\sigma(i)}}}}}^{q}N^{[1]}_{(f,a_{v})}(r)+o(T_{f}(r)).$
Summing-up both sides of the above inequality over $i=1,\ldots,q$ and by
Corollary 1.2(b), we have
$\displaystyle q(T_{f}(r)+T_{g}(r))\geq$ $\displaystyle
2\sum_{v=i}^{q}\left(N^{[n]}_{(f,a_{v})}(r)+N^{[n]}_{(g,a_{v})}(r)\right)$
$\displaystyle+(q-2n-2)\sum_{v=1}^{q}N^{[1]}_{(f,a_{v})}(r)+o(T_{f}(r))$
$\displaystyle\geq$
$\displaystyle(2+\frac{q-2n-2}{2n})\sum_{v=i}^{q}\left(N^{[n]}_{(f,a_{v})}(r)+N^{[n]}_{(g,a_{v})}(r)\right)+o(T_{f}(r))$
$\displaystyle\geq$
$\displaystyle(2+\frac{q-2n+2}{2n})\dfrac{2q}{3(n+1)}(T_{f}(r)+T_{g}(r))+o(T_{f}(r)).$
Letting $r\to\infty$, we get
$q\geq(2+\frac{q-2n-2}{2n})\dfrac{2q}{3(n+1)}\Leftrightarrow q\leq
3n^{2}+n+2.$ This is a contradiction.
Then $f=g$. This implies that
$\sharp\mathcal{F}(f,\\{a_{i}\\}_{i=1}^{q},1)=1$. The theorem is proved.
$\square$
## References
* [1] H. Fujimoto, Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into ${\mathbf{P}}^{N_{1}}({\mathbf{C}})\times\ldots\times{\mathbf{P}}^{N_{k}}({\mathbf{C}}),$ Japanese J. Math. 11 (1985), 233-264.
* [2] H. Fujimoto, Uniqueness problem with truncated multiplicities in value distribution theory, Nagoya Math. J. 152 (1998), 131-152.
* [3] J. Noguchi and T. Ochiai, Introduction to Geometric Function Theory in Several Complex Variables, Trans. Math. Monogr. 80, Amer. Math. Soc., Providence, Rhode Island, 1990.
* [4] M. Ru, A uniqueness theorem with moving targets without counting multiplicity, Proc. Amer. Math. Soc. 129 (2001), 2701-2707.
* [5] M. Ru and J. T-Y. Wang, Truncated second main theorem with moving targets, Trans. Amer. Math. Soc. 356 (2004), 557-571.
* [6] D. D. Thai and S. D. Quang, Uniqueness problem with truncated multiplicities of meromorphic mappings in several complex variables for moving targets, Internat. J. Math., 16 (2005), 903-939.
* [7] D. D. Thai and S. D. Quang, Second main theorem with truncated counting function in several complex variables for moving targets, Forum Mathematicum 20 (2008), 145-179.
* [8] B. Shiffman, Introduction to the Carlson - Griffiths equidistribution theory, Lecture Notes in Math. 981 (1983), 44-89.
|
arxiv-papers
| 2014-02-13T14:49:02 |
2024-09-04T02:49:58.198233
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Si Duc Quang",
"submitter": "Duc Quang Si",
"url": "https://arxiv.org/abs/1402.3156"
}
|
1402.3218
|
#
.
, . , .
, . . . . , . . , . . .
Annotation. The paper explores connection between the order and the type of an
entire function and the speed of the best polynomial approximation in the unit
disk. The relations which define the order and the type of an entire function
through the sequence of its best approximations, have been found. The results
were obtained by generalization previous results of Reddy, I.I. Ibragimov and
N. I. Shyhaliev, S. B. Vakarchyk, R. Mamadov.
2000 MSC. 41 10, 41 25, 41 58.
. , , , .
## 1
$X$, $\mathbb{D}$ , $\|\cdot\|$ . , $\|\cdot\|$
$i)\quad\,\|f(\cdot e^{it})\|\,\equiv\|f(\cdot)\|\,$ (1)
$t\in\mathbb{R}$ $f\in X;$
$ii)\quad\,\|f(\cdot)\|\,<\infty\,$ (2)
( . . $X$ );
$iii)\quad\,\|\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(ze^{it})g(t)\,dt\|\leq\,\frac{1}{2\pi}\int\limits_{0}^{2\pi}|g(t)|\,dt\,\,\|f(\cdot)\|$
(3)
$f\in X$ $g\in L[0;2\pi]$ ( , $f\in X$ $g\in L[0;2\pi]$ $\|f\ast
g\|\leq\|f\|\,\|g\|_{L[0;2\pi]}$).
, ( , i), ii) iii) ). .
1) $B$ , $\mathbb{D}$ $\overline{\mathbb{D}}$
$\|f\|=\max\limits_{z\in\overline{\mathbb{D}}}|f(z)|<\infty\,.$
2) $H_{p}$ ($p\geq 1$) , $\mathbb{D}$
$\|f\|=\sup\limits_{0<r<1}M_{p}\,(f,r),\quad
M_{p}\,(f,r):=\left(\frac{1}{2\pi}\int\limits_{0}^{2\pi}|f(re^{it})|^{p}\,dt\right)^{\frac{1}{p}}\;,\quad
p\in[1;\infty);$
$\|f\|=\sup\limits_{z\in\mathbb{D}}|f(z)|\,,\,\quad p=\infty.$
3) $H_{p}^{\prime}$ , $\mathbb{D}$ $p\in[1;\infty)$
$\|f\|=\left(\frac{1}{\pi}\int\limits_{\;\;z\in
D}\int|f(x+iy)|^{p}\;dxdy\right)^{\frac{1}{p}}\;,$
( ) $H_{p,\,\rho}^{\prime}$ , $\mathbb{D}$ $p\in[1;\infty)$
$\|f\|=\left(\frac{1}{\pi}\int\limits_{\;\;z\in
D}\int|f(x+iy)|^{p}\;\rho(|z|)dxdy\right)^{\frac{1}{p}}\;$
$\rho(|z|)$.
4) $A_{p},\quad p\in(0;1)\;$ , $\mathbb{D}$
$\|f\|=\int\limits_{0}^{1}(1-r)^{\frac{1}{p}-2}M_{1}\,(f,r)\,dr\,,$
[1] , [2].
5) $\mathcal{B}_{p,\,q,\,\lambda}\,,\quad 0<p<q\leq\infty,\quad\lambda>0,\;$ ,
$\mathbb{D}$
$\|f\|=\left\\{\int\limits_{0}^{1}(1-r)^{\lambda\,p\;q\,(q-p)^{-1}}M_{q}^{\lambda}(f,r)\,dr\right\\}^{\frac{1}{\lambda}},\,\quad\lambda<\infty,$
$\|f\|=\sup\limits_{0<r<1}\left\\{(1-r)^{\,p\;q\,(q-p)^{-1}}M_{q}\,(f,r)\right\\}\,,\quad\lambda=\infty,$
[1] ( . [3]).
6) $H^{p,\,q,\,\alpha},\,(p,q\geq 1,\quad\,\alpha>0),$ , $\mathbb{D}$
$\|f\|=\left\\{\int\limits_{0}^{1}(1-r)^{q\alpha-1}M_{p}^{q}(f,r)\,dr\right\\}^{\frac{1}{q}}\,,\quad
q<\infty,$
$\|f\|=\sup\limits_{0<r<1}\left\\{(1-r)^{\alpha}M_{p}\,(f,r)\right\\}\,,\quad
q=\infty,$
[1]. , $H^{p,\,q,\,\alpha}$ $\mathcal{B}_{p,\,q,\,\lambda}\,$ .
7) $\quad BMOA$ [4], $f\in H_{1}$
$\|f\|=\sup\limits_{I}\int\limits_{I}|f(\zeta)-f_{I}|d\sigma(\zeta)\,,$
$f(\zeta)$ \- $f(z)$ , $f_{I}$ \- $f(\zeta)$ $I$.
8) $\mathcal{B}_{\alpha}\,$, $\alpha\in(0,\infty)$, , $D$
$\|f\|=|f(0)|+\sup\limits_{z\in D}(1-|z|^{2})^{\alpha}|f^{\prime}(z)|\,.$
$\mathcal{B}_{\alpha}\,$ [5], $\alpha=1$ $\mathcal{B}_{\alpha}$ $\mathcal{B}$.
9) . . [6] $\mathcal{A}_{p,q}^{s}(\mathbb{D})\,$ , . .
$\mathcal{B}_{p,q}^{s}[-1;\;1]\,$. $f\in H_{p}$, $p\in[1;\infty]$
$\|f\|=\left\\{\int\limits_{0}^{1}\left(\frac{\omega_{m}(f,\,t)_{p}}{t^{s}}\right)^{q}\frac{dt}{t}\right\\}^{\frac{1}{q}}\,+\sup\limits_{0<r<1}M_{p}\,(f,r).$
$q\in[1;\infty],\,s>0,\,m>s$ \- , $\omega_{m}(f,\,t)_{p}$ \- $m$\- $L_{p}\,$
$f(e^{i\cdot})$, $f$. $q=\infty$ .
10) $\mathcal{D}_{p}(\alpha)\,$ , $\mathbb{D}$,
$\|f(z)\|=\left(\sum\limits_{k=0}^{\infty}|c_{k}|^{p}\,\alpha_{k}\right)^{1/p}\,,$
$c_{k}=c_{k}(f)$ \- $f$, $p\geq 1$, ${\alpha}=\\{\alpha_{k}\\}$ -
$\limsup\limits_{k\rightarrow\infty}\left(\alpha_{k}\right)^{\frac{1}{k}}<\infty,\quad\liminf\limits_{k\rightarrow\infty}\left(\alpha_{k}\right)^{\frac{1}{k}}\geq
1.$
, i), ii) iii) .
$E_{n}(f)\equiv E_{n}(f,\,L_{n})\;$ $f\in X$ $L_{n}$:
$E_{n}(f):=\inf\limits_{p\in L_{n}}\|f-p\,\|\;.$
$L_{n}$ $\mathcal{P}_{n}$ $(n-1)$, $p^{*}_{n}$ \- $(n-1)$ $f$, . . ,
$E_{n}(f)=\|f-p^{*}_{n}\|$. [7] , $E_{n}(f)$ $X=H_{2}^{\prime}$. 1976 . . . .
. $X=H_{p}^{\prime}$ $p\geq 1$. [8] ( . [9]).
###### 1.1.
, $f(z)\in H_{p}^{\prime}$, ($p\geq 1$, ) , ,
$\lim_{n{\rightarrow}\infty}(E_{n}(f))^{\frac{1}{n}}=0.$ (4)
###### 1.2.
, $f(z)\in H_{p}^{\prime}$, ($p\geq 1$, ) $\rho$, ,
$\limsup\limits_{n\rightarrow\infty}\,\frac{n\ln n}{-\ln E_{n}(f)}=\rho.$ (5)
###### 1.3.
, $f(z)\in H_{p}^{\prime}$, ($p\geq 1$, ) $\rho$ $\sigma$, ,
$\limsup\limits_{n\rightarrow\infty}\,n(E_{n}(f))^{\frac{\rho}{n}}=\sigma
e\rho.$ (6)
1990 . [10, 11] . . $\mathcal{B}_{p,\,q,\,\lambda}$. . ( , 2009 ., [12])
$\rho(|z|)$.
, , 1) $-$ 10). , .
$X$ $p_{n}(z)$ ( , ; $(n-1)$ ${\mathcal{P}}_{n}[\mathbb{Z}]$ ). 5\. , .
## 2 .
.
###### 2.1.
$f\in X$.
$\lim\limits_{n\rightarrow\infty}(E_{n}(f))^{\frac{1}{n}}=0$ (7)
, $f$ .
###### 2.2.
, $f\in X$ $\rho\in(0;\infty)$ ,
$\limsup\limits_{n\rightarrow\infty}\,\frac{n\ln
n}{\ln\frac{\|z^{n}\|}{E_{n}(f)}}=\rho.$ (8)
###### 2.3.
$\lim\limits_{n\rightarrow\propto}(\|z^{n}\|)^{\frac{1}{n}}=\mu$. , $f\in X$
$\rho\in(0;\infty)$ $\sigma\in(0;\infty)$ ,
$\limsup\limits_{n\rightarrow\infty}\,\frac{n}{e\rho}\left(\frac{E_{n}(f)}{\|z^{n}\|}\right)^{\frac{\rho}{n}}=\sigma.$
(9)
## 3 .
i) ii) . , iii).
###### 3.1.
iii) $B$; $H_{p}$ ($p\geq 1$); $H_{p,\,\rho}^{\prime}$ ($p\geq 1$);
$A_{p},\quad(p\in(0;1))$; $H^{p,\,q,\,\alpha}\,,\quad(p,q\geq
1,\quad\alpha>0)$; $\quad BMOA$;
$\mathcal{B}_{\alpha},\quad(\alpha\in(0;\infty))$;
$\mathcal{A}_{p,q}^{s}(\mathbb{D})\,\,(p,q\in[1;\infty],\,s>0)$;
$\mathcal{D}_{p}(\alpha)\,$ ($p\geq 1$).
###### Proof.
$B$ $A_{p}$ iii) ; $H_{p}$, $H_{p,\,\rho}^{\prime}$ $p\geq 1$ iii) $L_{p}$.
$\quad BMOA$
$\|\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(ze^{it})g(t)\,dt\|=$
$=\sup\limits_{I}\int\limits_{I}\left|\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(e^{i(t+\varphi)})g(t)dt-\frac{1}{|I|}\int\limits_{I}\left(\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(e^{i(t+u)})g(t)dt\right)du\right|d\varphi=$
$=\sup\limits_{I}\int\limits_{I}\left|\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(e^{i(t+\varphi)})g(t)\,dt-\frac{1}{2\pi}\int\limits_{0}^{2\pi}\left(\frac{1}{|I|}\int\limits_{I}f(e^{i(t+u)})g(t)du\right)dt\right|\,d\varphi=$
$=\sup\limits_{I}\int\limits_{I}\left|\frac{1}{2\pi}\int\limits_{0}^{2\pi}g(t)\left(f(e^{i(t+\varphi)})-\frac{1}{|I|}\int\limits_{I}f(e^{i(t+u)})du\right)dt\right|\,d\varphi\leq\frac{1}{2\pi}\int\limits_{0}^{2\pi}|g(t)|\,dt\,\,\|f(\cdot)\|$
i) $X.$
iii) $\mathcal{B}_{\alpha}$.
$\|f\ast
g\|=\left|\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(0)g(t)\,dt\right|+\sup\limits_{z\in
D}(1-|z|^{2})^{\alpha}\,\left|\frac{1}{2\pi}\int\limits_{0}^{2\pi}f^{\prime}(ze^{it})g(t)e^{it}\,dt\right|\,\leq$
$\leq|f(0)|\,\|g(t)\|_{L}\,+\sup\limits_{z\in
D}\,\frac{1}{2\pi}\int\limits_{0}^{2\pi}(1-|z|^{2})^{\alpha}\,|f^{\prime}(ze^{it})g(t)e^{it}|\,dt\,\leq$
$\leq\|g(t)\|_{L}(|f(0)|+\sup\limits_{z\in
D}\,(1-|z|^{2})^{\alpha}\,|f^{\prime}(ze^{it})|)=\|f\|\,\|g(t)\|_{L}.$
$f\in H^{p,\,q,\,\alpha}$, $g\in L_{[0;\;2\pi]},\quad z=re^{i\varphi}$.
$H^{p,\,q,\,\alpha}$
$\|f\ast
g\|=\left\\{\int\limits_{0}^{1}(1-r)^{q\alpha-1}\left(\frac{1}{2\pi}\int\limits_{0}^{2\pi}|\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(re^{i(\varphi+t)})g(t)\,dt|^{p}d\varphi\right)^{\frac{q}{p}}\right\\}^{\frac{1}{q}}\leq$
$\leq\left\\{\int\limits_{0}^{1}(1-r)^{q\alpha-1}\left(\frac{1}{2\pi}\int\limits_{0}^{2\pi}\,dt\left(\frac{1}{2\pi}\int\limits_{0}^{2\pi}|f(re^{i(\varphi+t)})g(t)|^{p}d\varphi\right)^{\frac{1}{p}}\right)^{q}\right\\}^{\frac{1}{q}}=\|f\|\,\|g\|_{L}.$
, iii) $\mathcal{A}_{p,q}^{s}(\mathbb{D})\,$.
$f\ast
g\,\,\,(e^{i\varphi})=\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(e^{i(\varphi+t)})g(t)\,dt,$
$m$\- $\Delta^{h}_{m}(\cdot,\,\varphi)$
$\Delta^{h}_{m}(f\ast
g,\varphi)=\frac{1}{2\pi}\int\limits_{0}^{2\pi}\Delta^{h}_{m}(f(e^{i(\cdot\,\,+t)}),\varphi)\,g(t)\,dt$
, ,
$\omega_{m}(f\ast g,\,h)_{p}\leq\omega_{m}(f,\,h)_{p}\,\|g\|_{L}.$
iii) $H_{p}\,$ iii) $\mathcal{A}_{p,q}^{s}(\mathbb{D})\,$.
iii) $\mathcal{D}_{p}(\alpha)\,$. $c_{k}$ \- $f\in\mathcal{D}_{p}(\alpha)$,
$b_{k}$ \- $g$, $z\in\mathbb{D}$.
$f\ast
g(z)=\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(ze^{it})g(t)\,dt=\sum\limits_{k=0}^{\infty}c_{k}b_{-k}z^{k}$
$\|f\ast
g\|=\left(\sum\limits_{k=0}^{\infty}|c_{k}b_{-k}|^{p}\,\alpha_{k}\right)^{1/p}\,\leq\sup\limits_{k}|b_{k}|\left(\sum\limits_{k=0}^{\infty}|c_{k}|^{p}\,\alpha_{k}\right)^{1/p}\leq\|f\|\,\|g\|_{L}.$
∎
, 2.3 $\|z^{n}\|$. ,
$\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=1$. $B$ $H_{p}$,
($p\geq 1$) $n\geq 0\quad$ $\|z^{n}\|=1$ , , 2.3 . $H_{p}^{\prime}\,$, ($p\geq
1$) $n\geq 0\quad$ $\|z^{n}\|=(np+2)^{\frac{-1}{p}}$ 2.3 , 2.1 \- 2.3 [8].
$\mathcal{B}_{p}$ $\,\,p\in(0;1)\;$ $\|z^{n}\|=(2\pi
B(\frac{1}{p}-1,\,np+1))^{\frac{1}{p}}$, $B(\cdot,\,\cdot)$ \- \- . - ,
$\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=1$
$\mathcal{B}_{p}$ 2.3 . $\mathcal{B}_{p,\,q,\,\lambda}\,$
$\lambda<\infty\quad$ $\|z^{n}\|=B(\lambda n+1,\,\frac{\lambda pq}{q-p}+1)$ ,
, $\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=1$.
$\lambda=\infty\quad$
$\|z^{n}\|=\sup\limits_{0<r<1}\,r^{n}(1-r)^{\frac{pq}{q-p}}$.
$(1-\frac{1}{n})^{n}n^{\frac{pq}{p-q}}\leq\|z^{n}\|<1$,
$\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=1$ 2.3
$\mathcal{B}_{p,\,q,\,\lambda}$. 2.1 - 2.3 . . [10]. $H_{p,\rho}^{\prime}$
$\|z^{n}\|=\left(2\int\limits_{0}^{1}\;t^{pn+1}\;\rho(t)dt\right)^{\frac{1}{p}}\;\leq\left(2\int\limits_{0}^{1}\;t\;\rho(t)dt\right)^{\frac{1}{p}}\;$
$\limsup\limits_{n\rightarrow\infty}\,(\|z^{n}\|)^{\frac{1}{n}}\leq 1$ ,
$t\rho(t)$ [0; 1]. , $\varepsilon\in(0;1)$
$\|z^{n}\|\geq\left(2\int\limits_{1-\varepsilon}^{1}\;t^{pn+1}\;\rho(t)dt\right)^{\frac{1}{p}}\;\geq(1-\varepsilon)^{n}\;\left(2\int\limits_{1-\varepsilon}^{1}\;t\;\rho(t)dt\right)^{\frac{1}{p}}\;$
$\liminf\limits_{n\rightarrow\infty}\,(\|z^{n}\|)^{\frac{1}{n}}\geq(1-\varepsilon)$,
$\rho(t)$ , $\int\limits_{1-\varepsilon}^{1}\;t\;\rho(t)dt>0$
$\varepsilon\in(0;1)$. $\varepsilon\in(0;1)$
$\limsup\limits_{n\rightarrow\infty}\,(\|z^{n}\|)^{\frac{1}{n}}\leq 1$
$\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=1$. , 2.3
$H_{p,\rho}^{\prime}$. . ( . [12]).
, 2.3 $\mathcal{A}_{p,q}^{s}(\mathbb{D})\,$.
$|\Delta^{h}_{m}(e^{in\cdot})|=|(1-e^{inh})^{m}|=(2\sin\frac{nh}{2})^{m}$,
$\|z^{n}\|=\left\\{\int\limits_{0}^{\frac{\pi}{n}}\left(\frac{(2\sin\frac{nt}{2})^{m}}{t^{s}}\right)^{q}\frac{dt}{t}+\int\limits_{\frac{\pi}{n}}^{1}2^{mq}t^{-sq-1}dt\right\\}^{\frac{1}{q}}\,+1\leq$
$\leq\left\\{\int\limits_{0}^{\frac{\pi}{n}}n^{mq}t^{(m-s)q-1}dt+\int\limits_{\frac{\pi}{n}}^{1}2^{mq}t^{-sq-1}dt\right\\}^{\frac{1}{q}}\,+1\leq
C\,n^{q},$
C - , $m,s,q$. , $\mathcal{A}_{p,q}^{s}(\mathbb{D})\,$
$\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=1$.
$\quad BMOA$. $f=z^{n}$, $f_{I}$ $\|z^{n}\|$ $\quad BMOA$. $I$ $e^{it_{1}}$
$e^{it_{2}},\quad t_{2}>t_{1},\quad x=\frac{t_{1}+t_{2}}{2},\quad
h=\frac{t_{2}-t_{1}}{2}$.
$f_{I}=\frac{1}{t_{2}-t_{1}}\,\int\limits_{t_{1}}^{t_{2}}e^{int}dt=\frac{\sin
nh}{nh}\,e^{inx}$
$A=\frac{1}{t_{2}-t_{1}}\,\int\limits_{t_{1}}^{t_{2}}|e^{int}-\frac{\sin
nh}{nh}\,e^{inx}|\,dt=\frac{1}{2h}\,\int\limits_{-h}^{h}|e^{int}-\frac{\sin
nh}{nh}\,|\,dt.$
$A\leq 2$, $\|z^{n}\|=\sup\limits_{h\in[0;\,\pi]}A\leq 2$. $\|z^{n}\|$ .
$\|z^{n}\|\geq\sup\limits_{h=\frac{\pi}{2n}}A=\sup\limits_{n}\frac{n}{\pi}\,\int\limits_{\frac{-\pi}{2n}}^{\frac{\pi}{2n}}\left(1-\frac{2}{\pi}\,\cos
nt+\frac{4}{\pi^{2}}\right)^{\frac{1}{2}}\,dt\geq\sup\limits_{n}\frac{2n}{\pi}\,\int\limits_{0}^{\frac{\pi}{2n}}\left(\frac{2}{\pi}\,\right)^{\frac{1}{2}}\,dt=\left(\frac{2}{\pi}\,\right)^{\frac{1}{2}}.$
$\,\|z^{n}\|\,$ ,
$\,BMOA\,\,\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=1$.
$\mathcal{B}_{\alpha}$ $n\geq 1$
$\|z^{n}\|=\sup\limits_{z\in
D}|z^{n}|(1-|z|^{2})^{\alpha}\,=\left(\frac{n}{n+\alpha}\right)^{\frac{n}{2}}\,\left(\frac{2\alpha}{n+2\alpha}\right)^{\alpha},$
, $\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=1$.
$\mathcal{D}_{p}(\alpha)\,$ $\|z^{n}\|=(\alpha_{n})^{\frac{1}{p}}$ e 2.3 ,
$\lim\limits_{n\rightarrow\infty}(\alpha_{n})^{\frac{1}{n}}=1$.
, 2.1-2.3 .
$BMOA,\,\mathcal{B}_{\alpha},\,\mathcal{D}_{p}(\alpha),\,\mathcal{A}_{p,q}^{s}(\mathbb{D})$
2.1-2.3 .
## 4 .
. .
###### 4.1.
$f\in X\quad$ $\quad f(z)=\sum\limits_{k=0}^{\infty}c_{k}z^{k}\quad$
$\quad\mathbb{D}.\quad$
$|c_{n}|\,\|z^{n}\|\leq E_{n}(f)\leq\|f\|.$
###### Proof.
$c_{n}z^{n}=\frac{1}{2\pi
i}\int\limits_{|\zeta|=1}\frac{f(z\zeta)-P_{n}(z\zeta)}{\zeta^{n+1}}d\zeta\quad,$
$P_{n}$ \- $f(z)$ $(n-1)$. $\quad|c_{n}|\|z^{n}\|\leq
E_{n}(f)\leq\|f(z)\|\quad$ iii) i) $X.$ ∎
###### 4.2.
$f\in X\quad$
$\,\mu_{1}:=\liminf\limits_{n\rightarrow\infty}\,(\|z^{n}\|)^{\frac{1}{n}}$,
$\mu_{2}:=\limsup\limits_{n\rightarrow\infty}\,(\|z^{n}\|)^{\frac{1}{n}}$.
$\mu_{1}\geq 1,\,\mu_{2}<\infty$.
###### Proof.
$\,\beta_{n}=(\|z^{n}\|)^{\frac{1}{n}}$. , $\,\mu_{2}<\infty$. ,
$\,\beta_{n_{k}}$ , $\lim\limits_{k\rightarrow\infty}\,\beta_{n_{k}}=\infty$.
$f_{0}$,
$f_{0}(z)=\sum\limits_{k=0}^{\infty}\,(\beta_{n_{k}})^{\frac{-n_{k}}{2}}z^{n_{k}}.\,$
$X$. 4.1
$k\quad(\beta_{n_{k}})^{\frac{-n_{k}}{2}}\|z^{n_{k}}\|\leq\|f_{0}\|<\infty\,$,
. , $\mu_{1}\geq 1,\,$ , . . $\mu_{1}<1$. $\varrho\in(\mu_{1};\,1)$ ,
$\quad f_{0}(z)=\sum\limits_{k=0}^{\infty}\varrho^{-n_{k}}z^{n_{k}},$ (10)
${n_{k}}$ ,
$\liminf\limits_{n\rightarrow\infty}\,\beta_{n}=\lim\limits_{k\rightarrow\infty}\,\beta_{n_{k}}=\mu_{1}.$
$f_{0}$ $|z|<\rho,$ $\,\mathbb{D}$. , $S_{n,f_{0}}(z)$ (10) $X$ , , $f_{1}\in
X$. , $f_{0}$ $f_{1}$ . $k\in\mathbb{N}\bigcup\\{0\\},\,n>k$
$c_{k}(f_{1})=c_{k}(S_{n,f_{0}})+c_{k}(f_{1}-S_{n,f_{0}})=c_{k}(f_{0})+c_{k}(f_{1}-S_{n,f_{0}}).$
$n\rightarrow\infty$ 2.1 $c_{k}(f_{1})=c_{k}(f_{0})$. , $f_{1}\in X$,
$\mathbb{D}$, $X$. , , $\mu_{1}<1$ .
∎
###### 4.3.
$f\in X$, $\,K$\- , $\,K\subset\mathbb{D}$. $\,z\in K$
$|f(z)|\leq C\,\|f\|,$
$C$ \- , $f$ $z$.
###### Proof.
$\quad d:=\sup\\{|z|:\quad z\in K\\},\,d<1$. $f$ , 4.1 , $\,z\in K$.
$\quad f(z)=\sum\limits_{k=0}^{\infty}c_{k}z^{k},$
$|f(z)|\leq\sum\limits_{k=0}^{\infty}|c_{k}|\,|z^{k}|\,\leq\,\|f(z)\|\sum\limits_{k=0}^{\infty}\frac{d^{k}}{\|z^{k}\|}\,\leq
C\,\|f\|$
, 4.2. ∎
###### 1.
4.3 , $\,z\in K$ $X$ $K\subset\mathbb{D}$. , $n$ $\,z\in K$ $f^{(n)}(z)$ $X$.
2.1.
###### Proof.
. $f(z)=\sum\limits_{k=0}^{\propto}c_{k}z^{k}$ $z\in\mathbb{D}$.
4.1 $\quad|c_{n}|\,\|z^{n}\|\leq E_{n}(f)$.
$|c_{n}|\leq\frac{E_{n}(f)}{\|z^{n}\|}\quad\mbox{
}\quad\lim\limits_{n\rightarrow\infty}|c_{n}|^{\frac{1}{n}}\leq\lim\limits_{n\rightarrow\infty}\left(\frac{E_{n}(f)}{\|z^{n}\|}\right)^{\frac{1}{n}}=0\,,\quad\mbox{
. . }\,f\mbox{- .}$
. $f\in X$ $f_{\zeta}(z):=f(z\zeta)$. $f$ \- , $f_{R}\in X$ $R>1$ , 2 2 [13] (
. [14], . 2.3),
$E_{n}(f)\leq R^{-n}E_{n}(f_{R})\leq R^{-n}\|f_{R}\|.$
4.2
$0\leq\lim\limits_{n\rightarrow\infty}\left(\frac{E_{n}(f)}{\|z^{n}\|}\right)^{\frac{1}{n}}\leq\frac{1}{R}\,\limsup\limits_{n\rightarrow\infty}\left(\frac{1}{\|z^{n}\|}\right)^{\frac{1}{n}}\leq\frac{1}{R}.$
4.2 $R>1$ ,
$\lim\limits_{n\rightarrow\infty}\left(E_{n}(f)\right)^{\frac{1}{n}}=0.$
∎
###### 2.
[13] [14] iii), , , . iii) 2 2 [13] .
2.2.
###### Proof.
. (8) , 2.1 , , $f$ \- . $\alpha$ .
$\alpha=\limsup\limits_{n\rightarrow\infty}\,\frac{n\ln
n}{-\ln|c_{n}|}\leq\limsup\limits_{n\rightarrow\infty}\,\frac{n\ln
n}{\ln\frac{\|z^{n}\|}{E_{n}(f)}}=\rho$ (11)
4.1. , $\alpha>0$. , . .
$\limsup\limits_{n\rightarrow\infty}\,\frac{n\ln n}{-\ln|c_{n}|}=0.$
$\varepsilon\in(0,\,1)$ $N_{\varepsilon}$ , $n>N_{\varepsilon}$ $n\ln
n<-\varepsilon\ln{|c_{n}|}$
$|c_{n}|<n^{\frac{-n}{\varepsilon}}.$
$E_{n}(f)$ $n>N_{\varepsilon}$. $N_{\varepsilon}$ ,
$\|z^{n}\|\leq(\mu_{2}+\varepsilon)^{n}$ $\|z^{n}\|\geq(1-\varepsilon)^{n}$
$n\geq N_{\varepsilon}$.
$E_{n}(f)\leq\|\sum\limits_{k=n}^{\infty}c_{k}z^{k}\|\leq\sum\limits_{k=n}^{\infty}k^{\frac{-k}{\varepsilon}}\,(\mu_{2}+\varepsilon)^{k}\leq\sum\limits_{k=n}^{\infty}n^{\frac{-k}{\varepsilon}}\,\left(\mu_{2}+\varepsilon\right)^{k}=$
$=n^{\frac{-n}{\varepsilon}}(\mu_{2}+\varepsilon)^{n}(1-\frac{\mu_{2}+\varepsilon}{n^{\frac{1}{\varepsilon}}})^{-1}$
(12)
$n>(\mu_{2}+\varepsilon)^{\varepsilon}.$ (10)
$\frac{\|z^{n}\|}{E_{n}(f)}\geq\left(\frac{1-\varepsilon}{\mu_{2}+\varepsilon}\right)^{n}\,n^{\frac{n}{\varepsilon}}\left(1-\frac{\mu_{2}+\varepsilon}{n^{\frac{1}{\varepsilon}}}\right),$
$\ln\left(\frac{\|z^{n}\|}{E_{n}(f)}\right)^{\frac{1}{n}}\geq\ln\frac{1-\varepsilon}{\mu_{2}+\varepsilon}+\,\frac{1}{\varepsilon}\ln
n+\frac{1}{n}\ln\left(1-\frac{\mu_{2}+\varepsilon}{n^{\frac{1}{\varepsilon}}}\right).$
$\liminf\limits_{n\rightarrow\infty}\,\frac{\ln\left(\frac{\|z^{n}\|}{E_{n}(f)}\right)^{\frac{1}{n}}}{\ln
n}\geq\frac{1}{\varepsilon},$
$\rho=\limsup\limits_{n\rightarrow\infty}\,\,\frac{n\ln
n}{\ln\frac{\|z^{n}\|}{E_{n}(f)}}\leq\varepsilon,$
.
$\varepsilon\in(0;\frac{1}{2})\cap(0;\alpha)$. ,
$\alpha=\limsup\limits_{n\rightarrow\infty}\,\frac{n\ln n}{-\ln|c_{n}|}$
, $N_{\varepsilon}\in\mathbb{N}$, $\varepsilon$ , $|c_{n}|\leq
n^{-\frac{n}{\alpha+\varepsilon}}$ $n\geq N_{\varepsilon}$. $N_{\varepsilon}$
, $\|z^{n}\|\leq(\mu_{2}+\varepsilon)^{n}$ $\|z^{n}\|\geq(1-\varepsilon)^{n}$
$n\geq N_{\varepsilon}$. $n>N_{\varepsilon}$
$E_{n}(f)\leq\|\sum\limits_{k=n}^{\propto}c_{k}z^{k}\|\leq\sum\limits_{k=n}^{\propto}|c_{k}|\,\|z^{k}\|\leq\sum\limits_{k=n}^{\propto}k^{\frac{-k}{\alpha+\varepsilon}}\,\|z^{k}\|\leq$
$\leq\sum\limits_{k=n}^{\propto}n^{\frac{-k}{\alpha+\varepsilon}}(\mu_{2}+\varepsilon)^{k}=\frac{(\mu_{2}+\varepsilon)^{n}}{n^{\frac{n}{\alpha+\varepsilon}}}\cdot\left(1-\frac{\mu_{2}+\varepsilon}{n^{\frac{1}{\alpha+\varepsilon}}}\right)^{-1}.$
(13)
,
$\frac{\|z^{n}\|}{E_{n}(f)}\geq\frac{\|z^{n}\|}{(\mu_{2}+\varepsilon)^{n}}\cdot
n^{\frac{n}{\alpha+\varepsilon}}\left(1-\frac{\mu_{2}+\varepsilon}{n^{\frac{1}{\alpha+\varepsilon}}}\right),$
$\alpha+\varepsilon\geq\frac{n\ln
n}{\ln\frac{\|z^{n}\|}{E_{n}(f)}}\cdot\left(1+\frac{\alpha+\varepsilon}{n\ln
n}\ln\left(1-\frac{\mu_{2}+\varepsilon}{n^{\frac{1}{\alpha+\varepsilon}}}\right)+\frac{\alpha+\varepsilon}{n\ln
n}\ln\frac{\|z^{n}\|}{(\mu_{2}+\varepsilon)^{n}}\right).$ (14)
( 14) $n\rightarrow\infty$, $\alpha+\varepsilon\geq\rho$, $\varepsilon>0$ ,
$\alpha\geq\rho$. , $\alpha=\rho$ .
. $f\in X$ \- $\rho$, . .
$\limsup\limits_{n\rightarrow\infty}\,\frac{n\ln n}{-\ln|c_{n}|}=\rho.$ (15)
$\alpha=\limsup\limits_{n\rightarrow\infty}\,\frac{n\ln
n}{\ln\frac{\|z^{n}\|}{E_{n}(f)}}$
($\alpha$ $\rho$ ) , $\alpha=\rho$. 4.1 (11) , $\alpha\geq\rho$. ,
$\varepsilon$, $0<\varepsilon<1$ $N_{\varepsilon}$ , $|c_{n}|\leq
n^{-\frac{n}{\rho+\varepsilon}}$
$(1-\varepsilon)^{n}\leq\|z^{n}\|\leq(\mu_{2}+\varepsilon)^{n}$
$n>N_{\varepsilon}$.
( 13) ( 14) ( $\alpha$ $\rho$ )
$\rho+\varepsilon\geq\frac{n\ln
n}{\ln\frac{\|z^{n}\|}{E_{n}(f)}}\cdot\left(1+\frac{\rho+\varepsilon}{n\ln
n}\ln\left(1-\frac{\mu_{2}+\varepsilon}{n^{\frac{1}{\rho+\varepsilon}}}\right)+\frac{\rho+\varepsilon}{n\ln
n}\ln\frac{\|z^{n}\|}{(\mu_{2}+\varepsilon)^{n}}\right),$
$\rho+\varepsilon\geq\alpha$ , , $\rho\geq\alpha$. . ∎
2.3.
###### Proof.
. $f\in X$ 2.3 $\rho$ $\sigma$. (9) (8) 2.2, $f$ \- $\rho$. $f$ $\alpha$. ,
$\alpha=\sigma$.
$\alpha=\limsup\limits_{n\rightarrow\infty}\,\frac{n}{e\rho}|c_{n}|^{\frac{\rho}{n}}$
(16)
4.1 $\alpha\leq\sigma$. . (16) , $\varepsilon>0$
$N_{\varepsilon}\in\mathbb{N}$ , $n>N_{\varepsilon}$
$|c_{n}|<\left(\frac{\rho e(\alpha+\varepsilon)}{n}\right)^{\frac{n}{\rho}}.$
(17)
(17) (13), (14)
$E_{n}(f)\leq\sum\limits_{k=n}^{\infty}\left(\frac{\rho
e(\alpha+\varepsilon)}{k}\right)^{\frac{k}{\rho}}\|z^{k}\|\leq\left(\frac{\rho
e(\alpha+\varepsilon)}{n}\right)^{\frac{n}{\rho}}(\mu+\varepsilon)^{n}\times$
$\times\left(1-\frac{C}{n^{\frac{1}{\rho}}}\right)^{-1}\quad,$ (18)
$C=(\mu+\varepsilon)(\rho e(\alpha+\varepsilon))^{\frac{1}{\rho}}$. (18)
$\alpha+\varepsilon\geq\frac{n}{e\rho}\left(\frac{E_{n}(f)}{\|z^{n}\|}\right)^{\frac{\rho}{n}}\frac{\|z^{n\|^{\frac{\rho}{n}}}}{(\mu+\varepsilon)^{\rho}}\left(1-\frac{c}{n^{\frac{1}{\rho}}}\right)^{\frac{\rho}{n}}.$
(19)
(19),
$\alpha+\varepsilon\geq\sigma\left(\frac{\mu}{\mu+\varepsilon}\right)^{\rho},$
(20)
, $\varepsilon$ , $\alpha\geq\sigma$ , .
. $f\in X$ \- , . $\rho$ ( 2.2 (8)) $\alpha$ . , $\alpha=\sigma$. (16) 4.1
$\alpha\leq\sigma$. $\alpha\geq\sigma$ . ∎
2.1 - 2.3 , $f\in X$ . , . , $X$ \- , $\mathbb{D}$, i), ii) iii) ( $f\in X$
$\mathbb{D}$), 2.1 - 2.3.
###### 4.1.
$f\in X$
$\liminf\limits_{n\rightarrow\infty}\,(\|z^{n}\|)^{\frac{1}{n}}=\mu_{1}>0$.
$f$ \- ,
$\lim\limits_{n\rightarrow\infty}\,\left\\{\frac{E_{n}(f)}{\|z^{n}\|}\right\\}^{\frac{1}{n}}=0.$
(21)
, (21), $\\{p^{*}_{n}\\}$ $f$ $X$ $|z|<r,\,r\in(0;\mu_{1})$ .
###### Proof.
2.1. . $f\in X$ (21). , $\,E_{n}(f)\rightarrow 0$ $\,n\rightarrow\infty.$ ,
$m$ $n,\,m\geq n$ $\\{P^{*}_{n}(z)\\}$ $\|P^{*}_{n}(z)-P^{*}_{m}(z)\|\leq
2E_{n}(f)$. 4.3 , $sup$ \- $|z|\leq r$ $r\in(0;\mu_{1})$
$|P^{*}_{n}(z)-P^{*}_{m}(z)|\leq 2CE_{n}(f)$ $C$, $r$ $\mu_{1}$.
$\\{P^{*}_{n}(z)\\}$ $|z|<\mu_{1}$ $g(z)$, $|z|<\mu_{1}$,
$|P^{*}_{n}(z)-g(z)|\leq 2CE_{n}(f)$ $|z|\leq r$. $\gamma_{n}$ $g(z)$
$|\gamma_{n}|\leq 2CE_{n}r^{-n}(f)$. $g(z)$ \- .
∎
2.2 2.3 .
###### 4.2.
$\liminf\limits_{n\rightarrow\infty}\,(\|z^{n}\|)^{\frac{1}{n}}=\mu_{1}>0\,,\quad\limsup\limits_{n\rightarrow\infty}\,(\|z^{n}\|)^{\frac{1}{n}}=\mu_{2}<\infty$,
$f\in X$.
$f$ \- $\rho\in(0;\infty)$,
$\limsup\limits_{n\rightarrow\infty}\,\frac{n\ln
n}{\ln\frac{\|z^{n}\|}{E_{n}(f)}}=\rho.$ (22)
, (22), $\\{p^{*}_{n}\\}$ $f$ $X$ $|z|<r,\,r\in(0;\mu_{1})$
$\rho\in(0;\infty)$, (22).
###### 4.3.
$\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=\mu>0$, $f\in X$.
$f$ \- $\rho\in(0;\infty)$ $\sigma\in(0;\infty)$,
$\limsup\limits_{n\rightarrow\infty}\,\frac{n}{e\rho}\left(\frac{E_{n}(f)}{\|z^{n}\|}\right)^{\frac{\rho}{n}}=\sigma.$
(23)
, (23), $\\{p^{*}_{n}\\}$ $f$ $X$ $|z|<r,\,r\in(0;\mu_{1})$
$\rho\in(0;\infty)$ $\sigma\in(0;\infty)$ ( $\sigma,\rho$ (23)).
###### 3.
$\mu_{1}>0$, $\mu_{2}<\infty$ 4.1-4.2 ( , , ).
$\lim\limits_{n\rightarrow\infty}(\|z^{n}\|)^{\frac{1}{n}}=\mu>0$ 4.3? .
## 5 $f\in X$ ${\mathcal{P}}_{n}[\mathbb{Z}]$
, $\mathbb{C}$
. . . [15] ( . [16]). , $z=0$ , , .
${\mathcal{P}}_{n}[\mathbb{Z}]$ .
$X$. , , 4.1 - 4.4, $X$ , $\mathbb{D}$.
###### 5.1.
$f\in X$ $\inf\limits_{n\in\mathbb{N}}\,\|z^{n}\|>0$.
$p_{n}\in{\mathcal{P}}_{n}[\mathbb{Z}]$ ,
$\lim\limits_{n\rightarrow\infty}\,\|f-p_{n}\|=0,$
$f$ .
$B$ $H_{p}$ [17]. 5.1.
###### Proof.
$\|p_{n+1}-p_{n}\|\leq\|p_{n+1}-f\|+\|f-p_{n}\|\rightarrow 0$
$n\rightarrow\infty$. , $p_{n+1}\neq p_{n}$ , $p_{n+1}-p_{n}$ 4.1
$\|p_{n+1}-p_{n}\|\geq\inf\limits_{n\in\mathbb{N}}\,\|z^{n}\|,$
. . $\|p_{n+1}-p_{n}\|$ . , $n\quad p_{n+1}\equiv p_{n}\equiv f$.
∎
$B$ ( . , [17]).
###### 5.2.
$f\in X$. $p_{n}\in{\mathcal{P}}_{n}[\mathbb{Z}]$ ,
$\lim\limits_{n\rightarrow\infty}\,\|f-p_{n}\|=0,$
$c_{k}:=\frac{f^{k}(0)}{k!}$ \- .
###### Proof.
$p_{n}$
$p_{n}(z)=\sum\limits_{k=0}^{n}a_{k,n}z^{k},$
$a_{k,n}$ . 4.1
$|c_{k}-a_{k,n}|\,\|z^{k}\|\leq\|f(z)-p_{n}(z)\|\rightarrow 0,\quad
n\rightarrow\infty.$
,
$c_{k}=\lim\limits_{n\rightarrow\infty}\,a_{k,n}\in\mathbb{Z}.$
∎
###### 1.
5.2 , $\,z$ , , .
###### 5.3.
$f\in X\quad$, $\quad f(z)=\sum\limits_{k=0}^{\infty}c_{k}z^{k}\quad$
$\quad\mathbb{D}\quad$
$\lim\limits_{n\rightarrow\infty}\,\|f(z)-S_{n}(f,z)\|=0,$
$S_{n}(f,z)$ \- $f$ $n$.
$p_{n}\in{\mathcal{P}}_{n}[\mathbb{Z}]$ ,
$\lim\limits_{n\rightarrow\infty}\,\|f(z)-p_{n}(z)\|=0,$
, $c_{k}$ .
###### Proof.
5.2, - , $S_{n}(f,z)$ .
∎
$H_{p}$ $H^{\prime}_{p},\quad p\in(1,\infty)$, . [17] ( $H^{\prime}_{p},\quad
p\in(1,\infty)$
${\mathcal{P}}_{n}[\mathbb{Z}]$, $H_{p}$ \- ).
###### 5.4.
$f\in X,\quad$ , $p_{n}\in{\mathcal{P}}_{n}[\mathbb{Z}]$ ,
$\lim\limits_{n\rightarrow\infty}\,\|f(z)-p_{n}(z)\|=0.$
$\liminf\limits_{n\rightarrow\infty}\,\|z^{n}\|=0.$
###### Proof.
$p_{n}$ $z^{m_{n}}$, $m_{n}$ , $m_{n}>deg\,p_{n}$, $c_{m_{n}}$ $m_{n}$ ( , ;
). 5.2 $c_{m_{n}}$ ,
$\|z^{m_{n}}\|\leq|c_{m_{n}}|\,\|z^{m_{n}}\|\leq\|f(z)-p_{n}(z)\|\rightarrow
0$
$n\rightarrow\infty$.
∎
$\liminf\limits_{n\rightarrow\infty}\,\|z^{n}\|=0$ 5.4 , .
###### 5.5.
, $X$ $f$,
$p_{n}\in{\mathcal{P}}_{n}[\mathbb{Z}]$, , $X$
$\liminf\limits_{n\rightarrow\infty}\,\|z^{n}\|=0.$
###### Proof.
. . $\liminf\limits_{n\rightarrow\infty}\,\|z^{n}\|=0$ , $n_{k}$ ,
$\|z^{n_{k}}\|\leq 2^{-k}$ $k=1,2,3,...\,$.
$f(z)=\sum\limits_{k=0}^{\infty}z^{n_{k}}\quad.$
, $f$ , $f\in X\,$ $f$ ( $S_{n}(f,z)$ ).
∎
###### 4.
, , iii)
$iii^{\prime})\quad\,\|\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(ze^{it})g(t)\,dt\|\leq\,C\,\frac{1}{2\pi}\int\limits_{0}^{2\pi}|g(t)|\,dt\,\,\|f(\cdot)\|,$
(24)
$C$ $f\in X$ $g\in L_{[0;\,2\pi]}$.
## References
* [1] Hardy G.H., Littlewood J.E. _Some properties of fractional integrals II_ Math. Z., 1931, 34, N 3, 403–439.
* [2] Duren P.L., Romberg B.W., Shields A.L. _Linear functionals in $H_{p}$ spaces with $0<p<1$_ // J. reine und angew. Math. - 1969. - 238, s. 4–60.
* [3] . . _._ . // - 1977. 21, N 2, . 141–150.
* [4] . . _._ // . . . , , - 1985. 23 - . 3–124.
* [5] K. Zhu _Bloch type spaces of analytic functions_ // Rocky mountain J. Math. - 1993. - 23, N 3, p. 1143–1177.
* [6] . . _. . . . ._ // . - - 1981. - 155, . 41–76.
* [7] Reddy A.R. _A Constribution to best approximation in the $L^{2}$ norm._ // J. Approxim. Theory. 1974. - 11, N 11, p. 110–117.
* [8] . ., . . _._ // . - 1976. - 227, N 2, . 280–283.
* [9] . ., . . _ $A_{p}(|z|<1).$ _ // . .- : -1977. N 1, . 84–96.
* [10] . . _ $\mathcal{B}_{p,\,q,\,\lambda}.$_ // , . .- . . . - 1989. - N 8, . 6–9.
* [11] . . _._ // . -1990. - 42, N 6, . 838–843.
* [12] . _._ \- , . 01.01.01 - . - , 2009. - 14 .
* [13] . ., . . _._ // . - , . - 1983, . 63–73.
* [14] . . _,_ // . . . - 2006\. - 3, N 3, . 315–330.
* [15] . . _._ // . , . . - 1964. - 28, N 5 - . 1173–1186.
* [16] . . _._ // . - , . - 1971, . 267–333.
* [17] Igor E. Pritsker _An areal analog of Mahler’s measure._ // Illinois J. Math. - 2008. - 52, N 2 - p. 347-363. : , . , 83055 . , 24 -mail: [email protected] . +38(062)-2972953, +380667075599
|
arxiv-papers
| 2014-02-13T17:03:00 |
2024-09-04T02:49:58.208634
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Dveyrin",
"submitter": "Mykhaylo Dveyrin Z",
"url": "https://arxiv.org/abs/1402.3218"
}
|
1402.3275
|
# Automorphism groups of simplicial complexes of infinite type surfaces
Jesús Hernández Hernández Aix-Marseille Université, 39 rue F. Joliot Curie,
13453 Marseille Cedex 13, France
_e-mail:_ [email protected] José Ferrán Valdez Lorenzo Centro de Ciencias
Matemáticas, UNAM, Campus Morelia
58190, Morelia, Michoacán, México
_e-mail:_ [email protected]
###### Abstract
Let $S$ be any orientable surface of infinite genus with a finite number of
boundary components. In this work we consider the curve complex
$\mathcal{C}(S)$, the nonseparating curve complex $\mathcal{N}(S)$ and the
Schmutz graph $\mathcal{G}(S)$ of $S$. When all the topological ends of $S$
carry genus, we show that all elements in the automorphism groups
$\mathrm{Aut}(\mathcal{C}(S))$, $\mathrm{Aut}(\mathcal{N}(S))$ and
$\mathrm{Aut}(\mathcal{G}(S))$ are _geometric_ , _i.e._ these groups are
naturally isomorphic to the _extended_ mapping class group
$\mathrm{MCG}^{*}(S)$ of the infinite surface $S$. Finally, we study rigidity
phenomena within $\mathrm{Aut}(\mathcal{C}(S))$ and
$\mathrm{Aut}(\mathcal{N}(S))$.
## 1 Introduction
The mapping class group and the extended mapping class group of a given
surface $S$, that we will denote by $\mathrm{MCG}(S)$ and
$\mathrm{MCG}^{*}(S)$ respectively, have been studied mostly when $S$ has
_finite topological type_ , that is, when its fundamental group is finitely
generated. The main purpose of this article is the study of the natural
(simplicial) action of the group $\mathrm{MCG}^{*}(S)$ on two abstract
simplicial complexes and one simplicial graph associated to $S$ when the
surface $S$ has infinite genus. These complexes and graph are:
1. 1.
The _curve complex_ $\mathcal{C}(S)$. This is the abstract simplicial complex
whose vertices are the (isotopy classes of) essential curves in $S$, and whose
simplexes are multicurves of finite cardinality. It was introduced by Harvey
in 1978.
2. 2.
The _nonseparating curve complex_ $\mathcal{N}(S)$. This is the simplicial
subcomplex of $\mathcal{C}(S)$ formed by all nonseparating curves, that is,
all the (isotopy classes of) essential curves $\alpha$ such that
$S\setminus\alpha$ is connected. This was first introduced by Schmutz in
[Sch].
3. 3.
The _Schmutz graph_ $\mathcal{G}(S)$. Introduced by Paul Schmutz Schaller in
[Sch], this is the simplicial graph whose vertex set is the same as the vertex
set of $\mathcal{N}(S)$, and two vertices span an edge whenever their
geometric intersection number is 1. It is also known as _a modified complex of
nonseparating curves_ (see [Farb]), for it can be thought as a 1-dimensional
simplicial complex.
Recall that any _orientable_ surface of infinite topological type is
completely determined, up to homeomorphism, by its genus
$g(S)\in\mathbb{N}\cup\\{\infty\\}$, and a nested pair of topological spaces
$\mathrm{Ends}^{*}(S)\subset\mathrm{Ends}(S)$. Roughly speaking,
$\mathrm{Ends}(S)$ are the _topological ends_ of $S$ and
$\mathrm{Ends}^{*}(S)$ is formed by those that carry (infinite) genus. We will
focus our attention on infinite genus surfaces $S$ for which the boundary
$\partial S$ has finitely many connected components (possibly none) and
$\mathrm{Ends}^{*}(S)=\mathrm{Ends}(S)$.
To each of the aforementioned simplicial complexes one can associate its
automorphism group. We denote these groups by $\mathrm{Aut}(\mathcal{C}(S))$,
$\mathrm{Aut}(\mathcal{N}(S))$ and $\mathrm{Aut}(\mathcal{G}(S))$
respectively. For surfaces of finite topological type of positive genus (with
the exception of the two-holed torus), every element in
$\mathrm{Aut}(\mathcal{C}(S))$, $\mathrm{Aut}(\mathcal{N}(S))$ and
$\mathrm{Aut}(\mathcal{G}(S))$ is _geometric_. That is, if
$X=\mathcal{C}(S),\mathcal{N}(S)$ or $\mathcal{G}(S)$, then the natural map:
$\Psi_{X}:$ | $\mathrm{MCG}^{*}(S)$ | $\longrightarrow$ | $\mathrm{Aut}(X)$
---|---|---|---
| $[h]$ | $\mapsto$ | $h_{*}$
(1)
where $h_{*}$ is given by $h_{*}([\alpha])=[h(\alpha)]$ is an isomorphism.
This result is due to Ivanov [Ivanov] for $X=\mathcal{C}(S)$, to Irmak and
Schmutz [Irmak], [Sch] for $X=\mathcal{N}(S)$ and to Schmutz [Sch] when
$X=\mathcal{G}(S)$. The main purpose of this article is to extend this result
for surfaces of infinite genus:
###### Theorem 1.
Let $S$ be an infinite genus surface with finitely many boundary components
such that $\mathrm{Ends}^{*}(S)=\mathrm{Ends}(S)$. Then the natural map
$\Psi_{X}:\mathrm{MCG}^{*}(S)\longrightarrow\mathrm{Aut}(X)$ is an isomorphism
for $X=\mathcal{C}(S),\mathcal{N}(S)$ or $\mathcal{G}(S)$.
The techniques that we use to prove this result rely heavily on the hypothesis
$\mathrm{Ends}^{*}(S)=\mathrm{Ends}(S)$. However, we suspect that this theorem
remains valid for surfaces with arbitrarily many planar ends. In addition to
the study of the action of $\mathrm{MCG}^{*}(S)$ on simplicial complexes, we
study rigidity phenomena within the curve complex and the nonseparating curve
complex. More precisely:
###### Theorem 2.
Let $S_{1}$ and $S_{2}$ be infinite genus surfaces with finitely many boundary
components, such that $\mathrm{Ends}(S_{i})=\mathrm{Ends}^{*}(S_{i})$ for
$i=1,2$ and let $\phi:\mathcal{C}(S_{1})\rightarrow\mathcal{C}(S_{2})$ be an
isomorphism. Then $S_{1}$ is homeomorphic to $S_{2}$.
As we will see in section §4.2 this result is not valid if we allow the
infinite genus surface $S$ to have planar ends. In §4.2 we will also see that
the tools used in the proof of this theorem work for nonseparating curves.
Hence, we have the following:
###### Corollary 1.
Let $S_{1}$ and $S_{2}$ be infinite genus surfaces with finitely many boundary
components, such that $\mathrm{Ends}(S_{i})=\mathrm{Ends}^{*}(S_{i})$ for
$i=1,2$ and let $\phi:\mathcal{N}(S_{1})\rightarrow\mathcal{N}(S_{2})$ be an
isomorphism. Then $S_{1}$ is homeomorphic to $S_{2}$.
As we will see in section §5, contrary to the compact case, this kind of
rigidity results cannot be extended to injective simplicial maps when $S$ is
an infinite genus surface.
We must remark that while the results of this article are highly inspired by
those of the compact case, many proofs have been either modified or outright
rewritten to accommodate for the infinite type surfaces. We have also
stablished new results and techniques on which the main results of this
article rely. Among these we underline the relation between $\mathrm{Ends}(S)$
and the space of ends of the adjacency graph of a pants decomposition of $S$
(see §3, theorem 4) and a variant of the Alexander method for infinite type
surfaces (see §5, theorem 4).
We refer the reader to [Fou1], [Fou2] and [Fuji] for previous work on groups
formed by mapping classes of infinite type surfaces. We want to stress,
however, that the cited authors focus their work on several subgroups of what
we here call the mapping class group (_e.g._ those with assymptotic qualities
for a specific surface or quasiconformal automorphisms of a Riemann surface)
and on their action on the Teichmüller space.
_Acknowledgements_. We want to thank Camilo Ramírez Maluendas for the question
that lead to the creation of this article. We are greateful to Hamish Short
and Javier Aramayona for carefully reading preliminary versions of this text.
The first author would like to thanks Daniel Juan Pineda for his support
during the realization of this project. The second author was generously
supported by LAISLA, CONACYT CB-2009-01 127991 and PAPIIT projects IN103411 &
IB100212 during the realization of this project.
## 2 Preliminaries
### 2.1 Topological invariants for infinite type surfaces
Let $X$ be a locally compact, locally connected, connected Hausdorff space.
###### Definition 2.1.
[F] Let $U_{1}\supseteq U_{2}\supseteq\ldots$ be an infinite sequence of non-
empty connected open subsets of $X$ such that for each $i\in\mathbb{N}$ the
boundary $\partial U_{i}$ is compact and
$\bigcap\limits_{i\in\mathbb{N}}\overline{U_{i}}=\emptyset$. Two such
sequences $U_{1}\supseteq U_{2}\supseteq\ldots$ and $U^{\prime}_{1}\supseteq
U^{\prime}_{2}\supseteq\ldots$ are said to be equivalent if for every
$i\in\mathbb{N}$ there exist $j,k$ such that $U_{i}\supseteq U^{\prime}_{j}$
and $U^{\prime}_{i}\supseteq U_{k}$. The corresponding equivalence class is
called a topological end of $X$.
The set of ends ${\rm Ends}(X)$ of $X$ can be endowed with a topology in the
following way. For any set $U$ in $X$ whose boundary is compact, we define
$U^{*}$ to be the set of all ends $[U_{1}\supseteq U_{2}\supseteq\ldots]$ for
which there is a representative such that $U_{n}\subset U$ for $n$
sufficiently large. With respect to this topology, ${\rm Ends}(X)$ is a
compact, closed, totally disconnected space without interior points (see for
example Theorem 1.5, [Ray]).
The genus of a surface is the maximum of the genera of its compact
subsurfaces. A surface is said to be planar if all of its compact subsurfaces
are of genus zero. We define $\mathrm{Ends}^{*}(S)\subset{\rm Ends}(S)$ as the
set of all ends which are not planar. As stated in the following theorem, any
orientable surface is determined, up to homeomorphism, by its genus, boundary
and space of ends. Henceforth all surfaces in this text are connected.
###### Theorem 3.
Let $S$ and $S^{\prime}$ be two orientable surfaces of the same genus. Then
$S$ and $S^{\prime}$ are homeomorphic if and only if they have the same number
of boundary components, and $\mathrm{Ends}^{*}(S)\subset{\rm Ends}(S)$ and
$\mathrm{Ends}^{*}(S^{\prime})\subset{\rm Ends}(S^{\prime})$ are homeomorphic
as nested topological spaces.
The proof of this theorem for the case when $S$ and $S^{\prime}$ have no
boundary can be found in [R]. The case for surfaces with boundary was proven
in [PM].
### 2.2 Complexes and graphs of curves
There are several curve complexes that one can associate to a surface of
finite genus, with finitely many boundary components and punctures. In this
section we extend the definitions of these complexes to noncompact surfaces of
infinite topological type and explore some of their basic properties.
Abusing language and notation, we will call curve, a topological embedding
$S^{1}\hookrightarrow S$, the isotopy class of this embedding and its image on
$S$. A curve is said to be _essential_ if it is neither homotopic to a point
nor to a boundary component. Hereafter all curves are considered essential
unless otherwise stated. An essential curve is said to be _separating_ if the
surface obtained by cutting $S$ along its image is disconnected. It is said to
be nonseparating otherwise. A separating curve $\alpha$ is said to be an
_outer separating_ curve if by cutting $S$ along $\alpha$ one of the resulting
connected components is a pair of pants (_i.e._ a genus 0 surface with three
boundary components). A non-outer separating curve is a separating curve which
is not an outer separating curve. Two curves are _disjoint_ if they are
distinct and their (geometric) intersection number is $0$.
###### Definition 2.2 (Multicurves).
A multicurve is either a set of just one curve, or a pairwise disjoint and
locally finite set of curves of $S$. We allow multicurves to consist of an
infinite set of curves. If $M$ is a multicurve of $S$, the surface obtained by
cutting $S$ along pairwise disjoint representatives of the elements of $M$
will be denoted by $S_{M}$.
Infinite _countable_ multicurves arise in surfaces with nonfinitely generated
fundamental group. Take for example the Loch Ness Monster, that is, a surface
with infinite genus and one end. If $S$ is a compact surface of genus $g$ with
$n$ boundary components, the _complexity_ of $S$, denoted by $\kappa(S)$, is
equal to $3g-3+n$. This is the cardinality of a maximal multicurve in $S$.
###### Definition 2.3 (The Curve Complex).
The Curve complex of $S$, $\mathcal{C}(S)$, is the abstract simplicial complex
whose vertices are the isotopy classes of essential curves in $S$, and whose
simplexes are multicurves of finite cardinality. We denote the set of vertices
of $\mathcal{C}(S)$ by $\mathcal{V}(\mathcal{C}(S))$.
The $1$-skeleton of $\mathcal{C}(S)$ will be denoted by $\mathcal{C}^{1}(S)$.
Since every automorphism of $\mathcal{C}(S)$ is determined uniquely by a
function of its vertices, and the same statement is true for automorphisms of
$\mathcal{C}^{1}(S)$, then the groups $\mathrm{Aut}(\mathcal{C}(S))$ and
$\mathrm{Aut}(\mathcal{C}^{1}(S))$ are isomorphic.
###### Definition 2.4 (The Nonseparating Curve Complex).
The Nonseparating curve complex of $S$, $\mathcal{N}(S)$, is the subcomplex of
$\mathcal{C}(S)$ whose vertices are the isotopy classes of essential
_nonseparating_ curves in $S$. We denote the set of vertices of
$\mathcal{N}(S)$ by $\mathcal{V}(\mathcal{N}(S))$.
###### Definition 2.5 (The Schmutz graph).
The Schmutz graph of $S$, $\mathcal{G}(S)$, is the simplicial graph whose
vertices are the isotopy classes of essential nonseparating curves in $S$, and
two vertices span an edge if their geometric intersection number is $1$.
###### Proposition 2.6.
Let $S$ be a surface of infinite genus. Then $\mathcal{C}(S)$,
$\mathcal{N}(S)$ and $\mathcal{G}(S)$ are connected. In particular
$\mathcal{C}^{1}(S)$ and $\mathcal{N}^{1}(S)$ have diameter 2 while
$\mathcal{G}(S)$ has diameter 4.
###### Proof.
Given any two distinct curves $\alpha$ and $\beta$ (either in
$\mathcal{V}(\mathcal{C}(S))$ or in $\mathcal{V}(\mathcal{N}(S))$), we can
always find a compact (finite genus) subsurface $S^{\prime}$ such that
contains $\alpha$ and $\beta$. Hence we can take an essential nonseparating
curve $\gamma$ on $S$ contained in $S\backslash S^{\prime}$ and not isotopic
to $\alpha$ and $\beta$. Therefore $\mathcal{C}^{1}(S)$ and
$\mathcal{N}^{1}(S)$ are connected,
$\mathrm{diam}(\mathcal{C}^{1}(S))=\mathrm{diam}(\mathcal{N}^{1}(S))=2$.
If $\alpha$ and $\beta$ are two distinct nonseparating curves, as in the
paragraph above, we can always find a curve $\gamma$ such that
$i(\alpha,\gamma)=i(\gamma,\beta)=0$; then we can always find curves
$\delta_{1}$ and $\delta_{2}$ such that
$i(\alpha,\delta_{1})=i(\delta_{1},\gamma)=i(\gamma,\delta_{2})=i(\delta_{2},\beta)=1$.
Hence $\mathcal{G}(S)$ is connected, $\mathrm{diam}(\mathcal{G}(S))\leq 4$. ∎
###### Remark 1.
Number 2 as diameter for $\mathcal{C}^{1}(S)$ and $\mathcal{N}^{1}(S)$ is
optimal, but 4 as diameter for $\mathcal{G}(S)$ is not necessarily optimal.
### 2.3 Mapping Class Group
Through this article, we will be working with the mapping class group of a
surface $S$. When $S$ is compact, this group has different (equivalent)
definitions, see for example [Farb], §2.1. In this paper we will be working
with the following definition.
###### Definition 2.7 (Mapping Class Group).
Let $S$ be a surface. Then $\mathrm{Homeo}^{+}(S,\partial S)$ is the group of
orientation-preserving homeomorphisms of $S$ that restrict to the identity on
the boundary, and $\mathrm{Homeo}(S)$ is the group of _all_ homeomorphisms of
$S$. The mapping class group of $S$, $\mathrm{MCG}(S)$ is the group
$\mathrm{Homeo}^{+}(S)/\thicksim$, where $\thicksim$ represents the isotopy
relation relative to the boundary. The extended mapping class group of $S$ is
the group $\mathrm{MCG}^{*}(S)\coloneqq\mathrm{Homeo}(S)/\thicksim$, where
$\thicksim$ represents the isotopy relation.
The group $\mathrm{MCG}^{*}(S)$ is incredibly big. As evidence for this we
have the following lemma and corollaries.
###### Lemma 2.8.
Let $S$ be an infinite genus surface and $F$ a subsurface of $S$ such that
$S\backslash F$ has genus at least $1$ and the boundary components of $F$ are
either boundary components of $S$ or essential curves of $S$. Then there
exists a subgroup of $\mathrm{MCG}^{*}(S)$ isomorphic to $\mathrm{MCG}(F)$,
with infinite index in $\mathrm{MCG}^{*}(S)$.
###### Proof.
The subgroup of $\mathrm{MCG}^{*}(S)$ formed by those orientation-preserving
elements $[h]\in\mathrm{MCG}^{*}(S)$ that have a representative $h$ with
support on $F$, is isomorphic to $\mathrm{MCG}(F)$. This subgroup will have
index greater or equal to the number of different elements in
$\mathrm{MCG}^{*}(S)$ that have its support in the interior of the complement
of $F$, thus it will have infinite index. ∎
###### Corollary 2.
Let $S$ be an infinite genus surface and $S_{g,n}$ be a compact surface of
genus $g$ and $n$ boundary components. Then
$\mathrm{MCG}^{*}(S_{g,n})<\mathrm{MCG}^{*}(S)$.
###### Corollary 3.
Let $S$ be an infinite genus surface,
$\\{(g_{i},n_{i})\\}_{i\in\mathbb{N}}\subset(\mathbb{N}\times\mathbb{Z}^{+})\backslash\\{(0,1)\\}$
be a sequence and $S_{i}$ be a compact orientable surface of genus $g_{i}$ and
$n_{i}$ boundary components. Then $\mathrm{MCG}^{*}(S)$ contains a subgroup
isomorphic to $\prod_{i\in\mathbb{N}}\mathrm{MCG}^{*}(S_{i})$.
## 3 Ends of adjacency graphs and surfaces
In this section we prove that, under the hypotheses
$\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$, one can determine topologically
$\mathrm{Ends}(S)$ using the adjacency graph of a pants decomposition of $S$.
###### Definition 3.1 (Pants decomposition and the adjacency graph).
A _pants decomposition_ is a multicurve $P$ of maximal cardinality. We say
$\alpha,\beta\in P$ are adjacent with respect to $P$ if they bound the same
pair of pants in $S_{P}$. The adjacency graph of $P$, $\mathcal{A}(P)$, is the
simplicial graph whose vertex set is $P$ and two vertices span an edge if and
only if they are adjacent with respect to $P$. We say two nonseparating curves
form a _peripheral pair_ if they bound, along with a boundary component of
$S$, a pair of pants.
If $P$ is a pants decomposition, $S_{P}$ is the disjoint union of surfaces
homeomorphic to a pair of pants, for otherwise we contradict maximality. As an
abstract graph, $\mathcal{A}(P)$ is a subgraph of $\mathcal{C}^{1}(S)$, but we
have to keep in mind that adjacency of vertices in $\mathcal{A}(P)$ and
$\mathcal{C}^{1}(S)$ means different things for the corresponding curves in
$S$.
###### Remark 2.
It can be easily checked that the only cut points of an adjacency graph
$\mathcal{A}(P)$ are non-outer separating curves, and non-outer separating
curves are always cut points of any adjacency graph in which they are
vertices. Also, we can easily check outer separating curves always have degree
less or equal to two.
###### Theorem 4.
Let $S$ be an infinite genus surface such that
$\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$ and $P$ be a pants decomposition of
$S$. Then $\mathrm{Ends}(\mathcal{A}(P))$ is homeomorphic to
$\mathrm{Ends}(S)$.
###### Proof.
For every pants decomposition there is a natural, but not canonical,
topological embedding:
$f:\mathcal{A}(P)\hookrightarrow S$ (2)
This embedding is illustrated in figure 1. Let $\Gamma$ be a subgraph of
$\mathcal{A}(P)$ whose boundary $\partial\Gamma$ is compact. We define
$S(\Gamma)$ as the subsurface of $S$ formed by all pants in $S$ (defined by
the multicurve $P$) that intersect $f(\Gamma)$, _deprived of its boundary_. By
definition $S(\Gamma)$ is an open subsurface of $S$ whose boundary is formed
by a finite collection of curves $\\{C_{1},\ldots,C_{n}\\}\subset P$. Remark
that if the graph $\Gamma$ is connected, so is $S(\Gamma)$. Moreover if
$\Gamma\supset\Gamma^{\prime}$ are two connected subgraphs of $\mathcal{A}(P)$
with compact boundaries we have that $S(\Gamma)\supset S(\Gamma^{\prime})$.
For every $[\Gamma_{1}\supseteq\Gamma_{2}\supseteq\ldots]$ in
$\mathrm{Ends}(\mathcal{A}(P))$ we define:
$f_{*}[\Gamma_{1}\supseteq\Gamma_{2}\supseteq\ldots]=[S(\Gamma_{1})\supseteq
S(\Gamma_{2})\supseteq\ldots]\in\mathrm{Ends}(S)$ (3)
Figure 1: A natural embedding of $\mathcal{A}(P)$ into $S$.
It follows directly from definition 2.1 that $f_{*}$ is well defined. We claim
that $f_{*}:\mathrm{Ends}(\mathcal{A}(P))\to\mathrm{Ends}(S)$ is an
homeomorphism. The injectivity of $f_{*}$ follows from the following general
lemma:
###### Lemma 3.2.
[F] Let $[U_{1}\supseteq U_{2}\supseteq\ldots]$ and $[U^{\prime}_{1}\supseteq
U^{\prime}_{2}\supseteq\ldots]$ be two different points in $\mathrm{Ends}(X)$.
Then there exists $i\in\mathbb{N}$ such that $U_{i}\cap
U^{\prime}_{i}=\emptyset$.
Indeed, let us suppose that
$[\Gamma_{1}\supseteq\Gamma_{2}\supseteq\ldots]\neq[\Gamma^{\prime}_{1}\supseteq\Gamma^{\prime}_{2}\supseteq\ldots]$.
Then there exists an $i\in\mathbb{N}$ such that
$\Gamma_{i}\cap\Gamma^{\prime}_{i}=\emptyset$. Let us suppose that
$[S(\Gamma_{1})\supseteq
S(\Gamma_{2})\supseteq\ldots]=[S(\Gamma^{\prime}_{1})\supseteq
S(\Gamma^{\prime}_{2})\supseteq\ldots]$. Hence, for the previous
$i\in\mathbb{N}$ there exist $l,k\in\mathbb{N}$ such that
$S(\Gamma_{i})\supseteq S(\Gamma^{\prime}_{l})$ and
$S(\Gamma^{\prime}_{i})\supseteq S(\Gamma_{k})$. Without loss of generality
suppose that $S(\Gamma^{\prime}_{l})\supseteq S(\Gamma^{\prime}_{i})$, hence
$S(\Gamma_{i})\cap S(\Gamma^{\prime}_{i})=S(\Gamma^{\prime}_{i})$ and, since
both $\partial\Gamma_{i}$ and $\partial\Gamma^{\prime}_{i}$ have compact
boundary we conclude that $\Gamma_{i}\cap\Gamma^{\prime}_{i}\neq\emptyset$.
This contradicts our initial assumption. The case where
$S(\Gamma^{\prime}_{i})\supseteq S(\Gamma^{\prime}_{l})$ is analogous.
We address now surjectivity. Let $[S_{1}\supseteq
S_{2}\supseteq\ldots]\in\mathrm{Ends}(S)$. Since there are no planar ends,
that is $\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$, we can consider, for each
$S_{i}$ the surface $\mathfrak{S}_{i}$ formed by all pants in the pants in the
decomposition defined by $P$ that intersect $S_{i}$. Since $S_{i}$ is
connected, then $\mathfrak{S}_{i}$ must be connected. Also, since $\partial
S_{i}$ is compact, so is $\partial\mathfrak{S}_{i}$. Moreover, by definition,
if $i\leq j$ then $\mathfrak{S}_{i}\supseteq\mathfrak{S}_{j}$. Hence we have a
well defined end
$[\mathfrak{S}_{1}\supseteq\mathfrak{S}_{2}\supseteq\ldots]\in\mathrm{Ends}(S)$.
By construction, for every $i\in\mathbb{N}$ we have that
$S_{i}\subset\mathfrak{S}_{i}$. On the other hand, given $i\in\mathbb{N}$ we
can find $S_{j}$ such that $S_{i}\setminus S_{j}$ contains (properly) a
connected surface formed by pants in the pant decomposition defined by $P$.
This implies that there exists $k\in\mathbb{N}$ such that
$S_{j}\subset\mathfrak{S_{k}}$. Therefore
$[\mathfrak{S}_{1}\supseteq\mathfrak{S}_{2}\supseteq\ldots]=[S_{1}\supseteq
S_{2}\supseteq\ldots]$. Now define $\Gamma_{i}$ as the maximal subgraph of $S$
such that $f(\Gamma_{i})\subset\mathfrak{S}_{i}$. The graph $\Gamma_{i}$ has
compact boundary for $\mathfrak{S}_{i}$ has compact boundary and by definition
$f_{*}[\Gamma_{1}\supset\Gamma_{2}\supset\ldots]=[S_{1}\supseteq
S_{2}\supseteq\ldots]$. This proves that $f_{*}$ is a bijection.
Now we prove that $f_{*}$ is an homeomorphism. Let $\Gamma$ be a subgraph of
$\mathcal{A}(P)$ with compact boundary as before. We define
$\Gamma^{*}\coloneqq\\{[\Gamma_{1}\supseteq\Gamma_{2}\supseteq\ldots]\hskip
2.84526pt|\hskip 2.84526pt\Gamma\supseteq\Gamma_{i}\hskip 2.84526pt\text{for i
sufficiently big}\\}$ (4)
The collection of all $\Gamma^{*}$’s generates the topology of
$\mathrm{Ends}(\mathcal{A}(P))$. On the other hand we know from [R] that the
topology of $\mathrm{Ends}(S)$ is generated by
$U^{*}:=\\{[\hat{S}_{1}\supseteq\hat{S}_{2}\supseteq\ldots]\hskip
2.84526pt|\hskip 2.84526ptU\supseteq\hat{S}_{i}\hskip 2.84526pt\text{for i
sufficiently big}\\},$ (5)
where $U\subset S$ is an open subset with compact boundary. Clearly
$f_{*}\Gamma_{*}=S(\Gamma)^{*}$ , hence $f_{*}$ is open. From [Ray] we know
that both $\mathrm{Ends}(\mathcal{A}(P))$ and $\mathrm{Ends}(S)$ are compact
Hausdorff topological spaces. Hence $f_{*}$ is an homeomorphism. ∎
###### Remark 3.
We can think of punctures on a surface as planar ends, and hence the preceding
result is not true if we allow the surface $S$ to have them.
## 4 Proof of main results.
### 4.1 Injectivity.
In this section we will prove the following result:
###### Theorem 5.
Let $S$ be an infinite genus surface such that
$\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$. The natural map:
$\Psi_{\mathcal{C}(S)}:{\rm MCG}^{*}(S)\to{\rm Aut}(\mathcal{C}(S))$ (6)
is injective.
Most of the proof of this theorem will rely in the following lemma and a
variant of the Alexander method (see [Farb] for details on this method).
###### Lemma 4.1.
Let $S$ be an infinite genus surface possibly with marked points and possibly
a finite number of boundary components. Let $\gamma_{1},\ldots,\gamma_{n}$ be
a collection of simple closed curves and simple proper arcs in $S$ such that
satisfy the three following properties:
1. 1.
The $\gamma_{i}$ are in pairwise minimal position. That is, for $i\neq j$, the
(geometric) intersection of $\gamma_{i}$ with $\gamma_{j}$ is minimal within
their homotopy classes.
2. 2.
The $\gamma_{i}$ are pairwise nonisotopic.
3. 3.
For distinct $i,j,k$, at least one of $\gamma_{i}\cap\gamma_{j}$,
$\gamma_{i}\cap\gamma_{k}$, or $\gamma_{j}\cap\gamma_{k}$ is empty.
If $\gamma_{1}^{\prime},\ldots,\gamma_{n}^{\prime}$ is another such collection
so that $\gamma_{i}$ is isotopic to $\gamma_{i}^{\prime}$ for each $i$, then
there is an isotopy of $S$ that takes $\gamma_{i}^{\prime}$ to $\gamma_{i}$
for all $i$ simultaneously, and hence takes $\cup\gamma_{i}$ to
$\cup\gamma_{i}^{\prime}$.
A collection of curves $\gamma_{1},\ldots,\gamma_{n}$ satisfying (1)-(3) in
the preceding lemma will be called an _Alexander system_ in S. The proof of
this lemma is exactly the same as the proof of lemma 2.9 in [Farb].
Proof theorem 5. Let $h:S\rightarrow S$ be an homeomorphism such that
$h(\alpha)$ is isotopic to $\alpha$ for all
$\alpha\in\mathcal{V}(\mathcal{C}(S))$. For every infinite genus surface such
that $\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$ we can find a family of compact
subsurfaces $\\{K_{i}\\}_{i\in\mathbf{N}}$ such that:
* •
$S=\bigcup_{i\in\mathbf{N}}K_{i}$,
* •
$K_{i}\subset K_{j}$ if $i<j$.
* •
$K_{i}$ has genus at least $3$ for all $i\in\mathbf{N}$.
* •
$K_{j}\setminus K_{i}$ admits at least one curve nonisotopic to any boundary
curve of $K_{j}$ for $i<j$.
* •
Every boundary component of $K_{i}$ that is not a boundary component of $S$ is
an essential separating curve of $S$.
For each $i\in\mathbf{N}$ let us write $\partial K_{i}$ for the boundary of
$K_{i}$, $\partial_{S}K_{i}$ for all curves in $\partial K_{i}$ that are part
of the boundary of $S$ and $\partial_{i}K_{i}$ for $\partial
K_{i}\setminus\partial_{S}K_{i}$. Given such a family
$\\{K_{i}\\}_{i\in\mathbb{N}}$ of compact subsurfaces we can find
$\\{\Gamma_{i}\\}_{i\in\mathbb{N}}$ a collection of finite subsets of
$\mathcal{V}(\mathcal{C}(S))$ such that:
* •
Every boundary component of $K_{i}$ that is not a boundary component of $S$,
is in $\Gamma_{j}$ for $i<j$ and is disjoint from every other curve in
$\cup_{i\in\mathbf{N}}\Gamma_{i}$.
* •
$\Gamma_{0}$ fills $K_{0}$ and $\Gamma_{j}\setminus\Gamma_{j-1}$ fills
$K_{j}\setminus K_{j-1}$ for all $j>0$. In addition
$\Gamma_{i}\subset\Gamma_{j}$ for $i<j$.
* •
If we cut $K_{j}\setminus K_{i}$ along $\Gamma_{j}\setminus\Gamma_{i}$ we
obtain either discs or annuli with one boundary component in $\partial K_{k}$,
for $i<j$ and some $k$ with $i\leq k\leq j$.
* •
For all $\gamma\in(\Gamma_{j}\backslash\Gamma_{i})$ and
$\gamma^{\prime}\in\Gamma_{i}$, we have that $i(\gamma,\gamma^{\prime})=0$.
Moreover, if we define for each $i\in\mathbf{N}$
$\Gamma_{i}^{\prime}=\Gamma_{i}\cup\partial_{i}K_{i}$ (7)
then, for all $\gamma\in(\Gamma_{j}\backslash\Gamma_{i}^{\prime})$ and
$\gamma^{\prime}\in\Gamma_{i}^{\prime}$ we have $i(\gamma,\gamma^{\prime})=0$.
* •
Both $\Gamma_{i}$ and $\Gamma_{i}^{\prime}$ are Alexander systems in $S$.
Figure 2 shows an example of $\\{K_{i}\\}_{i\in\mathbf{N}}$ and its
corresponding $\\{\Gamma_{j}\\}_{j\in\mathbf{N}}$.
$K_{0}$$K_{1}$ Figure 2: Example for $K_{0},K_{1},\ldots$ and
$\Gamma_{0},\Gamma_{1},\ldots$.
###### Lemma 4.2.
There exist homotopies $H_{i}:S\times[0,1]\rightarrow S$ such that:
1. 1.
$H_{i}|_{S\times\\{0\\}}$ is the identity for $i\in\mathbf{N}$.
2. 2.
$H_{i}|_{K_{i}\times\\{1\\}}=h|_{K_{i}}$ for $i\in\mathbf{N}$.
3. 3.
$H_{i}|_{K_{i}\times[0,1]}=H_{j}|_{K_{i}\times[0,1]}$ for $i<j$.
The proof of this lemma is rather technical, the main difficulty being to
prove that $H_{i}|_{K_{i}\times[0,1]}=H_{j}|_{K_{i}\times[0,1]}$ for $i<j$. We
leave it for later. We will use the lemma to finish the proof of theorem 5.
For every $x\in S$ there exist $i\in\mathbf{N}$ such that $x\in K_{i}$ and
$x\notin\partial_{i}K_{i}$. Define $H:S\times[0,1]\rightarrow S$ as
$H(x,t)=H_{i}(x,t)$. From (3) in the preceding lemma we deduce that $H$ is
well-defined. The function $H$ is clearly continuous, $H|_{S\times\\{0\\}}$ is
the identity and $H|_{S\times\\{1\\}}=h$. Thus $H$ is an homotopy from the
identity to $h$. This, modulo the proof of lemma 4.2, finishes the proof of
theorem 5.
Proof of lemma 4.2. The idea of the proof is a variant of the Alexander method
(see [Farb] for details on this method). By hypothesis, for every
$\gamma\in\mathcal{C}(S)$ , the curves $\gamma$ and $h(\gamma)$ are isotopic.
Using lemma 4.1 we can assure the existence, for each $i\in\mathbf{N}$, of an
isotopy $\tilde{H}_{i}:S\times[0,1]\rightarrow S$, that takes $\gamma$ to
$h(\gamma)$ for all $\gamma\in\Gamma_{i}^{\prime}$ simultaneously. Moreover,
since for all $\gamma\in(\Gamma_{j}\backslash\Gamma_{i}^{\prime})$ and
$\gamma^{\prime}\in\Gamma_{i}^{\prime}$ we have $i(\gamma,\gamma^{\prime})=0$,
we can ask
$\displaystyle\tilde{H}_{i|\hskip
2.84526pt_{\Gamma_{i}^{\prime}\times[0,1]}}=\tilde{H}_{j|\hskip
2.84526pt_{\Gamma_{i}^{\prime}\times[0,1]}},\hskip 22.76219pt\text{for
$i<j$}.$ (8)
In other words, the homotopies can be chosen so that $\tilde{H}_{i}$ moves the
curves in $\Gamma_{i}^{\prime}$ at exactly the same time as $\tilde{H}_{j}$
moves the curves in $\Gamma_{i}^{\prime}$ for $i<j$. Let us define
$f_{i}:={H}_{i|S\times\\{1\\}}$. Remark that $h^{-1}\circ f_{i}$ fixes all the
points in $\Gamma_{i}^{\prime}$. On the other hand, $h$ has to be orientation-
preserving, since otherwise for every compact subsurface
$S^{\prime}\hookrightarrow S$ we could find an homeomorphism that reverses
orientation and at the same time acts trivially on $\mathcal{C}(S^{\prime})$,
which is not possible if $S^{\prime}$ has genus bigger than 3 and at least one
boundary component. Hence $h^{-1}\circ f_{i}$ is orientation-preserving and,
by the same argument used by Farb and Margalit (see proof proposition 2.8, p.
62-63, [Farb]), we have that $h^{-1}\circ f_{i}$ sends each connected region
in $S\setminus\Gamma_{i}^{\prime}$ to itself. By hypotheses $\Gamma_{0}$ fills
$K_{0}$ and $\Gamma_{j}\setminus\Gamma_{j-1}$ fills $K_{j}\setminus K_{j-1}$
for all $j>1$. Hence:
$S\setminus\Gamma_{i}^{\prime}=\left(\bigsqcup_{k=1}^{n_{i}}A_{k}\right)\sqcup\left(\bigsqcup_{k=1}^{m_{i}}D_{k}\right)\sqcup
S_{i}$ (9)
where each $D_{k}$ is homeomorphic to a disc, each $A_{k}$ is homeomorphic to
an annulus and $S_{i}=S\setminus K_{i}$ is an infinite genus surface.
Furthermore:
1. 1.
The boundary of each disc $D_{k}$ is formed by segments contained in
$\Gamma_{i}$.
2. 2.
The boundary of each annulus $A_{k}$ is either contained in
$\Gamma_{i}^{\prime}$ or one of its connected components is also a connected
component of the boundary of $S$.
From Alexander’s lemma, we deduce that $h^{-1}\circ f_{i}$ restricted to
$D_{k}$ is isotopic to $Id_{|D_{k}}$. When $A_{k}$ shares a boundary component
with $S$, the restriction of $h^{-1}\circ f_{i}$ to $A_{k}$ is isotopic to the
identity, for we are allowed to perform isotopies on $A_{k}$ that do not fix
the boundary of $S$ pointwise. Finally, when $A_{k}$ shares no boundary
component with the boundary of $S$ the restriction of $h^{-1}\circ f_{i}$ to
$A_{k}$ is also isotopic to the identity for else this restriction will be a
non-trivial Dehn twist and we could then find a curve
$\gamma\in\mathcal{C}(S)$ intersecting the interior of $A_{k}$ which is not
fixed by $h^{-1}$. From this three facts we conclude that $h^{-1}\circ f_{i}$
is isotopic to the identity in $K_{i}$ and hence $f_{i}$ is isotopic to $h$ in
$K_{i}$. The composition of these two isotopies form the desired isotopy
$H_{i}$.∎
### 4.2 Rigidity.
In this section we give the proof of theorem 2 and corollary 1. This requires
some auxiliary facts and lemmas, that we state and prove in the following
paragraphs.
Through this section $S_{1}$ and $S_{2}$ will denote (connected) infinite
genus surfaces with a finite number of boundary components and
$\phi:\mathcal{C}(S_{1})\rightarrow\mathcal{C}(S_{2})$ an isomorphism. We
remark that the image via $\phi$ of any pants decomposition of $S_{1}$ is a
pants decomposition of $S_{2}$. Moreover, if $P$ is a pants decomposition of
$S_{1}$, then $\alpha,\beta\in P$ are adjacent with respect to $P$ if and only
if $\phi(\alpha)$ and $\phi(\beta)$ are adjacent with respect to $\phi(P)$.
The sufficiency of this statement can be found in [shackleton] and the
necessity follows from the fact that we are dealing with an isomorphism of the
curve complex. Therefore
$\phi:\mathcal{C}(S_{1})\rightarrow\mathcal{C}(S_{2})$ induces a map
$\varphi:\mathcal{A}(P)\rightarrow\mathcal{A}(\phi(P))$ (10)
as follows: $\alpha\mapsto\varphi(\alpha):=\phi(\alpha)$. Moreover, $\varphi$
is an isomorphism. For this reason cut points of $\mathcal{A}(P)$ go to cut
points under $\phi$ and this isomorphism sends:
1. 1.
Non-outer separating curves to non-outer separating curves.
2. 2.
Nonseparating curves to nonseparating curves.
3. 3.
Outer curves to outer curves.
The proof of (1) and (2) can be found in [shackleton], where as (3) follows
from (1), (2) and the fact that $\phi$ is an isomorphism. The following lemmas
can be deduced from the work of Irmak (see [Irmak]), but since we use them
several times later, we present elementary and simple proofs.
###### Lemma 4.3.
Let $S_{1}$ and $S_{2}$ be infinite genus surfaces and let
$\phi:\mathcal{C}(S_{1})\rightarrow\mathcal{C}(S_{2})$ be an isomorphism. If
$\alpha$, $\beta$ and $\gamma$ are curves that bound a pair of pants on
$S_{1}$, then their images bound a pair of pants on $S_{2}$.
###### Proof.
If $\alpha\neq\beta=\gamma$, then $\beta$ cannot be an outer curve and hence
its image is not an outer curve. Also, in any pants decomposition $P$ the
curve $\beta$ will have degree one as a vertex of $\mathcal{A}(P)$. Hence
$\phi(\beta)$ will also have degree one as vertex of $\mathcal{A}(\phi(P))$,
given that (10) is an isomorphism. Then, the only option left is for $\beta$
to be the boundary of a pair of pants twice, as in the option of the left in
figure 3. Therefore $\phi(\alpha)$ and $\phi(\beta)=\phi(\gamma)$ bound a pair
of pants on $S_{2}$.
$v$$u$$v$$u$ Figure 3: The two options for $\deg(v)=1$.
It is impossible to bound a pair of pants using two separating curves and one
nonseparating curve. Hence, if $\alpha\neq\beta\neq\gamma\neq\alpha$, we only
have the following cases according to the number of separating curves:
1. 1.
Three separating curves. In this case, $\phi(\alpha)$, $\phi(\beta)$ and
$\phi(\gamma)$ are three different separating curves, since separating curves
go to separating curves as mentioned before. If these curves did not bound a
pair of pants on $S_{2}$ we would have a pair of pants bounded by
$\phi(\alpha)$ and $\phi(\beta)$ but not bounded by $\phi(\gamma)$, another
pair of pants bounded by $\phi(\beta)$ and $\phi(\gamma)$ but not bounded by
$\phi(\alpha)$ and another pair of pants bounded by $\phi(\gamma)$ and
$\phi(\alpha)$ but not bounded by $\phi(\beta)$, as in figure 4. But then none
of these curves would be separating, leading us to a contradiction. Hence,
$\phi(\alpha)$, $\phi(\beta)$ and $\phi(\gamma)$ bound a pair of pants on
$S_{2}$.
2. 2.
One separating curve. Let $\alpha$ and $\gamma$ be nonseparating curves and
let $\beta$ be a separating curve. Then $\phi(\alpha)$ and $\phi(\gamma)$ are
nonseparating curves and $\phi(\beta)$ is a separating curve, given the
properties of $\phi$ mentioned before. If these curves did not bound a pair of
pants on $S_{2}$, we would have a pair of pants bounded by $\phi(\alpha)$ and
$\phi(\beta)$ but not bounded by $\phi(\gamma)$, another pair of pants bounded
by $\phi(\beta)$ and $\phi(\gamma)$ but not bounded by $\phi(\alpha)$, but
since $\phi(\beta)$ is a separating curve there cannot exist a pair of pants
bounded by both $\phi(\alpha)$ and $\phi(\gamma)$, given that they are on
different connected components of $S_{2}\backslash\\{\phi(\beta)\\}$, which
leads us to a contradiction ($\phi(\alpha)$ and $\phi(\gamma)$ must be
adjacent). Then $\phi(\alpha)$, $\phi(\beta)$ and $\phi(\gamma)$ bound a pair
of pants on $S_{2}$.
3. 3.
Three nonseparating curves. Given that $\alpha$, $\beta$ and $\gamma$ are
nonseparating curves, we can always find a pants decomposition $P$ such that
all their neighbours in $\mathcal{A}(P)$ are nonseparating, $\alpha$ and
$\beta$ have degree three in $\mathcal{A}(P)$, $\gamma$ has degree four in
$\mathcal{A}(P)$ and $\alpha$ and $\gamma$ only have one common neighbour
$\beta$ in $\mathcal{A}(P)$. For an example consider figure 5. Then,
$\phi(\alpha)$ and $\phi(\beta)$ have degree three, $\phi(\gamma)$ has degree
four, and all their neighbours are nonseparating. If $\phi(\alpha)$,
$\phi(\beta)$ and $\phi(\gamma)$ do not bound a pair of pants on $S_{2}$ then
there exist a pair of pants bounded by $\phi(\alpha)$, $\phi(\beta)$ and
$\delta_{1}\neq\phi(\gamma)$, another pair of pants bounded by $\phi(\beta)$,
$\phi(\gamma)$ and $\delta_{2}\neq\phi(\alpha)$, and another pair of pants
bounded by $\phi(\alpha)$, $\phi(\gamma)$ and $\delta_{3}\neq\phi(\beta)$.
Since $\phi(\beta)$ is the only common neighbour of $\phi(\alpha)$ and
$\phi(\gamma)$, then $\delta_{3}$ is not an essential curve, which means it is
isotopic to a boundary component, but this leads us to a contradiction, since
$\phi(\gamma)$ would then have degree at most $3$.
$\phi(\gamma)$$\phi(\alpha)$$\phi(\beta)$$\delta_{3}$$\delta_{2}$$\delta_{1}$
Figure 4: If $\phi(\alpha)$, $\phi(\beta)$ and $\phi(\gamma)$ do not bound a
pair of pants. $\alpha$$\gamma$$\beta$ Figure 5: Three nonseparating curves
bounding a pair of pants.
∎
###### Remark 4.
If $P$ is a pants decomposition and $\alpha\in P$ is a nonseparating curve of
degree $2$ in $\mathcal{A}(P)$ such that its neighbours are also nonseparating
curves, then $\alpha$ forms part of two peripheral pairs, namely one with each
neighbour (otherwise either of its neighbours or $\alpha$ itself would become
separating).
###### Lemma 4.4.
Let $S_{1}$ and $S_{2}$ be infinite genus surfaces and let
$\phi:\mathcal{C}(S_{1})\rightarrow\mathcal{C}(S_{2})$ be an isomorphism. If
$\alpha$ and $\beta$ form a peripheral pair, then their images form a
peripheral pair. In particular, $S_{1}$ and $S_{2}$ have the same number of
boundary components.
###### Proof.
If $S_{1}$ admits at least $2$ peripheral pairs such that their curves are
pairwise disjoint as in figure 6, then we can always find a pants
decomposition $P$ of $S_{1}$ such that all the neighbours of $\beta$ are
nonseparating, $\deg(\alpha)=3$ and $\deg(\beta)=2$. Then all the neighbours
of $\phi(\beta)$ are nonseparating, and $\phi(\beta)$ has degree $2$, hence it
has to form a peripheral pair with $\phi(\alpha)$ by the previous remark.
If for any two peripheral pairs in $S_{1}$ at least one curve of each pair
intersect each other, we can always find a pants decomposition $P$ of $S_{1}$
such that all the neighbours of $\alpha$ and $\beta$ are nonseparating,
$\deg(\alpha)=\deg(\beta)=3$, and there is only one pair of pants in
$S_{1}\backslash P$ that is bounded by $\alpha$ and $\beta$ at the same time,
namely the one formed by $\alpha$ and $\beta$ being a peripheral pair. Then
$\phi(P)$ is a pants decomposition with all the neighbours of $\phi(\alpha)$
and $\phi(\beta)$ being nonseparating,
$\deg(\phi(\alpha))=\deg(\phi(\beta))=3$ and there exists a pair of pants in
$S_{2}$ bounded by $\phi(\alpha)$, $\phi(\beta)$ and $\delta$. Due to lemma
4.3 applied to $\phi$ and $\phi^{-1}$, $\delta$ cannot be an essential curve
different to both $\phi(\alpha)$ and $\phi(\beta)$, but if
$\delta=\phi(\alpha)$ or $\delta=\phi(\beta)$ then either $\phi(\beta)$ of
$\phi(\alpha)$, respectively, becomes separating. Then $\delta$ is isotopic to
a boundary component and so, $\phi(\alpha)$ and $\phi(\beta)$ form a
peripheral pair.
This result implies that $S_{2}$ has at least as many boundary components as
$S_{1}$, and applying the same result to $\phi^{-1}$ we get that they have the
same number of boundary components, even if this number is infinite. ∎
$\beta$$\alpha$ Figure 6: Example of a convenient pants decomposition.
Proof of theorem 2. Let $P$ be a pants decomposition of $S_{1}$. From the fact
that (10) is an isomorphism and theorem 4 we have
$\mathrm{Ends}(S_{1})\cong\mathrm{Ends}(\mathcal{A}(P))\cong\mathrm{Ends}(\mathcal{A}(\phi(P)))\cong\mathrm{Ends}(S_{2})$.
From the surface classification theorem for infinite surfaces by Richards,
Prishlyak and Mischenko in [Ray] and [PM], it is sufficient to prove that
$S_{1}$ and $S_{2}$ have the same number of boundary components to guarantee
that they are homeomorphic. But this is guaranteed by lemma 4.4.∎
###### Remark 5.
Theorem 2 cannot be extended for infinite genus surfaces with punctures.
Indeed, let $S$ be an infinite genus surface with $n>0$ boundary components
and without planar ends. Let $S^{\prime}$ be the infinite genus surface
obtained from $S$ by glueing one punctured disc to $S$ along a boundary
component. Clearly $S$ and $S^{\prime}$ are not homeomorphic, but
$\mathcal{C}(S)\cong\mathcal{C}(S^{\prime})$.
Proof of corollary 1. The statement is immediate for all arguments given in
the proof of theorem 2 remain valid if we change $\mathcal{C}(S)$ for
$\mathcal{N}(S)$ and take all pants decompositions to be formed just by
nonseparating curves. ∎
### 4.3 Surjectivity.
At the end of this section we give a proof for theorem 1. We begin by proving
the following theorems:
###### Theorem 6.
Let $S$ be an infinite genus surface. Then
$\mathrm{Aut}(\mathcal{G}(S))\cong\mathrm{Aut}(\mathcal{N}(S))$.
###### Theorem 7.
Let $S$ be an infinite genus surface such that
$\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$. The natural map:
$\Psi_{\mathcal{G}(S)}:{\rm MCG}^{*}(S)\to{\rm Aut}(\mathcal{G}(S))$ (11)
is surjective.
This two results imply that the natural map:
$\Psi_{\mathcal{N}(S)}:{\rm MCG}^{*}(S)\to{\rm Aut}(\mathcal{N}(S))$ (12)
is surjective. Using the surjectivity of this map, we can deduce the
following:
###### Theorem 8.
Let $S$ be an infinite genus surface such that
$\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$. The natural map:
$\Psi_{\mathcal{C}(S)}:{\rm MCG}^{*}(S)\to{\rm Aut}(\mathcal{C}(S))$ (13)
is surjective.
#### 4.3.1 Proof of theorems 6 and 7.
The proofs of theorems 6 and 7 require some auxiliary lemmas given in [Irmak]
and [Sch] but adapted to the context of infinite type surfaces. When the
proofs of these lemmas can be easily deduced from the cited works we just
state them without a proof. When this is not the case elementary and simple
proofs are provided. We recall first the different components that a curve
might have.
###### Definition 4.5 (Curve components).
Let $\alpha$ and $\beta$ be nonseparating curves such that
$i(\alpha,\beta)\geq 2$. Let $\beta_{1}$ be a connected component of $\beta$
in $S_{\alpha}$. If the surface resulting from cutting $S_{\alpha}$ along
$\beta_{1}$ is connected, then $\beta_{1}$ is called a nonseparating component
of $\beta$ (with respect to $\alpha$). Otherwise, $\beta_{1}$ is called a
separating component of $\beta$ (with respect ot $\alpha$). If $\beta_{1}$
connects the two different boundary components of $S_{\alpha}$ induced by
$\alpha$, then $\beta_{1}$ is called a two-sided component. Otherwise it is
called one-sided.
###### Lemma 4.6.
[Sch] Let $S$ be an infinite genus surface and
$\alpha,\beta\in\mathcal{V}(\mathcal{N}(S))$ such that $i(\alpha,\beta)\geq
2$. If $\beta$ has a nonseparating component $\beta_{1}$ with respect to
$\alpha$, then there exists
$\gamma,\gamma^{\prime}\in\mathcal{V}(\mathcal{N}(S))\backslash\\{\alpha,\beta\\}$
such that $N(\alpha,\beta)\subset(N(\gamma)\cup N(\gamma^{\prime}))$.
Moreover, if $\beta_{1}$ is one-sided, then $\alpha,\gamma,\gamma^{\prime}$
are mutually disjoint; if $\beta_{1}$ is two-sided, then
$\\{\alpha,\gamma,\gamma^{\prime}\\}$ is a triple with
$i(\alpha,\beta)=i(\beta,\gamma)+i(\beta,\gamma^{\prime})$
and $\min\\{i(\beta,\gamma),i(\beta,\gamma^{\prime})\\}>0$.
###### Lemma 4.7.
_[_Ibid._]_ Let $S_{1}$ and $S_{2}$ be infinite genus surfaces and let
$\phi:\mathcal{G}(S_{1})\rightarrow\mathcal{G}(S_{2})$ be an isomorphism. Then
for any disjoint curves $\alpha$ and $\beta$, their images under $\phi$ will
also be disjoint.
Proof theorem 6. Let $\phi\in\mathrm{Aut}(\mathcal{N}(S))$. Since any
automorphism of $\mathcal{N}(S)$ (and $\mathcal{G}(S)$ respectively) is
uniquely determined by the function on its vertices and
$\mathcal{V}(\mathcal{N}(S))=\mathcal{V}(\mathcal{G}(S))$, then $\phi$ induces
a bijection $\phi^{*}:\mathcal{G}(S)\rightarrow\mathcal{G}(S)$. From the work
of Irmak [Irmak] on the characterization of two curves that intersect once,
one can deduce that if $S_{1}$ and $S_{2}$ are infinite genus surfaces, and
$\phi_{1}:\mathcal{C}(S_{1})\rightarrow\mathcal{C}(S_{2})$ and
$\phi_{2}:\mathcal{N}(S_{1})\rightarrow\mathcal{N}(S_{2})$ are isomorphisms,
then for any curves $\alpha_{1}$ and $\alpha_{2}$ such that
$i(\alpha_{1},\alpha_{2})=1$ we have that
$i(\phi_{1}(\alpha_{1}),\phi_{1}(\alpha_{2}))=i(\phi_{2}(\alpha_{1}),\phi_{2}(\alpha_{2}))=1$.
This fact applied to $\phi$ and $\phi^{-1}$ implies that $\phi^{*}$ must
preserve adjacency and non-adjacency. Hence we can define the function
$\Phi:\mathrm{Aut}(\mathcal{N}(S))\rightarrow\mathrm{Aut}(\mathcal{G}(S))$
(14)
as $\phi\mapsto\phi^{*}$. This function is clearly an injective group
homomorphism.
In the same way, for any automorphism of $\mathcal{G}(S)$ we can induce a
bijection from $\mathcal{N}(S)$ to itself, and due to lemma 4.7 this bijection
will become an automorphism of $\mathcal{N}(S)$. Therefore $\Phi$ is an
isomorphism. ∎
###### Remark 6.
From the proof of theorem 6 and the proof of corollary 1 we conclude that the
statements of lemmas 4.3 and 4.4 remain valid if we change $\mathcal{C}(S)$
for $\mathcal{G}(S)$.
The following four lemmas are used in the proof of theorem 7. Let us recall
first the notion of triple of curves.
###### Definition 4.8 (Triples of curves).
Let $\alpha$, $\beta$ and $\gamma$ be nonseparating curves of $S$. We will say
$\\{\alpha,\beta,\gamma\\}$ is a triple if
$i(\alpha,\beta)=i(\alpha,\gamma)=i(\beta,\gamma)=1$ and there exists a
subsurface of $S$ which contains $\alpha$, $\beta$ and $\gamma$, and is
homeomorphic to a torus with one boundary component.
###### Lemma 4.9.
Let $S$ be infinite genus surface and
$\alpha,\beta\in\mathcal{V}(\mathcal{N}(S))$ be such that $i(\alpha,\beta)\geq
2$. If $\beta$ does not have two-sided components with respect to $\alpha$,
then there exists
$\gamma,\gamma^{\prime}\in\mathcal{V}(\mathcal{N}(S))\backslash\\{\alpha,\beta\\}$
such that $\\{\alpha,\gamma,\gamma^{\prime}\\}$ is a triple with
$i(\alpha,\beta)=i(\beta,\gamma)+i(\beta,\gamma^{\prime})$
and $\min\\{i(\beta,\gamma),i(\beta,\gamma^{\prime})\\}>0$.
###### Proof.
Let $\alpha_{1}$ and $\alpha_{2}$ be the boundary components on $S_{\alpha}$
induced by $\alpha$. Since $\beta$ does not have two-sided components then it
only has one-sided components and therefore we can choose a curve $\gamma$
that intersects $\alpha$ once, does not intersect any one sided component of
$\beta$ based on $\alpha_{1}$ and intersects $\beta$ in such a way that
$0<i(\gamma,\beta)\leq\frac{1}{2}i(\alpha,\beta)$. This can be done by drawing
$\gamma$ disjoint from every one-sided component of $\beta$ based on
$\alpha_{1}$, we keep on going “following” a convenient one-sided component of
$\beta$ based on $\alpha_{2}$ until before we reach $\alpha_{2}$, then we
intersect $\alpha_{2}$ in the corresponding point seeking the desired
inequality. See figure 7 for examples.
$\beta$$\gamma$$\beta$$\beta$$\gamma$$\beta$ Figure 7: Examples of $\beta$ and
$\gamma$ in $S_{\alpha}$
Then let $N$ be a regular neighbourhood of $\alpha$ and $\gamma$; since
$i(\alpha,\gamma)=1$ then $N$ is homeomorphic to a torus with one boundary
component. Let $\gamma^{\prime}$ be the image of $\gamma$ under a Dehn twist
along $\alpha$ on $N$. See figure 8 for the corresponding diagram.
$\alpha$$\gamma$$\alpha$$\gamma^{\prime}$$\beta$ Figure 8: Diagram of $N$.
Thus $\\{\alpha,\gamma,\gamma^{\prime}\\}$ form a triple and by construction
$i(\beta,\gamma^{\prime})=i(\alpha,\beta)-i(\beta,\gamma)$, with both curves
intersecting $\beta$ at least once. ∎
###### Lemma 4.10.
Let $S$ be an infinite genus surface and let
$\phi:\mathcal{G}(S)\rightarrow\mathcal{G}(S)$ be an automorphism. Then
$i(\alpha,\beta)=i(\phi(\alpha),\phi(\beta))$ for all
$\alpha,\beta\in\mathcal{V}(\mathcal{G}(S))$.
###### Proof.
Let $\alpha,\beta\in\mathcal{V}(\mathcal{G}(S))$. If $i(\alpha,\beta)=0$, then
due to lemma 4.7 we have that $i(\phi(\alpha),\phi(\beta))=0$. If
$i(\alpha,\beta)=1$, then due to $\phi$ being an automorphism
$i(\phi(\alpha),\phi(\beta))=1$. For $i(\alpha,\beta)\geq 2$, we will proceed
by induction on the geometric intersection number.
Let us suppose the geometric intersection number is preserved under
automorphisms for curves which intersect at most $k$ times for a $k\geq 1$.
Let $i(\alpha,\beta)=k+1$. Due to lemmas 4.6 and 4.9, we know there exists
$\gamma,\gamma^{\prime}\in\mathcal{V}(\mathcal{G}(S))\backslash\\{\alpha,\beta\\}$
such that $\\{\alpha,\gamma,\gamma^{\prime}\\}$ is a triple,
$i(\alpha,\beta)=i(\beta,\gamma)+i(\beta,\gamma^{\prime})$ and
$\min\\{i(\beta,\gamma),i(\beta,\gamma^{\prime})\\}>0$.
Since $i(\beta,\gamma),i(\beta,\gamma^{\prime})<k+1$, then
$i(\beta,\gamma)=i(\phi(\beta),\phi(\gamma))$ and
$i(\beta,\gamma^{\prime})=i(\phi(\beta),\phi(\gamma^{\prime}))$. From the work
of Schmutz [_Ibid_.] one can deduce that if $S$ is an infinite genus surface
and $\phi:\mathcal{G}(S)\rightarrow\mathcal{G}(S)$ an automorphism, then for
every triple $\\{\alpha,\beta,\gamma\\}$ we have that
$\\{\phi(\alpha),\phi(\beta),\phi(\gamma)\\}$ form a triple. Therefore
$\\{\phi(\alpha),\phi(\gamma),\phi(\gamma^{\prime})\\}$ form a triple. Using a
diagram of the torus with one boundary component which contains this triple
(see figure 8), we can see that each time $\phi(\beta)$ intersects
$\phi(\alpha)$ then either $\phi(\beta)$ intersects $\phi(\gamma)$ or
$\phi(\beta)$ intersects $\phi(\gamma^{\prime})$. Therefore
$i(\phi(\beta),\phi(\gamma))+i(\phi(\beta),\phi(\gamma^{\prime}))\geq
i(\phi(\alpha),\phi(\beta))$. Thus $i(\alpha,\beta)\geq
i(\phi(\alpha),\phi(\beta))$. Applying the same argument on $\phi^{-1}$ we
obtained the symmetric inequality, therefore
$i(\alpha,\beta)=i(\phi(\alpha),\phi(\beta))$. ∎
###### Lemma 4.11.
Let $S$ be an infinite genus surface and let
$\phi:\mathcal{G}(S)\rightarrow\mathcal{G}(S)$ be an automorphism. If $P$ is a
pants decomposition of $S$, then there exist an homeomorphism
$h\in\mathrm{MCG}^{*}(S)$ such that $h(\alpha)=\phi(\alpha)$ for all
$\alpha\in P$.
###### Proof.
From remark 6, we know that $\phi(P)$ is a pants decomposition and the
boundaries of pair of pants in $S_{\phi(P)}$ induced by curves of $\phi(P)$
are boundaries of pair of pants in $S_{P}$ induced by curves of $P$. Then we
can define an homeomorphism of $S$ by parts using homeomorphisms from the
connected components of $S_{P}$ to the corresponding connected components of
$S_{\phi(P)}$; this homeomorphism by construction will agree with $\phi$ for
every element in $P$. ∎
###### Remark 7.
It is clear, using theorem 6, that this lemma remains valid if we substitute
$\mathcal{G}(S)$ by $\mathcal{N}(S)$.
###### Lemma 4.12.
[Sch] Let $S^{\prime}$ be a surface of genus zero and four boundary
components. Let $\alpha,\beta\in\mathcal{V}(\mathcal{C}(S^{\prime}))$ with
$i(\alpha,\beta)=2$.
1. 1.
Let $\gamma\in\mathcal{V}(\mathcal{C}(S^{\prime}))$ such that
$i(\alpha,\gamma)=2$. Then there exists $h\in\mathrm{MCG}^{*}(S^{\prime})$
such that $h(\alpha)=\alpha$ and $h(\beta)=\gamma$.
2. 2.
There are exactly two curves
$\gamma_{1},\gamma_{2}\in\mathcal{V}(\mathcal{C}(S^{\prime}))$ such that
$i(\alpha,\gamma_{i})=i(\beta,\gamma_{i})=2$ for $i=1,2$. Moreover, there
exists $h\in\mathrm{MCG}^{*}(S^{\prime})$ such that $h(\alpha)=\alpha$,
$h(\beta)=\beta$, and $h(\gamma_{1})=\gamma_{2}$.
###### Remark 8.
The homeomorphism of part (1) in the preceding lemma is just a Dehn twist
about $\alpha$, where as the homeomorphism from part (2) is an orientation-
reversing involution that leaves invariant each connected component in the
boundary of $S_{0,4}$.
The proof theorem 7 uses the notion of Dehn-Thurston coordinates. Therefore we
recall it and discuss it briefly in the context of infinite surfaces in the
following paragraphs.
###### Definition 4.13 (Dehn-Thurston coordinates).
A Dehn-Thurston coordinates system of curves is a set $D$ of curves that
parametrize every curve $\alpha\in\mathcal{V}(\mathcal{C}(S))$ using the
geometric intersection number, i.e. for
$\alpha,\beta\in\mathcal{V}(\mathcal{C}(S))$ if
$i(\alpha,\gamma)=i(\beta,\gamma)$ for all $\gamma\in D$, then $\alpha=\beta$.
For compact surface, it is well known that Dehn-Thurston coordinate systems
exist, see [HP]. For noncompact surfaces such a system of curves can be
realized in the following way. Let $\\{\alpha_{i}\\}_{i\in\mathbf{N}}$ be a
pants decomposition, $\\{\beta_{i}\\}_{i\in\mathbf{N}}$ be curves such that
$i(\alpha_{i},\beta_{i})=2$ and $i(\alpha_{i},\beta_{j})=0$ for $i\neq j$, and
$\\{\gamma_{i}\\}_{i\in\mathbf{N}}$ be curves such that
$i(\alpha_{i},\gamma_{i})=i(\beta_{i},\gamma_{i})=2$ and
$i(\alpha_{i},\gamma_{j})=0$ for $i\neq j$. Then the set of curves $D$ formed
by the union of elements in $\\{\alpha_{i}\\}_{i\in\mathbf{N}}$,
$\\{\beta_{i}\\}_{i\in\mathbf{N}}$ and $\\{\gamma_{i}\\}_{i\in\mathbf{N}}$ is
a Dehn-Thurston coordinate system. Indeed, any curve $\delta$ in $S$ will only
intersect finitely many curves in $D$, hence we can take any compact
subsurface $S^{\prime}$, such that it contains $\delta$ and there is a
(finite) subset $D^{\prime}$ of $D$ that is a Dehn-Thurston coordinate system
of $S^{\prime}$. Any other curve in $S$ with the same Dehn-Thurston
coordinates as $\delta$ on the system $D$, would have to be isotopic to a
curve contained in $S^{\prime}$ and thus would have the same Dehn-Thurston
coordinates as $\delta$ on the system $D^{\prime}$, therefore it would be
isotopic to $\delta$. We must remark that, when $S$ is an infinite genus
surface such that $\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$, we can also
construct the Dehn-Thurston coordinate system $D$ with families
$\\{\alpha_{i}\\}_{i\in\mathbf{N}}$, $\\{\beta_{i}\\}_{i\in\mathbf{N}}$ and
$\\{\gamma_{i}\\}_{i\in\mathbf{N}}$ formed exclusively by nonseparating
curves.
Proof theorem 7. Given that $\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$ we can
construct $P=\\{\alpha_{i}\\}_{i\in\mathbf{N}}$ a pants decomposition of $S$
formed by nonseparating curves. Let $\phi:\mathcal{G}(S)\to\mathcal{G}(S)$ an
automorphism. Due to lemma 4.11 there exists an homeomorphism
$h_{1}:S\rightarrow S$ such that $h_{1}(\alpha_{i})=\phi(\alpha_{i})$ for all
$\alpha_{i}\in P$.
Again, since $\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$ we can construct
$\\{\beta_{i}\\}_{i\in\mathbf{N}}$ a collection of nonseparating curves such
that $i(\alpha_{i},\beta_{i})=2$ for all $i$ and $i(\alpha_{i},\beta_{j})=0$
for $i\neq j$. We can define an homeomorphism $h_{2}:S\rightarrow S$ such that
$h_{2}(h_{1}(\alpha_{i}))=h_{1}(\alpha_{i})=\phi(\alpha_{i})$ and
$h_{2}(h_{1}(\beta_{i}))=\phi(\beta_{i})$ in the following way. For every
$i\in\mathbf{N}$ the curves $\alpha=h_{1}(\alpha_{i})$,
$\beta=h_{1}(\beta_{i})$ and $\gamma=\phi(\beta_{i})$ satisfy the hypotheses
of part (1) in lemma 4.12 and lie in a subsurface $S_{i}$ homeomorphic to
$S_{0,4}$ that does not contain any element in
$h_{1}(P)\backslash\\{h(\alpha_{i})\\}$, $S_{i}$ contains $h_{1}(\beta_{i})$
and $\phi(\beta_{i})$, and its boundary components are isotopic to the curves
adjacent to $h_{1}(\alpha_{i})$ with respect to $h_{1}(P)$. Let
$h_{2,i}:S_{i}\rightarrow S_{i}$ be the homeomorphism from $(1)$ in lemma
4.12. This homeomorphism is just a Dehn twist about $\alpha$, therefore it
preserves orientation and its support $K_{i}\subset S_{i}$ satisfies that
$K_{i}\cap K_{j}=\emptyset$ for $i\neq j$ for all $i,j\in\mathbf{N}$. Hence
$h_{2}$ can be defined by parts using $\\{h_{2,i}\\}_{i\in\mathbf{N}}$.
Let $\\{\gamma_{i}\\}_{i\in\mathbf{N}}$ be a collection of curves such that
$i(\alpha_{i},\gamma_{i})=i(\beta_{i},\gamma_{i})=2$ and
$i(\alpha_{i},\gamma_{j})=0$ for $i\neq j$. We can define an homeomorphism
$h_{3}:S\rightarrow S$ such that
$h_{3}(h_{2}(h_{1}(\alpha_{i})))=h_{2}(h_{1}(\alpha_{i})=\phi(\alpha_{i})$,
$h_{3}(h_{2}(h_{1}(\beta_{i})))=h_{2}(h_{1}(\beta_{i}))=\phi(\beta_{i})$ and
$h_{3}(h_{2}(h_{1}(\gamma_{i})))=\phi(\gamma_{i})$ in the following way. For
every $i\in\mathbf{N}$, let now $\alpha=h_{1}(h_{2}(\alpha_{i}))$,
$\beta=h_{1}(h_{2}(\beta_{i}))$, $\gamma_{1}=h_{1}(h_{2}(\gamma_{i}))$ and
$\gamma_{2}=\phi(\gamma_{i})$. Analogously to the preceding case, these curves
satisfy the hypotheses of part (2) in lemma 4.12. Let
$h_{3,i}:R_{i}\rightarrow R_{i}$ be the (orientation-reversing) homeomorphism
from part (2) in lemma 4.12, where $R_{i}$ is homeomorphic to $S_{0,4}$ and
contains the curves $\alpha$, $\beta$, $\gamma_{1}$ and $\gamma_{2}$. It is
not difficult to see that if $i\neq j$ and $R_{i}\cap R_{j}\neq\emptyset$,
then $R_{i,j}=R_{i}\cap R_{j}\cong S_{0,3}$. Moreover $h_{3,i}$ and $h_{3,j}$
coincide in $R_{i,j}$ and hence we can define $h_{3}$ by parts using
$\\{h_{3,i}\\}_{i\in\mathbf{N}}$.
Let $h=h_{3}\circ h_{2}\circ h_{1}$. Since
$P^{\prime}=P\cup\\{\beta_{i}\\}\cup\\{\gamma_{i}\\}$ form a Dehn-Thurston
coordinates system of curves, then $h(P^{\prime})$ is a Dehn-Thurston
coordinates system of curves, and by construction
$h(\varepsilon)=\phi(\varepsilon)$ for all $\varepsilon\in P^{\prime}$.
Therefore, due to lemma 4.10, for all $\delta\in\mathcal{V}(\mathcal{G}(S))$
and all $\varepsilon\in P^{\prime}$:
$i(\phi(\delta),\phi(\varepsilon))=i(\delta,\varepsilon)=i(h(\delta),h(\varepsilon))=i(h(\delta),\phi(\varepsilon)),$
(15)
then $\phi(\delta)=h(\delta)$ for all $\delta\in\mathcal{V}(\mathcal{G}(S))$,
which implies $\Psi_{\mathcal{G}(S)}$ is surjective. ∎
###### Corollary 4.
Let $S_{1}$ and $S_{2}$ be infinite genus surfaces, such that
$\mathrm{Ends}(S_{i})=\mathrm{Ends}^{*}(S_{i})$ for $i=1,2$ and let
$\phi:\mathcal{G}(S_{1})\rightarrow\mathcal{G}(S_{2})$ be an isomorphism. Then
$S_{1}$ and $S_{2}$ are homeomorphic and $\phi$ is induced by a mapping class
in ${\rm MCG}^{*}(S_{1})$.
###### Proof.
Every isomorphism $\phi:\mathcal{G}(S_{1})\rightarrow\mathcal{G}(S_{2})$
induces an isomorphism $\phi:\mathcal{N}(S_{1})\rightarrow\mathcal{N}(S_{2})$.
Indeed, take $u,v$ two curves such that $i(u,v)=0$. Suppose
$i(\phi(u),\phi(v))\geq 2$ and remark that, as in the proof of theorem 7,
lemmas 1, 3 and 5 in [Sch] remain valid in the context of this corollary.
Hence we obtain a contradiction. On the other hand it is clear that
$i(\phi(u),\phi(v))\neq 1$, for $\phi$ is an isomorphism. Hence the only
possibility left is that $i(\phi(u),\phi(v))=0$. By corollary 1 we obtain that
$S_{1}$ is homemorphic to $S_{2}$. The rest of the proof follows from theorems
1 and 7. ∎
#### 4.3.2 Proof of theorem 8.
Any $\phi\in\mathrm{Aut}(\mathcal{C}(S))$ sends nonseparating curves to
nonseparating curves, hence
$\phi|_{\mathcal{N}(S)}\in\mathrm{Aut}(\mathcal{N}(S))$ and then due to
theorem 7 there exists $h\in\mathrm{MCG}^{*}(S)$ such that
$\phi|_{\mathcal{N}(S)}(\alpha)=h(\alpha)$ for all
$\alpha\in\mathcal{V}(\mathcal{N}(S))$. Hence we only need to check that
$\phi$ and $h$ coincide in the separating curves of $S$. Let $\alpha$ be a
separating curve of $S$; we consider three cases.
1. 1.
If both connected components of $S_{\alpha}$ have positive genus, then we can
find a pants decomposition $P$ such that $\alpha\in P$,
$(P\backslash\\{\alpha\\})\subset\mathcal{V}(\mathcal{N}(S))$ and
$\deg(\alpha)=4$ in $\mathcal{A}(P)$; let $\beta_{1}$, $\gamma_{1}$,
$\beta_{2}$ and $\gamma_{2}$ be the neighbours of $\alpha$ in $\mathcal{A}(P)$
such that $\beta_{i}$ and $\gamma_{i}$ are in the same connected component of
$S_{\alpha}$ for $i=1,2$. Let also $\delta_{1}$ and $\delta_{2}$ be
nonseparating curves such that $i(\alpha,\delta_{i})=0$ and
$i(\beta_{i},\delta_{i})=i(\gamma_{i},\delta_{i})=1$ for $i=1,2$. See figure 9
for an example.
$\gamma_{1}$$\gamma_{2}$$\beta_{1}$$\beta_{2}$$\delta_{1}$$\delta_{2}$$\alpha$
Figure 9: Catching $\alpha$ in a $S_{0,4}$.
By construction and lemma 4.3, $\phi(\alpha)$ and $h(\alpha)$ are contained in
the $S_{0,4}$ subsurface bounded by $\phi(\beta_{1})$, $\phi(\gamma_{1})$,
$\phi(\beta_{2})$ and $\phi(\gamma_{2})$ (recall that
$\phi(\beta_{i})=h(\beta_{i})$ and $\phi(\gamma_{i})=h(\gamma_{i})$ for
$i=1,2$ since they are nonseparating curves). Even more, since
$i(\alpha,\delta_{i})=0$ for $i=1,2$ then $\phi(\alpha)$ and $h(\alpha)$ must
be contained in the annulus formed by cutting the aforementioned $S_{0,4}$
subsurface along the arcs of $\phi(\delta_{i})=h(\delta_{i})$ for $i=1,2$;
therefore $\phi(\alpha)=h(\alpha)$.
2. 2.
If $\alpha$ is an outer curve, then let $P$ be a pants decomposition such that
the peripheral pairs of $P$ bounding the same boundary components as $\alpha$,
are consecutive to one another (similar to the proof of lemma 4.4), and
$\alpha$ intersects only one curve in $P$ (namely $\beta$); let also $\gamma$
be a nonseparating curve that intersects each curve in the peripheral pairs
bounding the same boundary component as $\alpha$ only once while being
disjoint from $\alpha$. Figure 10 illustrates this situation.
$\beta$$\alpha$$\gamma$ Figure 10: Catching $\alpha$ again in a $S_{0,4}$.
Due to $\phi$ being an isomorphism, $\phi(\alpha)$ will intersect
$\phi(\beta)$ and be disjoint of every other curve in $P$. Using that and
lemma 4.4, we know that $\phi(\alpha)$ and $h(\alpha)$ are contained in the
$S_{0,4}$ subsurface bounded by two boundary components of $S$ and the images
of the adjacent curves in $\mathcal{A}(P)$ of $\beta$; even more,
$\phi(\alpha)$ and $h(\alpha)$ must be contained in the pair of pants
resulting from cutting the aforementioned $S_{0,4}$ subsurface that contains
them along the arc of $\phi(\gamma)=h(\gamma)$. Since there is only one curve
in this pair of pants which is an essential curve of $S$, then
$\phi(\alpha)=h(\alpha)$.
3. 3.
Let $S_{1}$ and $S_{2}$ be the two connected components of $S_{\alpha}$ and
suppose that $S_{1}$ has genus zero and $n^{\prime}\geq 3$ boundary
components. We can find the following: a finite sequence
$\\{\beta_{i}\\}_{i=1}^{n^{\prime}-1}$ composed of outer curves, such that
$i(\beta_{i},\alpha)=0$ for $i=1,\ldots,n^{\prime}-1$,
$i(\beta_{i},\beta_{i+1})=2$ for $i=1,\ldots,n^{\prime}-2$ and
$i(\beta_{i},\beta_{j})=0$ for $j\notin\\{i-1,i+1\\}$; a pants decomposition
$P$ (composed solely of nonseparating curves) of the infinite genus connected
component of $S\backslash\\{\alpha\\}$; and finally, a curve $\gamma$ which
intersects once the curves $\delta_{1}$ and $\delta_{2}$ forming the
peripheral pair that bounds the boundary of $S_{2}$ induced by $\alpha$.
Figure 11 illustrates this situation.
$\alpha$$\gamma$$\delta_{1}$$\delta_{2}$$\beta_{1}$$\beta_{2}$$\beta_{3}$
Figure 11: Catching $\alpha$ in an annulus.
Given that isomorphism of $\mathcal{C}(S)$ send outer curves to outer curves,
part $(2)$ of this proof, the fact that $\phi(\alpha)$ and $h(\alpha)$ must
both be essential curves and they must be different from every element of
$\phi(\\{\beta_{i}\\}_{i=1}^{n^{\prime}})\cup\phi(P)\cup\\{\phi(\gamma)\\}$;
we can conclude that $\phi(\alpha)$ and $h(\alpha)$ must be contained in the
annulus obtained by cutting $S$ along
$\phi(\\{\beta_{i}\\}_{i=1}^{n^{\prime}})\cup\phi(P)\cup\\{\phi(\gamma)\\}$.
The boundary components of this annulus are formed by arcs of
$\phi(\beta_{i})$ for $i=1,\ldots,n^{\prime}-1$, $\phi(\gamma)$,
$\phi(\delta_{1})$ and $\phi(\delta_{2})$. Therefore $\phi(\alpha)=h(\alpha)$.
#### 4.3.3 Proof of theorem 1.
Theorems 5 and 8 imply that $\Psi_{\mathcal{C}(S)}$ is an isomorphism. From
theorem 7 we know that the natural map:
$\Psi_{\mathcal{N}(S)}:{\rm MCG}^{*}(S)\to{\rm Aut}(\mathcal{N}(S))$ (16)
is surjective. Let us suppose $h_{1},h_{2}\in\mathrm{MCG}^{*}(S)$ are such
that $h_{1}\neq h_{2}$ and
$\Psi_{\mathcal{N}(S)}(h_{1})=\Psi_{\mathcal{N}(S)}(h_{2})$. Then since
$\Psi_{\mathcal{C}(S)}$ is injective we have that
$\Psi_{\mathcal{C}(S)}(h_{1})\neq\Psi_{\mathcal{C}(S)}(h_{2})$ even though
their restrictions to $\mathcal{N}(S)$ are the same. This implies that
$\Psi_{\mathcal{C}(S)}(h_{1})$ and $\Psi_{\mathcal{C}(S)}(h_{2})$ differ in
some separating curves. But given that the restrictions of
$\Psi_{\mathcal{C}(S)}(h_{1})$ and $\Psi_{\mathcal{C}(S)}(h_{2})$ to
$\mathcal{N}(S)$ are the same, we can use the same technique as in the proof
of theorem 8, for catching the separating curves in an annulus (or a pair of
pants), which means
$\Psi_{\mathcal{C}(S)}(h_{1})(\alpha)=\Psi_{\mathcal{C}(S)}(h_{2})(\alpha)$
for every separating curve $\alpha$. Thus we have reached a contradiction and
therefore $\Psi_{\mathcal{N}(S)}$ is injective, hence it is an isomorphism. We
finish the proof by remarking that
$\Psi_{\mathcal{G}(S)}=\Phi\circ\Psi_{\mathcal{N}(S)}$, where $\Phi$ is the
isomorphism between $\mathrm{Aut}(\mathcal{N}(S))$ and
$\mathrm{Aut}(\mathcal{G}(S))$ defined in (14). ∎
###### Remark 9.
Using theorem 1 we can deduce that, for an infinite genus surface $S$ such
that $\mathrm{Ends}(S)=\mathrm{Ends}^{*}(S)$, every automorphism $\varphi$ of
${\rm MCG}^{*}(S)$ sending Dehn twists to Dehn twist must be an inner
automorphism. The proof of this fact is taken verbatim from the proof of
theorem 2, in [Ivanov]. However, it is still unknown if, as in the compact
case, every automorphism of ${\rm MCG}^{*}(S)$ sends Dehn twists to Dehn
twists.
## 5 Counterexamples
In this section we show that theorem 2 is not valid if the morphism between
curve complexes is not an isomorphism. For that, let us first recall the
notion of _superinjective map_.
###### Definition 5.1 (Superinjectivity).
A simplicial map $f:\mathcal{C}(S_{1})\rightarrow\mathcal{C}(S_{2})$ is called
superinjective if for any two vertices $\alpha$ and $\beta$ in
$\mathcal{C}(S_{1})$ such that $i(\alpha,\beta)\neq 0$ we have that
$i(f(\alpha),f(\beta))\neq 0$.
Every superinjective map is injective. For compact surfaces, we have the
following theorem concerning superinjective maps.
###### Theorem 9.
[Irmak] Let $S$ be a closed, connected, orientable surface of genus at least
3. A simplicial map, $f:\mathcal{C}(S)\to\mathcal{C}(S)$, is superinjective if
and only if $f$ is induced by an homeomorphism of $S$.
The following lemma shows that this result is not true for a large class of
surfaces of infinite genus and, in this sense, theorem 2 is optimal.
###### Lemma 5.2.
Let $S$ be a surface such that $\mathrm{Ends}^{*}(S)\neq\emptyset$. Then there
exist a simplicial superinjective map $f:\mathcal{C}(S)\to\mathcal{C}(S)$
which is not surjective.
###### Proof.
This proof makes reference to figure 12. Let
$\alpha\in\mathcal{V}(\mathcal{C}(S))$ be a separating curve. Without loss of
generality we can think that $\alpha$ is contained in a subsurface $S_{i}$ in
$[S_{1}\supseteq S_{2}\supseteq\ldots]\in\mathrm{Ends}^{*}(S)$ where $i$ is
large enough.
$\alpha$$\beta$$\beta^{\prime}$$\alpha$$\gamma$$f(\beta)$$f(\beta^{\prime})$$f$
Figure 12: A superinjective but not surjective simplicial map.
We describe $f$ topologically. Let $S_{1}$ and $S_{2}$ be the two connected
components of $S_{\alpha}$. Cut $S$ along $\alpha$ and then glue in a copy of
$S_{1,2}$. This operation produces a new surface $S^{\prime}=S_{1}\cup
S_{2}\cup S_{1,2}$. Remark that $S$ is homeomorphic to $S^{\prime}$ and that
there is a natural inclusion map $f_{i}:S_{i}\hookrightarrow S^{\prime}$, for
$i=1,2$. If $\beta\in\mathcal{V}(\mathcal{C}(S_{i}))$, then we define
$f(\beta)=f_{i}(\beta)$ for $i=1,2$. On the other hand, if $\beta^{\prime}$
intersects the curve $\alpha$ we define $f(\beta^{\prime})$ as depicted in
figure 12. Clearly, $f$ is superinjecive but no essential curve properly
contained in the copy of $S_{1,2}$ that we introduced is in the image of $f$.
Hence $f$ is not surjective and, in particular, $f$ cannot be induced by a
class in ${\rm MCG}^{*}(S)$. ∎
We think that this result can be optimized in the following way.
###### Conjecture 1.
Let S be a surface such that $\mathrm{Ends}^{*}(S)\neq\emptyset$ and
$\\{\alpha_{1},\ldots,\alpha_{n}\\}\subset\mathcal{C}(S)$ be simplex. Then
there exists a simplicial superinjective map
$f:\mathcal{C}(S)\to\mathcal{C}(S)$ whose image does not intersect
$\\{\alpha_{1},\ldots,\alpha_{n}\\}$.
The following result shows that the statement of theorem 2 is not valid for
superinjective maps.
$\alpha$$\beta$$\beta^{\prime}$$\alpha$$f(\beta^{\prime})$$f(\beta)$$f$ Figure
13: A superinjecive map between two nonhomeomorphic surfaces.
###### Lemma 5.3.
There exist uncountably many examples of pairs of nonhomeomorphic infinite
genus surfaces $S_{1}$ and $S_{2}$ for which there exists a superinjective map
$f:\mathcal{C}(S_{1})\to\mathcal{C}(S_{2})$.
###### Proof.
The arguments are similar to those of the proof of lema 5.2. Let $S_{1}$ be
the Loch Ness monster and $\alpha\in\mathcal{C}(S_{1})$ be a separating curve.
Let $S$ be your favorite infinite genus surface and suppose that $S$ has at
least two boundary components. We describe $f$ topologically. Cut $S_{1}$
along $\alpha$ and then glue in a copy of $S$ as indicated in figure 13. This
produces $S_{2}$. The rest of the proof is analogous to the proof of lemma
5.2.
∎
## References
|
arxiv-papers
| 2014-02-13T20:20:15 |
2024-09-04T02:49:58.219060
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jes\\'us Hern\\'andez Hern\\'andez, Jos\\'e Ferr\\'an Valdez Lorenzo",
"submitter": "Ferran Valdez",
"url": "https://arxiv.org/abs/1402.3275"
}
|
1402.3322
|
# $p$\-
. .
, . , 29, , 100125, . [email protected]
###### Abstract.
$p$-
. - $p$\- . $p$\- .
: , , , , - , $p$\- .
## 1\.
. .
[2, 4].
[5, 8, 14], $p$\- , . [5] $p$\- . $p$\- ( . [1, 15, 6, 11, 9, 10, 12]). [3].
[6] $p$-
. , $p$\- \- . , . [9, 10] \- $p$\- . , . [6] \- $p$\- . , $J<0$, \- $p$\- .
[6]. \- $p$\- . $p$\- .
## 2\.
### 2.1. $p$\- .
$x\neq 0$ $x=p^{r}\frac{n}{m}$, $r,n\in\mathbb{Z},m$– , $(n,m)=1$, $m$ $n$ $p$
$p$ – . $p$\- $|x|_{p}$
$|x|_{p}=\left\\{\begin{array}[]{ll}p^{-r},&\text{ }x\neq 0,\\\ 0,&\text{
}x=0.\end{array}\right.$
:
$|x+y|_{p}\leq\max\\{|x|_{p},|y|_{p}\\}.$
.
:
1) $|x|_{p}\neq|y|_{p}$, $|x-y|_{p}=\max\\{|x|_{p},|y|_{p}\\}$;
2) $|x|_{p}=|y|_{p}$, $|x-y|_{p}\leq|x|_{p}$;
$\mathbb{Q}$ $p$\- $p$\- $\mathbb{Q}_{p}$ $p$ ( . [7]).
$\mathbb{Q}$, $\mathbb{R}$, $p$\- $\mathbb{Q}_{p}$ ( ).
$p$\- $x\neq 0$
$x=p^{\gamma(x)}(x_{0}+x_{1}p+x_{2}p^{2}+\dots),$ (2.1)
$\gamma=\gamma(x)\in\mathbb{Z}$ $x_{j}$ , $0\leq x_{j}\leq p-1$, $x_{0}>0$,
$j=0,1,2,...$ ( [7, 13, 14]). $|x|_{p}=p^{-\gamma(x)}$.
###### 1.
[14] $x^{2}=a$, $0\neq a=p^{\gamma(a)}(a_{0}+a_{1}p+...),0\leq a_{j}\leq p-1$,
$a_{0}>0$ $x\in\mathbb{Q}_{p}$ , :
i) $\gamma(a)$ ;
ii) $y^{2}=a_{0}(\operatorname{mod}p)$ , $p\neq 2$; $a_{1}=a_{2}=0$, $p=2$.
###### 1.
[14] $x^{2}=-1$ $\mathbb{Q}_{p}$, , $p\equiv 1(\operatorname{mod}4)$.
$a\in\mathbb{Q}_{p}$ $r>0$
$B(a,r)=\\{x\in\mathbb{Q}_{p}:|x-a|_{p}<r\\}.$
$p$-
$\log_{p}(x)=\log_{p}(1+(x-1))=\sum_{n=1}^{\infty}(-1)^{n+1}{(x-1)^{n}\over
n},$
$x\in B(1,1)$; $p$-
$\exp_{p}(x)=\sum^{\infty}_{n=0}{x^{n}\over n!},$
$x\in B(0,p^{-1/(p-1)})$.
###### 1.
$x\in B(0,p^{-1/(p-1)})$.
$|\exp_{p}(x)|_{p}=1,\ \ |\exp_{p}(x)-1|_{p}=|x|_{p},\ \
|\log_{p}(1+x)|_{p}=|x|_{p},$ $\log_{p}(\exp_{p}(x))=x,\ \
\exp_{p}(\log_{p}(1+x))=1+x.$
$p$\- $p$\- [7, 13, 14].
$(X,\mathcal{B})$ , $\mathcal{B}$ $X$. $\mu:\mathcal{B}\to\mathbb{Q}_{p}$
$p$\- , $A_{1},...,A_{n}\in\mathcal{B}$ , $A_{i}\cap A_{j}=\varnothing,\ i\neq
j$
$\mu\bigg{(}\bigcup_{j=1}^{n}A_{j}\bigg{)}=\sum_{j=1}^{n}\mu(A_{j}).$
$p$\- , $\mu(X)=1$ ( . [3]).
### 2.2.
$\Gamma^{k}=(V,L)$ $k\geq 1$ ( ), $k+1$ , $V\ -$ $L\ -$ . $x$ $y$ , $l\in L$
$l=\langle x,y\rangle$. $d(x,y)\ -$ , $x$ $y$.
$x^{0}\in V$ . :
$W_{n}=\\{x\in V|d(x,x^{0})=n\\},\qquad V_{n}=\bigcup_{m=0}^{n}W_{m},$
$S(x)=\\{y\in W_{n+1}:d(x,y)=1\\},\quad x\in W_{n}.$
, $S(x)$ $x$. $y$ $z$ , $x\in V$ , $y,z\in S(x)$ $\rangle y,z\langle$.
### 2.3.
$p$\- .
$\mathbb{Q}_{p}$ $p$\- $\Phi=\\{-1;1\\}$. $\sigma$ $V$ $x\in
V\to\sigma(x)\in\Phi$; $\sigma_{n}$ $\sigma^{(n)}$ $V_{n}$ $W_{n}$, . $V$ (
$V_{n},\ W_{n}$) $\Omega=\Phi^{V}$ ( $\Omega_{V_{n}}=\Phi^{V_{n}},\
\Omega_{W_{n}}=\Phi^{W_{n}}$). $\sigma_{n-1}\in\Omega_{V_{n}}$
$\varphi^{(n)}\in\Omega_{W_{n}}$
$(\sigma_{n-1}\vee\varphi^{(n)})(x)=\left\\{\begin{array}[]{ll}\sigma_{n-1}(x),&\text{
}\ x\in V_{n-1},\\\ \varphi^{(n)}(x),&\text{ }\ x\in W_{n}.\end{array}\right.$
, $\sigma_{n-1}\vee\varphi^{(n)}\in\Omega_{V_{n}}.$
$H_{n}:\Omega_{V_{n}}\to\mathbb{Q}_{p}$ $p$-
$H_{n}(\sigma)=J_{1}\sum_{\langle x,y\rangle\in
L_{n}}\sigma(x)\sigma(y)+J_{2}\sum_{\rangle x,y\langle\atop{x,y\in
V_{n}}}\sigma(x)\sigma(y).$ (2.2)
$J_{1},J_{2}\in\mathbb{Q}_{p}$.
###### 1.
, . $J_{2}=0$, .
[12].
### 2.4. $p$\- .
[9, 10] $p$\- (2.2). , .
$h:x\to h_{x}\in\mathbb{Q}_{p}$ $p$\- $V$. $p$\- $\mu_{h}^{(n)}$
$\Omega_{V_{n}}$,
$\mu_{h}^{(n)}(\sigma_{n})=Z_{n,h}^{-1}p^{H_{n}(\sigma_{n})}\prod_{x\in
W_{n}}h_{x}^{\sigma(x)},\qquad n=1,2,...,$ (2.3)
$Z_{n,h}$
$Z_{n,h}=\sum_{\varphi\in\Omega_{V_{n}}}p^{H_{n}(\varphi)}\prod_{x\in
W_{n}}h_{x}^{\varphi(x)}.$ (2.4)
, $p$\- $\mu_{h}^{(n)}$ , $\mbox{ }\ n\geq 1$
$\sigma_{n-1}\in\Omega_{V_{n-1}},$
$\sum_{\varphi\in\Omega_{W_{n}}}\mu_{h}^{(n)}(\sigma_{n-1}\vee\varphi){\bf
1}(\sigma_{n-1}\vee\varphi\in\Omega_{V_{n}})=\mu_{h}^{(n-1)}(\sigma_{n-1}).$
(2.5)
[3] $\mu_{h}$ $\Omega$ ,
$\mu_{h}(\\{\sigma\big{|}_{V_{n}}=\sigma_{n}\\})=\mu_{h}^{(n)}(\sigma_{n})$
$n\in\mathbb{N}$ $\sigma_{n}\in\Omega_{V_{n}}$.
###### 1.
$p$\- $\mu$ $p$\- , $p$\- $h$ $x\in V$ ,
$\mu(\sigma\in\Omega:\sigma|_{V_{n}}=\sigma_{n})=\mu_{h}^{(n)}(\sigma_{n}),\qquad\mbox{
}\ \sigma_{n}\in\Omega_{V_{n}},\qquad n\in\mathbb{N}.$
$\mu_{h}^{(n)}$ (2.3),(2.4).
$\mathcal{QG}(H)$ $p$\- , $h=\\{h_{x},\ x\in V\\}$. (2.2)
$J=J_{1}=J_{2}\in\mathbb{Z}$.
###### 2.
, $\mu_{h}$ $\mu_{-h}$ $h$ $-h$ .
###### 1.
[6] $p$\- $\mu_{h}^{(n)},\ n=1,2,...$ (2.5) , $x\in V$ :
$u_{x}=\frac{\theta^{2}u_{y}u_{z}+u_{y}+u_{z}+1}{u_{y}u_{z}+u_{y}+u_{z}+\theta^{2}},$
(2.6)
$\theta=p^{2J},\ u_{x}=h_{x}^{2}$ $S(x)=\\{y,z\\}$.
###### 3.
, . ” ”. $p$\- $p$\- ” ” $\exp_{p}(x)$. $\exp_{p}(x)$ . , $p$\- . , $p$\- [9]
$p$\- , $p^{x}$. [9, 10] [6] , $\mathcal{QG}(H)$ , $p$\- . , $p$\- ( . [10]).
## 3\. \-
(2.6) $u_{x}=u\in\mathbb{Q}_{p},\ x\neq x_{0}$ \- . $p$\- \- .
$u$ $u_{x}$ $x\neq x_{0}$, (2.6)
$u=\frac{\theta^{2}u^{2}+2u+1}{u^{2}+2u+\theta^{2}}.$ (3.1)
, $u_{0}=1$ (3.1). (3.1) , ( )
$u_{1,2}=\frac{\theta^{2}-3\pm\sqrt{(1-\theta^{2})(5-\theta^{2})}}{2}.$ (3.2)
[6] :
###### 2.
$J>0$. :
(i) $p\in\\{2,3,5\\}$ \- $p$\- $\mu_{h_{0}}$;
(ii) $p>5$ $x_{0}$ $x^{2}\equiv 5\,(\operatorname{mod}p)$. $x^{2}+6\equiv
2x_{0}\,(\operatorname{mod}p)$ , \- $p$\- : $\mu_{h_{0}},\ \mu_{h_{1}},\
\mu_{h_{2}}$.
$h_{0}=1,\ h_{1}=\sqrt{u_{1}},\ h_{2}=\sqrt{u_{2}}$.
###### 3.
$J<0$. - $p$\- $\mu_{h_{0}},\ \mu_{h_{1}},\ \mu_{h_{2}}$.
### 3.1. \- $p$\-
###### 2.
$h$ (2.6) $\mu_{h}$ $p$\- . $Z_{n,h}$ ( . (2.4))
$Z_{n+1,h}=A_{n,h}Z_{n,h},$ (3.3)
$A_{n,h}$ (3.6).
###### Proof.
$h$ (2.6), $x\in V$ $a_{h}(x)\in\mathbb{Q}_{p}$ ,
$\sum_{\varphi\in\Omega_{W_{n+1}}}p^{J(\sigma(x)(\varphi(y)+\varphi(z))+\varphi(y)\varphi(z))}h_{y}^{\varphi(y)}h_{z}^{\varphi(z)}=a_{h}(x)h_{x}^{\sigma(x)},$
(3.4)
$S(x)=\\{y,z\\}$ $\sigma\in\Omega_{V_{n}}$.
$\prod_{x\in
W_{n}}\sum_{\varphi\in\Omega_{W_{n+1}}}p^{J(\sigma(x)(\varphi(y)+\varphi(z))+\varphi(y)\varphi(z))}h_{y}^{\varphi(y)}h_{z}^{\varphi(z)}=\prod_{x\in
W_{n}}a_{h}(x)h_{x}^{\sigma(x)}=A_{n,h}\prod_{x\in W_{n}}h_{x}^{\sigma(x)},$
(3.5)
$A_{n,h}=\prod_{x\in W_{n}}a_{h}(x).$ (3.6)
(2.3) (3.5)
$\sum_{\sigma\in\Omega_{V_{n}}}\sum_{\varphi\in\Omega_{W_{n+1}}}\mu_{h}^{(n+1)}(\sigma\vee\varphi)=\sum_{\sigma\in\Omega_{V_{n}}}\sum_{\varphi\in\Omega_{W_{n+1}}}\frac{1}{Z_{n+1,h}}p^{H(\sigma\vee\varphi)}\prod_{x\in
W_{n+1}}h_{x}^{\varphi(x)}$
$=\frac{A_{n,h}}{Z_{n+1,h}}\sum_{\sigma\in\Omega_{V_{n}}}p^{H(\sigma)}\prod_{x\in
W_{n}}h_{x}^{\sigma(x)}=\frac{A_{n,h}}{Z_{n+1,h}}Z_{n,h}=1.$
∎
$h$ (2.6). $h$ $a_{h}(x)$. $x\in V$ (3.4) $\sigma(x)=1$ $\sigma(x)=-1$.
$\sigma(x)=1$ $\sigma(x)=-1$
$p^{3J}h_{y}h_{z}+p^{-J}h_{y}^{-1}h_{z}+p^{-J}h_{y}h_{z}^{-1}+p^{-J}h_{y}^{-1}h_{z}^{-1}=a(x)h_{x}$
$p^{-J}h_{y}h_{z}+p^{-J}h_{y}^{-1}h_{z}+p^{-J}h_{y}h_{z}^{-1}+p^{3J}h_{y}^{-1}h_{z}^{-1}=a(x)h_{x}^{-1}.$
,
$a_{h}(x)=\frac{\left(\left(p^{4J}h_{y}^{2}h_{z}^{2}+h_{y}^{2}+h_{z}^{2}+1)(h_{y}^{2}h_{z}^{2}+h_{y}^{2}+h_{z}^{2}+p^{4J}\right)\right)^{\frac{1}{2}}}{p^{J}h_{y}h_{z}}.$
(3.7)
\- $h$ (3.7)
$a_{h}=\frac{\left(\left(p^{4J}h^{4}+2h^{2}+1)(h^{4}+2h^{2}+p^{4J}\right)\right)^{\frac{1}{2}}}{p^{J}h^{2}}.$
(3.8)
#### 3.1.1. C $J>0$.
###### 3.
$\sigma\in\Omega_{V_{n}}$ $n\geq 1$
$\left|p^{H_{n}(\sigma)}\right|_{p}\leq p^{J(2^{n}-1)}.$
###### Proof.
, $H_{n}(\sigma)\geq-J(2^{n}-1)$. , . , $\sigma\in\Omega_{V_{n}}$
$\sigma(y)\sigma(z)=-1,\ \mbox{ }\ x\in V_{n-1},\ S(x)=\\{y,z\\}$
. ∎
###### 4.
$\left|h_{0}\right|_{p}=\left|h_{1}\right|_{p}=\left|h_{2}\right|_{p}=1$.
###### Proof.
, $|h_{0}|_{p}=1$, $h_{0}=1$. 2 $h_{1},\ h_{2}$ $p>5$. , 1) 2.1
$|h_{1}|_{p}=\left|\sqrt{\frac{p^{4J}-3+\sqrt{p^{8J}-6p^{4J}+5}}{2}}\right|_{p}=\left|\sqrt{2\sqrt{5}-6}\right|_{p}=1.$
$|h_{2}|_{p}=1$. ∎
###### 5.
$Z_{n,h_{i}},\ i=0,1,2$ :
i) $|Z_{n,h_{1}}|_{p}=|Z_{n,h_{2}}|_{p}=p^{J(2^{n}-2)}$;
ii) $|Z_{n,h_{0}}|_{p}=\left\\{\begin{array}[]{ll}p^{J(2^{n}-2)},&\text{
}p\neq 3,\\\ p^{(J-1)(2^{n}-2)},&\text{ }p=3.\end{array}\right.$
###### Proof.
i) (3.8) $h_{1}$
$|a_{h_{1}}|_{p}=\left|\frac{\left(\left(p^{4J}h_{1}^{4}+2h_{1}^{2}+1)(h_{1}^{4}+2h_{1}^{2}+p^{4J}\right)\right)^{\frac{1}{2}}}{p^{J}h_{1}^{2}}\right|_{p}=$
$\left|p^{-J}\sqrt{(2\sqrt{5}-4)(\sqrt{5}+1)}\right|_{p}=\left|p^{-J}\sqrt{6-2\sqrt{5}}\right|_{p}=p^{J}$
, $Z_{n,h}=a_{h}^{|V_{n-1}|}$ $|V_{n-1}|=2^{n}-2$,
$|Z_{n,h_{1}}|_{p}=p^{J(2^{n}-2)}.$
$|Z_{n,h_{2}}|_{p}=p^{J(2^{n}-2)}$.
ii) $h_{0}=1$, (3.8)
$|a_{h_{0}}|_{p}=\left|\frac{\left(\left(p^{4J}+3)(3+p^{4J}\right)\right)^{\frac{1}{2}}}{p^{J}}\right|_{p}=\left|3p^{-J}\right|_{p}=\left\\{\begin{array}[]{ll}p^{J},&\text{
}p\neq 3,\\\ p^{J-1},&\text{ }p=3.\end{array}\right.$
,
$|Z_{n,h_{0}}|_{p}=\left\\{\begin{array}[]{ll}p^{J(2^{n}-2)},&\text{ }p\neq
3,\\\ p^{(J-1)(2^{n}-2)},&\text{ }p=3.\end{array}\right.$
∎
###### 4.
i) $p\neq 3$, - $p$\- .
ii) $p=3$, - $p$\- $\mu_{h_{0}}$. .
###### Proof.
i) $p\neq 3$. 5 $|Z_{n,h_{i}}|_{p}=p^{J(2^{n}-2)},\ i=0,1,2$. 3,4
$\sigma\in\Omega_{V_{n}}$ $n=1,2,...$
$\left|\mu_{h_{i}}^{(n)}(\sigma)\right|_{p}=\left|\frac{p^{H_{n}(\sigma)}\prod_{x\in
W_{n}}h_{i}^{\sigma(x)}}{Z_{n,h_{i}}}\right|_{p}\leq\frac{p^{J(2^{n}-2)}}{p^{J(2^{n}-2)}}=1,\qquad
i=0,1,2.$
, - $p$\- $\mu_{h_{i}},\ i=0,1,2$ .
ii) $p=3$. 2 \- $p$\- $\mu_{h_{0}}$. , . $\sigma$
$\sigma(y)\sigma(z)=-1,\ \mbox{ }\ x\in V_{n-1},\ S(x)=\\{y,z\\}$
4,5 $\mu_{h_{0}}$
$\left|\mu_{h_{0}}^{(n)}(\sigma)\right|_{p}=\left|\frac{p^{H_{n}(\sigma)}\prod_{x\in
W_{n}}h_{0}}{Z_{n,h_{0}}}\right|_{p}=\frac{p^{J(2^{n}-2)}}{p^{(J-1)(2^{n}-2)}}=p^{2^{n}-2}.$
$\left|\mu_{h_{0}}^{(n)}(\sigma)\right|_{p}\to\infty\qquad\mbox{ }\
n\to\infty.$
∎
#### 3.1.2. $J<0$.
###### 6.
$\left|p^{H_{n}(\sigma)}\right|_{p}\leq p^{-J(3\cdot 2^{n}-5)}$
$\sigma\in\Omega_{V_{n}}$ $n\geq 1$.
###### Proof.
, $\sigma\in\Omega_{V_{n}}$, 1 $x\in V_{n}$. ∎
###### 7.
$|h_{0}|_{p}=1,\qquad|h_{1}|_{p}=p^{-2J},\qquad|h_{2}|_{p}=p^{2J}$.
###### Proof.
, $|h_{0}|_{p}=1$. $h_{1}$
$|h_{1}|_{p}=\left|p^{2J}\sqrt{\frac{1-3p^{-4J}+\sqrt{1-6p^{-4J}+5{p^{-4J}}}}{2}}\right|_{p}=p^{-2J}.$
$h_{i}=\sqrt{u_{i}},\ i=1,2$ $u_{1}\cdot u_{2}=1$, $|h_{2}|_{p}=p^{2J}$. ∎
###### 8.
$Z_{n,h_{i}},\ i=0,1,2$
$|Z_{n,h_{i}}|_{p}=p^{-J(5\cdot 2^{n}-10)},\
i=1,2\qquad|Z_{n,h_{0}}|_{p}=p^{-J(3\cdot 2^{n}-6)}.$
###### Proof.
7 $h_{1}=p^{2J}\varepsilon$ $|\varepsilon|_{p}=1$. ,
$|a_{h_{1}}|_{p}=\left|\frac{\left(\left(p^{12J}\varepsilon^{4}+2p^{4J}\varepsilon^{2}+1)(p^{8J}\varepsilon^{4}+2p^{4J}\varepsilon^{2}+p^{4J}\right)\right)^{\frac{1}{2}}}{p^{5J}\varepsilon^{2}}\right|_{p}=p^{-5J}.$
,
$|Z_{n,h_{1}}|_{p}=p^{-J(5\cdot 2^{n}-10)}.$
$|Z_{n,h_{2}}|_{p}=p^{-J(5\cdot 2^{n}-10)}$ $|Z_{n,h_{0}}|_{p}=p^{-J(3\cdot
2^{n}-6)}$. ∎
###### 5.
\- $p$\- .
6,7,8.
## 4\. $p$\-
:
$u=f(f(u)),\qquad\mbox{ }\
f(u)=\frac{\theta^{2}u^{2}+2u+1}{u^{2}+2u+\theta^{2}}$ (4.1)
, (4.1) $u=f(u)$. ( - ) .
$\frac{f(f(u))-u}{f(u)-u}=0,$
:
$\theta^{2}u^{2}+(\theta^{2}+1)u+\theta^{2}=0.$ (4.2)
$\sqrt{1+2\theta^{2}-3\theta^{4}}$ $\mathbb{Q}_{p}$,
$u_{3,4}=\frac{-1-\theta^{2}\pm\sqrt{1+2\theta^{2}-3\theta^{4}}}{2\theta^{2}}.$
(4.3)
(4.2). $D(\theta)=1+2\theta^{2}-3\theta^{4}$. $\sqrt{D(\theta)}$
$\mathbb{Q}_{p}$. $\sqrt{u_{3}}$ $\sqrt{u_{4}}$. , . , , $\sqrt{u_{3}}$
$\mathbb{Q}_{p}$.
$u_{3}\cdot
u_{4}=\frac{(1+\theta^{2})^{2}-(1+2\theta^{2}-3\theta^{4})}{4\theta^{4}}=1.$
(4.4)
$\sqrt{u_{3}}\in\mathbb{Q}_{p}$, (4.4) $\sqrt{u_{4}}\in\mathbb{Q}_{p}$.
###### 4.
$\sqrt{u_{3}}$ $\sqrt{u_{4}}$ , , 2- $p$\- , 2- $p$\- .
$\mu^{per}_{1}$ ( . $\mu^{per}_{2}$) $p$\- $(h_{3},h_{4})$ ( .
$(h_{4},h_{3})$).
### 4.1. $J>0$
1 $\sqrt{D(\theta)}$ $p$. $\sqrt{u_{3}}$ $\mathbb{Q}_{p}$.
$p=2$.
$u_{3}=\frac{-1-2^{4J}+\sqrt{1+2^{4J+1}-3\cdot
2^{8J}}}{2^{4J+1}}=\frac{-1-2^{4J}+1+2+2^{2}+\dots}{2^{4J+1}}=2^{-4J}(1+2+\dots)$
1 , $\sqrt{u_{3}}$ $\mathbb{Q}_{p}$.
$p\neq 2$.
$u_{4}=\frac{-1-p^{4J}-\sqrt{1+2p^{4J}-3p^{8J}}}{2p^{4J}}=\frac{-1-p^{4J}-1-p^{4J}-\dots}{2p^{4J}}=\frac{-1+a_{1}p+a_{2}p^{2}+\dots}{p^{4J}}.$
, $\sqrt{u_{3}}$ $\sqrt{-1}$. 1 $\sqrt{-1}$ $\mathbb{Q}_{p}$ , $p\equiv
1(\operatorname{mod}4)$.
###### 6.
$p\equiv 1(\operatorname{mod}4)$, (2.2) 2- $p$\- : $\mu^{per}_{1}$
$\mu^{per}_{2}$.
### 4.2. C $J<0$
$|\theta|_{p}>1$. $D(\theta)=\theta^{4}(-3+2\theta^{-2}+\theta^{-4})$ ,
$\sqrt{D(\theta)}$ $\sqrt{-3}$ . 1 $p$ , $\sqrt{D(\theta)}$
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$
---|---|---|---|---|---|---|---|---
$\sqrt{D(\theta)}$ | $-$ | $-$ | $-$ | $+$ | $-$ | $+$ | $-$ | $-$
1.
###### 7.
i) $p\in\\{2,3\\}$, $p$\- .
ii) $p>3$. $x^{2}+3\equiv 0\,(\operatorname{mod}p)$ $\mathbb{Q}_{p}$, $p$\- .
iii) $p>3$ $x_{0}$ $x^{2}+3\equiv 0\,(\operatorname{mod}p)$. 2- $p$\- ,
$x^{2}-2x_{0}+2\equiv 0\,(\operatorname{mod}p)$ $\mathbb{Q}_{p}$.
###### Proof.
$\sqrt{D(\theta)}$ $\sqrt{-3}$ , , $x^{2}+3\equiv 0\,(\operatorname{mod}p)$
$\mathbb{Q}_{p}$. , $\sqrt{-3}\not\in\mathbb{Q}_{p}$ $p\leq 3$.
$p>3$ $x_{0}$ $x^{2}+3\equiv 0\,(\operatorname{mod}p)$.
$u_{3}=\frac{-1-p^{4J}+\sqrt{1+2p^{4J}-3p^{8J}}}{2p^{4J}}=\frac{x_{0}-1+p^{-4J}\varepsilon}{2},\quad|\varepsilon|_{p}\leq
1$
1 , $\sqrt{u_{3}}$ $x^{2}-2x_{0}+2\equiv 0\,(\operatorname{mod}p)$.
4 2- $p$\- , $x^{2}-2x_{0}+2\equiv 0\,(\operatorname{mod}p)$ $\mathbb{Q}_{p}$.
∎
. . . . . .
## References
* [1] Albeverio S., Karwowski W., Stochastic Processes Appl. 53 (1994), 1-22.
* [2] Bleher P.M., Ruiz J., Zagrebnov V.A. Journ. Statist. Phys. 79 (1995), 473-482.
* [3] . ., . ., . ., . . ., No. 4, (1998), 23-29.
* [4] Georgii H.-O., Gibbs Measures and Phase Transitions (W. de Gruyter, Berlin, 1988).
* [5] Khrennikov A. Yu., Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, Dordrecht, 1997).
* [6] Khakimov O.N., $p$-Adic Numbers, Ultr.Anal.Appl.5:3 (2013), 194-203.
* [7] Koblitz N., $p$-Adic Numbers, $p$-adic Analysis, and Zeta-Functions (Springer, Berlin, 1977).
* [8] Marinari E., Parisi G., Phys. Lett. B 203, (1988) 52-54.
* [9] Mukhamedov F.M., Math.Phys.Anal.Geom, 16 (2013), 49-87.
* [10] Mukhamedov F.M., $p$-Adic Numbers, Ultr.Anal.Appl., 2 (2010), 241-251.
* [11] . ., O. ., . 175:1 (2013), 84-92.
* [12] Rozikov U.A., Gibbs Measures on Cayley Trees.World Sci. Publ. Singapore. 2013, 404 pp.
* [13] Schikhof W.H., Ultrametric Calculus (Cambridge Univ. Press, Cambridge, 1984).
* [14] Vladimirov V.S., Volovich I. V., Zelenov E. V., $p$-Adic Analysis and Mathematical Physics (World Sci., Singapore, 1994).
* [15] Yasuda K., Osaka J. Math. 37, (2000), 967-985.
|
arxiv-papers
| 2014-02-12T17:20:23 |
2024-09-04T02:49:58.231073
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Otabek Khakimov",
"submitter": "Khakimov Otabek Norbuta ugli",
"url": "https://arxiv.org/abs/1402.3322"
}
|
1402.3513
|
# _Ab initio_ design of charge-mismatched ferroelectric superlattices
Claudio Cazorla Institut de Ci$\grave{e}$ncia de Materials de Barcelona
(ICMAB-CSIC), 08193 Bellaterra, Spain Massimiliano Stengel Institut de
Ci$\grave{e}$ncia de Materials de Barcelona (ICMAB-CSIC), 08193 Bellaterra,
Spain ICREA - Institució Catalana de Recerca i Estudis Avançats, 08010
Barcelona, Spain [email protected]
###### Abstract
We present a systematic approach to modeling the electrical and structural
properties of charge-mismatched superlattices from first principles. Our
strategy is based on bulk calculations of the parent compounds, which we
perform as a function of in-plane strain and out-of-plane electric
displacement field. The resulting two-dimensional phase diagrams allow us to
accurately predict, without performing further calculations, the behavior of a
layered heterostructure where the aforementioned building blocks are
electrostatically and elastically coupled, with an arbitrary choice of the
interface charge (originated from the polar discontinuity) and volume ratio.
By using the [PbTiO3]m/[BiFeO3]n system as test case, we demonstrate that
interface polarity has a dramatic impact on the ferroelectric behavior of the
superlattice, leading to the stabilization of otherwise inaccessible bulk
phases.
###### pacs:
71.15.-m, 77.22.Ej, 77.55.+f, 77.84.Dy
## I Introduction
When layers of perovskite oxides are epitaxially stacked to form a
periodically repeated heterostructure, new intriguing functionalities can
emerge in the resulting superlattice [ghosez08, ; junquera11, ]. These are
further tunable via applied electric fields and thermodynamic conditions, and
thus attractive for nanoelectronics and energy applications. An excellent
example is the [PbTiO3]m/[SrTiO3]n system, where the polarization,
tetragonality, piezoelectric response, and ferroelectric transition
temperature strongly change with the volume ratio of the parent compounds
[dawber05, ; dawber07, ; dawber12, ]. Such a remarkable tunability is usually
rationalized in terms of epitaxial strains [dawber05b, ], electrostatic
coupling (see Fig. 1a) [zubko12, ; wu12, ], and local interface effects
[junquera12, ; bousquet08, ].
While perovskite titanates with ATiO3 formula (A=Sr, Pb, Ba or Ca) have
traditionally been the most popular choice as the basic building blocks, a
much wider range of materials (e.g., BiFeO3) is currently receiving increasing
attention by the community. The motivation for such an interest is clear: a
superlattice configuration provides the unique opportunity of enhancing
materials properties via “strain engineering”, and a multifunctional compound
such as BiFeO3 appears to be a natural candidate in this context. (For
example, strain has been predicted to enhance the magnetoelectric response of
BiFeO3 by several orders of magnitude compared to bulk samples [wojdel09, ;
wojdel10, ].) Also, a superlattice geometry can alleviate the leakage issues
of pure BiFeO3 films [ranjith07, ; ranjith08, ].
Combining a III–III perovskite like BiFeO3 (or I–V, like KNbO3) with a II–IV
titanate appears, however, problematic from the conceptual point of view. In
fact, the charge-family mismatch inevitably leads to polar (and hence
electrostatically unstable) interfaces between layers murray09 . This is not
necessarily a drawback, though: recent research has demonstrated that polar
interfaces can be, rather than a nuisance to be avoided, a rich playground to
be exploited for exploring exciting new phenomena. The prototypical example is
the LaAlO3/SrTiO3 system, where a metallic two-dimensional electron gas
appears at the heterojunction to avoid a “polar catastrophe” nakagawa06 ;
ohtomo04 . Remarkably, first-principles calculations have shown that
interfaces in oxide superlattices can remain insulating provided that the
layers are thin enough, and produce rather dramatic effects on the respective
polarization of the individual components bristowe09 ; murray09 . This means
that, in a superlattice, polar discontinuities need not be compensated by
electronic or ionic reconstructions; they can, instead, be used as an
additional, powerful materials-design tool to control the behavior of the
polar degrees of freedom therein. Such a control may be realized, for
instance, by altering the stoichiometry at the interfaces (see Fig. 1b). To
fully explore the potential that this additional degree of freedom (the
interface built-in polarity) provides, and guide the experimental search for
the most promising materials combinations, one clearly needs to establish a
general theoretical framework where the “compositional charge” murray09 is
adequately taken into account.
Figure 1: (Color online) (a) Description of the electrostatic coupling in a
ferroelectric (orange)/paraelectric (blue) bilayer; $P$, $\mathcal{E}$, and
$D$ represent the component of the polarization, electric field and electric
displacement vectors along the stacking direction, and $\sigma_{\rm int}$ is
the interface charge density. (b) Intermixed AO-type interfaces in a
[BiFeO3]m/[PbTiO3]n superlattice and the resulting interface charge densities.
(c) Illustration of the $20$-atom simulation cell used in our calculations;
red, blue and black spheres represent O, B, and A atoms in the ABO3
perovskite.
In this Letter, we present a general first-principles approach to predict the
behavior of charge-mismatched perovskite oxide superlattices based exclusively
on the properties of their individual bulk constituents. Our formalism
combines the constrained-$D$ strategies of Wu et al. wu08 , which are key to
decomposing the total energy of the system into the contributions of the
individual layers, with the rigorous description of the interface polarity
proposed in Ref. stengel11 . As a result, we are able to exactly describe the
electrostatic coupling and mechanical boundary conditions, enabling a clear
separation between genuine interface and bulk effects. Crucially, the present
method allows one to quantify, in a straightforward way, the impact that
interface polarity has on the equilibrium (and metastable) phases of the
superlattice. As a proof of concept we apply our formalism to the study of
[PbTiO3]m/[BiFeO3]n (PTO/BFO) heterostructures. We find that (i) our _bulk_
model accurately matches earlier first-principles predictions obtained for
ultrashort-period superlattices (i.e., $m=n=3$) by using explicit _supercell_
simulations stengel12 , and (ii) by assuming interface terminations with
different nominal charge, we obtain a radical change in the overall
ferroelectric properties of the superlattice, which demonstrates the crucial
role played by the polar mismatch.
Figure 2: (Color online) Energy of PTO/BFO superlattices with $a=3.81$ Å
expressed as a function of $D$, for selected values of $\lambda$ and
$\sigma_{\rm int}$. Equilibrium and metastable superlattice states are
represented with solid and empty dots. Red (green) vertical lines indicate
phase transitions occurring in bulk BFO (PTO) under different $D$ conditions.
(a) and (b) represent the cases of neutral and polar interfaces, respectively.
We start by expressing the total energy of a monodomain two-color superlattice
(i.e., composed of species A and B) as,
$U_{\rm tot}(D,\lambda,a)=\lambda\cdot U_{\rm
A}(D,a)+\left(1-\lambda\right)\cdot U_{\rm B}(D,a)~{}.$ (1)
Here $U_{\rm A}$ and $U_{\rm B}$ are the internal energies of the individual
constituents, $D$ is the electric displacement along the out-of-plane stacking
direction (i.e., $D\equiv{\cal E}+4\pi P$ where ${\cal E}$ is the electric
field and $P$ is the _effective_ polarization, relative to the centrosymmetric
reference configuration), $\lambda$ is the relative volume ratio of material A
(i.e, $\lambda\equiv m/(n+m)$ where $m$ and $n$ are the thicknessess of layers
A and B, respectively), and $a$ is the in-plane lattice parameter (we assume
heterostructures that are coherently strained to the substrate). Note that
short-range interface effects have been neglected. (While it is certainly
possible to incorporate the latter in the model, e.g. along the guidelines
described in Ref. wu08 , we believe these would have been an unnecessary
complication in the context of the present study.) By construction, Eq. (1)
implicitly enforces the continuity of $D$ along the out-of-plane stacking
direction (which we label as $z$ henceforth), which is appropriate for
superlattices where the interfaces are nominally uncharged [ghosez08, ;
junquera11, ].
In presence of a polar mismatch, one has a net “external” interface charge, of
compositional origin murray09 , $\sigma_{\rm int}$ (see Fig. 1a), which is
localized at the interlayer junctions. In such a case, Eq. (1) needs to be
revised as follows,
$U_{\rm tot}(D,\sigma_{\rm int},\lambda,a)=\lambda\cdot U_{\rm
A}(D,a)+\left(1-\lambda\right)\cdot U_{\rm B}(D-\sigma_{\rm int},a)~{},$ (2)
i.e. the $U_{\rm B}$ curve is shifted in $D$-space to account for the jump in
$D$ produced by $\sigma_{\rm int}$. (Recall the macroscopic Maxwell equation,
$\nabla\cdot{\bf D}=\rho_{\rm ext}$, where $\rho_{\rm ext}$, the “external”
charge, encompasses all contributions of neither dielectric nor ferroelectric
origin.) Once the functions $U_{\rm A}$ and $U_{\rm B}$ are computed and
stored (e.g. by using the methodology of Ref. stengel09b ), one can predict
the ground-state of a hypothetical A/B superlattice by simply finding the
global minimum of $U_{\rm tot}$ with respect to $D$ at fixed values of
$\sigma_{\rm int}$, $\lambda$ and $a$. The advantage of this procedure is
that, for a given choice of A and B, the aforementioned four-dimensional
parameter space can be explored very efficiently, as no further _ab initio_
calculations are needed.
It is useful, before going any further, to specify the physical origin of
$\sigma_{\rm int}$ in the context of this work. Consider, for example, a
periodic BiFeO3/PbTiO3 superlattice, which we assume (i) to be stoichiometric
(and therefore charge-neutral) as a whole, (ii) to have an ideal AO-BO2-AO-BO2
stacking along the (001) direction, and (iii) to form (say) AO-type interfaces
(see Fig. 1b). (The same arguments can be equally well applied to the case of
BO2-type interfaces.) Depending on the growth conditions, one can have a
certain degree of intermixing in the boundary AO layers, which will adopt an
intermediate composition Bix Pb(1-x)O. As a pure BiO layer is formally charged
$+1$ and PbO is neutral, we can readily write $\sigma_{\rm
int}=\pm\left(x-\frac{1}{2}\right)$ (expressed in units of $e/S$ with $S$
being the surface of the corresponding 5-atom perovskite cell), where the
choice of plus or minus depends on the arbitrary assignment of BiFeO3 and
PbTiO3 as the A or B component in Eq. (2) [see Fig. 1b]. In the following we
shall illustrate the crucial role played by $\sigma_{\rm int}$ (and hence, by
the interface stoichiometry) on the ferroelectric properties of a BFO/PTO
superlattice, by combining Eq. (2) with the bulk $U_{\rm BFO}(D,a)$ and
$U_{\rm PTO}(D,a)$ curves that we calculate from first principles.
Our calculations are performed with the “in-house” LAUTREC code within the
local spin density approximation to density-functional theory. (We
additionally apply a Hubbard $U=3.8$ eV to Fe ions kornev07 ; yang12 .) We use
the $20$-atom simulation cell depicted in Fig. 1c for both BFO and PTO, which
allows us to describe the ferroelectric and anti-ferrodistortive (AFD) modes
of interest (i.e. in-phase AFDzi and out-of-phase AFDzo and AFDxy, see Ref.
[bousquet08, ]). Atomic and cell relaxations are performed by constraining the
out-of-plane component of $D$ stengel09b and the in-plane lattice constant
$a$ to a given value. [Calculations are repeated many times in order to span
the physically relevant two-dimensional $(D,a)$ parameter space.]
We start by illustrating the results obtained at fixed strain, $a=3.81$ Å (see
Fig. 2), by assuming $\sigma_{\rm int}=0$, which corresponds to fully
intermixed junctions ($x=0.5$), and we vary the BFO volume ratio, $\lambda$.
At the extreme values of $\lambda$, the results are consistent with the
expectations: the equilibrium configuration of BFO (i.e., the minimum of
$U_{tot}$ with $\lambda=1$) at this value of $a$ is the well-known R-type
$Cc$-I phase alison10 , derived from the bulk ground state via the application
of epitaxial compression; PTO ($\lambda=0$), on the other hand, is in a
tetragonal $P4mm$ phase with the polarization vector oriented out of plane.
Intermediate values of $\lambda$ yield a linear combination of the two single-
component $U(D)$ curves, where the spontaneous $P_{z}$ at equilibrium
gradually moves from the pure PTO to the pure BFO value.
Unfortunately, the possible equilibrium states that can be attained by solely
varying $\lambda$ (at this value of $a$ and $\sigma_{\rm int}$) lie far from
any physically “interesting” region of the phase diagram. For example, note
the kink at $|D|\sim$0.3 C/m2 in the pure BFO case, which corresponds to a
first-order transition to an orthorhombic $Pna2_{1}$ phase (a close relative
of the higher-symmetry $Pnma$ phase, occurring at $D=0$). A huge piezoelectric
and dielectric response is expected in BFO in a vicinity of the transition
cazorla14 , raising the question of whether one could approach this region by
playing with $\sigma_{\rm int}$, in addition to $\lambda$.
The answer is yes: when oxide superlattices with $\sigma_{\rm int}=0.5$ are
considered [corresponding to “ideal” (BiO)+/TiO2 and (FeO2)-/PbO interfaces],
the stable minimum of the system favors a smaller spontaneous polarization in
the BFO layers, approaching the aforementioned ($Cc{\rm-I}\to Pna2_{1}$) phase
boundary in the limit of small $\lambda$. Interestingly, the $U_{\rm tot}(D)$
curve becomes asymmetric (the interfacial charge breaks inversion symmetry),
and a secondary, metastable minimum appears. Overall, the resulting phase
diagram turns out to be much richer, with new combinations of phases emerging
(e.g. in region II’, where BFO exists in the orthorhombic $Pna2_{1}$ phase and
PTO in the tetragonal $P4mm$ phase), and highly non-trivial changes in the
electrical properties occurring as a function of $\lambda$.
Figure 3: Total energy (a) and out-of-plane electric displacement $D$ (b) of
the equilibrium (solid symbols) and metastable (empty symbols) states of
PTO/BFO superlattices with $\lambda=\frac{1}{2}$ and $\sigma_{\rm int}=0.5$,
expressed as a function of the in-plane lattice parameter. Regions in which
PTO and BFO exist in different phases are delimited with vertical dashed
lines; the corresponding space groups and AFD distortion patterns in Glazer’s
notation are shown in (a), and the components of the ferroelectric
polizarization in (b).
In order to further illustrate the power of our approach, we shall now fix the
volume ratio to $\lambda=0.5$ (corresponding to alternating BFO and PTO layers
of equal thickness) and vary the in-plane lattice parameter in the range
$3.6\leq a\leq 4.2$ Å . We shall first consider the case of charged interfaces
with $\sigma_{\rm int}=0.5$, as this choice allows for a direct comparison
with the results of Yang _et al._ (obtained via standard supercell
simulations) [stengel12, ]. In Fig. 3 we show the energy and spontaneous
electric displacement of the equilibrium and metastable states as a function
of $a$. Four regions can be identified in the diagrams depending on the phases
adopted by BFO and PTO at each value of the in-plane strain. (Their crystal
space groups, AFD pattern and in-plane / out-of-plane ferroelectric
polarization, respectively $P_{xy}$ and $P_{z}$, are specified in compact form
in the figure.) In region I’ both PTO and BFO adopt a monoclinic $Pc$ phase
characterized by large in-phase AFDz distortions and non-zero $P_{xy}$ and
$P_{z}$. Such a monoclinic $Pc$ phase is closely related to the orthorhombic
$Pmc2_{1}$ structure which has been recently predicted in PTO and BFO at large
tensile strains [yang12, ]. In region II’ PTO adopts an orthorhombic $Ima2$
phase, characterized by vanishing AFD distortions and a large in-plane ${\bf
P}$ (we neglect the small out-of-plane $P_{z}$), while BFO is in its well-
known $Cc$-I state. In region III, BFO remains $Cc$-I, while PTO adopts a
$P4mm$ phase, both with _opposite_ out-of-plane polarization with respect to
region II’. These structures switch back to a positively oriented $P_{z}$ in
region IV’, respectively transforming into a monoclinic $Cc$-II and a
tetragonal $I4cm$ phase. The $I4cm$ phase is characterized by anti-phase AFDz
distorsions and an out-of-plane ${\bf P}$, while the $Cc$-II corresponds to
the “supertetragonal” T-type phase of BFO zeches09 . Note that, as observed
already while discussing Fig. 2, the net interface charge leads to an
asymmetric double-well potential, and consequently to an energy difference
(typically of $\sim 20$ meV/f.u. or less, see Fig. 3a) between the two
oppositely polarized states. (Only one minimum survives at large tensile
strains, where the superlattice is no longer ferroelectric.) At the phase
boundaries such energy difference vanishes; the obvious kinks in the $U_{\rm
tot}$ curve shown in Fig. 3(a) indicate that the transitions (at $a=3.71$,
$3.87$, and $4.05$ Å) are all of first-order type.
The above results are in remarkable agreement with those of Yang _et al._
stengel12 . The only apparent discrepancy concerns the ordering of the
stable/metastable states in region III, which anyway involves a very subtle
energy difference (and is therefore sensitive to short-range interface
effects, not considered here). Obtaining such an accurate description of
superlattices where the individual layers are as thin as three perovskite
units stengel12 provides a stringent benchmark for our method, and validates
it as a reliable modeling tool. From the physical point of view this
comparison suggest that, even in the ultrathin limit, PTO/BFO superlattices
can be well understood in terms of macroscopic bulk effects, i.e., short-range
interface-specific phenomena appear to play a relatively minor role.
Figure 4: Same as in Fig. 3, but considering neutral interfaces. The out-of-
plane polarization is the same in PTO and BFO layers.
Having gained confidence in our method, we can use it to predict the behavior
of a hypothetical superlattice with $\sigma_{\rm int}=0$, corresponding to a
centrosymmetric reference structure with fully intermixed Pb0.5Bi0.5O
interface layers (see Fig. 4). Note the symmetry of the two opposite
polarization states, and the common value of the spontaneous electric
displacement adopted by BFO and PTO. The resulting phase diagram consists,
again, in four regions, with a first-order and two second-order phase
transitions occurring at $a=4.07$, $3.88$ and $3.73$ Å , respectively (see
Fig. 4a). In three of these regions, the individual layers display structures
which are different from those obtained in the $\sigma_{\rm int}=0.5$ case: in
region I both PTO and BFO stabilize in an orthorhombic $Pmc2_{1}$ phase
[yang12, ], characterized by a vanishing $P_{z}$; in region II PTO adopts a
monoclinic $Cm$ phase with the polarization roughly oriented along (111)
($P_{z}\neq P_{xy}\neq 0$) and no AFD, while BFO stabilizes in the already
discussed $Cc$-I phase; finally, in region IV, PTO is tetragonal $P4mm$ and
BFO is monoclinic $Cc$-II. These findings quantitatively demonstrates that the
interface charge mismatch can have a tremendous impact on the physical
properties of oxide superlattices. Our simple and general method allows one to
model and quantify accurately these effects, and most importantly to
rationalize them in terms of intuitive physical concepts.
In summary, we have discussed a general theoretical framework to predict the
behavior of charge-mismatched superlattices. We have showed that the effect of
the interface stoichiometry, which we describe via the “compositional”
interface charge $\sigma_{\rm int}$, is quite dramatic, and needs to be
properly accounted for in the models. More generally, we argue that
$\sigma_{\rm int}$ can be regarded, in addition to $\lambda$ and $a$, as a
further degree of freedom in designing oxide heterostructures with tailored
functionalities, opening exciting new avenues for future research.
###### Acknowledgements.
This work was supported by MICINN-Spain [Grants No. MAT2010-18113 and No.
CSD2007-00041], and the CSIC JAE-DOC program (C.C.). We thankfully acknowledge
the computer resources, technical expertise and assistance provided by RES and
CESGA.
## References
* (1) P. Ghosez and J. Junquera, J. Comp. Theor. Nanosci. 5, 2071 (2008).
* (2) C. Lichtensteiger et al., in Oxides Ultrathin Films: Science and Technology, edited by G. Pacchioni and S. Valeri, Ch. 12, 265 (Wiley-VCH, Germany, 2011).
* (3) M. Dawber et al., Phys. Rev. Lett. 95, 177601 (2005).
* (4) M. Dawber et al., Adv. Mater. 19, 4153 (2007).
* (5) J. Sinsheimer et al., Phys. Rev. Lett. 109, 167601 (2012).
* (6) M. Dawber, K. M. Rabe, and J. F. Scott, Rev. Mod. Phys. 77, 1083 (2005).
* (7) P. Zubko et al., Nano Letters 12, 2846 (2012).
* (8) C. W. Swartz and X. Wu, Phys. Rev. B 85, 054102 (2012).
* (9) P. Aguado-Puente, P. García-Fernández, and J. Junquera, Phys. Rev. Lett. 107, 217601 (2011).
* (10) E. Bousquet, M. Dawber, N. Stucki, C. Lichtensteiger, P. Hermet, S. Gariglio, J.-M. Triscone, and P. Ghosez, Nature (London) 452, 732 (2008).
* (11) J. C. Wojdeł and J. ${\rm\acute{I}}$${\rm\tilde{n}}$iguez, Phys. Rev. Lett. 103, 267205 (2009).
* (12) J. C. Wojdeł and J. ${\rm\acute{I}}$${\rm\tilde{n}}$iguez, Phys. Rev. Lett. 105, 037208 (2010).
* (13) R. Ranjith, B. Kundys, and W. Prellier, Appl. Phys. Lett. 91, 222904 (2007).
* (14) R. Ranjith et al., Appl. Phys. Lett. 92, 232905 (2008).
* (15) E. D. Murray and D. Vanderbilt, Phys. Rev. B 79, 100102 (2009).
* (16) N. Nakagawa, H. Y. Hwang, and D. A. Muller, Nature Mater. 5, 204 (2006).
* (17) A. Ohtomo and H. Y. Hwang, Nature (London) 427, 423 (2004).
* (18) N. C. Bristowe, E. Artacho, and P. B. Littlewood, Phys. Rev. B 80, 045425 (2009).
* (19) X. Wu, M. Stengel, K. M. Rabe, and D. Vanderbilt, Phys. Rev. Lett. 101, 087601 (2008).
* (20) M. Stengel and D. Vanderbilt, Phys. Rev. B 80, 241103(R) (2009).
* (21) Y. Yang, M. Stengel, W. Ren, X. H. Yan, and L. Bellaiche, Phys. Rev. B 86, 144114 (2012).
* (22) M. Stengel, N. A. Spaldin, and D. Vanderbilt, Nature Physics 5, 304 (2009).
* (23) I. A. Kornev, S. Lisenkov, R. Haumont, B. Dkhil, and L. Bellaiche, Phys. Rev. Lett. 99, 227602 (2007).
* (24) Y. Yang, W. Ren, M. Stengel, X. H. Yan, and L. Bellaiche, Phys. Rev. Lett. 109, 057602 (2012).
* (25) A. J. Hatt, N. A. Spaldin, and C. Ederer, Phys. Rev. B 81, 054109 (2010).
* (26) C. Cazorla and M. Stengel, to be published.
* (27) R. J. Zeches et al., Science 326, 977 (2009).
|
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| 2014-02-14T16:08:50 |
2024-09-04T02:49:58.246930
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Claudio Cazorla and Massimiliano Stengel",
"submitter": "Claudio Cazorla",
"url": "https://arxiv.org/abs/1402.3513"
}
|
1402.3618
|
arxiv-papers
| 2014-02-14T23:02:43 |
2024-09-04T02:49:58.258133
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Satya Mandal",
"submitter": "Satya Mandal",
"url": "https://arxiv.org/abs/1402.3618"
}
|
|
1402.3742
|
# Geometrical Approximation to the AdS/CFT Correspondence
M. A. Martin Contreras, J. M. R. Roldan Giraldo
High Energy Group, Department of Physics, University of los Andes
[email protected]@uniandes.edu.co
###### Abstract
In this paper an analysis of the geometrical construction of the AdS/CFT
Correspondence is made. A geometrical definition of the configuration manifold
and the boundary manifold in terms of the conformal compactification scheme is
given. As a conclusion, it was obtained that the usual definition of the
correspondence [2] is strongly dependent of the unicity of the conformal class
of metrics on the boundary. Finally, a summary of some of the geometrical
issues of the correspondence is made, along with a possible way to avoid them.
## 1 Introduction
Gravity/Gauge duality is maybe one of the most important developments of the
latest times in String Theory. From its very begining, dual models have been
applied in many areas different from High Energy Physics or Black Hole
Physics. Any branch of Physics that exhibits phase transitions can be modeled
using dual models[1].
The central idea of Gravity/Gauge duality is the geometrical connection
existing between any Gravity Theory (Superstrings, for example) in $d+1$
dimensions to a QFT living in $d$ dimensions. In fact, it can be said that _we
can extract information about QFT from spacetime_ , and viceversa. This is
just a conjecture, and it still needs a proof. Once the connection between
bulk and boundary is stablished, the next step is to write of a proper
holographic dictionary, allowing to switch between gravity and QFT.
AdS/CFT Correspondence [2] is the most relevant realization of the
Gravity/Gauge duality, but is not the only successful one. Some examples of
this kind of duality are the Klebanov–Strassler duality [3] or the
NS5–branes/LST [4]. In all the three cases mentioned above, the bulk is a
non–compact manifold endowed with gravity, such that the dual gauge theory is
encoded in its asymptotic behavior.
## 2 AdS/CFT Correspondence in a Nutshell
The most representative holographic duality is the AdS/CFT Correspondence
(Maldacena 1998). In this duality we link gravity in a weakly curved
AdS${}_{5}\,\times\,S^{5}$ with a CFT in $3+1$ dimensions, which is in the
conformal boundary of AdS. AdS/CFT Correspondence has strong/weak duality too,
which relates SUGRA backgrounds at strong coupling with CFT at weak coupling.
Thanks to this, it has been possible to construct toy models for thermal (non
perturbative) QCD, as for example, the dual models of QGP using Dp/Dq branes
as gravitational background.
The idea behind the AdS/CFT Correspondence is the geometrical connection
between the isomeries of AdS and the conformal group. To be more precise,
since AdS is a maximally symmetric space, its isomeries are holomorphic to the
Poincare Group. This implies that, at inner level, AdS and any CFT are
_essentially_ the same thing. The statement of the correspondence is
$Z_{\text{String}}\left[\phi,\mathcal{M}\right]=Z_{CFT}\left[\phi_{0},\mathcal{O};\partial\,\mathcal{M},\eta\right],$
(1)
where $\mathcal{M}$ is the manifold where gravity lies, $\phi$ is a bulk field
with $\phi_{0}$ as the value at the conformal boundary $\partial\mathcal{M}$.
The conformal boundary carries a metric in a fixed conformal class $[\eta]$.
The conjecture stablishes that $\phi_{0}$ acts as a Schwinger source for any
CFT operator $\mathcal{O}$ living on $\partial\,\mathcal{M}$. This is the
essence of the correspondence.
Some remarks. The conjecture in principle can be made with any background
$\mathcal{M}$ that satisfies string equations of motion and has a the pair
$\left(\partial\,\mathcal{M}\,,\eta\right)$. Since the solution is not unique,
i.e., the charts over $M$ are not trivial, the correlation functions are
dependent from the choice of coordinates. As a conclusion, it is possible to
obtain different holographies according to the choice of chart. For example,
in the Maldacena’s original proposal, the Anti de Sitter space is covered
partially with a Poincare chart $AdS_{5}$ that picks up one of the two folds
of the hyperbolic space, fixing a conformal boundary at the origin of the
radial coordinate of $AdS_{5}$. This conformal boundary has a topology of
$\mathbb{R}^{1,3}$. Choices of different charts on Anti de Sitter space lead
to boundaries as $\mathbb{R}^{4}$, $S^{1}\times S^{3}$,
$S^{1}\times\mathbb{R}^{3}$ or $S^{1}\times\mathbb{H}^{3}$ [6]. All of these
topologies are diffeomorphical between each other. This has a deeper
implication in the foundations of the correspondence, because different charts
could lead to different dualities.
The utility of the correspondence comes in the calculation procedure, encoded
in the holographic dictionary, which is the relation between the bulk and the
boundary physics. Since the Anti de Sitter radius and the string lenght are
free parameters, it is possible to take a low energy limit in (1) in order to
reduce the string generating function to a supergravity one,
$W_{CFT}\left[\phi_{0},\mathcal{O};\partial\,\mathcal{M},\eta\right]=-\text{ln}\,Z_{CFT}\left[\phi_{0},\mathcal{O};\partial\,\mathcal{M},\eta\right]=\sum_{i}Z_{\text{SUGRA}}\left[\phi,\mathcal{M}_{i}\right]+O\left(\frac{1}{N}\right)+O\left(\frac{1}{\sqrt{\lambda}}\right),$
(2)
where the sum in the supergravity action appears to take into account the
chart dependence. The supergravity description is valid only for the large $N$
and large ’tHooft coupling $\lambda$. Note that the supergravity action can
carry divergences due to infinite volume or IR behaviour. These divergences
must be renormalized [7] and could lead to anomalies.
The dictionary is obtained following the saddle point approximation and the
functional standard techniques from the supergravity on-shell action:
$\langle\mathcal{O}\left(x_{1}\right)\,\mathcal{O}\left(x_{2}\right)...\mathcal{O}\left(x_{n}\right)\rangle_{CFT}=\left.\frac{\delta^{n}S_{\text{SUGRA}}^{\text{on-
shell}}\left[\phi_{0},...\right]}{\delta\phi_{0}\left(x_{1}\right)...\phi_{0}\left(x_{n}\right)}\right|_{\text{Sources}=0}.$
(3)
Expresion (3) tells how to connect fields in both sides. For example, the
dilaton is related with the string coupling. For each possible supergravity
action a dictionary can be constructed. This is the path followed, for
example, in AdS/QCD models [8].
## 3 Geometrical Approximation to the Correspondence
### 3.1 Formal Aspects
Geometrically speaking, the correspondence is build up using the complex
geometry language. Consider a open $n+1$-dimensional manifold
$\left(M,g\right)$. This manifold $M$ will be the configuration space for the
possible physical states on the bulk. Along with this manifold, we define a
closed $n+1$-dimensional manifold $(\tilde{M},\tilde{g})$ with no empty
$n$-dimensional boundary $\partial\tilde{M}$, such that $M\subset\tilde{M}$. A
complete Riemmann metric $g$ on $M$ is called _conformally compact_
111Conformally compact is equivalent to Penrose compact. if a function
$f\in\,\Omega_{0}(\tilde{M})$ on $\tilde{M}$ exist such that
$\tilde{g}=f^{2}\,g,$ (4)
with $f^{-1}\left(0\right)=\partial\tilde{M}$ and $df$ is not zero on
$\partial\tilde{M}$. Such a function is called a _defining_ function [9]. The
metric $\tilde{g}$ is called _compactification_ of the metric $g$. The
compactification defines an induced metric
$\eta=\tilde{g}|_{\partial\tilde{M}}$ on $\partial\tilde{M}$.
There are many defining functions, and hence many conformal compactifications
of a given metric $g$, then the choice of $\eta$ is not unique. This problem
can be avoided using the conformal class $[\eta]$ (called conformal infinity)
of $\eta$ on $\partial\tilde{M}$ defined by conformal transformations of
$\eta$. Recall that $[\eta]$ is uniquely determined by the pair $(M,g)$.
Physically, the choice of $[\eta]$ implies that the causal structure of
spacetime is conserved under conformal transformations. The pair
$(\partial\tilde{M},\eta)$ with $\eta\in[\eta]$, defines the _conformal
boundary_ , where the CFT operators are constructed.
Since the symmetries of $M$ and $\partial\tilde{M}$ must be the same222Both
manifolds must have the same causal structure., the moduli space of
$\partial\tilde{M}$, $\mathcal{M}_{\partial\tilde{M}}$ is defined by
$\mathcal{M}_{\partial\tilde{M}}=\text{Teich}(M)/\text{MCG}(M)$, since both
manifolds must have the same conformal stucture because they are
diffeomorphic. Following the discussions above, the entire moduli space of
$M$, $\mathcal{M}$, is restricted by the choice of a conformal class $[\eta]$,
thus not all the metrics $g$ on $M$ will contribute to the partition function
on the bulk. The restricted moduli space of $M$,
$\mathcal{M}_{(\partial\tilde{M},[\eta])}$, is defined as the set of all the
conformally compact metrics $g$ on $M$ [10].
Under these ideas, the AdS/CFT Correspondence can be summarized saying that
given any bulk data $(M,g)$, it is possible to construct (or obtain) a
boundary $(\partial\tilde{M},[\eta])$ by means of the conformal
compactification scheme (4), i.e,
$\overbrace{Z\left(\partial\tilde{M},[\eta]\right)}^{\text{Boundary}}=\overbrace{\sum_{g\in\mathcal{M}_{(\partial\tilde{M},[\eta])}}Z\left(g,M\right).}^{\text{Bulk}}$
(5)
Following physical arguments from AdS/CFT Correspondence, $M$ must be
10-dimensional.Thus, in order to have a conformal boundary as
$\mathbb{R}^{(1,3)}$, $M$ has to be decomposed into $M=\mathbb{H}^{5}\times
X^{5}$, with $X^{5}$ some compact space, such that in the compactification
limit $M\backsim\mathbb{R}^{2,4}\subset\mathbb{H}^{5}$, as in the AdS/CFT
Correspondence, in which $M$ is factorized as $AdS_{5}\times S^{5}$. All the
metrics $g$ that satisfies these conditions are the so called _asymptotically
hyperbolic Einstein metrics_.
### 3.2 Geodesic Compactifications
Any compactification (4) with a defining function given by
$f_{g}=\text{Dist}_{g}\left(x,M\right)$ is called _geodesic_ [9, 10]. These
compactifications are useful for computational purposes, and because given a
conformal infinity $[\eta]$ of $(M,g)$ exists a unique geodesic defining
function $f_{[\eta]}$ that has $\eta\in[\eta]$ as a boundary metric.
Following the Gauss lemma, the compactificacion $\tilde{g}$ can be expanded
into
$\tilde{g}=dr^{2}+g_{f},$ (6)
where $g_{f}$ is a family of metrics on $\partial\tilde{M}$. The
Fefferman–Graham expansion [11] of $g$ is a truncated Taylor-type expansion of
the family of metrics $g_{f}$, that depends on the dimensionality $n$ of $M$.
The exact form of the series depends on whether $n$ is even or odd. In a
general case, the series can be written as
$g_{f}=g_{\left(0\right)}+r\,g_{\left(1\right)}+r^{2}\,g_{\left(2\right)}+\ldots+r^{n}\,g_{\left(n\right)}+\text{terms
depending of even or odd }n,$ (7)
where $\eta:=g_{\left(0\right)}$ and the coefficients $g_{\left(k\right)}$
with $1<k<n$ are locally fixed by the curvature of $\eta$ and its covariant
derivatives. The extra terms depending on the even–odd character of $n$ are
calculated from the Einstein equations for $\eta$.
The $g_{\left(n\right)}$ term is a little more complex. For even dimensions,
$g_{\left(n\right)}$ is transverse traceless, but is determined by global
properties of $M$. In odd dimensions, $g_{\left(n\right)}$ is not traceless
but is still determined by global aspects of $M$. The $g_{\left(n\right)}$
factor corresponds to the stress–energy tensor of the CFT living in
$\partial\tilde{M}$.
Mathematically, these expansions can be obtained by compactifying the Einstein
equations and taking Lie derivatives of $\tilde{g}$ with $f_{g}=0$:
$g_{\left(k\right)}=\frac{1}{k!}\,\mathcal{L}_{\tilde{\nabla}f_{g}}^{\left(k\right)}\tilde{g}.$
(8)
If the metric is Hoelder, all the expansions hold up to order $m+\alpha$, with
$\alpha$ the Hoelder exponent.
As a conclusion, knowing $g_{\left(0\right)}$ and $g_{\left(n\right)}$ allow
to construct the bulk metric field $g$ from the expansion (7). The real
problem here is to know the convergence of the series and how its inclusion
may introduce anomalies [12].
### 3.3 General Decomposition of $M$
Until now, all of the approach to the conjecture was classical, i.e, real
manifolds only. A quantum approach (thinking on strings instead of
supergravity) requires a more general factorization $M=X\times Y$, where
$X\in\mathbb{H}^{5}$ and $Y$ is a 5-dimensional Calabi–Yau manifold. The
Calabi-Yau manifold can be justified on the grounds that classical mechanics
requires a simplectic structure while quantum mechanics requires complex
structure to implement unitarity. The main problem with these structures lies
on the construction of Calabi-Yau metrics. This problem can be partially
avoided by considering $Y$ as a 5-dimensional Sasaki-Einstein manifold [13].
## 4 Geometrical Issues
As it was said above, the central idea for the construction of the conjecture
is the existence of a conformal infinty $[\eta]$ that fixes a conformal
boundary $\partial\tilde{M}$. This process is highly depending on the
convergence of the Fefferman-Graham expansion (7), which could introduce
undesirable anomalies due to the holographic renormalization. But this is not
the only problem.
In a realistic approximation, Gravity/Gauge duality suggests that any quantum
field theory must have a string dual. The large $N$ and large $\lambda$ limits
restrict the possible dual models to the AdS/CFT Correspondence, that is no
realistic. Leaving aside the limits, to obtain a non-conformal holography
would imply the naive idea of taking a different background from Type IIB
supergravity.
Advances in this scenario were given by Skenderis and Taylor with their
precision holography [14]. The idea is to categorize all the possible $X$
manifolds in the decomposition $M=X\times Y$ into spaces that are
_asymptotically AdS_ and those which are not. Asymptotically AdS spaces are
related to the usual 10-dimensional $AdS_{n+2}\times S^{8-n}$ through a Weyl
transformation. This transformation redefines the coupling constant of the QFT
on the conformal boundary including an energy scale with no trivial running.
As a conclusion from [14], only on asymptotically AdS spaces it is possible to
do non conformal holography. This implies that holographic extension only can
be made on AdS-like spaces.
Another issue arises when the index theory comes into play. Following [9, 13],
the conjecture is build up in conformal boundaries, where the index of any
pseudodifferential operator is well defined. When closed and compact manifolds
are considered, the index theorem fails. This problem leads to the
consideration of the definition and the role of the boundary in AdS/CFT
Correspondence [10].
## 5 Conclusions
AdS/CFT Correspondence is strongly related to the concept of conformal
boundary. The construction of this boundary is dependent on the chosen charts,
thus the holographic dictionary (3) is not univocally. The usual chart used to
do holography is the Poincare chart. Non-conformal extensions are made
relaxing the conformal symmetry of $AdS_{5}\times S^{5}$.
The choice of a Calabi-Yau (or a Sasaki–Einstein) manifold as the compact
space in the factorization $AdS_{5}\times Y^{5}$, besides the relaxation of
the large $N$ and large $\lambda$ limits could lead to string/QFT duality.
## References
* [1] E. Papantonopoulos, Editor. _From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence_. Springer. 2011.
* [2] J. M. Maldacena._The large N limit of superconformal field theories and supergravity_ Adv. Theor. Math. Phys. 2, 231 (1998). [arXiv:hep-th/9711200].
* [3] I. R. Klebanov, M. J. Strassler, _Supergravity and confining gauge theory: Duality Cascades and $\chi$SB-resolution of Naked Singularities_, JHEP 0008:052 (2000) [arXiv:hep-th/0007191].
* [4] O. Aharony, M. Berkooz, D. Kustasov, N. Seiberg. _Linear Dilatons, NS5-branes and holography_ , JHEP 9810:004 (1998) [arXiv:hep-th/9808149]
* [5] E. Witten, _Anti-de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories_. arXiv:hep-th/9803131v2. 1998.
* [6] C. A. Ballon Bayona, N. R. F. Braga, _Anti-de Sitter boundary in Poincare coordinates_. Gen. Rel. Grav. 39: 1367-1379,(2007).
* [7] I. Papadimitriou, K. Skenderis. _AdS/CFT Correspondence and Geometry_. IRMA Lectures in Mathematics and Theoretical Physics: AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, (2005).
* [8] U. Gursoy, E. Kiritsis. _Exploring improved holographic theories for QCD Part 1_. JHEP 0802, 019 (2008).
* [9] M. T. Anderson. _Geometrical Aspects of the AdS/CFT Correspondence_. IRMA Lectures in Mathematics and Theoretical Physics: AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, (2005).
* [10] M. Sanchez. _Causal boundaries and holography on wave type spacetimes_. Nonlinear Analysis 71, e1744e1764 (2009).
* [11] C. Fefferman, C. Graham. _Conformal invariants_. Ellie Cartan et les mathematiques d’aujourd’hui (1984), Asterisque (1985), 95 - 116.
* [12] C. R. Graham, E. Witten. _Conformal Anomaly Of Submanifold Observables In AdS/CFT Correspondence_. Nucl. Phys. B 546: 52-64 (1999).
* [13] J. P. Gaunttlet, D. Martelli, J. Sparks, D. Waldram. _Supersymmetric AdS backgrounds in String and M-theory_. IRMA Lectures in Mathematics and Theoretical Physics: AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, (2005).
* [14] I. Kanitscheider, K. Skenderis, M. Taylor. _Precision Holography for non-conformal branes_. JHEP 09, 094 (2008).
|
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| 2014-02-16T01:26:53 |
2024-09-04T02:49:58.268265
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Miguel Angel Martin Contreras, Jose Maria Rolando Roldan",
"submitter": "Miguel Angel Martin Contreras",
"url": "https://arxiv.org/abs/1402.3742"
}
|
1402.3847
|
# Towards the reproducibility in soil erosion modeling:
a new Pan-European soil erosion map
Claudio Bosco European Commission, Joint Research Centre, Institute for
Environment and Sustainability
Via E. Fermi 2749, I-21027 Ispra (VA), Italy Daniele de Rigo European
Commission, Joint Research Centre, Institute for Environment and
Sustainability
Via E. Fermi 2749, I-21027 Ispra (VA), Italy Politecnico di Milano,
Dipartimento di Elettronica e Informazione
Via Ponzio 34/5, I-20133 Milano, Italy Olivier Dewitte European Commission,
Joint Research Centre, Institute for Environment and Sustainability
Via E. Fermi 2749, I-21027 Ispra (VA), Italy Luca Montanarella European
Commission, Joint Research Centre, Institute for Environment and
Sustainability
Via E. Fermi 2749, I-21027 Ispra (VA), Italy
This is the authors’ version of the work. It is based on a poster presented at
the Wageningen Conference on Applied Soil Science,
http://www.wageningensoilmeeting.wur.nl/UK/ Cite as: Bosco, C., de Rigo, D.,
Dewitte, O., Montanarella, L., 2011. Towards the reproducibility in soil
erosion modeling: a new Pan-European soil erosion map. Wageningen Conference
on Applied Soil Science “Soil Science in a Changing World”, 18 - 22 September
2011, Wageningen, The Netherlands. Author’s version DOI:
10.6084/m9.figshare.936872 Abstract Soil erosion by water is a widespread
phenomenon throughout Europe and has the potentiality, with his on-site and
off-site effects, to affect water quality, food security and floods. Despite
the implementation of numerous and different models for estimating soil
erosion by water in Europe, there is still a lack of harmonization of
assessment methodologies. Often, different approaches result in soil erosion
rates significantly different. Even when the same model is applied to the same
region the results may differ. This can be due to the way the model is
implemented (i.e. with the selection of different algorithms when available)
and/or to the use of datasets having different resolution or accuracy.
Scientific computation is emerging as one of the central topic of the
scientific method, for overcoming these problems there is thus the necessity
to develop reproducible computational method where codes and data are
available. The present study illustrates this approach. Using only public
available datasets, we applied the Revised Universal Soil loss Equation
(RUSLE) to locate the most sensitive areas to soil erosion by water in Europe.
A significant effort was made for selecting the better simplified equations to
be used when a strict application of the RUSLE model is not possible. In
particular for the computation of the Rainfall Erosivity factor (R) the
reproducible research paradigm was applied. The calculation of the R factor
was implemented using public datasets and the GNU R language. An easily
reproducible validation procedure based on measured precipitation time series
was applied using MATLAB language. Designing the computational modelling
architecture with the aim to ease as much as possible the future reuse of the
model in analysing climate change scenarios is also a challenging goal of the
research.
## Introduction
Despite the implementation of a variety of models for estimating soil erosion
by water in Europe [1], there is still a lack of harmonization of assessment
methodologies.
Often, distinct approaches lead to significantly different soil erosion rates
and even when the same model is applied to the same region the results may
differ. This can be due to the way the model is implemented (i.e. with the
selection of different algorithms when available) and/or to the use of
datasets having distinct resolution or accuracy.
Scientific computation is emerging as one of the central topic within
environmental modelling [2], to overcome these problems there is thus the
necessity to develop reproducible computational methods based on free software
and data [3, 4], and to also reuse – in a controlled way – empirical equations
for compensating the lack of detailed data.
The present study illustrates such an approach. Using only public available
datasets (SGDBE [5], SRTM [6], CLC and E-OBS [7]) , we applied a derived
version of the Revised Universal Soil loss Equation (RUSLE) [8] to locate the
most sensitive areas to soil erosion in Europe. We decided to use a RUSLE-
based approach because of the flexibility and least data demanding of the
model [10, 9].
A significant effort was made [11, 12] toward reproducibility and to select
the better simplified equations to be used when a strict application of the
model is not possible. In particular for the computation of the Rainfall
Erosivity factor (R) the reproducible research paradigm was applied.
## The model
The Revised Universal Soil Loss Equation (RUSLE) has been extended by
including a correction factor $St_{c,Y}$ able to consider the stoniness:
$\begin{array}[]{lcl}Er_{c,Y}&=&R_{c,Y}\>\cdot\>K_{c,Y}\>\cdot\>L_{c,Y}\>\cdot\>S_{c,Y}\>\cdot\\\\[2.84526pt]
&&C_{c,Y}\>\cdot\>St_{c,Y}\>\cdot\>P_{c,Y}\end{array}$
where the factors refer to a specific grid cell $c$ and represent the annual
average for a certain set of years $Y={y_{1},\cdots,y_{i},\cdots,y_{n_{Y}}}$
(R factor) or – where data are stable or missing – the values corresponding to
a temporally more localized set of data:
$\begin{array}[]{lcl}Er_{c,Y}&=&\text{average annual soil loss }\\\
&&(t\>ha^{-1}\>yr^{-1}).\\\\[4.2679pt] R_{c,Y}&=&\text{rainfall erosivity
factor }\\\ &&(MJ\>mm\>ha^{-1}\>h^{-1}\>yr^{-1}).\\\\[4.2679pt]
K_{c,Y}&=&\text{soil erodibility factor }\\\
&&(t\>ha\>h\>ha^{-1}\>MJ^{-1}\>mm^{-1}).\\\\[4.2679pt] L_{c,Y}&=&\text{slope
length factor}\\\ &&\text{(dimensionless).}\\\\[4.2679pt]
S_{c,Y}&=&\text{slope steepness factor}\\\
&&\text{(dimensionless).}\\\\[4.2679pt] C_{c,Y}&=&\text{cover management
factor}\\\ &&\text{(dimensionless).}\\\\[4.2679pt] St_{c,Y}&=&\text{stoniness
correction factor}\\\ &&\text{(dimensionless).}\\\\[4.2679pt]
P_{c,Y}&=&\text{support practice aimed at}\\\ &&\text{erosion control
(dimensionless).}\\\\[5.69054pt] \end{array}$
Advantages: simplicity and robustness.
Limits: at this resolution and according to the uncertainties associated with
the input data, this model is only relevant to locate the areas prone to soil
erosion.
Table 1: Public available datasets used for running the extended RUSLE model Factor | Data | Database
---|---|---
R [8, 13, 14, 15, 16] | Average daily precipitation | The European daily gridded dataset – E-OBS
K [8] | Topsoil silt, clay, sand % | The database of European soils – SGDBE
L [17] | Elevation | SRTM 90 m
S [17] | Elevation | SRTM 90 m
C [18, 19, 20] | Land cover classes | CORINE Land Cover
St [21] | Percentage of stoniness | The database of European soils – SGDBE
P | Set equal to 1 | —
### The implemented reproducible part of the model
Rainfall erosivity factor. One of the main factors influencing soil erosion by
water is the rainfall intensity. The $R$ factor measures the erosivity of
precipitations. The composite parameter $EI^{30}$ has been identified by
Wischmeier [22] as the best indicator of precipitation erosivity. For
determining $EI^{30}$ the kinetic energy $E$ of rain is multiplied by the
maximum rainfall intensity $I^{30}$ occurred in 30 minutes in every $k$-th
precipitation event of the $i$-th year.
The R factor represents the average, on a consistent set of data, of $n_{Y}$
sums of $EI^{30}$ values. Each sum is computed for the whole set of
$n_{y_{i}}^{\text{event}}$ precipitation events in the $i$-th year:
$\begin{array}[]{ll}R_{c,Y}&=\quad\displaystyle\frac{1}{n_{Y}}\cdot\sum_{i=1}^{n_{Y}}\sum_{k_{i}=1}^{n_{y_{i}}^{\text{event}}}E_{c,k_{i}}\cdot
I_{c,k_{i}}^{30}\\\\[11.38109pt]
&=\quad\displaystyle\frac{1}{n_{Y}}\cdot\sum_{i=1}^{n_{Y}}\sum_{k_{i}=1}^{n_{y_{i}}^{\text{event}}}EI_{c,k_{i}}^{30}\end{array}$
Within the framework, the complete equation has been fully implemented to
accurately estimate R where detailed time series of measured precipitation (10
to 15 minutes of time-step) have been made available across Europe.
However, the scarcity of these accurate datasets and the desire to design a
reusable framework for assessing water soil erosion at regional scale with
only limited and approximated information motivated the creation of a
climatic-based ensemble model for estimating erosivity from multiple available
empirical relationships.
The array programming paradigm [23, 24] was applied using MATLAB language [25]
and GNU Octave [26] computational environment. Within that paradigm, a
semantic-constraint oriented support was adopted by exploiting the Mastrave
library [27, 28].
Figure 1: Soil erosion rate by water $(tha^{-1}yr^{-1})$ estimated applying
the extended RUSLE model. Figure 2: Climatic similarity estimated applying the
Relative Distance Similarity (RDS) to the Bollinne equation (Belgium) for
rainfall erosivity. The similarity of 26 climatic indicators over the whole
Europe is shown (red: maximum similarity; blue: maximum dissimilarity) and
aggregated computing respectively the mean (A1), median (A2), minimum (A3) and
geometric mean (A4).
Multiple layers of geospatial data over a wide spatial extent may naturally be
modelled as corresponding arrays (e.g. here raster grids of heterogeneous -
coarser or denser - spatial resolution have been used). Geoprocessing is
required for the layers to be transformed in arrays with harmonised projection
and datum.
Array programming has been introduced by Iverson [23] in order for the gap
between algorithm implementation and mathematical notation to be mitigated. As
Iverson underlined, “the advantages of executability and universality found in
programming languages can be effectively combined, in a single coherent
language, with the advantages offered by mathematical notation” [23].
Following this approach, prototyping complex algorithms can benefit from a
compact array-based mathematical semantics. This way, the mathematical
reasoning is relocated directly into the source code, actually the only place
where the mathematical description is completely formalised and reproducible.
The semantic array programming paradigm [27, 28] (here applied [29]) has been
designed to support nontrivial scientific modelling with the help of two
additional design concepts:
* •
modularizing complex data-transformations in autonomous tasks by means of
general and concise sub-models, possibly suitable of reuse in other context. A
harmonised predictable convention in module interfaces also relies on self-
documenting the code;
* •
semantically constraining the information flow in each module (input and
output variables and parameters) instead of relying on external assumptions
(e.g. instead of assuming the correctness of input information structured as
an object).
In the present application, the R factor climatic-based ensemble model was
implemented using public datasets and a novel methodology was applied for
merging together multiple empirical equations. This was done by extending the
original geographical domain of validity of each equation to similar areas.
The climatic similarity has been based on the relative-distance similarity
methods of Mastrave [27]. The climatic layers have been computed by using GNU
R language [30] and GNU Octave. The R factor computational framework will be
available as free software [31].
### Climatic ensemble modelling using Relative-Distance Similarity
The ensemble modelling procedure was applied to 7 empirical equations based on
significant correlations between climatic information (such as average annual
precipitation, Fournier modified index, monthly rainfall for days with $\geq
10.0\,mm\,$, …) and locally measured erosivity of 4 geographical areas:
Algarve (Portugal), Belgium, Bavaria (Germany) and Sicily (Italy) [13, 14, 15,
16].
Similarity maps with respect to the climatic conditions of each equation’s
geographical domain have been computed based on the relative distance
(dimensionless) between pan-European maps of 26 climatic indicators and the
corresponding indicators’ values of the equation area of validity. The
behaviour of each empirical equation outside its definition domain was also
assessed for preventing meaningless out-of-range values to degrade the
ensemble estimation.
The aggregated similarities for each equation have been normalized for
estimating the ensemble erosivity map using weighted median [32, 27] of the 7
empirical models.
The contribution of each empirical equation based on its aggregated similarity
was accounted to estimate a qualitative trustability map of the ensemble
generalization. As a whole, the ensemble model is therefore a reproducible,
unsupervised data-transformation model applied to climatic data to reconstruct
erosivity.
Figure 3: Climatic similarity estimated applying the Relative Distance
Similarity (RDS) to the equation of de Santos Loureiro and de Azevedo Coutinho
(Algarve) for rainfall erosivity. The similarity of 26 climatic indicators
over the whole Europe is shown (red: maximum similarity; blue: maximum
dissimilarity) and aggregated computing respectively the mean (B1), median
(B2), minimum (B3) and geometric mean (B4).
## Conclusions
A lightweight architecture has been proposed to support environmental
modelling within the paradigm of semantic array programming [27, 28]. The
applied methodology benefits from the array programming paradigm with semantic
constraints to concisely implement models as semantically enhanced composition
of interoperable modules.
An application for estimating the pan-European soil erosion by water, using a
revised version of the RUSLE model, has been carried out merging existing
empirical rainfall-erosivity equations within a climatic ensemble model based
on the novel relative-distance similarity. An accurate estimation of the
rainfall erosivity factor, applying the proposed architecture, has been
implemented and will be used for validating simplified R-factor equations.
## Next Steps
The proposed architecture is designed to ease the future integration, within
the same lightweight framework, of erosion-related natural resources models
[11, 29]. In particular, forest resources and wildfires[33], natural
vegetation [34] and agriculture will be considered as key land cover factors
under different climate change scenarios.
Acknowledgments. We acknowledge the E-OBS dataset from the EU-FP6 project
ENSEMBLES (http://ensembles-eu.metoffice.com) and the data providers in the
ECA&D project (http://eca.knmi.nl).
## References
* [1] Rusco, E., Montanarella, L., Bosco, C., 2008. Soil erosion: a main threats to the soils in Europe. In: Tóth, G., Montanarella, L., Rusco, E. (Eds.), Threats to Soil Quality in Europe. No. EUR 23438 EN in EUR - Scientific and Technical Research series. Office for Official Publications of the European Communities, pp. 37-45. Google Scholar: 16771305971362909763
* [2] Casagrandi, R. and Guariso, G., 2009. Impact of ICT in Environmental Sciences: A citation analysis 1990-2007. Environmental Modelling & Software 24 (7), 865-871. DOI:10.1016/j.envsoft.2008.11.013 Google Scholar: 10214045160670186637
* [3] Stallman, R. M., 2005. Free community science and the free development of science. PLoS Med 2 (2), e47+. DOI:10.1371/journal.pmed.0020047 Google Scholar: 8322326185901188530
* [4] Waldrop, M. M., 2008. Science 2.0. Scientific American 298 (5), 68-73. DOI:10.1038/scientificamerican0508-68 Google Scholar: 11153245314321360836
* [5] Heineke, H. J., Eckelmann, W., Thomasson, A. J., Jones, R. J. A., Montanarella, L., and Buckley, B., 1998. Land Information Systems: Developments for planning the sustainable use of land resources. Office for Official Publications of the European Communities, Luxembourg. EUR 17729 EN
* [6] Farr, T. G., Rosen, P. A., Caro, E., Crippen, R., Duren, R., Hensley, S., Kobrick, M., Paller, M., Rodriguez, E., Roth, L., Seal, D., Shaffer, S., Shimada, J., Umland, J., Werner, M., Oskin, M., Burbank, D., Alsdorf, D., 2007. The Shuttle Radar Topography Mission. Review of Geophysics 45, RG2004, DOI:10.1029/2005RG000183 Google Scholar: 1127486706629895695
* [7] Haylock, M. R., Hofstra, N., Klein Tank, A. M. G., Klok, E. J., Jones, P. D., and New, M., 2008. A European daily high-resolution gridded dataset of surface temperature and precipitation. Journal of Geophysical Research 113, (D20) D20119+ DOI:10.1029/2008jd010201 Google Scholar: 8877766981889885612
* [8] Renard, K. G., Foster, G. R., Weesies, G. A., McCool, D. K., and Yoder, D. C., 1997. Predicting Soil Erosion by Water: A Guide to Conservation Planning with the Revised Universal Soil Loss Equation (RUSLE). Agriculture handbook 703. US Dept Agric., Agr. Handbook, 703 Google Scholar: 10835008925140666547
* [9] Bosco, C., Rusco, E., Montanarella, L., Panagos, P., 2009. Soil erosion in the alpine area: risk assessment and climate change. Studi Trentini di scienze naturali 85, 119-125. Google Scholar: 11887285678357495770
* [10] Bosco, C., Rusco, E., Montanarella, L., Oliveri, S., 2008. Soil erosion risk assessment in the alpine area according to the IPCC scenarios. In: Tóth, G., Montanarella, L., Rusco, E. (Eds.), Threats to Soil Quality in Europe. No. EUR 23438 EN in EUR - Scientific and Technical Research series. Office for Official Publications of the European Communities, pp. 47-58. Google Scholar: 15631759978034062180
* [11] de Rigo, D. and Bosco, C., 2011. Architecture of a Pan-European Framework for Integrated Soil Water Erosion Assessment. IFIP Advances in Information and Communication Technology 359 (34), 310-31. DOI:10.1007/978-3-642-22285-6_34 Google Scholar: 3950024085016158193
* [12] Bosco, C., de Rigo, D., Dewitte, O., and Montanarella, L., 2011. Towards a Reproducible Pan-European Soil Erosion Risk Assessment - RUSLE. Geophys. Res. Abstr. 13, 3351. Google Scholar: 9379475719708413586
* [13] Bollinne, A., Laurant, A., and Boon, W., 1979. L’érosivité des précipitations a Florennes. Révision de la carte des isohyétes et de la carte d’erosivite de la Belgique. Bulletin de la Société géographique de Liége 15, 77-99. Google Scholar: 12744078003335983139
* [14] Ferro, V., Porto, P., and Yu, B., 1999. A comparative study of rainfall erosivity estimation for southern Italy and southeastern Australia. Hydrolog. Sci. J. 44 (1), 3-24. DOI:10.1080/02626669909492199 Google Scholar: 5176842359285038632
* [15] de Santos Loureiro, N. S. and de Azevedo Coutinho, M., 2001. A new procedure to estimate the RUSLE EI30 index, based on monthly rainfall data and applied to the Algarve region, Portugal. J. Hydrol. 250, 12-18. DOI:10.1016/S0022-1694(01)00387-0 Google Scholar: 11524802670684946891
* [16] Rogler, H., and Schwertmann, U., 1981. Erosivität der Niederschläge und Isoerodentkarte von Bayern (Rainfall erosivity and isoerodent map of Bavaria). Zeitschrift fur Kulturtechnik und Flurbereinigung 22, 99-112. Google Scholar: 4595552058193254891
* [17] Nearing, M. A., 1997. A single, continuous function for slope steepness influence on soil loss. Soil Sci. Soc. Am. J. 61 (3), 917-919. DOI:10.2136/sssaj1997.03615995006100030029x Google Scholar: 16720789317381314176
* [18] Morgan, R. P. C., 2005. Soil Erosion and Conservation, 3rd ed. Blackwell Publ., Oxford, pp. 304. Google Scholar: 8117246297833593898
* [19] Šúri, M., Cebecauer, T., Hofierka, J., Fulajtár, E., 2002. Erosion Assessment of Slovakia at regional scale using GIS. Ecology 21 (4), 404-422. Google Scholar: 14071101661888298673
* [20] Cebecauer, T. and Hofierka, J., 2008. The consequences of land-cover changes on soil erosion distribution in Slovakia. Geomorphology 98, 187-198. DOI:10.1016/j.geomorph.2006.12.035 Google Scholar: 8092124721939638642
* [21] Poesen, J., Torri, D., and Bunte, K., 1994. Effects of rock fragments on soil erosion by water at different spatial scales: a review. Catena 23, 141-166. DOI:10.1016/0341-8162(94)90058-2 Google Scholar: 6109269939395759682
* [22] Wischmeier, W. H., 1959. A rainfall erosion index for a universal Soil-Loss Equation. Soil Sci. Soc. Amer. Proc. 23, 246-249. Google Scholar: 15852287544762057099
* [23] Iverson, K. E., 1980. Notation as a tool of thought. Commun. ACM 23 (8), 444-465. DOI:10.1145/358896.358899 Google Scholar: 15203139354397204728
* [24] Quarteroni, A., Saleri, F., 2006. Scientific Computing with MATLAB and Octave. Texts in Computational Science and Engineering. Milan, Springer-Verlag.
* [25] The MathWorks, 2011. MATLAB. http://www.mathworks.com/help/techdoc/ref/
* [26] Eaton, J. W., Bateman, D., and Hauberg, S., 2008. GNU Octave Manual Version 3. A high-level interactive language for numerical computations. Network Theory Limited, ISBN: 0-9546120-6-X
* [27] de Rigo, D., 2011. Semantic Array Programming with Mastrave - Introduction to Semantic Computational Modeling. The Mastrave project. http://mastrave.org/doc/MTV-1.012-1 Google Scholar: 6848554969929557252
* [28] de Rigo, D., (exp.) 2012. Semantic array programming for environmental modelling: application of the Mastrave library. In prep. Google Scholar: 6628751141895151391
* [29] Bosco, C., de Rigo, D., Dewitte, O., Poesen, J., Panagos, P.: Modelling Soil Erosion at European Scale. Towards Harmonization and Reproducibility. In prep.
* [30] R Development Core Team, 2005. R: A language and environment for statistical computing. R Foundation for Statistical Computing. Google Scholar: 6471212841525457824
* [31] Stallman, R. M., 2009. Viewpoint: Why “open source” misses the point of free software. Commun. ACM 52 (6), 31–33. DOI:10.1145/1516046.1516058 Google Scholar: 17751536887456926788
* [32] de Rigo, D. 2011. Multi-dimensional weighted median: the module "wmedian" of the Mastrave modelling library. Mastrave project technical report. http://mastrave.org/doc/mtv_m/wmedian
* [33] Shakesby, R. A., 2011. Post-wildfire soil erosion in the Mediterranean: Review and future research directions. Earth-Science Reviews 105 (3-4), 71-100. DOI:10.1016/j.earscirev.2011.01.001 Google Scholar: 14888605507911518120
* [34] Zuazo, V. H., Pleguezuelo, C. R., 2009. Soil-Erosion and runoff prevention by plant covers: A review. In: Lichtfouse, E., Navarrete, M., Debaeke, P., Véronique, S., Alberola, C. (Eds.), Sustainable Agriculture. Springer Netherlands, pp. 785-811. DOI:10.1007/978-90-481-2666-8_48 Google Scholar: 9825776700428271277
|
arxiv-papers
| 2014-02-16T22:10:42 |
2024-09-04T02:49:58.277566
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Claudio Bosco, Daniele de Rigo, Olivier Dewitte and Luca Montanarella",
"submitter": "Daniele de Rigo",
"url": "https://arxiv.org/abs/1402.3847"
}
|
1402.3933
|
# A Directed Continuous Time Random Walk Model with Jump Length Depending on
Waiting Time
Long Shi1,2, Zuguo Yu1,3, Zhi Mao1, Aiguo Xiao1
1Hunan Key Laboratory for Computation and Simulation in Science and
Engineering and Key
Laboratory of Intelligent Computing and Information Processing of Ministry of
Education,
Xiangtan University, Xiangtan, Hunan 411105, China.
2Institute of Mathematics and Physics, Central South University of Forest and
Technology, Changsha,
Hunan 410004, China.
3School of Mathematical Sciences, Queensland University of Technology, GPO Box
2434,
Brisbane, Q4001, Australia. Corresponding author, e-mail: [email protected]
or [email protected]
###### Abstract
In continuum one-dimensional space, a coupled directed continuous time random
walk model is proposed, where the random walker jumps toward one direction and
the waiting time between jumps affects the subsequent jump. In the proposed
model, the Laplace-Laplace transform of the probability density function
$P(x,t)$ of finding the walker at position $x$ at time $t$ is completely
determined by the Laplace transform of the probability density function
$\varphi(t)$ of the waiting time. In terms of the probability density function
of the waiting time in the Laplace domain, the limit distribution of the
random process and the corresponding evolving equations are derived.
## 1\. Introduction
The continuous time random walk (CTRW) theory, which was introduced by
Montroll and Weiss [1] to study random walks on a lattice, has been applied
successfully in many fields (see, e.g., the reviews [2-4] and references
therein).
In continuum one-dimensional space, a CTRW process is generated by a sequence
of independent identically distributed (IID) positive waiting times
$T_{1},T_{2},T_{3},\cdots$, and a sequence of IID random jump lengths
$X_{1},X_{2},X_{3},\cdots$. Each waiting time has the same probability density
function (PDF) $\varphi(t),t\geq 0$, and each jump length has the same PDF
$\lambda(x)$ (usually chosen to be symmetric $\lambda(x)=\lambda(-x)$).
Setting $t_{0}=0,t_{n}=T_{1}+T_{2}+\cdots+T_{n}$ for $n\in N$ and
$x_{0}=0,x_{n}=X_{1}+X_{2}+\cdots+X_{n},x(t)=x_{n}$ for $t_{n}\leq t<t_{n+1}$,
we get a microscopic description of the diffusion process [5]. If
$\\{X_{n}\\}$ and $\\{T_{n}\\}$ are independent, the CTRW is called decoupled.
Otherwise it is called coupled CTRW [6]. The decoupled CTRW, which is
completely determined by mutually independent random jump length and random
waiting time, has been widely studied in recent years [3-20].
In some applications it becomes important to consider coupled CTRW [7-8]. The
coupled CTRW can be described by the joint PDF $\phi(x,t)$ of jump length and
waiting time. Because $\phi(x,t)dxdt$ is the probability of a jump to be in
the interval $(x,x+dx)$ in the time interval $(t,t+dt)$, the waiting time PDF
$\varphi(t)=\int_{-\infty}^{+\infty}\phi(x,t)dx$ and the jump length PDF
$\lambda(x)=\int_{0}^{+\infty}\phi(x,t)dt$ can be deduced. Some kinds of
couplings and correlations were proposed in [21-25], where the symmetric jump
length PDF is chosen. For the coupled CTRW, there exist two coupled forms:
$\phi(x,t)=\lambda(x)\varphi(t|x)$ and $\phi(x,t)=\varphi(t)\lambda(x|t)$. The
first coupled form has been studied sufficiently in many literatures [8,
21-23]. The famous model is Lévy walk. Recently, we considered the second
coupled form, discussed the asymptotic behaviors of the coupled jump
probability density function in the Fourier-Laplace domain, and derived the
corresponding fractional diffusion equations from the given asymptotic
behaviors [25].
In this work, we introduce a directed CTRW model with jump length depending on
waiting time (i.e. $\phi(x,t)=\varphi(t)\lambda(x|t),x>0,t>0$). In our model,
the Laplace-Laplace transform [26] of $P(x,t)$ of finding the walker at
position $x$ at time $t$ is completely determined by the Laplace transform of
$\varphi(t)$. Generally, CTRW processes can be categorised by the mean waiting
time $T=\int_{0}^{+\infty}t\varphi(t)dt$ being finite or infinite. Here we
find that the long-time limit distributions of the PDF $P(x,t)$ are a Dirac
delta function for finite $T$ and a beta-like density for infinite $T$, the
corresponding evolving equations are a standard advection equation for finite
$T$ and a pseudo-differential equation with fractional power of coupled space
and time derivative for infinite $T$.
This paper is organized as follows. In section 2, we introduce the basic
concepts of the coupled CTRW. In section 3, a coupled directed CTRW model is
introduced. In section 4, the limit distributions and the corresponding
evolving equations of the coupled directed CTRW model are derived. The
conclusions are given in section 5.
## 2\. The coupled continuous time random walk
Now we recall briefly the general theory of CTRW [3]. Let $\eta(x,t)$ is the
PDF of just having arrived at position $x$ at time $t$. It can be expressed by
$\eta(x^{\prime},t^{\prime})$ (the PDF of just having arrived at position
$x^{\prime}$ at time $t^{\prime}<t$) as:
$\eta(x,t)=\int_{-\infty}^{+\infty}dx^{\prime}\int_{0}^{+\infty}dt^{\prime}\eta(x^{\prime},t^{\prime})\phi(x-x^{\prime},t-t^{\prime})+\delta(x)\delta(t).$
(1)
Then, the PDF $P(x,t)$ with the initial condition $P(x,0)=\delta(x)$ can be
described by the following integral equation [3]
$P(x,t)=\int_{0}^{t}\eta(x,t^{\prime})\omega(t-t^{\prime})dt^{\prime},$ (2)
where $\omega(t)=1-\int_{0}^{t}\varphi(\tau)d\tau$ is the probability of not
having made a jump until time $t$.
Let $\widehat{f}(k)$ and $\widetilde{g}(s)$ be the transforms of Fourier and
Laplace of sufficiently well-behaved (generalized) functions $f(x)$ and $g(t)$
respectively, defined by
$\widehat{f}(k)={\cal
F}\\{f(x);k\\}=\int_{-\infty}^{+\infty}f(x)e^{ikx}dx,\hskip 14.22636ptk\in R,$
(3)
$\widetilde{g}(s)={\cal L}\\{g(t);s\\}=\int_{0}^{+\infty}g(t)e^{-st}dt,\hskip
14.22636pts>s_{0}.$ (4)
After using the Fourier-Laplace transforms and the convolution theorems for
integral equation (2), one can obtain the following famous algebraic relation
[3]
$\widehat{\widetilde{P}}(k,s)=\frac{1-\widetilde{\varphi}(s)}{s}\cdot\frac{1}{1-\widehat{\widetilde{\phi}}(k,s)}.$
(5)
## 3\. A coupled directed CTRW model
In Ref. [23], the author considered a CTRW model with waiting time depending
on the preceding jump length, where the author supposed that the PDF of the
waiting time is a function of a preceding jump length. In that model, the
author introduced a natural ”physiological” analogy: after making a jump one
needs time to rest and recover. The longer the jump distance is, the recovery
and the waiting time needed are longer. This is an interesting hypothetical
physiological example. Motivated by this, we consider a directed CTRW model
with jump length depending on the waiting time and give an analogue
physiological explanation.
A directed CTRW model with jump length depending on the waiting time can be
generated by a sequence of IID positive waiting times
$T_{1},T_{2},T_{3},\cdots$, and a sequence of jumps
$X_{1},X_{2},X_{3},\cdots$. each waiting time has the same PDF
$\varphi(t),t\geq 0$. Every time jump has the same direction and each jump
length has the same conditional PDF $\lambda(x|t),x\geq 0$, which is the PDF
of the random walker making a jump of length $x$ following a waiting time $t$.
A natural assumption is that the jump length is proportional to the waiting
time. So we can take the simplest jump length PDF as
$\lambda(x|t)=\delta(x-vt),v>0$. Without the loss of generality, we take $v=1$
in the following discussion. Setting $t_{0}=0,t_{n}=T_{1}+T_{2}+\cdots+T_{n}$
for $n\in N$ and $x_{0}=0,x_{n}=X_{1}+X_{2}+\cdots+X_{n},x(t)=x_{n}$ for
$t_{n}\leq t<t_{n+1}$, we get a directed CTRW process, where the joint PDF
$\phi(x,t)$ can be expressed by $\phi(x,t)=\varphi(t)\delta(x-t)$. A
physiological explanation can be made as follows: the walker has a random time
for a rest to supplement energy, then makes a jump. The longer the rest time
is, the jump length can be longer.
Since the variable $x$ takes positive values in proposed directed CTRW model,
it is convenient to replace the Fourier transform for variable $x$ in the
formula (5) by the Laplace transform (i.e. $\widetilde{f}(k)={\cal
L}\\{f(x);k\\}=\int_{0}^{+\infty}f(x)e^{-kx}dx$) to obtain the following
Laplace-Laplace relation [26]:
$\widetilde{\widetilde{P}}(k,s)=\frac{1-\widetilde{\varphi}(s)}{s}\cdot\frac{1}{1-\widetilde{\widetilde{\phi}}(k,s)}.$
(6)
Since
$\begin{array}[]{lll}\widetilde{\widetilde{\phi}}(k,s)&=&\int_{0}^{+\infty}dt\int_{0}^{+\infty}\phi(x,t)e^{-kx-
st}dx\\\ \\\
&=&\int_{0}^{+\infty}dt\int_{0}^{+\infty}\varphi(t)\delta(x-t)e^{-kx-st}dx\\\
\\\ &=&\int_{0}^{+\infty}\varphi(t)e^{-(s+k)t}dt\\\ \\\
&=&\widetilde{\varphi}(s+k),\end{array}$ (7)
Eq.(6) is recast into
$\widetilde{\widetilde{P}}(k,s)=\frac{1-\widetilde{\varphi}(s)}{s}\cdot\frac{1}{1-\widetilde{\varphi}(s+k)}.$
(8)
The $n$th ($n=1,2$) moment of $P(x,t)$ is given by
$\begin{array}[]{lll}\langle
x^{n}\rangle(t)&=&\int_{0}^{+\infty}x^{n}(t)P(x,t)dx\\\ \\\
&=&(-1)^{n}\frac{\partial^{n}}{\partial k^{n}}\widetilde{P}(k,t)\mid_{k=0}\\\
\\\ &=&{\cal
L}^{-1}\\{\frac{1-\widetilde{\varphi}(s)}{s}\cdot(-1)^{n}\frac{\partial^{n}}{\partial
k^{n}}\frac{1}{1-\widetilde{\varphi}(s+k)}\mid_{k=0}\\}.\end{array}$ (9)
In the following section, we will study the possible behaviors of $P(x,t)$ and
its $n$th ($n=1,2$) moment.
## 4\. The limit distributions of the coupled directed CTRW model
From Eq.(8), we can see that the Laplace-Laplace transform of PDF $P(x,t)$ is
completely determined by the Laplace transform of the waiting time PDF
$\varphi(t)$. Usually, the random waiting time is characterized by its mean
value $T$. It may be finite or infinite.
For finite mean waiting time $T$, the Laplace transform of $\varphi(t)$ is of
the form
$\widetilde{\varphi}(s)=1-sT+o(s),\hskip 14.22636pts\rightarrow 0.$ (10)
Substituting Eq.(10) into Eq.(8), in the limit $(k,s)\rightarrow(0,0)$, we get
the asymptotic relation
$\widetilde{\widetilde{P}}(k,s)\sim\frac{1-(1-sT)}{s}\cdot\frac{1}{1-(1-(s+k)T)}=\frac{1}{s+k}.$
(11)
After taking the inverse Laplace transforms for Eq.(11) about $k$ and $s$, we
have
$P(x,t)=\delta(x-t).$ (12)
For long times
$\langle x\rangle(t)=t,$ (13) $\langle x^{2}\rangle(t)=t^{2}.$ (14)
From Eq.(11), we get
$s\widetilde{\widetilde{P}}(k,s)-1+k\widetilde{\widetilde{P}}(k,s)=0.$ (15)
Using ${\cal L}\\{\frac{\partial P(x,t)}{\partial
t};s\\}=s\widetilde{P}(x,s)-P(x,0)$, ${\cal L}\\{\frac{\partial
P(x,t)}{\partial x};k\\}=k\widetilde{P}(k,t)-P(0,t)$, initial condition
$P(x,0)=\delta(x)$ and natural boundary conditions, we obtain the partial
differential equation
$\frac{\partial P(x,t)}{\partial t}+\frac{\partial P(x,t)}{\partial x}=0,$
(16)
which is the standard advection equation.
In many applications, one needs to consider a long waiting time (i.e. $T$ is
infinite), it is natural to generalize Eq.(10) to the following form:
$\widetilde{\varphi}(s)=1-s^{\beta}+o(s^{\beta}),\hskip 14.22636pts\rightarrow
0,0<\beta\leq 1.$ (17)
Inserting Eq.(17) into Eq.(8), in the limit $(k,s)\rightarrow(0,0)$, we get
the asymptotic relation
$\widetilde{\widetilde{P}}(k,s)\sim\frac{1-(1-s^{\beta})}{s}\cdot\frac{1}{1-(1-(s+k)^{\beta})}=\frac{s^{\beta-1}}{(s+k)^{\beta}}.$
(18)
After taking the Laplace inverse transform for Eq.(18) about $s$, one has
$\begin{array}[]{lll}\widetilde{P}(k,t)&=&\frac{t^{-\beta}}{\Gamma(1-\beta)}\ast[e^{-kt}\frac{t^{\beta-1}}{\Gamma(\beta)}]\\\
\\\
&=&\int_{0}^{t}e^{-k\tau}\frac{\tau^{\beta-1}(t-\tau)^{-\beta}}{\Gamma(\beta)\Gamma(1-\beta)}d\tau,\end{array}$
(19)
where we use the formulas ${\cal
L}\\{t^{\beta-1};s\\}=\frac{\Gamma(\beta)}{s^{\beta}}$ for $\beta>0$, ${\cal
L}\\{e^{-at}g(t);s\\}=\widetilde{g}(s+a)$ and ${\cal L}\\{(f\ast
g)(t);s\\}=\widetilde{f}(s)\widetilde{g}(s)$.
According to the formula (9) and Eq.(19), for long times, one gets
$\langle x\rangle(t)=\beta t,$ (20) $\langle
x^{2}\rangle(t)=\frac{\beta(\beta+1)}{2}t^{2}.$ (21)
Then taking the Laplace inverse transform for Eq.(19) about $k$, the following
form is obtained
$\begin{array}[]{lll}P(x,t)&=&\int_{0}^{t}\delta(x-\tau)\frac{\tau^{\beta-1}(t-\tau)^{-\beta}}{\Gamma(\beta)\Gamma(1-\beta)}d\tau\\\
\\\
&=&\frac{x^{\beta-1}(t-x)^{-\beta}}{\Gamma(\beta)\Gamma(1-\beta)},\end{array}$
(22)
which is the density of a random variable $tB$, where $B$ has a Beta
distribution with parameters $\beta$ and $1-\beta$.
From Eq.(19), we can also obtain
$(s+k)^{\beta}\widetilde{\widetilde{P}}(k,s)=s^{\beta-1},$ (23)
which leads to the pseudo-differential equation [27-28]
$(\frac{\partial}{\partial t}+\frac{\partial}{\partial
x})^{\beta}P(x,t)=\delta(x)\frac{t^{-\beta}}{\Gamma(1-\beta)}$ (24)
with a coupled space-time fractional derivative operator on the left-hand
side.
Eq.(24) is useful to model flow in porous media and other physical systems
characterized by a link between the waiting time and the jump length.
## 5\. Conclusions
In this work, we introduce a directed CTRW model with jump lengths depending
on waiting times. By the Laplace-Laplace transform technique, we find that the
PDF $P(x,t)$ is determined only by the waiting times PDF $\varphi(t)$. For
finite and infinite mean waiting time, we deduce the limit distributions of
$P(x,t)$ from the asymptotic behaviors of $\varphi(t)$ in the Laplace domain
respectively. The corresponding evolving equations are also derived. For
finite mean waiting time, the limit behavior of the PDF $P(x,t)$ is governed
by a standard advection equation. For infinite mean waiting time, the limit
behavior of the PDF $P(x,t)$ is governed by a pseudo-differential equation
with coupled space-time fractional derivative. We also calculate the first
order moment $\langle x\rangle(t)$ and the second order moment $\langle
x^{2}\rangle(t)$ of $P(x,t)$. An interesting phenomenon is obtained: there
exist the relations $\langle x\rangle(t)\sim t$, $\langle x^{2}\rangle(t)\sim
t^{2}$, whether the mean waiting time is finite or not.
## Acknowledgements
This project was supported by the Natural Science Foundation of China (Grant
Nos. 11371016 and 11271311), the Chinese Program for Changjiang Scholars and
Innovative Research Team in University (PCSIRT) (Grant No. IRT1179), the
Research Foundation of Education Commission of Hunan Province of China (grant
no. 11A122), the Lotus Scholars Program of Hunan province of China.
## References
* [1] E.W. Montroll, G.H. Weiss, Random walks on lattices II, J. Math. Phys. 6 (1965), 167-181.
* [2] J.P. Bouchaud, A. Georges, Anomalous diffusion in disordered media: statistical mechanisms models and physical applications. Phys. Rep. 195 (1990), 127-293.
* [3] R. Metzler, J. Klafter, The random walk s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1-77.
* [4] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37 (2004), R161-R208.
* [5] R. Gorenflo, A. Vivoli, F. Mainardi, Discrete and continuous random walk models for space-time fractional diffusion. Nonlinear Dynamics 38 (2004), 101-116.
* [6] M.M. Meerschaert, E. Nane, Y. Xiao, Correlated continuous time random walks. Stat. Prob. Lett. 79 (2009), 1194-1202.
* [7] M.F. Shlesinger, J. Klafter, Y.M. Wong, Random walks with infinite spatial and temporal moments. J. Stat. Phys. 27 (1982), 499-512.
* [8] J. Klafter, A. Blumen, M.F. Shlesinger, Stochastic pathway to anomalous diffusion. Phys. Rev. A 35 (1987), 3081-3085.
* [9] G.H. Weiss, Aspects and Applications of the Random Walk. North Holland, Amsterdam, 1994.
* [10] H.E. Roman, P.A. Alemany, Continuous-time random walks and the fractional diffusion equation. J. Phys. A: Math. Gen. 27 (1994), 3407-3410.
* [11] R. Hilfer, L. Anton, Fractional master equations and fractal time random walks. Phys. Rev. E 51 (1995), R848-R851.
* [12] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance. Physica A 284 (2000), 376-384.
* [13] E. Barkai, CTRW pathways to the fractional diffusion equation. Chem. Phys. 284 (2002), 13-27.
* [14] R. Hilfer, On fractional diffusion and continuous time random walks. Physica A 329 (2003), 35-40.
* [15] E. Scalas, R. Gorenflo, F. Mainardi, Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. Phys. Rev. E 69 (2004), 011107.
* [16] E. Scalas, The application of continuous-time random walks in finance and economics. Physica A 362 (2006), 225-239.
* [17] R. Gorenflo, F. Mainardi, A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion. Chaos, Solitons and Fractals 34 (2007), 87-103.
* [18] A.V. Chechkin, M. Hofmann I.M. Sokolov, Continuous-time random walk with correlated waiting time. Phys. Rev. E 80 (2009), 031112.
* [19] V. Tejedor, R. Metzler, Anomalous diffusion in correlated continuous time random walks. J. Phys. A: Math. Theor. 43 (2010), 082002.
* [20] K.S. Fa, Uncoupled continuous-time random walk: finite jump length probability density function. J. Phys. A: Math. Theor. 45 (2012), 195002.
* [21] A. Blumen, G. Zumofen, J. Klafter, Transport aspects in anomalous diffusion: Levy walks. Phys. Rev. A 40 (1989), 3964-3973.
* [22] G. Zumofen, J. Klafter, Scale-invariant motion in intermittent chaotic systems. Phys. Rev. E 47 (1993), 851-863.
* [23] V.Y. Zaburdaev, Random walk model with waiting times depending on the preceding jump length. J. Stat. Phys. 123 (2006), 871-881.
* [24] J. Liu, J.D. Bao, Continuous time random walk with jump length correlated with waiting time. Physica A 392 (2013), 612-617.
* [25] L. Shi, Z. Yu, Z. Mao, A. Xiao, H. Huang, Space-time fractional diffusion equations and asymptotic behaviors of a coupled continuous time random walk model. Physica A 392 (2013), 5801-5807.
* [26] R. Gorenflo, F. Mainardi, Laplace-Laplace analysis of the fractional Poisson process, in Analytical Methods of Analysis and Differential Equations (Editor: S.Rogosin),Minsk, 2012, 43-58.
* [27] P. Becher-Kern, M.M. Meerschaert, H.P. Scheffler, Limit theorems for coupled continuous time random walks. The Annals of Probability 32 (2004), 730-756.
* [28] A. Jurlewicz, P. Kern, M.M. Meerschaert, H.P. Scheffler, Fractional governing equations for coupled random walks. Comput. Math. Appl. 64 (2012), 3021-3036.
|
arxiv-papers
| 2014-02-17T09:12:39 |
2024-09-04T02:49:58.286149
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Long Shi, Zuguo Yu, Zhi Mao, Aiguo Xiao",
"submitter": "Zu-Guo Yu",
"url": "https://arxiv.org/abs/1402.3933"
}
|
1402.3951
|
# Trends in Computer Network Modeling Towards the Future Internet
Jeroen van der Ham Mattijs Ghijsen Paola Grosso Cees de Laat
E-mail: {vdham, m.ghijsen, p.grosso, delaat}@uva.nl
###### Abstract
This article provides a taxonomy of current and past network modeling efforts.
In all these efforts over the last few years we see a trend towards not only
describing the network, but connected devices as well. This is especially
current given the many Future Internet projects, which are combining different
models, and resources in order to provide complete virtual infrastructures to
users.
An important mechanism for managing complexity is the creation of an abstract
model, a step which has been undertaken in computer networks too. The fact
that more and more devices are network capable, coupled with increasing
popularity of the Internet, has made computer networks an important focus area
for modeling. The large number of connected devices creates an increasing
complexity which must be harnessed to keep the networks functioning.
Over the years many different models for computer networks have been proposed,
and used for different purposes. While for some time the community has moved
away from the need of full topology exchange, this requirement resurfaced for
optical networks. Subsequently, research on topology descriptions has seen a
rise in the last few years. Many different models have been created and
published, yet there is no publication that shows an overview of the different
approaches.
## 1 Introduction
Communication networks, such as the Internet, play a fundamental role in
modern societies and economies. It is nearly superfluous to remind anybody of
the many changes that have occurred in the last twenty years since the
invention of the World Wide Web and the wide adoption of the TCP/IP protocol
suite.
Less known is that the role of networks is becoming even more central in
emerging ICT architectures. In these new infrastructures, which are labeled as
Future Internet, there is a much more integrated operation of networking,
computing and storage devices. All these components are being managed and
monitored in a coordinated manner in order to deliver services to applications
and end users.
One basic rule holds for both the current Internet and the upcoming Future
Internet platforms: the design, planning, management and monitoring of the
network rely on the knowledge of its topology. A network topology provides in
fact information on the location of devices and on the connections between
them; this information in turn gives a view of the physical and logical
structure of the network. Topologies are expressed as network models, and we
use these two terms interchangeably in this article.
Topology information needs to be available to all devices within the network
to operate properly, to external tools that act on the network and to
applications that use the network. We see three main challenges for network
models.
* •
_Handling different abstraction levels:_ From a devices perspective there is a
wide range of topology details needed: at the edges of the network knowledge
can be as minimal as knowing where the next hop is, while within the core
devices require much more information.
* •
_Managing multi-domain communication and path setup:_ External tools that
operate on the network need to be aware of the network or to provide metadata
of the network; monitoring tools require a comprehensive model to describe all
relevant details of computer networks and the connections through them, while
bandwidth-on-demand tools used in circuit switched networks will only need to
exchange some detail of network topology to be able to efficiently plan
connections.
* •
_Integration with computing-network-storage-planning services:_ Once
applications become more dependent on performance of the computer network they
need more detailed models to be able to express their requirements, and
closely monitor network performance.
In this article we provide an overview of some of the most used and well known
network models. It is our intention to guide the reader through a historical
journey that ultimately clarifies the need for new modeling approaches to
support the Future Internet. To this end we first look at network descriptions
in the history of the Internet in section 2. We then provide a categorisation
of network models (section 3).
Following our model categorisation we present management models (section 4),
monitoring models (section 5) and generic models (section 6). We also
introduce the existing Future Internet model (section 7). Section 8 provides
an overview and discussion on the current state of network models research. We
conclude the article with a summary and the upcoming research challenges in
section 9.
## 2 Historical Role of Network Descriptions
Figure 1: ARPANET logical map, March 1977, an example of early network models
Before delving into current network modelling efforts that aim to support the
Future Internet, it is helpful to understand the role and evolution that
network descriptions have had in the past years.
We will show that networks have evolved from the original packet-switched
architectures, to use optical circuit-switched designs to finally converge
towards the Future Internet hybrid models, i.e. networks offering both packet
and circuit switching services. We will also show that during this evolution
there is one constant requirement that has not changed: the need to exchange
information about the network topology. For packet-switched networks topology
information is needed for the operation of routing protocols, for circuit-
switched and hybrid networks it is required for the creation of dedicated
connections among end-points.
### Packet-switched networks
Topology descriptions have been used to support computer networking activities
since the start of the Internet. The most commonly used technique to capture a
network topology is of course a graphical representation. One of the obvious
drawback of this method is that it does not scale well as the network becomes
larger, making automated tools necessary. Fig. 1 shows an early representation
of the ARPANET[62]. This network started out with just four nodes in 1969, but
quickly grew larger. The figure shows a large network with many devices and
connections which is hard for humans to grasp in its entirety.
The ARPANET originally used Interface Message Processors (IMPs) to route
messages through the network[44]. These IMPs performed regular delay
measurements to all destinations, and then broadcasted the result. These
results were combined and then stored to function as a sort of distance-vector
protocol. Over the years the routing between IMPs was gradually improved,
until in 1983 the ARPANET switched over to TCP/IP.
Research on TCP/IP had already been going on during the seventies on several
test networks[26]. During this time the Routing Information Protoocol[45] was
also developed, implementing a distance-vector protocol. Distance-vector
protocols form an abstract view of the network, using the distance and general
direction as a way to select the forwarding interface. Similar to this is the
path-vector Border Gateway Protocol [53, 58] which rely on operator defined
paths in the network, serving most backbone networks in the Internet. These
protocols no longer need a complete picture of the network. Instead, each
router has a (different) aggregated view of the network, gathered from
exchanging aggregated information with others.
During the late 70s and early 80s several different link-state routing
protocols were developed, among them IS-IS and OSPF[57, 59]. Link-state
routing protocols broadcast messages containing the states of links, and where
they directly connect to. Traditionally this broadcasting is limited to
smaller areas, and not the whole network. Within an area, all routers do form
a complete view of the topology, and use this to calculate the shortest path
tree. Link-state networks are mostly used in local networks.
When relying on distance vector protocols full topology distribution is no
longer a requirement. However, having a full network description available is
still needed by link-state protocols and it can in general still be helpful in
monitoring or problem detection. In both cases the transfer of information
regarding the topology is done directly by the network nodes.
### Circuit-switched and hybrid networks
The methods to derive knowledge on the network topology we just described have
been driven by the routing protocols, and as such they are only applicable to
packet-switched networks. They are not very useful in the context of circuit-
switched or hybrid networks. Models better suited for these latter situations
have emerged in the past years.
For circuit-switched networks full topology distribution is still required. In
order to send data from a source to a destination in a circuit-switched
network, a circuit must be configured. In telephony networks dynamic
provisioning is achieved by using strict addressing, aggregated static routes,
and large capacity[46]. For circuits in data networks this is not feasible,
since there is no strict numbering plan, and the overall capacity compared to
the circuits is not that great.
Asynchronous Transfer Mode tried to merge the world of circuit-switched
networking with packet-switched networking. There the Private Network-to-
Network Interface [60] was used to relay topology information, and also
included some ideas on topology aggregations. In the end, ATM never became
very widely adopted, and is not currently in mainstream use.
GMPLS with its Path Computation Element[28] takes a different approach for
inter-domain path computation. Instead of sharing topology information, every
request in the network is broadcast to peers. The route of the request is
recorded and replied along the same path to implement a circuit reservation
request. While technically feasible, this approach poses problems as the
number of requests goes up. While GMPLS is implemted intra-domain, we have not
seen inter-domain deployments.
A different approach is seen in hybrid networking[29]. Many research and
educational networks are currently offering circuits on their own network, and
recently also started experimenting with inter-domain circuits[61]. Here the
topology of a domain needs to be exported in full or in an abstracted way to
the neighboring domains. The representation of the network needs to be
consistent and agreed upon, such that inter-domain circuit provisioning tools
can take decisions on how to engineer a circuit.
While the ARPANET and Internet have moved away from the need of full topology
exchange, the need for topology description and exchange has risen again for
optical networks. Subsequently, research on topology descriptions has also
seen a rise in the last few years. Many different models have been created and
published, yet there is no publication that shows an overview of the different
approaches.
## 3 Topology Categories
The historical perspective we just gave provides a sense of why models are
needed, and how they have been used concretely. But it is also useful to
categorise the various models in a more general way. We can, in fact, analyse
and compare different computing models suitable for Future Internet
infrastructures based on the following three features:
1. 1.
their purpose from an application perspective,
2. 2.
the range of infrastructure layers covered by the models,
3. 3.
the functional scope covered by the models.
An overview of these features and how they relate to each other is shown in
Figure 2, where we provide two main blocks, i.e application and future
internet infrastructures, and we position models in them according to their
characteristics.
Figure 2: Computing Infrastructure Models
From an application perspective, we distinguish between three different models
in terms of the type of application they support.
* •
_Management_ models are used in network-management applications or to restrict
actions that can be taken on a network.
* •
_Monitoring_ models are used for external applications to describe the dynamic
aspects of a computing infrastructure.
* •
_General_ infrastructure models are used for applications that require a
static view of computing infrastructures.
When starting from the infrastructure perspective, different models cover a
different range of layers in the infrastructure. In this paper we distinguish
between models that focus on a single technology layer of the infrastructure
and models that cover multiple layers of technology. We also identify two
different functional aspects of a Future Internet infrastructure that can be
covered by a model. Most of the models discussed are focused on the network
infrastructure that connects the different resources in the computing
infrastructure while other models also include computing and storage
capabilities of the Future Internet infrastructure.
Besides the content, we will also take the modeling approach into account for
comparing and analyzing different models. For this purpose we identify the
following types:
1. 1.
byte format, used in communication protocols and aimed at compact
descriptions;
2. 2.
database schema, used to describe the content of the database in which the
instances of the infrastructure are stored;
3. 3.
Unified Modeling Language, used to describe the classes and relations in an
object oriented model;
4. 4.
Extensible Markup Language (XML), used to provide a schema for the model and
syntax that is application and programming language independent;
5. 5.
Semantic-Web based models, i.e. models based on the Resource Description
Framework or the Web Ontology Language, used to provide semantic models of
future internet infrastructures.
Table 1: Overview of model characteristics. Model | Main Purpose | Scope | Type | Standard Organization | References
---|---|---|---|---|---
SNMP | Management | Network | DB schema | IETF | [25][71]
NetConf | Management | Network | XML | IETF | [34][35]
OSPF(-TE)/GMPLS | Management | Network | byte format | IETF | [54] [38]
CIM | Management | Network + Comp & Storage | UML + XML | DMTF | [33]
DEN-ng | Management | Network + Comp & Storage | UML | DMTF | [63]
perfSONAR/NMC | Monitoring | Network | XML | OGF | [22][43]
cNIS | Monitoring | Network | DB schema | - | [11][20]
MOMENT | Monitoring | Network + Comp. & Storage | OWL | - | [55]
G.805/G.809/G.800 | General | Network | None | ITU | [47, 48, 49, 31]
NDL | General | Network | RDF | - | [66][32][65]
NML | General | Network | XML + OWL | OGF | [67]
RSpec | Request | Comp & Network | XML | - | [9]
VxDL | Request | Comp & Network | XML | - | [50]
NDL-OWL | General | Network + Comp & Storage | OWL | - | [21, 69]
NOVI/GEYSERS/INDL | General | Network + Comp & Storage | OWL | - | [68][64][40][41]
In Table 1 we provide an overview of the models and their main characteristics
discussed in the following sections.
## 4 Management Models
Network management has used several different information models over the
years, and newer models are being proposed. These models are mainly used for
management of devices, or in protocols to exchange necessary topology
information. They are generally aimed at specific applications, the
information expressed in the protocols is not meant to be generically
available nor extensible.
### 4.1 SNMP
The Simple Network Management Protocol111Technically, the information model is
formed by the MIBs, Management Information Bases, and SNMP denotes the whole
set: protocol, information and data model.[25, 71] is a set of standards
describing a protocol, a database schema, and data objects. The whole suite
was originally created as a way of both monitoring and managing network
resources. In current networks it is mainly used for monitoring purposes.
Diagnostic, performance and configuration information of network devices can
be retrieved from the Management Information Base (MIB) of devices using
Simple Network Management Protocol (SNMP) messages. The MIB is a tree of name
– value pairs, which can be requested and changed. The values are restricted
to three different types of datatypes: integer, string and sequence of
datatypes. A large part of the MIB tree is standardised, but vendors also have
their own private part of the tree. This vendor space is used to store most
configuration and performance data of their devices in a proprietary format.
Virtually all networking devices support SNMP, with different levels of detail
in their MIB.
The network description provided by SNMP is distributed over the devices.
Depending on the layer the device is operating on, it may have a pointer
(address or identifier) to its neighbours on that layer. A view of the whole
topology can be created by combining the information gathered from all the
devices.
### 4.2 NetConf
The Internet Engineering Task Force has recently worked to replace SNMP with a
new standard, NetConf[34][35]. While SNMP uses its own protocol and only
allows for three data-types, NetConf uses XML, allowing for many more data-
types. NetConf defines a way of transporting monitoring data and change
requests over a small set of existing protocols. NetConf is aimed at
distributing diagnostic, performance and configuration information, but also
for managing devices. NetConf is currently being introduced in networking
devices.
As NetConf follows similar principles as its predecessor, the network
description provided by NetConf is similarly distributed over the managed
devices. Each device will have information about the neighbour it connects to
on the layer it operates on. The network topology can be created by combining
the information of the devices in the network.
### 4.3 GMPLS
GMPLS, Generalized Multi-Protocol Label Switching[54][38], is a protocol suite
developed by the IETF for the provisioning and management of label-switched
paths through multi-technology networks. It provides a unified control and
management plane for the management of multi-layer networks. Networking
devices use the Open Shortest Path First - Traffic Engineering (OSPF-TE)
protocol to exchange topology data with their neighbours. Devices broadcast
the received topology data to their other neighbours, so that in the end all
the devices in the domain have the same view of the network topology.
The topology data in OSPF-TE is exchanged in Link State Announcements packets
inside network domains. The topology data contained therein is encoded in a
compact byte format, using specifically defined header fields and Type-Length-
Value containers. This format is designed to be easy to process and store for
participating network devices, but it is hard to export to external
applications. The message format is somewhat extensible, there is specific
room for other applications to add data to the messages. The data must fit in
the Type-Length-Value container, and can be processed by agents participating
or listening to the OSPF-TE process.
Since OSPF-TE is only used intra-domain, there is no inter-domain exchange of
messages or information. In order to allow for inter-domain provisioning, the
Path Computation Element architecture [37] has been defined. Generalized
Multi-Protocol Label Switching (GMPLS) operators have expressed a desire to
keep network topology data confidential, so the path computation architecture
works by broadcasting requests, rather than by distributing topology
information[28][23].
### 4.4 CIM and DEN-ng
The Common Information Model (CIM)[33] is a network device information model
commonly used in enterprise settings. CIM is developed by the Distributed
Management Task Force[1] and it is an object-oriented information model
described using the Unified Modeling Language. This information model captures
descriptions of computer systems, operating systems, networks and other
related diagnostic information. CIM is a very broad and complex model, the
current UML schemata of the network model span over 40 pages, the total model
is over 200 pages.
A mapping from CIM to XML is also defined, which is mainly used in Web-Based
Enterprise Management. This is mainly implemented in enterprise-oriented
computing equipment, and operating systems such as Windows and Solaris. The
CIM model is highly expressive, and is still actively developped. There have
been many significant changes in the infrastructure part of CIM over the past
two years, both introducing new elements, as well as deprecating or changing
existing elements. The CIM model is capable of capturing the complete physical
setup, and almost everything with regards to the configuration of devices. The
model is capable of capturing the information with a very high level of
detail, yet provides almost no abstraction layer above this, making it very
hard to reason generically using this model.
A successor to CIM is the Directory Enabled Networking - next generation (DEN-
ng) model, Directory Enabled Networking – next generation[63], which extends
the CIM model also with description of business rules. The idea behind the
model is that with the right software, the business rules combined with the
capabilities of the devices can be automatically transformed into
configurations of firewalls, user restrictions, et cetera. This requires that
all configuration management is managed centrally, or at least by the same
tools.
## 5 Monitoring Models
The previous section provided an overview of management models, which are
usually aimed at specific tools for network and device management. Many
communities like to provide more generic access to monitoring data, so
monitoring models have been created. These models can take output from
different tools and combine them into a single model.
### 5.1 perfSONAR / NM and NMC
An early model for network topology description is the perfSONAR[22][43]
model. perfSONAR is a network monitoring architecture. It stores data from
different measurement tools which are then made available publicly. This is
particularly intended for inter-domain network connection debugging[70].
The perfSONAR architecture has been implemented by different partners,
providing two different, compatible implementations. The model has later been
brought to the Open Grid Forum (then Global Grid Forum)[7] for
standardisation. This resulted in the Network Measurements Working Group (NM-
WG)[6] which produced a standardised schema in 2009[3].
The NM-WG schema contains a base schema to describe network measurement tools,
and their results. There is also a time schema to accurately describe time
values in these measurements. Of particular interest here is the topology
schema, which provides a basic representation of network topologies using
hierarchical constructs in XML. This schema allows for a simple description of
domains, nodes, ports and their connections.
This schema is also used in the Inter-Domain Controller Protocol[30], which is
currently in use in many circuit provisioning tools, e.g. OSCARS[42]. The
OSCARS tool allows users to make circuit requests for the Energy Sciences
Network (ESnet[18]), and has also been implemented on the Internet2 ION
network[14].
The Network Measurement and Control WG has currently taken over the activities
of the activities of the NM-WG and is continuing development of the
measurements schema. The topology schema development has moved to the Network
Markup Language, which we discuss later.
### 5.2 cNIS and AutoBAHN
cNIS is the network topology description format for GÉANT network[19] and is
used as basis for the AutoBAHN[20] bandwidth on demand system. The data model
is implemented in a database schema[11]. This schema includes fixed
descriptions of a set of layers used in the GÉANT network, such as Ethernet,
and MPLS.
The AutoBAHN bandwidth on demand system at first started with the cNIS, but
later extended it towards their own model[24]. The AutoBAHN system uses a
Domain Manager which maintains the local topology. This Domain Manager does
automatic topology aggregation before exporting a topology to the Inter-Domain
Manager. Interestingly, the Inter-Domain Manager uses extensible OSPF messages
to exchange inter-domain topology information.
The Stitching Framework[36] is also a GÉANT activity, and it describes a
framework for ‘stitching’ together different technologies in bandwidth-on-
demand systems in a multi-domain and multi-layer environment. It provides a
framework to define the required information for creating connections across
multi-domain multi-layer networks. The Stitching Framework has been integrated
into the latest version of cNIS where it can stitch together the technologies
defined there. It should be noted that the Stitching Framework is built
generically, and could also be applied to other more expressive models.
### 5.3 Monitoring and Measurement Ontology
The perfSONAR and NM-WG work served as an important inspiration for the
Monitoring and Measurement Ontology (MOMENT) developed by ELTE[55]. This
ontology has taken the initial concepts from NM-WG and implemented them into
an Web Ontology Language (OWL)-based ontology. This ontology is mostly aimed
at measurement tools and results, which using their OWL ontology, can both be
expressed in great detail.
The ontology allows an application to describe the exact circumstances of a
measurement. For example that a traceroute command was performed at a certain
time, the parameters of that command, a description of the network at that
time, and the results of the command itself. These kinds of measurements can
then be recorded in a database, where they can be easily correlated and
analyzed using the generic description of the data.
The MOMENT ontology has served as a way of describing data for the ETOMIC[56]
infrastructure. This infrastructure consists of several nodes together forming
a network measurement virtual observatory. The OWL-based ontology then makes
it possible to easily share and reuse measurement data with others.
The experiences of the MOMENT ontology have been used also in the development
of the NOVI monitoring ontology.
## 6 General Models
In the previous sections described management and monitoring models, which are
aimed at management and monitoring applications respectively. Another category
is the set of general models, which aim to provide a more general description
of the network topology so that other applications can use them.
### 6.1 G.805, G.809 and G.800
A very generic set of models are the network models defined by the
International Telecommunication Union (ITU). These models are theoretical
models, in the sense that they have no explicit data model defined for them.
However they are important to discuss here as they have identified and defined
important terminology for network topology description, especially concerning
multi-layer networks.
In 2000 the ITU published the G.805 network model[47]. This model allows the
description of all kinds of transport networks, and especially different
layers and adaptations in that network. It is a very comprehensive, but also
complex model. A more readable introduction is available[31]. The G.805 model
allows the modelling of circuit-switched networks, and in 2003 the model was
extended in G.809[48] to also model connection-less networks. Then in 2007
these models were combined, along with some others into G.800: ‘Unified
functional architecture of transport networks’[49].
These models are very extensive and generic, allowing to describe any kind of
existing network, but also future network technologies. The models have
identified some fundamental concepts, such as:
* •
_Layers_ is defined as the set of connection points of the same technology,
* •
_Adaptations_ are the functions performed on data to transform it from one
layer to another,
* •
_Labels_ identify different flows of data in a Layer.
So as a simple example, VLAN tagged traffic is a specific Layer, the adding of
a VLAN tag to a packet is an Adaptation, and the VLAN tag is used to identify
a data flow among the other traffic.
However, G.805, G.809 and G.800 are only graphical models, there is no data
model underlying these information models, making them hard to use in
practice. The models do provide a very fundamental theoretical groundwork,
which is why NDL and NML have taken it as a source of inspiration.
### 6.2 Network Description Language
In 2006 the University of Amsterdam published a method of using RDF to
describe networks[66], called the Network Description Language (NDL). This
uses a simple model to describe devices, interfaces and their connections. The
descriptions would then be available to applications in a standard format. The
initial idea was also to apply the distributed description capability of the
semantic web, similar to the Friend of a Friend network[4]. This allows
networks to independently describe their network topologies and link them
together so that they together form a global description of the network.
The initial model of NDL (v1) was simple, and in some ways similar to the
model used by PerfSonar, but implemented in Resource Description Framework
(RDF). Using ideas from G.805 we extended NDL to version 2, which describes
multi-layer networks generically[32][65]. This model introduces a notation for
the G.805 concepts of Layers, Adaptations and Labels. This allows for
descriptions of any kind of network topology, ranging from physical networks
to completely virtualised networks, and also the relations between those
network layers.
NDL has been used as one of the models on which the Network Markup Language is
based, and also heavily influenced the design of the NOVI and GEYSERS
information models.
### 6.3 Network Markup Language
During 2007 efforts have been combined from PerfSonar, NM-WG and NDL to create
a standard network topology information model. A new working-group was formed
at the OGF called the Network Markup Language[2]. This group aims to create a
generic network model that can be used for describing measurements,
monitoring, describing topologies, and also requests.
The Network Markup Language (NML) schema describes networks using uni-
directional constructs. The unidirectional Port objects can be connected
together, externally through Links or internally through a Node’s
Crossconnect. The model also includes the capability of describing multi-layer
networks based on the ideas from G.805 and NDL as described earlier. The
unidirectional model causes the network model to be very verbose, however this
allows the model to be more generic, as a unidirectional model can describe
bidirectional networks, but vice versa this is not possible.
The standardisation process has recently resulted in the publication of the
first NML base schema[67]. To support different applications, NML has two
different data models, one in XML and one in OWL.
## 7 Future Internet Models
In recent years several initiatives have started to work on so-called Future
Internet platforms. Examples are the GENI[12] initiative in the United States,
and the FIRE[5] initiative in the Europe. From these several different
projects have started, which we discuss below.
#### 7.0.1 RSpec & RSpec v2
The GENI project[12] in the United States has been working on very large
distributed virtualization infrastructures, such as PlanetLab[27, 16], and
ProtoGENI[17]. These testbeds contain nodes distributed over different
locations, connected to the Internet, where users can request virtual machines
and conduct network experiments.
Initially PlanetLab developed the Slice-based Federated Architecture (SFA)
format to provide infrastructure and request descriptions. The first version
of this format have been defined in Resource Specification (RSpec)[9]. This
later evolved into ProtoGENI RSpec v2[10], which has been chosen as the
standard interchange format for PlanetLab, and all other Global Environment
for Network Innovations (GENI) platforms.
The RSpec v2 format is a simple XML based format geared towards the specific
use in virtual environments. It allows platforms and users to describe nodes,
their virtualisation properties, and a very limited form of network
connectivity. The format works very well with PlanetLab and compatible
systems, but it is very hard to use when describing any other kind of network
or infrastructure.
#### 7.0.2 Virtual private eXecution infrastructure Description Language
The Virtual private Execution infrastructure Description Language (VxDL) has
been developed by INRIA and Lyatiss[50, 15]. VxDL uses an XML syntax to
express infrastructure requests in varying levels of detail. Such a request
consists of four parts: a general description, a description of non-network
resources, a network topology, and the time interval for this reservation.
VxDL is used in GRID5000[13], the GEYSERS project (see section 7.2, as well as
a commercial product developed by Lyatiss.
### 7.1 Network Description Language OWL
RENCI[8], a GENI participant, has also built an infrastructure, called ORCA-
BEN[21, 69]. This infrastructure contains several locations with
virtualisation capabilities, and a completely controllable optical network. In
order to control and manage this they have extended NDLv2 to the OWL syntax,
creating Network Description Language (NDL-OWL). This also extends NDL with
more virtualisation and service description features to describe their
infrastructure. These descriptions are then used in the client software to
describe requests, but also in the management software to match the requests
with the available infrastructure.
The development of NDL-OWL and ORCA-BEN has been performed in the context of
the GENI project, which means that ORCA-BEN is able to communicate with other
GENI platforms, including platforms speaking RSpec v2. NDL-OWL is thus a
superset of RSpec v2.
### 7.2 NOVI, GEYSERS and INDL
The NOVI project aims to federate Future Internet platforms and one of the
challenges of the NOVI Information Model is to interact with different
platforms[68, 64]. Using NML in the information model provides the basis for
interaction between NOVI and the FEDERICA and PlanetLab platforms. Not only
does the information model have to map to concepts used in these platforms, it
also needs to be able to accommodate interaction with other platforms that may
be added to the federation in the future. By adding concepts from the MOMENT
ontology also to the NOVI ontologies, users can easily use monitoring tools
and data to get a comprehensive view of their requested infrastructure. The
NOVI ontology suite allows a complete semantic description of a Future
Internet federation. NOVI has ontologies for the infrastructure, but also for
monitoring tools and results, as well as policy aspects and rules. Of special
interest in the NOVI model is the _unit_ ontology, which generically describes
the units used for capacity, measurements, et cetera.
One of the key innovations of GEYSERS is to enable virtualisation of optical
infrastructures. The GEYSERS Information Modeling Framework (IMF), is
currently under development to provide an information model for the Logical
Infrastructure Composition Layer [39]. This layer is the element responsible
of managing physical resource virtualisation and composing Virtual
Infrastructures. These are then offered as a service within the GEYSERS
architecture.
The information models in both NOVI and GEYSERS are used to both describe the
infrastructure and also to allow users to express requests. Once an
infrastructure request is handled by either system, the result is also
described in the same information model and made available to the user. This
description can then also be used to correlate data from the active monitoring
tools.
These platforms show that infrastructure provisioning is a complex interplay
of different hardware and software tools, which benefits greatly from having
an interoperable semantic model to exchange information. These models combine
many aspects of the previous models, providing users with a single
semantically compatible model for describing requests, physical and virtual
infrastructure, as well as directly related monitoring information.
The Infrastructure and Network Description Language (INDL)[40, 41] is an
evolution of the Network Description Language, combined with the experiences
in NOVI and GEYSERS. We have taken the general model from NML, and added
capabilities to describe the virtualisation of nodes and infrastructure. The
model is actually not that different from the model in NOVI and GEYSERS, but
provides a more reusable model available for other Future Internet platforms.
## 8 Discussion
Figure 3: An overview of different information models
This article presented an overview of the current state-of-the-art of network
description models, with the goal to show how these models are suitable to the
needs of Future Internet platforms. Figure 3 shows an overview of the
described information models and how they have influenced each other. This
figure groups the models by intended usage: at the top of the figure we have
the models more related to management; we then show the monitoring models, and
below them the general models. The Future Internet models which combine the
ideas of the monitoring and general models to form a complete ontology for
future infrastructures are on the bottom and right side of the figure.
The information models described in section 4 are aimed at describing purely
functional topology and diagnostic information, making these management
models. For example the GMPLS information model is aimed switches and routers.
The data model is designed for compactness and is therefore not easy for other
applications to understand, nor is it human readable. CIM and DEN-ng are also
management models, albeit at a higher level, combining all the information of
different low-level management models. This creates an aggregated management
model at the enterprise level. The information models in these categories are
both aimed at management, informing the direct operators of those machines.
These management models are aimed specifically at a single task, which they
perform very well. The models are used in isolated contexts and domains, and
the models are not generic enough to be used in applications not specifically
aimed at these contexts. Most of the other information models described in
this article have some form of an XML data model and are thus more generally
usable.
The monitoring models, PerfSonar/NM-WG, cNIS and MOMENT, have been defined
specifically to capture data from many different tools, and store and share
them in a generic way. These monitoring models are targeted at capturing
monitoring information, network measurement data along with topology data of
those measurements. Unlike the management models, the measurement models aim
to make the data as portable as possible so that different tools and
applications can interpret the data, instead of a single management
application. The network topology description elements of these models support
the description of results, and are not that advanced in describing different
technologies, or the dynamics between the technologies.
The general models are aimed specifically at describing network topologies.
The initial model of NDL was also not capable of describing multi-layer
networks, but this changed due the influence of the ITU G.805, G.809 and G.800
models. The ITU models have very clearly identified and extensively defined a
terminology for multi-layer networks. Using the generic (de)adaptation and
labelling concepts it becomes possible to describe any kind of technology,
without being dependent on a predefined notation for that technology. The NDL,
and NML models aim at generic network descriptions that can be extended or
embedded in other models. The intention of the generic models is to provide
applications using the data enough information to act on the network, either
by provisioning circuits or by adapting the applications behavior to the
capabilities of the network.
The general models have been very influential in the creation of most models
of the Future Internet. The initial models, SFA and VxDL, created for the
Future Internet have been limited models to allow users to easily describe
their requests for virtual infrastructures. The later Future Internet models,
NDL-OWL, NOVI, GEYSERS and INDL, have built on both these simpler request-like
models, as well as the general models to support the management of the Future
Internet testbeds. This support is both for users in clearly defining their
requests and the resulting topology. But the model also supports the
management of the testbed to describe in a single model the physical
resources, as well as the reserved virtual infrastructures.
The semantic web nature of the general models allow them to be easily
incorporated in other models. Which is what we see happening somewhat in the
NDL-OWL model, but even more so in the NOVI and GEYSERS models. The have taken
the basic network models of NDL/NDL-OWL/NML and extended these ideas towards
virtual infrastructures and also adapted the request models to form a single
information model for Future Internet infrastructures. The NOVI model takes
this another step further by also integrating the MOMENT monitoring and
measurement ontology, forming a complete semantic network model.
## 9 Conclusion
Our article documents a clear evolution in the modeling of networks and
infrastructure toward supporting Future Internet operations.
On one hand we have shown that management models have changed less, given they
are all aimed at specific applications, and target very specific use-cases or
tools. The hardware or chosen management software limits the choice for an
information model in this case.
On the other hand, monitoring, general, and Future Internet models have all
evolved significantly. The evolution we have documented shows that from
several different initiatives at first, there has been a convergence on the
newly defined Network Markup Language standard. Many of the models were of
direct influence to NML, so the standard is suitable for use in monitoring,
provisioning as well as request modelling. The Future Internet models we are
interested in have taken NML as their base model and extended it where
necessary to describe resources beyond the network topologies. These
extensions are also again converging in an extended model, INDL.
### 9.1 Challenges for Network Models
Computer networks have become complex systems over the years and interactions
with the network, especially circuit switched networks, should not be taken
for granted. Our overview of the different models we presented demonstrates
that creating an information model for computer networks is not a simple feat.
This is even more true for Future Internet platforms: there, networking is
becoming more and more ubiquitous and more integrated in the computing-storage
fabric, making the management of computer networks a much more difficult task.
We have identified three challenging areas for network models in the coming
years:
* •
handling abstraction levels appropriately;
* •
managing multi-domain communication and path setups;
* •
integration with computing-networking-storage planning services.
In 35 years we have moved from a situation where the entire Internet could be
captured in a single figure (see Figure 1) to a situation where we are running
out of IPv4 address, with many more devices hidden behind NAT solutions.
Network management has no choice but to move with this pace, requiring higher
abstraction levels. Network information models are a necessary prerequisite
for creating these abstraction levels. Current models do not adequately handle
different abstraction levels in the same models.
Network descriptions are important in supporting path selection tools.
Consider the architecture described by Lehman et al.[52] which points to the
fact that an interoperable inter-domain topology description is necessary in
order to allow path selection for multi-domain multi-layer circuit-based
networking. Path selection in single layer networks is trivial, however in
multi-layer networks it is much harder, and often NP-complete[51]. The generic
way of describing network technologies enabled by the abstract models of G.805
and G.809 makes it possible to create generic path selection algorithms which
will be able to handle many if not all existing and future network
technologies. The way that network topologies are represented are an important
factor in supporting the path selection process.
The problem of multi-layer path selection has many similarities with matching
requests with (virtual) infrastructures. The nodes and services that are part
of the request can be seen as special kinds of links connected to the network,
similar to multi-layer network requests. By using generic models the
application can choose to solve this problem directly, or it can choose to
carve the problem up and delegate subproblems to the relevant planning
services. This will lead the way towards a complete Future Internet
infrastructure.
## Acknowledgments
This research was financially supported by SURFnet in the GigaPort-NG Research
on Networks project and the Dutch national program COMMIT.
## List of Abbreviations
BGP
Border Gateway Protocol
CIM
Common Information Model
DEN-ng
Directory Enabled Networking - next generation
DMTF
Distributed Management Task Force
FIRE
Future Internet Research and Experimentation
GENI
Global Environment for Network Innovations
GEYSERS
Generalized Architecture for Dynamic Infrastructure ServicesAn FP7 EU Project
GLIF
Global Lambda Integrated Facility
GMPLS
Generalized Multi-Protocol Label Switching
IEEE
Institute of Electrical and Electronics Engineers
IETF
Internet Engineering Task Force
IS-IS
Intermediate System to Intermediate System
ITU-T
Telecommunication Standardization Sector (coordinates standards on behalf of the ITU)
ITU
International Telecommunication Union
MIB
Management Information Base
MOMENT
Monitoring and Measurement Ontology
NDL-OWL
Network Description Language OWL
NDL
Network Description Language
NM-WG
Network Measurements Working Group
NMC
Network Measurement and Control WG
NML
Network Markup Language
NOVI
Networking Over Virtualised InfrastructuresAn FP7 EU Project
OGF
Open Grid Forum
OSPF-TE
Open Shortest Path First - Traffic Engineering (An extension of OSPF)
OSPF
Open Shortest Path First
OWL
Web Ontology Language
PNNI
Private Network-to-Network Interface
RDF
Resource Description Framework
RFC
Request For Comments (an Internet Engineering Task Force (IETF) memorandum on Internet systems and standards)
RIP
Routing Information Protoocol
RSpec
Resource Specification
SFA
Slice-based Federated Architecture
SNMP
Simple Network Management Protocol
VxDL
Virtual private Execution infrastructure Description Language
INDL
Infrastructure and Network Description Language
## References
* [1] Distributed Management Task Force (DMTF), 2006.
* [2] The network markup language, 2007.
* [3] An extensible schema for network measurement and performance data. Technical report, Open Grid Forum, 2009.
* [4] Friend of a friend (FOAF) project, 2010.
* [5] Future internet research and experimentation, 2010.
* [6] OGF Network Measurement Working Group (NMWG), 2010.
* [7] Open grid forum (ogf), 2010.
* [8] RENCI (renaissance computing institute), 2010.
* [9] Rspec, 2011.
* [10] Slice federation architecture, 2011.
* [11] cnis database documentation, 2012.
* [12] GENI project, 2012.
* [13] Grid5000, 2012.
* [14] Internet2 interoperable on-demand network (ION), 2012.
* [15] Lyatiss resources, 2012.
* [16] Planetlab, 2012.
* [17] Protogeni, 2012.
* [18] Energy science network (ESnet), 2013.
* [19] GÉANT project website, 2013.
* [20] G. Alyfantis, M. Balcerkiewicz, J. Łukasik, R.Krzywania, and G. Priggouris. Technical documentation for the inter- domain bod service manager (idm). Technical Report DJ3.4.1,2, GÉANT2, 2007.
* [21] I. Baldine, Y. Xin, D. Evans, C. Heerman, J. Chase, V. Marupadi, and A. Yumerefendi. The missing link: Putting the network in networked cloud computing. In International Conference on the Virtual Computing Initiative (ICVCI 2009), October 2009.
* [22] J.W. Boote, E.L. Boyd, J. Durand, A. Hanemann, L. Kudarimoti, R. Łapacz, N. Simar, and S. Trocha. Towards multi-domain monitoring for the European research networks. Computational Methods in Science and Technology, 11(2):91–100, 2005\.
* [23] R. Bradford, JP. Vasseur, and A. Farrel. Preserving Topology Confidentiality in Inter-Domain Path Computation Using a Path-Key-Based Mechanism. RFC 5520 (Proposed Standard), April 2009.
* [24] Mauro Campanella, Radek Krzywania, Afrodite Sevasti, and Stella-Maria Thomas. Functional specification and design of a generic domain-centric bandwidth on demand service manager. Technical Report DJ3.3.4, GÉANT2, 2008.
* [25] J. Case, R. Mundy, D. Partain, and B. Stewart. Introduction and Applicability Statements for Internet-Standard Management Framework. RFC 3410 (Informational), December 2002.
* [26] V. Cerf and R. Kahn. A Protocol for Packet Network Intercommunication. IEEE Transactions on Communications, 22(5):637–648, May 1974.
* [27] Brent Chun, David Culler, Timothy Roscoe, Andy Bavier, Larry Peterson, Mike Wawrzoniak, and Mic Bowman. Planetlab: an overlay testbed for broad-coverage services. SIGCOMM Comput. Commun. Rev., 33(3):3–12, July 2003.
* [28] S. Dasgupta, J. C. de Oliveira, and J. P. Vasseur. Path-Computation-Element-Based Architecture for Interdomain MPLS/GMPLS Traffic Engineering: Overview and Performance. Network, IEEE, 21(4):38–45, July 2007.
* [29] Cees de Laat, Erik Radius, and Steven Wallace. The rationale of the current optical networking initiatives. Future Generation Computer Systems, 19(6):999–1008, August 2003\.
* [30] DICE. Interdomain controller protocol, 2010.
* [31] Freek Dijkstra, Bert Andree, Karst Koymans, and Jeroen van der Ham. Introduction to ITU-T recommendation G.805. Technical Report UVA-SNE-2007-01, Unversiteit van Amsterdam, December 2007\.
* [32] Freek Dijkstra, Bert Andree, Karst Koymans, Jeroen van der Ham, and Cees de Laat. A multi-layer network model based on itu-t g.805. Computer Networks, June 2007.
* [33] Distributed Management Task Force DMTF. Common Information Model (CIM).
* [34] R. Enns. NETCONF Configuration Protocol. RFC 4741 (Proposed Standard), December 2006. Obsoleted by RFC 6241.
* [35] R. Enns, M. Bjorklund, J. Schoenwaelder, and A. Bierman. Network Configuration Protocol (NETCONF). RFC 6241 (Proposed Standard), June 2011.
* [36] Alberto Escolano, Andrew Mackarel, Damir Regvart, Victor Reijs, Guy Roberts, and Hrvoje Popovski. Report on testing of technology stitching. Technical Report DJ3.5.3, GÉANT2, 2007.
* [37] A. Farrel, J.-P. Vasseur, and J. Ash. A Path Computation Element (PCE)-Based Architecture. RFC 4655 (Informational), August 2006.
* [38] Adrian Farrel and Igor Bryskin. GMPLS: Architecture and Applications. Morgan Kaufmann, first edition, 2006.
* [39] Joan A. Garcia-Espin, Jordi Ferrer Riera, Sergi Figuerola, Mattijs Ghijsen, Yuri Demchenko, Jens Buysse, Marc de Leenheer, Chris Develder, Fabienne Anhalt, and Sebastien Soudan. Logical Infrastructure Composition Layer, the GEYSERS Holistic Approach for Infrastructure Virtualisation. In Terena Networking Conference (TNC2012), 2012.
* [40] M. Ghijsen, J. van der Ham, P. Grosso, and C. de Laat. Towards an Infrastructure Description Language for Modeling Computing Infrastructures. In Parallel and Distributed Processing with Applications (ISPA), 2012 IEEE 10th International Symposium on, pages 207–214. IEEE, July 2012.
* [41] M. Ghijsen, J. van der Ham, P. Grosso, Cosmin Dumitru, Hao Zhu, Zhiming Zhao, and C. de Laat. A semantic-web approach for modeling computing infrastructures. Technical Report UVA-SNE-2013-01, University of Amsterdam, SNE group, March 2013.
* [42] Chin Guok, D. Robertson, M. Thompson, J. Lee, B. Tierney, and W. Johnston. Intra and Interdomain Circuit Provisioning Using the OSCARS Reservation System. In Broadband Communications, Networks and Systems, 2006. BROADNETS 2006. 3rd International Conference on, pages 1–8. IEEE, October 2006\.
* [43] Andreas Hanemann, Jeff Boote, Eric Boyd, Jérôme Durand, Loukik Kudarimoti, Roman Łapacz, Martin Swany, Szymon Trocha, and Jason Zurawski. Perfsonar: A service oriented architecture for multi-domain network monitoring. In Boualem Benatallah, Fabio Casati, and Paolo Traverso, editors, Service-Oriented Computing - ICSOC 2005, volume 3826 of Lecture Notes in Computer Science, pages 241–254. Springer Berlin / Heidelberg, 2005\. 10.1007/11596141_19.
* [44] F. E. Heart, R. E. Kahn, S. M. Ornstein, W. R. Crowther, and D. C. Walden. The interface message processor for the ARPA computer network. In Proceedings of the May 5-7, 1970, spring joint computer conference on - AFIPS ’70 (Spring), page 551, New York, New York, USA, May 1970\. ACM Press.
* [45] C.L. Hedrick. Routing Information Protocol. RFC 1058 (Historic), June 1988. Updated by RFCs 1388, 1723.
* [46] International Telecommunication Union (ITU). Signalling network functions and messages. Recommendation ITU-T Q.704, International Telecommunication Union (ITU), July 1996.
* [47] International Telecommunication Union (ITU). Generic functional architecture of transport networks. Recommendation ITU-T G.805, International Telecommunication Union (ITU), March 2000.
* [48] International Telecommunication Union (ITU). Functional architecture of connectionless layer networks. Recommendation ITU-T G.809, International Telecommunication Union (ITU), 2003.
* [49] International Telecommunication Union (ITU). Unified functional architecture of transport networks. Recommendation ITU-T G.800, International Telecommunication Union (ITU), 2007.
* [50] Guilherme Piegas Koslovski, Pascale Vicat-Blanc Primet, and Andrea Schwertner Charão. Vxdl: Virtual resources and interconnection networks description language. In Networks for Grid Applications, volume 2 of Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, pages 138–154. Springer Berlin Heidelberg, 2009\.
* [51] Fernando Kuipers and Freek Dijkstra. Path selection in multi-layer networks. Computer Communications, 32(1):78 – 85, 2009.
* [52] T. Lehman, Xi Yang, N. Ghani, Feng Gu, Chin Guok, I. Monga, and B. Tierney. Multilayer networks: an architecture framework. Communications Magazine, IEEE, 49(5):122 –130, may 2011.
* [53] K. Lougheed and Y. Rekhter. Border Gateway Protocol (BGP). RFC 1105 (Experimental), June 1989. Obsoleted by RFC 1163.
* [54] E. Mannie. Generalized Multi-Protocol Label Switching (GMPLS) Architecture. RFC 3945 (Proposed Standard), October 2004. Updated by RFC 6002.
* [55] Peter Mátray, I. Csabai, and P. Hága. A semantic extension of the network measurement virtual observatory. 2009\.
* [56] Peter Matray, Istvan Csabai, Peter Haga, Jozsef Steger, Laszlo Dobos, and Gabor Vattay. Building a prototype for network measurement virtual observatory. In Proceedings of the 3rd annual ACM workshop on Mining network data, MineNet ’07, pages 23–28, New York, NY, USA, 2007. ACM.
* [57] John M. McQuillan, Ira Richer, and Eric C. Rosen. An overview of the new routing algorithm for the ARPANET. In Proceedings of the sixth symposium on Data communications - SIGCOMM ’79, pages 63–68, New York, New York, USA, November 1979. ACM Press.
* [58] D. Meyer and K. Patel. BGP-4 Protocol Analysis. RFC 4274 (Informational), January 2006.
* [59] Radia Perlman. A comparison between two routing protocols: Ospf and is-is. Ieee Network, 5(5):18–24, 1991.
* [60] Private network-network interface specification. Technical report, ATM Forum, 1996.
* [61] Guy Roberts, Tomohiro Kudoh, Inder Monga, Jerry Sobieski, and John Vollbrecht. Network Services Framework v1.0 , March 2010.
* [62] Peter H. Salus. Casting the Net: From ARPANET to Internet and Beyond… Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1995.
* [63] J. Strassner. DEN-ng: achieving business-driven network management. In NOMS 2002. IEEE/IFIP Network Operations and Management Symposium. ’ Management Solutions for the New Communications World’(Cat. No.02CH37327), pages 753–766. IEEE, August 2002.
* [64] Jeroen van der Ham, Mauro Campanella, Alejandro Chuang, Fabio Farina, Paola Grosso, Yiannos Kryftis, Péter Mátray, Alvaro Monje, Chrysa Papagianni, Chariklis Pittaras, Celia Velayos József Stéger, Adianto Wibisono, and Klaas Wierenga. D2.2: First information and data models. Technical report, NOVI Consortium, October 2011.
* [65] Jeroen van der Ham, Freek Dijkstra, Paola Grosso, Ronald van der Pol, Andree Toonk, and Cees de Laat. A distributed topology information system for optical networks based on the semantic web. Optical Switching and Networking, 5(2–3):85 – 93, 2008. ¡ce:title¿Advances in IP-Optical Networking for IP Quad-play Traffic and Services¡/ce:title¿.
* [66] Jeroen van der Ham, Freek Dijkstra, Franco Travostino, Hubertus Andree, and Cees de Laat. Using rdf to describe networks. Future Generation Computer Systems, Feature topic iGrid 2005, October 2006.
* [67] Jeroen van der Ham, Freek Dijkstra, Roman Łapacz, and Jason Zurawski. Network Markup Language Base Schema version 1, June 2013.
* [68] Jeroen van der Ham, Chrysa Papagianni, Jozsef Steger, Peter Matray, Yiannos Kryftis, Paola Grosso, and Leonidas Lymberopoulos. Challenges of an information model for federating virtualized infrastructures. In 5th International DMTF Academic Alliance Workshop on Systems and Virtualization Management: Standards and the Cloud, 2011.
* [69] Yufeng Xin, I. Baldine, J. Chase, T. Beyene, B. Parkhurst, and A. Chakrabortty. Virtual smart grid architecture and control framework. In Smart Grid Communications (SmartGridComm), 2011 IEEE International Conference on, pages 1 –6, oct. 2011.
* [70] M. Yampolskiy and M.K. Hamm. Management of multidomain end-to-end links - a federated approach for the pan-european research network geant 2. In Integrated Network Management, 2007. IM ’07. 10th IFIP/IEEE International Symposium on, pages 189 –198, may 2007.
* [71] Dave Zeltserman. Practical Guide to SNMPv3 and Network Management. Prentice Hall PTR, 1999.
*[IS-IS]: Intermediate System to Intermediate System
*[OSPF]: Open Shortest Path First
*[GMPLS]: Generalized Multi-Protocol Label Switching
*[SNMP]: Simple Network Management Protocol
*[IETF]: Internet Engineering Task Force
*[CIM]: Common Information Model
*[DMTF]: Distributed Management Task Force
*[DEN-ng]: Directory Enabled Networking - next generation
*[NMC]: Network Measurement and Control WG
*[OGF]: Open Grid Forum
*[MOMENT]: Monitoring and Measurement Ontology
*[OWL]: Web Ontology Language
*[ITU]: International Telecommunication Union
*[NDL]: Network Description Language
*[RDF]: Resource Description Framework
*[NML]: Network Markup Language
*[RSpec]: Resource Specification
*[VxDL]: Virtual private Execution infrastructure Description Language
*[NDL-OWL]: Network Description Language
*[MIB]: Management Information Base
*[OSPF-TE]: Open Shortest Path First - Traffic Engineering (An extension of )
*[NM-WG]: Network Measurements Working Group
*[NOVI]: Networking Over Virtualised InfrastructuresAn FP7 EU Project
*[GEYSERS]: Generalized Architecture for Dynamic Infrastructure ServicesAn FP7 EU Project
*[GENI]: Global Environment for Network Innovations
*[FIRE]: Future Internet Research and Experimentation
*[SFA]: Slice-based Federated Architecture
*[]:
*[INDL]: Infrastructure and Network Description Language
|
arxiv-papers
| 2014-02-17T10:15:16 |
2024-09-04T02:49:58.293972
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jeroen van der Ham and Mattijs Ghijsen and Paola Grosso and Cees de\n Laat",
"submitter": "Jeroen van der Ham PhD",
"url": "https://arxiv.org/abs/1402.3951"
}
|
1402.4015
|
# Diagrammatic Monte Carlo study of the Fermi polaron in two dimensions
Jonas Vlietinck Department of Physics and Astronomy, Ghent University,
Proeftuinstraat 86, 9000 Gent, Belgium Jan Ryckebusch Department of Physics
and Astronomy, Ghent University, Proeftuinstraat 86, 9000 Gent, Belgium Kris
Van Houcke Department of Physics and Astronomy, Ghent University,
Proeftuinstraat 86, 9000 Gent, Belgium Laboratoire de Physique Statistique,
Ecole Normale Supérieure, UPMC, Université Paris Diderot, CNRS, 24 rue
Lhomond, 75231 Paris Cedex 05, France
###### Abstract
We study the properties of the two-dimensional Fermi polaron model in which an
impurity attractively interacts with a Fermi sea of particles in the zero-
range limit. We use a diagrammatic Monte Carlo (DiagMC) method which allows us
to sample a Feynman diagrammatic series to very high order. The convergence
properties of the series and the role of multiple particle-hole excitations
are discussed. We study the polaron and molecule energy as a function of the
coupling strength, revealing a transition from a polaron to a molecule in the
ground state. We find a value for the critical interaction strength which
complies with the experimentally measured one and predictions from variational
methods. For all considered interaction strengths, the polaron $Z$ factor from
the full diagrammatic series almost coincides with the one-particle-hole
result. We also formally link the DiagMC and the variational approaches for
the polaron problem at hand.
###### pacs:
05.30.Fk, 03.75.Ss, 02.70.Ss
## I Introduction
Experiments with ultracold gases are a powerful tool to investigate the
(thermo)dynamics of quantum many-body systems under controlled circumstances.
With Feshbach resonances feshbach , for example, one has the ability to tune
the interaction strength. Optical potentials potential can be exploited to
modify the dimensionality of the studied systems. The properties of a single
impurity that interacts strongly with a background gas, for example, can be
addressed with ultracold atoms.
The so-called Fermi polaron problem refers to a single spin-down impurity that
is coupled to a non-interacting spin-up Fermi sea (FS). This problem
corresponds to the extreme limit of spin imbalance in a two-component Fermi
gas Partridge06 ; Shin08 ; Nascim09 and has implications on the phase diagram
of the strongly spin-polarized Fermi gas Pilati08 ; pietro ; Bertaina . At
weak attraction, one expects a “polaron” state Chevy06 , in which the impurity
is dressed with density fluctuations of the spin-up Fermi gas. Recent
experiments have observed indications of a transition from this polaronic
state to a molecular state (a two-body bound state of the impurity and an atom
of the sea) upon increasing the attraction strength in three dimensions (3D)
polaronMIT and in two dimensions (2D) kos . Experimentally, the 2D regime can
be accomplished by means of a transverse trapping potential
$V(z)=\frac{1}{2}m\omega_{z}^{2}z^{2}$ (here, $\omega_{z}$ is the frequency
and $z$ is the transverse direction) that fulfills the condition
$k_{B}T\ll\epsilon_{F}\ll\hbar\omega_{z}$ ($T$ is the temperature and
$\epsilon_{F}$ is the Fermi energy of the FS). When excitations in the $z$
dimension are possible, one reaches the so-called quasi-2D regime jesper2 ;
dyke . The purely 2D limit is reached for
$\epsilon_{F}/\hbar\omega_{z}\rightarrow 0$ and will be the subject of this
paper.
The existence of a polaron-molecule transition in 3D has been predicted with
the aid of the diagrammatic Monte Carlo (DiagMC) method polaron1 ; polaron2 ;
polaron and of variational methods Chevy06 ; Punk09 ; Combescot09 ; Mora .
For the latter, the maximum number of particle-hole (p-h) excitations of the
FS is limited to one or two Chevy06 ; Punk09 ; Combescot09 ; Mora . One might
naively expect that the role of quantum fluctuations increases in importance
with decreasing dimensionality and that high-order p-h excitations could
become more important in one and two dimensions. For the one-dimensional (1D)
Fermi polaron the known analytical solution displays no polaron-molecule
transition oneDexact . Like for the 3D polaron, the approximate method in
which the truncated Hilbert space contains one p-h and two p-h excitations of
the FS gives results for the 1D polaron approaching the exact solution oneD1 ;
oneD2 . In 2D, the Fermi polaron properties have been studied with variational
wave functions Zollner11 ; Parish11 ; Levinsen13 . To observe a polaron-
molecule transition in 2D it is crucial to include particle-hole excitations
in both the polaron and molecule wave functions Parish11 . In the limit of
weak interactions, the 1p-h and 2p-h variational Ansätze for the polaron
branch provide similar results. Surprisingly, this is also the situation for
strong correlations Levinsen13 .
In this work we focus on the 2D Fermi polaron for attractive interactions and
study the role of multiple particle-hole (mp-h) excitations for the ground-
state properties of the system. The quasiparticle properties of the polaron
are computed with the DiagMC method. This technique evaluates stochastically
to high order a series of Feynman diagrams for the one-particle and two-
particle self-energies. For the details of the DiagMC method and the adopted
method for determining the ground-state energies from the computed self-
energies, we refer to Refs. polaron ; polaron2 . In this work we present
DiagMC predictions for the interaction-strength dependence of the polaronic
and molecular ground-state properties in 2D. We first briefly discuss the
model and the diagrammatic method. We then discuss the results of the
simulations, with particular emphasis on the role of the mp-h excitations. We
also discuss how variational results for the polaron problem can be obtained
within the DiagMC formalism.
## II Formalism
We consider a two-component Fermi gas confined to 2D at temperature $T=0$.
Even though we will consider the zero-range interaction in continuous space,
we start from a lattice model to avoid ultraviolet divergences from the onset.
The corresponding Hamiltonian reads
$\hat{H}=\sum_{\mathbf{k}\in\mathcal{B},\sigma=\uparrow\downarrow}\epsilon_{\mathbf{k}\sigma}~{}\hat{c}^{\dagger}_{\mathbf{k}\sigma}\hat{c}^{\phantom{\dagger}}_{\mathbf{k}\sigma}\\\
+g_{0}\sum_{\mathbf{r}}b^{2}~{}\hat{\Psi}^{\dagger}_{\uparrow}(\mathbf{r})\hat{\Psi}^{\dagger}_{\downarrow}(\mathbf{r})\hat{\Psi}^{\phantom{\dagger}}_{\downarrow}(\mathbf{r})\hat{\Psi}^{\phantom{\dagger}}_{\uparrow}(\mathbf{r})\;,$
(1)
with $\hat{\Psi}^{\phantom{\dagger}}_{\sigma}(\mathbf{r})$ and
$\hat{c}^{\phantom{\dagger}}_{\mathbf{k},\sigma}$ being the operators for
annihilating a spin-$\sigma$ fermion with mass $m_{\sigma}$ and dispersion
$\epsilon_{\mathbf{k}\sigma}=k^{2}/2m_{\sigma}$ in position and momentum
space. The components of the position vector $\mathbf{r}$ are integer
multiples of the finite lattice spacing $b$. Further, $g_{0}$ is the bare
interaction strength. The wave vectors $\mathbf{k}$ are in the first Brillouin
zone $\mathcal{B}=]-\pi/b,\pi/b]$. The continuum limit is reached for
$b\rightarrow 0$. We adopt the convention $\hbar=1$ and consider the mass-
balanced case $m_{\uparrow}=m_{\downarrow}=m$. lattice with spacing $b$. We
make use of the $T$ matrix landau for a single spin-$\uparrow$ and
spin-$\downarrow$ fermion in vacuum,
$-\frac{1}{g_{0}}=\frac{1}{\mathcal{V}}\sum_{\mathbf{k}\in\mathcal{B}}\frac{1}{\varepsilon_{B}+\epsilon_{\mathbf{k}\uparrow}+\epsilon_{\mathbf{k}\downarrow}}\;,$
(2)
where $\mathcal{V}$ is the area of the system and $\varepsilon_{B}$ is the
two-body binding energy [which depends on $m$, $g_{0}$, and $b$ and
$\varepsilon_{B}(m,g,b)>0$] of a weakly bound state. Such a state always
exists for an attractive interaction in 2D. With the above relation we
eliminate the bare interaction strength $g_{0}$ in favor of the quantity
$\varepsilon_{B}$. Moreover, the diagrammatic approach allows us to take the
continuum limit $b\to 0$ and $g_{0}\to 0^{-}$ while keeping $\varepsilon_{B}$
fixed. Summing all ladder diagrams gives a partially dressed interaction
vertex $\Gamma^{0}$:
$\mbox{\raisebox{0.0pt}{\psfig{scale=0.3,clip={true}}}~{}}\;,$ (3)
where the dot represents the bare interaction vertex $g_{0}$ and the lines
represent bare-particle propagators for the spin-down impurity (dashed lines)
and the spin-up Fermi sea (solid lines). In momentum-imaginary frequency this
graphical representation corresponds to
$[\Gamma^{0}(p,i\Omega)]^{-1}=g_{0}^{-1}-\Pi^{0}(p,i\Omega)\;,$ (4)
with
$\displaystyle\Pi^{0}(p,i\Omega)=\frac{1}{\mathcal{V}}\sum_{\mathbf{k}\in\mathcal{B}}\frac{H(|\frac{\mathbf{p}}{2}+\mathbf{k}|-k_{F})}{i\Omega-\epsilon_{\frac{\mathbf{p}}{2}-\mathbf{k}\downarrow}-\epsilon_{\frac{\mathbf{p}}{2}+\mathbf{k}\uparrow}+\mu+\varepsilon_{F}}\;,$
(5)
with $H(x)$ being the Heaviside step function and $\mu<0$ being a free
parameter representing an energy offset of the impurity dispersion. Further,
$k_{F}$ and $\varepsilon_{F}=\frac{k_{F}^{2}}{2m}$ are the Fermi momentum and
the Fermi energy of the spin-up sea. The combination of Eqs. (2) and (4) gives
$\displaystyle\frac{1}{\Gamma^{0}(p,i\Omega)}=$ $\displaystyle-$
$\displaystyle\frac{1}{\mathcal{V}}\sum_{\mathbf{k}\in\mathcal{B}}\left[\frac{1}{\varepsilon_{B}+\epsilon_{\mathbf{k}\uparrow}+\epsilon_{\mathbf{k}\downarrow}}\right.$
(6) $\displaystyle+$
$\displaystyle\left.\frac{H(|\frac{\mathbf{p}}{2}+\mathbf{k}|-k_{F})}{i\Omega-\epsilon_{\frac{\mathbf{p}}{2}-\mathbf{k}\downarrow}-\epsilon_{\frac{\mathbf{p}}{2}+\mathbf{k}\uparrow}+\mu+\varepsilon_{F}}\right]\;.$
The relevant parameter that characterizes the interaction in Equation. (6) is
$\epsilon_{B}$. Eq. (6) is well defined in the thermodynamic and $b\to 0$
limits. One finds
$\displaystyle\frac{1}{\Gamma^{0}(p,i\Omega)}=\frac{m}{4\pi}~{}{\rm
ln}\bigg{[}\frac{2\varepsilon_{B}}{-z+\sqrt{(z-\epsilon_{\mathbf{p}})^{2}-4\varepsilon_{F}\epsilon_{\mathbf{p}}}}\bigg{]}\;,$
(7)
with $z\equiv i\Omega+\mu-\varepsilon_{F}$. In deriving the above expression
for $\Gamma^{0}(p,i\Omega)$ we have taken $\mu<-\varepsilon_{F}$. Since
Feynman diagrams for the self-energy will be evaluated in the momentum-
imaginary-time representation $(p,\tau)$, we need to evaluate the Fourier
transform
$\Gamma^{0}(p,\tau)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}~{}d\Omega~{}e^{-i\Omega\tau}~{}\Gamma^{0}(p,i\Omega)\;.$
(8)
In order to determine the leading behavior of $\Gamma^{0}(p,\tau)$ for small
$\tau$, we introduce the vertex function $\tilde{\Gamma}^{0}$, which differs
from $\Gamma^{0}$ by ignoring the Fermi surface when integrating out the
internal momenta. This amounts to ignoring the Heaviside function in Eq. (5).
We obtain
$\displaystyle\frac{1}{\tilde{\Gamma}^{0}(p,i\Omega)}=\frac{m}{4\pi}~{}{\rm
ln}\bigg{[}-\frac{\varepsilon_{B}}{i\Omega+\mu+\varepsilon_{F}-\frac{\epsilon_{\mathbf{p}}}{2}}\bigg{]}\;.$
(9)
In the $(p,i\Omega)$-representation,
$\displaystyle\frac{1}{\Gamma^{0}}-\frac{1}{\tilde{\Gamma}^{0}}=\frac{m}{4\pi}{\rm
ln}\bigg{[}\frac{-2(z+2\varepsilon_{F})+\epsilon_{\mathbf{p}}}{-z+\sqrt{(z-\epsilon_{\mathbf{p}})^{2}-4\varepsilon_{F}\epsilon_{\mathbf{p}}}}\bigg{]}\;.$
(10)
The $(p,\tau)$ representation of $\tilde{\Gamma}^{0}$ is
$\displaystyle\tilde{\Gamma}^{0}(p,\tau)$ $\displaystyle=$
$\displaystyle-\frac{4\pi\varepsilon_{B}}{m}e^{-(\frac{\epsilon_{\mathbf{p}}}{2}-\varepsilon_{F}-\mu)\tau}~{}\bigg{[}\int_{0}^{+\infty}dx\frac{e^{-x\varepsilon_{B}\tau}}{\pi^{2}+{\rm
ln}^{2}(x)}$ (11)
$\displaystyle+~{}e^{\varepsilon_{B}\tau}H(\frac{\epsilon_{\mathbf{p}}}{2}-\varepsilon_{F}-\varepsilon_{B}-\mu)\bigg{]}~{}H(\tau)$
for $\mu<-\varepsilon_{F}-\varepsilon_{B}$, ensuring that only $\tau>0$
contributes for all momenta $p$. The integral in Eq. (11) can be computed
numerically, but converges poorly for $\tau\rightarrow 0^{+}$. Under those
conditions we make use of the asymptotic behavior:
$\int_{0}^{+\infty}dx\frac{e^{-x\varepsilon_{B}\tau}}{\pi^{2}+{\rm
ln}^{2}(x)}\underset{\tau\rightarrow
0}{\sim}\frac{1}{\varepsilon_{B}\tau}\frac{1}{{\rm
ln}^{2}(\varepsilon_{B}\tau)}\;.$ (12)
To obtain $\Gamma^{0}(p,\tau)$ we computed numerically the following Fourier
transform:
$\displaystyle\Gamma^{0}(p,\tau)-\tilde{\Gamma}^{0}(p,\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{+\infty}d\Omega~{}e^{-i\Omega\tau}$
(13) $\displaystyle\times$
$\displaystyle\left[\Gamma^{0}(p,i\Omega)-\tilde{\Gamma}^{0}(p,i\Omega)\right]\;.$
The left-hand side of Eq. (13) can be computed more easily than
$\tilde{\Gamma}^{0}(p,\tau)$ as it contains no singularities. Next, the
function $\Gamma^{0}(p,\tau)$ is obtained as
$\tilde{\Gamma}^{0}(p,\tau)+\left[\Gamma^{0}(p,\tau)-\tilde{\Gamma}^{0}(p,\tau)\right]$.
Although the functions $\tilde{\Gamma}^{0}(p,\tau)$ and $\Gamma^{0}(p,\tau)$
are extremely sharp and divergent for $\tau\to 0$, they are integrable.
Special care should be taken when using these functions in the Monte Carlo
code. It is important to correctly sample very short times, and one needs to
make sure there is no loss of accuracy when keeping track of imaginary time
differences of the $\Gamma^{0}$ lines in the diagrams.
Figure 1: Graphical representation of the Dyson equation. The free (dressed)
one-body impurity propagator is denoted by $G_{\downarrow}^{0}$
($G_{\downarrow}$). $\Sigma$ and $\Pi$ are the one-body and two-body self-
energies, respectively. $\Gamma$ is the fully dressed interaction, whereas
$\Gamma^{0}$ is the partially dressed interaction as shown in Eq. (3).
Just like for the 3D polaron problem polaron1 ; polaron2 ; polaron , we
consider a diagrammatic series for the self-energy built from the free one-
body propagators for the impurity and the spin-up Fermi sea and from the
renormalized interaction $\Gamma^{0}$. We refer to this series as the _bare
series_ , which we evaluate with the DiagMC method. The diagram topologies in
2D and 3D are exactly the same. The major differences between the
diagrammatic-series evaluations in 2D and 3D are the renormalized interaction
$\Gamma^{0}(p,\tau)$ and the phase-space volume elements. The one and two-body
self-energies are related to the one-particle propagator $G$ and the fully
dressed interaction $\Gamma$ by means of a Dyson equation, as illustrated
schematically in Fig. 1. From the poles of $G$ and $\Gamma$ we can extract the
polaron and the molecule energy, respectively. The fully dressed interaction
is closely related to the two-particle propagator polaron .
For the 3D Fermi polaron problem there are two dominant diagrams at each given
order that emerge next to many diagrams with a much smaller contribution
polaron . These dominant diagrams contribute almost equally but have opposite
sign. In 2D, however, the numerical calculations indicate that at a given
order the very same two diagrams dominate, but to a lesser extent; that is,
the nondominant diagrams have a larger weight in the final 2D result. By
_weight_ of a given diagram we mean the absolute value of its contribution to
the self-energy. We note that the sign of a single diagram at fixed internal
and external variables depends only on its topology and not on the values of
the internal and external variables. We stress that this is not true for a
Fermi system with two interacting components with finite density vanhoucke12 ;
vanhoucke13 . In 2D the total weight of a given order (i.e., the sum of the
absolute values of the contributions of diagrams) is distributed over more
diagrams than in 3D. Because the sign alternation occurs over a broader
distribution of the weights, we get more statistical noise in sampling the
self-energy in 2D compared to 3D. In 3D we can evaluate the diagrammatic
series for the one-body self-energy accurately up to order $12$, whereas in 2D
we can reach order $8$.
In principle, other choices for the propagators (“bare” versus “dressed”
propagators) are possible, and this was discussed in detail for the 3D Fermi
polaron in our previous paper polaron . Replacing the bare propagators by
dressed ones reduces the number of diagrams at each given order. One may
expect that this replacement could allow one to reach higher orders. For the
3D polaron, however, the most favorable conditions of cancellations between
the contributions from the various diagrams were met in the bare scheme
polaron . In the DiagMC framework a higher accuracy can be reached under
conditions of strong cancellations between the various contributions. From
numerical investigations with various propagators for the 2D Fermi polaron we
could draw similar conclusions as in the 3D studies. Accordingly, all
numerical results for the quasiparticle properties presented below are
obtained for a series expansion with bare propagators.
Figure 2: (Color online) Dependence of the polaron energy $E^{N}_{p}$ on the
cutoff diagram order $N$. $E_{p}$ is the value obtained after extrapolation to
$N\to+\infty$ (and resummation for $\eta=-0.25$). Results are shown for
$\eta=-0.25,\eta=0.5,\eta=1.5$. The lines represent an exponential fit.
To characterize the magnitude of the interaction strength we use the
dimensionless parameter
$\eta\equiv\text{ln}[k_{F}a_{2D}]=\text{ln}[2\varepsilon_{F}/\varepsilon_{B}]/2$.
Here, $a_{2D}>0$ is the 2D scattering length, related to the dimer binding
energy by $\varepsilon_{B}=1/(2m_{r}a_{2D}^{2})$ with
$m_{r}=m_{\uparrow}m_{\downarrow}/(m_{\uparrow}+m_{\downarrow})$ being the
reduced mass. The BCS regime corresponds to $\eta\gg 1$ while the Bose-
Einstein condensate (BEC) regime corresponds to $\eta\ll-1$. The system is
perturbative in the regimes $|\eta|\gg 1$, while the strongly correlated
regime corresponds to $|\eta|\lesssim 1$. bloom In the weak-coupling regime
[small interaction strengths $g_{0}$ in the Hamiltonian of Eq. (1) or large
positive $\eta$ in the zero-range limit], we find that the one-body and the
two-body self-energy $\Sigma$ and $\Pi$ converge absolutely as a function of
the maximum diagram order. This is demonstrated in Fig. 2 for $\eta=1.5$,
where the polaron energy $E^{N}_{p}$ converges exponentially as a function of
the cutoff diagram order $N$. Similar convergence is also found for the
molecule energy. Under conditions of convergence with diagram order,
extrapolation to order infinity can be done in a trivial way. Similar
convergence is also seen for $\eta=0.5$. In the strongly correlated regime the
series starts oscillating with order when $\eta\lesssim 0$, and the
oscillations get stronger the deeper we go into the BEC regime. The
oscillations in the extracted polaron energy are illustrated in Fig. 2 for
$\eta=-0.25$. To obtain meaningful results we rely on Abelian resummation
techniques polaron ; vanhoucke12 . We evaluate the series
$\sigma_{\epsilon}=\sum_{N}\sigma^{(N)}e^{-\epsilon\lambda_{N}}$, with
$\sigma^{N}$ being the one-body self-energy for diagram order $N$ and
$\lambda_{N}$ being a function that depends on the diagram order $N$. For each
$\epsilon$ the polaron energy $E_{p}$ is calculated from $\sigma_{\epsilon}$
and an extrapolation is done by taking the limit $\epsilon\rightarrow 0$. The
whole procedure is illustrated in Fig. 3. To estimate the systematic error of
the extrapolation procedure, different resummation functions $\lambda_{N}$ are
used. As becomes clear from Fig. 3 the whole resummation procedure is a stable
one and induces uncertainties on the extracted energies of the order of a few
percent. All the results of Fig. 4 are obtained with the Abelian resummation
technique. The stronger the coupling constant is the larger the size of the
error attributed to the resummation. An accurate extrapolation to infinite
diagram order could be achieved for all values of $\eta$.
Figure 3: (Color online) Abelian resummation of the bare series for the one-
body self-energy diagrams at $\eta=-0.25$. We evaluate
$\sigma_{\epsilon}=\sum_{N}\sigma^{(N)}e^{-\epsilon\lambda_{N}}$, with
$\sigma^{N}$ being the one-body self-energy for diagram order $N$. We use the
following functions $\lambda_{N}$ : (i) Gauss 1: $\lambda_{N}=(N-1)^{2}$ for
$N>1$ and $\lambda_{N}=0$ for $N=1$, (ii) Lindelöf 1: $\lambda_{N}$ $=$
$(N-1)~{}{\log}(N-1)$ for $N>2$ and $\lambda_{N}=0$ for $N\leq 2$, and (iii)
Gauss 2: $\lambda_{N}=(N-3)^{2}$ for $N>3$ and $\lambda_{N}=0$ for $N\leq 3$.
The polaron energy $E_{p}/\epsilon_{F}$ is extracted in the limit $\epsilon$
$=$ $0^{+}$ for various choices of $\lambda_{N}$. Figure 4: (Color online)
Polaron and molecule ground-state energies $E$ in units of the Fermi energy
$\varepsilon_{F}$ as a function of $\eta$. The momentum of the impurity is
equal to zero. Energies are shifted by the two-body binding energy
$\varepsilon_{B}/\varepsilon_{F}=2e^{-2\eta}$ to magnify the details. The
solid line is the DiagMC result for $N=1$. The symbols are the result of the
full DiagMC calculations (including diagrams up to order 8).
## III Results and discussion
In Fig. 4, polaron and molecule energies are displayed for a wide range of the
parameter $\eta$. DiagMC results include all diagrams up to order 8 and
extrapolation to the infinite diagram order. In the region $\eta\lesssim 0$ a
small discrepancy (of the order of $0.1\%$ of the ground-state energy) is
found with the variational results Levinsen13 of Parish and Levinsen based on
the wave-function Ansatz up to 2p-h excitations. Clearly, a phase transition
appears at the critical value $\eta_{c}=-0.95\pm 0.15$. A variational result
which includes 2p-h excitations for the polaron and 1p-h excitations for the
molecule, gives $\eta=-0.97$. Levinsen13 Both mentioned calculations are in
agreement with the experimental result $\eta=-0.88(0.20)$ kos .
The DiagMC method allows one to include a large number of particle-hole
excitations that dress the impurity. Truncation of the Hilbert space to a
maximum number of p-h pairs can nonetheless be achieved within the DiagMC
approach. This allows one to arrive at the variational formulation. Previous
variational studies using a wave function Ansatz up to 1p-h or 2p-h
excitations showed that these truncations give remarkably accurate results
Combescot .
To understand why the truncation is possible within a Feynman diagrammatic
approach for the self-energy, we first remark that a variational approach is
easily established within a path-integral formalism. Path integrals with
continuous imaginary time, for example, are based on an expansion of the
evolution operator,
$\displaystyle e^{-\beta\hat{K}}$ $\displaystyle=$ $\displaystyle
e^{-\beta\hat{K}_{0}}\big{(}1-\int_{0}^{\beta}d\tau_{1}K_{1}(\tau_{1})$ (14)
$\displaystyle+$
$\displaystyle\int_{0}^{\beta}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}~{}\hat{K}_{1}(\tau_{1})\hat{K}_{1}(\tau_{2})-\ldots\big{)}\;,$
where $\hat{K}=\hat{H}-\mu\hat{N}=\hat{K}_{0}+\hat{K}_{1}-\mu\hat{N}$, with
$\left[\hat{K}_{0},\hat{K}_{1}\right]\neq 0$. The operator
$\hat{K}_{1}(\tau)=e^{\hat{K}_{0}\tau}\hat{K}_{1}e^{-\hat{K}_{0}\tau}$, which
defines the series expansion, is expressed in the interaction picture.
Further, $\beta=1/k_{B}T$, with $k_{B}$ being Boltzmann’s constant and $T$
being temperature, $\hat{H}$ is the Hamiltonian, $\hat{N}$ the number
operator, and $\mu$ is the chemical potential. The imaginary-time evolution
operator in Eq. (14) can be used as a ground-state projection operator: for
sufficiently long imaginary time $\beta$ the excited-state components of a
trial state are exponentially suppressed. One typically evaluates all the
terms in the expansion equation (14) in the eigenbasis of $\hat{K}_{0}$. This
procedure forms the basis of path-integral Monte Carlo simulation of lattice
models, where $\hat{K}_{1}$ is usually the kinetic energy term worm . A
discretized time version is used in path-integral Monte Carlo methods in
continuous space ceperley ; boninsegni . Either way, the contributions to the
path integral have the direct physical interpretation of a time history of the
many-particle system. At each instant of time, one can constrain the
accessible states of the Hilbert space, in line with what is done in a
variational approach. Within the standard Feynman diagrammatic formalism for
Green’s functions, however, this truncation of the Hilbert space is not easy
to accomplish for an arbitrary system, as one expands in powers of the two-
body interaction term of the Hamiltonian. This will be explained in the next
paragraph.
It turns out to be formally easier to start from finite $T$ and to take the
$\beta\to\infty$ limit in the end. For a many-fermion system, the finite-
temperature Green’s function in position and imaginary-time representation
$(\mathbf{x},\tau)$ is defined as
$G_{\alpha\sigma}(\mathbf{x},\tau)=-\frac{{\rm
Tr}[e^{-\beta\hat{K}}T_{\tau}\hat{\psi}^{\phantom{\dagger}}_{H\alpha}(\mathbf{x},\tau)\hat{\psi}^{\dagger}_{H\sigma}(\mathbf{x},0)]}{{\rm
Tr}[e^{-\beta\hat{K}}]}\;,$ (15)
with $\alpha$ and $\sigma$ denoting an appropriate set of quantum numbers
(such as spin) and $T_{\tau}$ being the time-ordering operator. The field
operator in the Heisenberg picture
$\hat{\psi}^{\phantom{\dagger}}_{H\alpha}(\mathbf{x},\tau)=e^{\hat{K}\tau}\hat{\psi}^{\phantom{\dagger}}_{\alpha}(\mathbf{x})e^{-\hat{K}\tau}$
annihilates a fermion in state $\alpha$ at position $\mathbf{x}$ and time
$\tau$. To arrive at the Feynman diagrammatic expansion, one makes a
perturbative expansion for the evolution operator $e^{-\beta\hat{K}}$ in both
the numerator _and_ the denominator of Eq. (15) (the finite $T$ ensures that
both exist). The expansion of the partition function $Z$ in the denominator
can be represented graphically by the series of all fully closed diagrams
(connected and disconnected). When $\beta$ approaches $+\infty$, the
denominator is proportional to $\langle\Psi_{0}|\Psi_{0}\rangle$
($|\Psi_{0}\rangle$ is the ground state of the interacting many-body system),
and the disconnected diagrams correspond to all possible vacuum fluctuations
of the system at hand. The expansion in the numerator factorizes into an
expansion of connected diagrams for $G_{\alpha\sigma}$ and disconnected
diagrams for $Z$. So the sum of disconnected diagrams drops out, as expected
for an intensive quantity like $G_{\alpha\sigma}(\mathbf{x},\tau)$. It is
exactly this factorization that prevents one from truncating the Hilbert space
at any instant of time in the evolution. In other words, variational
calculations based on Feynman diagrams for the self-energy are generally not
feasible.
In the polaron problem vacuum fluctuations are absent since $|\Psi_{0}\rangle$
corresponds to the spin-down vacuum and a non-interacting spin-up Fermi sea.
In other words, the vacuum cannot be polarized in the absence of an impurity.
As a consequence, we face a situation similar to the path integral with a
direct physical interpretation of the time history of the impurity. This
peculiar feature allows one to restrict the Hilbert space at each given time.
If we allow at most 1p-h excitations at each instant of time, only one diagram
survives: the lowest-order self-energy diagram built from $\Gamma^{0}$ and the
free spin-up single-particle propagator $G^{0}_{\uparrow}$. The equivalence
between this diagram and the 1p-h variational approach had already been
pointed out in Ref. oneD1 . An $n$p-h variational approach is achieved by
allowing at most $n$ backward spin-up lines at each step in the imaginary-time
evolution.
For large $\eta$ it is obvious from Fig. 4 that the polaron energy from the
full series expansion becomes equal to the 1p-h result. Even for stronger
interactions (smaller $\eta$) the first-order results remain close to the full
DiagMC one. Within the statistical accuracy of the numerical calculations,
convergence for the one-body self-energy is already reached after inclusion of
2p-h excitations. Indeed, for all values of $\eta$, we find agreement between
our 2p-h variational DiagMC approach and the full DiagMC approach within
statistical error bars. For the molecular branch, we retrieve the result for
the two-body self-energy from the full series expansion after including 1p-h
excitations. For the 3D Fermi-polaron a similar conclusion was drawn. Also in
3D, the first-order result is a very good approximation polaron . Going up to
2p-h pairs gives a perfect agreement with full DiagMC results. From the above
considerations it follows, however, that the diagrammatic truncations which
provide good results for the polaron problem may not be appropriate for the
more complex many-body problem with comparable densities for both components.
The quasiparticle residue or $Z$ factor of the polaron gives the overlap of
the noninteracting wave function and the fully interacting one. This overlap
is very small for a molecular ground state of the fully interacting system
polaron . The $Z$ factor as a function of $\eta$ is shown in Fig. 5. Note that
the polaron $Z$ factor does not vanish in the region $\eta\lesssim-1$ where
the ground state is a molecule. The $Z$ factor is, however, still meaningful
since the polaron is a well-defined (metastable) excited state of the 2D
system. Again, the first-order result gives a good approximation to the full
result. The measured $Z$ factor for the 3D situation has been reported in Ref.
pietro ; polaronMIT . The 2D experimental data are reported in Ref. kos , and
the $\eta$ dependence of the quasiparticle weight $Z$ is presented in
arbitrary units. We reproduce the observation that $Z$ strongly increases
between $\eta_{c}\lesssim\eta\lesssim 1$ and saturates to a certain value for
$\eta>1$.
## IV Conclusion
Summarizing, we have developed a framework to study with the DiagMC method the
ground-state properties of the 2D Fermi polaron for attractive interactions.
We have shown that the framework allows one to select an arbitrary number of
$n$p-h excitations of the FS, thereby making a connection with typical
variational approaches which are confined to $n$=1 and $n$=2. We have studied
the quasiparticle properties of the ground state for a wide range of
interaction strengths. A phase transition between the polaron and molecule
states is found at interaction strengths compatible with experimental values
and with variational predictions. To a remarkable degree, it is observed that
for all interaction strengths the full DiagMC results (which include all
$n$p-h excitations) for the ground-state properties can be reasonably
approximated by n=1 truncations. In a n=2 truncation scheme the full result is
already reached within the error bars. This lends support for variational
approaches to the low-dimensional polaron problem, for which one could have
naively expected a large sensitivity to quantum fluctuations.
Figure 5: (Color online) The quasiparticle residue $Z$ of the polaron as a
function of $\eta$. The solid line represents the 1p-h result ($N=1$ diagram).
## Acknowledgments
This work is supported by the Fund for Scientific research - Flanders. We
would like to thank C. Lobo, N. Prokof’ev, B. Svistunov, and F. Werner for
helpful discussions and suggestions, and we thank J. Levinsen and M. Parish
for sending us their data.
## References
* (1) C. Chin, R. Grimm, P. Julienne and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
* (2) I. Bloch, Nat. Phys. 1, 23 (2005).
* (3) G.B. Partridge, W. Li, Y.A. Liao, R.G. Hulet, M. Haque and H.T.C. Stoof, Phys. Rev. Lett. 97, 190407 (2006).
* (4) Y. Shin, C.H. Schunck, A. Schirotzek and W. Ketterle, Nature (London)451, 689 (2008).
* (5) S. Nascimbène, N. Navon, K.J. Jiang, L. Tarruell, M. Teichmann, J. McKeever, F. Chevy and C. Salomon, Phys. Rev. Lett. 103, 170402 (2009).
* (6) S. Pilati and S. Giorgini, Phys. Rev. Lett. 100, 030401 (2008).
* (7) P. Massignan, Z. Yu and G.M. Bruun, Phys. Rev. Lett. 110, 230401 (2013).
* (8) G. Bertaina and S. Giorgini, Phys. Rev. Lett. 106, 110403 (2011).
* (9) F. Chevy, Phys. Rev. A 74, 063628 (2006).
* (10) A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwierlein, Phys. Rev. Lett. 102, 230402 (2009).
* (11) M. Koschorreck, D. Pertot, E. Vogt, B. Fröhlich, M. Feld and M. Köhl, Nature (London) 485, 619 (2012).
* (12) P. Dyke ,E. D. Kuhnle, S. Whitlock, H. Hu ,M. Mark, S. Hoinka, M. Lingham, P. Hannaford and C.J. Vale, Phys. Rev. Lett.106, 105304 (2011).
* (13) J. Levinsen and S. K. Baur, Phys. Rev. A 86, 041602 (2012).
* (14) N.V. Prokof’ev and B.V. Svistunov, Phys. Rev. B 77, 020408 (2008).
* (15) N.V. Prokof’ev and B.V. Svistunov, Phys. Rev. B 77, 125101 (2008).
* (16) J. Vlietinck, J. Ryckebusch and K. Van Houcke, Phys. Rev. B 87, 115133 (2013).
* (17) M. Punk, P.T. Dumitrescu and W. Zwerger, Phys. Rev. A 80, 053605 (2009).
* (18) R. Combescot, S. Giraud and X. Leyronas, EPL 88, 60007 (2009).
* (19) C. Mora and F. Chevy, Phys. Rev. A 80, 033607 (2009).
* (20) J.B. McGuire, J. Math. Phys. 7, 123 (1966).
* (21) R. Combescot, A. Recati, C. Lobo and F. Chevy, Phys. Rev. Lett. 98, 180402 (2007).
* (22) S. Giraud and R. Combescot, Phys. Rev. A 79, 043615 (2009).
* (23) S. Zöllner, G.M. Bruun and C.J. Pethick, Phys. Rev. A 83, 021603(R) (2011).
* (24) M.M. Parish, Phys. Rev. A 83, 051603(R) (2011).
* (25) M.M. Parish and J. Levinsen, Phys. Rev. A 87, 033616 (2013).
* (26) L.D. Landau and E.M. Lifshitz, _Statistical Physics_ (Pergamon, New York, 1980), Vol.2.
* (27) K. Van Houcke, F. Werner, E. Kozik, N. Prokof’ev, B. Svistunov, M.J.H. Ku, A.T. Sommer, L.W. Cheuk, A. Schirotzek and M.W. Zwierlein, Nat. Phys. 8, 366 (2012).
* (28) K. Van Houcke, F. Werner, N. Prokof’ev and B. Svistunov, arXiv:1305.3901.
* (29) P. Bloom, Phys. Rev. B 12, 125 (1975).
* (30) R. Combescot and S. Giraud, Phys. Rev. Lett. 101, 050404 (2008).
* (31) N.V. Prokof’ev, B.V. Svistunov and I.S. Tupitsyn, J. Exp. Theor. Phys. 87, 310 (1998).
* (32) D.M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).
* (33) M. Boninsegni, N. Prokof’ev and B. Svistunov, Phys. Rev. Lett. 96, 070601 (2006).
* (34) C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder, P. Massignan, G.M. Bruun, F. Schreck, and R. Grimm, Nature (London) 485, 615 (2012)
|
arxiv-papers
| 2014-02-17T14:34:08 |
2024-09-04T02:49:58.306408
|
{
"license": "Public Domain",
"authors": "Jonas Vlietinck, Jan Ryckebusch, Kris Van Houcke",
"submitter": "Jonas Vlietinck",
"url": "https://arxiv.org/abs/1402.4015"
}
|
1402.4020
|
# Fractal and multifractal properties of a family of fractal networks
Bao-Gen Li1, Zu-Guo Yu1,2 and Yu Zhou1
1 Hunan Key Laboratory for Computation and Simulation in Science and
Engineering and
Key Laboratory of Intelligent Computing and Information Processing of Ministry
of Education,
Xiangtan University, Xiangtan, Hunan 411105, China.
2School of Mathematical Sciences, Queensland University of Technology,
GPO Box 2434, Brisbane, Q4001, Australia. Corresponding author, email:
[email protected]
###### Abstract
In this work, we study the fractal and multifractal properties of a family of
fractal networks introduced by Gallos et al. ( Proc. Natl. Acad. Sci. U.S.A.,
2007, 104: 7746). In this fractal network model, there is a parameter $e$
which is between $0$ and $1$, and allows for tuning the level of fractality in
the network. Here we examine the multifractal behavior of these networks,
dependence relationship of fractal dimension and the multifractal parameters
on the parameter $e$. First, we find that the empirical fractal dimensions of
these networks obtained by our program coincide with the theoretical formula
given by Song et al. ( Nat. Phys, 2006, 2: 275). Then from the shape of the
$\tau(q)$ and $D(q)$ curves, we find the existence of multifractality in these
networks. Last, we find that there exists a linear relationship between the
average information dimension $<D(1)>$ and the parameter $e$.
Key words: Complex network, scale-free, multifractality, box covering
algorithm.
PACS: 89.75.Hc, 05.45.Df, 47.53.+n
## 1 Introduction
Complex networks have caused extensive attention due to their close connection
with so many real-world systems, such as the world-wide web, the internet,
energy landscapes, and biological and social systems [1]. The fractality and
percolation transition [2], fractal transition [3] in complex networks, and
properties of a scale-free Koch networks [6, 5, 4] have turned to be hot
topics in recent years.
Fractal analysis (using the fractal dimension) is a useful method to describe
global properties of complex fractal sets [7, 8, 9]. Song et al. [1, 10]
proposed an algorithm to calculate the fractal dimension of complex networks
which can unfold their self-similar property. They mentioned that the box
counting fractal analysis is an effective tool for the further study of
complex networks. But the fractal analysis is not enough when the object
studied can not be described by a single fractal dimension. It has been found
that the multifractal analysis (MFA) is a powerful tool in both the theory and
practice to describe the spatial heterogeneity of fractal object
systematically [11, 12]. The MFA was originally raised to handle turbulence
data, and now it has been successfully applied in many fields, such as
financial modelling [13, 14], biological systems [15, 16, 17, 18, 19, 20, 21,
22, 23, 24] and geophysical systems [25, 26, 27, 28, 29, 30, 31]. Lee and Jung
[32] found that MFA is the best tool to describe the probability distribution
of the clustering coefficient of a complex network. Furuya and Yakubo [33]
analytically and numerically demonstrated the possibility that the fractal
property of a scale-free network cannot be characterized by a single fractal
dimension when the network takes a multifractal structure. Almost at the same
time, Wang et al. [34] proposed a modified fixed-size box-counting algorithm
to study the multifractal property of complex networks.
In this paper, we study the fractal and multifractal properties of a family of
complex networks introduced by Gallos et al. [35]. In order to imitate the
fractal property of many scale-free networks found in nature, Song et al. [36]
developed a network model to describe the fractality of networks. The main
characteristic of this model is the introduction of a parameter $e$ which
could be used to control the original hubs whether continue to form
connections between the nodes in the process of the growth of complex
networks. The authors [36] pointed out that the parameter $e$ can be regarded
as a level of fractality of the network. The network corresponds to a pure
fractal network which is a pure fractal set (defined by Mandelbrot [7]) when
$e=0$, and a pure small world network when $e=1$ [2]. Later on, Gallos et al.
[35] proposed a generalized version of this network model.
In Section 2, we introduce the generalized version of the network model in
Ref. [35] and some of its topological properties. In Section 3, we examine the
fractal dimension of these networks . A new fixed-size box-counting algorithm
for MFA of networks modified from the one proposed in Ref. [34] is given in
Section 4. The multifractal properties of the model networks and their results
are also given in this Section. Some conclusions are presented in Section 5.
## 2 Network model
A graph (or network) is a collection of nodes which denote the elements of a
system, and links or edges which identify the relations or interactions among
these elements. In this section, the algorithm of the generalized version of
the network model in Ref. [35] is presented. The network could be obtained by
a method described as follows.
First we give a real number $0\leq e\leq 1$, and two positive integers $m$ and
$x$ ($x\leq m$). In the generation $n=0$, we start with only two nodes and one
edge between them. In order to get the network of the generation $n+1$, every
endpoint of each edge $L$ in the network of the generation $n$ is attached to
$m$ new nodes. Then we generate a random number $p$ which obeys the uniform
distribution between 0 and 1. If $0\leq p<e$, each edge $L$ of the generation
$n$ is kept and $x-1$ new edges are appended to connect pairs of the new nodes
attached to the endpoints of $L$; otherwise, for each edge $L$ of the
generation $n$, we add $x$ new edges matching new nodes at the ends of $L$ and
remove $L$ (see Fig. 1). As shown in Fig. 2, if we take $e=0,m=2,x=2$, we add
$m=2$ new nodes to the two endpoints of the sole edge in the generation $n=0$
to get the network of the generation $n=1$. Due to $e=0$, we take away the
edge of $n=0$ and add $x=2$ edges between the new nodes. Notice that when
$x=1$, we get tree structure without loop for any value of $e$.
Figure 1: Construction of network. The link between hub remains with
probability $e$, otherwise, it is replaced by another link between new nodes
with probability $1-e$.
Figure 2: Construction of a pure fractal network. Example of the network
modelof generations $n=0,1,2$ with parameters $m=2,x=2,e=0$.
According to above description, if we denote $M_{n}$ the number of edges in
the network of the generation $n$, we can have $M_{n+1}=(2m+x)*M_{n}$ in the
generation $n+1$. Hence ${{M_{n}}}={(2m+x)^{n}}$. Meanwhile in the growth of
the network from the generation $n-1$ to the generation $n$, each edge in the
network of the generation $n-1$ produces $2m$ new nodes. Hence we have
$N_{n}=2mM_{n-1}+N_{n-1}$, where $N_{n}$ is the number of nodes in the network
of the generation $n$. Therefore,
${N_{n}}=\frac{2m}{2m+x-1}(2m+x)^{n}+2-\frac{2m}{2m+x-1}$ (1)
It was proved in Song et al. [36] that the degree distribution $P(k)$ of the
network model satisfies a power law relationship $P(k)\approx{k^{-\gamma}}$
with $\gamma=1+\ln b/\ln s$, where $b$ is the scaling of the node number and
$s$ is the scaling of the node degrees between two adjacent generations in the
process of the growth of the network. From the algorithm described above, we
know that if the degree of a node in the network of the generation $n$ is
$K_{n}$, then it should be $mK_{n}+K_{n}$ with probability $e$, or $mK_{n}$
with probability $1-e$ in generation $n+1$, therefore
$K_{n+1}=e(mK_{n}+K_{n})+(1-e)mK_{n}=(m+e)K_{n}$. It is easy to know that
$b=2m+x$ from Eq. (1). So for the above model, we have
$\gamma=1+\frac{{\ln(2m+x)}}{{\ln(m+e)}}$ (2)
In addition, a simple analysis shows that the clustering coefficient of the
network model is $0$ for any value of $e$.
## 3 The fractal dimension
We find that if the distance between two nodes of the generation $n$ is
$L_{n}$, then in the network of the generation $n+1$, the distance $L_{n+1}$
would be $L_{n}$ with probability $e$, or $3L_{n}$ with probability $1-e$.
Hence $L_{n+1}=eL_{n}+3(1-e)L_{n}=(3-2e)L_{n}$. From Ref. [36], we know that
the theoretical fractal dimension of the model networks is ${d_{f}^{T}}=\ln
b/\ln a$, where $a=L_{n+1}/L_{n}$. Therefore,
${d_{f}^{T}}=\frac{{\ln(2m+x)}}{{\ln(3-2e)}}$ (3)
If $x=2,m=2$, we have $d_{f}^{T}=\ln(6)/\ln(3-2e)$.
We can also numerically calculate the fractal dimension of the model networks
using some algorithms (e.g. [10, 37]). Here we adopt the random sequential
box-counting algorithm proposed by Kim et al. [37] to estimate the fractal
dimension of networks (two examples for estimating fractal dimension are shown
in Fig. 3). We denote $d_{f}^{N}$ the fractal dimension of the network
obtained numerically. First we want to check whether values of $d_{f}^{N}$
coincide with the theoretical values of fractal dimension $d_{f}^{T}$. If the
numerical and theoretical fractal dimensions coincide with each other, we will
have confidence on our process and program to estimate the multifractal curves
$\tau(q)$ and $D(q)$ of these networks. Due to the limit of computational
capacity of our computers, we only generate the networks up to the generation
$n=5$.
For each value of $e$, we generate 100 networks (100 realizations) and
calculate the average value of $d_{f}^{N}$ over the 100 realizations. The $e$
vs $<d_{f}^{N}>$ plot is presented in Fig. 4. From Fig. 4, we can see that the
numerical $<d_{f}^{N}>$ coincides with the theoretical $<d_{f}^{T}>$
perfectly.
Figure 3: Two examples to estimate the fractal dimension of networks for
$e=0.1$ and $0.5$, here parameters $n=5,m=2,x=2$. We can see the estimated
fractal dimension is very close to the theoretical result 2.5850 (for $e=0.5$)
and 1.7402 (for $e=0.1$) respectively.
Figure 4: Numerical result of the relationship between the fractal dimension
$<d_{f}^{N}>$ of the networks and $e$ with parameters $n=5,m=2,x=2$. Here
$<\cdot>$ means the average over 100 realizations.
## 4 Multifractal analysis
In this section, we first introduce a new algorithm for MFA of networks
modified from the one proposed in Ref. [34] and then apply it to the model
networks presented in Section 2.
Two networks which have the same fractal dimension may look completely
different. In addition, when the networks have rich scale and self-similar
structures, they exhibit different dimensions in different scales. MFA is a
powerful method to study the networks with such characteristics.
At present, the fixed-size box-counting algorithm is the most common algorithm
for MFA [12]. For a given probability measure $0\leq\mu\leq 1$ with support
set $E$ in metric space, we consider the partition function
${Z_{r}}(q)=\sum\limits_{\mu(B)\neq 0}{{{[\mu(B)]}^{q}}}$ (4)
where $q\in R$ , the result is the sum of all the different non-overlapping
boxes $B$ with a given size $r$ in the covering of the support set $E$. It is
easy to know that ${Z_{r}}(q)\geq 0$ and ${Z_{r}}(0)=1$. We define the mass
exponent function $\tau(q)$ of the measure $\mu$ as
$\tau(q)=\mathop{\lim}\limits_{r\to 0}\frac{{\ln{Z_{r}}(q)}}{{\ln r}}$ (5)
Then we get the generalized fractal dimensions of the measure $\mu$ by
${D(q)}=\frac{{\tau(q)}}{{q-1}},q\neq 1$ (6)
and
${D(1)}=\mathop{\lim}\limits_{r\to 0}\frac{{{Z_{(1,r)}}}}{{\ln r}},q=1$ (7)
where${Z_{(1,r)}}=\sum\limits_{\mu(B)\neq 0}{\mu(B)\ln\mu(B)}$.
In practice, the generalized fractal dimensions are usually obtained by linear
regression. Specifically, $D(0)$ is the fractal dimension of the support set
of the measure $\mu$, ${D(1)}$ and ${D(2)}$ are called the information
dimension and the correlation dimension respectively.
For a network, the measure $\mu$ of each box can be defined as the ratio of
the number of nodes covered by the box to the total number of nodes in the
network [34, 37]. We need to complete the following two steps before we
proceed MFA.
i) Map a network to an adjacent matrix ${A_{N\times N}}$, where $N$ is the
total number of nodes in the network. It is easy to know that ${A_{N\times
N}}$ is a symmetric matrix where the elements ${a_{ij}}=1$ when there is an
edge between the nodes $i$ and $j$, otherwise ${a_{ij}}=0$. Here, the edge
from node $i$ to node $i$ is not considered, so ${a_{ii}}=0$.
ii) Use ${A_{N\times N}}$ to calculate the shortest distance between any two
nodes in the network and store them into another matrix ${B_{N\times N}}$.
Here, in our study, we use Dijkstra s algorithm of MatLab toolbox to calculate
the shortest distance between two nodes in the network.
After finishing the two steps presented above, we can use matrix ${B_{N\times
N}}$ as the input of MFA of the network model described in Section 2 based on
a new algorithm for MFA of networks modified from the one proposed in Ref.
[34] as follows.
(I) Ensure that all nodes in the network are not covered, and no node has been
selected as the center of a box.
(II) According to the size of our networks $N=6222$ (with the parameters
$n=5;m=2,x=2$), we set $t=1,2,\ldots,T$. Here we take $T=1000$, then we
rearrange the nodes number into $T=1000$ different random orders. That is to
ensure that the nodes of a network are randomly chosen as center nodes.
(III) Set the radius $r$ of boxes which will be used to cover the nodes in the
range $[1,d]$, where $d$ is the diameter of the network (i.e. the longest
distance between nodes in the network).
(IV) Treat the nodes of the $t$th kind of random orders that we have got in
(II) as the center of a box successively, then search all the other nodes. If
a node has a distance to the center node within $r$ and has not been covered
yet, then cover it.
(V) If no more new nodes can be covered by this box, then we abandon this box.
(VI) Repeat (IV) - (VI) until all the nodes are covered by the corresponding
boxes. We denote the number of boxes in this box covering as $N(t,r)$.
(VII) Repeat steps (III) and (VI) for all the random orders to find a box
covering with minimal number of boxes $N(t,r)$.
(VIII) For each nonempty box $B$ in the first box covering with minimal number
of boxes, we define its measure as ${\mu(B)={N_{B}}/6222}$, where ${N_{B}}$ is
the number of nodes covered by the box $B$. For each $r$, we calculate the
partition sum $Z_{r}(q)=\sum\limits_{\mu(B)\neq 0}{{{[\mu(B)]}^{q}}}$.
(IX) For different $r$, we repeat (III)-(VIII). Then we use $Z_{r}(q)$ for
linear regression.
Remark 1: In the algorithm of MFA of networks proposed in Ref. [34], we use
$\overline{Z}_{r}(q)$ (the average of $Z_{r}(q)$ for all $T=1000$ different
random orders of the nodes) for linear regression to get $\tau(q)$ (hence
$D(q)$). But when $q=0$, $D(0)$ got in this way is not the box-counting
dimension of the network because there requires minimum number of boxes which
cover the fractal set (network here) [9]. Here we modify to use $Z_{r}(q)$ of
a covering with minimum number of boxes for linear regression to get $\tau(q)$
(hence $D(q)$). So when $q=0$, $D(0)$ is exact the box-count dimension of the
network. It is a more reasonable extension from the traditional MFA.
In order to get the range $r\in[{r_{\min,}}{r_{\max}}]$ in which the networks
obey the power law and then to get the mass exponents $\tau(q)$ and the
generalized fractal dimensions $D_{q}$, linear regression is an important
step. In our calculation, we run the linear regression of $\ln{Z_{r}}(q)$
against $\ln r$ to get $\tau(q)$, and then get $D(q)$ through formula
$D(q)=\tau(q)/(q-1)$ for $q\neq 1$ and $D(1)$ through the linear regression of
$Z_{(1,r)}=\sum\limits_{\mu(B)\neq 0}{\mu(B)\ln\mu(B)}$ against $\ln r$ for
$q=1$.
By applying the new fixed-size box-counting algorithm described above on the
model networks, we get the following results:
First, for each value of $e$ (here we take $e=0.1,0.2,...,0.8$), we generate
100 networks (we take 100 realizations because the MFA for networks is very
time consuming when the network is large), and calculate the $\tau(q)$ and
$D(q)$ curves for each network using the new fixed-size box-counting
algorithm. Then we take average for these $\tau(q)$ and $D(q)$ curves over the
100 realizations. The shape of the $<\tau(q)>$ curves shown in Fig. 5 and the
$<D_{q}>$ curves shown in Fig. 6 are all nonlinear, which indicate that all
the networks we studied have multifractal property. We also find that the
value of $\Delta(<D(q)>)$ defined by $\max(<D(q)>)-\min(<D(q)>)$ increases
with the increase of the parameter $e$, which indicates that the multifractal
property of the model networks becomes more obvious when the value of the
parameter $e$ becomes larger.
The multifractal property of the model networks revealed by our work indicates
that the model networks are very complicated which cannot be characterized by
a single fractal dimension. The MFA algorithm proposed here can be used to
provide a more accurate characterization for the model networks, even for some
other complicated networks.
Second, we find that the average information dimension $<D(1)>$ has a linear
relation with the parameter $e$, i.e. $<D(1)>=1.5053*e+1.4735$ as shown in
Fig. 7, which is different from that of $D(0)$ shown in Eq. (3).
Figure 5: The $\tau(q)$ curves of the network model, here $<\cdot>$ means the
average over 100 realizations.
Figure 6: The $D(q)$ curves of the network model, here $<\cdot>$ means the
average over 100 realizations.
Figure 7: The relationship of $<D(1)>$ against parameter $e$ respectively,
here $<\cdot>$ means the average over 100 realizations.
Remark 2: Furuya and Yakubo [33] also proposed an algorithm for MFA of complex
networks. The difference between the algorithm in Ref. [33] and our algorithm
is the definition of the measure $\mu$. In the algorithm in Ref. [33], it
allows that any two boxes in the box covering have overlap and defines the
measure $\mu_{i}$ by counting the times of overlaps of each node, hence it is
not a natural extension of the traditional MFA (see Eq. (4)). In our
algorithm, overlap of any two boxes in the box covering is not allowed, so it
is a natural extension of the traditional MFA. Our network model with $e>0$ is
different from the ($u,v$)-flower network model. Only the network model with
$e=0$ corresponds to the deterministic ($u,v$)-flower network model with
$u=v=3$. Furuya and Yakubo [33] also gave a theoretical formula for the
$\tau(q)$ function of ($u,v$)-flower network model (Eq. (11) of Ref. [33]).
When $u=v=3$, $\tau(q)$ has the formula [33]: $\tau(q)=q$ if
$q\geq\ln(6)/\ln(2)=2.5850$, and $\tau(q)=(q-1)\frac{\ln(6)}{\ln(3)}$ if
$q<\ln(6)/\ln(2)=2.5850$. We compared this formula with our numerical result
for $e=0$ in Fig.5, and found that we also have $\tau(q)\simeq q$ if $q\geq
2.5850$, but the $\tau(q)$ values are different from
$(q-1)\frac{\ln(6)}{\ln(3)}$ if $q<2.5850$.
## 5 Conclusion
We have studied the fractal and multifractal properties of a family of model
networks that were originally proposed to explain the origin of fractality in
complex networks. This model introduces a parameter $e$, which can be used to
tune the fractality level of the network. One can get a pure fractal network
when $e=0$ and obtain a small-world network when $e=1$. We investigated the
fractal and multifractal properties through numerical calculation. To make the
calculation feasible and accurate, we calculated the model with parameters
${n=5;m=2,x=2}$; ${e=0}$, ${0.1,0.2,\ldots,0.8}$; ${q=-10,\ldots,10}$. The
result of the $\tau(q),D(q)$ (including $D(0)$ and $D(1)$) are averaged over
$100$ realizations (networks). The shape of $\tau(q)$ and $D(q)$ curves are
all nonlinear, which indicates that all the networks we studied have
multifractal property. We also found that the value of
$\Delta(<D(q)>)=\max(<D(q)>)-\min(<D(q)>)$ increases with the increase of the
parameter $e$, which indicates that the multifractal property of the model
becomes more obvious when the value of the parameter $e$ becomes larger.
We also found that the average information dimension $<D(1)>$ has a linear
relation with the parameter $e$, i.e. $<D(1)>=1.5053*e+1.4735$.
The MFA algorithm proposed here can be used to provide a more accurate
characterization for the model networks, even for some other complicated
networks.
## Acknowledgments
This project was supported by the Natural Science Foundation of China (Grant
Nos. 11071282 and 11371016), the Chinese Program for Changjiang Scholars and
Innovative Research Team in University (PCSIRT) (Grant No. IRT1179); the
Research Foundation of Education Commission of Hunan Province of China (Grant
No. 11A122); the Lotus Scholars Program of Hunan province of China. The
authors would like to thank the editor and the reviewers for their insights,
comments and suggestions to improve this paper.
## References
* [1]
* [1] Song C, Havlin S and Makse H A, Self-similarity of complex networks, Nature 433 (2005) 392-395.
* [2] Rozenfeld H D and Makse H A, Fractality and the percolation transition in complex networks, Chem. Eng. Sci. 64 (2009) 4572-4575.
* [3] Rozenfeld H D, Song C and Makse H A, Small-World to Fractal Transition in Complex Networks: A Renormalization Group Approach, Phys. Rev. Lett. 104 (2010) 025701.
* [4] Zhang Z., Zhou S., Zou T. and Chen G.,Fractal scale-free networks resistant to disease spread, J. Stat. Mech.: Theor. Exp. 9 (2008) P09008.
* [5] Zhang Z, Zhou S, Xie W, Chen L, Lin Y, and Guan J, Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect, Phys. Rev. E 79 (2009) 061113.
* [6] Liu J X and Kong X M, Establishment and structure properties of the scale-free Koch network (in Chinese), Acta Phy. Sin. 59 (2010) 2244-2249.
* [7] Mandelbrot B B, 1983 The Fractal Geometry of Nature (New York: Academic Press).
* [8] Feder J, 1988 Fractals (New York: Plenum).
* [9] Falconer K, 1997 Techniques in Fractal Geometry (New York: Wiley).
* [10] Song C, Gallos L K, Havlin S and Makse H A, How to calculate the fractal dimension of a complex network: the box covering algorithm, J. Stat. Mech.: Theor. Exp. 3 (2007) P03006.
* [11] Grassberger P and Procaccia I, Characterization of Strange Attractors, Phys. Rev. Lett 50 (1983) 346-349.
* [12] Halsey T C, Jensen M H, Kadanoff L P, Procaccia I and Shraiman B I, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33 (1986) 1141-1151.
* [13] Canessa E, Multifractality in time series, J. Phys. A 33 (2000) 3637-3651.
* [14] Anh V V, Tieng Q M and Tse Y K, Cointegration of stochastic multifractals with application to foreign exchange rates, Int. Trans. Oper. Res. 7 (2000) 349-363.
* [15] Yu Z G, Anh V and Lau K S, Multifractal characterisation of length sequences of coding and noncoding segments in a complete genome, Physica A 31 (2001) 351-361.
* [16] Yu Z G, Anh V and Lau K S, Measure representation and multifractal analysis of complete genomes, Phys. Rev. E 64 (2001) 031903\.
* [17] Yu Z G, Anh V and Lau K S, Multifractal and correlation analyses of protein sequences from complete genomes, Phys. Rev. E 68 (2003) 021913\.
* [18] Yu Z G, Anh V and Lau K S, Chaos game representation of protein sequences based on the detailed HP model and their multifractal and correlation analyses, J. Theor. Biol. 226 (2004) 341-348.
* [19] Yu Z G, Anh V V, Lau K S and Zhou L Q, Clustering of protein structures using hydrophobic free energy and solvent accessibility of proteins, Phys. Rev. E 73 (2006) 031920.
* [20] Yu Z G, Xiao Q J, Shi L, Yu J W, and Anh V, Chaos game representation of functional protein sequences, and simulation and multifractal analysis of induced measures, Chin. Phys. B 19 (2010) 068701.
* [21] Anh V V, Lau K S and Yu Z G, Recognition of an organism from fragments of its complete genome, Phys. Rev. E 66 (2002) 031910\.
* [22] Zhou L Q,Yu Z G, Deng J Q, Anh V and Long S C, A fractal method to distinguish coding and non-coding sequences in a complete genome based on a number sequence representation, J. Theor. Biol. 232 (2005) 559-567.
* [23] Han J J and Fu W J, Wavelet-based multifractal analysis of DNA sequences by using chaos-game representation, Chin. Phys. B 19 (2010) 010205.
* [24] Zhu S M, Yu Z G, Ahn V, Protein structural classification and family identification by multifractal analysis and wavelet spectrum, Chin. Phys. B 20 (2011) 010505\.
* [25] Kantelhardt J W, Koscielny-Bunde E, Rybski D, Braun P, Bunde A and Havlin S, Long-term persistence and multifractality of precipitation and river runoff records, J. Geophys. Res. 111 (2006) D01106.
* [26] Veneziano D, Langousis A and Furcolo P, Multifractality and rainfall extremes: A review, Water Resour. Res. 42 (2006) W06D15.
* [27] Venugopal V, Roux S G, Foufoula-Georgiou E and Arneodo A, Revisiting multifractality of high-resolution temporal rainfall using a wavelet-based formalism, Water Resour. Res. 42 (2006) W06D14.
* [28] Yu Z G, Anh V V, Wanliss J A, and Watson S M, Chaos game representation of the Dst index and prediction of geomagnetic storm events, Chaos, Solitons and Fractals 31 (2007) 736-746.
* [29] Yu Z G, Anh V and Eastes R, 2009 J. Geophys. Res. 114 A05214
* [30] Yu Z G, Anh V, Wang Y, Mao D and Wanliss J, Multifractal analysis of geomagnetic storm and solar flare indices and their class dependence, J. Geophys. Res. 115 (2010) A10219
* [31] Zang B J and Shang P J, Multifractal analysis of the Yellow River flow, Chin. Phys. 16 (2007) 565-569.
* [32] Lee C Y and Jung S H, Statistical self-similar properties of complex networks, Phys. Rev. E 73 (2006) 066102\.
* [33] Furuya S and Yakubo K, Multifractality of complex networks, Phys. Rev. E 84 (2011) 036118
* [34] Wang D L, Yu Z G and Anh V, Multifractal analysis of complex networks, Chin. Phys. B 21 (2012) 080504\.
* [35] Gallos L K, Song C, Havlin S and Makse H A, Proc. Natl. Acad. Sci. U.S.A. 104 (2007) 7746-7751.
* [36] Song C, Havlin S and Makse H A, Origins of fractality in the growth of complex networks, Nat. Phys. 2 (2006) 275-281.
* [37] Kim J S, Goh K I, Salvi G, Oh E, Kahng B and Kim D, Fractality in complex networks: Critical and supercritical skeletons, Phys. Rev. E 75 (2007) 016110\.
* [2]
|
arxiv-papers
| 2014-02-17T14:47:37 |
2024-09-04T02:49:58.314580
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bao-Gen Li, Zu-Guo Yu and Yu Zhou",
"submitter": "Zu-Guo Yu",
"url": "https://arxiv.org/abs/1402.4020"
}
|
1402.4030
|
# Multifractal analyses of daily rainfall time series in Pearl River basin of
China
Zu-Guo Yu1,2, Yee Leung3, Yongqin David Chen3, Qiang Zhang4, Vo Anh2 and Yu
Zhou1
1 Hunan Key Laboratory for Computation and Simulation in Science and
Engineering and
Key Laboratory of Intelligent Computing and Information Processing of Ministry
of Education,
Xiangtan University, Xiangtan, Hunan 411105, China.
2School of Mathematical Sciences, Queensland University of Technology,
GPO Box 2434, Brisbane, Q4001, Australia.
3Department of Geography and Resource Management, and Institute of
Environment,
Energy and Sustainability, The Chinese University of Hong Kong, Hong Kong,
China.
4Department of Water Resources and Environment, and Key Laboratory of
Water Cycle and Water Security in Southern China of Guangdong High Education
Institute,
Sun Yat-sen University, Guangzhou 510275, China. Corresponding author, email:
[email protected]
###### Abstract
The multifractal properties of daily rainfall time series at the stations in
Pearl River basin of China over periods of up to 45 years are examined using
the universal multifractal approach based on the multiplicative cascade model
and the multifractal detrended fluctuation analysis (MF-DFA). The results from
these two kinds of multifractal analyses show that the daily rainfall time
series in this basin have multifractal behavior in two different time scale
ranges. It is found that the empirical multifractal moment function $K(q)$ of
the daily rainfall time series can be fitted very well by the universal
mulitifractal model (UMM). The estimated values of the conservation parameter
$H$ from UMM for these daily rainfall data are close to zero indicating that
they correspond to conserved fields. After removing the seasonal trend in the
rainfall data, the estimated values of the exponent $h(2)$ from MF-DFA
indicate that the daily rainfall time series in Pearl River basin exhibit no
long-term correlations. It is also found that $K(2)$ and elevation series are
negatively correlated. It shows a relationship between topography and rainfall
variability.
Key words: Daily rainfall time series; multifractal property; universal
multifractal model; multifractal detrended fluctuation analysis.
## 1 Introduction
Rainfall is one of the most important variables studied because its non-
homogenous behavior in event and intensity, leading to drought, water runoff
and soil erosion with negative environmental and social consequences [1, 2].
Analysis and modelling of rainfall are significant research problems in
applied hydro-meteorology [3]. Rainfall time series often exhibit strong
variability in time and space.
Rainfall also exhibits scaling behavior in time and space (e.g. [3-7]). There
is thus a need to characterize and model rainfall variability at a range of
scales which goes beyond the scales that can be directly resolved from
observations [8]. Investigation of the existence of fractal behavior in
rainfall processes has been an active area of research for many years [9].
Some recent experiments have shown that scale invariance, in time and space,
does exist in rainfall fields [10]. Olsson et al. [11] investigated the
rainfall time series by calculating the box and correlation dimensions via a
monodimensional fractal approach (simple scaling). Their results indicate
scaling but with different dimensions for different time aggregation periods.
Hence the investigated rainfall time series display a multidimensional fractal
behavior. Venugopal et al. [12] employed the wavelet-based multifractal
analysis to reexamine the scaling structure of rainfall over time. Molnar and
Burlando [13] used the exponent of correlation function, a multifractal
parameter, to study the seasonal and spatial variabilities. Using
2-dimensional Fourier series analysis and spectral analysis, Boni et al. [14]
proposed a methodology to study the estimated index factor for rainfall in
mountainous regions. During the past two decades, stochastic models of
rainfall have increasingly exploited the property of multifractal scale
invariance, resulting in multifractal models that are more advantageous over
conventional models in rainfall representations [15-17].
The multiplicative cascade model has been widely used to study the
multifractal properties of the rainfall data (e.g. [2, 4-8, 17-29]). Schertzer
and Lovejoy [4] showed that statistically scaled invariant processes are
stable and converge to some universal attractor, and thus can be defined by a
small number of relevant parameters, specifically three with the universal
multifractal framework.
The simple multifractal analysis (MFA) is based upon the standard partition
function multifractal formalism [30], developed for the multifractal
characterization of normalized, stationary measurements. Unfortunately, this
standard formalism does not give correct results for non-stationary time
series that are affected by trends or that cannot be normalized [31]. Thus,
two generalizations of simple MFA were developed. One is the wavelet-based MFA
which has been used to study rainfall data (e.g. [12]). Another generalization
is the multifractal detrended fluctuation analysis (MF-DFA) [31] which is an
extension of the standard detrended fluctuation analysis (DFA) introduced by
Peng et al. [32, 33]. DFA can be employed to detect long-range correlations in
stationary and noisy nonstationary time series. It intends to avoid the
unravelling of spurious correlations in time series. The DFA method has been
successfully applied to problems in fields such as DNA and protein sequences
(e.g. [32, 34, 35]) and hydrology (e.g. [36-40]). The MF-DFA is a modified
version of DFA for the detection of multifractal properties of time series. It
renders a reliable multifractal characterization of nonstationary time series
encountered in phenomena such as those in geophysics [31, 37, 38, 41-46]. The
MF-DFA has also been successfully applied to problems in hydrology (e.g.
[37-39]). The relationship between topography and rainfall variability is a
very important issue in the study of rainfall.
Our work in this paper focuses on the multifractal properties of daily
rainfall time series and possible relationships between the multifractal
exponents and landscape properties. We use the universal multifractal model
(UMM) proposed by Schertzer and Lovejoy [4] to fit the multifractal moment
function $K(q)$ of the rainfall data and propose a method to estimate the
parameters. We also adopt the MF-DFA approach to detect the correlation and
multifractal properties of daily rainfall data in this paper.
As the largest watershed in South China, the Pearl River (Zhujiang in Chinese)
delta is a composite drainage basin with a total area of 45.4$\times 10^{4}$
km2, consisting of three major rivers (i.e., West River, North River, and East
River) and several independent rivers in the downstream and delta regions (see
Figure 1). The Asian monsoon and moisture transport are the important
influencing factors on precipitation patterns in this region. Given its large
size and dominance of a sub-tropical humid monsoon climate, the Pearl River
basin is under the influence of rainfall variability which is a highly
complicated process in space and time. Zhang et al. [47] reported an increased
high-intensity rainfall over the basin in conjunction with the decreased rainy
days and low-intensity rainfall. It was also found that the abrupt changes of
the precipitation totals (for annual, winter, and summer precipitation)
occurred in the late 1970s, 1980s, and early 1990s, and the precipitation
intensity basically increased after the change points [47, 48]. In this paper,
we study the daily rainfall data over the period from 1 January 1960 to 31
December 2005 at 41 locations in Pearl River basin using the UMM and MF-DFA
methods. Parameters from the above MFAs are used to infer the spatial
relationship of rainfall in Pearl River basin of China.
## 2 Multifractal analyses
### 2.1 Universal multifractal approach based on the multiplicative cascade
model
Let $T(t)$ be a positive stationary stochastic process at a bounded interval
of ${\bf R}$, assumed to be the unit interval (0, 1) for simplicity, with
$E(T(t))=1$ (For a time series $x_{i}$, $i=1,\cdots,L$, we can define
$t_{i}=i/L$, and $T(t_{i})=x_{i}/(\sum_{k=1}^{L}x_{k})$ ). The smoothing of
$T(t)$ at scale $r>0$ is defined as
$T_{r}(t)=\frac{1}{r}\int_{t-r/2}^{t+r/2}T(s)ds$. We consider the processes
$X_{r}(t)=\frac{T_{r}(t)}{T_{1}(t)}$, $t\in[0,1]$. The empirical multifractal
function $K(q)$ can be defined as the power exponents if the following
expectation behaves like [49]
$E(X_{r}^{q}(t))\ \propto\ r^{K(q)}.$ (1)
If we consider smoothing at discrete scales $r_{j}$, $j=1,2,\cdots$, then from
Eq. (1), the empirical $K(q)$ function (denoted as $K_{d}(q)$) for the data
can be obtained by
$K_{d}(q)=\lim_{j\rightarrow\infty}\frac{\ln E(X_{r_{j}}^{q})}{-\ln r_{j}}.$
(2)
Hence the empirical $K(q)$ function $K_{d}(q)$ can be estimated from the
slopes of $E(X_{r}^{q})$ against the scale ratio 1/r in a log-log plane. In
this paper, we adopt Eq. (2) to obtain $K_{d}(q)$ of our rainfall data. If the
curve $K_{d}(q)$ versus $q$ is a straight line, the data set is monofractal.
However, if this curve is convex, the data set is multifractal [30].
The universal multifractal model (UMM) proposed by Schertzer and Lovejoy [4]
assumes that the generator of multifractals was a random variable with an
exponentiated extremal Lévy distribution. Thus, the theoretical scaling
exponent function $K(q)$ for the moments $q\geq 0$ of a cascade process is
obtained according to [4, 18, 28, 29]:
$K(q)=qH+\left\\{\begin{array}[]{ll}C_{1}(q^{\alpha}-q)/(\alpha-1),&\alpha\neq
1,\\\ C_{1}q\log(q),&\alpha=1,\end{array}\right.$ (3)
in which the most significant parameter $\alpha\in[0,2]$ is the Lévy index,
which indicates the degree of multifractality (i.e. the deviation from
monofractality). $C_{1}\in[0,d]$, with $d$ being the dimension of the support
($d=1$ in our case), describes the sparseness or inhomogeneity of the mean of
the process [28]. The parameter $H$ is called the non-conservation parameter
since $H\neq 0$ implies that the ensemble average statistics depend on the
scale, while $H=0$ is a quantitative statement of ensemble average
conservation across the scales (e.g., [29]).
Although the double trace moment (DTM) technique [50, 51] has been widely used
to estimate the parameters $H$, $C_{1}$ and $\alpha$ in geophysical research,
it is complicated and the goodness of fit of the empirical $K(q)$ functions
depends on that of exponent $\beta$ of the power spectrum, and sometimes the
fitting of $K(q)$ is not satisfactory (e.g., [19, 28, 29]). In this paper, we
adopt a method in [52] and is similar to that proposed in [49]. If we denote
$K_{T}(q)$ the $K(q)$ function defined by Eq. (3), we estimate the parameters
by solving the least-squares optimization problem [52]
$\min_{H,C_{1},\alpha}\sum_{j=1}^{J}[K_{T}(q_{j})-K_{d}(q_{j})]^{2}.$ (4)
In our analysis, we take $q_{j}=j/3$ for $j=1,2,...,30$.
### 2.2 Multifractal detrended fluctuation analysis
We outline the MF-DFA procedure used here according to the procedure described
in [31].
Suppose that $x_{k}$ is a series of length $N$. First we determine the
’profile’ $Y(i)=\sum_{k=1}^{i}[x_{k}-\langle x\rangle],\ i=1,\cdots,N$, where
$\langle x\rangle$ is the mean of $\\{x_{k}\\}$. For an integer $s>0$, we
divide the profile $Y(i)$ into $N_{s}=int(N/s)$ non-overlapping segments of
equal lengths $s$, where $int(N/s)$ is the integer part of $N/s$. Since the
length $N$ of the series is often not a multiple of the timescale $s$ under
consideration, there may remain a slack at the end of the profile. In order
not to disregard this short part of the series, the same procedure is repeated
starting from the opposite end. Thus, $2N_{s}$ segments are obtained
altogether. Now we can calculate the local trend for each of the $2N_{s}$
segments by a least squares linear fit of the series, then determine the
variance $F^{2}(s,\nu)$ for $\nu=1,\cdots,2N_{s}$ [31]. Then the $q$th-order
fluctuation function is defined as
$F_{q}(s)=\left[\frac{1}{2N_{s}}\sum_{\nu=1}^{2N_{s}}(F^{2}(s,\nu))^{q/2}\right]^{1/q}$,
where $q\neq 0$. Finally we determine the scaling behavior
$F_{q}(s)\ \propto\ s^{h\left(q\right)}.$ (5)
of the fluctuation functions by analyzing the log-log plot of $F_{q}(s)$
versus $s$ for each value of $q$. The exponent $h(q)$ is commonly referred to
as the generalized Hurst exponent. The MF-DFA is suitable for both stationary
and nonstationary time series [31]. We denote $\tilde{H}$ the Hurst exponent
of time series. The range $0.5<\overline{H}<1$ indicates long memory or
persistence; and the range $0<\overline{H}<0.5$ indicates short memory or
anti-persistence. For uncorrelated series, the scaling exponent $\overline{H}$
is equal to 0.5. Assuming the setting of fractional Brownian motion, Movahed
et al. [53] proved the relation $\overline{H}=h(2)-1$ between $\overline{H}$
and the exponent $h(2)$ for small scales. In the case of fractional Gaussian
noise, it was shown that $h(2)=\overline{H}$ [53]. Hence we can use the value
of $\overline{H}$ calculated from $h(2)$ to detect the nature of memory in
time series under the assumption of fractional Gaussian noise or fractional
Brownian motion.
In the case of a power law, the power spectrum $S(f)$ is related to the
frequency $f$ by $S(f)\propto(1/f)^{\beta}$. The exponents $h(2)$ and $\beta$
are related to each other by the equation $h(2)=(1+\beta)/2$ [36, 54]. As
pointed out by Lovejoy et al. [26], the relationship between mass exponent
$\tau(q)$, which is based upon the standard partition function multifractal
formalism [30], and $K(q)$ is
$\tau(q)=(q-1)-K(q),$ (6)
for 1-dimensional data. For a conservative process, Koscielny-Bunde et al.
[39] pointed out the relationship between $h(q)$ and $K(q)$ as
$qh(q)=qh(1)-K(q).$ (7)
By combining Eqs. (6) and (7), we get [55]
$\tau(q)=q(h(q)-h(1))+q-1.$ (8)
## 3 Results and discussion
In this study, we apply the above methods to examine the multifractal
properties of daily rainfall data in Pearl River basin over time as a regional
case study. At each of the 41 stations in Pearl River basin, daily rainfall
data over the period from 1 January 1960 to 31 December 2005 consist of 16,802
observations. The information on location and elevation of the 41 stations in
Pearl River basin is given in Table 1 (we list the stations according to the
deceasing order of their elevations). According to the elevation, we can
divide the stations into three groups (Group 1 with elevation higher than
1000m, Group 2 with elevation between 200m to 1000m, Group 3 with elevation
lower than 200m). The daily rainfall data of Station 56691 and Station 57922
(in the Pearl River basin) over the entire study period are shown in Figure 2
as examples.
First, we computed the empirical $K(q)$ curves of all daily rainfall data via
Eq.(2) by taking values for $r_{j}$ from 0.0010 to 0.056 (corresponding to
time scale from 180-960 days) for data in Pearl River basin because the power-
law relation in Eq.(2) in these time scale ranges becomes linear. An example
for obtaining the empirical $K(q)$ curves is given in Figure 3. The empirical
$K(q)$ curves of the rainfall data in two stations are shown in Figure 4 (the
dotted lines) as examples. We observed that all the empirical $K(q)$ curves of
the rainfall data in all stations are not straight lines (i.e. are convex
lines) like those in Figure 4. This suggests that all daily rainfall time
series have multifractal behavior in the time scale range from 180 to 960
days. In order to use the UMM (i.e. Eq. (3)) to fit the empirical $K(q)$
curves, we use the function fminsearch in MATLAB to solve the optimization
problem (Eq.(4)) and obtain the estimates of parameters $H$, $\alpha$ and $C1$
(we set 0.5, 0.5, 0.5 as the initial values of these three parameters,
respectively). The estimated values of parameter $\alpha$ for stations in the
Pearl River basin are given in Table 1. We found that the theoretical $K(q)$
curves based on the UMM fit exceedingly well the empirical $K(q)$ curves of
the rainfall data in all stations. We plot two fitted theoretical $K(q)$
curves in Figure 4 (the continuous lines) as illustrations. From the estimated
values of $H$, $C_{1}$ and $\alpha$, we find that $H\in[-0.0459,0.0196]$ with
mean value $-0.0085\pm 0.0126$, $C_{1}\in[0.0867,0.2665]$ with mean value
$0.1631\pm 0.0385$, and $\alpha\in[0.6213,1.6072]$ with mean value $1.0236\pm
0.2141$ for stations in Pearl River basin. The values of $H$ with mean value
$-0.0085\pm 0.0126$ for these daily rainfall data are close to zero,
indicating that they correspond to conserved fields which is consistent with
previously published results (e.g., [26-29]). Since the values of $\alpha$ are
fairly large (far from the monofractal value of zero), it again confirms that
all daily rainfall time series in Pearl River basin have multifractal behavior
in the time scale range from 180 to 960 days. The values of $C_{1}$ with mean
value $0.1631\pm 0.0385$ indicate that the conserved multifractal daily
rainfall is not too sparse [18], which can be compared with previously
published results [19, 23].
Second, we employed the MF-DFA to analyze the rainfall data. There are usually
seasonal variations in rainfall data. In order to get the long term
correlations correctly, the data need to be deseasonalized before we can
perform the MF-DFA [39, 40, 56-58]. In this paper, the deseasonalized rainfall
$z_{i}$ ($i=1,2,\cdots,N$, $N$ is the total number of data points) are
obtained by subtracting the mean daily rainfall $\overline{x_{i}}$ from the
original rainfall $x_{i}$ and normalized by variance at each calendar date
[40, 56-58], i.e.,
$z_{i}=(x_{i}-\overline{x_{i}})/(\overline{x_{i}^{2}}-\overline{x_{i}}^{2}).$
(9)
The deseasonalized rainfall was analyzed with MF-DFA. Here we calculated
$h(q)$ over the scale range of 10 to 87 days for all values of $q$ because the
log-log plot of $F_{q}(s)$ versus $s$ for each value of $q$ in this time scale
range becomes linear. An example for obtaining the empirical $h(q)$ curve is
given in Figure 5. The empirical $h(q)$ curves of the rainfall data in two
stations are shown in Figure 6 as examples. We observed that all the empirical
$h(q)$ curves of the rainfall data in all stations we considered are not
straight lines (i.e. are convex lines) like those in Figure 6. This suggests
that all daily rainfall time series have multifractal behavior in the time
scale range from 10 to 87 days. Usually the value of $\Delta h(q)$ (defined as
$\max\\{h(q)\\}-\max\\{h(q)\\}$) is used to characterize the multifractality
of time series. The estimated values of $h(1)$, $h(2)$ and $\Delta h(q)$ for
stations in Pearl River basin are given in Table 1. From Table 1, we find that
$h(2)\in[0.5248,0.6436]$ with mean value $0.5891\pm 0.0275$, $\Delta
h(q)\in[0.3724,0.8851]$ with mean value $0.5681\pm 0.1210$ for stations in
Pearl River basin. The values of $\Delta h(q)\in[0.3724,0.8851]$ obtained by
us with mean value $0.5681\pm 0.1210$ (far from the monofractal value of zero)
for stations in Pearl River basin again confirms that all daily rainfall time
series in Pearl River basin have multifractal behavior in the time scale range
from 10 to 87 days. It was reported that the scaling exponents of rainfall
obtained by DFA for the intermediate time scales (10.0 to 100.0-300.0 days)
range in values from 0.62 to 0.89 [36] without removing the seasonal trend in
the data. Later on, after removing the seasonal trend in the rainfall data,
Kantelhardt et al. [38] found that most precipitation records exhibit no long-
term correlations ($h(2)\approx 0.55$), the mean value is $h(2)=0.53\pm 0.04$.
The values of $h(2)\in[0.5248,0.6436]$ obtained by us with mean value
$0.5891\pm 0.0275$ for stations in Pearl River basin consists with the result
that precipitations are mainly uncorrelated reported in [38].
It is also interesting to test the relationship between $K(2)$ and $h(2)$
given by Eq. (7), i.e. whether $K(2)=2[h(1)-h(2)]$ holds. We denote
$K^{\prime}(2)$ to be $2[h(1)-h(2)]$. The estimated values of $K^{\prime}(2)$
for stations in Pearl River basin are given in Table 1. From Table 1, we find
that $K^{\prime}(2)\in[0.1960,0.5300]$ with mean value $0.2980\pm 0.0728$ for
stations in Pearl River basin. We find from Table 1 that the values of
$K^{\prime}(2)$ are quite different from those of $K(2)$, this because that
they are estimated for different time scale ranges.
Last, we want to see whether the parameters from these MFAs of daily rainfall
can reflect some spatial or geographical characteristics of the stations in
Pearl River basin. In other words, we would like to explore the spatial
dimension of rainfall variability in the basin. In particular, we are
interested in finding out whether rainfall variations over time are related
to, for example, the topography of the basin. A scrutiny of the parameters
$H$, $\alpha$ and $C_{1}$ in UMM, $K(2)$ in the $K(q)$ curves, and $h(2)$ from
MF-DFA show that there exhibit some correlations between rainfall regime and
basin characteristics such as topography. In fact, we found that the parameter
$K(2)$, which is related to the correlation dimension $D(2)$ via
$D(2)=1-K(2)$, of the daily rainfall data reflects some spatial and
geographical features of the stations in the basin. First, K(2) and elevation
series are negatively correlated. The value of the correlation coefficient
between $K(2)$ and elevation is up to -0.4995 in the Pearl River basin as
shown in Figure 7. The possible trend is that the higher the elevation at
which a station is located, the smaller the value of $K(2)$ becomes and the
closer it is to 0.0 (so also the larger the value of $D(2)$ becomes and the
closer it is to 1.0). According to the elevation, we can divide the stations
into three groups (Group 1 with elevation higher than 1000m, Group 2 with
elevation between 200m to 1000m, Group 3 with elevation lower than 200m). We
found that $K(2)$ of Group 1 have mean value $0.1927\pm 0.0110$, that of Group
2 have mean value $0.2000\pm 0.0181$ and that of Group 3 have mean value
$0.2155\pm 0.0202$. One can see that the mean value of $K(2)$ of these three
groups become larger with decreasing of elevation. We also notice that
rainfall stations at higher elevations in the northwestern side of the basin
similarly tend to have smaller $K(2)$ values in comparison with stations at
lower elevations in the southeastern side. Using the wavelet analysis on the
monthly precipitation data in Pearl River basin, Niu [59] recently found that,
apart from the high variability for the less than 1-year period, the high
wavelet power in the dominant band (0.84-4.8 years) for the first and second
modes (especially for northwest part and east part of Pearl River basin)
reflects long-term precipitation variability. Niu [59] explained that the
northwest region has the highest altitudes, and therefore it is influenced by
the topographic rain shadow with respect to the prevailing storm tracks; while
the east region is close to the South China Sea which is subjected to
convective movement of water by semitropical hurricanes and typhoons.
## 4 Conclusion
Multifractal analysis is a useful method to characterize the heterogeneity of
both theoretical and experimental fractal patterns. As a regional case study,
numerical results obtained from the universal multifractal approach and MF-DFA
on the daily rainfall data in Pearl River basin show that these time series
have multifractal behavior in two different time scale ranges. It is found
that the empirical $K(q)$ curves of the daily rainfall time series can be
fitted very well by the UMM. The estimated values of $H$ for these daily
rainfall data are close to zero, indicating a correspondence to the conserved
fields.
After removing the seasonal trend in the rainfall data, the estimated values
of $h(2)$ indicate that the daily rainfall time series in Pearl River basin
exhibit no long-term correlations.
It is found that $K(2)$ and elevation series are negatively correlated. It
shows a relationship between topography and rainfall variability.
## Acknowledgements
This project was supported by Geographical Modelling and Geocomputation
Program under the Focused Investment Scheme of The Chinese University of Hong
Kong, and the Earmarked grant CUHK405308 of the Research Grants Council of the
Hong Kong Special Administrative Region; the Natural Science Foundation of
China (Grant no. 11071282 and 11371016), the Chinese Program for Changjiang
Scholars and Innovative Research Team in University (PCSIRT) (Grant No.
IRT1179), the Research Foundation of Education Commission of Hunan Province of
China (grant no. 11A122), the Lotus Scholars Program of Hunan province of
China.
## References
* [1]
* [1] G.R. Walther, E. Post, P. Convey, A. Menzel, C. Parmesank, T.J.C. Beebee, J.-M. Fromentin, O. Hoegh-Guldberg and F. Bairlein, Ecological responses to recent climate change. Nature 416 (2002) 389-395.
* [2] J.L. Valencia, A.S. Requejo, J.M. Gasco, A.M. Tarquis, A universal multifractal description applied to precipitation patterns of the Ebro River Basin, Spain. Clim. Res. 44 (2010) 17-25.
* [3] Q. Zhang, C.-Y. Xu, Z. Zhang, Y.D. Chen, C.-L. Liu, Spatial and temporal variability of precipitation maxima during 1960-2005 in the Yangtze River basin and possible association with large-scale circulation. J. Hydrol. 353 (2008) 215-227.
* [4] D. Schertzer and S. Lovejoy, Physical modeling and analysis of rain and clouds by anisotropic scaling of multiplicative processes. J. Geophys. Res. 92(D8) (1987) 9693-9714.
* [5] V.K. Gupta and E. Waymire, Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res. 95(D3) (1990) 1999-2009.
* [6] T.M. Over and V.K. Gupta, Statistical analysis of masoscale rainfall: dependence of a random cascade generator on large-scale forcing. J. Appl. Meteorol. 33 (1994) 1526-1542.
* [7] F. Schmitt, S. Vannitsem and A. Barbosa, Modeling of rainfall time series using two-state renewal processes and multifractals. J. Geophys. Res. 103(D18) (1998) 23181-23193.
* [8] C. Svensson, J. Olsson and R. Berndtsson, Multifractal properties of daily rainfall in two different climates. Water Resour. Res. 32(8) (1996) 2463-2472.
* [9] B. Sivakumar, Fractal analysis of rainfall observed in two different climatic regions. Hydrol. Sci. J. 45(5) (2000) 727-738.
* [10] C. De Michele and P. Bernardara, Spectral analysis and modeling of space-time rainfall fields. Atmos. Res. 77 (2005) 124-136.
* [11] J. Olsson, J. Niemczynowicz, and R. Berndtsson, Fractal analysis of high-resolution rainfall time series. J. Geophys. Res. 98(D12) (1993) 23265-23274.
* [12] V. Venugopal, S.G Roux., E. Foufoula-Georgiou and A. Arneodo, Revisiting multifractality of high-resolution temporal rainfall using a wavelet-based formalism. Water Resour. Res. 42 (2006) W06D14.
* [13] P. Molnar and P. Burlando, Variability in the scale properties of high resolution precipitation data in the Alpine climate of Switzerland. Water Resour. Res. 44(10) (2008) W10404.
* [14] G. Boni, A. Parodi and F. Siccardi, A new parsimonious methodology of mapping the spatial variability of annual maximum rainfall in mountainous environments. J. Hydrometeorol. 9(3) (2008) 492-506.
* [15] D. Veneziano and P. Furcolo, Multifractality of rainfall and scaling of intensity-duration-frequency curves. Water Resour. Res. 38(12) (2002) 1306.
* [16] D. Veneziano, A. Langousis and P. Furcolo, Multifractality and rainfall extremes: A review. Water Resour. Res. 42 (2006) W06D15.
* [17] D. Veneziano and C. Lepore, The scaling of temporal rainfall. Water Resour. Res. 48 (2012) W08516.
* [18] Y. Tessier, S. Lovejoy and D. Schertzer, Universal multifractals: theory and observations for rain and clouds. J. Appl. Meteorol. 32(2) (1993) 223-250.
* [19] J. Olsson and J. Niemczynowicz, Multifractal analysis of daily spatial rainfall distributions. J. Hydrol. 187(1-2) (1996) 29-43.
* [20] S. Perica and E. Foufoula-Georgiou, Model for multiscale disaggregation of spatial rainfall based on coupling meteorological and scaling descriptions. J. Geophys. Res. 101 (D21) (1996) 26347-26341.
* [21] M. Menabde, D. Harris, A. Seed, G. Austin and D. Stow, Multiscaling properties of rainfall and bounded random cascades. Water Resour. Res. 33(12) (1997) 2823-2830.
* [22] G. Pandey, S. Lovejoy and D. Schertzer, Multifractal analysis of daily river flows including extremes for basins of five to two million square kilometres, one day to 75 years. J. Hydrol. 208(1-2) (1998) 62-81.
* [23] I. De Lima and J. Grasman, Multifractal analysis of 15-min and daily rainfall from a semi-arid region in Portugal. J Hydrol., 220 (1999) 1-11.
* [24] R. Deidda, R. Benzi and F. Siccardi, Multifractal modeling of anomalous scaling laws in rainfall. Water Resour. Res. 35 (6) (1999) 1853-1867.
* [25] M. Lilley, S. Lovejoy, N. Desaulniers-Soucy, D. Schertzer, Multifractal large number of drops limit in rain. J. Hydrol. 328 (2006) 20-37.
* [26] S. Lovejoy, D. Schertzer and V. Allaire, The remarkable wide range scaling of TRMM precipitation. Atmos. Res. 90 (2008) 10-32.
* [27] S. Lovejoy, J. Pinel, D. Schertzer, The global space-time cascade structure of precipitation: Satellites, gridded gauges and reanalyses. Adv. Water Resour. 45 (2012) 37-50.
* [28] A.P. Garcia-Marin, F.J. Jimenez-Hornero, and J.L. Ayuso-Munoz, Universal multifractal description of an hourly rainfall time series from a location in southern Spain. Atmosfera 21(4) (2008) 347-355.
* [29] F. Serinaldi, Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models. Nonlin. Processes Geophys. 17 (2010) 697-714.
* [30] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.I. Schraiman, Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33 (1986) 1141-1151.
* [31] J.W. Kantelhardt, S.A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde and H.E. Stanley, Multifractal detrended fluctuation analysis of nonstationary time series. Physica A 316 (2002) 87-114.
* [32] C.K. Peng, S.V. Buldyrev, A.L. Goldberger, S. Havlin, F. Sciortino, M. Simons, and H.E. Stanley, Long-range correlations in nucleotide sequences. Nature 356 (1992) 168-170.
* [33] C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley and A.L. Goldberger, Mosaic organization of DNA nucleotides. Phys. Rev. E 49 (1994) 1685-1689.
* [34] Z.G. Yu, V. Anh and B. Wang, Correlation property of length sequences based on global structure of complete genome. Phys. Rev. E 63 (2001) 011903\.
* [35] Z.G. Yu, V.V. Anh, K.S. Lau and L.Q. Zhou, Fractal and multifractal analysis of hydrophobic free energies and solvent accessibilities in proteins. Phys. Rev. E 73 (2006) 031920.
* [36] C. Matsoukas, S. Islam, I. Rodriguez-Iturbe, Detrended fluctuation analysis of rainfall and streamflow time series. J. Geophys. Res. 105 (D23) (2000) 29165-29172.
* [37] J.W. Kantelhardt, D. Rybski, S.A. Zschiegner, P. Braun, E. Koscielny-Bunde, V. Livina, S. Havlin and A. Bunde, Multifractality of river runoff and precipitation: comparison of fluctuation analysis and wavelet methods. Physica A 330 (1-2) (2003) 240-245.
* [38] J.W. Kantelhardt, E. Koscielny-Bunde, D. Rybski, P. Braun, A. Bunde and S. Havlin, Long-term persistence and multifractality of precipitation and river runoff records. J. Geophys. Res. 111(D1) (2006) D01106.
* [39] E. Koscielny-Bunde, J.W. Kantelhardt, P. Braun, A. Bunde and S. Havlin, Longterm persistence and multifractality of river runoff records: Detrended fluctuation studies. J. Hydrol. 322(1-4) (2006) 120-137.
* [40] Z.W. Li and Y.K. Zhang, Quantifying fractal dynamics of groundwater systems with detrended fluctuation analysis, J. Hydrol. 336(1-2) (2007) 139-146.
* [41] V. Anh, Z.G. Yu and J.A. Wanliss, Analysis of global geomagnetic variability. Nonlin. Processes Geophys. 14(6) (2007) 701-708.
* [42] V.V. Anh, J.M. Yong and Z.G. Yu, Stochastic modeling of the auroral electrojet index. J. Geophys. Res. 113 (2008) A10215.
* [43] Q. Zhang, C.-Y. Xu, Z.G. Yu, C.-L. Liu, Y.D. Chen, Multifractal analysis of streamflow records of the East River basin (Pearl River), China. Physica A 388 (2009) 927-934.
* [44] Q. Zhang, Z.G. Yu, C.-Y. Xu, V. Anh, Multifractal analysis of measure representation of flood/drought grade series in the Yangtze Delta, China, during the past millennium and their fractal model simulation. Int. J. Climatol. 30 (2010) 450-457.
* [45] Z.G. Yu, V. Anh and R. Eastes, Multifractal analysis of geomagnetic storm and solar flare indices and their class dependence. J. Geophys. Res. 114 (2009) A05214.
* [46] Z.G. Yu, V. Anh, Y. Wang, D. Mao and J. Wanliss, Modeling and simulation of the horizontal component of the geomagnetic field by fractional stochastic differential equations in conjunction with empirical mode decomposition. J. Geophys. Res. 115 (2010) A10219.
* [47] Q. Zhang, C.Y. Xu, S. Becker, Z.X. Zhang, Y.Q. Chen, M. Coulibaly, Trends and abrupt changes of precipitation maxima extremes in the Pearl River basin, China. Atmos. Sci. Lett. 10 (2009) 132 C144.
* [48] M. Gemmer, T. Fischer, B. Su, L.L. Liu, Trends of precipitation extremes in the Zhujiang River Basin, South China. J. Clim. 24 (2011)750 C761
* [49] V. Anh, K.S. Lau and Z.G. Yu, Multifractal characterisation of complete genomes. J. Phys. A: Math. Gen. 34(36) (2001) 7127-1739.
* [50] F. Schmitt, D. Lavallee, D. Schertzer and S. Lovejoy, Empirical determination of universal multifractal exponents in turbulent velocity fields. Phys. Rev. Lett., 68 (1992) 305-308.
* [51] D. Lavallee, S. Lovejoy, D. Schertzer and P. Ladoy, Nonlinear variability and landscape topography: analysis and simulation. In: Fractals in Geography (N. Lam and L. De Cola, Eds.) Prentice Hall, Englewood Cliffs, p158-192, 1993.
* [52] Z.G. Yu, V. Anh, R. Eastes and D.L. Wang, Multifractal analysis of solar flare indices and their horizontal visibility graphs. Nonlin. Processes Geophys. 19 (2012) 657-669.
* [53] M.S. Movahed, G.R. Jafari, F. Ghasemi, S. Rahvar and M.R.R. Tabar, Multifractal detrended fluctuation analysis of sunspot time series. J. Stat. Mech.: Theory Exper. 2 (2006) P02003.
* [54] S. Havlin, R. Selinger, M. Schwartz, H.E. Stanley, and A. Bunde, Random multiplicative processes and transport in structures with correlated spatial disorder. Phys. Rev. Lett. 61(13) (1988) 1438-1441.
* [55] Y. Zhou, Y. Leung and Z.G. Yu, Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis. Chin. Phys. B 20(9) (2011) 090507.
* [56] A.J. Lawrence, N.T. Kottegota, Stochastic modeling of riverflow time series. J. R. Stat. Soc., Ser. A (General) 140 (1) (1977) 1-47.
* [57] E. Kocielny-Bunde, A. Bunde, S. Havlin, Y. Goldreich, Analysis of daily temperature fluctuations. Physica A 231 (1996) 393-396.
* [58] V. Livina, Y. Ashkenazy, Z. Kizner, V. Strygin, A. Bunde, S. Halvin, A stochastic model of river discharge fluctuations. Physica A 330 (2003) 283-290.
* [59] J. Niu, Precipitation in the Pearl River basin, South China: scaling, regional patters, and influence of large-scale climate anomalies. Stoch. Environ. Res. Risk Assess. 27 (2013) 1253-1268.
Table 1: The geographical information of the rainfall stations and estimated multifractal parameters of the daily rainfall data in Pearl River basin. We list the stations according to the deceasing order of their elevations. | Station | Long. | Lat. | Elev. | | | | | |
---|---|---|---|---|---|---|---|---|---|---
Group | name | ( ∘) | ( ∘) | (m) | $\alpha$ | $h(1)$ | $\Delta h(q)$ | $h(2)$ | $K(2)$ | $K^{\prime}(2)$
| 56691 | 104.28 | 26.87 | 2237.5 | 0.9905 | 0.7239 | 0.4602 | 0.6106 | 0.1780 | 0.2266
| 56786 | 103.83 | 25.58 | 1998.7 | 1.2019 | 0.7279 | 0.6758 | 0.5666 | 0.1970 | 0.3226
| 56875 | 102.55 | 24.33 | 1716.9 | 0.9807 | 0.7898 | 0.8851 | 0.5248 | 0.1951 | 0.5300
Group 1 | 56886 | 103.77 | 24.53 | 1704.3 | 0.9563 | 0.7560 | 0.7340 | 0.5615 | 0.1877 | 0.3890
(with Elev. | 57806 | 105.90 | 26.25 | 1431.1 | 1.1437 | 0.6958 | 0.3929 | 0.5978 | 0.1886 | 0.1960
$\geq 1000$m | 57902 | 105.18 | 25.43 | 1378.5 | 1.0583 | 0.6908 | 0.4169 | 0.5797 | 0.1874 | 0.2222
| 56985 | 103.38 | 23.38 | 1300.7 | 0.9451 | 0.7567 | 0.4517 | 0.5805 | 0.2159 | 0.3524
| 57922 | 107.55 | 25.83 | 1013.3 | 1.1786 | 0.6932 | 0.3905 | 0.5902 | 0.1919 | 0.2060
| 59209 | 105.83 | 23.42 | 794.10 | 1.0300 | 0.7271 | 0.5094 | 0.5717 | 0.1745 | 0.3108
| 59218 | 106.42 | 23.13 | 739.90 | 1.1541 | 0.7188 | 0.6016 | 0.5779 | 0.1697 | 0.2818
Group 2 | 57906 | 106.08 | 25.18 | 566.80 | 0.9873 | 0.7125 | 0.4786 | 0.5816 | 0.2072 | 0.2618
(with Elev. | 59021 | 107.03 | 24.55 | 484.60 | 0.8460 | 0.6927 | 0.5752 | 0.5515 | 0.2009 | 0.2824
between | 57916 | 106.77 | 25.43 | 440.30 | 0.9970 | 0.7324 | 0.6086 | 0.5714 | 0.2105 | 0.3220
200m to | 59102 | 115.65 | 24.95 | 303.90 | 1.0614 | 0.7619 | 0.6160 | 0.6034 | 0.2184 | 0.3170
1000m) | 57932 | 108.53 | 25.97 | 285.70 | 1.0908 | 0.7102 | 0.5882 | 0.5807 | 0.2045 | 0.2590
| 59096 | 114.48 | 24.37 | 214.80 | 1.0230 | 0.7622 | 0.5354 | 0.6279 | 0.2145 | 0.2686
| 59211 | 106.60 | 23.90 | 173.50 | 0.7283 | 0.7395 | 0.5026 | 0.5893 | 0.2222 | 0.3004
| 59037 | 108.10 | 23.93 | 170.80 | 1.2448 | 0.7338 | 0.5106 | 0.6168 | 0.2316 | 0.2340
| 57957 | 110.30 | 25.32 | 164.40 | 0.8394 | 0.7308 | 0.7217 | 0.5804 | 0.2192 | 0.3008
| 59058 | 110.52 | 24.20 | 145.70 | 1.0215 | 0.7190 | 0.3825 | 0.6176 | 0.1916 | 0.2028
| 57996 | 114.32 | 25.13 | 133.80 | 0.8723 | 0.7465 | 0.5241 | 0.6200 | 0.1991 | 0.2530
| 59417 | 106.85 | 22.33 | 128.80 | 1.4062 | 0.7411 | 0.5340 | 0.6131 | 0.1880 | 0.2560
| 59431 | 108.22 | 22.63 | 121.60 | 1.1208 | 0.7404 | 0.4751 | 0.6208 | 0.2169 | 0.2392
| 57947 | 109.40 | 25.22 | 121.30 | 0.8441 | 0.7265 | 0.5512 | 0.5933 | 0.2393 | 0.2664
Group 3 | 59265 | 111.30 | 23.48 | 114.80 | 1.4781 | 0.7321 | 0.5647 | 0.5843 | 0.2116 | 0.2956
(with Elev. | 59065 | 111.53 | 24.42 | 108.80 | 0.8759 | 0.7425 | 0.4271 | 0.6272 | 0.1996 | 0.2306
$\leq 200$m) | 59072 | 112.38 | 24.78 | 98.30 | 1.0028 | 0.7403 | 0.4047 | 0.6287 | 0.1949 | 0.2232
| 59046 | 109.40 | 24.35 | 96.80 | 0.8041 | 0.7196 | 0.6851 | 0.5592 | 0.2191 | 0.3208
| 59242 | 109.23 | 23.75 | 84.90 | 0.7554 | 0.7342 | 0.5614 | 0.6077 | 0.2200 | 0.2530
| 59087 | 113.53 | 23.87 | 68.60 | 1.1332 | 0.7595 | 0.6136 | 0.6126 | 0.2201 | 0.2938
| 59082 | 113.60 | 24.68 | 61.00 | 0.9246 | 0.7466 | 0.6232 | 0.6155 | 0.2060 | 0.2622
| 59271 | 112.43 | 23.63 | 57.30 | 1.2472 | 0.7295 | 0.5859 | 0.5820 | 0.1889 | 0.2950
| 59462 | 111.57 | 22.77 | 53.30 | 1.3551 | 0.7411 | 0.5056 | 0.5901 | 0.2097 | 0.3020
| 59254 | 110.08 | 23.40 | 42.50 | 0.7011 | 0.7496 | 0.3724 | 0.6436 | 0.1978 | 0.2120
| 59278 | 112.45 | 23.03 | 41.00 | 1.6072 | 0.7472 | 0.7785 | 0.5413 | 0.1945 | 0.4118
| 59287 | 113.33 | 23.17 | 41.00 | 1.1680 | 0.7467 | 0.6313 | 0.5675 | 0.2113 | 0.3584
| 59293 | 114.68 | 23.73 | 40.60 | 0.9206 | 0.7711 | 0.5821 | 0.6040 | 0.2570 | 0.3342
| 59294 | 113.83 | 23.33 | 38.90 | 0.6213 | 0.7641 | 0.7561 | 0.5460 | 0.2104 | 0.4362
| 59478 | 112.78 | 22.25 | 32.70 | 0.8213 | 0.7829 | 0.6593 | 0.6031 | 0.2476 | 0.3596
| 59298 | 114.42 | 23.08 | 22.40 | 0.7185 | 0.7566 | 0.6993 | 0.5558 | 0.2308 | 0.4016
| 59493 | 114.10 | 22.55 | 18.20 | 1.1102 | 0.7695 | 0.7200 | 0.5552 | 0.2600 | 0.4286
mean | | | | | 1.0236 | 0.7381 | 0.5681 | 0.5891 | 0.2080 | 0.2980
$\pm$ std | | | | | $\pm$0.2141 | $\pm$0.0234 | $\pm$0.1210 | $\pm$0.0275 | $\pm$0.0205 | $\pm$0.0728
Figure 1: Location of the rain gauge stations in the Pearl River basin, China.
Figure 2: The daily rainfall data of station 56691 and Station 57922 in the
Pearl River basin over the entire study period.
Figure 3: An example for obtaining the empirical $K(q)$ curve.
Figure 4: The $K(q)$ curves of daily rainfall data in two stations (the dotted
curves), and their fitted curves (continuous lines) by the universal
multifractal model.
Figure 5: An example for obtaining the empirical $h(q)$ curve.
Figure 6: The $h(q)$ curves of daily rainfall data in two stations.
Figure 7: The correlation relationship between the elevation of the rainfall
stations and the $K(2)$ value of the rainfall time series for the Pearl River
basin.
|
arxiv-papers
| 2014-02-17T15:23:52 |
2024-09-04T02:49:58.322397
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zu-Guo Yu, Yee Leung, Yongqin David Chen, Qiang Zhang, Vo Anh and Yu\n Zhou",
"submitter": "Zu-Guo Yu",
"url": "https://arxiv.org/abs/1402.4030"
}
|
1402.4205
|
Method of Studying $\mathchar 28931\relax^{0}_{b}$ decays with one missing
particle
Sheldon Stone and Liming Zhang
Physics Department Syracuse University, Syracuse, NY, USA 13244-1130
A new technique is discussed that can be applied to $\mathchar
28931\relax^{0}_{b}$ baryon decays where decays with one missing particle can
be discerned from background and their branching fractions determined, along
with other properties of the decays. Applications include measurements of the
CKM elements $|V_{ub}|$ and $|V_{cb}|$, and detection of any exotic objects
coupling to $b\rightarrow s$ decays, such as the inflaton. Potential use of
$\overline{B}^{0**}\rightarrow\pi^{+}B^{-}$ and
$\overline{B}_{s}^{0**}\rightarrow K^{+}B^{-}$ to investigate $B^{-}$ decays
is also commented upon.
Submitted to Advances in High Energy Physics
## 1 Introduction
Detection of $b$-flavored hadron decays with one missing neutral particle,
such as a neutrino, is important for many measurements and searches. These
include semileptonic decays, such as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow D\mu^{-}\overline{\nu}$,
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow\pi\mu^{-}\overline{\nu}$,
and any exotic long lived particles that could be produced in decays such as
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow X\chi$, where the $X$
is any combination of detected particles and the $\chi$ escapes the
detector.111In this paper mention of a particular decay mode implies the use
of the charge-conjugated mode as well. These measurements are possible at an
$e^{+}e^{-}$ collider operating at the $\Upsilon(4S)$. Since
$\Upsilon(4S)\rightarrow B\overline{B}$, fully reconstructing either the $B$
or the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ determines the negative
of the initial four-momentum of the other. With this information it is
possible to measure final states where one particle is not detected, such as a
neutrino. To implement this procedure, taking the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ to be fully reconstructed, the missing
mass-squared, $m_{x}^{2}$, is calculated including the information on the
initial $B$ four-momentum and measurements of the found $X$ particles as
$m_{x}^{2}=(E_{B}-E_{X})^{2}-(\overrightarrow{p}\\!\\!_{B}-\overrightarrow{p}\\!\\!_{X})^{2},$
(1)
where $E$ and $\overrightarrow{p}$ indicate energy and three-momentum,
respectively. Peaks in $m_{x}^{2}$ would be indicative of single missing
particles in the $B$ decay.
A related example is charm semileptonic decays with a missing neutrino.
Determinations of branching fractions and form-factors have been carried out
in fixed target experiments, exploiting the measured direction of the charmed
hadron and assuming that the missing particle has zero mass, which leads to a
two-fold ambiguity in the neutrino momentum calculation [1]. If the charm
decay particle is a $D^{0}$, extra constraints can be imposed on its decay
requiring it to be produced from a $D^{*+}$ in the decay
$D^{*+}\rightarrow\pi^{+}D^{0}$. This leads to more constraints than unknowns,
and is quite useful for rejecting backgrounds [2, *Agostino:2004na].
Interesting decays of the $\mathchar 28931\relax^{0}_{b}$ baryon also exist,
but investigations are not feasible in the $\Upsilon(4S)$ energy region.
Potential studies include determination of the CKM matrix element $|V_{cb}|$,
possible using $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax_{c}^{+}\ell^{-}\overline{\nu}$ decays and $|V_{ub}|$ using the
$\mathchar 28931\relax^{0}_{b}\rightarrow p\ell^{-}\overline{\nu}$ mode.
Neutral particles that have not yet been seen could be searched for, even if
they are stable or have long enough lifetimes that they would have only a very
small fraction of their decays inside the detection apparatus. One example of
such a possibly long-lived particle is the “inflaton.” This particle couples
to a scalar field and is responsible for cosmological inflaton. Bezrukov and
Gorbunov predicted branching fractions and decay modes of inflatons, $\chi$,
in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ meson decays [4] using a
specific model, which is a particular version of the simple chaotic inflation
with a quartic potential and having the inflaton field coupled to the SM Higgs
boson via a renormalizable operator. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow\chi X_{s}$ decays the
branching fraction is
$\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow\chi X_{s})$
$\displaystyle\simeq
0.3\frac{\left|V_{ts}V_{tb}^{*}\right|^{2}}{\left|V_{cb}\right|^{2}}\left(\frac{m_{t}}{M_{W}}\right)^{4}\left(1-\frac{m_{\chi}^{2}}{m_{b}^{2}}\right)^{2}\theta^{2}$
(2) $\displaystyle\simeq
10^{-6}\cdot\left(1-\frac{m_{\chi}^{2}}{m_{b}^{2}}\right)^{2}\left(\frac{\beta}{\beta_{0}}\right)\left(\frac{\rm
300~{}MeV}{m_{\chi}}\right)^{2},$
where $X_{s}$ stands for strange meson channels mostly saturated by a sum of
$K$ and $K^{*}(890)$ mesons, $m_{\chi}$ and $m_{t}$, the inflaton and top
quark masses, respectively. The model parameters are $\theta$, $\beta$ and
$\beta_{0}$, where $\beta/\beta_{0}\approx{\cal{O}}(1).$ Their inflaton
branching fraction predictions are shown in Fig. 1(a). The branching fractions
are quite similar for $\mathchar 28931\relax^{0}_{b}$ decays. The $\mathchar
28931\relax^{0}_{b}\rightarrow pK^{-}\chi$ channel would seem to be the most
favorable, since the $\mathchar 28931\relax^{0}_{b}$ decay point could be
accurately determined from the $pK^{-}$ vertex. The mass dependent inflaton
branching fraction predictions for different decay modes are shown in Fig.
1(a). Collider searches that rely on directly detecting the inflaton decay
products may not be sensitive to lifetimes much above a few times 1 ns [5],
because the particles mostly decay outside of the detector, while searches
that could be done inclusively, e.g., without detecting the inflaton decay
products, would be independent of this restriction.
Figure 1: Predictions from Ref. [4]. (a) Inflaton branching ratios to various
two-body final states as functions of inflaton mass for $m_{\chi}<1.5$ GeV.
Above 2.5 GeV only quark-antiquark and dilepton modes are predicted. In the
intermediate region no reliable prediction is given. (b) Inflaton lifetime
$\tau_{\chi}$ as a function of the inflaton mass $m_{\chi}$. The lifetime can
be up to two times smaller, depending on model-dependent parameters.
Use of $\mathchar 28931\relax^{0}_{b}$ decays in measuring CKM matrix elements
as well as new particle searches has been not as fruitful as in $B$ meson
decays because $e^{+}e^{-}$ machines have access only to the lighter $B$
mesons. In addition, absolute branching fraction determinations have been made
difficult by the relatively large uncertainty on ${\cal{B}}(\mathchar
28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$. Recently, the Belle
collaboration reduced this uncertainty from 25% to about 5%, allowing for
measurements with much better precision [6].
Inclusive decay searches using $\mathchar 28931\relax^{0}_{b}$ baryons can be
made at high energy colliders if it were possible to find a way to estimate
the $\mathchar 28931\relax^{0}_{b}$ momentum. The $\mathchar
28931\relax^{0}_{b}$ direction is measured by using its finite decay distance.
To get an estimate of the $\mathchar 28931\relax^{0}_{b}$ energy we can use
$\mathchar 28931\relax^{0}_{b}$’s that come from $\mathchar
28934\relax_{b}^{\pm}\rightarrow\pi^{\pm}\mathchar 28931\relax^{0}_{b}$ and
$\mathchar 28934\relax_{b}^{*\pm}\rightarrow\pi^{\pm}\mathchar
28931\relax^{0}_{b}$ decays. The $\mathchar 28934\relax_{b}^{(*)\pm}$ states
were found by the CDF collaboration [7]. Their masses and widths are
consistent with theoretical predictions [8].
The $\mathchar 28931\relax^{0}_{b}$ energy is determined from the measurement
of the $\pi^{\pm}$ from the $\mathchar 28934\relax_{b}^{(*)}$ decay along with
the $\mathchar 28931\relax^{0}_{b}$ direction. Let us assume we have a pion
from the $\mathchar 28934\relax_{b}^{(*)}$ decay. Then
$m_{\mathchar 28934\relax_{b}^{(*)}}^{2}=(E_{\pi}+E_{\mathchar
28931\relax^{0}_{b}})^{2}-(\overrightarrow{p}\\!\\!_{\pi}+\overrightarrow{p}\\!\\!_{\mathchar
28931\relax^{0}_{b}})^{2},$ (3)
and after some algebraic manipulations we find
$\displaystyle|p_{\mathchar 28931\relax^{0}_{b}}|$ $\displaystyle=$
$\displaystyle(-b\pm\sqrt{b^{2}-4ac})/(2a)$ (4) $\displaystyle a$
$\displaystyle=$ $\displaystyle 4(E^{2}_{\pi}-p^{2}_{\pi}\cos^{2}\theta)$
$\displaystyle b$ $\displaystyle=$
$\displaystyle-4p_{\pi}\Delta^{2}_{m}\cos\theta$ $\displaystyle c$
$\displaystyle=$ $\displaystyle 4E^{2}_{\pi}m^{2}_{\mathchar
28931\relax^{0}_{b}}-\Delta^{4}_{m}$ $\displaystyle\Delta^{2}_{m}$
$\displaystyle=$ $\displaystyle m^{2}_{\mathchar
28934\relax_{b}^{(*)}}-m^{2}_{\pi}-m^{2}_{\mathchar 28931\relax^{0}_{b}},$
where $\cos\theta$ is the measured angle between the pion and the $\mathchar
28931\relax^{0}_{b}$, and $m_{\mathchar 28934\relax_{b}^{(*)}}$ indicates
either the $\mathchar 28934\relax_{b}$ or $\mathchar 28934\relax_{b}^{*}$
mass. With the measured $\mathchar 28931\relax^{0}_{b}$ direction and
$\mathchar 28931\relax^{0}_{b}$ energy Eq. (1) can now be used to find decays
with any number of detected and one missing particle. Two possible solutions
result because of the $\pm$ sign in the first line of Eq. 4. In similar
studies one solution is often unphysical. This was seen, for example, using
$D^{*+}\rightarrow\pi^{+}D^{0}$, $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$
decays with one missing pion [9]. The resolution in $m_{x}^{2}$ depends on
several quantities including the measurement uncertainties on momentum of the
final state particles and the $\mathchar 28931\relax^{0}_{b}$ direction, so it
may be advantageous for analyses to select long-lived decays at the expense of
statistics. The relatively long $\mathchar 28931\relax^{0}_{b}$ lifetime of
about 1.5 ps is helpful in this respect [10, *Aaij:2014owa].
The $\mathchar 28934\relax_{b}^{(*)\pm}$ states have only been seen by CDF
[7]. Their data are shown in Fig. 2, and listed in Table 1.
Figure 2: The ${Q=M(\mathchar 28931\relax^{0}_{b}\pi^{\pm})-M(\mathchar 28931\relax^{0}_{b})-m_{\pi}}$ spectrum for candidates with the projection of the corresponding unbinned likelihood fit superimposed, (a) for $\pi^{+}\mathchar 28931\relax^{0}_{b}$ and (b) for $\pi^{-}\mathchar 28931\relax^{0}_{b}$ candidates. (From Ref. [7]). Table 1: Summary of the results of the fits to the ${Q=M(\mathchar 28931\relax^{0}_{b}\pi^{\pm})-M(\mathchar 28931\relax^{0}_{b})-m_{\pi}}$ spectra from CDF [7]. State | $Q$ value, | Natural width, | Yield
---|---|---|---
| MeV | $\Gamma_{0}$, MeV |
${\mathchar 28934\relax_{b}^{-}}$ | ${56.2}_{-0.5}^{+0.6}$ | ${4.9}_{-2.1}^{+3.1}$ | $340_{-70}^{+90}$
${\mathchar 28934\relax_{b}^{*-}}$ | ${75.8}\pm{0.6}$ | ${7.5}_{-1.8}^{+2.2}$ | $540_{-80}^{+90}$
${\mathchar 28934\relax_{b}^{+}}$ | ${52.1}_{-0.8}^{+0.9}$ | ${9.7}_{-2.8}^{+3.8}$ | $470_{-90}^{+110}$
${\mathchar 28934\relax_{b}^{*+}}$ | ${72.8}\pm{0.7}$ | ${11.5}_{-2.2}^{+2.7}$ | $800_{-100}^{+110}$
## 2 Potential measurements
Although there is no measurement of the relative $\mathchar
28934\relax_{b}^{(*)\pm}/\mathchar 28931\relax^{0}_{b}$ production cross-
section, $r_{\mathchar 28934\relax\Lambda}$, one might imagine that the
production ratio would be close to unity. The pions from the $\mathchar
28934\relax_{b}^{(*)\pm}$ decays have relatively low momenta, so their
detection efficiencies could be small. Although CDF does not report a value
for the production ratio, the number of seen signal events gives an observed
value of $r_{\mathchar 28934\relax\Lambda}$ equal to 13%. This is certainly a
useful sample. Backgrounds will be an issue, however, as the CDF data do show
a substantial amount of non-resonant combinations under the signal peaks, but
this will not prevent searches, just limit their sensitivities with a given
data sample.
Measurement of $|V_{cb}|$, determined using $\mathchar
28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax_{c}^{+}\ell^{-}\overline{\nu}$ decays with $\mathchar
28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+}$ would provide an important
cross-check on this important fundamental parameter, especially when updated
lattice gauge calculations become available [12]. This measurement is not
subject to the uncertainty on ${\cal{B}}(\mathchar
28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$ provided that the total number
of $\mathchar 28931\relax^{0}_{b}$ events in the event sample is determined
using the same branching fraction [13]. The LHCb determination of the ratio of
$\mathchar 28931\relax^{0}_{b}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production, for example, uses the
$\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+}$ decay mode [14], and
then the absolute number of $\mathchar 28931\relax^{0}_{b}$ events produced is
found by measuring the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rate
in a channel with a known branching fraction. The branching ratio for the
channel $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax_{c}^{+}\ell^{-}\overline{\nu}$ can be determined using Eq. (1)
using the measured value for the $\mathchar 28931\relax^{0}_{b}$ energy
determined by using Eq. (4); a signal would appear near $m_{x}^{2}$ equal to
zero. To determine the four-momentum transfer squared from the $\mathchar
28931\relax^{0}_{b}$ to the $\mathchar 28931\relax_{c}^{+}$ a similar
procedure as used in the decay sequence $D^{*+}\rightarrow D^{0}\pi^{+}$,
$D^{0}\rightarrow K^{*-}\ell^{+}\nu$ can be implemented [2, *Agostino:2004na].
In this procedure, the neutrino mass is set to zero,
$(E_{\mathchar 28931\relax^{0}_{b}}-E_{X})^{2}-(\vec{p}_{\mathchar
28931\relax^{0}_{b}}-\vec{p}_{X})^{2}=m_{x}^{2}=0,$ (5)
where $X$ represents the sum of $\mathchar 28931\relax_{c}^{+}$ and $\ell^{-}$
energies and momenta. Eq. (3) and Eq. (5) can be used as two constraint
equations with one unknown variable $|p_{\mathchar 28931\relax^{0}_{b}}|$.
Measurement of $|V_{ub}|$ using $\mathchar 28931\relax^{0}_{b}\rightarrow
p\ell^{-}\overline{\nu}$ decays is subject to the uncertainty on
${\cal{B}}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$ , but here
the current precision of 5% on this branching fraction is sufficient.
Theoretical calculations of the decay width from the lattice gauge
calculations done in a limited four-momentum transfer range [15], light cone
sum rules [16, 17, *Azizi:2009wn, *Huang:2004vf], and QCD sum rules [20,
*Huang:1998rq] can be used to extract $|V_{ub}|$. The
$p\ell^{-}\overline{\nu}$ final state is subject to backgrounds from
$N^{*}\ell^{-}\overline{\nu}$, where $N^{*}\rightarrow p\pi^{0}$, that are
difficult to eliminate and thus the use of the $\mathchar
28934\relax_{b}^{(*)\pm}\rightarrow\pi^{\pm}\mathchar 28931\relax^{0}_{b}$
decay sequence may be crucial. The decay sequence constraint can also possibly
help measure the branching fraction for $\mathchar
28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax_{c}^{(*)+}\tau^{-}\overline{\nu}$ decays as measurements in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ meson system of analogous
decays are somewhat larger than Standard Model predictions [22,
*Adachi:2009qg, *Bozek:2010xy].
Particles characteristic of scalar fields such as inflatons or dilatons can be
searched for in $\mathchar 28931\relax^{0}_{b}$ decays. It is also possible to
search for Majorana neutrinos through a process similar to that used for
searches in $B^{-}\rightarrow\mu^{-}\mu^{-}\pi^{+}$ decays [5, 25,
*Aaij:2011ex], where the Majorana neutrino, $\nu_{M}$, decays into a
$\mu^{-}\pi^{+}$ pair. The initial quark content of the $\mathchar
28931\relax^{0}_{b}$ is $bud$. The $b$-quark can annihilate with a
$\overline{u}$-quark from a $u\overline{u}$ pair arising from the vacuum into
a virtual $W^{-}$ leaving a $uud$ system that can form a $p$. The virtual
$W^{-}$ then can decay into $\mu^{-}$ in association with a Majorana neutrino
that can transform to its own anti-particle and decay into $\mu^{-}$ and a
virtual $W^{+}$. In the analogous case to the
$B^{-}\rightarrow\mu^{-}\mu^{-}\pi^{+}$ decay, the $W^{+}$ would decay into a
$\pi^{+}$, however here we do not have to detect the Majorana decays, so we
can look for the decay $\mathchar 28931\relax^{0}_{b}\rightarrow
p\mu^{-}\nu_{M}$ independently of the $\nu_{M}$ decay mode or lifetime. Other
mechanisms for Majorana neutrino production discussed in Ref. [27] for $B^{-}$
decays when adopted to $\mathchar 28931\relax^{0}_{b}$ decays, would also lead
to the $p\mu^{-}\nu_{M}$ final state.
A more mundane search can be considered for $\mathchar 28931\relax^{0}_{b}$
decays into non-charmed final states containing $\mathchar 28934\relax^{\pm}$
light baryons; these have been proposed for flavor SU(3) tests [28]. Since the
largest decay modes are $\mathchar 28934\relax^{-}\rightarrow n\pi^{-}$, and
$\mathchar 28934\relax^{+}\rightarrow n\pi^{+}$ or $p\pi^{0}$, there is always
a missing neutron in the $\mathchar 28934\relax^{-}$ decay, while the
$\mathchar 28934\relax^{+}$, in principle, can be detected in the $p\pi^{0}$
mode. The method suggested here can be adopted to search for both $\mathchar
28934\relax^{-}$ and $\mathchar 28934\relax^{+}$ baryons in $\mathchar
28931\relax^{0}_{b}$ decays.
Note that similar methods can be applied to $B^{-}$ decays by the use of the
$\overline{B}^{0**}\rightarrow\pi^{+}B^{-}$ decay sequence. The measured
production ratio of
$\left(\overline{B}^{0**}\rightarrow\pi^{+}B^{-}\right)/B^{-}$ is about 15%,
but the $B^{0**}$’s have widths of about 130 MeV which introduces very large
backgrounds that have thus far precluded their use. Another possible source of
tagged $B^{-}$ events is the decays of $\overline{B}_{s}^{0**}$ mesons into a
$K^{+}B^{-}$ that would have the advantage of a charged kaon tag, and have a
much narrower width.
In conclusion, we propose a new method of analyzing $\mathchar
28931\relax^{0}_{b}$ decays into one missing particle, where the $\mathchar
28931\relax^{0}_{b}$ is part of a detected $\mathchar
28934\relax_{b}^{(*)\pm}\rightarrow\pi^{\pm}\mathchar 28931\relax^{0}_{b}$
decay that provides additional kinematic constraints. This method may be
useful for studies of CKM elements and searches for new particles such as
inflatons, dilatons or Majorana neutrinos. Thus, investigations of $\mathchar
28931\relax^{0}_{b}$ decays may present a unique opportunity in the study of
$b$-flavored hadron decays.
## Acknowledgements
We are grateful for the support of the National Science Foundation, and
discussions with Jon Rosner, and many of our LHCb colleagues.
## References
* [1] E791 Collaboration, E. Aitala et al., Measurement of the form-factor ratios for $D^{+}\rightarrow\overline{K}^{*0}\ell^{+}\nu_{\ell}$, Phys. Lett. B440 (1998) 435, arXiv:hep-ex/9809026
* [2] FOCUS Collaboration, J. Link et al., Analysis of the semileptonic decay $D^{0}\rightarrow\overline{K}^{0}\pi^{-}\mu^{+}\nu$, Phys. Lett. B607 (2005) 67, arXiv:hep-ex/0410067
* [3] L. Agostino, Pseudoscalar Semileptonic Decays of the $D^{0}$ Meson, 2004. FERMILAB-THESIS-2004-59, UMI-31-53801-MC
* [4] F. Bezrukov and D. Gorbunov, Light inflaton Hunter’s Guide, JHEP 1005 (2010) 010, arXiv:0912.0390
* [5] LHCb Collaboration, R. Aaij et al., Search for Majorana neutrinos in $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$ decays, arXiv:1401.5361
* [6] Belle Collaboration, A. Zupanc et al., Measurement of the Branching Fraction ${\cal{B}}(\mathchar 28931\relax_{c}\rightarrow pK^{-}\pi^{+})$, arXiv:1312.7826
* [7] CDF Collaboration, T. Aaltonen et al., Measurement of the masses and widths of the bottom baryons $\mathchar 28934\relax_{b}^{\pm}$ and $\mathchar 28934\relax_{b}^{*\pm}$, Phys. Rev. D85 (2012) 092011, arXiv:1112.2808
* [8] M. Karliner, B. Keren-Zur, H. J. Lipkin, and J. L. Rosner, The Quark Model and $b$ Baryons, Annals Phys. 324 (2009) 2, arXiv:0804.1575
* [9] LHCb Collaboration, R. Aaij et al., Measurement of the $D_{s}^{+}-D_{s}^{-}$ production asymmetry in 7 TeV $pp$ collisions, Phys. Lett. B713 (2012) 186, arXiv:1205.0897
* [10] LHCb collaboration, R. Aaij et al., Precision measurement of the $\mathchar 28931\relax^{0}_{b}$ baryon lifetime, Phys. Rev. Lett. 111 (2013) 102003, arXiv:1307.2476
* [11] LHCb collaboration, R. Aaij et al., Measurements of the $B^{+}$, $B^{0}$, $B_{s}^{0}$ meson and $\mathchar 28931\relax^{0}_{b}$ baryon lifetimes, arXiv:1402.2554
* [12] UKQCD Collaboration, K. Bowler et al., First lattice study of semileptonic decays of $\mathchar 28931\relax^{0}_{b}$ and $\mathchar 28932\relax_{b}$ baryons, Phys. Rev. D57 (1998) 6948, arXiv:hep-lat/9709028
* [13] LHCb Collaboration, R. Aaij et al., Study of the kinematic dependences of $\mathchar 28931\relax^{0}_{b}$ production in $pp$ collisions and a measurement of the $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax_{c}^{+}\pi^{-}$ branching fraction, arXiv:1405.6842
* [14] LHCb Collaboration, Updated average $f_{s}/f_{d}$ $b$-hadron production fraction ratio for 7 TeV $pp$ collisions, 2013\. LHCb-CONF-2013-011, CERN-LHCb-CONF-2013-011
* [15] W. Detmold, C. J. D. Lin, S. Meinel, and M. Wingate, $\mathchar 28931\relax^{0}_{b}\rightarrow pl^{-}\bar{\nu}$ form factors from lattice QCD with static $b$ quarks, Phys. Rev. D88 (2013) 014512, arXiv:1306.0446
* [16] A. Khodjamirian, C. Klein, T. Mannel, and Y.-M. Wang, Form Factors and Strong Couplings of Heavy Baryons from QCD Light-Cone Sum Rules, JHEP 1109 (2011) 106, arXiv:1108.2971
* [17] Y.-M. Wang, Y.-L. Shen, and C.-D. Lu, $\mathchar 28931\relax^{0}_{b}\rightarrow p$, $\mathchar 28931\relax$ transition form factors from QCD light-cone sum rules, Phys. Rev. D80 (2009) 074012, arXiv:0907.4008
* [18] K. Azizi, M. Bayar, Y. Sarac, and H. Sundu, Semileptonic $\mathchar 28931\relax_{b,c}$ to Nucleon Transitions in Full QCD at Light Cone, Phys. Rev. D80 (2009) 096007, arXiv:0908.1758
* [19] M.-q. Huang and D.-W. Wang, Light cone QCD sum rules for the semileptonic decay $\mathchar 28931\relax^{0}_{b}\rightarrow p\mu^{-}\overline{\nu}$, Phys. Rev. D69 (2004) 094003, arXiv:hep-ph/0401094
* [20] R. Marques de Carvalho et al., Form-factors and decay rates for heavy Lambda semileptonic decays from QCD sum rules, Phys. Rev. D60 (1999) 034009, arXiv:hep-ph/9903326
* [21] C.-S. Huang, C.-F. Qiao, and H.-G. Yan, Decay $\mathchar 28931\relax^{0}_{b}\rightarrow p\mu^{-}\overline{\nu}$ in QCD sum rules, Phys. Lett. B437 (1998) 403, arXiv:hep-ph/9805452
* [22] BaBar Collaboration, J. Lees et al., Measurement of an excess of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow D^{(*)}\tau^{-}\overline{\nu}$ decays and implications for charged Higgs bosons, Phys. Rev. D88 (2013) 072012, arXiv:1303.0571
* [23] Belle Collaboration, I. Adachi et al., Measurement of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow D^{(*)}\tau^{-}\overline{\nu}$ using full reconstruction tags, arXiv:0910.4301
* [24] Belle Collaboration, A. Bozek et al., Observation of $B^{+}\rightarrow\overline{D}^{*0}\tau^{+}\nu$ and evidence for $B^{+}\rightarrow\overline{D}^{0}\tau^{+}\nu$ at Belle, Phys. Rev. D82 (2010) 072005, arXiv:1005.2302
* [25] LHCb Collaboration, R. Aaij et al., Searches for Majorana neutrinos in $B^{-}$ decays, Phys. Rev. D85 (2012) 112004, arXiv:1201.5600
* [26] LHCb Collaboration, R. Aaij et al., Search for the lepton number violating decays $B^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$ and $B^{+}\rightarrow K^{-}\mu^{+}\mu^{+}$, Phys. Rev. Lett. 108 (2012) 101601, arXiv:1110.0730
* [27] G. L. Castro and N. Quintero, Bounding resonant Majorana neutrinos from four-body $B$ and $D$ decays, Phys. Rev. D87 (2013) 077901, arXiv:1302.1504
* [28] M. Gronau and J. L. Rosner, Flavor SU(3) and $\mathchar 28931\relax^{0}_{b}$ decays, Phys. Rev. D89 (2014) 037501, arXiv:1312.5730
|
arxiv-papers
| 2014-02-18T02:13:46 |
2024-09-04T02:49:58.337696
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sheldon Stone and Liming Zhang",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1402.4205"
}
|
1402.4221
|
# Blow-up formulae of high genus Gromov-Witten invariants in dimension six
Weiqiang He1 Department of Mathematics
Sun Yat-Sen University
Guangzhou, 510275
China [email protected] , Jianxun Hu2 Department of Mathematics
Sun Yat-Sen University
Guangzhou, 510275
China [email protected] , Hua-Zhong Ke Department of Mathematics
Sun Yat-Sen University
Guangzhou, 510275
China [email protected] and Xiaoxia Qi Sino-French Institute of
Nuclear and Technology
Sun Yat-sen University
Tang Jia Wan, Zhuhai 519082
China [email protected]
###### Abstract.
Using the degeneration formula and absolute/relative correspondence, one
studied the change of Gromov-Witten invariants under blow-up for six
dimensional symplectic manifolds and obtained closed blow-up formulae for high
genus Gromov-Witten invariants. Our formulae also imply some relations among
generalized BPS numbers introduced by Pandharipande.
Key words: Gromov-Witten invariant, Blow-up, Degeneration formula,
Absolute/relative correspondence, Degenerate contribution
1Partially supported by China Scholarship Council
2Partially supported by NSFC Grant 11228101 and 11371381
###### Contents
1. 1 Introduction
2. 2 Preliminaries
3. 3 Formulae for Blow-up at a point
4. 4 Formulae for Blow-up along a smooth curve
5. 5 Generalized BPS numbers
## 1\. Introduction
Gromov-Witten invariants count stable pseudo-holomorphic curves in a
symplectic manifold. The Gromov-Witten invariants for semi-positive symplectic
manifolds were first defined by Ruan [R1] and Ruan-Tian [RT1, RT2]. Gromov-
Witten invariants can be applied to define a quantum product on the cohomology
groups of a symplectic manifold in [RT1] and have many applications in
symplectic geometry and symplectic topology, see [MS] and references therein.
Using the virtual moduli cycle technique, Li-Tian [LT1] defined the Gromov-
Witten invariants purely algebraically for smooth projective varieties. During
last two decades, there were a great deal of activities to remove the semi-
positivity condition, see [B, FO, R2, S, LT2]. After its mathematical
foundation was established, the study of Gromov-Witten theory focused on its
computation and applications. We now know a lot about genus zero invariants
of, say, toric manifolds, homogeneous spaces, etc. Some of the higher genus
computations have also been done, but the understanding of higher genus
Gromov-Witten invariants is still far from complete.
The computation of the Gromov-Witten invariants is known to be a difficult
problem in geometry and physics. There are two major techniques: the
degeneration formula and localization. Li-Ruan [LR] first obtained the
degeneration formula, see [IP] for a different version and [Li] for an
algebraical version. It used to be applied to the situations that a symplectic
or Kahler manifold $X$ degenerates into a union of two pieces $X^{\pm}$ glued
along a common divisor $Z$. The idea of degeneration formula is to express the
Gromov-Witten invariants of $X$ in terms of relative Gromov-Witten invariants
of the pairs $(X^{\pm},Z)$. Localization played a very important role in the
computation of Gromov-Witten invariants. Kontsevich [Ko2] first introduced
this technique into this field, then Givental [Gi] and Lian-Liu-Yau [LLY]
applied this technique to prove the mirror theorem in the genus zero case. So
far the computation of high genus invariants is still a difficult task. The
difficulty is that the localization technique often transfers the computation
of high genus invariants into that of some Hodge integrals over
$\bar{\mathscr{M}}_{g,n}$, which so far one does not have effective methods to
compute. To obtain some general structures or close formulae of Gromov-Witten
theory in many applications, we degenerate a symplectic or Kahler manifold
into two toric relative pairs $(X^{\pm},Z)$ and then use the localization
technique to compute the associated relative invariants, see [HLR, MP]. The
combination of the degeneration technique and localization technique has
proven to be very powerful.
Ruan [R3] speculated that there should be a deep relation between quantum
cohomology and birational geometry. The birational symplectic geometry program
requires a thorough understanding of blow-up type formula of Gromov-Witten
invariants and quantum cohomology, because blow-up is the elementary
birational surgery. Actually, it is rare to be able to obtain a general blow-
up formula. For the last twenty years, only a few limited case were known, see
[H1, H2, G]. Hu-Li-Ruan [HLR] studied the change of Gromov-Witten invariants
under blow-up and obtained a blow-up correspondence of absolute/relative
Gromov-Witten invariants. The second named author [H1, H2] obtained some blow-
up formulae for genus zero Gromov-Witten invariants. In this paper, we try to
apply the degeneration formula to study the change of Gromov-Witten invariants
under blow-ups and generalize a genus zero formula in [H1] to all genera case
in dimension six.
Throughout this paper, let $X$ be a connected, closed, smooth symplectic
manifold of real dimension six, and $p:\tilde{X}\rightarrow X$ the natural
projection of the symplectic blow-up $\tilde{X}$ of $X$ along a connected
smooth symplectic submanifold of $X$. Let $E$ be the exceptional divisor of
the blow-up, and $e\in H_{2}(\tilde{X},{\mathbb{Z}})$ the class of a line in
the fiber of $E$. Note that $p$ induces a natural injection via ’pullback’ of
$2$-cycles
$p^{!}=PD_{\tilde{X}}\circ p^{*}\circ PD_{X}:H_{2}(X,{\mathbb{Z}})\rightarrow
H_{2}(\tilde{X},{\mathbb{Z}}),$
where the image of $p^{!}$ is the subset of $H_{2}(\tilde{X},{\mathbb{Z}})$
consisting of $2$-cycles having intersection number zero with $E$.
We first consider blow-up at a point. Given a nonzero class $A\in
H_{2}(X,{\mathbb{Z}})$, from the viewpoint of geometry, we could express the
condition of counting curves with homology class $A$ passing through a generic
point in $X$ in two ways: adding a point class, or blowing up $X$ at the point
and counting curves in $\tilde{X}$ with homology class $p^{!}A-e$. One would
expect that the two methods give the same Gromov-Witten invariants, which was
proved by the second named author [H1] in all dimensions for $g=0$, and by the
fourth named author [Q] in real dimension four for all genera. In this paper,
we study the dimension six case for all genera:
###### Theorem 1.1.
Let $p:\tilde{X}\rightarrow X$ be the blow-up at a point. Suppose that
$\alpha_{1},\cdots,\alpha_{m}\in H^{>0}(X,{\mathbb{Q}})$, $1\leq i\leq m$, and
$d_{1},\cdots,d_{m}\in{\mathbb{Z}}_{\geqslant 0}$. Then for nonzero $A\in
H_{2}(X,{\mathbb{Z}})$ and $g\geq 0$, we have
$\langle[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}=\sum_{g_{1}+g_{2}=g}\frac{(-1)^{g_{1}}\cdot
2}{(2g_{1}+2)!}\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g_{2},p^{!}A-e}.$
###### Theorem 1.2.
Under the same assumptions as in Theorem 1.1, we have
$\displaystyle\langle\tau_{1}[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum_{g_{1}+g_{2}=g}\frac{(-1)^{g_{1}}}{(2g_{1}+1)!}\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g_{2},p^{!}A-e}.$
Through studying the proof of Theorem 1.2 carefully, we obtain the following
result, which seems to be nontrivial when compared with divisor equation and
dilaton equation.
###### Theorem 1.3.
Under the same assumptions as in Theorem 1.1, we have
$\displaystyle\langle\tau_{1}E,\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$ $\displaystyle
3\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle\qquad-2\sum_{g_{1}+g_{2}=g}\frac{(-1)^{g_{1}}}{(2g_{1}+1)!}\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g_{2},p^{!}A-e}.$
We also consider the blow-up along a curve.
###### Theorem 1.4.
Let $p:\tilde{X}\rightarrow X$ be the blow-up along a smooth curve $C$ with
$\int_{C}c_{1}(X)>0$. Suppose that $\alpha_{1},\cdots,\alpha_{m}\in
H^{>2}(X,{\mathbb{Q}})$, $1\leq i\leq m$, support away from the curve $C$, and
$d_{1},\cdots,d_{m}\in{\mathbb{Z}}_{\geqslant 0}$. Then for nonzero $A\in
H_{2}(X,{\mathbb{Z}})$ and $g\geq 0$, we have
$\langle[C],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}=\sum_{g_{1}+g_{2}=g}\frac{(-1)^{g_{1}}}{(2g_{1}+1)!\cdot
2^{2g_{1}}}\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g_{2},p^{!}A-e}.$
The above blow-up formulae relate Gromov-Witten invariants of $X$ and those of
$\tilde{X}$ in a nontrivial way. Theorem 1.1 and 1.4 imply the following
simple relations among generalized BPS numbers
$n^{X}_{g,A}(\alpha_{1},\dots,\alpha_{m})$ introduced by Pandharipande [P1,
P2].
###### Proposition 1.5.
Suppose that $\alpha_{1},\cdots,\alpha_{m}\in H^{>2}(X,{\mathbb{Q}})$, $A\in
H_{2}(X,{\mathbb{Z}})$ is nonzero and $g\in{\mathbb{Z}}_{\geqslant 0}$.
* (a)
If $p:\tilde{X}\rightarrow X$ is the blow-up at a point, then we have
$n_{g,A}^{X}([pt],\alpha_{1},\cdots,\alpha_{m})=n_{g,p^{!}A-e}^{\tilde{X}}(p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m}).$
* (b)
If $p:\tilde{X}\rightarrow X$ is the blow-up along a smooth curve $C$ with
$\int_{C}c_{1}(X)>0$, then we have
$n_{g,A}^{X}([C],\alpha_{1},\cdots,\alpha_{m})=n_{g,p^{!}A-e}^{\tilde{X}}(p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m}).$
Our proof of the above blow-up formulae is inspired by the absolute/relative
correspondence obtained by Hu-Li-Ruan [HLR], which is a generalization of the
idea of Maulik-Pandharipande [MP]. This correspondence partially describes the
change of Gromov-Witten invarians under blow-ups. We first use degeneration
formula to obtain comparison results between absolute and relative Gromov-
Witten invariants, and then use these comparison results to prove our blow-up
formulae.
The rest of the paper is organized as follows. In Section 2, we briefly review
basic materials of absolute/relative Gromov-Witten invariants and the
degeneration formula. In Section 3, we consider the case of blow-up at a point
and prove Theorem 1.1, 1.2 and 1.3. In Section 4, we consider the case of
blow-up along a smooth curve and prove Theorem 1.4. In Section 5, we review
the definition of generalized BPS numbers and prove Corollary 1.5.
## 2\. Preliminaries
In this section, we briefly review absolute/relative Gromov-Witten invariants
and the degeneration formula and fix notations throughout. We use [LR] as our
general reference.
Recall that we always let $X$ be a connected compact smooth symplectic
manifold of real dimension six. For $A\in H_{2}(X,{\mathbb{Z}})$, let
$\overline{\mathscr{M}}_{g,m}(X,A)$ be the moduli space of connected
$m$-pointed stable maps to $X$ of arithmetic genus $g$ and degree $A$. Let
$e_{i}:\overline{\mathscr{M}}_{g,m}(X,A)\longrightarrow X$ be the evaluation
map at the $i^{th}$ marked point. The Gromov-Witten invariants of $X$ are
defined as
$\langle\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}:=\int_{[\overline{\mathscr{M}}_{g,m}(X,A)]^{vir}}\prod\limits_{i=1}^{m}\psi_{i}^{d_{i}}e_{i}^{*}\alpha_{i},$
where $\alpha_{1},\cdots,\alpha_{m}\in H^{*}(X,{\mathbb{Q}})$,
$d_{1},\cdots,d_{m}\in{\mathbb{Z}}_{\geqslant 0}$, $\psi_{i}$ is the first
Chern class of the cotangent line bundle, and
$[\overline{\mathscr{M}}_{g,m}(X,A)]^{vir}$ is the virtual fundamental cycle.
The degeneration formula [LR, IP, Li] provides a rigorous formulation about
the change of Gromov-Witten invariants under semi-stable degeneration, or
symplectic cutting. The formula relates the absolute Gromov-Witten invariant
of $X$ to the relative Gromov-Witten invariants of two smooth pairs.
Now we recall the relative invariants of a smooth relative pair $(X,Z)$ with
$Z\hookrightarrow X$ a connected smooth symplectic divisor. Let $A\in
H_{2}(X,{\mathbb{Z}})$ with $A\cdot Z\geqslant 0$, and $\mu$ a partition of
$A\cdot Z$. We customarily use relative graphs to describe the topological
type of relative stable maps. A connected relative graph $\Gamma=(g,m,A,\mu)$
is defined to be a connected decorated graph consisting of the following data:
1. (1)
a vertex decorated by $A$ and genus $g$;
2. (2)
$m$ tails with no decoration;
3. (3)
$\ell(\mu)$ tails decorated by entries of $\mu$.
A connected relative stable map has topological type $\Gamma$ if it has
arithmetic genus $g$, degree $A$, $m$ absolute marked points and $\ell(\mu)$
relative marked points with contact order given by $\mu$. Let
$\overline{\mathscr{M}}_{\Gamma}(X,Z)$ be the moduli space of connected
relative stable maps with topological type $\Gamma$. Let
$e_{i}:\overline{\mathscr{M}}_{\Gamma}(X,Z)\longrightarrow X$ be the
evaluation map at the $i^{th}$ absolute marked point, and
$e_{j}^{Z}:\overline{\mathscr{M}}_{\Gamma}(X,Z)\longrightarrow Z$ the
evaluation map at the $j^{th}$ relative marked point. The relative Gromov-
Witten invariants of $(X,Z)$ are of the form
$\langle\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\mid\delta_{1},\cdots,\delta_{\ell(\mu)}\rangle_{\Gamma}^{X,Z}:=\int_{[\overline{\mathscr{M}}_{\Gamma}(X,Z)]^{vir}}\prod_{i=1}^{m}\psi_{i}^{d_{i}}e_{i}^{*}\alpha_{i}\cdot\prod\limits_{j=1}^{\ell(\mu)}(e^{Z}_{j})^{*}\delta_{i},$
where $\alpha_{1},\cdots,\alpha_{m}\in
H^{*}(X,{\mathbb{Q}}),d_{1},\cdots,d_{m}\in{\mathbb{Z}}_{\geqslant
0},\delta_{1},\cdots,\delta_{\ell(\mu)}\in H^{*}(Z,{\mathbb{Q}})$, and
$[\overline{\mathscr{M}}_{\Gamma}(X,Z)]^{vir}$ is the virtual fundamental
cycle of dimension:
$\dim[\overline{\mathscr{M}}_{\Gamma}(X,Z)]^{vir}=2\int_{A}c_{1}(X)+2m+2\ell(\mu)-2|\mu|.$
The relative invariants with disconnceted domains are defined by the usual
product rule, and the invariants will be denoted by
$\langle\cdots|\cdots\rangle_{\Gamma}^{\bullet X,Z}$.
Next, we shall introduce the degeneration formula. Let
$\pi:\chi\longrightarrow D$ be a connected, smooth symplectic manifold of real
dimension eight over a disk $D$ such that $\chi_{t}=\pi^{-1}(t)\cong X$ for
$t\not=0$ and $\chi_{0}$ is a union of two connected compact smooth symplectic
manifolds $X_{1}$ and $X_{2}$ intersecting transversally along a symplectic
divisor $Z$. We write $\chi_{0}=X_{1}\cup_{Z}X_{2}$.
Consider the natural inclusion maps
$i_{t}:X=\chi_{t}\longrightarrow\chi,\,\,\,\,\,\,\,\,i_{0}:\chi_{0}\longrightarrow\chi,$
and the gluing map
$g=(j_{1},j_{2}):X_{1}\coprod X_{2}\longrightarrow\chi_{0}.$
We have
$H_{2}(X,{\mathbb{Z}})\stackrel{{\scriptstyle
i_{t*}}}{{\longrightarrow}}H_{2}(\chi,{\mathbb{Z}})\stackrel{{\scriptstyle
i_{0_{*}}}}{{\longleftarrow}}H_{2}(\chi_{0},{\mathbb{Z}})\stackrel{{\scriptstyle
g_{*}}}{{\longleftarrow}}H_{2}(X_{1},{\mathbb{Z}})\oplus
H_{2}(X_{2},{\mathbb{Z}}),$
where $i_{0*}$ is an isomorphism since there exists a deformation retract from
$\chi$ to $\chi_{0}$(see [C]). Also, since the family $\chi\longrightarrow D$
comes from a trivial family, it follows that each $\alpha\in
H^{*}(X,{\mathbb{Q}})$ has global liftings such that the restriction
$\alpha(t)$ on $\chi_{t}$ is defined for all $t$.
Fix a basis $\\{\delta_{i}\\}$ of $H^{*}(Z,{\mathbb{Q}})$ and denote by
$\\{\delta^{i}\\}$ its dual basis. The degeneration formula expresses the
absolute invariants of $X$ in terms of the relative invariants of the two
smooth pairs $(X_{1},Z)$ and $(X_{2},Z)$:
$\displaystyle\langle\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum_{\mu}{\mathfrak{z}}(\mu)\sum\limits_{i_{1},\cdots,i_{\ell(\mu)}}\sum_{\eta\in\Omega_{\mu}}\langle\tau_{d_{i^{-}_{1}}}j_{1}^{*}\alpha_{i^{-}_{1}}(0),\cdots,\tau_{d_{i^{-}_{k_{1}}}}j_{1}^{*}\alpha_{i^{-}_{k_{1}}}(0)\mid\delta_{i_{1}},\cdots,\delta_{i_{\ell(\mu)}}\rangle^{\bullet
X_{1},Z}_{\Gamma_{1}}$
$\displaystyle\,\,\cdot\,\,\langle\tau_{d_{i^{+}_{1}}}j_{2}^{*}\alpha_{i^{+}_{1}}(0),\cdots,\tau_{d_{i^{+}_{k_{2}}}}\alpha_{i^{+}_{k_{2}}}(0)\mid{\delta}^{i_{1}},\cdots,\delta^{i_{\ell(\mu)}}\rangle^{\bullet
X_{2},Z}_{\Gamma_{2}},$
where
${\mathfrak{z}}(\mu)=|\mbox{Aut}\mu|\prod\limits_{i=1}^{\ell(\mu)}\mu_{i}$,
and $\eta=(\Gamma_{1},\Gamma_{2},I)$ is an admissible triple, which consists
of (possibly disconnected) topological types $\Gamma_{1},\Gamma_{2}$ with the
same partition $\mu$ under the identification $I$ of relative marked points,
satisfying the following requirements:
* (1)
the gluing of $\Gamma_{1}$ and $\Gamma_{2}$ under $I$ is connected;
* (2)
let $g_{i}$ be the total genus of $\Gamma_{i}$, and we have
$g=g_{1}+g_{2}+\ell(\mu)+1-|\Gamma_{1}|-|\Gamma_{2}|$, where $|\Gamma_{i}|$ is
the number of connected components of $\Gamma_{i}$;
* (3)
let $A_{i}\in H_{2}(X_{i},{\mathbb{Z}})$ be the total degree of $\Gamma_{i}$,
and we have $i_{t*}A=i_{0*}(j_{1*}A_{1}+j_{2*}A_{2})$ and $|\mu|=A_{1}\cdot
Z=A_{2}\cdot Z$;
* (4)
the absolute marked points of $\Gamma_{1},\Gamma_{2}$ are indexed by
$\\{i_{1}^{-},\cdots,i_{k_{1}}^{-}\\}$ and $\\{i_{1}^{+},\cdots$,
$i_{k_{2}}^{+}\\}$ respectively, the disjoint union of which is exactly
$\\{1,2,\cdots,m\\}$.
We denote by $\Omega_{\mu}$ the equivalence class of all admissible triples
with fixed partition $\mu$. For $\eta\in\Omega_{\mu}$ having nonzero
contribution in the degeneration formula, we have the following important
dimension constraint (Theorem 5.1 in [LR]):
(2)
$\dim\overline{\mathscr{M}}_{\Gamma_{1}}(X_{1},Z)+\dim\overline{\mathscr{M}}_{\Gamma_{2}}(X_{2},Z)=\dim\overline{\mathscr{M}}_{g,m}(X,A)+4\ell(\mu).$
###### Remark 2.1.
Symplectic cutting is a kind of surgery in symplectic geometry which is
suitable for the above degeneration formula (see [LR]). Suppose that
$X_{0}\subset X$ is an open codimension zero submanifold with Hamiltonian
$S^{1}$-action. Let $H:X_{0}\longrightarrow{\mathbb{R}}$ be a Hamiltonian
function with $0$ as a regular value. If $H^{-1}(0)$ is a separating
hypersurface of $X_{0}$, then we obtain two connected manifolds $X^{\pm}_{0}$
with boundary $\partial X^{\pm}_{0}=H^{-1}(0)$, where the $+$ side corresponds
to $H<0$. Suppose further that $S^{1}$ acts freely on $H^{-1}(0)$. Then the
symplectic reduction $Z=H^{-1}(0)/S^{1}$ is canonically a symplectic manifold.
Collapsing the $S^{1}$-action on $\partial X^{\pm}=H^{-1}(0)$, we obtain two
closed smooth manifolds $\bar{X}^{\pm}$ containing respectively real
codimension 2 submanifolds $Z^{\pm}=Z$ with opposite normal bundles.
Furthermore $\bar{X}^{\pm}$ admits a symplectic structure $\bar{\omega}^{\pm}$
which agrees with the restriction of $\omega$ away from $Z$, and whose
restriction to $Z^{\pm}$ agrees with the canonical symplectic structure
$\omega_{Z}$ on $Z$ from symplectic reduction. The pair of symplectic
manifolds $(\bar{X}^{\pm},\bar{\omega}^{\pm})$ is called the symplectic cut of
$X$ along $H^{-1}(0)$.
Suppose that $Y\subset X$ is a submanifold of $X$ of codimension $2k$. Denote
by $N_{Y}$ the normal bundle. By the symplectic neighborhood theorem ,and by
possibly taking a smaller $\epsilon_{0}$, a tubular neighborhood
${\mathscr{N}}_{\epsilon_{0}}(Y)$ of $Y$ in $X$ is symplectomorphic to the
disc bundle $N_{Y}(\epsilon_{0})$ of $N_{Y}$. Denote by
$\phi:{\mathscr{N}}_{\epsilon_{0}}(Y)\longrightarrow N_{Y}(\epsilon_{0})$ be
such a symplectomorphism. Consider the Hamiltonian $S^{1}$-action on
$X_{0}={\mathscr{N}}_{\epsilon_{0}}(Y)$ by complex multiplication. Fix
$\epsilon$ with $0<\epsilon<\epsilon_{0}$ and consider the moment map
$H(u)=|\phi(u)|^{2}-\epsilon,\,\,\,\,\,u\in{\mathscr{N}}_{Y}(\epsilon_{0}),$
where $|\phi(u)|$ is the norm of $\phi(u)$ considered as a vector in a fiber
of the Hermitian bundle $N_{Y}$. We cut $X$ along $H^{-1}(0)$ to obtain two
closed symplectic manifolds $\bar{X}^{\pm}$. Notice that
$\bar{X}^{+}\cong{\mathbb{P}}_{Y}(N_{Y}\oplus{\mathscr{O}}_{Y})$.
$\bar{X}^{-}$ is called the blow-up of $X$ along $Y$, denoted by $\tilde{X}$.
## 3\. Formulae for Blow-up at a point
In this section, we prove Theorem 1.1, 1.2 and 1.3. We always assume that
total degrees of insertions match the virtual dimension of the moduli spaces,
since otherwise the required equalities are trivial.
First of all, we will divide the proof of Theorem 1.1 into some comparison
theorems of Gromov-Witten invariants as follows.
###### Lemma 3.1.
Under the same assumptions as in Theorem 1.1, we have
(3)
$\displaystyle{\langle[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle}^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle[pt]|[pt]\rangle}^{\mathbb{P}^{3},H}_{g^{+},L,(1)}{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$
where $H$ is the hyperplane at infinity, and $L\in
H_{2}(\mathbb{P}^{3};{\mathbb{Z}})$ is the class of a line.
###### Proof.
We first perform the symplectic cutting along a point as in Remark 2.1. Here
we have assumed that the class $[pt]$ has support in $X^{+}$ and $\alpha_{i}$
has support in $X^{-}$. By the degeneration formula (2),we have
$\displaystyle\langle[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum{\mathfrak{z}}(\mu)\langle[pt]|\delta_{j_{1}},\cdots,\delta_{j_{\ell(\mu)}}\rangle^{\mathbb{P}^{3},H}_{g^{+},A^{+},\mu}$
$\displaystyle\cdot\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\delta^{j_{1}},\cdots,\delta^{j_{\ell(\mu)}}\rangle^{\tilde{X},E}_{g^{-},A^{-},\mu}.$
By our assumption that total degrees of insertions match the virtual dimension
of the moduli space, we have
$\dim\bar{\mathscr{M}}_{g,m+1}(X,A)=\sum_{i=1}^{m}\deg\alpha_{i}+2\sum_{i=1}^{m}d_{i}+6.$
Suppose that $(\Gamma^{+},\Gamma^{-})$ has nonzero contribution in the
degeneration formula. Then
$\displaystyle\dim\bar{\mathscr{M}}_{\Gamma^{+}}(\mathbb{P}^{3},H)$
$\displaystyle=$ $\displaystyle
2\int_{A^{+}}c_{1}({\mathbb{P}}^{3})+2+2\ell(\mu)-2|\mu|,$
$\displaystyle\dim\bar{\mathscr{M}}_{\Gamma^{-}}(\tilde{X},E)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\deg\alpha_{i}+2\sum_{i=1}^{m}d_{i}+\sum\limits_{i=1}^{\ell(\mu)}\deg\delta^{j_{i}}.$
So by the dimension constraint (2),
$\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{A^{+}}c_{1}({\mathbb{P}}^{3})-|\mu|=2+\ell(\mu).$
Note that $A^{+}\cdot H=|\mu|$, and hence $A^{+}=|\mu|L$, which implies that
$\displaystyle\int_{A^{+}}c_{1}({\mathbb{P}}^{3})=4|\mu|.$
Now the dimension constraint becomes
$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+3|\mu|=2+\ell(\mu).$
So the dimension constraint holds only if
$\displaystyle\mu=(1),\quad deg\delta^{j_{1}}=0,$
which implies the required equality. ∎
###### Lemma 3.2.
Under the same assumptions as in Theorem 1.1, we have
$\displaystyle{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle\,\,|[pt]\rangle}_{g^{+},F,(1)}^{\tilde{\mathbb{P}}^{3},H}$
$\displaystyle\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$
where $F\in H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})$ is the class of a
fiber in
$\tilde{\mathbb{P}}^{3}\cong{\mathbb{P}}_{{\mathbb{P}}^{2}}(\mathscr{O}\oplus\mathscr{O}(-1))$.
###### Proof.
We perform symplectic cutting along $E$ in $\tilde{X}$ as in Remark 2.1. Here
we also assumed that the class $p^{*}\alpha_{i}$ has support away from $E$. By
the degeneration formula (2), we have
$\displaystyle{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle\sum{\mathfrak{z}}(\mu){\langle\,\,|\delta_{j_{1}},\cdots,\delta_{j_{\ell(\mu)}}\rangle}_{g^{+},(p!(A)-e)^{+},\nu}^{\tilde{\mathbb{P}}^{3},H}$
$\displaystyle\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\delta^{j_{1}},\cdots,\delta^{j_{\ell(\mu)}}\rangle}^{\tilde{X},E}_{g^{-},(p^{!}A-e)^{-},\mu}.$
By our assumption that degrees match the virtual dimension, we have
$\displaystyle\dim_{\mathbb{C}}\bar{\mathscr{M}}_{g,m}(\tilde{X},p^{!}A-e)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{m}deg\alpha_{i}+\sum_{i=1}^{m}d_{i}.$
Suppose that a term with $(\Gamma^{+},\Gamma^{-})$ has nonzero contribution in
RHS of the degeneration formula (3). Then
$\dim_{\mathbb{C}}\bar{\mathscr{M}}_{\Gamma^{+}}(\tilde{\mathbb{P}}^{3},H)=\int_{(p^{!}A-e)^{+}}c_{1}(\tilde{{\mathbb{P}}}^{3})+\ell(\mu)-|\mu|,$
$\dim_{\mathbb{C}}\bar{\mathscr{M}}_{\Gamma^{-}}(\tilde{X},E)=\frac{1}{2}\sum_{i=1}^{m}deg\alpha_{i}+\sum_{i=1}^{m}d_{i}+\frac{1}{2}\sum_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}.$
So by the dimension constraint (2),
$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{(p^{!}A-e)^{+}}c_{1}(\tilde{{\mathbb{P}}}^{3})-|\mu|=\ell(\mu).$
Let $L\in H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})$ be the class of the
total transform of a line in ${\mathbb{P}}^{3}$. Then we have the following
natural decomposition
$\displaystyle
H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})={\mathbb{Z}}F\oplus{\mathbb{Z}}L.$
We have the following constraints for $(p^{!}A-e)^{+}$:
${(p!(A)-e)}^{+}\cdot H=|\mu|,\,\,\,\,\,(p^{!}(A)-e)^{+}\cdot E=1.$
So we have $(p^{!}A-e)^{+}=F+(|\mu|-1)L$, and hence
$\int_{(p^{!}A-e)^{+}}c_{1}(\tilde{\mathbb{P}}^{3})=4|\mu|-2$. Now the
dimension constraint becomes
$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+3|\mu|=2+\ell(\mu).$
So the dimension constraint holds only if
$\displaystyle\mu=(1),\quad deg\delta^{j_{1}}=0,$
which implies the required equality. ∎
Using the above comparison results we may obtain the following
absolute/relative correspondence for Gromov-Witten invariants under blow-up.
###### Lemma 3.3.
Under the same assumptions as in Theorem 1.1, denote
$\langle[pt],\tau_{d_{1}}\alpha_{1},$
$\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$ and
$\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
by $H_{g}$ and $P_{g}$ respectively. Then
(6) $H_{g}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}P_{g_{2}},$
where $C_{g}$’s can be determined by relative invariants
${\langle[pt]|[pt]\rangle}^{\mathbb{P}^{3},H}_{g,L,(1)}$ and
${\langle\,\,|[pt]\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$.
###### Proof.
Denote
$\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle^{\tilde{X},E}_{g,p!(A)-e,(1)}$,
${\langle[pt]|[pt]\rangle}^{\mathbb{P}^{3},H}_{g,L,(1)}$ and
${\langle\,\,|[pt]\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$ by $K_{g}$,
$I_{g}$ and $J_{g}$ respectively. Then for $g\geq 0$, we may rewrite our
comparison results (3) and (3.2) as
$\displaystyle H_{g}$ $\displaystyle=$ $\displaystyle
I_{g}K_{0}+I_{g-1}K_{1}+\cdots+I_{0}K_{g}$ $\displaystyle P_{g}$
$\displaystyle=$ $\displaystyle J_{g}K_{0}+J_{g-1}K_{1}+\cdots+J_{0}K_{g},$
or in matrix form
$\left(\begin{array}[]{l}H_{0}\\\ H_{1}\\\ \vdots\\\
H_{g}\end{array}\right)=\left(\begin{array}[]{llll}I_{0}&&\lx@intercol\hfil
0\hfil\lx@intercol\\\ I_{1}&I_{0}&&\\\ \vdots&&\ddots&\\\
I_{g}&I_{g-1}&\cdots&I_{0}\end{array}\right)\left(\begin{array}[]{l}K_{0}\\\
K_{1}\\\ \vdots\\\ K_{g}\end{array}\right),$
$\left(\begin{array}[]{l}P_{0}\\\ P_{1}\\\ \vdots\\\
P_{g}\end{array}\right)=\left(\begin{array}[]{llll}J_{0}&&\lx@intercol\hfil
0\hfil\lx@intercol\\\ J_{1}&J_{0}&&\\\ \vdots&&\ddots&\\\
J_{g}&J_{g-1}&\cdots&J_{0}\end{array}\right)\left(\begin{array}[]{l}K_{0}\\\
K_{1}\\\ \vdots\\\ K_{g}\end{array}\right).$
This is a special form of absolute/relative correspondence for Gromov-Witten
invariants (Theorem 5.15 in [HLR]). In particular, $I_{0}\neq 0$ and
$J_{0}\neq 0$, which implies that both matrices with entries $I_{g}$ and
$J_{g}$ are invertible (one can also use virtual localization [GP] to check
that $I_{0}=J_{0}=1$). Write
$\left(\begin{array}[]{llll}C_{0}&&\lx@intercol\hfil 0\hfil\lx@intercol\\\
C_{1}&C_{0}&&\\\ \vdots&&\ddots&\\\
C_{g}&C_{g-1}&\cdots&C_{0}\end{array}\right)=\left(\begin{array}[]{llll}I_{0}&&\lx@intercol\hfil
0\hfil\lx@intercol\\\ I_{1}&I_{0}&&\\\ \vdots&&\ddots&\\\
I_{g}&I_{g-1}&\cdots&I_{0}\end{array}\right)\left(\begin{array}[]{llll}J_{0}&&\lx@intercol\hfil
0\hfil\lx@intercol\\\ J_{1}&J_{0}&&\\\ \vdots&&\ddots&\\\
J_{g}&J_{g-1}&\cdots&J_{0}\end{array}\right)^{-1},$
and we obtain the required equality. ∎
To get Theorem 1.1, we need to compute $C_{g}$’s in (6). A crucial observation
from the proof of Lemma 3.3 is that $C_{g}$’s are determined by relative
invariants ${\langle[pt]|[pt]\rangle}^{\mathbb{P}^{3},H}_{g,L,(1)}$ and
${\langle\,\,|[pt]\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$, and are
independent of the choice of $X,m,\alpha_{i},A$. Therefore, to compute these
universal coefficients, we may choose
$X=\mathbb{P}^{3},m=1,\alpha_{1}=[pt],A=L$. Then (6) becomes
(7)
$\langle[pt],[pt]\rangle_{g,L}^{\mathbb{P}^{3}}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}\cdot\langle[pt]\rangle_{g_{2},F}^{\tilde{\mathbb{P}}^{3}},$
where $F$ is the class of a fiber in
$\tilde{\mathbb{P}}^{3}\cong{\mathbb{P}}_{{\mathbb{P}}^{2}}({\mathscr{O}}\oplus{\mathscr{O}}(-1))$.
To get $C_{g}$’s by solving the equation (7), we need to compute the absolute
Gromov-Witten invariants $\langle[pt],[pt]\rangle_{g,L}^{\mathbb{P}^{3}}$ and
$\langle[pt]\rangle_{g_{2},F}^{\tilde{\mathbb{P}}^{3}}$. From this, we have
###### Lemma 3.4.
$\displaystyle\langle[pt],[pt]\rangle_{g,L}^{\mathbb{P}^{3}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{g}\cdot 2}{(2g+2)!},$
$\displaystyle\langle[pt]\rangle_{g,F}^{\tilde{\mathbb{P}}^{3}}$
$\displaystyle=$ $\displaystyle\delta_{g,0}.$
These equalities can be proved either directly by virtual localization [GP] or
by degenerate contribution computation [P2]. In fact, Theorem 3 in [P2] may
specialize to the case of ${\mathbb{P}}^{3}$ and obtain these invariants. Here
we omit the proof.
Proof of Theorem 1.1: We first perform symplectic cutting at a point in $X$
and get equation (3). Then we perform symplectic cutting along the exceptional
divisor $E$ in $\tilde{X}$ and get (3.2). Finally, we can solve the equation
(7) to get the universal coefficients
$C_{g}=\frac{(-1)^{g}\cdot 2}{(2g+2)!}.$
This proves Theorem 1.1.
###### Remark 3.5.
One can relax the requirement in Theorem 1.1 to $m\geqslant 0$, which can be
checked by going through the proof of Lemma 3.1, 3.2 and 3.3. This also holds
for Theorem 1.2 and 1.3.
It is illuminating to rephrase this using a genus $g$ gravitational Gromov-
Witten generating function. Suppose that $T_{0}=1,T_{1},\cdots,T_{m}$ is a
basis for $H^{*}(X,{\mathbb{Q}})$. We introduce supercommuting variables
$t_{d}^{j}$ for $d\geq 0$ and $0\leq j\leq m$ with $\deg t^{j}_{d}=\deg
T_{j}$. Set
$\gamma=\sum_{d=0}^{\infty}\sum_{j=1}^{m}t^{j}_{d}\tau_{d}T_{j}.$
Define the genus $g$ gravitational Gromov-Witten generating function as
$F_{g}^{X}(t_{d}^{j})=\sum_{n=0}^{\infty}\sum_{A\in
H_{2}(X,{\mathbb{Z}})}\frac{1}{n!}\langle\gamma^{n},[pt]\rangle_{g,A}^{X}q^{A},$
$F_{g}^{\tilde{X}}(t_{d}^{j})=\sum_{n=0}^{\infty}\sum_{A\in
H_{2}(X,{\mathbb{Z}})}\frac{1}{n!}\langle(p^{*}\gamma^{n}\rangle_{g,p^{!}(A)-e}^{\tilde{X}}q^{p^{!}(A)-e}.$
Set
$F^{X}(u,t_{d}^{j})=\sum_{g\geq 0}u^{2g-2}F_{g}^{X}(t_{d}^{j})$
and
$F^{\tilde{X}}(u,t_{d}^{j})=\sum_{g\geq
0}u^{2g-2}F_{g}^{\tilde{X}}(t_{d}^{j}).$
Then from Theorem 1.1, we have
###### Corollary 3.6.
$F^{X}(u,\gamma)=\bigg{(}\frac{\sin\frac{u}{2}}{\frac{u}{2}}\bigg{)}^{2}\cdot
F^{\tilde{X}}(u,p^{*}\gamma),$
where we need to change the variable $q^{A}$ to $q^{p^{!}(A)-e}$.
Similar to the proof of Theorem 1.1 above, we may divide the proof of Theorem
1.2 into the following Lemma 3.7, 3.8 and 3.9, the proof of which is analogous
to that of Lemma 3.1, 3.2 and 3.3 respectively.
###### Lemma 3.7.
Under the same assumptions as in Theorem 1.1, we have
$\displaystyle{\langle\tau_{1}[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle}^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle\tau_{1}[pt]|\xi\rangle}^{\mathbb{P}^{3},H}_{g^{+},L,(1)}{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\xi\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$
where $H$ is the hyperplane at infinity, $L$ is the class of a line in
${\mathbb{P}}^{3}$, and $\xi$ is the cohomology class of a line in $H\cong
E\cong{\mathbb{P}}^{2}$.
###### Proof.
The same argument as in the proof of Lemma 3.1 leads to the dimension
constraint
$\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}\deg\delta^{j_{i}}+3|\mu|=3+\ell(\mu).$
This constraint holds only if
$\mu=(1),\,\,\,\,\deg\delta_{j_{1}}=2.$
This implies Lemma 3.7. ∎
###### Lemma 3.8.
Under the same assumptions as in Theorem 1.1, we have
$\displaystyle{\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle-E^{2}|\xi\rangle}_{g^{+},F,(1)}^{\tilde{\mathbb{P}}^{3},H}\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\xi\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$
where $F\in H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})$ is the class of a
fiber in
$\tilde{\mathbb{P}}^{3}\cong{\mathbb{P}}_{{\mathbb{P}}^{2}}(\mathscr{O}\oplus\mathscr{O}(-1))$,
and $\xi$ is the cohomology class of a line in $H\cong
E\cong{\mathbb{P}}^{2}$.
###### Proof.
The same dimension calculation as in the proof of Lemma 3.1 gives rise to the
dimension constraint
$\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}\deg\delta^{j_{i}}+3|\mu|=3+\ell(\mu).$
This also implies
$\mu=(1),\,\,\,\,\deg\delta_{j_{1}}=2,$
which proves Lemma 3.8. ∎
###### Lemma 3.9.
Under the same assumptions as in Theorem 1.1, denote
$\langle\tau_{1}[pt],\tau_{d_{1}}\alpha_{1}$,
$\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$ and
$\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
by $H_{g}$ and $P_{g}$ respectively. Then
(8) $H_{g}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}P_{g_{2}},$
where $C_{g}$’s can be determined by relative invariants
${\langle\tau_{1}[pt]|\xi\rangle}^{\mathbb{P}^{3},H}_{g,L,(1)}$ and
${\langle-E^{2}|\xi\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$. Here $\xi$
is the cohomology class of a line in $H\cong E\cong{\mathbb{P}}^{2}$.
The proof of Lemma 3.9 is identical to that of Lemma 3.3 with Lemma 3.1 and
3.2 replaced by Lemma 3.7 and 3.8 respectively.
Proof of Theorem 1.2: Similar to the proof of Theroem 1.1, we only need to
compute the universal coefficients $C_{g}$’s in (8). Similarly, we choose
$X=\mathbb{P}^{3},m=1,\alpha_{1}=[L],A=L$. Then (8) becomes
(9)
$\langle\tau_{1}[pt],[L]\rangle_{g,L}^{\mathbb{P}^{3}}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}\cdot\langle-E^{2},[L]\rangle_{g_{2},F}^{\tilde{\mathbb{P}}^{3}}.$
By virtual localization [GP], we have
$\langle\tau_{1}[pt],[L]\rangle_{g,L}^{\mathbb{P}^{3}}=\frac{(-1)^{g}}{(2g+1)!},$
$\langle-E^{2},[L]\rangle_{g,F}^{\tilde{\mathbb{P}}^{3}}=\delta_{g,0}.$
We solve (9) to obtain the universal coefficients:
$C_{g}=\frac{(-1)^{g}}{(2g+1)!},$
which gives Theorem 1.2.
In the rest of this section, we will prove Theorem 1.3. Similar argument to in
the proof of Lemma 3.2 and 3, we may prove the following Lemmas.
###### Lemma 3.10.
Under the same assumptions as in Theorem 1.1, we have
$\displaystyle{\langle\tau_{1}E,\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle\tau_{1}E|\xi\rangle}_{g^{+},F,(1)}^{\tilde{\mathbb{P}}^{3},H}\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\xi\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$
where $F\in H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})$ is the class of a
fiber in
$\tilde{\mathbb{P}}^{3}\cong{\mathbb{P}}_{{\mathbb{P}}^{2}}(\mathscr{O}\oplus\mathscr{O}(-1))$,
and $\xi$ is the cohomology class of a line in $H\cong
E\cong{\mathbb{P}}^{2}$.
###### Lemma 3.11.
Under the same assumptions as in Theorem 1.1, denote
$\langle\tau_{1}E,\tau_{d_{1}}p^{*}\alpha_{1}$,
$\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$ and
$\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
by $H_{g}$ and $P_{g}$ respectively. Then
$H_{g}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}P_{g_{2}},$
where $C_{g}$’s can be determined by relative invariants
${\langle\tau_{1}E|\xi\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$ and
${\langle-E^{2}|\xi\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$. Here $\xi$
is the cohomology class of a line in $H\cong E\cong{\mathbb{P}}^{2}$.
Proof of Theorem 1.3: Similar to the proof of Theorem 1.2, we choose
$X=\tilde{\mathbb{P}}^{3},m=1,\alpha_{1}=[L],A=F$ and obtain:
$\langle\tau_{1}E,L\rangle_{g,F}^{\tilde{\mathbb{P}}^{3}}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}\cdot\langle-E^{2},L\rangle_{g_{2},F}^{\tilde{\mathbb{P}}^{3}}.$
By virtual localization [GP], we have
$\langle\tau_{1}E,L\rangle_{g,F}^{\tilde{\mathbb{P}}^{3}}=\delta_{g,0}\cdot
3-\frac{(-1)^{g}\cdot 2}{(2g+1)!}.$
So
$C_{g}=\delta_{g,0}\cdot 3-\frac{(-1)^{g}\cdot 2}{(2g+1)!},$
which gives Theorem 1.3.
## 4\. Formulae for Blow-up along a smooth curve
In this section, we give a detailed proof of Theorem 1.4. We always assume
that total degrees of insertions match the virtual dimension of the moduli
spaces, since otherwise the required equalities are trivial.
###### Lemma 4.1.
Under the same assumptions as in Theorem 1.4, we have
$\displaystyle\langle[C],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle_{g,A}^{X}$
$\displaystyle=$
$\displaystyle\sum\limits_{g_{1}+g_{2}=g}\langle[C]|[pt]\rangle^{\bar{X}^{+},Z}_{g_{1},F,(1)}\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle}^{\tilde{X},E}_{g_{2},p^{!}A-e,(1)},$
where $F\in H_{2}(\bar{X}^{+},{\mathbb{Z}})$ is the class of a line in the
fiber of $\bar{X}^{+}={\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C})$.
###### Proof.
We first perform symplectic cutting along $C$ and assume that the support of
$[C]$ is in $X^{+}$ and the support of $\alpha_{i}$ is away from $C$. By the
degeneration formula (2), we have:
$\displaystyle\langle[C],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle_{g,A}^{X}$
$\displaystyle=$
$\displaystyle\sum\limits\mathfrak{z}(\mu)\langle[C]|\delta_{j_{1}},\cdots,\delta_{j_{\ell(\mu)}}\rangle_{\Gamma_{+}}^{\bullet,\bar{X}^{+},Z}$
$\displaystyle\qquad\cdot\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\delta^{j_{1}},\cdots,\delta^{j_{\ell(\mu)}}\rangle_{\Gamma_{-}}^{\bullet,\tilde{X},E}.$
Recall that we have assumed that
$\dim\overline{M}_{g,m+1}(X,A)=\sum\limits_{i=1}^{m}deg\alpha_{i}+2\sum\limits_{i=1}^{m}d_{i}+4.$
Assume that A term with $(\Gamma^{+},\Gamma^{-})$ in RHS of (4) has nonzero
contribution, and then
$\displaystyle\dim\overline{M}_{\Gamma_{+}}(\bar{X}^{+},Z)$ $\displaystyle=$
$\displaystyle 2\int_{A^{+}}c_{1}(X^{+})+2+2\ell(\mu)-2|\mu|,$
$\displaystyle\dim\overline{M}_{\Gamma_{-}}(\tilde{X},E)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}deg\alpha_{i}+2\sum\limits_{i=1}^{m}d_{i}+\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}.$
So by the dimension constraint (2) for the degeneration formula, we have
$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{A^{+}}c_{1}(X^{+})-|\mu|=1+\ell(\mu).$
Let $\xi^{+}$ be the tautological line bundle of
$\bar{X}^{+}={\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C})$, and we have
$\displaystyle c_{1}(\bar{X}^{+})=\pi^{*}c_{1}(X)|_{C}-3c_{1}(\xi^{+}),$
where $\pi:\bar{X}^{+}\rightarrow C$ is the natural projection. Note that
$-c_{1}(\xi^{+})$ is the Poincaré dual of the divisor $Z$ in $\bar{X}^{+}$.
Since $|\mu|=A^{+}\cdot Z$, it follows that
$\displaystyle\int_{A^{+}}c_{1}(\bar{X}^{+})=\int_{\pi_{*}A^{+}}c_{1}(X)|_{C}+3|\mu|.$
Therefore, dimension constraint becomes
$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{\pi_{*}A^{+}}c_{1}(X)|_{C}+2|\mu|=1+\ell(\mu).$
Since $m\geqslant 1$, it follows that $\mu\neq\emptyset$ by the connectedness
of the stable maps to $X$, and the dimension constraint holds only if
$\displaystyle\mu=(1),\quad\textrm{deg}\delta^{j_{1}}=0,\quad\int_{\pi_{*}A^{+}}c_{1}(X)|_{C}=0,$
which implies Lemma 4.1. ∎
###### Lemma 4.2.
Under the same assumptions as in Theorem 1.4, we have
$\displaystyle\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle_{g,p^{!}A-e}^{\tilde{X}}$
$\displaystyle=$
$\displaystyle\sum\limits_{g_{1}+g_{2}=g}\langle|[pt]\rangle^{\bar{\tilde{X}}^{+},Z}_{g_{1},F,(1)}{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle}^{\tilde{X},E}_{g_{2},p^{!}A-e,(1)},$
where $F\in H_{2}(\tilde{X}^{+},{\mathbb{Z}})$ is the class of a line in the
fiber of $\bar{\tilde{X}}^{+}={\mathbb{P}}_{E}(N_{E}\oplus{\mathscr{O}}_{E})$.
###### Proof.
We first degenerate $\tilde{X}$ along $E$, and assume that the support of
$p^{*}\alpha_{i}$ is away from $E$. By the degeneration formula (2), we have
$\displaystyle{\langle
E,\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$ $\displaystyle\sum\limits\mathfrak{z}(\mu)\langle
E|\delta_{j_{1}},\cdots,\delta_{j_{\ell(\mu)}}\rangle_{\Gamma_{+}}^{\bullet,\bar{\tilde{X}}^{+},Z}$
$\displaystyle\qquad\cdot\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\delta^{j_{1}},\cdots,\delta^{j_{\ell(\mu)}}\rangle_{\Gamma_{-}}^{\bullet,\tilde{X},E}.$
Recall that we have assumed that
$\dim\overline{M}_{g,m+1}(\tilde{X},p^{!}A-e)=\sum\limits_{i=1}^{m}deg\alpha_{i}+2\sum\limits_{i=1}^{m}d_{i}+2.$
Assume that a term with $(\Gamma^{+},\Gamma^{-})$ in RHS of (4) has nonzero
contribution. Then
$\displaystyle\dim\overline{M}_{\Gamma_{+}}(\bar{\tilde{X}}^{+},Z)$
$\displaystyle=$ $\displaystyle
2\int_{(p^{!}A-e)^{+}}c_{1}(\bar{\tilde{X}}^{+})+2+2\ell(\mu)-2|\mu|,$
$\displaystyle\dim\overline{M}_{\Gamma_{-}}(\tilde{X},E)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}deg\alpha_{i}+2\sum\limits_{i=1}^{m}d_{i}+\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}.$
So by the dimension constraint (2) for the degeneration formula, we have
$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}\textrm{deg}\delta^{j_{i}}+\int_{(p^{!}A-e)^{+}}c_{1}(\tilde{X}^{+})-|\mu|=\ell(\mu).$
Let $\xi^{+}$ be the tautological line bundle of
$\bar{\tilde{X}}^{+}={\mathbb{P}}_{E}(N_{E}\oplus{\mathscr{O}}_{E})$. Then
Euler exact sequence gives
$\displaystyle
c_{1}(\bar{\tilde{X}}^{+})=\pi^{*}c_{1}(E)+\pi^{*}c_{1}(N_{E})-2c_{1}(\xi^{+}),$
where $\pi:\bar{\tilde{X}}^{+}\rightarrow E$ is the natural projection. Note
that $N_{E}$ is the tautological line bundle of
$E\cong{\mathbb{P}}_{C}(N_{C})$, and so
$\displaystyle c_{1}(E)=\pi_{E}^{*}c_{1}(X)|_{C}-2c_{1}(N_{E}),$
where $\pi_{E}:E\rightarrow C$ is the natural projection. Therefore,
$\displaystyle
c_{1}(\bar{\tilde{X}}^{+})=(\pi_{E}\circ\pi)^{*}c_{1}(X)|_{C}-\pi^{*}c_{1}(N_{E})-2c_{1}(\xi^{+}).$
Note that we have the following natural decomposition
$\displaystyle H_{2}(\bar{\tilde{X}}^{+},{\mathbb{Z}})\cong{\mathbb{Z}}F\oplus
H_{2}(E,{\mathbb{Z}}),$
and we can write
$\displaystyle(p^{!}A-e)^{+}=aF+\pi_{*}(p^{!}A-e)^{+},\textrm{ for some
}a\in{\mathbb{Z}}_{\geqslant 0}.$
We have the following constraints for $(p^{!}A-e)^{+}$:
$\displaystyle\left\\{\begin{array}[]{ccl}(p^{!}A-e)^{+}\cdot Z&=&|\mu|,\\\
(p^{!}A-e)^{+}\cdot E&=&(p^{!}A-e)\cdot E=1,\end{array}\right.$
and this gives
$\displaystyle\pi_{*}(p^{!}A-e)^{+}\cdot E=-(|\mu|-1).$
Note that $-c_{1}(\xi^{+})$ is the Poincaré dual of the divisor $Z$ in
$\bar{\tilde{X}}^{+}$, and therefore
$\displaystyle\int_{(p^{!}A-e)^{+}}c_{1}(\bar{\tilde{X}}^{+})=\int_{(\pi_{E}\circ\pi)_{*}(p^{!}A-e)^{+}}c_{1}(X)|_{C}+3|\mu|-1.$
Hence the dimension constraint becomes
$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{(\pi_{E}\circ\pi)_{*}(p^{!}A-e)^{+}}c_{1}(X)|_{C}+2|\mu|=1+\ell(\mu).$
Since $m\geqslant 1$, it follows that $\mu\neq\emptyset$ by the connectedness
of the stable maps to $\tilde{X}$. So the dimension constraint holds only if
$\displaystyle\mu=(1),\quad
deg\delta^{j_{1}}=0,\quad\int_{(\pi_{E}\circ\pi)_{*}(p^{!}A-e)^{+}}c_{1}(X)|_{C}=0,$
which implies Lemma 4.2. ∎
Using the above comparison results, the same argument as in the proof of Lemma
3.3 shows that the following lemma holds.
###### Lemma 4.3.
Under the same assumptions as in Theorem 1.4, denote
$\langle[C],\tau_{d_{1}}\alpha_{1},\cdots$ ,
$\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$ and
$\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
by $H_{g}$ and $P_{g}$ respectively. Then
(13) $H_{g}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}P_{g_{2}},$
where $C_{g}$’s can be determined by relative invariants
${\langle[C]|[pt]\rangle}^{\bar{X}^{+},Z}_{g,F,(1)}$ and
${\langle|[pt]\rangle}_{g,F,(1)}^{\bar{\tilde{X}}^{+},Z}$.
Similar to Theorem 1.1, we only need to determine the universal coefficients
$C_{g}$’s in (13). For this, we choose
$X={\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C}),m=1,\alpha_{1}=[pt],A=F$ and
rewrite (13) as follows
(14)
$\langle[C],[pt]\rangle_{g,F}^{{\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C})}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}\cdot\langle[pt]\rangle_{g_{2},F}^{{\mathbb{P}}_{E}(N_{E}\oplus{\mathscr{O}}_{E})}.$
About the two absolute invariants in (14), we have
###### Lemma 4.4.
$\displaystyle\langle[C],[pt]\rangle_{g,F}^{{\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C})}$
$\displaystyle=$ $\displaystyle\frac{(-1)^{g}}{(2g+1)!\cdot 2^{2g}},$
$\displaystyle\langle[pt]\rangle_{g,F}^{{\mathbb{P}}_{E}(N_{E}\oplus{\mathscr{O}}_{E})}$
$\displaystyle=$ $\displaystyle\delta_{g,0}.$
###### Proof.
In the first equality, let
$C={\mathbb{P}}_{C}(\\{0\\}\oplus{\mathscr{O}}_{C})$ and
$P_{0}\in{\mathbb{P}}_{C}(N_{C}\oplus\\{0\\})$ be the Poincaré duals of $[C]$
and $[pt]$ respectively. There is a unique connected smooth embedded curve
with homology class $F$ passing through $C$ and $P_{0}$, which is the line, in
the fiber containing $P_{0}$, passing through $P_{0}$ and the intersection of
$C$ and the fiber. Now LHS is equal to the degenerate contribution of the
line. So Theorem 1.5 in [Z] can be specialized to the first equality. The
proof of the second equality is similar. ∎
###### Remark 4.5.
Theorem 1.5 in [Z] is the symplectic version of degenerate contribution
computation for Fano case in [P1].
Proof of Theorem 1.4: Using Lemma 4.4 and solving the equaiton (14), we obtain
the universal coefficients $C_{g}=\frac{(-1)^{g}}{(2g+1)!\cdot 2^{2g}}$, which
gives Theorem 1.4.
###### Remark 4.6.
If $\int_{A}c_{1}(X)>1$, then we can relax the condition $m>0$ to $m\geqslant
0$. One can check this by going through the proof of Lemma 4.1 and 4.2.
## 5\. Generalized BPS numbers
Gromov-Witten invariants are only rational numbers in general, and hidden
integrality for these invariants of projective $3$-folds has been studied
since the very beginning of Gromov-Witten theory. For example, the
mathematically non-rigorous computation of genus zero invariants of quintic
$3$-folds [COGP] inspired the famous multiple covering formula [AM]. Based on
M-theory consideration, Gopakumar and Vafa [GV1, GV2] conjectured that
countings of BPS states give hidden integrality for Gromov-Witten invariants
of Calabi-Yau $3$-folds in all genera, which are multiplicities of certain
representations of $SL(2)$ in the cohomology of moduli space of sheaves. Based
on degenerate contribution computation, Pandharipande [P1, P2] generalized the
working definition of BPS numbers to arbitrary $3$-folds, and he also
conjectured that these generalized BPS numbers are integers which are counts
of curves satisfying incidence conditions given by insertions.
Let us review the definition of generalized BPS numbers and Pandharipande’s
conjecture. Let $X$ be a connected closed symplectic manifold of real
dimension $6$ and $A\in H_{2}(X,{\mathbb{Z}})$ a nonzero class. Note that by
dimension consideration, $A$ carries nonzero Gromov-Witten invariants only if
$\int_{A}c_{1}(X)\geqslant 0$. Suppose that $\alpha_{1},\cdots,\alpha_{m}\in
H^{>2}(X,{\mathbb{Q}})$. When $\int_{A}c_{1}(X)>0$, the generalized BPS number
$n_{g,A}^{X}(\alpha_{1},\cdots,\alpha_{m})$ (at least one insertion) is given
by
$\displaystyle\sum\limits_{g=0}^{\infty}u^{2g}\langle\alpha_{1},\cdots,\alpha_{m}\rangle_{g,A}^{X}=\sum\limits_{g=0}^{\infty}u^{2g}n_{g,A}^{X}(\alpha_{1},\cdots,\alpha_{m})\cdot(\frac{\sin\frac{u}{2}}{\frac{u}{2}})^{2g-2+\int_{A}c_{1}(X)},$
and when $\int_{A}c_{1}(X)=0$, then generalized BPS number $n_{g,A}^{X}$ (no
insertion) is given by
$\displaystyle\sum\limits_{g=0}^{\infty}u^{2g}\langle\rangle_{g,A}^{X}=\sum\limits_{g=0}^{\infty}u^{2g}n_{g,A}^{X}\cdot\sum\limits_{\begin{subarray}{c}d\in{\mathbb{Z}}_{>0}\\\
\frac{A}{d}\in
H_{2}(X,{\mathbb{Z}})\end{subarray}}\frac{1}{d}(\frac{\sin\frac{du}{2}}{\frac{u}{2}})^{2g-2}.$
In general, the generalized BPS numbers are defined to satisfy the divisor
equation and defined to vanish if degree $0$ and $1$ classes are inserted. So
these invariants can be extended to include all cohomology classes.
If $X$ is a projective $3$-fold, and $\alpha_{i}$ is the Poincaré dual of a
subvariety $X_{i}\subset X$ in general position, then Pandharipande
conjectured that $n_{g,A}^{X}(\alpha_{1},\cdots,\alpha_{m})$ is the number of
irreducible embedded curves in $X$ of geometric genus $g$, with homology class
$A$ and intersecting all $X_{i}$’s. An important corollary of this conjecture
is the integrality of generalized BPS numbers, which was proved by Zinger in
the Fano case [Z], and by Ionel and Parker in the Calabi-Yau case [IP3].
Proof of Proposition 1.5: We first prove Part (a) in Proposition 1.5.
From Corollary 3.6, we have
$\displaystyle\sum_{g=0}^{\infty}u^{2g-2}\langle[pt],\alpha_{1},\cdots,\alpha_{m}\rangle^{X}_{g,A}=(\frac{\sin(u/2)}{u/2})^{2}\sum_{g=0}^{\infty}u^{2g-2}\langle
p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}.$
Now by the definition of generalized BPS numbers, we have
$\displaystyle\sum_{g\geq
0}u^{2g-2}n^{X}_{g,A}([pt],\alpha_{1},\cdots,\alpha_{m})(\frac{\sin(u/2)}{u/2})^{2g-2+\int_{A}c_{1}(X)}$
$\displaystyle=$
$\displaystyle\sum_{g=0}^{\infty}u^{2g-2}\langle[pt],\alpha_{1},\cdots,\alpha_{m}\rangle^{X}_{g,A}$
$\displaystyle=$
$\displaystyle(\frac{\sin(u/2)}{u/2})^{2}\sum_{g=0}^{\infty}u^{2g-2}\langle
p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle(\frac{\sin(u/2)}{u/2})^{2}\sum_{g=0}^{\infty}u^{2g-2}n^{\tilde{X}}_{g,p^{!}A-e}(p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m})(\frac{\sin(u/2)}{u/2})^{2g-2+\int_{p^{!}A-e}c_{1}(\tilde{X})}$
$\displaystyle=$
$\displaystyle\sum_{g=0}^{\infty}u^{2g-2}n^{\tilde{X}}_{g,p^{!}A-e}(p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m})(\frac{\sin(u/2)}{u/2})^{2g-2+\int_{A}c_{1}(X)}.$
This gives Part (a) of Corollary 1.5. The proof of Part (b) is analogous.
###### Remark 5.1.
In the proof above, we only consider the case $m\geqslant 1$. The case $m=0$
can be treated similarly.
Acknowledgements. The authors would like to thank Prof. Pandharipande for
pointing out his early results about the degenerate contributions to us. We
would also like to thank Prof. Yongbin Ruan and Pedro Acosta for their useful
comments on earlier drafts. Huazhong would like to thank Prof. Jian Zhou for
sharing his ideas on Gromov-Witten theory generously, and Xiaowen Hu and
Hanxiong Zhang for helpful discussions. Weiqiang and Huazhong would like to
thank Department of Mathematics of University of Michigan for its hospitality
during their visiting.
## References
* [AM] Aspinwall, P. S., Morrison, D. R., Topological field theory and rational curves, Comm. Math. Phys. 151 (1993), no. 2, 245-262.
* [B] Behrend, K., Gromov-Witten invariants in algebraic geometry, Invent. Math. 127(1997), 601-617.
* [C] Clemens, C. H., Degeneration of Kähler manifolds, Duke Math. J., 44 (1977), no. 2, 215-290.
* [COGP] Candelas, P., de la Ossa, X. C., Green, P. S., Parkes, L., A pair of Calabi-Yau manifolds as as exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21-74.
* [FO] Fukaya, K., Ono, K., Arnold conjecture and Gromov-Witten invariant, Topology,38(5)(1999), 933-1048.
* [FP] Faber, C., Pandharipande, R., Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), 173-199.
* [G] Gathmann, A., Gromov-Witten invariants of blow-ups, J. Algebraic Geom. 10 (2001), no. 3, 399-432.
* [Gi] Givental, A., Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices (1996), 613-663.
* [GV1] Gopakumar, R., Vafa, C., M-theory and topological strings I, hep-th/9809187.
* [GV2] Gopakumar, R., Vafa, C., M-theory and topological strings II, hep-th/9812127.
* [GP] Graber, T., Pandharipande, R., Localization of virtual classes, Invent.math. 135, 487-518(1999).
* [GV] Graber, T., Vakil, R., Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J. 130 (1) (2005) 1-37.
* [H1] Hu, J., Gromov-Witten invariants of blow-ups along points and curves, Math.Z. 233, 709-739(2000).
* [H2] Hu, J., Gromov-Witten invariants of blow-ups along surfaces, Compositio Math. 125 (2001), no. 3, 345-352.
* [HLR] Hu, J., Li, T.-J., Ruan, Y., Birational cobordism invariance of uniruled symplectic manifolds, Invent. Math., 172(2008), 231-275.
* [IP] Ionel, E., Parker, T., The Symplectic Sum Formula for Gromov-Witten Invariants, Ann. of Math., 159(3), 2004, 935-1025.
* [IP3] Ionel, E., Parker, T., The Gopakumar-Vafa formula for symplectic manifolds, arXiv:1306.1516v2.
* [Ko1] Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., 147(1992), no. 1, 1-23.
* [Ko2] Kontsevich, M., Enumeration of rational curves via torus actions, in The Moduli Space of Curves, Progress in Math. 129 Boston-Basek-Berlin, 1995, 335-368.
* [Li] Li, J., A degeneration formula of GW-invariants, J. Diff. Geom., 60(2002),199-293.
* [LLY] Lian, B., Liu, K., Yau, S.-T., Mirror principle I, Asian J. Math. 1(1997),729-763.
* [LR] Li, A.-M., Ruan, Y., Symplectic surgery and Gromov-Witten invariants of Calabi 3-folds, Invent.Math. 145 (2001), no. 1, 151-218.
* [LT1] Li, J., Tian, G., Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc., 11(1)(1998), 119-174.
* [LT2] Li, J., Tian, G., Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic 4-manifolds(Irvine, CA, 1996), 47-83, First Int. Press Lect. Ser. I, Internat. Press, Cambridge, MA, 1998.
* [MP] Maulik, D., Pandharipande, R., A topological view of Gromov-Witten theory, Topology 45 (2006) 887-918.
* [MS] McDuff, D., Salamon, D., J-holomorphic curves and symplectic topology, AMS Colloquium Publications, vol. 52.
* [P1] Pandharipande, R., Hodge integrals and degenerate contributions, Commum. Math. Phys. 208(1999), 489-506.
* [P2] Pandharipande, R., Three questions in Gromov-Witten theory, International Congress of Mathematics, Vol. II (Beijing, 2002), 503-512, Higher Ed. Press, Beijing, 2002.
* [Q] Qi, X., A blowup formula of high-genus Gromov-Witten invariants of symplectic 4-manifolds, to appear in Advances in Mathematics (China).
* [R1] Ruan, Y., Sympletic topology on algebraic 3-folds, J. Diff. Geom., 39(1994), 215-227.
* [R2] Ruan, Y., Virtual neighborhoods and pseudo-holomorphic curves, Proceedings of 6th Gokova Geometry-Topology Conference, Turkish J. Math., 23(1)(1999),161-231.
* [R3] Ruan, Y., Surgery, quantum cohomology and birational geometry, Northern California Symplectic Geometry Seminar AMS Translations, Series 2, 1999 (196), 183-198.
* [RT1] Ruan, Y., Tian, G., A mathematical theory of quantum cohomology, J. Diff. Geom., 42(2)(1995), 259-367.
* [RT2] Ruan, Y., Tian, G., Higher genus symplectic invariants and sigma model coupled with gravity, Invent. Math. 130(1997), 455-516.
* [S] Siebert, B., Gromov-Witten invariants for general symplectic manifolds, preprint.
* [W] Witten, E., Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990), 243-310, Lehigh Univ., Bethlehem, PA, 1991.
* [Z] Zinger, A., A comparison theorem for Gromov-Witten invariants in the symplectic category, Adv. Math., 228 (2011), no. 1, 535-574.
|
arxiv-papers
| 2014-02-18T03:53:13 |
2024-09-04T02:49:58.345540
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Weiqiang He, Jianxun Hu, Hua-Zhong Ke, Xiaoxia Qi",
"submitter": "Jianxun Hu",
"url": "https://arxiv.org/abs/1402.4221"
}
|
1402.4252
|
# A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow
Structure
José A. Carrillo, Alina Chertock, and Yanghong Huang Department of
Mathematics, Imperial College London, London SW7 2AZ, UK;
[email protected] of Mathematics, North Carolina State
University, Raleigh, NC 27695, USA; [email protected] of
Mathematics, Imperial College London, London SW7 2AZ, UK;
[email protected]
###### Abstract
We propose a positivity preserving entropy decreasing finite volume scheme for
nonlinear nonlocal equations with a gradient flow structure. These properties
allow for accurate computations of stationary states and long-time asymptotics
demonstrated by suitably chosen test cases in which these features of the
scheme are essential. The proposed scheme is able to cope with non-smooth
stationary states, different time scales including metastability, as well as
concentrations and self-similar behavior induced by singular nonlocal kernels.
We use the scheme to explore properties of these equations beyond their
present theoretical knowledge.
## 1 Introduction
In this paper, we consider a finite-volume method for the following problem:
$\left\\{\begin{aligned}
&\rho_{t}=\nabla\cdot\big{[}\rho\nabla\big{(}H^{\prime}(\rho)+V(\mathbf{x})+W\ast\rho\big{)}\big{]},\quad\mathbf{x}\in\mathbb{R}^{d},\
t>0,\\\ &\rho(\mathbf{x},0)=\rho_{0}(\mathbf{x}),\end{aligned}\right.$ (1.1)
where $\rho(\mathbf{x},t)\geq 0$ is the unknown probability measure,
$W(\mathbf{x})$ is an interaction potential, which is assumed to be symmetric,
$H(\rho)$ is a density of internal energy, and $V(\mathbf{x})$ is a
confinement potential.
Equations such as (1.1) appear in various contexts. If $W$ and $V$ vanishes,
and $H(\rho)=\rho\log\rho-\rho$ or $H(\rho)=\rho^{m}$, it is the classical
heat equation or porous medium/fast diffusion equation [38]. If mass-
conserving, self-similar solutions of these equations are sought, the
quadratic term $V(\mathbf{x})=|\mathbf{x}|^{2}$ is added, leading to new
equations in similarity variables. More generally, $V$ usually appears as a
confining potential in Fokker-Planck type equations [19, 31]. Finally, $W$ is
related to the interaction energy, and can be as singular as the Newtonian
potential in chemotaxis system [25] or as smooth as
$W(\mathbf{x})=|\mathbf{x}|^{\alpha}$ with $\alpha>2$ in granular flow [4].
The free energy associated to equation (1.1) is given by (see [17, 18, 40]):
$E(\rho)=\int_{\mathbb{R}^{d}}H(\rho)\,d\mathbf{x}+\int_{\mathbb{R}^{d}}V(\mathbf{x})\rho(\mathbf{x})\,d\mathbf{x}+\frac{1}{2}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}W(\mathbf{x}-\mathbf{y})\rho(\mathbf{x})\rho(\mathbf{y})\,d\mathbf{x}\,d\mathbf{y}\,.$
(1.2)
This energy functional is the sum of internal energy, potential energy and
interaction energy, corresponding to the three terms on the right-hand side of
(1.2), respectively. A simple computation shows that, at least for classical
solutions, the time-derivative of $E(\rho)$ along solutions of (1.1) is
$\frac{d}{dt}E(\rho)=-\int_{\mathbb{R}^{d}}\rho|\mathbf{u}|^{2}\,d\mathbf{x}:=-I(\rho),$
(1.3)
where
$\mathbf{u}=-\nabla\xi,\quad\xi:=\frac{\delta
E}{\delta\rho}=H^{\prime}(\rho)+V(\mathbf{x})+W\ast\rho.$ (1.4)
The functional $I$ will henceforth be referred to as the entropy dissipation
functional.
The equation (1.1) and its associated energy $E(\rho)$ are the subjects of
intensive study during the past fifteen years, see e.g. [29, 1, 40, 17] and
the references therein. The general properties of (1.1) are investigated in
the context of interacting gases [29, 40, 17], and are common to a wide
variety of models, including granular flows [4, 3, 36, 27], porous medium
flows [19, 31], and collective behavior in biology [35]. The gradient flow
structure, in the sense of (1.3), is generalized from smooth solutions to
measure-valued solutions [1]. Certain entropy-entropy dissipation inequalities
between $E(\rho)$ and $I(\rho)$ are also recognized to characterize the fine
details of the convergence to steady states [19, 31, 17].
The steady state of (1.1), if it exists, usually verifies the form
$\xi=H^{\prime}(\rho)+V(\mathbf{x})+W\ast\rho=C,\quad\mbox{on supp }\rho,$
(1.5)
where the constant $C$ could be different on different connected components of
$\mbox{supp }\rho$. In many cases, especially in the presence of the
interaction potential $W$, there are multiple steady states, whose explicit
forms are available only for particular $W$. Most of studies of these steady
states are based on certain assumptions on the support and the characterizing
equation (1.5).
In this work, we propose a positivity preserving finite-volume method to treat
the general nonlocal nonlinear PDE (1.1). Moreover, we show the existence of a
discrete free energy that is dissipated for the semi-discrete scheme (discrete
in space only). A related method was already proposed in [5] for the case of
nonlinear degenerate diffusions in any dimension. We generalize this method to
cover the nonlocal terms for both 1D and 2D cases in Section 2. In fact, the
first order scheme generalizes easily to cover unstructured meshes. However,
it is an open problem how to obtain entropy decreasing higher order schemes in
this setting in 2D. Let us remark that other numerical methods based on finite
element approximations have been proposed in the literature which are
positivity preserving and entropy decreasing at the expense of constructing
them by an implicit discretization in time but continuous in space, see [9].
Section 3 is devoted to numerical experiments, in which the performance of the
developed numerical approach is tested. In Section 3.1, we conduct the
convergence study of stationary states, where the order of accuracy depends on
the regularity at free boundaries. We then showcase the performance of this
method for finding stable stationary states with nonlocal terms and their
equilibration rate in time for different nonlocal models. In Section 3.2, we
emphasize how this method is useful to explore different open problems in the
analysis of these nonlocal nonlinear models such as the Keller-Segel model for
chemotaxis in its different versions. We continue in Section 3.3 with
aggregation equations with repulsive-attractive kernels and address the issue
of singular kernels and discontinuous steady states. Finally, in Section 3.4,
we demonstrate the performance of the scheme in a number of 2-D experiments
showcasing numerical difficulties and interesting asymptotics.
## 2 Numerical Method
In this section, we describe both one- (1-D) and two-dimensional (2-D) finite-
volume schemes for (1.1) and prove their positivity preserving and entropy
dissipation properties. We also establish error estimates and convergence
results for the proposed methods. We start in §2.1 with the 1-D case and then
generalize it to the 2-D case in §2.2, both on uniform meshes. The extension
to higher dimensions and non-uniform structured meshes is straightforward.
### 2.1 One-Dimensional Case
We begin with the derivation of the 1-D second-order finite-volume method for
equation (1.1). For simplicity, we divide the computational domain into
finite-volume cells $C_{j}=[x_{j-\frac{1}{2}},x_{j+\frac{1}{2}}]$ of a uniform
size $\Delta x$ with $x_{j}=j\Delta x$, $j\in\\{-M,\cdots,M\\}$, and denote by
$\overline{\rho}_{j}(t)=\frac{1}{\Delta x}\int_{C_{j}}\rho(x,t)\,dx,$
the computed cell averages of the solution $\rho$, which we assume to be known
or approximated at time $t\geq 0$. A semi-discrete finite-volume scheme is
obtained by integrating equation (1.1) over each cell $C_{j}$ and is given by
the following system of ODEs for $\overline{\rho}_{j}$:
$\frac{d\overline{\rho}_{j}(t)}{dt}=-\frac{F_{j+\frac{1}{2}}(t)-F_{j-\frac{1}{2}}(t)}{\Delta
x},$ (2.1)
where the numerical flux $F_{j+\frac{1}{2}}$ approximate the continuous flux
$-\rho\xi_{x}=-\rho(H^{\prime}(\rho)+V(x)+W\ast\rho)_{x}$ at cell interface
$x_{j+\frac{1}{2}}$ and is constructed next. For simplicity, we will omit the
dependence of the computed quantities on $t\geq 0$ in the rest. As in the case
of degenerate diffusion equations treated in [5], we use the upwind numerical
fluxes. To this end, we first construct piecewise linear polynomials in each
cell $C_{j}$,
$\widetilde{\rho}_{j}(x)=\overline{\rho}_{j}+(\rho_{x})_{j}(x-x_{j}),\quad
x\in C_{j},$ (2.2)
and compute the right (“east”), $\rho_{j}^{\rm E}$, and left (“west”),
$\rho_{j}^{\rm W}$, point values at the cell interfaces $x_{j-\frac{1}{2}}$
and $x_{j+\frac{1}{2}}$, respectively:
$\displaystyle\rho_{j}^{\rm
E}=\widetilde{\rho}_{j}(x_{j+\frac{1}{2}}-0)=\overline{\rho}_{j}+\frac{\Delta
x}{2}(\rho_{x})_{j},$ (2.3) $\displaystyle\rho_{j}^{\rm
W}=\widetilde{\rho}_{j}(x_{j-\frac{1}{2}}+0)=\overline{\rho}_{j}-\frac{\Delta
x}{2}(\rho_{x})_{j}.$
These values will be second-order accurate provided the numerical derivatives
$(\rho_{x})_{j}$ are at least first-order accurate approximations of
$\rho_{x}(x,\cdot)$. To ensure that the point values (2.3) are both second-
order and nonnegative, the slopes $(\rho_{x})_{j}$ in (2.2) are calculated
according to the following adaptive procedure. First, the centered-difference
approximations
$(\rho_{x})_{j}=(\overline{\rho}_{j+1}-\overline{\rho}_{j-1})/(2\Delta x)$ is
used for all $j$. Then, if the reconstructed point values in some cell $C_{j}$
become negative (i.e., either $\rho_{j}^{\rm E}<0$ or $\rho_{j}^{\rm W}<0$),
we recalculate the corresponding slope $(\rho_{x})_{j}$ using a slope limiter,
which guarantees that the reconstructed point values are nonnegative as long
as the cell averages $\overline{\rho}_{j}$ are nonnegative. In our numerical
experiments, we have used a generalized minmod limiter [28, 30, 34, 37]:
$(\rho_{x})_{j}={\rm
minmod}\Big{(}\theta\,\frac{\overline{\rho}_{j+1}-\overline{\rho}_{j}}{\Delta
x},\,\frac{\overline{\rho}_{j+1}-\overline{\rho}_{j-1}}{2\Delta
x},\,\theta\,\frac{\overline{\rho}_{j}-\overline{\rho}_{j-1}}{\Delta
x}\Big{)},$
where
${\rm
minmod}(z_{1},z_{2},\ldots):=\left\\{\begin{array}[]{ll}\min(z_{1},z_{2},\ldots),&\mbox{if}~{}z_{i}>0\quad\forall\
i,\\\ \max(z_{1},z_{2},\ldots),&\mbox{if}~{}z_{i}<0\quad\forall\ i,\\\
0,&~{}\mbox{otherwise},\end{array}\right.$
and the parameter $\theta$ can be used to control the amount of numerical
viscosity present in the resulting scheme. In all the numerical examples
below, $\theta=2$ is used.
Equipped with the piecewise linear reconstruction $\widetilde{\rho}_{j}(x)$
and point values $\rho_{j}^{\rm E},\ \rho_{j}^{\rm W}$, the upwind fluxes in
(2.1) are computed as
$F_{j+\frac{1}{2}}=u_{j+\frac{1}{2}}^{+}\rho_{j}^{\rm
E}+u_{j+\frac{1}{2}}^{-}\rho_{j+1}^{\rm W},$ (2.4)
where the discrete values $u_{j+\frac{1}{2}}$ of the velocities are obtained
using the centered-difference approach,
$u_{j+\frac{1}{2}}=-\frac{\xi_{j+1}-\xi_{j}}{\Delta x},$ (2.5)
and the positive and negative parts of $u_{j+\frac{1}{2}}$ are denoted by
$u_{j+\frac{1}{2}}^{+}=\max(u_{j+\frac{1}{2}},0),\qquad
u_{j+\frac{1}{2}}^{-}=\min(u_{j+\frac{1}{2}},0).$ (2.6)
The discrete velocity field $\xi_{j}$ is calculated by discretizing (1.4):
$\xi_{j}=\Delta
x\sum\limits_{i}W_{j-i}\overline{\rho}_{i}+H^{\prime}(\overline{\rho}_{j})+V_{j},$
(2.7)
where $W_{j-i}=W(x_{j}-x_{i})$ and $V_{j}=V(x_{j})$. The formula (2.7) is a
second-order approximation of
$\sum\limits_{i}\int_{C_{i}}W(x_{j}-s)\widetilde{\rho}_{i}(s)\,ds+H^{\prime}(\widetilde{\rho}_{j}(x_{j}))+V(x_{j}).$
Indeed, the reconstruction (2.2) yields
$H^{\prime}(\widetilde{\rho}_{j}(x_{j}))=H^{\prime}(\overline{\rho}_{j})$ and
$\displaystyle\sum\limits_{i}\int_{C_{i}}W(x_{j}-s)\widetilde{\rho}_{i}(s)\,ds$
$\displaystyle=\sum\limits_{i}\overline{\rho}_{i}\int_{C_{i}}W(x_{j}-s)\,ds+\sum\limits_{i}(\rho_{x})_{i}\int_{C_{i}}W(x_{j}-s)(s-x_{i})\,ds$
(2.8) $\displaystyle=\Delta
x\sum\limits_{i}W_{j-i}\overline{\rho}_{i}+\mathcal{O}(\Delta x^{2}),$ (2.9)
Here $W_{j-i}$ can be any approximation of the local integral $\frac{1}{\Delta
x}\int_{C_{i}}W(x_{j}-s)ds$ with error $O(\Delta x^{2})$. If $W$ has a bounded
second order derivative near $x_{j-i}$, $W_{j-i}$ can be chosen to be
$W(x_{j-i})$ (the middle point rule) or
$\big{(}W(x_{j-i-1/2})+W(x_{j-i+1/2})\big{)}/2$ (the trapezoidal rule). The
integral $\int_{C_{i}}W(x_{j}-s)(s-x_{i})\,ds$ in the second summation is of
$O(\Delta x^{3})$ because of the anti-symmetric factor $s-x_{i}$, leading to
overall error $O(\Delta x^{2})$.
The case with non-smooth or singular interaction potential $W$ has to be
treated more carefully. First, the last integral
$\int_{C_{i}}W(x_{j}-s)(s-x_{i})\,ds$ in the above formula vanishes as soon as
$i=j$ due to the symmetry of $W$ independently of any possible singularity at
$x=x_{j}$. If $W$ has a locally integrable singularity (usually at the
origin), $\frac{1}{\Delta x}\int_{C_{i}}W(x_{j}-s)ds$ can still be
approximated by a higher order quadrature scheme with an error $O(\Delta
x^{2})$ or smaller. Actually, in the particular case of powers or logarithm
kernels, it can be explicitly computed. However, the second sum above may have
a slightly larger error. For instance, if $W(x)\sim|x|^{-\alpha}$ for
$0<\alpha<1$, then $\int_{C_{i}}W(x_{j}-s)(s-x_{i})\,ds\sim O(\Delta
x^{2-\alpha})$ by direct computation when $|i-j|$ is close to zero.
Finally, the semi-discrete scheme (2.1) is a system of ODEs, which has to be
integrated numerically using a stable and accurate ODE solver. In all
numerical examples reported in next section, the third-order strong preserving
Runge-Kutta (SSP-RK) ODE solver [24] is used.
###### Remark 2.1.
The computational bottleneck is the discrete convolution in (2.7). This is a
classical problem in scientific computing that can be effectively evaluated
using fast convolution algorithms, mainly based Fast Fourier Transforms [41].
###### Remark 2.2.
The second-order finite-volume scheme (2.1), (2.4)–(2.7), reduces to the
first-order one if the piecewise constant reconstruction is used instead of
(2.2), in which case one has
$\widetilde{\rho}_{j}(x)=\overline{\rho}_{j},\quad x_{j}\in
C_{j},\quad\mbox{and therefore}\quad\rho_{j}^{\rm E}=\rho_{j}^{\rm
W}=\overline{\rho}_{j},\quad\forall j.$
#### Positivity Preserving.
The resulting scheme preserves positivity of the computed cell averages
$\overline{\rho}_{j}$ as stated in the following theorem. The proof is based
on the forward Euler integration of the ODE system (2.1), but will remain
equally valid if the forward Euler method were replaced by a higher-order SSP
ODE solver [24], whose time step can be expressed as a convex combination of
several forward Euler steps.
###### Theorem 2.3.
Consider the system (1.1) with initial data $\rho_{0}(x)\geq 0$ and the semi-
discrete finite-volume scheme (2.1), (2.4)–(2.7) with a positivity preserving
piecewise linear reconstruction (2.2) for $\rho$. Assume that the system of
ODEs (2.1) is discretized by the forward Euler method. Then, the computed cell
averages $\overline{\rho}_{j}\geq 0,\ \forall\ j$, provided that the following
CFL condition is satisfied:
$\Delta t\leq\frac{\Delta x}{2a},\quad\mbox{where}\quad
a=\max\limits_{j}\left\\{u_{j+\frac{1}{2}}^{+},-u_{j+\frac{1}{2}}^{-}\right\\},$
(2.10)
with $u_{j+\frac{1}{2}}^{+}$ and $u_{j+\frac{1}{2}}^{-}$ defined in (2.6).
###### Proof.
Assume that at a given time $t$ the computed solution is known and positive:
$\overline{\rho}_{j}\geq 0,\ \forall j$. Then the new cell averages are
obtained from the forward Euler discretization of equation (2.1):
$\overline{\rho}_{j}(t+\Delta
t)=\overline{\rho}_{j}(t)-\lambda\left[F_{j+\frac{1}{2}}(t)-F_{j-\frac{1}{2}}(t)\right],$
(2.11)
where $\lambda:={\Delta t}/{\Delta x}$. As above, the dependence of all terms
on the RHS of (2.11) on $t$ is suppressed in the following to simplify the
notation. Using (2.4) and the fact that
$\overline{\rho}_{j}=\frac{1}{2}\left(\rho_{j}^{\rm E}+\rho_{j}^{\rm
W}\right)$ (see (2.3)), we obtain
$\displaystyle\overline{\rho}_{j}(t+\Delta t)$
$\displaystyle=\frac{1}{2}\left(\rho_{j}^{\rm E}+\rho_{j}^{\rm
W}\right)-\lambda\left[u_{j+\frac{1}{2}}^{+}\rho_{j}^{\rm
E}+u_{j+\frac{1}{2}}^{-}\rho_{j+1}^{\rm
W}-u_{j-\frac{1}{2}}^{+}\rho_{j-1}^{\rm E}-u_{j-\frac{1}{2}}^{-}\rho_{j}^{\rm
W}\right]$ (2.12) $\displaystyle=\lambda u_{j-\frac{1}{2}}^{+}\rho_{j-1}^{\rm
E}+\left(\frac{1}{2}-\lambda u_{j+\frac{1}{2}}^{+}\right)\rho_{j}^{\rm
E}+\left(\frac{1}{2}+\lambda u_{j-\frac{1}{2}}^{-}\right)\rho_{j}^{\rm
W}-\lambda u_{j+\frac{1}{2}}^{-}\rho_{j+1}^{\rm W}.$
It follows from (2.12) that the new cell averages
$\overline{\rho}_{j}(t+\Delta t)$ are linear combinations of the nonnegative
reconstructed point values $\rho_{j-1}^{\rm E},\rho_{j}^{\rm E},\rho_{j}^{\rm
W}$ and $\rho_{j+1}^{\rm W}$. Since $u_{j-\frac{1}{2}}^{+}\geq 0$ and
$u_{j+\frac{1}{2}}^{-}\leq 0$, we conclude that $\overline{\rho}_{j}(t+\Delta
t)\geq 0,\ \forall j$, provided that the CFL condition (2.10) is satisfied. ∎
###### Remark 2.4.
Similar result holds for the first-order finite-volume scheme with the CFL
condition reduced to
$\Delta t\leq\frac{\Delta
x}{2\max\limits_{j}\left(u_{j+\frac{1}{2}}^{+}-u_{j-\frac{1}{2}}^{-}\right)}.$
#### Discrete Entropy Dissipation.
A discrete version of the entropy $E$ defined in (1.2) is given by
$E_{\Delta}(t)=\Delta x\sum\limits_{j}\left[\frac{1}{2}\Delta
x\sum\limits_{i}W_{j-i}\overline{\rho}_{i}\overline{\rho}_{j}+H(\overline{\rho}_{j})+{V}_{j}\overline{\rho}_{j}\right].$
(2.13)
We also introduce the discrete version of the entropy dissipation
$I_{\Delta}(t)=\Delta
x\sum\limits_{j}(u_{j+\frac{1}{2}})^{2}\min\limits_{j}(\rho_{j}^{\rm
E},\rho_{j+1}^{\rm W}).$ (2.14)
In the following theorem, we prove that the time derivative of $E_{\Delta}(t)$
is less or equal than the negative of $I_{\Delta}(t)$, mimicking the
corresponding property of the continuous relation.
###### Theorem 2.5.
Consider the system (1.1) with no flux boundary conditions on $[-L,L]$ with
$L>0$ and with initial data $\rho_{0}(x)\geq 0$. Given the semi-discrete
finite-volume scheme (2.1) with $\Delta x=L/M$, (2.4)–(2.7) with a positivity
preserving piecewise linear reconstruction (2.2) for $\rho$ and discrete
boundary conditions $F_{M+\frac{1}{2}}=F_{-M-\frac{1}{2}}=0$. Then,
$\frac{d}{dt}E_{\Delta}(t)\leq-I_{\Delta}(t),\quad\forall t>0.$
###### Proof.
We start by differentiating (2.13) with respect to time to obtain:
$\displaystyle\frac{d}{dt}E_{\Delta}(t)$ $\displaystyle=\Delta
x\sum\limits_{j}\left[\Delta
x\sum\limits_{i}W_{j-i}\overline{\rho}_{i}\frac{d\overline{\rho}_{j}}{dt}+H^{\prime}(\overline{\rho}_{j})\frac{d\overline{\rho}_{j}}{dt}+{V}_{j}\frac{d\overline{\rho}_{j}}{dt}\right]$
$\displaystyle=\Delta x\sum\limits_{j}\left[\Delta
x\sum\limits_{i}W_{j-i}\overline{\rho}_{i}+H^{\prime}(\overline{\rho}_{j})+{V}_{j}\right]\frac{d\overline{\rho}_{j}}{dt}.$
Using the definition (2.7) and the numerical scheme (2.1), we have
$\frac{d}{dt}E_{\Delta}(t)=-\Delta
x\sum\limits_{j}\xi_{j}\,\frac{F_{j+\frac{1}{2}}-F_{j-\frac{1}{2}}}{\Delta
x}.$
A discrete integration by parts using the no flux discrete boundary conditions
along with (2.5) yields
$\frac{d}{dt}E_{\Delta}(t)=-\sum\limits_{j}(\xi_{j}-\xi_{j+1})F_{j+\frac{1}{2}}=-\Delta
x\sum\limits_{j}u_{j+\frac{1}{2}}F_{j+\frac{1}{2}}.$
Finally, using the definition of the upwind fluxes (2.4) and formulas (2.6)
and (2.14), we conclude
$\frac{d}{dt}E_{\Delta}(t)=-\Delta
x\sum\limits_{j}u_{j+\frac{1}{2}}\left[u_{j+\frac{1}{2}}^{+}\rho_{j}^{\rm
E}+u_{j+\frac{1}{2}}^{-}\rho_{j+1}^{\rm W}\right]\leq-\Delta
x\sum\limits_{j}(u_{j+\frac{1}{2}})^{2}\min\limits_{j}(\rho_{j}^{\rm
E},\rho_{j+1}^{\rm W})=-I_{\Delta}(t).$
∎
### 2.2 Two-Dimensional Case
In this subsection, we quickly describe a semi-discrete second-order finite-
volume method for the 2-D equation (1.1). We explain the main ideas in 2D for
the sake of the reader. As already mentioned, the first order scheme
generalizes easily to unstructured meshes. However, higher order schemes with
the desired entropy decreasing property are harder to obtain in this setting
for higher dimensions. We introduce a Cartesian mesh consisting of the cells
$C_{j,k}:=[x_{j-\frac{1}{2}},x_{j+\frac{1}{2}}]\times[y_{k-\frac{1}{2}},y_{k+\frac{1}{2}}]$,
which for the sake of simplicity are assumed to be of the uniform size $\Delta
x\Delta y$, that is, $x_{j+\frac{1}{2}}-x_{j-\frac{1}{2}}\equiv\Delta x,\
\forall\ j$, and $y_{k+\frac{1}{2}}-y_{k-\frac{1}{2}}\equiv\Delta y,\ \forall\
k$.
A general semi-discrete finite-volume scheme for equation (1.1) can be written
in the following form:
$\frac{d\overline{\rho}_{j,k}}{dt}=-\frac{F^{x}_{j+\frac{1}{2},k}-F^{x}_{j-\frac{1}{2},k}}{\Delta
x}-\frac{F^{y}_{j,k+\frac{1}{2}}-F^{y}_{j,k-\frac{1}{2}}}{\Delta y}.$ (2.15)
Here, we define
$\bar{\rho}_{j,k}(t)\approx\dfrac{1}{\Delta x\Delta
y}\iint_{C_{j,k}}\rho(x,y,t)dxdy$
as the cell averages of the computed solution and $F^{x}_{j+\frac{1}{2},k}$
and $F^{y}_{j,k+\frac{1}{2}}$ are upwind numerical fluxes in the $x$ and $y$
directions, respectively.
As in the 1-D case, to obtain formulae for numerical fluxes, we first compute
$\rho_{j,k}^{\rm E},\rho_{j,k}^{\rm W},\rho_{j,k}^{\rm N}$ and
$\rho_{j,k}^{\rm S}$, which are one-sided point values of the piecewise linear
reconstruction
$\widetilde{\rho}(x,y)=\overline{\rho}_{j,k}+(\rho_{x})_{j,k}(x-x_{j})+(\rho_{y})_{j,k}(y-y_{k}),\quad(x,y)\in
C_{j,k},$ (2.16)
at the cell interfaces $(x_{j+\frac{1}{2}},y_{k})$,
$(x_{j-\frac{1}{2}},y_{k})$, $(x_{j},y_{k+\frac{1}{2}})$,
$(x_{j},y_{k-\frac{1}{2}})$, respectively. Namely,
$\displaystyle\rho_{j,k}^{\rm
E}:=\widetilde{\rho}(x_{j+\frac{1}{2}}-0,y_{k})=\overline{\rho}_{j,k}+\frac{\Delta
x}{2}(\rho_{x})_{j,k},\quad\rho_{j,k}^{\rm
W}:=\widetilde{\rho}(x_{j-\frac{1}{2}}+0,y_{k})=\overline{\rho}_{j,k}-\frac{\Delta
x}{2}(\rho_{x})_{j,k},$ (2.17) $\displaystyle\rho_{j,k}^{\rm
N}:=\widetilde{\rho}(x_{j},y_{k+\frac{1}{2}}-0)=\overline{\rho}_{j,k}+\frac{\Delta
y}{2}(\rho_{y})_{j,k},\quad\rho_{j,k}^{\rm
S}:=\widetilde{\rho}(x_{j},y_{k-\frac{1}{2}}+0)=\overline{\rho}_{j,k}-\frac{\Delta
y}{2}(\rho_{y})_{j,k}.$
To ensure the point values in (2.17) are both second-order and nonnegative,
the slopes in (2.16) are calculated according to the adaptive procedure
similarly to the 1-D case. First, the centered-difference approximations,
$(\rho_{x})_{j,k}=\frac{\overline{\rho}_{j+1,k}-\overline{\rho}_{j-1,k}}{2\Delta
x}\quad\mbox{and}\quad(\rho_{y})_{j,k}=\frac{\overline{\rho}_{j,k+1}-\overline{\rho}_{j,k-1}}{2\Delta
y}$
are used for all $j,k$. Then, if the reconstructed point values in some cell
$C_{j,k}$ become negative, we recalculate the corresponding slopes
$(\rho_{x})_{j,k}$ or $(\rho_{y})_{j,k}$ using a monotone nonlinear limiter,
which guarantees that the reconstructed point values are nonnegative as long
as the cell averages of $\overline{\rho}_{j,k}$ are nonnegative for all $j,k$.
In our numerical experiments, we have used the one-parameter family of the
generalized minmod limiters with $\theta\in[1,2]$:
$\displaystyle(\rho_{x})_{j,k}=\textrm{minmod}\left(\theta\frac{\overline{\rho}_{j,k}-\overline{\rho}_{j-1,k}}{\Delta
x},\frac{\overline{\rho}_{j+1,k}-\overline{\rho}_{j-1,k}}{2\Delta
x},\theta\frac{\overline{\rho}_{j+1,k}-\overline{\rho}_{j,k}}{\Delta
x}\right),$
$\displaystyle(\rho_{y})_{j,k}=\textrm{minmod}\left(\theta\frac{\overline{\rho}_{j,k}-\overline{\rho}_{j,k-1}}{\Delta
y},\frac{\overline{\rho}_{j,k+1}-\overline{\rho}_{j,k-1}}{2\Delta
y},\theta\frac{\overline{\rho}_{j,k+1}-\overline{\rho}_{j,k}}{\Delta
y}\right).$
Given the polynomial reconstruction (2.16) and its point values (2.17), the
upwind numerical fluxes in (2.15) are defined as
$F_{j+\frac{1}{2},k}^{x}=u_{j+\frac{1}{2},k}^{+}\rho_{j,k}^{\rm
E}+u_{j+\frac{1}{2},k}^{-}\rho_{j+1,k}^{\rm W},\qquad
F_{j,k+\frac{1}{2}}^{y}=v_{j,k+\frac{1}{2}}^{+}\rho_{j,k}^{\rm
N}+v_{j,k+\frac{1}{2}}^{-}\rho_{j,k+1}^{\rm S},$ (2.18)
where
$u_{j+\frac{1}{2},k}=-\frac{\xi_{j+1,k}-\xi_{j,k}}{\Delta x},\qquad
v_{j,k+\frac{1}{2}}=-\frac{\xi_{j,k+1}-\xi_{j,k}}{\Delta y},$
the values of $u_{j+\frac{1}{2},k}^{\pm}$ and $v_{j,k+\frac{1}{2}}^{\pm}$ are
defined according to (2.6), and
$\xi_{j,k}=\Delta x\Delta
y\sum\limits_{i,l}W_{j-i,k-l}\overline{\rho}_{i,l}+H^{\prime}(\overline{\rho}_{j,k})+V_{j,k}.$
(2.19)
Here, $W_{j-i,k-l}=W(x_{j}-x_{i},y_{k}-y_{l})$ and $V_{j,k}=V(x_{j},y_{k})$.
Similarly to the 1-D case, the formula (2.19) for $\xi_{j,k}$ is obtained by
using the reconstruction formula (2.16) and applying the midpoint quadrature
rule to the first integral in
$\xi_{j,k}=\sum\limits_{i,l}\iint_{C_{i,l}}W(x-s,y-r)\widetilde{\rho}_{i,l}(s,r)\,ds\,dr\\\
+H^{\prime}(\widetilde{\rho}_{j,k}(x,y))+V(x_{j},y_{k}).$
As in the 1-D case, the ODE system (2.15) is to be integrated numerically by a
stable and sufficiently accurate ODE solver such as the third-order SSP-RK ODE
solver [24].
###### Remark 2.6.
As in the 1-D case, the first-order finite-volume method is obtained by taking
$\widetilde{\rho}_{j,k}(x,y)=\overline{\rho}_{j,k}\quad{\rm
and}\quad\rho_{j,k}^{\rm E}=\rho_{j,k}^{\rm W}=\rho_{j,k}^{\rm
N}=\rho_{j,k}^{\rm S}=\overline{\rho}_{j,k},\quad\forall j,k.$
#### Positivity Preserving.
The resulting 2-D finite-volume scheme will preserve positivity of the
computed cell averages $\overline{\rho}_{j,k},\ \forall j,k$, as long as an
SSP ODE solver, whose time steps are convex combinations of forward Euler
steps, is used for time integration. We omit the proof of the positivity
property of the scheme as it follows exactly the lines of Theorem 2.3. The
only difference is that in the 2-D case
$\overline{\rho}_{j,k}=\tfrac{1}{4}\left(\rho_{j,k}^{\rm E}+\rho_{j,k}^{\rm
W}+\rho_{j,k}^{\rm N}+\rho_{j,k}^{\rm S}\right)$, which leads to a slightly
modified CFL condition. We thus have the following theorem.
###### Theorem 2.7.
Consider the system (1.1) with initial data $\rho_{0}(x)\geq 0$ and the semi-
discrete finite-volume scheme (2.15), (2.18)–(2.19) with a positivity
preserving piecewise linear reconstruction (2.16) for $\rho$. Assume that the
system of ODEs (2.15) is discretized by the forward Euler (or a strong
stability preserving Runge-Kutta) method. Then, the computed cell averages
$\overline{\rho}_{j,k}\geq 0,\ \forall j,k$, provided the following CFL
condition is satisfied:
$\Delta t\leq\min\left\\{\frac{\Delta x}{4a},\frac{\Delta
y}{4b}\right\\},\quad
a=\max\limits_{j,k}\left\\{u_{j+\frac{1}{2},k}^{+},-u_{j+\frac{1}{2},k}^{-}\right\\},\quad
b=\max\limits_{j,k}\left\\{v_{j,k+\frac{1}{2}}^{+},-v_{j,k+\frac{1}{2}}^{-}\right\\},$
where $u_{j+\frac{1}{2},k}^{\pm}$ and $v_{j,k+\frac{1}{2}}^{\pm}$ are defined
according to (2.6).
#### Discrete Entropy Dissipation.
We define the discrete entropy
$E_{\Delta}(t)=\Delta x\Delta y\sum\limits_{j,k}\left[\frac{1}{2}\Delta
x\Delta
y\sum\limits_{i,l}W_{j-i,k-l}\overline{\rho}_{i,l}\overline{\rho}_{j,k}+H(\overline{\rho}_{j,k})+\overline{V}_{j,k}\overline{\rho}_{j,k}\right],$
and discrete entropy dissipation
$I_{\Delta}(t)=\Delta x\Delta
y\sum\limits_{j,k}\left[(u_{j+\frac{1}{2},k})^{2}+(v_{j,k+\frac{1}{2}})^{2}\right]\min\limits_{j,k}\left(\rho_{j,k}^{\rm
E},\rho_{j+1,k}^{\rm W},\rho_{j,k}^{\rm N},\rho_{j,k+1}^{\rm S}\right).$
Similarly to the 1-D case, we can show the following dissipative property of
the scheme.
###### Theorem 2.8.
Consider the system (1.1) with no flux boundary conditions in the domain
$[-L,L]^{2}$ with $L>0$ and with initial data $\rho_{0}(x)\geq 0$. Given the
semi-discrete finite-volume scheme (2.15), (2.18)–(2.19) with a positivity
preserving piecewise linear reconstruction (2.16) for $\rho$, with $\Delta
x=L/M$, and with discrete no-flux boundary conditions
$F^{x}_{M+\frac{1}{2},k}={F^{x}_{-M-\frac{1}{2},k}}=F^{y}_{j,M+\frac{1}{2}}={F^{y}_{j,-M-\frac{1}{2}}}=0$.
Then,
$\frac{d}{dt}E_{\Delta}(t)\leq-I_{\Delta}(t),\quad\forall t>0.$
## 3 Numerical Experiments
In this section, we present several numerical examples, focusing mainly on the
steady states or long time behaviors of the solutions to the general equation
$\rho_{t}=\nabla\cdot\big{[}\rho\nabla\big{(}H^{\prime}(\rho)+V(\mathbf{x})+W\ast\rho\big{)}\big{]},\quad\mathbf{x}\in\mathbb{R}^{d},\
t>0.$
A previous detailed study in [5] for the degenerate parabolic and drift-
diffusion equations demonstrated the good performance of the method (with
small variants) in dealing with exponential rates of convergence toward
compactly supported Barenblatt solutions. Here we will concentrate mostly on
cases with the interaction potential $W$, and show that key properties like
non-negativity and entropy dissipation are preserved. We will first start our
discussion by using some test cases to validate the order of convergence of
the scheme in space and its relation to the regularity of the steady states.
If the solution $\rho$ is smooth, the spatial discretization given in Section
2 is shown to be of second order. However, in practice, the steady states of
(1.1) are usually compactly supported, with discontinuities in the derivatives
or even in the solutions themselves near the boundaries. This loss of
regularity of the steady states usually leads to degeneracy in the order of
convergence, as shown in Examples 1–4. Then, we will illustrate with several
examples that the presented finite-volume scheme can be used for a numerical
study of many challenging questions in which theoretical analysis has not yet
been fully developed.
### 3.1 Steady states: Spatial Order and Time Stabilization
###### Example 1 (Attractive-repulsive kernels).
We first consider equation (1.1) in 1-D with only the interaction kernel
$W(x)=|x|^{2}/2-\log|x|$ (i.e., $H(\rho)=0,V(x)=0$). The corresponding unit-
mass steady state is given by (see [32]):
$\rho_{\infty}(x)=\begin{cases}\frac{1}{\pi}\sqrt{2-x^{2}},\qquad&|x|\leq\sqrt{2},\cr
0,&\mbox{otherwise}\,.\end{cases}$
and is Hölder continuous with exponent $\alpha=\frac{1}{2}$. This steady state
is the unique global minimizer of the free energy (1.2) and it approached by
the solutions of (1.1) with an exponential convergence rate as shown in [16].
We compute $\rho_{\infty}$ by numerically solving (1.1) at large time, with
the initial condition $\rho(x,0)=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$. In Figure
3.1(a), we plot the numerical steady state obtained on a very coarse grid with
$\Delta x=\sqrt{2}/5$. As one can see, even on such a coarse grid, the
numerical steady state is in good agreement with the exact one, except near
the boundary $x=\pm\sqrt{2}$. The spatial convergence error of the steady
states in $L^{1}$ norm and $L^{\infty}$ norm is shown in Figure 3.1(b). As a
general rule, the practical convergence error of the numerical steady state is
$\alpha$ in $L^{\infty}$ norm and $\alpha+1$ in $L^{1}$ norm, if the exact
steady state is $C^{\alpha}$-Hölder continuous.
Figure 3.1: (a) The numerical steady state with grid size $\Delta
x=\sqrt{2}/5$, compared with the exact one. (b) The convergence of error in
$L^{1}$ and $L^{\infty}$ norms. Here the $L^{1}$ norm is computed by taking
the numerical steady state piecewise constant inside each cell and
$L^{\infty}$ norm is evaluated only at the cell centers.
###### Example 2 (Nonlinear diffusion with nonlocal attraction kernel).
Next, we consider the equation (1.1) in 1-D with
$H(\rho)=\frac{\nu}{m}\rho^{m}$, $W(x)=W(|x|)$ and $V(x)=0$, where $\nu>0,m>1$
and $W\in\mathcal{W}^{1,1}(\mathbb{R})$ is an increasing function on
$[0,\infty)$, i.e.,
$\rho_{t}=\big{(}\rho(\nu\rho^{m-1}+W\ast\rho)_{x}\big{)}_{x}.$ (3.1)
This equation arises in some physical and biological modelling with competing
nonlinear diffusion and nonlocal attraction, see [35] for instance. The
attraction represented by convolution $W*\rho$ is relatively weak (compared to
that in the Keller-Segel model discussed below), and the solution does not
blow up with bounded initial data, while the long time behavior of the
solution is characterized by an extensive study of the steady states in [11].
When $m>2$, the attraction dominates the nonlinear diffusion, leading to a
compactly supported steady state. When $m<2$, the behavior depends on the
diffusion coefficient $\nu$: there is a local steady state for small $\nu$
with localized initial data and the solution always decays to zero for large
$\nu$. The borderline case $m=2$ is investigated in [10] for non-compactly
supported kernels, where the evolution depends on the coefficient $\nu$, the
total conserved mass, and $\|W\|_{1}$.
We begin by numerically calculating the solutions to the 1-D equation (3.1)
with nonlinear diffusion and $W(x)=-e^{-|x|^{2}/2\sigma}/\sqrt{2\pi\sigma}$,
for some constants $\sigma>0$. The corresponding steady states can also be
obtained by implementing an iterative procedure proposed in [11]. Here, we
compute the steady state solutions $\rho_{\infty}$ by the time evolution of
(3.1) subject to Gaussian-type initial data
$\rho_{0}(x)=\frac{1}{\sqrt{8\pi}}\left[e^{-0.5(x-3)^{2}}+e^{-0.5(x+3)^{2}}\right].$
The simulations are run on the computational domain $[-6,6]$ with the mesh
size $\Delta x=0.02$ for large time until stabilization and the results are
plotted in Figure 3.2(a) for different values of $m$. As one can observe, the
boundary behavior of the compactly supported steady states has a similar
dependence on $m$ as the Barenblatt solutions of the classical porous medium
equation $\rho_{t}=\nu\big{(}\rho(\rho^{m-1})_{x}\big{)}_{x}$, that is, only
Hölder continuous with exponent $\alpha=\min\big{(}1,1/(m-1)\big{)}$. Using
the steady states of (1.1) computed by the iterative scheme proposed in [11],
we can check the spatial convergence error of our scheme on different grid
sizes $\Delta x$. As shown in Figure 3.2(b), the spatial convergence error of
the steady states is $\min\big{(}2,m/(m-1)\big{)}$ in $L^{1}$ norm and is
$\min\big{(}1,1/(m-1)\big{)}$ in $L^{\infty}$ norm.
Figure 3.2: (a) The steady states with unit total mass for different $m$ have
Hölder exponent $\alpha=\min\big{(}1,1/(m-1)\big{)}$ and $\sigma=1$, where
$\nu$ is chosen such that the corresponding $\rho_{\infty}$ is supported on
$[-2,2]$. (b) The convergence of the steady states $\rho_{\infty}$ on
different grid size $\Delta x$, which is $\min\big{(}2,m/(m-1)\big{)}$ in
$L^{1}$ norm and is $\min\big{(}1,1/(m-1)\big{)}$ in $L^{\infty}$ norm.
Now let us turn our attention to the time evolution and the stabilization in
time toward equilibria and show that the convergence in time toward
equilibration can be arbitrarily slow. This is due to the fact that the effect
of attraction is very small for large distances. Actually, different bumps at
large distances will slowly diffuse and take very long time to attract each
other. However, once they reach certain distance, the convexity of the
Gaussian well will lead to equilibration exponentially fast in time. These two
different time scales can be observed in Figure 3.3, where the time energy
decay and density evolution are plotted to the solution corresponding to
$m=3,\sigma=1$, and $\nu=1.48$ (see also Figure 3.2).
Figure 3.3: (a) The two timescales in the decay towards the unique equilibrium
solution corresponding: very slow energy decay followed by an exponential
decay. (b) Time evolution of the density. Here, $m=3,\sigma=1$ and $\nu=1.48$.
###### Example 3 (Nonlinear diffusion with compactly supported attraction
kernel).
The dynamics of the solution of the 1-D equation (1.1) with characteristic
functions as initial data is shown in Figure 3.4, for the compactly supported
interaction kernel $W(x)=-(1-|x|)_{+}$. For $\rho_{0}(x)=\chi_{[-2,2]}(x)$,
the solution forms two bumps and then merges to a single one, which is the
global minimizer of the energy. When $\rho_{0}(x)=\chi_{[-3,3]}(x)$, the
solution converges to three non-interacting bumps (in the sense that
$\tfrac{\partial\xi}{\partial x}\rho\equiv 0$), each of which is a steady
state.
Figure 3.4: The dynamics of (3.1) starting with the initial data
$\rho_{0}=\chi_{[-2,2]}$ and $\rho_{0}=\chi_{[-3,3]}$.
The decay of the energy for these two cases is shown in Figure 3.5(a). After
the initial transient disappears, the energy decreases significantly at later
times only when the topology changes, i.e. the merge of disconnected
components. Although there is a steady state with one single component with
all the mass, the three-bump solutions with $\rho_{0}(x)=\chi_{[-3,3]}$ seems
to be the correct final stable steady state. This can be confirmed from Figure
3.5(b), as $\xi$ is a constant on each connected component of the support.
This example shows a very interesting effect in this equation, which is the
appearance in the long time asymptotics of steady states with disconnected
support. It should be observed that each bump is at distance larger than 1
from the other bumps, and thus the interaction force exerted between them is
zero. This together with the finite speed of propagation of the degenerate
diffusion are the reasons why the steady state with the total mass and
connected support is not achieved in the long time asymptotics. This fact is
related to the existence of local minimizers of the functional (1.2) in
certain weak topologies, infinity Wasserstein distance, not allowing for large
perturbations of the support, see [2, Section 5] and [22] for related
questions.
Figure 3.5: (a) The decay of the entropy of the equation (3.1) with initial
data $\rho_{0}(x)=\chi_{[-2,2]}$ and $\rho_{0}(x)=\chi_{[-3,3]}$. After an
initial transient behavior, there is a significant decrease in the entropy
only when the topology of the solution changes. (b) The final steady state of
(3.1) with initial data $\rho_{0}(x)=\chi_{[-3,3]}$ and the corresponding
$\xi$. Here $\xi$ assumes different constant values on different connected
components of the support.
For other non-compactly supported kernels like $W(x)=-\frac{1}{2}e^{-|x|}$ or
the Gaussian as in Example 2, there is a unique steady state with one single
connected component in its support, though it exhibits the same slow-fast
behavior in its convergence in time as shown in Figure 3.3. This metastability
and other decaying solutions when $m<2$ are discussed in more details in [11].
###### Example 4 (Nonlinear diffusion with double well external potential).
In this example, we elaborate more on stationary states which are not global
minimizers of the total energy. More precisely, we consider nonlinear
diffusion equation for particles under an external double-well potential of
the form
$\rho_{t}=\big{(}\rho(\nu\rho^{m-1}+V)_{x}\big{)}_{x},\quad
V(x)=\frac{x^{4}}{4}-\frac{x^{2}}{2}.$ (3.2)
Actually, the steady states of (3.2) are of the form
$\rho_{\infty}(x)=\left(\frac{C(x)-V(x)}{\nu}\right)_{+}^{\frac{1}{m-1}}$
with $C(x)$ piecewise constant possibly different in each connected component
of the support.
We run the computation with $\nu=1$, $m=2$ and initial data of the form
$\rho_{0}(x)=\frac{M}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-x_{c})^{2}}{2\sigma^{2}}},\quad
M=0.1,\ \sigma^{2}=0.2,$ (3.3)
corresponding to the symmetric ($x_{c}=0$) and asymmetric ($x_{c}=0.2$) cases,
respectively. It is obvious that for small mass, we can get infinitely many
stationary states with two connected components in its support by perturbing
the value of $C$ defining a symmetric steady state. Actually, each of them has
a non zero basin of attraction depending on the distribution of mass initially
as shown in Figure 3.6(b). While the global minimizer of the free energy is
the symmetric steady state, the non symmetric ones are local minimizers in the
infinity Wassertein distance or informally for small perturbations in the
sense of its support. It is interesting to observe that even if the long time
asymptotics is different for each initial data, the rate of convergence to
stabilization seems uniformly 2, see Figure 3.6(a).
Figure 3.6: (a) The decay of the entropy of the equation (3.2) with initial
data (3.3), for the symmetric ($x_{c}=0$) and asymmetric ($x_{c}=0.2$) cases,
respectively. A uniform rate of convergence of order 2 is observed towards the
stationary states. (b) The evolution of the asymmetric initial data
($x_{c}=0.2$) towards the corresponding asymmetric stationary state.
### 3.2 Generalized Keller-Segel model
Another related diffusion equation with nonlocal attraction is the generalized
Keller-Segel model,
$\rho_{t}=\nabla\cdot\big{(}\rho\nabla(\nu\rho^{m-1}+W\ast\rho)\big{)},$ (3.4)
with the kernel $W(\mathbf{x})=|\mathbf{x}|^{\alpha}/\alpha$ with $-d<\alpha$
or the convention $W(\mathbf{x})=\ln|\mathbf{x}|$ for $\alpha=0$. The bound
from below in $\alpha$ due to local integrability of the kernel $W$. When
$\alpha=2-d$, $W$ is the Newtonian potential in $\mathbb{R}^{d}$, and the
equation reduces to the Keller-Segel model for chemotaxis with nonlinear
diffusion:
$\rho_{t}=\nabla\cdot\big{(}\rho\nabla(\nu\rho^{m-1}-c)\big{)},\quad-\Delta
c=\rho.$ (3.5)
Compared with Example 2 where the interaction potential $W$ is integrable, the
long tail for $W(\mathbf{x})=|\mathbf{x}|^{\alpha}/\alpha$ has non-trivial
consequences. In certain parameter regimes, the solution can even blow up in
finite time with smooth initial data. To clarify the different regimes, we can
easily evaluate the balance between the attraction due to the nonlocal kernel
$W$ and the repulsion due to diffusion by scaling arguments. In fact, taking
the corresponding energy functional (1.2) and checking the scaling under
dilations of each term, we can find three different regimes:
* •
Diffusion-dominated regime: $m>(d-\alpha)/d$. Here, the intuition is that
solutions exist globally in time and the aggregation effect only shows in the
long-time behavior where we observe nontrivial compactly supported stationary
states.
* •
Balanced regime: $m=(d-\alpha)/d$. Here the mass of the system is the critical
quantity. There is a critical mass, separating the diffusive behavior from the
blow-up behavior.
* •
Aggregation-dominated regime: $m<(d-\alpha)/d$. Blow-up and diffusive behavior
coexist for all values of the mass depending on the initial concentration.
Figure 3.7: (a) The critical mass $M_{c}$ when $m+\alpha=1$, $\nu=1$ for
different exponents $m$. (b) The blowing up solution for $m=1.5$,
$\alpha=-0.5$ and $\nu=1$ with initial data
$\rho_{0}(x)=M(e^{-4(x+2)^{2}}+e^{-4(x-2)^{2}})/\sqrt{\pi}$, where the total
mass $M=0.057>M_{c}\approx 0.055$. (c) The decaying solution for $m=1.5$,
$\alpha=-0.5$ and $\nu=1$ with initial data
$\rho_{0}(x)=Me^{-x^{2}}/\sqrt{\pi}$, where $M=0.53<M_{c}=0.55$.
The classical 2-D Keller-Segel system corresponds to $m=1$, $\alpha=0$, see
[39, 12, 8, 6] and the references therein for the different behaviors. The
nonlinear diffusion model for the balanced case with the Newtonian potential
in $d\geq 3$ was studied in detail in [7]. Finite time blow-up solutions for
general kernel $W(\mathbf{x})=|\mathbf{x}|^{\alpha}/\alpha$ in the
aggregation-dominated regime were also investigated, combined with numerical
simulations [42].
###### Example 5 (Generalized Keller-Segel model in the balanced regime).
Let us start with the 1-D example when $m+\alpha=1$ corresponding to the
balanced case. Here, the behavior of the dynamics depends on the total
conserved mass. The solutions blow up if the mass is greater than the
threshold $M_{c}$ and otherwise the solutions decay to zero. This threshold
mass can be determined by solving the equation with different initial
conditions and is shown in Figure 3.7(a) for different values of $m$. For
example, when $m=1.5$ and $\alpha=1-m=-0.5$, the threshold mass $M_{c}$ is
about $0.055$. If the initial data has a larger mass as in Figure 3.7(b), the
solution blows up. Since the numerical method is conservative, the density
concentrates inside one cell instead of being infinity. Otherwise, if the
initial data has a smaller mass as in Figure 3.7(c), the solution decays to
zero.
We have also checked the self-similar behavior for subcritical mass cases
$(M<M_{c})$ in the sense of solving (3.4) with $V(x)=|x|^{2}/2$. That is in
the similarity variables, the solution of
$\rho_{t}=\nabla\cdot\big{(}\rho\nabla(\nu\rho^{m-1}+W\ast\rho+|x|^{2}/2)\big{)}$
converges to the self-similar profile. The decay rate in time is computed for
several subcritical masses and is shown in Figure 3.8(a), illustrating that
this rate is independent of the mass and is exactly $O(e^{-2t})$ as proven in
the classical 2-D Keller-Segel model in [13]. We also observe in Figure 3.8(b)
how the self-similar profiles become concentrated as a Dirac Delta at the
origin as $M\to M_{c}$.
Figure 3.8: (a) The uniformly exponential decay towards equilibrium (in
similarity variables) for subcritical mass in self-similar variables when
$m=1.5$, $\alpha=-0.5$, $\nu=1$ for different values of the mass $M<M_{c}$.
(b) The equilibrium profiles for different $M<M_{c}$.
###### Example 6 (Generalized Keller-Segel model in the other regimes).
The general behaviors of solutions to the 1-D version of (3.4) in other
parameter regimes are also known to some extent. When $m>1-\alpha$
corresponding to the diffusion-dominated regime, a compact steady state is
always expected, which is the global minimizer of the energy (1.2) as in [33].
If the nonlinearity of the diffusion is increased to be $m=1.6$ with the same
total mass $(=0.057)$ and the exponent $\alpha=-0.5$, the solution converges
to a steady state as in Figure 3.9 instead of blowing up as in Figure 3.7(b).
When $\alpha+m<1$ corresponding to the aggregation-dominated regime, the small
initial data decays to zero while large initial data blows up in finite time
(see Figure 3.10). The size of the initial data determining the distinct
behaviors is usually measured in a norm different from $L^{1}$ (the conserved
mass), and no critical value in this norm as in the case $m+\alpha=1$ is
expected.
Figure 3.9: The evolution of the generalized Keller-Segel equation in the
diffusion dominated regime ($m=1.6$, $\alpha=-0.5$) with $\nu=1.0$. The
initial condition $\rho_{0}(x)=M(e^{-4(x+2)^{2}}+e^{-4(x-2)^{2}})/\sqrt{\pi}$
($M=0.057$) is the same as that in Figure 3.7 (b).
Figure 3.10: The evolution of the generalized Keller-Segel equation in the
aggregation-dominated regime ($m=1.6$, $\alpha=-0.5$) with $\nu=1.0$. The
initial condition is
$\rho_{0}(x)=M(e^{-4(x+2)^{2}}+e^{-4(x-2)^{2}})/\sqrt{\pi}$, with $M=0.047$
for decaying solution in (a) and $M=0.048$ for blowup solutions in (b).
### 3.3 Aggregation equation with repulsive-attractive kernels
In the absence of diffusion from $H(\rho)$ or confinement from $V$, steady
states of the general equation (1.1) are still expected when the kernel $W$
incorporates both short range repulsion and long range attraction. This type
of kernels arises in the continuum formulation of moving flocks of self-
propelled particles [26, 20], and the popular ones are the Morse potential
$W(x)=Ce^{-|x|/\ell}-e^{-|x|},\quad C>1,\ell<1$
and the power-law type
$W(x)=\frac{|x|^{a}}{a}-\frac{|x|^{b}}{b},\quad a>b,$
with the convention that $|x|^{0}/0=\ln|x|$ below.
###### Example 7 (Quadratic attractive and Newtonian repulsive kernels).
The regularity of the solution depends on the singularity of the repulsion
force. If this force is small at short distance (or equivalently $b$ is
relatively large), the solution can concentrate at a lower dimension subset,
while more singular forces lead to smooth steady states except possible
discontinuities near the boundary [2]. The case $a=2$ and $b=0$ is shown in
Example 1, whose steady state is a semi-circle [32, 16], while the case $a=2$
and $b=1$ leads to a steady state which is a constant supported on an interval
[21, 23].
We remind that the discrete convolution for the velocity field in (2.9) is
discretized using the coefficients $W_{j-i}$, chosen as approximations of the
local integral
$W_{j-i}=\frac{1}{\Delta x}\int_{C_{i}}W(x_{j}-s)ds.$ (3.6)
In the case of smooth kernels $(b>0)$, we can either use the mid-point rule or
a direct computation of the integral if available. We show the numerically
computed stationary state with both options in Figure 3.11 (a) and (b)
respectively. As one can observe, the first choice is oscillation free while
the second choice with exact integrals shows an overshoot of the density near
the boundary of the support. The difference between the two cases can be
explained by carefully writing down the characterization of the discrete
stationary states based on the discrete entropy inequality in Theorem 2.5. The
mid-point rule performs better due to its symmetry that induces some numerical
diffusion.
Figure 3.11: The steady states computed with: (a) mid-point quadrature rule
for (3.6); (b) exact computation of $W_{j-i}$ in (3.6); (c) Same as (b) but
adding small nonlinear diffusion.
In case we would be dealing with singular kernels, we cannot use simple
quadrature formulas like middle-point rule but rather we need to implement
either quadrature formulas for singular integrals or perform exact evaluations
of the integrals in (3.6). To avoid the oscillations as in Figure 3.11(b), we
added a small nonlinear diffusion term, i.e.,
$\rho_{t}=\big{(}\rho(\epsilon\rho+W*\rho)_{x})_{x}$. Here quadratic nonlinear
diffusion is used, respecting the same nonlinearity and scaling as in the
original equation $\rho_{t}=(\rho(W*\rho)_{x})_{x}$. Numerical experiments as
in Figure 3.11(c) indicate that $\epsilon=0.25(\Delta x)^{2}$ is close to
optimal, in the sense that $\epsilon$ is just large enough to prevent the
overshoot. This near optimal diffusion coefficient has been further confirmed
by numerical experiments with different $\Delta x$.
Figure 3.12: The steady states computed with on a finer mesh with: (a) mid-
point rule for (3.6); (b) exact computation of $W_{j-i}$; (c) the convergence
of $L^{1}$ errors for both options.
For the sake of clarity, we show in Figure 3.12(a)-(b) the steady-state
solutions computed on a finer mesh for the same cases as in Figure 3.11(a)-(b)
along with the $O(\Delta x)$ decay of $L^{1}$ errors for different grid sizes
$\Delta x$ in Figure 3.12(c). The $L^{\infty}$ errors is almost constant and
not decaying with mesh refinement. They clearly indicate that the overshoot
amplitude seen in Figures 3.11(b) and 3.12(b) is not reduced by mesh
refinement and it needs the fix of small diffusion regularization. This will
be further discussed in 2-D simulations below.
### 3.4 Two-dimensional simulations
Now, let us illustrate the performance of the scheme in 2-D with some selected
examples showcasing different numerical difficulties and interesting
asymptotics.
Figure 3.13: The evolution of the 2d aggregation equation with nonlinear
diffusion with $\nu=0.1$, $m=3$, $W(\mathbf{x})=\exp(-|\mathbf{x}|^{2})/\pi$
and initial condition
$\rho_{0}(\mathbf{x})=\frac{1}{4}\chi_{[-3,3]\times[-3,3]}(\mathbf{x})$. The
computational domain is $[-4,4]\times[-4,4]$, with grid size $\Delta x=\Delta
y=0.1$ and time step $\Delta t=0.001$.
###### Example 8 (Nonlinear diffusion with nonlocal attraction in 2-D).
For the equation with $H(\rho)=\frac{\nu}{m}\rho^{m}$,
$W(\mathbf{x})=-\exp(-|\mathbf{x}|^{2})/\pi$ and $V\equiv 0$, the dynamics is
similar to that in 1-D, being the result of the competition between the
nonlinear diffusion $\nabla\cdot\big{(}\rho\nabla(\nu\rho^{m-1})\big{)}$ and
the nonlocal attraction $\nabla\cdot\big{(}\rho\nabla W*\rho)\big{)}$. The
evolution starting from the rescaled characteristic function supported on the
square $[-3,3]\times[-3,3]$ is shown in Figure 3.13. Because the interaction
represented by the kernel $W(\mathbf{x})$ is nonzero for any
$\mathbf{x}=(x,y)$, the final steady state consists of one single component;
however, four clumps are formed in the evolution, as the attraction dominates
the relatively weak diffusion.
###### Example 9 (Quadratic attractive and Newtonian repulsive kernel with
small nonlinear diffusion).
Similarly, overshoots may appear near the boundary of discontinuous solutions
of $\rho_{t}=\nabla\cdot\big{(}\rho\nabla W*\rho\big{)}$ with repulsive-
attractive kernels $W$. These overshoots can not be eliminated as easily as in
one dimension, either by a careful choice of grid to align with the boundary
or by a special numerical quadrature for $W_{i-j}$. However, stable solutions
can be obtained by adding small nonlinear diffusion as in Example 7.
Therefore, we consider the equation
$\rho_{t}=\nabla\cdot\big{(}\rho\nabla(\epsilon\rho+W*\rho)\big{)}.$
For quadratic attractive and Newtonian repulsive kernel
$W(\mathbf{x})=|\mathbf{x}|^{2}/2-\ln|\mathbf{x}|$, the steady states are
shown in Figure 3.14, without $(\epsilon=0)$ or with the diffusion. The near
optimal coefficient $\epsilon$ is numerically shown to be close to
$0.4((\Delta x)^{2}+(\Delta y)^{2})$, exhibiting a similar mesh dependence as
in Example 7. Since $W$ is singular in this (and next) example, $W_{j,k}$ is
computed using Gaussian quadrature with four points in each dimension, to
avoid the evaluation of $W$ at the origin.
(a) $\epsilon=0$ (b) $\epsilon=0.4((\Delta x)^{2}+(\Delta y)^{2})$
Figure 3.14: (a) the steady state of the equation with
$W(\mathbf{x})=|\mathbf{x}|^{2}/2-\ln|\mathbf{x}|$; (b) the steady state with
the same $W(\mathbf{x})$, regularized by quadratic diffusion
$\nabla\cdot\big{(}\rho\nabla(\epsilon\rho)\big{)}$. The exact steady state
without diffusion is the characteristic function of the unit disk with density
$\frac{1}{\pi}$.
###### Example 10 (Steady mill solutions).
Another common pattern observed for the self-propelled particle systems with
an attractive-repulsive kernel in 2-D is the rotating mill [15], and the
steady pattern can be obtained from the equation
$\rho_{t}=\nabla\cdot\big{(}\rho\nabla(W\ast\rho-\frac{\alpha}{\beta}\log|\mathbf{x}|)\big{)},\quad\mathbf{x}\in\mathbb{R}^{2},$
with some positive constants $\alpha$ and $\beta$. For the kernel
$W(\mathbf{x})=\frac{1}{2}|\mathbf{x}|^{2}-\ln|\mathbf{x}|$, the steady state
is still a constant $\rho_{\infty}=2$ on an annulus, whose inner and outer
radius are given by
$R_{0}=\sqrt{\frac{\alpha}{\beta}},\quad
R_{1}=\sqrt{\frac{\alpha}{\beta}+\frac{M}{2\pi}},$
with the total conserved mass $M=\int_{\mathbb{R}^{d}}\rho d{\bf x}$. For
other more realistic kernels like the Morse type [15] or Quasi-Morse type
[14], the radial density is in general more concentrated near the inner
radius, but the explicit form of $\rho_{\infty}$ can not be obtained in
general. Numerical diffusion, in the form of
$\epsilon\nabla\cdot(\rho\nabla\rho)$, is still needed to prevent the
overshoot and the resulting steady states with $\epsilon=0.2((\Delta
x)^{2}+(\Delta y)^{2})$ are shown in Figure 3.15 for two different potentials.
(a) $W(\mathbf{x})=|\mathbf{x}|^{2}/2-\ln|\mathbf{x}|$ (b)
$W(\mathbf{x})=\lambda\big{(}V(|\mathbf{x}|)-CV(|\mathbf{x}|/\ell)\big{)}$
Figure 3.15: The steady density $\rho_{\infty}$ for the rotating mill with
$\Delta x=\Delta y=0.05$. (a) $\alpha=0.25$, $\beta=2\pi$; (b)
$V(r)=-K_{0}(kr)/2\pi$, where $K_{0}(r)$ is the modified Bessel function of
the second kind and the parameters
$C=10/9,\ell=0.75,k=0.5,\lambda=100,\alpha=1.0,\beta=40$ are taken from[14].
Acknowledgment:
JAC acknowledges support from projects MTM2011-27739-C04-02, 2009-SGR-345 from
Agència de Gestió d’Ajuts Universitaris i de Recerca-Generalitat de Catalunya,
and the Royal Society through a Wolfson Research Merit Award. JAC and YH were
supported by Engineering and Physical Sciences Research Council (UK) grant
number EP/K008404/1. The work of AC was supported in part by the NSF Grant
DMS-1115682. The authors also acknowledge the support by NSF RNMS grant
DMS-1107444.
## References
* [1] L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008.
* [2] D. Balagué, J. A. Carrillo, T. Laurent, and G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), pp. 1055–1088.
* [3] D. Benedetto, E. Caglioti, J. A. Carrillo, and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media, J. Statist. Phys., 91 (1998), pp. 979–990.
* [4] D. Benedetto, E. Caglioti, and M. Pulvirenti, A kinetic equation for granular media, RAIRO Modél. Math. Anal. Numér., 31 (1997), pp. 615–641.
* [5] M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput., 34 (2012), pp. B559–B583.
* [6] A. Blanchet, E. A. Carlen, and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Funct. Anal., 262 (2012), pp. 2142–2230.
* [7] A. Blanchet, J. A. Carrillo, and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), pp. 133–168.
* [8] A. Blanchet, J. A. Carrillo, and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb{R}^{2}$, Comm. Pure Appl. Math., 61 (2008), pp. 1449–1481.
* [9] M. Burger, J. A. Carrillo, and M.-T. Wolfram, A mixed finite element method for nonlinear diffusion equations, Kinet. Relat. Models, 3 (2010), pp. 59–83.
* [10] M. Burger, M. di Francesco, and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2013), pp. 709–738.
* [11] M. Burger, R. Fetecau, and Y. Huang, Stationary states and asymptotic behaviour of aggregation models with nonlinear local repulsion, 2013\.
* [12] V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), pp. 155–175.
* [13] J. F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic keller-segel model in the plane, preprint, (2013).
* [14] J. Carrillo, S. Martin, and V. Panferov, A new interaction potential for swarming models, Physica D: Nonlinear Phenomena, 260 (2013), pp. 112 – 126\.
* [15] J. A. Carrillo, M. R. D’Orsogna, and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), pp. 363–378.
* [16] J. A. Carrillo, L. C. F. Ferreira, and J. C. Precioso, A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math., 231 (2012), pp. 306–327.
* [17] J. A. Carrillo, R. J. McCann, and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), pp. 971–1018.
* [18] , Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), pp. 217–263.
* [19] J. A. Carrillo and G. Toscani, Asymptotic $L^{1}$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), pp. 113–142.
* [20] M. R. D’Orsogna, Y.-L. Chuang, A. L. Bertozzi, and L. S. Chayes, Self-propelled particles with soft-core interactions: patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), p. 104302.
* [21] K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), pp. 2267–2291.
* [22] , Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling, 53 (2011), pp. 1436–1450.
* [23] R. C. Fetecau, Y. Huang, and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), pp. 2681–2716.
* [24] S. Gottlieb, C.-W. Shu, and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), pp. 89–112.
* [25] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), pp. 399–415.
* [26] H. Levine, W.-J. Rappel, and I. Cohen, Self-organization in systems of self-propelled particles, Phys. Rev. E, 63 (2000), p. 017101.
* [27] H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal., 172 (2004), pp. 407–428.
* [28] K.-A. Lie and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput., 24 (2003), pp. 1157–1174.
* [29] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), pp. 153–179.
* [30] H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp. 408–463.
* [31] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), pp. 101–174.
* [32] E. B. Saff and V. Totik, Logarithmic potentials with external fields, vol. 316 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1997\. Appendix B by Thomas Bloom.
* [33] G. Ströhmer, Stationary states and moving planes, in Parabolic and Navier-Stokes equations. Part 2, vol. 81 of Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 2008, pp. 501–513.
* [34] P. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21 (1984), pp. 995–1011.
* [35] C. M. Topaz, A. L. Bertozzi, and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), pp. 1601–1623.
* [36] G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 1277–1291.
* [37] B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., 32 (1979), pp. 101–136.
* [38] J. L. Vázquez, The porous medium equation, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007. Mathematical theory.
* [39] J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), pp. 1198–1223.
* [40] C. Villani, Topics in optimal transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003.
* [41] J. von zur Gathen and J. Gerhard, Modern computer algebra, Cambridge University Press, Cambridge, second ed., 2003.
* [42] Y. Yao and A. L. Bertozzi, Blow-up dynamics for the aggregation equation with degenerate diffusion, Physica D: Nonlinear Phenomena, 260 (2013), pp. 77 – 89.
|
arxiv-papers
| 2014-02-18T09:08:04 |
2024-09-04T02:49:58.355514
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jos\\'e A. Carrillo and Alina Chertock and Yanghong Huang",
"submitter": "Yanghong Huang",
"url": "https://arxiv.org/abs/1402.4252"
}
|
1402.4266
|
# Decay-assisted collinear resonance ionization spectroscopy:
Application to neutron-deficient francium
K. M. Lynch [email protected] School of Physics and Astronomy, The
University of Manchester, Manchester, M13 9PL, United Kingdom ISOLDE, PH
Department, CERN, CH-1211 Geneva-23, Switzerland Instituut voor Kern- en
Stralingsfysica, KU Leuven, B-3001 Leuven, Belgium J. Billowes School of
Physics and Astronomy, The University of Manchester, Manchester, M13 9PL,
United Kingdom M. L. Bissell Instituut voor Kern- en Stralingsfysica, KU
Leuven, B-3001 Leuven, Belgium I. Budinc̆ević Instituut voor Kern- en
Stralingsfysica, KU Leuven, B-3001 Leuven, Belgium T.E. Cocolios School of
Physics and Astronomy, The University of Manchester, Manchester, M13 9PL,
United Kingdom ISOLDE, PH Department, CERN, CH-1211 Geneva-23, Switzerland
R.P. De Groote Instituut voor Kern- en Stralingsfysica, KU Leuven, B-3001
Leuven, Belgium S. De Schepper Instituut voor Kern- en Stralingsfysica, KU
Leuven, B-3001 Leuven, Belgium V.N. Fedosseev EN Department, CERN, CH-1211
Geneva 23, Switzerland K.T. Flanagan School of Physics and Astronomy, The
University of Manchester, Manchester, M13 9PL, United Kingdom S. Franchoo
Institut de Physique Nucléaire d’Orsay, F-91406 Orsay, France R.F. Garcia
Ruiz Instituut voor Kern- en Stralingsfysica, KU Leuven, B-3001 Leuven,
Belgium H. Heylen Instituut voor Kern- en Stralingsfysica, KU Leuven, B-3001
Leuven, Belgium B.A. Marsh EN Department, CERN, CH-1211 Geneva 23,
Switzerland G. Neyens Instituut voor Kern- en Stralingsfysica, KU Leuven,
B-3001 Leuven, Belgium T.J. Procter Present address: TRIUMF, Vancouver,
British Columbia, V6T 2A3, Canada School of Physics and Astronomy, The
University of Manchester, Manchester, M13 9PL, United Kingdom R.E. Rossel EN
Department, CERN, CH-1211 Geneva 23, Switzerland Institut für Physik,
Johannes Gutenberg-Universität Mainz, D-55128 Mainz, Germany S. Rothe EN
Department, CERN, CH-1211 Geneva 23, Switzerland Institut für Physik,
Johannes Gutenberg-Universität Mainz, D-55128 Mainz, Germany I. Strashnov
School of Physics and Astronomy, The University of Manchester, Manchester, M13
9PL, United Kingdom H.H. Stroke Department of Physics, New York University,
NY, New York 10003, USA K.D.A. Wendt Institut für Physik, Johannes
Gutenberg-Universität Mainz, D-55128 Mainz, Germany
###### Abstract
This paper reports on the hyperfine-structure and radioactive-decay studies of
the neutron-deficient francium isotopes 202-206Fr performed with the Collinear
Resonance Ionization Spectroscopy (CRIS) experiment at the ISOLDE facility,
CERN. The high resolution innate to collinear laser spectroscopy is combined
with the high efficiency of ion detection to provide a highly-sensitive
technique to probe the hyperfine structure of exotic isotopes. The technique
of decay-assisted laser spectroscopy is presented, whereby the isomeric ion
beam is deflected to a decay spectroscopy station for alpha-decay tagging of
the hyperfine components. Here, we present the first hyperfine-structure
measurements of the neutron-deficient francium isotopes 202-206Fr, in addition
to the identification of the low-lying states of 202,204Fr performed at the
CRIS experiment.
## I Introduction
Recent advances in high-resolution laser spectroscopy have resulted in the
ability to measure short-lived isotopes with yields of less than 100 atoms per
second Cheal and Flanagan (2010); Blaum _et al._ (2013). The Collinear
Resonance Ionization Spectroscopy (CRIS) experiment Procter and Flanagan
(2013), located at the ISOLDE facility, CERN, aims to push the limits of laser
spectroscopy further, performing hyperfine-structure measurements on isotopes
at the edges of the nuclear landscape. It provides a combination of high-
detection efficiency, high resolution and ultra-low background, allowing
measurements to be performed on isotopes with yields below, in principle, one
atom per second.
The first optical measurements of francium were performed in 1978. Liberman
identified the 7s 2S${}_{1/2}\rightarrow$ 7p 2P3/2 atomic transition,
performing hyperfine-structure and isotope-shift measurements first with low-
resolution Liberman _et al._ (1978) and later with high-resolution laser
spectroscopy Liberman _et al._ (1980). The wavelength of this transition
$\lambda$(D2) = 717.97(1) nm was in excellent agreement with the prediction of
Yagoda Yagoda (1932), made in 1932 before francium was discovered. Further
measurements of francium followed in the next decade. High-resolution optical
measurements were performed on both the 7s 2S${}_{1/2}\rightarrow$ 7p 2P3/2
atomic transition Coc _et al._ (1985); Touchard _et al._ (1984); Bauche _et
al._ (1986), as well as the 7s 2S${}_{1/2}\rightarrow$ 8p 2P3/2 transition
Duong _et al._ (1987), along with transitions into high-lying Rydberg states
Andreev _et al._ (1987). The CRIS technique, a combination of collinear laser
spectroscopy and resonance ionization was originally proposed by Kudriavtsev
in 1982 Kudriavtsev and Letokhov (1982), but the only experimental realization
of the technique was not performed until 1991 on ytterbium atoms Schulz _et
al._ (1991).
The ability to study the neutron-deficient francium (Z = 87) isotopes at the
CRIS beam line offers the unique opportunity to answer questions arising from
the study of the nuclear structure in this region of the nuclear chart. As the
isotopes above the Z = 82 shell closure become more neutron deficient, a
decrease in the excitation energy of the ($\pi$1i13/2)${}_{13/2^{+}}$,
($\pi$2f7/2)${}_{7/2^{+}}$, ($\nu$1i13/2)${}_{13/2^{+}}$ and
($\pi$3s${}^{-1}_{1/2}$)${}_{1/2^{+}}$ states is observed. In 185Bi (Z = 83)
and 195At (Z = 85), the ($\pi$3s${}^{-1}_{1/2}$)${}_{1/2^{+}}$ deformed
intruder state has been observed to be the ground state Davids _et al._
(1996); Kettunen _et al._ (2003).
Recent radioactive-decay measurements suggest the existence of a
($\pi$3s${}^{-1}_{1/2}$)${}_{1/2^{+}}$ proton intruder state for 203Fr and,
with a lower excitation energy, for 201Fr, suggesting that this state may
become the ground state in 199Fr Uusitalo _et al._ (2005); Jakobsson _et
al._ (2012). The intruder configurations polarize the nucleus, creating
significant deformation. From the study of the nuclear structure of the
neutron-deficient francium isotopes towards 199Fr (by measuring the magnetic
dipole moments and change in mean-square charge radii of the ground and
isomeric states), the quantum configuration of the states and the shape of the
nuclei can be investigated.
Radioactive-decay measurements on the neutron-deficient francium isotopes have
aimed to determine the level structure of the low-lying nuclear states, but
their exact nature is still unknown Huyse _et al._ (1992); Uusitalo _et al._
(2005); Jakobsson _et al._ (2012, 2013). High-resolution collinear laser
spectroscopy has allowed determination of the ground-state properties of
204,205,206Fr Voss _et al._ (2013), confirming the tentative spin
assignments. The spin of 205Fr was measured to be 9/2-, the ground-state spins
of 204,206Fr were confirmed as 3(+), but the low-lying spin (7+) and (10-)
isomers are still under investigation.
General methods of isomer identification have already been achieved with in-
source laser spectroscopy Fedosseev _et al._ (2012a) (and references
therein). In the case of 68,70Cu Weissman _et al._ (2002), following the
selection of isomeric beams, experiments such as Coulomb excitation Stefanescu
_et al._ (2007) and mass measurements Van Roosbroeck _et al._ (2004) have
been performed. However, these experiments suffered from isobaric
contamination, as well as significant ground-state contamination due to the
Doppler broadening of the hyperfine resonances of each isomer Cheal and
Flanagan (2010). One way of addressing the difficulties of in-source laser
spectroscopy (isobaric contamination, Doppler broadening, pressure broadening)
is selecting the ground or isomeric state of interest by resonance ionization
in a collinear geometry.
In a sub-Doppler geometry, the process of isomer selective resonance laser
ionization Fedosseev _et al._ (2012a) can result in a high-purity isomeric
beam. Deflection of the pure-state ion beam to the decay spectroscopy station
allows identification of the hyperfine component with alpha-decay
spectroscopy.
## II Experimental technique
Radioactive ion beams of francium were produced at the ISOLDE facility, CERN
Kugler (2000) by impinging 1.4 GeV protons onto a thick UCx target (up to 2
$\mu$A integrated proton current). The radioisotopes were surface ionized
through interaction with the rhenium coating on the hot (2400 K) tantalum
transfer line and extracted from the target-ion source at 50 keV. The isotope
of interest was mass selected using the high-resolution HRS separator and
bunched (at 31.25 Hz) with the radio-frequency cooler-buncher ISCOOL Jokinen
_et al._ (2003); Mané _et al._ (2009). The bunched-ion beam was deflected
into the CRIS beam line and transported through a potassium-vapour charge
exchange cell (CEC) (420 K, $\sim$10-6 mbar chamber pressure, 6$\times
10^{-4}$ mbar vapour pressure Haynes (2013)) to be neutralized. In the 1.2 m
long interaction region, the arrival of the atomic bunch was synchronized with
two co-propagating pulsed laser beams to excite the state of interest followed
by ionization in a step-wise scheme. The temporal length of the atomic bunch
was 2-3 $\mu$s, corresponding to a spatial length of 45-70 cm. To reduce the
background signal resulting from non-resonant collisional ionization, the
interaction region aims at ultra-high vacuum (UHV) conditions. A pressure of
$<$10-8 mbar was achieved during this experiment. A schematic diagram of the
CRIS beam line is shown in Fig. 1.
### II.1 Collinear resonance ionization spectroscopy
The resonant excitation step from the 7s 2S1/2 electronic ground state to the
8p 2P3/2 state was probed with 422.7-nm light. The laser light of this
resonant step was provided by a narrow-band titanium:sapphire (Ti:Sa) laser of
the ISOLDE RILIS installation Fedosseev _et al._ (2012b); Rothe _et al._
(2013), pumped by the second harmonic output of a Nd:YAG laser (Model:
Photonics Industries DM-60-532, 10 kHz). The fundamental output from the
tuneable Ti:Sa laser was frequency doubled using a BBO crystal to produce the
required 422.7-nm laser light. The light was fibre-coupled into the CRIS beam
line through 35 m of multimode optical fibre ($\sim$100 mW output). The laser
linewidth of 1.5 GHz limited the resolution achieved in the present
experiment, allowing only the lower-state (7s 2S1/2) splitting to be fully
resolved. The second (non-resonant) transition from the 8p 2P3/2 state to the
continuum was driven using 1064-nm light. This light was produced by a
fundamental Nd:YAG laser (Model: Quanta-Ray LAB 130, operated at 31.25 Hz)
next to the CRIS setup, temporally overlapped with the 422.7-nm laser beam and
aligned through the laser/atom interaction region. The standard repetition
rate of the RILIS lasers (10 kHz) limited the repetition rate of the 1064-nm
laser light to 31.25 Hz (one out of every 320 pulses of 422.7-nm laser light
was utilized). The bunching of the ion beam with ISCOOL was matched to the
lower repetition rate of 31.25 Hz to overlap the atom bunch with the two laser
pulses every 32 ms.
Figure 1: Schematic diagram of the CRIS beam line. Laser ions can be deflected
to a copper plate and the corresponding secondary electrons detected by the
MCP, or implanted into a carbon foil for alpha-decay spectroscopy. (Inset) The
decay spectroscopy station (DSS) ‘windmill’ system for alpha-decay tagging.
The synchronization of the first- and second-step laser pulses and the release
of the ion bunch from ISCOOL was controlled by a Quantum Composers digital
delay generator (Model: QC9258). The 10 kHz pulse generator of the Ti:Sa pump
laser acted as the master clock, triggering the delay generator to output a
sequence of TTL pulses to synchronize the 1064-nm laser light and the ion
bunch with the 422.7-nm light, allowing resonance ionization of the francium
atoms to occur. The laser ions were detected by a micro-channel plate (MCP)
housed in the decay spectroscopy station (DSS). The electronic signal from the
MCP was digitized by a LeCroy oscilloscope (Model: WavePro 725Zi, 2 GHz
bandwidth, 8 bit ADC, 20 GS/s), triggered by the digital delay generator. The
data were transferred from the oscilloscope using a LabVIEW™ program.
The frequency of the resonant excitation step, the 422.7-nm laser light, was
scanned to study the 7s 2S${}_{1/2}\rightarrow$ 8p 2P3/2 atomic transition.
The scanning and stabilization of the frequency was controlled by the RILIS
Equipment Acquisition and Control Tool (REACT), a LabVIEW control program
package that allows for remote control, equipment monitoring and data
acquisition Rossel _et al._ (2013). This was achieved by controlling the
etalon tilt angle inside the Ti:Sa laser resonator to adjust the laser
wavelength, which was measured with a HighFinesse wavemeter (Model: WS7),
calibrated with a frequency stabilised HeNe laser. The francium experimental
campaign at CRIS marked the first implementation of the REACT framework for
external users. The remote control LabVIEW interface for the Ti:Sa laser ran
locally at the CRIS setup, allowing independent laser scanning and control.
### II.2 Decay-assisted laser spectroscopy
The technique of decay-assisted collinear laser spectroscopy was further
developed at the CRIS beam line to take advantage of the ultra-pure ion beams
produced by resonance ionization in a collinear geometry Lynch _et al._
(2013a). The selectivity from resonance ionization of an isotope is a result
of the selectivity of the Lorentzian profile of the natural linewidth
($\sim$12.5 MHz) of the state and the Gaussian profile of the laser linewidth
($\sim$1.5 GHz). At a frequency separation of 4 GHz, the Gaussian component
falls to 1% of its peak intensity and the selectivity is dominated by the
natural linewidth of the state. Thus, the maximum selectivity from resonance
ionization is given by Eq. (1),
$S=\prod_{n=1}^{N}\Big{(}\frac{\Delta\omega_{\textnormal{AB},n}}{\Gamma_{n}}\Big{)}^{2}=\prod_{n=1}^{N}S_{n},$
(1)
where $\Delta\omega_{\textnormal{AB}}$ is the separation in frequency of the
two states (A and B), $\Gamma_{n}$ is the FWHM of the natural linewidth of the
state, $S_{n}$ is the selectivity of the transition and $N$ is the number of
transitions used. The total selectivity of a resonance ionization process is
given by the product of the individual selectivities. In the case of the two
states being the ground state and isomer, the selectivity can be calculated
from Eq. (1). When the two states are the isotope of interest and
contamination from a neighbouring isotope, additional selectivity can be
gained from the kinematic shift since the laser is overlapped with an
accelerated beam.
In addition to hyperfine-structure studies with ion detection, the decay
spectroscopy station can be used to identify the hyperfine components of
overlapping structures. This allows the hyperfine structure of two states to
be separated by exploiting their characteristic radioactive-decay mechanisms.
This results in a smaller error associated with the hyperfine parameters, and
a better determination of the extracted nuclear observables.
The decay spectroscopy station (DSS) consists of a rotatable wheel
implantation system Rajabali _et al._ (2013). It is based on the design from
KU Leuven Dendooven (1992) (Fig. 1 of Ref. Andreyev _et al._ (2010)), which
has provided results in a number of successful experiments Elseviers _et al._
(2013) (and references therein). The wheel holds 9 carbon foils, produced at
the GSI target laboratory Lommel _et al._ (2002), with a thickness of 20(1)
$\mu$g cm-2 ($\sim$90 nm) into which the ion beam is implanted (at a depth of
$\sim$25 nm).
Two Canberra Passivated Implanted Planar Silicon (PIPS) detectors for charged-
particle detection (e.g. alpha, electron, fission fragments) are situated on
either side of the implantation carbon foil, as shown in Fig. 1. One PIPS
detector (Model: BKA 300-17 AM, thickness 300 $\mu$m) sits behind the carbon
foil and another annular PIPS (APIPS) (Model: BKANPD 300-18 RM, thickness 300
$\mu$m, with an aperture of 4 mm) is placed in front of the carbon foil. The
detectors are connected to charge sensitive Canberra preamplifiers (Model:
2003BT) via a UHV type-C sub-miniature electrical feed-through.
Laser-produced ions from the interaction region in the CRIS beam line are
deflected to the DSS by applying a potential difference between a pair of
vertical electrostatic plates, see Fig. 1. The deflected ion beam is implanted
into the carbon foil, after passing though a collimator with a 4 mm aperture
and the APIPS detector. The collimator shields the APIPS detector from direct
implantation of radioactive ions into the silicon wafer, see Fig. 1. Decay
products from the carbon foil can be measured by either the APIPS or PIPS
detector, with a total solid-angle coverage of 63% (simulated, assuming a
uniform distribution of implanted activity). Operation of the single APIPS
detector during the experiment gave an alpha-detection efficiency of 25%. An
electrical contact is made to the collimator, allowing the current generated
by the ion beam when it strikes the collimator to be measured and the plate to
be used as a beam monitoring device. When it is not in use, it is electrically
grounded to avoid charge build-up. A Faraday cup is installed in the location
of one of the carbon foils. This copper plate (thickness 0.5 mm, diameter 10
mm) is electrically isolated from the steel wheel by PEEK rings and connected
by a spot-welded Kapton cable attached to a rotatable BNC connection in the
centre of the wheel Rajabali _et al._ (2013).
The alpha-decay spectroscopy data is acquired with a digital data acquisition
system (DAQ), consisting of XIA digital gamma finder (DGF) revision D modules
Hennig _et al._ (2007). Each module has four input channels with a 40 MHz
sampling rate. Signals fed into the digital DAQ are self-triggered with no
implementation of master triggers.
Due to the reflective surface of the inside of the vacuum chambers, a
significant fraction of 1064-nm laser light was able to scatter into the
silicon detectors. Despite the collimator in front of the APIPS detector to
protect it from ion implantation (and laser light), the infra-red light caused
a shift in the baseline of the signal from the silicon detector. This required
the parameters for the DGF modules to be adjusted to account for this effect
online, since the reflections were due to the particular setup of the
experiment (power and laser-beam path).
The low energy resolution of the APIPS detector was associated with the
necessity of optimizing the DGF parameters online with the radioactive 221Fr
(t1/2 = 4.9(2) min). In addition, a fluctuating baseline resulting from the
changing power of the 1064-nm laser light meant that only a resolution of 30
keV at 6.341 MeV was achieved. This however was sufficient to identify the
characteristic alpha decays of the neutron-deficient francium isotopes under
investigation.
## III Results
The hyperfine structures of the neutron-deficient francium isotopes 202-206Fr
were measured with collinear resonance ionization spectroscopy, with respect
to the reference isotope 221Fr. This paper follows the recent publication
reporting the hyperfine-structure studies of 202,203,205Fr Flanagan _et al._
(2013). During the experimental campaign, the neutron-rich francium isotopes
218m,219,229,231Fr were also studied. A detailed description of the nature of
these isotopes will be the topic of a future publication Budinc̆ević _et al._
.
The resonance spectrum of the 7s 2S1/2 $\rightarrow$ 8p 2P3/2 transition was
fit with a $\chi^{2}$-minimization routine. The hyperfine $A_{P_{3/2}}$ factor
was fixed to the ratio of the 7s 2S1/2 $\rightarrow$ 8p 2P3/2 transition of
$A_{P_{3/2}}/A_{S_{1/2}}=+22.4/6209.9$, given in literature Duong _et al._
(1987). For the 8p 2P3/2 state, the hyperfine $B_{P_{3/2}}$ factor is small
enough to have no impact on the fit to the data, and was consequently set to
zero Cocolios _et al._ (2013).
Figure 2: Collinear resonance ionization spectroscopy of 204Fr relative to
221Fr. The hyperfine structure of the 3(+) ground state of 204gFr is shown in
blue, the 7+ state of 204m1Fr is shown in green and the (10-) state of 204m2Fr
is shown in red. Figure 3: The radioactive decay of 204Fr and its isomers
Huyse _et al._ (1992); Uusitalo _et al._ (2005); Jakobsson _et al._ (2013).
The intensities of the hyperfine transitions $S_{FF^{\prime}}$ between
hyperfine levels $F$ and $F^{\prime}$ (with angular momentum $J$ and
$J^{\prime}$ respectively) are related to the intensity of the underlying fine
structure transition $S_{JJ^{\prime}}$ Blaum _et al._ (2013). The relative
intensities of the hyperfine transitions are given by
$\frac{S_{FF^{\prime}}}{S_{JJ^{\prime}}}=(2F+1)(2F^{\prime}+1)\begin{Bmatrix}F&F^{\prime}&1\\\
J^{\prime}&J&I\end{Bmatrix}^{2},$ (2)
where $\\{\ldots\\}$ denotes the Wigner 6-$j$ coefficient. Although these
theoretical intensities are only strictly valid for closed two-level systems,
and there was jitter on the temporal overlap of the two laser pulses in the
interaction region, they were used as currently the most reliable estimate.
The $A_{S_{1/2}}$ factor and the centroid frequency of the hyperfine structure
were determined for each scan individually. For isotopes with multiple scans,
a weighted mean for the $A_{S_{1/2}}$ factor and the centroid frequency were
calculated based on the error of the fits. The uncertainty attributed to the
$A_{S_{1/2}}$ factor was calculated as the weighted standard deviation of the
values. The isotope shifts were determined relative to 221Fr, with the
uncertainty propagated from the error of the fits, the scatter and the drift
in centroid frequency of the 221Fr reference scans Cocolios _et al._ (2013).
### III.1 Spectroscopic studies of 204Fr
Figure 4: Alpha-particle spectroscopy of (blue) 204gFr, (green) 204m1Fr and
(red) 204m2Fr allowed the hyperfine peaks in Fig. 2 to be identified. The
laser was detuned by 11.503 GHz (peak A, 204gFr), 8.508 GHz (peak B, 204m1Fr)
and 18.693 GHz (peak C, 204m2Fr) relative to the centroid frequency of 221Fr.
The hyperfine structure of 204Fr is shown in Fig. 2, measured by detecting the
laser ions with the MCP detector as a function of the scanned first-step laser
frequency. Five peaks are observed in the spectrum. Considering that only the
lower-state splitting is resolved (associated with the 1.5 GHz linewidth of
the scanning laser), two hyperfine resonances are expected per nuclear (ground
or isomeric) state. Consequently, Fig. 2 contains the hyperfine structure of
three long-lived states in 204Fr, with one of the resonances unresolved
(labeled E). In order to identify the states of the hyperfine resonances,
laser assisted alpha-decay spectroscopy was used.
Figure 5: Alpha-particle spectroscopy of the (10-) state of 204m2Fr. The
decay of 204m2Fr to 204m1Fr via an E3 IT is observed through the presence of
204m1Fr alpha particles of 6969 keV (denoted by $\star$). The laser was
detuned by 43.258 GHz relative to the centroid frequency of 221Fr.
The radioactive decay of the low-lying states in 204Fr is presented in Fig. 3.
The characteristic alpha decay of each nuclear state in 204Fr was utilized to
identify the hyperfine-structure resonances of Fig. 2. The laser was tuned on
resonance with each of the first three hyperfine resonances (labeled A, B and
C) and alpha-decay spectroscopy was performed on each. The alpha-particle
energy spectrum of these three states is illustrated in Fig. 4. The energy of
the alpha particles emitted when the laser was on resonance with an atomic
transition of the hyperfine spectrum characteristic of 204gFr is shown in
blue. This transition occurred at 11.503 GHz (peak A of Fig. 2) relative to
the centroid frequency of 221Fr. Similarly, the alpha-particle energy spectra
for 204m1Fr and 204m2Fr are shown in green and red, when the laser was detuned
by 8.508 GHz and 18.693 GHz (peak B and C) from the reference frequency,
respectively. Present in the alpha-particle energy spectrum are the alpha
particles emitted from the decay of the 204Fr states (6950-7050 keV) in
addition to those emitted from the nuclear states in the daughter isotope
200At (6400-6500 keV). Each state in 204Fr has a characteristic alpha-particle
emission energy: 7031 keV for 204gFr, 6969 keV for 204m1Fr and 7013 keV for
204m2Fr. This was confirmed by the presence of the corresponding daughter
decays of 200gAt (6464 keV), 200m1Fr (6412 keV), and 200m2At (6537 keV) in the
alpha-particle energy spectrum.
Figure 6: A two-dimensional histogram of the alpha-particle energy as the
hyperfine structure of 204Fr is probed. (Top) Projection of the frequency
axis. The total number of alpha particles detected at each laser frequency
reveals the hyperfine structure of 204Fr.
An additional alpha-decay measurement was performed on peak D in the hyperfine
spectrum of 204Fr (see Fig. 2) at 43.258 GHz relative to the centroid
frequency of 221Fr. The observation of 7013 keV alpha particles allowed this
state to be identified as 204m2Fr. This meant the identity of all five
hyperfine-structure peaks could be allocated to a state in 204Fr (hence the
hyperfine structure peak E is the overlapping structure of 204gFr and
204m1Fr), allowing analysis of the hyperfine structure of each state.
In addition to the 7031 keV alpha particles of 204m2Fr, alpha particles of
6969 keV from the decay of 204m1Fr were also observed when the laser was on
resonance with the 204m2Fr state. The decay of the (10-) state to 204m1Fr via
an E3 internal transition (IT) has been predicted Huyse _et al._ (1992) but
only recently observed Jakobsson _et al._ (2012), see Fig. 3. This was
achieved by tagging the conversion electron from the internal conversion of
204m2Fr with the emitted 6969 keV alpha particles of 204m1Fr that followed
(with a 5 s correlation time). This allowed the predicted energy of the 275
keV isomeric transition to be confirmed. During the CRIS experiment, an
additional alpha-decay measurement was performed on 204m2Fr, with the laser
detuned by 43.258 GHz relative to the centroid frequency of 221Fr, see Fig. 5.
This spectrum confirms the presence of the 6969 keV alpha particle (denoted by
$\star$), emitted from the decay of 204m1Fr. The ultra-pure conditions of this
measurement allowed the first unambiguous extraction of the branching ratios
in the decay of 204m2Fr: $B_{\alpha}=53(10)$% and $B_{IT}=47(10)$% Lynch
(2013).
Figure 7: Alpha-tagged hyperfine structure of (a) 204gFr spin 3(+) ground
state, (b) 204m1Fr spin (7+) isomer, and (c) 204m2Fr spin (10-) isomer.
Additional peaks are discussed in the text.
Decay-assisted laser spectroscopy was also performed on the hyperfine
structure of the low-lying states of 204Fr. Just as the laser frequency of the
resonant 422.7-nm ionization step was scanned and resonant ions were detected
in the collinear resonance ionization spectroscopy of 204Fr, the same
technique was repeated with the measurement of alpha particles. At each laser
frequency, a radioactive-decay measurement of 60 s was made at the DSS,
measuring the alpha particles emitted from the implanted ions. Fig. 6 (Top)
shows the hyperfine peaks associated with each state in 204Fr. Measurement of
the alpha decay as a function of laser frequency allowed production of a
matrix of alpha-particle energy versus laser frequency, see Fig. 6.
In order to separate hyperfine structures for each states, an alpha-energy
gating was used to maximize the signal-to-noise ratio for the alpha particle
of interest. The alpha-energy gates were chosen to be 7031-7200 keV for
204gFr, 6959-6979 keV for 204m1Fr and 7003-7023 keV for 204m2Fr.
By gating on the characteristic alpha-particle energies of the three states in
204Fr, the hyperfine structures of individual isomers become enhanced in the
hyperfine spectrum. Fig. 7(a) shows the hyperfine structure of 204gFr, (b)
204m1Fr, and (c) 204m2Fr. The presence of 204m2Fr can be observed in the
spectra of 204gFr due to the overlapping peaks of the alpha energies: the tail
of the 7013 keV alpha peak is present in the gate of the 204gFr alpha peak.
The presence of 204m2Fr in the hyperfine structure spectrum of 204m1Fr is
attributed to the E3 IT decay of 204m2Fr to 204m1Fr: alpha particles of energy
6969 keV are observed when on resonance with 204m2Fr. Additionally, 204gFr is
present in the 204m2Fr spectra due to the similar energies of the 7031 keV and
7013 keV alpha particles. However, despite the contamination in the hyperfine
spectra, each peak is separated sufficiently in frequency to be analysed
independently.
From the resulting hyperfine structures of Fig. 7 produced by the alpha-
tagging process (in comparison to the overlapping ion data of Fig. 2), each
state of 204Fr can be analysed individually and the hyperfine factors
extracted with better accuracy and reliability. The estimated error of the
$A_{S_{1/2}}$ factors was 30 MHz on account of the scatter of $A_{S_{1/2}}$
values for 221Fr. Likewise, an error of 100 MHz was assigned to the isotope
shifts.
### III.2 Identification of the hyperfine structure of 202Fr
Figure 8: Collinear resonance ionization spectroscopy of 202Fr relative to
221Fr. The hyperfine structure of the (3+) ground state of 202gFr is shown in
blue and the (10-) state of 202mFr is shown in red. Figure 9: The radioactive
decay of 202Fr and its isomer De Witte _et al._ (2005).
The hyperfine structure of 202Fr obtained with collinear resonance ionization
spectroscopy is presented in Fig. 8. The four hyperfine resonances illustrate
the presence of the ground (202gFr) and isomeric (202mFr) states.
Identification of these two states was performed with laser-assisted alpha-
decay spectroscopy. According to literature, the radioactive decay of 202gFr
(t1/2 = 0.30(5) s) emits an alpha particle of energy 7241(8) keV, whereas
202mFr (t1/2 = 0.29(5) s) emits an alpha particle of energy 7235(8) keV Zhu
and Kondev (2008). The radioactive decay of the ground and isomeric state of
202Fr is presented in Fig. 9.
Figure 10: Alpha-particle spectroscopy of (blue) 202gFr and (red) 202mFr
allowed the hyperfine peaks in Fig. 8 to be identified. The laser was detuned
by (a) 13.760 GHz (peak A, 202gFr) and (b) 20.950 GHz (peak B, 202mFr)
relative to the centroid frequency of 221Fr.
The laser was tuned onto resonance with peak A (202gFr, 13.760 GHz relative to
the centroid frequency of 221Fr) and peak B (202mFr, 20.950 GHz relative to
the centroid frequency of 221Fr) of Fig. 8 obtained from ion detection. For
each position, an alpha-decay measurement was performed, shown in Fig. 10. The
alpha particles emitted when the laser was on resonance with an atomic
transition characteristic to 202gFr are shown in blue, and 202mFr in red. Due
to the limited statistics of our measurement, and the similarity in energies
of the alpha particles (within error), it is impossible to say that alpha
particles of different energies are observed in Fig. 10.
Firm identification of the hyperfine components can be achieved however by
studying the alpha particles emitted by the daughter isotopes 198g,mAt.
Evident in the spectrum of 202gFr are the alpha particles emitted from the
decay of the daughter nucleus 198gAt with an energy of 6755 keV. Similarly,
present in the 202mFr spectrum are the alpha particles from the decay of
198mAt with an energy of 6856 keV. The difference in energy of these two alpha
peaks illustrates the ability of the CRIS technique to separate the two states
and provide pure ground state and isomeric beams for decay spectroscopy.
### III.3 Isomer identification of the resonance spectrum of 206Fr
Figure 11: Collinear resonance ionization spectroscopy of 206Fr relative to
221Fr. (a) Option 1: Peak A is assigned to 206m2Fr and peak B to 206m1Fr. (b)
Option 2: Peak A is assigned to 206m1Fr and peak B to 206m2Fr.
Two sets of data were used in the determination of the nuclear observables
from the hyperfine structure scans. The data for the francium isotopes
202-206,221Fr were taken in Run I and the data for 202-205,221Fr were taken in
Run II. Consistency checks were carried out, allowing 206Fr to be evaluated
with respect to the rest of the data set from Run II. A detailed description
of this analysis can be found in Ref. Lynch (2013). In Run I, no alpha-tagging
was available and consequently the peaks in the ion-detected hyperfine
spectrum needed to be identified in a different way. Recent measurements of
the ground-state hyperfine structure of 206gFr provided the $A_{S_{1/2}}$
factor for the splitting of the 7s 2S1/2 state Voss _et al._ (2013). One peak
of the (7+) isomeric state was also identified in this experiment (see Fig.
1(c) of Ref. Voss _et al._ (2013)), allowing the positions of the overlapping
resonances to be determined. This left only the identity of peaks A and B
(shown in Fig. 11) unknown. Fig. 11(a) presents the hyperfine structures when
peak A is assigned to 206m2Fr and peak B to 206m1Fr. Fig. 11(b) shows the fit
when peak A is 206m1Fr and peak B is 206m2Fr. The suggested identity of the
two resonances (based on mean-square charge radii and $g$-factor systematics)
is discussed in Sec. IV.
### III.4 Yield measurements
Table 1: Yields of the neutron-deficient francium isotopes at the ISOLDE facility (1.4 GeV protons on a UCx target). The nuclear-state composition of the radioactive beams for 202,204,206Fr are presented. $A$ | Yield | Proportion of beam
---|---|---
| (ions/s) | Spin 3(+) | Spin (7+) | Spin (10-)
202 | $1\times 10^{2}$ | 76(14)% | | 24(6)%
203 | $1\times 10^{3}$ | | |
204 | $1\times 10^{4}$111Estimate based on yield systematics. | 63(3)% | 27(3)% | 10(1)%
205 | $2\times 10^{5}$ | | |
206 | $3\times 10^{6}$ | 63(7)% | 27(5)% | 9(1)%
The yields of the neutron-deficient francium isotopes 202-206Fr are presented
in Table 1. The quoted yields, scaled ISOLDE-database yields based on an
independent yield measurement of 202Fr, can be expected to vary by a factor of
two due to different targets. The quoted value for 204Fr is estimated based on
francium yields systematics. The composition of the beam for 202,204,206Fr was
calculated from the ratio of hyperfine-peak intensities (based on the
strongest hyperfine-structure resonance) from the CRIS ion data. The
composition of the beam for 202Fr was confirmed with the alpha-decay data
measured with the DSS.
### III.5 King-plot analysis
Figure 12: A King plot for the extraction of atomic factors $F$ and $M$ for the 422.7-nm transition. See text for details. Table 2: Spins, half-lives, $A_{S_{1/2}}$ factors, isotope shifts, magnetic moments and change in mean-square charge radii of the neutron-deficient francium isotopes 202-206Fr with reference to 221Fr for the 7s 2S1/2 $\rightarrow$ 8p 2P3/2 atomic transition. All $A_{S_{1/2}}$-factor and magnetic-moment values were deduced using the nuclear spins presented. The half-life values are taken from Refs. Lourens (1967); Ritchie _et al._ (1981); Huyse _et al._ (1992); Uusitalo _et al._ (2005); Kondev and Lalkovski (2011); Singh _et al._ (2013); Browne and Tuli (2011). $A$ | $I$ | $t_{1/2}$ (s) | $A_{S_{1/2}}$ (GHz) | $\mu$ ($\mu_{N}$) | $\delta\nu^{A,221}$ (GHz) | $\delta\langle r^{2}\rangle$ (fm2)
---|---|---|---|---|---|---
| | Lit. | Exp. | Lit. | Exp. | Lit. | Exp. | Exp. | Lit.
202g | (3+) | 0.30(5) | +12.80(5) | | +3.90(5) | | 32.68(10) | -1.596(18) |
202m | (10-) | 0.29(5) | +2.30(3) | | +2.34(4) | | 32.57(13) | -1.591(19) |
203 | (9/2-) | 0.53(2) | +8.18(3) | | +3.73(4) | | 31.32(10) | -1.530(18) |
204g222Calculated from the alpha-decay gated hyperfine structure scan of 204Fr. See text for details. | 3(+) | 1.9(5) | +12.99(3) | +13.1499(43)333Literature value taken from Ref. Voss _et al._ (2013). | +3.95(5) | +4.00(5)22footnotemark: 2,444Literature magnetic-moment values re-calculated in reference to $\mu$(210Fr) Gomez _et al._ (2008) | 32.19(10) | -1.571(18) | -1.5542(4)22footnotemark: 2
204m111footnotemark: 1 | (7+) | 1.6(5) | +6.44(3) | | +4.57(6) | | 32.32(10) | -1.577(18) |
204m211footnotemark: 1 | (10-) | 0.8(2) | +2.31(3) | | +2.35(4) | | 30.99(10) | -1.513(17) |
205 | 9/2- | 3.96(4) | +8.40(3) | +8.3550(11)22footnotemark: 2 | +3.83(5) | +3.81(4)22footnotemark: 2,33footnotemark: 3 | 30.21(10) | -1.475(17) | -1.4745(4)22footnotemark: 2
206g | 3(+) | 15.9(3) | +13.12(3) | +13.0522(20)22footnotemark: 2 | +3.99(5) | +3.97(5)22footnotemark: 2,33footnotemark: 3 | 30.04(12) | -1.465(17) | -1.4768(4)22footnotemark: 2
206m1555Based on the isomeric identity of the hyperfine resonances of Option 1. See text for details. | (7+) | 15.9(3) | +6.61(3) | | +4.69(6) | | 30.23(16) | -1.475(18) |
206m244footnotemark: 4 | (10-) | 0.7(1) | +3.50(3) | | +3.55(5) | | 23.57(12) | -1.153(14) |
206m1666Based on the isomeric identity of the hyperfine resonances of Option 2. See text for details. | (7+) | 15.9(3) | +6.74(4) | | +4.79(6) | | 29.69(15) | -1.449(17) |
206m255footnotemark: 5 | (10-) | 0.7(1) | +3.40(3) | | +3.45(5) | | 24.13(12) | -1.180(14) |
207 | 9/2- | 14.8(1) | +8.48(3) | +8.484(1)777Literature value taken from Ref. Coc _et al._ (1985). | +3.87(5) | +3.87(4)33footnotemark: 3,66footnotemark: 6 | 28.42(10) | -1.386(16) | -1.386(3)888Literature value taken from Ref. Dzuba _et al._ (2005).
211 | 9/2- | 186(1) | +8.70(6) | +8.7139(8)66footnotemark: 6 | +3.97(5) | +3.98(5)33footnotemark: 3,66footnotemark: 6 | 24.04(10) | -1.171(13) | -1.1779(4)77footnotemark: 7
220 | 1+ | 27.4(3) | -6.50(4) | -6.5494(9)66footnotemark: 6 | -0.66(1) | -0.66(1)33footnotemark: 3,66footnotemark: 6 | 2.75(10) | -0.134(5) | -0.133(10)77footnotemark: 7
221 | 5/2- | 294(12) | +6.20(3) | +6.2046(8)66footnotemark: 6 | +1.57(2) | +1.57(2)33footnotemark: 3,66footnotemark: 6 | 0 | 0 |
The atomic factors $F$ and $M$ were evaluated by the King-plot method King
(1963). This combines the previously measured isotope shifts by Coc Coc _et
al._ (1985) of the 7s 2S1/2 $\rightarrow$ 7p 2P3/2 transition with 718-nm
laser light, with those made by Duong Duong _et al._ (1987) of the 7s 2S1/2
$\rightarrow$ 8p 2P3/2 transition (422.7 nm). The isotope shifts of
$\delta\nu^{207,221}$ and $\delta\nu^{211,221}$ from this work were combined
with $\delta\nu^{220,221}$ and $\delta\nu^{213,212}$ from Duong. These values
were plotted against the corresponding isotope shifts from Coc Coc _et al._
(1985), shown in Fig. 12. From the linear fit of the data, and using
$\mu^{A,A^{\prime}}\delta\nu_{422}^{A,A^{\prime}}=\frac{F_{422}}{F_{718}}\mu^{A,A^{\prime}}\delta\nu_{718}^{A,A^{\prime}}+M_{422}-\frac{F_{422}}{F_{718}}M_{718},$
(3)
where $\mu^{A,A^{\prime}}=AA^{\prime}/(A^{\prime}-A)$, enabled the evaluation
of $F_{422}/F_{718}=+0.995(3)$ and
$M_{422}-(F_{422}/F_{718})M_{718}=+837(308)~{}\text{GHz~{}amu}$ respectively.
From these values, the atomic factors for the 422.7-nm transition were
calculated to be
$F_{422}=-20.67(21)\text{~{}GHz/fm}^{2},$
$M_{422}=+750(330)\text{~{}GHz~{}amu}.$
For comparison, the atomic factors evaluated for the 718-nm transition were
determined by Dzuba to be $F_{718}=-$20.766(208) GHz/fm2 and
$M_{718}=-$85(113) GHz amu Dzuba _et al._ (2005).
The mass factor is the linear combination of two components: the normal mass
shift, $K^{\textnormal{NMS}}$, and the specific mass shift,
$K^{\textnormal{SMS}}$,
$M_{422}=K^{\textnormal{NMS}}_{422}+K^{\textnormal{SMS}}_{422},$ (4)
and is dependent on the frequency of the transition probed. Subtraction of the
normal mass shift of the 422.7-nm transition
($K^{\textnormal{NMS}}_{422}=+389$ GHz amu) from the mass factor $M_{422}$
allows for calculation of the specific mass shift, giving
$K^{\textnormal{SMS}}_{422}=+360(330)$ GHz amu. The specific mass shift for
the 718-nm line was determined by Dzuba to be
$K^{\textnormal{SMS}}_{718}=-314(113)$ GHz amu Dzuba _et al._ (2005).
### III.6 Hyperfine structure observables
Table 2 presents the hyperfine $A_{S_{1/2}}$ factor, isotope shift, change in
mean-square charge radius and magnetic moment values extracted from the CRIS
data for the francium isotopes 202-206Fr with reference to 221Fr. Additional
data for 207,211,220Fr (used in the creation of the King plot of Fig. 12) in
included for completeness. All values were deduced using the nuclear spins
presented.
The hyperfine $A_{S_{1/2}}$ factor is defined as
$A=\frac{\mu_{I}B_{e}}{I\cdot J},$ (5)
with $\mu_{I}$ the magnetic dipole moment of the nucleus and $B_{e}$ the
magnetic field of the electrons at the nucleus. For each isotope, it was
calculated from the weighted mean of $A_{S_{1/2}}$ values for isotopes where
more than one hyperfine structure scan is present. A minimum error of 30 MHz
was attributed to the $A_{S_{1/2}}$ factor values due to the scatter of the
measured $A_{S_{1/2}}$ for the reference isotope 221Fr Cocolios _et al._
(2013); Lynch _et al._ (2013b).
The isotope shift, $\delta\nu^{A,A^{\prime}}$, between isotopes $A$ and
$A^{\prime}$ is expressed as
$\delta\nu^{A,A^{\prime}}=M\frac{A^{\prime}-A}{AA^{\prime}}+F\delta\langle
r^{2}\rangle^{A,A^{\prime}}.$ (6)
As with the $A_{S_{1/2}}$ values, the isotope shifts were calculated as the
weighted mean of all isotope shifts for a given nucleus. The error on the
isotope shift was determined to be 100 MHz due to the long-term drift of the
centroid frequency of 221Fr as the experiment progressed, and the scan-to-scan
scatter in centroid frequency. When the calculated weighted standard deviation
of the isotope shift was higher than 100 MHz, this error is quoted instead.
Combining the extracted $F$ and $M$ atomic factors from the King-plot analysis
with the measured isotope shifts, evaluation of the change in mean-square
charge radii, $\delta\langle r^{2}\rangle^{A,A^{\prime}}$, between francium
isotopes can be performed, see Eq. (6).
The magnetic moment of the isotopes under investigation can be extracted from
the known moment of another isotope of the element, using the ratio
$\mu=\mu_{ref}\frac{IA}{I_{ref}A_{ref}}.$ (7)
In this work, calculation of the magnetic moments was evaluated in reference
to the magnetic moment of 210Fr, measured by Gomez ($\mu=+4.38(5)~{}\mu_{N}$,
$I^{\pi}=6^{+}$, $A_{S_{1/2}}=+7195.1(4)$ MHz Gomez _et al._ (2008); Coc _et
al._ (1985)). This represents the most accurate measurement of the magnetic
moment of a francium isotope to date, due to probing the 9s 2S1/2 hyperfine
splitting which has reduced electron-correlation effects than that of the
ground state. The current evaluated magnetic moments of the francium isotopes
are made in reference to the magnetic moment of 211Fr of Ekström Ekström _et
al._ (1986).
The hyperfine anomaly for the francium isotopes is generally considered to be
of the order of 1% and is included as a contribution to the error of the
hyperfine $A_{S_{1/2}}$ factors and magnetic moments Stroke _et al._ (1961).
Table 2 presents the experimental results alongside comparison to literature
of the hyperfine $A_{S_{1/2}}$ factor, change in mean-square charge radius and
magnetic moment values. The literature values for 204-206Fr have been taken
from Ref. Voss _et al._ (2013) and 207,211,220,221Fr from Ref. Coc _et al._
(1985). The magnetic-moment values from literature have been re-calculated in
reference to $\mu$(210Fr) Gomez _et al._ (2008), the most accurate
measurement to date. The change in mean-square charge radii values for
207,211,220Fr have been taken from Ref. Dzuba _et al._ (2005). All
experimental results are in broad agreement with those of literature.
## IV Discussion
### IV.1 Charge radii of the neutron-deficient francium
Figure 13: Mean-square charge radii of the francium (circle) isotopes Ekström
_et al._ (1986) presented alongside the lead (diamond) isotopes Anselment _et
al._ (1986). The dashed lines represent the prediction of the droplet model
for given iso-deformation Myers and Schmidt (1983). The data were calibrated
by using $\beta_{2}$(213Fr)$=0.062$, evaluated from the energy of the
$2^{+}_{1}$ state in 212Rn Raman _et al._ (2001). Option 1 and 2 for the (7+)
and (10-) states in 206Fr are based on the isomeric identification given in
Fig. 11.
Located between radon and radium, francium (Z = 87) has 5 valence protons
occupying the $\pi$1h9/2 orbital, according to the shell model of spherical
nuclei. Below the N = 126 shell closure, the neutron-deficient francium
isotopes were studied down to 202Fr (N = 115). The change in mean-square
charge radii for the francium and lead isotopes are presented in Fig. 13. The
data of francium show the charge radii of 207-213Fr re-evaluated by Dzuba
Dzuba _et al._ (2005) alongside the CRIS values which extends the data set to
202Fr. The blue data points show the francium ground states, while the (7+)
isomeric states are in green and the (10-) states in red. The error bars
attributed to the CRIS values are propagated from the experimental error of
the isotope shift and the systematic error associated with the atomic factors
$F_{422}$ and $M_{422}$. The systematic error is the most significant
contribution to the uncertainty associated with the mean-square charge radii,
and not that arising from the isotope shift. The francium data is presented
with the lead data of Anselment Anselment _et al._ (1986) to illustrate the
departure from the spherical nucleus. The change in mean-square charge radii
of the francium isotopes have been overlapped with the charge radii of the
lead isotopes, by using 213Fr (N = 126) and 208Pb (N = 126) as reference
points. The dashed iso-deformation lines represent the prediction of the
droplet model for the francium isotopes Myers and Schmidt (1983). The data
were calibrated using $\beta_{2}$(213Fr)$=0.062$, evaluated from the energy of
the $2^{+}_{1}$ state in 212Rn Raman _et al._ (2001).
The doubly-magic 208Pb represents a model spherical nucleus, with the shape of
the nucleus remaining spherical with the removal of neutrons from the closed N
= 126 shell. This trend is observed until N = 114, where a small deviation
from the spherical droplet model (isodeformation line $\beta_{2}=0.0$) is
interpreted as enhanced collectivity due to the influence of particle-hole
excitations across the Z = 82 shell closure De Witte _et al._ (2007). The
change in mean-square charge radii for the francium isotopes shows agreement
with the lead data as the $\nu$3p3/2, $\nu$2f5/2 and $\nu$3p1/2 orbitals are
progressively depleted. The deviation from sphericity at N = 116 with 203Fr
marks the onset of collective behaviour. The spectroscopic quadrupole moments
were not measured in this work, since they require a laser linewidth of $<$100
MHz. Measurement of the quadrupole moment will provide information on the
static deformation component of the change in mean-square charge radii,
allowing better understanding of this transition region.
Figure 14: Mean-square charge radii of the francium (circle) isotopes Ekström
_et al._ (1986) presented alongside the radon (diamond) isotopes Borchers _et
al._ (1987). The dashed lines represent the prediction of the droplet model
for given iso-deformation Myers and Schmidt (1983). The data were calibrated
using $\beta_{2}$(213Fr)$=0.062$, evaluated from the energy of the $2^{+}_{1}$
state in 212Rn Raman _et al._ (2001). Option 1 and 2 for the (7+) and (10-)
states in 206Fr are based on the isomeric identification given in Fig. 11.
Recent laser spectroscopy measurements on the ground-state properties of
204,205,206Fr suggest this deviation occurs earlier, at 206Fr (N = 119) Voss
_et al._ (2013). In Ref. Voss _et al._ (2013), a more pronounced odd-even
staggering is observed in relation to the lead isotopes, where the mean-square
charge radius of 205Fr is larger than that of 206Fr. The CRIS experiment
observed a smaller mean-square charge radius of 205Fr in comparison to 206Fr,
the deviation from the lead isotopes occurring at 203Fr instead. However, both
experiments are in broad agreement within errors down to N = 117.
Figure 13 presents the two options of the mean-square charge radii of 206m1Fr
and 206m2Fr (as defined by their hyperfine peak identity in Fig. 11). Option 1
is favoured over option 2 due to the smaller mean-square charge radii of
206m1Fr (compared to 206gFr) agreeing with the systematics of the states in
204Fr. As seen in Fig. 13, 206gFr (N = 119) overlaps with the lead data within
errors. The large change in the mean-square charge radius of 206m2Fr suggests
a highly deformed state for the (10-) isomer.
The mean-square charge radii of francium are overlaid with the radon (Z = 86)
charge-radii of Borchers (down to N = 116, with the exception of N = 117)
Borchers _et al._ (1987) in Fig. 14. The mean-square charge-radii of radon
have been calibrated to the francium pair $\delta\langle
r^{2}\rangle^{211,213}$ to account for the uncertainty in $F$ and $M$ for the
optical transition probed (the original isotope shifts are presented
graphically). Despite this, the agreement between the mean-square charge radii
of the francium and radon data is clear. The addition of a single $\pi$1h9/2
proton outside the radon even-Z core does not affect the charge-radii trend,
suggesting the valence proton acts as a spectator particle.
Table 3 presents a comparison of $\beta_{2}$ values with literature. The
droplet model Myers and Schmidt (1983) was used to extract the rms values for
$\beta_{2}$ (column 3) from the change in mean-square charge radii (calibrated
using $\beta_{2}$(213Fr)$=0.062$, as before). Column 4 presents $\beta_{2}$
values extracted from the quadrupole moments of Ref. Voss _et al._ (2013).
The larger $\beta_{2}$ values extracted from the mean-square charge radii,
compared to those extracted from the quadrupole moments, suggest that the
enhanced collectivity observed in Figs. 13 and 14 is due to a large dynamic
component of the nuclear deformation.
Table 3: Extracted $\beta_{2}$ values. (Exp.) The droplet model Myers and Schmidt (1983) was used to extract the rms values for $\beta_{2}$ from the change in mean-square charge radii. The charge-radii values were calibrated using $\beta_{2}$(213Fr)$=0.062$, as before. (Lit.) $\beta_{2}$ values were extracted from the quadrupole moments of Ref. Voss _et al._ (2013). See text for details. $A$ | $I$ | $\langle\beta_{2}^{2}\rangle^{1/2}$ | $\beta_{2}$
---|---|---|---
| | Exp. | Lit.
202g | (3+) | 0.11 |
202m | (10-) | 0.11 |
203 | (9/2-) | 0.11 |
204g | 3(+) | 0.06999Calculated from the alpha-decay gated hyperfine structure scan of 204Fr. See text for details. | -0.0140(14)
204m1 | (7+) | 0.0611footnotemark: 1 |
204m2 | (10-) | 0.0911footnotemark: 1 |
205 | 9/2- | 0.08 | -0.0204(2)
206g | 3(+) | 0.05 | -0.0269(8)
206m1 | (7+) | 0.04101010Based on the isomeric identity of the hyperfine resonances of Option 1. See text for details. |
206m2 | (10-) | 0.1722footnotemark: 2 |
202m1 | (7+) | 0.07111111Based on the isomeric identity of the hyperfine resonances of Option 2. See text for details. |
202m2 | (10-) | 0.1733footnotemark: 3 |
### IV.2 Interpretation of the nuclear $g$-factors
Figures 15 and 17 show the experimental $g$-factors for odd-A and even-A
francium isotopes, respectively. These plots present the CRIS data alongside
the data from Ekström Ekström _et al._ (1986). The Ekström data has been re-
evaluated with respect to the $\mu$(210Fr) measurement of Gomez Gomez _et
al._ (2008).
In Fig. 15, the blue line represents the empirical $g$-factor ($g_{emp}$) of
the odd-A isotopes for the single-particle occupation of the valence proton in
the $\pi$1$h_{9/2}$ orbital. $g_{\text{emp}}$($\pi$1h9/2) was determined from
the magnetic moment of the single-particle state in 209Bi Bastug _et al._
(1996). Similarly, $g_{\text{emp}}$($\pi$3s1/2) was estimated from the
magnetic moment of the single-hole ground-state in 207Tl Neugart _et al._
(1985). From N = 126 to 116, every isotope has a $g$-factor consistent with
the proton occupying the $\pi$1h9/2 orbital. This indicates that the 9/2-
state remains the ground state, and the ($\pi$3s${}^{-1}_{1/2}$)${}_{1/2^{+}}$
proton intruder state has not yet inverted. This lowering in energy of the
$\pi$3s1/2 state to become the ground state would be apparent in the sudden
increase in $g$-factor of the ground state, as illustrated by the black
$g_{\text{emp}}$($\pi$3s1/2) line.
Figure 15: $g$-factors for francium (blue) Ekström _et al._ (1986); Gomez
_et al._ (2008) and thallium (red) isotopes with odd-A Stone (2011). The
$g$-factors for the $\pi$3s1/2 and $\pi$1h9/2 proton orbitals have been
calculated empirically. See text for details.
Figure 15 highlights the robustness of the Z = 82 and N = 126 shell closure
with a shell-model description valid over a range of isotopes. A close-up of
$g_{\text{emp}}$(1$\pi$1h9/2) in Fig. 16 illustrates that the $g$-factor is
sensitive to bulk nuclear effects. The departure from the
$g_{\text{emp}}$(1$\pi$1h9/2) line shows the sensitivity of the $g$-factor to
second-order core polarization in the odd-A thallium, bismuth and francium
isotopes. The systematic decrease in $g$-factor of francium is attributed to
second-order core polarization associated with the presence of five valence
particles, compared to one-particle (hole) in the bismuth (thallium) isotopes,
enough to significantly weaken the shell closure. The linear trend observed in
bismuth, thallium and francium (until N = 118) is suggested to be related to
the opening of the neutron shell, yet allowing for more neutron and proton-
neutron correlations.
Figure 16: Close up of the $g$-factors for francium (blue) Ekström _et al._
(1986); Gomez _et al._ (2008), bismuth (green) Stone (2011) and thallium
(red) isotopes Barzakh _et al._ (2012) with odd-A. The $g$-factor for the
$\pi$1h9/2 proton orbital has been calculated empirically. See text for
details.
Further measurements towards the limit ./of stability are needed to better
understand the prediction of the inversion of the $\pi$3s1/2 intruder orbital
with the $\pi$1h9/2 ground state. A re-measurement of 203Fr could determine
the presence of the spin 1/2+ isomer (t1/2 = 43(4) ms Jakobsson _et al._
(2013)), which was not observed during this experiment.
The $g$-factors for the odd-odd francium isotopes are presented in Fig. 17.
With the coupling of the single valence proton in the $\pi$1$h_{9/2}$ orbital
with a valence neutron, a large shell model space is available. The
empirically calculated $g$-factors for the coupling of the $\pi$1h9/2 proton
with the valence neutrons are denoted by the colored lines. These $g$-factors
were calculated from the additivity relation
$g=\frac{1}{2}\Big{[}g_{p}+g_{n}+(g_{p}-g_{n})\frac{j_{p}(j_{p}+1)-j_{n}(j_{n}+1)}{I(I+1)}\Big{]},$
(8)
as outlined by Neyens Neyens (2003). The empirical $g$-factors of the odd
valence neutrons were calculated from the magnetic moments of neighbouring
nuclei: 201Po for the blue $g_{\text{emp}}$($\pi$1h${}_{9/2}\otimes\nu$3p3/2)
and red $g_{\text{emp}}$($\pi$1h${}_{9/2}\otimes\nu$1i13/2) line Wouters _et
al._ (1991); 213Ra for the black
$g_{\text{emp}}$($\pi$1h${}_{9/2}\otimes\nu$3p1/2) line; and 211Ra for the
green $g_{\text{emp}}$($\pi$1h${}_{9/2}\otimes\nu$2f5/2) line Ahmad _et al._
(1983). The empirical $g$-factors for the valence proton in the $\pi$1h9/2
orbital were calculated from the magnetic moment of the closest odd-A francium
isotope (203Fr and 213Fr respectively) from the CRIS data.
The ground state of 202,204,206Fr display similar $g$-factors, with the
valence proton and neutron coupling to give a spin 3(+) state. The tentative
configuration in literature of ($\pi$1h${}_{9/2}\otimes\nu$2f5/2)${}_{3^{+}}$
for 202gFr is based on the configuration of the (3+) state in 194Bi (from
favoured Fr-At-Bi alpha-decay chain systematics) Zhu and Kondev (2008).
Similarly, the assignment of the same configuration for 204gFr and 206gFr is
based on the alpha-decay systematics of neighbouring nuclei 196,198Bi.
However, the initial assignment of 194gBi was declared to be either
($\pi$1h${}_{9/2}\otimes\nu$2f5/2)${}_{3^{+}}$ or
($\pi$1h${}_{9/2}\otimes\nu$3p3/2)${}_{3^{+}}$ Duppen _et al._ (1991). From
the $g$-factors of the ground states of 202,204,206Fr, it is clear that the
configuration of these states is indeed
($\pi$1h${}_{9/2}\otimes\nu$3p3/2)${}_{3^{+}}$.
Figure 17 also presents the $g$-factors of 206m1Fr and 206m2Fr for option 1
and 2 (see Fig. 11). The first isomeric states of 204,206Fr (7+) have a
valence neutron that occupies the $\nu$2$f_{5/2}$ state. This coupling of the
proton-particle neutron-hole results in a
($\pi$1h${}_{9/2}\otimes\nu$2f5/2)${}_{7^{+}}$ configuration Kondev (2008).
For 202mFr, 204m2Fr and 206m2Fr, the particle proton-neutron hole coupling
result in a tentative ($\pi$1h${}_{9/2}\otimes\nu$1i13/2)${}_{10^{-}}$
configuration assignment for each isomer Chiara and Kondev (2010). However,
while the agreement of the $g$-factors of the spin (10-) state in 202,204Fr
point to a $\nu$1$i_{13/2}$ occupancy, the observed value for 206m2Fr is in
disagreement with the $g$-factor of such a (10-) state. The charge radius of
206m2Fr indicates a highly deformed configuration, where the single-particle
description of the nucleus is no longer valid. This is consistent with the
$g$-factor of this state: it is no longer obeying a simple shell-model
description. This leads to the conclusion, that while a
($\pi$1h${}_{9/2}\otimes\nu$1i13/2)${}_{10^{-}}$ configuration for 206m2Fr is
suggested, the charge radii and magnetic moment point to a drastic change in
the structure of this isomeric state.
Figure 17: $g$-factors for francium isotopes with even-A Ekström _et al._
(1986): ground state (blue), spin (7+) state (green) and spin (10-) state
(red). The $g$-factor for the coupling of the proton and neutron orbitals have
been calculated empirically. See text for details.
For completeness, the configurations of the odd-odd francium isotopes
208,210,212Fr are presented. The coupling of the valence proton and neutron in
the $\pi$1h9/2 and $\nu$2f5/2 orbital in 208Fr and 210Fr leads to a
($\pi$1h${}_{9/2}\otimes\nu$2f5/2)${}_{7^{+}}$ and
($\pi$1h${}_{9/2}\otimes\nu$2f5/2)${}_{6^{+}}$ configuration respectively
Martin (2007). With the $\nu$2f5/2 neutron orbital fully occupied, the valence
neutron in 212Fr occupies the $\nu$3p1/2 orbital, resulting in a
($\pi$1h${}_{9/2}\otimes\nu$3p1/2)${}_{5^{+}}$ configuration Browne (2005).
The agreement of the experimental and empirical $g$-factors, as shown in Figs.
15-17, illustrates the suitability of the single-particle description of the
neutron-deficient francium isotopes, with the exception of the (10-) state in
206m2Fr. A model-independent spin and spectroscopic quadrupole moment
determination is needed to clarify the nature of this isomeric state. The
neutron-deficient francium isotopes display a single-particle nature where the
additivity relation is still reliable.
## V Conclusion and Outlook
The hyperfine structures and isotope shifts of the neutron-deficient francium
isotopes 202-206Fr with reference to 221Fr were measured with collinear
resonance ionisation spectroscopy, and the change in mean-square charge radii
and magnetic moments extracted. The selectivity of the alpha-decay patterns
allowed the unambiguous identification of the hyperfine components of the low-
lying isomers of 202,204Fr for the first time.
The resonant atomic transition of 7s 2S${}_{1/2}\rightarrow$ 8p 2P3/2 was
probed, and the hyperfine $A_{S_{1/2}}$ factor measured. A King plot analysis
of the 422.7-nm transition in francium allowed the atomic factors to be
calibrated. The field and mass factors were determined to be F422 =
$-$20.670(210) GHz/fm2 and M422 = +750(330) GHz amu, respectively.
The novel technique of decay-assisted laser spectroscopy in a collinear
geometry was performed on the isotopes 202,204Fr. The decay spectroscopy
station was utilized to identify the peaks in the hyperfine spectra of
202,204Fr. Alpha-tagging the hyperfine structure scan of 204Fr allowed the
accurate determination of the nuclear observables of the three low-lying
isomeric states and the determination of the branching ratios in the decay of
204m2Fr.
Analysis of the change in mean-square charge radii suggests an onset of
collectivity that occurs at 203Fr (N = 116). However, measurement of the
spectroscopic quadrupole moment is required to determine the nature of the
deformation (static or dynamic). The magnetic moments suggest that the single-
particle description of the neutron-deficient francium isotopes still holds,
with the exception of the (10-) isomeric state of 206m2Fr. Based on the
systematics of the region, the tentative assignment of the hyperfine structure
peaks in 206Fr result in magnetic moments and mean-square charge radii that
suggest a highly deformed state. Laser assisted nuclear decay spectroscopy of
206Fr would unambiguously determine their identity.
The occupation of the valence proton in the $\pi$1h9/2 orbital has been
suggested for all measured isotopes down to 202Fr, indicating the
($\pi$1s${}_{1/2}^{-1}$)${}_{1/2^{+}}$ intruder state does not yet invert with
the $\pi$1h9/2 orbital as the ground state. Further measurements of the very
neutron-deficient francium isotopes towards 199Fr are required to fully
determine the nature of the proton-intruder state. A laser linewidth of 1.5
GHz was enough to resolve the lower-state (7s 2S1/2) splitting of the
hyperfine structure and measure the $A_{S_{1}/2}$ factor. In the future, the
inclusion of a narrow-band laser system for the resonant-excitation step will
enable the resolution of the upper-state (8p 2P3/2) splitting, providing the
hyperfine $B_{P_{3/2}}$ factor. This will allow extraction of the
spectroscopic quadrupole moment and determination of the nature of the
deformation.
Successful measurement of 202Fr was performed during this experiment, with a
yield of 100 atoms per second. By pushing the limits of laser spectroscopy,
further measurements of 201Fr (with a yield of 1 atom per second) and 200Fr
(less than 1 atom per second) are thought to be possible. The ground state
(9/2-) of 201Fr has a half life of 53 ms and its isomer (1/2+) a half life of
19 ms. By increasing the sensitivity of the CRIS technique, the presence of
the 1/2+ isomers in 201,203Fr can be confirmed. A positive identification will
lead to nuclear-structure measurements that will determine (along with the
verification of nuclear spin) the magnetic moments which are sensitive to the
single-particle structure and thus to the ($\pi$3s1/2)${}_{1/2^{+}}$ proton
intruder nature of these states. With sufficient resolution ($<$100 MHz), the
spectroscopic quadrupole moment of these neutron-deficient states (with $I\geq
1$) will be directly measurable and the time-averaged static deformation can
be determined.
The successful measurements performed by the CRIS experiment demonstrates the
high sensitivity of the collinear resonance ionization technique. The decay
spectroscopy station provides the ability to identify overlapping hyperfine
structure and eventually perform laser assisted nuclear decay spectroscopy
measurements on pure ground and isomeric-state beams Lynch _et al._ (2012,
2013a).
## Acknowledgements
The authors extend their thanks to the ISOLDE team for providing the beam, the
GSI target lab for producing the carbon foils, and IKS-KU Leuven and The
University of Manchester machine shops for their work. This work was supported
by the IAP project P7/23 of the OSTC Belgium (BRIX network) and by the FWO-
Vlaanderen (Belgium). The Manchester group was supported by the STFC
consolidated grant ST/F012071/1 and continuation grant ST/J000159/1. K.T.
Flanagan was supported by STFC Advanced Fellowship Scheme grant number
ST/F012071/1. The authors would also like to thank Ed Schneiderman for
continued support through donations to the Physics Department at NYU.
## References
* Cheal and Flanagan (2010) B. Cheal and K. T. Flanagan, J. Phys. G 37, 113101 (2010).
* Blaum _et al._ (2013) K. Blaum, J. Dilling, and W. Nörtershäuser, Physica Scripta 2013, 014017 (2013).
* Procter and Flanagan (2013) T. J. Procter and K. T. Flanagan, Hyperfine Interact. 216, 89 (2013).
* Liberman _et al._ (1978) S. Liberman _et al._ , C. R. Acad. Sci. Paris Ser. B 286, 353 (1978).
* Liberman _et al._ (1980) S. Liberman _et al._ , Phys. Rev. A 22, 2732 (1980).
* Yagoda (1932) H. Yagoda, Phys. Rev. 40, 1017 (1932).
* Coc _et al._ (1985) A. Coc _et al._ , Phys. Lett. B 163, 66 (1985).
* Touchard _et al._ (1984) F. Touchard _et al._ , Atomic Masses and Fundamental Constants 7 , 353 (1984).
* Bauche _et al._ (1986) J. Bauche _et al._ , J. Phys. B: At. Mol. Opt. Phys. 19, L593 (1986).
* Duong _et al._ (1987) H. T. Duong _et al._ , Europhys. Lett. 3, 175 (1987).
* Andreev _et al._ (1987) S. V. Andreev, V. I. Mishin, and V. S. Letokhov, Phys. Rev. Lett. 59, 1274 (1987).
* Kudriavtsev and Letokhov (1982) Y. Kudriavtsev and V. Letokhov, Applied Physics B 29, 219 (1982).
* Schulz _et al._ (1991) C. Schulz _et al._ , J. Phys. B: At. Mol. Opt. Phys. 24, 4831 (1991).
* Davids _et al._ (1996) C. N. Davids _et al._ , Phys. Rev. Lett. 76, 592 (1996).
* Kettunen _et al._ (2003) H. Kettunen _et al._ , Eur. Phys. J. A 16, 457 (2003).
* Uusitalo _et al._ (2005) J. Uusitalo _et al._ , Phys. Rev. C 71, 024306 (2005).
* Jakobsson _et al._ (2012) U. Jakobsson _et al._ , Phys. Rev. C 85, 014309 (2012).
* Huyse _et al._ (1992) M. Huyse, P. Decrock, P. Dendooven, G. Reusen, P. Van Duppen, and J. Wauters, Phys. Rev. C 46, 1209 (1992).
* Jakobsson _et al._ (2013) U. Jakobsson _et al._ , Phys. Rev. C 87, 054320 (2013).
* Voss _et al._ (2013) A. Voss, M. R. Pearson, J. Billowes, F. Buchinger, B. Cheal, J. E. Crawford, A. A. Kwiatkowski, C. D. P. Levy, and O. Shelbaya, Phys. Rev. Lett. 111, 122501 (2013).
* Fedosseev _et al._ (2012a) V. N. Fedosseev, Y. Kudryavtsev, and V. I. Mishin, Phys. Scripta 85, 058104 (2012a).
* Weissman _et al._ (2002) L. Weissman _et al._ , Phys. Rev. C 65, 024315 (2002).
* Stefanescu _et al._ (2007) I. Stefanescu _et al._ , Phys. Rev. Lett. 98, 122701 (2007).
* Van Roosbroeck _et al._ (2004) J. Van Roosbroeck _et al._ , Phys. Rev. Lett. 92, 112501 (2004).
* Kugler (2000) E. Kugler, Hyperfine Interact. 129, 23 (2000).
* Jokinen _et al._ (2003) A. Jokinen, M. Lindroos, E. Molin, and M. Petersson, Nucl. Instrum. Methods Phys. Res. B 204, 86 (2003).
* Mané _et al._ (2009) E. Mané _et al._ , Eur. Phys. J. A 42, 503 (2009).
* Haynes (2013) W. H. Haynes, _Handbook of Chemistry and Physics_ , 94th ed. (CRC, 2013).
* Fedosseev _et al._ (2012b) V. N. Fedosseev _et al._ , Rev. Sci. Instrum. 83, 02A903 (2012b).
* Rothe _et al._ (2013) S. Rothe, V. N. Fedosseev, T. Kron, B. A. Marsh, R. E. Rossel, and K. D. A. Wendt, Nucl. Instrum. Methods Phys. Res. B 317, Part B, 561 (2013).
* Rossel _et al._ (2013) R. E. Rossel, V. N. Fedosseev, B. A. Marsh, D. Richter, S. Rothe, and K. D. A. Wendt, Nucl. Instrum. Methods Phys. Res. B 317, Part B, 557 (2013).
* Lynch _et al._ (2013a) K. M. Lynch, T. E. Cocolios, and M. M. Rajabali, Hyperfine Interact. 216, 95 (2013a).
* Rajabali _et al._ (2013) M. M. Rajabali _et al._ , Nucl. Instrum. Methods Phys. Res. A 707, 35 (2013).
* Dendooven (1992) P. Dendooven, Ph.D. thesis, IKS, KU Leuven (1992).
* Andreyev _et al._ (2010) A. N. Andreyev _et al._ , Phys. Rev. Lett. 105, 252502 (2010).
* Elseviers _et al._ (2013) J. Elseviers _et al._ , Phys. Rev. C 88, 044321 (2013).
* Lommel _et al._ (2002) B. Lommel, W. Hartmann, B. Kindler, J. Klemm, and J. Steiner, Nucl. Instrum. Methods Phys. Res. A 480, 199 (2002).
* Hennig _et al._ (2007) W. Hennig, H. Tan, M. Walby, P. Grudberg, A. Fallu-Labruyere, W. K. Warburton, C. Vaman, K. Starosta, and D. Miller, Nucl. Instrum. Methods Phys. Res. B 261, 1000 (2007).
* Flanagan _et al._ (2013) K. T. Flanagan _et al._ , Phys. Rev. Lett. 111, 212501 (2013).
* (40) I. Budinc̆ević _et al._ , (unpublished).
* Cocolios _et al._ (2013) T. E. Cocolios _et al._ , Nucl. Instrum. Methods Phys. Res. B 317, Part B, 565 (2013).
* Lynch (2013) K. M. Lynch, Ph.D. thesis, The University of Manchester (2013).
* De Witte _et al._ (2005) H. De Witte _et al._ , Eur. Phys. J. A 23, 243 (2005).
* Zhu and Kondev (2008) S. Zhu and F. G. Kondev, Nuclear Data Sheets 109, 699 (2008).
* Lourens (1967) W. Lourens, Ph.D. thesis, Technische Hogeschool Delft (1967).
* Ritchie _et al._ (1981) B. G. Ritchie, K. S. Toth, H. K. Carter, R. L. Mlekodaj, and E. H. Spejewski, Phys. Rev. C 23, 2342 (1981).
* Kondev and Lalkovski (2011) F. Kondev and S. Lalkovski, Nuclear Data Sheets 112, 707 (2011).
* Singh _et al._ (2013) B. Singh, D. Abriola, C. Baglin, V. Demetriou, T. Johnson, E. McCutchan, G. Mukherjee, S. Singh, A. Sonzogni, and J. Tuli, Nuclear Data Sheets 114, 661 (2013).
* Browne and Tuli (2011) E. Browne and J. Tuli, Nuclear Data Sheets 112, 1115 (2011).
* Gomez _et al._ (2008) E. Gomez, S. Aubin, L. A. Orozco, G. D. Sprouse, E. Iskrenova-Tchoukova, and M. S. Safronova, Phys. Rev. Lett. 100, 172502 (2008).
* Dzuba _et al._ (2005) V. A. Dzuba, W. R. Johnson, and M. S. Safronova, Phys. Rev. A 72, 022503 (2005).
* King (1963) W. H. King, J. Opt. Soc. Am. 53, 638 (1963).
* Lynch _et al._ (2013b) K. M. Lynch _et al._ , EPJ Web of Conferences 63, 01007 (2013b).
* Ekström _et al._ (1986) C. Ekström, L. Robertsson, and A. Rosén, Phys. Scripta 34, 624 (1986).
* Stroke _et al._ (1961) H. H. Stroke, R. J. Blin-Stoyle, and V. Jaccarino, Phys. Rev. 123, 1326 (1961).
* Anselment _et al._ (1986) M. Anselment, W. Faubel, S. Göring, A. Hanser, G. Meisel, H. Rebel, and G. Schatz, Nucl. Phys. A 451, 471 (1986).
* Myers and Schmidt (1983) W. D. Myers and K.-H. Schmidt, Nucl. Phys. A 410, 61 (1983).
* Raman _et al._ (2001) S. Raman, C. N. Jr., and P. Tikkanen, Atomic Data and Nuclear Data Tables 78, 1 (2001).
* De Witte _et al._ (2007) H. De Witte _et al._ , Phys. Rev. Lett. 98, 112502 (2007).
* Borchers _et al._ (1987) W. Borchers, R. Neugart, E. W. Otten, H. T. Duong, G. Ulm, and K. Wendt, Hyperfine Interact. 34, 25 (1987).
* Bastug _et al._ (1996) T. Bastug, B. Fricke, M. Finkbeiner, and W. R. Johnson, Z. Phys. D 37, 281 (1996).
* Neugart _et al._ (1985) R. Neugart, H. H. Stroke, S. A. Ahmad, H. T. Duong, H. L. Ravn, and K. Wendt, Phys. Rev. Lett. 55, 1559 (1985).
* Stone (2011) N. Stone, _Table of Nuclear Magnetic Dipole and Electric Quadrupole Moments_ (Nuclear Data Services, International Atomic Energy Agency, Vienna, Austria, 2011).
* Barzakh _et al._ (2012) A. E. Barzakh, L. K. Batist, D. V. Fedorov, V. S. Ivanov, K. A. Mezilev, P. L. Molkanov, F. V. Moroz, S. Y. Orlov, V. N. Panteleev, and Y. M. Volkov, Phys. Rev. C 86, 014311 (2012).
* Neyens (2003) G. Neyens, Rep. Prog. Phys. 66, 633 (2003).
* Wouters _et al._ (1991) J. Wouters, N. Severijns, J. Vanhaverbeke, and L. Vanneste, J. Phys. G: Nucl. Part. Phys. 17, 1673 (1991).
* Ahmad _et al._ (1983) S. A. Ahmad, W. Klempt, R. Neugart, E. W. Otten, K. Wendt, and C. Ekström, Physics Letters B 133, 47 (1983).
* Duppen _et al._ (1991) P. V. Duppen, P. Decrock, P. Dendooven, M. Huyse, G. Reusen, and J. Wauters, Nucl. Phys. A 529, 268 (1991).
* Kondev (2008) F. G. Kondev, Nuclear Data Sheets 109, 1527 (2008).
* Chiara and Kondev (2010) C. J. Chiara and F. G. Kondev, Nuclear Data Sheets 111, 141 (2010).
* Martin (2007) M. J. Martin, Nuclear Data Sheets 108, 1583 (2007).
* Browne (2005) E. Browne, Nuclear Data Sheets 104, 427 (2005).
* Lynch _et al._ (2012) K. M. Lynch _et al._ , J. Phys.: Conf. Ser. 381, 012128 (2012).
|
arxiv-papers
| 2014-02-18T09:50:18 |
2024-09-04T02:49:58.367386
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K. M. Lynch, J. Billowes, M. L. Bissell, I. Budin\\v{c}evi\\'c, T.E.\n Cocolios, R.P. De Groote, S. De Schepper, V.N. Fedosseev, K.T. Flanagan, S.\n Franchoo, R.F. Garcia Ruiz, H. Heylen, B.A. Marsh, G. Neyens, T.J. Procter,\n R.E. Rossel, S. Rothe, I. Strashnov, H.H. Stroke, K.D.A. Wendt",
"submitter": "Kara Lynch",
"url": "https://arxiv.org/abs/1402.4266"
}
|
1402.4430
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-023 LHCb-PAPER-2013-070 April 30, 2014
Measurement of charged particle multiplicities and densities in $pp$
collisions at $\sqrt{s}=7\;$TeV in the forward region
The LHCb collaboration†††Authors are listed on the following pages.
Charged particle multiplicities are studied in proton-proton collisions in the
forward region at a centre-of-mass energy of $\sqrt{s}=7\;$TeV with data
collected by the LHCb detector. The forward spectrometer allows access to a
kinematic range of $2.0<\eta<4.8$ in pseudorapidity, momenta greater than
$2\;\mbox{GeV/}c$ and transverse momenta greater than $0.2\;\mbox{GeV/}c$. The
measurements are performed using events with at least one charged particle in
the kinematic acceptance. The results are presented as functions of
pseudorapidity and transverse momentum and are compared to predictions from
several Monte Carlo event generators.
Submitted to European Physical Journal C
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25, S. Amato2, S. Amerio22, Y. Amhis7, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel39, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson50, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30, W.
Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,r, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez36,p, A. Camboni36, P. Campana18,38, D. Campora
Perez38, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,j, A. Cardini15, H.
Carranza-Mejia50, L. Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20,
L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph.
Charpentier38, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38,
X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47,
J. Closier38, C. Coca29, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A.
Comerma-Montells36, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G.
Corti38, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz
Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J. Dalseno46, P.
David8, P.N.Y. David41, A. Davis57, I. De Bonis4, K. De Bruyn41, S. De
Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D.
Derkach55, O. Deschamps5, F. Dettori42, A. Di Canto11, H. Dijkstra38, S.
Donleavy52, F. Dordei11, M. Dorigo39, P. Dorosz26,o, A. Dosil Suárez37, D.
Dossett48, A. Dovbnya43, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49, U. Egede53, V. Egorychev31, S. Eidelman34,
S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5,
Ch. Elsasser40, S. Esen11, A. Falabella16,f, C. Färber11, C. Farinelli41, S.
Farry52, D. Ferguson50, V. Fernandez Albor37, F. Ferreira Rodrigues1, M.
Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M. Fiorini16,f, C. Fitzpatrick38,
M. Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C.
Frei38, M. Frosini17,38,g, J. Fu21, E. Furfaro24,l, A. Gallas Torreira37, D.
Galli14,d, M. Gandelman2, P. Gandini59, Y. Gao3, J. Garofoli59, J. Garra
Tico47, L. Garrido36, C. Gaspar38, R. Gauld55, L. Gavardi9, E. Gersabeck11, M.
Gersabeck54, T. Gershon48, Ph. Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47,
L. Giubega29, V.V. Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38,
A. Gomes1,a, H. Gordon38, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A.
Granado Cardoso38, E. Graugés36, G. Graziani17, A. Grecu29, E. Greening55, S.
Gregson47, P. Griffith45, L. Grillo11, O. Grünberg61, B. Gui59, E. Gushchin33,
Yu. Guz35,38, T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, T.W.
Hafkenscheid64, S.C. Haines47, S. Hall53, B. Hamilton58, T. Hampson46, S.
Hansmann-Menzemer11, N. Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann61,
J. He38, T. Head38, V. Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A.
Hernando Morata37, E. van Herwijnen38, M. Heß61, A. Hicheur1, D. Hill55, M.
Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55, N. Hussain55, D.
Hutchcroft52, D. Hynds51, V. Iakovenko44, M. Idzik27, P. Ilten56, R.
Jacobsson38, A. Jaeger11, E. Jans41, P. Jaton39, A. Jawahery58, F. Jing3, M.
John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N. Jurik59, M.
Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, M.
Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C. Khurewathanakul39, S.
Klaver54, O. Kochebina7, I. Komarov39, R.F. Koopman42, P. Koppenburg41, M.
Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11, M. Kreps48, G.
Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k, V.
Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31,38, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, T. Latham48, C. Lazzeroni45,
R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32, J.
Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11, Y. Li3, M.
Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G. Liu38, S.
Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40, H. Lu3, D.
Lucchesi22,r, J. Luisier39, H. Luo50, E. Luppi16,f, O. Lupton55, F.
Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O. Maev30,38, S. Malde55, G.
Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5, U. Marconi14, P.
Marino23,t, R. Märki39, J. Marks11, G. Martellotti25, A. Martens8, A. Martín
Sánchez7, M. Martinelli41, D. Martinez Santos42, F. Martinez Vidal63, D.
Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A.
Mazurov16,38,f, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
McSkelly52, B. Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,t, R. Mountain59, F.
Muheim50, K. Müller40, R. Muresan29, B. Muryn27, B. Muster39, P. Naik46, T.
Nakada39, R. Nandakumar49, I. Nasteva1, M. Needham50, N. Neri21, S. Neubert38,
N. Neufeld38, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,q, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-
Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R.
Oldeman15,e, G. Onderwater64, M. Orlandea29, J.M. Otalora Goicochea2, P.
Owen53, A. Oyanguren36, B.K. Pal59, A. Palano13,c, F. Palombo21,u, M.
Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51, L. Pappalardo16,
C. Parkes54, C.J. Parkinson9, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, C. Pavel-Nicorescu29, A. Pazos Alvarez37, A. Pearce54, A.
Pellegrino41, G. Penso25,m, M. Pepe Altarelli38, S. Perazzini14,d, E. Perez
Trigo37, P. Perret5, M. Perrin-Terrin6, L. Pescatore45, E. Pesen65, G.
Pessina20, K. Petridis53, A. Petrolini19,j, E. Picatoste Olloqui36, B.
Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50, M. Plo
Casasus37, F. Polci8, G. Polok26, A. Poluektov48,34, E. Polycarpo2, A.
Popov35, D. Popov10, B. Popovici29, C. Potterat36, A. Powell55, J.
Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39,
G. Punzi23,s, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Redford55, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S.
Ricciardi49, A. Richards53, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts58, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez
Perez37, S. Roiser38, V. Romanovsky35, A. Romero Vidal37, M. Rotondo22, J.
Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P. Ruiz Valls36, G.
Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B. Saitta15,e,
V. Salustino Guimaraes2, B. Sanmartin Sedes37, R. Santacesaria25, C.
Santamarina Rios37, E. Santovetti24,l, M. Sapunov6, A. Sarti18, C.
Satriano25,n, A. Satta24, M. Savrie16,f, D. Savrina31,32, M. Schiller42, H.
Schindler38, M. Schlupp9, M. Schmelling10, B. Schmidt38, O. Schneider39, A.
Schopper38, M.-H. Schune7, R. Schwemmer38, B. Sciascia18, A. Sciubba25, M.
Seco37, A. Semennikov31, K. Senderowska27, I. Sepp53, N. Serra40, J. Serrano6,
P. Seyfert11, M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52,
L. Shekhtman34, O. Shevchenko43, V. Shevchenko62, A. Shires9, R. Silva
Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, T. Skwarnicki59, N.A.
Smith52, E. Smith55,49, E. Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D.
Sokoloff57, F.J.P. Soler51, F. Soomro39, D. Souza46, B. Souza De Paula2, B.
Spaan9, A. Sparkes50, F. Spinella23, P. Spradlin51, F. Stagni38, S. Stahl11,
O. Steinkamp40, S. Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S.
Stracka23,38, M. Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38,
L. Sun57, W. Sutcliffe53, S. Swientek9, V. Syropoulos42, M. Szczekowski28, P.
Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, E. Teodorescu29, F. Teubert38, C. Thomas55, E. Thomas38, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D.
Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4,53, S. Tourneur39,
M.T. Tran39, M. Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M.
Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin62, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez
Sierra37, S. Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M.
Vesterinen11, B. Viaud7, D. Vieira2, X. Vilasis-Cardona36,p, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß61, H.
Voss10, J.A. de Vries41, R. Waldi61, C. Wallace48, R. Wallace12, S.
Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, A.D. Webber54, D.
Websdale53, M. Whitehead48, J. Wicht38, J. Wiechczynski26, D. Wiedner11, L.
Wiggers41, G. Wilkinson55, M.P. Williams48,49, M. Williams56, F.F. Wilson49,
J. Wimberley58, J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A.
Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y. Xie50,38, Z. Xing59, Z. Yang3,
X. Yuan3, O. Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, F. Zhang3, L.
Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3, A.
Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
62National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
63Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
64KVI - University of Groningen, Groningen, The Netherlands, associated to 41
65Celal Bayar University, Manisa, Turkey, associated to 38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
## 1 Introduction
The phenomenology of soft quantum chromodynamic (QCD) processes such as light
particle production in proton-proton ($pp$) collisions cannot be predicted
using perturbative calculations, but can be described by models implemented in
Monte Carlo event generators. The calculation of the fragmentation and
hadronization processes as well as the modelling of the final states [1, 2]
arising from the soft component of a collision (underlying event) are treated
differently in the various event generators. The phenomenological models
contain parameters that need to be tuned depending on the collision energy and
colliding particles species. This is typically achieved using soft QCD
measurements. The LHCb collaboration reported measurements on energy flow [3],
production cross-sections [4, 5] and production ratios of various particle
species [6] in the forward region, all of which provide information for event
generator optimization.
A fundamental input used for the tuning process is the measurement of prompt
charged particle multiplicities. In combination with the study of the
corresponding momentum spectra and angular distributions, these measurements
can be used to gain a better understanding of hadron collisions. An accurate
description of the underlying event is vital for understanding backgrounds in
beyond the Standard Model searches or precision measurements of the Standard
Model parameters. Previous measurements of charged particle multiplicities
performed with $pp$ collisions at the Large Hadron Collider (LHC) were
reported by the ATLAS [7, 8], CMS [9] and ALICE [10, 11] collaborations. All
of these measurements were performed in the central pseudorapidity region. The
forward region was studied with the LHCb detector, where an inclusive
multiplicity measurement without momentum information was performed [12].
In this paper, $pp$ interactions at a centre-of-mass energy of
$\sqrt{s}=7\;$TeV that produce at least one prompt charged particle in the
pseudorapidity range of $2.0<\eta<4.8$, with a momentum of
$\mbox{$p$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and transverse momentum of
$\mbox{$p_{\rm T}$}>0.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, are studied. A
prompt particle is defined as a particle that either originates directly from
the primary vertex or from a decay chain in which the sum of mean lifetimes
does not exceed $10{\rm\,ps}$. As a consequence, decay products of beauty and
charm hadrons are treated as prompt particles. The information from the full
tracking system of the LHCb detector is used, which permits the measurement of
the momentum dependence of charged particle multiplicities. Multiplicity
distributions, $P(n)$, for prompt charged particles are reported for the total
accessible phase space region as well as for $\eta$ and $p_{\rm T}$ ranges. In
addition, mean particle densities are presented as functions of transverse
momentum, $dn/d\mbox{$p_{\rm T}$}$, and of pseudorapidity, $dn/d\eta$.
The paper is organised as follows. In Sect. 2 a brief description of the LHCb
detector and an overview of track reconstruction algorithms are provided. The
recorded data set and Monte Carlo simulations are described in Sect. 3,
followed by a discussion of the definition of visible event and the data
selection in Sect. 4. The analysis method is described in Sect. 5, and
systematic uncertainties are given in Sect. 6. The final results are compared
to event generator predictions in Sects. 7 and 8, before summarising in Sect.
9.
## 2 LHCb detector and track reconstruction
The LHCb detector [13] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream.
The combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4 % at 2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6 % at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
resolution of 20$\,\upmu\rm m$ for tracks with large transverse momentum. The
direction of the magnetic field of the spectrometer dipole magnet is reversed
regularly. Different types of charged hadrons are distinguished by information
from two ring-imaging Cherenkov detectors. Photon, electron and hadron
candidates are identified by a calorimeter system consisting of scintillating-
pad and preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers. The trigger consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies full event reconstruction.
The reconstruction algorithms provide different track types depending on the
sub-detectors considered. Only two types of tracks are used in this analysis.
VELO tracks are only reconstructed in the VELO sub-detector and provide no
momentum information. Long tracks are reconstructed by extrapolating VELO
tracks through the magnetic dipole field and matching them with hits in the
downstream tracking stations, providing momentum information. This is the
highest-quality track type and is used for most physics analyses. Requiring
charged particles to stay within the geometric acceptance of the LHCb detector
after deflection by the magnetic field further restricts the accessible phase
space to a minimum momentum of around $2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
The LHCb detector design minimizes the material of the tracking detectors and
allows a high track-reconstruction efficiency even for particles with low
momenta. However, the limited number of tracking stations results in the
presence of misreconstructed (fake) tracks. A reconstructed track is
considered as fake if it does not correspond to the trajectory of a genuine
charged particle. The fraction of fake long tracks is non-negligible as the
extrapolation of a track through the magnetic field is performed over a
distance of several metres, resulting in wrong association between VELO tracks
and track segments reconstructed downstream. Another source of wrong track
assignment arises from duplicate tracks. These track pairs either share a
certain number of hits or consist of different track segments originating from
a single particle.
## 3 Data set and simulation
The measurements are performed using a minimum-bias data sample of $pp$
collisions at a centre-of-mass energy of $\sqrt{s}$
=7$\mathrm{\,Te\kern-1.00006ptV}$ collected during 2010. In this low-
luminosity running period, the average number of interactions in the detector
acceptance per recorded bunch crossing was less than $0.1$. The contribution
from bunch crossings with more than one collision (pile-up events) is
determined to be less than $4\,\%$ and is considered as a correction in the
analysis. The data consists of 3 million events recorded in equal proportion
for both magnetic field polarities. The low luminosity and interaction rate of
the proton beams allowed the LHCb detector to be operated with a simplified
trigger scheme. For the minimum-bias data set of this analysis, the hardware
stage of the trigger system accepted all events, which were then reconstructed
by the higher-level software trigger. Events with at least one reconstructed
track segment in the VELO were selected.
Fully simulated minimum-bias $pp$ collisions are generated using the Pythia
6.4 event generator [14] with a specific LHCb configuration [15] using CTEQ6L
[16] parton density functions (PDFs). This implementation, called the LHCb
tune, contains contributions from elastic and inelastic processes, where the
latter also include single and double diffractive components. Decays of
hadrons are performed by EvtGen [17], in which final-state radiation is
generated using Photos [18]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [19,
*Agostinelli:2002hh], as described in Ref. [21]. Processing, reconstruction
and selection are identical for simulated events and data. The simulation is
used to determine correction factors for the detector acceptance and
resolution as well as for quantifying background contributions and
reconstruction performance.
The measurements are compared to predictions of two classes of generators,
those that have not been optimized using LHC data and those that have. The
former includes the Perugia 0 and Perugia NOCR [22] tunes of Pythia 6, both of
which rely on CTEQ5L [23] PDFs, and the Phojet event generator [24]. Phojet
describes soft-particle production by relying on the dual-parton model [2],
which comprises semi-hard processes modelled by parton scattering and soft
processes modelled by pomeron exchange. Pythia 8 [25] is available in both
classes. An early version of Pythia 8 is represented by version 8.145. In more
recent versions, the default configuration has been changed to Tune 4C, which
is based on LHC measurements in the central rapidity region. Both Pythia 8
versions utilize the CTEQ5L PDFs. The results of the latest available version,
Pythia 8.180, are used to represent Tune 4C. Pythia 8.180, together with
recent versions of Herwig++ [26], represent the class of recent event
generators. In contrast to the Pythia generator, where hadronisation is
described by the Lund string fragmentation, the Herwig++ generator relies on
cluster fragmentation and the preconfinement properties of parton showers.
Predictions of two versions of Herwig++ are chosen, each operated in the
minimum-bias configuration, which uses the respective default underlying-event
tune. For Herwig++ version 2.6.3, this corresponds to tune UE-EE-4-MRST
(UE-4), while version 2.7.0 [27] relies on tune UE-EE-5-MRST (UE-5). Both
tunes were also optimized to reproduce LHC measurements in the central
rapidity region and rely on the MRST LO** [28] PDF set.
## 4 Event definition and data selection
In analogy with similar approaches adopted in previous measurements [8, 11],
an event is defined as visible if it contains at least one charged particle in
the pseudorapidity range of $2.0<\eta<4.8$ with $\mbox{$p_{\rm
T}$}>0.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$\mbox{$p$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. These criteria correspond
to the typical kinematic requirements for particles traversing the magnetic
field and reaching the downstream tracking stations. In order to compare the
data directly to predictions from Monte Carlo generators without having a full
detector simulation, the visibility definition is based on the actual presence
of real charged particles, regardless of whether they are reconstructed as
tracks or not.
The tracks are corrected for detector and reconstruction effects to obtain the
distribution of charged particles produced in $pp$ collisions. Only tracks
traversing the full tracking system are considered. The kinematic criteria are
explicitly applied to all tracks to restrict the measurement to a kinematic
range in which reconstruction efficiency is high. The track reconstruction
requires a minimum number of detector hits and a successful track fit. To
retain high reconstruction efficiency, no additional quality requirement for
suppressing the contribution from misreconstructed tracks is applied. To
ensure that tracks originate from the primary interaction, it is required that
the smallest distance of the extrapolated track to the beam line is less than
$2\rm\,mm$. The position of the beam line is determined independently for each
data taking period from events with reconstructed primary vertices.
Additionally, a track is required to originate from the luminous region; the
distance $z_{0}$ of the track to the centre of this region has to fulfil
$z_{0}<3\sigma_{\text{L}}$, where the width $\sigma_{\text{L}}$ is of the
order of $40\rm\,mm$, determined from a Gaussian fit to the longitudinal
position of primary vertices. This restriction also suppresses the
contamination from beam-gas background interactions to a negligible amount.
The distribution of the $z$-position of tracks at the closest point to the
beam line shows that in both high-multiplicity and single-track events, beam-
gas interactions are distributed over the entire $z$-range of the VELO,
whereas the distribution of tracks originating from $pp$ collisions peaks in
the luminous region. There is no explicit requirement for a reconstructed
primary vertex in this analysis. Together with the chosen definition of a
visible event, this allows the measurement to also be performed for events
with only single particles in the acceptance.
## 5 Analysis
The measured particle multiplicity distributions and mean particle densities
are corrected in four steps: (1) reconstructed events are corrected on an
event-by-event basis by weighting each track according to a purity factor to
account for the contamination from reconstruction artefacts and non-prompt
particles; (2) the event sample is further corrected for unobserved events
that fulfil the visibility criteria but in which no tracks are reconstructed;
(3) in order to obtain measurements for single $pp$ collisions, a correction
to remove pile-up events is applied; (4) the effects of various sources of
inefficiencies, such as track reconstruction, are addressed.
While correction factors for the multiplicity distributions and mean particle
densities are the same, their implementation differs and is discussed in the
following.
### 5.1 Correction for reconstruction artefacts and non-prompt particles
The selected track sample includes three significant categories of impurities:
approximately $6.5\,\%$ are fake tracks, less than $1\,\%$ are duplicate
tracks and about $4.5\,\%$ are tracks from non-prompt particles. The
individual contributions are determined using fully simulated events.
Henceforth, all impurity categories are collectively referred to as background
tracks.
The probability of reconstructing a fake track, $\mathcal{P}_{\text{fake}}$,
is dependent on the occupancy of the tracking detectors and on the track
parameters. The occupancy dependence is determined as a function of the track
multiplicity measured by the VELO and as a function of the number of hits in
the downstream tracking stations. This accounts for the increasing probability
of reconstructing a fake track depending on the number of hits in each of the
tracking devices involved. $\mathcal{P}_{\text{fake}}$ also depends on $\eta$
and $p_{\rm T}$; this is taken into account in an overall four-dimensional
parametrisation.
Duplicate tracks are reconstruction artefacts, they have only a weak
dependence on tracking-detector occupancy but exhibit a pronounced kinematic
dependence. The probability of reconstructing a duplicate track,
$\mathcal{P}_{\text{dup}}$, is estimated as a function of $\eta$, $p_{\rm T}$
and VELO track multiplicity.
The probability that a non-prompt particle is selected,
$\mathcal{P}_{\text{sec}}$, is also estimated as a function of the same
variables as for duplicate tracks. The predominant contribution is due to
material interaction, such as photon conversion, and depends on the amount of
material traversed in the detector. Low $p_{\rm T}$ particles are more
affected.
For each track, a combined impurity probability, $\mathcal{P}_{\text{bkg}}$,
is calculated, which is the sum of the three contamination types,
$\mathcal{P}_{\text{bkg}}=\mathcal{P}_{\text{fake}}+\mathcal{P}_{\text{dup}}+\mathcal{P}_{\text{sec}}$,
and depends on the kinematic properties of the track, the occupancy of the
tracking detectors and the track multiplicity. When measuring the mean
particle densities, it is sufficient to assign a per-track weighting factor of
$(1-\mathcal{P}_{\text{bkg}})$ to correct for the impurities mentioned above.
However, correcting particle multiplicity distributions in the same way would
lead to non-physical fractional event multiplicities. To obtain the
background-subtracted multiplicity distributions, the procedure described
below is applied. The description only corresponds to the full kinematic
range, but the procedure is performed in each of the $\eta$ and $p_{\rm T}$
sub-ranges separately. The impurity probability, $\mathcal{P}_{\text{bkg},i}$,
of each track, is summed for all tracks in an event to obtain a total event
impurity correction, $\mu_{\text{ev}}$. This corresponds to a mean number of
expected background tracks in the event and permits to calculate the
probability to reconstruct a certain number of background tracks in each
event, assuming Poisson statistics. The number of background tracks $k$ in an
event with $n_{\text{ev}}$ observed tracks obeys the probability distribution
$\mathcal{P}_{\text{bkg}}(\mu_{\text{ev}},k)=\frac{\mu_{\text{ev}}^{k}}{k!}e^{-\mu_{\text{ev}}},\;\;\text{with}\;\;\mu_{\text{ev}}=\sum_{i=1}^{n_{\text{ev}}}\mathcal{P}_{\text{bkg},i}.$
(1)
From this relation we derive the probability that an event contains a given
number of real prompt particles. Summing the normalized probability
distribution of all events we obtain the multiplicity distribution corrected
for background tracks.
### 5.2 Correction for undetected events
Defining a visible event based on the properties of the actual charged
particles present in the event rather than on the reconstructed tracks
introduces a fraction of spuriously undetected events. These are events that
should be visible but contain no reconstructed tracks and thus remain
undetected. These unobserved events are most likely to occur when few charged
particles are within the kinematic acceptance. The reconstruction of a track
can fail due to multiple scattering, material interaction, or inefficiencies
of the detector or of the reconstruction algorithms. In order to determine the
amount of undetected events that nevertheless fulfil the visibility
definition, a data-driven approach is adopted.
The true multiplicity distribution for visible events, $T(n)$, where $n$ is
the number of charged particles, starts at $n=1$. Since some of these events
have no reconstructed tracks, they follow a multiplicity distribution $U(n)$
starting from $n=0$. As an event can only be detected if at least one track is
reconstructed, $U(0)$ cannot be determined directly. However, the number of
undetected events can be estimated from the observed uncorrected distribution
$U(n)$, if the average survival probability, $\mathcal{P}_{sur}$, for a single
particle in the kinematic acceptance is known. Assuming that the survival
probability, which is determined from simulation, is independent for two or
more particles, the observed distribution is approximated in terms of the
still unknown actual multiplicity distribution $T$
$U(k)=\sum\limits_{n\geq
k}\binom{n}{k}\mathcal{P}_{sur}^{k}(1-\mathcal{P}_{sur})^{n-k}T(n).$ (2)
This equation is only valid under the assumption that reconstruction
artefacts, such as fake tracks, which increase the number of observed tracks
with respect to the number of true tracks, can be ignored. Following this
approach, an event with a certain number of particles is only reconstructed
with the same number of tracks or fewer, but not with more tracks. The
uncertainties due to these assumptions are evaluated in simulation and are
accounted for as systematic uncertainties. Equation 2 allows $U(0)$ to be
estimated from the true distribution $T$. All actual elements $T(k)$ can also
be expressed using the corresponding uncorrected measured bin $U(k)$ and
correction terms of $T(n)$ at higher values of $n>k$,
$\begin{split}&U(0)\approx\displaystyle\sum\limits_{k=1}^{\rm{r}}(1-\mathcal{P}_{sur})^{k}T(k)\text{
\; with}\\\
&T(k)\approx\frac{U(k)}{\mathcal{P}_{sur}^{k}}-\displaystyle\sum\limits_{n=k+1}^{k+\rm{r}}\binom{n}{k}(1-\mathcal{P}_{sur})^{n-k}T(n).\\\
\end{split}$ (3)
Combining the formulas in Eq. 3 results in a recursive expression for $U(0)$,
which can be calculated numerically up to a given order $r$. The procedure is
tested in simulation, where the estimated and actual fractions of undetected
events agree within an uncertainty of $13\,\%$. This is considered as a
systematic uncertainty related to the assumptions made in the calculation. The
fraction of undetected events obtained for data is $2.3\,\%$ compared to
$3.1\,\%$ in simulation. The fraction estimated in data is added to the
measured multiplicity distributions and is also considered in the event
normalisation of the mean particle density measurement.
### 5.3 Pile-up correction
The average number of interactions per bunch crossing in the selected data
taking period is small, resulting in a limited bias from pile-up. The measured
particle multiplicity distributions are mainly composed of single $pp$
collisions and a small fraction of additional second $pp$ collisions.
Therefore events with larger pile-up can be neglected. To obtain the particle
multiplicity distribution of single $pp$ collisions the iterative approach
used in Ref. [12] is applied. The procedure typically converges after two
iterations when the change of the multiplicity distribution is of the order of
the statistical uncertainty. The pile-up correction changes the mean value of
the multiplicity distribution by $3.3\,\%$. The measurements of the mean
particle density are normalised to the total number of $pp$ collisions.
### 5.4 Efficiency correction and unfolding procedure
The final correction step accounts for limited efficiencies due to detector
acceptance $(\epsilon_{\text{acc}})$ in the kinematic range of $2.0<\eta<4.8$
and track reconstruction $(\epsilon_{\text{tr}})$. For particles fulfilling
the kinematic requirements, the detector acceptance describes the fraction
that reach the end of the downstream tracking stations and are unlikely to
interact with material or to be deflected out of the detector by the magnetic
field. This fraction and the overall reconstruction efficiency are evaluated
independently using simulated events. Correction factors are determined as
functions of pseudorapidity and transverse momentum. No multiplicity
dependence is observed. The mean particle densities are corrected by applying
a combined correction factor of
$1/(\epsilon_{\text{acc}}\epsilon_{\text{tr}})$ to each track in the same way
as described in Sect. 5.1.
In order to correct the particle multiplicity distributions, an unfolding
technique based on a detector response matrix is employed. The response
matrix, $R_{m,n}$, accounts for inefficiencies due to the detector acceptance
and track reconstruction. It is constructed from the relation between the
distribution of true prompt charged-particles $T(n)$ and the distribution of
measured tracks $M(m)$, subtracted for background and pile-up,
$M(m)=\sum_{n}R_{m,n}T(n).$ (4)
The matrix is obtained from simulated events. The simulated number of charged
particles per event, $n$, is compared to the corresponding number of
reconstructed and background subtracted tracks, $m$. Thus each possible value
of simulated particle multiplicity is mapped to a distribution of
reconstructed tracks. For very high multiplicities, the available number of
events from the Monte Carlo sample is not sufficient to populate the entire
matrix. The mapping is well described by a Gaussian distribution with mean
value $\bar{m}$ and standard deviation $\sigma_{m}$. The distribution of
$\bar{m}$ and $\sigma_{m}$ for a true multiplicity bin $n$ can be parametrized
by combinations of polynomial and logarithmic functions. This allows an
extrapolation of the matrix up to large values of $n$ and simultaneously
suppresses the effect of statistical fluctuations in the entries of the
matrix. For further information the reader is referred to the Appendix, where
an example of the detector response matrix is shown in Fig. 8.
To extract the true particle multiplicity distribution $T(n)$ from the
measured distribution $M(m)$, a procedure based on $\chi^{2}$-minimization
[29, 30] of the measured distribution $M(m)$ and the folded distribution
$R_{m,n}\tilde{T}(n)$ for different hypotheses of the true distribution,
$\tilde{T}(n)$, is adopted. The range of variation of $\tilde{T}(n)$ is
constrained by parametrising the multiplicity distributions. To avoid
introducing model dependencies to the unfolded result, six different models
with up to eight floating parameters are used. Five models are based on sums
of exponential functions combined with polynomial functions of various order
in the exponent and as a multiplier. In addition, a model based on a sum of
negative binomial distributions is used. While particle multiplicities in
$\eta$ and $p_{\rm T}$ bins can be well described by two negative binomial
distributions, this is not sufficient for the multiplicity distribution in the
full kinematic range, where this model has not been employed. All the
parametrisations used are capable of describing the simulated multiplicity
distributions. The floating parameters of the hypothesis $\tilde{T}(n)$ are
varied in order to minimise the $\chi^{2}$-function
$\chi^{2}(\tilde{T})=\displaystyle\sum\limits_{m}\frac{1}{E(m)^{2}}\left(M(m)-\displaystyle\sum\limits_{n}R_{mn}\tilde{T}(n)\right)^{2},$
(5)
where $E(m)$ represents the uncertainty of the measured distribution $M(m)$.
The parametrisation model yielding the best $\chi^{2}$-value is chosen as the
central result, the other models are considered in the systematic uncertainty
determination. Both the binned and total event unfolding procedures using
simulated data are found to reproduce the generated distributions
satisfactorily. The uncertainty of the unfolded distribution is determined
through pseudo-experiments. Each pseudo-experiment is generated from the
analytical model with the parameters randomly perturbed according to the best
fit and the correlation matrix.
As a consistency check, a Bayesian unfolding technique [31] is used. The
unfolded distributions of both methods in all kinematic bins are found to be
in agreement.
The unfolded distribution for the total event is truncated at a value of 50
particles and the binned distributions at a value of 20 particles. This
corresponds to the limit where, even with the extended detector-response
matrix, larger particle multiplicities cannot be fully mapped to the range of
the measured track-multiplicity distribution and where systematic
uncertainties become large.
## 6 Systematic uncertainties
The precision of the measurements of charged particle multiplicities and mean
particle densities are limited by systematic effects. The bin contents of the
particle multiplicity distribution for the full event typically have a
relative statistical uncertainty in the range of $10^{-4}$ to $10^{-2}$ for
low and high multiplicities, respectively. The systematic uncertainties are
typically around $1-10\,\%$, the largest contribution arising from the
uncertainty of the amount of detector material. All individual contributions
are discussed below.
The properties of fake tracks are studied in detail by using fully simulated
events. The agreement between data and simulation is verified by estimating
the fake-track fraction in both samples by probing the matching probability of
track segments in the long-track reconstruction algorithm. The results are in
good agreement and the differences amount to an overall $2\,\%$ systematic
uncertainty on the applied correction factors.
The systematic uncertainty introduced by differences in the fraction of
duplicate tracks in data and simulation is determined by studying the number
of track pairs with small opening angles. The observed excess of duplicate
tracks in data results in a relative systematic uncertainty on the duplicate-
track fraction of $9\,\%$. As the total amount of this type of reconstruction
artefacts is small, this results in an overall $0.1\,\%$ systematic
uncertainty on the final result.
Uncertainties introduced by the correction for non-prompt particles depend
predominantly on the knowledge of the amount of material within the detector.
The agreement with the amount of material modelled in the simulation, on
average, is found to be within $10\,\%$. In order to estimate the effects of
non-prompt particles still passing the track selection, their composition is
studied. Around $40\,\%$ of the wrongly selected particles arise from photon
conversion and is related to the uncertainty of the amount of material.
Another third of the particles are decay products of $K_{\text{S}}^{\text{0}}$
mesons, whose production cross-section has previously been measured by LHCb
[4] to be in good agreement with simulation. Around $20\,\%$ of the particles
originate from decays of $\Lambda$ baryons and hyperons. These are measured to
disagree by approximately $40\,\%$ with the production cross-sections used in
the simulation. Combining these contributions results in a $12\,\%$ systematic
uncertainty on the fraction of non-prompt particles.
To account for differences between the actual track reconstruction efficiency
and that estimated from simulation, a global systematic uncertainty of $4\,\%$
in average is assigned [32, 33].
The uncertainty on the detector acceptance can be split in two components: the
uncertainty on the knowledge of the detector material and the uncertainty
related to the requirement for particles to have trajectories within the
acceptance of the downstream tracking stations. The momentum distributions of
charged particles in data and in simulation are in good agreement, therefore
the second effect is negligible. The remaining uncertainty related to material
interaction leads to a relative systematic uncertainty on the correction
factors of $3\,\%$ and is assigned as an individual factor for each track.
A modified response matrix is used to estimate the impact on the multiplicity
distributions of systematic uncertainties due to the track reconstruction and
detector acceptance. The systematic uncertainties of both efficiencies are
combined quadratically and result in a $5\,\%$ uncertainty on the response
matrix. A response matrix with an efficiency decreased by this value is
generated. The whole unfolding procedure (Sect. 5.4) is repeated with this
matrix and the full difference to the nominal result is assigned as
uncertainty.
Model dependencies due to the parametrisations used to unfold the true
particle multiplicity distributions are determined by sampling six different
parametrisation models for each of the multiplicity distributions. The model
corresponding to the minimum $\chi^{2}$ value of the unfolding fit is taken as
the central result, while the maximum difference in each bin between all
models and the central result is taken as the systematic uncertainty. This
difference is small compared to the uncertainty due to the modified response
matrix.
Uncertainties related to the correction for undetected events (Sect. 5.2) are
dominated by the $13\,\%$ systematic uncertainty arising from the assumptions
made in the calculation model. In addition, the average survival probability
used in this model is affected by uncertainties of the amount of detector
material, detector acceptance and track reconstruction efficiency. This sums
to a maximum uncertainty of $15\,\%$ on the number of undetected events. Only
bins from one to three tracks are affected, where the variation is dominated
by this uncertainty. For the particle densities, the impact is negligible with
respect to other uncertainties. For the particle multiplicity distributions it
results in a small change of $0.4\,\%$ of the truncated mean.
Uncertainties related to the pile-up fraction are evaluated to be negligible
compared to all other contributions as the total size of the corrections is
already small.
The effect of non-zero beam crossing angles is determined to be insignificant,
as well as the background induced by beam gas interactions.
## 7 Charged particle densities
Figure 1: Charged particle density as a function of $\eta$. The LHCb data are
shown as points with statistical error bars (smaller than the marker size) and
combined systematic and statistical uncertainties as the grey band. The
measurement is compared to several Monte Carlo generator predictions, (a)
Pythia 6 and Phojet, (b) Pythia 8 and Herwig++. Both plots show predictions of
the LHCb tune of Pythia 6, which is used in the analysis.
The fully corrected measurement of mean particle densities in the kinematic
region of $p>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $\mbox{$p_{\rm
T}$}>0.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<\eta<4.8$ is presented
as a function of pseudorapidity in Fig. 1 and as a function of transverse
momentum in Fig. 2; the corresponding numbers are presented in the Appendix.
The data points show a characteristic drop towards larger pseudorapidities but
also a falling edge for $\eta<3$, which is caused by the minimum momentum
requirement in this analysis. This is qualitatively described by all
considered Monte Carlo event generators and their tunes.
The first group of generators that are compared to our measurements are
different tunes of Pythia 6 and Phojet and are shown in Figs. 1a and 2a. The
default configuration of Pythia 6.426 underestimates the amount of charged
particles from roughly $20\,\%$ at large $\eta$ up to $50\,\%$ at small
$\eta$. The descending slopes towards small and large pseudorapidities are
also insufficiently modelled. The Perugia NOCR tune shows a slight improvement
in shape and in the amount of charged particles; Perugia 0 predicts an even
smaller mean particle density over the whole kinematic range. Predictions of
the Phojet generator are similar to the tunes of Pythia 6\. In this group of
predictions, the LHCb tune of Pythia 6 provides the best agreement with the
data but still underestimates the charged-particle production rate by
$10-40\,\%$. This behaviour is also observed in the $p_{\rm T}$ dependence,
where all configurations underestimate the number of charged particles. The
aforementioned generator predictions were optimized without input of LHC
measurements.
Figure 2: Charged particle density as a function of $p_{\rm T}$. The LHCb data
are shown as points with statistical error bars (smaller than the marker size)
and combined systematic and statistical uncertainties as the grey band. The
measurement is compared to several Monte Carlo generator predictions, (a)
Pythia 6 and Phojet, (b) Pythia 8 and Herwig++. Both plots show predictions of
the LHCb tune of Pythia 6, which is used in the analysis.
Predictions from the more recent generators Pythia 8 and Herwig++ are shown in
Figs. 1b and 2b. Pythia 8.145 with default parameters was released without
tuning to LHC measurements and is not better than the LHCb tune of Pythia 6\.
In contrast, Pythia 8.180, which was optimized on LHC data, describes the
measurements significantly better than the previous version. The predictions
of Herwig++ are also in reasonably good agreement with data, although the
charged-particle production rate is underestimated at small pseudorapidities.
The Herwig++ generator version 2.7.0, which uses tune UE-5, overestimates the
number of prompt charged particles in the low $p_{\rm T}$ range but
underestimates it at larger transverse momenta. The predictions of Herwig++ in
version 2.6.3, which relies on tune UE-4, show a more complete description of
the data. Both event generators, Pythia 8 and Herwig++, describe the data over
a wide range.
Figure 3: Observed charged particle multiplicity distribution in the full
kinematic range of the analysis. The error bars represent the statistical
uncertainty, the error band shows the combined statistical and systematic
uncertainties. The data are compared to several Monte Carlo predictions, (a)
Pythia 6 and Phojet, (b) Pythia 8 and Herwig++. Both plots show predictions of
the LHCb tune of Pythia 6, which is used in the analysis.
## 8 Multiplicity distributions
The charged particle multiplicity distribution in the full kinematic range of
the analysis is shown in Fig. 3, compared to the predictions from the event
generators. The corresponding mean value, $\mu$, and the root-mean-square
deviation, $\sigma$, of the distribution, truncated in the range from 1 to 50
particles, is measured to be $\mu=11.304\pm 0.008\pm 0.091$ and
$\sigma=9.496\pm 0.006\pm 0.021$, where the uncertainties are statistical and
systematic, respectively. Using the full range gives consistent results with
the value obtained from the particle densities. All generators that do not use
LHC data input underestimate the multiplicity distributions. In this
comparison, the Phojet generator predicts the smallest probabilities to
observe a large multiplicity event, being in disagreement with the
measurement. This can be understood since Phojet mostly contains soft
scattering events. All Pythia 6 tunes underestimate the charged particle
production cross-section significantly. The prediction from the LHCb tune is
closest to the data, but the mean value of the distribution is still about
$15\,\%$ too small. Calculations from more recent generators are in better
agreement with the measurement. While Pythia 8.145 gives the same insufficient
description of the data as its predecessor, the prediction of version 8.180
using Tune 4C shows a reasonable agreement. The Herwig++ event generator using
the underlying event tune UE-4 shows good agreement with the measurement and
reproduces the data better than the more recent UE-5 tune.
Figure 4: Observed charged particle multiplicity distribution in different
$\eta$ bins. Error bars represent the statistical uncertainty, the error bands
show the combined statistical and systematic uncertainties. The data are
compared to Monte Carlo predictions, (a,b) Pythia 6 and Phojet, (c,d) Pythia 8
and Herwig++. All plots show predictions of the LHCb tune of Pythia 6, which
is used in the analysis.
Figure 5: Observed charged particle multiplicity distribution in different
$\eta$ bins. Error bars represent the statistical uncertainty, the error bands
show the combined statistical and systematic uncertainties. The data are
compared to Monte Carlo predictions, (a-c) Pythia 6 and Phojet, (d-f) Pythia 8
and Herwig++. All plots show predictions of the LHCb tune of Pythia 6, which
is used in the analysis.
Charged particle multiplicity distributions for bins in pseudorapidity are
displayed in Figs. 4 and 5. The comparison with the predictions from Monte
Carlo generators shows the same general features as discussed for the
integrated distribution. The predictions of Phojet and Pythia 6 all
underestimate the particle multiplicity. The difference in particle production
is most prominent at small $\eta$, where the minimum $p$ requirement in this
analysis significantly reduces the amount of particles. Even though the LHCb
tune is in better agreement with the data, the difference remains large.
Recent generator predictions match the data better. Both Pythia 8 and Herwig++
show good agreement with data at larger pseudorapidity, only the range from
$2<\eta<3$ being still underestimated.
Figure 6: Observed charged particle multiplicity distribution in different
$p_{\rm T}$ bins. Error bars represent the statistical uncertainty, the error
bands show the combined statistical and systematic uncertainties. The data are
compared to Monte Carlo predictions, (a,b) Pythia 6 and Phojet, (c,d) Pythia 8
and Herwig++. All plots show predictions of the LHCb tune of Pythia 6, which
is used in the analysis.
Figure 7: Observed charged particle multiplicity distribution in different
$p_{\rm T}$ bins. Error bars represent the statistical uncertainty, the error
bands show the combined statistical and systematic uncertainties. The data are
compared to Monte Carlo predictions, (a-c) Pythia 6 and Phojet, (d-f) Pythia 8
and Herwig++. All plots show predictions of the LHCb tune of Pythia 6, which
is used in the analysis.
Charged particle multiplicities for bins of transverse momentum are shown in
Figs. 6 and 7. The LHCb tune describes the data better than the other tunes.
It is interesting to note that at large transverse momenta, where the
discrepancies are most prominent, Pythia 6.426 in the default configuration
matches the shape of the distribution. Pythia 8 in the recent configuration
shows a reasonably good agreement to the measurement in the mid- and
high-$p_{\rm T}$ range, where also the Herwig++ generator describes the data.
Predictions using the UE-4 tune are closer to the measurement than using the
UE-5 tune. Towards larger $p_{\rm T}$, Herwig++ predictions underestimate the
amount of particles while the Pythia 8 prediction is slightly better. Pythia 8
underestimates the data towards lower $p_{\rm T}$, while Herwig++
overestimates it.
The mean value and the root-mean-square deviation for the multiplicity
distributions in $\eta$ and $p_{\rm T}$ bins are tabulated in the Appendix.
## 9 Summary
The charged particle multiplicities and the mean particle densities are
measured in inclusive $pp$ interactions at a centre-of-mass energy of
$\sqrt{s}=7\;$TeV with the LHCb detector. The measurement is performed in the
kinematic range $\mbox{$p$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$\mbox{$p_{\rm T}$}>0.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<\eta<4.8$, in which at least one charged particle per event is required.
By using the full spectrometer information, it is possible to extend the
previous LHCb results [12] to include momentum dependent measurements. The
comparison of data with predictions from several Monte Carlo event generators
shows that predictions from recent generators, tuned to LHC measurements in
the central rapidity region, are in better agreement than predictions from
older generators. While the phenomenology in some kinematic regions is well
described by recent Pythia and Herwig++ simulations, the data in the higher
$p_{\rm T}$ and small $\eta$ ranges of the probed kinematic region are still
underestimated. None of the event generators considered are able to describe
the entire range of measurements.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages we depend on. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia).
Appendix
Figure 8: Example of the parametrized detector response matrix in the full kinematic range. The matrix is obtained from fully simulated events showing the relation between the true charged particle multiplicity and the reconstructed and background subtracted track multiplicity. Pseudorapidity range | $dn/d\eta$
---|---
$2.0\leq\eta<2.2$ | $3.600\pm 0.048\pm 0.463$
$2.2\leq\eta<2.4$ | $4.032\pm 0.050\pm 0.460$
$2.4\leq\eta<2.6$ | $4.428\pm 0.055\pm 0.367$
$2.6\leq\eta<2.8$ | $4.754\pm 0.056\pm 0.277$
$2.8\leq\eta<3.0$ | $4.943\pm 0.057\pm 0.285$
$3.0\leq\eta<3.2$ | $4.977\pm 0.055\pm 0.267$
$3.2\leq\eta<3.4$ | $4.734\pm 0.052\pm 0.213$
$3.4\leq\eta<3.6$ | $4.500\pm 0.050\pm 0.207$
$3.6\leq\eta<3.8$ | $4.267\pm 0.049\pm 0.200$
$3.8\leq\eta<4.0$ | $4.026\pm 0.047\pm 0.194$
$4.0\leq\eta<4.2$ | $3.845\pm 0.046\pm 0.186$
$4.2\leq\eta<4.4$ | $3.613\pm 0.047\pm 0.263$
$4.4\leq\eta<4.6$ | $3.358\pm 0.043\pm 0.179$
$4.6\leq\eta<4.8$ | $3.281\pm 0.045\pm 0.174$
Table 1: Charged particle density as a function of pseudorapidity. The first quoted uncertainty is statistical and the second systematic. Transverse momentum range [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | $dn/d\mbox{$p_{\rm T}$}\;[0.1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]^{-1}$
---|---
$0.20\leq\mbox{$p_{\rm T}$}<0.30$ | $1.908\pm 0.024\pm 0.116$
$0.30\leq\mbox{$p_{\rm T}$}<0.40$ | $1.866\pm 0.026\pm 0.099$
$0.40\leq\mbox{$p_{\rm T}$}<0.50$ | $1.678\pm 0.022\pm 0.093$
$0.50\leq\mbox{$p_{\rm T}$}<0.60$ | $1.347\pm 0.009\pm 0.092$
$0.60\leq\mbox{$p_{\rm T}$}<0.70$ | $1.082\pm 0.007\pm 0.091$
$0.70\leq\mbox{$p_{\rm T}$}<0.80$ | $0.817\pm 0.006\pm 0.064$
$0.80\leq\mbox{$p_{\rm T}$}<0.90$ | $0.617\pm 0.006\pm 0.042$
$0.90\leq\mbox{$p_{\rm T}$}<1.00$ | $0.481\pm 0.005\pm 0.044$
$1.00\leq\mbox{$p_{\rm T}$}<1.10$ | $0.366\pm 0.005\pm 0.019$
$1.10\leq\mbox{$p_{\rm T}$}<1.20$ | $0.290\pm 0.004\pm 0.015$
$1.20\leq\mbox{$p_{\rm T}$}<1.30$ | $0.228\pm 0.004\pm 0.012$
$1.30\leq\mbox{$p_{\rm T}$}<1.40$ | $0.180\pm 0.004\pm 0.009$
$1.40\leq\mbox{$p_{\rm T}$}<1.50$ | $0.144\pm 0.003\pm 0.007$
$1.50\leq\mbox{$p_{\rm T}$}<1.60$ | $0.113\pm 0.002\pm 0.007$
$1.60\leq\mbox{$p_{\rm T}$}<1.70$ | $0.092\pm 0.002\pm 0.006$
$1.70\leq\mbox{$p_{\rm T}$}<1.80$ | $0.075\pm 0.001\pm 0.005$
$1.80\leq\mbox{$p_{\rm T}$}<1.90$ | $0.061\pm 0.001\pm 0.004$
$1.90\leq\mbox{$p_{\rm T}$}<2.00$ | $0.053\pm 0.001\pm 0.003$
Table 2: Charged particle density as a function of transverse momentum. The first quoted uncertainty is statistical and the second systematic. Pseudorapidity range | Mean value | Root-mean-square
---|---|---
$2.0\leq\eta<2.5$ | $2.010\pm 0.002\pm 0.118$ | $2.460\pm 0.002\pm 0.115$
$2.5\leq\eta<3.0$ | $2.424\pm 0.002\pm 0.097$ | $2.736\pm 0.002\pm 0.094$
$3.0\leq\eta<3.5$ | $2.409\pm 0.002\pm 0.100$ | $2.668\pm 0.002\pm 0.113$
$3.5\leq\eta<4.0$ | $2.121\pm 0.002\pm 0.087$ | $2.396\pm 0.001\pm 0.117$
$4.0\leq\eta<4.5$ | $1.852\pm 0.002\pm 0.069$ | $2.093\pm 0.001\pm 0.073$
Table 3: Truncated mean value and root-mean-square deviation for charged particle multiplicities in different $\eta$-bins. The range is from 0 to 20 particles. The first quoted uncertainty is statistical and the second systematic. Transverse momentum range [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | Mean value | Root-mean-square
---|---|---
$0.2\leq\mbox{$p_{\rm T}$}<0.3$ | $1.928\pm 0.002\pm 0.073$ | $2.083\pm 0.001\pm 0.067$
$0.3\leq\mbox{$p_{\rm T}$}<0.4$ | $1.865\pm 0.002\pm 0.065$ | $1.971\pm 0.001\pm 0.050$
$0.4\leq\mbox{$p_{\rm T}$}<0.6$ | $2.988\pm 0.002\pm 0.098$ | $2.855\pm 0.002\pm 0.069$
$0.6\leq\mbox{$p_{\rm T}$}<1.0$ | $2.881\pm 0.003\pm 0.103$ | $3.029\pm 0.002\pm 0.090$
$1.0\leq\mbox{$p_{\rm T}$}<2.0$ | $1.580\pm 0.002\pm 0.096$ | $2.195\pm 0.001\pm 0.093$
Table 4: Truncated mean value and root-mean-square deviation for charged
particle multiplicities in different $p_{\rm T}$-bins. The range is from 0 to
20 particles. The first quoted uncertainty is statistical and the second
systematic.
## References
* [1] A. Kaidalov and K. Ter-Martirosyan, Multihadron production at high energies in the model of quark gluon strings, Sov. J. Nucl. Phys. 40 (1984) 135
* [2] A. Capella, U. Sukhatme, C.-I. Tan, and J. T. T. Van, Dual parton model, Physics Reports 236 (1994), no. 4–5 225
* [3] LHCb collaboration, R. Aaij et al., Measurement of the forward energy flow in $pp$ collisions at $\sqrt{s}=7~{}T\kern-0.50003pteV$, Eur. Phys. J. C73 (2013) 2421, arXiv:1212.4755
* [4] LHCb collaboration, R. Aaij et al., Prompt $K^{0}_{\rm\scriptscriptstyle S}$ production in $pp$ collisions at $\sqrt{s}=0.9~{}T\kern-0.50003pteV$, Phys. Lett. B693 (2010) 69, arXiv:1008.3105
* [5] LHCb collaboration, R. Aaij et al., Measurement of the inclusive $\phi$ cross-section in $pp$ collisions at $\sqrt{s}=7~{}T\kern-0.50003pteV$, Phys. Lett. B703 (2011) 267, arXiv:1107.3935
* [6] LHCb collaboration, R. Aaij et al., Measurement of prompt hadron production ratios in $pp$ collisions at $\sqrt{s}=$ 0.9 and $7~{}T\kern-0.50003pteV$, Eur. Phys. J. C72 (2012) 2168, arXiv:1206.5160
* [7] ATLAS Collaboration, G. Aad et al., Charged-particle multiplicities in $pp$ interactions at $\sqrt{s}=900$ GeV measured with the ATLAS detector at the LHC, Phys. Lett. B688 (2010) 21, arXiv:1003.3124
* [8] ATLAS collaboration, G. Aad et al., Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHC, New J. Phys. 13 (2011) 053033, arXiv:1012.5104
* [9] CMS collaboration, V. Khachatryan et al., Charged particle multiplicities in pp interactions at $\sqrt{s}$ = 0.9, 2.36, and 7 TeV, JHEP 01 (2011) 079, arXiv:1011.5531
* [10] ALICE Collaboration, K. Aamodt et al., Charged-particle multiplicity measurement in proton-proton collisions at $\sqrt{s}=0.9$ and 2.36 TeV with ALICE at LHC, Eur. Phys. J. C68 (2010) 89, arXiv:1004.3034
* [11] ALICE collaboration, K. Aamodt et al., Charged-particle multiplicity measurement in proton-proton collisions at $\sqrt{s}$ = 7 TeV with ALICE at LHC, Eur. Phys. J. C68 (2010) 345, arXiv:1004.3514
* [12] LHCb collaboration, R. Aaij et al., Measurement of charged particle multiplicities in $pp$ collisions at $\sqrt{s}=7~{}T\kern-0.50003pteV$ in the forward region, Eur. Phys. J. C72 (2012) 1947, arXiv:1112.4592
* [13] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [14] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [15] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [16] J. Pumplin et al., New generation of parton distributions with uncertainties from global QCD analysis, JHEP 07 (2002) 012, arXiv:hep-ph/0201195
* [17] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [18] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [19] GEANT4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [20] GEANT4 collaboration, S. Agostinelli et al., GEANT4: a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [21] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. Phys. : Conf. Ser. 331 (2011) 032023
* [22] P. Z. Skands, The Perugia tunes, arXiv:0905.3418
* [23] CTEQ collaboration, H. L. Lai et al., Global QCD analysis of parton structure of the nucleon: CTEQ5 parton distributions, Eur. Phys. J. C12 (2000) 375, arXiv:hep-ph/9903282
* [24] R. Engel, Photoproduction within the two-component dual parton model: amplitudes and cross-sections, Z. Phys. C66 (1995) 203
* [25] T. Sjöstrand, S. Mrenna, and P. Skands, A brief introduction to PYTHIA 8.1, Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
* [26] M. Bahr et al., Herwig++ Physics and Manual, Eur. Phys. J. C58 (2008) 639, arXiv:0803.0883
* [27] J. Bellm et al., Herwig++ 2.7 Release Note, arXiv:1310.6877
* [28] A. Sherstnev and R. Thorne, Parton Distributions for LO Generators, Eur. Phys. J. C55 (2008) 553, arXiv:0711.2473
* [29] V. Blobel, Unfolding methods in high-energy physics experiments, DESY-84-118 (1984) 40 p, arXiv:0208022v1
* [30] G. Zech, Comparing statistical data to Monte Carlo simulation: Parameter fitting and unfolding, DESY-95-113 (1995)
* [31] G. D’Agostini, A Multidimensional unfolding method based on Bayes’ theorem, Nucl. Instrum. Meth. A362 (1995) 487
* [32] A. Jaeger et al., Measurement of the track finding efficiency, Tech. Rep. LHCb-PUB-2011-025. CERN-LHCb-PUB-2011-025, CERN, Geneva, Apr, 2012
* [33] LHCb collaboration, R. Aaij et al., Measurement of $\sigma(pp\rightarrow b\overline{}bX)$ at $\sqrt{s}=7~{}T\kern-0.50003pteV$ in the forward region, Phys. Lett. B694 (2010) 209, arXiv:1009.2731
|
arxiv-papers
| 2014-02-18T18:30:25 |
2024-09-04T02:49:58.383003
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z. Ajaltouni, J.\n Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle,\n A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. Anderlini, J. Anderson, R.\n Andreassen, M. Andreotti, J.E. Andrews, R.B. Appleby, O. Aquines Gutierrez,\n F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, M. Baalouch,\n S. Bachmann, J.J. Back, A. Badalov, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, V. Batozskaya, Th. Bauer, A. Bay, J. Beddow,\n F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, M. Borsato, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, R. Calabrese, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Cassina, L.\n Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph.\n Charpentier, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid\n Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C.\n Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A.\n Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, I. Counts, B. Couturier,\n G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie, C.\n D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, S. Donleavy, F.\n Dordei, M. Dorigo, P. Dorosz, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger, R. Ekelhof,\n L. Eklund, I. El Rifai, Ch. Elsasser, S. Esen, A. Falabella, C. F\\\"arber, C.\n Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues,\n M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, J. Fu, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P.\n Gandini, Y. Gao, J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld,\n L. Gavardi, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, A. Gianelle, S.\n Giani', V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, P. Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz,\n T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, T.W. Hafkenscheid, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, L. Henry, J.A. Hernando Morata, E. van Herwijnen, M.\n He\\ss, A. Hicheur, D. Hill, M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt,\n N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R.\n Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, N. Jurik, M. Kaballo, S. Kandybei, W.\n Kanso, M. Karacson, T.M. Karbach, M. Kelsey, I.R. Kenyon, T. Ketel, B.\n Khanji, C. Khurewathanakul, S. Klaver, O. Kochebina, I. Komarov, R.F.\n Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev,\n K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai,\n D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T.\n Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A.\n Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, M. Liles, R. Lindner, C. Linn, F. Lionetto, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, N. Lopez-March, P. Lowdon, H. Lu, D. Lucchesi, J.\n Luisier, H. Luo, E. Luppi, O. Lupton, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, M. Manzali, J. Maratas,\n U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A.\n Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, F. Martinez Vidal, D.\n Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov,\n M. McCann, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F.\n Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez,\n S. Monteil, D. Moran, M. Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, R.\n Mountain, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik,\n T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neri, S. Neubert, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M.\n Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano,\n F. Palombo, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, L.\n Pappalardo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pearce, A. Pellegrino,\n G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, P. Perret, M.\n Perrin-Terrin, L. Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini,\n E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, A. Pistone, S.\n Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A.\n Popov, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A.\n Pritchard, C. Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B.\n Rachwal, J.H. Rademacker, B. Rakotomiaramanana, M. Rama, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, S. Reichert, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, A.B. Rodrigues, E. Rodrigues, P. Rodriguez\n Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T.\n Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, M. Schiller, H. Schindler, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, G. Simi, M. Sirendi, N. Skidmore,\n T. Skwarnicki, N.A. Smith, E. Smith, E. Smith, J. Smith, M. Smith, H. Snoek,\n M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B.\n Spaan, A. Sparkes, F. Spinella, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S. Stracka, M.\n Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah, L. Sun, W. Sutcliffe, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, D. Szilard, T. Szumlak,\n S. T'Jampens, M. Teklishyn, G. Tellarini, E. Teodorescu, F. Teubert, C.\n Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, L.\n Tomassetti, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti, A.\n Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez Sierra, S.\n Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D.\n Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, J.A. de Vries, R. Waldi, C.\n Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley,\n J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin (LHCb collaboration)",
"submitter": "Marco Meissner",
"url": "https://arxiv.org/abs/1402.4430"
}
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1402.4518
|
# Cherenkov friction on a neutral particle moving parallel to a dielectric
Gregor Pieplow and Carsten Henkel
Institute of Physics and Astronomy, Universität Potsdam, Germany
(18 Feb 2014)
###### Abstract
Based on a fully relativistic framework and the assumption of local
equilibrium, we describe a simple mechanism of quantum friction for a particle
moving parallel to a dielectric. The Cherenkov effect explains how the bare
ground state becomes globally unstable and how fluctuations of the
electromagnetic field and the particle’s dipole are converted into pairs of
excitations. Modelling the particle as a silver nano-sphere, we investigate
the spectrum of the force and its velocity dependence. We find that the
damping of the plasmon resonance in the silver particle has a relatively
strong impact near the Cherenkov threshold velocity. We also present an
expansion of the friction force near the threshold velocity for both damped
and undamped particles.
Key words: radiation force, quantum friction, Cherenkov, quantum fluctuations.
– PACS: 12.20.-m, 42.50.Lc, 05.30.-d, 03.65.-w, 03.50.De, 42.50.Wk, 03.30.+p
## Introduction
The conversion of mechanical energy into heat is referred to as friction in
most cases. Numerous mechanisms can be identified that cause friction, but it
is still a challenge to infer macroscopic observations from microscopic
phenomena. So far only very simple scenarios permit a detailed analysis of the
fundamental aspects of friction. A prominent example is the theory of the
quantized electromagnetic field applied to the case of two parallel moving
plates separated by a small vacuum gap [1, 2, 3, 4, 5]; see Refs.[6, 7, 8, 9]
for reviews. Friction arises due to the spontaneous creation of particle pairs
that propagate away into the plates or are dissipated there. A similar
treatment can be applied to a body moving above a flat surface at constant
speed [10, 11, 12, 13]. Taking advantage of Lorentz invariance, one achieves
treatments consistent with special relativity [14, 15], as required for the
archetypal situation that high-energy charges are stopped in a medium. In a
recent paper, we described such a formalism for a neutral, polarizable
particle moving parallel to a flat interface [16]. At a typical distance of at
least a few nanometers (larger than the atomic scale), the interaction depends
on a few macroscopic parameters (refractive index, conductivity, surface
impedance …). In the present paper we discuss a special configuration of this
setting with a dielectric medium below the surface, and with both particle and
medium at zero temperature. These conditions make the friction a pure quantum-
mechanical drag and closely relates it to the realm of Casimir phenomena [17].
A friction force appears when the speed of the particle relative to the
surface exceeds the velocity of light in the medium ($c/n$): this drag can
thus be attributed to the Cherenkov effect. The situation is somewhat unusual
because neither the surface nor the particle have to be dissipative. All that
is required are spectral mode densities for the medium field and the particle.
For a moving charge the Cherenkov drag is well known and is described easily
with classical electromagnetic theory [18]. A neutral body requires a more
refined treatment, as quantum fluctuations have to be treated accordingly. As
in previous work [3, 19], we use the fluctuation-dissipation theorem to do
just that. Because of the growing interest in this field and some controversy
surrounding it (see Ref.[13] for a review), the simple situation studied here
might provide another test bed to compare current results and ideas in detail.
In this paper, we analyze in detail a spectral representation of the friction
force that must be applied to move a small particle parallel to a flat
dielectric surface. While this setup has obvious applications for micro- and
nano-machines, our focus is on illustrating the underlying mechanisms. The
basic physics is very similar to the seminal explanation of the Cherenkov
effect [20] by Tamm and Frank [21]: for a certain sector of field modes, the
Doppler shift flips the sign of the mode frequency (anomalous Doppler effect).
This leads to scattering relations (S-matrix) in the form of a Bogoliubov
transformation [7, 9]: incident waves get amplified, and pairs of elementary
excitations (photon-polaritons) can be created out of the quantum fluctuations
in the field and in the particle’s dipole moment. The frictional force arises
from the power carried away by these excitations as they are absorbed or as
they propagate into the bulk of the body. The recent paper by Barton [13]
provides a particularly transparent calculation of these processes in a
simplified setting (only surface plasmon modes are considered). The starting
point we use here is based on the fluctuation electrodynamics developed by
Rytov and co-workers [19]: the basic assumption is that both the solid surface
and the moving particle are in _local thermodynamic equilibrium_. This is a
good approximation for a mesoscopic particle made from thousands of atoms, at
least over time scales where its temperature can be considered constant (large
heat capacity). The approximation is much more questionable for microscopic
particles like atoms or molecules because these may settle into a non-thermal
state due to spontaneous excitation.
We structure our analysis in the following way: some results of previous work
are summoned to provide the basis for the Cherenkov effect. The quantum
(Cherenkov) friction is then calculated and its physical properties are
discussed. We link the friction force to an absorbed power that has to be
provided to move the particle at constant speed. A relativistic argument put
forward by Polevoi [3] attributes this power to an increase in mass-energy.
After the analytics, we numerically investigate a silver nano-particle moving
at relativistic speed above a dielectric surface. We quantify the magnitude of
the friction, and provide a geometric picture of most of the features that
determine the friction spectrum. We then present an expansion of frequency
spectrum of the force and of the force itself near the threshold in $(v-c/n)$.
This further illustrates the relative importance of the resonance and the low,
off-resonant frequencies in the particle polarizability. We find a remarkable
agreement with the numerical results close to the threshold. The main result
is that the Cherenkov friction is linked to composite modes at the vacuum-
dielectric interface [22, 23] which couple to the particle via their
evanescent vacuum tail; their plane-wave component in the medium can be seen
as carrying away the dissipated power.
## 1 The formalism
### 1.1 Friction force
In an earlier paper [16] we presented a covariant approach to the force on a
particle that moves with arbitrary speed parallel to a flat surface that
responds linearly to electromagnetic waves. We recovered the results of
Refs.[24, 14]. The formalism allows for different temperatures of particle and
surface, assuming a state of local equilibrium. The relative motion leads to
Doppler shifts that are handled by Lorentz transforming an incident field into
the frame co-moving with the particle or the surface. The Doppler-shifted
frequency distribution of the equilibrium distributions are responsible for a
non-equilibrium force that persists even when both temperatures $T\to 0$.
Let us fix coordinates such that the $x$-axis points along the motion of the
particle (velocity ${\mathbf{v}}$), while the half-space $z\leq 0$ coincides
with the medium. According to Refs.[14, 16], the force component $F_{x}$
acting on the particle (at distance $z$ from the surface) is
$\displaystyle F_{x}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2\gamma}\int\frac{{\rm d}\omega}{2\pi}\frac{{\rm
d}^{2}k_{\parallel}}{(2\pi)^{2}}[{\rm sign}(\omega)-{\rm sign}(\omega-
vk_{x})]\times$ (1) $\displaystyle\qquad k_{x}\,\mathop{\rm
Im}\,\alpha[\gamma(\omega-
vk_{x})]\sum_{\sigma=s,p}\phi_{\sigma}(\omega,{\mathbf{k}}_{\parallel})\mathop{\rm
Im}\left(\frac{r_{\sigma}{\rm e}^{-2\kappa z}}{\kappa}\right)~{}.$
The frequency $\omega$ and parallel wave numbers
${\mathbf{k}}_{\parallel}=(k_{x},k_{y})$ are measured in the rest frame of the
medium; the integral boundaries are $(-\infty,\infty)$. The difference of sign
functions arises from the thermal factors $\coth[\hbar\omega/(2k_{\rm B}T)]$
in the zero-temperature limit, evaluated in the respective rest frames of
medium and particle ($\omega^{\prime}=\gamma(\omega-vk_{x})$). The particle
polarizability is $\alpha$, $\gamma$ is the Lorentz factor, and for the weight
functions $\phi_{\sigma}$ we have (setting $c=1$)
$\displaystyle\phi_{s}(\omega,{\mathbf{k}}_{\parallel})$ $\displaystyle=$
$\displaystyle\omega^{\prime
2}+2\gamma^{2}({\mathbf{v}}\times{\mathbf{k}}_{\parallel})^{2}\left(1-\frac{\omega^{2}}{k_{\parallel}^{2}}\right)~{},$
(2) $\displaystyle\phi_{p}(\omega,{\mathbf{k}}_{\parallel})$ $\displaystyle=$
$\displaystyle\omega^{\prime
2}+2\gamma^{2}(k_{\parallel}^{2}-({\mathbf{v}}\cdot{\mathbf{k}}_{\parallel})^{2})\left(1-\frac{\omega^{2}}{k_{\parallel}^{2}}\right)~{}.$
(3)
The reflection coefficients for $p$\- and $s$-polarized light are
$r_{\rm s}=\frac{{\rm i}\kappa-\kappa_{n}}{{\rm
i}\kappa+\kappa_{n}}\quad,\quad r_{\rm p}=\frac{{\rm
i}n^{2}\kappa-\kappa_{n}}{{\rm i}n^{2}\kappa+\kappa_{n}}~{},$ (4)
where $\kappa=\sqrt{k_{\parallel}^{2}-(\omega+i0)^{2}}$ and
$\kappa_{n}=\sqrt{n^{2}(\omega+i0)^{2}-k_{\parallel}^{2}}$. Here, $n$ is the
refractive index of the medium.
Using symmetries and other properties we can further simplify the integral in
Eq.(1). The integrand is even under the transformations
$(\omega,k_{x})\mapsto(-\omega,-k_{x})$ and $k_{y}\mapsto-k_{y}$ so that it is
sufficient to integrate over the domain $\omega>0$, $k_{y}>0$. The difference
of the ${\rm sign}$-functions reduces to a factor of two for
$0<\omega<vk_{x}$. This wedge-shaped domain in the $k_{x},\omega$-plane is
below the projected light cone $\omega=k_{\parallel}$, so that only fields
that are evanescent at the particle’s location contribute to the force
[Fig.1.1(_left_)]. We conclude that the factor ${\rm e}^{-2\kappa z}/\kappa$
is real-valued.
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Figure 1. Sketch of the system and visualization of the relevant photon modes.
(_left_) Kinematics of Cherenkov (quantum) friction: the moving particle is spontaneously excited and a photon is emitted into the medium beyond the critical angle. | (_right_) Cherenkov friction arises from a domain in the $\omega,{\mathbf{k}}_{\parallel}$-space that is enclosed by the projected light cone in the medium $\omega=k_{\parallel}/n$ (dark red) and the plane $\omega=vk_{x}$ (pink). Below this plane, the Doppler shift is anomalous and in the frame moving with the particle $\omega^{\prime}<0$. All points below the vacuum light cone (blue) correspond to evanescent waves bound to the medium surface. We take $n=2$ and $v=0.8\,c>c/n$.
Another crucial insight is contained in the reflection coefficients (4): their
imaginary part is nonzero only in the annulus $\omega<k_{\parallel}<n\omega$
[see Fig.2.1(_right_) below]. With the condition derived from the ${\rm sign}$
functions we get $\omega<vk_{\parallel}\cos\phi<vn\omega\cos\phi$, so that the
condition for Cherenkov radiation follows
$1<vn\cos\phi$ (5)
where $\phi$ is the angle between ${\mathbf{v}}$ and
${\mathbf{k}}_{\parallel}$. The expression (1) for the force thus becomes:
$\displaystyle F_{x}$ $\displaystyle=$
$\displaystyle\frac{4\hbar}{\gamma(2\pi)^{3}}\int^{\infty}_{0}{\rm
d}\omega\int^{n\omega}_{\omega/v}{\rm
d}k_{x}\int^{\sqrt{n^{2}\omega^{2}-k_{x}^{2}}}_{0}{\rm d}k_{y}$ (6)
$\displaystyle\quad k_{x}\,\mathop{\rm Im}\,\alpha[\gamma(\omega-
vk_{x})]\sum_{\sigma=s,p}\phi_{\sigma}(\omega,{\mathbf{k}}_{\parallel})\mathop{\rm
Im}\left(r_{\sigma}\right)\frac{{\rm e}^{-2\kappa z}}{\kappa}~{}.$
### 1.2 Photon emission and anomalous Doppler shift
The manipulations performed so far have a clear physical meaning within the
theory of the Cherenkov effect [20, 21, 18] which is well understood. A
kinematic explanation of the friction above the Cherenkov threshold can be
given following the equations outlined in Ref.[25]. We start with the
conservation of 4-momentum
$p^{\mu}_{1}=\hbar k^{\mu}+p^{\mu}_{2}~{}.$ (7)
The momenta $p^{\nu}_{a}$ describe the particle before and after the emission
of a photon with momentum $\hbar k^{\nu}$, where $a=1,2$ labels the internal
states (energy levels $\epsilon_{1,2}$). Although Eq.(7) and Ref.[25] deal
with a particle moving through a medium, the physics is the same for the
motion parallel to the dielectric medium. We have for the particle and the
photon (recall that $c=1$)
$\displaystyle p^{\mu}_{a}$ $\displaystyle=$ $\displaystyle(E_{a}\,,\gamma
m_{a}{\mathbf{v}})~{},\quad m_{a}=M+\epsilon_{a}~{},$ (8) $\displaystyle
E_{a}$ $\displaystyle=$
$\displaystyle\sqrt{m_{a}^{2}+\gamma^{2}m_{a}^{2}{\mathbf{v}}^{2}}=\gamma
m_{a}~{},$ (9) $\displaystyle k^{\mu}$ $\displaystyle=$
$\displaystyle(\omega,\,{\mathbf{k}})~{},\quad k=n\omega~{}.$ (10)
The Greek indices run from $0$ to $3$, and toggling between co- and
contravariant indices is done with the metric $g_{\mu\nu}=\mathop{\rm
diag}(1,-1,-1,-1)$. It is understood that $k=\sqrt{{\mathbf{k}}^{2}}$. The
masses $m_{a}$ are associated with the particle’s energy levels. The photon is
supposed to be emitted into the medium, hence the dispersion relation in
eq.(10). Because the particle is pushed by an “invisible hand”, the velocity
${\mathbf{v}}$ does not change during the emission. This is equivalent to
neglecting the recoil [25] of the particle. Squaring eq.(7) leads to
$(\epsilon_{1}-\epsilon_{2})(2M+\epsilon_{1}+\epsilon_{2})=2E_{1}\hbar\omega(1-vn\cos\phi)~{}.$
(11)
with the same notation as in Eq.(5) above. We can reasonably make the
approximation $\epsilon_{1,2}\ll M$ so that we recover
$\hbar\omega=-\frac{\epsilon_{2}-\epsilon_{1}}{\gamma(1-vn\cos\phi)}~{}.$ (12)
If the particle is faster than the speed of light inside the medium, $1/n$,
the denominator is negative [Cherenkov condition (5)]. This is an illustration
of the so-called anomalous Doppler effect where the photon frequency, as seen
from the moving particle, $\omega^{\prime}=\gamma(\omega-vk_{x})$, is
negative. The authors of Ref.[25] point out that this allows for the
_excitation of the particle to a higher energy level_ ,
$\epsilon_{2}>\epsilon_{1}$, _while emitting a photon into the medium_ ,
inside the Cherenkov cone [see Fig.1.1(_left_)]. The power lost into the
emission must be supplied by the force that keeps the particle on its track.
In other words, considering quantum electrodynamics at a dielectric interface
coupled to a polarizable particle moving faster than the Cherenkov threshold,
it turns out that this is an example of an unstable field theory [7, 26],
similar to electron-positron production in strong electric fields and Hawking
radiation in a strong gravitational field.
### 1.3 Heating and frictional power
This simple kinematic analysis corresponds neatly to the integration domain in
eqs.(1, 6). Note in particular that the particle’s response function is
evaluated at the Doppler-shifted frequency and yields $\mathop{\rm
Im}\alpha(\gamma(\omega-vk_{x}))<0$ in the domain. This is a clear indicator
that the anomalous Doppler effect in combination with the photon emission of
photons into the Cherenkov cone indeed slows down the particle. Another
quantity of interest is the rate of mass change in the particle’s co-moving
frame. This is given by $\dot{m}=u^{\mu}F_{\mu}$ where $u_{\mu}$ is the
particle’s 4-velocity. The full 4-vector of force $F_{\mu}$ can be found in
[16], and for our particle moving in the $x$-direction, we find
$\displaystyle\dot{m}$ $\displaystyle=$
$\displaystyle\gamma\left(F_{0}-vF_{x}\right)$ (13)
$\displaystyle\left({F_{0}\atop vF_{x}}\right)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\\!{\rm
d}\omega\,\int^{n\omega}_{\omega/v}\\!{\rm
d}k_{x}\,\int^{\sqrt{n^{2}\omega^{2}-k_{x}^{2}}}_{0}\\!{\rm
d}k_{y}\,\left({-\hbar\omega\atop-\hbar
vk_{x}}\right)\Gamma(\omega,{\mathbf{k}}_{\parallel})~{},$ (14)
where the positive quantity
$\displaystyle\Gamma(\omega,{\mathbf{k}}_{\parallel})$
$\displaystyle=\frac{4}{\gamma(2\pi)^{3}}{\rm
Im}\,\alpha[\gamma(vk_{v}-\omega)]\sum_{\sigma=s,p}\phi_{\sigma}(\omega,{\mathbf{k}}_{\parallel})\mathop{\rm
Im}(r_{\sigma})\frac{{\rm e}^{-2\kappa z}}{\kappa}$ (15)
can be identified as a spectrally resolved photon emission rate. (We exploited
the fact that $\mathop{\rm Im}\alpha(\omega^{\prime})$ is an odd function.)
Note that the proper mass increases, $\dot{m}>0$, because per emission event,
a positive energy $-\hbar\omega^{\prime}=\hbar\gamma(vk_{x}-\omega)$ is dumped
into the particle’s internal mass-energy, as discussed in the previous
section. Indeed, we shall see, through a simple oscillator model for the
polarizability, that the frequency $\omega^{\prime}$ in the co-moving frame is
essentially fixed by the particle’s resonance.
To summarize this section, let us re-write the power balance as a sum of two
positive terms:
$-vF_{x}=-F_{0}+\frac{{\rm d}m}{\gamma{\rm d}\tau}$ (16)
On the left-hand side, we see the frictional power spent to maintain the
constant speed of the particle. The first term on the right-hand side gives
the power of photon emission (energy $\hbar\omega$ at rate
$\Gamma(\omega,{\mathbf{k}}_{\parallel})$, see Eq.(14)), while the second
gives the power absorbed in the particle. (The factor $1/\gamma$ gives the
relativistic time dilation between the particle’s proper time $\tau$ and the
laboratory time $t$.)
## 2 Case study: relativistic nanoparticle
### 2.1 Numerical investigations
To illustrate further the physical features of the Cherenkov friction force,
we provide some numerical estimates for a metallic nano-particle. We chose a
silver nano-sphere with radius $a=3\,{\rm nm}$ that moves at a distance
$z=10\,{\rm nm}$ above a dielectric medium with refractive index $n=2$. For
simplicity, frequency dispersion is neglected in the medium [23]. For the
particle, we adopt a Drude model with parameters for silver: plasma frequency
$\hbar\omega_{\rm pl}=9.01{\rm eV}$ and damping rate $\hbar/\tau=16\,{\rm
meV}$ (not to be confused with the proper time coordinate $\tau$ above). For
such a small particle, the first term of the Mie series will suffice so that
its response is given by the electric dipole polarizability (for SI units,
multiply with the Coulomb constant $\varepsilon_{0}$)
$\alpha(\omega)=4\pi
a^{3}\frac{\varepsilon(\omega)-1}{\varepsilon(\omega)+2}=4\pi
a^{3}\frac{\Omega^{2}}{\Omega^{2}-\omega^{2}-{\rm i}\omega/\tau}$ (17)
where $\varepsilon(\omega)$ is the metal permittivity. The resonance at
$\Omega=\omega_{\rm pl}/\sqrt{3}$ corresponds to a plasmon mode localized on
the particle. The calculations simplify considerably in the no-damping limit
$\tau\to\infty$ which gives
$\lim_{\tau\rightarrow\infty}\mathop{\rm
Im}\alpha(\omega)=2\pi^{2}a^{3}\omega\,[\delta(\omega-\Omega)+\delta(\omega+\Omega)]~{}.$
(18)
We have checked that at this distance and for velocities above the Cherenkov
threshold, both polarizations contribute roughly the same amount to the force.
This is at variance with the more familiar regime of short (non-retarded)
distances and slow (non-relativistic) atoms where the p-polarization dominates
and an electrostatic calculation suffices.
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Figure 2. Impact of the particle plasmon resonance.
(_left_) Spectral density $F_{x}(\omega,{\mathbf{k}}_{\parallel})$ of the friction force plotted in the plane $\omega^{\prime}=-\Omega$ where the particle’s oscillator strength $\mathop{\rm Im}\alpha(\omega^{\prime})$ peaks. The inclination of the plane is determined by the velocity of the particle. The intersection with the medium light cone (pink) is a hyperbola whose apex has the coordinates given in Eq.(20). | (_right_) Density plot of $F_{x}(\omega,{\mathbf{k}}_{\parallel})$ in the $k_{x},k_{y}$-plane (cut through the left figure at constant frequency). The dashed lines give the outline of the light cones in vacuum and the medium. The relevant integration domain is inside the larger circle and to the right of the orange line $k_{x}=\omega/v$. The wide blue line in this area illustrates the absorption resonance $\omega^{\prime}=-\Omega$ of the particle polarizability $\alpha(\omega^{\prime})$. If $T\neq 0$, the orange line blurs, and the inner circle contributes to the integral as well. Red/blue colors represent positive/negative values. Parameters: $\omega=\sqrt{3}\,\Omega$, where $\Omega$ is the particle’s resonance. We took a relatively short damping time with $\Omega\tau=32.5$.
Fig.1.1(_right_) above illustrates the simple appearance of the integration
volume, which determines most of the features of the force spectrum: it lies
between zero-frequency plane $\omega^{\prime}=0$ and the medium light cone
$\omega=k_{\parallel}/n$. The opening angle $\phi_{\max}$ of the intersection
(measured in the ${\mathbf{k}}_{\parallel}$-plane, relative to the direction
of the velocity ${\bf v}$) is given by the Cherenkov formula
$\cos\phi_{\max}=\frac{\omega/v}{n\omega}=\frac{1}{nv}$ (19)
where $v$ is the particle velocity (scaled to $c$).
Fig.2.1(_left_) shows the impact of the particle’s plasmon resonance: the
plane $\omega^{\prime}=-\Omega$ and the medium light cone intersect in a
hyperbola whose opening angle (projected onto the
${\mathbf{k}}_{\parallel}$-plane) is again given by the Cherenkov formula
(19)—the higher the speed of the particle, the more inclined the plane. The
integrand roughly peaks near the apex of the hyperbola whose position is
easily calculated to be
$\omega_{\rm a}=\frac{\Omega}{\gamma(nv-1)}~{},\qquad k_{x{\rm
a}}=\frac{n\Omega/c}{\gamma(nv-1)}~{},\qquad k_{y{\rm a}}=0~{}.$ (20)
In Fig.2.1(_right_), we plot a slice at constant frequency through the
spectral density $F_{x}(\omega,{\mathbf{k}}_{\parallel})$ of the friction
force given by
$\displaystyle F_{x}(\omega,{\mathbf{k}}_{\parallel})$ $\displaystyle=$
$\displaystyle\frac{4\hbar k_{x}}{\gamma(2\pi)^{3}}\mathop{\rm
Im}\alpha[\gamma(\omega-vk_{x})]{\rm e}^{-2\kappa
z}\sum_{\sigma=s,p}\frac{2k_{\parallel}q_{\sigma}\phi_{\sigma}(\omega,{\mathbf{k}}_{\parallel})}{q_{\sigma}^{2}\kappa^{2}+k_{\parallel}^{2}}~{},$
(21)
where the imaginary part of the reflection amplitudes $r_{\sigma}$ was worked
out from Eqs.(4), and $q_{s}=1$ and $q_{p}=n^{2}$. The density plot reveals
how the resonances of the polarizability $\alpha(\omega^{\prime})$ select
narrow stripes in the ${\mathbf{k}}_{\parallel}$ plane. Only the resonance
$\omega^{\prime}=-\Omega$ (blue) lies in the integration domain relevant for
quantum friction.
These geometric considerations carry over when we integrate over $k_{x}$ and
$k_{y}$ and consider the force spectrum. This is illustrated in Fig.2.1.
Photon emission resonant with the particle plasmon resonance becomes dominant
at velocities well above the Cherenkov threshold [Fig.2.1(_left_)]. Closer to
the threshold, contributions at lower frequency arise from photons that are
off-resonant, more precisely quasi-static, in the frame co-moving with the
particle. Similar to Cherenkov radiation, they are boosted into the visible
range by the Doppler shift. These photons arise from the nonzero value of the
polarizability at low frequencies
$\omega^{\prime}\ll\Omega:\quad\mathop{\rm Im}\alpha(\omega^{\prime})\approx
4\pi a^{3}\frac{\omega^{\prime}}{\Omega^{2}\tau}$ (22)
Note that the only material parameter in this regime is the metal conductivity
$\sigma=\varepsilon_{0}\Omega^{2}\tau$, see also Refs.[4, 27]. Our
interpretation is confirmed in Fig.2.1(_right_) where the spectrum is also
calculated in the lossless limit, using the approximate polarizability (18).
Off-resonant photon emission is suppressed, and the frequency $\omega_{\rm a}$
from Eq.(20) provides a sharp threshold.
Finally, the total friction force is plotted as a function of the particle
velocity in Fig.4. Note again the relatively large difference between finite
damping and the lossless limit near the Cherenkov threshold.
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---|---
Figure 3. Impact of particle velocity and plasmon damping on quantum friction.
(_left_) Frequency spectrum $F_{x}(\omega)$ of the friction force for a silver nano-particle at different velocities above the Cherenkov threshold $c/n=0.5$, obtained by integrating $F_{x}(\omega,{\mathbf{k}}_{\parallel})$ over ${\mathbf{k}}_{\parallel}$. The arrows give the apex of the hyperbola [Eq.(20)] shown in the left plot. We used the quite arbitrary normalization factor $4\hbar(4\pi a^{3})(2\pi)^{-3}10^{-4}(\omega_{\rm pl}/c)^{4}\approx 3.4\,{\rm aN}/\omega_{\rm pl}$ for the force spectrum. We took a damping time fixed by $\Omega\tau=32.5$, which is shorter than in a bulk due to electron scattering at the nano particle surface [28, 29, 30, 31]. | (_right_) Comparison of the lossless case $1/\tau=0$ and a particle resonance with a finite width (same parameters as in Fig.2.1). The arrow indicates the frequency $\omega_{\rm a}$ [Eq.(20)] where the particle resonance $\omega^{\prime}=-\Omega$ intersects the light cone in the medium (apex of the hyperbola in Fig.2.1(_left_)). Same normalization as in Fig.2.1(_left_).
### 2.2 Approximations near threshold
The integrals can be calculated approximately when the opening angle of the
Cherenkov cone is very narrow ($v\approx 1/n$). The main features are captured
by the reflection coefficient in p-polarization, expanded for small
$\kappa_{n}$ [see Eq.(4)]. (See the Appendix for more details.) The formulas
of this section are represented in dashed (gray) lines on Figs.2.1, 4: the
agreement is quite remarkable.
For a particle polarizability with a very narrow resonance, we find the
approximate spectrum
$\displaystyle\omega\geq\omega_{\rm a}:\quad$ $\displaystyle F_{x}(\omega){\rm
d}\omega\approx-\frac{4\hbar(4\pi
a^{3})}{(2\pi)^{3}}\frac{\pi^{2}\Omega}{8nv\gamma^{2}}{\rm d}\omega\,\omega
k_{y{\rm max}}^{2}\,{\rm e}^{-2\omega z\sqrt{n^{2}-1}}\times$ (24)
$\displaystyle\qquad{}\times\left(4+\frac{3k_{y{\rm
max}}^{2}z}{\omega\sqrt{n^{2}-1}}\right)$ $\displaystyle k_{y{\rm
max}}^{2}=(nv-1)\frac{\omega-\omega_{\rm
a}}{v^{2}}\left[2\omega+(nv-1)(\omega+\omega_{\rm a})\right]$
where $\omega_{\rm a}$ is given by Eq.(20), and $k_{y{\rm max}}$ parametrizes
the width of the hyperbola in Fig.2.1(_left_). This spectrum has a sharp
threshold (dashed gray lines in Figs.2.1). If the polarizability includes
damping, the contribution from quasi-static frequencies can be computed
similarly, using the approximation (22). The resulting spectrum is
$\displaystyle\omega\sim 0:\quad F_{x}(\omega){\rm d}\omega$
$\displaystyle\approx$ $\displaystyle-\frac{4\hbar(4\pi
a^{3})}{(2\pi)^{3}}\frac{\pi(nv-1)^{3}}{n^{2}v^{6}\Omega^{2}\tau}{\rm
d}\omega\,\omega^{5}\,{\rm e}^{-2\omega z\sqrt{n^{2}-1}}$
$\displaystyle\qquad{}\times\left(\frac{v^{2}}{20}+\frac{3nv^{3}}{20}+\frac{2n^{2}v^{4}}{15}\right.$
$\displaystyle\qquad\left.+\frac{(nv-1)\omega
z}{280\sqrt{n^{2}-1}}(5+20nv+29n^{2}v^{2}+16n^{3}v^{3})\right)$
and peaks roughly at the inverse roundtrip time $1/(z\sqrt{n^{2}-1})$ (dashed
lines in Fig.2.1(_right_)). As illustrated in the figure above, this
approximation becomes quite poor away from the threshold, as frequencies above
the validity of the low-frequency approximation (22) for $\mathop{\rm
Im}\alpha(\omega^{\prime})$ become relevant.
From both approximations for the spectra, the velocity-dependent friction
force can be calculated, leading to:
$\displaystyle\mbox{no damping}:$ $\displaystyle
F_{x}\approx-\frac{4\hbar(4\pi
a^{3})}{(2\pi)^{3}}\frac{\pi^{2}n^{3}}{8(n^{2}-1)^{3/2}}\frac{\omega_{\rm
a}}{z^{4}}(v-1/n)^{2}{\rm e}^{-2\sqrt{n^{2}-1}\,\omega_{\rm a}z}$ (26)
$\displaystyle\qquad{}\times\left(3+4\sqrt{n^{2}-1}\,\omega_{\rm
a}z+2(n^{2}-1)(\omega_{\rm a}z)^{2}\right)$ $\displaystyle\mbox{with
damping}:\quad$ $\displaystyle F_{x}\approx-\frac{4\hbar(4\pi
a^{3})}{(2\pi)^{3}}\frac{5\pi
n^{5}}{8(n^{2}-1)^{3}}\frac{(v-1/n)^{3}}{z^{6}\Omega^{2}\tau}$ (27)
$\displaystyle\qquad{}\times\left(1+(v-1/n)\frac{11n-2n^{3}}{4(n^{2}-1)}+{\cal
O}(v-1/n)^{5}\right)$
In both cases, we have simplified the complicated polynomial in $v$ to the
lowest order above $1/n$. The dependence on the threshold frequency
$\omega_{\rm a}\sim(v-1/n)^{-1}$ makes the no-damping result exponentially
small at threshold, while damping leads to a cubic power law
$\sim(v-1/n)^{3}$. We also emphasize the different power laws with distance
$z$ from the surface; the corrections to the no-damping case in Eq.(26) are
quite significant for our parameters, as we have the relatively large value
$\omega_{\rm a}z\approx 2.2$ at $v=0.55\,c$. The numerical calculation for a
particle with damping agrees quite well with formula (27) close to the
threshold velocity. Around $v\sim 0.53$, the contribution from the resonance
takes over and the dependence on the damping constant become negligible.
Figure 4: Total friction force vs. particle velocity (log scale). The arrow
indicates the Cherenkov threshold $v=c/n$. Similar to Fig.2.1, the force is
normalized to the value $(2/\pi^{2})10^{-4}\hbar(\omega_{\rm
pl}^{5}/c^{4})a^{3}$, i.e., an acceleration of $\approx 2900\,{\rm m/s}^{2}$.
## Conclusion
We investigated a neutral particle moving in close proximity parallel to a
dielectric. Studying the expression (1) that was derived from a fully
relativistic extension of the fluctuation-dissipation theorem, we provided a
connection to a fundamental and simple friction mechanism. If the particle
moves faster than the speed of light inside the medium (Cherenkov condition),
it can dissipate energy by creating pairs of excitations. Unlike as in
Ref.[13], the pairs are excitations of the particle and photon modes
propagating in the medium. These modes change the sign of their frequency
under the Doppler shift (anomalous Doppler effect [21, 32]). This leads to an
S-matrix in the form of a Bogoliubov transformation that spontaneously excites
the particle and generates a photon emitted into the medium [25, 7, 9]. The
mechanism we described is another example of an unstable vacuum state in a
quantum field theory [26]. The main features of Cherenkov friction were
explained in geometrical terms by analyzing the frequency spectrum of the
force. In order to provide a concrete example, we considered a metallic nano
particle whose polarizability is dominated by a plasmon resonance. We found a
remarkable agreement of the numerical data with an expansion of the force and
the force spectrum in $(v-c/n)$ near the threshold. The approximate
expressions further illustrated the role of the low frequency behavior of the
particles polarizability and its plasmon resonance.
In order to connect with the current discussion on quantum friction [5, 12,
13, 33], we note that in its simplest form, Cherenkov friction does not
require damping in either the particle or the surface. We studied the impact
of dissipation in the particle (as described by a damped plasmon mode) and
found that this significantly changes the friction force just above the
Cherenkov velocity, while maintaining strictly zero friction below the
threshold. This changes, however, when finite temperatures are introduced, or
absorption is allowed for in the surface. Our general result for the radiative
force is identical to that of Ref.[15]. The general setting for the field
quantization (lossless and non-dispersive dielectric) is the same as in
Ref.[23], however, a different particle is considered there (self-energy of a
moving electron). The approach of Ref.[34] is limited to friction forces
linear in the relative velocity of two systems which are both at the same
temperature. The vanishing of linear friction at $T=0$ is consistent with our
analysis. The investigation in Ref.[13] uses a different model for the
particle’s polarizability: a microscopic two-level system with radiative
damping only. In the description of the field modes near the surface, damping
(absorption) is allowed for, and only electrostatic fields are considered
(non-relativistic limit). We emphasize in particular that the excitations that
lead to frictional losses are pairs of surface plasmons in Ref.[13]. A
comprehensive picture where the weight of this excitation process can be
compared directly to the spontaneous particle excitation studied here still
needs to be developed. The simple setting put forward in this paper may
provide a route towards such a picture.
### Acknowledgements
We thank V. E. Mkrtchian, H. R. Haakh, and J. Schiefele for helpful
discussions in various stages of this work.
## Appendix
For the sake of convenience we repeat the general expression for the force at
$T=0$:
$\displaystyle F_{x}$ $\displaystyle=$
$\displaystyle\frac{4\hbar}{\gamma(2\pi)^{3}}\int\limits^{\infty}_{0}{\rm
d}\omega\int\limits^{n\omega}_{\omega/v}{\rm
d}k_{x}\int\limits^{\sqrt{n^{2}\omega^{2}-k_{x}^{2}}}_{0}{\rm d}k_{y}$ (28)
$\displaystyle\quad{}\times k_{x}\,{\rm Im}\alpha[\gamma(\omega-
vk_{x})]\sum_{\sigma=s,p}\phi_{\sigma}(\omega,{\mathbf{k}}_{\parallel})\mathop{\rm
Im}\left(r_{\sigma}\right)\frac{{\rm e}^{-2\kappa z}}{\kappa}~{},$
Close to the threshold ($v-1/n$ becomes small) the range in $k_{x}$ becomes
narrow ($c=1$):
$n\omega-\frac{\omega}{v}=\frac{\omega}{v}(nv-1)\approx n\omega(nv-1)~{}.$
(29)
For the range in $k_{y}$ we find:
$|k_{y}|\leq\sqrt{(n\omega)^{2}-k_{x}^{2}}=\sqrt{(n\omega-
k_{x})(n\omega+k_{x})}\leq\omega\sqrt{(2n/v)(nv-1)}~{}.$ (30)
Hence the wave vectors in the reflection coefficients become
$\displaystyle n^{2}\omega^{2}-\frac{\omega^{2}(n^{2}v^{2}-1)}{v^{2}}\leq
k_{\parallel}^{2}\leq n^{2}\omega^{2}~{},$ (31)
$\displaystyle(n^{2}-1)\omega^{2}-\frac{\omega^{2}(n^{2}v^{2}-1)}{v^{2}}\leq\kappa^{2}\leq(n^{2}-1)\omega^{2}~{},$
(32) $\displaystyle
0\leq\kappa_{n}^{2}\leq\frac{\omega^{2}(n^{2}v^{2}-1)}{v^{2}}~{}.$ (33)
For small values of $\kappa_{n}$ the reflection coefficients thus scale like
$\mathop{\rm Im}r_{\sigma}\sim\sqrt{nv-1}$. For the integration domain in
frequency, we distinguish whether the particle is lossy or lossless:
lossy: $\displaystyle 0\leq\omega<{\cal O}[1/(z\sqrt{n^{2}-1})]<\infty~{},$ no
loss: $\displaystyle\omega_{\rm a}=\frac{\Omega/\gamma}{nv-1}\leq\omega<{\cal
O}[1/(z\sqrt{n^{2}-1})]<\infty$ (34)
where the upper limit arises from the exponential ${\rm e}^{-2\kappa z}$. For
the comoving frequency $\omega^{\prime}=\gamma(\omega-vk_{x})$ we find:
lossy: $\displaystyle 0\leq-\omega^{\prime}\leq\gamma\omega(nv-1)<{\cal
O}[\gamma(nv-1)/(z\sqrt{n^{2}-1})]~{},$ no loss:
$\displaystyle-\omega^{\prime}=\Omega~{}.$ (35)
Hence we see how the relevant frequency ranges are separated by a considerable
margin between the lossy and the lossless case. This suggests that we can
capture both from two different approximations. Note that the lossy case is
dominated by a range of quasi-static frequencies which becomes narrower, the
closer the velocity gets to the Cherenkov threshold (difference $v-1/n$).
## Particle with no losses
In the case of no losses we approximate the particle’s resonance with a
$\delta$-distribution. Formally this is done by taking the limit
$1/\tau\rightarrow 0$ in Eq.(17). Because we only have to focus on $\omega<0$
we find [see also Eq.(18)]:
$\displaystyle(4\pi a^{3})\mathop{\rm
Im}\frac{\Omega^{2}}{\Omega^{2}-\omega^{2}-{\rm i}\omega/\tau}\approx-(4\pi
a^{3})\frac{\pi}{2}\delta(\omega+\Omega)~{}.$ (36)
The oscillator strength $\mathop{\rm Im}\alpha(\omega^{\prime})$ thus fixes
the $k_{x}$-wave vector to (label ‘a’ from ‘apex’ of hyperbola)
$\mbox{no loss:}\quad k_{x}=k_{x{\rm
a}}=\frac{\omega+\Omega/\gamma}{v}\,,\qquad\omega>\omega_{\rm
a}=\frac{\Omega/\gamma}{nv-1}~{}.$ (37)
Note that $\omega$ becomes ‘large’ near the threshold $nv=1$. The expressions
for wave vectors $\kappa$ and $\kappa_{n}$ read
$\displaystyle k_{y{\rm max}}^{2}$ $\displaystyle=$
$\displaystyle(n\omega)^{2}-k_{x{\rm a}}^{2}~{},$ (38)
$\displaystyle\kappa_{n}^{2}$ $\displaystyle=$ $\displaystyle k_{y{\rm
max}}^{2}-k_{y}^{2}~{},$ (39) $\displaystyle\kappa$ $\displaystyle\approx$
$\displaystyle\sqrt{n^{2}-1}\,\omega-\frac{k_{y{\rm
max}}^{2}-k_{y}^{2}}{2\sqrt{n^{2}-1}\,\omega}~{}.$ (40)
For the reflection coefficients this yields the approximation (keeping only
the lowest order in $\kappa$)
$\mathop{\rm Im}r_{p}\approx\frac{2}{n^{2}\sqrt{n^{2}-1}}\frac{k_{y{\rm
max}}}{\omega}(1-q_{y}^{2})^{1/2}$ (41)
where $0\leq q_{y}\leq 1$ is a scaled version of $k_{y}=q_{y}k_{y{\rm max}}$.
For the exponential, we include the next order of $\kappa$ and take
$\frac{{\rm e}^{-2\kappa z}}{\kappa}\approx\frac{{\rm e}^{-2\omega
z\sqrt{n^{2}-1}}}{\omega\sqrt{n^{2}-1}}\left[1+\frac{zk_{y{\rm
max}}^{2}}{\sqrt{n^{2}-1}\,\omega}(1-q_{y}^{2})\right]~{}.$ (42)
We find that the p-polarization yields the leading contribution and use the
lowest-order approximation
$\displaystyle\phi_{p}(\omega,{\mathbf{k}}_{\parallel})$ $\displaystyle=$
$\displaystyle 2\omega^{2}(n^{2}-1)+{\cal O}[\omega^{2}(nv-1)]~{},$ (43)
$\displaystyle\phi_{s}(\omega,{\mathbf{k}}_{\parallel})$ $\displaystyle=$
$\displaystyle{\cal O}[\omega^{2}(nv-1)]~{}.$ (44)
Putting everything together in the leading order, we get the approximate
expression
$\displaystyle F_{x}$ $\displaystyle\approx$ $\displaystyle-\frac{4\hbar(4\pi
a^{3})}{(2\pi)^{3}}\frac{2\pi\Omega}{nv\gamma^{2}}\int\limits_{\omega_{\rm
a}}^{\infty}\\!{\rm d}\omega\,\omega k_{y{\rm max}}^{2}\,{\rm e}^{-2\omega
z\sqrt{n^{2}-1}}$ (45) $\displaystyle\qquad{}\times\int\limits^{1}_{0}\\!{\rm
d}q_{y}\,(1-q_{y}^{2})^{1/2}\left[1+\frac{zk_{y{\rm
max}}^{2}}{\omega\sqrt{n^{2}-1}}(1-q_{y}^{2})\right]~{}.$
A convenient formulation for
$k_{y{\rm max}}^{2}=(nv-1)\frac{\omega-\omega_{\rm
a}}{v^{2}}\left[2\omega+(nv-1)(\omega+\omega_{\rm a})\right]$ (46)
shows the scaling above threshold. The $q_{y}$-integral can be performed and
produces Eq.(24).
## Particle with losses
As outlined in the estimates above, we can use the low-frequency approximation
$\omega^{\prime}\ll\Omega:\quad\mathop{\rm Im}\alpha(\omega^{\prime})\approx
4\pi a^{3}\frac{\omega^{\prime}}{\Omega^{2}\tau}~{}.$ (47)
for the polarizability. This assumes that the frequency range for
$\omega^{\prime}$ near zero is sufficient to capture the integral and ignores
the resonant peak. The integrals that must be performed are
$\displaystyle F_{x}$ $\displaystyle\approx$ $\displaystyle-\frac{4\hbar(4\pi
a^{3})}{(2\pi)^{3}}\frac{4n^{2}}{\Omega^{2}\tau}\int\limits_{0}^{\infty}\\!{\rm
d}\omega\,\omega^{5}\,{\rm e}^{-2\omega z\sqrt{n^{2}-1}}$ (48)
$\displaystyle\qquad\int\limits_{1/(nv)}^{1}\\!{\rm
d}q_{x}\,q_{x}(1-q_{x}^{2})(vnq_{x}-1)$
$\displaystyle\qquad\int\limits_{0}^{1}\\!{\rm
d}q_{y}\,\sqrt{1-q_{y}^{2}}\left[1+\frac{\omega
zn^{2}(1-q_{x}^{2})}{\sqrt{n^{2}-1}}(1-q_{y}^{2})\right]~{}.$
The $q_{y}$-integral is the same as before [Eq.(24)] and gives
$\displaystyle F_{x}$ $\displaystyle\approx$ $\displaystyle-\frac{4\hbar(4\pi
a^{3})}{(2\pi)^{3}}\frac{4n^{2}}{\Omega^{2}\tau}\int\limits_{0}^{\infty}\\!{\rm
d}\omega\,\omega^{5}\,{\rm e}^{-2\omega z\sqrt{n^{2}-1}}$ (49)
$\displaystyle\qquad\int\limits_{1/(nv)}^{1}\\!{\rm
d}q_{x}\,q_{x}(1-q_{x}^{2})(vnq_{x}-1)\frac{\pi}{4}\left(1+\frac{3\omega
zn^{2}(1-q_{x}^{2})}{4\sqrt{n^{2}-1}}\right)~{}.$
The $q_{x}$-integral yields the force spectrum in eq.(2.2).
## References
* [1] E.V. Teodorovich. On the contribution of macroscopic Van Der Waals interactions to frictional force. Proc. Roy. Soc. (London) A, 362:71–77, 1978.
* [2] L.S. Levitov. Van der Waals’ friction. Europhys. Lett., 8:499, (1989).
* [3] V. G. Polevoi. Tangential molecular forces caused between moving bodies by a fluctuating electromagnetic field. Sov. Phys. JETP, 71(6):1119–24, 1990.
* [4] J.B. Pendry. Shearing the vacuum—quantum friction. J. Phys.: Cond. Matt., 9:10301, 1997.
* [5] T.G. Philbin and U. Leonhardt. No quantum friction between uniformly moving plates. New J. Phys., 11:033035, 2009.
* [6] Mehran Kardar and Ramin Golestanian. The “friction” of vacuum, and other fluctuation-induced forces. Rev. Mod. Phys., 71(4):1233–45, 1999.
* [7] P. C. W. Davies. Quantum vacuum friction. J. Opt. B: Quant. Semiclass. Opt., 7(3):S40, 2005.
* [8] A.I. Volokitin and B.N.J. Persson. Near-field radiative heat transfer and noncontact friction. Rev. Mod. Phys., 79(4):1291–1329, 2007.
* [9] Mohammad F. Maghrebi, Ramin Golestanian, and Mehran Kardar. Quantum Cherenkov radiation and non-contact friction. Phys. Rev. A, 88:042509, 2013.
* [10] W L Schaich and J Harris. Dynamic corrections to Van der Waals potentials. J. Phys. F: Metal Phys., 11(1):65–78, 1981.
* [11] G.V. Dedkov and A.A. Kyasov. Electromagnetic friction forces on the scanning probe asperity moving near surface. Phys. Lett. A, 259(1):38–42, 1999.
* [12] Stefan Scheel and Stefan Yoshi Buhmann. Casimir-Polder forces on moving atoms. Phys. Rev. A, 80(4):042902, 2009.
* [13] G. Barton. On van der Waals friction. II: Between atom and half-space. New J. Phys., 12:113045, 2010.
* [14] A.A. Kyasov and G.V. Dedkov. Relativistic theory of fluctuating electromagnetic slowing down of neutral spherical particles moving in close vicinity to a flat surface. Nucl. Instr. Meth. Phys. Res. B, 195(3-4):247–58, 2002.
* [15] G.V. Dedkov and A.A. Kyasov. Fluctuation-electromagnetic interaction of a moving neutral particle with a condensed-medium surface: relativistic approach. Phys. Sol. State, 51(1):1–26, 2009.
* [16] Gregor Pieplow and Carsten Henkel. Fully covariant radiation force on a polarizable particle. New J. Phys., 15(2):023027, 2013.
* [17] Diego A. R. Dalvit, Paulo A. Maia Neto, and Francisco Diego Mazzitelli. Fluctuations, dissipation and the dynamical Casimir effect, volume 834 of Lecture Notes in Physics, chapter 13, pages 419–57. Springer, Berlin Heidelberg, 2011.
* [18] B.M. Bolotovskii. The theory of the Vavilov-Cherenkov effect. Uspekhi Fiz. Nauk, 62:201–46, 1957. (in Russian); see also Sov. Phys. Uspekhi 4 (1962) 781–811.
* [19] Sergei M. Rytov, Yurii A. Kravtsov, and Valeryan I. Tatarskii. Elements of Random Fields, volume 3 of Principles of Statistical Radiophysics. Springer, Berlin, 1989.
* [20] P A Cherenkov. Visible radiation produced by electrons moving in a medium with velocities exceeding that of light. Phys. Rev., 52(4):378, 1937.
* [21] I.E. Tamm and I.M. Frank. Coherent radiation of fast electrons in a medium. Dokl. Akad. Nauk SSSR, 14(3):107–12, 1937.
* [22] C. K. Carnaglia and L. Mandel. Quantization of evanescent electromagnetic waves. Phys. Rev. D, 3:280–96, 1971.
* [23] C. Eberlein and D. Robaschik. Quantum electrodynamics near a dielectric half-space. Phys. Rev. D, 73(2):025009, 2006.
* [24] V.E. Mkrtchian. Interaction between moving macroscopic bodies: viscosity of the electromagnetic vacuum. Phys. Lett. A, 207(5):299–302, 1995.
* [25] V.P. Frolov and V.L. Ginzburg. Excitation and radiation of an accelerated detector and anomalous Doppler effect. Phys. Lett. A, 116(9):423–426, 1986.
* [26] Mario G. Silveirinha. Quantization of the electromagnetic field in non-dispersive polarizable moving media above the Cherenkov threshold. Phys. Rev. A, 88:043846, 2013.
* [27] R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry. Rotational quantum friction. Phys. Rev. Lett., 109:123604, 2012.
* [28] H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer. Width of cluster plasmon resonances: Bulk dielectric functions and chemical interface damping. Phys. Rev. B, 48(24):18178–88, 1993.
* [29] Uwe Kreibig and Michael Vollmer. Optical Properties of Metal Clusters, volume 25 of Springer Series in Materials Science. Springer, Berlin Heidelberg New York, 1995.
* [30] Stéphane Berciaud, Laurent Cognet, Philippe Tamarat, and Brahim Lounis. Observation of intrinsic size effects in the optical response of individual gold nanoparticles. Nano Lett., 5(3):515–18, 2005.
* [31] Lucía B Scaffardi and Jorge O Tocho. Size dependence of refractive index of gold nanoparticles. Nanotechnology, 17(5):1309–15, 2006.
* [32] Vitalii L Ginzburg. Radiation by uniformly moving sources (Vavilov–Cherenkov effect, transition radiation, and other phenomena). Phys. Uspekhi, 39(10):973–82, 1996.
* [33] F. Intravaia, R. O. Behunin, and Diego A. R. Dalvit. Quantum friction and non-equilibrium fluctuation theorems. arXiv:1308.0712, comment by P. Milonni, arXiv:1309.1490, 2013.
* [34] J.S. Høye and I. Brevik. Casimir friction force between polarizable media. Eur. Phys. J. D, 66(6):1–5, 2012.
|
arxiv-papers
| 2014-02-18T22:41:09 |
2024-09-04T02:49:58.400009
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gregor Pieplow and Carsten Henkel",
"submitter": "Gregor Pieplow",
"url": "https://arxiv.org/abs/1402.4518"
}
|
1402.4525
|
11institutetext: Department of Computer Science
University of Miami
1365 Memorial Drive, Coral Gables, FL, 33146 USA
{saminda,aseek,visser}@cs.miami.edu
# Off-Policy General Value Functions to Represent Dynamic Role Assignments in
RoboCup 3D Soccer Simulation
Saminda Abeyruwan Andreas Seekircher Ubbo Visser
###### Abstract
Collecting and maintaining accurate world knowledge in a dynamic, complex,
adversarial, and stochastic environment such as the RoboCup 3D Soccer
Simulation is a challenging task. Knowledge should be learned in real-time
with time constraints. We use recently introduced Off-Policy Gradient Descent
algorithms within Reinforcement Learning that illustrate learnable knowledge
representations for dynamic role assignments. The results show that the agents
have learned competitive policies against the top teams from the RoboCup 2012
competitions for three vs three, five vs five, and seven vs seven agents. We
have explicitly used subsets of agents to identify the dynamics and the
semantics for which the agents learn to maximize their performance measures,
and to gather knowledge about different objectives, so that all agents
participate effectively and efficiently within the group.
ynamic Role Assignment Function, Reinforcement Learning, GQ($\lambda$),
Greedy-GQ($\lambda$), Off-PAC, Off-Policy Prediction and Control, and RoboCup
3D Soccer Simulation.
###### Keywords:
D
## 1 Introduction
The RoboCup 3D Soccer Simulation environment provides a dynamic, real-time,
complex, adversarial, and stochastic multi-agent environment for simulated
agents. The simulated agents formalize their goals in two layers: 1. the
physical layers, where controls related to walking, kicking etc. are
conducted; and 2. the decision layers, where high level actions are taken to
emerge behaviors. In this paper, we investigate a mechanism suitable for
decision layers to use recently introduced Off-Policy Gradient Decent
Algorithms in Reinforcement Leaning (RL) that illustrate learnable knowledge
representations to learn about a dynamic role assignment function.
In order to learn about an effective dynamic role assignment function, the
agents need to consider the dynamics of agent-environment interactions. We
consider these interactions as the agent’s knowledge. If this knowledge is
represented in a formalized form (e.g., first-order predicate logic) an agent
could infer many aspects about its interactions consistent with that
knowledge. The knowledge representational forms show different degrees of
computational complexities and expressiveness [22]. The computational
requirements increase with the extension of expressiveness of the
representational forms. Therefore, we need to identify and commit to a
representational form, which is scalable for on-line learning while preserving
expressivity. A human soccer player knows a lot of information about the game
before (s)he enters onto the field and this prior knowledge influences the
outcome of the game to a great extent. In addition, human soccer players
dynamically change their knowledge during games in order to achieve maximum
rewards. Therefore, the knowledge of the human soccer player is to a certain
extent either predictive or goal-oriented. Can a robotic soccer player collect
and maintain predictive and goal-oriented knowledge? This is a challenging
problem for agents with time constraints and limited computational resources.
We learn the role assignment function using a framework that is developed
based on the concepts of Horde, the real-time learning methodology, to express
knowledge using General Value Functions (GVFs) [22]. Similar to Horde’s sub-
agents, the agents in a team are treated as independent RL sub-agents, but the
agents take actions based on their belief of the world model. The agents may
have different world models due to noisy perceptions and communication delays.
The GVFs are constituted within the RL framework. They are predictions or off-
policy controls that are answers to questions. For example, in order to make a
prediction a question must be asked of the form “If I move in this formation,
would I be in a position to score a goal?”, or “What set of actions do I need
to block the progress of the opponent agent with the number 3?”. The question
defines what to learn. Thus, the problem of prediction or control can be
addressed by learning value functions. An agent obtains its knowledge from
information communicated back and forth between the agents and the agent-
environment interaction experiences.
There are primarily two algorithms to learn about the GVFs, and these
algorithms are based on Off-Policy Gradient Temporal Difference (OP-GTD)
learning: 1. with action-value methods, a prediction question uses
GQ($\lambda$) algorithm [8], and a control or a goal-oriented question uses
Greedy-GQ($\lambda$) algorithm [9]. These algorithms learned about a
deterministic target policies and the control algorithm finds the greedy
action with respect to the action-value function; and 2. with policy-gradient
methods, a goal-oriented question can be answered using Off-Policy Actor-
Critic algorithm [24], with an extended state-value function, GTD($\lambda$)
[7], for GVFs. The policy gradient methods are favorable for problems having
stochastic optimal policies, adversarial environments, and problems with large
action spaces. The OP-GTD algorithms possess a number of properties that are
desirable for on-line learning within the RoboCup 3D Soccer Simulation
environment: 1. off-policy updates; 2. linear function approximation; 3. no
restrictions on the features used; 4. temporal-difference learning; 5. on-
line and incremental; 6. linear in memory and per-time-step computation
costs; and 7. convergent to a local optimum or equilibrium point [23, 9].
In this paper, we present a methodology and an implementation to learn about a
dynamic role assignment function considering the dynamics of agent-environment
interactions based on GVFs. The agents ask questions and approximate value
functions answer to those questions. The agents independently learn about the
role assignment functions in the presence of an adversary team. Based on the
interactions, the agents may have to change their roles in order to continue
in the formation and to maximize rewards. There is a finite number of roles
that an agent can commit to, and the GVFs learn about the role assignment
function. We have conducted all our experiments in the RoboCup 3D Soccer
Simulation League Environment. It is based on the general purpose multi-agent
simulator SimSpark111http://svn.code.sf.net/p/simspark/svn/trunk/. The robot
agents in the simulation are modeled based on the Aldebaran
NAO222http://www.aldebaran-robotics.com/ robots. Each robot has 22 degrees of
freedom. The agents communicate with the server through message passing and
each agent is equipped with noise free joint perceptors and effectors. In
addition to this, each agent has a noisy restricted vision cone of $120^{o}$.
Every simulation cycle is limited to $20~{}ms$, where agents perceive noise
free angular measurements of each joint and the agents stimulate the necessary
joints by sending torque values to the simulation server. The vision
information from the server is available every third cycle ($60~{}ms$), which
provides spherical coordinates of the perceived objects. The agents also have
the option of communicating with each other every other simulation cycle
($40~{}ms$) by broadcasting a $20~{}bytes$ message. The simulation league
competitions are currently conducted with 11 robots on each side (22 total).
The remainder of the paper is organized as follows: In Section 2, we briefly
discuss knowledge representation forms and existing role assignment
formalisms. In Section 3, we introduce GVFs within the context of robotic
soccer. In Section 4, we formalize our mechanisms of dynamic role assignment
functions within GVFs. In Section 5, we identify the question and answer
functions to represent GVFs, and Section 6 presents the experiment results and
the discussion. Finally, Section 7 contains concluding remarks, and future
work.
## 2 Related Work
One goal of multi-agent systems research is the investigation of the prospects
of efficient cooperation among a set of agents in real-time environments. In
our research, we focus on the cooperation of a set of agents in a real-time
robotic soccer simulation environment, where the agents learn about an optimal
or a near-optimal role assignment function within a given formation using
GVFs. This subtask is particularly challenging compared to other simulation
leagues considering the limitations of the environment, i.e. the limited
locomotion capabilities, limited communication bandwidth, or crowd management
rules. The role assignment is a part of the hierarchical machine learning
paradigm [20, 19], where a formation defines the role space. Homogeneous
agents can change roles flexibly within a formation to maximize a given reward
function.
RL framework offerers a set of tools to design sophisticated and hard-to-
engineer behaviors in many different robotic domains (e.g., [4]). Within the
domain of robotic soccer, RL has been successfully applied in learning the
keep-away subtask in the RoboCup 2D [18] and 3D [16] Soccer Simulation
Leagues. Also, in other RoboCup leagues, such as the Middle Size League, RL
has been applied successfully to acquire competitive behaviors [2]. One of the
noticeable impact on RL is reported by the Brainstormers team, the RoboCup 2D
Simulation League team, on learning different subtasks [14]. A comprehensive
analysis of a general batch RL framework for learning challenging and complex
behaviors in robot soccer is reported in [15]. Despite convergence guarantees,
Q($\lambda$) [21] with linear function approximation has been used in role
assignment in robot soccer [5] and faster learning is observed with the
introduction of heuristically accelerated methods [3]. The dynamic role
allocation framework based on dynamic programming is described in [6] for
real-time soccer environments. The role assignment with this method is tightly
coupled with the agent’s low-level abilities and does not take the opponents
into consideration. On the other hand, the proposed framework uses the
knowledge of the opponent positions as well as other dynamics for the role
assignment function.
Sutton et al. [22] have introduced a real-time learning architecture, Horde,
for expressing knowledge using General Value Functions (GVFs). Our research is
built on Horde to ask a set of questions such that the agents assign optimal
or near-optimal roles within formations. In addition, following researches
describe methods and components to build strategic agents: [1] describes a
methodology to build a cognizant robot that possesses vast amount of situated,
reversible and expressive knowledge. [11] presents a methodology to “next” in
real time predicting thousands of features of the world state, and [10]
presents methods predict about temporally extended consequences of a robot’s
behaviors in general forms of knowledge. The GVFs are successfully used (e.g.,
[13, 25]) for switching and prediction tasks in assistive biomedical robots.
## 3 Learnable knowledge representation for Robotic Soccer
Recently, within the context of the RL framework [21], a knowledge
representation language has been introduced, that is expressive and learnable
from sensorimotor data. This representation is directly usable for robotic
soccer as agent-environment interactions are conducted through perceptors and
actuators. In this approach, knowledge is represented as a large number of
approximate value functions each with its 1. own policy; 2. pseudo-reward
function; 3. pseudo-termination function; and 4. pseudo-terminal-reward
function[22]. In continuous state spaces, approximate value functions are
learned using function approximation and using more efficient off-policy
learning algorithms. First, we briefly introduce some of the important
concepts related to the GVFs. The complete information about the GVFs are
available in [22, 8, 9, 7]. Second, we show its direct application to
simulated robotic soccer.
### 3.1 Interpretation
The interpretation of the approximate value function as a knowledge
representation language grounded on information from perceptors and actuators
is defined as:
###### Definition 1
The knowledge expressed as an approximate value function is true or accurate,
if its numerical values matches those of the mathematically defined value
function it is approximating.
Therefore, according to the Definition (1), a value function asks a question,
and an approximate value function is the answer to that question. Based on
prior interpretation, the standard RL framework extends to represent learnable
knowledge as follows. In the standard RL framework [21], let the agent and the
world interact in discrete time steps $t=1,2,3,\ldots$. The agent senses the
state at each time step $S_{t}\in\mathcal{S}$, and selects an action
$A_{t}\in\mathcal{A}$. One time step later the agent receives a scalar reward
$R_{t+1}\in\mathbb{R}$, and senses the state $S_{t+1}\in\mathcal{S}$. The
rewards are generated according to the reward function
$r:S_{t+1}\rightarrow\mathbb{R}$. The objective of the standard RL framework
is to learn the stochastic action-selection policy
$\pi:\mathcal{S}\times\mathcal{A}\rightarrow[0,1]$, that gives the probability
of selecting each action in each state,
$\pi(s,a)=\pi(s|a)=\mathcal{P}(A_{t}=a|S_{t}=s)$, such that the agent
maximizes rewards summed over the time steps. The standard RL framework
extends to include a terminal-reward-function,
$z:\mathcal{S}\rightarrow\mathbb{R}$, where $z(s)$ is the terminal reward
received when the termination occurs in state $s$. In the RL framework,
$\gamma\in[0,1)$ is used to discount delayed rewards. Another interpretation
of the discounting factor is a constant probability of $1-\gamma$ termination
of arrival to a state with zero terminal-reward. This factor is generalized to
a termination function $\gamma:\mathcal{S}\rightarrow[0,1]$, where
$1-\gamma(s)$ is the probability of termination at state $s$, and a terminal
reward $z(s)$ is generated.
### 3.2 Off-Policy Action-Value Methods for GVFs
The first method to learn about GVFs, from off-policy experiences, is to use
action-value functions. Let $G_{t}$ be the complete return from state $S_{t}$
at time $t$, then the sum of the rewards (transient plus terminal) until
termination at time $T$ is:
$G_{t}=\sum_{k=t+1}^{T}r(S_{k})+z(S_{T}).$
The action-value function is:
$Q^{\pi}(s,a)=\mathbb{E}(G_{t}|S_{t}=s,A_{t}=a,A_{t+1:T-1}\sim\pi,T\sim\gamma),$
where, $Q^{\pi}:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$. This is
the expected return for a trajectory started from state $s$, and action $a$,
and selecting actions according to the policy $\pi$, until termination occurs
with $\gamma$. We approximate the action-value function with
$\hat{Q}:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$. Therefore, the
action-value function is a precise grounded question, while the approximate
action-value function offers the numerical answer. The complete algorithm for
Greedy-GQ($\lambda$) with linear function approximation for GVFs learning is
as shown in Algorithm (1).
1:Initialize $w_{0}$ to $0$, and $\theta_{0}$ arbitrary.
2:Choose proper (small) positive values for $\alpha_{\theta}$, $\alpha_{w}$,
and set values for $\gamma(.)\in(0,1]$, $\lambda(.)\in[0,1]$.
3:repeat
4: Initialize $e=0$.
5: Take $A_{t}$ from $S_{t}$ according to $\pi_{b}$, and arrive at $S_{t+1}$.
6: Observe sample, ($S_{t},A_{t},r(S_{t+1}),z(S_{t+1}),S_{t+1},$) at time step
$t$ (with their corresponding state-action feature vectors), where
$\hat{\phi}_{t+1}=\phi(S_{t+1},A_{t+1}^{*}),A_{t+1}^{*}=\operatornamewithlimits{argmax}_{b}{\bf\theta}_{t}^{\mathrm{T}}\phi(S_{t+1},b)$.
7: for each observed sample do
8: $\delta_{t}\leftarrow
r(S_{t+1})+(1-\gamma(S_{t+1}))z(S_{t+1})+\gamma(S_{t+1})\theta_{t}^{\mathrm{T}}\hat{\phi}_{t+1}-\theta_{t}^{\mathrm{T}}\phi_{t}$.
9: If $A_{t}=A_{t}^{*}$, then
$\rho_{t}\leftarrow\frac{1}{\pi_{b}(A_{t}^{*}|S_{t})}$; otherwise
$\rho_{t}\rightarrow 0$.
10: $e_{t}\leftarrow
I_{t}\phi_{t}+\gamma(S_{t})\lambda(S_{t})\rho_{t}e_{t-1}$.
11:
$\theta_{t+1}\leftarrow\theta_{t}+\alpha_{\theta}[\delta_{t}e_{t}-\gamma(S_{t+1})(1-\lambda(S_{t+1}))(w_{t}^{\mathrm{T}}e_{t})\hat{\phi}_{t+1}]$.
12: $w_{t+1}\leftarrow
w_{t}+\alpha_{w}[\delta_{t}e_{t}-(w_{t}^{\mathrm{T}}\phi_{t})\phi_{t})]$.
13: end for
14:until each episode.
Algorithm 1 Greedy-GQ($\lambda$) with linear function approximation for GVFs
learning [7].
The GVFs are defined over four functions: $\pi,\gamma,r,\mbox{and }z$. The
functions $r\mbox{ and }z$ act as pseudo-reward and pseudo-terminal-reward
functions respectively. Function $\gamma$ is also in pseudo form as well.
However, $\gamma$ function is more substantive than reward functions as the
termination interrupts the normal flow of state transitions. In pseudo
termination, the standard termination is omitted. In robotic soccer, the base
problem can be defined as the time until a goal is scored by either the home
or the opponent team. We can consider a pseudo-termination has occurred when
the striker is changed. The GVF with respect to a state-action function is
defined as:
$Q^{\pi,\gamma,r,z}(s,a)=\mathbb{E}(G_{t}|S_{t}=s,A_{t}=a,A_{t+1:T-1}\sim\pi,T\sim\gamma).$
The four functions, $\pi,\gamma,r,\mbox{and }z$, are the question functions to
GVFs, which in return defines the general value function’s semantics. The RL
agent learns an approximate action-value function, $\hat{Q}$, using the four
auxiliary functions $\pi,\gamma,r$ and $z$. We assume that the state space is
continuous and the action space is discrete. We approximate the action-value
function using a linear function approximator. We use a feature extractor
$\mathcal{\phi}:S_{t}\times A_{t}\rightarrow\\{0,1\\}^{N},N\in\mathbb{N}$,
built on tile coding [21] to generate feature vectors from state variables and
actions. This is a sparse vector with a constant number of “1” features,
hence, a constant norm. In addition, tile coding has the key advantage of
real-time learning and to implement computationally efficient algorithms to
learn approximate value functions. In linear function approximation, there
exists a weight vector, $\theta\in\mathbb{R}^{N},N\in\mathbb{N}$, to be
learned. Therefore, the approximate GVFs are defined as:
$\hat{Q}(s,a,\theta)=\theta^{\mathrm{T}}\phi(s,a),$
such that,
$\hat{Q}:\mathcal{S}\times\mathcal{A}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$.
Weights are learned using the gradient-descent temporal-difference Algorithm
(1) [7]. The Algorithm learns stably and efficiently using linear function
approximation from off-policy experiences. Off-policy experiences are
generated from a behavior policy, $\pi_{b}$, that is different from the policy
being learned about named as target policy, $\pi$. Therefore, one could learn
multiple target policies from the same behavior policy.
### 3.3 Off-Policy Policy Gradient Methods for GVFs
The second method to learn about GVFs is using the off-policy policy gradient
methods with actor-critic architectures that use a state-value function
suitable for learning GVFs. It is defined as:
$V^{\pi,\gamma,r,z}(s)=\mathbb{E}(G_{t}|S_{t}=s,A_{t:T-1}\sim\pi,T\sim\gamma),$
where, $V^{\pi,\gamma,r,z}(s)$ is the true state-value function, and the
approximate GVF is defined as:
$\hat{V}(s,v)=v^{\mathrm{T}}\phi(s),$
where, the functions $\pi,\gamma,r,\mbox{and }z$ are defined as in the
subsection (3.2). Since our the target policy $\pi$ is discrete stochastic, we
use a Gibbs distribution of the form:
$\pi(a|s)=\frac{e^{u^{\mathrm{T}}\phi(s,a)}}{\sum_{b}e^{u^{\mathrm{T}}\phi(s,b)}},$
where, $\phi(s,a)$ are state-action features for state $s$, and action $a$,
which are in general unrelated to state features $\phi(s)$, that are used in
state-value function approximation.
$u\in\mathbb{R}^{N_{u}},N_{u}\in\mathbb{N}$, is a weight vector, which is
modified by the actor to learn about the stochastic target policy. The log-
gradient of the policy at state $s$, and action $a$, is:
$\frac{\nabla_{u}\pi(a|s)}{\pi(a|s)}=\phi(s,a)-\sum_{b}\pi(b|s)\phi(s,b).$
The complete algorithm for Off-PAC with linear function approximation for GVFs
learning is as shown in Algorithm (2).
1:Initialize $w_{0}$ to $0$, and $v_{0}$ and $u_{0}$ arbitrary.
2:Choose proper (small) positive values for $\alpha_{v}$, $\alpha_{w}$,
$\alpha_{u}$, and set values for $\gamma(.)\in(0,1]$, $\lambda(.)\in[0,1]$.
3:repeat
4: Initialize $e^{v}=0,\mbox{and }e^{u}=0$.
5: Take $A_{t}$ from $S_{t}$ according to $\pi_{b}$, and arrive at $S_{t+1}$.
6: Observe sample, ($S_{t},A_{t},r(S_{t+1}),z(S_{t+1}),S_{t+1}$) at time step
$t$ (with their corresponding state ($\phi_{t},\phi_{t+1}$) feature vectors,
where $\phi_{t}=\phi(S_{t})$).
7: for each observed sample do
8: $\delta_{t}\leftarrow
r(S_{t+1})+(1-\gamma(S_{t+1}))z(S_{t+1})+\gamma(S_{t+1})v_{t}^{\mathrm{T}}\phi_{t+1}-v_{t}^{\mathrm{T}}\phi_{t}$.
9: $\rho_{t}\leftarrow\frac{\pi(A_{t}|S_{t})}{\pi_{b}(A_{t}|S_{t})}$.
10: Update the critic (GTD($\lambda$) algorithm for GVFs).
11:
$e^{v}_{t}\leftarrow\rho_{t}(\phi_{t}+\gamma(S_{t})\lambda(S_{t})e^{v}_{t-1})$.
12: $v_{t+1}\leftarrow
v_{t}+\alpha_{v}[\delta_{t}e^{v}_{t}-\gamma(S_{t+1})(1-\lambda(S_{t+1}))({e^{v}_{t}}^{\mathrm{T}}w_{t})\phi_{t+1}]$.
13: $w_{t+1}\leftarrow
w_{t}+\alpha_{w}[\delta_{t}e_{t}-(w_{t}^{\mathrm{T}}\phi_{t})\phi_{t})]$.
14: Update the actor.
15:
$e^{u}_{t}\leftarrow\rho_{t}\left[\frac{\nabla_{u}\pi(A_{t}|S_{t})}{\pi(A_{t}|S_{t})}+\gamma(S_{t})\lambda(S_{t+1})e^{u}_{t-1}\right]$.
16: $u_{t+1}\leftarrow u_{t}+\alpha_{u}\delta_{t}e^{u}_{t}$.
17: end for
18:until each episode.
Algorithm 2 Off-PAC with linear function approximation for GVFs learning [7,
24].
We are interested in finding optimal policies for the dynamic role assignment,
and henceforth, we use Algorithms (1), and (2) for control purposes333We use
an C++ implementation of Algorithm (1) and (2) in all of our experiments. An
implementation is available in https://github.com/samindaa/RLLib. We use
linear function approximation for continuous state spaces, and discrete
actions are used within options. Lastly, to summarize, the definitions of the
question functions and the answer functions are given as:
###### Definition 2
The question functions are defined by:
1. 1.
$\pi:S_{t}\times A_{t}\rightarrow[0,1]$ 35mm (target policy is greedy w.r.t.
learned value function);
2. 2.
$\gamma:S_{t}\rightarrow[0,1]$ 35mm (termination function);
3. 3.
$r:S_{t+1}\rightarrow\mathbb{R}$ 35mm (transient reward function); and
4. 4.
$z:S_{t+1}\rightarrow\mathbb{R}$ 35mm (terminal reward function).
###### Definition 3
The answer functions are defined by:
1. 1.
$\pi_{b}:S_{t}\times A_{t}\rightarrow[0,1]$ 35mm (behavior policy);
2. 2.
$I_{t}:S_{t}\times A_{t}\rightarrow[0,1]$ 35mm (interest function);
3. 3.
$\phi:S_{t}\times A_{t}\rightarrow\mathbb{R}^{N}$ 35mm (feature-vector
function); and
4. 4.
$\lambda:S_{t}\rightarrow[0,1]$ 35mm (eligibility-trace decay-rate function).
## 4 Dynamic Role Assignment
A role is a specification of an internal or an external behavior of an agent.
In our soccer domain, roles select behaviors of agents based on different
reference criteria: the agent close to the ball becomes the striker. Given a
role space, $\mathcal{R}=\\{r_{1},\ldots,r_{n}\\}$, of size $n$, the
collaboration among $m\leq n$ agents, $\mathcal{A}=\\{a_{1},\dots,a_{m}\\}$,
is obtained through formations. The role space consists of active and reactive
roles. For example, the striker is an active role and the defender could be a
reactive role. Given a reactive role, there is a function, $R\mapsto T$, that
maps roles to target positions, $T$, on the field. These target positions are
calculated with respect to a reference pose (e.g., ball position) and other
auxiliary criteria such as crowd management rules. A role assignment function,
$R\mapsto A$, provides a mapping from role space to agent space, while
maximizing some reward function. The role assignment function can be static or
dynamic. Static role assignments often provide inferior performance in robot
soccer [6]. Therefore, we learn a dynamic role assignment function within the
RL framework using off-policy control.
Figure 1: Primary formation, [17]
### 4.1 Target Positions with the Primary Formation
Within our framework, an agent can choose one role among thirteen roles. These
roles are part of a primary formation, and an agent calculates the respective
target positions according to its belief of the absolute ball position and the
rules imposed by the 3D soccer simulation server. We have labeled the role
space in order to describe the behaviors associated with them. Figure (1)
shows the target positions for the role space before the kickoff state. The
agent closest to the ball takes the striker role (SK), which is the only
active role. Let us assume that the agent’s belief of the absolute ball
position is given by $(x_{b},y_{b})$. Forward left (FL) and forward right (FR)
target positions are offset by $(x_{b},y_{b})\pm(0,2)$. The extended forward
left (EX1L) and extended forward right ((EX1R)) target positions are offset by
$(x_{b},y_{b})\pm(0,4)$. The stopper (ST) position is given by
$(x_{b}-2.0,y_{b})$. The extended middle (EX1M) position is used as a blocking
position and it is calculated based on the closest opponent to the current
agent. The other target positions, wing left (WL), wing right (WR), wing
middle (WM), back left (BL), back right (BR), and back middle (BM) are
calculated with respect to the vector from the middle of the home goal to the
ball and offset by a factor which increases close to the home goal. When the
ball is within the reach of goal keeper, the (GK) role is changed to goal
keeper striker (GKSK) role. We slightly change the positions when the ball is
near the side lines, home goal, and opponent goal. These adjustments are made
in order to keep the target positions inside the field. We allow target
positions to be overlapping. The dynamic role assignment function may assign
the same role during the learning period. In order to avoid position conflicts
an offset is added; the feedback provides negative rewards for such
situations.
### 4.2 Roles to RL Action Mapping
The agent closest to the ball becomes the striker, and only one agent is
allowed to become the striker. The other agents except the goalie are allowed
to choose from twelve roles. We map the available roles to discrete actions of
the RL algorithm. In order to use Algorithm 1, an agent must formulate a
question function using a value function, and the answer function provides the
solution as an approximate value function. All the agents formulate the same
question: What is my role in this formation in order to maximize future
rewards? All agents learn independently according to the question, while
collaboratively aiding each other to maximize their future reward. We make the
assumption that the agents do not communicate their current role. Therefore,
at a specific step, multiple agents may commit to the same role. We discourage
this condition by modifying the question as What is my role in this formation
in order to maximize future rewards, while maintaining a completely different
role from all teammates in all time steps?
Figure 2: State variable representation and the primary function. Some field
lines are omitted due to clarity.
### 4.3 State Variables Representation
Figure 2 shows the schematic diagram of the state variable representation. All
points and vectors in Figure 2 are defined with respect to a global coordinate
system. $h$ is the middle point of the home goal, while $o$ is the middle
point of the opponent goal. $b$ is the ball position. $\parallel$.$\parallel$
represents the vector length, while $\angle pqr$ represents the angle among
three points $p,~{}q,\mbox{ and }r$ pivoted at $q$. $a_{i}$ represents the
self-localized point of the $i=1,\ldots,11$ teammate agent. $y_{i}$ is some
point in the direction of the robot orientation of teammate agents. $c_{j}$,
$j=1,\ldots,11$, represents the mid-point of the tracked opponent agent. $x$
represents a point on a vector parallel to unit vector $e_{x}$. Using these
labels, we define the state variables as:
$\displaystyle\\{\parallel\vec{v}_{hb}\parallel,\parallel\vec{v}_{bo}\parallel,\angle
hbo,\\{\parallel\vec{v}_{a_{i}b}\parallel,\angle y_{i}a_{i}b,\angle
a_{i}bx\\}_{i=n_{start}}^{n_{end}},\\{\parallel\vec{v}_{c_{j}b}\parallel,\angle
c_{j}bx,\\}_{j=1}^{m_{max}}\\}.$
$n_{start}$ is the teammate starting id and $n_{end}$ the ending id. $m_{max}$
is the number of opponents considered. Angles are normalized to
[$-\frac{\pi}{2},\frac{\pi}{2}$].
## 5 Question and Answer Functions
There are twelve actions available in each state. We have left out the striker
role from the action set. The agent nearest to the ball becomes the striker.
All agents communicate their belief to other agents. Based on their belief,
all agents calculate a cost function and assign the closest agent as the
striker. We have formulated a cost function based on relative distance to the
ball, angle of the agent, number of teammates and opponents within a region
near the ball, and whether the agents are active. In our formulation, there is
a natural termination condition; scoring goals. With respect to the striker
role assignment procedure, we define a pseudo-termination condition. When an
agent becomes a striker, a pseudo-termination occurs, and the striker agent
does not participate in the learning process unless it chooses another role.
We define the question and answer functions as follows:
### 5.1 GVF Definitions for State-Action Functions
Question functions:
1. 1.
$\pi=$ greedy w.r.t. $\hat{Q}$,
2. 2.
$\gamma(.)=0.8$,
3. 3.
$r(.)=$ (a) the change of $x$ value of the absolute ball position; (b) a
small negative reward of $0.01$ for each cycle; (c) a negative reward of $5$
is given to all agents within a radius of 1.5 meters;
4. 4.
$z(.)=$ (a) $+100$for scoring against opponent; (b) $-100$for opponent
scoring; and
5. 5.
$\mbox{time step}=2$ seconds.
Answer functions:
1. 1.
$\pi_{b}=$ $\epsilon$-greedy w.r.t. target state-action function,
2. 2.
$\epsilon=0.05$,
3. 3.
$I_{t}(.)=1$,
4. 4.
$\phi(.,.)=$ (a) we use tile coding to formulate the feature vector.
$n_{start}=2$ and $n_{end}=3,5,7$. $m_{max}=3,5,7$. Therefore, there are
$18,28,30$ state variables. (b) state variable is independently tiled with 16
tilings with approximately each with $\frac{1}{16}$ generalization. Therefore,
there are $288+1,448+1,608+1$ active tiles (i.e., tiles with feature 1) hashed
to a binary vector dimension $10^{6}+1$. The bias feature is always active,
and
5. 5.
$\lambda(.)=0.8$.
Parameters:
• 6mm 1. $\parallel{\bf{\theta}}\parallel=\parallel{\bf w}\parallel=10^{6}+1$;
2. $\parallel{\bf e}\parallel=2000$(efficient trace implementation); 6mm 3.
$\alpha_{\theta}=\frac{0.01}{289},\frac{0.01}{449},\frac{0.01}{609}$; and 4.
$\alpha_{w}=0.001\times\alpha_{\theta}$.
### 5.2 GVF for Gradient Descent Functions
Question functions:
1. 1.
$\pi=$ Gibbs distribution,
2. 2.
$\gamma(.)=0.9$,
3. 3.
$r(.)=$ (a) the change of $x$ value of the absolute ball position; (b) a
small negative reward of $0.01$ for each cycle; (c) a negative reward of $5$
is given to all agents within a radius of 1.5 meters;
4. 4.
$z(.)=$ (a) $+100$for scoring against opponent; (b) $-100$for opponent
scoring; and
5. 5.
$\mbox{time step}=2$ seconds.
Answer functions:
1. 1.
$\pi_{b}=$ the learned Gibbs distribution is used with a small perturbation.
In order to provide exploration, with probability $0.01$, Gibbs distribution
is perturbed using some $\beta$ value. In our experiments, we use $\beta=0.5$.
Therefore, we use a behavior policy:
$\frac{e^{u^{\mathrm{T}}\phi(s,a)+\beta}}{\sum_{b}e^{u^{\mathrm{T}}\phi(s,b)+\beta}}$
2. 2.
$\phi(.)=$ (a) the representations for the state-value function, we use tile
coding to formulate the feature vector. $n_{start}=2$ and $n_{end}=3,5,7$.
$m_{max}=3,5,7$. Therefore, there are $18,28,30$ state variables. (b) state
variable is independently tiled with 16 tilings with approximately each with
$\frac{1}{16}$ generalization. Therefore, there are $288+1,448+1,608+1$ active
tiles (i.e., tiles with feature 1) hashed to a binary vector dimension
$10^{6}+1$. The bias feature is always set to active;
3. 3.
$\phi(.,.)=$ (a) the representations for the Gibbs distribution, we use tile
coding to formulate the feature vector. $n_{start}=2$ and $n_{end}=3,5,7$.
$m_{max}=3,5,7$. Therefore, there are $18,28,30$ state variables. (b) state
variable is independently tiled with 16 tilings with approximately each with
$\frac{1}{16}$ generalization. Therefore, there are $288+1,448+1,608+1$ active
tiles (i.e., tiles with feature 1) hashed to a binary vector dimension
$10^{6}+1$. The hashing has also considered the given action. The bias feature
is always set to active; and
4. 4.
$\lambda_{\mbox{critic}}(.)=\lambda_{\mbox{actor}}(.)=0.3$.
Parameters:
• 6mm 1. $\parallel{\bf{u}}\parallel=10^{6}+1$; 2.
$\parallel{\bf{\theta}}\parallel=\parallel{\bf w}\parallel=10^{6}+1$; 6mm 3.
$\parallel{\bf e^{v}}\parallel=\parallel{\bf e^{u}}\parallel=2000$(efficient
trace implementation); 6mm 4.
$\alpha_{v}=\frac{0.01}{289},\frac{0.01}{449},\frac{0.01}{609}$; 5.
$\alpha_{w}=0.0001\times\alpha_{v}$; and 6.
$\alpha_{v}=\frac{0.001}{289},\frac{0.001}{449},\frac{0.001}{609}$.
## 6 Experiments
We conducted experiments against the teams Boldhearts and MagmaOffenburg, both
semi-finalists of the RoboCup 3D Soccer Simulation competition in Mexico
2012444The published binary of the team UTAustinVilla showed unexpected
behaviors in our tests and is therefore omitted.. We conducted knowledge
learning according to the configuration given in Section (5). Subsection (6.1)
describes the performance of the Algorithm (1), and Subsection (6.2) describes
the performance of the Algorithm (2) for the experiment setup.
### 6.1 GVFs with Greedy-GQ($\lambda$)
The first experiments were done using a team size of five with the RL agents
against Boldhearts. After 140 games our RL agent increased the chance to win
from 30% to 50%. This number does not increase more in the next games, but
after 260 games the number of lost games (initially 35%) is reduced to 15%. In
the further experiments we used the goal difference to compare the performance
of the RL agent. Figure (3) shows the average goal differences that the hand-
tuned role assignment and the RL agents archive in games against Boldhearts
and MagmaOffenburg using different team sizes. With only three agents per team
the RL agent only needs 40 games to learn a policy that outperforms the hand-
coded role selection (Figure (3(a))). Also with five agents per team, the
learning agent is able to increase the goal difference against both opponents
(Figure (3(b))). However, it does not reach the performance of the manually
tuned role selection. Nevertheless considering the amount of time spent for
fine-tuning the hand-coded role selection, these results are promising.
Furthermore, the outcome of the games depends a lot on the underlying skills
of the agents, such as walking or dribbling. These skills are noisy, thus the
results need to be averaged over many games (std. deviations in Figure (3) are
between 0.5 and 1.3).
((a)) Three vs three agents.
((b)) Five vs five agents.
((c)) Seven vs seven agents.
Figure 3: Goal difference in games with (a) three; (b) five; and (c) seven
agents per team using Greedy-GQ($\lambda$) algorithm.
The results in Figure (3(c)) show a bigger gap between RL and the hand-coded
agent. However, using seven agents the goal difference is generally decreased,
since the defense is easily improved by increasing the number of agents. Also
the hand-coded role selection results in a smaller goal difference.
Furthermore, considering seven agents in each team the state space is already
increased significantly. Only 200 games seem to be not sufficient to learn a
good policy. Sometimes the RL agents reach a positive goal difference, but it
stays below the hand-coded role selection. In Section 7, we discuss some of
the reasons for this inferior performances for the team size seven. Even
though the RL agent did not perform well considering only the goal difference,
it has learned a moderately satisfactory policy. After 180 games the amount of
games won is increased slightly from initially 10% to approximately 20%.
### 6.2 GVFs with Off-PAC
With Off-PAC, we used a similar environment to that of Subsection (6.1), but
with a different learning setup. Instead of learning individual policies for
teams separately, we learned a single policy for both teams. We ran the
opponent teams in a round robin fashion for 200 games and repeated complete
runs for multiple times. The first experiments were done using a team size of
three with RL agents against both teams. Figure (4(a)) shows the results of
bins of 20 games averaged between two trials. After 20 games, the RL agents
have learned a stable policy compared to the hand-tuned policy, but the
learned policy bounded above the hand-tuned role assignment function. The
second experiments were done using a team size of five with the RL agents
against opponent teams. Figure (4(b)) shows the results of bins of 20 games
averaged among three trials. After 100 games, our RL agent increased the
chance of winning to 50%. This number does not increase more in the next
games. As Figures (4(a)) and (4(b)) show, the three and five agents per team
are able to increase the goal difference against both opponents. However, it
does not reach the performance of the manually tuned role selection. Similar
to Subsection (6.1), the amount of time spent for fine-tuning the hand-coded
role selection, these results are promising, and the outcome of the experiment
heavily depends on the underlying skills of the agents.
((a)) Three vs three agents.
((b)) Five vs five agents.
((c)) Seven vs seven agents.
Figure 4: Goal difference in games with (a) three; (b) five; and (c) seven
agents per team using Off-PAC algorithm.
The final experiments were done using a team size of seven with the RL agents
against opponent teams. Figure (4(c)) shows the results of bins of 20 games
averaged among two trials. Similar to Subsection (6.1), with seven agents per
team, the results in Figure (4(c)) show a bigger gap between RL and the hand-
tuned agent. However, using seven agents the goal difference is generally
decreased, since the defense is easily improved by increasing the number of
agents. Also the hand-tuned role selection results in a smaller goal
difference. Figure 4(c) shows an increase in the trend of winning games. As
mentioned earlier, only 200 games seem to be not sufficient to learn a good
policy. Even though the RL agents reach a positive goal difference, but it
stays below the hand-tuned role selection method. Within the given setting,
the RL agents have learned a moderately satisfactory policy. Whether the
learned policy is satisfactory for other teams needs to be further
investigated.
The RoboCup 3D soccer simulation is inherently a dynamic, and a stochastic
environment. There is an infinitesimal chance that a given situation (state)
may occur for many games. Therefore, it is paramount important that the
learning algorithms extract as much information as possible from the training
examples. We use the algorithms in the on-line incremental setting, and once
the experience is consumed it is discarded. Since, we learned from off-policy
experiences, we can save the tuples,
$(S_{t},A_{t},S_{t+1},r(S_{t+1}),\rho_{t},z(S_{t+1}))$, and learn the policy
off-line. The Greedy-GQ($\lambda$) learns a deterministic greedy policy. This
may not be suitable for complex and dynamic environments such as the RoboCup
3D soccer simulation environment. The Off-PAC algorithm is designed for
stochastic environment. The experiment shows that this algorithm needs careful
tuning of learning rates and feature selection, as evident from Figure (4(a))
after 160 games.
## 7 Conclusions
We have designed and experimented RL agents that learn to assign roles in
order to maximize expected future rewards. All the agents in the team ask the
question “What is my role in this formation in order to maximize future
rewards, while maintaining a completely different role from all teammates in
all time steps?”. This is a goal-oriented question. We use Greedy-
GQ($\lambda$) and Off-PAC to learn experientially grounded knowledge encoded
in GVFs. Dynamic role assignment function is abstracted from all other low-
level components such as walking engine, obstacle avoidance, object tracking
etc. If the role assignment function selects a passive role and assigns a
target location, the lower-layers handle this request. If the lower-layers
fail to comply to this request, for example being reactive, this feedback is
not provided to the role assignment function. If this information needs to be
included; it should become a part of the state representation, and the reward
signal should be modified accordingly. The target positions for passive roles
are created w.r.t. the absolute ball location and the rules imposed by the 3D
soccer simulation league. When the ball moves relatively quickly, the target
locations change more quickly. We have given positive rewards only for the
forward ball movements. In order to reinforce more agents within an area close
to the ball, we need to provide appropriate rewards. These are part of reward
shaping [12]. Reward shaping should be handled carefully as the agents may
learn sub-optimal policies not contributing to the overall goal.
The experimental evidences show that agents are learning competitive role
assignment functions for defending and attacking. We have to emphasize that
the behavior policy is $\epsilon$-greedy with a relatively small exploration
or slightly perturbed around the target policy. It is not a uniformly
distributed policy as used in [22]. The main reason for this decision is that
when an adversary is present with the intention of maximizing its objectives,
practically the learning agent may have to run for a long period to observe
positive samples. Therefore, we have used the off-policy Greedy-GQ($\lambda$)
and Off-PAC algorithms for learning goal-oriented GVFs within on-policy
control setting. Our hypothesis is that with the improvements of the
functionalities of lower-layers, the role assignment function would find
better policies for the given question and answer functions. Our next step is
to let the RL agent learn policies against other RoboCup 3D soccer simulation
league teams. Beside the role assignment, we also contributed with testing
off-policy learning in high-dimensional state spaces in a competitive
adversarial environment. We have conducted experiments with three, five, and
seven agents per team. The full game consists of eleven agents. The next step
is to extend learning to consider all agents, and to include methods that
select informative state variables and features.
## References
* [1] Degris, T., Modayily, J.: Scaling-up Knowledge for a Cognizant Robot. In Notes of the AAAI Spring Symposium on Designing Intelligent Robots: Reintegrating AI (2012)
* [2] Gabel, T., Lange, S., Lauer, M., Riedmiller, M.: Bridging the Gap: Learning in the Robocup Simulation and Midsize League. In: Proceedings of the 7th Portuguese Conference on Automatic Control (Controlo) (2006)
* [3] Gurzoni, Jr., J.A., Tonidandel, F., Bianchi, R.A.C.: Market-Based Dynamic Task Allocation using Heuristically Accelerated Reinforcement Learning. In: Proceedings of the 15th Portugese Conference on Progress in Artificial Intelligence. pp. 365–376. EPIA’11, Springer-Verlag, Berlin, Heidelberg (2011)
* [4] Kober, J., Bagnell, J.A.D., Peters, J.: Reinforcement Learning in Robotics: A Survey. International Journal of Robotics Research (July 2013)
* [5] Köse, H., Tatladede, U., Mericli, C., Kaplan, K., Akan, H.L.: Q-Learning Based Market-Driven Multi-Agent Collaboration in Robot Soccer. In: Proceedings of the Turkish Symposium on Artificial Intelligence and Neural Networks (TAINN). pp. 219–228 (2004)
* [6] MacAlpine, P., Urieli, D., Barrett, S., Kalyanakrishnan, S., Barrera, F., Lopez-Mobilia, A., Ştiurcă, N., Vu, V., Stone, P.: UT Austin Villa 2011: A Champion Agent in the RoboCup 3D Soccer Simulation Competition. In: Proceedings of 11th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2012) (June 2012)
* [7] Maei, H.R.: Gradient Temporal-Difference Learning Algorithms. PhD Thesis, University of Alberta. (2011), phD Thesis
* [8] Maei, H.R., Sutton, R.S.: GQ($\lambda$): A General Gradient Algorithm for Temporal-Difference Prediction Learning with Eligibility Traces. Proceedings of the 3rd Conference on Artificial General Intelligence (AGI-10) pp. 1–6 (2010)
* [9] Maei, H.R., Szepesvári, C., Bhatnagar, S., Sutton, R.S.: Toward Off-Policy Learning Control with Function Approximation. In: Proceedings of the 27th International Conference on Machine Learning (ICML 2010). pp. 719–726 (2010)
* [10] Modayil, J., White, A., Pilarski, P.M., Sutton, R.S.: Acquiring a Broad Range of Empirical Knowledge in Real Time by Temporal-Difference Learning. In: Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics (SMC). pp. 1903–1910. IEEE (2012)
* [11] Modayil, J., White, A., Sutton, R.S.: Multi-timescale Nexting in a Reinforcement Learning Robot. In: From Animals to Animats 12 - 12th International Conference on Simulation of Adaptive Behavior (SAB). pp. 299–309 (2012)
* [12] Ng, A.Y., Harada, D., Russell, S.J.: Policy Invariance Under Reward Transformations: Theory and Application to Reward Shaping. In: Proceedings of the Sixteenth International Conference on Machine Learning (ICML). pp. 278–287. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (1999)
* [13] Pilarski, P., Dawson, M., Degris, T., Carey, J., Sutton, R.: Dynamic Switching and Real-Time Machine Learning for Improved Human Control of Assistive Biomedical Robots. In: 4th IEEE RAS EMBS International Conference on Biomedical Robotics and Biomechatronics (BioRob). pp. 296 –302 (June 2012)
* [14] Riedmiller, M., Gabel, T.: On Experiences in a Complex and Competitive Gaming Domain: Reinforcement Learning Meets RoboCup. In: Third IEEE Symposium on Computational Intelligence and Games. pp. 17–23. IEEE (2007)
* [15] Riedmiller, M., Gabel, T., Hafner, R., Lange, S.: Reinforcement Learning for Robot Soccer. Autonomous Robots 27, 55–73 (July 2009)
* [16] Seekircher, A., Abeyruwan, S., Visser, U.: Accurate Ball Tracking with Extended Kalman Filters as a Prerequisite for a High-Level Behavior with Reinforcement Learning. In: The 6th Workshop on Humanoid Soccer Robots at Humanoid Conference, Bled (Slovenia) (2011)
* [17] Stoecker, J., Visser, U.: Roboviz: Programmable Visualization for Simulated Soccer. In: Röfer, T., Mayer, N.M., Savage, J., Saranli, U. (eds.) RoboCup. pp. 282–293. Lecture Notes in Computer Science, Springer (2011)
* [18] Stone, P., Sutton, R.S., Kuhlmann, G.: Reinforcement Learning for RoboCup-Soccer Keepaway. Adaptive Behavior 13(3), 165–188 (2005)
* [19] Stone, P., Veloso, M.: Layered Learning. In: Proceedings of the Eleventh European Conference on Machine Learning. pp. 369–381. Springer Verlag (1999)
* [20] Stone, P., Veloso, M.: Task Decomposition, Dynamic Role Assignment, and Low-Bandwidth Communication for Real-Time Strategic Teamwork. Artificial Intelligence 110(2), 241–273 (June 1999)
* [21] Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. MIT Press (1998)
* [22] Sutton, R.S., Modayil, J., Delp, M., Degris, T., Pilarski, P.M., White, A., Precup, D.: Horde: A Scalable Real-Time Architecture for Learning Knowledge from Unsupervised Sensorimotor Interaction. In: The 10th International Conference on Autonomous Agents and Multiagent Systems. pp. 761–768. AAMAS ’11, International Foundation for Autonomous Agents and Multiagent Systems (2011)
* [23] Sutton, R.S., Szepesvári, C., Maei, H.R.: A Convergent O(N) Algorithm for Off-Policy Temporal-Difference Learning with Linear Function Approximation. In: Advances in Neural Information Processing Systems (NIPS). pp. 1609–1616. MIT Press (2008)
* [24] Thomas Degris, Martha White, R.S.S.: Off-Policy Actor-Critic. In: Proceedings of the Twenty-Ninth International Conference on Machine Learning (ICML) (2012)
* [25] White, A., Modayil, J., Sutton, R.: Scaling Life-Long Off-Policy Learning. In: International Conference on Development and Learning and Epigenetic Robotics (ICDL), 2012 IEEE. pp. 1–6 (2012)
|
arxiv-papers
| 2014-02-18T23:01:13 |
2024-09-04T02:49:58.408997
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Saminda Abeyruwan and Andreas Seekircher and Ubbo Visser",
"submitter": "Saminda Abeyruwan",
"url": "https://arxiv.org/abs/1402.4525"
}
|
1402.4568
|
# Linear Receding Horizon Control with Probabilistic System Parameters
Raktim Bhattacharya James Fisher Aerospace Engineering, Texas A&M
University, USA. Raytheon Missile Systems.
###### Abstract
In this paper we address the problem of designing receding horizon control
algorithms for linear discrete-time systems with parametric uncertainty. We do
not consider presence of stochastic forcing or process noise in the system. It
is assumed that the parametric uncertainty is probabilistic in nature with
known probability density functions. We use generalized polynomial chaos
theory to design the proposed stochastic receding horizon control algorithms.
In this framework, the stochastic problem is converted to a deterministic
problem in higher dimensional space. The performance of the proposed receding
horizon control algorithms is assessed using a linear model with two states.
## 1 Introduction
Receding horizon control (RHC), also known as model predictive control (MPC),
has been popular in the process control industry for several years Qin and
Badgwell (1996); Bemporad and Morari (1999), and recently gaining popularity
in aerospace applications, see Bhattacharya et al. (2002). It is based on the
idea of repetitive solution of an optimal control problem and updating states
with the first input of the optimal command sequence. The repetitive nature of
the algorithm results in a state dependent feedback control law. The
attractive aspect of this method is the ability to incorporate state and
control limits as constraints in the optimization formulation. When the model
is linear, the optimization problem is quadratic if the performance index is
expressed via a $\mathcal{L}_{2}$-norm, or linear if expressed via a
$\mathcal{L}_{1}/\mathcal{L}_{\infty}$-norm. Issues regarding feasibility of
online computation, stability and performance are largely understood for
linear systems and can be found in refs. Kwon (1994); Bitmead et al. (1990).
For nonlinear systems, stability of RHC methods is guaranteed by Primbs
(1999); Jadbabaie et al. (1999), by using an appropriate control Lyapunov
function . For a survey of the state-of-the-art in nonlinear receding horizon
control problems the reader is directed to Mayne et al. (2000a).
Traditional RHC laws perform best when modeling error is small. Fisher et al.
(2007) has shown that system uncertainty can lead to significant oscillatory
behavior and possibly instabilty. Furthermore, Grimm et al. (2004) showed that
in the presence of modeling uncertainty RHC strategy may not be robust with
RHC designs. Many approaches have been taken to improve robustness of RHC
strategy in the presence of unknown disturbances and bounded uncertainty, see
work of Raković et al. (2006); Lee and Yu (1997); Kouvaritakis et al. (2000);
Mayne et al. (2000b). These approaches involve the computation of a feedback
gain to ensure robustness. The difficulty with this approach is that, even for
linear systems, the problem becomes difficult to solve, as the unknown
feedback gain transforms the quadratic programming problem into a nonlinear
programming problem.
In this paper we address the problem of RHC design for linear systems with
probabilistic uncertainty in system parameters. Parametric uncertainty arises
in systems when the physics governing the system is known and the system
parameters are either not known precisely or are expected to vary in the
operational lifetime. Such uncertainty also occurs when system models are
build from experimental data using system identification techniques. As a
result of experimental measurements, the values of the parameters in the
system model have a range of variations with quantifiable likelihood of
occurrence. In either case, the range of variation of these parameters and the
likelihood of their occurrence are assumed to be known and it is desired to
design controllers that achieve specified performance for these variations.
While the area of robust RHC is not new, approaching the problem from a
stochastic standpoint is only recently receiving attention, for example van
Hessem and Bosgra (2002); Batina et al. (2002). These approaches however
suffered from either computational complexity, high degree of conservativeness
or do not address closed-loop stability. The key difficulty in stochastic RHC
is the propagation of uncertainty over the prediction horizon. More recently,
Cannon et al. (2009) avoid this difficulty by using an autonomous augmented
formulation of the prediction dynamics. Constraint satisfaction and stability
is achieved in Cannon et al. (2009) by extending ellipsoid invariance theory
to invariance with a given probability. The cost function minimized was the
expected value of a quadratic function of random state and control
trajectories. Additionally, the uncertainty in the system parameters were
assumed to have normal distribution.
This paper presents formulation of robust RHC design problems in polynomial
chaos framework, where parametric uncertainty can be governed by any
probability density function. In this approach the solution, not the dynamics,
of the random process is approximated using a series expansion. It is assumed
that the random process to be controlled has finite second moment, which is
the assumption of the polynomial chaos framework. The polynomial chaos based
approach predicts the propagation of uncertainty more accurately, is
computationally cheaper than methods based on Monte-Carlo or series
approximation of the dynamics, and is less conservative than the invariance
based methods.
The paper is organized as follows. We first present a brief introduction to
polynomial chaos and its application in transforming linear stochastic
dynamics to linear deterministic dynamics in higher dimensional state-space.
Next stability of stochastic linear dynamics in the polynomial chaos framework
is presented. This is followed by formulation of RHC design for discrete-time
stochastic linear systems. Stability of the proposed RHC algorithm is then
analyzed. The paper concludes with numerical examples that assesses the
performance of the proposed method.
## 2 Background on Polynomial Chaos
Recently, use of polynomial chaos to study stochastic differential equations
is gaining popularity. It is a non-sampling based method to determine
evolution of uncertainty in dynamical system, when there is probabilistic
uncertainty in the system parameters. Polynomial chaos was first introduced by
Wiener (1938) where Hermite polynomials were used to model stochastic
processes with Gaussian random variables. It can be thought of as an extension
of Volterra’s theory of nonlinear functionals Schetzen (2006) for stochastic
systems Ghanem and Spanos (1991). According to Cameron and Martin (1947) such
an expansion converges in the $\mathcal{L}_{2}$ sense for any arbitrary
stochastic process with finite second moment. This applies to most physical
systems. Xiu and Karniadakis (2002) generalized the result of Cameron-Martin
to various continuous and discrete distributions using orthogonal polynomials
from the so called Askey-scheme Askey and Wilson (1985) and demonstrated
$\mathcal{L}_{2}$ convergence in the corresponding Hilbert functional space.
This is popularly known as the generalized polynomial chaos (gPC) framework.
The gPC framework has been applied to applications including stochastic fluid
dynamics Hou et al. (2006),stochastic finite elements Ghanem and Spanos
(1991), and solid mechanics Ghanem and Red-Horse (1999). It has been shown in
Xiu and Karniadakis (2002) that gPC based methods are computationally far
superior than Monte-Carlo based methods. However, application of gPC to
control related problems has been surprisingly limited and is only recently
gaining popularity. See Prabhakar et al. (2008); Fisher and Bhattacharya
(2008a, b) for control related application of gPC theory.
### 2.1 Wiener-Askey Polynomial Chaos
Let $(\Omega,\mathcal{F},P)$ be a probability space, where $\Omega$ is the
sample space, $\mathcal{F}$ is the $\sigma$-algebra of the subsets of
$\Omega$, and $P$ is the probability measure. Let
$\Delta(\omega)=(\Delta_{1}(\omega),\cdots,\Delta_{d}(\omega)):(\Omega,\mathcal{F})\rightarrow(\mathbb{R}^{d},\mathcal{B}^{d})$
be an $\mathbb{R}^{d}$-valued continuous random variable, where
$d\in\mathbb{N}$, and $\mathcal{B}^{d}$ is the $\sigma$-algebra of Borel
subsets of $\mathbb{R}^{d}$. A general second order process
$X(\omega)\in\mathcal{L}_{2}(\Omega,\mathcal{F},P)$ can be expressed by
polynomial chaos as
$X(\omega)=\sum_{i=0}^{\infty}x_{i}\phi_{i}({\Delta}(\omega)),$ (1)
where $\omega$ is the random event and $\phi_{i}({\Delta}(\omega))$ denotes
the gPC basis of degree $p$ in terms of the random variables $\Delta(\omega)$.
The functions $\\{\phi_{i}\\}$ are a family of orthogonal basis in
$\mathcal{L}_{2}(\Omega,\mathcal{F},P)$ satisfying the relation
$\langle\phi_{i}\phi_{j}\rangle:=\int_{\mathcal{D}_{\Delta(\omega)}}{\phi_{i}\phi_{j}w(\Delta(\omega))\,d\Delta(\omega)}=h_{i}^{2}\delta_{ij}$
(2)
where $\delta_{ij}$ is the Kronecker delta, $h_{i}$ is a constant term
corresponding to
$\int_{\mathcal{D}_{\Delta}}{\phi_{i}^{2}w(\Delta)\,d\Delta}$,
$\mathcal{D}_{\Delta}$ is the domain of the random variable $\Delta(\omega)$,
and $w(\Delta)$ is a weighting function. Henceforth, we will use $\Delta$ to
represent $\Delta(\omega)$. For random variables $\Delta$ with certain
distributions, the family of orthogonal basis functions $\\{\phi_{i}\\}$ can
be chosen in such a way that its weight function has the same form as the
probability density function $f(\Delta)$. When these types of polynomials are
chosen, we have $f(\Delta)=w(\Delta)$ and
$\int_{\mathcal{D}_{\Delta}}{\phi_{i}\phi_{j}f(\Delta)\,d\Delta}=\mathbf{E}\left[\phi_{i}\phi_{j}\right]=\mathbf{E}\left[\phi_{i}^{2}\right]\delta_{ij},$
(3)
where $\mathbf{E}\left[\cdot\right]$ denotes the expectation with respect to
the probability measure $dP(\Delta(\omega))=f(\Delta(\omega))d\Delta(\omega)$
and probability density function $f(\Delta(\omega))$. The orthogonal
polynomials that are chosen are the members of the Askey-scheme of polynomials
(Askey and Wilson (1985)), which forms a complete basis in the Hilbert space
determined by their corresponding support. Table 1 summarizes the
correspondence between the choice of polynomials for a given distribution of
$\Delta$. See Xiu and Karniadakis (2002) for more details.
Random Variable $\Delta$ | $\phi_{i}(\Delta)$ of the Wiener-Askey Scheme
---|---
Gaussian | Hermite
Uniform | Legendre
Gamma | Laguerre
Beta | Jacobi
Table 1: Correspondence between choice of polynomials and given distribution
of $\Delta(\omega)$ Xiu and Karniadakis (2002).
### 2.2 Approximation of Stochastic Linear Dynamics Using Polynomial Chaos
Expansions
Here we derive a generalized representation of the deterministic dynamics
obtained from the stochastic system by approximating the solution with
polynomial chaos expansions.
Define a linear discete-time stochastic system in the following manner
$x(k+1,\Delta)=A(\Delta)x(k,\Delta)+B(\Delta)u(k,\Delta),$ (4)
where $x\in\mathbb{R}^{n},u\in\mathbb{R}^{m}$. The system has probabilistic
uncertainty in the system parameters, characterized by $A(\Delta),B(\Delta)$,
which are matrix functions of random variable
$\Delta\equiv\Delta(\omega)\in\mathbb{R}^{d}$ with certain stationary
distributions. Due to the stochastic nature of $(A,B)$, the system trajectory
$x(k,\Delta)$ will also be stochastic.
By applying the Wiener-Askey gPC expansion of finite order to
$x(k,\Delta),A(\Delta)$ and $B(\Delta)$, we get the following approximations,
$\displaystyle\hat{x}(k,\Delta)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{p}x_{i}(k)\phi_{i}(\Delta),\,x_{i}(k)\in\mathbb{R}^{n}$
(5) $\displaystyle\hat{u}(k,\Delta)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{p}u_{i}(k)\phi_{i}(\Delta),\,u_{i}(k)\in\mathbb{R}^{m}$
(6) $\displaystyle\hat{A}(\Delta)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{p}A_{i}\phi_{i}(\Delta),\,A_{i}=\frac{\langle
A(\Delta),\phi_{i}(\Delta)\rangle}{\langle\phi_{i}(\Delta)^{2}\rangle}\in\mathbb{R}^{n\times
n}$ (7) $\displaystyle\hat{B}(\Delta)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{p}B_{i}\phi_{i}(\Delta),\,B_{i}=\frac{\langle
B(\Delta),\phi_{i}(\Delta)\rangle}{\langle\phi_{i}(\Delta)^{2}\rangle}\in\mathbb{R}^{n\times
m}.$ (8)
The inner product or ensemble average $\langle\cdot,\cdot\rangle$, used in the
above equations and in the rest of the paper, utilizes the weighting function
associated with the assumed probability distribution, as listed in table 1.
The number of terms $p$ is determined by the dimension $d$ of $\Delta$ and the
order $r$ of the orthogonal polynomials $\\{\phi_{k}\\}$, satisfying
$p+1=\frac{(d+r)!}{d!r!}$. The $n(p+1)$ time varying coefficients,
$\\{x_{i}(k)\\};k=0,\cdots,p$, are obtained by substituting the approximated
solution in the governing equation (eqn.(4)) and conducting Galerkin
projection on the basis functions $\\{\phi_{k}\\}_{k=0}^{p}$, to yield
$n(p+1)$ deterministic linear system of equations, which given by
$\mathbf{X}(k+1)=\mathbf{A}\mathbf{X}(k)+\mathbf{B}\mathbf{U}(k),$ (9)
where
$\displaystyle\mathbf{X}(k)$ $\displaystyle=$
$\displaystyle[x_{0}(k)^{T}\;x_{1}(k)^{T}\;\cdots x_{p}(k)^{T}]^{T},$ (10)
$\displaystyle\mathbf{U}(k)$ $\displaystyle=$
$\displaystyle[u_{0}(k)^{T}\;u_{1}(k)^{T}\;\cdots u_{p}(k)^{T}]^{T}.$ (11)
Matrices $\mathbf{A}\in\mathbb{R}^{n(p+1)\times n(p+1)}$ and
$\mathbf{B}\in\mathbb{R}^{n(p+1)\times m}$ are defined as
$\displaystyle\mathbf{A}$ $\displaystyle=$ $\displaystyle(W\otimes
I_{n})^{-1}\left[\begin{array}[]{c}H_{A}(E_{0}\otimes I_{n})\\\ \vdots\\\
H_{A}(E_{p}\otimes I_{n})\end{array}\right],$ (16) $\displaystyle\mathbf{B}$
$\displaystyle=$ $\displaystyle(W\otimes
I_{n})^{-1}\left[\begin{array}[]{c}H_{B}(E_{0}\otimes I_{m})\\\ \vdots\\\
H_{B}(E_{p}\otimes I_{m})\end{array}\right],$ (20)
where $H_{A}=\left[A_{0}\,\cdots\,A_{p}\right]$,
$H_{B}=\left[B_{0}\,\cdots\,B_{p}\right]$,
$W=diag(\langle\phi_{0}^{2}\rangle,\cdots,\langle\phi_{p}^{2}\rangle)$, and
$E_{i}=\left[\begin{array}[]{ccc}\langle\phi_{i},\phi_{0},\phi_{0}\rangle&\cdots&\langle\phi_{i},\phi_{0},\phi_{p}\rangle\\\
\vdots&&\vdots\\\
\langle\phi_{i},\phi_{p},\phi_{0}\rangle&\cdots&\langle\phi_{i},\phi_{p},\phi_{p}\rangle\end{array}\right],$
with $I_{n}$ and $I_{m}$ as the identity matrix of dimension $n\times n$ and
$m\times m$ respectively. It can be easily shown that
$\mathbf{E}\left[x(k)\right]=x_{0}(k)$, or
$\mathbf{E}\left[x(k)\right]=\left[I_{n}\;0_{n\times np}\right]\mathbf{X}(k).$
Therefore, transformation of a stochastic linear system with
$x\in\mathbb{R}^{n},u\in\mathbb{R}^{m}$, with $p^{th}$ order gPC expansion,
results in a deterministic linear system with increased dimensionality equal
to $n(p+1)$.
## 3 Stochastic Receding Horizon Control
Here we develop a RHC methodology for stochastic linear systems similar to
that developed for deterministic systems, presented by Goodwin et al. (2005).
Let $x(k,\Delta)$ be the solution of the system in eqn.(4) with control
$u(k,\Delta)$. Consider the following optimal control problem defined by,
$\displaystyle V_{N}^{*}=\min\,V_{N}(\\{x(k+1,\Delta)\\},\\{u(k,\Delta)\\})$
(21) $\displaystyle{\rm subject\,to:}$ $\displaystyle
x(k+1,\Delta)=A(\Delta)x(k,\Delta)+B(\Delta)u(k,\Delta),$ (22)
$\displaystyle\textrm{Initial Condition: }x(0,\Delta);$ (23)
$\displaystyle\mu(u(k,\Delta))\in\mathbb{U}\subset\mathbb{R}^{m},$ (24)
$\displaystyle\mu(x(k,\Delta))\in\mathbb{X}\subset\mathbb{R}^{n},$ (25)
$\displaystyle\mu(x(N,\Delta))\in\mathbb{X}_{f}\subset\mathbb{X},$ (26)
for $k=0,\cdots,N-1$; where $N$ is the horizon length, $\mathbb{U}$ and
$\mathbb{X}$ are feasible sets for $u(k,\Delta)$ and $x(k,\Delta)$ with
respect to control and state constraints. $\mu(\cdot)$ represents moments
based constraints on state and control.The set $\mathbb{X}_{f}$ is a terminal
constraint set. The cost function $V_{N}$ is given by
$\begin{split}&V_{N}=\sum_{k=1}^{N}\mathbf{E}\left[x^{T}(k,\Delta)Qx(k,\Delta)+\right.\\\
&\left.u^{T}(k-1,\Delta)Ru(k-1,\Delta)\right]+C_{f}(x(N),\Delta),\end{split}$
(27)
where $C_{f}(x(N),\Delta)$ is a terminal cost function, and $Q=Q^{T}>0$,
$R=R^{T}>0$ are matrices with appropriate dimensions.
### 3.1 Control Structure
Here we consider the control structure,
$u(k,\Delta)=\bar{u}(k)+K(k)\left(x(k,\Delta)-\mathbf{E}\left[x(k,\Delta)\right]\right),$
(28)
where $\bar{u}(k)$, and $K(k)$ are unknown deterministic quantities. This is
similar to that proposed by Primbs et al.Primbs and Sung (2009) and enables us
to regulate the mean trajectory using open loop control and deviations about
the mean using a state-feedback control.
In terms of gPC coefficients, the system dynamics in eqn.(4) with the first
control structure is given by eqn.(9). The system dynamics in term of the gPC
expansions, with the second control structure is given by
$\mathbf{X}(k+1)=(\mathbf{A}+\mathbf{B}(\mathbf{M}\otimes
K(k)))\mathbf{X}(k)+\mathbf{B}\bar{U}(k),$ (29)
where $\bar{U}(k)=[1\,\,0_{1\times p}]^{T}\otimes\bar{u}(k)$ and
$\mathbf{M}=\left[\begin{array}[]{cc}0&0_{1\times p}\\\ 0_{p\times
1}&I_{p\times p}\end{array}\right]$.
### 3.2 Cost Functions
Here we derive the cost function in eqn.(27) is derived in terms of the gPC
coefficients $\mathbf{X}$ and $\mathbf{U}$. For scalar $x$, the quantity
$\mathbf{E}\left[x^{2}\right]$ in terms of its gPC expansions is given by
$\mathbf{E}\left[x^{2}\right]=\sum_{i=0}^{p}\sum_{j=0}^{p}x_{i}x_{j}\int_{\mathcal{D}_{\Delta}}\phi_{i}\phi_{j}fd\Delta=\mathbf{x}^{T}W\mathbf{x},$
(30)
where $\mathcal{D}_{\Delta}$ is the domain of $\Delta$ , $x_{i}$ are the gPC
expansions of $x$, $f\equiv f(\Delta)$ is the probability distribution of
$\Delta$. Here we use the notation $\mathbf{x}$ to represent the gPC state
vector for scalar $x$. The expression $\mathbf{E}\left[x^{2}\right]$ can be
generalized for $x\in\mathbb{R}^{n}$ where $\mathbf{E}\left[x^{T}Qx\right]$ is
given by
$\mathbf{E}\left[x^{T}Qx\right]=\mathbf{X}^{T}(W\otimes Q)\mathbf{X}.$ (31)
The expression for the cost function in eqn.(27), in terms of gPC states and
control is
$\begin{split}&V_{N}=\sum_{k=0}^{N-1}[\mathbf{X}^{T}(k)\bar{Q}\mathbf{X}(k)+\\\
&(\bar{U}^{T}(k)+\mathbf{X}^{T}(k)(\mathbf{M}\otimes
K^{T}(k)))\bar{R}(\bar{U}(k)+\\\ &(\mathbf{M}\otimes
K(k)))\mathbf{X}(k))]+C_{f}(x(N),\Delta),\end{split}$ (32)
where $\bar{Q}=W\otimes Q$ and $\bar{R}=W\otimes R$.
In deterministic RHC, the terminal cost is the cost-to-go from the terminal
state to the origin by the local controller Goodwin et al. (2005). In the
stochastic setting, a local controller can be synthesized using methods
presented in our previous work Fisher and Bhattacharya (2008a). The cost-to-go
from a given stochastic state variable $x(N,\Delta)$ can then be written as
$C_{f}(x(N),\Delta)=\mathbf{X}^{T}(N)P\mathbf{X}(N),$ (33)
where $\mathbf{X}(N)$ are gPC states corresponding to $x(N,\Delta)$ and
$P=P^{T}>0$ is a $n(p+1)\times n(p+1)$-dimensional matrix, obtained from the
synthesis of the terminal control law Fisher and Bhattacharya (2008a). In the
current stochastic RHC literature, the terminal cost function has been defined
on the expected value of the final state Lee and Cooley (1998); de la Penad et
al. (2005); Primbs and Sung (2009); Bertsekas (2005) or using a combination of
mean and variance Darlington et al. (2000); Nagy and Braatz (2003). The
terminal cost function in eqn.(33) is more general than the terminal cost
functions used in the literature because it penalizes all the moments of the
random variable $x(N,\Delta)$, as they are functions of $\mathbf{X}(N)$. This
can be shown as follows.
To avoid tensor notation and without loss of generality, we consider
$x(k,\Delta)\in\mathbb{R}$ and let
$\mathbf{X}(k)=[x_{0}(k),x_{1}(k),\cdots,x_{p}(k)]^{T}$ be the gPC expansion
of $x(k,\Delta)$. The $p^{th}$ moment in terms of $x_{i}(k)$ are then given by
$\begin{split}&m_{p}(k)=\sum_{i_{1}=0}^{P}\cdots\sum_{{i_{p}}=0}^{P}x_{i_{1}}(k)\cdots
x_{i_{p}}(k)\int_{\mathcal{D}_{\Delta}}\phi_{i_{1}}(\Delta)\cdots\\\
&\phi_{i_{p}}(\Delta)f(\Delta)d\Delta.\end{split}$ (34)
Thus, minimizing $C_{f}(x(N),\Delta)$ in eqn.(33) minimizes all moments of
$x(N,\Delta)$. Consequently, constraining the probability density function of
$x(N,\Delta)$.
### 3.3 State and Control Constraints
In this section we present the state and control constraints for the receding
horizon policy.
#### 3.3.1 Expectation Based Constraints
Here we first consider constraints of the following form,
$\displaystyle\mathbf{E}\left[x(k,\Delta)^{T}H_{x}x(k,\Delta)+G_{x}x(k,\Delta)\right]$
$\displaystyle\leq$ $\displaystyle\alpha_{i,x},$ (35)
$\displaystyle\mathbf{E}\left[u(k,\Delta)^{T}H_{u}u(k,\Delta)+G_{u}u(k,\Delta)\right]$
$\displaystyle\leq$ $\displaystyle\alpha_{i,u},$ (36)
for $k=0\ldots N$. These constraints are on the expected value of the
quadratic functions. Thus, instead of requiring that the constraint be met for
all trajectories, they instead imply that the constraints should be satisfied
on average. These constraints can be expressed in terms of the gPC states as
$\displaystyle\mathbf{X}(k)^{T}\bar{H}_{x}\mathbf{X}(k)+\bar{G}_{x}\mathbf{X}(k)$
$\displaystyle\leq$ $\displaystyle\alpha_{i,x},$ (37)
$\displaystyle\mathbf{U}(k)^{T}\bar{H}_{u}\mathbf{U}(k)+\bar{G}_{u}\mathbf{U}(k)$
$\displaystyle\leq$ $\displaystyle\alpha_{i,u},$ (38)
where $\bar{H}_{x}=W\otimes H_{x}$, $\bar{H}_{u}=W\otimes H_{u}$,
$\bar{G}_{x}=G_{x}\left[I_{n}\;0_{n\times np}\right]$, and
$\bar{G}_{u}=G_{u}\left[I_{n}\;0_{n\times np}\right]$.
#### 3.3.2 Variance Based Constraints
In many practical applications, it may be desirable to constrain the second
moment of the state trajectories, either at each time step or at final time.
One means of achieving this is to use a constraint of the form
$\mathbf{Tr}\left[\mathbf{E}\left[(x(k)-\mathbf{E}\left[x(k)\right])(x(k)-\mathbf{E}\left[x(k)\right])^{T}\right]\right]\leq\alpha_{\sigma^{2}}.$
(39)
For scalar $x$, the variance $\sigma^{2}(x)$ in terms of the gPC expansions
can be shown to be
$\sigma^{2}=\mathbf{E}\left[x-\mathbf{E}\left[x\right]\right]^{2}=\mathbf{E}\left[x^{2}\right]-\mathbf{E}\left[x\right]^{2}=\mathbf{x}^{T}W\mathbf{x}-\mathbf{E}\left[x\right]^{2},$
where
$\begin{split}&\mathbf{E}\left[x\right]=\mathbf{E}\left[\sum_{i=0}^{p}x_{i}\phi_{i}\right]=\sum_{i=0}^{p}x_{i}\mathbf{E}\left[\phi_{i}\right]=\sum_{i=0}^{p}x_{i}\int_{\mathcal{D}_{\Delta}}\phi_{i}fd\Delta\\\
&=\mathbf{x}^{T}F,\end{split}$
and $F=\left[\begin{array}[]{cccc}1\;0\;\cdots\;0\end{array}\right]^{T}$.
Therefore, $\sigma^{2}$ for scalar $x$ can be written in a compact form as
$\sigma^{2}=\mathbf{x}^{T}(W-FF^{T})\mathbf{x}.$ (40)
In order to represent the covariance for $x\in\mathbb{R}^{n}$, in terms of the
gPC states, let us define $\Phi=[\phi_{0}\cdots\phi_{p+1}]^{T}$ and write
$G=\int_{\mathcal{D}_{\Delta}}\Phi\Phi^{T}fd\Delta$. Let us represent a sub-
vector of $\mathbf{X}$, defined by elements $n_{1}$ to $n_{2}$, as
$X_{n_{1}\cdots n{2}}$, where $n_{1}$ and $n_{2}$ are positive integers. Let
us next define matrix $M_{\mathbf{X}}$, with subvectors of $\mathbf{X}$, as
$M_{\mathbf{X}}=[\mathbf{X}_{1\cdots n}\;\mathbf{X}_{n+1\cdots
2n}\;\cdots\mathbf{X}_{np+1\cdots n(p+1)}]$. For $x\in\mathbb{R}^{n}$, it can
be shown that
$\mathbf{E}\left[x\right]=(F\otimes I_{n})\mathbf{X},$ (41)
and the covariance can then be shown to be
$\textbf{Cov}(x)=M_{\mathbf{X}}GM_{\mathbf{X}}^{T}-(F\otimes
I_{n})\mathbf{X}\mathbf{X}^{T}(F^{T}\otimes I_{n}).$ (42)
The trace of the covariance matrix $\textbf{Cov}(x)$ can then be written as
$\mathbf{Tr}\left[\textbf{Cov}(x)\right]=\mathbf{X}^{T}((W-FF^{T})\otimes
I_{n})\mathbf{X}.$
Therefore, a constraint of the type
$\mathbf{Tr}\left[\textbf{Cov}(x(k))\right]\leq\alpha_{\sigma^{2}}$
can be written in term of gPC states as
$\mathbf{X}^{T}Q_{\sigma^{2}}\mathbf{X}\leq\alpha_{\sigma^{2}},$ (43)
where $Q_{\sigma^{2}}=(W-FF^{T})\otimes I_{n}$.
## 4 Stability of the RHC Policy
Here we show the stability properties of the receding horizon policy when it
is applied to the system in eqn.(9). Using gPC theory we can convert the
underlying stochastic RHC formulation in $x(t,\Delta)$ and $u(t,\Delta)$ to a
deterministic RHC formulation in $\mathbf{X}(k)$ and $\mathbf{U}(k)$. The
stability of $\mathbf{X}(k)$ in an RHC setting, with suitable terminal
controller, can be proved using results by Goodwin et al. (2005), which shows
that $\lim_{k\rightarrow\infty}\mathbf{X}(k)\rightarrow 0$, when a receding
horizon policy is employed. To relate this result to the stability of
$x(k,\Delta)$, we first present the following known result in stochastic
stability in terms of the moments of $x(k,\Delta)$. For stochastic dynamical
systems in general, stability of moments is a weaker definition of stability
than the almost sure stability definition. However, the two definitions are
equivalent for linear autonomous systems (pg. 296, Khas’minskii (1969) also
pg. 349 Chen and Hsu (1995)). Here we present the definition of asymptotic
stability in the $p^{th}$ moment for discrete-time systems.
###### Definition 1
The zero equilibrium state is said to be stable in the $p^{th}$ moment if
$\forall\epsilon>0,\,\exists\delta>0$ such that
$\sup_{k\geq 0}\mathbf{E}\left[x(k,\Delta)^{p}\right]\leq\epsilon,\;\forall
x(0,\Delta):||x(0,\Delta)||\leq\delta,\forall\Delta\in\mathcal{D}_{\Delta}.$
(44)
###### Definition 2
The zero equilibrium state is said to be asymptotically stable in the $p^{th}$
moment if it is stable in $p^{th}$ moment and
$\lim_{k\rightarrow\infty}\mathbf{E}\left[x(k,\Delta)^{p}\right]=0,$ (45)
for all $x(0,\Delta)$ in the neighbourhood of the zero equilibrium.
###### Proposition 1
For the system in eqn.(4), $\lim_{k\rightarrow\infty}\mathbf{X}(k)\rightarrow
0$ is a sufficient condition for the asymptotic stability of the zero
equilibrium state, in all moments.
###### Proof 4.1.
To avoid tensor notation and without loss of generality, we consider
$x(k,\Delta)\in\mathbb{R}$ and let
$\mathbf{X}(k)=[x_{0}(k),x_{1}(k),\cdots,x_{p}(k)]^{T}$ be the gPC expansion
of $x(k,\Delta)$. The moments in terms of $x_{i}(k)$ are given by eqn.(34).
Therefore, if $\lim_{k\rightarrow\infty}\mathbf{X}(k)\rightarrow 0$
$\implies\lim_{k\rightarrow\infty}x_{i}(k)\rightarrow 0$. Consequently,
$\lim_{k\rightarrow\infty}m_{i}(k)\rightarrow 0$ for $i=1,2,\cdots$, and
eqn.(45) is satisfied. This completes the proof. $\square$
## 5 Numerical Example
Here we consider the following linear system, similar to that considered in
Primbs and Sung (2009),
$x(k+1)=(A+G(\Delta))x(k)+Bu(k)$ (46)
where
$A=\left[\begin{array}[]{cc}1.02&-0.1\\\
.1&.98\end{array}\right],\,B=\left[\begin{array}[]{c}0.1\\\
0.05\end{array}\right],\,G=\left[\begin{array}[]{cc}0.04&0\\\
0&0.04\end{array}\right]\Delta.$
The system in consideration is open-loop unstable and the uncertainty appears
linearly in the $G$ matrix. Here, $\Delta\in[-1,1]$ and is governed by a
uniform distribution, that doesn’t change with time. Consequently, Legendre
polynomials is used for gPC approximation and polynomials up to $4^{th}$ order
are used to formulate the control. Additionally, we assume that there is no
uncertainty in the initial condition. The expectation based constraint is
imposed on $x(k,\Delta))$ as
$\mathbf{E}\left[\;[1\;\;0]x(k,\Delta)\;\right]\geq-1,$
which in terms of the gPC states, this corresponds to
$\left[\begin{array}[]{cc}1&\mathbf{0}_{1\times
2p+1}\end{array}\right]\mathbf{X}(k)\geq-1.$
The terminal controller is designed using probabilistic LQR design techniques
described by Fisher and Bhattacharya (2008a). The cost matrices used to
determine the terminal controller are
$Q=\left[\begin{array}[]{cc}2&0\\\ 0&5\end{array}\right],\,R=1.$
Figure (1) illustrates the performance of the proposed RHC policy The
resulting optimization problem is a nonlinear programming problem which has
been solved using MATLAB’s fmincon(...) function. From the figure, we see that
the constraint on the expected value of $x_{1}$ has been satisfied and the RHC
algorithm was able to stabilize the system. These plots have been obtained
using $4^{th}$ order gPC approximation of the stochastic dynamics.
## 6 Summary
In this paper we present a RHC strategy for linear discrete time systems with
probabilistic system parameters. We have used the polynomial chaos framework
to design stochastic RHC algorithms in an equivalent deterministic setting.
The controller structure has an open loop component that controls the mean
behavior of the system, and a state-feedback component that controls
deviations about the mean trajectory. This controller structure results in a
polynomial optimization problem with polynomial constraints that is solved in
the general nonlinear programming framework. Theoretical guarantees for the
stability of the proposed algorithm has also been presented. Performance of
the RHC algorithm has been assessed using a two dimensional dynamical system.
Figure 1: State trajectories with expectation constraints.
## References
* Askey and Wilson [1985] R. Askey and J. Wilson. Some basic hypergeometric polynomials that generalize jacobi polynomials. _Memoirs Amer. Math. Soc._ , 319, 1985.
* Batina et al. [2002] I. Batina, A. A. Stoorvogel, and S. Weiland. Optimal control of linear, stochastic systems with state and input constraints. _Proceedings of the 41st IEEE Conference on Decision and Control_ , 2:1564– 1569, 2002.
* Bemporad and Morari [1999] A. Bemporad and M. Morari. Robust model predictive control: A survey. Technical report, Automatic Control Laboratory, Swiss Federal Institute of Technology (ETH), Physikstrasse 3, CH-8092 Zürich, Switzerland, www.control.ethz.ch, 1999.
* Bertsekas [2005] D. Bertsekas. Dynamic programming and suboptimal control: A survey from ADP to MPC. _European Journal of Control_ , 11(4-5):310–334, 2005.
* Bhattacharya et al. [2002] Raktim Bhattacharya, Gary J. Balas, M. Alpay Kaya, and Andy Packard. Nonlinear receding horizon control of an f-16 aircraft. _Journal of Guidance, Control, and Dynamics_ , 25(5):924–931, 2002.
* Bitmead et al. [1990] R.R. Bitmead, M. Gevers, and V. Wertz. _Adaptive Optimal Control: The Thinking Man’s GPC_. International Series in Systems and Control Engineering. Prentice Hall, Englewood Cliffs, NJ, 1990.
* Cameron and Martin [1947] R. H. Cameron and W. T. Martin. The orthogonal development of non-linear functionals in series of fourier-hermite functionals. _The Annals of Mathematics_ , 48(2):385–392, 1947\.
* Cannon et al. [2009] M. Cannon, B. Kouvaritakis, and X. Wu. Model predictive control for systems with stochastic multiplicative uncertainty and probabilistic constraints. _Automatica_ , 45(1):167 – 172, 2009.
* Chen and Hsu [1995] By G. Chen and S. H. Hsu. _Linear Stochastic Control Systems_. CRC Press, 1995.
* Darlington et al. [2000] J. Darlington, CC Pantelides, B. Rustem, and BA Tanyi. Decreasing the sensitivity of open-loop optimal solutions in decision making under uncertainty. _European Journal of Operational Research_ , 121(2):343–362, 2000.
* de la Penad et al. [2005] DM de la Penad, A. Bemporad, and T. Alamo. Stochastic programming applied to model predictive control. In _44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05_ , pages 1361–1366, 2005.
* Fisher and Bhattacharya [2008a] J. Fisher and R. Bhattacharya. On stochastic LQR design and polynomial chaos. In _American Control Conference, 2008_ , pages 95–100, 2008a.
* Fisher and Bhattacharya [2008b] J. Fisher and R. Bhattacharya. Optimal Trajectory Generation with Probabilistic System Uncertainty Using Polynomial Chaos. _In Press Journal of Dynamic Systems, Measurement and Control_ , 2008b.
* Fisher et al. [2007] James Fisher, Raktim Bhattacharya, and S. R. Vadali. Spacecraft momentum management and attitude control using a receding horizon approach. In _Proceedings of the 2007 AIAA Guidance, Navigation, and Control Conference and Exhibit_ , Hilton Head, SC, August 2007. AIAA.
* Ghanem and Red-Horse [1999] Roger Ghanem and John Red-Horse. Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach. _Phys. D_ , 133(1-4):137–144, 1999. ISSN 0167-2789. http://dx.doi.org/10.1016/S0167-2789(99)00102-5.
* Ghanem and Spanos [1991] Roger G. Ghanem and Pol D. Spanos. _Stochastic Finite Elements: A Spectral Approach_. Springer-Verlag Inc., New York, NY, 1991. ISBN 0-387-97456-3.
* Goodwin et al. [2005] G.C. Goodwin, M. Seron, and J. De Dona. _Constrained control and estimation: an optimisation approach_. Springer, 2005.
* Grimm et al. [2004] Gene Grimm, Michael J. Messina, Sezai E. Tuna, and Andrew R. Teel. Examples when nonlinear model predictive control is nonrobust. _Automatica_ , 40:1729–1738, 2004.
* Hou et al. [2006] Thomas Y. Hou, Wuan Luo, Boris Rozovskii, and Hao-Min Zhou. Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics. _J. Comput. Phys._ , 216(2):687–706, 2006. ISSN 0021-9991. http://dx.doi.org/10.1016/j.jcp.2006.01.008.
* Jadbabaie et al. [1999] A. Jadbabaie, J. Yu, and J. Hauser. Stabilizing receding horizon control of nonlinear systems: A control lyapunov function approach. _Proceedings of the 1999 American Control Conference_ , 3:1535–1539, 1999.
* Khas’minskii [1969] R. Z. Khas’minskii. _Stability of Systems of Differential Equations in the Presence of Random Disturbances (in Russian)_. Nauka, Moscow, 1969.
* Kouvaritakis et al. [2000] B. Kouvaritakis, J. A. Rossiter, and J. Schuurmans. Efficient robust predictive control. _IEEE Transactions on Automatic Control_ , 45(8):1545–1549, 2000.
* Kwon [1994] W.H. Kwon. Advances in predictive control: Theory and application. _1st Asian Control Conference, Tokyo_ , 1994.
* Lee and Yu [1997] J. H. Lee and Zhenghong Yu. Worst-case formulations of model predictive control for systems with bounbded parameters. _Automatica_ , 33(5):763–781, 1997.
* Lee and Cooley [1998] JH Lee and BL Cooley. Optimal feedback control strategies for state-space systems with stochastic parameters. _IEEE Transactions on Automatic Control_ , 43(10):1469–1475, 1998.
* Mayne et al. [2000a] D. Mayne, J. Rawlings, C. Rao, and P. Scokaert. Constrained model predictive control, stability and optimality. _Automatica_ , 36:789–814, 2000a.
* Mayne et al. [2000b] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and PO Scokaert. Constrained model predictive control: Stability and optimality. _AUTOMATICA-OXFORD-_ , 36:789–814, 2000b.
* Nagy and Braatz [2003] Z.K. Nagy and R.D. Braatz. Robust nonlinear model predictive control of batch processes. _AIChE Journal_ , 49(7):1776–1786, 2003.
* Prabhakar et al. [2008] A. Prabhakar, J. Fisher, and R. Bhattacharya. Polynomial Chaos Based Analysis of Probabilistic Uncertainty in Hypersonic Flight Dynamics. _submitted AIAA Journal of Guidance, Control, and Dynamics_ , 2008\.
* Primbs [1999] J.A. Primbs. _Nonlinear Optimal Control: A Receding Horizon Approach_. PhD thesis, California Institute of Technology, Pasadena, CA, 1999.
* Primbs and Sung [2009] J.A. Primbs and C.H. Sung. Stochastic receding horizon control of constrained linear systems with state and control multiplicative noise. _Automatic Control, IEEE Transactions on_ , 54(2):221–230, 2009.
* Qin and Badgwell [1996] S.J. Qin and T. Badgwell. An overview of industrial model predictive control technology. _AIChE Symposium Series_ , 93:232–256, 1996.
* Raković et al. [2006] S. V. Raković, A. R. Teel, D. Q. Mayne, and A. Astolfi. Simple robust control invariant tubes for some classes of nonlinear discrete time systems. In _Proceedings of the 45th IEEE Conference on Decision and Control_ , pages 6397–6402, San Diego, CA, December 2006. IEEE.
* Schetzen [2006] M. Schetzen. _The Volterra and Wiener Theories of Nonlinear Systems_. Krieger Pub., 2006.
* van Hessem and Bosgra [2002] D. H. van Hessem and O. H. Bosgra. A conic reformulation of model predictive control including bounded and stochastic disturbances and input constraints. In _Proceedings of the 2002 Conference on Decision and Control_ , volume 4, pages 4643–4648, Las Vegas, NV, December 2002. IEEE.
* Wiener [1938] N. Wiener. The homogeneous chaos. _American Journal of Mathematics_ , 60(4):897–936, 1938.
* Xiu and Karniadakis [2002] Dongbin Xiu and George Em Karniadakis. The wiener–askey polynomial chaos for stochastic differential equations. _SIAM J. Sci. Comput._ , 24(2):619–644, 2002\. ISSN 1064-8275.
|
arxiv-papers
| 2014-02-19T07:09:10 |
2024-09-04T02:49:58.419625
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Raktim Bhattacharya and James Fisher",
"submitter": "Raktim Bhattacharya",
"url": "https://arxiv.org/abs/1402.4568"
}
|
1402.4601
|
# Effective representations of Path Semigroups
Love Forsberg
###### Abstract.
We give a formula which determines the minimal effective dimensions of path
semigroups and truncated path semigroups over an uncountable field of
characteristic zero.
## 1\. Introduction and preliminaries
Let $S$ be a semigroup and $\Bbbk$ a fixed field. In the paper [MS] Mazorchuk
and Stienberg addressed the question of dermining the so-called minimal
effective dimension $\mathrm{eff.dim}_{\Bbbk}(S)$ of $S$ over $\Bbbk$, that is
the minimal $m$ (a positive integer or infinity) for which there is an
injective homomorphism from $S$ to the semigroup $\mathrm{Mat}_{m\times
m}(\Bbbk)$ of all $m\times m$ matrices with coefficients in $\Bbbk$. If $S$ is
finite, it is clear that $\mathrm{eff.dim}_{\Bbbk}(S)<\infty$, more precisely,
$\mathrm{eff.dim}_{\Bbbk}(S)\leq|S|+1$, which is the dimension of the regular
representation of the semigroup $S^{1}$ obtained from $S$ by formally
adjoining an identity element $1$. An effective representation of a semigroup
$S$ with $\mathrm{eff.dim}_{\Bbbk}(S)=m$ is an injective homomorphism
$S\to\mathrm{Mat}_{m\times m}(\Bbbk)$.
One of the examples considered in [MS] was that of truncated path semigroups
which we now define. Let $Q=(Q_{0},Q_{1},h,t)$ be a quiver, where $Q_{0}$ is
the set of vertices, $Q_{1}$ is the set of arrows, $h:Q_{1}\to Q_{0}$ is the
function assigning to each arrow its head and $t:Q_{1}\to Q_{0}$ is the
function assigning to each arrow its tail. Denote by $\mathcal{P}$ the set of
all oriented paths in $Q$ (including the trivial path $\varepsilon_{x}$ at
each vertex $x\in Q_{0}$ and the zero path $\mathtt{z}$). Then $\mathcal{P}$
has the natural structure of a semigroup under the usual concatenation of
oriented paths (in case two paths cannot be concatenated, their product is
postulated to be the path $\mathtt{z}$ and the latter is the zero element of
$\mathcal{P}$). The semigroup $\mathcal{P}$ is called the path semigroup of
$Q$. The semigroup $\mathcal{P}$ is finite if and only if the quiver $Q$ is
finite and does not have oriented cycles. We write $\mathcal{P}^{*}$ for the
set $\mathcal{P}\setminus\\{\mathtt{z}\\}$ of non-zero paths.
There is a unique function $\mathfrak{l}:\mathcal{P}^{*}\to\\{0,1,2,\dots\\}$,
called the path length, having the properties that the length of each arrow is
$1$ and $\mathfrak{l}(pq)=\mathfrak{l}(p)+\mathfrak{l}(q)$ whenever
$p,q,pq\in\mathcal{P}^{*}$ (note that $\mathfrak{l}(\varepsilon_{x})=0$ for
each $x\in Q_{0}$). Elements of $Q_{1}$ thus can be identified with all paths
of length $1$. Let $J\subset\mathcal{P}$ be the two-sided ideal of
$\mathcal{P}$ generated by $Q_{1}$. For every $N\in\\{1,2,3,\dots\\}$ we
define the _truncated path semigroup_ as $\mathcal{P}_{N}:=\mathcal{P}/J^{N}$.
Note that the semigroup $\mathcal{P}_{N}$ is finite whenever $Q$ is. In [MS,
Subsection 8.1] one finds a formula for the effective dimension of
$\mathcal{P}_{N}$ in the case when every vertex in $Q$ appears in some
oriented cycle (or loop). The aim of this paper is give a formula for the
effective dimension of $\mathcal{P}_{N}$ for any $Q$.
From now on we assume that $Q$ is finite and set $n=|Q_{0}|$. Consider the
path algebra $\Bbbk[Q]$ of $Q$ which is the $\mathtt{z}$-reduced semigroup
algebra of $\mathcal{P}$ over $\Bbbk$. The algebra $\Bbbk[Q]$ is unital with
unit element $1=\sum_{x\in Q_{0}}\varepsilon_{x}$ where $\varepsilon_{x}$ are
pairwise orthogonal idempotents. This implies that any $\Bbbk[Q]$-module $V$
splits as a direct sum of vector spaces
$V=\bigoplus_{x\in Q_{0}}V_{x},$
where $V_{x}=\varepsilon_{x}V$. Given a $\Bbbk[Q]$-module $V$, we set
$D_{x}=\dim(V_{x})$. Each arrow $\alpha:x\to y$ acts as zero on all $V_{z}$
such that $z\neq x$ and hence is uniquely determined by the induced linear map
from $V_{x}$ to $V_{y}$. Hence we can make the following convention: A matrix
representation of $\mathcal{P}$ or $\mathcal{P}_{N}$ is an assignment to each
arrow $\alpha\in Q_{1}$ a $D_{y}\times D_{x}$-matrix with coefficients in
$\Bbbk$ representing the action of $\alpha$ in fixed bases of $V_{x}$ and
$V_{y}$. For more details on representations of quivers we refer the reader to
[GR].
Note that $\mathcal{P}_{N}$-modules are exactly $\mathcal{P}$-modules
annihilated by $J^{N}$. We will usually denote $\mathcal{P}$\- or
$\mathcal{P}_{N}$-modules by $V$ and the corresponding representation by $R$.
## 2\. Path semigroups
In [MS, Section 8] one finds formulae for effective dimension of path
semigroups over $\Bbbk$ in case of acyclic $Q$ and algebraically closed
$\Bbbk$. In this section we determine the effective dimension of path
semigroups for all finite quivers at the expense of assuming $\Bbbk$ to be a
field containing an infinite purely transcendental extension of its prime
subfield (for example, $\mathbb{R}\subset\Bbbk$). We denote by $\mathbb{N}$
the set of positive integers and by $\mathbb{N}_{0}$ the set of non-negative
integers.
For $x\in Q_{0}$ let $\mathcal{P}_{x}$ denote the set of all paths in
$\mathcal{P}^{*}$ which start and terminate at $x$. Then $\mathcal{P}_{x}$ is
a subsemigroup of $\mathcal{P}$, in fact, $\mathcal{P}_{x}$ is a monoid with
identity $\varepsilon_{x}$. Denote by $A$ the set of all vertices $x\in Q_{0}$
for which $\mathcal{P}_{x}$ is not commutative and set $B:=Q_{0}\setminus A$.
###### Lemma 1.
Let $x\in Q_{0}$.
1. $($i$)$
The monoid $\mathcal{P}_{x}$ has a unique irreducible generating system (which
we denote by $M_{x}$).
2. $($ii$)$
The monoid $\mathcal{P}_{x}$ is free over $M_{x}$.
3. $($iii$)$
The monoid $\mathcal{P}_{x}$ is commutative if and only if $|M_{x}|\leq 1$.
###### Proof.
Define $N_{i}$ and $\tilde{N}_{i}$ for $i\in\mathbb{N}$ recursively as
follows:
* •
$N_{1}$ is the set of all paths of length $1$ in $\mathcal{P}_{x}$;
* •
$\tilde{N}_{1}$ is the subsemigroup of $\mathcal{P}_{x}$ generated by $N_{1}$;
* •
$N_{2}$ is the set of all paths of length $2$ in
$\mathcal{P}_{x}\setminus\tilde{N}_{1}$;
* •
$\tilde{N}_{2}$ is the subsemigroup of $\mathcal{P}_{x}$ generated by
$N_{1}\cup N_{2}$;
* •
$N_{3}$ is the set of all paths of length $3$ in
$\mathcal{P}_{x}\setminus\tilde{N}_{2}$;
* •
$\tilde{N}_{3}$ is the subsemigroup of $\mathcal{P}_{x}$ generated by
$N_{1}\cup N_{2}\cup N_{3}$;
* •
and so on.
From this definition it is clear that the set $M_{x}=N_{1}\cup N_{2}\cup\dots$
is a generating system of $\mathcal{P}_{x}$ (as a monoid) and that it is
included in every generating system of $\mathcal{P}_{x}$ (as a monoid). Claim
(i) follows.
Assume that $\mathcal{P}_{x}$ is not free over $M_{x}$. Then there exist
$a_{1},a_{2},\dots,a_{k},b_{1},b_{2},\dots,b_{l}\in M_{x}$ such that
$a_{1}a_{2}\cdots a_{k}=b_{1}b_{2}\cdots b_{l}$. This can be chosen such that
$(k,l)$ is minimal possible with respect to the lexicographic order (note
that, obviously, both $k,l>0$). If $\mathfrak{l}(a_{k})<\mathfrak{l}(b_{l})$,
then $b_{l}=ta_{k}$ for some $t\in\mathcal{P}_{x}$ which contradicts $b_{l}\in
M_{x}$. Therefore this case is not possible. Similarly
$\mathfrak{l}(a_{k})>\mathfrak{l}(b_{l})$ is not possible. This means that
$\mathfrak{l}(a_{k})=\mathfrak{l}(b_{l})$ and hence $a_{k}=b_{l}$. Therefore
$a_{1}a_{2}\dots a_{k-1}=b_{1}b_{2}\dots b_{l-1}$ which contradicts minimality
of $(k,l)$. This proves claim (ii) and claim (iii) follows directly from claim
(ii). ∎
A generator of $\mathcal{P}_{x}$ will be called a minimal oriented cycle
starting at $x$
###### Lemma 2.
Let $V$ be an effective $S$-module and $x\in A$. Then $D_{x}\geq 2$ and
$\mathrm{eff.dim}_{\Bbbk}(S)\geq 2|A|+|B|=|A|+n.$
###### Proof.
It is clear that $D_{x}\geq 1$ for all $x\in Q_{0}$ (for otherwise the actions
of $\varepsilon_{x}$ and $\mathtt{z}$ would coincide). Assume $x\in A$ and
$D_{x}=1$. Then $\mathcal{P}_{x}$ acts effectively acts on the $1$-dimensional
vector space $V_{x}$. However, the semigroup of linear endomorphisms of
$V_{x}$ is commutative (as $V_{x}$ is one dimensional), while
$\mathcal{P}_{x}$ is not (as $x\in A$), a contradiction. This implies that
$D_{x}>1$ for $x\in A$, that is $D_{x}\geq 2$. As $|A|+|B|=|Q_{0}|=n$, the
claim of the lemma follows. ∎
To prove that the bound given by Lemma 2 is sharp, we will need the following
construction: For a fixed positive integer $k$ consider the alphabet
$A=\\{a_{1},a_{2},\dots,a_{k}\\}$ and the free monoid $A^{*}$ of all finite
words over $A$ with respect to concatenation of words. Let $\Bbbk_{k}$ be the
purely transcendental extension of its prime subfield $\mathbb{K}$ with basis
$\mathbf{B}:=\\{\tau_{i},\eta_{i},\zeta_{i}\,|\,i=1,2,\dots,k\\}$.
###### Lemma 3.
There is a unique representation $R:A^{*}\to\mathrm{Mat}_{2\times
2}(\Bbbk_{k})$ such that
$R(a_{i})=\left(\begin{array}[]{cc}\tau_{i}&\eta_{i}\\\
0&\zeta_{i}\end{array}\right),$
moreover, the map $R$ is injective.
###### Proof.
Existence and uniqueness of $R$ follows from the fact that $A^{*}$ is free
over $A$. For $a_{i_{1}}a_{i_{2}}\dots a_{i_{l}}\in A^{*}$ the coefficient in
the first row and second column of the matrix $R(a_{i_{1}}a_{i_{2}}\dots
a_{i_{l}})$ equals
$\sum_{i=1}^{l}\tau_{1}\tau_{2}\cdots\tau_{i-1}\eta_{i}\zeta_{i+1}\zeta_{i+2}\cdots\zeta_{l}.$
This uniquely determines the sequence $i_{1},i_{2},\dots,i_{l}$ and the claim
about injectivity follows. ∎
For a fixed $Q$ let $\Bbbk_{Q}$ be the purely transcendental extension of its
prime subfield $\mathbb{K}$ with basis
$\mathbf{B}:=\\{\tau_{\alpha},\eta_{\alpha},\zeta_{\alpha}\,|\,\alpha\in
Q_{1}\\}$. For $\alpha\in Q_{1}$ set
$\mathbf{B}_{\alpha}:=\\{\tau_{\alpha},\eta_{\alpha},\zeta_{\alpha}\\}$. For
$x,y\in Q_{0}$ write $x\sim y$ if $x=y$ or there is an oriented path from $x$
to $y$ as well as an oriented path from $y$ to $x$.
###### Lemma 4.
Let $x,y\in Q_{0}$ be such that $x\sim y$. Then $x\in A$ if and only if $y\in
A$.
###### Proof.
As $x\sim y$, there exist paths $\omega_{xy}:x\to y$ and $\omega_{yx}:y\to x$.
Assume $x\in A$. Let $\omega_{1}$ and $\omega_{2}$ be two different minimal
oriented cycles in $\mathcal{P}_{x}$. Then $\omega_{xy}\omega_{1}\omega_{yx}$
and $\omega_{xy}\omega_{2}\omega_{yx}$ are two noncommuting elements in
$\mathcal{P}_{x}$, proving $y\in A$. Claim now follows by symmetry. ∎
The following is our first main result.
###### Theorem 5.
Let $Q$ be a finite quiver and $\mathcal{P}$ the corresponding path semigroup.
Then
$\mathrm{eff.dim}_{\Bbbk_{Q}}(\mathcal{P})=|A|+n.$
###### Proof.
We only need to show that the bound given by Lemma 2 is sharp. To do this we
construct an effective matrix representation $V$ of $\mathcal{P}$ as follows:
set
$D_{x}=\dim(V_{x})=\begin{cases}2,&x\in A;\\\ 1,&x\in B;\end{cases}$
with a fixed basis in each $V_{x}$. To each $\alpha\in Q_{1}$ we assign a
$\Bbbk_{Q}$-matrix with $D_{h(\alpha)}$ rows and $D_{t(\alpha)}$ columns by
the following rule:
* •
If $h(\alpha_{i}),t(\alpha_{i})\in A$, then we assign to $\alpha$ the matrix
$\left(\begin{array}[]{cc}\tau_{\alpha}&\eta_{\alpha}\\\
0&\zeta_{\alpha}\end{array}\right)$.
* •
If $h(\alpha_{i})\in A$ and $t(\alpha_{i})\in B$, then we assign to $\alpha$
the matrix $(\tau_{\alpha}\,\,\,\zeta_{\alpha})$.
* •
If $h(\alpha_{i})\in B$ and $t(\alpha_{i})\in A$, then we assign to $\alpha$
the matrix $\left(\begin{array}[]{c}\tau_{\alpha}\\\
\zeta_{\alpha}\end{array}\right)$.
* •
If $h(\alpha_{i}),t(\alpha_{i})\in B$, then we assign to $\alpha$ the matrix
$(\tau_{\alpha})$.
Finally, to each $\varepsilon_{x}$ we assign the identity matrix of size
$D_{x}$ and to each path of length more than $1$ the corresponding product of
the matrices assigned to arrows which this path consists of. It is obvious
that this gives a well-defined representation of $\mathcal{P}$. It remains to
show that this representation sends different elements of $\mathcal{P}$ to
different linear operators.
Let $x,y\in Q_{0}$ and $\omega$ be an oriented paths from $x$ to $y$. Directly
from the above construction it follows that each coefficient of the matrix
representing $\omega$ is a homogeneous polynomial in elements from
$\mathbf{B}$. If this coefficient is nonzero (which is the case for all
diagonal entries and all entries above the diagonal, in case the latter
exist), this polynomial has degree $\mathfrak{l}(\omega)$ and depends on at
least one element from $\\{\tau_{\alpha},\eta_{\alpha},\zeta_{\alpha}\\}$ for
each arrow $\alpha$ in $\omega$.
Let $x,y\in Q_{0}$ and $\omega,\omega^{\prime}$ be two paths from $x$ to $y$.
We have to show that $\omega$ and $\eta$ are represented by different linear
operators. From the previous paragraph it follows that this is clear in the
case when $\omega$ and $\omega^{\prime}$ have different lengths and in the
case when one of these paths contains an arrow which is not contained in the
other path.
Assume that there exists $x,y\in Q_{0}$ and $\omega,\omega^{\prime}$ two
different paths from $x$ to $y$ such that $R(\omega)=R(\omega^{\prime})$.
Without loss of generality we may assume that the pair
$(\mathfrak{l}(x),\mathfrak{l}(y))$ is minimal with respect to the
lexicographic order.
Write $\omega$ in the form
$\omega_{1}\beta_{1}\omega_{2}\beta_{2}\cdots\omega_{k-1}\beta_{k-1}\omega_{k}$
where $\omega_{i}$ are (possibly trivial) paths inside an equivalence class of
the relation $\sim$ and $\beta_{i}$ are arrow between equivalence classes.
From the above it then follows that $\omega^{\prime}$ can similarly be written
as
$\omega^{\prime}_{1}\beta_{1}\omega^{\prime}_{2}\beta_{2}\cdots\omega^{\prime}_{k-1}\beta_{k-1}\omega^{\prime}_{k}$.
Assume $\omega_{1}$ is a trivial path. Then $\omega$ has no arrow starting
from the $\sim$-equivalence class of $h(\beta_{1})$. From the above we get
that $\omega^{\prime}$ has no arrow starting from the $\sim$-equivalence class
of $h(\beta_{1})$ and hence $\omega^{\prime}_{1}$ is a trivial path as well.
We claim that this implies
(2.1)
$R(\omega_{2}\beta_{2}\cdots\omega_{k-1}\beta_{k-1}\omega_{k})=R(\omega^{\prime}_{2}\beta_{2}\cdots\omega^{\prime}_{k-1}\beta_{k-1}\omega^{\prime}_{k})$
which would then contradict the minimality of
$(\mathfrak{l}(x),\mathfrak{l}(y))$. To prove (2.1), the only non-trivial case
to consider is when $R(\beta_{1})$ is not injective, that is $t(\beta_{1})\in
A$ and $h(\beta_{1})\in B$. Assume
$R(\omega_{2}\beta_{2}\cdots\omega_{k-1}\beta_{k-1}\omega_{k})=\left(\begin{array}[]{cc}a&b\\\
0&c\end{array}\right)\neq\left(\begin{array}[]{cc}a^{\prime}&b^{\prime}\\\
0&c^{\prime}\end{array}\right)=R(\omega^{\prime}_{2}\beta_{2}\cdots\omega^{\prime}_{k-1}\beta_{k-1}\omega^{\prime}_{k}).$
Then none of $a,b,c,a^{\prime},b^{\prime},c^{\prime}$ depends on
$\tau_{\beta_{1}}$ or $\zeta_{\beta_{1}}$ and hence we have
$R(\omega)=\left(\begin{array}[]{cc}\tau_{\beta_{1}}&\zeta_{\beta_{1}}\end{array}\right)\left(\begin{array}[]{cc}a&b\\\
0&c\end{array}\right)=\left(\begin{array}[]{cc}\tau_{\beta_{1}}a&\tau_{\beta_{1}}b+\zeta_{\beta_{1}}c\end{array}\right)\neq\\\
\neq\left(\begin{array}[]{cc}\tau_{\beta_{1}}a^{\prime}&\tau_{\beta_{1}}b^{\prime}+\zeta_{\beta_{1}}c^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}\tau_{\beta_{1}}&\zeta_{\beta_{1}}\end{array}\right)\left(\begin{array}[]{cc}a^{\prime}&b^{\prime}\\\
0&c^{\prime}\end{array}\right)=R(\omega^{\prime}),$
a contradiction.
Therefore $\omega_{1}$ is non-trivial and thus $R(\omega_{1})$ is invertible
by construction and Lemma 4 as both the starting point and the ending point of
$\omega_{1}$ belong to the same $\sim$-equivalence class. Multiplying with
$R(\omega_{1})^{-1}$ we get
$R(\beta_{1}\omega_{2}\beta_{2}\cdots\omega_{k-1}\beta_{k-1}\omega_{k})=R(\omega_{1})^{-1}R(\omega^{\prime}_{1})R(\beta_{1}\omega^{\prime}_{2}\beta_{2}\cdots\omega^{\prime}_{k-1}\beta_{k-1}\omega^{\prime}_{k}).$
Note that the left hand side does not depend on elements in
$\mathbf{B}_{\alpha}$ for $\alpha$ occurring in $\omega_{1}$. Hence the right
hand side does not depend on these elements either which forces the injective
linear map $R(\omega_{1})^{-1}R(\omega^{\prime}_{1})$ to be the identity
linear map as the image of the linear map
$R(\beta_{1}\omega^{\prime}_{2}\beta_{2}\cdots\omega^{\prime}_{k-1}\beta_{k-1}\omega^{\prime}_{k})$
is nonzero by construction. Therefore in this case we have the equality
$R(\omega_{1})=R(\omega^{\prime}_{1})$. If
$\beta_{1}\omega_{2}\beta_{2}\cdots\omega_{k-1}\beta_{k-1}\omega_{k}$ or
$\beta_{1}\omega^{\prime}_{2}\beta_{2}\cdots\omega^{\prime}_{k-1}\beta_{k-1}\omega^{\prime}_{k}$
is non-trivial, the above gives
$R(\beta_{1}\omega_{2}\beta_{2}\cdots\omega_{k-1}\beta_{k-1}\omega_{k})=R(\beta_{1}\omega^{\prime}_{2}\beta_{2}\cdots\omega^{\prime}_{k-1}\beta_{k-1}\omega^{\prime}_{k})$
which contradicts minimality of $(\mathfrak{l}(x),\mathfrak{l}(y))$. Hence
$\omega=\omega_{1}$ and $\omega^{\prime}=\omega^{\prime}_{1}$.
If $x\in A$, then $R(\omega)=R(\omega^{\prime})$ implies
$\omega=\omega^{\prime}$ by Lemma 3, a contradiction. Therefore $x,y\in B$. In
this case there is a unique minimal oriented cycle $q$ from $x$ to $x$ ($q$
may be a trivial path) and hence a unique path $p$ of minimal length from $x$
to $y$ (for otherwise, composing two different such minimal paths from $x$ to
$y$ with a minimal path from $y$ to $x$ we would get two minimal oriented
cycles from $x$ to $x$). Any path from $x$ to $y$ has thus the form $pq^{l}$
for some positive integer $l$. In particular, two paths of the same length
from $x$ to $y$ must coincide, which contradicts our choice of $\omega$ and
$\omega^{\prime}$. This final contradiction completes the proof of the
theorem. ∎
## 3\. Truncated path semigroups
As truncated path semigroups are obtained by adding some relations to usual
path semigroups, it is reasonable to expect that the effective dimension
increases, e.g. compare the statements of Theorem 5 above with the results of
[MS, Subsection 8.2].
Let $\Bbbk$ be any field, $N\in\mathbb{N}$ and $V$ a representation of
$\mathcal{P}_{N}$. For every $k\in\mathbb{N}_{0}$ let
$W^{(k)}=\mathrm{span}\\{\omega
V\,|\,\omega\in\mathcal{P},\,\,l(\omega)=k\\}$. By convention,
$\omega=\mathtt{z}$ when $\mathfrak{l}(\omega)\geq N$, which gives
$W^{(N)}=0$. Thus we get the chain of subspaces
$V=W^{(0)}\supset W^{(1)}\supset\cdots\supset W^{(N-1)}\supset W^{(N)}=0.$
For every $x\in Q_{0}$ set $W_{x}^{(k)}:=V_{x}\cap W^{(k)}$ and choose _some_
$V_{x}^{(k)}\subset W^{(k)}_{x}$ such that $W_{x}^{(k)}=V_{x}^{(k)}\oplus
W_{x}^{(k+1)}$. Set $\displaystyle V^{(k)}:=\bigoplus_{x\in
Q_{0}}V_{x}^{(k)}$. This gives the vector space decompositions
$V=\bigoplus_{i=0}^{N-1}V^{(i)}=\bigoplus_{x\in
Q_{0}}V_{x}=\bigoplus_{\begin{subarray}{c}0\leq i\leq N-1\\\ x\in
Q_{0}\end{subarray}}V_{x}^{(i)}.$
In any module $V$, let $D_{x}^{(i)}:=\dim(V_{x}^{(i)})$, which gives
$D_{x}=\sum_{i=0}^{N-1}D_{x}^{(i)}$. From the definition of $W^{(i)}$ for any
$\omega\in\mathcal{P}$ we have $\omega W^{(i)}\subset
W_{h(\omega)}^{(i+\mathfrak{l}(\omega))}$.
###### Lemma 6.
Let $x\in Q_{0}$ be such that there are paths $\omega_{l}$, $\omega_{r}$ and
$0\leq k<N$ such that $\mathfrak{l}(\omega_{l})=k$,
$\mathfrak{l}(\omega_{r})=N-1-k$ and $h(\omega_{l})=t(\omega_{r})=x$. Then
$D_{x}^{(k)}\geq 1$ for every effective $\mathcal{P}$-module $V$.
###### Proof.
Assume $D_{x}^{(k)}=0$, that is $V_{x}^{(k)}=0$, and let $y=t(\omega_{l})$ and
$z=h(\omega_{r})$. Then
$\omega_{l}(V)=\omega_{l}(V_{y})=\omega_{l}(W_{y}^{(0)})\subset
W_{x}^{(l(\omega_{l}))}=W_{x}^{(k)}=V_{x}^{(k)}\oplus
W_{x}^{(k+1)}=W_{x}^{(k+1)}\mathrm{\ and}$ $\omega_{r}(W_{x}^{(k+1)})\subset
W_{z}^{(k+1+l(\omega_{r}))}=W_{z}^{(k+1+N-1-k)}=W_{z}^{(N)}=0.$
Thus $\omega_{r}\omega_{l}(V)=0$ and $\omega_{r}\omega_{l}$ acts as
$\mathtt{z}$ on $V$ contradicting effectiveness. ∎
For $x\in Q_{0}$ define
$K(x):=\big{\\{}k\in\\{0,\cdots,N-1\\}\,|\,\text{ there are paths
}\omega_{l},\omega_{r}\text{ such that }\\\
\mathfrak{l}(\omega_{l})=k,\mathfrak{l}(\omega_{r})=N-1-k\text{ and
}h(\omega_{l})=t(\omega_{r})=x\big{\\}}.$
Set $B:=\\{x\in Q_{0}\,|\,K(x)=\varnothing\\}$ and $A:=Q_{0}\setminus B$. For
$x\in A$ set
$\underline{k}_{x}:=\min(K(x))\quad\text{ and
}\quad\overline{k}_{x}:=\max(K(x)).$
For $x\in Q_{0}$ define
$l_{x}^{-}:=\sup\\{\mathfrak{l}(\omega)\,|\,\omega\in\mathcal{P}\text{ and
}h(\omega)=x\\}\text{ and
}l_{x}^{+}:=\sup\\{\mathfrak{l}(\omega)\,|\,\omega\in\mathcal{P}\text{ and
}t(\omega)=x\\}.$
We are now ready to state our second main result.
###### Theorem 7.
Define
$d_{x}:=\min\big{\\{}l_{x}^{-}+1,l_{x}^{+}+1,N,\max\\{l_{x}^{-}+l_{x}^{+}+2-N,1\\}\big{\\}}$.
1. $($i$)$
For every effective $\mathcal{P}_{N}$-module $V$ over any field $\Bbbk$ we
have $D_{x}\geq d_{x}$.
2. $($ii$)$
If $\Bbbk$ has characteristic zero or is uncountable, then $D_{x}=d_{x}$ for
some effective $\mathcal{P}_{N}$-module (over $\Bbbk)$ and
$\mathrm{eff.dim}_{\Bbbk}(\mathcal{P}_{N})=\sum_{x\in Q_{0}}d_{x}$.
###### Proof.
First we prove claim (i). Let $x\in Q_{0}$. Then $x\in A$ or $x\in B$. In any
case, $D_{x}\geq|K(x)|$ by Lemma 6.
Assume first that $x\in A$. Then $K(x)\neq\varnothing$ and it suffices to show
that $|K(x)|\geq d_{x}$. As $K(x)\neq\varnothing$, there is some path of
length $N-1$ passing through $x$, which means that $l_{x}^{-}+l_{x}^{+}\geq
N-1$, in particular, $l_{x}^{-}+l_{x}^{+}-N+2\geq 1$ and thus
$\max\\{l_{x}^{-}+l_{x}^{+}+2-N,1\\}=l_{x}^{-}+l_{x}^{+}-N+2$.
Pick some paths $\omega_{-},\omega_{+}$ such that
$h(\omega_{-})=t(\omega_{+})=x$ and $l(\omega_{\pm})=\min(l_{x}^{\pm},N-1)$.
Let $\omega_{\min(l_{x}^{-},N-1)}$ be a path of length $N-1$ that starts with
$\omega_{-}$ and continues into $\omega_{+}$ (if needed). From Lemma 6 we get
$\min(l_{x}^{-},N-1)\in K(x)$. Now we repeat recursively the following
procedure as long as possible: Change $\omega_{k}$ to $\omega_{k-1}$ by
removing the tail arrow and adding a new head arrow from $\omega_{+}$. On each
step of this procedure we get a new $\omega_{k-1}$ with $k-1\in K(x)$. This
procedure can stop for two reasons:
* •
There are no more arrows from $\omega_{-}$ to remove.
* •
There are no more arrows from $\omega_{+}$ to add.
The first case (there are no more arrows from $\omega_{-}$ to remove) can only
happen if the latest $k-1$ created is equal to $0$. In this case
$K(x)\supset\\{0,1,\cdots,\min(l_{x}^{-},N-1)\\}$ and hence
$|K(x)|\geq\min(l_{x}^{-}+1,N)$ and we are done.
We split the second case (there are no more arrows from $\omega_{+}$ to add)
into two subcases. The first subcase is that
$\omega_{\min(l_{x}^{-},N-1)}=\omega_{-}$, that is $l_{x}^{-}\geq N-1$. In
this subcase we have $K(x)\supset\\{N-1,N-2,\cdots,N-1-\min(l_{x}^{+},N-1)\\}$
which implies that $|K(x)|\geq\min(l_{x}^{+}+1,N)$ and we are done.
The second subcase is when $\omega_{\min(l_{x}^{-},N-1)}\neq\omega_{-}$. In
this subcase we have $l_{x}^{-}<N-1$ and
$K(x)\supset T:=\\{l_{x}^{-},l_{x}^{-}-1,\cdots,N-1-\min(l_{x}^{+},N-1)\\}.$
Hence $|K(x)|\geq|T|=(l_{x}^{-}+\min(l_{x}^{+},N-1)+2-N)$. If
$\min(l_{x}^{+},N-1)=l_{x}^{+}$, this gives $|K(x)|\geq
l_{x}^{-}+l_{x}^{+}+2-N$ and we are done. If $\min(l_{x}^{+},N-1)=N-1$, this
gives $|K(x)|\geq l_{x}^{-}+1$ and we are done. This completes verification of
$D_{x}\geq|K(x)|\geq d_{x}$ for $x\in A$.
Assume now that $x\in B$. In this case $l_{x}^{-}+l_{x}^{+}+2-N\leq 0$ and
$d_{x}=1$. The fact that $D_{x}\geq 1$ is clear as $\varepsilon_{x}$ acts as
the identity on $V_{x}$ and this should be different from the action of
$\mathtt{z}$ which acts as zero. This completes the proof of claim (i) and
implies
$\mathrm{eff.dim}_{\Bbbk}(\mathcal{P}_{N})\geq\sum_{x\in Q_{0}}d_{x}.$
To prove claim (ii) we assume that $\Bbbk$ has characteristic zero or is
uncountable. We have to construct an effective representation $V$ such that
$D_{x}=d_{x}$ for every $x\in Q_{0}$. To do this we define the following:
* •
for $x\in A$ and $k\in K(x)$ let $V_{x}^{(k)}$ be the one-dimensional vector
space with basis $\\{v_{x}^{(k)}\\}$;
* •
for $x\in A$ and $k\not\in K(x)$ let $V_{x}^{(k)}$ be the zero vector space;
* •
for $x\in B$ let $V_{x}$ be the one-dimensional vector space with basis
$\\{v_{x}\\}$.
Set
$V:=\big{(}\bigoplus_{\begin{subarray}{c}0\leq i\leq N-1\\\ x\in
A\end{subarray}}V_{x}^{(i)}\big{)}\oplus\big{(}\bigoplus_{x\in
B}V_{x}\big{)}.$
Fix an injective map $(\alpha,k)\mapsto p_{\alpha,k}$ from the set of all
pairs $(\alpha,k)$ where $\alpha\in Q_{1}$ and $0\leq k\leq N$ to the set of
positive integer prime numbers if $\Bbbk$ has charachteristic 0. In case
$\Bbbk$ is uncountable we choose the codomain as a basis of a purely
transcendental extension over its prime subfield by sufficiently many base
elements. Define the action of $\mathcal{P}_{N}$ on $V$ as follows:
* •
the zero element of $\mathcal{P}_{N}$ acts as zero;
* •
$\varepsilon_{x}$ acts as the identity on $V_{x}$ and as zero on $V_{y}$,
$y\neq x$;
* •
for every arrow $\alpha:x\to y$ with $x,y\in A$ we have
$\alpha:v_{x}^{(N-1)}\to 0$ and for each $k\in K(x)$ we have
$\alpha:v_{x}^{(k)}\to p_{\alpha,k}v_{y}^{(j)}$, where
$j=\min\\{i\in\\{k+1,k+2,\dots,N-1\\}\,|\,V_{y}^{(i)}\neq 0\\};$
* •
for every arrow $\alpha:x\to y$ with $x\in A$ and $y\in B$ and for each $k\in
K(x)$ we have $\alpha:v_{x}^{(k)}\to p_{\alpha,k}v_{y}$;
* •
for every arrow $\alpha:x\to y$ with $x\in B$ and $y\in A$ we have
$\alpha:v_{x}\to p_{\alpha,0}v_{y}^{(\underline{k}_{y})}$;
* •
for every arrow $\alpha:x\to y$ with $x,y\in B$ we have $\alpha:v_{x}\to
p_{\alpha,0}v_{y}$;
* •
actions of paths of length greater than one are defined using composition of
maps.
Assume that $x,y\in A$ and $k\in K(x)$. Let $\omega_{-}$ and $\omega_{+}$ be
two paths such that $\mathfrak{l}(\omega_{-})=k$,
$\mathfrak{l}(\omega_{+})=N-1-k$ and $h(\omega_{-})=t(\omega_{+})=x$. Assume
further that there is an arrow $\alpha$ from $x$ to $y$. If $N-1-k\leq
l_{y}^{+}+1$, then without loss of generality we may assume that $\alpha$ is
the first arrow in $\omega_{+}$. In this case we directly get $k+1\in K(y)$.
If $l_{y}^{+}+1<N-1-k$, then any $k^{\prime}\in K(y)$ satisfies
$N-1-k^{\prime}\leq l_{y}^{+}<N-1-k$ which implies $k^{\prime}>k$. Since
$K(y)$ is not empty (as $y\in A$), we get that the set
$\\{i\in\\{k+1,k+2,\dots,N-1\\}|V_{y}^{(i)}\neq 0\\}$ is non-empty. Therefore
the above definitions make sense.
The only non-trivial relation to check is the fact that any path $\omega$ with
$\mathfrak{l}(\omega)\geq N$ acts as zero. From the definition of $B$ it
follows that each arrow in $\omega$ is an arrow between two vartices in $A$.
From the definition of the action we then see that
$\omega(V_{t(\omega)}^{(i)})\subset V^{(i+\mathfrak{l}(\omega))}_{h(\omega)}.$
This implies $\omega(V)\subset 0$ and thus $V$ is a $\mathcal{P}_{N}$-module.
It remains to show that our module is effective. For this we need to show that
paths of length at most $N-1$ act in a non-zero way and pairwise differently.
A path $\omega$ is said to be maximal if there is no arrow $\alpha$ such that
$\alpha\omega$ or $\omega\alpha$ is nonzero. Note that if a path $\omega$ acts
in a nonzero way, then $h(\omega)$ can be recovered as the unique $y$ such
that $\omega(V)\subset V_{y}$. Moreover, $t(\omega)$ can be recovered as the
unique $x$ such that $\omega(V_{x})\neq 0$. Thus if two different paths
$\omega_{1}$ and $\omega_{2}$ act equally and in a nonzero way, then they
share the same head and the same tail. Furtheremore, the action of each
maximal path $\omega_{l}\omega_{1}\omega_{r}$ coincides with the action of
$\omega_{l}\omega_{2}\omega_{r}$. Thus it suffices to show that all maximal
paths act nonzero and differently.
To simplify notation let
$\hat{v}_{x}:=\begin{cases}v_{x}^{(\underline{k}_{x})},&x\in A;\\\ v_{x},&x\in
B;\end{cases}\qquad\qquad\check{v}_{y}:=\begin{cases}v_{y}^{(\overline{k}_{x})},&y\in
A;\\\ v_{y},&y\in B.\end{cases}$
Let $\omega=\alpha_{N-1}\alpha_{N-2}\cdots\alpha_{2}\alpha_{1}$ be a path of
length $N-1$ and set $x_{i}:=h(\alpha_{i})=t(\alpha_{i+1})$ with
$x_{0}:=t(\alpha_{1})$. Then from Lemma 6 and our construction we get that
$V_{x_{i}}^{(i)}$ is nonzero for all $i$ and
$\omega(\hat{v}_{x})=p_{\alpha_{1},0}p_{\alpha_{2},1}\cdots
p_{\alpha_{N-1},N-2}\check{v}_{y}$. Injectivity of the map $(\alpha,k)\mapsto
p_{\alpha,k}$ guarantees that the coefficient at $\check{v}_{y}$ uniquely
determines the sequence
$(\alpha_{1},0),(\alpha_{2},1),\dots,(\alpha_{N-1},N-2)$ which uniquely
deretmines $\omega$.
Finally, assume that $\omega=\alpha_{k}\alpha_{k-1}\cdots\alpha_{2}\alpha_{1}$
for some $k<N-1$. Set $w_{0}=\hat{v}_{x}$ and
$w_{i}=\alpha_{i}\cdots\alpha_{2}\alpha_{1}(\hat{v}_{x})$ for $i=1,2,\dots,k$.
Let us prove that $w_{i}$ is nonzero for all $i=0,1,2,\dots,k$ by induction.
The basis is obvious. Assume $w_{i}\neq 0$. If $\alpha_{i+1}$ is adjacent to
at least one vertex in $B$, we have $\alpha_{i+1}(w_{i})\neq 0$ directly by
construction. Assume now that $\alpha_{i+1}$ is an arrow between two vertices
in $A$. By construction, the only basis element in $V_{t(\alpha_{i+1})}$ which
$\alpha_{i+1}$ annihilates is the one which is in the image of some path of
length $N-2$. We have $i<N-2$. Hence $\alpha_{i+1}w_{i}\neq 0$ if
$t(\alpha_{j})\in A$ for all $j\leq i$. Otherwise let $j$ be maximal such that
$j\leq i$ and $t(\alpha_{j})\in B$. Then, by construction,
$\alpha_{j}(w_{j-1})$ is a non-zero multiple of $\hat{v}_{h(\alpha_{j})}$,
which implies that $w_{i}$ is not in the image of a path of length $N-2$ and
therefore $\alpha_{i+1}(w_{i})\neq 0$ again. This shows that $\omega$ acts in
a nonzero way on $V$. As $\omega$ is a maximal path of length strictly less
than $N-1$, it is uniquely determined by the arrows it consists of.
Injectivity of the map $(\alpha,k)\mapsto p_{\alpha,k}$ thus implies that
$\omega$ is uniquely determined by the prime decomposition of the coefficients
in its matrix. This completes the proof. ∎
Theorem 7 implies the following stabilization property for
$\mathrm{eff.dim}_{\Bbbk}(\mathcal{P}_{N})$:
###### Corollary 8.
Assume that $\Bbbk$ has characteristic zero. Then there exist
$a,b\in\mathbb{N}_{0}$ such that
$\mathrm{eff.dim}_{\Bbbk}(\mathcal{P}_{N})=aN+b\quad\text{ for all }\quad
N\geq n.$
###### Proof.
For each $x$ the numbers $l_{x}^{-}$ and $l_{x}^{+}$ satisfy
$l_{x}^{-}+l_{x}^{+}\in\\{0,1,\cdots,n-1,\infty\\}$ as any path of length at
least $n$ must contain a subcycle. This means that we always have one of the
following three cases:
* •
Both $l_{x}^{-}$ and $l_{x}^{+}$ are finite, and thus $l_{x}^{-}+l_{x}^{+}\leq
n-1$. Then for all $N>n$ we have $l_{x}^{-}+l_{x}^{+}+2-N\leq 1$ and
$d_{x}=1$.
* •
Exactly one of $l_{x}^{-},l_{x}^{+}$ is finite. Then
$d_{x}=\min\\{l_{x}^{-},l_{x}^{+}\\}+1$ for all $N\geq n$.
* •
Both $l_{x}^{-},l_{x}^{+}$ are infinite. Then $d_{x}=N$ for all $N\geq 1$.
Therefore we can take $a$ to be the number of $x$ such that both
$l_{x}^{-},l_{x}^{+}$ are infinite. As $b$ we take the sum of $1$’s over all
$x$ such that both $l_{x}^{-}$ and $l_{x}^{+}$ are finite plus the sum of
$\min\\{l_{x}^{-},l_{x}^{+}\\}+1$ over all $x$ such that exactly one of
$l_{x}^{-}$ and $l_{x}^{+}$ is finite. The claim follows. ∎
From Corollary 8 it follows that to calculate
$\mathrm{eff.dim}_{\Bbbk}(\mathcal{P}_{N})$ for all $N\in\mathbb{N}$ it is
enough to consider the cases $N=1,2,\cdots,n,n+1$.
## 4\. Examples
### 4.1. Quivers with cycles at each vertex
Let $Q$ be a quiver in which every vertex is part of some (nontrivial) cycle
or loop. Then $\mathrm{eff.dim}_{\Bbbk}(\mathcal{P_{N}})=Nn$ for $\Bbbk$
uncountable or of characteristic 0. Proof: Let $x\in Q_{0}$. Then
$l_{x}^{-}=l_{x}^{+}=\infty$ and hence $d_{x}=N$. Sum over all vertices. This
result is similar to [MS, Theorem 31], but the set of fields $\Bbbk$ differ.
### 4.2. Quivers of type $A_{n}$
A quiver $Q$ is said to be of type $A_{n}$ if the underlying unoriented graph
is the Dynkin diagram $A_{n}$. Let $Q$ be of type $A_{n}$ and let
$(n_{1},n_{2},\cdots,n_{k})$ be the the number of vertices in the ordered
segments. Then
$\mathrm{eff.dim}_{\Bbbk}(\mathcal{P})=1+\sum_{N<n_{i}}(N(n_{i}+1-N)-1)+\sum_{n_{i}\leq
N}(n_{i}-1).$
Proof: Because local dimensions $d_{x}$ only depend on maximal paths in and
out of $x$, different ordered segments can be counted independently, if we
subtract the overlaps. Thus we need only to consider the case when $Q$ has one
ordered segment. For a quiver of type $A_{n}$ with only one ordered segment
(with vertices from $\mathbf{1}$ to $\mathbf{n}$) the picture is as follows,
when $N<n$. When $n\leq N$ each $V_{x}$ is one-dimensional.
|
---|---
$\textstyle{\mathbf{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{n-(N-1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{n}}$$\textstyle{V_{\mathbf{1}}^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{\mathbf{N}}^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{\mathbf{n-(N-1)}}^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{\mathbf{N}}^{(N-1)}}$$\textstyle{\cdots}$$\textstyle{V_{\mathbf{n-(N-1)}}^{(N-1)}}$$\textstyle{\cdots}$$\textstyle{V_{\mathbf{n}}^{(N-1)}}$
## References
* [MS] V. Mazorchuk, B. Steinberg. Effective dimension of finite semigroups. J. Pure Appl. Algebra 216 (2012), no. 12, 2737–2753.
* [GR] Gabriel, P.; Roiter, A. V. Representations of finite-dimensional algebras. With a chapter by B. Keller. Springer-Verlag, Berlin, 1997.
Department of Math., Uppsala University, Box 480, SE-751 06, Uppsala, Sweden;
e-mail: [email protected]
|
arxiv-papers
| 2014-02-19T09:35:18 |
2024-09-04T02:49:58.428569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Love Forsberg",
"submitter": "Love Forsberg",
"url": "https://arxiv.org/abs/1402.4601"
}
|
1402.4602
|
# New Quantitative Deformation Lemma and New Mountain Pass Theorem
Liang Ding1, 3, Fode Zhang2, and Shiqing Zhang1,
1Department of Mathematics and Yangtze Center of Mathematics,
Sichuan University
Chengdu 610064, People’s Republic of China
2Department of Mathematics
Kunming University of Science and Technology
Yunnan 650093, People’s Republic of China
3Department of Basis Education
Dehong Vocational College
Mangshi, Yunnan, 678400
People’s Republic of China
Email: [email protected](Liang Ding), Email: [email protected](Fode
Zhang)Corresponding author’s email: [email protected](Shiqing Zhang),
Abstract In this paper, we obtain a new quantitative deformation Lemma so that
we can obtain more critical points, especially for supinf critical value
$c_{1}$, $x=\varphi^{-1}(c_{1})$ is a new critical point. For $infmax$
critical value $c_{2}$, we can obtain two new critical points $x=0$ (valley
point) and $x=e$(peak point) ,comparing with Willem’s variant of the mountain
pass theorem of Ambrosetti-Rabinowitz,in which
$\varphi(e)\leq\varphi(0)<c_{2}$, but in our new mountain pass theorem,
$\varphi(e)=c_{2}$.
Key words Critical Points; Quantitative Deformation Lemma; Mountain Pass Lemma
2010 MR Subject Classification 47H10, 47J30, 39A10.
## 1 Introduction
In 1973, Ambrosetti and Rabinowitz [1] presented the famous Mountain Pass
Theorem. Later, there were many variants and generalizations([2]-[14]).
Specially, Willem [11] gave the Quantitative Deformation Lemma and the
corresponding mountain pass theorem. It is well known that quantitative
deformation lemma is a very powerful tool to obtain mountain pass theorem, and
the mountain pass theorem has proved to be a power tool in many areas of
analysis. But to our best knowledge, very few works have been done for
quantitative deformation lemma or mountain pass theorem in the past thirty
years.
In this paper, we extend the quantitative deformation lemma in [11] so that we
can obtain more critical points, especially for supinf critical value $c_{1}$,
$x=\varphi^{-1}(c_{1})$ is a new critical point. Moreover,as an application of
our deformation lemma,a new mountain pass theorem is given. Comparing with the
mountain pass type theorem in [11], $\varphi(e)\leq\varphi(0)<c_{2}$, but in
our new mountain pass theorem, $\varphi(e)=c_{2}$, so that our new mountain
pass theorem can not be obtained by the quantitative deformation lemma in
[11];besides, in our theorem, if $\varphi$ satisfies $(PS)_{c_{2}}$ condition,
we can obtain two new critical points $x=0$ (valley point) and $x=e$ (peak
point).
The organization of this paper is as following. In section $2$, the
quantitative deformation lemma in [11] and the corresponding mountain pass
theorem in [11] are given. In section $3$, on the basis of the quantitative
deformation lemma in [11], we prove the new quantitative deformation lemma. In
section $4$, as an application of our deformation lemma, our new mountain pass
theorem is given.
## 2 Preliminaries
For convenience, we introduce the Quantitative Deformation Lemma in [11] and
the corresponding Mountain Pass Type Theorem in [11] as the following:
###### Lemma 2.1.
(Quantitative deformation lemma) Let $X$ be a Hilbert space, $\varphi\in
C^{2}(X,\mathbb{R})$, $c\in\mathbb{R}$, $\varepsilon>0$. Assume that
$\big{(}\forall
u\in\varphi^{-1}([c-2\varepsilon,c+2\varepsilon])\big{)}:\|\varphi^{\prime}(u)\|\geq
2\varepsilon.$
Then there exists $\eta\in$ $C(X,X)$, such that
* $(a)$
$\eta(u)=u$, $\forall
u\notin\varphi^{-1}\big{(}[c-2\varepsilon,c+2\varepsilon]\big{)}$.
* $(b)$
$\eta(\varphi^{c+\varepsilon})\subset\varphi^{c-\varepsilon}$, where
$\varphi^{c-\varepsilon}:=\varphi^{-1}\big{(}(-\infty,c-\varepsilon]\big{)}$.
###### Theorem 2.1.
(Mountain pass type theorem) Let $X$ be a Hilbert space, $\varphi\in
C^{2}(X,\mathbb{R})$, $e\in X$ and $r>0$ be such that $\|e\|>r$ and
$\displaystyle b:=\inf_{\|u\|=r}\varphi(u)>\varphi(0)\geq\varphi(e).$ (2.1)
Then, for each $\varepsilon>0$, there exists $u\in X$ such that
* $(i)$
$c-2\varepsilon\leq\varphi(u)\leq c+2\varepsilon$,
* $(ii)$
$\|\varphi^{\prime}(u)\|<2\varepsilon$,
where
$c:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)}$
and
$\Gamma:=\\{\gamma\in C\big{(}[0,1],X\big{)}:\gamma(0)=0,\gamma(1)=e\\}.$
###### Definition 2.1.
([14])Let $X$ be a Banach space, $\varphi\in C^{1}(X,\mathbb{R})$ and
$c\in\mathbb{R}$. The function $\varphi$ satisfies the $(PS)_{c}$ condition if
any sequence $(u_{n})\subset X$ such that
$\varphi(u_{n})\rightarrow c,\varphi^{\prime}(u_{n})\rightarrow 0$
has a convergent subsequence.
## 3 New Quantitative Deformation Lemma
###### Theorem 3.1.
Let $X$ be a Hilbert space, $\varphi\in C^{2}(X,\mathbb{R})$,
$c\in\mathbb{R}$, $\varepsilon>0$. Assume that
$\displaystyle\big{(}\forall
u\in\varphi^{-1}([c-2\varepsilon,c+2\varepsilon])\big{)}:\|\varphi^{\prime}(u)\|\geq
2\varepsilon.$
Then there exists $\eta\in$ $C(X,X)$, such that
* $(a\,^{\prime})$
$\eta(u)=u$, $\forall
u\notin\varphi^{-1}\big{(}[c-2\varepsilon,c+2\varepsilon]\big{)}\backslash D$,
where $D\subseteq\varphi^{-1}\big{(}[c-0.5\varepsilon,c+\varepsilon]\big{)}$.
* $(b\,^{\prime})$
$\eta\big{(}\varphi^{-1}[c-\varepsilon,c-0.6\varepsilon]\big{)}\subset\varphi_{*}^{c+\varepsilon}$,
where $\varphi_{*}^{c+\varepsilon}$ denotes
$\varphi^{-1}\big{(}[c+\varepsilon,+\infty)\big{)}$.
* $(c\,^{\prime})$
$\eta\big{(}\varphi^{-1}[c+0.6\varepsilon,c+\varepsilon]\big{)}\subset\varphi^{c-\varepsilon}$,
where $\varphi^{c-\varepsilon}$ denotes
$\varphi^{-1}\big{(}(-\infty,c-\varepsilon]\big{)}$.
###### Proof.
Let us define
$\displaystyle A$
$\displaystyle:=\varphi^{-1}\big{(}[c-2\varepsilon,c+2\varepsilon]\big{)}\backslash
D,$ $\displaystyle B$
$\displaystyle:=\varphi^{-1}\big{(}[c-\varepsilon,c-0.6\varepsilon]\big{)},$
$\displaystyle C$
$\displaystyle:=\varphi^{-1}\big{(}[c+0.6\varepsilon,c+\varepsilon]\big{)},$
$\displaystyle\psi(u)$
$\displaystyle:=\frac{[dist(u,C)-dist(u,B)]dist(u,X\backslash
A)}{[dist(u,C)+dist(u,B)]dist(u,X\backslash A)+dist(u,B)dist(u,C)},$
so that $\psi$ is locally Lipschitz continuous, $\psi=1$ on $B$, $\psi=-1$ on
$C$ and $\psi=0$ on $X\backslash$ $A$.
Let us also define the locally Lipschitz continuous vector field
$\displaystyle f(u)$ $\displaystyle:=$
$\displaystyle\psi(u)\|\nabla\varphi(u)\|^{-2}\nabla\varphi(u),\quad\ \ u\in
A,$ $\displaystyle:=$ $\displaystyle 0,\quad\ \ u\in X\backslash A.$
It is clear that $\|f(u)\|\leq(2\varepsilon)^{-1}$ on $X$. For each $u\in$
$X$, the Cauchy problem
$\displaystyle\frac{d}{dt}\sigma(t,u)$ $\displaystyle=$ $\displaystyle
f\big{(}\sigma(t,u)\big{)},$ $\displaystyle\sigma(0,u)$ $\displaystyle=$
$\displaystyle u,$
has a unique solution $\sigma(\cdot,u)$ defined on $\mathbb{R}$. Moreover,
$\sigma$ is continuous on $\mathbb{R}\times X$(see e.g. [15]). The map $\eta$
defined on $X$ by $\eta(u):=\sigma(2\varepsilon,u)$ satisfies
$(a\,^{\prime})$. Since
$\displaystyle\frac{d}{dt}\varphi\big{(}\sigma(t,u)\big{)}$ $\displaystyle=$
$\displaystyle\bigg{(}\nabla\varphi\big{(}\sigma(t,u)\big{)},\frac{d}{dt}\sigma(t,u)\bigg{)}$
(3.1) $\displaystyle=$
$\displaystyle\bigg{(}\nabla\varphi\big{(}\sigma(t,u)\big{)},f\big{(}\sigma(t,u)\big{)}\bigg{)}$
$\displaystyle=$ $\displaystyle\psi\big{(}\sigma(t,u)\big{)},$
If
$\displaystyle\sigma(t,u)\in\varphi^{-1}\big{(}[c-\varepsilon,c-0.6\varepsilon]\big{)}=B,\quad\
\ \forall t\in[0,2\varepsilon],$
then
$\psi(\sigma(t,u))=1.$
So, we obtain from (3.1),
$\displaystyle\varphi\big{(}\sigma(2\varepsilon,u)\big{)}$ $\displaystyle=$
$\displaystyle\varphi(u)+\int_{0}^{2\varepsilon}\frac{d}{dt}\varphi\big{(}\sigma(t,u)\big{)}dt$
$\displaystyle=$
$\displaystyle\varphi(u)+\int_{0}^{2\varepsilon}\psi\big{(}\sigma(t,u)\big{)}dt$
$\displaystyle\geq$ $\displaystyle c-\varepsilon+2\varepsilon=c+\varepsilon,$
and $(b\,^{\prime})$ is also satisfied.
Finally, similar to prove $(b\,^{\prime})$, we prove $(c\,^{\prime})$.
If
$\displaystyle\sigma(t,u)\in\varphi^{-1}\big{(}[c+0.6\varepsilon,c+\varepsilon]\big{)}=C,\quad\
\ \forall t\in[0,2\varepsilon],$
then
$\psi(\sigma(t,u))=-1.$
So, we obtain from (3.1),
$\displaystyle\varphi\big{(}\sigma(2\varepsilon,u)\big{)}$ $\displaystyle=$
$\displaystyle\varphi(u)+\int_{0}^{2\varepsilon}\frac{d}{dt}\varphi\big{(}\sigma(t,u)\big{)}dt$
$\displaystyle=$
$\displaystyle\varphi(u)+\int_{0}^{2\varepsilon}\psi\big{(}\sigma(t,u)\big{)}dt$
$\displaystyle\leq$ $\displaystyle c+\varepsilon-2\varepsilon=c-\varepsilon,$
and $(c\,^{\prime})$ is also satisfied.
###### Remark 3.1.
By Theorem 3.1, we get more critical points than the Quantitative Deformation
Lemma in [11]. All the domain $D$ in Theorem 3.1, especially for $supinf$
critical value $c$, $x=\varphi^{-1}(c)$ are all new critical points.
###### Remark 3.2.
In Lemma $2.1$, there are two conclusions, but in Theorem 3.1, there are three
conclusions.
## 4 An Example (New Mountain Pass Theorem)
Let $X$ be a Hilbert space, $\varphi\in C^{2}(X,\mathbb{R})$, $e\in X$ and
$r>0$ be such that $\|e\|>r$ and
$\varphi(0)=c_{1},\quad\ \ \varphi(e)=c_{2},\quad\ \ c_{1}\neq c_{2},$
and
$c_{1}:=\sup_{\gamma\in\Gamma}\min_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},\,\,\,\,c_{2}:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},$
where
$\Gamma:=\\{\gamma\in
C\big{(}[0,1],X\big{)}:\gamma(\frac{1}{4})=0,\gamma(\frac{1}{2})=e\\}.$
Then, for each $\varepsilon>0$, there exists $u^{\ast}\in X$ and
$u^{\triangle}\in X$ such that
* $(\mathrm{I})$
$c_{1}-2\varepsilon\leq\varphi(u^{\ast})\leq c_{1}+2\varepsilon$,
* $(\mathrm{II})$
$\|\varphi^{\prime}(u^{\ast})\|<2\varepsilon$.
* $(\mathrm{III})$
$c_{2}-2\varepsilon\leq\varphi(u^{\triangle})\leq c_{2}+2\varepsilon$,
* $(\mathrm{IV})$
$\|\varphi^{\prime}(u^{\triangle})\|<2\varepsilon$.
###### Proof.
Obviously, for each $\varepsilon>0$, (I) and (III) are easy to get. Next, we
prove (II) and (IV).
Suppose that at least one of (II) and (IV) is not true. Then, we can get the
contradiction:
Case 1. We assume that (II) is not true. It means that there exists
$\varepsilon$ such that
$\|\varphi^{\prime}(u^{\ast})\|\geq 2\varepsilon.$
From
$c_{1}:=\sup_{\gamma\in\Gamma}\min_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},\,\,\,\,c_{2}:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},$
and $c_{1}\neq c_{2}$, we get $c_{1}<c_{2}$ or $c_{1}>c_{2}$. Then, Case 1 can
be divided into two parts.
Firstly, when $c_{1}<c_{2}$, let
$\varepsilon_{1}=\min\\{\frac{c_{2}-c_{1}}{4},\,\varepsilon\\}$. It is clear
that
$\|\varphi^{\prime}(u^{\ast})\|\geq 2\varepsilon_{1}.$
and for $\varepsilon_{1}$, (I) is still easy to get.
From $\varepsilon_{1}=\min\\{\frac{c_{2}-c_{1}}{4},\,\varepsilon\\}$, we
obtain
$c_{1}+2\varepsilon_{1}\leq
c_{1}+2\times\frac{c_{2}-c_{1}}{4}=c_{1}+\frac{c_{2}-c_{1}}{2}=\frac{c_{2}}{2}+\frac{c_{1}}{2}<c_{2}.$
It means that
$c_{2}>c_{1}+2\varepsilon_{1}.$
In Theorem 3.1, we can take $D=\\{u\in X\mid\varphi(u)=c_{1}\\}$. Consider
$\beta=\eta\circ\gamma$, where $\eta$ is given by Theorem 3.1. Using
$(a\,^{\prime})$, we have,
$\displaystyle\beta(\frac{1}{4})$ $\displaystyle=$
$\displaystyle\eta\big{(}\gamma(\frac{1}{4})\big{)}=\eta(0)=0.$
$\displaystyle\beta(\frac{1}{2})$ $\displaystyle=$
$\displaystyle\eta\big{(}\gamma(\frac{1}{2})\big{)}=\eta(e)=e,$
so that $\beta\in\Gamma$. From
$c_{1}:=\sup_{\gamma\in\Gamma}\min_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},$
there exist $\gamma\in\Gamma$ and $\varepsilon_{2}>0$ such that
$c_{1}-\varepsilon_{2}\leq\min_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)}\leq
c_{1}-0.6\varepsilon_{2}.$
Then, from $(b\,^{\prime})$, we have
$\min_{t\in[0,1]}\varphi\bigg{(}\eta\big{(}\gamma(t)\big{)}\bigg{)}\geq
c_{1}+\varepsilon_{2}.$
It means that
$\min_{t\in[0,1]}\varphi\big{(}\beta(t)\big{)}\geq c_{1}+\varepsilon_{2}.$
So
$c_{1}+\varepsilon_{2}\leq\min_{t\in[0,1]}\varphi\big{(}\beta(t)\big{)}\leq
c_{1}.$
This is a contradiction. Therefore, (II) is true.
Secondly, when $c_{1}>c_{2}$, let
$\varepsilon_{1}=\min\\{\frac{c_{1}-c_{2}}{4},\,\varepsilon\\}$. It is clear
that
$\|\varphi^{\prime}(u^{\ast})\|\geq 2\varepsilon_{1},$
and for $\varepsilon_{1}$, (I) is still easy to get.
From $\varepsilon_{1}=\min\\{\frac{c_{1}-c_{2}}{4},\,\varepsilon\\}$, we
obtain
$c_{1}-2\varepsilon_{1}\geq
c_{1}-2\times\frac{c_{1}-c_{2}}{4}=\frac{c_{1}}{2}+\frac{c_{2}}{2}>c_{2}.$
It means that
$c_{2}<c_{1}-2\varepsilon_{1}.$
In Theorem 3.1, we can take $D=\\{u\in X\mid\varphi(u)=c_{1}\\}$. Consider
$\beta=\eta\circ\gamma$, where $\eta$ is given by Theorem 3.1. Using
$(a\,^{\prime})$, we have,
$\displaystyle\beta(\frac{1}{4})$ $\displaystyle=$
$\displaystyle\eta\big{(}\gamma(\frac{1}{4})\big{)}=\eta(0)=0.$
$\displaystyle\beta(\frac{1}{2})$ $\displaystyle=$
$\displaystyle\eta\big{(}\gamma(\frac{1}{2})\big{)}=\eta(e)=e,$
so that $\beta\in\Gamma$. From
$c_{1}:=\sup_{\gamma\in\Gamma}\min_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},$
there exist $\gamma\in\Gamma$ and $\varepsilon_{2}>0$ such that
$c_{1}-\varepsilon_{2}\leq\min_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)}\leq
c_{1}-0.6\varepsilon_{2}.$
Then, from $(b\,^{\prime})$, we have
$\min_{t\in[0,1]}\varphi\bigg{(}\eta\big{(}\gamma(t)\big{)}\bigg{)}\geq
c_{1}+\varepsilon_{2}.$
It means that
$\min_{t\in[0,1]}\varphi\big{(}\beta(t)\big{)}\geq c_{1}+\varepsilon_{2}.$
So
$c_{1}+\varepsilon_{2}\leq\min_{t\in[0,1]}\varphi\big{(}\beta(t)\big{)}\leq
c_{1}.$
This is a contradiction. Therefore, (II) is true.
Case 2. We assume that (IV) is not true. It means that there exists
$\varepsilon$ such that
$\|\varphi^{\prime}(u^{\triangle})\|\geq 2\varepsilon.$
From
$c_{1}:=\sup_{\gamma\in\Gamma}\min_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},\,\,\,\,c_{2}:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},$
and $c_{1}\neq c_{2}$ we get $c_{1}<c_{2}$ or $c_{1}>c_{2}$. Then, Case 2 can
be divided into two parts.
Firstly, when $c_{1}<c_{2}$, let
$\varepsilon_{1}=\min\\{\frac{c_{2}-c_{1}}{4},\,\varepsilon\\}$. It is clear
that
$\|\varphi^{\prime}(u^{\triangle})\|\geq 2\varepsilon_{1}.$
and for $\varepsilon_{1}$, (III) is still easy to get.
From $\varepsilon_{1}=\min\\{\frac{c_{2}-c_{1}}{4},\,\varepsilon\\}$, we
obtain
$c_{2}-2\varepsilon_{1}\geq
c_{2}-2\times\frac{c_{2}-c_{1}}{4}=c_{2}-\frac{c_{2}-c_{1}}{2}=\frac{c_{2}}{2}+\frac{c_{1}}{2}>c_{1}.$
It means that
$c_{1}<c_{2}-2\varepsilon_{1}.$
In Theorem 3.1, we can take $D=\\{u\in X\mid\varphi(u)=c_{2}\\}$. Consider
$\beta=\eta\circ\gamma$, where $\eta$ is given by Theorem 3.1. Using
$(a\,^{\prime})$, we have,
$\displaystyle\beta(\frac{1}{4})$ $\displaystyle=$
$\displaystyle\eta\big{(}\gamma(\frac{1}{4})\big{)}=\eta(0)=0.$
$\displaystyle\beta(\frac{1}{2})$ $\displaystyle=$
$\displaystyle\eta\big{(}\gamma(\frac{1}{2})\big{)}=\eta(e)=e,$
so that $\beta\in\Gamma$. From
$c_{2}:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},$
there exist $\gamma\in\Gamma$ and $\varepsilon_{3}>0$ such that
$c_{2}+0.6\varepsilon_{3}\leq\max_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)}\leq
c_{2}+\varepsilon_{3}.$
Then, from $(c_{1}\,^{\prime})$, we have
$\max_{t\in[0,1]}\varphi\bigg{(}\eta\big{(}\gamma(t)\big{)}\bigg{)}\leq
c_{2}-\varepsilon_{3}.$
It means that
$\max_{t\in[0,1]}\varphi\big{(}\beta(t)\big{)}\leq c_{2}-\varepsilon_{3}.$
So
$c_{2}\leq\max_{t\in[0,1]}\varphi\big{(}\beta(t)\big{)}\leq
c_{2}-\varepsilon_{3}.$
This is a contradiction. Therefore, (IV) is true.
Secondly, when $c_{2}<c_{1}$, let
$\varepsilon_{1}=\min\\{\frac{c_{1}-c_{2}}{4},\,\varepsilon\\}$ and take
$D=\\{u\in X\mid\varphi(u)=c_{2}\\}$, the rest of the proof is similar to the
first part of Case 2. Therefore, (IV) is true.
From Case 1 and Case 2, our new mountain pass theorem is proved.
###### Remark 4.1.
In Theorem 2.1 (Mountain pass theorem), $c$ is defined as
$c:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)}$
where
$\Gamma:=\\{\gamma\in C\big{(}[0,1],X\big{)}:\gamma(0)=0,\gamma(1)=e\\}.$
But in our new mountain pass theorem, $c_{1}$ and $c_{2}$ are defined as
$c_{1}:=\sup_{\gamma\in\Gamma}\min_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},\quad\
\
c_{2}:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\varphi\big{(}\gamma(t)\big{)},$
where
$\Gamma:=\\{\gamma\in
C\big{(}[0,1],X\big{)}:\gamma(\frac{1}{4})=0,\gamma(\frac{1}{2})=e\\}.$
###### Remark 4.2.
In fact, in Theorem 2.1 (Mountain pass theorem),
$c_{2}>\varphi(0)\geq\varphi(e).$
But in our new mountain pass theorem,
$\varphi(0)=c_{1},\quad\ \ \varphi(e)=c_{2},\quad\ \ c_{1}\neq c_{2}.$
and in the proof of our new mountain pass theorem, we take $D=\\{u\in
X\mid\varphi(u)=c_{1}\\}$ in Case 1, and take $D=\\{u\in
X\mid\varphi(u)=c_{2}\\}$ in Case 2.
###### Remark 4.3.
In the example, if we do not use our Theorem 3.1 (New quantitative deformation
lemma), we can not obtain
$\beta(\frac{1}{4})=\eta\big{(}\gamma(\frac{1}{4})\big{)}=\eta(0)=0.$
Moreover, we can not obtain $\beta\in\Gamma$.
###### Remark 4.4.
An interesting point in the example is that, if $\varphi$ satisfies $(PS)_{c}$
condition, it is easy to obtain two new critical points $x=0$ and $x=e$ which
have not been obtained before.
## References
* [1] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.
* [2] G. Barletta and S.A. Marano, Some remarks on critical point theory for locally Lipschitz functions, Glasgow Math. J. 45 (2003), 131-141.
* [3] H. Brezis, J.M. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and theorem of P. Rabinowitz, Comm. Pure Appl. Math, 33 (1980), 667-684.
* [4] K.C.,Chang, Infinite dimensional Morse theory,Birkhäuser, 1993.
* [5] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Math. 107, Cambridge Univ. Press, Cambridge, 1993.
* [6] H. Hofer, A geometric description of the neighbourhood of a critical point given by the Mountain Pass Theorem, J. London Math. Soc. 31 (1985), 566-570.
* [7] I. Peral, Beyond the mountain pass: some applications. Adv. Nonlinear Stud. 12 (2012), no. 4, 819-850
* [8] P. Pucci and J. Serrin, A Mountain Pass Theorem, J. Differential Equations 60 (1985), 142-149.
* [9] P. Pucci and J. Serrin, Extensions of the Mountain Pass Theorem, J. Funct. Anal. 59 (1984), 185-210.
* [10] P. Pucci, J. Serrin, The structure of the critical set in the mountain pass theorem, Trans. Amer. Math. Soc. 299, (1987), no. 1, 115-132.“
* [11] M. Willem, Minimax Theorems, Birkhäuser,Boston, 1996.
* [12] R. Livrea and S.A. Marano, Existence and classification of critical points for nondifferentiable functions, Adv. Differential Equations 9 (2004), 961-978.
* [13] S.A. Marano and D. Motreanu, A deformation theorem and some critical points results for non- differentiable functions, Topol. Methods Nonlinear Anal. 22 (2003), 139-158.
* [14] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence, RI, 1986\.
* [15] Schwartz L., Cours $d^{,}$analyse, Hermann, Paris, 1991-1994.
|
arxiv-papers
| 2014-02-19T09:38:11 |
2024-09-04T02:49:58.436957
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liang Ding and Fode Zhang and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1402.4602"
}
|
1402.4630
|
# New Periodic Solutions for Second Order Hamiltonian Systems with Local
Lipschitz Potentials
Li Bingyu and Li Fengying and Zhang Shiqing
$\begin{array}[]{c}{\rm
MathematicalDepartment,SichuanUniversity,Chengdu610064,China}\\\
{\rm(DedicatedtotheMemoryofProfessorShiShuzhong)}\end{array}$
Abstract Firstly,we generalize the classical Palais-Smale-Cerami condition for
$C^{1}$ functional to the local Lipschitz case,then generalize the famous
Benci-Rabinowitz’s and Rabinowitz’s Saddle Point Theorems with classical
Cerami-Palais-Smale condition to the local Lipschitz functional, then we apply
these Theorems to study the existence of new periodic solutions for second
order Hamiltonian systems with local Lipschitz potentials which are weaker
than Rabinowitz’s original conditions .The key point of our proof is proving
Cerami-Palais-Smale condition for local Lipschitz case,which is difficult
since no smooth and symmetry for the potential.
Key Words: Second order Hamiltonian systems, Cerami-Palais-Smale condition for
local Lipschitz functional,Periodic solutions, Saddle Point Theorems.
2000 Mathematical Subject Classification: 34C15, 34C25, 58F.
## 1\. Introduction
In the critical point theory,the compactness condition is a key for proving
the existence of critical points for some functionals.In 1964,R.Palais and
S.Smale [13]introduced the famous $(PS)_{c}$ condition:
Definition 1.1 Let $X$ is a Banach space, $f\in C^{1}(X,R)$, if
$\\{x_{n}\\}\subset X$ s.t.
$f(x_{n})\rightarrow c,$ $f^{\prime}(x_{n})\rightarrow 0,$
and $\\{x_{n}\\}$ has a strongly convergent subsequence, then we say $f$
satisfies $(PS)_{c}$ condition.
In 1978,Cerami[4] presented a weaker compactness condition than the above
classical $(PS)_{c}$ condition:
Definition 1.2 Let $X$ be a Banach space, $\Phi$ be defined on $X$ is Gateaux-
differentiable, if the sequence $\\{x_{n}\\}\subset X$ such that
$\Phi(x_{n})\rightarrow c,$
$(1+\|x_{n}\|)\|\Phi^{{}^{\prime}}(x_{n})\|\rightarrow 0,$
then $\\{x_{n}\\}$ has a strongly convergent subsequence in $X$. Then we call
$f$ satisfies $(CPS)_{c}$ condition in $X$.
For the functional $f(x)$ in locally Lipschitz functional space
$C^{1-0}(X,R)$,Clarke [6] define the generalized gradient $\partial f(x)$
which is the subset of $X^{*}$ defined by
$\partial f(x)=\\{x^{*}\in X^{*}|\langle x^{*},v\rangle\leq f^{0}(x,v),\forall
v\in X\\},$
where
$f^{0}(x,v)=\lim_{y\rightarrow x,\lambda\downarrow 0}sup\frac{f(y+\lambda
v)-f(y)}{\lambda}.$
In 1981,K.C.Chang[5] introduced the (PS) condition for locally Lipschitz
function:
Definition 1.3 Let $X$ is a Banach space, $f\in C^{1-0}(X,R)$, if
$\\{x_{n}\\}\subset X$ s.t.$f(x_{n})$ is bounded and
$min_{x^{*}\in\partial f(x_{n})}||x^{*}||\rightarrow 0,$
and $\\{x_{n}\\}$ has a strongly convergent subsequence, then we say $f$
satisfies $(PSC)$ condition.
if $\\{x_{n}\\}\subset X$ s.t.$f(x_{n})\rightarrow c$ and
$min_{x^{*}\in\partial f(x_{n})}||x^{*}||\rightarrow 0,$
and $\\{x_{n}\\}$ has a strongly convergent subsequence, then we say $f$
satisfies $(PSC)_{c}$ condition.
Ekeland [8],Ghoussoub-Preiss[9] used Ekeland’s variational principle to prove
Lemma1.1Let $X$ be a Banach space, suppose that $\Phi$ defined on $X$ is
Gateaux-differentiable and lower semi-continuous and bounded from below.Then
there is a sequence $\\{x_{n}\\}$ such that
$\Phi(x_{n})\rightarrow\inf\Phi$
$(1+\|x_{n}\|)\|\Phi^{{}^{\prime}}(x_{n})\|\rightarrow 0.$
Motivated by the above Definitions and Lemma,we introduce the following
(CPS)-type condition for the locally Lipschitz functional:
Definition 1.4 Let $X$ is a Banach space, $f\in C^{1-0}(X,R)$, we say $f$
satisfies $(CPSC)_{c}$ condition if $\\{x_{n}\\}\subset X$ s.t.
$f(x_{n})\rightarrow c,$ $(1+||x_{n}||)min_{x^{*}\in\partial
f(x_{n})}||x^{*}||\rightarrow 0,$
then $\\{x_{n}\\}$ has a strongly convergent subsequence.
K.C.Chang[5] and Shi S.Z.[16] use the $(PSC)$ condition for the local
Lipschitz functional to generalize the classical Mountain Pass Lemma[2] and
general minimax Theorems[12]. Here we can generalize the classical Benci-
Rabinowitz’s and Rabinowitz’s Saddle Point Theorems to the local Lipschitz
functional cases with the Cerami-Palais-Smale-Chang-type conditions:
Theorem1.1 Let $X$ be a Banach space, $f\in C^{1-0}(X,R)$. Let
$X=X_{1}\bigoplus X_{2},\rm{dim}X_{1}<+\infty$,$X_{2}$ is closed in $X$. Let
$\displaystyle B_{a}=\\{x\in X|\|x\|\leq a\\},$ $\displaystyle S=\partial
B_{\rho}\cap X_{2},\rho>0,$ $\displaystyle Q$ $\displaystyle=$
$\displaystyle\\{x_{1}+se|(x_{1},s)\in X_{1}\times R^{1},\|x_{1}\|\leq
r_{1},0\leq s\leq r_{2},r_{2}>\rho\\},$ $\displaystyle\partial
Q=(B_{r_{1}}\cap X_{1})\cup\partial\\{x_{1}\bigoplus se,\|x_{1}\|\leq
r_{1},0<s\leq r_{2}\\},$
where $e\in X_{2},\|e\|=1$. If
$f|_{S}\geq\alpha,$
and
$f|_{\partial Q}\leq\beta<\alpha.$
Then $c=\inf\limits_{\phi\in\Gamma}\sup\limits_{x\in Q}f(\phi(x))\geq\alpha$
,if $f(q)$ satisfies $(CPSC)_{c}$ ,then $c$ is a critical value for $f$.
Theorem1.2 Let $X$ be a Banach space and let $f\in C^{1-0}(X,R)$, let
$X=X_{1}\bigoplus X_{2}$ with
$\rm{dim}X_{1}<+\infty$
and
$\sup\limits_{S^{1}_{R}}f<\inf\limits_{X_{2}}f,$
where $S^{1}_{R}=\\{u\in X_{1}||u|=R\\}$.
Let $B^{1}_{R}=\\{u\in X_{1},|u|\leq R\\},M=\\{g\in C(B^{1}_{R},X)|g(s)=s$,
$s\in S^{1}_{R}\\}$
$c=\inf\limits_{g\in M}\max\limits_{s\in B_{R}^{1}}(g(s)).$
Then $c\geq\inf\limits_{X_{2}}f$, if $f$ satisfies $(CPSC)_{c}$ condition,
then $c$ is a critical value of $f$.
In 1978, Rabinowitz [14] firstly used mini-max methods with the classical
Palais-Smale condition to study the periodic solutions for second order
Hamiltonian systems with the super-quadratic potential:
$\ddot{q}+V^{\prime}(q)=0$ (1.1)
He proved that
Theorem 1.3([14]) Suppose $V$ satisfies
$(V_{1})\ V\in C^{1}(R^{n},R)$
$(V_{2})$ There exist constants $\mu>2,r_{0}>0$ such that
$0<\mu V(x)\leq V^{\prime}(x)\cdot x,\ \ \ \ \forall|x|\geq r_{0},$
$(V_{3})\ V(x)\geq 0,\ \ \ \ \forall x\in R^{n},$
$(V_{4})\ V(x)=o(|x|^{2}),$ as $|x|\rightarrow 0$.
Then for any $T>0,$ (1.1) has a non-constant $T$-periodic solution.
In the last 30 years, there were many works for (1.1), we can refer
([3]-[12],[15,17] etc.), and the references there. In this paper, we try to
generalize the result of Rabinowitz to local Lipschitz potential, we get the
following Theorem:
Theorem 1.4 Suppose $V$ satisfies
$(V1)\ V\in C^{1-0}(R^{n},R);$
$(V2)$ There exist constants ${\mu}_{1}>2,\mu_{2}\in R$ such that
$\langle y,x\rangle\geq\mu_{1}V(x)+\mu_{2},\ \ \ \ \forall x\in
R^{n},y\in\partial V(x);$
$(V3)$ There are $a_{1}>0,a_{2}\in R$ such that
$V(x)\geq a_{1}|x|^{\mu_{1}}+a_{2},\ \ \ \ \forall x\in R^{n},$
$(V4)$
$0\leq V(x)\leq A|x|^{2},|x|\rightarrow 0.$
Then for any $T<(\frac{2}{A})^{1/2}\pi,$ the following system
$0\in\ddot{q}+\partial V(q)$ (1.2)
has at least one non-zero $T$-periodic solution.
For sub-quadratic second order Hamiltonian system,we can get
Theorem 1.5 Suppose $V$ satisfies
$(V1)\ V\in C^{1-0}(R^{n},R);$
$(V2^{\prime})$ There exist constants ${\mu}_{1}<2,\mu_{2}\in R$ such that
$\langle y,x\rangle\leq\mu_{1}V(x)+\mu_{2},\ \ \ \ \forall x\in
R^{n},y\in\partial V(x);$
$(V3^{\prime})$
$V(x)\rightarrow+\infty,|x|\rightarrow+\infty;$
$(V4^{\prime})$
$V(x)\leq A|x|^{2}+a.$
Then for any $T<(\frac{2}{A})^{1/2}\pi,$ (1.2) has at least one $T$-periodic
solution.
## 2\. Some Lemmas
In order to prove Theorem 1.1, we define functional:
$f(q)=\frac{1}{2}\int^{T}_{0}|\dot{q}|^{2}dt-\int^{T}_{0}V(q)dt,\ \ \ \
\forall q\in H^{1}$ (2.1)
where
$H^{1}=W^{1,2}(R/TZ,R^{n}).$ (2.2)
Lemma 2.1([6]) Let $\widetilde{q}\in H^{1}$ be such that $\partial
f(\widetilde{q})=0.$
Then $\widetilde{q}(t)$ is a $T$-periodic solution for (1.2).
Lemma2.2(Sobolev-Rellich-Kondrachov, Compact Imbedding Theorem [1])
$W^{1,2}(R/TZ,R^{n})\subset C(R/TZ,R^{n})$
and the imbedding is compact.
Lemma 2.3(Eberlein-Shmulyan [18]) A Banach space $X$ is reflexive if and only
if any bounded sequence in $X$ has a weakly convergent subsequence.
Lemma 2.4([11],[19]) Let $q\in W^{1,2}(R/TZ,R^{n})$ and $q(0)=q(T)=0$
We have Friedrics-Poincare’s inequality:
$\int^{T}_{0}|\dot{q}(t)|^{2}dt\geq\left(\frac{\pi}{T}\right)^{2}\int^{T}_{0}|q(t)|^{2}dt.$
Let $q\in W^{1,2}(R/TZ,R^{n})$ and $\int^{T}_{0}q(t)dt=0,$ then
(i) We have Poincare-Wirtinger’s inequality
$\int^{T}_{0}|\dot{q}(t)|^{2}dt\geq\left(\frac{2\pi}{T}\right)^{2}\int^{T}_{0}|q(t)|^{2}dt$
(ii) We have Sobolev’s inequality
$\max_{0\leq t\leq
T}|q(t)|=\|q\|_{\infty}\leq\sqrt{\frac{T}{12}}\left(\int^{T}_{0}|\dot{q}(t)|^{2}dt\right)^{1/2}$
We define the equivalent norm in $H^{1}=W^{1,2}(R/TZ,R^{n})$
$\|q\|_{H^{1}}=\left(\int^{T}_{0}|\dot{q}|^{2}dt\right)^{1/2}+|q(0)|$
Shi Shuzhong[16] generalized the classical Mini-max Theorems including Benci-
Rabinowitz’s Generalized Mountain-Pass Lemma and Rabinowitz’s Saddle Point
Theorem to the local Lipschitz functionals with Chang’s compactness condition:
Lemma 2.5 Let $X$ be a Banach space, $f\in C^{1-0}(X,R)$. Let
$X=X_{1}\bigoplus X_{2},\rm{dim}X_{1}<+\infty$,$X_{2}$ is closed in $X$.Let
$\displaystyle B_{a}=\\{x\in X|\|x\|\leq a\\},$ $\displaystyle S=\partial
B_{\rho}\cap X_{2},\rho>0,$ $\displaystyle Q$ $\displaystyle=$
$\displaystyle\\{x_{1}+se|(x_{1},s)\in X_{1}\times R^{1},\|x_{1}\|\leq
r_{1},0\leq s\leq r_{2},r_{2}>\rho\\},$ $\displaystyle\partial
Q=(B_{r_{1}}\cap X_{1})\cup\partial\\{x_{1}\bigoplus se,\|x_{1}\|\leq
r_{1},0<s\leq r_{2}\\},$
where $e\in X_{2},\|e\|=1$.If
$f|_{S}\geq\alpha,$
and
$f|_{\partial Q}\leq\beta<\alpha,$
Then $c=\inf\limits_{\phi\in\Gamma}\sup\limits_{x\in Q}f(\phi(x))\geq\alpha$
,if $f(q)$ satisfies $(PSC)_{c}$ ,then $c$ is a critical value for $f$.
Lemma 2.6 Let $X$ be a Banach space and let $f\in C^{1}(X,R)$, let
$X=X_{1}\bigoplus X_{2}$ with
$\rm{dim}X_{1}<+\infty$
and
$\sup\limits_{S^{1}_{R}}f<\inf\limits_{X_{2}}f,$
where $S^{1}_{R}=\\{u\in X_{1}||u|=R\\}$.
Let $B^{1}_{R}=\\{u\in X_{1},|u|\leq R\\},M=\\{g\in C(B^{1}_{R},X)|g(s)=s$,
$s\in S^{1}_{R}\\}$
$c=\inf\limits_{g\in M}\max\limits_{s\in B_{R}^{1}}(g(s))$
Then $c\geq\inf\limits_{X_{2}}f$, if $f$ satisfies $(PSC)_{c}$ condition, then
$c$ is a critical value of $f$.
Lemma 2.7 Let $X$ be a Banach space, suppose that $F$ defined on $X$ is local
Lipschitz functional and lower semi-continuous and bounded from below.Then
$\forall\epsilon_{n}\downarrow 0$, there is a sequence $\\{g_{n}\\}$ such that
$F(g_{n})\rightarrow\inf F,$
$(1+\|g_{n}\|)F^{0}(g_{n},h)|\geq-\epsilon_{n}\|h\|.$
Proof Applying Ekeland’s variational principle ([7,8]),we can get a sequence
$g_{n}$ such that
$F(g_{n})\leq\inf F+\epsilon_{n}^{2},$ $F(g)\geq
F(g_{n})-\epsilon_{n}\delta(g,g_{n}).$
Let $g=g_{n}+th,t>0,h\in X$,then we have
$F(g_{n}+th)-F(g_{n})\geq-\epsilon_{n}\delta(g_{n}+th,g_{n}),$
where $\delta$ is the geodesic distance.
$F(g_{n}+th)-F(g_{n})\geq-\epsilon_{n}\int_{0}^{t}\frac{||h||ds}{1+||g_{n}+sh||},$
then
$\frac{1}{t}F(g_{n}+th)-F(g_{n})\geq-\epsilon_{n}\frac{1}{t}\int_{0}^{t}\frac{||h||ds}{1+||g_{n}+sh||},$
let $t\rightarrow 0$,we have
$F^{0}(g_{n},h)\geq\lim_{t\rightarrow 0}\frac{1}{t}(F(g_{n}+th)-F(g_{n}))$
$\geq-\epsilon_{n}\lim_{t\rightarrow
0}\frac{1}{t}\int_{0}^{t}\frac{||h||ds}{1+||g_{n}+sh||}$
$=-\epsilon_{n}||h||(1+||g_{n}||)^{-1}.$
## 3\. The Proof of Theorems 1.1,1.2,1.4 and 1.5
By Lemma 2.7 and similar arguments of Shi Shuzhong [16],we can prove Theorem
1.1 and 1.2.
Lemma 3.1 If $(V1)-(V3)$ in Theorem 1.4 hold, then $f(q)$ satisfies the
$(Cerami-Palais-Smale-Chang)$ condition on $H^{1}$.
Proof Let $\\{q_{n}\\}\subset H^{1}$ satisfy
$f(q_{n})\rightarrow c,\ \ \ \ (1+||q_{n}||)min_{x^{*}\in\partial
f(q_{n})}||x^{*}||\rightarrow 0,$ (3.1)
Then we claim $\\{q_{n}\\}$ is bounded. In fact,by $f(q_{n})\rightarrow c$, we
have
$\frac{1}{2}\|\dot{q}_{n}\|^{2}_{L^{2}}-\int^{T}_{0}V(q_{n})dt\rightarrow c$
(3.2)
By the definition ,we have
$<\partial f(q_{n}),q_{n}>=\|\dot{q}_{n}\|^{2}_{L^{2}}-\int^{T}_{0}(<\partial
V(q_{n}),q_{n}>)dt$
By $(V2)$,for any $v\in\partial V(q_{n})$,we have
$\displaystyle\|\dot{q}_{n}\|^{2}_{L^{2}}-\int^{T}_{0}<v,q_{n}>dt\leq\|\dot{q}_{n}\|^{2}_{L^{2}}-\int^{T}_{0}[\mu_{2}+\mu_{1}V(q_{n})]dt$
(3.3)
By (3.2) and (3.3), $\forall x^{*}\in\partial f(q_{n})$,we have
$\displaystyle<x^{*},q_{n}>$ $\displaystyle\leq$ $\displaystyle
a\|\dot{q_{n}}\|^{2}_{L^{2}}+C_{1}+\delta,n\rightarrow+\infty,$ (3.4)
where $C_{1}=c\mu_{1}-T\mu_{2}+\delta,\delta>0,a=1-\frac{\mu_{1}}{2}<0.$
By the above inequality (3.4) and (3.1),we have $\|\dot{q}_{n}\|_{L^{2}}\leq
M_{1}$. Then we claim $|q_{n}(0)|$ is also bounded. Otherwise, there a
subsequence, still denoted by $q_{n}$, s.t.
$|q_{n}(0)|\rightarrow+\infty$,since $\|\dot{q}_{n}\|\leq M_{1}$,then
$\displaystyle\min_{0\leq t\leq 1}|q_{n}(t)|$ $\displaystyle\geq$
$\displaystyle|q_{n}(0)|-\|\dot{q}_{n}\|_{2}\rightarrow+\infty,\rm{as}\
n\rightarrow+\infty$ (3.5)
We notice that
$\displaystyle\langle\partial
f(q_{n}),q_{n}\rangle=\int^{T}_{0}[|\dot{q}_{n}|^{2}dt-\langle\partial
V(q_{n}),q_{n}\rangle]dt$ (3.6)
$\displaystyle=2f(q_{n})+\int_{0}^{T}[2V(q_{n})-\langle\partial
V(q_{n}),q_{n}\rangle]dt$ (3.7)
By $(V2)-(V3)$,$\forall y\in\partial V(x)$ we have
$\langle
y,x\rangle-2V(x)\geq(\mu_{1}-2)V+\mu_{2}\rightarrow+\infty,|x|\rightarrow+\infty$
By (3.1) and (3.7),we get a contradiction,so
$\|q_{n}\|=\|\dot{q}_{n}\|_{L^{2}}+|q_{n}(0)|$ is bounded.
By the embedding theorem, $\\{q_{n}\\}$ has a weakly convergent subsequence
which uniformly converges to $q\in H^{1}$.
Furthermore, by $V\in C^{1-0}$ and the $w^{*}-upper$ semi-continuity, it’s
standard step for the rest proof that the weakly convergent subsequence is
also strongly convergent to $q\in H^{1}$.
Now we prove Theorem 1.4. In Theorem1.1, we take
$X_{1}=R^{n},X_{2}=\\{q\in W^{1,2}(R/TZ,R^{n}),\int^{T}_{0}q(t)dt=0\\}$
$S=\left\\{q\in
X_{2}|\left(\int^{T}_{0}|\dot{q}|^{2}dt\right)^{1/2}=\rho>0\right\\},$
$\partial Q=\\{x_{1}\in R^{n}||x_{1}|\leq r_{1}\\}\cup$
$\left\\{q=x_{1}+se,x_{1}\in R^{n},e\in
X_{2},\|e\|=1,s>0,\|q\|=(r_{1}^{2}+r_{2}^{2})^{1/2}>\rho\right\\}.$
When $q\in X_{2}$,by Sobolev’s
inequality,$\int^{T}_{0}|\dot{q}|^{2}dt\rightarrow 0$ implies
$max|q(t)|\rightarrow 0$.So when$\int^{T}_{0}|\dot{q}|^{2}dt\rightarrow 0$ ,
$(V4)$ implies
$V(q)\leq A|q|^{2}$
When $q\in X_{2}$,we have Poincare-Wirtinger inequality, so when
$\rho=[\int^{T}_{0}|\dot{q}|^{2}dt]^{\frac{1}{2}}\rightarrow 0$
We have
$f(q)\geq\frac{1}{2}\int^{T}_{0}|\dot{q}|^{2}dt-A\int^{T}_{0}|q|^{2}dt$
$\geq[\frac{1}{2}-A(2\pi)^{-2}T^{2}]\rho^{2},$
On the other hand, if $q\in X_{1}$,and we take $|x_{1}|\leq r_{1}$ very small,
then by $(V_{4})$, we have
$f(q)=-\int^{T}_{0}V(q)dt\leq 0,|q|\rightarrow 0.$
If
$q\in\left\\{q=x_{1}+se,x_{1}\in R^{n},e\in
X_{2},\|e\|=1,s>0,\|q\|=(|x_{1}|^{2}+s^{2})^{1/2}=R=(r_{1}^{2}+r_{2}^{2})^{1/2}>\rho\right\\},$
then by $(V3)$ and Jensen’s inequality,we have
$f(q)=\frac{1}{2}s^{2}-\int^{T}_{0}V(x_{1}+se)dt$
$\leq\frac{1}{2}s^{2}-\int^{T}_{0}(a|x_{1}+se|^{\mu_{1}}+b)dt$
$\leq\frac{1}{2}s^{2}-[aT^{1-\frac{\mu_{1}}{2}}(\int^{T}_{0}|x_{1}+se|^{2}dt)^{\frac{\mu_{1}}{2}}+bT]$
$=\frac{1}{2}s^{2}-aT^{1-\frac{\mu_{1}}{2}}[T|x_{1}|^{2}+s^{2}\int^{T}_{0}|e(t)|^{2}dt]^{\frac{\mu_{1}}{2}}-bT$
Notice that we can take $r_{2}$ large enough,then
$(|x_{1}|^{2}+s^{2})^{1/2}=R=(r_{1}^{2}+r_{2}^{2})^{1/2}$ is large enough,then
$|x_{1}|$ or $s$ must be large,so $T|x_{1}|^{2}+s^{2}\int^{T}_{0}|e(t)|^{2}dt$
must be large since $\int^{T}_{0}|e(t)|^{2}>0$, so that in such case $f(q)<0.$
The rest of the proof for Theorem 1.4 is obvious.
Using Theorem 1.2 and similar methods for proving Theorem 1.4,we can prove
Theorem 1.5,here we omit it
## Acknowledgements
The author Zhang Shiqing sincerely thank the supports of NSF of China and the
Grant for the Advisors of Ph.D students.
## References
* [1] R.A.Adams and J.F.Fournier, Sobolev spaces, Second edition, Academic Press, 2003.
* [2] A.Ambrosetti and P.Rabinowitz,Dual variational methods in critical point theory and applications,J.Funct. Anal.14(1973),349-381.
* [3] V.Benci and P.Rabinowitz,Critical point theorem for indefinite functionals,Inv.Math.52(1979),241-273.
* [4] Cerami G.,Un criterio di esistenza per i punti critici so variete illimitate, Rend. dell academia di sc.lombardo112(1978),332-336.
* [5] K.C.Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations,JMAA 80(1981),102-129.
* [6] Clarke,F.H.,Optimization and nonsmooth analysis,Wiley-Interscience,New York,1983.
* [7] I.Ekeland,On the variational principle,JMAA 47(1974),324-353.
* [8] I.Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, 1990.
* [9] N.Ghoussoub, D.Preiss, A general mountain pass principle for locating and clasifying critical points, Ann. Inst. Henri Poincare Anal. NonLineaire 6(1984), 321-330.
* [10] Y.Long, Index theory for symplectic paths with applications, Birkhauser Verlag, 2002.
* [11] J. Mawhin and M.Willem, Critical point theory and Hamiltonian system, Springer, Berlin ,1989.
* [12] L.Nirenberg,Variational and toplogical methods in nonlinear problems,Bull.AMS,New Series 4(1981),267-302.
* [13] Palais,R. and Smale S.,A generalized Morse theory,Bull.AMS 70(1964),165-171.
* [14] P.H.Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31(1978), 157-184.
* [15] P.H.Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. in Math. 65, AMS, 1986.
* [16] Shuzhong Shi,Ekeland variational principle and Mountain Pass Lemma,Acta Math.Sinica,New Series 1(1985),348-355.
* [17] M.Struwe,Variational methods, Springer, Berlin, 1990\.
* [18] K.Yosida, Functional analysis, 5th ed., Springer, Berlin, 1978.
* [19] W.P.,Ziemer,Weakly differentiable functions,Springer,1989.
|
arxiv-papers
| 2014-02-19T11:58:44 |
2024-09-04T02:49:58.444097
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li Bingyu and Li Fengying and Zhang Shiqing",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1402.4630"
}
|
1402.4701
|
# Contiguous $3d-$ and $4f-$magnetism:
towards strongly correlated $3d-$electrons in YbFe2Al10
P. Khuntia Max Planck Institute for Chemical Physics of Solids, 01187
Dresden, Germany P. Peratheepan Highly Correlated Matter Research Group,
Physics Department, University of Johannesburg, P.O. Box 524, Auckland Park
2006, South Africa Department of Physics, Eastern University, Vantharumoolai,
Chenkalady 30350, Sri Lanka A. Strydom Max Planck Institute for Chemical
Physics of Solids, 01187 Dresden, Germany Highly Correlated Matter Research
Group, Physics Department, University of Johannesburg, P.O. Box 524, Auckland
Park 2006, South Africa Y. Utsumi Max Planck Institute for Chemical Physics
of Solids, 01187 Dresden, Germany K.-T. Ko Max Planck POSTECH Center for
Complex Phase Materials, 01187 Dresden, Germany and Pohang 790-784, Korea
K.-D. Tsuei National Synchrotron Radiation Research Center, 101 Hsin-Ann
Road, Hsinchu 30077, Taiwan L. H. Tjeng Max Planck Institute for Chemical
Physics of Solids, 01187 Dresden, Germany F. Steglich Max Planck Institute
for Chemical Physics of Solids, 01187 Dresden, Germany M. Baenitz Max Planck
Institute for Chemical Physics of Solids, 01187 Dresden, Germany
([; date; date; date; date)
###### Abstract
We present magnetization, specific heat, and 27Al NMR investigations on
YbFe2Al10 over a wide range in temperature and magnetic field. The magnetic
susceptibility at low temperatures is strongly enhanced at weak magnetic
fields, accompanied by a $\ \ln(T_{0}/T)$ divergence of the low$-T$ specific
heat coefficient in zero field, which indicates a ground state of correlated
electrons. From our hard X-ray photoemission spectroscopy (HAXPES) study, the
Yb valence at 50 K is evaluated to be 2.38. The system displays valence
fluctuating behavior in the low to intermediate temperature range, whereas
above 400 K, Yb3+ carries a full and stable moment, and Fe carries a moment of
about 3.1 $\mu_{B}.$ The enhanced value of the Sommerfeld Wilson ratio and the
dynamic scaling of spin-lattice relaxation rate divided by T $\
[^{27}$($1/T_{1}T)]$ with static susceptibility suggests admixed ferromagnetic
correlations. 27($1/T_{1}T)$ simultaneously tracks the valence fluctuations
from the 4f -Yb ions in the high temperature range and field dependent
antiferromagnetic correlations among partially Kondo screened Fe 3d moments at
low temperature, the latter evolve out of an Yb 4f admixed conduction band.
NFL, NMR, Spin Fluctuations, QCP.
###### pacs:
71.27.+a, 74.40.Kb, 76.60.-k, 76.60.Es, 71.10.Hf
††preprint: APS/123-QED
year number number identifier Date text]date LABEL:FirstPage1
LABEL:LastPage#12
Novel phases ranging from unconventional superconductivity and spin liquid to
quantum criticality in correlated electron systems result from competing
interactions between magnetic, charge, orbital and lattice degrees of freedom
SS ; FS . Competing interactions such as the mostly antiferromagnetic (AFM)
Rudermann-Kittel-Kasuya-Yosida (RKKY) exchange and the Kondo effect on a
localized spin may lead to a magnetic instability which generates unusual
temperature ($T$) and magnetic field ($H$) scaling behavior of bulk and
microscopic observables. The competing magnetic interactions frequently
produce generalized non-Fermi liquid (nFL) scaling in the thermal behavior of
physical properties. If the RKKY spin exchange succeeds in overcoming the
thermal energy of the spin system conducive to a paramagnetic-to-AFM
transition, the addition of a competing Kondo spin exchange with the
conduction electrons achieves a curbing effect on the phase transition.
Moreover, under favorable conditions such as applied pressure or magnetic
field the phase transition may become confined to temperatures arbitrarily
close to zero, which in turn leads to remarkable thermal scaling in the realm
of quantum criticalityRMP1 ; RMP2 ; LH ; MB ; QS . In exceptional cases
quantum criticality presents itself under ambient conditions, such as in
U2Pt2In ams96 ; estrela99 or in the superconductor $\beta-$YbAlB4 SN .
Quantum criticality stemming from ferromagnetic exchange on the other hand is
a rare occurrence, and has been discussed among $5f-$electron systems such as
UGe2 SSC ; HK or UCoGeES ; TH , $4f$ systems like YbNi${}_{4}($P1-xAsx)2 RS1
; AS , Ce(Ru1-xFex)POSK1 ; SK2 , and in weak itinerant ferromagnets like ZrZn2
PRS and NbFe2 MB2 . YFe2Al10, an isostructural version of YbFe2Al10 with no
$4f-$electrons, is reported to be a plausible candidate for a FM quantum
critical magnetPK ; AM ; per10 ; KP ; str13 .
The ternary orthorhombic aluminides of $RM_{2}$Al10 type ($R=$rare earth
element, $M=$Fe, Ru, Os) have been the subject of considerable debate in view
of a fascinating conundrum of physical properties. Most notable are the
extremes of magnetic interactions found in the Ce series ranging from
unprecedently high AFM order at $27~{}$K in CeRu2Al10 CS ; SC ; AM1 ; DD to
the Kondo insulating state in CeFe2Al10 YM . In the present study to further
unravel the nature of the $3d-$electrons in this class of material, we assess
the response of Fe-based magnetism in the presence of localized magnetism,
namely the rare earth element Yb, and we use a combination of bulk and
microscopic probes due to the anticipated complexity of an admixture of
different types of magnetic exchange. A comparable situation can be found in
CeFe2Al10 in which the confluence of the two types of magnetic species has the
surprising effect of producing the non-magnetic Kondo insulating state YM ,
which is an extreme case of local-moment hybridization with the conduction
electrons. Recently, there has been a resurgence of research activities in
intermediate valence systems following the discovery of superconductivity and
quantum critical behavior in an intermediate valence (IV) heavy fermion
$\beta-$YbAlB4SW ; DTA ; SP ; SN ; PC ; MO ; PC1 ; LMH ; WS .
In this Letter, we present comprehensive magnetic susceptibility, specific
heat, and 27Al NMR investigations on polycrystalline YbFe2Al10. Furthermore,
hard X-ray photoemission spectroscopy (HAXPES) at SPring-8, Japan was carried
out as a direct probe of the valence state of Yb. Magnetic susceptibility and
specific heat display low temperature divergences, yet without any signature
of magnetic ordering down to $0.35~{}$K. In order to understand the low energy
spin dynamics governing the underlying magnetism of the title compound, we
have carried out NMR investigations with special attention to the spin-lattice
relaxation measurements. The low field spin-lattice relaxation rate shows a
divergence towards low temperatures, which is consistent with magnetization
and specific heat data. The observed deviations from the FL behavior is
associated with correlated $3d$ Fe moments strongly coupled via the conduction
band, which is hybridized with the Yb derived 4$\mathit{f}$ states.
Polycrystalline samples of YbFe2Al10 have been sythesized following a method
discussed elsewhereAM ; AM1 . The dc magnetic susceptibility $\chi(T)$
(=$M(T)/H)$ and thermopower data were obtained using a QD PPMS.
In a recent work, a Kondo-like electrical resistivity accompanied by
divergences in magnetic susceptibility, $\chi(T)$ and the Sommerfeld
coefficient $\gamma$($T$) = Cp(T)/T (where Cp($T$) is the electronic specific
heat) in zero field were reported per10 on YbFe2Al10. The magnetism of Yb in
this compound was demonstrated PKSS to be subject to an unstable valence and
to recover its full trivalent state at $T$ $>$ 400 K, which is in agreement
with earlier reportsMV .
Shown in Fig. 1 (a) is the field dependent magnetic susceptibility, $\chi(T)$
of YbFe2Al10. The values of $\chi(T)$ are enhanced by one order of magnitude
in comparison with the non-$4f$ electron homologue YFe2Al10 PK , which
indicates a strong hybridization of the Yb derived $4f$ states with the
conduction electron states. A modified band structure is therefore expected
with subsequent effects on the itinerant $3d$ magnetism of Fe. Towards
elevated temperatures, Yb tends to reach its full trivalent state and this
temperature-driven evolution is appropriately reflected in the thermopower
$S(T)$ i.e., by a broad peak centered at $T$* ($\approx$100 K) (see Fig.
1(d)), which we use to denote the temperature scale of the valence change of
Yb. Such a peak in thermopower is typical for IV Yb compounds, and our HAXPES
study on YbFe2Al10 confirmed such an IV state of Yb in this compound and
valence of Yb is evaluated to be 2.38 (see Supplemental Material for more
details). At the high temperature end, the consequence of this peak is played
out by a change in the sign of $S(T)$ at about $300~{}$K, which likely implies
a temperature-driven change in the relative weights and participation of both
holes and electrons in the underlying bandstructure. This could also be due to
the asymmetry of the density of states (DOS) or the scattering rate at the
Fermi energy for a single band. However, the negative sign in $S(T)$ of
YbFe2Al10 signals stable and local-moment magnetic character of Yb above 300
K, because S($T$) native to the weakly hybridized $4f^{13+\delta}$ state of Yb
is expected to be negative beh04 .
The small upturn in S($T$) below 10 K is consistent with the incoherent Kondo
like restistivity $\rho(T)$per10 . A peak in $\rho(T)$ at T $\simeq$ 4.5 K
(see Supplemental) is reminiscent of Kondo-lattice behaviorper10 .The Kondo
type upturn in $\rho(T)$ as well as the low$-T$ divergence in $\chi(T)$ is
quenched by applying magnetic fields of a few teslas. The initial
susceptibility $\chi(H\rightarrow 0)$ at $2~{}$K as well as the high-field
magnetization $M(H)$ yield extremely small values of the magnetic moment in
YbFe2Al10. Following a weak curvature in the M($H$) in low fields, there is
however no saturation achieved in $M(H)$ at $2~{}K$ even up to 7 T , Fig.
1(c), where a quasi-linear in field magnetization is found.
Figure 1: (Color online) Temperature dependence of $\chi$ in various applied
magnetic fields (b) $1/\chi$ _vs_. $T$ at $0.1~{}$T with Curie-Weiss fits. (c)
Magnetization isotherm at $2~{}$K and the inset shows the $\chi^{-1}$ vs. T in
5 kOe and 10 kOe with Curie Weiss fit as discussed in the text (d) Temperature
dependence of thermopower.
A detailed analysis of $\chi(T)$ (see Fig 1b) reveals an intermediate valence
(IV) state of Yb at low and intermediate temperatures, but Yb recovers its
full moment (4.54 $\mu_{B}$) with a high spin state Fe (3.1 $\mu_{B}$) at
$T>$400 K with predominant AFM correlations. A similar scenario has been
discussed in the IV Yb-based skutterudites YbFe4Sb12 sch05 and YbFe4P12AY .
The deconvolution of the Fe$-3d$ contribution and the more localized Yb$-4f$
contribution is a daunting task and beyond the scope of this manuscript.
Nonetheless, based on the model of RajanVTR , for the Yb$-4f$ part a constant
and $T$-independent susceptibility could be expected towards low temperatures.
The HAXPES measurement performed with h$\upsilon$=6.5 keV at 50 K confirms the
IV state of Yb with valence 2.38 (see Supplemental). Therefore, we assume that
the magnetism below 50 K is solely driven by the Fe 3$d$ moments in YbFe2Al10
and we speculate that the Curie-Weiss behavior of $\chi(T)$ in the
intermediate temperature range 80$\leq$T$\leq$370 K is associated to Fe-3d
moments with an effective moment of 3.1/$\sqrt{2}$= 2.2 $\mu_{B}.$This is in
contrast to its non-4$f$ analog YFe2Al10 where Fe carries a much smaller
magnetic moment of 0.35 $\mu_{B}$ per FePK ; KP . This might be related to the
difference in charge transfer from the divalent Yb2+ to the Fe2Al10 host
lattice in comparision to the trivalent Y3+. The Curie-Weiss behavior of
$\chi(T)$ at $T\leq$10 K (see Fig. 1(c) inset) reveals a small Fe moment
0.89/$\sqrt{2}$=0.63 $\mu_{B}$ per Fe in YbFe2Al${}_{10}.$ The Weiss
temperature of $\theta\simeq$ –2 K yields an on-site Kondo temperature of the
Fe moments, which amounts to several KelvinsAH .
Shown in Fig. 2(a) is the specific heat coefficient $\gamma$($T$) in different
magnetic fields measured using the 3He option of QD PPMS. The specific heat
coefficient ($\gamma)$ is enhanced towards low temperatures and follows a
$\ln(T_{0}/T)$ behavior with $T_{0}$= 2 K in zero field, which suggests a
correlated behavior of electrons. This may be attributed to entropy of
unquenched spin degrees of freedom, or to impending cooperative behavior at
much lower temperatures. Applied magnetic fields achieve a suppression and
eventual saturation into a constant value of $C_{\mathrm{p}}/T$ and thus the
recovery of the Fermi liquid ground state. The ratio of the enhanced
$\gamma_{0}$ value at zero field to the fully quenched value
$\gamma_{\mathrm{H}}$ in $9~{}$T at $0.35~{}$K is about $2.5$. Surprisingly,
this enhancement factor is qualitatively similar to that of the non-$4f$
compound YFe2Al10 PKSS . Despite the fact that the relative enhancements
$\Delta\gamma/\gamma_{H}=(\gamma_{0}-\gamma_{H})/\gamma_{H}$ are similar, it
should be mentioned that the $T$ dependencies of Cp/$T$ are dissimilar:
(ln($T_{\mathit{0}}$/$T$) for YbFe2Al10 and power law behavior in case of
YFe2Al10). For YbFe2Al10 the magnetic entropy (0.014Rln2) below 2 K is about
three times larger than its non-4f counter part YFe2Al10ESR . Therefore, we
relate the low-temperature divergence of the Sommerfeld coefficient to the
emergence of correlations among Fe moments amplified by the strong
hybridization between Yb$-4f$ states and $s+d$ conduction band states at the
Fermi level. The field dependence of the Sommerfeld coefficient at $0.5~{}$K
follows a $H^{-0.35}$ behavior (Fig. 2b), and the transition to a constant in
temperature regime provides the crossover scale between FL and nFL behavior
(inset of Fig. 2b).
Figure 2: (Color online) (a) Temperature dependence of specific heat
coefficients taken in different applied magnetic fields. The solid line is a
fit to $\ln(T_{0}/T)$ with T0 = 2 K. (b) $C_{\mathrm{P}}(T)/T$ _vs._
$\mu_{0}H$ at $0.5~{}$ and $2~{}$K, with solid line is a fit to H-0.35. The
inset shows the field dependence of the cross over temperature (to FL
behavior) obtained from $C_{\mathrm{P}}(T)/T$.
The residual quenched Sommerfeld coefficient of $\gamma_{H}$ = 75.3 mJ/mol K2
in YbFe2Al10 exceeds that of the La equivalent YM by a factor $\sim 3$, which
indicates that the Fermi level in YbFe2Al10 is occupied predominantly by heavy
charge carriers. An enhanced value of the Sommerfeld-Wilson ratio
$R_{\mathrm{W}}=\pi^{2}k_{\mathrm{B}}^{2}/\mu_{0}\mu_{\mathrm{eff}}(\chi/\gamma)\approx
12$ at $2~{}$K indicates the presence of FM correlations. It is worth to
mention that there is a striking similarity of YbFe2Al10 specific heat data
shown in Fig. 2a to those of $\beta-$YbAlB4 SN , which is a rare example of an
IV system with local moment low-T electron correlationsLMH . Another prominent
example in that context is the IV metal YbAl3ZF .
27Al-NMR ($I=5/2$) measurements have been performed using a standard _Tecmag_
NMR spectrometer in the temperature range $1.8\leq T\leq 300~{}$K and in the
field range $0.98\leq\mu_{0}H\leq 7.27~{}$T. The orthorhombic crystal
structure of YbFe2Al10 hosts five inequivalent Al sites. Usually this results
in rather broad NMR spectra with a clear central transition and superimposed
first order satellite transitions. Surprisingly, we found a rather well-
resolved central transition with a small field dependent anisotropy, which
implies that the different Al sites are rather equal in their magnetic
environmentPK ; PKSS .
Figure 3: (Color online) 27Al NMR spectra at $80~{}$MHz for different
temperatures.The inset shows the simulation of the $4.3~{}$K spectra.
The sharp central transition enables us to perform 27Al spin-lattice
relaxation rate (SLRR) measurements consistently following saturation recovery
method with suitable _rf_ pulses and the results are shown in Fig.4. The
relaxation rate divided by $T$, _i.e._ $1/T_{1}~{}T$, shows a divergence
towards low temperatures (Fig. 4a) with a proportionality $\chi(T)/\sqrt{T}$
in the lowest magnetic fields. Such a dynamic scaling is frequently found in
heavy fermion systems with AFM correlations and even with admixed FM
correlations like in CeFePO DC ; NB ; MB1 . In addition, the relative change
$\left[(\gamma_{0}-\gamma_{\mathrm{H}})/\gamma_{\mathrm{H}}\right]^{2}=1.7$
underestimates the SLRR enhancement found in the experiment ($\simeq 4.6$).
The stronger enhancement in the SLRR points towards the presence of dominant
$q=0$ contributions, as a response to FM correlations. Usually the specific
heat is more sensitive to finite $q$ excitations which explains the difference
in the enhancement factors. In contrast with the discrepancy in the $T$
enhancement, the field dependence of the SLRR is in agreement with the Fermi
liquid theory exhibiting $1/T_{1}T\sim(C/T)^{2}\sim\gamma^{2}$ behavior. Here
a power law $H^{-0.35}$ is found for the Sommerfeld coefficient which implies
a power law $H^{-0.7}$ for the SLRR. This field dependence is indeed found for
the SLRR at $2.5~{}$K (see Fig. 4b with $\mathit{H}^{-0.77}$), which is
further supported by $1/T_{1}T\sim\chi^{2}$ behavior commonly found in local-
moment metals and is in contrast to that observed in YFe2Al10 PK .
Independent of the magnetic field a peak (Fig.4a) in the SLRR at
$T^{\ast}$($\simeq$100 K) signals the onset of valence fluctuations in the
SLRR of the Al nuclei at high temperature. In general the SLRR probes the
$q-$averaged low lying excitations in the spin fluctuation spectra and
$1/T_{1}T$ can be expressed as; $\frac{1}{T_{1}T}\propto\sum_{q}\mid
A_{hf}(q)\mid^{2}\frac{\chi^{\prime\prime}(q,\omega_{n})}{\omega_{n}}$ ,where
$A_{hf}(q)$ is the $q-$dependent form factor of the hyperfine interactions and
$\chi^{\prime\prime}(q,\omega_{n})$ is the imaginary part of dynamic electron
spin susceptibility TM ; YK . In the presence of $q-$isotropic $4f$
fluctuations of IV Yb coexisting with $q=0$ FM $3d$ correlations, the SLRR
could be approximated by $\frac{1}{T_{1}T}\simeq
A_{hf}{}^{2}\chi_{0}\left[\tau_{3d}+\tau_{4f}\right]$, where
$\tau_{4f}(=1/\Gamma_{\mathrm{4f}}$) is the effective fluctuation time of the
$4f$ ion, $\tau_{3d}(=1/\Gamma_{\mathrm{3}\mathit{d}}$ ) is the effective
fluctuation time of the $3d$ ion ($\Gamma_{4f},_{3d}$ are corresponding
dynamic relaxation rates), and $\chi_{0}$ is the uniform bulk suceptibility.
It has to be mentioned that in case of large valence variations (like in Eu
systems where the valence could vary between 2+ and 3+) the electronic
structure may be perturbed which changes $A_{\mathrm{hf}}$, but we omit this
detail for YbFe2Al10 and assume that $A_{\mathrm{hf}}$ is not varying with
temperature.
Figure 4: (Color online) (a)27($1/T_{1}T$ ) _vs._ $T$ in different applied
magnetic fields. The solid line is the calculated value as discussed in the
text. (b) The field dependence of 27($1/T_{1}T)$ at 2.5 K with a fit to
H-0.77.(c) The temperature dependence of $\tau_{4f}$ at 7.27 T.
The beauty of these results is that 27 Al NMR simultaneously senses the
valence fluctuations from the 4$\mathit{f}$\- Yb ions in the high temperature
range and the low temperature field dependent Kondo-like correlations
associated to the 3$\mathit{d}$\- Fe ions. Upon the application of high
magnetic fields these fluctuations are quenched (here $\tau_{3d}\ll$
$\tau_{4f}$ for entire temperature range). Therefore, the relaxation rate at
7.27 T allows for the determination of the effective fluctuation time
$\tau_{4f}=1/\Gamma_{4f},$which is plotted as a function of temperature in
Fig. 4(c). The step like change of $\tau_{4f}$ at about 100 K signals more a
charge gap scenario (like in Kondo insulators) than an intermediate valence
system with a smooth variation in $\tau_{4f}$. With the knowledge of the T
dependent (but not $\mathit{H}$-dependent) relaxation time $\tau_{4f},$ we now
proceed to fit the 1/$T_{1}T$ vs. $T$ results in low magnetic field.
Surprisingly, the assumption of $\tau_{3d}=1/\sqrt{T}\propto\chi$ results in a
very good agreement with the experimental data (red line in Fig. 4(a)), which
also explains the 27(1/$T_{1}T$)$\propto\chi^{2}$ behavior at $T\rightarrow$0
limit. With this approach we have convincingly shown that NMR is able to probe
both energy regimes; i) the high-temperature IV regime where
$\Gamma_{\mathrm{4f}}$ is changing strongly and ii) the low-T regime where
$\Gamma_{\mathrm{4f}}$ is constant and $\Gamma_{\mathrm{3d}}$ shows a local
moment behavior with $\Gamma_{\mathrm{3d}}\propto\sqrt{T}$.
In conclusion, we have found an unexpected localization of Fe-derived 3$d$
states upon cooling YbFe2Al10 to helium temperatures. As in this material the
Yb-derived 4$f$ electrons form a non-magnetic, intermediate-valent state at
low temperatures (with Yb valence 2.38), the observed Kondo-lattice behavior
has to be attributed to the localized 3$d$ electrons. Because of the low on-
site Kondo scale of $\mathit{T}_{0}$ $\approx$ 2 K, one would expect the 3$d$
magnetic moments to be subject to some kind of long-range orderingMI .
However, this appears to be avoided, at least above 0.4 K, by a competition
between ferro- and antiferromagnetic correlations, which have been inferred
from a strongly enhanced Sommerfeld Wilson ratio on the one hand and the field
dependencies of the specific heat and spin-lattice relaxation rate on the
other.
We thank C. Geibel, A. P. Mackenzie, H. Yasuoka, M. C. Aronson, M. Brando, and
M. Garst for fruitful discussions. We thank C. Klausnitzer for technical
support concerning specific heat measurements. We thank the DFG for financial
support (project OE-511/1-1). AMS acknowledges support from SA-NRF (78832).
*[email protected]
## References
* (1) S. Sachdev, Quantum Phase Transitions (Cambridge University Press, 1999).
* (2) P. Gegenwart, Q. Si, and F. Steglich, Nature Physics 4, 186 (2008).
* (3) G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001).
* (4) G. R. Stewart, Rev. Mod. Phys. 78, 743 (2006).
* (5) H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Rev. Mod. Phys. 79, 1015 (2007).
* (6) M. B. Maple et al., J. Low. Temp. Phys. 161, 4 (2010).
* (7) Q. Si and F. Steglich, Science 329, 1161(2010).
* (8) A. M. Strydom et al., Physica B 223& 224, 222 (1996).
* (9) P. Estrela et al., Physica B 259–261, 409 (1999).
* (10) S. Nakatsuji et al., Nature Phys. 4, 303 (2008).
* (11) S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, I. Sheikin, D. Braithwaite, and J. Flouquet, Nature 406, 587 (2000).
* (12) H. Kotegawa et al., J. Phys. Soc. Jpn. 80, 083703 (2011).
* (13) E. Slooten, T. Naka, A. Gasparini, Y. K. Huang, and A. de Visser, Phys. Rev. Lett. 103, 097003 (2009).
* (14) T. Hattori et al., Phys. Rev. Lett. 108, 066403 (2012).
* (15) R. Sarkar et al, Phys. Rev. B 85, 140409(R) (2012).
* (16) A. Steppke et al., Science 339, 933 (2013).
* (17) S. Kitagawa, K. Ishida, T. Nakamura, M. Matoba, and Y. Kamihara, Phys. Rev. Lett. 109, 227004 (2012).
* (18) S. Kitagawa et al., J. Phys. Soc. Jpn. 82,033704 (2013).
* (19) R. P. Smith et al., Nature 455, 1220 (2008).
* (20) M. Brando et al., Phys. Rev. Lett. 101, 026401(2008).
* (21) P. Khuntia et al., Phys. Rev. B 86, 220401(R) (2012).
* (22) A. M. Strydom et al., Phys. Stat. Solidi 4, 356 (2010).
* (23) A. M. Strydom et al., Phys. Status Solidi RRL 4, 356 (2010).
* (24) K. Park, L. S. Wu, Y. Janssen, M. S. Kim, C. Marques, and M. C. Aronson, Phys. Rev. B 84, 094425 (2011).
* (25) A.M.Strydom et al., Phys. Status Solidi B 250 630 (2013).
* (26) C. S. Lue et al., Phys. Rev. B 82, 045111 (2010).
* (27) S. C. Chen and C. S. Lue, Phys. Rev. B 81, 075113 (2010).
* (28) A. M. Strydom, Physica B 404, 2981 (2009).
* (29) D. D. Khalyavin, A. D. Hillier, D. T. Adroja, A. M. Strydom, P. Manuel, L. C. Chapon, P. Peratheepan, K. Knight, P. Deen, C. Ritter, Y. Muro, and T. Takabatake, Phys. Rev. B 82, 100405 (R) (2010).
* (30) Y. Muro et al., J. Phys. Soc. Japan 78, 083707 (2009).
* (31) S. Watanabe et al., J. Phys.: Condens. Matter 24, 294208 (2012).
* (32) D. T. Adroja et al., J. Magn. Magn. Mater. 100, 126 (1991).
* (33) S. Patil et al., Phys. Rev. B. 47, 8794 (1993).
* (34) Y. Matsumoto, S. Nakatsuji, K.Kuga, Y. Karaki, N. Horie, Y.Shimura, T. Sakakibara, A.H. Nevidomskyy, and P. Coleman, Science 331, 316 (2011).
* (35) M. Okawa et al., Phys. Rev. Lett. 104, 247201(2010).
* (36) A. H. Nevidomskyy and P. Coleman, Phys. Rev. Lett. 102, 077202 (2009).
* (37) L. M. Holanda, J. M. Vargas, W. Iwamoto, C. Rettori, S. Nakatsuji, K. Kuga, Z. Fisk, S. B. Oseroff, and P. G. Pagliuso, Phys. Rev. Lett. 107, 026402 (2011).
* (38) W. Schnelle et al., Phys. Rev. B. 77, 094421 (2008).
* (39) P. Khuntia et al., Phys. Status Solidi B 250, 525 (2013).
* (40) V. M. T. Thiede et al., J. Mater. Chem. 8, 125 (1998).
* (41) K. Behnia et al., J. Phys. Condens. Matter 16, 5187 (2004).
* (42) W. Schnelle, A. Leithe-Jasper, M. Schmidt, H. Rosner, H. Borrmann, U. Burkhardt, J. A. Mydosh, and Y. Grin, Phys. Rev. B 72, 020402(R) (2005).
* (43) A. Yamamoto et al., J. Phys. Soc. Jpn. 75, 063703 (2006).
* (44) V.T. Rajan, Phys. Rev. Lett. 51, 308 (1983).
* (45) A. J. Heeger, Solid State Phys. 23, 283 (1969).
* (46) E. S. R. Gopal, Specific Heat at Low Temperatures (Plenum Press, New York, 1966).
* (47) U. Walter, E. Holland-Moritz and Z. Fisk, Phys. Rev. B 43, 320 (1991).
* (48) D. L. Cox et al., J. Appl. Phys. 57, 3166 (1985).
* (49) N. Büttgen, R. Böhmer, A. Krimmel, and A. Loidl, Phys. Rev. B 53, 5557 (1996).
* (50) E. Bruening et al., Phys. Rev. Lett. 101, 117206 (2008).
* (51) T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springer-Verlag: Berlin, 1985).
* (52) Y. Kuramoto and Y. Kitaoka, Dynamics of Heavy Electrons (Oxford, 2000).
* (53) V.Yu. Irkhin and M.I. Katsnelson, Phys. Rev. B 56, 8109 (1997). SUPPLEMENTAL MATERIALS
## .1 I. Magnetic susceptibility
The temperature dependence of the dc magnetic susceptibility $\chi$($T$) was
measured in different magnetic fields in the temperature range 1.8$\leq T\leq
300$ K using Quantum Design (QD) SQUID magnetometer. In addition, measurements
up to 800 K were carried out using the SQUID-Vibrating Sample Magnetometer.
The Curie-Weiss (CW) fit of the inverse magnetic susceptibility 1/$\chi$($T$)
in the temperature range 80 $\leq T\leq$ 370 K yields an effective Fe moment
of $\mu_{3d}$ =3.1/$\sqrt{2=}$ 2.2 $\mu_{B}$ whereas the CW fit in the high
temperature range 400 $\leq$ $T\leq$ 800 K results in an effective moment of
6.3 $\mu_{B}$ which is too large to assign to Yb3+ exclusively (Fig. 1b of the
manuscript). This is due to the combined role of Fe and Yb magnetic moments on
the underlying magnetism of this system. Hence we associate this net magnetic
moment to arise from two Fe (two identical octahedral sites occupied by Fe in
the lattice) and one Yb moments. This scenario is most likely in view of the
intermediate valence state of Yb in YbFe2Al10, which is confirmed by more
definitive probe for valence transition i.e., HAXPES and the results are
discussed in the following section. The negative values of the Weiss
temperatures obtained from the CW fit in both temperature ranges suggest the
dominant antiferromagnetic correlationsPK1 ; VMT . The observed magnetic
susceptibility is strikingly different from the recently reported
non-4$\mathit{f}$ homologue critical ferromagnet YFe2Al10 PK2 . In addition
the CW fit of the magnetic susceptibility data in the $T\leq$ 10K unveil a
very low Weiss temp, $\theta_{CW}\simeq$ – 2 K, which signals an on-site Kondo
screening between Fe 3$\mathit{d}$ moment with Kondo temperature of a few
Kelvin. Furthermore, an evaluation of the stability of the FM phenomena in
YFe2Al10 revealed that the quantum criticality found in YFe2Al10 is not pliant
with small variation in the Fe contentAMS1 . So the present compound YbFe2Al10
offers a fertile ground to study the low temperature correlation among
3$\mathit{d}$ Fe along with high temperature fluctuations due to 4$\mathit{f}$
Yb moments.
## .2 II. Electrical resistivity
The temperature dependence of resistivity $\rho$($T$) in YbFe2Al10 was
obtained in the temperature range 1.8$\leq T\leq 50$ K in three applied
magnetic fields for the first sample and in an extended temperature down to
0.4 K in zero field and 9T for the second sample. The magnetic contributions
to the resistivity were obtained by subtracting the resistivity of LaRu2Al10
as a non magnetic reference. The resulting normalized [
$\rho$($T$)/$\rho(40K)]$ data are shown in Fig.5. The $\rho$(T)/$\rho(40K)$
data exhibits a logarithmic divergence in the intermediate temperature regime
and passes through a maximum at about 4.5 K and decreases at low
temperature.The crossover from incoherent Kondo scattering to coherent Kondo
scattering behavior of the resistivity below 4.5 K can be interpreted in the
framework of the Kondo impurity model with $\mathit{S}$ = 1/2DR ; TAC ; TAC2 .
Figure 5: (Color online) (a) Temperature dependence of $\rho$(T)/$\rho(40K)$
in various applied magnetic fields for sample 1. (b) Temperature dependence of
$\rho$(T)/$\rho(40K)$ in various applied magnetic fields for sample 2.
The field dependence of electrical resistivity is consistent with the
interpretation of Costi for dilute magnetic alloysTAC ; TAC2 ; WF and could
be explained as well in good agreement with our experimental data. The
observed behavior in resistivity is reminiscent of a Kondo-lattice behavior
and therefore clearly evidences the presence of strong electron correlations
among Fe ($\mathit{S}$ = 1/2) moments at low temperatures. These results are
consistent with magnetic susceptibility data wherein the low temperature
admixed FM correlations are dominated by Fe moments and Yb displays an
intermediate valence state.
## .3 III. Specific heat
As a further measure of the low temperature correlations, specific heat
studies on polycrystalline samples have been performed in various applied
fields in the temperature range 0.35 $\leq T\leq$ 10 K using the 3He option of
QD PPMS. It may be noted that in high magnetic fields $C_{p}$($T$)/$T$
slightly increases at very low temperature, which is attributed to the high
temperature part of the nuclear Schottky contributions. This part could be
described as $C_{N}$=$\beta$/$T^{2}$ (where $\beta$ depends on magnetic
field), which appears because of the Zeeman interaction and the electric field
gradient at the nuclear site responsible for lifting the degeneracy of the
nuclear energy levels. The nuclear Schottky contributions to the specific
heats in magnetic fields are subtractedESR1 and the resulting plot is shown
in Fig. 2a in the manuscript.
## .4 IV. 27Al (I = 5/2) Nuclear Magnetic Resonance (NMR)
27Al-NMR ($I=5/2$) spectra and spin-lattice relaxation measurements have been
performed using a standard _Tecmag_ NMR spectrometer in the temperature range
$1.8\leq T\leq 300~{}$K and in the field range $0.98\leq\mu_{0}H\leq
7.27~{}$T. YbFe2Al10 hosts five inequivalent Al sites but different Al sites
are rather equal in their magnetic environment, which are very similar to our
previous NMR results on the structural homologue YFe2Al10PK . There are no
sharp features assigned to the first order quadrupolar transitions and no
appreciable shift observed in the 27Al-NMR line (Fig. 3 of the manuscript),
but instead a broadening of the central transition with decreasing
temperatures is found. The line width (FWHM) increases with decreasing
temperature and scales with the bulk susceptibility yielding a Curie-Weiss
like behavior. At the lowest $T$ and in small magnetic fields the scaling of
FWHM with $\chi(T)$ breaks down, which is in-line with the expected behavior
at the onset of electronic correlations. Furthermore, the narrow central
transition evidences high purity of the sample studied here and indicates the
absence of onsite disorder and Al-Fe site exchange.
The recoveries of longitudinal magnetization at time $\mathit{t}$,
$\mathit{M}_{z}$($\mathit{t}$) after the saturation pulse in the temperature
and field range of the present investigation were fitted with a single
component appropriate for $\mathit{I}$= 5/2 nuclei
1-$\mathit{M}_{\mathit{z}}$($\mathit{t}$)/$\mathit{M}$($\infty$)
=0.0291e${}^{-t/T_{1}}$+0.178e${}^{-6t/T_{1}}$+0.794e${}^{-15t/T_{1}},$where
M($\infty$) is the equilibrium magnetizationPK2 ; AN .
## .5 V. Hard x-ray photoemission spectroscopy(HAXPES)
In order to gain further insights into the electronic states and to determine
the valence state of Yb in YbFe2Al10, hard x-ray photoemission spectroscopy
(HAXPES) with $h\nu=6.5$ keV were performed at BL12XU of SPring-8, Japan. The
HAXPES spectra were taken by using a hemispherical analyzer (MB Scientific
A1-HE) and the overall energy resolution was set to about 0.2 eV. Clean
surface of the sample was obtained by fracturing in situ and the spectra were
measured at 50 K. The binding energy of the spectra was calibrated by the
Fermi edge of a gold film.
Figure 6: Wide scan of YbFe2Al10 measured at 50 K. No O 1$s$ and C 1$s$ were
observed.
A full-range energy HAXPES spectrum of YbFe2Al10 revealing the well-resolved
three elemental contributions from the title compound is shown in Fig. 6. We
note the absence of any extraneous contributions such as oxygen or elemental
carbon.
Figure 7: Yb 3$d$ spectrum of YbFe2Al10 measured at 50 K. The Al $1s$ peak
with its plasmon peaks are also indicated.
Figure 7 depicts the Yb $3d$ spectra of YbFe2Al10 measured at 50 K. The Yb
3$d$ spectrum is split into 3$d_{5/2}$ region at 1515-1540 eV and 3$d_{3/2}$
region at 1565-1580 eV due to the spin-orbit interaction. A strong Al 1$s$
peak can be observed around 1560 eV and its plasmon peaks are visible at 16.3
eV higher binding energies and multiples thereof.
Figure 8: The Yb 3$d_{5/2}$ spectra of YbFe2Al10 measured at 50 K and its
simulation consisting of the Yb2+ and Yb3+ $3d_{5/2}$ multiplets and their
plasmon satellites as well as an integral backgound.
Figure 8 shows in more detail the Yb $3d_{5/2}$ spectra of YbFe2Al10 measured
at 50 K. Here, we will evaluate only the 3$d_{5/2}$ part of the Yb 3$d$
spectra because the tail of the enormous Al 1$s$ peak and its plasmon
structure are overlapping with the 3$d_{3/2}$ region. The Yb2+ component is
observed as a prominent peak at 1520 eV and the Yb3+ component shows up at
1525-1535 eV as a multiplet structure arising from the Coulomb interaction
between the 3$d$ and 4$f$ holes in the electronic configuration of the
3$d^{9}4f^{13}$ final states. The structures at higher binding energies can be
attributed to the plasmon satellites of these Yb core levels. Coexistence of
the Yb2+ and Yb3+ structures indicates directly the intermediate valence
states of Yb in YbFe2Al10.
To extract a number for the Yb valence in YbFe2Al10 we performed a simulation
of the spectrum by taking into account not only the Yb3+ line and the Yb2+
multiplet structure, but also their respective plasmon satellites, the
relative intensities and energy positions of which were calibrated using the
Al 1$s$ and its plasmons. Including also the standard integral background, we
were able to obtain a satisfactory simulation of the experimental spectrum,
see Fig.8. The Yb valence is then estimated to be 2.38. We note that a similar
HAXPES study was performed recently on the quantum critical intermediate
valent compound $\beta$-YbAlB4 by M. Okawa et alMOX . There, the much higher
prevalence of the magnetic Yb3+ state is consistent with the heavy fermion
ground state in $\beta$-YbAlB4, whereas in YbFe2Al10 the Yb3+ is found to play
a much more subdued role.
*[email protected]
## References
* (1) P. Khuntia et al., Phys. Status Solidi B 250, 525 (2013).
* (2) V. M. T. Thiede et al., J. Mater. Chem. 8, 125 (1998).
* (3) P. Khuntia et al., Phys. Rev. B 86, 220401(R) (2012).
* (4) A.M.Strydom et al., Phys. Status Solidi B 250, 630 (2013).
* (5) D. R Hamann, Phys. Rev. 158, 570 (1967).
* (6) T. A. Costi, Phys. Rev. Lett. 85, 1504 (2000).
* (7) T. A. Costi, A. C. Hewson, and V. Zlatic, J. Phys. Condens. Matter 6, 2519 (1994).
* (8) W. Felsch and K. Winzer, Solid State Commun. 13, 569 (1973).
* (9) E. S. R. Gopal, Specific Heat at Low Temperatures (Plenum Press, New York, 1966).
* (10) A. Narath, Phys. Rev. 162, 320 (1967).
* (11) M. Okawa et al., Phys. Rev. Lett. 104, 247201(2010).
|
arxiv-papers
| 2014-02-19T15:35:22 |
2024-09-04T02:49:58.457401
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. Khuntia, P. Peratheepan, A. Strydom, Y. Utsumi, K.-T. Ko, K.-D.\n Tsuei, L. H. Tjeng, F. Steglich, and M. Baenitz",
"submitter": "Panchanan Khuntia",
"url": "https://arxiv.org/abs/1402.4701"
}
|
1402.5083
|
# Fully Three-dimensional Simulation and Modeling of a Dense Plasma Focus
B. T. Meehan, J. H. J. Niederhaus B. T. Meehan is with National Security
Technologies, LLC, a Department of Energy Contractor, e-mail:
[email protected]. H. J. Niederhaus is a Computer Scientist at Sandia
National Laboratories, email: [email protected] received December
1, 2012; revised January 11, 2013.
###### Abstract
A Dense Plasma Focus (DPF) is a pulsed-power machine that electromagnetically
accelerates and cylindrically compresses a shocked plasma in a Z-pinch. The
pinch results in a brief ($\sim$100 nanosecond) pulse of X-rays, and, for some
working gases, also a pulse of neutrons. A great deal of experimental research
has been done into the physics of DPF reactions, and there exist mathematical
models describing its behavior during the different time phases of the
reaction. Two of the phases, known as the inverse pinch and the rundown, are
approximately governed by magnetohydrodynamics, and there are a number of
well-established codes for simulating these phases in two dimensions or in
three dimensions under the assumption of axial symmetry. There has been little
success, however, in developing fully three-dimensional simulations. In this
work we present three-dimensional simulations of DPF reactions and demonstrate
that 3D simulations predict qualitatively and quantitatively different
behavior than their 2D counterparts. One of the most important quantities to
predict is the time duration between the formation of the gas shock and
Z-pinch, and the 3D simulations more faithfully represent experimental results
for this time duration and are essential for accurate prediction of future
experiments.
###### Index Terms:
Dense Plasma Focus, Magnetohydrodynamics, Simulation and Modeling, Controlled
Fusion
## I Introduction
There is an extensive literature on experiments performed with Dense Plasma
Focus (DPF) machines, exploring both fundamental Z-pinch physics [1, 2, 3] and
applications of the DPF to fields like X-ray lithography [4], fusion energy
[5], and modeling of astrophysics [6], to name a few. The vast majority of
research has been driven by theory and experimentation, but there is a shift
towards developing new experiments informed by computational models and
simulations. Many of the simulations that have been used to design DPF
experiments are 1-dimensional, in the sense that current, temperature, or
expected neutron yield are computed as a function of a single parameter being
varied [7]. There has also been work to develop codes that are capable of full
magnetohydrodynamic (MHD) modeling of the inverse pinch and run-down phases of
the DPF reaction on spatial domains (see Section II for descriptions of the
reaction phases), and there exist 2D simulations modeling the physics of the
DPF, which are often extended to 3D assuming axial symmetry [8]. Despite some
success with 1D and 2D simulations, fully 3D simulations of the MHD phases of
the DPF – which do not impose symmetry on the physics of the reaction – are
difficult, and the literature presenting the results of such simulations is
sparse. There are a number of challenges in the 3D modeling, including the
computational complexity of the problem, a need for appropriate initial
conditions to ignite the inverse pinch, insufficient equation of state (EOS)
data for the working gas and the electrodes, and incorporating radiative
effects into the MHD model.
In a DPF, a working gas is charged, forming a plasma, that travels down an
anode surrounded by a cathode, and the speed that the plasma shock travels
down the anode is determined by the initial pressure and voltage in the
system. The Z-pinch occurs when the plasma shock gets to the end of the anode.
The energy available to the Z-pinch is maximized when the current through the
DPF is maximized, which means that the rundown time is one of the most
important quantities to properly simulate. If simulations correctly predict
rundown for a given initial pressure and voltage configuration, those settings
can be used in actual experimentation ensuring that the Z-pinch occurs with
maximum energy, which in turn results in maximum neutron yield. The neutron
yield can be determined experimentally using a Beryllium activation detector
[9] (for deuterium fusion), or a Praseodymium activation detector [10] (for
deuterium-tritium fusion). Further, when deuterium, or deuterium-tritium
mixtures are used as a fill gas, the neutron yield has a power-law
relationship [11] with the maximum current, which means that it is also
important to be able to faithfully simulate the maximum current. For these
reasons, most of our analysis centers on comparing experimentally-measured
current waveforms to the simulated current waveform.
In this work, we present simulation results of a fully 3D MHD model of a DPF
using the ALEGRA [12] multiphysics code developed at Sandia National
Laboratories. The simulations are run in both 2D and 3D, and the results are
compared to each other as well as to experiments run in the DPF lab at
National Security Technologies, LLC. The predictions of the 2D and 3D
computations are qualitatively and quantitatively different, as the 2D
simulations show systematically lower inductance, which results in
systematically higher currents but unrealistically fast rundown times. The 3D
simulations predict lower maximum current values but accurately represent the
true rundown time shown in experimental data. This demonstrates that, despite
the symmetric geometry of the machine, there are three-dimensional effects
present in the MHD physics that must be accounted for in order to faithfully
predict the outcome of DPF experiments.
## II DPF Physics and the Experimental Setup
The DPF used in our experiments, and the geometry of which was modeled in the
simulations, is formed of coaxial electrodes in a rarified deuterium
atmosphere (about 7 Torr). A two-stage Marx capacitor bank is charged, and
when discharged, breaks down the gas, forming a shock and starting the
“inverse pinch” phase of the reaction, in which the gas expands outward from
the anode to the cathode bars (see Fig. 1). Once the gas touches the cathode,
the “run-down” phase begins, and the plasma moves up the anode until it
Z-pinches at the top of the anode. These two phases, the inverse pinch and
rundown, are approximately governed by magnetohydrodynamics and are the
components of the reaction that are studied and simulated.
### II-A DPF Geometry and Setup
Fig. 1 shows the DPF setup used in our experiments. The outer electrode
(cathode) is formed of 24 copper bars, 0.375 inches thick and 30.75 inches
tall, in a ring with an inside diameter of 6 inches. The cathode is at ground
potential, and its bars are shorted at the top with a ring of copper. The
anode is a hollow copper tube with an outer diameter of 4 inches that stands
23.6 inches above the ground plane, capped with a hemisphere. The vacuum
chamber is 1 foot in diameter, and roughly 6 inches taller than the cathode
cage. A Pyrex insulator tube, which is about 0.5 inches thick and stands 8.63
inches above the cathode base, separates the anode and cathode.
Figure 1: A rendering of the DPF used for the models and experiments. The
anode is the dome-topped cylinder in the center; the cathode “cage” is the
collection of rectangular bars that surrounds the anode; and the insulator is
visible through the bars, at the bottom of the cathode cage. The vacuum
envelope is represented as the tall cylinder that surrounds the cathode and
anode, and the ground plate is the flat cylinder at the bottom. Some support
features, such as the cathode top support ring, have been left out of the
drawing.
The DPF is driven by a two-stage Marx capacitor bank, which is connected to
the plasma focus tube by 36 coaxial cables. The total capacitance of the bank
(when configured for discharge) is 432 $\mu$F, and the maximum total voltage
in discharge configuration is 70 kV, which makes the maximum stored energy of
the bank 1 MJ. The plasma shock is driven by an external circuit, and the
circuit model used in the MHD simulations is shown in Fig. 2. The discharge
switch in the circuit represents a collection of eight rail-gap switches that
are simultaneously triggered by a single spark gap. The series inductance
represents the transmission lines that feed power to the plasma focus tube and
was determined empirically by fitting exponentially-dampened sine waves to the
experimental data. The 10 nF capacitor in series with the small resistor
represents the imperfect capacitance of the terminal plates and the
transmission lines that supply the power to the plasma focus tube. The 120
$\Omega$ resistor in parallel with the plasma focus tube is the equivalent
parallel resistance of the safety resistors.
Figure 2: The equivalent DPF discharge circuit used in the MHD models. The top
connection of the DPF is to the anode, and the bottom connection is to the
cathode. The switch shown is a collection of eight triggered rail-gap
switches.
### II-B Faraday Current Diagnostic
The discharge current diagnostic is important for comparing simulations to
experiment, because it allows for the measurement of both the maximum current
through and rundown time of the DPF. As was noted above, these are two of the
most important quantities for simulations in order to accurately predict
neutron yield and to ensure that the maximum current runs through the system
at the time of the Z-pinch. On the NSTec DPF, the discharge current is
measured with a Faraday rotator [13], which uses the magnetically-induced
linear polarization rotation in quartz fibers to measure the current in a
circuit. The fiber is wound in a circular fashion around the anode, an
orientation that causes the fiber to follow the direction of the magnetic
field, allowing it to accurately measure the current. In (1), the polarization
rotation angle, $\Theta$ (in radians), is related to the permeability of the
vacuum, $\mu_{0}$, the current, $I$, the Verdet constant of the fiber, $V$,
and the number of loops the fiber makes around the anode $n$. The interaction
of the magnetic field, $\vec{B}$, with an element of the fiber’s length,
$\vec{d\ell}$, is then integrated over the path of the fiber around the anode.
In our arrangement, the fiber wraps around the anode $n=5.25$ turns. This path
is denoted by $\xi$, and since the fiber is either parallel with the magnetic
field or perpendicular to it, the rotation angle is
$\Theta=V\int_{\xi}\vec{B}\cdot\vec{d\ell}=\mu_{0}nVI,$ (1)
in MKSA units. The Faraday rotator current diagnostic is perferred over other
discharge current measurements, such as the Rogowski coil, because the Faraday
rotator gives current measurements that are not dependent on calibration
factors, but rather on an easily measurable geometric quantitiy: the number of
turns around the current to be measured. The only other factor that must be
determined is the Verdet constant for the fiber, which can either be measured
or obtained from a datasheet on the fiber. When properly set up, the Faraday
rotator is a reliable diagnostic.
Figure 3: Shown is a comparison of the current profiles for thirty-seven DPF
shots, all at the same voltage and pressure (37.5 kV and 7.28 Torr,
respectively). This demonstrates the consistency of the current produced by
the machine, as well as representative Faraday rotator data.
Fig. 3 shows experimentally-measured results from the Faraday rotator detector
for 37 DPF shots initiated with the same voltage and pressure. As can be seen,
the profiles are all nearly identical, which demonstrates both the shot-to-
shot consistency of the DPF and the reliability of the Faraday diagnostic. The
placement of the Faraday loop is important for understanding the current that
it measures. This is easier to show than it is to describe, so this is
included as fig. 4.
Figure 4: Shown is a diagram of where the Faraday coil is placed on the Gemini
DPF. In the cutaway, the red (darker) area represents the parts electrically
connected to the anode, and the violet (lighter) areas are considered to be at
cathode potential. The Faraday loop is shown as a hoop that is under the
center center conductor wires, and above the ground plate.
## III Modeling and Simulation of the DPF
The modeling for this project was performed with Sandia National Laboratories’
ALEGRA-MHD code. ALEGRA is a finite-element, multi-material, arbitrary
Lagrangian-Eulerian (ALE) shock hydrodynamic code designed for parallel
computing. It uses an operator-split edge-element formulation to simulate
resistive MHD in 2D and 3D high-deformation shocked media and pulsed power
systems.
ALEGRA provides fine control over how the simulation is performed through a
text file known as the “input deck.” The primary purpose of the input deck is
to define the simulation geometry, material composition, and physics to be
modeled, though it also allows a user to request output and provide runtime
controls. Part of defining the physics of a simulation is setting up the
equations of state for the gases, which was among the most challenging aspects
of the problem. Tabular equations of state provide the most effective means of
modeling the thermodynamic state of material in the simulations, which may be
in the solid, liquid, gaseous, or ionized state. Tabular EOS models were made
available through ALEGRA’s interface to Los Alamos National Laboratory’s
SESAME [14] data. Tabular representations of the Lee-More-Desjarlais (LMD)
electrical and thermal conductivity model [15] were also used here. The LMD
model combines empirical data with inferences from quantum molecular dynamics
modeling and density functional theory (QMD-DFT) to provide a conductivity
representation that spans the transition between conducting and insulating
conditions and has proven quite successful in this “warm dense matter” regime
[16, 17].
There are two approaches to defining the geometry on which to simulate the
reaction using ALGERA. The first is using ALEGRA’s built-in functionality, by
declaring pre-defined volumetric shapes (like spheres, prisms, pyramids, etc.)
and defining the material composition of each shape. For example, many DPF
machines have cylindrical copper cathode bars, and it is possible to define in
the input deck a cylinder made of copper. The code then generates a 2D or 3D
unstructured mesh that overlays the defined shapes, allowing for mesh elements
to intersect more than one user-defined shape. The second approach to defining
the geometry is by performing the computation on a body-fitted mesh generated
using an external meshing tool. The body-fitted mesh method is a more accurate
method of describing the material in the simulation volume, but is more time-
consuming to set up than the geometric method. ALEGRA’s built-in functionality
was used to define the geometry and material composition in all of the
simulations shown in what follows, and boundary conditions are specified on
subsets of nodes or faces within the simulation volume.
In any simulation, it is important that the geometry and the physics being
simulated reflect the important geometries and physics of the experiment, and
both raise several concerns in our simulations. For example, it is not
necessary to include a faithful model of the vacuum chamber in our
simulations, since the plasma does not interact with the top of the chamber
during the simulation. Evaluating the physics being modeled, it is important
to understand that the plasma in the DPF reaction is driven by an external
electrical network, and a great strength of the ALEGRA-MHD code is that it has
a sophisticated built-in circuit solver that can be used to couple electrical
energy from user-specified circuits into the simulation volume.
Near the end of the simulation, just prior to the Z-pinch, the MHD simulation
begins to become unphysical because of its inability to represent certain
phenomena, such as, the kinetic instabilities which raise the plasma
resistivity. The simulation may run past the point of Z-pinch without
crashing, but the time-evolution of the simulation would be unphysical. Once
the MHD simulation begins to approach the Z-pinch, it is possible to transfer
the model state information to a particle-in-cell model to accurately simulate
the Z-pinch, which has been demonstrated by researchers at LLNL[18] and
SNL[8].
Setting up the initial condition for the simulation can also be tricky, since
the DPF’s starting state happens when the gas in the chamber breaks down in
the vicinity of the insulating sheath and becomes a ring-shaped plasma shock.
The breakdown of the gas is not covered by MHD physics, so the gas near the
insulator has to be initialized in an artificial state that will quickly
transition to the plasma shock known experimentally to exist in the DPF. We
have found that setting up a thin layer of extremely hot ($\sim 10^{6}$ K, and
therefore conductive) gas on the surface of the insulator results in the
simulation initiating a plasma shock without causing observable artifacts in
the time evolution of the simulation. Our experience with this initial
condition is that the thin layer of hot gas should be about as thin as the
Pyrex insulator and should touch both the anode and the cathode. Using this
initiation of the plasma shock results in temperatures that match data from
particle-in-cell calculations [8]. The artificially hot gas layer should
stabilize its temperature near the shock temperature, $\sim 10^{4}K$, within a
few solver timesteps (typically, about 20 nanoseconds).
### III-A Two-dimensional Modeling
One and two dimensional models of the DPF are the most common in the
literature, frequently coming in two generic types: empirical models and
finite-element MHD models. The primary strength of empirical models, such as
RADPFV5.5de [19], or Scat95 [20] is that they give results that are often very
close to experimental data. A significant drawback is that they require
existing data to fit the model to, and can therefore only be used to simulate
devices that are quite similar to the devices that one has experimental data
for already. The second class are finite-element codes such as MHRDR [21] and
Mach2 [22] that perform MHD modeling in one or two dimensions or in axis-
symmetric 3D. Fully 3D MHD codes are not new, but can be hard to obtain and
are usually more difficult to operate than 2D codes. Results of fully-3D DPF
modeling are not well represented in the literature.
In many experimental realizations of the DPF, the cathode is a cylinder
composed of metallic bars. While the plasma shock is traveling down the tube
during rundown, the plasma spills outside of the anode-cathode gap through the
spaces between the bars, a phenomenon that can be seen in the framing camera
picture in Fig. 5. This can be approximated in 2D simulations, but in practice
it is difficult to predict the time evolution of an actual DPF using these
approximations.
Figure 5: Shown is a framing camera picture of the plasma rising up the
electrodes in a DPF. The top of the anode can be seen as the dark disk in the
middle of the bars, which can be seen at the top. The lower region of the
chamber is bright due to the plasma, which has escaped the anode-cathode gap
and surrounded the bars.
Simulations in 2D also impose other geometric constraints: the current density
must either be completely in the radial-axial plane, or perpendicular to it,
but not both at the same time. This precludes modeling situations that may
have currents flowing helically, or situations in which the current may be
flowing asymmetrically or off-axis. These restrictions on 2D simulations are
often not of concern to investigators, who may want to ensure symmetry and
simplicity in their experiments in order to simplify the data they collect.
Nonetheless, the most general, physically realizable simulations are essential
for complete understanding.
The 2D simulation that was run in ALEGRA made the ad hoc assumption that there
was a lower density floor below which the material was assumed to have no
electrical or thermal conductivity. The floor was set at a density of
2.5$\times$10-4 kg/m3, which was was necessary in order to eliminate
unphysical behavior in the simulation. The LMD model for deuterium plasma,
shown in Figure 6, attributes moderate electrical conductivity to the plasma
at this density for temperatures higher than approximately 1 eV, and this
behavior is suppressed here, in order to cause the plasma shock to travel
properly down the anode of the DPF.
Figure 6: The Lee-More-Desjarlais (LMD) electrical conductivity model for
deuterium.
### III-B Three-dimensional Modeling
Fully 3D magnetohydrodynamic codes are available, such as ALEGRA and NIMROD
[23], among others, and the benefit of using these codes is that the electrode
geometry in the DPF can be modeled and simulated. The ability to investigate
the effects of electrode structures without presupposing symmetries allows
investigators to gain deeper understanding into how the plasma shock evolves
over time. The primary drawback of 3D MHD modeling is the computational
complexity of solving the MHD equations on large meshes, requiring computer
clusters to perform simulations in a reasonable amount of time. Similar to the
2D modeling, the 3D modeling in ALEGRA required the imposition of a density
floor at 2.5$\times$10-4 kg/m3 in the electrical and thermal conductivity
models.
Fig. 7 shows a representation of the material density midway through the
rundown phase of the DPF in both the 2D and 3D simulations. The 2D modeling
assumes axial symmetry, and the symmetry axis in both simulation volumes is on
the far left-hand side. In the both simulations, the cathode bars are
represented as rectangles on the right of the simulation domain, and the
plasma shock travels from the bottom of the image to the top of the image,
where it Z-pinches slightly above the hemispherical anode top. The plasma
cannot flow around the cathode bars in the 2D simulation as it can in the 3D
simulation. The 3D simulation simulates the entire gas volume of the chamber,
and it can be seen in Fig. 7 that the plasma is slightly slower and less dense
in the 3D simulation than in the 2D simulation. The larger inductance results
in longer rundown times and lower maximum current as compared to the 2D
simulation, if all other system parameters are equal.
Figure 7: This is a comparison of the density on a slice through the
simulation volume at about 3 microseconds. The image on the left shows the 2D
simulation where the plasma cannot flow around the bars, and the image on the
right shows the 3D simulation, where the plasma can flow around the bars.
Since the plasma flows around the bars in the 3D simulation, it also affects
the external impedance as a function of time for the simulation volume.
## IV Experimental and Predicted Results Comparison
Fig. 8 shows the current profiles from a 3D simulation, two 2D simulations,
and from the Faraday diagnostic of an actual experimental run. The “simulated
current” in 2D (red, dashed line) and 3D (yellow, dash-dotted line) were both
initiated using the same voltage and pressure as the experimental data. The 2D
simulation predicts a peak current of 2.08 MA, which differs from the peak
current measured by the Faraday diagnostic (2.17 MA) by only 4%. The 3D
simulation systematically predicts lower peak currents, due to the higher
inductance of the plasma escaping the cathode bars, and estimates peak current
at 1.82 MA, an error of approximately 16%. Thus, for estimating peak current,
the 2D simulation is more accurate than the corresponding 3D simulation.
The more important quantity of interest, however, is the duration time of the
rundown phase of the reaction. The end of the rundown phase is defined at the
point of maximum derivative of the current profile, which is 6.96 $\mu$sec for
the experimental data. The 3D simulation predicts 6.69 $\mu$sec, an error of
less than 4%, whereas the 2D simulation predicts a rundown time of 5.59
$\mu$sec, an error of almost 20%, showing that the 3D simulation vastly
outperforms the 2D simulation in predicting this quantity.
The two primary inputs into the ALEGRA simulation that we have discussed so
far are the initial voltage and pressure, since these are the adjustable
initial conditions of the DPF machine. The simulation allows for other
quantities to be tweaked as well, though, and it is natural to adjust the
model parameters to attempt to match the experimental data more closely. This
is difficult with the 3D simulations, since they are too computationally
intensive to “tweak” parameters one at a time and analyze the changes in
output. For the 2D simulations, however, this is possible, and Fig. 8 shows
the results of a 2D simulation (“tweaked current,” pink dotted line) that was
designed to match the rundown time of the experimental results. This required
that the series inductance be adjusted from 25 nH to 28.2 nH. Note that there
is no experimental justification for this change, it is done just to show that
the true rundown time can be achieved with a 2D simulation. This simulation
does not outperform the 3D, however, since the 2D simulation matching the
experimental rundown time results in a far inferior peak current measurement.
Thus a small improvement in rundown time over the 3D gives a large degradation
in peak current.
Figure 8: Shown is a comparison of the simulated 2D current (red, dashed line)
to the current measured by the Faraday probe (blue, solid line) and to the
simulated 3D current (yellow, dash-dotted line) and a Scat95 simulation
(green, dotted line). The 2D simulation underestimates peak current and
severely underestimates rundown time. The 3D simulation also underestimates
peak current but more faithfully predicts rundown time. Also shown is a 2D
simulation (pink, dotted line) whose input parameters are adjusted to give a
rundown time similar to the experimental data. In order that the 2D simulation
match the experimental rundown time, the peak current is severely
underestimated. The Scat95 simualtion shows better agreement, however,
requires iterative adjustment of parameters to already existing experimental
data.
Naturally one should use simulation input parameters that represent the
experiment being simulated as faithfully as possible, and the story of Fig. 8
is that 2D simulations of a DPF can give quite good results when the peak
current is the quantity of interest. When the rundown time is of interest,
which is more often the case, it is necessary to use the fully 3D simulation
to accurately predict the rundown time of an experiment. For comparison, a
Scat95 simulation was iteratively adjusted to match the experimental data.
While the agreement is good for this simulation, the parameters used in the
match are only good matches for geometries and setups that are close to this
particular case. Otherwise Scat95 achieves results that are similar to the 2D
MHD simulations.
## V Conclusions
In this work we have presented results of fully 3D predictive simulations of a
Dense Plasma Focus, using the ALEGRA MHD code from Sandia National
Laboratories. As opposed to 2D or axis-symmetric 3D simulations, the fully 3D
models more faithfully predict the duration of the rundown phase of the DPF,
which is essential for ensuring that the maximum current runs through the
system at the time of Z-pinch, which is required to accurately predict neutron
yield. The 2D simulations are appropriate for predicting the peak current in
the DPF, but are not capable of matching both the peak current and rundown
time simultaneously.
## Acknowledgment
The authors would like to thank Aaron Luttman for helpful comments and
suggestions on the manuscript. We would also like to thank Chris Hagen for
providing support and encouragement for this project, as well as his insight
into the theoretical and experimental operation of the DPF.
Sandia National Laboratories is a multi-program laboratory managed and
operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin
Corporation, for the U.S. Department of Energy’s National Nuclear Security
Administration under contract DE-AC04-94AL85000.
This manuscript has been authored in part by National Security Technologies,
LLC, under Contract No. DE-AC52-06NA25946 with the U.S. Department of Energy
and supported by the Site-Directed Research and Development Program. The
United States Government retains and the publisher, by accepting the article
for publication, acknowledges that the United States Government retains a non-
exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this manuscript, or allow others to do so, for United
States Government purposes.
## References
* [1] J. W. Mather, “Investigation of the High Energy Acceleration Mode in the Coaxial Gun,” Phys. Fluids. Suppl., vol. 7, S28, 1964.
* [2] J. W. Mather, “Formation of a High-Density Deuterium Plasma Focus,” Phys. Fluids, vol. 8, no. 2, Feb., 1965, pp. 366-377.
* [3] N. V. Filipov, T. I. Filipova, and V. P. Vinogradov, “Dense, High Temperature Plasma in a Noncylindrical Z-pinch Compression,” Nuc. Fusion, vol. 2, 577, 1962.
* [4] V. A. Gribkov, L. Mahe, P. Lee, S. Lee, A. Srivastava, “Dense Plasma Focus radiation source for microlithography & micro-machining,” Proc. SPIE Microlithographic Techniques Integrated Circuit Fabrication II, vol. 4226, 2000, pp. 151-159.
* [5] E. J. Lerner, S. K. Murali and A. Haboub, “Theory and Experimental Program for p-B11 Fusion with the Dense Plasma Focus,” J. Fusion Energy, vol. 30, no. 5, Jan., 2012, pp. 367-376.
* [6] J. O. Pouzo, “Applications of the dense plasma focus to nuclear fusion and plasma astrophysics,” Plasma Science, IEEE Trans., vol. 31, no. 6, Dec., 2003, pp. 1237-1242.
* [7] J. H. Gonzalez, F. R. Brollo, and A. Clausse, “Modeling of the Dynamic Plasma Pinch in Plasma Focus Discharges Based in Von Karman Approximations,” Plasma Science, IEEE Transactions on, vol. 37, no. 11, pp. 2178-2185, Nov., 2009.
* [8] C. S. Kueny, D. G. Flicker, and D. V. Rose, “ALEGRA-HEDP Simulations of the Dense Plasma Focus,” Sandia National Laboratories, Albuquerque, NM, SAND2009-6373, 2009.
* [9] T. J. Murphy, “A practical Beryllium activation detector for measuring DD neutron yield from ICF targets,” LA-UR-96-1649 Los Alamos Unclassified Report, 1996.
* [10] B. T. Meehan, E. C. Hagen, C. L. Ruiz, G. W. Cooper, “Praseodymium activation detector for measuring bursts of 14 MeV neutrons,” Nucl. Instr. and Meth. A, vol. 620, 2010, pp. 397-400.
* [11] O. Zucker, et. al., “The plasma focus as a large fluence neutron source,” Nucl. Instrm. and Meth., vol. 145, issue 1, Aug., 1977, pp. 185-190.
* [12] W. J. Rider, A. C. Robinson, et al., “ALEGRA: An Arbitrary Lagrangian-Eulerian Multimaterial, Multiphysics Code,” Proc. 46th AIAA Aero. Sci. Meeting., Reno, NV, Jan., 2008.
* [13] L.R. Veeser, G.W. Day, “Fiber-Optic, Faraday Rotation Current Sensor,” LA-UR-86-2084 Los Alamos Unclassified Report, 1984.
* [14] S. P. Lyon, J. D. Johnson, “SESAME: The Los Alamos National Laboratory equation of state database.” LA-UR-92-3407 Los Alamos Unclassified Report, 1992.
* [15] M. P. Desjarlais, “Practical Improvements to the Lee-More Conductivity Near the Metal-Insulator Transition,” Contrib. Plasm. Phys., vol. 41, 2001, pp. 267-270.
* [16] M. K. Matzen, M. A. Sweeney, R. G. Adams, J. R. Asay, J. E. Bailey, et al., “Pulsed-power-driven high energy density physics and inertial confinement fusion research,” Phys. Plasmas, vol. 12, no. 055503, 2005.
* [17] R. M. Lemke, M. D. Knudson, C. A. Hall, T. H. Haill, M. P. Desjarlais, J. R. Asay, “Characterization of magneticaly accelerated flyer plates,” Phys. Plasmas, vol. 10, no. 4, 2003, pp. 1092-1099.
* [18] A. Schmidt, V. Tang, D. Welch, “Fully Kinetic Simulations of Dense Plasma Focus Z-Pinch Devices,” Phys. Rev. Letters, vol. 109, 205003 (2012).
* [19] S. Lee, S. H. Saw, et al., “Characterizing Plasma Focus Devices - Role of the Static Inductance - Instability Phase Fitted by Anomalous Resistances,” J. Fusion Energy, vol. 30, no. 4, Aug. 2011pp. 277-282.
* [20] R. Gribble, M. Yapuncich, W. Deninger, “SCAT95,” 2.0 Los Alamos National Laboratory, 1997.
* [21] V. Makhin, B. Bauer, et al., “Numerical Modeling of a Magnetic Flux Compression Experiment,” Journal of Fusion Energy, vol. 26, 109-112, 2007
* [22] M. H. Freese, “MACH2: A Two-Dimensional Magnetohydrodynamic Simulation Code for Complex Experimental Configurations,” AMRC-R-874, (1987).
* [23] C. R. Sovinec, A. H. Glasser, et al., “Nonlinear Magnetohydrodynamics with High-order Finite Elements,” J. Comp. Phys., 195, 335 (2004).
B. T. Meehan received a B.S. in Physics from the United States Naval Academy
in 1995 and an M.S. in Applied Physics from Stanford University in 1997.
Currently he works with the Dense Plasma Focus Group at National Security
Technologies, LLC, modeling DPFs with magnetohydrodynamics codes.
---
J. H. J. Niederhaus holds a B.S. in Physics from the Virginia Military
Institute (2001), an M.S. in Nuclear Engineering from the Pennsylvania State
University (2003), and a Ph.D. in Engineering Physics from the University of
Wisconsin-Madison (2007). He is a computer scientist in the Computational
Shock and Multiphysics Department at Sandia National Laboratories.
---
|
arxiv-papers
| 2014-02-20T17:32:21 |
2024-09-04T02:49:58.478914
|
{
"license": "Public Domain",
"authors": "B.T. Meehan, J.H.J. Niederhaus",
"submitter": "Bernard Meehan",
"url": "https://arxiv.org/abs/1402.5083"
}
|
1402.5147
|
# Degenerate complex Hessian equations on compact Kähler manifolds
Chinh H. Lu Chalmers University of Technology
Mathematical Sciences
412 96 Gothenburg
Sweden [email protected] and Van-Dong Nguyen Department of Mathematics-
Informatics
Ho Chi Minh city University of Pedagogy
280 An Duong Vuong
Ho Chi Minh city, Vietnam [email protected]
(Date:
The first-named author is partially supported by the french ANR project MACK)
###### Abstract.
Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$ and fix
$m\in\mathbb{N}$ such that $1\leq m\leq n$. We prove that any $(\omega,m)$-sh
function can be approximated from above by smooth $(\omega,m)$-sh functions. A
potential theory for the complex Hessian equation is also developed which
generalizes the classical pluripotential theory on compact Kähler manifolds.
We then use novel variational tools due to Berman, Boucksom, Guedj and Zeriahi
to study degenerate complex Hessian equations.
###### Contents
1. 1 Introduction
2. 2 Preliminaries
3. 3 Approximation of $(\omega,m)$-subharmonic functions
4. 4 The Hessian $m$-capacity
5. 5 Energy classes
6. 6 The variational method
7. 7 Resolution of the degenerate complex Hessian equation
## 1\. Introduction
Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$. Let
$m$ be a natural number between $1$ and $n$. Denote by $d,d^{c}$ the usual
real differential operators $d:=\partial+\bar{\partial}$,
$d^{c}=\frac{\sqrt{-1}}{2\pi}(\bar{\partial}-\partial)$ so that
$dd^{c}=\frac{i}{\pi}\partial\bar{\partial}$.
The complex $m$-Hessian equation can be considered as an interpolation between
the classical Poisson equation (corresponds to the case when $m=1$) and the
complex Monge-Ampère equation (corresponds to $m=n$) which has been studied
intensively in the recent years with many applications to complex geometry.
For recent developments of the latter, see [44, 7, 20, 21, 27, 28] and the
references therein.
The study of complex Hessian equations was initiated by Li in [29] where he
solved the Dirichlet problem with smooth data on a smooth strictly
$m$-pseudoconvex domain in $\mathbb{C}^{n}$. Błocki [7] developed the first
steps of a potential theory for this equation and suggested a study of the
corresponding equation on compact Kähler manifolds which is analogous to the
complex Monge-Ampère equation.
The non-degenerate complex Hessian equation is of the following form
(1.1) $(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=f\omega^{n},$
where $0<f\in\mathcal{C}^{\infty}(X)$ satisfies the necessary condition
$\int_{X}f\omega^{n}=\int_{X}\omega^{n}$. It has been studied by Kokarev [26],
Jbilou [25] and Hou-Ma-Wu [23],[24]. In [25] and [23] the authors
independently proved that equation (1.1) has a unique (up to an additive
constant) smooth admissible solution provided the metric $\omega$ has non-
negative holomorphic bisectional curvature. This technical assumption turned
out to be very strong since manifolds carrying such metrics are very
restrictive thanks to a uniformization theorem of Mok (see [33]). Another
effort from Hou-Ma-Wu [24] provided an a priori almost
$\mathcal{C}^{2}$-estimate without any curvature assumption. The authors also
mentioned that their estimate can also be used in a blow-up analysis which
actually reduced the problem of solving the complex Hessian equation on a
compact Kähler manifold to proving a Liouville-type theorem for
$m$-subharmonic functions in $\mathbb{C}^{n}$. The latter has been recently
solved by Dinew and Kołodziej [16] which confirmed the smooth resolution of
equation (1.1) in full generality. Dinew and Kołodziej also gave a very
powerful $\mathcal{C}^{0}$-estimate in [15] which allows one to find
continuous weak solution of the degenerate complex Hessian equation with $f\in
L^{p}(X)$ for some $p>n/m$.
The real Hessian equation has been studied intensively with many geometric
applications. For a survey of this theory we refer the reader to [39], [40],
[45], [13] and the references therein. Some similar non-linear elliptic
equations of have appeared in the study of geometric deformation flows such as
the $J$-flow (see [38]). From the point of view of non-linear elliptic partial
differential equations the complex Hessian equation is an interesting and
important object. Recently another general (and powerful)
$\mathcal{C}^{2,\alpha}$ estimate for equations of this type has been obtained
in [41]. In [1] equations of complex Hessian type appear in the study of
quaternionic geometry. Thus it is expected that the complex Hessian equation
we considered here will have some interesting geometric applications.
The notion of $(\omega,m)$-subharmonic functions has been introduced in [15]
and studied by the first-named author in a systematic way in [31]. It was not
clear how to define the complex Hessian operator for any bounded
$(\omega,m)$-sh function due to a lack of regularization process. Recently
Plis [36] proved that one can approximate continuous strictly $(\omega,m)$-sh
functions by smooth ones. In this paper we show that one can globally
approximate any $(\omega,m)$-subharmonic functions from above by a sequence of
smooth $(\omega,m)$-subharmonic functions. In particular, the class
$\mathcal{P}_{m}(X,\omega)$ introduced in [30] consists of all
$(\omega,m)$-subharmonic functions.
For any upper semicontinuous function $f$ we define the projection of $f$ on
the space of $(\omega,m)$-subharmonic functions by:
$P(f):=\sup\left\\{u\in\mathcal{SH}_{m}(X,\omega)\ \big{|}\ u\leq f\right\\}.$
Using Berman’s technique [4] combined with the viscosity method by Eyssidieux,
Guedj and Zeriahi [17] we can prove that the projection of a smooth function
is continuous. Moreover, we can also prove the orthogonal relation without
solving the local Dirichlet problem. Let us stress that our method is new even
in the case of complex Monge-Ampère equations ($n=m$).
###### Theorem 1.
Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$ and fix
$m\in\mathbb{N}$ such that $1\leq m\leq n$. Let $h$ be a continuous function
on $X$. The $(\omega,m)$-subharmonic function $P(h)$ is continuous and its
Hessian measure is a non-negative measure supported on the contact set
$\\{P(h)=h\\}$.
Following [18] we can approximate any $(\omega,m)$-sh function from above by a
sequence of smooth $(\omega,m)$-sh functions.
###### Theorem 2.
For every $u\in\mathcal{SH}_{m}(X,\omega)$ we can find a decreasing sequence
of smooth $(\omega,m)$-subharmonic functions on $X$ which decreases to $u$ on
$X$.
The approximation theorem (Theorem 2) was known to hold for continuous
$(\omega,m)$-sh functions by a recent paper of Plis ([36]) while the same
question for any $(\omega,m)$-sh function is still open until now. One can
easily figure out that we only need to regularize any $(\omega,m)$-sh function
by continuous functions and then apply Plis’s result. Let us emphasize that we
prove the approximation theorem independently by combining the
”$\beta$-convergence” method of Berman [4] and the envelope method of
Eyssidieux-Guedj-Zeriahi [18].
Being able to regularize bounded $(\omega,m)$-subharmonic functions we can
define the complex Hessian operator for these functions following the
pluripotential method of Bedford and Taylor [3]. We then can adapt many
arguments in pluripotential theory for complex Monge-Ampère equations to a
potential theory for complex Hessian equations on compact Kähler manifolds. We
can also mimic the definition of the class $\mathcal{E}(X,\omega)$ for
$\omega$-psh functions introduced in [21] to define a similar class of
$(\omega,m)$-subharmonic functions $\mathcal{E}(X,\omega,m)$. Using the
variational approach inspired by Berman, Boucksom, Guedj and Zeriahi [6] we
can solve very degenerate complex Hessian equations with right-hand sides
being positive measures vanishing on $m$-polar sets. The two major steps to
apply this method are the regularization process and the orthogonal relation
which have been established in the previous results. We also give simpler
proof of the differentiability of the energy functional composed with the
projection (see Lemma 6.13).
###### Theorem 3.
Let $\mu$ be a positive Radon measure on $X$ satisfying the compatibility
condition $\mu(X)=\int_{X}\omega^{n}$. Assume that $\mu$ does not charge
$m$-polar subsets of $X$. Then there exists a solution
$\varphi\in\mathcal{E}(X,\omega,m)$ to
$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=\mu.$
Acknowledgements. We thank Robert Berman for many useful discussions. We are
indebted to Eleonora Di Nezza and Ahmed Zeriahi for a very careful reading of
a previous draft version of this paper.
## 2\. Preliminaries
In this section we recall basic facts about $(\omega,m)$-subharmonic
functions. We refer the readers to [7], [15], [16], [30], [31], [34], [35],
[37], [12] for more details. We always denote by $(X,\omega)$ a compact Kähler
manifold. By $(M,\omega)$ we denote a Kähler manifold which is not necessary
compact. Let $n$ be the complex dimension of the manifold and fix an integer
$m$ such that $1\leq m\leq n$. We denote $\mathcal{C}^{\uparrow}$ the space of
upper semicontinuous functions.
### 2.1. $m$-Subharmonic functions
###### Definition 2.1.
A real $(1,1)$-form $\alpha$ is called $m$-positive on $M$ if the following
inequalities hold in the classical sense:
$\alpha^{k}\wedge\omega^{n-k}\geq 0,\ \forall k=1,...,m.$
A function $\varphi\in\mathcal{C}^{2}(M)$ is called $m$-subharmonic ($m$-sh
for short) on $M$ if the $(1,1)$-form $dd^{c}\varphi$ is $m$-positive on $M$.
A current $T$ of bidegree $(n-p,n-p)$, $p\leq m$, is called $m$-positive on
$M$ if for any $m$-positive $(1,1)$-forms $\alpha_{1},...,\alpha_{p}$ the
following inequality holds in the sense of currents :
$\alpha_{1}\wedge\cdots\wedge\alpha_{p}\wedge T\geq 0.$
For each $k\geq 1$ the symmetric polynomial of degree $k$ on $\mathbb{R}^{n}$
is defined by
$S_{k}(\lambda):=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}\lambda_{i_{1}}\cdots\lambda_{i_{k}},\ \ \
\lambda:=(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{R}^{n}.$
Let $\varphi\in\mathcal{C}^{2}(M)$ and set
$\lambda:=\lambda_{\varphi}(x)\in\mathbb{R}^{n}$ the vector of eigenvalues of
$dd^{c}\varphi$ at $x$ with respect to $\omega$. Then $\varphi$ is
$m$-subharmonic in $\Omega$ if and only if
$S_{k}(\lambda_{\varphi}(x))\geq 0,\ \forall x\in M,\ \forall k=1,...,m.$
The following lemma follows immediately from Gårding’s inequality [19] (see
also [7]).
###### Lemma 2.2.
Let $u\in\mathcal{C}^{2}(M)$. Then $u$ is $m$-subharmonic in $M$ if and only
if for every $m$-positive $(1,1)$-forms $(\alpha_{1},...,\alpha_{m-1})$ on $M$
the following inequality holds in the classical sense:
$dd^{c}u\wedge\alpha_{1}\wedge\cdots\wedge\alpha_{m-1}\wedge\omega^{n-m}\geq
0.$
This lemma suggests a way to extend Definition 2.1 for non-smooth functions.
###### Definition 2.3.
Assume that $u\in\mathcal{C}^{\uparrow}(M)$ is locally integrable on $M$. Then
$u$ is called $m$-sh on $M$ if
(i) for any $m$-positive $(1,1)$-forms $\alpha_{1},...,\alpha_{m-1}$ on $M$,
the following inequality holds in the weak sense of currents on $M$ :
$dd^{c}u\wedge\alpha_{1}\wedge\cdots\wedge\alpha_{m-1}\wedge\omega^{n-m}\geq
0.$
(ii) if $v\in\mathcal{C}^{\uparrow}(M)$ is locally integrable, satisfies (i)
and $u=v$ almost everywhere on $M$ then $u\leq v$.
We denote by $\mathcal{SH}_{m}(M)$ the class of all $m$-sh functions on $M$.
If $M$ is compact this class contains only constant functions. We will study
instead the class of $(\omega,m)$-subharmonic functions (see the next
section).
###### Definition 2.4.
A function $u$ is called strictly $m$-sh on $M$ if for every function
$\chi\in\mathcal{C}^{2}(M)$ there exists $\varepsilon>0$ such that
$u+\varepsilon\chi$ is $m$-sh on $M$.
When $M=\Omega$ a bounded $m$-hyperconvex domain in $\mathbb{C}^{n}$, we
recover the definition of $m$-subharmonic functions introduced in [7], [37],
[34], [35],[32].
### 2.2. $(\omega,m)$-subharmonic functions
###### Definition 2.5.
Let $u\in\mathcal{C}^{\uparrow}(X)$ be an integrable function. Then $u$ is
called $(\omega,m)$-subharmonic on $X$ if for every local chart $\Omega$ the
function $u+\rho$ is $m$-sh in $\Omega$, where $\rho$ is a local potential of
$\omega$. Here, we regard $(\Omega,\omega|_{\Omega})$ as an open Kähler
manifold. The notion of $m$-sh functions on $\Omega$ is defined in the
previous subsection.
When $m=1$, $(\omega,1)$-sh functions on $X$ are just $\omega$-subharmonic
functions on $X$. When $m=n$, $(\omega,m)$-sh functions are exactly
$\omega$-plurisubharmonic functions which have been studied intensively by
many authors in the recent years.
It follows from Lemma 2.2 that a function $u\in\mathcal{C}^{2}(X)$ is
$(\omega,m)$-sh if and only if the associated $(1,1)$-form
$\omega+dd^{c}\varphi$ is $m$-positive on $X$. In general $u$ is
$(\omega,m)$-sh if the current $\omega+dd^{c}\varphi$ is $m$-positive on $X$.
###### Definition 2.6.
A function $u$ is called strictly $(\omega,m)$-sh on $X$ if for every function
$\chi\in\mathcal{C}^{2}(X)$ there exists $\varepsilon>0$ such that
$u+\varepsilon\chi$ is $(\omega,m)$-sh on $X$.
Continuous strictly $(\omega,m)$-sh functions on $X$ can also be approximated
from above by smooth ones. This is the content of the next theorem due to Plis
[36]:
###### Theorem 2.7.
[36] Let $(X,\omega)$ be a compact Kähler manifold and $u$ be a continuous
strictly $(\omega,m)$-subharmonic function on $X$. Let $h$ be a continuous
function on $X$ such that $h>0$. Then there exists a smooth strictly
$(\omega,m)$-sh function $\varphi$ on $X$ such that
$u\leq\varphi\leq u+h.$
### 2.3. The complex Hessian operator
We briefly recall basic facts about the class $\mathcal{P}_{m}(X,\omega)$ and
the complex Hessian operator introduced in [30].
Let $U\subset X$ be an open subset. The Hessian $m$-capacity of $U$ is defined
by
${\rm Cap}_{\omega,m}(U):=\sup\left\\{\int_{U}H_{m}(u)\ \big{|}\
u\in\mathcal{SH}_{m}(X,\omega)\cap\mathcal{C}^{2}(X),\ -1\leq u\leq
0\right\\},$
where for a smooth function $u$, we denote
$H_{m}(u):=(\omega+dd^{c}u)\wedge\omega^{n-m}$.
###### Definition 2.8.
Let $\varphi\in\mathcal{SH}_{m}(X,\omega)$. By definition $\varphi$ belongs to
$\mathcal{P}_{m}(X,\omega)$ if there exists a sequence
$(\varphi_{j})\subset\mathcal{SH}_{m}(X,\omega)\cap\mathcal{C}^{2}(X)$ which
converges to $\varphi$ quasi-uniformly on $X$, i.e. for any $\varepsilon>0$
there exists an open subset $U$ such that ${\rm
Cap}_{\omega,m}(U)<\varepsilon$ and $\varphi_{j}$ converges uniformly to
$\varphi$ on $X\setminus U$.
We will show in the next section that
$\mathcal{P}_{m}(X,\omega)=\mathcal{SH}_{m}(X,\omega)$. It follows from the
definition that every $\varphi\in\mathcal{P}_{m}(X,\omega)$ is quasi-
continuous, i.e. for any $\varepsilon>0$ we can find an open subset $U$ such
that ${\rm Cap}_{\omega,m}(U)<\varepsilon$ and the restriction of $\varphi$ on
$X\setminus U$ is continuous.
Assume that $\varphi\in\mathcal{P}_{m}(X,\omega)$ is bounded. Let
$(\varphi_{j})$ be a sequence of functions in
$\mathcal{SH}_{m}(X,\omega)\cap\mathcal{C}^{2}(X)$ which converges quasi-
everywhere on $X$ to $\varphi$. Then the sequence $H_{m}(\varphi_{j})$
converges weakly to some positive Radon measure $\mu$. This measure $\mu$ does
not depend on the choice of the sequence $(\varphi_{j})$ and is defined to be
the Hessian measure of $\varphi$:
$(\omega+dd^{c}\varphi_{j})^{m}\wedge\omega^{n-m}=:H_{m}(\varphi_{j})\rightharpoonup
H_{m}(\varphi).$
The class $\mathcal{P}_{m}(X,\omega)$ is stable under taking maximum and under
decreasing sequence.
### 2.4. Viscosity vs potential sub-solution
Let $0\leq F$ be a continuous function on $X$ and $u$ be an upper
semicontinuous function on $X$. Let $x_{0}\in X$ and $q$ be a
$\mathcal{C}^{2}$ function in a small neighborhood $V$ of $x_{0}$. We say that
$q$ touches $u$ from above (in $V$) at $x_{0}$ if $q\geq u$ in $V$ with
equality at $x_{0}$.
We say that $u$ is a viscosity sub-solution of equation
(2.1) $F\omega^{n}-(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=0$
if for any $x_{0}\in X$ and any $\mathcal{C}^{2}$ function $q$ in a
neighborhood of $x_{0}$ which touches $u$ from above at $x_{0}$ then the
following inequality holds at $x_{0}$
$F\omega^{n}-(\omega+dd^{c}q)^{m}\wedge\omega^{n-m}\leq 0.$
The following result has been proved by Plis ([36]) using [22]. This will play
an important role in our regularization theorem.
###### Lemma 2.9.
[36] Let $u$ be a $(\omega,m)$-subharmonic function on $X$. Then $u$ is a
viscosity sub-solution of (2.1) with $F\equiv 0$. More precisely, for any
$x_{0}\in X$ and any $\mathcal{C}^{2}$ function $q$ in a neighborhood of
$x_{0}$ which touches $u$ from above at $x_{0}$ then the following
inequalities hold at $x_{0}$:
$(\omega+dd^{c}q)^{k}\wedge\omega^{n-k}\geq 0,\ \forall k=1,...,m.$
We also need a generalized version of the above result. The idea is to adapt
some useful tricks in [22].
###### Lemma 2.10.
Let $F$ be a non-negative continuous function on $X$. Let
$u\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}(X)$ be a potential solution of
equation (2.1). Then $u$ is also a viscosity sub-solution of (2.1).
###### Proof.
We argue by contradiction. Assume that there exists $x_{0}\in X$,
$B=\bar{B}(x_{0},r)$ a small open ball centered at $x_{0}$ and
$q\in\mathcal{C}^{2}(B)$ such that $q$ touches $u$ from above in $B$ at
$x_{0}$ but
(2.2) $F\omega^{n}-(\omega+dd^{c}q)^{m}\wedge\omega^{n-m}>\varepsilon,$
at $x_{0}$ for some positive constant $\varepsilon>0$. Since $F$ is
continuous, by shrinking $B$ a little bit we can assume that (2.2) holds for
every point in $B$.
Fix $\delta>0$. It follows from [15], [16] that we can find
$u_{\delta}\in\mathcal{SH}_{m}(X,\omega)\cap\mathcal{C}^{\infty}(X)$ such that
$\sup_{X}|u_{\delta}-u|<\delta r^{2}/2\ ,\sup_{X}|F_{\delta}-F|<\delta/2\ ,\
{\rm and}\
(\omega+dd^{c}u_{\delta})^{m}\wedge\omega^{n-m}=F_{\delta}\omega^{n}.$
Consider the function
$\phi_{\delta}(x):=u_{\delta}(x)-q(x)-\delta|x-x_{0}|^{2},\ \ x\in
B=\bar{B}(x_{0},r).$
Let $x_{\delta}\in B$ be a maximum point of $\phi_{\delta}$ in $B$. Observe
that if $x\in\partial B$ we have
$\phi_{\delta}(x)<u(x)+\delta r^{2}/2-q(x)-\delta r^{2}\leq-\delta r^{2}/2,$
while $\phi_{\delta}(x_{0})>u(x_{0})-\delta r^{2}/2-q(x_{0})=-\delta r^{2}/2$.
Thus $x_{\delta}$ is in the interior of $B$ and hence the maximum principle
yields that
$(\omega+dd^{c}(q+\delta|x-x_{0}|^{2}))^{m}\wedge\omega^{n-m}\geq
F_{\delta}\omega^{n}$
holds at $x_{\delta}$. Letting $\delta\downarrow 0$ we can find $\bar{x}\in B$
such that the following holds at $\bar{x}$
$(\omega+dd^{c}q)^{m}\wedge\omega^{n-m}\geq F\omega^{n}.$
This yields a contradiction since (2.2) holds in $B$.
∎
## 3\. Approximation of $(\omega,m)$-subharmonic functions
In this section we prove the approximation theorem. Recall that it follows
from the recent work of Plis [36] (Theorem 2.7) that any continuous
$(\omega,m)$-sh function can be uniformly approximated by smooth ones. Thus
one needs only to approximate any $(\omega,m)$-sh function by continuous ones.
Let us stress that our proof is independent of Plis’s result. We immediately
regularize $(\omega,m)$-sh functions by using recent methods of Berman [4] and
Eyssidieux-Guedj-Zeriahi [18].
We first define the projection of any function on the class of $(\omega,m)$-sh
functions. Let $f$ be any upper semicontinuous function such that there is a
$(\omega,m)$-sh function lying below $f$. We define
$P(f):=\sup\left\\{v\in\mathcal{SH}_{m}(X,\omega)\ \big{|}\ v\leq f\ {\rm on}\
X\right\\}.$
It is clear that $P(f)^{*}$ is again a candidate defining $P(f)$. Then
$P(f)^{*}\leq P(f)$ which implies that $P(f)=P(f)^{*}$ is a $(\omega,m)$-sh
function. This is the maximal $(\omega,m)$-sh function lying below $f$. The
following observation follows straightforward from the definition:
###### Lemma 3.1.
Let $f,g$ be two bounded upper semicontinuous functions on $X$. Then
$\sup_{X}|P(f)-P(g)|\leq\sup_{X}|f-g|.$
Let $f$ be any function of class $\mathcal{C}^{2}$ on $X$. We define
$H_{m}(f)^{+}$ to be the non-negative part of $H_{m}(f)$, i.e.
$H_{m}(f)^{+}(z):=\max\left[(\omega+dd^{c}f)^{m}\wedge\omega^{n-m}(z),0\right].$
Observe that for any $f\in\mathcal{C}^{2}(X)$, the non-negative part of the
Hessian measure of $f$ is a non-negative measure having continuous density.
This measure also has positive mass. It follows from [30] that for every
$\beta>0$ the following equation has a unique continuous solution in the class
$\mathcal{P}_{m}(X,\omega)$:
(3.1)
$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=e^{\beta(\varphi-f)}\left(H_{m}(f)^{+}+\frac{\omega^{n}}{\beta}\right).$
###### Theorem 3.2.
Let $f\in\mathcal{C}^{2}(X)$. For each $\beta>1$, let
$u_{\beta}\in\mathcal{P}_{m}(X,\omega)$ be the unique solution to equation
(3.1). Then $u_{\beta}\leq f$ on $X$. Moreover, when $\beta$ goes to
$+\infty$, $u_{\beta}$ converges uniformly on $X$ to the upper envelope
$P(f):=\sup\left\\{v\in\mathcal{SH}_{m}(X,\omega)\ \big{|}\ v\leq f\ {\rm on}\
X\right\\}.$
In particular $P(f)$ belongs to $\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}(X)$
and satisfies
$H_{m}(P(f))\leq{\bf 1}_{\\{P(f)=f\\}}H_{m}(f).$
This Theorem is the principal result of our paper. Right after we know how to
regularize singular functions the other parts of the potential theory can be
easily adapted from the classical pluripotential theory for Monge-Ampère
equations. Let us also stress that our proof is new even in the case of
complex Monge-Ampère equations.
###### Proof.
Fix $\beta>1$. To simplify the notation we set $\varphi:=u_{\beta}$. It
follows from Lemma 2.10 that $\varphi$ is also a viscosity sub-solution of
equation
$e^{\beta(\varphi-f)}\left(H_{m}(f)^{+}+\frac{\omega^{n}}{\beta}\right)-(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=0.$
Let $x_{0}$ be a point where $\varphi-f$ attains its maximum on $X$. Then
$f-f(x_{0})+\varphi(x_{0})$ touches $\varphi$ from above at $x_{0}$. By
definition of viscosity sub-solutions we get
$e^{\beta(\varphi(x_{0})-f(x_{0}))}\left(H_{m}(f)^{+}+\frac{\omega^{n}}{\beta}\right)(x_{0})-(\omega+dd^{c}f)^{m}\wedge\omega^{n-m}(x_{0})\leq
0.$
We then infer that $\varphi(x_{0})\leq f(x_{0})$ which proves that
$\varphi-f\leq 0$ on $X$.
Now, fix $\beta>\gamma>1$. Since $u_{\beta}\leq f$, it is easy to see that
$(\omega+dd^{c}u_{\beta})^{m}\wedge\omega^{n-m}\leq
e^{\gamma(u_{\beta}-f)}\left(H_{m}(f)^{+}+\frac{\omega^{n}}{\gamma}\right).$
It thus follows from the comparison principle (see [30, Corollary 3.15]) that
$u_{\beta}\geq u_{\gamma}$. Therefore the sequence $(u_{\beta})$ converges.
Observe also that the right-hand side of (3.1) has uniformly bounded density.
It then follows from [15] and [30] that $(u_{\beta})$ converges uniformly on
$X$ to $u\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}(X)$ which satisfies
$(\omega+dd^{c}u)^{m}\wedge\omega^{n-m}\leq{\bf 1}_{\\{u=f\\}}H_{m}(f)^{+}.$
Now, we prove that $u=P(f)$. Let us fix $h\in\mathcal{SH}_{m}(X,\omega)$ such
that $h\leq f$. We need to show that $h\leq u$. The idea behind the proof is
quite simple: since $H_{m}(u)$ vanishes on $\\{u<f\\}$, $u$ must be maximal
there, hence $u$ dominates any candidate defining $P(f)$.
Fix $\varepsilon>0$ and set $U:=\\{u<f-\varepsilon\\}$. Write
$(\omega+dd^{c}u_{\beta})^{m}\wedge\omega^{n-m}=f_{\beta}\omega^{n},$
where $f_{\beta}$ is a non-negative continuous function on $X$. Since
$u_{\beta}$ converges uniformly on $X$ to $u$ and $f_{\beta}$ converges
uniformly to $0$ on $U$, we can find $\beta>0$ very big such that
$\sup_{U}f_{\beta}<\varepsilon^{m}/2\ \ {\rm and}\ \
\sup_{X}|u_{\beta}-u|<\varepsilon/2.$
Now, since $f_{\beta}$ is continuous on $X$ there is a sequence of smooth
positive functions $g_{\beta}^{j}$ converging uniformly to $f_{\beta}$ on $X$
such that $\int_{X}g_{\beta}^{j}\omega^{n}=\int_{X}\omega^{n}$. Let
$v_{\beta}^{j}$ be the corresponding smooth solutions to the complex Hessian
equations $H_{m}(v_{\beta}^{j})=g_{\beta}^{j}\omega^{n}$. Then it follows from
[15] that $v_{\beta}^{j}$ converges uniformly to $u_{\beta}$ on $X$. Now for
$j$ large enough we have found $v_{\beta}$ (we drop the index $j$) a smooth
$(\omega,m)$-sh functions on $X$ such that
$H_{m}(v_{\beta})=g_{\beta}\omega^{n},\ \
\sup_{X}|v_{\beta}-u_{\beta}|<\varepsilon/2\ \ {\rm and}\ \
\sup_{X}|g_{\beta}-f_{\beta}|<\varepsilon^{m}/2.$
In particular, we have
$H_{m}(v_{\beta})=g_{\beta}\omega^{n},\ \ \sup_{X}|v_{\beta}-u|<\varepsilon\ \
{\rm and}\ \ \sup_{U}g_{\beta}<\varepsilon^{m}.$
Consider the function
$\phi:=h-(1+\delta)v_{\beta},$
where $\delta=\varepsilon/(1-\varepsilon)$. Since $\phi$ is upper
semicontinuous on $X$ compact, it attains its maximum on $X$ at some $y_{0}\in
X$.
Assume that $y_{0}\in U$. Then the function
$(1+\delta)v_{\beta}-(1+\delta)v_{\beta}(y_{0})+h(y_{0})$ touches $h$ from
above at $y_{0}$. Then by definition of viscosity sub-solutions and by Lemma
2.9 we get
$\left[\omega+(1+\delta)dd^{c}v_{\beta}\right]^{k}\wedge\omega^{n-k}(y_{0})\geq
0,\forall k=1\cdots m.$
This yields
$(1+\delta)^{m}H_{m}(v_{\beta})(y_{0})\geq\delta^{m}\omega^{n}(y_{0}),$
which is a contradiction since in $U$,
$H_{m}(v_{\beta})<\varepsilon^{m}\omega^{n}$ and
$\delta=\varepsilon(1+\delta)$.
Therefore, $y_{0}\notin U$. Since $u\geq f-\varepsilon$ on $X\setminus U$ and
since $\sup_{X}|v_{\beta}-u|<\varepsilon$ we get
$\phi(y)\leq\phi(y_{0})\leq-\delta f(y_{0})+2\varepsilon(1+\delta),\ \forall
y\in X.$
We thus obtain
$h\leq(1+\delta)u+\delta\sup_{X}|f|+3\varepsilon(1+\delta).$
By letting $\varepsilon\downarrow 0$ (and hence $\delta$ also goes to $0$) we
obtain $h\leq u$.
∎
###### Corollary 3.3.
For any $\varphi\in\mathcal{SH}_{m}(X,\omega)$ there exists a sequence
$(\varphi_{j})\subset\mathcal{SH}_{m}(X,\omega)\cap\mathcal{C}^{\infty}(X)$
decreasing to $\varphi$ on $X$. In particular
$\varphi\in\mathcal{P}_{m}(X,\omega)$ and the two classes coincide.
###### Proof.
Let $\varphi$ be a $(\omega,m)$-subharmonic function on $X$. Since $\varphi$
is in particular upper semicontinuous we can find a sequence $(f_{j})$ of
smooth functions on $X$ decreasing to $\varphi$. Note that $(f_{j})$ is a
priori not $(\omega,m)$-sh on $X$. Let $P(f_{j})$ be the upper envelope of
$(\omega,m)$-sh functions lying below $f_{j}$. It follows from Theorem 3.2
that $P(f_{j})$ is a continuous $(\omega,m)$-sh function on $X$ which belongs
to $\mathcal{P}_{m}(X,\omega)$ and satisfies
$H_{m}(P(f_{j}))\leq{\bf 1}_{\\{P(f_{j})=f_{j}\\}}H_{m}(f_{j}).$
On the above the right-hand side has bounded density. Thus it follows from
[15] (see also [30]) that for each $j$, $P(f_{j})$ is the uniform limit of a
sequence of smooth $(\omega,m)$-sh functions. Therefore, for each $j$ we can
find $\varphi_{j}\in\mathcal{SH}_{m}(X,\omega)\cap\mathcal{C}^{\infty}(X)$
such that
$P(f_{j})+\frac{1}{j+1}\leq\varphi_{j}\leq P(f_{j})+\frac{1}{j}.$
Then it is clear that $\varphi_{j}$ decreases to $\varphi$. Now, it follows
from [30, Proposition 3.2] that $\varphi$ belongs to
$\mathcal{P}_{m}(X,\omega)$ and hence
$\mathcal{SH}_{m}(X,\omega)=\mathcal{P}_{m}(X,\omega)$. ∎
We immediately get the following:
###### Corollary 3.4.
If $h\in\mathcal{C}(X)$ then $P(h)$ is a continuous $(\omega,m)$-sh function.
Its Hessian measure $H_{m}(P(h))$ vanishes on $\\{P(h)<h\\}$.
###### Proof.
To prove the first statement it suffices to approximate $h$ by smooth
functions and apply Lemma 3.1. To prove the second statement we first assume
that $h$ is smooth on $X$. It follows from Theorem 3.2 that
$P(h)=\lim_{\beta\to+\infty}u_{\beta}$ is the uniform limit of continuous
$(\omega,m)$-sh functions on $X$. For each $\varepsilon>0$, $H_{m}(u_{\beta})$
converges uniformly to $0$ on $\\{P(h)<h-\varepsilon\\}$. This coupled with
convergence results in [30] explain why $H_{m}(P(h))$ vanishes on
$\\{P(h)<h\\}$. The general case follows by approximating $h$ uniformly by
smooth functions. ∎
## 4\. The Hessian $m$-capacity
In Section 2.3 we define Hessian $m$-capacity of any open subset. Now, we know
that the Hessian operator is well-defined for any bounded $(\omega,m)$-sh
function. It turns out that in the definition of the Hessian $m$-capacity one
can take the supremum over all $(\omega,m)$-sh functions whose values vary
from $-1$ to $0$ and not only $\mathcal{C}^{2}$ functions. We then extend this
definition to any Borel subset.
For each Borel subset $E\subset X$ we define the Hessian $m$-capacity of $E$
by
${\rm Cap}_{\omega,m}(E):=\sup\left\\{\int_{E}H_{m}(u)\ \big{|}\
u\in\mathcal{SH}_{m}(X,\omega),\ -1\leq u\leq 0\right\\}.$
The following properties of ${\rm Cap}_{\omega,m}$ follow directly from the
definition:
###### Proposition 4.1.
(i) If $E_{1}\subset E_{2}\subset X$ then ${\rm Cap}_{\omega,m}(E_{1})\leq{\rm
Cap}_{\omega,m}(E_{2})$ .
(ii) If $E_{1},E_{2},\cdots$ are Borel subsets of $X$ then
${\rm
Cap}_{\omega,m}\left(\bigcup_{j=1}^{\infty}E_{j}\right)\leq\sum_{j=1}^{+\infty}{\rm
Cap}_{\omega,m}(E_{j}).$
(iii) If $E_{1}\subset E_{2}\subset\cdots$ are Borel subsets of $X$ then
${\rm
Cap}_{\omega,m}\left(\bigcup_{j=1}^{\infty}E_{j}\right)=\lim_{j\to+\infty}{\rm
Cap}_{\omega,m}(E_{j}).$
For each Borel subset $E$ set
$h_{m,E}:=\sup\\{u\in\mathcal{SH}_{m}(X,\omega)\ \big{|}\ u\leq-1\ {\rm on}\
E\ {\rm and}\ u\leq 0\ {\rm on}\ X\\}.$
Let $h^{*}_{m,E}$ be the upper semicontinuous regularization of $h_{m,E}$. We
call it the relative $m$-extremal function of $E$.
###### Theorem 4.2.
Let $E$ be any Borel subset of $X$ and denote by $h_{E}$ the relative
$m$-extremal function of $E$. Then $h_{E}$ is a bounded
$(\omega,m)$-subharmonic function. Its Hessian measure vanishes on the open
subset $\\{h_{E}<0\\}\setminus\bar{E}$.
###### Proof.
The first statement is obvious. The second statement follows from the standard
balayage argument since it follows from [36] that we can locally solve the
Dirichlet problem on any small ball.
∎
The following convergence result can be proved in the same way as in [28].
###### Lemma 4.3.
Let $(\varphi_{j}^{k})_{j=1}^{+\infty}$ be a uniformly bounded sequence of
$(\omega,m)$-sh functions for $k=1,...,m$, which increases almost everywhere
to $\varphi^{1},...,\varphi^{m}\in\mathcal{SH}_{m}(X,\omega)$ respectively.
Then
$u(\omega+dd^{c}\varphi_{j}^{1})\wedge(\omega+dd^{c}\varphi_{j}^{2})\wedge...\wedge(\omega+dd^{c}\varphi_{j}^{m})\wedge\omega^{n-m}$
converges weakly in the sense of Radon measures to
$u(\omega+dd^{c}\varphi^{1})\wedge(\omega+dd^{c}\varphi^{2})\wedge...\wedge(\omega+dd^{c}\varphi^{m})\wedge\omega^{n-m},$
for every quasi-continuous function $u$.
###### Lemma 4.4.
Let $\mathcal{U}$ be a family of $(\omega,m)$-sh functions which are uniformly
bounded from above. Define
$\varphi(z):=\sup\\{u(z)\ \big{|}\ u\in\mathcal{U}\\}.$
Then the Borel subset $\\{\varphi<\varphi^{*}\\}$ has zero Hessian
$m$-capacity.
###### Proof.
By Choquet’s lemma we can find a sequence $(\varphi_{j})\subset\mathcal{U}$
which increases to $\varphi$.
Step 1: Assume that $u\in\mathcal{SH}_{m}(X,\omega)$ and $-1\leq u\leq 0$. We
prove by induction on $m$ that
(4.1)
$\lim_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(u)=\int_{X}\varphi^{*}H_{m}(u).$
The result holds for $m=0$ since $\varphi=\varphi^{*}$ almost everywhere (with
respect to the Lebesgue measure). Assume that it holds for $k=1,...,m-1$. By
setting $T:=(\omega+dd^{c}u)^{m-1}\wedge\omega^{n-m}$ and integrating by parts
we get
$\displaystyle\int_{X}\varphi_{j}H_{m}(u)$ $\displaystyle=$
$\displaystyle\int_{X}\varphi_{j}H_{m-1}(u)+\int_{X}\varphi_{j}dd^{c}u\wedge
T$ $\displaystyle=$
$\displaystyle\int_{X}\varphi_{j}H_{m-1}(u)+\int_{X}udd^{c}\varphi_{j}\wedge
T$ $\displaystyle=$
$\displaystyle\int_{X}\varphi_{j}H_{m-1}(u)+\int_{X}u(\omega+dd^{c}\varphi_{j})\wedge
T-\int_{X}uH_{m-1}(u).$
Since $u$ is quasi-continuous on $X$, by letting $j\to+\infty$ and using Lemma
4.3 and using the induction hypothesis we obtain
$\displaystyle\lim_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(u)$ $\displaystyle=$
$\displaystyle\int_{X}\varphi^{*}H_{m-1}(u)+\int_{X}u(\omega+dd^{c}\varphi^{*})\wedge
T-\int_{X}uH_{m-1}(u)$ $\displaystyle=$
$\displaystyle\int_{X}\varphi^{*}H_{m}(u).$
Step 2: Since ${\rm Cap}_{\omega,m}$ is $\sigma$-subadditive, it suffices to
prove that for each pair $(\alpha,\beta)$ of rational numbers such that
$\alpha<\beta$ the compact subset
$K_{\alpha,\beta}:=\\{x\in X\ \big{|}\
\varphi(x)\leq\alpha<\beta\leq\varphi^{*}\\}$
has zero Hessian $m$-capacity. This is an easy consequence of Step 1. The
proof is thus complete.
∎
The outer Hessian $m$-capacity of a Borel subset $E$ is defined by
${\rm Cap}_{\omega,m}^{*}(E):=\inf\left\\{{\rm Cap}_{\omega,m}(U)\ \big{|}\
E\subset U\subset X,\ U\ {\rm is\ open\ in}\ X\right\\}.$
The following properties of ${\rm Cap}_{\omega,m}^{*}$ follow directly from
the definition:
###### Proposition 4.5.
(i) If $E_{1}\subset E_{2}\subset X$ then ${\rm
Cap}_{\omega,m}^{*}(E_{1})\leq{\rm Cap}_{\omega,m}^{*}(E_{2})$ .
(ii) If $E_{1},E_{2},\cdots$ are Borel subsets of $X$ then
${\rm
Cap}_{\omega,m}^{*}\left(\bigcup_{j=1}^{\infty}E_{j}\right)\leq\sum_{j=1}^{+\infty}{\rm
Cap}_{\omega,m}^{*}(E_{j}).$
Now we give a formula for the outer Hessian $m$-capacity of any Borel subset
of $X$ in terms of its relative $m$-extremal function.
###### Theorem 4.6.
Let $E\subset X$ be a Borel subset and $h$ denote the relative $m$-extremal
function of $E$. Then the outer Hessian $m$-capacity of $E$ is given by
${\rm Cap}_{\omega,m}^{*}(E)=\int_{X}(-h)H_{m}(h).$
The Hessian $m$-capacity satisfies the following continuity properties:
(i) If $(E_{j})_{j\geq 0}$ is an increasing sequence of arbitrary subsets of
$X$ and $E:=\cup_{j\geq 0}E_{j}$ then
${\rm Cap}_{\omega,m}^{*}(E)=\lim_{j\to+\infty}{\rm
Cap}_{\omega,m}^{*}(E_{j}).$
(ii) If $(K_{j})_{j\geq 0}$ is a decreasing sequence of compact subsets of $X$
and $K:=\cap_{j\geq 0}K_{j}$
$\lim_{j\to+\infty}{\rm Cap}_{\omega,m}^{*}(K_{j})={\rm
Cap}_{\omega,m}^{*}(K)={\rm Cap}_{\omega,m}(K).$
In particular, ${\rm Cap}_{\omega,m}^{*}$ is an outer regular Choquet capacity
and we have
${\rm Cap}_{\omega,m}^{*}(B)={\rm Cap}_{\omega,m}(B)$
for every Borel set $B$.
###### Proof.
It follows from Lemma 4.4 that the subset $\\{h>-1\\}\cap E$ has zero Hessian
$m$-capacity. Thus we can copy from lines to lines the proof of Theorem 4.2 in
[20]. For the last statement let us briefly recall the arguments in [6]. Since
${\rm Cap}_{\omega,m}^{*}$ is an (outer regular) Choquet capacity it then
follows from Choquet’s capacitability theorem that ${\rm Cap}_{\omega,m}^{*}$
is also inner regular on Borel sets. We thus get
${\rm Cap}_{\omega,m}(B)\leq{\rm Cap}_{\omega,m}^{*}(B)=\sup_{K\subset B}{\rm
Cap}_{\omega,m}^{*}(K)=\sup_{K\subset B}{\rm Cap}_{\omega,m}(K)={\rm
Cap}_{\omega,m}(B).$
∎
###### Definition 4.7.
Let $E$ be a Borel subset of $X$. The global $(\omega,m)$-subharmonic extremal
function of $E$ is $V^{*}_{m,E}$, where
$V_{m,E}:=\sup\left\\{u\in\mathcal{SH}_{m}(X,\omega)\ \big{|}\ u\leq 0\ {\rm
on}\ E\right\\}.$
###### Definition 4.8.
A subset $E\subset X$ is called $m$-polar if ${\rm Cap}_{\omega,m}^{*}(E)=0$.
###### Lemma 4.9.
If $E\subset\\{\varphi=-\infty\\}$ for some
$\varphi\in\mathcal{SH}_{m}(X,\omega)$ then $E$ is $m$-polar.
###### Proof.
It is easy to see that the relative $m$-extremal function of $E$ is $0$. Then
the result is obtained by invoking Theorem 4.6. ∎
We now prove the Josefson theorem for $(\omega,m)$-sh functions which
generalize [20, Theorem 7.2]. Let us stress that our proof is more direct
without using the local capacity which has not been available yet. Recall that
a local $m$-capacity has been studied in [37] and [32] where the metric is
flat. For a general Kähler metric we believe that similar study can be done.
###### Theorem 4.10.
If ${\rm Cap}_{\omega,m}^{*}(E)=0$ then $E\subset\\{\varphi=-\infty\\}$ for
some $\varphi\in\mathcal{SH}_{m}(X,\omega)$.
###### Proof.
Without loss of generality we can assume that $\bar{E}\neq X$. Observe that
${\rm Cap}_{\theta,m}^{*}(E)=0$ for any Kähler form $\theta$ since we have
$C^{-1}\omega\leq\theta\leq C\omega$ for some positive constant $C$. Let
$V:=V^{*}_{m,E}$ be the global $m$-extremal function of $E$.
We prove that $V\equiv+\infty$. Assume by contradiction that it is not the
case. Then $V$ is a bounded $(\omega,m)$-sh function on $X$. Using a balayage
argument as in the proof of Theorem 4.2 we can prove that $H_{m}(V)$ vanishes
on $X\setminus\bar{E}$. Thus $V$ can not be constant.
Let $M=\sup_{X}V<+\infty$. We claim that $\psi:=(V-M)/M$ is the relative
$(\theta,m)$-extremal function of $E$ with $\theta=\omega/M$. Indeed, let $u$
be any non-positive $(\theta,m)$-sh function on $X$ such that $u\leq-1$ on
$E$. Then $M(u+1)$ is a $(\omega,m)$-sh function on $X$ which is non-positive
on $E$. By definition of the global $(\omega,m)$-sh extremal function we
deduce that $M(u+1)\leq V$, which implies what we have claimed.
Now, since ${\rm Cap}_{\theta,m}^{*}(E)=0$ it follows from Theorem 4.6 that
$\int_{\\{\psi<0\\}}H_{m}(\psi)=0.$
This coupled with the comparison principle reveal that $\psi=0$ which implies
that $V\equiv M$. The latter is a contradiction since $H_{m}(V)$ vanishes on
the open non-empty set $X\setminus\bar{E}$.
Therefore, $V_{m,E}$ is not bounded from above. We then can find a sequence
$(\varphi_{j})\subset\mathcal{SH}_{m}(X,\omega)$ such that $\varphi_{j}\equiv
0$ on $E$ and $\sup_{X}\varphi_{j}\geq 2^{j}$. Consider
$\varphi:=\sum_{j=1}^{+\infty}2^{-j}(\varphi_{j}-\sup_{X}\varphi_{j}).$
Then since $\varphi_{j}=0$ on $E$ it is easy to see that $\varphi=-\infty$ on
$E$. It follows from Hartogs’ lemma (see [30]) that
$\int_{X}(u-\sup_{X}u)\ \omega^{n}\geq-C,\ \ \forall
u\in\mathcal{SH}_{m}(X,\omega),$
for some positive constant $C$. It follows that $\varphi$ is not identically
$-\infty$. Hence $\varphi\in\mathcal{SH}_{m}(X,\omega)$ satisfies our
requirement. ∎
## 5\. Energy classes
For convenience we rescale $\omega$ so that $\int_{X}\omega^{n}=1$. It follows
from Corollary 3.3 that
$\mathcal{SH}_{m}(X,\omega)=\mathcal{P}_{m}(X,\omega)$. Therefore the complex
Hessian operator is well-defined for any bounded $(\omega,m)$-sh function. We
will follow [21] to extend the definition of $H_{m}$ to unbounded
$(\omega,m)$-sh functions. Almost all results about the weighted energy
classes $\mathcal{E}_{\chi}(X,\omega)$ can be extended without effort to the
corresponding classes of $(\omega,m)$-subharmonic functions. For this reason
we only state the result without proof.
Let $\varphi\in\mathcal{SH}_{m}(X,\omega)$ and denote by $\varphi_{j}$ the
canonical approximation of $\varphi$ by bounded functions, i.e.
$\varphi_{j}:=\max(\varphi,-j)$. It follows from the comparison principle (see
[30]) that
${\bf 1}_{\\{\varphi>-j\\}}H_{m}(\varphi_{j})$
is a non-decreasing sequence of positive Borel measures on $X$. We define
$H_{m}(\varphi)$ to be its limit. Note that the total mass of $H_{m}(\varphi)$
varies from $0$ to $1$ and it does not charge $m$-polar sets.
###### Definition 5.1.
We let $\mathcal{E}(X,\omega,m)$ denote the class of $(\omega,m)$-sh functions
having full Hessian mass, i.e.
$\mathcal{E}(X,\omega,m):=\left\\{u\in\mathcal{SH}_{m}(X,\omega)\ \big{|}\
\int_{X}H_{m}(u)=1\right\\}.$
###### Lemma 5.2.
A function $u\in\mathcal{SH}_{m}(X,\omega)$ belongs to
$\mathcal{E}(X,\omega,m)$ if and only if
$\lim_{j\to+\infty}\int_{\\{u\leq-j\\}}H_{m}(\max(u,-j))=0.$
The sequence $H_{m}(\max(\varphi,-j))$ converges to $H_{m}(\varphi)$ in the
sense of Borel measures, i.e. for any Borel subset $E\subset X$,
$\lim_{j\to+\infty}\int_{E}H_{m}(\max(\varphi,-j))=\int_{E}H_{m}(\varphi).$
###### Definition 5.3.
Let $\chi$ be an increasing function $\mathbb{R}^{-}\to\mathbb{R}^{-}$ such
that $\chi(0)=0$ and $\chi(-\infty)=-\infty$. We let
$\mathcal{E}_{\chi}(X,\omega,m)$ denote the class of functions $\varphi$ in
$\mathcal{E}(X,\omega,m)$ such that $\chi\circ\varphi$ is integrable with
respect to $H_{m}(\varphi)$. When $\chi(t)=-(-t)^{p}$, $p>0$ we use the
notation $\mathcal{E}_{m}^{p}(X,\omega)$
$\mathcal{E}^{p}(X,\omega,m):=\left\\{u\in\mathcal{E}(X,\omega,m)\ \big{|}\
\int_{X}|u|^{p}H_{m}(u)<+\infty\right\\}.$
###### Lemma 5.4.
Let $\varphi\in\mathcal{E}(X,\omega,m)$ and
$h:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a continuous increasing
function such that $h(+\infty)=+\infty$. Then
$\int_{X}h\circ|\varphi|H_{m}(\varphi)<+\infty\Longleftrightarrow\sup_{j\geq
0}\int_{X}h\circ|\varphi_{j}|H_{m}(\varphi_{j})<+\infty,$
where $\varphi_{j}:=\max(\varphi,-j)$.
###### Lemma 5.5.
If $\varphi\in\mathcal{E}(X,\omega,m)$ and $\varphi\leq 0$ there exists a
convex increasing function $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ such
that $\chi(-\infty)=0$ and $\varphi\in\mathcal{E}_{\chi}(X,\omega,m)$.
###### Theorem 5.6.
Let $\varphi\in\mathcal{SH}_{m}(X,\omega)$ be such that
$\sup_{X}\varphi\leq-1$. Let $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$ be
a smooth convex increasing function such that $\chi^{\prime}(-1)\leq 1$ and
$\chi^{\prime}(-\infty)=0$. Then $\chi\circ\varphi\in\mathcal{E}(X,\omega,m)$.
The maximum principle and the comparison principle hold for
$\mathcal{E}(X,\omega,m)$:
###### Theorem 5.7.
Let $\varphi,\psi$ be two functions in $\mathcal{E}(X,\omega,m)$. Then
${\bf 1}_{\\{\varphi<\psi\\}}H_{m}(\max(\varphi,\psi))={\bf
1}_{\\{\varphi<\psi\\}}H_{m}(\psi)$
and
$\int_{\\{\varphi<\psi\\}}H_{m}(\psi)\leq\int_{\\{\varphi<\psi\\}}H_{m}(\varphi).$
###### Proposition 5.8.
Assume that $\varphi,\psi\in\mathcal{E}(X,\omega,m)$ such that
$H_{m}(\varphi)\geq\mu$ and $H_{m}(\psi)\geq\mu$ for some positive Borel
measure $\mu$ on $X$. Then
$H_{m}(\max(\varphi,\psi))\geq\mu.$
###### Theorem 5.9.
Let $(\varphi_{j})$ be a monotone sequence of functions in
$\mathcal{E}(X,\omega,m)$ converging to $\varphi\in\mathcal{E}(X,\omega,m)$.
Then $H_{m}(\varphi_{j})$ converges weakly to $H_{m}(\varphi)$.
###### Proposition 5.10.
The set $\mathcal{E}(X,\omega,m)$ is convex. It is stable under the max
operation: if $\varphi,\psi\in\mathcal{SH}_{m}(X,\omega)$ are such that
$\varphi\leq\psi$ and $\varphi\in\mathcal{E}(X,\omega,m)$, then
$\psi\in\mathcal{E}(X,\omega,m)$.
When $m=n$, the class $\mathcal{E}(X,\omega,n)$ is exactly
$\mathcal{E}(X,\omega)$, the class of $\omega$-psh functions having full
Monge-Ampère mass, introduced and studied in [21].
One can follow the lines in [14] to prove the ”partial comparison principle”:
###### Lemma 5.11.
Let $T$ be a positive current of type
$T=(\omega+dd^{c}\phi_{1})\wedge\cdots\wedge(\omega+dd^{c}\phi_{k})\wedge\omega^{n-m},\
\ k<m,$
where the $\phi_{j}$’s are functions in $\mathcal{E}(X,\omega,m)$. Let
$u,v\in\mathcal{E}(X,\omega,m)$. Then
$\int_{\\{u<v\\}}(\omega+dd^{c}v)^{m-k}\wedge
T\leq\int_{\\{u<v\\}}(\omega+dd^{c}u)^{m-k}\wedge T.$
###### Theorem 5.12.
$\mathcal{E}(X,\omega,n)\subset\mathcal{E}(X,\omega,n-1)\subset\cdots\subset\mathcal{E}(X,\omega,1)$.
###### Proof.
Fix $p<m$ and $\varphi\in\mathcal{E}(X,\omega,m)$. Let
$\varphi_{j}:=\max(\varphi,-j)$ be the canonical approximation sequence of
$\varphi$. We are to prove that
$\int_{\\{\varphi>-j\\}}H_{m-1}(\varphi_{j})\longrightarrow 1.$
From the partial comparison principle above we get
$\int_{\\{\varphi_{j}>-j\\}}(\omega+dd^{c}\varphi_{j})^{p}\wedge\omega^{m-p}\wedge\omega^{n-m}\geq\int_{\\{\varphi_{j}>-j\\}}(\omega+dd^{c}\varphi_{j})^{p}\wedge(\omega+dd^{c}\varphi_{j})^{m-p}\wedge\omega^{n-m}.$
From this and since $\varphi\in\mathcal{E}(X,\omega,m)$ we get the conclusion.
∎
###### Example 5.13.
Let $z$ be a local coordinate of $X$ and consider
$\varphi:=\varepsilon\theta\log|z|,$
where $\theta$ is a cut-off function and $\varepsilon>0$ is a very small
constant so that $\varphi\in\mathcal{SH}_{m}(X,\omega)$. Then
$\varphi\in\mathcal{E}(X,\omega,m)$ for any $m<n$ but
$\varphi\notin\mathcal{E}(X,\omega,n)$.
## 6\. The variational method
The variational method has first introduced in [6] to solve degenerate complex
Monge-Ampère equations on compact Kähler manifolds. A local version of this
approach has been developed in [2].
Due to some similar structure one expects that this method can also be applied
for the complex Hessian equation. In the local setting with a standard Kähler
metric the first-named author [32] has used this method to solve degenerate
complex Hessian equations in $m$-hyperconvex domains of $\mathbb{C}^{n}$. To
make it available for the compact setting the principal steps are: first to
smoothly regularize singular $(\omega,m)$-sh functions and then to prove an
othorgonal relation. Both of them have been proved in Section 3. In the sequel
we briefly recall the techniques of [6]. Most of the proof will be omitted due
to similarity and repetition.
### 6.1. The energy functional
###### Definition 6.1.
Let $\varphi$ be a bounded $(\omega,m)$-sh function on $X$. We define
$E(\varphi):=\frac{1}{m+1}\sum_{k=0}^{m}\int_{X}\varphi(\omega+dd^{c}\varphi)^{k}\wedge\omega^{n-k}$
to be the energy of $\varphi$. For any $u\in\mathcal{SH}_{m}(X,\omega)$ the
energy of $u$ is defined by
$E(u):=\inf\left\\{E(\varphi)\ \big{|}\
\varphi\in\mathcal{SH}_{m}(X,\omega)\cap L^{\infty}(X),\
u\leq\varphi\right\\}.$
###### Lemma 6.2.
For any $\varphi\in\mathcal{E}^{1}(X,\omega,m)$ such that $\varphi\leq 0$ we
have
$\int_{X}\varphi H_{m}(\varphi)\leq E(\varphi)\leq\frac{1}{m+1}\int_{X}\varphi
H_{m}(\varphi).$
The class $\mathcal{E}^{1}(X,\omega,m)$ consists of finite energy
$(\omega,m)$-subharmonic functions. If $(\varphi_{j})$ is a sequence in
$\mathcal{E}^{1}(X,\omega,m)$ decreasing to $\varphi$ such that
$\inf_{j}E(\varphi_{j})>-\infty$
then $\varphi\in\mathcal{E}^{1}(X,\omega,m)$ and
$E(\varphi)=\lim_{j\to+\infty}E(\varphi_{j})$
###### Lemma 6.3.
The functional $E$ is a primitive of the complex Hessian operator. More
precisely, whenever $\varphi+tv$ belongs to $\mathcal{E}^{1}(X,\omega,m)$ for
small $t$,
$\frac{dE(\varphi+tv)}{dt}|_{t=0}=\int_{X}vH_{m}(\varphi).$
The functional $E$ is concave increasing, satisfies
$E(\varphi+c)=E(\varphi)+c$ for all
$c\in\mathbb{R},\varphi\in\mathcal{E}^{1}(X,\omega,m)$ , and the cocycle
condition
$E(\varphi)-E(\psi)=\frac{1}{m+1}\sum_{j=0}^{m}\int_{X}(\varphi-\psi)(\omega+dd^{c}\varphi)^{j}\wedge(\omega+dd^{c}\psi)^{m-j}\wedge\omega^{n-m},$
for all $\varphi,\psi\in\mathcal{E}^{1}(X,\omega,m)$. Moreover,
$\int_{X}(\varphi-\psi)H_{m}(\varphi)\leq
E(\varphi)-E(\psi)\leq\int_{X}(\varphi-\psi)H_{m}(\psi).$
###### Proof.
The proof is a trivial adaptation of [6]. ∎
###### Lemma 6.4.
The functional $E$ is upper semicontinuous with respect to the $L^{1}$
topology on $\mathcal{SH}_{m}(X,\omega)$.
###### Proof.
Assume that $(\varphi_{j})$ is a sequence in $\mathcal{SH}_{m}(X,\omega)$
converging to $\varphi\in\mathcal{SH}_{m}(X,\omega)$ in $L^{1}$. We are to
prove that
$\limsup_{j\to+\infty}E(\varphi_{j})\leq E(\varphi).$
If the limsup is $-\infty$ there is nothing to do. Thus we can assume that
$E(\varphi_{j})$ is uniformly bounded from below. Then since
$E(\varphi_{j})\leq\int_{X}\varphi_{j}\omega^{n}$
the sequence $(\varphi_{j})$ stays in a compact subsets of
$\mathcal{SH}_{m}(X,\omega)$. Assume that
$\varphi_{j}\rightarrow\varphi\in\mathcal{SH}_{m}(X,\omega)$ in $L^{1}(X)$.
Set
$\psi_{j}:=(\sup_{k\geq j}\varphi_{k})^{*}.$
Then $\psi_{j}$ decreases to $\varphi$. Since $E$ is increasing we get a
uniform lower bound for $E(\psi_{j})$. Thus $\varphi$ belongs to
$\mathcal{E}(X,\omega,m)$ and
$E(\varphi)=\lim_{j\to+\infty}E(\psi_{j})\geq\limsup_{j\to+\infty}E(\varphi_{j}).$
∎
###### Lemma 6.5.
For each $C>0$ the set
$\mathcal{E}^{1}_{C}(X,\omega,m):=\\{\varphi\in\mathcal{E}^{1}(X,\omega,m)\
\big{|}\ \sup_{X}\varphi\leq 0,\ E(\varphi)\geq-C\\}$
is a compact convex subset of $\mathcal{SH}_{m}(X,\omega)$.
###### Proof.
The convexity of $\mathcal{E}^{1}_{C}(X,\omega,m)$ follows from the concavity
of $E$. The compactness follows from the upper semicontinuity of $E$. ∎
The following volume-capacity estimate is due to Dinew and Kołodziej [15]:
###### Lemma 6.6.
Let $1<p<\frac{n}{n-m}.$ There exists a constant $C=C(p,\omega)$ such that for
every Borel subset $K$ of $X$, we have
$V(K)\leq C\cdot{\rm Cap}_{\omega,m}(K)^{p},$
where $V(K):=\int_{K}\omega^{n}$.
###### Corollary 6.7.
Let $\varphi\in\mathcal{SH}_{m}(X,\omega)$. Then $\varphi\in
L^{p}(X,\omega^{n})$ for any $p<\frac{n}{n-m}$.
###### Proof.
We can assume that $\sup_{X}\varphi=1$. Fix $p<n/(n-m)$ and $q$ such that
$p<q<n/(n-m)$. It follows from [30, Corollary 3.19] and the previous volume-
capacity estimate that
$\displaystyle\int_{X}(-\varphi)^{p}\omega^{n}$ $\displaystyle=$
$\displaystyle 1+p\int_{1}^{+\infty}t^{p-1}V(\varphi<-t)dt$
$\displaystyle\leq$ $\displaystyle 1+C_{q}p\int_{1}^{+\infty}t^{p-1}\left[{\rm
Cap}_{\omega,m}(\varphi<-t)\right]^{q}dt$ $\displaystyle\leq$ $\displaystyle
1+C_{q}Cp\int_{1}^{+\infty}t^{p-q-1}dt<+\infty.$
∎
One expects that Corollary 6.7 holds for any $p<\frac{nm}{n-m}$. In the local
context where $\omega$ is the standard Kähler metric, it was known as Błocki’s
conjecture.
###### Lemma 6.8.
Fix $\varphi\in\mathcal{SH}_{m}(X,\omega)$. If
$\int_{0}^{+\infty}t^{m}{\rm Cap}_{\omega,m}(\varphi<-t)dt<+\infty$
then $\varphi\in\mathcal{E}^{1}(X,\omega,m)$. Conversely for each $C>0$,
$\sup\left\\{\int_{0}^{+\infty}t{\rm Cap}_{\omega,m}(\varphi<-t)dt\ \big{|}\
\varphi\in\mathcal{E}_{m,C}^{1}(X,\omega)\right\\}<+\infty.$
###### Proof.
Fix $\varphi\in\mathcal{SH}_{m}(X,\omega)$. We can assume that
$\sup_{X}\varphi=-1$. Observe that for $t\geq 1$, the function
$1+t^{-1}\max(\varphi,−t)$ is $(\omega,m)$-sh with values in $[0,1]$, hence
$H_{m}(\max(\varphi,-t))\leq t^{m}{\rm Cap}_{\omega,m}.$
Let us prove the first assertion. If
$\int_{0}^{+\infty}t^{m}{\rm Cap}_{\omega,m}(\varphi<-t)dt<+\infty$
then in particular $t^{m}{\rm Cap}_{\omega,m}(\varphi<-t)$ converges to $0$ as
$t\to+\infty$. This coupled with the above observation yields
$\int_{\\{\varphi\leq-t\\}}H_{m}(\max(\varphi,-t))\longrightarrow 0,$
which implies that $\varphi\in\mathcal{E}(X,\omega,m)$. Now by the comparison
principle $H_{m}(\max(\varphi,-t))$ coincides with $H_{m}(\varphi)$ on the
Borel set $\\{\varphi>-t\\}$. We thus get
$\displaystyle\int_{X}(-\varphi)H_{m}(\varphi)$ $\displaystyle=$
$\displaystyle 1+\int_{1}^{+\infty}H_{m}(\varphi)(\varphi\leq-t)dt$
$\displaystyle=$ $\displaystyle
1+\int_{1}^{+\infty}\left[1-H_{m}(\varphi)(\varphi>-t)\right]dt$
$\displaystyle=$ $\displaystyle
1+\int_{1}^{+\infty}\left[1-H_{m}(\max(\varphi,-t))(\varphi>-t)\right]dt$
$\displaystyle\leq$ $\displaystyle
1+\int_{1}^{+\infty}H_{m}(\max(\varphi,-t))(\varphi\leq-t))dt$
$\displaystyle\leq$ $\displaystyle 1+\int_{1}^{+\infty}{\rm
Cap}_{\omega,m}(\varphi\leq-t)dt$ $\displaystyle<$ $\displaystyle+\infty,$
which yields $\varphi\in\mathcal{E}(X,\omega,m)$.
We now prove the second assertion. The proof is slightly different from the
classical Monge-Ampère equation due to a lack of integrability (it is not very
clear that $\int_{X}\varphi^{2}\omega^{n}<+\infty$). Fix
$u\in\mathcal{SH}_{m}(X,\omega)$ with values in $[-1,0]$. Observe that
$(\varphi<-2t)\subset(t^{-1}\varphi<u-1)\subset(\varphi<-t).$
It follows from the comparison principle that
$\int_{\\{\varphi<-2t\\}}H_{m}(u)\leq\int_{\\{\varphi<-t\\}}H_{m}(t^{-1}\varphi).$
Expanding
$H_{m}(t^{-1}\varphi)\leq(t^{-1}(\omega+dd^{c}\varphi)+\omega)^{m}\wedge\omega^{n-m}$
yields
$\int_{2}^{+\infty}t{\rm Cap}_{\omega,m}(\varphi<-t)=4\int_{1}^{+\infty}t{\rm
Cap}_{\omega,m}(\varphi<-2t)dt\\\ \leq 4\int_{1}^{+\infty}t{\rm
Vol}(\varphi<-t)dt+4\sum_{j=1}^{m}\binom{m}{j}\int_{X}(-\varphi)\omega_{\varphi}^{j}\wedge\omega^{n-j}.$
The last term is finite and uniformly bounded in
$\varphi\in\mathcal{E}^{1}_{C}(X,\omega,m)$. Fix $1<p<\frac{n}{n-m}$ and
$0<\gamma<1$. Using Hölder inequality we get
$\int_{1}^{+\infty}t{\rm Vol}(\varphi<-t)dt=\int_{1}^{+\infty}t{\rm
Vol}(\varphi<-t)^{\gamma}{\rm Vol}(\varphi<-t)^{1-\gamma}dt\\\
\leq\left[\int_{1}^{+\infty}t{\rm
Vol}(\varphi<-t)^{q\gamma}dt\right]^{1/q}\left[\int_{1}^{+\infty}t{\rm
Vol}(\varphi<-t)^{r(1-\gamma)}dt\right]^{1/r}\\\ \leq
A\left[\int_{1}^{+\infty}t{\rm
Cap}_{\omega,m}(\varphi<-t)^{pq\gamma}dt\right]^{1/q}\left[\int_{1}^{+\infty}t{\rm
Cap}_{\omega,m}(\varphi<-t)^{pr(1-\gamma)}dt\right]^{1/r}\\\ \leq
A\left[\int_{1}^{+\infty}t{\rm
Cap}_{\omega,m}(\varphi<-t)dt\right]^{1/q}\left[\int_{1}^{+\infty}t^{1-pr(1-\gamma)}dt\right]^{1/r}.$
Here, $1/q+1/r=1$ and we have chosen $\gamma$ so that $pq\gamma=1$ and
$pr(1-\gamma)>2$. Such a choice is always possible. The constant $A$ is also
uniform in $\varphi\in\mathcal{E}^{1}_{C}(X,\omega,m)$ since
$\sup_{X}\varphi\geq E(\varphi)\geq-C$ and
${\rm Cap}_{\omega,m}(u<-t)\leq C/t$
for a uniform constant $C$ as follows from [30].
By considering $\varphi_{j}:=\max(\varphi,-j)$ and applying what we have done
so far we get
$C_{j}\leq A\cdot C_{j}^{1/q}+B,$
where $C_{j}:=\int_{1}^{+\infty}t{\rm Cap}_{\omega,m}(\varphi_{j}<-t)$ and
$A,B$ are universal constant. Letting $j\to+\infty$ we get the result. ∎
### 6.2. Upper semicontinuity
Let $\mu$ be a probability measure on $X$. The functional $\mathcal{F}_{\mu}$
is defined by
$\mathcal{F}_{\mu}(\varphi):=E(\varphi)-\int_{X}\varphi d\mu.$
###### Lemma 6.9.
Let $\mu$ be a probability measure which does not charge $m$-polar sets. Let
$(u_{j})\subset\mathcal{SH}_{m}(X,\omega)$ be a sequence which converges in
$L^{1}(X)$ towards $u\in\mathcal{SH}_{m}(X,\omega)$. If $\sup_{j\geq
0}\int_{X}u_{j}^{2}d\mu<+\infty$ then
$\int_{X}u_{j}d\mu\longrightarrow\int_{X}ud\mu.$
###### Proof.
Since $\int_{X}u_{j}d\mu$ is bounded it suffices to prove that every cluster
point is $\int_{X}ud\mu.$ Without loss of generality we can assume that
$\int_{X}u_{j}d\mu$ converges. Since the sequence $u_{j}$ is bounded in
$L^{2}(\mu)$, one can apply Banach-Saks theorem to extract a subsequence
(still denoted by $u_{j}$) such that
$\varphi_{N}:=\frac{1}{N}\sum_{j=1}^{N}u_{j}$
converges in $L^{2}(\mu)$ and $\mu$-almost everywhere to $\varphi.$ Observe
also that $\varphi_{N}\to u$ in $L^{1}(X)$. For each $j\in\mathbb{N}$ set
$\psi_{j}:=(\sup_{k\geq j}\varphi_{k})^{*}.$
Then $\psi_{j}\downarrow u$ in $X$. But $\mu$ does not charge the $m$-polar
set
$\left\\{(\sup_{k\geq j}\varphi_{k})^{*}>\sup_{k\geq j}\varphi_{k}\right\\}.$
We thus get $\psi_{j}=\sup_{k\geq j}\varphi_{k}$ $\mu$-almost everywhere.
Therefore, $\psi_{j}$ converges to $\varphi$ $\mu$-almost everywhere hence
$u=\varphi$ $\mu$-almost everywhere. This yields
$\lim_{j}\int_{X}u_{j}d\mu=\lim_{j}\int_{X}\varphi_{j}d\mu=\int_{X}ud\mu.$
∎
###### Lemma 6.10.
Let $\mu$ be a probability measure on $X$ such that
$\mu(K)\leq A{\rm Cap}_{\omega,m}(K),\ \forall K\subset X,$
for some positive constant $A$. Then the functional $\mathcal{F}_{\mu}$ is
upper semicontinuous on each compact subset $\mathcal{E}^{1}_{C}(X,\omega,m)$,
$C>0$.
###### Proof.
Let $(\varphi_{j})$ be a sequence in $\mathcal{E}^{1}_{C}(X,\omega,m)$
converging in $L^{1}(X)$ to $\varphi\in\mathcal{E}^{1}_{C}(X,\omega,m)$. We
can assume that $\varphi_{j}\leq 0$. It follows from Lemma 6.8 that
$\int_{X}(-\varphi_{j})^{2}d\mu\leq
2\int_{0}^{+\infty}t\mu(\varphi_{j}<-t)dt\leq 2A\int_{0}^{+\infty}t{\rm
Cap}_{\omega,m}(\varphi_{j}<-t)dt\leq 2AC^{\prime},$
for a positive constant $C^{\prime}$. From Lemma 6.9 we thus get
$\int_{X}\varphi_{j}d\mu\longrightarrow\int_{X}\varphi d\mu.$
This coupled with the upper semicontinuity of $E$ yield the result. ∎
###### Definition 6.11.
We say that the functional $\mathcal{F}_{\mu}$ is proper if whenever
$\varphi_{j}\in\mathcal{E}^{1}(X,\omega,m)$ are such that
$E(\varphi_{j})\rightarrow-\infty$ and $\int_{X}\varphi_{j}=0$ then
$\mathcal{F}_{\mu}(\varphi_{j})\to-\infty$.
###### Lemma 6.12.
Let $\mu$ be a probability measure on $X$ such that
$\mathcal{E}^{1}_{C}(X,\omega,m)\subset L^{1}(\mu)$. The functional
$\mathcal{F}_{\mu}$ is proper: there exists $C>0$ such that for all
$\varphi\in\mathcal{E}^{1}(X,\omega,m)$ with $\int_{X}\varphi\omega^{n}=0$ we
have
$\mathcal{F}_{\mu}(\varphi)\leq E(\varphi)+C|E(\varphi)|^{1/2}.$
###### Proof.
Arguing by contradiction we can prove that
$\sup\left\\{\int_{X}(-\psi)d\mu\ \big{|}\
\psi\in\mathcal{E}^{1}_{C}(X,\omega,m)\right\\}<+\infty,\forall C>0.$
Now we can repeat the arguments in [6]. ∎
### 6.3. The projection theorem
Let $f$ be an upper semicontinuous function on $X$. Recall that the projection
of $f$ on $\mathcal{SH}_{m}(X,\omega)$ is defined by
$P(f):=\sup\left\\{u\in\mathcal{SH}_{m}(X,\omega)\ \big{|}\ u\leq f\right\\}.$
###### Lemma 6.13.
Let $u,v$ be continuous function on $X$. Then
$E\circ P(u+v)-E\circ
P(u)=\int_{0}^{1}\left[\int_{X}vH_{m}(P(u+tv))\right]dt.$
###### Proof.
One could prove the lemma by following [5]. But we give here a slightly
different (and simpler) proof using the same ideas.
Observe that it is equivalent to showing that
(6.1) $\frac{dE\circ P(u+tv)}{dt}\big{|}_{t=0}=\int_{X}vH_{m}(P(u)).$
By changing $v$ to $-v$ it suffices to take care of the right derivative. Fix
$t>0$. It follows from Lemma 6.3 that
$\displaystyle\int_{X}\frac{P(u+tv)-P(u)}{t}H_{m}(P(u+tv))$
$\displaystyle\leq$ $\displaystyle\frac{E\circ P(u+tv)-E(P(u))}{t}$
$\displaystyle\leq$ $\displaystyle\int_{X}\frac{P(u+tv)-P(u)}{t}H_{m}(P(u)).$
Since $\int_{X}(u-P(u))H_{m}(P(u))=0$ as follows from Theorem 1 the second
inequality above yields the inequality ”$\leq$” in (6.1). On the other hand
the first inequality above coupled with the orthogonal relation gives
$\displaystyle\frac{E\circ P(u+tv)-E(P(u))}{t}$ $\displaystyle\geq$
$\displaystyle\int_{X}\frac{P(u+tv)-P(u)}{t}H_{m}(P(u+tv))$ $\displaystyle=$
$\displaystyle\int_{X}\frac{u+tv-P(u)}{t}H_{m}(P(u+tv))$ $\displaystyle\geq$
$\displaystyle\int_{X}vH_{m}(P(u+tv)).$
By letting $t\to 0^{+}$ we get the inequality ”$\geq$” in (6.1) since $H_{m}$
is continuous under uniform convergence. The proof is thus complete. ∎
###### Theorem 6.14.
Fix $\varphi\in\mathcal{E}^{1}(X,\omega,m)$ and
$v\in\mathcal{C}(X,\mathbb{R})$. Then the function $t\mapsto E\circ
P(\varphi+tv)$ is differentiable at zero, with
$\frac{dE\circ P(\varphi+tv)}{dt}\big{|}_{t=0}=\int_{X}vH_{m}(\varphi).$
###### Proof.
As in the previous lemma it suffices to prove that
$E\circ P(\varphi+v)-E\circ
P(\varphi)=\int_{0}^{1}\left[\int_{X}vH_{m}(P(\varphi+tv))\right]dt,$
for every $\varphi\in\mathcal{E}^{1}(X,\omega,m)$ and
$v\in\mathcal{C}(X,\mathbb{R})$. It follows from our approximation theorem
(Theorem 2) that we can find a sequence of smooth $(\omega,m)$-sh functions
decreasing to $\varphi$. By the continuity of $H_{m}$ we thus can assume that
$\varphi$ is smooth. The result now follows from Lemma 6.13. ∎
## 7\. Resolution of the degenerate complex Hessian equation
Let $\mu$ be a probability measure on $X$ which does not charge $m$-polar
sets. We study the following degenerate complex Hessian equation
(7.1) $(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=\mu.$
###### Theorem 7.1.
Let $\mu$ be a probability measure such that $\mu\leq A{\rm Cap}_{\omega,m}$
for some positive constant $A$ . If $\mathcal{F}_{\mu}$ is proper, then there
exists $\varphi\in\mathcal{E}^{1}(X,\omega,m)$ which solves (7.1) and such
that
$\mathcal{F}_{\mu}(\varphi)=\sup_{\mathcal{E}^{1}(X,\omega,m)}\mathcal{F}_{\mu}.$
###### Proof.
The proof is a word-by-word copy of [6]. We recall the arguments below.
Since $\mathcal{F}_{\mu}$ is invariant by translations and proper, we can find
$C>0$ so large that
$\sup_{\mathcal{E}^{1}(X,\omega,m)}\mathcal{F}_{\mu}=\sup_{\mathcal{E}^{1}_{C}(X,\omega,m)}\mathcal{F}_{\mu}.$
Recall that by definition
$\mathcal{E}^{1}_{C}(X,\omega,m):=\\{\varphi\in\mathcal{E}^{1}(X,\omega,m)\
\big{|}\ \sup_{X}\varphi\leq 0,\ E(\varphi)\geq-C\\}$
is a compact convex subset of $\mathcal{SH}_{m}(X,\omega)$. It follows from
Lemma 6.10 that $\mathcal{F}_{\mu}$ is upper semi-continuous on
$\mathcal{E}^{1}_{C}(X,\omega,m)$, thus we can find
$\varphi\in\mathcal{E}^{1}_{C}(X,\omega,m)$ which maximizes the functional
$\mathcal{F}_{\mu}$ on $\mathcal{E}^{1}(X,\omega,m)$.
Fix $v\in\mathcal{C}(X,\mathbb{R})$ an arbitrary continuous function on $X$
and consider
$g(t):=E\circ P(\varphi+tv)-\int_{X}(\varphi+tv)d\mu,t\in\mathbb{R}.$
Then for every $t\in\mathbb{R}$,
$g(t)\leq E\circ
P(\varphi+tv)-\int_{X}P(\varphi+tv)d\mu=\mathcal{F}_{\mu}(P(\varphi+tv))\leq\mathcal{F}_{\mu}(\varphi)=g(0).$
Thus $g$ attains its maximum at $0$ and hence by differentiability of $g$ at
$0$ we have $g^{\prime}(0)=0$ which implies
$\int_{X}vd\mu=\int_{X}vH_{m}(\varphi).$
Since $v$ has been chosen arbitrarily the conclusion follows. ∎
###### Theorem 7.2.
Let $\mu$ be a probability measure on $X$. Then
$\mathcal{E}^{1}(X,\omega,m)\subset L^{1}(\mu)$ if and only if
$\mu=H_{m}(\varphi)$ for some $\varphi\in\mathcal{E}^{1}(X,\omega,m)$.
###### Proof.
If $\mu=H_{m}(\varphi)$ for some $\varphi\in\mathcal{E}(X,\omega,m)$ then for
any $\psi\in\mathcal{E}(X,\omega,m)$,
$\int_{X}\psi H_{m}(\varphi)>-\infty,$
since by the comparison principle we can prove that (see [21, Proposition
2.5])
$\int_{X}\psi H_{m}(\varphi)\geq 2(m+1)(E(\varphi)+E(\psi))>-\infty.$
Assume now that $\mathcal{E}^{1}(X,\omega,m)\subset L^{1}(\mu)$. In
particular, $\mu$ does not charge $m$-polar sets. Observe first that the set
$\mathcal{M}:=\\{\nu\in\mathcal{P}(X)\ \big{|}\ \nu\leq{\rm
Cap}_{\omega,m}\\},$
where $\mathcal{P}(X)$ is the space of probability measures on $X$, is a
compact convex subset of $\mathcal{P}(X)$. Indeed, the convexity is clear
while the compactness follows from the outer regularity of the $m$-capacity
(see Theorem 4.6). Using [43], we project $\mu$ on this compact convex set
(the original idea of this proof is due to Cegrell [11])
$\mu=f\nu+\sigma,$
where $\nu\in\mathcal{M}$, $0\leq f\in L^{1}(\nu)$ and
$\sigma\perp\mathcal{M}$. Since $\mu$ vanishes on $m$-polar sets one has
$\sigma\equiv 0$. Set $\mu_{j}:=c_{j}\min(f,j)\nu$ where $c_{j}$ is a
normalization constant so that $\mu_{j}$ is a probability measure. Since
$\mu_{j}\leq jc_{j}{\rm Cap}_{\omega,m}$ it follows from Theorem 7.1 that
there exists $\varphi_{j}\in\mathcal{E}(X,\omega,m)$ such that
$\mu_{j}=H_{m}(\varphi_{j})$. We normalize $\varphi_{j}$ so that
$\sup_{X}\varphi_{j}=0$. We can also assume that
$\varphi_{j}\rightarrow\varphi\in\mathcal{SH}_{m}(X,\omega)$ in $L^{1}(X)$.
Now
$|E(\varphi_{j})|\leq\int_{X}(-\varphi_{j})H_{m}(\varphi_{j})\leq
c_{j}\int_{X}(-\varphi_{j})d\nu\leq C|E(\varphi_{j})|^{1/2},$
as follows from Lemma 6.12. It follows that $E(\varphi_{j})$ is uniformly
bounded and hence $\varphi\in\mathcal{E}^{1}(X,\omega,m)$. Now consider
$\phi_{j}:=(\sup_{k\geq j}\varphi_{k})^{*}.$
Then $\phi_{j}\downarrow\varphi$ and it follows from Proposition 5.8 that
$H_{m}(\phi_{j})\geq\min(f,j)\nu.$
Hence $H_{m}(\varphi)\geq\mu$ whence equality since both of them are
probability measures. ∎
###### Theorem 7.3.
Let $\mu$ be a probability measure on $X$ which does not charge $m$-polar
sets. Then there exists $\varphi\in\mathcal{E}(X,\omega,m)$ such that
$H_{m}(\varphi)=\mu$.
###### Proof.
One can repeat the arguments in [6]. ∎
### Concluding remarks
The principal result of this paper is the regularization theorem. It is
amazing that we can directly regularize any $(\omega,m)$-sh functions by
solving appropriate complex Hessian equations. On the way to regularize
singular functions we also proved the orthogonal relation is the second
amazing thing. The classical method to prove such a thing is to use the
balayage argument which is now possible thanks to the resolution of the
corresponding local Dirichlet problem [36].
One can also carry a similar study of a potential theory for
$(\omega,m)$-subharmonic functions in $\mathbb{C}^{n}$ with $\omega$ being any
Kähler metric.
## References
* [1] S. Alekser, M. Verbitsky, Quaternionic Monge-Ampère equations and Calabi problem for HKT-manifolds, Israel J. Math. 176 (2010), 109-138.
* [2] P. Ahag, U. Cegrell, R. Czyz, On Dirichlet’s principle and problem, Math. Scand. 110 (2012), no. 2, 235-250.
* [3] E. Bedford, B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), no. 1, 1-44.
* [4] R. J. Berman, From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit, arXiv:1307.3008.
* [5] R. Berman, S. Boucksom, Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math., 181 (2010), 337-394.
* [6] R. J. Berman, S. Boucksom, V. Guedj, A. Zeriahi, A variational approach to complex Monge-Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179-245.
* [7] Z. Błocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1735-1756.
* [8] Z. Błocki, The Monge-Ampère equation on compact Kähler manifolds, Lect. Notes in Mathematics 238 (2012).
* [9] Z. Błocki, S. Kołodziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2089-2093.
* [10] L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261-301.
* [11] U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998), no. 2, 187-217.
* [12] M. Charabati, Modulus of continuity of solutions to complex Hessian equations, preprint arXiv:1401.8254.
* [13] K.-S. Chou and X.-J. Wang, Variational theory for Hessian equations, Comm. Pure Appl. Math., 54 (2001), 1029-1064.
* [14] S. Dinew, Uniqueness in $\mathcal{E}(X,\omega)$, J. Funct. Anal. 256 (2009), no. 7, 2113-2122.
* [15] S. Dinew, S. Kołodziej, A priori estimates for complex Hessian equations, arXiv:1112.3063v1.
* [16] S. Dinew, S. Kołodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, arXiv:1203.3995v.
* [17] P. Eyssidieux, V. Guedj, A. Zeriahi, Viscosity solutions to degenerate complex Monge-Ampère equations, Comm. Pure Appl. Math. 64 (2011), no. 8, 1059-1094.
* [18] P. Eyssidieux, V. Guedj, A. Zeriahi, Continuous approximation of quasi-plurisubharmonic functions, arXiv:1311.2866.
* [19] L. Gårding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959) 957-965.
* [20] V. Guedj, A. Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), no. 4, 607-639.
* [21] V. Guedj, A. Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), no. 2, 442-482.
* [22] F. R. Harvey, H. B. Lawson, The equivalence of viscosity and distributional subsolutions for convex subequations - a strong Bellman principle, arXiv:1301.4914.
* [23] Z. Hou, Complex Hessian equation on Kähler manifold, Int. Math. Res. Not. IMRN (2009), no. 16, 3098-3111.
* [24] Z. Hou, X. Ma, D.-M. Wu, A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett. 17 (2010), no. 3, 547-561.
* [25] A. Jbilou, Équations hessiennes complexes sur des variétés kählériennes compactes, C. R. Math. Acad. Sci. Paris 348 (2010), no. 1-2, 41-46.
* [26] V. N. Kokarev, Mixed volume forms and a complex equation of Monge-Ampère type on Kähler manifolds of positive curvature, Izv. RAN. Ser. Mat. 74:3 (2010), 65-78.
* [27] S. Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998) 69-117.
* [28] S. Kołodziej, The complex Monge-Ampère equation and theory, Memoirs Amer. Math. Soc. 178 (2005) 64p.
* [29] S.-Y. Li, On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian, Asian J. Math. 8 (2004),no. 1, 87-106.
* [30] H. C. Lu, Solutions to degenerate complex Hessian equations, Journal de mathématiques pures et appliquées 100 (2013) pp. 785-805.
* [31] H. C. Lu, Viscosity solutions to complex Hessian equations, J. Funct. Anal. 264 (2013) pp. 1355-1379.
* [32] H.C. Lu, A variational approach to complex Hessian equations in $\mathbb{C}^{n}$, arXiv:1301.6502.
* [33] N. Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988), no. 2, 179-214.
* [34] N. C. Nguyen, Subsolution theorem for the complex Hessian equation, to appear on Universitatis Iagellonicae Acta Mathematica.
* [35] N. C. Nguyen, Hölder continuous solutions to complex Hessian equations, arXiv:1301.0710.
* [36] S. Plis, The smoothing of $m$-subharmonic functions, arXiv:1312.1906.
* [37] A. S. Sadullaev, B. I. Abdullaev, Capacities and Hessians in a class of m-subharmonic functions, Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Vol. 279, pp. 166-192.
* [38] J. Song, B. Weinkove, On the convergence and singularities of the J-flow with applications to the Mabuchi energy, Comm. Pure. Appl. Math. 61 (2008), 210-229.
* [39] N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151-164.
* [40] N. S. Trudinger, X.-J. Wang, Hessian measures II, Ann. of Math. 150 (1999), 579-604.
* [41] V. Tosatti, Y. Wang, B. Weinkove, X. Yang, $C^{2,\alpha}$ estimates for nonlinear elliptic equations in complex and almost complex geometry, arXiv:1402.0554.
* [42] J. Urbas, An interior second derivative bound for solutions of Hessian equations, Calc. Var. PDE (12) (2001), 417-431.
* [43] J. Rainwater, A note on the preceding paper, Duke Math. J. 36 (1969) 799-800.
* [44] S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411.
* [45] X.-J. Wang, The k-Hessian equation, Lecture Notes in Math., 1977, Springer, Dordrecht, 2009.
* [46] Y. Wang, A Viscosity Approach to the Dirichlet Problem for Complex Monge-Ampère Equations, Math. Z. 272 (2012), no. 1-2, 497-513.
|
arxiv-papers
| 2014-02-20T21:06:44 |
2024-09-04T02:49:58.495537
|
{
"license": "Public Domain",
"authors": "Chinh H. Lu and Van-Dong Nguyen",
"submitter": "Chinh Lu Hoang",
"url": "https://arxiv.org/abs/1402.5147"
}
|
1402.5188
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arxiv-papers
| 2014-02-21T02:22:57 |
2024-09-04T02:49:58.510240
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chao Wang",
"submitter": "Chao Wang",
"url": "https://arxiv.org/abs/1402.5188"
}
|
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1402.5197
|
# An $L_{p}$-theory for a class of non-local elliptic equations related to
nonsymmetric measurable kernels
Ildoo Kim Department of Mathematics, Korea University, 1 Anam-dong, Sungbuk-
gu, Seoul, 136-701, Republic of Korea [email protected] and Kyeong-Hun Kim
Department of Mathematics, Korea University, 1 Anam-dong, Sungbuk-gu, Seoul,
136-701, Republic of Korea [email protected]
###### Abstract.
We study the integro-differential operators $L$ with kernels $K(y)=a(y)J(y)$,
where $J(y)dy$ is a Lévy measure on $\mathbb{R}^{d}$ (i.e.
$\int_{\mathbb{R}^{d}}(1\wedge|y|^{2})J(y)dy<\infty$) and $a(y)$ is an only
measurable function with positive lower and upper bounds. Under few additional
conditions on $J(y)$, we prove the unique solvability of the equation
$Lu-\lambda u=f$ in $L_{p}$-spaces and present some $L_{p}$-estimates of the
solutions.
###### Key words and phrases:
Non-local elliptic equations, Integro-differential equations, Lévy processes,
non-symmetric measurable kernels
###### 2010 Mathematics Subject Classification:
35R09, 47G20
The research of the second author was supported by Basic Science Research
Program through the National Research Foundation of Korea(NRF) funded by the
Ministry of Education, Science and Technology (2013020522)
## 1\. introduction
There has been growing interest in the integro-differential equations related
to pure jump processes owing to their applications in various models in
physics, biology, economics, engineering and many others involving long-range
jumps and interactions. In this article we study the non-local elliptic
equations having the operators
$Lu:=\int_{{\mathbb{R}}^{d}}\Big{(}u(x+y)-u(x)-y\cdot\nabla
u(x)\chi(y)\Big{)}\,K(x,y)dy,$
and
$\tilde{L}u:=\int_{{\mathbb{R}}^{d}}\Big{(}u(x+y)-u(x)-y\cdot\nabla
u(x)1_{|y|<1}\Big{)}\,K(x,y)dy,$
where the kernel $K(x,y)=a(y)J(y)$ depends only on $y$,
$\displaystyle\chi(y)=0~{}\text{if}~{}\sigma\in(0,1),\quad\chi(y)=1_{|y|<1}~{}\text{if}~{}\sigma=1,\quad\chi(y)=1~{}\text{if}~{}\sigma\in(1,2].$
The constant $\sigma$ depends on $J(y)$ and is defined in (2.10). In
particular, if $J(y)=c(d,\alpha)|y|^{-d-\alpha}$ for some $\alpha\in(0,2)$
then $\sigma=\alpha$. Note that if $a(y)$ is symmetric then $\tilde{L}=L$, and
in general we (formally) have
$\tilde{L}u=Lu+b\cdot\nabla u,$
where
$b^{i}=-\int_{B_{1}}y^{i}a(y)J(y)dy\quad\text{if}\,\,\sigma\in(0,1),\quad\quad
b^{i}=\int_{\mathbb{R}^{d}\setminus
B_{1}}y^{i}a(y)J(y)dy\quad\text{if}\,\,\sigma\in(1,2].$
The main goal of this article is to prove the unique solvability of the
equations
$\displaystyle Lu-\lambda u=f\quad\text{and}\quad\tilde{L}u-\lambda
u=f,\quad\lambda>0$ (1.1)
in appropriate $L_{p}$-spaces and present some $L_{p}$-estimates of the
solutions. Here $p>1$. If $p=2$, the only condition we are assuming is that
$a(y)$ has positive lower and upper bounds and $J(y)$ is rotationally
invariant. If $p\neq 2$, we assume some additional conditions on $J(y)$, which
are described in (1.5) and (1.6) below (also see Assumption 2.18).
Below is a short description on related $L_{p}$-theories. For other results
such as the Harnack inequality and Hölder estimates we refer the readers to
[4], [5], [8], [10] and [14]. If $K(x,y)=c(d,\alpha)|y|^{-d-\alpha}$, where
$\alpha\in(0,2)$ and $c(d,\alpha)$ is some normalization constant, then $L$
becomes the fractional Laplacian operator
$\Delta^{\alpha/2}:=-(-\Delta)^{\alpha/2}$. For the fractional Laplacian
operator, $L_{p}$-estimates can be easily obtained by the Fourier multiplier
theory (for instance, [16]). In [2] $L_{p}$-estimates were obtained for
elliptic equations with “symmetric” kernels, and an $L_{p}$-theory for the
equation $Lu-\lambda u=f$ with measurable nonsymmetric kernel
$K(x,y)=a(y)|y|^{-d-\alpha}$ was recently introduced in [9]. For parabolic
equations, the authors of [12] handled the equations with the kernel
$K(x,y)=a(x,y)|y|^{-d-\alpha}$ under the condition that the coefficient
$a(x,y)$ is homogeneous of order zero in $y$ and sufficiently smooth in $y$,
but it is allowed that $a$ also depends on $x$. Lately in [17], an
$L_{p}$-regularity theory for parabolic equations was constructed for $J(y)$
satisfying
$\nu_{1}^{\alpha}(B)\leq\int I_{B}(y)J(y)dy\leq\nu_{2}^{\alpha}(B)\quad\forall
B\in\mathcal{B}(\mathbb{R}^{d}),$
where $\nu_{i}^{(\alpha)}$ are Lévy measures taking the form
$\displaystyle\nu_{i}^{(\alpha)}(B):=\int_{\mathbb{S}^{d-1}}\Big{(}\int_{0}^{\infty}\frac{1_{B}(r\theta)dr}{r^{1+\alpha}}\Big{)}S_{i}(d\theta),$
(1.2)
with finite surface measures $dS_{i}$ on $\mathbb{S}^{d-1}$. Since the same
constant $\alpha$ is used for both $\nu_{1}^{(\alpha)}$ and
$\nu_{2}^{(\alpha)}$, even the Lévy measure $J(y)$ related to the operator
$\Delta^{\alpha_{1}/2}+\Delta^{\alpha_{2}/2}$ is not of type (1.2) if
$\alpha_{1}\neq\alpha_{2}$.
From the probabilistic point of view, the fractional Laplacian operator can be
described as the infinitesimal generator of $\alpha$-stable processes. That
is,
$\Delta^{\alpha/2}f(x)=\lim_{t\to
0^{+}}\frac{1}{t}\mathbb{E}[f(x+X_{t})-f(x)],\quad f\in C^{\infty}_{0}$
where $X_{t}$ is an $\mathbb{R}^{d}$-valued Lévy process in a probability
space $(\Omega,P)$ with the characteristic function
$\mathbb{E}e^{i\lambda\cdot X_{t}}:=\int_{\Omega}e^{i\lambda\cdot
X_{t}}\,dP=e^{-t|\lambda|^{\alpha}}$. More generally, for any Bernstein
function $\phi$ with $\phi(0+)=0$ (equivalently,
$\phi(\lambda)=\int^{\infty}_{0}(1-e^{-\lambda t})\mu(dt)$ for some measure
$\mu$ satisfying $\int^{\infty}_{0}(1\wedge 1)\mu(dt)<\infty$), the operator
$\phi(\Delta)$ is the infinitesimal generator of the process
$X_{t}:=W_{S_{t}}$, where $S_{t}$ is a subordinator (i.e. an increasing Lévy
process satisfying $S_{0}=0$) with Laplace exponent $\phi$ (i.e.
$\mathbb{E}e^{\lambda S_{t}}=\exp\\{t\phi(\lambda)\\}$) and $W_{t}$ is a
$d$-dimensional Brwonian motion independent of $S_{t}$. Such process is called
the subordinate Brownian motion. Actually $\phi$ is a Bernstein function with
$\phi(0+)=0$ if and only if it is a Laplace exponent of a subordinator.
Furthermore, the relation
$\displaystyle\phi(\Delta)f:=-\phi(-\Delta)f=\int_{\mathbb{R}^{d}}\left(f(x+y)-f(x)-\nabla
f(x)\cdot y\chi(y)\right)J(y)~{}dy$ (1.3)
holds with $j(|y|):=J(y)$ given by
$\displaystyle j(r)=\int_{0}^{\infty}(4\pi
t)^{-d/2}e^{-r^{2}/(4t)}~{}\mu(dt).$ (1.4)
For the equations with the kernel $K(x,y)=a(y)J(y)$, an $L_{p}$-estimate is
obtained in aforementioned article [2] if $a(y)$ is symmetric. However to the
best of our knowledge, if the coefficient $a(y)$ is only measurable and
$J(y)\neq|y|^{-d-\alpha}$ then the $L_{p}$-estimate has not been known yet. In
this article we extend [9] to the class of Lévy measures $J(y)$ satisfying the
following two conditions: (i) there exists a constant $\alpha_{0}$, where
$\alpha_{0}\in(0,1]$ if $\sigma\leq 1$ and $\alpha_{0}\in(1,2)$ if $\sigma>1$,
so that
$\frac{j(t)}{j(s)}\leq N(\frac{s}{t})^{d+\alpha_{0}},\quad\forall\,0<s\leq t,$
(1.5)
and, (ii) for any $t>0$
$1_{\sigma<1}\int_{|y|\leq 1}|y|j(t|y|)~{}dy+1_{\sigma\geq 1}\int_{|y|\leq
1}|y|^{2}j(t|y|)~{}dz\leq Nj(t).$ (1.6)
See Section 2 for few remarks on these conditions. It is easy to check that
(1.5) and (1.6) are satisfied if there exists $\alpha\geq\alpha_{0}$ so that
$(s/t)^{d+\alpha}j(s)\leq N_{1}j(t)\leq
N_{2}(s/t)^{d+\alpha_{0}}j(s),\quad\forall\,\,\,0<s\leq t.$ (1.7)
One can construct many interesting jump functions $j(t)$ satisfying (1.7). For
example, (1.7) holds if $J(y)$ is defined from (1.3) and $\phi$ is one of the
following (see Example 2.12 for details):
* (1)
$\phi(\lambda)=\sum_{i=1}^{n}\lambda^{\alpha_{i}}$, $0<\alpha_{i}<1$;
* (2)
$\phi(\lambda)=(\lambda+\lambda^{\alpha})^{\beta}$, $\alpha,\beta\in(0,1)$;
* (3)
$\phi(\lambda)=\lambda^{\alpha}(\log(1+\lambda))^{\beta}$, $\alpha\in(0,1)$,
$\beta\in(0,1-\alpha)$;
* (4)
$\phi(\lambda)=\lambda^{\alpha}(\log(1+\lambda))^{-\beta}$, $\alpha\in(0,1)$,
$\beta\in(0,\alpha)$;
* (5)
$\phi(\lambda)=(\log(\cosh(\sqrt{\lambda})))^{\alpha}$, $\alpha\in(0,1)$;
* (6)
$\phi(\lambda)=(\log(\sinh(\sqrt{\lambda}))-\log\sqrt{\lambda})^{\alpha}$,
$\alpha\in(0,1)$.
In these cases, the jump function $j(r)$ is comparable to
$r^{-d}\phi(r^{-2})$.
Our approach is borrowed from [9]. We estimate the sharp functions of the
solutions and apply the Hardy-Littlewod theorem and the Fefferman-Stein
theorem. This approach is typically used to treat the second-order PDEs with
small BMO or VMO coefficients (for instance, see [11]). In [9] this method is
applied to a non-local operator with the kernel $K(x,y)=a(y)|y|^{-d-\alpha}$.
As in [9], our sharp function estimates are based on some Hölder estimates of
solutions. The original idea of obtaining Hölder estimates is from [3].
Nonetheless, since we are considering much general $J(y)$ rather then
$c(d,\alpha)|y|^{-d-\alpha}$, many new difficulties arise. In particular, our
operators do not have the nice scaling property which is used in [11] and [9],
and this cause many difficulties in the estimates.
The article is organized as follows. In Section 2 we introduce the main
results. Section 3 contains the unique solvability in the $L_{2}$-space. In
Section 4 we establish some Hölder estimates of solutions. Using these
estimates we obtain the sharp function and maximal function estimates in
Section 5. In Section 6, the proofs of main results are given.
We finish the introduction with some notation. As usual $\mathbb{R}^{d}$
stands for the Euclidean space of points $x=(x^{1},...,x^{d})$,
$B_{r}(x):=\\{y\in\mathbb{R}^{d}:|x-y|<r\\}$ and $B_{r}:=B_{r}(0)$. For
$i=1,...,d$, multi-indices $\beta=(\beta_{1},...,\beta_{d})$,
$\beta_{i}\in\\{0,1,2,...\\}$, and functions $u(x)$ we set
$u_{x^{i}}=\frac{\partial u}{\partial x^{i}}=D_{i}u,\quad
D^{\beta}u=D_{1}^{\beta_{1}}\cdot...\cdot
D^{\beta_{d}}_{d}u,\quad|\beta|=\beta_{1}+...+\beta_{d}.$
For an open set $U\subset\mathbb{R}^{d}$ and a nonnegative non-integer
constant $\gamma$, by $C^{\gamma}(U)$ we denote the usual Hölder space. For a
nonnegative integer $n$, we write $u\in C^{n}(U)$ if $u$ is $n$-times
continuously differentiable in $U$. By $C^{n}_{0}(U)$ (resp.
$C^{\infty}_{0}(U)$) we denote the set of all functions in $C^{n}(U)$ (resp.
$C^{\infty}(U)$) with compact supports. Similarly by $C^{n}_{b}(U)$ (resp.
$C^{\infty}_{b}(U)$) we denote the set of functions in $C^{n}(U)$ (resp.
$C^{\infty}(U)$) with bounded derivatives. The standard $L_{p}$-space on $U$
with Lebesgue measure is denoted by $L_{p}(U)$. We simply use $L_{p}$,
$C^{n}$, $C_{b}^{n}$, $C_{0}^{n}$, $C_{b}^{\infty}$, and $C_{0}^{\infty}$ when
$U=\mathbb{R}^{d}$. We use “$:=$” to denote a definition. $a\wedge
b=\min\\{a,b\\}$ and $a\vee b=\max\\{a,b\\}$. If we write $N=N(a,\ldots,z)$,
this means that the constant $N$ depends only on $a,\ldots,z$. The constant
$N$ may change from location to location, even within a line. By $\mathcal{F}$
and $\mathcal{F}^{-1}$ we denote the Fourier transform and the inverse Fourier
transform, respectively. That is,
$\mathcal{F}(f)(\xi):=\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}f(x)dx$ and
$\mathcal{F}^{-1}(f)(x):=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{i\xi\cdot
x}f(\xi)d\xi$. For a Borel set $A\subset\mathbb{R}^{d}$, we use $|A|$ to
denote its Lebesgue measure and by $I_{A}(x)$ we denote the indicator of $A$.
## 2\. Setting and main results
Throughout this article, we assume that $J(y)$ is rotationally invariant,
$\nu\leq a(y)\leq\Lambda$ (2.8)
for some constants $\nu,\Lambda>0$, and
$\int_{\mathbb{R}^{d}}(1\wedge|y|^{2})J(y)~{}dy<\infty.$ (2.9)
Let $e_{1}$ be a unit vector. Obviously, the condition that $J(y)$ is
rotationally invariant can be replaced by the condition that $J(y)$ is
comparable to $j(|y|):=J(|y|e_{1})$, because $J(y)a(y)=j(|y|)\cdot
a(y)J(y)j^{-1}(|y|):=j(|y|)\tilde{a}(y)$ and $\tilde{a}$ also has positive
lower and upper bounds.
Denote
$\sigma:=\inf\\{\delta>0:\int_{|y|\leq 1}\,|y|^{\delta}J(y)~{}dy<\infty\\},$
(2.10)
$\displaystyle\chi(y)=0~{}\text{if}~{}\sigma\in(0,1),\quad\chi(y)=1_{B_{1}}~{}\text{if}~{}\sigma=1,\quad\chi(y)=1~{}\text{if}~{}\sigma\in(1,2].$
Note that if $J(y)=c(d,\alpha)|y|^{-d-\alpha}$ for some $\alpha\in(0,2)$ then
we have $\sigma=\alpha$.
For $u\in C^{2}_{b}$ we introduce the non-local elliptic operators
$\displaystyle\mathcal{A}u=\int_{{\mathbb{R}}^{d}}\big{(}u(x+y)-u(x)-y\cdot\nabla
u(x)\chi(y)\big{)}\,J(y)~{}dy,$ $\displaystyle
Lu=\int_{{\mathbb{R}}^{d}}\big{(}u(x+y)-u(x)-y\cdot\nabla
u(x)\chi(y)\big{)}\,a(y)J(y)~{}dy,$
$\tilde{L}u=\int_{{\mathbb{R}}^{d}}\big{(}u(x+y)-u(x)-y\cdot\nabla
u(x)I_{B_{1}}(y)\big{)}\,a(y)J(y)~{}dy,$ $\displaystyle
L^{\ast}u=\int_{{\mathbb{R}}^{d}}\big{(}u(x+y)-u(x)-y\cdot\nabla
u(x)\chi(y)\big{)}\,a(-y)J(-y)~{}dy,$
and
$\tilde{L}^{\ast}u=\int_{{\mathbb{R}}^{d}}\big{(}u(x+y)-u(x)-y\cdot\nabla
u(x)I_{B_{1}}(y)\big{)}\,a(-y)J(-y)~{}dy.$
We start with a simple but interesting result, which will be used later in the
proof of Theorem 2.21.
###### Lemma 2.1.
For any $p>1$ and $\lambda>0$,
$\|u\|_{L_{p}}\leq\frac{1}{\lambda}\|\tilde{L}u-\lambda
u\|_{L_{p}},\quad\forall\,u\in C^{\infty}_{0}.$
###### Proof.
Put
$\Phi(\xi):=-\int_{\mathbb{R}^{d}}(e^{i\xi\cdot
y}-1-i(y\cdot\xi)I_{B_{1}})a(-y)J(-y)~{}dy$
and
$f:=\tilde{L}u-\lambda u.$
Since $a(-y)J(-y)$ is a Lévy measure (i.e.
$\int_{\mathbb{R}^{d}}(1\wedge|y|^{2})a(-y)J(-y)~{}dy<\infty$), there exists a
Lévy process whose characteristic exponent is $-t\Phi(\xi)$ (for instance, see
Corollary 1.4.6 of [1]). Denoting by $p_{\Phi}(t,dx)$ its law at $t$, we have
$\int_{\mathbb{R}^{d}}e^{-i\xi\cdot
x}p_{\Phi}(t,dx)=\int_{\mathbb{R}^{d}}e^{i(-\xi)\cdot
x}p_{\Phi}(t,dx)=e^{-t\Phi(-\xi)}.$ (2.11)
In non-probabilistic terminology it can be rephrased that if
$\int_{\mathbb{R}^{d}}(1\wedge|y|^{2})a(-y)J(-y)~{}dy<\infty$ then there
exists a continuous measure-valued function $p_{\Phi}(t,dx)$ such that
$p_{\Phi}(t,\mathbb{R}^{d})=1$ and (2.11) holds. Since
$(-\Phi(-\xi)-\lambda)\mathcal{F}u=\mathcal{F}f$
and $\text{Re}\,\Phi(-\xi)\geq 0$, we have
$\displaystyle\mathcal{F}u(\xi)$ $\displaystyle=$
$\displaystyle-\frac{1}{\Phi(-\xi)+\lambda}\mathcal{F}f(\xi)$ $\displaystyle=$
$\displaystyle-\Big{(}\int_{0}^{\infty}e^{-t\Phi(-\xi)-\lambda
t}~{}dt~{}\mathcal{F}f(\xi)\Big{)}$ $\displaystyle=$
$\displaystyle-\Big{(}\int_{0}^{\infty}\int_{\mathbb{R}^{d}}e^{-i\xi\cdot
x}p_{\Phi}(t,dx)e^{-\lambda t}~{}dt~{}\mathcal{F}f(\xi)\Big{)}$
$\displaystyle=$
$\displaystyle-\mathcal{F}\Big{(}\int_{0}^{\infty}(p_{\Phi}(t,\cdot)\ast
f(x))e^{-\lambda t}~{}dt\Big{)}(\xi).$
Therefore,
$u(x)=-\int_{0}^{\infty}(p_{\Phi}(t,\cdot)\ast f)e^{-\lambda t}~{}dt$
and by Young’s inequality,
$\displaystyle\|u\|_{L_{p}}\leq\int_{0}^{\infty}\int_{\mathbb{R}^{d}}p_{\Phi}(t,dx)e^{-\lambda
t}~{}dt\|f\|_{L_{p}}\leq\frac{1}{\lambda}\|f\|_{L_{p}}.$
Hence the lemma is proved. $\Box$
###### Definition 2.2.
We write $u\in\mathcal{H}_{p}^{\mathcal{A}}$ if and only if there exists a
sequence of functions $u_{n}\in C_{0}^{\infty}$ such that $u_{n}\to u$ in
$L_{p}$ and $\\{\mathcal{A}u_{n}:n=1,2,\cdots\\}$ is a cauchy sequence in
$L_{p}$. By $\mathcal{A}u$ we denote the limit of $\mathcal{A}u_{n}$ in
$L_{p}$.
###### Lemma 2.3.
$\mathcal{H}_{p}^{\mathcal{A}}$ is a Banach space equipped with the norm
$\displaystyle\|u\|_{\mathcal{H}_{p}^{\mathcal{A}}}:=\|u\|_{L_{p}}+\|\mathcal{A}u\|_{L_{p}}.$
###### Proof.
It is obvious. $\Box$
###### Definition 2.4.
We say that $u\in\mathcal{H}_{p}^{\mathcal{A}}$ is a solution of the equation
$\displaystyle Lu-\lambda u=f\quad\quad~{}\text{in}\,\,~{}\mathbb{R}^{d}$
(2.12)
if and only if there exists a sequence $\\{u_{n}\in C_{0}^{\infty}\\}$ such
that $u_{n}$ converges to $u$ in $\mathcal{H}_{p}^{\mathcal{A}}$ and
$Lu_{n}-\lambda u_{n}$ converges to $f$ in $L_{p}$. Similarly, we consider the
equation
$\displaystyle\tilde{L}u-\lambda
u=f\quad\quad~{}\text{in}\,\,~{}\mathbb{R}^{d}$ (2.13)
in the same sense.
###### Lemma 2.5 (Maximum principle).
Let $\lambda>0$, $b(x)$ be an $\mathbb{R}^{d}$-valued bounded function on
$\mathbb{R}^{d}$ and $u$ be a function in $C^{2}_{b}$ satisfying $u(x)\to 0$
as $|x|\to\infty$. If $Lu+b(x)\cdot\nabla u-\lambda u=0$ in $\mathbb{R}^{d}$,
then $u\equiv 0$. Also, the same statement is true with $\tilde{L}$ in place
of $L$.
###### Proof.
Suppose that $u$ is not identically zero. Without loss of generality, assume
$\sup_{\mathbb{R}^{d}}u>0$ (otherwise consider $-u$). Since $u$ goes to zero
as $|x|\to\infty$, there exists $x_{0}\in\mathbb{R}^{d}$ such that
$u(x_{0})=\sup_{\mathbb{R}^{d}}u$. Thus $\nabla u(x_{0})=0$ and
$\displaystyle
Lu(x_{0})=\int_{{\mathbb{R}}^{d}}\left(u(x_{0}+y)-u(x_{0})-y\cdot\nabla
u(x_{0})\chi(y)\right)a(y)J(y)\,dy\leq 0.$
Therefore we reach the contradiction. Indeed,
$\displaystyle Lu(x_{0})+b(x_{0})\cdot\nabla u(x_{0})-\lambda u(x_{0})<0.$
The proof for $\tilde{L}$ is almost identical. The lemma is proved. $\Box$
This maximum principle yields the denseness of
$(L+b\cdot\nabla-\lambda)C_{0}^{\infty}$ and
$(\tilde{L}+b\cdot\nabla-\lambda)C_{0}^{\infty}$ in $L_{p}$.
###### Lemma 2.6.
Let $\lambda>0$ and $b\in\mathbb{R}^{d}$ be independent of $x$. Then
$(L+b\cdot\nabla-\lambda)C_{0}^{\infty}:=\\{Lu+b\cdot\nabla u-\lambda u:u\in
C_{0}^{\infty}\\}$ is dense in $L_{p}$ for any $p\in(1,\infty)$. Also, the
same statement holds with $\tilde{L}$ in place of $L$.
###### Proof.
Due to the similarity we only prove the first statement. Suppose that the
statement is false. Then by the Hahn-Banach theorem and Riesz’s representation
theorem, there exists a nonzero $v\in L_{p/(p-1)}$ such that
$\displaystyle\int_{\mathbb{R}^{d}}\left(Lu(x)+b\cdot\nabla u(x)-\lambda
u(x)\right)v(x)~{}dx=0$ (2.14)
for all $u\in C_{0}^{\infty}$.
Fixing $y\in\mathbb{R}^{d}$, we apply (2.14) with $u(y-\cdot)$. Then, due to
Fubini’s Theorem,
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\left(L^{\ast}u(y-x)-b\cdot\nabla
u(y-x)-\lambda u(y-x)\right)v(x)~{}dx$ $\displaystyle=$ $\displaystyle
L^{\ast}u\ast v(y)-b\cdot(\nabla u\ast v(y))-\lambda u\ast
v(y)=(L^{\ast}-b\cdot\nabla-\lambda)(u\ast v)(y).$
Therefore from the previous lemma, we have $u\ast v=0$ for any $u\in
C^{\infty}_{0}$. Therefore, $v=0$ $(a.e.)$ and we have a contradiction. $\Box$
###### Corollary 2.7 (Uniqueness).
Let $\lambda>0$. Suppose that there exist
$u,v\in\mathcal{H}^{\mathcal{A}}_{p}$ satisfying
$Lu-\lambda u=0,\quad\tilde{L}v-\lambda v=0.$
Then $u=v=0$.
###### Proof.
By the definition of a solution and the assumption of this corollary, there
exists a sequence $\\{u_{n}\in C_{0}^{\infty}\\}$ such that for all $w\in
C_{0}^{\infty}$
$\displaystyle 0=\int_{\mathbb{R}^{d}}\lim_{n\to\infty}(Lu_{n}-\lambda
u_{n})w~{}dx=\int_{\mathbb{R}^{d}}u(L^{\ast}w-\lambda w)~{}dx.$
Since $\\{L^{\ast}w-\lambda w:w\in C_{0}^{\infty}\\}$ is dense in
$L_{p/(p-1)}$ owing to Lemma 2.6, we conclude $u=0$, and by the same argument
we have $v=0$. $\Box$
Here is our $L_{2}$-theory. We emphasize that only (2.8) and (2.9) are assumed
for the $L_{2}$-theory. The proof of Theorem 2.8 is given in Section 3.
###### Theorem 2.8.
Let $\lambda>0$. Then for any $f\in L_{2}$ there exist unique solutions
$u,v\in\mathcal{H}_{2}^{\mathcal{A}}$ of equation (2.12) and (2.13)
respectively, and for these solutions we have
$\displaystyle\|\mathcal{A}u\|_{L_{2}}+\lambda\|u\|_{L_{2}}\leq
N(d,\nu,\Lambda)\|f\|_{L_{2}},$ (2.15)
$\displaystyle\|\mathcal{A}v\|_{L_{2}}+\lambda\|v\|_{L_{2}}\leq
N(d,\nu,\Lambda)\|f\|_{L_{2}}.$ (2.16)
The issue regarding the continuity of $L$ (or $\tilde{L}$) :
$\mathcal{H}^{\mathcal{A}}_{p}\to L_{p}$ will be discussed later.
For the case $p\neq 2$, we consider the following conditions on $J(y)=j(|y|)$
:
(H1): There exist constants $\kappa_{1}>0$ and $\alpha_{0}>0$ such that
$\displaystyle
j(t)\leq\kappa_{1}(s/t)^{d+\alpha_{0}}j(s),\quad\forall\,\,0<s\leq t.$ (2.17)
Moreover, $\alpha_{0}\leq 1$ if $\sigma\leq 1$ and $1<\alpha_{0}<2$ if
$\sigma>1$.
(H2): There exists a constant $\kappa_{2}>0$ such that for all $t>0$,
$\displaystyle\int_{|y|\leq
1}|y|j(t|y|)~{}dy\leq\kappa_{2}j(t)\quad\quad\text{if}\,\,\sigma\in(0,1),$
(2.18) $\displaystyle\int_{|y|\leq
1}|y|^{2}j(t|y|)~{}dz\leq\kappa_{2}j(t)\quad\quad\text{if}\,\,\sigma\geq 1.$
(2.19)
###### Remark 2.9.
(i) By taking $t=1$ in (2.17),
$j(1)\kappa^{-1}_{1}s^{-d-\alpha_{0}}\leq j(s),\quad\forall\,\,s\in(0,1).$
(2.20)
An upper bound of $j(s)$ near $s=0$ is obtained in the following lemma.
(ii) H1 and H2 are needed even to guarantee the continuity of the operator
$L:\mathcal{H}^{A}_{2}\to L_{2}$ (see Lemma 3.1).
###### Lemma 2.10.
Suppose
$\displaystyle j(s)\geq Cj(t),\quad\forall\,s\leq t,$ (2.21)
and H2 hold. Then there exists a constant $N(d,\kappa_{2},C)>0$ such that for
all $0<s\leq t$
$\displaystyle j(t)\geq N(s/t)^{d+1}j(s)\quad(\text{if}\,\,\sigma<1),\quad
j(t)\geq N(s/t)^{d+2}j(s)\quad(\text{if}\,\,\sigma\geq 1).$ (2.22)
On the other hand, if there exists $\alpha>0$ so that $\alpha<1$ if
$\sigma<1$, $\alpha<2$ if $\sigma\geq 1$, and
$\displaystyle j(t)\geq N(s/t)^{d+\alpha}j(s),\quad\forall\,0<s\leq t,$ (2.23)
then H2 holds.
###### Remark 2.11.
By Lemma 2.10, both H1 and H2 hold if $0<\alpha_{0}\leq\alpha$ and
$N^{-1}(s/t)^{d+\alpha}j(s)\leq j(t)\leq
N(s/t)^{d+\alpha_{0}}j(s),\quad\forall\,\,\,0<s\leq t.$
###### Example 2.12.
Let $J(y)=j(|y|)$ be defined as in (1.4), that is for a Bernstein function
$\phi(\lambda)=\int_{\mathbb{R}}(1-e^{-\lambda t})\mu(dt)$ and $u\in
C^{2}_{0}$,
$j(r)=\int_{0}^{\infty}(4\pi t)^{-d/2}e^{-r^{2}/(4t)}~{}\mu(dt),$
and
$\displaystyle\phi(\Delta)u$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\left(u(x+y)-u(x)-\nabla u(x)\cdot
yI_{|y|\leq 1}\right)J(y)~{}dy$ $\displaystyle=$
$\displaystyle-\mathcal{F}(\phi(|\xi|^{2})\mathcal{F}(u)(\xi)).$
Then, H1 and H2 are satisfied if $\phi$ is given, for instance, by any one of
* (1)
$\phi(\lambda)=\sum_{i=1}^{n}\lambda^{\alpha_{i}}$, $0<\alpha_{i}<1$;
* (2)
$\phi(\lambda)=(\lambda+\lambda^{\alpha})^{\beta}$, $\alpha,\beta\in(0,1)$;
* (3)
$\phi(\lambda)=\lambda^{\alpha}(\log(1+\lambda))^{\beta}$, $\alpha\in(0,1)$,
$\beta\in(0,1-\alpha)$;
* (4)
$\phi(\lambda)=\lambda^{\alpha}(\log(1+\lambda))^{-\beta}$, $\alpha\in(0,1)$,
$\beta\in(0,\alpha)$;
* (5)
$\phi(\lambda)=(\log(\cosh(\sqrt{\lambda})))^{\alpha}$, $\alpha\in(0,1)$;
* (6)
$\phi(\lambda)=(\log(\sinh(\sqrt{\lambda}))-\log\sqrt{\lambda})^{\alpha}$,
$\alpha\in(0,1)$.
This is because all these functions satisfy the conditions
* A:
$\exists\,0<\delta_{1}\leq\delta_{2}<1$,
$N^{-1}\lambda^{\delta_{1}}\phi(t)\leq\phi(\lambda t)\leq
N\lambda^{\delta_{2}}\phi(t),\quad\forall\,\lambda\geq 1,t\geq 1$
* B:
$\exists\,0<\delta_{3}\leq\delta_{4}<1$,
$N^{-1}\lambda^{\delta_{3}}\phi(t)\leq\phi(\lambda t)\leq
N\lambda^{\delta_{4}}\phi(t),\quad\forall\,\lambda\leq 1,t\leq 1,$
and under these condition one can prove (see [10])
$\displaystyle
N^{-1}\Big{(}\frac{R}{r}\Big{)}^{\delta_{1}\wedge\delta_{3}}\leq\frac{\phi(R)}{\phi(r)}\leq
N\Big{(}\frac{R}{r}\Big{)}^{\delta_{2}\vee\delta_{4}}$
and
$\displaystyle N^{-1}\phi(|y|^{-2})|y|^{-d}\leq J(y)\leq
N\phi(|y|^{-2})|y|^{-d},$ (2.24)
and consequently our conditions H1 and H2 hold. One can easily construct
concrete examples of $j(r)$ using (2.24) and $(1)$-$(6)$ (just replace
$\lambda$ by $r^{-2}$). See the tables at the end of [13] for more examples
satisfying A and B.
###### Remark 2.13.
If $p\neq 2$, our $L_{p}$-theory does not cover the case when the jump
function $J(y)$ is related to the relativistic $\alpha$-stable process with
mass $m>0$ (i.e. a subordinate Brownian motion with the infinitesimal
generator $\phi(\Delta)=m-(m^{2/{\alpha}}-\Delta)^{\alpha/2}$). This is
because the related jump function decreases exponentially fast at the infinity
(for instance, see [7]) and thus condition H2 fails (see (2.22)).
Proof of Lemma 2.10. Assume (2.21) and H2 hold. We put
$B_{1}=\cup_{n=0}^{\infty}B_{(n)}$, where $B_{(n)}=B_{2^{-n}}\setminus
B_{2^{-(n+1)}}$. Due to (2.21) for each $n\geq 0$,
$\displaystyle\kappa_{2}j(t)$ $\displaystyle\geq$ $\displaystyle\int_{|y|\leq
1}|y|^{2}j(t|y|)~{}dy=\sum_{n=0}^{\infty}\int_{B(n)}|y|^{2}j(t|y|)~{}dy$
$\displaystyle\geq$ $\displaystyle
N\sum_{n=0}^{\infty}2^{-(n+1)(d+2)}j(t2^{-n})\geq N2^{-(n+1)(d+2)}j(t2^{-n}).$
Put $s=t\lambda$, where $\lambda\in(0,1)$, and take an integer $m(\lambda)\geq
0$ such that $2^{-(m+1)}\leq\lambda\leq 2^{-m}$. Then by (2.21),
$\displaystyle j(t)\geq N2^{-(m+2)(d+2)}j(2^{-(m+1)}t)\geq
N\lambda^{d+2}j(\lambda t).$
Similarly, $j(\lambda t)\leq\lambda^{-d-1}j(t)$ if $\sigma<1$.
For the other direction, put $s=t|y|$ in (2.23). If $\sigma<1$ then
$\displaystyle\int_{|y|\leq 1}|y|j(t|y|)~{}dy$ $\displaystyle\leq$
$\displaystyle Nj(t)\int_{|y|\leq 1}|y|\frac{j(t|y|)}{j(t)}~{}dy$
$\displaystyle\leq$ $\displaystyle Nj(t)\int_{|y|\leq
1}|y|^{-d-\alpha_{1}+1}~{}dy\leq Nj(t)$
and otherwise, that is, if $\sigma\geq 1$ then
$\displaystyle\int_{|y|\leq 1}|y|^{2}j(t|y|)~{}dy$ $\displaystyle\leq$
$\displaystyle Nj(t)\int_{|y|\leq 1}|y|^{2}\frac{j(t|y|)}{j(t)}~{}dy$
$\displaystyle\leq$ $\displaystyle Nj(t)\int_{|y|\leq
1}|y|^{-d-\alpha_{2}+2}~{}dy\leq Nj(t).$
The lemma is proved. $\Box$
Define
$\Psi(\xi):=-\int_{\mathbb{R}^{d}}(e^{i\xi\cdot
y}-1-i(y\cdot\xi)\chi(y))J(y)dy=\int_{\mathbb{R}^{d}}(1-\cos\xi\cdot
y)J(y)dy.$
Then
$\mathcal{A}u=\mathcal{F}^{-1}(-\Psi(\xi)\mathcal{F}u),\quad\quad\forall\,u\in
C^{\infty}_{0}.$
By abusing the notation, we also use $\Psi(|\xi|)$ instead of $\Psi(\xi)$
because $\Psi(\xi)$ is rotationally invariant.
The following result will be used to prove the continuity of the operator $L$.
###### Lemma 2.14.
Suppose that (2.21) holds. Then there exists a constant $N(d,C)>0$ such that
for all $\xi\in\mathbb{R}^{d}$
$\displaystyle j(|\xi|)\leq N|\xi|^{-d}\Psi(|\xi|^{-1}).$ (2.25)
###### Proof.
By (2.21),
$\displaystyle\Psi(|\xi|^{-1})$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}(1-\cos(y^{1}/|\xi|))J(y)~{}dy=|\xi|^{d}\int_{\mathbb{R}^{d}}(1-\cos(y^{1}))J(|\xi|y)~{}dy$
$\displaystyle\geq$ $\displaystyle|\xi|^{d}\int_{|y|\leq
1}(1-\cos(y^{1}))J(|\xi|y)dy$ $\displaystyle\geq$ $\displaystyle
Cj(|\xi|)|\xi|^{d}\int_{|y|\leq 1}(1-\cos(y^{1}))~{}dy\geq
Nj(|\xi|)|\xi|^{d}.$
Hence the lemma is proved. $\Box$
The following condition will be considered for the case $\sigma=1$. This
condition is needed even to prove the continuity of $L$.
###### Assumption 2.15.
If $\sigma=1$ then
$\displaystyle\int_{\partial B_{r}}y^{i}a(y)J(y)dS_{r}(y)=0,\quad\forall
r\in(0,\infty),\,i=1,\cdots,d,$ (2.26)
where $dS_{r}$ is the surface measure on $\partial B_{r}$.
Here is our $L_{p}$-theory for equation (2.27) below.
###### Theorem 2.16.
Suppose that H1 and H2 hold and Assumption 2.15 also holds if $\sigma=1$. Let
$\lambda>0$ and $p>1$. Then for any $f\in L_{p}$ there exists a unique
solution $u\in\mathcal{H}_{p}^{\mathcal{A}}$ of the equation
$Lu-\lambda u=f,$ (2.27)
and for this solution we have
$\displaystyle\|\mathcal{A}u\|_{L_{p}}+\lambda\|u\|_{L_{p}}\leq
N(d,p,\nu,\Lambda,J)\|f\|_{L_{p}}.$ (2.28)
Moreover, $L$ is a continuous operator from $\mathcal{H}_{p}^{\mathcal{A}}$ to
$L_{p}$, and (2.28) holds for all $u\in\mathcal{H}_{p}^{\mathcal{A}}$ with
$f:=Lu-\lambda u$.
The proof of this theorem will be given in Section 6.
###### Remark 2.17.
Since the constant $N$ in (2.28) does not depend on $\lambda$, for any
$u\in\mathcal{H}_{p}^{\mathcal{A}}$
$\displaystyle\|\mathcal{A}u\|_{L_{p}}\leq N\|Lu\|_{L_{p}}.$
To study the equations with the operator $\tilde{L}$, we consider an
additional condition, which always holds when $\sigma=1$.
###### Assumption 2.18 (H3).
Any one of the following (i)-(iv) holds:
(i) $\mathcal{A}$ is a higher order differential operator than $I_{\sigma\neq
1}\nabla u$, that is for any $\varepsilon>0$ there exists $N(\varepsilon)>0$
so that for any $u\in C^{\infty}_{0}$
$I_{\sigma\neq 1}\|\nabla
u\|_{p}\leq\varepsilon\|\mathcal{A}u\|_{p}+N(\varepsilon)\|u\|_{p}.$ (2.29)
(ii) $\sigma<1$ and
$\int_{r\leq|y|\leq 1}y^{i}\Big{(}a(y)-[a(y)\wedge
a(-y)]\Big{)}\,J(y)dy=0,\quad\forall\,r\in(0,1),\,i=1,\cdots,d.$ (2.30)
(iii) $\sigma<1$ and there exists a constant $\kappa_{3}>0$ such that for all
$0<t<1$,
$\displaystyle\int_{|z|\geq 1}|z|j(t|z|)~{}dz\leq\kappa_{2}j(t).$ (2.31)
(iv) $\sigma>1$ and
$\int_{1\leq|y|\leq r}y^{i}\Big{(}a(y)-[a(y)\wedge
a(-y)]\Big{)}\,J(y)dy=0,\quad\forall\,r>1\,\,i=1,\cdots,d.$ (2.32)
###### Remark 2.19.
(i) Note that (2.29) is satisfied if for some $\alpha>1$,
$\|\Delta^{\alpha/2}u\|_{p}\leq N(\|u\|_{p}+\|\mathcal{A}u\|_{p}),\quad\forall
u\in C^{\infty}_{0},$ (2.33)
or, equivalently $|\xi|^{\alpha}(1+\Psi(\xi))^{-1}$ is a $L_{p}$-Fourier
multiplier. Thus, certain differentiability of $J(y)$ is required (see Lemma
2.20 below).
(ii) It is easy to check that (2.31) holds if for a $\alpha>1$,
$\displaystyle j(\lambda t)\leq
N\lambda^{-d-\alpha}j(t),\quad\forall\,\,\lambda\in(1,\infty),\,\,0<t<1.$
(2.34)
(iii) Obviously, (2.30) holds if $a(y)=a(-y)$ for $|y|\leq 1$, and (2.32)
holds if $a(y)=a(-y)$ for $|y|\geq 1$.
Below we give a sufficient condition for (2.29).
###### Lemma 2.20.
(i) H3-(i) holds if $\mathcal{A}=\phi(\Delta)$ for some Bernstein function
$\phi$ satisfying
$1+\phi(|\xi|^{2})\geq N|\xi|^{\alpha},\quad\forall\xi\in\mathbb{R}^{d},$
(2.35)
where $\alpha>1$ and $N>0$.
(ii) All of H1, H2 and H3 hold if $\sigma>1$, $\mathcal{A}=\phi(\Delta)$ and
$\phi$ satisfies conditions A and B described in Example 2.12.
###### Proof.
(i). Let $\phi(\lambda)=\int_{\mathbb{R}}(1-e^{-\lambda t})\mu(dt)$, where
$\int_{\mathbb{R}}(1\wedge|t|)\mu(dt)<\infty$. Then from $t^{n}e^{-t}\leq
N(n)(1-e^{-t})$, we get
$|\lambda|^{n}|D^{n}\phi(\lambda)|\leq N\phi(\lambda).$ (2.36)
For any $u\in C^{\infty}_{0}$,
$\mathcal{A}u=\mathcal{F}^{-1}(\phi(|\xi|^{2})\mathcal{F}(u)(\xi)),$
$\Delta^{\alpha/2}u=\mathcal{F}^{-1}(|\xi|^{\alpha}\mathcal{F}(u)(\xi))=\mathcal{F}^{-1}(\eta(\xi)(1+\phi(|\xi|^{2})\mathcal{F}(u)(\xi)),$
where $\eta(\xi)=|\xi|^{\alpha}(1+\phi(|\xi|^{2}))^{-1}$. Using (2.35) and
(2.36), one can easily check
$|D^{n}\eta(\xi)|\leq N(n)|\xi|^{-n},\quad\forall\,\xi,$
and therefore $\eta$ is a Fourier multiplier (see Theorem IV.3.2 of [16]) and
$\|\Delta^{\alpha/2}u\|\leq N(\|u\|_{p}+\|\mathcal{A}u\|_{p}),$ $\|\nabla
u\|_{p}\leq\varepsilon\|\Delta^{\alpha/2}u\|_{p}+N(\varepsilon)\|u\|_{p}\leq
N\varepsilon\|\mathcal{A}u\|_{p}+N\|u\|_{p}.$
(ii) If A and B hold, then as explained before both H1, H2 hold, and we also
have (see (2.24)),
$N^{-1}\phi(|y|^{-2})|y|^{-d}\leq J(y)\leq N\phi(|y|^{-2})|y|^{-d}.$
Thus if $|\xi|\geq 1$, then
$\phi(|\xi|^{2})\geq N|\xi|^{-d}J(|\xi|^{-1})\geq N|\xi|^{\alpha_{0}},$
where (2.20) is used for the last inequality. Hence the lemma is proved.
$\Box$
Here is our $L_{p}$-theory for equation (2.37) below.
###### Theorem 2.21.
Suppose that H1, H2 and H3 hold and Assumption 2.15 also holds if $\sigma=1$.
Let $\lambda>0$ and $p>1$. Then for any $f\in L_{p}$ there exists a unique
solution $u\in\mathcal{H}_{p}^{\mathcal{A}}$ of the equation
$\tilde{L}u-\lambda u=f,$ (2.37)
and for this solution we have
$\displaystyle\|\mathcal{A}u\|_{L_{p}}+\lambda\|u\|_{L_{p}}\leq
N(d,\nu,\Lambda,\lambda,J)\|f\|_{L_{p}}.$ (2.38)
The proof of this theorem will be given in Section 6. Actually the constant
$N$ in (2.38) is independent of $\lambda$ except the case when H3(i) is
assumed.
## 3\. $L_{2}$-theory
In this section we prove (2.15) and (2.16). These estimates and Lemma 2.6
yield the unique solvability of equations (2.12) and (2.13). The Fourier
transform and Parseval’s identity are used to prove these estimates.
###### Lemma 3.1.
Let $\lambda\geq 0$ be a constant.
(i) For any $u\in C_{0}^{\infty}$
$\displaystyle\|\mathcal{A}u\|_{L_{2}}+\lambda\|u\|_{L_{2}}\leq
N(d,\nu)\|Lu-\lambda u\|_{L_{2}}$ (3.39)
and
$\displaystyle\|\mathcal{A}u\|_{L_{2}}+\lambda\|u\|_{L_{2}}\leq
N(d,\nu)\|\tilde{L}u-\lambda u\|_{L_{2}}.$ (3.40)
(ii) Let H1 hold and $\sigma>1$. Then both $L$ and $\tilde{L}$ are continuous
operators from $\mathcal{H}_{2}^{\mathcal{A}}$ to $L_{2}$, and for any $u\in
C^{\infty}_{0}$,
$\|Lu\|_{L_{2}}\leq
N\|\mathcal{A}u\|_{L_{2}},\quad\quad\|\tilde{L}u\|_{L_{2}}\leq
N\|u\|_{\mathcal{H}_{2}^{\mathcal{A}}},$ (3.41)
where $N=N(d,\nu,J)$. Moreover, (3.39) and (3.40) hold for any
$u\in\mathcal{H}_{2}^{\mathcal{A}}$.
(iii) Let H1 and H2 hold, and Assumption 2.15 also hold if $\sigma=1$. Then
the claims of (ii) hold for $L$ (not for $\tilde{L}$) for any
$\sigma\in(0,1]$.
###### Proof.
(i). Let $u\in C^{\infty}_{0}$. Taking the Fourier transform, we get
$\displaystyle\mathcal{F}(Lu)(\xi)=\mathcal{F}u(\xi)\int_{\mathbb{R}^{d}}(e^{i\xi\cdot
y}-1-iy\cdot\xi\chi(y))a(y)J(y)dy.$ (3.42)
By Parseval’s identity,
$\displaystyle\int_{\mathbb{R}^{d}}|Lu(x)|^{2}dx=(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}(Lu)(\xi)|^{2}d\xi$
$\displaystyle\geq$
$\displaystyle(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\text{Re}\int_{\mathbb{R}^{d}}(e^{i\xi\cdot
y}-1-iy\cdot\xi\chi(y))a(y)J(y)dy\right|^{2}d\xi$ $\displaystyle=$
$\displaystyle(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{\mathbb{R}^{d}}(1-\cos(\xi\cdot
y))a(y)J(y)dy\right|^{2}d\xi$ $\displaystyle\geq$
$\displaystyle(2\pi)^{-d}\nu^{2}\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{\mathbb{R}^{d}}(1-\cos(\xi\cdot
y))J(y)dy\right|^{2}d\xi$ $\displaystyle=$
$\displaystyle\nu^{2}\int_{\mathbb{R}^{d}}|\mathcal{A}u|^{2}dx,$
where the facts that $1-\cos(\xi\cdot y)$ is nonnegative and $a(y)\geq\nu$ are
used above.
Similarly, since $uLu$ is real,
$\displaystyle-\int_{\mathbb{R}^{d}}uLu~{}dx=-(2\pi)^{-d}\int_{\mathbb{R}^{d}}\mathcal{F}(Lu)(\xi)\overline{\mathcal{F}(u)(\xi)}~{}d\xi$
$\displaystyle=$
$\displaystyle-(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}(u)(\xi)|^{2}\text{Re}\int_{\mathbb{R}^{d}}\left(e^{i\xi\cdot
y}-1-iy\cdot\xi\chi^{(\sigma)}(y)\right)a(y)J(y)~{}dyd\xi$ $\displaystyle=$
$\displaystyle(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}(u)(\xi)|^{2}\int_{\mathbb{R}^{d}}\left(1-\cos(\xi\cdot
y)\right)a(y)J(y)~{}dyd\xi$ $\displaystyle\geq$
$\displaystyle\frac{\nu}{2}(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}(u)(\xi)|^{2}\int_{\mathbb{R}^{d}}\left(1-\cos(\xi\cdot
y)\right)J(y)~{}dyd\xi$ $\displaystyle=$
$\displaystyle-\frac{\nu}{2}\int_{\mathbb{R}^{d}}u\mathcal{A}u~{}dx.$
Hence,
$\displaystyle\int_{\mathbb{R}^{d}}|Lu-\lambda u|^{2}~{}dx$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}|Lu|^{2}~{}dx-2\lambda\int_{\mathbb{R}^{d}}uLu~{}dx+\lambda^{2}\int_{\mathbb{R}^{d}}|u|^{2}~{}dx$
$\displaystyle\geq$
$\displaystyle\nu^{2}\int_{\mathbb{R}^{d}}|\mathcal{A}u|^{2}~{}dx-\lambda\nu\int_{\mathbb{R}^{d}}u\mathcal{A}u~{}dx+\lambda^{2}\int_{\mathbb{R}^{d}}|u|^{2}~{}dx$
$\displaystyle\geq$
$\displaystyle\nu^{2}\int_{\mathbb{R}^{d}}|\mathcal{A}u|^{2}~{}dx-\frac{\nu^{2}}{2}\int_{\mathbb{R}^{d}}u^{2}~{}dx-\frac{\lambda^{2}}{2}\int_{\mathbb{R}^{d}}|\mathcal{A}u|^{2}~{}dx+\lambda^{2}\int_{\mathbb{R}^{d}}|u|^{2}~{}dx$
$\displaystyle=$
$\displaystyle\frac{\nu^{2}}{2}\int_{\mathbb{R}^{d}}|\mathcal{A}u|^{2}~{}dx+\frac{\lambda^{2}}{2}\int_{\mathbb{R}^{d}}|u|^{2}~{}dx.$
Thus (3.39) holds. Also, (3.40) is proved similarly.
(ii)-(iii). Next, we prove (3.41) for any $u\in C^{\infty}_{0}$. Unlike the
case $j(r)=r^{-d-\alpha}$, the proof is not completely trivial. Condition H1
is needed if $\sigma>1$, and H2 is additionally needed if $\sigma\leq 1$.
By using (3.42) and Parseval’s identity again,
$\displaystyle\int_{\mathbb{R}^{d}}|Lu(x)|^{2}dx=(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}(Lu)(\xi)|^{2}d\xi$
$\displaystyle=$
$\displaystyle(2\pi)^{-d}\Big{[}\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\text{Re}\int_{\mathbb{R}^{d}}(e^{i\xi\cdot
y}-1-iy\cdot\xi\chi(y))a(y)J(y)~{}dy\right|^{2}d\xi$
$\displaystyle+\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\text{Im}\int_{\mathbb{R}^{d}}(e^{i\xi\cdot
y}-1-iy\cdot\xi\chi(y))a(y)J(y)~{}dy\right|^{2}d\xi\Big{]}$
$\displaystyle\leq$
$\displaystyle(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{\mathbb{R}^{d}}(1-\cos(\xi\cdot
y))a(y)J(y)~{}dy\right|^{2}d\xi$
$\displaystyle+(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{|y||\xi|\geq
1}(\sin(\xi\cdot y)-y\cdot\xi\chi(y))a(y)J(y)~{}dy\right|^{2}d\xi$
$\displaystyle+(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{|y||\xi|<1}(\sin(\xi\cdot
y)-y\cdot\xi\chi(y))a(y)J(y)~{}dy\right|^{2}d\xi$ $\displaystyle:=$
$\displaystyle\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}.$
Similarly,
$\int_{\mathbb{R}^{d}}|\tilde{L}u|^{2}dx=\tilde{\mathcal{I}}_{1}+\tilde{\mathcal{I}}_{2}+\tilde{\mathcal{I}}_{3},$
where $\tilde{\mathcal{I}}_{i}$ are obtained by replacing $\chi(y)$ in
$\mathcal{I}_{i}$ with $I_{B_{1}}(y)$. Here $\mathcal{I}_{1}$ and
$\tilde{\mathcal{I}}_{1}$ are easily controlled by
$N\|\mathcal{A}u\|_{L_{2}}^{2}$.
Due to H1, (2.26), the definition of $\chi$, and the change of variables
$y\to\frac{y}{|\xi|}$,
$\displaystyle\mathcal{I}_{2}$ $\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}|\xi|^{-2d}\left|\int_{|y|\geq
1}(\sin(\frac{\xi}{|\xi|}\cdot
y)-y\cdot\frac{\xi}{|\xi|}\chi(\frac{y}{|\xi|}))a(\frac{y}{|\xi|})J(\frac{y}{|\xi|})~{}dy\right|^{2}d\xi$
$\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}|\xi|^{-2d}j(1/|\xi|)^{2}$
$\displaystyle\quad\quad\quad\quad\times\left(\int_{|y|\geq
1}\left|\sin(\frac{\xi}{|\xi|}\cdot y)-I_{\sigma\neq
1}y\cdot\frac{\xi}{|\xi|}\chi(\frac{y}{|\xi|})\right|a(\frac{y}{|\xi|})|y|^{-d-\alpha_{0}}~{}dy\right)^{2}d\xi$
$\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}|\xi|^{-2d}j(1/|\xi|)^{2}~{}d\xi.$
Hence, by Lemma 2.14,
$\displaystyle\mathcal{I}_{2}$ $\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}(\Psi(\xi))^{2}~{}d\xi=N\int_{\mathbb{R}^{d}}|\mathcal{A}u|^{2}~{}dx.$
Similarly, if $\sigma>1$,
$\displaystyle\tilde{\mathcal{I}}_{2}$ $\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}|\xi|^{-2d}j(1/|\xi|)^{2}$
$\displaystyle\quad\quad\times\left(\int_{|y|\geq
1}\left|\sin(\frac{\xi}{|\xi|}\cdot
y)-I_{\sigma>1}y\cdot\frac{\xi}{|\xi|}I_{|y|\leq|\xi|}\right|\,\,a(\frac{y}{|\xi|})|y|^{-d-\alpha_{0}}~{}dy\right)^{2}d\xi$
$\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}|\xi|^{-2d}j(1/|\xi|)^{2}~{}d\xi\leq
N\int_{\mathbb{R}^{d}}|\mathcal{A}u|^{2}~{}dx.$
Also, using the fundamental theorem of calculus, the definition of $\chi$ and
(2.26),
$\displaystyle\mathcal{I}_{3}$ $\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{|y||\xi|<1}(\sin(\xi\cdot
y)-y\cdot\xi\chi(y))a(y)J(y)~{}dy\right|^{2}d\xi$ $\displaystyle=$
$\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{|y||\xi|<1}\int_{0}^{1}\frac{d}{dt}(\sin(t\xi\cdot
y)-ty\cdot\xi\chi(y))~{}dt~{}a(y)J(y)~{}dy\right|^{2}d\xi$ $\displaystyle=$
$\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{|y||\xi|<1}(\xi\cdot
y)\int_{0}^{1}(\cos(t\xi\cdot y)-\chi(y))~{}dt~{}a(y)J(y)~{}dy\right|^{2}d\xi$
$\displaystyle=$ $\displaystyle I_{\sigma\leq
1}N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{|y||\xi|<1}(\xi\cdot
y)\int_{0}^{1}\cos(t\xi\cdot y)~{}dt~{}a(y)J(y)~{}dy\right|^{2}d\xi$
$\displaystyle+I_{\sigma>1}N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{|y||\xi|<1}(\xi\cdot
y)\int_{0}^{1}(\cos(t\xi\cdot y)-1)~{}dt~{}a(y)J(y)~{}dy\right|^{2}d\xi.$
Observe that by H1, for any $t\in(0,1)$,
$\Psi(t|\xi|)=\int_{\mathbb{R}^{d}}(1-\cos(ty\cdot\xi))J(y)dy=t^{-d}\int_{\mathbb{R}^{d}}(1-\cos(y\cdot\xi)J(t^{-1}y)dy\leq
Nt^{\alpha_{0}}\Psi(|\xi|).$
Thus, if $\sigma>1$,
$\mathcal{I}_{3}\leq
N\int_{\mathbb{R}^{d}}|\mathcal{F}(u)|^{2}\left(\int^{1}_{0}\Psi(t|\xi|)dt\right)^{2}\,d\xi\leq
N\|\mathcal{A}u\|^{2}_{L_{2}}.$
Also, if $\sigma>1$,
$\displaystyle\tilde{\mathcal{I}}_{3}$ $\displaystyle\leq$
$\displaystyle(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{|y||\xi|<1}(\xi\cdot
y)\int_{0}^{1}\cos(t\xi\cdot y)I_{|y|\geq
1}~{}dt~{}a(y)J(y)~{}dy\right|^{2}d\xi$
$\displaystyle+(2\pi)^{-d}\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left|\int_{|y||\xi|<1}(\xi\cdot
y)\int_{0}^{1}(1-\cos(t\xi\cdot y))~{}dt~{}a(y)J(y)~{}dy\right|^{2}d\xi$
$\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left(\int_{|y|\geq
1}J(y)dy\right)^{2}\,d\xi+N\int_{\mathbb{R}^{d}}|\mathcal{F}(u)|^{2}\left(\int^{1}_{0}\Psi(t|\xi|)dt\right)^{2}\,d\xi$
$\displaystyle\leq$ $\displaystyle
N\|u\|^{2}_{\mathcal{H}_{2}^{\mathcal{A}}}.$
Thus (3.41) is proved if $\sigma>1$, and (3.39) and (3.40) are obtained for
general $u\in\mathcal{H}_{2}^{\mathcal{A}}$ owing to (3.41). Therefore (ii) is
proved.
Now assume $\sigma\leq 1$. To estimate $\mathcal{I}_{3}$ we use the Fubini’s
Theorem, the change of variable $|\xi|ty\to y$, H1, H2, and Lemma 2.14
$\displaystyle\mathcal{I}_{3}$ $\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}$
$\displaystyle\quad\quad\quad\times\left|\int_{0}^{1}t^{-d-1}|\xi|^{-d}\int_{|y|<t}(\frac{\xi}{|\xi|}\cdot
y)\cos(\frac{\xi}{|\xi|}\cdot
y)a(\frac{y}{|\xi|t})J(\frac{y}{|\xi|t})~{}dydt\right|^{2}d\xi$
$\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left||\xi|^{-d}\int_{0}^{1}t^{-d-1}\int_{|y|<1}|y|J(\frac{y}{|\xi|t})~{}dydt\right|^{2}d\xi$
$\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left||\xi|^{-d}\int_{0}^{1}t^{\alpha_{0}-1}~{}dt\int_{|y|<1}|y|J(y/|\xi|)~{}dy\right|^{2}d\xi$
$\displaystyle\leq$ $\displaystyle
N\int_{\mathbb{R}^{d}}|\mathcal{F}u(\xi)|^{2}\left||\xi|^{-d}j(1/|\xi|)\right|^{2}d\xi\leq
N\|\mathcal{A}u\|^{2}_{L_{2}}.$
Therefore the lemma is proved. $\Box$
Corollary 2.7 and Lemmas 2.6 and 3.1 easily prove Theorem 2.8.
## 4\. Some Hölder estimates
In this section obtain some Hölder estimates for functions
$u\in\mathcal{H}_{2}^{\mathcal{A}}\cap C_{b}^{\infty}$. The estimates will be
used later for the estimates of the mean oscillation. Throughout this section
we assume Assumption 2.15 holds if $\sigma=1$.
###### Lemma 4.1.
For any $\alpha\in(0,1)$, $b\in\mathbb{R}^{d}$, and a nonnegative measurable
function $\mathcal{K}(z)$, there exist $\eta_{1},\eta_{2}\in(0,1/4)$,
depending only on $\alpha$, such that
$\displaystyle\int_{\mathcal{C}}[\left(|b+2z|^{\alpha}+|b-2z|^{\alpha}-2|b|^{\alpha}\right)\mathcal{K}(z)]~{}dz$
(4.43) $\displaystyle\leq$
$\displaystyle-2^{\alpha-3}\alpha(1-\alpha)\int_{\mathcal{C}}|b|^{\alpha-2}|z|^{2}\mathcal{K}(z)dz,$
where
$\mathcal{C}=\\{|z|<\eta_{1}|b|:|z\cdot b|\geq(1-\eta_{2})|b||z|\\}.$
###### Proof.
We repeat the proof of Lemma 4.2 in [9] with few minor changes. Put
$\eta(t):=b+2tz$ and $\varphi(t):=|b+2tz|^{\alpha}=|\eta(t)|^{\alpha}$ for
$z\in\mathcal{C}$. Then
$\displaystyle\varphi^{\prime\prime}(t)$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{d}\left(\alpha(\alpha-2)(\eta_{i}(t))(\eta_{j}(t))|\eta(t)|^{\alpha-4}+I_{i=j}\alpha|\eta(t)|^{\alpha-2}\right)4z_{i}z_{j}$
$\displaystyle=$ $\displaystyle
4\alpha(\alpha-2)|\eta(t)|^{\alpha-4}|\eta(t)\cdot
z|^{2}+4\alpha|\eta(t)|^{\alpha-2}|z|^{2}$ $\displaystyle=$ $\displaystyle
4\alpha|b+2tz|^{\alpha-4}[(\alpha-2)|(b+2tz)\cdot z|^{2}+|b+2tz|^{2}|z|^{2}].$
For $t\in[-1,1]$ and $z\in\mathcal{C}$, observer that,
$|b+2tz|^{2}\leq(1+2\eta_{1})^{2}|b|^{2}$
and
$\displaystyle|(b+2tz)\cdot z|$ $\displaystyle=$ $\displaystyle|b\cdot
z+2t|z|^{2}|\geq|b\cdot z|-2|z|^{2}$ $\displaystyle\geq$
$\displaystyle(1-\eta_{2})|b||z|-2|z|^{2}\geq(1-2\eta_{1}-\eta_{2})|z||b|.$
Thus
$\displaystyle\varphi^{\prime\prime}(t)\leq
4\alpha|a+2tz|^{\alpha-4}[(\alpha-2)(1-2\eta_{1}-\eta_{2})^{2}+(1+2\eta_{1})^{2}]|b|^{2}|z|^{2}.$
(4.44)
Since $(1-2\eta_{1}-\eta_{2})^{2}\to 1$ and $(1+2\eta_{1})^{2}\to 1$ as
$\eta_{1},\eta_{2}\downarrow 0$, one can choose sufficiently small
$\eta_{1},\eta_{2}\in(0,1/4)$, depending only on $\alpha\in(0,1)$, such that
$(\alpha-2)(1-2\eta_{1}-\eta_{2})^{2}+(1+2\eta_{1})^{2}\leq(\alpha-1)/2.$
By combining this with (4.44)
$\displaystyle\varphi^{\prime\prime}(t)\leq-2\alpha(1-\alpha)|b+2tz|^{\alpha-4}|b|^{2}|z|^{2}.$
(4.45)
Furthermore observe that
$\displaystyle|b+2tz|^{\alpha-4}\geq(1+2\eta_{1})^{\alpha-4}|b|^{\alpha-4}\geq
2^{\alpha-4}|b|^{\alpha-4}.$
Therefore, from (4.45)
$\displaystyle\varphi^{\prime\prime}(t)\leq-2^{\alpha-3}\alpha(1-\alpha)|b|^{\alpha-2}|z|^{2},\quad
t\in[-1,1],~{}z\in\mathcal{C}.$
In addition to this, to prove (4.43), it is enough to use the fact that there
exists $t_{0}\in(-1,1)$ satisfying
$\varphi(1)+\varphi(-1)-2\varphi(0)=\varphi^{\prime\prime}(t_{0}),$
which can be shown by the mean value theorem. The lemma is proved. $\Box$
###### Theorem 4.2.
Let $R>0,\lambda\geq 0$ and H1 hold. Suppose $f\in L_{\infty}(B_{1})$ and
$u,\tilde{u}\in C_{b}^{2}(B_{R})\cap L_{1}(\mathbb{R}^{d},w_{R})$, where
$w_{R}(x)=\frac{1}{1/j(R)+1/J(x/2)}$. Also assume
$\displaystyle Lu-\lambda
u=f,\quad\quad\tilde{L}\tilde{u}-\lambda\tilde{u}=f\quad\text{in}~{}\,\,B_{R}.$
(4.46)
(i) For any $\alpha\in(0,\min\\{1,\alpha_{0}\\})$ and $0<r<R$, it holds that
$\displaystyle[u]_{C^{\alpha}(B_{r})}$ $\displaystyle\leq$ $\displaystyle
Nr_{1}^{-\alpha}\|u\|_{L_{\infty}(B_{R})}$ (4.47)
$\displaystyle+N\frac{\|u\|_{L_{\infty}(B_{R})}}{j(r_{1})r_{1}^{d+\alpha}}\Big{(}r_{1}^{-2}\int_{B_{r_{1}}}|z|^{2}J(z)~{}dz+I_{\sigma<1}r_{1}^{-1}\int_{B_{r_{1}}}|z|J(z)~{}dz\Big{)}$
$\displaystyle+N\Big{(}\frac{1}{r^{d+\alpha}_{1}j(R)}\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}+\frac{1}{j(r_{1})r_{1}^{d+\alpha}}\text{osc}_{B_{R}}f\Big{)},$
where $r_{1}=(R-r)/2$ and $N=N(d,\nu,\Lambda,\kappa_{1},\alpha_{0},\alpha)$.
Consequently, if H2 is additionally assumed, then
$[u]_{C^{\alpha}(B_{r})}\leq
N\left(r_{1}^{-\alpha}\|u\|_{L_{\infty}(B_{R})}+\frac{1}{r_{1}^{d+\alpha}j(R)}\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}+\frac{\text{osc}_{B_{R}}f}{j(r_{1})r_{1}^{d+\alpha}}\right).$
(4.48)
(ii) In addition to H1, let one of H3(ii)- H3(iv) hold. Then (4.47) holds for
$\tilde{u}$. Consequently, if H2 additionally holds, (4.48) holds for
$\tilde{u}$.
###### Proof.
We adopt the method used in [9] (cf. [3]). Assume that $u$ is not identically
zero in $B_{r}$. Set
$r_{1}=(R-r)/2,\quad r_{2}=(R+r)/2,\quad w(t,x)=I_{B_{R}}(x)u(t,x).$
For $x\in B_{r_{2}}$, $u(x)=v(x)$ and $\nabla u(x)=\nabla w(x)$. Thus
$\displaystyle Lu(x)=Lw(t,x)+\int_{|z|\geq
r_{1}}\left(u(t,x+z)-w(t,x+z)\right)a(z)J(z)dz.$
So in $B_{r_{2}}$
$\displaystyle Lw(x)-\lambda w=g(x)+f(x),$ (4.49)
where
$g(x)=-\int_{|z|\geq r_{1}}\left(u(x+z)-w(x+z)\right)a(z)J(z)dz.$
Note that by H1
$\displaystyle\|g\|_{L_{\infty}(B_{R})}\leq
N\frac{j(r_{1})}{j(R)}\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})},$ (4.50)
where $N=N(d,\Lambda)$. Indeed, this comes from the fact that for all $|z|\geq
r_{1}$, $x\in B_{R}$, and $|x+z|\leq R$
$\displaystyle|j(z)|\leq
Nj(r_{1})\leq\frac{j(r_{1})}{j(R)}\cdot\frac{N}{1/j(R)+1/j(|x+z|/2)}.$
For $x_{0}\in B_{r}$ and $\alpha\in(0,\min\\{1,\alpha_{0}\\})$, we define
$M(x,y):=w(x)-w(y)-C|x-y|^{\alpha}-8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|x-x_{0}|^{2},$
where $C$ is a positive constant which will be chosen later so that it is
independent of $x_{0}$ and
$\displaystyle\sup_{x,y\in\mathbb{R}^{d}}M(x,y)\leq 0.$ (4.51)
For $x\in\mathbb{R}^{d}\setminus B_{r_{1}/2}(x_{0})$,
$\displaystyle w(x)-w(y)\leq 2\|u\|_{L_{\infty}(B_{R})}\leq
8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|x-x_{0}|^{2}.$ (4.52)
This shows
$M(x,y)\leq 0,\quad x\in\mathbb{R}^{d}\setminus B_{r_{1}/2}(x_{0}).$
Assume that there exist $x,y\in\mathbb{R}^{d}$ such that $M(x,y)>0$. We will
get the contradiction by choosing an appropriate constant $C$. Due to (4.52),
$x\in B_{r_{1}/2}(x_{0})$. Moreover
$\displaystyle w(x)-w(y)>C|x-y|^{\alpha},$
which implies
$\displaystyle|x-y|^{\alpha}<\frac{2\|u\|_{L_{\infty}(B_{R})}}{C}.$ (4.53)
If we take $C$ large enough so that $C\geq
2(r_{1}/2)^{-\alpha}\|u\|_{L_{\infty}(B_{R})}$, then
$y\in B_{r+r_{1}}.$
Therefore, there exist $\bar{x},\bar{y}\in B_{r+r_{1}}$ satisfying
$\sup_{x,y\in\mathbb{R}^{d}}M(x,y)=M(\bar{x},\bar{y})>0.$
Moreover, from (4.49)
$\displaystyle-2\|g\|_{L_{\infty}(B_{R})}-\text{osc}_{B_{R}}f$
$\displaystyle\leq$ $\displaystyle(Lw(\bar{x})-\lambda
w(\bar{x}))-(Lw(\bar{y})-\lambda w(\bar{y}))$ (4.54) $\displaystyle=$
$\displaystyle(Lw(\bar{x})-Lw(\bar{y}))+\lambda(w(\bar{y})-w(\bar{x}))$
$\displaystyle\leq$ $\displaystyle Lw(\bar{x})-Lw(\bar{y}):=\mathcal{I}.$
Put $K(z):=a(z)J(z)$ and
$K_{1}(z):=K(z)\wedge K(-z),\quad K_{2}(z):=K(z)-K_{1}(z).$
By $L_{1}$ and $L_{2}$, respectively, we denote the operators with kernels
$K_{1}$ and $K_{2}$. Then
$\mathcal{I}=\mathcal{I}_{1}+\mathcal{I}_{2},$
where
$\mathcal{I}_{1}:=L_{1}w(\bar{x})-L_{1}w(\bar{y})\quad\text{and}\quad\mathcal{I}_{2}:=L_{2}w(\bar{x})-L_{2}w(\bar{y}).$
Since $K_{1}$ is symmetric (i.e. $K_{1}(z)=K_{1}(-z)$),
$\mathcal{I}_{1}=\frac{1}{2}\int_{\mathbb{R}^{d}}\mathcal{J}(\bar{x},\bar{y},z)K_{1}(z)dz,$
where
$\mathcal{J}(\bar{x},\bar{y},z)=w(\bar{x}+z)+w(\bar{x}-z)-2w(\bar{x})-w(\bar{y}+z)-w(\bar{y}-z)+2w(\bar{y}).$
Also, since $M(x,y)$ attains its maximum at $(\bar{x},\bar{y})$,
$\displaystyle
w(\bar{x}+z)-w(\bar{y}+z)-C|\bar{x}-\bar{y}|^{\alpha}-8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|\bar{x}+z-x_{0}|^{2}$
(4.55) $\displaystyle\leq$ $\displaystyle
w(\bar{x})-w(\bar{y})-C|\bar{x}-\bar{y}|^{\alpha}-8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|\bar{x}-x_{0}|^{2}$
and
$\displaystyle
w(\bar{x}-z)-w(\bar{y}-z)-C|\bar{x}-\bar{y}|^{\alpha}-8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|\bar{x}-z-x_{0}|^{2}$
(4.56) $\displaystyle\leq$ $\displaystyle
w(\bar{x})-w(\bar{y})-C|\bar{x}-\bar{y}|^{\alpha}-8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|\bar{x}-x_{0}|^{2}$
for all $z\in\mathbb{R}^{d}$. By combining these two inequalities,
$\displaystyle\mathcal{J}(\bar{x},\bar{y},z)\leq
8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\left(|\bar{x}+z-x_{0}|^{2}+|\bar{x}-z-x_{0}|^{2}-2|\bar{x}-x_{0}|^{2}\right).$
(4.57)
Similarly,
$\displaystyle
w(\bar{x}+z)-w(\bar{y}-z)-C|\bar{x}-\bar{y}+2z|^{\alpha}-8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|\bar{x}+z-x_{0}|^{2}$
$\displaystyle\leq$ $\displaystyle
w(\bar{x})-w(\bar{y})-C|\bar{x}-\bar{y}|^{\alpha}-8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|\bar{x}-x_{0}|^{2},$
$\displaystyle
w(\bar{x}-z)-w(\bar{y}+z)-C|\bar{x}-\bar{y}-2z|^{\alpha}-8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|\bar{x}-z-x_{0}|^{2}$
$\displaystyle\leq$ $\displaystyle
w(\bar{x})-w(\bar{y})-C|\bar{x}-\bar{y}|^{\alpha}-8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}|\bar{x}-x_{0}|^{2}.$
It follows that, for any $z\in\mathbb{R}^{d}$,
$\displaystyle\mathcal{J}(\bar{x},\bar{y},z)$ $\displaystyle\leq$
$\displaystyle
C\left(|\bar{x}-\bar{y}+2z|^{\alpha}+|\bar{x}-\bar{y}-2z|^{\alpha}-2|\bar{x}-\bar{y}|^{\alpha}\right)$
$\displaystyle+8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\left(|\bar{x}+z-x_{0}|^{2}+|\bar{x}-z-x_{0}|^{2}-2|\bar{x}-x_{0}|^{2}\right).$
Put $b=\bar{x}-\bar{y}$. Since $(\bar{x},\bar{y})$ satisfy (4.53),
$|b|<r_{1}/2$ if $C\geq 2(r_{1}/2)^{-\alpha}\|u\|_{L_{\infty}(B_{R})}$. Also
set for $\eta_{1},\eta_{2}\in(0,1/4)$ specified in Lemma 4.1,
$\mathcal{C}=\\{|z|<\eta_{1}|b|:|z\cdot b|\geq(1-\eta_{2})|b||z|\\}.$
Then
$\displaystyle 2\mathcal{I}_{1}$ $\displaystyle=$ $\displaystyle\int_{|z|\geq
r_{1}/2}\mathcal{J}(\bar{x},\bar{y},z)K_{1}(z)~{}dz+\int_{B_{r_{1}/2}\setminus\mathcal{C}}\mathcal{J}(\bar{x},\bar{y},z)K_{1}(z)~{}dz$
(4.59)
$\displaystyle+\int_{\mathcal{C}}\mathcal{J}(\bar{x},\bar{y},z)K_{1}(z)~{}dz:=\mathcal{I}_{11}+\mathcal{I}_{12}+\mathcal{I}_{13}.$
Note that by H1,
$\mathcal{I}_{11}\leq Nj(r_{1}/2)r_{1}^{d}\|u\|_{L_{\infty}(B_{R})}.$
Indeed,
$\displaystyle\mathcal{I}_{11}$ $\displaystyle\leq$ $\displaystyle
N\|u\|_{L_{\infty}(B_{R})}\int_{|z|\geq r_{1}/2}J(z)~{}dz$ $\displaystyle\leq$
$\displaystyle Nr_{1}^{d}\|u\|_{L_{\infty}(B_{R})}\int_{|z|\geq
1}J(r_{1}z/2)~{}dz$ $\displaystyle\leq$ $\displaystyle
Nj(r_{1}/2)r_{1}^{d}\|u\|_{L_{\infty}(B_{R})}\int_{|z|\geq
1}|z|^{-d-\alpha_{0}}~{}dz.$
On the other hand from (4.57), it follows that
$\displaystyle\mathcal{I}_{12}$ $\displaystyle\leq$ $\displaystyle
8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}\setminus\mathcal{C}}\left(|\bar{x}+z-x_{0}|^{2}+|\bar{x}-z-x_{0}|^{2}-2|\bar{x}-x_{0}|^{2}\right)K_{1}(z)~{}dz$
$\displaystyle\leq$ $\displaystyle
Nr_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}}|z|^{2}J(z)~{}dz.$
Next using (4) we obtain
$\displaystyle\mathcal{I}_{13}\leq
C\int_{\mathcal{C}}\left(|\bar{x}-\bar{y}+2z|^{\alpha}+|\bar{x}-\bar{y}-2z|^{\alpha}-2|\bar{x}-\bar{y}|^{\alpha}\right)K_{1}(z)~{}dz$
$\displaystyle+8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\int_{\mathcal{C}}\left(|\bar{x}+z-x_{0}|^{2}+|\bar{x}-z-x_{0}|^{2}-2|\bar{x}-x_{0}|^{2}\right)K_{1}(z)~{}dz$
$\displaystyle:=\mathcal{I}_{131}+\mathcal{I}_{132}.$
The term $\mathcal{I}_{132}$ is again bounded by
$Nr_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}}|z|^{2}J(z)~{}dz.$
Furthermore, from lemma 4.1
$\mathcal{I}_{131}\leq-2^{\alpha-3}C\alpha(1-\alpha)\int_{\mathcal{C}}|b|^{\alpha-2}|z|^{2}K_{1}(z)dz.$
Combining all these facts above, we obtain
$\displaystyle\mathcal{I}_{1}$ $\displaystyle\leq$ $\displaystyle
N\|u(\cdot)\|_{L_{\infty}(B_{R})}\left(j(r_{1}/2)r_{1}^{d}+r_{1}^{-2}\int_{B_{r_{1}/2}}|z|^{2}J(z)~{}dz\right)$
(4.60)
$\displaystyle-2^{\alpha-3}C\alpha(1-\alpha)\int_{\mathcal{C}}|b|^{\alpha-2}|z|^{2}K_{1}(z)dz.$
For $\mathcal{I}_{2}$, we first consider the case $\sigma<1$. In this case,
$\displaystyle\mathcal{I}_{2}$ $\displaystyle=$ $\displaystyle\int_{|z|\geq
r_{1}/2}\left(w(\bar{x}+z)-w(\bar{x})-w(\bar{y}+z)+w(\bar{y})\right)K_{2}(z)~{}dz$
$\displaystyle+~{}\int_{B_{r_{1}/2}}\left(w(\bar{x}+z)-w(\bar{x})-w(\bar{y}+z)+w(\bar{y})\right)K_{2}(z)~{}dz:=\mathcal{I}_{21}+\mathcal{I}_{22}.$
Analogously to $\mathcal{I}_{11}$, we bound $\mathcal{I}_{21}$ by
$Nj(r_{1}/2)r_{1}^{d}\|u\|_{L_{\infty}(B_{R})}$. For the other term
$\mathcal{I}_{22}$, since $|\bar{x}-x_{0}|<r_{1}/2$, from (4.55)
$\displaystyle\mathcal{I}_{22}$ $\displaystyle\leq$ $\displaystyle
Nr_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}}\left(|\bar{x}+z-x_{0}|^{2}-|\bar{x}-x_{0}|^{2}\right)K_{2}(z)~{}dz$
$\displaystyle\leq$ $\displaystyle
Nr_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}}\left(|z|^{2}+2|z||\bar{x}-x_{0}|\right)J(z)~{}dz$
$\displaystyle\leq$ $\displaystyle
Nr_{1}^{-1}\|u\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}}|z|J(z)~{}dz.$
So
$\displaystyle\mathcal{I}_{2}\leq
N\|u\|_{L_{\infty}(B_{R})}\Big{(}\,j(r_{1}/2)r_{1}^{d}+r_{1}^{-1}\int_{B_{r_{1}/2}}|z|J(z)~{}dz\Big{)}.$
(4.61)
By combining (4.50), (4.54), (4.60) and (4.61),
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
N_{1}\Big{(}\text{osc}_{B_{R}}f+\frac{j(r_{1})}{j(R)}\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}$
$\displaystyle\quad\quad+\|u\|_{L_{\infty}(B_{R})}\big{[}j(r_{1}/2)r_{1}^{d}+r_{1}^{-1}\int_{B_{r_{1}/2}}|z|J(z)~{}dz\big{]}\Big{)}$
$\displaystyle-2^{\alpha-3}C\alpha(1-\alpha)\int_{\mathcal{C}}|b|^{\alpha-2}|z|^{2}K_{1}(z)~{}dz.$
Thus, if $C\geq C_{1}:=2(r_{1}/2)^{-\alpha}\|u\|_{L_{\infty}(B_{R})}$ and
$\displaystyle C$ $\displaystyle\geq$ $\displaystyle
C_{2}:=N_{1}C_{3}\Big{(}\text{osc}_{B_{R}}f+\frac{j(r_{1})}{j(R)}\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}$
$\displaystyle\quad\quad+\|u\|_{L_{\infty}(B_{R})}\Big{[}j(r_{1}/2)r_{1}^{d}+r_{1}^{-1}\int_{B_{r_{1}/2}}|z|J(z)~{}dz\Big{]}\Big{)},$
then
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
N_{1}\Big{(}\text{osc}_{B_{R}}f+\frac{j(r_{1})}{j(R)}\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}$
$\displaystyle\quad\quad+\|u\|_{L_{\infty}(B_{R})}\Big{[}j(r_{1}/2)r_{1}^{d}+r_{1}^{-1}\int_{B_{r_{1}/2}}|z|J(z)~{}dz\Big{]}\Big{)}$
$\displaystyle\times\Big{(}1-C_{3}2^{\alpha-3}\alpha(1-\alpha)\int_{\mathcal{C}}|b|^{\alpha-2}|z|^{2}K_{1}(z)~{}dz\Big{)}$
$\displaystyle:=$ $\displaystyle(1-C_{3}C_{4}(b)).$
If we take $C_{3}$ so that $C_{3}=1/C_{5}$ for a
$C_{5}=C_{5}(r_{1},\alpha)<C_{4}(b)$ which does not depend on $b$ and will be
chosen below, we get the contradiction. To select $C_{5}$, observe that with
H1 and the fact $|b|\leq r_{1}/2$
$\displaystyle C_{4}(b)$ $\displaystyle=$ $\displaystyle
2^{\alpha-3}\alpha(1-\alpha)\int_{\mathcal{C}}|b|^{\alpha-2}|z|^{2}K_{1}(z)dz$
$\displaystyle\geq$ $\displaystyle\nu
2^{\alpha-3}\alpha(1-\alpha)\int_{\mathcal{C}}|b|^{\alpha-2}|z|^{2}J(z)dz$
$\displaystyle\geq$ $\displaystyle\kappa_{1}^{-1}\nu
2^{\alpha-3}\alpha(1-\alpha)j(\eta_{1}|b|)\int_{\mathcal{C}}|b|^{\alpha-2}|z|^{2}dz$
$\displaystyle\geq$ $\displaystyle\kappa_{1}^{-1}\nu
2^{\alpha-3}\alpha(1-\alpha)j(\eta_{1}|b|)|b|^{\alpha-2}|\eta_{1}b|^{d+2}\int_{\mathcal{C}_{\eta_{2}}}|z|^{2}dz$
$\displaystyle\geq$
$\displaystyle\kappa_{1}^{-1}\nu\eta_{1}^{d+2}2^{\alpha-3}\alpha(1-\alpha)j(|b|)|b|^{d+\alpha}\int_{\mathcal{C}_{\eta_{2}}}|z|^{2}dz$
$\displaystyle\geq$ $\displaystyle\kappa_{1}^{-2}\nu
j(r_{1}/2)(r_{1}/2)^{d+\alpha}\eta_{1}^{d+2}2^{\alpha-3}\alpha(1-\alpha)\int_{\mathcal{C}_{\eta_{2}}}|z|^{2}dz$
$\displaystyle=$ $\displaystyle
j(r_{1}/2)r_{1}^{d+\alpha}N(\alpha,\eta_{1},\eta_{2}):=C_{5},$
where $\mathcal{C}=\\{|z|<\eta_{1}|b|:|z\cdot b|\geq(1-\eta_{2})|b||z|\\}$ and
$\mathcal{C}_{\eta_{2}}=\\{|z|<1:\frac{|z\cdot
b|}{|b||z|}\geq(1-\eta_{2})\\}$. Therefore, (4.51) holds with $C=C_{1}+C_{2}$.
Since $C$ is independent of $x_{0}$, (4.47) is proved.
Next we consider the case $\sigma=1$. Note that, because $K_{1}$ is symmetric,
both $K_{1}$ and $K_{2}$ satisfy (2.26). Therefore, we can replace $1_{B_{1}}$
with $I_{B_{r_{1}}}$ in the definition of $L_{2}$, and get
$\mathcal{I}_{2}=\mathcal{I}_{21}+\mathcal{I}_{22}$, where
$\displaystyle\mathcal{I}_{21}=\int_{|z|\geq
r_{1}/2}\left(w(\bar{x}+z)-w(\bar{x})-w(\bar{y}+z)+w(\bar{y})\right)K_{2}(z)~{}dz,$
$\displaystyle\mathcal{I}_{22}=\int_{B_{r_{1}/2}}\left(w(\bar{x}+z)-w(\bar{x})-w(\bar{y}+z)+w(\bar{y})-z\cdot(\nabla
w(\bar{x})-\nabla w(\bar{y}))\right)K_{2}(z)~{}dz.$
$\mathcal{I}_{21}$ is already estimated in the previous case. Thus we only
consider $\mathcal{I}_{22}$. Since $M(x,y)$ attains its maximum at the
interior point $(\bar{x},\bar{y})$, we have
$\nabla_{x}M(\cdot,\bar{y})(\bar{x})=0$,
$\nabla_{y}M(\bar{x},\cdot)(\bar{y})=0$, and therefore
$\displaystyle\nabla w(\bar{x})-\nabla
w(\bar{y})=16r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}(\bar{x}-x_{0}).$ (4.62)
We use (4.55) and (4.62) to get
$\displaystyle\mathcal{I}_{22}$ $\displaystyle\leq$ $\displaystyle
8r_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}}|z|^{2}K_{2}(z)~{}dz$
$\displaystyle\leq$ $\displaystyle
8r_{1}^{-2}\int_{B_{r_{1}/2}}|z|^{2}J(z)~{}dz\|u\|_{L_{\infty}(B_{R})}.$
Therefore, (4.47) is proved following the argument in the case $\sigma<1$.
Finally, let $\sigma>1$. Now we have
$\mathcal{I}_{2}=\mathcal{I}_{21}+\mathcal{I}_{22}$, where
$\mathcal{I}_{21}=\int_{|z|\geq
r_{1}/2}[w(\bar{x}+z)-w(\bar{x})-w(\bar{y}+z)+w(\bar{y})-z\cdot(\nabla
w(\bar{x})-\nabla w(\bar{y}))]K_{2}(z)~{}dz,$
$\mathcal{I}_{22}=\int_{B_{r_{1}/2}}[w(\bar{x}+z)-w(\bar{x})-w(\bar{y}+z)+w(\bar{y})-z\cdot(\nabla
w(\bar{x})-\nabla w(\bar{y}))]K_{2}(z)~{}dz.$
Since $\sigma>1$, $|\bar{x}-x_{0}|<r_{1}/2$, by (4.62) and H1
$\displaystyle\mathcal{I}_{21}$ $\displaystyle\leq$
$\displaystyle\int_{|z|\geq
r_{1}/2}[4\|u\|_{L_{\infty}(B_{R})}+4(r_{1}/2)^{-1}\|u\|_{L_{\infty}(B_{R})}|z|]K_{2}(z)~{}dz~{}$
$\displaystyle\leq$ $\displaystyle
Nr_{1}^{d}j(r_{1}/2)\|u\|_{L_{\infty}(B_{R})}.$
For $\mathcal{I}_{22}$, we apply (4.55) and (4.62) to get
$\displaystyle\mathcal{I}_{22}$ $\displaystyle\leq$ $\displaystyle
Nr_{1}^{-2}\|u\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}}|z|^{2}J(z)~{}dz.$
So we again argue as in the first case to get the contradiction. Hence (i) is
proved.
The proof of (ii) is quite similar to that of (i). Denote the counter parts of
$w$ and $g$ by $\tilde{w}$ and $\tilde{g}$, respectively. Also we introduce
$\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ similarly. That is $\mathcal{I}_{1}$
is same as before, and $\mathcal{I}_{2}$ is given by
$\displaystyle\mathcal{I}_{2}$ $\displaystyle=$ $\displaystyle\int_{|z|\geq
r_{1}/2}\Big{[}\tilde{w}(\bar{x}+z)-\tilde{w}(\bar{x})-\tilde{w}(\bar{y}+z)+\tilde{w}(\bar{y})$
$\displaystyle\quad-
I_{B_{1}}(z)z\cdot\nabla(\tilde{w}(\bar{x})-\tilde{w}(\bar{y}))\Big{]}K_{2}(z)~{}dz$
$\displaystyle+\int_{B_{r_{1}/2}}\Big{[}\tilde{w}(\bar{x}+z)-\tilde{w}(\bar{x})-\tilde{w}(\bar{y}+z)+\tilde{w}(\bar{y})$
$\displaystyle\quad-
I_{B_{1}}(z)z\cdot\nabla(\tilde{w}(\bar{x})-\tilde{w}(\bar{y}))\Big{]}K_{2}(z)~{}dz$
$\displaystyle:=$ $\displaystyle\mathcal{I}_{21}+\mathcal{I}_{22}.$
All of the differences are as follows. If $r_{1}/2\geq 1$, then by using
(4.55) and (4.62),
$\displaystyle\mathcal{I}_{22}$ $\displaystyle\leq$ $\displaystyle
Nr_{1}^{-2}\|\tilde{u}\|_{L_{\infty}(B_{R})}\Big{[}\int_{B_{1}}|z|^{2}K_{2}(z)~{}dz+\int_{1\leq|z|\leq
r_{1}/2}(|z|^{2}+(\bar{x}-x_{0})\cdot z)K_{2}(z)~{}dz\Big{]}$
$\displaystyle\leq$ $\displaystyle
NI_{\sigma<1}r_{1}^{-1}\|\tilde{u}\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}}|z|J(z)~{}dz$
$\displaystyle+NI_{\sigma>1}r_{1}^{-2}\|\tilde{u}\|_{L_{\infty}(B_{R})}\int_{B_{r_{1}/2}}|z|^{2}J(z)~{}dz.$
In the above, we also used $\int_{1\leq|z|\leq r_{1}/2}z^{i}K_{2}(z)dz=0$ if
$\sigma>1$ (due to H3(iv)).
Let $\sigma<1$ and $r_{1}/2<1$. If H3(ii) hold, then by (2.17),
$\displaystyle\mathcal{I}_{21}$ $\displaystyle\leq$ $\displaystyle
N\|u\|_{L_{\infty}(B_{R})}\int_{|z|\geq r_{1}/2}J(z)~{}dz$ $\displaystyle=$
$\displaystyle Nr_{1}^{d}\int_{|z|\geq 1}J(r_{1}z/2)dz\leq
Nj(r_{1}/2)r_{1}^{d}\|u\|_{L_{\infty}(B_{R})}.$
Also, if H3(iii) holds, then by using (4.62),
$\displaystyle\mathcal{I}_{21}$ $\displaystyle\leq$
$\displaystyle\|u\|_{L_{\infty}(B_{R})}\int_{|z|\geq
r_{1}/2}[1+8r_{1}^{-1}|z|]K_{2}(z)~{}dz$ $\displaystyle\leq$ $\displaystyle
Nj(r_{1}/2)r_{1}^{d}\|u\|_{L_{\infty}(B_{R})}.$
This completes the proof of the theorem. $\Box$
We remove $\sup_{B_{R}}u$ on the right hand side of (4.48) in the following
corollary. Recall $w_{R}(x)=\frac{1}{1/j(R)+1/J(x/2)}$.
###### Corollary 4.3.
Suppose that H1 and H2 hold. Let $\lambda\geq 0$, $f\in L_{\infty}(B_{1})$,
and $u,\tilde{u}\in C_{b}^{2}(B_{R})\cap L_{1}(\mathbb{R}^{d},w_{R})$ satisfy
$\displaystyle Lu-\lambda
u=f,\quad\quad\tilde{L}\tilde{u}-\lambda\tilde{u}=f\quad\quad\text{in}\quad
B_{R}.$ (4.63)
(i) For any $\alpha\in(0,\min\\{1,\alpha_{0}\\})$, it holds that
$\displaystyle[u]_{C^{\alpha}(B_{R/2})}\leq\frac{N}{j(R)R^{d+\alpha}}\left(\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}+\text{osc}_{B_{R}}f\right),$
(4.64)
where $N=N(d,\nu,\Lambda,\kappa_{1},\alpha_{0},\alpha)$.
(ii) If one of H3 (ii)-(iv) is additionally assumed, then (4.64) holds for
$\tilde{u}$.
###### Proof.
For $n=1,2,\ldots$, set
$r_{n}:=R(1-2^{-n}).$
Observe that $(r_{n+1}-r_{n})/2=R2^{-n-2}\leq R$ and by H1
$\displaystyle\frac{1}{j(r_{n+1})}\|u\|_{L_{1}(\mathbb{R}^{d},w_{r_{n+1}})}$
$\displaystyle\leq$
$\displaystyle\left(\int_{|z|<2R}u(z)~{}dz+\frac{1}{j(r_{n+1})}\int_{|z|\geq
2R}u(z)j(z/2)~{}dz\right)$ $\displaystyle\leq$ $\displaystyle
N\left(\int_{|z|<2R}u(z)~{}dz+\frac{1}{j(R)}\int_{|z|\geq
2R}u(z)j(z/2)~{}dz\right)$ $\displaystyle\leq$ $\displaystyle
N\frac{1}{j(R)}\int_{\mathbb{R}^{d}}u(z)w_{R}(z)~{}dz.$
Then by Theorem 4.2 (i) and H1,
$\displaystyle[u]_{C^{\alpha}(B_{r_{n}})}$ $\displaystyle\leq$ $\displaystyle
NR^{-\alpha}2^{\alpha n}\sup_{B_{r_{n+1}}}|u|$
$\displaystyle+N\frac{2^{(d+\alpha)n}}{j(R2^{-n-2})R^{d+\alpha}}\left(\frac{j(R2^{-n-2})}{j(r_{n+1})}\|u\|_{L_{1}(\mathbb{R}^{d},w_{r_{n+1}})}+\text{osc}_{B_{r_{n+1}}}f\right)$
$\displaystyle\leq$ $\displaystyle N\Big{[}R^{-\alpha}2^{\alpha
n}\sup_{B_{r_{n+1}}}|u|+\frac{2^{(d+\alpha)n}}{j(R)R^{d+\alpha}}\Big{(}\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}+\text{osc}_{B_{R}}f\Big{)}\Big{]}.$
In order to estimate the term $\sup_{B_{r_{n+1}}}|u|$ above, we use the
following :
$\sup_{B_{r_{n+1}}}|u|\leq(\varepsilon{r_{n+1}})^{\alpha}[u]_{C^{\alpha}(r_{n+1})}+N(\varepsilon
r_{n+1})^{-d}\|u\|_{L_{1}(B_{r_{n+1}})},\quad\varepsilon\in(0,1).$ (4.66)
Actually this inequality can be easily obtained as follows. For all
$\varepsilon\in(0,1)$, $x\in B_{r_{n+1}}$ and $y\in B_{r_{n+1}}\cap
B_{\varepsilon r_{n+1}}(x)$,
$\displaystyle|B_{r_{n+1}}\cap B_{\varepsilon r_{n+1}}(x)|\cdot|u(x)|$
$\displaystyle\leq$ $\displaystyle\int_{B_{r_{n+1}}\cap B_{\varepsilon
r_{n+1}}(x)}\left(|u(x)-u(y)|+|u(y)|\right)~{}dy$ $\displaystyle\leq$
$\displaystyle|B_{r_{n+1}}\cap B_{\varepsilon
r_{n+1}}(x)|\cdot(\varepsilon{r_{n+1}})^{\alpha}[u]_{C^{\alpha}(B_{r_{n+1}})}+\int_{B_{r_{n+1}}\cap
B_{\varepsilon r_{n+1}}(x)}|u(y)|~{}dy.$
Now it is enough to note that $|B_{r_{n+1}}\cap B_{\varepsilon
r_{n+1}}(x)|\sim(\varepsilon r_{n+1})^{d}$ because $\varepsilon\in(0,1)$ and
$x\in B_{r_{n+1}}$.
Take $N$ from (LABEL:3221) and define $\varepsilon$ so that
$\varepsilon^{\alpha}=N^{-1}2^{-\alpha n}2^{-3d}.$
Then by combining (LABEL:3221) and (4.66),
$\displaystyle[u]_{C^{\alpha}(B_{r_{n}})}$ $\displaystyle\leq$ $\displaystyle
2^{-3d}[u]_{C^{\alpha}(B_{r_{n+1}})}+NR^{-d-\alpha}2^{2dn}\|u\|_{L_{1}(B_{r_{n+1}})}$
(4.67)
$\displaystyle+N\frac{2^{(d+\alpha)n}}{j(R)R^{d+\alpha}}(\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}+\text{osc}_{B_{R}}f)$
$\displaystyle\leq$ $\displaystyle
2^{-3d}[u]_{C^{\alpha}(B_{r_{n+1}})}+NR^{-d-\alpha}2^{2dn}\|u\|_{L_{1}(B_{r_{n+1}})}$
$\displaystyle+N\frac{2^{2dn}}{j(R)R^{d+\alpha}}(\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}+\text{osc}_{B_{R}}f).$
Multiply both sides of (4.67) by $2^{-3dn}$ and take the sum over $n$ to get
$\displaystyle\sum_{n=1}^{\infty}2^{-3dn}[u]_{C^{\alpha}(B_{r_{n}})}$
$\displaystyle\leq$
$\displaystyle\sum_{n=1}^{\infty}2^{-3d(n+1)}[u]_{C^{\alpha}(B_{r_{n+1}})}+N\sum_{n=1}^{\infty}2^{-dn}R^{-d-\alpha}\|u\|_{L_{1}(B_{r_{n+1}})}$
$\displaystyle+N\Big{(}\sum_{n=1}^{\infty}2^{-dn}\Big{)}\frac{1}{j(R)R^{d+\alpha}}(\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}+N\text{osc}_{B_{R}}f).$
Since $[u]_{C^{\alpha}(B_{r_{n}})}\leq[u]_{C^{\alpha}(B_{R})}<\infty$ and by
H1
$\displaystyle\|u\|_{L_{1}(B_{r_{n+1}})}\leq\|u\|_{L_{1}(B_{R})}=\frac{j(R)}{j(R)}\|u\|_{L_{1}(B_{R})}\leq\frac{N}{j(R)}\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})},$
(i) is proved.
(ii) is proved similarly by following the proof of (i) with Theorem 4.2 (ii).
$\Box$
## 5\. Some sharp function and maximal function estimates
For $g\in L_{1,{\rm loc}}(\mathbb{R}^{d})$, the maximal function and sharp
function are defined as follows :
$\mathcal{M}g(x):=\sup_{r>0}-\int_{B_{r}(x)}|g(y)|~{}dy:=\sup_{r>0}\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}|g(y)|~{}dy,$
and
$g^{\\#}(x):=\sup_{r>0}-\int_{B_{r}(x)}|g(y)-(g)_{B_{r}(x)}|~{}dy:=\sup_{r>0}\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}|g(y)-(g)_{B_{r}(x)}|~{}dy,$
where $(g)_{B_{r}(x)}=\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}g(y)~{}dy$ the
average of $g$ on $B_{r}(x)$.
###### Lemma 5.1.
Suppose that H1 and H2 hold. Let $\lambda\geq 0$, $R>0$, $f\in
C_{0}^{\infty}$, and $f=0$ in $B_{2R}$. Assume that $u,\tilde{u}\in
H_{2}^{\mathcal{A}}\cap C_{b}^{\infty}$ satisfy
$Lu-\lambda u=f,\quad\quad\tilde{L}\tilde{u}-\lambda\tilde{u}=f.$ (5.68)
(i) Then for all $\alpha\in(0,\min\\{1,\alpha_{0}\\})$,
$[u]_{C^{\alpha}(B_{R/2})}\leq
NR^{-\alpha}\sum_{k=1}^{\infty}2^{-\alpha_{0}k}(|u|)_{B_{2^{k}R}},$ (5.69)
$[\mathcal{A}u]_{C^{\alpha}(B_{R/2})}\leq
NR^{-\alpha}\left(\sum_{k=1}^{\infty}2^{-\alpha_{0}k}(|\mathcal{A}u|)_{B_{2^{k}R}}+\mathcal{M}f(0)\right),$
(5.70)
where $N$ depends only on $d,\nu,\Lambda,\kappa_{1},\kappa_{2},\alpha_{0}$,
and $\alpha$.
(ii) If one of H3(ii)-(iv) is additionally assumed, then (5.69) and (5.70)
hold for $\tilde{u}$.
###### Proof.
By Corollary 4.3 and the assumption that $f=0$ in $B_{2R}$,
$\displaystyle[u]_{C^{\alpha}(B_{R/2})}\leq
N\frac{1}{j(R)R^{d+\alpha}}\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}.$ (5.71)
Set
$B_{(0)}=B_{R},\quad B_{(k)}=B_{2^{k}R}\setminus B_{2^{k-1}R},\quad k\geq 1.$
Observe that
$\displaystyle\|u\|_{L_{1}(\mathbb{R}^{d},w_{R})}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}|u(y)|\frac{1}{1/j(R)+1/j(|y|/2)}~{}dy$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\int_{B_{(k)}}|u(y)|\frac{1}{1/j(R)+1/j(|y|/2)}dy$
$\displaystyle\leq$ $\displaystyle
2j(R)\int_{B_{2R}}|u(y)|~{}dy+N\sum_{k=2}^{\infty}j(2^{k-2}R)\int_{B_{2^{k}R}}|u(y)|~{}dy$
$\displaystyle\leq$ $\displaystyle
N\left(j(R)R^{d}(|u|)_{B_{2R}}+\sum_{k=2}^{\infty}2^{-(k-2)(d+\alpha_{0})}j(R)\int_{B_{2^{k}R}}|u(y)|~{}dy\right)$
$\displaystyle\leq$ $\displaystyle
N\left(j(R)R^{d}(|u|)_{B_{2R}}+\sum_{k=2}^{\infty}2^{-(k-2)(d+\alpha_{0})}2^{kd}j(R)R^{d}(|u|)_{B_{2^{k}R}}\right)$
$\displaystyle\leq$ $\displaystyle
Nj(R)R^{d}\left(\sum_{k=1}^{\infty}2^{-\alpha_{0}k}(|u|)_{B_{2^{k}R}}\right),$
where the first and second inequalities come from H1. Therefore we get (5.69).
To prove (5.70), we apply the operator $\mathcal{A}$ to both sides of
$Lu-\lambda u=f$ and obtain
$(L-\lambda)(\mathcal{A}u)=\mathcal{A}f.$
By applying Corollary 4.3 again,
$\displaystyle[\mathcal{A}u]_{C^{\alpha}(B_{R/2})}\leq
N\frac{1}{j(R)R^{d+\alpha}}\left(\|\mathcal{A}u\|_{L_{1}(\mathbb{R}^{d},w_{R})}+\sup_{B_{R}}|\mathcal{A}f|\right).$
(5.72)
The first term on the right hand side of (5.72) is bounded by
$NR^{-\alpha}\left(\sum_{k=0}^{\infty}2^{-\alpha_{0}k}(|\mathcal{A}u|)_{B_{2^{k}R}}\right).$
In order to estimate the second term, we recall the definition of
$\mathcal{A}$. For $|x|<R$,
$\displaystyle|\mathcal{A}f(x)|$ $\displaystyle=$
$\displaystyle\left|\int_{\mathbb{R}^{d}}[f(x+y)-f(x)]J(y)~{}dt\right|$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{\infty}\int_{B_{(k)}}|f(x+y)|j(|y|)~{}dy$
$\displaystyle\leq$ $\displaystyle
N\sum_{k=1}^{\infty}j(2^{k-1}R)\int_{B_{(k)}}|f(x+y)|~{}dy$
$\displaystyle\leq$ $\displaystyle
N\sum_{k=1}^{\infty}2^{-(k-1)(d+\alpha_{0})}j(R)\int_{B_{2^{k}R}}|f(x+y)|~{}dy$
$\displaystyle\leq$ $\displaystyle
N\sum_{k=1}^{\infty}2^{-(k-1)(d+\alpha_{0})}j(R)\int_{B_{2^{k+1}R}}|f(y)|~{}dy$
$\displaystyle\leq$ $\displaystyle
Nj(R)R^{d}\left(\sum_{k=1}^{\infty}2^{-\alpha_{0}k}(|f|)_{B_{2^{k+1}R}}\right)\leq
Nj(R)R^{d}\mathcal{M}f(0),$
where the first inequality is due to the assumption $f(x)=0$ if $|x|<2R$ and
both the second and the third inequality are owing to H1. Therefore (i) is
proved. Also, (ii) is proved similarly with Corollary 4.3 (ii). $\Box$
The above lemma easily yields the following mean oscillation estimate.
###### Corollary 5.2.
Suppose that H1 and H2 hold. Let $\lambda\geq 0$ an $r,\kappa>0$. Asume $f\in
C_{0}^{\infty}$, $f=0$ in $B_{2kr}$, and $u,\tilde{u}\in
H_{2}^{\mathcal{A}}\cap C_{b}^{\infty}$ satisfy
$\displaystyle Lu-\lambda
u=f,\quad\quad\tilde{L}\tilde{u}-\lambda\tilde{u}=f.$
(i) Then for all $\alpha\in(0,\min\\{1,\alpha_{0}\\})$,
$\displaystyle(|u-(u)_{B_{r}}|)_{B_{r}}\leq
N\kappa^{-\alpha}\sum_{k=1}^{\infty}2^{-\alpha_{0}k}|u|_{B_{2^{k}\kappa r}},$
(5.73) $\displaystyle(|\mathcal{A}u-\mathcal{A}u)_{B_{r}}|)_{B_{r}}\leq
N\kappa^{-\alpha}\left(\sum_{k=1}^{\infty}2^{-\alpha_{0}k}|\mathcal{A}u|_{B_{2^{k}\kappa
r}}+\mathcal{M}f(0)\right),$ (5.74)
where $N$ depends only on $d,\nu,\Lambda,\kappa_{1},\kappa_{2},\alpha_{0}$,
and $\alpha$.
(ii) If one of H3 (ii)-(iv) is additionally assumed, then (5.73) and (5.74)
hold for $\tilde{u}$.
###### Proof.
It is enough to use the following inequality
$\displaystyle(|u-(u)_{B_{r}}|)_{B_{r}}\leq
2^{\alpha}r^{\alpha}[u]_{C^{\alpha}(r)}\leq
2^{\alpha}r^{\alpha}[u]_{C^{\alpha}(\kappa r/2)}$
and apply Lemma 5.1 with $R=\kappa r$. $\Box$
Next we show that the mean oscillation of $u$ is controlled by the maximal
functions of $u$ and $Lu-\lambda u$.
###### Lemma 5.3.
Suppose that H1 and H2 hold. Let $\lambda>0$, $\kappa\geq 2$, $r>0$, and $f\in
C_{0}^{\infty}$. Assume $u,\tilde{u}\in H_{2}^{\mathcal{A}}\cap
C_{b}^{\infty}$ satisfy
$Lu-\lambda u=f,\quad\quad\tilde{L}u-\lambda u=f.$ (5.75)
(i) Then for all $\alpha\in(0,\min\\{1,\alpha_{0}\\})$,
$\displaystyle\lambda(|u-(u)_{B_{r}}|)_{B_{r}}+(|\mathcal{A}u-(\mathcal{A}u)_{B_{r}}|)_{B_{r}}$
$\displaystyle\leq
N\kappa^{-\alpha}\left(\lambda\mathcal{M}u(0)+\mathcal{M}(\mathcal{A}u)(0)\right)+N\kappa^{d/2}(\mathcal{M}(f^{2})(0))^{1/2},$
(5.76)
where $N$ depends only on $d,\nu,\Lambda$, and $J$.
(ii) If one of H3 (ii)-(iv) is additionally assumed, then (5.76) holds for
$\tilde{u}$.
###### Proof.
Due to the similarity of the proof, we only prove the assertion (i).
Take a cut-off function $\eta\in C_{0}^{\infty}(B_{4\kappa r})$ satisfying
$\eta=1$ in $B_{2\kappa r}$. By Theorem 2.8, there exists a unique solution
$u$ in $H_{2}^{\mathcal{A}}$ satisfying
$\displaystyle Lw-\lambda w=\eta f$ (5.77)
and
$\displaystyle\lambda\|w\|_{L_{2}}+\|\mathcal{A}w\|_{L_{2}}\leq N\|\eta
f\|_{L_{2}}.$ (5.78)
From (5.78), Jensen’s inequality, and the fact $\eta f$ has its support within
$B_{4\kappa r}$, for any $R>0$,
$\displaystyle\lambda(|w|)_{B_{R}}+(|\mathcal{A}w|)_{B_{R}}$
$\displaystyle\leq$ $\displaystyle
NR^{-d/2}\left(\lambda\|w\|_{L_{2}}+\|\mathcal{A}w\|_{L_{2}}\right)$ (5.79)
$\displaystyle\leq$ $\displaystyle NR^{-d/2}\|\eta f\|_{L_{2}}$
$\displaystyle\leq$ $\displaystyle NR^{-d/2}(\kappa
r)^{d/2}(\mathcal{M}(f^{2})(0))^{1/2}.$
Furthermore, taking $(1-\Delta)^{\gamma}$ to both sides of (5.77) and using
the fact $(1-\Delta)^{\gamma}Lw=L(1-\Delta)^{\gamma}w$, we can easily check
that $w\in C_{b}^{\infty}$ by Sobolev’s inequality. By setting $v:=u-w$, from
(5.77) and (5.75)
$Lv-\lambda v=(1-\eta)f,\quad v\in C_{b}^{\infty}\cap H_{2}^{\mathcal{A}}.$
By applying Corollary 5.2 to $v$,
$\displaystyle(\lambda|v-(v)_{B_{r}}|)_{B_{r}}+(|\mathcal{A}v-(\mathcal{A}v)_{B_{r}}|)_{B_{r}}$
(5.80) $\displaystyle\leq$ $\displaystyle
N\kappa^{-\alpha}\left(\sum_{k=1}^{\infty}2^{-\alpha_{0}k}[\lambda(|v|)_{B_{2^{k}\kappa
r}}+(|\mathcal{A}v|)_{B_{2^{k}\kappa r}}]+\mathcal{M}f(0)\right)$
$\displaystyle\leq$ $\displaystyle
N\kappa^{-\alpha}\left(\sum_{k=1}^{\infty}2^{-\alpha_{0}k}[\lambda(|u|)_{B_{2^{k}\kappa
r}}+(|\mathcal{A}u|)_{B_{2^{k}\kappa r}}]\right)$
$\displaystyle+N\kappa^{-\alpha}\left(\sum_{k=0}^{\infty}2^{-\alpha_{0}k}[\lambda(|w|)_{B_{2^{k}\kappa
r}}+(|\mathcal{A}w|)_{B_{2^{k}\kappa r}}]+\mathcal{M}f(0)\right)$
$\displaystyle\leq$ $\displaystyle
N\kappa^{-\alpha}\left(\sum_{k=1}^{\infty}2^{-\alpha_{0}k}[\lambda(|u|)_{B_{2^{k}\kappa
r}}+(|\mathcal{A}u|)_{B_{2^{k}\kappa r}}]\right)$
$\displaystyle+N\kappa^{-\alpha}\left(\sum_{k=1}^{\infty}2^{-\alpha_{0}k}[2^{-dk/2}(\mathcal{M}(f^{2})(0))^{1/2}]+\mathcal{M}f(0)\right)$
$\displaystyle\leq$ $\displaystyle
N\kappa^{-\alpha}\left(\lambda\mathcal{M}u(0)+\mathcal{M}(\mathcal{A}u)(0)+(\mathcal{M}(f^{2})(0))^{1/2}\right),$
where (5.79) is used for the third inequality with $R=2^{k}\kappa r$, and for
the last inequality we use $\mathcal{M}f(0)\leq(\mathcal{M}(f^{2})(0))^{1/2}$.
By combining (5.79) and (5.80),
$\displaystyle\lambda(|u-(u)_{B_{r}}|)_{B_{r}}+(|\mathcal{A}u-(\mathcal{A}u)_{B_{r}}|)_{B_{r}}$
$\displaystyle\leq$ $\displaystyle
N\left(\lambda(|v-(v)_{B_{r}}|)_{B_{r}}+(|\mathcal{A}v-(\mathcal{A}v)_{B_{r}}|)_{B_{r}}+\lambda(|w|)_{B_{r}}+(|\mathcal{A}w|)_{B_{r}}\right)$
$\displaystyle\leq$ $\displaystyle
N\kappa^{-\alpha}\left(\lambda\mathcal{M}u(0)\right)+N\mathcal{M}(\mathcal{A}u)(0)+N(\mathcal{M}(f^{2})(0))^{1/2}.$
Therefore, the lemma is proved. $\Box$
We make full use of Lemma 5.1 to get the mean oscillation of $Lu$.
###### Lemma 5.4.
Suppose that H1 and H2 hold. Let $\lambda>0$, $\kappa\geq 2$, $r>0$, and $f\in
C_{0}^{\infty}$. Assume $u\in H_{2}^{\mathcal{A}}\cap C_{b}^{\infty}$ satisfy
$\displaystyle\mathcal{A}u-\lambda u=f.$
Then for all $\alpha\in(0,\min\\{1,\alpha_{0}\\})$,
$\displaystyle\lambda(|u-(u)_{B_{r}}|)_{B_{r}}+(|Lu-(Lu)_{B_{r}}|)_{B_{r}}$
$\displaystyle\leq
N\kappa^{-\alpha}\left(\lambda\mathcal{M}u(0)+\mathcal{M}(Lu)(0)\right)+N\kappa^{d/2}(\mathcal{M}(f^{2})(0))^{1/2},$
where $N$ depends only on $d,\nu,\Lambda$, and $J$.
###### Proof.
Exchanging the roles of $\mathcal{A}$ and $L$ in the proof of Lemma 5.1, we
easily get
$\displaystyle[Lu]_{C^{\alpha}(B_{R/2})}\leq
NR^{-\alpha}\left(\sum_{k=0}^{\infty}2^{-\alpha_{0}k}(Lu)_{B_{2^{k}R}}+\mathcal{M}f(0)\right).$
Therefore, the lemma is proved as we follow the proof of Lemma 5.3. $\Box$
## 6\. Proof of Theorems 2.16 and 2.21
Proof of Theorem 2.16
The case $p=2$ was already proved in Theorem 2.8. Due to Corollary 2.7 and
Lemmas 2.6, it is sufficient to prove
$\|\mathcal{A}u\|_{L_{p}}+\lambda\|u\|_{L_{p}}\leq N\|Lu-\lambda
u\|_{L_{p}},\quad\forall\,u\in C_{0}^{\infty},$ (6.81)
where $N=N(d,\nu,\Lambda,\kappa_{1},\kappa_{2},\alpha_{0})$.
First, assume $p>2$. Put $f:=Lu-\lambda u$. From Lemma 5.3, for all
$\alpha\in(0,\min\\{1,\alpha_{0}\\})$
$\displaystyle\lambda(|u-(u)_{B_{r}}|)_{B_{r}}+(|\mathcal{A}u-(\mathcal{A}u)_{B_{r}}|)_{B_{r}}$
$\displaystyle\leq
N\kappa^{-\alpha}\left(\lambda\mathcal{M}u(0)+\mathcal{M}(\mathcal{A}u)(0)\right)+N\kappa^{d/2}(\mathcal{M}(f^{2})(0))^{1/2}.$
By translation, it is easy to check that the above inequality holds for all
$B_{r}(x)$ with $x\in\mathbb{R}^{d}$ and $r>0$. By the arbitrariness of $r$,
$\displaystyle\lambda u^{\\#}(x)+(\mathcal{A}u)^{\\#}(x)$ $\displaystyle\leq
N\kappa^{-\alpha}\left(\lambda\mathcal{M}u(x)+\mathcal{M}(\mathcal{A}u)(x)\right)+N\kappa^{d/2}(\mathcal{M}(f^{2})(x))^{1/2}.$
Therefore, by the Fefferman-Stein theorem and Hardy-Littlewood maximal theorem
(see, for instance, chapter 1 of [15]), we get
$\displaystyle\lambda\|u\|_{L_{p}}+\|\mathcal{A}u\|_{L_{p}}\leq
N\kappa^{-\alpha}\left(\lambda\|u\|_{L_{p}}+\|\mathcal{A}u\|_{L_{p}}\right)+N\kappa^{d/2}\|f\|_{L_{p}}.$
By choosing $\kappa>2$ large enough so that $N\kappa^{-\alpha}<1/2$,
$\displaystyle\lambda\|u\|_{L_{p}}+\|\mathcal{A}u\|_{L_{p}}\leq
N\|f\|_{L_{p}}.$
We use the duality argument for $p\in(1,2)$. Put $q:=p/(p-1)$. Then since
$q\in(2,\infty)$, for any $g\in C_{0}^{\infty}$ there is a unique $v_{g}\in
H_{q}^{\mathcal{A}}$ satisfying
$L^{\ast}v_{g}-\lambda v_{g}=g\quad\text{in}~{}\mathbb{R}^{d}.$
Therefore, by applying (6.81) with $q\in(2,\infty)$, for any $u\in
C_{0}^{\infty}$,
$\displaystyle\|\mathcal{A}u\|_{L_{p}}$ $\displaystyle\leq$
$\displaystyle\sup_{\|g\|_{L_{q}}=1,~{}g\in
C_{0}^{\infty}}\int_{\mathbb{R}^{d}}|g\mathcal{A}u|~{}dx$ $\displaystyle=$
$\displaystyle\sup_{\|g\|_{L_{q}}=1,~{}g\in
C_{0}^{\infty}}\int_{\mathbb{R}^{d}}|(L^{\ast}v_{g}-\lambda
v_{g})\mathcal{A}u|~{}dx$ $\displaystyle=$
$\displaystyle\sup_{\|g\|_{L_{q}}=1,~{}g\in
C_{0}^{\infty}}\int_{\mathbb{R}^{d}}|\mathcal{A}v_{g}(Lu-\lambda u)|~{}dx$
$\displaystyle\leq$ $\displaystyle\sup_{\|g\|_{L_{q}}=1,~{}g\in
C_{0}^{\infty}}\|\mathcal{A}v_{g}\|_{L_{q}}\|Lu-\lambda u\|_{L_{p}}$
$\displaystyle\leq$ $\displaystyle\sup_{\|g\|_{L_{q}}=1,~{}g\in
C_{0}^{\infty}}N\|g\|_{L_{q}}\|Lu-\lambda u\|_{L_{p}}=N\|Lu-\lambda
u\|_{L_{p}}.$
Similarly,
$\lambda\|u\|_{L_{p}}\leq N\|Lu-\lambda u\|_{L_{p}}.$
Finally, we prove the continuity of the operator $L$ by showing
$\displaystyle\|Lu\|_{L_{p}}\leq N\|\mathcal{A}u\|_{p},\quad\forall\,\,u\in
C_{0}^{\infty}.$ (6.82)
Recall that we proved (6.81) based on Lemma 5.3. Similarly, using Lemma 5.4,
one can prove
$\displaystyle\|Lu\|_{L_{p}}\leq N\|\mathcal{A}u-\lambda
u\|_{L_{p}}\quad\forall u\in C_{0}^{\infty},\quad\forall\lambda>0.$
Since $N$ is independent of $\lambda$, this leads to (6.82). The theorem is
proved.
Proof of Theorem 2.21
The proof is identical to that of Theorem 2.16 if one of H3(ii)-(iv) holds. So
it only remains to prove
$\|\mathcal{A}u\|_{L_{p}}+\lambda\|u\|_{L_{p}}\leq N\|\tilde{L}u-\lambda
u\|_{L_{p}},\quad\forall\,u\in C_{0}^{\infty}$
under the condition H3(i). Define
$b^{i}=-\int_{B_{1}}y^{i}a(y)J(y)dy\quad\text{if}\,\,\sigma\in(0,1),\quad\quad
b^{i}=\int_{\mathbb{R}^{d}\setminus
B_{1}}y^{i}a(y)J(y)dy\quad\text{if}\,\,\sigma\in(1,2).$
Then under H1 and H2, $|b|<\infty$ and for for any $u\in C_{0}^{\infty}$, we
have
$\tilde{L}u=Lu+b\cdot\nabla u,$
and therefore
$\displaystyle\|u\|_{L_{p}}+\|\mathcal{A}u\|_{L_{p}}\leq N\|Lu-\lambda
u\|_{L_{p}}\leq N\big{(}\|\tilde{L}u-\lambda u\|_{L_{p}}+\|\nabla
u\|_{L_{p}}\big{)}.$
Take $\varepsilon=1/(2N)$ in H3(i) and apply Lemma 2.1. Then, the theorem is
proved. $\Box$
## References
* [1] D. Applebaum, Lévy processes and stochastic calculus, Cambridge University Press, 2009.
* [2] R. Bañuelos and K. Bogdan, Lévy processes and Fourier multipliers, J. Funct. Anal. 250 (2007), no. 1, 197–213.
* [3] G. Barles, E. Chasseigne, and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations.(English summary) J. Eur. Math. Soc. (JEMS) 13 (2011), no. 1, 1–26.
* [4] R.F. Bass, Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc. 357 (2005), no. 2, 837–850.
* [5] R.F. Bass, Hölder continuity of harmonic functions with respect to operators of variable order, Comm. Partial Differential Equations 30 (2005), 1249–1259.
* [6] J. Bertoin, Lévy processes , Cambridge University Press, Cambridge, 1996\.
* [7] Z. Q. Chen, P. Kim, and R. Song, Sharp heat kernel estimates for relativistic stable processes in open sets, Ann. Probab. 40 (2012), no. 1, 213–244.
* [8] H. Dong and D. Kim, Schauder estimates for a class of non-local elliptic equations, Discrete Contin. Dyn. Syst. 33 (2013), no. 6, 2319–2347.
* [9] H. Dong and D. Kim, On $L_{p}$-estimates for a class of non-local ellipitic equations, J. Funct. Anal. 262 (2012), no. 3, 1166–1199.
* [10] P. Kim, R. Song, and Z. Vondraček, Global uniform boundary Harnack principle with explicit decay rate and its application, arXiv:1212.3092 [math.PR] (2012)
* [11] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, American Mathematical Society, 2008.
* [12] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Liet. Mat. Rink. 32 (1992), no. 2, 299-331.
* [13] R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions: Theory and Applications, de Gruyter Studies in Mathematics 37. Berlin: Walter de Gruyter, 2010.
* [14] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 (2006), no. 3, 1155–1174.
* [15] E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993.
* [16] E. Stein, Singular integrals and differentiability properties of functions, Princeton. N.J, 1970.
* [17] X. Zhang, $L^{p}$-maximal regularity of nonlocal parabolic equation and applications, arXiv:1109.0816v4.
|
arxiv-papers
| 2014-02-21T03:36:24 |
2024-09-04T02:49:58.516102
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ildoo Kim and Kyeong-Hun Kim",
"submitter": "Kyeong-Hun Kim",
"url": "https://arxiv.org/abs/1402.5197"
}
|
1402.5280
|
# Direct CP asymmetries of three-body $B$ decays in perturbative QCD
Wen-Fei Wang1 [email protected] Hao-Chung Hu2,3 [email protected]
Hsiang-nan Li3,4,5 [email protected] Cai-Dian Lü1 [email protected]
1Institute of High Energy Physics and Theoretical Physics Center for Science
Facilities, Chinese Academy of Sciences, Beijing 100049, People’s Republic of
China, 2Department of Physics, National Taiwan University, Taipei, Taiwan
106, Republic of China, 3Institute of Physics, Academia Sinica, Taipei,
Taiwan 115, Republic of China, 4Department of Physics, National Tsing-Hua
University, Hsinchu, Taiwan 300, Republic of China, 5Department of Physics,
National Cheng-Kung University, Tainan, Taiwan 701, Republic of China
###### Abstract
We propose a theoretical framework for analyzing three-body hadronic $B$ meson
decays based on the perturbative QCD approach. The crucial nonperturbative
input is a two-hadron distribution amplitude for final states, whose time-like
form factor and rescattering phase are fit to relevant experimental data.
Together with the short-distance strong phase from the $b$-quark decay kernel,
we are able to make predictions for direct CP asymmetries in, for example, the
$B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}$ and $\pi^{+}\pi^{-}K^{\pm}$ modes, which
are consistent with the LHCb data in various localized regions of phase space.
Applications of our formalism to other three-body hadronic and radiative $B$
meson decays are mentioned.
###### pacs:
13.20.He, 13.25.Hw, 13.30.Eg
Three-body hadronic $B$ meson decays have been studied for many years CL02 ;
CY02 ; FPP04 ; BIL13 . They attracted much attention recently, after the LHCb
Collaboration measured sizable direct CP asymmetries in localized regions of
phase space LHCb1 ; LHCb2 ; LHCb3 , such as
$\displaystyle A_{CP}^{\rm reg}(\pi^{+}\pi^{-}\pi^{+})=0.584\pm 0.082\pm
0.027\pm 0.007,$ (1)
for $m^{2}_{\pi^{+}\pi^{-}\rm high}>15$ GeV2 and $m^{2}_{\pi^{+}\pi^{-}\rm
low}<0.4$ GeV2, and
$\displaystyle A_{CP}^{\rm reg}(\pi^{+}\pi^{-}K^{+})=0.678\pm 0.078\pm
0.032\pm 0.007,$ (2)
for $m^{2}_{K^{+}\pi^{-}\rm high}<15$ GeV2 and $0.08<m^{2}_{\pi^{+}\pi^{-}\rm
low}<0.66$ GeV2. Theoretical attempts to understand these data were made: The
above CP asymmetries were attributed to the interference between a light
scalar and intermediate resonances in ZGY13 ; the relations among the above CP
asymmetries in the U-spin symmetry limit were examined in BGR13 ; SU(3) and
U-spin symmetry breaking effects were included in the amplitude
parametrization in XLH13 ; in CT13 the non-resonant contributions were
parameterized in the framework of heavy meson chiral perturbation theory LLW92
; and the resonant contributions were estimated by means of the usual Breit-
Wigner formalism.
Viewing the experimental progress, it is important to construct a
corresponding framework based on the factorization theorem, in which
perturbative evaluation can be performed systematically with controllable
nonperturbative inputs. Motivated by its theoretical self-consistency and
phenomenological success, we shall generalize the perturbative QCD (PQCD)
approach KLS ; Lu:pqcd to three-body hadronic $B$ meson decays. A direct
evaluation of hard $b$-quark decay kernels, which contain two virtual gluons
at leading order (LO), is not practical because of the enormous number of
diagrams. Besides, the contribution from two hard gluons is power-suppressed
and is not important. In this region all three final-state mesons carry
momenta of $O(m_{B})$, and all three pairs of them have invariant masses of
$O(m_{B}^{2})$, $m_{B}$ being the $B$ meson mass. The dominant contribution
comes from the region, where at least one pair of light mesons has an
invariant mass below $O(\bar{\Lambda}m_{B})$ CL02 ,
$\bar{\Lambda}=m_{B}-m_{b}$ being the $B$ meson and $b$ quark mass difference.
The configuration involves two energetic mesons almost collimating to each
other, in which the dynamics associated with the pair of mesons can be
factorized into a two-meson distribution amplitude $\phi_{h_{1}h_{2}}$ MP . It
is evident that $\phi_{h_{1}h_{2}}$ appropriately describes the
nonperturbative dynamics of a two-meson system in the localized region of
phase space, say, $m^{2}_{\pi^{+}\pi^{-}\rm low}<0.4$ GeV2.
With the introduction of a two-meson distribution amplitude, the LO diagrams
for three-body hadronic $B$ meson decays reduce to those for two-body decays,
as displayed in Figs. 4-4. The PQCD factorization formula for a $B\to
h_{1}h_{2}h_{3}$ decay amplitude is then written as CL02
$\displaystyle\mathcal{A}=\phi_{B}\otimes
H\otimes\phi_{h_{1}h_{2}}\otimes\phi_{h_{3}},$ (3)
where the hard kernel $H$ contains only a single hard gluon. The $B$ meson
($h_{1}$-$h_{2}$ pair, $h_{3}$ meson) distribution amplitude $\phi_{B}$
($\phi_{h_{1}h_{2}}$, $\phi_{h_{3}}$) absorbs nonperturbative dynamics
characterized by the soft scale $\bar{\Lambda}$ (the invariant mass of the
meson pair, the $h_{3}$ meson mass). Figure 4 involves the transition of the
$B$ meson into two light mesons. The amplitude from Fig. 4 is expressed as a
product of a heavy-to-light form factor and a time-like light-light form
factor in the heavy-quark limit. In Figs. 4 and 4, a $B$ meson annihilates
completely, and three light mesons are produced.
Figure 1: Single-pion emission diagrams for the
$B^{+}\to\pi^{+}\pi^{-}\pi^{+}$ decay, where $Ms$ stands for the pion pair.
Figure 2: Two-pion emission diagrams, where $q$ denotes a $u$ or $d$ quark.
Figure 3: Annihilation diagrams.
Figure 4: More annihilation diagrams.
Take Fig. 1(a) for the $B^{+}\to\pi^{+}\pi^{-}\pi^{+}$ decay as an example, in
which the $B^{+}$ meson momentum $p_{B}$, the total momentum $p=p_{1}+p_{2}$
of the pion pair, and the momentum $p_{3}$ of the second $\pi^{+}$ meson are
chosen, in light-cone coordinates, as
$\displaystyle p_{B}=\frac{m_{B}}{\sqrt{2}}(1,1,0_{\rm T}),~{}\quad
p=\frac{m_{B}}{\sqrt{2}}(1,\eta,0_{\rm T}),~{}\quad
p_{3}=\frac{m_{B}}{\sqrt{2}}(0,1-\eta,0_{\rm T}),$ (4)
with the variable $\eta=\omega^{2}/m^{2}_{B}$, $\omega^{2}=p^{2}$ being the
invariant mass squared. The momenta $p_{1}$ and $p_{2}$ of the $\pi^{+}$ and
$\pi^{-}$ mesons in the pair, respectively, have the components
$\displaystyle p^{+}_{1}=\zeta\frac{m_{B}}{\sqrt{2}},\quad
p^{-}_{1}=(1-\zeta)\eta\frac{m_{B}}{\sqrt{2}},\quad
p^{+}_{2}=(1-\zeta)\frac{m_{B}}{\sqrt{2}},\quad
p^{-}_{2}=\zeta\eta\frac{m_{B}}{\sqrt{2}},$ (5)
with the $\pi^{+}$ meson momentum fraction $\zeta$. The momenta of the
spectators in the $B$ meson, the pion pair, and the $\pi^{+}$ meson read,
respectively, as
$\displaystyle k_{B}=\left(0,\frac{m_{B}}{\sqrt{2}}x_{B},k_{B{\rm
T}}\right),\quad k=\left(\frac{m_{B}}{\sqrt{2}}z,0,k_{\rm T}\right),\quad
k_{3}=\left(0,\frac{m_{B}}{\sqrt{2}}(1-\eta)x_{3},k_{3{\rm T}}\right).$ (6)
The definitions of the two-pion distribution amplitudes in terms of hadronic
matrix elements of nonlocal quark operators up to twist 3 can be found in CL02
; MP ; DGP00 . We parameterize them at the leading partial waves as
$\displaystyle\phi^{v,t}_{\pi\pi}(z,\zeta,\omega^{2})=\frac{3F_{\pi,t}(\omega^{2})}{\sqrt{2N_{c}}}z(1-z)(2\zeta-1),$
(7)
$\displaystyle\phi^{s}_{\pi\pi}(z,\zeta,\omega^{2})=\frac{3F_{s}(\omega^{2})}{\sqrt{2N_{c}}}z(1-z),$
(8)
with the number of colors $N_{c}$, where the factor $2\zeta-1$ arises from the
Legendre polynomial $P_{l}(2\zeta-1)$ for $l=1$. The PQCD power counting
indicates the scaling of the vector-current form factor in the asymptotic
region, $F_{\pi}(w^{2})\sim 1/w^{2}$, and the relative importance of the
scalar-current and tensor-current form factors,
$F_{s,t}(w^{2})/F_{\pi}(w^{2})\sim m_{0}^{\pi}/w$, where
$m_{0}^{\pi}=m_{\pi}^{2}/(m_{u}+m_{d})$ is the chiral scale associated with
the pion, $m_{\pi}$, $m_{u}$, and $m_{d}$ being the masses of the pion, the
$u$ quark, and the $d$ quark, respectively. To evaluate the nonresonant
contribution in the arbitrary range of $w^{2}$, we propose the parametrization
for the complex time-like form factors
$\displaystyle
F_{\pi}(w^{2})=\frac{m^{2}\exp[i\delta^{1}_{1}(w)]}{w^{2}+m^{2}},\;\;\;\;F_{t}(w^{2})=\frac{m_{0}^{\pi}m^{2}\exp[i\delta^{1}_{1}(w)]}{w^{3}+m_{0}^{\pi}m^{2}},\;\;\;\;F_{s}(w^{2})=\frac{m_{0}^{\pi}m^{2}\exp[i\delta^{0}_{0}(w)]}{w^{3}+m_{0}^{\pi}m^{2}},$
(9)
in which the parameter $m=1$ GeV is determined by the fit to the experimental
data $m_{J/\psi}^{2}|F_{\pi}(m_{J/\psi}^{2})|^{2}\sim 0.9$ GeV2 PDG ,
$m_{J/\psi}$ being the $J/\psi$ meson mass. The resultant $w^{2}$ dependence
of $F_{\pi}(w^{2})$ also agrees with the low-energy data of the time-like pion
electromagnetic form factor for $w<1$ GeV Whalley2003aaa , and with the next-
to-leading-order (NLO) PQCD calculation HL13 . The strong phases
$\delta^{I}_{l}$ are chosen as the phase shifts for the $S$ wave ($I=0$,
$l=0$) and $P$ wave ($I=1$, $l=1$) of elastic $\pi\pi$ scattering DGP00
according to Watson’s theorem. We simply parameterize the data of these strong
phases Proto1973 ; EM74 ; KS90 for $2m_{\pi}<w<0.7$ GeV as
$\displaystyle\delta^{0}_{0}(w)=\pi(w-2m_{\pi}),\;\;\;\;\delta^{1}_{1}(w)=1.4\pi(w-2m_{\pi})^{2},$
(10)
in which $2m_{\pi}$ represents the $\pi\pi$ threshold. The increase of
$\delta_{1}^{1}$ with $w$ in the above expression is consistent with the NLO
PQCD result of the time-like pion electromagnetic form factor HL13 .
The $B$ meson, pion, and kaon distribution amplitudes are the same as those
widely adopted in the PQCD approach to two-body hadronic $B$ meson decays. We
have the $B$ meson distribution amplitude
$\displaystyle\phi_{B}(x,b)$ $\displaystyle=$ $\displaystyle
N_{B}x^{2}(1-x)^{2}\exp\left[-\frac{1}{2}\left(\frac{xm_{B}}{\omega_{B}}\right)^{2}-\frac{\omega_{B}^{2}b^{2}}{2}\right],$
(11)
with the shape parameter $\omega_{B}=0.45\pm 0.05$ GeV, and the normalization
constant $N_{B}=73.67$ GeV being related to the $B$ meson decay constant
$f_{B}=0.21$ GeV via $\lim_{b\to 0}\int
dx\phi_{B}(x,b)=f_{B}/(2\sqrt{2N_{c}})$. The pion and kaon distribution
amplitudes up to twist 3, $\phi_{i}^{A}(x)$ and $\phi_{i}^{P,T}(x)$ for
$i=\pi,K$, are chosen as refs-pball
$\displaystyle\phi_{i}^{A}(x)$ $\displaystyle=$
$\displaystyle\frac{3f_{i}}{\sqrt{6}}\,x(1-x)\left[1+a_{1}C_{1}^{3/2}(t)+a_{2}C_{2}^{3/2}(t)+a_{4}C_{4}^{3/2}(t)\right],$
(12) $\displaystyle\phi^{P}_{i}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{i}}{2\sqrt{6}}\,\left[1+\left(30\eta_{3}-\frac{5}{2}\rho_{i}^{2}\right)C_{2}^{1/2}(t)-\,3\left\\{\eta_{3}\omega_{3}+\frac{9}{20}\rho_{i}^{2}(1+6a_{2})\right\\}C_{4}^{1/2}(t)\right],$
(13) $\displaystyle\phi^{\sigma}_{i}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{i}}{2\sqrt{6}}\,x(1-x)\left[1+\left(5\eta_{3}-\frac{1}{2}\eta_{3}\omega_{3}-\frac{7}{20}\rho_{i}^{2}-\frac{3}{5}\rho_{i}^{2}a_{2}\right)C_{2}^{3/2}(t)\right],$
(14)
with the pion (kaon) decay constant $f_{\pi}=0.13$ ($f_{K}=0.16$) GeV, the
variable $t=2x-1$, the Gegenbauer polynomials
$\displaystyle C_{1}^{3/2}(t)\,$ $\displaystyle=$ $\displaystyle 3\,t\;,\quad
C_{2}^{1/2}(t)=\frac{1}{2}\left(3\,t^{2}-1\right),\quad
C_{2}^{3/2}(t)\,=\,\frac{3}{2}\left(5\,t^{2}-1\right),$ $\displaystyle
C_{4}^{1/2}(t)\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{8}\left(3-30\,t^{2}+35\,t^{4}\right),\quad
C_{4}^{3/2}(t)\,=\,\frac{15}{8}\left(1-14\,t^{2}+21\,t^{4}\right),$ (15)
and the mass ratio $\rho_{\pi(K)}=m_{\pi(K)}/m_{0}^{\pi(K)}$, where
$m_{0}^{K}=m_{K}^{2}/(m_{s}+m_{d})$ is the chiral scale associated with the
kaon, $m_{K}$ and $m_{s}$ being the masses of the kaon and the $s$ quark,
respectively. The Gegenbauer moments $a^{\pi,K}$ are set to refs-pball
$\displaystyle a_{1}^{\pi}$ $\displaystyle=$ $\displaystyle 0,\quad
a_{1}^{K}=0.06\pm 0.03,\quad a_{2}^{\pi,K}=0.25\pm 0.15,$ $\displaystyle
a_{4}^{\pi}$ $\displaystyle=$
$\displaystyle-0.015,\quad\eta_{3}^{\pi,K}=0.015,\quad\omega_{3}^{\pi,K}=-3.$
(16)
The above set of meson distribution amplitudes corresponds to the $B\to\pi$
transition form factors at maximal recoil
$F_{+}^{B\pi}(0)=F_{0}^{B\pi}(0)=0.23$ in LO PQCD, which are consistent with
the results derived from other approaches refs-pball ; bpi .
The $B^{+}\to\pi^{+}\pi^{-}\pi^{+}$ decay width in the localized region of
$m^{2}_{\pi^{+}\pi^{-}\min}<m^{2}_{\min}=0.4$ GeV2 and
$m^{2}_{\pi^{+}\pi^{-}\max}>m^{2}_{\max}=15$ GeV2 is written as
$\displaystyle\Gamma=\frac{G_{F}^{2}m_{B}}{512\pi^{4}}\int_{\eta_{\min}}^{\eta_{\max}}d\eta(1-\eta)\int_{0}^{\zeta_{\max}}d\zeta|\mathcal{A}|^{2},$
(17)
with the Fermi constant $G_{F}=1.16639^{-5}$ GeV-2 and the bounds
$\displaystyle\eta_{\max}=\frac{m^{2}_{\min}}{m_{B}^{2}},\;\;\;\;\eta_{\min}=\frac{4m^{2}_{\pi}}{m_{B}^{2}},\;\;\;\;\zeta_{\max}=1-\frac{m^{2}_{\max}}{(1-\eta)m_{B}^{2}},$
(18)
where the upper bound $\zeta_{\max}$ is derived from the invariant mass
squared $(p_{2}+p_{3})^{2}$. The contributions from all the diagrams in Figs.
4-4 to the decay amplitude $\mathcal{A}$ are collected in the Appendix. The
corresponding formulas for the $B^{+}\to\pi^{+}\pi^{-}K^{+}$ decay can be
obtained straightforwardly.
Employing the input parameters $\Lambda^{(f=4)}_{\overline{MS}}=0.25$ GeV,
$m_{\pi^{\pm}}=0.1396$ GeV, $m_{K^{\pm}}=0.4937$ GeV, $m_{B^{\pm}}=5.279$ GeV
PDG ; prd76-074018 , and the Wolfenstein parameters in PDG , we derive the
direct CP asymmetries in the region of $m^{2}_{\pi^{+}\pi^{-}{\rm low}}<0.4$
GeV2 and $m^{2}_{\pi^{+}\pi^{-}\;{\rm or}\;K^{+}\pi^{-}{\rm high}}>15$ GeV2,
$\displaystyle A_{CP}(B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm})$ $\displaystyle=$
$\displaystyle
0.519^{+0.124}_{-0.219}(\omega_{B})^{+0.108}_{-0.091}(a_{2}^{\pi})^{+0.027}_{-0.032}(m^{\pi}_{0}),$
(19) $\displaystyle A_{CP}(B^{\pm}\to\pi^{+}\pi^{-}K^{\pm})$ $\displaystyle=$
$\displaystyle-0.018^{+0.024}_{-0.044}(\omega_{B})^{+0.006}_{-0.009}(a_{2}^{\pi}\;\&\;a_{2}^{K})^{+0.002}_{-0.003}(m_{0}^{\pi}\;\&\;m_{0}^{K}).$
(20)
The first and second errors come from the variation of $\omega_{B}=0.45\pm
0.05$ GeV and $a_{2}^{\pi,K}=0.25\pm 0.15$, respectively, and the third errors
are induced by $m_{0}^{\pi}=1.4\pm 0.1$ GeV and $m_{0}^{K}=1.6\pm 0.1$ GeV.
The uncertainties caused by the variation of the Wolfenstein parameters
$\lambda,A,\rho,\eta$, and of the Gegenbauer moment $a^{K}_{1}=0.06\pm 0.03$
are very small, and have been neglected. While the decay widths are
quadratically proportional to the decay constants $f_{B},f_{\pi}$ and/or
$f_{K}$, the CP asymmetries are independent of them.
Obviously, our prediction for $A_{CP}(B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm})$
agrees well with the LHCb data. Since the emission contribution and the
imaginary annihilation contribution depend on the $B$ meson distribution
amplitude in different ways, the variation of $\omega_{B}$ explores the
relevance of the short-distance strong phase from the $b$-quark decay kernel.
The sensitivity of the predicted CP asymmetries to $\omega_{B}$ then implies
the importance of this strong phase. As the $P$-wave rescattering phase
associated with the pion electromagnetic form factor decreases to half, the
predicted CP asymmetries are also reduced by half. The change of the phases
associated with the scalar and tensor form factors does not modify the CP
asymmetries much. Therefore, we conclude that the short-distance and long-
distance $P$-wave strong phases are equally crucial for the direct CP
asymmetries in the localized region of phase space. The LHCb data in Eq. (2)
are dominated by the resonant channel $B^{\pm}\to\rho^{0}K^{\pm}$. It is
encouraging that the data confirm the NLO PQCD prediction
$A_{CP}(B^{\pm}\to\rho^{0}K^{\pm})=0.71^{+0.25}_{-0.35}$ LM06 . We have
checked that our prediction in Eq. (20) for the localized region of phase
space is consistent with the LHCb data in Fig. 2 of LHCb1 . Moreover, we have
predicted larger $A_{CP}(B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm})=0.631$ in the
region of $m^{2}_{\pi^{+}\pi^{-}{\rm low}}<0.4$ GeV2 and
$m^{2}_{\pi^{+}\pi^{-}{\rm high}}>20.5$ GeV2 for the central values of the
input parameters, which also matches the data LHCb2 .
In this paper we have proposed a promising formalism for three-body hadronic
$B$ meson decays based on the PQCD approach. The calculation is greatly
simplified with the introduction of the nonperturbative two-hadron
distribution amplitude for final states. The time-like form factors and the
rescattering phases involved in the two-pion distribution amplitudes have been
fixed by experiments, and the $B$ meson, pion, and kaon distribution
amplitudes are the same as in the previous PQCD analysis of two-body hadronic
$B$ meson decays. Without any free parameters, our results for
$A_{CP}(B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm})$ and
$A_{CP}(B^{\pm}\to\pi^{+}\pi^{-}K^{\pm})$ accommodate well the recent LHCb
data in various localized regions of phase space. It has been observed that
the short-distance strong phase from the $b$-quark decay kernel and the final-
state rescattering phase are equally important for explaining the measured
direct CP asymmetries. The success indicates that our formalism has potential
applications to other three-body hadronic and radiative $B$ meson decays CL04
, if phase shifts from meson-meson scattering can be derived in
nonperturbative methods LZL ; Doring:2013wka .
###### Acknowledgements.
We thank Wei Wang for helpful discussions. This work was partly supported by
the National Science Council of R.O.C. under Grant No.
NSC-101-2112-M-001-006-MY3, by the National Center for Theoretical Sciences of
R.O.C., and by the National Science Foundation of China under Grants No.
11375208, No. 11228512 and No. 11235005.
## Appendix A Decay amplitudes
In this appendix we present the PQCD factorization formulas for the diagrams
in Figs. 4-4. The sum of the contributions from Figs. 4(a) and 4(b) gives
$\displaystyle\mathcal{A}_{1(a,b)}$ $\displaystyle=$ $\displaystyle
V^{*}_{ub}V_{ud}F^{LL}_{B\to\pi\pi}-V^{*}_{tb}V_{td}\left(F^{\prime
LL}_{B\to\pi\pi}+F^{SP}_{B\to\pi\pi}\right),$ (21)
where the amplitudes for the $B$ meson transition into two pions are written
as
$\displaystyle F^{LL}_{B\to\pi\pi}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{\pi}\int dx_{B}dz\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})(1-\eta)$ (22)
$\displaystyle\times\bigg{\\{}\left[\sqrt{\eta}(1-2z)(\phi_{s}+\phi_{t})+(1+z)\phi_{v}\right]a_{1}(t_{1a})E_{1ab}(t_{1a})h_{1a}(x_{B},z,b_{B},b)$
$\displaystyle+\sqrt{\eta}\left(2\phi_{s}-\sqrt{\eta}\phi_{v}\right)a_{1}(t_{1b})E_{1ab}(t_{1b})h_{1b}(x_{B},z,b_{B},b)\bigg{\\}},$
$\displaystyle F^{\prime LL}_{B\to\pi\pi}$ $\displaystyle=$ $\displaystyle
F^{LL}_{B\to\pi\pi}|_{a_{1}\to a_{3}}$ (23) $\displaystyle
F^{SP}_{B\to\pi\pi}$ $\displaystyle=$ $\displaystyle-16\pi
C_{F}m^{4}_{B}rf_{\pi}\int dx_{B}dz\int b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$
(24)
$\displaystyle\times\bigg{\\{}\left[\sqrt{\eta}(2+z)\phi_{s}-\sqrt{\eta}z\phi_{t}+(1+\eta(1-2z))\phi_{v}\right]a_{5}(t_{1a})E_{1ab}(t_{1a})h_{1a}(x_{B},z,b_{B},b)$
$\displaystyle+\left[2\sqrt{\eta}(1-x_{B}+\eta)\phi_{s}+(x_{B}-2\eta)\phi_{v}\right]a_{5}(t_{1b})E_{1ab}(t_{1b})h_{1b}(x_{B},z,b_{B},b)\bigg{\\}},$
with $r=m^{\pi}_{0}/m_{B}$ and
$\phi_{s,t,v}\equiv\phi_{s,t,v}(z,\zeta,\omega^{2})$. The Wilson coefficients
in the above expressions are defined as $a_{1}=C_{1}/N_{c}+C_{2}$,
$a_{3}=C_{3}/N_{c}+C_{4}+C_{9}/N_{c}+C_{10}$, and
$a_{5}=C_{5}/N_{c}+C_{6}+C_{7}/N_{c}+C_{8}$. The spectator diagrams in Figs.
4(c) and 4(d) lead to
$\displaystyle\mathcal{A}_{1(c,d)}=V^{*}_{ub}V_{ud}M^{LL}_{B\to\pi\pi}-V^{*}_{tb}V_{td}\left(M^{\prime
LL}_{B\to\pi\pi}+M^{LR}_{B\to\pi\pi}\right),$ (25)
with the amplitudes
$\displaystyle M^{LL}_{B\to\pi\pi}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})\phi^{A}_{\pi}(1-\eta)$ (26)
$\displaystyle\times\bigg{\\{}\left[\sqrt{\eta}z(\phi_{s}+\phi_{t})+((1-\eta)(1-x_{3})-x_{B}+z\eta)\phi_{v}\right]C_{1}(t_{1c})E_{1cd}(t_{1c})h_{1c}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle-\left[z(\sqrt{\eta}(\phi_{s}-\phi_{t})+\phi_{v})+(x_{3}(1-\eta)-x_{B})\phi_{v}\right]C_{1}(t_{1d})E_{1cd}(t_{1d})h_{1d}(x_{B},z,x_{3},b_{B},b_{3})\bigg{\\}},$
$\displaystyle M^{\prime LL}_{B\to\pi\pi}$ $\displaystyle=$ $\displaystyle
M^{LL}_{B\to\pi\pi}|_{C_{1}\to a_{9}}$ (27) $\displaystyle
M^{LR}_{B\to\pi\pi}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}rm^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})$ (28)
$\displaystyle\times\bigg{\\{}\bigg{[}\sqrt{\eta}z(\phi^{P}_{\pi}+\phi^{T}_{\pi})(\phi_{s}-\phi_{t})+\sqrt{\eta}((1-x_{3})(1-\eta)-x_{B})(\phi^{P}_{\pi}-\phi^{T}_{\pi})$
$\displaystyle\times(\phi_{s}+\phi_{t})-((1-x_{3})(1-\eta)-x_{B})(\phi^{P}_{\pi}-\phi^{T}_{\pi})\phi_{v}-\eta
z(\phi^{P}_{\pi}+\phi^{T}_{\pi})\phi_{v}\bigg{]}$ $\displaystyle\times
a_{7}(t_{1c})E_{1cd}(t_{1c})h_{1c}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle+\big{[}\sqrt{\eta}z(\phi^{P}_{\pi}-\phi^{T}_{\pi})((\phi_{t}-\phi_{s})+\sqrt{\eta}\phi_{v})+(x_{B}-x_{3}(1-\eta))(\phi^{P}_{\pi}+\phi^{T}_{\pi})$
$\displaystyle\times(\sqrt{\eta}(\phi_{s}+\phi_{t})-\phi_{v})\big{]}a_{7}(t_{1d})E_{1cd}(t_{1d})h_{1d}(x_{B},z,x_{3},b_{B},b_{3})\bigg{\\}},$
and the Wilson coefficients $a_{7}=C_{5}+C_{7}$ and $a_{9}=C_{3}+C_{9}$.
For Figs. 4(a) and 4(b), we have
$\displaystyle\mathcal{A}^{q=u}_{2(a,b)}$ $\displaystyle=$ $\displaystyle
V^{*}_{ub}V_{ud}F^{LL}_{B\to\pi}-V^{*}_{tb}V_{td}\left(F^{\prime
LL}_{B\to\pi}+F^{LR}_{B\to\pi}\right),$ (29)
$\displaystyle\mathcal{A}^{q=d}_{2(a,b)}$ $\displaystyle=$
$\displaystyle-V^{*}_{tb}V_{td}\left(F^{\prime\prime LL}_{B\to\pi}+F^{\prime
LR}_{B\to\pi}+F^{SP}_{B\to\pi}\right).$ (30)
The amplitudes involving the $B\to\pi$ transition form factors are expressed
as
$\displaystyle F^{LL}_{B\to\pi}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}F_{\pi}(\omega^{2})\int dx_{B}dx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})(2\zeta-1)$ (31)
$\displaystyle\times\bigg{\\{}\left[(1+x_{3}(1-\eta))(1-\eta)\phi^{A}_{\pi}+r(1-2x_{3})(1-\eta)\phi^{P}_{\pi}+r(1+\eta-2x_{3}(1-\eta))\phi^{T}_{\pi}\right]$
$\displaystyle\times
a_{2}(t_{2a})E_{2ab}(t_{2a})h_{2a}(x_{B},x_{3},b_{B},b_{3})$
$\displaystyle+\left[x_{B}(1-\eta)\eta\phi^{A}_{\pi}+2r(1-\eta(1+x_{B}))\phi^{P}_{\pi}\right]a_{2}(t_{2b})E_{2ab}(t_{2b})h_{2b}(x_{B},x_{3},b_{B},b_{3})\bigg{\\}},$
$\displaystyle F^{LR}_{B\to\pi}$ $\displaystyle=$ $\displaystyle
F^{LL}_{B\to\pi}|_{a_{2}\to a_{6}},$ (32) $\displaystyle F^{\prime
LL}_{B\to\pi}$ $\displaystyle=$ $\displaystyle F^{LL}_{B\to\pi}|_{a_{2}\to
a_{4}},$ (33) $\displaystyle F^{\prime LR}_{B\to\pi}$ $\displaystyle=$
$\displaystyle F^{LL}_{B\to\pi}|_{a_{2}\to a_{8}},$ (34) $\displaystyle
F^{\prime\prime LL}_{B\to\pi}$ $\displaystyle=$ $\displaystyle
F^{LL}_{B\to\pi}|_{a_{2}\to a_{10}},$ (35) $\displaystyle F^{SP}_{B\to\pi}$
$\displaystyle=$ $\displaystyle 16\pi
C_{F}m^{4}_{B}\sqrt{\eta}F_{\pi}(\omega^{2})\int dx_{B}dx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})$ (36)
$\displaystyle\times\bigg{\\{}\left[(1-\eta)\phi^{A}_{\pi}+r(2+x_{3}(1-\eta))\phi^{P}_{\pi}-rx_{3}(1-\eta)\phi^{T}_{\pi}\right]a_{8}^{\prime}(t_{2a})E_{2ab}(t_{2a})h_{2a}(x_{B},x_{3},b_{B},b_{3})$
$\displaystyle+\left[x_{B}(1-\eta)\phi^{A}_{\pi}+2r(1-x_{B}-\eta)\phi^{P}_{\pi}\right]a_{8}^{\prime}(t_{2b})E_{2ab}(t_{2b})h_{2b}(x_{B},x_{3},b_{B},b_{3})\bigg{\\}},$
in which the Wilson coefficients are given by $a_{2}=C_{1}+C_{2}/N_{c}$,
$a_{4}=C_{3}+C_{4}/N_{c}+C_{9}+C_{10}/N_{c}$,
$a_{6}=C_{5}+C_{6}/N_{c}+C_{7}+C_{8}/N_{c}$,
$a_{8}=C_{5}+C_{6}/N_{c}-C_{7}/2-C_{8}/(2N_{c})$,
$a_{8}^{\prime}=C_{5}/N_{c}+C_{6}-C_{7}/(2N_{c})-C_{8}/2$, and
$a_{10}=\left[C_{3}+C_{4}-C_{9}/2-C_{10}/2\right](N_{c}+1)/N_{c}$. We derive
from Figs. 4(c) and 4(d)
$\displaystyle\mathcal{A}^{q=u}_{2(c,d)}$ $\displaystyle=$ $\displaystyle
V^{*}_{ub}V_{ud}M^{LL}_{B\to\pi}-V^{*}_{tb}V_{td}\left(M^{\prime
LL}_{B\to\pi}+M^{SP}_{B\to\pi}\right),$ (37)
$\displaystyle\mathcal{A}^{q=d}_{2(c,d)}$ $\displaystyle=$
$\displaystyle-V^{*}_{tb}V_{td}\left(M^{\prime\prime
LL}_{B\to\pi}+M^{LR}_{B\to\pi}+M^{\prime SP}_{B\to\pi}\right),$ (38)
with the amplitudes
$\displaystyle M^{LL}_{B\to\pi}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})\phi_{v}$ (39)
$\displaystyle\times\bigg{\\{}\big{[}(1-x_{B}-z)(1-\eta^{2})\phi^{A}_{\pi}+rx_{3}(1-\eta)(\phi^{P}_{\pi}-\phi^{T}_{\pi})+r(x_{B}+z)\eta(\phi^{P}_{\pi}+\phi^{T}_{\pi})$
$\displaystyle-2r\eta\phi^{P}_{\pi}\big{]}C_{2}(t_{2c})E_{2cd}(t_{2c})h_{2c}(x_{B},z,x_{3},b_{B},b)$
$\displaystyle-\left[(z-x_{B}+x_{3}(1-\eta))(1-\eta)\phi^{A}_{\pi}+r(x_{B}-z)\eta(\phi^{P}_{\pi}-\phi^{T}_{\pi})-rx_{3}(1-\eta)(\phi^{P}_{\pi}+\phi^{T}_{\pi})\right]$
$\displaystyle\times
C_{2}(t_{2d})E_{2cd}(t_{2d})h_{2d}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$
$\displaystyle M^{LR}_{B\to\pi}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}\sqrt{\eta}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})$ (40)
$\displaystyle\times\bigg{\\{}\big{[}(1-x_{B}-z)(1-\eta)(\phi_{s}+\phi_{t})\phi^{A}_{\pi}+r(1-x_{B}-z)(\phi_{s}+\phi_{t})(\phi^{P}_{\pi}-\phi^{T}_{\pi})$
$\displaystyle+r(x_{3}(1-\eta)+\eta)(\phi_{s}-\phi_{t})(\phi^{P}_{\pi}+\phi^{T}_{\pi})\big{]}a_{5}^{\prime}(t_{2c})E_{2cd}(t_{2c})h_{2c}(x_{B},z,x_{3},b_{B},b)$
$\displaystyle-\big{[}(z-x_{B})(1-\eta)(\phi_{s}-\phi_{t})\phi^{A}_{\pi}+r(z-x_{B})(\phi_{s}-\phi_{t})(\phi^{P}_{\pi}-\phi^{T}_{\pi})$
$\displaystyle+rx_{3}(1-\eta)(\phi_{s}+\phi_{t})(\phi^{P}_{\pi}+\phi^{T}_{\pi})\big{]}a_{5}^{\prime}(t_{2d})E_{2cd}(t_{2d})h_{2d}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$
$\displaystyle M^{SP}_{B\to\pi}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}bdb\phi_{B}(x_{B},b_{B})\phi_{v}$ (41)
$\displaystyle\times\bigg{\\{}\big{[}(1+\eta-
x_{B}-z+x_{3}(1-\eta))(1-\eta)\phi^{A}_{\pi}+r\eta(x_{B}+z)(\phi^{P}_{\pi}-\phi^{T}_{\pi})$
$\displaystyle-
rx_{3}(1-\eta)(\phi^{P}_{\pi}+\phi^{T}_{\pi})-2r\eta\phi^{P}_{\pi}\big{]}a_{6}^{\prime}(t_{2c})E_{2cd}(t_{2c})h_{2c}(x_{B},z,x_{3},b_{B},b)$
$\displaystyle-\left[(z-x_{B})(1-\eta^{2})\phi^{A}_{\pi}-rx_{3}(1-\eta)(\phi^{P}_{\pi}-\phi^{T}_{\pi})+r(x_{B}-z)\eta(\phi^{P}_{\pi}+\phi^{T}_{\pi})\right]$
$\displaystyle\times
a_{6}^{\prime}(t_{2d})E_{2cd}(t_{2d})h_{2d}(x_{B},z,x_{3},b_{B},b)\bigg{\\}},$
$\displaystyle M^{\prime LL}_{B\to\pi}$ $\displaystyle=$ $\displaystyle
M^{LL}_{B\to\pi}|_{C_{2}\to a_{4}^{\prime}},$ (42) $\displaystyle
M^{\prime\prime LL}_{B\to\pi}$ $\displaystyle=$ $\displaystyle
M^{LL}_{B\to\pi}|_{C_{2}\to a_{10}^{\prime}},$ (43) $\displaystyle M^{\prime
SP}_{B\to\pi}$ $\displaystyle=$ $\displaystyle
M^{SP}_{B\to\pi}|_{a_{6}^{\prime}\to a_{6}^{\prime\prime}},$ (44)
where the Wilson coefficients are defined as $a_{4}^{\prime}=C_{4}+C_{10}$,
$a_{5}^{\prime}=C_{5}-C_{7}/2$, $a_{6}^{\prime}=C_{6}+C_{8}$,
$a_{6}^{\prime\prime}=C_{6}-C_{8}/2$, and
$a_{10}^{\prime}=C_{3}+C_{4}-C_{9}/2-C_{10}/2$.
The factorizable annihilation diagrams in Figs. 4(a) and 4(b) lead to
$\displaystyle\mathcal{A}_{3(a,b)}$ $\displaystyle=$ $\displaystyle
V^{*}_{ub}V_{ud}F^{LL}_{a\pi}-V^{*}_{tb}V_{td}\left(F^{\prime
LL}_{a\pi}+F^{SP}_{a\pi}\right),$ (45)
with the three-pion production amplitudes
$\displaystyle F^{LL}_{a\pi}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{B}\int dzdx_{3}\int bdbb_{3}db_{3}$ (46)
$\displaystyle\times\bigg{\\{}\left[(x_{3}(1-\eta)-1)(1-\eta)\phi^{A}_{\pi}\phi_{v}+2r\sqrt{\eta}(x_{3}(1-\eta)(\phi^{P}_{\pi}-\phi^{T}_{\pi})-2\phi^{P}_{\pi})\phi_{s}\right]$
$\displaystyle\times a_{1}(t_{3a})E_{3ab}(t_{3a})h_{3a}(z,x_{3},b,b_{3})$
$\displaystyle+\left[z(1-\eta)\phi^{A}_{\pi}\phi_{v}+2r\sqrt{\eta}\phi^{P}_{\pi}((1-\eta)(\phi_{s}-\phi_{t})+z(\phi_{s}+\phi_{t}))\right]$
$\displaystyle\times
a_{1}(t_{3b})E_{3ab}(t_{3b})h_{3b}(z,x_{3},b,b_{3})\bigg{\\}},$ $\displaystyle
F^{\prime LL}_{a\pi}$ $\displaystyle=$ $\displaystyle F^{LL}_{a\pi}|_{a_{1}\to
a_{3}},$ (47) $\displaystyle F^{SP}_{a\pi}$ $\displaystyle=$ $\displaystyle
16\pi C_{F}m^{4}_{B}f_{B}\int dzdx_{3}\int bdbb_{3}db_{3}$ (48)
$\displaystyle\times\bigg{\\{}\left[2\sqrt{\eta}(1-\eta)\phi^{A}_{\pi}\phi_{s}+r(1-x_{3})(\phi^{P}_{\pi}+\phi^{T}_{\pi})\phi_{v}+r\eta((1+x_{3})\phi^{P}_{\pi}-(1-x_{3})\phi^{T}_{\pi})\phi_{v}\right]$
$\displaystyle\times a_{5}(t_{3a})E_{3ab}(t_{3a})h_{3a}(z,x_{3},b,b_{3})$
$\displaystyle+\left[2r(1-\eta)\phi^{P}_{\pi}\phi_{v}+z\sqrt{\eta}((1-\eta)\phi^{A}_{\pi}(\phi_{s}-\phi_{t})+2r\sqrt{\eta}\phi^{P}_{\pi}\phi_{v})\right]$
$\displaystyle\times
a_{5}(t_{3b})E_{3ab}(t_{3b})h_{3b}(z,x_{3},b,b_{3})\bigg{\\}}.$
The nonfactorizable annihilation diagrams in Figs. 4(c) and 4(d) give
$\displaystyle\mathcal{A}_{3(c,d)}=V^{*}_{ub}V_{ud}M^{LL}_{a\pi}-V^{*}_{tb}V_{td}\left(M^{\prime
LL}_{a\pi}+M^{LR}_{a\pi}\right),$ (49)
with the amplitudes
$\displaystyle M^{LL}_{a\pi}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})$ (50)
$\displaystyle\times\bigg{\\{}\big{[}(1-\eta)(\eta-(1+\eta)(x_{B}+z))\phi^{A}_{\pi}\phi_{v}+r\sqrt{\eta}(x_{3}(1-\eta)+\eta)(\phi^{P}_{\pi}+\phi^{T}_{\pi})(\phi_{s}-\phi_{t})$
$\displaystyle-r\sqrt{\eta}(1-x_{B}-z)(\phi^{P}_{\pi}-\phi^{T}_{\pi})(\phi_{s}+\phi_{t})+4r\sqrt{\eta}\phi^{P}_{\pi}\phi_{s}\big{]}C_{1}(t_{3c})E_{3cd}(t_{3c})h_{3c}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle+\big{[}(1-\eta)(1-x_{3}(1-\eta)-\eta(1+x_{B}-z))\phi^{A}_{\pi}\phi_{v}-r\sqrt{\eta}(x_{B}-z)(\phi^{P}_{\pi}+\phi^{T}_{\pi})(\phi_{s}-\phi_{t})$
$\displaystyle+r\sqrt{\eta}(1-\eta)(1-x_{3})(\phi^{P}_{\pi}-\phi^{T}_{\pi})(\phi_{s}+\phi_{t})\big{]}C_{1}(t_{3d})E_{3cd}(t_{3d})h_{3d}(x_{B},z,x_{3},b_{B},b_{3})\bigg{\\}},$
$\displaystyle M^{\prime LL}_{a\pi}$ $\displaystyle=$ $\displaystyle
M^{LL}_{a\pi}|_{C_{1}\to a_{9}},$ (51) $\displaystyle M^{LR}_{a\pi}$
$\displaystyle=$ $\displaystyle 32\pi C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int
dx_{B}dzdx_{3}\int b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})$ (52)
$\displaystyle\times\bigg{\\{}\big{[}\sqrt{\eta}(1-\eta)(2-x_{B}-z)\phi^{A}_{\pi}(\phi_{s}+\phi_{t})-r(1+x_{3})(\phi^{P}_{\pi}-\phi^{T}_{\pi})\phi_{v}$
$\displaystyle-r\eta[(1-x_{B}-z)(\phi^{P}_{\pi}+\phi^{T}_{\pi})-x_{3}(\phi^{P}_{\pi}-\phi^{T}_{\pi})+2\phi^{P}_{\pi}]\phi_{v}\big{]}a_{7}(t_{3c})E_{3cd}(t_{3c})h_{3c}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle-\big{[}r(1-\eta)(1-x_{3})(\phi^{P}_{\pi}-\phi^{T}_{\pi})\phi_{v}-\sqrt{\eta}(x_{B}-z)[r\sqrt{\eta}(\phi^{P}_{\pi}+\phi^{T}_{\pi})\phi_{v}$
$\displaystyle-(1-\eta)\phi^{A}_{\pi}(\phi_{s}+\phi_{t})]\big{]}a_{7}(t_{3d})E_{3cd}(t_{3d})h_{3d}(x_{B},z,x_{3},b_{B},b_{3})\bigg{\\}}.$
Similarly, we derive from Figs. 4(a) and 4(b)
$\displaystyle\mathcal{A}_{4(a,b)}$ $\displaystyle=$ $\displaystyle
V^{*}_{ub}V_{ud}F^{LL}_{a\pi\pi}-V^{*}_{tb}V_{td}\left(F^{\prime
LL}_{a\pi\pi}+F^{SP}_{a\pi\pi}\right),$ (53)
with the three-pion production amplitudes
$\displaystyle F^{LL}_{a\pi\pi}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m^{4}_{B}f_{B}\int dzdx_{3}\int bdbb_{3}db_{3}$ (54)
$\displaystyle\times\bigg{\\{}\left[2r\sqrt{\eta}\phi^{P}_{\pi}((2-z)\phi_{s}+z\phi_{t})-(1-\eta)(1-z)\phi^{A}_{\pi}\phi_{v}\right]a_{1}(t_{4a})E_{4ab}(t_{4a})h_{4a}(z,x_{3},b,b_{3})$
$\displaystyle+\big{[}2r\sqrt{\eta}[(1-x_{3})(1-z)\phi^{T}_{\pi}-(1+x_{3}+(1-x_{3})\eta)\phi^{P}_{\pi}]\phi_{s}$
$\displaystyle+(x_{3}(1-\eta)+\eta)(1-\eta)\phi^{A}_{\pi}\phi_{v}\big{]}a_{1}(t_{4b})E_{4ab}(t_{4b})h_{4b}(z,x_{3},b,b_{3})\bigg{\\}},$
$\displaystyle F^{\prime LL}_{a\pi\pi}$ $\displaystyle=$ $\displaystyle
F^{LL}_{a\pi\pi}|_{a_{1}\to a_{3}}$ (55) $\displaystyle F^{SP}_{a\pi\pi}$
$\displaystyle=$ $\displaystyle 16\pi C_{F}m^{4}_{B}f_{B}\int dzdx_{3}\int
bdbb_{3}db_{3}$ (56)
$\displaystyle\times\bigg{\\{}\left[\sqrt{\eta}(1-\eta)(1-z)\phi^{A}_{\pi}(\phi_{s}+\phi_{t})-2r(1+(1-z)\eta)\phi^{P}_{\pi}\phi_{v}\right]$
$\displaystyle\times a_{5}(t_{4a})E_{4ab}(t_{4a})h_{4a}(z,x_{3},b,b_{3})$
$\displaystyle+\left[2\sqrt{\eta}(1-\eta)\phi^{A}_{\pi}\phi_{s}-r(2\eta+x_{3}(1-\eta))\phi^{P}_{\pi}\phi_{v}+rx_{3}(1-\eta)\phi^{T}_{\pi}\phi_{v}\right]$
$\displaystyle\times
a_{5}(t_{4b})E_{4ab}(t_{4b})h_{4b}(z,x_{3},b,b_{3})\bigg{\\}},$
and from Figs. 4(c) and 4(d)
$\displaystyle\mathcal{A}_{4(c,d)}=V^{*}_{ub}V_{ud}M^{LL}_{a\pi\pi}-V^{*}_{tb}V_{td}\left(M^{\prime
LL}_{a\pi\pi}+M^{LR}_{a\pi\pi}\right),$ (57)
with the amplitudes
$\displaystyle M^{LL}_{a\pi\pi}$ $\displaystyle=$ $\displaystyle 32\pi
C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int dx_{B}dzdx_{3}\int
b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})$ (58)
$\displaystyle\times\bigg{\\{}\big{[}(\eta-1)[x_{3}(1-\eta)+x_{B}+\eta(1-z)]\phi^{A}_{\pi}\phi_{v}+r\sqrt{\eta}(x_{3}(1-\eta)+x_{B}+\eta)(\phi^{P}_{\pi}+\phi^{T}_{\pi})$
$\displaystyle\times(\phi_{s}-\phi_{t})+r\sqrt{\eta}(1-z)(\phi^{P}_{\pi}-\phi^{T}_{\pi})(\phi_{s}+\phi_{t})+2r\sqrt{\eta}(\phi^{P}_{\pi}\phi_{s}+\phi^{T}_{\pi}\phi_{t})\big{]}$
$\displaystyle\times
C_{1}(t_{4c})E_{4cd}(t_{4c})h_{4c}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle+\big{[}(1-\eta^{2})(1-z)\phi^{A}_{\pi}\phi_{v}+r\sqrt{\eta}(x_{B}-x_{3}(1-\eta)-\eta)(\phi^{P}_{\pi}-\phi^{T}_{\pi})(\phi_{s}+\phi_{t})$
$\displaystyle-r\sqrt{\eta}(1-z)(\phi^{P}_{\pi}+\phi^{T}_{\pi})(\phi_{s}-\phi_{t})\big{]}C_{1}(t_{4d})E_{4cd}(t_{4d})h_{4d}(x_{B},z,x_{3},b_{B},b_{3})\bigg{\\}},$
$\displaystyle M^{\prime LL}_{a\pi\pi}$ $\displaystyle=$ $\displaystyle
M^{LL}_{a\pi\pi}|_{C_{1}\to a_{9}},$ (59) $\displaystyle M^{LR}_{a\pi\pi}$
$\displaystyle=$ $\displaystyle-32\pi C_{F}m^{4}_{B}/\sqrt{2N_{c}}\int
dx_{B}dzdx_{3}\int b_{B}db_{B}b_{3}db_{3}\phi_{B}(x_{B},b_{B})$ (60)
$\displaystyle\times\bigg{\\{}\big{[}\sqrt{\eta}(1-\eta)(1+z)\phi^{A}_{\pi}(\phi_{s}-\phi_{t})+r(2-x_{B}-x_{3}(1-\eta))(\phi^{P}_{\pi}+\phi^{T}_{\pi})\phi_{v}$
$\displaystyle+r\eta(z\phi^{P}_{\pi}-(2+z)\phi^{T}_{\pi})\phi_{v}\big{]}a_{7}(t_{4c})E_{4cd}(t_{4c})h_{4c}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle+\big{[}\sqrt{\eta}(1-\eta)(1-z)\phi^{A}_{\pi}(\phi_{s}-\phi_{t})+r(x_{3}(1-\eta)-x_{B})(\phi^{P}_{\pi}+\phi^{T}_{\pi})\phi_{v}$
$\displaystyle+r\eta((2-z)\phi^{P}_{\pi}+z\phi^{T}_{\pi})\phi_{v}\big{]}a_{7}(t_{4d})E_{4cd}(t_{4d})h_{4d}(x_{B},z,x_{3},b_{B},b_{3})\bigg{\\}}.$
The threshold resummation factor $S_{t}(x)$ follows the parametrization in
prd65-014007
$\displaystyle
S_{t}(x)=\frac{2^{1+2c}\Gamma(3/2+c)}{\sqrt{\pi}\Gamma(1+c)}[x(1-x)]^{c},$
(61)
in which the parameter is set to $c=0.3$. The hard functions are written as
$\displaystyle h_{1a}(x_{B},z,b_{B},b)$ $\displaystyle=$ $\displaystyle
K_{0}(m_{B}\sqrt{x_{B}z}b_{B})\big{[}\theta(b_{B}-b)K_{0}(m_{B}\sqrt{z}b_{B})I_{0}(m_{B}\sqrt{z}b)+(b\leftrightarrow
b_{B})\big{]}S_{t}(z),$ $\displaystyle h_{1b}(x_{B},z,b_{B},b)$
$\displaystyle=$ $\displaystyle K_{0}(m_{B}\sqrt{x_{B}z}b_{2})S_{t}(x_{B})$
(64)
$\displaystyle\times\left\\{\begin{array}[]{ll}\frac{i\pi}{2}\left[\theta(b-b_{B})H_{0}^{(1)}(m_{B}\sqrt{\eta-
x_{B}}b)J_{0}(m_{B}\sqrt{\eta-x_{B}}b_{B})+(b\leftrightarrow
b_{B})\right],~{}x_{B}<\eta,\\\
\left[\theta(b-b_{B})K_{0}(m_{B}\sqrt{x_{B}-\eta}b)I_{0}(m_{B}\sqrt{x_{B}-\eta}b_{B})+(b\leftrightarrow
b_{B})\right],\quad\quad~{}~{}x_{B}\geq\eta,\\\ \end{array}\right.$
$\displaystyle h_{1c}(x_{B},z,x_{3},b_{B},b_{3})$ $\displaystyle=$
$\displaystyle\big{[}\theta(b_{B}-b_{3})K_{0}(m_{B}\sqrt{x_{B}z}b_{B})I_{0}(m_{B}\sqrt{x_{B}z}b_{3})+(b_{B}\leftrightarrow
b_{3})\big{]}$ (67)
$\displaystyle\times\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{z[(1-\eta)(1-x_{3})-x_{B}]}b_{3}),~{}\quad\quad(1-\eta)(1-x_{3})>x_{B},\\\
K_{0}(m_{B}\sqrt{z[x_{B}-(1-\eta)(1-x_{3})]}b_{3}),~{}~{}~{}\quad\quad\quad(1-\eta)(1-x_{3})\leq
x_{B},\end{array}\right.$ $\displaystyle h_{1d}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle=$
$\displaystyle\big{[}\theta(b_{B}-b_{3})K_{0}(m_{B}\sqrt{x_{B}z}b_{B})I_{0}(m_{B}\sqrt{x_{B}z}b_{3})+(b_{B}\leftrightarrow
b_{3})\big{]}$ (70)
$\displaystyle\times\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{z[x_{3}(1-\eta)-x_{B}]}b_{3}),~{}\quad\quad
x_{3}(1-\eta)>x_{B},\\\
K_{0}(m_{B}\sqrt{z[x_{B}-x_{3}(1-\eta)]}b_{3}),~{}~{}~{}\quad\quad\quad
x_{3}(1-\eta)\leq x_{B},\end{array}\right.$ $\displaystyle
h_{2a}(x_{B},x_{3},b_{B},b_{3})$ $\displaystyle=$ $\displaystyle
K_{0}(m_{B}\sqrt{x_{B}x_{3}(1-\eta)}b_{B})\big{[}\theta(b_{B}-b_{3})K_{0}(m_{B}\sqrt{x_{3}(1-\eta)}b_{B})$
$\displaystyle\times
I_{0}(m_{B}\sqrt{x_{3}(1-\eta)}b_{3})+(b_{3}\leftrightarrow
b_{B})\big{]}S_{t}(x_{3}),$ $\displaystyle h_{2b}(x_{B},x_{3},b_{B},b_{3})$
$\displaystyle=$ $\displaystyle h_{2a}(x_{3},x_{B},b_{3},b_{B}),$
$\displaystyle h_{2c}(x_{B},z,x_{3},b_{B},b)$ $\displaystyle=$
$\displaystyle\big{[}\theta(b_{B}-b)K_{0}(m_{B}\sqrt{x_{B}x_{3}(1-\eta)}b_{B})I_{0}(m_{B}\sqrt{x_{B}x_{3}(1-\eta)}b)$
(73) $\displaystyle+(b_{B}\leftrightarrow
b)\big{]}\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{(1-x_{B}-z)[x_{3}(1-\eta)+\eta]}b),~{}\quad\quad
x_{B}+z<1,\\\
K_{0}(m_{B}\sqrt{(x_{B}+z-1)[x_{3}(1-\eta)+\eta]}b),~{}~{}~{}\quad\quad\quad
x_{B}+z\geq 1,\end{array}\right.$ $\displaystyle
h_{2d}(x_{B},z,x_{3},b_{B},b)$ $\displaystyle=$
$\displaystyle\big{[}\theta(b_{B}-b)K_{0}(m_{B}\sqrt{x_{B}x_{3}(1-\eta)}b_{B})I_{0}(m_{B}\sqrt{x_{B}x_{3}(1-\eta)}b)$
(76) $\displaystyle+(b_{B}\leftrightarrow
b)\big{]}\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{x_{3}(z-x_{B})(1-\eta)}b),~{}\quad\quad
x_{B}<z,\\\
K_{0}(m_{B}\sqrt{x_{3}(x_{B}-z)(1-\eta)}b),~{}~{}~{}\quad\quad\quad x_{B}\geq
z,\end{array}\right.$ $\displaystyle h_{3a}(z,x_{3},b,b_{3})$ $\displaystyle=$
$\displaystyle\left(\frac{i\pi}{2}\right)^{2}H_{0}^{(1)}(m_{B}\sqrt{(1-x_{3})z(1-\eta)}b)S_{t}(x_{3})$
$\displaystyle\times\big{[}\theta(b-b_{3})H_{0}^{(1)}(m_{B}\sqrt{1-x_{3}(1-\eta)}b)J_{0}(m_{B}\sqrt{1-x_{3}(1-\eta)}b_{3})+(b\leftrightarrow
b_{3})\big{]},$ $\displaystyle h_{3b}(z,x_{3},b,b_{3})$ $\displaystyle=$
$\displaystyle\left(\frac{i\pi}{2}\right)^{2}H_{0}^{(1)}(m_{B}\sqrt{(1-x_{3})z(1-\eta)}b_{3})S_{t}(z)$
$\displaystyle\times\big{[}\theta(b-b_{3})H_{0}^{(1)}(m_{B}\sqrt{z(1-\eta)}b)J_{0}(m_{B}\sqrt{z(1-\eta)}b_{3})+(b\leftrightarrow
b_{3})\big{]},$ $\displaystyle h_{3c}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle=$
$\displaystyle\frac{i\pi}{2}K_{0}(m_{B}\sqrt{1-x_{3}(1-x_{B}-z)(1-\eta)+(x_{B}+z-1)\eta}b_{B})$
$\displaystyle\times\big{[}\theta(b_{B}-b_{3})H_{0}^{(1)}(m_{B}\sqrt{(1-x_{3})z(1-\eta)}b_{B})J_{0}(m_{B}\sqrt{(1-x_{3})z(1-\eta)}b_{3})$
$\displaystyle+(b_{B}\leftrightarrow b_{3})\big{]},$ $\displaystyle
h_{3d}(x_{B},z,x_{3},b_{B},b_{3})$ $\displaystyle=$
$\displaystyle\frac{i\pi}{2}\big{[}\theta(b_{B}-b_{3})H_{0}^{(1)}(m_{B}\sqrt{(1-x_{3})z(1-\eta)}b_{B})J_{0}(m_{B}\sqrt{(1-x_{3})z(1-\eta)}b_{3})+(b_{B}\leftrightarrow
b_{3})\big{]}$ (79)
$\displaystyle\times\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{(1-x_{3})(z-x_{B})(1-\eta)}b_{B}),~{}\quad\quad
x_{B}<z,\\\
K_{0}(m_{B}\sqrt{(1-x_{3})(x_{B}-z)(1-\eta)}b_{B}),~{}~{}~{}\quad\quad\quad
x_{B}\geq z,\end{array}\right.$ $\displaystyle h_{4a}(z,x_{3},b,b_{3})$
$\displaystyle=$
$\displaystyle\left(\frac{i\pi}{2}\right)^{2}H_{0}^{(1)}(m_{B}\sqrt{(1-z)(\eta+x_{3}(1-\eta))}b_{3})S_{t}(z)$
$\displaystyle\times\big{[}\theta(b-b_{3})H_{0}^{(1)}(m_{B}\sqrt{1-z}b)J_{0}(m_{B}\sqrt{1-z}b_{3})+(b\leftrightarrow
b_{3})\big{]},$ $\displaystyle h_{4b}(z,x_{3},b,b_{3})$ $\displaystyle=$
$\displaystyle\left(\frac{i\pi}{2}\right)^{2}H_{0}^{(1)}(m_{B}\sqrt{(1-z)(\eta+x_{3}(1-\eta))}b)S_{t}(x_{3})$
$\displaystyle\times\big{[}\theta(b-b_{3})H_{0}^{(1)}(m_{B}\sqrt{\eta+x_{3}(1-\eta)}b)J_{0}(m_{B}\sqrt{\eta+x_{3}(1-\eta)}b_{3})+(b\leftrightarrow
b_{3})\big{]},$ $\displaystyle h_{4c}(x_{B},z,x_{3},b_{B},b_{3})$
$\displaystyle=$
$\displaystyle\frac{i\pi}{2}K_{0}(m_{B}\sqrt{1-z((1-x_{3})(1-\eta)-x_{B})}b_{B})$
$\displaystyle\times\big{[}\theta(b_{B}-b_{3})H_{0}^{(1)}(m_{B}\sqrt{(1-z)(\eta+x_{3}(1-\eta))}b_{B})J_{0}(m_{B}\sqrt{(1-z)(\eta+x_{3}(1-\eta))}b_{3})$
$\displaystyle+(b_{B}\leftrightarrow b_{3})\big{]},$ $\displaystyle
h_{4d}(x_{B},z,x_{3},b,b_{3})$ $\displaystyle=$
$\displaystyle\frac{i\pi}{2}\big{[}\theta(b_{B}-b_{3})H_{0}^{(1)}(m_{B}\sqrt{(1-z)(\eta+x_{3}(1-\eta))}b_{B})$
(82) $\displaystyle\times
J_{0}(m_{B}\sqrt{(1-z)(\eta+x_{3}(1-\eta))}b_{3})+(b_{B}\leftrightarrow
b_{3})\big{]}$
$\displaystyle\times\left\\{\begin{array}[]{ll}\frac{i\pi}{2}H_{0}^{(1)}(m_{B}\sqrt{(1-z)(\eta+x_{3}(1-\eta)-x_{B})}b_{B}),~{}\quad
x_{B}<\eta+x_{3}(1-\eta),\\\ K_{0}(m_{B}\sqrt{(1-z)(x_{B}-\eta-
x_{3}(1-\eta))}b_{B}),~{}~{}~{}\quad\quad
x_{B}\geq\eta+x_{3}(1-\eta),\end{array}\right.$
with the Hankel function $H_{0}^{(1)}(x)=J_{0}(x)+iY_{0}(x)$.
The evolution factors in the above factorization formulas are given by
$\displaystyle E_{1ab}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{Ms}(t)],$ $\displaystyle
E_{1cd}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{Ms}(t)-S_{\pi}]|_{b=b_{B}},$
$\displaystyle E_{2ab}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{\pi}(t)],$ $\displaystyle
E_{2cd}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{Ms}(t)-S_{\pi}]|_{b_{3}=b_{B}},$
$\displaystyle E_{3ab}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{Ms}-S_{\pi}(t)],$ $\displaystyle
E_{3cd}(t)$ $\displaystyle=$
$\displaystyle\alpha_{s}(t)\exp[-S_{B}(t)-S_{Ms}(t)-S_{\pi}]|_{b_{3}=b},$
$\displaystyle E_{4ab}(t)$ $\displaystyle=$ $\displaystyle E_{3ab}(t),$
$\displaystyle E_{4cd}(t)$ $\displaystyle=$ $\displaystyle E_{3cd}(t),$ (83)
in which the Sudakov exponents are defined as
$\displaystyle S_{B}$ $\displaystyle=$ $\displaystyle
s\left(x_{B}\frac{m_{B}}{\sqrt{2}},b_{B}\right)+\frac{5}{3}\int^{t}_{1/b_{B}}\frac{d\bar{\mu}}{\bar{\mu}}\gamma_{q}(\alpha_{s}(\bar{\mu})),$
$\displaystyle S_{Ms}$ $\displaystyle=$ $\displaystyle
s\left(z\frac{m_{B}}{\sqrt{2}},b\right)+s\left((1-z)\frac{m_{B}}{\sqrt{2}},b\right)+2\int^{t}_{1/b}\frac{d\bar{\mu}}{\bar{\mu}}\gamma_{q}(\alpha_{s}(\bar{\mu})),$
$\displaystyle S_{\pi}$ $\displaystyle=$ $\displaystyle
s\left(x_{3}\frac{m_{B}}{\sqrt{2}},b_{3}\right)+s\left((1-x_{3})\frac{m_{B}}{\sqrt{2}},b_{3}\right)+2\int^{t}_{1/b_{3}}\frac{d\bar{\mu}}{\bar{\mu}}\gamma_{q}(\alpha_{s}(\bar{\mu})),$
(84)
with the quark anomalous dimension $\gamma_{q}=-\alpha_{s}/\pi$. The explicit
expressions of the functions $s(Q,b)$ can be found, for example, in Appendix A
of Ref. prd76-074018 . The involved hard scales are chosen in the PQCD
approach as
$\displaystyle t_{1a}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{z},1/b_{B},1/b\right\\},$ $\displaystyle
t_{1b}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{|x_{B}-\eta|},1/b_{B},1/b\right\\},$
$\displaystyle t_{1c}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{B}z},m_{B}\sqrt{z|(1-\eta)(1-x_{3})-x_{B}|},1/b_{B},1/b_{3},\right\\},$
$\displaystyle t_{1d}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{B}z},m_{B}\sqrt{z|x_{B}-x_{3}(1-\eta)|},1/b_{B},1/b_{3}\right\\},$
$\displaystyle t_{2a}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{3}(1-\eta)},1/b_{B},1/b_{3}\right\\},$
$\displaystyle t_{2b}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{B}(1-\eta)},1/b_{B},1/b_{3}\right\\},$
$\displaystyle t_{2c}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{B}x_{3}(1-\eta)},m_{B}\sqrt{|1-x_{B}-z|[x_{3}(1-\eta)+\eta]},1/b_{B},1/b,\right\\},$
$\displaystyle t_{2d}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{x_{B}x_{3}(1-\eta)},m_{B}\sqrt{|x_{B}-z|x_{3}(1-\eta)},1/b_{B},1/b\right\\},$
$\displaystyle t_{3a}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{1-x_{3}(1-\eta)},1/b,1/b_{3}\right\\},$
$\displaystyle t_{3b}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{z(1-\eta)},1/b,1/b_{3}\right\\},$
$\displaystyle t_{3c}$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}m_{B}\sqrt{(1-x_{3})z(1-\eta)},m_{B}\sqrt{1-x_{3}(1-x_{B}-z)(1-\eta)+(x_{B}+z-1)\eta},$
$\displaystyle\quad\quad\quad 1/b_{B},1/b_{3},\bigg{\\}},$ $\displaystyle
t_{3d}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{(1-x_{3})z(1-\eta)},m_{B}\sqrt{|x_{B}-z|(1-x_{3})(1-\eta)},1/b_{B},1/b_{3}\right\\},$
$\displaystyle t_{4a}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{1-z},1/b,1/b_{3}\right\\},$
$\displaystyle t_{4b}$ $\displaystyle=$
$\displaystyle\max\left\\{m_{B}\sqrt{\eta+x_{3}(1-\eta)},1/b,1/b_{3}\right\\},$
$\displaystyle t_{4c}$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}m_{B}\sqrt{(1-z)(\eta+x_{3}(1-\eta))},m_{B}\sqrt{1-z((1-x_{3})(1-\eta)-x_{B})},$
$\displaystyle\quad\quad\quad 1/b_{B},1/b_{3},\bigg{\\}},$ $\displaystyle
t_{4d}$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}m_{B}\sqrt{(1-z)(\eta+x_{3}(1-\eta))},m_{B}\sqrt{(1-z)|x_{B}-\eta-
x_{3}(1-\eta)|},$ (85) $\displaystyle\quad\quad\quad
1/b_{B},1/b_{3}\bigg{\\}}.$
## References
* (1) C.H. Chen and H.-n. Li, Phys. Lett. B 561, 258 (2003).
* (2) H.Y. Cheng and K.C. Yang, Phys. Rev. D 66, 054015 (2002); H.-Y. Cheng, C.-K. Chua, and A. Soni, Phys. Rev. D 76, 094006 (2007).
* (3) S. Fajfer, T.N. Pham, and A. Prapotnik, Phys. Rev. D 70, 034033 (2004).
* (4) B. Bhattacharya, M. Imbeault, and D. London, Phys. Lett. B 728, 206 (2014); N.R.-L. Lorier, M. Imbeault, and D. London, Phys. Rev. D 84, 034040 (2011); M. Imbeault, N.R.-L. Lorier, and D. London, Phys. Rev. D 84, 034041 (2011); N.R.-L Lorier and D. London, Phys. Rev. D 85, 016010 (2012).
* (5) R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 111, 101801 (2013).
* (6) R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 112, 011801 (2014).
* (7) I. Nasteva (LHCb Collaboration), arXiv:1308.0740; J. M. de Miranda (LHCb Collaboration), arXiv:1301.0283.
* (8) Z.H. Zhang, X.H. Guo, and Y.D. Yang, Phys. Rev. D 87, 076007 (2013).
* (9) B. Bhattacharya, M. Gronau, and J.L. Rosner, Phys. Lett. B 726, 337 (2013).
* (10) D. Xu, G.N. Li, and X.G. He, Int. J. Mod. Phys. A29, 1450011 (2014); Phys. Lett. B 728, 579 (2014).
* (11) H.Y. Cheng and C.K. Chua, Phys. Rev. D 88, 114014 (2013); Phys. Rev. D 89, 074025 (2014); Y. Li, arXiv:1401.5948.
* (12) C.L.Y. Lee, M. Lu, and M.B. Wise, Phys. Rev. D 46, 5040 (1992).
* (13) Y.Y. Keum, H.-n. Li, and A.I. Sanda, Phys Lett. B 504, 6 (2001); Phys. Rev. D 63, 054008 (2001).
* (14) C.D. Lu, K. Ukai, and M.Z. Yang, Phys. Rev. D 63, 074009 (2001).
* (15) D. Muller et al., Fortschr. Physik. 42, 101 (1994); M. Diehl, T. Gousset, B. Pire, and O. Teryaev, Phys. Rev. Lett. 81, 1782 (1998); M.V. Polyakov, Nucl. Phys. B555, 231 (1999).
* (16) M. Diehl, T. Gousset, and B. Pire, Phys. Rev. D 62, 073014 (2000).
* (17) J. Beringer, et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
* (18) M.R. Whalley, J. Phys. G 29, A1 (2003); J. Milana, S. Nussinov, and M.G. Olsson, Phys. Rev. Lett. 71, 2533 (1993); T.K. Pedlar et al.,Phys. Rev. Lett. 95, 261803 (2005).
* (19) H.C. Hu and H.-n. Li, Phys. Lett. B 718, 1351 (2013).
* (20) S.D. Protopopescu, Phys. Rev. D 7, 1279 (1973).
* (21) P. Estabrooks and A.D. Martin, Nucl. Phys. B79, 301 (1974).
* (22) J.H. Kuhn and A. Santamaria, Z. Phys. C 48, 445 (1990).
* (23) P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005); P. Ball, V.M. Braun, and A. Lenz, J. High Energy Phys. 0605, 004 (2006).
* (24) P. Colangelo, F. De Fazio, P. Santorelli, and E. Scrimieri, Phys. Rev. D 53, 3672 (1996); A. Khodjamirian, T. Mannel, and N. Offen, Phys. Rev. D 75, 054013 (2007); M.A. Ivanov, J.G. Korner, S.G. Kovalenko, and C.D. Roberts, Phys. Rev. D 76, 034018 (2007); D. Ebert, R.N. Faustov, and V.O. Galkin, Phys. Rev. D 75, 074008 (2007); X.G. Wu and T. Huang, Phys. Rev. D 79, 034013 (2009); W.F. Wang and Z.J. Xiao, Phys. Rev. D 86, 114025 (2012).
* (25) A. Ali, G. Kramer, Y. Li, C.D. Lü, Y.L. Shen, W. Wang, and Y.M. Wang, Phys. Rev. D 76, 074018 (2007).
* (26) H.-n. Li and S. Mishima, Phys. Rev. D 74, 094020 (2006).
* (27) C.H. Chen and H.-n. Li, Phys. Rev. D 70, 054006 (2004).
* (28) L. Li, B.S. Zou, and G.L. Li, Phys. Rev. D 67, 034025 (2003).
* (29) M. Doring, U.G. Meisner, and W. Wang, J. High Energy Phys. 1310, 011 (2013); U.G. Meisner and W. Wang, Phys. Lett. B 730, 336 (2014).
* (30) T. Kurimoto, H.-n. Li, and A.I. Sanda, Phys. Rev. D 65, 014007 (2001).
|
arxiv-papers
| 2014-02-21T12:41:05 |
2024-09-04T02:49:58.530729
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wen-Fei Wang, Hao-Chung Hu, Hsiang-nan Li and Cai-Dian L\\\"u",
"submitter": "Wen-Fei Wang",
"url": "https://arxiv.org/abs/1402.5280"
}
|
1402.5301
|
]http://www.chalmers.se/rss/
# Gyrokinetic modelling of stationary electron and impurity profiles in
tokamaks
A. Skyman [email protected] H. Nordman [email protected] D.
Tegnered [email protected] Euratom–VR Association, Department of Earth and
Space Sciences,
Chalmers University of Technology, SE-412 96 Göteborg, Sweden [
###### Abstract
Particle transport due to Ion Temperature Gradient/Trapped Electron (ITG/TE)
mode turbulence is investigated using the gyrokinetic code GENE. Both a
reduced quasilinear (QL) treatment and nonlinear (NL) simulations are
performed for typical tokamak parameters corresponding to ITG dominated
turbulence.
A selfconsistent treatment is used, where the stationary local profiles are
calculated corresponding to zero particle flux simultaneously for electrons
and trace impurities. The scaling of the stationary profiles with magnetic
shear, safety factor, electron-to-ion temperature ratio, collisionality,
toroidal sheared rotation, triangularity, and elongation is investigated. In
addition, the effect of different main ion mass on the zero flux condition is
discussed.
The electron density gradient can significantly affect the stationary impurity
profile scaling. It is therefore expected, that a selfconsistent treatment
will yield results more comparable to experimental results for parameter scans
where the stationary background density profile is sensitive. This is shown to
be the case in scans over magnetic shear, collisionality, elongation, and
temperature ratio, for which the simultaneous zero flux electron and impurity
profiles are calculated.
A slight asymmetry between hydrogen, deuterium and tritium with respect to
profile peaking is obtained, in particular for scans in collisionality and
temperature ratio.
###### pacs:
28.52.Av, 52.25.Vy, 52.30.Ex, 52.30.Gz, 52.35.Ra, 52.55.Fa, 52.65.Tt
## I Introduction
It is well known that the shape of the main ion density and impurity profiles
are crucial for the performance of a fusion device. Inward peaking of the main
ion (electron) density profile is beneficial for the fusion performance since
it enhances the fusion power production. For impurities on the other hand, a
flat or hollow profile is preferred, since impurity accumulation in the core
leads to fuel dilution and radiation losses which degrades performance.
The particle profiles are determined by a balance between particle sources and
particle fluxes, a subject which historically has been given much less
attention than energy transport and the associated temperature profiles.
Hence, electron density profiles are often treated as a parameter in
theoretical studies of transport rather than being selfconsistently
calculated.
Turbulent transport in the core of tokamaks is expected to be driven mainly by
Ion Temperature Gradient (ITG) and Trapped Electron (TE) modes. Impurity
transport driven by ITG/TE mode turbulence has been investigated in a number
of theoretical studies.Frojdh1992 ; Basu2003 ; Angioni2005 ; Estrada-Mila2005
; Naulin2005 ; Angioni2006 ; Guirlet2006 ; Bourdelle2007 ; Dubuit2007 ;
Nordman2007a ; Angioni2008 ; Nordman2008 ; Angioni2009a ; Angioni2009c ;
Camenen2009 ; Moradi2009 ; Fable2010 ; Fulop2010 ; Futatani2010 ; Hein2010 ;
Angioni2011 ; Nordman2011 ; Skyman2011a ; Skyman2011b ; Casson2013 ;
Skyman2014 Most work in this area has been focused on either scalings of
stationary electron profiles or on impurity transport using prescribed
electron density profiles.
It is well established theoretically that turbulent particle transport in
tokamaks has contributions from both diagonal (diffusive) and non-diagonal
(convective) terms. The non-diagonal transport contributions may give rise to
an inward pinch which can support an inwardly peaked profile even in the
absence of particle sources in the core. The stationary peaked profile is then
obtained from a balance between diffusion and convection.
It is known that the electron density gradient can significantly affect the
stationary impurity profile scaling.Skyman2011b In the present work,
therefore, the background electron density and impurity peaking is treated
selfconsistently, by simultaneously calculating the local profiles
corresponding to zero turbulent particle flux of both electrons and
impurities.
Linear and nonlinear gyrokinetic simulations using the code
GENE111http://www.ipp.mpg.de/~fsj/gene/ are employed.Jenko2000 ; Merz2008a
The scaling of the stationary profiles with key plasma parameters like
magnetic shear, temperature ratio and temperature gradient, toroidal sheared
rotation, safety factor, and collisionality is investigated for a deuterium
(D) plasma. The isotope scaling of stationary profiles, for hydrogen (H) and
tritium (T) plasmas, is also studied.
The parameters are taken from the Cyclone Base Case (CBCDimits2000 ), but with
deuterium as main ions; see Tab. 1 for the main parameters. It is an ITG mode
dominated scenario and, though set far from marginal stability, is an
interesting case for study, and is widely used as a testing ground and
benchmark for theoretical and numerical studies.
The rest of the paper is organised as follows: In section II the theoretical
background is given, including considerations regarding analysis and numerics;
the main results are presented in section III, where scalings of the
stationary profiles for electrons and impurities are presented; results for
background peaking for different main ion isotopes is presented and discussed
in section IV; finally, in section V, follow the concluding remarks.
Table 1: Parameters for the Cyclone Base Case (CBC). † denotes derived parameters $r/R$ | $0.18$
---|---
$\hat{s}$ | $0.796$
$q_{0}$ | $1.4$
$R/L_{n_{i,e}}$ | $2.22$
$R/L_{T_{i,e}}$ | $6.96$
$T_{i}/T_{e}$ | $1.0$
$T_{e}$ | $2.85\,\mathrm{keV}$
$n_{e}$ | $3.51\cdot 10^{19}\,\mathrm{m^{-3}}$
$B_{0}$ | $3.1\,\mathrm{T}$
$R$ | $1.65\,\mathrm{m}$
$\beta$ | $0$
$\nu_{ei}$† | $0.05$ $c_{s}/R$
## II Background
The local particle transport for species $j$ can be formally divided into its
diagonal and off-diagonal parts
$\frac{R\Gamma_{j}}{n_{j}}=D_{j}\frac{R}{L_{n_{j}}}+D_{T_{j}}\frac{R}{L_{T_{j}}}+RV_{p,j}.$
(1)
Here, the first term on the right hand side is the diffusion and the second
and third constitute the off-diagonal pinch. The first of the pinch terms is
the particle transport due to the temperature gradient (thermo-diffusion) and
the second is the convective velocity, which includes contributions from
curvature, parallel compression and roto-diffusion. In equation (1),
$R/L_{X_{j}}\equiv-R\nabla X_{j}/X_{j}$ are the local gradient scale lengths
of density and temperature, normalised to the major radius ($R$). In general,
the transport coefficients dependent on the gradients, though in the trace
impurity limit the transport is linear in both $R/L_{n_{Z}}$ and
$R/L_{T_{Z}}$. A review of the off-diagonal contributions is given in Ref.
Angioni2012, .
At steady state, the contributions from the different terms in the particle
transport will tend to cancel, resulting in zero particle flux. Solving
equation (1) for zero particle flux, with $V_{j}=D_{T_{j}}1/L_{T_{j}}+V_{p,j}$
yields
$PF_{j}\equiv\left.\frac{R}{L_{n_{j}}}\right|_{\Gamma=0}=-\frac{R\,V_{j}}{D_{j}},$
(2)
which is the steady state gradient of zero particle flux for species $j$. This
measure quantifies the balance between diffusion and advection, and gives a
measure of how “peaked” the local density profile is at steady state. It is
therefore referred to as the “peaking factor” and denoted $PF_{j}$.
(1(a)) electron particle flux for different density gradients (1(b))
poloidal wavenumber spectra of the electron particle flux, evaluated at the
gradient of zero flux
Figure 1: Timeseries and spectra of electron particle transport for CBC with D
as main ions near to the zero flux gradient ($R/L_{n_{e}}=2.77$); obtained
from NL GENE simulations.
In order to investigate the transport, nonlinear (NL) GENE simulations were
performed from which $PF_{e}$ for the stationary electron profiles were
calculated. The results were compared with quasilinear (QL) results, also
obtained using GENE. The background peaking factor was found by explicitly
seeking the gradient of zero particle flux by calculating the electron flux
for several values of the density gradient. A typical set of simulations is
displayed in Fig. 1(a), where the time evolution of the electron flux for
three density gradients near the gradient of zero particle flux is shown
(fluxes are in gyro-Bohm units, with
$D_{\text{GB}}=c_{\text{s}}\rho_{\text{s}}^{2}/R$). A second order polynomial
$p$ was then fitted to the data closest to the zero flux gradient and then the
$PF_{e}$ was found as the appropriate root of $p$. The error for $PF_{e}$ was
approximated by finding the corresponding roots of
$p\pm\text{max}\left[\sigma_{\Gamma}\right]$, and using the difference between
these roots as a measure of the error. In Fig. 1(b) the particle flux spectrum
for a NL simulation for CBC near this gradient is shown. The figure
illustrates that the total flux is zero due to a balance of inward and outward
transport occurring at different wavenumbers. The method for finding $PF_{e}$
from the QL simulations is the same, but here a reduced treatment was used,
including only the dominant mode, which is an ITG mode for CBC-like
parameters. This was done for a range of values of several key plasma
parameters.
In the trace impurity limit, i.e. when the fraction of impurities is
sufficiently small, the impurity dynamics do not affect the turbulence
dynamics. Therefore, when finding the simultaneous peaking factor of the
background and impurities, the former can be found first and used in the
simulations of the latter without loss of generality. Furthermore, in the
trace impurity limit, the transport coefficients of Eq. (1) for trace
impurities do not depend on the species’ gradients of density and temperature,
meaning that (1) is a linear relation in those gradients. This means that the
impurity peaking, as well as the contribution to $PF_{Z}$ from the
thermodiffusion ($PF_{T}$) and the convective velocity ($PF_{p}$), can be
found from simulations with appropriately chosen gradients using the method
outlined in Casson2010 . The peaking factors are calculated for several
impurity species, using the reduced QL model. The difference in impurity
peaking factors between NL and QL models has been covered in previous
work.Nordman2011 ; Skyman2011a ; Skyman2011b ; Skyman2014
The simulations have been performed in a circular equilibrium with aspect
ratio $R/a=3$, using kinetic ions, electrons and impurities, except when
studying the effects of shaping. Then the Miller equilibrium model was used
instead.Miller1998 Impurities were included at trace amounts
($n_{Z}/n_{e}=10^{-6}$), so as not to affect the turbulent dynamics. The
impurities mass was assumed to be $A_{Z}=2Z$, where $Z$ is the charge number.
The dynamics were further assumed to be electrostatic ($\beta\approx 0$).
For the simulation domain, a flux tube with periodic boundary conditions in
the perpendicular plane was used. The nonlinear simulations were performed
using a $96\times 96\times 32$ grid in the normal, bi-normal, and parallel
spatial directions respectively; in the parallel and perpendicular momentum
directions, a $48\times 12$ grid was used. For the linear and quasilinear
computations, a typical resolution was $12\times 24$ grid points in the
parallel and normal directions, with $64\times 12$ grid points in momentum
space. The nonlinear simulations were typically run up to
$t=300\,R/c_{\text{s}}$ for the experimental geometry scenario, where $R$ is
the major radius and $c_{\text{s}}=\sqrt{T_{e}/m_{i}}$.
## III Simultaneous stationary profiles of electrons and impurities
First, we examine the dependence of the transport and of $PF_{e}$ on the ion
temperature gradient. The result is shown in Fig. 2, where the ion energy
transport from NL simulations is displayed, together with electron peaking
factors from NL and QL simulations. Though the ion energy transport shows a
stiff increase with the driving gradient, only a moderate reduction is seen in
the peaking factor.
Figure 2: Scaling of $PF_{e}$ and ion heat flux with ion temperature gradient
$\nabla T_{i}$.
(3(a)) NL and QL scalings of $PF_{e}$ (3(b)) simultaneous QL scalings of
$PF_{e}$ and $PF_{Z}$ (3(c)) contributions to $PF_{Z}$ from thermopinch
($PF_{T}$) and pure convection ($PF_{p}$) vs. impurity charge (3(d)) scaling
of ITG growthrates ($\gamma$) and real frequencies ($\omega_{r}$), normalised
to $c_{\text{s}}/R$
Figure 3: Scaling of background electron peaking, impurity peaking and linear
eigenvalues with $T_{i}/T_{e}$.
(4(a)) NL and QL scalings of $PF_{e}$ (4(b)) simultaneous QL scalings of
$PF_{e}$ and $PF_{Z}$ (4(c)) contributions to $PF_{Z}$ from thermopinch
($PF_{T}$) and pure convection ($PF_{p}$) vs. impurity charge
Figure 4: Scaling of background electron and impurity peaking with $\hat{s}$.
(5(a)) simultaneous QL scaling of $PF_{e}$ and $PF_{Z}$ (5(b)) contributions
to $PF_{Z}$ from thermopinch ($PF_{T}$) and pure convection ($PF_{p}$) vs.
impurity charge
Figure 5: Scaling of background electron and impurity peaking with $\nu_{ei}$.
(6(a)) simultaneous QL scaling of $PF_{e}$ and $PF_{Z}$ (6(b)) contributions
to $PF_{Z}$ from thermopinch ($PF_{T}$) and pure convection ($PF_{p}$) vs.
impurity charge
Figure 6: Scaling of background electron and impurity peaking with
$\gamma_{E}$.
(7(a)) simultaneous QL scaling of $PF_{e}$ and $PF_{Z}$ (7(b)) contributions
to $PF_{Z}$ from thermopinch ($PF_{T}$) and pure convection ($PF_{p}$) vs.
impurity charge
Figure 7: Scaling of background electron and impurity peaking with $\kappa$.
It is worth noting that the steady state peaking found in the simulations is
considerably higher than that in the original CBC experiment
($R/L_{n_{e,i}}=2.22$). As is knownAngioni2005 ; Angioni2009b ; Angioni2009c ;
Fable2010 , this is due to the neglect of collisions, as they normally are in
the CBC. The collisionality for the CBC parameters is $\nu_{ei}\approx
0.05\,c_{\text{s}}/R$, which is of the same order as the growthrates and real
frequencies observed, and collisions can be expected to have a notable impact
on the transport. When collisions are added, the background peaking factor is
indeed lowered to a level consistent with the prescribed background gradient
for the CBC, as seen in Fig. 2. The QL peaking factor shows a stronger
decrease than its NL counterpart. Below $R/L_{T_{i}}\approx 4.5$, the ITG mode
is stable, and the TE mode dominates. In the following, focus is on the
collisionless case, but the simulations have been complemented with scalings
including collisions.
The electron peaking factor is reduced with increasing ion–electron
temperature ratio ($T_{i}/T_{e}$) for CBC parameters, as can be seen in Fig.
3(a). As with the temperature gradient, the NL results show only a weak
scaling, while the trend is more pronounced for the QL simulations. This may
be a result of the QL treatment, which only includes the dominant mode, while
the contribution from the subdominant TE mode is non-negligible for low values
of $T_{i}/T_{e}$. A more complete QL treatment may give a better
agreement.Bourdelle2007 ; Fable2010 In Fig. 3(b), the selfconsistently
obtained quasilinear peaking of electrons and impurities (Be ($Z=4$), C
($Z=6$), Ne ($Z=10$), and Ni ($Z=28$)) is shown. Impurities with lower charge
numbers ($Z$), as well as the background, show the same dependence on
$T_{i}/T_{e}$, with a decrease in the peaking as the ion temperature is
increased, and a weaker tendency for smaller wavenumbers. For the impurities
with higher $Z$, on the other hand, increased ion temperature leads to
slightly more peaked impurity profiles. In Fig. 3(c) it is shown that the
effect for the impurities is mainly due to an increase in the relative
contribution from the outward thermopinch ($\sim 1/Z$) with increased ion
temperature, which affects the low $Z$ impurities more strongly. To first
order, the thermopinch is proportional to the real frequency. As seen in Fig.
3(d) it increases with increasing $T_{i}/T_{e}$, which explains its increasing
importance for higher ion temperatures.
In Fig. 4(a), the scaling with magnetic shear ($\hat{s}$) is studied. The
electron peaking shows a strong and near linear dependence on $\hat{s}$. This
is similar to the results reported in Ref. Fable2010, and is due to the shear
dependence of the curvature pinch. This trend is as strong in both the QL and
NL simulations. The effect of shear on the linear eigenvalues is not
monotonous, with a destabilisation in the low to medium shear region followed
by stabilisation as $\hat{s}$ is increased further. The selfconsistent results
are shown in Fig. 4(b). For the impurities, the change in peaking factors due
to magnetic shear follows the trend seen for the electrons, and impurities
with higher $Z$ are more strongly affected. This is seen in Fig. 4(c) to be
due mainly to a stronger inward convective pinch with increasing shear.
Next we cover the effect of electron–ion collisions on the peaking factors.
Collisionality is known to affect the background by reducing the peaking
factor.Angioni2005 ; Angioni2009a ; Angioni2009c ; Fable2010 In Fig. 5(a),
the selfconsistent results for a range of collisionalities are shown. The
reduction in peaking factors with collisionality is also seen for low $Z$
impurities, while the high-$Z$ impurities show little or no change in peaking
due to collisions. The effect on the impurities is mainly due to an increase
in the outward thermopinch ($\sim 1/Z$) with increased collisionality (Fig.
5(b)), due to a change of the real frequency.
The influence of sheared toroidal flows on the selfconsistent impurity peaking
was also studied. Only purely toroidal rotation was considered, included
through the $\boldsymbol{E}\times\boldsymbol{B}$ shearing rate, defined as
$\gamma_{E}=-\frac{r}{q}\frac{1}{R}\frac{\partial v_{\text{tor}}}{\partial
r}$. Hence, we flow shear in the limit where the flow is small, neglecting
effects of centrifugal and Coriolis forces. These may, however, be important
for heavier impurities.Camenen2009 The results are shown in Fig. 6(a), where
it can be seen that impurities are much more strongly affected by the rotation
than the electrons, due to the difference in thermal velocity. For large
values of $\gamma_{E}$, a strong decrease in impurity peaking is seen. The
effect is due to the outward roto-pinch which becomes important for large
values of $\gamma_{E}$, as shown in Fig. 6(b). As with the shearing rate, this
effect is more pronounced for high-$Z$ impurities, since the thermopinch
dominates for low $Z$ values. In ASDEX U roto-diffusion has been found to be a
critical ingredient to include in order to reproduce the Boron profiles seen
in experiments.Angioni2011 ; Casson2013
Finally, shaping effects were studied using the Miller equilibrium model. The
quasi-linear electron peaking factor as well as the self-consistent impurity
peaking factors increase with higher elongation $(\kappa)$ as shown in Fig.
7(a). For impurities with low charge number the increase in peaking is mainly
due to a larger inward thermopinch while for high-$Z$ impurities it is caused
by an increased pure convection, as seen in Fig. 7(b).
The dependence of the selfconsistent peaking factors on the safety factor
($q_{0}$) and triangularity ($\delta$) was also studied, and the scalings were
found to be very weak.
## IV Isotope effects on the background peaking
The CBC prescribes hydrogen ions as the main ions, however, for future fusion
power plants, a deuterium/tritium mixture will be used. Due to the difference
in mass, it is known that D and T plasmas will behave differently from pure H
plasmas. Differences in steady state peaking factors are expected, since both
collisions and non-adiabatic electrons can break the gyro-Bohm
scaling.Pusztai2011 To get an insight into the effect of the main ion
isotope, the scalings for the normal CBC were compared with simulations where
D was substituted for H and T.
Figure 8: Eigenvalue spectra for CBC parameters (Tab. 1), for H, D and T as
main ions, with $k_{\theta}\rho_{\text{s}i}$ and eigenvalues in species units
($c_{\text{s}i}/R$).
(9(a)) scaling with collisionality
(9(b)) scaling with temperature ratio
Figure 9: Scaling of main ion peaking with different parameters for the CBC
(Tab. 1), for H, D and T as main ions with $k_{\theta}\rho_{\text{s}i}=0.3$ in
species units. Figure 10: Timeseries of D, T and $e$ particle flux for CBC
(Tab. 1) with a 50/50 mixture of D and T as main ions. Evaluated at the zero
flux gradient for the pure D case ($R/L_{n_{e}}=2.77$).
First, we review the known isotope effects on linear eigenvalues. Figure 8
displays the ITG eigenvalues in the collisionless case for H, D, and T in
species units. The slight difference in eigenvalues obtained is due to the
non-adiabatic electron response into which the mass ratio $\sqrt{m_{i}/m_{e}}$
enters, as discussed in Ref. Pusztai2011, .
The QL background peaking versus collisionality is displayed in Fig. 9(a) for
$k_{\theta}\rho_{\text{s}i}=0.3$ in species units, corresponding to the peaks
in the growthrate spectra. For $\nu_{ei}=0$, a slight difference in $PF$ is
observed, with $PF_{\text{T}}>PF_{\text{D}}>PF_{\text{H}}$. This is consistent
with the asymmetry in D and T transport reported in Refs. Estrada-Mila2005, ;
Nordman2005, . For larger values of the collisionality, however, the order is
reversed.
Next, the effect of the ion mass on the stationary profile scaling with ion to
electron temperature ratio ($T_{i}/T_{e}$) is studied. In Fig. 9(b), the
peaking factor is seen to decrease with increasing ion temperature, but in
this case the lighter isotopes are more sensitive, showing a stronger decrease
with $T_{i}/T_{e}$. The other parameter scalings discussed in section III show
only a very weak isotope effect.
The scenario with a 50/50 mixture of D and T was also studied, and the
simultaneous peaking of D and T calculated. The results were seen to follow
the pure D and pure T results closely, albeit with the T profile approximately
10% more peaked than the D profile for all values of the collisionality; see
Fig. 9(a). For the scan with $T_{i}/T_{e}$, the self-consistent case gave a
larger difference in D and T peaking than the corresponding pure cases, as
seen in Fig. 9(b). These results were corroborated by NL simulations using the
standard CBC parameters, with the background electron density gradient
corresponding to zero flux for the pure D case ($R/L_{n_{e}}=2.77$). The
results are shown in Fig. 10. For these parameters, the electron particle flux
remained close to zero, while the deuterium flux was postivive and the tritium
flux negative, indicating a more peaked steady state D profile, and a less
peaked T profile in the mixed scenario.
The effect of main ion mass on the stationary profiles discussed here are
weak, but may result in a D–T fuel separation in a fusion plasma.Estrada-
Mila2005
## V Conclusions
In the present paper electron and impurity particle transport due to Ion
Temperature Gradient/Trapped Electron (ITG/TE) mode turbulence was studied
using gyrokinetic simulations. A reduced quasilinear (QL) treatment was used
together with nonlinear (NL) simulations using the code GENE. Neoclassical
contributions to the impurity transport, which may be relevant for high-$Z$
impurities, were neglected. The impurities, with impurity charge in the region
$3\leq Z\leq 42$, were included in low concentrations as trace species. The
focus was on a selfconsistent treatment of particle transport, where the
stationary local profiles of electrons and impurities are calculated
simultaneously corresponding to zero particle flux. The zero flux condition is
relevant to the core region of tokamaks where the particle sources are absent
or small. The parameters were taken from the Cyclone Base Case, corresponding
to ITG dominated turbulence with a subdominant TE mode relevant for the core
region of tokamaks, and scalings of the stationary profiles with magnetic
shear, safety factor, electron-to-ion temperature ratio, collisionality,
sheared toroidal rotation, elongation and triangularity were investigated.
It was shown that the stationary background density profile was sensitive in
scans over magnetic shear, collisionality, elongation, and temperature ratio,
for which the simultaneous zero flux electron and impurity profiles are
calculated. The selfconsistent treatment mainly tended to enhance these
parameter scalings of the impurity profile peaking. For safety factor, sheared
toroidal rotation and triangularity on the other hand, the effects on the
electron profile were weak and hence a selfconsistent treatment did not add
significant new results to the previous investigations in this area. For all
considered cases, both the electron profile and the impurity profile were
found to be inwardly peaked, with peaking factors $R/L_{n_{Z}}$ typically in
the range $1.0$–$4.0$, i.e. substantially below neoclassical expectations. For
large sheared toroidal rotation ($\gamma_{E}\gtrsim 0.4$), a flux reversal
resulting in outwardly peaked impurity profiles was seen. Furthermore, the
electrons were consistently more peaked than the impurities.
In addition, a slight asymmetry between hydrogen, deuterium and tritium with
respect to profile peaking was obtained. The effect was more pronounced for
high collisionality plasmas and large ion to electron temperature ratios. The
effect may have consequences for fuel separation in D–T fusion plasmas.
## Acknowledgements
This work was funded by a grant from The Swedish Research Council (C0338001).
The main simulations were performed on resources provided on the
Lindgren222See http://www.pdc.kth.se/resources/computers/lindgren/ for details
on Lindgren high performance computer, by the Swedish National Infrastructure
for Computing (SNIC) at Paralleldatorcentrum (PDC).
The authors would like to thank F Jenko, T Görler, MJ Püschel, D Told, and the
rest of the GENE team at IPP–Garching for their valuable support and input.
## References
* (1) M. Fröjdh, M. Liljeström, and H. Nordman, Nucl. Fusion 32, 419 (1992).
* (2) R. Basu, T. Jessen, V. Naulin, and J. J. Rasmussen, Phys. Plasmas 10, 2696 (2003).
* (3) C. Angioni, A. G. Peeters, F. Jenko, and T. Dannert, Phys. Plasmas 12, 112310 (2005).
* (4) C. Estrada-Mila, J. Candy, and R. W. Waltz, Phys. Plasmas 12, 022305 (2005).
* (5) V. Naulin, Phys. Rev. E 71, 015402 (2005).
* (6) C. Angioni and A. G. Peeters, Phys. Rev. Lett. 96, 095003 (2006).
* (7) R. Guirlet, C. Giroud, T. Parisot, M. E. Puiatti, C. Bourdelle, L. Carraro, N. Dubuit, X. Garbet, and P. R. Thomas, Plasma Phys. Contr. F. 48, B63 (2006).
* (8) C. Bourdelle, X. Garbet, F. Imbeaux, A. Casati, N. Dubuit, R. Guirlet, and T. Parisot, Phys. Plasmas 14, 112501 (2007).
* (9) N. Dubuit, X. Garbet, T. Parisot, R. Guirlet, and C. Bourdelle, Phys. Plasmas 14, 042301 (2007).
* (10) H. Nordman, T. Fülöp, J. Candy, P. Strand, and J. Weiland, Phys. Plasmas 14, 052303 (2007).
* (11) C. Angioni and A. G. Peeters, Phys. Plasmas 15, 052307 (2008).
* (12) H. Nordman, R. Singh, T. Fülöp, L.-G. Eriksson, R. Dumont, J. Anderson, P. Kaw, P. Strand, M. Tokar, and J. Weiland, Phys. Plasmas 15, 042316 (2008).
* (13) C. Angioni, A. G. Peeters, G. V. Pereverzev, A. Bottino, J. Candy, R. Dux, E. Fable, T. Hein, and R. E. Waltz, Nucl. Fusion 49, 055013 (2009).
* (14) C. Angioni, E. Fable, M. Greenwald, M. Maslov, A. G. Peeters, H. Takenaga, and H. Weisen, Plasma Phys. Contr. F. 51, 124017 (2009).
* (15) Y. Camenen, A. G. Peeters, C. Angioni, F. J. Casson, W. A. Hornsby, A. P. Snodin, and D. Strintzi, Phys. Plasmas 16, 012503 (2009).
* (16) S. Moradi, M. Z. Tokar, R. Singh, and B. Weyssow, Nucl. Fusion 49, 085007 (2009).
* (17) E. Fable, C. Angioni, and O. Sauter, Plasma Phys. Contr. F. 52, 015007 (2010).
* (18) T. Fülöp, S. Braun, and I. Pusztai, Phys. Plasmas 17, 062501 (2010).
* (19) S. Futatani, X. Garbet, S. Benkadda, and N. Dubuit, Phys. Rev. Lett. 104, 015003 (2010).
* (20) T. Hein and C. Angioni, Phys. Plasmas 17, 012307 (2010).
* (21) C. Angioni, R. McDermott, E. Fable, R. Fischer, T. Pütterich, F. Ryter, G. Tardini, and the ASDEX Upgrade Team, Nucl. Fusion 51, 023006 (2011).
* (22) H. Nordman, A. Skyman, P. Strand, C. Giroud, F. Jenko, F. Merz, V. Naulin, T. Tala, and the JET–EFDA contributors, Plasma Phys. Contr. F. 53, 105005 (2011).
* (23) A. Skyman, H. Nordman, and P. Strand, Phys. Plasmas 19, 032313 (2012).
* (24) A. Skyman, H. Nordman, and P. Strand, Nucl. Fusion 52, 114015 (2012).
* (25) F. J. Casson, R. M. McDermott, C. Angioni, Y. Camenen, R. Dux, E. Fable, R. Fischer, B. Geiger, P. Manas, L. Menchero, G. Tardini, and the ASDEX Upgrade Team, Nucl. Fusion 53, 063026 (2013).
* (26) A. Skyman, L. Fazendeiro, D. Tegnered, H. Nordman, J. Anderson, and P. Strand, Nucl. Fusion 54, 013009 (2014).
* (27) http://www.ipp.mpg.de/~fsj/gene/.
* (28) F. Jenko, W. Dorland, M. Kotschenreuther, and B. N. Rogers, Phys. Plasmas 7, 1904 (2000).
* (29) F. Merz, _Gyrokinetic Simulation of Multimode Plasma Turbulence_ , Ph.d. thesis (monography), Westfälischen Wilhelms-Universität Münster (2008).
* (30) A. M. Dimits, G. Bateman, M. A. Beer, B. I. Cohen, W. Dorland, G. W. Hammett, C. Kim, J. E. Kinsey, M. Kotschenreuther, A. H. Kritz, L. L. Lao, J. Mandrekas, W. M. Nevins, S. E. Parker, A. J. Redd, D. E. Shu-maker, R. Sydora, and J. Weiland, Phys. Plasmas 7, 969 (2000).
* (31) C. Angioni, Y. Camenen, F. J. Casson, E. Fable, R. M. McDermott, A. G. Peeters, and J. E. Rice, Nucl. Fusion 52, 114003 (2012).
* (32) F. J. Casson, A. G. Peeters, C. Angioni, Y. Camenen, W. A. Hornsby, A. P. Snodin, and G. Szepesi, Phys. Plasmas 17, 102305 (2010).
* (33) R. L. Miller, M. S. Chu, J. M. Greene, Y. R. Lin-Liu, and R. E. Waltz, Phys. Plasmas 5, 973 (1998).
* (34) C. Angioni, J. Candy, E. Fable, M. Maslov, A. G. Peeters, R. E. Waltz, and H. Weisen, Phys. Plasmas 16, 060702 (2009).
* (35) I. Pusztai, J. Candy, and P. Gohil, Phys. Plasmas 18, 122501 (2011).
* (36) H. Nordman, P. Strand, A. Eriksson, and J. Weiland, Plasma Phys. Contr. F. 47, L11 (2005).
* (37) See http://www.pdc.kth.se/resources/computers/lindgren/ for details on Lindgren.
|
arxiv-papers
| 2014-02-21T14:29:28 |
2024-09-04T02:49:58.543760
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andreas Skyman, Hans Nordman, Daniel Tegnered",
"submitter": "Andreas Skyman",
"url": "https://arxiv.org/abs/1402.5301"
}
|
1402.5382
|
###### Contents
1. Preface
2. 1\. Historical introduction: I
3. 2\. Some early works on cosmic rays
4. 3\. Historical introduction: II
1. 3.1 On Landé separation factors
2. 3.2 On Field Theory aspects of AMM
3. 3.3 Experimental determinations of the lepton AMM: a brief historical sketch
1. 3.3.1 On the early 1940s experiences
2. 3.3.2 Some previous theoretical issues
3. 3.3.3 Further experimental determinations of the lepton AMM
5. 4\. Towards the first exact measurements of the anomalous magnetic moment of the muon
6. List of some publications of A. Zichichi
7. List of some publications of R.L. Garwin
8. Bibliography
## Preface
Most of the work of leading scientists has always been characterized both by
an initial theoretical setting and analysis of the given problem under
examination and by the related experimental arrangement, and vice versa,
taking into account the main Galileian paradigm of scientific knowledge,
essentially given by the dialectic and inseparable relationships between
experimental bases and theoretical-formal structures from which arise the
rational thought. These scientists have always been interested both to
theoretical aspects and experimental data, like Jun John Sakuari (1933-1982)
as remembered by John S. Bell in (Sakurai 1985, Foreword). On the other hand,
just due to its Galileian nature, no history of theoretical physics can be
disjoined from experimental context, and vice versa. We have herein tried to
adopt a new way of doing history of science: namely, trying to delineate a
technical (or internal) history of a certain field of knowledge through the
life and the work of those people who have, at international level,
significantly and permanently contributed to it, along their life. Amongst
them, we shall consider, for example, some of the main works of A. Zichichi
and R. Garwin, namely those which have led to the first exact measurements of
the anomalous magnetic moment of the muon, one of the first precise test of
QED.
To be precise, in drawing up this work, we adopt that unique possible
historiographical methodology which has to be followed to pursue a correct and
objective historic-biographical report of the work of a given author under
examination, that is to say, the one consisting in giving primary and absolute
priority to the study and to the analysis of the original papers and works of
the author under examination (primary literature). Only subsequently it will
be then possible also to take into account the related already existent
secondary literature. This for trying to minimize, as much as possible, the
distortions and mystifications due to the unavoidable personal equation111This
concept has its own history which starts from Astronomy to Freudian
Psychoanalysis and Jungian Analytical Psychology. Here, we shall mean such a
term in the latter wider psychological meaning (see (Galimberti 2006) and
(Thomä and Kächele 1990, Vol. 1., Chap. 3, § 3.1)). Following (Carotenuto
1991, Chapter X), the personal equation is an unavoidable subjective factor
which influences on the evaluation of objective data, leading to different
visions of the phenomenological fields under examination. It is determined by
the individual history, by constitutional and typological elements and by
social-cultural factors. It acts as a perceptive filter, or rather as an
internal transducer, which redefines, according to personal parameters, the
reality, shaping the knowledge’s act. For instance, the various mythological
deformations and biases are mainly due to its action, so that, as regards
historical sciences, we agree with that historiographical method which gives
priority to the study of the primary literature of a given author, like her or
his works. (biases, complexes, mythicizations, etc) which is implicitly
presents in everyone of us. As picturesquely recalled by Vittorio de Alfaro in
(De Alfaro 1993, Introduction, p. 3), <<the historical reconstruction is
everything except a ’fractal’: indeed, the level of enlargement with which we
treat a historical process, greatly shall influence the conclusions that we
deduce>>. A similar case is besides also recalled by Bruno Rossi in (Rossi
1964, Preface), in which he warns on the impartiality with which himself has
written the history of cosmic rays, since he was directly involved, as a
leading actor, in the international research on this field, so that he does
not exclude to have given a major load to the work made by his research group.
On the other hand, this historiographical methodology is just that advocated
by Benedetto Croce himself (see (Croce 1938)) to obtain, through a rational
analysis of the sources, an impartial historical judgement devoid of any
biased or partial mystification.
There are non-negligible historiographical questions about the general history
of science, which we wish to outline too as an apology for the method used in
carrying on this work; in exposing such a historiographical problematic, we
mainly follow (Piattelli Palmarini 1992, Chap. I, §§ 2.I, 2.II and 2.III). Let
us say it immediately: such a problematic derives from the far from being
trivial questions existing between myth and science, which embed their roots
in the crucial historical passage from mythological to philosophical thought.
The relationships between myth and science are far from being ancient and
negligible: in this regards, it is enough to remember as Wolfgang Pauli
himself, after a long period of collaboration with Carl Gustav Jung, put much
attention to the possible links and intersections between epistemology and
analytical psychology, writing many interesting works on these arguments (see
(Jacob 2000), (Tagliagambe and Malinconico 2011) and references therein). The
French anthropologist Pierre Smith claims that the myth is always in nuce
(that is to say, implicitly) presents in the way in which each of us tells of
herself or himself, above all as regards her or him own past, this, in turn,
implying unavoidable distortions and mystifications which give rise to a
mythic production. In short, the myth is an efficient way to organize and to
coordinate the individual and collective memory. The Smith’s schoolmaster,
Claude Lévi-Strauss, said that the myth is a story continuously transformed by
who believe only of repeating it and to which, instead, he or she gives ”an
excess of meaning” whenever it is re-evoked.
Also in science there is an organization of collective and individual memory
where the mythical element may appear. For instance, Thomas S. Kuhn,
collecting a great number of interviews carried out with the founders of
modern physics, frequently noticed many inaccuracies and inconsistences as
concerns their biographies which resulted to be strangely logical, linear and
educationally edifying, but contrasting with the real facts and the original
sources; hence, Kuhn finished to conclude that the real history of science is
not so perfectly constructed and have not the direct pedagogical function of
those a little mythical stories told by handbooks and protagonists. Also
Gerald Holton has experienced an emblematic case of the same type: precisely,
interviewing in old age, Einstein convinced himself to have developed his
special theory of relativity on the basis of the results of the famous
Michelson-Morley experiment, building up, in such a manner, a logical
motivation to the birth of his theory222Also (Brown and Hoddeson 1983) confirm
that ¡¡people cannot be totally objective about the events in which they
participate; we tend unconsciously to reinterpret history in terms of present-
day values¿¿. But, in doing so, often the historical reality may go lost..
Indeed, such an experiment was yes carried out few years before the
Einsteinian publications, but Einstein did not know it when he formulated his
ideas. The Einstein reconstruction was logic and educationally efficacy even
if historically false; bona fide, he self-convinced himself that the things
were just gone so. In these cases, we should consider, according to Claude
Lévi-Strauss and François Jacob, the myth as an excess of meaning needed for
organizing the memory and for giving a logical and instructive meaning to its
contents, often to detriment of the real historical and chronological truth.
All this makes particularly difficult to do history of science; it may be
included in the wider unavoidable problematic concerning the already mentioned
personal equation, which would shed a certain shadow of discredit on the
history of science if it weren’t taken into the right account. For these
reasons, we think that the more correct historiographical method for carrying
out a scientific biography is that consisting at first in analyzing directly
the primary related literature, trying to prevent the non-objective
deformations given by the effects of this personal equation, almost desiring
to aspire to emulate the coldness or indifference of a psychoanalyst which
must simply reflect like a clean mirror (see (Thomä and Kächele 1990, Vol. 1.,
Chap. 3, § 3.1)) with the highest objectivity degree333The psychoanalysts try
to attain this by means of the so-called didactic analysis, which is strictly
correlated to the dualistic and dialectic interaction between transfer and
countertransfer phenomena (see (Thomä and Kächele 1990, Vol. 1., Chap. 3, §
3.1))..
This methodology, moreover, is the only way which permits us to may infer the
formation and evolution of the thought of a given author, undergoing his
creative process (as in this case), along her or his social-cultural and
scientific career. Such a historiographical method resembles, in a certain
sense, that already adopted by Francesco Giacomo Tricomi in (Tricomi 1967)
from the mathematics history side. On the other hand, all this had already
been highlighted by Sir Patrick M. S. Blackett (1897-1974) who remembered to
what distortions may lead the above recalled mythical production, by the so-
called scientific divulgation, if one does not take the right position respect
to the author under examination. The first and the most frequent
methodological error made by the historians of science concerns the location
of the own Ego, in the sense that often he puts herself or himself as main
subject rather than the author under examination. This decentralization of the
Ego is a primary epistemological process whose importance has been highlighted
by Sigmund Freud himself: indeed, following (Vegetti Finzi 1986, Chap. I, §
2), the scientific knowledge has reached its highest levels in concomitance to
real narcissistic wounds, as those occurred with the Copernican revolution,
the Darwinism and the Freudian psychoanalysis, each of these having just
reappraised the human Ego, self-limiting this. Such a self-limitation of the
Ego therefore corresponds to a general criterion of further improvement and
completion of knowledge, as highlighted too by Max Planck himself in (Planck
1964) (see also (Straneo 1947, Introduction)).
This Ego decentralization also plays a non-trivial role in historiography as
regards the position of the historian compared to the object under attention.
This fact, for instance, has been emblematically recalled by one of the most
important Italian mathematicians of the last century, Bruno Pini444For some
brief biobibliographical notes on the life and works of Bruno Pini
(1918-2007), see (Cavallucci and Lanconelli 2011) and (Lanconelli 2012)., who
had to say that <<sometimes, when one is called to commemorate someone, it
goes end up to overly speak of herself or himself>> (see (Lanconelli 2008));
this simple consideration may be extended to the general biographical
studies555The opposite case to this is that related to the deifications, like
those present in many hagiographies.. This is simply due to the human
weakness, scientist or not who he or she be, even turned toward the own
egocentric accomplishments (from which it follows, for some respects, the
well-known Latin maxim <<tot capita, tot sententiæ>>). Therefore, it turns out
clear what methodological importance has the examination of the original
scientific production of every author under examination, as we just hope to do
in any case herein analyzed, to avoid any possible mystification.
Finally, since, according to Chen Ning Yang (see (Yang 1961, Preface)), <<a
concept, especially a scientific one, have not full meaning if it is not
defined respect to that knowledge context from which it derived and has
developed>>, each examined original work or paper of the author under
consideration shall be even contextually laid out into the related theoretical
framework of the time, so that, where possible, a brief historical recognition
should be mentioned as a contextual story meant as follows. In a certain
sense, we might say to follow an epistemological path analogous to that
outlined by Stephan Hartmann in (Hartmann 1999) where he claims on the
importance, above all in hadron physics, to consider a theoretical model as
the result of an interpreted formalism plus a story, this last being meant
both as a narrative but rigorous told around the formalism of the given model
and as a complement of it, hence an integral part of the model; the
relationships between formalism and story are then placeable out into the
wider class of relationships subsisting between the syntactic and semantic
parts of a general physical model which are unavoidable just in Physics.
Therefore, our work might be considered as a sort of making a story to certain
groups of works of the author under examination in order to get an overall
historical view of the subject matter in which her or him worked. For these
reasons, it is also indispensable refers us to the general technical-
scientific literature to support what said. Only doing so, it will be possible
to pursue the highest objectivity degree and historical correctness in
descrying the scientific figure of an author, trying to avoid the above
mentioned irrationality elements. At the same time, in dependence on the
scientific level of the treated author, with this method it shall be possible
to outline a history of the related work area.
Another confirmation of the validity of the above mentioned work program
follows from some epistemological considerations about the foundations of
science, due to the modern French school which goes from G. Bachelard, A.
Koiré and G. Canguilhem to the structuralists J. Lacan, C. Lévi-Strauss, L.
Althusser, M. Foucault, F. Regnault, A. Badiou and F. Wahl. Indeed, following
(Cressant 1971, Introduction), the scientific activity should be looked at as
a constructive process which pulls out the truth or the essence of the real
objects that will constitute the central core upon which building up the
corresponding knowledge object, trying to separate ideological questions from
the mere scientific contents. Read an arbitrary work just means try to
separate the general ideological and philosophical context from the scientific
one; it means to analyze the problematic frame within which this work has been
conceived, rebuilding up the prime structural causes from which it shall
develop. In doing so, a passive and sterile lecture will be replaced by an
active and productive re-enact (see (Wahl 1971)), almost analogously to what
foreseen both by the Robin Collingwood historicism (see (Kragh 1990) and
(Iurato 2013)) and by Wilhelm Dilthey methodological hermeneutics, according
to which any written source should be laid out into the proper original
historical context, according to the Zeitgeist of the time. Following (Schultz
1969, Chapter I) and (Wertheimer 1979, Chapter 1), the ideology is always an
unavoidable judgment component of human being, hence also of every historian,
since it is a common perspective to conceive the history as chiefly due to the
subjective idiosyncrasies and to the preconceptions which will play the role
of selective mental grid of what to consider or not and of how to interpret
this. Contrarily to what one could thought, the ideology is also an
unavoidable component of the normal scientific context: in this regards, see
(Boudon 1991, Chapter VIII).
## 1\. Historical introduction: I
Mario Gliozzi, in666The Enciclopedia delle Matematiche Elementari e
Complementi has been the most important and notable Italian encyclopedic
handbook on mathematical sciences and their applications, reviewed abroad as
one of the main encyclopedic work made in this context, as valuably remarked
by (Archibald 1950) and (Miller 1932). This article of Mario Gliozzi was the
first systematic attempt to outline a brief history of physics. It was later
retaken as a first core for drawing up another more extended article published
in the 1962 Nicola Abbagnano treatise on the history of science, in turn
posthumously enlarged and revised by the sons of Mario Gliozzi, in the new and
definitive 2005 edition (Gliozzi 2005), which is one of the most complete
textbook on the history of physics. Herein, we have mainly followed (Gliozzi
1949) because of its conciseness which is functional to the aims of this
section, referring to (Gliozzi 2005) for a more complete and in-depth view.
(Gliozzi 1949, § 30), outlines the main features of the experimental physics
through the last 19th Century decades to the 1940s. This was an almost unique
period for the history of physics since, from the new results of atomic
physics of the 19th Century end, appears, in all its complexity, the new
submicroscopic Weltbild to whose knowledge inextricably taken part
philosophical, theoretical and experimental physical questions above all
characterized by the crucial passage from the classical determinism to the
modern probabilism as recalled by (Pignedoli 1968, Chap. I) which gives a
clear and synthetic historical summary of this critical epistemological step.
Above all, the experimental physics had needed for new methods, techniques and
tools to approach and to examine this unexpected world so closed to our direct
perceptions, this, in turn, implying the formulation of new theories to
explain it at the light of these experimental results which arose from the
discovery of cathodic and anodic emissions, channel and X rays, and
radioactivity777The spontaneous radioactivity was discovered by H. Becquerel
in 1896 under advice of H.J. Poincaré. Indeed, the latter suggested to the
former to investigate on the possible relationships between optical
fluorescence and X rays, which revealed to be fake, but that led, for
serendipity, to the discovery of radioactivity (see (Segrè 1999, Chap. 1)).
(see (Born 1976, Chap. 2)). In this regards, in 1897, Charles T.R. Wilson
discovered that the ions produced in air by ultraviolet and X rays as well as
by radioactive radiations, acted as condensation nuclei of water steam
suitably supersaturated by rapid adiabatic expansion. This notable discovery
was at the basis of the so-called cloud chamber, one of the first valuable
displaying particle detector, first set up at the Cambridge Cavendish
Laboratory in 1896 and subsequently improved by Wilson (see (Wilson and
Littauer 1965, Chap. 3) and (Yang 1969, Chap. 1)), so that it is often called
too Wilson chamber; it will play a fundamental role in experimental atomic
physics, even to be said ”an open window on the world” (E. Persico). The
particle detectors may be classified into two main categories, namely the
displaying detectors and the optical (or electronic) detectors; the first ones
comprise the Čerenkov and scintillation counters, the Wilson (or cloud),
bubble, spark and photographic emulsion chambers, whereas the second ones
include the ionization chamber, the Geiger-Müller, the proportional and solid-
state counters (see (Segrè 1999, Chap. 3), (Tolansky 1966, Chap. 17) and
(Chiavassa, Ramello and Vercellin 1991, Chap. 2)).
Following (Segrè 1999, Chap. 1), after the discovery of electron in 1897, the
first atomic models due to J.J. Thomson, E. Rutherford and N.H. Bohr at the
beginnings of 20th Century, together with the introduction of quanta by M.
Planck and A. Einstein as regards the electromagnetic radiation, led to the
formulation of quantum mechanics which succeeded to explain many atomic
phenomena. At the same time, after the discovery of atomic nucleus in 1911,
the new quantum theories gradually opened the way to nuclear physics with the
first $\alpha$ particle bombardment phenomena which led, after the 1919
pioneering Rutherford discovery of the proton, to the definitive 1924-25
experimental ascertainment of such a particle by P.M.S. Blackett, who was a
Rutherford’s pupil (see (Gliozzi 1949, § 30, footnote 239)) and (Gamow 1963,
Chap. VIII)). Thereafter, on the basis of the previous works made by R.J. Van
de Graaff888Nevertheless, the principle of the method upon which relies the
running of such machines is quite similar to one already studied by A. Righi
at the end of 19th Century (see (Gliozzi 1949, § 30, footnote 241)))., J.D.
Cockcroft, E.T.S. Walton, H. Greinacher and R. Wilderöe, in 1933 E.O. Lawrence
and S. Livingston built up the first particle accelerator, the so-called
cyclotron (see (Wilson and Littauer 1965) and (Segrè 1976, Chap. XI)), based
on the resonant acceleration method. Independently of each other, in 1944 the
Russian physicist V.I. Veksler proposed a new particle accelerator based on
the phase stability method, while in 1945 E.M. MacMillan proposed an analogous
particle accelerator which will be called synchrotron. See (Lee 2004) for a
complete and masterful updated knowledge on accelerator physics, in which
there are also interesting historical notes.
Retaking into account some previous experiences made by W. Bethe and H.
Becker, the spouses I. Curie and F. Joliot discovered a new particle, already
suggested by Rutherford in 1920 and whose exact nature was subsequently
experimentally ascertained by J. Chadwick who called it neutron (see (Hughes
1960)); the Curie’s experiences given rise to the first artificial
radioactivity phenomena. In the years 1932-1934, a new particle was observed,
almost at the same time, by many scientists: amongst them, by I. Curie and F.
Joliot in collision phenomena with $\alpha$ particles, by C.D. Anderson in the
United States and by P.M.S. Blackett with G. Occhialini in England, in
experiences concerning cosmic rays (see (Gliozzi 1949, § 30, footnote 243))),
which was called, by C.D. Anderson, positive electron, or positron. Such a
particle had already been theoretically provided by P.A.M. Dirac with his
elegant 1930s electron theory, which, inter alia, established too the so-
called charge conjugation invariance principle; this new particle was
experimentally determined having mass almost equal to the electron one but
with positive charge. The discovery of positron was a celebrated experimental
confirmation of Dirac’s electron theory, which was besides unknown to Anderson
but not to Blackett and Occhialini which made their above researches at the
Cavendish Laboratory of which Dirac was a member, at that time (see (Rossi
1964, Chap. VI)).
In the years 1933-1934, taking into account the previous works of the Curie-
Joliot spouses, E. Fermi was the first to use neutrons as collision particles,
in place of $\alpha$ particles: indeed, he rightly argued that neutrons were
more suitable to this, due to the lack of electrostatic repulsion respect to
an atomic nucleus; slow neutrons turned out to be very efficient in breaking
the atomic nucleus. Such ingenious intuitions were put in practice in Rome, by
E. Amaldi, O. D’Agostino, B. Pontecorvo, F. Rasetti and E. Segrè, where it was
carried out the celebrated experiences with slow neutrons (see (Gliozzi 1949,
§ 30, footnotes 245), 246)) which will lead to the discovery of nuclear
fission and to the subsequent chain reactions, all this at the World War II
eve (see (Gliozzi 1949, § 30, footnotes 247)-251))). It was the beginning of
the nuclear physics with the use and applications of the nuclear energy by E.
Fermi in 1942, in this, the Italian school having been leader in the
international research framework of the time. In this regards, from a
historical viewpoint, it is enough to give a glance to the fundamental works
(Wick 1945; 1946) to witness all this, which represented the first treatise on
the new neutron physics; this unique two-volume treatise is the most valuable
historical source which exposes the ”state of the art” of that time as regards
this new chapter of nuclear physics.
In the decade 1920s to 1930s, the building of quantum mechanics was achieved,
with the elegant and rigorous formulation given by P.A.M. Dirac in his
celebrated textbook (Dirac 1958), whose first edition date back to 1930 and
that is still the classical and definitive treatise on the subject with its
last 1958 fourth edition; in it, the chapters on the new quantum
electrodynamics were updated till the results of 1950s. Once discovered the
neutron, one of the main problem of the new nuclear physics was to determine
the interaction forces among the constituents of the atomic nucleus, that is
to say (d’après W. Heisenberg, D. Ivanenko and E. Majorana) protons and
neutrons, which have been interpreted as two states of the same particle,
called nucleons, having different values of a well-determined numerical
parameter called isospin. This last quantum number is related to the formal
description of the notion of isotopic (or isobaric) invariance, that was first
introduced by W. Heisenberg in 1932, then used by B. Cassen and E.U. Condon in
1936 and by E.P. Wigner in 1937 (see (Landau and Lifšits 1982, Chap. XVI, §
116)) and subsequently applied to the classification of other subnuclear
particles, as we will see later. The next twenty years will see the birth of
the so-called quantum field theory (QFT), before all with the new quantum
electrodynamics999For this fascinating story, see (Schweber 1983; 1994), (De
Alfaro 1993) and (Weinberg 1999, Vol. I, Chap. 1). In (Schweber 1983; 1994)
there is also an extensive history of quantum field theory of 20th Century,
both from an internal and external historical standpoint. (QED), which
develops, according to the Galileian scientific method, in close concomitance
with the related experimental physics contexts, above all those concerning the
radioactive emissions and the cosmic radiation, which will play a fundamental
role in developing the nuclear and subnuclear physics; within the theoretical
framework given by the incoming QFT, they will flow into the dawning of
particle physics. The quantum electrodynamics started with the works of W.
Heisenberg, W. Pauli and P.A.M. Dirac, culminating in the Dirac’s radiation
theory in which the photons (already experimentally determined by E. Mayer and
W. Gerlach in 1914 - see (Born 1976, Chap. 4, § 24)) are the quanta of the
electromagnetic field, this theory having been taken as main model for
building up any further quantum field theory, like the electronic-positronic
field, the nucleonic and the mesonic ones, and so on (see (Fermi 1963, Chap.
1, § 1)). In the decade from 1940s to 1950s, the electromagnetic field has
been successfully quantized starting from the Maxwell’s equations, while the
electronic-positronic field has been treated starting from the Dirac’s
electron theory with a new formal process introduced by E.P. Wigner and W.
Pauli in 1928, called second quantization, which is a modification of the
previous quantization procedures to account for supervened spin statistic
problems. The situation concerning the electronic-positronic and nucleonic
fields was instead much more complex (see also (Weinberg 1999, Chap. 1, §
1.2)).
For our historical ends, we are more interested towards those aspects of
particle physics history regarding both radioactive decays and cosmic rays,
which, as already said, have played a very fundamental role in the dawning of
particle physics and whose historical paths often have intertwined each other.
Indeed, following (Weinberg 1999, Chap. 1, § 1.2), despite significant
successes achieved by QFT (in primis, the Dirac’s ones), a certain
dissatisfaction held towards it for all the 1930s, above all due to its
apparent incapacity to explain many new phenomena coming from the cosmic
radiation as well as all the new type of particles contained in it. On the
other hand, as regards the various proposed theories explaining the
radioactive emissions, above all that regarding the $\beta$ decay to have
played a fundamental role both in understanding the nuclear structure and in
developing QFT. Following (Persico 1959, Chaps. XI, XII and XIII), (Segrè
1999, Chap. 8), (Castagnoli 1975, Chap. 3, § 3.7), (Tolansky 1966, Chap. XV),
(Born 1976, Chap. 7, § 53) and (Friedlander and Kennedy 1965, Chaps. 6 and 7),
the theory of $\alpha$ emission was successfully achieved by G. Gamow, R.
Gurney and E. Condon in 1928-29, as the first attempt to apply the new quantum
theories to nuclear structure, while the development of the theory of $\gamma$
emission was parallel to that of quantum theory of radiation which started
with an interpretative theory analogous to the first atomic radiative
emissions and continued, through 1920s to 1950s, with the works of E.
Rutherford, H. Robinsson, W.F. Rowlison, E.N. Da Costa Andrade, L. Meitner,
H.J. Von Baeyer, O. Hahn, C.F. Von Weizsäcker, H.A. Bethe, W. Heitler, O.
Klein, Y. Nishina, J.P. Thibaud, E. Feenberg, H. Primakoff, E.P. Wigner, M.
Goldhaber, J.M. Blatt, V.F. Weisskopf, E. Wilson, K.T. Bainbrige, A.W. Sunyar,
P.B. Moon, R.L. Mössbauer, and others (see (Heitler 1953)).
Instead, the $\beta$ emission, in both its $\beta^{-}$ and $\beta^{+}$
components, shown to have particular difficulties to be laid out into a
coherent theoretical description, above all in relation to the interpretation
of the related continuum electron velocity spectrum which was one of the most
serious theoretical nuclear physics problems of the time. The main theoretical
problem concerned an apparent non-validity of the energy and spin conservation
laws at every elementary emission act, that Bohr attempted to justify invoking
a sort of mean validity of it. Nevertheless, in analogy with the case of
$\gamma$ emission (which was experimentally excluded to be associated with a
$\beta$ emission), E. Fermi proposed an alternative and more valid
quantitative interpretation based on the possible contemporaneous emission of
a new particle, called neutrino, together the electron involved in each
elementary $\beta$ emission act. The neutrino was a particle first
theoretically predicted by H. Weyl in 1929 (see (Weyl 1931) and references
therein) on the basis of Dirac’s electron theory, but his hypothesis was at
once refused since it did not verify the parity symmetry. It however will be
reconsidered later in 1957 after the work of T.D. Lee and C.N. Yang on parity
violation. Thereafter, in 1930, W. Pauli proposed to consider such a new
particle to explain the lack of validity of the above mentioned conservation
laws as regards $\beta$ decay, which was later so named by E. Fermi in 1934.
It was supposed to have zero mass and spin one-half. The quantitative theory
of $\beta$ emission, first proposed by E. Fermi in 1934 and later improved by
E.J. Konopinski, H.J. Lipkin, G. Uhlenbeck and H. Yukawa, is a very general
one which may be also applied to other type of interactions. Taking into
account previous theoretical studies, as already said mainly due to W.
Heisenberg, E.U. Condon and E.P. Wigner, this theory assumes proton and
neutron as two distinct quantum states of a unique particle, the quantum of
the nucleonic field, called nucleon, which can go either into one, or into the
other, of these two states just through a $\beta$ decay, emitting one
positive/negative electron and one neutrino101010In this historical account,
it is no possible to consider the problem of helicity of the neutrino as well
as the related questions inherent the existence or not of the antineutrino,
the alternative theory of E. Majorana, and so on. See (Fermi 1963, Chap. 1)
and (Yang 1969, Chap. 1).. To be precise, we have a $\beta^{-}$ emission in
the transformation of a neutron ($n$) into a proton ($p$) according to a decay
process of the type $n\rightarrow p+e^{-}+\nu$, with the emission of one
(negative) electron ($e^{-}$) and one neutrino ($\nu$); we have a $\beta^{+}$
emission in the transformation of a proton into a neutron according to a decay
process of the type $p\rightarrow n+e^{+}+\nu$, with the emission of one
positive electron ($e^{+}$) and of one neutrino. Nevertheless, as it will be
proved later by L.M. Lederman and co-workers as well as by other workers,
there exists another type of neutrino different from the one produced by the
above proton and neutron decays (denoted by $\nu_{e}$), namely the neutrino
produced by $\mu$ and $\pi$ meson decays (denoted by $\nu_{\mu}$). On this
last point, we shall return later.
For a certain time, it was supposed that the nuclear forces could be explained
through a nucleonic field associated to the pair electron-neutrino (see
(Polara 1949, Chap. IV, § 7), (Segrè 1976, Chap. X) and (Weinberg 1999, Chap.
1, § 1.2)) and whose quantum was the neutrino. It was hypothesized that proton
and neutron were linked together by means of an exchange of one neutrino, like
in case of the ionized molecule $H_{2}^{+}$ where the force between the atom
$H$ and the ion $H^{+}$ became attractive at distances of the magnitude of
$10^{-15}$ just thanks to a periodic exchange of the unique available
electron111111Indeed, the notion of exchange force comes from the quantum
theory of chemical bond (see (Slater 1980)). It will be later extended first
to nuclear physics thanks to the work of E.P. Wigner (see (Eisenbud and Wigner
1960, Chaps. 5 and 11)), then to the particle physics context.. Nevertheless,
in this way it wasn’t possible to account for nucleus stability questions, so
that this hypothesis (which had also been considered by E. Fermi in his 1934
theory) had to be rejected. All that, however, will be one of the starting
points of the subsequent pioneering 1935 Yukawa’s work (see (Yukawa 1935)).
Following (Castelfranchi 1959, Chaps. XX and XXI, §§ 231, 262), the positive
electrons $e^{+}$ are emitted only in artificial radioactive decays and were,
almost at the same time, discovered in 1932-33, by many researches,
independently of each other, amongst whom C.D. Anderson, R.A. Millikan and
P.M.S. Blackett with G. Occhialini in experiences with cosmic rays, as well as
by I. Curie with F. Joliot, by L. Meitner, C.Y. Chao, H.H. Hupfeld, J.R.
Richardson and J. Chadwick in experiments on induced radioactivity. It was
later observed, above all by Blackett and Occhialini, that these positive
electrons just were the antielectrons expected by the Dirac’s electron theory.
At the same time in which it was carried out the above experiences on $\beta$
decay, a prominent role began to have the study of a new type of very high
energy radiation, most highly penetrating, that is to say, the cosmic
radiation. Between the 1940s and 1950s, the unique available high energy
sources were the cosmic rays, until the coming of particle accelerators in the
1950s which allowed more controllable energetic sources. In that period, there
was a research competition between experiences made on cosmic rays and those
through particle accelerators.
Following (Polara 1949, Chap. V), (Castelfranchi 1959, Chap. XXIII), (Rossi
1964), (Tolanski 1966, Chap. 18), (Born 1976, Chap. 2, § 15), (Brown and
Hoddeson 1983), (Schlaepfer 2003) and (Carlson and De Angelis 2010), the
cosmic radiation was discovered, in the early years of 20th Century, by J.
Elster with H. Geitel in German and by C.T.R. Wilson in England, from the
observation of a weak residual electrification in perfectly isolated
electroscopes. Some year later, E. Rutherford with H. Lester and H.L. Cooke,
together to J.C. McLennan and E.F. Burton, showed that 5 cm of lead reduced
this mysterious radiation by 30% while an additional 5 tonnes of unrefined
lead failed to reduce the radiation further. Such a phenomenon was immediately
attributed to a not well identified external strong penetrating radiation,
maybe coming from the Earth. Thanks to a new electroscope made by T. Wulf in
1907, it was possible to observe that this external radiation did not decrease
with the altitude but, in some cases, even increased, so that it could not
come from the Earth as it was later confirmed first by A. Gockel in 1909-10,
then both by V. Hess with W. Kolhörster in 1911-14 and by D. Pacini in 1911,
the former with a series of experiments made with flight balloons equipped
with electroscopes, and the latter121212Very few are the textbooks which quote
the Italian physicist Domenico Pacini (1878-1934) as one of the pioneers of
cosmic-ray research; amongst them (Castelfranchi 1959, Chap. XXIII) - where,
inter alia, it is also possible to find interesting historical notes
throughout the text - and (Gliozzi 2005, Chap. 16, Section 16.12). For this
historical case, and for a modern general historical revisitation of the
cosmic ray story, see (Carlson and De Angelis 2010). by means of deep sea
immersions of electroscopes. The World War I interrupted the researches on
this strange type of penetrating radiation, then retaken later in 1920s and
1930s with the experiences of A. Millikan, E. Regener, G. Pfotzer, I.S. Bowen,
H.V. Neher, H. Tizard, A. Piccard, M. Cosyns, W. Bothe, J. Clay, A. Corlin, D.
Hoffmann, D.V. Skobeltzyn, E. Steinke, G.H. Cameron, P.S. Gill, G.L. Locher,
E.J. Williams, C.F. Von Weizsäcker, L. Nordheim, J.B. Street, H. Kulenkampff,
E.C. Stevenson and others, continued until the 1940s and 1950s pioneering
experimental works of T.H. Johnson, L.W. Alvarez, A.H. Compton, C.D. Anderson,
D.A. Glaser, S. Neddermeyer, G.E. Roberts, R.E. Marshak, H.A. Bethe, B. Rossi,
F. Rasetti, G. Bernardini, S. De Benedetti, C. Størmer, G. Lemaître, M.S.
Vallarta, V. Bush, G. Clark, P. Bassi, M. Schein, H.L. Bradt, B. Peters,
P.M.S. Blackett, G. Occhialini, C.F. Powell, C.M.G. Lattes, M. Conversi, E.
Pancini, O. Piccioni, H. Muirhead, J.F. Carlson, J.R. Oppenheimer, P. Auger,
P. Ehrenfest, L. Leprince-Ringuet, S.I. Tomonaga, G. Araki, G.D. Rochester,
C.C. Butler and others (see (Brown and Hoddeson 1983, Part III) and (Rossi
1964)). As regards the related experimental techniques employed, the first
group of researches were conducted by means of ionization chambers and, above
all, Wilson chambers, these last first used by D. Skobeltzyn in 1929, in the
version improved by P.M.S. Blackett and under the action of strong magnetic
fields. In the second group of researches, instead, besides Wilson chambers,
sequential Geiger-Müller counters were also used as well as photographic
emulsion chambers in the version improved by C.F. Powell and G. Occhialini on
the basis of the previous 1937 works made by M. Blau and H. Wambacher on
nuclear emulsions.
From the experimental data provided by all these notable works, in particular
from the various absorption curves related to this cosmic radiation and
related geomagnetic effects, it was possible to identify two secondary
components, departing from a primary one, which have different nature
according to the results of the azimuthal and latitude effects, both then
characterized, but in a different manner, by the so-called east-west asymmetry
phenomenon which is closely connected with an asymmetry related to cosmic ray
intensity distributions, in turn related to the geometry of the so-called
Størmer cones which give the allowed trajectories of cosmic rays under the
action of the geomagnetic field. For this, it was identified both a hard
component, much penetrating, and a soft component, little penetrating. The
terrestrial magnetic field and the atmosphere, constitute two protective
layers against the reaching of cosmic rays on the Earth’s surface: once that
high energy primary cosmic rays hit terrestrial magnetosphere (involving too
the Van Allen belts - see (Rossi 1964, Chap. XIII)) and the upper atmosphere,
they interact with the encountered atoms (above all, the nitrogen and oxygen
ones), the resulting collisions producing fragment’s stars (that is to say,
multiple traces outgoing from a same collision point) and atmospheric showers
of many particles, at that time most unknown, according to multiple production
processes theorized by W. Heisenberg and G. Wataghin. Theoretical attempts to
explain the related phenomenology led to the so-called cascade theory of
cosmic showers which was worked out, in 1930s and 1940s, mainly by J.F.
Carlson with J.R. Oppenheimer in the United States and by M. Blau, H.
Wambacher, L. Jánossy, H.A. Bethe, W. Heitler, J. Hamilton, H. Peng and H.J.
Bhabha in Europe, but also with notable contributions by L.D. Landau, I.E.
Tamm, V.L. Ginzburg, S.Z. Belenky, H.S. Snyder, R. Serber, W.H. Furry and S.K.
Chakrabarty. Thanks to this theory, it was possible to ascertain that the soft
component of cosmic radiation was mainly made by high energy electrons and
photons, whereas the determination of the particle composition of the hard
component was more difficult to achieve; and, at this point, the cosmic
radiation and $\beta$ decay research pathways meet.
Following (Polara 1949, Chaps. V and VI), (Rossi 1964), (Muirhead 1965, Chap.
1), (Yang 1969, Chap. 2), (Segrè 1976, Chap. XII), (Born 1976, Chap. 2),
(Zichichi 1981), (Brown et al., 1989), (Segrè 1999, Part III), (Zichichi 2000,
II.1-3a-II.1-4b) and (Gliozzi 2005, Chap. 16, Section 16.12), whilst the
(local) cosmic radiation soft component was ascertained to be mainly made by
high energy electrons and photons, many perplexities yet held as regards the
hard component whose constituents seemed do not belong to the set of
elementary particles then known. Indeed, the latter appeared to possess either
positive or negative electric charge, while the analysis of experimental data
strongly indicated the existence of a particle having mass intermediate
between that of the proton and of the electron, and probably in the region of
100-200 electron masses ($m_{e}$). This suspicion was verified by S.H.
Neddermeyer with C.D. Anderson and by E.C. Stevenson with J.C. Street, in the
years between 1936 and 1938, who photographed quite instable particles with
masses estimated (above all by R.B. Brode and co-workers) to be about 200-240
$m_{e}$, stopping in a cloud chamber. These particles were generically called
mesotrons by C.D. Anderson or mesons by W. Heisenberg, because of their mass
value; it ended then to prevail the second name131313Following (Gamow 1966,
Chap. VIII), the name mesotron was discarded because, under advise of
Heisenberg father, a professor of classical language, the right etymology of
the term was inclined towards the term meson.. As said above, notwithstanding
many difficulties subsisted in the research field devoted to cosmic radiation,
the tenacity of researchers led to the conclusion that this type of radiation
(namely, the hard one) was formed by new particles both positively and
negatively charged with mass intermediate between the electron and proton
ones. Nevertheless, these authors did not know the 1935 Yukawa work and what
there was predicted141414The discovery of the meson, as well as that of the
positron, has been preceded by theoretical forecasts respectively due to H.
Yukawa in the first case, and to P.A.M. Dirac in the second one., mainly due
to the fact that it was published in a journal not widely known outside Japan
(see (Kragh 2002, Chap. 13)). In such a paper, starting from the Heisenberg
work on nuclear structure and from the Fermi theory on $\beta$ decay, Yukawa
supposed that proton and neutron could interact through a quantized field (the
mesonic one) of which, in analogy with the electromagnetic case, he computed
too the main physical properties of the related quantum.
Yukawa made a further suggestion about the properties of his hypothetical
particle: indeed, in order to simultaneously account for nuclear $\beta$ decay
and for the fact that the meson had not been observed (at that time), he
suggested that it decayed spontaneously into one electron and one neutrino in
a time which was estimated to be about $10^{-7}$ sec. In 1938, with some first
experiences made by H. Kuhlenkampff, the question related to meson decay was
one of the most debated of the period between the 1930s and the 1940s, which
had, as main protagonists, W. Heisenberg, P.M.S. Blackett, H. Euler, A.H.
Compton, B. Rossi, D.B. Hall, N. Hilberry, J.B. Hoag, W.M. Nielsen, H.V.
Neher, M.A. Pomerantz, G. Bernardini, G. Cocconi, O. Piccioni, M. Conversi and
others. An apparent verification of this property was obtained by E.J.
Williams and G.E. Roberts in 1940, observing a $\beta$ decay of a particle of
mass about 250 $m_{e}$ into a cloud chamber. In this period, attempts to
identify the generic meson observed in the cosmic radiation by C.D. Anderson
and co-workers, with the Yukawa’s particle were done, notwithstanding that
will reveal out to be false151515The occurred mistaken particle’s
identifications will be mainly due to the experimental difficulties to
identify the related spin values which are the only ones that allow to discern
between particles having equal mass and charge values (see (Villi et al. 1971,
Introduction)).. Furthermore, an apparent experimental evidence for such an
identification was provided by the measurements of the meson lifetimes by F.
Rasetti in 1941 and by many others, amongst whom B. Rossi, N.G. Nereson, K.I.
Greiser, R. Chaminade, A. Freon and R. Maze. At the same time, the comparison
of the Yukawa nucleonic theory with the cosmic radiation one (mainly due to
J.R. Oppenheimer and J.F. Carlson) in the light of the obtained experimental
data, above all those made by M. Conversi, E. Pancini and O. Piccioni in Italy
in the years 1943-1945 and by R. Chaminade, A. Freon and R. Maze in France in
1945, led R.E. Marshak and H.A. Bethe to suggest in 1947 the possible
existence of two different types of mesons, also on the wake of what
previously envisaged by E. Fermi, E. Teller and V.F. Weisskopf. Nevertheless,
due to the World War II circumstances, the Japanese physicists worked in an
almost full isolation and most of their researches of that time were
recognized only later. Indeed, S.I. Tomonaga, Y. Tanikawa, S. Sakata and T.
Inoue, already in 1943 had proposed the hypothesis of the possible existence
of two different types of meson. The main conclusion of the above mentioned
experiences was that the negative and positive mesons differently interacted
with matter: in fact, measuring the related capture rates $\lambda_{c}$, the
positive ones decayed as they were more or less free, whereas the negative
ones were attracted by the nuclei, reacting in a strong manner with heavy
nuclei and in a weak manner with the light ones, and this wasn’t what
predicted by S.I. Tomonaga and G. Araki in 1940 on the basis of Yukawa’s
theory.
The clarification of such a question came from the technological developments
which have even been historically connected with the scientific progress of
ideas. Indeed, starting from the previous experimental techniques due to S.
Kinoshita and C. Waller, it was set up new nuclear emulsion detectors in 1940s
by the Bristol group made by C.F. Powell, G. Occhialini, C.M.G. Lattes and H.
Muirhead, thanks to which it was possible to effectively identify two types of
mesons. This conclusion was further confirmed, in 1948, by other experiments
run both by the above Bristol group with also Y. Goldschmidt-Clermont, D.T.
King and D.M. Ritson, and, at Berkeley, by E. Gardner and C.M.G. Lattes with a
particle accelerator. These two types of mesons, detected by the above
fundamental experiences, led to identify two first classes of mesons: in one,
it was included those mesons at first called primary mesons, then meson $\mu$
or muon; in the other, it was included lighter mesons at first called
secondary mesons, then meson $\pi$, or pion. In the former falls the meson
foreseen by C.D. Anderson with S.D. Neddermeyer and detected by one of the
celebrated Conversi-Pancini-Piccioni experiences, while in the latter it
should fall the Yukawa’s one. Thus, the $\pi$ meson provided the glue for
nuclear forces and undergone to the following main decay-chain-reaction
$\pi\rightarrow\mu\rightarrow e$ where the first decay scheme
$\pi\rightarrow\mu+\nu_{\mu}$ was first studied, in 1948, by U. Camerini, H.
Muirhead, C.F. Powell and D.M. Ritson as well as by J.R. Richardson. Then, the
various experiences performed on negative and positive counterparts of cosmic
rays led to the conclusion according to which both these two types of meson
may be either positively or negatively charged, denoted by $\mu^{\pm}$ and
$\pi^{\pm}$, the Yukawa’s one being of the type $\pi^{-}$. The $\pi$ meson
significatively and strongly interacts with atomic nuclei, contrarily to the
$\mu$ meson which is mainly subjected to weak interactions; the former has
mass about 270-300 $m_{e}$ while the latter has mass about 200-210 $m_{e}$.
The $\pi$ mesons decay in $\mu$ mesons and these, in turn, decay in electrons,
by means of reactions of the type
$\pi^{+}\rightarrow\mu^{+}+\nu_{\mu},\ \pi^{-}\rightarrow
e^{-}+\bar{\nu}_{\mu},\ \mu^{+}\rightarrow e^{+}+\nu_{\mu}+\bar{\nu}_{\mu},\
\mu^{-}\rightarrow e^{-}+\nu_{\mu}+\bar{\nu}_{\mu}$
where $\nu_{\mu}$ denotes the neutrino and $\bar{\nu}_{\mu}$ the related
antineutrino. As said above, the neutrino $\nu_{e}$ was determined to be
different from the neutrino $\nu_{\mu}$ by experiences made by G. Danby, J.M.
Gaillard, K. Goulianos, L.M. Lederman, N. Minstry, M. Schwartz and J.
Steinberger in 1962 at Brookhaven. Nevertheless, the possible existence of two
different types of neutrino was first theoretically proposed by G. Puppi in
1948 in studying the universality of Fermi weak interactions (Puppi triangle)
on the basis of decay processes involving $\mu$ and $\pi$ mesons: indeed,
following (Hughes and Wu 1977, Vol. I, Chapter I), the most important
contribution resulting from the study of the muon so far, was probably the
revelation of the close relationships between muon decay, muon capture and
nuclear beta decay, just known as the three sides of the Puppi triangle.
Moreover, further experiences made by F. Reines and C.L. Cowan Jr. in 1959, by
M.G. Inghram and J.H. Reynolds in 1950, by C.S. Wu in 1960 and by G.
Bernardini161616See (Zichichi 2008) for brief recalls on the work of Gilberto
Bernardini. and co-workers in 1964, showed that
$\nu_{e}\neq\bar{\nu}_{e},\nu_{\mu}\neq\bar{\nu}_{\mu},\nu_{e}\neq\nu_{\mu}$.
Afterwards, to explain the experimental evidence for the charge independence
of nuclear forces as well as to account for the soft component in cosmic
radiation, independently of each other, N. Kemmer in 1938, H. Tamaki in 1942
and H.W. Lewis, J.R. Oppenheimer with S.A. Wouthuysen in 1947, pointed out
that in addition to Yukawa’s charged meson, a neutral meson had to exist,
whose experimental evidence was obtained in the years 1950-1951 by A.G.
Carlson, J.E. Hooper and D.T. King, by R. Bjorkland, W.E. Crandall, B.J. Moyer
and H.F. York and by W.K.H. Panofsky, R.L. Aamodt, H.F. York and J. Hadley.
Such a meson, denoted by $\pi^{0}$, decays according to a law of the type
$\pi^{0}\rightarrow\gamma+\gamma$, being $\gamma$ a photon. The $\mu$ and
$\pi$ mesons will be generically called too L mesons. At this point, it was
generally felt that the neutral pion discovery marked the end of particle
searches, whereas the decay $\pi^{0}\rightarrow\gamma+\gamma$ marked instead
the opening of new horizons in subnuclear and theoretical physics: for
instance, on the basis of this decay process, J. Schwinger formulated the so-
called partial conservation of the axial current (PCAC) hypothesis in quantum
electrodynamics and quantum chromodynamics, opening new fruitful chapters in
current algebra theory. Indeed, studying in-depth the nuclear interactions of
the particles of cosmic rays, it was possible to discover other elementary
particles. As said above, the $\mu$ mesons weakly interact with matter whereas
the $\pi$ mesons are nucleary active particles together other ones which were
discovered at high altitudes in many mountain laboratories located in
different world areas, amongst which those at Aiguille du Midi in the French
Pyrenees (Chamonix), at Testa Grigia and Plateau Rosa in the Italian Alps, at
Chacaltaya in the Bolivian Andes, at Mount Evans in Colorado, at Jungfraujoch
in the Bernese Alps, and so on. Thanks to nuclear emulsion techniques set up
by the Bristol group headed by C.F. Powell and by further experiences made by
R.H. Brown, U. Camerini, P.H. Fowler, H. Muirhead, C.F. Powell and D.M.
Ritson, in the years 1947-1950 new particles having mass intermediate between
the $\pi$ meson mass and the proton one, were detected. They observed the
decay of a charged particle into three charged mesons, one of these appearing
to be a $\pi$-particle. The parent particle was called a $\tau$ meson and its
mass was estimated to be about 1000 $m_{e}$, the first heavy meson. So, these
researchers identified two classes of new heavy unstable particles, that of
heavy mesons (lighter than the protons and heavier than the $\pi$ mesons) and
that of hyperons (heavier than the protons), which may be electrically charged
or neutral and never isolated. The heavy mesons were also generically called K
mesons or kaons, while the hyperons were also generically called Y mesons, so
that a new hierarchy of mesons had to present. It was also customary to
indicate the nature and number of the decay products by subscripts; thus, for
example, the $\tau$ meson was also called a $K_{\pi 3}$ due to one of its
decay schemes $\tau^{+}\rightarrow\pi^{+}+\pi^{+}+\pi^{-}$. In the 1950s,
besides the usual experimental research on cosmic radiation, with the advent
of particle accelerators new experiences begun too, in such a manner that new
particles were discovered and considerable further researches were
accomplished to classify these new particles according to masses, lifetimes
and decay schemes. All this represented one of the most important period of
the physics of 1950s.
The first heavy mesons and hyperons were observed by L. Jánossy in the years
1943-1946 at Dublin and by G.D. Rochester and C.C. Butler in 1947 at
Manchester. At first, such new particles were variously called k particles or
$V$ particles, due to the V-shaped tracks leaved by the non-neutral decay
particles observed into cloud chambers. Amongst these, there were those
particles which will be later called $K^{0},\Lambda^{0}$, $\tau$ and $\theta$
particles, these last two could be either neutral or electrically charged. As
recalled above, the Bristol group, headed by C.F. Powell, detected in 1949 the
first positively charged heavy meson, at first called $\tau^{+}$ meson, then
$K^{+}$ meson, that undergo different decay processes, amongst which
$K^{+}\rightarrow\pi^{+}+\pi^{0}$, whose experimental evidences were obtained,
in the years 1951-1954, by C. O’Ceallaigh, by the Paris group of B.P. Gregory,
A. Laggarigue, L. Leprince-Ringuet, F. Muller and Ch. Peyrou, by J. Crussard,
M.F. Kaplon, J. Klarmann and J.H. Noon and by A.L. Hodson, J. Ballam, W.H.
Arnold, D.R. Harris, R.R. Rau, G.T. Reynolds and S.B. Treiman. Further
researches made in cloud chambers with magnetic fields will show that both
$K^{+}$ and $K^{-}$ mesons exist. Then, the neutral heavy meson, at first
called $\theta^{0}$ particle, then $K^{0}$ particle, was first observed by C.
O’Ceallaigh in 1950. At the same time, R. Armenteros, K.H. Barker, C.C. Butler
and A. Cachon as well as L.M. Lederman K. Lande, E.T. Booth, J. Impeduglia and
W. Chinowsky in 1956, were able to show that at least two types of neutral
particles existed, one is the $\Lambda^{0}$ hyperon decaying according to the
scheme $\Lambda^{0}\rightarrow p+\pi^{-}$, and the other probably decayed as
follows $\theta^{0}\rightarrow\pi^{+}+\pi^{-}$. At that time, the Bristol
research group discovered two heavy mesons, called $\tau$ and $\theta$ mesons,
which initially seemed to be the same particle because they had same mass and
mean lifetime, but underwent distinct decay processes and had different
parity, so that, in the years 1953-1956, d’après R.H. Dalitz, it spoke of a
$\theta-\tau$ puzzle. The analysis of further experimental data led T.D. Lee
and C.N. Yang to assume in 1956 a parity violation of weak interactions,
thanks to which it was possible to establish that $\tau$ and $\theta$ mesons
are the same particle, thereafter called K particle. The discovery of the
breaking of the symmetry operators parity (P) and charge (C) received first
experimental evidence by C.S. Wu, E. Ambler, R.W. Hayward and D.D. Hoppes in
1957. Subsequent 1957 works made by R. Garwin, L. Lederman and M. Weinrich and
by J.J. Friedman and V.L. Telegdi, showed further evidence for a non-
conservation of parity and charge in the decay of kaons and hyperons,
attaining a deeper theoretical knowledge on the C, P and T invariance
properties. Subsequently, many decay modes for kaons were found even if, at
first, it was not realised that they represented alternative decay modes of
the same particle. Then, other types of hyperons were also found in cosmic
radiation: amongst these, C.M. York, R.B. Leighton and E.K. Bjornerund, in
1952, were able to experimentally ascertain a new type of hyperon, called
$\Sigma^{+}$ particle, which decays according to a reaction of the type
$\Sigma^{+}\rightarrow n+\pi^{0}$. A further confirmation of the existence of
$\Sigma$-hyperons was obtained by A. Bonetti, R. Levi-Setti, M. Panetti and G.
Tomasini in 1953, who also identified the alternative decay mode
$\Sigma^{+}\rightarrow n+\pi^{+}$, while the negative counterpart $\Sigma^{-}$
was observed by W.B. Fowler, R.P. Shutt, A.M. Thorndike and W.L. Whittemore at
Brookhaven in 1954. Another hyperon of mass $\sim 2600m_{e}$, called $\Xi^{-}$
meson, was detected by E.W. Cowan in 1954, which decays according to the
scheme $\Xi^{-}\rightarrow\Lambda^{0}+\pi^{-}$. The neutral $\Xi^{0}$ and
$\Sigma^{0}$ hyperons were then experimentally revealed by L.W. Alvarez, P.
Eberhard, M.L. Good, W. Graziano, H.K. Ticho and S.G. Wojcicki in 1959, only
after having been theoretically predicted as follows. This was the situation
around middle of 1950s, where the emphasis shifted from work using cosmic
radiations to work on large accelerators. The attention was also focused on
the classification of the various particles so far discovered according to
their masses, lifetimes and decay schemes.
It was observed that the various kaons and hyperons discovered in the 1950s
(above all the $\Lambda^{0}$), had a strange behavior respect to their decay
and production processes, so that they were given the collective appellation
of strange particles. Namely, following (Yang 1961, Chapter III) and (Muirhead
1965, Chapter 1, Section 1.4), it was experimentally found that these strange
particles have production times of about $10^{-23}$ sec and decay times of
about $10^{-10}$ sec, so that the forces involved in their production
processes were stronger than those present in the decay ones; furthermore,
this strange fact did not occur when such particles were isolated, in which
only weak interactions taken place. To explain this incompatibility between
experimental data and theoretical framework, A. Pais proposed in 1952 the
hypothesis of associated production according to which the decay and
production processes are not inverses of each other, but rather they differ
for the presence of another (associated) particle, or rather, at least two
strange particles should be involved in the production process in order that a
strong interaction could occur, whilst a weak interaction occurs if only one
strange particle is presents, as in the decay process. The Pais hypothesis
received experimental confirmation with the 1953-55 works of W.B. Fowler, R.P.
Shutt, A.M. Thorndike and W.L. Whittemore. At this point, it is necessary to
reconsider the above mentioned notion of isospin and related conservation law.
Following (Muihread 1965, Chapter 1, Sections 1.3 and 1.4), the concept of
conservation of isotopic spin (isospin) is associated with the experimental
evidence for the principle of charge independence of nuclear forces according
to which, at identical energies, the forces between any of the pairs of
nucleons n-n, n-p and p-p, depend only on the total angular momentum and
parity of the pair and not upon their charge state. So, B. Cassen and E.U.
Condon, in 1936, showed that the principle of charge independence could be
elegantly expressed by the isospin concept. The isospin of a system is
formally similar to angular momentum but is linked to the charge states of the
system. If a group of nuclei or particles exist in n charge multiplets, then
the isospin number T for this group is given by $2T+1=n$. The charge state of
a particle or nucleus in the multiplet is related to the third (along the z
axis) component of an isospin operator via the relations
$Z={Q}/{e}=(T_{3}+\frac{1}{2}A)$ for nucleons and nuclei, and $Q/e=T_{3}$ for
pions, where $Z$ is the atomic number, $Q$ the total charge (hypercharge) and
$A$ the atomic weight of the system. For instance, if $\chi_{p}$ and
$\chi_{n}$ denote the isospin functions for the proton and the neutron
respectively, then $T_{3}$ has eigenvalues $1/2$ and $-1/2$ respectively, that
is to say $T_{3}\chi_{p}=(1/2)\chi_{p}$ and $T_{3}\chi_{n}=-(1/2)\chi_{n}$.
The isospin quantum numbers were assigned to the strange particles produced,
independently by M. Gell-Mann and by T. Nakano and K. Nishijima, in the years
between 1952 and 1956. In regards to a cluster of elementary particles, they
observed invariance properties when the charge center of the multiplet of
strange particles was displaced respect to the center of the multiplet of non-
strange particles. Furthermore, they considered a scheme assuming that the
conservation laws for $T$ and $T_{3}$ were conserved or broken in dependence
on the given interaction: to be precise, $T$ and $T_{3}$ were conserved in
strong interactions, $T$ was broken whilst $T_{3}$ was conserved for
electromagnetic interactions and, finally, $T$ and $T_{3}$ were broken in weak
interactions. The satisfactory nature of Gell-Mann, Nakano and Nishijima
scheme lays in the fact that it predicted the existence of two new elementary
particles which were later experimentally found.
Later, again following (Muirhead 1965, Chapter 1, Section 1.4), it was pointed
out by Gell-Mann and Nishijima, independently of each other in the years
1955-56, that a more elegant classification of the strongly interacting
particles than that based on isospin alone, could be made if a new parameter
$S$, called the strangeness number, was introduced and defined by the relation
$Q/e=T_{3}+(B+S)/2$ where $B$ is the baryon number; baryon is a generic name
for nucleons and hyperons. Such a relation shows that the associated
production phenomena and the isospin symmetry are related together. As it has
been said above, the success of the Gell-Mann, Nakano and Nishijima scheme
lies in the fact that it predicted, on the basis of isospin and strangeness
conservation laws, various charge multiplets; in particular, new particles
were predicted like the $\Sigma^{0}$, $\Xi^{0}$, $K^{+}$, $K^{-}$ and related
antiparticles. All these theoretical assumptions received experimental
confirmation by the works of Y. Nambu, K. Nishilima, Y. Yamaguchi and W.B.
Fowler in the years 1953-1955, as well as many of the above predicted
particles (besides the $\Xi^{0}$ and $\Sigma^{0}$ above remembered) were
detected by W.B. Fowler, R.P. Shutt, A.M. Thorndike, W.L. Whittemore and W.D.
Walker, in the years 1955-1959. However, the elegant semi-empirical
classification scheme of Gell-Mann, Nakano and Nishijima, envisaged the
existence of many other new particles which will be later discovered, whose
static and dynamic properties will be of fundamental importance for the
subsequent 1960s theoretical work of M. Gell-Mann and Y. Ne’eman (eihgtfold
way). In conclusion, following171717Such a leading textbook has been one of
main references herein followed. (Muirhead 1965, Chapter 1, Section 1.5), the
discoveries of new particles have occurred sometimes as a result of
theoretical insights and sometimes by accident, the most strange particle
falling into the latter category: from what has been said above, at this
stage, we have that the weak interactions are associated with electrons, muons
and neutrinos $(e,\mu,\nu)$, collectively called leptons (which are not
subject to strong interactions) and with certain decay processes for mesons
and hyperons; the nucleons and hyperons $(n,p,\Lambda,\Sigma,\Xi)$ are then
collectively called baryons. Following (Roman 1960, Introduction), the
particles recalled so far may be also classified into two classes according to
their spin values, identifying a first class in which fall particles with
integer spin $(\gamma,\pi,K)$, said to be bosons, and a second one in which
fall particles with half-integer spin $(\nu,e,\mu,p,n,\Lambda,\Sigma,\Xi)$
said to be fermions. The fermions again fall into two rather distinct groups,
namely leptons and baryons. Apart from the photons, the bosons fall too into
two groups: the lighter $\pi$-mesons (or pions) and the heavier $K$-mesons (or
kaons). Thus, our classification scheme is tentatively Photon-Leptons-Pions-
Kaons-Baryons, and represents one of the main achievement of the physics in
the decade 1950-1960.
## 2\. Some early works on cosmic rays
In this section, according to what has been said in the Preface, in
delineating some early works on cosmic rays we consider the first research
papers of A. Zichichi and co-workers, dating back to 1950s, which concerned
physics of $K$ mesons, with particular attention to the experimental context.
To be precise, as a research fellow at CERN of Geneva, he joined the heavy
meson decay research group which, in turn, belonged to the wider investigation
activity on cosmic rays, according to the research program policy of that
period of this worldwide renowned research institution. The first papers of
Zichichi on cosmic rays mainly concern with the various properties of strange
particles.
As said in the previous section 1, in that period there were still many
unsolved questions concerning the so-called strange or $V$ particles. In 1.,
it is just discussed, together the related experimental arrangements, certain
peculiar features showed by only two out of twenty events observed into a
cloud chamber located at Testa Grigia laboratories (3,500 m.a.s.l.) in which
about 150 $V^{0}$-particles related to decay of a single charged unstable
fragment (like an unstable hydrogen isotope, or an excited deuteron or triton)
have been found. From a deep phenomenology analysis of the experimental data
and taking into account the already existent related literature, it emerged
that, very likely, one of these two observed peculiar events, say the Event I,
could be due to a certain $\Lambda^{0}$-decay of an unstable hydrogen isotope
rather than a nuclear interaction (which perhaps has would been quite
unusual), whereas, instead, the second one, say the Event II, led to the
conclusion according to which, under certain further hypotheses, it could be a
$\tau$-decay, even supposing that it is not a nuclear interaction.
The second work 2. is a contribution to the experimental evidence for the
existence of the neutral $\tau^{0}$-meson which had been predicted but not
established. To be precise, it is reported and discussed the result of an
experiment made at Jungfraujoch Research Station and consisting of four tracks
originating at a point in the cloud chamber gas which may be interpreted
either as the radiative decay of a $\theta^{0}$-meson or as the decay of a
$\tau^{0}$-meson. Two out of these are tracks of positive and negative
electrons electrons, while the other two are tracks of fast particles
resembling a typical $V$-event which is most readily explained as the decay of
a neutral $\tau$-meson, rather than a $\theta^{0}$-meson, with the subsequent
decay of the secondary $\pi^{0}$-meson into an electron pair and a
$\gamma$-ray, which may be respectively written as
$\tau^{0}\rightarrow\pi^{+}+\pi^{-}+\pi^{0}$ and $\pi^{0}\rightarrow
e^{+}+e^{-}+\gamma$. After having discussed on the various possibilities, this
four-pronged event is deemed to be geometrically associated with a small
nuclear interaction with, in turn, can be interpreted, on the basis of the
experimental data, as a charge exchange reaction of the type
$K^{+}+n\rightarrow\tau^{0}+p$, albeit it is also not excluded a possible
double production according to the scheme
$\pi^{+}+n\rightarrow\tau^{0}+\Sigma^{+}$ in which, in turn, the
$\Sigma^{+}$-particle decays into one proton and one yet undetected
$\pi^{0}$-meson.
Taking into account the previous work 2. discussed above, the work 3. mainly
shows some results coming from certain emulsion experiments which provide the
first two remarkable examples of $K$-meson pairs of the type
$(K^{0},\bar{K}^{0})$ and $(K^{+},\bar{K}^{0})$, produced in elementary
neutron-proton interactions whose production reactions respectively are
$n+p\rightarrow K^{0}+\bar{K}^{0}+n+p$ and $n+p\rightarrow
K^{+}+\bar{K}^{0}+n+n$; these allowed to extend the knowledge on the
phenomenology of heavy mesons, to further confirmation of some of the various
hypotheses suggested by M. Gell-Mann and A. Pais in the years 1952-54 about
associated productions, in particular, the prediction according to which the
$K^{0}$-mesons should exist in two states as particle and antiparticle with
$S=+1$ and $S=-1$. To be precise, from a systematic study of the associated
production of heavy mesons and hyperons in a cosmic-ray cloud chamber, in 3.
examples of very simple nuclear interactions giving rise to pairs of
$K$-mesons have been found. The importance of these observations is that they
provide experimental evidence to support the theoretical prediction that
$K^{0}$-mesons should exist in two states with opposite strangeness $S=+1$ and
$S=-1$, that is to say, these events are evidence that $K^{0}$-mesons with
both positive and negative strangeness exist. In relation to the well-known
$\theta-\tau$ puzzle, the authors also argued on the possible identification
or not of the produced $K^{0}$-mesons with the $\theta^{0}$-particles, but the
experimental measurements weren’t of great usefulness for this.
The work 4. is a brief research note in which it is determined, on the basis
of previous works made by J.A. Newth and M.S. Bartlett, the mean lifetime
estimates of the two decaying particles $\Lambda_{0}$ and $\theta_{0}$,
isolated amongst 115 $V^{0}$-events observed in a multi-plate cloud chamber
triggered for penetrating showers, and respectively interested to the
following main decay reactions $\Lambda^{0}\rightarrow p+\pi^{-}+37$ MeV and
$\theta^{0}\rightarrow\pi^{+}+\pi^{-}+214$ MeV. Moreover, neglecting the
existence of other types of unstable neutral particles with a two-body decay,
it has been possible to classify the above 115 $V^{0}$-particles.
The paper 5. is a research report, presented at CERN Scientific Policy
Committee on 21 October 1957, after the CERN research activity on cosmic rays
taken the decision to stop the so-called Geneva experiment on $K$-meson
decays, whose related motivations were exposed in the subsequent CERN report
No. CERN/SPC/52 (B). In 5., the director-general of the Jungfraujoch Research
Station in detail proposed a new experimental apparatus specifically designed
to prosecute the research activity on high energy interactions with a mountain
experiment based on the nuclear interaction of protons at energies in the
neighborhood of 100 GeV, through a large magnet cloud chamber. The novel
feature of this apparatus was a magnetic spectrometer which measured the
momenta of the primary particles. One of the main aims of this experiment was
also that to fathom new directions on particle physics as, for instance, the
search for new unstable strange particles having very short lifetimes.
As it has been said in section 1, in 1950s gradually started to run the first
particle accelerators of synchrotron type which will be intended to replace
the cosmic-ray researches. But this conclusion did not yet apply to the study
of the production processes of strange particles. Now, the results described
in the work 6. come from an experiment designed to study the production of
strange particles in materials of low and high atomic weight, precisely carbon
and copper consecutively used, through the interaction of energetic
secondaries, sprung out by nuclear interactions in passing through the targets
(of carbon and copper), which produce 79 neutral $V$-particles. The division
of all the $V^{0}$-events into $\Lambda^{0}$\- and $\theta^{0}$\- decays was
made in order to determine their lifetime estimates. If one denotes with
$N(\Lambda^{0})$ and $N(\theta^{0})$ the numbers of $\Lambda^{0}$\- and
$\theta^{0}$-mesons so produced, then a significant difference between the
values of the ratio $N(\Lambda^{0}):N(\theta^{0})$ for their production in
carbon and copper has been found; this asymmetry’s fact occurred in the decay
of the $\Lambda^{0}$ particles respect to the short-lived $\theta^{0}$ ones,
was also explained by H. Blumenfeld, E.T. Booth, L.M. Ledermann and W.
Chinowsky, who conducted similar experiences in 1956 with carbon and lead,
reaching to almost equal results, through the associated production of pairs
of $K$-mesons through which it is possible to increase the number of
$\Lambda^{0}$ particles so slowly produced, with also non-conservation of
strangeness. Thus, the results achieved in 6. as well as by Blumenfeld and co-
workers, may be taken as further evidence for the great importance of the pair
production of $K$-mesons in cosmic-ray experiments.
The decay asymmetry detected in the previous work 6. will be deeper studied in
the next work 7. where many other properties of $\Lambda^{0}$\- and
$\theta^{0}$\- particles, like for example spin, mean lifetime, behavior with
respect to inversion operators and anisotropy effects on geometrical
distributions, have carried out on 107 $\Lambda^{0}$ and $\theta^{0}$
particles produced in iron plates of a multiple cloud chamber exposed to
cosmic radiation at an altitude of 3,500 m.a.s.l. Likewise, the work 8.
reports the first results of an experimental study of the nuclear interaction
of cosmic rays (mainly of the type proton-proton) with a magnet cloud chamber
based at an altitude of 3,500 m.a.s.l. and operating at energy of about 100
GeV, showing that such a type of nuclear interaction study is feasible.
## 3\. Historical introduction: II
In this section, we recall the main events and facts of that historical path
which goes from the introduction of the spin to the notion of anomalous
magnetic moment, with particular attention to the leptonic case. The
necessarily limited historical framework so outlined in this section, covers a
temporal period which roughly goes up from early 1920s to 1960s.
### 3.1 On Landé separation factors
Following (Muirhead 1965, Chapter 2) and (Tomonaga 1997), when a fundamental
interaction is taken into account then the experimental determination of the
basic particle data, like masses, lifetimes, spins and magnetic moments, is
necessarily required. The most accurately known properties of the particles
are those which can be associated with their magnetic moments. Magnetic
properties of elementary particles have been and yet are of paramount
importance both to theoretical and experimental high energy physics. One of
the main intrinsic properties of the elementary particles is the spin, which
can be inferred from the conservation laws for angular momentum. Following
(Landau 1982, Chapter VIII), in both classical and quantum mechanics, the laws
of conservation of angular momentum are a consequence of the isotropy of space
respect to a closed system, so that it depends on the transformation
properties under rotation of the coordinate system. Therefore, all quantum
systems, like atomic nuclei or composite systems of elementary particles,
besides the orbital angular momentum, show to have as well an intrinsic
angular momentum, called spin, which is unconnected with its motion in space
and to which it is also associated a magnetic moment whose strengths are not
quantized and may assume any value. The spin disappears in the classical limit
$\hbar\rightarrow 0$ so that it has no classical counterpart. The spin must be
meant as fully distinct from the angular momentum due to the motion of the
particle in space, that is to say, the orbital angular momentum. The particle
concerned may be either elementary or composite but behaving in some respect
as an elementary particle (e.g. an atomic nucleus). The spin of a particle
(measured, like the orbital angular momentum, in units of $\hbar$) will be
denoted by $\vec{s}$. Following (Rich and Wesley 1972), (Bertolotti 2005,
Chapter 8), (Miller et al. 2007) and (Roberts and Marciano 2010, Chapter 1),
the physical idea that an electron has an intrinsic angular momentum was first
put forward independently of each other by A.H. Compton in 1921 to explain
ferromagnetism181818Furthermore, Compton acknowledges A.L. Parson for having
first proposed the electron as a spinning ring of charge. Compton modified
this idea considering a much smaller distribution of charge mainly
concentrated near the center of the electron. The Compton’s paper is almost
unknown (see (Compton 1921)) albeit it is quoted by the 1926 Uhlenbeck and
Goudsmit paper. Following (Roberts and Marciano 2010, Chapter 3, Section
3.2.1), also R. Kronig proposed, in 1925, the spin as an internal angular
momentum responsible for the electron forth’s quantum number (see (Bertolotti
2005, Chapter 8). and by G. Uhlenbeck and S. Goudsmit in 1925 to explain
spectroscopic observations in relation to the anomalous Zeeman effect, while
spin was introduced into quantum mechanics by W. Pauli in 1927 as an
additional term to the Pauli equation which is obtained by the non-
relativistic representation of the Dirac equation to small velocities (see
(Jegerlehner 2008, Part I, Chapter 3, Section 3.2)) to account for the quantum
mechanical treatment of the spin-orbit coupling of the anomalous Zeeman effect
(see also (Haken & Wolf 2005, Chapter 14, Section 3)). An equation similar to
the Pauli’s one, was also introduced by C.G. Darwin in 1927 (see (Roberts and
Marciano 2010, Chapter 3, Section 3.2.1)).
Following (Jegerlehner 2008, Part I, Chapter 1), (Melnikov and Vainshtein
2006, Chapter 1) and (Shankar 1994, Chapter 14), leptons have interesting
static (classical) electromagnetic and weak properties like the magnetic and
electric dipole moments. Classically, dipole moments may arise either from
electrical charges or currents. In this regards, an important example which
may turns out to be useful to our purposes is the circulating current, due to
an orbiting particle with electric charge $Q$ and mass $m$, which exhibits the
following orbital magnetic dipole moment
(1)
$\vec{\mu}_{L}=\frac{Q}{2c}\vec{r}\wedge\vec{v}=\frac{Q}{2mc}\vec{L}=\Gamma\vec{L}$
where $\Gamma=Q/2mc$ is the classical gyromagnetic ratio191919Usually, the
gyromagnetic ratio is denoted by lower case $\gamma$, but here we prefer to
use the upper case $\Gamma$ to distinguish it by the well-known Lorentz factor
$\gamma=1/\sqrt{1-\beta^{2}}$ with $\beta=v^{2}/c^{2}$. and
$\vec{L}=m\vec{r}\wedge\vec{v}=\vec{r}\wedge\vec{p}$ is the orbital angular
momentum whose corresponding quantum observable is the operator
$-i\hbar\vec{r}\wedge\nabla=\hbar\vec{l}$, so that we have the following
orbital magnetic dipole moment operator (see (Jegerlehner 2008, Part I,
Chapter 3) and (Shankar 1994, Chapter 14))
(2) $\vec{\mu}_{l}=g_{l}\frac{Q\hbar}{2mc}\vec{l}$
where $g_{l}$ is a constant introduced by the usual quantization transcription
rules. For $Q=e$, the quantity $\mu_{0}=e\hbar/2mc$ is normally used as a unit
for the magnetic moments and is called the Bohr magneton. The electric charge
$Q$ is usually measured in units of $e$, so that $Q=-1$ for leptons and $Q=+1$
for antileptons; therefore, we also may rewrite (2) in the following form
(3) $\vec{\mu}_{l}=g_{l}\frac{Qe\hbar}{2mc}\vec{l}=g_{l}Q\mu_{0}\vec{l}.$
Both electric and magnetic properties have their origin in the electrical
charges and their currents, apart from the existence or not of magnetic
charges. Following (Jegerlehner 2008, Part I, Chapter 1) and (Muirhead 1965,
Chapter 9, Section 9.2(d)), whatever the origin of magnetic and electric
moments are, they contribute to the electromagnetic interaction Hamiltonian
(interaction energy) of the particle with magnetic and electric fields which,
in the non-relativistic limit, is given by
(4) $\mathcal{H}_{em}=-(\vec{\mu}_{m}\cdot\vec{B}+\vec{d}_{e}\cdot\vec{E})$
where $\vec{\mu}_{m}$ and $\vec{d}_{e}$ are respectively the magnetic and
electric dipole moments (see (Jegerlehner 2008, Part I, Chapter 1)).
If one replaces the orbital angular momentum $\vec{L}$ with the spin
$\vec{s}$, then we might search for an analogous (classical) magnetic dipole
moment, say $\vec{\mu}_{s}$, associated with it and, therefore, given by
$(Q/2mc)\vec{s}$. Nevertheless, following (Born 1969, Chapter 6, Section 38)
and (Muirhead 1965, Chapter 2, Section 2.5)), to fully account for the
anomalous Zeeman effect, we should consider this last expression multiplied by
a certain scalar factor, say $g_{s}$ (often simply denoted by $g$), so that
(5) $\vec{\mu}_{s}=g_{s}\frac{Q}{2mc}\vec{s}$
which is said to be the spin magnetic moment. Now, introducing, as a
corresponding quantum observable, the spin operator defined by
$\vec{S}=\hbar\vec{s}=\hbar\vec{\sigma}/2$, where $\vec{\sigma}$ is the Pauli
spin operator, it is possible to consider both the spin magnetic moment
operator and the electric dipole moment operator (see (Jegerlehner 2008, Part
I, Chapter 1)), respectively defined as follows
(6) $\vec{\mu}_{s}\doteq g_{s}Q\mu_{0}\frac{\vec{\sigma}}{2},\ \ \ \ \ \ \ \ \
\ \vec{d}_{e}\doteq\eta Q\mu_{0}\frac{\vec{\sigma}}{2},$
where $\eta$ is a constant, the electric counterpart of $g_{s}$. Following
(Caldirola et al. 1982, Chapter XI, Section 3), the attribution of a $s=1/2$
spin value to the electron, led to the formulation of the so-called vectorial
model of the atom. In such a model, amongst other things, the electron orbital
angular moment $\vec{L}$ composes with the spin $\vec{s}$ through well-defined
spin-orbit coupling rules (like the Russell-Saunders ones) to give the
(classical) total angular moment defined to be $\vec{j}\doteq\vec{L}+\vec{s}$,
while the (classical) total magnetic moment is defined to be
$\vec{\mu}_{total}\doteq\vec{\mu}_{L}+\vec{\mu}_{s}$, so that, taking into
account (3) and (6), the corresponding quantum observable counterpart, in this
vectorial model, is
(7)
$\vec{\mu}_{total}\doteq\vec{\mu}_{l}+\vec{\mu}_{s}=g_{l}Q\mu_{0}\vec{l}+g_{s}Q\mu_{0}\frac{\vec{\sigma}}{2}=Q\mu_{0}(g_{l}\vec{l}+g_{s}\vec{S})$
which is said to be the total magnetic moment of the given elementary particle
with charge $Q$ and mass $m$; since $g_{s}\neq 1$, it follows that it is not,
in general, parallel to the total angular moment operator
$\vec{J}\doteq\vec{l}+\vec{S}$, so that it undergoes to precession phenomena
when magnetic fields act.
The existence of the various above constants $g_{l},g_{s}$ and $\eta$ is
mainly due to the fact that, in the vectorial model of anomalous Zeeman
effect, the direction of total angular moment $\vec{j}$ does not coincide with
the direction of total magnetic moment, so that these scalar factors just take
into account the related non-zero angles which are called Landé separation
factors because first introduced by A. Landé (1888-1976) in the early 1920s
(see (Born 1969, Chapter 6, Section 38)). To be precise, only the parallel
component of $\vec{\mu}_{tot}$ to $\vec{j}$, say $\vec{\mu}_{tot}^{\|}$, is
efficacious, so that we should have
(8) $\vec{\mu}_{tot}^{\|}=g_{j}\frac{Q\hbar}{2mc}\vec{j}$
where the scalar factor $g_{j}$ (or simply $g$) takes into account the
difference between the vectorial model of anomalous Zeeman effect and the
theory of the normal one. To may computes this factor, we start from the
relation
(9)
${\mu}_{tot}^{\|}=\mu_{l}\cos(\widehat{\vec{l},\vec{j}})+\mu_{s}\cos(\widehat{\vec{s},\vec{j}})$
with
(10) $\mu_{l}=g_{l}\frac{Q\hbar}{2mc}l,\ \ \ \ \ \ \ \ \ \
\mu_{s}=g_{s}\frac{Q\hbar}{2mc}s$
where $g_{l}$ and $g_{s}$ are known to be respectively the orbital and spin
factors, which, in turn, represent the ratios respectively between the orbital
and spin magnetic and mechanic moments. Replacing (10) into (9), we have
(11)
$g_{j}=g_{l}\frac{l}{j}\cos(\widehat{\vec{l},\vec{j}})+g_{s}\frac{s}{j}\cos(\widehat{\vec{s},\vec{j}})$
from which (see (Born 1969, Chapter 6, Section 38)) it is possible to reach to
the following relation
(12)
$g_{j}=g_{l}\frac{(j^{2}+l^{2}-s^{2})}{2j^{2}}+g_{s}\frac{(j^{2}+s^{2}-l^{2})}{2j^{2}}$
Experimental evidences dating back to 1920s and mainly related to the
anomalous Zeeman effect, seemed suggesting that $g_{l}=1$ and $g_{s}=2$ for
the electron, that is, the atomic vectorial model explains the fine structure
features of alkali metals and the anomalous Zeeman effect if one supposes to
be $g_{s}\neq 1$, that is to say, a spin intrinsic gyromagnetic ratio
anomalous respect to the orbital one ($g_{l}=1$), so speaking of a spin
anomaly. Following (Bohm 1993, Chapter IX, Section 3), the deviations from the
$g_{s}=2$ value for the electron comes from the radiative corrections of
quantum electrodynamics and is of the same order as, and of analogous origin
to, the Lamb shift. The value $g_{s}=2$ was first established as far back as
1915 by a celebrated experiment of A. Einstein and W.J. de Haas which led to
the formulation of the so-called Einstein-de Hass effect and that was also
incorporated in the spin hypothesis put forward in the 1920s (see (S̆polskij
1986, Volume II, Chapter VII, Section 70)). Following (Jegerlehner 2008, Part
I, Chapter 1), the anomalous magnetic moment is an observable which may be
studied through experimental analysis of the motion of leptons. The story
started in 1925 when Uhlenbeck and Goudsmit put forward the hypothesis that an
electron had an intrinsic angular momentum of $\hbar/2$ and that associated
with this there were a magnetic dipole moment equal to $e\hbar/2mc$, i.e. the
Bohr magneton $\mu_{0}$. According to E. Back and A. Landé, the question which
naturally arose was whether the magnetic moment of the electron
$(\mu_{m})_{e}$ is precisely equal to $\mu_{0}$, or else $g_{s}=1$ in
$(10)_{2}$, to which them tried to answer through a detailed study of numerous
experimental investigations on the Zeeman effect made in 1925, reaching to the
conclusion that the Uhlenbeck and Goudsmit hypothesis was consistent although
they did not really determine the value of $g_{s}$. In 1927, Pauli formulated
the quantum mechanical treatment of the electron spin in which $g_{s}$
remained a free parameter, whilst Dirac presented his revolutionary
relativistic theory of electron in 1928, which, instead, unexpectedly
predicted $g_{s}=2$ and $g_{l}=1$ for a free electron. The first experimental
evidences for the Dirac’s theoretical foresights for electrons came from L.E.
Kinster and W.V. Houston in 1934, albeit with large experimental errors at
that time. Following (Kusch 1956), it took many more years of experimental
attempts to descry that the electron magnetic moment could exceed 2 by about
0.12, the first clear indication of the existence of a certain anomalous
contribution to the magnetic moment given by
(13) $a_{i}\doteq\frac{(g_{s})_{i}-2}{2},\ \ \ \ \ \ \ \ \ \ i=e,\mu,\tau.$
With the new results on renormalization of QED by J. Schwinger, S.I. Tomonaga
and R.P. Feynman of 1940s, the notion of anomalous magnetic moment (AMM) falls
into the wider class of QED radiative corrections.
### 3.2 On Field Theory aspects of AMM
Following (Jegerlehner 2008, Part I, Chapter 3), for the measurement of the
anomalous magnetic moment of a lepton, it is necessary to consider the motion
of a relativistic point-particle $i$ (or Dirac particle202020That is to say, a
particle without internal structure.) of charge $Q_{i}e$ and mass $m_{i}$ in
an external electromagnetic field $A_{\mu}^{ext}(x)$. The equations of motion
of a charged Dirac particle in an external field are given by the Dirac
equation
(14)
$\big{(}i\hbar\gamma^{\mu}\partial_{\mu}+Q_{i}\frac{e}{c}\gamma^{\mu}(A_{\mu}+A_{\mu}^{ext}(x))-m_{i}c\big{)}\psi_{i}(x)=0,$
and by the second order wave equation
(15) $\big{(}\Box
g^{\mu\nu}-(1-\xi^{-1}\big{)}\partial^{\mu}\partial^{\nu})A_{\nu}(x)=-Q_{i}e\bar{\psi}_{i}(x)\gamma^{\mu}\psi_{i}(x).$
The first step is now to find a solution to the relativistic one-particle
problem given by the Dirac equation (14) in the presence of an external field,
neglecting the radiation field in first approximation. In such a case, the
equation (14) reduces to
(16) $i\hbar\frac{\partial\psi_{i}}{\partial
t}=\big{(}-c{\vec{\alpha}}(i\hbar{\vec{\nabla}}-Q_{i}\frac{e}{c}{\vec{A}})-Q_{i}e\Phi+\beta
m_{i}c^{2}\big{)}\psi_{i}$
where
(17) $\beta=\gamma^{0}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right),\ \ \
\vec{\alpha}=\gamma^{0}\vec{\gamma}=\left(\begin{array}[]{cc}0&{\vec{\sigma}}\\\
\vec{\sigma}&0\\\ \end{array}\right)$
are the Dirac matrices, $A^{\mu\ ext}=(\Phi,\vec{A})$ is the electromagnetic
four-potential with scalar and vector potential respectively given by $\Phi$
and $\vec{A}$ (of the external electromagnetic field) and $i=e,\mu,\tau$. For
the interpretation of the solution to the last Dirac equation (16), the non-
relativistic limit plays an important role because many relativistic QFT
problems may be most easily understood and solved in terms of the non-
relativistic problem as a starting point. To this end, it is helpful and more
transparent to work in natural units, the general rules of transcription being
the following: $p^{\mu}\rightarrow
p^{\mu},d\mu(p)\rightarrow\hbar^{-3}d\mu(p),m\rightarrow mc,e\rightarrow
e/(\hbar c),\exp(ipx)\rightarrow\exp(ipx/\hbar)$ and, for spinors,
${}^{t}(u,v)\rightarrow{{}^{t}(u/\sqrt{c},v/\sqrt{c})}$; furthermore, we shall
consider a generic lepton $e^{-},\mu^{-},\tau^{-}$ with charge $Q_{i}$,
dropping the index $i$. Moreover, to get, from the Dirac spinor $\psi$, the
two-component Pauli spinors $\varphi$ and $\chi$ in the non-relativistic
limit, one has to perform an appropriate unitary transformation, the so-called
Foldy-Wouthuysen transformation212121It is a unitary transformation introduced
around the late 1940s by L.L. Foldy and S.A. Wouthuysen to study the non-
relativistic limits of Dirac equation as well as to overcome certain
conceptual and theoretical problems arising from the relativistic
interpretations of position and momentum operators. Following (Foldy and
Wouthuysen 1950), in the non-relativistic limit, where the momentum of the
particle is small compared to $m$, it is well known that a Dirac particle
(that is, one with spin 1/2) can be described by a two-component wave function
in the Pauli theory. The usual method of demonstrating that the Dirac theory
goes into the Pauli theory in this limit makes use of the fact that two of the
four Dirac-function components become small when the momentum is small. One
then writes out the equations satisfied by the four components and solves,
approximately, two of the equations for the small components. By substituting
these solutions in the remaining two equations, one obtains a pair of
equations for the large components which are essentially the Pauli spin
equations. See (Bjorken and Drell 1964, Chapter 4)., upon the Dirac equation
(16) rewritten as follows
(18) $i\hbar\frac{\partial\psi}{\partial t}=\vec{H}\psi,\ \ \ \ \
\vec{H}=c\vec{\alpha}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}+\beta
mc^{2}+Q\Phi,$
with $\vec{\alpha}$ and $\beta$ given by (17) (see (Bjorken and Drell 1964,
Chapter 1, Section 4, Formula (1.26)).
Then, following (Bjorken and Drell 1964, Chapter 1, Section 4) and
(Jegerlehner 2008, Part I, Chapter 3), in order to obtain the non-relativistic
representation for small velocities, we should split off the phase of the
Dirac field $\psi$, which is due to the rest energy of the lepton
(19) $\psi=\tilde{\psi}\exp\big{(}-i\frac{mc^{2}}{\hbar}t\big{)},\ \ \ \ \
\tilde{\psi}=\left(\begin{array}[]{c}\tilde{\varphi}\\\
\tilde{\chi}\end{array}\right)$
so that the Dirac equation takes the form
(20) $i\hbar\frac{\partial\tilde{\psi}}{\partial
t}=(\vec{H}-mc^{2})\tilde{\psi}$
and describes the following coupled system of equations
(21) $\big{(}i\hbar\frac{\partial}{\partial
t}-Q\Phi\big{)}\tilde{\varphi}=c\vec{\sigma}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}\tilde{\chi},$
(22) $\big{(}i\hbar\frac{\partial}{\partial
t}-Q\Phi+2mc^{2}\big{)}\tilde{\chi}=c\vec{\sigma}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}\tilde{\varphi}$
which, respectively, provide the Pauli description in the non-relativistic
limit and the one of the negative-energy states. As $c\rightarrow\infty$, it
is possible to prove that
(23)
$\tilde{\chi}\cong\frac{1}{2mc}\vec{\sigma}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}\tilde{\varphi}+O(1/c^{2}),$
by which we have
(24) $\big{(}i\hbar\frac{\partial}{\partial
t}-Q\Phi\big{)}\tilde{\varphi}\cong\frac{1}{2m}\big{(}\vec{\sigma}(\vec{p}-\frac{Q}{c}\vec{A})\big{)}^{2}\tilde{\varphi}$
and since $\vec{p}$ does not commute with $\vec{A}$, we may use the relation
(25)
$(\vec{\sigma}\vec{a})(\vec{\sigma}\vec{b})=\vec{a}\vec{b}+i\vec{\sigma}(\vec{a}\wedge\vec{b})$
to obtain
(26)
$\big{(}\vec{\sigma}(\vec{p}-\frac{Q}{c}\vec{A})\big{)}^{2}=\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}^{2}-\frac{Q\hbar}{c}\vec{\sigma}\cdot\vec{B}$
where $\vec{B}=rot\ \vec{A}$, so finally reaching to the following 1927 Pauli
equation
(27) $i\hbar\frac{\partial\tilde{\varphi}}{\partial
t}=\tilde{H}\tilde{\varphi}=\Big{(}\frac{1}{2m}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}^{2}+Q\Phi-\frac{Q\hbar}{2mc}\vec{\sigma}\cdot\vec{B}\Big{)}$
which, up to the spin term, is nothing but the non-relativistic Schrödinger
equation. Following too (Muirhead 1965, Chapter 3, Section 3.3(f)), the last
term of (27) has the form of an additional potential energy. Now, by (4),
since the potential energy of a magnet of moment $\vec{\mu}_{m}$, in a field
of strength $B$, is $-\vec{\mu}_{m}\cdot\vec{B}$, equation (27) shows that a
Dirac particle with electric charge $Q$ should possess a magnetic moment equal
to $(Q\hbar/2mc)\vec{\sigma}=2Q\mu_{0}\vec{\sigma}/2$ that, compared with
$(6)_{1}$, would imply $g_{s}=2$. This is what Dirac theory historically
provided for an electron. Later, Pauli showed as the Dirac equation could be
little modified to account for leptons of arbitrary magnetic moment by adding
a suitable term.
Indeed, in222222See also (Pauli 1973, Chapter 6, Section 29). (Pauli 1941,
Section 5)), the author concludes his report with some simple applications of
the theories discussed in (Pauli 1941, Part II, Sections 1, 2(d) and 3(a)),
concerning the interaction of particles of spin 0, 1, and 1/2 with the
electromagnetic field. In the last two cases we denote the value $e\hbar/2mc$
of the magnetic moment as the normal one, where $m$ is the rest mass of the
particle. The assumption of a more general value $g_{s}(e\hbar/2mc)$ for the
magnetic moment demands the introduction of additional terms, proportional to
$g_{s}-1$, into the Lagrangian or Hamiltonian. Pauli concludes his report with
some simple applications of the theories discussed in (Pauli 1941, Part II,
Sections 1, 2(d) and 3(a)) concerning the interaction of particles of spin 0,
1, and 1/2 with the electromagnetic field. In the last two cases, Pauli
denotes the value $e\hbar/2mc$ of the magnetic moment as the normal one, where
$m$ is the rest mass of the particle. The assumption of a more general value
$g(e\hbar/2mc)$ for the magnetic moment demands the introduction of additional
terms, proportional to $g-1$, in the Lagrangian or Hamiltonian. To be precise,
following (Dirac 1958, Chapter 11, Section 70), (Corinaldesi and Strocchi
1963, Chapter VII, Section 4), (Muirhead 1965, Chapter 3, Section 3.3(f)) and
(Levich et al. 1973, Chapter 8, Section 63 and Chapter 13, Section 118), Pauli
modified the basic Dirac equation, written in scalar form as follows
(28) $i\hbar\gamma_{\mu}\frac{\partial}{\partial
x_{\mu}}\psi+mc^{2}\psi-i\hbar\frac{Q}{c}\gamma_{\mu}A_{\mu}\psi=0,$
to get the following Lorentz invariant Dirac-Pauli equation
$\displaystyle i\hbar\gamma_{\mu}\frac{\partial}{\partial
x_{\mu}}\psi+mc^{2}\psi-i\hbar\frac{Q}{c}\gamma_{\mu}A_{\mu}\psi-i\hbar
a_{\mu}\gamma_{\mu}\gamma_{\nu}(A_{\mu,\nu}-A_{\nu,\mu})$ (29)
$\displaystyle=i\hbar\gamma_{\mu}\frac{\partial}{\partial
x_{\mu}}\psi+mc^{2}\psi-i\hbar\frac{Q}{c}\gamma_{\mu}A_{\mu}\psi-i\hbar
a_{\mu}\sigma_{\mu\nu}q_{\nu}A_{\mu}=0$
replacing the gauge invariant interaction term
$-i\hbar\sigma_{\mu\nu}q_{\nu}A_{\mu}$ with the following phenomenological
term (see also (Sakurai 1967, Chapter 3, Section 3-5) $-i\hbar
a_{\mu}\sigma_{\mu\nu}q_{\nu}A_{\mu}$ called an anomalous moment interaction
(or Pauli moment), where $a_{\mu}$ represents the anomalous part of the
magnetic moment of the particle, $q$ is the momentum transfer and
$\hat{\sigma}=-(i/2)[\vec{\gamma},\vec{\gamma}]$ is the spin $1/2$ momentum
tensor. In the non-relativistic limit, this last expression reduces to the
following equation (compare with (27))
(30) $i\hbar\frac{\partial\psi}{\partial
t}=\Big{(}\frac{1}{2m}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}^{2}+Q\psi-\big{(}a_{\mu}+\frac{Q\hbar}{2mc}\big{)}\vec{\sigma}\cdot\vec{B}\Big{)}$
so justifying the use of the term ’anomalous’ to denote a deviation from the
classical results. Thus, the transition from the non-relativistic
approximation of the Dirac equation goes over into the Pauli equation;
furthermore, from this reduction there results not only the existence of the
spin of particles but also the existence of the intrinsic magnetic moment of
particle and its anomalous part. Namely, we should have $g_{s}=2(1+a_{\mu})$,
where its higher order part $a_{\mu}=(g_{s}-2)/2\geq 0$ just measures the
deviation’s degree respect to the value $g_{s}=2$ (Dirac moment) as predicted
by the 1928 Dirac theory for electron232323Following (Roberts and Marciano
2010) and (Miller et al. 2007, Section 1), the non-relativistic reduction of
the Dirac equation for an electron in a weak magnetic field $\vec{B}$, is as
follows $i\hbar(\partial\psi/\partial
t)=[(p^{2}/2m)-(e/2m)(\vec{L}+2\vec{S})\cdot\vec{B}]\psi$, by which it follows
that $g_{s}=2$. as well as by H.A. Kramers in 1934 (see (Farley and
Semertzidis 2004, Section 1)) developing Lorentz covariant equations for spin
motion in a moving system. Later, this Pauli ansatz was formally improved and
generalized by L.L. Foldy and S.A. Wouthuysen in the forties to obtain a
generalized Pauli equation which will be the theoretical underpinning of
further experiments. Indeed, at the first order in $1/c$, the lepton behaves
as a particle which has, other than a charge, also a magnetic moment given by
$\mu_{m}=(Q\hbar/2mc)\vec{\sigma}=(Q/mc)\vec{S}$, as said above. Following
(Corinaldesi and Strocchi 1963, Chapter VII, Section 5), (Bjorken and Drell
1964, Chapter 4, Section 3) and (Jegerlehner 2008, Part I, Chapter 3), from an
expansion in $1/c$ of the Dirac Hamiltonian given by $(18)_{2}$, we have the
following effective third order Hamiltonian obtained applying a third
canonical Foldy-Wouthuysen transformation to $(18)_{2}$
(31) $\displaystyle\vec{H}^{\prime\prime\prime}_{FW}$ $\displaystyle=$
$\displaystyle\beta\Big{(}mc^{2}+\frac{\big{(}\vec{p}-(Q/c)\vec{A}\big{)}^{2}}{2m}-\frac{\vec{p}^{4}}{8m^{3}c^{2}}\Big{)}+Q\Phi-\beta\frac{Q\hbar}{2mc}\vec{\sigma}\cdot\vec{B}+$
$\displaystyle-\frac{Q\hbar^{2}}{8m^{2}c^{2}}div\vec{E}-\frac{Q\hbar}{4m^{2}c^{2}}\vec{\sigma}\cdot\big{[}(\vec{E}\wedge\vec{p}+\frac{i}{2}rot\vec{E})\big{]}+O(1/c^{3})$
where each term of it, has a direct physical meaning: see (Bjorken and Drell
1964, Chapter 4, Section 3) for more details. In particular, the last term
takes into account the spin-orbit coupling interaction energy and will play a
fundamental role in setting up the experimental apparatus of many $g-2$ later
experiments. The last Hamiltonian, to the third order, gives rise to the
following generalized Pauli equation $i\hbar(\partial\tilde{\varphi}/\partial
t)=\vec{H}^{\prime\prime\prime}_{FW}\tilde{\varphi}$, which is a generalized
version, including high relativistic terms via the application of a Foldy-
Wouthuysen transformation, of the first form proposed by Pauli in 1941 (see
(Pauli 1941)) and that leads to the second approximation Schrödinger-Pauli
equation as a non-relativistic limit of the Dirac equation (see (Corinaldesi
and Strocchi 1963, Chapter VIII, Section 1)).
Our particular interest is the motion of a lepton in an external field under
consideration of the full relativistic quantum behavior which is ruled by the
QED equations of motions (14) and (15) that, in turn, under the action of an
external field, reduce to (16). For slowly varying field, the motion is
essentially determined by the generalized Pauli equation which besides also
serves as a basis for understanding the role of the magnetic moment of a
lepton at the classical level. The anomalous magnetic moment roughly estimates
the deviations from the exact value $g_{s}=2$, because of certain relativistic
quantum fluctuations in the electromagnetic field (initially called
Zitterbewegung) around the leptons and mainly due, besides weak and strong
interaction effects, to QED higher order effects as a consequence of the
interaction of the lepton with the external (electromagnetic) field and which
are usually eliminated through the so-called radiative corrections. At
present, we are interested to QED contributions only. Following (Muirhead
1965, Chapter 11, Section 11.4), (Jegerlehner 2008, Part I, Chapter 3) and
(Melnikov and Vainshtein 2006, Chapter 2), the QED Lagrangian of interaction
of leptons and photons is (see also (Muirhead 1965, Chapter 8, Section
8.3(a)))
(32)
$\mathcal{L}^{QED}_{int}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i{\gamma_{\mu}\partial_{\mu}}-m)\psi-
QJ^{\mu}A_{\mu}$
where $\psi$ is the lepton field, $A^{\mu}=(\Phi,\vec{A})$ is the vector
potential of the electromagnetic field,
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ is the field-strength
tensor of the electromagnetic field,
$J^{\mu}(x)=\bar{\psi}(x)\gamma^{\mu}\psi(x)$ is the electric current and $Q$
is the lepton charge. Let us consider an incoming lepton
$l(p_{1}^{\mu},r_{1})$, with 4-momentum $p_{1}^{\mu}$, rest mass $m$, charge
$Q$ and $r_{1}$ as third component of spin, which scatters off the external
electromagnetic potential $A_{\mu}$ towards a lepton $l(p_{2}^{\mu},r_{2})$ of
4-momentum $p_{2}^{\mu}$ and third component of spin $r_{2}$. To the first
order in the external field and in the classical limit of
$q^{2}=p_{2}^{2}-p_{1}^{2}\rightarrow 0$, the interaction is described by the
following scattering amplitude
(33) $\mathcal{M}(x;p)=\langle
l(p_{2}^{\mu},r_{2})|J^{\mu}(x)|l(p_{1}^{\mu},r_{1})\rangle$
where $\vec{q}=\vec{p}_{2}-\vec{p}_{1}$ is the momentum transfer. In practice,
it will be more convenient to work, through Fourier transforms, with invariant
momentum transfers rather than spatial functions. So, in momentum space, due
to space-time translation invariance for which
$J^{\mu}(x)=\exp(iPx)J^{\mu}(0)\exp(-iPx)$, and to the fact that the lepton
states are eigenstates of 4-momentum, that is to say
$\exp(-iPx)|l(p_{i},r_{i})\rangle=\exp(-ip_{i}x)|l(p_{i};r_{i})\rangle,i=1,2$,
we find the following Fourier transform of the scattering matrix
(34) $\displaystyle\tilde{\mathcal{M}}(q;p)$ $\displaystyle=$
$\displaystyle\int\exp(iqx)\langle
l(p_{2},r_{2})|J^{\mu}(x)|l(p_{1},r_{1})\rangle d^{4}x=$ $\displaystyle=$
$\displaystyle\int\exp[i(p_{2}-p_{1}-q)x]\langle
l(p_{2},r_{2})|J^{\mu}(0)|l(p_{1},r_{1})\rangle d^{4}x=$ $\displaystyle=$
$\displaystyle(2\pi)^{4}\delta^{(4)}(q-p_{2}+p_{1})\langle
l(p_{2},r_{2})|J^{\mu}(0)|l(p_{1},r_{1})\rangle$
which is proportional to the Dirac $\delta$-function of 4-momentum
conservation. Therefore, the $T$-matrix element is given by
(35) $\langle l(p_{2},r_{2})|J^{\mu}(0)|l(p_{1},r_{1})\rangle.$
Via the current conservation law $\partial_{\mu}J^{\mu}(\vec{x})=0$ and the
parity conservation in QED, the most general parametrization of the $T$-matrix
element has the following QED relativistically covariant decomposition
(36) $\langle
l(p_{2})|J^{\mu}(0)|l(p_{1})\rangle=\bar{u}(p_{2})\Gamma^{\mu}(p_{2},p_{1})u(p_{1})$
where $\Gamma^{\mu}$, called lepton-photon vertex function, is any expression
(or group of expression) which has the transformation properties of a 4-vector
and is also a $4\times 4$ matrix in the spin space of the lepton. Following
(Muirhead 1965, Chapter 11, Section 11.4(c)) and (Roberts and Marciano 2010,
Chapter 2, Section 2.2; Chapter 3, Section 3.2.2), we shall have the following
Lorentz structure for the scattering amplitude
(37)
$\bar{u}(p_{2})\Gamma^{\mu}(p_{2},p_{1})u(p_{1})=-iQ\bar{u}(p_{2})\Big{(}F_{D}(q^{2})\gamma^{\mu}+F_{P}(q^{2})\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}\Big{)}u(p_{1})$
where $u(p)$ denotes the Dirac spinors, while
$\sigma^{\mu\nu}=(i/2)(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})=(i/2)[\gamma^{\mu},\gamma^{\nu}]$
are the components of the Dirac spin operator
$\hat{\sigma}=-(i/2)\vec{\gamma}\wedge\vec{\gamma}$ or else the spin $1/2$
angular momentum tensor. $F_{D}(q^{2})$ (or $F_{E}(q^{2})$) is the Dirac (or
electric charge) form factor, while $F_{P}(q^{2})$ (or $F_{M}(q^{2})$) is the
Pauli (or magnetic) form factor, which roughly are connected respectively with
the distribution of charge over the lepton and with the anomalous magnetic
moment to the interaction lepton-electromagnetic field. We now need to know
the relationships between these form factors and the anomalous part of the
lepton magnetic moment.
In the non-relativistic quantum mechanics, a lepton interacting with an
electromagnetic field is described by the Hamiltonian
(38) $H=\frac{(\vec{p}-Q\vec{A})^{2}}{2m}-\vec{\mu}_{s}\cdot\vec{B}+Q\Phi,\ \
\ \ \ \ \vec{B}=rot\vec{A}$
which is nothing that $\tilde{H}$ of (27). To find the relations between the
lepton magnetic moment $\mu_{s}$ and the Dirac and Pauli form factors, we
consider the scattering of the lepton off the external vector potential
$A_{\mu}$ in the non-relativistic approximation, using the Hamiltonian (38)
and comparing the results with (33). Following (Melnikov and Vainshtein 2006,
Chapter 2), the non-relativistic scattering amplitude in the first order Born
approximation is given by
(39)
$\Omega=-\frac{m}{2\pi}\int{\bar{\psi}}(\vec{p}_{2})V\psi(\vec{p}_{1})d^{3}\vec{r}$
where $\psi(\vec{p}_{1})=\tilde{\varphi}\exp(i\vec{p}_{1}\cdot\vec{r})$ and
$\psi(\vec{p}_{2})=\tilde{\chi}\exp(i\vec{p}_{2}\cdot\vec{r})$ are the wave
functions of the lepton described by the two components of Pauli spinors (see
(19)) $\tilde{\varphi}$ and $\tilde{\chi}$, and
(40)
$V=-\frac{Q}{2m}(\vec{p}\cdot\vec{A}+\vec{A}\cdot\vec{p})-\mu_{s}\vec{\sigma}\cdot\vec{B}+Q\Phi.$
By a Fourier transform, we have
(41)
$\Omega=-\frac{m}{2\pi}\tilde{\chi}\Big{(}-\frac{Q}{2m}\vec{A}_{q}\cdot(\vec{p}_{2}+\vec{p}_{1})+Q\Phi_{q}-i\mu_{s}\vec{\sigma}\cdot(\vec{q}\wedge\vec{A}_{q})\Big{)}\tilde{\varphi}$
where $\Phi_{q}$ and $\vec{A}_{q}$ stands for the Fourier transforms of the
electric potential $\Phi$ and of the vector potential $\vec{A}$. Therefore, we
will derive (41) starting from the relativistic expression for the scattering
amplitude (33) and taking then the non-relativistic limit. If the Dirac
spinors are normalized to $2m$, the relation between the two oscillating
amplitudes in the non-relativistic limit, is given by
(42) $-i\lim_{|\vec{p}|\ll m}\mathcal{M}(x;p)=4\pi\Omega.$
To derive the non-relativistic limit of the scattering amplitude
$\mathcal{M}$, we use the explicit representation of the Dirac matrices, given
by
(43) $\gamma^{0}=\left(\begin{array}[]{cc}I&0\\\ 0&-I\\\ \end{array}\right),\
\ \ \ \ \gamma^{i}=\left(\begin{array}[]{cc}0&{{\sigma}_{i}}\\\
-{\sigma}_{i}&0\\\ \end{array}\right)\ \ \ i=1,2,3,$
and the Dirac spinors $u(p)$. Using these expressions in $\mathcal{M}$ and
working at first order in $|\vec{p}_{i}|/m\ i=1,2$, we obtain
(44) $\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle-2iem\tilde{\chi}\Big{[}F_{D}(0)\Big{(}\Phi_{q}-\frac{\vec{A}_{q}\cdot(\vec{p}_{1}+\vec{p}_{2})}{2m}\Big{)}+$
$\displaystyle-i\frac{F_{D}(0)+F_{P}(0)}{2m}\vec{\sigma}\cdot(\vec{q}\wedge\vec{A}_{q})\Big{]}\tilde{\varphi}.$
Using (41), (42) and (44), we find
(45) $F_{D}(0)=1,\ \ \ \ \ \ \ \ \ \
\mu_{s}=\frac{Q}{2m}\big{(}F_{D}(0)+F_{P}(0)\big{)}$
which compared with (5) and (6), give
(46) $g_{s}=2(1+F_{P}(0))$
so that, if the Pauli form factor $F_{P}(q^{2})$ does not vanish for $q=0$,
then $g_{s}$ is different from 2, the value predicted by Dirac theory of
electron. It is conventional to call this difference the muon anomalous
magnetic moment and write it as
(47) $a_{\mu}=F_{P}(0)=\frac{g_{s}-2}{2}$
so that, in the static (classical) limit we have too
(48) $F_{D}(0)=1,\ \ \ \ \ \ \ \ \ \ F_{P}(0)=a_{\mu}$
where the first relation is the so-called charge renormalization condition (in
units of $Q$), while the second relation is the finite prediction for
$a_{\mu}$ in terms of the pauli form factor. In QED, $a_{\mu}$ may be computed
in the perturbative expansion in the fine structure constant242424Following
(Muirhead 1965, Chapter 1, Section 1.3(b)), the interaction of the elementary
particles with each other can be separated into three main classes, each with
its own coupling strength. To be precise, the common parameter appearing in
the electromagnetic processes is the fine structure constant
$\alpha=e^{2}/4\pi\hbar c$; the strength of strong interactions is
characterized by the dimensionless coupling term $g^{2}/4\pi\hbar c$, while
the weak interactions are ruled by the Fermi coupling constant $G_{F}$.
$\alpha=Q^{2}/4\pi$ as follows
(49)
$a_{\mu}^{QED}=\sum_{i=1}^{\infty}a_{\mu}^{(i)}=\sum_{i=1}^{\infty}c_{i}\Big{(}\frac{\alpha}{\pi}\Big{)}^{i}.$
The first term in the series is $O(\alpha)$ since, when radiative corrections
are neglected, the Pauli form factor vanishes. This is easily seen from the
QED Lagrangian $\mathcal{L}_{int}^{QED}$ given by (32), which implies that,
through leading order in $\alpha$, the interaction between the external
electromagnetic field and the lepton, is given by
$-iQ\bar{u}(p_{2})\gamma^{\mu}u(p_{1})A_{\mu}$. A consequence of the current
conservation, is the fact that the Dirac form factor satisfies the condition
$F_{D}(0)=1$ to all orders in the perturbation expansion. The renormalization
constants influence the Pauli form factor only indirectly, through the mass,
the charge and the fermion wave function renormalization, because there is no
corresponding tree-level operator in QED Lagrangian. Therefore, the anomalous
magnetic moment is the unique prediction of QED; moreover, the $O(\alpha)$
contribution to $a_{\mu}$ has to be finite without any renormalization. The
QED radiative corrections provide the largest contribution to the lepton
anomalous magnetic moment. The one-loop result was computed by J. Schwinger in
1948 (see (Schwinger 1948)), who found the following lowest-order radiative
(or one-loop) correction to the electron anomaly (see (Rich and Wesley 1972)
and (Roberts and Marciano 2010, Chapter 3, Section 3.2.2.1))
(50) $a_{e}^{(2)}=F_{P}(0)=\alpha/2\pi\cong 0.00116.$
In 1949, F.J. Dyson showed that Schwinger’s theory could be extended to allow
calculation of higher-order corrections to the properties of quantum systems.
Since Dyson showed too that the one-loop QED contribution to the anomalous
magnetic moment did not depend on the mass of the fermion, the Schwinger’s
result turned out to be valid for all leptons, so that we have
$a_{i}^{(2)}=F_{P}(0)=\alpha/2\pi,\ i=e,\mu,\tau$. Currently, QED calculations
have been extended to the four-loop order and even some estimates of the five-
loop contribution exist. It is interesting however to remark that Schwinger’s
calculation was performed before the renormalizability of QED were understood
in details; historically, this provided a first interesting example of a
fundamental physics result derived from a theory that was considered to be
quite ambiguous at that time. Therefore, the anomalous magnetic moment of a
lepton is a dimensionless quantity which may be computed order by order as a
perturbative expansion in the fine structure constant $\alpha$ in QED and
beyond this, in the Standard Model (SM) of elementary particles or extensions
of it. As an effective interaction term, the anomalous magnetic moment is
mainly induced by the interaction of the lepton with photons or other
particles, so that it has a pure QED origin. It corresponds to a dimension 5
operator (see (51)) and since any renormalizable theory is constrained to
exhibit terms of dimension 4 or less only, it follows that such a term must be
absent for any fermion in any renormalizable theory at tree (or zero-loop)
level. It is the absence of such a Pauli term that leads to the prediction
$g_{s}=2+O(\alpha)$. Therefore, at that time, it was necessary looking for
other theoretical tools and techniques to experimentally approach the
determination of the anomalous magnetic moment of leptons. Following
(Jegerlehner 2008, Part I, Chapter 3), in higher orders the form factors for
the muon in general acquires an imaginary part. Indeed, if one considers the
following effective dipole moment Lagrangian with complex coupling
(51)
$\mathcal{L}_{eff}^{DM}=-\frac{1}{2}\Big{[}\bar{\psi}\sigma^{\mu\nu}\Big{(}D_{\mu}\frac{1+\gamma_{5}}{2}+\bar{D}_{\mu}\frac{1-\gamma_{5}}{2}\Big{)}\psi\Big{]}F_{\mu\nu}$
with $\psi$ the muon field, we have
(52) $\Re D_{\mu}=a_{\mu}\frac{Q}{2m_{\mu}},\ \ \ \ \ \Im
D_{\mu}=d_{\mu}=\frac{\eta}{2}\frac{Q}{2m_{\mu}},$
so that the imaginary part of $F_{P}(0)$ corresponds to an electric dipole
moment (EDM) which is non-vanishing only if we have $T$ violation. The
equation (51) provides as well the connection between the magnetic and
electric dipole moments through the dipole operator $D$. As we will see later,
the incoming new ideas on symmetry in QFT will turn out to be of extreme
usefulness to approach and to analyze the problem of determination of the
anomalous magnetic moment of the leptons, the equation (51) being just one of
these important results.
### 3.3 Experimental determinations of the lepton AMM: a brief historical
sketch
#### 3.3.1 On the early 1940s experiences
Following (Kusch 1956), (Rich and Wesley 1972), (Farley and Picasso 1979),
(Hughes 2003) and (Jegerlehner 2008, Part I, Chapter 1), in the same period in
which appeared the famous 1948 Schwinger seminal research note, thanks to the
new molecular-beams magnetic resonance spectroscopy methods mainly worked out
by the research group leaded by I.I. Rabi in the late of 1930s, P. Kusch and
H.M. Foley detected, in 1947, a small anomalous $g_{L}$-value for the electron
within a 4% accuracy (see also (Weisskopf 1949)), analyzing the
${}^{2}P_{3/2}$ and ${}^{2}P_{1/2}$ state transition of Gallium: to be precise
they found the values $g_{s}=2.00229\pm 0.00008$ and $g_{l}=0.99886\pm
0.00004$; later, J.E. Nafe, E.B. Nelson and Rabi himself were able, in May
1947, to detect a discrepancy between theoretical and predicted values of
about 0.26% by the measurements of the hyperfine structure level splitting of
hydrogen and deuterium in the ground state on the accepted Dirac $g$-factor of
2, which was quickly confirmed in the same year by D.E. Nagle, R.S. Julian and
J.R. Zacharias (see also (Schweber 1961, Chapter 15, Section d)). In this
regards, in September 1947, G. Breit (1947a,b) suggested that such
discordances between theoretical expectations and experimental evidences could
be overcome if one had supposed $g\neq 2$. Independently by Breit, also J.M.
Luttinger (1948) (as well as T.A. Welton and Z. Koba - see (Rich and Wesley
1972) and references therein - between 1948 and 1949) stated that some
experiments of then, seemed to require a modification in the $g$-factor of the
electron. In this regards, Schwinger suggested that the coupling between the
electron and the radiation field could be the responsible of this, calculating
the effect on the basis of a general subtraction formalism for the infinities
of quantum electrodynamics. Luttinger, instead, shown that the possible change
in the electron magnetic moment could be derived very simply without any
reference to an elaborate subtraction formalism. Soon after, P. Kusch, E.B.
Nelson and H.M. Foley presented, in 1948, another precision measurement of the
magnetic moment of the electron, just before Schwinger’s theoretical result
whose 1948 paper besides quotes them, which together the discovery of the fine
structure of hydrogen spectrum (Lamb shift) by W.E. Lamb Jr. and R.C.
Retherford in 1947, as well as the corresponding calculations by H.A. Bethe,
N.M. Kroll, V. Weisskopf, J.B. French and W.E. Lamb Jr. in the same period,
were the main triumphs of testing the new level of QED theoretical
understanding with precision experiments. All that was therefore a stimulus
for the development of modern QED. These successes had a strong impact in
establishing the QFT as a general formal framework for the theory of
elementary particles and for our understanding of fundamental interactions.
The late 1940s were characterized by a close intertwinement between theory and
experiment which greatly stimulated the rise of the new QED. On the
theoretical side, a prominent role was gradually undertaken by the new non-
Abelian gauge theory proposed by C.N. Yang and R.L. Mills in 1954 as well as
by the various relativistic local QFT symmetries amongst which the discrete
ones of charge conjugation $(C)$, parity $(P)$ and time-reversal $(T)$
reflection which are related amongst them by the well-known $CPT$ theorem,
according to which the product of the these three discrete transformations,
taken in any order, is a symmetry of any relativistic QFT (see (Streater and
Wightman 1964)). Actually, in contrast to the single transformations $C$, $P$
and $T$, which are symmetries of the electromagnetic and strong interactions
only (d’après T.D. Lee and C.N. Yang celebrated work), $CPT$ is a universal
symmetry and it is this symmetry which warrants that particles and
antiparticles have identical masses as well as equal lifetimes; but also the
dipole moments are very interesting quantities for the study of the discrete
symmetries mentioned above.
#### 3.3.2 Some previous theoretical issues
The celebrated 1956 paper of T.D. Lee and C.N. Yang (see (Lee and Yang 1956))
on parity violation, has been an invaluable source of theoretical insights.
The paper discusses the question of the possible failure of parity
conservation in weak interactions taking into account what experimental
evidences existed then as well as possible proposal of experiments for testing
this hypothesis. Amongst these last, they discuss, since the beginning, on
some experiments concerning polarized proton beams which would have led to an
electric dipole moment if the parity violation were occurred. The related
important consequences were too discussed, like the proton and neutron EDM,
taking into consideration the previous early 1950s experiences made by E.M.
Purcell, N.F. Ramsey and J.H. Smith for the proton who made an experimental
measurement of the electric dipole moment of the neutron by a neutron-beam
magnetic resonance method, finding a value less than $10^{-20}$ $e$-cm ca. in
agreement with parity conservation for strong and electromagnetic
interactions. Nevertheless, Lee and Yang argued that yet lacked valid
experimental confirmations of parity conservation for weak interactions
suggesting, to this end, to consider the measure of the angular distribution
of the electrons coming from $\beta$ decays of oriented nuclei like those of
$Co^{60}$, thing that will be immediately done, with success, by C.S. Wu and
co-workers, furnishing a first experimental evidence for a lack of parity
conservation in $\beta$ decays. Subsequently, Lee and Yang also argue on the
question of parity conservation in meson and hyperon decays, as well as in
those strange particle decays having the following features: 1) the strange
particle involved has a non-vanishing spin and (2) it decays into two
particles at least one of which has a non-vanishing spin or rather it decays
into three or more particles. Thus, what conjectured by Lee and Yang could be
also applied to the decay processes a) $\pi\rightarrow\mu+\nu$ and b)
$\mu\rightarrow e+2\nu$. So, in the sequential decay
$\pi\rightarrow\mu\rightarrow e$, starting from a $\pi$ meson at rest, one
might study the distribution of the angle $\theta$ between the $\mu$-meson
momentum and the electron momentum, the latter being in the center-of-mass
system of the $\mu$ meson. The decay b) is then a pure leptonic one, so no
hadronic phenomenon is involved, this making easier the related calculations
(see (Okun 1986, Chapter 3)). Lee and Yang then argue that, if parity is
conserved in neither a) nor b), then the distribution will not in general be
identical for $\theta$ and $\pi-\theta$ directions. To understand this, one
may consider first the orientation of the muon spin. If a) violates parity
conservation, then the muon would be in general polarized along its direction
of motion. In the subsequent decay b), the angular distribution problem with
respect to $\theta$ is therefore closely similar to the angular distribution
problem of $\beta$ rays from oriented nuclei, as discussed before, so that, in
this way, it will be also possible to detect possible parity violations in
this type of decays. These last remarks on $\pi\mu e$ sequence will be
immediately put in practice in the celebrated 1956 experiences pursued by R.L.
Garwin, L.M. Lederman with M. Weinrich and by J.L. Friedman with V.L. Telegdi,
which will further confirm Lee and Yang hypothesis of parity violation in weak
interactions. Following (Sakurai 1964, Chapter 7, Section 2) and (Schwartz
1972, Chapter 4, Section 11), polarized muons slow down and stop before they
decay, but depending on the material (graphite, aluminium, etc.) the muon spin
direction is still preserved, so we have a source of polarized muons. Negative
muons are emitted with their angular momenta pointing along their directions
of motion, whereas positive muons are emitted with their angular momenta
pointing opposite to their directions of motion. Furthermore, if these
positive muons were stopped in matter and allowed to decay, then the direction
of this angular momentum (or spin) at the moment of decay could be determined
by the distribution in directions of the emitted decay electron which follow
the former. If parity is not conserved in muon decay either, then there will
be a forward-backward asymmetry in the positron distribution with respect to
the original $\mu^{+}$ direction. The just above mentioned experiences showed
more positrons emitted backward with respect to the $\mu^{+}$ direction,
showing that parity is not conserved in both $\pi$ and $\mu$ decays.
As it has said above, Lee and Yang already argued on electric dipole moments
in relation to parity conservation law for fundamental interactions, in some
respects enlarging the discussion to the general framework of discrete
symmetry transformations. To understand about the properties of the dipole
moments under the action of such transformations, in particular the behavior
under parity and time-reversal, we have to look at the interaction Hamiltonian
(4) and, above all, at the equations (6) which both depend on the axial vector
$\vec{\sigma}$, so that also $\vec{\mu}_{m}$ and $\vec{d}_{e}$ will be also
axial vectors. On the other hand, the electric field $\vec{E}$ and the
magnetic one $\vec{B}$ transform respectively as a (polar) vector and as an
axial vector. Then, an axial vector changes sign under $T$ but not under $P$,
while a (polar) vector changes sign under $P$ but not under $T$. Furthermore,
since electromagnetic and strong interactions are the two dominant
contributions to the dipole moments, and since both preserve $P$ and $T$, it
follows that the corresponding contributions to (4) must conserve these
symmetries as well. Indeed, following (Muirhead 1965, Chapter 9, Section
9.2(d)), we have
(53) $\displaystyle P\vec{\sigma}P^{-1}=\sigma,\ \ \ \ \
T\vec{\sigma}T^{-1}=-\vec{\sigma},\ \ \ \ \ P\vec{H}P^{-1}=\vec{H},$
$\displaystyle T\vec{H}T^{-1}=-\vec{H},\ \ \ \ \ P\vec{E}P^{-1}=-\vec{E},\ \ \
\ \ T\vec{E}T^{-1}=\vec{E},$
whence it follows that
(54) $\displaystyle
P(\vec{\sigma}\cdot\vec{H})P^{-1}=\vec{\sigma}\cdot\vec{H},\ \ \ \ \ \ \ \ \ \
T(\vec{\sigma}\cdot\vec{H})T^{-1}=\vec{\sigma}\cdot\vec{H},$ $\displaystyle
P(\vec{\sigma}\cdot\vec{E})P^{-1}=-\vec{\sigma}\cdot\vec{E},\ \ \ \ \ \ \ \ \
\ T(\vec{\sigma}\cdot\vec{E})T^{-1}=-\vec{\sigma}\cdot\vec{E}.$
Therefore, as L.D. Landau and Ya.B. Zel’dovich pointed out (see (Landau 1957)
and (Zel’dovich 1961)), due to these symmetry rules on $P$ and $T$, the
magnetic term $-\vec{\mu}_{m}\cdot\vec{B}$ is allowed, while an electric
dipole term $-\vec{d}_{e}\cdot\vec{E}$ is forbidden so that we should have
$\eta=0$ in $(6)_{2}$. Now, $T$ invariance (that, by $CPT$ theorem, is
equivalent to $CP$ invariance) is also violated by weak interactions, which
however are very small for light leptons. Nevertheless, for non-negligible
second order weak interactions (as for heavier leptons \- see (Chanowitz et
al. 1978) and (Tsai 1981)), an approximate $T$ invariance will require the
suppression of electric dipole moments, i.e. $d_{e}\rightarrow 0$. Thus,
electric dipole interaction cannot occur unless both $P$ and $T$ invariance
breaks down in electrodynamics. Following (Roberts and Marciano 2010, Chapter
1, Section 1.3), P.A.M. Dirac discovered, in 1928, an electric dipole moment
term in the relativistic equations involved in his electron theory. Like the
magnetic dipole moment, the electric dipole moment had to be aligned with
spin, so that we have an expression of the type
$\vec{d}=\eta(Q\hbar/2mc)\vec{s}$ (see $(6)_{2}$) where, as already said,
$\eta$ is a dimensionless constant which is the analogous to $g_{s}$. Whilst
the magnetic dipole moment is a natural property of charged particles with
spin, electric dipole moment are forbidden both by parity and time reversal
symmetries as said above. Nevertheless, from a historical viewpoint, the
search for an EDM dates back to suggestions due to E.M. Purcell and N.F.
Ramsey since 1950 who however pointed out that the usual parity arguments for
the non-existence of electric dipole moments for nuclei and elementary
particles, albeit appealing from the standpoint of symmetry, weren’t
necessarily valid. They questioned about these arguments based on parity and
tried, in 1957, to experimentally measure the EDM of the neutron through a
neutron-beam magnetic resonance method, finding a value for $d$ of about
$(-0.1\pm 2.4)\cdot 10^{-20}$ $e$-cm. This result was published only after the
discovery of parity violation although their arguments were provided in
advance of the celebrated 1956 T.D. Lee and C.N. Yang paper on parity
violation for weak interactions. Once parity violation received experimental
evidence, other than L.D. Landau, soon after also N.F. Ramsey, in 1958,
pointed out that an EDM would violate both $P$ and $T$ symmetries.
#### 3.3.3 Further experimental determinations of the lepton AMM
A) Some introductory theoretical topics
i) On resonance spectroscopy methods. Amongst special devices and techniques
of experimental physics, a fundamental role is played by magnetic resonance
spectroscopic techniques through which Zeeman level transitions are induced by
magnetic dipole radiations by means of the application of an external static
magnetic field $\vec{B}$. The spontaneous transitions with $\Delta l=\pm 1$
(electric dipole) are more probable than those with $\Delta l=0$ and $\Delta
m=\pm 1$ (magnetic dipole). Nevertheless, the presence of a resonant
electromagnetic field increases the latter. With the action of this perturbing
field the probability of induced transitions is proportional to the square of
the intensity of the electromagnetic field, so that magnetic dipole
transitions may be easily induced through suitable radio-frequency (RF) values
provided by a RF oscillator with an imposed constant magnetic field which has
the main role to select the desired RF frequencies to be put in resonance with
the precession ones. As an extension of the original method of the famous
Stern-Gerlach experiment, the above mentioned technique was first proposed by
I.I. Rabi, together his research group at Chicago around the late 1930s, who
made important experiments on atomic beams that, amongst other things, led to
the precise determination of the atomic hyperfine structure; in particular,
the Lamb shift between hydrogen $2S_{1/2}$ and $2P_{1/2}$ gave an accurate
measurement of the electron anomalous magnetic moment. Independently by Rabi’s
research group works, also L.W. Alvarez and F. Bloch set up, in 1940, a
similar technique. The nuclear magnetic moments have been measured through
nuclear magnetic resonance (NMR) techniques that, thanks to relaxation
mechanisms which release thermal energy in such a manner to warrant a weak
thermal contact between nuclear spins and liquid or solid systems to which
they belong, allow to determine fundamental physical properties of the latter.
The electron paramagnetic resonance (EPR) or electron spin resonance (ESR)
refers to induced transitions between Zeeman levels of almost free electrons
in liquids and solids. It has been first observed by E.K. Zavoiskij in 1945
and usually runs into the microwaves frequencies and it has been applied to
determine anomalous magnetic moment values. Both in NMR and EPR, in which an
external inhomogeneous magnetic field $\vec{B}_{0}$ is acting, the transitions
between Zeeman levels are induced by an additional homogeneous alternating
weak magnetic field $\vec{B}_{1}$ (for instance, acting upon a $x$-$y$ plane),
oscillating transversally to $\vec{B}_{0}$ (for instance, directed along the
$z$ axis) with an angular frequency $\omega_{1}$ which may be, or not, in
phase with Larmor precession frequency; for instance, if $\vec{B}_{1}$ acts
along the $x$ axis, then an induced e.m.f. will be detectable along the $y$
axis. Thanks to the 1949 N.F. Ramsey works, it is also possible to apply a
second alternating static magnetic field $\vec{B}_{2}$, even perpendicularly
to $\vec{B}_{0}$ (double resonance techniques), and so on (multiple resonance
techniques); the possible reciprocal geometrical dispositions of the various
involved magnetic fields $\vec{B}_{0},\vec{B}_{1},\vec{B}_{2}$ and so on, give
rise to different resonance experimental methods also in dependence on the
adopted relaxation methods and related detected times: amongst them, the Bloch
decay and the spin echoes. In single resonance techniques, the perturbing
alternating field $\vec{B}_{1}$ must be in resonance with the separation
between two adjacent Zeeman levels (i.e. with $\Delta m=\pm 1$). The resulting
statistical coherence will imply a macroscopic value (roughly $N\mu_{ct}$)
quite high to may be detected by a coil, with the symmetry axis belonging in
the equatorial plane and, for instance, oriented along the $y$ axis, also
thanks to electronic devices which will amplify the initial value.
Following (Dekker 1958, Chapter 20), (Kittel 1966, Chapter 16), (Kastler 1976,
Part III, Chapter V), (Cohen-Tannoudij et al. 1977, Volume I, Complement
$F_{IV}$), (Bauer et al. 1978, Chapters 12 and 13), (Pedulli et al. 1996,
Chapters 7, 8 and 9), (Humphreis 1999, Chapter 14), (Bertolotti 2005, Chapter
9) and (Haken and Wolf 2005, Chapter 12), for particles having a non-zero
spin, the application of the field $\vec{B}_{0}$ only, implies a torque acting
upon the cyclotron (or orbital) magnetic moment $\vec{\mu}_{L}$ so giving rise
to two non-zero components, namely a longitudinal component $\vec{\mu}_{cl}$
(directed along $\vec{B}_{0}$) and a transversal one $\vec{\mu}_{ct}$
(belonging to the plane having $\vec{B}_{0}$ as normal vector). This torque
will imply too a Larmor precession, with angular frequency given by
$\omega_{0}=g(eB_{0}/2mc)$ (for elementary particles with rest mass $m$), that
causes a rotation of $\vec{\mu}_{ct}$ in the equatorial plane around the $z$
axis. Nevertheless, in general there is no statistical coherence amongst these
transversal components, also due to the thermal excitation. But, as showed by
F. Bloch, W.W. Hansen and M. Packard as well as by E.M. Purcell, H.C. Torrey,
N. Bloembergen and R.V. Pound in the years 1945-46, the application of a
perturbing (alternating) magnetic field $\vec{B}_{1}$, transversally arranged
respect to $\vec{B}_{0}$ and usually induced by the passage, along a
transmissive spire, of a direct current (DC) into a variable RF oscillator,
gives rise to a coherent and ordered precession of the transversal components
of magnetic moment when the frequency of the perturbing field, say
$\omega_{1}$, is equal to $\omega_{0}$ (magnetic resonance condition or
resonance equation); this, in turn, will imply either spin-orbit decouplings
as well as resonating Zeeman magnetic level transitions, in agreement with the
well-known Bohr’s correspondence principle according to which the concept of
quantum level transition should correspond, in the classical electrodynamics,
to the periodic variation either of an atomic electric or magnetic moment (in
our case, the rotation of $\vec{\mu}_{ct}$ in the equatorial plane). The weak
perturbing magnetic field $\vec{B}_{1}$ is usually applied, above all in NMR
techniques, in such a manner that its values verify $B_{1}\ll B_{0}$ which
nevertheless imply long storage times; often, as in the original (Chicago)
I.I. Rabi research group experiences, a second opposed (to $\vec{B}_{0}$)
inhomogeneous magnetic field is also applied next to the RF oscillator group,
to refocalize the particle beam until the receiver device. In such a manner, a
very weak rotating magnetic field is able to reverse the spin direction of the
beam particles, whilst $\vec{\mu}_{L}$ precesses (Rabi’s precession), in the
rotating frame, about a well-precise ’effective’ magnetic field
$\vec{B}_{eff}$, given by the superposition of the various applied magnetic
fields, according to particular equations of motion called Bloch’s equations.
In dependence on the RF oscillator chosen as an energy source, we have either
continuous wave (CW) or pulsed wave (PW) resonance techniques: the intensity
of the resulting signal is measured in function of the magnetic field or
frequency values for the former and in function of the time for the latter. As
we shall see later, the resonance spectroscopy methods have played a
fundamental role in determining magnetic ed electric properties of atomic and
nuclear systems (see, for instance, (Bloch 1946)): for instance, through a
suitable formulation of a resonance condition, it will be possible to
experimentally determine the anomalous magnetic moment of elementary
constituents as electrons, neutrons, protons and muons.
ii) On spin precession motion. Following (Schwartz 1972, Chapter 4, Section
10), (Rich and Wesley 1972, Section 3.1.1), (Cohen-Tannoudij et al. 1977,
Volume I, Complement $F_{IV}$), (Ohanian 1988, Chapter 11, Section 11.1),
(Kinoshita 1990, Chapter 11, Sections 1-4), (Picasso 1996, Section 2), (Farley
and Semertzidis 2004, Section 3) and (Barone 2004, Chapter 6, Section 6.10), a
general precession problem is identified by a kinematical equation of the form
$d\vec{\Phi}/dt=\vec{\Omega}(t)\wedge\vec{\Phi}$, where $\vec{\Phi}$ is the
vectorial quantity that precesses around the given vector $\vec{\Omega}$; for
instance, $\vec{\Phi}$ may be a magnetic moment, an angular momentum or the
spin, which precesses around the direction given by the force lines of the
perturbing field $\vec{\Omega}$ (as, for example, a magnetic field), with
angular velocity $\Omega(t)$. The related experienced torque $\vec{\tau}$, is
given by $\vec{\Omega}(t)\wedge\vec{\Phi}$. In case of an elementary spinning
particle having charge $Q$ and mass $m$, in a (uniform) magnetic field
$\vec{B}$, we may put $\vec{\Phi}=\vec{\mu}_{s}$, where $\vec{\mu}_{s}$ is the
spin magnetic moment given by $g_{s}Q\mu_{0}\vec{\sigma}/2$ the $(6)_{1}$. In
this case, $\vec{\Omega}=k\vec{\mu}_{s}=(gQ/2mc)\vec{\mu}_{s}$, so that we
have, in the particle rest frame, the following Larmor precession equation
$d\vec{\mu}_{s}/dt=k\vec{\mu}_{s}\wedge\vec{B}$ (see (Cohen-Tannoudij et al.
1977, Volume I, Complement $F_{IV}$), (Bloch 1946, Equation (11)) and (Bargman
et al. 1959, Equation (3))) related to the precession of $\vec{\mu}_{s}(t)$
around $\vec{B}$; $\vec{\sigma}$ is said to be the polarization vector. The
relativistic generalization of the last precession equation will lead to the
so-called Bargman-Michel-Telegdi equation (see (Bargman et al. 1959)).
Following (Gottfried 1966, Chapter VI, Section 49), for beams of elementary
particles, said $\vec{\sigma}$ the Pauli operator whose components are the
Pauli matrices, the beam polarization is defined to be
$\langle\vec{\sigma}\rangle$ and shall often be written as $\vec{P}$; it is
zero for an incoherent and equal mixture of $|1/2\rangle$ and $|-1/2\rangle$,
whereas $|\vec{P}|=1$ for pure spin states.
B) The first experimental determinations of the electron AMM
Following (Kusch 1956), (Rich and Wesley 1972), (Crane 1976), (Farley and
Picasso 1979), (Combley et al. 1981), (Kinoshita 1990, Chapters 8 and 11) and
(Jegerlehner 2008, Part I, Chapter 1) and as it has already said above, P.
Kusch and H.M. Foley, in November 1947, measured $a_{e}$ for the electron with
a precision of about 5%, obtaining the value $a_{e}=0.00119(5)=0.00119\pm
0.00005$ at one standard deviation. The establishment of the reality of the
anomalous magnetic moment of the electron and the precision determination of
its magnitude, was part of an intensive programme of postwar research with
atomic and molecular beams which seen actively involved P. Kusch at Columbia,
together to I.I. Rabi research group. All that was crowned by success with the
assignment of Nobel Prize for Physics in 1955, shared with W.E. Lamb, whose
related Nobel lecture is reprinted in (Kusch 1956). Other attempts to estimate
the anomalous magnetic moment either of the electron and of the proton were
carried out by J.H. Gardner and E.M Purcell in 1949 and 1951, by R. Karplus
and N.M. Kroll in 1950, by S.H. Koenig, A.G. Prodell with P. Kusch in 1952, by
R. Beringer with M.A. Heald and by J.B. Wittke and R.H. Dicke in 1954, by P.A.
Franken and S. Liebes Jr. in 1956 as well as by W.A. Hardy and E.M. Purcell in
1958, in any case reaching to an accuracy of about 1% for the various
anomalous moment values. The Gardner and Purcell experiments (see (Gardner and
Purcell 1949) and (Gardner 1951)) introduced, for the first time, a new
experimental method to determine $a_{e}$, based on a comparison of the
cyclotron frequency of free electrons with the nuclear magnetic resonance
(NMR) frequency of protons, so opening the way to the application of resonance
techniques to measure the lepton anomalous moments on the wake of the
pioneering Rabi’s molecular beam resonance method for measuring nuclear
magnetic moments (see (Rabi et al. 1938, 1939)) recalled above. To be precise,
an experimental determination of the ratio of the precession frequency of the
proton, $\omega_{p}=\mu_{p}H_{0}$, to the cyclotron frequency,
$\omega_{e}=eH_{0}/mc$, of a free electron in the same magnetic field, was
carried out. The result, $\omega_{p}/\omega_{e}$, is the magnitude of the
proton magnetic moment, $\mu_{p}$, in Bohr magnetons $\mu_{0}$. Finally, by
the comparison between $\mu_{p}/\mu_{0}$ and $\mu_{e}/\mu_{p}$, it was
possible to determine $\mu_{e}/\mu_{0}$. Possible sources of systematic error
were carefully investigated and in view of the results of this investigation
and the high internal consistency of the data, it was felt that the true
ratio, uncorrected for diamagnetism, lie within the range
$\omega_{e}/\omega_{p}=657.475\pm 0.008$. If the diamagnetic correction to the
field at the proton for the hydrogen molecule was applied, the proton moment
in Bohr magnetons became $\mu_{p}=(1.52101\pm 0.00002)\times
10^{-3}(e\hbar/2mc)$. In (Koenig et al. 1952), the ratio of the electron spin
$g_{e}$ value and the proton $g_{p}$ value was measured with high precision.
It was found that $g_{e}/g_{p}=658.2288\pm 0.0006$, where $g_{p}$ is the $g$
value of the proton measured in a spherical sample of mineral oil. This
result, when combined with the previous measurement by Gardner and Purcell of
the ratio of the electron orbital $g_{e}$ value and the proton $g_{p}$ value,
yielded for the experimental value of the magnetic moment of the electron
$\mu_{s}=(1.001146\pm 0.000012)\mu_{0}$. The result was in excellent agreement
with the theoretical value calculated by Karplus and Kroll, namely
$\mu_{s}=(1.0011454)\mu_{0}$. However, all these methods were related to
electrons bound in atoms, this implying, amongst other things, a lower
accuracy level due to the corrections necessary to account for atomic binding
effects. Thus, anomalous moment experimental determinations on free electrons
were more suitable.
Following (Rich and Wesley 1972), (Kinoshita 1990, Chapter 8), in the years
1953-54, H.R. Crane, W.H. Louisell and R.W. Pidd at Michigan, for the first
time, determined $a_{e}$ for free electrons from measurements of $g-2$ (not
$g$ itself) by means of the precession of the electron spin in a uniform
magnetic field, obtaining the result $g=2.00\pm 0.01$, that is to say, $g$
must be within 10% of 2.00. They introduced, on the basis of the previous
basic work made by N.F. Mott in 1930s, a new pioneering technique which will
be later called the $(g-2)$ precession method, so opening the way to the
precession methods for determining lepton g factors. Following (Louisell et
al. 1954), (Hughes and Schultz 1967, Chapter 3), (Rich and Wesley 1972),
(Combley and Picasso 1974) and (Crane 1976), we briefly recall the main stages
which led to the experimental methods for measuring the magnetic moment of the
free electron according to this $(g-2)$ precession method. A first attempt was
based, after a N.H. Bohr argument252525Arguing upon the unobservability of the
magnetic moment of a single electron on the basis of the well-known Heisenberg
indetermination principle. Therefore, we must consider a statistical approach
in such a manner that the average behavior of the spins of a large ensemble of
particles can be treated, to a large extent, as a classical collection of
spinning bar magnets., on a statistical fashion of the well-known 1924 Stern-
Gerlach experiment on the atomic magnetic moments, applied to free electrons
and consisting in sending a large number of electrons through a magnetic field
and by attempting to use the detailed line shape to reveal the effects of the
magnetic moment. Nevertheless, such a method appeared particularly unpromising
in connection to a precise solution to the electron moment problem. A second
attempt, instead, was based on the previous 1929 N.F. Mott double-scattering
method for studying the polarization of particles beams. The Louisell, Pidd
and Crane principle of the method employed a Mott double-scattering method
roughly consisting in producing polarized electrons by shooting high-energy
electrons upon a gold foil; hence, the part of the electron bunch which is
scattered at right angles, is then partially polarized and trapped in a
constant magnetic field where spin precession takes place for some time. The
bunch is afterwards released from the trap and allowed to strike a second gold
foil, which allows to analyze the relative polarization. To be precise, this
method depend on the fact that a beam of electrons is partially polarized
along a direction normal to the plane defined by the incident beam and the
emerging scattering direction. Furthermore, a second scattering process
exhibited an azimuthal asymmetry in scattering intensity, if measured in the
same plane, mainly due to polarization perpendicular to the plane of the
incident and scattered beams. Mott defined the amplitude of this asymmetry as
$\delta$ and provided some its estimates. To explain this effect, both on the
basis of the above Bohr’ argument and in taking into account the Stern-Gerlach
results, Mott put forward the hypothesis that electron spins had to be thought
of as precessing around the direction of a magnetic field rather than as
aligned parallel or anti-parallel to this, like in the Stern-Gerlach
experiment262626Following (Miller et al. 2007) and (Roberts and Marciano 2010,
Chapter 1), the study of atomic and subatomic magnetic moments began in 1921
first with a paper by O. Stern then with the famous 1924 O. Stern and W.
Gerlach experiment in which a beam of silver atoms was done pass through a
gradient magnetic field to separate the different magnetic quantum states.
From this separation, the magnetic moment of the silver atom was determined to
be one Bohr magneton $\mu_{0}$ within 10%. This experiment was carried out to
test the Bohr-Sommerfeld quantum theory. In 1927, T.E. Phipps and J.B. Taylor
repeated the experiment with a hydrogen beam and they also observed two bands
from whose splitting they concluded that, like silver, the magnetic moment of
the hydrogen atom was too one $\mu_{0}$. Subsequently, in 1933, R.O. Frisch
and O. Stern determined the anomalous magnetic moment of the proton, while in
1940, L.W. Alvarez and F. Bloch determined the anomalous magnetic moment of
the neutron, and both turned out to be quite different from the value 2,
because of their internal structure.. Therefore, the asymmetry observed along
the second scattering should be due to this precession because, if the spin
were aligned parallel and anti-parallel to the direction of a magnetic field
parallel to the beam incident on the scatterer of the experimental apparatus,
then it would be enough to apply a weak magnetic field to remove such an
asymmetry effect. In this sense, the spin had to be meant as a physical
observable rather than a mathematical device (d’après Pauli). Furthermore,
since this 1954 Louisell-Pidd-Crane method essentially requires a simultaneous
measurement of the electron position and of a single spin component, it
follows that the uncertainty principle is not violated. Crane says that Mott’s
way out of his dilemma was, perhaps, the first break toward thinking of
electrons as precessing magnets. Nevertheless, this far seeing Mott’s hint
didn’t took by nobody at that time until the 1953-54 pioneering works of
Louisell, Pidd and Crane. They extended this Mott double-scattering method
inserting, between the first and second scatterers, a constant magnetic field,
parallel to the path to the path between the scatterers, in the form of a
magnetic mirror trap which permitted the electrons to undergo several hundred
$(g-2)$ precessions between scatterings. This causes the electron to precess
and rotates the polarization plane of maximum asymmetry after the second
scattering no longer coincides with the plane of the first scattering. By
measuring the angle of rotation and knowing the magnetic field, the electron
energy and the distance, the gyromagnetic ratio for the electron may be found.
A fact which had a dominating influence was that the orbital, or cyclotron,
angular frequency of the electron in the magnetic field differs from the
angular frequency of precession of the spin direction although in higher-order
correction terms, these respectively being given by $\omega_{o}=eB/(2mc)$ and
$\omega_{s}=g(eB/(2mc))$ with $g=2(1+\alpha/2\pi+...)$ (d’après Schwinger).
This fact turns out to be useful to determine $g$ whose value may be therefore
determined from a direct comparison of the rotation of the plane of
polarization and the cyclotron rotation. Moreover, all observed asymmetries in
the beam, whether they are associated with the spin or not, rotate around
together, so that it was needed for discriminating amongst them. Certain
sources of asymmetry have nothing to do with the polarization effect
notwithstanding they follow the polarization asymmetry itself as it rotates
around. However, Louisell, Pidd and Crane were able to determine and isolate
the non-spin asymmetry, mainly due to scattering nonlinearities, from spin
asymmetry that was experimentally detected with very small measurement errors.
Due to the action of the Lorentz force, if $\phi_{c}$ (or $\phi_{o}$) is the
cyclotron (or orbital) rotation angle between scatterers, $\phi_{d}$ is the
sum of deflection angles at entry and exit to the solenoid field, and
$\phi_{s}$ is the angle through which the spin asymmetry was rotated relative
to the direction of the beam before entry into the solenoid field, then an
estimate to $g$ is given by $2(\phi_{s}-\phi_{d})/\phi_{c}$, whose
experimentally detected values were reported in Table I of (Louisell et al.
1954), computed at different values of $B$. Nevertheless, Louisell, Pidd and
Crane concluded that the precision of which their method is capable (they
obtained an accuracy of 1%) was not enough to reveal the correction to the $g$
factor at about one part in a thousand, so that their result wasn’t
sufficiently precise to be useful in comparison with the theoretical
prediction. Meanwhile, or in parallel, the results so found have been
ascertained to be coherent with Dirac theory of electron by H. Mendlowitz with
K.M. Case, who also calculated the possible effects of a uniform magnetic
field on a Mott double-scattering experiment showing that they can be used to
measure $a_{e}$ as in the Louisell-Pidd-Crane experience. Coherence with Dirac
theory also came from a previous 1951 work of H.A. Tolhoek and S.R. De Groot
which concerned another parallel research area on hyperfine structures
oriented towards precision measurements on $g$ of the free electron; the
latter proposed, in 1951, a scheme in which a magnetic field and a RF field
were interposed between the first and second Mott scatterers, and in which
destruction of the asymmetry indicated resonance. A notable research group
based at the University of Columbia and directed by I.I. Rabi since 1940s,
followed another line of attack to measure the gyromagnetic ratio for the free
electron, based upon the magnetic resonance method, proposing new experiments
in two somewhat different forms respect to the previous research line based on
Mott scattering method. In both these forms, polarized electrons are trapped
in stable orbits into a magnetic field. A radio-frequency (RF) perturbing
field is then applied and the frequency which destroys the polarization is
determined. From the frequency which destroys the polarization and the
strength of the magnetic field, the value of the gyromagnetic ratio is
obtained. Since 1956, H.G. Dehmelt group at Washington demonstrated that spin-
exchange collisions between oriented sodium atoms and free, thermal energy
electrons could be used to measure $a_{e}$ via a direct RF resonance
technique, so contributing to the first determinations of the free electron
anomalous magnetic moment.
The two above mentioned methods mainly differ in the way in which the
electrons are polarized, giving priority to trapping, and in the way in which
the presence or absence of polarization is determined after the application of
the magnetic or RF perturbing field and the subsequent escaping from the
trapping phase carried out by the latter. The essence of the method consists
essentially in finding the frequency of the feeble beat between the rotation
of the spin direction (in the trap or well) and the orbital, or cyclotron,
rotation when the particles are trapped in a well-determined magnetic well.
Afterwards, a careful determination of electron energies as well as a precise
control of fields and potentials are also demanded. Forerunners of resonance
methods, other than the above mentioned one, may be also retraced in some
previous experiences made by R.H. Dicke and F. Bloch in the early 1940s. In
any case, following (Louisell et al. 1954), in both methods in which resonance
is involved, the strong coupling to the cyclotron motion due to the fact that
the required perturbing frequency is almost identical to the cyclotron one
with consequent transfer of energy from the perturbing field to the cyclotron
motion, might introduce serious difficulties in order to achieve the right
accuracy with the increasing of the cyclotron revolutions. Furthermore, it is
very difficult to control the particle while it is into the trap inside which
it oscillates (along the $Z$ direction, parallel to the perturbing field).
Nevertheless, Louisell, Pidd and Crane state that the magnetic resonance
methods, together their experimental extension to the Mott double-scattering
method, seem to be the only ones272727Besides some other experimental attempts
to get polarized beams of electrons, by F.E. Myers and R.T. Cox as well as by
E. Fues and H. Hellman, at the end of 1930s. able to give really quantitative
results of sufficient accuracy to reveal the correction to the electron
moment. Some problems occur when we consider electrons and positrons which
both require to be previously polarized: for the former, the above mentioned
Mott scattering method is used, while for the latter, a suitable radioactive
source is used for their initial polarization whereas the final one is found
through a clever scheme first proposed by V.L. Telegdi (see …. (Grodzins 1959)
and references therein). As regards muons, instead, this last problem does not
subsist since them born already polarized and reveal their final polarization
through the direction of the related decay products. Following (Crane 1976)
and (Hughes and Schultz 1967, Chapter 3, Section 3.5.3.1), in 1958, P.S.
Farago proposed a method282828Besides also quoted by (Bargmann et al. 1959,
Case (E))). for comparing the orbital and the spin precession of electrons
moving in a magnetic field, which will turn out to be useful to directly
measure radiative corrections to the free-electron magnetic moment. Indeed,
the Farago’s principle of the method consisted in considering initially
polarized electrons, emitted by a $\beta$ active source and moving
perpendicular to a strong uniform magnetic field $\vec{B}$, hence using a Mott
scattering for analysis. A uniform weak vector field $\vec{E}$ is also applied
perpendicularly to $\vec{B}$ in such a manner that the beam walks enough to
miss the back of the source of the first turn. The beam continues walking
towards right for a distance almost equal to the orbital diameter. After the
order of about some hundreds of revolutions, it then encounters a Mott
scattering foil at which the final direction of polarization perpendicular is
determined from the intensity asymmetry in the direction perpendicular to the
orbit plane. If the final polarization direction is measured as a function of
the transit time between source and target (consisting of about 250 orbital
revolutions or turns), then a sine curve is obtained whose frequency is equal
to the difference between the spin precession frequency and the orbital
frequency of the circulating electrons. To the extent that $E/B\ll 1$
(electron trochoidal motion), this difference frequency is proportional to
$(\mu_{e}/\mu_{0}-1)=g/2-1=a_{e}$, so that the Farago’s method measures
directly the radiative correction to the free electron magnetic moment
$\mu_{e}$, hence $a_{e}$ (see (Farago 1958)). The Farago’s method was later
improved and experimentally realized by his research group at the University
of Edinburgh (see (Farago et al. 1963)); it constituted, at that time, the
first method that allowed a continuous measurement rather than by pulses.
Nevertheless, the Farago’s method couldn’t compete in accuracy with
experiments in which the particles are trapped and allowed to make a far
larger number of revolutions. In any case, its principle of the method, in
some respects, has preempted certain basic methods underpinning some later
storage techniques (amongst which the one based on polynomial magnetic
fields). Other determinations of $a_{e}$ were later realized, in the early
1960s, by D.T. Wilkinson, D.F. Nelson, A.A. Schupp, R.W. Pidd and H.R. Crane
(Michigan group) even improving their principle of the method of 1954 and
mainly based upon the remark that, if polarized electrons were caused to move
with their velocities perpendicular to a uniform magnetic field, then, at a
fixed azimuth on the cyclotron orbits, one would observe the polarization
precessing at a rate equal to the difference between the spin precession rate
($\omega_{s}$) and the orbital cyclotron rate ($\omega_{c}$), just this
difference precession rate (anomalous or spin-cyclotron-beat frequency
$\omega_{a}=\omega_{s}-\omega_{c}$) being directly proportional to $a_{e}$.
This method will be generically called the (Michigan) principle of $(g-2)$
spin motion, or simply spin precession method (or also free-precession
method), and will lead to the next basic equation (59).
Following (Rich and Wesley 1972) and (Crane 1976), meanwhile the spin
precession methods were further pursued as a result of the pioneering works
made by the above Michigan group, other techniques were employed to approach
$g-2$, above all for electrons. As it has already said above, H.A. Tolhoek and
S.R. De Groot proposed, since 1951, a scheme in which a magnetic field,
coupled with a RF field, would be interposed between the first and the second
Mott scatterers, even if themselves were aware that such an apparatus wasn’t
able to provide enough cycles of the spin precession to give a well defined
frequency, mainly because of the absence of a trap. In 1953, F. Bloch proposed
a novel resonance-type experiment to measure $a_{e}$ using electrons occupying
the lowest Landau level in a magnetic field. In the years 1956-58, H.G.
Dehmelt performed an experiment in which free thermal electrons in argon
buffer gas, at the mean temperature of 400${}^{o}K$, become polarized in
detectable numbers by undergoing exchange collisions with oriented sodium
atoms during which the atom orientation is transferred to the electrons. Such
collisions establish interrelated equilibrium values among the atom and the
electron polarizations which depend on the balance between the polarizing
agency acting upon the atoms (optical pumping) and the disorienting relaxation
effects acting both on atoms and electrons. When the electrons were
furthermore artificially disoriented by gyromagnetic spin resonance, an
additional reduction of the atom polarization ensued, which was detected by an
optical monitoring technique (with an optical pumping cell rather than a
quadrupole trap), so allowing to the determination of the free-electron spin
$g$ factor and opening the way to experimentally use the so-called Penning
trap consisting of a uniform axial magnetic field $\vec{B}=B_{0z}\hat{z}$ and
a superimposed electric quadrupole field generated by a pair of hyperbolic
electrodes surrounding the storage region. The magnetic field confines the
electrons radially, while the electric field confines them axially. The
essential novel feature of the this Dehmelt’s techniques consisted, following
an idea of V.L. Telegdi and co-workers (see (Ford et al. 1972)), in the fact
that a RF induced pulse (or beat) frequency, rather than a spin precession
frequency, was the main responsible to rotate the polarization. The principle
of the method is quite similar to the known spin echoes of E.L. Hahn (1950) in
which an intense RF power in the form of pulses is applied to an ensemble of
spins in a large static magnetic field. The frequency of the pulsed RF power
is applied through a RF current circulating in a wire stretched along the
center axis of the trapping chamber, producing lines of force that are circles
concentric with the orbits. If the RF is held on for the right length of time,
then the polarization is turned from the plane perpendicular to the applied
magnetic field towards the direction parallel to it. Afterwards, it comes back
again if the RF pulse is held on twice as long, just like spin echoes.
Following (Gräff 1971), (Rich and Wesley 1972) and (Holzscheiter 1995), the
precision measurements of lepton $g$-factor anomalies can be classified as
being either precession experiments and resonance experiments in dependence on
the technique employed, in both of which the main involved problem being that
concerning the trapping of polarized charged particles. The main dynamical
features of the problem are as follows: the momentum $\vec{p}$ of the
particle, which is exactly perpendicular to $\vec{B}$, revolves with the
cyclotron (or orbital) angular frequency $\omega_{c}=QB/mc$, the spin
precesses about $\vec{B}$ with Larmor angular frequency
$\omega_{s}=(1+a_{l})\omega_{c}$ with $a_{l}=(g-2)/2$, while the difference
between these angular frequencies is the one at which the spin rotates about
the momentum, that is to say
$\omega_{a_{l}}=\omega_{s}-\omega_{c}=a_{l}QB/mc=\theta/T$ where $\theta$ is
the angle between spin and momentum and $T$ the time. Consequently, to get the
lepton anomaly $a_{l}$, it is thus necessary to measure the quantities
$\omega_{a_{l}}$ and $B$, assuming $Q/mc$ to be known. Thus, we have
$a_{l}=\omega_{a_{l}}/\omega_{c}$ (see also (Kinoshita 1990, Chapter 11,
Section 4.1, Equation (4.8)). If the particle velocity has a small angle
relative to the orbital plane $x$-$y$ of motion particle, then the particle
will follow a spiral path, along the axial direction given by the $z$-axis,
with pitch angle $\psi$, spiralling in the main (not necessarily constant)
magnetic field $B_{z}$; the $(g-2)$ frequency is consequently altered. In any
real storage system, the pitch angle is corrected by suitable vertical
focusing forces which prevent the particles to be lost. Furthermore, the pitch
angle changes periodically between positive and negative values, so that the
correction to the $(g-2)$ frequency become more complex. All the $(g-2)$
experiments for electrons and muons are in principle subject to a pitch
correction and, as we will see later, this problem will be successfully
overcome, for the first time, with the introduction of the so-called
polynomial magnetic fields. An arbitrary experiment which attempts to measure
the anomalous magnetic moment of a free lepton necessarily encounters the
following problems: a) trapping of the particle; b) measurement of the
trapping field either by nuclear magnetic resonance (NMR) or by measuring
$\omega_{s}$ or $\omega_{c}$; c) polarization of the spin of the particle; d)
determination of the anomaly frequency either i) by detection of the spin
polarization vector relative to the momentum vector of the particle as a
function of the time in a magnetic field, calling this type of experiment a
geometrical experiment292929Roughly corresponding to the above precession
experiment type., or, alternatively, ii) by induction and detection of the
relevant RF transition $\omega_{s}$ and $\omega_{c}$ or, if possible,
$\omega_{s}$ or $\omega_{c}$ and the difference angular frequency $\omega_{a}$
directly, calling this type of experiment a RF spectroscopic
experiment303030Roughly corresponding to the above resonance experiment type..
To trap particles, it has been used: 1) the magnetic bottle method consisting
in imposing a homogeneous magnetic field with a superimposed relatively weak
inhomogeneous magnetic field as first used by the above mentioned Michigan
group; 2) a RF quadrupole trap starting from the first studies on electric
quadrupole mass separator made by F. v. Bush, W. Paul, H.P. Reinhard with U.
v. Zahn and by E. Fisher, in the 1950s, for separating isotopes. To detect the
ions, a resonance detection technique is used, taking advantage of the fact
that for given parameters of the trap each charge-to-mass ratio exhibits a
certain unique ”eigenfrequency”. In addition to the radio-frequency quadruple
field, a RF dipole field at the frequency $\omega_{res}$ is applied as well to
the end caps. If through proper choice of the parameters $a$ and $q$,
respectively representing the amplitudes of the RF component and the direct
current (DC) component of the quadruple field, the ions are brought to
resonance with this dipole field, then the amplitude of the ion motion is
increased, absorbing energy from the drive field, and can be detected. The
important fact is that different ions will have different frequencies for a
given set of $a$ and $q$, or, that at a fixed frequency, one can bring all
different ion species to resonance subsequently by slowly varying the DC
potential at a constant RF amplitude. This made the quadruple trap an ideal
tool for precision mass spectrometry or residual gas analysis, areas in which
RF traps have gained high respect over the last decades. At first glance, the
RF drive field seems to be a disturbance to the system, and in effect it is.
Due to the continuously applied drive force stored particles are heated
permanently, leading to 2nd order doppler broadening of spectral lines. This
effect can be counteracted by cooling mechanisms, either collisions with
residual gas molecules, or far more powerful and selective than this, by laser
cooling. Nevertheless, due to this ”micromotion”, the Paul’s research group
trap has always been a second choice respect to the so-called Penning trap if
one desired an ultrahigh precision work. Based on this last new device, dating
back to the late 1930s F.M. Penning works, D.H. Dehmelt group at Seattle
(Washington), P.S. Farago group at Edinburgh and G. Gräff group at Bonn/Mainz
have performed various electron $g-2$ experiments. As concerns, instead, the
polarization problem, in experiments of geometrical type, polarized muons are
produced by the forward decay of pions, polarized electrons by Mott double-
scattering and polarized positrons by beta decays, while, as regards
experiments of RF spectroscopic type, electrons are polarized by means of spin
exchange with a polarized atomic beam as well as electrons of low energy are
created in pulses in a high magnetic field. Finally, as regards the
determination of the lepton anomaly, in the geometrical experiments the angle
$\theta$ between the spin vector and momentum of the particle is measured at a
fixed orbital point as a function of time. The polarization of electrons is
detected by Mott double-scattering, the polarization of positrons by
exploiting the spin dependence by ortho- and para-positronium formation,
whilst the muon polarization is measured using the fact that in the rest
frame, the decay electrons are preferentially emitted along the spin
direction. As the momentum of a particle in a magnetic bottle is no longer
perpendicular to the magnetic field, the Bargmann-Michel-Telegdi (BMT) formula
for $\omega_{a}$ (see (Bargmann et al. 1959, Equation (9))) has to be used.
Instead, in the RF spectroscopic measurements, the transition at frequency
$\omega_{a}$ has to be induced and observed. Nevertheless, this level
transition corresponds to a combination of a magnetic and electric dipole
transition with $\Delta n=\pm 1$ and $\Delta m_{s}=\pm 1$ at the same
time313131For instance, a quantum state transition from $|n,m_{s}=-1/2\rangle$
to $|n-1,m_{s}=+1/2\rangle$ is forbidden being a second order (two-photon)
transition because it involves a simultaneous change of the spin quantum
number ($m_{s}$) and of the orbital (or cyclotron) quantum number ($n$). But,
with a proper choice of the electromagnetic configuration by means of the
application of a suitable perturbing field, this transition can be driven.;
such a transition if forbidden to first order, but it can be enforced by an
inhomogeneous magnetic RF field which, in turn, necessarily must be
accompanied by a homogeneous magnetic RF field. This last field, nevertheless,
may produce line shifts and line asymmetries. Furthermore, the transition at
frequency $\omega_{a}$ involves a jump from one cyclotron orbit to another
with a spin flip at the same time; likewise for the induction of the Larmor
frequency. The main limitations of RF spectroscopic experiments lie just in
this transition prohibition and in the presence of unwanted homogeneous
magnetic RF fields; another limitation is also provided by the limited energy
of the trapped particles. In conclusion, the principle of the method of almost
all $g-2$ experiments roughly consists in measuring the interaction between
the magnetic moment of the particle and a homogeneous magnetic field
superimposed by an inhomogeneous magnetic or electric trapping field. The
latter, nevertheless reduces the accuracy of the experiments which may be
improved decreasing the relative inhomogeneity even if, for technical reasons,
this is not possible in the $g-2$ experiments of the muons through further
substantial increase of the homogeneous magnetic field.
Therefore, to sum up (following (Rich and Wesley 1972)), the precession
experiments include measurements of the electron, positron and muon anomalies,
the distinguishing feature of these experiments (as those made at Michigan for
electrons and at CERN for muons) being a direct observation of the spin
precession motion of polarized leptons in region of static magnetic field. The
resonance technique instead has mainly been used to measure lepton anomaly
(prior to electrons), its characteristic feature being the presence of an
oscillating electromagnetic field used to induce transitions between the
energy eigenstates of a lepton interacting with a static magnetic field by
applying a microwave field at the spin precession frequency $\omega_{c}$ and
subsequently a RF field at the spin-cyclotron difference frequency
$\omega_{a}$.
c) Towards the first experimental determinations of muon AMM
In the same period in which the above mentioned electron AMM determinations
were achieved, many further experimental evidences were also accumulated in
confirming that the muon behaved as a heavy electron of spin $1/2$, so that
the former were taken as models to set up possible experiences for the latter.
But, before to outline these, what were the theoretical motivations underlying
the researches towards muon? In 1956, V.B. Berestetskii, O.N. Krokhin and A.X.
Klebnikov, in providing, through processes involving photons and leptons, a
sensitive test of the limit for the (R.P. Feynman) UV cut-off (or QED-
breaking) $\Lambda_{l}$, which represents a measure for the distance at which
QED breaks down, pointed out that the measurement of the muon anomalous
magnetic moment could accomplish this in a more sensitive manner than that of
the electron. Indeed, if one supposes that the muon is not completely point-
like in its behavior, but has a form factor323232The dependence on $q^{2}$ of
the form factors, experimentally enables us to get information about charge
radial distributions and magnetic moments of charged leptons (see (Povh et al.
1995, Part I, Chapter 6, Section 6.1)). For instance, for a generic Dirac
particle, we have $F(q^{2})=1$.
$F_{\mu}(q^{2})=\Lambda_{\mu}^{2}/(q^{2}+\Lambda_{\mu}^{2})$, then it can be
show that an expression for the sensitivity of $a_{\mu}$ is given by
(55) $\frac{\delta a_{\mu}}{a_{\mu}}=-\frac{4m_{\mu}^{2}}{3\Lambda_{\mu}^{2}}$
which may be generalized for leptons as follows
(56) $\frac{\delta a_{l}}{a_{l}}\sim\frac{m_{l}^{2}}{\Lambda_{l}^{2}},\ \ \ \
\ \ l=e,\mu,\tau.$
Berestetskii, Krokhin and Klebnikov emphasized that the high muon mass could
imply a significant correction to $a_{\mu}$ even when $\Lambda_{\mu}$ is
large. Therefore, due to its high mass, the muon allows to explore very small
distances (of the order of $10^{-15}$ cm) because of the simple fact that
$q^{2}\sim m$ and the higher it is the momentum $q^{2}$, the higher it is the
energy involved and, therefore, the shorter it is the involved distance scale
due to uncertainty principle. Furthermore, mainly because of the vastly
different behavior of the three charged leptons mainly due to the very
different masses $m_{l}$ implying completely different lifetimes
$\tau_{e}\simeq\infty$ and $\tau_{l}=1/\Gamma_{l}\varpropto
1/(G_{F}^{2}m_{l}^{5})\ l=\mu,\tau$, as well as vastly different decay
patterns, it was clear that the anomalous magnetic moment of the muon would be
a much better probe for possible deviations from QED. In 1957, J. Schwinger
thought that the muon could have an extra interaction which distinguished it
from the electron and gave it its higher mass. This could be a coupling with a
new massive field or some specially mediated coupling to the nucleon. Whatever
the source be, the new field would have had its own quantum fluctuations, and
therefore gives rise to an extra contribution to the anomalous moment of the
muon. Thus, the principle of $(g-2)$ spin motion was also recognized as a very
sensitive test of the existence of such fields and potentially a crucial
signpost to the so-called $\mu-e$ puzzle (see later). But, at that time, there
wasn’t any possibility to descry some useful principle of the method for
pursuing this333333For instance, the parity violation of weak interactions was
not yet known at that time., so that nobody had an idea how to measure
$a_{\mu}$. Albeit the $(g-2)$ spin motion principle will turn out to be, a
priori, very similar to those later developed to measure $a_{\mu}$,
nevertheless it was immediately realized that handling the muons in a similar
way was impossible, and this raised the difficult task of how to may polarize
such short lived particles like muons, in comparison with the long lifetimes
of electrons which allowed to measure $a_{e}$ directly by atomic spectroscopy
in magnetic fields. As we shall see later, this was pursued, for the first
time, by the pioneering works of the first CERN research groups on $g-2$ since
the late 1950s, above all thanks to new magnetic storage techniques set up
just to this end. Nevertheless, behind this last pioneering research work,
there was a great and considerable previous work of which a brief outline we
are however historically obliged to remember.
The principle of the method of the Michigan group experiments has been applied
to determine the muon $g$-factor in some experiments performed, since the
middle 1950s, by a notable research group of the Columbia University headed by
L.M. Lederman in the wake of the previous work of his maestro I.I. Rabi (see
(Lederman 1992)). In 1958, T. Coffin, R.L. Garwin, S. Penman, L.M. Lederman
and A.M. Sachs (see (Coffin et al. 1958)) made a RF spectroscopic experiment
with stopped muons in which the magnetic moment of the positive $\mu$ meson
was measured in several target materials by means of a solid-state nuclear
magnetic resonance technique with perturbing RF pulses. Muons were brought to
rest with their spins parallel to a magnetic field. A radio-frequency (RF)
pulse was applied to produce a spin reorientation which was detected by
counting the decay electrons emerging after the pulse in a fixed direction.
The experimental results were expressed in terms of a $g$-factor which for a
spin 1/2 particle is the ratio of the actual moment to $e\hbar/2m\mu c$. The
most accurate result obtained in a $CHBr_{3}$ target, was $g=2(1.0026\pm
0.0009)$ compared to the theoretical prediction of $g=2(1.0012)$, while less
accurate measurements yielded $g=2.005\pm 0.005$ in a copper target and
$g=2.00\pm 0.01$ in a lead target. After the well-known above mentioned 1956
proposal of parity violation in weak transitions by T.D. Lee and C.N. Yang, it
was immediately realized that muons produced in weak decays of the pion
$\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ (see Section 1) could be longitudinally
polarized, while the decay positron of the muon $\mu^{+}\rightarrow
e^{+}+2\nu_{\mu}$343434Only after 1960, it was ascertained that
$\nu_{\mu}\neq\bar{\nu}_{\mu}$, whereupon we might more correctly write
$\mu^{+}\rightarrow e^{+}+\nu_{\mu}+\bar{\nu}_{\mu}$ (see Section 1). could
indicate the muon spin direction. This was confirmed by R.L. Garwin, L.M.
Lederman and M. Weinrich (see (Garwin et al. 1957)), as well as by J.I.
Friedman and V.L. Telegdi (see (Friedman and Telegdi 1957)), in the same year
of353535For technical reasons, the paper of Friedman and Telegdi was delayed
to the Physical Review Letters issue next to the one in which was published
the paper of Garwin, Lederman and Weinrich, notwithstanding both papers were
received almost contemporaneously, the former on January 17, 1957 and the
latter on January 15, 1957. Nevertheless, following (Cahn and Goldhaber 2009,
Chapter 6), the Friedman and Telegdi emulsion experiment at Chicago was
started before others but has employed more time to be completed because of
the laborious scanning procedure. 1957\. The first researchers, who achieved
an accuracy of 5%, started from certain suggestions, made in the remarkable
works of T.D. Lee, R. Oehme and C.N. Yang, according to which their hypotheses
on violation of $C$, $P$ and $T$ symmetries had to be sought in the study of
the successive reactions $1)\ \pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ and $2)\
\mu^{+}\rightarrow e^{+}+\nu_{\mu}+\bar{\nu}_{\mu}$. To be precise, they
pointed out that the parity violation would have implied a polarization of the
spin of the muon emitted from stopped pions in the first decay reaction along
the direction of the motion; furthermore, the angular distribution of
electrons in the second decay reaction could serve as an analyzer for the muon
polarization. Moreover, in a private communication, Lee and Yang also
suggested to Garwin, Lederman and Weinrich that the longitudinal polarization
of the muons could offer a natural way of determining their magnetic moment,
partial confirmations of the validity of this idea having already been
provided by the preliminary results of the celebrated C.S. Wu and co-workers
experiments on $Co^{60}$ nuclei. By stopping, in a carbon target puts inside a
magnetic shield, the polarized $\mu^{+}$ beam formed by forward decay in
flight of $\pi^{+}$ mesons inside the cyclotron, Garwin and co-workers
established the following facts: i) a large asymmetry was found for electrons
in 2), establishing that the $\mu^{+}$ beam was strongly polarized; ii) the
angular distribution of the electrons was given by $1+a\cos\theta$ where
$\theta$ was measured from the velocity vector of the incident muons, founding
$\theta=100^{o}$ $a=-1/3$ with an estimated error of 10%; iii) in both
reactions, parity was violated; iv) by a theorem of Lee, Oheme and Yang (see
(Lee et al. 1957)), the observed asymmetry proves that invariance under charge
conjugation is not conserved; v) the $g$ value for free $\mu^{+}$ particles
was found to be $+2.00\pm 0.10$; and vi) the measured $g$ value and the
angular distribution in 2), led to the very strong probability that the
$\mu^{+}$ spin was 1/2. The magnetizing current, induced by applying a uniform
small vertical field in the magnetic shielded enclosure about the target,
produced as a main effect the precession of muon spins, so that a road based
on muon spin precession principle to seriously think about the experimental
investigation of $a_{\mu}$, was finally descried. Amongst other things, the
work of Garwin, Lederman and Weinrich opened the way to the so-called muon
spin resonance ($\mu$SR), a widespread tool in solid state physics and
chemical physics. In 1957, their result was improved to an accuracy of about
4% by J.M. Cassels, T.W. O’Keele, M. Rigby, A.M. Wetherall and J.R. Wormald.
Likewise, following the celebrated suggestion of Lee and Yang on non-
conservation of parity in weak interactions, Friedman and Telegdi (1957)
investigated the correlation between the initial direction of motion of the
muon and the direction of emission of the positron in the main decay chain
$\pi^{+}\rightarrow\mu^{+}\rightarrow e^{+}$ produced in nuclear emulsions
just to detect a possible parity non-conservation in the latter decay
interactions. Following Lee and Yang arguments, violation of parity
conservation may be inferred essentially by the measurement of the probability
distribution of some pseudoscalar quantity, like the projection of a polar
vector along an axial vector. For instance, Lee and Yang themselves suggested
several experiments in which a spin direction is available as a suitable axial
vector; in particular, they pointed out that the initial direction of motion
of the muon in the decay process $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ can
serve for this purpose, as the muon will be produced with its spin axis along
its initial line of motion if the Hamiltonian responsible for this process
does not have the customary invariance properties. If parity is further not
conserved in the decay process $\mu^{+}\rightarrow e^{+}+2\nu_{\mu}$, then a
forward-backward asymmetry in the distribution of angles, say $W(\theta)$,
between this initial direction of motion and the moment of the decay electron,
is predicted. To this end, positive pions from the University of Chicago
synchrocyclotron were brought to rest in emulsion carefully shielded from
magnetic fields, as well as over 1300 complete decay events were measured. A
correlation $W(\theta)=1+a\cos\theta$ was found, with $a=-0.174\pm 0.038$,
clearly indicating a backward-forward asymmetry, that is to say a violation of
parity conservation in both decay processes. Following an argument of T.D.
Lee, R. Oehme363636Reinhard Oehme (1928-2010) was an influential theoretical
physicist who gave notable contributions mainly in mathematical and
theoretical physics. Amongst these, Oehme was the first to realize that every
time the $CPT$ symmetry must be obeyed, then if $P$ was violated, $C$ and/or
$T$ had to be violated as well. He proved that if the various experiments
suggested by Lee and Yang showed a $P$ violation, then $C$ had to be violated
too. In this regards, Oehme sent a letter to Yang and Lee explaining this
insight, and they immediately suggested that all three together would have
written a paper (Lee et al. 1957)). See above all (Yang 2005) where this
historical event, often misunderstood, has definitively been clarified. and
C.N. Yang, this asymmetry would have implied a non-invariance of either decay
reactions with respect to both space inversion $P$ and charge conjugation $C$,
taken separately. Furthermore, Friedman and Telegdi given a detailed
discussion of a depolarization process specific to $\mu^{+}$ mesons, i.e. the
possible formation of muonium $(\mu^{+}e^{-})$. The results of this and
similar experiments were also compared with those obtained with muons
originating from $p^{+}$ decays in flight and the implications of such a
comparison were discussed too. Therefore, the Friedman and Telegdi work, for
the first time, pointed out, also thanks to a private communication with R.
Oehme, that $P$ and $C$ were violated simultaneously, or rather, to be
precise, $P$ was normally violated while $CP$ was to very good approximation
conserved, in the decay processes analyzed by them.
Following (Farley and Picasso 1979) and (Jegerlehner 2008, Part I, Chapter 1),
it should be mentioned that until the end of 1950s, the nature of the muon was
quite a mystery. In that period, the possible deviations from the Dirac moment
$g=2$ were ascribed to the interaction of leptonic particle with its own
electromagnetic field. Any other field coupled to the particle would produce a
similar effect and, in this regards, the calculations have been made for
scalar, pseudoscalar, vector and axial-vector fields, using an assumed small
coupling constant $f$ to a certain boson of mass $M$. For example, for the
case of a vector field, the above mentioned work of Berestetskii, Krokhin and
Klebnikov as well as the 1958 work of W.S. Cowland, provided the estimate
$\delta a_{\mu}^{Vec}=(1/3\pi)(f^{2}/M^{2})m^{2}_{\mu}$ so that a precise
measurement of $a_{\mu}$ could therefore reveal the presence of a new field,
but, before this, it had to be discovered all the known fields, comprising the
weak and strong interactions, and hereupon taken into account. Following
(Picasso 1996) and references quoted therein, the theoretical value for
$a_{\mu}$ can be expressed as follows
$a_{\mu}^{(th)}=a_{\mu}^{QED(th)}+a_{\mu}^{QCD(th)}+a_{\mu}^{Weak(th)}$. In
the 1950s, the only contribution which could be measured with a certain
precision was the QED one, while both the strong and weak interaction
contributions will be determined only later373737The first ones who pointed
out on the importance of hadronic vacuum-polarization contributions to
$a_{\mu}$ were C. Bouchiat and L. Michel in 1961 as well as L. Durand in 1962
(see (Roberts and Marciano 2010, Chapter 3, Section 3.2.2.2)).. In any case,
the QED contribution turns out to be the dominant one for $a_{e}$ while as of
today, good estimates have been achieved for weak interaction contributions to
$a_{\mu}$ but not for the hadronic ones. While today it is well-known that
there exist three lepton-quark families with identical basic properties except
for differences in their masses, decay times and patterns, at that time it was
very hard to believe that the muon is just a heavier version of the electron,
so giving rise to the so-called $\mu-e$ puzzle, paraphrasing the previous
well-known $\theta-\tau$ puzzle which was brilliantly solved by the celebrated
work of T.D. Lee and C.N. Yang on the parity violation for weak interactions.
For instance, it was expected that the muon exhibited some unknown kind of
interaction, not shared by electron and that would have due to explain the
much higher mass. All this motivated and stimulated the experimental research
to explore $a_{\mu}$. As it has already been said above, the big interest in
the muon anomalous magnetic moment was motivated by the above mentioned
Berestetskii, Krokhin and Klebnikov argument in relation to the main fact
according to which the anomalous magnetic moment of leptons mediates spin-flip
transitions whose amplitudes are proportional to the masses of particles, so
that they are particularly appreciable for heavier ones via a generalization
of (55) given by
(57) $\frac{\delta a_{l}}{a_{l}}\varpropto\frac{m_{l}^{2}}{M_{l}^{2}}\ \ \ \ \
(M_{l}\gg m_{l})$
where $M_{l}$ is a parameter which may be either an energy scale or an
ultraviolet cut-off where QED ceases to be valid (QED-breaking) or as well the
mass of a hypothetical heavy state or of a new heavier particle. The relation
(57) also allows us to ascertain whether an elementary particle has an
internal structure: indeed, if the lepton $l$ is made by hypothetical
components of mass $M_{l}$, then the anomaly $a_{l}$ would be modified by a
quantity $\delta a_{l}$ given by the relation $\delta
a_{l}=O(m_{l}^{2}/M_{l}^{2})$ so that the measurements of $a_{l}$ might
provide a lower limit for $M_{l}$ which, at the current state of research, has
a magnitude of about 1 TeV, which imply strong limitations to the possible
hypotheses on the internal structure of a lepton (see (Picasso 1985)). On the
other hand, the relation (57) also implies that the heavier the new state or
scale, the harder it is to see. Therefore, from (57), it follows that the
sensitivity to high-energy physics grows quadratically with the mass of the
lepton, which means that the interesting effects are magnified in $a_{\mu}$
compared to $a_{e}$ by a factor of about $(m_{\mu}/m_{e})^{2}\sim 4\cdot
10^{4}$, and this is just what has made and still makes $a_{\mu}$ the elected
monitoring fundamental parameter for the new physics also because of the fact
that the measurements of $a_{\tau}$ go out of the present experimental
possibilities due to the very short lifetime of $\tau$.
As also reported in (Garwin et al. 1957), if $g=2$ then the direction of muon
polarization would remain fixed relatively to the direction of motion
throughout the trajectory, while if $g\neq 2$ then a phase angle $\delta$
opens up between these two directions. Following (Muirhead 1965, Chapter 2,
Section 2.5(a,e)), (Farley and Picasso 1979) and (Picasso 1996), to estimate
$\delta$, let us assume that we have longitudinally polarized charged leptons
slowly moving in a magnetic field and we know their direction of polarization.
If they are allowed to pass into a system with a magnetic field of strength
$B$, they experience a torque given by $\vec{\tau}=\vec{\mu}_{s}\wedge\vec{B}$
which, in turn, implies the execution of helical orbits about the direction of
$\vec{B}$ which lead to a Larmor precession about the direction of $\vec{B}$
with the following angular velocity (in natural units) calculated in the
particle rest frame
(58) $\omega_{s}=g\frac{Q}{2mc}B=\Gamma B$
where $\Gamma=g(Q/2mc)$ is the gyromagnetic ratio. If the charged particle is
also in motion, then it will execute spiral orbits about $\vec{B}$ which
possess the characteristic cyclotron frequency $\nu_{c}$ given by
$\omega_{c}=2\pi\nu_{c}=(Q/mc)B$. In one defines the laboratory rotation
frequency of the spin relative to the momentum vector as
$\omega_{a_{i}}\doteq\omega_{s}-\omega_{c}$, then the phase angle $\delta$,
after a time $t$, is given by
(59)
$\delta=\omega_{a_{i}}t=(\omega_{s}-\omega_{c})t=\frac{g-2}{2}\frac{Q}{mc}Bt=a_{i}\frac{Q}{mc}Bt$
where $g=2(1+a_{i})\ i=e,\mu,\tau$. Hence, if $g=2$, then
$\omega_{s}=\omega_{c}$ and the charged leptons will always remain
longitudinally polarized. But if $g>2$ as predicted, then the spin starts to
precess and turns faster than the momentum vector. Therefore, it is
immediately realized that a measurement of the phase angle $\delta$ after a
time $t$, may estimate the magnitude of the deviation of the $g$-value from 2.
Equation (59) will be the basic formal tool for the so-called $(g-2)$
experiments and that will be carried out later: if the charged lepton is kept
turning in a known magnetic field $\vec{B}$ and the angle between the spin and
the direction of motion is measured as a function of time $t$, then $a_{i}$
may be estimated. The value of $Q/mc$ is obtained from the precession
frequency of the charged leptons at rest, via equation (58). Furthermore, the
fundamental equation (59) has been derived only in the limit of low velocities
but it has been proved to be exactly true as well at any speed as, for
example, made in (Bargmann et al. 1959) using a covariant classical
formulation of spin-motion. It has also been proved that the $(g-2)$
precession is not slowed down by time dilation even for high-velocity muons.
Following (Farley and Picasso 1979) and (Brown and Hoddeson 1983, Part III,
Chapter 8), after the celebrated experience made by Garwin, Lederman and
Weinrich in 1957, the possibility of a $(g-2)$ experiment for muon was finally
envisaged. In 1959, as recalled by (Jegerlehner 2008, Part I, Chapter 1), the
Columbia research group made by L.M. Lederman, R.L. Garwin, D.P. Hutchinson,
S. Penman and G. Shapiro, performed a measurement of $a_{\mu}$ with a
precision of about 5%, even using a precession technique applied to a
polarized muon beam whose directions are determined by means of their
asymmetric decay modes. In the same years, many other research groups at
Berkeley, Chicago, Liverpool and Dubna started as well to study the problem.
If the muon had a structure giving a form factor less than one for photon
interactions, then the value of $a_{\mu}$ should be less than predicted.
Nevertheless, compared with the measurement on the electron, the muon $(g-2)$
experiment was much more difficult because of the low intensity, diffusive
nature and high momentum of available muon sources. All this, together the
possibility to get a reasonable number of precession cycles, entailed, amongst
other things, the need to have large volumes of magnetic field. One solution,
adopted by A.A. Schupp, R.W. Pidd and H.R. Crane in 1961, was to scale up the
original Michigan $(g-2)$ method for electrons whose spin directions was
established with the aid of a double scattering experiment in which the first
and second scatterings were performed respectively before and after the
passage of the electrons through a solenoid. However, out of the many attempts
to approach such a problem (see also (Garwin 2003)), the first valuable
results were achieved by the first CERN $(g-2)$ team composed in alphabetic
order by G. Charpak, F.J.M. Farley, T. Muller, J.C. Sens and A. Zichichi
(credit by CERN-BUL-PHO-2009-017), formalized the 1st of January 1959 but
already operative since 1958\. As recall (Combley and Picasso 1974), (Farley
and Picasso 1979), (Combley et al. 1981) and (Jegerlehner 2008, Part I,
Chapter 1), the breakthrough experiment which made the direct attack on the
magnetic moment anomaly of muons was performed at CERN synchrocyclotron (SC)
by the first $(g-2)$ team mentioned above. As a result of this measurement,
the experimental accuracy in the value of the muon anomalous magnetic moment
was reduced to 0.4% from the level of 15% at which it had previously stood.
Following (Brown and Hoddeson 1983, Part III, Chapter 8), the CERN experiments
performed from 1961 to 1965, have been based on the main idea according to
which, roughly speaking, the muons produced by a beam of pions decaying in
flight are longitudinally polarized; furthermore, in the subsequent decays,
the electrons reveal the direction of the muon spins because they are
preferentially emitted along the spin direction at the momentum of decay.
Hence, a $(g-2)$ experiment may be performed trapping the longitudinally
polarized muons in a uniform magnetic field and then measuring the precession
frequency of the spins. It has only to be added that, due to the very short
muon lifetime, it was necessary to use high-energy muons in order to lengthen
their decay times using the relativistic time dilation effect. The results
reduced the error in the measure of $(g-2)$ from the previous 15% to 0.4%.
Following (Jegerlehner 2008, Part I, Chapter 1), surprisingly nothing of
special was observed even within 0.4% level of accuracy of the experiment; it
was the first real evidence that the muon was just a heavy electron, so
reaching to another celebrated experimental evidence of the validity of QED.
In particular, this meant that the muon was point-like and no extra short
distance effects could be seen. This latter point was however a matter of
accuracy and therefore the challenge to go further was quite evident; in this
regards, see the reviews (Farley and Semertzidis 2004) and (Garwin 2003). As
recalled in (Cabibbo 1994, Part I), G. Bernardini, then research director
responsible for the SC at CERN, remembers as, around the end of 1950s, there
were many ideas for the high precision measurements of the anomalous magnetic
moment of the muon, two of them having been that of the screw magnet and that
of the flat magnet. Gilberto Bernardini consulted the greatest magnet
specialist, Dr. Bent Hedin, who said that would have been necessary some years
to fully carried out one of this project, the flat magnet one, so that it was
initially chosen the screw magnet project. In the meanwhile, A. Zichichi had
the ingenious idea to trying a new very simple technique consisting in shaping
a flat pole with very thin iron sheets, glued together by means of the
simplest possible method, the scotch tape. In this way, instead of six years,
a few months of hard work allowed Zichichi to built up particular high
accuracy magnetic fields, based on the theoretical notion of Garwin-Panofsky-
Zichichi polynomial magnetic fields, which constitute just those experimental
tools that needed for attaining high measurements of $a_{\mu}$. The so-called
six-meters long flat magnet providing an injection field, followed by two
transitions, hence a storage, then another transition and finally an ejection
field, became the core of the first high precision measurement of the muon
$(g-2)$. Likewise, R.L. Garwin, in (Cabibbo 1994, Part I), remembers that, in
achieving this, it was determinant the special responsibility of Zichichi
profused by him in producing the bizarre magnetic field in their storage
magnetic system, accomplished with imagination, energy and efficiency. Again,
in (Garwin 1986, 1991, 2001) and (Garwin 2003), the author recalls that the
80-ton magnet six-meters long was shimmed in a wondrous fashion under the
responsibility of Nino Zichichi who did a wonderful job in doing this, while
the polarization was measured as the muons emerged from the static magnetic
field thanks a system perfected by G. Charpak; F.J.M. Farley was instead in
charge to develop the computer program which would take the individual counts
from the polarization analyzer done by Charpak, while T. Muller played the
electronic work with the help of C. York. Following (Jones 2005), the six-
meters magnet came to CERN as the heart of the first $g-2$ experiment, the aim
of which was to measure accurately the anomalous magnetic moment, or
$g$-factor, of the muon. This experiment was one of CERN outstanding
contributions to fundamental physics and for many years was unique to the
laboratory.
To this point, it is need to retake the equations of motion of a charged
particle in a magnetic field $\vec{B}$ from a relativistic viewpoint.
Following (Combley et al. 1981), (Picasso 1996) and (Jegerlehner 2008, Part
II, Chapter 6), the cyclotron (or orbital) frequency is given by
(60) $\vec{\omega}_{c}=\frac{Q}{\gamma mc}\vec{B}$
where $\gamma=1/\sqrt{1-\beta^{2}}$ and $\vec{\beta}=\vec{v}/c$. When a
relativistic particle is subject to a circular motion, then it is also need to
take into account the so-called Thomas precession, which may be computed as
follows. The particle rest frame of muon rotates around the laboratory frame
with angular velocity $\vec{\omega}_{T}$ given by
(61) $\vec{\omega}_{T}=\Big{(}1-\frac{1}{\gamma}\Big{)}\frac{Q\vec{B}}{mc}$
and it is different from the direction of the angular velocity with which the
muon’s spin rotates in the rest frame, so that the angular velocity of spin
rotation in the laboratory frame is given by
(62)
$\vec{\omega}_{s}\doteq\vec{\omega}_{L}-\vec{\omega}_{T}=\Big{(}a_{\mu}+\frac{1}{\gamma}\Big{)}\frac{Q\vec{B}}{mc}$
which shows that the angular frequency of anomalous magnetic moment is, in
relativistic regime, equal to the angular frequency at very low energies, that
is to say
(63)
$\vec{\omega}_{a_{\mu}}=\vec{\omega}_{s}-\vec{\omega}_{c}=a_{\mu}\frac{Q\vec{B}}{mc}.$
To argue upon the electric dipole moment of the muon, we should consider the
relativistic equations of the muon in the laboratory system in presence of an
electric field $\vec{E}$ and of a magnetic field $\vec{B}$. In this case,
under the conditions of purely transversal fields
$\vec{\beta}\cdot\vec{E}=\vec{\beta}\cdot\vec{B}=0$, following (Bargmann et
al. 1959), the cyclotron angular velocity is given by
(64)
$\vec{\omega}_{c}=\frac{Q}{mc}\Big{(}\frac{\vec{B}}{\gamma}-\frac{\gamma}{\gamma^{2}-1}\vec{\beta}\wedge\vec{E}\Big{)}$
while the spin angular velocity is given by
(65)
$\vec{\omega}_{s}=\frac{Q}{mc}\Big{(}\frac{\vec{B}}{\gamma}-\frac{1}{1+\gamma}\vec{\beta}\wedge\vec{E}+(\vec{B}-\vec{\beta}\wedge\vec{E})\Big{)}$
so that the angular frequency of the muon anomalous magnetic moment, related
to the spin precession, is given by
(66)
$\vec{\omega}_{a_{\mu}}=\vec{\omega}_{s}-\vec{\omega}_{c}=\frac{Q}{mc}\Bigg{(}a_{\mu}\vec{B}+\Big{(}\frac{1}{\gamma^{2}-1}-a_{\mu}\Big{)}\vec{\beta}\wedge\vec{E}\Bigg{)}$
which is the key formula for measuring $a_{\mu}$;
$\omega_{a}=|\vec{\omega}_{a}|=\omega_{s}-\omega_{c}$ is the anomalous
frequency difference or spin-flip transition. If a large enough electric
dipole moment given by $(6)_{2}$ there exists, then either the applied field
$\vec{E}$ (which is zero at the equilibrium beam position) and the motional
electric field induced in the muon rest frame, say
$\vec{E}^{*}=\gamma\vec{\beta}\wedge\vec{B}$, will add an extra precession of
the spin with a component along $\vec{E}$ and one around an axis perpendicular
to $\vec{B}$, that is to say
(67)
$\vec{\omega}=\vec{\omega}_{a_{\mu}}+\vec{\omega}_{EDM}=\vec{\omega}_{a_{\mu}}+\frac{\eta
Q}{2mc}\Big{(}{\vec{E}}+\vec{\beta}\wedge\vec{B}\Big{)}$
or else
(68) $\Delta\omega_{a_{\mu}}\cong d_{e}(\vec{E}+\vec{\beta}\wedge\vec{B})$
which, for $\beta\sim 1$ and $d_{e}\vec{E}\sim 0$, yields
(69) $\omega_{a_{\mu}}\cong
B\sqrt{\Big{(}\frac{Q}{mc}a_{\mu}\Big{)}^{2}+(d_{e})^{2}}.$
The result is that the plane of precession is no longer horizontal but tilted
at an angle
(70)
$\theta\equiv\arctan\frac{\omega_{EDM}}{\omega_{a_{\mu}}}=\arctan\frac{\eta\beta}{2a_{\mu}}\cong\frac{\eta}{2a_{\mu}}$
and the precession frequency is increased by a factor
(71) $\omega^{\prime}_{a_{\mu}}=\omega_{a_{\mu}}\sqrt{1+\delta^{2}}.$
The angle $\theta$ produces a phase difference in the $(g-2)$ oscillation. It
is therefore important to determine whether there is a vertical component to
the precession in order to separate out the effect of an electric dipole
moment from the determination of $\omega_{a_{\mu}}$. The angle of tilt
$\theta$ given, in the small angle approximation, by (70), may be detected by
looking for the time variation of the vertical component of the muon
polarization with the same frequency as the $(g-2)$ precession of the
horizontal polarization. Therefore, in order to eliminate the electric dipole
moment as a source of any discrepancy which might appear in $(g-2)$ direct
measurements of higher precision is preliminarily required. In any case, the
main determination in the electric dipole moment of the muon is not merely
this last clarification of the $(g-2)$ measurements. Indeed, it is also of
fundamental importance in itself since the existence of such a static property
for any particle would imply the lack of invariance for the electromagnetic
interaction under both $P$ and $T$, as recalled above. Some of the theories
unifying the weak and electromagnetic interactions predict a small electric
dipole moment for some particles including the muon and a precise measurement
of this property would tighten the constrains within which such theories might
operate, so that precise measurements of the electric dipole moment of the
muon as of other particles were and still are highly desirable.
## 4\. Towards the first exact measurements of the anomalous magnetic moment
of the muon
In Section 2, we have outlined the first works of A. Zichichi and co-workers
on cosmic rays carried out until the end of 1950s. From this period onwards,
A. Zichichi was involved, as briefly said above, in some crucial experiments
concerning the muon $(g-2)$ measurements and carried out at CERN of Geneva.
The first work on muon anomalous magnetic moment in which he was involved is
9. where a precise measurement of the electric dipole moment of the muon was
obtained within the QED context only. The work starts from the above mentioned
Michigan spin precession method used to measure $a_{e}$ which exploits the
possibility to have beams of polarized leptons underwent to asymmetric decay.
With this method, i.e. the spin precession methods (see previous Section), one
can measure $(g-2)$ by storing the particles for some time in a magnetic field
and then measuring the relative precession angle between the spin and the
angular momentum which serves as a reference vector. As in the electron
experiments, the primary requirement was in being able in injecting the muons
into a magnetic field so that they could circulate on essentially periodic
orbits, hence to trap them in this field for a large number of orbit periods
as possible. Nevertheless, at that time, the available muon beams exhibited,
in comparison with the electron case, very low fluxes, high momenta and large
extensions in position and momentum space (hence, low density in phase space)
which implied many other new difficulties besides the above mentioned primary
requirement. On the other hand, the muons did not require the analysis of the
spin polarization by scattering since the asymmetric electron decay reveals
the spin deviation; indeed, as said above, the electrons were emitted along
the spin direction at the moment of decay. Starting from the principle of the
method of the experimental apparatus used in (Garwin et al. 1957), the essence
of this idea had already been established in (Berley et al. 1958) where the
existence of longitudinally polarized beams of $\mu$ mesons and the
availability of muon decay electron asymmetry as a polarization analyzer
suggested this method by means of which one may search for a muon electric
dipole moment. A discussion of the results achieved in (Berley et al. 1958)
was then made in (Garwin and Lederman 1959) from which turns out that several
practical methods for overcoming these difficulties were either experimentally
and theoretically undertaken before this work of Charpak, Lederman, Sens and
Zichichi, but without succeed in the enterprize. Instead, this research group
was able, for the first time, to trap 85 MeV/c momentum muons for 28 turns,
i.e. orbit periods, with no pulse magnets. Their results clearly suggested too
that minor modifications in their method were enough to enable one in
achieving storage for several hundreds of turns. Well, all this was made
possible, as also recalled in the previous section, just thanks to the
ingenious technical and experimental ability of A. Zichichi in building up
suitable polynomial magnetic fields of high precision and thanks to which it
was possible to obtain thousand muon turns (see also (Farley 2005)); in turn,
all this was carried out on the basis of the theoretical framework mainly
worked out on previous remarkable studies made by R. Garwin and W.K.H.
Panofsky, upon which we shall in-depth return later. The extreme importance
and innovativeness of this experimental technique was successfully carried out
later, at a technical level, in producing the so-called six-meters long flat
magnet which, in turn, was mainly built up by A. Zichichi starting from a
suitable modification of a previous magnet provided by the University of
Liverpool (see (Zichichi 2010) and (CERN 1960)). Seen the fundamental
importance of this event, it is necessary to outline the early works and ideas
which came before the dawning of this experimental apparatus, and mainly
worked out, for the first time, in the paper 9. on whose content we now will
briefly argue.
The principle of the method consists in injecting, say along the $Y$ axis, a
muon beam into a median $(X,Y)$ plane of a flat magnet gap. A moderator (or
absorber) $M$, centered on the origin of the $(X,Y)$ plane, will contain such
a beam through a suitable reduction of the momentum beam $p$ and of the mean
vertical (i.e. along $Z$ direction) field value $B_{z0}$. So, the muons lost
energy and consequently follow small and more sharply orbits which will be
contained within the magnetic field region, and to prevent a reabsorption by
moderator after one turn, a small transverse linear gradient of the magnetic
field is inserted, causing an orbit drift along the $X$ axis in the direction
opposed to $sign\ a$. The magnetic field configuration is therefore planned to
produce such a drift of the muon orbits along the $X$ axis away from the
moderator $M$, focusing the muon beam in the median $(X,Y)$ plane. The
magnetic field therein used has the following polynomial form
(72) $B_{z}=B_{z0}(1+aY+bY^{2})$
along the median plane, where $a,b\in\mathbb{R}$ have to be small (Garwin-
Panofsky). If $r$ is the distance from the origin and $ar\ll 1$ and $br^{2}\ll
1$, then the muons emerging from $M$ will move on nearly circular orbits of
radius $r$. A linear gradient alone leads to a step-size drift of these orbits
along the $X$-direction by an amount equal to
(73) $s=\pi r^{2}\langle grad_{Y}\frac{B_{z}}{B_{z0}}\rangle=\pi r^{2}a\ \
\mbox{\rm per\ turn}$
where $\langle\ ,\ \rangle$ denotes average over one orbit loop. This drift
will enable some muons to get over $M$ after their first turn, whereupon they
go on along a trochoidal orbit. Moreover, following previous basic and notable
studies made by R.L. Garwin and W.K.H. Panofsky383838See R.L. Garwin,
Numerical calculations of the stability bands and solutions of a Hill
differential equation, CERN Internal Report (October 1959) and W.K.H.
Panofsky, Orbits in the linear magnet, CERN Internal Report (October 1959).,
the linear gradient also produces a weak vertical focusing with wavelength
given by
(74) $\frac{\lambda_{\nu}}{2\pi}\cong\frac{0.76}{a}.$
Taking into account equation (73), because we want to be $r/s\gg 1$ in order
to store as large as possible a number of turns in a magnet of given finite
size, it follows that this focusing is very weak either because of sensitive
variations of the field index $n$ and since $(r/s\gg
1)\Rightarrow(\lambda_{\nu}/2\pi r\gg 1)$ which implies low frequencies and
consequently a weak focusing, hence a poor storage. Nevertheless, as was
pointed out by R.L. Garwin (see his 1959 CERN Internal Report), one can
improve the vertical focusing while maintaining a given large value of $r/s$
by the addition of a quadratic term of the type $by^{2}$ and indeed, for a
polynomial magnetic field of the type (72) with $a$ and $b$ small, one has
(75)
$\frac{\lambda_{\nu}}{2\pi}\cong\frac{1}{\sqrt{b+1.74a^{2}}}\sim\frac{1}{\sqrt{b}}$
while the drift step-size is still given by (73), so that we can handle $a$
and $b$ in such a manner to have high values of the former and low values of
the latter. For example, by taking $b=50a^{2}$, one can, while maintaining the
same $r/s$ of above (for such orbits), improve the focusing to 1 oscillation
per 7 turns. Therefore, the intensity of stored muons is increased by a factor
$38/7\sim 5$ by the addition of the quadratic term to the magnetic field.
Thus, to sum up, the term $ay$ produces the $X$ axis drift of an orbit of
radius $r$ in step-sizes of magnitude $a\pi r^{2}$ per turn393939According to
a principle of the method almost similar to the one proposed by P.S. Farago in
(Farago 1958) for the free electron case.. The next $by^{2}$ term adds
vertical focusing in such a manner that the wavelength of the vertical
oscillations are about $2\pi/\sqrt{b}$; it has as well the useful function to
fix more firmly the magnetic median plane around the center of the magnet gap
because just the median plane begins to touch the poles, then all the
particles will go lost. In any case, it is not allowed to choose $b$
arbitrarily large for vertical defocusings minimizing $\lambda_{\nu}$ because
this would lead to a spread in the drift step-size and hence in storage times.
Indeed, orbits emerging at an angle $\phi$ with respect to the $Y$ axis would
have a step-size given by
(76) $s(\phi)=\pi r^{2}(a-2br\phi)$
so that the magnitude of $b$ may be chosen in order to maximize the number of
particle stored for a given number of turns.
Once having established these fundamental theoretical points, mainly due, as
recalled above, to previous works of R.L. Garwin and W.K.H. Panofsky, the next
step was to practically realize such polynomial magnetic fields, far from
being an easy task. This primary work was masterfully and cleverly
accomplished by A. Zichichi starting from a previous magnet provided by the
University of Liverpool for whose technical details we refer to the Section 2
- Injection and Trapping, of the original work 9. He was very able to set up a
complex but efficient experimental framework that provided suitable polynomial
magnetic fields for the magnetic storage of muon beams. The experimental
results are of historical importance and were represented in the Figures 2.
and 3.a)-b) of 9. whose characteristics were adequately theoretically
explained in the above mentioned Section 2 of 9. These results were the first
valuable experimental evidence of the fact that particles turning several
times inside a small magnetic arrangement was pursuable, so endorsing that
presentiment according to which longer magnetic systems of this type could
give further and more precise measurements. All this was in fact done in the
subsequent experiments made by A. Zichichi and co-workers and that will be
described later. The final section of the work 9. deals then with attempts to
measure the electric dipole moment of the muon starting from the experimental
results achieved by the previous works (Berley et al. 1959) and (Garwin and
Lederman 1959) and whose principle of the method was mainly based on the
determination of the phase angle given by (59) through the so-called up-down
asymmetry parameter404040It is given by
$\alpha=(N_{up}-N_{down})/(N_{up}+N_{down})$ respect to the median plane.
$\alpha$, taking into account the original theoretical treatment given by
(Bargmann et al. 1959) and briefly recalled in the previous Section 3. To this
end, Charpak, Lederman, Sens and Zichichi used their innovative experimental
arrangement to storage polarized muon beams, just to determine this EDM of the
muon. The related value so found was consistent with time reversal invariance
and could be considered equal to zero within the experimental errors which
have been considerably reduced respect to those of the above mentioned
previous works on muon EDM determination. To be precise, their formal
treatment is that of (Bargmann et al. 1959) in which are considered the
covariant classical equations of motion of a particle of arbitrary spin moving
in a homogeneous electromagnetic field. As it has already been said, the
theoretical considerations made in (Bargmann et al. 1959) include too the
relativistic case because of a remark due to F. Bloch. We consider
longitudinally polarized muons possessing an EDM given by $(6)_{2}$, which
move in a magnetic field $\vec{B}$ in a plane perpendicular to the latter. In
their instantaneous rest frame, they experience an electric field given by
$\vec{E}^{*}=\gamma\vec{\beta}\wedge\vec{B}$ which causes a precession of the
EDM. In the laboratory frame, the spin precesses around $\vec{v}\wedge\vec{B}$
(hence, out of the orbit plane in which relies $\vec{v}\wedge\vec{B}$) by an
angle $\Theta_{s}=\omega_{s}t$ when the orbit has gone through an angle
$\Theta_{o}=\omega_{c}t$ (or $\Theta_{c}$) on its orbital plane (see Equation
(59)). The polarization (perpendicular to the orbit) thus produced, is
detected by stopping the muons after a known $\Theta_{o}$ and measuring the
up-down asymmetry of the electrons emerging from the muon decay with respect
to the orbit plane (placed in the median plane of the storage magnet set up in
9. and detected by the scintillator No. 4 of their apparatus). This
determination, successfully achieved by Charpak, Lederman, Sens and Zichichi,
was different from the previous ones only in the magnitude of $\Theta_{o}$, in
which it was assumed to be $\Theta_{o}\in]0,2\pi[$, whereas they used the new
storage device based on polynomial magnetic fields to get $\Theta_{o}=2n\pi$
with $n\geq 28$, just thanks to the multiple turns that their arrangement was
able to provide. The principle of the method consisted in analyzing two range
of flight times of particles, a group A of early particles having made few
turns in the storage magnet and which are used for calibration, and a group B
of late particles which have made many revolutions. In turn, the measurements
were divided into three groups in dependence on the mean turn index $\langle
n\rangle$ of late particles, this being fixed for the early ones and equal to
$\langle n\rangle\approx 1$. The Group $I$ concerns late muons with $\langle
n\rangle\approx 11.5$; the Group $II$ concerns late muons with $\langle
n\rangle\approx 16.5$, while Group $III$ concerns muons with $\langle
n\rangle\approx 19.5$. For each of these groups, the difference in up-down
asymmetry, say $\Delta^{(i)}=a_{early}^{(i)}-a_{late}^{(i)}\ \ i=I,II,III$,
between the early and late ones, is evaluated. The values so found are
reported in the Table I of 9. and from these it is then possible to estimate
the angle $\Theta_{s}^{(i)}$, through which the spin has rotated out of the
median plane, as $\Delta^{(i)}/a_{max}^{(i)}$ where $a_{max}^{(i)}$ is the
maximal obtainable value of asymmetry in the given $i$th group. Then
$\Theta_{o}^{(i)}\approx\omega_{c}\langle t^{(i)}\rangle$ where $t^{(i)}$ is
the beam flight time detected by the final median plane scintillator.
Furthermore, to improve distribution calculations and to reduce systematic
errors, the EDM telescope was also symmetrically displaced at different
heights with respect to the magnet median plane. Finally, combining the three
values of $\Theta_{s}^{(i)}/\Theta_{o}^{(i)}\ \ i=I,II,III$ (listed in the
above mentioned Table I), it was possible to estimate $\eta$ of $(6)_{2}$,
whence to deduce the upper limit for the EDM of the muon.
Following (Lee 2004, Chapter 2), the accelerator physics principles involved
in the work 9. mainly concern with transverse particle motion in the sense as
first outlined in the 1941 seminal paper (Kerst and Serber 1941) for the
betatron case. In Frenet-Serret coordinates $(x,s,z)$ ($s$ is oriented as the
tangent, $x$ as the normal and $z$ as the binormal respect to the orbit plane)
and in zero electric potential, we have a two-dimensional magnetic field given
by $\vec{B}=B_{x}(x,z)\hat{x}+B_{z}(x,z)\hat{z}$ where
$\hat{z}=\hat{x}\wedge\hat{z}$. In straight geometries, we have a magnetic
flux density given by
(77) $B_{z}+iB_{x}=B_{0}\sum_{n\in\mathbb{N}_{0}}(b_{n}+ia_{n})(x+iz)^{n}$
where $a_{n},b_{n}$ are called $2(n+1)$th multipole coefficients and are given
by (Lee 2004, Chapter 2, Section I.3, Equations (2.26)). The expression (77)
is said to be the Beth representation (see (Beth 1966, 1967)). For example, in
discussing the focusing of atomic beams, the sextupole terms are show to be
able to make high spin focusings (see (Lee 2004, Chapter 2, Exercise 2.2.18)).
In such a case, some historical predecessors of these techniques to obtain
polarized ions may be found in (Haeberli 1967) where, among other things, are
discussed too some previous experiences with separate magnets operating at the
quadrupolar or sextupole order, due to H. Friedburg, W. Paul and H.G.
Bennewitz in the early 1950s. In certain sense, looking at the (77), the
Garwin-Panofsky-Zichichi polynomial magnetic fields might be considered as
special cases forerunner of such Beth representations. One of the main aims of
this historical paper has just been that pointing out the following remarkable
fact: the first exact measurements of muon AMM will be possible thanks to the
use of these Garwin-Panofsky-Zichichi polynomial magnetic fields which were
masterfully used, for the first time, in 9. to measure the muon EDM; then, the
principle of the method there worked out will be gradually improved both
theoretically and experimentally through further pioneering works until the
seminal paper 10. in which the first exact measurement of the muon AMM was
finally achieved with success. This marked a milestone of fundamental physics
of the second half of 20th-century, achieved at CERN of Geneva, upon which we
shall return later in a deeper manner. Nevertheless, we must point out as
nobody, including the authors themselves of these pioneering researches, have
recognized the right primary role played by polynomial magnetic fields in
achieving these, whose history is utterly neglected. In this regards, the
unpublished theoretical work made by Richard L. Garwin (together to the one
made by W.K.H. Panofsky) has been of fundamental importance in setting up the
theoretical bases for these polynomial magnetic fields; later, the genial
technical ability of Antonino Zichichi will be determinant in providing an
experimental version of these fields which were very basilar to get the first
exact measurement of the muon AMM. In another place, however, we will deal
with this last historical question, also thanks to precious unpublished
bibliographical material which has been kindly provided to me by Professor
Richard L. Garwin to whom I bear my thankful acknowledgements, and that will
be historically in-depth analyzed in another forthcoming paper.
## List of some publications of A. Zichichi
1\. G. Alexander, J.P. Astbury, G. Ballario, R. Bizzarri, B. Brunelli, A. De
Marco, A. Michelini, G.C. Moneti, E. Zavattini and A. Zichichi, A Cloud
Chamber Observation of a Singly Charged Unstable Fragment, Il Nuovo Cimento,
Serie X, Vol. 2 (1955) pp. 365-369 [Received on 18 July 1955 and Published in
August 1955 - Registered Preprint No.].
2\. W.A. Cooper, H. Filthuth, J.A. Newth, G. Petrucci, R.A. Salmeron and A.
Zichichi, A Probable Example of the Production and Decay of a Neutral Tau-
Meson, Il Nuovo Cimento, Serie X, Vol. 4 (1956) pp. 1433-1444 [Received on 09
September 1956 and Published in December 1956 - Registered Preprint No.].
3\. W.A. Cooper, H. Filthuth, J.A. Newth, G. Petrucci, R.A. Salmeron and A.
Zichichi, Example of the Production of $(K^{0},\bar{K}^{0})$ and
$(K^{+},\bar{K}^{0})$ Pairs of Heavy Mesons, Il Nuovo Cimento, Serie X, Vol. 5
(1957) pp. 1388-1397 [Received on 14 January 1957 and Published in June 1957 -
Registered Preprint No.].
4\. C. Ballario, R. Bizzarri, B. Brunelli, A. De Marco, E. Di Capua, A.
Michelini, G.C. Moneti, E. Zavattini and A. Zichichi, Life Time Estimate of
$\Lambda^{0}$ and $\theta^{0}$ Particles, Il Nuovo Cimento, Serie X, Vol. 6
(1957) pp. 994-996 [Received on 01 August 1957 and Published in October 1957 -
Registered Preprint No.].
5\. H. Filthuth, J.A. Newth, G. Petrucci, R.A. Salmeron and A. Zichichi,
Cosmic Ray Research: Proposal for a New High Energy Experiment, CERN
Scientific Policy Committee, Seventh Meeting, Document No.
CERN/SPC/52(A)-3784/e, Geneva, 21-29 October 1957.
6\. W.A. Cooper, H. Filthuth, L. Montanet, J.A. Newth, G. Petrucci, R.A.
Salmeron and A. Zichichi, Neutral $V$-Particle from Copper and Carbon, Il
Nuovo Cimento, Serie X, Vol. 8 (1958) pp. 471-481 [Received on 11 February
1958 and Published on May 1958 - Registered Preprint No.].
7\. G. Alexander, C. Ballario, R. Bizzarri, B. Brunelli, E. Di Capua, A.
Michelini, G.C. Moneti and A. Zichichi, $\Lambda^{0}$\- and $\theta^{0}$\-
Particles Produced in Iron, Il Nuovo Cimento, Serie X, Vol. 9 (1958) pp.
624-646 [Received on 23 April 1958 and Published in August 1958 - Registered
Preprint No.].
8\. L. Montanet, J.A. Newth, G. Petrucci, R.A. Salmeron and A. Zichichi, A
Cloud Chamber Study of Nuclear Interactions with Energies of about 100 GeV, Il
Nuovo Cimento, Serie X, Vol. 17 (1960) pp. 166-188 [Received on 18 March 1960
and Published on 18 July 1960 - Registered Preprint No.].
9\. G. Charpak, L.M. Lederman, J.C. Sens and A. Zichichi, A Method for
Trapping Muons in Magnetic Fields, and Its Application to a Redetermination of
the EDM of the Muon, Il Nuovo Cimento, Serie X, Vol. 17 (1960) pp. 288-303
[Received on 04 April 1960 and Published on 01 August 1960 - Registered
Preprint No. CERN-SC/8431/nc].
10\. G. Charpak, F.J.M.Farley, R.L. Garwin, T. Muller, J.C. Sens and A.
Zichichi, The Anomalous Magnetic Moment of the Muon, Il Nuovo Cimento, Serie
X, Vol. 37 (1965) pp. 1241-1363.
## List of some publications of R.L. Garwin
1’. R.L. Garwin, L.M. Lederman and M. Weinrich, Observations of the Failure of
Conservation of Parity and Charge Conjugation in Meson Decays: the Magnetic
Moment of the Free Muon, Physical Review, 105, No. 4, pp. 1415-1417, February
15, 1957.
2’. Columbia University Physics Department announcement of parity experiments
by C.S. Wu, E. Ambler, R.L. Garwin, L.M. Lederman, et al., January 15, 1957.
3’. D. Berley, T. Coffin, R.L. Garwin, L. Lederman, and M. Weinrich,
Depolarization of Positive Muons in Matter, Bulletin of the American Physical
Society, Series II, 2, No. 4, p. 204, April 25, 1957.
4’. Berley, T. Coffin, R.L. Garwin, L. Lederman, and M. Weinrich, Energy
Dependence of the Asymmetry in Polarized Muon Decay, Bulletin of the American
Physical Society, Series II, 2, No. 4, p. 204, April 25, 1957.
5’. T. Coffin, R.L. Garwin, L.M. Lederman, S. Penman, and A.M. Sachs, Magnetic
Resonance Determination of the Magnetic Moment of the Mu Meson, Physical
Review, 106, pp. 1108-1110, May 1957.
6’. D. Berley, T. Coffin, R.L. Garwin, L.M. Lederman and M. Weinrich, Energy
Dependence of the Asymmetry in the Beta Decay of Polarized Muons, Physical
Review, 106, pp. 835-837, May 1957.
7’. R.L. Garwin, S. Penman, L.M. Lederman, and A.M. Sachs, Magnetic Moment of
the Free Muon, Physical Review, 109, No. 3, pp. 973-979, February 1, 1958.
8’. T. Coffin, R.L. Garwin, S. Penman, L.M. Lederman, and A.M. Sachs, Magnetic
Moment of the Free Muon, Bulletin of the American Physical Society, Series II,
3, No. 1, p. 34, January 29, 1958.
9’. D. Berley, R.L. Garwin, G. Gidal and L.M. Lederman, Electric Dipole Moment
of the Muon, Physical Review Letters, 1, No. 4, pp. 144-146, August 15, 1958.
10’. R.L. Garwin and L.M. Lederman, The Electric Dipole Moment of Elementary
Particles, ll Nuovo Cimento, Serie X, 11, pp. 776-780, 1959.
11’. D. Berley, R.L. Garwin, G. Gidal, and L.M. Lederman, Electric Dipole
Moment of the Muon, Bulletin of the American Physical Society, Series II, 4,
No. 1, Part 1, p. 81, January 28, 1959.
12’. D. Berley, R.L. Garwin, G. Gidal, and L.M. Lederman, Electric Dipole
Moment of the Muon, Bulletin of the American Physical Society, Series II, 5,
No. 1 Part 2, p. 81, January 27, 1960.
13’. Garwin, R.L. (1986, 1991, 2001), Interviews of Richard L. Garwin by Finn
Aaserud and W. Patrick McCray, made on October 23, 1986 in Yorktown Heights,
NY (in three sessions), on June 24, 1991 at the IBM Research Laboratory,
Croton-Harmon, NY, and on June 7, 2001 in New York City, Oral History
Transcripts, Center for History of Physics of the American Institute of
Physics, Niels Bohr Library & Archives, American Institute of Physics (AIP),
College Park (MD), USA.
## Bibliography
Alberico, W.M. (1992), Introduzione alla Fisica Nucleare, Torino: La
Scientifica Editrice..
Araki, H. (1999), Mathematical Theory of Quantum Fields, New York: Oxford
University Press, Inc.
Archibald, R. C. (1950), Book Reviews: Luigi Berzolari, Enciclopedia delle
matematiche elementari e complementi con estensione alle principali teorie
analitiche, geometriche e fisiche, loro applicazione e notizie storico-
bibliografiche, Vol. 3, Bulletin of the American Mathematical Society (N.S.),
56 (6): 517-518.
Bacry, H. (1967), Leçons sur la Théorie des Groupes et les Symétries des
Particules Élémentaires, Paris: Gordon & Breach Science Publishers, Inc.
Bargmann, V., Michel, L. and Telegdi, V.L. (1959), Precession of the
polarization of particle moving in a homogeneous electromagnetic field,
Physical Review Letters, 2 (10): 435-436.
Barone, V. (2004), Relatività. Princìpi e applicazioni, Torino: Bollati
Boringhieri editore.
Barut, A.O. and Ra̧czka, R. (1977), Theory of Group Representations and
Applications, Warszawa: PWN-Polish Scientific Publishers.
Bauer, H.H., Christian, G.D. and O’Reilly, J.E. (Eds.) (1978), Instrumental
Analysis, Boston (MA): Allyn & Bacon, Inc. (Italian Translation: (1985),
Analisi Strumentale, Padova: Piccin Nuova Libraria).
Berezin, F.A. and Shubin, M.A. (1991) The Schrödinger Equation, Dordrecht:
Kluwer Academic Publishers.
Berley, D., Garwin, R.L., Gidal, G. and Lederman, L.M. (1958), Electric Dipole
Moment of the Muon, Physical Review Letters, 1 (4): 144-146.
Beth, R.A. (1966), Complex Representation and Computation of Two Dimensional
Magnetic Fields, Journal of Applied Physics, 37 (7): 2568-2571.
Bertolotti, M. (2005), The History of the Laser, London: IoP - Institute of
Physics Publishing, Ltd. (Italian Edition: (1999), Storia del laser, Torino:
Bollati Boringhieri editore).
Beth, R.A. (1967), An Integral Formula for Two-Dimensional Fields, Journal of
Applied Physics, 38 (12): 4689-4692.
Bjorken, J.D. and Drell, S.D. (1964), Relativistic Quantum Mechanics, New
York: McGraw-Hill Book Company, Inc.
Bjorken, J.D. and Drell, S.D. (1965), Relativistic Quantum Fields, New York:
McGraw-Hill Book Company, Inc.
Bloch, F. (1946), Nuclear Induction, Physical Review, 70 (7-8): 460-474.
Bogolubov, N.N., Logunov, A.A. and Todorov, I.T. (1975), Introduction to
Axiomatic Quantum Field Theory, Reading (MA): W.A. Benjamin, Inc.
Bogoliubov, N.N. and Shirkov, D.V. (1980), Introduction to the Theory of
Quantized Fields, third edition, New York: A Wiley-Interscience Publication,
John Wiley & Sons, Inc.
Bogolubov, N.N., Logunov, A.A., Oksak, A.I. and Todorov, I.T. (1990), General
Principles of Quantum Field Theory, Dordrecht: Kluwer Academic Publishers.
Bohr, A. and Mottelson, B.R. (1969, 1975), Nuclear Structure, Volume I,
Single-Particle Motion, Volume II, Nuclear Deformations, London: W.A.
Benjamin, Inc.
Bohm, A. (1993), Quantum Mechanics. Foundations and Applications, third
edition, New York: Springer-Verlag.
Born, M. (1969), Atomic Physics, 8th edition, London-Glasgow: Blackie & Son,
Ltd. (Italian Translation: (1976), Fisica atomica, Torino: Bollati Boringhieri
Editore).
Boudon, R. (1986) L’Ideologie. L’origine des idées reçues, Paris: Éditions
Fayard (Italian Translation: (1991), L’ideologia. Origine dei pregiudizi,
Torino: Giulio Einaudi editore).
Breit, G. (1947a), Relativistic Corrections to Magnetic Moments of Nuclear
Particles, Physical Review, 71 (7): 400-402.
Breit, G. (1947b), Does the Electron Have an Intrinsic Magnetic Moment?,
Physical Review, 72 (10): 984.
Brown, L.M. and Hoddeson L. (Eds), The Birth of Particle Physics, Based on the
lectures and round-table discussion of the International Symposium on the
History of Particle Physics, held at Fermilab in May, 1980, New York:
Cambridge University Press.
Brown, L.M., Dresden, M. and Hoddeson, L. (Eds) (1989), Pions to Quarks:
Particle Physics in the 1950s, Based on the lectures and discussions of
historians and physicists at the Second International Symposium on the History
of Particle Physics, held at Fermilab on May 1-4, 1985, Cambridge (UK):
Cambridge University Press.
Cabibbo, N. (Ed.) (1994), Lepton Physics at Cern and Frascati, Singapore:
World Scientific Publishing Company.
Cahn, R.N. and Goldhaber, G. (2009), The Experimental Foundations of Particle
Physics, 2nd Edition, Cambridge (UK): Cambridge University Press.
P. Caldirola, Dalla microfisica alla macrofisica, Biblioteca EST, Arnoldo
Mondadori Editore, Milano, 1974.
P. Caldirola, R. Cirelli, G.M. Prosperi, Introduzione alla Fisica Teorica,
UTET, Torino, 1982.
P. Carlson, A. De Angelis, Nationalism and internationalism in science: the
case of the discovery of cosmic rays, The European Physical Journal H, 35 (4)
(2010) pp. 309-329.
Carotenuto, A. (1991), Trattato di psicologia della personalità, Milano:
Raffaello Cortina Editore.
P.A. Carruthers, Introduction to Unitary Symmetry, Interscience Publishers, a
division of John Wiley & Sons, Inc., New York, 1966.
P.A. Carruthers, Spin and Isospin in Particle Physics, Gordon and Breach
Science Publishers, New York, 1971.
C. Castagnoli, Lezioni di struttura della materia, seconda edizione riveduta
ed ampliata, Libreria editrice universitaria Levrotto & Bella, Torino, 1975.
G. Castelfranchi, Fisica moderna, atomica e nucleare, decima edizione
completamente rinnovata, Editore Ulrico Hoepli, Milano, 1959.
L. Castellani, R. D’Auria, P. Fré, Supergravity and Superstrings. A Geometric
Perspective, Volume 1, Mathematical Foundations, Volume 2, Supergravity,
Volume 3, Superstrings, World Scientific Publishing Company, Ltd., 1991.
A. Cavallucci, E. Lanconelli, Commemorazione di Bruno Pini, La matematica
nella Società e nella Cultura. Rivista della Unione Matematica Italiana, Serie
I, Vol. 4, No. 2 (2011) pp. 261-274.
CERN 1960 Annual Report, Geneva, 1961.
Chanowitz, M.S., Furman, M.A. and Hinchliffe, I. (1978), Weak interactions of
ultra heavy fermions, Physics Letters B, 78 (2-3): 285-289.
T-P. Cheng, L-F. Li, Gauge theory of elementary particle physics, Oxford at
Clarendon Press, New York, 1984.
E. Chiavassa, L. Ramello, E. Vercellin, Rivelatori di particelle. Appunti
dalle lezioni di Fisica dei Neutroni, La Scientifica Editrice, Torino, 1991.
Coffin, T., Garwin, R.L., Penman, S., Lederman, L.M. and Sachs, A.M. (1958),
Magnetic Moment of the Free Muon, Physical Review, 109 (3): 973-979.
Cohen-Tannoudji, C., Diu, B. and Laloë, F. (1977), Quantum Mechanics, Volume
I, II, Paris-New York: Èditions Hermann and John Wiley & Sons, Inc.
C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons et atomes.
Introduction à l’électrodynamique quantique, Savoirs Actuels,
InterEditions/Editions du CNRS, Paris, 1987.
S. Coleman, Aspects of symmetry. Selected Erice lectures, Cambridge University
Press, Cambridge (UK), 1985.
P.D.B. Collins, A.D. Martin, E.J. Squires, Particle Physics and Cosmology,
John Wiley & Sons, Inc., New York, 1989.
Combley, F. and Picasso, E. (1974), The Muon $(g-2)$ Precession Experiments:
Past, Present and Future, Physics Reports, 14 (1): 1-58.
Combley, F., Farley, F.J.M. and Picasso, B. (1981), The CERN Muon $(g-2)$
Experiments, Physics Reports, 68 (2): 93-119.
Compton, A.H. (1921), The magnetic electron, Journal of the Franklin
Institute, 192 (2): 145-155.
E. Corinaldesi, F. Strocchi, Relativistic Wave Mechanics, North-Holland
Publishing Company, Amsterdam, 1963.
J.F. Cornwell, Group Theory in Physics, Volumes I, II, III, Academic Press,
Ltd., London, 1984, 1989.
Crane, H.R. (1976), $g-2$ techniques: past evolution and future prospects, AIP
Conference Proceedings, 35 (1): 306-314.
Cressant, P. (1970), Lévi Strauss, Paris: Éditions Universitaires (Italian
Translation: (1971), Lévi Strauss, Firenze: C/E Giunti - G. Barbèra).
Croce, B. (1938), La storia come pensiero e come azione, Bari: Editori
Laterza.
P. Davies (Ed), The New Physics, Cambridge University Press, Cambridge (UK),
1989 (Italian Translation: La Nuova Fisica, Bollati Boringhieri editore,
Torino, 1992).
A.S. Davydov, Teoria del nucleo atomico, Nicola Zanichelli Editore, Bologna,
1966.
A.S. Davydov, Meccanica quantistica, Edizioni Mir, Mosca, 1981.
V. De Alfaro, T. Regge, Potential Scattering, North-Holland Publishing
Company, Amsterdam, 1965.
V. De Alfaro, S. Fubini, G. Furlan, C. Rossetti, Currents in Hadron Physics,
North-Holland Publishing Company, Amsterdam and London, 1973.
V. De Alfaro, Introduzione alla teoria dei campi, Parte I, Lezioni date alla
Facoltà di Scienze dell’Università di Torino, Anno Accademico 1993-1994,
Edizioni Cooperativa Libraria Universitaria - CLU, Torino, 1993.
S.R. de Groot, L.G. Suttorp, Foundations of Electrodynamics, North-Holland
Publishing Company, Amsterdam, 1972.
Dekker, A.J. (1958), Solid State Physics, 6th Edition, Englewood Cliffs (NJ):
Prentice-Hall, Inc. (Italian Translation: (1965), Fisica dello stato solido,
Milano: CEA - Casa Editrice Ambrosiana).
P. Deligne, P. Etingof, D.S. Freed, L.C. Jeffrey, D. Kazhdan, J.W. Morgan,
D.R. Morrison, E. Witten (Eds), Quantum Fields and Strings: A Course for
Mathematicians, Volumes 1, 2, American Mathematical Society and Institute for
Advanced Study, Providence, Rhode Island, 1999.
A. Derdzinski, Geometry of the Standard Model of Elementary Particles,
Springer-Verlag, Berlin and Heidelberg, 1992.
A. Di Giacomo, Lezioni di Fisica Teorica, Edizioni ETS, Pisa, 1992.
P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edition, Oxford
University Press at Clarendon, Oxford (UK), 1958 (Italian Translation: I
principi della meccanica quantistica, Editore Boringhieri, Torino, 1959).
P.A.M. Dirac, Lectures on Quantum Mechanics and Relativistic Field Theory,
Tata Institute of Fundamental Research, Bombay, 1955.
P.A.M. Dirac, Lectures on Quantum Field Theory, Belfer Graduate School of
Science, Yeshiva University, Academic Press, Inc., New York, 1966.
Ducrot, O., Todorov, T., Sperber, D., Safouan, M. and Wahl, F. (1968), Qu’est-
ce que le structuralisme?, Paris: Éditions du Seuil (Italian Translation:
(1971), Che cos’è lo strutturalismo, Milano: ISEDI).
L. Eisenbud, E.P. Wigner, Nuclear Structure, Princeton University Press,
Princeton (NJ), 1958 (Italian Translation: La struttura del nucleo, Editore
Boringhieri, Torino, 1960).
Farago, P.S. (1958), Proposed Method for Direct Measurement of the $g$-Factor
of Free Electrons, Proceedings of the Physical Society (London), 72 (5):
891-894.
Farago, P.S., Gardiner, R.B. and Rae, A.G.A. (1963), Direct Measurement of the
$g$-Factor Anomaly of Free Electrons, Proceedings of the Physical Society
(London), 82 (4): 493-500.
Farley, F.J.M. and Picasso, E. (1979), The Muon $(g-2)$ Experiments, Annual
Review of Nuclear and Particle Science, 29 (1): 243-282 [Registered Preprint:
The Muon $(g-2)$ Experiments at CERN, Report No. CERN-EP/79-20, 13 March
1979].
Farley, F.J.M. and Semertzidis, Y.K. (2004), The 47 years of muon $g-2$,
Progress in Particle and Nuclear Physics, 52 (1): 1-83.
Farley, F.J.M. (2005), A new method of measuring the muon $g-2$, talk
presented in the Section: ”Role of innovative detectors” of the Symposium The
Golden Age of Particle Physics and its Legacy: A Festschrift in honour of
Larry Sulak, Boston University, October 21-22, 2005.
E. Fermi, Notes on Quantum Mechanics. A Course Given by Enrico Fermi at the
University of Chicago, Phoenix Books, The University of Chicago Press, Chicago
& London, 1962.
E. Fermi, Elementary Particles, Yale University Press, Inc., New Haven, 1951
(Italian Translation: Particelle elementari, Editore Boringhieri, Torino,
1963).
P. Fleury, J.P. Mathieu, Cours de physique générale et expérimentale, 8.
Atomes, Molécules, Noyaux, Éditions Eyrolles, Paris, 1963 (Italian
Translation: Trattato di fisica generale e sperimentale, 8. Atomi, Molecole,
Nuclei, Nicola Zanichelli editore, Bologna, 1965).
Foldy, L.L. and Wouthuysen, S.A. (1950), On the Dirac Theory of Spin 1/2
Particles and its Non-Relativistic Limit, Physical Review, 78 (1): 29-36.
L. Fonda, G.C. Ghirardi, Symmetry Principles in Quantum Physics, Marcel
Dekker, Inc., New York, 1970.
Ford, J.W., Luxon, J.L., Rich, A., Wesley, J.C. and Telegdi, V.L. (1972),
Resonant Spin Rotation - a New Lepton $g-2$ Technique, Physical Review
Letters, 29 (25): 1691-1695.
K.W. Ford, The World of Elementary Particles, Blaisdell Publishing Company,
1963 (Italian Translation: La fisica delle particelle, Biblioteca EST, Arnoldo
Mondadori Editore, Milano, 1965).
A. Frank, P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure
Physics, John Wiley & Sons, Inc., New York, 1994.
G. Friedlander, J.W. Kennedy, Nuclear and Radiochemistry, John Wiley & Sons,
Inc., New York, 1960 (Italian Translation: Chimica nucleare e radiochimica,
Carlo Manfredi Editore, Milano, 1965).
Friedman, J.I. and Telegdi, V.L. (1957), Nuclear Emulsion Evidence for Parity
Nonconservation in the Decay Chain $p^{+}\rightarrow\mu^{+}\rightarrow e^{+}$,
Physical Review, 106 (6): 1290-1293.
K.O. Friedrichs, Mathematical Aspects of the Quantum Theory of Fields,
Interscience Publishers, Inc., New York, 1953.
U. Galimberti, Dizionario di Psicologia, UTET Libreria, Torino, 2006.
G. Gamow, Biography of Physics, Harper & Row, New York, 1961 (Italian
Translation: Biografia della fisica, Biblioteca EST, Arnoldo Mondadori
Editore, Milano, 1963).
G. Gamow, Thirty Years That Shook Physics. The Story of Quantum Theory, Anchor
Books Doubleday & Company, Inc., Garden City, New York, 1966 (Italian
Translation: Trent’anni che sconvolsero la fisica. La storia della Teoria dei
Quanti, Nicola Zanichelli editore, Bologna, 1966).
Gardner, J.H. and Purcell, E.M. (1949), A Precise Determination of the Proton
Magnetic Moment in Bohr Magnetons, Physical Review, 76 (8): 1262-1263.
Gardner, J.H. (1951), Measurement of the Magnetic Moment of the Proton in Bohr
Magnetons, Physical Review, 83 (5): 996-1004.
Garwin, R.L., Lederman, L.M. and Weinrich, M. (1957), Observation of the
Failure of Conservation of Parity and Charge Conjugation in Meson Decays: the
Magnetic Moment of the Free Muon, Physical Review, 105 (4): 1415-1417.
Garwin, R.L., Hutchinson, D.P., Penman, S. and Shapiro, G. (1960), Accurate
Determination of the $\mu^{+}$ Magnetic Moment, Physical Review, 118 (1):
271-283.
Garwin, R.L. and Lederman, L.M. (1959), The Electric Dipole Moment of
Elementary Particles, Il Nuovo Cimento, 11 (6): 776-780.
Garwin, R.L. (2003), The first muon spin rotation experiment, Physica B, 326
(1): 1-10.
H. Georgi, Lie Algebras in Particle Physics. From Isospin to Unified Theories,
Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1982.
Germain, P. (1989), Introduction aux accélérateurs de particules, Cours édité
par D. Dekkers et D. Manglunki, CERN 89-07, 7 July 1989 - Rév. 15/28.02.2005,
Genève.
V.L. Ginzburg, Questioni di fisica e astrofisica, Editori Riuniti-Edizioni
Mir, Roma-Mosca, 1983.
J. Glimm, A. Jaffe, Quantum Physics. A Functional Integral Point of View,
Springer-Verlag, New York, 1981.
M. Gliozzi, Storia del pensiero fisico, Articolo LX in: L. Berzolari (Ed),
Enciclopedia delle Matematiche Elementari e Complementi, con estensione alle
principali teorie analitiche, geometriche e fisiche, loro applicazioni e
notizie storico-bibliografiche, Volume III, Parte 2a, Editore Ulrico Hoepli,
Milano, 1949 (ristampa anastatica 1972).
M. Gliozzi, Storia della fisica, a cura di Alessandra e Ferdinando Gliozzi,
Bollati Boringhieri Editore, Torino, 2005.
Gottfried, K. (1966), Quantum Mechanics, Volume I, Foundations, New York: W.A.
Benjamin, Inc.
Gräff, G. (1971), Methods for Lepton $g$-Factor Anomaly Measurement, in:
Langenberg, D.N. and Taylor, B.N. (Eds.) (1971), Precision Measurement and
Fundamental Constants. Proceedings of the International Conference held at the
National Bureau of Standards in Gaithersburg, Maryland, August 3-7, 1970,
Washington (D.C.): U.S. Government Printing Office, U.S. Department of
Commerce, National Bureau of Standards Special Publication No. 343, CODEN:
XNBSA.
M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, Volume 1,
Introduction, Volume 2, Loop amplitudes, anomalies and phenomenology,
Cambridge University Press, Cambridge (UK), 1987.
Grodzins, L. (1959), Measurements of helicity, in: Frisch, R.O. (Ed.) (1959),
Progress in Nuclear Physics, Volume 7, London: Pergamon Press, Ltd., pp.
163-241.
E. Guadagnini, Fisica Teorica, Lezioni per il Corso di Dottorato di Ricerca,
Edizioni ETS, Pisa, 1999.
R. Haag, Local Quantum Physics. Fields, Particles, Algebras, Springer-Verlag,
Berlin and Heidelberg, 1992.
Haeberli, W. (1967), Sources of Polarized Ions, Annual Review of Nuclear
Science, 17: 373-426.
Hahn, E.L. (1950), Spin Echoes, Physical Review, 80 (4): 580-594.
Haken, H. and Wolf, H.C. (2005), The Physics of Atoms and Quanta. Introduction
to Experiments and Theory, 7th Edition, Berlin and Heidelberg: Springer-Verlag
(Italian Translation of the 1987 2nd Edition: (1990), Fisica atomica e
quantistica. Introduzione ai fondamenti sperimentali e teorici, Torino:
Bollati Boringhieri Editore).
S. Hartmann, Models and stories in hadron physics, in: M.S. Morgan, M.
Morrison (Eds), Models as Mediators. Perspectives on Natural and Social
Science, Cambridge University Press, Cambridge (UK), 1999, Chapter 11, pp.
326-346.
W.P. Healy, Non-Relativistic Quantum Electrodynamics, Academic Press, Ltd.,
London, 1982.
W. Heisenberg, Die Physikalischen Prinzipien der Quantentheorie, S. Hirzel
Verlag, Lipsia, 1930 (Italian Translation: I principi fisici della teoria dei
quanti, Editore Boringhieri, Torino, 1976).
W. Heisenberg, Wandlungen in den Grundlagen der Naturwissenshaft, S. Hirzel
Verlag, Stuttgart (FRG), 1959 (Italian Translation: Mutamenti nelle basi della
scienza, Paolo Boringhieri editore, Torino, 1978).
W. Heitler, The Quantum Theory of Radiation, Third Edition, Oxford University
Press at Clarendon, Oxford (UK), 1954.
Holzscheiter, M.H. (1995), A brief history in time of ion traps and their
achievements in science, Physica Scripta, Supplement T59: 69-76, Special
volume devoted to the Proceedings of the Nobel Symposium 91: Trapped Charged
Particles and Related Fundamental Physics, 19-26 August 1994, Lysekil, Sweden.
K. Huang, Quarks, Leptons & Gauge Fields, 2nd edition, World Scientific
Publishing Co., Singapore, 1992.
D.J. Hughes, The Neutron Story, Doubleday & Company, Inc., Garden City, New
York, 1959 (Italian Translation: Fisica del Neutrone, Piccola Biblioteca
Einaudi, Torino, Giulio Einaudi editore, Torino, 1960).
Hughes, V.W. and Schultz, H.L. (Eds) (1967), Methods of Experimental Physics,
Vol. 4, Part B: Atomic and Electron Physics. Atomic Sources and Detectors, New
York: Academic Press, Inc.
Hughes, V.W. and Wu, C.S. (Eds.) (1977), Muon Physics, Vol. I. Electromagnetic
Interactions, Vol. II, Weak Interactions, Vol. III, Chemistry and Solids, New
York: Academic Press, Inc.
Hughes, V.W. and Sichtermann, E.P. (2003), The Anomalous Magnetic Moment of
the Muon, International Journal of Modern Physics A, 18 (S1): 215-272.
Humphries, S. Jr. (1999), Principles of Charged Particle Acceleration, New
York: John Wiley & Sons, Inc.
F. Iachello, A. Arima, The interacting boson model, Cambridge University
Press, Cambridge (UK), 1987.
F. Iachello, P. Van Isacker, The interacting boson-fermion model, Cambridge
University Press, Cambridge (UK), 1991.
Iurato, G. (2013), On Collingwood’s historicism, hal.archives-
ouvertes.fr/hal-00921948-version 1.
Jackson, J.D. (1975), Classical Electrodynamics, 2nd Edition, New York: John
Wiley & Sons, Inc. (Italian Translation: (1984), Elettrodinamica Classica,
Bologna: Nicola Zanichelli Editore).
M. Jacob, Pas d’exclusion pour Wolfgang Pauli, CERN Courier - International
Journal of High-Energy Physics, Volume 40, No. 9, September 2000, pp. 30-32.
J.M. Jauch, F. Rohrlich, The Theory of Photons and Electrons. The Relativistic
Quantum Field Theory of Charged Particles with Spin One-half, Second Expanded
Edition, Springer-Verlag, New York, Berlin and Heidelberg, 1976.
C.J. Joachain, Quantum Collision Theory, North-Holland Publishing Company,
Amsterdam, 1975.
R. Jost, The General Theory of Quantized Fields, American Mathematical
Society, Providence, Rhode Island, 1965.
M. Kaku, Introduction to Superstrings, Springer-Verlag, New York, 1988.
M. Kaku, Quantum Field Theory. A Modern Introduction, Oxford University Press,
New York, 1993.
D. Kastler, Introduction a l’électrodynamique quantique, Dunod, Paris, 1961.
Kastler, A. (1976), Cette Éstrange Matière, Paris: Editions Stock (Italian
Translation: (1977), Questa strana materia, Milano: Arnoldo Mondandori
Editore).
E.C. Kemble, The Fundamental Principles of Quantum Mechanics with Elementary
Applications, Dover Publications, Inc., New York, 1958.
Kerst, D.W. and Serber, R. (1941), Electronic Orbits in the Induction
Accelerators, Physical Review, 60 (1): 53-58.
Kinoshita, T. (Ed.) (1990), Quantum Electrodynamics, Singapore: World
Scientific Publishing Company.
E. Kiritsis, Introduction to Superstring Theory, preprint CERN-TH/97-218 (hep-
th/9709062), March 1997.
Kittel, C. (1966), Introduction to Solid State Physics, 3rd Edition, New York:
John Wiley & Sons, Inc. (Italian Translation: (1971), Introduzione alla fisica
dello stato solido, Torino: Editore Boringhieri).
Koenig, S.H., Prodell, A.G. and Kusch, P. (1952), The Anomalous Magnetic
Moment of the Electron, Physical Review, 88 (2): 191-199.
H.S. Kragh, An Introduction to Historiography of Science, Cambridge University
Press, Cambridge (UK), 1987 (Italian Translation: Introduzione alla
storiografia della scienza, Nicola Zanichelli Editore, Bologna, 1990).
H.S. Kragh, (1990), Dirac. A Scientific Biography, Cambridge (UK): Cambridge
University Press.
H.S. Kragh, (2002), Quantum Generations. A History of Physics in the Twentieth
Century, Princeton (NJ): Princeton University Press.
Kusch, P. (1956), Magnetic Moment of the Electron, Science, 123 (3189):
207-211.
E. Lanconelli, Commemorazione di Bruno Pini, necrologio tenuto all’Accademia
Nazionale dei Lincei, Roma, 2007.
E. Lanconelli, Bruno Pini and the Parabolic Harnack Inequality: The Dawning of
Parabolic Potential Theory, in: S. Coen (Ed), Mathematicians in Bologna:
1861-1960, Springer Basel AG, Basel, 2012, pp. 317-332.
Landau, L.D. (1957), On the conservation laws for weak interactions, Nuclear
Physics, 3 (1): 127-131.
L.D. Landau, E.M. Lifšits, Fisica Teorica, 3. Meccanica Quantistica. Teoria
non relativistica, Editori Riuniti-Edizioni Mir, Roma-Mosca, 1982.
L.D. Landau, E.M. Lifšits, Fisica teorica, 4. Teoria quantistica
relativistica, a cura di V.B. Berestetskij, E.M. Lifšits, L.P. Pitaevskij,
Editori Riuniti-Edizioni Mir, Roma-Mosca, 1978.
Lederman, L.M. (1992), Observations in Particle Physics from Two Neutrinos to
the Standard Model, Fermilab Golden Book Collection, Batavia, Illinois: Fermi
National Accelerator Laboratory.
Lee, T.D. and Yang, C.N. (1956), Question of Parity Conservation in Weak
Interactions, Physical Review, 104 (1): 254-258.
Lee, T.D., Oehme, R. and Yang, C.N. (1957), Remarks on Possible Noninvariance
under Time Reversal and Charge Conjugation, Physical Review, 106 (2): 340-345.
T.D. Lee, Particle Physics and Introduction to Field Theory, Harwood Academic
Publishers, New York, 1981.
Lee, S.J. (2004), Accelerator Physics, Second Edition, Singapore: World
Scientific Publishing Company, Ltd.
B. Lehnert, Dynamics of Charged Particles, North-Holland Publishing Company,
Amsterdam, 1964.
J. Leite Lopes, Gauge Field Theories. An Introduction, Pergamon Press, Ltd.,
Oxford (UK), 1981.
B.G. Levich, V.A. Myamlin, Yu.A. Vdovin, Theoretical Physics. An Advanced
Text, Volume 3, Quantum Mechanics, North-Holland Publishing Company,
Amsterdam, 1973.
J. Łopuszànski, An Introduction to Symmetry and Supersymmetry in Quantum Field
Theory, World Scientific Publishing Company, Ltd., Singapore, 1991.
Louisell, W.H., Pidd, R.W. and Crane, H.R. (1954), An Experimental Measurement
of the Gyromagnetic Ratio of the Free Electron, Physical Review, 94 (1): 7-16.
Luttinger, J.M. (1948), A Note on the Magnetic Moment of the Electron,
Physical Review, 74 (8): 893-898.
L. Maiani and R.A. Ricci (Eds), Symposium in honour of Antonino Zichichi to
celebrate the 30th anniversary of The Discovery of Nuclear Antimatter,
Conference Proceedings of the Italian Physical Society, Volume 53, Bologna, 18
December 1995, Published for the Italian Physical Society by Editrice
Compositori, Bologna, 1996.
Mandl, F. and Shaw, G. (1984), Quantum Field Theory, Chichester (WS-UK): John
Wiley and Sons, Ltd.
K.B. Marathe, G. Martucci, The Mathematical Foundations of Gauge Theories,
North-Holland Elsevier Science Publishers B.V., Amsterdam, 1992.
P.T. Matthews, The Nuclear Apple. Recent discoveries in fundamental physics,
Chatto and Windus, London, 1971 (Italian Translation: Nel nucleo dell’atomo.
Le più recenti scoperte della fisica fondamentale, Biblioteca EST, Arnoldo
Mondadori Editore, Milano, 1972).
P.T. Matthews, Introduction to Quantum Mechanics, McGraw-Hill Publishing
Company, Ltd., Maidenhead, Berkshire (UK), 1974 (Italian Translation:
Introduzione alla meccanica quantistica, Nicola Zanichelli Editore, Bologna,
1978.
Melnikov, K. and Vainshtein, A. (2006), Theory of the Muon Anomalous Magnetic
Moment, Berlin and Heidelberg: Springer-Verlag.
G.A. Miller, Book Reviews: Enciclopedia delle Matematiche Elementari, Vol. 1,
Bulletin of the American Mathematical Society, 38 (3) (1932) pp. 157-158.
Miller, J.P., de Rafael, E. and Roberts, L.B. (2007), Muon $(g-2)$: experiment
and theory, Reports on Progress in Physics, 70 (5): 795-881.
G. Morpurgo, Lezioni sulle forze nucleari, A.A. 1954-1955, Scuola di
Perfezionamento in Fisica Nucleare, Istituto di Fisica dell’Università di
Roma, Litografia Marves, Roma, 1955.
G. Morpurgo, Introduzione alla fisica delle particelle, Nicola Zanichelli
Editore, Bologna, 1987.
H. Muirhead, The Physics of Elementary Particles, Pergamon Press, Ltd., Oxford
(UK), 1965.
H.J.W. Müller-Kirsten, A. Wiedemann, Supersymmetry. An Introduction with
Conceptual and Calculational Details, World Scientific Publishing Company,
Ltd., Singapore, 1987.
Y. Ne’eman, Algebraic Theory of Particle Physics. Hadron Dynamics in terms of
Unitary Spin Currents, W.A. Benjamin, Inc., New York, 1967.
R.G. Newton, Scattering Theory of Waves and Particles, Second Edition,
Springer-Verlag, New York, Heidelberg and Berlin, 1982.
R.G. Newton, The Complex j-Plane. Complex Angular Momentum in Nonrelativistic
Quantum Scattering Theory, W.A. Benjamin, Inc., New York, 1964.
Ohanian, H.C. (1988), Classical Electrodynamics, Boston (MA): Allyn & Bacon,
Inc.
L.B. Okun, Leptons and Quarks, Elsevier Science Publishers, Ltd., Amsterdam
and New York, 1982 (Italian Edition: Leptoni e quark, Editori Riuniti-Edizioni
Mir, Roma-Mosca, 1986).
L. O’Raifeartaigh, Group Structure of Gauge Theories, Cambridge University
Press, Cambridge (UK), 1986.
Panofsky, W.K.H. and Phillips, M. (1962), Classical Electricity and Magnetism,
Reading (MA): Addison-Wesley Publishing Company, Inc. (Italian Translation:
(1966), Elettricità e magnetismo, Milano: CEA - Casa Editrice Ambrosiana).
J.C. Parikh, Group Symmetries in Nuclear Structure, Plenum Press, New York and
London, 1978.
Pauli, W. (1941), Relativistic Field Theory of Elementary Particles, Review of
Modern Physics, 13 (3): 203-232.
W. Pauli, Wellenmechanik, Verlag des Vereins der Mathematiker und Physiker an
der ETH, Zürich, 1959 (Italian Translation: Meccanica ondulatoria, Editore
Boringhieri, Torino, 1962).
W. Pauli, Pauli Lectures on Physics, 6. Selected Topics in Field Quantization,
Edited by C. Enz, The MIT Press, Cambridge, Massachusetts, 1973.
W. Pauli, Lectures on continuous groups and reflections in quantum mechanics,
Notes by R.J. Riddell jr., Radiation Laboratory UCRL-8213, University of
California, Berkeley, Printed for the U.S. Atomic Energy Commission, 1958.
Pedulli, G.F., Alberti, A. and Lucarini, M. (1996), Metodi Fisici in Chimica
Organica. Princìpi ed applicazioni di tecniche spettroscopiche, Padova: Piccin
Nuova Libraria.
E. Persico, Fondamenti della Meccanica Atomica, Casa Editrice Nicola
Zanichelli, Bologna, 1936.
E. Persico, Gli atomi e la loro energia, Nicola Zanichelli Editore, Bologna,
1959.
M. Piattelli-Palmarini, Scienza come Cultura. Protagonisti, Luoghi e Idee
delle Scienze Contemporanee, edizione paperback a cura di Simone Piattelli,
Saggi Oscar Mondadori Editore, Milano, 1992.
Picasso, E. (1985), Le misure del momento magnetico del muone, Il Nuovo
Saggiatore. Bollettino della Società Italiana di Fisica, 4: 22-30.
Picasso, E. (1996), The anomalous magnetic moment of the muon and related
topics, Atti dell’Accademia Nazionale dei Lincei, Rendiconti della classe di
scienze fisiche, matematiche e naturali, 7 (9): 209-241.
A. Pignedoli, Alcune teorie meccaniche ”superiori”, CEDAM, Padova, 1969.
M. Planck, La conoscenza del mondo fisico, Paolo Boringhieri Editore, Torino,
1964.
J. Polchinski, String Theory, Volume I, An Introduction to the Bosonic String,
Volume II, Superstring Theory and Beyond, Cambridge University Press,
Cambridge (UK), 1998.
S. Pokorski, Gauge Field Theories, Cambridge University Press, Cambridge,
1987.
V. Polara, L’atomo e il suo nucleo. Struttura dell’atomo e disintegrazione
spontanee ed artificiali del nucleo, Perrella Editore, Roma, 1949.
A.M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers GmbH,
Chur, Switzerland, 1987.
Povh, B., Rith, K., Scholz, C. and Zetsche, F. (1995), Particles and Nuclei.
An Introduction to the Physical Concepts, Berlin and Heidelberg: Springer-
Verlag (Italian Translation: (1998), Particelle e nuclei. Un’introduzione ai
concetti fisici, Torino: Bollati Boringhieri editore).
Rabi, I.I., Zacharias, J.R., Millman, S. and Kusch, P. (1938), A New Method of
Measuring Nuclear Magnetic Moment, Physical Review, 53 (4): 318(L).
Rabi, I.I., Millman, S., Kusch, P. and Zacharias, J.R. (1939), The Molecular
Beam Resonance Method for Measuring Nuclear Magnetic Moments. The Magnetic
Moments of ${}_{3}Li^{6},_{3}Li^{7}$ and ${}_{9}Fe^{19}$, Physical Review, 55
(6): 526-535.
M. Reed, B. Simon, Methods of Modern Mathematical Physics, Volume I,
Functional Analysis, Volume II, Fourier Analysis, Self-Adjointness, Volume
III, Scattering Theory, Volume IV, Analysis of Operators, Academic Press,
Inc., New York, 1980, 1975, 1979, 1978.
Rich, A. and Wesley, J. (1972), The Current Status of the Lepton $g$ Factors,
Reviews of Modern Physics, 44 (2): 250-283.
P. Ring, P. Schuck, The Nuclear Many-Body Problem, Springer-Verlag, New York,
1980.
R.G. Roberts, The Structure of the Proton, Cambridge University Press,
Cambridge (UK), 1990.
Roberts, B.L. and Marciano, W.J. (Eds.) (2010), Lepton Dipole Moments,
Singapore: World Scientific Publishing Company, Ltd.
P. Roman, Theory of Elementary Particles, North-Holland Publishing Company,
Amsterdam, 1960.
P. Roman, Advanced Quantum Theory. An outline of the fundamental ideas,
Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1965.
C. Rossetti, Elementi di teoria dell’urto, Libreria editrice universitaria
Levrotto & Bella, Torino, 1985.
P.A. Rowlatt, Group Theory and Elementary Particles, Longmans, Green and
Company, Ltd., London, 1966.
Rossi, B. (1964), Cosmic Rays, New York: McGraw-Hill Book Company, Inc.
(Italian Translation: (1971), I raggi cosmici, Torino: Giulio Einaudi
editore).
J.J. Sakurai, Invariance Principles and Elementary Particles, Princeton
University Press, Princeton (NJ), 1964.
J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley Publishing Company,
Inc., Reading, Massachusetts, 1967.
J.J. Sakurai, S.F. Tuan, Modern Quantum Mechanics, The Benjamin/Cummings
Publishing Company, Inc., Menlo Park (CA), 1985 (Italian Translation:
Meccanica quantistica moderna, Nicola Zanichelli Editore, Bologna, 1990).
F. Scheck, H. Upmeier, W. Werner (Eds), Noncommutative Geometry and the
Standard Model of Elementary Particle Physics, Springer-Verlag, Berlin and
Heidelberg, 2002.
L.I. Schiff, Quantum Mechanics, McGraw-Hill Book Company, Inc., New York, 1952
(Italian Translation: Meccanica quantistica, Edizioni Scientifiche Einaudi,
Torino, 1952).
H. Schlaepfer, Cosmic Rays, Spatium, Published by International Space Science
Institute of Bern, 11 (2003) pp. 3-15.
Schultz, D.P. (1969), A History of Modern Psychology, New York: Academic
Press, Inc. (Italian Translation: (1974), Storia della psicologia moderna,
Firenze: Giunti-Barbèra).
L. Schwartz, Application of Distributions to the Theory of Elementary
Particles in Quantum Mechanics, Gordon & Breach Science Publishers, Inc., New
York, 1968.
Schwartz, M. (1972), Principles of Electrodynamics, New York: McGraw-Hill Book
Company, Inc.
A.S. Schwarz, Quantum Field Theory and Topology, Springer-Verlag, Berlin and
Heidelberg, 1993.
S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row,
Peterson and Company, Evanston, Illinois and Elmsford, New York, 1961.
Schweber, S.S. (1983), Some chapters for a history of quantum field theory:
1938-1952, in: DeWitt, B. and Stora, R. (Eds) (1984), Relativity, Groups and
Topology, II, Parts 1, 2, 3, Les Houches, Session XL, 27 June - 4 August 1983,
Amsterdam (The Netherlands): North-Holland Physics Publishing Company, Inc.,
pp. 37-220.
S.S. Schweber, QED And The Men Who Made It: Dyson, Feynman, Schwinger, and
Tomonaga, Princeton University Press, Princeton (NJ), 1994.
J. Schwinger, On Quantum-Electrodynamics and the Magnetic Moment of the
Electron, Physical Review, 73(4): 416-417.
J. Schwinger, Quantum Mechanics. Symbolism of Atomic Measurements, edited by
B-G. Englert, Springer-Verlag, Berlin and Heidelberg, 2001.
I.E. Segal, Mathematical Problems of Relativistic Physics, American
Mathematical Society, Providence, Rhode Island, 1963.
E. Segrè, Nuclei and Particles. An Introdduction to Nuclear and Subnuclear
Physics, W.A. Benjamin, Inc., New York, 1964 (Italian Translation: Nuclei e
particelle. Introduzione alla fisica nucleare e subnucleare, Nicola Zanichelli
Editore, Bologna, 1966; seconda edizione, 1999).
E. Segrè, Personaggi e scoperte della fisica contemporanea, Biblioteca EST,
Arnoldo Mondadori Editore, Milano, 1976.
Shankar, R. (1994), Principles of Quantum Mechanics, 2nd Edition, New York:
Kluwer Academic/Plenum Publishers.
A.G. Sitenko, V.K. Tartakovskij, Lezioni di teoria del nucleo, Edizioni Mir,
Mosca, 1981.
Slater, J.C. (1968), Quantum Theory of Matter, New York: McGraw-Hill Book
Company, Inc. (Italian Translation: (1980), Teoria quantistica della materia,
Bologna: Nicola Zanichelli Editore).
S̆polskij, E.D. (1986), Fisica atomica, Volume I, Introduzione alla fisica
atomica, Volume II, Fondamenti della meccanica quantistica e struttura del
guscio elettronico dell’atomo, Mosca-Roma: Edizioni Mir.
Fl. Stancu, Group Theory in Subnuclear Physics, Oxford University Press at
Clarendon, Oxford (UK), 1996.
S. Sternberg, Group Theory and Physics, Cambridge University Press, Cambridge
(UK), 1994.
P. Straneo, Materia, irraggiamento e fisica quantica, Articolo LII in: L.
Berzolari (Ed), Enciclopedia delle Matematiche Elementari e Complementi, con
estensione alle principali teorie analitiche, geometriche e fisiche, loro
applicazioni e notizie storico-bibliografiche, Volume III, Parte 1a, Editore
Ulrico Hoepli, Milano, 1947 (ristampa anastatica 1975).
R.F. Streater, A.S. Wightman, PCT, Spin and Statistics, and all that, W.A.
Benjamin, Inc., New York, 1964.
Sundermeyer, K. (1982) Contrained Dynamics, with Applications to Yang-Mills
Theory, General Relativity, Classical Spin and Dual String Model, Berlin and
Heidelberg: Springer-Verlag.
Tagliagambe, S. and Malinconico, A. (2011), Pauli e Jung. Un confronto su
materia e psiche, Milano: Raffaello Cortina Editore.
J.R. Taylor, Scattering Theory. The Quantum Theory on Nonrelativistic
Collisions, John Wiley & Sons, Inc., New York, 1972.
W. Thirring, A Course in Mathematical Physics, Volume 1, Classical Dynamical
Systems, Volume 2, Classical Field Theory, Volume 3, Quantum Mechanics of
Atoms and Molecules, Volume 4, Quantum Mechanics of Large Systems, Springer-
Verlag New York, Inc., 1978, 1979, 1981, 1983.
H. Thomä, H. Kächele, Psychoanalytic Practice, Volume 1, Principles, Volume 2,
Praxis, Springer-Verlag, Berlin and Heidelberg, 1987, 1992 (Italian
Translation: Trattato di terapia psicoanalitica, Volume 1, Fondamenti teorici,
Volume 2, Pratica clinica, Bollati Boringhieri editore, Torino, 1990, 1993).
I.T. Todorov, M.C. Mintchev, V.B. Petrova, Conformal Invariance in Quantum
Field Theory, Pubblicazioni della Classe di Scienze della Scuola Normale
Superiore, SNS, Pisa, 1978.
S. Tolansky, Introduction to Atomic Physics, 5th edition, Longmans, Green and
Company, Inc., London, 1963 (Italian Translation: Introduzione alla fisica
atomica, 2 voll., Editore Boringhieri Torino, 1966).
Tomonaga, S.I. (1997), The Story of Spin, Chicago: The University of Chicago
Press.
F.G. Tricomi, La mia vita di matematico attraverso la cronistoria dei miei
lavori (bibliografia commentata 1916-1967), Casa Editrice Dott. Antonio Milani
- CEDAM, Padova, 1967.
Tsai, S.Y. (1981), Universal weak interactions involving heavy leptons,
Lettere al Nuovo Cimento, 2 (18): 949-952.
C. Villi, G. Pisent, V. Vanzani, Lezioni di Istituzioni di Fisica Nucleare,
Anno Accademico 1970-1971, Istituto di Fisica dell’Università di Padova,
Padova, 1971.
C. Villi, F. Zardi, Appunti di lezioni di fisica nucleare, Anno Accademico
1974-1975, Istituto di Fisica dell’Università di Padova, Padova, 1975.
A. Visconti, Théorie quantique des champs, Tomes I, II, Gauthier-Villars,
Paris, 1961, 1965.
A.H. Völkel, Field, Particles and Currents, Springer-Verlag, Berlin and
Heidelberg, 1977.
J.D. Walecka, Theoretical Nuclear and Subnuclear Physics, Oxford University
Press, New York, 1995.
S. Weinberg, The Quantum Theory of Fields, Volume I, Foundations, Volume II,
Modern Applications, Volume III, Supersymmetry, Cambridge University Press,
Cambridge (UK), 1995, 1996, 2000 (Italian Translation of Volume I: La teoria
quantistica dei campi, Nicola Zanichelli editore, Bologna, 1999).
Weisskopf, V.F. (1949), Recent Developments in the Theory of the Electron,
Reviews of Modern Physics, 21 (2): 305-315.
G. Wentzel, Quantum Theory of Fields, Interscience Publishers, Inc., New York,
1949.
Wertheimer, M. (1979), A Brief History of Psychology, New York: Holt, Rinehart
and Wilson, Inc. (Italian Translation: (1983), Breve storia della psicologia,
Bologna: Nicola Zanichelli editore).
H. Weyl, Gruppentheorie und quantenmechanik, Verlag Von S. Hirzel, Leipzig,
1928; second edition, 1931 (English Translation: The Theory of Groups and
Quantum Mechanics, translated from the second revised German edition by H.P.
Robertson, A. Methuen & Co., Ltd., London, 1931; reprinted by Dover
Publications, Inc., New York, 1950).
E.V.H. Wichmann, Quantum Physics, McGraw-Hill Book Company, Inc., New York,
1971 (Italian Translation: La Fisica di Berkeley, 4. Fisica Quantistica,
Nicola Zanichelli Editore, Bologna, 1973).
Wick, G.C. (1945), Appunti di Fisica Nucleare, parte I, Anno Accademico
1944-1945, redatti da E. Amaldi, Roma: Tipo-litografia Romolo Piola.
Wick, G.C. (1946), Appunti di Fisica Nucleare, parte II, Anno Accademico
1945-1946, redatti a cura di E. Amaldi, Roma: Tipo-litografia Romolo Piola.
E.P. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der
Atomspektren, Friedrich Vieweg und Sohn Akt. Ges., Braunschweig, 1931 (English
Translation: Group Theory and Its Application To The Quantum Mechanics of
Atomic Spectra, expanded and improved edition of the first one translated from
the German by J.J. Griffin, Academic Press, Inc., New York, 1959).
R.R. Wilson, R. Littauer, Accelerators. Machines of Nuclear Physics, Anchor
Books Doubleday & Company, Inc., Garden City, New York, 1960 (Italian
Translation: Acceleratori di particelle. Macchine della fisica nucleare,
Nicola Zanichelli editore, Bologna, 1965).
C.N. Yang, Elementary Particles. A Short History of Some Discoveries in Atomic
Physics - 1959 Vanuxem Lectures, Princeton University Press, Princeton (NJ),
1961 (Italian Translation: La scoperta delle particelle elementari, Editore
Boringhieri, Torino, 1969).
Yang, C.N. (2005), Selected Papers (1945-1980) With Commentary, World
Scientific Series in 20th Century Physics, Vol. 36, Singapore: World
Scientific Publishing Company, Ltd.
H. Yukawa, On the Interaction of Elementary Particles. I, Proceedings of the
Physico-Mathematical Society of Japan, 17 (1935) pp. 48-57 (reprinted in
Progress of Theoretical Physics, Supplement, 1 (1955) pp. 1-10).
Zel’dovich, Ya.B. (1961), Dipole moment of unstable elementary particles,
Soviet Physics - JETP (Journal of Experimental and Theoretical Physics), 12:
1030-1031.
Zichichi, A. (1981), Struttura delle particelle, in: Enciclopedia del
Novecento, Vol. V, Roma: Istituto della Enciclopedia Italiana fondata da G.
Treccani, pp. 125-215.
A. Zichichi, Scienza ed emergenze planetarie, 6a edizione Superbur saggi,
Biblioteca Universale Rizzoli, RCS Libri, Milano, 1997.
A. Zichichi, Creativity in Science, 6th International Zermatt Symposium on
Creativity in Economics, Arts and Science, Zermatt, Switzerland, 12-16 January
1996, World Scientific Publishing Company, Ltd., Singapore, 1999.
A. Zichichi, Subnuclear Physics. The first 50 years: highlights from Erice to
ELN, edited by O. Bernabei, P. Pupillo, F. Roversi Monaco, World Scientific
Publishing Company, Ltd., Singapore, 2001.
Zichichi, A. (2008), The 40th Anniversary of EPS: Gilberto Bernardini’s
contributions to the Physics of the XX Century, Il Nuovo Saggiatore.
Bollettino della Società Italiana di Fisica, 24 (5-6): 77-94.
Zichichi, A. (2010), In ricordo di George Charpak (1924-2010), Il Nuovo
Saggiatore. Bollettino della Società Italiana di Fisica, 26 (5-6): 109.
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University
Press at Clarendon, New York, 1989.
|
arxiv-papers
| 2014-02-01T11:58:37 |
2024-09-04T02:49:58.561487
|
{
"license": "Public Domain",
"authors": "Giuseppe Iurato",
"submitter": "Giuseppe Iurato",
"url": "https://arxiv.org/abs/1402.5382"
}
|
1402.5406
|
MENU 2013
11institutetext: Theory Center, Thomas Jefferson National Accelerator
Facility, 12000 Jefferson Avenue, Newport News, VA 23606, U.S.A.
# SPQR — Spectroscopy: Prospects, Questions & Results
M. R. Pennington 11 [email protected]
###### Abstract
Tremendous progress has been made in mapping out the spectrum of hadrons over
the past decade with plans to make further advances in the decade ahead.
Baryons and mesons, both expected and unexpected, have been found, the results
of precision experiments often with polarized beams, polarized targets and
sometimes polarization of the final states. All these hadrons generate poles
in the complex energy plane that are consequences of the strong coupling
regime of QCD. They reveal how this works.
## 1 Why Spectroscopy?
The spectrum of states of any system is fundamental: reflecting the
constituents that make up that system and the interactions between them. The
rich spectrum of hadrons reveals the workings of QCD in the strong coupling
regime. There are two ways to study this. One is wholly theoretical. Knowing
the QCD Lagrangian as we do, one can, in principle, compute its consequences.
This turns out to be only just within our capabilities, and only in simpler
cases can definitive results be obtained. The alternative is to use experiment
as our guide, and learn from there. In experiment quarks know how to solve the
field equations of QCD in the strong coupling regime even without the help of
a BlueGene computer. Nevertheless, extracting the spectrum from complex data
is often far from straightforward, requiring close interaction between theory
and experiment.
Substantial progress has been made in both the baryon and meson sectors during
the past ten years with increasingly precise experiments, measuring not just
differential cross-sections, but all manner of polarization observables too.
Even more results are to come from BESIII, COMPASS, LHCb, MAMI, ELSA, and
Jefferson Lab experiments, with PANDA to follow.
## 2 The Hadron Spectrum: Baryons and Mesons
Figure 1: $N^{*}$ and $\Delta^{*}$ spectra, labeled by their spin and parity
as $J^{P}$ along the abscissa, and the real part of the resonance pole
positions along the ordinate, from the Bonn-Gatchina bn-ga and ANL-Osaka
ebac2 analyses of experimental data. For the ANL-Osaka (aka EBAC) analysis
all the states have $3^{*}-4^{*}$ provenance, while Bonn-Gatchina also include
those with $1^{*}-2^{*}$ ratings, according to the legend shown. Note the
tendency of some $N^{*}$’s and $\Delta^{*}$’s to appear in parity pairs as
their mass increases above 1800 MeV bn-ga-doublet .
Baryons have a special place in the firmament of quark bound states. First it
was their multiplet structure that led to the proposal of the quark model, and
the discovery of the triply strange $\Omega^{-}$ that confirmed this was on
the right lines. The inclusion of quarks in the dynamics of QCD made baryons
special too. They most obviously reflect the non-Abelian nature of the theory,
since a minimum of three quarks each with different colour charges is required
to build a colour neutral hadron with half-integer spin. To learn about the
spectrum of excited baryons we first fired pion beams at proton targets and
measured the cross-section and polarization for the production of $\pi N$ and
$\pi\pi N$ final states. Since states in the spectrum of hadrons have definite
quantum numbers, to find these the $\pi N$ cross-sections and asymmetries are
decomposed into underlying amplitudes with definite spin. However, these only
provided a glimpse of a limited part of the spectrum. A more complete picture
is provided by detecting strange, as well as non-strange, final states (like
$K\Lambda$, $K\Sigma$, etc) sarantsev ; bn-ga2 ; menu and by more recent
studies with photon beams, in different polarization states scattering on
polarized targets bn-ga2 . This has been enabled by a wonderful set of
experiments at ELSA@Bonn, MAMI@Mainz and CLAS6@JLab. The outcome of two
Amplitude Analyses of all these data is shown in Fig. 1. One is a
sophisticated, but traditional Amplitude Analysis, by the Bonn-Gatchina team
bn-ga , and the other which attempts to learn about the underlying dynamics
directly is that by the ANL-Osaka group ebac2 . This uses the Sato-Lee
effective Lagrangian sato-lee as its basis, and relies on computing the
contribution of many Feynman diagrams as the energy increases. While these
approaches satisfy unitarity for two-body channels, three- and higher-body
interactions are more complicated. Consequently, it is the more flexible Bonn-
Gatchina analysis that can fit the $\pi\pi N$ final states and determine the
spectrum to higher masses. The results in Fig. 1 show that the $N^{*}$’s and
$\Delta^{*}$’s from these two analyses have much in common, but there are some
key differences that need to be resolved. The measurement of double
polarization asymmetries, like the so called $G$-function with linearly
polarized photons on a longitudinally polarized target open a unique window on
to the higher partial waves krusche ; bn-ga3 . They show that the need for
important spin-3 components above 1.55 GeV, seen in the top right corner of
Fig. 1. Many of these new results from Bonn and Mainz are being presented at
this conference krusche ; thiel . More data are to come. Beam and target
technology are providing detailed access to this spectrum up to 2.2 GeV.
The aim is not just to assemble hadron states like a stamp collection, but to
determine their masses and widths (given by their poles in the complex energy
plane), and their couplings to all the channels in which they appear (given by
the appropriate residues of these poles), and from these to learn about the
composition of these states. By virtue of the uncertainty principle, the
proton and neutron inevitably have a meson cloud, which has detectable effects
— much like the Lamb shift in QED. However, for excited states this cloud is
even more tangible. It is real. $\pi N$ and $\pi\pi N$ configurations are an
essential part of the Fock space of the $N^{*}(1440)$ and all the many excited
states shown in Fig. 1. It is through these components that each decays. The
degrees of freedom are not just three quarks, but all the decay channels too.
They are not just objects with a $qqq$-core of the constituent quark model
capstick , but they must have additional ${\overline{q}}q$, or even
${\overline{q}}gq$ components. The aim is to determine this structure for each
of the lower lying excited states, and then to understand from this the
detailed workings of strong coupling QCD. Studying in electroproduction
experiments how these compositions change as the virtuality of the probing
photon increases, may yet confirm these insights mokeev .
In the constituent quark model, decays were often treated as some
“perturbative” addition, as in the ${}^{3}P_{0}$ scheme barnes . However, more
recently, it has been appreciated that decays actually change the dynamics of
the spectrum pennington-wilson ; santopinto . This complexity can bring new
states into view, for which the opening of decay channels are essential, while
making others merge into the continuum as they no longer bind but just fall
apart. Such hadronic components are there in modern lattice calculations too
edwards1 . However at present with $up$ and $down$ quarks having 10 times
their physical mass, and so pions of $400$ MeV, only to a limited extent. As
computations advance towards pions of 140 MeV, these hadronic components are
likely to shift the masses of the resulting baryons and change their couplings
mrp-LEAP , hopefully, approaching those that appear in experiment.
That decay channels are essential to hadron states has long been suspected for
mesons: the enigmatic scalars mrp-FSU $f_{0}(980)$ and $a_{0}(980)$ clearly
have ${\overline{K}}K$ channels at the heart of their existence. The discovery
of the new $X,\,Y,\,Z$ states in the heavy quark sectors have highlighted this
too. The $X(3872)$ is closely associated to the $D{\overline{D}}^{*}$ channel.
The charged $Z_{c}(4430)$ clearly must be more complex than simply
${\overline{c}}c$. New states with hidden strangeness have been found too,
like the $Y(2175)$ in the $\phi f_{0}(980)$ channel. These all have the
feature that $S$-wave coupling to nearby hadron channels brings binding.
Indeed, it is in the meson sector where some of the previously unconfirmed QCD
configurations of colour singlets are to be found: glueballs and hybrids. A
world of pure glue, while theoretically most interesting, doesn’t exist in the
real world. Light glueball configurations inevitably mix with channels in
which ${\overline{q}}q$ states appear through their common $\pi\pi$,
${\overline{K}}K$, $\eta\eta$, etc., decay channels. However, hybrids, states
in which glue contributes not just binding but to their quantum numbers, can
arise with $J^{PC}$’s not possible for simpler ${\overline{q}}q$ systems. Such
states like $1^{-+}$ are called “exotic”, but they are only exotic in the
quark model, not in QCD, where their appearance is to be expected.
Figure 2: Lattice QCD results for the meson spectrum labeled by their spin and
parity as $J^{PC}$ along the abscissa, and their masses along the ordinate,
from the Hadron Spectrum Collaboration dudek with a pion mass of 400 MeV,
showing their flavour structure. The calculations are for states constructed
from operators with $q$, $\overline{q}$ and $g$ configurations. The results
are grouped into those with natural and unnatural parity allowed by simple
${\overline{q}}q$ states. Those labeled exotic do not appear in the quark
model, and in the lattice calculation are dominated by ${\overline{q}}gq$
components.
The latest lattice calculations dudek , shown in Fig. 2, predict multiplets of
such states around 2 GeV. Since these computations are in a world with 400 MeV
pions, they are expected to be shifted in the real world, just as we discussed
for baryons. Nevertheless, the calculations are robust enough for a whole new
program of exploration to be the focus of the Hall D program at Jefferson Lab
whitepaper . There polarized photons scattering on a nucleon target will be
studied in many final states: $\pi\pi N$, $3\pi N$, $\eta N$, $4\pi N$, $5\pi
N$, $\eta^{\prime}N$, etc, with a detector designed to have a close to perfect
acceptance. To this will be added kaon identification. With millions of
events, the aim is to perform precision partial wave analyses. Hybrids, and
other new states involving light flavours of quark, are unlikely to be narrow,
and appear as simple “bumps”, but only by performing Amplitude Analyses of
many channels simultaneously will poles in the complex energy plane be
definitively revealed. This requires close cooperation between theorists and
experimentalists. To facilitate this, the JLab Physics Analysis Center has
been set up, led by Adam Szczepaniak.
## 3 JLab Physics Analysis Center
The states that first populated the Particle Data Tables were those that
naturally were those that lived longest and so appearing as narrow(ish) peaks
in the appropriate integrated cross-section: the $\rho$, $\omega$, $\phi$,
$N^{*}(1520)$, $\cdots$. This gave the impression that determining the hadron
spectrum was just a matter of bump-hunting. However, it soon became clear that
many states were highly inelastic, appearing in several channels, often not
creating more than a wiggle in any one cross-section. Nevertheless, these
correspond to poles in the complex energy plane, which is the true signature
of a state in the spectrum of states. Others, like the $f_{0}(980)$, couple
strongly to a threshold that is just about to open above their notional mass.
Such a state appears as a peak in some reactions and as a dip in others.
Nevertheless, these too are described by a pole in the complex energy plane,
regardless of the way they appear in experiment on the real energy axis. All
this makes it clear that one must have the right framework in which to
describe the amplitudes in which resonances appear and the right tools to
continue the amplitudes into the complex energy plane. This framework is
provided by Reaction Theory. This requires that the Scattering (or $S$-)
matrix that describes each reaction satisfies the consequences of causality,
relativity and the conservation of probability. These are the basics of no
particular theory, but every theory. These require that the $S$-matrix
elements possess the correct analyticity, crossing and unitarity properties.
Amplitudes are complex functions. Experiment can sometimes determine both
their modulus and phase, or at least their relative phase. To connect these
from one energy to another demands the use of dispersion relations, or other
analytic mapping techniques. Our experimental colleagues, who conceive and
build the detectors and understand their acceptances, write the data
acquisition software, connect up the electronics and a thousand myriad things
to turn pulses into cross-sections, need the help of theorists to provide the
translation of these results into the physics of hadrons. Theorists are an
integral part of the analysis team, increasingly embedded within the
collaborations. The aim here is not to prove some particular favourite model,
whether based on constituent quarks, or some modelling of interactions in the
bound state equations, or even to validate a lattice calculation, but rather
to input essential truths of scattering theory. Testing models has a role, but
that comes later, once definitive results have been obtained from experiment.
Figure 3: Two of the reaction mechanisms at work in $\gamma N\to 3\pi N$. (a)
represents Regge exchange (R) creating intermediate states that decay to
$\,\rho\pi$ or $\sigma\pi$, that might include $1^{-+}$ quantum numbers. (b)
Deck production of the same final state. These mechanisms interfere. The
consequences of this have to be understood across the kinematic range of the
reaction to determine the production mode of any partial wave.
The mission of the JLab Physics Analysis Center is to network with appropriate
theorists and experimentalists in different collaborations to achieve this
goal, whether with CLAS12 or GlueX@JLab, COMPASS@CERN, BESIII@BEPC or
PANDA@Fair. The purpose of this networking for spectroscopy is to share the
$S$-matrix technology that is required and to make this a practical tool. To
this end, various working groups have been set up for the first year to study
reaction mechanisms and final state interactions, in particular. As prompted
by the discussions of the $a_{1}$ years ago, multi-hadron production is far
from simple. To establish that the $a_{1}$ was indeed a state in the spectrum
required a detailed understanding of how the different mechanisms for three
pion production contributed, Fig. 3; whether the behaviour of the relevant
$J^{PC}\,=\,1^{++}$ $3\pi$ partial wave requires a resonance like that
generated by the graph in Fig. 3a, or can it be wholly understood in terms of
the Deck effect of Fig. 3b. Multi-body final state interactions play a key
role in searching for new states that may point to glue as an essential
contributor to their $J^{PC}$ quantum numbers. Heavy flavour factories, like
BaBar and Belle are rich sources of information about such decays. This has to
be combined with practical methods for implementing two and three-body
unitarity to be used in Amplitude Analyses of the precision data to come.
COMPASS is confronting all these issues compass1 and is a key experiment from
which we hope to learn. To meet these demands the JLab Physics Analysis Center
is not just working with experimentalists but establishing close connections
with other theory consortia like the NABIS group nabis and the Haspect
project haspect . To make the most of the precision data that modern
experiments deliver, with much more to come, we must have tools of comparable
precision to extract the detailed physics required to understand how the
dynamics of QCD, with its properties of colour confinement and chiral symmetry
breaking, really works. That is the challenge.
It is a pleasure to thank the organisers, especially Annalisa D’Angelo, for
the invitation to give this talk in such an auspicious venue. Discussions with
Reinhard Beck on the latest experimental results were much appreciated. This
paper has been authored by Jefferson Science Associates, LLC under U.S. DOE
Contract No. DE-AC05-06OR23177.
## References
* (1) A. V. Anisovich, R. Beck, E. Klempt, V. A. Nikonov, A. V. Sarantsev and U. Thoma, Eur. Phys. J. A 48, 15 (2012).
* (2) H. Kamano, S. X. Nakamura, T. -S. H. Lee and T. Sato, Phys. Rev. C 88, 035209 (2013).
* (3) A. V. Anisovich, E. Klempt, V. A. Nikonov, A. V. Sarantsev, H. Schmieden and U. Thoma, Phys. Lett. B 711, 162 (2012).
* (4) A. V. Sarantsev, Acta Phys. Polon. Supp. 3, 891 (2010); V. D. Burkert, EPJ Web Conf. 37 (2012) 01017.
* (5) A. V. Anisovich, R. Beck, E. Klempt, V. A. Nikonov, A. V. Sarantsev and U. Thoma, Eur. Phys. J. A 48, 88 (2012).
* (6) R. Schumacher, Strange Photoproduction (Excited States), these Proceedings.
* (7) A. Matsuyama, T. Sato and T. S. Lee, Phys. Rept. 439, 193 (2007).
* (8) B. Krusche, Latest results from meson photoproduction off neutrons, these proceedings.
* (9) A. Thiel, A. V. Anisovich, D. Bayadilov, B. Bantes, R. Beck, Y. Beloglazov, M. Bichow and S. Bose et al., Phys. Rev. Lett. 109, 102001 (2012).
* (10) e.g., A. Thiel, The Double Polarization Program of Crystal Barrel at ELSA, these Proceedings.
* (11) S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986); S. Capstick and W. Roberts, Phys. Rev. D 49, 4570 (1994).
* (12) I. G. Aznauryan and V. D. Burkert, Prog. Part. Nucl. Phys. 67, 1 (2012); P. L. Cole, V. D. Burkert, R. W. Gothe, V. I. Mokeev and CLAS Collaboration, Nucl. Phys. Proc. Suppl. 233, 247 (2012); V. I. Mokeev and I. G. Aznauryan, arXiv:1310.1101 [nucl-ex].
* (13) E. S. Ackleh, T. Barnes and E. S. Swanson, Phys. Rev D 54, 6811 (1996).
* (14) M. R. Pennington and D. J. Wilson, Phys. Rev. D 76, 077502 (2007).
* (15) J. Ferretti, G. Galata and E. Santopinto, arXiV:1302.6857 [hep-ph].
* (16) R. G. Edwards, J. J. Dudek, D. G. Richards and S. J. Wallace, Phys. Rev. D 84, 074508 (2011).
* (17) M. R. Pennington, Proceedings of LEAP 2013, Uppsala, Sweden, June 2013.
* (18) M. R. Pennington, AIP Conf. Proc. 1257, 27 (2010) [arXiv:1003.2549 [hep-ph]].
* (19) J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards and C. E. Thomas, Phys. Rev. D 82, 034508 (2010); J. J. Dudek, R. G. Edwards, B. Joo, M. J. Peardon, D. G. Richards and C. E. Thomas, Phys. Rev. D 83, 111502 (2011).
* (20) J. J. Dudek, R. Ent, R. Essig, K. S. Kumar, C. Meyer, R. D. McKeown, Z. E. Meziani and G. A. Miller, M. R. Pennington, D. G. Richards, L. Weinstein, G. Young and S. Brown, Eur. Phys. J. A 48, 187 (2012).
* (21) F. Haas [COMPASS], AIP Conf. Proc. 1257, 293 (2010); F. Nerling [COMPASS], EPJ Web Conf. 37, 01016 and 09025 (2012); T. Schlüter et al. [COMPASS], PoS QNP2012, 074 (2012); S. Paul, Meson Spectroscopy in the $3\pi$ Final States using COMPASS Data, these Proceedings.
* (22) I. Bigi, I. Bediaga, et al, [Les NABIS Collab.]; B. Kubis, F. Niecknig and S. P. Schneider, Nucl. Phys. Proc. Suppl. 225-227, 75 (2012).
* (23) M. Battaglieri, [HaSpect] https:agenda.infn.it/getFile.py/access?contribid=&&resid=0&materialId=6561
|
arxiv-papers
| 2014-02-21T20:50:31 |
2024-09-04T02:49:58.595654
|
{
"license": "Public Domain",
"authors": "M.R. Pennington",
"submitter": "Michael R. Pennington",
"url": "https://arxiv.org/abs/1402.5406"
}
|
1402.5435
|
# Understanding the baryon and meson spectra
M.R. Pennington
###### Abstract
A brief overview is given of what we know of the baryon and meson spectra,
with a focus on what are the key internal degrees of freedom and how these
relate to strong coupling QCD. The challenges, experimental, theoretical and
phenomenological, for the future are outlined, with particular reference to a
program at Jefferson Lab to extract hadronic states in which glue
unambiguously contributes to their quantum numbers.
###### Keywords:
Baryons, mesons, spectrum, decays, coupled channels, QCD
###### :
14.20.Gk, 13.30.Eg, 14.40.Be, 14.40.Df, 12.38.-t, 11.55.-m, 11.80.Et
## 1 Revealing the workings of strong QCD
With eyes fixed on the wonders of the LHC at the TeV scale, one may question
why is the physics of the strong interaction at 1 GeV of interest any longer.
Is this not all ancient history? However, it is at the GeV scale that we
already know the scalar sector that gives mass to most of the visible
universe. A GeV is the energy scale at which we have discovered half the
particles of a possible supersymmetric world. New strong interactions may
await discovery, but QCD is the only strong interaction we already know. We
should study it in as much detail as we can. After all it determines the
properties of the nuclear matter of which we are made. It is the strength of
this interaction that brings a complexity of phenomena that outshines those of
perturbative electroweak physics. The richness of the tapestry of strong QCD
is to be seen in the hadrons, their properties and structure, that it creates.
The paradigm for what can be learnt from spectroscopy is provided by atoms.
Even if we did not have enough energy to separate electrons from the nucleus,
we would know by studying the spectrum that though atoms are electrically
neutral, they behave as though they are made of electrically charged objects
held together by an electromagnetic force governed by the rules of Quantum
Electrodynamics. In a similar way color neutral hadrons are built of
constituents carrying color charge, bound by the rules of QCD. But what are
these rules? While asymptotic freedom provides a well exploited simplification
for hard scattering processes, it is the fact that over a distance of a fermi
the interaction is strong that makes QCD so challenging and why we look to
experiment for guidance on how it really works. Strong coupling confines
quarks and breaks chiral symmetry, and so defines the world of light hadrons.
Quark confinement is reflected in the spectrum and properties of hadrons, and
we can learn from what experiment teaches about these. We ask: what are the
internal degrees of freedom of hadron states? The quark model, that was of
course the seed from which the idea of QCD first germinated, suggests these
are constituent quarks (and anti-quarks). But is that all there are? What is
the role of glue? Do gluons just stick the quarks together, and nothing more?
It is in the spectrum of charmonium that we have a working template from which
to judge complexity most readily. Below ${\overline{D}}D$ threshold it all
appears simple. We have the tightly bound systems of $J/\psi$,
$\psi^{\prime}$, $\eta_{c}\,\cdots$, as given by non-relativistic potential
models. Above the open charm threshold, we once thought the (almost) stable
charmonia are replaced by states with 1-50 MeV widths decaying to
${\overline{D}}D$, ${\overline{D}}D^{*}$, ${\overline{D}^{*}}D^{*}$,
${\overline{D}_{s}}D_{s}^{*},\cdots$, as their mass increases. What we find is
that the states predicted by potential models are shifted by tens of MeV
themselves: the decays affect their dynamics barnes-swanson ; wilson .
Hadronic decay channels are an essential degree of freedom. These not only
shift predominantly ${\overline{c}}c$ states, but generate states that would
not have existed without these hadron channels. The first discovery of a state
of this type is the $X(3872)$, whose very existence is tied to the dynamics of
the ${\overline{D}}^{0}D^{*0}$ channel tornqvist . More new states, a string
of $X,\,Y$ and $Z$ states perhaps only exist because of their hadronic decays,
sometimes these channels binding in molecular (or multiquark) configurations.
As dynamically coupled channel models have long suggested vanbev-lutz ,
hadrons and their decays are intimately related. Only for ground states may
one think of them as having minimal quark configurations.
## 2 What are the degrees of freedom in each hadron?
Baryons have a special place in the study of hadrons, as their structure is
most obviously related to the color degree of freedom. While a color singlet
quark-antiquark system is basically the same however many colors there are,
the minimum number of quarks in a baryon is intimately tied to the number of
colors. If $N_{c}$ were some other number than 3, the world would be quite
different. Recognizing the flavor pattern of the ground state baryons was the
key step in the development of the quark model. Consequently, this model with
three independent quark degrees of freedom isgur ; capstick has naturally
served as the paradigm for what we expect the spectrum of excited baryons,
both nucleons and $\Delta$’s, to look like too. While experiment has long
confirmed the lower lying states, many of the heavier ones seemed to be
missing above 1.6 GeV.
If baryons were diquark–quark systems, as noted more than 40 years ago
lichtenberg , the number of states would be restricted and in fact be very
like that observed uptil a year or so ago. However most of the early evidence
on the baryon spectrum was accumulated from $\pi N$ scattering, and decays
into the same channel. Perhaps the missing states are just dark in these
channels, and “shine” most in $\pi\pi N$ and $KY$. Consequently, the
experimental program has concentrated more recently on these channels, which
are an increasing part of the $\pi N$ total cross-section as the energy goes
up.
Figure 1: The imaginary parts of the $I=1/2$ $\pi N\to\pi N$ partial wave
amplitudes, labeled by the quantum numbers $L_{2I\,2J}\,=\,D_{13}$ and
$P_{11}$ from the SAID analysis said as functions of the $\pi N$ c.m. energy,
$E$. The arrows mark the real part of the resonance pole positions.
But first how do we identify states in the spectrum of hadrons? Since states
have definite quantum numbers, spin, parity, isospin etc, we have to decompose
the observed data, integrated and differential cross-sections, into partial
waves that specify these quantum numbers. To do this completely for processes
with spin requires measurements with polarized beams and polarized targets.
Having separated the partial waves, one finds it is only for the lowest mass
state with a given quantum numbers that the partial wave looks anything like a
simple Breit-Wigner resonance, see, as an example, the $D_{13}$ wave in Fig. 1
said . Higher mass states are much less obvious. For instance in the $P_{11}$
wave of Fig. 1, while the $N^{*}(1440)$ (the Roper) appears as a bump in the
imaginary part (and modulus), the higher mass $N^{*}(1710)$ can barely be seen
in the same $\pi N\to\pi N$ amplitude. It is highly inelastic. A state in the
spectrum is then only identifiable by its pole in the complex energy plane on
some nearby unphysical sheet. It is the poles that are the universal outcome
of any modern amplitude analysis, as recognized by the PDG pdg2012 .
By now a vast amount of data has been accumulated, and is being accumulated,
on a wide range of baryonic processes, most recently initiated by real and
virtual photon beams. The presence of many decay channels and the large widths
to each of these demands coupled-channel amplitude analyses be performed. This
requires a rich supply of input data if the richness of the spectrum is to be
exposed. Thus from JLab jlab-photo ; jlab-data and from ELSA elsa-photo , we
have thousands of data on $\gamma p\to\pi^{0}p$ and $KY$, differential cross-
sections and polarizations. These feature prominently in the latest analyses.
The most ambitious analysis is that by the Excited Baryon Analysis Center
(EBAC) team led by Harry Lee ebac . Not only does this fit a very wide range
of data on baryonic channels, but it does this in terms of an effective field
theory of hadronic interactions developed by Sato and Lee ebac . Their
calculational procedure ensures unitarity is fulfilled, and their Lagrangian
provides a framework in which to consider the nature and structure of each
resonant state, and its “core” revealed. “Bare” or “core” states are those
with no decays ebacpoles . While for heavy quark systems one might reasonably
define such bare states as those that arise in a potential model for
charmonium or bottomonium, for light quark systems the model template is not
so obvious. Here it is the Sato-Lee Lagrangian. How are such “bare” states
connected to QCD? In fact are these connected to QCD at all? Perhaps there is
no limit of QCD in which the hadronic decays of bound states can be turned
off. Notwithstanding such interpretations, the results for the $N^{*}$ and
$\Delta^{*}$ spectra of EBAC up to 1800 MeV have now been finalized ebac2 ,
and are shown in Fig. 2. Their analysis of the detailed nature of these states
is to come.
Figure 2: $N^{*}$ and $\Delta^{*}$ spectra, labeled by their spin and parity
as $J^{P}$ along the abscissa, and the real part of the resonance pole
positions along the ordinate, from the EBAC ebac2 and Bonn-Gatchina bn-ga
analyses. For the EBAC analysis all the states have $3^{*}-4^{*}$ provenance,
while Bonn-Gatchina also include those with $1^{*}-2^{*}$ ratings, according
to the legend shown.
A more computationally flexible amplitude analysis program has been carried
out by the Bonn-Gatchina team sarantsev . They fit an even more extensive
range of multi-hadron final states and so are able to present results up to a
higher energy bn-ga . Their states up to 2.1 GeV are shown in Fig. 2 too, with
their assignment of their 1-4 star confidence pdg2012 . The EBAC and Bonn-
Gatchina spectra and couplings are very similar, but not identical. The larger
mass range fitted includes the JLab data on channels such as $\gamma p\to KY$
jlab-data and this has enabled a number of the “dark” or “missing” baryons at
last to be revealed, like the $1/2^{+}$ $N^{*}(1880)$ and the $3/2^{+}$
$N^{*}(1900)$ bn-ga .
Experiments on $\gamma p\to K^{+}\Lambda$ with polarized beam and polarized
target, together with the spin information from the weak decay $\Lambda\to\pi
N$, allow more observables to be measured than the minimum needed to determine
all the independent amplitudes (up to an overall phase) tiator ; sandorfi .
These over-complete experiments hold out the prospect of checking that the
partial wave solution that results in the spectrum shown in Fig. 2 is indeed
the correct one. The development of polarized targets, such as FROST and HDice
at JLab jlab-targets , have allowed neutron scattering data to be determined
too. These results are eagerly awaited as they are an essential component in
securing the partial wave solution and its isospin decomposition.
Fig. 2 only shows the spectrum with zero strangeness. Within a simple quark
model picture (which we have stressed may not be a realistic guide for highly
excited states with their complex multi-hadron decays), baryons form flavor
multiplets. Consequently, searching for baryons in the
$\Sigma^{*},\,\Lambda^{*},\,\Xi^{*}$ families is a key part of the future
experimental program. Such states have fewer (or better separated) hadronic
decay channels and so may be narrower and more easily identifiable.
Figure 3: Cartoon of the possible Fock components (a-d) of some excited
baryon, for instance the $N^{*}(1440)$. It almost certainly has components (a)
and (b), but the relative amounts of (a-d) awaits to be determined for the
Roper, or any other excited, baryon.
Such results will teach us the Fock space decomposition of each resonant
state. All but the ground states are inevitably complicated. As an example,
the Roper, the $N^{*}(1440)$, cannot just be a three quark state, as depicted
in Fig. 3a. It must have an explicit $\pi N$ component in its Fock space, Fig.
3b, since it is through this component (amongst others) that it decays. Its
Fock space might then be thought to include a nucleon and a pion (or even a
multi-pion) cloud (Fig. 3c), but might also contain a pentaquark
configuration, like that in Fig. 3d. Dynamical models, and eventually QCD,
will tell us what are the proportion of these components for each physical
state. Such compositions are also probed experimentally in photo-transition
processes. Once the data on these from the final running of CEBAF at 6 GeV are
analyzed appropriately in terms of pole quantities mokeev we may have a
better idea.
How is the spectrum of Fig. 2 related to QCD? The lattice provides the most
consistent theoretical connection. The four-dimensional world is modeled as a
discrete space-time to make the problem computationally feasible. The baryon
spectrum computed most recently edwards reveals a pattern very like that of
the quark model: certainly not that of a pointlike diquark–quark system. The
“missing” states are there. However, one essential ingredient is clearly
missing in such calculations. While continuum hadronic effects are included,
they are not yet those of the physical world. Though great computational
strides have been made, the up and down quark masses are 8-15 times their
physical value and so the pion mass is still 3 or 4 times too heavy.
Consequently, the Fock space decomposition of the excited baryons is not
physical. In terms of the pictures in Fig. 3, components (b) and (c) are much
much smaller than those of the real world, and so it’s perhaps not surprising
that the quark model-like Fig. 3a dominates. However, calculational progress
towards a 140 MeV pion mass continues.
A continuum approach to QCD with physical mass quarks is provided by the
solution of the Schwinger-Dyson/Bethe-Salpeter (SD/BS) system of equations
sdbsreviews . There has been steady progress over decades in solving this
complex system self-consistently. However, speedier computations are made
possible by modeling the gluon by a simple contact interaction and presuming
that baryons are bound states of a quark with an extended (not pointlike)
diquark. Detailed calculations of the $N^{*}$ spectrum have then been made cdr
. These include no decays and so no hadronic components. Amusingly there is a
“bare” $P_{11}$ state that can be identified with the EBAC “core” state
ebacpoles . The physical Roper is $\sim 500$ MeV lighter. As with the more
ambitious SD/BS approach treating baryons as full three quark systems eichmann
, these calculations must include decays if a meaningful comparison of excited
states with experiment is to be achieved.
## 3 Mesons: is this where glue is to be found?
We now turn to mesons, first in the quark model. The ${\overline{q}}q$ pair
can have spin, $S_{qq}$, equal to 0 or 1. When combined with units of orbital
angular momentum $L_{qq}$, they make a series of flavor multiplets, with each
unit of $L_{qq}$ adding $\sim 700$ MeV of mass. The ground states with
$L_{qq}=0$ have $J^{PC}\,=\,0^{-+}$ and $1^{--}$ quantum numbers. While the
light pseudoscalars, being the Goldstone bosons of chiral symmetry breaking,
have atypical dynamics, the vector multiplet gives the ideally mixed paradigm,
replicated by the mesons with higher $J=L_{qq}+1$.
The scalar ${\overline{q}}q$ multiplet is part of the $L_{qq}=S_{qq}=1$
family. There are at least 19 scalars below 2 GeV pdg2012 , far more than can
fit into one nonet mrp-scalars . It was Jaffe in his seminal work on
multiquark states jaffe that recognized that the scalars below 1 GeV might be
tetraquark states, while the more conventional ${\overline{q}}q$ $\,0^{++}$
mesons would be up close to their $2^{++}$ companions around 1.3 GeV. Such an
interpretation naturally explains how the isosinglet $f_{0}(980)$ and
isotriplet $a_{0}(980)$ can be degenerate in mass and both couple strongly to
${\overline{K}}K$: each is a $\overline{sn}sn$ state, with $n=u,d$. However,
recent studies menu2009 ; wilson2 , making use of the fine energy binning
possible with BaBar data marco , have shown that the $f_{0}(980)$ is dominated
by long range ${\overline{K}}K$ components, rather than a tighter bound
tetraquark configuration. Similarly, the $\sigma$ and the $\kappa$ seem to be
dominated by $\pi\pi$ and $\pi K$ components: their masses depending far more
on their couplings to these channels than related to any simple quark mixing
scheme. Indeed long ago, the dynamical calculation by van Beveren and Rupp
vanbev highlighted how scalar ${\overline{q}}q$ seeds up at 1.3 GeV can give
rise to two multiplets of hadrons, when their strong couplings to di-meson
channels are included: an explicit example of dynamical coupled channel
effects.
Figure 4: The isovector meson spectrum from the lattice calculations of Dudek
et al. dudek with $m_{\pi}\,=\,396$ MeV, arranged according to their $J^{PC}$
quantum numbers. Those found with ${\overline{q}}q$ operators are shown as
black blocks, the size of which denote the statistical uncertainties. States
from ${\overline{q}}qg$ operators are shown as grey blocks. Some of these have
spin-exotic quantum numbers. These are shown on the right.
Ever since the QCD Lagrangian was written down, it was recognized that there
may exist hadrons with more complicated configurations than those of the
simple quark model: states in which gluons pay a role in determining their
quantum numbers. At first, calculations and experimental searches were for
states made purely of glue. While many sightings were claimed, they never
stood up to challenge mrp-lund . Indeed, it was quickly realized that any
meson made of glue (viz. glueballs) must couple to quarks in order to decay
into pions and kaons, and so mixing with these quark configurations is
inevitable and could easily be large. Thus in the scalar sector discussed
above, several states between 1.3 and 1.8 GeV might have sizeable admixtures
of glue, viz. $gg$, without any being predominantly a glueball. That is a
detail of dynamics that we do not yet understand, except in unrealistically
simple mixing schemes. Consequently, attention has turned to other meson
quantum numbers than those of the vacuum.
Lattice computations of the ${\overline{q}}q$ spectrum are approaching a
maturity that includes all the states we know of from experiment, as shown in
first two columns of Fig. 4. There is displayed the results of the present
state-of-the art computations for isovector mesons from Dudek et al. dudek .
By using an inventive and ingenious set of operators, they have also been able
to compute the spectrum of states that are ${\overline{q}}qg$. The grey blocks
in Fig. 4 denote these hybrid states. On the left are seen hybrids with
conventional quantum numbers, where exciting glue is found to require an extra
$\sim 800$ MeV of mass. In addition, states with spin-exotic quantum numbers
appear on the right of Fig. 4. The lightest is that with $J^{PC}=1^{-+}$, as
long had been expected. At a pion mass of 400 MeV, this hybrid is found to be
up around 2 GeV. Of course, a real mass pion is expected to affect this: in
general making it lighter and broader.
Possible states with $1^{-+}$ quantum numbers were claimed in a series of
searches starting more than 35 years ago with GAMS gams , then (as shown in
Fig. 5) BNL-E852 chung and VES ves ten years later. All find enhancements in
the relevant partial wave. However, these signals only constitute a few
percent of the integrated cross-section, and inevitably have $1^{-+}$ waves
with sizeable uncertainties dzierba . Consequently, these experiments were
never able to show that the underlying partial waves were resonant with a pole
in the complex energy plane. The phase variation observed was always rather
weak.
Figure 5: On the left is the $J^{PC}\,=\,1^{-+}$ signal from BNL-E852 data
chung on $\pi N\to(3\pi)N$. The grey histogram is the calculated “leakage”
into this channel from other partial waves. The enhancement at $\sim 1.4$ GeV
is thereby explained dzierba , but leaves a clean $\sim 1.6$ GeV enhancement.
The graph on the right displays the VES results ves on $\,\eta\pi\,$ and
$\,\eta^{\prime}\pi\,$ production as a function of the di-meson mass in
$\,\pi^{-}Be\,$ collsions at 28 GeV$/c$, again with enhancements at 1.4 and
1.6 GeV, respectively. Whether any of these is resonant is unclear.
A much more ambitious program is that of COMPASS@CERN. This studies multi-
hadron production at small momentum transfers with a 192 GeV pion beam on
nucleon and nuclear targets, in particular studying $\pi\eta^{\prime}$ and
$3\pi$ final states. The $\pi\eta^{\prime}$ data show a significant broad
enhancement in $1^{-+}$ waves around 1600 MeV, but with little relative phase
variation compared with the reference $2^{++}$ wave with its pronounced
(conventional ${\overline{q}}q$) $\,a_{2}(1320)$ signal compass1 . In the
$3\pi$ channel, the first runs in 2004 showed a very crude enhancement in
$1^{-+}$ waves, which was fitted to a Breit-Wigner form with doubtful
significance compass0 . However, now COMPASS are studying 96 million events in
the $3\pi$ channel. With these statistics, one has to have a good
understanding of the reaction mechanisms involved: simple Pomeron exchange
with possibly important Deck effect backgrounds. At last report the data
require at least 52 partial waves to obtain a stable set. Only the dominant
$2^{++}$ and $1^{+-}$ waves have been shown in talks. This meeting will
elaborate more on this compass2 . However, further work is needed to establish
that there really is a $1^{-+}$ hybrid to add to the spectrum of physical
hadrons.
A complementary effort is underway at Jefferson Lab with the instalation of
magnets to increase the CEBAF energy to 12 GeV, a photon beam line and new
detectors. A prime motivation for this upgrade is the search for hybrid mesons
in all their quantum numbers, $J^{PC}$ and flavor: not just $1^{-+}$, but the
$0^{+-}$, $2^{+-}$, etc., expected at higher mass (Fig. 4). GlueX is the new
detector dedicated to studying multi-hadron final states created by an 11 GeV
polarized photon beam on a proton target gluex . This is due to start taking
data in 2016. Statistics comparable to COMPASS are expected, i.e. $10^{8}$
events. With wonderful angular coverage, this should allow small partial waves
to be disentangled. Complementary (and occasionally competing) data on the low
multiplicity final states will be taken by the CLAS12 detector at JLab too.
The task of extracting small signals with certainty is a real challenge to
experiment, phenomenology and theory. One most go beyond the simple isobar
picture that was good enough, when one had even $10^{4}$ events. However, in
the era of precision data one needs precision analyses too. This demands
detailed knowledge of the reaction mechanisms involved, and importantly all
the contributing final state interactions of $\pi$’s, $K$’s and $N$’s to be
well-represented in terms of amplitudes that respect all the key properties of
scattering theory. This requires a pooling of world expertise on partial wave
analyses and $S$-matrix technology to ensure multichannel unitarity is
fulfilled ASI . We have to learn from the experience of EBAC, Bonn-Gatchina,
COMPASS and others, working with all the relevant analysis and experimental
groups in the world. This will not just underpin the effort at JLab, but the
same technology is required for comprehensive analyses of BESIII, LHCb and
PANDA data. Steps are under way to bring this together. It is only by such
collective efforts that we can be sure that signals of hybrids at the few
percent level can be reliably extracted, and the poles of the $S$-matrix
determined. It is not enough to confirm some putative $\pi_{1}(1600)$ signal
(suggested by VES and BNL-E852), we must find the whole multiplet structure.
It is only then that we can know that such “exotic” states are really hybrids
of quarks and glue, and not states with additional ${\overline{q}}q$ pairs, or
hadronic molecules. The flavor multiplet structure is the guide bali . An
understanding of the role of glue in QCD is the prize.
Unless some real surprises happen, these experiments are likely to be the last
in light hadron spectroscopy. If we are going to claim a real understanding of
the detailed consequences of confinement, we had better get this right. That
is the challenge for the next 10-15 years.
It is pleasure to thank the CIPANP organizers, particularly Wim van Oers and
Martin Comyn, for inviting me to give this talk. The work was authored in part
by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-
AC05-06OR23177.
## References
* (1) T. Barnes and E. S. Swanson, Phys. Rev. C77, 055206 (2008).
* (2) M. R. Pennington and D. J. Wilson, Phys. Rev. D 76, 077501 (2007) [arXiv:0704.3384 [hep-ph]].
* (3) See, for instance, N. A. Tornqvist, Phys. Lett. B590, 209 (2004).
* (4) See, for instance, E. van Beveren, C. Dullemond and T. A. Rijken, Z. Phys. C19, 275 (1983); M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A755, 29 (2005).
* (5) S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986).
* (6) S. Capstick and W. Roberts, Phys. Rev. D 49, 4570 (1994) [arXiv:nucl-th/9310030].
* (7) D. B. Lichtenberg and L. J. Tassie, Phys. Rev. 155, 1601 (1967); D. B. Lichtenberg, L. J. Tassie and P. J. Keleman, Phys. Rev. 167, 1535 (1968).
* (8) R. A. Arndt, W. J. Briscoe, M. W. Paris, I. I. Strakovsky and R. L. Workman, Chin. Phys. C 33, 1063 (2009) [arXiv:0906.3709 [nucl-th]].
* (9) J. Beringer et al, J. Phys. G 86, 010001 (2012).
* (10) M. Aghasyan et al. [CLAS], Phys. Lett. B704, 397 (2011).
* (11) M. E. McCracken et al. [CLAS], Phys. Rev. C81, 025201 (2010); B. Dey et al. [CLAS], Phys. Rev. C82, 025202 (2010).
* (12) N. Sparks et al. [CBELSA/TAPS], Phys. Rev. C81, 065210 (2010); V. Crede et al. [CBELSA/TAPS], Phys. Rev. C84, 055203 (2011).
* (13) A. Matsuyama, T. Sato and T. S. Lee, Phys. Rept. 439, 193 (2007) [arXiv:nucl-th/0608051].
* (14) H. Kamano, S. X. Nakamura, T. S. Lee and T. Sato [EBAC], Phys. Rev. C 81, 065207 (2010) [arXiv:1001.5083 [nucl-th]].
* (15) H. Kamano and T.-S. H. Lee [EBAC], AIP Conf. Proc. 1432, 74 (2012); H. Kamano [EBAC], AIP Conf. Proc. 1388, 396 (2011) [arXiv:1103.2693 [nucl-th]], arXiv: 1206.3374 [nucl-th].
* (16) A. V. Sarantsev, Acta Phys. Polon. Supp. 3, 891 (2010).
* (17) A. V. Anisovich, R. Beck, E. Klempt, V. A. Nikonov, A. V. Sarantsev and U. Thoma, Eur. Phys. J. A48 15 (2012) [arXiv:1112.4937 [hep-ph]], Eur. Phys. J. A48 88 (2012) [arXiv:1205.2255 [nucl-th]], Phys. Lett. B711, 167 (2012) [arXiv: 1116.6150 [hep-ex]].
* (18) L. Tiator, AIP Conf. Proc. 1432, 162 (2012).
* (19) A. M. Sandorfi, S. Hoblit, H. Kamano and T. S. Lee, J. Phys. G 38, 053001 (2011) [arXiv:1010.4555 [nucl-th]].
* (20) see http://userweb.jlab.org/ keith/Frozen/Frozen.html www.jlab.org/Hall-B/HDIce/talks/g14_Lab_Users_mtg_Jun5_12.pdf
* (21) I. G. Aznauryan et al. [CLAS], Phys. Rev. C80, 055203 (2009); V. I. Mokeev, I. G. Aznauryan and V. D. Burkert, arXiv:1109.1294 [nucl-ex]; I. G. Aznauryan, V. D. Burkert and V. I. Mokeev, AIP Conf. Proc. 1432, 68 (2012), [arXiv:1108.1125 [nucl-ex]].
* (22) R. G. Edwards, J. J. Dudek, D. G. Richards and S. J. Wallace, Phys. Rev. D84, 074508 (2011), AIP Conf. Proc. 1432, 33 (2012).
* (23) A. Bashir, L. Chang, I. C. Cloet, B. El-Bennich, Y.-X. Liu, C. D. Roberts and P. C. Tandy, Commun. Theor. Phys. 58, 79 (2012); P. C. Tandy, AIP Conf.Proc. 1374 (2011) 139-144 [arXiv:1011.5250 [nucl-th]].
* (24) G. Eichmann, I. C. Cloet, R. Alkofer, A. Krassnigg and C. D. Roberts, Phys. Rev. C 79, 012202 (2009); I. C. Cloet, C. D. Roberts and D. J. Wilson, AIP Conf. Proc. 1388, 121 (2011).
* (25) G. Eichmann, R. Alkofer, A. Krasnigg and D. Nicmorus, Phys. Rev. Lett. 104, 201601 (2010); H. Sanchis-Alepaz, G. Eichmann, S. Villalba-Chavez and R. Alkofer, Phys. Rev. D84, 096003 (2011).
* (26) M. R. Pennington, AIP Conf. Proc. 1257, 27 (2010) [arXiv:1003.2549 [hep-ph]].
* (27) R. L. Jaffe, Phys. Rev. D 15, 267 (1977).
* (28) M. R. Pennington, AIP Conf. Proc. 1432, 176 (2012) [arXiv:1109.3690 [nucl-th]].
* (29) M. R. Pennington and D. J. Wilson, in preparation.
* (30) B. Aubert et al. [BaBar], Phys. Rev. D79, 032003 (2009); P. del Amo Sanchez et al. [BaBar], Phys. Rev. D83, 052001 (2011).
* (31) E. van Beveren, T. A. Rijken, K. Metzger, C. Dullemond, G. Rupp and J. E. Ribeiro, Z. Phys. C 30, 615-620 (1986).
* (32) M. R. Pennington, “Glueballs: the naked truth” Proc. Workshop on Photon Interactions and Photon Structure, Lund, Sweden, Sept. 1998 (ed. G. Jarlskog and T. Sjostrand; pub. Lund, 1999) pp. 313-328.
* (33) J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards and C. E. Thomas, Phys. Rev. D82, 034508 (2010).
* (34) D. Alde D et al [GAMS], Phys. Lett. 205B, 397 (1988).
* (35) S-U. Chung et al. [BNL-E852], Phys. Rev. D60, 092001 (1999) [hep-ex/9902003].
* (36) G. M. Beladidze et al. [VES], Phys. Lett. B313, 276 (1993).
* (37) A. R. Dzierba et al., Phys. Rev. D67, 094015 (2003).
* (38) B. Grube [COMPASS], PoS HQL2010,034 (2011) [arXiv: 1011.6615[hep-ex]]; F. Nerling [COMPASS], PoS EPS-HEP2011, 303 (2011) [arXiv: 1111.0259 [hep-ex]].
* (39) M. G. Alekseev et al. [COMPASS], Phys. Rev. Lett. 104 241803 (2010) [arXiv:1001.4654[hep-ex]].
* (40) F. Haas [COMPASS], these proceedings.
* (41) See: http://www.gluex.org
* (42) See, for instance, lectures at the Jefferson Lab Advanced Study Institute on Techniques for Amplitude Analysis, Williamsburg, June 2012, http://www.jlab.org/conferences/asi2012
* (43) To answer a question from Gunnar Bali at this conference.
|
arxiv-papers
| 2014-02-21T22:14:23 |
2024-09-04T02:49:58.605727
|
{
"license": "Public Domain",
"authors": "M.R. Pennington",
"submitter": "Michael R. Pennington",
"url": "https://arxiv.org/abs/1402.5435"
}
|
1402.5500
|
#
Handbook of Network Analysis
KONECT – the Koblenz Network Collection
Jérôme Kunegis
## 1 Introduction
Everything is a network – whenever we look at the interactions between things,
a network is formed implicitly. In the areas of data mining, machine learning,
information retrieval, etc., networks are modeled as _graphs_. Many, if not
most problem types can be applied to graphs: clustering, classification,
prediction, pattern recognition, and others. Networks arise in almost all
areas of research, commerce and daily life in the form of social networks,
road networks, communication networks, trust networks, hyperlink networks,
chemical interaction networks, neural networks, collaboration networks and
lexical networks. The content of text documents is routinely modeled as
document–word networks, taste as person–item networks and trust as
person–person networks. In recent years, whole database systems have appeared
specializing in storing networks. In fact, a majority of research projects in
the areas of web mining, web science and related areas uses datasets that can
be understood as networks. Unfortunately, results from the literature can
often not be compared easily because they use different datasets. What is
more, different network datasets have slightly different properties, such as
allowing multiple or only single edges between two nodes. In order to provide
a unified view on such network datasets, and to allow the application of
network analysis methods across disciplines, the KONECT project defines a
comprehensive network taxonomy and provides a consistent access to network
datasets. To validate this approach on real-world data from the Web, KONECT
also provides a large number (180+) of network datasets of different types and
different application areas.
KONECT, the Koblenz Network Collection, contains 168 network datasets as of
April 2013. In addition to these datasets, KONECT consists of Matlab code to
generate statistics and plots about them, which are shown on the KONECT
website111konect.uni-koblenz.de. KONECT contains networks of all sizes, from
small classical datasets from the social sciences such as Kenneth Read’s
Highland Tribes network with 16 vertices and 58 edges (HT), to the Twitter
social network with 52 million nodes and 1.9 billion edges (TF). Figure 1
shows a scatter plot of all networks by the number of nodes and the average
degree in the network. Each network in KONECT is represented by a unique two-
or three-character code which we write in a sans-serif font, and is indicated
in parentheses as used previously in this paragraph. The full list of codes is
given online.222konect.uni-koblenz.de/networks
Figure 1: All networks in KONECT arranged by the size (the number of nodes)
and the average number of neighbors of all nodes. Each network is represented
by a two- or three-character code. The color of each code corresponds to the
network category as given in Table 3.
This handbook first describes the different network types covered by KONECT in
Section 2, gives important mathematical definitions in Section 3, lists the
numerical network statistics in Section 4, lists node features in Section 5,
lists the plot types in Section 6, reviews graph characteristic matrices and
their decompositions in Section 7, documents the KONECT Toolbox in Section 8
and describes KONECT’s file formats in Section 9. ††margin: ⟨name⟩ Throughout
the handbook, we will use margin notes to give the internal names of various
parameters.
## 2 Taxonomy of Networks
Datasets in KONECT represent networks, i.e., a set of nodes connected by
links. Networks can be classified by their format
(directed/undirected/bipartite), by their edge weight types and
multiplicities, by the presence of metadata such as timestamps and node
labels, and by the types of objects represented by nodes and links. The full
list of networks is given online.333 konect.uni-koblenz.de/networks
The format of a network is always one of the following. The network formats
are summarized in Table 1.
* •
In undirected networks (U), ††margin: sym edges are undirected. That is,
there is no difference between the edge from $u$ to $v$ and the edge from $v$
to $u$; both are the edge $\\{u,v\\}$. An example of an undirected network is
the social network of Facebook (Ow), in which there is no difference between
the statements “A is a friend of B” and “B is a friend of A.”
* •
In a directed network (D), ††margin: asym the links are directed. That is,
there is a difference between the edge $(u,v)$ and the edge $(u,v)$. Directed
networks are sometimes also called _digraphs_ (for _directed graphs_), and
their edges _arcs_. An example of a directed social network is the follower
network of Twitter (TF), in which the fact that user A follows user B does not
imply that user B follows user A.
* •
Bipartite networks (B) ††margin: bip include two types of nodes, and all
edges connect one node type with the other. An example of a bipartite network
is a rating graph, consisting of the node types _user_ and _movie_ , and each
rating connects a user and a movie (M3). Bipartite networks are always
undirected in KONECT.
Table 1: The network formats allowed in KONECT. Each network dataset is
exactly of one type.
# Symbol Type Edge partition Edge types Internal name 1 U Undirected
Unipartite Undirected sym 2 D Directed Unipartite Directed asym 3 B Bipartite
Bipartite Undirected bip
The edge weight and multiplicity types of networks are represented by one of
the following six types. The types of edge weights and multiplicities are
summarized in Table 2.
* •
An unweighted network ($-$) ††margin: unweighted has edges that are
unweighted, and only a single edge is allowed between any two nodes.
* •
In a network with multiple edges ($=$), ††margin: positive two nodes can be
connected by any number of edges, and all edges are unweighted. This type of
network is also called a multigraph.
* •
In a positive network ($+$), ††margin: posweighted edges are annotated with
positive weights, and only a single edge is allowed between any node pair. The
weight zero identified with the lack of an edge and thus, we require that each
edge has a weight strictly larger than zero.
* •
In a signed network ($\pm$), ††margin: signed both positive and negative
edges are allowed. Positive and negative edges are represented by positive and
negative edge weights. Many networks of this type have only the weights $\pm
1$, but in the general case we allow any nonzero weight.
* •
Rating networks ($*$) ††margin: weighted have arbitrary real edge weights.
They differ from positive and signed networks in that the edge weights are
interpreted as an interval scale, and thus the value zero has no special
meaning. Adding a constant to all edge weights does not change the semantics
of a rating network. Ratings can be discrete, such as the one-to-five star
ratings, or continuous, such as a rating given in percent. This type of
network allows only a single edge between two nodes.
* •
Networks with multiple ratings (${}_{*}{}^{*}$) ††margin: multiweighted have
edges annotated with rating values, and allow multiple edges between two
nodes.
* •
Dynamic networks ($\rightleftarrows$) are networks in ††margin: dynamic which
edges can appear and disappear. They are always temporal. Individual edges are
not weighted.
Metadata of networks are further properties that go beyond the formats and
weights listed abive.
* •
Temporal networks (⏲) include a timestamp for each edge, and thus the network
can be reconstructed for any moment in the past.
* •
Networks with loops ($\circlearrowright$) are unipartite networks in which
edges of the form $\\{u,u\\}$ are allowed, i.e., edges connecting a node with
itself.
Table 2: The edge weight and multiplicity types allowed in KONECT. Each
network dataset is exactly of one type. Note that due to historical reasons,
networks with multiple unweighted edges have the internal name positive, while
positively weighted networks have the internal posweighted. For signed
networks and positive edge weights, weights of zero are only allowed when the
tag #zeroweight is set.
# Symbol Type Multiple Edge weight Edge weight Internal name edges range scale
1 $-$ Unweighted No $\\{1\\}$ – unweighted 2 $=$ Multiple unweighted Yes
$\\{1\\}$ – positive 3 $+$ Positive weights No $(0,\infty)$ Ratio scale
posweighted 4 $\pm$ Signed No $(-\infty,+\infty)$ Ratio scale signed 5
$\stackrel{{\scriptstyle+}}{{=}}$ Multiple signed Yes $(-\infty,+\infty)$
Ratio scale multisigned 6 $*$ Rating No $(-\infty,+\infty)$ Interval scale
weighted 7 ${}_{*}{}^{*}$ Multiple ratings Yes $(-\infty,+\infty)$ Interval
scale multiweighted 8 $\rightleftarrows$ Dynamic Yes $\\{1\\}$ – dynamic 9
Multiple positive weights Yes $(0,\infty)$ Ratio scale multiposweighted
Finally, the network categories classify networks by the type of data they
represent. An overview of the categories is given in Table 3.
Table 3: The network categories in KONECT. Each category is assigned a color,
which is used in plots, for instance in Figure 1. The property symbols are
defined in Table 2. U: Undirected network, D: Directed network, B: Bipartite
network.
Category Vertices Edges Properties Count $\newmoon$ Affiliation Actors, groups
Membership U D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$
${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 11 $\newmoon$ Animal Animals Tie U D B
$-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 1 $\newmoon$ Authorship Authors, works Authorship U D
B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 18 $\newmoon$ Citation Documents Citation U D B $-$
$=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 6 $\newmoon$ Coauthorship Authors Coauthorship U D B
$-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 5 $\newmoon$ Communication Persons Message U D B $-$
$=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 11 $\newmoon$ Computer Computers Connection U D B $-$
$=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 5 $\newmoon$ Feature Items, features Property U D B
$-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 9 $\newmoon$ Folksonomy Users, tags, items Tag
assignment U D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$
${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 18 $\newmoon$ HumanContact Persons
Real-life contact U D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$
$*$ ${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 4 $\newmoon$ HumanSocial Persons
Real-life tie U D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$
${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 3 $\newmoon$ Hyperlink Web page
Hyperlink U D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$
${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 28 $\newmoon$ Infrastructure Location
Connection U D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$
${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 9 $\newmoon$ Interaction Persons, items
Interaction U D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$
${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 6 $\newmoon$ Lexical Words Lexical
relationship U D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$
${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 6 $\newmoon$ Metabolic Metabolites
Interaction U D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$
${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 6 $\newmoon$ Misc Various Various U D B
$-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 6 $\newmoon$ OnlineContact Users Online interaction U
D B $-$ $=$ $+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 5 $\newmoon$ Rating Users, items Rating U D B $-$ $=$
$+$ $\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$
$\rightleftarrows$ $++$ 15 $\newmoon$ Social Persons Tie U D B $-$ $=$ $+$
$\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$ $\rightleftarrows$
$++$ 30 $\newmoon$ Software Software Component Dependency U D B $-$ $=$ $+$
$\pm$ $\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$ $\rightleftarrows$
$++$ 3 $\newmoon$ Text Documents, words Occurrence U D B $-$ $=$ $+$ $\pm$
$\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 5
$\newmoon$ Trophic Species Carbon exchange U D B $-$ $=$ $+$ $\pm$
$\stackrel{{\scriptstyle+}}{{=}}$ $*$ ${}_{*}{}^{*}$ $\rightleftarrows$ $++$ 3
Affiliation networks
are bipartite networks denoting the ††margin: Affiliation membership of
actors in groups. Groups can be defined as narrowly as individual online
communities in which users have been active (FG) or as broadly as countries
(CN). The actors are mainly persons, but can also be other actors such as
musical groups. Note that in all affiliation networks we consider, each actor
can be in more than one group, as otherwise the network cannot be connected.
Animal networks
are networks of contacts between animals. ††margin: Animal They are the
animal equivalent to human social networks. Note that datasets of websites
such as Dogster (Sd) are _not_ included here but in the Social (online social
network) category, since the networks are generated by humans.
Authorship networks
are unweighted bipartite networks consisting ††margin: Authorship of links
between authors and their works. In some authorship networks such as that of
scientific literature (Pa), works have typically only few authors, whereas
works in other authorship networks may have many authors, as in Wikipedia
articles (en).
Citation networks
consist of documents that reference each ††margin: Citation other. The
primary example are scientific publications, but the category also allow
patents and other types of documents that reference each other.
Coauthorship networks
are unipartite network connecting authors ††margin: Coauthorship ††margin: w
ho have written works together, for instance academic literature, but also
other types of works such as music or movies.
Communication networks
contain edges that represent ††margin: Communication individual messages
between persons. Communication networks are directed and allow multiple edges.
Examples of communication networks are those of emails (EN) and those of
Facebook messages (Ow). Note that in some instances, edge directions are not
known and KONECT can only provide an undirected network.
Computer networks
are networks of connected computers. ††margin: Computer Nodes in them are
computers, and edges are connections. When speaking about _networks_ in a
computer science context, one often means only computer networks. An example
is the internet topology network (TO).
Feature networks
are bipartite, and denote any kind of feature ††margin: Feature assigned to
entities. Feature networks are unweighted and have edges that are not
annotated with edge creation times. Examples are songs and their genres (GE).
Folksonomies
consist of tag assignments connecting a user, an ††margin: Folksonomy item
and a tag. For folksonomies, we follow the 3-bipartite projection approach and
consider the three possible bipartite networks, i.e., the user–item, user–tag
and item–tag networks. This allows us to apply methods for bipartite graphs to
hypergraphs, which is not possible otherwise. Items that are tagged in
folksonomies include bookmarks (Dui), scientific publications (Cui) and movies
(Mui).
Human contact networks
are unipartite networks of actual contact ††margin: HumanContact between
persons, i.e., talking with each other, spending time together, or at least
being physically close. Usually, these datasets are collected by giving out
RFID tags to people with chips that record which other people are in the
vicinity. Determining when an actual contact has happened (as opposed to for
instance to persons standing back to back) is a nontrivial research problem.
An example is the Reality Mining dataset (RM).
Human social networks
are real-world social networks between humans. ††margin: HumanSocial The ties
must be offline, and not from an online social network. Also, the ties
represent a state, as opposed to human contact networks, in which each edge
represents an event.
Hyperlink networks
are the networks of web pages connected by hyperlinks.
Infrastructure networks
are networks of physical infrastructure. ††margin: Infrastructure Examples
are road networks (RO), airline connection networks (OF), and power grids
(UG).
Interaction networks
are bipartite networks consisting of people ††margin: Interaction and items,
where each edge represents an interaction. In interaction networks, we always
allow multiple edges between the same person–item pair. Examples are people
writing in forums (UF), commenting on movies (Fc) or listening to songs (Ls).
Lexical networks
consist of words from natural ††margin: Lexical languages and the
relationships between them. Relationships can be semantic (i.e, related to the
meaning of words) such as the synonym relationship (WO), associative such as
when two words are associated with each other by people in experiments (EA),
or denote cooccurrence, i.e., the fact that two words co-occur in text (SB).
Note that lexical cooccurrence networks are explicitly not included in the
broader Cooccurrence category.
Metabolic networks
model metabolic pathways. ††margin: Metabolic
Miscellaneous networks
are any networks that do not fit into one ††margin: Misc of the other
categories.
Online Contact networks
consist of people and interactions between ††margin: OnlineContact them.
Contact networks are unipartite and allow multiple edges, i.e., there can
always be multiple interactions between the same two persons. They can be both
directed or undirected. Examples are people that meet each other (RM), or
scientists that write a paper together (Pc).
Physical networks
represent physically existing network ††margin: Physical structures in the
broadest sense. This category covers such diverse data as physical computer
networks (TO), transport networks (OF) and biological food networks (FD).
Rating networks
consist of assessments given to items by users, ††margin: Rating weighted by
a rating value. Rating networks are bipartite. Networks in which users can
rate other users are not included here, but in the Social category instead. If
only a single type of rating is possible, for instance the “favorite”
relationship, then rating networks are unweighted. Examples of items that are
rated are movies (M3), songs (YS), jokes (JE), and even sexual escorts (SX).
Online social networks
represent ties between ††margin: Social persons in online social networking
platforms. Certain social networks allow negative edges, which denote enmity,
distrust or dislike. Examples are Facebook friendships (FSG), the Twitter
follower relationship (TF), and friends and foes on Slashdot (SZ). Note that
some social networks can be argued to be rating networks, for instance the
user–user rating network of a dating site (LI). These networks are all
included in the Social category.
Software networks
are networks of interacting software ††margin: Software component. Node can
be software packages connected by their dependencies, source files connected
by includes, and classes connected by imports.
Text networks
consist of text documents containing words. They ††margin: Text are bipartite
and their nodes are documents and words. Each edge represents the occurrence
of a word in a document. Document types are for instance newspaper articles
(TR) and Wikipedia articles (EX).
Trophic networks
consist of biological species connected by edges denotes ††margin: Trophic
which pairs of species are subject to carbon exchange, i.e., which species
eats which. The term _food chain_ describes such relation ships, but note that
in the general case, a trophic network is not a chain, i.e., it is not linear.
Trophic networks are directed.
Note that the category system of KONECT is in flux. As networks are added to
the collection, large categories are split into smaller ones.
We do not include certain kinds of networks that lack a complex structure.
This includes networks without a giant connected component, in which most
nodes are not reachable from each other, and trees, in which there is only a
single path between any two nodes. Note that bipartite relationships extracted
from n-to-1 relationships are therefore excluded, as they lead to a disjoint
network. For instance, a bipartite person–city network containing _was-born-
in_ edges would not be included, as each city would form its own component
disconnected from the rest of the network. On the other hand, a band–country
network where edges denote the country of origin of individual band members is
included, as members of a single band can have different countries of origin.
In fact the Countries network (CN) is of this form. Another example is a
bipartite song–genre network, which would only be included in KONECT when
songs can have multiple genres. As an example of the lack of complex structure
when only a single genre is allowed, the degree distribution in such a
song–genre network is skewed because all song nodes have degree one, the
diameter cannot be computed since the network is disconnected, and each
connected component trivially has a diameter of two or less.
## 3 Definitions
The areas of graph theory and network analysis are young, and many concepts
within them do not have a single established notation. The notation chosen for
KONECT represents a compromise between familiarity with the most common
conventions, and the need to use an unambigous choice of letters and symbols.
Graphs will be denoted as $G=(V,E)$, in which $V$ is the set of vertices, and
$E$ is the set of edges [Bol98]. Without loss of generality, we assume that
the vertices $V$ are consecutive natural numbers, i.e.,
$\displaystyle V$ $\displaystyle=\\{1,2,3,\dotsc,|V|\\}.$ (1)
Edges $e\in E$ will be denoted as sets of two vertices, i.e., $e=\\{u,v\\}$.
We say that two vertices are adjacent if they are connected by an edge; this
will be written as $u\sim v$. We say that an edge is incident to a vertex if
the edge touches the vertex.
We also allow loops, i.e., edges of the form $\\{u,u\\}=\\{u\\}$. Loops appear
for instance in email networks, where it is possible to send an email to
oneself, and therefore an edge may connect a vertex with itself. Most networks
however do not contain loops, and therefore networks that allow loops are
annotated in KONECT with the #loop tag, as described in Section 9.
Most of the time, we work with only one given graph, and therefore it is
unambigous with node and edge set are meant by $V$ and $E$. When ambiguity is
possible, we will however use the notation $V[G]$ and $E[G]$ to denote the
vertex and edge sets of a graph $G$. This notation may occasionally be
extended to other graph characteristics.
In directed networks, edges are pairs instead of sets, i.e., $e=(u,v)$. In
directed networks, edges are sometimes called _arcs_ ; in KONECT, we use the
term _edge_ for them.
In bipartite graphs, we can partition the set of nodes $V$ into two disjoint
sets $V_{1}$ and $V_{2}$, which we will call the left and right set
respectively. Although the assignment of a bipartite network’s two node types
to left and right sides is mathematically arbitrary, it is chosen in KONECT
such that the left nodes are _active_ and the right nodes are _passive_. For
instance, a rating graph with users and items will always have users on the
left since they are active in the sense that it is they who give the ratings.
Such a distinction is sensible in most networks [Ops12]. The number of left
and right nodes will be denoted $n_{1}=|V_{1}|$ and $n_{2}=|V_{2}|$.
Networks with multiple edges will be written as $G=(V,E)$, where $E$ is a
multiset. The degree of nodes in such networks takes into account multiple
edges. Thus, the degree does not equal the number of adjacent nodes but the
number of incident edges. When $E$ is a multiset, it can contain the edge
$\\{u,v\\}$ multiple times. Mathematically, we may write $\\{u,v\\}_{1}$,
$\\{u,v\\}_{2}$, etc. Note that we will be lax with this notation. In
expressions valid for all types of networks, we will use sums such as
$\sum_{\\{u,v\\}\in E}$ and understand that the sum is over all edges.
In positively weighted networks, we define $w$ as the weight function,
returning the edge weight when given an edge. In such networks, the weights
are not taken into account when computing the degree.
In a signed network, each edge is assigned a signed weight such as $+1$ or
$-1$. In such networks, we define $w$ to be the signed weight function. In the
general case, we allow arbitrary nonzero real numbers, representing degrees of
positive and negative edges.
In rating networks, we define $r$ to be the rating function, returning the
rating value when given an edge. Note that rating values are interpreted to be
invariant under shifts, i.e., adding a real constant to all ratings in the
network must not change the semantics of the network. Thus, we will often make
use of the mean rating defined as
$\displaystyle\mu$ $\displaystyle=\frac{1}{|E|}\sum_{e\in E}r(e).$ (2)
For consistency, we also define the edge weight function $w$ for unweighted
and rating networks:
$\displaystyle w(e)$ $\displaystyle=\left\\{\begin{array}[]{ll}1&\text{when
$G$ is unweighted}\\\ r(e)-\mu&\text{when $G$ is a rating network}\\\
\end{array}\right.$ (5)
We also define a weighting function for node pairs, also denoted $w$. This
function takes into account both the weight of edges and edge multiplicities.
It is defined as $w(u,v)=0$ when the nodes $u$ and $v$ are not connected and
if they are connected as
$\displaystyle w(u,v)$ $\displaystyle=\left\\{\begin{array}[]{ll}1&\text{when
$G$ is $-$}\\\ |\\{k\mid\\{u,v\\}_{k}\in E\\}|&\text{when $G$ is $=$}\\\
w(\\{u,v\\})&\text{when $G$ is $+$}\\\ w(\\{u,v\\})&\text{when $G$ is
$\pm$}\\\ r(\\{u,v\\})-\mu&\text{when $G$ is $*$}\\\ \sum_{\\{u,v\\}_{k\in
E}}[r(\\{u,v\\}_{k})-\mu]&\text{when $G$ is ${}_{*}{}^{*}$}\end{array}\right.$
(12)
Dynamic networks are special in that they have a set of events (edge addition
and removal) instead of a set of edges. In most cases, we will model dynamic
networks as unweighted networks $G=(V,E)$ representing their state at the
latest known timepoint. For analyses that are performed over time, we consider
the graph at different time points, with the graph always being an unweighted
graph.
In an unweighted graph $G=(V,E)$, the degree of a vertex is the number of
neighbors of that node
$\displaystyle d(u)$ $\displaystyle=\\{v\in V\mid\\{u,v\\}\in E\\}.$ (13)
In networks with multiple edges, the degree takes into account multiple edges,
and thus to be precise, it equals the number of incident edges and not the
number of adjacent vertices.
$\displaystyle d(u)$ $\displaystyle=\\{\\{u,v\\}_{k}\in E\mid v\in V\\}$ (14)
In directed graphs, the sum is over all of $u$’s neighbors, regardless of the
edge orientation. Note that the sum of the degrees of all nodes always equals
twice the number of edges, i.e.,
$\displaystyle\sum_{v\in V}d(u)$ $\displaystyle=2|E|.$ (15)
In a directed graph we define the outdegree $d_{1}$ of a node as the number of
outgoing edges, and the indegree $d_{2}$ as the number of ingoing edges.
$\displaystyle d_{1}(u)$ $\displaystyle=\\{v\in V\mid(u,v)\in E\\}$ (16)
$\displaystyle d_{2}(u)$ $\displaystyle=\\{v\in V\mid(v,u)\in E\\}$ (17)
The sum of all outdegrees, and likewise the sum of all indegrees always equals
the number of nodes in the network.
$\displaystyle\sum_{u\in V}d_{1}(u)=\sum_{u\in V}d_{2}(u)=|E|$ (18)
We also define the weight of a node, also denoted by the symbol $w$, as the
sum of the absolute weights of incident edges
$\displaystyle w(u)$ $\displaystyle=\sum_{\\{u,v\\}\in E}|w(\\{u,v\\})|.$ (19)
The weight of a node coincides with the degree of a node in unweighted
networks and networks with multiple edges. The weight of a node may also be
called its strength [OAS10].
### 3.1 Graph Transformations
Sometimes, it is necessary to construct a graph out of another graph. In the
following, we briefly review such constructions.
Let $G=(V,E,w)$ be any weighted, signed or rating graph, regardless of edge
multiplicities. Then, $\bar{G}$ will denote the corresponding unweighted
graph, i.e.,
$\displaystyle\bar{G}$ $\displaystyle=(V,E).$ (20)
Note that the graph $\bar{G}$ may still contain multiple edges.
Let $G=(V,E,w)$ be any graph with multiple edges. We define the corresponding
unweighted simple graphs as
$\displaystyle\bar{\bar{G}}=(V,\bar{\bar{E}}),$ (21)
where $\bar{\bar{E}}$ is the set underlying the multiset $E$. For simple
graphs, we define $\bar{\bar{G}}=G$.
Let $G=(V,E,w)$ be a signed or rating network. Then, $|G|$ will denote the
corresponding unsigned graph defined by
$\displaystyle|G|$ $\displaystyle=(V,E,w^{\prime})$ (22) $\displaystyle
w^{\prime}(e)$ $\displaystyle=|w(e)|.$
Let $G=(V,E,w)$ be any network with weight function $w$. The negative network
to $G$ is then defined as
$\displaystyle-G$ $\displaystyle=(V,E,w^{\prime})$ (23) $\displaystyle
w^{\prime}(e)$ $\displaystyle=-w(e).$
This construction is possible for all types of networks. For unweighted and
positively weighted networks, it leads to signed networks.
### 3.2 Characteristic Matrices
A very useful representation of graph is using matrices. In fact, a subfield
of graph theory, _algebraic graph theory_ , is devoted to this representation
[GR01]. When a graph is represented as a matrix, operations on graphs can
often be expressed as simple algebraic expressions. For instance, the number
of common friends of two people in a social network can be expressed as the
square of a matrix.
An unweighted graph $G=(V,E)$ can be represented by a $|V|$-by-$|V|$ matrix
containing the values 0 and 1, denoting whether a certain edges between two
nodes is present. This matrix is called the adjacency matrix of $G$ and will
be denoted $\mathbf{A}$. Remember that we assume that the vertices are the
natural numbers $1,2,\dotsc,|V|$. Then the entry $\mathbf{A}_{uv}$ is one when
$\\{u,v\\}\in E$ and zero when not. This makes $\mathbf{A}$ square and
symmetric for undirected graphs, generally asymmetric (but still square) for
directed graphs.
For a bipartite graph $G=(V_{1}\cup V_{2},E)$, the adjacency matrix has the
form
$\displaystyle\mathbf{A}$
$\displaystyle=\left[\begin{array}[]{cc}&\mathbf{B}\\\
\mathbf{B}^{\mathrm{T}}&\end{array}\right].$ (26)
The matrix $\mathbf{B}$ is a $|V_{1}|$-by-$|V_{2}|$ matrix, and thus generally
rectangular. $\mathbf{B}$ will be called the biadjacency matrix.
In weighted networks, the adjacency matrix takes into account edge weights. In
networks with multiple edges, the adjacency matrix takes into account edge
multiplicities. Thus, the general definition of the adjacency matrix is given
by
$\displaystyle\mathbf{A}_{uv}$ $\displaystyle=w(u,v).$ (27)
The degree matrix $\mathbf{D}$ is a diagonal $|V|$-by-$|V|$ matrix containing
the absolute weights of all nodes, i.e.,
$\displaystyle\mathbf{D}_{uu}$ $\displaystyle=|w(u)|.$ (28)
Note that we define the degree matrix explicitly to contain node weights
instead of degrees, to be consistent with the definition of $\mathbf{A}$.
The normalized adjacency matrix $\mathbf{N}$ is a $|V|$-by-$|V|$ matrix given
by
$\displaystyle\mathbf{N}$
$\displaystyle=\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}.$ (29)
Finally the Laplacian matrix $\mathbf{L}$ is an $|V|$-by-$|V|$ matrix defined
as
$\displaystyle\mathbf{L}=\mathbf{D}-\mathbf{A}.$ (30)
Note that in some disciplines the Laplacian matrix may be defined as
$\mathbf{A}-\mathbf{D}$, making it negative-semidefinite.
Other matrices used in KONECT include the normalized Laplacian matrix, the
stochastic adjacency matrix and the signless Laplacian.
The normalized Laplacian $\mathbf{Z}$ is a normalized version of the Laplacian
matrix $\mathbf{L}$. Just as the ordinary Laplacian, $\mathbf{Z}$ capture
aspects of the graph that are useful for clustering.
$\displaystyle\mathbf{Z}=\mathbf{I}-\mathbf{N}=\mathbf{D}^{-1/2}\mathbf{L}\mathbf{D}^{-1/2}$
(31)
The equation $\mathbf{Z}=\mathbf{I}-\mathbf{N}$ shows that $\mathbf{Z}$ has
the same eigenvectors as $\mathbf{N}$, and its eigenvalues are those of
$\mathbf{N}$, but shifted and inverted.
The consideration of random walks on a graph leads to the definition of the
stochastic adjacency matrix $\mathbf{P}$. Imagine a random walker on the nodes
of a graph, who can walk from node to node by following edges. If, at each
edge, the probability that the random walker will go to each neighboring node
with equal probability, then the random walk can be described be the
transition probability matrix defined as
$\displaystyle\mathbf{P}=\mathbf{D}^{-1}\mathbf{A}=\mathbf{D}^{-1/2}\mathbf{N}\mathbf{D}^{1/2}.$
(32)
The matrix $\mathbf{P}$ is right stochastic, since its row sums are one.
A further variant of Laplacian matrix is the signless Laplacian $\mathbf{K}$.
$\displaystyle\mathbf{K}=\mathbf{D}+\mathbf{A}.$ (33)
The signless Laplacian $\mathbf{K}$ corresponds to the ordinary Laplacian
$\mathbf{L}$ of the graph with inverted edge weights, i.e.,
$\mathbf{K}[G]=\mathbf{L}[-G]$.
Note that in most cases, we work on just a single graph, and it is implicit
that the characteristic matrices apply to this graph. In a few cases, we may
need to consider the characteristic matrices of multiple graphs. In these
cases, we will write
$\displaystyle\mathbf{A}[G],\mathbf{D}[G],\mathbf{L}[G],\dotsc$
to denote the characteristic matrices of the graph $G$.
## 4 Statistics
A network statistic is a numerical value that characterizes a network.
Examples of network statistics are the number of nodes and the number of edges
in a network, but also more complex measures such as the diameter and the
clustering coefficient. Statistics are the basis of most network analysis
methods; they can be used to compare networks, classify networks, detect
anomalies in networks and for many other tasks. Network statistics are also
used to map a network’s structure to a simple numerical space, in which many
standard statistical methods can be applied. Thus, network statistics are
essential for the analysis of almost all network types. All statistics
described in KONECT are real numbers.
This section gives the definitions for the statistics supported by KONECT, and
briefly reviews their uses. All network statistics can be computed using the
KONECT Toolbox using the function konect_statistic(). Each statistic has an
internal name that must be passed as the first argument to konect_statistic().
The internal names are given in the margin in this section. Additionally, the
KONECT Toolbox includes functions named konect_statistic_<NAME>() which
compute a single statistic <NAME>.
The values of selected statistics are shown for the KONECT networks on the
website444konect.uni-koblenz.de/statistics.
### 4.1 Basic Network Statistics
Some statistics are simple to define, trivial to compute, and are reported
universally in studies about networks. These include the number of nodes, the
number of edges, and statistics derived from them such as the average number
of neighbors a node has.
The size of a network is the number of nodes it contains, and is almost
universally denoted $n$. The size of a graph is sometimes also called the
order of the graph.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{size}}n\@add@raggedright$
$\displaystyle=|V|$ (34)
In a bipartite graph, the size can be decomposed as $n=n_{1}+n_{2}$ with
$n_{1}=|V_{1}|$ and $n_{2}=|V_{2}|$. The size of a network is not necessarily
a very meaningful number. For instance, adding a node without edges to a
network will increase the size of the network, but will not change anything in
the network. In the case of an online social network, this would correspond to
creating a user account and not connecting it to any other users – this adds
an inactive user, which are often not taken into account. Therefore, a more
representative measure of the _size_ of a network is actually given by the
number of edges, giving the volume of a network.
The volume of a network equals the number of edges and is defined as
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{volume}}m\@add@raggedright$
$\displaystyle=|E|.$ (35)
Note that in mathematical contexts, the number of edges may be called the
_size_ of the graph, in which case the number of nodes is called the _order_.
In this text, we will consistently use _size_ for the number of nodes and
_volume_ for the number of edges.
The volume can be expressed in terms of the adjacency or biadjacency matrix of
the underlying unweighted graph as
$\displaystyle m$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{1}{2}\|\mathbf{A}[\bar{G}]\|_{\mathrm{F}}^{2}&\text{when
$G$ is undirected}\\\ \|\mathbf{A}[\bar{G}]\|_{\mathrm{F}}^{2}&\text{when $G$
is directed}\\\ \|\mathbf{B}[\bar{G}]\|_{\mathrm{F}}^{2}&\text{when $G$ is
bipartite}\end{array}\right.$ (39)
The number of edges in network is often considered a better measure of the
_size_ of a network than the number vertices, since a vertex unconnected to
any other vertices may often be ignored. On the practical side, the volume is
also a much better indicator of the amount of memory needed to represent a
network.
We will also make use of the number of edges without counting multiple edges.
We will call this the unique volume of the graph.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{uniquevolume}}\bar{\bar{m}}\@add@raggedright$
$\displaystyle=m[\bar{\bar{G}}]$ (40)
The weight $w$ of a network is defined as the sum of absolute edge weights.
For unweighted networks, the weight equals the volume. For rating networks,
remember that the weight is defined as the sum over ratings from which the
overall mean rating has been subtracted, in accordance with the definition of
the adjacency matrix for these networks.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{weight}}w\@add@raggedright$
$\displaystyle=\sum_{e\in E}|w(e)|$ (41)
The average degree is defined as
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{avgdegree}}d\@add@raggedright$
$\displaystyle=\frac{1}{|V|}\sum_{u\in V}d(u)=\frac{2m}{n}.$ (42)
The average degree is sometimes called the density. We avoid the term
_density_ in KONECT as it is sometimes used for the fill, which denotes the
probability that an edge exists. In bipartite networks, we additionally define
the left and right average degree
$\displaystyle d_{1}$ $\displaystyle=\frac{1}{|V_{1}|}\sum_{u\in
V_{1}}d(u)=\frac{m}{n_{1}}$ (43) $\displaystyle d_{2}$
$\displaystyle=\frac{1}{|V_{2}|}\sum_{u\in V_{2}}d(u)=\frac{m}{n_{2}}$ (44)
Note that in directed networks, the average outdegree equals the average
indegree, and both are equal to $m/n$.
The fill of a network is the proportion of edges to the total number of
possible edges. The fill is used as a basic parameter in the Erdős–Rényi
random graph model [ER59], where it denotes the probability that an edge is
present between two randomly chosen nodes, and is usually called $p$, which is
the notation we also use in KONECT.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{fill}}p\@add@raggedright$
$\displaystyle=\left\\{\begin{array}[]{ll}2m/[n(n-1)]&\text{when $G$ is
undirected without loop}\\\ 2m/[n(n+1)]&\text{when $G$ is undirected with
loops}\\\ m/[n(n-1)]&\text{when $G$ is directed without loops}\\\
m/n^{2}&\text{when $G$ is directed with loops}\\\ m/(n_{1}n_{2})&\text{when
$G$ is bipartite}\end{array}\right.$ (50)
In the undirected case, the expression is explained by the fact that the total
number of possible edges is $n(n-1)/2$ excluding loops. The fill is sometimes
also called the density of the network, in particular in a mathematical
context, or the connectance of the network555Used for instance in this blog
entry: proopnarine.wordpress.com/2010/02/11/graphs-and-food-webs.
The maximum degree equals the highest degree value attained by any node.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{maxdegree}}d_{\max}\@add@raggedright$
$\displaystyle=\max_{u\in V}d(u)$ (51)
The maximum degree can be divided by the average degree to normalize it.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{relmaxdegree}}d_{\mathrm{MR}}\@add@raggedright$
$\displaystyle=\frac{d_{\max}}{d}$ (52)
In a directed network, the reciprocity equals the proportion of edges for
which an edge in the opposite direction exists, i.e., that are reciprocated.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{reciprocity}}y=\frac{1}{m}|\\{(u,v)\in
E\mid(v,u)\in E\\}|\@add@raggedright$ (53)
The reciprocity can give an idea of the type of network. For instance,
citation networks only contain only few pairs of papers that mutually cite
each other. On the other hand, an email network will contain many pairs of
people who have sent emails to each other. Thus, citation networks typically
have low reciprocity, and communnication networks have high reciprocity.
In networks that allow negative edges such as signed networks and rating
networks, we may be interested in the proportion of edges that are actually
negative. We call this the _negativity_ of the network.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{negativity}}\zeta\@add@raggedright$
$\displaystyle=\frac{|\\{e\in E\mid w(e)<0\\}|}{m}$ (54)
### 4.2 Connectivity Statistics
Connectivity statistics measure to what extent a network is connected. Two
nodes are said to be connected when they are either directly connected through
an edge, or indirectly through a path of several edges. A connected component
is a set of vertices all of which are connected, and unconnected to the other
nodes in the network. The largest connected component in a network is usually
very large and called the giant connected component. When it contains all
nodes, the network is connected.
The size of the largest connected component is denoted $N$.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{coco}}N\@add@raggedright$
$\displaystyle=\max_{F\subseteq\mathcal{C}}|F|$ (55)
$\displaystyle\mathcal{C}$ $\displaystyle=\\{C\subseteq V\mid\forall u,v\in
C:\exists w_{1},w_{2},\ldots\in V:u\sim w_{1}\sim w_{2}\sim\cdots\sim v\\}$
In bipartite networks, the number of left and right nodes in the largest
connected components are denoted $N_{1}$ and $N_{2}$, with $N_{1}+N_{2}=N$.
The relative size of the largest connected component equals the size of the
largest connected component divided by the size of the network
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{cocorel}}N_{\mathrm{rel}}\@add@raggedright$
$\displaystyle=\frac{N}{n}.$ (56)
We also use an inverted variant of the relative size of the largest connected
component, which makes it easier to plot the values of a logarithmic scale.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{cocorelinv}}N_{\mathrm{inv}}\@add@raggedright$
$\displaystyle=1-\frac{N}{n}$ (57)
In directed networks, we additionally define the size of the largest ††margin:
cocos strongly connected component $N_{\mathrm{s}}$. A strongly connected
component is a set of vertices in a directed graph such that any node is
reachable from any other node using a path following only directed edges in
the forward direction. We always have $N_{\mathrm{s}}\leq N$.
### 4.3 Count Statistics
The fundamental building block of a network are the edges. Thus, the number of
edges is a basic statistic of any network. To understand the structure of a
network, it is however not enough to analyse edges individually. Instead,
larger patterns such as triangles must be considered. These patterns can be
counted, and give rise to count statistics, i.e., statistics that count the
number of ocurrences of specific patterns.
Table 4 gives a list of fundamental patterns in networks, and their
corresponding count statistics.
Table 4: Patterns that occur in networks. Each pattern can be counted, giving rise to a count statistic. Pattern | Name(s) | Count statistic | Internal name
---|---|---|---
| Edge, 1-path, 1-star, 2-clique | Volume $m$ | volume
| Wedge, 2-star, 2-path | Wedge count $s$ | twostars
| Claw, 3-star | Claw count $z$ | threestars
| Cross, 4-star | cross count $x$ | fourstars
| Triangle, 3-cycle, 3-clique | Triangle count $t$ | triangles
| Square, 4-cycle | Square count $q$ | squares
A star is defined as a graph in which a central node is connected to all other
nodes, and no other edges are present. Specifically, a $k$-star is defined as
a star in which the central node is connected to $k$ other nodes. Thus, a
2-star consists of a node connected to two other nodes, or equivalently two
incident edges, or a path of length 2. The specific name for 2-stars is
_wedges_. The number of wedges can be defined as
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{twostars}}s=\sum_{u\in
V}{d(u)\choose 2}=\sum_{u\in V}\frac{1}{2}d(u)(d(u)-1),\@add@raggedright$ (58)
where $d(u)$ is the degree of node $u$. Wedges have many different names:
2-stars, 2-paths and hairpins [GO12].
The number of triangles defined in the following way is independent of the
orientation of edges when the graph is directed. Loops in the graph, as well
as edge multiplicities, are ignored.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{triangles}}t=|\\{\\{u,v,w\\}\mid
u\sim v\sim w\sim u\\}|\;/\;6\@add@raggedright$ (59)
Three-stars are defined analogously to two-stars, and their count denoted $z$.
Three-stars are also called _claws_ and _tripins_ [GO12].
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{threestars}}z=\sum_{u\in
V}{d(u)\choose 3}=\sum_{u\in
V}\frac{1}{6}d(u)(d(u)-1)(d(u)-2)\@add@raggedright$ (60)
A square is a cycle of length four, and the number of squares in a graph is
denoted $q$.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{squares}}q=|\\{u,v,w,x\mid
u\sim v\sim w\sim x\sim u\\}|\;/\;8\@add@raggedright$ (61)
The factor 8 ensures that squares are counted regardless of their edge
labeling.
Multiple edges are ignored in these count statistics, and edges in patterns
are not allowed to overlap.
Triangles and squares are both cycles – which we can generalize to $k$-cycles,
sequences of $k$ distinct vertices that are cyclically linked by edges. We
denote the number of $k$-cycles by $C_{k}$. For small $k$, we note the
following equivalences:
$\displaystyle C_{1}$ $\displaystyle=0$ $\displaystyle C_{2}$
$\displaystyle=m$ $\displaystyle C_{3}$ $\displaystyle=t$ $\displaystyle
C_{4}$ $\displaystyle=q$
for graphs without loops. Cycles of length three and four have special
notation: $C_{3}=t$ and $C_{4}=q$ and are called triangles and squares.
A cycle cannot the same node twice. Due to this combinatorial restriction,
$C_{k}$ is quite complex to compute for large $k$. Therefore, we may use
_tours_ instead, defined as cyclical lists of connected vertices in which we
allow several vertices to overlap. The number of $k$-tours will be denoted
$T_{k}$. For computational conveniance, we will define labeled tours, where
two tours are not equal when they are identical up to shifts or inversions. We
note the following equalities:
$\displaystyle T_{1}$ $\displaystyle=0$ $\displaystyle T_{2}$
$\displaystyle=2m$ $\displaystyle T_{3}$ $\displaystyle=6t$
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{tour4}}T_{4}\@add@raggedright$
$\displaystyle=8q+4s+2m$ (62)
Again, these are true when the graph is loopless. The last equality shows that
trying to divide the tour count by $2k$ to count them up to shifts and
inversions is a bad idea, since it cannot be implemented by dividing the
present definition by $2k$.
As mentioned before, counting cycles is a complex problem. Counting tours is
however much easier. The number of tours of length $k$ can be expressed as the
trace of a power of the graph’s adjacency matrix, and thus also as a moment of
the adjacency matrix’s spectrum when $k>2$.
$\displaystyle T_{k}$
$\displaystyle=\mathrm{Tr}(\mathbf{A}^{k})=\sum_{i}\lambda_{i}[\mathbf{A}]^{k}$
This remains true when the graph includes loops.
### 4.4 Degree Distribution Statistics
The distribution of degree values $d(u)$ over all nodes $u$ is often taken to
characterize a network. Thus, a certain number of network statistics are based
solely on this distribution, regardless of overall network structure.
The power law exponent is a number that characterizes the degrees of the nodes
in the network. In many circumstances, networks are modeled to follow a degree
distribution power law, i.e., the number of nodes with degree $n$ is taken to
be proportional to the power $n^{-\gamma}$, for a constant $\gamma$ larger
than one [BA99]. This constant $\gamma$ is called the power law exponent.
Given a network, its degree distribution can be used to estimate a value
$\gamma$. There are multiple ways of estimating $\gamma$, and thus a network
does not have a single definite value of it. In KONECT, we estimate $\gamma$
using the robust method given in [New06, Eq. 5]
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{power}}\gamma\@add@raggedright$
$\displaystyle=1+n\left(\sum_{u\in V}\ln\frac{d(u)}{d_{\min}}\right)^{-1},$
(63)
in which $d_{\min}$ is the minimal degree.
The Gini coefficient is a measure of inequality from economics, typically
applied to distributions of wealth or income. In KONECT, we apply it to the
degree distribution, as described in [KP12]. The Gini coefficient can either
be defined in terms of the Lorenz curve, a type of plot that visualizes the
inequality of a distribution, or using the following expression. Let
$d_{1}\leq d_{2}\leq\dotsb\leq d_{n}$ be the sorted list of degrees in the
network. Then, the Gini coefficient is defined as
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{gini}}G\@add@raggedright$
$\displaystyle=\frac{2\sum_{i=1}^{n}id_{i}}{n\sum_{i-1}^{n}d_{i}}-\frac{n+1}{n}.$
(64)
The Gini coefficient takes values between zero and one, with zero denoting
total equality between degrees, and one denoting the dominance of a single
node.
The relative edge distribution entropy is a measure of the equality of the
degree distribution, and equals one when all degrees are equal, and attains
the limit value of zero when all edges attach to a single node [KP12]. It is
defined as
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{dentropyn}}H_{\mathrm{er}}\@add@raggedright$
$\displaystyle=\frac{1}{\ln n}\sum_{u\in
V}-\frac{d(u)}{2m}\ln\frac{d(u)}{2m}.$ (65)
Another statistic for ††margin: own measuring the inequality in the degree
distribution is associated with the Lorenz curve (see Section 6.3), and is
given by the intersection point of the Lorenz curve with the antidiagonal
given by $y=1-x$ [KP12]. By construction, this point equals $(1-P,P)$ for some
$0<P<1$, where the value $P$ corresponds exactly to the number “25%” in the
statement “25% of all users account for 75% of all friendship links on
Facebook”. By construction, we can expect $P$ to be smaller when $G$ is large.
The analysis of degrees can be generalized to pairs of nodes: What is the
distribution of degrees for pairs of connected edges? In some networks, high-
degree nodes are connected to other high-degree nodes, while low-degree nodes
are connected to low-degree nodes. This property is called assortativity.
Inversely, in a network with dissortativity, high-degree nodes are typically
connected to low-degree and vice versa. ††margin: assortativity The amount of
assortativity can be measured by the Pearson correlation $\rho$ between the
degree of connected nodes.
### 4.5 Clustering Statistics
The term _clustering_ refers to the observation that in almost all networks,
nodes tend to form small groups within which many edges are present, and such
that only few edges connected different clusters with each other. In a social
network for instance, people form groups in which almost every member known
the other members. Clustering thus forms one of the primary characteristics of
real-world networks, and thus many statistics for measuring it have been
defined. The main method for measuring clustering numerically is the
clustering coefficient, of which there exist several variants. As a general
rule, the clustering coefficient measures to what extent edges in a network
tend to form triangles. Since it is based on triangles, it can only be applied
to unipartite networks, because bipartite networks do not contain triangles.
The number of triangles $t$ itself as defined in Section 4.3 is however not a
statistic that can be used to measure the clustering in a network, since it
correlates with the size and volume of the network. Instead, the clustering
coefficients in all its variants can be understood as a count of triangles,
normalized in different ways in order to compare several networks with it.
The local clustering coefficient $c(u)$ of a node $u$ is defined as the
probability that two randomly chosen (but distinct) neighbors of $u$ are
connected [WS98].
$\displaystyle c(u)$ $\displaystyle=\left\\{\begin{array}[]{ll}\frac{\\{v,w\in
V\mid u\sim v\sim w\sim u\\}}{\\{v,w\in V\mid u\sim v\neq w\sim
u\\}}&\text{when }d(u)>1\\\ 0&\text{when }d(u)\leq 1\end{array}\right.$ (68)
The global clustering of a network can be computed in two ways. The first way
defines it as the probability that two incident edges are completed by a third
edge to form a triangle [NWS02]. This is also called the transitivity ratio,
or simply the transitivity.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{clusco}}c\@add@raggedright$
$\displaystyle=\frac{|\\{u,v,w\in V\mid u\sim v\sim w\sim u\\}|}{|\\{u,v,w\in
V\mid u\sim v\neq w\sim u\\}|}=\frac{3t}{s}$ (69)
This variant of the global clustering coefficient has values between zero and
one, with a value of one denoting that all possible triangles are formed
(i.e., the network consists of disconnected cliques), and zero when it is
triangle free. Note that the clustering coefficient is trivially zero for
bipartite graphs. This clustering coefficient is however not defined when each
node has degree zero or one, i.e., when the graph is a disjoint union of edges
and unconnected nodes. This is however not a problem in practice.
The second variant variant of the clustering coefficient uses the average of
the local clustering coefficients. This second variant was historically the
first to be defined. In was defined in 1998 [WS98] and precedes the first
variant by four years.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{clusco2}}c_{2}\@add@raggedright$
$\displaystyle=\frac{1}{|V|}\sum_{u\in V}c(u)$ (70)
This second variant of the global clustering coefficient is zero when a graph
is triangle-free, and one when the graph is a disjoint union of cliques of
size at least three. This variant of the global clustering coefficient is
defined for all graphs, except for the empty graph, i.e., the graph with zero
nodes. A slightly different definition of the second variant computes the
average only over nodes with a degree of at least two, as seen for instance in
[BKM08].
Because of the arbitrary decision to define $c(u)$ as zero when the degree of
$c$ is zero or one, we recommend to use the first variant of the clustering
coefficient. In the following, the extensions to the clustering coefficient we
present are all based on the first variant, $c$.
For signed graphs, we may define the clustering coefficient to take into
account the sign of edges. The signed clustering coefficient is based on
balance theory [KLB09]. In a signed network, edges can be positive or
negative. For instance in a signed social network, positive edges represent
friendship, while negative edges represent enmity. In such networks, balance
theory stipulates than triangles tend to be balanced, i.e., that three people
are either all friends, or two of them are friends with each other, and
enemies with the third. On the other hand, a triangle with two positive and
one negative edge, or a triangle with three negative edges is unbalanced. In
other words, we can define the sign of a triangle as the product of the three
edge signs, which then leads to the stipulation that triangles tend to have
positive weight. To extend the clustering coefficient to signed networks, we
thus distinguis between balanced and unbalanced triangles, in a way that
positive triangles contribute positively to the signed clustering coefficient,
and negative triangles contribute negatively to it. For a triangle
$\\{u,v,w\\}$, let $\sigma(u,v,w)=w(u,v)w(v,w)w(w,u)$ be the sign of the
triangle, then the following definition captures the idea:
$\displaystyle c_{\mathrm{s}}$ $\displaystyle=\frac{\sum_{u,v,w\in
V}\sigma(u,v,w)}{|\\{u,v,w\in V\mid u\sim v\neq w\sim u\\}|}$ (71)
Here, the sum is over all triangles $\\{u,v,w\\}$, but can also be taken over
all triples of vertices, since $w(u,v)=0$ when $\\{u,v\\}$ is not an edge.
The signed clustering coefficient is bounded by the clustering coefficient:
$\displaystyle|c_{\mathrm{s}}|\leq c$ (72)
The relative signed clustering coefficient can then be defined as
$\displaystyle c_{\mathrm{r}}=\frac{c_{\mathrm{s}}}{c}=\frac{\sum_{u,v,w\in
V}\sigma(u,v,w)}{|\\{u,v,w\in V\mid u\sim v\sim w\sim u\\}|}$ (73)
which also equals the proportion of all triangles that are balanced, minus the
proportion of edges that are unbalanced.
### 4.6 Distance Statistics
The distance between two nodes in a network is defined as the number of edges
needed to reach one node from another, and serves as the basis for a class of
network statistics.
A path in a network is a sequence of incident edges, or equivalently, a
sequence of nodes $P=(u_{0},u_{2},\dotsc,u_{k})$, such that
$(u_{i},u_{i+1})\in E$ for all $i\in\\{0,\dotsc,k-1\\}$. The number $k$ is
called the length of the path, and will also be denoted $l(P)$. A further
restriction can be set on the visited nodes, definining that each node can
only be visited at most once. If the distinction is made, the term _path_ is
usually reserved for sequences of non-repeating nodes, and general sequence of
adjacent nodes are then called _walks_. We will not make this distinction
here.
Paths in networks can be used to model browsing behavior of people in
hyperlink networks, navigation in transport networks, and other types of
movement-like activities in a network. When considering navigation and
browsing, an important problem is the search for shortest paths. Since the
length of a path determines the number of steps needed to reach one node from
another, it can be used as a measure of distance between nodes of a network.
The distance defined in this way may also be called the shortest-path distance
to distinguish it from other distance measures between nodes of a network.
$\displaystyle d(u,v)$
$\displaystyle=\left\\{\begin{array}[]{ll}\min_{P=(u,\dotsc,v)}l(P)&\text{when
$u$ and $v$ are connected}\\\ \infty&\text{when $u$ and $v$ are not
connected}\end{array}\right.$ (76)
In the case that a network is not connected, the distance is defined as
infinite. In practice, only the largest connected component of a network may
be used, making it unnecessary to deal with infinite values. The distribution
of all $|V|^{2}$ values $d(u,v)$ for all $u,v\in V$ is called the distance
distribution, and it too characterizes the network.
The eccentricity of a node can then be defined as the maximal distance from
that node to any other node, defining a measure of _non-centrality_ :
$\displaystyle\epsilon(u)$ $\displaystyle=\max_{v\in V}d(u,v)$ (77)
The diameter $\delta$ of a graph equals the longest shortest path in the
network. It can be equivalently defined as the largest eccentricity of all
nodes.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{diam}}\delta\@add@raggedright$
$\displaystyle=\max_{u\in V}\epsilon(u)=\max_{u,v\in V}d(u,v)$ (78)
Note that the diameter is undefined (or infinite) in unconnected networks, and
thus in numbers reported for actual networks in KONECT we consider always the
diameter of the network’s largest connected component. Du to the high runtime
complexity of computing the diameter, it may be estimated by various methods,
in which case it is noted noted $\tilde{\delta}$.
A statistic related to the diameter is the radius, defined as the smallest
eccentricity
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{radius}}r\@add@raggedright$
$\displaystyle=\min_{u\in V}\epsilon(u)=\min_{u\in V}\max_{v\in V}d(u,v)$ (79)
The diameter is bounded from below by the radius, and from above by twice the
radius.
$\displaystyle r\leq\delta\leq 2r$
The first inequality follows directly from the definition of $r$ and $\delta$
as the minimal and maximal eccentricity. The second inequality follows from
the fact that between any two nodes, the path joining them cannot be longer
that the path joining them going through a node with minimal eccentricity,
which has length of at most $2r$.
The radius and the diameter are not very expressive statistics: Adding or
removing an edge will, in many cases, not change their values. Thus, a better
statistic that reflects the typical distances in a network in given by the
mean and average distance.
The mean path length $\delta_{\mathrm{m}}$ in a network is defined as as the
mean distance over all node pairs, including the distance between a node and
itself:
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{meandist}}\delta_{\mathrm{m}}\@add@raggedright$
$\displaystyle=\frac{1}{n^{2}}\sum_{u\in V}\sum_{v\in V}d(u,v)$ (80)
The mean path length defined in this way is undefined when a graph is
disconnected.
††margin: mediandist
Likewise, the median path length $\delta_{\mathrm{M}}$ is the median length of
shortest paths in the network. In KONECT, both the median and mean path
lengths are computed taking into account node pairs of the form $(u,u)$.
Both the mean and median path length can be called the _characteristic path
length_ of the network.
A related statistic is the 90-percentile effective diameter $\delta_{0.9}$,
which equals the number of edges needed on average to reach 90% of all other
nodes.
### 4.7 Algebraic Statistics
Algebraic statistics are based on a network’s characteristic matrices. They
are motivated by the broader field of spectral graph theory, which
characterizes graphs using the spectra of these matrices [Chu97].
In the following we will denote by $\lambda_{k}[\mathbf{X}]$ the $k$th
dominant eigenvalue of the matrix $\mathbf{X}$. For the adjacency matrix
$\mathbf{A}$, the dominant eigenvalues are the largest absolute ones; for the
Laplacian $\mathbf{L}$ they are the smallest ones.
Also, the matrix $\mathbf{L}$ will only be considered for the network’s
largest connected component.
The spectral norm of a network equals the spectral norm (i.e., the largest
absolute eigenvalue) of the network’s adjacency matrix
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{snorm}}\left\|\mathbf{A}\right\|_{2}.\@add@raggedright$
$\displaystyle=|\lambda_{1}[\mathbf{A}]|$ (81)
The spectral norm can be understood as an alternative measure of the size of a
network.
The algebraic connectivity equals the second smallest nonzero eigenvalue of
$\mathbf{L}$ [Fie73]
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{alcon}}a\@add@raggedright$
$\displaystyle=\lambda_{2}[\mathbf{L}].$ (82)
The algebraic connectivity is zero when the network is disconnected – this is
one reason why we restrict the matrix $\mathbf{L}$ to each network’s giant
connected component. The algebraic connectivity is larger the better the
network’s largest connected component is connected.
In signed and ratings networks, i.e., networks in which the weights of node
pairs can be negative, the smallest eigenvalue of $\mathbf{L}$ can be larger
than zero. (In other networks, it is always zero.) The algebraic conflict
equals this smallest eigenvalue
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{conflict}}\xi\@add@raggedright$
$\displaystyle=\lambda_{1}[\mathbf{L}].$ (83)
The algebraic conflict measures the amount of conflict in the network, i.e.,
the tendency of the network to contain cycles with an odd number of negatively
weighted edges.
### 4.8 Bipartivity Statistics
Some unipartite networks are almost bipartite. Almost-bipartite networks
include networks of sexual contact [LEA+01] and ratings in online dating sites
[BP07, KGG12]. Other, more subtle cases, involve online social networks. For
instance, the follower graph of the microblogging service Twitter is by
construction unipartite, but has been observed to reflect, to a large extent,
the usage of Twitter as a news service [KLPM10]. This is reflected in the fact
that it is possible to indentify two kinds of users: Those who primarily get
followed and those who primarily follow. Thus, the Twitter follower graph is
almost bipartite. Other social networks do not necessarily have a near-
bipartite structure, but the question might be interesting to ask to what
extent a network is bipartite. To answer this question, measures of
bipartivity have been developed.
Instead of defining measures of bipartivity, we will instead consider measures
of non-bipartivity, as these can be defined in a way that they equal zero when
the graph is zero. Given an (a priori) unipartite graph, a measure of non-
bipartivity characterizes the extent to which it fails to be bipartite. These
measures are defined for all networks, but are trivially zero for bipartite
networks. For non-bipartite networks, they are larger than zero.
A first measure of bipartivity consists in counting the minimum number of
_frustrated edges_ [HLEK03]. Given a bipartition of vertices $V=V_{1}\cup
V_{2}$, a frustrated edge is an edge connecting two nodes in $V_{1}$ or two
nodes in $V_{2}$. Let $f$ be the minimal number of frustrated edges in any
bipartition of $V$, or, put differently, the minimum number of edges that have
to be removed from the graph to make it bipartite. Then, a measure of non-
bipartivity is given by
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{frustration}}F\@add@raggedright$
$\displaystyle=\frac{f}{|E|}.$ (84)
This statistic is always in the range $[0,1/2]$. It attains the value zero if
and only if $G$ is bipartite.
The minimal number of frustrated edges $f$ can be approximated by algebraic
graph theory. First, we represent a bipartition $V=V_{1}\cup V_{2}$ by its
characteristic vector $\mathbf{x}\in\mathbb{R}^{|V|}$ defined as
$\displaystyle\mathbf{x}_{u}$
$\displaystyle=\left\\{\begin{array}[]{ll}+1/2&\text{when $u\in V_{1}$}\\\
-1/2&\text{when $u\in V_{2}$}\end{array}\right.$
Note that the number of edges connecting the sets $V_{1}$ and $V_{2}$ is then
given by
$\displaystyle\left\\{\\{u,v\\}\mid u\in V_{1},v\in
V_{2}\right\\}=\frac{1}{2}\mathbf{x}^{\mathrm{T}}\mathbf{K}[\bar{G}]\mathbf{x}$
$\displaystyle=\frac{1}{2}\sum_{(u,v)\in
E}(\mathbf{x}_{u}+\mathbf{x}_{v})^{2},$
where $\mathbf{K}[\bar{G}]=\mathbf{D}[\bar{G}]+\mathbf{A}[\bar{G}]$ is the
signless Laplacian matrix of the underlying unweighted graph. Thus, the
minimal number of frustrated edges $f$ is given by
$\displaystyle f$ $\displaystyle=\min_{\mathbf{x}\in\\{\pm
1/2\\}^{|V|}}\frac{1}{2}\mathbf{x}^{\mathrm{T}}\mathbf{K}[\bar{G}]\mathbf{x}.$
By relaxing the condition $\mathbf{x}\in\\{\pm 1/2\\}^{|V|}$, we can express
$f$ in function of $\mathbf{K}[\bar{G}]$’s minimal eigenvalue, using the fact
that the norm of all vectors $\mathbf{x}\in\\{\pm 1/2\\}^{|V|}$ equals
$\sqrt{|V|/4}$, and the property that the minimal eigenvalue of a matrix
equals its minimal Rayleigh quotient.
$\displaystyle\frac{2f}{|V|/4}$
$\displaystyle\approx\min_{\mathbf{x}\neq\mathbf{0}}\frac{\mathbf{x}^{\mathrm{T}}\mathbf{K}[\bar{G}]\mathbf{x}}{\left\|\mathbf{x}\right\|^{2}}=\lambda_{\min}[\mathbf{K}[\bar{G}]]$
We can thus approximate the previous measure of non-bipartivity by
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{anticonflict}}\tilde{F}\@add@raggedright$
$\displaystyle=\frac{|V|}{8|E[\bar{G}]|}\lambda_{\min}[\mathbf{K}[\bar{G}]]$
(85)
The eigenvalue $\lambda_{\min}[\mathbf{K}[\bar{G}]]$ can also be interpreted
as the algebraic conflict in $G$ interpreted as a signed graph in which all
edges have negative weight.
A further measure of bipartivity exploits the fact that the adjacency matrix
$\mathbf{A}$ of a bipartite graph has eigenvalues symmetric around zero, i.e.,
all eigenvalues of a bipartite graph come in pairs $\pm\lambda$. Thus, the
ratio of the smallest and largest eigenvalues can be used as a measure of non-
bipartivity
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{nonbip}}b_{\mathrm{A}}\@add@raggedright$
$\displaystyle=1-\left|\frac{\lambda_{\min}[\mathbf{A}[\bar{G}]]}{\lambda_{\max}[\mathbf{A}[\bar{G}]]}\right|,$
(86)
where $\lambda_{\min}$ and $\lambda_{\max}$ are the smallest and largest
eigenvalue of the given matrix, and $\bar{G}$ is the unweighted graph
underlying $G$. Since the largest eigenvalue always has a larger absolute
value than the smallest eigenvalue (due to the Perron–Frobenius theorem, and
from the nonnegativity of $\mathbf{A}[\bar{G}]$), it follows that this measure
of non-bipartivity is always in the interval $[0,1)$, with zero denoting a
bipartite network.
Another spectral measure of non-bipartivity is based on considering the
smallest eigenvalue of the matrix $\mathbf{N}[\bar{G}]$. This eigenvalue is
$-1$ exactly when $G$ is bipartite. Thus, this value minus one is a measure of
non-bipartivity. Equivalently, it equals two minus the largest eigenvalue of
the normalized Laplacian matrix $\mathbf{Z}$.
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{nonbipn}}b_{\mathrm{N}}\@add@raggedright$
$\displaystyle=\lambda_{\min}[\mathbf{N}[\bar{G}]]+1=2-\lambda_{\max}[\mathbf{Z}[\bar{G}]]$
(87)
## 5 Features
A feature is a numerical characteristic of a node, such as the degree and the
eccentricity.
The degree is defined as the number ††margin: degree of neighbors of a node.
## 6 Plots
Plots are drawn to visualize a certain aspect of a dataset. These plots can be
used to compare several network visually, or to illustrate the definition of a
certain numerical statistic.
As a running example, we show the plots for the Wikipedia elections network
(EL). Plots for all networks (in which computation was feasible) are shown on
the KONECT website666konect.uni-koblenz.de/plots. The KONECT Toolbox contains
Matlab code for generating these plot types.
### 6.1 Temporal Distribution
The temporal distributions shows the distribution of edge creation times. It
is only defined for networks with known edge creation times. The X axis is the
time, and the Y axis is the number of edges added during each time interval.
Figure 2: The temporal distribution of edges for the Wikipedia elections
network.
### 6.2 Edge Weight and Multiplicity Distribution
The edge weight and multiplicity distribution plots show the distribution of
edge weights and of edge multiplicities, respectively. They are not generated
for unweighted networks. The X axis shows values of the edge weights or
multiplicities, and the Y axis shows frequencies. Edge multiplicity
distributions are plotted on doubly logarithmic scales.
(a) Edge weight distribution (b) Edge multiplicity distribution
Figure 3: The distribution of (a) edge weights for the MovieLens rating
network (M2) and (b) edge multiplicities for the German Wikipedia edit network
(de).
### 6.3 Degree Distribution
The distribution of degree values $d(u)$ over all vertices $u$ characterizes
the network as a whole, and is often used to visualize a network. In
particular, a power law is often assumed, stating that the number of nodes
with $n$ neighbors is proportional to $n^{-\gamma}$, for a constant $\gamma$
[BA99]. This assumption can be inspected visually by plotting the degree
distribution on a doubly logarithmic scale, on which a power law renders as a
straight line. KONECT supports two different plots: The degree distribution,
and the cumulative degree distribution. The degree distribution shows the
number of nodes with degree $n$, in function of $n$. The cumulative degree
distribution shows the probability that the degree of a node picked at random
is larger than $n$, in function of $n$. Both plots use a doubly logarithmic
scale.
Another visualization of the degree distribution supported by KONECT is in the
form of the Lorenz curve, a type of plot to measure inequality originally used
in economics (not shown).
(a) Degree distribution (b) Cumulative degree distribution
Figure 4: The degree distribution and cumulative degree distribution for the
Wikipedia election network (EL).
The Lorenz curve is a tool originally from economics that visualizes
statements of the form “X% of nodes with smallest degree account for Y% of
edges”. The set of values $(X,Y)$ thus defined is the Lorenz curve. In a
network the Lorenz curve is a straight diagonal line when all nodes have the
same degree, and curved otherwise [KP12]. The area between the Lorenz curve
and the diagonal is half the Gini coefficient (see above).
Figure 5: The Lorenz curve for the Wikipedia election network (EL).
### 6.4 Out/indegree Comparison
The out/indegree comparison plots show the joint distribution of outdegrees
and indegrees of all nodes of directed graphs. The plot shows, for one
directed network, each node as a point, which the outdegree on the X axis and
the indegree on the Y axis.
An example is shown in Figure 6 for the Wikipedia elections network.
Figure 6: The out/indegree comparison plot of the Wikipedia election network
(EL).
### 6.5 Assortativity Plot
In some networks, nodes with high degree are more often connected with other
nodes of high degree, while nodes of low degree are more often connected with
other nodes of low degree. This property is called assortativity, i.e., such
networks are said to be assortativity. On the other hand, some networks, are
dissortative, i.e., in them nodes of high degree are more often connected to
nodes of low degree and vice versa. In addition to the assortativity $\rho$
defined as the Pearson correlation coefficient between the degrees of
connected nodes, the assortativity or dissortativity of networks may be
analyse by plotting all nodes of a network by their degree and the average
degree of their neighbors. Thus, the assortativity plot of a network shows all
nodes of a network with the degree on the X axis, and the average degree of
their neighbors on the Y axis.
An example of the assortativity plot is shown for the Wikipedia elections
network in Figure 7.
Figure 7: The assortativity plot of the Wikipedia election network (EL).
### 6.6 Clustering Coefficient Distribution
In Section 4.5, we defined the clustering coefficient of a node in a graph as
the propotion of that node’s neighbors that are connected, and proceeded to
define the clustering coefficient as the corresponding measure applied to the
whole network. In some case however, we may be interested in the distribution
of the clustering coefficient over the nodes in the network. For instance, a
network could have some very clustered parts, and some less clustered parts,
while another network could have many nodes with a similar, average clustering
coefficient. Thus, we may want to consider the distribution of clustering
coefficient. This distribution can be plotted as a cumulated plot.
Figure 8: The clustering coefficient distribution for Facebook link network
(Ol).
### 6.7 Spectral Plot
The eigenvalues of a network’s characteristic matrices $\mathbf{A}$,
$\mathbf{N}$ and $\mathbf{L}$ are often used to characterize the network as a
whole. KONECT supports computing and visualizing the spectrum (i.e., the set
of eigenvalues) of a network in multiple ways. Two types of plots are
supported: Those showing the top-$k$ eigenvalues computed exactly, and those
showing the overall distribution of eigenvalues, computed approximately. The
eigenvalues of $\mathbf{A}$ are positive and negative reals, the eigenvalues
of $\mathbf{N}$ are in the range $[-1,+1]$, and the eigenvalues of
$\mathbf{L}$ are all nonnegative. For $\mathbf{A}$ and $\mathbf{N}$, the
largest absolute eigenvalues are used, while for $\mathbf{L}$ the smallest
eigenvalues are used. The number of eigenvalue shown $k$ depends on the
network, and is chosen by KONECT such as to result in reasonable runtimes for
the decomposition algorithms.
(a) Top-$k$ eigenvalues of $\mathbf{A}$ (b) Cumulative eigenvalue distribution
of $\mathbf{N}$
Figure 9: The top-$k$ eigenvalues of $\mathbf{A}$ and the cumulative spectral
distribution of $\mathbf{N}$ for the Wikipedia election network (EL). In the
first plot (a), positive eigenvalues are shown in green and negative ones in
red.
Two plots are generated: the non-cumulative eigenvalue distribution, and the
cumulative eigenvalue distribution. For the non-cumulative distribution, the
absolute $\lambda_{i}$ are shown in function of $i$ for $1\leq i\leq k$. The
sign of eigenvalues (positive and negative) is shown by the color of the
points (green and red). For the cumulated eigenvalue plots, the range of all
eigenvalues is computed, divided into 49 bins (an odd number to avoid a bin
limit at zero for the matrix $\mathbf{N}$), and then the number of eigenvalues
in each bin is computed. The result is plotted as a cumulated distribution
plot, with boxes indicating the uncertainty of the computation, due to the
fact that eigenvalues are not computed exactly, but only in bins.
### 6.8 Complex Eigenvalues Plot
The adjacency matrix of an undirected graph is symmetric and therefore its
eigenvalues are real. For directed graphs however, the adjacency matrix
$\mathbf{A}$ is asymmetric, and in the general case its eigenvalues are
complex. We thus plot, for directed graphs, the top-$k$ complex eigenvalues by
absolute value of the adjacency matrix $\mathbf{A}$.
Three properties can be read off the complex eigenvalues: whether a graph is
nearly acyclic, whether a graph is nearly symmetric, and whether a graph is
nearly bipartite. If a directed graph is acyclic, its adjacency matrix is
nilpotent and therefore all its eigenvalues are zero. The complex eigenvalue
plot can therefore serve as a test for networks that are nearly acyclic: the
smaller the absolute value of the complex eigenvalues of a directed graph, the
nearer it is to being acyclic. When a directed network is symmetric, i.e., all
directed edges come in pairs connecting two nodes in opposite direction, then
the adjacency matrix $\mathbf{A}$ is symmetric and therefore all its
eigenvalues are complex. Thus, a nearly symmetric directed network has complex
eigenvalues that are near the real line. Finally, the eigenvalues of a
bipartite graph are symmetric around the imaginary axis. In other words, if
$a+bi$ is an eigenvalue, then so is $-a+bi$ when the graph is bipartite. Thus,
the amount of symmetric along the imaginary axis is an indicator for
bipartivity. Note that bipartivity here takes into account edge directions:
There must be two groups such that all (or most) directed edges go from the
first group to second. Figure 10 shows two examples of such plots.
(a) Wikipedia elections (b) UC Irvine messages
Figure 10: The top-$k$ complex eigenvalues $\lambda_{i}$ of the asymmetric
adjacency matrix $\mathbf{A}$ of the directed Wikipedia election (EL) and UC
Irvine messages (UC) networks.
### 6.9 Distance Distribution Plot
Distance statistics can be visualized in the distance distribution plot. The
distance distribution plot shows, for each integer $k$, the number of node
pairs at distance $k$ from each other, divided by the total number of node
pairs. The distance distribution plot can be used to read off the diameter,
the median path length, and the 90-percentile effective diameter (see Section
4.6). For temporal networks, the distance distribution plot can be shown over
time.
The non-temporal distance distribution plot shows the cumulated distance
distribution function between all node pairs $(u,v)$ in the network, including
pairs of the form $(u,u)$, whose distance is zero.
The temporal distance distribution plot shows the same data in function of
time, with time on the X axis, and each colored curve representing one
distance value.
(a) Distance distribution plot (b) Temporal distance distribution plot
Figure 11: The distance distribution plot and temporal distance distribution
plot of the Wikipedia election network (EL).
### 6.10 Graph Drawings
A graph drawing is a representation of a graph, showing its vertices and egdes
laid out in two (or three) dimensions in order for the graph structure to
become visible. Graph drawings are easy to produce when a graph is small, and
become harder to generate and less useful when a graph is larger.
Given a graph, a graph drawing can be specified by the placement of its
vertices in the plane. To determine such a placement is a non-trivial problem,
for which many algortihms exist, depending on the required properties of the
drawing. For instance, each vertex should be placed near to its neighbors,
vertices should not be drawn to near to each other, and edges should, if
possible, not cross each other. It is clear that it is impossible to fulfill
all these requirements at once, and thus no best graph drawing exists.
In KONECT, we show drawings of small graphs only, such that vertices and edges
remain visible. The graph drawings in KONECT are spectral graph drawings,
i.e., they are based on the eigenvectors of characteric graph matrices. In
particular, KONECT included graph drawings based on the adjacency matrix
$\mathbf{A}$, the normalized adjacency matrix $\mathbf{N}$ and the Laplacian
matrix $\mathbf{L}$. Let $\mathbf{x}$ and $\mathbf{y}$ be the two chosen
eigenvector of each matrix, then the coordinate of the node $u\in V$ is given
by $\mathbf{x}_{u}$ and $\mathbf{y}_{u}$.
For the adjacency matrix $\mathbf{A}$ and the normalized adjacency matrix
$\mathbf{N}$, we use the two eigenvector with largest absolute eigevalue. For
the Laplacian matrix $\mathbf{L}$, we use the two eigenvectors with smallest
nonzero eigenvalue. Examples for the Zachary karate club social network (ZA)
are shown in Figure 12.
(a) Adjacency matrix $\mathbf{A}$ (b) Normalized adjacency matrix $\mathbf{N}$
(c) Laplacian $\mathbf{L}$
Figure 12: Drawings of the Zachary karate club social network (ZA) using (a)
the adjacency matrix $\mathbf{A}$, (b) the normalized adjacency matrix
$\mathbf{N}$, (c) the Laplacian matrix $\mathbf{L}$.
## 7 Matrices and Matrix Decompositions
In this section, we review characteristic graph matrices, their
decompositions, and their uses.
Matrix decompositions are implemented in the KONECT Toolbox by the
konect_decomposition() function. Each decomposition has a name, which is given
in the margin in the following.
### 7.1 Undirected Graphs
These matrices and decompositions apply to undirected graphs natively.
#### 7.1.1 Symmetric Adjacency matrix
The symmetric adjacency matrix $\mathbf{A}$ is the most basic graph
characteristic matrix. It is a symmetric $n\times n$ matrix defined as
$\mathbf{A}_{uv}=1$ when the nodes $u$ and $v$ are connected, and
$\mathbf{A}_{uv}=0$ when $u$ and $v$ are not connected.
The eigenvalue decomposition of the matrix $\mathbf{A}$ for undirected graphs
is widely used to analyse graphs:
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{sym}}\mathbf{A}\@add@raggedright$
$\displaystyle=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{\mathrm{T}}$ (88)
$\mathbf{\Lambda}$ is an $n\times n$ real diagonal matrix containing the
eigenvalues of $\mathbf{A}$, i.e.,
$\mathbf{\Lambda}_{ii}=\lambda_{i}[\mathbf{A}]$. $\mathbf{U}$ is an $n\times
n$ orthogonal matrix having the corresponding eigenvectors as columns.
The largest absolute eigenvalue of $\mathbf{A}$ is the networks spectral norm,
i.e.,
$\displaystyle\max_{i}|\Lambda_{ii}|$
$\displaystyle=\left\|\mathbf{A}\right\|_{2}.$
The sum of all eigenvalues $\lambda_{i}$ equal the trace of $\mathbf{A}$,
i.e., the sum of its diagonal elements. The sum of the eigenvalues of
$\mathbf{A}$ thus equals the number of loops in the graphs. In particular,
when a graph has no loops, then the sum of the eigenvalues of its adjacency
matrix is zero.
Higher moments the eigenvalues of $\mathbf{A}$ give the number of tours in the
graph. Remember that a tour of length $k$ is defined as a sequence of $k$
connected nodes, such that the first and the last node are connected, such
that two tours are considered as distinct when they have a different starting
node or orientation. The sum of $k$th powers of the eigenvalues of
$\mathbf{A}$ then equals the number of $k$-tours $T_{k}$. We thus have in a
loopless graph, that the traces of powers of $\mathbf{A}$ are related to the
number of edges $m$, the number of triangles $t$, the number of squares $q$
and the number of wedges $s$ by:
$\displaystyle\mathrm{Tr}(\mathbf{A})$ $\displaystyle=0$
$\displaystyle\mathrm{Tr}(\mathbf{A}^{2})$ $\displaystyle=2m$
$\displaystyle\mathrm{Tr}(\mathbf{A}^{3})$ $\displaystyle=6t$
$\displaystyle\mathrm{Tr}(\mathbf{A}^{4})$ $\displaystyle=8q+4s+2m$
The traces of $\mathbf{A}$ can also be expressed as sums of powers (moments)
of the eigenvalues of $\mathbf{A}$:
$\displaystyle\mathrm{Tr}(\mathbf{A}^{k})$
$\displaystyle=\sum_{i=1}^{n}\lambda_{i}^{k}$
The spectrum of $\mathbf{A}$ can also be characterized in terms of graph
bipartivity. When the graph is bipartite, then all eigenvalues come in pairs
$\\{\pm\lambda\\}$, i.e., they are distributed around zero symmetrically. When
the graph is not bipartite, then their distribution is not symmetric. It
follows that when the graph is bipartite, the smallest and largest eigenvalues
have the same absolute value.
#### 7.1.2 Laplacian Matrix
The Laplacian matrix of an undirected graph is defined as
$\displaystyle\mathbf{L}$ $\displaystyle=\mathbf{D}-\mathbf{A},$
i.e., the diagonal degree matrix from which we subtract the adjacency matrix.
We consider the eigenvalue decomposition of the Laplacian:
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{lap}}\mathbf{L}\@add@raggedright$
$\displaystyle=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{\mathrm{T}}$
The Laplacian matrix of positive-semidefinite, i.e., all eigenvalues are
nonnegative. When the graph is unsigned, the smallest eigenvalue is zero and
its multiplicity equals the number of connected components in the graph.
The second-smallest eigenvalue is called the algebraic connectivity of the
graph, and is denoted $a=\lambda_{2}[\mathbf{L}]$ [Fie73]. If the graph is
unconnected, that value is zero, i.e., an unconnected graph has an algebraic
connectivity of zero.
When the graph is connected, the eigenvector corresponding to eigenvalue zero
is a constant vector, i.e., a vector with all entries equal. The eigenvector
corresponding the the second-smallest eigenvalue is called the Fiedler vector,
and can be used to cluster nodes in the graph. Together with further
eigenvectors, it can be used to draw graphs [KSLL10].
When the graph is signed, i.e., when the grpah admits edges with negative
weights, then the smallest eigenvalue of $\mathbf{L}$ is called the algebraic
conflict $\xi$. It is zero if and only if the graph is balanced, i.e., when
the nodes can be divided into two groups such that all positive edges connect
nodes within the same group, and all negative edges connect nodes of different
groups. Equivalently, $\xi$ is larger than zero if and only if each connected
component contains at least one cycle with an odd number of negative edges.
#### 7.1.3 Normalized Adjacency Matrix
The normalized adjacency matrix $\mathbf{N}$ of an undirected graph is defined
as
$\displaystyle\mathbf{N}$
$\displaystyle=\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2},$
where we remind the reader that the diagonal matrix $\mathbf{D}$ contains the
node degrees, i.e., $\mathbf{D}_{uu}=d(u)$. The matrix $\mathbf{N}$ is
symmetric and its eigenvalue decomposition can be considered:
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{sym-n}}\mathbf{N}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{\mathrm{T}}\@add@raggedright$
(89)
The eigenvalues $\lambda_{i}$ of $\mathbf{N}$ can be used to characterize thge
graph, in analogy with those of the nonnormalized adjacency matrix. All
eigenvalues of $\mathbf{N}$ are contained in the range $[-1,+1]$. When the
graph is insigned, the largest eigenvalue is one. Minus one is the smallest
eigenvalue if and only iff the graph is bipartite. As with the nonnormalized
adjacency matrix, the eigenvalues of $\mathbf{N}$ are distributed
symmetrically around zero if and only if the graph is bipartite.
When the graph is connected, the eigenvector corresponding to eigenvalue one
has entries proportional to the square root of node degrees, i.e.
$\displaystyle\mathbf{U}_{u1}$ $\displaystyle=\sqrt{\frac{d(u)}{2m}}$
#### 7.1.4 Normalized Laplacian Matrix
The Laplacian matrix too, can be normalized. It turns out that the normalized
Laplacian and the normalized adjacency matrix are tighly related to each
other: They share the same set of eigenvectors, and their eigenvalues are
reflectopms of each other.
The normalized Laplacian matrix of an undirected graph is defined as
$\displaystyle\mathbf{Z}$
$\displaystyle=\mathbf{D}^{-1/2}\mathbf{L}\mathbf{D}^{-1/2}.$
As opposed to $\mathbf{A}$, $\mathbf{L}$ and $\mathbf{N}$, there is no
standardized notation of the normalized Laplacian. The notation $\mathbf{Z}$
is specific to KONECT.
The normalized Laplacian is related to the normalized adjacency matrix by
$\displaystyle\mathbf{Z}$ $\displaystyle=\mathbf{I}-\mathbf{N},$
as can be derived directly from their definitions. It follows that
$\mathbf{Z}$ and $\mathbf{N}$ have the same set of eigenvectors, and that
their eigenvalues are related by the transformation $1-\lambda$. Thus, the
properties of $\mathbf{Z}$ can be derived from those of $\mathbf{N}$. For
instance, all eigenvalues of $\mathbf{Z}$ are contained in the range $[0,2]$,
and the multiplicity of the eigenvalue zero equals the number of connected
components (when the graph is unsigned). If the graph is connected, the
eigenvector of eigenvalue zero contains entries proportional to the square
root of the node degrees.
In KONECT, the decomposition of the normalized Laplacian is not included,
since it can be derived from that of the normalized adjacency matrix.
#### 7.1.5 Stochastic Adjacency Matrix
The matrix
$\displaystyle\mathbf{P}$ $\displaystyle=\mathbf{D}^{-1}\mathbf{A}$
is called the stochastic adjacency matrix. This matrix is asymmetric, even
when the graph is undirected. ††margin: stoch1 Thus, its eigenvalue
decomposition is not always defined, and in any case may not involve
orthogonal matrices.
For directed graphs we may distinguish the right-stochastic (or row-
stochastic) matrix $\mathbf{D}^{-1}\mathbf{A}$ and the left-stochastic (or
column-stochastic) matrix $\mathbf{A}\mathbf{D}^{-1}$. Note the subtle
terminology: $\mathbf{D}^{-1}\mathbf{A}$ is left-normalized but right-
stochastic.
The eigenvalues of $\mathbf{P}$ are contained in the range $[-1,+1]$. This
matrix is related to the normalized adjacency matrix $\mathbf{N}$ by
$\displaystyle\mathbf{P}$
$\displaystyle=\mathbf{D}^{-1/2}\mathbf{N}\mathbf{D}^{1/2}$
and therefore both matrices have the same set of eigenvalues, and the
eigenvectors of $\mathbf{P}$ are related to those of $\mathbf{N}$ by factors
of the diagonal elements of $\mathbf{D}^{1/2}$, i.e., the square roots of node
degrees. As a consequence, $\mathbf{P}$ has all-real eigenvalues for
undirected graphs (even though it is asymmetric). Since $\mathbf{P}$ is
asymmetric, its left eigenvectors differ from its right eigenvectors. When the
graph is undirected, the left eigenvector corresponding to the eigenvalue one
has entries proportional to the degree of nodes, while the right eigenvector
corresponding to the eigenvalue one is the constant vector.
The matrix $\mathbf{P}$ is the state transition matrix of a random walk on the
graph, and thus its largest eigenvector is one if the graph is (strongly)
connected. The matrix $\mathbf{P}$ is also related to the PageRank matrix,
which equals $(1-\alpha)\mathbf{P}+\alpha\mathbf{J}$ for some number
$0<\alpha<1$, where $\mathbf{J}$ is the matrix containing all ones. The left
eigenvalues of the PageRank matrix give the PageRank values, and thus we see
that (ignoring the teleportation term), the PageRank of nodes in an undirected
network equals the degrees of the nodes.
The alternative matrix $\mathbf{A}\mathbf{D}^{-1}$ can also be considered.
††margin: stoch2 It is left-stochastic, and can be derived by considering
random walks that tranverse edges in a backward direction.
#### 7.1.6 Stochastic Laplacian Matrix
A further variant of the Laplacian exists, based on the stochastic adjacency
matrix:
$\displaystyle\mathbf{S}$
$\displaystyle=\mathbf{I}-\mathbf{P}=\mathbf{I}-\mathbf{D}^{-1}\mathbf{A}=\mathbf{I}-\mathbf{D}^{-1/2}\mathbf{N}\mathbf{D}^{1/2}=\mathbf{D}^{-1/2}\mathbf{Z}\mathbf{D}^{1/2}$
This matrix shares much properties with $\mathbf{P}$ and thus with
$\mathbf{N}$ and $\mathbf{Z}$.
#### 7.1.7 Signless Laplacian
The signless Laplacian of a graph is defined as the Laplacian of the
corresponding graph in which all edges are interpreted as negative. It thus
equals
$\displaystyle\marginpar{\normalcolor\raggedright\texttt{lapq}}\mathbf{K}\@add@raggedright$
$\displaystyle=\mathbf{D}+\mathbf{A}$ (90)
This matrix is positive-semidefinite, and its smallest eigenvalue is zero if
and only if the graph is bipartite. Thus, $\mathbf{K}$ is used in measures of
bipartivity.
## 8 KONECT Toolbox
The KONECT Toolbox777konect.uni-koblenz.de/toolbox for Matlab is a set of
functions for the Matlab programming
language888www.mathworks.com/products/matlab containing implementations of
statistics, plots and other network analysis methods. The KONECT Toolbox is
used to generate the numerical statistics and plots in this handbook as well
as on the KONECT website.
##### Installation
The KONECT Toolbox is provided as a directory containing *.m files. The
directory can be added to the Matlab path using addpath() to be used.
##### Usage
All functions have names beginning with konect_.
### 8.1 Examples
This section gives short example for using the toolbox. The examples can be
executed in Matlab.
##### Load a unipartite dataset
This example loads the Slashdot signed social network.
T = load(’out.slashdot-zoo’);
n = max(max(T(:,1:2)));
A = sparse(T(:,1), T(:,2), T(:,3), n, n);
This loads the weighted adjacency matrix of the Slashdot Zoo into the matrix
A.
### 8.2 Variables
Naming variables can be quite complicated and hard to read in Matlab.
Therefore KONECT code follows these rules.
Long variable names (containing full words) are in all-lowercase. Words are
separated by underscore. When refering to a variable in comments, the variable
is written in all-uppercase. Short variable names (letters) are lowercase for
numbers and vectors, and uppercase for matrices.
#### 8.2.1 Strings
Table 5 shows common variable names used for string variables.
Table 5: Long variable names of string type used in KONECT. network | The internal network name, e.g., “advogato”. The internal network name is used in the names of files related to the network.
---|---
class | The internal name for a set of networks, e.g., “test”, “1”, “2”, “3”. The class “N” includes the $10\times N$ smallest networks.
code | The 1/2/3-character code for a network, e.g., “EN” for Enron.
curve | The internal name of a curve fitting method.
decomposition | The internal of a matrix decomposition, as passed to the function konect_decomposition(), e.g., “sym”, “asym” and “lap”.
feature | The internal name of a feature, e.g., “degree” and “decomp.sym”.
filename | A filename.
format | The network format in lower case as defined in the function konect_consts(), e.g., “sym” and “bip”.
label | The readable name of things used in plots, tables, etc.
measure | The internal name of a measure of link prediction accuracy, e.g., “map” and “auc”.
method | The internal name of a link prediction method.
statistic | The internal name of a network statistic, e.g., “power” and “alcon”.
transform | The name of a transform, e.g. “simple” and “lcc”.
type | The internal name of the computation type. This can be “split” or “full”. This decides which version of a network gets used, in particular for time-dependent analyses.
weights | The edge weight type as defined in the function konect_consts(), e.g., “unweighted” and “signed”.
#### 8.2.2 Scalars
Table 6 shows variable names used for scalar values.
Table 6: Variable names used for scalars in KONECT. n, n1, n2 | Row/column count in matrices, left/right vertex count
---|---
r | Rank of a decomposition
m | Edge count
i, j | Vertices as integer, i.e., indexes in rows and columns.
prediction | A link prediction score, i.e., a value returned by a link prediction algorithm for a given node pair.
precision | The prediction accuracy value, typically between 0 and 1.
means | Values used for additive (de)normalization, as a structure.
#### 8.2.3 Matrices
Table 7 shows variable names used for matrix-valued variables.
Note that when the adjacency matrix of an undirected graph is stored in a
variable, each edge is usually stored just once, instead of twice. In other
words, the variable A for undirected networks does not equal the matrix
$\mathbf{A}$, instead the expression A + A’ does.
Table 7: Variable names used for matrices in KONECT. A | ($n\times n$) Adjacency matrix (in code where the adjacency and biadjacency matrix are distinguished)
---|---
A | ($n\times n$ or $n_{1}\times n_{2}$) Adjacency or biadjacency matrix (in code where the two are not distinguished)
B | ($n_{1}\times n_{2}$) Biadjacency matrix (in code where the adjacency and biadjacency matrix are distinguished)
D | ($r\times r$) Central matrix; e.g., eigenvalues; as matrix
dd | ($r\times 1$) Diagonal of the central matrix
L | ($n\times n$) Laplacian matrix
M, N | Normalized (bi)adjacency matrix
T | ($m\times 2$ or $m\times 3$ or $m\times 4$) Compact adjacency matrix, as stored in out.* files, and such that it can be converted to a sparse matrix using konect_spconvert().
| First column: row IDs
| Second column: column IDs
| Third column (optional): edge weights (1 if not present)
| Fourth column (optional): timestamps in Unix time
U | ($n\times r$ or $n_{1}\times r$) Left part of decomposition; e.g., left eigenvectors
V | ($n\times r$ or $n_{2}\times r$) Right part of decomposition; e.g., right eigenvectors
X | ($r\times r$) Central matrix, when explicitly nondiagonal
Z | ($n\times n$) Normalized Laplacian matrix
#### 8.2.4 Compound Types
A struct containing elements whose names are of a specific type are named
[VALUETYPE]s_[KEYTYPE]. For instance, a struct with labels used for methods is
named as follows:
labels_method.(’auc’) = ’Area under the curve’;
Note:
* •
The first element is the name of the content type.
* •
The plural is used only for the content type.
#### 8.2.5 IDs
Variables named method, decomposition, etc. are always strings. If a method,
decomposition or any other type is represented as an integer (e.g., as an
index into an array), then _id is appended to the variable name. For instance:
decomposition = ’sym’; decomposition_id = 2;
This means that an array of values by ID of keys is called for instance:
labels_decomposition_id{1} = ’Eigenvalue decomposition’;
labels_decomposition_id{2} = ’Singular value decomposition’;
## 9 File Formats
Due to the ubiquity of networks in many areas, there are a large number of
file formats for storing graphs and graph-like structures. Some of these are
well-suited for accessibility from many different programming languages
(mostly line-oriented text formats), some are well-suited for integration with
other formats (semantic formats such as RDF and XML-based ones), while other
formats are optimized for efficient access (binary formats). In KONECT, we
thus use three file formats covering the three cases:
* •
Text format: This format is text-based and uses tab-separated values. This is
the main KONECT data format from which the two others are derived. The format
has the advantage that it can be read easily from many different programming
languages and environment.
* •
RDF format: Datasets are also available as RDF. This is intended for easy
integration with other datasets.
* •
Matlab format: To compute statistics and plots and perform experiments, we use
Matlab’s own binary format, which can be accessed efficiently from within
Matlab.
In the following, we describe KONECT’s text format. Each network $NETWORK is
represented by the following files:
* •
out.$NETWORK: The edges stored as tab separated values (TSV). The file is a
text file, and each line contains information about one edge. Each line
contains two, three or four numbers represented textually, and separated by
any sequence of whitespace (most KONECT code uses a single tab character when
generating such files). The first two columns are mandatory and contain the
source and destination node ID of the edge. The third column is optional and
contains the edge weight. When the network is dynamic, the third column
contains $+1$ for added edges and $-1$ for removed edges. For unweighted, non-
temporal networks, multiple edges may be aggregated into a single line
containing, in the third column, the number of aggregated edges. The fourth
column is optional and contains the edge creation time, and is stored as UNIX
time, i.e., the number of seconds since 1 January 1970. The fourth column is
usually an integer, but may contain floating point numbers. If the fourth
column is present, the third column must also be given. The beginning of the
file contains additional comment lines with the following information:
% FORMAT WEIGHTS
% RELATIONSHIP-COUNT SUBJECT-COUNT OBJECT-COUNT
where FORMAT is the internal name for the format as given in Table 1, WEIGHTS
is the internal name for the weight types as given in Table 2, RELATIONSHIP-
COUNT is the number of data lines in the file, and SUBJECT-COUNT and OBJECT-
COUNT both equal the number of nodes $n$ in unipartite networks, and the
number of left and right nodes $n_{1}$ and $n_{2}$ in bipartite networks. The
first line is mandatory; the second line is optional.
* •
meta.$NETWORK: This file contains metadata about the network that is
independent of the mathematical structure of the network. The file is a text
file coded in UTF-8. Each line contains one key/value pair, written as the
key, a colon and the value. The following metadata are used:
* –
name: The name of the dataset (usually only the name of the source, without
description the type or category, e.g., “YouTube”, “Wikipedia elections”). The
name uses sentence case. For networks with the same name the source (e.g., the
conference) is added in parentheses. Within each category, all names must be
distinct.
* –
code: The short code used in plots and narrow tables. The code consists of two
or three characters. The first two characters are usually uppercase letters
and denote the data source. The last character, if present, usually
distinguishes the different networks from one source.
* –
url: (optional) The URL(s) of the data sources, as a comma separated list.
Most datasets have a single URL.
* –
category: The name of the category, as given in the column “Category” in Table
3.
* –
description: (deprecated) A short description of the form “User–movie
ratings”. Note that the file should contain an actual en dash, coded in UTF-8.
* –
cite: (optional) The bibtex code(s) for this dataset, as a comma separated
list. Most dataset have a single bibtex entry.
* –
fullname: (optional) A longer name to disambiguate different datasets from the
same source, e.g., “Youtube ratings” and “Youtube friendships”. Uses sentence
case. All networks must have different fullnames.
* –
long-description: (optional) A long descriptive text consisting of full
sentences, and describing the dataset in a verbose way. HTML markup may be
used sparingly.
* –
entity-names: A comma-seperated list of entity names (e.g., “user, movie”).
Unipartite networks give a single name; bipartite networks give two.
* –
relationship-names: The name of the relationship represented by edges, as a
substantive (e.g., “friendship”).
* –
extr: (optional) The name of the subdirectory that contains the extraction
code for this dataset.
* –
timeiso: (optional) A single ISO timestamp denoting the date of the dataset or
two timestamps separated by a slash(/) for a time range. The format is:
YYYY[-MM[-DD]][/YYYY[-MM[-DD]]], e.g., “2005-10-08/2006-11-03” or “2007”.
* –
tags: (optional) A space-separated list of hashtags describing the network.
The following tags are used:
* *
#acyclic: The network is acyclic. Can only be set for directed networks. If
this is not set, a directed network must contain at least two pairs of
reciprocal edges of the form $(u,v)$ and $(v,u)$.
* *
#incomplete: The network is incomplete, i.e., not all edges or nodes are
included. This implies that for instance its degree distribution is not
meaningful.
* *
#join: The network is actually the join of more fundamental networks. For
instance, a co-authorship network is a join of the authorship network with
itself. Networks that have this tag may have skewed properties, such as skewed
degree distributions.
* *
#kcore: The network contains only nodes with a certain minimal degree $k$. In
other words, the nodes with degree less than a certain number $k$ were removed
from the dataset. This changes a network drastically, and is called the
“$k$-core” of a network. The is sometimes done to get a less sparse network
applications that do not perform well on sparse networks. This tag implies the
#incomplete tag.
* *
#missingorientation: This tag is used for undirected networks which are based
on an underlying directed network. For instance, in a citation network, we may
only know that the documents A and B are linked, but not which one cites the
other. In such a case, the network in KONECT is undirected, although the
underlying network is actually directed.
* *
#lcc: The dataset actually contains only the largest connected component of
the actual network. Implies #incomplete. This tag is not used when the network
is connected for other reasons.
* *
#loop: The network may contain loops, i.e., egdes connecting a vertex to
itself. This tag is only allowed for unipartite networks. When this tag is not
present, loops are not allowed, and the presence of loops will be considered
an error by analysis code.
* *
#nonreciprocal: For directed networks only. The network does not contain
reciprocal edges.
* *
#regenerate: The network can be regenerated periodically and may be updated
when a more recent dataset becomes available.
* *
#zeroweight: Must be set if it is allowed for edge weights to be zero. Only
used for networks with positive edge weights and signed networks.
* –
n3-*: (optional) Metadata which is used for the generation of RDF files. The
symbol {n} in the name of the meta key represents an order by unique,
sequential numbers starting at 1.
* *
n3-add-prefix{n} (optional): Used to define additional N3 prefixes. The
default prefixes are specified in this way.
* *
n3-comment-{n} (optional): Add commentary lines which are placed at the
beginning of the N3 file.
* *
n3-edgedata-{n} (optional): Additional N3-data, to be displayed with each
edge.
* *
n3-nodedata-m-{n} (optional): Additional N3-data, to be displayed with the
first occurence of the source ID.
* *
n3-nodedata-n-{n} (optional): Additional N3-data, to be displayed with the
first occurence of the target ID.
* *
n3-prefix-m: N3-prefix for the source IDs.
* *
n3-prefix-n (optional): N3-prefix for the target IDs. If this field is left
out, the value of {n3-prefix-m} is used.
* *
n3-prefix-j (optional): Additional prefix which can be used with the source
id, if there is an entity to be represented with the same id.
* *
n3-prefix-k (optional): Additional prefix which can be used with the target
id, if there is an entity to be represented with the same id. This is used for
example in meta.facebook-wosn-wall for the representation of users walls.
* *
n3-prefix-l (optional): N3-prefix for the edges, if they are to be represented
by some N3-entity.
* *
n3-type-l (optional): RDF-type for the edges.
* *
n3-type-m: RDF-type for source IDs.
* *
n3-type-n (optional): RDF-type for target IDs.
The following symbols are used in the n3-expressions for edgedata and
nodedata:
* $m
: n3-prefix-m + source ID
* $n
: n3-prefix-n (or n3-prefix-m if the other is undefined) + target ID
* $j
: source ID
* $k
: target ID
* $l
: edge ID
* $timestamp
: edge timestamp
## Acknowledgments
The Koblenz Network Collection would not have been possible without the effort
of many people who have published network datasets. KONECT is maintained by
Jérôme Kunegis, Daniel Dünker and Holger Heinz. KONECT was also support by
funding from multiple research projects. The research leading to these results
has received funding from the European Community’s Seventh Frame Programme
under grant agreement no 257859, ROBUST and 287975, SocialSensor.
## References
* [BA99] Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286(5439):509–512, 1999.
* [BKM08] Shweta Bansal, Shashank Khandelwal, and Lauren Ancel Meyers. Evolving clustered random networks. CoRR, abs/0808.0509, 2008.
* [Bol98] Béla Bollobás. Modern Graph Theory. Springer, 1998.
* [BP07] Lukáš Brožovský and Václav Petříček. Recommender system for online dating service. In Proc. Conf. Znalosti, pages 29–40, 2007.
* [Chu97] Fan Chung. Spectral Graph Theory. American Math. Soc., 1997.
* [ER59] Paul Erdős and Alfréd Rényi. On random graphs I. Publ. Math. Debrecen, 6:290–297, 1959.
* [Fie73] Miroslav Fiedler. Algebraic connectivity of graphs. Czechoslovak Math. J., 23(98):298–305, 1973.
* [GO12] David Gleich and Art Owen. Moment-based estimation of stochastic Kronecker graph parameters. Internet Math., 8(3):232–256, 2012.
* [GR01] Chris D. Godsil and Gordon Royle. Algebraic Graph Theory. Springer, 2001.
* [HLEK03] Petter Holme, Fredrik Liljeros, Christofer R. Edling, and Beom Jun Kim. Network bipartivity. Phys. Rev. E, 68(5):056107, 2003.
* [KGG12] Jérôme Kunegis, Gerd Gröner, and Thomas Gottron. Online dating recommender systems: The split-complex number approach. In Proc. Workshop on Recommender Systems and the Social Web, pages 37–44, 2012.
* [KL09] Jérôme Kunegis and Andreas Lommatzsch. Learning spectral graph transformations for link prediction. In Proc. Int. Conf. on Machine Learning, pages 561–568, 2009.
* [KLB09] Jérôme Kunegis, Andreas Lommatzsch, and Christian Bauckhage. The Slashdot Zoo: Mining a social network with negative edges. In Proc. Int. World Wide Web Conf., pages 741–750, 2009.
* [KLPM10] Haewoon Kwak, Changhyun Lee, Hosung Park, and Sue Moon. What is Twitter, a social network or a news media? In Proc. Int. World Wide Web Conf., pages 591–600, 2010.
* [KP12] Jérôme Kunegis and Julia Preusse. Fairness on the web: Alternatives to the power law. In Proc. Web Science Conf., pages 175–184, 2012.
* [KSLL10] Jérôme Kunegis, Stephan Schmidt, Andreas Lommatzsch, and Jürgen Lerner. Spectral analysis of signed graphs for clustering, prediction and visualization. In Proc. SIAM Int. Conf. on Data Mining, pages 559–570, 2010.
* [LEA+01] Fredrik Liljeros, Christofer R. Edling, Luís A. Nunes Amaral, H. Eugene Stanley, and Yvonne Åberg. The web of Human sexual contact. Nature, 411:907–908, June 2001.
* [New06] M. E. J. Newman. Power laws, Pareto distributions and Zipf’s law. Contemporary Phys., 46(5):323–351, 2006.
* [NWS02] M. E. J. Newman, D. J. Watts, and S. H. Strogatz. Random graph models of social networks. Proc. Natl. Acad. Sci. USA, 99:2566–2572, 2002.
* [OAS10] Tore Opsahl, Filip Agneessens, and John Skvoretz. Node centrality in weighted networks: Generalizing degree and shortest paths, 2010. Preprint submitted to Social Networks.
* [Ops12] Tore Opsahl. Triadic closure in two-mode networks: Redefining the global and local clustering coefficients. Social Networks, 34, 2012.
* [WS98] Duncan J. Watts and Steven H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393(1):440–442, 1998.
## Appendix A Glossary of Terms
The area of graph theory and network analysis is still recent enough that
there is no unified glossary across the literature. The choices made in this
work are those of the authors, and were chosen to reflect best practices and
to avoid confusion.
Arc
A directed edge. In general, we consider arcs to be a special cases of edges,
and thus we rarely use the term _arc_ in favor of _directed edge_.
Category
Networks have a category, which describes the domain they apply to: social
networks, transport networks, citation networks, etc.
Central matrix
The matrix $\mathbf{X}$ in the decomposition
$\mathbf{U}\mathbf{X}\mathbf{V}^{\mathrm{T}}$, not necessarily diagonal or
symmetric; a generalization of the diagonal eigenvalue matrix
Class
The networks of KONECT are divided into classes by their volume: Class 1
contains the ten smallest networks, Class 2 contains the next ten smallest
networks, etc.
Claw
Three edges sharing a single vertex. A claw can be understood as a 3-star.
Code
The two- or three-character code representation of a network. These are used
in scatter plots that show many networks.
Cross
Four edges sharing a single endpoint. Also called a 4-star.
Curve
A curve fitting method used for link prediction, when using the link
prediction method described in [KL09] (learning spectral transformations).
Cycle
A cyclic sequence of connected edges, not containing any edge twice. A cycle
contrasts with a tour, in which a single vertex can appear multiple times.
Decomposition
In KONECT the word _decomposition_ is used to denote the combination of a
characteristic graph matrix (e.g. the adjacency matrix or Laplacian) with a
matrix decomposition. As an extension, some other constructions are also
called _decomposition_ , such as LDA.
Density
This word is avoided in KONECT. In the literature, it may refer to either the
fill (probability that a node exists), or to the average degree. The former
definition is used in mathematical contexts, while the latter is used in
computer science contexts.
Edge
A connection between two nodes.
Feature
A node feature. I.e., a number assigned to each node. Examples are the degree,
PageRank and the eccentricity. Equivalently, a node vector.
Fill
The probability that two randomly chosen nodes are connected. Also called the
_density_ , in particular in a mathematical context.
Format
The format of a network determines its general structure, and whether edges
are directed. There are three possible formats: unipartite and undirected;
unipartite and directed; and bipartite. Directed bipartite networks are not
possible. Possible future extensions would include hypergraphs (e.g.,
tripartite networks).
Half-adjacency matrix
The adjacency matrix $\mathbf{A}$ of an undirected graph contains two nonzero
entries for each edge $\\{i,j\\}$: $\mathbf{A}_{ij}$ and $\mathbf{A}_{ji}$. To
avoid this, KONECT code uses the half-adjacency matrix, which contains only
one of the two nonzero entries. The half-adjacency matrix is therefore not
unique. In code, the half-adjacency matrix is denoted A.
Measure
A measure of the accuracy of link prediction methods, for instance the area
under the curve or the mean average precision.
Method
A link prediction method.
Path
A sequence of connected nodes, in which each node can appear only once. The
extension that allows multiple nodes is called a walk.
Score
A numerical value given to a node pair. Usually used for link prediction, but
can also measure distance or similary between nodes.
Size
The number of nodes in a network.
Statistic
A statistic is a numerical measure of a network, i.e., a number that describes
a network, such as the clustering coefficient, the diameter or the algebraic
connectivity. All statistics are real numbers.
Tour
A cyclic sequence of connected nodes which may contain a single vertex
multiple times. It can be considered a walk that returns to it starting point,
or a generalization of a cycle that allows to visit nodes multiple times.
Transform
A transform is an operation that applies to a graph and that gives another
graph. Examples are taking the largest connected component, removing multiple
edges, and making a bipartite graph unipartite. Certain graph properties can
be expressed at other graph properties applied to graph transforms. For
instance, the size of the largest connected component is the size of the
transform which keeps only the largest connected component.
Triangle
Three nodes all connected with each other. The number of triangles in a
network is a very commonly used statistic, used for instance as the basis to
compute the clustering coefficient. Counting the triangles in a network is a
very common computational problem.
Volume
The number of edges in a network.
Walk
A sequence of connected nodes, which may contain a single node multiple times.
The restriction to include a single node only once is called a path. If the
endpoints of a walk are identical, then the walk is also a tour.
Wedge
Two edges sharing a common node, i.e., two adjacent edges. The number of
wedges in a network is an important network statistic, which characterizes
that skewness of the degree distribution, and which can be easily calculated.
A wedge can be seen as a 2-star or a 2-path.
Weights
(always in the plural) The weights of a network describe the range of edge
weights it allows. The list of possible edge weights is given in Table 2.
## Appendix B Glossary of Mathematical Symbols
The following symbols are used in mathematical expessions throughout KONECT.
Due to the large number of different measures used in graph theory and network
analysis, many common symbols for measures overlap. For many measures, there
is more than one commonly-used notation; the following tables shows a
reasonable balance between using established notation when it exists, and
having distinct symbols for different measures.
$a$ | algebraic connectivity
---|---
$b$ | non-bipartivity
$c$ | global clustering coefficient
$c(u)$ | local clustering coefficient
$d$ | average degree
$d(u)$ | degree of a vertex
$d(u,v)$ | shortest-path distance
$e$ | edge
$g$ | line count, data volume
$l$ | loop count
$m$ | volume, edge count
$n$ | size, node count
$p$ | fill
$q$ | square count
$r$ | rank of a decomposition
$r$ | rating value
$r$ | radius of a graph
$s$ | wedge count
$t$ | triangle count
$u,v,w$ | vertices
$w$ | edge weight
$w$ | network weight
$w(\ldots)$ | weight function
$x$ | cross count
$y$ | reciprocity
$z$ | claw count
$\gamma$ | power law exponent
---|---
$\delta$ | diameter
$\epsilon$ | eccentricity
$\zeta$ | negativity
$\lambda$ | eigenvalue
$\mu$ | average edge weight
$\rho$ | assortativity
$\rho$ | spectral radius
$\xi$ | algebraic conflict
$\sigma$ | singular value
$C_{k}$ | $k$-cycle count
---|---
$E$ | edge set
$F$ | frustration
$G$ | graph
$G$ | Gini coefficient
$H$ | entropy
$N$ | size of largest connected component
$S_{k}$ | $k$-star count
$T_{k}$ | $k$-tour count
$V$ | vertex set
$\mathbf{A}$ | adjacency matrix
---|---
$\mathbf{B}$ | biadjacency matrix
$\mathbf{D}$ | degree matrix
$\mathbf{K}$ | signless Laplacian matrix
$\mathbf{L}$ | Laplacian matrix
$\mathbf{M}$ | normalized biadjacency matrix
$\mathbf{N}$ | normalized adjacency matrix
$\mathbf{P}$ | stochastic adjacency matrix
$\mathbf{S}$ | stochastic Laplacian matrix
$\mathbf{U},\mathbf{V}$ | eigenvector matrices
$\mathbf{Z}$ | normalized Laplacian matrix
$\mathbf{\Lambda}$ | eigenvalue matrix
---|---
$\mathbf{\Sigma}$ | singular value matrix
$\bar{G}$ | unweighted graph
---|---
$\bar{\bar{G}}$ | graph with unique edges
$|G|$ | unsigned graph
|
arxiv-papers
| 2014-02-22T11:31:04 |
2024-09-04T02:49:58.619907
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "J\\'er\\^ome Kunegis",
"submitter": "J\\'er\\^ome Kunegis",
"url": "https://arxiv.org/abs/1402.5500"
}
|
1402.5533
|
††footnotetext: 2010 Mathematics Subject Classification: Primary 32H30,
32A22; Secondary 30D35.
Key words and phrases: second main theorem, uniqueness problem, meromorphic
mapping, truncated multiplicity.
# Degeneracy and finiteness theorems for meromorphic mappings in several
complex variables
Si Duc Quang Department of Mathematics, Hanoi National University of
Education
136-Xuan Thuy, Cau Giay, Hanoi, Vienam. [email protected]
###### Abstract.
In this article, we prove that there are at most two meromorphic mappings of
${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})\ (n\geqslant 2)$
sharing $2n+2$ hyperplanes in general position regardless of multiplicity,
where all zeros with multiplicities more than certain values do not need to be
counted. We also show that if three meromorphic mappings $f^{1},f^{2},f^{3}$
of ${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})\ (n\geqslant 5)$
share $2n+1$ hyperplanes in general position with truncated multiplicity then
the map $f^{1}\times f^{2}\times f^{3}$ is linearly degenerate.
## 1\. Introduction
In $1926$, R. Nevanlinna [3] showed that two distinct nonconstant meromorphic
functions $f$ and $g$ on the complex plane ${\mathbf{C}}$ cannot have the same
inverse images for five distinct values, and that $g$ is a special type of
linear fractional transformation of $f$ if they have the same inverse images
counted with multiplicities for four distinct values [3]. These results are
usually called the five values and the values theorems of R. Nevanlinna.
After that, many authors extended and improved the results of Nevanlinna to
the case of meromorphic mappings into complex projective sapces. The
extensions of the five values theorem are usually called the uniqueness
theorems, and the extensions of the four values theorem are usually called the
finiteness theorems. Here we formulate some recent results on this problem.
To state some of them, first of all we recall the following.
Let $f$ be a nonconstant meromorphic mapping of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$ and $H$ a hyperplane in
${\mathbf{P}}^{n}({\mathbf{C}}).$ Let $k$ be a positive integer or $k=\infty$.
Denote by $\nu_{(f,H)}$ the map of ${\mathbf{C}}^{m}$ into $\mathbf{Z}$ whose
value $\nu_{(f,H)}(a)\ (a\in{\mathbf{C}}^{m})$ is the intersection
multiplicity of the images of $f$ and $H$ at $f(a).$ For every
$z\in{\mathbf{C}}^{m}$, we set
$\displaystyle\nu_{(f,H),\leqslant k}(z)=\begin{cases}0&\text{ if
}\nu_{(f,H)}(z)>k,\\\ \nu_{(f,H)}(z)&\text{ if }\nu_{(f,H)}(z)\leqslant
k,\end{cases}$ and $\displaystyle\nu_{(f,H),\geqslant
k}(z)=\begin{cases}0&\text{ if }\nu_{(f,H)}(z)<k,\\\ \nu_{(f,H)}(z)&\text{ if
}\nu_{(f,H)}(z)\geqslant k,\end{cases}$
Take a meromorphic mapping $f$ of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$ which is linearly nondegenerate over
${\mathbf{C}}$, a positive integer $d$ and $q$ hyperplanes
$H_{1},\ldots,H_{q}$ of ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position
with
$\dim f^{-1}(H_{i}\cap H_{j})\leqslant m-2\quad(1\leqslant i<j\leqslant q)$
and consider the set $\mathcal{F}(f,\\{H_{i}\\}_{i=1}^{q},d)$ of all linearly
nondegenerate over ${\mathbf{C}}$ meromorphic maps
$g:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ satisfying the following
two conditions:
(a) $\min\ (\nu_{(f,H_{j})},d)=\min\ (\nu_{(g,H_{j})},d)\quad(1\leqslant
j\leqslant q),$
(b) $f(z)=g(z)$ on $\bigcup_{j=1}^{q}f^{-1}(H_{j})$.
We see that conditions a) and b) mean the sets of all intersecting points
(counted with multiplicity to level $d$) of $f$ and $g$ with each hyperplane
are the same, and two mappings $f$ and $g$ agree on these sets. If $d=1$, we
will say that $f$ and $g$ share $q$ hyperplanes $\\{H_{j}\\}_{j=1}^{q}$
regardless of multiplicity.
Denote by $\sharp\ S$ the cardinality of the set $S.$ In 1983, L. Smiley [7]
proved the following uniqueness theorem.
Theorem A. If $q=3n+2$ then $\sharp\
\mathcal{F}(f,\\{H_{i}\\}_{i=1}^{q},1)=1.$
In 1998, H. Fujimoto [2] proved a finiteness theorem for meromorphic mappings
as follows.
Theorem B. If $q=3n+1$ then $\sharp\
\mathcal{F}(f,\\{H_{i}\\}_{i=1}^{q},2)\leqslant 2.$
In 2009, Z. Chen-Q. Yan [1] considered the case of $2n+3$ hyperplanes and
proved the following uniqueness theorem.
Theorem C. If $q=2n+3$ then $\sharp\
\mathcal{F}(f,\\{H_{i}\\}_{i=1}^{q},1)=1.$
After that, in 2011 S. D. Quang [5] improved the result of Z. Chen-Q. Yan by
omitting all zeros with multiplicity more than a certain number in the
conditions on sharing hyperplanes of meromorphic mappings. As far as we known,
there is still no uniqueness theorem for meromorphic mappings sharing less
than $2n+3$ hyperplanes regardless of multiplicities. In 2011 Q. Yan-Z. Chen
[8] also proved a degeneracy theorem as follows.
Theorem D. If $q=2n+2$ then the map $f^{1}\times f^{2}\times f^{3}$ of
${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{N}({\mathbf{C}})\times{\mathbf{P}}^{N}({\mathbf{C}})\times{\mathbf{P}}^{N}({\mathbf{C}})$
is linearly degenerate for every three maps
$f^{1},f^{2},f^{3}\in\mathcal{F}(f,\\{H_{i}\\}_{i=1}^{q},2)$.
The first finiteness theorem for the case of meromorphic mappings sharing
$2n+2$ hyperplanes regardless of multiplicities are given by S. D. Quang [6]
in 2012 as follows.
Theorem E. If $n\geqslant 2$ and $q=2n+2$ then $\sharp\
\mathcal{F}(f,\\{H_{i}\\}_{i=1}^{q},1)\leqslant 2.$
However we note that there is a gap in the proof of [6, Theorem 1.1]. For
detail, the inequality (3.26) in [6, Lemma 3.20] does not holds. Hence the
inequality of [6, Lemma 3.20(ii)] may not hold. In order to fix this gap, we
need to slightly change the estimate of this inequality by adding
$N^{(1)}_{(f,H_{j})}(r)$ to its right-hand side. The rest of the proof is
still valid for the case where $N\geqslant 3$. In this paper, we will show a
correction for [6, Lemma 3.20] (see Lemma 3.9 below). Also this theorem
(including the case where $N=2$) will be corrected and improved (see Theorem
1.1 below) by another approach.
We would also like to emphasize that in the above results, all intersecting
points of the mappings and the hyperplanes are considered. It seems to us that
the technique used in the proof of the above results do not work for the case
where all such points with multiplicities more than a certain number are not
taken to count. Our first purpose in this paper is to improve the above result
by omitting all such intersecting points. In order to states the main results,
we give the following definition.
Let $f$ be a linearly nondegenerate meromorphic mapping of ${\mathbf{C}}^{m}$
into ${\mathbf{P}}^{n}({\mathbf{C}})$ and let $H_{1},\ldots,H_{q}$ be $q$
hyperplanes of ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position. Let
$k_{1},\ldots,k_{q}$ be $q$ positive integers or $+\infty$. Assume that
$\dim\\{z;\nu_{(f,H_{i}),\leqslant k_{i}}(z)\cdot\nu_{(f,H_{j}),\leqslant
k_{j}}(z)>0\\}\leqslant m-2\quad(1\leqslant i<j\leqslant q).$
Let $d$ be an integer. We consider the set
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},d)$ of all meromorphic maps
$g:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ satisfying the
conditions:
* (a)
$\min\ (\nu_{(f,H_{i}),\leqslant k_{i}},d)=\min\ (\nu_{(g,H_{i}),\leqslant
k_{i}},d)\quad(1\leqslant j\leqslant q),$
* (b)
$f(z)=g(z)$ on $\bigcup_{i=1}^{q}\\{z;\nu_{(f,H_{i}),\leqslant
k_{i}}(z)>0\\}$.
Then we see that
$\mathcal{F}(f,\\{H_{i}\\}_{i=1}^{q},d)=\mathcal{F}(f,\\{H_{i},\infty\\}_{i=1}^{q},d)$
###### Theorem 1.1.
Let $f$ be a linearly nondegenerate meromorphic mapping of ${\mathbf{C}}^{m}$
into ${\mathbf{P}}^{n}({\mathbf{C}})$ $(n\geqslant 2)$. Let
$H_{1},\ldots,H_{2n+2}$ be $2n+2$ hyperplanes of
${\mathbf{P}}^{n}({\mathbf{C}})$ in general position and let
$k_{1},\ldots,k_{n+2}$ be positive integers or $+\infty$. Assume that
$\dim\\{z;\nu_{(f,H_{i}),\leqslant k_{i}}(z)\cdot\nu_{(f,H_{j}),\leqslant
k_{j}}(z)>0\\}\leqslant m-2\quad(1\leqslant i<j\leqslant 2n+2),$ $\text{ and
}\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}<\min\left\\{\dfrac{n+1}{3n^{2}+n},\dfrac{5n-9}{24n+12},\dfrac{n^{2}-1}{10n^{2}+8n}\right\\}.$
Then $\sharp\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)\leqslant 2.$
Then we see that in the case $n\geqslant 2$, Theorems D and E are corollaries
of Theorem 1.1 when $k_{1}=\cdots=k_{2n+2}=+\infty$.
The last purpose of this paper is to prove a degeneracy theorem for three
mappings sharing $2n+1$ hyperplanes. Namely, we will proved the following.
###### Theorem 1.2.
Let $f$ be a linearly nondegenerate meromorphic mapping of ${\mathbf{C}}^{m}$
into ${\mathbf{P}}^{n}({\mathbf{C}})\ (n\geqslant 5)$. Let
$H_{1},\ldots,H_{2n+1}$ be $2n+1$ hyperplanes of
${\mathbf{P}}^{n}({\mathbf{C}})$ in general position and let
$k_{1},\ldots,k_{2n+1}$ be positive integers or $+\infty$ such that
$\dim\\{z;\nu_{(f,H_{i}),\leqslant k_{i}}(z)\cdot\nu_{(f,H_{j}),\leqslant
k_{j}}(z)>0\\}\leqslant m-2\quad(1\leqslant i<j\leqslant 2n+2).$
If there exists a positive integer $p$ with $p\leqslant n$ and
$\sum_{i=1}^{2n+1}\dfrac{1}{k_{i}+1}<\dfrac{np-3n-p}{4n^{2}+3np-n}.$
then the map $f^{1}\times f^{2}\times f^{3}$ of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})$
is linearly degenerate for every three maps
$f^{1},f^{2},f^{3}\in\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+1},p)$
## 2\. Basic notions in Nevanlinna theory
### 2.1. Counting functions of divisors.
We set $||z||=\big{(}|z_{1}|^{2}+\dots+|z_{m}|^{2}\big{)}^{1/2}$ for
$z=(z_{1},\dots,z_{n})\in{\mathbf{C}}^{m}$ and define
$B(r):=\\{z\in{\mathbf{C}}^{m}:||z||<r\\},\quad
S(r):=\\{z\in{\mathbf{C}}^{m}:||z||=r\\}\ (0<r<\infty).$
Define
$v_{m-1}(z):=\big{(}dd^{c}||z||^{2}\big{)}^{m-1}\quad\quad\text{and}$
$\sigma_{m}(z):=d^{c}\text{log}||z||^{2}\land\big{(}dd^{c}\text{log}||z||^{2}\big{)}^{m-1}\text{on}\quad{\mathbf{C}}^{m}\setminus\\{0\\}.$
We mean by a divisor divisor $\nu$ on a domain $\Omega$ in ${\mathbf{C}}^{m}$
a formal sum
$\nu=\sum_{\lambda\in\Lambda}a_{\lambda}Z_{\lambda},$
where $a_{\lambda}\in\mathbf{Z}$ and $\\{Z_{\lambda}\\}_{\lambda\in\Lambda}$
is a locally finite family of distinct irreducible hypersurfaces of $\Omega$.
Then, we may consider the divisor $\nu$ as a function on $\Omega$ with values
in $\mathbf{Z}$ as follows
$\nu(z)=\sum_{Z_{\lambda}\ni z}a_{\lambda}.$
The support of $\nu$ is defined by
$\mathrm{Supp}\,\nu=\bigcup_{a_{\lambda}\neq 0}Z_{\lambda}$.
For a nonzero meromorphic function $\varphi$ on a domain $\Omega$ in
${\mathbf{C}}^{m}$, we denote by $\nu^{0}_{\varphi}$ (resp.
$\nu^{\infty}_{\varphi}$) the divisor of zeros (resp. divisor of poles) of
$\varphi$, and denote by
$\nu_{\varphi}=\nu^{0}_{\varphi}-\nu^{\infty}_{\varphi}$ the divisor generated
by $\varphi$.
For a divisor $\nu$ on ${\mathbf{C}}^{m}$ and for positive integers $k,M$ (or
$M=\infty$), we define the counting functions of $\nu$ as follows. Set
$\nu^{(M)}(z)=\min\ \\{M,\nu(z)\\},$ $\displaystyle\nu_{\leqslant
k}^{(M)}(z)=\begin{cases}0&\text{ if }\nu(z)>k,\\\ \nu^{(M)}(z)&\text{ if
}\nu(z)\leqslant k,\end{cases}$
$\displaystyle\nu_{>k}^{(M)}(z)=\begin{cases}\nu^{(M)}(z)&\text{ if
}\nu(z)>k,\\\ 0&\text{ if }\nu(z)\leqslant k.\end{cases}$
We define $n(t)$ by
$\displaystyle n(t)=\begin{cases}\int\limits_{|\nu|\,\cap
B(t)}\nu(z)v_{n-1}&\text{ if }n\geqslant 2,\\\ \sum\limits_{|z|\leqslant
t}\nu(z)&\text{ if }n=1.\end{cases}$
Similarly, we define $n^{(M)}(t),\ n_{\leqslant k}^{(M)}(t),\
n_{>k}^{(M)}(t).$
Define
$N(r,\nu)=\int\limits_{1}^{r}\dfrac{n(t)}{t^{2n-1}}dt\quad(1<r<\infty).$
Similarly, we define $N(r,\nu^{(M)}),\ N(r,\nu_{\leqslant k}^{(M)}),\
N(r,\nu_{>k}^{(M)})$ and denote them by $N^{(M)}(r,\nu)$, $N_{\leqslant
k}^{(M)}(r,\nu)$, $N_{>k}^{(M)}(r,\nu)$ respectively.
Let $\varphi:{\mathbf{C}}^{m}\longrightarrow{\mathbf{C}}$ be a meromorphic
function. Define
$N_{\varphi}(r)=N(r,\nu^{0}_{\varphi}),\
N_{\varphi}^{(M)}(r)=N^{(M)}(r,\nu^{0}_{\varphi}),$ $N_{\varphi,\leqslant
k}^{(M)}(r)=N_{\leqslant k}^{(M)}(r,\nu^{0}_{\varphi}),\
N_{\varphi,>k}^{(M)}(r)=N_{>k}^{(M)}(r,\nu^{0}_{\varphi}).$
For brevity we will omit the superscript (M) if $M=\infty$.
For a set $S\subset{\mathbf{C}}^{m}$, we define the characteristic function of
$S$ by
$\chi_{S}(z)=\begin{cases}1&\text{ if }z\in S,\\\ 0&\text{ if }z\not\in
S.\end{cases}$
If the closure $\bar{S}$ of $S$ is an analytic subset of ${\mathbf{C}}^{m}$
then we denote by $N(r,S)$ the counting function of the reduced divisor whose
support is the union of all irreducible components of $\bar{S}$ with
codimension one.
### 2.2. Characteristic and Proximity functions.
Let $f:{\mathbf{C}}^{m}\longrightarrow{\mathbf{P}}^{n}({\mathbf{C}})$ be a
meromorphic mapping. For arbitrarily fixed homogeneous coordinates
$(w_{0}:\dots:w_{n})$ on ${\mathbf{P}}^{n}({\mathbf{C}})$, we take a reduced
representation $f=(f_{0}:\dots:f_{n})$, which means that each $f_{i}$ is a
holomorphic function on ${\mathbf{C}}^{m}$ and
$f(z)=\big{(}f_{0}(z):\cdots:f_{n}(z)\big{)}$ outside the analytic set
$\\{f_{0}=\cdots=f_{n}=0\\}$ of codimension $\geqslant 2$. Set
$\|f\|=\big{(}|f_{0}|^{2}+\dots+|f_{n}|^{2}\big{)}^{1/2}$.
The characteristic function of $f$ is defined by
$\displaystyle
T_{f}(r)=\int\limits_{S(r)}\text{log}\|f\|\sigma_{m}-\int\limits_{S(1)}\text{log}\|f\|\sigma_{m}.$
Let $H$ be a hyperplane in ${\mathbf{P}}^{n}({\mathbf{C}})$ given by
$H=\\{a_{0}\omega_{0}+\cdots+a_{n}\omega_{n}\\},$ where
$(a_{0},\ldots,a_{n})\neq(0,\ldots,0)$. We set
$(f,H)=\sum_{i=0}^{n}a_{i}f_{i}$. Then we see that the divisor $\nu_{(f,H)}$
does not depend on the reduced representation of $f$ and presentation of $H$.
We define the proximity function of $H$ by
$m_{f,H}(r)=\int_{S(r)}\log\dfrac{||f||\cdot||H||}{|(f,H)|}\sigma_{m}-\int_{S(1)}\log\dfrac{||f||\cdot||H||}{|(f,H)|}\sigma_{m},$
where $||H||=(\sum_{i=0}^{N}|a_{i}|^{2})^{\frac{1}{2}}.$
Let $\varphi$ be a nonzero meromorphic function on ${\mathbf{C}}^{m}$, which
are occasionally regarded as a meromorphic mapping into
${\mathbf{P}}^{1}({\mathbf{C}})$. The proximity function of $\varphi$ is
defined by
$m(r,\varphi):=\int_{S(r)}\log\max\ (|\varphi|,1)\sigma_{n}.$
As usual, by the notation “$||\ P$” we mean the assertion $P$ holds for all
$r\in[0,\infty)$ excluding a Borel subset $E$ of the interval $[0,\infty)$
with $\int_{E}dr<\infty$.
### 2.3. Some lemmas.
The following results play essential roles in Nevanlinna theory (see [4]).
###### Theorem 2.1 (The first main theorem).
Let $f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a linearly
nondegenerate meromorphic mapping and $H$ be a hyperplane in
${\mathbf{P}}^{n}({\mathbf{C}})$. Then
$N_{(f,H)}(r)+m_{f,H}(r)=T_{f}(r)\ (r>1).$
###### Theorem 2.2 (The second main theorem).
Let $f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a linearly
nondegenerate meromorphic mapping and $H_{1},\ldots,H_{q}$ be hyperplanes in
general position in ${\mathbf{P}}^{n}({\mathbf{C}}).$ Then
$||\ \
(q-n-1)T_{f}(r)\leqslant\sum_{i=1}^{q}N_{(f,H_{i})}^{(n)}(r)+o(T_{f}(r)).$
For meromorphic functions $F,G,H$ on ${\mathbf{C}}^{m}$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbf{Z}_{+}^{m}$, we put
$\Phi^{\alpha}(F,G,H):=F\cdot G\cdot H\cdot\left|\begin{array}[]{cccc}1&1&1\\\
\frac{1}{F}&\frac{1}{G}&\frac{1}{H}\\\
\mathcal{D}^{\alpha}(\frac{1}{F})&\mathcal{D}^{\alpha}(\frac{1}{G})&\mathcal{D}^{\alpha}(\frac{1}{H})\\\
\end{array}\right|$
###### Lemma 2.3 ([2, Proposition 3.4]).
If $\Phi^{\alpha}(F,G,H)=0$ and
$\Phi^{\alpha}(\frac{1}{F},\frac{1}{G},\frac{1}{H})=0$ for all $\alpha$ with
$|\alpha|\leq 1$, then one of the following assertions holds :
(i) $F=G,G=H$ or $H=F$
(ii) $\frac{F}{G},\frac{G}{H}$ and $\frac{H}{F}$ are all constant.
###### Lemma 2.4.
Let $f^{1},f^{2},f^{3}$ be three maps in
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},p)$. Assume that $f^{i}$ has a
representation $f^{i}=(f^{i}_{0}:\cdots:f^{i}_{n})$, $1\leqslant i\leqslant
3$. Suppose that there exist $s,t,l\in\\{1,\cdots,q\\}$ such that
$P:=Det\left(\begin{array}[]{ccc}(f^{1},H_{s})&(f^{1},H_{t})&(f^{1},H_{l})\\\
(f^{2},H_{s})&(f^{2},H_{t})&(f^{2},H_{l})\\\
(f^{3},H_{s})&(f^{3},H_{t})&(f^{3},H_{l})\end{array}\right)\not\equiv 0.$
Then we have
$\displaystyle T(r)\geqslant$
$\displaystyle\sum_{i=s,t,l}(N(r,\min\\{\nu_{(f^{u},H_{i}),\leqslant
k_{i}};1\leqslant u\leqslant 3\\})$ $\displaystyle-N^{(1)}_{(f,H_{i}),\leq
k_{i}}(r))+2\sum_{i=1}^{q}N^{(1)}_{(f,H_{i}),\leq k_{i}}(r)+o(T(r)),$
where $T(r)=\sum_{u=1}^{3}T_{f^{u}}(r)$.
###### Proof.
Denote by $S$ the closure of $\bigcup_{1\leqslant u\leqslant
3}I(f^{u})\cup\bigcup_{1\leqslant i<j\leqslant
2n+2}\\{z;\nu_{(f,H_{i}),\leqslant k_{i}}(z)\cdot\nu_{(f,H_{j}),\leqslant
k_{j}}(z)>0\\}$. Then $S$ is an analytic subset of codimension two of
${\mathbf{C}}^{m}$.
For $z\not\in S$, we consider the following two cases:
Case 1. $z$ is a zero of $(f,H_{i})$ with multiplicity at most $k_{i}$, where
$i\in\\{s,t,l\\}$. For instance, we suppose that $i=s$. We set
$m=\min\\{\nu_{(f^{1},H_{s}),\leqslant k_{s}}(z),\nu_{(f^{2},H_{s}),\leqslant
k_{s}}(z),\nu_{(f^{3},H_{s}),\leqslant k_{s}}(z)\\}.$
Then there exist a neighborhood $U$ of $z$ and a holomorphic function $h$
defined on $U$ such that $\mathrm{Zero}(h)=U\cap\mathrm{Zero}(f,H_{s})$ and
$dh$ has no zero. Then the functions $\varphi_{u}=\frac{(f^{u},H_{s})}{h^{m}}\
(1\leqslant u\leqslant 3)$ are holomorphic in a neighborhood of $z$. On the
other hand, since $f^{1}=f^{2}=f^{3}$ on
$\mathrm{Supp}\,\nu_{(f,H_{s}),\leqslant k_{s}}$, we have
$P_{uv}:=(f^{u},H_{t})(f^{v},H_{l})-(f^{u},H_{l})(f^{v},H_{t})=0\text{ on
}\mathrm{Supp}\,\nu_{(f,H_{s}),\leqslant k_{s}},\ 1\leqslant u<v\leqslant 3.$
Therefore, there exist holomorphic functions $\psi_{uv}$ on a neighborhood of
$z$ such that $P_{uv}=h\psi_{uv}.$ Then we have
$P=h^{m+1}(\varphi_{1}\psi_{23}-\varphi_{2}\psi_{13}+\varphi_{3}\psi_{12})$
on a neighborhood of $z$. This yeilds that
$\nu_{P}(z)\geqslant m+1=\sum_{i=s,t,l}(\min\\{\nu_{(f^{u},H_{i}),\leqslant
k_{i}}(z);1\leqslant u\leqslant 3\\}-\nu^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(z))+2\sum_{i=1}^{q}\nu^{(1)}_{(f,H_{i}),\leqslant k_{i}}(z).$
Case 2. $z$ is a zero point of $(f,H_{i})$ with multiplicity at most $k_{i}$,
where $i\not\in\\{s,t,l\\}$. There exist an index $v$ such that
$(f^{1},H_{v})(z)\neq 0$. Since $f^{1}(z)=f^{2}(z)=f^{3}(z),$ we have
$(f^{u},H_{v})(z)\neq 0\ (1\leqslant u\leqslant 3)$ and
$\displaystyle P$
$\displaystyle=\prod_{u=1}^{3}(f^{u},H_{v})\cdot\det\left(\begin{array}[]{ccc}\dfrac{(f^{1},H_{s})}{(f^{1},H_{v})}&\dfrac{(f^{1},H_{t})}{(f^{1},H_{v})}&\dfrac{(f^{1},H_{l})}{(f^{1},H_{v})}\\\
\dfrac{(f^{2},H_{1})}{(f^{2},H_{l})}&\dfrac{(f^{2},H_{t})}{(f^{2},H_{l})}&\dfrac{(f^{2},H_{s})}{(f^{2},H_{l})}\\\
\dfrac{(f^{3},H_{1})}{(f^{3},H_{l})}&\dfrac{(f^{3},H_{t})}{(f^{3},H_{l})}&\dfrac{(f^{3},H_{s})}{(f_{3},H_{l})}\end{array}\right)$
$\displaystyle=\prod_{u=1}^{3}(f^{u},H_{l})\cdot\det\left(\begin{array}[]{ccc}\dfrac{(f^{1},H_{1})}{(f^{1},H_{l})}&\dfrac{(f^{1},H_{t})}{(f^{1},H_{l})}&\dfrac{(f^{1},H_{s})}{(f^{1},H_{l})}\\\
\\\
\frac{(f^{2},H_{1})}{(f^{2},H_{l})}-\frac{(f^{1},H_{1})}{(f^{1},H_{l})}&\frac{(f^{2},H_{t})}{(f^{2},H_{l})}-\frac{(f^{1},H_{t})}{(f^{1},H_{l})}&\frac{(f^{2},H_{s})}{(f^{2},H_{l})}-\frac{(f^{1},H_{s})}{(f^{1},H_{l})}\\\
\\\
\frac{(f^{3},H_{1})}{(f^{3},H_{l})}-\frac{(f^{1},H_{1})}{(f^{1},H_{l})}&\frac{(f^{3},H_{t})}{(f^{3},H_{l})}-\frac{(f^{1},H_{t})}{(f^{1},H_{l})}&\frac{(f^{3},H_{s})}{(f^{3},H_{l})}-\frac{(f^{1},H_{s})}{(f^{1},H_{l})}\end{array}\right).$
vanishes at $z$ with multiplicity at least two. Therefore, we have
$\nu_{P}(z)\geqslant 2=\sum_{i=s,t,l}(\min\\{\nu_{(f^{u},H_{i}),\leqslant
k_{i}}(z);1\leqslant u\leqslant 3\\}-\nu^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(z))+2\sum_{i=1}^{q}\nu^{(1)}_{(f,H_{i}),\leqslant k_{i}}(z).$
Thus, from the above two cases we have
$\displaystyle\nu_{P}(z)\geqslant\sum_{i=s,t,l}(\min\\{\nu_{(f^{u},H_{i}),\leqslant
k_{i}}(z);1\leqslant u\leqslant 3\\}-\nu^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(z))+2\sum_{i=1}^{q}\nu^{(1)}_{(f,H_{i}),\leqslant k_{i}}(z),$
for all $z$ outside the analytic set $S$. Integrating both sides of the above
inequality, we get
$\displaystyle N_{P}(r)\geqslant$
$\displaystyle\sum_{i=s,t,l}(N(r,\min\\{\nu_{(f^{u},H_{i}),\leqslant
k_{i}};1\leqslant u\leqslant 3\\})-N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r))$
$\displaystyle+2\sum_{i=1}^{q}N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)+o(T(r)).$
On the other hand, by Jensen’s formula and the definition of the
characteristic function we have
$\displaystyle N_{P}(r)=$ $\displaystyle\int_{S(r)}\log|P|\sigma_{m}+O(1)$
$\displaystyle\leqslant$
$\displaystyle\sum_{u=1}^{3}\int_{S(r)}\log(|(f^{u},H_{1})|^{2}+|(f^{u},H_{t})|^{2}+|(f^{u},H_{s})|)^{\frac{1}{2}}\sigma_{m}+O(1)$
$\displaystyle\leqslant$
$\displaystyle\sum_{u=1}^{3}\int_{S(r)}\log||f^{u}||\sigma_{m}+O(1)=T(r)+o(T(r)).$
Thus, we have
$\displaystyle T(r)\geqslant$
$\displaystyle\sum_{i=s,t,l}(N(r,\min\\{\nu_{(f^{u},H_{i}),\leqslant
k_{i}};1\leqslant u\leqslant 3\\})-N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r))$
$\displaystyle+2\sum_{i=1}^{q}N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)+o(T(r)).$
The lemma is proved. ∎
## 3\. Proof of Main Theorems
Let $f$ be a linearly nondegenerate meromorphic mapping of ${\mathbf{C}}^{m}$
into ${\mathbf{P}}^{n}({\mathbf{C}})$. Let $H_{1},\ldots,H_{2n+2}$ be $2n+2$
hyperplanes of ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position and let
$k_{i}\geqslant n\ (1\leqslant i\leqslant 2n+2)$ be positive integers or
$+\infty$ with
$\dim\\{z;\nu_{(f,H_{i}),\leqslant k_{i}}(z)\cdot\nu_{(f,H_{j}),\leqslant
k_{j}}(z)>0\\}\leqslant m-2\quad(1\leqslant i<j\leqslant 2n+2).$
In order to prove Theorem 3.2, we need the following lemmas.
###### Lemma 3.1.
If $\sum_{i=1}^{2n+2}\frac{1}{k_{i}+1}<\frac{1}{n},$ then every mapping $g$ in
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)$ is linearly nondegenerate
and
$||\ T_{g}(r)=O(T_{f}(r))\mathrm{\ and\ }||\ T_{f}(r)=O(T_{g}(r)).$
###### Proof.
Suppose that there exists a hyperplane $H$ satisfying
$g({\mathbf{C}}^{m})\subset H$. We assume that $f$ and $g$ have reduce
representations $f=(f_{0}:\cdots:f_{n})$ and $g=(g_{0}:\cdots:g_{n})$
respectively. Assume that $H=\\{(\omega_{0}:\cdots:\omega_{n})\ |\
\sum_{i=0}^{n}a_{i}\omega_{i}=0\\}$. Since $f$ is linearly nondegenerate,
$(f.H)\not\equiv 0$. On the other hand $(f,H)(z)=(g,H)(z)=0$ for all
$z\in\bigcup_{i=1}^{2n+2}\\{\nu_{(f,H_{i}),\leqslant k_{i}}\\}$, hence
$N_{(f,H)}(r)\geqslant\sum_{i=1}^{2n+2}N_{(f,H_{i}),\leqslant
k_{i}}^{(1)}(r).$
It yields that
$\displaystyle||\ T_{f}(r)$ $\displaystyle\geqslant
N_{(f,H)}(r)\geqslant\sum_{i=1}^{2n+2}N_{(f,H_{i}),\leqslant
k_{i}}^{(1)}(r)=\sum_{i=1}^{2n+2}\bigl{(}N_{(f,H_{i})}^{(1)}(r)-N_{(f,H_{i}),>k_{i}}^{(1)}(r)\bigl{)}$
$\displaystyle\geqslant\sum_{i=1}^{2n+2}\dfrac{1}{n}N_{(f,H_{i})}^{(n)}(r)-\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}T_{f}(r)\geqslant\big{(}\dfrac{n+1}{n}-\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}\big{)}T_{f}(r)+o(T_{f}(r)).$
Letting $r\longrightarrow+\infty$, we get
$\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}\geqslant\dfrac{1}{n}.$
This is a contradiction. Hence $g({\mathbf{C}}^{m})$ can not be contained in
any hyperplanes of ${\mathbf{P}}^{n}({\mathbf{C}})$. Therefore $g$ is linearly
nondegenerate.
Also by the Second Main Theorem, we have
$\displaystyle||\quad(n+1)T_{g}(r)\leqslant$
$\displaystyle\sum_{i=1}^{2n+2}N_{(g,H_{i})}^{(n)}(r)+o(T_{g}(r))$
$\displaystyle\leqslant$ $\displaystyle\sum_{i=1}^{2n+2}n\
N_{(g,H_{i})}^{(1)}(r)+o(T_{g}(r))$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2n+2}n\bigl{(}N_{(g,H_{i}),\leqslant
k_{i}}^{(1)}(r)+N_{(g,H_{i}),>k_{i}}^{(1)}(r)\bigl{)}+o(T_{g}(r))$
$\displaystyle\leqslant$
$\displaystyle\sum_{i=1}^{2n+2}n\bigl{(}N_{(f,H_{i}),\leqslant
k_{i}}^{(1)}(r)+\dfrac{1}{k_{i}+1}T_{g}(r)\bigl{)}+o(T_{g}(r))$
$\displaystyle\leqslant$
$\displaystyle\sum_{i=1}^{2n+2}n\bigl{(}T_{f}(r)+\dfrac{1}{k_{i}+1}T_{g}(r)\bigl{)}+o(T_{f}(r)+T_{g}(r)).$
Thus
$\bigl{(}n+1-\sum_{i=1}^{2n+2}\dfrac{n}{k_{i}+1}\bigl{)}T_{g}(r)\leqslant
n(2n+2)T_{f}(r)+o(T_{f}(r)+T_{g}(r)).$
We note that
$n+1-\sum_{i=1}^{2n+2}\dfrac{n}{k_{i}+1}>n>0.$
Hence $||T_{g}(r)=O(T_{f}(r)).$ Similarly, we get $||T_{f}(r)=O(T_{g}(r)).$ ∎
###### Lemma 3.2.
Assume that $n\geqslant 2$ and
$\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}<\dfrac{n+1}{n(3n+1)}.$
Then for three maps
$f^{1},f^{2},f^{3}\in\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)$ we have
$f^{1}\wedge f^{2}\wedge f^{3}=0.$
###### Proof.
By Lemma 3.1, we have that $f^{s}$ is linearly nondegenerate and
$||T_{f^{s}}(r)=O(T_{f}(r))$ and $||T_{f}(r)=O(T_{f^{s}}(r))$ for all
$s=1,2,3.$
Suppose that $f^{1}\wedge f^{2}\wedge f^{3}\not\equiv 0$. For each $1\leqslant
i\leqslant 2n+2$, we set
$N_{i}(r)=\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)-(2n+1)N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r).$
Here, we note that for positive integers $a,b,c$ we have
$(\min\\{a,b,c\\}-1)\geqslant\min\\{a,n\\}+\min\\{a,n\\}+\min\\{a,n\\}-2n-1.$
Then
$\min\\{\nu_{(f^{u},H_{i}),\leqslant k_{i}}(z);1\leqslant u\leqslant
3\\}-\nu^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(z)\geqslant\sum_{u=1}^{3}\nu^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(z)-(2n+1)\nu^{(1)}_{(f,H_{i}),\leqslant k_{i}}(z)$
for all $z\in\mathrm{Supp}\,\nu_{(f,H_{i}),\leqslant k_{i}}$. This yeilds that
$\displaystyle N(r,\min\\{\nu_{(f^{u},H_{i}),\leqslant k_{i}}(z)$
$\displaystyle;1\leqslant u\leqslant 3\\})-N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r)$
$\displaystyle\geqslant\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)-(2n+1)N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)=N_{i}(r).$
We denote by $\mathcal{I}$ the set of all permutations of the $(2n+2)-$tuple
$(1,\ldots,2n+2)$, that means
$\mathcal{I}=\\{I=(i_{1},\ldots,i_{2n+2})\ :\
\\{i_{1},\ldots,i_{2n+2}\\}=\\{1,\ldots,{2n+2}\\}\\}.$
For each $I=(i_{1},\ldots,i_{2n+2})\in\mathcal{I}$ we define a subset $E_{I}$
of $[1,+\infty)$ as follows
$E_{I}=\\{r\geqslant 1\ :\ N_{i_{1}}(r)\geqslant\cdots\geqslant
N_{i_{2n+2}}(r)\\}.$
It is clear that $\bigcup_{I\in\mathcal{I}}E_{I}=[1,+\infty).$ Therefore,
there exists an element of $\mathcal{I},$ for instance it is
$I_{0}=(1,2,\ldots,2n+2)$, satisfying
$\int\limits_{E_{i_{0}}}dr=+\infty.$
Then, we have $N_{1}(r)\geqslant N_{2}(r)\geqslant\cdots\geqslant N_{2n+2}(r)$
for all $r\in E_{i_{0}}.$
We consider $\mathcal{M}^{3}$ as a vector space over the field $\mathcal{M}$.
For each $i=1,\ldots,2n+2,$ we set
$V_{i}=\left((f^{1},H_{i}),(f^{2},H_{i}),(f^{3},H_{i})\right)\in\mathcal{M}^{3}.$
We put
$s=\min\\{i\ :\ V_{1}\wedge V_{i}\not\equiv 0\\}.$
Since $f^{1}\wedge f^{2}\wedge f^{3}\not\equiv 0$, we have $1<s<n+1.$ Also by
again $f^{1}\wedge f^{2}\wedge f^{3}\not\equiv 0$, there exists an index
$t\in\\{s+1,\ldots,n+1\\}$ such that $V_{1}\wedge V_{s}\wedge V_{t}\not\equiv
0$. This means that
$P:=\det(V_{1},V_{s},V_{t})=\det\left(\begin{array}[]{ccc}(f^{1},H_{1})&(f^{1},H_{s})&(f^{1},H_{t})\\\
(f^{2},H_{1})&(f^{2},H_{s})&(f^{2},H_{t})\\\
(f^{3},H_{1})&(f^{3},H_{s})&(f^{3},H_{t})\end{array}\right)\not\equiv 0.$
Set $T(r)=\sum_{u=1}^{3}T_{f^{u}}(r)$. By Lemma 2.4, for $r\in E_{I_{0}}$ we
have
$\displaystyle T(r)$
$\displaystyle\geqslant\sum_{i=1,s,t}(N(r,\min\\{\nu_{(f^{u},H_{i}),\leqslant
k_{i}};1\leqslant u\leqslant 3\\})-N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r))$
$\displaystyle+2\sum_{i=1}^{q}N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)+o(T(r))$
$\displaystyle\geqslant
N_{1}(r)+N_{s}(r)+2\sum_{i=1}^{q}N^{(1)}_{(f,H_{i})}(r)+o(T(r))$
$\displaystyle\geqslant\dfrac{1}{n+1}\sum_{i=1}^{2n+2}N_{i}(r)+2\sum_{i=1}^{2n+2}N_{(f,H_{i}),\leqslant
k_{i}}^{(1)}(r)+o(T(r)).$
$\displaystyle=\dfrac{1}{n+1}\sum_{i=1}^{2n+2}\biggl{(}\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(z)-(2n+1)N^{(1)}_{(f,H_{i})}(z)\biggl{)}+2\sum_{i=1}^{2n+2}N_{(f,H_{i}),\leqslant
k_{i}}^{(1)}(r)$
$\displaystyle=\dfrac{1}{n+1}\sum_{i=1}^{2n+2}\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(z)+\dfrac{1}{3(n+1)}\sum_{i=1}^{2n+2}\sum_{u=1}^{3}N^{(1)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)$
$\displaystyle\geqslant(1+\dfrac{1}{3n})\dfrac{1}{n+1}\sum_{i=1}^{2n+2}\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)$
$\displaystyle\geqslant(1+\dfrac{1}{3n})\dfrac{1}{n+1}\sum_{i=1}^{2n+2}\sum_{u=1}^{3}\biggl{(}N^{(n)}_{(f^{u},H_{i})}(r)-N^{(n)}_{(f^{u},H_{i}),>k_{i}}(r)\biggl{)}$
$\displaystyle\geqslant(1+\dfrac{1}{3n})\dfrac{1}{n+1}\sum_{u=1}^{3}\biggl{(}n+1-\sum_{i=1}^{2n+2}\dfrac{n}{k_{i}+1}\biggl{)}T_{f^{u}}(r)+o(T(r))$
$\displaystyle=\bigl{(}1+\dfrac{1}{3n}-\dfrac{3n+1}{3(n+1)}\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}\bigl{)}T(r)+o(T(r)).$
Letting $r\rightarrow+\infty$ $(r\in E_{i_{0}})$ we get
$\displaystyle 1\geqslant
1+\dfrac{1}{3n}-\dfrac{3n+1}{3(n+1)}\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}.$
Thus
$\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}\geqslant\dfrac{n+1}{n(3n+1)}.$
This is a contradiction. Hence, $f^{1}\wedge f^{2}\wedge f^{3}\equiv 0.$ The
lemma is proved. ∎
Now for three mappings
$f^{1},f^{2},f^{3}\in\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)$, we
define:
$\displaystyle F_{k}^{ij}$
$\displaystyle=\dfrac{(f^{k},H_{i})}{(f^{k},H_{j})}\ (0\leqslant k\leqslant
2,\ 1\leqslant i,j\leqslant 2n+2),$ $\displaystyle V_{i}$
$\displaystyle=((f^{1},H_{i}),(f^{2},H_{i}),(f^{3},H_{i}))\in\mathcal{M}_{m}^{3},$
$\displaystyle T_{i}$ $\displaystyle=\\{z;\nu_{(f,H_{i}),\leqslant
k_{i}}(z)>0\\},S_{i}=\bigcup_{u=1}^{3}\\{z;\nu_{(f_{u},H_{i}),>k_{i}}(z)>0\\},$
$\displaystyle R_{i}$
$\displaystyle=\bigcap_{u=1}^{3}\\{z;\nu_{(f_{u},H_{i}),>k_{i}}(z)>0\\},$
$\displaystyle\nu_{i}$
$\displaystyle=\\{z;k_{i}\geqslant\nu_{(f^{u},H_{i})}(z)\geqslant\nu_{(f^{v},H_{i})}(z)=\nu_{(f^{t},H_{i})}(z)\text{
for a permutation }(u,v,t)\text{ of }(1,2,3)\\}.$
We write $V_{i}\cong V_{j}$ if $V_{i}\wedge V_{j}\equiv 0$, otherwise we write
$V_{i}\not\cong V_{j}.$ For $V_{i}\not\cong V_{j}$, we wirte $V_{i}\sim V_{j}$
if there exist $1\leqslant u<v\leqslant 3$ such that $F_{u}^{ij}=F_{v}^{ij}$,
otherwise we write $V_{i}\not\sim V_{j}$.
###### Lemma 3.3.
With the assumption of Theorem 1.1. Let $h$ and $g$ be two elements of the
family $\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)$. If there exist a
constant $\lambda$ and two indices $i,j$ such that
$\dfrac{(h,H_{i})}{(h,H_{j})}=\lambda\dfrac{(g,H_{i})}{(g,H_{j})}$ then
$\lambda=1$.
Proof. By Lemma 3.1, we see that $h$ and $g$ are linearly nondegenerate and
have the characteristic functions of the same order with the characteristic
function of $f$. Setting $H=\dfrac{(h,H_{i})}{(h,H_{j})}\text{ and
}G=\dfrac{(g,H_{i})}{(g,H_{j})}$ and
$\displaystyle
S_{t}^{\prime}=\\{z;\nu_{(h,H_{t}),>k_{t}}(z)>0\\}\cup\\{z;\nu_{(g,H_{t}),>k_{t}}(z)>0\\}\quad(1\leqslant
t\leqslant 2n+2).$
Then $H=\lambda G$. Supposing that $\lambda\neq 1$, since $H=G$ on the set
$\bigcup_{t\neq i,j}T_{t}\setminus(S_{i}^{\prime}\cup S_{j}^{\prime})$, we
have $\bigcup_{t\neq i,j}T_{t}\subset S_{i}^{\prime}\cup S_{j}^{\prime}$. Thus
$\displaystyle 0\geq$ $\displaystyle\sum_{t\neq
i,j}N^{(1)}_{(f,H_{t}),\leqslant
k_{t}}(r)-(N(r,S_{i}^{\prime})+N(r,S_{j}^{\prime}))$ $\displaystyle\geq$
$\displaystyle\dfrac{1}{2}\sum_{t\neq i,j}(N^{(1)}_{(h,H_{t}),\leqslant
k_{t}}(r)+N^{(1)}_{(g,H_{t}),\leqslant
k_{t}}(r))-(N(r,S_{i}^{\prime})+N(r,S_{j}^{\prime}))$ $\displaystyle\geq$
$\displaystyle\dfrac{1}{2n}\sum_{t\neq
i,j}(N^{(n)}_{(h,H_{t})}(r)+N^{(n)}_{(g,H_{t})}(r))-\sum_{t=1}^{2n+2}(N^{(1)}_{(h,H_{t}),>k_{t}}(r)+N^{(1)}_{(g,H_{t}),>k_{t}}(r))$
$\displaystyle\geq$
$\displaystyle\dfrac{n-1}{2n}(T_{h}(r)+T_{g}(r))-\sum_{t=1}^{2n+2}\dfrac{1}{k_{t}+1}(T_{h}(r)+T_{g}(r))+o(T_{f}(r)).$
Letting $r\longrightarrow+\infty$, we get
$\dfrac{n-1}{2n}\leqslant\sum_{t=1}^{2n+2}\dfrac{1}{k_{t}+1}.$
This is a contradiction. Therefore $\lambda=1$. The lemma is proved $\square$
###### Lemma 3.4.
Let $f^{1},f^{2},f^{3}$ be three elements of
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)$, where $k_{i}\ (1\leqslant
i\leqslant 2n+2)$ are positive integers or $+\infty$. Suppose that
$f^{1}\wedge f^{2}\wedge f^{3}\equiv 0$ and $V_{i}\sim V_{j}$ for some
distinct indices $i$ and $j$. Then $f^{1},f^{2},f^{2}$ are not distinct.
Proof. Suppose $f^{1},f^{2},f^{2}$ are distinct. Since $V_{i}\sim V_{j}$, we
may suppose that $F_{1}^{ij}=F_{2}^{ij}\neq F_{3}^{ij}$. Since $f^{1}\wedge
f^{2}\wedge f^{3}\equiv 0$ and $f^{1}\neq f^{2}$, there exists a meromorphic
function $\alpha$ such that
$F_{3}^{tj}=\alpha F_{1}^{tj}+(1-\alpha)F_{2}^{tj}\ (1\leqslant t\leqslant
2n+2).$
This implies that $F_{3}^{ij}=F_{1}^{ij}=F^{ij}_{2}$. This is a contradiction.
Hence $f^{1},f^{2},f^{3}$ are not distinct. The lemma is proved $\square$
###### Lemma 3.5.
With the assumption of Theorem 1.1. Let $f^{1},f^{2},f^{3}$ be three maps in
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)$. Suppose that
$f^{1},f^{2},f^{3}$ are distinct and there are two indices
$i,j\in\\{1,2,\ldots,2n+2\\}\ (i\neq j)$ such that $V_{i}\not\cong V_{j}$ and
$\Phi_{ij}^{\alpha}:=\Phi^{\alpha}(F_{1}^{ij},F_{2}^{ij},F_{3}^{ij})\equiv 0$
for every $\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbf{Z}^{m}_{+}$ with
$|\alpha|=1$. Then for every $t\in\\{1,\ldots,2n+2\\}\setminus\\{i\\}$, the
following assertion hold:
* (i)
$\Phi^{\alpha}_{it}\equiv 0$ for all $|\alpha|\leqslant 1,$
* (ii)
if $V_{i}\not\cong V_{t}$ then $F_{1}^{ti},F_{2}^{ti},F_{3}^{ti}$ are distinct
and
$\displaystyle N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)$
$\displaystyle\geqslant\sum_{s\neq i,t}N^{(1)}_{(f,H_{s}),\leqslant
k_{s}}(r)-N^{(1)}_{(f,H_{t}),\leqslant k_{t}}(r)-2(N(r,S_{i})+N(r,S_{t}))$
$\displaystyle\geqslant\sum_{s\neq i,t}N^{(1)}_{(f,H_{s}),\leqslant
k_{s}}(r)-N^{(1)}_{(f,H_{t}),\leqslant
k_{t}}(r)-2\sum_{u=1}^{3}\sum_{s=i,t}N_{(f^{u},H_{s}),\leqslant k_{s}}(r).$
Proof. By the supposition $V_{i}\not\cong V_{j}$, we may assume that
$F_{2}^{ji}-F_{1}^{ji}\neq 0$.
(a) For all $\alpha\in\mathbf{Z}^{m}_{+}$ with $|\alpha|=1$, we have
$\Phi_{ij}^{\alpha}=0$, and hence
$\displaystyle\mathcal{D}^{\alpha}\biggl{(}\dfrac{F_{3}^{ji}-F_{1}^{ji}}{F_{2}^{ji}-F_{1}^{ji}}\biggl{)}=$
$\displaystyle\dfrac{1}{(F_{2}^{ji}-F_{1}^{ji})^{2}}\cdot\biggl{(}(F_{2}^{ji}-F_{1}^{ji})\cdot\mathcal{D}^{\alpha}(F_{3}^{ji}-F_{1}^{ji})$
$\displaystyle\hskip
90.0pt-(F_{3}^{ji}-F_{1}^{ji})\cdot\mathcal{D}^{\alpha}(F_{2}^{ji}-F_{1}^{ji})\biggl{)}$
$\displaystyle=$
$\displaystyle\dfrac{1}{{(F_{2}^{ji}-F_{1}^{ji})^{2}}}\cdot\left|\begin{array}[]{cccc}1&1&1\\\
F_{1}^{ji}&F_{2}^{ji}&F_{3}^{ji}\\\
\mathcal{D}^{\alpha}(F_{1}^{ji})&\mathcal{D}^{\alpha}(F_{2}^{ji})&\mathcal{D}^{\alpha}(F_{3}^{ji})\end{array}\right|=0.$
Since the above equality hold for all $|\alpha|=1$, then there exists a
constant $c\in{\mathbf{C}}$ such that
$\displaystyle\dfrac{F_{3}^{ji}-F_{1}^{ji}}{F_{2}^{ji}-F_{1}^{ji}}=c$
By Theorem 3.2, we have $f^{1}\wedge f^{2}\wedge f^{3}=0.$ Then for each index
$t\in\\{1,\ldots,2n+2\\}\setminus\\{i,j\\}$ we have
$\displaystyle 0$
$\displaystyle=\det\left(\begin{array}[]{ccc}(f_{1},H_{i})&(f_{1},H_{j})&(f_{1},H_{t})\\\
(f_{2},H_{i})&(f_{2},H_{j})&(f_{2},H_{t})\\\
(f_{3},H_{i})&(f_{3},H_{j})&(f_{3},H_{t})\end{array}\right)=\prod_{u=1}^{3}(f^{u},H_{i})\cdot\det\left(\begin{array}[]{ccc}1&F_{1}^{ji}&F_{1}^{ti}\\\
1&F_{2}^{ji}&F_{2}^{ti}\\\ 1&F_{3}^{ji}&F_{3}^{ti}\\\ \end{array}\right)$
$\displaystyle=\prod_{u=1}^{3}(f^{u},H_{i})\cdot\det\left(\begin{array}[]{ccc}F_{2}^{ji}-F_{1}^{ji}&F_{2}^{ti}-F_{1}^{ti}\\\
F_{3}^{ji}-F_{1}^{ji}&F_{3}^{ti}-F_{1}^{ti}\\\ \end{array}\right).$
Thus
$(F_{2}^{ji}-F_{1}^{ji})\cdot(F_{3}^{ti}-F_{1}^{ti})=(F_{3}^{ji}-F_{1}^{ji})\cdot(F_{2}^{ti}-F_{1}^{ti}).$
If $F_{2}^{ti}-F_{1}^{ti}=0$ then $F_{3}^{ti}-F_{1}^{ti}=0$, and hence
$\Phi^{\alpha}_{it}=0$ for all $\alpha\in\mathbf{Z}^{m}_{+}$ with
$|\alpha|<1$. Otherwise, we have
$\dfrac{F_{3}^{ti}-F_{1}^{ti}}{F_{2}^{ti}-F_{1}^{ti}}=\dfrac{F_{3}^{ji}-F_{1}^{ji}}{F_{2}^{ji}-F_{1}^{ji}}=c.$
This also implies that
$\displaystyle\Phi^{\alpha}_{it}$ $\displaystyle=F_{1}^{it}\cdot
F_{2}^{it}\cdot F_{3}^{it}\cdot\left|\begin{array}[]{ccc}1&1&1\\\
F_{1}^{ti}&F_{2}^{ti}&F_{3}^{ti}\\\
\mathcal{D}^{\alpha}(F_{1}^{ti})&\mathcal{D}^{\alpha}(F_{2}^{ti})&\mathcal{D}^{\alpha}(F_{3}^{ti})\\\
\end{array}\right|$ $\displaystyle=F_{1}^{it}\cdot F_{2}^{it}\cdot
F_{3}^{it}\cdot\left|\begin{array}[]{cc}F_{2}^{ti}-F_{1}^{ti}&F_{3}^{ti}-F_{1}^{ti}\\\
\mathcal{D}^{\alpha}(F_{2}^{ti}-F_{1}^{ti})&\mathcal{D}^{\alpha}(F_{3}^{ti}-F_{1}^{ti})\\\
\end{array}\right|$ $\displaystyle=F_{1}^{it}\cdot F_{2}^{it}\cdot
F_{3}^{it}\cdot\left|\begin{array}[]{cc}F_{2}^{ti}-F_{1}^{ti}&c(F_{2}^{ti}-F_{1}^{ti})\\\
\mathcal{D}^{\alpha}(F_{2}^{ti}-F_{1}^{ti})&c\mathcal{D}^{\alpha}(F_{2}^{ti}-F_{1}^{ti})\end{array}\right|=0.$
Then one always has $\Phi^{\alpha}_{it}=0$ for all
$t\in\\{1,\ldots,2n+2\\}\setminus\\{i\\}$. The first assertion is proved.
(b) We suppose that $V_{i}\not\cong V_{t}$. From the above part, we have
$cF_{2}^{si}+(1-c)F_{1}^{si}=F_{3}^{si}\ (s\neq i).$
By the supposition $f^{1},f^{2},f^{3}$ are distinct, we have
$c\not\in\\{0,1\\}$. This implies that $F_{1}^{ti},F_{2}^{ti},F_{3}^{ti}$ are
distinct.
We see that the second inequality is clear, then we prove the remain first
inequality. We consider the meromorphic mapping $F^{t}$ of ${\mathbf{C}}^{m}$
into ${\mathbf{P}}^{1}({\mathbf{C}})$ with a reduced representation
$F^{t}=(F_{1}^{ti}h_{t}:F_{2}^{ti}h_{t}),$
where $h_{t}$ is a meromorphic function on ${\mathbf{C}}^{m}$. We see that
$\displaystyle T_{F^{t}}(r)=$ $\displaystyle
T\biggl{(}r,\dfrac{F_{1}^{ti}}{F_{2}^{ti}}\biggl{)}\leqslant
T(r,F_{1}^{ti})+T\biggl{(}r,\dfrac{1}{F_{2}^{ti}}\biggl{)}+O(1)$
$\displaystyle\leqslant T(r,F_{1}^{ti})+T(r,F_{2}^{ti})+O(1)\leqslant
T_{f^{1}}(r)+T_{f^{2}}(r)+O(1)=O(T_{f}(r)).$
For a point $z\not\in I(F^{t})\cup S_{i}\cup S_{t}$ which is a zero of some
functions $F_{u}^{ti}h_{t}\ (1\leqslant u\leqslant 3)$, then $z$ must be
either zero of $(f,H_{i})$ with multiplicity at most $k_{i}$ or zero of
$(f,H_{t})$ with multiplicity at most $k_{t}$, and hence
$\sum_{u=1}^{3}\nu^{(1)}_{F_{u}^{ti}h_{t}}(z)=1\leqslant\nu^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(z)+\nu^{(1)}_{(f,H_{t}),\leqslant k_{t}}(z).$
This implies that
$\sum_{u=1}^{3}\nu^{(1)}_{F_{u}^{ti}h_{t}}(z)\leqslant\nu^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(z)+\nu^{(1)}_{(f,H_{t}),\leqslant
k_{t}}(z)+\chi_{S_{i}}(z)+\chi_{S_{t}}(z)$
outside an analytic subset of codimension two. By integrating both sides of
this inequality, we get
(3.6) $\displaystyle\sum_{u=1}^{3}N^{(1)}_{F_{u}^{ti}h_{t}}(r)\leqslant
N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)+N^{(1)}_{(f,H_{t}),\leqslant
k_{t}}(r)+N(r,S_{i})+N(r,S_{t}).$
By the second main theorem, we also have
(3.7) $\displaystyle||\
T_{F^{t}}(r)\leqslant\sum_{u=1}^{3}N^{(1)}_{F_{u}^{ti}h_{t}}(r)+o(T(r)).$
On the other hand, applying the first main theorem to the map $F^{t}$ and the
hyperplane $\\{w_{0}-w_{1}=0\\}$ in ${\mathbf{P}}^{1}({\mathbf{C}}),$ we have
(3.8) $\displaystyle T_{F^{t}}(r)$ $\displaystyle\geqslant
N_{(F_{1}^{ti}-F_{2}^{ti})h_{t}}(r)\geqslant\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,t}}}}}^{2n+2}N^{(1)}_{(f,H_{v}),\leqslant k_{v}}(r)-N(r,S_{i})-N(r,S_{t}).$
Therefore, from (3.6), (3.7) and (3.8) we have
$\displaystyle||\ N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r)\geqslant\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,t}}}}}^{2n+2}N^{(1)}_{(f,H_{v}),\leqslant
k_{v}}(r)-N^{(1)}_{(f,H_{t}),\leqslant
k_{t}}(r)-2(N(r,S_{i})+N(r,S_{t}))+o(T(r)).$
The second assertion of the lemma is proved. $\square$
###### Lemma 3.9.
With the assumption of Theorem 1.1, let $f^{1},f^{2},f^{3}$ be three
meromorphic mappings in $\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)$.
Assume that there exist $i,j\in\\{1,2,\ldots,2n+2\\}\ (i\neq j)$ and
$\alpha\in\mathbf{Z}^{m}_{+}$ with $|\alpha|=1$ such that
$\Phi^{\alpha}_{ij}\not\equiv 0.$ Then we have
$\displaystyle T(r)$
$\displaystyle\geqslant\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)+\sum_{k=1}^{3}N^{(n)}_{(f^{k},H_{j}),\leqslant
k_{j}}(r)+2\sum_{{\mathrel{\mathop{{t\neq
i,j}}\limits^{t=1}}}}^{2n+2}N_{(f,H_{t}),\leqslant k_{t}}^{(1)}(r)$
$\displaystyle\ \ \ -(2n+1)N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r)-(n+1)N^{(1)}_{(f,H_{j}),\leqslant k_{j}}(r)+N(r,\nu_{j})$
$\displaystyle\ \ \
-N(r,S_{i})-N(r,S_{j})-(2n-2)N(r,R_{i})-(n-1)N(r,R_{j})+o(T(r))$
$\displaystyle\geqslant\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)+\sum_{k=1}^{3}N^{(n)}_{(f^{k},H_{j}),\leqslant
k_{j}}(r)+2\sum_{{\mathrel{\mathop{{t\neq
i,j}}\limits^{t=1}}}}^{2n+2}N_{(f,H_{t}),\leqslant k_{t}}^{(1)}(r)$
$\displaystyle\ \ \ -(2n+1)N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r)-(n+1)N^{(1)}_{(f,H_{j}),\leqslant k_{j}}(r)+N(r,\nu_{j})$
$\displaystyle\ \ \ \
-\sum_{u=1}^{3}\bigl{(}(1+\dfrac{n-1}{3})N^{(1)}_{(f^{u},H_{j}),>k_{j}}-(1+\dfrac{2n-2}{3})N_{(f^{u},H_{i}),>k_{i}}\bigl{)}+o(T(r)).$
Proof. The second inequality is clear. We remain prove the first inequality.
We have
$\displaystyle\Phi^{\alpha}$ $\displaystyle=F_{1}^{ij}\cdot F_{2}^{ij}\cdot
F_{3}^{ij}\cdot\left|\begin{array}[]{cccc}1&1&1\\\
F_{1}^{ji}&F_{2}^{ji}&F_{3}^{ji}\\\
\mathcal{D}^{\alpha}(F_{1}^{ji})&\mathcal{D}^{\alpha}(F_{2}^{ji})&\mathcal{D}^{\alpha}(F_{3}^{ji})\\\
\end{array}\right|$
$\displaystyle=\left|\begin{array}[]{cccc}F_{1}^{ij}&F_{2}^{ij}&F_{3}^{ij}\\\
1&1&1\\\
F_{1}^{ij}\mathcal{D}^{\alpha}(F_{2}^{ji})&F_{2}^{ij}\mathcal{D}^{\alpha}(f^{ji})&F_{3}^{ij}\mathcal{D}^{\alpha}(g^{ji})\end{array}\right|$
Thus
(3.10)
$\displaystyle\begin{split}\Phi_{ij}^{\alpha}&=F_{1}^{ij}\biggl{(}\dfrac{\mathcal{D}^{\alpha}(F_{3}^{ji})}{F^{ji}_{3}}-\dfrac{\mathcal{D}^{\alpha}(F_{2}^{ji})}{F^{ji}_{2}}\biggl{)}+F^{ij}_{2}\biggl{(}\dfrac{\mathcal{D}^{\alpha}(F_{1}^{ji})}{F^{ji}_{1}}-\dfrac{\mathcal{D}^{\alpha}(F_{3}^{ji})}{F^{ji}_{3}}\biggl{)}\\\
&\ \ \
+F^{ij}_{3}\biggl{(}\dfrac{\mathcal{D}^{\alpha}(F_{2}^{ji})}{F^{ji}_{2}}-\dfrac{\mathcal{D}^{\alpha}(F_{1}^{ji})}{F^{ji}_{1}}\biggl{)}.\end{split}$
By the Logarithmic Derivative Lemma, it follows that
$\displaystyle
m(r,\Phi^{\alpha}_{ij})\leqslant\sum_{u=1}^{3}m(r,F_{u}^{ij})+2\sum_{u=1}^{3}m\biggl{(}\dfrac{\mathcal{D}^{\alpha}(F_{u}^{ji})}{F_{v}^{ji}}\biggl{)}+O(1)\leqslant\sum_{u=1}^{3}m(r,F_{u}^{ij})+o(T_{f}(r)).$
Therefore, we have
$\displaystyle T(r)$
$\displaystyle\geqslant\sum_{u=1}^{3}T(r,F_{u}^{ij})=\sum_{u=1}^{3}(m(r,F_{u}^{ij})+N_{\frac{1}{F_{u}^{ij}}}(r))=m(r,\Phi^{\alpha}_{ij})+\sum_{u=1}^{3}N_{\frac{1}{F_{u}^{ij}}}(r)+o(T(r))$
$\displaystyle\geqslant
T(r,\Phi^{\alpha}_{ij})-N_{\frac{1}{\Phi^{\alpha}_{ij}}}+\sum_{u=1}^{3}N_{\frac{1}{F_{u}^{ij}}}(r)+o(T(r))$
$\displaystyle\geqslant
N_{\Phi^{\alpha}_{ij}}(r)-N_{\frac{1}{\Phi^{\alpha}_{ij}}}+\sum_{u=1}^{3}N_{\frac{1}{F_{u}^{ij}}}(r)+o(T(r))$
$\displaystyle=N(r,\nu_{\Phi^{\alpha}_{ij}})+\sum_{u=1}^{3}N_{\frac{1}{F_{u}^{ij}}}(r)+o(T(r)).$
Then, in order to prove the lemma, it is sufficient for us to prove
$\displaystyle N(r,\nu_{\Phi^{\alpha}_{ij}})$
$\displaystyle\geqslant\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)+\sum_{k=1}^{3}N^{(n)}_{(f^{k},H_{j}),\leqslant
k_{j}}(r)+2\sum_{{\mathrel{\mathop{{t\neq
i,j}}\limits^{t=1}}}}^{2n+2}N_{(f,H_{t}),\leqslant k_{t}}^{(1)}(r)$
$\displaystyle\ \ \ -(2n+1)N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r)-(n+1)N^{(1)}_{(f,H_{j}),\leqslant
k_{j}}(r)-\sum_{u=1}^{3}N_{\frac{1}{F_{u}^{ij}}}(r)+N(r,\nu_{j})$ (3.11)
$\displaystyle\ \ \
-N(r,S_{i})-N(r,S_{j})-(2n-2)N(r,R_{i})-(n-1)N(r,R_{j})+o(T(r)).$
Denote by $S$ the set of all singularities of $f^{-1}(H_{t})\ (1\leqslant
t\leqslant q)$. Then $S$ is an analytic subset of codimension at least two in
${\mathbf{C}}^{m}$. We set
$I=S\cup\bigcup_{s\neq t}\\{z;\nu_{(f,H_{s}),\leqslant
k_{s}}(z)\cdot\nu_{(f,H_{t}),\leqslant k_{t}}(z)>0\\}.$
Then $I$ is also an analytic subset of codimension at least two in
${\mathbf{C}}^{m}$.
In order to prove the inequality (3.11), it is sufficient for us to show that
the following inequality
(3.12) $\displaystyle P:{\mathrel{\mathop{{=}}\limits^{Def}}}$
$\displaystyle\sum_{u=1}^{3}\nu^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}+\sum_{u=1}^{3}\nu^{(n)}_{(f^{k},H_{j}),\leqslant
k_{j}}+2\sum_{{\mathrel{\mathop{{t\neq
i,j}}\limits^{t=1}}}}^{2n+2}\chi_{T_{t}}-(2n+1)\chi_{T_{i}}-(n+1)\chi_{T_{j}}$
$\displaystyle-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}+\chi_{\nu_{j}}-\chi_{S_{i}}-\chi_{S_{j}}-2(n-1)\chi_{R_{i}}-(n-1)\chi_{R_{j}}\leqslant\nu_{\Phi^{\alpha}_{ij}}.$
holds outside the set $I$.
Indeed, for $z\not\in I$, we distinguish the following cases:
Case 1: $z\in T_{t}\setminus S_{i}\cup S_{j}\ (t\neq i,j)$. We see that
$P(z)=2$. We write $\Phi^{\alpha}_{ij}$ in the form
$\Phi^{\alpha}_{ij}=F_{1}^{ij}\cdot F_{2}^{ij}\cdot
F_{3}^{ij}\times\left|\begin{array}[]{cccc}\bigl{(}F_{1}^{ji}-F_{2}^{ji}\bigl{)}&\bigl{(}F_{1}^{ji}-F_{3}^{ji}\bigl{)}\\\
\mathcal{D}^{\alpha}\bigl{(}F_{1}^{ji}-F_{2}^{ji}\bigl{)}&\mathcal{D}^{\alpha}\bigl{(}F_{1}^{ji}-F_{3}^{ji}\bigl{)}\end{array}\right|.$
Then by the assumption that $f^{1},f^{2},f^{3}$ are identify on $T_{t}$, we
have $F_{1}^{ji}=F_{2}^{ji}=F_{3}^{ji}$ on $T_{t}\setminus S_{i}$. The
property of the wronskian implies that $\nu_{\Phi^{\alpha}_{ij}}(z)\geqslant
2=P(z)$.
Case 2: $z\in T_{t}\cap(S_{i}\cup S_{j})\ (t\neq i,j)$. We see that
$P(z)\leqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)-1.$
From (3.10) we see that
$\nu_{\Phi^{\alpha}_{ij}}(z)\geqslant\min\\{\nu_{F_{1}^{ij}}(z)-1,\nu_{F_{2}^{ij}}(z)-1,\nu_{F_{3}^{ij}}(z)-1\\}\geqslant
P(z).$
Case 3: $z\in T_{i}\setminus S_{j}$. We have
$P(z)=\sum_{u=1}^{3}\nu^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(z)-(2n+1)\leqslant\min_{1\leqslant u\leqslant
3}\\{\nu^{(n)}_{(f^{u},H_{i}),\leqslant k_{i}}(z)\\}-1.$
We may assume that
$\nu_{(f^{1},H_{i})}(z)\leqslant\nu_{(f^{2},H_{i})}(z)\leqslant\nu_{(f^{3},H_{i})}(z)$.
We write
$\displaystyle\Phi^{\alpha}_{ij}=F_{1}^{ij}\biggl{[}F_{2}^{ij}(F_{1}^{ji}-F_{2}^{ji})F_{3}^{ij}\mathcal{D}^{\alpha}(F_{1}^{ji}-F_{3}^{ji})-F_{3}^{ij}(F_{1}^{ji}-F_{2}^{ji})F_{2}^{ij}\mathcal{D}^{\alpha}(F_{1}^{ji}-F_{2}^{ji})\biggl{]}$
It is easy to see that $F_{2}^{ij}(F_{1}^{ji}-F_{2}^{ji})$ and
$F_{3}^{ij}(F_{1}^{ji}-F_{3}^{ji})$ are holomorphic on a neighborhood of $z$
and
$\nu^{\infty}_{F_{3}^{ij}\mathcal{D}^{\alpha}(F_{1}^{ji}-F_{3}^{ji})}(z)\leqslant
1$
and
$\nu^{\infty}_{F_{2}^{ij}\mathcal{D}^{\alpha}(F_{1}^{ji}-F_{2}^{ji})}(z)\leqslant
1.$
Therefore, it implies that
$\displaystyle\nu_{\Phi^{\alpha}_{ij}}(z)$
$\displaystyle\geqslant\nu^{(n)}_{(f^{1},H_{i}),\leqslant k_{i}}(z)-1\geqslant
P(z).$
Case 4: $z\in T_{i}\cap S_{j}$. The assumption that $f^{1},f^{2},f^{3}$ are
identity on $T_{i}$ yields that $z\in R_{j}$. We have
$P(z)\leqslant\sum_{u=1}^{3}\nu_{(f^{u},H_{i}),\leqslant
k_{i}}^{(n)}(z)-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)-(2n+1)-n\leqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)-1.$
We have
$\displaystyle\nu_{\Phi^{\alpha}_{ij}}(z)$
$\displaystyle\geqslant\min\\{\nu_{F_{1}^{ij}}(z)-1,\nu_{F_{2}^{ij}}(z)-1,\nu_{F_{3}^{ij}}(z)-1\\}\geqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)-1\geqslant
P(z).$
Case 5: $z\in T_{j}$. We may assume that
$\nu_{F_{1}^{ji}}(z)=d_{1}\geqslant\nu_{F_{2}^{ji}}(z)=d_{2}\geqslant\nu_{F_{3}^{ji}}(z)=d_{3}.$
Choose a holomorphic function $h$ on ${\mathbf{C}}^{m}$ with multiplicity $1$
at $z$ such that $F_{u}^{ji}=h^{d_{u}}\varphi_{u}\ (1\leqslant u\leqslant 3),$
where $\varphi_{u}$ are meromorphic on ${\mathbf{C}}^{m}$ and holomorphic on a
neighborhood of $z$. Then
$\displaystyle\Phi^{\alpha}_{ij}$ $\displaystyle=F_{1}^{ij}\cdot
F_{2}^{ij}\cdot
F_{3}^{ij}\cdot\left|\begin{array}[]{ccc}F_{2}^{ji}-F_{1}^{ji}&F_{3}^{ji}-F_{1}^{ji}\\\
\mathcal{D}^{\alpha}(F_{2}^{ji}-F_{1}^{ji})&\mathcal{D}^{\alpha}(F_{3}^{ji}-F_{1}^{ji})\\\
\end{array}\right|$ $\displaystyle=F_{1}^{ij}\cdot F_{2}^{ij}\cdot
F_{3}^{ij}\cdot
h^{d_{2}+d_{3}}\cdot\left|\begin{array}[]{ccc}\varphi_{2}-h^{d_{1}-d_{2}}\varphi_{1}&\varphi_{3}-h^{d_{1}-d_{3}}\varphi_{1}\\\
\dfrac{\mathcal{D}^{\alpha}(h^{d_{2}-d_{3}}\varphi_{2}-h^{d_{1}-d_{3}}\varphi_{1})}{h^{d_{2}-d_{3}}}&\mathcal{D}^{\alpha}(\varphi_{3}-h^{d_{1}-d_{3}}\varphi_{1})\\\
\end{array}\right|.$
This yields that
$\displaystyle\nu_{\Phi^{\alpha}_{ij}}(z)$
$\displaystyle\geqslant\sum_{u=1}^{3}\nu_{F_{u}^{ij}}(z)+d_{2}+d_{3}-\max\\{0,\min\\{1,d_{2}-d_{3}\\}\\}.$
If $z\not\in S_{i}$ then
$\displaystyle P(z)$
$\displaystyle=-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)+\sum_{u=1}^{3}\min\\{n,d_{u}\\}-(n+1)+\chi_{\nu_{j}},$
and
$\displaystyle\nu_{\Phi^{\alpha}_{ij}}(z)$
$\displaystyle\geqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)+\sum_{u=1}^{3}\nu^{0}_{F_{u}^{ij}}(z)+d_{2}+d_{3}-1+\chi_{\nu_{j}}$
$\displaystyle\geqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)+d_{2}+d_{3}-1+\chi_{\nu_{j}}\geqslant
P(z).$
Otherwise, if $z\in S_{i}$ then $z\in R_{i}$, and hence
$P(z)\leqslant\sum_{u=1}^{3}\nu^{(n)}_{(f^{u},H_{j}),\leqslant
k_{j}}-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)-3n-1+\chi_{\nu_{j}}\leqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)-3n,$
$\displaystyle\text{and \ \ }\nu_{\Phi^{\alpha}_{ij}}(z)$
$\displaystyle\geqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)+\sum_{u=1}^{3}\nu^{0}_{F_{u}^{ij}}(z)+d_{2}+d_{3}-1$
$\displaystyle\geqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)+\max\\{0,-d_{1}\\}+\max\\{d_{2},0\\}+\max\\{d_{3},0\\}-1\geqslant
P(z).$
Case 6: $z\in(S_{i}\cup S_{j})\setminus(\bigcup_{t=1}^{2n+2}T_{t})$. Similarly
as Case 5, we have
$\displaystyle\nu_{\Phi^{\alpha}_{ij}}(z)$
$\displaystyle\geqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)+\max\\{0,-d_{1}\\}+\max\\{d_{2},0\\}+\max\\{d_{3},0\\}-1$
$\displaystyle\geq-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)-1\geqslant-\sum_{u=1}^{3}\nu^{\infty}_{F_{u}^{ij}}(z)-\chi_{S_{i}}-\chi_{S_{j}}\geqslant
P(z).$
From the above six cases, we see that the inequality (3.12) holds. Hence the
lemma is proved $\square$
###### Proof of theorem 1.1.
Suppose that there exits three distinct meromorphic mappings
$f^{1},f^{2},f^{3}$ in $\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)$. By
Lemma 3.2, we have $f^{1}\wedge f^{2}\wedge f^{3}\equiv 0$. Without loss of
generality, we may assume that
$\underbrace{V_{1}\cong\cdots\cong V_{l_{1}}}_{\text{ group
}1}\not\cong\underbrace{V_{l_{1}+1}\cong\cdots\cong V_{l_{2}}}_{\text{ group
}2}\not\equiv\underbrace{V_{l_{2}+1}\cong\cdots\cong V_{l_{3}}}_{\text{ group
}3}\not\cong\cdots\not\cong\underbrace{V_{l_{s}+1}\cong\cdots\cong
V_{l_{s+1}}}_{\text{ group }s},$
where $l_{s}=2n+2.$
Denote by $P$ the set of all $i\in\\{1,\ldots,2n+2\\}$ satisfying there exist
$j\in\\{1,\ldots,2n+2\\}\setminus\\{i\\}$ such that $V_{i}\not\cong V_{j}$ and
$\Phi^{\alpha}_{ij}\equiv 0$ for all $\alpha\in\mathbf{Z}^{m}_{+}$ with
$|\alpha|\leqslant 1.$ We consider the following three cases.
Case 1: $\sharp P\geqslant 2$. Then $P$ contains two elements $i,j$. Then we
have $\Phi^{\alpha}_{ij}=\Phi^{\alpha}_{ji}=0$ for all
$\alpha\in\mathbf{Z}^{m}_{+}$ with $|\alpha|\leqslant 1.$ By Lemma 2.3, there
exist two functions, for instance they are $F_{1}^{ij}$ and $F^{2}_{ij}$, and
a constant $\lambda$ such that $F_{1}^{ij}=\lambda F_{2}^{ij}$. This yields
that $F_{1}^{ij}=F^{ij}_{2}$ (by Lemma 3.3). Then by Lemma 3.5 (ii), we easily
see that $V_{i}\cong V_{j}$, i.e., $V_{i}$ and $V_{j}$ belong to the same
group in the above partition.
Without loss of generality, we may assum that $i=1$ and $j=2$. Since
$f^{1},f^{2},f^{3}$ are supposed to be distinct, the number of each group in
the above partition is less than $n+1$. Hence we have $V_{1}\cong
V_{2}\not\cong V_{t}$ for all $t\in\\{n+1,\ldots,2n+2\\}$. Then by Lemma 3.5
(ii), we have
$\displaystyle N^{(1)}_{(f,H_{i}),\leqslant
k_{1}}(r)+N^{(1)}_{(f,H_{t}),\leqslant k_{t}}(r)$
$\displaystyle\geqslant\sum_{s\neq 1,t}N^{(1)}_{(f,H_{s}),\leqslant
k_{s}}(r)-2\sum_{u=1}^{3}\sum_{s=1,t}N^{(1)}_{(f^{u},H_{s}),>k_{s}}(r),$
$\displaystyle\text{and }N^{(1)}_{(f,H_{2}),\leqslant
k_{2}}(r)+N^{(1)}_{(f,H_{t}),\leqslant k_{t}}(r)$
$\displaystyle\geqslant\sum_{s\neq 2,t}N^{(1)}_{(f,H_{s}),\leqslant
k_{s}}(r)-2\sum_{u=1}^{3}\sum_{s=2,t}N^{(1)}_{(f^{u},H_{s}),>k_{s}}(r).$
Summing-up both sides of the above two inequalities, we get
$\displaystyle 2N^{(1)}_{(f,H_{t}),\leqslant k_{t}}(r)\geq$ $\displaystyle
2\sum_{s\neq 1,2,t}N^{(1)}_{(f,H_{s}),\leqslant
k_{s}}(r)-2\sum_{u=1}^{3}(N^{(1)}_{(f^{u},H_{1}),>k_{1}}(r)+N^{(1)}_{(f^{u},H_{2}),>k_{2}}(r)$
$\displaystyle+2N^{(1)}_{(f^{u},H_{t}),>k_{t}}(r)).$
After summing-up both sides of the above inequalities over all
$t\in\\{n+1,2n+2\\}$, we easily obtain
$\displaystyle||\ \sum_{u=1}^{3}((n+2)$
$\displaystyle(N^{(1)}_{(f^{u},H_{1}),>k_{1}}(r)+N^{(1)}_{(f^{u},H_{2}),>k_{2}}(r))+2\sum_{t=n+1}^{2n+2}N^{(1)}_{(f^{u},H_{t}),>k_{t}}(r))$
$\displaystyle\geqslant(n+2)\sum_{t=3}^{n}N^{(1)}_{(f,H_{t}),\leqslant
k_{t}}(r)+n\sum_{t=n+1}^{2n+2}N^{(1)}_{(f,H_{t}),\leqslant k_{t}}(r)$
$\displaystyle\geqslant n\sum_{t=3}^{2n+2}N^{(1)}_{(f,H_{t}),\leqslant
k_{t}}(r)\geqslant\dfrac{n}{3}\sum_{u=1}^{3}\sum_{t=3}^{2n+2}N^{(1)}_{(f_{u},H_{t}),\leqslant
k_{t}}(r)$
$\displaystyle\geqslant\dfrac{n}{3}\sum_{u=1}^{3}\sum_{t=3}^{2n+2}N^{(1)}_{(f_{u},H_{t})}(r)-\dfrac{n}{3}\sum_{u=1}^{3}\sum_{t=3}^{2n+2}N^{(1)}_{(f^{u},H_{t}),>k_{t}}(r)$
$\displaystyle\geqslant\dfrac{1}{3}\sum_{u=1}^{3}\sum_{t=3}^{2n+2}N^{(n)}_{(f_{u},H_{t})}(r)-\dfrac{n}{3}\sum_{u=1}^{3}\sum_{t=3}^{2n+2}N^{(1)}_{(f^{u},H_{t}),>k_{t}}(r)$
$\displaystyle\geqslant\dfrac{n-1}{3}T(r)-\dfrac{n}{3}\sum_{u=1}^{3}\sum_{t=3}^{2n+2}N^{(1)}_{(f^{u},H_{t}),>k_{t}}(r)+o(T(r)).$
Therefore, we have
$\displaystyle\dfrac{n-1}{3}T(r)$
$\displaystyle\leqslant(n+2)\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N^{(1)}_{(f^{u},H_{t}),>k_{t}}(r)\leqslant(n+2)\sum_{u=1}^{3}\sum_{t=1}^{2n+2}\dfrac{1}{k_{t}+1}N_{(f^{u},H_{t}),>k_{t}}(r)$
$\displaystyle\leqslant(n+2)\sum_{t=1}^{2n+2}\dfrac{1}{k_{t}+1}T(r).$
Letting $r\longrightarrow+\infty$, we get
$\dfrac{n-1}{3(n+2)}\leqslant\sum_{t=1}^{2n+2}\dfrac{1}{k_{t}+1}.$
This is a contradiction.
Case 2: $\sharp P=1$. We assume that $P=\\{1\\}$. We easily see that
$V_{1}\not\cong V_{i}$ for all $i=2,\ldots,2n+2$ (otherwise $i\in P$, this
contradict to $\sharp P=1$). Then by Lemma 3.5 (ii), we have
$\displaystyle N^{(1)}_{(f,H_{1}),\leqslant k_{1}}(r)\geqslant\sum_{s\neq
1,i}N^{(1)}_{(f,H_{s}),\leqslant k_{s}}(r)-N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r)-2\sum_{u=1}^{3}\sum_{s=1,i}N^{(1)}_{(f^{u},H_{s}),>k_{s}}(r)+o(T(r)).$
Summing-up both sides of the above inequality over all $i=2,\ldots,2n+2$, we
get
(3.13) $\displaystyle\begin{split}(2n+1)N^{(1)}_{(f,H_{1}),\leqslant
k_{1}}(r)\geq&(2n-1)\sum_{i=2}^{2n+2}N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r)-2\sum_{u=1}^{3}\sum_{i=2}^{2n+2}N^{(1)}_{(f^{u},H_{i}),>k_{s}}(r)\\\
&-2(2n+1)\sum_{u=1}^{3}N^{(1)}_{(f^{u},H_{1}),>k_{1}}(r)+o(T(r)).\end{split}$
We also see that $i\not\in P$ for all $2\leqslant i\leqslant 2n+2$. We set
$\sigma(i)=\begin{cases}i+n&\text{ if }i\leqslant n+2,\\\ i-n&\text{ if
}n+2<i\leqslant 2n+2.\end{cases}$
Then we easily see that $i$ and $\sigma(i)$ belong to two distinct groups,
i.e, $V_{i}\not\cong V_{\sigma(i)}$, for all $i\in\\{2,\ldots,2n+2\\}$, and
hence $\Phi^{\alpha}_{i\sigma(i)}\not\equiv 0$ for all
$\alpha\in\mathbf{Z}^{m}_{+}$ with $|\alpha|\leqslant 1$. By Lemma 3.6 we have
$\displaystyle T(r)\geq$
$\displaystyle\sum_{u=1}^{3}\sum_{t=i,\sigma(i)}N^{(n)}_{(f^{u},H_{t}),\leqslant
k_{t}}(r)-(2n+1)N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r)-(n+1)N^{(1)}_{(f,H_{\sigma(i)}),\leqslant k_{\sigma(i)}}(r)$
$\displaystyle+2\sum_{{\mathrel{\mathop{{t\neq
i,{\sigma(i)}}}\limits^{t=1}}}}^{2n+2}N_{(f,H_{t}),\leqslant
k_{t}}^{(1)}(r)-\sum_{u=1}^{3}\biggl{(}\dfrac{2n+1}{3}N^{(1)}_{(f^{u},H_{i}),>k_{i}}(r)+\dfrac{n+2}{3}N^{(1)}_{(f^{u},H_{\sigma(i)}),>k_{\sigma(i)}}\biggl{)}$
$\displaystyle+o(T(r)).$
Summing-up both sides of the above inequalities over all
$i\in\\{2,\ldots,2n+2\\}$, we get
$\displaystyle(2n+1)$ $\displaystyle T(r)\geqslant
2\sum_{i=2}^{2n+2}\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)+(n-4)\sum_{i=2}^{2n+2}N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)$
$\displaystyle\ \ \ \ \ \ +2(2n+1)N^{(1)}_{(f,H_{1}),\leqslant
k_{1}}(r)-(n+1)\sum_{u=1}^{3}\sum_{i=2}^{2n+2}N^{(1)}_{(f^{u},H_{i}),>k_{i}}+o(T(r))$
$\displaystyle\geqslant
2\sum_{i=2}^{2n+2}\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)+\dfrac{5n-6}{3}\sum_{u=1}^{3}\sum_{i=2}^{2n+2}N^{(1)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)$
$\displaystyle-(8n+4)\sum_{u=1}^{3}N^{(1)}_{(f^{u},H_{1}),>k_{1}}(r)-(n+5)\sum_{u=1}^{3}\sum_{i=2}^{2n+2}N^{(1)}_{(f^{u},H_{i}),>k_{i}}+o(T(r))+o(T(r))$
$\displaystyle\geqslant\dfrac{11n-6}{3n}\sum_{u=1}^{3}\sum_{i=1}^{2n+2}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)$
$\displaystyle-\dfrac{4n+2}{3}\sum_{u=1}^{3}N^{(1)}_{(f^{u},H_{1}),>k_{1}}(r)-(n+1)\sum_{u=1}^{3}\sum_{i=2}^{2n+2}N^{(1)}_{(f^{u},H_{i}),>k_{i}}+o(T(r))+o(T(r))$
$\displaystyle\geqslant\dfrac{11n-6}{3n}\sum_{u=1}^{3}\sum_{i=2}^{2n+2}N^{(n)}_{(f^{u},H_{i})}(r)$
$\displaystyle-(8n+4)\sum_{u=1}^{3}N^{(1)}_{(f^{u},H_{1}),>k_{1}}(r)-\dfrac{14n+3}{3}\sum_{u=1}^{3}\sum_{i=2}^{2n+2}N^{(1)}_{(f^{u},H_{i}),>k_{i}}+o(T(r))+o(T(r))$
$\displaystyle\geqslant\dfrac{11n-6}{3}T(r)-(8n+4)\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}T(r)+o(T(r)).$
Letting $r\longrightarrow+\infty$, we get
$\displaystyle\dfrac{5n-9}{24n+12}\leqslant\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}.$
This is a contradiction.
Case 3: $P=\emptyset$. Then for all $i\neq j$, by Lemma 3.6 we have
$\displaystyle T(r)$
$\displaystyle\geqslant\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)+\sum_{k=1}^{3}N^{(n)}_{(f^{k},H_{j}),\leqslant
k_{j}}(r)+2\sum_{{\mathrel{\mathop{{t\neq
i,j}}\limits^{t=1}}}}^{2n+2}N_{(f,H_{t}),\leqslant k_{t}}^{(1)}(r)$
$\displaystyle\ \ -(2n+1)N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r)-(n+1)N^{(1)}_{(f,H_{j}),\leqslant k_{j}}(r)+N(r,\nu_{j})$
$\displaystyle\ \
-\sum_{u=1}^{3}\bigl{(}(1+\dfrac{n-1}{3})N^{(1)}_{(f^{u},H_{j}),>k_{j}}(r)+(1+\dfrac{2n-2}{3})N^{(1)}_{(f^{u},H_{i}),>k_{i}}(r)\bigl{)}+o(T(r)).$
Summing-up both sides of the above inequalities over all pairs $(i,j)$ we get
$\displaystyle(2n+2)T(r)\geqslant$ $\displaystyle
2\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N^{(n)}_{(f^{u},H_{t}),\leqslant
k_{t}}(r)+(n-2)\sum_{t=1}^{2n+2}N^{(1)}_{(f,H_{t}),\leqslant
k_{t}}(r)+\sum_{t=1}^{2n+2}N(r,\nu_{t})$ (3.14)
$\displaystyle-(n+1)\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N_{(f^{u},H_{i}),>k_{i}}+o(T(r)).$
On the other hand, by Lemma 3.4, we see that $V_{j}\not\sim V_{l}$ for all
$j\neq l$. Hence, we have
$P_{st}^{jl}:{\mathrel{\mathop{{=}}\limits^{Def}}}(f^{s},H_{j})(f^{t},H_{l})-(f^{t},H_{l})(f^{s},H_{j})\not\equiv
0\ (s\neq t,j\neq l).$
###### Claim 3.15.
With $i\neq j\neq l\neq i$, for every $z\in T_{i}$ we have
$\sum_{1\leqslant s<t\leqslant 3}\nu_{P_{st}^{jl}}(z)\geqslant
4\chi_{T_{i}}(z)-\chi_{\nu_{i}}(z).$
Indeed, for $z\in T_{i}\setminus\nu_{i}$, we may assume that
$\nu_{(f^{1},H_{i})}(z)<\nu_{(f^{2},H_{i})}(z)\leqslant\nu_{(f^{3},H_{i})}(z)$.
Since $f^{1}\wedge f^{2}\wedge f^{3}\equiv 0$, we have
$\det(V_{i},V_{j},V_{l})\equiv 0$, and hence
$\displaystyle(f^{1},H_{i})P_{23}^{jl}=(f^{2},H_{i})P_{13}^{jl}-(f^{3},H_{i})P_{12}^{jl}.$
This yields that
$\nu_{P_{23}^{jl}}(z)\geqslant 2$
and hence $\sum_{1\leqslant s<t\leqslant 3}\nu_{P_{st}^{jl}}(z)\geqslant
4=4\chi_{T_{i}}(z)-\chi_{\nu_{i}}(z)$ .
Now, for $z\in\nu_{i}$, we have $\sum_{1\leqslant s<t\leqslant
3}\nu_{P_{st}^{jl}}(z)\geqslant 3=4\chi_{T_{i}}(z)-\chi_{\nu_{i}}(z)$. Hence,
the claim is proved.
On the other hand, with $i=j$ or $i=l$, for every
$z\in\\{\nu_{(f,H_{i}),\leqslant k_{i}}(z)>0\\}$ we see that
$\displaystyle\nu_{P_{st}^{jl}}(z)\geq$
$\displaystyle\min\\{\nu_{(f^{s},H_{i}),\leqslant
k_{i}}(z),\nu_{(f^{t},H_{i}),\leqslant k_{i}}(z)\\}$ $\displaystyle\geq$
$\displaystyle\nu^{(n)}_{(f^{s},H_{i}),\leqslant
k_{i}}(z)+\nu^{(n)}_{(f^{t},H_{i}),\leqslant
k_{i}}(z)-n\nu^{(1)}_{(f,H_{i}),\leqslant k_{i}}(z).$ $\text{and hence \
}\sum_{1\leqslant s<t\leqslant 3}\nu_{P_{st}^{jl}}(z)\geqslant
2\sum_{u=1}^{3}\nu^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(z)-3n\nu^{(1)}_{(f,H_{i}),\leqslant k_{i}}(z).$
Combining this inequality and the above claim, we have
$\sum_{1\leqslant s<t\leqslant
3}\nu_{P_{st}^{jl}}(z)\geqslant\sum_{i=j,l}(2\sum_{u=1}^{3}\nu^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(z)-3n\nu^{(1)}_{(f,H_{i}),\leqslant k_{i}}(z))+\sum_{i\neq
j,l}(4\nu^{(1)}_{(f,H_{i}),\leqslant k_{i}}(z)-\chi_{\nu_{i}}(z)).$
This yields that
(3.16) $\displaystyle\begin{split}\sum_{1\leqslant s<t\leqslant
3}N_{P_{st}^{jl}}(z)\geqslant&\sum_{i=j,l}(2\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)-3nN^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r))\\\ &+\sum_{i\neq
j,l}(4N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)-N(r,\nu_{i})).\end{split}$
On the other hand, be Jensen formula, we easily see that
$N_{P_{st}^{jl}}(z)\leqslant T_{f^{s}}(r)+T_{f^{t}}(r)+o(T(r))\ (1\leqslant
s<t\leqslant 3).$
Then the inequality (3.16) implies that
$\displaystyle 2T(r)\geqslant$
$\displaystyle\sum_{i=j,l}(2\sum_{u=1}^{3}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)-3nN^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r))+\sum_{i\neq
j,l}(4N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)-N(r,\nu_{i})).$
Summing-up both sides of the above inequalities over all pair $(j,l)$, we
obtain
$\displaystyle 2T(r)\geq$
$\displaystyle\dfrac{2}{n+1}\sum_{u=1}^{3}\sum_{i=1}^{2n+2}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)+\dfrac{n}{3\times(n+1)}\sum_{u=1}^{3}\sum_{i=1}^{2n+2}N^{(1)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)$
$\displaystyle-\dfrac{n}{n+1}\sum_{i=1}^{2n+2}N(r,\nu_{i})+o(T(r)).$
Thus
$\displaystyle\sum_{i=1}^{2n+2}N(r,\nu_{i})\geq$
$\displaystyle\dfrac{2}{n}\sum_{u=1}^{3}\sum_{i=1}^{2n+2}N^{(n)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)+\dfrac{1}{3}\sum_{u=1}^{3}\sum_{i=1}^{2n+2}N^{(1)}_{(f^{u},H_{i}),\leqslant
k_{i}}(r)$ $\displaystyle-\dfrac{2(n+1)}{n}T(r)+o(T(r)).$
Using this estimate, from (3.14) we have
$\displaystyle(2n+2)T(r)\geqslant$
$\displaystyle(2+\dfrac{2}{n})\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N^{(n)}_{(f^{u},H_{t}),\leqslant
k_{t}}(r)+\dfrac{n-1}{3}\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N^{(1)}_{(f_{u},H_{t}),\leqslant
k_{t}}(r)$
$\displaystyle-\dfrac{2(n+1)}{n}T(r)-(n+1)\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N_{(f^{u},H_{i}),>k_{i}}+o(T(r)).$
$\displaystyle\geqslant(2+\dfrac{2}{n}+\dfrac{n-1}{3n})\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N^{(n)}_{(f^{u},H_{t}),\leqslant
k_{t}}(r)-\dfrac{2(n+1)}{n}T(r)$
$\displaystyle-(n+1)\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N_{(f^{u},H_{i}),>k_{i}}+o(T(r)).$
$\displaystyle\geqslant(2+\dfrac{2}{n}+\dfrac{n-1}{3n})\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N^{(n)}_{(f^{u},H_{t})}(r)-\dfrac{2(n+1)}{n}T(r)$
$\displaystyle-(3n+3+\dfrac{n-1}{3})\sum_{u=1}^{3}\sum_{t=1}^{2n+2}N_{(f^{u},H_{i}),>k_{i}}+o(T(r)).$
$\displaystyle\geqslant(2+\dfrac{2}{n}+\dfrac{n-1}{3n})(n+1)T(r)-\dfrac{2(n+1)}{n}T(r)$
$\displaystyle-(3n+3+\dfrac{n-1}{3})\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}T(r)+o(T(r)).$
Letting $r\longrightarrow+\infty$, we get
$2n+2\geqslant(2+\dfrac{2}{n}+\dfrac{n-1}{3n})(n+1)-\dfrac{2(n+1)}{n}-(3n+3+\dfrac{n-1}{3})\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}.$
Thus
$\sum_{i=1}^{2n+2}\dfrac{1}{k_{i}+1}\geqslant\dfrac{n^{2}-1}{10n^{2}+8n}$
This is a contradiction.
Hence the supposition is impossible. Therefore,
$\sharp\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)\leqslant 2$. We
complete the proof of the theorem. ∎
###### Proof of Theorem 1.4.
Let $f^{1},f^{2},f^{3}\in\mathcal{F}(f,\\{H_{i},k_{i}\\}^{2n+1}_{i=1},p)$.
Suppose that $f^{1}\times f^{2}\times
f^{3}:{\mathbf{C}}^{m}\rightarrow{\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})$
is linearly nondegenerate, where
${\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})$
is embedded into ${\mathbf{P}}^{(n+1)^{3}-1}({\mathbf{C}})$ by Seger
imbedding. Then for every $s,t,l$ we have
$P:=Det\left(\begin{array}[]{ccc}(f^{1},H_{s})&(f^{1},H_{t})&(f^{1},H_{l})\\\
(f^{2},H_{s})&(f^{2},H_{t})&(f^{2},H_{l})\\\
(f^{3},H_{s})&(f^{3},H_{t})&(f^{3},H_{l})\end{array}\right)\not\equiv 0.$
By Lemma 2.4 we have
$\displaystyle T(r)\geqslant$
$\displaystyle\sum_{i=s,t,l}(N(r,\min\\{\nu_{(f^{u},H_{i}),\leqslant
k_{i}};1\leqslant u\leqslant 3\\})$
$\displaystyle-N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r))+2\sum_{i=1}^{2n+1}N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)+o(T(r)),$
where $T(r)=\sum_{u=1}^{3}T_{f^{u}}(r)$. Summing-up both sides of the above
inequality over all $(s,t,l)$, we obtain
$\displaystyle T(r)\geqslant\dfrac{1}{2n+1}\sum_{i=1}^{2n+1}$
$\displaystyle(3N(r,\min\\{\nu_{(f^{u},H_{i}),\leqslant k_{i}};1\leqslant
u\leqslant 3\\})$ (3.17) $\displaystyle+(4n-1)N^{(1)}_{(f,H_{i}),\leqslant
k_{i}}(r))+o(T(r)).$
It is easy to see that for positive integers $a,b,c$ with
$\min\\{a,p\\}=\min\\{b,p\\}=\min\\{c,p\\}$, we have
$3\min\\{a,b,c\\}+(4n-1)\geqslant\dfrac{4n-1+3p}{2n+p}(\min\\{a,n\\}+\min\\{b,n\\}+\min\\{c,n\\}).$
Hence
$\displaystyle 3N(r,\min\\{\nu_{(f^{u},H_{i}),\leqslant k_{i}};1\leqslant
u\leqslant 3\\})$ $\displaystyle+(4n-1)N^{(1)}_{(f,H_{i}),\leqslant k_{i}}(r)$
$\displaystyle\geqslant\dfrac{4n-1+3p}{2n+p}\sum_{u=1}^{3}N^{(n)}_{(f,H_{i}),\leqslant
k_{i}}(r),\ (1\leqslant i\leqslant 2n+1).$
Therefore, the inequality (3.17) implies that
$\displaystyle T(r)$
$\displaystyle\geqslant\dfrac{1}{2n+1}\sum_{i=1}^{2n+1}\dfrac{4n-1+3p}{2n+p}\sum_{u=1}^{3}N^{(n)}_{(f,H_{i}),\leqslant
k_{i}}(r)+o(T(r))$
$\displaystyle\geqslant\dfrac{4n-1+3p}{(2n+1)(2n+p)}\sum_{i=1}^{2n+1}\sum_{u=1}^{3}(N^{(n)}_{(f,H_{i})}(r)-N^{(n)}_{(f,H_{i}),>k_{i}}(r))+o(T(r))$
$\displaystyle\geqslant\dfrac{4n-1+3p}{(2n+1)(2n+p)}(n-\sum_{i=1}^{2n+1}\dfrac{n}{k_{i}+1})T(r)+o(T(r)).$
Letting $r\longrightarrow+\infty$, we get
$1\geqslant\dfrac{4n-1+3p}{(2n+1)(2n+p)}(n-\sum_{i=1}^{2n+1}\dfrac{n}{k_{i}+1}),$
$i.e.,\sum_{i=1}^{2n+1}\dfrac{1}{k_{i}+1}\geqslant\dfrac{np-3n-p}{4n^{2}+3np-n}.$
This is a contradiction.
Hence, the map $f^{1}\times f^{2}\times f^{3}$ is linearly degenerate. The
theorem is proved. ∎
## References
* [1] Z. Chen and Q. Yan, Uniqueness theorem of meromorphic mappings into ${\mathbf{P}}^{n}({\mathbf{C}})$ sharing $2N+3$ hyperplanes regardless of multiplicities, Internat. J. Math., 20 (2009), 717-726.
* [2] H. Fujimoto, Uniqueness problem with truncated multiplicities in value distribution theory, Nagoya Math. J., 152 (1998), 131-152.
* [3] R. Nevanlinna, Einige Eideutigkeitssätze in der Theorie der meromorphen Funktionen, Acta. Math., 48 (1926), 367-391.
* [4] J. Noguchi and T. Ochiai, Introduction to Geometric Function Theory in Several Complex Variables, Trans. Math. Monogr. 80, Amer. Math. Soc., Providence, Rhode Island, 1990.
* [5] S. D. Quang, Unicity of meromorphic mappings sharing few hyperplanes, Ann. Polon. Math., 102 No. 3 (2011), 255-270.
* [6] S. D. Quang, A finiteness theorem for meromorphic mappings sharing few hyperplanes, Kodai Math. J., 102 No. 35 (2012), 463-484.
* [7] L. Smiley, Geometric conditions for unicity of holomorphic curves, Contemp. Math. 25 (1983), 149-154.
* [8] Q. Yan and Z. Chen, Degeneracy theorem for meromorphic mappings with truncated multiplicity, Acta Math. Scientia, 31B (2011) 549-560.
|
arxiv-papers
| 2014-02-22T17:33:30 |
2024-09-04T02:49:58.641503
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Si Duc Quang",
"submitter": "Si Duc Quang",
"url": "https://arxiv.org/abs/1402.5533"
}
|
1402.5578
|
# KINETIC EVOLUTION OF THE GLASMA AND
THERMALIZATION IN HEAVY ION COLLISIONS
XU-GUANG HUANG Physics Department and Center for Particle Physics and Field
Theory, Fudan University, Shanghai 200433, China.
[email protected] JINFENG LIAO Physics Department and Center for
Exploration of Energy and Matter, Indiana University,
2401 N Milo B. Sampson Lane, Bloomington, IN 47408, USA.
RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory,
Upton, NY 11973, USA.
[email protected]
(Day Month Year; Day Month Year)
###### Abstract
In relativistic heavy ion collisions, a highly occupied gluonic matter is
created shortly after initial impact, which is in a non-thermal state and
often referred to as the Glasma. Successful phenomenology suggests that the
glasma evolves rather quickly toward the thermal quark-gluon plasma and a
hydrodynamic behavior emerges at very early time $\sim\hat{o}(1)\,\rm fm/c$.
Exactly how such “apparent thermalization” occurs and connects the initial
conditions to the hydrodynamic onset, remains a significant challenge for
theory as well as phenomenology. We briefly review various ideas and recent
progress in understanding the approach of the glasma to the thermalized quark-
gluon plasma, with an emphasis on the kinetic theory description for the
evolution of such far-from-equilibrium and highly overpopulated, thus weakly-
coupled yet strongly interacting glasma.
###### keywords:
Thermalization; Glasma; Heavy Ion Collisions; Kinetic Theory.
PACS numbers:12.38.Mh, 25.75.-q, 05.60.-k, 03.75.Nt
## 1 Introduction
Relativistic heavy ion collisions provide the unique way for creating and
measuring new forms of strongly interacting matter. Such experiments are now
carried out at both the Relativistic Heavy Ion Collider (RHIC) [1, 2, 3, 4]
and the Large Hadron Collider (LHC) [5]. In such collisions, two large nuclei
(e.g. gold or lead ions) are accelerated to move at nearly the speed of light
and collide with each other, creating a domain of matter with extremely high
energy density well exceeding the expected energy density for the transition
from confined hadronic matter to deconfined strongly interacting quark-gluon
matter. This matter subsequently evolves toward a thermal quark-gluon plasma
(QGP) [6, 7] experiencing hydrodynamic expansion [8, 9, 10, 11]. The QGP
expands in a viscous-hydrodynamic way and eventually cools down enough to
hadronize into the hadronic gas that further expands and ultimately freezes
out into thousands of individual hadrons measured by detectors. The evolution
during such collisions is highly dynamical and involves the thermal, near-
thermal (transport), as well as far-from-thermal properties of the created
strongly interacting matter.
The focus of this brief review is the so-called “thermalization” problem [12,
13, 14, 15], which is about how the system evolves from the initial condition
to the nearly thermal QGP, in a relatively short time at the order of
$\sim\hat{o}(1)\,\rm fm/c$. In order to understand the context, let us first
discuss what is before and after this short transient period. Before
collision, the initial state of the large nuclei at very high energy can be
relatively well understood and described by the so-called color glass
condensate (CGC) effective theory, with an inherent momentum scale called the
saturation scale $Q_{s}$. The scale $Q_{s}$ grows with collisional beam energy
and with the size of nuclei, and for RHIC and LHC collisions the relevant
scale is on the order of a few $\rm GeV$, which implies that the relevant QCD
coupling $\alpha_{s}$ is not large and weak coupling description should be
feasible at least initially. On the other hand, successful phenomenology based
on hydrodynamic simulations of the fireball evolution have provided accurate
and detailed descriptions of an incredible amount of data from RHIC to LHC.
Essentially all such simulations require the hydrodynamic stage to start at a
very short time, typically $0.5\sim 1\,\rm fm/c$, after the initial collision.
The onset of hydrodynamic behavior is usually assumed as an indication of
nearly local thermal equilibration, so the system seems to have a very short
pre-equilibrium evolution stage in between the initial state and the
hydrodynamic onset. Normally a short relaxation time $\tau\sim
1/(\alpha_{s}^{2}Q_{s})$ toward equilibrium would indicate large coupling,
which apparently is in tension with the relatively high scale $Q_{s}$ thus
small coupling from the initial state. To make it even more complicated, there
is strong longitudinal expansion from the very beginning of the evolution
which constantly drives system toward significant anisotropy between
longitudinal and transverse pressures: to what degree the system maintains
(an)isotropy and how, remain as open questions. To date, a precise
understanding of evolution toward the (apparent and approximate)
thermalization in such pre-equilibrium matter, called the “glasma” (in between
the color glass condensate and the thermal quark-gluon plasma), is still
lacking. The thermalization problem thus presents both an outstanding
theoretical challenge and a significant phenomenological gap.
With more than a decade’s study on this problem, many different ideas and
approaches have been proposed and developed and varied mechanisms are found to
play certain roles. These include e.g. the kinetic evolutions emphasizing
either elastic or inelastic or both processes, the various plasma
instabilities, the real time lattice simulations based on classical
statistical field theories, and more recently the strongly coupled scenarios
based on gauge/gravity duality framework. It is difficult to sufficiently
discuss all these in the present paper, and there already exist excellent
sources covering one or more aspects of these. For recent reviews, see e.g.
[12, 13, 14, 15]. Instead, we will focus on the discussions of the transport
approach, with an emphasis on some recent nontrivial results in the kinetic
evolution of the far-from-equilibrium and highly overpopulated glasma.
The rest of the paper is organized as follows. In Section 2, we will briefly
survey the general context and the key issues in the pre-equilibrium
evolution, including discussions on approaches other than the kinetic one. In
Section 3, the kinetic framework for describing the glasma will be introduced
and different scattering processes will be discussed. The Section 4 will
summarize some recent results on the kinetic evolution of the glasma that make
the nontrivial link from the initial overpopulation to possible dynamical,
transient, Bose-Einstein Condensation. The Section 5 will discuss results from
other kinetic approaches. Finally the summary and some concluding remarks will
be given in Section 6.
## 2 Pre-Equilibrium Evolution in Heavy Ion Collisions
The pre-equilibrium evolution in a heavy ion collision is complicated and may
involve a few stages, from the CGC [16, 17, 18, 19, 20, 21, 22, 23, 24],
through an anisotropic strong field stage [25, 26], toward initial
isotropization via instabilities of various kinds [27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
53] till the time scale $\sim 1/Q_{\rm s}$ (up to factors logarithmic in
coupling). These plasma instabilities are “triggered” by the very initial
anisotropy and the rapid growth of unstable models frees up the quanta from
classical fields and redistributes momentum among different directions thus
bringing the system back to be near isotropy. From thereon the system could
also be considered as a dense system of gluons that becomes amenable to
kinetic evolution toward local equilibration based upon the Boltzmann
transport approach [54, 55, 56, 57, 58, 59, 60, 32, 61, 62, 63, 64, 65, 66].
In this section, we will briefly discuss these different stages, some of the
key issues, and various approaches (other than the kinetic one which will be
the focus of the next two sections).
### 2.1 Pre-collision: the Color Glass Condensate
Unlike the Big Bang, the one-shot start of our Universe, for which we can not
know what preceded it, for the heavy ion collisions known as the Little Bang,
we have a good understanding of the high energy nuclei coming into the
collisions and we can have such collisions repeatedly in laboratories. As it
turns out, the gluonic content of a nucleon or nucleus at very high energy
enters the so-called saturation regime, described by the Color Glass
Condensate effective theory (see reviews in e.g. [13, 21, 22, 23, 24]). To see
how the saturation arises, consider the gluon distribution of a nucleon moving
at extremely high energy $E=\sqrt{s}/2$ with respect to the lab frame. The
extreme high energy brings in significant Lorentz dilation effect (with
$\gamma=E/M$, $M$ the nucleon mass) on all the intrinsic time scales of the
nucleon, in particular 1) the lifetime of a quantum fluctuation
$\delta\tau\sim 1/\delta E\to\gamma\delta\tau$ from which all sea gluons (and
quarks) originate, and 2) the time scale of interactions among all these
valence and sea partons. Therefore when viewed at very high energy, the
nucleon looks like a very dense system of nearly free “wee” partons. In
addition, going to very high energy allows probing partons that carry very
small fraction of longitudinal momentum of the nucleon $x=p_{z}/E\to 0$, i.e.
the small-$x$ region of parton distributions. Measurements from Deep Inelastic
Scatterings (e.g. at HERA [68]) have shown indeed that there is a very rapid
growth of gluon numbers in the small-$x$ region that overwhelmingly dominates
over all other parton species. So there is a very dense system of gluons
emerging at small $x$ inside the high energy nucleon/nucleus.
The growth of gluon numbers at small-$x$, however, can not continue forever.
At high enough density of these gluons, the recombination starts to become
important and ultimately brings such growth to stop at a maximal density $\sim
1/\alpha_{s}$, i.e. saturation. Suppose the gluon density (on the transverse
plane of the highly contracted nucleon/nucleus) probed at given $x$ and
transverse resolution scale $Q$ is $xG(x,Q^{2})$ then an intrinsic saturation
scale $Q_{s}$ emerges and the onset of saturation, $xG/Q^{2}\to 1/\alpha_{s}$
happens for all scales $Q\leq Q_{s}$, with
$\displaystyle Q_{s}^{2}\equiv\ \alpha_{s}\,xG(x,Q_{s}^{2})$ (1)
This saturation scale changes with the nuclear size, $Q_{s}^{2}\sim A^{1/3}$,
as well as with $x$, $Q_{s}^{2}\sim 1/x^{-0.3}$. Therefore for very large
nucleus colliding at very high energy, the $Q_{s}$ becomes very large
$Q_{s}\gg\Lambda_{QCD}$ thus allowing a weak-coupling based description of the
saturated dense gluon system. Estimates suggest that $Q_{s}\sim 1-2\rm GeV$
for RHIC AuAu collisions and $Q_{s}\sim 2-3\rm GeV$ for LHC PbPb collisions.
In the CGC description based on the McLerran-Venugopalan model [16, 17], one
separates the fast and slow partons in the fast-moving nucleon with certain
cutoff scale in longitudinal momentum. By virtue of Lorentz dilation the fast
partons can be treated as approximately independent color sources with certain
color charge distribution $\rho^{a}$ and only subject to local correlations.
The slow partons (dominantly gluons) as a dense saturated system with the
phase space density $f\sim xG/Q_{s}^{2}\sim 1/\alpha_{s}$ can then be treated
as classical fields from solving Yang-Mills field equations with the presence
of such color charge distribution. The cutoff dependence is governed by proper
evolution equations. Of course, quantum fluctuations dictate that such source
distribution $\rho^{a}$ differs in each collision event. So one needs to
specify a whole ensemble of the charge density distribution based on certain
probability distribution $W[\rho]$ (e.g. Guassian) together with the classical
field configurations solved for each specific $\rho$. With this machinery,
proper initial conditions from the colliding nuclei for the heavy ion
collisions can be provided.
Let us just emphasize two important features of the pre-collision color glass
condensate: first, the emergence of an intrinsic scale, $Q_{s}$; second, the
saturated phase space density $f\sim 1/\alpha_{s}$ for gluons seen at
$Q<Q_{s}$. One may naturally imagine these two features being inherited by the
very initial stage of the glasma, which is true albeit through a rather
indirect way as we discuss next.
### 2.2 The initial Glasma fields, instability, and isotropization
With the initial states of colliding nuclei described by the GCC framework,
let us then examine the system in the collision zone just after collision
($\tau=0^{+}$). This can be done by numerically solving the Yang-Mills
equations in the forward light-cone with the given sources in that collision,
i.e.
$\displaystyle[D_{\mu},F^{\mu\nu}]=J^{\nu}$ (2)
with $J^{\nu}$ given by the fast moving color charge densities
$\rho_{1,2}(x_{\perp})$ on the two light cones from the two initial nuclei. A
striking finding is that the classical color fields are basically electric and
magnetic fields in parallel to the collision beam axis (in $\hat{z}$
direction) with vanishing transverse components, i.e. ${\bf
E}^{a}=E^{a}\hat{z}$ and ${\bf B}^{a}=B^{a}\hat{z}$, just like color flux
tubes stretching between (random) color sources in the two sheets of nuclei
moving apart from the collision point. The corresponding stress tensor
associated with such field configurations takes the form
$T^{\mu\nu}={\rm{diag}}(\epsilon,\epsilon,\epsilon,-\epsilon)$, i.e. with
negative longitudinal pressure that is obviously far from a hydrodynamic form.
So, the initial glasma fields are highly anisotropic. Such anisotropy however
does not last for long, due to the instabilities [29, 30, 31, 39, 38, 40, 41,
48, 43].
The various plasma instabilities generically arise as a consequence of
anisotropy and leads to exponential growth of modes that help restore the
isotropy. In a sense, just like particle scatterings in a gas always tend to
randomize and thus isotropize the momentum distribution, the classical fields
have interactions built in and it is not surprising that the fluctuations on
top of the fields “know” which direction to involve toward. The interesting
feature, however, is the exponential behavior which is significantly more
efficient than usual scattering processes. This has been quantitatively
studied in several approaches, such as the semi-classical transport in the
hard-loop framework [43, 44] or the classical-statistical lattice simulations
of the field evolution [49, 50, 51, 52]. For simplicity let us take the
classical-statistical field theory approach as the example here. On top of the
purely longitudinal boost-invariant initial fields, one may introduce
rapidity-dependent quantum fluctuations that evolve in the background initial
fields. The solutions exhibit exponential growth of such fluctuations for
characteristic modes (with $(\\#)\sim\hat{o}(1)$ constant)
$\displaystyle\delta A\sim e^{(\\#)\,\sqrt{(\\#)\,Q_{s}\tau}}$ (3)
This sets a limiting time scale at which the quantum fluctuations become as
large as the initial background classic fields, i.e. $\delta A(\tau_{s})\to
A_{0}\sim 1/g$: $\tau_{s}\sim(1/Q_{s})\,\ln^{2}(1/g)$. The evolution from the
initial collision till this limiting time scale for instabilities is of course
rather complicated, but is in principle computable with ab initio first-
principle approach at sufficiently weak coupling. For the purpose of our
discussions, let us just mention the following important features regarding
the system at the time scale $\tau_{s}$: 1) as the result of (primary and
secondary) instabilities, a wide range of modes, up to the order of saturation
momentum, grow until reaching the saturated regime with non-perturbatively
large occupation number $\sim 1/\alpha_{s}$ — this in a sense inherits the
characteristics of the saturated initial gluon distribution in an indirect
way; 2) the growth of these modes builds up the longitudinal pressure and
isotropizes the stress tensor
$T^{\mu\nu}={\rm{diag}}(\epsilon,P_{T},P_{T},P_{L})$ to the extent of possible
remaining $\hat{o}(1)$ anisotropy between $P_{L}$ and $P_{T}$.
### 2.3 Field evolution from classical-statistical lattice simulations
From this point on, i.e. $\tau>\tau_{s}\sim(1/Q_{s})\,\ln^{2}(1/g)$, the
system becomes a very dense system of fluctuation modes (in the statistical-
field language) or equivalently gluons (in the kinetic language) with high
occupation $\sim 1/\alpha_{s}$. The initial high anisotropy in glasma fields
has by now been reduced to be rather mild. Studies on the further evolution of
such a system toward equilibration may be divided into two categories. One
category deals with the system’s evolution in a fixed volume i.e. without
expansion (which is often referred to as “static box case”). This is certainly
of great theoretical interest and in fact quite challenging. For example, a
final conclusion is yet to be achieved even regarding the seemingly simple
parametric question of thermalization time $\tau_{\rm
th}\sim\alpha_{s}^{(\\#?)}/Q_{s}$ in the static box case. The studies of
static box case also provide extremely useful insights on the roles of various
driving mechanisms toward thermalization. The other category deals with the
evolution of the system undergoing boost-invariant longitudinal expansion
(often referred to as “expanding case”), which is a more realistic setting
relevant to heavy ion collisions.
Both the static box case and the expanding case have been thoroughly studied
using the classical-statistical lattice simulations, for the scalar field
theories as well as the Yang-Mills theories: see e.g. most recent results in
[50, 51, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 67] . Within this
approach’s regime of validity i.e. weak coupling $\alpha_{s}\ll 1$ and high
occupation $f\gg 1$, these studies have provided fairly detailed pictures of
the field evolution, with the stress tensor components and spectrum of their
correlators also evaluated. Among other interesting results, it was found that
the evolution is characterized by a pair of dual cascades: the particle
cascade toward the infrared modes, and the energy cascade toward the
ultraviolet modes. On the ultraviolet end, the simulations running toward very
large time (which implies very small coupling to ensure the validity of this
approach at late time) appears to show the system’s evolution onto non-thermal
fixed point with a self-similar form for which the scaling exponents could be
understood via turbulent scaling arguments [77, 78, 79]. On the infrared end,
simulations (for the scalar field case) starting with overpopulated initial
conditions appear to show clear evidences [69, 71] for the onset of a
dynamically formed out-of-equilibrium Bose-Einstein Condensate as predicted in
[62, 63] .
Let us discuss a little bit about the complications in the expanding case. In
the static box case, there is a well-defined thermal fixed point to which the
system will eventually equilibrate, and conservation laws (e.g. for energy,
and for particle number in purely elastic case) are straightforward. When the
system undergoes boost-invariant longitudinal expansion, the situation is
quite different as strictly speaking there is no well-defined static thermal
fixed point. Both the energy and the particle number are dropping with time
due to expansion that dilutes the system. What’s more the expansion is
constantly bringing the system out of isotropy, i.e. even if the system starts
in an isotropic state it will quickly become anisotropic due to “shrinking of
longitudinal momenta”. Note this is a dynamical issue quite separated and
different from the high anisotropy from glasma fields in the very initial
condition. If there is no interaction, then the system will continuously free-
stream with less and less average longitudinal momentum compared with the
transverse one, $\langle p^{2}_{L}\rangle\,\ll\,\langle p^{2}_{T}\rangle$. Of
course interactions, e.g. scatterings, help re-distribute the momentum between
the longitudinal and the transverse and try to restore isotropy to some
extent. The issue is, whether the scatterings are strong enough to compete
with the longitudinal expansion. It could be that the expansion wins and the
system anisotropy constantly grows till falling apart [56, 57]. It could also
be that the scatterings are able to keep the anisotropy below or at most of
order one for a long time [62] . To complicate it further, how the energy
density decreases with time depends on the degree of anisotropy, while on the
other hand the energy density dictates the hard scale that would affect the
efficiency of scatterings against expansion: so this is a highly dynamical,
and nonlinear issue. Classical-statistical lattice simulations at extremely
small coupling (e.g. $\alpha_{s}\sim 10^{-5}$) appear to suggest the scenario
of growing anisotropy [77, 78, 79]. However while moving to the relatively
larger coupling regime (e.g. $\alpha_{s}\sim 10^{-2}$) but still within the
applicability of the approach, there appears to be a plausible transition to a
different behavior of the evolution in which the pressure anisotropy is
maintained at order one and able to be matched to viscous hydrodynamics [51] .
At the moment a final conclusion is yet to be reached, and in particular a lot
of future efforts will be required to push the classical field approach toward
the physically more relevant regime (with $\alpha_{s}\sim 10^{-1}$) which is a
highly nontrivial challenge. As a final comment, it seems quite clear that
with the presence of longitudinal expansion, a full isotropization may never
have been reached, and an anisotropy at least of order one may be present for
a considerable window in the early time dynamics of heavy ion collisions. Such
“fixed anisotropy” for microscopically long but macroscopically short time
scale may provide an underlying basis for the recently developed anisotropic
hydrodynamics (aHydro) [80, 81, 82, 83] framework that explicitly accounts for
the sizable anisotropy at early times.
### 2.4 From classical fields to kinetic quanta
From the discussions above it is clear that the classical field theory
description, valid for large occupation number $f\gg 1$, will come to an end
at some point as the occupation number $f$ of various modes decreases rapidly
with time due to longitudinal expansion. This is of course the process of
field decoherence that frees up individual gluons. A natural framework to
describe the weakly coupled gluon system is the kinetic theory based on
Boltzmann transport equation. In such an approach, one uses a distribution
function $f(t,{\mathbf{x}},{\mathbf{p}})$ to effectively describe the system
and dynamics (e.g. various scattering processes) enters via collision kernel
${\cal C}[f]$, and with provided an initial condition then it evolves via
transport equation ${\cal D}_{t}f={\cal C}[f]$ in a definitive manner. The
transport approach is good for weak coupling and not too large occupation,
$\alpha_{s}\ll 1$ and $f\leq 1/\alpha_{s}$. A complete description of the
glasma evolution at weak coupling shall plausibly involve a proper switch from
the classical fields to the kinetic quanta. Many works [54, 55, 56, 57, 58,
59, 60, 32, 61, 62, 63, 64, 65, 66] have been done to study the pre-
equilibrium evolution in the kinetic approach, which will be thoroughly
discussed in the next two Sections.
One may notice that there is an overlapping region for the validity of the
classical field versus kinetic approaches: for occupation in the regime $1\ll
f\leq 1/\alpha_{s}$, both descriptions are feasible, and therefore should be
connected with each other. Indeed, the equivalence of the two has been
demonstrated in various theories, see e.g. [84, 85, 60, 86]. Roughly speaking,
the equation of motion for the Green’s function in classical field theory can
be suitably mapped to the evolution equation for distribution function, with
the interaction terms in the former becoming the collision terms in the
transport equation. This is particularly interesting as one may expect dual
descriptions in the two approaches for the same physical phenomenon occurring
in their common valid regime. The kinetic approach can oftentimes help develop
intuitive pictures for understanding results from classical field simulations.
For example, the interesting turbulent scaling exponents from late time non-
thermal fixed point found in classical field simulations could be easily
understood via the pertinent kinetic description [77]. Another example is the
occurrence of BEC starting with overpopulated initial condition, which was
easier to be predicted first from the kinetic evolution [62], while less
obvious in the classical field description. It is extremely useful to have
such dual descriptions as an important tool to develop deeper understanding
and have mutual confirmation for interesting results from each other.
### 2.5 Thermalization in strongly-coupled theories
Finally let us also briefly discuss an “orthogonal” approach for understanding
thermalization with strong coupling for the underlying microscopic theory.
This is different from what has been discussed so far, where we consider the
system to be weakly coupled but strongly interacting due to high phase space
density. In strong coupling regime it is difficult to have a direct “attack”,
and instead one utilizes the tool of gauge/gravity duality [87, 88, 89, 90,
91, 92, 93, 94, 95, 96, 97, 98, 99, 100]. This holographic correspondence
provides a powerful framework to study the evolution of 4-dimensional strongly
coupled quantum fluid toward equilibration subject to varied far-from-
equilibrium initial conditions, via solving 5-dimensional classical gravity
problems. The time dependence of the 4-D theory is translated into the general
relativity dynamics in the 5-D (asymptotically) AdS space with proper boundary
conditions. Apart from the issue of to what extent such strongly coupled
descriptions are applicable to the early time system in heavy ion collisions,
these studies have certainly provided interesting insights into the far-from-
equilibrium evolution in quantum field theories. Not being able to give a
detailed discussion here on many interesting results from the holographic
approach (which is not the primary focus of the present review), let us
emphasize one particularly important point clearly demonstrated in such
studies: by analyzing the evolution of the energy-momentum tensor one sees a
viscous-hydrodynamic behavior that emerges quickly, well before the full
equilibration, and upon the onset of such hydrodynamic regime the system still
bears significant anisotropy between the longitudinal and transverse
pressures. That is, an “apparent macroscopic thermalization” may occur long
preceding the true and complete microscopic thermalization.
## 3 Kinetic Description of the Pre-Equilibrium Evolution
We now turn to an elementary discussion on the kinetic description of the pre-
equilibrium evolution, namely, a description based on the transport equation
of gluons with various scattering processes. Standard textbooks for kinetic
theory include [101, 102, 103].
As discussed previously, in heavy ion collisions gluons are freed at a time
scale $t_{0}\sim Q_{s}^{-1}$, with momentum typically of order $Q_{s}$ and
phase space occupation number of order $1/\alpha_{s}$. For large nuclei and/or
high collision energies $Q_{s}\gg\Lambda_{QCD}$ so that $\alpha_{s}\ll 1$.
After that (i.e., for $Q_{s}t>1$), one may then treat the gluons as on-shell
quanta, and effectively describe the system with a phase space distribution
function
$\displaystyle
f(t,{\mathbf{x}},{\mathbf{p}})\equiv\frac{(2\pi)^{3}}{N_{g}}\frac{dN}{d^{3}{\mathbf{x}}d^{3}{\mathbf{p}}},$
(4)
where $N_{g}=2(N_{c}^{2}-1)=16$ is the gluon degeneracy. The kinetic evolution
of $f(t,{\mathbf{x}},{\mathbf{p}})$ is described by the Boltzmann equation
which schematically reads
$\displaystyle{\cal D}_{t}f(t,{\mathbf{x}},{\mathbf{p}})={\cal C}[f],$ (5)
where
$\displaystyle{\cal
D}_{t}f(t,{\mathbf{x}},{\mathbf{p}})\equiv\frac{p^{\mu}}{E_{p}}\partial_{\mu}f(t,{\mathbf{x}},{\mathbf{p}})=(\partial_{t}+{\bf
v}_{p}\cdot\nabla_{\mathbf{x}})f(t,{\mathbf{x}},{\mathbf{p}})$ (6)
with ${\bf v}_{p}\equiv{\mathbf{p}}/E_{p}$ being the velocity of gluon with
momentum ${\mathbf{p}}$ and $E_{p}=|{\mathbf{p}}|$ the energy of the gluon.
Note that on the left-hand side of the equation we have neglected the force
term ${\bf F}\cdot\nabla_{\bf p}$ which need to be included if external color
fields may be applied. In the following, unless explicitly stated, we will
restrict ourselves to the spatially homogeneous systems so that $f$ has no
${\mathbf{x}}$ dependence and we can ignore the drift term ${\bf
v}_{p}\cdot\nabla_{\mathbf{x}}f_{p}$ on the left-hand side of the equation.
With the general structure of kinetic equation above, there are two important
ingredients that govern the solutions to it: (1) the initial condition, i.e.
the $f(t_{0},{\mathbf{x}},{\mathbf{p}})$ at initial time $t_{0}$; (2) the
dynamics of the underlying microscopic theory, i.e. various collisional
processes, that will enter through the collision kernel on the right hand side
of the equation. Both play nontrivial roles in the evolution, as we shall see
in later sections with more concrete examples.
### 3.1 Conservation laws
The conservation laws play important roles in studies of kinetic evolution.
These conservation laws originate from the conservation laws observed by the
corresponding microscopic interactions. Let us discuss a number of such
examples.
First of all, let us consider the particle number conservation when there are
only elastic scatterings, i.e. the collision term only has contributions from
$m\to m$ processes. The number density is given by $n=\int
d^{3}{\mathbf{p}}/(2\pi)^{3}f_{p}=\int_{{\mathbf{p}}}2E_{p}f_{p}$ where
$f_{p}\equiv f(t,{\mathbf{p}})$ and
$\displaystyle\int_{\mathbf{p}}$ $\displaystyle\equiv$
$\displaystyle\int\frac{d^{3}{\mathbf{p}}}{(2\pi)^{3}2E_{p}}.$ (7)
Therefore the changing rate of number density is given by (considering all
particles being bosons)
$\displaystyle{\cal D}_{t}n=\int\frac{d^{3}{\mathbf{p}}_{1}}{(2\pi)^{3}}{\cal
C}[f_{1}]\propto\int_{{1,2,...,m}}\int_{{m+1,m+2,...,2m}}|M_{m\rightarrow
m}|^{2}\delta^{(4)}\left(\Sigma_{i=1}^{m}p_{i}-\Sigma_{j=m+1}^{2m}p_{j}\right)$
$\displaystyle\quad\times\\{[\Pi_{i=1}^{m}(1+f_{i})][\Pi_{j=m+1}^{2m}f_{j}]-[\Pi_{i=1}^{m}f_{i}][\Pi_{j=m+1}^{2m}(1+f_{j})]\\}\quad$
$\displaystyle\quad=0,$ (8)
where the right hand side vanishes by symmetry (i.e. with the same number of
particles in the initial and final states in the microscopic scatterings).
Note also that ${\cal D}_{t}n=\partial_{\mu}\langle
nu_{\mu}\rangle=\partial_{\mu}\int d^{3}{\mathbf{p}}/(2\pi)^{3}f_{p}u_{\mu}$
with $u_{\mu}=p_{\mu}/E_{p}$ the four-velocity.
Now let us consider the energy-momentum conservation. The energy momentum
tensor is given by $T^{\mu\nu}=\int_{{\mathbf{p}}}{p^{\mu}p^{\nu}}f_{p}$ and
for generic $m\to n$ processes
$\displaystyle\partial_{\mu}T^{\mu\nu}=\int\frac{d^{3}{\mathbf{p}}_{1}}{(2\pi)^{3}}p_{1}^{\nu}{\cal
C}[f_{1}]\propto\int_{{1,...,m}}\int_{{m+1,...,m+n}}p_{1}^{\nu}|M_{m\rightarrow
n}|^{2}\delta^{(4)}\left(\Sigma_{i=1}^{m}p_{i}-\Sigma_{j=m+1}^{m+n}p_{j}\right)$
$\displaystyle\quad\times\\{[\Pi_{i=1}^{m}(1+f_{i})][\Pi_{j=m+1}^{m+n}f_{j}]-[\Pi_{i=1}^{m}f_{i}][\Pi_{j=m+1}^{m+n}(1+f_{j})]\\}$
$\displaystyle\quad\propto\int_{{1,..,m}}\int_{{m+1,...,m+n}}|M_{m\rightarrow
n}|^{2}(\Sigma_{i=1}^{m}p_{i}^{\nu}-\Sigma_{j=m+1}^{m+n}p_{j}^{\nu})\delta^{(4)}(\Sigma_{i=1}^{m}p_{i}-\Sigma_{j=m+1}^{m+n}p_{j})$
$\displaystyle\quad\times\\{[\Pi_{i=1}^{m}(1+f_{i})][\Pi_{j=m+1}^{m+n}f_{j}]-[\Pi_{i=1}^{m}f_{i}][\Pi_{j=m+1}^{m+n}(1+f_{j})]\\}$
$\displaystyle\quad=0,$ (9)
where the right-hand side vanishes by the cyclic symmetry for re-labeling all
particles and the microscopic conservation (via the delta function).
If one considers homogeneous systems with the origin at the whole system’s
center of mass (thus zero total momentum), then the energy conservation
$\partial_{t}\epsilon=0$ and particle number conservation $\partial_{t}n=0$
(in the pure elastic case) are the two important global constraints.
More directly relevant to application toward heavy ion collisions is the
situation with boost-invariant longitudinal expansion. In that case, one has
to take into account the drift term on the left-hand side (LHS), i.e.
$\displaystyle{\cal
D}_{t}f(t,{\mathbf{x}},{\mathbf{p}})=(\partial_{t}+v_{z}\partial_{z})f(t,z,{\mathbf{p}})={\cal
C}[f].$ (10)
By assuming boost-invariance $f(t,z,{\mathbf{p}})\to
f(\tau,y-\eta,{\mathbf{p}}_{\perp})$ (where $y$ is momentum rapidity while
$\eta$ the spatial rapidity) and by focusing on the system at mid-rapidity
$\eta\to 0$ or $z\to 0$, one can simplify the above equation into the
following form
$\displaystyle{\cal
D}_{t}f(t,{\mathbf{x}},{\mathbf{p}})=\left(\partial_{\tau}-\frac{p_{z}}{\tau}\partial_{p_{z}}\right)f(\tau,{\mathbf{p}})=\left(\partial_{t}-\frac{p_{z}}{t}\partial_{p_{z}}\right)f(t,{\mathbf{p}})={\cal
C}[f].$ (11)
We notice that the above kinetic equation can be rewritten as
$\displaystyle\left(\partial_{t}-\frac{p_{z}}{t}\partial_{p_{z}}\right)f(t,{\mathbf{p}})=\frac{\partial(t\,f)}{t\partial_{t}}-\nabla_{{\mathbf{p}}}\cdot\left[\frac{p_{z}}{t}f\,\hat{z}\right].$
(12)
By integrating the above equation one can easily get the corresponding version
for the time evolution of number density $n$ and energy density $\epsilon$:
$\displaystyle\partial_{t}n+n/t=0\quad\to\quad n=n_{0}\times\frac{t_{0}}{t}$
(13)
$\displaystyle\partial_{t}\epsilon+(1+\delta)\epsilon/t=0\quad\to\quad\epsilon=\epsilon_{0}\times\left(\frac{t_{0}}{t}\right)^{1+\delta}$
(14)
where $\delta=P_{L}/\epsilon$ is the ratio of longitudinal pressure to the
energy density that characterizes the system’s degree of anisotropy: in
isotropic case $\delta=1/3$ and $\epsilon\sim 1/t^{4/3}$ as in ideal
hydrodynamics while in free-streaming case $\delta\to 0$ and $\epsilon\to
1/t$.
### 3.2 The elastic collision kernel
The collision kernel ${\cal C}[f]$ controls the rate at which the gluons
change their momentum state by collision processes. With gluon scatterings in
QCD, the leading order contributions to ${\cal C}[f]$ consist of two
essentially different processes, the elastic $2\rightarrow 2$ process to be
discussed in this subsection, as well as the inelastic effective $1\rightarrow
2$ process to be discussed in the next subsection.
Let us first look at the $2\rightarrow 2$ elastic collision kernel, given by
$\displaystyle{\cal C}_{2\rightarrow 2}[f_{1}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{234}\frac{1}{2E_{1}}|M_{12\rightarrow
34}|^{2}(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}-p_{3}-p_{4})$ (15)
$\displaystyle\times[(1+f_{1})(1+f_{2})f_{3}f_{4}-f_{1}f_{2}(1+f_{3})(1+f_{4})].$
In Eq. (15), the factor $1/2$ in front of the integral is a symmetry factor,
which takes into account the fact that exchanging the gluons 3 and 4 leads to
identical configuration being doubly counted in both the matrix element and
the full 3 and 4 momentum integration. The $(1+f_{i})$ factors represent the
final state Bose enhancement arising from quantum nature which is vitally
important when $f_{i}$ becomes large. In the dilute limit $f_{i}\ll 1$ then
$(1+f_{i})\to 1$ reducing to the classical Boltzmann regime.
$|M_{12\rightarrow 34}|^{2}$ is the $2\rightarrow 2$ matrix element (in the
vacuum),
$\displaystyle|M_{12\rightarrow 34}|^{2}$ $\displaystyle=$ $\displaystyle
8g^{4}N_{c}^{2}\left(3-\frac{tu}{s^{2}}-\frac{su}{t^{2}}-\frac{ts}{u^{2}}\right),$
(16)
with $s,t,u$ the usual Mandelstam variables
$\displaystyle
s=(p_{1}+p_{2})^{2},\;\;t=(p_{1}-p_{3})^{2},\;\;u=(p_{1}-p_{4})^{2},$ (17)
and $p_{i}=(E_{i},{\mathbf{p}}_{i})$. As is well known for Coulomb type long
range interaction, the dominant contribution in the elastic collision integral
comes from the small angle scatterings, where there is very small momentum
transfer $q\ll p_{i}$ between the colliding gluons and the incoming states’
momenta get deflected only with small angles. This is evident from the matrix
element $|M_{12\rightarrow 34}|^{2}$ with divergence in the $u$ and $t$
channels at $u\rightarrow 0$ or $t\rightarrow 0$ which ultimately lead to
logarithmic contributions. Thus under the small angle approximation, the
particle’s momentum receives a small but random deflection in each of a series
of collisions and experiences a “random walk” in the momentum space. Following
the Landau-Lifshitz approach [101], the Boltzmann equation with the
$2\rightarrow 2$ collision kernel can then be reduced to a Fokker-Planck
equation describing such momentum space diffusion, by rewriting ${\cal
C}_{2\rightarrow 2}[f_{1}]$ as the divergence of a current ${\bf\cal
J}({\mathbf{p}}_{1})$ in momentum space,
$\displaystyle\partial_{t}f_{1}=-{\bm\nabla}_{1}\cdot{\bf\cal
J}({\mathbf{p}}_{1}),$ (18)
where $\nabla_{1}=\partial/\partial{\mathbf{p}}_{1}$. A standard calculation
yields [62, 63]
$\displaystyle{\bf\cal
J}({\mathbf{p}})=-\frac{g^{4}N_{c}^{2}L}{8\pi^{3}}\left[I_{a}{\bm\nabla}f_{p}+I_{b}{\bf
v}_{p}f_{p}(1+f_{p})\right].$ (19)
Here $L=\int d|{\mathbf{q}}|/|{\mathbf{q}}|$ is the Coulomb logarithm which
needs to be regularized by the medium screening effect, $L\sim\ln(q_{\rm
max}/q_{\rm min})$ where ${q_{\rm max}}$ is typically of order of the hard
scale of the problem (e.g., the temperature if equilibrium is achieved) and
$q_{\rm min}$ is determined by the screening mass, $q_{\rm min}\sim m_{D}$
with $m_{D}$ the Debye screening mass, $m_{D}^{2}\sim\alpha_{s}\int
d^{3}{\mathbf{p}}(f_{p}/E_{p})$.
It shall be noted that the thermal fixed point of Eq.(18) is precisely the
Bose-Einstein distribution $f_{\rm
eq}({\mathbf{p}})=\frac{1}{\exp{[(E_{p}-\mu)/T}]-1}$. It is also
straightforward to show that the derived elastic collision kernel on the right
hand side of (18) conserves both particle number and energy.
The two integrals $I_{a}$ and $I_{b}$ are defined as follows
$\displaystyle I_{a}$ $\displaystyle\equiv$ $\displaystyle
2\pi^{2}\int\frac{d^{3}{\mathbf{p}}}{(2\pi)^{3}}f_{p}(1+f_{p}),$ (20)
$\displaystyle I_{b}$ $\displaystyle\equiv$ $\displaystyle
2\pi^{2}\int\frac{d^{3}{\mathbf{p}}}{(2\pi)^{3}}\frac{2f_{p}}{E_{p}}.$ (21)
The integral $I_{a}$ plays the role of a diffusion constant. For a gluon
undergoing successive random small angle scatterings over a time window $t$,
its momentum will undergo a random walk acquiring a total of final momentum
square transfer $\langle\Delta p^{2}\rangle\sim\hat{q}_{\rm el}t$ where the
parameter $\hat{q}_{\rm el}$ characterizes the momentum diffusion. The
$\hat{q}_{\rm el}$ is relates to $I_{a}$ simply by (up to pre-factor in
logarithm of $\alpha_{s}$)
$\displaystyle\hat{q}_{\rm el}\sim\alpha_{s}^{2}I_{a}.$ (22)
The integral $I_{b}$ is proportional to the Debye screening mass,
$\displaystyle m^{2}_{D}\sim\alpha_{s}I_{b}.$ (23)
To give concrete examples: in the thermal equilibrium with Bose-Einstein
distribution, the two integrals become $I_{a}\sim T^{3}$ and $I_{b}\sim T^{2}$
and in fact $I_{a}=T\,I_{b}$; away from equilibrium this gets changed, e.g. in
a glasma-type distribution with $f\sim 1/\alpha_{s}$ up to $Q_{s}$, one gets
$I_{a}\sim Q_{s}^{3}/\alpha_{s}^{2}$ while $I_{b}\sim Q_{s}^{2}/\alpha_{s}$
with different power dependence on coupling.
### 3.3 The inelastic collision kernel
We now turn to the inelastic collision kernel. The lowest-order inelastic
process of gluon scatterings is the $2\rightarrow 3$ process which in naive
power counting is $\alpha_{s}$ suppressed as compared with the elastic
process. This however is rather tricky due to strong infrared divergences
present in the corresponding matrix element. In fact, a careful analysis would
reveal that its contribution to the collision kernel ${\cal C}[f]$ is at the
same parametric order of coupling constant as the elastic process, due to the
strong soft and collinear enhancement in the $2\rightarrow 3$ matrix element,
as will become transparent later. In general, the $2\rightarrow 3$ collision
kernel takes the following form:
$\displaystyle{\cal C}_{2\rightarrow 3}[f_{1}]$ $\displaystyle=$
$\displaystyle{\cal C}^{a}_{2\rightarrow 3}[f_{1}]+{\cal C}_{2\rightarrow
3}^{b}[f_{1}],$ (24) $\displaystyle{\cal C}_{2\rightarrow 3}^{a}[f_{1}]$
$\displaystyle=$
$\displaystyle\frac{1}{6}\int_{2345}\frac{1}{2E_{1}}|M_{12\rightarrow
345}|^{2}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4}-p_{5})$
$\displaystyle\times[(1+f_{1})(1+f_{2})f_{3}f_{4}f_{5}-f_{1}f_{2}(1+f_{3})(1+f_{4})(1+f_{5})],$
$\displaystyle{\cal C}_{2\rightarrow 3}^{b}[f_{1}]$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{2345}\frac{1}{2E_{1}}|M_{34\rightarrow
125}|^{2}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}+p_{5}-p_{3}-p_{4})$ (25)
$\displaystyle\times[(1+f_{1})(1+f_{2})(1+f_{5})f_{3}f_{4}-f_{1}f_{2}f_{5}(1+f_{3})(1+f_{4})].$
The two terms ${\cal C}^{a}$ and ${\cal C}^{b}$ differ in that the momentum
$p_{1}$ (that one is “watching”) is on the two-particle side in the former
while on the three-particle side in the latter. The general form of the
leading-order $|M_{12\rightarrow 345}|^{2}$ is known [104, 105, 106]:
$\displaystyle|M_{12\rightarrow 345}|^{2}$ $\displaystyle=$ $\displaystyle
g^{6}N_{c}^{3}\frac{\cal N}{\cal
D}[(12345)+(12354)+(12435)+(12453)+(12534)+(12543)$ (26)
$\displaystyle+(13245)+(13254)+(13425)+(13524)+(14235)+(14325)],$
where
$\displaystyle{\cal N}$ $\displaystyle=$
$\displaystyle(12)^{4}+(13)^{4}+(14)^{4}+(15)^{4}+(23)^{4}$
$\displaystyle+(24)^{4}+(25)^{4}+(34)^{4}+(35)^{4}+(45)^{4},$
$\displaystyle{\cal D}$ $\displaystyle=$
$\displaystyle(12)(13)(14)(15)(23)(24)(25)(34)(35)(45),$
$\displaystyle(ijklm)$ $\displaystyle=$ $\displaystyle(ij)(jk)(kl)(lm)(mi),$
$\displaystyle(ij)$ $\displaystyle\equiv$ $\displaystyle p_{i}\cdot p_{j}.$
Note that $|M_{12\rightarrow 345}|^{2}$ itself is completely symmetric in
permutation of $p_{i}$.
Like the $2\rightarrow 2$ matrix element $|M_{12\rightarrow 34}|^{2}$, the
$2\rightarrow 3$ matrix element $|M_{12\rightarrow 345}|^{2}$ also contains
the small angle singularity. In addition, it possesses a more severe IR
singularity, the collinear singularity which occurs when the softest gluon of
the five particles, say, gluon $5$ moves in a collinear way with one of the
other gluons during either absorption or emission. The collinear singularity
is also well known in perturbation theory, associated with the massless
kinematics of gluons. By collecting the most singular contributions in
$|M_{12\rightarrow 345}|^{2}$, one arrives at the Gunion-Bertsch formula (e.g.
for the piece of $t$-channel and soft $p_{5}$) [107, 108, 109, 110, 64, 111]:
$\displaystyle|M_{12\rightarrow 345}|_{\rm GB}^{2}$ $\displaystyle=$
$\displaystyle 16g^{6}N_{c}^{3}\frac{(p_{1}\cdot p_{2})^{3}}{(p_{1}\cdot
p_{3})(p_{2}\cdot p_{4})(p_{1}\cdot p_{5})(p_{2}\cdot p_{5})}.$ (27)
When applying the above (specific channel) Gunion-Bertsch formula to ${\cal
C}_{2\rightarrow 3}^{a}$, one needs to multiply the kernel by a symmetry
factor of $2\times 3=6$ to account for the other five identical contributions
(from $u$-channel and soft $p_{3},p_{4}$ singularities). When applying it to
${\cal C}_{2\rightarrow 3}^{b}$, there is a symmetry factor of $2\times 2=4$
to take into account the other three identical contributions (with one factor
$2$ from the degeneracy of $u,t$-channel, another factor $2$ from the
degeneracy of soft $p_{2},p_{5}$). As a commonly adopted strategy, one needs
to regularize various IR singularities that survive to the end results of the
collision kernel, by appropriate medium screening mass as a IR cutoff in order
to obtain finite results. It should also be noted that the GB approximation
can be systematically extended by including less singular terms order by order
through expanding the exact $|M_{12\rightarrow 345}|^{2}$ in $p_{5}/\sqrt{s}$
and $t/s$, see Ref. [112, 113, 111, 64] for details.
Under the small angle and collinear approximations, the inelastic kernel can
be much simplified. Physically there are two types of contributions that can
be seen by examining the softest scale among the external gluons and the
internal (exchanging) gluons, say, $q$ and $p_{5}$. If $p_{5}\ll q$, the
$2\rightarrow 3$ process can be regarded as an effective $2\rightarrow 2$
process with a slight modification due to final state emission of a very soft
gluon $p_{5}$. On the other hand, if $q\ll p_{5}$, the $2\rightarrow 3$
process can be considered as an effective $1\rightarrow 2$ process with one
“hard” gluon getting a small “kick” and experiencing “bremsstrahlung”. With
this in mind, an analytic inelastic kernel can be derived (see details in
[64]) and the final result reads:
$\displaystyle{\cal C}_{2\rightarrow 3}[f_{1}]$ $\displaystyle=$
$\displaystyle{\cal C}^{\rm eff}_{2\rightarrow 2}[f_{1}]+{\cal
C}_{1\rightarrow 2}^{\rm eff}[f_{1}],$ (28)
where
$\displaystyle{\cal C}_{2\rightarrow 2}^{\rm eff}[f_{1}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{234}\frac{1}{2E_{1}}|M_{12\rightarrow
34}|^{2}_{\rm eff}(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}-p_{3}-p_{4})$ (29)
$\displaystyle\times[(1+f_{1})(1+f_{2})f_{3}f_{4}-f_{1}f_{2}(1+f_{3})(1+f_{4})],$
and
$\displaystyle{\cal C}_{1\rightarrow 2}^{\rm eff}[f_{1}]$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dz|M_{1\rightarrow 2}|^{2}_{\rm
eff}\Big{\\{}\frac{1}{2}[g_{p_{1}}f_{(1-z){p_{1}}}f_{z{p_{1}}}-f_{p_{1}}g_{(1-z){p_{1}}}g_{z{p_{1}}}]$
(30)
$\displaystyle+\frac{1}{z^{3}}[f_{p_{1}/z}g_{1}g_{(1-z){p_{1}}/z}-g_{p_{1}/z}f_{p_{1}}f_{(1-z){p_{1}}/z}]\Big{\\}}.$
In the above we have introduced
$|M_{12\rightarrow 34}|^{2}_{\rm eff}={\cal D}(q)|M_{12\rightarrow 34}|^{2}$
and
$|M_{1\rightarrow 2}|^{2}_{\rm
eff}=\frac{6g^{6}N_{c}^{3}CQI_{a}}{(2\pi)^{5}}\frac{1}{z(1-z)}$
where $C=\int_{-1}^{1}dx/(1-x)$, $Q=\int dq/q^{3}$, and
$\displaystyle{\cal D}(q)$ $\displaystyle=$
$\displaystyle\int_{k<q}\frac{2g^{2}N_{c}}{|{\mathbf{k}}|^{2}}\left[\frac{1+2f_{k}}{1-{\bf
v}_{k}\cdot{\bf v}_{1}}+\frac{1+2f_{k}}{1-{\bf v}_{k}\cdot{\bf
v}_{p}}\right].$ (31)
Again, the IR divergence in $C$, $Q$ and ${\cal D}$ should be appropriately
regularized by screening effects and up to the logarithm of $\alpha_{s}$
$\displaystyle C\sim 1,\;\;\;Q\sim 1/m_{D}^{2},$ (32)
and ${\cal D}$ is at most $\hat{o}(1)$ order for $f\lesssim 1/\alpha_{s}$.
Thus the main new, number-changing, contribution of the $2\rightarrow 3$
process to the collision kernel, is the effective splitting/joining
$1\rightarrow 2$ kernel ${\cal C}_{1\rightarrow 2}^{\rm eff}$ which can be
concisely written as follows by collecting all the order $\hat{o}(1)$
constants into a parameter $R$:
$\displaystyle{\cal C}_{1\rightarrow 2}^{\rm eff}[f_{1}]$ $\displaystyle=$
$\displaystyle
R\frac{\alpha_{s}^{3}I_{a}}{m_{D}^{2}}\int_{0}^{1}dz\frac{1}{z(1-z)}\Big{\\{}\frac{1}{2}[g_{p_{1}}f_{(1-z){p_{1}}}f_{z{p_{1}}}-f_{p_{1}}g_{(1-z){p_{1}}}g_{z{p_{1}}}]$
(33)
$\displaystyle+\frac{1}{z^{3}}[f_{p_{1}/z}g_{1}g_{(1-z){p_{1}}/z}-g_{p_{1}/z}f_{p_{1}}f_{(1-z){p_{1}}/z}]\Big{\\}}.$
It is not difficult to see that: first, the thermal fixed point of Eq.(33) is
the Bose-Einstein distribution with zero chemical potential $f_{\rm
eq}({\mathbf{p}})=1/[\exp{(E_{p}/T)}-1]$; second, the kernel conserves energy
while does not conserve particle number.
Let us now examine the power counting and compare the elastic kernel (18)
versus the inelastic kernel (33). By noting parametrically the
$m_{D}^{2}\sim\alpha_{s}f^{2}$ one realizes that both kernels are at the same
order. To be more concrete, let us examine both the thermal case and the
highly off-equilibrium glasma case. In the thermal case, we have $f\sim 1$,
$m_{D}^{2}\sim\alpha_{s}T^{2}$, $I_{a}\sim T^{3}$ and $I_{b}\sim T^{2}$: thus
the collision rate for both types of processes are parametrically
$\Gamma\sim\alpha_{s}^{2}T$. In the glasma case we have $f\sim 1/\alpha_{s}$
up to $Q_{s}$, $m_{D}^{2}\sim Q_{s}^{2}$, $I_{a}\sim Q_{s}^{3}/\alpha_{s}^{2}$
and $I_{b}\sim Q_{s}^{2}/\alpha_{s}$: thus the collision rate for both types
of processes are parametrically $\Gamma\sim Q_{s}$. We therefore see that the
two types of processes are indeed contributing to the kinetic evolution at the
same parametric order, and hence both need to be included at this order.
### 3.4 Higher order kernels and the LPM suppression
From last subsection, we have seen that the leading contribution from the the
$2\rightarrow 3$ matrix element benefits from soft and collinear enhancement
and becomes an effective $1\rightarrow 2$ process parametrically at the same
order in $\alpha_{s}$ as the small-angle $2\rightarrow 2$ scattering. This
contribution, essentially a bremsstrahlung process, may however bear further
complication. Let us consider the emission of a soft gluon with momentum
$p_{5}$ by a parent gluon upon one small “kick”. The emission needs a
formation time $t_{\rm form}(p_{5})$ to be completed, though, and during that
time it is likely the parent gluon may experience yet another “kick”: in fact
this is inevitable if the formation time $t_{\rm form}(p_{5})$ becomes bigger
than the “mean-free-path” of the parent gluon in the medium, and the emission
is “blended” together with multiple scatterings. This is of course the well
know and well studied Landau-Pomeranchuk-Migdal (LPM) effect [115, 116, 117]
initially found in QED. A proper treatment requires resuming the
$1+n\rightarrow 2+n$ ($n\geq 1$) multiple scatterings, and has been understood
in the context of QCD particularly for the jet energy loss [118, 119, 120,
121, 122].
To take into account the LPM effect in the kinetic equation is a nontrivial
task, and a thorough treatment in this aspect has been developed by Arnold,
Moore and Yaffe [123, 124, 125, 126, 127, 128]. As a schematic approach, one
can encode the LPM effect into the Boltzmann equation in the following way.
Let the differential splitting/merging rate be $d\Gamma/dk$, then the
$1\rightarrow 2$ collision kernel in the collinear limit would be
$\displaystyle{\cal C}_{1\rightarrow 2}[f_{p}]\sim\int
dk\frac{d\Gamma}{dk}\left[(g_{p}f_{p-k}f_{k}-f_{p}g_{p-k}g_{k})+\left(\frac{p+k}{p}\right)^{\eta}(g_{p}g_{k}f_{p+k}-f_{p}f_{k}g_{p+k})\right],$
where $p=|{\mathbf{p}}|,k=|{\mathbf{k}}|$, and
${\mathbf{k}}\parallel{\mathbf{p}}$. The parameter $\eta$ will be fixed by
enforcing energy conservation to be preserved by the kernel. If the formation
time $t_{\rm form}$ is shorter than the duration $t_{\rm el}$ between two
successive elastic scatterings, the spitting/merging rate is of the Bethe-
Heitler type which parametrically reads [129]
$\displaystyle\frac{d\Gamma_{\rm BH}}{dk}\sim\frac{\alpha_{s}}{k}\Gamma_{\rm
el}\sim\frac{\alpha_{s}}{k}\int_{pp^{\prime}}|M|^{2}_{2\rightarrow
2}f_{p}(1+f_{p^{\prime}})\sim\frac{\alpha_{s}}{k}\frac{\hat{q}_{\rm
el}}{m_{D}^{2}},$ (35)
where $\Gamma_{\rm el}$ is the rate of soft elastic scattering. Substituting
this into Eq. (3.4), one arrives at a kernel bearing the structure of Eq. (33)
for a single scattering case. On the other hand, if the formation time $t_{\rm
form}$ is longer than $t_{\rm el}$, then the emission process cannot resolve
individual collision and “feels” the coherent superposition of multiple
scatterings during the formation time: in this case, one has $\Gamma_{\rm
LPM}(k)\sim\alpha_{s}t_{\rm form}^{-1}(k)$. During this formation time, the
emitted gluon obtains a transverse momentum $\Delta
k_{\perp}\sim\sqrt{\hat{q}_{\rm el}t_{\rm form}}$ while its transverse size is
$\Delta l_{\perp}\sim 1/\Delta k_{\perp}$ and its transverse velocity is
$v_{\perp}\sim\Delta k_{\perp}/k$. For the emission to be completed, the
emitted gluon must separate from its parent gluon, which implies the condition
$\displaystyle t_{\rm form}\sim\frac{\Delta
l_{\perp}}{v_{\perp}}\sim\frac{k}{\hat{q}_{\rm el}t_{\rm form}},$ (36)
from which we obtain $t_{\rm form}(k)\sim\sqrt{k/\hat{q}_{\rm el}}$. Thus the
LPM suppressed splitting/merging rate would have the form
$\displaystyle\frac{d\Gamma_{\rm
LPM}}{dk}\sim\frac{\alpha_{s}}{k}\sqrt{\frac{\hat{q}_{\rm el}}{k}}.$ (37)
Substituting it into Eq. (3.4), one obtains the following form of the kernel
at the end:
$\displaystyle{\cal C}_{1\rightarrow 2}^{\rm LPM}[f_{p}]$ $\displaystyle\sim$
$\displaystyle\alpha_{s}^{2}\sqrt{\frac{I_{a}}{p}}\Big{\\{}\int_{0}^{1}\frac{dz}{z^{3/2}}[g_{p}f_{(1-z){p}}f_{z{p}}-f_{p}g_{(1-z){p}}g_{z{p}}]$
(38)
$\displaystyle+\int_{0}^{1}\frac{dz}{z^{2}}\frac{1}{[z(1-z)]^{3/2}}[f_{p/z}g_{p}g_{(1-z){p}/z}-g_{p/z}f_{p}f_{(1-z){p}/z}]\Big{\\}}.$
Note that the LPM effect plays important role only when $t_{\rm
form}(k)\gtrsim t_{\rm el}\sim m_{D}^{2}/\hat{q}_{\rm el}$, i.e, when $k$ is
larger than $m_{D}^{4}/\hat{q}_{\rm el}$. It is clear the above kernel
preserves a similar loss/gain structure to the kernel in Eq. (33), thus also
having the same thermal fixed point. It is also at the same parametric order
as (Eq. (33)) in coupling constant.
## 4 Kinetic Evolution in Overpopulated Regime and Possible Bose-Einstein
Condensation in the Glasma
With the kinetic framework set up in the previous Section, we now turn to
discuss the application of this framework to the description of the kinetic
evolution of the glasma that is pertinent to the early stage in heavy ion
collisions. As already discussed previously, since the initial scale $Q_{s}$
in the glasma is large and thus the coupling is weak, the kinetic theory seems
to be a natural and plausible framework to investigate the detailed evolution
of the phase space distribution in the dense gluon system starting from the
time scale $\sim 1/Q_{\rm s}$. Such efforts were initiated long ago [54, 55,
56, 57, 58, 59, 60, 61] and some of these will be discussed in the next
Section. An apparent tension in such approaches exists in that in a naive
counting the scattering rate (of leading elastic processes)
$\sim\alpha_{s}^{2}$ may not be able to bring the system back to
thermalization quickly enough. A number of past kinetic works suggest that the
inelastic processes may play more significant role as compared with the
elastic ones in speeding up the thermalization process, especially in
populating the very soft momentum region. This may be true in the dilute
regime (close to the Boltzmann limit), however may not be the accurate picture
when the system under consideration is in the highly overpopulated regime with
$f\sim 1/\alpha_{s}$. As shown in a number of recent kinetic studies [62, 63],
the elastic scatterings with highly overpopulated initial conditions can lead
to order $\sim{\alpha_{s}^{0}}$ evolution and develop strong infrared cascade
with the Bose enhancement, and in fact may even induce a dynamical Bose-
Einstein Condensation. In this Section we focus on discussing some of these
most recent developments.
### 4.1 The highly overpopulated Glasma
To see a few nontrivial features associated with high overpopulation, let us
consider the kinetic evolution in a weakly coupled gluon system initially
described by the following glasma-type distribution as inspired by the CGC
picture:
$\displaystyle f(p\leq Q_{\rm s})=f_{0}\quad,\quad f(p>Q_{\rm s})=0\,.$ (39)
For the glasma in heavy ion collisions, the phase space is maximally filled:
$f_{0}\sim 1/\alpha_{s}$ (with $\alpha_{\rm s}\ll 1$). As first emphasized in
a recent paper [62], such high occupation coherently amplifies scattering and
changes usual power counting of scattering rate: the resulting collision term
from the $2\leftrightarrow 2$ gluon scattering process will scale as
$\sim\alpha_{\rm s}^{2}f^{2}\sim\hat{o}(1)$ despite smallish $\alpha_{\rm s}$.
This is a natural consequence of the essential Bose enhancement factor $(1+f)$
which would scale as $f$ in the dense regime while scale as $1$ in the dilute
regime. It becomes more obvious if one examines the momentum diffusion
parameter in Eq.(22): $I_{a}\sim\hat{o}(1/\alpha_{s}^{2})\,Q_{s}^{3}$ and
$\hat{q}_{el}\sim\hat{o}(1)\,Q_{s}^{3}$, and therefore the time for order one
change of typical momentum via scatterings scales as
$\tau\sim\hat{o}(1)\,Q_{s}^{-1}$. With the coupling constant dropping out of
the problem, the system behaves as an emergent strongly interacting matter,
even though the elementary coupling is small.
A novel finding [62, 63], hitherto unrealized, is that a system with such
initial condition is highly overpopulated: that is, the gluon occupation
number is parametrically large when compared to a system in thermal
equilibrium with the same energy density. To illustrate this point, consider
the energy and particle number densities with the initial distribution (39),
we have
$\displaystyle\epsilon_{0}=f_{0}\,\frac{Q_{s}^{4}}{8\pi^{2}},\qquad
n_{0}=f_{0}\,\frac{Q_{s}^{3}}{6\pi^{2}},\qquad
n_{0}\,\epsilon_{0}^{-3/4}=f_{0}^{1/4}\,\frac{2^{5/4}}{3\,\pi^{1/2}},$ (40)
with $\epsilon_{0}$ and $n_{0}$ the initial energy density and number density,
respectively. The energy is always conserved during the evolution while the
particle number would also be conserved if only elastic scatterings are
involved. The value of the parameter $n\,\epsilon^{-3/4}$ that corresponds to
the onset of Bose-Einstein condensation, i.e., to an equilibrium state with
vanishing chemical potential, is obtained by taking for $f(p)$ the ideal
distribution for massless particles at temperature $T$. One gets then
$\epsilon_{SB}=(\pi^{2}/30)\,T^{4}$ and $n_{SB}=(\zeta(3)/\pi^{2})\,T^{3}$, so
that
$\displaystyle
n\,\epsilon^{-3/4}|_{SB}=\frac{30^{3/4}\,\zeta(3)}{\pi^{7/2}}\approx 0.28.$
(41)
Comparing with $n_{0}\,\epsilon_{0}^{-3/4}$ in Eq. (40), one sees that when
$f_{0}$ exceeds the value $f_{0}^{c}\approx 0.154$, the initial distribution
(39) contains too many gluons to be accommodated in an equilibrium Bose-
Einstein distribution, i.e. overpopulated: in this case the equilibrium state
will have to contain a Bose-Einstein condensate if there are only elastic
scatterings. It is worth emphasizing that over-occupation does not require
necessarily large values of $f_{0}$, in fact the values just quoted are
smaller than unity. It follows therefore that the situation of over-occupation
will be met for generic values of $\alpha_{s}$. For instance, for
$\alpha_{s}\simeq 0.3$, $f_{0}=1/\alpha_{s}$ is significantly larger than
$f_{0}^{c}$ for a wide class of initial conditions (and even more so if the
coupling is smaller). One though may also notice that in theories like QCD
there are inelastic processes: this removes in principle the possibility of
any condensate in the equilibrated state as the inelastic, number changing
processes (no matter slow or fast) will eventually remove all the excessive
particles. This however leaves open an even more interesting question:
starting with overpopulated initial conditions, will the system dynamically
evolve and develop a transient Bose-Einstein Condensate?
We therefore see that a Bose system in highly overpopulated regime bears
distinctive features that may play key roles in the glasma evolution,
including the parametrically enhanced soft elastic scatterings with order one
rate and the possibility of a transient Bose condensate during the course of
thermalization. Significant interests and intensive investigations have been
triggered recently in understanding such overpopulated regime with a variety
of approaches [65, 66, 69, 70, 71, 72, 73, 74, 75, 76, 12, 130, 131]. There
are strong evidences for Bose condensation reported for similar overpopulated
systems in the classical-statistical lattice simulation of scalar field theory
[69, 70, 71], with the case for non-Abelian gauge theory still under
investigation [72, 73, 74, 75, 76]. In the rest of this Section, we will first
discuss a number of interesting results on the kinetic evolution in such
overpopulated systems, including the scaling solutions and the dynamical onset
of kinetic BEC with the purely elastic scattering, and in the last part also
discuss the effects of inelastic collisions.
### 4.2 The two scales and scaling solutions for elastic scattering
To qualitatively describe the kinetic evolution, one may introduce two scales
for characterizing a general distribution: a soft scale $\Lambda_{\rm s}$
below which the occupation reaches $f(p<\Lambda_{\rm s})\sim 1/\alpha_{\rm
s}\gg 1$ and a hard cutoff scale $\Lambda$ beyond which the occupation is
negligible $f(p>\Lambda)\ll 1$. For the glasma initial distribution in (39),
there is essentially only one scale i.e. the saturation scale $Q_{\rm s}$
which divides the phase space into two regions, one with $f\gg 1$ and the
other with $f\ll 1$, i.e. with the two scales overlapping $\Lambda_{\rm
s}\sim\Lambda\sim Q_{\rm s}$. The thermalization is a process of maximizing
the entropy (with the given amount of energy). The entropy density for an
arbitrary distribution function is given by $s\sim\int
d^{3}{\mathbf{p}}\left[(1+f)\,\ln(1+f)-f\,\ln(f)\right]$: this implies that
with the total energy constrained, it is much more beneficial to have as wide
as possible a phase space region with $f\sim 1$. Indeed, for a thermal Bose
gas one has the soft scale $\Lambda_{\rm s}^{th}\sim\alpha_{\rm s}T$ and the
hard scale $\Lambda^{th}\sim T$ separated by the coupling $\alpha_{\rm s}$. By
this general argument, one shall expect the separation of the two scales along
the thermalization process: from the $\Lambda_{s}\sim\Lambda$ in the initial
glasma toward the $\Lambda^{th}_{s}\sim\alpha_{\rm s}\Lambda^{th}$ in the
thermal situation.
To be more quantitative, one may define the two scales $\Lambda$ and
$\Lambda_{\rm s}$ as follows:
$\displaystyle\Lambda\left({{\Lambda_{\rm s}}\over{\alpha_{\rm
s}}}\right)^{2}\equiv I_{a}\quad$ ,
$\displaystyle\quad\Lambda\left({{\Lambda_{\rm s}}\over{\alpha_{\rm
s}}}\right)\equiv I_{b}$ (42) $\displaystyle{\rm
or}\;\;\;\;\;\;\;\Lambda=\frac{I_{b}^{2}}{I_{a}}\quad$ ,
$\displaystyle\quad\Lambda_{s}=\alpha_{s}\frac{I_{a}}{I_{b}}$ (43)
With the above definition we indeed have $\Lambda_{\rm s}\sim\Lambda\sim
Q_{\rm s}$ for the glasma distribution while $\Lambda_{\rm
s}^{th}\sim\alpha_{\rm s}\Lambda^{th}\sim\alpha_{\rm s}T$ for thermal
distribution. Again one can see that with the overpopulated glasma
distribution the collision term ${\cal
C}\sim\Lambda_{s}^{2}\Lambda\sim\hat{o}(1)$ in coupling, in contrast to the
thermal case with ${\cal
C}\sim{\Lambda^{th}_{s}}^{2}\Lambda^{th}\sim\hat{o}(\alpha_{s}^{2})$.
Let us now discuss possible scaling solution for the evolution of the two
scales in the static box case. With the glasma distribution the scattering
time from the collision integral on the RHS of the transport equation (18)
scales as $t_{\rm sca}\sim\Lambda/\Lambda_{\rm s}^{2}$. To find scaling
solution for the time evolution of $\Lambda$ and $\Lambda_{s}$, we use two
conditions — that the energy must be conserved and that the scattering time
shall scale with the time itself, i.e.:
$\displaystyle t_{\rm sca}\sim\frac{\Lambda}{\Lambda_{\rm s}^{2}}\sim
t\quad,\quad\epsilon\sim\frac{\Lambda_{\rm s}\Lambda^{3}}{\alpha_{\rm s}}={\rm
constant}$ (44)
The particle number also must be conserved, albeit with a possible component
in the condensate: $n=n_{g}+n_{c}\sim(\Lambda_{\rm s}\Lambda^{2}/\alpha_{\rm
s})+n_{c}={\rm constant}$. The condensate plays a vital role with little
contribution to energy while unlimited capacity to accommodate excessive
gluons. With these two conditions we thus obtain:
$\displaystyle\Lambda_{\rm s}\sim Q_{\rm
s}\left(\frac{t_{0}}{t}\right)^{3/7}\quad,\quad\Lambda\sim Q_{\rm
s}\left(\frac{t_{0}}{t}\right)^{-1/7}$ (45)
From this solution, the gluon density $n_{g}$ decreases as
$\sim(t_{0}/t)^{1/7}$, and therefore the condensate density is growing with
time, $n_{c}\sim(Q_{\rm s}^{3}/\alpha_{\rm s})[1-(t_{0}/t)^{1/7}]$. A
parametric thermalization time could be identified by the required
$\Lambda_{\rm s}/\Lambda\sim\alpha_{\rm s}$:
$\displaystyle t_{\rm th}\sim\frac{1}{Q_{\rm s}}\,\left(\frac{1}{\alpha_{\rm
s}}\right)^{7/4}$ (46)
At the same time scale the overpopulation parameter $n\epsilon^{-3/4}$ indeed
also reduces from the initial value of order $\sim 1/\alpha_{s}^{1/4}$ to be
of the order one.
What would change if one considers the more realistic evolution with boost-
invariant longitudinal expansion? First of all the conservation laws will be
manifest differently: the total number density will decrease as $n\sim
n_{0}t_{0}/t$, while the time-dependence of energy density depends upon the
momentum space anisotropy $\epsilon\sim\epsilon_{0}(t_{0}/t)^{1+\delta}$ for a
fixed anisotropy $\delta\equiv P_{L}/\epsilon$ (with $P_{L}$ the longitudinal
pressure). Along similar line of analysis as before with the new condition of
energy evolution we obtain the following scaling solution in the expanding
case:
$\displaystyle\Lambda_{\rm s}\sim Q_{\rm
s}\left(t_{0}/t\right)^{(4+\delta)/7}\,,\,\Lambda\sim Q_{\rm
s}\left(t_{0}/t\right)^{(1+2\delta)/7}\,.$ (47)
With this solution, we see the gluon number density $n_{g}\sim(Q_{\rm
s}^{3}/\alpha_{\rm s})(t_{0}/t)^{(6+5\delta)/7}$, and therefore with any
$\delta>1/5$ the gluon density would drop faster than $\sim t_{0}/t$ and there
will be formation of the condensate, i.e. $n_{c}\sim(Q_{\rm s}^{3}/\alpha_{\rm
s})(t_{0}/t)[1-(t_{0}/t)^{(5\delta-1)/7}]$. Similarly a thermalization time
scale can be identified through the separation of scales to be:
$\displaystyle t_{\rm th}\sim\frac{1}{Q_{\rm s}}\,\left(\frac{1}{\alpha_{\rm
s}}\right)^{7/(3-\delta)}\,.$ (48)
The possibility of maintaining a fixed anisotropy during the glasma evolution
is not obvious but quite plausible due to the large scattering rate
$\sim\Lambda_{\rm s}^{2}/\Lambda\sim 1/t$ that is capable of competing with
the $\sim 1/t$ expansion rate and may reach a dynamical balance. In such a
scenario a complete isotropization may never be reached due to longitudinal
expansion, while the system may yet evolve for a long time with a fixed
anisotropy between average longitudinal and transverse momenta.
### 4.3 Dynamical onset of Bose-Einstein Condensation
To more quantitatively understand the kinetic evolution of overpopulated
glasma, one needs to numerically solve the transport equation which in the
pure elastic case is given by Eqs.(18)(19). This has recently been reported in
[63]. The solutions of course depend on the initial conditions. In general,
one expects two types of solutions: evolution from underpopulated initial
conditions leads at late time to a thermal Bose-Einstein distribution
function; while evolution starting with overpopulated initial conditions shows
a transition, in a finite time, to a Bose-Einstein condensate. If the initial
distribution is specified as the glasma type in (39), then as discussed above,
which solution occurs depends on whether $f_{0}$ is greater or smaller than
the critical value $f_{0}^{c}$. For simplicity we focus our discussions here
mostly on the static box case and will briefly comment on the expanding case
at the end.
So how does the thermalization proceed in such a system? Numerical solutions
in both the underpopulated and the overpopulated cases suggest two generic
features in the kinetic evolution driven by elastic scatterings. First, two
cascades in momentum space will quickly develop: a particle cascade toward the
IR momentum region that quickly populates the soft momentum modes to high
occupation, and a energy cascade toward the UV momentum region that spreads
the energy out. The two cascades are of course interrelated as per the
particle number and energy conservation. This can be clearly seen by the plots
of the momentum space current (19) for the underpopulated (Fig.1 right panel)
as well as the overpopulated (Fig.2 right panel) cases: the negative current
at low momenta is the IR cascade and the positive current at high momenta is
the UV cascade. It worths emphasizing that the Bose statistical factors play a
key role in the strong particle cascade toward IR, amplifying the rapid growth
of the population of the soft modes. As a consequence a high occupation number
at IR is quickly achieved, thus with very fast scattering rate, leads to the
second interesting feature: an almost instantaneous local “equilibrium” form
for the distribution near the origin $p\to 0$:
$\displaystyle f^{*}(p\to 0)=\frac{1}{{\rm e}^{(p-\mu^{*})/T^{*}}-1},$ (49)
Analytically this follows from the requirement that as long as the $f(p\to 0)$
is finite then the current (19) has to vanish linearly in $p$ toward the
origin. This can be easily seen by integrating the transport equation (18) in
an arbitrarily small sphere around the origin [63]. The quick emergence of
such local IR thermal form has also been numerically verified in both the
underpopulated (Fig.1 left panel) and the overpopulated (Fig.2 left panel)
cases. Note that the $T^{*}$ and $\mu^{*}$ are only parameters characterizing
the small momentum shape of the distribution and not to be confused with a
true thermal temperature and chemical potential.
Figure 1: The distribution function $f(p)$ (left) and the current
${\mathcal{J}}(p)$ (right) for various times, from an early time till the time
where thermalization is nearly completed, starting with the underpopulated
initial condition $f_{0}=0.1$.
Figure 2: The distribution function $f(p)$ (left) and the current
${\mathcal{J}}(p)$ (right) for various times, from an early time till the time
where thermalization is nearly completed, starting with the overpopulated
initial condition $f_{0}=1$.
The above picture naturally leads to the next question: how the local IR
thermal form eventually evolves into the global thermal form? In the
underpopulated case the answer is simple (as explicitly shown by numerical
solutions [63]): the distribution will take time to adjust the whole
distribution toward Bose-Einstein distribution (as the proper fixed point of
the collision term), with the local parameters $T^{*}$ and $\mu^{*}$
approaching the final thermal $T$ and $\mu$ determined by energy and particle
number conservation. In the overpopulated case, however, the condensate will
need to be formed before the ultimate thermalization. As is well known in the
kinetic study of BEC literature [132, 133], one has to separately describe the
evolution prior to the onset of condensation (with the usual transport
equation) and the evolution afterwards (with a coupled set of two equations
explicitly for condensate and regular distribution). Of particular
significance is to understand dynamically how and when the condensation occurs
starting from an overpopulated initial condition. So here let us focus on the
pre-BEC stage, and with the equations (18)(19), such question could be
answered by numerically solving it till the time of BEC onset.
In [63] this problem has been thoroughly studied with varied initial
conditions and firm evidence has been found that initially overpopulated
systems are driven by coherently amplified soft elastic scatterings to reach
the onset of Bose-Einstein condensation in a finite time, approaching the
onset with a scaling behavior. Different from the underpopulated case, in the
overpopulated case the IR cascade persists to drive the local thermal
distribution near $p=0$ to increase rapidly in a self-similar form (see Fig.2
left panel). The associated negative local “chemical potential” is driven to
approach zero, i.e. $(-\mu^{*})\to 0^{+}$ (see Fig.3 left panel) and
ultimately vanishes in a finite time, marking the onset of the condensation.
The approaching toward onset is well described by a scaling behavior:
$\displaystyle|\mu^{*}|=C(\tau_{c}-\tau)^{\eta}$ (50)
with a universal exponent $\eta\approx 1$ for varied values of
$f_{0}>f_{0}^{c}$. One may analytically show that the exponent is expected to
be unity via similar scaling arguments used in the famous turbulent wave
scaling analysis. The onset time $\tau_{c}$ and the coefficient $C$ is shown
in Fig.3 (middle and right panels). Such evolution toward onset is robust
against different initial distribution shapes, e.g. the same behavior was
found with a Guassian initial distribution in the overpopulated regime.
Figure 3: The approach of $\mu^{*}$ toward zero in a scaling way i.e.
$\mu^{*}\approx C(\tau_{c}-\tau)$ (left panel) for a variety choice of
$f_{0}$. One may further extract the value $\tau_{c}$ (middle panel) at which
Bose condensation sets in (left panel) as well as the slope $C$ (right panel)
as a function of $f_{0}$.
These results, obtained by using kinetic theory, with a quantum Boltzmann
equation in the small angle approximation, have therefore provided numerical
evidence that a system of gluons with an initial distribution that mimic that
expected in heavy ion collisions reaches the onset of Bose-Einstein
condensation in a finite time. The role of Bose statistical factors in
amplifying the rapid growth of the population of the soft modes is essential.
With these factors properly taken into account, one finds that elastic
scattering alone provides an efficient mechanism for populating soft modes,
that could be competitive with the radiation mechanism invoked in the scenario
of Ref. [56]. Ongoing efforts have extended studies of such kinetic evolution
toward more general situations, including the effect of longitudinal expansion
and possible initial momentum space anisotropy, as well as the effect of
finite medium-generated mass. The general link from initial overpopulation to
the onset of BEC in a finite time with a scaling behavior appears to be very
robust.
There is one particularly important issue, though. It is a prior unclear
whether this picture of dynamics BEC onset will be significantly altered,
should there be inelastic processes. One may even wonder if such onset
(manifested as the development of an infrared singularity in the kinetic
evolution) would happen anymore provided any inelastic processes could in
principle remove excess particles from overpopulation. To answer this, one
needs to study the kinetic evolution including both processes: a first attempt
has been done, recently in [64], to be discussed in the next subsection.
### 4.4 The effects of inelastic processes
As we have already seen in Sec. 3.3 and Sec. 3.4, the peculiar IR enhancement
of the QCD makes the effective $1\rightarrow 2$ process be comparable to the
$2\rightarrow 2$ process in the medium. One therefore needs to include both
processes in the kinetic evolution. The inclusion of inelastic, number
changing process has the immediate consequence that the ultimate thermal
equilibrium state can not have any condensate: provided long enough time all
excessive gluons can be removed. This however leaves the interesting question:
what changes the inelastic collisions bring to the dynamical evolution of the
system, and in particular, whether the elastic-driven dynamical onset of
condensation from overpopulated initial conditions (as shown in the previous
subsection) would still occur or not. To answer such question, an explicit
evaluation including both elastic and inelastic collisions becomes mandatory.
The key issue is the competition between the two kernels: the elastic that
drives overpopulated system toward onset of condensation, while the inelastic
that tends to reduce the total number density down toward the underpopulation.
This problem has recently been addressed in [64], with surprising finding that
is quite different from naive expectations.
Figure 4: (Left)The distribution function $f(p)$ at different time moments.
(Right) The occupation near zero momentum as a function of time for different
values of parameter $R$.
Starting from overpopulated initial condition in (39) and including both
kernels (18)(33), one can numerically solve the kinetic equation: see [64] for
detailed results and analysis. Let us just highlight the most interesting
finding. There is one parameter $R$ that controls the relative strength
between the two kernels. As shown in Fig. 4 (left panel), when the inelastic
processes are turned on, the gluon distribution function at small $p$ region
grows very fast (much faster than that with purely elastic process) and
quickly becomes a local thermal form $f^{*}(p)=1/[e^{(p-\mu^{*})/T^{*}}-1]$
with the small $p$ part becoming steeper and steeper with time (meaning
decreasing $|\mu^{*}|$). This IR evolution proceeds despite that the
distribution in the wide range of larger momentum region is still far from
equilibrium shape and despite that the overall particle number is indeed
dropping. As a result, the rapid filling of IR modes is enhanced by the
inelastic process and the onset of the BEC will occur faster than the purely
elastic case. Furthermore as shown in Fig. 4 (right panel), the stronger
(i.e., larger $R$) the inelastic kernel is, the faster the occupation at
vanishing momentum will “explode” toward the onset of condensation. At first
sight this may sound counter-intuitive. To better understand this IR local
effect of the inelastic kernel, let us examine the low momentum behavior of
the inelastic kernel:
$\displaystyle{\cal C}^{\rm eff}_{1\leftrightarrow 2}(p\to 0)\to
R\frac{I_{a}}{I_{b}}\left[A_{0}f_{0}(1+f_{0})+A_{1}f^{\prime}_{0}(1+2f_{0})\,p+\hat{O}(p^{2})\right],$
(51)
where we have introduced the constants
$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\ln\frac{1}{1-z_{c}}+\frac{1}{6}\frac{z_{c}(11z_{c}^{2}-27z_{c}+18)}{(1-z_{c})^{3}},$
$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\ln\frac{1}{1-z_{c}}-\frac{1}{12}\frac{z_{c}(25z_{c}^{3}-88z_{c}^{2}+108z_{c}-48)}{(1-z_{c})^{4}},$
(52)
with $z_{c}$ is an upper cutoff for the integral over $z$. All these $A$’s are
positive for $0<z_{c}<1$. Clearly for sufficiently small $p$ the leading term
in the inelastic kernel $\sim Rf_{0}(1+f_{0})A_{0}$ is always positive and
becomes bigger and bigger with increasing $f_{0}$ (which is a kind of “self-
amplification”). This leads to extremely rapid growth of the particle number
near $p=0$ and the effect becomes stronger with increasing values of $R$,
which explains the behavior seen in Fig. 4.
Physically this behavior may be understood in two ways. First note that the
inelastic kernel has its fixed point to be $1/(e^{p/T}-1)$ which at small $p$
is $\sim 1/p$ so as long as $f(p=0)$ is finite yet the inelastic kernel will
try to fill it up toward $1/p$. Second, this is also related to the quantum
effect from Bosonic nature: if all involved particles are from small $p$, then
the merging rate is like $\sim f_{0}^{2}(1+f_{0})$ while the splitting rate is
like $\sim f_{0}(1+f_{0})^{2}$ so the splitting “wins” due to Bose enhancement
for the final state and it increases particle number at small $p$.
Our finding may sound counter-intuitive at first, as the usual conception
would suggest that increasing the strength of the inelastic collisions tends
to obstruct more effectively the formation of any condensate. It should
however be emphasized that the evolution toward onset that has been studied
thus far is not the end of the story. Our analysis addresses the evolution up
to the onset of BEC while does not treat the evolution afterwards. As is well
known in the BEC literature (see e.g. [132, 133]), in order to describe the
kinetic evolution of the system with the presence of condensate, a new set of
kinetic equations is needed for an explicit description of the coupled
evolution for a condensate plus a regular distribution. Efforts are underway
to derive these equations, and so far a kinetic study of the stage after BEC
onset for the Glasma system has not been achieved to our best knowledge.
However, it appears very plausible that the subsequent evolutions may develop
as follows: immediately after onset, the strong IR flux will not cease right
away but continue for a while and thus drive the condensate to grow in time;
at certain point, the time would be long enough to allow the inelastic
processes to decrease the total number density adequately and cause the
condensate to decay thus decreasing in time; eventually the inelastic
processes will be able to remove all excess gluons and lead to the thermal
equilibrium state with neither condensate nor any chemical potential. While
the detailed understanding of such dynamic processes can only be achieved
through solving the new set of kinetic equations, one can reasonably expect
that with increasing strength of the inelastic processes the whole evolution
would be faster. Thus the following overall picture may likely be the case:
with increasing strength, the inelastic processes on one hand catalyze the
onset of condensation initially, while on the other hand eliminate the fully
formed condensate faster, thus limiting the time duration for the presence of
condensate to be shorter. A schematic picture of such conjectured full
evolution is shown in Fig. 5, which is in line with the usual conception. It
is worth mentioning that recent analysis in [134] has shown that the the
$2\leftrightarrow 3$ inelastic cross section from exact matrix element becomes
significantly smaller than that from the Gunion-Bertsch formula, and amounts
to $\sim 20\%$ of the $2\leftrightarrow 2$ cross section. It therefore seems
very plausible that a realistic choice of $R$ value would be rather modest,
which may imply a considerable time window for the condensate to be sizable
and play an important role for the evolution. A complete investigation of the
evolution including the condensate will be an interesting problem to be
pursued in the future.
Figure 5: Conjectured evolution of the condensate with both elastic and
inelastic processes.
A final remark concerns the inclusion of quarks (and anti-quarks) into the
kinetic evolution of the glasma. So far our discussions have included only
gluons, while in reality the quarks and anti-quarks must be there. Even the
starting glasma may be overwhelmingly gluonic, quarks and anti-quarks will
surely be produced with time via e.g. gluon annihilations into $q\bar{q}$
pairs. The consequences of adding them are interesting to know. One important
change is that the thermal state will have to include the gas of quarks and
anti-quarks which change the composition and take a share of the total energy
of the system: this will necessarily change the condition for the
overpopulation. Another important change is that the individual number
conservation for gluons is evaded even without going to higher order
complicated multi-gluon scatterings: essentially quarks and gluons can
mutually serve as sources via identity-changing processes. On the other hand,
one may realize that fermions (subject to Pauli exclusion), unlike bosons,
will contribute no more than order $\hat{o}(1)$ to the thermodynamic extensive
quantities with each single flavor. Of course one might evade this by dialing
large number of flavors, while in reality one has $N_{f}=3$ which is much
smaller than $1/\alpha_{s}$ provided $\alpha_{s}$ is small. With these general
considerations in mind, one may expect that starting with a pure-gluonic,
highly-overpopulated initial condition, the gluons may still necessarily reach
onset of condensation provided large enough overpopulation, despite that part
of the gluons (about $\sim\hat{o}(N_{f})$ ) will be converted into quarks and
anti-quarks. Regarding the dynamical evolution of the gluonic sector, one may
anticipate a competition between the gluonic elastic scatterings (that drive
toward condensation) and the gluon-to-quark conversions (that tend to reduce
the gluon overpopulation). A nice and thoroughly quantitative study of this
problem has been done very recently by Blaizot, Wu, and Yan [114] . With a set
of kinetic equations that govern the evolution of distributions of both
sectors and couple them together, they have found three distinctive behaviors
in the solutions from different initial conditions: starting from sufficiently
high initial overpopulation, the solution necessarily runs into onset of BEC;
starting from initial occupation below certain threshold, the gluon-to-quark
conversion is fast enough to completely avoid onset of BEC; while with initial
occupation in between the previous two limits, the system reaches a thermal
state without gluon condensate but along its evolution runs into a transient
stage with gluon condensate. These interesting findings provide further non-
trivial evidences for the robustness of the gluon elastic-scattering driven
kinetic evolution from overpopulated initial condition toward the dynamical
onset of condensation.
## 5 Discussions on other kinetic approaches
While the previous Section has discussed the recent developments emphasizing
the role of overpopulation and possible condensation phenomenon, in this
Section we also give a brief survey of a number of other interesting studies
on the thermalization process in the kinetic framework.
### 5.1 The “bottom-up” scenario
A pioneering study in applying the kinetic framework to understand the
thermalization in heavy ion collisions was done by Baier, Mueller, Schiff, and
Son in [56], where the so-called “bottom-up” scenario was proposed. In this
scenario, one considers a gluon system resulting from the collision between
two very large nuclei at extremely high energy, which is approximately (i)
homogeneous in the transverse plane (set as $x$ and $y$ directions), (ii)
expanding along the beam direction (set as $z$ axis) in a boost-invariant way,
and (iii) having an initial distribution that is highly occupied $f\sim
1/\alpha_{s}$ and dominated by “hard” gluons with momenta $p$ of the order
saturation scale $Q_{s}\gg\Lambda_{\rm QCD}$ and thus $\alpha_{s}\ll 1$.
The thermalization in this scenario is achieved through three stages. In the
first stage, $1\ll Q_{s}t\ll\alpha_{s}^{-3/2}$, the longitudinal expansion
dilutes the system and also anisotropizes the distribution according to
$p_{x,y}\sim Q_{s}$, $p_{z}\sim Q_{s}/t$ if there were no interactions
presented i.e. in free-streaming case. However the small-angle elastic
scatterings between hard gluons weaken this anisotropization process via
broadening $p_{z}$ at a rate $dp_{z}/dt\sim\hat{q}_{\rm el}/p_{z}$ and as a
result the longitudinal momentum is diluted at a slower rate, $p_{z}\sim
Q_{s}(Q_{s}t)^{-1/3}$. In this case the hard gluon distribution evolves like
$f_{h}\sim\alpha_{s}^{-1}(Q_{s}t)^{-2/3}$. The soft gluons are generated by
soft splitting induced by small angle collisions between hard gluons. Once
generated, they also “suffer” from dilution due to expansion. The combination
of these two effects gives the evolution of the soft gluon distribution like
$f_{s}\sim\alpha_{s}^{-1}(Q_{s}t)^{-1/3}$ (while overall this whole stage the
number density of soft gluons $N_{s}$ is still much lower than that of hard
gluons $N_{h}$ because they occupy much smaller phase space). At the moment
$Q_{s}t\sim\alpha_{s}^{-3/2}$ the hard gluon distribution $f_{h}$ drops from
the order $1/\alpha_{s}$ initially to the order one by dilution and the system
proceeds to the second stage.
In the second stage, the hard sector of the system becomes underpopulated and
the hard gluons continue to split into softer ones via inelastic scatterings.
The system builds up two scales: one is the hard scale $Q_{s}$, and the other
is the soft scale $k_{s}\sim\sqrt{\alpha_{s}}Q_{s}$ determined by the
screening mass $m_{D}$ as well as the hard collision rate. The number density
of hard gluons continues to drop mainly due to the longitudinal expansion,
$N_{h}\sim(Q_{s}^{3}/\alpha_{s})(Q_{s}t)^{-1}$, while the number density of
the soft gluons decreases more slowly by virtue of the generation from hard
gluon splittings, $N_{s}\sim\alpha_{s}^{1/4}Q_{s}^{3}(Q_{s}t)^{-1/2}$. The
distribution of the soft gluons evolves according to
$f_{s}\sim\alpha_{s}^{-5/4}(Q_{s}t)^{-1/2}$ which becomes order $1$ at the
time $Q_{s}t\sim\alpha_{s}^{-5/2}$. After that moment, the system evolves into
the third stage.
In the third stage, the soft sector becomes dominant over the hard one while
the occupation in both regimes drops below order one. The soft gluons collide
frequently and they can isotropize and thermalize fast with small angle
scatterings. These then form a “thermal bath” with a characteristic
temperature $T$ which initially (at the moment $Q_{s}t\sim\alpha_{s}^{-5/2}$)
is $T\sim k_{s}$. The hard gluons behave like “jets” with energy $Q_{s}$
propagating through this thermal bath and constantly loose their energy into
the latter. Therefore the energy is transferred from the hard to soft sector
via the LPM-suppressed splitting upon multiple scatterings. The scatterings
between the hard gluons themselves are rare due to already low phase space
density and can be neglected. Thus the splitting rate is $t_{\rm
split}^{-1}\sim\alpha_{s}\sqrt{\hat{q}_{\rm el}/k_{\rm split}}$ where $k_{\rm
split}$ is the momentum of the emitted gluon and $\hat{q}_{\rm
el}\sim\alpha_{s}^{2}T^{3}$. The temperature of the soft bath increases until
the hard gluons loose all of their energy, which happens when $k_{\rm
split}\sim Q_{s}$. At this point the system is nearly thermalized. By equating
$t_{\rm split}$ with $t$ and imposing the energy conservation condition
$T^{4}\sim Q_{s}^{4}/[\alpha_{s}(Q_{s}t)]$, one arrives at a thermalizatoin
time $Q_{s}t_{\rm th}\sim\alpha_{s}^{-13/5}$ and an equilibrium temperature
$T_{\rm eq}\sim\alpha_{s}^{2/5}Q_{s}$.
In the “bottom-up” scenario, the overall picture is that soft modes (which can
be easily thermalized) will be filed up by hard gluon bremsstrahlung and
thermalize first, which then further drains the energy from the hard gluons
and make them thermalized, thus the thermalization proceeds from bottom to top
in energy scale. We note this scenario differs in two main points from the
mostly elastic-driven scenario discussed in the previous Section: first, the
elastic scatterings alone are extremely efficient in developing a strong IR
flux and provide a mechanism of quickly filling up soft modes, which is absent
in the “bottom-up” scenario; second, (at least in the static box case) the
elastic scatterings also drive a strong UV energy cascade to adjust and
thermalize the high momentum tail beyond $Q_{s}$ scale (a region not discussed
in the “bottom-up” which may be justified due to expansion) — how such
elastic-driven UV cascade may change by expansion remains to be understood.
### 5.2 Instability modified “bottom-up” approach
Shortly after the development of the “bottom-up” scenario, it was realized
that there may be complication in the reasoning. This is related to the
delicate role of momentum space anisotropy, induced by longitudinal expansion.
When the momentum distribution becomes anisotropic, a mechanism completely
different from usual scatterings, namely the “plasma instability”, will occur
and play important role. To see that, one may examine the one-loop self-energy
tensor $\Pi^{\mu\nu}(\omega,{\mathbf{k}})$ in a medium with momentum
anisotropy, and in turn the effective propagator of soft gluons is also
anisotropic. As it turns out, the dispersion relation obtained from this
effective propagator contains branches with negative mass square, or
${\rm{Im}}(\omega({\mathbf{k}}))>0$ for certain soft momentum region. This
implies that such soft modes become unstable and their occupation would grow
exponentially. As was first emphasized by Mrowczynski and studied in many
later papers [28, 29, 30, 36, 37], the particularly important instability is
the non-Abelian equivalence of the Weibel instability [27]. As a result of
such instabilities, a set of chromo-magnetic modes at scale $k_{\rm inst}$
will exponentially grow to be strong and subsequently diffuse the momenta of
hard gluons via Lorentz force to drive the system toward isotropization and
thermalizaton. Shortly after the “bottom-up” scenario, Arnold, Lenaghan, and
Moore [32] argued that the plasma instability could be a more efficient
mechanism for filling up soft modes and for isotropizing momentum distribution
at least for the first stage of the “bottom-up” scenario and can lead to a
faster thermalization at time $Q_{s}t\sim\alpha_{s}^{-5/2}$. The roles of
plasma instabilities have subsequently been thoroughly analyzed by analytical
method [46], modified kinetic approaches [58, 59, 60, 35], classical field
simulations [39, 38, 41, 40], hybrid approaches [44, 45, 34, 42, 47], etc.
More recently Kurkela and Moore [65, 66, 75, 67] has carefully analyzed again
the roles of plasma instability versus scatterings, particularly in the
longitudinally expanding case. An interesting new feature they proposed is
that the plasma instability is not only important at the very early stage but
may also dominate the thermalization dynamics in all the three stages of the
“bottom-up” scenario. In the first stage, $1\ll Q_{s}t\ll\alpha_{s}^{-8/7}$,
the occupancy of both hard and soft gluons decrease and the expansion causes
the anisotropy to increase as a function of time, $\langle
p_{z}\rangle/\langle p_{\perp}\rangle\sim(Q_{s}t)^{-1/8}(Q_{s}/p)^{2/3}$.
However, the instability causes very fast isotropization for gluons with
$p<k_{\rm iso}\sim(Q_{s}t)^{-3/16}Q_{s}$ and in more infrared region $p<p_{\rm
max}\sim(Q_{s}t)^{-1/4}Q_{s}$ the distribution quickly forms a thermal-like
tail which evolves like $f(p)\sim\alpha_{s}^{-1}(Q_{s}t)^{-7/8}(Q_{s}/p)$. In
the second stages, $\alpha_{s}^{-8/7}\ll Q_{s}t\ll\alpha_{s}^{-12/5}$, the
system is highly anisotropic but the hard modes are underpopulated. The hard
gluons begin to emit daughter gluons and the anisotropy of hard modes
“propagates” into the soft region. The plasma instability driven by the
anisotropy from these emitted gluons is argued to dominate and control the
evolution of $k_{\rm iso}$ as well as $p_{\rm max}$. At the moment
$Q_{s}t\sim\alpha_{s}^{-56/25}$, $f(p_{\rm max})$ drops to $\sim 1$ and the
soft sector now forms a nearly-thermal bath with temperature $T\sim p_{\rm
max}$. This soft bath does not dominate either energy or scattering at this
stage, but it grows to be more and more important and eventually begins to
dominate the physics at $Q_{s}t\sim\alpha_{s}^{-12/5}$. Then the system enters
the third stage $\alpha_{s}^{-12/5}\ll Q_{s}t\ll\alpha_{s}^{-5/2}$. In this
stage, the soft sector (which is weakly anisotropic as a result of expansion
as well as anisotropy passed along from hard gluon splittings) and the
resulting plasma instabilities control the broadening of the hard primary
gluons. The instabilities give $\hat{q}_{\rm inst}\sim\alpha_{s}^{3}Q_{s}^{3}$
(see [65, 66]) and thus the splitting scale is given by $k_{\rm
split}\sim\alpha_{s}^{2}\hat{q}_{\rm inst}t^{2}$. Combining this with the
energy conservation condition and letting $k_{\rm split}\sim Q_{s}$ (when the
energy cascade stops), one arrives at a thermalization time $Q_{s}t_{\rm
th}\sim\alpha_{s}^{-5/2}$ and equilibrium temperature scale $T_{\rm
eq}\sim\alpha_{s}^{3/8}Q_{s}$. In general, the instabilities would be present
due to the inevitable anisotropy brought by the longitudinal expansion and
play a role in the IR filing and isotropization. Whether they play a dominant
role as compared with various other driving mechanisms, remains to be sorted
out.
### 5.3 The BAMPS approach
Quantitative simulations on how the inelastic processes contribute to the
thermalization of the gluon system have been carried out by Xu, Greiner and
collaborators within the BAMPS (for Boltzmann Approach to MultiParton
Scatterings) model [61, 109, 135, 136, 137]. BAMPS is a microscopic transport
model based on the kinetic equation Eq. (5) for on-shell partons with the
collision kernel including both the $2\rightarrow 2$ elastic and the
$2\rightarrow 3$ inelastic processes. The main feature of BAMPS is based on
the stochastic interpretation of the transition rates which ensure full
detailed balance for $2\rightarrow 3$ scatterings. BAMPS subdivides space into
small cells in which the transition rates are calculated and the gluon
distribution function $f(t,{\mathbf{p}},{\mathbf{x}})$ is then extracted [61].
In BAMPS framework, the matrix elements Eq. (16) for elastic $2\rightarrow 2$
process and Gunion-Bertsch formula Eq. (27) for inelastic $2\rightarrow 3$
process are used while all the infrared divergences due to small-angle and
soft collinear singularities are regularized by introducing the Debye
screening mass $m_{D}$ as an infrared cutoff. This Debye mass is calculated
locally in space and is an angle-averaged one so that it is always positive
even for anisotropic distribution and no instabilities would be present. The
LPM effect is approximately encoded in BAMPS by introducing an infrared cutoff
to the transverse momentum $k_{\perp}$ of the emitted gluon which is
determined by requiring the formation time of the emitted gluon to be smaller
than the gluon in-medium mean-free path [138, 139].
The simulation results of BAMPS have clearly shown the important contribution
of the inelastic processes in filling up the infrared modes and in speeding up
the system’s evolution toward equilibrium. Different initial conditions, both
wounded-nucleons initial condition and CGC-inspired initial condition, have
been explored and it has been found that the thermalization time is relatively
insensitive to such different choices: for either initial condition, for
coupling constant $\alpha_{s}\sim 0.3$, the gluons in the central region of
the collision can be effectively isotropized and kinetically thermalized at a
time on the order $t_{\rm eq}\sim 1$ fm. One nontrivial feature found in the
BAMPS simulation with the CGC-inspired initial condition is that the soft and
hard gluons appear to thermalize at almost the same time scale $Q_{s}t_{\rm
eq}\sim[\alpha_{s}(\ln\alpha_{s})]^{-2}$ in contrast to the “bottom-up”
scenario in which the thermalization first occurs in the IR region and
proceeds up to the UV region. It would be of great interest to utilize the
BAMPS framework and explore the kinetic evolution incorporating full quantum
Bose statistics (beyond the classical Boltzmann limit) with overpopulated
initial conditions. In particular, it is tempting to see whether a
condensation phenomenon may occur or not. For more direct applications to
heavy ion collision phenomenology, one may explore within such comprehensive
simulation framework the full physical evolution from the initial condition
through the thermalization toward the dynamical evolution in the thermal QGP
stage, as has been explored in the BAMPS framework as well as in other new
transport framework recently developed in e.g.[140].
### 5.4 Turbulent thermalization and non-thermal fixed point
As already discussed in Section 2, the classical-statistical lattice
simulations provide a first-principle method to explore the thermalization
process in the weak coupling and high occupancy limit. Many studies have been
done in this framework with a lot of interesting results found [77, 78, 60,
141, 142, 143, 73, 50, 76, 144]. A very interesting recent finding in [77, 78]
is that in the simulation for Yang-Mills gauge theory with very small coupling
$\alpha_{s}\sim 10^{-4}$ (which allows exploring late time behavior within the
classical-statistical framework) and for both the non-expanding and
longitudinally expanding cases, the system, after a short transient regime,
exhibits universal self-similar scaling solutions with wave turbulence
characteristic.
As previously discussed, the classical-statistical lattice method and the
kinetic method have an overlap in the range of validity for occupation $1\ll
f\ll 1/\alpha_{s}$. With such interesting self-similar solutions found
directly from real-time lattice situations [77, 78], it is tempting to see
whether such solutions could at least be approximately explained in the more
intuitive kinetic picture with microscopic scatterings as scaling solutions to
the transport equation. In fact, similar turbulent cascade and self-similar
evolutions were found in scalar field theory studies in the context of early
universe evolution [145, 146] where the appearance of the wave turbulence
corresponds to a self-similar non-thermal fixed-point solution of the kinetic
equation. Following a similar strategy, the authors of [77, 78] has indeed
shown that the self-similar solutions found from simulations can be well
approximated by solutions from a kinetic theory of the Fokker-Planck type. To
see that, let us for a moment neglect the inelastic processes and examine a
kinetic equation of the structure given in Eq. (18). In the non-expanding
case, it is straightforward to show that the equation allows a scaling
solution of the general form
$f(t,{\mathbf{p}})=(Q_{s}t)^{\alpha}f_{S}((Q_{s}t)^{\beta}{\mathbf{p}})$
provided two conditions: the stationary function $f_{S}({\mathbf{p}})$
satisfying $\alpha
f_{S}({\mathbf{p}})+\beta{\mathbf{p}}\cdot\nabla_{\mathbf{p}}f_{S}({\mathbf{p}})+Q_{s}^{-1}\nabla_{\mathbf{p}}\cdot{\bf\cal
J}[f_{S}({\mathbf{p}})]=0$, and the scaling parameters satisfying a relation
$\alpha-1=3\alpha-\beta$. Furthermore, the energy conservation implies an
additional relation, $\alpha=4\beta$. Combining these two relations, one
obtains $\alpha=-4/7$ and $\beta=-1/7$, which turn out to nicely reproduce the
exponents extracted from their classical-statistical simulations.
In the expanding case, the analysis is less straightforward due to the
expansion and the competing effect of scatterings in kinetic theory. Since
such self-similar solution emerges at relatively later time in the evolution,
the system becomes much diluter and the effect of scatterings may be plausibly
approximated by pure elastic momentum broadening in the $z$ (beam) direction,
${\cal C}[f]=\hat{q}_{\rm el}\partial^{2}_{p_{z}}f$ with $\hat{q}_{\rm el}$
given by Eq. (22). Such a highly simplified Boltzmann equation
$[\partial_{t}-(p_{z}/t)\partial_{p_{z}}]f={\cal C}[f]$ does allow a scaling
solution of the form
$f(t,{\mathbf{p}}_{\perp},p_{z})=(Q_{s}t)^{\alpha}f_{S}((Q_{s}t)^{\beta}{\mathbf{p}}_{\perp},(Q_{s}t)^{\gamma}p_{z})$
with the scaling parameters satisfying $2\alpha-2\beta+\gamma+1=0$.
Furthermore when the system becomes diluter at late time, its evolution may
approach free-streaming case, with the energy and particle number densities
both dropping as $\sim 1/t$. Under such assumptions, one arrives at the
solution with exponents $\alpha=-2/3,\beta=0,\gamma=1/3$. As the authors [77,
78] have shown, these scaling parameters obtained in the kinetic equation are
in surprisingly excellent agreement with the self-similar behavior seen in
their classical-statistical simulations. It has also been numerically checked
that with the given coupling constant regime such self-similar evolution is
insensitive to the initial condition and at very late time it approaches
toward the original “bottom-up” scenario. In general, the appearance of the
non-thermal fixed point will delay the thermalization of the system toward the
true thermal fixed point. It is worth commenting on the roles of the very
small value of coupling used in these studies: technically it allows the
classical field approach to be a better controlled approximation with much
longer evolution time; physically it opens a very wide window for the
occupation (in the kinetic picture) in between the saturated limit $f\sim
1/\alpha_{s}$ and the quantum limit $f\sim 1$, likely maximizing the
manifestation of nonlinear effects such as turbulent cascade. These findings
are extremely interesting, and leave open a number of questions to be explored
further, in particular, what change may happen to this scenario when one
gradually moves toward the coupling constant regime $\alpha_{s}\sim 10^{-1}$
that may be more directly relevant to the glasma in heavy ion collisions.
## 6 Summary
In summary, we have given a brief review of the thermalization problem in
heavy ion collisions, with emphasis on recent progress in understanding the
kinetic evolution of the glasma. A short discussion has been given on the
general context of the thermalization problem and a number of approaches other
than the kinetic one. We have then provide an elementary introduction on the
transport framework to be used for describing the pre-equilibrium evolution,
including both the elastic and inelastic collisions. Recent interesting
developments on the kinetic evolution in the overpopulated regime, as in the
case for the glasma, and the possibility of dynamical Bose-Einstein
Condensation in such system, have been discussed in details. Finally a number
of other approaches within the kinetic framework have been surveyed.
Though there have been a lot of interesting developments and some nontrivial
ideas in the last few years, it may be fair to say that we are still far from
a detailed understanding of the kinetic evolution for the pre-equilibrium
stage in heavy ion collisions. For the kinetic approach in the overpopulated
regime, a number of pressing issues need to be understood, including the co-
evolution of the condensate and regular distribution (after the onset of
condensate), the far-from-equilibrium medium effects (e.g. the dressing of
internal/external gluons involved in a scattering), the roles of higher order
processes, etc. It is also of great importance to further explore the relation
(the overlap in their applicability and their complementarity) between the
kinetic description and the classical field description, in particular how
certain behavior (e.g. condensation and turbulent scaling) observed in one
description would be manifested in the other description. Toward more
phenomenological end, it is crucial to implement and investigate the effects
of longitudinal expansion as well as the roles of anisotropy (both that from
initial condition and that dynamically generated from expansion). It is also
highly interesting to study the kinetic evolution with more realistic
transverse distributions e.g. by introducing transverse-position dependent
initial conditions (via saturation scale) which would allow determining
possible early transverse flow generation. The condensate, if formed, would
play nontrivial roles in many aspects of phenomenology from pA to AA
collisions, as explored by a number recent studies along this direction [147,
148, 149, 150], and there are certainly many more possibilities to be fully
investigated.
Let us end with a discussion on the interesting evolution of the very
conception of the problem itself. The initially perceived “thermalization”
problem , as the name suggests it, has the implicit picture of two distinctive
stages: a pre-thermal stage with the system evolving to a (relatively)
complete local thermalization (and of course isotropization) and a thermal
stage which then expands in a nearly ideal hydrodynamic fashion, with the
switch between the two stages at rather early time $\sim 1$ fm/c. This was
largely motivated by the phenomenological success of the ideal hydrodynamic
simulations at the early RHIC era (see the nice discussion in the recent
review article [14]), along with the conventional wisdom that the
applicability of hydrodynamics requires local thermal equilibration. However
there has been no direct evidence for full thermalization (and not even for
isotropization). In fact, the later developments of viscous hydrodynamic
studies have demonstrated that even with extremely small dissipation (i.e.
$\eta/s$ close to the conjectured lower bound) the stress tensor bears sizable
anisotropy between longitudinal/transverse pressures over several fm/c time
window [14]. There have also been interesting works from both strong coupling
approach (via the holographic models) [95, 96] and weak coupling approach (via
real time lattice simulations) [51] that show the emergence of
(viscous)hydrodynamic behaviors without reaching either isotropization or full
thermalization. To add to the complications, most recent experimental
measurements of high multiplicity pPb collisions at LHC and dAu collisions at
RHIC show very interesting patterns in the soft particle productions and
correlations, which seem to be accountable by collective expansions akin to
viscous hydrodynamic simulations applied to such systems much smaller in size
and much shorter-lived in time as compared with the bulk matter in AA
collisions [151] (noting though whether this is indeed so is still under
intensive debate [152, 153, 154, 155] and subject to conclusion in the
future). All these may call for a change in our very identification of the
“thermalization” problem, splitting into two closely related but clearly
different aspects: theoretically how and when a full thermalization is
achieved in a quark-gluon system starting with initial conditions close to
that in the heavy ion collisions; phenomenologically, how and when an apparent
hydrodynamic behavior emerges from the pertinent initial conditions and how
far one can push the limit (e.g. in the system size, in the anisotropy, in the
dissipation, in the microscopic coupling, etc) for the system to stay amenable
to a collective expansion. It will require significant future efforts to fully
explore both of these issues and make progress in understanding the
“thermalization” problem.
## Acknowledgements
The authors are grateful to J. Berges, J.-P. Blaizot, F. Gelis, L. McLerran,
R. Venugopalan, Q. Wang, B. Wu, Z. Xu, and P. Zhuang for discussions and
communications. XGH is supported by Fudan University grants EZH1512519 and
EZH1512600. The research of JL is supported by the National Science Foundation
under Grant No. PHY-1352368. JL thanks the RIKEN BNL Research Center for
partial support. JL is also grateful to the Yukawa Institute for Theoretical
Physics, Kyoto University, where this work was partly completed during the
YITP-T-13-05 on “New Frontiers in QCD”.
## References
* [1] J. Adams et al. [STAR Collaboration], Nucl. Phys. A 757, 102 (2005) [nucl-ex/0501009].
* [2] I. Arsene et al. [BRAHMS Collaboration], Nucl. Phys. A 757, 1 (2005) [nucl-ex/0410020].
* [3] B. B. Back, M. D. Baker, M. Ballintijn, D. S. Barton, B. Becker, R. R. Betts, A. A. Bickley and R. Bindel et al., Nucl. Phys. A 757, 28 (2005) [nucl-ex/0410022].
* [4] K. Adcox et al. [PHENIX Collaboration], Nucl. Phys. A 757, 184 (2005) [nucl-ex/0410003].
* [5] B. Muller, J. Schukraft and B. Wyslouch, Ann. Rev. Nucl. Part. Sci. 62, 361 (2012) [arXiv:1202.3233 [hep-ex]].
* [6] M. Gyulassy and L. McLerran, Nucl. Phys. A 750, 30 (2005) [nucl-th/0405013].
* [7] E. V. Shuryak, Nucl. Phys. A 750, 64 (2005) [hep-ph/0405066].
* [8] U. W. Heinz, in ’Relativistic Heavy Ion Physics’, Landolt-Boernstein New Series, I/23, edited by R. Stock (Springer Verlag, New York,2010) Chap. 5 [arXiv:0901.4355 [nucl-th]].
* [9] U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. 63, 123 (2013) [arXiv:1301.2826 [nucl-th]].
* [10] P. Huovinen, Int. J. Mod. Phys. E 22, 1330029 (2013) [arXiv:1311.1849 [nucl-th]].
* [11] P. Romatschke, Int. J. Mod. Phys. E 19, 1 (2010) [arXiv:0902.3663 [hep-ph]].
* [12] J. Berges, J. -P. Blaizot and F. Gelis, J. Phys. G 39, 085115 (2012) [arXiv:1203.2042 [hep-ph]].
* [13] F. Gelis, Int. J. Mod. Phys. A 28, 1330001 (2013) [arXiv:1211.3327 [hep-ph]].
* [14] M. Strickland, arXiv:1312.2285 [hep-ph].
* [15] P. B. Arnold, Int. J. Mod. Phys. E 16, 2555 (2007) [arXiv:0708.0812 [hep-ph]].
* [16] L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994) [hep-ph/9309289].
* [17] L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 3352 (1994) [hep-ph/9311205].
* [18] L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rept. 100, 1 (1983).
* [19] A. H. Mueller and J. -w. Qiu, Nucl. Phys. B 268, 427 (1986).
* [20] J. P. Blaizot and A. H. Mueller, Nucl. Phys. B 289, 847 (1987).
* [21] E. Iancu and R. Venugopalan, In *Hwa, R.C. (ed.) et al.: Quark gluon plasma* 249-3363 [hep-ph/0303204].
* [22] E. Iancu, A. Leonidov and L. McLerran, hep-ph/0202270.
* [23] H. Weigert, Prog. Part. Nucl. Phys. 55, 461 (2005) [hep-ph/0501087].
* [24] F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, Ann. Rev. Nucl. Part. Sci. 60, 463 (2010) [arXiv:1002.0333 [hep-ph]].
* [25] T. Lappi and L. McLerran, Nucl. Phys. A 772, 200 (2006) [hep-ph/0602189].
* [26] F. Gelis, T. Lappi and L. McLerran, Nucl. Phys. A 828, 149 (2009) [arXiv:0905.3234 [hep-ph]].
* [27] E. S. Weibel, Phys. Rev. Lett. 2, 83 (1959).
* [28] S. Mrowczynski, Phys. Lett. B 214, 587 (1988) [Erratum-ibid. B 656, 273 (2007)].
* [29] S. Mrowczynski, Phys. Lett. B 314, 118 (1993).
* [30] S. Mrowczynski and M. H. Thoma, Phys. Rev. D 62, 036011 (2000) [hep-ph/0001164].
* [31] S. Mrowczynski, Acta Phys. Polon. B 37, 427 (2006) [hep-ph/0511052].
* [32] P. B. Arnold, J. Lenaghan and G. D. Moore, JHEP 0308, 002 (2003) [hep-ph/0307325].
* [33] P. B. Arnold, J. Lenaghan, G. D. Moore and L. G. Yaffe, Phys. Rev. Lett. 94, 072302 (2005) [nucl-th/0409068].
* [34] P. B. Arnold and J. Lenaghan, Phys. Rev. D 70, 114007 (2004) [hep-ph/0408052].
* [35] P. B. Arnold and G. D. Moore, Phys. Rev. D 76, 045009 (2007) [arXiv:0706.0490 [hep-ph]].
* [36] P. Romatschke and M. Strickland, Phys. Rev. D 68, 036004 (2003) [hep-ph/0304092].
* [37] P. Romatschke and M. Strickland, Phys. Rev. D 70, 116006 (2004) [hep-ph/0406188].
* [38] P. Romatschke and R. Venugopalan, Eur. Phys. J. A 29, 71 (2006) [hep-ph/0510292].
* [39] P. Romatschke and R. Venugopalan, Phys. Rev. Lett. 96, 062302 (2006) [hep-ph/0510121].
* [40] P. Romatschke and A. Rebhan, Phys. Rev. Lett. 97, 252301 (2006) [hep-ph/0605064].
* [41] P. Romatschke and R. Venugopalan, Phys. Rev. D 74, 045011 (2006) [hep-ph/0605045].
* [42] A. Rebhan, P. Romatschke and M. Strickland, Phys. Rev. Lett. 94, 102303 (2005) [hep-ph/0412016].
* [43] A. Rebhan, P. Romatschke and M. Strickland, JHEP 0509, 041 (2005) [hep-ph/0505261].
* [44] A. Rebhan, M. Strickland and M. Attems, Phys. Rev. D 78, 045023 (2008) [arXiv:0802.1714 [hep-ph]].
* [45] A. Rebhan and D. Steineder, Phys. Rev. D 81, 085044 (2010) [arXiv:0912.5383 [hep-ph]].
* [46] D. Bodeker, JHEP 0510, 092 (2005) [hep-ph/0508223].
* [47] D. Bodeker and K. Rummukainen, JHEP 0707, 022 (2007) [arXiv:0705.0180 [hep-ph]].
* [48] A. Dumitru, Y. Nara and M. Strickland, Phys. Rev. D 75, 025016 (2007) [hep-ph/0604149].
* [49] J. Berges, D. Gelfand, S. Scheffler and D. Sexty, Phys. Lett. B 677, 210 (2009) [arXiv:0812.3859 [hep-ph]].
* [50] J. Berges and S. Schlichting, Phys. Rev. D 87, 014026 (2013) [arXiv:1209.0817 [hep-ph]].
* [51] T. Epelbaum and F. Gelis, Phys. Rev. Lett. 111, 232301 (2013) [arXiv:1307.2214 [hep-ph], arXiv:1307.2214 [hep-ph]].
* [52] T. Epelbaum and F. Gelis, Phys. Rev. D 88, 085015 (2013) [arXiv:1307.1765].
* [53] K. Dusling, T. Epelbaum, F. Gelis and R. Venugopalan, Phys. Rev. D 86, 085040 (2012) [arXiv:1206.3336 [hep-ph]].
* [54] A. H. Mueller, Nucl. Phys. B 572, 227 (2000) [hep-ph/9906322].
* [55] A. H. Mueller, Phys. Lett. B 475, 220 (2000) [hep-ph/9909388].
* [56] R. Baier, A. H. Mueller, D. Schiff and D. T. Son, Phys. Lett. B 502, 51 (2001) [hep-ph/0009237].
* [57] R. Baier, A. H. Mueller, D. T. Son and D. Schiff, Nucl. Phys. A 698, 217 (2002).
* [58] A. H. Mueller, A. I. Shoshi and S. M. H. Wong, Phys. Lett. B 632, 257 (2006) [hep-ph/0505164].
* [59] A. H. Mueller, A. I. Shoshi and S. M. H. Wong, Eur. Phys. J. A 29, 49 (2006) [hep-ph/0512045].
* [60] A. H. Mueller, A. I. Shoshi and S. M. H. Wong, Nucl. Phys. B 760, 145 (2007) [hep-ph/0607136].
* [61] Z. Xu and C. Greiner, Phys. Rev. C 71, 064901 (2005) [hep-ph/0406278].
* [62] J. -P. Blaizot, F. Gelis, J. -F. Liao, L. McLerran and R. Venugopalan, Nucl. Phys. A 873, 68 (2012) [arXiv:1107.5296 [hep-ph]].
* [63] J. -P. Blaizot, J. Liao and L. McLerran, Nucl. Phys. A 920, 58 (2013) [arXiv:1305.2119 [hep-ph]].
* [64] X. -G. Huang and J. Liao, arXiv:1303.7214 [nucl-th].
* [65] A. Kurkela and G. D. Moore, JHEP 1112, 044 (2011) [arXiv:1107.5050 [hep-ph]].
* [66] A. Kurkela and G. D. Moore, JHEP 1111, 120 (2011) [arXiv:1108.4684 [hep-ph]].
* [67] M. C. A. York, A. Kurkela, E. Lu and G. D. Moore, arXiv:1401.3751 [hep-ph].
* [68] F. D. Aaron et al. [H1 and ZEUS Collaboration], JHEP 1001, 109 (2010) [arXiv:0911.0884 [hep-ex]].
* [69] T. Epelbaum and F. Gelis, Nucl. Phys. A 872, 210 (2011) [arXiv:1107.0668 [hep-ph]].
* [70] F. Gelis, J. Phys. Conf. Ser. 381, 012021 (2012) [arXiv:1110.1544 [hep-ph]].
* [71] J. Berges and D. Sexty, Phys. Rev. Lett. 108, 161601 (2012) [arXiv:1201.0687 [hep-ph]].
* [72] J. Berges, S. Scheffler, S. Schlichting and D. Sexty, Phys. Rev. D 85, 034507 (2012) [arXiv:1111.2751 [hep-ph]].
* [73] J. Berges, S. Schlichting and D. Sexty, Phys. Rev. D 86, 074006 (2012) [arXiv:1203.4646 [hep-ph]].
* [74] J. Berges, K. Boguslavski and S. Schlichting, Phys. Rev. D 85, 076005 (2012) [arXiv:1201.3582 [hep-ph]].
* [75] A. Kurkela and G. D. Moore, Phys. Rev. D 86, 056008 (2012) [arXiv:1207.1663 [hep-ph]].
* [76] S. Schlichting, Phys. Rev. D 86, 065008 (2012) [arXiv:1207.1450 [hep-ph]].
* [77] J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan, arXiv:1303.5650 [hep-ph].
* [78] J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan, arXiv:1311.3005 [hep-ph].
* [79] J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan, arXiv:1312.5216 [hep-ph].
* [80] M. Martinez and M. Strickland, Nucl. Phys. A 848, 183 (2010) [arXiv:1007.0889 [nucl-th]].
* [81] M. Martinez and M. Strickland, Nucl. Phys. A 856, 68 (2011) [arXiv:1011.3056 [nucl-th]].
* [82] M. Martinez, R. Ryblewski and M. Strickland, Phys. Rev. C 85, 064913 (2012) [arXiv:1204.1473 [nucl-th]].
* [83] D. Bazow, U. W. Heinz and M. Strickland, arXiv:1311.6720 [nucl-th].
* [84] J. -P. Blaizot and E. Iancu, Nucl. Phys. B 557, 183 (1999) [hep-ph/9903389].
* [85] A. H. Mueller and D. T. Son, Phys. Lett. B 582, 279 (2004) [hep-ph/0212198].
* [86] S. Jeon and L. G. Yaffe, Phys. Rev. D 53, 5799 (1996) [hep-ph/9512263].
* [87] P. M. Chesler and L. G. Yaffe, Phys. Rev. D 82, 026006 (2010) [arXiv:0906.4426 [hep-th]].
* [88] B. Wu, JHEP 1210, 133 (2012) [arXiv:1208.1393 [hep-th]].
* [89] B. Wu, JHEP 1304, 044 (2013) [arXiv:1301.3796 [hep-th]].
* [90] P. M. Chesler and L. G. Yaffe, arXiv:1309.1439 [hep-th].
* [91] S. Lin and E. Shuryak, Phys. Rev. D 78, 125018 (2008) [arXiv:0808.0910 [hep-th]].
* [92] S. Lin and E. Shuryak, Phys. Rev. D 79, 124015 (2009) [arXiv:0902.1508 [hep-th]].
* [93] Y. V. Kovchegov and S. Lin, JHEP 1003, 057 (2010) [arXiv:0911.4707 [hep-th]].
* [94] G. Beuf, M. P. Heller, R. A. Janik and R. Peschanski, JHEP 0910, 043 (2009) [arXiv:0906.4423 [hep-th]].
* [95] M. P. Heller, R. A. Janik and P. Witaszczyk, Phys. Rev. Lett. 108, 201602 (2012) [arXiv:1103.3452 [hep-th]].
* [96] M. P. Heller, R. A. Janik and P. Witaszczyk, Phys. Rev. D 85, 126002 (2012) [arXiv:1203.0755 [hep-th]].
* [97] S. Caron-Huot, P. M. Chesler and D. Teaney, Phys. Rev. D 84, 026012 (2011) [arXiv:1102.1073 [hep-th]].
* [98] P. M. Chesler and D. Teaney, arXiv:1112.6196 [hep-th].
* [99] V. Balasubramanian, A. Bernamonti, J. de Boer, B. Craps, L. Franti, F. Galli, E. Keski-Vakkuri and B. M ller et al., Phys. Rev. Lett. 111, 231602 (2013) [arXiv:1307.1487 [hep-th]].
* [100] V. Balasubramanian, A. Bernamonti, J. de Boer, B. Craps, L. Franti, F. Galli, E. Keski-Vakkuri and B. M ller et al., arXiv:1307.7086.
* [101] E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Pergamon Press, Oxford, 1981.
* [102] S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, Relativistic Kinetic Theory: Principles and Applications, North-Holland Inc.,1980.
* [103] R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions,3rd Edt, Spinger-Verlag, New York, 2003.
* [104] F. A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T. T. Wu, Phys. Lett. B 103, 124 (1981).
* [105] R. K. Ellis and J. C. Sexton, Nucl. Phys. B 269, 445 (1986).
* [106] T. Gottschalk and D. W. Sivers, Phys. Rev. D 21, 102 (1980).
* [107] J. F. Gunion and G. Bertsch, Phys. Rev. D 25, 746 (1982).
* [108] P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0011, 001 (2000) [hep-ph/0010177].
* [109] Z. Xu, C. Greiner and H. Stocker, Phys. Rev. Lett. 101, 082302 (2008) [arXiv:0711.0961 [nucl-th]].
* [110] J. -W. Chen, H. Dong, K. Ohnishi and Q. Wang, Phys. Lett. B 685, 277 (2010) [arXiv:0907.2486 [nucl-th]].
* [111] O. Fochler, J. Uphoff, Z. Xu and C. Greiner, Phys. Rev. D 88, 014018 (2013) [arXiv:1302.5250 [hep-ph]].
* [112] R. Abir, C. Greiner, M. Martinez and M. G. Mustafa, Phys. Rev. D 83, 011501 (2011) [arXiv:1011.4638 [nucl-th]].
* [113] T. Bhattacharyya, S. Mazumder, S. KDas and J. -eAlam, Phys. Rev. D 85, 034033 (2012) [arXiv:1106.0609 [nucl-th]].
* [114] J. -P. Blaizot, B. Wu and L. Yan, arXiv:1402.5049 [hep-ph].
* [115] L. D. Landau and I. Pomeranchuk, Dokl. Akad. Nauk Ser. Fiz. 92, 535 (1953).
* [116] L. D. Landau and I. Pomeranchuk, Dokl. Akad. Nauk Ser. Fiz. 92, 735 (1953).
* [117] A. B. Migdal, Phys. Rev. 103, 1811 (1956).
* [118] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl. Phys. B 484, 265 (1997) [hep-ph/9608322].
* [119] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl. Phys. B 483, 291 (1997) [hep-ph/9607355].
* [120] B. G. Zakharov, JETP Lett. 63, 952 (1996) [hep-ph/9607440].
* [121] B. G. Zakharov, JETP Lett. 65, 615 (1997) [hep-ph/9704255].
* [122] R. Baier, D. Schiff and B. G. Zakharov, Ann. Rev. Nucl. Part. Sci. 50, 37 (2000) [hep-ph/0002198].
* [123] P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0111, 057 (2001) [hep-ph/0109064].
* [124] P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0112, 009 (2001) [hep-ph/0111107].
* [125] P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0206, 030 (2002) [hep-ph/0204343].
* [126] P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0301, 030 (2003) [hep-ph/0209353].
* [127] P. B. Arnold, G. DMoore and L. G. Yaffe, JHEP 0305, 051 (2003) [hep-ph/0302165].
* [128] G. Baym, J. -P. Blaizot, F. Gelis and T. Matsui, Phys. Lett. B 644, 48 (2007) [hep-ph/0604209].
* [129] H. Bethe and W. Heitler, Proc. Roy. Soc. Lond. A 146, 83 (1934).
* [130] M. Attems, A. Rebhan and M. Strickland, Phys. Rev. D 87, 025010 (2013) [arXiv:1207.5795 [hep-ph]].
* [131] A. Iwazaki, Phys. Rev. C 87, no. 2, 024903 (2013) [arXiv:1208.5320 [hep-ph]].
* [132] D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995) [hep-ph/9409202].
* [133] D. V. Semikoz and I. I. Tkachev, Phys. Rev. D 55, 489 (1997) [hep-ph/9507306].
* [134] B. Zhang, J. Phys. Conf. Ser. 420, 012035 (2013) [arXiv:1208.1224 [nucl-th]]; B. Zhang, arXiv:1307.6234 [nucl-th].
* [135] Z. Xu and C. Greiner, Phys. Rev. Lett. 100, 172301 (2008) [arXiv:0710.5719 [nucl-th]].
* [136] Z. Xu and C. Greiner, Phys. Rev. C 76, 024911 (2007) [hep-ph/0703233].
* [137] A. El, Z. Xu and C. Greiner, Nucl. Phys. A 806, 287 (2008) [arXiv:0712.3734 [hep-ph]].
* [138] T. S. Biro, E. van Doorn, B. Muller, M. H. Thoma and X. N. Wang, Phys. Rev. C 48, 1275 (1993) [nucl-th/9303004].
* [139] S. M. H. Wong, Nucl. Phys. A 607, 442 (1996) [hep-ph/9606305].
* [140] M. Ruggieri, F. Scardina, S. Plumari and V. Greco, arXiv:1312.6060 [nucl-th].
* [141] J. Berges, S. Scheffler and D. Sexty, Phys. Lett. B 681, 362 (2009) [arXiv:0811.4293 [hep-ph]].
* [142] V. Khachatryan, Nucl. Phys. A 810, 109 (2008) [arXiv:0803.1356 [hep-ph]].
* [143] K. Fukushima and F. Gelis, Nucl. Phys. A 874, 108 (2012) [arXiv:1106.1396 [hep-ph]].
* [144] K. Fukushima, arXiv:1307.1046.
* [145] R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003) [hep-ph/0210202].
* [146] R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004) [hep-ph/0403101].
* [147] S. Floerchinger and C. Wetterich, arXiv:1311.5389 [hep-ph].
* [148] A. Dumitru, T. Lappi and Y. Nara, arXiv:1401.4124 [hep-ph].
* [149] A. Dumitru, T. Lappi and L. McLerran, Nucl. Phys. A 922, 140 (2014) [arXiv:1310.7136 [hep-ph]].
* [150] A. Majumder, arXiv:1402.1912 [nucl-th].
* [151] P. Bozek, Phys. Rev. C 85, 014911 (2012) [arXiv:1112.0915 [hep-ph]]; P. Bozek and W. Broniowski, Phys. Rev. C 88, 014903 (2013) [arXiv:1304.3044 [nucl-th]]; P. Bozek, W. Broniowski and G. Torrieri, Phys. Rev. Lett. 111, 172303 (2013) [arXiv:1307.5060 [nucl-th]]; P. Bozek and W. Broniowski, arXiv:1401.2367 [nucl-th].
* [152] A. Bzdak, B. Schenke, P. Tribedy and R. Venugopalan, Phys. Rev. C 87 (2013) 6, 064906 [arXiv:1304.3403 [nucl-th]].
* [153] K. Dusling and R. Venugopalan, Phys. Rev. D 87, no. 9, 094034 (2013) [arXiv:1302.7018 [hep-ph]]; K. Dusling and R. Venugopalan, Phys. Rev. D 87, no. 5, 054014 (2013) [arXiv:1211.3701 [hep-ph]].
* [154] X. Zhang and J. Liao, arXiv:1311.5463 [nucl-th].
* [155] E. Shuryak and I. Zahed, Phys. Rev. C 88, 044915 (2013) [arXiv:1301.4470 [hep-ph]]; E. Shuryak and I. Zahed, arXiv:1311.0836 [hep-ph].
|
arxiv-papers
| 2014-02-23T04:57:42 |
2024-09-04T02:49:58.654177
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xu-Guang Huang, Jinfeng Liao",
"submitter": "Xu-Guang Huang",
"url": "https://arxiv.org/abs/1402.5578"
}
|
1402.5605
|
# On the Dirichlet problem associated with Dunkl Laplacian
Ben Chrouda Mohamed
High Institute of Informatics and Mathematics
5000 Monastir, Tunisia
E-mail: [email protected]
###### Abstract
This paper is devoted to the study of the Dirichlet problem associated with
the Dunkl Laplacian $\Delta_{k}$. We establish, under some condition on a
bounded domain $D$ of $\mathbb{R}^{d}$, the existence of a unique continuous
function $h$ on $\mathbb{R}^{d}$ such that $\Delta_{k}h=0$ on $D$ and $h=f$ on
$\mathbb{R}^{d}\setminus D$ the complement of $D$ in $\mathbb{R}^{d}$, where
the function $f$ is asumed to be continuous. We also give an analytic formula
characterizing the solution $h$.
## 1 Introduction
Let $R$ be a root system in $\mathbb{R}^{d}$, $d\geq 1$, and we fix a positive
subsystem $R_{+}$ of $R$ and a nonnegative multiplicity function
$k:R\to\mathbb{R}_{+}$. For every $\alpha\in R$, let $H_{\alpha}$ be the
hyperplane orthogonal to $\alpha$ and $\sigma_{\alpha}$ be the reflection with
respect to $H_{\alpha}$, that is, for every $x\in\mathbb{R}^{d}$,
$\sigma_{\alpha}x=x-2\frac{\langle x,\alpha\rangle}{|\alpha|^{2}}\alpha$
where $\langle\cdot,\cdot\rangle$ denotes the Euclidean inner product of
$\mathbb{R}^{d}$. The Dunkl Laplacian $\Delta_{k}$ is defined [3], for $f\in
C^{2}(\mathbb{R}^{d})$, by
$\Delta_{k}f(x)=\Delta f(x)+2\sum_{\alpha\in
R_{+}}k(\alpha)\left(\frac{\langle\nabla
f(x),\alpha\rangle}{\langle\alpha,x\rangle}-\frac{|\alpha|^{2}}{2}\frac{f(x)-f(\sigma_{\alpha}x)}{\langle\alpha,x\rangle^{2}}\right),$
where $\nabla$ denotes the gradient on $\mathbb{R}^{d}$. Obviously,
$\Delta_{k}=\Delta$ when $k\equiv 0$.
Given a bounded open subset $D$ of $\mathbb{R}^{d}$, we consider the following
Dirichlet problem :
$\displaystyle\left\\{\begin{array}[]{rcll}\Delta_{k}h&=&0&\mbox{on }\;D,\\\
h&=&f&\mbox{on }\;\mathbb{R}^{d}\setminus D,\end{array}\right.$ (1)
where $f$ is a continuous function on $\mathbb{R}^{d}\setminus D$. When $D$ is
invariant under all reflections $\sigma_{\alpha}$, it was shown in [1], using
probabilistic tools from potential theory, that there exists a unique
continuous function $h$ on $\mathbb{R}^{d}$, twice differentiable on $D$ and
such that both equations in (1) are pointwise fulfilled. In this paper, we
shall investigate problem (1) for a bounded domain $D$ which is not invariant.
Let $D$ be a bounded open set such that its closure $\overline{D}$ is in some
Domain of $\mathbb{R}^{d}\setminus\cup_{\alpha\in R_{+}}H_{\alpha}$. We mean
by a solution of problem (1), every function $h:\mathbb{R}^{d}\to\mathbb{R}$
which is continuous on $\mathbb{R}^{d}$ such that $h=f$ on
$\mathbb{R}^{d}\setminus D$ and
$\int_{\mathbb{R}^{d}}h(x)\Delta_{k}\varphi(x)w_{k}(x)dx=0\quad\textrm{ for
every }\;\varphi\in C^{\infty}_{c}(D),$
where $C^{\infty}_{c}(D)$ denotes the space of infinitely differentiable
functions on $D$ with compact support and $w_{k}$ is the invariant weight
function defined on $\mathbb{R}^{d}$ by
$w_{k}(x)=\prod_{\alpha\in R_{+}}\langle x,\alpha\rangle^{2k(\alpha)}.$
The set $D$ is called $\Delta_{k}$-regular if, for every continuous function
$f$ on $\mathbb{R}^{d}\setminus D$, problem (1) admits one and only one
solution; this solution will be denoted by $H_{D}^{\Delta_{k}}f$. By
transforming problem (1) to a boundary value problem associated with
Schrödinger’s operator $\Delta-q$, we show that $D$ is $\Delta_{k}$-regular
provided it is $\Delta$-regular. We also give an analytic formula
characterizing the solution $H_{D}^{\Delta_{k}}f$ (see Theorem 1 below). We
derive from this formula that, for every $x\in D$, $H_{D}^{\Delta_{k}}f(x)$
depends only on the values of $f$ on $\cup_{\alpha\in
R_{+}}\sigma_{\alpha}(D)$ and on $\partial D$ the Euclidean boundary of $D$.
If, in addition, we assume that $f$ is locally Hölder continuous on
$\cup_{\alpha\in R_{+}}\sigma(D)$ then $H_{D}^{\Delta_{k}}f$ is continuously
twice differentiable on $D$ and therefore the first equation in (1) is
fulfilled by $H_{D}^{\Delta_{k}}f$ not only in the sense of distributions but
also pointwise.
It was shown in [5, 6] that the operator $\Delta_{k}$ is hypoelliptic on all
invariant open subset $D$ of $\mathbb{R}^{d}$. However, if $D$ is not
invariant, the question whether $\Delta_{k}$ is hypoelliptic on $D$ or not
remaind open. For $\Delta_{k}$-regular open set $D$, we show that if $D$ is
not invariant then $\Delta_{k}$ is not hypoelliptic in $D$. Hence the
condition ” $D$ is invariant” is necessary and sufficient for the
hypoellipticity of $\Delta_{k}$ on $D$.
## 2 Main results
We first present various facts on the Dirichlet boundary value problem
associated with Schrödinger’s operator which are needed for our approach. We
refer to [2, 4] for details. Let $G$ be the Green function on
$\mathbb{R}^{d}$, but without the constant factors :
$G(x,y)=\left\\{\begin{array}[]{ll}|x-y|^{2-d}&\hbox{if}\;d\geq 3;\\\
\ln\frac{1}{|x-y|}&\hbox{if}\;d=2;\\\ |x-y|&\hbox{if}\;d=1.\end{array}\right.$
Let $D$ be a bounded domain of $\mathbb{R}^{d}$ and let $q\in J(D)$ the Kato
class on $D$, i.e., $q$ is a Borel measurable function on $\mathbb{R}^{d}$
such that $G(1_{D}|q|)$ the Green potential of $1_{D}|q|$ is continuous on
$\mathbb{R}^{d}$. Note that the Kato class $J(D)$ contains all bounded Borel
measurable functions on $D$. Assume that $D$ is $\Delta$-regular. Then, for
every continuous function $f$ on $\partial D$, there exists a unique
continuous function $h$ on $\overline{D}$ such that $h=f$ on $\partial D$ and
$\int h(x)(\Delta-q)\varphi(x)dx=0\quad\textrm{ for every}\;\varphi\in
C^{\infty}_{c}(D).$ (2)
In the sequel, we denote $H_{D}^{\Delta-q}f$ the unique continuous extension
on $\overline{D}$ of $f$ which satisfies the Schrödinger’s equation (2). Let
$G_{D}^{\Delta}$ and $G_{D}^{\Delta-q}$ denotes, respectively, the Green
potential operator in $D$ of $\Delta$ and $\Delta-q$. The operator
$G_{D}^{\Delta-q}$ acts as a right inverse of the Schrödinger’s operator
$-(\Delta-q)$, i.e., for every Borel bounded function $g$ on $D$, we have
$\int G_{D}^{\Delta-q}g(x)(\Delta-q)\varphi(x)dx=-\int
g(x)\varphi(x)dx\quad\textrm{ for every}\;\varphi\in C^{\infty}_{c}(D).$
Then the unique continuous function $h$ on $\overline{D}$ such that $h=f$ on
$\partial D$ and
$\int h(x)(\Delta-q)\varphi(x)dx=-\int g(x)\varphi(x)dx\quad\textrm{ for
every}\;\varphi\in C^{\infty}_{c}(D)$ (3)
is given, for $x\in D$, by
$h(x)=H_{D}^{\Delta-q}f(x)+G_{D}^{\Delta-q}g(x).$ (4)
The function $G_{D}^{\Delta-q}g$ is continuous on $\overline{D}$, vanishing on
$\mathbb{R}^{d}\setminus D$ and, for every $x\in D$,
$G_{D}^{\Delta-q}g(x)=G_{D}^{\Delta}g(x)-G_{D}^{\Delta}(qG_{D}^{\Delta-q}g)(x).$
(5)
Moreover, if, in addition, we assume that $q\in C^{\infty}(D)$ then,
proceeding by induction, it follows from (5) that $G_{D}^{\Delta-q}g\in
C^{n}(D)$ if and only if $G_{D}^{\Delta}g\in C^{n}(D),\;n\in\mathbb{N}$.
Now we are ready to establish our first main result giving a characterization
of solutions of the Dirichlet boundary value problem associated with the Dunkl
Laplacian $\Delta_{k}$.
###### Theorem 1.
Let $D$ be a bounded open set such that $\overline{D}$ is in some Domain of
$\mathbb{R}^{d}\setminus\cup_{\alpha\in R_{+}}H_{\alpha}$. If $D$ is
$\Delta$-regular then $D$ is $\Delta_{k}$-regular. Moreover, for every
continuous function $f$ on $\mathbb{R}^{d}\setminus D$ and for every $x\in D$,
$H_{D}^{\Delta_{k}}f(x)=\frac{1}{\sqrt{w_{k}(x)}}\left(H_{D}^{\Delta-q}(f\sqrt{w_{k}})(x)+G_{D}^{\Delta-q}\left(\sqrt{w_{k}}Nf\right)(x)\right),$
(6)
where $q$ and $Nf$ are the functions defined, for $x\in D$, by
$q(x):=\sum_{\alpha\in R_{+}}\left(\frac{|\alpha|k(\alpha)}{\langle
x,\alpha\rangle}\right)^{2}$
and
$Nf(x):=\sum_{\alpha\in R_{+}}\frac{|\alpha|^{2}k(\alpha)}{\langle
x,\alpha\rangle^{2}}f(\sigma_{\alpha}x).$
###### Proof.
Let $f$ be a continuous function on $\mathbb{R}^{d}\setminus D$. We intend to
prove existence and uniqueness of a continuous function $h$ on $D$ such that
$h=f$ on $\mathbb{R}^{d}\setminus D$ and
$\int h(x)\Delta_{k}\varphi(x)w_{k}(x)dx=0\quad\textrm{ for every}\;\varphi\in
C^{\infty}_{c}(D).$ (7)
It is clear that
$\nabla\left(\sqrt{w_{k}}\right)(x)=\sqrt{w_{k}(x)}\sum_{\alpha\in
R_{+}}\frac{k(\alpha)}{\langle x,\alpha\rangle}\alpha.$
Then, using the fact that [3]
$\sum_{\alpha,\beta\in
R_{+}}k(\alpha)k(\beta)\frac{\langle\alpha,\beta\rangle}{\langle
x,\alpha\rangle\;\langle x,\beta\rangle}=\sum_{\alpha\in
R_{+}}\frac{|\alpha|^{2}k^{2}(\alpha)}{\langle x,\alpha\rangle^{2}},$
direct computation shows that
$\Delta\left(\sqrt{w_{k}}\right)(x)=\sqrt{w_{k}(x)}\sum_{\alpha\in
R_{+}}|\alpha|^{2}\frac{k^{2}(\alpha)-k(\alpha)}{\langle
x,\alpha\rangle^{2}}.$
Thus, for every $\varphi\in C^{\infty}_{c}(D)$,
$\displaystyle\Delta\left(\varphi\sqrt{w_{k}}\right)(x)$ $\displaystyle=$
$\displaystyle\sqrt{w_{k}(x)}\left(\Delta\varphi(x)+2\sum_{\alpha\in
R_{+}}k(\alpha)\left(\frac{\langle\nabla\varphi(x),\alpha\rangle}{\langle\alpha,x\rangle}-\frac{|\alpha|^{2}}{2}\frac{\varphi(x)}{\langle\alpha,x\rangle^{2}}\right)\right)$
$\displaystyle+\;q(x)\varphi(x)\sqrt{w_{k}(x)},$
and thereby
$\sqrt{w_{k}(x)}\Delta_{k}\varphi(x)=\left(\Delta\left(\varphi\sqrt{w_{k}}\right)(x)-q(x)\varphi(x)\sqrt{w_{k}(x)}\right)+\sqrt{w_{k}(x)}N\varphi(x).$
(8)
Since the map $\varphi\to\varphi\sqrt{w_{k}}$ is invertible on the space
$C^{\infty}_{c}(D)$ and the function $x\to\frac{w_{k}(x)}{\langle
x,\alpha\rangle^{2}}$ is invariant under the reflection $\sigma_{\alpha}$,
equation (7) is equivalent to the following Schrödinger’s equation : For every
$\psi\in C^{\infty}_{c}(D)$,
$\int
h(x)\sqrt{w_{k}(x)}\left(\Delta-q\right)\psi(x)dx=-\int\sqrt{w_{k}(x)}Nf(x)\psi(x)dx.$
Finally, since $q$ is bounded on $D$ and therefore is in $J(D)$, the
statements follow from (3) and (4). ∎
To construct a $\Delta$-regular set $D$, it suffices to choose $D$ such that
its Euclidean boundary $\partial D$ satisfies the the geometric assumption
known as ” cone condition”, i.e., for every $z\in\partial D$ there exists a
cone $C$ of vertex $z$ such that $C\cap B(z,r)\subset\mathbb{R}^{d}\setminus
D$ for some $r>0$, where $B(z,r)$ is the ball of center $z$ and radius $r$
(see, for example, [4]).
###### Remark 2.
Note that, in order to obtain $q\in J(D)$, the hypothesis of the above theorem
”$\overline{D}\subset\mathbb{R}^{d}\setminus\cup_{\alpha\in R_{+}}H_{\alpha}$”
is nearly optimal. Indeed, assume that there exists a cone $C_{z}$ of vertex
$z\in\overline{D}\cap H_{\alpha}$ for some $\alpha\in R_{+}$ with
$k(\alpha)\neq 0$ such that $C_{z}^{r}:=C_{z}\cap B(z,r)\subset D$ for some
$r>0$. Then,
$\displaystyle G(1_{D}q)(z)$ $\displaystyle\geq$
$\displaystyle|\alpha|^{2}k^{2}(\alpha)\int_{C_{z}^{r}}G(z,y)\frac{1}{\langle
y,\alpha\rangle^{2}}dy$ $\displaystyle=$
$\displaystyle|\alpha|^{2}k^{2}(\alpha)\int_{C_{z}^{r}}G(z,y)\frac{1}{\langle
z-y,\alpha\rangle^{2}}dy$ $\displaystyle\geq$ $\displaystyle
k^{2}(\alpha)\int_{C_{z}^{r}-z}G(0,y)\frac{1}{|y|^{2}}dy$ $\displaystyle=$
$\displaystyle\infty.$
It is easy to see that for every $x\in D$ the map $f\to
H_{D}^{\Delta_{k}}f(x)$ defines a positive Radon measure on
$\mathbb{R}^{d}\setminus D$. We denote this measure by
$H_{D}^{\Delta_{k}}(x,dy)$. The following results are obtained in a convenient
way by using formula (6) of the above theorem.
###### Corollary 3.
For every $x\in D$, $H_{D}^{\Delta_{k}}(x,dy)$ is a probability measure
supported by
$\partial D\cup\left(\cup_{\alpha\in R_{+}}\sigma_{\alpha}(D)\right)$
and satisfies
$\frac{\sqrt{w_{k}(x)}}{\sqrt{w_{k}(y)}}H_{D}^{\Delta_{k}}(x,dy)=H_{D}^{\Delta-q}(x,dy)+\sum_{\alpha\in
R_{+}}\frac{|\alpha|^{2}k(\alpha)}{\langle
y,\alpha\rangle^{2}}G_{D}^{\Delta-q}(x,\sigma_{\alpha}y)dy.$
###### Corollary 4.
Let $D$ be a $\Delta$-regular bounded open set such that $\overline{D}$ is in
some Domain of $\mathbb{R}^{d}\setminus\cup_{\alpha\in R_{+}}H_{\alpha}$. Let
$f$ be a continuous function on $\partial D\cup\left(\cup_{\alpha\in
R_{+}}\sigma_{\alpha}(D)\right)$. If $f$ is locally Hölder continuous on
$\cup_{\alpha\in R_{+}}\sigma(D)$ then $H_{D}^{\Delta_{k}}f\in C^{2}(D)$ and,
for every $x\in D$,
$\Delta_{k}\left(H_{D}^{\Delta_{k}}f\right)(x)=0.$
###### Proof.
Since $H_{D}^{\Delta-q}(f\sqrt{w_{k}})$ is a solution of the Schrödinger’s
equation (2), the hypoellipticity of the operator $\Delta-q$ on $D$ implies
that $H_{D}^{\Delta-q}(f\sqrt{w_{k}})\in C^{\infty}(D)$. Moreover, since $Nf$
is locally Hölder continuous on $D$,
$G_{D}^{\Delta}\left(\sqrt{w_{k}}Nf\right)\in C^{2}(D)$ and consequently
$G_{D}^{\Delta-q}\left(\sqrt{w_{k}}Nf\right)\in C^{2}(D)$. Then it follows
from (6) that $H_{D}^{\Delta_{k}}f\in C^{2}(D)$. For every $\varphi\in
C^{\infty}_{c}(D)$, direct computation using (8) yields
$\int\Delta_{k}\left(H_{D}^{\Delta_{k}}f\right)(x)\varphi(x)w_{k}(x)dx=\int
H_{D}^{\Delta_{k}}f(x)\Delta_{k}\varphi(x)w_{k}(x)dx.$
This completes the proof. ∎
Let $D$ be an open subset of $\mathbb{R}^{d}$. The operator $\Delta_{k}$ is
said to be hypoelliptic on $D$ if, for every $f\in C^{\infty}(D)$, every
continuous function $h$ on $\mathbb{R}^{d}$ which satisfies
$\int_{\mathbb{R}^{d}}h(x)\Delta_{k}\varphi(x)w_{k}(x)dx=\int
f(x)\varphi(x)w_{k}(x)dx\quad\textrm{ for every }\;\varphi\in
C^{\infty}_{c}(D)$
is infinitely differentiable on $D$. We note that the problem of the
hypoellipticity of $\Delta_{k}$ is discussed in [5, 6], where the authors show
that $\Delta_{k}$ is hypoelliptic on $D$ provided $D$ is invariant under all
reflections $\sigma_{\alpha}$. However, if $D$ is not invariant, the question
whether $\Delta_{k}$ is hypoelliptic on $D$ or not remaind open.
###### Theorem 5.
Let $D$ be a $\Delta_{k}$-regular open set. Then $\Delta_{k}$ is hypoelliptic
on $D$ if and only if $D$ is invariant.
###### Proof.
It is obviously sufficient to prove that $\Delta_{k}$ is not hypoelliptic on
$D$ provided $D$ is not invariant. Assume that $D$ is not invariant. Since the
open set $D\setminus\cup_{\alpha\in R_{+}}H_{\alpha}$ is also not invariant,
there exists a nonempty open ball $B$ such that
$\overline{B}\subset D\setminus\cup_{\alpha\in
R_{+}}H_{\alpha}\quad\textrm{and}\quad\sigma_{\alpha}(B)\subset\mathbb{R}^{d}\setminus
D\;\textrm{ for some }\;\alpha\in R_{+}.$
We also choose the ball $B$ small enough such that, for every $\alpha\in
R_{+}$,
$\sigma_{\alpha}(B)\subset D\quad\textrm{ or
}\quad\sigma_{\alpha}(B)\subset\mathbb{R}^{d}\setminus D.$
Let $I:=\\{\alpha\in R_{+};\;\sigma_{\alpha}(B)\subset\mathbb{R}^{d}\setminus
D\\}$ and $J:=R_{+}\setminus I$. Let $f$ be a continuous function on
$\mathbb{R}^{d}\setminus D$ and denote $H_{D}^{\Delta_{k}}f$ by $h$. Since $B$
is $\Delta$-regular and $h$ satisfies
$\int h(x)\Delta_{k}\varphi(x)w_{k}(x)dx=0\quad\textrm{ for every}\;\varphi\in
C^{\infty}_{c}(B),$
it follows from Theorem 1 that $B$ is $\Delta_{k}$-regular and, for every
$x\in B$,
$h(x)=\frac{1}{\sqrt{w_{k}(x)}}\left(H_{B}^{\Delta-q}(h\sqrt{w_{k}})(x)+G_{B}^{\Delta-q}\left(\sqrt{w_{k}}Nh\right)(x)\right).$
(9)
Let $g_{1}$ and $g_{2}$ be the functions defined on $B$ by
$g_{1}(x)=\sum_{\alpha\in J}\frac{|\alpha|^{2}k(\alpha)}{\langle
x,\alpha\rangle^{2}}h(\sigma_{\alpha}x)\quad\textrm{and}\quad
g_{2}(x)=\sum_{\alpha\in I}\frac{|\alpha|^{2}k(\alpha)}{\langle
x,\alpha\rangle^{2}}f(\sigma_{\alpha}x).$
It is clear that the function $g_{2}$ is not trivial and $Nh=g_{1}+g_{2}$.
Now, assume that $h\in C^{\infty}(D)$. Then $g_{1}\in C^{\infty}(B)$ and
therefore $G_{B}^{\Delta-q}\left(\sqrt{w_{k}}g_{1}\right)\in C^{\infty}(B)$.
Furthermore, since $H_{B}^{\Delta-q}(h\sqrt{w_{k}})\in C^{\infty}(B)$, it
follows from (9) that $G_{B}^{\Delta-q}\left(\sqrt{w_{k}}g_{2}\right)\in
C^{\infty}(B)$. Thus
$-(\Delta-q)G_{B}^{\Delta-q}\left(\sqrt{w_{k}}g_{2}\right)=\sqrt{w_{k}}g_{2}\in
C^{\infty}(B)$ and therefore $g_{2}\in C^{\infty}(B)$, a contradiction. Hence
$h$ is not infinitely differentiable on $D$ and consequently the Dunkl
Laplacian $\Delta_{k}$ is not hypoelliptic on $D$. ∎
## References
* [1] M. Ben Chrouda and K. El Mabrouk, Dirichlet problem associated with Dunkl Laplacian on $W$-invariant open sets, Preprint. arxiv: 1402.1597 (2014).
* [2] A. Boukricha, W. Hansen and H. Hueber, Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces, Expo. Math. 5 (1987) 97–135.
* [3] C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Am. Math. Soc. 311 (1989) 167–183.
* [4] K. L. Chung and Z. Zhao, From Brownian motion to Schrödinger’s equation, Springer-Verlag, 1995.
* [5] K. Hassine, Mean value property associated with the Dunkl Laplacian, Preprint. arxiv: 1401.1949 (2014).
* [6] H. Mejjaoli and K. Trimèche, Hypoellipticity and hypoanalyticity of the Dunkl Laplacian operator, Integral Transforms Spec. Funct. 15 (2004) 523–548.
|
arxiv-papers
| 2014-02-23T13:31:29 |
2024-09-04T02:49:58.672146
|
{
"license": "Public Domain",
"authors": "Mohamed Ben Chrouda",
"submitter": "Mohamed Ben Chrouda",
"url": "https://arxiv.org/abs/1402.5605"
}
|
1402.5715
|
# Variational Particle Approximations
Ardavan Saeedi111First two authors contributed equally.
[email protected]
CSAIL
Massachusetts Institute of Technology
Tejas D. Kulkarni*
[email protected]
Department of Brain & Cognitive Sciences
Massachusetts Institute of Technology
Vikash K. Mansinghka
[email protected]
Department of Brain & Cognitive Sciences
Massachusetts Institute of Technology
Samuel J. Gershman
[email protected]
Department of Psychology and Center for Brain Science
Harvard University
###### Abstract
Approximate inference in high-dimensional, discrete probabilistic models is a
central problem in computational statistics and machine learning. This paper
describes discrete particle variational inference (DPVI), a new approach that
combines key strengths of Monte Carlo, variational and search-based
techniques. DPVI is based on a novel family of particle-based variational
approximations that can be fit using simple, fast, deterministic search
techniques. Like Monte Carlo, DPVI can handle multiple modes, and yields exact
results in a well-defined limit. Like unstructured mean-field, DPVI is based
on optimizing a lower bound on the partition function; when this quantity is
not of intrinsic interest, it facilitates convergence assessment and
debugging. Like both Monte Carlo and combinatorial search, DPVI can take
advantage of factorization, sequential structure, and custom search operators.
This paper defines DPVI particle-based approximation family and partition
function lower bounds, along with the sequential DPVI and local DPVI algorithm
templates for optimizing them. DPVI is illustrated and evaluated via
experiments on lattice Markov Random Fields, nonparametric Bayesian mixtures
and block-models, and parametric as well as non-parametric hidden Markov
models. Results include applications to real-world spike-sorting and
relational modeling problems, and show that DPVI can offer appealing
time/accuracy trade-offs as compared to multiple alternatives.
## 1 Introduction
Monte Carlo methods are based on the idea that one can approximate a complex
distribution with a set of stochastically sampled particles. The flexibility
and variety of Monte Carlo methods have made them the workhorse of statistical
computation (Robert and Casella, 2004). However, their success relies
critically on having available a good sampler, and designing such a sampler is
often challenging.
In this paper, we rethink particle approximations over discrete hypothesis
spaces from a different perspective. Suppose we got to pick where to place the
particles in the hypothesis space; where would we put them? Intuitively, we
would want to distribute them in such a way that they cover high probability
regions of the target distribution, but without the particles all devolving
onto the mode of the distribution. This problem can be formulated precisely
within the framework of variational inference (Wainwright and Jordan, 2008),
which treats probabilistic inference as an optimization problem over a set of
distributions. We derive a coordinate ascent update for particle
approximations that iteratively minimizes the Kullback-Leibler (KL) divergence
between the particle approximation and the target distribution.
After introducing our general framework, we describe how it can be applied to
filtering and smoothing problems. We then show experimentally that variational
particle approximations can overcome a number of problems that are challenging
for conventional Monte Carlo methods. In particular, our approach is able to
produce a diverse, high probability set of particles in situations where Monte
Carlo and mean-field variational methods sometimes degenerate.
## 2 Background
Consider the problem of approximating a probability distribution $P(x)$ over
discrete latent variables
$x=\\{x_{1},\ldots,x_{N}\\},x_{n}\in\\{1,\ldots,M^{k}_{n}\\}$, where the
target distribution is known only up to a normalizing constant $Z$:
$P(x)=f(x)/Z$. We will refer to $f(x)\geq 0$ as the _score_ of $x$ and $Z$ as
the _partition function_. We further assume that $P(x)$ is a Markov network
defined on a graph $G$, so that $f(x)$ factorizes according to:
$\displaystyle f(x)=\prod_{c}f_{c}(x_{c}),$ (1)
where $c\subseteq\\{1,\ldots,N\\}$ indexes the maximal cliques of $G$.
### 2.1 Importance sampling and sequential Monte Carlo
A general way to approximate $P(x)$ is with a weighted collection of $K$
particles, $\\{x^{1},\ldots,x^{K}\\}$:
$\displaystyle P(x)\approx Q(x)=\sum_{k=1}^{K}w^{k}\delta[x,x^{k}],$ (2)
where
$x^{k}=\\{x^{k}_{1},\ldots,x^{k}_{N}\\},x^{k}_{n}\in\\{1,\ldots,M^{k}_{n}\\}$
and $\delta[\cdot,\cdot]=1$ if its arguments are equal and 0 otherwise.
Importance sampling is a Monte Carlo method that stochastically generates
particles from a proposal distribution, $x^{k}\sim\phi(\cdot)$, and computes
the weight according to $w^{k}\propto f(x^{k})/\phi(x^{k})$. Importance
sampling has the property that the particle approximation converges to the
target distribution as $K\rightarrow\infty$ (Robert and Casella, 2004).
Sequential Monte Carlo methods such as particle filtering (Doucet et al.,
2001) apply importance sampling to stochastic dynamical systems (where $n$
indexes time) by sequentially sampling the latent variables at each time point
using a proposal distribution $\phi(x_{n}|x_{n-1})$. This procedure can
produce conditionally low probability particles; therefore, most algorithms
include a resampling step which replicates high probability particles and
kills off low probability particles. The downside of resampling is that it can
produce degeneracy: the particles become concentrated on a small number of
hypotheses, and consequently the effective number of particles is low.
### 2.2 Variational inference
Variational methods (Wainwright and Jordan, 2008) define a parametrized family
of probability distributions $\mathcal{Q}$ and then choose $Q\in\mathcal{Q}$
that maximizes the _negative variational free energy_ :
$\displaystyle\mathcal{L}[Q]=\sum_{x}Q(x)\log\frac{f(x)}{Q(x)}.$ (3)
The negative variational free energy is related to the partition function $Z$
and the KL divergence through the following identity:
$\displaystyle\log Z=\mbox{KL}[Q||P]+\mathcal{L}[Q],$ (4)
where
$\displaystyle\mbox{KL}[Q||P]=\sum_{x}Q(x)\log\frac{Q(x)}{P(x)}.$ (5)
Since $\mbox{KL}[Q||P]\geq 0$, the negative variational free energy is a lower
bound on the log partition function, achieving equality when the KL divergence
is minimized to 0. Maximizing $\mathcal{L}[Q]$ with respect to $Q$ is thus
equivalent to minimizing the KL divergence between $Q$ and $P$.
Unlike the Monte Carlo methods described in the previous section, variational
methods do not in general converge to the target distribution, since typically
$P$ is not in $\mathcal{Q}$. The advantage of variational methods is that they
guarantee an improved bound after each iteration, and convergence is easy to
monitor (unlike most Monte Carlo methods). In practice, variational methods
are also often more computationally efficient.
We next consider particle approximations from the perspective of variational
inference. We then turn to the application of particle approximations to
inference in stochastic dynamical systems.
## 3 Variational particle approximations
Variational inference can be connected to Monte Carlo methods by viewing the
particles as a set of variational parameters parameterizing $Q$. For the
particle approximation defined in Eq. 2, the negative variational free energy
takes the following form:
$\displaystyle\mathcal{L}[Q]=\sum_{k=1}^{K}w^{k}\log\frac{f(x^{k})}{w^{k}V^{k}},$
(6)
where $V^{k}=\sum_{j=1}^{K}\delta[x^{j},x^{k}]$ is the number of times an
identical replica of $x^{k}$ appears in the particle set. We wish to find the
set of $K$ particles and their associated weights that maximize
$\mathcal{L}[Q]$, subject to the constraint that $\sum_{k=1}^{K}w^{k}=1$. This
constraint can be implemented by defining a new functional with Lagrange
multiplier $\lambda$:
$\displaystyle\tilde{\mathcal{L}}[Q]=\mathcal{L}[Q]+\lambda\left(\sum_{k=1}^{K}w^{k}-1\right).$
(7)
Taking the functional derivative of the Lagrangian with respect to $w^{k}$ and
equating to zero, we obtain:
$\displaystyle\frac{\partial\tilde{\mathcal{L}}[Q]}{\partial w^{k}}$
$\displaystyle=\log f(x^{k})-\log w^{k}-\log V^{k}+\lambda-1=0$
$\displaystyle\Longrightarrow w^{k}=Z_{Q}^{-1}f(x^{k})/V^{k},$ (8)
where
$\displaystyle
Z_{Q}=\exp(\lambda-1)^{-1}=\sum_{k=1}^{K}\frac{f(x^{k})}{V^{k}}.$ (9)
We can plug the above result back into the definition of $\mathcal{L}[Q]$:
$\displaystyle\mathcal{L}[Q]$
$\displaystyle=Z^{-1}_{Q}\sum_{k=1}^{K}\frac{f(x^{k})}{V^{k}}\log\frac{f(x^{k})V^{k}}{Z^{-1}_{Q}f(x^{k})V^{k}}$
$\displaystyle=Z^{-1}_{Q}\sum_{k=1}^{K}\frac{f(x^{k})}{V^{k}}\log Z_{Q}$
$\displaystyle=\log Z_{Q}$ (10)
Thus, $\mathcal{L}[Q]$ is maximized by choosing the $K$ values of $x$ with the
highest score. The following theorem shows that allowing $V^{k}>1$ (i.e.,
having replica particles) can never improve the bound.
Theorem: Let $Q$ and $Q^{\prime}$ denote two particle approximations, where
$Q$ consists of unique particles ($V^{k}=1$ for all $k$) and $Q^{\prime}$ is
identical to $Q$ except that particle $x^{j}$ is replicated $V^{j}$ times
(displacing $V^{j}$ other particles with cumulative score $F$). For any choice
of particles, $\mathcal{L}[Q]\geq\mathcal{L}[Q^{\prime}]$.
Proof: We first apply Jensen’s inequality to obtain an upper bound on
$\mathcal{L}[Q^{\prime}]$:
$\displaystyle\mathcal{L}[Q^{\prime}]\leq\log\sum_{k=1}^{K}w^{k}Z_{Q}=\log\sum_{k=1}^{K}\frac{f(x^{k})}{V^{k}}.$
(11)
Since $\mathcal{L}[Q]=\log Z_{Q}$, we wish to show that
$Z_{Q}\geq\sum_{k=1}^{K}\frac{f(x^{k})}{V^{k}}$. All the particles in $Q$ and
$Q^{\prime}$ are identical except for $x^{j}$ and the $V^{j}$ particles in $Q$
that were displaced by replicas of $x^{j}$ in $Q^{\prime}$; thus we only need
to establish that $f(x^{j})+F\geq\frac{V^{j}f(x^{j})}{V^{j}}=f(x^{j})$. Since
the score can never be negative, $F\geq 0$ and the inequality holds for any
choice of particles. $\blacksquare$
Algorithm 1 Discrete particle variational inference
1: /*$N$ is the number of latent variables */
2: /*$x^{k}$ is the set of all latent variables for the $k$th particle:
$x^{k}=\\{x^{k}_{1},\ldots,x^{k}_{N}\\}$ */
3: /*$M^{k}_{n}$ is the support of latent variable $x^{k}_{n}$ */
4: Input: initial particle approximation $Q$ with $K$ particles, tolerance
$\epsilon$
5: while $|\mathcal{L}[Q]-\mathcal{L}[Q^{\prime}]|>\epsilon$ do
6: for $n=1$ to $N$ do
7: $\mathcal{X}=\emptyset$
8: for $k=1$ to $K$ do
9: Copy particle $k$: $\tilde{x}^{k}\leftarrow x^{k}$
10: for $m=1$ to $M^{k}_{n}$ do
11: Modify particle: $\tilde{x}_{n}^{k}\leftarrow m$
12: Score $\tilde{x}^{k}$ using Eq. 12
13: $\mathcal{X}\leftarrow\mathcal{X}\cup(\tilde{x}^{k},f(\tilde{x}^{k}))$
14: end for
15: end for
16: Select the $K$ particles from $\mathcal{X}$ with the largest scores
17: Construct new particle approximation
$Q^{\prime}(x)=\sum_{k=1}^{K}w^{k}\delta[x,x^{k}]$
18: Compute variational bound $\mathcal{L}[Q^{\prime}]$ using Eq. 10
19: end for
20: end while
21: return particle approximation $Q^{\prime}$
The variational bound can be optimized by coordinate ascent, as specified in
Algorithm 1, which we refer to as _discrete particle variational inference_
(DPVI). This algorithm takes advantage of the fact that when optimizing the
bound with respect to a single variable, only potentials local to that
variable need to be computed. In particular, let $\tilde{x}^{k}$ be a replica
of $x^{k}$ with a single-variable modification, $\tilde{x}_{n}^{k}=m$. We can
compute the unnormalized probability of this particle efficiently using the
following equation:
$\displaystyle
f(\tilde{x}^{k})=f(x^{k})\frac{\mathcal{F}_{n}(\tilde{x}^{k})}{\mathcal{F}_{n}(x^{k})}$
(12)
where $\mathcal{F}_{n}(x)=\prod_{c:n\in c}f_{c}(x_{c})$. The variational bound
for the modified particle can then be computed using Eq. 10. Particles can be
initialized arbitrarily. When repeatedly iterated, DPVI will converge to a
local maximum of the negative variational free energy. Note that in principle
more sophisticated methods can be used to find the top $K$ modes (e.g.,
Flerova et al., 2012; Yanover and Weiss, 2003); however, we have found that
this coordinate ascent algorithm is fast, easy to implement, and very
effective in practice (as our experiments below demonstrate).
An important aspect of this framework is that it maintains one of the same
asymptotic guarantees as importance sampling: $Q$ converges to $P$ as
$K\rightarrow\infty$, since in this limit DPVI is equivalent to exact
inference. Thus, DPVI combines advantages of variational methods
(monotonically decreasing KL divergence between $Q$ and $P$) with the
asymptotic correctness of Monte Carlo methods. The asymptotic complexity of
DPVI in the sequential setting is $O(SNK)$ where $S$ is the maximum support
size of the latent variables. For the iterative update of the particles the
complexity is $O(TCSK)$, where $T$ is the maximum number of iterations until
convergence and $C$ is the maximum clique size. In our experiments, we
empirically observed that we only need a small number of iterations and
particles in order to outperform our baselines.
## 4 Filtering and smoothing in hidden Markov models
We now describe how variational particle approximations can be applied to
filtering and smoothing in hidden Markov models (HMMs). Consider a hidden
Markov model with observations $y=\\{y_{1},\ldots,y_{N}\\}$ generated by the
following stochastic process:
$\displaystyle
P(y,x,\theta)=P(\theta)\prod_{n}P(y_{n}|x_{n},\theta)P(x_{n}|x_{n-1},\theta),$
(13)
where $\theta$ is a set of transition and emission parameters. We are
particularly interested in _marginalized_ HMMs where the parameters are
integrated out: $P(y,x)=\int_{\theta}P(y,x,\theta)d\theta$. This induces
dependencies between observation $n$ and all previous observations, making
inference challenging.
Filtering is the problem of computing the posterior over the latent variables
at time $n$ given the history $y_{1:n}$. To construct the variational particle
approximation of the filtering distribution, we need to compute the product of
potentials for variable $n$:
$\displaystyle\mathcal{F}_{n}(x)=P(y_{n}|x_{1:n},y_{1:n-1})P(x_{n}|x_{1:n-1}).$
(14)
We can then apply the coordinate ascent update described in the previous
section. This update is simplified in the filtering context due to the
underlying Markov structure:
$\displaystyle f(\tilde{x}^{k})=$ $\displaystyle
f(x^{k})P(y_{n}|x_{n}^{k}=m,x_{1:n-1},y_{1:n-1})P(x^{k}_{n}=m|x_{1:n-1}).$
(15)
At each time step, the algorithm selects the $K$ continuations (new variable
assignments of the current particle set) that maximize the negative
variational free energy.
Smoothing is the problem of computing the posterior over the latent variables
at time $n$ given data from both the past and the future, $y_{1:N}$. The
product of potentials is given by:
$\displaystyle\mathcal{F}_{n}(x)=P(y_{n}|x_{1:n},y_{-n})P(x_{n}|x_{-n}),$ (16)
where $x_{-n}$ refers to all the latent variables except $x_{n}$ (and likewise
for $y_{-n}$). This potential can be plugged into the updates described in the
previous section.
To understand DPVI applied to filtering problems, it is helpful to contemplate
three possible fates for a particle at time $n$ (illustrated in Figure 1):
* •
Selection: A single continuation of particle $k$ has non-zero weight. This can
be seen as a deterministic version of particle filtering, where the sampling
operation is replaced with a max operation.
* •
Splitting: Multiple continuations of particle $k$ have non-zero weight. In
this case, the particle is split into multiple particles at the next
iteration.
* •
Deletion: No continuations of particle $k$ have non-zero weight. In this case,
the particle is deleted from the particle set.
Similar to particle filtering with resampling, DPVI deletes and propagates
particles based on their probability. However, as we show later, DPVI is able
to escape some of the problems associated with resampling.
|
---|---
(A) DPVI | (B) Particle Filtering
Figure 1: Schematic of DPVI versus particle filtering for filtering problems.
Illustration of different filtering scenarios over 2 time steps in a binary
state space with $K=3$ particles. The number in each circle indicates the
binary value of the corresponding variable. Arrows indicate the evolution of
the particles. (A) DPVI: The size of the putative particles represents the
score of the particle. The $K$ continuations with highest score are selected
for propagation to the next time step. The size of the new particle set
corresponds to the normalized score. Particle $P1$ is split, $P2$ is deleted
and one putative particle from $P3$ is selected. (B) Particle filtering: The
size of the node represents the weight of the particle for the resampling
step.
## 5 Related work
DPVI is related to several other ideas in the statistics literature:
* •
DPVI is a special case of a _mixture mean-field variational approximation_
(Jaakkola and Jordan, 1998; Lawrence, 2000):
$\displaystyle Q(x)=\sum_{k=1}^{K}Q(k)\prod_{n=1}^{N}Q(x_{n}|k).$ (17)
In DPVI, $Q(k)=w^{k}$ and $Q(x_{n}|k)=\delta[x_{n},x_{n}^{k}]$. A distinct
advantage of DPVI is that the variational updates do not require the
additional lower bound used in general mixture mean-field, due to the
intractability of the mean-field updates.
* •
When $K=1$, DPVI is equivalent to _iterated conditional modes_ (ICM; Besag,
1986), which iteratively maximizes each latent variable conditional on the
rest of the variables.
* •
DPVI is conceptually similar to nonparametric variational inference (Gershman
et al., 2012), which approximates the posterior over a continuous state space
using a set of particles convolved with a Gaussian kernel.
* •
Frank et al. (2009) used particle approximations within a variational message
passing algorithm. The resulting approximation is “local” in the sense that
the particles are used to approximate messages passed between nodes in a
factor graph, in contrast to the “global” approximation produced by DPVI,
which attempts to capture the distribution over the entire set of variables.
* •
Ionides (2008) described a truncated version of importance sampling in which
weights falling below some threshold are set to the threshold value. This is
similar (though not equivalent) to the DPVI setting where latent variables are
sampled exhaustively and without replacement.
* •
Finally, DPVI is closely related to the problem of finding the $K$ most
probable latent variable assignments (Flerova et al., 2012; Yanover and Weiss,
2003). We view this problem through the lens of particle approximations,
connecting it to both Monte Carlo and variational methods.
## 6 Experiments
In this section, we compare the performance of DPVI to several widely used
approximate inference algorithms, including particle filtering and variational
methods. We first present a didactic example to illustrate how DPVI can
sometimes succeed where particle filtering fails. We then apply DPVI to four
probabilistic models: the Dirichlet process mixture model (DPMM; Antoniak,
1974; Escobar and West, 1995), the infinite HMM (iHMM; Beal et al., 2002; Teh
et al., 2006), the infinite relational model (IRM; Kemp et al., 2006) and the
Ising model.
### 6.1 Didactic example: binary HMM
(A)
---
(B)
Figure 2: Comparison of approximate inference schemes. (_A_) Approximating
families for DPVI, Monte Carlo and mean-field. (_B_) Illustration of the
differences between schemes in (A) on a binary HMM.
As a didactic example, we use a simple HMM with binary hidden states ($x$) and
observations ($y$):
$\displaystyle P(x_{n+1}=0|x_{n}=0)=\alpha_{0}$ $\displaystyle
P(x_{n+1}=1|x_{n}=1)=\alpha_{1}$ $\displaystyle P(y_{n}=0|x_{n}=0)=\beta_{0}$
$\displaystyle P(y_{n}=1|x_{n}=1)=\beta_{1},$ (18)
with $\alpha_{0}$, $\alpha_{1}$, $\beta_{0}$, and $\beta_{1}$ all less than
0.5. We will use this model to illustrate how DPVI differs from particle
filtering. Figure 2 compares several inference schemes for this model.
For illustration, we use the following parameters: $\alpha_{0}=0.2$,
$\alpha_{1}=0.1$, $\beta_{0}=0.3$, and $\beta_{1}=0.2$. Suppose you observe a
sequence generated from this model. For a sufficiently long sequence, a
particle filter with resampling will eventually delete all conditionally
unlikely particles, and thus suffer from degeneracy. On the other hand,
without resampling the approximation will degrade over time because
conditionally unlikely particles are never replaced by better particles. For
this reason, it is sometimes suggested that resampling only be performed when
the effective sample size (ESS) falls below some threshold. The ESS is
calculated as follows:
$\displaystyle\mbox{ESS}=\frac{1}{\sum_{k=1}^{K}(w^{k})^{2}}.$ (19)
A low ESS means that most of the weight is being placed on a small number of
particles, and hence the approximation may be degenerate (although in some
cases this may mean that the target distribution is peaky). We evaluated
particle filtering with multinomial resampling on synthetic data generated
from the HMM described above. Approximation accuracy was measured by using the
forward-backward algorithm to compute the hidden state posterior marginals
exactly and then comparing these marginals to the particle approximation.
Figure 3 shows performance as a function of ESS threshold, demonstrating that
there is a fairly narrow range of thresholds for which performance is good.
Thus in practice, successful applications of particle filtering may require
computationally expensive tuning of this threshold.
Figure 3: HMM with binary hidden states and observations. Total marginal error
computed for a sequence of length 200. For particle filtering the total error
for every ESS value is averaged over 5 sequences generated from the HMM; in
addition, for each sequence we reran the particle filter 5 times (thus 25 runs
total). Note the logarithmic scale of the x-axis. Error bars and the thin
black lines correspond to standard error of the mean.
In contrast, DPVI achieves performance comparable to the optimal particle
filter, but without a tunable threshold. This occurs because DPVI uses an
implicit threshold that is automatically tuned to the problem. Instead of
resampling particles, DPVI deletes or propagates particles deterministically
based on their relative contribution to the variational bound.
### 6.2 Dirichlet process mixture model
A DPMM generates data from the following process (Antoniak, 1974; Escobar and
West, 1995):
$\displaystyle G\sim\mbox{DP}(\alpha,G_{0}),\quad\quad\theta_{n}|G\sim
G,\quad\quad y_{n}|\theta_{n}\sim F(\theta_{n}),$
where $\alpha\geq 0$ is a concentration parameter and $G_{0}$ is a base
distribution over the parameter $\theta_{n}$ of the observation distribution
$F(y_{n}|\theta_{n})$. Since the Dirichlet process induces clustering of the
parameters $\theta$ into $K$ distinct values, we can equivalently express this
model in terms of a distribution over cluster assignments,
$x_{n}\in\\{1,\ldots,C\\}$. The distribution over $x$ is given by the Chinese
restaurant process (Aldous, 1985):
$\displaystyle P(x_{n}=c|x_{1:n-1})\propto\begin{cases}t_{c}&\text{if }k\leq
C_{+}\\\ \alpha&\text{if }c=C_{+}+1,\end{cases}$ (20)
where $t_{c}$ is the number of data points prior to $n$ assigned to cluster
$c$ and $C_{+}$ is the number of clusters for which $t_{c}>0$.
#### 6.2.1 Synthetic data
We first demonstrate our approach on synthetic datasets drawn from various
mixtures of bivariate Gaussians (see Table 1). The model parameters for each
simulated dataset were chosen to create a spectrum of increasingly overlapping
clusters. In particular, we constructed models out of the following building
blocks:
$\displaystyle\mu_{1}=\bigl{(}\begin{smallmatrix}0.0,&0.0\end{smallmatrix}\bigr{)},\quad\quad\mu_{2}=\bigl{(}\begin{smallmatrix}0.5,&0.5\end{smallmatrix}\bigr{)}$
$\displaystyle\Sigma_{1}=\bigl{(}\begin{smallmatrix}0.25,&0.0\\\
0.0,&0.25\end{smallmatrix}\bigr{)},\quad\quad\Sigma_{2}=\bigl{(}\begin{smallmatrix}0.5,&0.0\\\
0.0,&0.5\end{smallmatrix}\bigr{)}.$
For the DPMM, we used a Normal likelihood with a Normal-Inverse-Gamma prior on
the component parameters:
$\displaystyle
y_{nd}|x_{n}=k\sim\mathcal{N}(m_{kd},\sigma_{kd}^{2}),\quad\quad
m_{kd}\sim\mathcal{N}(0,\sigma_{kd}^{2}/\tau),\quad\quad\sigma_{kd}^{2}\sim\text{IG}(a,b),$
(21)
where $d\in\\{1,2\\}$ indexes observation dimensions and $\text{IG}(a,b)$
denotes the Inverse Gamma distribution with shape $a$ and scale $b$. We used
the following hyperparameter values: $\tau=25,a=1,b=1,\alpha=0.5$.
Dataset | Particle Filtering ($K=20$) | DPVI ($K=1$) | DPVI ($K=20$)
---|---|---|---
D1: $[\mu_{1},4\mu_{2},8\mu_{2}],\Sigma_{1}$ | 0.97$\pm$0.03 | 0.93$\pm$0.05 | 0.99$\pm$0.02
D2: $[\mu_{1},4\mu_{2},8\mu_{2}],\Sigma_{2}$ | 0.89$\pm$0.05 | 0.86$\pm$0.07 | 0.90$\pm$0.03
D3: $[\mu_{1},2\mu_{2},4\mu_{2}],\Sigma_{1}$ | 0.58$\pm$0.12 | 0.51$\pm$0.03 | 0.74$\pm$0.16
D4: $[\mu_{1},2\mu_{2},4\mu_{2}],\Sigma_{2}$ | 0.50$\pm$0.06 | 0.46$\pm$0.05 | 0.55$\pm$0.07
D5: $[\mu_{1},\mu_{2},2\mu_{2}],\Sigma_{1}$ | 0.05$\pm$0.05 | 0.014$\pm$0.02 | 0.14$\pm$0.10
D6: $[\mu_{1},\mu_{2},2\mu_{2}],\Sigma_{2}$ | 0.15$\pm$0.08 | 0.11$\pm$0.06 | 0.19$\pm$0.07
Table 1: Clustering accuracy (V-Measure) for DPMM. Each dataset consisted of
200 points drawn from a mixture of 3 Gaussians. For each dataset, we repeated
the experiment 150 times by iterating through random seeds. The left column
shows the ground truth mean for each cluster and the covariance matrix (shared
across clusters).
Clustering accuracy was measured quantitatively using V-measure (Rosenberg and
Hirschberg, 2007). Figure 4 graphically demonstrates the discovery of latent
clusters for both DPVI as well as particle filtering. As shown in Table 1, we
observe only marginal improvements when the means are farthest from each other
and variances are small, as these parameters leads to well-separated clusters
in the training set. However, the relative accuracy of DPVI increases
considerably when the clusters are overlapping, either due to the fact that
the means are close to each other or the variances are high.
An interesting special case is when $K=1$. In this case, DPVI is equivalent to
the greedy algorithm proposed by Daume (2007) and later extended by Wang and
Dunson (2011). In fact, this algorithm was independently proposed in cognitive
psychology by Anderson (1991). As shown in Table 1, DPVI with 20 particles
outperforms the greedy algorithm, as well as particle filtering with 20
particles.
Ground Truth Particle Filter DPVI
(D1)
(D2)
(D3)
(D4)
(D5)
(D6)
Figure 4: DPMM clustering of synthetic datasets. We treat DPMM as a filtering
problem, analyzing one randomly chosen data point at a time. Colors indicate
cluster assignments. Each row corresponds to one synthetic dataset; refer to
Table 1 for corresponding quantitative results. Column 1: Ground truth; Column
2: particle filtering; Column 3: DPVI. The DPVI filter scales similarly to the
particle filter but does not underfit as severely.
#### 6.2.2 Spike sorting
Spike sorting is an important problem in experimental neuroscience settings
where researchers collect large amounts of electrophysiological data from
multi-channel tetrodes. The goal is to extract from noisy spike recordings
attributes such as the number of neurons, and cluster spikes belonging to the
same neuron. This problem naturally motivates the use of DPMM, since the
number of neurons recorded by a single tetrode is unknown. Previously, Wood
and Black (2008) applied the DPMM to spike sorting using particle filtering
and Gibbs sampling. Here we show that DPVI can outperform particle filtering,
achieving high accuracy even with a small number of particles.
We used data collected from a multiunit recording from a human epileptic
patient (Quiroga et al., 2004). The raw spike recordings were preprocessed
following the procedure proposed by Quiroga et al. (2004), though we note that
our inference algorithm is agnostic to the choice of preprocessing. The
original data consist of an input vector with $D=10$ dimensions and 9196 data
points. Following Wood and Black (2008), we used a Normal likelihood with a
Normal-Inverse-Wishart prior on the component parameters:
$\displaystyle\mathbf{y}_{n}|x_{n}=k\sim\mathcal{N}(\mathbf{m}_{k},\Lambda_{k}),\quad\quad\mathbf{m}_{k}\sim\mathcal{N}(0,\Lambda_{k}/\tau),\quad\quad\Lambda_{k}\sim\text{IW}(\Lambda_{0},\nu),$
(22)
where $\text{IW}(\Lambda_{0},\nu)$ denotes the Inverse Wishart distribution
with degrees of freedom $\nu$ and scale matrix $\Lambda_{0}$. We used the
following hyperparameter values:
$\nu=D+1,\Lambda_{0}=\mathbf{I},\tau=0.01,\alpha=0.1$.
We compared our algorithm to the current best particle filtering baseline,
which uses stratified resampling (Wood and Black, 2008; Fearnhead, 2004). The
same model parameters were used for all comparisons. Qualitative results,
shown in Figure 5, demonstrate that DPVI is better able to separate the spike
waveforms into distinct clusters, despite running DPVI with 10 particles and
particle filtering with 100 particles. We also provide quantitative results by
calculating the held-out log-likelihood on an independent test set of spike
waveforms. The quantitative results (summarized in Table 2) demonstrate that
even with only 10 particles DPVI can outperform particle filtering with $1000$
particles.
(A) (B)
Figure 5: Spike Sorting using the DPMM. Each line is an individual spike waveform, colored according to the inferred cluster. (_A_) Result using particle filtering with 100 particles and stratified resampling as reported in Wood and Black (2008). (_B_) Result using DPVI. The same model parameters were used for both particle filtering and DPVI. Method | Held-out log-likelihood
---|---
DPVI ($K=10$) | -3.2474$\times 10^{5}$ ($\hat{C}=3$)
DPVI ($K=100$) | -1.3888$\mathbf{\times 10^{5}}$ ($\mathbf{\hat{C}=3}$)
Particle Filtering (Stratified) ($K=10$) | -1.4771$\pm 0.21\times 10^{6}$ ($\hat{C}=37$)
Particle Filtering (Stratified) ($K=100$) | -5.6757$\pm 1.14\times 10^{5}$ ($\hat{C}=13$)
Particle Filtering (Stratified) ($K=1000$) | -3.2965$\times 10^{5}$ ($\hat{C}=5$)
Table 2: Spike sorting held-out log-likelihood scores for 200 test points. The
best performance is achieved by DPVI with 100 particles. Shown in parentheses
is the _maximum a posteriori_ number of clusters, $\hat{C}$.
(A) Number of particles DPVI Particle Filtering $K=10$ 15.20s 14.71s $K=50$
153.75s 184.17s $K=100$ 567.84s 699.43s (B) Number of particles DPVI Particle
Filtering $K=10$ 36.20s 124s $K=50$ 144.6s 334.2s $K=100$ 313.8s 454.2s
Table 3: Run time comparison for DPMM. (_A_) Results using synthetic DPMM
dataset from Table 1 and (_B_) highlights results obtained by using the spike
sorting dataset. In both cases, the run time of DPVI is slightly better than
particle filtering.
### 6.3 Infinite HMM
An iHMM generates data from the following process (Teh et al., 2006):
$\displaystyle G_{0}\sim\mbox{DP}(\gamma,H),\quad\quad
G_{k}|G_{0}\sim\mbox{DP}(\alpha,G_{0}),$ $\displaystyle x_{n}|x_{n-1}\sim
G_{x_{n-1}},\quad\quad\theta_{k}\sim H,\quad\quad y_{n}|x_{n}\sim
F(\theta_{x_{n}}).$
Like the DPMM, the iHMM induces a sequence of cluster assignments. The
distribution over cluster assignments is given by the Chinese restaurant
franchise (Teh et al., 2006). Letting $t_{jc}$ denote the number of times
cluster $j$ transitioned to cluster $c$, $x_{n}$ is assigned to cluster $c$
with probability proportional to $t_{x_{n-1}c}$, or to a cluster never visited
from $x_{n-1}$ ($t_{x_{n-1}c}=0$) with probability proportional to $\alpha$.
If an unvisited cluster is selected, $x_{n}$ is assigned to cluster $c$ with
probability proportional to $\sum_{j}t_{jc}$, or to a new cluster (i.e., one
never visited from any state, $\sum_{j}t_{jc}=0$) with probability
proportional to $\gamma$.
(A) (B)
Figure 6: Infinite HMM results. (_A_) Results on 500 synthetic data points
generated from an HMM with 10 hidden states. Error is the Hamming distance
between the true hidden sequence and the sampled sequence, averaged over 50
datasets. M: multinomial resampling; S: stratified resampling. Lower bound is
the expected Hamming distance between data-generating distribution and ground
truth. Upper bound is the expected Hamming distance between uniform
distribution and ground truth. (_B_) Predictive log-likelihood for the “Alice
in Wonderland” dataset. Particle filtering (M) and (S) overlap in the figure.
The error bars in both parts show standard error.
#### 6.3.1 Synthetic data
We generated 50 sequences with length 500 from 50 different HMMs, each with 10
hidden and 5 observed states. For the rows of the transition and initial
probability matrices of the HMMs we used a symmetric Dirichlet prior with
concentration parameter 0.1; for the emission probability matrix, we used a
symmetric Dirichlet prior with concentration parameter 10.
Figure 6A illustrates the performance of DPVI and particle filtering (with
multinomial and stratified resampling) for varying numbers of particles
($K=1,10,100$). Performance error was quantified by computing the Hamming
distance between the true hidden sequence and the sampled sequence. The
Munkres algorithm was used to maximize the overlap between the two sequences.
The results show that DPVI outperforms particle filtering in all three cases.
When the data consist of long sequences, resampling at every step will produce
degeneracy in particle filtering; this tends to result in a smaller number of
clusters relative to DPVI. The superior accuracy of DPVI suggests that a
larger number of clusters is necessary to capture the latent structure of the
data. Not surprisingly, this leads to longer run times (Table 4), but it is
important to note that particle filtering and DPVI have comparable per-cluster
time complexity.
(A) Number of particles DPVI Particle Filtering $K=1$ 1.28 1.14s $K=10$ 3.56s
1.92s $K=100$ 204.42s 31.99s (B) Number of particles DPVI Particle Filtering
$K=1$ 4.73s 1.64s $K=10$ 41.62s 28.08s $K=100$ 1685s 211.66s
Table 4: Run time comparison for iHMM. (_A_) Results using the synthetic iHMM
dataset from Figure 5A and (_B_) results using the “Alice in Wonderland”
dataset.
#### 6.3.2 Text analysis
We next analyzed a real-world dataset, text taken from the beginning of “Alice
in Wonderland”, with 31 observation symbols (letters). We used the first 1000
characters for training, and the subsequent 4000 characters for test.
Performance was measured by calculating the predictive log-likelihood. We
fixed the hyperparameters $\alpha$ and $\gamma$ to 1 for both DPVI and the
particle filtering.
We ran one pass of DPVI (filtering) and particle filtering over the training
sequence. We then sampled 50 datasets from the distribution over the
sequences. We truncated the number of states and used the learned transition
and emission matrices to compute the predictive log-likelihood of the test
sequence. To handle the unobserved emissions in the test sequence we used
“add-$\delta$” smoothing with $\delta=1$. Finally, we averaged over all the 50
datasets.
We also compared DPVI to the beam sampler (Van Gael et al., 2008), a
combination of dynamic programming and slice sampling, which was previously
applied to this dataset. For the beam sampler, we followed the setting of Van
Gael et al. (2008). We run the sampler for 10000 iterations and collect a
sample of hidden state sequence every 200 iterations. Figure 6B shows the
predictive log-likelihood for varying numbers of particles. Even with a small
number of particles, DPVI can outperform both particle filtering and the beam
sampler.
### 6.4 Infinite relational model (IRM)
The IRM (Kemp et al., 2006) is a nonparametric model of relational systems.
The model simultaneously discovers the clusters of entities and the
relationships between the clusters. A key assumption of the model is that each
entity belongs to exactly one cluster.
Given a relation $R$ involving $J$ types of entities, the goal is to infer a
vector of cluster assignments $x^{j}$ for all the entities of each type
$j=1,\ldots,J$.222The IRM model can be defined for multiple relations but for
simplicity we only describe the single relation case. Assuming the cluster
assignments for each type are independent, the joint density of the relation
and the cluster assignment vectors can be written as:
$\displaystyle
P(R,x^{1},\ldots,x^{J})=P(R|x^{1},\ldots,x^{J})\prod_{j=1}^{J}P(x^{j}).$ (23)
The cluster assignment vectors are drawn from a $\mbox{CRP}(\alpha)$ prior.
Given the cluster assignment vectors, the relations are drawn from a Bernoulli
distribution with a parameter that depends on the clusters involved in that
relation. More formally, let us define a binary relation
$R:T^{d_{1}}\times\dots T^{d_{M}}\mapsto\\{0,1\\}$, where $d_{m}$ is the label
of the type occupying position $m$ in the relation. Each relational value is
generated according to:
$\displaystyle
R(i_{1},\ldots,i_{M})|x^{1},\ldots,x^{J}\sim\text{Bernoulli}(\eta(x_{i_{1}}^{d_{1}},\ldots,x_{i_{M}}^{d_{M}})),$
(24)
where $i_{m}$ denotes the entity (of type $d_{m}$) occupying position $m$.
Each entry of $\eta$ is drawn from a Beta($\beta,\beta$) distribution. By
using a conjugate Beta-Bernoulli model, we can analytically marginalize the
parameters $\eta$ (see Kemp et al., 2006), allowing us to directly compute the
likelihood of the relational matrix given the cluster assignments,
$P(R|x^{1},\ldots,x^{J})$.
We compared the performance of DPVI with Gibbs sampling, using predictive log-
likelihood on held-out data as a performance metric. Two datasets analyzed in
Kemp et al. (2006), “animals” and “Alyawarra”, were used for this task. The
animals dataset (Osherson et al., 1991) is a two type dataset $R:T_{1}\times
T_{2}\to\\{0,1\\}$ with animals and features as it types; it contains 50
animals and 85 features. The Alyawarra dataset (Denham, 1973) has a ternary
relation $R:T_{1}\times T_{1}\times T_{2}\to\\{0,1\\}$ where $T_{1}$ is the
set of 104 people and $T_{2}$ is the set of 25 kinship terms.
We removed 20% of the relations form each dataset and computed the predictive
log-likelihood for the held-out data. We ran DPVI with 1, 10 and 20 particles
for 100 iterations. Given the weights of the particles, we computed the
weighted log-likelihood. We also ran 20 independent runs of the Gibbs sampler
for 100 iterations and computed the average predictive log-likelihood. Every
iteration scans all the data points in all the types sequentially. We set the
hyperparameters $\alpha$ and $\beta$ to 1. Figure 7 illustrates the co-
clustering discovered by DPVI for the animals dataset, demonstrating
intuitively reasonable animal and feature clusters.
| Sample animal clusters:
---|---
A1: Hippopotamus, Elephant, Rhinoceros
A2: Seal, Walrus, Dolphins, Blue Whale,
Killer Whale, Humpback Whale
A3: Beaver, Otter, Polar Bear
Sample feature clusters:
F1: Hooves, Long neck, Horns
| F2: Inactive, Slow, Bulbous Body, Tough Skin
| F3: Lives in Fields, Lives in Plains, Grazer
| F4: Walks, Quadrupedal, Ground
| F5: Fast, Agility, Active, Tail
Figure 7: Co-clustering of animals (rows) and features (columns) after 50
iterations of DPVI with 10 particles.
The results after 100 iterations are presented in Table 5. The best
performance is achieved by DPVI with 20 particles. Figure 8 shows the
predictive log-likelihood for every iteration of DPVI and Gibbs sampling. For
the animals dataset, DPVI with 10 and 20 particles converge in 11 and 18
iterations, respectively. The number of iterations required for convergence in
the Alyawarra dataset is just 2 and 3 for 10 and 20 particles, respectively.
In terms of computation time per iteration of DPVI versus Gibbs, the only
difference for DPVI with one particle and Gibbs is the sorting cost. Hence,
for the multiple particle versus multiple runs of Gibbs sampling, the only
additional cost is the sorting cost for multiple particles (e.g. 10 or 20).
However, this insignificant additional cost is compensated for by a faster
convergence rate in our experiments.
| Predicitive log-likelihood
---|---
Method | Animals | Alyawarra
DPVI ($K=1$) | -418.498 | -8.452 $\times 10^{3}$
DPVI ($K=10$) | -382.543 | -8.450 $\times 10^{3}$
DPVI ($K=20$) | -370.674 | -8.450 $\times 10^{3}$
Gibbs (avg. of $20$ runs) | -374.986 | -8.453 $\times 10^{3}$
Table 5: Predictive log-likelihood after 100 iterations of DPVI and Gibbs for the animals and Alyawarra datasets (with 20 % held-out). The best performance is achieved by DPVI with 20 particles. |
---|---
(A) | (B)
Figure 8: Predictive log-likelihood vs iteration for (_A_) Animals and (_B_)
Alyawarra datasets. For DPVI the predictive log-likelihood is the weighted
average across all the particles. For Gibbs sampling the bold line corresponds
to the mean across samples, and the error bars correspond to the standard
error.
### 6.5 Ising model
So far, we have been studying inference in directed graphical models, but DPVI
can also be applied to undirected graphical models. We illustrate this using
the Ising model for binary vectors $x\in\\{-1,+1\\}^{N}$:
$\displaystyle f(x)=\frac{1}{2}xWx^{\top}+\theta x^{\top},$ (25)
where $W\in\mathbb{R}^{N\times N}$ and $\theta\in\mathbb{R}^{N}$ are fixed
parameters. In particular, we study a square lattice ferromagnet, where
$W_{ij}=\beta$ for neighboring nodes (0 otherwise) and $\theta_{i}=0$ for all
nodes. We refer to $\beta$ as the _coupling strength_. This model has two
global modes: when all the nodes are set to 1, and when all the nodes are set
to 0. As the coupling strength increases, the probability mass becomes
increasingly concentrated at the two modes.
We applied DPVI to this model, varying the number of particles and the
coupling strength. To quantify performance, we computed the DPVI variational
lower bound on the partition function and compared this to the lower bound
furnished by the mean-field approximation (see Wainwright and Jordan, 2008).
Figure 9A shows the results of this analysis for low coupling strength
($\beta=0.01$) and high coupling strength ($\beta=100$). DPVI consistently
achieves a better lower bound than mean-field, even with a single particle,
and this advantage is especially conspicuous for high coupling strength.
Adding more particles improves the results, but more than 3 particles does not
appear to confer any additional improvement for high coupling strength. These
results illustrate how DPVI is able to capture multimodal target
distributions, where mean-field approximations break down (since they cannot
effectively handle multimodality).
Figure 9: Ising model results. Difference between DPVI and mean-field lower
bounds on the partition function. Positive values indicates superior DPVI
performance. (_A_) Low coupling strength; (_B_) high coupling strength.
To illustrate the performance of DPVI further, we compared several posterior
approximations for the Ising model in Figure 10. In addition to the mean-field
approximation, we also compared DPVI with two other standard approximations:
the Swendsen-Wang Monte Carlo sampler (Swendsen and Wang, 1987) and loopy
belief propagation (Murphy et al., 1999). The sampler tended to produce noisy
results, whereas mean-field and BP both failed to capture the multimodal
structure of the posterior. In contrast, DPVI with two particles perfectly
captured the two modes.
Figure 10: Ising model simulations. Examples of posteriors for the
ferromagnetic lattice at low coupling strength. (_Top_) Two configurations
from a Swendsen-Wang sampler. (_Middle_) Two DPVI particles. (_Bottom left_)
Mean-field expected value. (_Bottom right_) Loopy belief propagation expected
value.
## 7 Conclusions
This paper introduced a variational framework for particle approximations of
discrete probability distributions. We described a practical algorithm for
optimizing the approximation, and showed empirically that it can outperform
widely-used Monte Carlo and variational algorithms. The key to the success of
this approach is an optimal selection of particles: Rather than generating
them randomly (as in Monte Carlo algorithms), we deterministically choose a
set of unique particles that optimizes the KL divergence between the
approximation and the target distribution. Because we are selecting particles
optimally, we can achieve good performance with a smaller number of particles
compared to Monte Carlo algorithms, thereby improving computational
efficiency. Another advantage of DPVI is that its deterministic nature
eliminates the contribution of Monte Carlo variance to estimation error.
A consistent problem vexing sequential Monte Carlo methods like particle
filtering is the double-edged sword of resampling: this step is necessary to
remove conditionally unlikely particles, but the resulting loss of particle
diversity can lead to degeneracy. As we showed in our experiments, tuning an
ESS threshold for resampling can improve performance, but requires finding a
relatively narrow sweet spot for the threshold. DPVI achieves comparable
performance to the best particle filter by using a deterministic strategy for
deleting and replacing particles, avoiding finicky tuning parameters. It is
also worth noting two other desirable properties of DPVI in this context: (1)
the particle set is guaranteed to be diverse because all particles are unique;
(2) all the particles have high probability and therefore the propagation of
conditionally unlikely particles is avoided, as happens when particle
filtering is run without resampling. We believe that this combination of
properties is a key to the superior performance of DPVI relative to particle
filtering.
An important task for future work is to consider how DPVI can be efficiently
applied to models with combinatorial latent structure (such as the factorial
HMM), which may have too many assignments to enumerate completely. In this
setting, it is desirable to use a proposal distribution to selectively sample
certain assignments. An interesting possibility is to use randomly seeded
optimization algorithms to generate high probability proposals. Since the
proposal mechanism does not play any role in the score function (unlike in
particle filtering, where samples have to be reweighted), we are free to
choose any deterministic or stochastic proposal mechanism without needing to
evaluate its probability density function.
In summary, DPVI harmoniously combines a number of ideas from Monte Carlo and
variational methods. The resulting algorithm can achieve performance superior
to widely used particle filtering, MCMC and mean-field methods, though more
work is needed to evaluate its performance on a wider range of probabilistic
models and to compare it to other inference algorithms.
## Acknowledgments
TDK is generously supported by the Leventhal Fellowship. VKM is supported by
the Army Research Office Contract Number 0010363131, Office of Naval Research
Award N000141310333, and the DARPA PPAML program. The views and conclusions
contained herein are those of the authors and should not be interpreted as
necessarily representing the official policies or endorsements, either
expressed or implied, of the U.S. Government. The U.S. Government is
authorized to reproduce and distribute reprints for Governmental purposes
notwithstanding any copyright annotation thereon.
## References
* Aldous (1985) David Aldous. Exchangeability and related topics. _École d’Été de Probabilités de Saint-Flour XIII 1983_ , pages 1–198, 1985.
* Anderson (1991) John R Anderson. The adaptive nature of human categorization. _Psychological Review_ , 98:409–429, 1991.
* Antoniak (1974) Charles E Antoniak. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. _The Annals of Statistics_ , pages 1152–1174, 1974.
* Beal et al. (2002) Matthew J Beal, Zoubin Ghahramani, and Carl E Rasmussen. The infinite hidden Markov model. In _Advances in Neural Information Processing Systems_ , pages 577–584, 2002.
* Besag (1986) Julian Besag. On the statistical analysis of dirty pictures. _Journal of the Royal Statistical Society. Series B (Methodological)_ , 48:259–302, 1986.
* Daume (2007) Hal Daume. Fast search for Dirichlet process mixture models. In _International Conference on Artificial Intelligence and Statistics_ , pages 83–90, 2007.
* Denham (1973) Woodrow W Denham. _The detection of patterns in Alyawara nonverbal behavior_. PhD thesis, University of Washington, 1973.
* Doucet et al. (2001) Arnaud Doucet, Nando De Freitas, Neil Gordon, et al. _Sequential Monte Carlo methods in practice_. Springer New York, 2001.
* Escobar and West (1995) Michael D Escobar and Mike West. Bayesian density estimation and inference using mixtures. _Journal of the American Statistical Association_ , 90:577–588, 1995.
* Fearnhead (2004) Paul Fearnhead. Particle filters for mixture models with an unknown number of components. _Statistics and Computing_ , 14:11–21, 2004.
* Flerova et al. (2012) Natalia Flerova, Emma Rollon, and Rina Dechter. Bucket and mini-bucket schemes for m best solutions over graphical models. In _Graph Structures for Knowledge Representation and Reasoning_ , pages 91–118. Springer, 2012.
* Frank et al. (2009) Andrew Frank, Padhraic Smyth, and Alexander T Ihler. Particle-based variational inference for continuous systems. In _Advances in Neural Information Processing Systems_ , 2009.
* Gershman et al. (2012) Samuel Gershman, Matt Hoffman, and David Blei. Nonparametric variational inference. _Proceedings of the 29th International Conference on Machine Learning_ , 2012.
* Ionides (2008) Edward L Ionides. Truncated importance sampling. _Journal of Computational and Graphical Statistics_ , 17:295–311, 2008.
* Jaakkola and Jordan (1998) Tommi S Jaakkola and Michael I Jordan. Improving the mean field approximation via the use of mixture distributions. In MI Jordan, editor, _Learning in Graphical Models_ , pages 163–173. Springer, 1998.
* Kemp et al. (2006) Charles Kemp, Joshua B Tenenbaum, Thomas L Griffiths, Takeshi Yamada, and Naonori Ueda. Learning systems of concepts with an infinite relational model. In _AAAI_ , volume 3, page 5, 2006.
* Lawrence (2000) Neil D Lawrence. _Variational inference in probabilistic models_. PhD thesis, University of Cambridge, 2000.
* Murphy et al. (1999) Kevin P Murphy, Yair Weiss, and Michael I Jordan. Loopy belief propagation for approximate inference: An empirical study. In _Proceedings of the Fifteenth conference on Uncertainty in artificial intelligence_ , pages 467–475. Morgan Kaufmann Publishers Inc., 1999\.
* Osherson et al. (1991) Daniel N Osherson, Joshua Stern, Ormond Wilkie, Michael Stob, and Edward E Smith. Default probability. _Cognitive Science_ , 15:251–269, 1991.
* Quiroga et al. (2004) R Quian Quiroga, Z Nadasdy, and Y Ben-Shaul. Unsupervised spike detection and sorting with wavelets and superparamagnetic clustering. _Neural computation_ , 16:1661–1687, 2004.
* Robert and Casella (2004) Christian P Robert and George Casella. _Monte Carlo Statistical Methods_. Springer, 2004.
* Rosenberg and Hirschberg (2007) Andrew Rosenberg and Julia Hirschberg. V-measure: A conditional entropy-based external cluster evaluation measure. In _EMNLP-CoNLL_ , volume 7, pages 410–420, 2007.
* Swendsen and Wang (1987) Robert H Swendsen and Jian-Sheng Wang. Nonuniversal critical dynamics in Monte Carlo simulations. _Physical Review Letters_ , 58:86–88, 1987.
* Teh et al. (2006) Yee Whye Teh, Michael I Jordan, Matthew J Beal, and David M Blei. Hierarchical Dirichlet processes. _Journal of the American Statistical Association_ , 101:1566–1581, 2006.
* Van Gael et al. (2008) Jurgen Van Gael, Yunus Saatci, Yee Whye Teh, and Zoubin Ghahramani. Beam sampling for the infinite hidden Markov model. In _Proceedings of the 25th international conference on Machine learning_ , pages 1088–1095. ACM, 2008.
* Wainwright and Jordan (2008) Martin J Wainwright and Michael I Jordan. Graphical models, exponential families, and variational inference. _Foundations and Trends in Machine Learning_ , 1:1–305, 2008.
* Wang and Dunson (2011) Lianming Wang and David B Dunson. Fast Bayesian inference in Dirichlet process mixture models. _Journal of Computational and Graphical Statistics_ , 20, 2011.
* Wood and Black (2008) Frank Wood and Michael J Black. A nonparametric Bayesian alternative to spike sorting. _Journal of Neuroscience Methods_ , 173:1–12, 2008.
* Yanover and Weiss (2003) Chen Yanover and Yair Weiss. Finding the M most probable configurations in arbitrary graphical models. In _Advances in Neural Information Processing Systems_ , page None, 2003.
|
arxiv-papers
| 2014-02-24T03:58:16 |
2024-09-04T02:49:58.681180
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Ardavan Saeedi, Tejas D Kulkarni, Vikash Mansinghka, Samuel Gershman",
"submitter": "Tejas Kulkarni",
"url": "https://arxiv.org/abs/1402.5715"
}
|
1402.5725
|
# Some Applications of Generalized Mountain Pass Lemma
Fengying Li111Email:[email protected]
The School of Economic and Mathematics, Southwestern University of Finance and
Economics,
Chengdu 611130, China
Bingyu Li and Shiqing Zhang
College of Mathematics, Sichuan University, Chengdu 610064, People’s Republic
of China
###### Abstract
The Ghoussoub-Preiss’s generalized Mountain Pass Lemma with Cerami-Palais-
Smale type condition is a generalization of classical MPL of Ambrosetti-
Rabinowitz, we apply it to study the existence of the periodic solutions with
a given energy for some second order Hamiltonian systems with symmetrical and
non-symmetrical potentials.
Key Words: Second order Hamiltonian systems, periodic solutions, Ghoussoub-
Preiss’s Generalized Mountain Pass Lemma, Cerami-Palais-Smale condition at
some levels for a closed subset.
2000 Mathematical Subject Classification: 34C15, 34C25, 58F.
## 1\. Introduction and Main Results
In 1948, Seifert([17]) studied the periodic solutions of the Hamiltonian
systems using geometrical and topological methods; in 1978 and 1979,
Rabinowitz([15,16])studied the periodic solutions of the Hamiltonian systems
using global variational methods; in 1980’s, Benci ([4])and Gluck-Ziller([9])
and Hayashi([11]) used Jacobi metric and very complicated geodesic methods and
algebraic topology to study the periodic solutions for second order
Hamiltonian systems with a fixed energy:
$\displaystyle\ddot{q}+V^{\prime}(q)=0$ (1.1)
$\displaystyle\frac{1}{2}|\dot{q}|^{2}+V(q)=h$ (1.2)
They proved the following very general theorem:
###### Theorem 1.1
Suppose $V\in C^{1}(R^{n},R)$ ,if
$\\{x\in R^{n}|V(x)\leq h\\}$
is bounded, and
$V^{\prime}(x)\not=0,\ \ \ \ \forall x\in\\{x\in R^{n}|V(x)=h\\},$
then the (1.1)-(1.2) has a periodic solution with energy h.
For the existence of multiple periodic solutions for (1.1)-(1.2), we can refer
Groessen([10]) and Long [12] and the references there.
Ambrosetti–Coti Zelati([1]) used Ljusternik-Schnirelmann theory with classical
$(PS)^{+}$ compact condition to get the following Theorem:
###### Theorem 1.2
Suppose $V\in C^{2}(\mathbb{R}^{n}\backslash\\{0\\},\mathbb{R})$ satisfies:
$(A1)$. $3V^{\prime}(x)\cdot{x}+V^{\prime\prime}(x)x\cdot{x}\neq
0,\>\forall\,x\in\Omega=\mathbb{R}^{n}\backslash\\{0\\}$;
$(A2)$. $V^{\prime}(x)\cdot{x}>0,\quad\forall\,x\in\Omega$;
$(A3^{\prime})$. $\exists\,\alpha\in(0,2)$, such that
$V^{\prime}(x)\cdot{x}\geq-\alpha V(x),\quad\forall\>x\in\Omega;$
$(A4^{\prime})$. $\exists\,\delta\in(0,2)$ and $r>0$, such that
$V^{\prime}(x)\cdot{x}\leq-\delta V(x),\quad\forall\>0<|x|\leq r;$
$(A5^{\prime})$.
$\underset{|x|\rightarrow+\infty}{\liminf}\left[V(x)+\dfrac{1}{2}V^{\prime}(x)\cdot{x}\right]\geq
0$.
Then $\forall\,h<0$ the system (1.1)-(1.2) has at least a non-constant weak
periodic solution which satisfies (1.1)-(1.2) pointwise except on a zero-
measurable set.
Ambrosetti-Coti Zelati ([2]) used a variant of the classical Mountain-Pass
Lemma and a constraint minimizing method to get the following Theorems:
###### Theorem 1.3
Suppose $V\in C^{1}(\mathbb{R}^{n}\backslash\\{0\\},\mathbb{R})$ satisfies:
$(V1)$.
$V(-\xi)=V(\xi),\>\forall\,\xi\in\Omega=\mathbb{R}^{n}\backslash\\{0\\}$;
$(V2)$. $\exists\,\alpha\in[1,2)$, such that
$\nabla V(\xi)\cdot{\xi}\geq-\alpha V(\xi)>0,\quad\forall\,\xi\in\Omega;$
$(V3)$. $\exists\,\delta\in(0,2)$ and $r>0$, such that
$\nabla V(\xi)\cdot{\xi}\leq-\delta V(\xi),\quad\forall\>0<|\xi|\leq r;$
$(V4)$. $V(\xi)\rightarrow 0$, as $|\xi|\rightarrow+\infty$.
Then $\forall\,h<0$, the problem $(1.1)-(1.2)$ has a weak periodic solution.
###### Theorem 1.4
Suppose $V$ satisfies $(V1),(V3),(V4)$ and
$(V2^{\prime})$. $\exists\,\alpha\in(0,2)$, such that
$\nabla V(\xi)\cdot{\xi}\geq-\alpha V(\xi)>0,\quad\forall\,\xi\in\Omega;$
$(V5)$. $V\in C^{2}(\Omega,\mathbb{R})$ and
$3\nabla V(\xi)\cdot{\xi}+V^{\prime\prime}(\xi)\xi\cdot{\xi}>0.$
Then $\forall h<0,(1.1)-(1.2)$ has a weak periodic solution.
Yuan-Zhang([19]) proved the following Theorem:
###### Theorem 1.5
Suppose $V\in C^{1}(\mathbb{R}^{n}\backslash\\{0\\},\mathbb{R})$ satisfies:
$(V_{1})$. $V(-q)=V(q);$
$(V_{2})$. There are constant $0<\alpha<2$ such that
$\langle V^{\prime}(q),q\rangle\geqslant-\alpha
V(q)>0,\quad\forall\,q\in\mathbb{R}^{n}\backslash\\{0\\};$
$(V_{3})$. $\exists\,\delta\in(0,2),r>0$, such that
$\langle V^{\prime}(q),q\rangle\leqslant-\delta V(q),\quad\forall\,0<|q|\leq
r;$
$(V_{4})$. $V(q)\rightarrow 0$, as $|q|\rightarrow+\infty$.
Then for any given $h<0$, the system (1.1)-(1.2) has at least a non-constant
weak periodic solution which can be obtained by Mountain Pass Lemma.
Motivated by these papers ,we use Ghoussoub-Preiss’s Generalized Mountain Pass
Lemma with Cerami-Palais-Smale condition at some levels for a closed subset to
study the new periodic solutions with symmetrical and non-symmetrical
potentials, we obtain the following Theorems:
Theorem 1.6 Suppose $V\in C^{1}(R^{n},R)$ and $h\in R$ satisfies
$(B_{1})$ $V(-q)=V(q).$
$(B_{2})$ $\exists\mu_{1}>0,\mu_{2}\geq 0,s.t.$ $V^{\prime}(q)\cdot
q\geq\mu_{1}V(q)-\mu_{2}.$
$(B_{3})$ $V(q)\geq h,|q|\rightarrow+\infty.$
$(B_{4})$ $\forall q\not=0,3V^{\prime}(q)\cdot q+V^{\prime\prime}(q)q\cdot
q\not=0.$
Then for any $h>\frac{\mu_{2}}{\mu_{1}},$ $(1.1)-(1.2)$ has at least one non-
constant periodic solution with the given energy h, which can be obtained by
the generalized MPL method.
Corollary1.1 Suppose $a>0,\mu_{1}\geq 2,\mu_{2}\geq
0,V(q)=a|q|^{\mu_{1}}+\frac{\mu_{2}}{\mu_{1}}$, then the conditions of
Theorem1.1 hold and for any $h>\frac{\mu_{2}}{\mu_{1}}$ , $(1.1)-(1.2)$ has at
least two non-constant periodic solution with the given energy h.
Theorem 1.7 Suppose $V\in C^{1}(R^{n},R)$ and $h\in R$ satisfies ($B_{2}$),
($B_{3}$) and ($B_{5}$) $\exists r>0$, s.t.
$\inf_{u\in F}\int_{0}^{1}(h-V(u))dt>0,$
where
$F\triangleq\\{u\in H^{1}|\|\dot{u}\|_{L^{2}}=r\\}.$
Then $\forall h>\frac{\mu_{2}}{\mu_{1}}$, (1.1)-(1.2) has at least one non-
constant periodic solution with energy $h$.
## 2 A Few Lemmas
Define Sobolev space:
$H^{1}=W^{1,2}(R/TZ,R^{n})=\\{u:R\rightarrow R^{n},u\in L^{2},\dot{u}\in
L^{2},u(t+1)=u(t)\\}$
Then the standard $H^{1}$ norm is equivalent to
$\|u\|=\|u\|_{H^{1}}=\left(\int^{1}_{0}|\dot{u}|^{2}dt\right)^{1/2}+|\int_{0}^{1}u(t)dt|.$
Lemma 2.1([1,10]) Let
$f(u)=\frac{1}{2}\int^{1}_{0}|\dot{u}|^{2}dt\int^{1}_{0}(h-V(u))dt$ and
$\widetilde{u}\in H^{1}$ be such that $f^{\prime}(\widetilde{u})=0$ and
$f(\widetilde{u})>0$. Set
$\frac{1}{T^{2}}=\frac{\int^{1}_{0}(h-V(\widetilde{u}))dt}{\frac{1}{2}\int^{1}_{0}|\dot{\widetilde{u}}|^{2}dt}$
(2.1)
Then $\widetilde{q}(t)=\widetilde{u}(t/T)$ is a non-constant $T$-periodic
solution for (1.1)-(1.2).
By symmetry condition $(B_{1})$, similar to Ambrosetti-Coti Zelati[2], let
$E_{1}=\\{u\in H^{1}=W^{1,2}(R/Z,R^{n}),u(t+1/2)=-u(t)\\},$ $E_{2}=\\{u\in
H^{1}=W^{1,2}(R/Z,R^{n}),u(-t)=-u(t)\\}.$
By the symmetrical condition $(B_{1})$ and Palais’s symmetrical
principle([14]) or similar proof of [1,2],we have
Lemma 2.2 If $\bar{u}\in E_{i}$ is a critical point of $f(u)$ and
$f(\bar{u})>0$, then $\bar{q}(t)=\bar{u}(t/T)$ is a non-constant $T$-periodic
solution of (1.1)-(1.2).
Using the famous Ekeland’s variational principle, Ekeland proved
Lemma 2.3(Ekeland[7]) Let $X$ be a Banach space, $F\subset X$ be a closed
(weakly closed) subset. Suppose that $\Phi$ defined on $F$ is Gateaux-
differentiable and lower semi-continuous (or weakly lower semi-continuous) and
bounded from below. Then there is a sequence $x_{n}\subset F$ such that
$\Phi(x_{n})\rightarrow\inf_{F}\Phi$
$(1+\|x_{n}\|)\|\Phi^{{}^{\prime}}(x_{n})\|\rightarrow 0.$
Motivated by the paper of Cerami[6], Ekeland [7], Ghoussoub-Preiss[8]
presented a weaker compact condition than the classical $(CPS)_{c}$ condition:
Definition 2.1([7,8]) Let $X$ be a Banach space, $F\subset X$ be a closed
subset, let $\delta(x,F)$ denotes the distance of $x$ to the set $F$. Suppose
that $\Phi$ defined on $X$ is Gateaux-differentiable, if sequence
$\\{x_{n}\\}\subset X$ such that
$\delta(x_{n},F)\rightarrow 0,$ $\Phi(x_{n})\rightarrow c,$
$(1+\|x_{n}\|)\|\Phi^{{}^{\prime}}(x_{n})\|\rightarrow 0,$
then $\\{x_{n}\\}$ has a strongly convergent subsequence.
Then we call $f$ satisfies $(CPS)_{c,F}$ condition at the level $c$ for the
closed subset $F\subset X$, we denote it as $(CPS)_{c,F}$
We can give a weaker condition than $(CPS)_{c}$ condition:
Definition 2.2 Let $X$ be a Banach space. $F\subset X$ be a weakly closed
subset. Suppose that $\Phi$ defined on $X$ is Gateaux-differentiable, if
sequence $x_{n}$ such that
$\delta(x_{n},F)\rightarrow 0,$ $\Phi(x_{n})\rightarrow\gamma,$
$(1+\|x_{n}\|)\|\Phi^{{}^{\prime}}(x_{n})\|\rightarrow 0,$
then $\\{x_{n}\\}$ has a weakly convergent subsequence.
Then we call $f$ satisfies $(WCPS)_{c,F}$ condition.
Now by $\bf Lemma2.3$, it’s easy to prove
Lemma 2.4 Let $X$ be a Banach space,
(i). Let $F\subset X$ be a closed subset. Suppose that $\Phi$ defined on $X$
is Gateaux-differentiable and lower semi-continuous and bounded from below, if
$\Phi$ satisfies $(CPS)_{\inf\Phi,F}$ condition, then $\Phi$ attains its
infimum on $F$.
(ii). Let $F\subset X$ be a weakly closed subset. Suppose that $\Phi$ defined
on $F$ is Gateaux-differentiable and weakly lower semi-continuous and bounded
from below, if $\Phi$ satisfies $(WCPS)_{inf\Phi,F}$ condition, then $\Phi$
attains its infimum on $F$.
Definition 2.3([7,8]) Let $X$ be a Banach space, $F\subset X$ be a closed
subset. If $z_{0},z_{1}$ belong different disjoint connected components in
$X\backslash F$, then we call $F$ separates $z_{0}$ and $z_{1}$.
Motivated by the famous classical Mountain Pass Lemma of Ambrosetti-Rabinowitz
[3], Ghoussoub-Preiss[8] gave a generalized MPL:
Lemma 2.5 (Ghoussoub-Preiss’s generalized MPL [8],[7]) Let $X$ be a Banach
space.Suppose that $\Phi(u):X\rightarrow R$ is a continuous Gateaux-
differentiable function with $\Phi^{\prime}:X\rightarrow X^{*}$ norm-to-weak∗
continuous. Take two points $z_{0},z_{1}$ in $X$, and define
$\Gamma=\\{c\in C^{0}([0,1];X)|c(0)=z_{0},c(1)=z_{1}\\}$
$\gamma=\inf_{c\in\Gamma}\max_{0\leq t\leq 1}\Phi(c(t))$
Let $F\subset X$ be a closed subset separating $z_{0}$ and $z_{1}$. Assume
that
$\Phi(x)>\max\\{\Phi(z_{0}),\Phi(z_{1})\\},\forall x\in F,$
$\Phi$ satisfies condition $(CPS)_{\gamma,F}$ on the level $\gamma$ for the
set $F$. Then there is a critical point of $\Phi$ on the level $\gamma.$
## 3 The Proof of Theorem 1.6
We define weakly closed subsets of $H^{1}$:
$F=\\{u\in H^{1}|\int_{0}^{1}(V(u)+\frac{1}{2}V^{\prime}(u)u)dt=h\\}.$
$F_{i}=\\{u\in
E_{i}|\int_{0}^{1}(V(u)+\frac{1}{2}V^{\prime}(u)u)dt=h\\},i=1,2.$
Lemma 3.1 If $(B_{2})-(B_{4})$ hold,then $F,F_{1},F_{2}\not=\emptyset$.
Proof Similar to the proof of [1].Let $u\in H^{1},u\not=0$ be fixed. For
$a>0$,let
$g_{u}(a)=g(au)=\int_{0}^{1}[V(au)+\frac{1}{2}V^{\prime}(au)au]dt$
By $(B_{4})$,$\frac{d}{da}g_{u}(a)\not=0$,so $g_{u}$ is strictly monotone.
Notice that
$g_{u}(0)=g(0)=V(0)\leq\frac{\mu_{2}}{\mu_{1}}$
When $a$ is large,we use $(B_{2})-(B_{3})$ to have
$g_{u}(a)=g(au)=\int_{0}^{1}[V(au)+\frac{1}{2}V^{\prime}(au)au]dt$
$\geq(1+\frac{\mu_{1}}{2})\int_{0}^{1}V(au)dt-\frac{\mu_{1}}{2}$
$\geq(1+\frac{\mu_{1}}{2})h-\frac{\mu_{1}}{2}$
Hence $\forall h>\frac{\mu_{2}}{\mu_{1}}$, we have
$g_{u}(+\infty)=g(+\infty)>h$
So for any given $u\in H^{1},u\not=0$,there is $a(u)>0$ such that $a(u)u\in
F$. Similarly we can prove that for any given $u\in E_{i},u\not=0$,there is
$a(u)>0$ such that $a(u)u\in F_{i}.$
Lemma 3.2 If $(B_{1}),(B_{2})$ and $(B_{4})$ hold , then for any given $c>0$,
$f(u)$ satisfies $(CPS)_{c,F_{i}}$ condition, that is : If $\\{u_{n}\\}\subset
H^{1}$ satisfies
$\delta(u_{n},F_{i})\rightarrow 0,f(u_{n})\rightarrow c>0,\ \ \ \
(1+\|u_{n}\|)f^{\prime}(u_{n})\rightarrow 0.$ (3.1)
Then $\\{u_{n}\\}$ has a strongly convergent subsequence.
Proof Notice that $\forall u\in E_{i},\int_{0}^{1}u(t)dt=0$, so we know
$\|u\|_{E_{i}}\triangleq(\int_{0}^{1}|\dot{u}|^{2}dt)^{1/2}$ is an equivalent
norm on $E_{i}$. Now from $f(u_{n})\rightarrow c$, we have
$-\frac{1}{2}\|u_{n}\|_{E_{i}}^{2}\cdot\int^{1}_{0}V(u_{n})dt\rightarrow
c-\frac{h}{2}\|u_{n}\|_{E_{i}}^{2}$ (3.2)
By $(B_{2})$ we have
$\displaystyle<f^{\prime}(u_{n}),u_{n}>$ $\displaystyle=$
$\displaystyle\|u_{n}\|_{E_{i}}^{2}\cdot\int^{1}_{0}(h-V(u_{n})-\frac{1}{2}<V^{\prime}(u_{n}),u_{n}>)dt$
(3.3) $\displaystyle\leq$
$\displaystyle\|u_{n}\|_{E_{i}}^{2}\int^{1}_{0}[h+\frac{\mu_{2}}{2}-(1+\frac{\mu_{1}}{2})V(u_{n})]dt$
By (3.2) and (3.3) we have
$\displaystyle<f^{\prime}(u_{n}),u_{n}>$ $\displaystyle\leq$
$\displaystyle(h+\frac{\mu_{2}}{2})\|u_{n}\|_{E_{i}}^{2}+(1+\frac{\mu_{1}}{2})(2c-h\|u_{n}\|_{E_{i}}^{2})$
(3.4) $\displaystyle=$
$\displaystyle(-\frac{\mu_{1}}{2}h+\frac{\mu_{2}}{2})\|u_{n}\|_{E_{i}}^{2}+C_{1}$
Where $C_{1}=2(1+\frac{\mu_{1}}{2})c$
Since $h>\frac{\mu_{2}}{\mu_{1}}$, then (3.1)and (3.4) imply
$\|u_{n}\|_{E_{i}}$ is bounded.
The rest for proving $\\{u_{n}\\}$ has a strongly convergent subsequence is
standard.
Remark 3.1 We notice that in our proof, we didn’t use the condition
$\delta(u_{n},F_{i})\rightarrow 0.$ (3.5)
It seems interesting to efficiently use this condition to weak our
assumptions.
Lemma 3.2 Let
$G=\\{u\in H^{1}|\int_{0}^{1}(V(u)+\frac{1}{2}V^{\prime}(u)u)dt<h\\},$ (3.6)
$G_{i}=\\{u\in E_{i}|\int_{0}^{1}(V(u)+\frac{1}{2}V^{\prime}(u)u)dt<h\\}.$
(3.7)
Then
(i).$F,F_{i},i=1,2$ are respectively the boundaries of $G,G_{i}$.
(ii).If $(B_{1})$ holds, then $F,F_{i},G,G_{i}$ are symmetric with respect to
the origin $0$.
(iii).If $V(0)<h$ holds, then $0\in G,G_{i},i=1,2.$
It’s not difficult to prove the following two Lemmas:
Lemma 3.3 $f(u)$ is weakly lower semi-continuous on $H^{1}$ and $F,F_{i}.$
Lemma 3.4 $F,F_{i},i=1,2.$ are weakly closed subsets in $H^{1}$.
Lemma 3.5 The functional $f(u)$ has positive lower bound on $F_{i}.$
Proof By the definitions of $f(u)$ and $F_{i}$, we have
$f(u)=\frac{1}{4}\int^{1}_{0}|\dot{u}|^{2}dt\int^{1}_{0}(V^{\prime}(u)u)dt,u\in
F_{i}.$ (3.8)
For $u\in F_{i}$ and $(B_{2})$ ,we have
$\frac{1}{2}V^{\prime}(u)u=h-V(u)\geq
h-\frac{1}{\mu_{1}}V^{\prime}(u)u-\frac{\mu_{2}}{\mu_{1}},$
$V^{\prime}(u)u\geq\frac{h-\frac{\mu_{2}}{\mu_{1}}}{\frac{1}{2}+\frac{1}{\mu_{1}}}>0.$
So we have the functional $f(u)\geq 0$. Furthermore, we claims that
$\inf f(u)>0,$ (3.9)
since otherwise, $u(t)=const$ attains the infimum 0.
If $u\in F_{i}$, then by the symmetry $u(t+1/2)=-u(t)$ or $u(-t)=-u(t)$, we
know $u(t)=0,\forall t$; by ($B_{2}$) we have
$V(0)\leq\frac{\mu_{2}}{\mu_{1}}$, by $h>\frac{\mu_{2}}{\mu_{1}}$, we have
$V(0)<h$. By the definition of $F_{i}$, $0\notin F_{i}$. So
$\inf_{F_{2}}f(u)>0.$ (3.10)
Now by Lemmas 3.1-3.5 and Lemma 2.4, we know $f(u)$ attains the infimum on
$F_{i}$, and we know that the minimizer is nonconstant .
Lemma3.6 $\exists z_{1}\not=0,z_{1}\in H^{1}$ s.t. $f(z_{1})\leq 0.$
Proof For any given $y_{1}\not=const$,$\dot{y}_{1}\not=0$,so
$min|y_{1}(t)|>0,$ we let $z_{1}(t)=Ry_{1}(t)$, then when R is large enough,
by condition $(B_{3})$, we have
$\int_{0}^{1}(h-V(z_{1}))dt\leq 0,$ (3.11)
that is,
$f(z_{1})\leq 0.$ (3.12)
Lemma3.7 $f(0)=0.$
Lemma3.8 $F_{i}$ separates $z_{1}$ and $0$.
Proof By $V(0)<h$, we have that $0\in G_{i}$. By $(B_{2})$ and $(B_{3})$ and
$h>\frac{\mu_{2}}{\mu_{1}}$, we can choose R large enough such that
$z_{1}=Ry_{1}\in\\{u\in H^{1}|\int_{0}^{1}(V(u)+\frac{1}{2}V^{\prime}(u)u)dt$
$\geq(1+\frac{\mu_{1}}{2})\int_{0}^{1}V(u)dt-\frac{\mu_{1}}{2}$
$\geq(1+\frac{\mu_{1}}{2})h-\frac{\mu_{1}}{2}>h\\}$
.
So $F_{i}$ separates $z_{1}$ and $0$.
Now by Lemmas 2.4-2.5, 3.1-3.8, we can prove Theorem 1.6.
## 4 The Proof of Theorem 1.7
Let
$F=\\{u\in H^{1}|\|\dot{u}\|_{L^{2}}=r\\},$ $G_{1}=\\{u\in
H^{1}|\|\dot{u}\|_{L^{2}}<r\\},$ $G_{2}=\\{u\in
H^{1}|\|\dot{u}\|_{L^{2}}>r\\}.$
Then $H^{1}\setminus F=G_{1}\cup G_{2}$.
Notice that we can use $(B_{5})$ to get that
$F\cap\\{u\in H^{1}|f(u)\geq c\\}=\\{u\in
H^{1},\frac{1}{2}r^{2}\int_{0}^{1}(h-V(u))dt\geq c\\},$ $\displaystyle H^{1}$
$\displaystyle\setminus(F\cap\\{u\in H^{1}|f(u)\geq c\\})$ $\displaystyle=$
$\displaystyle\\{u\in H^{1}|\|\dot{u}\|_{L^{2}}<r\\}\cup\\{u\in
H^{1}|\|\dot{u}\|_{L^{2}}>r\\}\cup\\{u\in H^{1}|f(u)<c\\}.$
It’s easy to see $u_{1}=0\in G_{1}$, we choose $u_{2}$ such that
$\|\dot{u_{2}}\|_{L^{2}}>r$, so $u_{2}\in G_{2}$. Now every path $g(t)$
connecting $u_{1}$ and $u_{2}$ must pass $F$, so we have
$\max_{0\leq t\leq 1}f(g(t))\geq\inf_{u\in F}f(u)=(\frac{1}{2}r^{2})\inf_{u\in
F}\int_{0}^{1}(h-V(u))\geq c>0.$
So from the above, in order to apply Ghoussoub-Preiss’s generalized MPL, now
we only need to prove the closed set $F$ separate $u_{1}$ and $u_{2}$ and $f$
satisfies $(CPS)_{c,F}$.
From the definitions of the set $F$ and $u_{1}$ and $u_{2}$, we know $F$
separate $u_{1}$ and $u_{2}$.
In order to prove $f$ satisfies $(CPS)_{c,F}$ for any $c>0$, firstly, from
$(B_{2})$, similar to the proof of Lemma 3.1, we can get
$(\int_{0}^{1}|\dot{u}_{n}|^{2}dt)^{1/2}$ is bounded, then by $(B_{3})$, we
prove that $|u_{n}(0)|$ is bounded. In fact, if otherwise, there exists a
subsequence, we still denote it as $\\{u_{n}(0)\\}$ satisfying
$|u_{n}(0)|\rightarrow+\infty.$
By Newton-Leibniz formula and Cauchy-Schwarz inequality, we have
$\displaystyle\min_{0\leq t\leq 1}|u_{n}(t)|$ $\displaystyle\geq$
$\displaystyle|u_{n}(0)|-\|\dot{u}_{n}\|_{2}\rightarrow+\infty$
So by $(B_{3})$ we have
$\int^{1}_{0}V(u_{n})dt\geq h,\ \ \ \ \rm{as}\ n\rightarrow+\infty,$ (4.14)
$\lim\limits_{n\rightarrow\infty}f(u_{n})=\lim\limits_{n\rightarrow\infty}\frac{1}{2}\int^{1}_{0}|\dot{u}_{n}|^{2}dt\int^{1}_{0}(h-V(u_{n}))dt\leq
0,$ (4.15)
which contradicts with $f(u_{n})\rightarrow c>0$.
We know that $H^{1}$ is a reflexive Banach space, so $\\{u_{n}\\}$ has a
weakly convergent subsequence. The rest that proving $\\{u_{n}\\}$ has a
strongly convergent subsequence is standard, we can refer to Ambrosetti-Coti
Zelati [2].
## Acknowledgements
We would like to thank the supports of NSF of China and a research fund for
the Doctoral program of higher education of China.
## References
* [1] A.Ambrosetti, V.Coti Zelati, Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Rat. Mech. Anal. 112(1990), 339-362.
* [2] A.Ambrosetti, V.Coti Zelati, Closed orbits of fixed energy for a class of $N$-body problems, Ann. Inst. H. Poincare, Analyse Non Lineare 9(1992), 187-200.
* [3] A.Ambrosetti, P.Rabinowitz, Dual variational methods in critical point theory and applications, J.of Functional Analysis, 14(1973), 349-381.
* [4] V.Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems, Ann. Inst. Henri Poincare Anal. NonLineaire 1(1984), 401-412.
* [5] K.C.Chang, Infinite dimensional Morse theory and mutiple solution problems, Birkhauser, 1993.
* [6] G.Cerami, Un criterio di esistenza per i punti critici so variete illimitate, Rend. dell academia di sc.lombardo112(1978), 332-336.
* [7] I.Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, 1990.
* [8] N.Ghoussoub, D.Preiss, A general mountain pass principle for locating and clasifying critical points, Ann. Inst. Henri Poincare Anal. NonLineaire 6(1984), 321-330.
* [9] H.Gluck and W.Ziller, Existence of periodic motions of conservative systems, in Seminar on minimal submanifolds, E.Bombieri Ed., Princeton Univ. Press,1983.
* [10] E.W.C.Van Groesen,Analytical mini-max methods for Hamiltonian break orbits with a prescribed energy, JMAA 132(1988), 1-12.
* [11] K.Hayashi, Periodic solutions of classical Hamiltonian systems, Tokyo J.Math., 1983.
* [12] Y. Long, Index Theory for Symplectic Paths with Applications, Basel: Birkhauser, 2002.
* [13] J.Mawhin, M.Willem,Critical Point Theory and Applications, Springer, 1989.
* [14] R.Palais,The principle of symmetric criticality,CMP69(1979),19-30.
* [15] P.H.Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31(1978), 157-184.
* [16] P.H.Rabinowitz, Periodic solutions of a Hamiltonian systems on a prescribed energy surface, JDE 33(1979), 336-352.
* [17] H.Seifert, Periodischer bewegungen mechanischer system, Math.Zeit51(1948), 197-216.
* [18] K.Yosida, Functional Analysis, Springer, Berlin, 1978.
* [19] P.F.Yuan,S.Q.Zhang,New periodic solutions for a class of singular Hamiltonian systems, Acta.Math.Sinica-New Series,29(2013),1205-1218.
* [20] W.P.Ziemer, Weakly differentiable functions, Springer, 1989.
|
arxiv-papers
| 2014-02-24T05:32:08 |
2024-09-04T02:49:58.691962
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fengying Li and Bingyu Li and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1402.5725"
}
|
1402.5746
|
# Maximal estimates for Schrödinger equation with inverse-square potential
Changxing Miao Institute of Applied Physics and Computational Mathematics, P.
O. Box 8009, Beijing, China, 100088 [email protected] , Junyong
Zhang Department of Mathematics, Beijing Institute of Technology, Beijing
100081, China [email protected] and Jiqiang Zheng The Graduate
School of China Academy of Engineering Physics, P. O. Box 2101, Beijing,
China, 100088 [email protected]
###### Abstract.
In this paper, we consider the maximal estimates for the solution to an
initial value problem of the linear Schrödinger equation with a singular
potential. We show a result about the pointwise convergence of solutions to
this special variable coefficient Schrödinger equation with initial data
$u_{0}\in H^{s}(\R^{n})$ for $s>1/2$ or radial initial data $u_{0}\in
H^{s}(\R^{n})$ for $s\geq 1/4$ and the solution does not converge when
$s<1/4$.
Key Words: Inverse square potential, Maximal estimate, Spherical harmonics
AMS Classification: 35B65, 35Q55, 47J35.
## 1\. Introduction and Statement of Main Result
We study the maximal estimates for the solution to an initial value problem of
the linear Schrödinger equation with an inverse square potential. More
precisely, we consider the following Schrödinger equation
(1.1) $\begin{cases}i\partial_{t}u-\Delta
u+\frac{a}{|x|^{2}}u=0,\qquad(t,x)\in\R\times\R^{n},~{}a>-(n-2)^{2}/4,\\\
u(x,0)=u_{0}(x).\end{cases}$
The scale-covariance elliptic operator $P_{a}:=-\Delta+\frac{a}{|x|^{2}}$
appearing in (1.1) plays a key role in many problems of physics and geometry.
The heat and wave flows for the elliptic operator $P_{a}$ have been studied in
the theory of combustion (see [28]), and in the wave propagation on conic
manifolds (see [8]). The Schrödinger equation (1.1) arises in the study of
quantum mechanics [10]. There has been a lot of interest in developing
Strichartz estimates both for the Schrödinger and wave equations with the
inverse square potential, we refer the reader to Burq etc.[3, 4, 16, 17] and
the authors [13]. However, as far as we known, there is few result about the
maximal estimates associated with the operator $P_{a}$, which arises in the
study of pointwise convergence problem for the Schrödinger and wave equations
with the inverse square potential. In this paper, we aim to address some
maximal estimates in the special settings associated with the operator
$P_{a}$. As a direct consequence, we obtain the pointwise convergence result
for $u_{0}\in H^{s}(\R^{n})$ with $s>1/2$.
In the case of the free Schrödinger equation without potential, i.e. $a=0$,
there are a large amount of literature in developing the maximal estimate for
its solution, which can be formally written as
$\begin{split}u(t,x)=e^{it\Delta}u_{0}(x)=\int_{\R^{n}}e^{2\pi
i(x\cdot\xi-t|\xi|^{2})}\hat{u}_{0}(\xi)\mathrm{d}\xi.\end{split}$
When $n=1$, Carleson [5] proved the convergence result holds in sense of that
$\lim\limits_{t\rightarrow 0}u(t)=u_{0},a.e.~{}x$ when $u_{0}\in H^{s}(\R)$
with $s\geq 1/4$. Dahlberg-Kenig [7] showed that the result is sharp in the
sense that the solution does not converge when $s<1/4$. When $n\geq 2$, Sjölin
[22] and Vega [27] independently proved the convergence results hold when
$u_{0}\in H^{s}(\R^{n})$ when $s>1/2$. It follows from the construction of
Dahlberg-Kenig [7], or alternatively Vega [27] that the solution does not
converge when $s<1/4$. When $n=2$, Bourgain [1] showed that there is a certain
$s<1/2$ such that the convergence result holds, and this result was improved
by Moyua-Vargas-Vega [12]. Having shown the bilinear restriction estimates for
paraboloids, Tao-Vargas [25] and Tao [24] showed the convergence result holds
for $s>15/32$ and $s>2/5$ respectively. The result was improved further to
$s>3/8$ by Lee [11] and Shao [23]. Very recently, Bourgain [2] made some
progress in high dimension $n\geq 2$ to show that the convergence result holds
for $s>1/2-1/(4n)$ when $n\geq 1$ and the convergence result needs
$s\geq(n-2)/(2n)$ when $n\geq 5$.
In the situation when $a\neq 0$, the equation (1.1) can be viewed as a special
Schrödinger equation with variable singular coefficients. The potential
prevents us from using the Fourier transform to give the expression of the
solution. With the motivation of regarding the potential term as a
perturbation on angular direction in [3, 16, 13], we express the solution by
using the Hankel transform of radial functions and spherical harmonics.
Instead of Fourier transform, we utilize the Hankel transform and modify the
argument of Vega [27] to show the pointwise convergence result holds when the
initial data $u_{0}\in H^{s}(\R^{n})$ for $s>1/2$, or radial initial data
$u_{0}\in H^{s}(\R^{n})$ for $s\geq 1/4$, and the solution does not converge
when $s<1/4$.
Let $u$ be the solution to (1.1), we define the maximal function by
(1.2) $\begin{split}u^{*}(x)=\sup_{|t|>0}|u(x,t)|.\end{split}$
Our main theorems are the following:
###### Theorem 1.1.
Let $\beta>1$, $n\geq 2$ and $s>\frac{1}{2}$. Then
(1.3)
$\begin{split}\int_{\R^{n}}|u^{*}(x)|^{2}\frac{\mathrm{d}x}{(1+|x|)^{\beta}}\leq
C\|u_{0}\|^{2}_{H^{s}(\R^{n})}.\end{split}$
As a direct consequence of Theorem 1.1, we have:
###### Corollary 1.1.
Let $u_{0}\in H^{s}(\R^{n})$ with $s>\frac{1}{2}$ and $n\geq 2$. Then
(1.4) $\lim_{t\rightarrow 0}u(t,x)=u_{0}(x),\quad a.e.~{}~{}x\in\R^{n}.$
###### Theorem 1.2.
Let $B^{n}$ be the open unit ball in $\R^{n}$. Assume that there exists a
constant $C$ independent of $u_{0}$ such that
(1.5) $\begin{split}\int_{B^{n}}|u^{*}(x)|^{2}\mathrm{d}x\leq
C\|u_{0}\|^{2}_{H^{s}(\R^{n})},\quad\forall~{}u_{0}(x)\in
H^{s}(\R^{n}).\end{split}$
Then $s\geq\frac{1}{4}$.
With this in mind, Theorem 1.1 is far from being sharp. Assuming that the
initial data possesses additional angular regularity, we have
###### Theorem 1.3.
Let $B^{n}$ be the open unit ball in $\R^{n}$ and $\epsilon>0$. Then there
exists a constant $C$ independent of $u_{0}$ such that
(1.6) $\begin{split}\int_{B^{n}}|u^{*}(x)|^{2}\mathrm{d}x\leq
C\|u_{0}\|^{2}_{H^{\frac{1}{4}}_{r}H^{\frac{n-1}{2}+\epsilon}_{\theta}},\end{split}$
where for $s,s^{\prime}\geq 0$
$\begin{split}H^{s}_{r}H^{s^{\prime}}_{\theta}=\Big{\\{}g:\|g\|_{H^{s}_{r}H^{s^{\prime}}_{\theta}}:=\big{\|}(1-\Delta_{\theta})^{\frac{s^{\prime}}{2}}\big{(}(1-\Delta)^{\frac{s}{2}}g\big{)}\big{\|}_{L^{2}_{r^{n-1}\mathrm{d}r}(\R^{+};L^{2}_{\theta}(\mathbb{S}^{n-1}))}\Big{\\}}.\end{split}$
Here $\Delta_{\theta}$ denotes the Laplace-Beltrami operator on
$\mathbb{S}^{n-1}$.
Remarks:
$\mathrm{i}).$ This result implies that the pointwise convergence of solutions
to (1.1) holds for radial initial data $u_{0}\in H^{s}(\R^{n})$ with $s\geq
1/4$.
$\mathrm{ii}).$ This result is an analogue of Theorem 1.1 in [6]. We remark
that the parameter $\epsilon$ in [6] should be corrected for $\epsilon>1/2$
while not $\epsilon>0$. Thus, we generalize and improve the result in [6] by
making use of a finer result proved in [9].
Now we introduce some notations. We use $A\lesssim B$ to denote the statement
that $A\leq CB$ for some large constant $C$ which may vary from line to line
and depend on various parameters, and similarly use $A\ll B$ to denote the
statement $A\leq C^{-1}B$. We employ $A\sim B$ to denote the statement that
$A\lesssim B\lesssim A$. If the constant $C$ depends on a special parameter
other than the above, we shall denote it explicitly by subscripts. We briefly
write $A+\epsilon$ as $A+$ or $A-\epsilon$ as $A-$ for $0<\epsilon\ll 1$.
Throughout this paper, pairs of conjugate indices are written as
$p,p^{\prime}$, where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ with $1\leq
p\leq\infty$.
This paper is organized as follows: In the section 2, we mainly revisit the
property of the Bessel functions and the Hankel transforms associated with
$-\Delta+\frac{a}{|x|^{2}}$. Section 3 is devoted to the proofs of the
theorems.
Acknowledgments: The authors thank the referee and the associated editor for
their invaluable comments and suggestions which helped improve the paper
greatly. This work was supported in part by the NSF of China under grant
No.11171033, No.11231006, and No.11371059. The second author was partly
supported by the Fundamental Research Foundation of BIT(20111742015) and
RFDP(20121101120044). C. Miao was also supported by Beijing Center for
Mathematics and Information Interdisciplinary Sciences.
## 2\. Preliminary
In this section, we first list some results about the Hankel transform and the
Bessel functions and then show a characterization of Sobolev norm in the
Hankel transform version.
We begin with recalling the expansion formula with respect to the spherical
harmonics. For more details, we refer to Stein-Weiss [21]. For the sake of
convenience, let
(2.1) $\xi=\rho\omega\quad\text{and}\quad
x=r\theta\quad\text{with}\quad\omega,\theta\in\mathbb{S}^{n-1}.$
For any $g\in L^{2}(\R^{n})$, the expansion formula with respect to the
spherical harmonics yields
$g(x)=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}a_{k,\ell}(r)Y_{k,\ell}(\theta)$
where
$\\{Y_{k,1},\ldots,Y_{k,d(k)}\\}$
is the orthogonal basis of the spherical harmonics space of degree $k$ on
$\mathbb{S}^{n-1}$, called $\mathcal{H}^{k}$, with the dimension
$d(k)=\frac{2k+n-2}{k}C^{k-1}_{n+k-3}\simeq\langle k\rangle^{n-2}.$
We remark that for $n=2$, the dimension of $\mathcal{H}^{k}$ is a constant,
which is independent of $k$. Obviously, we have the orthogonal decomposition
$L^{2}(\mathbb{S}^{n-1})=\bigoplus_{k=0}^{\infty}\mathcal{H}^{k}.$
By orthogonality, it gives
(2.2)
$\|g(x)\|_{L^{2}_{\theta}(\mathbb{S}^{n-1})}=\|a_{k,\ell}(r)\|_{\ell^{2}_{k,\ell}}.$
From $-\Delta_{\theta}Y_{k,\ell}(\theta)=k(k+n-2)Y_{k,\ell}(\theta)$, the
fractional power of $1-\Delta_{\theta}$ can be written explicitly [15]
(2.3)
$(1-\Delta_{\theta})^{\frac{s}{2}}g(x)=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}(1+k(k+n-2))^{\frac{s}{2}}a_{k,\ell}(r)Y_{k,\ell}(\theta).$
For our purpose, we need the Fourier transform of
$a_{k,\ell}(r)Y_{k,\ell}(\theta)$. Theorem 3.10 in [21] asserts the Hankel
transform formula
(2.4)
$\hat{g}(\rho\omega)\sim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}i^{k}Y_{k,\ell}(\omega)\rho^{-\frac{n-2}{2}}\int_{0}^{\infty}J_{k+\frac{n-2}{2}}(2\pi
r\rho)a_{k,\ell}(r)r^{\frac{n}{2}}\mathrm{d}r.$
Here the Bessel function $J_{k}(r)$ of order $k$ is defined by the integral
$J_{k}(r)=\frac{(r/2)^{k}}{\Gamma(k+\frac{1}{2})\Gamma(1/2)}\int_{-1}^{1}e^{isr}(1-s^{2})^{(2k-1)/2}\mathrm{d}s\quad\text{with}~{}k>-\frac{1}{2}~{}\text{and}~{}r>0.$
A simple computation gives the rough estimates
(2.5)
$|J_{k}(r)|\leq\frac{Cr^{k}}{2^{k}\Gamma(k+\frac{1}{2})\Gamma(1/2)}\left(1+\frac{1}{k+1/2}\right),$
where $C$ is a absolute constant. This estimate will be mainly used when
$r\lesssim 1$. Another well known asymptotic expansion about the Bessel
function is
(2.6)
$J_{k}(r)=r^{-1/2}\sqrt{\frac{2}{\pi}}\cos(r-\frac{k\pi}{2}-\frac{\pi}{4})+O_{k}(r^{-3/2}),\quad\text{as}~{}r\rightarrow\infty$
but with a constant depending on $k$ (see [21]). As pointed out in [20], if
one seeks a uniform bound for large $r$ and $k$, then the best one can do is
$|J_{k}(r)|\leq Cr^{-\frac{1}{3}}$. One will find that this decay doesn’t lead
to the desirable result. Moreover, we recall the properties of Bessel function
$J_{k}(r)$ in [18, 20], we refer the readers to [14] for the detailed proof.
###### Lemma 2.1 (Asymptotics of the Bessel function).
Assume that $k\in\N$ and $k\gg 1$. Let $J_{k}(r)$ be the Bessel function of
order $k$ defined as above. Then there exist a large constant $C$ and small
constant $c$ independent of $k$ and $r$ such that:
$\bullet$ when $r\leq\frac{k}{2}$
(2.7) $\begin{split}|J_{k}(r)|\leq Ce^{-c(k+r)};\end{split}$
$\bullet$ when $\frac{k}{2}\leq r\leq 2k$
(2.8) $\begin{split}|J_{k}(r)|\leq
Ck^{-\frac{1}{3}}(k^{-\frac{1}{3}}|r-k|+1)^{-\frac{1}{4}};\end{split}$
$\bullet$ when $r\geq 2k$
(2.9) $\begin{split}J_{k}(r)=r^{-\frac{1}{2}}\sum_{\pm}a_{\pm}(r,k)e^{\pm
ir}+E(r,k),\end{split}$
where $|a_{\pm}(r,k)|\leq C$ and $|E(r,k)|\leq Cr^{-1}$.
As a consequence of Lemma 2.1, we have
###### Lemma 2.2.
Let $R\gg 1$. Then there exists a constant $C$ independent of $k,R$ such that
(2.10) $\int_{R}^{2R}|J_{k}(r)|^{2}\mathrm{d}r\leq C.$
###### Proof.
To prove (2.10), we write
$\begin{split}\int_{R}^{2R}|J_{k}(r)|^{2}\mathrm{d}r=\int_{I_{1}}|J_{k}(r)|^{2}\mathrm{d}r+\int_{I_{2}}|J_{k}(r)|^{2}\mathrm{d}r+\int_{I_{3}}|J_{k}(r)|^{2}\mathrm{d}r\end{split}$
where $I_{1}=[R,2R]\cap[0,\frac{k}{2}],I_{2}=[R,2R]\cap[\frac{k}{2},2k]$ and
$I_{3}=[R,2R]\cap[2k,\infty]$. By (2.7) and (2.9), we have
(2.11) $\begin{split}\int_{I_{1}}|J_{k}(r)|^{2}\mathrm{d}r\leq
C\int_{I_{1}}e^{-cr}\mathrm{d}r\leq Ce^{-cR},\end{split}$
and
(2.12) $\begin{split}\int_{I_{3}}|J_{k}(r)|^{2}\mathrm{d}r\leq C.\end{split}$
On the other hand, one has by (2.8)
$\begin{split}\int_{[\frac{k}{2},2k]}|J_{k}(r)|^{2}\mathrm{d}r&\leq
C\int_{[\frac{k}{2},2k]}k^{-\frac{2}{3}}(1+k^{-\frac{1}{3}}|r-k|)^{-\frac{1}{2}}\mathrm{d}r\leq
C.\end{split}$
Observing $[R,2R]\cap[\frac{k}{2},2k]=\emptyset$ unless $R\sim k$, we obtain
(2.13) $\begin{split}\int_{I_{2}}|J_{k}(r)|^{2}\mathrm{d}r\leq C.\end{split}$
This together with (2.11) and (2.12) yields (2.10). ∎
For simplicity, we define
(2.14)
$\mu(k)=\frac{n-2}{2}+k,\quad\text{and}\quad\nu(k)=\sqrt{\mu^{2}(k)+a}\quad\text{with}\quad
a>-(n-2)^{2}/4.$
We sometime briefly write $\nu$ as $\nu(k)$. Let $f$ be Schwartz function
defined on $\R^{n}$, we define the Hankel transform of order $\nu$
(2.15)
$(\mathcal{H}_{\nu}f)(\xi)=\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu}(r\rho)f(r\omega)r^{n-1}\mathrm{d}r,$
where $\rho=|\xi|$, $\omega=\xi/|\xi|$ and $J_{\nu}$ is the Bessel function of
order $\nu$. In particular, the function $f$ is radial, then we have
(2.16)
$(\mathcal{H}_{\nu}f)(\rho)=\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu}(r\rho)f(r)r^{n-1}\mathrm{d}r.$
If
$f(x)=\sum\limits_{k=0}^{\infty}\sum\limits_{\ell=1}^{d(k)}a_{k,\ell}(r)Y_{k,\ell}(\theta)$,
it follows from (2.4) that
(2.17) $\begin{split}\hat{f}(\xi)=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}2\pi
i^{k}Y_{k,\ell}(\omega)\big{(}\mathcal{H}_{\mu(k)}a_{k,\ell}\big{)}(\rho).\end{split}$
The following properties of the Hankel transform are obtained in [3, 16]:
###### Lemma 2.3.
Let $\mathcal{H}_{\nu}$ be defined above and
$A_{\nu(k)}:=-\partial_{r}^{2}-\frac{n-1}{r}\partial_{r}+\big{[}\nu^{2}(k)-\big{(}\frac{n-2}{2}\big{)}^{2}\big{]}{r^{-2}}.$
Then
$(\rm{i})$ $\mathcal{H}_{\nu}=\mathcal{H}_{\nu}^{-1}$,
$(\rm{ii})$ $\mathcal{H}_{\nu}$ is self-adjoint, i.e.
$\mathcal{H}_{\nu}=\mathcal{H}_{\nu}^{*}$,
$(\rm{iii})$ $\mathcal{H}_{\nu}$ is an $L^{2}$ isometry, i.e.
$\|\mathcal{H}_{\nu}\phi\|_{L^{2}_{\xi}}=\|\phi\|_{L^{2}_{x}}$,
$(\rm{iv})$
$\mathcal{H}_{\nu}(A_{\nu}\phi)(\xi)=|\xi|^{2}(\mathcal{H}_{\nu}\phi)(\xi)$,
for $\phi\in L^{2}$.
We next recall the following almost orthogonality inequality. Denote by
$P_{j}$ and $\tilde{P}_{j}$ the usual dyadic frequency localization at
$|\xi|\sim 2^{j}$ and the localization with respect to
$\big{(}-\Delta+\frac{a}{|x|^{2}}\big{)}^{\frac{1}{2}}$. We define the
projectors $M_{jj^{\prime}}=P_{j}\tilde{P}_{j^{\prime}}$ and
$N_{jj^{\prime}}=\tilde{P}_{j}P_{j^{\prime}}$. More precisely, let $f$ be in
the $k$’th harmonic subspace, then
$P_{j}f=\mathcal{H}_{\mu(k)}\beta_{j}\mathcal{H}_{\mu(k)}f\quad\text{and}\quad\tilde{P}_{j}f=\mathcal{H}_{\nu(k)}\beta_{j}\mathcal{H}_{\nu(k)}f,$
where $\beta_{j}(\xi)=\beta(2^{-j}|\xi|)$ with $\beta\in
C_{0}^{\infty}(\R^{+})$ supported in $[\frac{1}{2},2]$. Then we have the
following almost orthogonality inequality [3]:
###### Lemma 2.4 (Almost orthogonality inequality).
Let $f\in L^{2}(\R^{n})$, then there exists a constant $C$ independent of
$j,j^{\prime}$ such that
(2.18)
$\|M_{jj^{\prime}}f\|_{L^{2}(\R^{n})},~{}~{}\|N_{jj^{\prime}}f\|_{L^{2}(\R^{n})}\leq
C2^{-\epsilon|j-j^{\prime}|}\|f\|_{L^{2}(\R^{n})},$
where
$\epsilon<1+\min\\{\frac{n-2}{2},(\frac{(n-2)^{2}}{4}+a)^{\frac{1}{2}}\\}$.
As a consequence, we have
###### Lemma 2.5.
Let $f\in L^{2}(\R^{n})$ such that
$f(x)=\sum\limits_{k=0}^{\infty}\sum\limits_{\ell=1}^{d(k)}a_{k,\ell}(r)Y_{k,\ell}(\theta)$.
Then for $0\leq
s<1+\min\\{\frac{n-2}{2},(\frac{(n-2)^{2}}{4}+a)^{\frac{1}{2}}\\}$ and
$s^{\prime}\geq 0$
(2.19) $\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{M\in
2^{\Z}}M^{2s}(1+k)^{2s^{\prime}}\|b_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|_{L_{\rho}^{2}}^{2}\sim\|f\|_{\dot{H}^{s}_{r}{H}^{s^{\prime}}_{\theta}}^{2},$
where $b_{k,\ell}(\rho)=(\mathcal{H}_{\nu(k)}a_{k,\ell})(\rho)$ and $\chi\in
C_{0}^{\infty}(\R^{n})$ such that $\text{supp}~{}\chi\subset[1/2,1]$.
###### Proof.
Note that $-\Delta_{\theta}Y_{k,\ell}=k(k+n-2)Y_{k,\ell}$, then we have by
Lemma 2.3
$\begin{split}\|f\|_{\dot{H}^{0}_{r}{H}^{s^{\prime}}_{\theta}}^{2}&\sim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}(1+k)^{2s^{\prime}}\|a_{k,\ell}(r)\|_{L^{2}_{r^{n-1}\mathrm{d}r}(\R^{+})}^{2}\|Y_{k,\ell}(\theta)\|_{L_{\theta}^{2}}^{2}\\\
&\sim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}(1+k)^{2s^{\prime}}\|b_{k,\ell}(\rho)\|_{L^{2}_{\rho^{n-1}\mathrm{d}\rho}(\R^{+})}^{2}.\end{split}$
By (2.3), it suffices to show (2.19) with $s^{\prime}=0$. By Lemma 2.3, we
have
$\displaystyle\|b_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|_{L_{\rho}^{2}}=$
$\displaystyle\big{\|}\chi(\frac{\rho}{M})\mathcal{H}_{\nu}\big{[}Y_{k,l}(\theta)a_{k,\ell}(r)\big{]}(\xi)\big{\|}_{L_{\xi}^{2}}$
$\displaystyle=$
$\displaystyle\Big{\|}\mathcal{H}_{\nu}\Big{[}\chi(\frac{\rho}{M})\mathcal{H}_{\nu}\big{(}Y_{k,l}(\theta)a_{k,\ell}(r)\big{)}(\xi)\Big{]}\Big{\|}_{L_{x}^{2}}.$
This yields that by letting let $j=\log_{2}M$
$\displaystyle\big{\|}b_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\big{\|}_{L_{\rho}^{2}}=$
$\displaystyle\big{\|}\big{[}\mathcal{H}_{\nu}\chi(\frac{\rho}{M})\mathcal{H}_{\nu}\big{]}\big{(}Y_{k,l}(\theta)a_{k,\ell}(r)\big{)}\big{\|}_{L_{x}^{2}}$
$\displaystyle=$
$\displaystyle\big{\|}\tilde{P}_{j}\big{(}Y_{k,l}(\theta)a_{k,\ell}(r)\big{)}\big{\|}_{L_{x}^{2}}.$
Let $g_{k,\ell}(x)=Y_{k,l}(\theta)a_{k,\ell}(r)$ and
$\overline{P_{j^{\prime}}}=P_{j^{\prime}-1}+P_{j^{\prime}}+P_{j^{\prime}+1}$.
We have by the triangle inequality and Lemma 2.4
$\displaystyle\text{L.H.S of}~{}\eqref{2.18}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{j\in\Z}2^{2sj}\big{\|}\tilde{P}_{j}g_{k,\ell}\big{\|}^{2}_{L_{x}^{2}}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{j\in\Z}2^{2sj}\big{(}\sum_{j^{\prime}}\big{\|}\tilde{P}_{j}\overline{P_{j^{\prime}}}P_{j^{\prime}}g_{k,\ell}\big{\|}_{L_{x}^{2}}\big{)}^{2}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{j\in\Z}2^{2sj}\big{(}\sum_{j^{\prime}}2^{-\epsilon|j-j^{\prime}|}\big{\|}P_{j^{\prime}}g_{k,\ell}\big{\|}_{L_{x}^{2}}\big{)}^{2},$
where
$s<\epsilon<1+\min\\{\frac{n-2}{2},(\frac{(n-2)^{2}}{4}+a)^{\frac{1}{2}}\\}$.
Let $0<\epsilon_{1}\ll 1$ such that $\epsilon_{2}:=\epsilon-\epsilon_{1}>s$,
then
$\begin{split}\text{L.H.S of}~{}\eqref{2.18}&\leq
C\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{j\in\Z}2^{2js}\sum_{j^{\prime}}2^{-2\epsilon_{2}|j-j^{\prime}|}\|P_{j^{\prime}}g_{k,\ell}\|^{2}_{L^{2}(\R^{n})}\sum_{j^{\prime}}2^{-2\epsilon_{1}|j-j^{\prime}|}\\\
&\leq
C\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{j^{\prime}}2^{2j^{\prime}s}\sum_{j\in\Z}2^{2js}2^{-2\epsilon_{2}|j|}\|P_{j^{\prime}}g_{k,\ell}\|^{2}_{L^{2}(\R^{n})}\\\
&\leq
C\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{j^{\prime}}2^{2j^{\prime}s}\|P_{j^{\prime}}g_{k,\ell}\|^{2}_{L^{2}(\R^{n})}.\end{split}$
By the definition of $P_{j^{\prime}}$, Lemma 2.3 and (2.17), we have
$\begin{split}\text{L.H.S of}~{}\eqref{2.18}&\leq
C\sum_{j^{\prime}}2^{2j^{\prime}s}\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\big{\|}\chi(\frac{\rho}{2^{j^{\prime}}})\big{[}\mathcal{H}_{\mu(k)}a_{k,\ell}\big{]}(\rho)\rho^{\frac{n-1}{2}}\big{\|}^{2}_{L^{2}(\R^{+})}\\\
&=C\sum_{j^{\prime}}2^{2j^{\prime}s}\big{\|}\chi(\frac{\rho}{2^{j^{\prime}}})\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}2\pi
i^{k}\big{[}\mathcal{H}_{\mu(k)}a_{k,\ell}\big{]}(\rho)Y_{k,\ell}(\omega)\big{\|}^{2}_{L^{2}(\R^{n})}\\\
&=C\sum_{j^{\prime}}2^{2j^{\prime}s}\big{\|}\chi(\frac{\rho}{2^{j^{\prime}}})\hat{f}\big{\|}^{2}_{L^{2}(\R^{n})}\sim\|f\|^{2}_{\dot{H}^{s}}.\end{split}$
We can use the similar argument to prove
$\begin{split}\text{L.H.S of}~{}\eqref{2.18}&\geq
c\|f\|^{2}_{\dot{H}^{s}}.\end{split}$
Therefore we conclude the proof of Lemma 2.4. ∎
## 3\. Proof of the Main Theorems
In this section, we first use the spherical harmonic expansion to write the
solution as a linear combination of products of the Hankel transform of radial
functions and spherical harmonics. We prove the main theorems by analyzing the
property of the Hankel transform. The key ingredients are to use the
stationary phase argument and to exploit the asymptotics behavior of the
Bessel function.
### 3.1. The expression of the solution.
Consider the following Cauchy problem:
(3.1) $\begin{cases}i\partial_{t}u-\Delta u+\frac{a}{|x|^{2}}u=0,\\\
u(x,0)=u_{0}(x).\end{cases}$
We use the spherical harmonic expansion to write
(3.2)
$u_{0}(x)=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}a^{0}_{k,\ell}(r)Y_{k,\ell}(\theta).$
Let us consider the equation (3.1) in polar coordinates. Write
$v(t,r,\theta)=u(t,r\theta)$ and $g(r,\theta)=u_{0}(r\theta)$. Then
$v(t,r,\theta)$ satisfies that
(3.3)
$\begin{cases}i\partial_{t}v-\partial_{rr}v-\frac{n-1}{r}\partial_{r}v-\frac{1}{r^{2}}\Delta_{\theta}v+\frac{a}{r^{2}}v=0\\\
v(0,r,\theta)=g(r,\theta).\end{cases}$
By (3.2), we have
$g(r,\theta)=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}a^{0}_{k,\ell}(r)Y_{k,\ell}(\theta).$
Using separation of variables, we can write $v$ as a linear combination of
products of radial functions and spherical harmonics
(3.4)
$v(t,r,\theta)=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}v_{k,\ell}(t,r)Y_{k,\ell}(\theta),$
where $v_{k,\ell}$ is given by
$\begin{cases}i\partial_{t}v_{k,\ell}-\partial_{rr}v_{k,\ell}-\frac{n-1}{r}\partial_{r}v_{k,\ell}+\frac{k(k+n-2)+a}{r^{2}}v_{k,\ell}=0,\\\
v_{k,\ell}(0,r)=a^{0}_{k,\ell}(r)\end{cases}$
for each $k,\ell\in\N,~{}1\leq\ell\leq d(k)$. Then it reduces to consider by
the definition of $A_{\nu(k)}$
(3.5) $\begin{cases}i\partial_{t}v_{k,\ell}+A_{\nu(k)}v_{k,\ell}=0,\\\
v_{k,\ell}(0,r)=a^{0}_{k,\ell}(r).\end{cases}$
Applying the Hankel transform to the equation (3.5), by $(\rm{iv})$ in Lemma
2.3, we have
(3.6)
$\begin{cases}i\partial_{t}\tilde{v}_{k,\ell}+\rho^{2}\tilde{v}_{k,\ell}=0\\\
\tilde{v}_{k,\ell}(0,\rho)=b^{0}_{k,\ell}(\rho),\end{cases}$
where
(3.7) $\tilde{v}_{k,\ell}(t,\rho)=(\mathcal{H}_{\nu}v_{k,\ell})(t,\rho),\quad
b^{0}_{k,\ell}(\rho)=(\mathcal{H}_{\nu}a^{0}_{k,\ell})(\rho).$
Solving this ODE and inverting the Hankel transform, we obtain
$\begin{split}v_{k,\ell}(t,r)&=\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)\tilde{v}_{k,\ell}(t,\rho)\rho^{n-1}\mathrm{d}\rho\\\
&=\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)e^{it\rho^{2}}b^{0}_{k,\ell}(\rho)\rho^{n-1}\mathrm{d}\rho.\end{split}$
Therefore we get
(3.8) $\begin{split}&u(x,t)=v(t,r,\theta)\\\
&=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}Y_{k,\ell}(\theta)\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)e^{it\rho^{2}}b^{0}_{k,\ell}(\rho)\rho^{n-1}\mathrm{d}\rho\\\
&=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}Y_{k,\ell}(\theta)\mathcal{H}_{\nu(k)}\big{[}e^{it\rho^{2}}b^{0}_{k,\ell}(\rho)\big{]}(r).\end{split}$
### 3.2. Proof of the Theorem 1.1.
In this subsection, we prove the Theorem 1.1. By the Sobolev embedding
$\dot{H}^{\frac{1}{2}-}(\R)\cap\dot{H}^{\frac{1}{2}+}(\R)\hookrightarrow
L^{\infty}(\R)$, it suffices to show
###### Proposition 3.1.
Let $\alpha\geq\frac{1}{2}-\frac{\beta}{4}$ and $\beta=1+$ such that
$2\alpha-1+\frac{\beta}{2}<1+\min\big{\\{}{(n-2)}/2,({(n-2)^{2}}/4+a)^{\frac{1}{2}}\big{\\}},$
then there exists a constant $C$ independent of $u_{0}$ such that
(3.9)
$\begin{split}\int_{\R^{n}}\int_{\R}|\partial_{t}^{\alpha}u(x,t)|^{2}\frac{\mathrm{d}t\mathrm{d}x}{(1+|x|)^{\beta}}\leq
C\|u_{0}\|^{2}_{\dot{H}^{2\alpha-1+\frac{\beta}{2}}(\R^{n})}.\end{split}$
###### Proof.
By the Plancherel theorem with respect to time $t$, we obtain
$\begin{split}&\int_{\R^{n}}\int_{\R}|\partial_{t}^{\alpha}u(x,t)|^{2}\frac{\mathrm{d}t\mathrm{d}x}{(1+|x|)^{\beta}}=\int_{\R^{n}}\int_{\R}\big{|}\tau^{\alpha}\int_{\R}e^{-it\tau}u(x,t)\mathrm{d}t\big{|}^{2}\frac{\mathrm{d}\tau\mathrm{d}x}{(1+|x|)^{\beta}}.\end{split}$
Using (3.8), we further have
$\begin{split}&\text{L.H.S of}~{}\eqref{3.9}\\\
\lesssim&\int_{\R^{n+1}}\big{|}\tau^{\alpha}\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}Y_{k,\ell}(\theta)\int_{\R}\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)e^{it(\rho^{2}-\tau)}b^{0}_{k,\ell}(\rho)\rho^{n-1}\mathrm{d}\rho\mathrm{d}t\big{|}^{2}\frac{\mathrm{d}\tau\mathrm{d}x}{(1+|x|)^{\beta}}\\\
\lesssim&\int_{\R^{n+1}}\big{|}\tau^{\alpha}\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}Y_{k,\ell}(\theta)\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)b^{0}_{k,\ell}(\rho)\rho^{n-1}\delta(\tau-\rho^{2})\mathrm{d}\rho\big{|}^{2}\frac{\mathrm{d}\tau\mathrm{d}x}{(1+|x|)^{\beta}}\\\
\lesssim&\int_{\R^{n}}\int_{0}^{\infty}\big{|}\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}Y_{k,\ell}(\theta)\rho^{\alpha}(r\sqrt{\rho})^{-\frac{n-2}{2}}J_{\nu(k)}(r\sqrt{\rho})b^{0}_{k,\ell}(\sqrt{\rho})\rho^{\frac{n-1}{2}}\rho^{-\frac{1}{2}}\big{|}^{2}\frac{\mathrm{d}\rho\mathrm{d}x}{(1+|x|)^{\beta}}.\end{split}$
By the orthogonality, we see that
(3.10) $\begin{split}&\text{L.H.S of}~{}\eqref{3.9}\\\
&\lesssim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\int_{0}^{\infty}\int_{0}^{\infty}\big{|}\rho^{2\alpha+\frac{1}{2}}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)b^{0}_{k,\ell}(\rho)\rho^{n-2}\big{|}^{2}\frac{\mathrm{d}\rho~{}r^{n-1}\mathrm{d}r}{(1+r)^{\beta}}.\end{split}$
Let $\chi$ be a smoothing function supported in $[1,2]$. For our purpose, we
make a dyadic decomposition to obtain
$\begin{split}&\text{L.H.S of}~{}\eqref{3.9}\\\
\lesssim&\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{M\in
2^{\Z}}\int_{0}^{\infty}\int_{0}^{\infty}\big{|}\rho^{2\alpha+\frac{1}{2}}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)b^{0}_{k,\ell}(\rho)\rho^{n-2}\chi(\frac{\rho}{M})\big{|}^{2}\frac{r^{n-1}\mathrm{d}r\mathrm{d}\rho}{(1+r)^{\beta}}\\\
\lesssim&\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{M\in
2^{\Z}}M^{2(n-2+2\alpha+\frac{1}{2})+1-n}\int_{0}^{\infty}\int_{0}^{\infty}\big{|}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)b^{0}_{k,\ell}(M\rho)\chi(\rho)\big{|}^{2}\frac{r^{n-1}\mathrm{d}r\mathrm{d}\rho}{(1+\frac{r}{M})^{\beta}}\\\
\lesssim&\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{M\in 2^{\Z}}\sum_{R\in
2^{\Z}}M^{n-2+4\alpha}R^{n-1}\int_{R}^{2R}\int_{0}^{\infty}\big{|}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)b^{0}_{k,\ell}(M\rho)\chi(\rho)\big{|}^{2}\frac{\mathrm{d}r\mathrm{d}\rho}{(1+\frac{r}{M})^{\beta}}.\end{split}$
Define
(3.11)
$\begin{split}G_{k,\ell}(R,M)=\int_{R}^{2R}\int_{0}^{\infty}\big{|}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)b^{0}_{k,\ell}(M\rho)\chi(\rho)\big{|}^{2}\frac{\mathrm{d}r\mathrm{d}\rho}{(1+\frac{r}{M})^{\beta}}.\end{split}$
###### Proposition 3.2.
We have the following inequality
(3.12)
$G_{k,\ell}(R,M)\lesssim\begin{cases}R^{2\nu(k)-n+3}M^{-n}\min\Big{\\{}1,\Big{(}\frac{M}{R}\Big{)}^{\beta}\Big{\\}}\|b^{0}_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|^{2}_{L^{2}},~{}R\lesssim
1;\\\
\min\Big{\\{}1,\Big{(}\frac{M}{R}\Big{)}^{\beta}\Big{\\}}R^{-(n-2)}M^{-n}\|b^{0}_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|^{2}_{L^{2}},~{}R\gg
1.\end{cases}$
###### Proof.
To prove (3.12), we break it into two cases.
$\bullet$ Case 1: $R\lesssim 1$. Since $\rho\sim 1$, we have $r\rho\lesssim
1$. By the property of the Bessel function (2.5), we obtain
(3.13)
$\begin{split}G_{k,\ell}(R,M)&\lesssim\int_{R}^{2R}\int_{0}^{\infty}\Big{|}\frac{(r\rho)^{\nu(k)}(r\rho)^{-\frac{n-2}{2}}}{2^{\nu(k)}\Gamma(\nu(k)+\frac{1}{2})\Gamma(\frac{1}{2})}b^{0}_{k,\ell}(M\rho)\chi(\rho)\Big{|}^{2}\mathrm{d}\rho\frac{\mathrm{d}r}{(1+\frac{r}{M})^{\beta}}\\\
&\lesssim
R^{2\nu(k)-n+3}M^{-n}\min\Big{\\{}1,\Big{(}\frac{M}{R}\Big{)}^{\beta}\Big{\\}}\|b^{0}_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|^{2}_{L^{2}}.\end{split}$
$\bullet$ Case 2: $R\gg 1$. Since $\rho\sim 1$, we have $r\rho\gg 1$. We
estimate
(3.14) $\begin{split}G_{k,\ell}(R,M)&\lesssim
R^{-(n-2)}\int_{0}^{\infty}\big{|}b^{0}_{k,\ell}(M\rho)\chi(\rho)\big{|}^{2}\int_{R}^{2R}\big{|}J_{\nu(k)}(r\rho)\big{|}^{2}\frac{\mathrm{d}r}{(1+\frac{r}{M})^{\beta}}\mathrm{d}\rho.\end{split}$
$(i)$ Subcase: $R\lesssim M$. Noting that $\rho\sim 1$, we obtain by Lemma 2.2
(3.15)
$\begin{split}\int_{R}^{2R}\big{|}J_{\nu(k)}(r\rho)\big{|}^{2}\frac{\mathrm{d}r}{(1+\frac{r}{M})^{\beta}}\lesssim\int_{R}^{2R}\big{|}J_{\nu(k)}(r\rho)\big{|}^{2}\mathrm{d}r\lesssim
1.\end{split}$
$(ii)$ Subcase: $R\gg M$. Noticing that $\rho\sim 1$ again, we obtain by Lemma
2.2
(3.16)
$\begin{split}\int_{R}^{2R}\big{|}J_{\nu(k)}(r\rho)\big{|}^{2}\frac{\mathrm{d}r}{(1+\frac{r}{M})^{\beta}}\lesssim\Big{(}\frac{M}{R}\Big{)}^{\beta}\int_{R}^{2R}\big{|}J_{\nu(k)}(r\rho)\big{|}^{2}\mathrm{d}r\lesssim\Big{(}\frac{M}{R}\Big{)}^{\beta}.\end{split}$
Putting (3.15) and (3.16) into (3.14), we have
$\begin{split}G_{k,\ell}(R,M)&\lesssim\min\Big{\\{}1,\Big{(}\frac{M}{R}\Big{)}^{\beta}\Big{\\}}R^{-(n-2)}\int_{0}^{\infty}\big{|}b^{0}_{k,\ell}(M\rho)\chi(\rho)\big{|}^{2}\mathrm{d}\rho\\\
&\lesssim\min\Big{\\{}1,\Big{(}\frac{M}{R}\Big{)}^{\beta}\Big{\\}}R^{-(n-2)}M^{-n}\|b^{0}_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|^{2}_{L^{2}}.\end{split}$
Thus we prove (3.12). ∎
Now we return to prove Proposition 3.1. By Proposition 3.2, we show
$\begin{split}&\int_{\R^{n}}\int_{\R}|\partial_{t}^{\alpha}u(x,t)|^{2}\frac{\mathrm{d}t\mathrm{d}x}{(1+|x|)^{\beta}}\\\
&\lesssim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{M\in
2^{\Z}}\sum_{\\{R\in 2^{\Z}:R\lesssim
1\\}}M^{4\alpha-2}R^{2(\nu(k)+1)}\min\Big{\\{}1,\Big{(}\frac{M}{R}\Big{)}^{\beta}\Big{\\}}\|b^{0}_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|^{2}_{L^{2}}\\\
&+\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{M\in 2^{\Z}}\sum_{\\{R\in
2^{\Z}:R\gg
1\\}}M^{4\alpha-2+\beta}R^{1-\beta}\|b^{0}_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|^{2}_{L^{2}}.\end{split}$
From $\beta=1+$, one has
$\begin{split}&\sum_{M\in 2^{\Z}}\sum_{\\{R\in 2^{\Z}:R\lesssim
1\\}}M^{4\alpha-2}R^{2(\nu(k)+1)}\min\Big{\\{}1,\Big{(}\frac{M}{R}\Big{)}^{\beta}\Big{\\}}\|b^{0}_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|^{2}_{L^{2}}\\\
&\lesssim\sum_{M\in
2^{\Z}}M^{4\alpha-2+\beta}\|b^{0}_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|^{2}_{L^{2}}.\end{split}$
Since $\alpha\geq\frac{1}{2}-\frac{\beta}{4}$, we have by Lemma 2.5
$\begin{split}\int_{\R^{n}}\int_{\R}|\partial_{t}^{\alpha}u(x,t)|^{2}\frac{\mathrm{d}t\mathrm{d}x}{(1+|x|)^{\beta}}&\lesssim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\sum_{M\in
2^{\Z}}M^{4\alpha-2+\beta}\|b^{0}_{k,\ell}(\rho)\chi(\frac{\rho}{M})\rho^{\frac{n-1}{2}}\|^{2}_{L^{2}}\\\
&\leq
C\|u_{0}\|^{2}_{\dot{H}^{2\alpha-1+\frac{\beta}{2}}(\R^{n})}.\end{split}$
∎
Finally, we apply Proposition 3.1 with $\alpha=\frac{1}{2}+$ and
$\alpha=\frac{1}{2}-$ to prove Theorem 1.1.
### 3.3. Proof of Theorem 1.2.
In this subsection, we construct an example to show Theorem 1.2. The main idea
is the stationary phase argument. By (3.8), we recall
(3.17)
$\begin{split}&u(x,t)=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}Y_{k,\ell}(\theta)\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)e^{it\rho^{2}}b^{0}_{k,\ell}(\rho)\rho^{n-1}\mathrm{d}\rho,\end{split}$
where
$b^{0}_{k,\ell}(\rho)=(\mathcal{H}_{\nu}a^{0}_{k,\ell})(\rho),\quad
u_{0}(x)=u_{0}(r\theta)=\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}a^{0}_{k,\ell}(r)Y_{k,\ell}(\theta).$
In particular we choose $u_{0}(x)$ to be a radial function such that
$(\mathcal{H}_{\nu(0)}u_{0})(\xi)=\chi_{N}(|\xi|)$ where $\chi_{N}$ is a
smooth positive function supported in $J_{N}$ (to be chosen later) and $N\gg
1$. Then
(3.18)
$\begin{split}&u(x,t)=\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu(0)}(r\rho)e^{it\rho^{2}}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho.\end{split}$
Recalling the asymptotic expansion about the Bessel function
$J_{\nu}(r)=r^{-1/2}\sqrt{\frac{2}{\pi}}\cos(r-\frac{\nu\pi}{2}-\frac{\pi}{4})+O_{\nu}(r^{-3/2}),\quad\text{as}~{}r\rightarrow\infty$
with a constant depending on $\nu$ (see [21]), then we can write
(3.19)
$\begin{split}u(x,t)&={C_{\nu}}\int_{0}^{\infty}(r\rho)^{-\frac{n-1}{2}}\big{(}e^{i(r\rho-\frac{\nu\pi}{2}-\frac{\pi}{4})}-e^{-i(r\rho-\frac{\nu\pi}{2}-\frac{\pi}{4})}\big{)}e^{it\rho^{2}}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho\\\
&+C_{\nu}\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}O_{\nu}\big{(}(r\rho)^{-\frac{3}{2}}\big{)}e^{it\rho^{2}}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho.\end{split}$
Let us define
(3.20)
$I_{1}(r)=C_{\nu}e^{i(\frac{\nu\pi}{2}+\frac{\pi}{4})}\int_{0}^{\infty}(r\rho)^{-\frac{n-1}{2}}e^{i(-r\rho+t\rho^{2})}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho,$
(3.21)
$I_{2}(r)=C_{\nu}e^{-i(\frac{\nu\pi}{2}+\frac{\pi}{4})}\int_{0}^{\infty}(r\rho)^{-\frac{n-1}{2}}e^{i(r\rho+t\rho^{2})}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho,$
and
(3.22)
$I_{3}(r)=C_{\nu}\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}O_{\nu}\big{(}(r\rho)^{-\frac{3}{2}}\big{)}e^{it\rho^{2}}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho.$
Let $\phi_{r}(\rho)=t\rho^{2}-r\rho$. The fundamental idea is to choose sets
$J_{N}$ and $E\subset B^{n}$, in which $t(r)$ can be chosen, so that
$\partial_{\rho}\phi_{r}(\rho)=2t(r)\rho-r$ almost vanishes for all $\rho\in
J_{N}$ and $r\in\\{|x|:x\in E\\}$. To this end, we choose
$E=\\{x:\frac{1}{100}\leq|x|\leq\frac{1}{8}\\}~{}~{}\text{and}~{}~{}J_{N}=[N,N+2N^{\frac{1}{2}}].$
Choose $t(r)=\frac{r}{2(N+\sqrt{N})}$, then
$\partial_{\rho}\phi_{r}(N+N^{\frac{1}{2}})=0$. Thus
(3.23)
$I_{1}(r)=C_{\nu}e^{i(\frac{\nu\pi}{2}+\frac{\pi}{4})}e^{i\phi_{r}(N+\sqrt{N})}\int_{0}^{\infty}(r\rho)^{-\frac{n-1}{2}}e^{\frac{ir[\rho-(N+\sqrt{N})]^{2}}{2(N+\sqrt{N})}}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho.$
Observe that
(3.24)
$|I_{1}(r)|\geq{c_{\nu}}\int_{0}^{\infty}(r\rho)^{-\frac{n-1}{2}}\cos{\big{(}\frac{r[\rho-(N+\sqrt{N})]^{2}}{2(N+\sqrt{N})}\big{)}}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho.$
Moreover, there exists a small constant $c>0$ such that
$\cos{\big{(}\frac{r[\rho-(N+\sqrt{N})]^{2}}{2(N+\sqrt{N})}\big{)}}\geq c,$
since $|\frac{r[\rho-(N+\sqrt{N})]^{2}}{2(N+\sqrt{N})}|\leq\frac{\pi}{4}$ for
all $\rho\in J_{N}$ with $N\gg 1$ and $r\in[\frac{1}{100},\frac{1}{8}]$.
Therefore,
(3.25) $|I_{1}(r)|\geq
c_{\nu}r^{-\frac{n-1}{2}}\int_{0}^{\infty}\chi_{N}(\rho)\rho^{\frac{n-1}{2}}\mathrm{d}\rho\geq
c_{\nu}r^{-\frac{n-1}{2}}N^{\frac{n}{2}}.$
On the other hand, let $\varphi_{r}(\rho)=t\rho^{2}+r\rho$, $t=t(r)$ as
before, then $\partial_{\rho}\varphi_{r}(\rho)=2t(r)\rho+r\geq\frac{1}{200}$
when $\rho\in J_{N}$ and $r\in[\frac{1}{100},\frac{1}{8}]$. From the integral
by parts, we obtain
(3.26) $|I_{2}(r)|\leq C_{\nu}r^{-\frac{n}{2}}N^{\frac{n-2}{2}}.$
Obviously, we have
(3.27) $|I_{3}(r)|\leq C_{\nu}r^{-\frac{n}{2}}N^{\frac{n-2}{2}}.$
Combining (3.25)-(3.27), we get for $N\gg 1$ and
$r\in[\frac{1}{100},\frac{1}{8}]$
(3.28) $u^{*}(x)\geq cN^{\frac{n}{2}}.$
On the other hand, let $j_{0}=\log_{2}N$, then we obtain by the definition of
$P_{j}$ and $\tilde{P}_{j}$
$\|u_{0}(x)\|^{2}_{H^{s}}=\sum_{j}2^{2js}\|P_{j}u_{0}\|^{2}_{L^{2}}=\sum_{j}2^{2js}\|P_{j}\tilde{P}_{j_{0}}u_{0}\|^{2}_{L^{2}}.$
By Lemma 2.4, we choose
$s<\epsilon<1+\min\\{\frac{n-2}{2},(\frac{(n-2)^{2}}{4}+a)^{\frac{1}{2}}\\}$
to obtain
(3.29) $\begin{split}\|u_{0}(x)\|^{2}_{H^{s}}&\leq
C\sum_{j}2^{2js-2\epsilon|j-j_{0}|}\|u_{0}\|^{2}_{L^{2}}\\\
&=CN^{2s}\sum_{j}2^{2js-2\epsilon|j|}\|\chi_{N}\|^{2}_{L^{2}}=N^{2s+n-\frac{1}{2}}.\end{split}$
Thus, by (1.5) and (3.28), we must have $s\geq 1/4$.
### 3.4. Proof of the Theorem 1.3.
In this subsection, we show Theorem 1.3. Even though there is a loss of the
angular regularity in Theorem 1.3, the result implies the sharp result for the
radial initial data. The key ingredient here is the following lemma proved in
[9]:
###### Lemma 3.1.
Let $\tilde{J}_{\nu}(s)=s^{\frac{1}{2}}J_{\nu}(s)$ with $s\geq 0$, and let
(3.30)
$\begin{split}T_{\nu}g(r)=\int_{I}\frac{e^{it(r)\rho^{2}}\tilde{J}_{\nu}(r\rho)}{\rho^{\frac{1}{4}}}g(\rho)\mathrm{d}\rho.\end{split}$
Then
(3.31) $\begin{split}\int_{0}^{1}\big{|}T_{\nu}g(r)\big{|}^{2}\mathrm{d}r\leq
C\int_{I}|g(\rho)|^{2}\mathrm{d}\rho,\end{split}$
where the constant $C$ is independent of $g\in L^{2}(I)$, of the interval $I$,
of the measurable function $t(r)$ and of the order $\nu\geq 0$.
We also can follow the Carleson approach [5] to linearize our maximal
operator, by making t into a function of $r$, $t(r)$. By the triangle
inequality, we estimate
$\begin{split}\|u^{*}(x)\|_{L^{2}(B^{n})}\leq
C\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\Big{\|}\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu(k)}(r\rho)e^{it(r)\rho^{2}}b^{0}_{k,\ell}(\rho)\rho^{n-1}\mathrm{d}\rho\Big{\|}_{L^{2}_{r^{n-1}\mathrm{d}r}}.\end{split}$
Let $g(\rho)=b^{0}_{k,\ell}(\rho)\rho^{\frac{n-1}{2}+\frac{1}{4}}$, then
(3.32)
$\begin{split}\|u^{*}(x)\|_{L^{2}(B^{n})}\lesssim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\Big{\|}\int_{0}^{\infty}\tilde{J}_{\nu(k)}(r\rho)e^{it(r)\rho^{2}}\rho^{-\frac{1}{4}}g(\rho)\mathrm{d}\rho\Big{\|}_{L^{2}_{r}([0,1])}.\end{split}$
Using Lemma 3.1, we obtain
(3.33) $\begin{split}\|u^{*}(x)\|_{L^{2}(B^{n})}\lesssim
C\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\Big{\|}b^{0}_{k,\ell}(\rho)\rho^{\frac{n-1}{2}+\frac{1}{4}}\Big{\|}_{L^{2}_{\rho}(\R^{+})}.\end{split}$
Let $\alpha=(n-1)/2+\epsilon$ with $\epsilon>0$, we have by the Cauchy-Schwarz
inequality
$\begin{split}\|u^{*}(x)\|_{L^{2}(B^{n})}\leq
C\Big{(}\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}(1+k)^{-2\alpha}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}(1+k)^{2\alpha}\Big{\|}b^{0}_{k,\ell}(\rho)\rho^{\frac{n-1}{2}+\frac{1}{4}}\Big{\|}^{2}_{L^{2}_{\rho}(\R^{+})}\Big{)}^{\frac{1}{2}}.\end{split}$
Since $d(k)\simeq\langle k\rangle^{n-2}$, we have by Lemma 2.5
$\begin{split}\|u^{*}(x)\|_{L^{2}(B^{n})}&\lesssim\Big{(}\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}(1+k)^{2\alpha}\Big{\|}b^{0}_{k,\ell}(\rho)\rho^{\frac{n-1}{2}+\frac{1}{4}}\Big{\|}^{2}_{L^{2}_{\rho}(\R^{+})}\Big{)}^{\frac{1}{2}}\lesssim\|u_{0}\|_{H^{\frac{1}{4}}_{r}H^{\alpha}_{\theta}}.\end{split}$
This completes the proof of Theorem 1.3.
## References
* [1] J. Bourgain, Some new estimates on oscillatory integrals, Essays on Fourier Analysis in Honor of E. M. Stein (Princeton, NJ, 1991), Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, (1995), 83-112.
* [2] J. Bourgain, On the Schrödinger maximal function in higher dimension, Proceedings of the Steklov Institute of Mathematics, 280(2013), 46-60.
* [3] N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal. 203 (2003), 519-549.
* [4] N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J. 53(2004), 1665-1680.
* [5] L. Carleson, Some analytic problems related to statistical mechanics, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), Lecture Notes in Math. Vol. 779, Springer Berlin, 1980, 5-45.
* [6] Y. Cho, S. Lee and Y. Shim, A maximal inequality associated to Schrödinger type equation, Hokkaido Mathematical J. 35 (2006) 767-778.
* [7] B. E. J. Dahlberg and C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic Analysis (Minneapolis, Minn, 1981), Lecture Notes in Math. vol. 908, Springer Berlin, 1982, 205-209.
* [8] J. Cheeger, M. Taylor, On the diffraction of waves by conical singularities I, Comm. Pure Appl. Math., 35(1982), 275-331.
* [9] G. Gigante and F. Soria, On the the boundedness in $H^{1/4}$ of the maximal square function associated with the Schrödinger equation, J. Lond. Math. Soc. 77 (2008), 51-68.
* [10] H. Kalf, U. W. Schmincke, J. Walter and R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials. In Spectral theory and differential equations, 182-226. Lect. Notes in Math., 448 (1975) Springer, Berlin.
* [11] S. Lee, On pointwise convergence of the solution to Schrödinger equations in $\R^{2}$, Int. Math. Res. Not. (2006) 1-21.
* [12] A. Moyua, A. Vargas and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, Int. Math. Res. Not. 16 (1996) 793-815.
* [13] C. Miao, J. Zhang and J. Zheng, Strichartz estimates for wave equation with inverse square potential, Commun. Contemp. Math., DOI: 10.1142/S0219199713500260.
* [14] C. Miao, J. Zhang and J. Zheng, Linear Adjoint Restriction Estimates for Paraboloid, Preprint.
* [15] S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Func. Anal. 219(2005) 1-20.
* [16] F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, $L^{p}$ estimates for the wave equation with the inverse-square potential, Discrete Contin. Dynam. Systems, 9(2003), 427-442.
* [17] F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Dispersive estimate for the wave equation with the inverse-square potential. Discrete Contin. Dynam. Systems, 9(2003), 1387-1400.
* [18] K. Stempak, A Weighted uniform $L^{p}$ estimate of Bessel functions: A note on a paper of Guo, Proceedings of the AMS. 128 (2000) 2943-2945.
* [19] E.M. Stein, Some problems in harmonic analysis, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll. Williamstown, Mass., 1978), 3-20.
* [20] E.M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Mathematical Series, 43(1993), Princeton University Press, Princeton, N.J.
* [21] E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, 32 (1971), Princeton University Press, Princeton, N. J.
* [22] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699-715.
* [23] S. Shao, On localization of the Schrödinger maximal operator, Arxiv: 1006.2787v1.
* [24] T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), 1359-1384.
* [25] T. Tao and A. Vargas, A bilinear approach to cone multipliers and applications. II, Geom. Funct. Anal. 10 (2000), 216-258.
* [26] G. N. Watson, A Treatise on the Theory of Bessel Functions. Second Edition Cambridge University Press, (1944).
* [27] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. of the AMS 102 (1988), 874-878.
* [28] J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Func. Anal., 173(2000) 103-153.
|
arxiv-papers
| 2014-02-24T08:31:59 |
2024-09-04T02:49:58.699530
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Changxing Miao, Junyong Zhang and Jiqiang Zheng",
"submitter": "Junyong Zhang",
"url": "https://arxiv.org/abs/1402.5746"
}
|
1402.5786
|
# Integrated and Differentiated Sequence Spaces
Murat Kirişci Department of Mathematical Education, Hasan Ali Yücel Education
Faculty, Istanbul University, Vefa, 34470, Fatih, Istanbul, Turkey
[email protected], [email protected]
###### Abstract.
In this paper, we investigate integrated and differentiated sequence spaces
which emerge from the concept of the space $bv$ of sequences of bounded
variation.The integrated and differentiated sequence spaces which was
initiated by Goes and Goes [4]. The main propose of the present paper, we
study of matrix domains and some properties of the integrated and
differentiated sequence spaces. In section 3, we compute the alpha-, beta- and
gamma duals of these spaces. Afterward, we characterize the matrix classes of
these spaces with well-known sequence spaces.
###### Key words and phrases:
Matrix transformations, sequence spaces, $BK$-space, dual spaces, Schauder
basis, $AK$-property
###### 2010 Mathematics Subject Classification:
Primary 46A45; Secondary 46A35, 46B15.
This work was supported by Scientific Projects Coordination Unit of Istanbul
University. Project number 34465.
## 1\. Introduction
The set of all sequences denotes with
$\omega:=\mathbb{C}^{\mathbb{N}}:=\\{x=(x_{k}):x:\mathbb{N}\rightarrow\mathbb{C},k\rightarrow
x_{k}:=x(k)\\}$ where $\mathbb{C}$ denotes the complex field and
$\mathbb{N}=\\{0,1,2,\ldots\\}$. Each linear subspace of $\omega$ (with the
induced addition and scalar multiplication) is called a _sequence space_. The
following subsets of $\omega$ are obviously sequence
spaces:$\ell_{\infty}=\\{x=(x_{k})\in\omega:\sup_{k}|x_{k}|<\infty\\}$,
$c=\\{x=(x_{k})\in\omega:\lim_{k}x_{k}~{}\textrm{ exists}~{}\\}$,
$c_{0}=\\{x=(x_{k})\in\omega:\lim_{k}x_{k}=0\\}$,
$bs=\\{x=(x_{k})\in\omega:\sup_{n}|\sum_{k=1}^{n}x_{k}|<\infty\\}$,
$cs=\\{x=(x_{k})\in\omega:(\sum_{k=1}^{n}x_{k})\in c\\}$ and
$\ell_{p}=\\{x=(x_{k})\in\omega:\sum_{k}|x_{k}|^{p}<\infty,\quad 1\leq
p<\infty\\}$. These sequence spaces are Banach space with the following norms;
$\|x\|_{\ell_{\infty}}=\sup_{k}|x_{k}|$,
$\|x\|_{bs}=\|x\|_{cs}=\sup_{n}|\sum_{k=1}^{n}x_{k}|$ and
$\|x\|_{\ell_{p}}=\left(\sum_{k}|x_{k}|^{p}\right)^{1/p}$ as usual,
respectively. And also the concept of integrated and differentiated sequence
spaces was employed as $\int X=\left\\{x=(x_{k})\in\omega:(kx_{k})\in
X\right\\}$ and $d(X)=\left\\{x=(x_{k})\in\omega:(k^{-1}x_{k})\in X\right\\}$
in [4].
A sequence, whose $k-th$ term is $x_{k}$, is denoted by $x$ or $(x_{k})$. _A
coordinate space_ (or _$K-$ space_) is a vector space of numerical sequences,
where addition and scalar multiplication are defined pointwise. That is, a
sequence space $X$ with a linear topology is called a $K$-space provided each
of the maps $p_{i}:X\rightarrow\mathbb{C}$ defined by $p_{i}(x)=x_{i}$ is
continuous for all $i\in\mathbb{N}$. A $BK-$space is a $K-$space, which is
also a Banach space with continuous coordinate functionals $f_{k}(x)=x_{k}$,
$(k=1,2,...)$. A $K-$space $K$ is called an _$FK-$ space_ provided $\lambda$
is a complete linear metric space. An _$FK-$ space_ whose topology is normable
is called a _$BK-$ space_. If a normed sequence space $X$ contains a sequence
$(b_{n})$ with the property that for every $x\in X$ there is unique sequence
of scalars $(\alpha_{n})$ such that
$\displaystyle\lim_{n\rightarrow\infty}\|x-(\alpha_{0}b_{0}+\alpha_{1}b_{1}+...+\alpha_{n}b_{n})\|=0$
then $(b_{n})$ is called _Schauder basis_ (or briefly basis) for $X$. The
series $\sum\alpha_{k}b_{k}$ which has the sum $x$ is then called the
expansion of $x$ with respect to $(b_{n})$, and written as
$x=\sum\alpha_{k}b_{k}$. An _$FK-$ space_ $X$ is said to have $AK$ property,
if $\phi\subset X$ and $\\{e^{k}\\}$ is a basis for $X$, where $e^{k}$ is a
sequence whose only non-zero term is a $1$ in $k^{th}$ place for each
$k\in\mathbb{N}$ and $\phi=span\\{e^{k}\\}$, the set of all finitely non-zero
sequences.
Let $X$ and $Y$ be two sequence spaces, and $A=(a_{nk})$ be an infinite matrix
of complex numbers $a_{nk}$, where $k,n\in\mathbb{N}$. Then, we say that $A$
defines a matrix mapping from $X$ into $Y$, and we denote it by writing
$A:X\rightarrow Y$ if for every sequence $x=(x_{k})\in X$. The sequence
$Ax=\\{(Ax)_{n}\\}$, the $A$-transform of $x$, is in $Y$; where
(1.1) $\displaystyle(Ax)_{n}=\sum_{k}a_{nk}x_{k}~{}\textrm{ for each
}~{}n\in\mathbb{N}.$
For simplicity in notation, here and in what follows, the summation without
limits runs from $1$ to $\infty$. By $(X:Y)$, we denote the class of all
matrices $A$ such that $A:X\rightarrow Y$. Thus, $A\in(X:Y)$ if and only if
the series on the right side of (1.1) converges for each $n\in\mathbb{N}$ and
each $x\in X$ and we have $Ax=\\{(Ax)_{n}\\}_{n\in\mathbb{N}}\in Y$ for all
$x\in X$. A sequence $x$ is said to be $A$-summable to $l$ if $Ax$ converges
to $l$ which is called the $A$-limit of $x$.
Let $X$ is a sequence space and $A$ is an infinite matrix. The sequence space
(1.2) $\displaystyle X_{A}=\\{x=(x_{k})\in\omega:Ax\in X\\}$
is called the matrix domain of $X$ which is a sequence space(for several
examples of matrix domains, see [2] p. 49-176). By $\mathcal{F}$, we will
denote the collection of all finite subsets on $\mathbb{N}$. In [3], Başar and
Altay have defined the sequence space $bv_{p}$ which consists of all sequences
such that $\Delta$-transforms of them are in $\ell_{p}$ where $\Delta$ denotes
the matrix $\Delta=(\delta_{nk})$
$\displaystyle\delta_{nk}=\left\\{\begin{array}[]{ccl}(-1)^{n-k}&,&\quad(n-1\leq
k\leq n)\\\ 0&,&\quad(0\leq k<n-1~{}\textrm{ or }~{}k>n)\end{array}\right.$
for all $k,n\in\mathbb{N}$. And also we define the matrices
$\Gamma=(\gamma_{nk})$ and $\Sigma=(\sigma_{nk})$ by
(1.7) $\displaystyle\gamma_{nk}=\left\\{\begin{array}[]{ccl}k&,&\quad(n=k)\\\
-k&,&\quad(n-1=k)\\\ 0&,&\quad(other)\end{array}\right.$
(1.11)
$\displaystyle\sigma_{nk}=\left\\{\begin{array}[]{ccl}\frac{1}{k}&,&\quad(n=k)\\\
-\frac{1}{k}&,&\quad(n-1=k)\\\ 0&,&\quad(other)\end{array}\right.$
The integrated and differentiated sequence spaces which was initiated by Goes
and Goes [4]. In the present paper, we study of matrix domains and some
properties of the integrated and differentiated sequence spaces. In section 3,
we compute the alpha-, beta- and gamma duals of these spaces. Afterward, we
characterize matrix classes of these spaces with well-known sequence spaces.
## 2\. The Sequence Spaces $\int bv$ and $d(bv)$
The integrated spaces defined by
$\displaystyle\int\ell_{1}$ $\displaystyle=$
$\displaystyle\left\\{x=(x_{k})\in\omega:\sum_{k}|k.x_{k}|<\infty\right\\}$
$\displaystyle\int bv$ $\displaystyle=$
$\displaystyle\left\\{x=(x_{k})\in\omega:\sum_{k}|k.x_{k}-(k-1).x_{k-1}|<\infty\right\\}$
and the differentiated spaces defined by
$\displaystyle d(\ell_{1})$ $\displaystyle=$
$\displaystyle\\{x=(x_{k})\in\omega:\sum_{k}|k^{-1}.x_{k}|<\infty\\}$
$\displaystyle d(bv)$ $\displaystyle=$
$\displaystyle\\{x=(x_{k})\in\omega:\sum_{k}|k^{-1}.x_{k}-(k-1)^{-1}.x_{k-1}|<\infty\\}.$
Consider the notation (1.2) and the matrices (1.7), (1.11). From here, we can
re-define the spaces $\int bv$ and $d(bv)$ by
(2.1) $\displaystyle\left(\int\ell_{1}\right)_{\Delta}=\int bv~{}\textrm{ or
}~{}\left(\ell_{1}\right)_{\Gamma}=\int bv$
and
(2.2) $\displaystyle\left[d(\ell_{1})\right]_{\Delta}=d(bv)~{}\textrm{ or
}~{}\left(\ell_{1}\right)_{\Sigma}=d(bv).$
Let $x=(x_{k})\in\int bv$ and $\Delta x_{k}=x_{k}-x_{k-1}$. The
$\Gamma-$transform of a sequence $x=(x_{k})$ is defined by
(2.5) $\displaystyle y_{k}=(\Gamma
x)_{k}=\left\\{\begin{array}[]{ccl}x_{1}&,&\quad k=1\\\ \Delta(kx_{k})&,&\quad
k\geq 2\end{array}\right.$
where $\Gamma$ is defined by (1.7). Let $x=(x_{k})\in d(bv)$ and $\Delta
x_{k}=x_{k}-x_{k-1}$. The $\Sigma-$transform of a sequence $x=(x_{k})$ is
defined by
(2.8) $\displaystyle y_{k}=(\Sigma
x)_{k}=\left\\{\begin{array}[]{ccl}x_{2}/2&,&\quad k=2\\\
\Delta(k^{-1}x_{k})&,&\quad k\geq 3\end{array}\right.$
where $\Sigma$ is defined by (1.11).
###### Theorem 2.1.
The spaces $\int\ell_{1}$ and $d(\ell_{1})$ are $BK-$spaces with the norms
$\|x\|_{\int\ell_{1}}=\sum_{k}|kx_{k}|$ and
$\|x\|_{d(\ell_{1})}=\sum_{k}|k^{-1}x_{k}|$, respectively.
###### Proof.
Let $x=(x_{k})\in\int\ell_{1}$. We define $f_{k}(x)=x_{k}$ for all
$k\in\mathbb{N}$. Then, we have
$\displaystyle\|x\|_{\int\ell_{1}}=1.|x_{1}|+2.|x_{2}|+3.|x_{3}|+\cdots+k.|x_{k}|+\cdots$
Hence $k.|x_{k}|\leq\|x\|_{\int\ell_{1}}\Rightarrow|x_{k}|\leq
K.\|x\|_{\int\ell_{1}}\Rightarrow|f_{k}(x)|\leq K.\|x\|_{\int\ell_{1}}$. Then,
$f_{k}$ is continuous linear functional for each $k$. Thus $\int\ell_{1}$ is a
$BK-$space.
In a similar way, we can prove that the space $d(\ell_{1})$ is a $BK-$spaces.
∎
###### Lemma 2.2.
[4] The space $\int bv$ is a $BK-$space with the norm $\|x\|_{\int
bv}=\sum_{k}|\Delta(kx_{k})|$.
###### Theorem 2.3.
The space $d(bv)$ is a $BK-$space with the norm
$\|x\|_{d(bv)}=\sum_{k}|\Delta(k^{-1}x_{k})|$.
###### Proof.
Since $d(bv)=\left[d(\ell_{1})\right]_{\Delta}$ holds, $d(\ell_{1})$ is a
$BK-$space with the norm $\|x\|_{d(\ell_{1})}$ and the matrix $\Delta$ is a
triangle matrix, then Theorem 4.3.2 of Wilansky[6] gives the fact that the
space $d(bv)$ is a $BK-$space. ∎
###### Theorem 2.4.
* (i).
The spaces $\int\ell_{1}$ and $d(\ell_{1})$ have $AK-$property.
* (ii).
The spaces $\int bv$ and $d(bv)$ have $AK-$property.
###### Proof.
The fact that of the space $\int bv$ has $AK-$property was given by Goes and
Goes[4]. Then, we will only prove that the space $d(bv)$ has $AK-$property in
(ii).
Let $x=(x_{k})\in d(bv)$ and
$x^{[n]}=\\{x_{1},x_{2},\cdots,x_{n},0,0,\cdots\\}$. Hence,
$\displaystyle
x-x^{[n]}=\\{0,0,\cdots,0,x_{n+1},x_{n+2},\cdots\\}\Rightarrow\|x-x^{[n]}\|_{d(bv)}=\|0,0,\cdots,0,x_{n+1},x_{n+2},\cdots\|$
and since $x\in d(bv)$,
$\displaystyle\|x-x^{[n]}\|_{d(bv)}=\sum_{k\geq
n+1}|\Delta(k^{-1}x_{k})|\rightarrow 0~{}\textrm{ as}~{}\ n\rightarrow\infty$
$\displaystyle\Rightarrow\lim_{n\rightarrow\infty}\|x-x^{[n]}\|_{d(bv)}=0\Rightarrow
x^{[n]}\rightarrow\infty~{}\textrm{ as}~{}\ n\rightarrow\infty~{}\textrm{
in}~{}\ d(bv).$
Then the space $d(bv)$ has $AK-$property. ∎
###### Theorem 2.5.
The spaces $\int bv$ and $d(bv)$ are norm isomorphic to $\ell_{1}$.
###### Proof.
We must show that a linear bijection between the spaces $\int bv$ and
$\ell_{1}$ exists. Consider the transformation $T$ defined, with the notation
(2.5), from $\int bv$ to $\ell_{1}$ by $x\mapsto y=Tx$. The linearity of $T$
is clear. Also, it is trivial that $x=\theta$ whenever $Tx=\theta$ and
therefore, $T$ is injective.
Let $y\in\ell_{1}$ and define the sequence $x=(x_{k})$ by
$x_{k}=\frac{1}{k}.\sum_{j=1}^{k}y_{j}$. Then
$\displaystyle\|x\|_{\int bv}$ $\displaystyle=$
$\displaystyle\sum_{k}|\Delta(kx_{k})|=\sum_{k}\left|k.\frac{1}{k}\sum_{j=1}^{k}y_{j}-(k-1).\frac{1}{k-1}\sum_{j=1}^{k-1}y_{j}\right|=\sum_{k}|y_{k}|=\|y\|_{\ell_{1}}<\infty.$
Then, we have that $x\in\int bv$. So, $T$ is surjective and norm preserving.
Hence $T$ is a linear bijection. It shown us that the space $\int bv$ is norm
isomorphic to $\ell_{1}$.
As similar, using the notation (2.8), we can define the transformation $S$
from $d(bv)$ and $\ell_{1}$ by $x\mapsto y=Sx$. And also, if we choose the
sequence $x=(x_{k})$ by $x_{k}=k.\sum_{j=2}^{k}y_{j}$ while $y\in\ell_{1}$,
then we obtain the space $d(bv)$ is norm isomorphic to $\ell_{1}$ with the
norm $\|x\|_{d(bv)}$. ∎
###### Theorem 2.6.
The spaces $\int bv$ and $d(bv)$ have monotone norm.
###### Proof.
Let $x=(x_{k})\in\int bv$. We define the norms $\|x\|_{\int
bv}=\sum_{k}|\Delta(kx_{k})|$ and $\|x^{[n]}\|_{\int
bv}=\sum_{k=1}^{n}|\Delta(kx_{k})|$, for all $x\in\int bv$. For $n<m$,
$\displaystyle\|x^{[n]}\|=\sum_{k=1}^{n}|\Delta(kx_{k})|\leq\sum_{k=1}^{m}|\Delta(kx_{k})=\|x^{[m]}\|,$
that is,
(2.9) $\displaystyle\|x^{[m]}\|\geq\|x^{[n]}\|.$
The sequence $\|x^{[n]}\|$ is monotonically increasing sequence and bounded
above.
(2.10)
$\displaystyle\sup\|x^{[n]}\|=\sup\left(\sum_{k=1}^{n}|\Delta(kx_{k})|\right)=\left(\sum_{k=1}^{n}|\Delta(kx_{k})|\right)=\|x\|.$
From (2.9) and (2.10), it follows that the space $\int bv$ has the monotone
norm.
In similar way, we can obtain to the space $d(bv)$ has the monotone norm. ∎
Because of the isomorphisms $T$ and $S$, defined in the proof of Theorem 2.5,
are onto the inverse image of the basis $\\{e^{(k)}\\}_{k\in\mathbb{N}}$ of
the space $\ell_{1}$ is the basis of the spaces $\int bv$ and $d(bv)$.
Therefore, we have the following:
###### Theorem 2.7.
* (i).
Define a sequence $t^{(k)}=\\{t_{n}^{(k)}\\}_{n\in\mathbb{N}}$ of elements of
the space $\int bv$ for every fixed $k\in\mathbb{N}$ by
$\displaystyle t_{n}^{(k)}=\left\\{\begin{array}[]{ccl}1/k&,&\quad(n\geq k)\\\
0&,&\quad(n<k)\end{array}\right.$
Therefore, the sequence $\\{t^{(k)}\\}_{k\in\mathbb{N}}$ is a basis for the
space $\int bv$ and if we choose $E_{k}=(Ax)_{k}$ for all $k\in\mathbb{N}$,
where the matrix $A$ defined by (1.7), then any $x\in\int bv$ has a unique
representation of the form
$\displaystyle x=\sum_{k}E_{k}b^{(k)}.$
* (ii).
Define a sequence $s^{(k)}=\\{s_{n}^{(k)}\\}_{n\in\mathbb{N}}$ of elements of
the space $d(bv)$ for every fixed $k\in\mathbb{N}$ by
$\displaystyle s_{n}^{(k)}=\left\\{\begin{array}[]{ccl}k&,&\quad(n\geq k)\\\
0&,&\quad(n<k)\end{array}\right.$
Therefore, the sequence $\\{s^{(k)}\\}_{k\in\mathbb{N}}$ is a basis for the
space $d(bv)$ and if we choose $F_{k}=(Bx)_{k}$ for all $k\in\mathbb{N}$,
where the matrix $B$ defined by (1.11), then any $x\in d(bv)$ has a unique
representation of the form
$\displaystyle x=\sum_{k}F_{k}b^{(k)}.$
The result follows from fact that if a space has a Schauder basis, then it is
separable. Hence, we can give following corollary:
###### Corollary 2.8.
The spaces $\int bv$ and $d(bv)$ are separable.
## 3\. The $\alpha-$, $\beta-$ and $\gamma-$ Duals of the spaces $\int bv$
and $d(bv)$
In this section, we state and prove the theorems determining the $\alpha$-,
$\beta$\- and $\gamma$-duals of the sequence spaces $\int bv$ and $d(bv)$.
Let $x$ and $y$ be sequences, $X$ and $Y$ be subsets of $\omega$ and
$A=(a_{nk})_{n,k=0}^{\infty}$ be an infinite matrix of complex numbers. We
write $xy=(x_{k}y_{k})_{k=0}^{\infty}$, $x^{-1}*Y=\\{a\in\omega:ax\in Y\\}$
and $M(X,Y)=\bigcap_{x\in X}x^{-1}*Y=\\{a\in\omega:ax\in Y~{}\textrm{ for all
}~{}x\in X\\}$ for the _multiplier space_ of $X$ and $Y$. In the special cases
of $Y=\\{\ell_{1},cs,bs\\}$, we write $x^{\alpha}=x^{-1}*\ell_{1}$,
$x^{\beta}=x^{-1}*cs$, $x^{\gamma}=x^{-1}*bs$ and $X^{\alpha}=M(X,\ell_{1})$,
$X^{\beta}=M(X,cs)$, $X^{\gamma}=M(X,bs)$ for the $\alpha-$dual, $\beta-$dual,
$\gamma-$dual of $X$. By $A_{n}=(a_{nk})_{k=0}^{\infty}$ we denote the
sequence in the $n-$th row of $A$, and we write
$A_{n}(x)=\sum_{k=0}^{\infty}a_{nk}x_{k}$ $n=(0,1,...)$ and
$A(x)=(A_{n}(x))_{n=0}^{\infty}$, provided $A_{n}\in x^{\beta}$ for all $n$.
###### Lemma 3.1.
[1, Theorem 2.1] Let $\lambda,\mu$ be the BK-spaces and $B_{\mu}^{U}=(b_{nk})$
be defined via the sequence $\alpha=(\alpha_{k})\in\mu$ and triangle matrix
$U=(u_{nk})$ by
$\displaystyle b_{nk}=\sum_{j=k}^{n}\alpha_{j}u_{nj}v_{jk}$
for all $k,n\in\mathbb{N}$. Then, the inclusion
$\mu\lambda_{U}\subset\lambda_{U}$ holds if and only if the matrix
$B_{\mu}^{U}=UD_{\alpha}U^{-1}$ is in the classes $(\lambda:\lambda)$, where
$D_{\alpha}$ is the diagonal matrix defined by $[D_{\alpha}]_{nn}=\alpha_{n}$
for all $n\in\mathbb{N}$.
###### Lemma 3.2.
[1, Theorem 3.1] $B_{\mu}^{U}=(b_{nk})$ be defined via a sequence
$a=(a_{k})\in\omega$ and inverse of the triangle matrix $U=(u_{nk})$ by
$\displaystyle b_{nk}=\sum_{j=k}^{n}a_{j}v_{jk}$
for all $k,n\in\mathbb{N}$. Then,
$\displaystyle\lambda_{U}^{\beta}=\\{a=(a_{k})\in\omega:B^{U}\in(\lambda:c)\\}.$
and
$\displaystyle\lambda_{U}^{\gamma}=\\{a=(a_{k})\in\omega:B^{U}\in(\lambda:\ell_{\infty})\\}.$
###### Lemma 3.3.
Let $A=(a_{nk})$ be an infinite matrix. Then, the following statements hold:
* (i)
$A\in(\ell_{1}:\ell_{\infty})$ if and only if
(3.1) $\displaystyle\sup_{k,n\in\mathbb{N}}|a_{nk}|<\infty.$
* (ii)
$A\in(\ell_{1}:c)$ if and only if (3.1) holds, and there are
$\alpha_{k},\in\mathbb{C}$ such that
(3.2) $\displaystyle\lim_{n\rightarrow\infty}a_{nk}=\alpha_{k}~{}\textrm{ for
each }~{}k\in\mathbb{N}.$
* (iii)
$A\in(\ell_{1}:\ell_{1})$ if and only if
(3.3) $\displaystyle\sup_{k\in\mathbb{N}}\sum_{n}|a_{nk}|<\infty.$
###### Theorem 3.4.
$\left[\int bv\right]^{\alpha}=d(\ell_{1})$
###### Proof.
We take the matrix $\Gamma$ as defined by (1.7) and $\Gamma_{n}$ denotes the
sequences in the $n$th rows of the matrices $\Gamma$. We define the matrix $C$
whose rows are the product of the rows of the matrix $\Gamma^{-1}$ and the
sequence $a=(a_{n})$, i.e., $C_{n}=(\Gamma^{-1})_{n}a$. From the relation
(2.5), we obtain
(3.4) $\displaystyle
a_{n}x_{n}=\sum_{k=1}^{n}\frac{a_{n}}{n}y_{k}=(Cy)_{n}\quad\quad(n\in\mathbb{N}).$
It follows from (3.4) that $ax=(a_{n}x_{n})\in\ell_{1}$ whenever
$x=(x_{k})\in\int bv$ if and only if $Cy\in\ell_{1}$ whenever $y\in\ell_{1}$.
By using Lemma 3.3 (iii), we obtain that $\left[\int
bv\right]^{\alpha}=d(\ell_{1})$. ∎
###### Theorem 3.5.
$[d(bv)]^{\alpha}=\int\ell_{1}$
###### Proof.
As similar way in proof of Theorem 3.4, if we take the matrix $\Sigma$ as
defined by (1.11) and define the matrix $D=(d_{nk})$ with
$a_{n}x_{n}=\sum_{k=2}^{n}n.a_{n}.y_{k}=(Dy)_{n}$ for all $n\in\mathbb{N}$,
using by the relation (2.8), this gives us that
$[d(bv)]^{\alpha}=\int\ell_{1}$. ∎
###### Theorem 3.6.
$\left[\int bv\right]^{\beta}=d(bs)$
###### Proof.
Consider the equation
(3.5)
$\displaystyle\sum_{k=1}^{n}a_{k}x_{k}=\sum_{k=1}^{n}a_{k}\left(k^{-1}\sum_{j=1}^{k}y_{j}\right)=\sum_{k=1}^{n}\left(\sum_{j=k}^{n}\frac{a_{j}}{j}\right)y_{k}=(Ey)_{n}$
where $E=(e_{nk})$ is defined by
(3.8) $\displaystyle
e_{nk}=\left\\{\begin{array}[]{ccl}\sum_{j=k}^{n}j^{-1}a_{j}&,&\quad(0\leq
k\leq n)\\\ 0&,&\quad(k>n)\end{array}\right.$
for all $n,k\in\mathbb{N}$. Then we deduce from Lemma 3.3 (ii) with (3.5) that
$ax=(a_{k}x_{k})\in cs$ whenever $x=(x_{k})\in\int bv$ if and only if $Ey\in
c$ whenever $y=(y_{k})\in\ell_{1}$. Thus, $(a_{k})\in cs$ and $(a_{k})\in
d(bs)$ by (3.1) and (3.2), respectively. Since the inclusion $d(bs)\subset cs$
holds, then, we have $(a_{k})\in d(bs)$, whence $\left[\int
bv\right]^{\beta}=d(bs)$. ∎
###### Lemma 3.7.
[4] $(cs)^{\beta}=bv\Rightarrow[d(cs)]^{\beta}=\int bv$
From Theorem 3.6 and Lemma 3.7, we have,
###### Theorem 3.8.
$(bv)^{\beta}=cs\Rightarrow[d(bv)]^{\beta}=\int cs$.
###### Theorem 3.9.
$\left[\int bv\right]^{\gamma}=d(bs)$
###### Proof.
This can be obtained by analogy with the proof of Theorem 3.6 with Lemma 3.3
(i) instead of Lemma 3.3 (ii). So we omit the details. ∎
###### Theorem 3.10.
$[d(bs)]^{\gamma}=\int bv$
## 4\. Matrix Mappings on the spaces $\int bv$ and $d(bv)$
In this section, we characterize some matrix transformations on the spaces
$\int bv$ and $d(bv)$.
We shall write throughout for brevity that
$\displaystyle\overline{a}_{nk}=k^{-1}\sum_{j=k}^{\infty}a_{nj},\quad\quad\widetilde{a}_{nk}=k\sum_{j=k}^{\infty}a_{nj},$
$\displaystyle\widehat{a}_{nk}=n.a_{nk}-(n-1).a_{n-1,k},\quad\quad\overrightarrow{a}_{nk}=n^{-1}.a_{nk}-(n-1)^{-1}.a_{n-1,k}$
for all $k,n\in\mathbb{N}$.
###### Lemma 4.1.
[1] Let $X,Y$ be any two sequence spaces, $A$ be an infinite matrix and $U$ a
triangle matrix matrix.Then, $A\in(X:Y_{U})$ if and only if $UA\in(X:Y)$.
###### Theorem 4.2.
Suppose that the entries of the infinite matrices $A=(a_{nk})$ and
$F=(f_{nk})$ are connected with the relation
(4.1) $\displaystyle f_{nk}=\overline{a}_{nk}$
for all $k,n\in\mathbb{N}$ and $Y$ be any given sequence space. Then,
$A\in(\int bv:Y)$ if and only if $\\{a_{nk}\\}_{k\in\mathbb{N}}\in[\int
bv]^{\beta}$ for all $n\in\mathbb{N}$ and $F\in(\ell_{1}:Y)$.
###### Proof.
Let $Y$ be any given sequence space. Suppose that (4.1) holds between
$A=(a_{nk})$ and $F=(f_{nk})$, and take into account that the spaces $\int bv$
and $\ell_{1}$ are norm isomorphic.
Let $A\in(\int bv:Y)$ and take any $y=(y_{k})\in\ell_{1}$. Then $\Gamma F$
exists and $\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ which
yields that (4.1) is necessary and
$\\{f_{nk}\\}_{k\in\mathbb{N}}\in(\ell_{1})^{\beta}$ for each
$n\in\mathbb{N}$. Hence, $Fy$ exists for each $y\in\ell_{1}$ and thus
$\displaystyle\sum_{k}f_{nk}y_{k}=\sum_{k}a_{nk}x_{k}~{}\textrm{ for all
}~{}n\in\mathbb{N},$
we obtain that $Fy=Ax$ which leads us to the consequence $F\in(\ell_{1}:Y)$.
Conversely, let $\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for
each $n\in\mathbb{N}$ and $F\in(\ell_{1}:Y)$ hold, and take any
$x=(x_{k})\in\int bv$. Then, $Ax$ exists. Therefore, we obtain from the
equality
$\displaystyle\sum_{k=1}^{m}a_{nk}x_{k}=\sum_{k=1}^{m}\left[k^{-1}\sum_{j=k}^{m}a_{nj}\right]y_{k}~{}\textrm{
for all }~{}m,n\in\mathbb{N}$
as $m\rightarrow\infty$ that $Ax=Fy$ and this shows that $F\in(\ell_{1}:Y)$.
This completes the proof.
∎
###### Theorem 4.3.
Suppose that the entries of the infinite matrices $A=(a_{nk})$ and
$G=(g_{nk})$ are connected with the relation
(4.2) $\displaystyle g_{nk}=\widetilde{a}_{nk}$
for all $k,n\in\mathbb{N}$ and $Y$ be any given sequence space. Then,
$A\in(d(bv):Y)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in[d(bv)]^{\beta}$ for all $n\in\mathbb{N}$ and
$G\in(\ell_{1}:Y)$.
###### Theorem 4.4.
Suppose that the entries of the infinite matrices $A=(a_{nk})$ and
$H=(h_{nk})$ are connected with the relation
(4.3) $\displaystyle h_{nk}=\widehat{a}_{nk}$
for all $k,n\in\mathbb{N}$ and $Y$ be any given sequence space. Then,
$A\in(Y:\int bv)$ if and only if $M\in(Y:\ell_{1})$.
###### Proof.
Let $z=(z_{k})\in Y$ and consider the following equality
$\displaystyle\sum_{k=0}^{m}\widehat{a}_{nk}z_{k}=\sum_{k=0}^{m}(n.a_{nk}-(n-1)a_{n-1,k})z_{k}\quad~{}\textrm{
for all, }~{}m,n\in\mathbb{N}$
which yields that as $m\rightarrow\infty$ that $(Hz)_{n}=\\{\Gamma(Az)\\}_{n}$
for all $n\in\mathbb{N}$. Therefore, one can observe from here that $Az\in\int
bv$ whenever $z\in Y$ if and only if $Hz\in\ell_{1}$ whenever $z\in Y$. ∎
###### Theorem 4.5.
Suppose that the entries of the infinite matrices $A=(a_{nk})$ and
$M=(m_{nk})$ are connected with the relation
(4.4) $\displaystyle m_{nk}=\overrightarrow{a}_{nk}$
for all $k,n\in\mathbb{N}$ and $Y$ be any given sequence space. Then,
$A\in(Y:d(bv))$ if and only if $F\in(Y:\ell_{1})$.
###### Lemma 4.6.
* (i)
$A\in(\ell_{1}:bs)$ if and only if
(4.5)
$\displaystyle\sup_{k,m\in\mathbb{N}}\left|\sum_{n=0}^{m}a_{nk}\right|<\infty.$
* (ii)
$A\in(\ell_{1}:cs)$ if and only if (4.5) holds, and
(4.6) $\displaystyle\sum_{n}a_{nk}~{}\textrm{ convergent for each
}~{}k\in\mathbb{N}.$
* (iii)
$A\in(\ell_{1}:c_{0}s)$ if and only if (4.5) holds, and
(4.7) $\displaystyle\sum_{n}a_{nk}=0~{}\textrm{ for each }~{}k\in\mathbb{N}.$
###### Lemma 4.7.
* (i)
$A\in(\ell_{\infty}:\ell_{1})=(c:\ell_{1})=(c_{0}:\ell_{1})$ if and only if
(4.8) $\displaystyle\sup_{N,K\in\mathcal{F}}\left|\sum_{n\in N}\sum_{k\in
K}(a_{nk}-a_{n,k+1})\right|<\infty$
* (ii)
$A\in(bs:\ell_{1})$ if and only if
(4.9) $\displaystyle\lim_{k}a_{nk}=0~{}\textrm{ for each }~{}n\in\mathbb{N}.$
(4.10) $\displaystyle\sup_{N,K\in\mathcal{F}}\left|\sum_{n\in N}\sum_{k\in
K}(a_{nk}-a_{n,k+1})\right|<\infty$
* (iii)
$A\in(cs:\ell_{1})$ if and only if
(4.11) $\displaystyle\sup_{N,K\in\mathcal{F}}\left|\sum_{n\in N}\sum_{k\in
K}(a_{nk}-a_{n,k-1})\right|<\infty$
* (iv)
$A\in(c_{0}s:\ell_{1})$ if and only if (4.10) holds.
Now, we can give the following results:
###### Corollary 4.8.
The following statements hold:
* (i)
$A=(a_{nk})\in(\int bv:\ell_{\infty})$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (3.1) holds with $\overline{a}_{nk}$ instead of
${a}_{nk}$.
* (ii)
$A=(a_{nk})\in(\int bv:c)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (3.1) and (3.2) hold with $\overline{a}_{nk}$ instead of
${a}_{nk}$.
* (iii)
$A\in(\int bv:c_{0})$ if and only if $\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int
bv\\}^{\beta}$ for all $n\in\mathbb{N}$ and (3.1) and (3.2) hold with
$\alpha_{k}=0$ as $\overline{a}_{nk}$ instead of $a_{nk}$.
* (iv)
$A=(a_{nk})\in(\int bv:bs)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (4.5) holds with $\overline{a}_{nk}$ instead of
${a}_{nk}$.
* (v)
$A=(a_{nk})\in(\int bv:cs)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (4.5), (4.6) hold with $\overline{a}_{nk}$ instead of
${a}_{nk}$.
* (vi)
$A=(a_{nk})\in(\int bv:c_{0}s)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (4.5), (4.7) hold with $\overline{a}_{nk}$ instead of
${a}_{nk}$.
###### Corollary 4.9.
The following statements hold:
* (i)
$A=(a_{nk})\in(d(bv):\ell_{\infty})$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (3.1) holds with $\widetilde{a}_{nk}$ instead of ${a}_{nk}$.
* (ii)
$A=(a_{nk})\in(d(bv):c)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (3.1) and (3.2) hold with $\widetilde{a}_{nk}$ instead of ${a}_{nk}$.
* (iii)
$A\in(d(bv):c_{0})$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (3.1) and (3.2) hold with $\alpha_{k}=0$ as $\widetilde{a}_{nk}$ instead
of $a_{nk}$.
* (iv)
$A=(a_{nk})\in(d(bv):bs)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (4.5) holds with $\widetilde{a}_{nk}$ instead of ${a}_{nk}$.
* (v)
$A=(a_{nk})\in(d(bv):cs)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{d(bv)\\}^{\beta}$ for all $n\in\mathbb{N}$
and (4.5), (4.6) hold with $\widetilde{a}_{nk}$ instead of ${a}_{nk}$.
* (vi)
$A=(a_{nk})\in(d(bv):c_{0}s)$ if and only if
$\\{a_{nk}\\}_{k\in\mathbb{N}}\in\\{\int bv\\}^{\beta}$ for all
$n\in\mathbb{N}$ and (4.5), (4.7) hold with $\widetilde{a}_{nk}$ instead of
${a}_{nk}$.
###### Corollary 4.10.
We have:
* (i)
$A=(a_{nk})\in(\ell_{\infty}:\int bv)=(c:\int bv)=(c_{0}:\int bv)$ if and only
if (4.8) hold with $\widehat{a}_{nk}$ instead of ${a}_{nk}$.
* (ii)
$A=(a_{nk})\in(bs:\int bv)$ if and only if (4.9) and (4.10) hold with
$\widehat{a}_{nk}$ instead of ${a}_{nk}$.
* (iii)
$A=(a_{nk})\in(cs:\int bv)$ if and only if (4.11) holds with
$\widehat{a}_{nk}$ instead of ${a}_{nk}$.
* (iv)
$A=(a_{nk})\in(c_{0}s:\int bv)$ if and only if (4.10) holds with
$\widehat{a}_{nk}$ instead of ${a}_{nk}$.
###### Corollary 4.11.
We have:
* (i)
$A=(a_{nk})\in(\ell_{\infty}:d(bv))=(c:d(bv))=(c_{0}:d(bv))$ if and only if
(4.8) hold with $\overrightarrow{a}_{nk}$ instead of ${a}_{nk}$.
* (ii)
$A=(a_{nk})\in(bs:d(bv))$ if and only if (4.9) and (4.10) hold with
$\overrightarrow{a}_{nk}$ instead of ${a}_{nk}$.
* (iii)
$A=(a_{nk})\in(cs:d(bv))$ if and only if (4.11) holds with
$\overrightarrow{a}_{nk}$ instead of ${a}_{nk}$.
* (iv)
$A=(a_{nk})\in(c_{0}s:d(bv))$ if and only if (4.10) holds with
$\widehat{a}_{nk}$ instead of ${a}_{nk}$.
## 5\. Conclusion
Goes and Goes [4] introduced the integrated and differentiated sequence
spaces. Subramanian et.al. [5] gave the integrated rate space $\int\ell_{\pi}$
and studied some properties of this space. And they also characterized the
matrix classes $\left(\int\ell_{\pi}:Y\right)$, where
$Y=\\{\ell_{\infty},c,c_{0},\ell_{p},bv,bv_{0},bs,cs,\ell_{\rho},\ell_{\pi}\\}$.
There are no studies on differentiated sequence spaces.
In this paper, we studied some properties of integrated and differentiated
sequence spaces. We compute the alpha-, beta- and gamma-duals of these spaces.
For $Y=\\{\ell_{\infty},c,c_{0},bs,cs,c_{0}s\\}$, we characterize matrix
classes $(\int bv:Y),(d(bv):Y)$ and $(Y:\int bv),(Y:d(bv))$ in the last
section.
We should note from now on that the investigation of the domain of some
particular limitation matrices, namely Cesàro means of order one, Euler means
of order r, Riesz means, Nörlund means, the double band matrix $B(r,s)$, the
triple band matrix $B(r,st),$etc., in the spaces $\int bv$ and $d(bv)$ will
lead us to new results which are not comparable with the present results. If
we can choose different sequence spaces for the space $Y$, it can study new
matrix characterizations of $(\int bv:Y),(d(bv):Y)$ and $(Y:\int
bv),(Y:d(bv))$. Also the spaces $\int bv$ and $d(bv)$ can be defined by a
index $p$ and paranormed sequence spaces as $p=(p_{k})$ is a sequence of
strictly positive numbers.
## References
* [1] B. Altay and F. Başar, Certain topological properties and duals of the domain of a triangle matrix in a sequence spaces, J. Math. Analysis and Appl., 336 (2007), 632–645.
* [2] F. Başar, Summability Theory and its Applications, Bentham Science Publishers, e-books, Monographs, (2011).
* [3] F. Başar and B. Altay On the space of sequences of $p$-bounded variation and related matrix mappings, Ukranian Math. J., 55(1) (2003), 136–147.
* [4] G. Goes and S., Goes, Sequences of bounded variation and sequences of Fourier coefficients I, Math. Z., 118(1970), 93–102.
* [5] N. Subramanian, K. C. Rao and N. Gurumoorthy, Integrated rate space $\int\ell_{\pi}$, Commun. Korean Math. Soc., 22 (2007), 527–534.
* [6] A. Wilansky, Summability through Functinal Analysis, North Holland, New York, (1984).
|
arxiv-papers
| 2014-02-24T10:57:51 |
2024-09-04T02:49:58.712845
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Murat Kiri\\c{s}ci",
"submitter": "Murat Kiri\\c{s}ci",
"url": "https://arxiv.org/abs/1402.5786"
}
|
1402.5788
|
# On the fine spectrum of the forward difference operator on the Hahn space
Medi̇ne Yeşi̇lkayagi̇l and Murat Ki̇ri̇şci̇ Department of Mathematics, Uşak
University, 1 Eylül Campus, 64200 - Uşak, Turkey
[email protected] Department of Mathematical Education, Hasan
Ali Yucel Education Faculty, İstanbul University, Vefa, 34470–İstanbul, Turkey
[email protected], [email protected]
###### Abstract.
The main purpose of this paper is to determine the fine spectrum with respect
to Goldberg’s classification of the difference operator over the sequence
space $h$. As a new development, we give the approximate point spectrum,
defect spectrum and compression spectrum of the difference operator on the
sequence space $h$.
###### Key words and phrases:
Spectrum of an operator, spectral mapping theorem, the Hahn sequence space,
Goldberg’s classification.
###### 2010 Mathematics Subject Classification:
47A10, 47B37.
**Corresponding author.
## 1\. Introduction
An important branch of mathematics due to its application in other branches of
science is a Spectral Theory. It has been proved to be a very useful tool
because of its convenient and easy applicability of the different fields. In
numerical analysis, the spectral values may determine whether a discretization
of a differential equation will get the right answer or how fast a conjugate
gradient iteration will converge. In aeronautics, the spectral values may
determine whether the flow over a wing is laminar or turbulent. In electrical
engineering, it may determine the frequency response of an amplifier or the
reliability of a power system. In quantum mechanics, it may determine atomic
energy levels and thus, the frequency of a laser or the spectral signature of
a star. In structural mechanics, it may determine whether an automobile is too
noisy or whether a building will collapse in an earthquake. In ecology, the
spectral values may determine whether a food web will settle into a steady
equilibrium. In probability theory, they may determine the rate of convergence
of a Markov process.
There are several studies about the spectrum of the linear operators defined
by some triangle matrices over certain sequence spaces. This long-time
behavior was intensively studied over many years, starting with the work by
Wenger [29], who established the fine spectrum of the integer power of the
Cesàro operator in c. The generalization of [29] to the weighted mean methods
is due to Rhoades [26]. The study of the fine spectrum of the operator on the
sequence space $\ell_{p}$, $(1<p<\infty)$ was initiated by Gonzàlez [14]. The
method of the spectrum of the Cesàro operator prepared by Reade [25], Akhmedov
and Başar [1], and Okutoyi [21], respectively, whose established to this idea
on the sequence spaces $c_{0}$ and $bv$.In [31], the fine spectrum of the
Rhaly operators on the sequence spaces $c_{0}$ and $c$ was given. The spectrum
and fine spectrum for p-Cesàro operator acting on the space $c_{0}$ was
studied by Coşkun [9]. The investigation of the spectrum and the fine spectrum
of the difference operator on the sequence spaces $s_{r}$ and $c_{0}$, $c$ was
made by Malafosse [20] and Altay and Başar [4], respectively, where $s_{r}$
denotes the Banach space of all sequences $x=(x_{k})$ normed by
$\|x\|_{s_{r}}=\underset{k\in\mathbb{N}}{\sup}|x_{k}|/r^{k}$, $(r>0)$. The
idea of the fine spectrum applied to the Zweier matrix which is a band matrix
as an operator over the sequence spaces $\ell_{1}$ and $bv$ by Altay and
Karakus [5]. Let $\Delta_{\nu}$ is double sequential band matrix on $\ell_{1}$
such that $(\Delta_{\nu})_{nn}=\nu_{n}$ and $(\Delta_{\nu})_{n+1,n}=-\nu_{n}$
for all $n\in\mathbb{N}$, under certain conditions on the sequence
$\nu=(\nu_{k})$. The spectra and the fine spectra of matrix $\Delta_{\nu}$
were determined by Srivastava and Kumar [27]. Afterwards, these results of the
double sequential band matrix $\Delta_{\nu}$ generalized to the double
sequential band matrix $\Delta_{uv}$ such that defined by
$\Delta_{uv}x=(u_{n}x_{n}+v_{n-1}x_{n-1})_{n\in\mathbb{N}}$ for all
$n\in\mathbb{N}$ (see [28]). In [6], the fine spectra of the Toeplitz
operators represented by an upper and lower triangular $n$-band infinite
matrices, over the sequence spaces $c_{0}$ and $c$ was computed. The fine
spectra of upper triangular double-band matrices over the sequence spaces
$c_{0}$ and $c$ was obtained by Karakaya and Altun [16]. Let $\Delta_{a,b}$ is
a double band matrix with the convergent sequences $\widetilde{a}=(a_{k})$ and
$\widetilde{b}=(b_{k})$ having certain properties, over the sequence space
$c$. The fine spectrum of the matrix $\Delta_{a,b}$ was examined by Akhmedov
and El-Shabrawy [3]. The approach to the fine spectrum with respect to
Goldberg’s classification studied with of the operator $B(r,s,t)$ defined by a
triple band matrix over the sequence spaces $\ell_{p}$ and $bv_{p}$,
$(1<p<\infty)$ by Furkan et al. [11]. Quite recently, the fine spectrum with
respect to Goldberg’s classification of the operator defined by the lambda
matrix over the sequence spaces $c_{0}$ and $c$ computed by Yeşilkayagil and
Başar [30].
Hahn sequence space is defined as $x=(x_{k})$ such that
$\sum_{k=1}^{\infty}k|x_{k}-x_{k+1}|$ converges and $x_{k}$ is a null sequence
and is denoted by $h$. Initially, this space was defined and studied to some
general properties by Hahn [15]. It was examined different properties of this
space by Goes and Goes [12] and Rao [22], [23], [24]. Quite recently, the
studies on Hahn sequence space has been compiled by Kirisci [17]. Also in
[18], it has been defined a new Hahn sequence space by Cesàro mean.
In the present paper, our propose is to investigate the fine spectrum of the
difference operator $\Delta$ on the sequence space $h$. And also, we define
the approximate point spectrum, defect spectrum and compression spectrum of
the difference operator on the sequence space $h$, as a new approach.
## 2\. Preliminaries and Definition
Let $X$ and $Y$ be Banach spaces, and also let $T:X\rightarrow Y$ be a bounded
linear operator. The range of the operator $T$ defined by
$\displaystyle R(T)=\\{y\in Y:y=Tx,\text{ }x\in X\\}.$
The set of all bounded linear operators on $X$ into itself denoted by $B(X)$.
We choose any Banach space $X$ and let $T\in B(X)$. Then we can define the
adjoint $T^{\ast}$ of $T$ is a bounded linear operator on the dual $X^{\ast}$
of $X$ such that $\left(T^{\ast}f\right)(x)=f(Tx)$ for all $f\in Y^{\ast}$ and
$x\in X$.
Let $X\neq\\{\theta\\}$ be a non-trivial complex normed space. A linear
operator $T$ defined by $T:D(T)\rightarrow X$ on a subspace $D(T)\subseteq X$.
We do not assume that $D(T)$ is dense in $X$ or that T has closed graph
$\\{(x,Tx):x\in D(T)\\}\subseteq X\times X$. We mean by the expression ”$T$
is invertible” that there exists a bounded linear operator $S:R(T)\rightarrow
X$ for which $ST=I$ on $D(T)$ and $\overline{R(T)}=X$; such that $S=T^{-1}$ is
necessarily uniquely determined, and linear; the boundedness of $S$ means that
$T$ must be bounded below, in the sense that there is $k>0$ for which
$\|Tx\|\geq k\|x\|$ for all $x\in D(T)$. The perturbed operator defined on the
same domain $D(T)$ as $T$ as follows:
$\displaystyle T_{\alpha}=\alpha I-T$
such that associated with each complex number $\alpha$. The spectrum
$\sigma(T,X)$ consists of those $\alpha\in\mathbb{C}$ for which $T_{\alpha}$
is not invertible, and the resolvent is the mapping from the complement
$\sigma(T,X)$ of the spectrum into the algebra of bounded linear operators on
X defined by $\alpha\mapsto T_{\alpha}^{-1}$.
Let $\omega$ is the space of all complex valued sequences and $\phi$ the set
of all infinitely nonzero sequences. A linear subspace of $\omega$ which
contain $\phi$ said a sequence space. We write $\ell_{\infty}$, $c$, $c_{0}$
and $bv$ for the spaces of all bounded, convergent, null and bounded variation
sequences which are the Banach spaces with the sup-norm
$\|x\|_{\infty}=\underset{k\in\mathbb{N}}{\sup}|x_{k}|$ and
$\|x\|_{bv}=\stackrel{{\scriptstyle\infty}}{{\underset{k=0}{\sum}}}|x_{k}-x_{k+1}|$,
respectively, while $\phi$ is not a Banach space with respect to any norm,
where $\mathbb{N}=\\{0,1,2,\ldots\\}$. Also by $\ell_{p}$, we denote the space
of all $p$–absolutely summable sequences which is a Banach space with the norm
$\|x\|_{p}=\left(\stackrel{{\scriptstyle\infty}}{{\underset{k=0}{\sum}}}|x_{k}|^{p}\right)^{1/p}$,
where $1\leq p<\infty$.
Let $\mu$ and $\nu$ be two sequence spaces, and $A=(a_{nk})$ be an infinite
matrix of complex numbers $a_{nk}$, where $k,n\in\mathbb{N}$. Then, we say
that $A$ defines a matrix mapping from $\mu$ into $\nu$, and we denote it by
writing $A:\mu\rightarrow\nu$ if for every sequence $x=(x_{k})\in\mu$, the
$A$-transform $Ax=\\{(Ax)_{n}\\}$ of $x$ is in $\nu$; where
(2.1) $\displaystyle(Ax)_{n}=\sum_{k=0}^{\infty}a_{nk}x_{k}~{}\textrm{ for all
}~{}n\in\mathbb{N}.$
By $(\mu:\nu)$, we denote the class of all matrices $A$ such that
$A:\mu\rightarrow\nu$. Thus, $A\in(\mu:\nu)$ if and only if the series on the
right side of (2.1) converges for each $n\in\mathbb{N}$ and each $x\in\mu$,
and we have $Ax=\\{(Ax)_{n}\\}_{n\in\mathbb{N}}\in\nu$ for all $x\in\mu$.
The $BK-$space $h$ of all sequences $x=(x_{k})$ such that
$\displaystyle h=\left\\{x:\sum_{k=1}^{\infty}k|\Delta
x_{k}|<\infty~{}\textrm{ and }~{}\lim_{k\rightarrow\infty}x_{k}=0\right\\}$
was defined by Hahn [15]. Here and after $\Delta$ denotes the forward
difference operator, that is, $\Delta x_{k}=x_{k}-x_{k+1}$, for all
$k\in\mathbb{N}$. The following norm
$\displaystyle\|x\|_{h}=\sum_{k}k|\Delta x_{k}|+\sup_{k}|x_{k}|$
was given on the space $h$ by Hahn [15] (and also [12]). Rao [22, Proposition
2.1] defined a new norm on $h$ as $\|x\|=\sum_{k}k|\Delta x_{k}|.$
Hahn proved following properties of the space $h$:
###### Lemma 2.1.
The following statements hold:
* (i)
$h$ is a Banach space.
* (ii)
$h\subset\ell_{1}\cap\int c_{0}$.
* (iii)
$h^{\beta}=\rho_{\infty}$,
where $\int\lambda=\big{\\{}x=(x_{k})\in\omega:(kx_{k})\in\lambda\big{\\}}$
and
$\rho_{\infty}=\big{\\{}x=(x_{k})\in\omega:\sup_{n}n^{-1}\big{|}\sum_{k=1}^{n}x_{k}\big{|}<\infty\big{\\}}$.
Functional analytic properties of the $BK-$space $bv_{0}\cap d\ell_{1}$ was
studied by Goes and Goes [12], where
$d\ell_{1}=\\{x=(x_{k})\in\omega:\sum_{k=1}^{\infty}\frac{1}{k}|x_{k}|<\infty\\}$.
Also, in [12], the arithmetic means of sequences in $bv_{0}$ and $bv_{0}\cap
d\ell_{1}$ were considered, and used the fact that the Cesàro transform
$(n^{-1}\sum_{k=1}^{n}x_{k})$ of order one $x\in bv_{0}$ is a quasiconvex null
sequence.
Rao [22] studied some geometric properties of Hahn sequence space and gave the
characterizations of some classes of matrix transformations. Also, in [23] and
[24], Rao examined the different properties of Hahn sequence space.
Balasubramanian and Pandiarani [8] defined the new sequence space $h(F)$
called the Hahn sequence space of fuzzy numbers and proved that $\beta-$ and
$\gamma-$duals of $h(F)$ is the Cesàro space of the set of all fuzzy bounded
sequences.
Until the new studies of Kirişçi [17, 18], there has not been any work
containing the Hahn sequence space.
## 3\. Subdivision of the Spectrum
In this section, we define the parts called point spectrum, continuous
spectrum, residual spectrum, approximate point spectrum, defect spectrum and
compression spectrum of the spectrum. There are many different ways to
subdivide the spectrum of a bounded linear operator. Some of them are
motivated by applications to physics, in particular, quantum mechanics.
### 3.1. The Point Spectrum, Continuous Spectrum and Residual Spectrum
The name resolvent is appropriate since $T_{\alpha}^{-1}$ helps to solve the
equation $T_{\alpha}x=y$. Thus, $x=T_{\alpha}^{-1}y$ provided that
$T_{\alpha}^{-1}$ exists. More importantly, the investigation of properties of
$T_{\alpha}^{-1}$ will be basic for an understanding of the operator $T$
itself. Naturally, many properties of $T_{\alpha}$ and $T_{\alpha}^{-1}$
depend on $\alpha$, and the spectral theory is concerned with those
properties. For instance, we shall be interested in the set of all $\alpha$’s
in the complex plane such that $T_{\alpha}^{-1}$ exists. Boundedness of
$T_{\alpha}^{-1}$ is another property that will be essential. We shall also
ask for what $\alpha$’s the domain of $T_{\alpha}^{-1}$ is dense in X, to name
just a few aspects. A regular value $\alpha$ of $T$ is a complex number such
that $T_{\alpha}^{-1}$ exists and is bounded whose domain is dense in X. For
our investigation of T, $T_{\alpha}$ and $T_{\alpha}^{-1}$, we need some basic
concepts in the spectral theory which are given, as follows (see Kreyszig [19,
pp. 370-371]):
The resolvent set $\rho(T,X)$ of $T$ is the set of all regular values $\alpha$
of $T$. Furthermore, the spectrum $\sigma(T,X)$ is partitioned into the
following three disjoint sets:
The point (discrete) spectrum $\sigma_{p}(T,X)$ is the set such that
$T_{\alpha}^{-1}$ does not exist. An $\alpha\in\sigma_{p}(T,X)$ is called an
eigenvalue of $T$.
The continuous spectrum $\sigma_{c}(T,X)$ is the set such that
$T_{\alpha}^{-1}$ exists and is unbounded, and the domain of $T_{\alpha}^{-1}$
is dense in $X$.
The residual spectrum $\sigma_{r}(T,X)$ is the set such that $T_{\alpha}^{-1}$
exists (and may be bounded or not) but the domain of $T_{\alpha}^{-1}$ is not
dense in $X$.
Therefore, these three subspectra form a disjoint subdivision such that
(3.1)
$\displaystyle\sigma(T,X)=\sigma_{p}(T,X)\cup\sigma_{c}(T,X)\cup\sigma_{r}(T,X).$
To avoid trivial misunderstandings, let us say that some of the sets defined
above may be empty. This is an existence problem which we shall have to
discuss. Indeed, it is well-known that
$\sigma_{c}(T,X)=\sigma_{r}(T,X)=\emptyset$ and the spectrum $\sigma(T,X)$
consists of only the set $\sigma_{p}(T,X)$ in the finite-dimensional case.
### 3.2. The Approximate Point Spectrum, Defect Spectrum and Compression
Spectrum
In this subsection, three more subdivision of the spectrum called the
approximate point spectrum, defect spectrum and compression spectrum have been
defined as in Appell et al. [7].
Let $X$ is a Banach space and $T$ is a bounded linear operator. A $(x_{k})\in
X$ Weyl sequence for $T$ defined by $\left\|x_{k}\right\|=1$ and
$\left\|Tx_{k}\right\|\rightarrow 0$, as $k\rightarrow\infty$.
In what follows, we call the set
(3.2) $\displaystyle\sigma_{ap}(T,X):=\\{\alpha\in\mathbb{C}:\text{there
exists a Weyl sequence for }\alpha I-T\\}$
the approximate point spectrum of $T$. Moreover, the subspectrum
(3.3) $\displaystyle\sigma_{\delta}(T,X):=\\{\alpha\in\mathbb{C}:\alpha
I-T\text{ is not surjective}\\}$
is called defect spectrum of $T$.
The two subspectra given by (3.2) and (3.3) form a (not necessarily disjoint)
subdivision
$\displaystyle\sigma(T,X)=\sigma_{ap}(T,X)\cup\sigma_{\delta}(T,X)$
of the spectrum. There is another subspectrum,
$\displaystyle\sigma_{co}(T,X)=\\{\alpha\in\mathbb{C}:\overline{R(\alpha
I-T)}\neq X\\}$
which is often called compression spectrum in the literature. The compression
spectrum gives rise to another (not necessarily disjoint) decomposition
$\displaystyle\sigma(T,X)=\sigma_{ap}(T,X)\cup\sigma_{co}(T,X)$
of the spectrum. Clearly, $\sigma_{p}(T,X)\subseteq\sigma_{ap}(T,X)$ and
$\sigma_{co}(T,X)\subseteq\sigma_{\delta}(T,X)$. Moreover, comparing these
subspectra with those in (3.1) we note that
$\displaystyle\sigma_{r}(T,X)$ $\displaystyle=$
$\displaystyle\sigma_{co}(T,X)\backslash\sigma_{p}(T,X),$
$\displaystyle\sigma_{c}(T,X)$ $\displaystyle=$
$\displaystyle\sigma(T,X)\backslash[\sigma_{p}(T,X)\cup\sigma_{co}(T,X)].$
Sometimes it is useful to relate the spectrum of a bounded linear operator to
that of its adjoint. Building on classical existence and uniqueness results
for linear operator equations in Banach spaces and their adjoints are also
useful.
###### Proposition 3.1.
[7, Proposition 1.3, p. 28] The following relations on the spectrum and
subspectrum of an operator $T\in B(X)$ and its adjoint $T^{\ast}\in
B(X^{\ast})$ hold:
1. (a)
$\sigma(T^{\ast},X^{\ast})=\sigma(T,X)$.
2. (b)
$\sigma_{c}(T^{\ast},X^{\ast})\subseteq\sigma_{ap}(T,X)$.
3. (c)
$\sigma_{ap}(T^{\ast},X^{\ast})=\sigma_{\delta}(T,X)$.
4. (d)
$\sigma_{\delta}(T^{\ast},X^{\ast})=\sigma_{ap}(T,X)$.
5. (e)
$\sigma_{p}(T^{\ast},X^{\ast})=\sigma_{co}(T,X)$.
6. (f)
$\sigma_{co}(T^{\ast},X^{\ast})\supseteq\sigma_{p}(T,X)$.
7. (g)
$\sigma(T,X)=\sigma_{ap}(T,X)\cup\sigma_{p}(T^{\ast},X^{\ast})=\sigma_{p}(T,X)\cup\sigma_{ap}(T^{\ast},X^{\ast})$.
The relations (c)-(f) show that the approximate point spectrum is in a certain
sense dual to the defect spectrum and the point spectrum is dual to the
compression spectrum. The equality (g) implies, in particular, that
$\sigma(T,X)=\sigma_{ap}(T,X)$ if $X$ is a Hilbert space and $T$ is normal.
Roughly speaking, this shows that normal (in particular, self-adjoint)
operators on Hilbert spaces are most similar to matrices in finite dimensional
spaces (see Appell et al. [7]).
### 3.3. Goldberg’s Classification of Spectrum
If $X$ is a Banach space and $T\in B(X)$, then there are three possibilities
for $R(T)$:
* (A)
$\quad R(T)=X$.
* (B)
$\quad R(T)\neq\overline{R(T)}=X$.
* (C)
$\quad\overline{R(T)}\neq X$.
and
* (1)
$\quad T^{-1}$ exists and is continuous.
* (2)
$\quad T^{-1}$ exists but is discontinuous.
* (3)
$\quad T^{-1}$ does not exist.
If these possibilities are combined in all possible ways, nine different
states are created. These are labelled by: $A_{1}$, $A_{2}$, $A_{3}$, $B_{1}$,
$B_{2}$, $B_{3}$, $C_{1}$, $C_{2}$, $C_{3}$. If an operator is in state
$C_{2}$ for example, then $\overline{R(T)}\neq X$ and $T^{-1}$ exists but is
discontinuous (see Goldberg [13]).
$C_{3}$$C_{2}$$C_{1}$$B_{3}$$B_{2}$$B_{1}$$A_{3}$$A_{2}$$A_{1}$$C_{3}$$C_{2}$$C_{1}$$B_{3}$$B_{2}$$B_{1}$$A_{3}$$A_{2}$$A_{1}$$T$$T^{*}$Table
1.1: State diagram for $B(X)$ and $B(X^{\ast})$ for a non-reflective Banach
space $X$
If $\alpha$ is a complex number such that $T_{\alpha}\in A_{1}$ or
$T_{\alpha}\in B_{1}$, then $\alpha\in\rho(T,X)$. All scalar values of
$\alpha$ not in $\rho(T,X)$ comprise the spectrum of $T$. The further
classification of $\sigma(T,X)$ gives rise to the fine spectrum of $T$. That
is, $\sigma(T,X)$ can be divided into the subsets
$A_{2}\sigma(T,X)=\emptyset$, $A_{3}\sigma(T,X)$, $B_{2}\sigma(T,X)$,
$B_{3}\sigma(T,X)$, $C_{1}\sigma(T,X)$, $C_{2}\sigma(T,X)$,
$C_{3}\sigma(T,X)$. For example, if $T_{\alpha}$ is in a given state, $C_{2}$
(say), then we write $\alpha\in C_{2}\sigma(T,X)$.
By the definitions given above, we can illustrate the subdivision (3.1) in the
following table:
| | 1 | 2 | 3
---|---|---|---|---
| | $T^{-1}_{\alpha}$ exists | $T^{-1}_{\alpha}$ exists | $T^{-1}_{\alpha}$
| | and is bounded | and is unbounded | does not exist
| | | | $\alpha\in\sigma_{p}(T,X)$
A | $R(\alpha I-T)=X$ | $\alpha\in\rho(T,X)$ | – | $\alpha\in\sigma_{ap}(T,X)$
| | | $\alpha\in\sigma_{c}(T,X)$ | $\alpha\in\sigma_{p}(T,X)$
B | $\overline{R(\alpha I-T)}=X$ | $\alpha\in\rho(T,X)$ | $\alpha\in\sigma_{ap}(T,X)$ | $\alpha\in\sigma_{ap}(T,X)$
| | | $\alpha\in\sigma_{\delta}(T,X)$ | $\alpha\in\sigma_{\delta}(T,X)$
| | $\alpha\in\sigma_{r}(T,X)$ | $\alpha\in\sigma_{r}(T,X)$ | $\alpha\in\sigma_{p}(T,X)$
C | $\overline{R(\alpha I-T)}\not=X$ | $\alpha\in\sigma_{\delta}(T,X)$ | $\alpha\in\sigma_{ap}(T,X)$ | $\alpha\in\sigma_{ap}(T,X)$
| | | $\alpha\in\sigma_{\delta}(T,X)$ | $\alpha\in\sigma_{\delta}(T,X)$
| | $\alpha\in\sigma_{co}(T,X)$ | $\alpha\in\sigma_{co}(T,X)$ | $\alpha\in\sigma_{co}(T,X)$
Table 1.2: Subdivision of spectrum of a linear operator
One can observe by the closed graph theorem that in the case $A_{2}$ cannot
occur in a Banach space $X$. If we are not in the third column of Table 1.2,
i.e., if $\alpha$ is not an eigenvalue of $T$, we may always consider the
resolvent operator $T^{-1}_{\alpha}$ (on a possibly thin domain of definition)
as algebraic inverse of $\alpha I-T$.
The forward difference operator $\Delta$ is represented by the matrix
$\displaystyle\Delta=\left[\begin{array}[]{cccccc}1&-1&0&0&\ldots\\\
0&1&-1&0&\ldots\\\ 0&0&1&-1&\ldots\\\ 0&0&0&1&\ldots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right].$
###### Corollary 3.2.
$\Delta:h\rightarrow h$ is a bounded linear operator.
## 4\. On the fine spectrum of the forward difference operator on the Hahn
space
In this section, we determine the spectrum and fine spectrum of the forward
difference operator $\Delta$ on the Hahn space $h$ and calculate the norm of
the operator
###### Theorem 4.1.
$\sigma(\Delta,h)=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|\leq
1\right\\}$.
###### Proof.
Let $\left|1-\alpha\right|>1$. Since $\Delta-\alpha I$ is triangle,
$(\Delta-\alpha I)^{-1}$ exists and solving the matrix equation
$(\Delta-\alpha I)x=y$ for $x$ in terms of $y$ gives the matrix
$(\Delta-\alpha I)^{-1}=B=(b_{nk})$, where
$\displaystyle
b_{nk}=\left\\{\begin{array}[]{ccl}\frac{1}{(1-\alpha)^{n+1}}&,&0\leq k\leq
n,\\\ 0&,&k>n\end{array}\right.$
for all $k,n\in\mathbb{N}$. Thus, we observe that
$\displaystyle\|(\Delta-\alpha I)^{-1}\|_{(h:h)}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}n|b_{nk}-b_{n+1,k}|$ $\displaystyle\leq$
$\displaystyle\sum_{n=1}^{\infty}n|b_{nk}|+\sum_{n=1}^{\infty}n|b_{n+1,k}|$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{n}{|1-\alpha|^{n+1}}+\sum_{n=1}^{\infty}\frac{n}{|1-\alpha|^{n+2}}.$
From the ratio test, we have
$\displaystyle\|(\Delta-\alpha I)^{-1}\|_{(h:h)}<\infty,$
that is, $(\Delta-\alpha I)^{-1}\in(h:h)$. But for $\left|1-\alpha\right|\leq
1$,
$\displaystyle\|(\Delta-\alpha I)^{-1}\|_{(h:h)}=\infty,$
that is, $(\Delta-\alpha I)^{-1}$ is not in $B(h)$. This completes the proof.
∎
###### Theorem 4.2.
$\sigma_{p}(\Delta,h)=\emptyset$.
###### Proof.
Suppose that $\Delta x=\alpha x$ for $x\neq\theta$ in $h$. Then, by solving
the system of linear equations
$\displaystyle\begin{array}[]{rcl}x_{0}&=&\alpha x_{0},\\\
-x_{0}+x_{1}&=&\alpha x_{1}\\\ -x_{1}+x_{2}&=&\alpha x_{2}\\\ &\vdots&\\\
-x_{n-1}+x_{n}&=&\alpha x_{n}\\\ \\\ &\vdots&\end{array}$
we find that if $x_{n_{0}}$ is the first nonzero entry of the sequence
$x=(x_{n})$, then $\alpha=1$. From the equality $-x_{n_{0}}+x_{n_{0}+1}=\alpha
x_{n_{0}+1}$ we have $x_{n_{0}}$ is zero. This contradicts the fact that
$x_{n_{0}}\neq 0$, which completes the proof. ∎
###### Theorem 4.3.
$\sigma_{p}(\Delta^{\ast},h^{\ast})=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|<1\right\\}$.
###### Proof.
Suppose that $\Delta^{\ast}x=\alpha x$ for $x\neq\theta$ in
$h^{\ast}\cong\sigma_{\infty}$. Then, by solving the system of linear
equations
$\displaystyle\begin{array}[]{rcl}x_{0}-x_{1}&=&\alpha x_{0},\\\
x_{1}-x_{2}&=&\alpha x_{1},\\\ &\vdots&\\\ x_{n-1}-x_{n}&=&\alpha x_{n},\\\
&\vdots&\end{array}$
we observe that $x_{n}=(1-\alpha)^{n}x_{0}$. Therefore,
$\stackrel{{\scriptstyle}}{{\underset{n}{\sup}}}\frac{|x_{0}|}{n}\stackrel{{\scriptstyle
n}}{{\underset{k=1}{\sum}}}|1-\alpha|^{k}<\infty$ if and only if
$|1-\alpha|<1$. This step concludes the proof. ∎
If $T\in B(h)$ with the matrix $A$, then it is known that the adjoint operator
$T^{\ast}:h^{\ast}\rightarrow h^{\ast}$ is defined by the transpose $A^{t}$ of
the matrix $A$. It should be noted that the dual space $h^{\ast}$ of $h$ is
isometrically isomorphic to the Banach space $\sigma_{\infty}$ of absolutely
summable sequences normed by
$\|x\|=\stackrel{{\scriptstyle\infty}}{{\underset{k=0}{\sum}}}k|x_{k}-x_{k+1}|$.
###### Lemma 4.4.
[13, p. 59] $T$ has a dense range if and only if $T^{\ast}$ is one to one.
###### Theorem 4.5.
$\sigma_{r}(\Delta,h)=\sigma_{p}(\Delta^{\ast},h^{\ast})$.
###### Proof.
For $\left|1-\alpha\right|<1$, the operator $\Delta-\alpha I$ is triangle, so
has an inverse. But $\Delta^{\ast}-\alpha I$ is not one to one by Theorem 4.3.
Therefore by Lemma 4.4, $\overline{R(\Delta-\alpha I)}\neq h$ and this step
concludes the proof. ∎
###### Theorem 4.6.
$\sigma_{c}(\Delta,h)=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|=1\right\\}$.
###### Proof.
For $\left|1-\alpha\right|<1$, the operator $\Delta-\alpha I$ is triangle, so
has an inverse but is unbounded. Also $\Delta^{\ast}-\alpha I$ is one to one
by Theorem 4.3. By Lemma 4.4, $\overline{R(\Delta-\alpha I)}=h$. Thus, the
proof is completed. ∎
###### Theorem 4.7.
$A_{3}\sigma(\Delta,h)=B_{3}\sigma(\Delta,h)=C_{3}\sigma(\Delta,h)=\emptyset$.
###### Proof.
From Theorem 4.2 and Table 1.2.,
$A_{3}\sigma(\Delta,h)=B_{3}\sigma(\Delta,h)=C_{3}\sigma(\Delta,h)=\emptyset$
is observed. ∎
###### Theorem 4.8.
$C_{1}\sigma(\Delta,h)=\emptyset$ and $\alpha\in\sigma_{r}(\Delta,h)\cap
C_{2}\sigma(\Delta,h)$.
###### Proof.
We know $C_{1}\sigma(\Delta,h)\cup C_{2}\sigma(\Delta,h)=\sigma_{r}(\Delta,h)$
from Table 1.2. For $\alpha\in\sigma_{r}(\Delta,h)$, the operator
$(\Delta-\alpha I)^{-1}$ is unbounded by Theorem 4.1. So
$C_{1}\sigma(\Delta,h)=\emptyset$. This completes the proof. ∎
###### Theorem 4.9.
The following results hold:
1. (a)
$\sigma_{ap}(\Delta,h)=\sigma(\Delta,h)$.
2. (b)
$\sigma_{\delta}(\Delta,h)=\sigma(\Delta,h)$.
3. (c)
$\sigma_{co}(\Delta,h)=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|<1\right\\}$.
###### Proof.
(a) Since $\sigma_{ap}(\Delta,h)=\sigma(\Delta,h)\backslash
C_{1}\sigma(\Delta,h)$ from Table 1.2. and $C_{1}\sigma(\Delta,h)=\emptyset$
by Theorem 4.8, we have $\sigma_{ap}(\Delta,h)=\sigma(\Delta,h)$.
(b) Since $\sigma_{\delta}(\Delta,h)=\sigma(\Delta,h)\backslash
A_{3}\sigma(\Delta,h)$ from Table 1.2 and $A_{3}\sigma(\Delta,h)=\emptyset$ by
Theorem 4.7, we have $\sigma_{\delta}(\Delta,h)=\sigma(\Delta,h)$.
(c) Since the equality $\sigma_{co}(\Delta,h)=C_{1}\sigma(\Delta,h)\cup
C_{2}\sigma(\Delta,h)\cup C_{3}\sigma(\Delta,h)$ holds from Table 1.2, we have
$\sigma_{co}(\Delta,h)=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|<1\right\\}$
by Theorems 4.8 and 4.9. ∎
The next corollary can be obtained from Proposition 2.1.
###### Corollary 4.10.
The following results hold:
1. (a)
$\sigma_{ap}(\Delta^{\ast},\ell_{1})=\sigma(\Delta,h)$.
2. (b)
$\sigma_{\delta}(\Delta^{\ast},\ell_{1})=\left\\{\alpha:\left|\alpha-(2-\delta)^{-1}\right|=(1-\delta)(2-\delta)\right\\}\cup
E$.
3. (c)
$\sigma_{p}(\Delta^{\ast},\ell_{1})=\left\\{\alpha\in\mathbb{C}:\left|\alpha-(2-\delta)^{-1}\right|<(1-\delta)/(2-\delta)\right\\}\cup
S$.
## 5\. Conclusion
Hahn [15] defined the space $h$ and gave its some general properties. Goes and
Goes [12] studied the functional analytic properties of the space $h$. The
study on the Hahn sequence space was initiated by Rao [22] with certain
specific purpose in Banach space theory. Also Rao [22] emphasized on some
matrix transformations. Rao and Srinivasalu [23] introduced a new class of
sequence space called the semi replete space. Rao and Subramanian [24] defined
the semi Hahn space and proved that the intersection of all semi Hahn spaces
is Hahn space. Balasubramanian and Pandiarani [8] defined the new sequence
space $h(F)$ called the Hahn sequence space of fuzzy numbers and proved that
$\beta-$ and $\gamma-$duals of $h(F)$ is the Cesàro space of the set of all
fuzzy bounded sequences. The sequence space $h$ was introduced by Hahn [15]
and Goes and Goes [12], and Rao [22, 23, 24] investigated some properties of
the space $h$. Quite recently, Kirişci [17] has defined a new Hahn sequence
space by using Cesàro mean, in [18].
The difference matrix $\Delta$ was used for determining the spectrum or fine
spectrum acting as a linear operator on any of the classical sequence spaces
$c_{0}$ and $c$, $\ell_{1}$ and $bv$, $\ell_{p}$ for $(1\leq p<\infty)$,
respectively in [4], [10] and [2].
As a natural continuation of this paper, one can study the spectrum and fine
spectrum of the Cesàro operator, Weighted mean operator or another known
operators in the sequence space $h$.
## Acknowledgement
We would like to thank Professor Feyzi Başar, Fatih University, Büyükçekmece
Campus, 34500 - İstanbul, Turkey, for his careful reading and valuable
suggestions on the earlier version of this paper.
## References
* [1] A.M. Akhmedov, F. Başar, The fine spectrum of the difference operator $\Delta$ over the sequence space $bv_{p}$, $(1\leq p<\infty)$, Acta Math. Sinica Eng. Ser. 23(10)(2007), 1757–1768.
* [2] A.M. Akhmedov, F. Başar, On the fine spectra of the difference operator $\Delta$ over the sequence space $\ell_{p}$, $(1\leq p<\infty)$, Demonstratio Math. 39(3)(2006), 585–595.
* [3] A.M. Akhmedov, S.R. El-Shabrawy, On the fine spectrum of the operator $\Delta_{a,b}$ over the sequence space $c$, Comput. Math. Appl. 61(10)(2011), 2994–3002.
* [4] B. Altay, F. Başar, On the fine spectrum of the difference operator $\Delta$ on $c_{0}$ and $c$, Inform. Sci. 168(2004), 217–224.
* [5] B. Altay, M. Karakuş, On the spectrum and fine spectrum of the Zweier matrix as an operator on some sequence spaces, Thai J. Math. 3(2005), 153–162.
* [6] M. Altun, On the fine spectra of triangular Toeplitz operators, Appl. Math. Comput. 217(20)(2011), 8044–8051.
* [7] J. Appell, E. Pascale, A. Vignoli, Nonlinear Spectral Theory, de Gruyter Series in Nonlinear Analysis and Applications 10, Walter de Gruyter, Berlin, New York, 2004.
* [8] T. Balasubramanian, A. Pandiarani, The Hahn sequence spaces of fuzzy numbers, Tamsui Oxf. J. Inf. Math. Sci. 27(2)(2011), 213–224.
* [9] C. Coşkun, The spectra and fine spectra for $p$-Cesàro operators, Turkish J. Math. 21(1997), 207–212.
* [10] H. Furkan, K. Kayaduman The fine spectra of the difference operator $\delta$ over the sequence spaces $\ell_{1}$ and $bv$, Internat. Math. Forum 1(24)(2006), 1153–1160.
* [11] H. Furkan, H. Bilgiç, F. Başar, On the fine spectrum of the operator $B(r,s,t)$ over the sequence spaces $\ell_{p}$ and $bv_{p}$, $(1<p<\infty)$, Comput. Math. Appl. 60(7)(2010), 2141–2152.
* [12] G. Goes, S. Goes, Sequences of bounded variation and sequences of Fourier coefficients I, Math. Z. 118(1970), 93–102.
* [13] S. Goldberg, Unbounded Linear Operators, Dover Publications, Inc. New York, 1985.
* [14] M. Gonzàlez, The fine spectrum of the Cesàro operator in $\ell_{p}~{}(1<p<\infty)$, Arch. Math. 44(1985), 355–358.
* [15] H. Hahn, Über folgen linearer operationen, Monatsh. Math. 32(1) (1922), 3-88.
* [16] V. Karakaya, M. Altun, Fine spectra of upper triangular double-band matrices, J. Comput. Appl. Math. 234(2010) 1387-1394.
* [17] M. Kirişci A survey on the Hahn sequence spaces, Gen. Math. Notes, 19(2)(2013), 37–58.
* [18] M. Kirişci The Hahn sequence spaces defined by Cesàro mean, Abstr. Appl. Anal. 2013, Article ID 817659, 6 pages, 2013. doi:10.1155/2013/817659
* [19] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc. New York, Chichester, Brisbane, Toronto, 1978.
* [20] B. de Malafosse, Properties of some sets of sequences and application to the spaces of bounded difference sequences of order $\mu$, Hokkaido Math. J. 31(2002), 283–299.
* [21] J.I. Okutoyi, On the spectrum of $C_{1}$ as an operator on $bv$, Commun. Fac. Sci. Univ. Ank. Ser. $A_{1}$. 41(1992), 197–207.
* [22] W.C. Rao, The Hahn sequence spaces I, Bull. Calcutta Math. Soc. 82(1990), 72–78.
* [23] W.C. Rao and T.G. Srinivasalu, The Hahn sequence spaces II, Y.Y.U. J. Fac. Edu. 2(1996), 43–45.
* [24] W.C. Rao, N. Subramanian, The Hahn sequence spaces III, Bull. Malaysian Math. Sci. Soc. 25(2002), 163–171.
* [25] J.B. Reade, On the spectrum of the Cesaro operator, Bull. Lond. Math. Soc. 17(1985), 263–267.
* [26] B.E. Rhoades, The fine spectra for weighted mean operators, Pacific J. Math. 104(1)(1983), 219–230.
* [27] P.D. Srivastava, S. Kumar, Fine spectrum of the generalized difference operator $\Delta_{\nu}$ on sequence space $\ell_{1}$, Thai J. Math. 8(2)(2010), 7–19.
* [28] P.D. Srivastava, S. Kumar, Fine spectrum of the generalized difference operator $\Delta_{uv}$ on sequence space $\ell_{1}$, Appl. Math. Comput. 218(11)(2012), 6407–6414.
* [29] R.B. Wenger,The fine spectra of Hölder summability operators, Indian J. Pure Appl. Math. 6(1975), 695–712.
* [30] M. Yeşilkayagil, F. Başar, On the Fine Spectrum of the Operator Defined by the Lambda Matrix over the Spaces of Null and Convergent Sequences, Abstr. Appl. Anal. 2013, Article ID 687393, 13 pages, 2013. doi:10.1155/2013/687393.
* [31] M. Yıldırım, On the spectrum and fine spectrum of the compact Rhally operators, Indian J. Pure Appl. Math. 27(8)(1996), 779–784.
|
arxiv-papers
| 2014-02-24T11:03:03 |
2024-09-04T02:49:58.719726
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Medine Ye\\c{s}ilkayagil, Murat Kiri\\c{s}ci",
"submitter": "Murat Kiri\\c{s}ci",
"url": "https://arxiv.org/abs/1402.5788"
}
|
1402.5826
|
# Depth and Stanley Depth of the Canonical Form of a factor of monomial ideals
Adrian Popescu Adrian Popescu, Department of Mathematics, University of
Kaiserslautern, Erwin-Schrödinger-Str., 67663 Kaiserslautern, Germany
[email protected]
###### Abstract.
We introduce a so called canonical form of a factor of two monomial ideals.
The depth and the Stanley depth of such a factor is invariant under taking the
canonical form. This can be seen using a result of Okazaki and Yanagawa [7].
In the case of depth we present in this paper a different proof. It follows
easily that the Stanley Conjecture holds for the factor if and only if it
holds for its canonical form. In particular, we construct an algorithm which
simplifies the depth computation and using the canonical form we massively
reduce the run time for the sdepth computation.
The support from the Department of Mathematics of the University of
Kaiserslautern is gratefully acknowledged.
## 1\. Introduction
Let $K$ be a field and $S=K[x_{1},\ldots,x_{n}]$ be the polynomial ring over
$K$ in $n$ variables. A Stanley decomposition of a graded $S-$module $M$ is a
finite family
$\mathcal{D}=(S_{i},u_{i})_{i\in I}$
in which $u_{i}$ are homogeneous elements of $M$ and $S_{i}$ are graded
$K-$algebra retract if $S$ for all $i\in I$ such that
$S_{i}\cap\operatorname{Ann}(u_{i})=0$ and
$M=\displaystyle\bigoplus_{i\in I}S_{i}u_{i}$
as a graded $K-$vector space. The Stanley depth of $\mathcal{D}$, denoted by
$\operatorname{sdepth}(\mathcal{D})$, is the depth of the $S-$module
$\displaystyle\bigoplus_{i\in I}S_{i}u_{i}$. The Stanley depth of $M$ is
defined as
$\operatorname{sdepth}\ (M):=\operatorname{max}\\{\operatorname{sdepth}\
({\mathcal{D}})\ |\ {\mathcal{D}}\;\text{is a Stanley decomposition
of}\;I\\}.$
Another definition of sdepth using partitions is given in [4].
Stanley’s Conjecture [12] states that the Stanley depth
$\operatorname{sdepth}(M)$ is $\geq\operatorname{depth}\ (M)$.
Let $J\subsetneq I\subset S$ be two monomial ideals in $S$. In [5], Ichim et.
al. studied the sdepth and depth of the factor $\nicefrac{{I}}{{J}}$ under
polarization and reduced the Stanley’s Conjecture to the case when the ideals
are monomial squarefree. This is possible the best result from the last years
concerning Stanley’s depth. It is worth to mention that this result is not
very useful for computing sdepth since it introduces a lot of new variables.
In the squarefree case there are not many known results about the Stanley
conjecture (see for example [9]).
Another result of [5] which helps in the sdepth computation is the following
proposition, which extends [2, Lemma 1.1], [6, Lemma 2.1].
###### Proposition 1.
[5, Proposition 5.1] Let $k\in{\mathbb{N}}$ and $I^{\prime\prime}$,
$J^{\prime\prime}$ be the monomial ideals obtained from $I$, $J$ in the
following way: Each generator whose degree in $x_{n}$ is at least $k$ is
multiplied by $x_{n}$ and all other generators are taken unchanged. Then
$\operatorname{sdepth}_{S}\nicefrac{{I}}{{J}}=\operatorname{sdepth}_{S}\nicefrac{{I^{\prime\prime}}}{{J^{\prime\prime}}}$.
Inspired by this proposition we introduce a canonical form of a factor
$\nicefrac{{I}}{{J}}$ of monomial ideals (see Definition 2) and we prove
easily that sdepth is invariant under taking the canonical form (see Theorem
1). This leads us to the idea to study also the depth case (see Theorem 2).
Theorem 3 says that Stanley’s Conjecture holds for a factor of monomial ideals
if and only if it holds for its canonical form. As a side result, in the depth
(respectively sdepth) computation algorithm for $\nicefrac{{I}}{{J}}$, one can
first compute the canonical form and use the algorithm on this new much more
simpler module (see the Appendix).
In Example 3 we conclude that the $\operatorname{depth}$ and
$\operatorname{sdepth}$ algorithms are faster when considering the canonical
form: using CoCoA[1], Singular[3] and Rinaldo’s $\operatorname{sdepth}$
computation algorithm [11] we see a small decrease in the
$\operatorname{depth}$ case timing, but in the $\operatorname{sdepth}$ case
the run time is massively reduced. We hope that our algorithm together with
the one from [8] will be used very often in problems concerning monomial
ideals.
We owe thanks to Y.-H. Shen who noticed our results in a previous arXiv
version and showed us the papers of Okazaki and Yanagawa [7] and [13], because
they are strongly connected with our topic. Indeed Proposition 1 and Corollary
1 follow from [7, Theorem 5.2] (see also [7, Section 2,3]). However, our
proofs of Lemma 2 and Corollary 1 are completely different from those appeared
in the quoted papers and we keep them for the sake of our completeness.
## 2\. The canonical form of a factor of monomial ideals
Let $R=K[x_{1},\ldots,x_{n-1}]$ be the polynomial $K$-algebra over a field $K$
and $S:=R[x_{n}]$. Consider $J\subsetneq I\subset R$ two monomial ideals and
denote by $G(I)$, respectively $G(J)$, the minimal (monomial) system of
generators of $I$, respectively $J$.
###### Definition 1.
The power $x_{n}^{r}$ enters in a monomial $u$ if $x_{n}^{r}|u$ but
$x_{n}^{r+1}\nmid u$.
We say that $I$ is of type $(k_{1},\ldots,k_{s})$ with respect to $x_{n}$ if
$x_{n}^{k_{i}}$ are all the powers of $x_{n}$ which enter in a monomial of
$G(I)$ for $i\in[s]$ and $1\leq k_{1}<\ldots<k_{s}$.
$I$ is in the canonical form with respect to $x_{n}$ if $I$ is of type
$(1,\ldots,s)$ for some $s\in{\mathbb{N}}$.
We simply say that $I$ is the canonical form if it is in the canonical form
with respect to all variables $x_{1},\ldots,x_{n}$.
###### Remark 1.
Suppose that $I$ is of type $(k_{1},\ldots,k_{s})$ with respect to $x_{n}$. It
is easy to get the canonical form $I^{\prime}$ of $I$ with respect to $x_{n}$:
replace $x_{n}^{k_{i}}$ by $x_{n}^{i}$ whenever $x_{n}^{k_{i}}$ enters in a
generators of $G(I)$. Applying by recurrence this procedure for other
variables we get the canonical form of $I$, that is with respect to all
variables. Note that a squarefree monomial ideal is of type $(1)$ with respect
to each $x_{i}$ and it is in the canonical form with respect to $x_{i}$, so in
this case $I^{\prime}=I$.
###### Definition 2.
Let $J\subsetneq I\subset S$ two monomial ideals. We say that
$\nicefrac{{I}}{{J}}$ is of type $(k_{1},\ldots,k_{s})$ with respect to
$x_{n}$ if $x_{n}^{k_{i}}$ are all the powers of $x_{n}$ which enter in a
monomial of $G(I)\cup G(J)$ for $i\in[s]$ and $1\leq k_{1}<\ldots<k_{s}$. All
the terminology presented in Definition 1 will extend automatically to the
factor case. Thus we may speak about the canonical form
$\overline{\nicefrac{{I}}{{J}}}$ of $\nicefrac{{I}}{{J}}$.
###### Remark 2.
In order to compute the canonical form with respect to $x_{n}$ of the
$(k_{1},\ldots,k_{s})-$type factor $\nicefrac{{I}}{{J}}$, one will replace
$x_{n}^{k_{i}}$ by $x_{n}^{i}$ whenever $x_{n}^{k_{i}}$ enters a generator of
$G(I)\cup G(J)$.
###### Example 1.
We present some examples where we compute the canonical form of a monomial
ideal, respectively a factor of two monomial ideals.
1. (1)
Consider $S=\mathbb{Q}[x,y]$ and the monomial ideal $I=(x^{4},x^{3}y^{7})$.
Then the canonical form of $I$ is $I^{\prime}=(x^{2},xy)$.
2. (2)
Consider $S=\mathbb{Q}[x,y,z]$, $I=(x^{10}y^{5},x^{4}yz^{7},z^{7}y^{3})$ and
$J=(x^{10}y^{20}z^{2},x^{3}y^{4}z^{13},x^{9}y^{2}z^{7})$.
The canonical form of $\nicefrac{{I}}{{J}}$ is
$\overline{\nicefrac{{I}}{{J}}}=\displaystyle\frac{(x^{4}y^{5},x^{2}yz^{2},y^{3}z^{2})}{(x^{4}y^{6}z,xy^{4}z^{3},x^{3}y^{2}z^{2})}$.
The canonical form of a factor of monomial ideals $\nicefrac{{I}}{{J}}$ is not
usually the factor of the canonical forms of $I$ and $J$ as shows the
following example.
###### Example 2.
Let $S=\mathbb{Q}[x,y]$, $I=(x^{4},y^{10},x^{2}y^{7})$ be and
$J=(x^{20},y^{30})$. The canonical form of $I$ is
$I^{\prime}=(x^{2},y^{2},xy)$ and the canonical form of $J$ is
$J^{\prime}=(x,y)$. Then $J^{\prime}\not\subset I^{\prime}$. But the canonical
form of the factor $\nicefrac{{I}}{{J}}$ is
$\overline{\nicefrac{{I}}{{J}}}=\displaystyle\frac{(x^{2},y^{2},xy)}{(x^{3},y^{3})}$.
Using Proposition 1, we see that the Stanley depth of a monomial ideal does
not change when considering its canonical form.
###### Theorem 1.
Let $I$, $J$ be monomial ideals in $S$ and $\overline{\nicefrac{{I}}{{J}}}$
the canonical form of $\nicefrac{{I}}{{J}}$. Then
$\operatorname{sdepth}_{S}\nicefrac{{I}}{{J}}=\operatorname{sdepth}_{S}\overline{\nicefrac{{I}}{{J}}}.$
The proof goes applying inductively the following lemma.
###### Lemma 1.
Suppose that $\nicefrac{{I}}{{J}}$ is of type $(k_{1},\ldots,k_{s})$ with
respect to $x_{n}$ and $k_{j}+1<k_{j+1}$ for some $0\leq j<s$ (we set
$k_{0}=0$). Let $G(I^{\prime})$ (resp. $G(J^{\prime})$) be the set of
monomials obtained from $G(I)$ (resp. $G(J)$) by substituting $x_{n}^{k_{i}}$
by $x_{n}^{k_{i}-1}$ for $i>j$ whenever $x_{n}^{k_{i}}$ enters in a monomial
of $G(I)$ (resp. $G(J)$). Let $I^{\prime}$ and $J^{\prime}$ be the ideals
generated by $G(I^{\prime})$ and $G(J^{\prime})$. Then
$\operatorname{sdepth}_{S}\nicefrac{{I}}{{J}}=\operatorname{sdepth}_{S}\nicefrac{{I^{\prime}}}{{J^{\prime}}}.$
The proof of Lemma 1 follows from the proof of [5, Proposition 5.1] (see here
Proposition 1).
Next we focus on the $\operatorname{depth}\nicefrac{{I}}{{J}}$ and
$\operatorname{depth}\overline{\nicefrac{{I}}{{J}}}$. The idea of the proof of
the following lemma is taken from [10, Section 2].
###### Lemma 2.
Let $I_{0}\subset I_{1}\subset\ldots\subset I_{e}\subset R$, $J\subset S$,
$U_{0}\subset U_{1}\subset\ldots\subset U_{e}\subset R$, $V\subset S$ be some
graded ideals of $S$, respectively $R$, such that $U_{i}\subset I_{i}$ for
$0\leq i\leq e$, $I_{e}\subset J$, $V\subset J$ and $U_{e}\subset V$. Consider
$T_{k}=\displaystyle\sum_{i=0}^{e}x_{n}^{i}I_{i}S+x_{n}^{k}J$ and
$W_{k}=\displaystyle\sum_{i=0}^{e}x_{n}^{i}U_{i}S+x_{n}^{k}V$ for $k>e$. Then
$\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W_{k}}$ is constant for all
$k>e$.
###### Proof.
Consider the following linear subspaces of $S$:
$I:=\displaystyle\sum_{i=0}^{e}x_{n}^{i}I_{i}$ and
$U:=\displaystyle\sum_{i=0}^{e}x_{n}^{i}U_{i}$. Note that $I$ and $U$ are not
ideals in $S$.
If $I=U$, then the claim follows easily from the next chain of isomorphisms
$\displaystyle\frac{T_{k}}{W_{k}}\cong\displaystyle\frac{x_{n}^{k}J}{x_{n}^{k}J\cap(I+x_{n}^{k}V)S}\cong\displaystyle\frac{x_{n}^{k}J}{x_{n}^{k}(I+V)S}\cong\displaystyle\frac{J}{(I+V)S}$
for all $k>e$, and hence
$\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W_{k}}$ is constant for all
$k>e$.
Assume now that $I\neq U$ and consider the following exact sequence
$0\rightarrow\displaystyle\frac{J}{V}\xrightarrow{\cdot
x_{n}^{k}}\displaystyle\frac{T_{k}}{W_{k}}\rightarrow\displaystyle\frac{T_{k}}{W_{k}+x_{n}^{k}J}\rightarrow
0,$
where the last term we denote by $H_{k}$. Note that
$H_{k}\cong\displaystyle\frac{IS}{IS\cap(U+x_{n}^{k}J)S}$ and
$IS\cap(U+x_{n}^{k}J)S=US+x_{n}^{k}IS$. Since $x_{n}^{k}H_{k}=0$, $H_{k}$ is a
$\nicefrac{{S}}{{(x_{n}^{k})}}-$module. Then
$\operatorname{depth}_{S}H_{k}=\operatorname{depth}_{\nicefrac{{S}}{{(x_{n}^{k})}}}H_{k}=\operatorname{depth}_{R}H_{k}$
because the graded maximal ideal $m$ of $R$ generates a zero dimensional ideal
in $\nicefrac{{S}}{{(x_{n}^{k})}}$. But $H_{k}$ over $R$ is isomorphic with
$\displaystyle\frac{\oplus_{i=0}^{k-1}I_{i}}{\oplus_{i=0}^{k-1}U_{i}}\cong\bigoplus_{i=0}^{k-1}\displaystyle\frac{I_{i}}{U_{i}}$,
where $I_{i}=I_{e}$ and $U_{i}=U_{e}$ for $e<i<k$. It follows that
$t:=\operatorname{depth}_{S}H_{k}=\operatorname{min}_{i}\left\\{\operatorname{depth}_{R}\displaystyle\frac{I_{i}}{U_{i}}\right\\}$.
If $\operatorname{depth}_{S}\displaystyle\frac{J}{V}=0$, then the Depth Lemma
gives us $\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W_{k}}=t=0$ for
all $k>e$ and hence we are done. Therefore we may suppose that
$\operatorname{depth}_{S}\displaystyle\frac{J}{V}>0$. Note that $t>0$ implies
$\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W_{k}}>0$ by the Depth
Lemma since otherwise
$\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W_{k}}=\operatorname{depth}_{S}\displaystyle\frac{J}{V}=0$,
which is false. Next we will split the proof in two cases.
$\circ$ Case $t=0$.
Let ${\mathcal{F}}=\big{\\{}i\in\\{0,\ldots,e\\}\ \big{|}\
\operatorname{depth}_{R}\nicefrac{{I_{i}}}{{U_{i}}}=0\big{\\}}$ and
$L_{i}\subset I_{i}$ be the graded ideal containing $U_{i}$ such that
$\nicefrac{{L_{i}}}{{U_{i}}}\cong H_{m}^{0}(\nicefrac{{I_{i}}}{{U_{i}}})$.
If $i\in{\mathcal{F}}$ and there exists $u\in(L\cap V)\setminus U_{i}$ then
$(m^{s},x_{n}^{k})x_{n}^{i}u\subset W_{k}$ for some $s\in{\mathbb{N}}$, that
is $\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W_{k}}=0$ for all $k>e$.
Now consider the case when $L_{i}\cap V=U_{i}$ for all $i\in{\mathcal{F}}$. If
$i\in{\mathcal{F}}$ then note that $L_{i}\subset L_{j}$ for $i<j\leq e$. Set
$V^{\prime}=V+L_{e}S$,
$U^{\prime}=U+\displaystyle\sum_{i\in{\mathcal{F}}}x_{n}^{i}L_{i}$ and
$W^{\prime}_{k}:=U^{\prime}S+x_{n}^{k}V^{\prime}=U^{\prime}S+x_{n}^{k}V$
because $x_{n}^{k}L_{e}S\subset U^{\prime}S$. Consider the following exact
sequence
$0\rightarrow\displaystyle\frac{W^{\prime}_{k}}{W_{k}}\rightarrow\displaystyle\frac{T_{k}}{W_{k}}\rightarrow\displaystyle\frac{T_{k}}{W_{k}^{\prime}}\rightarrow
0.$
For the last term we have $H_{m}^{0}(\nicefrac{{I_{j}}}{{U^{\prime}_{j}}})=0$,
$0\leq j\leq e$ and so the new $t>0$, which is our next case. Thus we get
$\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W^{\prime}_{k}}>0$ is
constant for $k>e$. The first term is isomorphic to
$\displaystyle\frac{U^{\prime}S}{U^{\prime}S\cap W_{k}}$. But $U^{\prime}S\cap
W_{k}=US+(U^{\prime}S\cap x_{n}^{k}V)$ since $US\subset U^{\prime}S$. Since
$U^{\prime}S\cap(x_{n}^{k}S)=x_{n}^{k}(U_{e}+L_{e})S$ and $U_{e}\subset V$ it
follows that $U^{\prime}S\cap x_{n}^{k}V=x_{n}^{k}US+(x_{n}^{k}L_{e}S\cap
x_{n}^{k}VS)=x_{n}^{k}US$. Consequently, the first term from the above exact
sequence is isomorphic with $\displaystyle\frac{U^{\prime}S}{US}$. Note that
the annihilator of the element induced by some $u\in L_{e}\setminus V$ in
$\nicefrac{{U^{\prime}S}}{{US}}$ contains a power of $m$ and so
$\operatorname{depth}_{S}\displaystyle\frac{U^{\prime}S}{US}\leq 1$. The
inequality is equality since $x_{n}$ is regular on
$\nicefrac{{U^{\prime}S}}{{US}}$. By the Depth Lemma we get
$\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W_{k}}=1$ for all $k>e$.
$\circ$ Case $t>0$.
If $\operatorname{depth}_{R}\displaystyle\frac{J}{V}\leq
t=\operatorname{depth}_{S}H_{k}$ then the Depth Lemma gives us again the
claim, i.e.
$\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W_{k}}=\operatorname{depth}_{S}\displaystyle\frac{J}{V}$
for all $k>e$.
Assume that $\operatorname{depth}_{S}\displaystyle\frac{J}{V}>t$. Apply
induction on $t$, the initial step $t=0$ being done in the first case. Suppose
that $t>0$. Then $\operatorname{depth}_{S}\displaystyle\frac{J}{V}>t>0$
implies that $\operatorname{depth}_{S}\displaystyle\frac{J}{V}\geq 2$ and so
we may find a homogeneous polynomial $f\in m$ that is regular on
$\displaystyle\frac{J}{V}$. Moreover we may find $f$ to be regular also on all
$\displaystyle\frac{I_{i}}{U_{i}}$, $i\leq e$. Then $f$ is regular on
$\displaystyle\frac{T_{k}}{W_{k}}$. Set $V^{\prime\prime}:=V+fJ$ and
$U^{\prime\prime}_{i}:=U_{i}+fI_{i}$ for all $i\leq e$ and set
$W^{\prime\prime}_{k}:=\displaystyle\sum_{i=0}^{e}x_{n}^{i}U^{\prime\prime}_{i}S+x_{n}^{k}V^{\prime\prime}$.
By Nakayama’s Lemma we get $U^{\prime\prime}\neq U$, and therefore
$\operatorname{depth}_{R}\displaystyle\frac{I}{U^{\prime\prime}}=t-1$ and by
induction hypothesis it results that
$\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W_{k}}=1+\operatorname{depth}_{S}\displaystyle\frac{T_{k}}{W^{\prime\prime}_{k}}=$
constant for all $k>e$.
Finally, note that we may pass from the first case to the second one and
conversely. In this way $U$ increases at each step. By Noetherianity at last
we may arrive in finite steps to the case $I=U$, which was solved at the
beginning. ∎
The next corollary is in fact [5, Proposition 5.1] (see Proposition 1) for
depth. It follows easily from Lemma 2 but also from [7, Proposition 5.2] (see
also [13, Sections 2, 3].
###### Corollary 1.
Let $e\in\mathbb{N}$, $I$ and $J$ monomial ideals in
$S:=K[x_{1},\ldots,x_{n}]$. Consider $I^{\prime}$ and $J^{\prime}$ be the
monomial ideals obtained from $I$ and $J$ in the following way: each generator
whose degree in $x_{n}$ $\geq e$ is multiplied by $x_{n}$ and all the other
generators are left unchanged. Then
$\operatorname{depth}_{S}\nicefrac{{I}}{{J}}=\operatorname{depth}_{S}\nicefrac{{I^{\prime}}}{{J^{\prime}}}.$
This leads us to the equivalent result of Theorem 1 for depth.
###### Theorem 2.
Let $I$ and $J$ be two monomial ideals in $S$ and
$\overline{\nicefrac{{I}}{{J}}}$ the canonical form of $\nicefrac{{I}}{{J}}$.
Then
$\operatorname{depth}_{S}\nicefrac{{I}}{{J}}=\operatorname{depth}_{S}\overline{\nicefrac{{I}}{{J}}}.$
###### Proof.
Assume that $\nicefrac{{I}}{{J}}$ is of type $(k_{1},\ldots,k_{s})$ with
respect to $x_{n}$ and obviously $\overline{\nicefrac{{I}}{{J}}}$ is of type
$(1,2,\ldots,s)$ with respect to $x_{n}$. Starting with
$\overline{\nicefrac{{I}}{{J}}}$, we apply Corollary 1 till we obtain an
$\nicefrac{{I^{\prime}_{1}}}{{J^{\prime}_{1}}}$ of type
$(k_{1},k_{1}+1,\ldots,k_{1}+s-1)$ having the same depth as
$\overline{\nicefrac{{I}}{{J}}}$. We repeat the process until we get
$\nicefrac{{I^{\prime}_{s}}}{{J^{\prime}_{s}}}$ of type
$(k_{1},k_{2},\ldots,k_{s})$ with respect to $x_{n}$ with the unchanged depth.
Now we iterate and take the next variable. At the very end the claim will
follow. ∎
Theorem 1 and Theorem 2 give us the following theorem
###### Theorem 3.
The Stanley conjecture holds for a factor of monomial ideals
$\nicefrac{{I}}{{J}}$ if and only if it holds for its canonical form
$\overline{\nicefrac{{I}}{{J}}}$.
Using Theorem 2, instead of computing the $\operatorname{depth}$ or the
$\operatorname{sdepth}$ of $\nicefrac{{I}}{{J}}$, $J\subsetneq I\subset S$, we
can compute it for the simpler module $\overline{\nicefrac{{I}}{{J}}}$.
###### Example 3.
We present the different timings for the depth and sdepth computation
algorithms with and without extracting the canonical form. Singular[3] was
used in the depth computations while CoCoA [1] and Rinaldo’s paper[11] were
used for the Stanley depth computation.
1. (1)
Consider the ideals from Example 1(2).
Timing for $\operatorname{sdepth}\nicefrac{{I}}{{J}}$ computation: 22s.
Timing for $\operatorname{sdepth}\overline{\nicefrac{{I}}{{J}}}$ computation:
74 ms.
2. (2)
Consider $R=\mathbb{Q}[x,y,z]$ and
$I=(x^{100}yz,x^{50}yz^{50},x^{50}y^{50}z)$. Then the canonical form is
$I^{\prime}=(x^{2}yz,xyz^{2},xy^{2}z)$.
Timing for $\operatorname{sdepth}I$ computation: 13m 3s.
Timing for $\operatorname{sdepth}I^{\prime}$ computation: 21 ms.
Notice that the difference in timings is very large. Therefore using the
canonical form in the $\operatorname{sdepth}$ computation is a very important
optimization step. On the other side, the $\operatorname{depth}$ computation
is immediate in both cases. In the last example, the timing difference can be
seen.
3. (3)
Consider $R=\mathbb{Q}[x,y,z,t,v,a_{1},\ldots,a_{5}]$,
$I=(v^{4}x^{12}z^{73},v^{87}t^{21}y^{13},x^{43}y^{18}z^{72}t^{28},vxy,vyz,vzt,vtx,a_{1}^{7000},a_{2}^{413};)$,
$J=(v^{5}x^{13}z^{74},v^{88}t^{22}y^{14},x^{44}y^{19}z^{73}t^{29},v^{2}x^{2}y^{2},v^{2}y^{2}z^{2},v^{2}z^{2}t^{2},v^{2}t^{2}x^{2})$.
Timing for $\operatorname{depth}\nicefrac{{I}}{{J}}$ computation: 16m 11s.
Timing for $\operatorname{depth}\overline{\nicefrac{{I}}{{J}}}$ computation:
11m.
## 3\. Appendix
We sketch the simple idea of the algorithm which computes the canonical form
of a monomial ideal $I$. This can easily be extended to compute the canonical
form of $\nicefrac{{I}}{{J}}$ by simple applying it for $G(I)\cup G(J)$ and
afterwards extracting the generators corresponding to $I$ and $J$. This was
used in Example 3.
The algorithm is based on Remark 2: for each variable $x_{i}$ we build the
list `gp` in which we save the pair $(g,p)$, were $p$ is chosen such that
$x_{i}^{p}$ enters the $g-$generator of the monomial ideal $I$. This list will
be sorted by the powers $p$ as in the following example
###### Example 4.
Consider the ideal
$I:=(x^{13},x^{4}y^{7},y^{7}z^{10})\subset\mathbb{Q}[x,y,z]$. Then for each
variable we will obtain a different `gp` as shown below:
* $\circ$
For the first variable $x$, `gp` is equal to 2 4 1 13 . Therefore $I$ is of
type $(4,13)$ with respect to $x$. Hence, in order to obtain the canonical
form with respect to $x$, one has to divide the second generator by
$x^{4-1}=x^{3}$ and the first generator by $x^{13-2}=x^{11}$. After these
computation we will get $I_{1}=(x^{2},xy^{7},y^{7}z^{10})$. Note that $I_{1}$
is in the canonical form w.r.t. $x$.
* $\circ$
For the second variable $y$, `gp` is equal to 3 7 2 7 . Similar as above, one
has to divide the second and the third generator by $y^{6}$, and hence it
results $I_{2}=(x^{2},xy,yz^{10})$. Again, $I_{2}$ is in the canonical form
w.r.t. $y$ and $x$.
* $\circ$
For the last variable $z$, `gp` is equal to 3 10 . We divide the third
generator of $I_{2}$ by $z^{9}$ and we get our final result
$I^{\prime}=(x^{2},xy,yz)$., which is in the canonical form with respect to
all variables.
Based on the above idea, we construct two procedures: `putIn` and `canonical`
$-$ the first one constructing the list `gp`, and the second one computing the
canonical form of a monomial ideal. The proof of correctness and termination
is trivial. The procedures were written in the Singular language.
proc putIn(intvec v, int power, int nrgen)
{
if(size(v) == 1)
{
v[1] = nrgen;
v[2] = power;
return(v);
}
int i,j;
if(power <= v[2])
{
for(j = size(v)+2; j >=3; j--)
{
v[j] = v[j-2];
}
v[1] = nrgen;
v[2] = power;
return(v);
}
if(power >= v[size(v)])
{
v[size(v)+1] = nrgen;
v[size(v)+1] = power;
return(v);
}
for(j = size(v) + 2; (j>=4) && (power < v[j-2]); j = j-2)
{
v[j] = v[j-2];
v[j-1] = v[j-3];
}
v[j] = power;
v[j-1] = nrgen;
return(v);
}
proc canonical(ideal I)
{
int i,j,k;
intvec gp;
ideal m;
intvec v;
v = 0:nvars(basering);
for(i = 1; i<=nvars(basering); i++)
{
gp = 0;
v[i] = 1;
for(j = 1; j<=size(I); j++)
{
if(deg(I[j],v) >= 1)
{
gp = putIn(gp,deg(I[j],v),j);
}
}
k = 0;
if(size(gp) == 2)
{
I[gp[1]] = I[gp[1]]/(var(i)^(gp[2]-1));
}
else
{
for(j = 1; j<=size(gp)-2;)
{
k++;
I[gp[j]] = I[gp[j]]/(var(i)^(gp[j+1]-k));
j = j+2;
while((j<=size(gp)-2) && (gp[j-1] == gp[j+1]) )
{
I[gp[j]] = I[gp[j]]/(var(i)^(gp[j+1]-k));
j = j + 2;
}
}
if(j == size(gp)-1)
{
if(gp[j-1] == gp[j+1])
{
I[gp[j]] = I[gp[j]]/(var(i)^(gp[j+1]-k));
}
else
{
k++;
I[gp[j]] = I[gp[j]]/(var(i)^(gp[j+1]-k));
}
}
}
v[i] = 0;
}
return(I);
}
## References
* [1] J. Abbott, A. M. Bigatti: CoCoALib: a C++ library for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it/cocoalib
* [2] M. Cimpoeas, Stanley depth of complete intersection monomial ideals, Bull. Math. Soc. Sc. Math. Roumanie 51(99), 205-211, (2008).
* [3] W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann: Singular 3-1-6 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2012).
* [4] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322, 3151-3169, (2009).
* [5] B. Ichim, L. Katthän, J. J. Moyano-Fernández, The behaviour of Stanley depth under polarization, arXiv:AC/1401.4309, (2014).
* [6] M. Ishaq, M. I. Qureshi, Upper and lower bounds for the Stanley depth of certain classes of monomial ideals and their residue class rings, Communications in Algebra, Volume 41, 1107-1116, (2013)
* [7] R. Okazaki, K. Yanagawa: Alexander duality and Stanley depth of multigraded modules, Journal of Algebra 340: 35-52, (2011).
* [8] A. Popescu: An algorithm to compute the Hilbert depth, Journal of Symbolic Computation, 10.1016/j.jsc.2014.03.002, (2013), arXiv.org/AC/1307.6084v3
* [9] A. Popescu, D. Popescu, Four generated, squarefree, monomial ideals, to appear in Proceedings of the International Conference ”Experimental and Theoretical Methods in Algebra, Geometry, and Topology, June 20-24, 2013”, Editors Denis Ibadula, Willem Veys, Springer-Verlag, arxiv:AC/1309.4986v4, (2014)
* [10] D. Popescu, An inequality between depth and Stanley depth, Bull. Math. Soc. Sc. Math. Roumanie 52(100), 377-382, (2009).
* [11] G. Rinaldo, An algorithm to compute the Stanley depth of monomial ideals, Le Matematiche, Vol. LXIII, 243-256, (2008).
* [12] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 175-193, (1982).
* [13] K. Yanagawa, Sliding functor and polarization functor for multigraded modules, Cummunications in Algebra, 40: 1151-1166, (2012).
|
arxiv-papers
| 2014-02-24T13:57:52 |
2024-09-04T02:49:58.726603
|
{
"license": "Public Domain",
"authors": "Adrian Popescu",
"submitter": "Adrian Popescu",
"url": "https://arxiv.org/abs/1402.5826"
}
|
1402.5827
|
# A new approach to the vakonomic mechanics
Jaume Llibre1, Rafael Ramírez2 and Natalia Sadovskaia3 1 Departament de
Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona,
Catalonia, Spain. [email protected] 2 Departament d’Enginyeria Informàtica
i Matemàtiques, Universitat Rovira i Virgili, Avinguda dels Països Catalans
26, 43007 Tarragona, Catalonia, Spain. [email protected] 3
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya,
C. Pau Gargallo 5, 08028 Barcelona, Catalonia, Spain.
[email protected]
###### Abstract.
The aim of this paper is to show that the Lagrange–d’Alembert and its
equivalent the Gauss and Appel principle are not the only way to deduce the
equations of motion of the nonholonomic systems. Instead of them, here we
consider the generalization of the Hamiltonian principle for nonholonomic
systems with nonzero transpositional relations.
By applying this variational principle which takes into the account
transpositional relations different from the classical ones we deduce the
equations of motion for the nonholonomic systems with constraints that in
general are nonlinear in the velocity. These equations of motion coincide,
except perhaps in a zero Lebesgue measure set, with the classical differential
equations deduced with d’Alembert–Lagrange principle.
We provide a new point of view on the transpositional relations for the
constrained mechanical systems: the virtual variations can produce zero or
non–zero transpositional relations. In particular the independent virtual
variations can produce non–zero transpositional relations. For the
unconstrained mechanical systems the virtual variations always produce zero
transpositional relations.
We conjecture that the existence of the nonlinear constraints in the velocity
must be sought outside of the Newtonian model.
All our results are illustrated with precise examples.
###### Key words and phrases:
variational principle, generalized Hamiltonian principle, d’Alembert–Lagrange
principle, constrained Lagrangian system, transpositional relations, vakonomic
mechanic, equation of motion, Vorones system, Chapligyn system, Newton model.
###### 2010 Mathematics Subject Classification:
Primary 14P25, 34C05, 34A34.
## 1\. Introduction
The history of nonholonomic mechanical systems is long and complex and goes
back to the 19 century, with important contribution by Hertz [16] (1894) ,
Ferrers [10] (1871), Vierkandt [51] (1892) and Chaplygin [6] (1897).
The nonholonomic mechanic is a remarkable generalization of the classical
Lagrangian and Hamiltonian mechanic. The birth of the theory of dynamics of
nonholonomic systems occurred when Lagrangian-Euler formalism was found to be
inapplicable for studying the simple mechanical problem of a rigid body
rolling without slipping on a plane.
A long period of time has been needed for finding the correct equations of
motion of the nonholonomic mechanical systems and the study of the deeper
questions associated with the geometry and the analysis of these equations. In
particular the integration theory of equations of motion for nonholonomic
mechanical systems is not so complete as in the case of holonomic systems.
This is due to several reasons. First, the equations of motion of nonholonomic
systems have more complex structure than the Lagrange one, which describes the
behavior of holonomic systems. Indeed, a holonomic systems can be described by
a unique function of its state and time, the Lagrangian function. For the
nonholonomic systems this is not possible. Second, the equations of motion of
nonholonomic systems in general have no invariant measure, as they have the
equations of motion of holonomic systems (see [21, 28, 30, 50]).
One of the most important directions in the development of the nonholonomic
mechanics is the research connected with the general mathematical formalism to
describe the behavior of such systems which differs from the Lagrangian and
Hamiltonian formalism. The main problem with the equations of motion of the
nonholonomic mechanics has been centered on whether or not these equations can
be derived from the Hamiltonian principle in the usual sense, such as for the
holonomic systems (see for instance [33]). But there is not doubt that the
correct equations of motion for nonholonomic systems are given by the
d’Alembert–Lagrange principle.
The general understanding of inapplicability of Lagrange equations and
variational Hamiltonian principles to the nonholonomic systems is due to
Hertz, who expressed it in his fundamental work Die Prinzipien der Mechanik in
neuem Zusammenhaange dargestellt [16]. Hertz’s ideas were developed by
Poincaré in [39]. At the same time various aspects of nonholonomic systems
need to be studied such as
(a) The problem of the realization of nonholonomic constraints (see for
instance [22, 23]).
(b) The stability of nonholonomic systems (see for instance [35, 43]).
(c) The role of the so called transpositional relations (see [19, 34, 35, 42])
(1)
$\delta\dfrac{d\textbf{x}}{dt}-\dfrac{d}{dt}\delta{\textbf{x}}=\left(\delta\dfrac{dx_{1}}{dt}-\dfrac{d}{dt}\delta{x_{1}},\ldots,\delta\dfrac{dx_{N}}{dt}-\dfrac{d}{dt}\delta{x_{N}}\right),$
where $\dfrac{d}{dt}$ denotes the differentiation with respect to the time,
$\delta$ is the virtual variation, and
$\textbf{x}=\left(x_{1},\ldots,x_{N}\right)$ is the vector of the generalized
coordinates.
Indeed the most general formulation of the Hamiltonian principle is the
Hamilton–Suslov principle
(2)
$\displaystyle\int_{t_{0}}^{t_{1}}\left(\delta\,{\tilde{L}}-\displaystyle\sum_{j=1}^{N}\dfrac{\partial{{\tilde{L}}}}{\partial\dot{x}_{j}}\left(\delta\dfrac{dx_{j}}{dt}-\dfrac{d}{dt}\delta{x_{j}}\right)\right)dt=0,$
suitable for constrained and unscontrained Lagrangian systems, where
$\tilde{L}$ is the Lagrangian of the mechanical system.. Clearly the equations
of motion obtained from the Hamilton–Suslov principle depend on the point of
view on the transpositional relations. This fact shows the importance of these
relations.
(d) The relation between nonholonomic mechanical systems and vakonomic
mechanical systems.
There was some confusion in the literature between nonholonomic mechanical
systems and variational nonholonomic mechanical systems also called vakonomic
mechanical systems. Both kinds of systems have the same mathematical
“ingredients”: a Lagrangian function and a set of constraints. But the way in
which the equations of motion are derived differs. As we observe the equations
of motion in nonholonomic mechanic are deduced using d’Alembert–Lagrange’s
principle. In the case of vakonomic mechanics the equations of motion are
obtained through the application of a constrained variational principle. The
term vakonomic (“variational axiomatic kind”) is due to Kozlov (see [24, 25,
26]), who proposed this mechanics as an alternative set of equations of motion
for a constrained Lagrangian systems.
The distinction between the classical differential equations of motion and the
equations of motion of variational nonholonomic mechanical systems has a long
history going back to the survey article of Korteweg (1899) [20] and discussed
in a more modern context in [9, 18, 29, 49]. In these papers the authors have
discussed the domain of the vakonomic and nonholonomic mechanics. In the paper
Critics of some mathematical model to describe the behavior of mechanical
systems with differential constraints [18], Kharlamov studied the Kozlov model
and in a concrete example showed that the subset of solutions of the studied
nonholonomic systems is not included in the set of vakonomic model and proved
that the principle of determinacy is not valid in the Kozlov model. In [27]
the authors put in evidence the main differences between the d’Alembertian and
the vakonomic approaches. From the results obtained in several papers it
follows that in general the vakonomic model is not applicable to the
nonholonomic constrained Lagrangian systems.
The equations of motion for the constrained mechanical systems deduced by
Kozlov (see for instance [2]) from the Hamiltonian principle with the
Lagrangian
$L:\mathbb{R}\times{T}\textsc{Q}\times\mathbb{R}^{M}\longrightarrow\mathbb{R}$
such that $L=L_{0}-\displaystyle\sum_{j=1}^{M}\lambda_{j}L_{j},$ where
$L_{j}=0$ for $j=1,\ldots,M<N$ are the given constraints, and $L_{0}$ is the
classical Lagrangian. These equations are
(3) $E_{k}L={\dfrac{d}{dt}\dfrac{\partial
L}{\partial\dot{x}_{k}}-\dfrac{\partial
L}{\partial{x}_{k}}}=0\Longleftrightarrow
E_{k}L_{0}=\displaystyle\sum_{j=1}^{M}\left(\lambda_{j}E_{k}\,L_{j}+\dfrac{d\lambda_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial\dot{x}_{k}}\right),$
for $k=1,\ldots,N,$ see for more details [2]. Clearly, equations (3) differ
from the classical equations by the presence of the terms
$\lambda_{j}E_{k}\,L_{j}.$ If the constraints are integrable, i.e.
$L_{j}=\dfrac{d}{dt}g_{j}(t,\textbf{x}),$ then the vakonomic mechanics reduces
to the holonomic one.
In this paper we give a modification of the vakonomic mechanics. This
modification is valid for the holonomic and nonholonomic constrained
Lagrangian systems. We apply the generalized constrained Hamiltonian principle
with non–zero transpositional relations. By applying this constrained
variational principle we deduce the equations of motion for the nonholonomic
systems with constraints which in general are nonlinear in the velocity. These
equations coincide, except perhaps in a zero Lebesgue measure set, with the
classical differential equations deduced from d’Alembert–Lagrange principle.
## 2\. Statement of the main results
In this paper we solve the following inverse problem of the constrained
Lagrangian systems (see [31])
We consider the constrained Lagrangian systems with configuration space Q and
phase space $T\textsc{Q}.$
Let
$L:\mathbb{R}\times{T\textsc{Q}}\times{\mathbb{R}^{M}}\longrightarrow\mathbb{R}$
be a smooth function such that
(4)
${L}\left(t,\textbf{x},\dot{\textbf{x}},\Lambda\right)=L_{0}\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=1}^{M}\lambda_{j}\,L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=M+1}^{N}\lambda^{0}_{j}L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right),$
where $\Lambda=\left(\lambda_{1},\ldots,\lambda_{M}\right)$ are the additional
coordinates (Lagrange multipliers),
$L_{j}:\mathbb{R}\times{T\textsc{Q}}\longrightarrow\mathbb{R},\quad\left(t,\textbf{x},\dot{\textbf{x}}\right)\longmapsto\,L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right),$
be smooth functions for $j=0,\ldots,N,$ where $L_{0}$ is the nonsingular
function i.e.
$\det\left(\dfrac{\partial^{2}L_{0}}{\partial{\dot{x}_{k}}\partial{\dot{x}_{j}}}\right)\neq{0},$
and $L_{j}=0,$ for $j=1,\ldots,M,$ are the constraints satisfying
(5)
$\mbox{rank}\left(\dfrac{\partial(L_{1},\ldots,L_{M})}{\partial(\dot{x}_{1},\ldots,\dot{x}_{N})}\right)=M$
in all the points of $\mathbb{R}\times T\textsc{Q},$ except perhaps in a zero
Lebesgue measure set, $L_{j}$ and $\lambda^{0}_{j}$ are arbitrary functions
and constants respectively, for $j=M+1,\ldots,N$.
We must determine the smooth functions $L_{j},$ constants $\lambda^{0}_{j}$
for $j=M+1,\ldots,N$ and the matrix $A$ in such a way that the differential
equations describing the behavior of the constrained Lagrangian systems and
obtained from the the Hamiltonian principle
(6)
$\displaystyle\displaystyle\int_{t_{0}}^{t_{1}}\delta\,{L}=\displaystyle\int_{t_{0}}^{t_{1}}\left(\dfrac{\partial
L}{\partial x_{j}}\delta x_{j}+\dfrac{\partial
L}{\partial\dot{x}_{j}}\dfrac{d}{dt}{\delta
x}_{j}+\displaystyle\sum_{j=1}^{N}\dfrac{\partial{{L}}}{\partial\dot{x}_{j}}\left(\delta\dfrac{dx_{j}}{dt}-\dfrac{d}{dt}\delta{x_{j}}\right)\right)dt=0,$
with transpositional relation given by
(7)
$\delta\dfrac{d\textbf{x}}{dt}-\dfrac{d}{dt}\delta{\textbf{x}}=A\left(t,\textbf{x},\dot{\textbf{x}},\ddot{\textbf{x}}\right)\delta{\textbf{x}},$
where
$A=A\left(t,\textbf{x},\dot{\textbf{x}},\ddot{\textbf{x}}\right)=\left(A_{\nu\,j}\left(t,\textbf{x},\dot{\textbf{x}},\ddot{\textbf{x}}\right)\right)$
is a $N\times N$ matrix,
We give the solutions of this problem in two steps. First we obtain the
differential equations along the solutions satisfying (6). Second we shall
contrast the obtained equations and classical differential equations which
described the behavior of the constrained mechanical systems. The solution of
this inverse problem is presented in section 4.
Note that the function $L$ is singular, due to the absence of $\dot{\lambda}.$
We observe that the arbitrariness of the functions $L_{j},$ of the constants
$\lambda^{0}_{j}$ for $j=M+1,\ldots,N,$ and of the matrix $A$ will play a
fundamental role in the construction of the mathematical model which we
propose in this paper.
Our main results are the following
###### Theorem 1.
We assume that $\delta{x_{\nu}(t)},\quad\nu=1,\ldots,N,$ are arbitrary
functions defined in the interval $[t_{0},\,t_{1}]$, smooth in the interior of
$[t_{0},\,t_{1}]$ and vanishing at its endpoints, i.e.,
$\delta{x_{\nu}}({t_{0}})=\delta{x_{\nu}}({t_{1}})=0.$ If (7) holds then the
path $\gamma(t)=(x_{1}(t),\ldots,x_{N}(t))$ compatible with the constraints
$L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)=0$, for $j=1,\ldots,M$
satisfies (6) with $L$ given by the formula (4) if and only if it is a
solution of the differential equations
(8)
$D_{\nu}L:={E_{\nu}\,L-\displaystyle\sum_{j=1}^{N}{A_{{\nu}j}\dfrac{\partial{L}}{\partial{\dot{x}_{j}}}}}=0,\quad\dfrac{\partial
L}{\partial\lambda_{k}}=-L_{k}=0,$
for $\nu=1,\ldots,N,$ and $k=1,\ldots,M,$ where
$E_{\nu}={\dfrac{d}{dt}\dfrac{\partial}{\partial\dot{x}_{\nu}}-\dfrac{\partial}{\partial{x}_{\nu}}}.$
System (8) is equivalent to the following two differential systems
(9)
$\begin{array}[]{rl}D_{\nu}L_{0}=&\displaystyle\sum_{j=1}^{M}\left(\lambda_{j}D_{\nu}L_{j}+\dfrac{d\lambda_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}\right)+\displaystyle\sum_{j=M+1}^{N}\lambda^{0}_{j}D_{\nu}\,L_{j},\quad
L_{k}=0\Longleftrightarrow\vspace{0.2cm}\\\
E_{\nu}L_{0}=&\displaystyle\sum_{k=1}^{N}A_{jk}\dfrac{\partial
L_{0}}{\partial\dot{x}_{k}}+\sum_{j=1}^{M}\left(\lambda_{j}D_{\nu}L_{j}+\dfrac{d\lambda_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}\right)+\displaystyle\sum_{j=M+1}^{N}\lambda^{0}_{j}D_{\nu}\,L_{j},\quad
L_{k}=0.\end{array}$
for $\nu=1,\ldots,N$ and $k=1,\ldots,M.$
###### Theorem 2.
Using the notation of Theorem 1 let
(10)
$L=L\left(t,\textbf{x},\dot{\textbf{x}},\Lambda\right)=L_{0}\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=1}^{M}\lambda_{j}\,L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=M+1}^{N}\lambda^{0}_{j}L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)$
be the Lagrangian and let $L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)=0$
be the independent constraints for $j=1,\ldots,M<N,$ and let $\lambda^{0}_{k}$
be the arbitrary constants for $k=M+1,\ldots,N,$
$L_{k}:\mathbb{R}\times\,T\textsc{Q}\longrightarrow\mathbb{R}$ for
$k=M+1,\ldots,N$ arbitrary functions such that
$|W_{1}|=\det{W_{1}}=\det{\left(\dfrac{\partial(L_{1},\ldots,L_{N})}{\partial(\dot{x}_{1},\ldots,\dot{x}_{N})}\right)}\neq
0,$
except perhaps in a zero Lebesgue measure set $|W_{1}|=0$. We determine the
matrix $A$ satisfying
(11)
$W_{1}A=\Omega_{1}:=\left(\begin{array}[]{ccc}E_{1}L_{1}&\ldots&E_{N}L_{1}\\\
\vdots&\ldots&\vdots\\\ \vdots&\ldots&\vdots\\\
E_{1}L_{N}&\ldots&E_{N}L_{N}\\\ \end{array}\right).$
Then the differential equations (9) become
(12)
$\begin{array}[]{rl}D_{\nu}L_{0}=&\displaystyle\sum_{\alpha=1}^{M}\dot{\lambda}_{\alpha}\dfrac{\partial{L_{\alpha}}}{\partial{\dot{x}_{\nu}}}\quad\mbox{for}\quad\nu=1,\ldots,N\vspace{0.30cm}\\\
\Longleftrightarrow&\dfrac{d}{dt}\dfrac{\partial
L_{0}}{\partial\dot{\textbf{x}}}-\dfrac{\partial
L_{0}}{\partial{\textbf{x}}}=\left(W^{-1}_{1}\Omega_{1}\right)^{T}\dfrac{\partial{L_{0}}}{\partial{\dot{\textbf{x}}}}+W^{T}_{1}\dfrac{d\lambda}{dt},\end{array}$
where
$\dfrac{\partial}{\partial\dot{\textbf{x}}}=\left(\dfrac{\partial}{\partial\dot{x_{1}}},\ldots,\dfrac{\partial}{\partial\dot{x_{N}}}\right)^{T},\,\,\dfrac{\partial}{\partial{\textbf{x}}}=\left(\dfrac{\partial}{\partial{x_{1}}},\ldots,\dfrac{\partial}{\partial{x_{N}}}\right)^{T},$
$\lambda=\left(\lambda_{1},\ldots,\lambda_{M},0,\ldots,0\right)^{T},$ and the
transpositional relation (7) becomes
(13)
$\delta\dfrac{d\textbf{x}}{dt}-\dfrac{d}{dt}\delta{\textbf{x}}=\left(W^{-1}_{1}\Omega_{1}\right)\delta{\textbf{x}}.$
###### Theorem 3.
Using the notation of Theorem 1 let
(14)
$L\left(t,\textbf{x},\dot{\textbf{x}},\Lambda\right)=L_{0}\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=1}^{M}\lambda_{j}\,L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=M+1}^{N-1}\lambda^{0}_{j}L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)$
be the Lagrangian and $L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)=0$ be
the independent constraints for $j=1,\ldots,M<N,$ and let $\lambda^{0}_{j}$ be
arbitrary constants, for $j=M+1,\ldots,N-1$ and $\lambda^{0}_{N}=0,$
$L_{j}:\mathbb{R}\times\,T\textsc{Q}\longrightarrow\mathbb{R}$ for
$j=M+1,\ldots,N-1$ arbitrary functions, and $L_{N}=L_{0}$ such that
$|W_{2}|=\det{W_{2}}=\det{\left(\dfrac{\partial(L_{1},\ldots,L_{N-1},L_{0})}{\partial(\dot{x}_{1},\ldots,\dot{x}_{N})}\right)}\neq
0,$
except perhaps in a zero Lebesgue measure set $|W_{2}|=0$. We determine the
matrix $A$ satisfying
(15)
$W_{2}A=\Omega_{2}:=\left(\begin{array}[]{ccc}E_{1}L_{1}&\ldots&E_{N}L_{1}\\\
\vdots&\ldots&\vdots\\\ E_{1}L_{N-1}&\ldots&E_{N}L_{N-1}\\\ 0&\ldots&0\\\
\end{array}\right).$
Then the differential equations (9) become
(16) $\dfrac{d}{dt}\dfrac{\partial
L_{0}}{\partial\dot{\textbf{x}}}-\dfrac{\partial
L_{0}}{\partial{\textbf{x}}}=W^{T}_{2}\dfrac{d}{dt}\tilde{\lambda},$
where
$\lambda:=\tilde{\lambda}=\left(\tilde{\lambda}_{1},\ldots,\tilde{\lambda}_{M},\,0,\ldots,0\right)^{T},$
and the transpositional relation (7) becomes
(17)
$\delta\dfrac{d\textbf{x}}{dt}-\dfrac{d}{dt}\delta{\textbf{x}}=\left(W^{-1}_{2}\Omega_{2}\right)\delta{\textbf{x}},$
The proofs of Theorems 1, 2 and 3 are given in section 5.
###### Theorem 4.
Under the assumptions of Theorem 2 and assuming that
$\begin{array}[]{rl}x_{\alpha}=&x_{\alpha},\quad
x_{\beta}=y_{\beta}\quad\textbf{x}=\left(x_{1},\ldots,x_{s_{1}}\right)\quad\textbf{y}=\left(y_{1},\ldots,y_{s_{2}}\right),\vspace{0.2cm}\\\
L_{\alpha}=&\dot{x}_{\alpha}-\Phi_{\alpha}\left(\textbf{x},\textbf{y},\dot{\textbf{x}},\,\dot{\textbf{y}}\right)=0,\quad
L_{\beta}=\dot{y}_{\beta},\end{array}$
for $\alpha=1,\ldots,s_{1}=M$ and $\beta=s_{1}+1,\ldots,s_{1}+s_{2}=N.$
Then $|W_{1}|=1$ and the differential equations (12) take the form
(18)
$\begin{array}[]{rl}E_{j}L_{0}=&\displaystyle\sum_{\alpha=1}^{s_{1}}\left(E_{j}L_{\alpha}\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}\right)+\dot{\lambda}_{j}\quad
j=1,\ldots,s_{1},\\\
E_{k}L_{0}=&\displaystyle\sum_{\alpha=1}^{s_{1}}\left(E_{k}L_{\alpha}\,\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}+\dot{\lambda}_{\alpha}\dfrac{\partial
L_{\alpha}}{\partial\dot{y}_{k}}\right)\quad k=1,\ldots,s_{2}.\end{array}$
or, equivalently (excluding the Lagrange multipliers)
(19)
$E_{k}L_{0}=\displaystyle\sum_{\alpha=1}^{s_{1}}\left(E_{k}L_{\alpha}\,\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}+\left(E_{\alpha}L_{0}-\displaystyle\sum_{\beta=1}^{s_{1}}\left(E_{\alpha}L_{\beta}\dfrac{\partial
L_{0}}{\partial\dot{x}_{\beta}}\right)\right)\dfrac{\partial\,L_{\alpha}}{\partial\dot{y}_{k}}\right),\quad
k=1,\ldots,s_{2}.$
In particular if we choose
$L_{0}=\tilde{L}\left(\textbf{x},\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right)-\tilde{L}\left(\textbf{x},\textbf{y},\Phi,\dot{\textbf{y}}\right)=\tilde{L}-L^{*},$
where $\Phi=\left(\Phi_{1},\ldots,\Phi_{s_{1}}\right),$ then (19) holds if
$E_{k}\tilde{L}=\displaystyle\sum_{\alpha=1}^{s_{1}}E_{\alpha}\tilde{L}\dfrac{\partial\,L_{\alpha}}{\partial\dot{y}_{k}},\quad
k=1,\ldots,s_{2},$
and
(20)
$E_{k}(L^{*})=\displaystyle\sum_{\alpha=1}^{s_{1}}\left(\dfrac{d}{dt}\left(\dfrac{\partial\Phi_{\alpha}}{\partial\dot{y}_{k}}\right)-\left(\dfrac{\partial{\Phi_{\alpha}}}{\partial\,{y}_{k}}+\displaystyle\sum_{\nu=1}^{s_{1}}\dfrac{\partial\,\Phi_{\alpha}}{\partial\,x_{\nu}}\dfrac{\partial\,\Phi_{\nu}}{\partial\dot{y}_{k}}\right)\right)\Psi_{\alpha}+\displaystyle\sum_{\nu=1}^{s_{1}}\dfrac{\partial
L^{*}}{\partial\,x_{\nu}}\dfrac{\partial\Phi\nu}{\partial\dot{y}_{k}},$
where
$\Psi_{\alpha}=\left.\dfrac{\partial\,\tilde{L}}{\partial\dot{x}_{\alpha}}\right|_{\dot{x}_{1}=\Phi_{1},\ldots,\dot{x}_{s_{1}}=\Phi_{s_{1}}}.$
The transpositional relations (13) in this case are
(21)
$\begin{array}[]{rl}\delta\dfrac{dx_{\alpha}}{dt}-\dfrac{d}{dt}\delta\,x_{\alpha}=&\displaystyle\sum_{k=1}^{s_{2}}\left(\displaystyle\sum_{j=1}^{s_{1}}E_{j}(L_{\alpha})\frac{\partial{L_{j}}}{\partial{\dot{y_{k}}}}+E_{k}(L_{\alpha})\right)\delta
y_{k},\quad\alpha=1,\ldots,s_{1},\vspace{0.2cm}\\\
\delta\dfrac{dy_{m}}{dt}-\dfrac{d}{dt}\delta\,y_{m}=&0,\quad
m=1,\ldots,s_{2}.\end{array}$
###### Proposition 5.
Differential equations (20) describe the motion of the nonholonomic systems
with the constraints
$L_{\alpha}=\dot{x}_{\alpha}-\Phi_{\alpha}(\textbf{x},\textbf{y},\dot{\textbf{y}})=0$
for $\alpha=1,\ldots,s_{1}.$ In particular if the constraints are given by the
formula
(22)
$\dot{x}_{j}=\sum_{k=1}^{s_{2}}a_{jk}(t,\textbf{x},\textbf{y})\dot{y}_{k}+a_{j}(t,\textbf{x}),\quad
j=1,\ldots,s_{1},$
then systems (20) becomes
$\begin{array}[]{rl}E_{k}(L^{*})=&\displaystyle\sum_{\alpha=1}^{s_{1}}\left(\dfrac{da_{\alpha\,k}}{dt}-\left(\dfrac{\partial{a_{\alpha\,m}}}{\partial\,{y}_{k}}+\displaystyle\sum_{\nu=1}^{s_{1}}\dfrac{\partial\,a_{\alpha\,m}}{\partial\,x_{\nu}}a_{\nu\,k}\right)\dot{y}_{m}\,\right)\Psi_{\alpha}+\displaystyle\sum_{\nu=1}^{s_{1}}\dfrac{\partial
L^{*}}{\partial\,x_{\nu}}a_{\nu\,k},\end{array}$
which are the classical Voronets differential equations. Consequently
equations (20) are an extension of the Voronets differential equations for the
case when the constraints are nonlinear in the velocities.
###### Proposition 6.
Differential equations (20) describe the motion of the constrained Lagrangian
systems with the constraints
$L_{\alpha}=\dot{x}_{\alpha}-\Phi_{\alpha}(\textbf{y},\dot{\textbf{y}})=0$ and
Lagrangian $L^{*}=L^{*}(\textbf{y},\dot{\textbf{y}}).$ Under these assumptions
equations (20) take the form
(23)
$E_{k}(L^{*})=\displaystyle\sum_{\alpha=1}^{s_{1}}\left(\dfrac{d}{dt}\left(\dfrac{\partial\Phi_{\alpha}}{\partial\dot{y}_{k}}\right)-\dfrac{\partial\Phi_{\alpha}}{\partial\,y_{k}}\right)\Psi_{\alpha}.$
In particular if the constraints are given by the formula
(24)
$\dot{x}_{\alpha}=\sum_{k=1}^{s_{2}}a_{\alpha\,k}(\textbf{y})\dot{y}_{k},\quad\alpha=1,\ldots,s_{1},$
then systems (23) becomes
(25)
$E_{k}L^{*}=\sum_{j=1}^{s_{1}}\sum_{r=1}^{s_{2}}\left(\dfrac{\partial\alpha_{jk}}{\partial{y_{r}}}-\dfrac{\partial\alpha_{jr}}{\partial{y_{k}}}\right)\dot{y}_{r}\Psi_{j},$
for $k=1,\ldots,s_{2},$ which are the equations which Chaplygin published in
the Proceeding of the Society of the Friends of Natural Science in 1897 .
Consequently equations (23) are an extension of the classical Chaplygin
equations for the case when the constraints are nonlinear.
From (5) and in view of the Implicit Function Theorem, we can locally express
the constraints (reordering coordinates if is necessary) as
(26)
$\dot{x}_{\alpha}=\Phi_{\alpha}\left(\textbf{x},\dot{x}_{M+1},\ldots,\dot{x}_{N}\right)$
for $\alpha=1,\ldots,M.$ We note that Propositions 5 and 6 are also valid for
every constrained mechanical systems with constraints locally given by (26),
this follows from Theorem 4 changing the notations, see Corollary 22.
The proofs of Theorem 4 and Propositions 5 and 6 is given in section 8.
The next result is the third point of view on the transpositional relations.
###### Corollary 7.
For the constrained mechanical systems the virtual variations can produce zero
or non–zero transpositional relations. For the unconstrained mechanical
systems the virtual variations always produce zero transpositional relations.
The proof of this corollary is given in section 9.
We have the following conjecture.
###### Conjecture 8.
The existence of mechanical systems with nonlinear constraints in the velocity
must be sought outside of the Newtonian model.
This conjecture is supported by several facts see section 9.
The results are illustrated with precise examples.
## 3\. Variational Principles. Transpositional relations
### 3.1. Hamiltonian principle
We introduce the following results, notations and definitions which we will
use later on (see [2]).
A Lagrangian system is a pair $(\textsc{Q},\tilde{L})$ consisting of a smooth
manifold $\textsc{Q},$ and a smooth function $\tilde{L}:\mathbb{R}\times
T\textsc{Q}\longrightarrow\mathbb{R},$ where $T\textsc{Q}$ is the tangent
bundle of $\textsc{Q}.$ The point ${\bf
x}=\left(x_{1},\ldots,x_{N}\right)\in\textsc{Q}$ denotes the position (usually
its components are called generalized coordinates) of the system and we call
each tangent vector $\dot{{\bf
x}}=\left(\dot{x}_{1},\ldots,\dot{x}_{N}\right)\in T_{\textbf{x}}\textsc{Q}$
the velocity (usually called generalized velocity) of the system at the point
${\bf x}.$ A pair $({\bf x},\dot{{\bf x}})$ is called a state of the system.
In Lagrangian mechanics it is usual to call $\textsc{Q},$ the configuration
space, the tangent bundle $T\textsc{Q}$ is called the phase space, $\tilde{L}$
is the Lagrange function or Lagrangian and the dimension $N$ of Q is the
number of degrees of freedom.
Let $a_{0}$ and $a_{1}$ be two points of $\textsc{Q}.$ The map
$\begin{array}[]{rl}\gamma:[t_{0},t_{1}]\subset\mathbb{R}&\longrightarrow\textsc{Q},\vspace{0.2cm}\\\
t&\longmapsto\gamma(t)=\left(x_{1}(t),\ldots,x_{N}(t)\right),\end{array}$
such that $\gamma(t_{0})=a_{0},\,\gamma(t_{1})=a_{1}$ is called a path from
$a_{0}$ to $a_{1}.$ We denote the set of all these path by
$\Omega(\textsc{Q},a_{0},a_{1},t_{0},t_{1}):=\Omega$.
We shall derive one of the most simplest and general variational principles
the Hamiltonian principle (see [40]).
The functional $F:\Omega\longrightarrow\mathbb{R}$ defined by
$F(\gamma(t))=\displaystyle\int_{\gamma(t)}\tilde{L}dt=\displaystyle\int_{t_{0}}^{t_{1}}\tilde{L}(t,\textbf{x}(t),\dot{\textbf{x}}(t))dt$
is called the action.
We consider the path
$\gamma(t)=\textbf{x}(t)=\left(x_{1}(t),\ldots,x_{N}(t)\right)\in\Omega.$
Let the variation of the path $\gamma(t)$ be defined as a smooth mapping
$\begin{array}[]{rl}\gamma^{*}:[t_{0},t_{1}]\times(-\tau,\tau)&\longrightarrow\textsc{Q},\vspace{0.2cm}\\\
(t,\varepsilon)&\longmapsto\gamma^{*}(t,\varepsilon)=\textbf{x}^{*}(t,\varepsilon)=\left(x_{1}(t)+\varepsilon\delta{x}_{1}(t),\ldots,x_{N}(t)+\varepsilon\delta{x}_{N}(t)\right),\end{array}$
satisfying
$\textbf{x}^{*}(t_{0},\varepsilon)=a_{0},\quad\textbf{x}^{*}(t_{1},\varepsilon)=a_{1},\quad\textbf{x}^{*}(t,0)=\textbf{x}(t).$
By definition we have
$\delta{\textbf{x}}(t)=\left.\dfrac{\partial\textbf{x}^{*}(t,\varepsilon)}{\partial\varepsilon}\right|_{\varepsilon=0}.$
This function is called the virtual displacement or virtual variation
corresponding to the variation of $\gamma(t)$ and it is a function of time,
all its components are functions of $t$ of class $C^{2}(t_{0},t_{1})$ and
vanish at $t_{0}$ and $t_{1}$ i.e.
$\delta\textbf{x}(t_{0})=\delta\textbf{x}(t_{1})=0.$
A varied path is a path which can be obtained as a variation path.
The first variation of the functional $F$ at $\gamma(t)$ is
$\delta{F}:=\left.\dfrac{\partial
F\left(\textbf{x}^{*}(t,\varepsilon)\right)}{\partial\varepsilon}\right|_{\varepsilon=0},$
and it is called the differential of the functional $F$ (see [2]). The path
$\gamma(t)\in\Omega$ is called the critical point of $F$ if $\delta
F(\gamma(t))=0.$
Let $\mathbb{L}$ be the space of all smooth functions $g:\mathbb{R}\times
T\textsc{Q}\longrightarrow\mathbb{R}.$ The operator
$\begin{array}[]{rl}E_{\nu}:\mathbb{L}&\longrightarrow\mathbb{R},\\\
g&\longmapsto E_{\nu}g={\dfrac{d}{dt}\dfrac{\partial
g}{\partial\dot{x}_{\nu}}-\dfrac{\partial
g}{\partial{x}_{\nu}}},\quad\mbox{for}\quad\nu=1,\ldots,N,\end{array}$
is known as the Lagrangian derivative.
It is easy to show the following property of the Lagrangian derivative
(27) $E_{\nu}\dfrac{df}{dt}=0,$
for arbitrary smooth function $f=f(t,\textbf{x}).$ We observe that in view of
(27) we obtain that the Lagrangian derivative is unchanged if we replace the
function $g$ by $g+\dfrac{df}{dt},$ for any function $f=f(t,\textbf{x}).$ This
reflects the gauge invariance. We shall say that the functions
$g=g\left(t,\textbf{x},\dot{\textbf{x}}\right)$ and
$\hat{g}=\hat{g}\left(t,\textbf{x},\dot{\textbf{x}}\right)$ are equivalently
if $g-\hat{g}=\dfrac{df(t,\textbf{x})}{dt},$ and we shall write
$g\simeq\hat{g}.$
###### Proposition 9.
The differential of the action can be calculated as follows
(28)
$\delta{F}=-\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{k=1}^{N}\left(E_{k}\tilde{L}\delta{x}_{k}-\dfrac{\partial\tilde{L}}{\partial\dot{{x_{k}}}}\left(\delta\dfrac{d{x_{k}}}{dt}-\dfrac{d}{dt}\delta{x_{k}}\right)\right)dt,$
where ${\bf{x}}={\bf{x}}(t),\,\dot{\bf{x}}=\dfrac{d{\bf{x}}}{dt},$ and
$\tilde{L}=\tilde{L}\left(t,{\bf{x}},\dfrac{d{{\bf{x}}}}{dt}\right).$
###### Proof.
We have that
$\begin{array}[]{rl}\delta{F}=&\left.\dfrac{\partial
F\left(\textbf{x}^{*}(t,\varepsilon)\right)}{\partial\varepsilon}\right|_{\varepsilon=0}\vspace{0.2cm}\\\
=&\displaystyle\int_{t_{0}}^{t_{1}}\left.\dfrac{\partial}{\partial\varepsilon}\right|_{\varepsilon=0}L\left(t,\textbf{x}^{*}(t,\varepsilon),\dfrac{d}{dt}\left(\textbf{x}^{*}(t,\varepsilon)\right)\right)\,dt=\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{k=1}^{N}\left(\dfrac{\partial
L}{\partial{x_{k}}}\delta{x_{k}}+\dfrac{\partial
L}{\partial\dot{{x_{k}}}}\delta{\dot{x_{k}}}\right)dt\vspace{0.2cm}\\\
=&\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{k=1}^{N}\left(\dfrac{\partial
L}{\partial{{x_{k}}}}\delta{x_{k}}+\dfrac{\partial
L}{\partial\dot{{x_{k}}}}\dfrac{d}{dt}\delta x_{k}+\dfrac{\partial
L}{\partial\dot{{x_{k}}}}\left(\delta\dfrac{d{x_{k}}}{dt}-\dfrac{d}{dt}\delta{x_{k}}\right)\right)dt\vspace{0.3cm}\\\
=&\left.\displaystyle\sum_{k=1}^{N}\dfrac{\partial
L}{\partial\dot{{x_{k}}}}\delta{x_{k}}\right|_{t=t_{0}}^{t=t_{1}}+\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{k=1}^{N}\left(\left(\dfrac{\partial
L}{\partial{{x_{k}}}}-\dfrac{d}{dt}\dfrac{\partial
L}{\partial\dot{{x_{k}}}}\right)\delta{{x_{k}}}+\dfrac{\partial
L}{\partial\dot{{x_{k}}}}\left(\delta\dfrac{d{x_{k}}}{dt}-\dfrac{d}{dt}\delta{x_{k}}\right)\right)dt.\end{array}$
Hence, by considering that the virtual variation vanishes at the points
$t=t_{0}$ and $t=t_{1}$ we obtain the proof of the proposition. ∎
###### Corollary 10.
The differential of the action for a Lagrangian system
$\left(\textsc{Q},\,\tilde{L}\right)$ can be calculated as follows
$\delta{F}=-\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{k=1}^{N}E_{k}\tilde{L}\left(t,{\bf{x}},\dfrac{d{\bf{x}}}{dt}\right)\,\delta{x}_{k}\,dt.$
###### Proof.
Indeed, for the Lagrangian system the transpositional relation is equal to
zero (see for instance [32] page 29), i.e.
(29) $\delta\dfrac{d\textbf{x}}{dt}-\dfrac{d}{dt}\delta\textbf{x}=0.$
Thus, from Proposition 9, it follows the proof of the corollary. ∎
The path $\gamma(t)\in\Omega$ is called a motion of the Lagrangian systems
$\left(\textsc{Q},\,\tilde{L}\right)$ if $\gamma(t)$ is a critical point of
the action $F,$ i.e.
$\delta{F}\left(\gamma(t)\right)=0\Longleftrightarrow\displaystyle\int_{t_{0}}^{t_{1}}\delta{\tilde{L}}\,dt=0.$
This definition is known as the Hamiltonian variational principle or
Hamiltonian variational principle of least action or simple Hamiltonian
principle.
Now we need the Lagrange lemma or fundamental lemma of calculus of variations
(see for instance [1])
###### Lemma 11.
Let $f$ be a continuous function of the interval $[t_{0},\,t_{1}]$ satisfying
the equation
$\displaystyle\int_{t_{0}}^{t_{1}}f(t)\zeta{(t)}dt=0,$
for arbitrary continuous function $\zeta(t)$ such that
$\zeta(t_{0})=\zeta(t_{1})=0.$ Then $f(t)\equiv 0.$
###### Corollary 12.
The Hamiltonian principle for Lagrangian systems is equivalent to the
Lagrangian equations
(30)
$E_{\nu}\tilde{L}=\displaystyle\dfrac{d}{dt}\left(\dfrac{\partial\tilde{L}}{\partial\dot{{x}_{\nu}}}\right)-\dfrac{\partial\tilde{L}}{\partial{{x}_{\nu}}}=0,$
for $\nu=1,\ldots,N.$
###### Proof.
Clearly, if (30) holds, by Corollary 10, $\delta{F}=0.$ The reciprocal result
follows from Lemma 11. ∎
From the formal point of view, the Hamiltonian principle in the form
(LABEL:LLag) is equivalent to the problem of variational calculus [13, 40].
However, despite the superficial similarity, they differ essentially. Namely,
in mechanics the symbol $\delta$ stands for the its virtual variation, i.e.,
it is not an arbitrary variation but a displacement compatible with the
constraints imposed on the systems. Thus only in the case of the holonomic
systems, for which the number of degrees of freedom is equal to the number of
generalized coordinates, the virtual variations are arbitrary and the
Hamiltonian principle (LABEL:LLag) is completely equivalent to the
corresponding problem of the variational calculus. An important difference
arises for the systems with nonholonomic constraints, when the variations of
the generalized coordinates are connected by the additional relations usually
called Chetaev conditions which we give later on.
### 3.2. D’Alembert–Lagrange principle
Let $L_{j}:\mathbb{R}\times{T\textsc{Q}}\longrightarrow\mathbb{R}$ be smooth
functions for $j=1,\ldots,M.$ The equations
$L_{j}=L_{j}\left(t,\bf{x},\dot{\bf{x}}\right)=0,\quad\mbox{for}\quad
j=1,\ldots,M<N,$
with
$\mbox{rank}\left(\dfrac{\partial(L_{1},\ldots,L_{M})}{\partial(\dot{x}_{1},\ldots,\dot{x}_{N})}\right)=M$
in all the points of $\mathbb{R}\times T\textsc{Q},$ except perhaps in a zero
Lebesgue measure set, define $M$ independent constraints for the Lagrangian
systems $(\textsc{Q},\tilde{L}).$
Let $\mathcal{M}^{*}$ be the submanifold of $\mathbb{R}\times T\textsc{Q}$
defined by the equations (LABEL:01111), i.e.
$\mathcal{M}^{*}=\\{\left({t,\bf x},\,\dot{{\bf x}}\right)\in\mathbb{R}\times
T\textsc{Q}:L_{j}({t,\bf x},\dot{\bf{x}})=0,\quad\mbox{for}\quad
j=1,\ldots,M\\}.$
A constrained Lagrangian system is a triplet
$(\textsc{Q},\tilde{L},\mathcal{M}^{*}).$ The number of degree of freedom is
$\kappa=dim{\textsc{Q}}-M=N-M.$
The constraint is called integrable if it can be written in the form
$L_{j}=\dfrac{d}{dt}\left(G_{j}(t,\textbf{x})\right)=0,$ for a convenient
function $G_{j}.$ Otherwise the constraint is called nonintegrable. According
to Hertz [16] the nonintegrable constraints are also called nonholonomic.
The Lagrangian systems with nonintegrable constraints are usually called (also
following to Hertz) the nonholonomic mechanical systems, or nonholonomic
constrained mechanical systems, and with integrable constraints are called the
holonomic constrained mechanical systems or holonomic constrained Lagrangian
systems. The systems free of constraints are called Lagrangian systems or
holonomic systems.
Sometimes it is also useful to distinguish between constraints that are
dependent on or independent of time. Those that are independent of time are
called scleronomic, and those that depend on time are called rheonomic. This
therminology can also be applied to the mechanical systems themselves. Thus we
say that the constrained Lagrangian systems is scleronomic (reonomic) if the
constraints and Lagrangian are time independent (dependent).
The constraints
(31)
$L_{k}=\displaystyle\sum_{j=1}^{N}a_{kj}\dot{x}_{j}+a_{k}=0,\quad\mbox{for}\quad
k=1,\ldots,M,$
where $a_{kj}=a_{kj}(t,\textbf{x}),\,\,a_{k}=a_{k}(t,\textbf{x}),$ are called
linear constraints with respect to the velocity. For simplicity we shall call
linear constraints.
We observe that (31) admits an equivalent representation as a Pfaffian
equations (for more details see [38])
$\omega_{k}:=\displaystyle\sum_{j=1}^{N}a_{kj}dx_{j}+a_{k}\,dt=0.$
We shall consider only two classes of systems of equations, the equations of
constraints linear with respect to the velocity
$(\dot{x}_{1},\ldots,\dot{x}_{N})$, or linear with respect to the differential
$(dx_{1},\ldots,dx_{N},dt).$ In order to study the integrability or
nonintegrability problem of the constraints the last representation, a
Pfaffian system is the more useful. This is related with the fact that for the
given 1-forms we have the Frobenius theorem which provides the necessary and
sufficient conditions under which the 1-forms are closed and consequently the
given set of constraints is integrable.
The constrains $L_{j}({t,\bf x},\dot{\bf{x}})=0$ are called perfect
constraints or ideal if they satisfy the Chetaev conditions (see [7])
(32) $\displaystyle\sum_{k=1}^{N}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{k}}\,\delta x_{k}=0,$
for $\alpha=1,\ldots,M.$
In what follows, we shall consider only perfect constraints.
If the constraints admit the representation (26) then the Chetaev conditions
takes the form
$\delta
x_{\alpha}=\displaystyle\sum_{k=M+1}^{N}\dfrac{\partial\Phi_{\alpha}}{\partial\dot{x}_{k}}\delta
x_{k}.$
The virtual variations of the variables $x_{\alpha}$ for $\alpha=1,\ldots,M$
are called dependent variations and for the variable $x_{\beta}$ for
$\beta=M+1,\ldots,N$ are called independent variations.
We say that the path $\gamma(t)=\textbf{x}(t)$ is admissible with the perfect
constraint if $L_{j}({t,\bf x}(t),\dot{\bf{x}}(t))\,\,=0.$
The admissible path is called the motion of the constrained Lagrangian systems
$(\textsc{Q},\tilde{L},\mathcal{M}^{*})$ if for all $t\in[t_{0},t_{1}]$
$\displaystyle\sum_{\nu=1}^{N}E_{\nu}\tilde{L}\left({t,\bf
x}(t),\dot{\bf{x}}(t)\right)\,\delta{x}_{\nu}(t)=0,$
for all virtual displacement $\delta{\textbf{x}}(t)$ of the path $\gamma(t).$
This definition is known as d’Alembert–Lagrange principle.
It is well known the following result (see for instance [2, 5, 14, 35]).
###### Proposition 13.
The d’Alembert–Lagrange principle for constrained Lagrangian systems is
equivalent to the Lagrangian differential equations with multipliers
(33)
$\begin{array}[]{rl}E_{j}\tilde{L}=&{\dfrac{d}{dt}\dfrac{\partial\tilde{L}}{\partial{\dot{x}_{j}}}-\dfrac{\partial\tilde{L}}{\partial{{x}_{j}}}}=\displaystyle\sum_{\alpha=1}^{M}{\mu_{\alpha}\dfrac{\partial
L_{\alpha}}{\partial{\dot{x_{j}}}}},\quad\mbox{for}\quad
j=1,\ldots,N,\vspace{0.2cm}\\\ L_{j}({t,\bf
x},\dot{\bf{x}})=&0,\quad\mbox{for}\quad j=1,\ldots,M,\end{array}$
where $\mu_{\alpha}$ for $\alpha=1,\ldots,M$ are the Lagrangian multipliers.
### 3.3. The varied path
The varied path produced in Hamiltonian’s principle is not in general an
admissible path if the perfect constraints are nonholonomic, i.e. the
mechanical systems cannot travel along the varied path without violating the
constraints. We prove the following result, which shall play an important role
in the all assertions below.
###### Proposition 14.
If the varied path is an admissible path then, the following relations hold
(34) $\displaystyle\sum_{k=1}^{N}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{k}}\left(\delta\dfrac{dx_{k}}{dt}-\dfrac{d}{dt}\delta
x_{k}\right)=\displaystyle\sum_{k=1}^{N}E_{k}L_{\alpha}\,\delta{x_{k}},$
for $\alpha=1,\ldots,M.$
###### Proof.
Indeed, the original path $\gamma(t)=\textbf{x}(t)$ by definition satisfies
the Chetaev conditions, and constraints, i.e.
$L_{j}\left(t,\textbf{x}(t),\dot{\textbf{x}}(t)\right)=0.$ If we suppose that
the variation path
$\gamma^{*}(t)=\textbf{x}(t)+\varepsilon\delta{\textbf{x}}(t),$ also satisfies
the constraints i.e.
$L_{j}\left(t,\textbf{x}+\varepsilon\delta\textbf{x},\dot{\textbf{x}}+\varepsilon\delta\dot{\textbf{x}}\right)=L_{j}\left(t,\textbf{x}(t),\dot{\textbf{x}}(t)\right)+\varepsilon\delta\,L_{\alpha}\left(t,\textbf{x}(t),\dot{\textbf{x}}(t)\right)+\ldots=0.$
Thus restricting only to the terms of first order with respect to
$\varepsilon$ and by the Chetaev conditions we have (for simplicity we omitted
the argument)
(35)
$\begin{array}[]{rl}0=&\delta\,L_{\alpha}=\displaystyle\sum_{k=1}^{N}\left(\dfrac{\partial
L_{\alpha}}{\partial\,{x}_{k}}\delta{{x}_{k}}+\dfrac{\partial
L_{\alpha}}{\partial\dot{{x}_{k}}}\delta{\dot{{x}_{k}}}\right),\vspace{0.2cm}\\\
0=&\displaystyle\sum_{k=1}^{N}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{k}}\,\delta x_{k},\end{array}$
for $\alpha=1,\ldots,M.$ The Chetaev conditions are satisfied at each instant,
so
$\dfrac{d}{dt}\left(\displaystyle\sum_{k=1}^{N}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{k}}\,\delta
x_{k}\right)=\displaystyle\sum_{k=1}^{N}\dfrac{d}{dt}\left(\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{k}}\right)\,\delta
x_{k}+\displaystyle\sum_{k=1}^{N}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{k}}\dfrac{d}{dt}\delta x_{k}=0.$
Subtracting these relations from (35) we obtain (34). Consequently if the
varied path is an admissible path, then relations (34) must hold. ∎
From (34) and (7) it follows that the elements of the matrix $A$ satisfy
(36)
$\displaystyle\sum_{m=1}^{N}\delta{x_{m}}\left(E_{m}L_{\alpha}-\displaystyle\sum_{k=1}^{N}A_{k\,m}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{k}}\right)=\displaystyle\sum_{m=1}^{N}\delta{x_{m}}D_{m}L_{\alpha}=0,\quad\mbox{for}\quad\alpha=1,\ldots,M.$
This property will be used below.
###### Corollary 15.
For the holonomic constrained Lagrangian systems the relations (34) hold if
and only if
(37) $\displaystyle\sum_{k=1}^{N}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{k}}\left(\delta\dfrac{dx_{k}}{dt}-\dfrac{d}{dt}\delta
x_{k}\right)=0,\quad\mbox{for}\quad\alpha=1,\ldots,M.$
###### Proof.
Indeed, for holonomic constrained Lagrangian systems the constraints are
integrable, consequently in view of (27) we have $E_{k}L_{\alpha}=0$ for
$k=1,\ldots,N$ and $\alpha=1,\ldots,M.$ Thus, from (34), we obtain (37). ∎
Clearly the equalities (37) are satisfied if (29) holds. We observe that in
general for holonomic constrained Lagrangian systems relation (29) cannot hold
(see example 2).
### 3.4. Transpositional relations
As we observe in the previous subsection for nonholonomic constrained
Lagrangian systems the curves, obtained doing a virtual variation in the
motion of the systems, in general are not kinematical possible trajectories
when (29) is not fulfilled. This leads to the conclusion that the Hamiltonian
principle cannot be applied to nonholonomic systems, as it is usually employed
for holonomic systems. The essence of the problem of the applicability of this
principle for nonholonomic systems remains unclarified (see [35]). In order to
clarify this situation, it is sufficient to note that the question of the
applicability of the principle of stationary action to nonholonomic systems is
intimately related to the question of transpositional relation.
The key point is that the Hamiltonian principle assumes that the operation of
differentiation with respect to the time $\dfrac{d}{dt}$ and the virtual
variation $\delta$ commute in all the generalized coordinate systems.
For the holonomic constrained Lagrangian systems relations (29) cannot hold
(see Corollary 15). For a nonholonomic systems the form of the Hamiltonian
principle will depend on the point of view adopted with respect to the
transpositional relations.
What are then the correct transpositional relations? Until now, does not exist
a common point of view concerning to the commutativity of the operation of
differentiation with respect to the time and the virtual variation when there
are nonintegrable constraints. Two points of view have been maintained.
According to one (supported, for example, by Volterra, Hamel, Hölder, Lurie,
Pars,…), the operations $\dfrac{d}{dt}$ and $\delta$ commute for all the
generalized coordinates, independently if the systems are holonomic or
nonholonomic, i.e.
$\delta\dfrac{dx_{k}}{dt}-\dfrac{d}{dt}\delta x_{k}=0,\quad\mbox{for}\quad
k=1,\ldots,N.$
According to the other point of view (supported by Suslov, Voronets, Levi-
Civita, Amaldi,…) the operations $\dfrac{d}{dt}$ and $\delta$ commute always
for holonomic systems, and for nonholonomic systems with the constraints
$\dot{x}_{\alpha}=\displaystyle\sum_{j=M+1}^{N}a_{\alpha
j}(t,\textbf{x})\dot{x}_{j}+a_{\alpha}(t,\textbf{x}),\quad\mbox{for}\quad\alpha=1,\ldots,M.$
the transpositional relations are equal to zero only for the generalized
coordinates $x_{M+1},\ldots,x_{N},$ ( for which their virtual variations are
independent). For the remaining coordinates $x_{1},\ldots,x_{M},$ (for which
their virtual variations are dependent), the transpositional relations must be
derived on the basis of the equations of the nonholonomic constraints, and
cannot be identically zero, i.e.
$\begin{array}[]{rl}\delta\dfrac{dx_{k}}{dt}-\dfrac{d}{dt}\delta
x_{k}=&0,\quad\mbox{for}\quad k=M+1,\ldots,N\\\
\delta\dfrac{dx_{k}}{dt}-\dfrac{d}{dt}\delta x_{k}\neq&0,\quad\mbox{for}\quad
k=1,\ldots,M.\end{array}$
The second point of view acquired general acceptance and the first point of
view was considered erroneous (for more details see [35]). The meaning of the
transpositional relations (1) can be found in [19, 32, 34, 35].
In the results given in the following section play a key role the equalities
(34). From these equalities and from the examples it will be possible to
observe that the second point of view is correct only for the so called
Voronets–Chaplygin systems, and in general for locally nonholonomic systems.
There exist many examples for which the independent virtual variations
generated non–zero transpositional relations. Thus we propose a third point of
view on the transpositional relations: the virtual variations can generate the
transpositional relations given by the formula (7) where the elements of the
matrix $A$ satisfies the conditions (see formula (36))
(38)
$D_{\nu}L_{\alpha}=E_{\nu}L_{\alpha}-\displaystyle\sum_{k=1}^{N}A_{k\,\nu}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{k}}=0,\quad\mbox{for}\quad\nu=1,\ldots,M,\quad\alpha=1,\ldots,M.$
we observe that here the $L_{\alpha}=0$ are constraints which in general are
nonlinear in the velocity.
### 3.5. Hamiltonian–Suslov principle
After the introduction of the nonholonomic mechanics by Hertz, it appeared the
question of extending to the nonholonomic mechanics the results of the
holonomic mechanics. Hertz [16] was the first in studying the problem of
applying the Hamiltonian principle to systems with nonintegrable constraints.
In [16] Hertz wrote: “Application of Hamilton’s principle to any material
systems does not exclude that between selected coordinates of the systems
rigid constraints exist, but it still requires that these relations could be
expressed by integrable constraints. The appearance of nonintegrable
constraints is unacceptable. In this case the Hamilton’s principle is not
valid.” Appell [3] in correspondence with Hertz’s ideas affirmed that it is
not possible to apply the Hamiltonian principle for systems with nonintegrable
constraints
Suslov [48] claimed that ”Hamilton’s principle is not applied to systems with
nonintegrable constraints, as derived based on this equation are different
from the corresponding equations of Newtonian mechanics”.
The applications of the most general differential principle, i.e. the
d’Alembert–Lagrange and their equivalent Gauss and Appel principle, is
complicated due to the presence of the terms containing the second order
derivative. On the other hand the most general variational integral principle
of Hamilton is not valid for nonholonomic constrained Lagrangian systems. The
generalization of the Hamiltonian principle for nonholonomic mechanical
systems was deduced by Voronets and Suslov (see for instance [48, 53]). As we
can observe later on from this principle follows the importance of the
transpositional relations to determine the correct equations of motion for
nonholonomic constrained Lagrangian systems.
###### Proposition 16.
The d’Alembert–Lagrangian principle for the contrained Lagrangian systems
$\displaystyle\sum_{k=1}^{N}\delta{x}_{k}E_{k}\tilde{L}=0$ is equivalent to
the Hamilton–Suslov principle (2) where we assume that
$\delta{x_{\nu}(t)},\quad\nu=1,\ldots,N,$ are arbitrary smooth functions
defined in the interior of the interval $[t_{0},\,t_{1}]$ and vanishing at its
endpoints, i.e., $\delta{x_{\nu}}({t_{0}})=\delta{x_{\nu}}({t_{1}})=0.$
###### Proof.
From the d’Alembert–Lagrangian principle we obtain the identity
$\begin{array}[]{rl}0=&-\displaystyle\sum_{k=1}^{N}\delta{x}_{k}E_{k}\tilde{L}=\displaystyle\sum_{k=1}^{N}\delta{x}_{k}\dfrac{\partial\tilde{L}}{\partial
x_{k}}-\displaystyle\sum_{k=1}^{N}\delta{x}_{k}\dfrac{d}{dt}\dfrac{\partial\tilde{L}}{\partial\dot{x}_{k}}\vspace{0.2cm}\\\
=&\displaystyle\sum_{k=1}^{N}\left(\delta{x}_{k}\dfrac{\partial\tilde{L}}{\partial
x_{k}}+\delta{\dot{x}}_{k}\dfrac{\partial\tilde{L}}{\partial\dot{x}_{k}}\right)-\displaystyle\sum_{k=1}^{N}\left(\left(\delta\dfrac{dx_{k}}{dt}-\dfrac{d}{dt}\delta{x_{k}}\right)\dfrac{\partial\tilde{L}}{\partial\dot{x}_{k}}-\dfrac{d}{dt}\left(\dfrac{\partial\tilde{L}}{\partial\dot{x}_{k}}\delta{x_{k}}\right)\right)\vspace{0.2cm}\\\
=&\delta\tilde{L}-\displaystyle\sum_{k=1}^{N}\left(\left(\delta\dfrac{dx_{k}}{dt}-\dfrac{d}{dt}\delta{x_{k}}\right)\dfrac{\partial\tilde{L}}{\partial\dot{x}_{k}}-\dfrac{d}{dt}\left(\dfrac{\partial\tilde{L}}{\partial\dot{x}_{k}}\delta{x_{k}}\right)\right),\end{array}$
where $\delta\tilde{L}$ is a variation of the Lagrangian $\tilde{L}$. After
the integration and assuming that $\delta x_{k}(t_{0})=0,\,\delta
x_{k}(t_{1})=0$ we easily obtain (2), which represent the most general
formulation of the Hamiltonian principle (Hamilton–Suslov principle) suitable
for constrained and unconstrained Lagrangian systems. ∎
Suslov determine the transpositional relations only for the case when the
constraints are of Voronets type, i.e. given by the formula (22). Assume that
$\delta\dfrac{dy_{k}}{dt}-\dfrac{d}{dt}\delta{y_{k}}=0,\quad\mbox{for}\quad
k=M+1,\ldots,N,$
Voronets and Suslov deduced that
$\delta\dfrac{dx_{k}}{dt}-\dfrac{d}{dt}\delta{x_{k}}=\displaystyle\sum_{k=1}^{N}B_{kr}\delta{y_{r}}-\delta{a_{k}}$
for convenient functions
$B_{kr}=B_{kr}\left(t,\textbf{x},\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right),$
for $r=M+1,\ldots,N$ and $k=1,\ldots,M.$
Thus we obtain
$\displaystyle\int_{t_{0}}^{t_{1}}\left(\delta\,\tilde{L}-\displaystyle\sum_{k=1}^{N}\dfrac{\partial{\tilde{L}}}{\partial\dot{x}_{j}}\left(\displaystyle\sum_{k=1}^{N}B_{kr}\delta{y_{r}}-\delta{a_{k}}\right)\right)dt=0,$
This is the Hamiltonian principle for nonholonomic systems in the Suslov form
(see for instance [48]). We observe that the same result was deduced by
Voronets in [53].
It is important to observe that Suslov and Voronets require a priori that the
independent virtual variations produce the zero transpositional relations. At
the sometimes these authors consider only linear constraints with respect to
the velocity of the type (22).
### 3.6. Modification of the vakonomic mechanics (MVM)
As we observe in the introduction, the main objective of this paper is to
construct the variational equations of motion describing the behavior of the
constrained Lagrangian systems in which the equalities (34) take place in the
most general possible way. We shall show that the d’Alembert–Lagrange
principle is not the only way to deduce the equations of motion for the
constrained Lagrangian systems. Instead of it we can apply the generalization
of the Hamiltonian principle, whereby the motions of such systems are
extremals of the variational Lagrange problem (see for instance [13]), i.e.
the problem of determining the critical points of the action in the class of
curves with fixed endpoints and satisfying the constraints. The solution of
this problem as we shall see will give the differential equations of second
order which coincide with the well–known classical equations of the mechanics
except perhaps in a zero Lebesgue measure set.
From the previous section we deduce that in order to generalize the
Hamiltonian principle to nonholonomic systems we must take into account the
following relations
$\begin{array}[]{rl}&\mbox{(A)}\qquad\delta
L_{\alpha}=\displaystyle\sum_{j=1}^{N}\left(\dfrac{\partial
L_{\alpha}}{\partial x_{j}}\delta x_{j}+\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{j}}\delta\dot{x}_{j}\right)=0\quad\mbox{for}\quad\alpha=1,\ldots,M,\vspace{0.2cm}\\\
&\mbox{(B)}\qquad\displaystyle\sum_{j=1}^{N}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{j}}\delta{x_{j}}=0\quad\mbox{for}\quad\alpha=1,\ldots,M,\vspace{0.2cm}\\\
&\mbox{(C)}\qquad\delta\dfrac{dx_{j}}{dt}-\dfrac{d}{dt}\delta{x_{j}}=0\quad\mbox{for}\quad
j=1,\ldots,N,\end{array}$
where $L_{\alpha}=0$ for $\alpha=1,\ldots,M$ are the constraints.
A lot of authors consider that (C) is always fulfilled (see for instance [32,
38]), together with the conditions (A) and (B). However these conditions are
incompatible in the case of the nonintegrable constrains. We observe that
these authors deduced that the Hamiltonian principle is not applicable to the
nonholonomic systems.
To obtain a generalization of the Hamiltonian principle for the nonholonomic
mechanical systems, some of these three conditions must be excluded.
In particular for the Hölder principle conditions (A) is excluded and keep (B)
and (C) (see [17]). For the Hamiltonian–Suslov principle condition (A) and (B)
hold, and (C) only holds for the independent variations.
In this paper we extend the Hamiltonian principle by supposing that conditions
(A) and (B) hold and (C) does not hold . Instead of (C) we consider that (7)
holds where elements of matrix $A$ satisfy the relations (38).
## 4\. Solution of the inverse problem of the constrained Lagrangian systems
We shall determine the equations of motion of the constrained Lagrangian
systems using the Hamiltonian principle with non zero transpositional
relations, whereby the motions of the systems are extremals of the variational
Lagrange’s problem (see for instance [13]), i.e. are the critical points of
the action functional
$\displaystyle\int_{t_{0}}^{t_{1}}L_{0}\left(t,\textbf{x},\dot{\textbf{x}}\right)\,dt,$
in the class of path with fixed endpoints satisfying the independent
constraints
$L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)=0,\quad\mbox{for}\quad
j=1,\ldots,M.$
In the classical solution of the Lagrange problem usually we apply the
Lagrange multipliers method which consists in the following. We introduce the
additional coordinates $\Lambda=\left(\lambda_{1},\ldots,\lambda_{M}\right),$
and Lagrangian
$\widehat{L}:\mathbb{R}\times{T\textsc{Q}}\times\mathbb{R}^{M}\longrightarrow\mathbb{R}$
given by
$\widehat{L}\left(t,\textbf{x},\dot{\textbf{x}},\Lambda\right)=L_{0}\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=1}^{M}\lambda_{j}\,L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right),$
Under this choice we reduce the Lagrange problem to a variational problem
without constraints, i.e. we must determine the extremal of the action
functional $\displaystyle\int_{t_{0}}^{t_{1}}\widehat{L}\,dt.$ We shall study
a slight modification of the Lagrangian multipliers method. We introduce the
additional coordinates $\Lambda=\left(\lambda_{1},\ldots,\lambda_{M}\right),$
and the Lagrangian on $\mathbb{R}\times{T\textsc{Q}}\times\mathbb{R}^{M}$
given by the formula (4), where we assume that $\lambda^{0}_{j}$ are arbitrary
constants, and $L_{j}$ are arbitrary functions for $j=M+1,\ldots,N.$
Now we determine the critical points of the action functional
$\displaystyle\int_{t_{0}}^{t_{1}}L\left(t,\textbf{x},\dot{\textbf{x}},\Lambda\right)dt,$
i.e. we determine the path $\gamma(t)$ such that
$\displaystyle\int_{t_{0}}^{t_{1}}\delta\left(L\left(t,\textbf{x},\dot{\textbf{x}},\Lambda\right)\right)dt=0$
under the additional condition that the transpositional relations are given by
the formula (7).
The solution of the inverse problem stated in section 2 is the following.
Differential equations obtained from (6) are given by the formula (8) (see
Theorem 1). We choose the arbitrary functions $L_{j}$ in such a away that the
matrix $W_{1}$ and $W_{2}$ given in Theorems 2 and 3 are nonsingular, except
perhaps in a zero Lebesgue measure set. The constants $\lambda^{0}_{j}$ for
$j=M+1,\ldots,N$ are arbitrary in Theorem 2, and $\lambda^{0}_{j}$ for
$j=1,\ldots,N-1$ are arbitrary and $\lambda^{0}_{N}=0$ in Theorem 3. The
matrix $A$ is determined from the equalities (11) and (15) of Theorems 2 and 3
respectively.
###### Remark 17.
It is interesting to observe that from the solutions of the inverse problem,
the constants $\lambda^{0}_{j}$ for $j=M+1,\ldots,N$ are arbitrary except in
Theorem 3 in which $\lambda^{0}_{N}=0.$ Clearly, if
$L_{j}\left(t,\textbf{x},\dot{\textbf{x}}\right)=\dfrac{d}{dt}f_{j}(t,\textbf{x})$
for $j=M+1,\ldots,N,$ then the $L\simeq\widehat{L}.$ Using the arbitrariness
of the constants $\lambda^{0}_{j}$ we can always take that $\lambda^{0}_{k}=0$
if
$L_{k}\left(t,\textbf{x},\dot{\textbf{x}}\right)\neq\dfrac{d}{dt}f_{k}(t,\textbf{x}).$
Consequently we can always suppose that $L\simeq\widehat{L}.$ Thus the only
difference between the classical and the modified Lagrangian multipliers
method consists only on the transpositional relations: for the classical
method the virtual variations produce zero transpositional relations (i.e. the
matrix $A$ is the zero matrix) and for the modified method in general it is
determined by the formulae (7) and (36).
A very important subscase is obtained when the constraints are given in the
form (Voronets-Chapliguin constraints type)
$\dot{x}_{\alpha}-\Phi_{\alpha}\left(t,\textbf{x},\dot{x}_{M+1},\ldots,\dot{x}_{N}\right)=0,$
for $\alpha=1,\ldots,M.$ As we shall show under these assumptions the
arbitrary functions are determined as follows: $L_{j}=\dot{x}_{j}$ for
$j=M+1,\ldots,N.$ Consequently the action of the modified Lagrangian
multipliers method and the action of the classical Lagrangian multipliers
method are equivalently. In view of (26) this equivalence always locally holds
for any constrained Lagrangian systems.
## 5\. Proof of Theorems 1, 2 and 3
###### Proof of Theorem 1.
In view of the equalities
$\begin{array}[]{rl}\displaystyle\int_{t_{0}}^{t_{1}}\delta{L}\,dt=&\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{k=1}^{M}\left(\dfrac{\partial
L}{\partial\lambda_{k}}\delta\lambda_{k}\right)dt+\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{j=1}^{N}\left(\dfrac{\partial
L}{\partial x_{j}}\delta x_{j}+\dfrac{\partial
L}{\partial\dot{x}_{j}}\delta\dfrac{d{x}_{j}}{dt}\right)dt\vspace{0.20cm}\\\
=&\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{k=1}^{M}\left(-L_{k}\delta\lambda_{k}\right)dt+\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{j=1}^{N}\left(\dfrac{\partial
L}{\partial x_{j}}\delta x_{j}+\dfrac{\partial
L}{\partial\dot{x}_{j}}\dfrac{d}{dt}\delta{x_{j}}+\dfrac{\partial
L}{\partial\dot{x}_{j}}\left(\delta\dfrac{dx_{j}}{dt}-\dfrac{d}{dt}\delta{x_{j}}\right)\right)dt\vspace{0.20cm}\\\
=&\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{k=1}^{M}\left(-L_{k}\delta\lambda_{k}\right)dt+\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{j=1}^{N}\dfrac{d}{dt}\left(\dfrac{\partial
T}{\partial\dot{x}_{j}}\delta x_{j}\right)dt\vspace{0.20cm}\\\
&-\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{j=1}^{N}\left(\left(-\dfrac{\partial
L}{\partial x_{j}}+\dfrac{d}{dt}\left(\dfrac{\partial
L}{\partial\dot{x}_{j}}\right)\right)\delta{x_{j}}+\dfrac{\partial
L}{\partial\dot{x}_{j}}\left(\delta\dfrac{dx_{j}}{dt}-\dfrac{d}{dt}\delta{x_{j}}\right)\right)dt.\end{array}$
Consequently
$\begin{array}[]{rl}\left.\displaystyle\int_{t_{0}}^{t_{1}}\delta{L}\,dt\right|_{L_{\nu}=0}=&\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{j=1}^{N}\left(\dfrac{d}{dt}\left(\dfrac{\partial
T}{\partial\dot{x}_{j}}\delta
x_{j}\right)-\left(E_{j}L-\displaystyle\sum_{k=1}^{N}A_{jk}\dfrac{\partial
L}{\partial\dot{x}_{k}}\right)\delta x_{j}\right)dt\vspace{0.20cm}\\\
=&\displaystyle\sum_{j=1}^{N}\left.\dfrac{\partial
T}{\partial\dot{x}_{j}}\delta
x_{j}\right|_{t=t_{0}}^{t=t_{1}}-\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{j=1}^{N}\left(E_{j}L-\displaystyle\sum_{k=1}^{N}A_{jk}\dfrac{\partial
L}{\partial\dot{x}_{k}}\right)\delta x_{j}dt\vspace{0.20cm}\\\
=&-\displaystyle\int_{t_{0}}^{t_{1}}\displaystyle\sum_{j=1}^{N}\left(E_{j}L-\displaystyle\sum_{k=1}^{N}A_{jk}\dfrac{\partial
L}{\partial\dot{x}_{k}}\right)\delta x_{j}dt=0,\end{array}$
where $\nu=1,\ldots,M.$ Here we use the equalities
$\delta{\textbf{x}}(t_{0})=\delta{\textbf{x}}(t_{1})=0.$ Hence if (8) holds
then (6) is satisfied. The reciprocal result is proved by choosing
$\delta x_{k}(t)=\begin{cases}\zeta(t)&\text{if}\,\,k=1,\\\
0&\text{otherwise},\end{cases}$
where $\zeta(t)$ is a positive function in the interval
$(t^{*}_{0},t^{*}_{1}),$ and it is equal to zero in the intervals
$[t_{0},\,t^{*}_{0}]$ and $[t^{*}_{1},\,t_{1}],$ and applying Corollary 11.
From the definition (8) we have that
$D_{\nu}(fg)=D_{\nu}f\,g+f\,D_{\nu}\,g+\dfrac{\partial
f}{\partial\dot{x}_{\nu}}\dfrac{dg}{dt}+\dfrac{df}{dt}\dfrac{\partial
g}{\partial\dot{x}_{\nu}},\quad D_{\nu}a=0,$
where $a$ is a constant.
Now we shall write (8) in a more convenient way
$\begin{array}[]{rl}0=D_{\nu}{L}=&D_{\nu}\left(L_{0}-\displaystyle\sum_{j=1}^{M}\lambda_{j}L_{j}-\displaystyle\sum_{j=M+1}^{N}\lambda^{0}_{j}L_{j}\right)\vspace{0.20cm}\\\
=&D_{\nu}\,L_{0}-\displaystyle\sum_{j=1}^{M}D_{\nu}\left(\lambda_{j}L_{j}\right)-\displaystyle\sum_{j=M+1}^{N}\lambda^{0}_{j}D_{\nu}\,L_{j}\vspace{0.20cm}\\\
=&D_{\nu}\,L_{0}-\displaystyle\sum_{j=M+1}^{N}\lambda^{0}_{j}D_{\nu}\,L_{j}-\vspace{0.20cm}\\\
&-\displaystyle\sum_{j=1}^{M}\left(D_{\nu}\,\lambda_{j}\,\,L_{j}+\lambda_{j}D_{\nu}\,L_{j}+\dfrac{d\lambda_{j}}{dt}\dfrac{\partial
L_{j}}{\partial\dot{x}_{\nu}}+\dfrac{dL_{j}}{dt}\dfrac{\partial\lambda_{j}}{\partial\dot{x}_{\nu}}\right).\end{array}$
From these relations and since the constraints $L_{j}=0$ for $j=1,\ldots,M,$
we easily obtain equations (9) or equivalently
(39) $E_{\nu}L_{0}=\displaystyle\sum_{k=1}^{N}A_{jk}\dfrac{\partial
L_{0}}{\partial\dot{x}_{k}}+\sum_{j=1}^{M}\left(\lambda_{j}D_{\nu}L_{j}+\dfrac{d\lambda_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}\right)+\displaystyle\sum_{j=M+1}^{N}\lambda^{0}_{j}D_{\nu}\,L_{j}.$
Thus the theorem is proved. ∎
Now we show that the differential equations (39) for convenient functions
$L_{j}$ constants $\lambda^{0}_{j}$ for $j=M+1,\ldots,N$ and for convenient
matrix $A$ describe the motion of the constrained Lagrangian systems.
###### Proof of Theorem 2.
The matrix equation (11) can be rewritten in components as follows
(40) $\displaystyle\sum_{j=1}^{N}A_{kj}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{j}}=E_{k}L_{\alpha}\Longleftrightarrow
D_{k}L_{\alpha}=0,$
for $\alpha,\,k=1,\ldots,N.$ Consequently the differential equations (39)
become
(41) $E_{\nu}L_{0}=\displaystyle\sum_{k=1}^{N}\left(A_{\nu k}\dfrac{\partial
L_{0}}{\partial\dot{x}_{k}}+\dfrac{d\lambda_{k}}{dt}\dfrac{\partial{L_{k}}}{\partial{\dot{x}_{\nu}}}\right)\Longleftrightarrow
D_{\nu}L_{0}=\displaystyle\sum_{j=1}^{M}\dfrac{d\lambda_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}},$
which coincide with the first systems (12).
In view of the condition $|W_{1}|\neq 0$ we can solve equation (11) with
respect to $A$ and obtain $A=W^{-1}_{1}\Omega_{1}.$ Hence, by considering (40)
we obtain the second systems from (12) and the transpositional relation (13).
∎
###### Proof of Theorem 3.
The matrix equation (15) is equivalent to the systems
$\begin{array}[]{rl}\displaystyle\sum_{j=1}^{N}A_{kj}\dfrac{\partial
L_{\alpha}}{\partial\dot{x}_{j}}=&E_{k}L_{\alpha}\Longleftrightarrow
D_{k}L_{\alpha}=0,\vspace{0.2cm}\\\
\displaystyle\sum_{j=1}^{N}{A_{kj}\dfrac{\partial{L_{0}}}{\partial\dot{x}_{j}}}=&0,\end{array}$
for $k=1,\ldots,N,$ and $\alpha=1,\ldots,N-1.$ Thus, by considering that
$\lambda^{0}_{N}=0$ we deduce that systems (39) takes the form
$E_{\nu}L_{0}=\displaystyle\sum_{j=1}^{M}\dfrac{d\tilde{\lambda}_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}.$
Hence we obtain systems (16). On the other hand from (15) we have that
$A=W^{-1}_{2}\Omega_{2}.$ Hence we deduce that the transpositional relation
(7) can be rewritten in the form (17). ∎
The mechanics basic on the Hamiltonian principle with non–zero transpositional
relations given by formula (7), Lagrangian (4) and equations of motion (8) are
called here the modification of the vakonomic mechanics and we shortly write
MVM.
From the proofs of Theorems 2 and 3 follows that the relations (36) holds
identically in MVM.
###### Corollary 18.
Differential equations (12) are invariant under the change
$L_{0}\longrightarrow L_{0}-\displaystyle\sum_{j=1}^{N}a_{j}L_{j},$
where the $a_{j}$’s are constants for $j=1,\ldots,N.$
###### Proof.
Indeed, from (41) and (40) it follows that
$D_{\nu}\left(L_{0}-\displaystyle\sum_{j=1}^{N}a_{j}L_{j}\right)=D_{\nu}\,L_{0}-\displaystyle\sum_{j=1}^{N}a_{j}D_{\nu}\,L_{j}=D_{\nu}\,L_{0}=\displaystyle\sum_{j=1}^{M}\dfrac{d{\lambda}_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}.$
∎
###### Remark 19.
The following interesting facts follow from Theorems 2 and 3.
* (1)
The equations of motion obtained from Theorem 2 are more general than the
equations obtained from Theorem 3. Indeed in (12) there are $N-M$ arbitrary
functions while in (16) are $N-M-1$ arbitrary functions.
* (2)
If the constraints are linear in the velocity then between the Lagrangian
multipliers $\mu,\,\,\dfrac{d\lambda}{dt}$ and $\dfrac{d\tilde{\lambda}}{dt}$
there is the following relation
$\mu=\dfrac{d\tilde{\lambda}}{dt}=\left(W^{-1}_{2}\right)^{T}\left(W^{T}_{1}\dfrac{d\lambda}{dt}+W^{-1}_{2}\Omega^{T}_{1}W^{-T}_{1}\dfrac{\partial
L_{0}}{\partial\dot{\textbf{x}}}\right),$
where $W_{1}$ and $W_{2}$ are the matrixes defined in Theorems 2 and 3.
* (3)
If the constraints are linear in the velocity then one of the important
question which appear in MVM is related with the arbitrariness functions
$L_{j}$ for $j=M+1,\ldots,N.$ The following question arise: Is it possible to
determine these functions in such a way that $|W_{1}|$ or $|W_{2}|$ is
non–zero everywhere in $\mathcal{M}^{*}$? If we have a positive answer to this
question, then the equations of motion of the MVM give a global behavior of
the constrained Lagrangian systems, i.e. the obtained motions completely
coincide with the motions obtained from the classical mathematical models.
Thus if $|W_{1}|\neq 0$ and $|W_{2}|\neq 0$ everywhere in $\mathcal{M}^{*}$
then we have the equivalence
(42)
$D_{\nu}L_{0}={\displaystyle\sum_{j=1}^{M}\dfrac{d\lambda_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}}\Longleftrightarrow
E_{\nu}L_{0}={\displaystyle\sum_{j=1}^{M}\dfrac{d\tilde{\lambda}_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}}\Longleftrightarrow
E_{\nu}L_{0}=\displaystyle\sum_{j=1}^{M}\mu_{j}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}$
If the constraints are nonlinear in the velocity and $|W_{2}|\neq 0$
everywhere in $\mathcal{M}^{*}$ then we have the equivalence
(43)
$E_{\nu}L_{0}={\displaystyle\sum_{j=1}^{M}\dfrac{d\tilde{\lambda}_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}}\Longleftrightarrow
E_{\nu}L_{0}=\displaystyle\sum_{j=1}^{M}\mu_{j}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}$
The equivalence with respect to the equations
$D_{\nu}L_{0}={\displaystyle\sum_{j=1}^{M}\dfrac{d\lambda_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}}$
in general is not valid in this case because the term
$\Omega^{T}_{1}W^{-T}_{1}\dfrac{\partial L_{0}}{\partial\dot{\textbf{x}}}$
depend on $\ddot{\textbf{x}}.$
### 5.1. Application of Theorems 2 and 3 to the Appell–Hamel mechanical
systems
As a general rule the constraints studied in classical mechanics are linear
with respect to the velocities, i.e. $L_{j}$ can be written as (31). However
Appell and Hamel (see [3, 15]) in 1911, considered an artificial example of
nonlinear nonholonomic constrains. A big number of investigations have been
devoted to the derivation of the equations of motion of mechanical systems
with nonlinear nonholonomic constraints see for instance [8, 15, 35, 36]. The
works of these authors do not contain examples of systems with nonlinear
nonholonomic constraints differing essentially from the example given by
Appell and Hamel.
###### Corollary 20.
The equivalence (42) also holds for the Appell – Hamel system i.e. for the
constrained Lagrangian systems
$\left(\mathbb{R}^{3},\quad\tilde{L}=\dfrac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-gz,\quad\\{\dot{z}-a\sqrt{\dot{x}^{2}+\dot{y}^{2}}=0\\}\right),$
where $a$ and $g$ are positive constants.
###### Proof.
The classical equations (33) for the Appell-Hamel system are
(44)
$\ddot{x}=-\dfrac{a\dot{x}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\mu,\qquad\ddot{y}=-\dfrac{a\dot{y}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\mu,\qquad\ddot{z}=-g+\mu,$
where $\mu$ is the Lagrangian multiplier.
Now we apply Theorem 3. Hence, in order to obtain that $|W_{2}|\neq 0$
everywhere we choose the functions $L_{j}$ for $j=1,2,3$ as follows
$L_{1}=\dot{z}-a\sqrt{\dot{x}^{2}+\dot{y}^{2}}=0,\quad
L_{2}=\arctan\dfrac{\dot{x}}{\dot{y}},\quad L_{3}=L_{0}=\tilde{L}.$
In this case the matrices $W_{2},\,\Omega_{2}$ and $A$ are
$\begin{array}[]{rl}W_{2}=&\left(\begin{array}[]{ccc}-\dfrac{a\dot{x}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}&-\dfrac{a\dot{y}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}&1\\\
\dfrac{\dot{y}}{\dot{x}^{2}+\dot{y}^{2}}&-\dfrac{\dot{x}}{\dot{x}^{2}+\dot{y}^{2}}&0\\\
\dot{x}&\dot{y}&\dot{z}\end{array}\right),\quad|W_{2}|_{L_{1}=0}=1+a^{2},\vspace{0.3cm}\\\
\Omega_{2}=&\left(\begin{array}[]{ccc}-\dot{y}q&\dot{x}q&0\vspace{0.2cm}\\\
\dfrac{\ddot{y}\left(\dot{x}^{2}-\dot{y}^{2}\right)-2\dot{x}\dot{y}\ddot{x}}{\left(\dot{x}^{2}+\dot{y}^{2}\right)^{2}}&\dfrac{\ddot{x}\left(\dot{x}^{2}-\dot{y}^{2}\right)+2\dot{x}\dot{y}\ddot{y}}{\left(\dot{x}^{2}+\dot{y}^{2}\right)^{2}}&0\\\
0&0&0\end{array}\right),\end{array}$
and the matrix $\left.A\right|_{L_{1}=0}$ is
$\left(\begin{array}[]{ccc}-\dfrac{\dot{y}\left(a^{2}\dot{y}\dot{x}\ddot{x}+\left((a^{2}+1)\dot{y}^{2}+\dot{x}^{2}\right)\ddot{y}\right)}{(1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)}&\dfrac{\left(a^{2}\dot{x}^{2}+(a^{2}+1)\left(\dot{y}^{2}+\dot{x}^{2}\right)^{2}\right)\dot{y}\ddot{x}-a^{2}\dot{x}^{3}\ddot{y}}{(1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)^{2}}&0\\\
\\\
\dfrac{\left(a^{2}\dot{y}^{2}+(a^{2}+1)\left(\dot{y}^{2}+\dot{x}^{2}\right)\right)\dot{x}\ddot{y}-a^{2}\dot{y}^{3}\ddot{x}}{(1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)}&-\dfrac{\dot{x}\left(a^{2}\dot{x}\dot{y}\ddot{y}+\left((a^{2}+1)\dot{x}^{2}+\dot{y}^{2}\right)\ddot{x}\right)}{(1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)^{2}}&0\\\
\\\
\dfrac{\dot{y}a\left(\dot{y}\ddot{x}-\dot{x}\ddot{y}\right)}{(1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)^{3/2}}&-\dfrac{\dot{x}a\left(\dot{y}\ddot{x}-\dot{x}\ddot{y}\right)}{(1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)^{3/2}}&0\end{array}\right).$
By considering that $|W_{2}|_{L_{1}=0}=1+a^{2},$ we obtain that the equations
(16) in this case describe the global behavior of the Appell–Hamel systems and
take the form
(45)
$\ddot{x}=-\dfrac{a\dot{x}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\dot{\tilde{\lambda}},\qquad\ddot{y}=-\dfrac{a\dot{y}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\dot{\tilde{\lambda}},\qquad\ddot{z}=-g+\dot{\tilde{\lambda}}.$
Clearly that this system coincide with classical differential equations (44)
with $\dot{\tilde{\lambda}}=\mu$.
After the derivation of the constraint
$\dot{z}-a\,\sqrt{\dot{x}^{2}+\dot{y}^{2}}=0$ along the solutions of (45), we
obtain
$0=\ddot{z}-a\dfrac{\ddot{x}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}+a\dfrac{\ddot{y}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}=-g+(1+a^{2})\dot{\tilde{\lambda}}.$
Therefore $\dot{\tilde{\lambda}}=\dfrac{g}{1+a^{2}}.$ Hence the equations of
motion (45) become
(46)
$\ddot{x}=-\frac{ag}{1+a^{2}}\frac{\dot{x}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}},\qquad\ddot{y}=-\frac{ag}{1+a^{2}}\frac{\dot{y}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}},\qquad\ddot{z}=-\frac{a^{2}g}{1+a^{2}}.$
In this case the Lagrangian (14) writes
$L=\dfrac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-gz-\dfrac{g\,(t+C)}{1+a^{2}}(\dot{z}-a\sqrt{\dot{x}^{2}+\dot{y}^{2}})-\lambda^{0}_{2}\arctan\dfrac{\dot{x}}{\dot{y}},$
where $C$ and $\lambda^{0}_{2}$ are an arbitrary constants.
Under the condition $L_{1}=0$ we obtain that the transpositional relations are
(47)
$\begin{array}[]{rl}\delta\dfrac{dx}{dt}-\dfrac{d}{dt}\delta{x}=&\dfrac{\dot{y}\left((1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)\left(\ddot{x}\delta{y}-\ddot{y}\delta{x}\right)+a^{2}\dot{x}\left(\dot{y}\ddot{x}-\dot{x}\ddot{y}\right)\left(\dot{x}\delta{y}-\dot{y}\delta{x}\right)\right)}{(1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)^{2}},\vspace{0.20cm}\\\
\delta\dfrac{dy}{dt}-\dfrac{d}{dt}\delta{y}=&\dfrac{\dot{x}\left((1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)\left(\ddot{y}\delta{x}-\ddot{x}\delta{y}\right)+a^{2}\dot{y}\left(\dot{y}\ddot{x}-\dot{x}\ddot{y}\right)\left(\dot{x}\delta{y}-\dot{y}\delta{x}\right)\right)}{(1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)^{2}},\vspace{0.20cm}\\\
\delta\dfrac{dz}{dt}-\dfrac{d}{dt}\delta{z}=&\dfrac{a\left(\dot{y}\ddot{x}-\dot{x}\ddot{y}\right)\left(\dot{x}\delta{y}-\dot{x}\delta{y}\right)}{(1+a^{2})\left(\dot{x}^{2}+\dot{y}^{2}\right)^{3/2}}.\end{array}$
From this example we obtain that the independent virtual variations $\delta x$
and $\delta y$ produce non–zero transpositional relations. This result is not
in accordance with with the Suslov point on view on the transpositional
relations.
Now we apply Theorem 2. The functions $L_{0},\,L_{1},\,L_{2}$ and $L_{3}$ are
determined as follows
$L_{0}=\tilde{L},\quad L_{1}=\dot{z}-a\,\sqrt{\dot{x}^{2}+\dot{y}^{2}},\quad
L_{2}=\dot{y},\quad L_{3}=\dot{x}.$
Thus the matrix $W_{1}$ and $\Omega_{1}$ are
$W_{1}=\left(\begin{array}[]{ccc}-\dfrac{a\dot{x}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}&-\dfrac{a\dot{y}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}&1\\\
0&1&0\\\
1&0&0\end{array}\right),\quad\Omega_{1}=\left(\begin{array}[]{ccc}\dot{y}q&-\dot{x}q&0\\\
0&0&0\\\ 0&0&0\end{array}\right),$
where
$q=\dfrac{a(\ddot{x}\dot{y}-\ddot{x}\dot{y})}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}^{3}}.$
Therefore $|W_{1}|=-1.$
Hence, after some computations from (11) we have that
$A=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
\dot{y}q&-\dot{x}q&0\end{array}\right).$
The equations of motion (12) becomes
(48)
$\begin{array}[]{rl}\ddot{x}=&-\dfrac{a^{2}\dot{y}}{{\dot{x}^{2}+\dot{y}^{2}}}(\dot{y}\ddot{x}-\dot{x}\ddot{y})-\dfrac{a\dot{{\lambda}}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\dot{x},\vspace{0.2cm}\\\
\ddot{y}=&-\dfrac{a^{2}\dot{x}}{{\dot{x}^{2}+\dot{y}^{2}}}(\dot{x}\ddot{y}-\dot{y}\ddot{x})-\dfrac{a\dot{{\lambda}}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\dot{y},\vspace{0.2cm}\\\
\ddot{z}=&-g+\dot{{\lambda}}.\end{array}$
By solving these equations with respect to $\ddot{x},\,\ddot{y}$ and
$\ddot{z}$ we obtain the equations
$\ddot{x}=-\dfrac{a\dot{x}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\dot{{\lambda}},\qquad\ddot{y}=-\dfrac{a\dot{y}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\dot{{\lambda}},\qquad\ddot{z}=-g+\dot{{\lambda}},$
We observe in this case that $|W_{1}|=-1,$ consequently these equations,
obtained from Theorem 2, give a global behavior of the Appell–Hamel systems,
i.e. coincide with the classical equations (44) with
$\dot{{\lambda}}=\dot{{\tilde{\lambda}}}=\mu=\dfrac{g}{1+a^{2}}.$
The transpositional relations (13) can be written as
(49)
$\delta\dfrac{dx}{dt}-\dfrac{d}{dt}\delta\,x=0,\quad\delta\dfrac{dy}{dt}-\dfrac{d}{dt}\delta\,y=0,\quad\delta\dfrac{dz}{dt}-\dfrac{d}{dt}\delta\,z=q\left(\dot{y}\delta\,x-\dot{x}\delta\,y\right).$
∎
From this corollary we observe that the independent virtual variations $\delta
x$ and $\delta y$ produce non–zero transpositional relations (47) and zero
transpositional relations (49).
The Lagrangian (10) in this case takes the form
$\begin{array}[]{rl}L=&\dfrac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-gz-\dfrac{g\,(t+C)}{1+a^{2}}(\dot{z}-a\sqrt{\dot{x}^{2}+\dot{y}^{2}})-\lambda^{0}_{2}\dot{y}-\lambda^{0}_{3}\dot{x}\vspace{0.2cm}\\\
\simeq&\dfrac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-gz-\dfrac{g\,(t+C)}{1+a^{2}}(\dot{z}-a\sqrt{\dot{x}^{2}+\dot{y}^{2}}).\end{array}$
From (34) it follows that
$\delta\dfrac{dz}{dt}-\dfrac{d}{dt}\delta\,z=q\left(\dot{y}\delta\,x-\dot{x}\delta\,y\right)+\dfrac{a\dot{x}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\left(\delta\dfrac{dx}{dt}-\dfrac{d}{dt}\delta\,x\right)+\dfrac{a\dot{y}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\left(\delta\dfrac{dy}{dt}-\dfrac{d}{dt}\delta\,y\right).$
Therefore this relation holds identically for (47) and (49).
In the next sections we show the importance of the equations of motion (12)
and (16) contrasting them with the classical differential equations of
nonholonomic mechanics.
## 6\. Modificated vakonomic mechanics versus vakonomic mechanics
Now we show that the equations of the vakonomic mechanics (3) can be obtained
from equations (9). More precisely, if in (7) we require that all the virtual
variations of the coordinates produce the zero transpositional relations, i.e.
the matrix $A$ is the zero matrix and we require that $\lambda^{0}_{j}=0$ for
$j=M+1,\ldots,N$, then from (9) by considering that $D_{k}L=E_{k}L,$ we obtain
the vakonomic equations (3), i.e.
$\begin{array}[]{rl}D_{\nu}L_{0}=&\displaystyle\sum_{j=1}^{M}\left(\lambda_{j}D_{\nu}L_{j}+\dfrac{d\lambda_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial{\dot{x}_{\nu}}}\right)+\displaystyle\sum_{j=M+1}^{N}\lambda^{0}_{j}D_{\nu}\,L_{j}{\Longrightarrow\vspace{0.2cm}}\\\
E_{\nu}\,L_{0}=&\displaystyle\sum_{j=1}^{M}\left(\lambda_{j}E_{\nu}\,L_{j}+\dfrac{d\lambda_{j}}{dt}\dfrac{\partial{L_{j}}}{\partial\dot{x}_{\nu}}\right),\quad{\nu=1,\ldots,N}\end{array}$
In the following example in order to contrast Theorems 2 with the vakonomic
model we study the skate or knife edge on an inclined plane.
Example 1. To set up the problem, consider a plane $\Xi$ with cartesian
coordinates $x$ and $y,$ slanted at an angle $\alpha$. We assume that the
$y$–axis is horizontal, while the $x$–axis is directed downward from the
horizontal and let $(x,y)$ be the coordinates of the point of contact of the
skate with the plane. The angle $\varphi$ represents the orientation of the
skate measured from the $x$–axis. The skate is moving under the influence of
the gravity. Here the the acceleration due to gravity is denoted by $g$. It
also has mass $m,$ and the moment inertia of the skate about a vertical axis
through its contact point is denoted by $J,$ (see page 108 of [35] for a
picture). The equation of nonintegrable constraint is
(50) $L_{1}=\dot{x}\sin\varphi-\dot{y}\cos\varphi=0.$
With these notations the Lagrangian function of the skate is
$\hat{L}=\dfrac{m}{2}\left(\dot{x}^{2}+\dot{y}^{2}\right)+\dfrac{J}{2}\dot{\varphi}^{2}+mg\,x\,\sin\alpha.$
Thus we have the constrained mechanical systems
$\left(\mathbb{R}^{2}\times\mathbb{S}^{1},\quad\hat{L}=\dfrac{m}{2}\left(\dot{x}^{2}+\dot{y}^{2}\right)+\dfrac{J}{2}\dot{\varphi}^{2}+mg\,x\,\sin\alpha,\quad\\{\dot{x}\sin\varphi-\dot{y}\cos\varphi=0\\}\right).$
For appropriate choice of mass, length and time units, we reduces the
Lagrangian $\hat{L}$ to
$L_{0}=\dfrac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{\varphi}^{2}\right)+x\,g\sin\alpha,$
here for simplicity we leave the same notations for the all variables. The
question is, what is the motion of the point of contact? To answer this
question we shall use the vakonomic equations (3) and the equations (12)
proposed in Theorem 2.
### 6.1. The study of the skate applying Theorem 2
We determine the motion of the point of contact of the skate using Theorem 2.
We choose the arbitrary functions $L_{2}$ and $L_{3}$ as follows
$L_{2}=\dot{x}\cos\varphi+\dot{y}\sin\varphi,\quad L_{3}=\dot{\varphi},$
in order that the determinant $|W_{1}|\neq 0$ everywhere in the configuration
space.
The Lagrangian (10) becomes
$\begin{array}[]{rl}L(x,y,\varphi,\dot{x},\dot{y},\dot{\varphi},\Lambda)=&\dfrac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{\varphi}^{2}\right)+g\sin\alpha
x-\lambda(\dot{x}\sin\varphi-\dot{y}\cos\varphi)-\lambda^{0}_{3}\dot{\varphi}\vspace{0.20cm}\\\
&\simeq\dfrac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{\varphi}^{2}\right)+g\sin\alpha
x-\lambda(\dot{x}\sin\varphi-\dot{y}\cos\varphi),\end{array}$
where $\lambda:=\lambda_{1}.$
The matrix $W_{1}$ and $\Omega_{1}$ are
$\begin{array}[]{rl}W_{1}=&\left(\begin{array}[]{cccc}\sin\varphi&-\cos{\varphi}&0\\\
\cos{\varphi}&\sin\varphi&0\\\ 0&0&1\\\
\end{array}\right),\quad|W_{1}|=1,\vspace{0.2cm}\\\
\Omega_{1}=&\left(\begin{array}[]{cccc}\dot{\varphi}\cos{\varphi}&\dot{\varphi}\sin{\varphi}&-L_{2}\\\
-\dot{\varphi}\sin{\varphi}&\dot{\varphi}\cos{\varphi}&-L_{1}\\\ 0&0&0\\\
\end{array}\right).\end{array}$
The matrix $A=W^{-1}_{1}\Omega_{1}$ becomes
$A=\left.\left(\begin{array}[]{cccc}0&\dot{\varphi}&-\sin\varphi
L_{2}-\cos\varphi L_{1}\\\ -\dot{\varphi}&0&\cos\varphi L_{2}-\sin\varphi
L_{1}\\\ 0&0&0\\\
\end{array}\right)\right|_{L_{1}=0}=\left(\begin{array}[]{cccc}0&\dot{\varphi}&-\dot{y}\\\
-\dot{\varphi}&0&\dot{x}\\\ 0&0&0\\\ \end{array}\right).$
Hence the equation (12) and transpositional relations (13) take the form
(51)
$\ddot{x}+\dot{\varphi}\dot{y}=g\sin\alpha+\dot{\lambda}\sin\varphi,\quad\ddot{y}-\dot{\varphi}\dot{x}=-\dot{\lambda}\cos\varphi,\quad\ddot{\varphi}=0,$
and
(52) $\begin{array}[]{rl}\delta\dfrac{dx}{dt}-\dfrac{d\delta
x}{dt}=&\dot{y}\delta\varphi-\dot{\varphi}\delta y,\\\
\delta\dfrac{dy}{dt}-\dfrac{d\delta y}{dt}=&\dot{\varphi}\delta
x-\dot{x}\delta{\varphi},\\\
\delta\dfrac{d\varphi}{dt}-\dfrac{d\delta\varphi}{dt}=&-L_{2}\left(\delta
x\sin\varphi-\delta y\cos\varphi\right)=0,\end{array}$
respectively, here we have applied the Lagrange–Chetaev’s condition
$\sin\varphi\,\delta x-\cos\varphi\,\delta y=0.$
The initial conditions
$x_{0}=\left.x\right|_{t=0},\quad
y_{0}=\left.y\right|_{t=0},\quad\varphi_{0}=\left.\varphi\right|_{t=0},\quad\dot{x}_{0}=\left.\dot{x}\right|_{t=0},\quad\dot{y}_{0}=\left.\dot{y}\right|_{t=0},\quad\dot{\varphi}_{0}=\left.\dot{\varphi}\right|_{t=0},$
satisfy the constraint, i.e.
(53) $\sin\varphi_{0}\dot{x}_{0}-\cos\varphi_{0}\dot{y}_{0}=0.$
After the derivation of the constraint along the solutions of the equation of
motion (51), and using (50) we obtain
$\begin{array}[]{rl}0=&\sin\varphi\ddot{x}-\cos\varphi\ddot{y}+\dot{\varphi}\left(\cos\varphi\dot{x}+\sin\varphi\dot{y}\right)\vspace{0.2cm}\\\
=&\sin\varphi\left(g\sin\alpha+\dot{\lambda}\sin\varphi-\dot{\varphi}\dot{y}\right)-\cos\varphi\left(-\dot{\lambda}\cos\varphi+\dot{\varphi}\dot{x}\right)+\dot{\varphi}\left(\cos\varphi\dot{x}+\sin\varphi\dot{y}\right).\end{array}$
Hence $\dot{\lambda}=-g\sin\alpha\sin\varphi.$ Therefore the differential
equations (51) can be written as
(54)
$\ddot{x}+\dot{\varphi}\dot{y}=g\sin\alpha\cos^{2}\varphi,\quad\ddot{x}-\dot{\varphi}\dot{x}=g\sin\alpha\sin\varphi\cos\varphi,\quad\ddot{\varphi}=0.$
We study the motion of the skate in the following three cases:
* (i)
$\left.\dot{\varphi}\right|_{t=0}=\omega=0.$
* (ii)
$\left.\dot{\varphi}\right|_{t=0}=\omega\neq 0.$
* (iii)
$\alpha=0.$
For the first case ($\omega=0$), after the change of variables
$X=\cos\varphi_{0}\,x-\sin\varphi_{0}\,y,\quad
Y=\cos\varphi_{0}\,x+\sin\varphi_{0}\,y,$
the differential equations (9) and the constraint become
$\ddot{X}=0,\quad\ddot{Y}=g\sin\alpha\cos\varphi_{0},\quad\varphi=\varphi_{0},\quad\dot{X}=0,$
respectively. Consequently
$X=X_{0},\quad
Y=g\sin\alpha\cos\varphi_{0}\dfrac{t^{2}}{2}+\dot{Y}_{0}t+Y_{0},\quad\varphi=\varphi_{0},$
thus the trajectories are straight lines.
For the second case ($\omega\neq 0$), we take
$\varphi_{0}=\dot{y}_{0}=\dot{x}_{0}=x_{0}=y_{0}=0$ in order to simplify the
computations. In view of the equality
$\dot{\varphi}=\left.\dot{\varphi}\right|_{t=0}=\omega$ and denoting by ′ the
derivation with respect $\varphi$ we get that (54) become
(55)
$x^{\prime\prime}+y^{\prime}=\dfrac{g\sin\alpha}{\omega^{2}}\cos^{2}\varphi,\quad
x^{\prime\prime}-{x}^{\prime}=\dfrac{g\sin\alpha}{\omega^{2}}\sin\varphi\cos\varphi,\quad\varphi^{\prime}=1.$
Which are easy to integrate and we obtain
$x=-\dfrac{g\sin\alpha}{4\omega^{2}}\cos{(2\varphi)},\quad
y=-\dfrac{g\sin\alpha}{4\omega^{2}}\sin{(2\varphi)}+\dfrac{g}{2\omega^{2}}\varphi,\quad\varphi=\omega
t,$
which correspond to the equation of the cycloid. Hence the point of contact of
the skate follows a cycloid along the plane, but do not slide down the plane.
For the third case ($\alpha=0$), if $\varphi_{0}=0,\,\omega\neq 0$ we obtain
that the solutions of the given differential systems (54) are
$x=\dot{y}_{0}\cos\varphi+\dot{x}_{0}\sin\varphi+a,\quad
y=\dot{y}_{0}\sin\varphi+\dot{y}_{0}\cos\varphi+b,\quad\varphi=\varphi_{0}+\omega
t,$
where
$a=x_{0}-\dfrac{\dot{y}_{0}}{\omega},\,b=y_{0}+\dfrac{\dot{x}_{0}}{\omega},$
which correspond to the equation of the circle with center at $(a,b)$ and
radius $\dfrac{\dot{x}^{2}_{0}+\dot{y}^{2}_{0}}{\omega^{2}}.$
If $\alpha=0$ and $\varphi_{0}=0,\,\omega=0$ then we obtain that the solutions
are
$x=\dot{x}_{0}t+x_{0},\quad y=\dot{y}_{0}t+y_{0}.$
All these solutions coincide with the solutions obtained from the Lagrangian
equations (33) with multipliers (see [2])
$\ddot{x}=g\sin\alpha+\mu\sin\varphi,\quad\ddot{y}-\dot{\varphi}\dot{x}=-\mu\cos\varphi,\quad\ddot{\varphi}=0,$
with $\mu=\dot{\lambda}=-g\sin\alpha\sin\varphi.$
### 6.2. The study of the skate applying vakonomic model
Now we consider instead of Theorem 2 the vakomic model for studying the motion
of the skate.
We consider the Lagrangian
$L(x,y,\varphi,\dot{x},\dot{y},\dot{\varphi},\Lambda)=\dfrac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{\varphi}^{2}\right)+g\,x\,\sin\alpha-\lambda(\dot{x}\sin\varphi-\dot{y}\cos\varphi).$
The equations of motion (3) for the skate are
$\dfrac{d}{dt}\left(\dot{x}-\lambda\sin\varphi\right)=0,\quad\dfrac{d}{dt}\left(\dot{y}+\lambda\cos\varphi\right)=0,\quad\ddot{\varphi}=-\lambda\left(\dot{x}\cos\varphi+\dot{y}\sin\varphi\right).$
We shall study only the case when $\alpha=0.$ After integration we obtain the
differential systems
(56)
$\begin{array}[]{rl}\dot{x}=&\lambda\sin\varphi+a=\cos\varphi\left(a\cos\varphi+b\sin\varphi\right),\\\
\dot{y}=&-\lambda\cos\varphi+b=\sin\varphi\left(a\cos\varphi+b\sin\varphi\right),\\\
\ddot{\varphi}=&\left(b\cos\varphi-a\sin\varphi\right)\left(a\cos\varphi+b\sin\varphi\right)=(b^{2}_{1}+a^{2}_{2})\sin(\varphi+\alpha)\cos(\varphi+\alpha),\\\
\lambda=&b\cos\varphi-a\sin\varphi,\end{array}$
where $a=\dot{x}_{0}-\lambda_{0}\sin\varphi_{0},$
$b=\dot{y}_{0}+\lambda_{0}\cos\varphi_{0}$ and $\lambda_{0}=\lambda|_{t=0}$ is
an arbitrary parameter. After the integration of the third equation we obtain
that
(57)
$\displaystyle\int_{0}^{\varphi}\dfrac{d\varphi}{\sqrt{1-\kappa^{2}\sin^{2}\varphi}}=t\sqrt{\dfrac{h+a^{2}+b^{2}}{2}},$
where $h$ is an arbitrary constant which we choose in such a way that
$\kappa^{2}=\dfrac{2(a^{2}+b^{2})}{h+a^{2}+b^{2}}<1.$
From (57) we get
$\sin\varphi=sn\left(t\sqrt{\dfrac{h+a^{2}+b^{2}}{2}}\right),\quad\cos\varphi=cn\left(t\sqrt{\dfrac{h+a^{2}+b^{2}}{2}}\right),$
where $sn$ and $cn$ are the Jacobi elliptic functions . Hence, if we take
$\dot{x}_{0}=1,\,\dot{y}_{0}=\varphi_{0}=0,$ then the solutions of the
differential equations (56) are
(58)
$\begin{array}[]{rl}x=&x_{0}+\displaystyle\int_{t_{0}}^{t}\left(cn\left(t\sqrt{\dfrac{h+1+\lambda^{2}_{0}}{2}}\right)sn\left(t\sqrt{\dfrac{h+1+\lambda^{2}_{0}}{2}}\right)+\lambda_{0}sn\left(t\sqrt{\dfrac{h+1+\lambda^{2}_{0}}{2}}\right)\right)dt,\vspace{0.2cm}\\\
y=&y_{0}+\displaystyle\int_{t_{0}}^{t}sn\left(t\sqrt{\dfrac{h+1+\lambda^{2}_{0}}{2}}\right)\,\lambda_{0}\,sn\left(t\sqrt{\dfrac{h+1+\lambda^{2}_{0}}{2}}\right)dt,\vspace{0.2cm}\\\
\varphi=&am\left(t\sqrt{\dfrac{h+1+\lambda^{2}_{0}}{2}}\right).\end{array}$
It is interesting to compare this amazing motions with the motions that we
obtained above. For the same initial conditions the skate moves sideways along
the circles. By considering that the solutions (58) depend on the arbitrary
parameter $\lambda_{0}$ we obtain that for the given initial conditions do not
exist a unique solution of the differential equations in the vakonomic model.
Consequently the principle of determinacy is not valid for vakonomic mechanics
with nonintegrable constraints (see the Corollary of page 36 in [2]).
## 7\. Modificated vakonomic mechanics versus Lagrangian and constrained
Lagrangian mechanics
### 7.1. MVM versus Lagrangian mechanics
The Lagrangian equations which describe the motion of the Lagrangian systems
can be obtained from Theorem 2 by supposing that $M=0,$ i.e. there is no
constraints We choose the arbitrary functions $L_{\alpha}$ for
$\alpha=1,\ldots,N$ as follows
$\quad L_{\alpha}=\dfrac{dx_{\alpha}}{dt},\quad\alpha=1,\ldots,N.$
Hence the Lagrangian (10) takes the form
$L=L_{0}-\sum_{j=1}^{N}\lambda^{0}_{j}\dfrac{dx_{j}}{dt}\simeq L_{0}.$
In this case we have that $|W_{1}|=1.$
By considering the property of the Lagrangian derivative (see (27)) we obtain
that $\Omega_{1}$ is a zero matrix . Hence the matrices $A_{1}$ is the zero
matrix. As a consequence the equations (12) become
$D_{\nu}L=E_{\nu}L=E_{\nu}\left(L_{0}-\displaystyle\sum_{j=1}^{N}\lambda^{0}_{j}\dot{x}_{j}\right)=E\nu
L_{0}=0$
because $L\simeq L_{0}.$ The transpositional relation (13) in this case are
$\delta\dfrac{d\textbf{x}}{dt}-\dfrac{d\delta{\textbf{x}}}{dt}=0,$ which are
the well known relations in the Lagrangian mechanics (see formula (29)).
### 7.2. MVM versus constrained Lagrangian systems
From the equivalences (42) we have that in the case when the constraints are
linear in the velocity the equations of motions of the MVM coincide with the
Lagrangian equations with multipliers (33) except perhaps in a zero Lebesgue
measure set $|W_{2}|=0$ or $|W_{1}|=0.$ When the constraints are nonlinear in
the velocity, we have the equivalence (43). Consequently equations of motions
of the MVM coincide with the Lagrangian equations with multipliers (33) except
perhaps in a zero Lebesgue measure set $|W_{2}|=0.$
We illustrate this result in the following example.
Example 2. Let
$\left(\mathbb{R}^{2},\quad
L_{0}=\dfrac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}\right)-U(x,y),\quad\\{2\left({x}\dot{{x}}+{y}\dot{{y}}\right)=0\\}\right),$
be the constrained Lagrangian systems.
In order to apply Theorem 2 we choose the arbitrary function $L_{1}$ and
$L_{2}$ as follow
* (a)
$L_{1}=2\left({x}\dot{{x}}+{y}\dot{{y}}\right),\quad
L_{2}=-y\dot{x}+x\dot{y}.$
Thus the matrices $W_{1}$ and $\Omega_{1}$ are
$W_{1}=\left(\begin{array}[]{cc}2x&2y\\\ -y&x\\\
\end{array}\right),\quad|W_{1}|=2x^{2}+2y^{2}=2,\quad\Omega_{1}=\left(\begin{array}[]{cc}0&0\\\
-2\dot{y}&2\dot{x}\\\ \end{array}\right).$
Consequently equations (12) describe the motion everywhere for the constrained
Lagrangian systems.
Equations (12) become
$\begin{array}[]{rl}\ddot{x}=\left.-\dfrac{\partial U}{\partial
x}+2\dot{y}\left(y\dot{x}-x\dot{y}\right)+2x\dot{\lambda}\right|_{L_{1}=0}=-\dfrac{\partial
U}{\partial
x}+x\left(\dot{\lambda}-2(\dot{x}^{2}+\dot{y}^{2})\right),\vspace{0.2cm}\\\
\ddot{y}=\left.-\dfrac{\partial U}{\partial
y}-2\dot{x}\left(y\dot{x}-x\dot{y}\right)+2y\dot{\lambda}\right|_{L_{1}=0}=-\dfrac{\partial
U}{\partial
y}+y\left(\dot{\lambda}-2(\dot{x}^{2}+\dot{y}^{2})\right),\end{array}$
Transpositional relations take the form
(59) $\delta\dfrac{d{x}}{dt}-\dfrac{d\delta{x}}{dt}=2y\left(\dot{y}\delta
x-\dot{x}\delta
y\right),\quad\delta\dfrac{d{y}}{dt}-\dfrac{d\delta{y}}{dt}=-2x\left(\dot{y}\delta
x-\dot{x}\delta y\right).$
* (b)
If we choose
$L_{2}=\dfrac{y\dot{x}}{x^{2}+y^{2}}-\dfrac{x\dot{y}}{x^{2}+y^{2}}=\dfrac{d}{dt}\arctan{\dfrac{x}{y}},$
then
$W_{1}=\left(\begin{array}[]{cc}2x&2y\\\
\dfrac{y}{x^{2}+y^{2}}&-\dfrac{x}{x^{2}+y^{2}}\\\
\end{array}\right),\quad|W_{1}|=-2,\quad\Omega_{1}=\left(\begin{array}[]{cc}0&0\\\
0&0\\\ \end{array}\right).$
Equations (12) and transpositional relations become
$\ddot{x}=-\dfrac{\partial
U}{\partial\,x}+2x\dot{\lambda},\quad\ddot{y}=-\dfrac{\partial U}{\partial
y}+2y\dot{\lambda},$ (60)
$\delta\dfrac{d{x}}{dt}-\dfrac{d\delta{x}}{dt}=0,\quad\delta\dfrac{d{y}}{dt}-\dfrac{d\delta{y}}{dt}=0.$
respectively.
From this example we obtain that for the holonomic constrained Lagrangian
systems the transpositional relations can be non–zero (see (59)), or can be
zero (see (60)). We observe that from condition (34) it follows the relation
$x\left(\delta\dfrac{d{x}}{dt}-\dfrac{d\delta{x}}{dt}\right)+y\left(\delta\dfrac{d{y}}{dt}-\dfrac{d\delta{y}}{dt}\right)=0.$
This equality holds identically if (60) and (59) takes place.
The equations of motions (33) in this case are
$\ddot{x}=-\dfrac{\partial
U}{\partial\,x}+2x\,\mu,\quad\ddot{y}=-\dfrac{\partial U}{\partial
y}+2y\,\mu,$
with $\mu=\dot{\lambda}-2(\dot{x}^{2}+\dot{y}^{2}).$
Example 3. To contrast the MVM with the classical model we apply Theorems 2 to
the Gantmacher’s systems (see for more details [11, 45]).
Two material points $m_{1}$ and $m_{2}$ with equal masses are linked by a
metal rod with fixed length $l$ and small mass. The systems can move only in
the vertical plane and so the speed of the midpoint of the rod is directed
along the rod. It is necessary to determine the trajectories of the material
points $m_{1}$ and $m_{2}.$
Let $(q_{1},\,r_{1})$ and $(q_{2},\,r_{2})$ be the coordinates of the points
$m_{1}$ and $m_{2},$ respectively. Clearly
$(q_{1}-q_{2})^{2}+(r_{1}-r_{2})^{2}=l^{2}.$ Thus we have a constrained
Lagrangian system in the configuration space $\mathbb{R}^{4}$ with the
Lagrangian function
$\textsc{L}=\dfrac{1}{2}\left(\dot{q}^{2}_{1}+\dot{q}^{2}_{2}+\dot{r}^{2}_{1}+\dot{r}^{2}_{2}\right)-g/2{(r_{1}+r_{2})},$
and with the linear constraints
$(q_{2}-q_{1})(\dot{q}_{2}-\dot{q}_{1})+(r_{2}-r_{1})(\dot{r}_{2}-\dot{r}_{1})=0,\quad(q_{2}-q_{1})(\dot{r}_{2}+\dot{r}_{1})-(r_{2}-r_{1})(\dot{q}_{2}+\dot{q}_{1})=0.$
Introducing the following change of coordinates:
$x_{1}=\dfrac{q_{2}-q_{1}}{2},\quad x_{2}=\dfrac{r_{1}-r_{2}}{2},\quad
x_{3}=\dfrac{r_{2}+r_{1}}{2},\quad x_{4}=\dfrac{q_{1}+q_{2}}{2},$
we obtain
$x^{2}_{1}+x^{2}_{2}=\dfrac{1}{4}\left((q_{1}-q_{2})^{2}+(r_{1}-r_{2})^{2}\right)=\dfrac{l^{2}}{4}.$
Hence we have the constrained Lagrangian mechanical systems
$\left(\mathbb{R}^{4},\quad\tilde{L}=\displaystyle\frac{1}{2}\sum_{j=1}^{4}\dot{x}^{2}_{j}-gx_{3},\quad\\{x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2}=0,\quad
x_{1}\dot{x}_{3}-x_{2}\dot{x}_{4}=0\\}\right).$
The equations of motion (33) obtained from the d’Alembert–Lagrange principle
are
(61)
$\ddot{x}_{1}=\mu_{1}x_{1},\quad\ddot{x}_{2}=\mu_{1}x_{2},\quad\ddot{x}_{3}=-g+\mu_{2}x_{1},\quad\ddot{x}_{4}=-\mu_{2}x_{2},$
where $\mu_{1},\,\mu_{2}$ are the Lagrangian multipliers such that
(62)
$\mu_{1}=-\dfrac{\dot{x}^{2}_{1}+\dot{x}^{2}_{2}}{x^{2}_{1}+x^{2}_{2}},\quad\mu_{2}=\dfrac{\dot{x}_{2}\dot{x}_{4}-\dot{x}_{1}\dot{x}_{3}+gx_{1}}{x^{2}_{1}+x^{2}_{2}}.$
For applying Theorem 2 we have the constraints
$L_{1}=x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2}=0,\quad
L_{2}=x_{1}\dot{x}_{3}-x_{2}\dot{x}_{4}=0,$
and we choose the arbitrary functions $L_{3}$ and $L_{4}$ as follows
$L_{3}=-x_{1}\dot{x}_{2}+x_{2}\dot{x}_{1},\quad
L_{4}=x_{2}\dot{x}_{3}+x_{1}\dot{x}_{4}.$
For the given functions we obtain that
$W_{1}=\left(\begin{array}[]{cccc}x_{1}&x_{2}&0&0\\\ 0&0&x_{1}&-x_{2}\\\
x_{2}&-x_{1}&0&0\\\
0&0&x_{2}&x_{1}\end{array}\right),\quad\Omega_{1}=\left(\begin{array}[]{cccc}0&0&0&0\\\
-\dot{x}_{3}&\dot{x}_{4}&\dot{x}_{1}&-\dot{x}_{2}\\\
-2\dot{x}_{2}&2\dot{x}_{1}&0&0\\\
-\dot{x}_{4}&-\dot{x}_{3}&\dot{x}_{2}&\dot{x}_{1}\end{array}\right).$
Therefore $|W_{1}|=(x^{2}_{1}+x^{2}_{2})^{2}=\dfrac{l^{2}}{4}\neq 0.$ The
matrix $A$ in this case is
$\left(\begin{array}[]{cccc}\dfrac{2x_{2}\dot{x}_{2}}{x^{2}_{1}+x^{2}_{2}}&-\dfrac{2x_{2}\dot{x}_{1}}{x^{2}_{1}+x^{2}_{2}}&0&0\vspace{0.2cm}\\\
-\dfrac{2x_{1}\dot{x}_{2}}{x^{2}_{1}+x^{2}_{2}}&\dfrac{2x_{1}\dot{x}_{1}}{x^{2}_{1}+x^{2}_{2}}&0&0\vspace{0.2cm}\\\
-\dfrac{x_{1}\dot{x}_{3}+x_{2}\dot{x}_{4}}{x^{2}_{1}+x^{2}_{2}}&\dfrac{x_{1}\dot{x}_{4}-x_{2}\dot{x}_{3}}{x^{2}_{1}+x^{2}_{2}}&\dfrac{x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2}}{x^{2}_{1}+x^{2}_{2}}&\dfrac{x_{2}\dot{x}_{1}-x_{1}\dot{x}_{2}}{x^{2}_{1}+x^{2}_{2}}\vspace{0.2cm}\\\
\dfrac{x_{1}\dot{x}_{4}-x_{2}\dot{x}_{3}}{x^{2}_{1}+x^{2}_{2}}&\dfrac{x_{1}\dot{x}_{3}-x_{2}\dot{x}_{4}}{x^{2}_{1}+x^{2}_{2}}&\dfrac{x_{2}\dot{x}_{1}-x_{1}\dot{x}_{2}}{x^{2}_{1}+x^{2}_{2}}&\dfrac{x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2}}{x^{2}_{1}+x^{2}_{2}}\end{array}\right).$
Consequently differential equations (12) take the form
(63)
$\begin{array}[]{rl}\ddot{x}_{1}=&\left.\left(\dfrac{2x_{2}\dot{x}_{1}\dot{x}_{2}-2x_{1}\dot{x}^{2}_{2}-x_{1}\dot{x}^{2}_{3}-x_{1}\dot{x}^{2}_{4}}{x^{2}_{1}+x^{2}_{2}}+x_{1}\dot{\lambda}_{1}\right)\right|_{L_{1}=L_{2}=0}\vspace{0.2cm}\\\
=&x_{1}\left(\dot{\lambda}_{1}-\dfrac{2\dot{x}^{2}_{1}+2\dot{x}^{2}_{2}+\dot{x}^{2}_{3}+\dot{x}^{2}_{4}}{x^{2}_{1}+x^{2}_{2}}\right),\vspace{0.2cm}\\\
\ddot{x}_{2}=&-\left.\left(\dfrac{-2x_{1}\dot{x}_{1}\dot{x}_{2}+2x_{2}\dot{x}^{2}_{2}+x_{2}\dot{x}^{2}_{3}+x_{2}\dot{x}^{2}_{4}}{x^{2}_{1}+x^{2}_{2}}+x_{2}\dot{\lambda}_{1}\right)\right|_{L_{1}=L_{2}=0}\vspace{0.2cm}\\\
=&x_{2}\left(\dot{\lambda}_{1}-\dfrac{2\dot{x}^{2}_{1}+2\dot{x}^{2}_{2}+\dot{x}^{2}_{3}+\dot{x}^{2}_{4}}{x^{2}_{1}+x^{2}_{2}}\right),\vspace{0.2cm}\\\
\ddot{x}_{3}=&\left.\left(\dfrac{\dot{x}_{3}\left(x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2}\right)-\dot{x}_{4}\left(x_{2}\dot{x}_{1}-x_{1}\dot{x}_{2}\right)}{x^{2}_{1}+x^{2}_{2}}+x_{1}\dot{\lambda}_{2}-g\right)\right|_{L_{1}=L_{2}=0}\vspace{0.2cm}\\\
=&\dfrac{\dot{x}_{4}\left(x_{2}\dot{x}_{1}-x_{1}\dot{x}_{2}\right)}{x^{2}_{1}+x^{2}_{2}}+x_{1}\dot{\lambda}_{2}-g,\vspace{0.2cm}\\\
\ddot{x}_{4}=&\left.\left(\dfrac{\dot{x}_{4}\left(x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2}\right)-\dot{x}_{3}\left(x_{2}\dot{x}_{1}-x_{1}\dot{x}_{2}\right)}{x^{2}_{1}+x^{2}_{2}}-x_{2}\dot{\lambda}_{2}\right)\right|_{L_{1}=L_{2}=0}\vspace{0.2cm}\\\
=&-\dfrac{\dot{x}_{3}\left(x_{2}\dot{x}_{1}-x_{1}\dot{x}_{2}\right)}{x^{2}_{1}+x^{2}_{2}}-x_{2}\dot{\lambda}_{2}.\end{array}$
Derivating the constraints we obtain that the multipliers $\dot{\lambda}_{1}$
and $\dot{\lambda}_{2}$ are
$\dot{\lambda}_{1}=\dfrac{\dot{x}^{2}_{1}+\dot{x}^{2}_{2}+\dot{x}^{2}_{3}+\dot{x}^{2}_{4}}{x^{2}_{1}+x^{2}_{2}}=\mu_{1}+\dfrac{\dot{x}^{2}_{3}+\dot{x}^{2}_{4}}{x^{2}_{1}+x^{2}_{2}},\quad\dot{\lambda}_{2}=\dfrac{gx_{1}}{x^{2}_{1}+x^{2}_{2}}=\mu_{2}+\dfrac{\dot{x}_{1}\dot{x}_{3}-\dot{x}_{2}\dot{x}_{4}}{x^{2}_{1}+x^{2}_{2}}.$
Inserting these values into (63) we deduce
$\begin{array}[]{ll}\ddot{x}_{1}=&-\dfrac{x_{1}\left(\dot{x}^{2}_{1}+\dot{x}^{2}_{2}\right)}{x^{2}_{1}+x^{2}_{2}},\qquad\ddot{x}_{2}=\dfrac{x_{2}\left(\dot{x}^{2}_{1}+\dot{x}^{2}_{2}\right)}{x^{2}_{1}+x^{2}_{2}},\vspace{0.2cm}\\\
\ddot{x}_{3}=&-g+\dfrac{x_{1}\left(\dot{x}_{2}\dot{x}_{4}-\dot{x}_{1}\dot{x}_{3}+gx_{1}\right)}{x^{2}_{1}+x^{2}_{2}},\quad\ddot{x}_{4}=-\dfrac{x_{2}\left(\dot{x}_{2}\dot{x}_{4}-\dot{x}_{1}\dot{x}_{3}+gx_{1}\right)}{x^{2}_{1}+x^{2}_{2}}.\end{array}$
These equations coincide with equations (61) everywhere because
$|W_{1}|=\dfrac{l^{2}}{4},$ where $l$ is the length of the rod.
The transpositional relations in this case are
(64)
$\begin{array}[]{ll}\delta\dfrac{d{x}_{1}}{dt}-\dfrac{d\delta{x}_{1}}{dt}=&-\dfrac{2x_{2}}{x^{2}_{1}+x^{2}_{2}}\left(\dot{x}_{1}\delta
x_{2}-\dot{x}_{2}\delta x_{1}\right),\vspace{0.2cm}\\\
\delta\dfrac{d{x}_{2}}{dt}-\dfrac{d\delta{x}_{2}}{dt}=&\dfrac{2x_{1}}{x^{2}_{1}+x^{2}_{2}}\left(\dot{x}_{1}\delta
x_{2}-\dot{x}_{2}\delta x_{1}\right),\vspace{0.2cm}\\\
\delta\dfrac{d{x}_{3}}{dt}-\dfrac{d\delta{x}_{3}}{dt}=&\dfrac{x_{1}}{x^{2}_{1}+x^{2}_{2}}\left(\dot{x}_{1}\delta
x_{3}-\dot{x}_{3}\delta x_{1}+\dot{x}_{4}\delta x_{2}-\dot{x}_{2}\delta
x_{4}\right),\vspace{0.2cm}\\\
&+\dfrac{x_{2}}{x^{2}_{1}+x^{2}_{2}}\left(\dot{x}_{1}\delta
x_{4}-\dot{x}_{4}\delta x_{1}+\dot{x}_{2}\delta x_{3}-\dot{x}_{3}\delta
x_{2}\right),\\\
\delta\dfrac{d{x}_{4}}{dt}-\dfrac{d\delta{x}_{4}}{dt}=&-\dfrac{x_{2}}{x^{2}_{1}+x^{2}_{2}}\left(\dot{x}_{1}\delta
x_{3}-\dot{x}_{3}\delta x_{1}+\dot{x}_{4}\delta x_{2}-\dot{x}_{2}\delta
x_{4}\right)\vspace{0.3cm}\\\
&+\dfrac{x_{1}}{x^{2}_{1}+x^{2}_{2}}\left(\dot{x}_{1}\delta
x_{4}-\dot{x}_{4}\delta x_{1}+\dot{x}_{2}\delta x_{3}-\dot{x}_{3}\delta
x_{2}\right).\end{array}$
From this example we again get that the virtual variations produce the
non–zero transpositional relations.
###### Remark 21.
From the previous example we observe that the virtual variations produce zero
or non–zero transpositional relations, depending on the arbitrary functions
which appear in the construction of the proposed mathematical model. Thus, the
following question arises: Can be choosen the arbitrary functions $L_{j}$ for
$j=M+1,\ldots,N$ in such a way that for the nonholonomic systems only the
independent virtual variations would generate zero transpositional relations?
The positive answer to this question is obtained locally for any constrained
Lagrangian systems and globally for the Chaplygin-Voronets mechanical systems,
and for the generalization of these systems studied in the next section.
## 8\. MVM and nonholonomic generalized Voronets–Chaplygin systems. Proofs
of Theorem 4 and Proposition 5 and 6.
It was pointed out by Chaplygin [6] that in many conservative nonholonomic
systems the generalized coordinates
$\left(\textbf{x},\textbf{y}\right):=\left(x_{1},\ldots,x_{s_{1}},y_{1},\ldots,y_{s_{2}}\right),\quad
s_{1}+s_{2}=N,$
can be chosen in such a way that the Lagrangian function and the constraints
take the simplest form. In particular Voronets in [53] studied the constrained
Lagrangian systems with Lagrangian
$\tilde{L}=\tilde{L}\left(\textbf{x},\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right)$
and constraints (22). This systems is called the Voronets mechanical systems.
We shall apply equations (12) to study the generalization of the Voronets
systems, which we define now.
The constrained Lagrangian mechanical systems
(65)
$\left(\textsc{Q},\quad\tilde{L}\left(t,\textbf{x},\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right),\quad\\{\dot{x}_{\alpha}-\Phi_{\alpha}\left(t,\textbf{x},\textbf{y},\dot{\textbf{y}}\right)=0,\quad\alpha=1,\ldots,s_{1}\\}\right),$
is called the generalized Voronets mechanical systems.
An example of generalized Voronets systems is Appell-Hamel systems analyzed in
the previous subsection.
###### Corollary 22.
Every Nonholonomic constrained Lagrangian mechanical systems locally is a
generalized Voronets mechanical systems.
###### Proof.
Indeed, the independent constraints can be locally represented in the form
(26). Thus by introducing the coordinates
$x_{j}=x_{j},\quad\mbox{for}\quad j=1,\ldots,M,\quad
x_{M+k}=y_{k},\quad\mbox{for}\quad k=1,\ldots,N-M,$
then we have that any constrained Lagrangian mechanical systems is locally a
generalized Voronets mechanical systems. ∎
###### Proof of Theorem 4.
For simplicity we shall study only scleronomic generalized Voronets systems.
To determine equations (12) we suppose that
(66)
$L_{\alpha}=\dot{x}_{\alpha}-\Phi_{\alpha}\left(\textbf{x},\textbf{y},\dot{\textbf{y}}\right)=0,\quad\alpha=1,\ldots,s_{1}.$
It is evident from the form of the constraint equations that the virtual
variations $\delta{\textbf{y}},$ are independent by definition. The remaining
variations $\delta{\textbf{x}},$ can be expressed in terms of them by the
relations (Chetaev’s conditions)
(67)
$\delta{x}_{\alpha}-\sum_{j=1}^{s_{2}}\frac{\partial{L_{\alpha}}}{\partial{\dot{y_{j}}}}\delta{y_{j}}=0,\quad\alpha=1,\ldots,s_{1}.$
We shall apply Theorem 2. To construct the matrix $W_{1}.$ We first determine
$L_{{s_{1}}+1},\ldots,L_{s_{1}+s_{2}}=L_{N}$ as follow:
$L_{s_{1}+j}=\dot{y}_{j},\quad j=1,\ldots,s_{2}.$
Hence, the Lagrangian (4) becomes
(68)
$L=L_{0}-\sum_{j=1}^{s_{1}}\lambda_{j}\left(\dot{x}_{\alpha}-\Phi_{\alpha}(x,y,\dot{y})\right)-\sum_{j=s_{1}+1}^{N}\lambda^{0}_{j}\dot{y}_{j}\simeq
L_{0}-\sum_{j=1}^{s_{1}}\lambda_{j}\left(\dot{x}_{\alpha}-\Phi_{\alpha}(x,y,\dot{y})\right).$
The matrices $W_{1}$ and $W^{-1}_{1}$ are
(69) $\left(\begin{array}[]{ccccccc}1&\ldots&0&0&a_{11}&\ldots&a_{{s_{2}}1}\\\
0&\ldots&0&0&a_{12}&\ldots&a_{{s_{2}}2}\\\
\vdots&\ldots&\vdots&\vdots&\vdots&\ldots&\vdots\\\
0&\ldots&\vdots&1&a_{1{s_{1}}}&\ldots&a_{{s_{2}}{s_{1}}}\\\
0&\ldots&0&0&1&\ldots&0\\\ \vdots&\ldots&\vdots&\vdots&\vdots&\ldots&\vdots\\\
0&\ldots&0&0&0&\ldots&1\end{array}\right),\quad\left(\begin{array}[]{ccccccc}1&\ldots&0&0&-a_{11}&\ldots&-a_{{s_{2}}1}\\\
0&\ldots&0&0&-a_{12}&\ldots&-a_{{s_{2}}2}\\\
\vdots&\ldots&\vdots&\vdots&\vdots&\ldots&\vdots\\\
0&\ldots&\vdots&1&a_{1{s_{1}}}&\ldots&-a_{{s_{2}}{s_{1}}}\\\
0&\ldots&0&0&1&\ldots&0\\\ \vdots&\ldots&\vdots&\vdots&\vdots&\ldots&\vdots\\\
0&\ldots&0&0&0&\ldots&1\end{array}\right),$
respectively, where
$a_{\alpha\,j}=\dfrac{\partial{L_{\alpha}}}{\partial\dot{y}_{j}},$ and the
matrices $\Omega_{1}$ and $A$ are
(70)
$A=\Omega_{1}:=\left(\begin{array}[]{ccccccc}E_{1}(L_{1})&\ldots&E_{s_{1}}(L_{1})&E_{s_{1}+1}(L_{1})&\ldots&E_{N}(L_{1})\\\
\vdots&\ldots&\vdots&\ldots&\ldots&\vdots\\\
E_{1}(L_{s_{1}})&\ldots&E_{s_{1}}(L_{s_{1}})&E_{s_{1}+1}(L_{s_{1}})&\ldots&E_{N}(L_{s_{1}})\\\
0&\ldots&0&\ldots&0&0\\\ \vdots&\ldots&\vdots&\ldots&\ldots&\vdots\\\
0&\ldots&0&\ldots&0&0\end{array}\right),$
respectively. Consequently the differential equations (12) take the form (18).
The transpositional relations (13) in view of (67) take the form (21). As we
can observe from (21) the independent virtual variations $\delta{\textbf{y}}$
for the systems with the constraints (66) produce the zero transpositional
relations. The fact that the transpositional relations are zero follows
automatically and it is not necessary to assume it a priori, and it is valid
in general for the constraints which are nonlinear in the velocity variables.
We observe that the relations (34) in this case take the form
$\delta\dfrac{dx_{\alpha}}{dt}-\dfrac{d}{dt}\delta\,x_{\alpha}+\displaystyle\sum_{m=1}^{s_{2}}\dfrac{\partial
L_{\alpha}}{\partial\dot{y}_{m}}\left(\delta\dfrac{dy_{m}}{dt}-\dfrac{d}{dt}\delta\,y_{m}\right)=\displaystyle\sum_{k=1}^{s_{1}}E_{k}(L_{\alpha})\delta
x_{k}+\displaystyle\sum_{k=1}^{s_{2}}E_{k}(L_{\alpha})\delta y_{k}.$
for $\alpha=1,\ldots,s_{1}.$ Clearly from (21) these relations hold
identically.
From differential equations (18), eliminating the Lagrangian multipliers we
obtain equations (19). After some computations we obtain
(71)
$\begin{array}[]{rl}\dfrac{d}{dt}\left(\dfrac{\partial\,L_{0}}{\partial\dot{y}_{k}}-\displaystyle\sum_{\alpha=1}^{s_{1}}\dfrac{\partial\,L_{\alpha}}{\partial\dot{y}_{k}}\dfrac{\partial\,L_{0}}{\partial\dot{x}_{\alpha}}\right)-&\left(\dfrac{\partial\,L_{0}}{\partial{y}_{k}}-\displaystyle\sum_{\alpha=1}^{s_{1}}\dfrac{\partial\,L_{\alpha}}{\partial{y}_{k}}\dfrac{\partial\,L_{0}}{\partial{\dot{x}}_{\alpha}}\right)+\vspace{0.2cm}\\\
&\displaystyle\sum_{\alpha=1}^{s_{1}}\left(\dfrac{\partial\,L_{0}}{\partial{x}_{\alpha}}-\displaystyle\sum_{\beta=1}^{s_{1}}\dfrac{\partial\,L_{\beta}}{\partial{x}_{\alpha}}\dfrac{\partial\,L_{0}}{\partial\dot{x}_{\beta}}\right)\dfrac{\partial\,L_{\alpha}}{\partial\dot{y}_{k}}=0,\end{array}$
for $k=1,\ldots,s_{2}.$
By introducing the function
$\Theta=\left.L_{0}\right|_{L_{1}=\ldots=L_{s_{1}}=0},$ equations (71) can be
written as
(72)
$\dfrac{d}{dt}\left(\dfrac{\partial\,\Theta}{\partial\dot{y}_{k}}\right)-\left(\dfrac{\partial\,\Theta}{\partial{y}_{k}}\right)+\displaystyle\sum_{\alpha=1}^{s_{1}}\left(\dfrac{\partial\,\Theta}{\partial{x}_{\alpha}}\right)\dfrac{\partial\,L_{\alpha}}{\partial\dot{y}_{k}}=0,$
for $k=1,\ldots,s_{2}.$ Here we consider that
$\dfrac{d}{dt}\left(\dfrac{\partial
L_{\beta}}{\partial\dot{x}_{\alpha}}\right)=0,$ for
$\alpha,\,\beta=1,\ldots,s_{1}.$
We shall study the case when equations (72) hold identically, i.e. $\Theta=0.$
We choose
(73)
$L_{0}=\tilde{L}\left(\textbf{x},\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right)-\tilde{L}\left(\textbf{x},\textbf{y},\Phi,\dot{\textbf{y}}\right)=\tilde{L}-L^{*},$
being $\tilde{L}$ the Lagrangian of (65). Now we establish the relations
between equations (18) and the classical Voronets differential equations with
the Lagrangian function
$L^{*}=\left.\tilde{L}\right|_{L_{1}=\ldots=L_{s_{1}}=0}.$ The functions
$\tilde{L}$ and $L^{*}$ are determined in such a way that equations (19) take
place in view of the equalities
$E_{k}\tilde{L}=\displaystyle\sum_{\alpha=1}^{s_{1}}E_{\alpha}\tilde{L}\dfrac{\partial\,L_{\alpha}}{\partial\dot{y}_{k}},$
and
$E_{k}L^{*}=-\displaystyle\sum_{\alpha=1}^{s_{1}}\left(-E_{k}(L_{\alpha}\,)+\displaystyle\sum_{\nu=1}^{s_{1}}E_{\nu}\,(L_{\alpha})\dfrac{\partial\,L_{\nu}}{\partial\dot{y}_{k}}\right)\dfrac{\partial\,\tilde{L}}{\partial\dot{x}_{\alpha}}-\displaystyle\sum_{\nu=1}^{s_{1}}E_{\nu}(L^{*})\dfrac{\partial
L_{\nu}}{\partial\dot{y}_{k}},$
for $k=1,\ldots,s_{2},$ which in view of equalities
$\dfrac{d}{dt}\left(\dfrac{\partial L^{*}}{\partial\dot{x}_{\nu}}\right)=0$
for $\nu=1,\ldots,s_{1},$ take the form (20). ∎
###### Proof of Proposition 5.
Equations (20) describe the motion of the constrained generalized Voronets
systems with Lagrangian $L^{*}$ and constraints (66). The classical Voronets
equations for scleronomic systems are easy to obtain from (20) with
$\Phi_{\alpha}=\displaystyle\sum_{k=1}^{s_{2}}a_{\alpha\,k}(\textbf{x},\textbf{y})\dot{y}_{k}.$
∎
Finally by considering Corollary 22 we get that differential equations (20)
describe locally the motions of any constrained Lagragian systems.
### 8.1. Generalized Chaplygin systems
The constrained Lagrangian mechanical systems with Lagrangian
$\tilde{L}=\tilde{L}\left(\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right),$
and constraints (24) is called the Chaplygin mechanical systems.
The constrained Lagrangian systems
$\left(\textsc{Q},\quad\tilde{L}\left(\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right),\qquad\\{\dot{x}_{\alpha}-\Phi_{\alpha}\left(\textbf{y},\,\dot{\textbf{y}}\right)=0,\quad\alpha=1,\ldots,s_{1}\\}\right)$
is called the generalized Chaplygin systems. Note that now the Lagrangian do
not depend on x and the constraints do not depend on x and $\dot{\textbf{x}}.$
So, the generalized Chaplygin systems are a particular case of the generalized
Voronets system.
###### Proof of Proposition 6.
To determine the differential equations which describe the behavior of the
generalized Chaplygin systems we apply Theorem 2, with
$L_{0}={L}_{0}\left(\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right),\quad
L_{\alpha}=\dot{x}_{\alpha}-\Phi_{\alpha}\left(\textbf{y}\dot{\textbf{y}}\right),\quad
L_{\beta}=\dot{y}_{\beta},$
for $\alpha=1,\ldots,s_{1}$ and $\beta=s_{1}+1,\ldots,s_{2}$ and consequently
the matrix $W_{1}$ is given by the formula (69) and
(74)
$\begin{array}[]{rl}A=\Omega_{1}:=&\left(\begin{array}[]{ccccccc}E_{1}(L_{1})&\ldots&E_{s_{1}}(L_{1})&E_{s_{1}+1}(L_{1})&\ldots&E_{N}(L_{1})\\\
\vdots&\ldots&\vdots&\ldots&\ldots&\vdots\\\
E_{1}(L_{s_{1}})&\ldots&E_{s_{1}}(L_{s_{1}})&E_{s_{1}+1}(L_{s_{1}})&\ldots&E_{N}(L_{s_{1}})\\\
0&\ldots&0&\ldots&0&0\\\ \vdots&\ldots&\vdots&\ldots&\ldots&\vdots\\\
0&\ldots&0&\ldots&0&0\end{array}\right)\vspace{0.30cm}\\\
=&\left(\begin{array}[]{ccccccc}0&\ldots&0&E_{s_{1}+1}(L_{1})&\ldots&E_{N}(L_{1})\\\
\vdots&\ldots&\vdots&\ldots&\ldots&\vdots\\\
0&\ldots&0&E_{s_{1}+1}(L_{s_{1}})&\ldots&E_{N}(L_{s_{1}})\\\
0&\ldots&0&\ldots&0&0\\\ \vdots&\ldots&\vdots&\ldots&\ldots&\vdots\\\
0&\ldots&0&\ldots&0&0\end{array}\right),\end{array}$
Therefore the differential equations (12) take the form
(75) $\begin{array}[]{rl}E_{j}L_{0}=&\dfrac{d}{dt}\left(\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}\right)=\dot{\lambda}_{j}\quad
j=1,\ldots,s_{1},\\\
E_{k}L_{0}=&\displaystyle\sum_{\alpha=1}^{s_{1}}\left(E_{k}L_{\alpha}\,\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}+\dot{\lambda}_{\alpha}\dfrac{\partial
L_{\alpha}}{\partial\dot{y}_{k}}\right)\quad k=1,\ldots,s_{2}.\end{array}$
The transpositional relations are
(76)
$\begin{array}[]{rl}&\delta\dfrac{dx_{\alpha}}{dt}-\dfrac{d}{dt}\delta\,x_{\alpha}=\displaystyle\sum_{k=1}^{s_{2}}E_{k}(L_{\alpha})\delta
y_{k},\quad\alpha=1,\ldots,s_{1},\\\
&\delta\dfrac{dy_{m}}{dt}-\dfrac{d}{dt}\delta\,y_{m}=0,\quad
m=1,\ldots,s_{2}.\end{array}$
By excluding the Lagrangian multipliers from (75) we obtain the equations
$E_{k}L_{0}=\displaystyle\sum_{\alpha=1}^{s_{1}}\left(E_{k}(L_{\alpha})\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}+\dfrac{d}{dt}\left(\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}\right)\dfrac{\partial
L_{\alpha}}{\partial\dot{y}_{k}}\right),$
for $k=1,\ldots,s_{2}.$
In this case equations (73) take the form
(77)
$\dfrac{d}{dt}\left(\dfrac{\partial\,\Theta}{\partial\dot{y}_{k}}\right)-\left(\dfrac{\partial\,\Theta}{\partial{y}_{k}}\right)=0,$
Analogously to the Voronets case we study the subcase when $\Theta=0.$ We
choose
$L_{0}=\tilde{L}\left(\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right)-\tilde{L}\left(\textbf{y},\Phi,\dot{\textbf{y}}\right):=\tilde{L}-L^{*}.$
We assume that the functions $\tilde{L}$ and $L^{*}$ are such that
(78)
$E_{k}L^{*}=-\displaystyle\sum_{\alpha=1}^{s_{1}}E_{k}(L_{\alpha})\dfrac{\partial\tilde{L}}{\partial\dot{x}_{\alpha}}\Psi_{\alpha},$
where
$\Psi_{\alpha}=\left.\dfrac{\partial\tilde{L}}{\partial\dot{x}_{\alpha}}\right|_{L_{1}=\ldots=L_{s_{1}}=0}$
and
$E_{k}(\tilde{L})=\displaystyle\sum_{\alpha=1}^{s_{1}}\dfrac{d}{dt}\left(\dfrac{\partial\tilde{L}}{\partial\dot{x}_{\alpha}}\right)\dfrac{\partial
L_{\alpha}}{\partial\dot{y}_{k}},$
for $k=1,\ldots,s_{2}.$
By inserting
$\dot{x}_{j}=\displaystyle\sum_{k=1}^{s_{2}}a_{j\,k}(\textbf{y})\dot{y}_{k},\quad
j=1,\ldots,s_{1},$ into equations (78) we obtain system (25). Consequently
system (78) is an extension of the classical Chaplygin equations when the
constraints are nonlinear. ∎
For the generalized Chaplygin systems the Lagrangian $L$ takes the form
(79)
$L=\tilde{L}(\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}})-\tilde{L}(\textbf{y},\Phi,\dot{\textbf{y}})-\displaystyle\sum_{j=1}^{s_{1}}\left(\dfrac{\partial
L^{*}}{\partial\dot{x}_{j}}+C_{j}\right)\left(\dot{x}_{j}-\Phi_{j}(\textbf{y},\dot{\textbf{y}})\right)-\displaystyle\sum_{j=}^{s_{2}}\lambda^{0}_{j}\dot{y}_{j},$
for $j=1,\ldots,s_{1}$ where the constants $C_{j}$ for $j=1,\ldots,s_{1}$ are
arbitrary. Indeed, from (75) follows that
$\lambda_{j}=\dfrac{\partial L_{0}}{\partial\dot{x}_{j}}+C_{j}=\dfrac{\partial
L^{*}}{\partial\dot{x}_{j}}+C_{j}.$
By inserting in (4) $L_{0}=\tilde{L}-L^{*}$ and $\lambda_{j}$ for
$j=1,\ldots,s_{1}$ we obtain function $L$ of (79).
We note that Vorones and Chaplygin equations with nonlinear constraints in the
velocity was also obtained by Rumiansev and Sumbatov $($see [44, 47]$)$.
Example 4. We shall illustrate the above results in the following example.
In the Appel’s and Hamel’s investigations the following mechanical system was
analyzed. A weight of mass $m$ hangs on a thread which passes around the
pulleys and is wound round the drum of radius $a$. The drum is fixed to a
wheel of radius $b$ which rolls without sliding on a horizontal plane,
touching it at the point $B$ with the coordinates $(x_{B},\,y_{B})$. The legs
of the frame that support the pulleys and keep the plane of the wheel vertical
slide on the horizontal plane without friction. Let $\theta$ be the angle
between the plane of the wheel and the $Ox$ axis; $\varphi$ the angle of the
rotation of the wheel in its own plane; and $(x,y,z)$ the coordinates of the
mass $m.$ Clearly,
$\dot{z}=b\dot{\varphi},\quad b>0.$
The coordinates of the point $B$ and the coordinates of the mass are related
as follows (see page 223 of [35] for a picture)
$x=x_{B}+\rho\cos\theta,\quad y=y_{B}+\rho\sin\theta.$
The condition of rolling without sliding leads to the equations of
nonholonomic constraints:
$\dot{x}_{B}=a\cos\theta\dot{\varphi},\quad\dot{y}_{B}=a\sin\theta\dot{\varphi}\quad
b>0.$
We observe that the constraints $\dot{z}=b\dot{\varphi}$ admits the
representation
$\dot{z}=\dfrac{b}{a}\sqrt{\dot{x}^{2}+\dot{y}^{2}-\rho^{2}\dot{\theta}^{2}}.$
Denoting by $m_{1},\,A$ and $C$ the mass and the moments of inertia of the
wheel and neglecting the mass of the frame, we obtain the following expression
for the Lagrangian function
$\tilde{L}=\dfrac{m+m_{1}}{2}\left(\dot{x}^{2}+\dot{y}^{2}\right)+\dfrac{m}{2}\dot{z}^{2}+m_{1}\rho\dot{\theta}\left(\sin\theta\dot{x}-\cos\theta\dot{y}\right)+\dfrac{A+m_{1}\rho^{2}}{2}\dot{\theta}^{2}+\dfrac{C}{2}\dot{\varphi}^{2}-mgz.$
The equations of the constraints are
$\dot{x}-a\cos\theta\dot{\varphi}+\rho\sin\theta\dot{\theta}=0,\quad\dot{y}-a\sin\theta\dot{\varphi}-\rho\cos\theta\dot{\theta}=0,\quad\dot{z}-b\dot{\varphi}=0,$
Now we shall study the motion of this constrained Lagrangian in the
coordinates
$x_{1}=x,\,x_{2}=y,\,x_{3}=\dot{\varphi},y_{1}=\theta,\,y_{2}=z.$
i.e., we shall study the nonholonomic system with Lagrangian
$\begin{array}[]{rl}\tilde{L}=&\tilde{L}\left(y_{1},\,y_{2},\,\dot{x}_{1},\,\dot{x}_{2},\,\dot{x}_{3},\,\dot{y}_{1},\,\dot{y}_{2}\right)\vspace{0.20cm}\\\
=&\dfrac{m+m_{1}}{2}\left(\dot{x}^{2}_{1}+\dot{x}^{2}_{2}\right)+\dfrac{C}{2}\dot{x}^{2}_{3}+\dfrac{J}{2}\dot{y}^{2}_{1}+\dfrac{m}{2}\dot{y}^{2}_{2}+m_{1}\rho\dot{y_{1}}\left(\sin\,y_{1}\dot{x}_{1}-\cos\,y_{1}\dot{x}_{2}\right)-\dfrac{mg}{b}y_{2},\end{array}$
and with the constraints
$\begin{array}[]{rl}l_{1}=&\dot{x}_{1}-\dfrac{a}{b}\,\dot{y}_{2}\cos
y_{1}-\rho\dot{y}_{1}\sin\,y_{1}=0,\vspace{0.2cm}\\\
l_{2}=&\dot{x}_{2}-\dfrac{a}{b}\dot{y}_{2}\sin
y_{1}+\rho\dot{y}_{1}\cos\,y_{1}=0,\vspace{0.2cm}\\\
l_{3}=&\dot{x}_{3}-\dfrac{1}{b}\dot{y}_{2}=0.\end{array}$
Thus we have a classical Chaplygin system. To determine differential equations
(78) and the transpositional relations (76) we define the functions:
$\begin{array}[]{rl}L^{*}=-&\tilde{L}|_{l_{1}=l_{2}=l_{3}=0}=\dfrac{m(a^{2}+b^{2})m+a^{2}m_{1}+C}{2b^{2}}\dot{y}^{2}_{2}+\dfrac{m\rho^{2}+J}{2}\dot{y}^{2}_{1}-\dfrac{mg}{b}y_{2},\vspace{0.20cm}\\\
L_{1}=&l_{1},\quad L_{2}=l_{2},\quad L_{3}=l_{3},\quad L_{4}=\dot{y}_{1},\quad
L_{5}=\dot{y}_{2}.\end{array}$
After some computations we obtain that the matrix $A$ (see formulae (74)) in
this case becomes
$A=\left(\begin{array}[]{cccccc}0&0&0&-\dfrac{a}{b}\dot{y}_{2}\sin\,y_{1}&\dfrac{a}{b}\dot{y}_{1}\sin\,y_{1}\vspace{0.20cm}\\\
0&0&0&\dfrac{a}{b}\dot{y}_{2}\cos\,y_{1}&-\dfrac{a}{b}\dot{y}_{1}\cos\,y_{1}\vspace{0.20cm}\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\end{array}\right),$
thus differential equations (78) take the form
$\begin{array}[]{rl}&\left(m\rho^{2}+J\right)\ddot{y}_{1}+\dfrac{a\rho
m}{b}\dot{y}_{1}\dot{y}_{2}=0,\vspace{0.20cm}\\\
&\left((m+m_{1})a^{2}+mb^{2}\right)\ddot{y}_{2}-{mab\rho}\dot{y}^{2}_{1}=-mgb.\end{array}$
Assuming that $(m+2m_{1})\rho^{2}+J\neq 0$ and by considering the existence of
the first integrals
$\begin{array}[]{rl}C_{2}=&\dot{y}_{1}\exp{\left(-\dfrac{a\varrho
my_{2}}{b\left(m\rho^{2}+J\right)}\right)},\vspace{0.2cm}\\\
h=&\dfrac{\left((m+m_{1})a^{2}+mb^{2}\right)}{2}\dot{y}^{2}_{2}+\dfrac{b^{2}\left(m\rho^{2}+J\right)}{2}\dot{y}^{2}_{1}+mgby_{2},\\\
\end{array}$
after the integration of these first integrals we obtain
$\begin{array}[]{rl}&\displaystyle\int\dfrac{\sqrt{(m+m_{1})a^{2}+mb^{2}}dy_{2}}{\sqrt{2h-2mgby_{2}-{b^{2}\left(m\rho^{2}+J\right)}C_{3}\exp{\left(\dfrac{a\rho\,my_{2}}{b{m\rho^{2}+J}}\right)}}}=t+C_{1},\vspace{0.30cm}\\\
&y_{1}(t)=C_{3}+C_{2}\displaystyle\int\exp{\left(2\dfrac{a\rho\,my_{2}(t)}{b{m\rho^{2}+J}}\right)}dt.\end{array}$
Consequently, if $\rho=0$ then
$y_{1}=C_{3}+C_{2}t,\qquad\displaystyle\int\dfrac{\sqrt{(m+m_{1})a^{2}+mb^{2}}dy_{2}}{\sqrt{2h-2mgby_{2}-{J}C_{3}}}=t+C_{1}.$
Hamel in [15] neglect the mass of the wheel ($m_{1}=J=C=0$). Under these
conditions the previous equations become
$\begin{array}[]{rl}&\rho^{2}\ddot{y}_{1}+\dfrac{a\rho}{b}\dot{y}_{1}\dot{y}_{2}=0,\vspace{0.20cm}\\\
&(a^{2}+b^{2})\ddot{y}_{2}-ab\rho\dot{y}^{2}_{1}=-gb\end{array}$
Appell and Hamel obtained the example of nonholonomic system with nonlinear
constraints by means of the passage to the limit $\rho\to 0.$ However, as a
result of this limiting process, the order of the system of differential
equations is reduced, i.e., they become degenerate. In [35] the authors study
the motion of the nondegenerate system for $\rho>0$ and $\rho<0.$ From these
studies it follows that the motion of the nondegenerate system ($\rho\neq 0$)
and degenerate system ($\rho\to 0$) differ essentially. Thus the Appell-Hamel
example with nonlinear constraints is incorrect.
The transpositional relations (76) become
$\begin{array}[]{rl}\delta\dfrac{dx_{1}}{dt}-\dfrac{d\delta\,x_{1}}{dt}=&\dfrac{a}{b}\sin\,y_{1}\left(\dfrac{dy_{1}}{dt}\delta{y_{2}}-\dfrac{dy_{2}}{dt}\delta{y_{1}}\right),\vspace{0.30cm}\\\
\delta\dfrac{dx_{2}}{dt}-\dfrac{d\delta\,x_{2}}{dt}=&\dfrac{a}{b}\cos\,y_{1}\left((\dfrac{dy_{1}}{dt}\delta{y_{2}}-\dfrac{dy_{2}}{dt}\delta{y_{1}}\right),\vspace{0.30cm}\\\
\delta\dfrac{dx_{3}}{dt}-\dfrac{d\delta\,x_{3}}{dt}=&0,\quad\delta\dfrac{dy_{1}}{dt}-\dfrac{d\delta\,y_{1}}{dt}=0,\quad\delta\dfrac{dy_{2}}{dt}-\dfrac{d\delta\,y_{2}}{dt}=0.\end{array}$
Clearly these relations are independent of $\varrho,\,A,\,C$ and $m_{1}.$
## 9\. Consequences of Theorems 2 and 3 and the proof of Corollary 7.
We observe the following important aspects from Theorems 2 and 3.
(I) Conjecture 8 is supported by the following facts. (a) As a general rule
the constraints studied in classical mechanics are linear in the velocities.
However Appell and Hamel in 1911, considered an artificial example with a
constraint nonlinear in the velocity . As it follows from [35] (see example 4)
this constraint does not exist in the Newtonian mechanics.
(b) The idea developed for some authors (see for instance [4]) to construct a
theory in Newtonian mechanics, by allowing that the field of force depends on
the acceleration, i.e. function of $\ddot{\textbf{x}}$ as well as of the
position $\textbf{x},$ velocity $\dot{\textbf{x}},$ and the time $t$ is
inconsistent with one of the fundamental postulates of the Newtonian
mechanics: when two forces act simultaneously on a particle the effect is the
same as that of a single force equal to the resultant of both forces (for more
details see [38] pages 11–12). Consequently the forces depending on the
acceleration are not admissible in Newtonian dynamics. This does not preclude
their appearance in electrodynamics, where this postulate does not hold.
(c) Let $T$ be the kinetic energy of the constrained Lagrangian systems. We
consider the generalization of the Newton law: the acceleration $($see [46,
37]$)$
$\dfrac{d}{dt}\dfrac{\partial T}{\partial\dot{\textbf{x}}}-\dfrac{\partial
T}{\partial{\textbf{x}}}$
is equal to the force $\textbf{F}.$ Then in the differential equations (12)
with $L_{0}=T$ we obtain that the field of force F generated by the
constraints is
$\textbf{F}=\left(W^{-1}_{1}\Omega_{1}\right)^{T}\dfrac{\partial{T}}{\partial{\dot{\textbf{x}}}}+W^{T}_{1}\dfrac{d}{dt}\lambda:=\textbf{F}_{1}+\textbf{F}_{2}.$
The field of force
$\textbf{F}_{2}=W^{T}_{1}\dfrac{d}{dt}\lambda=\left(F_{21},\ldots,F_{2N}\right)$
is called the reaction force of the constraints. What is the meaning of the
force
(80)
$\textbf{F}_{1}=\left(W^{-1}_{1}\Omega_{1}\right)^{T}\dfrac{\partial{T}}{\partial{\dot{\textbf{x}}}}\,?$
If the constraints are nonlinear in the velocity, then $\textbf{F}_{1}$
depends on $\ddot{\textbf{x}}.$ Consequently in Newtonian mechanics does not
exist a such field of force. Therefore, the existence of nonlinear constraints
in the velocity and the meaning of force $\textbf{F}_{1}$ must be sought
outside of the Newtonian model.
For example, for the Appel-Hamel constrained Lagrangian systems studied in the
previous subsection we have that
$\textbf{F}_{1}=\left(-\dfrac{a^{2}\dot{x}}{\dot{x}^{2}+\dot{y}^{2}}(\dot{x}\ddot{y}-\dot{y}\ddot{x}),\,\dfrac{a^{2}\dot{y}}{\dot{x}^{2}+\dot{y}^{2}}(\dot{x}\ddot{y}-\dot{y}\ddot{x}),\,0\right).$
For the generalized Voronets systems and locally for any nonholonomic
constrained Lagrangian systems from the equations (18) we obtain that the
field of force $\textbf{F}_{1}$ has the following components
(81)
$\begin{array}[]{rl}F_{k\,1}=&\displaystyle\sum_{\alpha=1}^{s_{1}}E_{k}L_{\alpha}\,\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}\\\
=&\displaystyle\sum_{j=1}^{N}\sum_{\alpha=1}^{s_{1}}\left(\dfrac{\partial^{2}L_{\alpha}}{\partial\dot{x}_{k}\dot{x}_{j}}\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}\ddot{x}_{j}+\dfrac{\partial^{2}L_{\alpha}}{\partial\dot{x}_{k}\partial{x}_{j}}\dfrac{\partial
L_{0}}{\partial\dot{x}_{\alpha}}\dot{x}_{j}\right)+\displaystyle\sum_{\alpha=1}^{s_{1}}\dfrac{\partial^{2}L_{\alpha}}{\partial\dot{x}_{k}\partial
t}\dfrac{\partial L_{0}}{\partial\dot{x}_{\alpha}},\quad\mbox{for}\quad
k=1\ldots,N,\quad s_{1}=M.\end{array}$
consequently such field of force does not exist in Newtonian mechanics if the
constraints are nonlinear in the velocity.
(II) Equations (12) can be rewritten in the form
(82) $G\ddot{\textbf{x}}+\textbf{f}(t,\textbf{x},\dot{\textbf{x}})=0,$
where $G=G(t,\textbf{x},\dot{\textbf{x}})$ is the matrix
$\left(G_{j,k}\right)$ given by
$G_{jk}=\dfrac{\partial^{2}L_{0}}{\partial\dot{x}_{j}\partial\dot{x}_{k}}-\displaystyle\sum_{n=1}^{N}\dfrac{\partial
A_{nk}}{\partial\ddot{x}_{j}}\dfrac{\partial L_{0}}{\partial\dot{x}_{n}},\quad
j,k=1,\ldots,N,$
and $\textbf{f}(t,\textbf{x},\dot{\textbf{x}})$ is a convenient vector
function. If $\det G\neq 0$ then equation (82) can be solved with respect to
$\ddot{\textbf{x}}.$ This implies, in particular that the motion of the
mechanical system at time $\overline{t}\in[t_{0},\,t_{1}]$ is uniquely
determined, i.e. the principle of determinacy (see for instance [2]) holds for
the mechanical systems with equation of motion given in (12).
In particular for the Appel-Hamel constrained Lagrangian systems we have (see
formula (48)) that
$\begin{array}[]{rl}\textbf{x}=&\left(x,\,y,\,z\right)^{T},\quad\textbf{f}=\left(\dfrac{a\dot{x}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\dot{\lambda},\,\dfrac{a\dot{y}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\dot{\lambda},\,g-\dot{\lambda}\right)^{T}\vspace{0.20cm}\\\
G=&\left(\begin{array}[]{cccc}1+\dfrac{a^{2}\dot{y}^{2}}{\dot{x}^{2}+\dot{y}^{2}}&-\dfrac{a^{2}\dot{x}\dot{y}}{\dot{x}^{2}+\dot{y}^{2}}&0\\\
-\dfrac{a^{2}\dot{x}\dot{y}}{\dot{x}^{2}+\dot{y}^{2}}&1+\dfrac{a^{2}\dot{x}^{2}}{\dot{x}^{2}+\dot{y}^{2}}&0\\\
0&0&1\end{array}\right),\quad|G|=1+a^{2}.\end{array}$
So in the Appel–Hamel system the principle of determinacy holds.
(III)
###### Proof of Corollary 7.
From Theorems 2 and 3 (see formulas (13) and (17)) and from all examples which
we gave in the previous sections we see that are examples with zero
transpositional relations and examples where all they are not zero. By
contrasting the MVM with the Lagrangian mechanics we obtain that for the
unconstrained Lagrangian systems the transpositional relations are always
zero. Thus we have the proof of the corollary. ∎
## Acknowledgements
The first author is partially supported by a MINECO/FEDER grant number
MTM2009-03437, an AGAUR grant number 2009SGR-410, ICREA Academia and
FPZ–PEOPLE–2012–IRSES–316338 and 318999. The second author was partly
supported by the Spanish Ministry of Education through projects
TSI2007-65406-C03-01 “E-AEGIS” and Consolider CSD2007-00004 “ARES”.
## References
* [1] V.M. Alekciev, V.M. Tixomirov and S.V. Fomin Optimal control, Ed. Nauka, 1979.
* [2] V.I. Arnold, V.V Kozlov, and A.I. Neishtadt , Mathematical aspects of classical mechanics, in Dynamical systems III, Springer, Berlin 1998.
* [3] P. Appell, Exemple de mouvement d’ un point assujettià une liaison exprimé par une relation non linéaire entre les composantes de la vitesse, Rend.Circ. Mat. Palermo 32 (1911), 48–50
* [4] G.D. Birkhoff, Dynamical systems, New York, 1927\.
* [5] A.M. Bloch, Nonholonomic Mechanics and Control, Springer, Berlin 2003.
* [6] S.A. Chaplygin, On the theory of motion of nonholonomic systems. Theorems on the reducing multiplier, Mat. Sb. 28 (1911), 303–314 (in Russian).
* [7] N.G. Chetaev, Izv. Fiz.Mat. Obshch. Kazan 6 (1932), 68–71 (in Russian).
* [8] N.G. Chetaev, On Gauss principle Izv. Fiz.Mat. Obshch. Kazan 6 (1941), 323–326 (in Russian).
* [9] M. Favretti, Equivalence of dynamics for nonholonomic systems with transverse constraints, J. Dynam. Differential Equations 10 (1998), 511–536.
* [10] N.M. Ferrers, Extension of Lagrange’s equations. Quart. J. of pure and applied mathematics 12 (1872), 1–5.
* [11] F.R. Gantmacher, , Lektsi po analitisheskoi mechanic, Ed. Nauka, Moscow, 1966 (in Russian).
* [12] X. Gracia, J. Marin–Solano, M. Muñoz–Lecanda, Some geometric aspects of variational calculus in constrained systems, Reports on Mathematical Physics 51 (2003), 127–148 .
* [13] I.M. Gelfand and S.V. Fomin, Calculus of variations, Ed. Prentice-Hall, INC., New Jersey 1963\.
* [14] P.A. Griffiths, Exterior differential systems and the calculus of variations, Birkhäuser Boston-Basel-Stuttgart 1983\.
* [15] G. Hamel, Teoretische Mechanik, Berlin, 1949\.
* [16] H. Hertz, Die Prinzipien der Mechanik in neuem Zusammenhaange dargestellt, Ges. Werke, Leipzig, Barth. 1910.
* [17] O. Hölder, Ueber die prinzipien von Hamilton und Maupertius, Nachtichten Kön. Ges. Wissenschaften zu Gottingen Math.–Phys. Kl. (1896), 122–157.
* [18] P.V. Kharlamov, A critique of some mathematical models of mechanical systems with differential constraints, J. Appl.Math. Mech. 54 (1992), 683–691 (in Russian)
* [19] V.I. Kirgetov, Transpositional relations in mechanics, J. Appl.Math. Mech. 22 (1958), 682–693 (in Russian).
* [20] D.J. Korteweg, Über eine ziemlich verbreitete unrichtige Behand–lungsweise eines. Problemes der rollenden Bewegung, Nieuw Archiv voor Wiskunde 4 (1899), 130–155.
* [21] V.V. Kozlov, Theory of integration of equations of nonholonomic mechanics, Uspekhi mekh. 8 (1985), 85–101.
* [22] V.V. Kozlov, Realization of nonintegrable constraints in classical mechanics, Dokl. Akad. Nauk SSSR 272 (1983), 550–554 (in Russian).
* [23] V.V. Kozlov, Gauss principle and realization of the constraints, Regular and Chaotic Dynamics 13, (2008), 431–434.
* [24] V.V. Kozlov, Dynamics of systems with non-integrable restrictions I, Vestn. Mosk. Univ., Ser.I Mat. Mekh. 3 (1982), 92–100 (in Russian).
* [25] V.V. Kozlov, Dynamics of systems with non-integrable restrictions II, Vestn. Mosk. Univ., Ser.I Mat. Mekh. 4 (1982), 70–76 (in Russian).
* [26] V.V. Kozlov, Dynamics of systems with non-integrable restrictions III, Vestn. Mosk. Univ., Ser.3 Mat. Mekh. 3 (1983), 102–111 (in Russian).
* [27] I. Kupka and W.M. Oliva The nonholonomic mechanics, J.Differential equations 169 (2001), 169–189.
* [28] M. de Leon and D. M. de Diego, On the geometry of generalized Chaplygin systems, Mathemathical Proceeding of the Cambridge Philosophical Society 132 (2002) 1389–1412.
* [29] A.D. Lewis and R.M. Murray, Variational principle for constrained mechanical systems: Theory and experiments,Internat. J. Non–linear Mech. 30 (1995), 793–815.
* [30] J. Llibre, R. Ramírez and N. Sadovskaia, Integrability of the constrained rigid body, preprint, (2012).
* [31] J. Llibre, R. Ramírez and N. Sadovskaia, Inverse problems in ordinary differential equations, preprint, (2012).
* [32] A.I. Lurie, Analytical dynamics, Ed. Fisiko–matematisheskoi literatury, 1961\.
* [33] C.M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Reports on Math. Physics 42 (1998), 211–229.
* [34] J.M. Marushin, A.M. Bloch, J.E. Marsden and D.V. Zenkov, A fiber bundle approach to the transpositional relations in nonholonomic mechanics, J. of Nonlinear Sci. 22 (2012), 431–461.
* [35] Ju.I. Neimark and N.A. Fufaev, Dynamics of Nonholonomic Systems, American Mathematical Society, Rhode Island, 1972\.
* [36] V.S.Novoselov, Example of a nonlinear nonholonomic constraints that is not of the type of N.G. Chetaev, Vestnik Leningrad Univ., 12 (1957) (in Russian).
* [37] W.Muniz Oliva, Geometric mechanics, Springer–Verlag, 2002.
* [38] L.A. Pars, A treatise on analytical dynamics, Heinemannn, London, 1968.
* [39] H. Poincaré, Hertz’s ideas in mechanics, in addition to H. Hertz, Die Prizipien der Mechanik in neum Zusammemhauge dargestellt, 1894.
* [40] L.S. Polak, Variation principle of mechanic, Ed. Fisico-matematicheskoi literature, 1960 (in Russian).
* [41] R. Ramirez and N. Sadovskaia, On the dynamics of nonholonomic systems, Reports on Math. Physics. 60 (2007), 427–451.
* [42] R. Ramirez, Dynamics of nonholonomic systems, Publisher VINITI 3878 (1985) (in Russian).
* [43] V.N. Rubanovskii and V.A. Samsonov, Stability of steady motions, in examples and problems, M.: Nauka 1998 (in Russian).
* [44] V.V. Rumiansev, O principe Hamiltona dlia niegolonomnix system, J. Appl.Math. Mech. 42 (1978), 407–419 (in Russian).
* [45] N. Sadovskaia, Inverse problem in theory of ordinary differential equations, Thesis Ph. D., Univ. Politécnica de Cataluña, 2002 (in Spanish).
* [46] J. L. Synge, On the geometry of dynamics, Phil. Trans. Roy. Soc. London ser. A 226 (1927), 31–106.
* [47] A.S. Sumbatov, Nonholonomic systems, Regular and chaotic dynamics 7 (2002), 221–238.
* [48] G.K.Suslov, On a particular variant of d’Alembert principle, Math. Sb. 22 (1901), 687–691 (in Russian).
* [49] G. Zampieri, Nonholonomic versus vakonomic dynamics, J. Differential Equations 163 (2000), 335–347.
* [50] A.M. Vershik and L.D. Faddeev, Differential geometry and Lagrangian mechanics with constraints, Soviet Physics–Doklady 17, (1972) (in Russian).
* [51] A. Vierkandt, Über gleitende und rollende, Bewegung Monatshefte der Mathh. und Phys. III (1982) 31–54.
* [52] V. Volterra,Sopra una classe di equazione dinamiche, Atti Accad. Sci. Torino 33 (1898), 451–475.
* [53] P. Voronets, On the equations of motion for nonholonomic systems Math. Sb. 22 (1901), 659–686 (in Russian).
|
arxiv-papers
| 2014-02-24T13:59:41 |
2024-09-04T02:49:58.735576
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Llibre, R. Ram\\'irez and N. Sadovskaia",
"submitter": "Rafael Ramirez Dr.",
"url": "https://arxiv.org/abs/1402.5827"
}
|
1402.5835
|
# Polcovar: Software for Computing the Mean and Variance of Subgraph Counts
in Random Graphs
Jérôme Kunegis
###### Abstract
The mean and variance of the number of appearances of a given subgraph $H$ in
an Erdős–Rényi random graph over $n$ nodes are rational polynomials in $n$
[2]. We present a piece of software named Polcovar (from _polynomial_ and
_covariance_) that computes the exact rational coefficients of these
polynomials in function of $H$.
## 1 Introduction
Large networks can be characterised by the number of times specific subgraphs
appear in them. For instance, the number of triangles measures the clustering
in a network and the number of wedges (i.e. 2-stars / 2-paths) characterises
the inequality of the degree distribution. Even the number of vertices and
edges in a graph can be characterised in this way as the number of times the
complete graphs on one and two nodes appear as subgraphs. To assess whether a
graph contains many or few subgraphs for its size, the subgraph count must be
compared to the expected subgraph count in random graphs. To do this, we must
know the distribution of subgraph counts. Under mild conditions which are
valid for all examples given above, the subgraph count becomes normal in the
large graph limit. Consequently, the distribution of subgraph counts is
characterised by its mean and average. Expressions for the mean and variance
of subgraph counts are given in [2], and are always rational polynomials in
the number of nodes $n$ of the network. In this paper, we present Matlab
software for computing the exact coefficients rational of these polynomials,
as their expressions are usually too unwieldy to be computed by hand.
## 2 Subgraph Counts
We consider a random graph $G$ on $n$ nodes distributed according to the
Erdős–Rényi model with parameter $p$, i.e., each edge in $G$ exists with
probability $p$ [1]. Let $H=(W,F)$ be a pattern, i.e. a small graph with
$k=|W|$ vertices and $l=|F|$ edges, and $c_{H}$ the number of times this
pattern appears as a subgraph of $G$. The mean of variance of $c_{H}$ are then
given by the following expressions [2]:
$\displaystyle\mathrm{E}[c_{H}]$
$\displaystyle=\frac{n^{\underline{k}}}{|\mathrm{Aut}(H)|}p^{-l}$
$\displaystyle\mathrm{Var}[c_{H}]$
$\displaystyle=\sum_{J}\left(\frac{n^{\underline{|V(J)|}}}{|\mathrm{Aut^{\prime}}(J)|}p^{-|E(J)|}\right)-\mathrm{E}[c_{H}]^{2}$
where $n^{\underline{i}}$ is the falling factorial111defined as
$n^{\underline{i}}=n(n-1)\cdots(n-i+1)$, the sum is over all graphs $J$
containing two differently colored copies of $H$ (which might overlap),
$\mathrm{Aut}(H)$ is the automorphism group of the graph $H$, and
$\mathrm{Aut^{\prime}}(J)$ is the group of automorphisms of $J$ that preserve
the edges of the underlying distinct copies of $H$.
Although the normal limit for $n\rightarrow\infty$ is only true when the graph
$H$ is _strictly balanced_ , the expressions for the mean and variance are
always correct. Note also that they are true exactly for any $n$, not just in
the large $n$ limit.
Alternatively, the following expression can be used, which gives the same
result. It is this expression that we implement in our code.
$\displaystyle\mathrm{Var}[c_{H}]$
$\displaystyle=-\mathrm{E}[c_{H}]^{2}+\frac{1}{\mathrm{Aut}(H)^{2}}\sum_{i=0}^{k}\frac{n^{\underline{2k-i}}}{i!(k-i)!^{2}}\sum_{P,Q}p^{-m(P,Q)}$
where the inner sum is over all pairs of $k$-permutations, and $m(P,Q)$
denotes the number of edges in the overlay of H permuted by $P$ and $H$
permuted by $Q$ which share $i$ nodes.
## 3 Special Cases
For specific small graphs $H$, we get the following exact results.
### 3.1 Node Count
Taking $H$ as the graph with one node gives the number of nodes. Plugging this
graph into the general form expression gives $\mathrm{E}[c_{H}]=n$ and
$\mathrm{Var}[c_{H}]=0$. In other words, the number of nodes is always exactly
$n$, as expected.
### 3.2 Edge Count
Edges are always independent of each other and therefore the binomial
approximation for the number of edges $m=c_{H}$ is exact.
$\displaystyle\mathrm{E}[m]$ $\displaystyle=\frac{1}{p}{n\choose
2}=\frac{n(n-1)}{2p}$ $\displaystyle\mathrm{Var}[m]$
$\displaystyle=\frac{n(n-1)}{8}\text{ when $p=1/2$}$
These expressions can be derived both by the general form we gave above, and
by the fact that the number of edges is a binomial distribution.
### 3.3 Triangle Count
In a random $n$-graph with parameter $p=1/2$, the number of triangles $t$ has
mean and variance given by
$\displaystyle\mathrm{E}[t]$ $\displaystyle=\frac{1}{8}{n\choose 3}$
$\displaystyle\mathrm{Var}[t]$
$\displaystyle=\frac{1}{128}n^{4}-\frac{11}{384}n^{3}+\frac{1}{32}n^{2}-\frac{1}{96}n$
The expressions follow from the general form given below.
### 3.4 Wedge Count
The number $s$ of wegdes (i.e., pairs of edges sharing one endpoint, also
known as 2-stars or 2-paths) has the following distributions when $p=1/2$:
$\displaystyle\mathrm{E}[s]$ $\displaystyle=\frac{n^{\underline{3}}}{8}$
$\displaystyle\mathrm{Var}[s]$
$\displaystyle=\frac{1}{8}n^{4}-\frac{19}{32}n^{3}+\frac{29}{32}n^{2}-\frac{7}{16}n$
The expressions follow from the general form given below.
### 3.5 Other Patterns
For the number $q$ of squares we get:
$\displaystyle\mathrm{E}[q]$
$\displaystyle=\frac{1}{128}n^{4}-\frac{3}{64}n^{3}+\frac{11}{128}n^{2}-\frac{3}{64}n$
$\displaystyle\mathrm{Var}[q]$
$\displaystyle=\frac{1}{512}n^{6}-\frac{5}{256}n^{5}+\frac{161}{2048}n^{4}-\frac{163}{1024}n^{3}+\frac{327}{2048}n^{2}-\frac{63}{1024}n$
For the number $c_{H}$ of 4-cliques we get:
$\displaystyle\mathrm{E}[c_{H}]$
$\displaystyle=\frac{1}{1536}n^{4}-\frac{1}{256}n^{3}+\frac{11}{1536}n^{2}-\frac{1}{256}n$
$\displaystyle\mathrm{Var}[c_{H}]$
$\displaystyle=\frac{1}{32768}n^{6}-\frac{17}{98304}n^{5}+\frac{19}{49152}n^{4}-\frac{73}{98304}n^{3}+\frac{115}{98304}n^{2}-\frac{11}{16384}n$
## 4 Proof Outline
A complete proof can be found in [2]. We here outline the proof as a starting
point. The total number of possible subgraphs $H$ in a graph with $n$ vertices
is
$\displaystyle\frac{n^{\underline{k}}}{|\mathrm{Aut}(H)|}.$
Define the random variables $x_{i}\in\\{0,1\\}$ to denote the presence or
absence of each possible pattern $i$. Then,
$\displaystyle c_{H}$ $\displaystyle=\sum_{i}x_{i}.$
The expected value of each $x_{i}$ is given by
$\displaystyle\mathrm{E}[x_{i}]=p^{-l}.$
Thus, the expected value of $c_{H}$ can be expressed as
$\displaystyle\mathrm{E}[c_{H}]=\mathrm{E}[\sum_{i}x_{i}]=\sum_{i}\mathrm{E}[x_{i}]=\frac{n^{\underline{k}}}{|\mathrm{Aut}(H)|}p^{-l}.$
To compute the variance we exploit the fact that the variance equals the
expected value of the square minus the square of the expected value:
$\displaystyle\mathrm{Var}[c_{H}]$
$\displaystyle=\mathrm{E}[c_{H}^{2}]-\mathrm{E}[c_{H}]^{2}$
$\displaystyle=\mathrm{E}[(\sum_{i}x_{i})(\sum_{i}x_{i})]-\mathrm{E}[c_{H}]^{2}$
$\displaystyle=\mathrm{E}[\sum_{ij}x_{i}x_{j}]-\mathrm{E}[c_{H}]^{2}$
$\displaystyle=\sum_{ij}\mathrm{E}[x_{i}x_{j}]-\mathrm{E}[c_{H}]^{2}$
Then, each possible pair corresponds to one possible pattern graph $J$, of
which the possible number is
$\frac{n^{\underline{|V(J)|}}}{|\mathrm{Aut^{\prime}}(J)|}$, and each exists
with independently with probability $p^{-|E(J)|}$. From this follows the given
expression.
## 5 Extension to Covariances
The method can be extended to covariances between the count statistics of
different patterns. As an example:
$\displaystyle\mathrm{Cov}[c_{\mathrm{edge}},c_{\mathrm{triangle}}]$
$\displaystyle=\frac{1}{32}n^{3}-\frac{3}{32}n^{2}+\frac{1}{16}n$
## 6 The Software
Our code is written in the programming language Matlab, and contains two entry
points, the function polcovar_mu() that computes the mean and the function
polcovar_sigma() that computes the variance or covariance.
r = polcovar_mu(H);
r = polcovar_sigma(H1, H2);
The input graphs H must be given as $k\times k$ adjacency matrices. The
function polcovar_sigma() expects two graphs H1 and H2 and returns the
covariance of their subgraph counts. To compute the variance, pass the same
adjacency matrix as both arguments. All input matrices must be symmetric 0/1
matrices with zero diagonals. All computations are valid for Erdős–Rényi
graphs with $p=1/2$.
The return values are rational polynomials in form of $2\times(m+1)$ matrices,
where $m$ is the degree, coded in the following way:
$\displaystyle r$
$\displaystyle=\left[\begin{array}[]{ccccc}a_{m}&a_{m-1}&\cdots&a_{1}&a_{0}\\\
b_{m}&b_{m-1}&\cdots&b_{1}&b_{0}\end{array}\right]$
representing the following rational polynomial in $n$:
$\displaystyle P_{r}(n)$
$\displaystyle=\sum_{i=0}^{m}\frac{a_{i}}{b_{i}}n^{i}$
All fractions $a_{i}/b_{i}$ are returned in simplified form.
### 6.1 Example
The following example uses Polcovar to compute the mean and standard deviation
of the number of triangles in a random graph with 1,000,000 nodes.
% Adjacency matrix of a triangle
H = [ 0 1 1; 1 0 1; 1 1 0]
% Compute polynomials
r_mu = polcovar_mu(H)
r_sigma = polcovar_sigma(H, H)
% Evaluate polynomials for a graph with 1,000,000 nodes
n = 1000000
mu = polyval(r_mu(1,:) ./ r_mu(2,:), n)
sigma = polyval(r_sigma(1,:) ./ r_sigma(2,:), n)
sigma_stddev = sqrt(sigma)
This will compute that a random graph with 1,000,000 nodes can be expected to
contain $2.0833\times 10^{16}\pm 8.8388\times 10^{10}$ triangles.
## Acknowledgements
We thank Thomas Sauerwald from the University of Cambridge.
## References
* [1] Paul Erdős and Alfréd Rényi. On random graphs I. Publ. Math. Debrecen, 6:290–297, 1959.
* [2] Andrzej Ruciński. When are small subgraphs of a random graph normally distributed? Prob. Th. Rel. Fields, 78:1–10, 1988.
|
arxiv-papers
| 2014-02-24T14:26:08 |
2024-09-04T02:49:58.746714
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "J\\'er\\^ome Kunegis",
"submitter": "J\\'er\\^ome Kunegis",
"url": "https://arxiv.org/abs/1402.5835"
}
|
1402.5868
|
# A Moments’ Analysis of Quasi-Exactly Solvable Systems: A New Perspective on
the Sextic Anharmonic and Bender-Dunne Potentials
Carlos R. Handy1, Daniel Vrinceanu1, and Rahul Gupta2 1Department of Physics,
Texas Southern University, Houston, Texas 77004;
2Lawrence E. Elkins High School, Missouri City, Texas 77459 [email protected]
###### Abstract
There continues to be great interest in understanding quasi-exactly solvable
(QES) systems. In one dimension, QES states assume the form
$\Psi(x)=x^{\gamma}P_{d}(x){\cal A}(x)$, where ${\cal A}(x)>0$ is known in
closed form, and $P_{d}(x)$ is a polynomial to be determined. That is
${{\Psi(x)}\over{x^{\gamma}{\cal A}(x)}}=\sum_{n=0}^{\infty}a_{n}x^{n}$
truncates. The extension of this “truncation” procedure to non-QES states
corresponds to the Hill determinant method, which is unstable when the
reference function assumes the physical asymptotic form (i.e. $x^{\gamma}{\cal
A}(x)$). Recently, Handy and Vrinceanu introduced the Orthogonal Polynomial
Projection Quantization (OPPQ) method which has non of these problems,
allowing for a unified analysis of QES and non-QES states ( 2013 J. Phys. A:
Math. Theor. 46 135202; 2013 J. Phys. B: 46 115002). OPPQ uses a non-
orthogonal basis constructed from the orthonormal polynomials of ${\cal A}$:
$\Psi(x)=\sum_{j=0}^{\infty}\Omega_{j}{\cal P}^{(j)}(x){\cal A}(x)$, where
$\langle{\cal P}^{(j_{1})}|{\cal A}|{\cal
P}^{(j_{2})}\rangle=\delta_{j_{1},j_{2}}$, and $\Omega_{j}=\langle{\cal
P}^{(j)}|\Psi\rangle$. For systems admitting a moment equation representation,
such as those considered here, these coefficients can be readily determined.
The OPPQ quantization condition, $\Omega_{j}=0$, is exact for QES states
(provided $j\geq d+1$); and is computationally stable, and exponentially
convergent, for non-QES states. OPPQ provides an alternate explanation to the
Bender-Dunne (BD) orthogonal polynomial formalism for identifying QES states:
they correlate with an anomalous kink behavior in the order of the finite
difference moment equation associated with the $\Phi=x^{\gamma}{\cal
A}(x)\Psi(x)$ Bessis-representation (i.e. a spontaneous change in the degrees
of freedom of the system). This was first noted by Handy and Bessis in their
implementation of the Eigenvalue Moment Method (EMM), the first application of
semidefinite programming analysis to quantum operators (1985 Phys. Rev. Lett.
55, 931 ). Additional properties ensue, such as $\Phi_{non-
QES}(x)=\partial_{x}^{d+2}\Upsilon(x)$, for states of the same symmetry as the
QES states. We study the above with respects to two sextic potentials of the
type $V(x)=gx^{6}+bx^{4}+mx^{2}+{\beta\over{x^{2}}}$.
###### pacs:
03.65.Ge, 02.30.Hq, 03.65.Fd
††: J. Phys. A: Math. Gen.
## 1 Introduction
### 1.1 Objectives and Overview
Inadequacies of the Hill determinant representation
The study of quasi-exactly solvable (QES) systems has continued to attract
much interest because of its relevance to physical systems and extensions to
quantum supersymmetry [1]. These correspond to Hamiltonians for which a subset
of the discrete spectrum, and corresponding wavefunctions, can be determined
in closed form. Their systematic study was initiated by Turbiner [2-5]. In one
space dimension, the typical QES state corresponds to a wavefunction of the
form $\Psi(x)=P_{d}(x){\cal A}(x)$, where the positive asymptotic
configuration ${\cal A}(x)>0$ is known in closed form, and $P_{d}(x)$ is some
polynomial of degree ‘$d$’ , to be determined. In other words, the ratio
${{\Psi(x)}\over{\cal A}(x)}$ truncates. This truncation philosophy does not
naturally extend, as given, to the non-QES states, for reasons given below.
This is the primary objective of this work: to develop a unified theoretical
and computational framework that can address the QES and non-QES states.
Additionally, our methods give a different interpretation for the existence of
QES states, from a moments’ representation perspective.
One may regard the QES truncation philosophy as a motivating factor for the
general Hill determinant quantization philosophy [6] which, in one dimension,
represents an arbitrary discrete state as $\Psi(x)=x^{\gamma}A(x){\cal R}(x)$,
where $\gamma$ is the (problem dependent) indicial exponent,
$A(x)=\sum_{j=0}^{\infty}a_{j}x^{j}$ is an analytic factor, and ${\cal R}(x)$
is some specified reference function, such as the Gaussian, $e^{-x^{2}}$. One
can relate the analytic properties of the wavefunction to the $a_{j}$’s, which
also acquire an energy dependence. The Hill determinant quantization
prescription determines those (approximate) energies leading to an effective
truncation of the power series expansion, $a_{N}(E)=0$, etc. In the limit
$N\rightarrow\infty$, these energy approximants (for the non-QES states)
usually converge to the true physical values, for the Gaussian reference
function. The major drawback of this approach, as is well known, is that if
the reference function is chosen to mimic the true asymptotic form of the
wavefunction, it leads to instabilities and erroneous energy convergence [7].
For the sextic anharmonic oscillator potential,
$V_{sa}(x)=gx^{6}+bx^{4}+mx^{2}$, and the Bender Dunne [8] sextic potential,
$V_{BD}(x)=x^{6}+mx^{2}+{b\over{x^{2}}}$, the physical reference functions are
${\cal R}_{sa}(x)=e^{-{{\sqrt{g}}\over 4}\big{(}x^{4}+{b\over g}x^{2}\big{)}}$
and ${\cal R}_{BD}(x)=e^{-{{x^{4}}\over 4}}$, respectively. However, the Hill
determinant approach proves unstable in either case, as suggested by the study
by Tate and Turbiner [9]. Thus, both QES and non-QES states can be
approximated, if a Gaussian type reference function is used. If the true
asymptotic form for the physical states could be used as reference functions,
then the same quantization approach would generate both the exact QES states
and approximate the non-QES states. However, this is inherently impossible
within the Hill determinant “truncation” philosophy. Nevertheless, the methods
introduced here can do precisely this.
Of relevance to the present formalism is the fact that the Hill determinant
method can also be implemented in Fourier space,
${\hat{\Psi}}(k)=k^{\gamma}A(k){\cal R}(k)$. It is referred to as the
Multiscale Reference Function (MRF) method, originally proposed by Tymczak,
Japaridze, Handy, and Wang [10]. The selection of an appropriate, positive,
reference function is somewhat limited, with the only viable choice usually
being the Gaussian, ${\cal R}(k)=e^{-k^{2}}$. However, even for this case, the
MRF method has better (faster and more monotonic) convergence properties than
the configuration space Hill determinant approach. A comparison, for the
sextic anharmonic oscillator, is given in Ref. [11]. The MRF method is only
implementable if the Schrodinger equation admits a moment equation
representation. If so, then the power moments of the (discrete states),
$\mu(p)=\int dx\ x^{p}\Psi(x)$, satisfy a linear, recursive relation that
involves the energy as a variable parameter. These are then used to generate
the power series coefficients of $A(k)$. The relevance of the MRF method to
the present analysis is that what is proposed here can be considered as a
merging of the Hill and the MRF into a new and much more powerful
representation particularly relevant for QES systems, as well as exactly
solvable systems (i.e. for which all discrete states are determinable in
closed form), which will be discussed in a subsequent work.
Orthogonal Polynomial Projection Quantization
Recently, Handy and Vrinceanu [11] proposed a new, multidimensional,
quantization formalism for systems admitting a moment equation representation.
It is referred to as the Orthogonal Polynomial Projection Quantization (OPPQ)
method. A motivatng factor was simply to improve upon the known limitations of
the Hill determinant analysis. These limitations not only include the
aforementioned instabilities when the reference function mimics the physical
asymptotic configuration (${\cal R}(x)\rightarrow{\cal A}(x)$), but also the
requirement that the reference function be analytic (of importance to the
Bender Dunne potential). Neither of these is a limitation within OPPQ.
The implementation of OPPQ requires working within a non-orthogonal basis,
$\\{{\cal P}^{(j)}(x){\cal R}(x)|j\geq 0\\}$, formed from the orthonormal
polynomials of the positive reference function, ${\cal R}(x)$: $\langle{\cal
P}^{(j_{1})}|{\cal R}|{\cal P}^{(j_{2})}\rangle=\delta_{j_{1},j_{2}}$. These
are used to generate the representation:
$\Psi(x)=\sum_{j=0}^{\infty}\Omega_{j}{\cal P}^{(j)}(x){\cal R}(x)$. The
expansion coefficients project out exactly, $\Omega_{j}=\langle{\cal
P}^{(j)}|\Psi\rangle$. They correspond to finite sums of the power moments of
$\Psi$. One can easily argue that for a broad class of reference functions,
including those asymptotic to the discrete physical states, we must have
$\lim_{n\rightarrow\infty}\Omega_{n}=0$. Assuming the existence of a moment
equation (of effective order $1+m_{s}$), the $\Omega_{j}$’s become linearly
dependent on the first $1+m_{s}$ power moments through known, energy
dependent, coefficients. Therefore, one can define the OPPQ quantization
procedure as taking $\Omega_{N+\ell}=0$, for $\ell=0,\ldots,m_{s}$. This
yields an energy dependent determinantal equation, $D_{N}(E)=0$, whose roots
exponentially converge to the physical states in the $N\rightarrow\infty$
limit.
If the asymptotic configuration of the physical states is known in closed
form, $x^{\gamma}{\cal A}(x)>0$ (i.e. if $\gamma\neq integer$, then
$x\rightarrow 0^{+}$ and $x\rightarrow\infty$ are necessary asymptotic
limits), then one can take it to be the reference function: ${\cal
R}(x)=x^{\gamma}{\cal A}(x)$. In this case, a subset of the OPPQ determinant’s
roots are the exact QES energies, for $N$ greater than a certain threshold.
Specifically, for one dimensional systems, since a QES state must assume the
form $\Psi_{QES}(x)=P_{d}(x)x^{\gamma}{\cal A}(x)$, and
$P_{d}(x)=\sum_{j=0}^{d}c_{j}{\cal P}^{(j)}(x)$, then $\Omega_{n}=\langle{\cal
P}^{(n)}|\Psi_{QES}\rangle=0$, for $n\geq d+1$. The QES energies must be the
exact roots, $D_{N}(E_{QES})=0$ for $N\geq d+1$, where each QES state will
have its own “$d$”. The non-QES states are approximated by the other roots of
the OPPQ determinant; and these approximations converge exponentially fast to
the physical values.
Although this work is limited to one dimensional systems, the entire OPPQ
formalism can be, and has been, applied in two dimensions. The original work
by Handy and Vrinceanu investigated several two dimensional quantum systems
including a particular pseudo-hermitian model. A subsequent work applied the
OPPQ formalism to the challenging two dimensional infinite dipole problem (two
oppositely charged line charges) [12] confirming a large basis Rayleigh Ritz
analysis by Amore and Fernandez [13].
In summary, the Hill determinant truncation philosophy (trivially) works for
QES states, but is unstable, or ineffective for the non-QES states. The OPPQ
analysis was developed independently of QES considerations, but turns out to
be the ideal, unifying quantization framework for both QES and non-QES, as
presented here.
Obtaining the QES (Bender Dunne), Energy, Polynomials
Within a configuration space Hill representation,
${{\Psi(x)}\over{x^{\gamma}{\cal A}(x)}}=\sum_{n}\ a_{n}(E)x^{n}$, if the
potential function parameters satisfy a particular constraint, then the
$a_{n}$’s will exhibit the defining truncation structure, $a_{n}(E)=0$, for
$n\geq n_{*}+1$, where $d\equiv n_{*}$; and for which $a_{n_{*}+1}(E)$ is a
polynomial of degree $n_{*}+1$ whose roots correspond to all the QES states
for that system. This polynomial is contained in the OPPQ determinant
expression; although its forms is not as immediately discernable as it is in
the Hill representation for the $a_{n}$’s. Whereas the Hill representation is
inadequate for determining the non-QES states, the OPPQ-$\Psi$ analysis is
able to generate both QES and non-QES states in a unified manner.
We would like a moments’ representation where the $a_{n_{*}+1}(E)$ polynomial
is also immediately transparent, and both QES and non-QES states can be
generated in a unified manner through OPPQ. This is possible within the Bessis
representation defined by $\Phi(x)=x^{\gamma}{\cal A}(x)\Psi(x)$. The
OPPQ-$\Phi$ analysis now requires working with reference functions of the type
${\cal R}(x)=x^{2\gamma}{\cal A}^{2}(x)$, and their orthonormal polynomials.
Note that contratry to the Hill representation, we are not stripping the
asymptotic form(s), but further enhancing the new representation by these
expressions. Within this representation, the corresponding moments,
$\nu(p)=\int dx\ x^{p}\Phi(x)$, have a moment equation that: (i) makes the
identification of the $a_{n_{*}+1}(E)$ polynomial very transparent; and (ii)
exhibits a structure that undelies the real reason for the anomalous behavior
of the Bender-Dunne (energy dependent) orthogonal polynomials.
Within the Hill representation, the expansion coefficients, $a_{n}(E)$,
satisfy a three term recursion relation, becoming polynomials in the energy
variable. This recursion relation remains of first order (i.e. $a_{n}$
generates $a_{n+1}$, etc.) for all ‘$n$’, regardless of the energy (QES or
non-QES). One can transform the $a_{n}$-recursion relation so that it
resembles the monic form of the usual three term relation for orthogonal
polynomials. Thus, $a_{n}(E)$, an $n$-th degree polynomial, transforms into an
$n$-th degree polynomial, $P_{bd}^{(n)}(E)\propto a_{n}(E)$, which satisfies
the manifestly monic three term relation
$P_{bd}^{(n)}(E)=(E-\alpha_{n})P_{bd}^{(n-1)}(E)-\gamma_{n-1}P_{bd}^{(n-2)}(E)$.
What Bender and Dunne did was to reinterpret the existence of QES states,
within the $P_{bd}$-representation, as corresponding to a breakdown of this
recursion relation, with $\gamma_{n_{*}+1}=0$. This relation implies that
these monic orthogonal polynomials have a signed weight, $w(E)$, and relative
to that weight the quantizing polynomial has zero norm: $\langle
P_{bd}^{(n_{*}+1)}|w|P_{bd}^{(n_{*}+1)}\rangle=0$.
From the moments’ perspective, the Bender and Dunne interpretation is not
necessary because the form of the moment equation for the $\nu$’s reveals the
true anomaly.
For the non-QES states of different symmetry to the QES states (i.e. the
sextic anharmonic oscillator case) , the $\nu$-moment equation is also a three
term, first order, recursion relation. However, for the QES states, the moment
equation is only of first order for the first $n_{*}+1$ moments,
$\\{\nu(p)|0\leq p\leq n_{*}\\}$. All the other moments satisfy a finite
difference equation of second order. Within the context of the moment
equation, the $\nu(n_{*}+1)$ moment decouples from the lower order moments;
although, it can be generated from the lower order moments through a relation
independent of the moment equation. For the QES states within the same
symmetry class as the QES states, the disruption is less severe since the
first $n_{*}+1$ moments must be zero (i.e. $\nu(p)=0,\ \leq p\leq n_{*}$);
whereas all the higher order moments define a first order finite difference
equation. This kink in the order of the underlying moment equation for QES
states is the real reason for the Bender-Dunne orthogonal polynomial anomaly.
For the QES states, the $\\{\nu(p)|0\leq p\leq n_{*}\\}$ moments are
polynomials in the energy, $E$, of degree corresponding to the moment-order
(‘$p$’). The BD polynomial $P_{bd}^{(n+1)}(E)$ corresponds to the linear sum
of the two moments $\\{\nu(n-1),\nu(n)\\}$, involving an energy dependent
coefficient that is a monomial in the energy (consistent with the monic form
for orthogonal polynomials). Because of the kink in the $\nu$-moment equation,
from the moments’ perspective, the $P_{bd}^{(n+1)}(E)$ polynomials have no
natural extension beyond $n\geq n_{*}$. For the non-QES states, for symmetry
different than the QES states, the corresponding monic orthogonal polynomials
exist (i.e. satisfy the monic orthogonal polynomial three term structure).
In summary, we do not have to look for kinks in the recursive structure of the
Bender Dunner orthogonal polynomial representation, it is easier to look for
kinks in the nature of the moment equation within the Bessis representation.
The latter would seem to be an easier analysis than the former, particularly
for multidimensional systems admiting a moment equation representation.
The Bessis Representation: Relevance of the Eigenvalue Moment Method
The existence of QES states as due to a nonuniform moment order structure was
known to Handy and Bessis (HB) in the context of their development of the
Eigenvalue Moment Method (EMM) [14-16], the first application of semidefinite
programming (SDP) [17-18] in quantum physics. We ouline the relevant history
since it impacts this work, and underscores the importance of moment
representations for quantizing physical systems.
In 1984 Handy [19] discovered that combining the moment equation
representation with a particular representation of the moment problem theorems
in mathematics (i.e. the nesting property of the Pade approximants of
Stieltjes measures) [20], yielded converging lower and upper bounds to the
(one dimensional) bosonic ground state energy, or any other quantum state
associated with a known nodal structure (i.e. the first excited state of
parity invariant systems). Handy and Bessis (HB) transformed this into a more
general, multidimensional, formulation through the use of the moment problem
Hankel Hadamard (HH) determinantal inequality constraints, $Det({\cal
H}_{n})>0$, where ${\cal H}_{i,j}=\mu(i+j),0\leq i,j\leq n$, is the Hankel
matrix [21]. Through the underlying moment equation of order $1+m_{s}$, these
moments depend linearly on the $1+m_{s}$ initialization moments (i.e. referred
to as the missing moments by HB), and nonlinearly on the energy parameter.
Unlike typical SDP problems that seek to optimize some objective function, the
EMM/SDP analysis requires that for each energy parameter value, $E$, one
determine the existence or nonexistence of the nonlinear convex solution set
to the HH inequalities (once a suitable normalization condition is chosen,
reducing the missing moment domain to $m_{s}$ dimensions), $Det\Big{(}{\cal
H}_{n}({\cal U}_{E}^{(N)})\Big{)}>0$, $n\leq N$, $N\rightarrow\infty$. The set
of admissible energy values define an interval $[E_{L}^{(N)},E_{U}^{(N)}]$,
such that if $E_{L}^{(N)}\leq E\leq E_{U}^{(N)}$, then ${\cal
U}_{E}^{(N)}\neq\emptyset$. As the dimension of the Hankel matrix increases,
$N\rightarrow\infty$, we have $E_{U}^{(N)}-E_{L}^{(N)}\rightarrow 0$,
exponentially with ‘$N$’, in most cases. This bounding procedure was
particularly effective for strongly coupled, singular perturbation type
systems for which conventional computational methods could be unreliable.
The EMM analysis, as originally formulated [14], is a nonlinear optimization
problem. There were no efficient SDP algorithms available in 1985, thus
limiting the class of problems HB could investigate. The hardest problem
amenable to a very basic computational strategy was the sextic anharmonic
oscillator. Motivated by this, Bessis realized that by multiplying the
wavefunction by its asymptotic form the computational complexity of the sextic
anharmonic oscillator problem became equivalent to that for the harmonic
oscillator problem. This revealed the anomalous kink behavior of the
$\nu$-moment equation for the QES states; however the focus of that first work
was on bounding the non-QES energies for the ground and first excited states.
In subsequent works [15,16] Handy was able to transform the nonlinear version
of EMM into an equivalent linear programming based formulation which allowed
for its implementation to a broad range of multidimensional systems, including
the notoriously difficult quadratic Zeeman effect for superstrong magnetic
fields [15,16]. The relevance of moment representations for quantizing
singular perturbation-strongly coupled systems had been previously noted by
Handy [22], in the context of finding a more rigorous alternative to lattice
high temperature expansions in field theory. This early work introduced the
scaling transform, whose perturbative structure in the inverse (lattice) scale
depends on the power moments. The relevance of this formalism to incorporating
wavelets into quantum mechanics was demonstrated in a subsequent work [23].
Coincidentally, one may characterize EMM as an affine map invariant
variational procedure, since it optimizes within the affine map invariant
space of polynomials. Indeed, the EMM bounds to the quadratic Zeeman effect
were highly correlated with the ground state binding energy estimates of an
order dependent, conformal, analysis by LeGuillou and Zinn-Justin [24]; and
similarly for the three dimensional quantum dot [25]. Beyond EMM, advances in
SDP code development have progressed rapidly since the 1990s with the impetus
coming from combinatorics [26] and reduced density matrix theory in quantum
chemistry [27-28].
A major focus of EMM analysis is to define new nonnegativity representations
for quantum systems. For arbitrary one dimensional systems with real
potentials, the probability density satisfies a third order, linear,
differential equation, enabling application of EMM bounding methods for all
states, depending on the nature of the potential [29]. That is, knowledge of
the nodes is not required. The same is true for complex, one dimensional,
potentials, because the Herglotz analytic continuation of $|\Psi(x)|^{2}$
satisfies a fourth order linear differential equation. One can then use EMM to
bound the complex quantum parameters. This was used to support the conjecture
on the reality of the discrete spectrum for the pseudo hermitian potential
$V(x)=ix^{3}$ (i.e. the Bessis conjecture) [30], as previously suggested by
Bender and Boetcher [31], and subsequently proved by Dorey et al [32]. It was
also used to computationally predict the correct onset of PT-symmetry breaking
for the pseudo-hermitian potential, $V(x)=ix^{3}+iax$ [33]. Other applications
include bounding Regge pole parameters relevant to atomic scattering [34,35].
Summary of problems and representations to be analyzed
The following sections will examine the sextic anharmonic oscillator
potential, $V_{sa}(x)=gx^{6}+bx^{4}+mx^{2}$, and the BD sextic potential,
$V_{BD}(x)=x^{6}+mx^{2}+{b\over{x^{2}}}$, both within the $\Psi$
representation and the Bessis $\Phi$ representation. In each we show how OPPQ
yields both the exact QES states and converging approximants to the non-QES
states. We also show how the $\nu$-moment equation within the Bessis
representation recovers the BD-polynomials and their recursion relation. It is
to be re-emphasized that within the Bessis representation, the QES states can
be generated two ways:(1) as the roots of an energy polynomial defined by the
$\nu$-moments (i.e. effectively the BD energy polynomials); (2) as the exact
roots of the OPPQ quantization determinant, for which the remaining roots
approximate the non-QES states.
The following discussion pertains to both potentials, but we limit our remarks
to the sextic anharmonic oscillator case. The sextic anharmonic oscillator
problem admits non-QES states and QES states, for particular potential
function parameter values. Within each parity class, there will be QES states.
There will be non-QES states of the same parity as the QES states; whereas all
states of the opposite parity will be non-QES states. These distinctions may
complicate our notation. These distinctions do not arise in the BD sextic
potential case because it is defined on the nonnegative real axis.
The sextic anharmonic oscillator Schrodinger equation, in the $\Psi$
representation is
$-\partial_{x}^{2}\Psi(x)+(gx^{6}+bx^{4}+mx^{2})\Psi(x)=E\Psi(x).$ (1)
It admits even and odd parity states as indexed by $\sigma=0,1$, respectively.
The QES states are represented by $\Psi(x)=P_{d}(x){\cal A}(x)$, for $d\equiv
n_{*}\geq 0$. Their corresponding parity will be denoted by $\sigma_{*}=0\ or\
1$. The asterisk notation is exclusively used to identify the QES states and
relations. The potential function parameter constraint admitting QES states
corresponds to
$g^{3\over 2}(16n_{*}+12+8\sigma_{*})+4mg-b^{2}=0.$ (2)
This condition can only be explicitly derived within two representations;
either from the Hill representation truncation analysis, or the $\nu$-moment
equation relation within the Bessis function representation. It can be
surmized within the $\Psi$-moment equation representation from a JWKB analysis
and then tested through OPPQ. All of these are implemented in the following
sections.
We will not give explicit algebraic forms for the QES energies, etc., because
the relations to be given are transparent and readily implementable by the
interested reader. Instead, we focus on the numerical consistency of our
results with the underlying OPPQ theory. We do not focus on wavefunction
reconstruction because this is also straightforward. Finally, we have
streamlined the OPPQ formalism from that originally presented by Handy and
Vrinceanu [11-12]. The present formalism emphasizes the orthonormal
polynomials of the physical reference function.
There are several types of polynomials considered here. First is the
polynomial factor, $P_{d}(x)$, defining the QES state. Second are the
orthonormal polynomials, ${\cal P}^{(j)}(x)$ corresponding to the positive
reference function, ${\cal A}(x)>0$. Third are the BD polynomials in the
energy space. Fouth will be the $\nu$-moments which correspond to polynomials
in the energy (and whose superposition defines the BD polynomials). The monic
form of the OPPQ orthonormal polynomials, ${\cal P}^{(j)}(x)$, will be denoted
by ${\tilde{\cal P}}^{(j)}(x)$.
## 2 Preliminaries
### 2.1 The $\Psi$-moment equation
Before continuing with the OPPQ generalities, we note that Eq.(1) can be
transformed into a moment equation for the discrete states. Define
$\mu(p)\equiv\int dxx^{p}\Psi(x)$ where we assume $\Psi(x)$ to be implicitly a
bound state, asymptotically vanishing at infinity. Multiplying both sides of
Eq.(1) by $x^{p}$ and performing the necessary integration by parts gives the
moment equation:
$\displaystyle g\mu(p+6)=-b\mu(p+4)-m\mu(p+2)+E\mu(p)+p(p-1)\mu(p-2),$ (3)
$p\geq 0$. The even and odd states separate, $\Psi_{\sigma}(x)$, $\sigma=0,1$,
respectively. Although not necessary, we prefer to explicitly work within the
even and odd representations, in order to reduce the dimensionality of the
OPPQ determinant matrix. We denote the power moments for the even or odd
states by
$\displaystyle u_{\sigma}(\rho)$
$\displaystyle=\mu(2\rho+\sigma)=\int_{-\infty}^{+\infty}dxx^{2\rho+\sigma}\Psi_{\sigma}(x)$
(4) $\displaystyle=\int_{0}^{\infty}d\xi\xi^{\rho}\psi_{\sigma}(\xi),\ where\
\psi_{\sigma}(\xi)={(\sqrt{\xi})^{\sigma-1}}\Psi_{\sigma}(\sqrt{\xi})\ ,$
and $\xi\equiv x^{2}$, $\sigma=0\ or\ 1$. We note that for the ground and
first excited state the $\psi_{\sigma}(\xi)$ configuration is nonnegative. The
corresponding $u_{\sigma}(\rho)$-moment equation becomes
$\displaystyle
gu_{\sigma}(\rho+3)=-bu_{\sigma}(\rho+2)-mu_{\sigma}(\rho+1)+Eu_{\sigma}(\rho)+2\rho(2(\rho+\sigma)-1))u_{\sigma}(\rho-1).$
The effective order of this homogeneous, linear, moment equation is $1+m_{s}$
where $m_{s}=2$, since the moments
$\\{u_{\sigma}(0),u_{\sigma}(1),u_{\sigma}(2)\\}$ must be specified, in
addition to the energy, before all the other moments can be generated.
Imposing an $L^{1}$ normalizaiton condition (i.e.
$\sum_{\ell=0}^{m_{s}}u_{\sigma}(\ell)=1$, within EMM) reduces these missing
moment, or initialization moment, variables, to two. We can represent the
moment-missing moment dependence by the relations
$u_{\sigma}(\rho)=\sum_{\ell=0}^{m_{s}}M_{E,\sigma}(\rho,\ell)u_{\sigma}(\ell),$
(6)
where $M_{E,\sigma}(\rho,\ell)$ are known energy dependent polynomials (or
more generally, rational fraction polynomials in $E$), satisfying the
corresponding moment equation subject to the initialization conditions,
$M_{E,\sigma}(\ell_{1},\ell_{2})=\delta_{\ell_{1},\ell_{2}}$.
### 2.2 Orthogonal Polynomial Projection Quantization
We review the Orthogonal Polynomial Projection Quantization method in general,
and then apply it to QES systems.
Suppose ${\cal A}(x)>0$ is a positive, bounded, configuration admitting an
infinite set of orthonormal polynomials (our bra-ket notation will omit
explicit reference to the underlying weight, for simplicity)
$\displaystyle\langle{\cal P}^{(j_{1})}|{\cal P}^{(j_{2}}\rangle\equiv\int
dx{\cal P}^{(j_{1})}(x){\cal P}^{(j_{2})}(x){\cal A}(x)=\delta_{j_{1},j_{2}},$
$\displaystyle{\cal P}^{(j)}(x)=\sum_{i=0}^{j}\Xi_{i}^{(j)}x^{i},\ where\
\Xi_{j}^{(j)}\neq 0.$ (7)
Assume that the quantum system under consideration admits a moment equation,
represented as
$\mu(p)=\sum_{\ell=0}^{m_{s}}M_{E}(p,\ell)\ \mu(\ell),p\geq 0.$ (8)
Consider expanding the desired discrete state in terms of the orthonormal
polynomial basis:
$\displaystyle\Psi(x)=\sum_{j=0}^{\infty}\Omega_{j}{\cal P}^{(j)}(x)\ {\cal
A}(x).$ (9)
One can then project out the expansion coefficients exactly:
$\displaystyle\Omega_{j}$ $\displaystyle=$ $\displaystyle\int dx\ {\cal
P}^{(j)}(x)\Psi(x),$ (10) $\displaystyle\Omega_{j}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{j}\Xi^{(j)}_{i}\mu(i),$ (11)
$\displaystyle\Omega_{j}$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{m_{s}}\Big{(}\sum_{i=0}^{j}\Xi^{(j)}_{i}M_{E}(i,\ell)\Big{)}\mu(\ell).$
(12)
Now consider the positive (and assumed finite) integral expression $\int
dx{{\Psi^{2}(x)}\over{{\cal A}(x)}}<\infty$. We obtain
$\int dx{{\Psi^{2}(x)}\over{{\cal
A}(x)}}=\sum_{j=0}^{\infty}\Omega_{j}^{2}<\infty,$ (13)
resulting in
$\lim_{j\rightarrow\infty}\Omega_{j}=0.$ (14)
The integral condition in Eq.(13) can be satisfied if ${\cal A}$ decays slower
than $\Psi^{2}(x)$ allowing the ratio to be integrable. A rougher statement
suggests that if ${\cal A}$ decreases as, or slower, than $\Psi$ then the
above is satisfied. Note that we do not want ${\cal A}$ decreasing so fast
that even the unphysical solutions have finite integrals. In this case, the
OPPQ quantization conditions will not work. These considerations are also
essential within the EMM analysis.
The asymptotic behavior of Eq.(14) suggests that we impose these conditions,
at finite order, on appropriate, successive, projection expressions as
represented in Eq.(12). In particular:
$\displaystyle\sum_{\ell=0}^{m_{s}}\Big{(}\sum_{i=0}^{j}\Xi^{(j)}_{i}M_{E}(i,\ell)\Big{)}\mu(\ell)=0,$
(15)
for $j=N,N+1,N+2,\ldots,N+m_{s}$, defining an $(m_{s}+1)\times(m_{s}+1)$
determinantal condition:
$\displaystyle D_{N}(E)=Det({\cal M}_{\eta,\ell}^{(N)}(E))=0,$ (16)
where ${\cal
M}_{\eta,\ell}^{(N)}(E)=\sum_{i=0}^{N+\eta}\Xi^{(N+\eta)}_{i}M_{E}(i,\ell)$.
We note that the degree of $D_{N}(E)$, generally a rational polynomial of the
energy , grows as $N\rightarrow\infty$, allowing for the generation of
converging approximants to all the discrete states.
The energy roots to Eq.(16) generally converge, exponentially fast to the
physical energies. The closer ${\cal A}$ describes the asymptotic behavior of
the desired physical state, the faster the convergence. We emphasize that
Eq.(16) is valid only if there is no symmetry related condition requiring the
any of the missing moments be zero. For parity invariant systems, the above
formalism should be implemented within the moment representation for that
symmetry class (i.e. Eq.(5)).
Several important features distinguish OPPQ with respect to other methods.
First of all, with respect to determining the energies, one does not need the
explicit form for ${\cal A}$. All that is required is that one be able to
generate the orthogonal polynomials accurately. Furthermore, this asymptotic
factor does not have to be differentiable. Indeed, for the quartic potential,
$V(x)=x^{4}$, using ${\cal A}(x)=exp(-{|x|^{3}}\over 3)$ gives better results
than using the gaussian.
### 2.3 Generating the Orthonormal Polynomials for ${\cal A}$
This subsection is included for completeness. The orthonormal polynomials of
${\cal A}(x)$ can be determined through the three term recursion relation for
their monic form. Let us denote the monic polynomials by ${\tilde{\cal
P}}^{(j)}(x)={1\over{\Xi_{j}^{(j)}}}{\cal P}^{(j)}(x)$. For simplicity, we
shall refer to the leading orthonormal coefficient as
$n_{j}\equiv\Xi_{j}^{(j)}$.
Let ${\tilde{\cal P}}^{(j)}(x)={1\over{n_{j}}}{\cal
P}^{(j)}(x)=x^{j}+b_{j}x^{j-1}+\ldots$, represent the monic form of the
orthonormal polynomial, ${\cal P}^{(j)}(x)$. We then have $\langle\tilde{\cal
P}^{(j)}|\tilde{\cal P}^{(j)}\rangle=\langle x^{j}|\tilde{\cal
P}^{(j)}\rangle$, and $\langle x\tilde{\cal P}^{(j)}|\tilde{\cal
P}^{(j)}\rangle=\langle x^{j+1}|\tilde{\cal P}^{(j)}\rangle+b_{j}\langle
x^{j}|\tilde{\cal P}^{(j)}\rangle$. The monic orthogonal polynomials satisfy
the well known three term recurrence relation,
$\displaystyle{\tilde{\cal
P}}^{(j+1)}(x)=(x-{\tilde{\alpha}}_{j+1}){\tilde{\cal
P}}^{(j)}(x)-{\tilde{\gamma}}_{j}{\tilde{\cal P}}^{(j-1)}(x)$ (17)
for$j\geq 0$, where ${\tilde{\cal P}}^{(-1)}(x)\equiv 0$,${\tilde{\cal
P}}^{(0)}(x)\equiv 1$, and ${\tilde{\alpha}}_{j+1}={\langle x{\tilde{\cal
P}}^{(j)}|{{\tilde{\cal P}}^{(j)}\rangle}\over{\langle x^{j}|{\tilde{\cal
P}}^{(j)}\rangle}}$ and ${\tilde{\gamma}}_{j}={\langle x^{j}|{\tilde{\cal
P}}^{(j)}\rangle}\over{\langle x^{j-1}|{\tilde{\cal P}}^{{(j-1)}}\rangle}$.
All these expressions depend on the power moments of the weight $m(p)=\int
dxx^{p}{\cal A}(x)$, which are assumed known. Specifically $\langle
x{\tilde{\cal P}}^{(j)}|{\tilde{\cal
P}}^{(j)}\rangle=\sum_{i_{1}=0}^{j}\sum_{i_{2}=0}^{j}\Xi^{(j)}_{i_{1}}\Xi_{i_{2}}^{(j)}m(i_{1}+i_{2}+1)$,
$\langle x^{j}|{\tilde{\cal
P}}^{(j)}\rangle=\sum_{i=0}^{j}\Xi_{i}^{(j)}m(i+j)$, and $\langle
x^{j-1}|{\tilde{\cal
P}}^{(j-1)}\rangle=\sum_{i=0}^{j-1}\Xi_{i}^{(j-1)}m(i+j-1)$. Given the monic
orthogonal polynomials, $\\{{\tilde{\cal P}}^{(j)}|j\leq J\\}$, the moments
$\\{m(2j+1),m(2j),m(2j-1),\ldots,m(0)|j\leq J\\}$ are required for generating
the ${\tilde{\alpha}}_{J+1}$ and ${\tilde{\gamma}}_{J}$ coefficients for
generating the next monic orthogonal polynomial, ${\tilde{P}}^{(J+1)}(x)$. The
three term recursion relation is usually the preferred procedure for
generating the monic orthogonal polynomials. The coefficient $n_{j}$ is then
obtained from $n_{j}^{2}\langle{\tilde{\cal P}}_{j}|{\tilde{\cal
P}}_{j}\rangle=1$. That is
${\tilde{\gamma}}_{j}=({{n_{j-1}}\over{n_{j}}})^{2}$, involving the ratio of
the norms.
We can transform the monic three term recursion relation into the counterpart
for the orthonormal polynomials:
$\displaystyle{{\cal P}}^{(j+1)}(x)=(x-{\tilde{\alpha}}_{j+1})\rho_{j}{{\cal
P}}^{(j)}(x)-{\tilde{\gamma}}_{j}\rho_{j}\rho_{j-1}{{\cal P}}^{(j-1)}(x),$
(18)
for$j\geq 0$, where ${\tilde{\cal P}}^{(-1)}(x)\equiv 0$, and
$\rho_{j}={{n_{j+1}}\over{n_{j}}}$.
An alternative representation for the orthogonal polynomials comes from Pade
analysis [20] which yields
$\displaystyle{\tilde{\cal
P}}^{(j)}(x)={1\over{{\Delta}_{0,j-1}(m)}}Det\pmatrix{&m(0)&m(1)&\ldots&m(j)\cr&m(1)&m(2)&\ldots&m(j+1)\cr&\ldots&\ldots&\ldots&\ldots\cr&m(j-1)&m(j)&\ldots&m(2j-1)\cr&1&x&\ldots&x^{j}\cr}\
$
$\displaystyle{\Delta}_{i,j-1}(m)=Det\pmatrix{&m(i)&m(i+1)&\ldots&m(i+j-1)\cr&m(i+1)&m(i+2)&\ldots&m(i+j)\cr&\ldots&\ldots&\ldots&\ldots\cr&m(i+j-1)&m(i+j)&\ldots&m(i+2j-2)}>0,\
for\ i=0,1.$
The latter correspond to the Hankel-Hadamard determinants, which must be
positive for a (non-atomic) nonnegative weight (although OPPQ requires ${\cal
A}$ to be positive). Note then that $\langle{\tilde{\cal
P}}^{(j)}|{\tilde{\cal P}}^{(j)}\rangle=\langle x^{j}|{\tilde{\cal
P}}^{(j)}\rangle={{\Delta_{0,j}(m)}\over{\Delta_{0,j-1}(m)}}=n_{j}^{-2}$.
We can project out, exactly, the $\Omega$ coefficients through
$\displaystyle\Omega_{j}=$ $\displaystyle\int_{-\infty}^{+\infty}dx\ {\cal
P}^{(j)}(x)\Psi(x),$ $\displaystyle=$
$\displaystyle{1\over\sqrt{\Delta_{0,j-1}(m)\Delta_{0,j}(m)}}Det\pmatrix{&m(0)&m(1)&\ldots&m(j)\cr&m(1)&m(2)&\ldots&m(j+1)\cr&\ldots&\ldots&\ldots&\ldots\cr&m(j-1)&m(j)&\ldots&m(2j-1)\cr&\mu(0)&\mu(1)&\ldots&\mu(j)\cr}.$
## 3 OPPQ and Quasi-Exactly Solvable Quantum Systems
This work solely focuses on QES systems; however, for completeness, we
contrast their structure with systems referred to as exactly solvable (ES),
for which all states are determinable in closed form. In one space dimension,
in some suitable coordinate transformed space if necessary, $s=s(x)$, the
wavefunction for an ES system will assume the form $\Psi(s)=s^{\gamma}{\cal
P}^{(n)}(s){\cal A}(s)$, where the positive asymptotic form is known in closed
form, ${\cal A}(s)>0$, and ${\cal P}^{(n)}(s)$ is the orthogonal polynomial
relative to some positive weight ${\cal W}(s)>0$. As before, $\gamma$ denotes
any required indicial exponent. The application of OPPQ to ES systems will be
discussed in a subsequent work.
Quasi-exactly solvable (QES) systems are those admitting wavefunctions of the
form
$\Psi(x)=x^{\gamma}P_{d}(x){\cal A}(x),$ (21)
(assuming $s(x)=x$), where $P_{d}(x)$ is a polynomial of degree “$d$”, to be
determined, and not necessarily the orthogonal polynomial of any weight.
Within OPPQ, such states will have exactly solvable energies and wavefunctions
(i.e. the roots of closed form algebraic functions of the energy, etc.). This
statement implicitly assumes the existence of a moment equation. For the
remainder of the subsequent presentation (i.e. the sextic anharmonic
oscillator), we will take $\gamma=0$.
From the discussion and definitions in the previous sections, since
$P_{d}(x)=\sum_{j=0}^{d}c_{j}{\cal P}^{(j)}(x)$, we know that for the QES
states:
$\displaystyle\int dx{\cal P}^{(j)}(x)\Psi(x)=0,\ j\geq d+1.$ (22)
That is, the OPPQ quantization condition in Eq.(14) is exactly satisfied at
all orders above a certain threshhold ($j\geq d+1$). Assuming that the
corresponding missing moments are not identically zero (for that particular
state), then Eq.(15) is satisfied for all $N\geq d+1$. Since the
$M_{E}(i,\ell)$ expressions are known in closed form (usually producing an
algebraic function of $E$ for the determinant in Eq.(16)) it means that the
discrete state energy would be determined in closed form, as the constant
roots of Eq.(16) for all orders $N\geq d+1$. That is, for QES systems, the
determinant quantization expression in Eq.(16) will admit two types of roots,
for $N\geq d+1$. There will be the varying roots that converge (exponentially
fast) to the true, non-QES states, of the system. The other roots, for
arbitrary $N\geq d+1$ will not vary and correspond to the exact energies.
Upon determining the QES energies, the corresponding missing moment values are
determined, thereby yielding the OPPQ projection coefficients (i.e.
$\Omega_{j}$’s), thereby generating the wavefunction.
### 3.1 Additional Moment Identities for QES Solutions
Although OPPQ is dependent on the existence of a moment equation, there is
another moment relation inherent to QES solutions that is independent of the
existence of such moment equations, but strongly suggest that these systems
must admit some form of moment equation.
If we take ${\cal P}^{(j)}(x)=\sum_{i=0}^{j}\Xi_{i}^{(j)}x^{i}$, where
$\Xi_{j}^{(j)}\neq 0$, and insert in Eq.(22), or $\langle{\cal
P}^{(j)}|\Psi\rangle=0$, for $j\geq d+1$, we obtain:
$\displaystyle\mu(j)=-{1\over{\Xi_{j}^{(j)}}}\sum_{i=0}^{j-1}\Xi_{i}^{(j)}\mu(i),j\geq
d+1.$ (23)
In particular, starting at $j=d+1$, this linear, recursive, relation connects
all the moments $\\{\mu(j)|j\geq d+1\\}$ to the lower order moments
$\\{\mu(j)|j\leq d\\}$. These relations are not valid for the non-QES states,
since $\langle{\cal P}^{(j)}|\Psi_{non-QES}\rangle\neq 0$. If the system in
question has a moment equation, represented as
$\mu(p)=\sum_{\ell=0}^{m_{s}}M_{E}(p,\ell)\mu(\ell)$, $p\geq 0$, then for
$p\geq d+1$, the moment equation and Eq.(23) must yield the same results once
the QES energy and corresponding missing moments have been determined.
If the system is parity invariant, the orthonormal polynomials will involve
polynomials of alternating even degrees and odd degrees. Therefore, for the
even states, if $d=2n$, then Eq.(23) holds for $j=d+2,d+4,\ldots$. If
$d=2n+1$, then $j=d+2,d+4,\ldots$.
### 3.2 QES-OPPQ Analysis of Sextic Anharmonic Oscillator
Consider the sextic anharmonic oscillator problem in Eq.(1). The leading
asymptotic form for the physical bound states corresponds to
${\cal A}(x)=exp\Big{(}-{{\sqrt{g}}\over 4}\big{(}x^{4}+{b\over
g}x^{2}\big{)}\Big{)}.$ (24)
We will illustrate the consistency of the OPPQ analysis applied within the
$\Psi$\- representations (i.e. OPPQ-$\Psi$), which corresponds, in this case,
to an $m_{s}=2$ moment equation representation. However, the ideal
representation that recovers the Bender-Dunne energy polynomials is that
defined by $\Phi(x)={\cal A}(x)\Psi(x)$, an $m_{s}=0$ problem, as discussed in
the next section, and referred to as the OPPQ-$\Phi$ analysis.
As will be seen in the next section, within the $\Phi$ representation, the
particular form of the corresponding moment equation will readily reveal the
existence of QES states. Within the $\Psi$ representation, this becomes more
difficult, unless one specifically implements a Hill representation analysis
and confirms the truncation of the $A(x)$ power series factor. However, a
systematic examination of the JWKB form for the wavefunction can suggest the
possible existence of QES states.
A simple, first order, JWKB approximation for the sextic anharmonic problem
suggests that there are QES discrete state wavefunctions of the form
$\Psi_{\sigma}(x)=P_{d}(x){\cal A}(x)$, where $d=2n_{*}+\sigma_{*}$, and
$\sigma_{*}=0\ or\ 1$, for the even or odd states, respectively. More
specifically, the first order JWKB asymptotic form of the discrete state
wavefunction gives $\Psi(x)\sim{1\over{({\partial_{x}S(x))^{1\over
2}}}}exp(-S(x))$, where
$S(x)={\sqrt{g}\over 4}(x^{4}+{b\over g}x^{2})+{1\over{\sqrt{g}}}({m\over
2}-{{b^{2}}\over{8g}})Ln(x).$ (25)
The asymptotic estimate becomes $\Psi_{\sigma}(x)\sim x^{d}{\cal A}(x)$, where
$d={1\over{\sqrt{g}}}({{b^{2}}\over{8g}}-{m\over 2})-{3\over 2}$. Since there
can only be even or odd solutions, the potential function parameters leading
to an integer form for $d=2n_{*}+\sigma_{*}$ correspond to the QES potential
function constraints in Eq.(2). We can test the validity of this by checking
that the OPPQ analysis yields constant QES energy values within the OPPQ
framework. As previously noted, the constraint in Eq.(2) can only be
explicitly confirmed either within a Hill representation (truncation)
analysis, or the $\nu$-moment analysis in the next section.
We will work within each parity symmetry class associated with the QES state.
The QES form for the wavefunction will be $\Psi_{\sigma_{*}}(x)=P_{d}(x){\cal
A}(x)$ , where $P_{d}(x)\rightarrow x^{\sigma_{*}}{P}_{n_{*}}(x^{2})$,
$d=\sigma_{*}+2n_{*}$, for the even or odd states ($\sigma_{*}=0,1$):
$\Psi_{\sigma_{*}}(x)=x^{\sigma_{*}}{P}_{n_{*}}(x^{2}){\cal A}(x).$ (26)
For notational simplicity, the following discussion implicitly assumes that
all references to $\sigma,n$ implicitly refer to the QES values
$\sigma_{*},n_{*}$. We expand the wavefunction in terms of
$\Psi_{\sigma}(x)=\sum_{j=0}^{n}\Omega_{j}x^{\sigma}{{\cal
P}}^{(j)}_{\sigma}(x^{2}){\cal A}(x),$ (27)
where $x^{\sigma}{\cal P}^{(j)}_{\sigma}(x^{2})$ are the even and odd
orthonormal polynomials of ${\cal A}$, satisfying $\langle{{\cal
P}}_{\sigma}^{(j_{1})}|x^{2{\sigma}}{\cal A}(x)|{{\cal
P}}_{\sigma}^{(j_{2})}\rangle=\delta_{j_{1},j_{2}}$.
Quantization via OPPQ involves
$\displaystyle\int dx\ x^{\sigma}{{\cal
P}}^{(j)}_{\sigma}(x^{2})\Psi_{\sigma}(x)=0,$
$\displaystyle\int_{0}^{\infty}d\xi\ {{\cal P}}^{(j)}_{\sigma}(\xi)\
\psi_{\sigma}(\xi)=0,\ for\ j\geq n_{*}+1,$ (28)
where $\psi_{\sigma}(\xi)\equiv\xi^{{\sigma-1}\over
2}\Psi_{\sigma}(\sqrt{\xi})$, from Eq.(4).
Writing ${{\cal P}}^{(j)}_{\sigma}(\xi)=\sum_{i=0}^{j}\Xi_{\sigma;i}^{(j)}\
\xi^{i}$, Eq.(27) transforms into
$\displaystyle\sum_{i=0}^{j}\Xi_{\sigma;i}^{(j)}\ u_{\sigma}(i)=0,$
$\displaystyle\sum_{\ell=0}^{m_{s}}\Big{(}\sum_{i=0}^{j}\Xi_{\sigma;i}^{(j)}M_{E,\sigma}(i,\ell)\Big{)}u_{\sigma}(\ell)=0,\
for\ j\geq n_{*}+1,$ (29)
using the $u_{\sigma}$-moment equation in Eq.(6). As suggested in Eq.(29),
this relation is exactly true for QES states. It becomes the OPPQ
approximation for non-QES states.
The missing moment order is $m_{s}=2$, therefore Eq.(29) must be valid for any
three successive $j$ values, provided they are greater than $n_{*}+1$. In
particular, for $j=N,N+1,N+2$, where $N\geq n_{*}+1$, we obtain the
determinantal relation
$\displaystyle D_{N}(E)=Det\pmatrix{{\cal M}_{(N,0)}(E)&{\cal
M}_{(N,1)}(E)&{\cal M}_{(N,2)}(E)\cr{\cal M}_{(N+1,0)}(E)&{\cal
M}_{(N+1,1)}(E)&{\cal M}_{(N+1,2)}(E)\cr{\cal M}_{(N+2,0)}(E)&{\cal
M}_{(N+2,1)}(E)&{\cal M}_{(N+2,2)}(E)}=0,\ for\ N\geq n_{*}+1,$
where ${\cal
M}_{(N+\ell_{1},\ell_{2})}(E)\equiv\sum_{i=0}^{N+\ell_{1}}\Xi_{\sigma;i}^{(N+\ell_{1})}M_{E,\sigma}(i,\ell_{2})$,
$0\leq\ell_{1,2}\leq 2$.
As stated before, the degree of the $D_{N}(E)$ polynomial increases with $N$.
Eq.(30) will be satisfied by all QES states for fixed index $n_{*}$. They will
be the exact roots of Eq.(30) for all $N\geq n_{*}+1$. The other roots
generated from Eq.(30) will approximate, and converge (exponentially fast) to,
the non-QES energies.
Once the QES energies are determined, the corresponding missing moments are
also determined $\\{u_{\sigma}(0),u_{\sigma}(1),u_{\sigma}(2)\\}$, subject to
a convenient normalization (i.e. $u_{\sigma}(0)=1$). The OPPQ expansion
coefficients in Eq.(27) are then obtained through
$\displaystyle\Omega_{j}=$ $\displaystyle\int dx\ x^{\sigma}{{\cal
P}}_{\sigma}^{(j)}(x^{2})\Psi_{\sigma}(x)$ $\displaystyle\Omega_{j}=$
$\displaystyle\sum_{i=0}^{j}\Xi_{\sigma;i}^{(j)}u_{\sigma}(i),$
$\displaystyle\Omega_{j}=$
$\displaystyle\sum_{i=0}^{j}\Xi_{\sigma;i}^{(j)}\big{(}\sum_{\ell=0}^{2}M_{E,\sigma}(i,\ell)u_{\sigma}(\ell)\big{)},\
for\ j\leq n_{*},$ (31)
generating the closed form expression for the wavefunction, as given in
Eq.(26).
The final component is generating the orthonormal polynomials of ${\cal A}$.
Since $\langle{{\cal P}}_{\sigma}^{(j_{1})}|\xi^{\sigma}{{{\cal
A}(\xi)}\over{\sqrt{\xi}}}|{{\cal
P}}_{\sigma}^{(j_{2})}\rangle=\int_{0}^{\infty}d\xi\ {\cal
P}_{\sigma}^{(j_{1})}(\xi){\cal P}_{\sigma}^{(j_{2})}(\xi)\xi^{\sigma-{1\over
2}}{\cal A}(\xi)=\delta_{j_{1},j_{2}}$, the respective orthonormal polynomials
are generated by different weights in the $\xi$-coordinate. We need the power
moments of these different weights,
$m_{\sigma}(\rho)=m(\rho+\sigma)=\int_{0}^{\infty}d\xi\
\xi^{\rho+\sigma}{{{\cal A}(\xi)}\over{\sqrt{\xi}}}$.
Anticipating the needs of the OPPQ-$\Phi$ representation, we define ${\cal
A}(s;x)=exp\big{(}-{{\sqrt{g}}\over s}\big{(}x^{4}+{b\over
g}x^{2}\big{)}\Big{)}$, where $s=4$ for OPPQ-$\Psi$ (i.e. the OPPQ-$\Phi$
works with ${\cal A}^{2}(x)$, thus requiring $s=2$). Since $\partial_{x}{\cal
A}(s;x)=-{{\sqrt{g}}\over s}\big{(}4x^{3}+{{2b}\over{{g}}}x\big{)}{\cal
A}(s;x)$, this generates the moment equation (upon multiplying both sides by
$x^{2\rho+1}$ and integrating by parts):
$\displaystyle
m(\rho+2)={s{(2\rho+1)}\over{4\sqrt{g}}}m(\rho)-{{b}\over{2{g}}}m(\rho+1),\rho\geq
0,$ (32)
where $m(\rho)=\int_{-\infty}^{+\infty}dxx^{2\rho}{\cal A}(s;x)$. We can use
Mathematica to determine the $m(0)$ and $m(1)$ moments in terms of the
modified Bessel function of the second kind $\int dx{\cal A}(x)=({e\over
2})^{1\over 4}K_{1\over 4}({1\over 4})$, and the Bessel function of the first
kind, $\int dxx^{2}{\cal A}(x)=-{\pi\over 2}({e\over 2})^{1\over
4}\Big{(}I_{-{1\over 4}}({1\over 4})-3I_{{1\over 4}}({1\over 4})+I_{{3\over
4}}({1\over 4})-I_{{5\over 4}}({1\over 4})\Big{)}$.
Consider the potential function parameters $g=1$, $b^{2}=8$, then the
potential function parameter becomes $m_{pot}=-(4n_{*}+2\sigma_{*}+1)$. Tables
1 and 2 give the OPPQ analysis for the corresponding QES and non-QES states
for $n_{*}=3$. It will be noted that as soon as $N\geq n_{*}+1$, the QES
states are exactly determined and remain the same constant roots for the
corresponding $D_{N}(E)$ function. The other OPPQ energy roots for $D_{N}(E)$
converge to the non-QES states. We emphasize that the numbers given for the
QES states represent the first six-seven decimal places of the exact energies
with no rounding off. We also give the OPPQ estimate for the non-QES states,
derived from a higher order OPPQ analysis using orthonormal polynomials of
$exp(-{{x^{4}}/4})$ as developed in Ref. [11].
For completenss, Tables 3 and 4 give both QES and non-QES states derived
without working in the explicit parity subspaces. That is, we work with the
$\mu$ moments directly ($m_{s}=5$), generating the corresponding ($6\times 6$)
OPPQ determinantal equation. The $N$ paramter quoted is different from that in
Talbes 1 and 2. For Tables 3 and 4, for the $n_{*}=3$ case, the QES states
have $P_{d}(x)$ with $d=2n_{*}+\sigma_{*}$, hence the exact QES energies
become manifest for $N\geq 7\ or\ 8$, depending on the even or odd states,
respectively.
Table 1: Convergence of OPPQ-$\Psi$ (QES* and non-QES) for the first six (even) energy levels of Eq.(1), $g=1$,$b=\sqrt{8}$, $m=-(4n_{*}+2\sigma_{*}+1)$,$n_{*}=3$, $\sigma_{*}=0$ $N$ | $E_{0}^{*}$ | $E_{2}^{*}$ | $E_{4}^{*}$ | $E_{6}^{*}$ | $E_{8}$ | $E_{10}$
---|---|---|---|---|---|---
$V(x)=gx^{6}+bx^{4}+mx^{2}$, ${\cal A}(x)=exp\big{(}-{{\sqrt{g}}\over
4}\big{(}x^{4}+{b\over g}x^{2}\big{)}\Big{)}$
1 | -3.500501 | | | | |
2 | -6.604075 | 0.507807 | | | |
3 | -4.538891 | 2.361563 | 8.006481 | | |
4 | -4.701631 | 2.289850 | 13.186912 | 28.822848 | |
5 | -4.701631 | 2.289850 | 13.186912 | 28.822848 | 61.179448 |
6 | -4.701631 | 2.289850 | 13.186912 | 28.822848 | 51.599563 | 102.816240
7 | -4.701631 | 2.289850 | 13.186912 | 28.822848 | 48.712815 | 82.421165
8 | -4.701631 | 2.289850 | 13.186912 | 28.822848 | 47.857837 | 74.249292
9 | -4.701631 | 2.289850 | 13.186912 | 28.822848 | 47.652156 | 70.817047
10 | -4.701631 | 2.289850 | 13.186912 | 28.822848 | 47.616909 | 69.545484
11 | -4.701631 | 2.289850 | 13.186912 | 28.822848 | 47.614022 | 69.227821
12 | -4.701631 | 2.289850 | 13.186912 | 28.822848 | 47.613850 | 69.232777
$\infty$ | | | | | 47.613209 | 69.043247
Table 2: Convergence of OPPQ-$\Psi$ (QES* and non-QES) for the first six (odd) energy levels of Eq.(1), $g=1$,$b=\sqrt{8}$, $m=-(4n_{*}+2\sigma_{*}+1)$,$n_{*}=3$, $\sigma_{*}=1$ $N$ | $E_{1}^{*}$ | $E_{3}^{*}$ | $E_{5}^{*}$ | $E_{7}^{*}$ | $E_{9}$ | $E_{11}$
---|---|---|---|---|---|---
$V(x)=gx^{6}+bx^{4}+mx^{2}$, ${\cal A}(x)=exp\big{(}-{{\sqrt{g}}\over
4}\big{(}x^{4}+{b\over g}x^{2}\big{)}\Big{)}$
1 | -8.086559 | | | | |
2 | -7.931590 | -0.067843 | | | |
3 | -6.466044 | 5.685330 | 10.297850 | | |
4 | –6.629227 | 4.618850 | 18.024593 | 34.897472 | |
5 | -6.629227 | 4.618850 | 18.024593 | 34.897472 | 70.431224 |
6 | -6.629227 | 4.618850 | 18.024593 | 34.897472 | 59.527051 | 114.774703
7 | -6.629227 | 4.618850 | 18.024593 | 34.897472 | 56.100032 | 92.408733
8 | -6.629227 | 4.618850 | 18.024593 | 34.897472 | 55.025448 | 83.209865
9 | -6.629227 | 4.618850 | 18.024593 | 34.897472 | 54.745576 | 79.194992
10 | -6.629227 | 4.618850 | 18.024593 | 34.897472 | 54.689889 | 77.548343
$\infty$ | | | | | 54.686459 | 76.977398
Table 3: Unified OPPQ-$\Psi$ analysis within $\mu$ representation for
$\sigma_{*}=0,n_{*}=3$, $b=\sqrt{8}$ and $m=-13$. QES states $E_{0}$, $E_{2}$,
$E_{4}$ and $E_{6}$ are “exact”, while the other states converge fast.
N | $E_{0}^{*}$ | $E_{1}$ | $E_{2}^{*}$ | $E_{3}$ | $E_{4}^{*}$ | $E_{5}$ | $E_{6}^{*}$ | $E_{7}$
---|---|---|---|---|---|---|---|---
1 | -3.50050190677 | | | | | | |
2 | -6.03169311724 | -3.50050190677 | | | | | |
3 | -6.60407548295 | -6.03169311724 | 0.507807963155 | | | | |
4 | -6.60407548295 | -4.72113209085 | 0.507807963155 | 2.25709085340 | | | |
5 | -4.72113209085 | -4.53889182150 | 2.25709085340 | 2.36156311823 | 8.00648129616 | | |
6 | -4.53889182150 | -4.21887392229 | 2.36156311823 | 7.10373221895 | 8.00648129616 | 16.7229518999 | |
7 | -4.70163122288 | -4.21887392229 | 2.28985002468 | 7.10373221895 | 13.1869125971 | 16.7229518999 | 28.8228483475 |
8 | -4.70163122288 | -4.25720912289 | 2.28985002468 | 6.71439942230 | 13.1869125971 | 20.2497283402 | 28.8228483475 | 43.7475563324
9 | -4.70163122288 | -4.25720912289 | 2.28985002468 | 6.71439942230 | 13.1869125971 | 20.2497283402 | 28.8228483475 | 43.7475563324
10 | -4.70163122288 | -4.25800570649 | 2.28985002468 | 6.71431004595 | 13.1869125971 | 20.5352883528 | 28.8228483475 | 39.2137668519
11 | -4.70163122288 | -4.25800570649 | 2.28985002468 | 6.71431004595 | 13.1869125971 | 20.5352883528 | 28.8228483475 | 39.2137668519
12 | -4.70163122288 | -4.25801075980 | 2.28985002468 | 6.71503370668 | 13.1869125971 | 20.5608997790 | 28.8228483475 | 38.1497221283
13 | -4.70163122288 | -4.25801075980 | 2.28985002468 | 6.71503370668 | 13.1869125971 | 20.5608997790 | 28.8228483475 | 38.1497221283
14 | -4.70163122288 | -4.25800781019 | 2.28985002468 | 6.71512397419 | 13.1869125971 | 20.5622044115 | 28.8228483475 | 37.9091389572
15 | -4.70163122288 | -4.25800781019 | 2.28985002468 | 6.71512397419 | 13.1869125971 | 20.5622044115 | 28.8228483475 | 37.9091389572
16 | -4.70163122288 | -4.25800743707 | 2.28985002468 | 6.71512989894 | 13.1869125971 | 20.5620795974 | 28.8228483475 | 37.8672556901
17 | -4.70163122288 | -4.25800743707 | 2.28985002468 | 6.71512989894 | 13.1869125971 | 20.5620795974 | 28.8228483475 | 37.8672556901
18 | -4.70163122288 | -4.25800741621 | 2.28985002468 | 6.71512975900 | 13.1869125971 | 20.5620354769 | 28.8228483475 | 37.8626239770
19 | -4.70163122288 | -4.25800741621 | 2.28985002468 | 6.71512975900 | 13.1869125971 | 20.5620354769 | 28.8228483475 | 37.8626239770
Table 4: Unified OPPQ-$\Psi$ analysis within $\mu$ representation for
$\sigma_{*}=1,n_{*}=3$,$b=\sqrt{8}$ and $m=-15$. QES states $E_{1}$, $E_{3}$,
$E_{5}$ and $E_{7}$ are “exact”, while the other states converge fast.
N | $E_{0}$ | $E_{1}^{*}$ | $E_{2}$ | $E_{3}^{*}$ | $E_{4}$ | $E_{5}^{*}$ | $E_{6}$ | $E_{7}^{*}$ | $E_{8}$
---|---|---|---|---|---|---|---|---|---
1 | -4.31962115162 | | | | | | | |
2 | -8.08655987811 | -4.31962115162 | | | | | | |
3 | -9.55087356079 | -8.08655987811 | -0.190781182520 | | | | | |
4 | -9.55087356079 | -7.93159013829 | -0.190781182520 | -0.0678433862236 | | | | |
5 | -7.93159013829 | -6.51760411099 | -0.0678433862236 | 0.0554777594537 | 4.59925352308 | | | |
6 | -6.51760411099 | -6.46604495681 | 0.0554777594537 | 4.59925352308 | 5.68533008405 | 10.2978504560 | | |
7 | -6.84097818474 | -6.46604495681 | 1.07749798410 | 5.68533008405 | 10.2978504560 | 11.7267374340 | 20.9221258143 | |
8 | -6.84097818474 | -6.62922791805 | 1.07749798410 | 4.61885026929 | 11.7267374340 | 18.0245932316 | 20.9221258143 | 34.8974726626 |
9 | -6.84901298988 | -6.62922791805 | 1.01868143168 | 4.61885026929 | 10.9341820352 | 18.0245932316 | 25.6172237107 | 34.8974726626 | 51.4874448871
10 | -6.84901298988 | -6.62922791805 | 1.01868143168 | 4.61885026929 | 10.9341820352 | 18.0245932316 | 25.6172237107 | 34.8974726626 | 51.4874448871
11 | -6.84939579745 | -6.62922791805 | 1.01724028394 | 4.61885026929 | 10.9283350058 | 18.0245932316 | 26.0186964925 | 34.8974726626 | 46.1617846405
12 | -6.84939579745 | -6.62922791805 | 1.01724028394 | 4.61885026929 | 10.9283350058 | 18.0245932316 | 26.0186964925 | 34.8974726626 | 46.1617846405
13 | -6.84941133696 | -6.62922791805 | 1.01721575801 | 4.61885026929 | 10.9291565335 | 18.0245932316 | 26.0597394744 | 34.8974726626 | 44.8467290999
14 | -6.84941133696 | -6.62922791805 | 1.01721575801 | 4.61885026929 | 10.9291565335 | 18.0245932316 | 26.0597394744 | 34.8974726626 | 44.8467290999
15 | -6.84941128633 | -6.62922791805 | 1.01721996310 | 4.61885026929 | 10.9292994586 | 18.0245932316 | 26.0625827763 | 34.8974726626 | 44.5272878301
16 | -6.84941128633 | -6.62922791805 | 1.01721996310 | 4.61885026929 | 10.9292994586 | 18.0245932316 | 26.0625827763 | 34.8974726626 | 44.5272878301
17 | -6.84941118433 | -6.62922791805 | 1.01722065544 | 4.61885026929 | 10.9293121992 | 18.0245932316 | 26.0625117133 | 34.8974726626 | 44.4658799515
18 | -6.84941118433 | -6.62922791805 | 1.01722065544 | 4.61885026929 | 10.9293121992 | 18.0245932316 | 26.0625117133 | 34.8974726626 | 44.4658799515
19 | -6.84941117246 | -6.62922791805 | 1.01722070755 | 4.61885026929 | 10.9293124794 | 18.0245932316 | 26.0624501552 | 34.8974726626 | 44.4579163980
## 4 The $\Phi$ Representation : An $m_{s}=0$ Perspective on the Bender-Dunne
Polynomials
### 4.1 Transformation of the sextic anharmonic oscillator to an $m_{s}=0$
moment equation representation
In general, if ${\cal A}(x)>0$ is the leading, positive, bounded, asymptotic
form for the discrete wavefunction, and it is known in closed form, then if
$\Phi(x)={\cal A}(x)\Psi(x)$ admits a moment equation, it will have an order
($m_{s})$ less than that in the $\Psi$ representation. For the sextic problem,
this corresponds to
$\displaystyle\Phi(x)=exp\Big{(}-{\sqrt{g}\over 4}\big{(}{{x^{4}}}+{b\over
g}{{x^{2}}}\big{)}\Big{)}\Psi(x),$
whose differential equation becomes
$\displaystyle-\partial_{x}^{2}\Phi-\big{(}{b\over{\sqrt{g}}}x+2\sqrt{g}x^{3}\big{)}\partial_{x}\Phi(x)$
$\displaystyle+\
\Big{(}(m-3\sqrt{g}-{{b^{2}}\over{4g}})x^{2}-(E+{b\over{2\sqrt{g}}})\Big{)}\Phi(x)=0.$
(34)
Let us now assume that $\Phi(x)$ is the exponentially decaying configuration
for a particular discrete state. Since it has to be continuously
differentiable, we can multiply both sides of Eq.(34) by $x^{p}$and integrate
by parts over the entire real axis. Define the power moments $\nu(p)=\int dx\
x^{p}\Phi(x)$, $p\geq 0$. These moments satisfy the moment equation:
$\displaystyle\Big{(}m+3\sqrt{g}-{{b^{2}}\over{4g}}+2\sqrt{g}p)\Big{)}\nu(p+2)=$
$\displaystyle(E-{b\over{\sqrt{g}}}(p+{1\over 2}))\nu(p)+p(p-1)\nu(p-2),\
p\geq 0.$ (35)
Given that the physical system admits parity invariant states, the moment
equation decouples into the corresponding even and odd order power moments.
Define $v_{\sigma}(\rho)=\nu(2\rho+\sigma)$, $\sigma=0,1$, corresponding to
the even and odd states, respectively. The corresponding moment equations
becomes:
$\displaystyle\Big{(}m+3\sqrt{g}-{{b^{2}}\over{4g}}+4\sqrt{g}\rho)\Big{)}v_{e}(\rho+1)=$
$\displaystyle(E-{b\over{\sqrt{g}}}(2\rho+{1\over
2}))v_{e}(\rho)+2\rho(2\rho-1)v_{e}(\rho-1),\ \rho\geq 0;$ (36)
and
$\displaystyle\Big{(}m+3\sqrt{g}-{{b^{2}}\over{4g}}+2\sqrt{g}(2\rho+1))\Big{)}v_{o}(\rho+1)=$
$\displaystyle(E-{b\over{\sqrt{g}}}(2p+{3\over
2}))v_{o}(\rho)+2\rho(2\rho+1)v_{o}(\rho-1),\ \rho\geq 0.$ (37)
We can express the above more compactly as
$\displaystyle\Big{(}m-{{b^{2}}\over{4g}}+\sqrt{g}(4\rho+3+2\sigma)\Big{)}v_{\sigma}(\rho+1)=$
$\displaystyle(E-{b\over{\sqrt{g}}}(2\rho+{1\over
2}+\sigma))v_{\sigma}(\rho)+2\rho(2\rho-1+2\sigma)v_{\sigma}(\rho-1),$
$\rho\geq 0$, and $\sigma=0,1$.
The above moments correspond to different Stieltjes measures. Specifically,
$\displaystyle v_{\sigma}(\rho)$
$\displaystyle=\nu(2\rho+\sigma)=\int_{-\infty}^{+\infty}dx\
x^{2\rho+\sigma}\Phi_{\sigma}(x)$ $\displaystyle v_{\sigma}(\rho)$
$\displaystyle=\int_{0}^{\infty}d\xi\ \xi^{\rho}\phi_{\sigma}(\xi),$ (39)
where $\
\phi_{\sigma}(\xi)\equiv{{\Phi_{\sigma}(\xi)}\over{({{\sqrt{\xi}}})^{1-\sigma}}}$,
$\xi\equiv x^{2}$, and $\Phi_{\sigma}(x)={\cal A}(x)\Psi_{\sigma}(x)$.
The QES-States
The moment equations in Eq.(35-38) are implicitly only valid for the physical
states. In general, except for the special QES states, they are of missing
moment order $m_{s}=0$ since if $v_{\sigma}(0)\neq 0$ (which is the case for
the ground and first excited states within EMM), all the higher order moments
are generated and become polynomials in the energy:
$\displaystyle v_{\sigma}(\rho)=Polynomial\ of\ degree\ \rho\ in\ the\
energy,\ E;$ $\displaystyle
v_{\sigma}(\rho)\equiv\Lambda_{\sigma}^{(\rho)}(E).$ (40)
We see that if the coefficient of the $v_{\sigma}(\rho+1)$ term in Eq.(38) is
never zero, for any integer $\rho$ and $\sigma$ value, then an infinite number
of such polynomials are generated.
If this coefficient is zero for some $\rho=n_{*}$ and $\sigma_{*}=0,1$, then
the potential function parameters are constrained to
$m-{{b^{2}}\over{4g}}+\sqrt{g}(4n_{*}+3+2\sigma_{*})=0,$ (41)
allowing only the first $n_{*}+1$ moments to be generated,
$\\{v_{\sigma_{*}}(\rho)|0\leq\rho\leq n_{*}\\}$. Eq.(41) is the QES parameter
condition in Eq.(2). We stress that if Eq.(41) is satisfied by the potential
function parameters, the states of opposite parity to the QES states,
$\sigma\neq\sigma_{*}$, will satisfy the corresponding version of Eq.(38), and
generate all the moments as polynomials in the energy
$\\{v_{\sigma}(\rho)|\rho\geq 0\\}$.
Define the $n+1$ degree polynomial
$\displaystyle{P}_{\sigma}^{(n+1)}(E)=(E-{b\over{\sqrt{g}}}(2n+{1\over
2}+\sigma))\Lambda_{\sigma}^{(n)}(E)+2n(2n-1+2\sigma)\Lambda_{\sigma}^{(n-1)}(E).$
Within the EMM framework, Handy and Bessis [14] realized that if the QES
parameter conditions in Eq.(41) are satisifed, then the fact that the ground
and first excited states must have nonzero, zeroth-order moments,
$v_{\sigma_{*}}(0)\neq 0$, makes them the roots of the respective polynomial
${P}_{\sigma_{*}}^{(n_{*}+1)}(E)=0.$ (43)
If any other excited state has its zeroth order moment also nonzero, then it
too must be a root of the above polynomial. The question is, will this
property also hold for all the first $n_{*}$ excited states? The answer is
yes. The proof, known to HB, is given below. That is, all the $n_{*}+1$ roots
of this polynomial correspond to the QES states.
### 4.2 Moments’ Proof that all the roots of
$P_{\sigma^{*}}^{(n_{*}+1)}(E)=0$, correspond to the QES states
We assume that the potential function parameters satisfy the constraint in
Eq.(41). Within the EMM framework, $P_{\sigma^{*}}^{(n_{*}+1)}(E)=0$ yields
the QES energy root corresponding to the lowest energy within the $\sigma^{*}$
(even/odd parity) symmetry class, since the corresponding zeroth order moment
is non-zero, $v_{\sigma^{*}}(0)\neq 0$, due to the underlying positivity (if
$\sigma_{*}=0$) or nonnegativity (if $\sigma_{*}=1$), for the ground or first
excited state Stieltjes measure (Eq.(4)), respectively. The other higher
energy states (i.e. second, third, etc.) in the $\sigma_{*}$\- symmetry class
must satisfy Eq.(38) for the moments $\\{v_{\sigma^{*}}(\rho)|0\leq\rho\leq
n_{*}\\}$. Given that this is an $m_{s}=0$ moment equation, there are only two
possibilities for any such excited state: $v_{\sigma^{*}}(0)=0$, or
$v_{\sigma^{*}}(0)\neq 0$. If the second option holds, then that energy must
be a root of the $n_{*}+1$ degree polynomial given in Eq.(43). Therefore, we
focus on the first option, which although a real possibility, will be shown
not to hold, for any of the first $n_{*}+1$ states (i.e. the QES states) . In
fact, there are two proofs for this. We give the original (unpublished)
analysis, followed by the proof based on assuming the states have the OPPQ/QES
form discussed previously: $\Psi(x)=P_{d}(x){\cal A}(x)$ or
$\Phi(x)=P_{d}(x){\cal A}^{2}(x)$.
If we assume that a particular excited state has $v_{\sigma_{*}}(0)=0$, then
the moment equation tells us that $v_{\sigma^{*}}(\rho)=0$, for $0\leq\rho\leq
n_{*}$ ,although not necessarily for $v_{\sigma_{*}}(n_{*}+1)$, since it is
not generated by the moment equation.
EMM-Moment Equation Proof
The Sturm-Liouville character of the sextic problem tells us that within the
symmetry class corresponding to $\sigma^{*}$, all states are uniquely
characterized by the number of nodes along the positive real axis,
$\xi=x^{2}>0$. The ground state has no nodes at all. The first excited state
has no nodes along the positive axis (its only node is at the origin). The
next higher energy state, within the even parity or odd parity states, will
have one node along the positive axis, etc. Let us denote the first $n_{*}+1$
states within the $\sigma^{*}$ symmetry class ( ordered in terms of energy or
number of nodes on the positive real axis) by $\phi_{\sigma^{*},j}(\xi)$,
$0\leq j\leq n_{*}$. The lowest energy state ($j=0$) is either the ground or
first excited state, depending on $\sigma^{*}$. Let
$\\{r_{\sigma^{*},j;i}|1\leq i\leq j\\}$ denote the nodes along the positive
$\xi$ \- real line, for the $j\geq 1$ state; therefore the configuration
$\pi_{\sigma^{*},j}(\xi)=\phi_{\sigma^{*},j}(\xi)\Pi_{i=1}^{j}(\xi-
r_{\sigma^{*},j;i})$ must be nonnegative (i.e. can be chosen as such).
However, this means that its power moments must be positive:
$\int_{0}^{\infty}d\xi\ \xi^{\rho}\pi_{\sigma^{*},j}(\xi)>0$. In particular,
the zeroth moment is the linear superposition of all the first $1+j$ moments
of $\phi_{\sigma^{*},j}(\xi)$ (i.e.
$\\{v_{\sigma^{*}}(0),\ldots,v_{\sigma^{*}}(j)\\}$). However, our starting
assumption is that all of these are zero, provided $j\leq n_{*}$. This is a
contradiction, so Eq.(43) is the quantization condition for all the first
$n_{*}+1$, QES states.
OPPQ-QES Proof
From OPPQ-$\Psi$ we argued that the QES states must have the form
$\Psi(x)=P_{d}(x){\cal A}(x)$, or
$\Phi_{\sigma_{*}}(x)=x^{\sigma_{*}}P_{n_{*}}(x^{2}){\cal A}^{2}(x)$. Let
${\cal O}^{(j)}_{\sigma_{*}}(x^{2})$ denote the orthonormal polynomials of the
respective even weights $x^{2\sigma_{*}}{\cal A}^{2}(x)$. We therefore have
$\int dx{\cal
O}^{(n_{*}+q)}_{\sigma_{*}}(x^{2})x^{\sigma_{*}}\Phi_{\sigma_{*}}(x)=0$, for
$q\geq 1$. However, these integrals correspond to a linear sum of the power
moments $\\{v_{\sigma_{*}}(0),\ldots,v_{\sigma_{*}}(n_{*}+q)\\}$. If all
$v_{\sigma_{*}}(\rho)=0$, for $\rho\leq n_{*}$, then so too must
$v_{\sigma_{*}}(n_{*}+1)$, and thereby all the higher order moments. This
essentially would imply that $\Phi_{\sigma_{*}}(x)=0$, a contradiction.
We note that both proofs rely on the existence of an $m_{s}=0$ moment equation
for the first $n_{*}+1$ moments. Neither makes use of the moment equation for
the moment of order higher than $n_{*}$.
The non-QES states of the Same Parity as the QES-State, must have
$v_{\sigma_{*}}(\rho)=0$, for $0\leq\rho\leq n_{*}$: $\Phi_{\sigma_{*}}^{(Non-
QES)}(x)=\partial_{x}^{2n_{*}+2+\sigma_{*}}\Upsilon(x)$
This is immediate. Since the sextic anharmonic potential is unbounded from
above, there are an infinite number of bound states of either parity. If the
potential function parameters satisfy Eq.(41), for some
$\\{\sigma_{*},n_{*}\\}$ pair, then only a finite number of the discrete
states correspond to the QES states, as determined by the $n_{*}+1$ roots of
the energy polynomial in Eq.(43). There are, therefore, an infinite number of
non-QES states of the same parity as the coresponding QES states,
$\sigma=\sigma_{*}$. These must satisfy the same moment equation as the QES
states.
Only the QES states can have $v_{\sigma_{*}}(0)\neq 0$ because this then means
that their energies are determined by Eq.(43). Therefore, the non-QES states
of the same parity as the QES states must have $v_{\sigma_{*}}(0)=0$, which
means all the first $n_{*}+1$ moments are zero. From a simple Fourier
analysis, one concludes that since $\nu(p)=\int
dxx^{p}\Phi_{\sigma_{*}}^{(non-QES)}(x)=0$, for $0\leq p\leq
2n_{*}+\sigma_{*}$ then $\Phi_{\sigma_{*}}^{(non-
QES)}(x)=\partial_{x}^{2n_{*}+2+\sigma_{*}}\Upsilon(x)$, where $\Upsilon$ has
the same parity as $\Phi_{\sigma_{*}}^{(non-QES)}$. This proves our claim, for
the sextic anharmonic oscillator case. The same result is true for the Bender
Dunne potential, with respects to the first $n_{*}+1$ moments being zero.
However, the implications for the form of the corresponding $\Phi(x)$
configuration is complicated by the singular (indicial factor) required.
Overview of the Moment Equation Structure for the QES and non-QES States
If the potential function parameters satisfy Eq.(41), we will refer to this as
“V(x) is of QES type”. Unless otherwize indicated, the following discussion
pertains to this case. The corresponding moment equation for the $\sigma_{*}$
parity states (QES or non-QES) will decouple the $v_{\sigma_{*}}(n_{*}+1)$
moment from the lower order moments. For the QES states, the first $n_{*}+1$
moments define an $m_{s}=0$ moment equation. From Eq.(38), taking
$\rho=n_{*}+1$, we see that the $\nu_{\sigma_{*}}(n_{*}+2)$ moment couples to
$\\{\nu_{\sigma_{*}}(n_{*}+1),\nu_{\sigma_{*}}(n_{*})\\}$, where the
$\nu_{\sigma_{*}}(n_{*})$ moment, in turn, is determined by the zeroth moment
$\nu_{\sigma_{*}}(0)$. Thus the QES states moments’
$\\{\nu_{\sigma_{*}}(\rho)|\rho\geq n_{*}+2\\}$ couple, linearly, to the
moments $\\{\nu_{\sigma_{*}}(n_{*}+1),\nu_{\sigma_{*}}(0)\\}$.
In summary, the first $n_{*}+1$ moments, for the QES states, satisfy an
$m_{s}=0$ moment equation. The higher order moments will satisfy an effective
$m_{s}=1$ moment equation.
We do not have to use the roots of the energy polynomial in Eq.(43), to
determine the QES energies. We can apply OPPQ to the
$\\{\nu_{\sigma_{*}}(\rho)|\rho\geq n_{*}+2\\}$ moments ( an $m_{s}=1$ moment
equation). It will generate the exact QES energies for $N$ above a certain
threshold. This is detailed in Sec. V. This same OPPQ analysis will also
generate many more roots to the OPPQ determinant. These will be (converging)
approximants to the non-QES energies, in the $N\rightarrow\infty$ limit. One
can verify that these OPPQ solutions correspond to solutions for which the
zeroth order moment vanishes asymptotically
($\lim_{N\rightarrow\infty}v_{\sigma_{*}}(0)=0$) corresponding to the non-QES
states. This is also discussed in Sec. V.
Continuing with the case of “V(x) of QES type”, the non-QES states of the same
symmetry as the QES states must satisfy the same moment equation. From
Eq.(38), if $\rho=n_{*}+1$, then $v_{\sigma_{*}}(n_{*}+2)$ is determined by
$v_{\sigma_{*}}(n_{*}+1)$ since $v_{\sigma_{*}}(n_{*})=0$, for these non-QES
states. In other words, the non-QES states of the same symmetry as the QES
states, satisfy an $m_{s}=0$ missing moment relation, with respect to the
moments of order $n_{*}+1$ and higher. They are all linearly dependent on
$v_{\sigma_{*}}(n_{*}+1)$. Tables 5 and 6 uses OPPQ on the higher order
moments to compute the non-QES states, of the same parity as the QES states,
assuming $\\{v_{\sigma_{*}}(\rho)=0|0\leq\rho\leq n_{*}\\}$. The details of
this analysis are also given in Sec. V.
If V(x) is of “QES type”, then there will be non-QES states of opposite parity
to the QES states. In this case, Eq.(38) is a full $m_{s}=0$ moment equation,
for all moments $\\{v_{\sigma}(\rho)|\rho\geq 0\\}$.
If V(x) is not of “QES type”, then all states satisfy Eq.(38) which is, again,
an $m_{s}=0$, moment equation.
We summarize all the above in Table 7.
Table 5: OPPQ-$\Phi$ determination of non-QES states (of $\sigma_{*}$
symmetry) computed by taking $v_{\sigma_{*}}(\rho)=0$, $0\leq\rho\leq n_{*}$,
and $\\{v_{\sigma_{*}}(\rho)|\rho\geq n_{*}+1\\}$ satisfy an effective
$m_{s}=0$ moment equation. Refer to Eq.(38).
$N$ $E_{8}$ $E_{10}$ $E_{12}$ $E_{14}$ $V(x)=x^{6}+\sqrt{8}x^{4}-13x^{2}$,
$n_{*}=3,\sigma_{*}=0$ 5 48.394656 6 47.671288 72.503581 7 47.617135 69.537633
101.036761 8 47.613461 69.101670 94.571123 133.727067 9 47.613226 69.049273
93.118330 122.972492 10 47.613211 69.044300 92.864698 119.986303 11 47.613211
69.044199 92.856805 119.850391
Table 6: OPPQ-$\Phi$ determination of non-QES states (of $\sigma_{*}$
symmetry) computed by taking $v_{\sigma_{*}}(\rho)=0$, $0\leq\rho\leq n_{*}$,
and $\\{v_{\sigma_{*}}(\rho)|\rho\geq n_{*}+1\\}$ satisfy an effective
$m_{s}=0$ moment equation. Refer to Eq.(38).
$N$ $E_{9}$ $E_{11}$ $E_{13}$ $E_{15}$ $V(x)=x^{6}+\sqrt{8}x^{4}-15x^{2}$,
$n_{*}=3,\sigma_{*}=1$ 5 55.531291 6 54.750410 80.668061 7 54.690840 77.512874
110.194217 8 54.686737 77.041110 103.388049 143.814626 9 54.686468 76.982602
101.807743 132.398420 10 54.686447 76.974990 101.398734 127.471438
Table 7: Moment Equation Structure for Sextic Anharmonic Potential for V(x) =
“QES Type” (QES) or “not of QES Type” (N-QES); $\Phi_{\sigma}(x)=$ QES or
N-QES. Refer to Eq.(38).
$V(x)$ $\Phi_{\sigma}$ $\Phi_{\sigma}(x)$-type $v_{\sigma}(\rho)$ $m_{s}$
$v_{\sigma}(\rho)$ $m_{s}$ N-QES $\sigma=0,1$ N-QES
$\\{v_{\sigma}(\rho)|\rho\geq 0\\}$ 0 QES $\sigma\neq\sigma_{*}$ N-QES
$\\{v_{\sigma}(\rho)|\rho\geq 0\\}$ 0 QES $\sigma=\sigma_{*}$ N-QES
$\\{v_{\sigma_{*}}(\rho)|\rho\geq n_{*}+1\\}^{a}$ 0
$\\{v_{\sigma_{*}}(\rho)=0|\rho\leq n_{*}\\}^{a}$ QES $\sigma=\sigma_{*}$ QES
$\\{v_{\sigma_{*}}(\rho)|\rho\geq n_{*}+1\\}^{b}$ 1
$\\{v_{\sigma_{*}}(\rho)\neq 0|\rho\leq n_{*}\\}^{b}$ 0
aApplication of OPPQ recovers non-QES energies given in Tables 5 and 6.
b Application of OPPQ (instead of Eq.(43)) gives exact QES energies, $N\geq
d+1$; and the non-QES energies in the $N\rightarrow\infty$ limit.
### 4.3 Defining the 3-Term Recursive Relation for the Bender-Dunne Energy-
Polynomials
For the $m_{s}=0$ cases indicated in Table 7, the indicated $v_{\sigma}(\rho)$
moments are polynomials in the energy and satisfy a three term recursion. One
can readily transform all of these cases into the Bender-Dunne (monic)
polynomials, although the more interesting case is for the QES states, for
comparison purposes within our formulation.
For future reference, we define the coefficient functions in Eq.(38):
$\displaystyle C_{\sigma;1}(m,b,g;\rho+1)$ $\displaystyle=$ $\displaystyle
m-{{b^{2}}\over{4g}}+\sqrt{g}(4\rho+3+2\sigma),$ $\displaystyle
C_{\sigma;0}(E,b,g;\rho)$ $\displaystyle=$ $\displaystyle
E-{b\over{\sqrt{g}}}(2\rho+{1\over 2}+\sigma),$ $\displaystyle
C_{\sigma:-1}(\rho-1)$ $\displaystyle=$ $\displaystyle
2\rho(2\rho-1+2\sigma).$
No potential function parameters can lead to $C_{\sigma;1}=0$ for two sets of
$(\rho,\sigma)$ values. To do so would require $(\rho_{2}-\rho_{1})=-{1\over
2}(\sigma_{2}-\sigma_{1})$, which is impossible for integer differences.
Define the (non-monic) polynomials.
$\displaystyle P_{\sigma}^{(\rho)}(E)\equiv C_{\sigma;1}(m,b,g;\rho)\
v_{\sigma}(\rho).$ (45)
If the potential function parameters satisfy $C_{\sigma;1}(m,b,g;\rho)\neq 0$,
for all $\rho$’s and $\sigma$’s, then these will satisfy the three term
recursion relation:
$\displaystyle P_{\sigma}^{(\rho+1)}(E)=$
$\displaystyle{{(E-{b\over{\sqrt{g}}}(2\rho+{1\over
2}+\sigma))}\over{C_{\sigma;1}(m,b,g;\rho)}}P_{\sigma}^{(\rho)}(E)$ (46)
$\displaystyle+{{2\rho(2\rho-1+2\sigma)}\over{C_{\sigma;1}(m,b,g;\rho-1)}}P_{\sigma}^{(\rho-1)}(E),$
for $0\leq\rho<\infty$. If the potential function parameters satisfy the QES
conditions for a particular $(n_{*},\sigma_{*})$ pair, then Eq.(46) is valid
only for $0\leq\rho\leq n_{*}$. Based on choosing $v_{\sigma}(0)=1$, the
corresponding zeroth order polynomial becomes
$P_{\sigma}^{(0)}(E)=C_{\sigma;1}(m,b,g;0)$. We can always choose
$v_{\sigma}(0)$ to give us the desired normalization for
$P_{\sigma}^{(0)}(E)$.
The three term recursion relation in Eq.(46) does not correspond to a three
term relation for monic (orthogonal ) polynomials. To do so requires the
modifications discussed in the context of Eq.(18). Specifically, let
${\tilde{P}}_{\sigma}^{(\rho)}=f_{\rho}P_{\sigma}^{(\rho)}$ denote the monic
form. Define $\beta_{\rho}\equiv{{f_{\rho+1}}\over f_{\rho}}$ and
$f_{\rho+1}=C_{\sigma;1}(m,b,g;\rho)f_{\rho}$. Let
${\tilde{\alpha}}_{\rho+1}={b\over{\sqrt{g}}}(2\rho+{1\over 2}+\sigma)$, and
${\tilde{\gamma}}_{\rho}=-2\rho(2\rho-1+2\sigma)\beta_{\rho}$. Then the monic
form of Eq.(46) becomes:
${\tilde{P}}^{(\rho+1)}_{\sigma}(E)=(E-{\tilde{\alpha}}_{\rho+1}){\tilde{P}^{(\rho)}}_{\sigma}(E)-{\tilde{\gamma}}_{\rho}{\tilde{P}}^{(\rho-1)}_{\sigma}(E).$
(47)
Since
$f_{n_{*}+1}=\big{(}\Pi_{i=0}^{n_{*}}C_{\sigma_{*};1}(m,b,g;i)\big{)}f_{0}$,
and the QES parameter conditions correspond to
$C_{\sigma_{*};1}(m,b,g;n_{*}+1)=0$, we see that the above monic form is valid
for the QES states. Also, ${\tilde{\gamma}}_{n_{*}+1}=0$ which tells us that
$\langle{\tilde{P}}^{(n_{*}+1)}_{\sigma_{*}}|{\tilde{P}}^{(n_{*}+1)}_{\sigma_{*}}\rangle=0$.
## 5 Quantization of QES and Non-QES States via OPPQ-$\Phi$
Whereas the OPPQ-$\Psi$ formulation involved an $m_{s}=2$ moment equation, its
structure does not change regardless of the QES or non-QES character of the
solution. That is, the same computational (numerical and algebraic) procedure
generates either type of state. However, in the present OPPQ-$\Phi$
formulation, the varying nature of the missing moment order, $m_{s}$, as given
in Table 7, results in various OPPQ representations, as detailed below.
Let $V_{sa}(x)=gx^{6}+bx^{4}+mx^{2}$ be the potential function. In the first
two cases given in Table 7, ($V_{sa}$ of non-QES type, or
$\sigma\neq\sigma_{*}$ non-QES solutions for $V_{sa}$ of QES type) the moment
equation is of uniform $m_{s}=0$ order for $\\{v_{\sigma}(\rho)|\rho\geq
0\\}$. The resulting OPPQ determinant is $1\times 1$, corresponding to a pure
energy polynomial whose roots generate all the discrete state energies in the
$N\rightarrow\infty$ limits.
If $V_{sa}(x)$ is of QES type, then for the QES parity class,
$\sigma=\sigma_{*}$, both non-QES and QES states have the same moment
equation. However, for the non-QES states, all moments of order no greater
than $n_{*}$ will be zero, $v_{\sigma_{*}}(\rho)=0$, if $\rho\leq n_{*}$. The
remaining moments, $\\{v_{\sigma_{*}}(\rho)|\rho\geq n_{*}+2\\}$, linearly
depend on $v_{\sigma_{*}}(n_{*}+1)$. This is also an effective $m_{s}=0$
relation; and the OPPQ determinant is a $1\times 1$ energy dependent
polynomial. Application of OPPQ yields the non-QES energies, as shown in
Tables 5 and 6.
For the QES states, the moments $\\{v_{\sigma_{*}}(\rho)|\rho\geq n_{*}+2\\}$
become linearly dependent on
$\\{v_{\sigma_{*}}(n_{*}+1),v_{\sigma_{*}}(n_{*})\\}$ defining an $m_{s}=1$
moment equation; however, since $v_{\sigma_{*}}(n_{*})$ is linearly dependent
on $v_{\sigma_{*}}(0)\neq 0$, the $\\{v_{\sigma_{*}}(\rho)|\rho\geq
n_{*}+2\\}$ become linearly dependent on
$\\{v_{\sigma_{*}}(n_{*}+1),v_{\sigma_{*}}(0)\\}$. We summarize the moment
equation structure in Table 8.
Table 8: Missing Moment Structure for Sextic Anharmonic Potential for V(x) =
“QES Type” (QES) or “not of QES Type” (N-QES); $\Phi_{\sigma}(x)=$ QES or
N-QES. Refer to Eq.(38).
$V(x)$ $\Phi_{\sigma}$ $\Phi_{\sigma}(x)$-type $v_{\sigma}(\rho)$
$\rho\in[a,b]$ N-QES $\sigma=0,1$ N-QES
$v_{\sigma}(\rho)=M_{E,\sigma}(\rho,0)v_{\sigma}(0)$ $[0,\infty)$ QES
$\sigma\neq\sigma_{*}$ N-QES
$v_{\sigma}(\rho)=M_{E,\sigma}(\rho,0)v_{\sigma}(0)$ $[0,\infty)$ QES
$\sigma=\sigma_{*}$ N-QES $v_{\sigma_{*}}(\rho)=0$ $[0,n_{*}]$ QES
$\sigma=\sigma_{*}$ N-QES
$v_{\sigma_{*}}(\rho)=M_{E,{\sigma_{*}}}(\rho,n_{*}+1)v_{\sigma_{*}}(n_{*}+1)$
$[n_{*}+1,\infty)$ QES $\sigma=\sigma_{*}$ QES
$v_{\sigma_{*}}(\rho)=M_{E,\sigma_{*}}(\rho,0)v_{\sigma_{*}}(0)$ $[0,n_{*}]$
QES $\sigma=\sigma_{*}$ QES
$v_{\sigma_{*}}(\rho)=\pmatrix{M_{E,{\sigma_{*}}}(\rho,n_{*}+1)v_{\sigma_{*}}(n_{*}+1)\cr+M_{E,{\sigma_{*}}}(\rho,0)v_{\sigma_{*}}(0)\cr}$
$[n_{*}+1,\infty)$
Within OPPQ-$\Phi$ we must work with the orthonormal polynomials of ${\cal
W}(x)\equiv{\cal A}^{2}(x)$, where ${\cal A}(x)$ is defined in Eq.(33). The
asymptotic exponential form of all the physical states, within the $\Phi$
representation, is given by ${\cal W}(x)$.
For the general even and odd parity states ($\sigma=0,1$), the OPPQ
representation becomes
$\displaystyle\Phi_{\sigma}(x)=\sum_{j=0}^{\infty}\Omega_{j}x^{\sigma}{\cal
O}^{(j)}_{\sigma}(x^{2}){\cal W}(x),$ (48)
where $x^{\sigma}{\cal O}^{(j)}_{\sigma}(x^{2})$ represent the even and odd
orthonormal polynomials of the weight ${\cal W}(x)$, $\langle x^{\sigma}{\cal
O}^{(j_{1})}_{\sigma}|{\cal W}|x^{\sigma}{\cal
O}^{(j_{2})}_{\sigma}\rangle=\delta_{j_{1},j_{2}}$. We represent them as
${\cal O}^{(j)}_{\sigma}(x^{2})=\sum_{i=0}^{j}\Xi_{\sigma;i}^{(j)}x^{2i}$. The
expansion coefficients are given by
$\displaystyle\Omega_{j}=$ $\displaystyle\int dx\ x^{\sigma}{\cal
O}^{(j)}_{\sigma}(x^{2})\Phi_{\sigma}(x),$ $\displaystyle\Omega_{j}=$
$\displaystyle\sum_{i=0}^{j}\Xi_{\sigma;i}^{(j)}v_{\sigma}(i).$ (49)
Depending on which case is considered, as summarized in Table 8, the
$\\{v_{\sigma}(i)\\}$ moments will be linearly dependent either on
$v_{\sigma}(0)$ (i.e. cases 1 and 2), $v_{\sigma}(n_{*}+1)$ (i.e. case 3, in
which $v_{\sigma_{*}}(\leq n_{*})=0$), or on both (for the QES states). We can
represent each of these by (using the notation in Table 7 and 8)
$\displaystyle\Omega_{j}=$
$\displaystyle\sum_{\ell=0}^{m_{s}=0,1}\Big{(}\sum_{i=0}^{j}\Xi_{\sigma;i}^{(j)}M_{E,\sigma}\big{(}i,\ell(n_{*}+1)\big{)}\Big{)}v_{\sigma}\big{(}\ell(n_{*}+1)\big{)}.$
(50)
Quantization corresponds to setting
$\Omega_{N+\ell}\big{(}v_{\sigma}(0),v_{\sigma}(m_{s}(n_{*}+1))\big{)}=0$, for
$0\leq\ell\leq m_{s}$, and $N\rightarrow\infty$, resulting in the
determinantal condition (either $1\times 1$ or $2\times 2$)
$\displaystyle D_{N}(E)=Det\Big{(}{\cal M}_{\ell_{1},\ell_{2}}(E;N)\Big{)}=0,$
(51)
where ${\cal
M}_{\ell_{1},\ell_{2}}(E,N)=\sum_{i=0}^{N+\ell_{1}}\Xi_{\sigma;i}^{(N+\ell_{1})}M_{E,\sigma}(i,\ell_{2}(n_{*}+1))$,
where $0\leq\ell_{1,2}\leq m_{s}\ (0\ or\ 1)$.
Non-QES-Potentials
For case 1 in Table 8, $m_{s}=0$ and $D_{N}(E)$ corresponds to the determinant
of a $1\times 1$ matrix. The OPPQ-$\Phi$ analysis will generate rapidly
converging approximants to the true physical values. The results of this are
not given here, but are in keeping with Tables 3 and 4.
Non-QES states for QES-Potentials
If the potential function parameters satisfy the QES conditions, let the non-
QES states be represented as $\Phi_{\sigma}(x)$. If $\sigma\neq\sigma_{*}$,
then the previous case applies and the corresponding moment equation is of
$m_{s}=0$ form. If $\sigma=\sigma_{*}$, then the non-QES states must have
$v_{\sigma_{*}}(\rho)=0$ for $0\leq\rho\leq n_{*}$. Only the moments
$\\{v_{\sigma_{*}}(\rho)|\rho\geq n_{*}+1\\}$ are nonzero, and satisfy an
effective $m_{s}=0$ relation. One can apply OPPQ on the nonzero moments:
$\displaystyle v_{\sigma_{*}}(\rho)=M_{E,\sigma_{*}}(\rho,n_{*}+1)\
v_{\sigma_{*}}(n_{*}+1),\ \rho\geq n_{*}+1;$
$\displaystyle\sum_{i=n_{*}+1}^{N}\Xi_{N;i}^{(N)}M_{E,\sigma}(i,n_{*}+1)=0,N\geq
n_{*}+2,\ the\ OPPQ\ condition.$
The results are given in Tables 5 and 6.
QES states: Approach- I
When the potential function parameters do satisfy the QES constraints, then
the QES state energies are determined from Eq.(43). The
$\\{v_{\sigma_{*}}(\rho)|0\leq\rho\leq n_{*}\\}$ moments are determined from
$v_{\sigma_{*}}(0)\neq 0$ (which can be normalized arbitrarily) . The higher
order moments $\\{v_{\sigma_{*}}(\rho)|\rho\geq n_{*}+1\\}$ are determined by
the exact (OPPQ) identities
$\displaystyle\int dx\ x^{\sigma_{*}}{\cal
O}_{\sigma}^{(n_{*}+q)}(x^{2})\Phi_{\sigma}(x)=0,\ for\ q\geq 1,$
$\displaystyle\sum_{i=0}^{n_{*}+q}\Xi_{\sigma_{*};i}^{(n_{*}+q)}v_{\sigma_{*}}(i)=0$
$\displaystyle
v_{\sigma_{*}}(n_{*}+q)=-{1\over{\Xi_{\sigma;n_{*}+q}^{(n_{*}+q)}}}\sum_{i=0}^{n_{*}+q-1}\Xi_{\sigma_{*};i}^{(n_{*}+q)}v_{\sigma_{*}}(i).$
(53)
QES states: Approach- II
An alternative approach is to not determine the QES states from Eq.(41) but
actually use the moment equation in Eq.(38) to generate linear constraints
amongst the $\\{v_{\sigma}(\rho)|0\leq\rho\leq n_{*}\\}$ (i.e. they all depend
on $v_{\sigma}(0)$), and amongst the $\\{v_{\sigma}(\rho)|\rho\geq n_{*}+2\\}$
with respect to the $\\{v_{\sigma}(0),v_{\sigma}(n_{*}+1)$ moments (i.e. the
linear dependence will be derived below). This is represented in Eq.(50). We
now focus on deriving its form and applying OPPQ to it.
### 5.1 QES Potential and Arbitrary (QES or non-QES) States: Generating the
$v_{\sigma_{*}}(\rho)$ moments for $\rho\geq n_{*}+2$
We now consider the moment equation for the QES-potential case and for all
states of the QES symmetry class $\sigma=\sigma_{*}$. Our primary motivation
is to show that OPPQ-$\Phi$ will recover the exact QES energies, for all
$N\geq d+1$, with respects to the $\\{v_{\sigma_{*}}(\rho)|\rho\geq
n_{*}+1\\}$ moments. This corresponds to an $m_{s}=1$ problem. However, in the
$N\rightarrow\infty$ limit the OPPQ determinant (of the underlying $2\times 2$
matrix) also generates other energy roots not related to the QES states. These
will correspond to the non-QES states, and exponentially converge to the true
energies in the infinite limit. In this approach, we are ignoring that Eq.(43)
also tells us that the QES states are the roots of the BD polynomials.
As previously noted, the moment equation under the QES-potential condition in
Eq.(41), and for the $\sigma_{*}$ parity states (QES or non-QES) does not have
a uniform $m_{s}$ index. The first $n_{*}+1$ moments are linearly connected to
$v_{\sigma_{*}}(0)$, thus defining an effective $m_{s}=0$ relationship;
whereas all the other moment are linearly related to
$\\{v_{\sigma_{*}}(n_{*}+1),v_{\sigma_{*}}(n_{*})\\}$, or equivalently
$\\{v_{\sigma_{*}}(n_{*}+1),v_{\sigma_{*}}(0)\\}$; thereby defining an
effective $m_{s}=1$ problem. We are explicitly not using the fact that for the
non-QES states: ($v_{\sigma_{*}}(0)=0$. For simplicity, we further abbreviate
the notation for the relevant coefficient functions:
$\displaystyle{C}_{1}(\rho+1)=m-{{b^{2}}\over{4g}}+\sqrt{g}(4\rho+3+\sigma_{*})$
$\displaystyle{C}_{0}(\rho)=E-{b\over{\sqrt{g}}}(2\rho+{1\over
2}+\sigma_{*}),$ $\displaystyle{C}_{-1}(\rho-1)=2\rho(2\rho-1+2\sigma_{*}).$
The moment equation for the QES-symmetry class states (QES and non-QES states)
then becomes:
$\displaystyle
v_{\sigma_{*}}(\rho+1)={{C_{0}(\rho)}\over{C_{1}(\rho+1)}}v_{\sigma_{*}}(\rho)+{{C_{-1}(\rho-1)}\over{C_{1}(\rho+1)}}v_{\sigma_{*}}(\rho-1),$
for $0\leq\rho\leq n_{*}-1$ and $\rho\geq n_{*}+1$, separately. The recursive
nature of Eq.(55) for $0\leq\rho\leq n_{*}-1$ defines the relation
$v_{\sigma_{*}}(\rho)=M_{E,\sigma_{*}}(\rho,0)v_{\sigma_{*}}(0),0\leq\rho\leq
n_{*}.$ (56)
From Eq.(55) we see that $v_{\sigma_{*}}(n_{*}+2)$ is generated through the
linear superposition of$\\{v_{\sigma_{*}}(n_{*}+1),v_{\sigma_{*}}(n_{*})\\}$.
In general, we can express all the $\\{v_{\sigma_{*}}(\rho)|\rho\geq n_{*}\\}$
moments in terms of the linear sum of
$\\{v_{\sigma_{*}}(n_{*}),v_{\sigma_{*}}(n_{*}+1)\\}$:
$\displaystyle
v_{\sigma^{*}}(\rho)=M_{E,\sigma_{*}}(\rho,n_{*})v_{\sigma^{*}}(n_{*})+M_{E,\sigma_{*}}(\rho,n_{*}+1)v_{\sigma^{*}}(n_{*}+1),$
for $\rho\geq n_{*}$, where
$\displaystyle M_{E,\sigma_{*}}(n_{*},n_{*})=1$ $\displaystyle
M_{E,\sigma_{*}}(n_{*},n_{*}+1)=0$ $\displaystyle
M_{E,\sigma_{*}}(n_{*}+1,n_{*})=0$ $\displaystyle
M_{E,\sigma_{*}}(n_{*}+1,n_{*}+1)=1.$
Inserting Eq.(57) into Eq.(55), and making use of the independence of
$\\{v_{\sigma^{*}}(n),v_{\sigma^{*}}(n+1)\\}$, gives:
$\displaystyle
M_{E,\sigma_{*}}(\rho+1,\ell)={{C_{0}(\rho)}\over{C_{1}(\rho+1)}}M_{E,\sigma_{*}}(\rho,\ell)+{{C_{-1}(\rho-1)}\over{C_{1}(\rho+1)}}M_{E,\sigma_{*}}(\rho-1,\ell),$
for $\rho\geq n_{*}+1$ and $\ell=n_{*},n_{*}+1$ subject to the initialization
conditions in Eq.(58). Thus
$M_{E,\sigma_{*}}(n_{*}+2,n_{*})={{C_{-1}(n_{*})}\over{C_{1}(n_{*}+2)}}$ and
$M_{E,\sigma_{*}}(n_{*}+2,n_{*}+1)={{C_{0}(n_{*}+1)}\over{C_{1}(n_{*}+2)}}$,
yielding
$v_{\sigma^{*}}(n_{*}+2)={{C_{-1}(n_{*})}\over{C_{1}(n_{*}+2)}}v_{\sigma^{*}}(n_{*})+{{C_{0}(n_{*}+1)}\over{C_{1}(n_{*}+2)}}v_{\sigma^{*}}(n_{*}+1)$.
Since $v_{\sigma_{*}}(n_{*})=M_{E,\sigma_{*}}(n_{*},0)v_{\sigma_{*}}(0)$, we
have that
$\displaystyle v_{\sigma_{*}}(\rho)$ $\displaystyle=$ $\displaystyle
M_{E,\sigma_{*}}(\rho,n_{*})M_{E,\sigma_{*}}(n_{*},0)v_{\sigma^{*}}(0)+M_{E,\sigma_{*}}(\rho,n_{*}+1)v_{\sigma_{*}}(n_{*}+1),$
or
$\displaystyle v_{\sigma_{*}}(\rho)$ $\displaystyle=$ $\displaystyle
M_{E,\sigma_{*}}(\rho,0)v_{\sigma^{*}}(0)+M_{E,\sigma_{*}}(\rho,n_{*}+1)v_{\sigma_{*}}(n_{*}+1),\rho\geq
n_{*}$
where
$M_{E,\sigma_{*}}(\rho,0)=M_{E,\sigma_{*}}(\rho,n_{*})M_{E,\sigma_{*}}(n_{*},0)$.
Also, it is implicitly understood that $M_{E,\sigma_{*}}(\rho,n_{*}+1)=0$ for
$0\leq\rho\leq n_{*}$.
Having defined Eq.(61), which effectively defines an $m_{s}=1$ moment
recursion relation, we want to implement Eq.(51), the OPPQ condition.
Let ${\cal
M}_{E,\sigma_{*}}(N,\ell)=\sum_{i=0}^{N}\Xi^{(N)}_{\sigma_{*},i}M_{E,\sigma_{*}}(i,\ell)$
for $\ell=0,n_{*}+1$. The OPPQ determinant condition becomes
$\displaystyle D_{N}(E)=Det\pmatrix{{\cal M}_{E,\sigma_{*}}(N,0)&{\cal
M}_{E,\sigma_{*}}(N,n_{*}+1)\cr{\cal M}_{E,\sigma_{*}}(N+1,0)&{\cal
M}_{E,\sigma_{*}}(N+1,n_{*}+1)}=0,$
for $N\geq n_{*}+1$.
We know that the QES energies given by Eq.(43) must also satisfy Eq.(62),
since it embodies the exact OPPQ conditions for these states. Therefore, the
OPPQ determinant must factorize according to
$\displaystyle D_{N}(E)=P^{(n_{*}+1)}_{\sigma_{*}}(E)\times
Poly_{N,\sigma_{*}}^{(Non-QES)}(E),$ (63)
for $N\geq n_{*}+1$. That is, the first polynomial factor is that for the QES
states in Eq.(43). The second polynomial factor’s roots become the OPPQ
converging approximants to the non-QES states. The numerical confirmation of
this is given in Tables 9 and 10 where we compare the (exact) QES and non-QES
energies generated through the above OPPQ analysis with the QES energies
generated from the BD energy polynomial.
Table 9: Comparison of QES and non-QES (of $\sigma_{*}$ symmetry) states
computed through exact root formula $P^{n_{*}+1}_{\sigma_{*}}(E^{*})=0$ in Eq.
(43) and OPPQ-$\Phi$ Applied to OPPQ-(polynomial) determinant in Eq.(62-63).
No rounding off for QES energies.
$N$ $E_{0}^{*}$ $E_{2}^{*}$ $E_{4}^{*}$ $E_{6}^{*}$ $E_{8}$ $E_{10}$ -4.701631
2.289850 13.186912 28.822848 NA NA $V(x)=x^{6}+\sqrt{8}x^{4}-13x^{2}$,
$n_{*}=3,\sigma_{*}=0$ 4 -4.701631 2.289850 13.186912 28.822848 5 -4.701631
2.289850 13.186912 28.822848 49.879720 6 -4.701631 2.289850 13.186912
28.822848 47.994447 76.381590 7 -4.701631 2.289850 13.186912 28.822848
47.679059 70.953850 8 -4.701631 2.289850 13.186912 28.822848 47.624584
69.527914 9 -4.701631 2.289850 13.186912 28.822848 47.615172 69.156251 10
-4.701631 2.289850 13.186912 28.822848 47.613408 69.058368 11 -4.701631
2.289850 13.186912 28.822848 47.612358 68.924938
Table 10: Comparison of QES and non-QES (of $\sigma_{*}$ symmetry) states
computed through exact root formula $P^{n_{*}+1}_{\sigma_{*}}(E^{*})=0$ in Eq.
(43) and OPPQ-$\Phi$ Applied to OPPQ-(polynomial) determinant in Eq.(62-63).
No rounding off of QES energies.
$N$ $E_{1}^{*}$ $E_{3}^{*}$ $E_{5}^{*}$ $E_{7}^{*}$ $E_{9}$ $E_{11}$ -6.629227
4.618850 18.024593 34.897472 NA NA $V(x)=x^{6}+\sqrt{8}x^{4}-15x^{2}$,
$n_{*}=3,\sigma_{*}=1$ 4 -6.629227 4.618850 18.024593 34.897472 5 -6.629227
4.618850 18.024593 34.897472 56.9465755 6 -6.629227 4.618850 18.024593
34.897472 55.0485775 84.3945915 7 -6.629227 4.618850 18.024593 34.897472
54.7454855 78.8505925 8 -6.629227 4.618850 18.024593 34.897472 54.6960975
77.4351185 9 -6.629227 4.618850 18.024593 34.897472 54.6881985 77.0885815 10
-6.629227 4.618850 18.024593 34.897472 54.6872425 77.0349865 11 -6.629227
4.618850 18.024593 34.897472 54.6873885 77.0356645
## 6 The Configuration Space QES Analysis
We want to contrast the previous moment QES formulation with the conventional
configuration space analysis. Although the configuration space analysis is
easier to implement, its major deficiency is that it does not immediately
transfer to the non-QES states. That is, the Bender-Dunne factorization
property for their polynomials does not give any immediate information about
the non-QES states, in contrast to the OPPQ factorization property expressed
in Eq.(63). This is primarily due to the inherent instability of the
configuration space Hill determinant approach, which tries to quantize by
imposing a truncation strategy to the ratio ${\Psi\over{\cal
A}}={{\Phi}\over{{{\cal A}^{2}}}}=\sum_{j=0}^{\infty}a_{j}x^{j}$. Although the
moment’s and configuration space representation generate the same QES-
polynomials, the moment’s formulation naturally truncates the polynomials of
degree greater than $n_{*}+1$ in Eq.(42) when defined in terms of the
$v_{\sigma_{*}}(n)$’s; however, if the recursion relation in Eq.(46) is used,
there is the misleading appearance that they can be defined up to degree
$n_{*}+2$ based on the discussion pertaining to Eq.(47) (although
$\tilde{\gamma}_{n_{*}+1}=0$). All this is because the moment equation
decouples the $v_{\sigma_{*}}(n_{*}+1)$ from the lower order moments; while,
all the higher order moments (i.e. $v_{\sigma_{*}}(\rho)$, $\rho\geq
n_{\sigma_{*}}+1$) couple to $v_{\sigma_{*}}(0)$ and
$v_{\sigma_{*}}(n_{*}+1)$. This is not the case for the configuration space
generated energy polynomials. One can generate them to all orders, as given by
the power series expansion $a_{j}$’s. The order of the recursion relation for
the $a_{j}$’s stays the same (i.e. order one) regardless of the QES or non-QES
nature of the state.
Define the analytic function $P(x)={\Psi\over{\cal A}}={{\Phi}\over{{{\cal
A}^{2}}}}=x^{\sigma_{*}}\sum_{i=0}c_{i}(E)x^{2i}$. The associated differential
equation is:
$\displaystyle-\partial_{x}^{2}P(x)+\big{(}{b\over{\sqrt{g}}}x+2\sqrt{g}x^{3}\big{)}\partial_{x}P(x)$
$\displaystyle+\
\Big{(}(m+3\sqrt{g}-{{b^{2}}\over{4g}})x^{2}-(E-{b\over{2\sqrt{g}}})\Big{)}P(x)=0,$
(64)
resuting in:
$\displaystyle(\sigma+2)(\sigma+1)c_{1}=\big{(}{b\over{\sqrt{g}}}({1\over
2}+\sigma)-E\big{)}c_{0},$ $\displaystyle
2(i+1)(2i+1+2\sigma)c_{i+1}=\big{(}{b\over{\sqrt{g}}}(2i+{1\over
2}+\sigma)-E\big{)}c_{i}$
$\displaystyle+\big{(}m-{{b^{2}}\over{4g}}+\sqrt{g}(4(i-1)+3+2\sigma)\big{)}c_{i-1},i\geq
1.$
The coefficients are polynomials in the energy. For the power series to
naturally truncate we want $c_{I}(E)=0$ and the coeffieicnt of $c_{I-1}$ to be
zero. This will make $c_{i+1}=0$ for all $i\geq I$. If we call $I=n_{*}+1$, we
recover the QES condition on the parameter and $c_{n_{*}+1}(E)$ becomes
proportional to $P_{\sigma_{*}}^{(n_{*}+1)}(E)$. We note that under the QES
condition, since the coefficient of $c_{i-1}$ is zero, for $i=n_{*}+1$, the
QES states correspond to $c_{n_{*}+1}(E)=0$. However, these will always be the
zeroes for the higher order polynomials, $c_{i+1}(E)$, for $i\geq n_{*}+1$.
More importantly, if the potential function parameters satisfy the QES
conditions, all the $\\{c_{i}(E)|i\geq 0\\}$ polynomials can be generated
through a recursive, first order, relation. This is not the case for the
$v_{\sigma_{*}}(\rho)$ energy-polynomials, since they naturally truncate at
$\rho=n_{*}$ . Furthermore, the finite order recursion relation for these
moments is not of uniform order, as argued in the previous sections.
If we move the $c_{i}$ term in Eq.(65) to the left hand side, we note that the
recursive structure is the reverse of the moment equation in Eq.(38), in the
sense defined below.
$\displaystyle\pmatrix{m-{{b^{2}}\over{4g}}+\sqrt{g}(4\rho+3+2\sigma)\cr
E-{b\over{\sqrt{g}}}(2\rho+{1\over 2}+\sigma)\cr
2\rho(2\rho-1+2\sigma)}\rightarrow\pmatrix{\rho\rightarrow&i-1\cr\rho\rightarrow&i\cr\rho\rightarrow&i+1}\rightarrow
Coeff\pmatrix{c_{i-1}\cr c_{i}\cr c_{i+1}}.$
That is, the recursive structure of the $c_{i}$’s, for $0\leq i\leq n_{*}+1$,
produces the polynomial $c_{n_{*}+1}(E)$, which is the same as that generated
by the $v_{\sigma_{*}}(\rho)$, for $0\leq\rho\leq n_{*}$, and combined to
produce the $P_{\sigma_{*}}^{(n_{*}+1)}(E)$ polynomial in Eqs.(42-43).
## 7 The Bender-Dunne Sextic Potential
We now consider the original Bender-Dunne Hamiltonian (with potential
$V_{BD}$)
$\displaystyle H=-\partial_{x}^{2}+{b\over{x^{2}}}+mx^{2}+x^{6},$
$\displaystyle b={1\over 4}(4s-1)(4s-3),$ $\displaystyle m=-(4s+4J-2).$ (67)
The wavefunction must assume the form $\Psi(x)=x^{\gamma}A(x^{2})$, near the
origin. Since the probability density must be integrable it follows that
$\gamma>-{1\over 2}$. The indicial equation gives $\gamma^{2}-\gamma-b=0$, or
$\gamma={{1\pm(4s-2)}\over 2}$. We take $\gamma=2s-{1\over 2}$. Note that
$A(x^{2})$ suggests an analytic function of $x^{2}$ whreas ${\cal A}(x)$, as
given below, corresponds to the leading asymptotic exponential form of the
solution.
The QES states should assume the form: $\Psi(x)=x^{\gamma}P_{d}(x^{2}){\cal
A}(x)$, where the physical asymptotic factor is ${\cal A}(x)=e^{-{{x^{4}}\over
4}}$. There are only two ways to confirm this, algebraically. One is to
implement the Hill representation truncation analysis to determine if such
solutions exist. The other is to establish the existence of a $\Phi(x)={\cal
A}(x)\Psi(x)$ representation whose $\nu$-moment equation confirms the
existence of such solutions, as was done for the sextic anharmonic oscillator
potential in the previous sections. Within the $\Psi$ representation, an
asymptotic analysis can suggest the potential function parameter constraints
consistent with a QES type of solution. Tailoring the asymptotic analysis in
Eq.(25) to the explict form of the BD potential yields $\Psi(x)\sim
x^{\delta}exp(-{1\over 4}x^{4})$, where $\delta=-{{(m+3)}\over 2}$. Here
$\delta=\gamma+2d$, since $P_{d}(x^{2})$ is a polynomial of degree $2d$. That
is,‘
A Hill representation truncation analysis for ${{\Psi(x)}\over{x^{\gamma}{\cal
A}(x)}}\equiv C(x^{2})=\sum_{i=0}^{\infty}c_{i}(E)x^{2i}$ gives
$\displaystyle c_{0}=1$ $\displaystyle c_{1}(E)=-{{E}\over{4\gamma+2}}c_{0}$
$\displaystyle
c_{i+1}(E)={{-ec_{i}(E)+(2\gamma+m+4i-1)c_{i-1}(E)}\over{(i+1)(4\gamma+4i+2)}},i\geq
1.$
We see that if
$\displaystyle 4n_{*}+2\gamma+m+3=0,\ or\ J=n_{*}+1,$ $\displaystyle
c_{n_{*}+1}(E)=0,$ (70)
determines the QES states; and $C(x^{2})=Polynomial\ of\ degree\
x^{2n_{*}}\equiv P_{n_{*}}(x^{2})$. That is $d=n_{*}$, or
$\gamma+2n_{*}=-{{(m+3)}\over 2}$, consistent with the $\Psi$-asymptotic
analysis above.
As in the OPPQ-$\Psi$ analysis, we could develop a moment equation for $\Psi$,
retaining the indicial exponent. However, one’s first inclination is to strip
the indicial factor, in order to generate a less complicated analysis. We will
do so for illustrative purposes, only. As we shall see, stripping the indicial
factor is incorrect: the Bessis representation is obtained by not only keeping
the indicial factor but further enhancing it by an additional indicial factor:
$\Phi(x)=\Psi(x)x^{\gamma}{\cal A}(x)$, or
$\Phi(x)=P_{d}(x^{2})x^{2\gamma}{\cal A}^{2}(x)$. We note that the latter is
multiplying $\Psi(x)$ by its leading asymptotic form as $x\rightarrow\infty$
as well as $x\rightarrow 0$.
Before examining the Bessis representation, we implement OPPQ on two
representations. The first of these involves stripping the wavefunction of the
indicial factor: $A(x^{2})=x^{-\gamma}\Psi(x)$. The second will be to enhance
this by multiplying by the physical (exponentially decaying) asymptotic form,
${\tilde{\Phi}}(x^{2})=x^{-\gamma}\Psi(x){\cal A}(x^{2})$.
For the first case, we work with the even power moments of $A(x^{2})$:
$u(\rho)\equiv\int_{0}^{\infty}dxx^{2\rho}A(x^{2})$. The relevant differential
equation is
$\displaystyle-\partial_{x}^{2}A-{{2\gamma}\over
x}\partial_{x}A+(mx^{2}+x^{6})A=EA.$ (71)
Upon multiplying both sides by $x^{2\rho+2}$ and integrating by parts we
obtain the moment equation
$\displaystyle
u(\rho+4)=-mu(\rho+2)+Eu(\rho+1)+2(\rho+1-\gamma)(2\rho+1)u(\rho),\rho\geq 0.$
The missing moment structure $m_{s}=3$, resulting in
$u(\rho)=\sum_{\ell=0}^{3}M_{E}(\rho,\ell)\ u(\ell)$. The OPPQ analysis is
done with respects to the representation
$A(x^{2})=\sum_{j=0}^{\infty}\Omega_{j}{\cal P}^{(j)}(x){\cal A}(x)$. The data
in Table 11 gives the results for $n_{*}=3,J=n_{*}+1=4,s=1$. We emphasize that
our objective is not to show the full convergence of the non-QES states, which
becomes manifest at higher orders (i.e. $N\rightarrow\infty$), but to suggest
the veracity of our OPPQ analysis as applied to both QES and non-QES states.
Table 11: Comparison of QES and non-QES states computed through exact root
formula $c_{4}(E)=0$ in Eq. (70) and OPPQ-($x^{-\gamma}\Psi$) for $m_{s}=3$
moment equation in Eq.(72). Parameters $s=1$, $J=n_{*}+1$, and $n_{*}=3$.
$N$ $E_{0}^{*}$ $E_{1}^{*}$ $E_{2}^{*}$ $E_{3}^{*}$ $E_{4}$ $E_{5}$ -20.926277
-6.487752 +6.487752 +20.926277 NA NA $V(x)=x^{6}+mx^{2}+{b\over{x^{2}}}$,
$b=3/2$, m = -18 1 -17.752051 2 -23.465769 -5.699531 3 -20.857859 -8.880996
4.319160 4 -20.926277 -6.487752 6.487752 20.926277 5 -20.926277 -6.487752
6.487752 20.926277 52.309013 6 -20.926277 -6.487752 6.487752 20.926277
41.490341 94.456407 7 -20.926277 -6.487752 6.487752 20.926277 38.426546
71.311307 8 -20.926277 -6.487752 6.487752 20.926277 37.787371 61.916009 9
-20.926277 -6.487752 6.487752 20.926277 37.839537 58.167011 36 38.002392718
57.536940282
The second OPPQ analysis is done on
${\tilde{\Phi}}(x^{2})=x^{-\gamma}\Psi(x)exp(-{{x^{4}}\over 4})$, which
involves the previous representation multiplied by an additional exponential
asymptotic form. We obtain the differential equation for
${\tilde{\Phi}}(x^{2})$
$\displaystyle
x\partial_{x}^{2}{\tilde{\Phi}}+2\big{(}\gamma+x^{4}\big{)}\partial_{x}{\tilde{\Phi}}+(2\gamma-m+3\big{)}x^{3}{\tilde{\Phi}}+Ex{\tilde{\Phi}}=0.$
Upon multiplying both sides by $x^{2\rho+1}$, and defining $u(\rho)=\int dx\
x^{2\rho}{\tilde{\Phi}}(x^{2})$, we obtain the moment equation:
$\displaystyle\big{(}4\rho-2\gamma+m+7\big{)}u(\rho+2)=Eu(\rho+1)+2(2\rho+1)(\rho+1-\gamma)u(\rho),\rho\geq
0.$
So long as $\gamma\neq integer$, we can generate all the power moments and
pursue OPPQ for generating the exact QES and (converging) approximate non-QES.
The OPPQ representation in this case is
${\tilde{\Phi}}(x)=\sum_{j=0}^{\infty}\Omega_{j}{\cal O}^{(j)}(x){\cal
A}^{2}(x)$, where ${\cal O}^{(j)}(x)$ are the orthonormal polynomials of
${\cal A}^{2}(x)$. The results are given in Table 12. The convergence of the
non-QES is much faster. The above moment equation almost suggests the manifest
existence of QES solutions. However it is not a three term recursion relation,
since the effective missing moment order is $m_{s}=1$.
A third OPPQ analysis (the Bessis representation) is possible on a somewhat
different moment equation formulation. Consider
$\Phi(x)=\Psi(x)x^{\gamma}exp(-{{x^{4}}\over 4})$. This is no longer an
analytic function at the origin: $\Phi(x)\approx O(x^{2\gamma})$, recall
$\gamma>-{1\over 2}$. The differential equation is that of Eq.(73) with
$\gamma\rightarrow-\gamma$, plus an additional term (due to a variant on the
indicial equation) yielding:
$\displaystyle
x\Phi^{\prime\prime}(x)+2\big{(}-\gamma+x^{4}\big{)}\Phi^{\prime}(x)+(-2\gamma-m+3\big{)}x^{3}\Phi(x)+Ex\Phi(x)+2{\gamma\over
x}\Phi(x)=0.$
If we multiply by $x^{2\rho+1}$ and integrate over the nonnegative real axis,
(i.e. $\int_{\epsilon}^{\infty}dx$ , $\epsilon\rightarrow 0^{+}$) we obtain
$\displaystyle\int_{\epsilon}^{\infty}dx\ x^{2\rho+1}\Big{(}Eq.(75))\Big{)}=$
$\displaystyle-\Big{(}\big{(}2\epsilon^{2\rho+5}-2(\gamma+\rho+1)\epsilon^{2\rho+1}\big{)}\Phi(\epsilon)+\epsilon^{2\rho+2}\Phi^{\prime}(\epsilon)\Big{)}$
$\displaystyle+\int_{\epsilon}^{\infty}dx\Big{(}(\rho+1)\big{(}4\rho+2+4\gamma\big{)}x^{2\rho}+Ex^{2\rho+2}\Big{)}\Phi(x)$
$\displaystyle-\int_{\epsilon}^{\infty}dx\Big{(}4\rho+2\gamma+m+7\Big{)}x^{2\rho+4}\Phi(x).$
Since $\Phi(\epsilon)=\epsilon^{2\gamma}(1+O(\epsilon^{2}))$,
$\Phi^{\prime}(\epsilon)=2\gamma\epsilon^{2\gamma-1}+2(\gamma+1)O(\epsilon^{2\gamma+1})$,
the first term in Eq.(76) vanishes in the zero limit :
$\lim_{\epsilon\rightarrow
0}\Big{(}\big{(}2\epsilon^{2\rho+5}-2(\gamma+\rho+1)\epsilon^{2\rho+1}\big{)}\Phi(\epsilon)+\epsilon^{2\rho+2}\Phi^{\prime}(\epsilon)\Big{)}=0$,
for $\rho\geq-1$ and $\gamma>-{1\over 2}$. Additionally, the integral
expressions are finite for $\rho\geq-1$. We therefore we have the following
moment equation, valid for $\gamma>-{1\over 2}$ and $\rho\geq-1$, where
$\nu(p)\equiv\int_{0}^{\infty}dx\ x^{p}\Phi$:
$\displaystyle\big{(}4\rho+2\gamma+m+7\big{)}\nu(\rho+2)=E\nu(\rho+1)+(\rho+1)\big{(}4\rho+2+4\gamma\big{)}\nu(\rho),\rho\geq-1.$
or $(\rho\rightarrow\rho+1)$:
$\displaystyle\big{(}4\rho+2\gamma+m+3\big{)}\nu(\rho+1)=E\nu(\rho)+\rho\big{(}4\rho-2+4\gamma\big{)}\nu(\rho-1),\rho\geq
0.$
This is also a three term recursion relation in which the QES potential
function conditions are manifest. That is, if $\gamma+{{m+3}\over 2}=-2n_{*}$,
where $\gamma=2s-{1\over 2}$, then only the first $n_{*}+1$ moments can be
generated $\\{\nu(\rho)|0\leq\rho\leq n_{*}\\}$, all defining an effective
$m_{s}=0$ missing moment problem in which the corresponding moments become
polynomials in the energy (i.e. $\nu(0)=1$), $\nu(\rho)=Polynomial\ of\
degree\ \rho\ in\ E$. The non-QES states must have these first $n_{*}+1$
moments identically zero:
$\displaystyle\nu_{QES}(\rho)\neq 0,0\leq\rho\leq n_{*},$
$\displaystyle\nu_{non-QES}(\rho)=0,0\leq\rho\leq n_{*},\ if\ V_{BD}\ admits\
QES\ states.$ (79)
As in the sextic anharmonic oscillator case, the $\nu(n_{*}+1)$ moment
decouples from the moment equation. We can repeat all the different types of
OPPQ computational implementations done for the sextic anharmonic oscillator;
however, we are only interested in repeating the OPPQ computational analysis
that uniformly generates the QES and the non-QES states.
As in the sextic anharmonic oscillator case, if the $V_{BD}$ potential admits
QES states, then the $\\{\nu(\rho)|\rho\geq n_{*}+2\\}$ moments couple to the
$\\{\nu(n_{*}+1),\nu(n_{*})\\}$ moments, through an effective $m_{s}=1$
recursion relation. However, $\nu(n_{*})$ couples to all the lower order
moments through an $m_{s}=0$ recursion relation. Therefore, the
$\\{\nu(\rho)|\rho\geq n_{*}+2\\}$ effectively couple, through an $m_{s}=1$
relation, to $\\{\nu(n_{*}+1),\nu(0)\\}$ . We can apply OPPQ on this relation
and uniformly obtain the QES and non-QES states. That is, we are not using the
BD energy polynomials,these are contained within the OPPQ conditions. The
following discussion defines the necessary relation connecting the
$\\{\nu(\rho)|\rho\geq n_{*}+2\\}$ moments to the $\\{\nu(n_{*}+1),\nu(0)\\}$
moments.
Within the Bessis representation $\Phi(x)=\Psi(x)x^{\gamma}{\cal A}(x)$,
${\cal A}(x)=e^{-{x^{4}}\over 4}$, the OPPQ representation becomes
$\Phi(x)=\sum_{j=0}^{\infty}\Omega_{j}{\cal Q}^{(j)}(x)x^{2\gamma}{\cal
A}^{2}(x)$, where the ${\cal Q}^{(j)}(x)$ are the orthonormal polynomials of
${\cal B}(x)\equiv x^{2\gamma}{\cal A}^{2}(x)$. We will work on the half real
axis in terms of the $x^{2}$ variable. We make explicit this $x^{2}$
dependence, $\Phi(x)\rightarrow\Phi(x^{2})$, ${\cal A}(x)\rightarrow{\cal
A}(x^{2})$, ${\cal B}(x)\rightarrow{\cal B}(x^{2})$, and ${\cal
Q}^{(j)}(x)\rightarrow{\cal Q}^{(j)}(x^{2})$.
Define ${\cal A}_{\sigma}(x^{2})=exp(-{{x^{4}}\over{\sigma}})$, where
$\sigma=2$. The orthonormality property for the ${\cal Q}^{(j)}(x^{2})$’s
becomes $\int_{0}^{\infty}dx{\cal Q}^{(j_{1})}(x^{2}){\cal
Q}^{(j_{2})}(x^{2}){\cal B}(x^{2})=\int_{0}^{\infty}\ d\xi{\cal
Q}^{(j_{1})}(\xi){\cal Q}^{(j_{2})}(\xi){{{\cal
B}(\xi)}\over{2\sqrt{\xi}}}=\delta_{j_{1},j_{2}}$, where $\xi=x^{2}$. These
orthonormal polynomials are generated from the moments
$m(\rho)=\int_{0}^{\infty}d\xi\xi^{\rho}{{{\cal B}(\xi)}\over{2\sqrt{\xi}}}$.
The weight becomes ${{{\cal B}(\xi)}\over{2\sqrt{\xi}}}={{\xi^{\gamma-{1\over
2}}}\over 2}\exp(-{{\xi^{2}}\over{\sigma}})$. Recalling that
$\gamma=2s-{1\over 2}$, we obtain $m(\rho)={1\over 2}\int
d\xi\xi^{\rho+2s-1}\exp(-{{\xi^{2}}\over{\sigma}})={1\over 4}\int
d\zeta\zeta^{{\rho\over 2}+s-1}exp(-{{\zeta}\over{\sigma}})={1\over
4}2^{{\rho\over 2}+s}\Gamma({\rho\over 2}+s)$, having set ${\sigma}=2$. This
enables us to generate the orthogonal polynomials, ${\cal
Q}^{(j)}(\xi)=\sum_{i=0}^{j}\Xi_{i}^{(j)}\xi^{i}$.
We now repeat the OPPQ analysis we did for the sextic anharmonic oscillator.
The $\\{\nu(\rho)|0\leq\rho\leq n_{*}\\}$ moments satisfy an $m_{s}=0$ moment
equation regardless of the nature of the discrete state. This is true for the
QES states. This is true for the non-QES states when the potential function
satisfies the QES condition; although in this case, they are identically zero
(in the following analysis we do not impose this, but it will be the result as
the OPPQ quantization order goes to infinity, $N\rightarrow\infty$). If the
potential function does not satisfy the QES conditions, then all the states
satisfy an $m_{s}=0$ moment equation, to all order. Accordingly, we have:
$\displaystyle\nu(\rho)=M_{E}(\rho,0)\nu(0),\ 0\leq\rho\leq n_{*},$
$\displaystyle M_{E}(0,0)\equiv 1.$ (80)
All the moments of order $n_{*}+2$ or higher, are linearly dependent on the
moments $\\{\nu(n_{*}),\nu(n_{*}+1)\\}$:
$\displaystyle\nu(\rho)=\sum_{\ell=n_{*}}^{n_{*}+1}{\cal
N}_{E}(\rho,\ell)\nu(\ell),\ n_{*}+2\leq\rho<\infty,$ $\displaystyle{\cal
N}_{E}(\ell_{1},\ell_{2})=\delta_{\ell_{1},\ell_{2}},\ n_{*}\leq\ell_{1,2}\leq
n_{*}+1,$ (81)
where ${\cal N}_{E}(\rho,\ell)$ satisfies Eq.(78) for
$n_{*}+2\leq\rho<\infty$.
Finally, we combine these to produce the representation
$\displaystyle\nu(\rho)=\sum_{\ell=0,n_{*}+1}M_{E}(\rho,\ell)\nu(\ell),\
0\leq\rho<\infty,$ (82)
where
$\displaystyle M_{E}(\rho,0)=\ determined\ from\ m_{s}\ =0\ moment\ equation\
for\ 0\leq\rho\leq n_{*},$ $\displaystyle M_{E}(\rho,n_{*}+1)\equiv 0,\
0\leq\rho\leq n_{*},$ $\displaystyle M_{E}(n_{*}+1,0)\equiv
0,M_{E}(n_{*}+1,n_{*}+1)\equiv 1,$ $\displaystyle M_{E}(\rho,0)={\cal
N}_{E}(\rho,n_{*})M_{E}(n_{*},0),\rho\geq n_{*}+2,$ $\displaystyle
M_{E}(\rho,n_{*}+1)={\cal N}_{E}(\rho,n_{*}+1),\ \rho\geq n_{*}+2.$
We implement OPPQ by demanding that
$\displaystyle\int d\xi{\cal Q}^{(N+\ell_{r})}(\xi)\Phi(\xi)=0,N\geq n_{*}+1,\
and\ \ell_{r}=0,1;$
$\displaystyle\sum_{i=0}^{N+\ell_{r}}\Xi_{i}^{(N+\ell_{r})}\nu(i)=0,$
$\displaystyle\sum_{\ell_{c}=0,n_{*}+1}\Big{(}\sum_{i=0}^{N+\ell_{r}}\Xi_{i}^{(N+\ell_{r})}M_{E}(i,\ell_{c})\Big{)}u(\ell_{c})=0.$
(84)
The latter results in a $2\times 2$ set of simultaneous equations whose
determinant exhibits the factorized form $D_{N}(E)=Poly_{QES}(E)\times
Poly_{nonQES}(E)$. The QES polynomial factor contains all the QES roots
consitent with the BD energy polynomial. The other polynomial factor generates
the approximate non-QES energies through its roots that converge,
exponentially fast, to the true non-QES values. The results of this analysis
are given in Table 13, with a much improved convergence compared to the case
reflected in Table 12.
## 8 Conclusion
We have presented an extensive OPPQ analysis of the QES and non-QES states for
the sextic anharmonic oscillator and the Bender and Dunne sextic potential.
The OPPQ analysis in either the $\Psi$ representation or $\Phi$ representation
yields the exact QES states and approximates the non-QES states (through
converging approximants). Within the Bessis function representation ($\Phi$)
we can recover the configuration space Bender and Dunne energy orthogonal
polynomials, leading to exact formulas for the energies, as well as the
wavefunctions. We have shown that the reason for the singular behavior
(breakdown) of the Bender and Dunne orthogonal polynomials is due to the
breakdown of the order of the moment equation in the Bessis representation.
This moments’ intepretation was known by Handy and Bessis within the context
of their formulation of the Eigenvalue Moment Method. This breakdwon in the
moment equation’s order can be interpreted as a spontaneous breakdown of the
implicit degree of freedom within the moment’s representation. The OPPQ
moments’ representation also reveals additional structure for the non-QES
states (i.e. lower order moments are zero within the Bessis representation).
We believe these propeties extend to multidimensional systems. Although we
have not proved that all one dimensional QES systems must have an $m_{s}=0$
moment equation for the QES states (within the Bessis representation), we
believe that the two examples presented here strongly argue in favor of this.
Table 12: Comparison of QES and non-QES states computed through exact root
formula $c_{4}(E)=0$ in Eq. (70) and OPPQ-${\tilde{\Phi}}$ for $m_{s}=1$
moment equation in Eq.(74). Parameters $s=1$, $J=n_{*}+1$, and $n_{*}=3$.
$N$ $E_{0}^{*}$ $E_{1}^{*}$ $E_{2}^{*}$ $E_{3}^{*}$ $E_{4}$ $E_{5}$ -20.926277
-6.487752 +6.487752 +20.926277 NA NA $V(x)=x^{6}+mx^{2}+{b\over{x^{2}}}$,
$b=3/2$, m = -18 1 -12.552595 2 -19.663222 -9.597580 3 -20.883219 -6.093770
9.002550 4 -20.926277 -6.487752 6.487752 20.926277 5 -20.926277 -6.487752
6.487752 20.926277 36.988059 6 -20.926277 -6.487752 6.487752 20.926277
37.544189 58.584676 7 -20.926277 -6.487752 6.487752 20.926277 37.887188
56.623863 8 -20.926277 -6.487752 6.487752 20.926277 37.976840 57.031923 9
-20.926277 -6.487752 6.487752 20.926277 37.996662 57.372312
Table 13: Comparison of QES and non-QES states computed through exact root
formula $c_{4}(E)=0$ in Eq. (70) and OPPQ-$\Phi$ for $m_{s}=1$ moment equation
in Eq.(78). Parameters $s=1$, $J=n_{*}+1$, and $n_{*}=3$.
$N$ $E_{0}^{*}$ $E_{1}^{*}$ $E_{2}^{*}$ $E_{3}^{*}$ $E_{4}$ $E_{5}$ -20.926277
-6.487752 +6.487752 +20.926277 NA NA $V(x)=x^{6}+mx^{2}+{b\over{x^{2}}}$,
$b=3/2$, m = -18 4 -20.926277 -6.487752 6.487752 20.926277 5 -20.926277
-6.487752 6.487752 20.926277 40.921277 6 -20.926277 -6.487752 6.487752
20.926277 38.584899 67.221602 7 -20.926277 -6.487752 6.487752 20.926277
38.122298 60.484136 8 -20.926277 -6.487752 6.487752 20.926277 38.026662
58.428088 9 -20.926277 -6.487752 6.487752 20.926277 38.007296 57.788594 10
-20.926277 -6.487752 6.487752 20.926277 38.003397 57.603593 11 -20.926277
-6.487752 6.487752 20.926277 38.002606 57.554137
## Acknowledgments
Discussions with Dr. D. Bessis are greatly appreciated. One of the authors
(DV) is grateful for the support received from the National Science Foundation
through a grant for the Center for Research on Complex Networks (HRD-1137732).
## References
## References
* [1] Cooper F, Khare A and Sukhatme U 1995 Phys. Rept. 251 267
* [2] Turbiner A, 1988 Sov. Phys. J.E.T.P. 67 230
* [3] Turbiner A, 1988 Comm. Math. Phys. 118 467
* [4] Morozov A, Perelomov A, Roslyi A, Shifman M and Turbiner A 1990 Int. J. Mod. Physi. A5 803
* [5] Turbiner A, 1994 AMS 160 263
* [6] Banerjee K 1979 Proc. R. Soc. Lond. A 368 155
* [7] Hautot A 1986 Phys. Rev. D 33 437
* [8] Bender C M and Dunne G V 1996 J. Math. Phys. 37 6
* [9] Tater M and Turbiner A V 1993 J. Phys. A: Math. Gen. 26 697
* [10] Tymczak C J, Japaridze G S, Handy C R, and Wang Xiao-Qian 1998 Phys. Rev. Lett. 80 3674; 1998 Phys. Rev. A 58 2708
* [11] Handy C R and Vrinceanu D 2013 J. Phys. A: Math. Theor. 46 135202
* [12] Handy C R and Vrinceanu D 2013 J. Phys. B:Atom. Mol. & Opt. Phys. 46 115002
* [13] Amore P and Fernandez F M 2012 J. Phys. B: Atom. Mol. & Opt. Phys. 45 235004
* [14] Handy C R and Bessis D 1985 Phys. Rev. Lett. 55, 931
* [15] Handy C R, Bessis D, Sigismondi G, and Morley T D 1988 Phys. Rev. A 37,4557
* [16] Handy C R, Bessis D, Sigismondi G, and Morley T D 1988 Phys. Rev. Lett. 60,253
* [17] Boyd S and Vandenberghe L 2004 Convex Optimization (New York: Cambridge University Press)
* [18] Lasserre J-B Moments, Positive Polynomials and Their Applications (London: Imperial College Press 2009)
* [19] Handy C R 1984 Clark Atlanta University unpublished
* [20] Baker G A Jr 1975 Essentials of Pade Approximants (New York; Academic)
* [21] Shohat J A and Tamarkin J D, 1963 The Problem of Moments (American Mathematical Society, Providence, RI)
* [22] Handy C R 1981 Phys. Rev. D 24 378
* [23] Handy C R and Murenzi R 1998 J. Phys. A: Math. Gen. 31 9897
* [24] Le Guillou J C and Zinn-Justin J 1983 Annals of Physics 147 57
* [25] Handy C R, Trallero-Giner C, and Rodriguez A H 2001 J. Phys. A34 10991
* [26] Goemans M 1997 Mathematical Programming 79 143
* [27] Greenman L and Mazziotti D A 2008 J. Chem. Phys 128 114109
* [28] Yasuda K 2002 Phys. Rev. A 65 052121
* [29] Handy C R 1987 Phys. Rev. A 36 4411; 1987 Phys. Lett. A 124 308
* [30] Handy C R 2001 J. Phys. A 34 L271
* [31] Bender C M and Boettcher S 1998 Phys. Rev. Lett. 80 5243
* [32] Dorey P, Dunning C, and Tateo R 2001 J. Phys. A Math. Gen. 34 L391
* [33] Handy C R 2001 J. Phys. A 34 5065; Handy C R, Khan D, Wang Xiao-Qian, and Tymczak C J 2001 J. Phys. A: Math. Gen. 34 5593
* [34] Handy C R and Msezane A Z 2001 J. Phys. A 34 L531
* [35] Handy C R, Msezane A Z, and Yan Z 2002 J. Phys. A 35 6359
|
arxiv-papers
| 2014-02-24T15:56:56 |
2024-09-04T02:49:58.753196
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos R. Handy, Daniel Vrinceanu and Rahul Gupta",
"submitter": "Daniel Vrinceanu",
"url": "https://arxiv.org/abs/1402.5868"
}
|
1402.6013
|
11institutetext: Joaquin Vanschoren 22institutetext: Leiden University,
Leiden, Netherlands, 22email: [email protected] 33institutetext: Mikio L. Braun
44institutetext: TU Berlin, Berlin, Germany, 44email: [email protected]
55institutetext: Cheng Soon Ong 66institutetext: National ICT Australia,
Melbourne, Austrialia, 66email: [email protected]
# Open science in machine learning
Joaquin Vanschoren and Mikio L. Braun and Cheng Soon Ong
###### Abstract
We present OpenML and mldata, open science platforms that provides easy access
to machine learning data, software and results to encourage further study and
application. They go beyond the more traditional repositories for data sets
and software packages in that they allow researchers to also easily share the
results they obtained in experiments and to compare their solutions with those
of others.
###### Keywords:
machine learning, open science
## 1 Introduction
Research in machine learning and data mining can be speeded up tremendously by
moving empirical research results “out of people’s heads and labs, onto the
network and into tools that help us structure and alter the information”
Nielsen2008 . The massive streams of experiments that are being executed to
benchmark new algorithms, test hypotheses or model new data sets have many
more uses beyond their original intent, but are often discarded or their
details are lost over time. In this paper, we present recently developed
infrastructures that aim to make machine learning research more open. They go
beyond the more traditional repositories111Well-known examples are the UCI
repository, (http://archive.ics.uci.edu/ml), myExperiment
(http://myexperiment.org) and MLOSS (http://mloss.org). for data sets,
implementations and workflows in that they allow researchers to also share
detailed results obtained in experiments and to compare their solutions with
those of others.
This _collaborative_ approach to experimentation allows researchers to share
all code and results that are possibly of interest to others, which may boost
their visibility, speed up further research and applications, and engender new
collaborations. Indeed, many questions about machine learning algorithms can
be answered on the fly by querying the combined results of thousands of
studies on all available data sets. This facilitates much larger-scale machine
learning studies, yielding more generalizable results Hand2006 . Last but not
least, these infrastructures keep track of experiment details, ensuring that
we can easily reproduce them later on, and confidently build upon earlier work
Hirsh2008 .
## 2 OpenML
OpenML (http://openml.org) is a website where researchers can share their data
sets, implementations and experiments in such a way that they can easily be
found and reused by others. It offers a web API through which new resources
and results can be submitted automatically, and is being integrated in a
number of popular machine learning and data mining platforms, such as Weka,
RapidMiner, KNIME, and data mining packages in R, so that new results can be
submitted automatically. Vice versa, it enables researchers to easily search
for certain results (e.g. evaluations of algorithms on a certain data set), to
directly compare certain techniques against each other, and to combine all
submitted data in advanced queries.
To make experiments from different researchers comparable, OpenML uses _tasks_
, well-described problems to be solved by a machine learning algorithm or
workflow. A typical task would be: _Predict (target) attribute X of data set Y
with maximal predictive accuracy_. Similar to a data mining challenge,
researchers are thus challenged to build algorithms or workflows that solve
these tasks. Tasks can be searched online, and will be generated on demand for
newly submitted data sets.
Tasks contain all necessary information to complete it, always including the
input data and what results should be submitted to the server. Some tasks
offers more structured input and output: predictive tasks, for instance,
include train and test splits for cross-validation, and a submission format
for all predictions. The server will evaluate the predictions and compute
scores for various evaluation metrics.
An attempt to solve a task is called a _run_ , and includes the task itself,
the algorithm or workflow (i.e., _implementation_) used, and a file detailing
the obtained results. These are all submitted to the server, where new
implementations will be registered. For each implementation, an online
overview page is generated summarising the results obtained over all tasks,
over various parameter settings. For each data set, a similar page is created,
containing a ranking of implementations that were run on tasks with that data
set as input.
OpenML provides a REST API for downloading tasks and uploading data sets,
implementations and results. This API is currently being integrated in various
machine learning platforms such as Weka, R packages, RapidMiner and KNIME
222Beta versions of these integrations can be downloaded from the OpenML
website..
To make the shared results maximally useful, OpenML links various bits of
information together in a single database. All results are stored in such a
way that implementations can directly be compared to each other (using various
evaluation measures), and parameter settings are stored so that the impact of
individual parameters can be tracked. Moreover, for all data sets, it
calculates meta-data about the features and the data distributionPeng2002 ,
and for all implementations, meta-data is stored about their (hyper)parameters
and properties such as what input data they can handle, what tasks they can
solve and, if possible, advanced properties such bias-variance profiles.
Finally, the OpenML website offers various search functionalities. data sets,
algorithms and implementations can be found through simple keyword searches,
linked to all results and meta-data. Runs can be aggregated to directly
compare many implementations over many data sets (e.g. for benchmarking).
Furthermore, the database can be queried directly through an SQL editor, or
through pre-defined advanced queries.333See the Advanced tab on
http://openml.org/search. The results of such queries are displayed as data
tables, scatterplots or line plots, which can be downloaded directly.
## 3 mldata
mldata (http://mldata.org) is a community-based website for the exchange of
machine learning data sets. Data sets can either be raw data files or
collections of files, or use one of the supported file formats like HDF5 or
ARFF in which case mldata looks at meta data contained in the files to display
more information. Similar to OpenML, mldata can define learning tasks based on
data sets, where mldata currently focuses on supervised learning data.
Learning tasks identify which features are used for input and output and also
which score is used to evaluate the functions. mldata also allows to create
learning challenges by grouping learning tasks together, and lets users submit
results in the form of predicted labels which are then automatically
evaluated.
mldata.org supports four kinds of information: raw data sets, learning tasks,
learning methods, and challenges. A raw data set is just some data, while the
learning task also specifies the input and output variables and the cost
function used in evaluation. A learning method is the description of a full
learning workflow, including feature extraction and learner. One can upload
predicted labels for a data set and a task to create a solution entry which
automatically evaluates the error on the predicted labels. Finally, a number
of learning tasks can be grouped to create a challenge.
Most of this data is text. mldata defines a general file exchange format for
supervised learning based on HDF5, a structured compressed file format. It is
similar to an archive of files but has additional structure on the level of
the files, such that users can directly store and access matrices, or
numerical arrays. Using this specified file format is not mandatory, but using
it unlocks a number of additional features like a summary of the data set, and
automatic conversion into a number of other formats.
Currently, OpenML is being integrated with mldata, so that data sets and
learning methods can be shared between both platforms.
## 4 Related work
There also exist platforms aimed at providing reproducible benchmarks. DELVE
(http://www.cs.utoronto.ca/~delve) was the first, but is currently in
abeyance. MLComp (http://mlcomp.org) allows users to upload their algorithms
and evaluate them on known data sets (or vice versa) on MLComp servers.
RunMyCode (http://runmycode.org) allows researchers to create _companion
websites_ for publications by uploading code and building an interface. Users
can then fill in all inputs online and get the result of the algorithm.
Compared to these systems, OpenML and mldata allow users more flexibility in
running experiments: new tasks can be introduced for novel types of
experiments and experiments can be run in any environment. OpenML also offers
clean integration in data mining platforms that researchers already use in
daily research, and closer data integration so that researchers can reuse
results in many ways beyond direct benchmark comparisons, such as meta-
learning studies Vanschoren2012 .
### Acknowledgments
This work is supported by grant 600.065.120.12N150 from the Dutch Fund for
Scientific Research (NWO), and by the IST Programme of the European Community,
under the PASCAL2 Network of Excellence, IST-2007-216886.
## References
* (1) Hand, D.: Classifier technology and the illusion of progress. Statistical Science (Jan 2006)
* (2) Hirsh, H.: Data mining research: Current status and future opportunities. Statistical Analysis and Data Mining 1(2), 104–107 (Jan 2008)
* (3) Nielsen, M.: The future of science: Building a better collective memory. APS Physics 17(10) (2008)
* (4) Peng, Y., Flach, P., Soares, C., Brazdil, P.: Improved dataset characterisation for meta-learning. Lecture Notes in Computer Science 2534, 141–152 (Jan 2002)
* (5) Vanschoren, J., Blockeel, H., Pfahringer, B., Holmes, G.: Experiment databases. A new way to share, organize and learn from experiments. Machine Learning 87(2), 127–158 (2012)
|
arxiv-papers
| 2014-02-24T23:12:42 |
2024-09-04T02:49:58.784144
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joaquin Vanschoren and Mikio L. Braun and Cheng Soon Ong",
"submitter": "Joaquin Vanschoren",
"url": "https://arxiv.org/abs/1402.6013"
}
|
1402.6178
|
# Iterative properties of birational rowmotion
Darij Grinberg
Department of Mathematics
Massachusetts Institute of Technology
Massachusetts, U.S.A.
[email protected]
! Supported by NSF grant 1001905. Tom Roby
Department of Mathematics
University of Connecticut
Connecticut, U.S.A.
[email protected] Supported by NSF grant 1001905.
(version 5.0 (7 September 2015).
This is the arXiv version, not the submitted version. The submitted version
has been abridged in several places and split into two papers.
Mathematics Subject Classifications: 06A07, 05E99)
###### Abstract
We study a birational map associated to any finite poset $P$. This map is a
far-reaching generalization (found by Einstein and Propp) of classical
rowmotion, which is a certain permutation of the set of order ideals of $P$.
Classical rowmotion has been studied by various authors (Fon-der-Flaass,
Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different
guises (Striker-Williams promotion and Panyushev complementation are two
examples of maps equivalent to it). In contrast, birational rowmotion is new
and has yet to reveal several of its mysteries. In this paper, we prove that
birational rowmotion has order $p+q$ on the $\left(p,q\right)$-rectangle poset
(i.e., on the product of a $p$-element chain with a $q$-element chain); we
furthermore compute its orders on some triangle-shaped posets and on a class
of posets which we call “skeletal” (this class includes all graded forests).
In all cases mentioned, birational rowmotion turns out to have a finite (and
explicitly computable) order, a property it does not exhibit for general
finite posets (unlike classical rowmotion, which is a permutation of a finite
set). Our proof in the case of the rectangle poset uses an idea introduced by
Volkov (arXiv:hep-th/0606094) to prove the $AA$ case of the Zamolodchikov
periodicity conjecture; in fact, the finite order of birational rowmotion on
many posets can be considered an analogue to Zamolodchikov periodicity. We
comment on suspected, but so far enigmatic, connections to the theory of root
posets. We also make a digression to study classical rowmotion on skeletal
posets, since this case has seemingly been overlooked so far.
Keywords: rowmotion; posets; order ideals; Zamolodchikov periodicity; root
systems; promotion; trees; graded posets; Grassmannian; tropicalization.
###### Contents
1. 0.1 Leitfaden
2. 0.2 Acknowledgments
3. 1 Linear extensions of posets
4. 2 Birational rowmotion
5. 3 Graded posets
6. 4 w-tuples
7. 5 Graded rescaling of labellings
8. 6 Homogeneous labellings
9. 7 Order
10. 8 The opposite poset
11. 9 Skeletal posets
12. 10 Interlude: Classical rowmotion on skeletal posets
13. 11 The rectangle: statements of the results
14. 12 Reduced labellings
15. 13 The Grassmannian parametrization: statements
16. 14 The Plücker-Ptolemy relation
17. 15 Dominance of the Grassmannian parametrization
18. 16 The rectangle: finishing the proofs
19. 17 The $\vartriangleright$ triangle
20. 18 The $\Delta$ and $\nabla$ triangles
21. 19 The quarter-triangles
22. 20 Negative results
23. 21 The root system connection
## Introduction
The present paper had originally been intended as a companion paper to David
Einstein’s and James Propp’s work [EiPr13], which introduced piecewise-linear
and birational rowmotion as extensions of the classical concept of rowmotion
on order ideals. While the present paper is mathematically self-contained (and
indeed gives some proofs on which [EiPr13] relies), it provides only a modicum
of motivation and applications for the results it discusses. For the latter,
the reader may consult [EiPr13].
Let $P$ be a finite poset, and $J\left(P\right)$ the set of the order
ideals111An order ideal of a poset $P$ is a subset $S$ of $P$ such that every
$s\in S$ and $p\in P$ with $p\leqslant s$ satisfy $p\in S$. of $P$. Rowmotion
is a classical map $J\left(P\right)\rightarrow J\left(P\right)$ which can be
defined in various ways, one of which is as follows: For every $v\in P$, let
$\mathbf{t}_{v}:J\left(P\right)\rightarrow J\left(P\right)$ be the map sending
every order ideal $S\in J\left(P\right)$ to
$\left\\{\begin{array}[c]{l}S\cup\left\\{v\right\\}\text{, if }v\notin S\text{
and }S\cup\left\\{v\right\\}\in J\left(P\right);\\\
S\setminus\left\\{v\right\\}\text{, if }v\in S\text{ and
}S\setminus\left\\{v\right\\}\in J\left(P\right);\\\ S\text{,
otherwise}\end{array}\right.$. These maps $\mathbf{t}_{v}$ are called
classical toggles222or just toggles in literature which doesn’t occupy itself
with birational rowmotion, since all they do is “toggle” an element into or
out of an order ideal. Let $\left(v_{1},v_{2},...,v_{m}\right)$ be a linear
extension of $P$ (see Definition 1.3 for the meaning of this). Then,
(classical) rowmotion is defined as the composition
$\mathbf{t}_{v_{1}}\circ\mathbf{t}_{v_{2}}\circ...\circ\mathbf{t}_{v_{m}}$
(which, as can be seen, does not depend on the choice of the particular linear
extension $\left(v_{1},v_{2},...,v_{m}\right)$). This rowmotion map has been
studied from various perspectives; in particular, it is isomorphic333By this,
we mean that there exists a bijection $\phi$ from $J\left(P\right)$ to the set
of all antichains of $P$ such that rowmotion is $\phi^{-1}\circ f\circ\phi$.
to the map $f$ of Fon-der-Flaass [Flaa93]444Indeed, let
$\mathcal{A}\left(P\right)$ denote the set of all antichains of $P$. Then, the
map $J\left(P\right)\rightarrow\mathcal{A}\left(P\right)$ which sends every
order ideal $I\in J\left(P\right)$ to the antichain of the maximal elements of
$I$ is a bijection which intertwines rowmotion and Fon-der-Flaass’ map $f$.,
the map $F^{-1}$ of Brouwer and Schrijver [BrSchr74], and the map $f^{-1}$ of
Cameron and Fon-der-Flaass [CaFl95]555This time, the intertwining bijection
from rowmotion to the map $f^{-1}$ of [CaFl95] is given by mapping every order
ideal $I$ to its indicator function. This is a bijection from
$J\left(P\right)$ to the set of Boolean monotonic functions
$P\rightarrow\left\\{0,1\right\\}$.. More recently, it has been studied (and
christened “rowmotion”) in Striker and Williams [StWi11], where further
sources and context are also given. Since so much has already been said about
this rowmotion map, we will only briefly touch on its properties in Section
10, while most of this paper will be spent studying a much more general
construction.
Among the questions that have been posed about rowmotion, the most prevalent
was probably that of its order: While it clearly has finite order (being a
bijective map from the finite set $J\left(P\right)$ to itself), it turns out
to have a much smaller order than what one would naively expect when the poset
$P$ has certain “special” forms (e.g., a rectangle, a root poset, a product of
a rectangle with a $2$-chain, or – apparently first considered in this paper –
a forest). Most strikingly, when $P$ is the rectangle
$\left[p\right]\times\left[q\right]$ (denoted
$\operatorname*{Rect}\left(p,q\right)$ in Definition 11.1), then the
$\left(p+q\right)$-th power of the rowmotion operator is the identity map.
This is proven in [BrSchr74, Theorem 3.6] and [Flaa93, Theorem 2]666Another
proof follows from two observations made in [PrRo14]: first, that the
rowmotion operator on the order ideals of the rectangle
$\left[p\right]\times\left[q\right]$ is equivalent to the operator named
$\Phi_{A}$ in [PrRo14] (i.e., there is a bijection between order ideals and
antichains of $\left[p\right]\times\left[q\right]$ which intertwines these two
operators), and second, that the $\left(p+q\right)$-th power of this latter
operator $\Phi_{A}$ is the identity map (this is proven in [PrRo14, right
after Proposition 26]). This argument can also be constructed from ideas given
in [PrRo13, §3.3.1].. We will (in Section 10) give a simple algorithm to find
the order of rowmotion on graded forests and similar posets.
In [EiPr13], David Einstein and James Propp have lifted the rowmotion map from
the set $J\left(P\right)$ of order ideals to the progressively more general
setups of:
(a) the order polytope $\mathcal{O}\left(P\right)$ of the poset $P$ (as
defined in [Stan11, Example 4.6.17] or [Stan86, Definition 1.1]), and
(b) even more generally, the affine variety of $\mathbb{K}$-labellings of $P$
for $\mathbb{K}$ an arbitrary infinite field.
In case (a), order ideals of $P$ are replaced by points in the order polytope
$\mathcal{O}\left(P\right)$, and the role of the map $\mathbf{t}_{v}$ (for a
given $v\in P$) is assumed by the map which reflects the $v$-coordinate of a
point in $\mathcal{O}\left(P\right)$ around the midpoint of the interval of
all values it could take without the point leaving $\mathcal{O}\left(P\right)$
(while all other coordinates are considered fixed). The operation of
“piecewise linear” rowmotion (inspired by work of Arkady Berenstein) is still
defined as the composition of these reflection maps in the same way as
rowmotion is the composition of the toggles $\mathbf{t}_{v}$. This “piecewise
linear” rowmotion extends (interpolates, even) classical rowmotion, as order
ideals correspond to the vertices of the order polytope
$\mathcal{O}\left(P\right)$ (see [Stan86, Corollary 1.3]). We will not study
case (a) here, since all of the results we could find in this case can be
obtained by tropicalization from similar results for case (b).
In case (b), instead of order ideals of $P$ one considers maps from the poset
$\widehat{P}:=\left\\{0\right\\}\oplus P\oplus\left\\{1\right\\}$ (where
$\oplus$ stands for the ordinal sum777More explicitly, $\widehat{P}$ is the
poset obtained by adding a new element $0$ to $P$, which is set to be lower
than every element of $P$, and adding a new element $1$ to $P$, which is set
to be higher than every element of $P$ (and $0$). We shall repeat this
definition in more formal terms in Definition 2.1.) to a given infinite field
$\mathbb{K}$ (or, to speak more graphically, of all labellings of the elements
of $P$ by elements of $\mathbb{K}$, along with two additional labels “at the
very bottom” and “at the very top”). The maps $\mathbf{t}_{v}$ are then
replaced by certain birational maps which we call birational $v$-toggles
(Definition 2.6); the resulting composition is called birational rowmotion and
denoted by $R$. By a careful limiting procedure (the tropical limit), we can
“degenerate” $R$ to the “piecewise linear” rowmotion of case (a), and thus it
can be seen as an even higher generalization of classical rowmotion. We refer
to the body of this paper for precise definitions of these maps. Note that
birational $v$-toggles (but not birational rowmotion) in the case of a
rectangle poset have also appeared in [OSZ13, (3.5)], but (apparently) have
not been composed there in a way that yields birational rowmotion.
As in the case of classical rowmotion on $J\left(P\right)$, the most
interesting question is the order of this map $R$, which in general no longer
has an obvious reason to be finite (since the affine variety of
$\mathbb{K}$-labellings is not a finite set like $J\left(P\right)$). Indeed,
for some posets $P$ this order is infinite. In this paper we will prove the
following facts:
* •
Birational rowmotion (i.e., the map $R$) on any graded poset (in the meaning
of this word introduced in Definition 3.3) has a very simple effect (namely,
cyclic shifting) on the so-called “w-tuple” of a labelling (a rather simple
fingerprint of the labelling). This does not mean $R$ itself has finite order
(but turns out to be crucial in proving this in several cases).
* •
Birational rowmotion on graded forests and, slightly more generally, skeletal
posets (Definition 9.5) has finite order (which can be bounded from above by
an iterative lcm, and also easily computed algorithmically). Moreover, its
order in these cases coincides with the order of classical rowmotion (Section
10).
* •
Birational rowmotion on a $p\times q$-rectangle has order $p+q$ and satisfies
a further symmetry property (Theorem 11.7). These results have originally been
conjectured by James Propp and the second author, and can be used as an
alternative route to certain properties of (Schützenberger’s) promotion map on
semistandard Young tableaux.
* •
Birational rowmotion on certain triangle-shaped posets (this is made precise
in Sections 17, 18, 19) also has finite order (computed explicitly below). We
show this for three kinds of triangle-shaped posets (obtained by cutting the
$p\times p$-square in two along either of its two diagonals) and conjecture it
for a fourth (a quarter of a $p\times p$-square obtained by cutting it along
both diagonals).
The proof of the most difficult and fundamental case – that of a $p\times
q$-rectangle – is inspired by Volkov’s proof of the “rectangular” (type-$AA$)
Zamolodchikov conjecture [Volk06], which uses a similar idea of parametrizing
(generic) $\mathbb{K}$-labellings by matrices (or tuples of points in
projective space). There is, of course, a striking similarity between the fact
itself and the Zamolodchikov conjecture; yet, we were not able to reduce
either result to the other.
Applications of the results of this paper (specifically Theorems 11.5 and
11.7) are found in [EiPr13]. Further directions currently under study of the
authors are relations to the totally positive Grassmannian and generalizations
to further classes of posets.
An extended (12-page) abstract [GrRo13] of this paper has been published in
the proceedings of the FPSAC 2014 conference.
For publication, this paper has been split into two parts (“Iterative
properties of birational rowmotion I: generalities and skeletal posets” and
“Iterative properties of birational rowmotion II: rectangles and triangles”),
which have been submitted to the Electronic Journal of Combinatorics. These
two parts have since received referee reports. All changes to the content (but
only few of the stylistic changes, and none of the shortenings) that were made
in response to the referee reports have been backported into the present
version of this paper.
### 0.1 Leitfaden
The following Hasse diagram shows how the sections of this paper depend upon
each other.
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{11\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{8\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{6\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{12\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{7\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{13\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{9\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{14\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{10}$$\textstyle{15\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{16\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{17\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{18\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{19}$
A section $n$ depends substantially on a section $m$ if and only if $m>n$ in
the poset whose Hasse diagram is depicted above. Only substantial dependencies
are shown; dependencies upon definitions do not count as substantial (e.g.,
many sections depend on Definition 7.1, but this does not make them
substantially dependent on Section 7), and dependencies which are only used in
proving inessential claims do not count (e.g., the proof of Theorem 11.5
relies on Proposition 7.3 in order to show that
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)=p+q$
rather than just
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)\mid
p+q$, but since the
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)\mid
p+q$ statement is in our opinion the only important part of the theorem, we do
not count this as a dependency on Section 7). Sections 20 and 21 are not
shown.
No section of this paper depends on the Introduction.
### 0.2 Acknowledgments
When confronted with the (then open) problem of proving what is Theorem 11.5
in this paper, Pavlo Pylyavskyy and Gregg Musiker suggested reading [Volk06].
This suggestion proved highly useful and built the cornerstone of this paper,
without which the latter would have ended at its “Skeletal posets” section.
The notion of birational rowmotion is due to James Propp and Arkady
Berenstein. This paper owes James Propp also for a constant flow of
inspiration and useful suggestions.
David Einstein found errors in our computations, and Hugh Thomas corrected
slips in the writing including an abuse of Zariski topology and some
accidental alternative history.
Nathan Williams noticed typos, too, and suggested a path connecting this
subject to the theory of minuscule posets (which we will not explore in this
paper).
The first author came to know birational rowmotion in Alexander Postnikov’s
combinatorics pre-seminar at MIT. Postnikov also suggested veins of further
study.
Jessica Striker helped the first author understand some of the past work on
this subject, in particular the labyrinthine connections between the various
operators (rowmotion, Panyushev complementation, Striker-Williams promotion,
Schützenberger promotion, etc.). The present paper explores merely one corner
of this labyrinth (the rowmotion corner).
We thank Dan Bump, Anne Schilling and the two referees of our FPSAC abstract
[GrRo13] for further helpful comments. We also owe a number of improvements in
this paper to the suggestions of two anonymous EJC referees.
Both authors were partially supported by NSF grant #1001905, and have utilized
the open-source CAS Sage ([S+09], [Sage08]) to perform laborious computations.
We thank Travis Scrimshaw, Frédéric Chapoton, Viviane Pons and Nathann Cohen
for reviewing Sage patches relevant to this project.
## 1 Linear extensions of posets
This first section serves to introduce some general notions concerning posets
and their linear extensions. In particular, we highlight that the set of
linear extensions of any finite poset is non-empty and connected by a simple
equivalence relation (Proposition 1.7). This will be used in subsequent
sections for defining the basic maps that we consider throughout the paper.
Let us first get a basic convention out of the way:
###### Convention 1.1.
We let $\mathbb{N}$ denote the set $\left\\{0,1,2,...\right\\}$.
We start by defining general notations related to posets:
###### Definition 1.2.
Let $P$ be a poset. Let $u\in P$ and $v\in P$. In this definition, we will use
$\leqslant$, $<$, $\geqslant$ and $>$ to denote the lesser-or-equal relation,
the lesser relation, the greater-or-equal relation and the greater relation,
respectively, of the poset $P$.
(a) The elements $u$ and $v$ of $P$ are said to be incomparable if we have
neither $u\leqslant v$ nor $u\geqslant v$.
(b) We write $u\lessdot v$ if we have $u<v$ and there is no $w\in P$ such that
$u<w<v$. One often says that “$u$ is covered by $v$” to signify that
$u\lessdot v$.
(c) We write $u\gtrdot v$ if we have $u>v$ and there is no $w\in P$ such that
$u>w>v$. (Thus, $u\gtrdot v$ holds if and only if $v\lessdot u$.) One often
says that “$u$ covers $v$” to signify that $u\gtrdot v$.
(d) An element $u$ of $P$ is called maximal if every $v\in P$ satisfying
$v\geqslant u$ satisfies $v=u$. It is easy to see that every nonempty finite
poset has at least one maximal element.
(e) An element $u$ of $P$ is called minimal if every $v\in P$ satisfying
$v\leqslant u$ satisfies $v=u$. It is easy to see that every nonempty finite
poset has at least one minimal element.
When any of these notations becomes ambiguous because the elements involved
belong to several different posets simultaneously, we will disambiguate it by
adding the words “in $P$” (where $P$ is the poset which we want to use).888For
instance, if $R$ denotes the poset $\mathbb{Z}$ endowed with the reverse of
its usual order, then we say (for instance) that “$1\lessdot 0$ in $R$” rather
than just “$1\lessdot 0$”.
###### Definition 1.3.
Let $P$ be a finite poset. A linear extension of $P$ will mean a list
$\left(v_{1},v_{2},...,v_{m}\right)$ of the elements of $P$ such that every
element of $P$ occurs exactly once in this list, and such that any
$i\in\left\\{1,2,...,m\right\\}$ and $j\in\left\\{1,2,...,m\right\\}$
satisfying $v_{i}<v_{j}$ (where $<$ is the smaller relation of $P$) must
satisfy $i<j$.
A brief remark on this definition is in order. Stanley, in [Stan11, one
paragraph below the proof of Proposition 3.5.2], defines a linear extension of
a poset $P$ as an order-preserving bijection from $P$ to the chain
$\left\\{1,2,...,\left|P\right|\right\\}$; this is equivalent to our
definition (indeed, our linear extension $\left(v_{1},v_{2},...,v_{m}\right)$,
whose length obviously is $m=\left|P\right|$, corresponds to the bijection
$P\rightarrow\left\\{1,2,...,\left|P\right|\right\\}$ which sends each $v_{i}$
to $i$). Another widespread definition of a linear extension of $P$ is as a
total order on $P$ compatible with the given order of the poset $P$; this is
equivalent to our definition as well (the total order is the one defined by
$v_{i}<v_{j}$ whenever $i<j$).
Notice that if $\left(v_{1},v_{2},...,v_{m}\right)$ is a linear extension of a
nonempty finite poset $P$, then $v_{1}$ is a minimal element of $P$ and
$v_{m}$ is a maximal element of $P$. The only linear extension of the empty
poset $\varnothing$ is the empty list $\left({}\right)$.
###### Theorem 1.4.
Let $P$ be a finite poset. Then, there exists a linear extension of $P$.
Theorem 1.4 is a well-known fact, and can be proven, e.g., by induction over
$\left|P\right|$ (with the induction step consisting of splitting off a
maximal element $u$ of $P$ and appending it to a linear extension of the
residual poset $P\setminus\left\\{u\right\\}$).
The following proposition can be easily checked by the reader:
###### Proposition 1.5.
Let $P$ be a finite poset. Let $\left(v_{1},v_{2},...,v_{m}\right)$ be a
linear extension of $P$. Let $i\in\left\\{1,2,...,m-1\right\\}$ be such that
the elements $v_{i}$ and $v_{i+1}$ of $P$ are incomparable. Then,
$\left(v_{1},v_{2},...,v_{i-1},v_{i+1},v_{i},v_{i+2},v_{i+3},...,v_{m}\right)$
(this is the tuple obtained from the tuple
$\left(v_{1},v_{2},...,v_{m}\right)$ by interchanging the adjacent entries
$v_{i}$ and $v_{i+1}$) is a linear extension of $P$ as well.
###### Definition 1.6.
Let $P$ be a finite poset. The set of all linear extensions of $P$ will be
called $\mathcal{L}\left(P\right)$. Thus,
$\mathcal{L}\left(P\right)\neq\varnothing$ (by Theorem 1.4).
In our approach to birational rowmotion, we will use the following fact (which
is folklore and has applications in various contexts, including Young tableau
theory):
###### Proposition 1.7.
Let $P$ be a finite poset. Let $\sim$ denote the equivalence relation on
$\mathcal{L}\left(P\right)$ generated by the following requirement: For any
linear extension $\left(v_{1},v_{2},...,v_{m}\right)$ of $P$ and any
$i\in\left\\{1,2,...,m-1\right\\}$ such that the elements $v_{i}$ and
$v_{i+1}$ of $P$ are incomparable, we set
$\left(v_{1},v_{2},...,v_{m}\right)\sim\left(v_{1},v_{2},...,v_{i-1},v_{i+1},v_{i},v_{i+2},v_{i+3},...,v_{m}\right)$
(noting that
$\left(v_{1},v_{2},...,v_{i-1},v_{i+1},v_{i},v_{i+2},v_{i+3},...,v_{m}\right)$
is also a linear extension of $P$, because of Proposition 1.5). 999Here is a
more formal way to restate this definition of $\sim$: We first introduce a
binary relation $\equiv$ on the set $\mathcal{L}\left(P\right)$ as follows: If
$\mathbf{v}$ and $\mathbf{w}$ are two linear extensions of $P$, then we set
$\mathbf{v}\equiv\mathbf{w}$ if and only if the list $\mathbf{w}$ can be
obtained from the list $\mathbf{v}$ by interchanging two adjacent entries $v$
and $v^{\prime}$ which are incomparable in $P$. It is clear that this binary
relation $\equiv$ is symmetric. It is also clear that for any linear extension
$\left(v_{1},v_{2},...,v_{m}\right)$ of $P$ and any
$i\in\left\\{1,2,...,m-1\right\\}$ such that the elements $v_{i}$ and
$v_{i+1}$ of $P$ are incomparable, the list
$\left(v_{1},v_{2},...,v_{i-1},v_{i+1},v_{i},v_{i+2},v_{i+3},...,v_{m}\right)$
is also a linear extension of $P$ (according to Proposition 1.5) and satisfies
$\left(v_{1},v_{2},...,v_{m}\right)\equiv\left(v_{1},v_{2},...,v_{i-1},v_{i+1},v_{i},v_{i+2},v_{i+3},...,v_{m}\right)$.
Now, we define $\sim$ as the reflexive and transitive closure of the binary
relation $\equiv$. Then, $\sim$ is an equivalence relation on
$\mathcal{L}\left(P\right)$. Then, any two elements of
$\mathcal{L}\left(P\right)$ are equivalent under the relation $\sim$.
This proposition is very basic (it generalizes the fact that the symmetric
group $S_{n}$ is generated by the adjacent-element transpositions) and is
classical, but a proof is hard to find in the literature. One proof is in
[AKSch12, Proposition 4.1 (for the $\pi^{\prime}=\pi\tau_{j}$ case)]; another
is sketched in [Rusk92, p. 79]. For the sake of completeness, we will prove it
too. Our proof is based on the following lemma (which is more or less a simple
particular case of Proposition 1.7):
###### Lemma 1.8.
Let $P$ be a finite poset. Define the equivalence relation $\sim$ on
$\mathcal{L}\left(P\right)$ as in Proposition 1.7. Let $a_{1}$, $a_{2}$,
$...$, $a_{k}$ be some elements of $P$. Let $b_{1}$, $b_{2}$, $...$,
$b_{\ell}$ be some further elements of $P$. Let $u$ be a maximal element of
$P$. Assume that
$\left(a_{1},a_{2},...,a_{k},u,b_{1},b_{2},...,b_{\ell}\right)$ is a linear
extension of $P$. Then,
$\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{\ell},u\right)$ is a linear
extension of $P$ satisfying
$\left(a_{1},a_{2},...,a_{k},u,b_{1},b_{2},...,b_{\ell}\right)\sim\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{\ell},u\right)$.
###### Proof of Lemma 1.8 (sketched)..
We will show that every $i\in\left\\{0,1,...,\ell\right\\}$ satisfies the
following assertion:
$\left(\begin{array}[c]{c}\text{The tuple
}\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{i},u,b_{i+1},b_{i+2},...,b_{\ell}\right)\text{
is a}\\\ \text{linear extension of }P\text{ satisfying}\\\
\left(a_{1},a_{2},...,a_{k},u,b_{1},b_{2},...,b_{\ell}\right)\sim\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{i},u,b_{i+1},b_{i+2},...,b_{\ell}\right)\end{array}\right).$
(1)
Proof of (1): We will prove (1) by induction over $i$:
Induction base: If $i=0$, then
$\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{i},u,b_{i+1},b_{i+2},...,b_{\ell}\right)=\left(a_{1},a_{2},...,a_{k},u,b_{1},b_{2},...,b_{\ell}\right)$.
Hence, (1) is a tautology for $i=0$, and the induction base is done.
Induction step: Let $I\in\left\\{1,2,...,\ell\right\\}$. Assume that (1) holds
for $i=I-1$. We need to prove that (1) holds for $i=I$.
We have assumed that (1) holds for $i=I-1$. In other words, the tuple
$\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{I-1},u,b_{I-1+1},b_{I-1+2},...,b_{\ell}\right)$
is a linear extension of $P$ satisfying
$\left(a_{1},a_{2},...,a_{k},u,b_{1},b_{2},...,b_{\ell}\right)\sim\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{I-1},u,b_{I-1+1},b_{I-1+2},...,b_{\ell}\right)$.
Denote the smaller relation of $P$ by $<$. Since the tuple
$\left(a_{1},a_{2},...,a_{k},u,b_{1},b_{2},...,b_{\ell}\right)$ is a linear
extension of $P$, we cannot have $u\geqslant b_{I}$ (because $u$ appears
strictly to the left of $b_{I}$ in this tuple). But we cannot have $u<b_{I}$
either (since $u$ is a maximal element of $P$). Thus, $u$ and $b_{I}$ are
incomparable.
Now,
$\displaystyle\left(a_{1},a_{2},...,a_{k},u,b_{1},b_{2},...,b_{\ell}\right)$
$\displaystyle\sim\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{I-1},u,b_{I-1+1},b_{I-1+2},...,b_{\ell}\right)$
$\displaystyle=\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{I-1},u,b_{I},b_{I+1},b_{I+2},...,b_{\ell}\right)$
$\displaystyle\sim\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{I-1},b_{I},u,b_{I+1},b_{I+2},...,b_{\ell}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{by the definition of the
relation }\sim\text{, since }u\text{ and }b_{I}\text{ are
incomparable}\right)$
$\displaystyle=\left(a_{1},a_{2},...,a_{k},b_{1},b_{2},...,b_{I},u,b_{I+1},b_{I+2},...,b_{\ell}\right).$
The proof of this equivalence also shows that its right hand side is a linear
extension of $P$. Thus, (1) holds for $i=I$. This completes the induction
step, whence (1) is proven.
Lemma 1.8 now follows by applying (1) to $i=\ell$.
∎
###### Proof of Proposition 1.7 (sketched)..
We prove Proposition 1.7 by induction over $\left|P\right|$. The induction
base $\left|P\right|=0$ is trivial. For the induction step, let $N$ be a
positive integer. Assume that Proposition 1.7 is proven for all posets $P$
with $\left|P\right|=N-1$. Now, let $P$ be a poset with $\left|P\right|=N$.
Let $\left(v_{1},v_{2},...,v_{N}\right)$ and
$\left(w_{1},w_{2},...,w_{N}\right)$ be two elements of
$\mathcal{L}\left(P\right)$. We are going to prove that
$\left(v_{1},v_{2},...,v_{N}\right)\sim\left(w_{1},w_{2},...,w_{N}\right)$.
Let $u=v_{N}$. Then, $u$ is a maximal element of $P$ (since it comes last in
the linear extension $\left(v_{1},v_{2},...,v_{N}\right)$). Let $i$ be the
index satisfying $w_{i}=u$.
Consider the poset $P\setminus\left\\{u\right\\}$. This poset has size
$\left|P\setminus\left\\{u\right\\}\right|=\underbrace{\left|P\right|}_{=N}-1=N-1$.
Define a relation $\sim$ on
$\mathcal{L}\left(P\setminus\left\\{u\right\\}\right)$ in the same way as the
relation $\sim$ on $\mathcal{L}\left(P\right)$ was defined. Recall that $u$ is
a maximal element of $P$. Hence,
$\left(\begin{array}[c]{c}\text{if }\left(a_{1},a_{2},...,a_{N-1}\right)\text{
is a linear extension of }P\setminus\left\\{u\right\\}\text{, then}\\\
\left(a_{1},a_{2},...,a_{N-1},u\right)\text{ is a linear extension of
}P\end{array}\right).$ (2)
Moreover, just by recalling how the relations $\sim$ were defined, we can
easily see that
$\left(\begin{array}[c]{c}\text{if two linear extensions
}\left(a_{1},a_{2},...,a_{N-1}\right)\text{ and
}\left(b_{1},b_{2},...,b_{N-1}\right)\text{ of
}P\setminus\left\\{u\right\\}\\\ \text{satisfy
}\left(a_{1},a_{2},...,a_{N-1}\right)\sim\left(b_{1},b_{2},...,b_{N-1}\right)\text{
in }\mathcal{L}\left(P\setminus\left\\{u\right\\}\right)\text{, then}\\\
\left(a_{1},a_{2},...,a_{N-1},u\right)\text{ and
}\left(b_{1},b_{2},...,b_{N-1},u\right)\text{ are two linear extensions}\\\
\text{of }P\text{ satisfying
}\left(a_{1},a_{2},...,a_{N-1},u\right)\sim\left(b_{1},b_{2},...,b_{N-1},u\right)\text{
in }\mathcal{L}\left(P\right)\end{array}\right)$ (3)
(here, the fact that $\left(a_{1},a_{2},...,a_{N-1},u\right)$ and
$\left(b_{1},b_{2},...,b_{N-1},u\right)$ are linear extensions of $P$ follows
from (2)).
It is rather clear that $\left(v_{1},v_{2},...,v_{N-1}\right)$ and
$\left(w_{1},w_{2},...,w_{i-1},w_{i+1},w_{i+2},...,w_{N}\right)$ are two
linear extensions of the poset $P\setminus\left\\{u\right\\}$ (since they are
obtained from the linear extensions $\left(v_{1},v_{2},...,v_{N}\right)$ and
$\left(w_{1},w_{2},...,w_{N}\right)$ of $P$ by removing $u$). Since we can
apply Proposition 1.7 to this poset $P\setminus\left\\{u\right\\}$ in lieu of
$P$ (by the induction hypothesis, since
$\left|P\setminus\left\\{u\right\\}\right|=N-1$), we thus see that
$\left(v_{1},v_{2},...,v_{N-1}\right)\sim\left(w_{1},w_{2},...,w_{i-1},w_{i+1},w_{i+2},...,w_{N}\right)$
in $\mathcal{L}\left(P\setminus\left\\{u\right\\}\right)$. By (3), this yields
that $\left(v_{1},v_{2},...,v_{N-1},u\right)$ and
$\left(w_{1},w_{2},...,w_{i-1},w_{i+1},w_{i+2},...,w_{N},u\right)$ are two
linear extensions of $P$ satisfying
$\left(v_{1},v_{2},...,v_{N-1},u\right)\sim\left(w_{1},w_{2},...,w_{i-1},w_{i+1},w_{i+2},...,w_{N},u\right)$
in $\mathcal{L}\left(P\right)$.
Now, we know that the tuple $\left(w_{1},w_{2},...,w_{N}\right)$ is a linear
extension of $P$. Since
$\displaystyle\left(w_{1},w_{2},...,w_{N}\right)$
$\displaystyle=\left(w_{1},w_{2},...,w_{i-1},\underbrace{w_{i}}_{=u},w_{i+1},w_{i+2},...,w_{N}\right)=\left(w_{1},w_{2},...,w_{i-1},u,w_{i+1},w_{i+2},...,w_{N}\right),$
this rewrites as follows: The tuple
$\left(w_{1},w_{2},...,w_{i-1},u,w_{i+1},w_{i+2},...,w_{N}\right)$ is a linear
extension of $P$. Hence, we can apply Lemma 1.8 to $k=i-1$, $\ell=N-i$,
$a_{j}=w_{j}$ and $b_{j}=w_{i+j}$. As a result, we see that
$\left(w_{1},w_{2},...,w_{i-1},w_{i+1},w_{i+2},...,w_{N},u\right)$ is a linear
extension of $P$ satisfying
$\left(w_{1},w_{2},...,w_{i-1},u,w_{i+1},w_{i+2},...,w_{N}\right)\sim\left(w_{1},w_{2},...,w_{i-1},w_{i+1},w_{i+2},...,w_{N},u\right)$.
Since the relation $\sim$ is symmetric (because $\sim$ is an equivalence
relation), this yields
$\left(w_{1},w_{2},...,w_{i-1},w_{i+1},w_{i+2},...,w_{N},u\right)\sim\left(w_{1},w_{2},...,w_{i-1},u,w_{i+1},w_{i+2},...,w_{N}\right).$
Altogether,
$\displaystyle\left(v_{1},v_{2},...,v_{N}\right)$
$\displaystyle=\left(v_{1},v_{2},...,v_{N-1},\underbrace{v_{N}}_{=u}\right)=\left(v_{1},v_{2},...,v_{N-1},u\right)$
$\displaystyle\sim\left(w_{1},w_{2},...,w_{i-1},w_{i+1},w_{i+2},...,w_{N},u\right)$
$\displaystyle\sim\left(w_{1},w_{2},...,w_{i-1},\underbrace{u}_{=w_{i}},w_{i+1},w_{i+2},...,w_{N}\right)$
$\displaystyle=\left(w_{1},w_{2},...,w_{i-1},w_{i},w_{i+1},w_{i+2},...,w_{N}\right)=\left(w_{1},w_{2},...,w_{N}\right).$
We thus have shown that any two elements $\left(v_{1},v_{2},...,v_{N}\right)$
and $\left(w_{1},w_{2},...,w_{N}\right)$ of $\mathcal{L}\left(P\right)$
satisfy
$\left(v_{1},v_{2},...,v_{N}\right)\sim\left(w_{1},w_{2},...,w_{N}\right)$. In
other words, Proposition 1.7 is proven for $\left|P\right|=N$, so the
induction step is complete, and Proposition 1.7 is proven. ∎
## 2 Birational rowmotion
In this section, we introduce the basic objects whose nature we will
investigate: labellings of a finite poset $P$ (by elements of a field) and a
birational map between them called “birational rowmotion”. This map
generalizes (in a certain sense) the notion of ordinary rowmotion on the set
$J\left(P\right)$ of order ideals of $P$ to the vastly more general setting of
field-valued labellings. We will discuss the technical concerns raised by the
definitions, and provide two examples and an alternative description of
birational rowmotion. A deeper study of birational rowmotion is deferred to
the following sections.
The concepts which we are going to define now go back to [EiPr13] and earlier
sources, and are often motivated there. The reader should be warned that the
notations used in [EiPr13] are not identical with those used in the present
paper (not to mention that [EiPr13] is working over $\mathbb{R}_{+}$ rather
than over fields as we do).
###### Definition 2.1.
Let $P$ be a poset. Then, $\widehat{P}$ will denote the poset defined as
follows: As a set, let $\widehat{P}$ be the disjoint union of the set $P$ with
the two-element set $\left\\{0,1\right\\}$. The smaller-or-equal relation
$\leqslant$ on $\widehat{P}$ will be given by
$\left(a\leqslant b\right)\Longleftrightarrow\left(\text{either }\left(a\in
P\text{ and }b\in P\text{ and }a\leqslant b\text{ in }P\right)\text{ or
}a=0\text{ or }b=1\right)$ 101010Here and in the following, the expression
“either/or” always has a non-exclusive meaning. (Thus, in particular,
$0\leqslant 1$ in $\widehat{P}$.)
. Here and in the following, we regard the canonical injection of the set $P$
into the disjoint union $\widehat{P}$ as an inclusion; thus, $P$ becomes a
subposet of $\widehat{P}$. In the terminology of Stanley’s [Stan11, section
3.2], this poset $\widehat{P}$ is the ordinal sum $\left\\{0\right\\}\oplus
P\oplus\left\\{1\right\\}$.
###### Convention 2.2.
Let $P$ is a finite poset, and let $u$ and $v$ be two elements of $P$. Then,
$u$ and $v$ are also elements of $\widehat{P}$ (since we are regarding $P$ as
a subposet of $\widehat{P}$). Thus, strictly speaking, statements like “$u<v$”
or “$u\lessdot v$” are ambiguous because it is not clear whether they are
referring to the poset $P$ or to the poset $\widehat{P}$. However, this
ambiguity is irrelevant, because it is easily seen that the truth of each of
the statements “$u<v$”, “$u\leqslant v$”, “$u>v$”, “$u\geqslant v$”,
“$u\lessdot v$”, “$u\gtrdot v$” and “$u$ and $v$ are incomparable” is
independent on whether it refers to the poset $P$ or to the poset
$\widehat{P}$. We are going to therefore omit mentioning the poset in these
statements, unless there are other reasons for us to do so.
###### Definition 2.3.
Let $P$ be a poset. Let $\mathbb{K}$ be a field. A $\mathbb{K}$-labelling of
$P$ will mean a map $f:\widehat{P}\rightarrow\mathbb{K}$. Thus,
$\mathbb{K}^{\widehat{P}}$ is the set of all $\mathbb{K}$-labellings of $P$.
If $f$ is a $\mathbb{K}$-labelling of $P$ and $v$ is an element of
$\widehat{P}$, then $f\left(v\right)$ will be called the label of $f$ at $v$.
###### Definition 2.4.
In the following, whenever we are working with a field $\mathbb{K}$, we are
going to tacitly assume that $\mathbb{K}$ is either infinite or at least can
be enlarged when necessity arises. This assumption is needed in order to
clarify the notions of rational maps and generic elements of algebraic
varieties over $\mathbb{K}$. (We will not require $\mathbb{K}$ to be
algebraically closed.)
We will use the terminology of algebraic varieties and rational maps between
them, although the only algebraic varieties that we will be considering are
products of affine and projective spaces, as well as their open subsets. We
use the punctured arrow $\dashrightarrow$ to signify rational maps (i.e., a
rational map from a variety $U$ to a variety $V$ is called a rational map
$U\dashrightarrow V$). A rational map $U\dashrightarrow V$ is said to be
dominant if its image is dense in $V$ (with respect to the Zariski topology).
The words “generic” and “almost” will always refer to the Zariski topology.
For example, if $U$ is a finite set, then an assertion saying that some
statement holds “for almost every point $p\in\mathbb{K}^{U}$” is supposed to
mean that there is a Zariski-dense open subset $D$ of $\mathbb{K}^{U}$ such
that this statement holds for every point $p\in D$. A “generic” point on an
algebraic variety $V$ (for example, this can be a “generic matrix” when $V$ is
a space of matrices, or a “generic $\mathbb{K}$-labelling of a poset $P$” when
$V$ is the space of all $\mathbb{K}$-labellings of $P$) means a point lying in
some fixed Zariski-dense open subset $S$ of $V$; the concrete definition of
$S$ can usually be inferred from the context (often, it will be the subset of
$V$ on which everything we want to do with our point is well-defined), but of
course should never depend on the actual point. (Note that one often has to
read the whole proof in order to be able to tell what this $S$ is. This is
similar to the use of the “for $\epsilon$ small enough” wording in analysis,
where it is often not clear until the end of the proof how small exactly the
$\epsilon$ needs to be.) We are sometimes going to abuse notation and say that
an equality holds “for every point” instead of “for almost every point” when
it is really clear what the $S$ is. (For example, if we say that “the equality
$\dfrac{x^{3}-y^{3}}{x-y}=x^{2}+xy+y^{2}$ holds for every $x\in\mathbb{K}$ and
$y\in\mathbb{K}$”, it is clear that $S$ has to be the set
$\mathbb{K}^{2}\setminus\left\\{\left(x,y\right)\in\mathbb{K}^{2}\mid
x=y\right\\}$, because the left hand side of the equality makes no sense when
$\left(x,y\right)$ is outside of this set.)
###### Remark 2.5.
Most statements that we make below work not only for fields, but also more
generally for semifields111111The word “semifield” here means a commutative
semiring in which each element other than $0$ has a multiplicative inverse.
(In contrast to other authors’ conventions, our semifields do have zeroes.) A
semiring is defined as a set with two binary operations called “addition” and
“multiplication” and two elements $0$ and $1$ which satisfies all axioms of a
ring (in particular, it must be associative and satisfy $0\cdot a=a\cdot 0=0$
and $1\cdot a=a\cdot 1=a$ for all $a$) except for having additive inverses.
such as the semifield $\mathbb{Q}_{+}$ of positive rationals or the tropical
semiring. Some (but not all!) statements actually simplify when the underlying
field is replaced by a semifield in which no two nonzero elements add to zero
(because in such cases, e.g., the denominators in (4) cannot become zero
unless some labels of $f$ are $0$). Thus, working with such semifields instead
of fields would save us the trouble of having things defined “almost
everywhere”. Moreover, applying our results to the tropical semifield would
yield some of the statements about order polytopes made in [EiPr13].
Nevertheless, we prefer to work with fields, for the following reasons:
– While most of our results can be formulated for semifields, not all of them
can (and sometimes, even when a result holds over semifields, its proof might
not work over semifields). In particular, Proposition 13.13 makes no sense
over semifields, because determinants involve subtraction. Also, if we were to
work in semifields which do contain two nonzero elements summing up to zero,
then we would still have the issue of zero denominators, but we are not aware
of a theoretical framework in the spirit of Zariski topology for fields to
reassure us in this case that these issues are negligible.
– If an identity between subtraction-free rational expressions (such as
$\dfrac{x^{3}+y^{3}}{x+y}+3xy=\left(x+y\right)^{2}$) holds over every field
(as long as the denominators involved are nonzero), then it must hold over
every semifield as well (again as long as the denominators involved are
nonzero), even if the identity has only been proven with the help of
subtraction (e.g., a proof of
$\dfrac{x^{3}+y^{3}}{x+y}+3xy=\left(x+y\right)^{2}$ over a field can begin by
simplifying $\dfrac{x^{3}+y^{3}}{x+y}$ to $x^{2}-xy+y^{2}$, a technique not
available over a semifield). This is simply because every true identity
between subtraction-free rational expressions can be verified by multiplying
by a common denominator (an operation which does not introduce any
subtractions) and comparing coefficients. Since our main results (such as
Theorem 11.7, or the
$p+q\mid\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)$
part of Theorem 11.5) can be construed as identities between subtraction-free
rational expressions, this yields that all these results hold over any
semifield (provided the denominators are nonzero) if they hold over every
field. So we are not losing any generality by restricting ourselves to
considering only fields.
###### Definition 2.6.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let $v\in P$. We
define a rational map
$T_{v}:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P}}$ by
$\left(T_{v}f\right)\left(w\right)=\left\\{\begin{array}[c]{l}f\left(w\right),\
\ \ \ \ \ \ \ \ \ \text{if }w\neq v;\\\
\dfrac{1}{f\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
v\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{f\left(u\right)}},\ \ \ \ \ \ \ \ \ \
\text{if }w=v\end{array}\right.\ \ \ \ \ \ \ \ \ \ \text{for all
}w\in\widehat{P}$ (4)
for all $f\in\mathbb{K}^{\widehat{P}}$. Note that this rational map $T_{v}$ is
well-defined, because the right-hand side of (4) is well-defined on a Zariski-
dense open subset of $\mathbb{K}^{\widehat{P}}$. (This follows from the fact
that for every $v\in P$, there is at least one $u\in\widehat{P}$ such that
$u\gtrdot v$ 121212Indeed, either there is at least one $u\in P$ such that
$u\gtrdot v$ in $P$ (and therefore also $u\gtrdot v$ in $\widehat{P}$), or
else $v$ is maximal in $P$ and then we have $1\gtrdot v$ in $\widehat{P}$..)
This rational map $T_{v}$ is called the $v$-toggle.
The map $T_{v}$ that we have just introduced (although defined over the
semifield $\mathbb{R}_{+}$ instead of our field $\mathbb{K}$) is called a
“birational toggle operation” in [EiPr13] (where it is denoted by $\phi_{i}$
with $i$ being a number indexing the elements $v$ of $P$; however, the same
notation is used for the “tropicalized” version of $T_{v}$). As is clear from
its definition, it only changes the label at the element $v$.
Note also the following almost trivial fact:
###### Proposition 2.7.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let $v\in P$. Then,
the rational map $T_{v}$ is an involution, i.e., the map $T_{v}^{2}$ is well-
defined on a Zariski-dense open subset of $\mathbb{K}^{\widehat{P}}$ and
satisfies $T_{v}^{2}=\operatorname*{id}$ on this subset.
We are calling this “almost trivial” because one subtlety is easily
overlooked: We have to check that the map $T_{v}^{2}$ is well-defined on a
Zariski-dense open subset of $\mathbb{K}^{\widehat{P}}$; this requires
observing that for every $v\in P$, there exists at least one $u\in\widehat{P}$
such that $u\lessdot v$.
Proposition 2.7 yields the following:
###### Corollary 2.8.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let $v\in P$. Then,
the map $T_{v}$ is a dominant rational map.
The reader should remember that dominant rational maps (unlike general
rational maps) can be composed, and their compositions are still dominant
rational maps. Of course, we are brushing aside subtleties like the fact that
dominant rational maps are defined only over infinite fields (unless we are
considering them in a sufficiently formal sense); as far as this paper is
concerned, it never hurts to extend the field $\mathbb{K}$ (say, by
introducing a new indeterminate), so when in doubt the reader can assume that
the field $\mathbb{K}$ is infinite.
The following proposition is trivially obtained by rewriting (4); we are
merely stating it for easier reference in proofs:
###### Proposition 2.9.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let $v\in P$. For
every $f\in\mathbb{K}^{\widehat{P}}$ for which $T_{v}f$ is well-defined, we
have:
(a) Every $w\in\widehat{P}$ such that $w\neq v$ satisfies
$\left(T_{v}f\right)\left(w\right)=f\left(w\right)$.
(b) We have
$\left(T_{v}f\right)\left(v\right)=\dfrac{1}{f\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
v\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{f\left(u\right)}}.$
It is very easy to check the following “locality principle”:
###### Proposition 2.10.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let $v\in P$ and $w\in
P$. Then, $T_{v}\circ T_{w}=T_{w}\circ T_{v}$, unless we have either
$v\lessdot w$ or $w\lessdot v$.
###### Proof of Proposition 2.10 (sketched)..
Assume that neither $v\lessdot w$ nor $w\lessdot v$. Also, WLOG, assume that
$v\neq w$, lest the claim of the proposition be obvious.
The action of $T_{v}$ on a labelling of $P$ merely changes the label at $v$.
The new value depends on the label at $v$, on the labels at the elements
$u\in\widehat{P}$ satisfying $u\lessdot v$, and on the labels at the elements
$u\in\widehat{P}$ satisfying $u\gtrdot v$. A similar thing can be said about
the action of $T_{w}$. Since we have neither $v\lessdot w$ nor $w\lessdot v$
nor $v=w$, it thus becomes clear that the actions of $T_{v}$ and $T_{w}$ don’t
interfere with each other, in the sense that the changes made by either of
them are the same no matter whether the other has been applied before it or
not. That is, $T_{v}\circ T_{w}=T_{w}\circ T_{v}$, so that Proposition 2.10 is
proven. ∎
###### Corollary 2.11.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let $v$ and $w$ be two
elements of $P$ which are incomparable. Then, $T_{v}\circ T_{w}=T_{w}\circ
T_{v}$.
###### .
This follows from Proposition 2.10 because incomparable elements never cover
each other. ∎
Combining Corollary 2.11 with Proposition 1.7, we obtain:
###### Corollary 2.12.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let
$\left(v_{1},v_{2},...,v_{m}\right)$ be a linear extension of $P$. Then, the
dominant rational map $T_{v_{1}}\circ T_{v_{2}}\circ...\circ
T_{v_{m}}:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P}}$ is
well-defined and independent of the choice of the linear extension
$\left(v_{1},v_{2},...,v_{m}\right)$.
###### Definition 2.13.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Birational rowmotion
is defined as the dominant rational map $T_{v_{1}}\circ T_{v_{2}}\circ...\circ
T_{v_{m}}:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P}}$,
where $\left(v_{1},v_{2},...,v_{m}\right)$ is a linear extension of $P$. This
rational map is well-defined (in particular, it does not depend on the linear
extension $\left(v_{1},v_{2},...,v_{m}\right)$ chosen) because of Corollary
2.12 (and also because a linear extension of $P$ always exists; this is
Theorem 1.4). This rational map will be denoted by $R$.
The reason for the names “birational toggle” and “birational rowmotion” is
explained in the paper [EiPr13], in which birational rowmotion (again, defined
over $\mathbb{R}_{+}$ rather than over $\mathbb{K}$) is denoted
(serendipitously from the standpoint of the second author of this paper) by
$\rho_{\mathcal{B}}$.
###### Example 2.14.
Let us demonstrate the effect of birational toggles and birational rowmotion
on a rather simple $4$-element poset. Namely, for this example, we let $P$ be
the poset $\left\\{p,q_{1},q_{2},q_{3}\right\\}$ with order relation defined
by setting $p<q_{i}$ for each $i\in\left\\{1,2,3\right\\}$. This poset has
Hasse diagram
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.81145pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\&&\crcr}}}\ignorespaces{\hbox{\kern-6.81145pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
30.81145pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.43434pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{q_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-23.0499pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
32.10727pt\raise-23.0499pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{p}$}}}}}}}{\hbox{\kern
72.24579pt\raise-23.0499pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
The extended poset $\widehat{P}$ has Hasse diagram
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.81145pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\&&\\\&&\\\&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 32.1229pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
72.24579pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-6.81145pt\raise-23.14713pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
30.81145pt\raise-23.14713pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.43434pt\raise-23.14713pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{q_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-46.19702pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
32.10727pt\raise-46.19702pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{p\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.24579pt\raise-46.19702pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-69.34415pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
32.1229pt\raise-69.34415pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern
72.24579pt\raise-69.34415pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
We can visualize a $\mathbb{K}$-labelling $f$ of $P$ by replacing, in the
Hasse diagram of $\widehat{P}$, each element $v\in\widehat{P}$ by the label
$f\left(v\right)$. Let $f$ be a $\mathbb{K}$-labelling sending $0$, $p$,
$q_{1}$, $q_{2}$, $q_{3}$, and $1$ to $a$, $w$, $x_{1}$, $x_{2}$, $x_{3}$, and
$b$, respectively (for some elements $a$, $b$, $w$, $x_{1}$, $x_{2}$, $x_{3}$
of $\mathbb{K}$); this $f$ is then visualized as follows:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.25763pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\&&\\\&&\\\&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 33.36943pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
74.03052pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-7.25763pt\raise-23.32713pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
31.25763pt\raise-23.32713pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
69.77289pt\raise-23.32713pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-45.33482pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
31.80113pt\raise-45.33482pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
74.03052pt\raise-45.33482pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-66.44029pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
32.87231pt\raise-66.44029pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a}$}}}}}}}{\hbox{\kern
74.03052pt\raise-66.44029pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
Now, recall the definition of birational rowmotion $R$ on our poset $P$. Since
$\left(p,q_{1},q_{2},q_{3}\right)$ is a linear extension of $P$, we have
$R=T_{p}\circ T_{q_{1}}\circ T_{q_{2}}\circ T_{q_{3}}$. Let us track how this
transforms our labelling $f$:
We first apply $T_{q_{3}}$, obtaining
$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{q_{3}}f=\
}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}\frac{bw}{x_{3}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a}$
(where we colored the label at $q_{3}$ red to signify that it is the label at
the element which got toggled). Indeed, the only label that changes under
$T_{q_{3}}$ is the one at $q_{3}$, and this label becomes
$\left(T_{q_{3}}f\right)\left(q_{3}\right)=\dfrac{1}{f\left(q_{3}\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
q_{3}\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot
q_{3}\end{subarray}}\dfrac{1}{f\left(u\right)}}=\dfrac{1}{f\left(q_{3}\right)}\cdot\dfrac{f\left(p\right)}{\left(\dfrac{1}{f\left(1\right)}\right)}=\dfrac{1}{x_{3}}\cdot\dfrac{w}{\left(\dfrac{1}{b}\right)}=\dfrac{bw}{x_{3}}.$
Having applied $T_{q_{3}}$, we next apply $T_{q_{2}}$, obtaining
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
26.68622pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 54.94385pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 91.15761pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
131.50735pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-26.68622pt\raise-24.9256pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T_{q_{2}}T_{q_{3}}f=\ }$}}}}}}}{\hbox{\kern
50.68622pt\raise-24.9256pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
89.20148pt\raise-24.9256pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}\frac{bw}{x_{2}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
127.4054pt\raise-24.9256pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{bw}{x_{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-48.53175pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
54.94385pt\raise-48.53175pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
89.58931pt\raise-48.53175pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
131.50735pt\raise-48.53175pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-69.63722pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
54.94385pt\raise-69.63722pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
90.66049pt\raise-69.63722pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a}$}}}}}}}{\hbox{\kern
131.50735pt\raise-69.63722pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
Next, we apply $T_{q_{1}}$, obtaining
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
32.45296pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 60.55492pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 96.613pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
136.96275pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-32.45296pt\raise-24.9256pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T_{q_{1}}T_{q_{2}}T_{q_{3}}f=\ }$}}}}}}}{\hbox{\kern
56.45296pt\raise-24.9256pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}\frac{bw}{x_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
94.65688pt\raise-24.9256pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{bw}{x_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
132.8608pt\raise-24.9256pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{bw}{x_{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-48.53175pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
60.55492pt\raise-48.53175pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
95.04471pt\raise-48.53175pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
136.96275pt\raise-48.53175pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-69.63722pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
60.55492pt\raise-69.63722pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
96.11589pt\raise-69.63722pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a}$}}}}}}}{\hbox{\kern
136.96275pt\raise-69.63722pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
Finally, we apply $T_{p}$, resulting in
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
37.47803pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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162.67487pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-37.47803pt\raise-24.9256pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T_{p}T_{q_{1}}T_{q_{2}}T_{q_{3}}f=\
}$}}}}}}}{\hbox{\kern 61.47803pt\raise-24.9256pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{bw}{x_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
110.02547pt\raise-24.9256pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{bw}{x_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
158.5729pt\raise-24.9256pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\frac{bw}{x_{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-51.49562pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
65.57999pt\raise-51.49562pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
99.68195pt\raise-51.49562pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}\frac{ab}{x_{1}+x_{2}+x_{3}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
162.67487pt\raise-51.49562pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-75.56496pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
65.57999pt\raise-75.56496pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
111.48448pt\raise-75.56496pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a}$}}}}}}}{\hbox{\kern
162.67487pt\raise-75.56496pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
since the birational $p$-toggle $T_{p}$ has changed the label at $p$ to
$\displaystyle\left(T_{p}T_{q_{1}}T_{q_{2}}T_{q_{3}}f\right)\left(p\right)$
$\displaystyle=\dfrac{1}{\left(T_{q_{1}}T_{q_{2}}T_{q_{3}}f\right)\left(p\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
p\end{subarray}}\left(T_{q_{1}}T_{q_{2}}T_{q_{3}}f\right)\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot
p\end{subarray}}\dfrac{1}{\left(T_{q_{1}}T_{q_{2}}T_{q_{3}}f\right)\left(u\right)}}$
$\displaystyle=\dfrac{1}{\left(T_{q_{1}}T_{q_{2}}T_{q_{3}}f\right)\left(p\right)}\cdot\dfrac{\left(T_{q_{1}}T_{q_{2}}T_{q_{3}}f\right)\left(0\right)}{\dfrac{1}{\left(T_{q_{1}}T_{q_{2}}T_{q_{3}}f\right)\left(q_{1}\right)}+\dfrac{1}{\left(T_{q_{1}}T_{q_{2}}T_{q_{3}}f\right)\left(q_{2}\right)}+\dfrac{1}{\left(T_{q_{1}}T_{q_{2}}T_{q_{3}}f\right)\left(q_{3}\right)}}$
$\displaystyle=\dfrac{1}{w}\cdot\dfrac{a}{\dfrac{1}{bw\diagup
x_{1}}+\dfrac{1}{bw\diagup x_{2}}+\dfrac{1}{bw\diagup
x_{3}}}=\dfrac{ab}{x_{1}+x_{2}+x_{3}}.$
We thus have computed $Rf$ (since
$R=T_{p}T_{q_{1}}T_{q_{2}}T_{q_{3}}T_{q_{4}}$). By repeating this procedure
(or just substituting the labels of $Rf$ obtained as variables), we can
compute $R^{2}f$, $R^{3}f$ etc.. Specifically, we obtain
$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.98781pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 95.08977pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
288.18465pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-18.98781pt\raise-24.85406pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Rf=\
}$}}}}}}}{\hbox{\kern 90.98781pt\raise-24.85406pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{bw}{x_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
187.53525pt\raise-24.85406pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{bw}{x_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
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0.0pt\hbox{$\textstyle{\frac{bw}{x_{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-51.35255pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
95.08977pt\raise-51.35255pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
177.19173pt\raise-51.35255pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{ab}{x_{1}+x_{2}+x_{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
288.18465pt\raise-51.35255pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-75.42189pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
95.08977pt\raise-75.42189pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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188.99426pt\raise-75.42189pt\hbox{\hbox{\kern 0.0pt\raise
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288.18465pt\raise-75.42189pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.38782pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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76.15541pt\raise-55.87341pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
137.923pt\raise-55.87341pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{ax_{1}x_{2}x_{3}}{w\left(x_{2}x_{3}+x_{3}x_{1}+x_{1}x_{2}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
257.22786pt\raise-55.87341pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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76.15541pt\raise-80.4775pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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164.04869pt\raise-80.4775pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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257.22786pt\raise-80.4775pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.38782pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R^{3}f=\
}$}}}}}}}{\hbox{\kern 56.38782pt\raise-26.45824pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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There are several patterns here that catch the eye, some of which are related
to the very simple structure of $P$ and don’t seem to generalize well.
However, the most striking observation here is that $R^{n}f=f$ for some
positive integer $n$ (namely, $n=6$ for this particular $P$). We will see in
Proposition 9.7 that this generalizes to a rather wide class of posets, which
we call “skeletal posets” (defined in Definition 9.5), a class of posets which
contain (in particular) all graded forests such as our poset $P$ here. (See
Definition 9.5 for the definitions of the concepts involved here.)
###### Example 2.15.
Let us demonstrate the effect of birational toggles and birational rowmotion
on another $4$-element poset. Namely, for this example, we let $P$ be the
poset $\left\\{1,2\right\\}\times\left\\{1,2\right\\}$ with order relation
defined by setting
$\left(i,k\right)\leqslant\left(i^{\prime},k^{\prime}\right)$ if and only if
$\left(i\leqslant i^{\prime}\text{ and }k\leqslant k^{\prime}\right)$. This
poset will later be called the “$2\times 2$-rectangle” in Definition 11.1. It
has Hasse diagram
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The extended poset $\widehat{P}$ has Hasse diagram
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16.51108pt\raise-78.622pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\left(1,1\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
58.24438pt\raise-78.622pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-103.64413pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
25.1222pt\raise-103.64413pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern
58.24438pt\raise-103.64413pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
We can visualize a $\mathbb{K}$-labelling $f$ of $P$ by replacing, in the
Hasse diagram of $\widehat{P}$, each element $v\in\widehat{P}$ by the label
$f\left(v\right)$. Let $f$ be a $\mathbb{K}$-labelling sending $0$,
$\left(1,1\right)$, $\left(1,2\right)$, $\left(2,1\right)$,
$\left(2,2\right)$, and $1$ to $a$, $w$, $y$, $x$, $z$, and $b$, respectively
(for some elements $a$, $b$, $x$, $y$, $z$, $w$ of $\mathbb{K}$); this $f$ is
then visualized as follows:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
15.15274pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 20.41034pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 33.23624pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
50.12695pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-22.42491pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
20.41034pt\raise-22.42491pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
32.83691pt\raise-22.42491pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
50.12695pt\raise-22.42491pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-15.15274pt\raise-45.82205pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{f=\
}$}}}}}}}{\hbox{\kern 17.5527pt\raise-45.82205pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
35.38206pt\raise-45.82205pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
47.49615pt\raise-45.82205pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-69.2192pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
20.41034pt\raise-69.2192pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
31.66794pt\raise-69.2192pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
50.12695pt\raise-69.2192pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-90.32466pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
20.41034pt\raise-90.32466pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
32.73912pt\raise-90.32466pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a}$}}}}}}}{\hbox{\kern
50.12695pt\raise-90.32466pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
Now, recall the definition of birational rowmotion $R$ on our poset $P$. Since
$\left(\left(1,1\right),\left(1,2\right),\left(2,1\right),\left(2,2\right)\right)$
is a linear extension of $P$, we have $R=T_{\left(1,1\right)}\circ
T_{\left(1,2\right)}\circ T_{\left(2,1\right)}\circ T_{\left(2,2\right)}$. Let
us track how this transforms our labelling $f$:
We first apply $T_{\left(2,2\right)}$, obtaining
$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}\frac{b\left(x+y\right)}{z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{\left(2,2\right)}f=\
}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a}$
(where we colored the label at $\left(2,2\right)$ red to signify that it is
the label at the element which got toggled). Indeed, the only label that
changes under $T_{\left(2,2\right)}$ is the one at $\left(2,2\right)$, and
this label becomes
$\displaystyle\left(T_{\left(2,2\right)}f\right)\left(2,2\right)$
$\displaystyle=\dfrac{1}{f\left(\left(2,2\right)\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot\left(2,2\right)\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot\left(2,2\right)\end{subarray}}\dfrac{1}{f\left(u\right)}}=\dfrac{1}{f\left(\left(2,2\right)\right)}\cdot\dfrac{f\left(\left(1,2\right)\right)+f\left(\left(2,1\right)\right)}{\left(\dfrac{1}{f\left(1\right)}\right)}$
$\displaystyle=\dfrac{1}{z}\cdot\dfrac{y+x}{\left(\dfrac{1}{b}\right)}=\dfrac{b\left(x+y\right)}{z}.$
Having applied $T_{\left(2,2\right)}$, we next apply $T_{\left(2,1\right)}$,
obtaining
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
33.89651pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 39.15411pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 59.05432pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
93.77687pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-25.27908pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
39.15411pt\raise-25.27908pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
50.41171pt\raise-25.27908pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{b\left(x+y\right)}{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
93.77687pt\raise-25.27908pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-33.89651pt\raise-53.42969pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T_{\left(2,1\right)}T_{\left(2,2\right)}f=\
}$}}}}}}}{\hbox{\kern 36.29648pt\raise-53.42969pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
61.20015pt\raise-53.42969pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
80.38855pt\raise-53.42969pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}\frac{b\left(x+y\right)w}{yz}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-78.72612pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
39.15411pt\raise-78.72612pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
57.48602pt\raise-78.72612pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
93.77687pt\raise-78.72612pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-99.83159pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
39.15411pt\raise-99.83159pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
58.5572pt\raise-99.83159pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{a}$}}}}}}}{\hbox{\kern
93.77687pt\raise-99.83159pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
Next, we apply $T_{\left(1,2\right)}$, obtaining
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
43.2684pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 59.05669pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 89.48758pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
124.21013pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-25.27908pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
59.05669pt\raise-25.27908pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
80.84497pt\raise-25.27908pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{b\left(x+y\right)}{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
124.21013pt\raise-25.27908pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-43.2684pt\raise-53.42969pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T_{\left(1,2\right)}T_{\left(2,1\right)}T_{\left(2,2\right)}f=\
}$}}}}}}}{\hbox{\kern 45.66837pt\raise-53.42969pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}\frac{b\left(x+y\right)w}{xz}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
91.6334pt\raise-53.42969pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
110.82181pt\raise-53.42969pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{b\left(x+y\right)w}{yz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-78.72612pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
59.05669pt\raise-78.72612pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
87.91928pt\raise-78.72612pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
124.21013pt\raise-78.72612pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-99.83159pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
59.05669pt\raise-99.83159pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
88.99046pt\raise-99.83159pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a}$}}}}}}}{\hbox{\kern
124.21013pt\raise-99.83159pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
Finally, we apply $T_{\left(1,1\right)}$, resulting in
$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{b\left(x+y\right)}{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf=T_{\left(1,1\right)}T_{\left(1,2\right)}T_{\left(2,1\right)}T_{\left(2,2\right)}f=\
}$$\textstyle{\frac{b\left(x+y\right)w}{xz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{b\left(x+y\right)w}{yz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}\frac{ab}{z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a}$
(after cancelling terms). We thus have computed $Rf$. By repeating this
procedure (or just substituting the labels of $Rf$ obtained as variables), we
can compute $R^{2}f$, $R^{3}f$ etc.. Specifically, we obtain
$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.98781pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 34.7761pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
34.7761pt\raise-25.27908pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
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There are two surprises here. First, it turns out that $R^{4}f=f$. This is not
obvious, but generalizes in at least two ways: On the one hand, our poset $P$
is a particular case of what we call a “skeletal poset” (Definition 9.5), a
class of posets which all share the property (Proposition 9.7) that
$R^{n}=\operatorname*{id}$ for some sufficiently high positive integer $n$
(which can be explicitly computed). On the other hand, our poset $P$ is a
particular case of rectangle posets, which turn out (Theorem 11.5) to satisfy
$R^{p+q}=\operatorname*{id}$ with $p$ and $q$ being the side lengths (here,
$2$ and $2$) of the rectangle. Second, on a more subtle level, the rational
functions appearing as labels in $Rf$, $R^{2}f$ and $R^{3}f$ are not as “wild”
as one might expect. The values
$\left(Rf\right)\left(\left(1,1\right)\right)$,
$\left(R^{2}f\right)\left(\left(1,2\right)\right)$,
$\left(R^{2}f\right)\left(\left(2,1\right)\right)$ and
$\left(R^{3}f\right)\left(\left(2,2\right)\right)$ each have the form
$\dfrac{ab}{f\left(v\right)}$ for some $v\in P$. This is a “reciprocity”
phenomenon which turns out to generalize to arbitrary rectangles (Theorem
11.7).
In the above calculation, we used the linear extension
$\left(\left(1,1\right),\left(1,2\right),\left(2,1\right),\left(2,2\right)\right)$
of $P$ to compute $R$ as $T_{\left(1,1\right)}\circ T_{\left(1,2\right)}\circ
T_{\left(2,1\right)}\circ T_{\left(2,2\right)}$. We could have just as well
used the linear extension
$\left(\left(1,1\right),\left(2,1\right),\left(1,2\right),\left(2,2\right)\right)$,
obtaining the same result. But we could not have used the list
$\left(\left(1,1\right),\left(1,2\right),\left(2,2\right),\left(2,1\right)\right)$
(for example), since it is not a linear extension (and indeed, the order of
$T_{\left(1,1\right)}\circ T_{\left(1,2\right)}\circ T_{\left(2,2\right)}\circ
T_{\left(2,1\right)}$ is infinite, as follows from the results of [EiPr13,
§12.2]).
Let us state another proposition, which describes birational rowmotion
implicitly:
###### Proposition 2.16.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let $v\in P$. Let
$f\in\mathbb{K}^{\widehat{P}}$. Then,
$\left(Rf\right)\left(v\right)=\dfrac{1}{f\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
v\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{\left(Rf\right)\left(u\right)}}.$ (5)
Here (and in statements further down this paper), we are taking the liberty to
leave assumptions such as “Assume that $Rf$ is well-defined” unsaid (for
instance, such an assumption is needed in Proposition 2.16) because these
assumptions are satisfied when the parameters belong to some Zariski-dense
open subset of their domains.
###### Proof of Proposition 2.16 (sketched)..
Fix a linear extension $\left(v_{1},v_{2},...,v_{m}\right)$ of $P$. Recall
that $R$ has been defined as the composition $T_{v_{1}}\circ
T_{v_{2}}\circ...\circ T_{v_{m}}$. Hence, $Rf$ can be obtained from $f$ by
traversing the linear extension $\left(v_{1},v_{2},...,v_{m}\right)$ from
right to left (thus starting with the largest element $v_{m}$, then proceeding
to $v_{m-1}$, etc.), and at every step toggling the element being traversed.
When an element $v$ is being toggled, the elements $u\in\widehat{P}$
satisfying $u\lessdot v$ have not yet been toggled (they are further left than
$v$ in the linear extension), whereas those satisfying $u\gtrdot v$ have been
toggled already. Denoting the state of the $\mathbb{K}$-labelling before the
$v$-toggle by $g$, we see that the state after the $v$-toggle will be $T_{v}g$
with
$\left(T_{v}g\right)\left(w\right)=\left\\{\begin{array}[c]{l}g\left(w\right),\
\ \ \ \ \ \ \ \ \ \text{if }w\neq v;\\\
\dfrac{1}{g\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
v\end{subarray}}g\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}},\ \ \ \ \ \ \ \ \ \
\text{if }w=v\end{array}\right.\ \ \ \ \ \ \ \ \ \ \text{for all
}w\in\widehat{P}.$ (6)
But $g\left(v\right)=f\left(v\right)$ (since $v$ has not yet been toggled at
the time of $g$) and
$\left(T_{v}g\right)\left(v\right)=\left(Rf\right)\left(v\right)$ (since $v$
has been toggled at the time of $T_{v}g$, and is not going to be toggled ever
again during the process of computing $Rf$); moreover, all $u\in\widehat{P}$
satisfying $u\lessdot v$ satisfy $g\left(u\right)=f\left(u\right)$ (since
these $u$ have not yet been toggled), whereas all $u\in\widehat{P}$ satisfying
$u\gtrdot v$ satisfy $g\left(u\right)=\left(Rf\right)\left(u\right)$ (since
these $u$ have already been toggled and will not be toggled ever again). Thus,
(6) (applied to $w=v$) transforms into (5). Proposition 2.16 is proven. ∎
Here is a little triviality to complete the picture of Proposition 2.16:
###### Proposition 2.17.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let
$f\in\mathbb{K}^{\widehat{P}}$. Then,
$\left(Rf\right)\left(0\right)=f\left(0\right)$ and
$\left(Rf\right)\left(1\right)=f\left(1\right)$.
###### .
This is clear since no toggle changes the labels at $0$ and $1$. ∎
We will often use Proposition 2.17 tacitly. A trivial corollary of Proposition
2.17 is:
###### Corollary 2.18.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let
$f\in\mathbb{K}^{\widehat{P}}$ and $\ell\in\mathbb{N}$. Then,
$\left(R^{\ell}f\right)\left(0\right)=f\left(0\right)$ and
$\left(R^{\ell}f\right)\left(1\right)=f\left(1\right)$.
(Recall that $\mathbb{N}$ denotes the set $\left\\{0,1,2,...\right\\}$ in this
paper.)
We will also need a converse of Propositions 2.16 and 2.17:
###### Proposition 2.19.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Let
$f\in\mathbb{K}^{\widehat{P}}$ and $g\in\mathbb{K}^{\widehat{P}}$ be such that
$f\left(0\right)=g\left(0\right)$ and $f\left(1\right)=g\left(1\right)$.
Assume that
$g\left(v\right)=\dfrac{1}{f\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
v\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}}\ \ \ \ \ \ \ \ \ \
\text{for every }v\in P.$ (7)
(This means, in particular, that we assume that all denominators in (7) are
nonzero.) Then, $g=Rf$. 131313More precisely, $Rf$ is well-defined and equal
to $g$.
###### Proof of Proposition 2.19 (sketched)..
It is clearly enough to show that
$g\left(v\right)=\left(Rf\right)\left(v\right)$ for every $v\in\widehat{P}$.
Since this is clear for $v=0$ (since
$g\left(0\right)=f\left(0\right)=\left(Rf\right)\left(0\right)$), we only need
to consider the case when $v\in\left\\{1\right\\}\cup P$. In this case, we can
prove $g\left(v\right)=\left(Rf\right)\left(v\right)$ by descending induction
over $v$ – that is, we assume as an induction hypothesis that
$g\left(u\right)=\left(Rf\right)\left(u\right)$ holds for all elements
$u\in\left\\{1\right\\}\cup P$ which are greater than $v$ in $\widehat{P}$.
The induction base ($v=1$) is clear (just like $v=0$), and the induction step
follows by comparing (5) with (7). We leave the details (including a check
that $Rf$ is well-defined, which piggybacks on the induction) to the reader. ∎
As an aside, at this point we could give an alternative proof of Corollary
2.12, foregoing the use of Proposition 1.7. In fact, the proofs of
Propositions 2.16, 2.17 and 2.19 only used that $R$ is a composition
$T_{v_{1}}\circ T_{v_{2}}\circ...\circ T_{v_{m}}$ for some linear extension
$\left(v_{1},v_{2},...,v_{m}\right)$ of $P$. Thus, starting with any linear
extension $\left(v_{1},v_{2},...,v_{m}\right)$ of $P$, we could have defined
$R$ as the composition $T_{v_{1}}\circ T_{v_{2}}\circ...\circ T_{v_{m}}$, and
then used Propositions 2.16, 2.17 and 2.19 to characterize the image $Rf$ of a
$\mathbb{K}$-labelling $f$ under this map $R$ in a unique way without
reference to $\left(v_{1},v_{2},...,v_{m}\right)$, and thus concluded that $R$
does not depend on $\left(v_{1},v_{2},...,v_{m}\right)$. The details of this
derivation are left to the reader.
On a related note, Proposition 2.16, Proposition 2.17 and Proposition 2.19
combined can be used as an alternative definition of birational rowmotion $R$,
which works even when the poset $P$ fails to be finite, as long as for every
$v\in P$, there exist only finitely many $u\in P$ satisfying $u>v$ and there
exist only finitely many $u\in P$ satisfying $u\lessdot v$ (provided that some
technicalities arising from Zariski topology on infinite-dimensional spaces
are dealt with).141414The asymmetry between the $>$ and $\lessdot$ signs in
this requirement is intentional. For instance, birational rowmotion can be
defined (but will not be invertible) for the poset
$\left\\{0,-1,-2,-3,\ldots\right\\}$ (with the usual order relation), but not
for the poset $\left\\{0,1,2,3,\ldots\right\\}$ (again with the usual order
relation). We will not dwell on this.
Another general property of birational rowmotion concerns the question of what
happens if the birational toggles are composed not in the “from top to bottom”
order as in the definition of birational rowmotion, but the other way round.
It turns out that the result is the inverse of birational rowmotion:
###### Proposition 2.20.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Then, birational
rowmotion $R$ is invertible (as a rational map). Its inverse $R^{-1}$ is
$T_{v_{m}}\circ T_{v_{m-1}}\circ...\circ
T_{v_{1}}:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P}}$,
where $\left(v_{1},v_{2},...,v_{m}\right)$ is a linear extension of $P$.
###### Proof of Proposition 2.20 (sketched)..
We know that $T_{w}$ is an involution for every $w\in P$. Thus, in particular,
for every $w\in P$, the map $T_{w}$ is invertible and satisfies
$T_{w}^{-1}=T_{w}$.
Let $\left(v_{1},v_{2},...,v_{m}\right)$ be a linear extension of $P$. Then,
$R=T_{v_{1}}\circ T_{v_{2}}\circ...\circ T_{v_{m}}$ (by the definition of
$R$), so that $R^{-1}=T_{v_{m}}^{-1}\circ T_{v_{m-1}}^{-1}\circ...\circ
T_{v_{1}}^{-1}$ (this makes sense since the map $T_{w}$ is invertible for
every $w\in P$). Since $T_{w}^{-1}=T_{w}$ for every $w\in P$, this simplifies
to $R^{-1}=T_{v_{m}}\circ T_{v_{m-1}}\circ...\circ T_{v_{1}}$. This proves
Proposition 2.20. ∎
## 3 Graded posets
In this section, we restrict our attention to what we call graded posets (a
notion that encompasses most of the posets we are interested in; see
Definition 3.1), and define (for this kind of posets) a family of “refined
rowmotion” operators $R_{i}$ which toggle only the labels of the $i$-th degree
of the poset. These each turn out to be involutions, and their composition
from top to bottom degree is $R$ on the entire poset. We will later on use
these $R_{i}$ to get a better understanding of $R$ on graded posets.
Let us first introduce our notion of a graded poset:
###### Definition 3.1.
Let $P$ be a finite poset. Let $n$ be a nonnegative integer. We say that the
poset $P$ is $n$-graded if there exists a surjective map
$\deg:P\rightarrow\left\\{1,2,...,n\right\\}$ such that the following three
assertions hold:
Assertion 1: Any two elements $u$ and $v$ of $P$ such that $u\gtrdot v$
satisfy $\deg u=\deg v+1$.
Assertion 2: We have $\deg u=1$ for every minimal element $u$ of $P$.
Assertion 3: We have $\deg v=n$ for every maximal element $v$ of $P$.
Note that the word “surjective” in Definition 3.1 is almost superfluous:
Indeed, whenever $P\neq\varnothing$, then any map
$\deg:P\rightarrow\left\\{1,2,...,n\right\\}$ satisfying the Assertions 1, 2
and 3 of Definition 3.1 is automatically surjective (this is easy to prove).
But if $P=\varnothing$, such a map exists (vacuously) for every $n$, whereas
requiring surjectivity forced $n=0$.
###### Example 3.2.
The poset $\left\\{1,2\right\\}\times\left\\{1,2\right\\}$ studied in Example
2.15 is $3$-graded. The poset $P$ studied in Example 2.14 is $2$-graded. The
empty poset is $0$-graded, but not $n$-graded for any positive $n$. A chain
with $k$ elements is $k$-graded.
###### Definition 3.3.
Let $P$ be a finite poset. We say that the poset $P$ is graded if there exists
an $n\in\mathbb{N}$ such that $P$ is $n$-graded. This $n$ is then called the
height of $P$.
The reader should be warned that the notion of a “graded poset” is not
standard across literature; we have found at least four non-equivalent
definitions of this notion in different sources.
###### Definition 3.4.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Then, there exists a
surjective map $\deg:P\rightarrow\left\\{1,2,...,n\right\\}$ that satisfies
the Assertions 1, 2 and 3 of Definition 3.1. A moment of thought reveals that
such a map $\deg$ is also uniquely determined by $P$ 151515In fact, if $v\in
P$, then it is easy to see that $\deg v$ equals the number of elements of any
maximal chain in $P$ with highest element $v$. This clearly determines $\deg
v$ uniquely.. Thus, we will call $\deg$ the degree map of $P$.
Moreover, we extend this map $\deg$ to a map
$\widehat{P}\rightarrow\left\\{0,1,...,n+1\right\\}$ by letting it map $0$ to
$0$ and $1$ to $n+1$. This extended map will also be denoted by $\deg$ and
called the degree map. Notice that this extended map $\deg$ still satisfies
Assertion 1 of Definition 3.1 if $P$ is replaced by $\widehat{P}$ in that
assertion.
For every $i\in\left\\{0,1,...,n+1\right\\}$, we will denote by
$\widehat{P}_{i}$ the subset $\deg^{-1}\left(\left\\{i\right\\}\right)$ of
$\widehat{P}$. For every $v\in\widehat{P}$, the number $\deg v$ is called the
degree of $v$.
The notion of an “$n$-graded poset” we just defined is identical with the
notion of a “graded finite poset of rank $n-1$” as defined in [Stan11, §3.1].
The degree of an element $v$ of $P$ as defined in Definition 3.4 is off by $1$
from the rank of $v$ in $P$ in the sense of [Stan11, §3.1], but the degree
$\deg v$ of an element $v$ of $\widehat{P}$ equals its rank in $\widehat{P}$
in the sense of [Stan11, §3.1].
The way we extended the map $\deg:P\rightarrow\left\\{1,2,...,n\right\\}$ to a
map $\deg:\widehat{P}\rightarrow\left\\{0,1,...,n+1\right\\}$ in Definition
3.4, of course, was not arbitrary. In fact, it was tailored to make the
following true:
###### Proposition 3.5.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let $u\in\widehat{P}$
and $v\in\widehat{P}$. Consider the map
$\deg:\widehat{P}\rightarrow\left\\{0,1,...,n+1\right\\}$ defined in
Definition 3.4.
(a) If $u\lessdot v$ in $\widehat{P}$, then $\deg u=\deg v-1$.
(b) If $u<v$ in $\widehat{P}$, then $\deg u<\deg v$.
(c) If $u<v$ in $\widehat{P}$ and $\deg u=\deg v-1$, then $u\lessdot v$ in
$\widehat{P}$.
(d) If $u\neq v$ and $\deg u=\deg v$, then $u$ and $v$ are incomparable in
$\widehat{P}$.
###### .
The rather simple proofs of these facts are left to the reader. (Note that
part (a) incorporates all three Assertions 1, 2 and 3 of Definition 3.1.) ∎
In words, Proposition 3.5 (d) states that any two distinct elements of
$\widehat{P}$ having the same degree are incomparable. We will use this
several times below.
One important observation is that any two distinct elements of a graded poset
having the same degree are incomparable. Hence:
###### Corollary 3.6.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let $i\in\left\\{1,2,...,n\right\\}$. Let
$\left(u_{1},u_{2},...,u_{k}\right)$ be any list of the elements of
$\widehat{P}_{i}$ with every element of $\widehat{P}_{i}$ appearing exactly
once in the list. Then, the dominant rational map $T_{u_{1}}\circ
T_{u_{2}}\circ...\circ
T_{u_{k}}:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P}}$ is
well-defined and independent of the choice of the list
$\left(u_{1},u_{2},...,u_{k}\right)$.
###### Proof of Corollary 3.6 (sketched)..
This is analogous to the proof of Corollary 2.12, because any two distinct
elements of $\widehat{P}_{i}$ are incomparable. (In place of the set
$\mathcal{L}\left(P\right)$ now serves the set of all lists of elements of
$\widehat{P}_{i}$ (with every element of $\widehat{P}_{i}$ appearing exactly
once in the list). Any two elements of this latter set are equivalent under
the relation $\sim$, because any two adjacent elements in such a list of
elements of $\widehat{P}_{i}$ are incomparable and can thus be switched.) ∎
###### Definition 3.7.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let $i\in\left\\{1,2,...,n\right\\}$. Then, let $R_{i}$ denote the
dominant rational map $T_{u_{1}}\circ T_{u_{2}}\circ...\circ
T_{u_{k}}:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P}}$,
where $\left(u_{1},u_{2},...,u_{k}\right)$ is any list of the elements of
$\widehat{P}_{i}$ with every element of $\widehat{P}_{i}$ appearing exactly
once in the list. This map $T_{u_{1}}\circ T_{u_{2}}\circ...\circ T_{u_{k}}$
is well-defined (in particular, it does not depend on the list
$\left(u_{1},u_{2},...,u_{k}\right)$) because of Corollary 3.6.
###### Proposition 3.8.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Then,
$R=R_{1}\circ R_{2}\circ...\circ R_{n}.$ (8)
###### Proof of Proposition 3.8 (sketched)..
For every $i\in\left\\{1,2,...,n\right\\}$, let
$\left(u_{1}^{\left[i\right]},u_{2}^{\left[i\right]},...,u_{k_{i}}^{\left[i\right]}\right)$
be a list of the elements of $\widehat{P}_{i}$ with every element of
$\widehat{P}_{i}$ appearing exactly once in the list. Then, every
$i\in\left\\{1,2,...,n\right\\}$ satisfies
$R_{i}=T_{u_{1}^{\left[i\right]}}\circ T_{u_{2}^{\left[i\right]}}\circ...\circ
T_{u_{k_{i}}^{\left[i\right]}}$.
But any listing of the elements of $P$ in order of increasing degree is a
linear extension of $P$ (because any two distinct elements of a graded poset
having the same degree are incomparable). Thus,
$\left(u_{1}^{\left[1\right]},u_{2}^{\left[1\right]},...,u_{k_{1}}^{\left[1\right]},\
\ \
u_{1}^{\left[2\right]},u_{2}^{\left[2\right]},...,u_{k_{2}}^{\left[2\right]},\
\ \ ...,\ \ \
u_{1}^{\left[n\right]},u_{2}^{\left[n\right]},...,u_{k_{n}}^{\left[n\right]}\right)$
is a linear extension of $P$. Thus, by the definition of $R$, we have
$\displaystyle R$ $\displaystyle=\left(T_{u_{1}^{\left[1\right]}}\circ
T_{u_{2}^{\left[1\right]}}\circ...\circ
T_{u_{k_{1}}^{\left[1\right]}}\right)\circ\left(T_{u_{1}^{\left[2\right]}}\circ
T_{u_{2}^{\left[2\right]}}\circ...\circ
T_{u_{k_{2}}^{\left[2\right]}}\right)\circ...\circ\left(T_{u_{1}^{\left[n\right]}}\circ
T_{u_{2}^{\left[n\right]}}\circ...\circ T_{u_{k_{n}}^{\left[n\right]}}\right)$
$\displaystyle=R_{1}\circ R_{2}\circ...\circ R_{n}$
(since every $i\in\left\\{1,2,...,n\right\\}$ satisfies
$T_{u_{1}^{\left[i\right]}}\circ T_{u_{2}^{\left[i\right]}}\circ...\circ
T_{u_{k_{i}}^{\left[i\right]}}=R_{i}$). This proves Proposition 3.8. ∎
We recall that birational rowmotion is a composition of toggle maps. As
Proposition 3.8 shows, the operators $R_{i}$ are an “intermediate” step
between these toggle maps and birational rowmotion as a whole, though they are
defined only when the poset $P$ is graded. They will be rather useful for us
in our understanding of birational rowmotion (and the condition on $P$ to be
graded doesn’t prevent us from using them, since most of our results concern
only graded posets anyway).
###### Proposition 3.9.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let $i\in\left\\{1,2,...,n\right\\}$. Then, $R_{i}$ is an involution
(that is, $R_{i}^{2}=\operatorname*{id}$ on the set where $R_{i}$ is defined).
###### Proof of Proposition 3.9 (sketched)..
We defined $R_{i}$ as the composition $T_{u_{1}}\circ T_{u_{2}}\circ...\circ
T_{u_{k}}$ of the toggles $T_{u_{i}}$ where
$\left(u_{1},u_{2},...,u_{k}\right)$ is any list of the elements of
$\widehat{P}_{i}$ with every element of $\widehat{P}_{i}$ appearing exactly
once in the list. These toggles are involutions and commute (the latter
because any two distinct elements of $\widehat{P}_{i}$ are incomparable,
having the same degree in $P$). Since a composition of commuting involutions
is always an involution, this shows that $R_{i}$ is an involution, qed. ∎
Similarly to Proposition 2.16, we have:
###### Proposition 3.10.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let
$i\in\left\\{1,2,...,n\right\\}$. Let $\mathbb{K}$ be a field. Let
$v\in\widehat{P}$. Let $f\in\mathbb{K}^{\widehat{P}}$.
(a) If $\deg v\neq i$, then
$\left(R_{i}f\right)\left(v\right)=f\left(v\right)$.
(b) If $\deg v=i$, then
$\left(R_{i}f\right)\left(v\right)=\dfrac{1}{f\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
v\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{f\left(u\right)}}.$ (9)
###### .
The proof of this proposition is very similar to that of Proposition 2.16 and
therefore left to the reader. ∎
Notice that using the proof of Proposition 3.10, it is easy to give an
alternative proof of Corollary 3.6 (in the same way as we saw that an
alternative proof of Corollary 2.12 could be given using the proofs of
Propositions 2.16, 2.17 and 2.19).
## 4 w-tuples
This section continues the study of birational rowmotion on graded posets by
introducing a “fingerprint” or “checksum” of a $\mathbb{K}$-labelling called
the w-tuple, defined by summing ratios of elements between successive degrees
(i.e., rows in the Hasse diagram). This w-tuple serves to extract some
information from a $\mathbb{K}$-labelling; we will later see how to make the
“rest” of the labelling more manageable.
###### Definition 4.1.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let $f\in\mathbb{K}^{\widehat{P}}$. Let
$i\in\left\\{0,1,...,n\right\\}$. Then, $\mathbf{w}_{i}\left(f\right)$ will
denote the element of $\mathbb{K}$ defined by
$\mathbf{w}_{i}\left(f\right)=\sum_{\begin{subarray}{c}x\in\widehat{P}_{i};\
y\in\widehat{P}_{i+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{f\left(x\right)}{f\left(y\right)}.$
(This element is not always defined, but is defined in the “generic” case when
$0\notin f\left(\widehat{P}\right)$.)
Intuitively, one could think of $\mathbf{w}_{i}\left(f\right)$ as a kind of
“checksum” for the labelling $f$ which displays how much its labels at degree
$i+1$ differ from those at degree $i$. Of course, in general, the knowledge of
$\mathbf{w}_{i}\left(f\right)$ for all $i\in\left\\{0,1,...,n\right\\}$ is far
from sufficient to reconstruct the whole labelling $f$; however, in Definition
6.2, we will introduce the so-called homogenization of $f$, which will provide
“complementary data” to these $\mathbf{w}_{i}\left(f\right)$. As for now, let
us show that the $\mathbf{w}_{i}\left(f\right)$ behave in a rather simple way
under the maps $R$ and $R_{j}$.
###### Definition 4.2.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let $f\in\mathbb{K}^{\widehat{P}}$. The $\left(n+1\right)$-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$
will be called the w-tuple of the $\mathbb{K}$-labelling $f$.
It is easy to see:
###### Proposition 4.3.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let $i\in\left\\{1,2,...,n\right\\}$. Then, every
$f\in\mathbb{K}^{\widehat{P}}$ satisfies
$\displaystyle\left(\mathbf{w}_{0}\left(R_{i}f\right),\mathbf{w}_{1}\left(R_{i}f\right),...,\mathbf{w}_{n}\left(R_{i}f\right)\right)$
$\displaystyle=\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{i-2}\left(f\right),\mathbf{w}_{i}\left(f\right),\mathbf{w}_{i-1}\left(f\right),\mathbf{w}_{i+1}\left(f\right),\mathbf{w}_{i+2}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right).$
In other words, the map $R_{i}$ changes the w-tuple of a
$\mathbb{K}$-labelling by interchanging its $\left(i-1\right)$-st entry with
its $i$-th entry (where the entries are labelled starting at $0$).
###### Proof of Proposition 4.3 (sketched)..
Let $f\in\mathbb{K}^{\widehat{P}}$. We need to show that every
$j\in\left\\{0,1,...,n\right\\}$ satisfies
$\mathbf{w}_{j}\left(R_{i}f\right)=\mathbf{w}_{\tau_{i}\left(j\right)}\left(f\right),$
(10)
where $\tau_{i}$ is the permutation of the set $\left\\{0,1,...,n\right\\}$
which transposes $i-1$ with $i$ (while leaving all other elements of this set
invariant).
Proof of (10): Let $j\in\left\\{0,1,...,n\right\\}$. We distinguish between
three cases:
Case 1: We have $j=i$.
Case 2: We have $j=i-1$.
Case 3: We have $j\notin\left\\{i-1,i\right\\}$.
Let us first consider Case 1. In this case, we have $j=i$. By the definition
of $\mathbf{w}_{i}\left(R_{i}f\right)$, we have
$\displaystyle\mathbf{w}_{i}\left(R_{i}f\right)$
$\displaystyle=\sum_{\begin{subarray}{c}x\in\widehat{P}_{i};\
y\in\widehat{P}_{i+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{\left(R_{i}f\right)\left(x\right)}{\left(R_{i}f\right)\left(y\right)}=\sum\limits_{x\in\widehat{P}_{i}}\left(R_{i}f\right)\left(x\right)\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i+1};\\\
y\gtrdot
x\end{subarray}}\left(\underbrace{\left(R_{i}f\right)\left(y\right)}_{\begin{subarray}{c}=f\left(y\right)\\\
\text{(by Proposition \ref{prop.Ri.implicit}
{(a)})}\end{subarray}}\right)^{-1}$
$\displaystyle=\sum\limits_{x\in\widehat{P}_{i}}\left(R_{i}f\right)\left(x\right)\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i+1};\\\
y\gtrdot x\end{subarray}}\left(f\left(y\right)\right)^{-1}.$ (11)
But every $x\in\widehat{P}_{i}$ satisfies
$\displaystyle\left(R_{i}f\right)\left(x\right)$
$\displaystyle=\dfrac{1}{f\left(x\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
x\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot x\end{subarray}}\dfrac{1}{f\left(u\right)}}\ \ \ \ \ \ \ \ \ \
\left(\text{by Proposition \ref{prop.Ri.implicit} {(b)}}\right)$
$\displaystyle=\dfrac{1}{f\left(x\right)}\cdot\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\lessdot
x\end{subarray}}f\left(u\right)\cdot\left(\sum\limits_{\begin{subarray}{c}u\in\widehat{P};\\\
u\gtrdot
x\end{subarray}}\left(f\left(u\right)\right)^{-1}\right)^{-1}=\dfrac{1}{f\left(x\right)}\cdot\sum\limits_{\begin{subarray}{c}y\in\widehat{P};\\\
y\lessdot
x\end{subarray}}f\left(y\right)\cdot\left(\sum\limits_{\begin{subarray}{c}y\in\widehat{P};\\\
y\gtrdot x\end{subarray}}\left(f\left(y\right)\right)^{-1}\right)^{-1}$
$\displaystyle=\dfrac{1}{f\left(x\right)}\cdot\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i-1};\\\
y\lessdot
x\end{subarray}}f\left(y\right)\cdot\left(\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i+1};\\\
y\gtrdot x\end{subarray}}\left(f\left(y\right)\right)^{-1}\right)^{-1}$
(here, we replaced $y\in\widehat{P}$ by $y\in\widehat{P}_{i-1}$ in the first
sum (because every $y\in\widehat{P}$ satisfying $y\lessdot x$ must belong to
$\widehat{P}_{i-1}$ 161616since $x\in\widehat{P}_{i}$) and we replaced
$y\in\widehat{P}$ by $y\in\widehat{P}_{i+1}$ in the second sum (for similar
reasons)) and thus
$\left(R_{i}f\right)\left(x\right)\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i+1};\\\
y\gtrdot
x\end{subarray}}\left(f\left(y\right)\right)^{-1}=\dfrac{1}{f\left(x\right)}\cdot\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i-1};\\\
y\lessdot
x\end{subarray}}f\left(y\right)=\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i-1};\\\
y\lessdot x\end{subarray}}\dfrac{f\left(y\right)}{f\left(x\right)}.$
Hence, (11) becomes
$\displaystyle\mathbf{w}_{i}\left(R_{i}f\right)$
$\displaystyle=\sum\limits_{x\in\widehat{P}_{i}}\underbrace{\left(R_{i}f\right)\left(x\right)\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i+1};\\\
y\gtrdot
x\end{subarray}}\left(f\left(y\right)\right)^{-1}}_{=\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i-1};\\\
y\lessdot
x\end{subarray}}\dfrac{f\left(y\right)}{f\left(x\right)}}=\sum\limits_{x\in\widehat{P}_{i}}\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i-1};\\\
y\lessdot
x\end{subarray}}\dfrac{f\left(y\right)}{f\left(x\right)}=\sum\limits_{\begin{subarray}{c}y\in\widehat{P}_{i-1};\
x\in\widehat{P}_{i};\\\ x\gtrdot
y\end{subarray}}\dfrac{f\left(y\right)}{f\left(x\right)}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}x\in\widehat{P}_{i-1};\
y\in\widehat{P}_{i}\\\ y\gtrdot
x\end{subarray}}\dfrac{f\left(x\right)}{f\left(y\right)}\ \ \ \ \ \ \ \ \ \
\left(\text{here, we switched the indices in the sum}\right)$
$\displaystyle=\mathbf{w}_{i-1}\left(f\right)\ \ \ \ \ \ \ \ \ \
\left(\text{by the definition of }\mathbf{w}_{i-1}\left(f\right)\right)$ (12)
$\displaystyle=\mathbf{w}_{\tau_{i}\left(i\right)}\left(f\right).$
In other words, (10) holds for $j=i$. Thus, (10) is proven in Case 1.
Let us now consider Case 2. In this case, $j=i-1$. Now, it can be shown that
$\mathbf{w}_{i-1}\left(R_{i}f\right)=\mathbf{w}_{i}\left(f\right)$. This can
be proven either in a similar way to how we proved
$\mathbf{w}_{i}\left(R_{i}f\right)=\mathbf{w}_{i-1}\left(f\right)$ (the
details of this are left to the reader), or by noticing that
$\displaystyle\mathbf{w}_{i}\left(f\right)$
$\displaystyle=\mathbf{w}_{i}\left(R_{i}^{2}f\right)\ \ \ \ \ \ \ \ \ \
\left(\begin{array}[c]{c}\text{since Proposition \ref{prop.Ri.invo} yields
that }R_{i}^{2}=\operatorname*{id}\text{,}\\\ \text{hence
}\mathbf{w}_{i}\left(R_{i}^{2}f\right)=\mathbf{w}_{i}\left(\operatorname*{id}f\right)=\mathbf{w}_{i}\left(f\right)\end{array}\right)$
$\displaystyle=\mathbf{w}_{i}\left(R_{i}\left(R_{i}f\right)\right)=\mathbf{w}_{i-1}\left(R_{i}f\right)\
\ \ \ \ \ \ \ \ \ \left(\text{by (\ref{pf.wi.Ri.short.main}), applied to
}R_{i}f\text{ instead of }f\right).$
Either way, we end up knowing that
$\mathbf{w}_{i-1}\left(R_{i}f\right)=\mathbf{w}_{i}\left(f\right)$. Thus,
$\mathbf{w}_{i-1}\left(R_{i}f\right)=\mathbf{w}_{i}\left(f\right)=\mathbf{w}_{\tau_{i}\left(i-1\right)}\left(f\right)$.
In other words, (10) holds for $j=i-1$. Thus, (10) is proven in Case 2.
Let us finally consider Case 3. In this case, $j\notin\left\\{i-1,i\right\\}$.
Hence, $\tau_{i}\left(j\right)=j$. On the other hand, by the definition of
$\mathbf{w}_{j}\left(R_{i}f\right)$, we have
$\displaystyle\mathbf{w}_{j}\left(R_{i}f\right)$
$\displaystyle=\sum_{\begin{subarray}{c}x\in\widehat{P}_{j};\
y\in\widehat{P}_{j+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{\left(R_{i}f\right)\left(x\right)}{\left(R_{i}f\right)\left(y\right)}=\sum_{\begin{subarray}{c}x\in\widehat{P}_{j};\
y\in\widehat{P}_{j+1};\\\ y\gtrdot
x\end{subarray}}\left(\underbrace{\left(R_{i}f\right)\left(y\right)}_{\begin{subarray}{c}=f\left(y\right)\\\
\text{(by Proposition \ref{prop.Ri.implicit}
{(a)})}\end{subarray}}\right)^{-1}\cdot\underbrace{\left(R_{i}f\right)\left(x\right)}_{\begin{subarray}{c}=f\left(x\right)\\\
\text{(by Proposition \ref{prop.Ri.implicit} {(a)})}\end{subarray}}$
$\displaystyle=\sum_{\begin{subarray}{c}x\in\widehat{P}_{j};\
y\in\widehat{P}_{j+1};\\\ y\gtrdot
x\end{subarray}}\left(f\left(y\right)\right)^{-1}\cdot
f\left(x\right)=\sum_{\begin{subarray}{c}x\in\widehat{P}_{j};\
y\in\widehat{P}_{j+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{f\left(x\right)}{f\left(y\right)}.$
Compared with
$\mathbf{w}_{j}\left(f\right)=\sum\limits_{\begin{subarray}{c}x\in\widehat{P}_{j};\
y\in\widehat{P}_{j+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{f\left(x\right)}{f\left(y\right)}$ (by the definition
of $\mathbf{w}_{j}\left(f\right)$), this yields
$\mathbf{w}_{j}\left(R_{i}f\right)=\mathbf{w}_{j}\left(f\right)$. Since
$j=\tau_{i}\left(j\right)$, this becomes
$\mathbf{w}_{j}\left(R_{i}f\right)=\mathbf{w}_{\tau_{i}\left(j\right)}\left(f\right)$.
Hence, (10) is proven in Case 3.
We have thus proven (10) in each of the three possible cases 1, 2 and 3. This
completes the proof of (10) and thus of Proposition 4.3. ∎
From Proposition 4.3, and (8), we conclude:
###### Proposition 4.4.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Then, every $f\in\mathbb{K}^{\widehat{P}}$ satisfies
$\left(\mathbf{w}_{0}\left(Rf\right),\mathbf{w}_{1}\left(Rf\right),...,\mathbf{w}_{n}\left(Rf\right)\right)=\left(\mathbf{w}_{n}\left(f\right),\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n-1}\left(f\right)\right).$
In other words, the map $R$ changes the w-tuple of a $\mathbb{K}$-labelling by
shifting it cyclically.
###### Proof of Proposition 4.4 (sketched)..
Proposition 3.8 yields $R=R_{1}\circ R_{2}\circ...\circ R_{n}$. But for every
$i\in\left\\{1,2,...,n\right\\}$, recall from Proposition 4.3 that the map
$R_{i}$ changes the w-tuple of a $\mathbb{K}$-labelling by interchanging its
$\left(i-1\right)$-st entry with its $i$-th entry (where the entries are
labelled starting at $0$). Hence, the effect of the compound map $R=R_{1}\circ
R_{2}\circ...\circ R_{n}$ on the w-tuple is that of first interchanging the
$\left(n-1\right)$-st entry with the $n$-th entry, then interchanging the
$\left(n-2\right)$-st entry with the $\left(n-1\right)$-st entry, and so on,
through to finally interchanging the $0$-th entry with the $1$-st entry. But
this latter sequence of interchanges is equivalent to a cyclic shift of the
w-tuple171717Indeed, the composition
$\left(0,1\right)\circ\left(1,2\right)\circ...\circ\left(n-1,n\right)$ of
transpositions in the symmetric group on the set $\left\\{0,1,...,n\right\\}$
is the $\left(n+1\right)$-cycle $\left(0,1,...,n\right)$.. Hence, the map $R$
changes the w-tuple of a $\mathbb{K}$-labelling by shifting it cyclically,
qed. ∎
As a consequence of Proposition 4.4, the map $R^{n+1}$ (for an $n$-graded
poset $P$) leaves the w-tuple of a $\mathbb{K}$-labelling fixed.
## 5 Graded rescaling of labellings
In general, birational rowmotion $R$ has something that one might call an
“avalanche effect”: If $f$ and $g$ are two $\mathbb{K}$-labellings of a poset
$P$ which differ from each other only in their labels at one single element
$v$, then the labellings $Rf$ and $Rg$ (in general) differ at all elements
covering $v$ and all elements beneath $v$, and further applications of $R$
make the labellings even more different. Thus, a change of just one label in a
labelling will often “spread” through a large part of the poset when $R$ is
repeatedly applied; the effect of such a change is hard to track in general.
Thus, knowing the behavior of one particular $\mathbb{K}$-labelling $f$ under
$R$ does not help us at understanding the behaviors of $\mathbb{K}$-labellings
obtained from $f$ by changing labels at particular elements. However, if $P$
is a graded poset and we simultaneously multiply the labels at all elements of
a given degree in a given labelling of $P$ with a given scalar, then the
changes this causes to the behavior of the labelling under $R$ are rather
predictable. We are going to formalize this observation in this section,
proving some explicit formulas for how birational rowmotion $R$ and its
iterates react to such rescalings. These explicit formulas will be subsumed
into slick conclusions in Section 6, where we will introduce a notion of
_homogeneous equivalence_ which formalizes the idea of a “labelling modulo
scalar factors at each degree”.
###### Definition 5.1.
Let $\mathbb{K}$ be a field. Then, $\mathbb{K}^{\times}$ denotes the
multiplicative group of nonzero elements of $\mathbb{K}$.
The following definition formalizes the idea of multiplying the labels at all
elements of a certain degree with one and the same scalar factor:
###### Definition 5.2.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. For every $\mathbb{K}$-labelling $f\in\mathbb{K}^{\widehat{P}}$ and any
$\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$,
we define a $\mathbb{K}$-labelling $\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\in\mathbb{K}^{\widehat{P}}$ by
$\left(\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\right)\left(v\right)=a_{\deg v}\cdot f\left(v\right)\ \ \ \ \ \ \ \ \ \
\text{for every }v\in\widehat{P}.$
Straightforward application of this definition and that of $R_{i}$ shows:
###### Proposition 5.3.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let us use the notation introduced in Definition 5.2.
Let $f\in\mathbb{K}^{\widehat{P}}$ be a $\mathbb{K}$-labelling. Let
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$.
Let $i\in\left\\{1,2,...,n\right\\}$. Then,
$\displaystyle R_{i}\left(\left(a_{0},a_{1},...,a_{n+1}\right)\flat f\right)$
$\displaystyle=\left(a_{0},a_{1},...,a_{i-1},\dfrac{a_{i+1}a_{i-1}}{a_{i}},a_{i+1},a_{i+2},...,a_{n+1}\right)\flat\left(R_{i}f\right)$
(provided that $R_{i}f$ is well-defined).
A similar result can be obtained for $R$ instead of $R_{i}$:
###### Proposition 5.4.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. For every $\mathbb{K}$-labelling $f\in\mathbb{K}^{\widehat{P}}$ and any
$\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$,
we define a $\mathbb{K}$-labelling $\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\in\mathbb{K}^{\widehat{P}}$ as in Definition 5.2.
Let $f\in\mathbb{K}^{\widehat{P}}$ be a $\mathbb{K}$-labelling. Let
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$.
Then,
$R\left(\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\right)=\left(a_{0},ga_{0},ga_{1},...,ga_{n-1},a_{n+1}\right)\flat\left(Rf\right),$
where $g=\dfrac{a_{n+1}}{a_{n}}$ (provided that $Rf$ is well-defined).
###### Proof of Proposition 5.4 (sketched)..
Let $g=\dfrac{a_{n+1}}{a_{n}}$. We claim that every
$j\in\left\\{1,2,...,n+1\right\\}$ satisfies
$\displaystyle\left(R_{j}\circ R_{j+1}\circ...\circ
R_{n}\right)\left(\left(a_{0},a_{1},...,a_{n+1}\right)\flat f\right)$
$\displaystyle=\left(a_{0},a_{1},a_{2},...,a_{j-1},ga_{j-1},ga_{j},...,ga_{n-1},a_{n+1}\right)\flat\left(\left(R_{j}\circ
R_{j+1}\circ...\circ R_{n}\right)f\right).$ (13)
Indeed, (13) is easily verified by reverse induction over $j$ (that is,
induction over $n+1-j$), using Proposition 5.3 in the step. Now, applying (13)
to $j=1$ and recalling that $R=R_{1}\circ R_{2}\circ...\circ R_{n}$, we obtain
the claim of Proposition 5.4.
∎
We can go further and generalize Proposition 5.4 to iterated birational
rowmotion:
###### Proposition 5.5.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. For every $\mathbb{K}$-labelling $f\in\mathbb{K}^{\widehat{P}}$ and any
$\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$,
we define a $\mathbb{K}$-labelling $\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\in\mathbb{K}^{\widehat{P}}$ as in Definition 5.2.
Let
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$.
For every $\ell\in\left\\{0,1,...,n+1\right\\}$ and
$k\in\left\\{0,1,...,n+1\right\\}$, define an element
$\widehat{a}_{k}^{\left(\ell\right)}\in\mathbb{K}^{\times}$ by
$\widehat{a}_{k}^{\left(\ell\right)}=\left\\{\begin{array}[c]{c}\dfrac{a_{n+1}a_{k-\ell}}{a_{n+1-\ell}},\
\ \ \ \ \ \ \ \ \ \text{if }k\geqslant\ell;\\\
\dfrac{a_{n+1+k-\ell}a_{0}}{a_{n+1-\ell}},\ \ \ \ \ \ \ \ \ \ \text{if
}k<\ell\end{array}\right..$
Let $f\in\mathbb{K}^{\widehat{P}}$ be a $\mathbb{K}$-labelling. Then, every
$\ell\in\left\\{0,1,...,n+1\right\\}$ satisfies
$R^{\ell}\left(\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\right)=\left(\widehat{a}_{0}^{\left(\ell\right)},\widehat{a}_{1}^{\left(\ell\right)},...,\widehat{a}_{n+1}^{\left(\ell\right)}\right)\flat\left(R^{\ell}f\right)$
(provided that $R^{\ell}f$ is well-defined).
###### Example 5.6.
For this example, let $n=3$, and let $P$ be a $3$-graded poset. Then,
$\displaystyle\left(\widehat{a}_{0}^{\left(0\right)},\widehat{a}_{1}^{\left(0\right)},\widehat{a}_{2}^{\left(0\right)},\widehat{a}_{3}^{\left(0\right)},\widehat{a}_{4}^{\left(0\right)}\right)$
$\displaystyle=\left(a_{0},a_{1},a_{2},a_{3},a_{4}\right);$
$\displaystyle\left(\widehat{a}_{0}^{\left(1\right)},\widehat{a}_{1}^{\left(1\right)},\widehat{a}_{2}^{\left(1\right)},\widehat{a}_{3}^{\left(1\right)},\widehat{a}_{4}^{\left(1\right)}\right)$
$\displaystyle=\left(a_{0},\dfrac{a_{4}a_{0}}{a_{3}},\dfrac{a_{4}a_{1}}{a_{3}},\dfrac{a_{4}a_{2}}{a_{3}},a_{4}\right);$
$\displaystyle\left(\widehat{a}_{0}^{\left(2\right)},\widehat{a}_{1}^{\left(2\right)},\widehat{a}_{2}^{\left(2\right)},\widehat{a}_{3}^{\left(2\right)},\widehat{a}_{4}^{\left(2\right)}\right)$
$\displaystyle=\left(a_{0},\dfrac{a_{3}a_{0}}{a_{2}},\dfrac{a_{4}a_{0}}{a_{2}},\dfrac{a_{4}a_{1}}{a_{2}},a_{4}\right);$
$\displaystyle\left(\widehat{a}_{0}^{\left(3\right)},\widehat{a}_{1}^{\left(3\right)},\widehat{a}_{2}^{\left(3\right)},\widehat{a}_{3}^{\left(3\right)},\widehat{a}_{4}^{\left(3\right)}\right)$
$\displaystyle=\left(a_{0},\dfrac{a_{2}a_{0}}{a_{1}},\dfrac{a_{3}a_{0}}{a_{1}},\dfrac{a_{4}a_{0}}{a_{1}},a_{4}\right);$
$\displaystyle\left(\widehat{a}_{0}^{\left(4\right)},\widehat{a}_{1}^{\left(4\right)},\widehat{a}_{2}^{\left(4\right)},\widehat{a}_{3}^{\left(4\right)},\widehat{a}_{4}^{\left(4\right)}\right)$
$\displaystyle=\left(a_{0},a_{1},a_{2},a_{3},a_{4}\right).$
More generally, we always have
$\left(\widehat{a}_{0}^{\left(0\right)},\widehat{a}_{1}^{\left(0\right)},...,\widehat{a}_{n+1}^{\left(0\right)}\right)=\left(a_{0},a_{1},...,a_{n+1}\right)$
and
$\left(\widehat{a}_{0}^{\left(n+1\right)},\widehat{a}_{1}^{\left(n+1\right)},...,\widehat{a}_{n+1}^{\left(n+1\right)}\right)=\left(a_{0},a_{1},...,a_{n+1}\right)$
(as can be verified directly).
###### Proof of Proposition 5.5 (sketched)..
This proof is a completely straightforward induction over $\ell$, with the
base case being trivial and the induction step relying on Proposition 5.4. It
is useful to notice that every $\ell\in\left\\{0,1,...,n+1\right\\}$ and
$k\in\left\\{0,1,...,n+1\right\\}$ satisfy
$\widehat{a}_{k}^{\left(\ell\right)}=\dfrac{a_{n+1+k-\ell}a_{0}}{a_{n+1-\ell}}\
\ \ \ \ \ \ \ \ \ \text{if }k\leqslant\ell$
to simplify the computations (this identity follows from the definition when
$k<\ell$ and can be easily checked for $k=\ell$). ∎
As a consequence of Proposition 5.5, we notice a very simple behavior of
rescaled labellings under $R^{n+1}$ for an $n$-graded poset $P$:
###### Corollary 5.7.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. For every $\mathbb{K}$-labelling $f\in\mathbb{K}^{\widehat{P}}$ and any
$\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$,
we define a $\mathbb{K}$-labelling $\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\in\mathbb{K}^{\widehat{P}}$ as in Definition 5.2.
Let
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$.
Let $f\in\mathbb{K}^{\widehat{P}}$ be a $\mathbb{K}$-labelling. Then,
$R^{n+1}\left(\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\right)=\left(a_{0},a_{1},...,a_{n+1}\right)\flat\left(R^{n+1}f\right)$
(provided that $R^{n+1}f$ is well-defined).
###### .
We leave deriving Corollary 5.7 from Proposition 5.5 to the reader. ∎
Let us furthermore record how the rescaling of labels according to their
degree affects their w-tuples (as defined in Definition 4.1):
###### Proposition 5.8.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. For every $\mathbb{K}$-labelling $f\in\mathbb{K}^{\widehat{P}}$ and any
$\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$,
we define a $\mathbb{K}$-labelling $\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\in\mathbb{K}^{\widehat{P}}$ as in Definition 5.2.
Let $f\in\mathbb{K}^{\widehat{P}}$ be a $\mathbb{K}$-labelling of $P$. Let
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$.
Then, the w-tuple of the $\mathbb{K}$-labelling
$\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$ is
$\left(\dfrac{a_{0}}{a_{1}}\mathbf{w}_{0}\left(f\right),\dfrac{a_{1}}{a_{2}}\mathbf{w}_{1}\left(f\right),...,\dfrac{a_{n}}{a_{n+1}}\mathbf{w}_{n}\left(f\right)\right).$
###### .
Proposition 5.8 follows by computation using just the definitions of the
notions involved. ∎
## 6 Homogeneous labellings
In the previous section, we have quantified how the rescaling of all labels at
a given degree affects a labelling (of a graded poset) under birational
rowmotion. In this section, we will introduce a notion of “homogeneous
labellings” which (roughly speaking) are “labellings up to rescaling at a
given degree” in the same way as a point in a projective space can be regarded
as (roughly speaking) “a point in the affine space up to rescaling the
coordinates”. To be precise, we will need to restrict ourselves to considering
only “zero-free” labellings (a Zariski-dense open subset of all labellings)
for the same reason as we need to exclude $0$ when defining a projective
space. Once done with the definitions, we will see that birational rowmotion
(and the maps $R_{i}$) can be defined on homogeneous labellings (it is here
that we will make use of the results of the previous section).
Let us begin with the definitions:
###### Definition 6.1.
Let $\mathbb{K}$ be a field.
(a) For every $\mathbb{K}$-vector space $V$, let $\mathbb{P}\left(V\right)$
denote the projective space of $V$ (that is, the set of equivalence classes of
vectors in $V\setminus\left\\{0\right\\}$ modulo proportionality).
(b) For every $n\in\mathbb{N}$, we let $\mathbb{P}^{n}\left(\mathbb{K}\right)$
denote the projective space $\mathbb{P}\left(\mathbb{K}^{n+1}\right)$.
###### Definition 6.2.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset.
(a) Denote by $\overline{\mathbb{K}^{\widehat{P}}}$ the product
$\prod\limits_{i=1}^{n}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)$ of
projective spaces. Notice that the product is just a Cartesian product of
algebraic varieties, and a reader unfamiliar with algebraic geometry can just
regard it as a Cartesian product of sets.181818The structure of algebraic
variety will only be needed to define the Zariski topology on
$\overline{\mathbb{K}^{\widehat{P}}}$, which is more or less obvious already
(e.g., when we say that something holds “for almost every element $x$ of
$\prod\limits_{i=1}^{n}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)$”,
we could equivalently say that it holds “for
$x=\operatorname*{proj}\left(X\right)$ for almost every element $X$ of
$\prod\limits_{i=1}^{n}\left(\mathbb{K}^{\widehat{P}_{i}}\setminus\left\\{0\right\\}\right)$”,
where $\operatorname*{proj}$ is the canonical map
$\prod\limits_{i=1}^{n}\left(\mathbb{K}^{\widehat{P}_{i}}\setminus\left\\{0\right\\}\right)\rightarrow\prod\limits_{i=1}^{n}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)$
defined as the product of the projections
$\mathbb{K}^{\widehat{P}_{i}}\setminus\left\\{0\right\\}\rightarrow\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)$).
We have
$\overline{\mathbb{K}^{\widehat{P}}}=\prod\limits_{i=1}^{n}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)\cong\prod\limits_{i=1}^{n}\mathbb{P}^{\left|\widehat{P}_{i}\right|-1}\left(\mathbb{K}\right)$
(since every $i\in\left\\{1,2,...,n\right\\}$ satisfies
$\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)\cong\mathbb{P}^{\left|\widehat{P}_{i}\right|-1}\left(\mathbb{K}\right)$).
We denote the elements of $\overline{\mathbb{K}^{\widehat{P}}}$ as homogeneous
labellings.
Notice that
$\overline{\mathbb{K}^{\widehat{P}}}=\prod\limits_{i=1}^{n}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)\cong\prod\limits_{i=0}^{n+1}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)$
(as algebraic varieties). This is because $\mathbb{K}^{\widehat{P}_{0}}$ and
$\mathbb{K}^{\widehat{P}_{n+1}}$ are $1$-dimensional vector spaces (since
$\left|\widehat{P}_{0}\right|=1$ and $\left|\widehat{P}_{n+1}\right|=1$), and
thus the projective spaces
$\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{0}}\right)$ and
$\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{n+1}}\right)$ each consist of a
single point.
(b) A $\mathbb{K}$-labelling $f\in\mathbb{K}^{\widehat{P}}$ is said to be
zero-free if for every $i\in\left\\{0,1,...,n+1\right\\}$, there exists some
$v\in\widehat{P}_{i}$ satisfying $f\left(v\right)\neq 0$. (In other words, a
$\mathbb{K}$-labelling $f\in\mathbb{K}^{\widehat{P}}$ is said to be zero-free
if there exists no $i\in\left\\{0,1,...,n+1\right\\}$ such that $f$ is
identically $0$ on all elements of $\widehat{P}$ having degree $i$.) Let
$\mathbb{K}_{\neq 0}^{\widehat{P}}$ be the set of all zero-free
$\mathbb{K}$-labellings. Clearly, this set $\mathbb{K}_{\neq 0}^{\widehat{P}}$
is a Zariski-dense open subset of $\mathbb{K}^{\widehat{P}}$.
(c) Identify the set $\mathbb{K}^{\widehat{P}}$ with
$\prod\limits_{i=0}^{n+1}\mathbb{K}^{\widehat{P}_{i}}$ in the obvious way
(since $\widehat{P}$, regarded as a set, is the disjoint union of the sets
$\widehat{P}_{i}$ over all $i\in\left\\{0,1,...,n+1\right\\}$).
Using the identifications
$\mathbb{K}^{\widehat{P}}\cong\prod\limits_{i=0}^{n+1}\mathbb{K}^{\widehat{P}_{i}}$
and
$\overline{\mathbb{K}^{\widehat{P}}}\cong\prod\limits_{i=0}^{n+1}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)$,
we now define a rational map
$\pi:\mathbb{K}^{\widehat{P}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$
as the product of the canonical projections
$\mathbb{K}^{\widehat{P}_{i}}\dashrightarrow\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)$
(which are defined everywhere outside of the $\left\\{0\right\\}$ subsets)
over all $i\in\left\\{0,1,...,n+1\right\\}$. Notice that the domain of
definition of this rational map $\pi$ is precisely $\mathbb{K}_{\neq
0}^{\widehat{P}}$. For every $f\in\mathbb{K}^{\widehat{P}}$, we denote
$\pi\left(f\right)$ as the homogenization of the $\mathbb{K}$-labelling $f$.
(d) Two zero-free $\mathbb{K}$-labellings $f\in\mathbb{K}^{\widehat{P}}$ and
$g\in\mathbb{K}^{\widehat{P}}$ are said to be homogeneously equivalent if and
only if they satisfy one of the following equivalent conditions:
Condition 1: For every $i\in\left\\{0,1,...,n+1\right\\}$ and any two elements
$x$ and $y$ of $\widehat{P}_{i}$, we have
$\dfrac{f\left(x\right)}{f\left(y\right)}=\dfrac{g\left(x\right)}{g\left(y\right)}$.
Condition 2: There exists an $\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$
such that every $x\in\widehat{P}$ satisfies $g\left(x\right)=a_{\deg x}\cdot
f\left(x\right)$.
Condition 3: We have $\pi\left(f\right)=\pi\left(g\right)$.
(The equivalence between these three conditions is very easy to check. We will
never actually use Condition 1.)
###### Remark 6.3.
Clearly, homogeneous equivalence is an equivalence relation on the set
$\mathbb{K}_{\neq 0}^{\widehat{P}}$ of all zero-free $\mathbb{K}$-labellings.
We can identify $\overline{\mathbb{K}^{\widehat{P}}}$ with the quotient of the
set $\mathbb{K}_{\neq 0}^{\widehat{P}}$ modulo this relation. Then, $\pi$
becomes the canonical projection map
$\mathbb{K}^{\widehat{P}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$.
One remark about the notion “zero-free”: Being zero-free is a very weak
condition on a $\mathbb{K}$-labelling (indeed the zero-free
$\mathbb{K}$-labellings form a Zariski-dense open subset of the space of all
$\mathbb{K}$-labellings), and the $\mathbb{K}$-labellings which don’t satisfy
this condition are rather useless for us (if $f$ is a $\mathbb{K}$-labelling
which is not zero-free, then $R^{2}f$ is not well-defined, and usually not
even $Rf$ is well-defined). We are almost never giving up any generality if we
require a labelling to be zero-free.
###### Remark 6.4.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. For every $\mathbb{K}$-labelling $f\in\mathbb{K}^{\widehat{P}}$ and any
$\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$,
we define a $\mathbb{K}$-labelling $\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\in\mathbb{K}^{\widehat{P}}$ as in Definition 5.2.
Let $f\in\mathbb{K}^{\widehat{P}}$ be a zero-free $\mathbb{K}$-labelling of
$P$. Let
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$.
Then, the $\mathbb{K}$-labelling $\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$
is also zero-free. (This follows immediately from the definitions.)
###### Definition 6.5.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. For every zero-free $f\in\mathbb{K}^{\widehat{P}}$ and every
$i\in\left\\{1,2,...,n\right\\}$, the image of the restriction of
$f:\widehat{P}\to\mathbb{K}$ to $\widehat{P}_{i}$ under the canonical
projection
$\mathbb{K}^{\widehat{P}_{i}}\dashrightarrow\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)$
will be denoted by $\pi_{i}\left(f\right)$. This image $\pi_{i}\left(f\right)$
encodes the values of $f$ at the elements of $\widehat{P}$ of degree $i$ up to
multiplying all these values by a common nonzero scalar. Notice that
$\pi\left(f\right)=\left(\pi_{1}\left(f\right),\pi_{2}\left(f\right),...,\pi_{n}\left(f\right)\right)$
(14)
for every $f\in\mathbb{K}^{\widehat{P}}$. (Here, the right hand side of (14)
is regarded as an element of $\overline{\mathbb{K}^{\widehat{P}}}$ because it
belongs to
$\prod\limits_{i=1}^{n}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)=\overline{\mathbb{K}^{\widehat{P}}}$.)
We are next going to see:
###### Corollary 6.6.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let $i\in\left\\{1,2,...,n\right\\}$. If $f\in\mathbb{K}^{\widehat{P}}$
and $g\in\mathbb{K}^{\widehat{P}}$ are two homogeneously equivalent zero-free
$\mathbb{K}$-labellings, then $R_{i}f$ is homogeneously equivalent to $R_{i}g$
(as long as $R_{i}f$ and $R_{i}g$ are zero-free).
###### Corollary 6.7.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. If $f\in\mathbb{K}^{\widehat{P}}$ and $g\in\mathbb{K}^{\widehat{P}}$
are two homogeneously equivalent zero-free $\mathbb{K}$-labellings, then $Rf$
is homogeneously equivalent to $Rg$ (as long as $Rf$ and $Rg$ are zero-free).
Notice that Corollary 6.6 would not be valid if we were to replace $R_{i}$ by
a single toggle $T_{v}$! So the operators $R_{i}$ in some sense combine the
nice properties of $T_{v}$ (like being an involution, cf. Proposition 3.9)
with the nice properties of $R$ (like having an easily describable action on
w-tuples, cf. Proposition 4.3, and respecting homogeneous equivalence, cf.
Corollary 6.6).
###### Proof of Corollary 6.6 (sketched)..
Let $f\in\mathbb{K}^{\widehat{P}}$ and $g\in\mathbb{K}^{\widehat{P}}$ be two
homogeneously equivalent zero-free $\mathbb{K}$-labellings.
We know that $f$ and $g$ are homogeneously equivalent. By Condition 2 in
Definition 6.2 (d), this means that there exists an $\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$
such that every $x\in\widehat{P}$ satisfies $g\left(x\right)=a_{\deg x}\cdot
f\left(x\right)$. In other words, there exists an $\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$
such that
$g=\left(a_{0},a_{1},...,a_{n+1}\right)\flat f.$
Consider this $\left(n+2\right)$-tuple $\left(a_{0},a_{1},...,a_{n+1}\right)$.
Since $g=\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$, we have
$\displaystyle R_{i}g=R_{i}\left(\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\right)$
$\displaystyle=\left(a_{0},a_{1},...,a_{i-1},\dfrac{a_{i+1}a_{i-1}}{a_{i}},a_{i+1},a_{i+2},...,a_{n+1}\right)\flat\left(R_{i}f\right)$
(by Proposition 5.3). Hence, there exists an $\left(n+2\right)$-tuple
$\left(b_{0},b_{1},...,b_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$
such that
$R_{i}g=\left(b_{0},b_{1},...,b_{n+1}\right)\flat\left(R_{i}f\right)$
(namely,
$\left(b_{0},b_{1},...,b_{n+1}\right)=\left(a_{0},a_{1},...,a_{i-1},\dfrac{a_{i+1}a_{i-1}}{a_{i}},a_{i+1},a_{i+2},...,a_{n+1}\right)$).
In other words, there exists an $\left(n+2\right)$-tuple
$\left(b_{0},b_{1},...,b_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$
such that every $x\in\widehat{P}$ satisfies
$\left(R_{i}g\right)\left(x\right)=b_{\deg
x}\cdot\left(R_{i}f\right)\left(x\right)$. But this is precisely Condition 2
in Definition 6.2 (d), stated for the labellings $R_{i}f$ and $R_{i}g$ instead
of $f$ and $g$. Hence, $R_{i}f$ and $R_{i}g$ are homogeneously equivalent.
This proves Corollary 6.6. ∎
###### Proof of Corollary 6.7 (sketched)..
Let $f\in\mathbb{K}^{\widehat{P}}$ and $g\in\mathbb{K}^{\widehat{P}}$ be two
homogeneously equivalent zero-free $\mathbb{K}$-labellings. By iterative
application of Corollary 6.6, we then conclude that the
$\mathbb{K}$-labellings $\left(R_{1}\circ R_{2}\circ...\circ R_{n}\right)f$
and $\left(R_{1}\circ R_{2}\circ...\circ R_{n}\right)g$ are homogeneously
equivalent (if they are well-defined). Since $R_{1}\circ R_{2}\circ...\circ
R_{n}=R$ (by Proposition 3.8), this rewrites as follows: The
$\mathbb{K}$-labellings $Rf$ and $Rg$ are homogeneously equivalent. This
proves Corollary 6.7. ∎
Let us introduce a general piece of notation:
###### Definition 6.8.
Let $S$ and $T$ be two sets. Let $\sim_{S}$ be an equivalence relation on the
set $S$, and let $\sim_{T}$ be an equivalence relation on the set $T$. Let
$\overline{S}$ be the quotient of the set $S$ modulo the equivalence relation
$\sim_{S}$, and let $\overline{T}$ be the quotient of the set $T$ modulo the
equivalence relation $\sim_{T}$. Let $\pi_{S}:S\rightarrow\overline{S}$ and
$\pi_{T}:T\rightarrow\overline{T}$ be the canonical projections of a set on
its quotient. Let $f:S\rightarrow T$ be a map. If
$\overline{f}:\overline{S}\rightarrow\overline{T}$ is a map for which the
diagram
$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}}$$\textstyle{\overline{T}}$
is commutative, then we say that “the map $f$ descends to the map
$\overline{f}$”. It is easy to see that there exists at most one map
$\overline{f}:\overline{S}\rightarrow\overline{T}$ such that the map $f$
descends to the map $\overline{f}$ (for given $S$, $T$, $\sim_{S}$, $\sim_{T}$
and $f$). Moreover, the existence of a map
$\overline{f}:\overline{S}\rightarrow\overline{T}$ such that the map $f$
descends to the map $\overline{f}$ is equivalent to the statement that every
two elements $x$ and $y$ of $S$ satisfying $x\sim_{S}y$ satisfy
$f\left(x\right)\sim_{T}f\left(y\right)$.
The above statements are not literally true if we replace the map
$f:S\rightarrow T$ by a partial map $f:S\dashrightarrow T$. However, when $S$
and $T$ are two algebraic varieties and $\sim_{S}$ and $\sim_{T}$ are
algebraic equivalences (i.e., equivalence relations defined by polynomial
relations between coordinates of points) and $f:S\dashrightarrow T$ is a
rational map, then the above statements still are true (of course, with
$\overline{f}$ being a partial map).
###### Definition 6.9.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let $i\in\left\\{1,2,...,n\right\\}$. Because of Corollary 6.6, the
rational map
$R_{i}:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P}}$
descends (through the projection
$\pi:\mathbb{K}^{\widehat{P}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$)
to a partial map
$\overline{\mathbb{K}^{\widehat{P}}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$.
We denote this partial map
$\overline{\mathbb{K}^{\widehat{P}}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$
by $\overline{R_{i}}$. Thus, the diagram
$\textstyle{\mathbb{K}^{\widehat{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{i}}$$\scriptstyle{\pi}$$\textstyle{\mathbb{K}^{\widehat{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathbb{K}^{\widehat{P}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{R_{i}}}$$\textstyle{\overline{\mathbb{K}^{\widehat{P}}}}$
(15)
is commutative.
###### Definition 6.10.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. We define the partial map
$\overline{R}:\overline{\mathbb{K}^{\widehat{P}}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$
by
$\overline{R}=\overline{R_{1}}\circ\overline{R_{2}}\circ...\circ\overline{R_{n}}.$
Then, the diagram
$\textstyle{\mathbb{K}^{\widehat{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\scriptstyle{\pi}$$\textstyle{\mathbb{K}^{\widehat{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathbb{K}^{\widehat{P}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{R}}$$\textstyle{\overline{\mathbb{K}^{\widehat{P}}}}$
(16)
is commutative191919Proof. We have $R=R_{1}\circ R_{2}\circ...\circ R_{n}$ and
$\overline{R}=\overline{R_{1}}\circ\overline{R_{2}}\circ...\circ\overline{R_{n}}$.
Hence, the diagram (16) can be obtained by stringing together the diagrams
(15) for all $i\in\left\\{1,2,...,n\right\\}$ and then removing the “interior
edges”. Therefore, the diagram (16) is commutative (since the diagrams (15)
are commutative for all $i$), qed.. In other words, $\overline{R}$ is the
partial map
$\overline{\mathbb{K}^{\widehat{P}}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$
to which the partial map
$R:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P}}$ descends
(through the projection
$\pi:\mathbb{K}^{\widehat{P}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$).
Next, we formulate a result which says something to the extent of “a zero-free
$\mathbb{K}$-labelling $f\in\mathbb{K}^{\widehat{P}}$ is almost always
uniquely determined by its w-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$,
its homogenization $\pi\left(f\right)$ and the value $f\left(0\right)$”. The
words “almost always” are required here because otherwise the statement would
be wrong; but they have to be made precise. Here is the exact statement we
want to make:
###### Proposition 6.11.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Let $f$ and $g$ be two zero-free $\mathbb{K}$-labellings in
$\mathbb{K}^{\widehat{P}}$ such that
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)=\left(\mathbf{w}_{0}\left(g\right),\mathbf{w}_{1}\left(g\right),...,\mathbf{w}_{n}\left(g\right)\right)$
and such that no $i\in\left\\{0,1,...,n\right\\}$ satisfies
$\mathbf{w}_{i}\left(f\right)=0$. Also assume that
$\pi\left(f\right)=\pi\left(g\right)$ and $f\left(0\right)=g\left(0\right)$.
Then, $f=g$.
Proposition 6.11 is easily proven by reconstructing $f$ and $g$ “bottom-up”
along $\widehat{P}$. Alternatively, we can prove Proposition 6.11 directly
using Proposition 5.8, as follows:
###### Proof of Proposition 6.11 (sketched)..
Since $\pi\left(f\right)=\pi\left(g\right)$, we know that $f$ and $g$ are
homogeneously equivalent. By Condition 2 in Definition 6.2 (d), this means
that there exists an $\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$
such that every $x\in\widehat{P}$ satisfies $g\left(x\right)=a_{\deg x}\cdot
f\left(x\right)$. In other words, there exists an $\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\left(\mathbb{K}^{\times}\right)^{n+2}$
such that
$g=\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$
(where $\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\in\mathbb{K}^{\widehat{P}}$ is defined as in Definition 5.2). Consider this
$\left(n+2\right)$-tuple $\left(a_{0},a_{1},...,a_{n+1}\right)$.
Since $g=\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$, we know that
$\displaystyle\left(\text{the w-tuple of }g\right)$
$\displaystyle=\left(\text{the w-tuple of
}\left(a_{0},a_{1},...,a_{n+1}\right)\flat f\right)$
$\displaystyle=\left(\dfrac{a_{0}}{a_{1}}\mathbf{w}_{0}\left(f\right),\dfrac{a_{1}}{a_{2}}\mathbf{w}_{1}\left(f\right),...,\dfrac{a_{n}}{a_{n+1}}\mathbf{w}_{n}\left(f\right)\right)$
(by Proposition 5.8). Compared with
$\left(\text{the w-tuple of
}g\right)=\left(\mathbf{w}_{0}\left(g\right),\mathbf{w}_{1}\left(g\right),...,\mathbf{w}_{n}\left(g\right)\right)=\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right),$
this yields
$\left(\dfrac{a_{0}}{a_{1}}\mathbf{w}_{0}\left(f\right),\dfrac{a_{1}}{a_{2}}\mathbf{w}_{1}\left(f\right),...,\dfrac{a_{n}}{a_{n+1}}\mathbf{w}_{n}\left(f\right)\right)=\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right).$
In other words,
$\dfrac{a_{i}}{a_{i+1}}\mathbf{w}_{i}\left(f\right)=\mathbf{w}_{i}\left(f\right)$
for every $i\in\left\\{0,1,...,n\right\\}$. Hence, $\dfrac{a_{i}}{a_{i+1}}=1$
for every $i\in\left\\{0,1,...,n\right\\}$ (here, we cancelled out
$\mathbf{w}_{i}\left(f\right)$, because by assumption we don’t have
$\mathbf{w}_{i}\left(f\right)=0$). In other words, $a_{i}=a_{i+1}$ for every
$i\in\left\\{0,1,...,n\right\\}$. Thus, $a_{0}=a_{1}=...=a_{n+1}$.
But since $g=\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$, we have
$g\left(0\right)=\left(\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\right)\left(0\right)=a_{\deg 0}\cdot f\left(0\right)=a_{0}\cdot
f\left(0\right)$ (since $\deg 0=0$), so that
$f\left(0\right)=g\left(0\right)=a_{0}\cdot f\left(0\right)$. Since
$f\left(0\right)\neq 0$ (because $f$ is zero-free, and the only element of
$\widehat{P}_{0}$ is $0$), we can cancel $f\left(0\right)$ here and obtain
$1=a_{0}$. In view of this, $a_{0}=a_{1}=...=a_{n+1}$ becomes
$a_{0}=a_{1}=...=a_{n+1}=1$. Thus,
$\left(a_{0},a_{1},...,a_{n+1}\right)=\left(\underbrace{1,1,...,1}_{n+2\text{
times}}\right)$, so that $g=\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f=\left(\underbrace{1,1,...,1}_{n+2\text{ times}}\right)\flat f=f$, proving
Proposition 6.11.
∎
###### Definition 6.12.
Let $\mathbb{K}$ be a field. In the following, if $S$ is a finite set, and $q$
is an element of a projective space $\mathbb{P}\left(\mathbb{K}^{S}\right)$ of
the free vector space with basis $S$, and $k$ is an integer, then $q^{k}$ will
denote the element of $\mathbb{P}\left(\mathbb{K}^{S}\right)$ obtained by
replacing every homogeneous coordinate of $q$ by its $k$-th power. This is
well-defined (and will mostly be used for $k=-1$). In particular, this
definition applies to $S=\left\\{1,2,...,n\right\\}$ for $n\in\mathbb{N}$ (in
which case $\mathbb{K}^{S}=\mathbb{K}^{n}$).
We can explicitly describe the action of the $\overline{R_{i}}$ when the
“structure of the poset $P$ between degrees $i-1$, $i$ and $i+1$” is
particularly simple:
###### Proposition 6.13.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Fix $i\in\left\\{1,2,...,n\right\\}$. Assume that every
$u\in\widehat{P}_{i}$ and every $v\in\widehat{P}_{i+1}$ satisfy $u\lessdot v$.
Assume further that every $u\in\widehat{P}_{i-1}$ and every
$v\in\widehat{P}_{i}$ satisfy $u\lessdot v$. Let
$f\in\mathbb{K}^{\widehat{P}}$. Then,
$\displaystyle\left(\pi_{1}\left(R_{i}f\right),\pi_{2}\left(R_{i}f\right),...,\pi_{n}\left(R_{i}f\right)\right)$
$\displaystyle=\left(\pi_{1}\left(f\right),\pi_{2}\left(f\right),...,\pi_{i-1}\left(f\right),\left(\pi_{i}\left(f\right)\right)^{-1},\pi_{i+1}\left(f\right),\pi_{i+2}\left(f\right),...,\pi_{n}\left(f\right)\right).$
From this proposition, we obtain two corollaries:
###### Corollary 6.14.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Fix $i\in\left\\{1,2,...,n\right\\}$. Assume that every
$u\in\widehat{P}_{i}$ and every $v\in\widehat{P}_{i+1}$ satisfy $u\lessdot v$.
Assume further that every $u\in\widehat{P}_{i-1}$ and every
$v\in\widehat{P}_{i}$ satisfy $u\lessdot v$. Let
$\widetilde{f}=\left(\widetilde{f}_{1},\widetilde{f}_{2},...,\widetilde{f}_{n}\right)\in\overline{\mathbb{K}^{\widehat{P}}}$.
Then,
$\overline{R_{i}}\left(\widetilde{f}\right)=\left(\widetilde{f}_{1},\widetilde{f}_{2},...,\widetilde{f}_{i-1},\widetilde{f}_{i}^{-1},\widetilde{f}_{i+1},\widetilde{f}_{i+2},...,\widetilde{f}_{n}\right).$
###### Corollary 6.15.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Assume that, for every $i\in\left\\{1,2,...,n-1\right\\}$, every
$u\in\widehat{P}_{i}$ and every $v\in\widehat{P}_{i+1}$ satisfy $u\lessdot v$.
Let $f\in\mathbb{K}^{\widehat{P}}$ be zero-free. Then,
$\left(\pi_{1}\left(Rf\right),\pi_{2}\left(Rf\right),...,\pi_{n}\left(Rf\right)\right)=\left(\left(\pi_{1}\left(f\right)\right)^{-1},\left(\pi_{2}\left(f\right)\right)^{-1},...,\left(\pi_{n}\left(f\right)\right)^{-1}\right).$
## 7 Order
In this short section, we will relate the orders of the maps $R$ and
$\overline{R}$ for a graded poset $P$. The relation will later be used to gain
knowledge on both of these orders.
We begin by defining the order of a partial map:
###### Definition 7.1.
Let $S$ be a set.
(a) If $\alpha$ and $\beta$ are two partial maps from the set $S$, then we
write “$\alpha=\beta$” if and only if every $s\in S$ for which both
$\alpha\left(s\right)$ and $\beta\left(s\right)$ are well-defined satisfies
$\alpha\left(s\right)=\beta\left(s\right)$. This is, per se, not a well-
behaved notation (e.g., it is possible that three partial maps $\alpha$,
$\beta$ and $\gamma$ satisfy $\alpha=\beta$ and $\beta=\gamma$ but not
$\alpha=\gamma$). However, we are going to use this notation for rational maps
and their quotients (and, of course, total maps) only; in all of these cases,
the notation is well-behaved (e.g., if $\alpha$, $\beta$ and $\gamma$ are
three rational maps satisfying $\alpha=\beta$ and $\beta=\gamma$, then
$\alpha=\gamma$, because the intersection of two Zariski-dense open subsets is
Zariski-dense and open).
(b) The order of a partial map $\varphi:S\dashrightarrow S$ is defined to be
the smallest positive integer $k$ satisfying
$\varphi^{k}=\operatorname*{id}\nolimits_{S}$, if such a positive integer $k$
exists, and $\infty$ otherwise. Here, we are disregarding the fact that
$\varphi$ is only a partial map; we will be working only with dominant
rational maps and their quotients (and total maps), so nothing will go wrong.
We denote the order of a partial map $\varphi:S\dashrightarrow S$ as
$\operatorname*{ord}\varphi$.
###### Convention 7.2.
In the following, we are going to occasionally make arithmetical statements
involving the symbol $\infty$. We declare that $0$ and $\infty$ are divisible
by $\infty$, but no positive integer is divisible by $\infty$. We further
declare that every positive integer (but not $0$) divides $\infty$. We set
$\operatorname{lcm}\left(a,\infty\right)$ and
$\operatorname{lcm}\left(\infty,a\right)$ to mean $\infty$ whenever $a$ is a
positive integer.
As a consequence of Proposition 6.11, we have:
###### Proposition 7.3.
Let $n\in\mathbb{N}$. Let $\mathbb{K}$ be a field. Let $P$ be an $n$-graded
poset. Then,
$\operatorname*{ord}R=\operatorname{lcm}\left(n+1,\operatorname*{ord}\overline{R}\right)$.
(Recall that $\operatorname{lcm}\left(n+1,\infty\right)$ is to be understood
as $\infty$.)
The proof of this boils down to considering the effect of $R$ on the w-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$
and on the homogenization $\pi\left(f\right)$ of a $\mathbb{K}$-labelling $f$.
The effect on the w-tuple is a cyclic shift (by Proposition 4.4), which has
order $n+1$. The effect on the homogenization is $\overline{R}$. It is now
easy to see (invoking Proposition 6.11) that the order of $R$ is the
$\operatorname{lcm}$ of the orders of these two actions. Here are the details
of this derivation:
###### Proof of Proposition 7.3 (sketched)..
1st step: The commutativity of the diagram (16) yields
$\overline{R}\circ\pi=\pi\circ R$. Hence,
$\text{every }\ell\in\mathbb{N}\text{ satisfies
}\overline{R}^{\ell}\circ\pi=\pi\circ R^{\ell}$ (17)
(this is clear by induction over $\ell$). Thus, if some $\ell\in\mathbb{N}$
satisfies $R^{\ell}=\operatorname*{id}$, then it satisfies
$\overline{R}^{\ell}=\operatorname*{id}$ as well202020Proof. Let
$\ell\in\mathbb{N}$ be such that $R^{\ell}=\operatorname*{id}$. Then,
$\overline{R}^{\ell}\circ\pi=\pi\circ\underbrace{R^{\ell}}_{=\operatorname*{id}}=\pi$.
Since $\pi$ is right-cancellable (since $\pi$ is surjective), this yields
$\overline{R}^{\ell}=\operatorname*{id}$, qed.. Hence,
$\operatorname*{ord}\overline{R}\mid\operatorname*{ord}R$ (recall that every
positive integer divides $\infty$, but only $0$ and $\infty$ are divisible by
$\infty$). In particular, if $\operatorname*{ord}\overline{R}=\infty$, then
$\operatorname*{ord}R=\infty$. Thus, Proposition 7.3 is obvious in the case
when $\operatorname*{ord}\overline{R}=\infty$. Hence, for the rest of the
proof of Proposition 7.3, we can WLOG assume that
$\operatorname*{ord}\overline{R}\neq\infty$. Assume this.
2nd step: Since $\operatorname*{ord}\overline{R}\neq\infty$, we know that
$\operatorname*{ord}\overline{R}$ is a positive integer. Let $m$ be this
positive integer. Then, $m=\operatorname*{ord}\overline{R}$, so that
$\overline{R}^{m}=\operatorname*{id}$.
Let $\ell=\operatorname{lcm}\left(n+1,m\right)$. Then, $n+1\mid\ell$ and
$m\mid\ell$. Since $\operatorname*{ord}\overline{R}=m\mid\ell$, we have
$\overline{R}^{\ell}=\operatorname*{id}$. But from (17), we have $\pi\circ
R^{\ell}=\underbrace{\overline{R}^{\ell}}_{=\operatorname*{id}}\circ\pi=\pi$.
We are now going to prove that $R^{\ell}=\operatorname*{id}$. In order to
prove this, it is clearly enough to show that almost every (in the sense of
Zariski topology) zero-free $\mathbb{K}$-labelling $f$ of $P$ satisfies
$R^{\ell}f=\operatorname*{id}f$ (because $R^{\ell}f=\operatorname*{id}f$ is a
polynomial identity in the labels of $f$). But it is easily shown that for
almost every (in the sense of Zariski topology) zero-free
$\mathbb{K}$-labelling $f$ of $P$, the w-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$
of $f$ consists of nonzero elements of $\mathbb{K}$. 212121Proof. We will
prove a slightly better result: Almost every $f\in\mathbb{K}^{\widehat{P}}$ is
a zero-free $\mathbb{K}$-labelling of $P$ with the property that
$\left(\mathbf{w}_{i}\left(f\right)\text{ is well-defined and nonzero for
every }i\in\left\\{0,1,...,n\right\\}\right).$ (18) In fact, the condition
(18) on an $f\in\mathbb{K}^{\widehat{P}}$ is a requirement saying that certain
rational expressions in the values of $f$ do not vanish (namely, the
denominators in $\mathbf{w}_{i}\left(f\right)$ and the sums
$\mathbf{w}_{i}\left(f\right)$ themselves). If we can prove that none of these
expressions is identically zero, then it will follow that for almost every
$f\in\mathbb{K}^{\widehat{P}}$, none of these expressions vanishes (because
there are only finitely many expressions whose vanishing we are trying to
avoid, and the infiniteness of $\mathbb{K}$ allows us to avoid them all if
none of them is identically zero); thus (18) will follow and we will be done.
Hence, it remains to show that none of these expressions is identically zero.
Assume the contrary. Then, one of our rational expressions – either a
denominator in one of the $\mathbf{w}_{i}\left(f\right)$, or one of the sums
$\mathbf{w}_{i}\left(f\right)$ – identically vanishes. It must be one of the
sums $\mathbf{w}_{i}\left(f\right)$, since the denominators in the
$\mathbf{w}_{i}\left(f\right)$ cannot identically vanish (they are simply
values $f\left(y\right)$). So there exists some
$i\in\left\\{0,1,...,n\right\\}$ such that every $\mathbb{K}$-labelling $f$ of
$P$ (for which $\mathbf{w}_{i}\left(f\right)$ is well-defined) satisfies
$\mathbf{w}_{i}\left(f\right)=0$. Consider this $i$. Notice that $i\leqslant
n$ and thus $1\notin\widehat{P}_{i}$. We have
$0=\mathbf{w}_{i}\left(f\right)=\sum_{\begin{subarray}{c}x\in\widehat{P}_{i};\
y\in\widehat{P}_{i+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{f\left(x\right)}{f\left(y\right)}=\sum_{x\in\widehat{P}_{i}}f\left(x\right)\sum_{\begin{subarray}{c}y\in\widehat{P}_{i+1};\\\
y\gtrdot x\end{subarray}}\dfrac{1}{f\left(y\right)}.$ This forces the sum
$\sum_{\begin{subarray}{c}y\in\widehat{P}_{i+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{1}{f\left(y\right)}$ to be identically $0$ for every
$x\in\widehat{P}_{i}$ (because these sums for different values of $x$ are
prevented from canceling each other by the completely independent
$f\left(x\right)$ coefficients in front of them). Fix some
$x\in\widehat{P}_{i}$ (such an $x$ clearly exists since
$\deg:\widehat{P}\rightarrow\left\\{0,1,...,n+1\right\\}$ is surjective), and
ponder what it means for the sum
$\sum_{\begin{subarray}{c}y\in\widehat{P}_{i+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{1}{f\left(y\right)}$ to be identically $0$. It means
that this sum is empty, i.e., that there exists no $y\in\widehat{P}_{i+1}$
satisfying $y\gtrdot x$. But this can only happen when $x=1$, which is not the
case in our situation (because $x\in\widehat{P}_{i}$ and
$1\notin\widehat{P}_{i}$). So we have obtained a contradiction.
Hence, in order to prove $R^{\ell}=\operatorname*{id}$, it is enough to show
that every zero-free $\mathbb{K}$-labelling $f$ of $P$ for which the w-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$
of $f$ consists of nonzero elements of $\mathbb{K}$ satisfies
$R^{\ell}f=\operatorname*{id}f$. This is what we are going to do now.
So let $f$ be a zero-free $\mathbb{K}$-labelling of $P$ for which the w-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$
of $f$ consists of nonzero elements of $\mathbb{K}$. We will prove that
$R^{\ell}f=\operatorname*{id}f$.
From Proposition 4.4, we know that the map $R$ changes the w-tuple of a
$\mathbb{K}$-labelling by shifting it cyclically. Hence, for every
$k\in\mathbb{N}$, the map $R^{k}$ changes the w-tuple of a
$\mathbb{K}$-labelling by shifting it cyclically $k$ times. If this $k$ is
divisible by $n+1$, then this obviously means that the map $R^{k}$ preserves
the w-tuple of a $\mathbb{K}$-labelling (because the w-tuple has $n+1$
entries, and thus shifting it cyclically for a multiple of $n+1$ times leaves
it invariant). Hence, the w-tuple of $f$ equals the w-tuple of $R^{\ell}f$.
Recalling the definition of a w-tuple, we can rewrite this as follows:
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)=\left(\mathbf{w}_{0}\left(R^{\ell}f\right),\mathbf{w}_{1}\left(R^{\ell}f\right),...,\mathbf{w}_{n}\left(R^{\ell}f\right)\right).$
Moreover, by assumption, the w-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$
of $f$ consists of nonzero elements of $\mathbb{K}$. In other words, no
$i\in\left\\{0,1,...,n\right\\}$ satisfies $\mathbf{w}_{i}\left(f\right)=0$.
Furthermore $\pi\left(R^{\ell}f\right)=\underbrace{\left(\pi\circ
R^{\ell}\right)}_{=\pi}f=\pi\left(f\right)$.
Also, Corollary 2.18 (applied to $k=\ell$) yields
$\left(R^{\ell}f\right)\left(0\right)=f\left(0\right)$.
We now can apply Proposition 6.11 to $g=R^{\ell}f$. As a result, we obtain
$R^{\ell}f=f$. In other words, $R^{\ell}f=\operatorname*{id}f$.
Now forget that we fixed $f$. We have thus shown that
$R^{\ell}f=\operatorname*{id}f$ for every zero-free $\mathbb{K}$-labelling $f$
of $P$ for which the w-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$
of $f$ consists of nonzero elements of $\mathbb{K}$. Therefore, we have shown
that $R^{\ell}=\operatorname*{id}$ (by what we have said above). Thus,
$\operatorname*{ord}R\mid\ell=\operatorname{lcm}\left(n+1,\underbrace{m}_{=\operatorname*{ord}\overline{R}}\right)=\operatorname{lcm}\left(n+1,\operatorname*{ord}\overline{R}\right)$.
3rd step: We now will show that
$\operatorname{lcm}\left(n+1,\operatorname*{ord}\overline{R}\right)\mid\operatorname*{ord}R$.
In order to do that, we assume WLOG that $\operatorname*{ord}R\neq\infty$
(because otherwise,
$\operatorname{lcm}\left(n+1,\operatorname*{ord}\overline{R}\right)\mid\operatorname*{ord}R$
is obvious). Hence, $\operatorname*{ord}R$ is a positive integer. Denote this
positive integer by $q$. So, $q=\operatorname*{ord}R$.
It is easy to see that for almost every (in the sense of Zariski topology)
zero-free $\mathbb{K}$-labelling $f$ of $P$, the entries of the w-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$
of $f$ are pairwise distinct. Hence, there exists a zero-free
$\mathbb{K}$-labelling $f$ of $P$ such that the entries of the w-tuple
$\left(\mathbf{w}_{0}\left(f\right),\mathbf{w}_{1}\left(f\right),...,\mathbf{w}_{n}\left(f\right)\right)$
of $f$ are pairwise distinct and such that $R^{k}f$ is well-defined for all
$k\in\left\\{0,1,...,q\right\\}$. Consider such an $f$.
Since $q=\operatorname*{ord}R$, we have $R^{q}=\operatorname*{id}$, so that
$R^{q}f=f$.
Recall once again (from the 2nd step) that for every $k\in\mathbb{N}$, the map
$R^{k}$ changes the w-tuple of a $\mathbb{K}$-labelling by shifting it
cyclically $k$ times. In particular, the map $R^{q}$ changes the w-tuple of
the $\mathbb{K}$-labelling $f$ by shifting it cyclically $q$ times. In other
words, the w-tuple of $R^{q}f$ is obtained from the w-tuple of $f$ by shifting
it cyclically $q$ times. Since $R^{q}f=f$, this rewrites as follows: The
w-tuple of $f$ is obtained from the w-tuple of $f$ by shifting it cyclically
$q$ times. In other words, the w-tuple of $f$ is fixed under a $q$-fold cyclic
shift. But since the w-tuple of $f$ is an $\left(n+1\right)$-tuple of pairwise
distinct entries, this can only happen if $n+1\mid q$. Hence, we have $n+1\mid
q$.
Combining $n+1\mid q=\operatorname*{ord}R$ with
$\operatorname*{ord}\overline{R}\mid\operatorname*{ord}R$, we obtain
$\operatorname{lcm}\left(n+1,\operatorname*{ord}\overline{R}\right)\mid\operatorname*{ord}R$.
Combining this with
$\operatorname*{ord}R\mid\operatorname{lcm}\left(n+1,\operatorname*{ord}\overline{R}\right)$,
we obtain
$\operatorname*{ord}R=\operatorname{lcm}\left(n+1,\operatorname*{ord}\overline{R}\right)$.
This proves Proposition 7.3. ∎
## 8 The opposite poset
Before we move on to the first interesting class of posets for which we can
compute the order of birational rowmotion, let us prove an easy “symmetry
property” of birational rowmotion.
###### Definition 8.1.
Let $P$ be a poset. Then, $P^{\operatorname*{op}}$ will denote the poset
defined on the same ground set as $P$ but with the order relation defined by
$\left(\left(a<_{P^{\operatorname*{op}}}b\text{ if and only if
}b<_{P}a\right)\text{ for all }a\in P\text{ and }b\in P\right)$
(where $<_{P}$ denotes the smaller relation of the poset $P$, and where
$<_{P^{\operatorname*{op}}}$ denotes the smaller relation of the poset
$P^{\operatorname*{op}}$ which we are defining). The poset
$P^{\operatorname*{op}}$ is called the opposite poset of $P$.
Note that $P^{\operatorname*{op}}$ is called the dual of the poset $P$ in
[Stan11].
###### Remark 8.2.
It is clear that $\left(P^{\operatorname*{op}}\right)^{\operatorname*{op}}=P$
for any poset $P$. Also, if $n\in\mathbb{N}$, and if $P$ is an $n$-graded
poset, then $P^{\operatorname*{op}}$ is an $n$-graded poset.
###### Definition 8.3.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. We denote the maps $R$
and $\overline{R}$ by $R_{P}$ and $\overline{R}_{P}$, respectively, so as to
make their dependence on $P$ explicit.
We can now state a symmetry property of $\operatorname*{ord}R$ (as defined in
Definition 7.1):
###### Proposition 8.4.
Let $P$ be a finite poset. Let $\mathbb{K}$ be a field. Then,
$\operatorname*{ord}\left(R_{P^{\operatorname*{op}}}\right)=\operatorname*{ord}\left(R_{P}\right)$
and
$\operatorname*{ord}\left(\overline{R}_{P^{\operatorname*{op}}}\right)=\operatorname*{ord}\left(\overline{R}_{P}\right)$.
###### Proof of Proposition 8.4 (sketched)..
Define a rational map
$\kappa:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P^{\operatorname*{op}}}}$
by
$\left(\kappa
f\right)\left(w\right)=\left\\{\begin{array}[c]{c}\dfrac{1}{f\left(w\right)},\
\ \ \ \ \ \ \ \ \ \text{if }w\in P;\\\ \dfrac{1}{f\left(1\right)},\ \ \ \ \ \
\ \ \ \ \text{if }w=0;\\\ \dfrac{1}{f\left(0\right)},\ \ \ \ \ \ \ \ \ \
\text{if }w=1\end{array}\right.\ \ \ \ \ \ \ \ \ \ \text{for every
}w\in\widehat{P^{\operatorname*{op}}}\text{ for every
}f\in\mathbb{K}^{\widehat{P}}.$
This map $\kappa$ is a birational map. (Its inverse map is defined in the same
way.)
We claim that $\kappa\circ R_{P}=R_{P^{\operatorname*{op}}}^{-1}\circ\kappa$.
Indeed, it is easy to see (by computation) that every element $v\in P$
satisfies
$\kappa\circ T_{v}=T_{v}\circ\kappa,$ (19)
where the $T_{v}$ on the left hand side is defined with respect to the poset
$P$, and the $T_{v}$ on the right hand side is defined with respect to the
poset $P^{\operatorname*{op}}$. Now, let $\left(v_{1},v_{2},...,v_{m}\right)$
be a linear extension of $P$. Then, $\left(v_{m},v_{m-1},...,v_{1}\right)$ is
a linear extension of $P^{\operatorname*{op}}$, so that Proposition 2.20
(applied to $P^{\operatorname*{op}}$ and
$\left(v_{m},v_{m-1},...,v_{1}\right)$ instead of $P$ and
$\left(v_{1},v_{2},...,v_{m}\right)$) yields that
$R_{P^{\operatorname*{op}}}^{-1}=T_{v_{1}}\circ T_{v_{2}}\circ...\circ
T_{v_{m}}:\mathbb{K}^{\widehat{P^{\operatorname*{op}}}}\dashrightarrow\mathbb{K}^{\widehat{P^{\operatorname*{op}}}}$.
On the other hand, the definition of $R_{P}$ yields $R_{P}=T_{v_{1}}\circ
T_{v_{2}}\circ...\circ
T_{v_{m}}:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P}}$.
Now, using (19), it is easy to see that
$\kappa\circ\left(T_{v_{1}}\circ T_{v_{2}}\circ...\circ
T_{v_{m}}\right)=\left(T_{v_{1}}\circ T_{v_{2}}\circ...\circ
T_{v_{m}}\right)\circ\kappa.$
Since the $T_{v_{1}}\circ T_{v_{2}}\circ...\circ T_{v_{m}}$ on the left hand
side equals $R_{P}$, and the $T_{v_{1}}\circ T_{v_{2}}\circ...\circ T_{v_{m}}$
on the right hand side equals $R_{P^{\operatorname*{op}}}^{-1}$, this rewrites
as $\kappa\circ R_{P}=R_{P^{\operatorname*{op}}}^{-1}\circ\kappa$. Since
$\kappa$ is a birational map, this shows that $R_{P}$ and
$R_{P^{\operatorname*{op}}}^{-1}$ are birationally equivalent, so that
$\operatorname*{ord}\left(R_{P}\right)=\operatorname*{ord}\left(R_{P^{\operatorname*{op}}}^{-1}\right)=\operatorname*{ord}\left(R_{P^{\operatorname*{op}}}\right)$.
Since $\kappa$ commutes with homogenization, we also obtain the birational
equivalence of the maps $\overline{R}_{P}$ and
$\overline{R}_{P^{\operatorname*{op}}}^{-1}$, whence
$\operatorname*{ord}\left(\overline{R}_{P}\right)=\operatorname*{ord}\left(\overline{R}_{P^{\operatorname*{op}}}^{-1}\right)=\operatorname*{ord}\left(\overline{R}_{P^{\operatorname*{op}}}\right)$.
This proves Proposition 8.4. ∎
## 9 Skeletal posets
We will now introduce a class of posets which we call “skeletal posets”.
Roughly speaking, these are graded posets built up inductively from the empty
poset by the operations of disjoint union (but only allowing disjoint union of
two $n$-graded posets for one and the same value of $n$) and “grafting” on an
antichain (generalizing the idea of grafting a tree on a new root). In
particular, all graded forests (oriented either away from the roots or towards
the roots) will belong to this class of posets, but also various other posets.
We begin by defining the notions involved:
###### Definition 9.1.
Let $n\in\mathbb{N}$. Let $P$ and $Q$ be two $n$-graded posets. We denote by
$PQ$ the disjoint union of the posets $P$ and $Q$. (This disjoint union is
denoted by $P+Q$ in [Stan11, §3.2]. Its poset structure is defined in such a
way that any element of $P$ and any element of $Q$ are incomparable, while $P$
and $Q$ are subposets of $PQ$.) Clearly, $PQ$ is again an $n$-graded poset.
###### Definition 9.2.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let $k$ be a positive
integer. We denote by $B_{k}P$ the result of adding $k$ new elements to the
poset $P$, and declaring these $k$ elements to be smaller than each of the
elements of $P$ (but incomparable with each other). Clearly, $B_{k}P$ is an
$\left(n+1\right)$-graded poset.
###### Definition 9.3.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let $k$ be a positive
integer. We denote by $B_{k}^{\prime}P$ the result of adding $k$ new elements
to the poset $P$, and declaring these $k$ elements to be larger than each of
the elements of $P$ (but incomparable with each other). Clearly,
$B_{k}^{\prime}P$ is an $\left(n+1\right)$-graded poset.
If $P$ is an $n$-graded poset and $k$ is a positive integer, then, in the
notations of Stanley ([Stan11, §3.2]), we have $B_{k}P=A_{k}\oplus P$ and
$B_{k}^{\prime}P=P\oplus A_{k}$, where $A_{k}$ denotes the $k$-element
antichain.
It is easy to see that $B_{k}P$ and $B_{k}^{\prime}P$ are “symmetric” notions
with respect to taking the opposite poset:
###### Proposition 9.4.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Then,
$B_{k}^{\prime}P=\left(B_{k}\left(P^{\operatorname*{op}}\right)\right)^{\operatorname*{op}}$.
(Here, we are using the notation introduced in Definition 8.1.)
We now define the notion of a skeletal poset:
###### Definition 9.5.
We define the class of skeletal posets inductively by means of the following
axioms:
– The empty poset is skeletal.
– If $P$ is an $n$-graded skeletal poset and $k$ is a positive integer, then
the posets $B_{k}P$ and $B_{k}^{\prime}P$ are skeletal.
– If $n$ is a nonnegative integer and $P$ and $Q$ are two $n$-graded skeletal
posets, then the poset $PQ$ is skeletal.
Notice that every skeletal poset is graded. Also, notice that every graded
rooted forest (made into a poset by having every node smaller than its
children) is a skeletal poset. (Indeed, every graded rooted forest can be
constructed from $\varnothing$ using merely the operations $P\mapsto B_{1}P$
and $\left(P,Q\right)\mapsto PQ$.) Also, every graded rooted arborescence
(i.e., the opposite poset of a graded rooted tree) is a skeletal poset (for a
similar reason).
###### Example 9.6.
The rooted forest
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$
is skeletal, and in fact can be written as
$\left(B_{1}\left(\left(B_{1}\left(B_{2}\varnothing\right)\right)\left(B_{1}\left(B_{1}\varnothing\right)\right)\right)\right)\left(B_{1}\left(B_{1}\left(B_{1}\varnothing\right)\right)\right)$.
(This form of writing is not unique, since
$B_{2}\varnothing=\left(B_{1}\varnothing\right)\left(B_{1}\varnothing\right)$.)
The tree
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$
can be written as
$B_{1}\left(\left(B_{1}\varnothing\right)\left(B_{1}\left(B_{1}\varnothing\right)\right)\right)$,
but is not skeletal because $B_{1}\varnothing$ and
$B_{1}\left(B_{1}\varnothing\right)$ are not $n$-graded with one and the same
$n$.
The poset
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$
is neither a tree nor an arborescence, but it has the form
$B_{1}\left(\left(B_{2}\left(B_{2}\varnothing\right)\right)\left(B_{1}^{\prime}\left(B_{2}\varnothing\right)\right)\right)$
and is skeletal.
Our main result on skeletal posets is the following:
###### Proposition 9.7.
Let $P$ be a skeletal poset. Let $\mathbb{K}$ be a field. Then,
$\operatorname*{ord}\left(R_{P}\right)$ and
$\operatorname*{ord}\left(\overline{R}_{P}\right)$ are finite.
In order to be able to prove this proposition, we first build up some
machinery for determining $\operatorname*{ord}\left(R_{P}\right)$ and
$\operatorname*{ord}\left(\overline{R}_{P}\right)$ given such orders in
smaller posets. Here is a very basic fact to get started:
###### Proposition 9.8.
Fix $n\in\mathbb{N}$. Let $P$ and $Q$ be two $n$-graded posets. Let
$\mathbb{K}$ be a field. Then,
$\operatorname*{ord}\left(R_{PQ}\right)=\operatorname{lcm}\left(\operatorname*{ord}\left(R_{P}\right),\operatorname*{ord}\left(R_{Q}\right)\right)$.
###### Proof of Proposition 9.8..
The proof of this is as easy as it looks: a $\mathbb{K}$-labelling of the
disjoint union $PQ$ can be regarded as a pair of a $\mathbb{K}$-labelling of
$P$ and a $\mathbb{K}$-labelling of $Q$ (with identical labels at $0$ and
$1$), and the map $R$ (as well as all $R_{i}$) acts on these labellings
independently. ∎
The analogue of Proposition 9.8 with all $R$’s replaced by $\overline{R}$’s is
false. Instead, $\operatorname*{ord}\left(\overline{R}_{PQ}\right)$ can be
computed as follows:222222The following proposition is, in some sense,
uninteresting, as it is a negative result (it merely serves to convince one
that $\operatorname*{ord}\left(\overline{R}_{PQ}\right)$ is not lower than
what is expected from Propositions 7.3 and 9.8).
###### Proposition 9.9.
Fix $n\in\mathbb{N}$. Let $P$ and $Q$ be two $n$-graded posets. Let
$\mathbb{K}$ be a field. Then,
$\operatorname*{ord}\left(\overline{R}_{PQ}\right)=\operatorname{lcm}\left(\operatorname*{ord}\left(R_{P}\right),\operatorname*{ord}\left(R_{Q}\right)\right)$.
###### Proof of Proposition 9.9 (sketched)..
Assume WLOG that $n\neq 0$ (else, everything is obvious). Hence, $P$ and $Q$
are nonempty (being $n$-graded).
Proposition 7.3 yields
$\operatorname*{ord}\left(R_{PQ}\right)=\operatorname{lcm}\left(n+1,\operatorname*{ord}\left(\overline{R}_{PQ}\right)\right)$.
WLOG assume that $\operatorname*{ord}\left(R_{P}\right)$ and
$\operatorname*{ord}\left(R_{Q}\right)$ are finite232323Otherwise,
$\operatorname{lcm}\left(\operatorname*{ord}\left(R_{P}\right),\operatorname*{ord}\left(R_{Q}\right)\right)$
is infinite, whence $\operatorname*{ord}\left(R_{PQ}\right)$ is infinite (by
Proposition 9.8), whence $\operatorname*{ord}\left(\overline{R}_{PQ}\right)$
is infinite (because
$\operatorname*{ord}\left(R_{PQ}\right)=\operatorname{lcm}\left(n+1,\operatorname*{ord}\left(\overline{R}_{PQ}\right)\right)$),
whence Proposition 9.9 is trivial.. Then, Proposition 9.8 shows that
$\operatorname*{ord}\left(R_{PQ}\right)=\operatorname{lcm}\left(\operatorname*{ord}\left(R_{P}\right),\operatorname*{ord}\left(R_{Q}\right)\right)$
is finite, so that $\operatorname*{ord}\left(\overline{R}_{PQ}\right)$ is
finite (because
$\operatorname*{ord}\left(R_{PQ}\right)=\operatorname{lcm}\left(n+1,\operatorname*{ord}\left(\overline{R}_{PQ}\right)\right)$).
Let $\ell$ be $\operatorname*{ord}\left(\overline{R}_{PQ}\right)$. Then,
$\ell$ is finite and satisfies $\overline{R}_{PQ}^{\ell}=\operatorname*{id}$.
We will show that $n+1\mid\ell$.
For every $\mathbb{K}$-labelling $f$ of $PQ$ and every
$i\in\left\\{0,1,...,n\right\\}$, define two elements
$\mathbf{w}_{i}^{\left(1\right)}\left(f\right)$ and
$\mathbf{w}_{i}^{\left(2\right)}\left(f\right)$ of $\mathbb{K}$ by
$\mathbf{w}_{i}^{\left(1\right)}\left(f\right)=\sum_{\begin{subarray}{c}x\in\widehat{P}_{i};\
y\in\widehat{P}_{i+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{f\left(x\right)}{f\left(y\right)}\ \ \ \ \ \ \ \ \ \
\text{and}\ \ \ \ \ \ \ \ \ \
\mathbf{w}_{i}^{\left(2\right)}\left(f\right)=\sum_{\begin{subarray}{c}x\in\widehat{Q}_{i};\
y\in\widehat{Q}_{i+1};\\\ y\gtrdot
x\end{subarray}}\dfrac{f\left(x\right)}{f\left(y\right)}$
(where, of course, $\widehat{P}_{j}$ and $\widehat{Q}_{j}$ are embedded into
$\widehat{PQ}_{j}$ for every $j\in\left\\{0,1,...,n+1\right\\}$ in the obvious
way). These elements $\mathbf{w}_{i}^{\left(1\right)}\left(f\right)$ and
$\mathbf{w}_{i}^{\left(2\right)}\left(f\right)$ are defined not for every $f$,
but for “almost every” $f$ in the sense of Zariski topology. We denote the
$\left(n+1\right)$-tuple
$\left(\mathbf{w}_{0}^{\left(1\right)}\left(f\right)\diagup\mathbf{w}_{0}^{\left(2\right)}\left(f\right),\
\mathbf{w}_{1}^{\left(1\right)}\left(f\right)\diagup\mathbf{w}_{1}^{\left(2\right)}\left(f\right),\
...,\
\mathbf{w}_{n}^{\left(1\right)}\left(f\right)\diagup\mathbf{w}_{n}^{\left(2\right)}\left(f\right)\right)$
as the comparative w-tuple of the labelling $f$. The advantage of comparative
w-tuples over usual w-tuples is the following fact: If $f$ and $g$ are two
homogeneously equivalent $\mathbb{K}$-labellings of $PQ$, then
$\left(\text{the comparative w-tuple of }f\right)=\left(\text{the comparative
w-tuple of }g\right).$ (20)
(This is easy to check and has no analogue for regular w-tuples.)
It is furthermore easy to see (in analogy to Proposition 4.4) that the map
$R_{PQ}$ changes the comparative w-tuple of a $\mathbb{K}$-labelling by
shifting it cyclically.
But it is also easy to see (the nonemptiness of $P$ and $Q$ must be used here)
that there exists some $f\in\mathbb{K}^{\widehat{PQ}}$ such that the ratios
$\mathbf{w}_{i}^{\left(1\right)}\left(f\right)\diagup\mathbf{w}_{i}^{\left(2\right)}\left(f\right)$
are well-defined and pairwise distinct for all
$i\in\left\\{0,1,...,n\right\\}$ and such that $R^{j}f$ is well-defined for
every $j\in\left\\{0,1,...,\ell\right\\}$. Consider such an $f$. The ratios
$\mathbf{w}_{i}^{\left(1\right)}\left(f\right)\diagup\mathbf{w}_{i}^{\left(2\right)}\left(f\right)$
are pairwise distinct for all $i\in\left\\{0,1,...,n\right\\}$; that is, the
comparative w-tuple of $f$ contains no two equal entries.
Since $\overline{R}_{PQ}^{\ell}=\operatorname*{id}$, we have
$\overline{R}_{PQ}^{\ell}\left(\pi\left(f\right)\right)=\pi\left(f\right)$.
The commutativity of the diagram (16) yields
$\overline{R}_{PQ}^{\ell}\circ\pi=\pi\circ R_{PQ}^{\ell}$. Now,
$\pi\left(f\right)=\overline{R}_{PQ}^{\ell}\left(\pi\left(f\right)\right)=\left(\underbrace{\overline{R}_{PQ}^{\ell}\circ\pi}_{=\pi\circ
R_{PQ}^{\ell}}\right)\left(f\right)=\left(\pi\circ
R_{PQ}^{\ell}\right)\left(f\right)=\pi\left(R_{PQ}^{\ell}f\right).$
In other words, the labellings $f$ and $R_{PQ}^{\ell}f$ are homogeneously
equivalent. Thus,
$\left(\text{the comparative w-tuple of }f\right)=\left(\text{the comparative
w-tuple of }R_{PQ}^{\ell}f\right)$ (21)
(by (20)).
Now, recall that the map $R_{PQ}$ changes the comparative w-tuple of a
$\mathbb{K}$-labelling by shifting it cyclically. Hence, for every
$k\in\mathbb{N}$, the map $R_{PQ}^{k}$ changes the comparative w-tuple of a
$\mathbb{K}$-labelling by shifting it cyclically $k$ times. Applying this to
the $\mathbb{K}$-labelling $f$ and to $k=\ell$, we see that the comparative
w-tuple of $R_{PQ}^{\ell}f$ is obtained from the comparative w-tuple of $f$ by
an $\ell$-fold cyclic shift. Due to (21), this rewrites as follows: The
comparative w-tuple of $f$ is obtained from the comparative w-tuple of $f$ by
an $\ell$-fold cyclic shift. In other words, the comparative w-tuple of $f$ is
invariant under an $\ell$-fold cyclic shift. But since the comparative w-tuple
of $f$ consists of $n+1$ pairwise distinct entries, this is impossible unless
$n+1\mid\ell$. Hence, we must have $n+1\mid\ell$.
Now,
$\operatorname*{ord}\left(R_{PQ}\right)=\operatorname{lcm}\left(n+1,\operatorname*{ord}\left(\overline{R}_{PQ}\right)\right)=\operatorname*{ord}\left(\overline{R}_{PQ}\right)$
(since $n+1\mid\ell=\operatorname*{ord}\left(\overline{R}_{PQ}\right)$).
Hence,
$\operatorname*{ord}\left(\overline{R}_{PQ}\right)=\operatorname*{ord}\left(R_{PQ}\right)=\operatorname{lcm}\left(\operatorname*{ord}\left(R_{P}\right),\operatorname*{ord}\left(R_{Q}\right)\right).$
This proves Proposition 9.9. ∎
Now, let us track the effect of $B_{k}$ on the order of $\overline{R}$:
###### Proposition 9.10.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let $\mathbb{K}$ be a
field.
(a) We have
$\operatorname*{ord}\left(\overline{R}_{B_{1}P}\right)=\operatorname*{ord}\left(\overline{R}_{P}\right)$.
(b) For every integer $k>1$, we have
$\operatorname*{ord}\left(\overline{R}_{B_{k}P}\right)=\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{R}_{P}\right)\right)$.
###### Proof of Proposition 9.10 (sketched)..
We will be proving parts (a) and (b) together. Let $k$ be a positive integer
(this has to be $1$ for proving part (a)). We need to prove that
$\operatorname*{ord}\left(\overline{R}_{B_{k}P}\right)=\left\\{\begin{array}[c]{l}\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{R}_{P}\right)\right),\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\
\operatorname*{ord}\left(\overline{R}_{P}\right),\ \ \ \ \ \ \ \ \ \ \text{if
}k=1\end{array}\right..$ (22)
Proving this clearly will prove both parts (a) and (b) of Proposition 9.10.
Let us make some conventions:
* •
For any $n$-tuple $\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right)$ and any
object $\beta$, let $\beta\rightthreetimes\alpha$ denote the
$\left(n+1\right)$-tuple
$\left(\beta,\alpha_{1},\alpha_{2},...,\alpha_{n}\right)$.
* •
We are going to identify $P$ with a subposet of $B_{k}P$ in the obvious way.
But of course, the degree map of $B_{k}P$ restricted to $P$ is not identical
with the degree map of $P$ (but rather differs from it by $1$), so we will
have to distinguish between “degree in $P$” and “degree in $B_{k}P$”. We
identify the elements $0$ and $1$ of $\widehat{P}$ with the elements $0$ and
$1$ of $\widehat{B_{k}P}$, respectively. Thus, $\widehat{P}$ becomes a
subposet of $\widehat{B_{k}P}$. However, it is not generally true that every
$u\lessdot v$ in $\widehat{P}$ must satisfy $u\lessdot v$ in
$\widehat{B_{k}P}$.
* •
We have a rational map
$\pi:\mathbb{K}^{\widehat{P}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$
and a rational map
$\pi:\mathbb{K}^{\widehat{B_{k}P}}\dashrightarrow\overline{\mathbb{K}^{\widehat{B_{k}P}}}$
denoted by the same letter. This is not problematic, because these two maps
can be distinguished by their different domains. We will also use the letter
$\pi$ to denote the rational map
$\mathbb{K}^{k}\dashrightarrow\mathbb{P}\left(\mathbb{K}^{k}\right)$ obtained
from the canonical projection
$\mathbb{K}^{k}\setminus\left\\{0\right\\}\rightarrow\mathbb{P}\left(\mathbb{K}^{k}\right)$
of the nonzero vectors in $\mathbb{K}^{k}$ onto the projective space.
Now, we recall that the construction of $B_{k}P$ from $P$ involved adding $k$
new (pairwise incomparable) elements smaller than all existing elements of $P$
to the poset. This operation clearly raises the degree of every element of $P$
by $1$ 242424In terms of the Hasse diagram, this can be regarded as the $k$
new elements “bumping up” all existing elements of $P$ by $1$ degree., whereas
the $k$ newly added elements all obtain degree $1$ in $B_{k}P$. Formally
speaking, this means that $\widehat{B_{k}P}_{i}=\widehat{P}_{i-1}$ for every
$i\in\left\\{2,3,...,n+1\right\\}$, while $\widehat{B_{k}P}_{1}$ is a
$k$-element set. Moreover, for any $i\in\left\\{2,3,...,n+1\right\\}$, any
$u\in\widehat{B_{k}P}_{i}=\widehat{P}_{i-1}$ and any
$v\in\widehat{B_{k}P}_{i+1}=\widehat{P}_{i}$, we have
$u\lessdot v\text{ in }\widehat{B_{k}P}\text{ if and only if }u\lessdot
v\text{ in }\widehat{P}\text{.}$
(This would not be true if we would allow $i=1$, $u\in\widehat{P}_{0}$ and
$v\in\widehat{P}_{1}$.)
We have $\mathbb{K}^{\widehat{B_{k}P}_{i}}=\mathbb{K}^{\widehat{P}_{i-1}}$ for
every $i\in\left\\{2,3,...,n+1\right\\}$ (since
$\widehat{B_{k}P}_{i}=\widehat{P}_{i-1}$ for every
$i\in\left\\{2,3,...,n+1\right\\}$), whereas
$\mathbb{K}^{\widehat{B_{k}P}_{1}}\cong\mathbb{K}^{k}$ (since
$\widehat{B_{k}P}_{1}$ is a $k$-element set). We will actually identify
$\mathbb{K}^{\widehat{B_{k}P}_{1}}$ with $\mathbb{K}^{k}$. Now,
$\displaystyle\overline{\mathbb{K}^{\widehat{B_{k}P}}}$
$\displaystyle=\prod\limits_{i=1}^{n+1}\mathbb{P}\left(\mathbb{K}^{\widehat{B_{k}P}_{i}}\right)=\mathbb{P}\left(\underbrace{\mathbb{K}^{\widehat{B_{k}P}_{1}}}_{=\mathbb{K}^{k}}\right)\times\prod\limits_{i=2}^{n+1}\mathbb{P}\left(\underbrace{\mathbb{K}^{\widehat{B_{k}P}_{i}}}_{=\mathbb{K}^{\widehat{P}_{i-1}}}\right)$
$\displaystyle=\mathbb{P}\left(\mathbb{K}^{k}\right)\times\prod\limits_{i=2}^{n+1}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i-1}}\right)=\mathbb{P}\left(\mathbb{K}^{k}\right)\times\underbrace{\prod\limits_{i=1}^{n}\mathbb{P}\left(\mathbb{K}^{\widehat{P}_{i}}\right)}_{=\overline{\mathbb{K}^{\widehat{P}}}}=\mathbb{P}\left(\mathbb{K}^{k}\right)\times\overline{\mathbb{K}^{\widehat{P}}}.$
(23)
Thus, the elements of $\overline{\mathbb{K}^{\widehat{B_{k}P}}}$ have the form
$\widetilde{p}\rightthreetimes\widetilde{g}$, where
$\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)$ and
$\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}$.
On the other hand, recall that $\widehat{P}$ is a subposet of
$\widehat{B_{k}P}$. More precisely, $\widehat{P}$ is the subposet
$\widehat{B_{k}P}\setminus\widehat{B_{k}P}_{1}$ of $\widehat{B_{k}P}$. Thus,
we can define a map
$\Phi:\mathbb{K}^{k}\times\mathbb{K}^{\widehat{P}}\rightarrow\mathbb{K}^{\widehat{B_{k}P}}$
by setting
$\left(\Phi\left(p,g\right)\right)\left(v\right)=\left\\{\begin{array}[c]{c}p\left(v\right),\
\ \ \ \ \ \ \ \ \ \text{if }v\in\widehat{B_{k}P}_{1};\\\ g\left(v\right),\ \ \
\ \ \ \ \ \ \ \text{if }v\notin\widehat{B_{k}P}_{1}\end{array}\right.\ \ \ \ \
\ \ \ \ \ \text{for every }v\in\widehat{B_{k}P}$
for every $\left(p,g\right)\in\mathbb{K}^{k}\times\mathbb{K}^{\widehat{P}}$.
Here, the term $p\left(v\right)$ is to be understood by means of regarding $p$
as an element of $\mathbb{K}^{\widehat{B_{k}P}_{1}}$ (since
$p\in\mathbb{K}^{k}=\mathbb{K}^{\widehat{B_{k}P}_{1}}$). Clearly, $\Phi$ is a
bijection. Moreover, it is easy to see that
$\pi\left(\Phi\left(p,g\right)\right)=\pi\left(p\right)\rightthreetimes\pi\left(g\right)\
\ \ \ \ \ \ \ \ \ \text{for all }p\in\mathbb{K}^{k}\text{ and
}g\in\mathbb{K}^{\widehat{P}}$ (24)
(where the $\pi$ on the left hand side is the map
$\pi:\mathbb{K}^{\widehat{B_{k}P}}\dashrightarrow\overline{\mathbb{K}^{\widehat{B_{k}P}}}$,
whereas the $\pi$ in “$\pi\left(p\right)$” is the map
$\pi:\mathbb{K}^{k}\dashrightarrow\mathbb{P}\left(\mathbb{K}^{k}\right)$, and
the $\pi$ in “$\pi\left(g\right)$” is the map
$\pi:\mathbb{K}^{\widehat{P}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$).
Now, we claim that every
$\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)$ and
$\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}$ satisfy
$\left(\overline{R_{i}}\right)_{B_{k}P}\left(\widetilde{p}\rightthreetimes\widetilde{g}\right)=\widetilde{p}\rightthreetimes\overline{R_{i-1}}_{P}\left(\widetilde{g}\right)\
\ \ \ \ \ \ \ \ \ \text{for all }i\in\left\\{2,3,...,n+1\right\\}$ (25)
and
$\left(\overline{R_{1}}\right)_{B_{k}P}\left(\widetilde{p}\rightthreetimes\widetilde{g}\right)=\widetilde{p}^{-1}\rightthreetimes\widetilde{g}.$
(26)
Proof of (25) and (26): In order to prove (25), it is clearly enough to show
that every $p\in\mathbb{K}^{k}$ and $g\in\mathbb{K}^{\widehat{P}}$ satisfy
$\left(R_{i}\right)_{B_{k}P}\left(p\rightthreetimes g\right)\sim
p\rightthreetimes\left(R_{i-1}\right)_{P}\left(g\right)\ \ \ \ \ \ \ \ \ \
\text{for all }i\in\left\\{2,3,...,n+1\right\\},$ (27)
where the sign $\sim$ stands for homogeneous equivalence.
It is easy to prove the relation (27) for $i>2$ (because if $i>2$, then the
elements of $\widehat{B_{k}P}$ having degrees $i-1$, $i$ and $i+1$ are
precisely the elements of $\widehat{P}$ having degrees $i-2$, $i-1$ and $i$,
and therefore toggling the elements of $\widehat{B_{k}P}_{i}$ in
$p\rightthreetimes g$ has precisely the same effect as toggling the elements
of $\widehat{P}_{i-1}$ in $g$ while leaving $p$ fixed, so that we even get the
stronger assertion that $\left(R_{i}\right)_{B_{k}P}\left(p\rightthreetimes
g\right)=p\rightthreetimes\left(R_{i-1}\right)_{P}\left(g\right)$). It is not
much harder to check that it also holds for $i=2$ (indeed, for $i=2$, the only
difference between toggling the elements of $\widehat{B_{k}P}_{i}$ in
$p\rightthreetimes g$ and toggling the elements of $\widehat{P}_{i-1}$ in $g$
while leaving $p$ fixed is a scalar factor which is identical across all
elements being toggled in either poset252525because every
$u\in\widehat{B_{k}P}_{1}$ and every $v\in\widehat{B_{k}P}_{2}$ satisfy
$u\lessdot v$; therefore the results are the same up to homogeneous
equivalence).
Finally, (26) is trivial to check (e.g., using Corollary 6.14).
But recall that
$\overline{R}=\overline{R_{1}}\circ\overline{R_{2}}\circ...\circ\overline{R_{n}}$
for any $n$-graded poset. Hence,
$\overline{R}_{B_{k}P}=\left(\overline{R_{1}}\right)_{B_{k}P}\circ\left(\overline{R_{2}}\right)_{B_{k}P}\circ\left(\overline{R_{3}}\right)_{B_{k}P}\circ...\circ\left(\overline{R_{n+1}}\right)_{B_{k}P}$
(because $B_{k}P$ is an $\left(n+1\right)$-graded poset) and
$\overline{R}_{P}=\overline{R_{1}}_{P}\circ\overline{R_{2}}_{P}\circ...\circ\overline{R_{n}}_{P}$
(because $P$ is an $n$-graded poset). Because of these equalities, and because
of (25) and (26), it is now easy to see that every
$\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)$ and
$\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}$ satisfy
$\overline{R}_{B_{k}P}\left(\widetilde{p}\rightthreetimes\widetilde{g}\right)=\widetilde{p}^{-1}\rightthreetimes\overline{R}_{P}\left(\widetilde{g}\right).$
(28)
Furthermore, every $\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)$ and
$\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}$ satisfy
$\overline{R}_{B_{k}P}^{\ell}\left(\widetilde{p}\rightthreetimes\widetilde{g}\right)=\widetilde{p}^{\left(-1\right)^{\ell}}\rightthreetimes\overline{R}_{P}^{\ell}\left(\widetilde{g}\right)\
\ \ \ \ \ \ \ \ \ \text{for all }\ell\in\mathbb{N}.$ (29)
(This is proven by induction over $\ell$, using (28).)
We know that the elements of $\overline{\mathbb{K}^{\widehat{B_{k}P}}}$ have
the form $\widetilde{p}\rightthreetimes\widetilde{g}$, where
$\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)$ and
$\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}$. Conversely, every
element $\widetilde{p}\rightthreetimes\widetilde{g}$ with
$\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)$ and
$\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}$ lies in
$\overline{\mathbb{K}^{\widehat{B_{k}P}}}$. Hence, for every
$\ell\in\mathbb{N}$, we have the following equivalence of assertions:
$\displaystyle\ \left(\text{we have
}\overline{R}_{B_{k}P}^{\ell}=\operatorname*{id}\right)$
$\displaystyle\Longleftrightarrow\ \left(\text{every
}\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)\text{ and
}\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}\text{ satisfy
}\overline{R}_{B_{k}P}^{\ell}\left(\widetilde{p}\rightthreetimes\widetilde{g}\right)=\widetilde{p}\rightthreetimes\widetilde{g}\right)$
$\displaystyle\Longleftrightarrow\ \left(\text{every
}\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)\text{ and
}\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}\text{ satisfy
}\widetilde{p}^{\left(-1\right)^{\ell}}\rightthreetimes\overline{R}_{P}^{\ell}\left(\widetilde{g}\right)=\widetilde{p}\rightthreetimes\widetilde{g}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{because of
(\ref{pf.Bk.ord.Rl})}\right)$ $\displaystyle\Longleftrightarrow\
\left(\text{every }\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)\text{
and }\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}\text{ satisfy
}\widetilde{p}^{\left(-1\right)^{\ell}}=\widetilde{p}\text{ and
}\overline{R}_{P}^{\ell}\left(\widetilde{g}\right)=\widetilde{g}\right)$
$\displaystyle\Longleftrightarrow\ \left(\underbrace{\text{every
}\widetilde{p}\in\mathbb{P}\left(\mathbb{K}^{k}\right)\text{ satisfies
}\widetilde{p}^{\left(-1\right)^{\ell}}=\widetilde{p}}_{\text{this is
equivalent to }\left(2\mid\ell\text{ if }k>1\right)}\text{, and
}\underbrace{\text{every
}\widetilde{g}\in\overline{\mathbb{K}^{\widehat{P}}}\text{ satisfies
}\overline{R}_{P}^{\ell}\left(\widetilde{g}\right)=\widetilde{g}}_{\text{this
is equivalent to }\overline{R}_{P}^{\ell}=\operatorname*{id}}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{since the sets
}\mathbb{P}\left(\mathbb{K}^{k}\right)\text{ and
}\overline{\mathbb{K}^{\widehat{P}}}\text{ are nonempty}\right)$
$\displaystyle\Longleftrightarrow\ \left(\text{we have }\left(2\mid\ell\text{
if }k>1\right)\text{ and
}\underbrace{\overline{R}_{P}^{\ell}=\operatorname*{id}}_{\text{this is
equivalent to
}\operatorname*{ord}\left(\overline{R}_{P}\right)\mid\ell}\right)$
$\displaystyle\Longleftrightarrow\ \left(\text{we have }\left(2\mid\ell\text{
if }k>1\right)\text{ and
}\operatorname*{ord}\left(\overline{R}_{P}\right)\mid\ell\right)$
$\displaystyle\Longleftrightarrow\ \left\\{\begin{array}[c]{c}\left(\text{we
have }2\mid\ell\text{ and
}\operatorname*{ord}\left(\overline{R}_{P}\right)\mid\ell\right),\ \ \ \ \ \ \
\ \ \ \text{if }k>1;\\\ \left(\text{we have
}\operatorname*{ord}\left(\overline{R}_{P}\right)\mid\ell\right),\ \ \ \ \ \ \
\ \ \ \text{if }k=1\end{array}\right.$ $\displaystyle\Longleftrightarrow\
\left\\{\begin{array}[c]{c}\left(\text{we have
}\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{R}_{P}\right)\right)\mid\ell\right),\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\ \left(\text{we have
}\operatorname*{ord}\left(\overline{R}_{P}\right)\mid\ell\right),\ \ \ \ \ \ \
\ \ \ \text{if }k=1\end{array}\right.$ $\displaystyle\Longleftrightarrow\
\left(\text{we have
}\left\\{\begin{array}[c]{l}\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{R}_{P}\right)\right),\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\
\operatorname*{ord}\left(\overline{R}_{P}\right),\ \ \ \ \ \ \ \ \ \ \text{if
}k=1\end{array}\right.\mid\ell\right).$
Hence, for every $\ell\in\mathbb{N}$, we have the following equivalence of
assertions:
$\displaystyle\left(\text{we have
}\operatorname*{ord}\left(\overline{R}_{B_{k}P}\right)\mid\ell\right)\ $
$\displaystyle\Longleftrightarrow\ \left(\text{we have
}\overline{R}_{B_{k}P}^{\ell}=\operatorname*{id}\right)$
$\displaystyle\Longleftrightarrow\ \left(\text{we have
}\left\\{\begin{array}[c]{l}\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{R}_{P}\right)\right),\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\
\operatorname*{ord}\left(\overline{R}_{P}\right),\ \ \ \ \ \ \ \ \ \ \text{if
}k=1\end{array}\right.\mid\ell\right).$
Consequently,
$\operatorname*{ord}\left(\overline{R}_{B_{k}P}\right)=\left\\{\begin{array}[c]{l}\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{R}_{P}\right)\right),\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\
\operatorname*{ord}\left(\overline{R}_{P}\right),\ \ \ \ \ \ \ \ \ \ \text{if
}k=1\end{array}\right.$. This is exactly what (22) claims. Thus, (22) is
proven, and with it Proposition 9.10. ∎
Here is an analogue of Proposition 9.10:
###### Proposition 9.11.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let $\mathbb{K}$ be a
field.
(a) We have
$\operatorname*{ord}\left(\overline{R}_{B_{1}^{\prime}P}\right)=\operatorname*{ord}\left(\overline{R}_{P}\right)$.
(b) For every integer $k>1$, we have
$\operatorname*{ord}\left(\overline{R}_{B_{k}^{\prime}P}\right)=\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{R}_{P}\right)\right)$.
###### .
The proof of this is very similar (though not exactly identical) to that of
Proposition 9.10. Alternatively, it is easy to deduce Proposition 9.11 from
Proposition 9.10 using Proposition 8.4 and Proposition 9.4. ∎
Proposition 9.7 is easily shown by induction using Propositions 9.8, 9.10,
9.11 and 7.3. Moreover, using Propositions 9.8, 9.9, 9.10, 9.11 and 7.3, we
can recursively compute (rather than just bound from the above) the orders of
$R_{P}$ and $\overline{R}_{P}$ for any skeletal poset $P$ without doing any
computations in $\mathbb{K}$. (This also shows that the orders of $R_{P}$ and
$\overline{R}_{P}$ don’t depend on the base field $\mathbb{K}$ as long as
$\mathbb{K}$ is infinite and $P$ is skeletal.)
In the case of forests and trees we can also use this induction to establish a
concrete bound:
###### Corollary 9.12.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let $\mathbb{K}$ be a
field. Assume that $P$ is a rooted forest (made into a poset by having every
node smaller than its children).
(a) Then,
$\operatorname*{ord}\left(R_{P}\right)\mid\operatorname{lcm}\left(1,2,...,n+1\right)$.
(b) Moreover, if $P$ is a tree, then
$\operatorname*{ord}\left(\overline{R}_{P}\right)\mid\operatorname{lcm}\left(1,2,...,n\right)$.
Corollary 9.12 is also valid if we replace “every node smaller than its
children” by “every node larger than its children”, and the proof is exactly
analogous.
###### Proof of Corollary 9.12 (sketched)..
(a) Corollary 9.12 (a) can be proven by strong induction over
$\left|P\right|$. Indeed, if $P$ is an $n$-graded poset and a rooted forest,
then we must be in one of the following three cases:
Case 1: We have $P=\varnothing$.
Case 2: The rooted forest $P$ is a tree.
Case 3: The rooted forest $P$ is a disjoint union of more than one tree.
The validity of Corollary 9.12 is trivial in Case 1, and in Case 3 it follows
from the induction hypothesis using Proposition 9.8. In Case 2, we have
$P=B_{1}Q$ for some rooted forest $Q$, which is necessarily
$\left(n-1\right)$-graded; thus, the induction hypothesis (applied to $Q$
instead of $P$) yields
$\operatorname*{ord}\left(R_{Q}\right)\mid\operatorname{lcm}\left(1,2,...,\left(n-1\right)+1\right)=\operatorname{lcm}\left(1,2,...,n\right)$,
and we obtain
$\displaystyle\operatorname*{ord}\left(\overline{R}_{P}\right)$
$\displaystyle=\operatorname*{ord}\left(\overline{R}_{B_{1}Q}\right)=\operatorname*{ord}\left(\overline{R}_{Q}\right)\
\ \ \ \ \ \ \ \ \ \left(\text{by Proposition \ref{prop.Bk.ord} {(a)}}\right)$
$\displaystyle\mid\operatorname{lcm}\left(1,2,...,n\right)$
and
$\displaystyle\operatorname*{ord}\left(R_{P}\right)$
$\displaystyle=\operatorname{lcm}\left(n+1,\underbrace{\operatorname*{ord}\left(\overline{R}_{P}\right)}_{\mid\operatorname{lcm}\left(1,2,...,n\right)}\right)\
\ \ \ \ \ \ \ \ \ \left(\text{by Proposition \ref{prop.ord-projord}}\right)$
$\displaystyle\mid\operatorname{lcm}\left(n+1,\operatorname{lcm}\left(1,2,...,n\right)\right)=\operatorname{lcm}\left(1,2,...,n+1\right).$
Thus, the induction step is complete in each of the three Cases.
(b) If $P$ is a tree, then we must be in Case 2 of the above case distinction,
and thus we have
$\operatorname*{ord}\left(\overline{R}_{P}\right)\mid\operatorname{lcm}\left(1,2,...,n\right)$
as shown above. Corollary 9.12 is therefore proven. ∎
## 10 Interlude: Classical rowmotion on skeletal posets
The above results concerning birational rowmotion on skeletal posets suggest
the question of what can be said about classical rowmotion (on the set of
order ideals) on this class of posets. Indeed, while the classical rowmotion
map (as opposed to the birational one) has been the object of several studies
(e.g., [StWi11] and [CaFl95]), it seems that this rather simple case has never
been explicitly covered. Let us therefore go on a tangent to bridge this gap
and derive the counterparts of Propositions 9.10 and 9.7 and Corollary 9.12
for classical rowmotion. Nothing of what we do in this Section 10 will be
relevant to later sections, so this section can be skipped.
First, we define the maps involved.
###### Definition 10.1.
Let $P$ be a poset.
(a) An order ideal of $P$ means a subset $S$ of $P$ such that every $s\in S$
and $p\in P$ with $p\leqslant s$ satisfy $p\in S$.
(b) The set of all order ideals of $P$ will be denoted by $J\left(P\right)$.
Here is the definition of (classical) toggles on order ideals (an analogue of
Definition 2.6):
###### Definition 10.2.
Let $P$ be a finite poset. Let $v\in P$. Define a map
$\mathbf{t}_{v}:J\left(P\right)\rightarrow J\left(P\right)$ by
$\mathbf{t}_{v}\left(S\right)=\left\\{\begin{array}[c]{l}S\cup\left\\{v\right\\}\text{,
if }v\notin S\text{ and }S\cup\left\\{v\right\\}\in J\left(P\right);\\\
S\setminus\left\\{v\right\\}\text{, if }v\in S\text{ and
}S\setminus\left\\{v\right\\}\in J\left(P\right);\\\ S\text{,
otherwise}\end{array}\right.\ \ \ \ \ \ \ \ \ \ \text{for every }S\in
J\left(P\right).$
(This is clearly well-defined.) This map $\mathbf{t}_{v}$ will be called the
classical $v$-toggle.
We can rewrite this definition in more “local” terms, by replacing the
conditions “$S\cup\left\\{v\right\\}\in J\left(P\right)$” and
“$S\setminus\left\\{v\right\\}\in J\left(P\right)$” by the respectively
equivalent conditions “every element $u\in P$ satisfying $u\lessdot v$ lies in
$S$” and “no element $u\in P$ satisfying $u\gtrdot v$ lies in $S$” (in fact,
the equivalence of these conditions is easily seen). Hence, we obtain the
following analogue to Proposition 2.9:
###### Proposition 10.3.
Let $P$ be a finite poset. Let $v\in P$. For every $S\in J\left(P\right)$, we
have:
(a) If $w$ is an element of $P$ such that $w\neq v$, then we have
$w\in\mathbf{t}_{v}\left(S\right)$ if and only if $w\in S$.
(b) We have $v\in\mathbf{t}_{v}\left(S\right)$ if and only if
$\displaystyle\left(v\in S\text{ and not }\left(\text{no element }u\in P\text{
satisfying }u\gtrdot v\text{ lies in }S\right)\right)$ $\displaystyle\text{or
}\left(v\notin S\text{ and }\left(\text{every element }u\in P\text{ satisfying
}u\lessdot v\text{ lies in }S\right)\right).$
While the complicated logical statement in Proposition 10.3 (b) can be
simplified, the form we have stated it in exhibits its similarity to
Proposition 2.9 particularly well. This, in fact, is more than a similarity:
If we allow $\mathbb{K}$ to be a semifield rather than a field, we can regard
the classical $v$-toggle $\mathbf{t}_{v}$ as a restriction of the birational
toggle $T_{v}$ (when $\mathbb{K}$ is chosen appropriately)262626Here are the
details: Let $\operatorname*{Trop}\mathbb{Z}$ be the tropical semiring over
$\mathbb{Z}$, that is, the semiring obtained by endowing the set
$\mathbb{Z}\cup\left\\{-\infty\right\\}$ with the binary operation
$\left(a,b\right)\mapsto\max\left\\{a,b\right\\}$ as “addition” and the binary
operation $\left(a,b\right)\mapsto a+b$ as “multiplication” (where the usual
rules for sums involving $-\infty$ apply). Then,
$\operatorname*{Trop}\mathbb{Z}$ is a semifield, with $\left(a,b\right)\mapsto
a-b$ serving as “subtraction”, with $-\infty$ serving as “zero” and with the
integer $0$ serving as “one”. Now, to every order ideal $S\in
J\left(P\right)$, we can assign a
$\left(\operatorname*{Trop}\mathbb{Z}\right)$-labelling
$\operatorname*{tlab}S\in\left(\operatorname*{Trop}\mathbb{Z}\right)^{\widehat{P}}$,
defined by
$\left(\operatorname*{tlab}S\right)\left(v\right)=\left\\{\begin{array}[c]{c}1,\text{
if }v\notin S\cup\left\\{0\right\\};\\\ 0,\ \text{if }v\in
S\cup\left\\{0\right\\}\end{array}\right..$ This yields a map
$\operatorname*{tlab}:J\left(P\right)\rightarrow\left(\operatorname*{Trop}\mathbb{Z}\right)^{\widehat{P}}$,
obviously injective. This map $\operatorname*{tlab}$ satisfies
$T_{v}\circ\operatorname*{tlab}=\operatorname*{tlab}\circ\mathbf{t}_{v}$ for
every $v\in P$. This allows us to regard the classical toggles
$\mathbf{t}_{v}$ as restrictions of the birational toggles $T_{v}$, if we
consider this map $\operatorname*{tlab}$ as an inclusion. This reasoning goes
back to Einstein and Propp [EiPr13].. Hence, some theorems about birational
toggles can be used to derive analogous theorems about classical
toggles272727For example, we could derive Proposition 10.5 from Proposition
2.10 using this tactic. However, we could not derive (say) Proposition 10.27
from Proposition 9.7 this way, because the order of a restriction of a
permutation could be a proper divisor of the order of the permutation.. We
will not use this tactic in the following, because often it will be easier to
study the classical $v$-toggles on their own. However, many of the properties
of classical toggles (and classical rowmotion) that we are going to discuss
will have proofs that are parallel to the proofs of the analogous results
about birational toggles. We will omit these proofs when the analogy is
glaring enough.
We have the following easily-verified analogues of Proposition 2.7,
Proposition 2.10 and Corollary 2.12:
###### Proposition 10.4.
Let $P$ be a finite poset. Let $v\in P$. Then, the map $\mathbf{t}_{v}$ is an
involution on $J\left(P\right)$ (that is, we have
$\mathbf{t}_{v}^{2}=\operatorname*{id}$).
###### Proposition 10.5.
Let $P$ be a finite poset. Let $v\in P$ and $w\in P$. Then,
$\mathbf{t}_{v}\circ\mathbf{t}_{w}=\mathbf{t}_{w}\circ\mathbf{t}_{v}$, unless
we have either $v\lessdot w$ or $w\lessdot v$.
###### Corollary 10.6.
Let $P$ be a finite poset. Let $\left(v_{1},v_{2},...,v_{m}\right)$ be a
linear extension of $P$. Then, the map
$\mathbf{t}_{v_{1}}\circ\mathbf{t}_{v_{2}}\circ...\circ\mathbf{t}_{v_{m}}:J\left(P\right)\rightarrow
J\left(P\right)$ is well-defined and independent of the choice of the linear
extension $\left(v_{1},v_{2},...,v_{m}\right)$.
The three results above are observations made on [CaFl95, page 546] (in
somewhat different notation).
Two convenient advantages of the classical setup are that we don’t have to
worry about denominators becoming zero, so our maps are actual maps rather
than partial maps, and that we don’t have to pass to the poset $\widehat{P}$.
We can now define rowmotion in analogy to Definition 2.13:
###### Definition 10.7.
Let $P$ be a finite poset. Classical rowmotion (simply called “rowmotion” in
existing literature) is defined as the map
$\mathbf{t}_{v_{1}}\circ\mathbf{t}_{v_{2}}\circ...\circ\mathbf{t}_{v_{m}}:J\left(P\right)\rightarrow
J\left(P\right)$, where $\left(v_{1},v_{2},...,v_{m}\right)$ is a linear
extension of $P$. This map is well-defined (in particular, it does not depend
on the linear extension $\left(v_{1},v_{2},...,v_{m}\right)$ chosen) because
of Corollary 10.6 (and also because of the fact that a linear extension of $P$
exists; this is Theorem 1.4). This map will be denoted by $\mathbf{r}$.
To highlight the similarities between the classical and birational cases, let
us state the analogue of Proposition 2.16:
###### Proposition 10.8.
Let $P$ be a finite poset. Let $v\in P$. Let $S\in J\left(P\right)$. Then,
$v\in\mathbf{r}\left(S\right)$ holds if and only if the following two
conditions hold:
Condition 1: Every $u\in P$ satisfying $u\lessdot v$ belongs to $S$.
Condition 2: Either $v\notin S$, or there exists an
$u\in\mathbf{r}\left(S\right)$ satisfying $u\gtrdot v$. (Recall that the
expression “either/or” is meant non-exclusively.)
This proposition is easily seen to be equivalent to the following well-known
equivalent description of rowmotion ([CaFl95, Lemma 1], translated into our
notation):
###### Proposition 10.9.
Let $P$ be a finite poset. Let $S\in J\left(P\right)$. Then, the maximal
elements of $\mathbf{r}\left(S\right)$ are precisely the minimal elements of
$P\setminus S$.
We record the analogue of Proposition 2.19:
###### Proposition 10.10.
Let $P$ be a finite poset. Let $S$ and $T$ be two order ideals of $P$. Assume
that for every $v\in P$, the relation $v\in T$ holds if and only if Conditions
1 and 2 of Proposition 10.8 hold with $\mathbf{r}\left(S\right)$ replaced by
$T$. Then, $T=\mathbf{r}\left(S\right)$.
In analogy to Proposition 2.20, we have:
###### Proposition 10.11.
Let $P$ be a finite poset. Then, classical rowmotion $\mathbf{r}$ is
invertible. Its inverse $\mathbf{r}^{-1}$ is
$\mathbf{t}_{v_{m}}\circ\mathbf{t}_{v_{m-1}}\circ...\circ\mathbf{t}_{v_{1}}:J\left(P\right)\rightarrow
J\left(P\right)$, where $\left(v_{1},v_{2},...,v_{m}\right)$ is a linear
extension of $P$.
We can study graded posets again. In analogy to Corollary 3.6, Definition 3.7,
Proposition 3.8 and Proposition 3.9, we have:
###### Corollary 10.12.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let
$i\in\left\\{1,2,...,n\right\\}$. Let $\left(u_{1},u_{2},...,u_{k}\right)$ be
any list of the elements of $\widehat{P}_{i}$ with every element of
$\widehat{P}_{i}$ appearing exactly once in the list. (Note that
$\widehat{P}_{i}$ is simply $\left\\{v\in P\ \mid\ \deg v=i\right\\}$, because
$i$ equals neither $0$ nor $n+1$. We are using the notation $\widehat{P}_{i}$
despite not working with $\widehat{P}$ merely to stress some analogies.) Then,
the map
$\mathbf{t}_{u_{1}}\circ\mathbf{t}_{u_{2}}\circ...\circ\mathbf{t}_{u_{k}}:J\left(P\right)\rightarrow
J\left(P\right)$ is well-defined and independent of the choice of the list
$\left(u_{1},u_{2},...,u_{k}\right)$.
###### Definition 10.13.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let
$i\in\left\\{1,2,...,n\right\\}$. Then, let $\mathbf{r}_{i}$ denote the map
$\mathbf{t}_{u_{1}}\circ\mathbf{t}_{u_{2}}\circ...\circ\mathbf{t}_{u_{k}}:J\left(P\right)\rightarrow
J\left(P\right)$, where $\left(u_{1},u_{2},...,u_{k}\right)$ is any list of
the elements of $\widehat{P}_{i}$ with every element of $\widehat{P}_{i}$
appearing exactly once in the list. This map
$\mathbf{t}_{u_{1}}\circ\mathbf{t}_{u_{2}}\circ...\circ\mathbf{t}_{u_{k}}$ is
well-defined (in particular, it does not depend on the list
$\left(u_{1},u_{2},...,u_{k}\right)$) because of Corollary 10.12.
###### Proposition 10.14.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Then,
$\mathbf{r}=\mathbf{r}_{1}\circ\mathbf{r}_{2}\circ...\circ\mathbf{r}_{n}.$
###### Proposition 10.15.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let
$i\in\left\\{1,2,...,n\right\\}$. Then, $\mathbf{r}_{i}$ is an involution on
$J\left(P\right)$ (that is, $\mathbf{r}_{i}^{2}=\operatorname*{id}$).
A parody of w-tuples can also be defined. The following is analogous to
Definition 4.1:
###### Definition 10.16.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let $S\in
J\left(P\right)$. Let $i\in\left\\{0,1,...,n\right\\}$. Then,
$\mathbf{w}_{i}\left(S\right)$ will denote the integer
$\left\\{\begin{array}[c]{l}1,\text{ if }P_{i}\subseteq S\text{ and
}P_{i+1}\cap S=\varnothing\\\ 0,\text{ otherwise}\end{array}\right..$
Here, we are using the notation $P_{j}$ for the subset
$\deg^{-1}\left(\left\\{j\right\\}\right)$ of $P$; this subset is empty if
$j=0$ and also empty if $j=n+1$.
Analogues of Proposition 4.3 and Proposition 4.4 are easily found:
###### Proposition 10.17.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let
$i\in\left\\{1,2,...,n\right\\}$. Then, every $S\in J\left(P\right)$ satisfies
$\displaystyle\left(\mathbf{w}_{0}\left(\mathbf{r}_{i}\left(S\right)\right),\mathbf{w}_{1}\left(\mathbf{r}_{i}\left(S\right)\right),...,\mathbf{w}_{n}\left(\mathbf{r}_{i}\left(S\right)\right)\right)$
$\displaystyle=\left(\mathbf{w}_{0}\left(S\right),\mathbf{w}_{1}\left(S\right),...,\mathbf{w}_{i-2}\left(S\right),\mathbf{w}_{i}\left(S\right),\mathbf{w}_{i-1}\left(S\right),\mathbf{w}_{i+1}\left(S\right),\mathbf{w}_{i+2}\left(S\right),...,\mathbf{w}_{n}\left(S\right)\right).$
###### Proposition 10.18.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Then, every $S\in
J\left(P\right)$ satisfies
$\left(\mathbf{w}_{0}\left(\mathbf{r}\left(S\right)\right),\mathbf{w}_{1}\left(\mathbf{r}\left(S\right)\right),...,\mathbf{w}_{n}\left(\mathbf{r}\left(S\right)\right)\right)=\left(\mathbf{w}_{n}\left(S\right),\mathbf{w}_{0}\left(S\right),\mathbf{w}_{1}\left(S\right),...,\mathbf{w}_{n-1}\left(S\right)\right).$
However, the $\left(n+1\right)$-tuple
$\left(\mathbf{w}_{0}\left(S\right),\mathbf{w}_{1}\left(S\right),...,\mathbf{w}_{n}\left(S\right)\right)$
obtained from an order ideal $S$ is not particularly informative. In fact, it
is $\left(0,0,...,0\right)$ for “most” order ideals; here is what this means
precisely:
###### Definition 10.19.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. An order ideal of $P$ is
said to be level if and only if it has the form $P_{1}\cup P_{2}\cup...\cup
P_{i}$ for some $i\in\left\\{0,1,...,n\right\\}$.
Easy properties of level order ideals are:
###### Proposition 10.20.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset.
(a) There exist precisely $n+1$ level order ideals of $P$, and those form an
orbit under classical rowmotion $\mathbf{r}$. Namely, one has
$\mathbf{r}\left(P_{1}\cup P_{2}\cup...\cup
P_{i}\right)=\left\\{\begin{array}[c]{l}P_{1}\cup P_{2}\cup...\cup P_{i+1},\ \
\ \ \ \ \ \ \ \ \text{if }i<n;\\\ \varnothing,\ \ \ \ \ \ \ \ \ \ \text{if
}i=n\end{array}\right..$
(b) If $S\in J\left(P\right)$, then
$\left(\mathbf{w}_{0}\left(S\right),\mathbf{w}_{1}\left(S\right),...,\mathbf{w}_{n}\left(S\right)\right)=\left(0,0,...,0\right)$
unless $S$ is level.
Now, we can define an (arguably toylike, but, as we will see, rather useful)
analogue of homogeneous equivalence. In somewhat questionable analogy with
Definition 6.2, we set:
###### Definition 10.21.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset.
Two order ideals $S$ and $T$ of $P$ are said to be homogeneously equivalent if
and only if either both $S$ and $T$ are level or we have $S=T$. Clearly, being
homogeneously equivalent is an equivalence relation. Let
$\overline{J\left(P\right)}$ denote the set of equivalence classes of elements
of $J\left(P\right)$ modulo this relation. Let $\pi$ denote the canonical
projection $J\left(P\right)\rightarrow\overline{J\left(P\right)}$. (We
distinguish this map $\pi$ from the map $\pi$ defined in Definition 6.2 by the
fact that they act on different objects.)
The following analogue of Corollary 6.7 is almost trivial:
###### Corollary 10.22.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. If $S$ and $T$ are two
homogeneously equivalent order ideals of $P$, then $\mathbf{r}\left(S\right)$
is homogeneously equivalent to $\mathbf{r}\left(T\right)$.
(An analogue of Corollary 6.6 exists as well.) We also have the following
analogue of Proposition 6.11:
###### Proposition 10.23.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let $S$ and $T$ be two
order ideals of $P$ such that
$\left(\mathbf{w}_{0}\left(S\right),\mathbf{w}_{1}\left(S\right),...,\mathbf{w}_{n}\left(S\right)\right)=\left(\mathbf{w}_{0}\left(T\right),\mathbf{w}_{1}\left(T\right),...,\mathbf{w}_{n}\left(T\right)\right)$.
Also assume that $\pi\left(S\right)=\pi\left(T\right)$. Then, $S=T$.
We can furthermore state analogues of Definitions 6.9 and 6.10:
###### Definition 10.24.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Let
$i\in\left\\{1,2,...,n\right\\}$. The map
$\mathbf{r}_{i}:J\left(P\right)\rightarrow J\left(P\right)$ descends (through
the projection $\pi:J\left(P\right)\rightarrow\overline{J\left(P\right)}$) to
a map $\overline{J\left(P\right)}\rightarrow\overline{J\left(P\right)}$. We
denote this map
$\overline{J\left(P\right)}\rightarrow\overline{J\left(P\right)}$ by
$\overline{\mathbf{r}_{i}}$. Thus, the diagram
$\textstyle{J\left(P\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{r}_{i}}$$\scriptstyle{\pi}$$\textstyle{J\left(P\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{J\left(P\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\mathbf{r}_{i}}}$$\textstyle{\overline{J\left(P\right)}}$
is commutative.
###### Definition 10.25.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. We define the map
$\overline{\mathbf{r}}:\overline{J\left(P\right)}\rightarrow\overline{J\left(P\right)}$
by
$\overline{\mathbf{r}}=\overline{\mathbf{r}_{1}}\circ\overline{\mathbf{r}_{2}}\circ...\circ\overline{\mathbf{r}_{n}}.$
Then, the diagram
$\textstyle{J\left(P\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{r}}$$\scriptstyle{\pi}$$\textstyle{J\left(P\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{J\left(P\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\mathbf{r}}}$$\textstyle{\overline{J\left(P\right)}}$
(30)
is commutative. In other words, $\overline{\mathbf{r}}$ is the map
$\overline{J\left(P\right)}\rightarrow\overline{J\left(P\right)}$ to which the
map $\mathbf{r}:J\left(P\right)\rightarrow J\left(P\right)$ descends (through
the projection $\pi:J\left(P\right)\rightarrow\overline{J\left(P\right)}$).
It might seem that the map $\overline{\mathbf{r}}$ is not worth considering,
since its cycle structure differs from the cycle structure of $\mathbf{r}$
only in the collapsing of an $\left(n+1\right)$-cycle (the one formed by all
level order ideals) to a point. However, triviality in combinatorics does not
preclude usefulness, and we will employ the “projective” version
$\overline{\mathbf{r}}$ of classical rowmotion as a stirrup in determining the
order of classical rowmotion $\mathbf{r}$ on skeletal posets.
We have the following simple relation between the orders of $\mathbf{r}$ and
$\overline{\mathbf{r}}$:
###### Proposition 10.26.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Then,
$\operatorname*{ord}\mathbf{r}=\operatorname{lcm}\left(n+1,\operatorname*{ord}\overline{\mathbf{r}}\right)$.
Notice that our convention to define
$\operatorname{lcm}\left(n+1,\infty\right)$ as $\infty$ is irrelevant for
Proposition 10.26: In fact, in the situation of Proposition 10.26, both
$\operatorname*{ord}\mathbf{r}$ and $\operatorname*{ord}\overline{\mathbf{r}}$
are clearly (finite) positive integers282828Indeed, the maps $\mathbf{r}$ and
$\overline{\mathbf{r}}$ are permutations of finite sets (namely, of the sets
$J\left(P\right)$ and $\overline{J\left(P\right)}$) and thus have finite
orders..
###### Proof of Proposition 10.26 (sketched)..
We know that $\mathbf{r}$ is an invertible map $J\left(P\right)\rightarrow
J\left(P\right)$, thus a permutation of the finite set $J\left(P\right)$.
Hence, $\operatorname*{ord}\mathbf{r}$ is the $\operatorname{lcm}$ of the
lengths of the cycles of this permutation $\mathbf{r}$. Similarly,
$\operatorname*{ord}\overline{\mathbf{r}}$ is the $\operatorname{lcm}$ of the
lengths of the cycles of the permutation $\overline{\mathbf{r}}$ of the finite
set $\overline{J\left(P\right)}$.
Let $Z_{1}$, $Z_{2}$, $...$, $Z_{k}$ be the cycles of the permutation
$\mathbf{r}$ of $J\left(P\right)$. We assume WLOG that $Z_{1}$ is the cycle
consisting of the $n+1$ level order ideals (because we know that they form a
cycle). Thus, $\left|Z_{1}\right|=n+1$. Since $\operatorname*{ord}\mathbf{r}$
is the $\operatorname{lcm}$ of the lengths of the cycles of the permutation
$\mathbf{r}$, we have
$\operatorname*{ord}\mathbf{r}=\operatorname{lcm}\left(\left|Z_{1}\right|,\left|Z_{2}\right|,...,\left|Z_{k}\right|\right)$.
Now, let us recall that $\overline{J\left(P\right)}$ is the quotient of
$J\left(P\right)$ modulo homogeneous equivalence. But homogeneous equivalence
merely identifies the $n+1$ level order ideals, while all other elements of
$J\left(P\right)$ are still pairwise non-equivalent. Hence, the cycles of the
permutation $\overline{\mathbf{r}}$ of $\overline{J\left(P\right)}$ are
$\pi\left(Z_{1}\right)$, $\pi\left(Z_{2}\right)$, $...$,
$\pi\left(Z_{k}\right)$, and while $\pi\left(Z_{2}\right)$,
$\pi\left(Z_{3}\right)$, $...$, $\pi\left(Z_{k}\right)$ are isomorphic to
$Z_{2}$, $Z_{3}$, $...$, $Z_{k}$, respectively, the first cycle
$\pi\left(Z_{1}\right)$ (being the projection of the cycle of the level order
ideals) now has length $1$. Now, $\operatorname*{ord}\overline{\mathbf{r}}$ is
the $\operatorname{lcm}$ of the lengths of the cycles of the permutation
$\overline{\mathbf{r}}$ of the finite set $\overline{J\left(P\right)}$. Since
these cycles are $\pi\left(Z_{1}\right)$, $\pi\left(Z_{2}\right)$, $...$,
$\pi\left(Z_{k}\right)$, this yields
$\displaystyle\operatorname*{ord}\overline{\mathbf{r}}$
$\displaystyle=\operatorname{lcm}\left(\left|\pi\left(Z_{1}\right)\right|,\left|\pi\left(Z_{2}\right)\right|,...,\left|\pi\left(Z_{k}\right)\right|\right)=\operatorname{lcm}\left(\underbrace{\left|\pi\left(Z_{1}\right)\right|}_{=1},\left|\pi\left(Z_{2}\right)\right|,\left|\pi\left(Z_{3}\right)\right|,...,\left|\pi\left(Z_{k}\right)\right|\right)$
$\displaystyle=\operatorname{lcm}\left(1,\left|\pi\left(Z_{2}\right)\right|,\left|\pi\left(Z_{3}\right)\right|,...,\left|\pi\left(Z_{k}\right)\right|\right)=\operatorname{lcm}\left(\left|\pi\left(Z_{2}\right)\right|,\left|\pi\left(Z_{3}\right)\right|,...,\left|\pi\left(Z_{k}\right)\right|\right)$
$\displaystyle=\operatorname{lcm}\left(\left|Z_{2}\right|,\left|Z_{3}\right|,...,\left|Z_{k}\right|\right)\
\ \ \ \ \ \ \ \ \ \left(\begin{array}[c]{c}\text{since
}\pi\left(Z_{2}\right)\text{, }\pi\left(Z_{3}\right)\text{, }...\text{,
}\pi\left(Z_{k}\right)\text{ are}\\\ \text{isomorphic to }Z_{2}\text{,
}Z_{3}\text{, }...\text{, }Z_{k}\text{, respectively}\end{array}\right).$
Now,
$\displaystyle\operatorname*{ord}\mathbf{r}$
$\displaystyle=\operatorname{lcm}\left(\left|Z_{1}\right|,\left|Z_{2}\right|,...,\left|Z_{k}\right|\right)=\operatorname{lcm}\left(\underbrace{\left|Z_{1}\right|}_{=n+1},\underbrace{\operatorname{lcm}\left(\left|Z_{2}\right|,\left|Z_{3}\right|,...,\left|Z_{k}\right|\right)}_{=\operatorname*{ord}\overline{\mathbf{r}}}\right)$
$\displaystyle=\operatorname{lcm}\left(n+1,\operatorname*{ord}\overline{\mathbf{r}}\right).$
This proves Proposition 10.26. ∎
Our goal is to make a statement about the order of classical rowmotion on
skeletal posets. Of course, the finiteness of these orders is obvious in this
case, because $J\left(P\right)$ is a finite set. However, we can make stronger
claims:
###### Proposition 10.27.
Let $P$ be a skeletal poset. Let $\mathbb{K}$ be a field. Then,
$\operatorname*{ord}\left(R_{P}\right)=\operatorname*{ord}\left(\mathbf{r}_{P}\right)$
and
$\operatorname*{ord}\left(\overline{R}_{P}\right)=\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)$.
Here, we are using the following convention:
###### Definition 10.28.
Let $P$ be a finite poset. We denote the maps $\mathbf{r}$ and
$\overline{\mathbf{r}}$ by $\mathbf{r}_{P}$ and $\overline{\mathbf{r}}_{P}$,
respectively, so as to make their dependence on $P$ explicit.
Proposition 10.27 yields (in particular) that the order of classical rowmotion
coincides with the order of birational rowmotion (whatever the base field) for
skeletal posets. This was conjectured by James Propp (private communication)
for the case of $P$ a tree. We are going to prove Proposition 10.27 by
exhibiting further analogies between classical and birational rowmotion. First
of all, the following proposition is just as trivial as its birational
counterpart (Proposition 9.8):
###### Proposition 10.29.
Let $n\in\mathbb{N}$. Let $P$ and $Q$ be two $n$-graded posets. Then,
$\operatorname*{ord}\left(\mathbf{r}_{PQ}\right)=\operatorname{lcm}\left(\operatorname*{ord}\left(\mathbf{r}_{P}\right),\operatorname*{ord}\left(\mathbf{r}_{Q}\right)\right)$.
We can show a simple counterpart of this proposition for
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{PQ}\right)$ (but still with
$\operatorname*{ord}\left(\mathbf{r}_{P}\right)$ and
$\operatorname*{ord}\left(\mathbf{r}_{Q}\right)$ on the right hand side!):
###### Proposition 10.30.
Let $n\in\mathbb{N}$. Let $P$ and $Q$ be two $n$-graded posets. Then,
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{PQ}\right)=\operatorname{lcm}\left(\operatorname*{ord}\left(\mathbf{r}_{P}\right),\operatorname*{ord}\left(\mathbf{r}_{Q}\right)\right)$.
###### Proof of Proposition 10.30 (sketched)..
WLOG, assume that $n\neq 0$ (else, the statement is trivial). Hence, $P$ and
$Q$ are nonempty.
Consider the order ideal $P$ of $PQ$. Then, one can easily see (by induction)
that every $i\in\left\\{0,1,...,n+1\right\\}$ satisfies
$\displaystyle\mathbf{r}_{PQ}^{i}\left(P\right)$
$\displaystyle=\left(\left\\{\begin{array}[c]{l}P_{1}\cup P_{2}\cup...\cup
P_{i-1},\ \ \ \ \ \ \ \ \ \ \text{if }i>0;\\\ P,\ \ \ \ \ \ \ \ \ \ \text{if
}i=0\end{array}\right.\right)\cup\left(\left\\{\begin{array}[c]{l}Q_{1}\cup
Q_{2}\cup...\cup Q_{i},\ \ \ \ \ \ \ \ \ \ \text{if }i\leqslant n;\\\
\varnothing,\ \ \ \ \ \ \ \ \ \ \text{if }i=n+1\end{array}\right.\right).$
From this, it follows that the smallest positive integer $k$ satisfying
$\mathbf{r}_{PQ}^{k}\left(P\right)=P$ is $n+1$. Since $P$ is not level, this
does not change under applying $\pi$; that is, the smallest positive integer
$k$ satisfying $\overline{\mathbf{r}}_{PQ}^{k}\left(P\right)=P$ is still
$n+1$. Hence,
$n+1\mid\operatorname*{ord}\left(\overline{\mathbf{r}}_{PQ}\right)$. But
Proposition 10.26 (applied to $PQ$ instead of $P$) yields
$\operatorname*{ord}\left(\mathbf{r}_{PQ}\right)=\operatorname{lcm}\left(n+1,\operatorname*{ord}\left(\overline{\mathbf{r}}_{PQ}\right)\right)=\operatorname*{ord}\left(\overline{\mathbf{r}}_{PQ}\right)$
(since $n+1\mid\operatorname*{ord}\left(\overline{\mathbf{r}}_{PQ}\right)$),
so that
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{PQ}\right)=\operatorname*{ord}\left(\mathbf{r}_{PQ}\right)=\operatorname{lcm}\left(\operatorname*{ord}\left(\mathbf{r}_{P}\right),\operatorname*{ord}\left(\mathbf{r}_{Q}\right)\right)$
(by Proposition 10.29). This proves Proposition 10.30. ∎
More interesting is the analogue of Proposition 9.10:
###### Proposition 10.31.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset.
(a) We have
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{1}P}\right)=\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)$.
(b) For every integer $k>1$, we have
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{k}P}\right)=\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)\right)$.
###### Proof of Proposition 10.31 (sketched)..
We will be proving parts (a) and (b) together. Let $k$ be a positive integer
(this has to be $1$ for proving part (a)). We need to prove that
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{k}P}\right)=\left\\{\begin{array}[c]{l}\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)\right),\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\
\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right),\ \ \ \ \ \ \ \ \ \
\text{if }k=1\end{array}\right..$ (31)
Proving this clearly will prove both parts (a) and (b) of Proposition 10.31.
Notice that $B_{k}P$ is an $\left(n+1\right)$-graded poset. For every
$\ell\in\left\\{1,2,...,n+1\right\\}$, let $\left(B_{k}P\right)_{\ell}$ be the
subset $\deg^{-1}\left(\left\\{\ell\right\\}\right)$ of $B_{k}P$. Thus,
$\left(B_{k}P\right)_{\ell}=\left\\{v\in B_{k}P\ \mid\ \deg v=\ell\right\\}$.
In particular, $\left(B_{k}P\right)_{1}$ is the set of all minimal elements of
$B_{k}P$, so that $\left(B_{k}P\right)_{1}$ is an antichain of size $k$ (by
the construction of $B_{k}P$). We also have $w<v$ for every
$w\in\left(B_{k}P\right)_{1}$ and $v\in P$.
For every finite poset $Q$, the map $\overline{\mathbf{r}}_{Q}$ is an
invertible map
$\overline{J\left(Q\right)}\rightarrow\overline{J\left(Q\right)}$, that is, a
permutation of the finite set $\overline{J\left(Q\right)}$. Hence, the order
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{Q}\right)$ is the
$\operatorname{lcm}$ of the lengths of the orbits of this map
$\overline{\mathbf{r}}_{Q}$. We are going to compare the orbits of the maps
$\overline{\mathbf{r}}_{B_{k}P}$ and $\overline{\mathbf{r}}_{P}$.
Define a map $\phi:J\left(P\right)\rightarrow J\left(B_{k}P\right)$ by
$\phi\left(S\right)=\left(B_{k}P\right)_{1}\cup S\ \ \ \ \ \ \ \ \ \ \text{for
every }S\in J\left(P\right).$
It is easy to see that this map $\phi$ is well-defined (that is,
$\left(B_{k}P\right)_{1}\cup S$ is an order ideal of $B_{k}P$ for every $S\in
J\left(P\right)$), and that it sends level order ideals of $P$ to level order
ideals of $B_{k}P$. Hence, it preserves homogeneous equivalence, so that it
induces a map
$\overline{J\left(P\right)}\rightarrow\overline{J\left(B_{k}P\right)}$. Denote
this map
$\overline{J\left(P\right)}\rightarrow\overline{J\left(B_{k}P\right)}$ by
$\overline{\phi}$. Thus, $\overline{\phi}\circ\pi=\pi\circ\phi$.
It is moreover easy to see that
$\overline{\mathbf{r}}_{B_{k}P}\circ\overline{\phi}=\overline{\phi}\circ\overline{\mathbf{r}}_{P}$
292929Proof. In order to prove this, it is enough to show that for every $S\in
J\left(P\right)$, the order ideals
$\left(\mathbf{r}_{B_{k}P}\circ\phi\right)\left(S\right)$ and
$\left(\phi\circ\mathbf{r}_{P}\right)\left(S\right)$ are homogeneously
equivalent. This is clear in the case when $S$ is level (because both
$\left(\mathbf{r}_{B_{k}P}\circ\phi\right)\left(S\right)$ and
$\left(\phi\circ\mathbf{r}_{P}\right)\left(S\right)$ are level in this case),
so let us WLOG assume that $S$ is not level. Then, we can actually show that
$\left(\mathbf{r}_{B_{k}P}\circ\phi\right)\left(S\right)$ and
$\left(\phi\circ\mathbf{r}_{P}\right)\left(S\right)$ are identical. Indeed, it
is easy to see that: • for every $T\in J\left(P\right)$ and every $v\in P$,
we have
$\left(\mathbf{t}_{v}\circ\phi\right)\left(T\right)=\left(\phi\circ\mathbf{t}_{v}\right)\left(T\right)$;
• for every nonempty $T\in J\left(P\right)$ and every
$w\in\left(B_{k}P\right)_{1}$, we have
$\left(\mathbf{t}_{w}\circ\phi\right)\left(T\right)=\phi\left(T\right)$. Using
these facts, and the definition of classical rowmotion as a composition of
classical toggle maps $\mathbf{t}_{v}$, we can then easily see that
$\left(\mathbf{r}_{B_{k}P}\circ\phi\right)\left(S\right)=\left(\phi\circ\mathbf{r}_{P}\right)\left(S\right)$.
This completes the proof of
$\overline{\mathbf{r}}_{B_{k}P}\circ\overline{\phi}=\overline{\phi}\circ\overline{\mathbf{r}}_{P}$..
Hence, the subset $\overline{\phi}\left(\overline{J\left(P\right)}\right)$ is
closed under application of the map $\overline{\mathbf{r}}_{B_{k}P}$.
The map $\overline{\phi}$ also is injective (this is very easy to see again,
since the only order ideals of $P$ which are mapped to level order ideals by
$\phi$ are themselves level). Thus,
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{k}P}\mid_{\overline{\phi}\left(\overline{J\left(P\right)}\right)}\right)=\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)$
(because the injectivity of $\overline{\phi}$ allows us to identify
$\overline{J\left(P\right)}$ with
$\overline{\phi}\left(\overline{J\left(P\right)}\right)$ along the map
$\overline{\phi}$, and then the equality
$\overline{\mathbf{r}}_{B_{k}P}\circ\overline{\phi}=\overline{\phi}\circ\overline{\mathbf{r}}_{P}$
rewrites as
$\overline{\mathbf{r}}_{B_{k}P}\mid_{\overline{\phi}\left(\overline{J\left(P\right)}\right)}=\overline{\mathbf{r}}_{P}$,
so that
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{k}P}\mid_{\overline{\phi}\left(\overline{J\left(P\right)}\right)}\right)=\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)$).
Let $H$ be the set of all nonempty proper subsets of
$\left(B_{k}P\right)_{1}$. It is clear that $H\subseteq J\left(B_{k}P\right)$.
Notice that $H=\varnothing$ if $k=1$. Every $T\in H$ satisfies
$\mathbf{r}_{B_{k}P}\left(T\right)=\left(B_{k}P\right)_{1}\setminus T$
(this is easy to see from any definition of classical rowmotion, or from
Proposition 10.8). Hence, the set $H$ is closed under application of the map
$\mathbf{r}_{B_{k}P}$, and this map $\mathbf{r}_{B_{k}P}$ maps every element
of $H$ to its complement in $\left(B_{k}P\right)_{1}$. In particular, this
shows that
$\operatorname*{ord}\left(\mathbf{r}_{B_{k}P}\mid_{H}\right)=\left\\{\begin{array}[c]{l}2,\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\ 1,\ \ \ \ \ \ \ \ \ \ \text{if
}k=1\end{array}\right.$.
We now use the map $\pi$ to identify the set $H$ with its projection
$\pi\left(H\right)$ under $\pi$ (this is allowed because $\pi$ is injective on
$H$). This identification entails
$\left.\overline{\mathbf{r}}_{B_{k}P}\mid_{H}\right.=\mathbf{r}_{B_{k}P}\mid_{H}$.
In particular, the set $H$ is closed under application of the map
$\overline{\mathbf{r}}_{B_{k}P}$.
But it is easy to see that $J\left(B_{k}P\right)$ is the union of the two
subsets $H$ and $\phi\left(J\left(P\right)\right)$ (because every order ideal
of $B_{k}P$ either contains the whole $\left(B_{k}P\right)_{1}$, or it does
not, in which case it cannot contain any element of degree $>1$). Hence,
$\overline{J\left(B_{k}P\right)}$ is the union of the two subsets
$\pi\left(H\right)=H$ and
$\pi\left(\phi\left(J\left(P\right)\right)\right)=\overline{\phi}\left(\overline{J\left(P\right)}\right)$.
Moreover, these two subsets are disjoint and each of them is closed under
application of the map $\overline{\mathbf{r}}_{B_{k}P}$. Hence,
$\displaystyle\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{k}P}\right)$
$\displaystyle=\operatorname{lcm}\left(\operatorname*{ord}\left(\underbrace{\overline{\mathbf{r}}_{B_{k}P}\mid_{H}}_{=\mathbf{r}_{B_{k}P}\mid_{H}}\right),\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{k}P}\mid_{\overline{\phi}\left(\overline{J\left(P\right)}\right)}\right)\right)$
$\displaystyle=\operatorname{lcm}\left(\underbrace{\operatorname*{ord}\left(\mathbf{r}_{B_{k}P}\mid_{H}\right)}_{=\left\\{\begin{array}[c]{l}2,\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\ 1,\ \ \ \ \ \ \ \ \ \ \text{if
}k=1\end{array}\right.},\underbrace{\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{k}P}\mid_{\overline{\phi}\left(\overline{J\left(P\right)}\right)}\right)}_{=\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)}\right)$
$\displaystyle=\operatorname{lcm}\left(\left\\{\begin{array}[c]{l}2,\ \ \ \ \
\ \ \ \ \ \text{if }k>1;\\\ 1,\ \ \ \ \ \ \ \ \ \ \text{if
}k=1\end{array}\right.,\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)\right)$
$\displaystyle=\left\\{\begin{array}[c]{l}\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)\right),\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\
\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right),\ \ \ \ \ \ \ \ \ \
\text{if }k=1\end{array}\right..$
This proves (31). Thus, the proof of Proposition 10.31 is complete. ∎
We can also formulate an analogue of Proposition 9.11:
###### Proposition 10.32.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset.
(a) We have
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{1}^{\prime}P}\right)=\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)$.
(b) For every integer $k>1$, we have
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{B_{k}^{\prime}P}\right)=\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)\right)$.
###### .
The proof of this is fairly similar to that of Proposition 10.31. ∎
We can now prove Proposition 10.27:
###### Proof of Proposition 10.27 (sketched)..
For any skeletal poset $T$, we can compute
$\operatorname*{ord}\left(R_{T}\right)$ and
$\operatorname*{ord}\left(\overline{R}_{T}\right)$ inductively using
Proposition 9.8, Proposition 9.9, Proposition 9.10 and Proposition 9.11 (and
the fact that $\operatorname*{ord}\left(R_{\varnothing}\right)=1$ and
$\operatorname*{ord}\left(\overline{R}_{\varnothing}\right)=1$). More
precisely:
* •
If $T$ is the empty poset $\varnothing$, then
$\operatorname*{ord}\left(R_{T}\right)=\operatorname*{ord}\left(R_{\varnothing}\right)=1$
and
$\operatorname*{ord}\left(\overline{R}_{T}\right)=\operatorname*{ord}\left(\overline{R}_{\varnothing}\right)=1$.
* •
If $T$ has the form $B_{k}P$ for some $n$-graded skeletal poset $P$ and some
positive integer $k$, then Proposition 9.10 yields
$\operatorname*{ord}\left(\overline{R}_{T}\right)=\operatorname*{ord}\left(\overline{R}_{B_{k}P}\right)=\left\\{\begin{array}[c]{l}\operatorname{lcm}\left(2,\operatorname*{ord}\left(\overline{R}_{P}\right)\right),\
\ \ \ \ \ \ \ \ \ \text{if }k>1;\\\
\operatorname*{ord}\left(\overline{R}_{P}\right),\ \ \ \ \ \ \ \ \ \ \text{if
}k=1\end{array}\right.,$
and Proposition 7.3 yields
$\operatorname*{ord}\left(R_{T}\right)=\operatorname{lcm}\left(n+1,\operatorname*{ord}\left(\overline{R}_{T}\right)\right)$.
* •
Analogously one can compute $\operatorname*{ord}\left(R_{T}\right)$ and
$\operatorname*{ord}\left(\overline{R}_{T}\right)$ if $T$ has the form
$B_{k}^{\prime}P$.
* •
If $T$ has the form $PQ$ for two WLOG nonempty $n$-graded skeletal posets $P$
and $Q$, then Proposition 9.8 yields
$\operatorname*{ord}\left(R_{PQ}\right)=\operatorname{lcm}\left(\operatorname*{ord}\left(R_{P}\right),\operatorname*{ord}\left(R_{Q}\right)\right)$,
and Proposition 9.9 yields
$\operatorname*{ord}\left(\overline{R}_{PQ}\right)=\operatorname{lcm}\left(\operatorname*{ord}\left(R_{P}\right),\operatorname*{ord}\left(R_{Q}\right)\right)$.
This gives an algorithm for inductively computing
$\operatorname*{ord}\left(R_{T}\right)$ and
$\operatorname*{ord}\left(\overline{R}_{T}\right)$ for a skeletal poset $T$.
Using Proposition 10.29, Proposition 10.30, Proposition 10.31 and Proposition
10.32 (and the fact that
$\operatorname*{ord}\left(\mathbf{r}_{\varnothing}\right)=1$ and
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{\varnothing}\right)=1$)
instead, we could similarly obtain an algorithm for inductively computing
$\operatorname*{ord}\left(\mathbf{r}_{T}\right)$ and
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{T}\right)$ for a skeletal
poset $T$. And these two algorithms are the same, because of the direct
analogy between the propositions that are used in the first algorithm and
those used in the second one. Therefore,
$\operatorname*{ord}\left(R_{P}\right)=\operatorname*{ord}\left(\mathbf{r}_{P}\right)$
and
$\operatorname*{ord}\left(\overline{R}_{P}\right)=\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)$.
This proves Proposition 10.27. ∎
Proposition 10.27 does not generalize to arbitrary graded posets.
Counterexamples to such a generalization can be found in Section 20.
Finally, in analogy to Corollary 9.12, we can now show:
###### Corollary 10.33.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Assume that $P$ is a
rooted forest (made into a poset by having every node smaller than its
children).
(a) Then,
$\operatorname*{ord}\left(\mathbf{r}_{P}\right)\mid\operatorname{lcm}\left(1,2,...,n+1\right)$.
(b) Moreover, if $P$ is a tree, then
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)\mid\operatorname{lcm}\left(1,2,...,n\right)$.
Corollary 10.33 is also valid if we replace “every node smaller than its
children” by “every node larger than its children”, and the proof is exactly
analogous.
Let us notice that the algorithm described in the proof of Proposition 10.27
can be turned into an explicit formula (not just an upper bound as in
Corollary 10.33), whose inductive proof we leave to the reader:
###### Proposition 10.34.
Let $n\in\mathbb{N}$. Let $P$ be an $n$-graded poset. Assume that $P$ is a
rooted forest (made into a poset by having every node smaller than its
children). Notice that
$\left|\widehat{P}_{i}\right|\leqslant\left|\widehat{P}_{i+1}\right|$ for
every $i\in\left\\{0,1,...,n-1\right\\}$ (where $\widehat{P}_{i}$ and
$\widehat{P}_{i+1}$ are defined as in Definition 3.4). Then,
$\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)=\operatorname{lcm}\left\\{n-i\
\mid\ i\in\left\\{0,1,...,n-1\right\\};\
\left|\widehat{P}_{i}\right|<\left|\widehat{P}_{i+1}\right|\right\\}.$
(Of course, $\operatorname*{ord}\left(\mathbf{r}_{P}\right)$ can now be
computed by
$\operatorname*{ord}\left(\mathbf{r}_{P}\right)=\operatorname{lcm}\left(n+1,\operatorname*{ord}\left(\overline{\mathbf{r}}_{P}\right)\right)$.)
The same property therefore holds for birational rowmotion $R_{P}$ and its
homogeneous version $\overline{R}_{P}$.
## 11 The rectangle: statements of the results
###### Definition 11.1.
Let $p$ and $q$ be two positive integers. The $p\times q$-rectangle will
denote the poset $\left\\{1,2,...,p\right\\}\times\left\\{1,2,...,q\right\\}$
with order defined as follows: For two elements $\left(i,k\right)$ and
$\left(i^{\prime},k^{\prime}\right)$ of
$\left\\{1,2,...,p\right\\}\times\left\\{1,2,...,q\right\\}$, we set
$\left(i,k\right)\leqslant\left(i^{\prime},k^{\prime}\right)$ if and only if
$\left(i\leqslant i^{\prime}\text{ and }k\leqslant k^{\prime}\right)$.
###### Example 11.2.
Here is the Hasse diagram of the $2\times 3$-rectangle:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.11111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\\&&&\\\&&&\\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.6222pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 47.13327pt\raise
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###### Remark 11.3.
Let $p$ and $q$ be positive integers. The $p\times q$-rectangle is denoted by
$\left[p\right]\times\left[q\right]$ in the papers [StWi11], [EiPr13],
[PrRo13] and [PrRo14].
###### Remark 11.4.
Let $p$ and $q$ be two positive integers. Let
$\operatorname*{Rect}\left(p,q\right)$ denote the $p\times q$-rectangle.
(a) The $p\times q$-rectangle is a $\left(p+q-1\right)$-graded poset, with
$\deg\left(\left(i,k\right)\right)=i+k-1$ for all
$\left(i,k\right)\in\operatorname*{Rect}\left(p,q\right)$.
(b) Let $\left(i,k\right)$ and $\left(i^{\prime},k^{\prime}\right)$ be two
elements of $\operatorname*{Rect}\left(p,q\right)$. Then,
$\left(i,k\right)\lessdot\left(i^{\prime},k^{\prime}\right)$ if and only if
either $\left(i^{\prime}=i\text{ and }k^{\prime}=k+1\right)$ or
$\left(k^{\prime}=k\text{ and }i^{\prime}=i+1\right)$.
We are going to use Remark 11.4 without explicit mention.
The following theorem was conjectured by James Propp and the second author:
###### Theorem 11.5.
Let $\operatorname*{Rect}\left(p,q\right)$ denote the $p\times q$-rectangle.
Let $\mathbb{K}$ be a field. Then,
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)=p+q$.
This is a birational analogue (and, using the reasoning of [EiPr13],
generalization) of the classical fact (appearing in [StWi11, Theorem 3.1] and
[Flaa93, Theorem 2]) that
$\operatorname*{ord}\left(\mathbf{r}_{\operatorname*{Rect}\left(p,q\right)}\right)=p+q$
(using the notations of Definition 10.7 and Definition 10.28).
Notice that Proposition 7.3 yields that
$p+q\mid\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)$,
so all that needs to be proven in order to verify Theorem 11.5 is showing that
$R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}=\operatorname*{id}$.
Notice also that in the case when $p\leqslant 2$ and $q\leqslant 2$, Theorem
11.5 follows rather easily from Propositions 9.10 (a), 9.11 (a) and 7.3
(because $\operatorname*{Rect}\left(p,q\right)$ is a skeletal poset in this
case), but this approach does not generalize to any interesting cases.
###### Remark 11.6.
Theorem 11.5 generalizes a well-known property of promotion on semistandard
Young tableaux of rectangular shape, albeit not in an obvious way. Here are
some details (which a reader unacquainted with Young tableaux can freely
skip):
Let $N$ be a nonnegative integer, and let $\lambda$ be a partition. Let
$\operatorname*{SSYT}\nolimits_{N}\lambda$ denote the set of all semistandard
Young tableaux of shape $\lambda$ whose entries are all $\leqslant N$. One can
define a map
$\operatorname*{Pro}:\operatorname*{SSYT}\nolimits_{N}\lambda\rightarrow\operatorname*{SSYT}\nolimits_{N}\lambda$
called jeu-de-taquin promotion (or Schützenberger promotion, or simply
promotion when no ambiguities can arise); see [Russ13, §5.1] for a precise
definition. (The definition in [Rhoa10, §2] is different – it defines the
inverse of this map. Conventions differ.) This map has some interesting
properties already for arbitrary $\lambda$, but the most interesting situation
is that of $\lambda$ being a rectangular partition (i.e., a partition all of
whose nonzero parts are equal). In this situation, a folklore theorem states
that $\operatorname*{Pro}\nolimits^{N}=\operatorname*{id}$. (The particular
case of this theorem when $\operatorname*{Pro}$ is applied only to standard
Young tableaux is well-known – see, e.g., [Haiman92, Theorem 4.4] –, but the
only proof of the general theorem that we were able to find in literature is
Rhoades’s – [Rhoa10, Corollary 5.6] –, which makes use of Kazhdan-Lusztig
theory.)
Theorem 11.5 can be used to give an alternative proof of this
$\operatorname*{Pro}\nolimits^{N}=\operatorname*{id}$ theorem. See a future
version of [EiPr13] (or, for the time being, [EiPr14, §2, pp. 4–5]) for how
this works.
Note that [Russ13, §5.1], [Rhoa10, §2] and [EiPr13] give three different
definitions of promotion. The definitions in [Russ13, §5.1] and in [EiPr13]
are equivalent, while the definition in [Rhoa10, §2] defines the inverse of
the map defined in the other two sources. Unfortunately, we were unable to
find the proofs of these facts in existing literature; they are claimed in
[KiBe95, Propositions 2.5 and 2.6], and can be proven using the concept of
tableau switching [Leeu01, Definition 2.2.1].
Besides Theorem 11.5, we can also state some kind of symmetry property of
birational rowmotion on the $p\times q$-rectangle (referred to as the “pairing
property” in [EiPr13]), which was also conjectured by Propp and the second
author:
###### Theorem 11.7.
Let $\operatorname*{Rect}\left(p,q\right)$ denote the $p\times q$-rectangle.
Let $\mathbb{K}$ be a field. Let
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$. Assume that
$R_{\operatorname*{Rect}\left(p,q\right)}^{\ell}f$ is well-defined for every
$\ell\in\left\\{0,1,...,i+k-1\right\\}$. Let
$\left(i,k\right)\in\operatorname*{Rect}\left(p,q\right)$. Then,
$f\left(\left(p+1-i,q+1-k\right)\right)=\dfrac{f\left(0\right)f\left(1\right)}{\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\left(\left(i,k\right)\right)}.$
This Theorem generalizes the “reciprocity phenomenon” observed on the $2\times
2$-rectangle in Example 2.15.
###### Remark 11.8.
While Theorem 11.5 only makes a statement about
$R_{\operatorname*{Rect}\left(p,q\right)}$, it can be used (in combination
with Proposition 9.10 and others) to derive upper bounds on the orders of
$R_{P}$ and $\overline{R}_{P}$ for some other posets $P$. Here is an example:
Let $\mathbb{K}$ be a field. For the duration of this remark, let us denote
the poset
$\operatorname*{Rect}\left(2,3\right)\setminus\left\\{\left(1,1\right),\left(2,3\right)\right\\}$
by $N$. (The Hasse diagram of this poset has the rather simple form
$\textstyle{\left(2,2\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(1,3\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(2,1\right)}$$\textstyle{\left(1,2\right)}$,
which explains why we have chosen to call it $N$ here.) Then,
$\operatorname*{ord}\left(R_{N}\right)\mid 15$ and
$\operatorname*{ord}\left(\overline{R}_{N}\right)\mid 5$. This can be proven
as follows: We have $\operatorname*{Rect}\left(2,3\right)\cong
B_{1}\left(B_{1}^{\prime}N\right)$ and therefore
$\displaystyle\operatorname*{ord}\left(\overline{R}_{\operatorname*{Rect}\left(2,3\right)}\right)$
$\displaystyle=\operatorname*{ord}\left(\overline{R}_{B_{1}\left(B_{1}^{\prime}N\right)}\right)=\operatorname*{ord}\left(\overline{R}_{B_{1}^{\prime}N}\right)\
\ \ \ \ \ \ \ \ \ \left(\text{by Proposition \ref{prop.Bk.ord} {(a)}}\right)$
$\displaystyle=\operatorname*{ord}\left(\overline{R}_{N}\right)\ \ \ \ \ \ \ \
\ \ \left(\text{by Proposition \ref{prop.B'k.ord} {(a)}}\right),$
so that
$\displaystyle\operatorname*{ord}\left(\overline{R}_{N}\right)$
$\displaystyle=\operatorname*{ord}\left(\overline{R}_{\operatorname*{Rect}\left(2,3\right)}\right)$
$\displaystyle\mid\operatorname{lcm}\left(4+1,\operatorname*{ord}\left(\overline{R}_{\operatorname*{Rect}\left(2,3\right)}\right)\right)=\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(2,3\right)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{by Proposition \ref{prop.ord-
projord}, since }\operatorname*{Rect}\left(2,3\right)\text{ is
}4\text{-graded}\right)$ $\displaystyle=2+3\ \ \ \ \ \ \ \ \ \ \left(\text{by
Theorem \ref{thm.rect.ord}}\right)$ $\displaystyle=5$
and thus
$\displaystyle\operatorname*{ord}\left(R_{N}\right)$
$\displaystyle=\operatorname{lcm}\left(\underbrace{2+1}_{=3},\underbrace{\operatorname*{ord}\left(\overline{R}_{N}\right)}_{\mid
5}\right)\ \ \ \ \ \ \ \ \ \ \left(\text{by Proposition \ref{prop.ord-
projord}, since }N\text{ is }2\text{-graded}\right)$
$\displaystyle\mid\operatorname{lcm}\left(3,5\right)=15.$
It can actually be shown that $\operatorname*{ord}\left(R_{N}\right)=15$ and
$\operatorname*{ord}\left(\overline{R}_{N}\right)=5$ by direct computation.
In the same vein it can be shown that
$\operatorname*{ord}\left(\overline{R}_{\operatorname*{Rect}\left(p,q\right)\setminus\left\\{\left(1,1\right),\left(p,q\right)\right\\}}\right)\mid
p+q$ and
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)\setminus\left\\{\left(1,1\right),\left(p,q\right)\right\\}}\right)\mid\operatorname{lcm}\left(p+q-2,p+q\right)$
for any integers $p>1$ and $q>1$. This doesn’t, however, generalize to
arbitrary posets obtained by removing some ranks from
$\operatorname*{Rect}\left(p,q\right)$ (indeed, sometimes birational rowmotion
doesn’t even have finite order on such posets, cf. Section 20).
## 12 Reduced labellings
The proof that we give for Theorem 11.5 and Theorem 11.7 is largely inspired
by the proof of Zamolodchikov’s conjecture in case $AA$ given by Volkov in
[Volk06]303030“Case $AA$” refers to the Cartesian product of the Dynkin
diagrams of two type-$A$ root systems. This, of course, is a rectangle, just
as in our Theorem 11.5.. This is not very surprising because the orbit of a
$\mathbb{K}$-labelling under birational rowmotion appears superficially
similar to a solution of a $Y$-system of type $AA$. Yet we do not see a way to
derive Theorem 11.5 from Zamolodchikov’s conjecture or vice versa. (It should
be noticed that Zamolodchikov’s Y-system has an obvious “reducibility
property”, namely consisting of two decoupled subsystems – a property at least
not obviously satisfied in the case of birational rowmotion.)
The first step towards our proof of Theorem 11.5 is to restrict attention to
so-called reduced labellings. Let us define these first:
###### Definition 12.1.
Let $\operatorname*{Rect}\left(p,q\right)$ denote the $p\times q$-rectangle.
Let $\mathbb{K}$ be a field. A labelling
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ is said to
be reduced if $f\left(0\right)=f\left(1\right)=1$. The set of all reduced
labellings $f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$
will be identified with $\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$ in
the obvious way.
Note that fixing the values of $f\left(0\right)$ and $f\left(1\right)$ like
this makes $f$ “less generic”, but still the operator
$R_{\operatorname*{Rect}\left(p,q\right)}$ restricts to a rational map from
the variety of all reduced labellings
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ to itself.
(This is because the operator $R_{\operatorname*{Rect}\left(p,q\right)}$ does
not change the values at $0$ and $1$, and does not degenerate from setting
$f\left(0\right)=f\left(1\right)=1$.)
Reduced labellings are not much less general than arbitrary labellings: In
fact, every zero-free $\mathbb{K}$-labelling $f$ of a graded poset $P$ is
homogeneously equivalent to a reduced labelling. Thus, many results can be
proven for all labellings by means of proving them for reduced labellings
first, and then extending them to general labellings by means of homogeneous
equivalence.313131A slightly different way to reduce the case of a general
labelling to that of a reduced one is taken in [EiPr13, §4]. We will use this
tactic in our proof of Theorem 11.5. Here is how this works:
###### Proposition 12.2.
Let $\operatorname*{Rect}\left(p,q\right)$ denote the $p\times q$-rectangle.
Let $\mathbb{K}$ be a field. Assume that almost every (in the Zariski sense)
reduced labelling
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ satisfies
$R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}f=f$. Then,
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)=p+q$.
###### Proof of Proposition 12.2 (sketched)..
Let $g\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ be any
$\mathbb{K}$-labelling of $\operatorname*{Rect}\left(p,q\right)$ which is
sufficiently generic for $R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}g$ to
be well-defined.
We use the notation of Definition 5.2. Recall that
$\operatorname*{Rect}\left(p,q\right)$ is a $\left(p+q-1\right)$-graded poset.
We can easily find a $\left(p+q+1\right)$-tuple
$\left(a_{0},a_{1},...,a_{p+q}\right)\in\left(\mathbb{K}^{\times}\right)^{p+q+1}$
such that $\left(a_{0},a_{1},...,a_{p+q}\right)\flat g$ is a reduced
$\mathbb{K}$-labelling (in fact, set $a_{0}=\dfrac{1}{g\left(0\right)}$ and
$a_{p+q}=\dfrac{1}{g\left(1\right)}$, and choose all other $a_{i}$
arbitrarily). Corollary 5.7 (applied to $p+q-1$,
$\operatorname*{Rect}\left(p,q\right)$ and $g$ instead of $n$, $P$ and $f$)
then yields
$R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}\left(\left(a_{0},a_{1},...,a_{p+q}\right)\flat
g\right)=\left(a_{0},a_{1},...,a_{p+q}\right)\flat\left(R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}g\right).$
(32)
We have assumed that almost every (in the Zariski sense) reduced labelling
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ satisfies
$R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}f=f$. Thus, every reduced
labelling $f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$
for which $R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}f$ is well-defined
satisfies $R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}f=f$ (because
$R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}f=f$ can be written as an
equality between rational functions in the labels of $f$, and thus it must
hold everywhere if it holds on a Zariski-dense open subset). Applying this to
$f=\left(a_{0},a_{1},...,a_{p+q}\right)\flat g$, we obtain that
$R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}\left(\left(a_{0},a_{1},...,a_{p+q}\right)\flat
g\right)=\left(a_{0},a_{1},...,a_{p+q}\right)\flat g$. Thus,
$\displaystyle\left(a_{0},a_{1},...,a_{p+q}\right)\flat g$
$\displaystyle=R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}\left(\left(a_{0},a_{1},...,a_{p+q}\right)\flat
g\right)$
$\displaystyle=\left(a_{0},a_{1},...,a_{p+q}\right)\flat\left(R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}g\right)\
\ \ \ \ \ \ \ \ \ \left(\text{by (\ref{pf.rect.reduce.short.2})}\right).$ (33)
We can cancel the “$\left(a_{0},a_{1},...,a_{p+q}\right)\flat$” from both
sides of this equality (because all the $a_{i}$ are nonzero), and thus obtain
$g=R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}g$.
Now, forget that we fixed $g$. We thus have proven that
$g=R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}g$ holds for every
$\mathbb{K}$-labelling
$g\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ of
$\operatorname*{Rect}\left(p,q\right)$ which is sufficiently generic for
$R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}g$ to be well-defined. In other
words, $R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}=\operatorname*{id}$ as
partial maps. Hence,
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)\mid
p+q$.
On the other hand, Proposition 7.3 (applied to
$P=\operatorname*{Rect}\left(p,q\right)$ and $n=p+q-1$) yields
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)=\operatorname{lcm}\left(\left(p+q-1\right)+1,\operatorname*{ord}\left(\overline{R}_{\operatorname*{Rect}\left(p,q\right)}\right)\right)$.
Hence,
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)$ is
divisible by $\left(p+q-1\right)+1=p+q$. Combined with
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)\mid
p+q$, this yields
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)=p+q$.
This proves Proposition 12.2. ∎
Let us also formulate the particular case of Theorem 11.7 for reduced
labellings:
###### Theorem 12.3.
Let $\operatorname*{Rect}\left(p,q\right)$ denote the $p\times q$-rectangle.
Let $\mathbb{K}$ be a field. Let
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ be reduced.
Assume that $R_{\operatorname*{Rect}\left(p,q\right)}^{\ell}f$ is well-defined
for every $\ell\in\left\\{0,1,...,i+k-1\right\\}$. Let
$\left(i,k\right)\in\operatorname*{Rect}\left(p,q\right)$. Then,
$f\left(\left(p+1-i,q+1-k\right)\right)=\dfrac{1}{\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\left(\left(i,k\right)\right)}.$
We will prove this before we prove the general form (Theorem 11.7), and in
fact we are going to derive Theorem 11.7 from its particular case, Theorem
12.3. We are not going to encumber this section with the derivation; its
details can be found in Section 16.
## 13 The Grassmannian parametrization: statements
In this section, we are going to introduce the main actor in our proof of
Theorem 11.5: an assignment of a reduced $\mathbb{K}$-labelling of
$\operatorname*{Rect}\left(p,q\right)$, denoted
$\operatorname*{Grasp}\nolimits_{j}A$, to any integer $j$ and almost any
matrix $A\in\mathbb{K}^{p\times\left(p+q\right)}$ (Definition 13.9). This
assignment will give us a family of $\mathbb{K}$-labellings of
$\operatorname*{Rect}\left(p,q\right)$ which is large enough to cover almost
all reduced $\mathbb{K}$-labellings of $\operatorname*{Rect}\left(p,q\right)$
(this is formalized in Proposition 13.14), while at the same time the
construction of this assignment makes it easy to track the behavior of the
$\mathbb{K}$-labellings in this family through multiple iterations of
birational rowmotion. Indeed, we will see that birational rowmotion has a very
simple effect on the reduced $\mathbb{K}$-labelling
$\operatorname*{Grasp}\nolimits_{j}A$ (Proposition 13.13).
###### Definition 13.1.
Let $\mathbb{K}$ be a commutative ring. Let $A\in\mathbb{K}^{u\times v}$ be a
$u\times v$-matrix for some nonnegative integers $u$ and $v$. (This means, at
least in this paper, a matrix with $u$ rows and $v$ columns.)
(a) For every $i\in\left\\{1,2,...,v\right\\}$, let $A_{i}$ denote the $i$-th
column of $A$.
(b) Moreover, we extend this definition to all $i\in\mathbb{Z}$ as follows:
For every $i\in\mathbb{Z}$, let
$A_{i}=\left(-1\right)^{\left(u-1\right)\left(i-i^{\prime}\right)\diagup
v}\cdot A_{i^{\prime}},$
where $i^{\prime}$ is the element of $\left\\{1,2,...,v\right\\}$ which is
congruent to $i$ modulo $v$. (Thus, $A_{v+i}=\left(-1\right)^{u-1}A_{i}$ for
every $i\in\mathbb{Z}$. Consequently, the sequence
$\left(A_{i}\right)_{i\in\mathbb{Z}}$ is periodic with period dividing $2v$,
and if $u$ is odd, the period also divides $v$.)
(c) For any four integers $a$, $b$, $c$ and $d$ satisfying $a\leqslant b$ and
$c\leqslant d$, we let $A\left[a:b\mid c:d\right]$ be the matrix whose columns
(from left to right) are $A_{a}$, $A_{a+1}$, $...$, $A_{b-1}$, $A_{c}$,
$A_{c+1}$, $...$, $A_{d-1}$. (This matrix has $b-a+d-c$ columns.)323232Some
remarks on this matrix $A\left[a:b\mid c:d\right]$ are appropriate at this
point. 1. We notice that we allow the case $a=b$. In this case, obviously, the
columns of the matrix $A\left[a:b\mid c:d\right]$ are $A_{c}$, $A_{c+1}$,
$...$, $A_{d-1}$, so the precise value of $a=b$ does not matter. Similarly,
the case $c=d$ is allowed. 2. The matrix $A\left[a:b\mid c:d\right]$ is not
always a submatrix of $A$. Its columns are columns of $A$ multiplied with $1$
or $-1$; they can appear several times and need not appear in the same order
as they appear in $A$. When $b-a+d-c=u$, this matrix $A\left[a:b\mid
c:d\right]$ is a square matrix (with $u$ rows and $u$ columns), and thus has a
determinant $\det\left(A\left[a:b\mid c:d\right]\right)$.
(d) We extend the definition of $\det\left(A\left[a:b\mid c:d\right]\right)$
to encompass the case when $b=a-1$ or $d=c-1$, by setting
$\det\left(A\left[a:b\mid c:d\right]\right)=0$ in this case (although the
matrix $A\left[a:b\mid c:d\right]$ itself is not defined in this case).
The reader should be warned that, for $\det\left(A\left[a:b\mid
c:d\right]\right)$ to be defined, we need $b-a+d-c=u$ (not just $b-a+d-c\equiv
u\operatorname{mod}v$, despite the apparent periodicity in the construction of
the matrix $A$.)
###### Example 13.2.
If $A$ is the $2\times 3$-matrix $\left(\begin{array}[c]{ccc}3&5&7\\\
4&1&9\end{array}\right)$, then Definition 13.1 (b) yields (for instance)
$A_{5}=\left(-1\right)^{\left(2-1\right)\left(5-2\right)\diagup 3}\cdot
A_{2}=-A_{2}=-\left(\begin{array}[c]{c}5\\\
1\end{array}\right)=\left(\begin{array}[c]{c}-5\\\ -1\end{array}\right)$ and
$A_{-4}=\left(-1\right)^{\left(2-1\right)\left(\left(-4\right)-2\right)\diagup
3}\cdot A_{2}=A_{2}=\left(\begin{array}[c]{c}5\\\ 1\end{array}\right)$.
If $A$ is the $3\times 2$-matrix $\left(\begin{array}[c]{cc}1&2\\\ 3&2\\\
-5&4\end{array}\right)$, then Definition 13.1 (b) yields (for instance)
$A_{0}=\left(-1\right)^{\left(3-1\right)\left(0-2\right)\diagup 2}\cdot
A_{2}=A_{2}=\left(\begin{array}[c]{c}2\\\ 2\\\ 4\end{array}\right)$.
###### Remark 13.3.
Some parts of Definition 13.1 might look accidental and haphazard; here are
some motivations and aide-memoires:
The choice of sign in Definition 13.1 (b) is not only the “right” one for what
we are going to do below, but also naturally appears in [Post06, Remark 3.3].
It guarantees, among other things, that if $A\in\mathbb{R}^{u\times v}$ is
totally nonnegative, then the matrix having columns $A_{1+i}$, $A_{2+i}$,
$...$, $A_{v+i}$ is totally nonnegative for every $i\in\mathbb{Z}$.
The notation $A\left[a:b\mid c:d\right]$ in Definition 13.1 (c) borrows from
Python’s notation $\left[x:y\right]$ for taking indices from the interval
$\left\\{x,x+1,...,y-1\right\\}$.
The convention to define $\det\left(A\left[a:b\mid c:d\right]\right)$ as $0$
in Definition 13.1 (d) can be motivated using exterior algebra as follows: If
we identify $\wedge^{u}\left(\mathbb{K}^{u}\right)$ with $\mathbb{K}$ by
equating with $1\in\mathbb{K}$ the wedge product $e_{1}\wedge
e_{2}\wedge...\wedge e_{u}$ of the standard basis vectors, then
$\det\left(A\left[a:b\mid c:d\right]\right)=A_{a}\wedge A_{a+1}\wedge...\wedge
A_{b-1}\wedge A_{c}\wedge A_{c+1}\wedge...\wedge A_{d-1}$; this belongs to the
product of $\wedge^{b-a}\left(\mathbb{K}^{u}\right)$ with
$\wedge^{d-c}\left(\mathbb{K}^{u}\right)$ in
$\wedge^{u}\left(\mathbb{K}^{u}\right)$. If $b=a-1$, then this product is $0$
(since
$\wedge^{b-a}\left(\mathbb{K}^{u}\right)=\wedge^{-1}\left(\mathbb{K}^{u}\right)=0$),
so $\det\left(A\left[a:b\mid c:d\right]\right)$ has to be $0$ in this case.
Before we go any further, we make several straightforward observations about
the notations we have just introduced.
###### Proposition 13.4.
Let $\mathbb{K}$ be a field. Let $A\in\mathbb{K}^{u\times v}$ be a $u\times
v$-matrix for some nonnegative integers $u$ and $v$. Let $a$, $b$, $c$ and $d$
be four integers satisfying $a\leqslant b$ and $c\leqslant d$ and $b-a+d-c=u$.
Assume that some element of the interval $\left\\{a,a+1,...,b-1\right\\}$ is
congruent to some element of the interval $\left\\{c,c+1,...,d-1\right\\}$
modulo $v$. Then, $\det\left(A\left[a:b\mid c:d\right]\right)=0$.
###### Proof of Proposition 13.4..
The assumption yields that the matrix $A\left[a:b\mid c:d\right]$ has two
columns which are proportional to each other by a factor of $\pm 1$. Hence,
this matrix has determinant $0$. ∎
###### Proposition 13.5.
Let $\mathbb{K}$ be a field. Let $A\in\mathbb{K}^{u\times v}$ be a $u\times
v$-matrix for some nonnegative integers $u$ and $v$. Let $a$, $b$, $c$ and $d$
be four integers satisfying $a\leqslant b$ and $c\leqslant d$ and $b-a+d-c=u$.
Then,
$\det\left(A\left[a:b\mid
c:d\right]\right)=\left(-1\right)^{\left(b-a\right)\left(d-c\right)}\det\left(A\left[c:d\mid
a:b\right]\right).$
###### Proof of Proposition 13.5..
This follows from the fact that permuting the columns of a matrix multiplies
its determinant by the sign of the corresponding permutation. ∎
###### Proposition 13.6.
Let $\mathbb{K}$ be a field. Let $A\in\mathbb{K}^{u\times v}$ be a $u\times
v$-matrix for some nonnegative integers $u$ and $v$. Let $a$, $b_{1}$, $b_{2}$
and $c$ be four integers satisfying $a\leqslant b_{1}\leqslant c$ and
$a\leqslant b_{2}\leqslant c$. Then,
$A\left[a:b_{1}\mid b_{1}:c\right]=A\left[a:b_{2}\mid b_{2}:c\right].$
###### Proof of Proposition 13.6..
Both matrices $A\left[a:b_{1}\mid b_{1}:c\right]$ and $A\left[a:b_{2}\mid
b_{2}:c\right]$ are simply the matrix with columns $A_{a}$, $A_{a+1}$, $...$,
$A_{c-1}$. ∎
###### Proposition 13.7.
Let $\mathbb{K}$ be a field. Let $A\in\mathbb{K}^{u\times v}$ be a $u\times
v$-matrix for some nonnegative integers $u$ and $v$. Let $c$ and $d$ be two
integers satisfying $c\leqslant d$. Then:
(a) Any integers $a_{1}$ and $a_{2}$ satisfy
$A\left[a_{1}:a_{1}\mid c:d\right]=A\left[a_{2}:a_{2}\mid c:d\right].$
(b) Any integers $a_{1}$ and $a_{2}$ satisfy
$A\left[c:d\mid a_{1}:a_{1}\right]=A\left[c:d\mid a_{2}:a_{2}\right].$
(c) If $a$ and $b$ are integers satisfying $c\leqslant b\leqslant d$, then
$A\left[c:b\mid b:d\right]=A\left[c:d\mid a:a\right].$
###### Proof of Proposition 13.7..
All six matrices in the above equalities are simply the matrix with columns
$A_{c}$, $A_{c+1}$, $...$, $A_{d-1}$. ∎
###### Proposition 13.8.
Let $\mathbb{K}$ be a field. Let $A\in\mathbb{K}^{u\times v}$ be a $u\times
v$-matrix for some nonnegative integers $u$ and $v$. Let $a$, $b$, $c$ and $d$
be four integers satisfying $a\leqslant b$ and $c\leqslant d$ and $b-a+d-c=u$.
(a) We have
$\det\left(A\left[v+a:v+b\mid v+c:v+d\right]\right)=\det\left(A\left[a:b\mid
c:d\right]\right).$
(b) We have
$\det\left(A\left[a:b\mid
v+c:v+d\right]\right)=\left(-1\right)^{\left(u-1\right)\left(d-c\right)}\det\left(A\left[a:b\mid
c:d\right]\right).$
(c) We have
$\det\left(A\left[a:b\mid v+c:v+d\right]\right)=\det\left(A\left[c:d\mid
a:b\right]\right).$
###### Proof of Proposition 13.8 (sketched)..
Nothing about this is anything more than trivial. Part (a) and (b) follow from
the fact that $A_{v+i}=\left(-1\right)^{u-1}A_{i}$ for every $i\in\mathbb{Z}$
(which is owed to Definition 13.1 (b)) and the multilinearity of the
determinant. The proof of part (c) additionally uses Proposition 13.5 and a
careful sign computation (notice that
$\left(-1\right)^{\left(d-c-1\right)\left(d-c\right)}=1$ because
$\left(d-c-1\right)\left(d-c\right)$ is even, no matter what the parities of
$c$ and $d$ are). All details can be easily filled in by the reader. ∎
###### Definition 13.9.
Let $\mathbb{K}$ be a field. Let $p$ and $q$ be two positive integers. Let
$A\in\mathbb{K}^{p\times\left(p+q\right)}$. Let $j\in\mathbb{Z}$.
(a) We define a map
$\operatorname*{Grasp}\nolimits_{j}A\in\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$
by
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,k\right)\right)$
$\displaystyle=\dfrac{\det\left(A\left[j+1:j+i\mid
j+i+k-1:j+p+k\right]\right)}{\det\left(A\left[j:j+i\mid
j+i+k:j+p+k\right]\right)}$ (34) $\displaystyle\ \ \ \ \ \ \ \ \ \
\left.\text{for every
}\left(i,k\right)\in\operatorname*{Rect}\left(p,q\right)=\left\\{1,2,...,p\right\\}\times\left\\{1,2,...,q\right\\}\right.$
(this is well-defined when the matrix $A$ is sufficiently generic (in the
sense of Zariski topology), since the matrix $A\left[j:j+i\mid
j+i+k:j+p+k\right]$ is obtained by picking $p$ distinct columns out of $A$,
some possibly multiplied with $\left(-1\right)^{u-1}$). This map
$\operatorname*{Grasp}\nolimits_{j}A$ will be considered as a reduced
$\mathbb{K}$-labelling of $\operatorname*{Rect}\left(p,q\right)$ (since we are
identifying the set of all reduced labellings
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ with
$\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$).
(b) It will be handy to extend the map $\operatorname*{Grasp}\nolimits_{j}A$
to a slightly larger domain by blindly following (34) (and using Definition
13.1 (d)), accepting the fact that outside
$\left\\{1,2,...,p\right\\}\times\left\\{1,2,...,q\right\\}$ its values can be
“infinity”:
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(0,k\right)\right)$
$\displaystyle=0\ \ \ \ \ \ \ \ \ \ \text{for all
}k\in\left\\{1,2,...,q\right\\};$
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(p+1,k\right)\right)$
$\displaystyle=\infty\ \ \ \ \ \ \ \ \ \ \text{for all
}k\in\left\\{1,2,...,q\right\\};$
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,0\right)\right)$
$\displaystyle=0\ \ \ \ \ \ \ \ \ \ \text{for all
}i\in\left\\{1,2,...,p\right\\};$
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,q+1\right)\right)$
$\displaystyle=\infty\ \ \ \ \ \ \ \ \ \ \text{for all
}i\in\left\\{1,2,...,p\right\\}.$
(We treat $\infty$ as a symbol with the properties $\dfrac{1}{0}=\infty$ and
$\dfrac{1}{\infty}=0$.)
The notation “$\operatorname*{Grasp}$” harkens back to “Grassmannian
parametrization”, as we will later parametrize (generic) reduced labellings on
$\operatorname*{Rect}\left(p,q\right)$ by matrices via this map
$\operatorname*{Grasp}\nolimits_{0}$. The reason for the word “Grassmannian”
is that, while we have defined $\operatorname*{Grasp}\nolimits_{j}$ as a
rational map from the matrix space $\mathbb{K}^{p\times\left(p+q\right)}$, it
actually is not defined outside of the Zariski-dense open subset
$\mathbb{K}_{\operatorname*{rk}=p}^{p\times\left(p+q\right)}$ of
$\mathbb{K}^{p\times\left(p+q\right)}$ formed by all matrices whose rank is
$p$, and on that subset
$\mathbb{K}_{\operatorname*{rk}=p}^{p\times\left(p+q\right)}$ it factors
through the quotient of
$\mathbb{K}_{\operatorname*{rk}=p}^{p\times\left(p+q\right)}$ by the left
multiplication action of $\operatorname*{GL}\nolimits_{p}\mathbb{K}$ (because
it is easy to see that $\operatorname*{Grasp}\nolimits_{j}A$ is invariant
under row transformations of $A$); this quotient is a well-known avatar of the
Grassmannian.
The formula (34) is inspired by the $Y_{ijk}$ of Volkov’s [Volk06]; similar
expressions (in a different context) also appear in [Kiri00, Theorem 4.21].
###### Example 13.10.
If $p=2$, $q=2$ and
$A=\left(\begin{array}[c]{cccc}a_{11}&a_{12}&a_{13}&a_{14}\\\
a_{21}&a_{22}&a_{23}&a_{24}\end{array}\right)$, then
$\left(\operatorname*{Grasp}\nolimits_{0}A\right)\left(\left(1,1\right)\right)=\dfrac{\det\left(A\left[1:1\mid
1:3\right]\right)}{\det\left(A\left[0:1\mid
2:3\right]\right)}=\dfrac{\det\left(\begin{array}[c]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right)}{\det\left(\begin{array}[c]{cc}-a_{14}&a_{12}\\\
-a_{24}&a_{22}\end{array}\right)}=\dfrac{a_{11}a_{22}-a_{12}a_{21}}{a_{12}a_{24}-a_{14}a_{22}}$
and
$\left(\operatorname*{Grasp}\nolimits_{1}A\right)\left(\left(1,2\right)\right)=\dfrac{\det\left(A\left[2:2\mid
3:5\right]\right)}{\det\left(A\left[1:2\mid
4:5\right]\right)}=\dfrac{\det\left(\begin{array}[c]{cc}a_{13}&a_{14}\\\
a_{23}&a_{24}\end{array}\right)}{\det\left(\begin{array}[c]{cc}a_{11}&a_{14}\\\
a_{21}&a_{24}\end{array}\right)}=\dfrac{a_{13}a_{24}-a_{14}a_{23}}{a_{11}a_{24}-a_{14}a_{21}}.$
We will see more examples of values of $\operatorname*{Grasp}\nolimits_{0}A$
in Example 15.1.
###### Proposition 13.11.
Let $\mathbb{K}$ be a field. Let $p$ and $q$ be two positive integers. Let
$A\in\mathbb{K}^{p\times\left(p+q\right)}$ be a matrix. Then,
$\operatorname*{Grasp}\nolimits_{j}A=\operatorname*{Grasp}\nolimits_{p+q+j}A$
for every $j\in\mathbb{Z}$ (provided that $A$ is sufficiently generic in the
sense of Zariski topology for $\operatorname*{Grasp}\nolimits_{j}A$ to be
well-defined).
###### Proof of Proposition 13.11 (sketched)..
We need to show that
$\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,k\right)\right)=\left(\operatorname*{Grasp}\nolimits_{p+q+j}A\right)\left(\left(i,k\right)\right)$
for every
$\left(i,k\right)\in\left\\{1,2,...,p\right\\}\times\left\\{1,2,...,q\right\\}$.
But we have
$\displaystyle A\left[p+q+j:p+q+j+i\mid p+q+j+i+k:p+q+j+p+k\right]$
$\displaystyle=A\left[j:j+i\mid j+i+k:j+p+k\right]$
(by Proposition 13.8 (a), applied to $u=p$, $v=p+q$, $a=j$, $b=j+i$, $c=j+i+k$
and $d=j+p+k$) and
$\displaystyle A\left[p+q+j+1:p+q+j+i\mid p+q+j+i+k-1:p+q+j+p+k\right]$
$\displaystyle=A\left[j+1:j+i\mid j+i+k-1:j+p+k\right]$
(by Proposition 13.8 (a), applied to $u=p$, $v=p+q$, $a=j+1$, $b=j+i$,
$c=j+i+k-1$ and $d=j+p+k$). Using these equalities, we immediately obtain
$\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,k\right)\right)=\left(\operatorname*{Grasp}\nolimits_{p+q+j}A\right)\left(\left(i,k\right)\right)$
from the definition of $\operatorname*{Grasp}\nolimits_{j}A$. Proposition
13.11 is proven. ∎
###### Proposition 13.12.
Let $\mathbb{K}$ be a field. Let $p$ and $q$ be two positive integers. Let
$A\in\mathbb{K}^{p\times\left(p+q\right)}$ be a matrix. Let
$\left(i,k\right)\in\operatorname*{Rect}\left(p,q\right)$ and
$j\in\mathbb{Z}$. Then,
$\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,k\right)\right)=\dfrac{1}{\left(\operatorname*{Grasp}\nolimits_{j+i+k-1}A\right)\left(\left(p+1-i,q+1-k\right)\right)}$
(provided that $A$ is sufficiently generic in the sense of Zariski topology
for
$\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,k\right)\right)$
and
$\left(\operatorname*{Grasp}\nolimits_{j+i+k-1}A\right)\left(\left(p+1-i,q+1-k\right)\right)$
to be well-defined).
###### Proof.
The proof of Proposition 13.12 is completely straightforward: one merely needs
to expand the definitions of
$\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,k\right)\right)$
and
$\left(\operatorname*{Grasp}\nolimits_{j+i+k-1}A\right)\left(\left(p+1-i,q+1-k\right)\right)$
and to apply Proposition 13.8 (c) twice. ∎
Now, let us state the two facts which will combine to a proof of Theorem 11.5:
###### Proposition 13.13.
Let $\mathbb{K}$ be a field. Let $p$ and $q$ be two positive integers. Let
$A\in\mathbb{K}^{p\times\left(p+q\right)}$ be a matrix. Let $j\in\mathbb{Z}$.
Then,
$\operatorname*{Grasp}\nolimits_{j}A=R_{\operatorname*{Rect}\left(p,q\right)}\left(\operatorname*{Grasp}\nolimits_{j+1}A\right)$
(provided that $A$ is sufficiently generic in the sense of Zariski topology
for the two sides of this equality to be well-defined).
###### Proposition 13.14.
Let $\mathbb{K}$ be a field. Let $p$ and $q$ be two positive integers. For
almost every (in the Zariski sense)
$f\in\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$, there exists a matrix
$A\in\mathbb{K}^{p\times\left(p+q\right)}$ satisfying
$f=\operatorname*{Grasp}\nolimits_{0}A$.
Once these propositions are proven, Theorems 11.5, 12.3 and 11.7 will be
rather easy to obtain. We delay the details of this until Section 16.
## 14 The Plücker-Ptolemy relation
This section is devoted to proving Proposition 13.13. Before we proceed to the
proof, we will need some fundamental identities concerning determinants of
matrices. Our main tool is the following fact, which we call the Plücker-
Ptolemy relation:
###### Theorem 14.1.
Let $\mathbb{K}$ be a field. Let $A\in\mathbb{K}^{u\times v}$ be a $u\times
v$-matrix for some nonnegative integers $u$ and $v$. Let $a$, $b$, $c$ and $d$
be four integers satisfying $a\leqslant b+1$ and $c\leqslant d+1$ and
$b-a+d-c=u-2$. Then,
$\displaystyle\det\left(A\left[a-1:b\mid
c:d+1\right]\right)\cdot\det\left(A\left[a:b+1\mid c-1:d\right]\right)$
$\displaystyle+\det\left(A\left[a:b\mid
c-1:d+1\right]\right)\cdot\det\left(A\left[a-1:b+1\mid c:d\right]\right)$
$\displaystyle=\det\left(A\left[a-1:b\mid
c-1:d\right]\right)\cdot\det\left(A\left[a:b+1\mid c:d+1\right]\right).$
Notice that the special case of this theorem for $v=u+2$, $a=2$, $b=p$,
$c=p+2$ and $d=p+q$ is the following lemma:
###### Lemma 14.2.
Let $\mathbb{K}$ be a field. Let $u\in\mathbb{N}$. Let
$B\in\mathbb{K}^{u\times\left(u+2\right)}$ be a
$u\times\left(u+2\right)$-matrix. Let $p$ and $q$ be two integers $\geqslant
2$ satisfying $p+q=u+2$. Then,
$\displaystyle\det\left(B\left[1:p\mid
p+2:p+q+1\right]\right)\cdot\det\left(B\left[2:p+1\mid p+1:p+q\right]\right)$
$\displaystyle+\det\left(B\left[2:p\mid
p+1:p+q+1\right]\right)\cdot\det\left(B\left[1:p+1\mid p+2:p+q\right]\right)$
$\displaystyle=\det\left(B\left[1:p\mid
p+1:p+q\right]\right)\cdot\det\left(B\left[2:p+1\mid p+2:p+q+1\right]\right).$
(35)
###### Proof of Theorem 14.1 (sketched)..
If $a=b-1$ or $c=d-1$, then Theorem 14.1 degenerates to a triviality (namely,
$0+0=0$). Hence, for the rest of this proof, we assume WLOG that neither
$a=b-1$ nor $c=d-1$. Hence, $a\leqslant b$ and $c\leqslant d$.
Now, Theorem 14.1 follows from the Plücker relations (see, e.g., [KlLa72,
(QR)]) applied to the $u\times\left(u+2\right)$-matrix $A\left[a-1:b+1\mid
c-1:d+1\right]$. But let us show an alternative proof of Theorem 14.1 which
avoids the use of the Plücker relations:
Let $p=b-a+2$ and $q=d-c+2$. Then, $p\geqslant 2$, $q\geqslant 2$ and
$p+q=u+2$.
Let $B$ be the matrix whose columns (from left to right) are $A_{a-1}$,
$A_{a}$, $...$, $A_{b}$, $A_{c-1}$, $A_{c}$, $...$, $A_{d}$. Then, $B$ is a
$u\times\left(u+2\right)$-matrix and satisfies
$\displaystyle A\left[a-1:b\mid c:d+1\right]$ $\displaystyle=B\left[1:p-1\mid
p+2:p+q+1\right];$ $\displaystyle A\left[a:b+1\mid c-1:d\right]$
$\displaystyle=B\left[2:p\mid p+1:p+q\right];$ $\displaystyle A\left[a:b\mid
c-1:d+1\right]$ $\displaystyle=B\left[2:p-1\mid p+1:p+q+1\right];$
$\displaystyle A\left[a-1:b+1\mid c:d\right]$ $\displaystyle=B\left[1:p\mid
p+2:p+q\right];$ $\displaystyle A\left[a-1:b\mid c-1:d\right]$
$\displaystyle=B\left[1:p-1\mid p+1:p+q\right];$ $\displaystyle
A\left[a:b+1\mid c:d+1\right]$ $\displaystyle=B\left[2:p\mid
p+2:p+q+1\right].$
Hence, the equality that we have to prove, namely
$\displaystyle\det\left(A\left[a-1:b\mid
c:d+1\right]\right)\cdot\det\left(A\left[a:b+1\mid c-1:d\right]\right)$
$\displaystyle+\det\left(A\left[a:b\mid
c-1:d+1\right]\right)\cdot\det\left(A\left[a-1:b+1\mid c:d\right]\right)$
$\displaystyle=\det\left(A\left[a-1:b\mid
c-1:d\right]\right)\cdot\det\left(A\left[a:b+1\mid c:d+1\right]\right),$
rewrites as
$\displaystyle\det\left(B\left[1:p\mid
p+2:p+q+1\right]\right)\cdot\det\left(B\left[2:p+1\mid p+1:p+q\right]\right)$
$\displaystyle+\det\left(B\left[2:p\mid
p+1:p+q+1\right]\right)\cdot\det\left(B\left[1:p+1\mid p+2:p+q\right]\right)$
$\displaystyle=\det\left(B\left[1:p\mid
p+1:p+q\right]\right)\cdot\det\left(B\left[2:p+1\mid p+2:p+q+1\right]\right).$
But this equality follows from Lemma 14.2. Hence, in order to complete the
proof of Theorem 14.1, we only need to verify Lemma 14.2. ∎
###### Proof of Lemma 14.2 (sketched)..
Let $\left(e_{1},e_{2},...,e_{u}\right)$ be the standard basis of the
$\mathbb{K}$-vector space $\mathbb{K}^{u}$.
Let $\alpha$ and $\beta$ be the $\left(p-1\right)$-st entries of the columns
$B_{1}$ and $B_{p+q}$ of $B$. Let $\gamma$ and $\delta$ be the $p$-th entries
of the columns $B_{1}$ and $B_{p+q}$ of $B$.
We need to prove (35). Since (35) is a polynomial identity in the entries of
$B$, let us WLOG assume that the columns $B_{2}$, $B_{3}$, $...$, $B_{p+q-1}$
of $B$ (these are the middle $u$ among the altogether $u+2=p+q$ columns of
$B$) are linearly independent (since $u$ vectors in $\mathbb{K}^{u}$ in
general position are linearly independent). Then, by applying row
transformations to the matrix $B$, we can transform these columns into the
basis vectors $e_{1}$, $e_{2}$, $...$, $e_{u}$ of $\mathbb{K}^{u}$. Since the
equality (35) is preserved under row transformations of $B$ (indeed, row
transformations of $B$ amount to row transformations of all six matrices
appearing in (35), and thus their only effect on the equality (35) is to
multiply the six determinants appearing in (35) by certain scalar factors, but
these scalar factors are all equal and thus don’t affect the validity of the
equality), we can therefore WLOG assume that the columns $B_{2}$, $B_{3}$,
$...$, $B_{p+q-1}$ of $B$ are the basis vectors $e_{1}$, $e_{2}$, $...$,
$e_{u}$ of $\mathbb{K}^{u}$. The matrix $B$ then looks as follows:
$\left(\begin{array}[c]{cccccccccccc}\ast&1&0&\cdots&0&0&0&0&0&\cdots&0&\ast\\\
\ast&0&1&\cdots&0&0&0&0&0&\cdots&0&\ast\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\ast\\\
\ast&0&0&\cdots&1&0&0&0&0&\cdots&0&\ast\\\
\ast&0&0&\cdots&0&1&0&0&0&\cdots&0&\ast\\\
\alpha&0&0&\cdots&0&0&1&0&0&\cdots&0&\beta\\\
\gamma&0&0&\cdots&0&0&0&1&0&\cdots&0&\delta\\\
\ast&0&0&\cdots&0&0&0&0&1&\cdots&0&\ast\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
\ast&0&0&\cdots&0&0&0&0&0&\cdots&1&\ast\end{array}\right),$
where asterisks ($\ast$) signify entries which we are not concerned with.
Now, there is a method to simplify the determinant of a matrix if some columns
of this matrix are known to belong to the standard basis
$\left(e_{1},e_{2},...,e_{u}\right)$. Indeed, such a matrix can first be
brought to a block-triangular form by permuting columns (which affects the
determinant by $\left(-1\right)^{\sigma}$, with $\sigma$ being the sign of the
permutation used), and then its determinant can be evaluated using the fact
that the determinant of a block-triangular matrix is the product of the
determinants of its diagonal blocks. Applying this method to each of the six
matrices appearing in (35), we obtain
$\displaystyle\det\left(B\left[1:p\mid p+2:p+q+1\right]\right)$
$\displaystyle=\left(-1\right)^{p+q}\left(\alpha\delta-\beta\gamma\right);$
$\displaystyle\det\left(B\left[2:p+1\mid p+1:p+q\right]\right)$
$\displaystyle=1;$ $\displaystyle\det\left(B\left[2:p\mid
p+1:p+q+1\right]\right)$ $\displaystyle=\left(-1\right)^{q-1}\beta;$
$\displaystyle\det\left(B\left[1:p+1\mid p+2:p+q\right]\right)$
$\displaystyle=\left(-1\right)^{p-1}\gamma;$
$\displaystyle\det\left(B\left[1:p\mid p+1:p+q\right]\right)$
$\displaystyle=\left(-1\right)^{p-2}\alpha;$
$\displaystyle\det\left(B\left[2:p+1\mid p+2:p+q+1\right]\right)$
$\displaystyle=\left(-1\right)^{q-2}\delta.$
Hence, (35) rewrites as
$\left(-1\right)^{p+q}\left(\alpha\delta-\beta\gamma\right)\cdot
1+\left(-1\right)^{q-1}\beta\cdot\left(-1\right)^{p-1}\gamma=\left(-1\right)^{p-2}\alpha\cdot\left(-1\right)^{q-2}\delta.$
Upon cancelling the signs, this simplifies to
$\left(\alpha\delta-\beta\gamma\right)+\beta\gamma=\alpha\delta$, which is
trivially true. Thus we have proven (35). Hence, Lemma 14.2 is proven. ∎
###### Remark 14.3.
Instead of transforming the middle $p+q$ columns of the matrix $B$ to the
standard basis vectors $e_{1}$, $e_{2}$, $...$, $e_{u}$ of $\mathbb{K}^{u}$ as
we did in the proof of Lemma 14.2, we could have transformed the first and
last columns of $B$ into the two last standard basis vectors $e_{u-1}$ and
$e_{u}$. The resulting identity would have been Dodgson’s condensation
identity (which appears, e.g., in [Zeil98, (Alice)]), applied to the matrix
formed by the remaining $u$ columns of $B$ and after some interchange of rows
and columns.
###### Proof of Proposition 13.13..
Let $f=\operatorname*{Grasp}\nolimits_{j+1}A$ and
$g=\operatorname*{Grasp}\nolimits_{j}A$.
Clearly, $f\left(0\right)=1=g\left(0\right)$ and
$f\left(1\right)=1=g\left(1\right)$.
We want to show that
$\operatorname*{Grasp}\nolimits_{j}A=R_{\operatorname*{Rect}\left(p,q\right)}\left(\operatorname*{Grasp}\nolimits_{j+1}A\right)$.
In other words, we want to show that
$g=R_{\operatorname*{Rect}\left(p,q\right)}\left(f\right)$ (because
$g=\operatorname*{Grasp}\nolimits_{j}A$ and
$f=\operatorname*{Grasp}\nolimits_{j+1}A$). According to Proposition 2.19
(applied to $P=\operatorname*{Rect}\left(p,q\right)$), this will follow once
we can show that
$g\left(v\right)=\dfrac{1}{f\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\lessdot
v\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}}\ \ \ \ \ \ \ \ \ \
\text{for every }v\in\operatorname*{Rect}\left(p,q\right).$ (36)
So let $v\in\operatorname*{Rect}\left(p,q\right)$. Thus, $v=\left(i,k\right)$
for some $i\in\left\\{1,2,...,p\right\\}$ and
$k\in\left\\{1,2,...,q\right\\}$. Consider these $i$ and $k$. We must prove
(36).
We are clearly in one of the following four cases:
Case 1: We have $v\neq\left(1,1\right)$ and $v\neq\left(p,q\right)$.
Case 2: We have $v=\left(1,1\right)$ and $v\neq\left(p,q\right)$.
Case 3: We have $v\neq\left(1,1\right)$ and $v=\left(p,q\right)$.
Case 4: We have $v=\left(1,1\right)$ and $v=\left(p,q\right)$.
Let us consider Case 1 first. In this case, we have $v\neq\left(1,1\right)$
and $v\neq\left(p,q\right)$. As a consequence, all elements
$u\in\widehat{\operatorname*{Rect}\left(p,q\right)}$ satisfying $u\lessdot v$
belong to $\operatorname*{Rect}\left(p,q\right)$, and the same holds for all
$u\in\widehat{\operatorname*{Rect}\left(p,q\right)}$ satisfying $u\gtrdot v$.
Due to the specific form of the poset $\operatorname*{Rect}\left(p,q\right)$,
there are at most two elements $u$ of
$\widehat{\operatorname*{Rect}\left(p,q\right)}$ satisfying $u\lessdot v$,
namely $\left(i,k-1\right)$ (which exists only if $k\neq 1$) and
$\left(i-1,k\right)$ (which exists only if $i\neq 1$). Hence, the sum
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\lessdot v\end{subarray}}f\left(u\right)$ takes one of the three forms
$f\left(\left(i,k-1\right)\right)+f\left(\left(i-1,k\right)\right)$,
$f\left(\left(i,k-1\right)\right)$ and $f\left(\left(i-1,k\right)\right)$. Due
to Definition 13.9 (b), all of these three forms can be rewritten uniformly as
$f\left(\left(i,k-1\right)\right)+f\left(\left(i-1,k\right)\right)$ (because
if $\left(i,k-1\right)\notin\operatorname*{Rect}\left(p,q\right)$ then
Definition 13.9 (b) guarantees that $f\left(\left(i,k-1\right)\right)=0$, and
similarly $f\left(\left(i-1,k\right)\right)=0$ if
$\left(i-1,k\right)\notin\operatorname*{Rect}\left(p,q\right)$). So we have
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\lessdot
v\end{subarray}}f\left(u\right)=f\left(\left(i,k-1\right)\right)+f\left(\left(i-1,k\right)\right).$
(37)
Similarly,
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\gtrdot
v\end{subarray}}\dfrac{1}{g\left(u\right)}=\dfrac{1}{g\left(\left(i,k+1\right)\right)}+\dfrac{1}{g\left(\left(i+1,k\right)\right)},$
(38)
where we set $\dfrac{1}{\infty}=0$ as usual.
But $f=\operatorname*{Grasp}\nolimits_{j+1}A$. Hence,
$\displaystyle f\left(\left(i,k-1\right)\right)$
$\displaystyle=\left(\operatorname*{Grasp}\nolimits_{j+1}A\right)\left(\left(i,k-1\right)\right)$
$\displaystyle=\dfrac{\det\left(A\left[\left(j+1\right)+1:\left(j+1\right)+i\mid\left(j+1\right)+i+\left(k-1\right)-1:\left(j+1\right)+p+\left(k-1\right)\right]\right)}{\det\left(A\left[j+1:\left(j+1\right)+i\mid\left(j+1\right)+i+\left(k-1\right):\left(j+1\right)+p+\left(k-1\right)\right]\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{by the definition of
}\operatorname*{Grasp}\nolimits_{j+1}A\right)$
$\displaystyle=\dfrac{\det\left(A\left[j+2:j+i+1\mid
j+i+k-1:j+p+k\right]\right)}{\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)}$
and
$\displaystyle f\left(\left(i-1,k\right)\right)$
$\displaystyle=\left(\operatorname*{Grasp}\nolimits_{j+1}A\right)\left(\left(i-1,k\right)\right)$
$\displaystyle=\dfrac{\det\left(A\left[\left(j+1\right)+1:\left(j+1\right)+\left(i-1\right)\mid\left(j+1\right)+\left(i-1\right)+k-1:\left(j+1\right)+p+k\right]\right)}{\det\left(A\left[j+1:\left(j+1\right)+\left(i-1\right)\mid\left(j+1\right)+\left(i-1\right)+k:\left(j+1\right)+p+k\right]\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{by the definition of
}\operatorname*{Grasp}\nolimits_{j+1}A\right)$
$\displaystyle=\dfrac{\det\left(A\left[j+2:j+i\mid
j+i+k-1:j+p+k+1\right]\right)}{\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)}.$
Due to these two equalities, (37) becomes
$\displaystyle\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\lessdot v\end{subarray}}f\left(u\right)$
$\displaystyle=\dfrac{\det\left(A\left[j+2:j+i+1\mid
j+i+k-1:j+p+k\right]\right)}{\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
+\dfrac{\det\left(A\left[j+2:j+i\mid
j+i+k-1:j+p+k+1\right]\right)}{\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)}$
$\displaystyle=\left(\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)\right)^{-1}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\left(\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)\right)^{-1}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\left(\det\left(A\left[j+1:j+i\mid j+i+k:j+p+k+1\right]\right)\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left.\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j+2:j+i+1\mid j+i+k-1:j+p+k\right]\right)\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left.+\det\left(A\left[j+2:j+i\mid
j+i+k-1:j+p+k+1\right]\right)\right.$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\left.\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)\right)$
$\displaystyle=\left(\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)\right)^{-1}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\left(\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)\right)^{-1}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j+1:j+i\mid j+i+k-1:j+p+k\right]\right)$ $\displaystyle\
\ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+2:j+i+1\mid
j+i+k:j+p+k+1\right]\right)$ (39)
(because applying Theorem 14.1 to $a=j+2$, $b=j+i$, $c=j+i+k$ and $d=j+p+k$
yields
$\displaystyle\det\left(A\left[j+1:j+i\mid j+i+k:j+p+k+1\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+2:j+i+1\mid
j+i+k-1:j+p+k\right]\right)$ $\displaystyle+\det\left(A\left[j+2:j+i\mid
j+i+k-1:j+p+k+1\right]\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j+1:j+i+1\mid j+i+k:j+p+k\right]\right)$
$\displaystyle=\det\left(A\left[j+1:j+i\mid j+i+k-1:j+p+k\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+2:j+i+1\mid
j+i+k:j+p+k+1\right]\right)$
).
On the other hand, $g=\operatorname*{Grasp}\nolimits_{j}A$, so that
$\displaystyle g\left(\left(i,k+1\right)\right)$
$\displaystyle=\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,k+1\right)\right)=\dfrac{\det\left(A\left[j+1:j+i\mid
j+i+\left(k+1\right)-1:j+p+\left(k+1\right)\right]\right)}{\det\left(A\left[j:j+i\mid
j+i+\left(k+1\right):j+p+\left(k+1\right)\right]\right)}$ $\displaystyle\ \ \
\ \ \ \ \ \ \ \left(\text{by the definition of
}\operatorname*{Grasp}\nolimits_{j}A\right)$
$\displaystyle=\dfrac{\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)}{\det\left(A\left[j:j+i\mid
j+i+k+1:j+p+k+1\right]\right)}$
and therefore
$\dfrac{1}{g\left(\left(i,k+1\right)\right)}=\dfrac{\det\left(A\left[j:j+i\mid
j+i+k+1:j+p+k+1\right]\right)}{\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)}.$ (40)
Also, from $g=\operatorname*{Grasp}\nolimits_{j}A$, we obtain
$\displaystyle g\left(\left(i+1,k\right)\right)$
$\displaystyle=\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i-1,k\right)\right)=\dfrac{\det\left(A\left[j+1:j+\left(i+1\right)\mid
j+\left(i+1\right)+k-1:j+p+k\right]\right)}{\det\left(A\left[j:j+\left(i+1\right)\mid
j+\left(i+1\right)+k:j+p+k\right]\right)}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\left(\text{by the definition of }\operatorname*{Grasp}\nolimits_{j}A\right)$
$\displaystyle=\dfrac{\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)}{\det\left(A\left[j:j+i+1\mid
j+i+k+1:j+p+k\right]\right)},$
so that
$\dfrac{1}{g\left(\left(i+1,k\right)\right)}=\dfrac{\det\left(A\left[j:j+i+1\mid
j+i+k+1:j+p+k\right]\right)}{\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)}.$ (41)
Due to (40) and (41), the equality (38) becomes
$\displaystyle\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}$
$\displaystyle=\dfrac{\det\left(A\left[j:j+i\mid
j+i+k+1:j+p+k+1\right]\right)}{\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
+\dfrac{\det\left(A\left[j:j+i+1\mid
j+i+k+1:j+p+k\right]\right)}{\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)}$ $\displaystyle=\left(\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)\right)^{-1}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\left(\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)\right)^{-1}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\left(\det\left(A\left[j:j+i\mid j+i+k+1:j+p+k+1\right]\right)\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left.\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j+1:j+i+1\mid j+i+k:j+p+k\right]\right)\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left.+\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)\right.$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \left.\
\ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j:j+i+1\mid
j+i+k+1:j+p+k\right]\right)\right)$
$\displaystyle=\left(\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)\right)^{-1}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\left(\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)\right)^{-1}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j:j+i\mid j+i+k:j+p+k\right]\right)$ $\displaystyle\ \ \
\ \ \ \ \ \ \ \cdot\det\left(A\left[j+1:j+i+1\mid
j+i+k+1:j+p+k+1\right]\right)$ (42)
(because applying Theorem 14.1 to $a=j+1$, $b=j+i$, $c=j+i+k+1$ and $d=j+p+k$
yields
$\displaystyle\det\left(A\left[j:j+i\mid j+i+k+1:j+p+k+1\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)$ $\displaystyle+\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j:j+i+1\mid j+i+k+1:j+p+k\right]\right)$
$\displaystyle=\det\left(A\left[j:j+i\mid j+i+k:j+p+k\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+1:j+i+1\mid
j+i+k+1:j+p+k+1\right]\right)$
).
Since $v=\left(i,k\right)$ and $g=\operatorname*{Grasp}\nolimits_{j}A$, we
have
$\displaystyle g\left(v\right)$
$\displaystyle=\left(\operatorname*{Grasp}\nolimits_{j}A\right)\left(\left(i,k\right)\right)=\dfrac{\det\left(A\left[j+1:j+i\mid
j+i+k-1:j+p+k\right]\right)}{\det\left(A\left[j:j+i\mid
j+i+k:j+p+k\right]\right)}$ (43) $\displaystyle\ \ \ \ \ \ \ \ \ \
\left(\text{by the definition of }\operatorname*{Grasp}\nolimits_{j}A\right).$
Since $v=\left(i,k\right)$ and $f=\operatorname*{Grasp}\nolimits_{j+1}A$, we
have
$\displaystyle f\left(v\right)$
$\displaystyle=\left(\operatorname*{Grasp}\nolimits_{j+1}A\right)\left(\left(i,k\right)\right)$
$\displaystyle=\dfrac{\det\left(A\left[\left(j+1\right)+1:\left(j+1\right)+i\mid\left(j+1\right)+i+k-1:\left(j+1\right)+p+k\right]\right)}{\det\left(A\left[j+1:\left(j+1\right)+i\mid\left(j+1\right)+i+k:\left(j+1\right)+p+k\right]\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{by the definition of
}\operatorname*{Grasp}\nolimits_{j+1}A\right)$
$\displaystyle=\dfrac{\det\left(A\left[j+2:j+i+1\mid
j+i+k:j+p+k+1\right]\right)}{\det\left(A\left[j+1:j+i+1\mid
j+i+k+1:j+p+k+1\right]\right)}.$ (44)
Now, we can rewrite the terms
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\lessdot v\end{subarray}}f\left(u\right)$,
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}$, $g\left(v\right)$ and
$f\left(v\right)$ in (36) using the equalities (39), (42), (43) and (44),
respectively. The resulting equation is a tautology because all determinants
cancel out (this can be checked by the reader). Hence, (36) is proven in Case
1.
Let us now consider Case 3. In this case, we have $v\neq\left(1,1\right)$ and
$v=\left(p,q\right)$. Hence, (39), (43) and (44) are still valid, whereas (42)
gets superseded by the simpler equality
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\gtrdot
v\end{subarray}}\dfrac{1}{g\left(u\right)}=\dfrac{1}{g\left(1\right)}=\dfrac{1}{1}=1.$
(45)
Now, we can rewrite the terms
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\lessdot v\end{subarray}}f\left(u\right)$,
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}$, $g\left(v\right)$ and
$f\left(v\right)$ in (36) using the equalities (39), (45), (43) and (44),
respectively. The resulting equation (after multiplying through with all
denominators and cancelling terms appearing on both sides) simplifies to
$\displaystyle\det\left(A\left[j+1:j+i+1\mid j+i+k:j+p+k\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+1:j+i\mid
j+i+k:j+p+k+1\right]\right)$ $\displaystyle=\det\left(A\left[j+1:j+i+1\mid
j+i+k+1:j+p+k+1\right]\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j:j+i\mid j+i+k:j+p+k\right]\right).$
Since $i=p$ and $k=q$ (because $\left(i,k\right)=v=\left(p,q\right)$), this
rewrites as
$\displaystyle\det\left(A\left[j+1:j+p+1\mid j+p+q:j+p+q\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+1:j+p\mid
j+p+q:j+p+q+1\right]\right)$ $\displaystyle=\det\left(A\left[j+1:j+p+1\mid
j+p+q+1:j+p+q+1\right]\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j:j+p\mid j+p+q:j+p+q\right]\right).$
But this follows from
$\displaystyle\det\left(A\left[j+1:j+p+1\mid j+p+q:j+p+q\right]\right)$
$\displaystyle=\det\left(A\left[j+1:j+p+1\mid j+p+q+1:j+p+q+1\right]\right)$
(which is clear from Proposition 13.7 (b)) and
$\displaystyle\det\left(A\left[j+1:j+p\mid j+p+q:j+p+q+1\right]\right)$
$\displaystyle=\det\left(A\left[j:j+p\mid j+p+q:j+p+q\right]\right)$
(which can be easily proven333333Proof. We have
$\displaystyle\det\left(A\left[j+1:j+p\mid j+p+q:j+p+q+1\right]\right)$
$\displaystyle=\det\left(A\left[j+1:j+p\mid
p+q+j:p+q+j+1\right]\right)=\det\left(\underbrace{A\left[j:j+1\mid
j+1:j+p\right]}_{\begin{subarray}{c}=A\left[j:j+p\mid j+p+q:j+p+q\right]\\\
\text{(by Proposition \ref{prop.minors.trivial} {(c)})}\end{subarray}}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{by Proposition
\ref{prop.minors.period} {(c)}, applied to }u=p\text{, }v=p+q\text{,
}a=j+1\text{, }b=j+p\text{, }c=j\text{ and }d=j+1\right)$
$\displaystyle=\det\left(A\left[j:j+p\mid j+p+q:j+p+q\right]\right),$ qed.).
Thus, (36) is proven in Case 3.
Let us next consider Case 2. In this case, we have $v=\left(1,1\right)$ and
$v\neq\left(p,q\right)$. Hence, (42), (43) and (44) are still valid, whereas
(39) gets superseded by the simpler equality
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\lessdot v\end{subarray}}f\left(u\right)=f\left(0\right)=1.$ (46)
Now, we can rewrite the terms
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\lessdot v\end{subarray}}f\left(u\right)$,
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,q\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}$, $g\left(v\right)$ and
$f\left(v\right)$ in (36) using the equalities (46), (42), (43) and (44),
respectively. The resulting equation (after multiplying through with all
denominators and cancelling terms appearing on both sides) simplifies to
$\displaystyle\det\left(A\left[j+1:j+i\mid j+i+k-1:j+p+k\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+2:j+i+1\mid
j+i+k:j+p+k+1\right]\right)$ $\displaystyle=\det\left(A\left[j+1:j+i+1\mid
j+i+k:j+p+k\right]\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j+1:j+i\mid j+i+k:j+p+k+1\right]\right).$
Since $i=1$ and $k=1$ (because $\left(i,k\right)=v=\left(1,1\right)$), this
rewrites as
$\displaystyle\det\left(A\left[j+1:j+1\mid j+1+1-1:j+p+1\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+2:j+1+1\mid
j+1+1:j+p+1+1\right]\right)$ $\displaystyle=\det\left(A\left[j+1:j+1+1\mid
j+1+1:j+p+1\right]\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j+1:j+1\mid j+1+1:j+p+1+1\right]\right).$
In other words, this rewrites as
$\displaystyle\det\left(A\left[j+1:j+1\mid j+1:j+p+1\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \cdot\det\left(A\left[j+2:j+2\mid
j+2:j+p+2\right]\right)$ $\displaystyle=\det\left(A\left[j+1:j+2\mid
j+2:j+p+1\right]\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\cdot\det\left(A\left[j+1:j+1\mid j+2:j+p+2\right]\right).$
But this trivially follows from
$\det\left(A\left[j+1:j+1\mid
j+1:j+p+1\right]\right)=\det\left(A\left[j+1:j+2\mid j+2:j+p+1\right]\right)$
(this is because of Proposition 13.6) and
$\det\left(A\left[j+2:j+2\mid
j+2:j+p+2\right]\right)=\det\left(A\left[j+1:j+1\mid j+2:j+p+2\right]\right)$
(this is because of Proposition 13.7 (a)). This proves (36) in Case 2.
We have now proven (36) in each of the Cases 1, 2 and 3\. We leave the proof
in Case 4 to the reader (this case is completely straightforward, since it has
$\left(p,q\right)=v=\left(1,1\right)$). Thus, we now know that (36) holds in
each of the four Cases 1, 2, 3 and 4. Since these four Cases cover all
possibilities, this yields that (36) always holds. As we have seen, this
completes the proof of Proposition 13.13. ∎
A remark seems in order, about why we paid so much attention to the
“degenerate” Cases 2, 3 and 4. Indeed, only in Cases 3 and 4 have we used the
fact that the sequence $\left(A_{n}\right)_{n\in\mathbb{Z}}$ is
“$\left(p+q\right)$-periodic up to sign” rather than just an arbitrary
sequence of length-$p$ column vectors. Had we left out these seemingly
straightforward cases, it would have seemed that the proof showed a result too
good to be true (because it is rather clear that the periodicity in the
definition of $A_{n}$ for general $n\in\mathbb{Z}$ is needed).
## 15 Dominance of the Grassmannian parametrization
Let us show an example before we start proving Proposition 13.14.
###### Example 15.1.
For this example, let $p=2$ and $q=2$. Let
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(2,2\right)}}$ be a generic
reduced labelling. We want to construct a matrix
$A\in\mathbb{K}^{2\times\left(2+2\right)}$ satisfying
$f=\operatorname*{Grasp}\nolimits_{0}A$.
Clearly, the condition $f=\operatorname*{Grasp}\nolimits_{0}A$ imposes $4$
equations on the entries of $A$ (one for every element of
$\operatorname*{Rect}\left(2,2\right)$). Since the matrix $A$ we want to find
has a total of $8$ entries, we are therefore trying to solve an
underdetermined system. However, we can get rid of the superfluous freedom if
we additionally try to ensure that our matrix $A$ has the form
$\left(I_{p}\mid B\right)$ for some $B\in\mathbb{K}^{2\times 2}$ (where
$\left(I_{p}\mid B\right)$ means the matrix obtained from the $p\times p$
identity matrix $I_{p}$ by attaching the matrix $B$ to it on the right). Let
us do this now. So we are looking for a matrix $B\in\mathbb{K}^{2\times 2}$
satisfying $f=\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid B\right)$.
This puts $4$ conditions on $4$ unknowns. Write
$B=\left(\begin{array}[c]{cc}x&y\\\ z&w\end{array}\right)$. Then,
$\left(I_{p}\mid B\right)=\left(\begin{array}[c]{cccc}1&0&x&y\\\
0&1&z&w\end{array}\right)$. Now,
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
B\right)\right)\left(\left(1,1\right)\right)$
$\displaystyle=\dfrac{\det\left(\left(I_{p}\mid B\right)\left[1:1\mid
1:3\right]\right)}{\det\left(\left(I_{p}\mid B\right)\left[0:1\mid
2:3\right]\right)}=\dfrac{\det\left(\begin{array}[c]{cc}1&0\\\
0&1\end{array}\right)}{\det\left(\begin{array}[c]{cc}-y&0\\\
-w&1\end{array}\right)}=\dfrac{-1}{y};$
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
B\right)\right)\left(\left(1,2\right)\right)$
$\displaystyle=\dfrac{\det\left(\left(I_{p}\mid B\right)\left[1:1\mid
2:4\right]\right)}{\det\left(\left(I_{p}\mid B\right)\left[0:1\mid
3:4\right]\right)}=\dfrac{\det\left(\begin{array}[c]{cc}0&x\\\
1&z\end{array}\right)}{\det\left(\begin{array}[c]{cc}-y&x\\\
-w&z\end{array}\right)}=\dfrac{-x}{wx-yz};$
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
B\right)\right)\left(\left(2,1\right)\right)$
$\displaystyle=\dfrac{\det\left(\left(I_{p}\mid B\right)\left[1:2\mid
2:3\right]\right)}{\det\left(\left(I_{p}\mid B\right)\left[0:2\mid
3:3\right]\right)}=\dfrac{\det\left(\begin{array}[c]{cc}1&0\\\
0&1\end{array}\right)}{\det\left(\begin{array}[c]{cc}-y&1\\\
-w&0\end{array}\right)}=\dfrac{1}{w};$
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
B\right)\right)\left(\left(2,2\right)\right)$
$\displaystyle=\dfrac{\det\left(\left(I_{p}\mid B\right)\left[1:2\mid
3:4\right]\right)}{\det\left(\left(I_{p}\mid B\right)\left[0:2\mid
4:4\right]\right)}=\dfrac{\det\left(\begin{array}[c]{cc}1&x\\\
0&z\end{array}\right)}{\det\left(\begin{array}[c]{cc}-y&1\\\
-w&0\end{array}\right)}=\dfrac{z}{w}.$
The requirement $f=\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid B\right)$
therefore translates into the system
$\left\\{\begin{array}[c]{lcl}f\left(\left(1,1\right)\right)&=&\dfrac{-1}{y};\\\
f\left(\left(1,2\right)\right)&=&\dfrac{-x}{wx-yz};\\\
f\left(\left(2,1\right)\right)&=&\dfrac{1}{w};\\\\[9.0pt]
f\left(\left(2,2\right)\right)&=&\dfrac{z}{w}\end{array}\right..$
This system can be solved by elimination: First, compute $w$ using
$f\left(\left(2,1\right)\right)=\dfrac{1}{w}$, obtaining
$w=\dfrac{1}{f\left(\left(2,1\right)\right)}$; then, compute $y$ using
$f\left(\left(1,1\right)\right)=\dfrac{-1}{y}$, obtaining
$y=\dfrac{-1}{f\left(\left(1,1\right)\right)}$; then, compute $z$ using
$f\left(\left(2,2\right)\right)=\dfrac{z}{w}$ and the already eliminated $w$,
obtaining
$z=\dfrac{f\left(\left(2,2\right)\right)}{f\left(\left(2,1\right)\right)}$;
finally, compute $x$ using $f\left(\left(1,2\right)\right)=\dfrac{-x}{wx-yz}$
and the already eliminated $w,y,z$, obtaining
$x=\dfrac{-f\left(\left(1,2\right)\right)f\left(\left(2,2\right)\right)}{\left(f\left(\left(1,2\right)\right)+f\left(\left(2,1\right)\right)\right)f\left(\left(1,1\right)\right)}$.
While the denominators in these fractions can vanish, leading to
underdetermination or unsolvability, this will not happen for generic $f$.
This approach to solving $f=\operatorname*{Grasp}\nolimits_{0}A$ generalizes
to arbitrary $p$ and $q$, and motivates the following proof.
We are now going to outline the proof of Proposition 13.14. As shouldn’t be
surprising after Example 15.1, the underlying idea of the proof is the
following: For any fixed
$f\in\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$, the equation
$f=\operatorname*{Grasp}\nolimits_{0}A$ (with $A$ an unknown matrix in
$\mathbb{K}^{p\times\left(p+q\right)}$) can be considered as a system of $pq$
equations on $p\left(p+q\right)$ unknowns (the entries of $A$). While this
system is usually underdetermined, we can restrict the entries of $A$ by
requiring that the leftmost $p$ columns of $A$ form the $p\times p$ identity
matrix. Upon this restriction, we are left with $pq$ unknowns only, and for
$f$ sufficiently generic, the resulting system will be uniquely solvable by
“triangular elimination” (i.e., there is an equation containing only one
unknown; then, when this unknown is eliminated, the resulting system again
contains an equation with only one unknown, and once this one is eliminated,
one gets a further system containing an equation with only one unknown, and so
forth) – like a triangular system of linear equations with nonzero entries on
the diagonal, but without the linearity. Of course, this is not a complete
proof because the applicability of “triangular elimination” has to be proven,
not merely claimed. We are only going to sketch the ideas of this proof,
leaving all straightforward details to the reader to fill in. For the sake of
clarity, we are going to word the argument using algebraic properties of
families of rational functions instead of using the algorithmic nature of
“triangular elimination” (similarly to how most applications of linear algebra
use the language of bases of vector spaces rather than talk about the process
of solving systems by Gaussian elimination). While this clarity comes at the
cost of a slight disconnect from the motivation of the proof, we hope that the
reader will still see how the wind blows.
We first introduce some notation to capture the essence of “triangular
elimination” without having to talk about actually moving around variables in
equations:
###### Definition 15.2.
Let $\mathbb{F}$ be a field. Let $\mathbf{P}$ be a finite set.
(a) Let $x_{\mathbf{p}}$ be a new symbol for every $\mathbf{p}\in\mathbf{P}$.
We will denote by $\mathbb{F}\left(x_{\mathbf{P}}\right)$ the field of
rational functions over $\mathbb{F}$ in the indeterminates $x_{\mathbf{p}}$
with $\mathbf{p}$ ranging over all elements of $\mathbf{P}$ (hence altogether
$\left|\mathbf{P}\right|$ indeterminates). We also will denote by
$\mathbb{F}\left[x_{\mathbf{P}}\right]$ the ring of polynomials over
$\mathbb{F}$ in the indeterminates $x_{\mathbf{p}}$ with $\mathbf{p}$ ranging
over all elements of $\mathbf{P}$. (Thus,
$\mathbb{F}\left(x_{\mathbf{P}}\right)=\mathbb{F}\left(x_{\mathbf{p}_{1}},x_{\mathbf{p}_{2}},...,x_{\mathbf{p}_{n}}\right)$
and
$\mathbb{F}\left[x_{\mathbf{P}}\right]=\mathbb{F}\left[x_{\mathbf{p}_{1}},x_{\mathbf{p}_{2}},...,x_{\mathbf{p}_{n}}\right]$
if $\mathbf{P}$ is written in the form
$\mathbf{P}=\left\\{\mathbf{p}_{1},\mathbf{p}_{2},...,\mathbf{p}_{n}\right\\}$.)
The symbols $x_{\mathbf{p}}$ are understood to be distinct, and are used as
commuting indeterminates. We regard $\mathbb{F}\left[x_{\mathbf{P}}\right]$ as
a subring of $\mathbb{F}\left(x_{\mathbf{P}}\right)$, and
$\mathbb{F}\left(x_{\mathbf{P}}\right)$ as the field of quotients of
$\mathbb{F}\left[x_{\mathbf{P}}\right]$.
(b) If $\mathbf{Q}$ is a subset of $\mathbf{P}$, then
$\mathbb{F}\left(x_{\mathbf{Q}}\right)$ can be canonically embedded into
$\mathbb{F}\left(x_{\mathbf{P}}\right)$, and
$\mathbb{F}\left[x_{\mathbf{Q}}\right]$ can be canonically embedded into
$\mathbb{F}\left[x_{\mathbf{P}}\right]$. We regard these embeddings as
inclusions.
(c) Let $\mathbb{K}$ be a field extension of $\mathbb{F}$. Let $f$ be an
element of $\mathbb{F}\left(x_{\mathbf{P}}\right)$. If
$\left(a_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\mathbb{K}^{\mathbf{P}}$
is a family of elements of $\mathbb{K}$ indexed by elements of $\mathbf{P}$,
then we let
$f\left(\left(a_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)$ denote
the element of $\mathbb{K}$ obtained by substituting $a_{\mathbf{p}}$ for
$x_{\mathbf{p}}$ for each $\mathbf{p}\in\mathbf{P}$ in the rational function
$f$. This
$f\left(\left(a_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)$ is
defined only if the substitution does not render the denominator equal to $0$.
If $\mathbb{K}$ is infinite, this shows that
$f\left(\left(a_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)$ is
defined for almost all
$\left(a_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\mathbb{K}^{\mathbf{P}}$
(with respect to the Zariski topology).
(d) Let $\mathbf{P}$ now be a finite totally ordered set, and let
$\vartriangleleft$ be the smaller relation of $\mathbf{P}$. For every
$\mathbf{p}\in\mathbf{P}$, let $\mathbf{p}\Downarrow$ denote the subset
$\left\\{\mathbf{v}\in\mathbf{P}\ \mid\
\mathbf{v}\vartriangleleft\mathbf{p}\right\\}$ of $\mathbf{P}$. For every
$\mathbf{p}\in\mathbf{P}$, let $Q_{\mathbf{p}}$ be an element of
$\mathbb{F}\left(x_{\mathbf{P}}\right)$.
We say that the family $\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$
is $\mathbf{P}$-triangular if and only if the following condition holds:
Algebraic triangularity condition: For every $\mathbf{p}\in\mathbf{P}$, there
exist elements $\alpha_{\mathbf{p}}$, $\beta_{\mathbf{p}}$,
$\gamma_{\mathbf{p}}$, $\delta_{\mathbf{p}}$ of
$\mathbb{F}\left(x_{\mathbf{p}\Downarrow}\right)$ such that
$\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$ and
$Q_{\mathbf{p}}=\dfrac{\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}}{\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}}$.
343434Notice that the fraction
$\dfrac{\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}}{\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}}$
is well-defined for any four elements $\alpha_{\mathbf{p}}$,
$\beta_{\mathbf{p}}$, $\gamma_{\mathbf{p}}$, $\delta_{\mathbf{p}}$ of
$\mathbb{F}\left(x_{\mathbf{p}\Downarrow}\right)$ such that
$\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$. (Indeed, $\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}\neq 0$ in
this case, as can easily be checked.)
We will use $\mathbf{P}$-triangularity via the following fact:
###### Lemma 15.3.
Let $\mathbb{F}$ be a field. Let $\mathbf{P}$ be a finite totally ordered set.
For every $\mathbf{p}\in\mathbf{P}$, let $Q_{\mathbf{p}}$ be an element of
$\mathbb{F}\left(x_{\mathbf{P}}\right)$. Assume that
$\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$ is a
$\mathbf{P}$-triangular family. Then:
(a) The family
$\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
is algebraically independent (over $\mathbb{F}$).
(b) There exists a $\mathbf{P}$-triangular family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
such that every $\mathbf{q}\in\mathbf{P}$ satisfies
$Q_{\mathbf{q}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=x_{\mathbf{q}}$.
###### Proof of Lemma 15.3 (sketched)..
As in the definition of $\mathbf{P}$-triangularity, we let
$\mathbf{p}\Downarrow$ denote the subset $\left\\{\mathbf{v}\in\mathbf{P}\
\mid\ \mathbf{v}\vartriangleleft\mathbf{p}\right\\}$ of $\mathbf{P}$ for every
$\mathbf{p}\in\mathbf{P}$.
(a) Assume that the family
$\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
is not algebraically independent (over $\mathbb{F}$). Then, some nonzero
polynomial $P\in\mathbb{F}\left[x_{\mathbf{P}}\right]$ satisfies
$P\left(\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=0$. Fix
such a $P$, and let $\mathbf{u}$ be the maximal (with respect to the order on
$\mathbf{P}$) element of $\mathbf{P}$ such that $x_{\mathbf{u}}$ appears in
$P$ (meaning that the degree of $P$ with respect to the variable
$x_{\mathbf{u}}$ is $>0$). Then, $P$ can be construed as a non-constant
polynomial in the variable $x_{\mathbf{u}}$ over the ring
$\mathbb{F}\left[x_{\mathbf{u}\Downarrow}\right]$. Hence,
$P\left(\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=0$ shows
that $Q_{\mathbf{u}}$ is algebraic over the subfield of
$\mathbb{F}\left(x_{\mathbf{P}}\right)$ generated by the elements
$Q_{\mathbf{v}}$ for $\mathbf{v}\in\left.\mathbf{u}\Downarrow\right.$.
Now, notice that every $\mathbf{v}\in\left.\mathbf{u}\Downarrow\right.$
satisfies $Q_{\mathbf{v}}\in\mathbb{F}\left(x_{\mathbf{u}\Downarrow}\right)$
353535Proof. Let $\mathbf{v}\in\left.\mathbf{u}\Downarrow\right.$. Then,
$\mathbf{v}\vartriangleleft\mathbf{u}$, so that
$\left.\mathbf{v}\Downarrow\right.\subseteq\left.\mathbf{u}\Downarrow\right.$,
hence
$\mathbb{F}\left(x_{\mathbf{v}\Downarrow}\right)\subseteq\mathbb{F}\left(x_{\mathbf{u}\Downarrow}\right)$.
By the algebraic triangularity condition, we know that there exist elements
$\alpha_{\mathbf{v}}$, $\beta_{\mathbf{v}}$, $\gamma_{\mathbf{v}}$,
$\delta_{\mathbf{v}}$ of $\mathbb{F}\left(x_{\mathbf{v}\Downarrow}\right)$
such that
$\alpha_{\mathbf{v}}\delta_{\mathbf{v}}-\beta_{\mathbf{v}}\gamma_{\mathbf{v}}\neq
0$ and
$Q_{\mathbf{v}}=\dfrac{\alpha_{\mathbf{v}}x_{\mathbf{v}}+\beta_{\mathbf{v}}}{\gamma_{\mathbf{v}}x_{\mathbf{v}}+\delta_{\mathbf{v}}}$.
These elements $\alpha_{\mathbf{v}}$, $\beta_{\mathbf{v}}$,
$\gamma_{\mathbf{v}}$, $\delta_{\mathbf{v}}$ belong to
$\mathbb{F}\left(x_{\mathbf{u}\Downarrow}\right)$ (by virtue of lying in
$\mathbb{F}\left(x_{\mathbf{v}\Downarrow}\right)\subseteq\mathbb{F}\left(x_{\mathbf{u}\Downarrow}\right)$),
and so does $x_{\mathbf{v}}$ (since
$\mathbf{v}\in\left.\mathbf{u}\Downarrow\right.$). Hence, the fraction
$\dfrac{\alpha_{\mathbf{v}}x_{\mathbf{v}}+\beta_{\mathbf{v}}}{\gamma_{\mathbf{v}}x_{\mathbf{v}}+\delta_{\mathbf{v}}}$
also lies in $\mathbb{F}\left(x_{\mathbf{u}\Downarrow}\right)$. Since this
fraction is $Q_{\mathbf{v}}$, we thus have shown
$Q_{\mathbf{v}}\in\mathbb{F}\left(x_{\mathbf{u}\Downarrow}\right)$, qed..
Hence, the subfield of $\mathbb{F}\left(x_{\mathbf{P}}\right)$ generated by
the elements $Q_{\mathbf{v}}$ for
$\mathbf{v}\in\left.\mathbf{u}\Downarrow\right.$ is a subfield of
$\mathbb{F}\left(x_{\mathbf{u}\Downarrow}\right)$. Since $Q_{\mathbf{u}}$ is
algebraic over the former field, we thus conclude that $Q_{\mathbf{u}}$ is
“all the more” algebraic over the latter field. But by the algebraic
triangularity condition, there exist elements $\alpha_{\mathbf{u}}$,
$\beta_{\mathbf{u}}$, $\gamma_{\mathbf{u}}$, $\delta_{\mathbf{u}}$ of
$\mathbb{F}\left(x_{\mathbf{u}\Downarrow}\right)$ such that
$\alpha_{\mathbf{u}}\delta_{\mathbf{u}}-\beta_{\mathbf{u}}\gamma_{\mathbf{u}}\neq
0$ and
$Q_{\mathbf{u}}=\dfrac{\alpha_{\mathbf{u}}x_{\mathbf{u}}+\beta_{\mathbf{u}}}{\gamma_{\mathbf{u}}x_{\mathbf{u}}+\delta_{\mathbf{u}}}$.
We can easily solve the equation
$Q_{\mathbf{u}}=\dfrac{\alpha_{\mathbf{u}}x_{\mathbf{u}}+\beta_{\mathbf{u}}}{\gamma_{\mathbf{u}}x_{\mathbf{u}}+\delta_{\mathbf{u}}}$
for $x_{\mathbf{u}}$ and obtain
$x_{\mathbf{u}}=\dfrac{Q_{\mathbf{u}}\delta_{\mathbf{u}}-\beta_{\mathbf{u}}}{\alpha_{\mathbf{u}}-Q_{\mathbf{u}}\gamma_{\mathbf{u}}}$
(and the denominator here does not vanish because of
$\alpha_{\mathbf{u}}\delta_{\mathbf{u}}-\beta_{\mathbf{u}}\gamma_{\mathbf{u}}\neq
0$). Therefore, $x_{\mathbf{u}}$ is algebraic over the field
$\mathbb{F}\left(x_{\mathbf{u}\Downarrow}\right)$ (because we know
$Q_{\mathbf{u}}$ to be algebraic over this field, whereas
$\alpha_{\mathbf{u}}$, $\beta_{\mathbf{u}}$, $\gamma_{\mathbf{u}}$,
$\delta_{\mathbf{u}}$ lie in that field). But this is absurd since
$\mathbf{u}\notin\left.\mathbf{u}\Downarrow\right.$. This contradiction shows
that our assumption was wrong, and Lemma 15.3 (a) is proven.
(b) We will construct the required family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
by induction. Of course, this is trivial if $\mathbf{P}=\varnothing$, so let
us assume that $\mathbf{P}$ is nonempty. Let $\mathbf{m}$ be the maximum
element of $\mathbf{P}$, and let us assume that we have already constructed a
$\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}$-triangular family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\in\left(\mathbb{F}\left(x_{\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)\right)^{\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}$
such that every $\mathbf{q}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}$
satisfies
$Q_{\mathbf{q}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)=x_{\mathbf{q}}$.
We now only need to find an element
$R_{\mathbf{m}}\in\mathbb{F}\left(x_{\mathbf{P}}\right)$ such that the
resulting family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
will be $\mathbf{P}$-triangular and satisfy
$Q_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=x_{\mathbf{m}}$.
Since $\mathbf{m}$ is maximum, we have
$\left.\mathbf{m}\Downarrow\right.=\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}$.
We know that the family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}$
is $\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}$-triangular. Hence, Lemma
15.3 (a) (applied to this family) yields that the family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}$
is algebraically independent. This yields that it can be substituted into any
rational function in
$\mathbb{F}\left(x_{\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)$
(without running the risk of denominators becoming $0$).
The family $\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$ is
$\mathbf{P}$-triangular, so that (by the algebraic triangularity condition)
there exist elements $\alpha_{\mathbf{m}}$, $\beta_{\mathbf{m}}$,
$\gamma_{\mathbf{m}}$, $\delta_{\mathbf{m}}$ of
$\mathbb{F}\left(x_{\mathbf{m}\Downarrow}\right)$ such that
$\alpha_{\mathbf{m}}\delta_{\mathbf{m}}-\beta_{\mathbf{m}}\gamma_{\mathbf{m}}\neq
0$ and
$Q_{\mathbf{m}}=\dfrac{\alpha_{\mathbf{m}}x_{\mathbf{m}}+\beta_{\mathbf{m}}}{\gamma_{\mathbf{m}}x_{\mathbf{m}}+\delta_{\mathbf{m}}}$.
Consider these $\alpha_{\mathbf{m}}$, $\beta_{\mathbf{m}}$,
$\gamma_{\mathbf{m}}$, $\delta_{\mathbf{m}}$. Now, define four elements
$\alpha_{\mathbf{m}}^{\prime}$, $\beta_{\mathbf{m}}^{\prime}$,
$\gamma_{\mathbf{m}}^{\prime}$, $\delta_{\mathbf{m}}^{\prime}$ of
$\mathbb{F}\left(x_{\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)$ by
$\displaystyle\alpha_{\mathbf{m}}^{\prime}$
$\displaystyle=\delta_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right),\
\ \ \ \ \ \ \ \ \
\beta_{\mathbf{m}}^{\prime}=-\beta_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right),$
$\displaystyle\gamma_{\mathbf{m}}^{\prime}$
$\displaystyle=-\gamma_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right),\
\ \ \ \ \ \ \ \ \
\delta_{\mathbf{m}}^{\prime}=\alpha_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right).$
Note that these are well-defined (because $\alpha_{\mathbf{m}}$,
$\beta_{\mathbf{m}}$, $\gamma_{\mathbf{m}}$, $\delta_{\mathbf{m}}$ belong to
$\mathbb{F}\left(x_{\mathbf{m}\Downarrow}\right)=\mathbb{F}\left(x_{\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)$
and because the family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}$
is algebraically independent) and belong to
$\mathbb{F}\left(x_{\mathbf{m}\Downarrow}\right)$ (since
$\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}=\left.\mathbf{m}\Downarrow\right.$).
They furthermore satisfy
$\displaystyle\alpha_{\mathbf{m}}^{\prime}\delta_{\mathbf{m}}^{\prime}-\beta_{\mathbf{m}}^{\prime}\gamma_{\mathbf{m}}^{\prime}$
$\displaystyle=\delta_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)\cdot\alpha_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)-\left(-\beta_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)\right)\left(-\gamma_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)\right)$
$\displaystyle=\underbrace{\left(\delta_{\mathbf{m}}\alpha_{\mathbf{m}}-\left(-\beta_{\mathbf{m}}\right)\left(-\gamma_{\mathbf{m}}\right)\right)}_{=\alpha_{\mathbf{m}}\delta_{\mathbf{m}}-\beta_{\mathbf{m}}\gamma_{\mathbf{m}}\neq
0}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)\neq
0$
(since
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}$
is algebraically independent). Let us now define
$R_{\mathbf{m}}=\dfrac{\alpha_{\mathbf{m}}^{\prime}x_{\mathbf{m}}+\beta_{\mathbf{m}}^{\prime}}{\gamma_{\mathbf{m}}^{\prime}x_{\mathbf{m}}+\delta_{\mathbf{m}}^{\prime}}$.
(This is easily seen to be well-defined because
$\alpha_{\mathbf{m}}^{\prime}\delta_{\mathbf{m}}^{\prime}-\beta_{\mathbf{m}}^{\prime}\gamma_{\mathbf{m}}^{\prime}\neq
0$ entails
$\left(\gamma_{\mathbf{m}}^{\prime},\delta_{\mathbf{m}}^{\prime}\right)\neq\left(0,0\right)$.)
Since the family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}$
is already $\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}$-triangular, and
because of the fact that $\alpha_{\mathbf{m}}^{\prime}$,
$\beta_{\mathbf{m}}^{\prime}$, $\gamma_{\mathbf{m}}^{\prime}$,
$\delta_{\mathbf{m}}^{\prime}$ are elements of
$\mathbb{F}\left(x_{\mathbf{m}\Downarrow}\right)$ satisfying
$\alpha_{\mathbf{m}}^{\prime}\delta_{\mathbf{m}}^{\prime}-\beta_{\mathbf{m}}^{\prime}\gamma_{\mathbf{m}}^{\prime}\neq
0$ and
$R_{\mathbf{m}}=\dfrac{\alpha_{\mathbf{m}}^{\prime}x_{\mathbf{m}}+\beta_{\mathbf{m}}^{\prime}}{\gamma_{\mathbf{m}}^{\prime}x_{\mathbf{m}}+\delta_{\mathbf{m}}^{\prime}}$,
we see that the family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
is $\mathbf{P}$-triangular. We are now going to prove that
$Q_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=x_{\mathbf{m}}$,
and then we will be done.
Since
$Q_{\mathbf{m}}=\dfrac{\alpha_{\mathbf{m}}x_{\mathbf{m}}+\beta_{\mathbf{m}}}{\gamma_{\mathbf{m}}x_{\mathbf{m}}+\delta_{\mathbf{m}}}$,
we have
$Q_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=\dfrac{\alpha_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)R_{\mathbf{m}}+\beta_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)}{\gamma_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)R_{\mathbf{m}}+\delta_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)}.$
(47)
But
$\alpha_{\mathbf{m}}\in\mathbb{F}\left(x_{\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)$,
so that the variable $x_{\mathbf{m}}$ does not appear in $\alpha_{\mathbf{m}}$
at all. Hence,
$\alpha_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=\alpha_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}\setminus\left\\{\mathbf{m}\right\\}}\right)=\delta_{\mathbf{m}}^{\prime}$.
Using this and the similarly proven equalities
$\beta_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=-\beta_{\mathbf{m}}^{\prime}$,
$\gamma_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=-\gamma_{\mathbf{m}}^{\prime}$
and
$\delta_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=\alpha_{\mathbf{m}}^{\prime}$,
we can rewrite the equality (47) as
$Q_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=\dfrac{\delta_{\mathbf{m}}^{\prime}R_{\mathbf{m}}-\beta_{\mathbf{m}}^{\prime}}{-\gamma_{\mathbf{m}}^{\prime}R_{\mathbf{m}}+\alpha_{\mathbf{m}}^{\prime}}.$
But the right hand side of this equality simplifies to $x_{\mathbf{m}}$ if we
recall that
$R_{\mathbf{m}}=\dfrac{\alpha_{\mathbf{m}}^{\prime}x_{\mathbf{m}}+\beta_{\mathbf{m}}^{\prime}}{\gamma_{\mathbf{m}}^{\prime}x_{\mathbf{m}}+\delta_{\mathbf{m}}^{\prime}}$
(the proof of this is mechanical, using no properties of
$\alpha_{\mathbf{m}}^{\prime}$, $\beta_{\mathbf{m}}^{\prime}$,
$\gamma_{\mathbf{m}}^{\prime}$, $\delta_{\mathbf{m}}^{\prime}$ and
$x_{\mathbf{m}}$ other than lying in a field). Hence, we have shown that
$Q_{\mathbf{m}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=x_{\mathbf{m}}$.
As explained above, this completes the (inductive) proof of Lemma 15.3 (b). ∎
We now can proceed to the proof of Proposition 13.14:
###### Proof of Proposition 13.14 (sketched)..
Let $\mathbb{F}$ be the prime field of $\mathbb{K}$. (This means either
$\mathbb{Q}$ or $\mathbb{F}_{p}$ depending on the characteristic of
$\mathbb{K}$.) In the following, the word “algebraically independent” will
always mean “algebraically independent over $\mathbb{F}$” (rather than over
$\mathbb{K}$ or over $\mathbb{Z}$).
Let $\mathbf{P}$ be a totally ordered set such that
$\mathbf{P}=\left\\{1,2,...,p\right\\}\times\left\\{1,2,...,q\right\\}\text{
as sets,}$
and such that
$\left(i,k\right)\trianglelefteq\left(i^{\prime},k^{\prime}\right)\text{ for
all }\left(i,k\right)\in\mathbf{P}\text{ and
}\left(i^{\prime},k^{\prime}\right)\in\mathbf{P}\text{ satisfying
}\left(i\geqslant i^{\prime}\text{ and }k\leqslant k^{\prime}\right),$
where $\trianglelefteq$ denotes the smaller-or-equal relation of $\mathbf{P}$.
Such a $\mathbf{P}$ clearly exists (in fact, there usually exist several such
$\mathbf{P}$, and it doesn’t matter which of them we choose). We denote the
smaller relation of $\mathbf{P}$ by $\vartriangleleft$. We will later see what
this total order is good for (intuitively, it is an order in which the
variables can be eliminated; in other words, it makes our system behave like a
triangular matrix rather than like a triangular matrix with permuted columns),
but for now let us notice that it is generally not compatible with
$\operatorname*{Rect}\left(p,q\right)$.
Let $Z:\left\\{1,2,...,q\right\\}\rightarrow\left\\{1,2,...,q\right\\}$ denote
the map which sends every $k\in\left\\{1,2,...,q-1\right\\}$ to $k+1$ and
sends $q$ to $1$. Thus, $Z$ is a permutation in the symmetric group $S_{q}$,
and can be written in cycle notation as $\left(1,2,...,q\right)$.
Consider the field $\mathbb{F}\left(x_{\mathbf{P}}\right)$ and the ring
$\mathbb{F}\left[x_{\mathbf{P}}\right]$ defined as in Definition 15.2.
Recall that we need to prove Proposition 13.14. In other words, we need to
show that for almost every
$f\in\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$, there exists a matrix
$A\in\mathbb{K}^{p\times\left(p+q\right)}$ satisfying
$f=\operatorname*{Grasp}\nolimits_{0}A$.
In order to prove this, it is enough to show that there exists a matrix
$\widetilde{D}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{p\times\left(p+q\right)}$
satisfying
$x_{\mathbf{p}}=\left(\operatorname*{Grasp}\nolimits_{0}\widetilde{D}\right)\left(\mathbf{p}\right)\
\ \ \ \ \ \ \ \ \ \text{for every }\mathbf{p}\in\mathbf{P}\text{.}$ (48)
Indeed, once the existence of such a matrix $\widetilde{D}$ is proven, we will
be able to obtain a matrix $A\in\mathbb{K}^{p\times\left(p+q\right)}$
satisfying $f=\operatorname*{Grasp}\nolimits_{0}A$ for almost every
$f\in\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$ simply by substituting
$f\left(\mathbf{p}\right)$ for every $x_{\mathbf{p}}$ in all entries of the
matrix $\widetilde{D}$ 363636Indeed, this matrix $A$ (obtained by substitution
of $f\left(\mathbf{p}\right)$ for $x_{\mathbf{p}}$) will be well-defined for
almost every $f\in\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$ (the
“almost” is due to the possibility of some denominators becoming $0$), and
will satisfy
$f\left(\mathbf{p}\right)=\left(\operatorname*{Grasp}\nolimits_{0}A\right)\left(\mathbf{p}\right)$
for every $\mathbf{p}\in\mathbf{P}$ (because $\widetilde{D}$ satisfies (48)),
that is, $f=\operatorname*{Grasp}\nolimits_{0}A$.. Hence, all we need to show
is the existence of a matrix
$\widetilde{D}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{p\times\left(p+q\right)}$
satisfying (48).
Define a matrix
$C\in\left(\mathbb{F}\left[x_{\mathbf{P}}\right]\right)^{p\times q}$ by
$C=\left(x_{\left(i,Z\left(k\right)\right)}\right)_{1\leqslant i\leqslant p,\
1\leqslant k\leqslant q}.$
This is simply a matrix whose entries are all the indeterminates
$x_{\mathbf{p}}$ of the polynomial ring
$\mathbb{F}\left[x_{\mathbf{P}}\right]$, albeit in a strange order. (The
order, again, is tailored to make the “triangularity” argument work nicely.
This matrix $C$ is not going to be directly related to the $\widetilde{D}$ we
will construct, but will be used in its construction.)
For every $\left(i,k\right)\in\mathbf{P}$, define an element
$\mathfrak{N}_{\left(i,k\right)}\in\mathbb{F}\left[x_{\mathbf{P}}\right]$ by
$\mathfrak{N}_{\left(i,k\right)}=\det\left(\left(I_{p}\mid
C\right)\left[1:i\mid i+k-1:p+k\right]\right).$ (49)
For every $\left(i,k\right)\in\mathbf{P}$, define an element
$\mathfrak{D}_{\left(i,k\right)}\in\mathbb{F}\left[x_{\mathbf{P}}\right]$ by
$\mathfrak{D}_{\left(i,k\right)}=\det\left(\left(I_{p}\mid
C\right)\left[0:i\mid i+k:p+k\right]\right).$ (50)
Our plan from here is the following:
Step 1: We will find alternate expressions for the polynomials
$\mathfrak{N}_{\left(i,k\right)}$ and $\mathfrak{D}_{\left(i,k\right)}$ which
will give us a better idea of what variables occur in these polynomials.
Step 2: We will show that $\mathfrak{N}_{\left(i,k\right)}$ and
$\mathfrak{D}_{\left(i,k\right)}$ are nonzero for all
$\left(i,k\right)\in\mathbf{P}$.
Step 3: We will define a
$Q_{\mathbf{p}}\in\mathbb{F}\left(x_{\mathbf{P}}\right)$ for every
$\mathbf{p}\in\mathbf{P}$ by
$Q_{\mathbf{p}}=\dfrac{\mathfrak{N}_{\mathbf{p}}}{\mathfrak{D}_{\mathbf{p}}}$,
and we will show that
$Q_{\mathbf{p}}=\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
C\right)\right)\left(\mathbf{p}\right)$.
Step 4: We will prove that the family
$\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
is $\mathbf{P}$-triangular.
Step 5: We will use Lemma 15.3 (b) and the result of Step 4 to find a matrix
$\widetilde{D}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{p\times\left(p+q\right)}$
satisfying (48).
Let us now go into detail on each specific step (although we won’t take that
detail very far).
Details of Step 1: Let us introduce three more pieces of notation pertaining
to matrices:
* •
If $\ell\in\mathbb{N}$, and if $A_{1}$, $A_{2}$, $...$, $A_{k}$ are several
matrices with $\ell$ rows each, then $\left(A_{1}\mid A_{2}\mid...\mid
A_{k}\right)$ will denote the matrix obtained by starting with an (empty)
$\ell\times 0$-matrix, then attaching the matrix $A_{1}$ to it on the right,
then attaching the matrix $A_{2}$ to the result on the right, etc., and
finally attaching the matrix $A_{k}$ to the result on the right. For example,
if $p$ is a nonnegative integer, and $B$ is a matrix with $p$ rows, then
$\left(I_{p}\mid B\right)$ means the matrix obtained from the $p\times p$
identity matrix $I_{p}$ by attaching the matrix $B$ to it on the right. (As a
concrete example, $\left(I_{2}\mid\left(\begin{array}[c]{cc}1&-2\\\
3&0\end{array}\right)\right)=\left(\begin{array}[c]{cccc}1&0&1&-2\\\
0&1&3&0\end{array}\right)$.)
* •
If $\ell\in\mathbb{N}$, if $B$ is a matrix with $\ell$ rows, and if $i_{1}$,
$i_{2}$, $...$, $i_{k}$ are some elements of $\left\\{1,2,...,\ell\right\\}$,
then $\operatorname*{rows}\nolimits_{i_{1},i_{2},...,i_{k}}B$ will denote the
matrix whose rows (from top to bottom) are the rows labelled $i_{1}$, $i_{2}$,
$...$, $i_{k}$ of the matrix $B$.
* •
If $u$ and $v$ are two nonnegative integers, and $A$ is a $u\times v$-matrix,
then, for any two integers $a$ and $b$ satisfying $a\leqslant b$, we let
$A\left[a:b\right]$ be the matrix whose columns (from left to right) are
$A_{a}$, $A_{a+1}$, $...$, $A_{b-1}$. This is a natural extension of the
notation introduced in Definition 13.1 (c) (or, rather, the latter notation is
a natural extension of the definition we just made) and has the obvious
property that if $a$, $b$ and $c$ are integers satisfying $a\leqslant
b\leqslant c$, then $A\left[a:c\right]=A\left[a:b\mid b:c\right]$.
We will use without proof a standard fact about determinants:
* •
Given a commutative ring $\mathbb{L}$, two nonnegative integers $a$ and $b$
satisfying $a\geqslant b$, and a matrix $U\in\mathbb{L}^{a\times b}$, we have
$\det\left(\left(\begin{array}[c]{c}I_{a-b}\\\
0_{b\times\left(a-b\right)}\end{array}\right)\mid
U\right)=\det\left(\operatorname*{rows}\nolimits_{a-b+1,a-b+2,...,a}U\right)$
(51)
and
$\det\left(\left(\begin{array}[c]{c}0_{b\times\left(a-b\right)}\\\
I_{a-b}\end{array}\right)\mid
U\right)=\left(-1\right)^{b\left(a-b\right)}\det\left(\operatorname*{rows}\nolimits_{1,2,...,b}U\right).$
(52)
(Here, $0_{u\times v}$ denotes the $u\times v$ zero matrix for all
$u\in\mathbb{N}$ and $v\in\mathbb{N}$, and
$\left(\begin{array}[c]{c}I_{a-b}\\\
0_{b\times\left(a-b\right)}\end{array}\right)$ and
$\left(\begin{array}[c]{c}0_{b\times\left(a-b\right)}\\\
I_{a-b}\end{array}\right)$ are to be read as block matrices.)
Now,
$\left(I_{p}\mid C\right)\left[1:i\mid
i+k-1:p+k\right]=\left(\left(\begin{array}[c]{c}I_{i-1}\\\
0_{\left(p-\left(i-1\right)\right)\times\left(i-1\right)}\end{array}\right)\
\mid\ \left(I_{p}\mid C\right)\left[i+k-1:p+k\right]\right),$
so that
$\displaystyle\det\left(\left(I_{p}\mid C\right)\left[1:i\mid
i+k-1:p+k\right]\right)$
$\displaystyle=\det\left(\left(\begin{array}[c]{c}I_{i-1}\\\
0_{\left(p-\left(i-1\right)\right)\times\left(i-1\right)}\end{array}\right)\
\mid\ \left(I_{p}\mid C\right)\left[i+k-1:p+k\right]\right)$
$\displaystyle=\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)\right)$
(by (51)). Thus,
$\displaystyle\mathfrak{N}_{\left(i,k\right)}$
$\displaystyle=\det\left(\left(I_{p}\mid C\right)\left[1:i\mid
i+k-1:p+k\right]\right)$
$\displaystyle=\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)\right).$ (53)
Also,
$\displaystyle\left(I_{p}\mid C\right)\left[0:i\mid i+k:p+k\right]$
$\displaystyle=\left(\underbrace{\left(I_{p}\mid
C\right)_{0}}_{\begin{subarray}{c}=\left(-1\right)^{p-1}C_{q}\\\ \text{(by
Definition \ref{def.minors} {(b)})}\end{subarray}}\ \mid\
\left(\begin{array}[c]{c}I_{i-1}\\\
0_{\left(p-\left(i-1\right)\right)\times\left(i-1\right)}\end{array}\right)\
\mid\ \left(I_{p}\mid C\right)\left[i+k:p+k\right]\right)$
$\displaystyle=\left(\left(-1\right)^{p-1}C_{q}\ \mid\
\left(\begin{array}[c]{c}I_{i-1}\\\
0_{\left(p-\left(i-1\right)\right)\times\left(i-1\right)}\end{array}\right)\
\mid\ \left(I_{p}\mid C\right)\left[i+k:p+k\right]\right),$
whence
$\displaystyle\det\left(\left(I_{p}\mid C\right)\left[0:i\mid
i+k:p+k\right]\right)$ $\displaystyle=\det\left(\left(-1\right)^{p-1}C_{q}\
\mid\ \left(\begin{array}[c]{c}I_{i-1}\\\
0_{\left(p-\left(i-1\right)\right)\times\left(i-1\right)}\end{array}\right)\
\mid\ \left(I_{p}\mid C\right)\left[i+k:p+k\right]\right)$
$\displaystyle=\left(-1\right)^{p-1}\det\left(C_{q}\ \mid\
\left(\begin{array}[c]{c}I_{i-1}\\\
0_{\left(p-\left(i-1\right)\right)\times\left(i-1\right)}\end{array}\right)\
\mid\ \left(I_{p}\mid C\right)\left[i+k:p+k\right]\right)$
$\displaystyle=\underbrace{\left(-1\right)^{p-1}\left(-1\right)^{i-1}}_{=\left(-1\right)^{p-i}}\det\left(\left(\begin{array}[c]{c}I_{i-1}\\\
0_{\left(p-\left(i-1\right)\right)\times\left(i-1\right)}\end{array}\right)\
\mid\ C_{q}\ \mid\ \left(I_{p}\mid C\right)\left[i+k:p+k\right]\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\begin{array}[c]{c}\text{since
permuting the columns of a matrix multiplies the}\\\ \text{determinant by the
sign of the permutation}\end{array}\right)$
$\displaystyle=\left(-1\right)^{p-i}\det\left(\left(\begin{array}[c]{c}I_{i-1}\\\
0_{\left(p-\left(i-1\right)\right)\times\left(i-1\right)}\end{array}\right)\
\mid\ C_{q}\ \mid\ \left(I_{p}\mid C\right)\left[i+k:p+k\right]\right)$
$\displaystyle=\left(-1\right)^{p-i}\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(C_{q}\
\mid\ \left(I_{p}\mid C\right)\left[i+k:p+k\right]\right)\right)$
(by (51)). Thus,
$\displaystyle\mathfrak{D}_{\left(i,k\right)}$
$\displaystyle=\det\left(\left(I_{p}\mid C\right)\left[0:i\mid
i+k:p+k\right]\right)$
$\displaystyle=\left(-1\right)^{p-i}\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(C_{q}\
\mid\ \left(I_{p}\mid C\right)\left[i+k:p+k\right]\right)\right).$ (54)
We have thus found alternative formulas (53) and (54) for
$\mathfrak{N}_{\left(i,k\right)}$ and $\mathfrak{D}_{\left(i,k\right)}$. While
not shorter than the definitions, these formulas involve smaller matrices
(unless $i=1$) and are more useful in understanding the monomials appearing in
$\mathfrak{N}_{\left(i,k\right)}$ and $\mathfrak{D}_{\left(i,k\right)}$.
Details of Step 2: We claim that $\mathfrak{N}_{\left(i,k\right)}$ and
$\mathfrak{D}_{\left(i,k\right)}$ are nonzero for all
$\left(i,k\right)\in\mathbf{P}$.
Proof. Let $\left(i,k\right)\in\mathbf{P}$. Let us first check that
$\mathfrak{N}_{\left(i,k\right)}$ is nonzero.
There are, in fact, many ways to do this. Here is probably the shortest one:
Assume the contrary, i.e., assume that $\mathfrak{N}_{\left(i,k\right)}=0$.
Then, every matrix $G\in\mathbb{F}^{p\times\left(p+q\right)}$ satisfies
$\det\left(G\left[1:i\mid i+k-1:p+k\right]\right)=0$ 373737Proof. Let
$\widetilde{\mathbb{F}}$ be a field extension of $\mathbb{F}$ such that
$\left|\widetilde{\mathbb{F}}\right|=\infty$. (We need this to make sense of
Zariski density arguments.) We are going to prove that every matrix
$G\in\widetilde{\mathbb{F}}^{p\times\left(p+q\right)}$ satisfies
$\det\left(G\left[1:i\mid i+k-1:p+k\right]\right)=0$; this will clearly imply
the same claim for $G\in\mathbb{F}^{p\times\left(p+q\right)}$. Let
$G\in\widetilde{\mathbb{F}}^{p\times\left(p+q\right)}$. We want to prove that
$\det\left(G\left[1:i\mid i+k-1:p+k\right]\right)=0$. Since this is a
polynomial identity in the entries of $G$, we can WLOG assume that $G$ is
generic enough that the first $p$ columns of $G$ are linearly independent
(since this just restricts $G$ to a Zariski-dense open subset of
$\widetilde{\mathbb{F}}^{p\times\left(p+q\right)}$). Assume this. Then, we can
write $G$ in the form $\left(U\mid V\right)$, with $U$ being the matrix formed
by the first $p$ columns of $G$, and $V$ being the matrix formed by the
remaining $q$ columns. Since the first $p$ columns of $G$ are linearly
independent, the matrix $U$ is invertible. Left multiplication by $U^{-1}$
acts on matrices column by column. This yields
$U^{-1}\cdot\left(G\left[1:i\mid
i+k-1:p+k\right]\right)=\left(U^{-1}G\right)\left[1:i\mid i+k-1:p+k\right].$
Also, $U^{-1}\underbrace{G}_{=\left(U\mid V\right)}=U^{-1}\left(U\mid
V\right)=\left(U^{-1}U\mid U^{-1}V\right)=\left(I_{p}\mid U^{-1}V\right)$.
Now, we have $\mathfrak{N}_{\left(i,k\right)}=0$. Since
$\mathfrak{N}_{\left(i,k\right)}=\det\left(\left(I_{p}\mid
C\right)\left[1:i\mid i+k-1:p+k\right]\right)$, this yields that
$\det\left(\left(I_{p}\mid C\right)\left[1:i\mid i+k-1:p+k\right]\right)=0$.
But the matrix $C$ is, in some sense, the “most generic matrix”: namely, the
entries of the matrix $C$ are pairwise distinct commuting indeterminates, and
therefore we can obtain any other matrix (over a commutative
$\mathbb{F}$-algebra) from $C$ by substituting the corresponding values for
the indeterminates. In particular, we can make a substitution that turns $C$
into $U^{-1}V$. Thus, from $\det\left(\left(I_{p}\mid C\right)\left[1:i\mid
i+k-1:p+k\right]\right)=0$, we obtain $\det\left(\left(I_{p}\mid
U^{-1}V\right)\left[1:i\mid i+k-1:p+k\right]\right)=0$. Now,
$\displaystyle\left(\det U\right)^{-1}\cdot\det\left(G\left[1:i\mid
i+k-1:p+k\right]\right)$
$\displaystyle=\det\left(\underbrace{U^{-1}\cdot\left(G\left[1:i\mid
i+k-1:p+k\right]\right)}_{=\left(U^{-1}G\right)\left[1:i\mid
i+k-1:p+k\right]}\right)=\det\left(\left(\underbrace{U^{-1}G}_{=\left(I_{p}\mid
U^{-1}V\right)}\right)\left[1:i\mid i+k-1:p+k\right]\right)$
$\displaystyle=\det\left(\left(I_{p}\mid U^{-1}V\right)\left[1:i\mid
i+k-1:p+k\right]\right)=0.$ Multiplying this with $\det U$ (which is nonzero
since $U$ is invertible), we obtain $\det\left(G\left[1:i\mid
i+k-1:p+k\right]\right)=0$, qed.. But this is absurd, because we can pick $G$
to have the $p$ columns labelled $1$, $2$, $...$, $i-1$, $i+k-1$, $i+k$,
$...$, $p+k-1$ linearly independent. This contradiction shows that our
assumption was wrong. Hence, $\mathfrak{N}_{\left(i,k\right)}$ is nonzero.
Similarly, $\mathfrak{D}_{\left(i,k\right)}$ is nonzero.
Details of Step 3: Define a
$Q_{\mathbf{p}}\in\mathbb{F}\left(x_{\mathbf{P}}\right)$ for every
$\mathbf{p}\in\mathbf{P}$ by
$Q_{\mathbf{p}}=\dfrac{\mathfrak{N}_{\mathbf{p}}}{\mathfrak{D}_{\mathbf{p}}}$.
This is well-defined because Step 2 has shown that $\mathfrak{D}_{\mathbf{p}}$
is nonzero. Moreover, it is easy to see that every
$\left(i,k\right)\in\mathbf{P}$ satisfies
$Q_{\left(i,k\right)}=\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
C\right)\right)\left(\left(i,k\right)\right).$ 383838Indeed, the definition of
$\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid C\right)$ yields
$\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
C\right)\right)\left(\left(i,k\right)\right)=\dfrac{\det\left(\left(I_{p}\mid
C\right)\left[1:i\mid i+k-1:p+k\right]\right)}{\det\left(\left(I_{p}\mid
C\right)\left[0:i\mid
i+k:p+k\right]\right)}=\dfrac{\mathfrak{N}_{\left(i,k\right)}}{\mathfrak{D}_{\left(i,k\right)}}$
(by (49) and (50)).
In other words, every $\mathbf{p}\in\mathbf{P}$ satisfies
$Q_{\mathbf{p}}=\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
C\right)\right)\left(\mathbf{p}\right).$ (55)
Details of Step 4: We are now going to prove that the family
$\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
is $\mathbf{P}$-triangular.
By the definition of $\mathbf{P}$-triangularity, this requires showing that
for every $\mathbf{p}\in\mathbf{P}$, there exist elements
$\alpha_{\mathbf{p}}$, $\beta_{\mathbf{p}}$, $\gamma_{\mathbf{p}}$,
$\delta_{\mathbf{p}}$ of $\mathbb{F}\left(x_{\mathbf{p}\Downarrow}\right)$
such that
$\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$ and
$Q_{\mathbf{p}}=\dfrac{\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}}{\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}}$
(where $\mathbf{p}\Downarrow$ is defined as in Definition 15.2 (d)). So fix
$\mathbf{p}\in\mathbf{P}$. Write $\mathbf{p}$ in the form
$\mathbf{p}=\left(i,k\right)$.
We will actually do something slightly better than we need. We will find
elements $\alpha_{\mathbf{p}}$, $\beta_{\mathbf{p}}$, $\gamma_{\mathbf{p}}$,
$\delta_{\mathbf{p}}$ of $\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]$
(not just of $\mathbb{F}\left(x_{\mathbf{p}\Downarrow}\right)$) such that
$\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$ and
$\mathfrak{N}_{\mathbf{p}}=\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}$
and
$\mathfrak{D}_{\mathbf{p}}=\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}$.
(Of course, the conditions
$\mathfrak{N}_{\mathbf{p}}=\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}$
and
$\mathfrak{D}_{\mathbf{p}}=\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}$
combined imply
$Q_{\mathbf{p}}=\dfrac{\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}}{\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}}$,
hence the yearned-for $\mathbf{P}$-triangularity.)
Let us first deal with two “boundary” cases: the case when $k=1$, and the case
when $k\neq 1$ but $i=p$.
The case when $k=1$ is very easy. In fact, in this case, it is easy to prove
that $\mathfrak{N}_{\mathbf{p}}=1$ (using (53)) and that
$\mathfrak{D}_{\mathbf{p}}=\left(-1\right)^{i+p}x_{\mathbf{p}}$ (using (54)).
Consequently, we can take $\alpha_{\mathbf{p}}=0$, $\beta_{\mathbf{p}}=1$,
$\gamma_{\mathbf{p}}=\left(-1\right)^{i+p}$ and $\delta_{\mathbf{p}}=0$, and
it is clear that all three requirements
$\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$ and
$\mathfrak{N}_{\mathbf{p}}=\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}$
and
$\mathfrak{D}_{\mathbf{p}}=\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}$
are satisfied.
The case when $k\neq 1$ but $i=p$ is not much harder. In this case, (53)
simplifies to $\mathfrak{N}_{\mathbf{p}}=x_{\mathbf{p}}$, and (54) simplifies
to $\mathfrak{D}_{\mathbf{p}}=x_{\left(p,1\right)}$. Hence, we can take
$\alpha_{\mathbf{p}}=1$, $\beta_{\mathbf{p}}=0$, $\gamma_{\mathbf{p}}=0$ and
$\delta_{\mathbf{p}}=x_{\left(p,1\right)}$ to achieve
$\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$ and
$\mathfrak{N}_{\mathbf{p}}=\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}$
and
$\mathfrak{D}_{\mathbf{p}}=\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}$.
Note that this choice of $\delta_{\mathbf{p}}$ is legitimate because
$x_{\left(p,1\right)}$ does lie in
$\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]$ (since
$\left(p,1\right)\in\left.\mathbf{p}\Downarrow\right.$).
Now that these two cases are settled, let us deal with the remaining case. So
we have neither $k=1$ nor $i=p$.
Consider the matrix
$\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)$ (this matrix appears on the right hand
side of (53)). Each entry of this matrix comes either from the matrix $I_{p}$
or from the matrix $C$. If it comes from $I_{p}$, it clearly lies in
$\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]$. If it comes from $C$, it
has the form $x_{\mathbf{q}}$ for some $\mathbf{q}\in\mathbf{P}$, and this
$\mathbf{q}$ belongs to $\left.\mathbf{p}\Downarrow\right.$ unless the entry
is the $\left(1,p-i+1\right)$-th entry. Therefore, each entry of the matrix
$\left(I_{p}\mid C\right)\left[i+k-1:p+k\right]$ apart from the
$\left(1,p-i+1\right)$-th entry lies in
$\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]$, whereas the
$\left(1,p-i+1\right)$-th entry is $x_{\mathbf{p}}$. Hence, if we use Laplace
expansion with respect to the first row to compute the determinant of this
matrix, we obtain a formula of the form
$\displaystyle\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)\right)$
$\displaystyle=x_{\mathbf{p}}\cdot\left(\text{some polynomial in entries lying
in }\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]\right)$ $\displaystyle\ \
\ \ \ \ \ \ \ \ +\left(\text{more polynomials in entries lying in
}\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]\right)$
$\displaystyle\in\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]\cdot
x_{\mathbf{p}}+\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right].$
In other words, there exist elements $\alpha_{\mathbf{p}}$ and
$\beta_{\mathbf{p}}$ of $\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]$ such
that
$\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)\right)=\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}$.
Consider these $\alpha_{\mathbf{p}}$ and $\beta_{\mathbf{p}}$. We have
$\displaystyle\mathfrak{N}_{\mathbf{p}}$
$\displaystyle=\mathfrak{N}_{\left(i,k\right)}=\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)\right)\ \ \ \ \ \ \ \ \ \ \left(\text{by
(\ref{pf.Grasp.generic.short.step1.N})}\right)$ (56)
$\displaystyle=\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}.$ (57)
We can similarly deal with the matrix
$\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(C_{q}\ \mid\ \left(I_{p}\mid
C\right)\left[i+k:p+k\right]\right)$ which appears on the right hand side of
(54). Again, each entry of this matrix apart from the
$\left(1,p-i+1\right)$-th entry lies in
$\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]$, whereas the
$\left(1,p-i+1\right)$-th entry is $x_{\mathbf{p}}$. Using Laplace expansion
again, we thus see that
$\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(C_{q}\ \mid\
\left(I_{p}\mid
C\right)\left[i+k:p+k\right]\right)\right)\in\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]\cdot
x_{\mathbf{p}}+\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right],$
so that
$\left(-1\right)^{p-i}\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(C_{q}\
\mid\ \left(I_{p}\mid
C\right)\left[i+k:p+k\right]\right)\right)\in\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]\cdot
x_{\mathbf{p}}+\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right].$
Hence, there exist elements $\gamma_{\mathbf{p}}$ and $\delta_{\mathbf{p}}$ of
$\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]$ such that
$\left(-1\right)^{p-i}\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(C_{q}\
\mid\ \left(I_{p}\mid
C\right)\left[i+k:p+k\right]\right)\right)=\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}$.
Consider these $\gamma_{\mathbf{p}}$ and $\delta_{\mathbf{p}}$. We have
$\displaystyle\mathfrak{D}_{\mathbf{p}}$
$\displaystyle=\mathfrak{D}_{\left(i,k\right)}=\left(-1\right)^{p-i}\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(C_{q}\
\mid\ \left(I_{p}\mid C\right)\left[i+k:p+k\right]\right)\right)\ \ \ \ \ \ \
\ \ \ \left(\text{by (\ref{pf.Grasp.generic.short.step1.D})}\right)$ (58)
$\displaystyle=\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}.$
We thus have found elements $\alpha_{\mathbf{p}}$, $\beta_{\mathbf{p}}$,
$\gamma_{\mathbf{p}}$, $\delta_{\mathbf{p}}$ of
$\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]$ satisfying
$\mathfrak{N}_{\mathbf{p}}=\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}$
and
$\mathfrak{D}_{\mathbf{p}}=\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}$.
In order to finish the proof of $\mathbf{P}$-triangularity, we only need to
show that
$\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$.
In order to achieve this goal, we notice that
$\alpha_{\mathbf{p}}\underbrace{\mathfrak{D}_{\mathbf{p}}}_{=\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}}-\underbrace{\mathfrak{N}_{\mathbf{p}}}_{=\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}}\gamma_{\mathbf{p}}=\alpha_{\mathbf{p}}\left(\gamma_{\mathbf{p}}x_{\mathbf{p}}+\delta_{\mathbf{p}}\right)-\left(\alpha_{\mathbf{p}}x_{\mathbf{p}}+\beta_{\mathbf{p}}\right)\gamma_{\mathbf{p}}=\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}.$
Hence, proving
$\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$ is equivalent to proving
$\alpha_{\mathbf{p}}\mathfrak{D}_{\mathbf{p}}-\mathfrak{N}_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$. It is the latter that we are going to do, because $\alpha_{\mathbf{p}}$,
$\mathfrak{D}_{\mathbf{p}}$, $\mathfrak{N}_{\mathbf{p}}$ and
$\gamma_{\mathbf{p}}$ are easier to get our hands on than $\beta_{\mathbf{p}}$
and $\delta_{\mathbf{p}}$.
So we need to prove that
$\alpha_{\mathbf{p}}\mathfrak{D}_{\mathbf{p}}-\mathfrak{N}_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$. To do so, we look back at our proof of
$\det\left(\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)\right)\in\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]\cdot
x_{\mathbf{p}}+\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right].$
This proof proceeded by applying Laplace expansion with respect to the first
row to the matrix
$\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)$. The only term involving
$x_{\mathbf{p}}$ was
$x_{\mathbf{p}}\cdot\left(\text{some polynomial in entries lying in
}\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]\right).$
Recalling the statement of Laplace expansion, we notice that “some polynomial
in entries lying in $\mathbb{F}\left[x_{\mathbf{p}\Downarrow}\right]$” in this
term is actually the $\left(1,p-i+1\right)$-th cofactor of the matrix
$\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)$. Hence,
$\displaystyle\alpha_{\mathbf{p}}$ $\displaystyle=\left(\text{the
}\left(1,p-i+1\right)\text{-th cofactor of
}\operatorname*{rows}\nolimits_{i,i+1,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k\right]\right)\right)$
$\displaystyle=\left(-1\right)^{p-i}\cdot\det\left(\operatorname*{rows}\nolimits_{i+1,i+2,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k-1\right]\right)\right).$ (59)
Similarly,
$\gamma_{\mathbf{p}}=\det\left(\operatorname*{rows}\nolimits_{i+1,i+2,...,p}\left(C_{q}\
\mid\ \left(I_{p}\mid C\right)\left[i+k:p+k-1\right]\right)\right)$ (60)
(note that we lost the sign $\left(-1\right)^{p-i}$ from (58) since it got
cancelled against the $\left(-1\right)^{p-\left(i+1\right)}$ arising from the
definition of a cofactor).
Now, recall that we have neither $k=1$ nor $i=p$. Hence,
$\left(i+1,k-1\right)$ also belongs to $\mathbf{P}$, so we can apply (53) to
$\left(i+1,k-1\right)$ in lieu of $\left(i,k\right)$, and obtain
$\mathfrak{N}_{\left(i+1,k-1\right)}=\det\left(\operatorname*{rows}\nolimits_{i+1,i+2,...,p}\left(\left(I_{p}\mid
C\right)\left[i+k-1:p+k-1\right]\right)\right).$
In light of this, (59) becomes
$\alpha_{\mathbf{p}}=\left(-1\right)^{p-i}\cdot\mathfrak{N}_{\left(i+1,k-1\right)}.$
Similarly, we can apply (54) to $\left(i+1,k-1\right)$ in lieu of
$\left(i,k\right)$, and use this to rewrite (60) as
$\gamma_{\mathbf{p}}=\left(-1\right)^{p-\left(i+1\right)}\cdot\mathfrak{D}_{\left(i+1,k-1\right)}.$
Hence,
$\displaystyle\underbrace{\alpha_{\mathbf{p}}}_{=\left(-1\right)^{p-i}\cdot\mathfrak{N}_{\left(i+1,k-1\right)}}\mathfrak{D}_{\mathbf{p}}-\mathfrak{N}_{\mathbf{p}}\underbrace{\gamma_{\mathbf{p}}}_{=\left(-1\right)^{p-\left(i+1\right)}\cdot\mathfrak{D}_{\left(i+1,k-1\right)}}$
$\displaystyle=\left(-1\right)^{p-i}\cdot\mathfrak{N}_{\left(i+1,k-1\right)}\cdot\mathfrak{D}_{\mathbf{p}}-\mathfrak{N}_{\mathbf{p}}\cdot\underbrace{\left(-1\right)^{p-\left(i+1\right)}}_{=-\left(-1\right)^{p-i}}\cdot\mathfrak{D}_{\left(i+1,k-1\right)}$
$\displaystyle=\left(-1\right)^{p-i}\cdot\left(\mathfrak{N}_{\left(i+1,k-1\right)}\mathfrak{D}_{\mathbf{p}}+\mathfrak{N}_{\mathbf{p}}\mathfrak{D}_{\left(i+1,k-1\right)}\right).$
Thus, we can shift our goal from proving
$\alpha_{\mathbf{p}}\mathfrak{D}_{\mathbf{p}}-\mathfrak{N}_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$ to proving
$\mathfrak{N}_{\left(i+1,k-1\right)}\mathfrak{D}_{\mathbf{p}}+\mathfrak{N}_{\mathbf{p}}\mathfrak{D}_{\left(i+1,k-1\right)}\neq
0$.
But this turns out to be surprisingly simple: Since
$\mathbf{p}=\left(i,k\right)$, we have
$\displaystyle\mathfrak{N}_{\left(i+1,k-1\right)}\mathfrak{D}_{\mathbf{p}}+\mathfrak{N}_{\mathbf{p}}\mathfrak{D}_{\left(i+1,k-1\right)}$
$\displaystyle=\mathfrak{N}_{\left(i+1,k-1\right)}\mathfrak{D}_{\left(i,k\right)}+\mathfrak{N}_{\left(i,k\right)}\mathfrak{D}_{\left(i+1,k-1\right)}=\mathfrak{D}_{\left(i,k\right)}\cdot\mathfrak{N}_{\left(i+1,k-1\right)}+\mathfrak{N}_{\left(i,k\right)}\cdot\mathfrak{D}_{\left(i+1,k-1\right)}$
$\displaystyle=\det\left(\left(I_{p}\mid C\right)\left[0:i\mid
i+k:p+k\right]\right)\cdot\det\left(\left(I_{p}\mid C\right)\left[1:i+1\mid
i+k-1:p+k-1\right]\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
+\det\left(\left(I_{p}\mid C\right)\left[1:i\mid
i+k-1:p+k\right]\right)\cdot\det\left(\left(I_{p}\mid C\right)\left[0:i+1\mid
i+k:p+k-1\right]\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\left(\begin{array}[c]{c}\text{here, we just substituted
}\mathfrak{D}_{\left(i,k\right)}\text{,
}\mathfrak{N}_{\left(i+1,k-1\right)}\text{,
}\mathfrak{N}_{\left(i,k\right)}\text{ and
}\mathfrak{D}_{\left(i+1,k-1\right)}\\\ \text{by their
definitions}\end{array}\right)$ (63) $\displaystyle=\det\left(\left(I_{p}\mid
C\right)\left[0:i\mid i+k-1:p+k-1\right]\right)\cdot\det\left(\left(I_{p}\mid
C\right)\left[1:i+1\mid i+k:p+k\right]\right)$ (64)
(by Theorem 14.1, applied to $p$, $p+q$, $\left(I_{p}\mid C\right)$, $1$, $i$,
$i+k$ and $p+k-1$ instead of $u$, $v$, $A$, $a$, $b$, $c$ and $d$). On the
other hand, $\left(i,k-1\right)$ and $\left(i+1,k\right)$ also belong to
$\mathbf{P}$ and satisfy
$\mathfrak{D}_{\left(i,k-1\right)}=\det\left(\left(I_{p}\mid
C\right)\left[0:i\mid i+k-1:p+k-1\right]\right)$
and
$\mathfrak{N}_{\left(i+1,k\right)}=\det\left(\left(I_{p}\mid
C\right)\left[1:i+1\mid i+k:p+k\right]\right)$
(by the respective definitions of $\mathfrak{D}_{\left(i,k-1\right)}$ and
$\mathfrak{N}_{\left(i+1,k\right)}$). Hence, (64) becomes
$\displaystyle\mathfrak{N}_{\left(i+1,k-1\right)}\mathfrak{D}_{\mathbf{p}}+\mathfrak{N}_{\mathbf{p}}\mathfrak{D}_{\left(i+1,k-1\right)}$
$\displaystyle=\underbrace{\det\left(\left(I_{p}\mid C\right)\left[0:i\mid
i+k-1:p+k-1\right]\right)}_{=\mathfrak{D}_{\left(i,k-1\right)}}\cdot\underbrace{\det\left(\left(I_{p}\mid
C\right)\left[1:i+1\mid
i+k:p+k\right]\right)}_{=\mathfrak{N}_{\left(i+1,k\right)}}$
$\displaystyle=\mathfrak{D}_{\left(i,k-1\right)}\cdot\mathfrak{N}_{\left(i+1,k\right)}\neq
0$
(since the result of Step 2 shows that $\mathfrak{D}_{\left(i,k-1\right)}$ and
$\mathfrak{N}_{\left(i+1,k\right)}$ are nonzero). This finishes our proof of
$\mathfrak{N}_{\left(i+1,k-1\right)}\mathfrak{D}_{\mathbf{p}}+\mathfrak{N}_{\mathbf{p}}\mathfrak{D}_{\left(i+1,k-1\right)}\neq
0$, thus also of
$\alpha_{\mathbf{p}}\mathfrak{D}_{\mathbf{p}}-\mathfrak{N}_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$, hence also of
$\alpha_{\mathbf{p}}\delta_{\mathbf{p}}-\beta_{\mathbf{p}}\gamma_{\mathbf{p}}\neq
0$, and ultimately of the $\mathbf{P}$-triangularity of the family
$\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$.
Details of Step 5: Recall that our goal is to prove the existence of a matrix
$\widetilde{D}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{p\times\left(p+q\right)}$
satisfying (48).
Since Step 4, we know that the family
$\left(Q_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
is $\mathbf{P}$-triangular. Hence, Lemma 15.3 (b) shows that there exists a
$\mathbf{P}$-triangular family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{\mathbf{P}}$
such that every $\mathbf{q}\in\mathbf{P}$ satisfies
$Q_{\mathbf{q}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=x_{\mathbf{q}}$.
Consider this $\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$.
Applying Lemma 15.3 (a) to this family
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$, we conclude that
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$ is algebraically
independent.
In Step 3, we have shown that
$Q_{\mathbf{p}}=\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
C\right)\right)\left(\mathbf{p}\right)$ for every $\mathbf{p}\in\mathbf{P}$.
Renaming $\mathbf{p}$ as $\mathbf{q}$, we rewrite this as follows:
$Q_{\mathbf{q}}=\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid
C\right)\right)\left(\mathbf{q}\right)\ \ \ \ \ \ \ \ \ \ \text{for every
}\mathbf{q}\in\mathbf{P}.$ (65)
Now, let
$\widetilde{C}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{p\times\left(p+q\right)}$
denote the matrix obtained from the matrix
$C\in\left(\mathbb{F}\left[x_{\mathbf{P}}\right]\right)^{p\times\left(p+q\right)}$
by substituting $\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$ for
the variables $\left(x_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$. Since
(65) is an identity between rational functions in the variables
$\left(x_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$, we thus can
substitute $\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$ for the
variables $\left(x_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$ in
(65)393939The substitution does not suffer from vanishing denominators because
$\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}$ is algebraically
independent., and obtain
$Q_{\mathbf{q}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid\widetilde{C}\right)\right)\left(\mathbf{q}\right)\
\ \ \ \ \ \ \ \ \ \text{for every }\mathbf{q}\in\mathbf{P}$
(since this substitution takes the matrix $C$ to $\widetilde{C}$). But since
$Q_{\mathbf{q}}\left(\left(R_{\mathbf{p}}\right)_{\mathbf{p}\in\mathbf{P}}\right)=x_{\mathbf{q}}$
for every $\mathbf{q}\in\mathbf{P}$, this rewrites as
$x_{\mathbf{q}}=\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid\widetilde{C}\right)\right)\left(\mathbf{q}\right)\
\ \ \ \ \ \ \ \ \ \text{for every }\mathbf{q}\in\mathbf{P}.$
Upon renaming $\mathbf{q}$ as $\mathbf{p}$ again, this becomes
$x_{\mathbf{p}}=\left(\operatorname*{Grasp}\nolimits_{0}\left(I_{p}\mid\widetilde{C}\right)\right)\left(\mathbf{p}\right)\
\ \ \ \ \ \ \ \ \ \text{for every }\mathbf{p}\in\mathbf{P}.$
Hence, there exists a matrix
$\widetilde{D}\in\left(\mathbb{F}\left(x_{\mathbf{P}}\right)\right)^{p\times\left(p+q\right)}$
satisfying (48) (namely, $\widetilde{D}=\left(I_{p}\mid\widetilde{C}\right)$).
Thus, as we know, Proposition 13.14 is proven. ∎
## 16 The rectangle: finishing the proofs
As promised, we now use Propositions 13.13 and 13.14 to derive our initially
stated results on rectangles. First, we formulate an easy consequence of
Proposition 13.13:
###### Corollary 16.1.
Let $\mathbb{K}$ be a field. Let $p$ and $q$ be two positive integers. Let
$A\in\mathbb{K}^{p\times\left(p+q\right)}$ be a matrix. Then, every
$i\in\mathbb{N}$ satisfies
$\operatorname*{Grasp}\nolimits_{-i}A=R_{\operatorname*{Rect}\left(p,q\right)}^{i}\left(\operatorname*{Grasp}\nolimits_{0}A\right)$
(provided that $A$ is sufficiently generic in the sense of Zariski topology
that both sides of this equality are well-defined).
###### Proof of Corollary 16.1 (sketched)..
We will prove Corollary 16.1 by induction over $i$:
Induction base: For $i=0$, the claim of Corollary 16.1 boils down to
$\operatorname*{Grasp}\nolimits_{0}A=R_{\operatorname*{Rect}\left(p,q\right)}^{0}\left(\operatorname*{Grasp}\nolimits_{0}A\right)$.
This is obvious, and so the induction base is complete.
Induction step: Let $j\in\mathbb{N}$. Assume that Corollary 16.1 holds for
$i=j$. We need to prove that Corollary 16.1 holds for $i=j+1$ as well.
Proposition 13.13 (applied to $-\left(j+1\right)$ instead of $j$) yields
$\displaystyle\operatorname*{Grasp}\nolimits_{-\left(j+1\right)}A$
$\displaystyle=R_{\operatorname*{Rect}\left(p,q\right)}\left(\operatorname*{Grasp}\nolimits_{-\left(j+1\right)+1}A\right)=R_{\operatorname*{Rect}\left(p,q\right)}\left(\underbrace{\operatorname*{Grasp}\nolimits_{-j}A}_{\begin{subarray}{c}=R_{\operatorname*{Rect}\left(p,q\right)}^{j}\left(\operatorname*{Grasp}\nolimits_{0}A\right)\\\
\text{(since Corollary \ref{cor.Grasp.GraspR}}\\\ \text{holds for
}i=j\text{)}\end{subarray}}\right)$
$\displaystyle=R_{\operatorname*{Rect}\left(p,q\right)}\left(R_{\operatorname*{Rect}\left(p,q\right)}^{j}\left(\operatorname*{Grasp}\nolimits_{0}A\right)\right)=R_{\operatorname*{Rect}\left(p,q\right)}^{j+1}\left(\operatorname*{Grasp}\nolimits_{0}A\right).$
In other words, Corollary 16.1 holds for $i=j+1$. This completes the induction
step. The induction proof of Corollary 16.1 is thus finished. ∎
###### Proof of Theorem 11.5 (sketched)..
We need to show that
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,q\right)}\right)=p+q$.
According to Proposition 12.2, it is enough to prove that almost every (in the
Zariski sense) reduced labelling
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ satisfies
$R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}f=f$. So let
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ be a
sufficiently generic reduced labelling. In other words, $f$ is a sufficiently
generic element of $\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$
(because the reduced labellings
$\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ are being
identified with the elements of
$\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$). Due to Proposition
13.14, there exists a matrix $A\in\mathbb{K}^{p\times\left(p+q\right)}$
satisfying $f=\operatorname*{Grasp}\nolimits_{0}A$. Consider this $A$. Due to
Corollary 16.1 (applied to $i=p+q$), we have
$\operatorname*{Grasp}\nolimits_{-\left(p+q\right)}A=R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}\left(\underbrace{\operatorname*{Grasp}\nolimits_{0}A}_{=f}\right)=R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}f.$
But Proposition 13.11 (applied to $j=-\left(p+q\right)$) yields
$\displaystyle\operatorname*{Grasp}\nolimits_{-\left(p+q\right)}A$
$\displaystyle=\operatorname*{Grasp}\nolimits_{p+q+\left(-\left(p+q\right)\right)}A=\operatorname*{Grasp}\nolimits_{0}A\
\ \ \ \ \ \ \ \ \ \left(\text{since
}p+q+\left(-\left(p+q\right)\right)=0\right)$ $\displaystyle=f.$
Hence,
$f=\operatorname*{Grasp}\nolimits_{-\left(p+q\right)}A=R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}f$.
In other words, $R_{\operatorname*{Rect}\left(p,q\right)}^{p+q}f=f$. This (as
we know) proves Theorem 11.5. ∎
###### Proof of Theorem 12.3 (sketched)..
Let us regard the reduced labelling
$f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ as an
element of $\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$ (because we
identify reduced labellings in
$\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ with elements of
$\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$). We assume WLOG that this
element $f\in\mathbb{K}^{\operatorname*{Rect}\left(p,q\right)}$ is generic
enough (among the reduced labellings) for Proposition 13.14 to apply. By
Proposition 13.14, there exists a matrix
$A\in\mathbb{K}^{p\times\left(p+q\right)}$ satisfying
$f=\operatorname*{Grasp}\nolimits_{0}A$. Consider this $A$. Due to Corollary
16.1 (applied to $i+k-1$ instead of $i$), we have
$\operatorname*{Grasp}\nolimits_{-\left(i+k-1\right)}A=R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}\left(\underbrace{\operatorname*{Grasp}\nolimits_{0}A}_{=f}\right)=R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f.$
But Proposition 13.12 (applied to $j=-\left(i+k-1\right)$) yields
$\displaystyle\left(\operatorname*{Grasp}\nolimits_{-\left(i+k-1\right)}A\right)\left(\left(i,k\right)\right)$
$\displaystyle=\dfrac{1}{\left(\operatorname*{Grasp}\nolimits_{-\left(i+k-1\right)+i+k-1}A\right)\left(\left(p+1-i,q+1-k\right)\right)}$
$\displaystyle=\dfrac{1}{f\left(\left(p+1-i,q+1-k\right)\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{since
}\operatorname*{Grasp}\nolimits_{-\left(i+k-1\right)+i+k-1}A=\operatorname*{Grasp}\nolimits_{0}A=f\right),$
so that
$f\left(\left(p+1-i,q+1-k\right)\right)=\dfrac{1}{\left(\operatorname*{Grasp}\nolimits_{-\left(i+k-1\right)}A\right)\left(\left(i,k\right)\right)}=\dfrac{1}{\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\left(\left(i,k\right)\right)}$
(since
$\operatorname*{Grasp}\nolimits_{-\left(i+k-1\right)}A=R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f$).
This proves Theorem 12.3. ∎
###### Proof of Theorem 11.7 (sketched)..
We will be using the notation $\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$
defined in Definition 5.2.
Let $f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,q\right)}}$ be
arbitrary. By genericity, we assume WLOG that $f\left(0\right)$ and
$f\left(1\right)$ are nonzero.
Let $n=p+q-1$. Then, $\operatorname*{Rect}\left(p,q\right)$ is an $n$-graded
poset. Also, $i+k-1\in\left\\{0,1,...,n\right\\}$. Moreover, $1\leqslant
n-i-k+2\leqslant n$.
Define an $\left(n+2\right)$-tuple
$\left(a_{0},a_{1},...,a_{n+1}\right)\in\mathbb{K}^{n+2}$ by
$a_{r}=\left\\{\begin{array}[c]{c}\dfrac{1}{f\left(0\right)},\ \ \ \ \ \ \ \ \
\ \text{if }r=0;\\\ 1,\ \ \ \ \ \ \ \ \ \ \text{if }1\leqslant r\leqslant
n;\\\ \dfrac{1}{f\left(1\right)},\ \ \ \ \ \ \ \ \ \ \text{if
}r=n+1\end{array}\right.\ \ \ \ \ \ \ \ \ \ \text{for every
}r\in\left\\{0,1,...,n+1\right\\}.$
Thus, $a_{n-i-k+2}=1$ (since $1\leqslant n-i-k+2\leqslant n$) and
$a_{0}=\dfrac{1}{f\left(0\right)}$ and $a_{n+1}=\dfrac{1}{f\left(1\right)}$.
Let $f^{\prime}=\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$. Then, it is easy
to see from the definition of $\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$
that $f^{\prime}\left(0\right)=1$ and $f^{\prime}\left(1\right)=1$. In other
words, $f^{\prime}$ is a reduced $\mathbb{K}$-labelling. Hence, Theorem 12.3
(applied to $f^{\prime}$ instead of $f$) yields
$f^{\prime}\left(\left(p+1-i,q+1-k\right)\right)=\dfrac{1}{\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}\left(f^{\prime}\right)\right)\left(\left(i,k\right)\right)}.$
(66)
On the other hand, again from the definition of
$f^{\prime}=\left(a_{0},a_{1},...,a_{n+1}\right)\flat f$, it is easy to see
that $f^{\prime}\left(v\right)=f\left(v\right)$ for every
$v\in\operatorname*{Rect}\left(p,q\right)$. This yields, in particular, that
$f^{\prime}\left(\left(p+1-i,q+1-k\right)\right)=f\left(\left(p+1-i,q+1-k\right)\right)$.
But let us define an element
$\widehat{a}_{\kappa}^{\left(\ell\right)}\in\mathbb{K}^{\times}$ for every
$\ell\in\left\\{0,1,...,n+1\right\\}$ and
$\kappa\in\left\\{0,1,...,n+1\right\\}$ as in Proposition 5.5. Then,
Proposition 5.5 (applied to $\ell=i+k-1$) yields
$R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}\left(\left(a_{0},a_{1},...,a_{n+1}\right)\flat
f\right)=\left(\widehat{a}_{0}^{\left(i+k-1\right)},\widehat{a}_{1}^{\left(i+k-1\right)},...,\widehat{a}_{n+1}^{\left(i+k-1\right)}\right)\flat\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right).$
Since $\left(a_{0},a_{1},...,a_{n+1}\right)\flat f=f^{\prime}$, this rewrites
as
$R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}\left(f^{\prime}\right)=\left(\widehat{a}_{0}^{\left(i+k-1\right)},\widehat{a}_{1}^{\left(i+k-1\right)},...,\widehat{a}_{n+1}^{\left(i+k-1\right)}\right)\flat\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right).$
Hence,
$\displaystyle\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}\left(f^{\prime}\right)\right)\left(\left(i,k\right)\right)$
$\displaystyle=\left(\left(\widehat{a}_{0}^{\left(i+k-1\right)},\widehat{a}_{1}^{\left(i+k-1\right)},...,\widehat{a}_{n+1}^{\left(i+k-1\right)}\right)\flat\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\right)\left(\left(i,k\right)\right)$
$\displaystyle=\widehat{a}_{\deg\left(\left(i,k\right)\right)}^{\left(i+k-1\right)}\cdot\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\left(\left(i,k\right)\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{by the definition of
}\left(\widehat{a}_{0}^{\left(i+k-1\right)},\widehat{a}_{1}^{\left(i+k-1\right)},...,\widehat{a}_{n+1}^{\left(i+k-1\right)}\right)\flat\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\right)$
$\displaystyle=\widehat{a}_{i+k-1}^{\left(i+k-1\right)}\cdot\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\left(\left(i,k\right)\right)\
\ \ \ \ \ \ \ \ \ \left(\text{since
}\deg\left(\left(i,k\right)\right)=i+k-1\right)$
$\displaystyle=\dfrac{1}{f\left(0\right)f\left(1\right)}\cdot\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\left(\left(i,k\right)\right)$
(since the definition of $\widehat{a}_{i+k-1}^{\left(i+k-1\right)}$ yields
$\displaystyle\widehat{a}_{i+k-1}^{\left(i+k-1\right)}$
$\displaystyle=\left\\{\begin{array}[c]{c}\dfrac{a_{n+1}a_{\left(i+k-1\right)-\left(i+k-1\right)}}{a_{n+1-\left(i+k-1\right)}},\
\ \ \ \ \ \ \ \ \ \text{if }i+k-1\geqslant i+k-1;\\\
\dfrac{a_{n+1+\left(i+k-1\right)-\left(i+k-1\right)}a_{0}}{a_{n+1-\left(i+k-1\right)}},\
\ \ \ \ \ \ \ \ \ \text{if }i+k-1<i+k-1\end{array}\right.$
$\displaystyle=\dfrac{a_{n+1}a_{\left(i+k-1\right)-\left(i+k-1\right)}}{a_{n+1-\left(i+k-1\right)}}\
\ \ \ \ \ \ \ \ \ \left(\text{since }i+k-1\geqslant i+k-1\right)$
$\displaystyle=\dfrac{a_{n+1}a_{0}}{a_{n-i-k+2}}=\underbrace{a_{n+1}}_{=\dfrac{1}{f\left(1\right)}}\underbrace{a_{0}}_{=\dfrac{1}{f\left(0\right)}}\
\ \ \ \ \ \ \ \ \ \left(\text{since }a_{n-i-k+2}=1\right)$
$\displaystyle=\dfrac{1}{f\left(0\right)f\left(1\right)}$
). Thus, (66) rewrites as
$f^{\prime}\left(\left(p+1-i,q+1-k\right)\right)=\dfrac{1}{\dfrac{1}{f\left(0\right)f\left(1\right)}\cdot\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\left(\left(i,k\right)\right)}=\dfrac{f\left(0\right)f\left(1\right)}{\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\left(\left(i,k\right)\right)}.$
This rewrites as
$f\left(\left(p+1-i,q+1-k\right)\right)=\dfrac{f\left(0\right)f\left(1\right)}{\left(R_{\operatorname*{Rect}\left(p,q\right)}^{i+k-1}f\right)\left(\left(i,k\right)\right)}$
(since we know that
$f^{\prime}\left(\left(p+1-i,q+1-k\right)\right)=f\left(\left(p+1-i,q+1-k\right)\right)$).
This proves Theorem 11.7. ∎
## 17 The $\vartriangleright$ triangle
Having proven the main properties of birational rowmotion $R$ on the rectangle
$\operatorname*{Rect}\left(p,q\right)$ and on skeletal posets, we now turn to
other posets. We will spend the next three sections discussing the order of
birational rowmotion on certain triangle-shaped posets obtained as subsets of
the square $\operatorname*{Rect}\left(p,p\right)$. We start with the easiest
case:
###### Definition 17.1.
Let $p$ be a positive integer. Define a subset
$\operatorname*{Tria}\left(p\right)$ of $\operatorname*{Rect}\left(p,p\right)$
by
$\operatorname*{Tria}\left(p\right)=\left\\{\left(i,k\right)\in\left\\{1,2,...,p\right\\}^{2}\
\mid\ i\leqslant k\right\\}.$
This subset $\operatorname*{Tria}\left(p\right)$ inherits a poset structure
from $\operatorname*{Rect}\left(p,p\right)$. In the following, we will
consider $\operatorname*{Tria}\left(p\right)$ as a poset using this structure.
This poset $\operatorname*{Tria}\left(p\right)$ is a
$\left(2p-1\right)$-graded poset. It has the form of a triangle (either of
$\vartriangleleft$ shape or of $\vartriangleright$ shape, depending on how you
draw the Hasse diagram).
###### Example 17.2.
Here is the Hasse diagram of the poset $\operatorname*{Rect}\left(4,4\right)$,
with the elements that belong to $\operatorname*{Tria}\left(4\right)$ marked
by underlines:
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And here is the Hasse diagram of the poset
$\operatorname*{Tria}\left(4\right)$ itself:
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###### Remark 17.3.
Let $p$ be a positive integer. The poset $\operatorname*{Tria}\left(p\right)$
appears in [StWi11, §6.2] under the guise of the poset of order ideals (under
inclusion) of the rectangle $\operatorname*{Rect}\left(2,p-1\right)$. In fact,
it is easily checked that the poset of order ideals just mentioned (denoted by
$J\left(\left[2\right]\times\left[p-1\right]\right)$ in [StWi11]) is
isomorphic to $\operatorname*{Tria}\left(p\right)$.
We could also consider the subset
$\left\\{\left(i,k\right)\in\left\\{1,2,...,p\right\\}^{2}\ \mid\ i\geqslant
k\right\\}$, but that would yield a poset isomorphic to
$\operatorname*{Tria}\left(p\right)$ and thus would not be of any further
interest.
###### Theorem 17.4.
Let $p$ be a positive integer. Let $\mathbb{K}$ be a field. Then,
$\operatorname*{ord}\left(R_{\operatorname*{Tria}\left(p\right)}\right)=2p$.
This theorem yields
$\operatorname*{ord}\left(\overline{R}_{\operatorname*{Tria}\left(p\right)}\right)\mid
2p$. It can be shown that actually
$\operatorname*{ord}\left(\overline{R}_{\operatorname*{Tria}\left(p\right)}\right)=2p$
for $p>3$, while
$\operatorname*{ord}\left(\overline{R}_{\operatorname*{Tria}\left(1\right)}\right)=1$,
$\operatorname*{ord}\left(\overline{R}_{\operatorname*{Tria}\left(2\right)}\right)=1$
and
$\operatorname*{ord}\left(\overline{R}_{\operatorname*{Tria}\left(3\right)}\right)=2$.
Again, Theorem 17.4 is the birational version of a known result on classical
rowmotion: From [StWi11, Theorem 6.2] (and our Remark 17.3), it follows that
$\operatorname*{ord}\left(\mathbf{r}_{\operatorname*{Tria}\left(p\right)}\right)=2p$
(using the notations of Definition 10.7 and Definition 10.28). Theorem 17.4
thus shows that birational rowmotion and classical rowmotion have the same
order for $\operatorname*{Tria}\left(p\right)$.
In order to prove Theorem 17.4, we need a way to turn labellings of
$\operatorname*{Tria}\left(p\right)$ into labellings of
$\operatorname*{Rect}\left(p,p\right)$ in a rowmotion-equivariant way. It
turns out that the obvious “unfolding” construction (with some fudge
coefficients) works:
###### Lemma 17.5.
Let $p$ be a positive integer. Let $\mathbb{K}$ be a field of characteristic
$\neq 2$.
(a) Let
$\operatorname*{vrefl}:\operatorname*{Rect}\left(p,p\right)\rightarrow\operatorname*{Rect}\left(p,p\right)$
be the map sending every
$\left(i,k\right)\in\operatorname*{Rect}\left(p,p\right)$ to
$\left(k,i\right)$. This map $\operatorname*{vrefl}$ is an involutive poset
automorphism of $\operatorname*{Rect}\left(p,p\right)$. (In intuitive terms,
$\operatorname*{vrefl}$ is simply reflection across the vertical axis.) We
have
$\operatorname*{vrefl}\left(v\right)\in\operatorname*{Tria}\left(p\right)$ for
every
$v\in\operatorname*{Rect}\left(p,p\right)\setminus\operatorname*{Tria}\left(p\right)$.
We extend $\operatorname*{vrefl}$ to an involutive poset automorphism of
$\widehat{\operatorname*{Rect}\left(p,p\right)}$ by setting
$\operatorname*{vrefl}\left(0\right)=0$ and
$\operatorname*{vrefl}\left(1\right)=1$.
(b) Define a map
$\operatorname*{dble}:\mathbb{K}^{\widehat{\operatorname*{Tria}\left(p\right)}}\rightarrow\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}$
by setting
$\left(\operatorname*{dble}f\right)\left(v\right)=\left\\{\begin{array}[c]{l}\dfrac{1}{2}f\left(1\right),\
\ \ \ \ \ \ \ \ \ \text{if }v=1;\\\ 2f\left(0\right),\ \ \ \ \ \ \ \ \ \
\text{if }v=0;\\\ f\left(v\right),\ \ \ \ \ \ \ \ \ \ \text{if
}v\in\operatorname*{Tria}\left(p\right);\\\
f\left(\operatorname*{vrefl}\left(v\right)\right),\ \ \ \ \ \ \ \ \ \
\text{otherwise}\end{array}\right.$
for all $v\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ for all
$f\in\mathbb{K}^{\widehat{\operatorname*{Tria}\left(p\right)}}$. This is well-
defined. We have
$\left(\operatorname*{dble}f\right)\left(v\right)=f\left(v\right)\ \ \ \ \ \ \
\ \ \ \text{for every }v\in\operatorname*{Tria}\left(p\right).$ (67)
Also,
$\left(\operatorname*{dble}f\right)\left(\operatorname*{vrefl}\left(v\right)\right)=f\left(v\right)\
\ \ \ \ \ \ \ \ \ \text{for every }v\in\operatorname*{Tria}\left(p\right).$
(68)
(c) We have
$R_{\operatorname*{Rect}\left(p,p\right)}\circ\operatorname*{dble}=\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}.$
The coefficients $\dfrac{1}{2}$ and $2$ in the definition of
$\operatorname*{dble}$ ensure that the equality
$R_{\operatorname*{Rect}\left(p,p\right)}\circ\operatorname*{dble}=\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}$ in part (c) of the Lemma holds on the
level of labellings and not just up to homogeneous equivalence.
###### Proof of Lemma 17.5 (sketched)..
(a) Obvious.
(b) The well-definedness of $\operatorname*{dble}$ is pretty obvious. The
relation (67) follows from the definition of $\operatorname*{dble}$. The
relation (68) follows from the fact that every
$v\in\operatorname*{Tria}\left(p\right)$ satisfies either
$\operatorname*{vrefl}\left(v\right)\notin\operatorname*{Tria}\left(p\right)\cup\left\\{0,1\right\\}$
(in which case the definition of $\operatorname*{dble}f$ yields
$\left(\operatorname*{dble}f\right)\left(\operatorname*{vrefl}\left(v\right)\right)=f\left(\underbrace{\operatorname*{vrefl}\left(\operatorname*{vrefl}\left(v\right)\right)}_{=v}\right)=f\left(v\right)$)
or $\operatorname*{vrefl}\left(v\right)=v$ (in which case
$\left(\operatorname*{dble}f\right)\left(\underbrace{\operatorname*{vrefl}\left(v\right)}_{=v}\right)=\left(\operatorname*{dble}f\right)\left(v\right)=f\left(v\right)$
again by the definition of $\operatorname*{dble}f$). This proves Lemma 17.5
(b).
(c) We need to check that $\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}=R_{\operatorname*{Rect}\left(p,p\right)}\circ\operatorname*{dble}$.
In other words, we have to prove that $\left(\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}\right)f=\left(R_{\operatorname*{Rect}\left(p,p\right)}\circ\operatorname*{dble}\right)f$
for every $f\in\mathbb{K}^{\widehat{\operatorname*{Tria}\left(p\right)}}$ for
which $R_{\operatorname*{Tria}\left(p\right)}\left(f\right)$ is well-defined.
So let $f\in\mathbb{K}^{\widehat{\operatorname*{Tria}\left(p\right)}}$ be such
that $R_{\operatorname*{Tria}\left(p\right)}\left(f\right)$ is well-defined.
Set $f^{\prime}=\operatorname*{dble}f$ and
$g=R_{\operatorname*{Tria}\left(p\right)}f$. Set
$g^{\prime}=\operatorname*{dble}g$. Then,
$\left(\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}\right)f=\operatorname*{dble}\left(\underbrace{R_{\operatorname*{Tria}\left(p\right)}f}_{=g}\right)=\operatorname*{dble}g=g^{\prime}$
and
$\left(R_{\operatorname*{Rect}\left(p,p\right)}\circ\operatorname*{dble}\right)f=R_{\operatorname*{Rect}\left(p,p\right)}\left(\underbrace{\operatorname*{dble}f}_{=f^{\prime}}\right)=R_{\operatorname*{Rect}\left(p,p\right)}f^{\prime}.$
Thus, our goal (namely, to prove that $\left(\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}\right)f=\left(R_{\operatorname*{Rect}\left(p,p\right)}\circ\operatorname*{dble}\right)f$)
is equivalent to showing that
$g^{\prime}=R_{\operatorname*{Rect}\left(p,p\right)}f^{\prime}$.
So we need to prove that
$g^{\prime}=R_{\operatorname*{Rect}\left(p,p\right)}f^{\prime}$. Since
$f^{\prime}\left(0\right)=g^{\prime}\left(0\right)$ (because the operation
$\operatorname*{dble}$ multiplies the label at $0$ with $2$, while the
operation $R_{\operatorname*{Tria}\left(p\right)}$ leaves it unchanged) and
$f^{\prime}\left(1\right)=g^{\prime}\left(1\right)$ (for a similar reason), we
know from Proposition 2.19 (applied to $\operatorname*{Rect}\left(p,p\right)$,
$f^{\prime}$ and $g^{\prime}$ instead of $P$, $f$ and $g$) that this will be
done if we can show that
$g^{\prime}\left(v\right)=\dfrac{1}{f^{\prime}\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\lessdot
v\end{subarray}}f^{\prime}\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g^{\prime}\left(u\right)}}\ \ \ \ \ \ \ \ \
\ \text{for every }v\in\operatorname*{Rect}\left(p,p\right).$ (69)
Our goal is therefore to prove (69).
But every $v\in\operatorname*{Tria}\left(p\right)$ satisfies
$g\left(v\right)=\left(R_{\operatorname*{Tria}\left(p\right)}f\right)\left(v\right)=\dfrac{1}{f\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Tria}\left(p\right)};\\\
u\lessdot
v\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Tria}\left(p\right)};\\\
u\gtrdot
v\end{subarray}}\dfrac{1}{\left(R_{\operatorname*{Tria}\left(p\right)}f\right)\left(u\right)}}$
(by Proposition 2.16, applied to $\operatorname*{Tria}\left(p\right)$ instead
of $P$). Since $R_{\operatorname*{Tria}\left(p\right)}f=g$, this rewrites as
$g\left(v\right)=\dfrac{1}{f\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Tria}\left(p\right)};\\\
u\lessdot
v\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Tria}\left(p\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}}.$ (70)
Now, let us prove (69). So fix $v\in\operatorname*{Rect}\left(p,p\right)$.
Write $v$ in the form $v=\left(i,k\right)\in\left\\{1,2,...,p\right\\}^{2}$.
We distinguish between three cases:
Case 1: We have $i<k$.
Case 2: We have $i=k$.
Case 3: We have $i>k$.
Let us first consider Case 1. In this case, $i<k$. As a consequence, every
$u\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ satisfying $u\lessdot v$
lies in $\operatorname*{Tria}\left(p\right)$. Hence, every
$u\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ satisfying $u\lessdot v$
satisfies
$\underbrace{f^{\prime}}_{=\operatorname*{dble}f}\left(u\right)=\left(\operatorname*{dble}f\right)\left(u\right)=f\left(u\right)$
(71)
(by (67)). Thus,
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\lessdot
v\end{subarray}}\underbrace{f^{\prime}\left(u\right)}_{=f\left(u\right)}=\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\lessdot
v\end{subarray}}f\left(u\right)=\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Tria}\left(p\right)};\\\
u\lessdot v\end{subarray}}f\left(u\right)$ (72)
(since the elements $u\in\widehat{\operatorname*{Tria}\left(p\right)}$ such
that $u\lessdot v$ are precisely the elements
$u\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ such that $u\lessdot v$).
Also, every $u\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ satisfying
$u\gtrdot v$ lies in $\operatorname*{Tria}\left(p\right)$. Hence, every
$u\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ satisfying $u\gtrdot v$
satisfies
$\underbrace{g^{\prime}}_{=\operatorname*{dble}g}\left(u\right)=\left(\operatorname*{dble}g\right)\left(u\right)=g\left(u\right)$
(73)
(by (67)). Hence,
$\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\gtrdot
v\end{subarray}}\dfrac{1}{g^{\prime}\left(u\right)}=\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\gtrdot
v\end{subarray}}\dfrac{1}{g\left(u\right)}=\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Tria}\left(p\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}$ (74)
(because the elements $u\in\widehat{\operatorname*{Tria}\left(p\right)}$ such
that $u\gtrdot v$ are precisely the elements
$u\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ such that $u\gtrdot v$).
Finally, from $i<k$, we have $v\in\operatorname*{Tria}\left(p\right)$, so that
$\underbrace{f^{\prime}}_{=\operatorname*{dble}f}\left(v\right)=\left(\operatorname*{dble}f\right)\left(v\right)=f\left(v\right)$
(by (67)) and similarly $g^{\prime}\left(v\right)=g\left(v\right)$.
Using the equalities (72) and (74) as well as
$f^{\prime}\left(v\right)=f\left(v\right)$ and
$g^{\prime}\left(v\right)=g\left(v\right)$, we can rewrite (69) as
$g\left(v\right)=\dfrac{1}{f\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Tria}\left(p\right)};\\\
u\lessdot
v\end{subarray}}f\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Tria}\left(p\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g\left(u\right)}}.$
But this follows from (70). Since (70) is known to hold, we thus have proven
(69) in Case 1.
Let us next consider Case 3. It is very easy to check that every
$h\in\operatorname*{dble}\left(\mathbb{K}^{\widehat{\operatorname*{Tria}\left(p\right)}}\right)$
satisfies $h\left(\operatorname*{vrefl}\left(w\right)\right)=h\left(w\right)$
for every $w\in\widehat{\operatorname*{Rect}\left(p,p\right)}$. Applied to
$h=f^{\prime}$ (which belongs to
$\operatorname*{dble}\left(\mathbb{K}^{\widehat{\operatorname*{Tria}\left(p\right)}}\right)$
because $f^{\prime}=\operatorname*{dble}f$), this yields
$f^{\prime}\left(\operatorname*{vrefl}\left(w\right)\right)=f^{\prime}\left(w\right)$
for every $w\in\widehat{\operatorname*{Rect}\left(p,p\right)}$. But applied to
$h=g^{\prime}$ (which belongs to
$\operatorname*{dble}\left(\mathbb{K}^{\widehat{\operatorname*{Tria}\left(p\right)}}\right)$
because $g^{\prime}=\operatorname*{dble}g$), the same property yields
$g^{\prime}\left(\operatorname*{vrefl}\left(w\right)\right)=g^{\prime}\left(w\right)$
for every $w\in\widehat{\operatorname*{Rect}\left(p,p\right)}$. We thus can
rewrite the equality (69) (which we desire to prove) by replacing each
$g^{\prime}\left(w\right)$ by
$g^{\prime}\left(\operatorname*{vrefl}\left(w\right)\right)$ and by replacing
each $f^{\prime}\left(w\right)$ by
$f^{\prime}\left(\operatorname*{vrefl}\left(w\right)\right)$. Additionally, we
can replace “$u\lessdot v$” by
“$\operatorname*{vrefl}\left(u\right)\lessdot\operatorname*{vrefl}\left(v\right)$”,
and replace “$u\gtrdot v$” by
“$\operatorname*{vrefl}\left(u\right)\gtrdot\operatorname*{vrefl}\left(v\right)$”.
Consequently, (69) rewrites as
$g^{\prime}\left(\operatorname*{vrefl}\left(v\right)\right)=\dfrac{1}{f^{\prime}\left(\operatorname*{vrefl}\left(v\right)\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
\operatorname*{vrefl}\left(u\right)\lessdot\operatorname*{vrefl}\left(v\right)\end{subarray}}f^{\prime}\left(\operatorname*{vrefl}\left(u\right)\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
\operatorname*{vrefl}\left(u\right)\gtrdot\operatorname*{vrefl}\left(v\right)\end{subarray}}\dfrac{1}{g^{\prime}\left(\operatorname*{vrefl}\left(u\right)\right)}}.$
(75)
This equality can be simplified further by substituting $u$ for
$\operatorname*{vrefl}\left(u\right)$ on its right hand side:
$g^{\prime}\left(\operatorname*{vrefl}\left(v\right)\right)=\dfrac{1}{f^{\prime}\left(\operatorname*{vrefl}\left(v\right)\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\lessdot\operatorname*{vrefl}\left(v\right)\end{subarray}}f^{\prime}\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\gtrdot\operatorname*{vrefl}\left(v\right)\end{subarray}}\dfrac{1}{g^{\prime}\left(u\right)}}.$
(76)
This is precisely the statement of (69) with
$\operatorname*{vrefl}\left(v\right)$ instead of $v$. But since we are in Case
3 with our element $v$, we have $i>k$, so that $k<i$, and thus the element
$\operatorname*{vrefl}\left(v\right)=\left(k,i\right)$ of
$\operatorname*{Rect}\left(p,p\right)$ is in Case 1. Having already verified
(69) in Case 1, we can thus apply (69) to
$\operatorname*{vrefl}\left(v\right)$ instead of $v$, and conclude that (76)
holds. This, as we know, is equivalent to (69), and so (69) is proven in Case
3.
Let us finally consider Case 2. In this case, $i=k$. Thus,
$v=\left(i,\underbrace{k}_{=i}\right)=\left(i,i\right)$. Hence,
$v\in\widehat{\operatorname*{Tria}\left(p\right)}$. Thus,
$\underbrace{f^{\prime}}_{=\operatorname*{dble}f}\left(v\right)=\left(\operatorname*{dble}f\right)\left(v\right)=f\left(v\right)$
(by (67), since $v\in\operatorname*{Tria}\left(p\right)$). Similarly,
$g^{\prime}\left(v\right)=g\left(v\right)$.
We should now consider four subcases, depending on whether
$i\notin\left\\{1,p\right\\}$ or $i=1\neq p$ or $i=p\neq 1$ or $i=1=p$. But we
are only going to deal with the first of these subcases here, leaving the
other three to the reader. So let us consider the subcase when
$i\notin\left\\{1,p\right\\}$.
We have $v=\left(i,i\right)$. Thus, the only element
$u\in\widehat{\operatorname*{Tria}\left(p\right)}$ such that $u\gtrdot v$ is
$\left(i,i+1\right)$, and the only element
$u\in\widehat{\operatorname*{Tria}\left(p\right)}$ such that $u\lessdot v$ is
$\left(i-1,i\right)$. Thus, (70) simplifies to
$g\left(v\right)=\dfrac{1}{f\left(v\right)}\cdot\dfrac{f\left(\left(i-1,i\right)\right)}{\left(\dfrac{1}{g\left(\left(i,i+1\right)\right)}\right)}.$
(77)
Now, recall that $g^{\prime}=\operatorname*{dble}g$. From the definition of
$\operatorname*{dble}g$, it therefore follows easily that
$g^{\prime}\left(\left(i,i+1\right)\right)=g\left(\left(i,i+1\right)\right)$
and
$g^{\prime}\left(\left(i+1,i\right)\right)=g\left(\left(i,i+1\right)\right)$.
Also, $f^{\prime}=\operatorname*{dble}f$. From the definition of
$\operatorname*{dble}f$, we thus obtain
$f^{\prime}\left(\left(i-1,i\right)\right)=f\left(\left(i-1,i\right)\right)$
and
$f^{\prime}\left(\left(i,i-1\right)\right)=f\left(\left(i-1,i\right)\right)$.
But the elements $u\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ such
that $u\gtrdot v$ are precisely $\left(i+1,i\right)$ and $\left(i,i+1\right)$,
and the elements $u\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ such
that $u\lessdot v$ are precisely $\left(i-1,i\right)$ and
$\left(i,i-1\right)$. Thus, the right hand side of (69) simplifies as follows:
$\displaystyle\dfrac{1}{f^{\prime}\left(v\right)}\cdot\dfrac{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\lessdot
v\end{subarray}}f^{\prime}\left(u\right)}{\sum\limits_{\begin{subarray}{c}u\in\widehat{\operatorname*{Rect}\left(p,p\right)};\\\
u\gtrdot v\end{subarray}}\dfrac{1}{g^{\prime}\left(u\right)}}$
$\displaystyle=\dfrac{1}{f^{\prime}\left(v\right)}\cdot\dfrac{f^{\prime}\left(\left(i-1,i\right)\right)+f^{\prime}\left(\left(i,i-1\right)\right)}{\dfrac{1}{g^{\prime}\left(\left(i+1,i\right)\right)}+\dfrac{1}{g^{\prime}\left(\left(i,i+1\right)\right)}}=\dfrac{1}{f\left(v\right)}\cdot\dfrac{f\left(\left(i-1,i\right)\right)+f\left(\left(i-1,i\right)\right)}{\dfrac{1}{g\left(\left(i,i+1\right)\right)}+\dfrac{1}{g\left(\left(i,i+1\right)\right)}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\begin{array}[c]{c}\text{since
}f^{\prime}\left(\left(i-1,i\right)\right)=f\left(\left(i-1,i\right)\right)\text{,
}f^{\prime}\left(\left(i,i-1\right)\right)=f\left(\left(i-1,i\right)\right)\text{,}\\\
g^{\prime}\left(\left(i+1,i\right)\right)=g\left(\left(i,i+1\right)\right)\text{,
}g^{\prime}\left(\left(i,i+1\right)\right)=g\left(\left(i,i+1\right)\right)\\\
\text{and }f^{\prime}\left(v\right)=f\left(v\right)\end{array}\right)$
$\displaystyle=\dfrac{1}{f\left(v\right)}\cdot\dfrac{2\cdot
f\left(\left(i-1,i\right)\right)}{2\cdot\dfrac{1}{g\left(\left(i,i+1\right)\right)}}=\dfrac{1}{f\left(v\right)}\cdot\dfrac{f\left(\left(i-1,i\right)\right)}{\left(\dfrac{1}{g\left(\left(i,i+1\right)\right)}\right)}=g\left(v\right)\
\ \ \ \ \ \ \ \ \ \left(\text{by (\ref{pf.Leftri.vrefl.5})}\right)$
$\displaystyle=g^{\prime}\left(v\right).$
In other words, (69) is proven in Case 2.
We have now proven (69) in all three cases (not counting the subcases which we
left to the reader to “enjoy”). Thus, (69) holds, and as we know this yields
that $g^{\prime}=R_{\operatorname*{Rect}\left(p,p\right)}f^{\prime}$. Lemma
17.5 (c) is thus proven. ∎
###### Proof of Theorem 17.4 (sketched)..
Applying Proposition 7.3 to $2p-1$ and $\operatorname*{Tria}\left(p\right)$
instead of $n$ and $P$, we obtain
$\operatorname*{ord}\left(R_{\operatorname*{Tria}\left(p\right)}\right)=\operatorname{lcm}\left(2p-1+1,\operatorname*{ord}\left(\overline{R}_{\operatorname*{Tria}\left(p\right)}\right)\right)$.
Hence,
$\operatorname*{ord}\left(R_{\operatorname*{Tria}\left(p\right)}\right)$ is
divisible by $2p-1+1=2p$. Now, if we can prove that
$\operatorname*{ord}\left(R_{\operatorname*{Tria}\left(p\right)}\right)\mid
2p$, then we will immediately obtain
$\operatorname*{ord}\left(R_{\operatorname*{Tria}\left(p\right)}\right)=2p$,
and Theorem 17.4 will be proven.
So let us show that
$\operatorname*{ord}\left(R_{\operatorname*{Tria}\left(p\right)}\right)\mid
2p$. This means showing that
$R_{\operatorname*{Tria}\left(p\right)}^{2p}=\operatorname*{id}$. Since this
statement boils down to a collection of polynomial identities in the labels of
an arbitrary $\mathbb{K}$-labelling of $\operatorname*{Tria}\left(p\right)$,
it is clear that it is enough to prove it in the case when $\mathbb{K}$ is a
field of rational functions in finitely many variables over $\mathbb{Q}$. So
let us WLOG assume that $\mathbb{K}$ is a field of rational functions in
finitely many variables over $\mathbb{Q}$. Then, the characteristic of
$\mathbb{K}$ is $\neq 2$ (it is $0$ indeed), so that we can apply Lemma 17.5.
Let us use the notations of Lemma 17.5. Lemma 17.5 (c) yields
$R_{\operatorname*{Rect}\left(p,p\right)}\circ\operatorname*{dble}=\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}.$
From this, it follows (by induction over $k$) that
$R_{\operatorname*{Rect}\left(p,p\right)}^{k}\circ\operatorname*{dble}=\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}^{k}$
for every $k\in\mathbb{N}$. Applied to $k=2p$, this yields
$R_{\operatorname*{Rect}\left(p,p\right)}^{2p}\circ\operatorname*{dble}=\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}^{2p}.$ (78)
But Theorem 11.5 (applied to $q=p$) yields
$\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,p\right)}\right)=p+p=2p$,
so that $R_{\operatorname*{Rect}\left(p,p\right)}^{2p}=\operatorname*{id}$.
Hence, (78) simplifies to
$\operatorname*{dble}=\operatorname*{dble}\circ
R_{\operatorname*{Tria}\left(p\right)}^{2p}.$
We can cancel $\operatorname*{dble}$ from this equation, because
$\operatorname*{dble}$ is an injective and therefore left-cancellable map. As
a consequence, we obtain
$\operatorname*{id}=R_{\operatorname*{Tria}\left(p\right)}^{2p}$. In other
words, $R_{\operatorname*{Tria}\left(p\right)}^{2p}=\operatorname*{id}$. This
proves Theorem 17.4. ∎
## 18 The $\Delta$ and $\nabla$ triangles
The next kind of triangle-shaped posets is more interesting.
###### Definition 18.1.
Let $p$ be a positive integer. Define three subsets $\Delta\left(p\right)$,
$\operatorname*{Eq}\left(p\right)$ and $\nabla\left(p\right)$ of
$\operatorname*{Rect}\left(p,p\right)=\left\\{1,2,...,p\right\\}\times\left\\{1,2,...,p\right\\}=\left\\{1,2,...,p\right\\}^{2}$
by
$\displaystyle\Delta\left(p\right)$
$\displaystyle=\left\\{\left(i,k\right)\in\left\\{1,2,...,p\right\\}^{2}\
\mid\ i+k>p+1\right\\};$ $\displaystyle\operatorname*{Eq}\left(p\right)$
$\displaystyle=\left\\{\left(i,k\right)\in\left\\{1,2,...,p\right\\}^{2}\
\mid\ i+k=p+1\right\\};$ $\displaystyle\nabla\left(p\right)$
$\displaystyle=\left\\{\left(i,k\right)\in\left\\{1,2,...,p\right\\}^{2}\
\mid\ i+k<p+1\right\\}.$
These subsets $\Delta\left(p\right)$, $\operatorname*{Eq}\left(p\right)$ and
$\nabla\left(p\right)$ inherit a poset structure from
$\operatorname*{Rect}\left(p,p\right)$. In the following, we will consider
$\Delta\left(p\right)$, $\operatorname*{Eq}\left(p\right)$ and
$\nabla\left(p\right)$ as posets using this structure.
Clearly, $\operatorname*{Eq}\left(p\right)$ is an antichain with $p$ elements.
(The name $\operatorname*{Eq}$ comes from “equator”.) The posets
$\Delta\left(p\right)$ and $\nabla\left(p\right)$ are
$\left(p-1\right)$-graded posets. They have the form of a “Delta-shaped
triangle” and a “Nabla-shaped triangle”, respectively (whence the names).
###### Example 18.2.
Here is the Hasse diagram of the poset $\operatorname*{Rect}\left(4,4\right)$,
where the elements belonging to $\Delta\left(4\right)$ have been underlined
and the elements belonging to $\operatorname*{Eq}\left(4\right)$ have been
boxed:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.11111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&&\\\&&&&&&\\\&&&&&&\\\&&&&&&\\\&&&&&&\\\&&&&&&\\\&&&&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.6222pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 58.24438pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 86.36658pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\underline{\left(4,4\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
119.48877pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 150.11096pt\raise
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And here is the Hasse diagram of the poset $\Delta\left(4\right)$ itself:
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Here, on the other hand, is the Hasse diagram of the poset
$\nabla\left(4\right)$:
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###### Remark 18.3.
Let $p$ be a positive integer. The poset $\Delta\left(p\right)$ is isomorphic
to the poset $\Phi^{+}\left(A_{p-1}\right)$ of [StWi11, §3.2].
###### Remark 18.4.
For every positive integer $p$, we have
$\nabla\left(p\right)\cong\left(\Delta\left(p\right)\right)^{\operatorname*{op}}$
as posets. This follows immediately from the poset antiautomorphism
$\displaystyle\operatorname*{hrefl}:\operatorname*{Rect}\left(p,p\right)$
$\displaystyle\rightarrow\operatorname*{Rect}\left(p,p\right),$
$\displaystyle\left(i,k\right)$ $\displaystyle\mapsto\left(p+1-k,p+1-i\right)$
sending $\nabla\left(p\right)$ to $\Delta\left(p\right)$.
Here we are using the following notions:
###### Definition 18.5.
(a) If $P$ and $Q$ are two posets, then a map $f:P\rightarrow Q$ is called a
poset antihomomorphism if and only if every $p_{1}\in P$ and $p_{2}\in P$
satisfying $p_{1}\leqslant p_{2}$ in $P$ satisfy $f\left(p_{1}\right)\geqslant
f\left(p_{2}\right)$ in $Q$. It is easy to see that the poset
antihomomorphisms $P\rightarrow Q$ are precisely the poset homomorphisms
$P\rightarrow Q^{\operatorname*{op}}$.
(b) If $P$ and $Q$ are two posets, then an invertible map $f:P\rightarrow Q$
is called a poset antiisomorphism if and only if both $f$ and $f^{-1}$ are
poset antihomomorphisms.
(c) If $P$ is a poset and $f:P\rightarrow P$ is an invertible map, then $f$ is
said to be a poset antiautomorphism if $f$ is a poset antiisomorphism.
We now state the main property of birational rowmotion $R$ on the posets
$\nabla\left(p\right)$ and $\Delta\left(p\right)$:
###### Theorem 18.6.
Let $p$ be an integer $\geqslant 1$. Let $\mathbb{K}$ be a field. For every
$\left(i,k\right)\in\nabla\left(p\right)$ and every
$f\in\mathbb{K}^{\widehat{\nabla\left(p\right)}}$, we have
$\left(R_{\nabla\left(p\right)}^{p}f\right)\left(\left(i,k\right)\right)=f\left(\left(k,i\right)\right).$
###### Theorem 18.7.
Let $p$ be an integer $\geqslant 1$. Let $\mathbb{K}$ be a field. For every
$\left(i,k\right)\in\Delta\left(p\right)$ and every
$f\in\mathbb{K}^{\widehat{\Delta\left(p\right)}}$, we have
$\left(R_{\Delta\left(p\right)}^{p}f\right)\left(\left(i,k\right)\right)=f\left(\left(k,i\right)\right).$
The following two corollaries follow easily from these two theorems:
###### Corollary 18.8.
Let $p$ be an integer $>1$. Let $\mathbb{K}$ be a field. Then:
(a) We have $\operatorname*{ord}\left(R_{\nabla\left(p\right)}\right)\mid 2p$.
(b) If $p>2$, then
$\operatorname*{ord}\left(R_{\nabla\left(p\right)}\right)=2p$.
###### Corollary 18.9.
Let $p$ be an integer $>1$. Let $\mathbb{K}$ be a field. Then:
(a) We have $\operatorname*{ord}\left(R_{\Delta\left(p\right)}\right)\mid 2p$.
(b) If $p>2$, then
$\operatorname*{ord}\left(R_{\Delta\left(p\right)}\right)=2p$.
Corollary 18.9 is analogous to a known result for classical rowmotion. In
fact, from [StWi11, Conjecture 3.6] (originally a conjecture of Panyushev,
then proven by Armstrong, Stump and Thomas) and our Remark 18.3, it can be
seen that (using the notations of Definition 10.7 and Definition 10.28) every
integer $p>2$ satisfies
$\operatorname*{ord}\left(\mathbf{r}_{\Delta\left(p\right)}\right)=2p$.
We now prepare for the proofs of Theorems 18.6 and 18.7.
First of all, Corollary 18.8 is clearly equivalent to Corollary 18.9 (because
of Remark 18.4 and Proposition 8.4). It is a bit more complicated to see that
Theorem 18.6 is equivalent to Theorem 18.7; we will show this later. But let
us first prove Theorem 18.7. The proof will use a mapping that transforms
labellings of $\Delta\left(p\right)$ into labellings of
$\operatorname*{Rect}\left(p,p\right)$ in a way that is rowmotion-equivariant
up to homogeneous equivalence. This mapping is similar in its function to the
mapping $\operatorname*{dble}$ of Lemma 17.5, but its definition is more
intricate:404040See also Lemma 18.12 further below for a generalization of
parts of this construction.
###### Lemma 18.10.
Let $p$ be a positive integer. Clearly, $\operatorname*{Rect}\left(p,p\right)$
is the disjoint union of the sets $\Delta\left(p\right)$,
$\nabla\left(p\right)$ and $\operatorname*{Eq}\left(p\right)$.
Let $\mathbb{K}$ be a field of characteristic $\neq 2$.
(a) Let
$\operatorname*{hrefl}:\operatorname*{Rect}\left(p,p\right)\rightarrow\operatorname*{Rect}\left(p,p\right)$
be the map sending every
$\left(i,k\right)\in\operatorname*{Rect}\left(p,p\right)$ to
$\left(p+1-k,p+1-i\right)$. This map $\operatorname*{hrefl}$ is an involution
and a poset antiautomorphism of $\operatorname*{Rect}\left(p,p\right)$. (In
intuitive terms, $\operatorname*{hrefl}$ is simply reflection across the
horizontal axis (i.e., the line $\operatorname*{Eq}\left(p\right)$).) We have
$\operatorname*{hrefl}\mid_{\operatorname*{Eq}\left(p\right)}=\operatorname*{id}$
and
$\operatorname*{hrefl}\left(\Delta\left(p\right)\right)=\nabla\left(p\right)$.
We extend $\operatorname*{hrefl}$ to an involutive poset antiautomorphism of
$\widehat{\operatorname*{Rect}\left(p,p\right)}$ by setting
$\operatorname*{hrefl}\left(0\right)=1$ and
$\operatorname*{hrefl}\left(1\right)=0$.
(b) Define a rational map
$\operatorname*{wing}:\mathbb{K}^{\widehat{\Delta\left(p\right)}}\dashrightarrow\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}$
by setting
$\left(\operatorname*{wing}f\right)\left(v\right)=\left\\{\begin{array}[c]{l}f\left(v\right),\
\ \ \ \ \ \ \ \ \ \text{if }v\in\Delta\left(p\right)\cup\left\\{1\right\\};\\\
1,\ \ \ \ \ \ \ \ \ \ \text{if }v\in\operatorname*{Eq}\left(p\right);\\\
\dfrac{1}{\left(R_{\Delta\left(p\right)}^{p-\deg
v}f\right)\left(\operatorname*{hrefl}v\right)},\ \ \ \ \ \ \ \ \ \ \text{if
}v\in\nabla\left(p\right)\cup\left\\{0\right\\}\end{array}\right.$
for all $v\in\widehat{\operatorname*{Rect}\left(p,p\right)}$ for all
$f\in\mathbb{K}^{\widehat{\Delta\left(p\right)}}$. This is well-defined.
(c) There exists a rational map
$\overline{\operatorname*{wing}}:\overline{\mathbb{K}^{\widehat{\Delta\left(p\right)}}}\dashrightarrow\overline{\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}}$
such that the diagram
$\textstyle{\mathbb{K}^{\widehat{\Delta\left(p\right)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\operatorname*{wing}}$$\textstyle{\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathbb{K}^{\widehat{\Delta\left(p\right)}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\operatorname*{wing}}}$$\textstyle{\overline{\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}}}$
(79)
commutes.
(d) The rational map $\overline{\operatorname*{wing}}$ defined in Lemma 18.10
(c) satisfies
$\overline{R}_{\operatorname*{Rect}\left(p,p\right)}\circ\overline{\operatorname*{wing}}=\overline{\operatorname*{wing}}\circ\overline{R}_{\Delta\left(p\right)}.$
(e) Consider the map
$\operatorname*{vrefl}:\operatorname*{Rect}\left(p,p\right)\rightarrow\operatorname*{Rect}\left(p,p\right)$
defined in Lemma 17.5. Define a map
$\operatorname*{vrefl}\nolimits^{\ast}:\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}\rightarrow\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}$
by setting
$\left(\operatorname*{vrefl}\nolimits^{\ast}f\right)\left(v\right)=f\left(\operatorname*{vrefl}\left(v\right)\right)\
\ \ \ \ \ \ \ \ \ \text{for all
}v\in\widehat{\operatorname*{Rect}\left(p,p\right)}$
for all $f\in\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}$.
Also, define a map
$\operatorname*{vrefl}\nolimits^{\ast}:\mathbb{K}^{\widehat{\Delta\left(p\right)}}\rightarrow\mathbb{K}^{\widehat{\Delta\left(p\right)}}$
by setting
$\left(\operatorname*{vrefl}\nolimits^{\ast}f\right)\left(v\right)=f\left(\operatorname*{vrefl}\left(v\right)\right)\
\ \ \ \ \ \ \ \ \ \text{for all }v\in\widehat{\Delta\left(p\right)}$
for all $f\in\mathbb{K}^{\widehat{\Delta\left(p\right)}}$. Then,
$\operatorname*{vrefl}\nolimits^{\ast}\circ
R_{\Delta\left(p\right)}=R_{\Delta\left(p\right)}\circ\operatorname*{vrefl}\nolimits^{\ast}$
(80)
(as rational maps
$\mathbb{K}^{\widehat{\Delta\left(p\right)}}\dashrightarrow\mathbb{K}^{\widehat{\Delta\left(p\right)}}$).
Furthermore,
$\operatorname*{vrefl}\nolimits^{\ast}\circ
R_{\operatorname*{Rect}\left(p,p\right)}=R_{\operatorname*{Rect}\left(p,p\right)}\circ\operatorname*{vrefl}\nolimits^{\ast}$
(81)
(as rational maps
$\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}\dashrightarrow\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}$).
Finally,
$\operatorname*{vrefl}\nolimits^{\ast}\circ\operatorname*{wing}=\operatorname*{wing}\circ\operatorname*{vrefl}\nolimits^{\ast}$
(82)
(as rational maps
$\mathbb{K}^{\widehat{\Delta\left(p\right)}}\dashrightarrow\mathbb{K}^{\widehat{\operatorname*{Rect}\left(p,p\right)}}$).
(f) Almost every (in the sense of Zariski topology) labelling
$f\in\mathbb{K}^{\widehat{\Delta\left(p\right)}}$ satisfying
$f\left(0\right)=2$ satisfies
$R_{\operatorname*{Rect}\left(p,p\right)}\left(\operatorname*{wing}f\right)=\operatorname*{wing}\left(R_{\Delta\left(p\right)}f\right).$
(g) If $f$ and $g$ are two homogeneously equivalent zero-free
$\mathbb{K}$-labellings of $\Delta\left(p\right)$, then
$\operatorname*{vrefl}\nolimits^{\ast}f$ is homogeneously equivalent to
$\operatorname*{vrefl}\nolimits^{\ast}g$.
###### Proof of Lemma 18.10 (sketched)..
We will not delve into the details of this tedious and yet straightforward
proof. Let us merely make some comments on the few interesting parts of it.
Parts (a), (b), (c) and (g) are obvious. Part (f) can be verified label-by-
label using Propositions 2.16 and 2.19 and some nasty casework. Part (d) won’t
be used in the following, but can easily be derived from part (f). Part (e)
more or less follows from the fact that the definitions of
$R_{\Delta\left(p\right)}$, $R_{\operatorname*{Rect}\left(p,p\right)}$ and
$\operatorname*{wing}$ are all “invariant” under the vertical reflection
$\operatorname*{vrefl}$; but proving part (e) in a pedestrian way might be
even more straightforward than formalizing this invariance
argument414141Again, Propositions 2.16 and 2.19 come in handy for proving (80)
and (81). Then, one can prove (by induction over $\ell$) that
$\operatorname*{vrefl}\nolimits^{\ast}\circ
R_{\Delta\left(p\right)}^{\ell}=R_{\Delta\left(p\right)}^{\ell}\circ\operatorname*{vrefl}\nolimits^{\ast}$
for all $\ell\in\mathbb{N}$. Using this, (82) is straightforward to check.. ∎
For easier reference, let us record a corollary of Lemma 18.10 (f):
###### Corollary 18.11.
Let $p$ be a positive integer. Let $\mathbb{K}$ be a field of characteristic
$\neq 2$. Consider the map $\operatorname*{wing}$ defined in Lemma 18.10. Let
$\ell\in\mathbb{N}$.
Then, almost every (in the sense of Zariski topology) labelling
$f\in\mathbb{K}^{\widehat{\Delta\left(p\right)}}$ satisfying
$f\left(0\right)=2$ satisfies
$R_{\operatorname*{Rect}\left(p,p\right)}^{\ell}\left(\operatorname*{wing}f\right)=\operatorname*{wing}\left(R_{\Delta\left(p\right)}^{\ell}f\right).$
###### Proof of Corollary 18.11 (sketched)..
The proof of Corollary 18.11 is an easy induction over $\ell$ (details left to
the reader), using Lemma 18.10 (f) and the fact that
$R_{\Delta\left(p\right)}$ does not change the label at $1$. ∎
We can now proceed to the proof of the theorems stated at the beginning of
this section:
###### Proof of Theorem 18.7 (sketched)..
The result that we are striving to prove is a collection of identities between
rational functions, hence boils down to a collection of polynomial identities
in the labels of an arbitrary $\mathbb{K}$-labelling of
$\Delta\left(p\right)$. Therefore, it is enough to prove it in the case when
$\mathbb{K}$ is a field of rational functions in finitely many variables over
$\mathbb{Q}$. So let us WLOG assume that we are in this case. Then, the
characteristic of $\mathbb{K}$ is $\neq 2$ (it is $0$ indeed), so that we can
apply Lemma 18.10 and Corollary 18.11.
Consider the maps $\operatorname*{hrefl}$, $\operatorname*{wing}$,
$\operatorname*{vrefl}$ and $\operatorname*{vrefl}\nolimits^{\ast}$ defined in
Lemma 18.10. Clearly, it will be enough to prove that
$R_{\Delta\left(p\right)}^{p}=\operatorname*{vrefl}\nolimits^{\ast}$
as rational maps
$\mathbb{K}^{\widehat{\Delta\left(p\right)}}\dashrightarrow\mathbb{K}^{\widehat{\Delta\left(p\right)}}$.
In other words, it will be enough to prove that
$R_{\Delta\left(p\right)}^{p}g=\operatorname*{vrefl}\nolimits^{\ast}g$ for
almost every $g\in\mathbb{K}^{\widehat{\Delta\left(p\right)}}$.
So let $g\in\mathbb{K}^{\widehat{\Delta\left(p\right)}}$ be any sufficiently
generic zero-free labelling of $\Delta\left(p\right)$. We need to show that
$R_{\Delta\left(p\right)}^{p}g=\operatorname*{vrefl}\nolimits^{\ast}g$.
Let us use Definition 5.2. The poset $\Delta\left(p\right)$ is
$\left(p-1\right)$-graded. We can find a $\left(p+1\right)$-tuple
$\left(a_{0},a_{1},...,a_{p}\right)\in\left(\mathbb{K}^{\times}\right)^{p+1}$
such that $\left(\left(a_{0},a_{1},...,a_{p}\right)\flat
g\right)\left(0\right)=2$ (by setting $a_{0}=\dfrac{2}{g\left(0\right)}$, and
choosing all other $a_{i}$ arbitrarily). Fix such a $\left(p+1\right)$-tuple,
and set $f=\left(a_{0},a_{1},...,a_{p}\right)\flat g$. Then,
$f\left(0\right)=2$. We are going to prove that
$R_{\Delta\left(p\right)}^{p}f=\operatorname*{vrefl}\nolimits^{\ast}f$. Until
we have done this, we can forget about $g$; all we need to know is that $f$ is
a sufficiently generic $\mathbb{K}$-labelling of $\Delta\left(p\right)$
satisfying $f\left(0\right)=2$.
Let $\left(i,k\right)\in\Delta\left(p\right)$ be arbitrary. Then, $i+k>p+1$
(since $\left(i,k\right)\in\Delta\left(p\right)$). Consequently,
$2p-\left(i+k-1\right)$ is a well-defined element of
$\left\\{1,2,...,p-1\right\\}$. Denote this element by $h$. Thus,
$h\in\left\\{1,2,...,p-1\right\\}$ and $i+k-1+h=2p$. Moreover,
$\left(k,i\right)=\operatorname*{vrefl}v\in\Delta\left(p\right)$.
Let $v=\left(p+1-k,p+1-i\right)$. Then,
$v=\operatorname*{hrefl}\left(\left(i,k\right)\right)\in\nabla\left(p\right)$
(since $\left(i,k\right)\in\Delta\left(p\right)$) and $\deg v=h$ (this follows
by simple computation). Moreover, $\operatorname*{hrefl}v=\left(i,k\right)$.
Applying Corollary 18.11 to $\ell=h$, we obtain
$R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)=\operatorname*{wing}\left(R_{\Delta\left(p\right)}^{h}f\right)$,
hence
$\displaystyle\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(v\right)$
$\displaystyle=\left(\operatorname*{wing}\left(R_{\Delta\left(p\right)}^{h}f\right)\right)\left(v\right)=\dfrac{1}{\left(R_{\Delta\left(p\right)}^{p-\deg
v}\left(R_{\Delta\left(p\right)}^{h}f\right)\right)\left(\operatorname*{hrefl}v\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{by the definition of
}\operatorname*{wing}\text{, since
}v\in\nabla\left(p\right)\subseteq\nabla\left(p\right)\cup\left\\{0\right\\}\right)$
$\displaystyle=\dfrac{1}{\left(R_{\Delta\left(p\right)}^{p-h}\left(R_{\Delta\left(p\right)}^{h}f\right)\right)\left(\left(i,k\right)\right)}\
\ \ \ \ \ \ \ \ \ \left(\text{since }\deg v=h\text{ and
}\operatorname*{hrefl}v=\left(i,k\right)\right)$
$\displaystyle=\dfrac{1}{\left(R_{\Delta\left(p\right)}^{p}f\right)\left(\left(i,k\right)\right)}\
\ \ \ \ \ \ \ \ \ \left(\text{since
}R_{\Delta\left(p\right)}^{p-h}\left(R_{\Delta\left(p\right)}^{h}f\right)=\underbrace{\left(R_{\Delta\left(p\right)}^{p-h}\circ
R_{\Delta\left(p\right)}^{h}\right)}_{=R_{\Delta\left(p\right)}^{p}}f=R_{\Delta\left(p\right)}^{p}f\right).$
(83)
But Theorem 11.7 (applied to $p$,
$R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)$
and $\left(k,i\right)$ instead of $q$, $f$ and $\left(i,k\right)$) yields
$\displaystyle\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(\left(p+1-k,p+1-i\right)\right)$
$\displaystyle=\dfrac{\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(0\right)\cdot\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(1\right)}{\left(R_{\operatorname*{Rect}\left(p,p\right)}^{i+k-1}\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\right)\left(\left(k,i\right)\right)}.$
Since $\left(p+1-k,p+1-i\right)=v$ and
$\displaystyle
R_{\operatorname*{Rect}\left(p,p\right)}^{i+k-1}\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)$
$\displaystyle=\left(\underbrace{R_{\operatorname*{Rect}\left(p,p\right)}^{i+k-1}\circ
R_{\operatorname*{Rect}\left(p,p\right)}^{h}}_{\begin{subarray}{c}=R_{\operatorname*{Rect}\left(p,p\right)}^{i+k-1+h}=R_{\operatorname*{Rect}\left(p,p\right)}^{2p}\\\
\text{(since
}i+k-1+h=2p\text{)}\end{subarray}}\right)\left(\operatorname*{wing}f\right)$
$\displaystyle=\underbrace{R_{\operatorname*{Rect}\left(p,p\right)}^{2p}}_{\begin{subarray}{c}=\operatorname*{id}\\\
\text{(since Theorem \ref{thm.rect.ord} (applied to }q=p\text{)}\\\
\text{yields
}\operatorname*{ord}\left(R_{\operatorname*{Rect}\left(p,p\right)}\right)=p+p=2p\text{)}\end{subarray}}\left(\operatorname*{wing}f\right)=\operatorname*{wing}f,$
this equality rewrites as
$\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(v\right)=\dfrac{\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(0\right)\cdot\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(1\right)}{\left(\operatorname*{wing}f\right)\left(\left(k,i\right)\right)}.$
Since
$\displaystyle\underbrace{\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(0\right)}_{\begin{subarray}{c}=\left(\operatorname*{wing}f\right)\left(0\right)\\\
\text{(by Corollary
\ref{cor.R.implicit.01})}\end{subarray}}\cdot\underbrace{\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(1\right)}_{\begin{subarray}{c}=\left(\operatorname*{wing}f\right)\left(1\right)\\\
\text{(by Corollary \ref{cor.R.implicit.01})}\end{subarray}}$
$\displaystyle=\underbrace{\left(\operatorname*{wing}f\right)\left(0\right)}_{\begin{subarray}{c}=\dfrac{1}{\left(R_{\Delta\left(p\right)}^{p-\deg
0}f\right)\left(\operatorname*{hrefl}0\right)}\\\ \text{(by the definition of
}\operatorname*{wing}\text{)}\end{subarray}}\cdot\underbrace{\left(\operatorname*{wing}f\right)\left(1\right)}_{\begin{subarray}{c}=f\left(1\right)\\\
\text{(by the definition of }\operatorname*{wing}\text{)}\end{subarray}}$
$\displaystyle=\dfrac{1}{\left(R_{\Delta\left(p\right)}^{p-\deg
0}f\right)\left(\operatorname*{hrefl}0\right)}\cdot f\left(1\right)=1$
(since Corollary 2.18 yields $\left(R_{\Delta\left(p\right)}^{p-\deg
0}f\right)\left(\operatorname*{hrefl}0\right)=f\left(\operatorname*{hrefl}0\right)=f\left(1\right)$),
this simplifies to
$\left(R_{\operatorname*{Rect}\left(p,p\right)}^{h}\left(\operatorname*{wing}f\right)\right)\left(v\right)=\dfrac{1}{\left(\operatorname*{wing}f\right)\left(\left(k,i\right)\right)}.$
Compared with (83), this yields
$\dfrac{1}{\left(R_{\Delta\left(p\right)}^{p}f\right)\left(\left(i,k\right)\right)}=\dfrac{1}{\left(\operatorname*{wing}f\right)\left(\left(k,i\right)\right)}$.
Taking inverses in this equality, we get
$\displaystyle\left(R_{\Delta\left(p\right)}^{p}f\right)\left(\left(i,k\right)\right)$
$\displaystyle=\left(\operatorname*{wing}f\right)\left(\left(k,i\right)\right)=f\left(\underbrace{\left(k,i\right)}_{=\operatorname*{vrefl}\left(i,k\right)}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{by the definition of
}\operatorname*{wing}\text{, since
}\left(k,i\right)\in\Delta\left(p\right)\subseteq\Delta\left(p\right)\cup\left\\{1\right\\}\right)$
$\displaystyle=f\left(\operatorname*{vrefl}\left(i,k\right)\right)=\left(\operatorname*{vrefl}\nolimits^{\ast}f\right)\left(\left(i,k\right)\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{since
}\left(\operatorname*{vrefl}\nolimits^{\ast}f\right)\left(\left(i,k\right)\right)=f\left(\operatorname*{vrefl}\left(i,k\right)\right)\text{
by the definition of }\operatorname*{vrefl}\nolimits^{\ast}\right).$
Now, we have shown this for every $\left(i,k\right)\in\Delta\left(p\right)$.
In other words, we have shown that
$R_{\Delta\left(p\right)}^{p}f=\operatorname*{vrefl}\nolimits^{\ast}f$.
Now, recall that $f=\left(a_{0},a_{1},...,a_{p}\right)\flat g$. Hence,
$R_{\Delta\left(p\right)}^{p}f=R_{\Delta\left(p\right)}^{p}\left(\left(a_{0},a_{1},...,a_{p}\right)\flat
g\right)=\left(a_{0},a_{1},...,a_{p}\right)\flat\left(R_{\Delta\left(p\right)}^{p}g\right)$
(84)
(by Corollary 5.7, applied to $\Delta\left(p\right)$, $p-1$ and $g$ instead of
$P$, $n$ and $f$). On the other hand,
$f=\left(a_{0},a_{1},...,a_{p}\right)\flat g$ yields
$\operatorname*{vrefl}\nolimits^{\ast}f=\operatorname*{vrefl}\nolimits^{\ast}\left(\left(a_{0},a_{1},...,a_{p}\right)\flat
g\right)=\left(a_{0},a_{1},...,a_{p}\right)\flat\left(\operatorname*{vrefl}\nolimits^{\ast}g\right)$
(85)
(this is easy to check directly using the definitions of $\flat$ and
$\operatorname*{vrefl}\nolimits^{\ast}$, since $\operatorname*{vrefl}$
preserves degrees). In light of (84) and (85), the equality
$R_{\Delta\left(p\right)}^{p}f=\operatorname*{vrefl}\nolimits^{\ast}f$ becomes
$\left(a_{0},a_{1},...,a_{p}\right)\flat\left(R_{\Delta\left(p\right)}^{p}g\right)=\left(a_{0},a_{1},...,a_{p}\right)\flat\left(\operatorname*{vrefl}\nolimits^{\ast}g\right)$.
We can cancel the “$\left(a_{0},a_{1},...,a_{p}\right)\flat$” from both sides
of this equation (since all $a_{i}$ are nonzero), and thus obtain
$R_{\Delta\left(p\right)}^{p}g=\operatorname*{vrefl}\nolimits^{\ast}g$. As we
have seen, this is all we need to prove Theorem 18.7. ∎
We can now obtain Theorem 18.6 from Theorem 18.7 using a construction from the
proof of Proposition 8.4:
###### Proof of Theorem 18.6 (sketched)..
The poset antiautomorphism $\operatorname*{hrefl}$ of
$\operatorname*{Rect}\left(p,p\right)$ defined in Remark 18.4 restricts to a
poset antiisomorphism
$\operatorname*{hrefl}:\nabla\left(p\right)\rightarrow\Delta\left(p\right)$,
that is, to a poset homomorphism
$\operatorname*{hrefl}:\nabla\left(p\right)\rightarrow\left(\Delta\left(p\right)\right)^{\operatorname*{op}}$.
We will use this isomorphism to identify the poset $\nabla\left(p\right)$ with
the opposite poset $\left(\Delta\left(p\right)\right)^{\operatorname*{op}}$ of
$\Delta\left(p\right)$.
Set $P=\Delta\left(p\right)$. Define a rational map
$\kappa:\mathbb{K}^{\widehat{P}}\dashrightarrow\mathbb{K}^{\widehat{P^{\operatorname*{op}}}}$
as in the proof of Proposition 8.4. Then, as in said proof, it can be shown
that the map $\kappa$ is a birational map and satisfies $\kappa\circ
R_{P}=R_{P^{\operatorname*{op}}}^{-1}\circ\kappa$. Since
$P=\Delta\left(p\right)$ and
$P^{\operatorname*{op}}=\left(\Delta\left(p\right)\right)^{\operatorname*{op}}=\nabla\left(p\right)$,
this rewrites as $\kappa\circ
R_{\Delta\left(p\right)}=R_{\nabla\left(p\right)}^{-1}\circ\kappa$. For the
same reason, we know that $\kappa$ is a rational map
$\mathbb{K}^{\widehat{\Delta\left(p\right)}}\dashrightarrow\mathbb{K}^{\widehat{\nabla\left(p\right)}}$.
From $\kappa\circ
R_{\Delta\left(p\right)}=R_{\nabla\left(p\right)}^{-1}\circ\kappa$, we can
easily obtain $\kappa\circ
R_{\Delta\left(p\right)}^{m}=R_{\nabla\left(p\right)}^{-m}\circ\kappa$ for
every $m\in\mathbb{N}$. In particular, $\kappa\circ
R_{\Delta\left(p\right)}^{p}=R_{\nabla\left(p\right)}^{-p}\circ\kappa$.
Now, consider the map
$\operatorname*{vrefl}\nolimits^{\ast}:\mathbb{K}^{\widehat{\Delta\left(p\right)}}\rightarrow\mathbb{K}^{\widehat{\Delta\left(p\right)}}$
defined in Lemma 18.10 (e), and also consider the similarly defined map
$\operatorname*{vrefl}\nolimits^{\ast}:\mathbb{K}^{\widehat{\nabla\left(p\right)}}\rightarrow\mathbb{K}^{\widehat{\nabla\left(p\right)}}$.
Both squares of the diagram
$\textstyle{\mathbb{K}^{\widehat{\Delta\left(p\right)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{\Delta\left(p\right)}^{p}}$$\scriptstyle{\kappa}$$\textstyle{\mathbb{K}^{\widehat{\Delta\left(p\right)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname*{vrefl}^{\ast}}$$\scriptstyle{\kappa}$$\textstyle{\mathbb{K}^{\widehat{\Delta\left(p\right)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{\mathbb{K}^{\widehat{\nabla\left(p\right)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{\nabla\left(p\right)}^{-p}}$$\textstyle{\mathbb{K}^{\widehat{\nabla\left(p\right)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname*{vrefl}^{\ast}}$$\textstyle{\mathbb{K}^{\widehat{\nabla\left(p\right)}}}$
commute (the left square does so because of $\kappa\circ
R_{\Delta\left(p\right)}^{p}=R_{\nabla\left(p\right)}^{-p}\circ\kappa$, and
the commutativity of the right square follows from a simple calculation), and
so the whole diagram commutes. In other words,
$\kappa\circ\left(\operatorname*{vrefl}\nolimits^{\ast}\circ
R_{\Delta\left(p\right)}^{p}\right)=\left(\operatorname*{vrefl}\nolimits^{\ast}\circ
R_{\nabla\left(p\right)}^{-p}\right)\circ\kappa.$ (86)
But the statement of Theorem 18.7 can be rewritten as
$R_{\Delta\left(p\right)}^{p}=\operatorname*{vrefl}\nolimits^{\ast}$. Since
$\operatorname*{vrefl}\nolimits^{\ast}$ is an involution (this is clear by
inspection), we have
$\operatorname*{vrefl}\nolimits^{\ast}=\left(\operatorname*{vrefl}\nolimits^{\ast}\right)^{-1}$,
so that
$\underbrace{\operatorname*{vrefl}\nolimits^{\ast}}_{=\left(\operatorname*{vrefl}\nolimits^{\ast}\right)^{-1}}\circ\underbrace{R_{\Delta\left(p\right)}^{p}}_{=\operatorname*{vrefl}\nolimits^{\ast}}=\left(\operatorname*{vrefl}\nolimits^{\ast}\right)^{-1}\circ\operatorname*{vrefl}\nolimits^{\ast}=\operatorname*{id}$.
Thus, (86) simplifies to
$\kappa\circ\operatorname*{id}=\left(\operatorname*{vrefl}\nolimits^{\ast}\circ
R_{\nabla\left(p\right)}^{-p}\right)\circ\kappa$. In other words,
$\kappa=\left(\operatorname*{vrefl}\nolimits^{\ast}\circ
R_{\nabla\left(p\right)}^{-p}\right)\circ\kappa$. Since $\kappa$ is a
birational map, we can cancel $\kappa$ from this identity, obtaining
$\operatorname*{id}=\operatorname*{vrefl}\nolimits^{\ast}\circ
R_{\nabla\left(p\right)}^{-p}$. In other words,
$R_{\nabla\left(p\right)}^{p}=\operatorname*{vrefl}\nolimits^{\ast}$. But this
is precisely the statement of Theorem 18.6. ∎
###### Proof of Corollary 18.9 (sketched)..
(a) Let $f\in\mathbb{K}^{\widehat{\Delta\left(p\right)}}$ be sufficiently
generic. Then, every $\left(i,k\right)\in\Delta\left(p\right)$ satisfies
$\displaystyle\left(\underbrace{R_{\Delta\left(p\right)}^{2p}}_{=R_{\Delta\left(p\right)}^{p}\circ
R_{\Delta\left(p\right)}^{p}}f\right)\left(\left(i,k\right)\right)$
$\displaystyle=\left(\left(R_{\Delta\left(p\right)}^{p}\circ
R_{\Delta\left(p\right)}^{p}\right)f\right)\left(\left(i,k\right)\right)=\left(R_{\Delta\left(p\right)}^{p}\left(R_{\Delta\left(p\right)}^{p}f\right)\right)\left(\left(i,k\right)\right)$
$\displaystyle=\left(R_{\Delta\left(p\right)}^{p}f\right)\left(\left(k,i\right)\right)\
\ \ \ \ \ \ \ \ \ \left(\text{by Theorem \ref{thm.Delta.halfway}, applied to
}R_{\Delta\left(p\right)}^{p}f\text{ instead of }f\right)$
$\displaystyle=f\left(\left(i,k\right)\right)\ \ \ \ \ \ \ \ \ \
\left(\text{by Theorem \ref{thm.Delta.halfway}, applied to
}\left(k,i\right)\text{ instead of }\left(i,k\right)\right).$
Hence, the two labellings $R_{\Delta\left(p\right)}^{2p}f$ and $f$ are equal
on every element of $\Delta\left(p\right)$. Since these two labellings are
also equal on $0$ and $1$ (because Corollary 2.18 yields
$\left(R_{\Delta\left(p\right)}^{2p}f\right)\left(0\right)=f\left(0\right)$
and
$\left(R_{\Delta\left(p\right)}^{2p}f\right)\left(1\right)=f\left(1\right)$),
this yields that the two labellings $R_{\Delta\left(p\right)}^{2p}f$ and $f$
are equal on every element of
$\Delta\left(p\right)\cup\left\\{0,1\right\\}=\widehat{\Delta\left(p\right)}$.
Hence, $R_{\Delta\left(p\right)}^{2p}f=f=\operatorname*{id}f$.
Now, forget that we fixed $f$. We thus have shown that
$R_{\Delta\left(p\right)}^{2p}f=\operatorname*{id}f$ for every sufficiently
generic $f\in\mathbb{K}^{\widehat{\Delta\left(p\right)}}$. Hence,
$R_{\Delta\left(p\right)}^{2p}=\operatorname*{id}$. In other words,
$\operatorname*{ord}\left(R_{\Delta\left(p\right)}\right)\mid 2p$. This proves
Corollary 18.9 (a).
(b) Proving Corollary 18.9 (b) is left to the reader. ∎
###### Proof of Corollary 18.8 (sketched)..
Corollary 18.8 can be deduced from Theorem 18.6 in the same way as Corollary
18.9 is deduced from Theorem 18.7. We won’t dwell on the details. ∎
Let us conclude this section by stating a generalization of parts (b), (c),
(d) and (f) of Lemma 18.10 that was pointed out by a referee. Rather than
restricting itself to $\operatorname*{Rect}\left(p,p\right)$, it is concerned
with an arbitrary $\left(2p-1\right)$-graded poset satisfying certain axioms
(which can be informally subsumed under the slogan “symmetric with respect to
degree $p$ and regular near the middle”):424242We choose to label the parts of
Lemma 18.12 by (b), (c), (d) and (f), since they generalize the parts (b),
(c), (d) and (f) of Lemma 18.10, respectively.
###### Lemma 18.12.
Let $p$ be a positive integer. Let $P$ be a $\left(2p-1\right)$-graded finite
poset. Let $\operatorname*{hrefl}:P\rightarrow P$ be an involution such that
$\operatorname*{hrefl}$ is a poset antiautomorphism of $P$. (This
$\operatorname*{hrefl}$ has nothing to do with the $\operatorname*{hrefl}$
defined in Lemma 18.10, although of course it is analogous to the latter.) We
extend $\operatorname*{hrefl}$ to an involutive poset antiautomorphism of
$\widehat{P}$ by setting $\operatorname*{hrefl}\left(0\right)=1$ and
$\operatorname*{hrefl}\left(1\right)=0$.
Assume that every $v\in\widehat{P}$ satisfies
$\deg\left(\operatorname{hrefl}v\right)=2p-\deg v.$ (87)
Let $N$ be a positive integer. Assume that, for every $v\in P$ satisfying
$\deg v=p-1$, there exist precisely $N$ elements $u$ of $P$ satisfying
$u\gtrdot v$.
Define three subsets $\Delta$, $\operatorname*{Eq}$ and $\nabla$ of $P$ by
$\displaystyle\Delta$ $\displaystyle=\left\\{v\in P\ \mid\ \deg v>p\right\\};$
$\displaystyle\operatorname*{Eq}$ $\displaystyle=\left\\{v\in P\ \mid\ \deg
v=p\right\\};$ $\displaystyle\nabla$ $\displaystyle=\left\\{v\in P\ \mid\ \deg
v<p\right\\}.$
Clearly, $\Delta$, $\operatorname*{Eq}$ and $\nabla$ become subposets of $P$.
The poset $\operatorname*{Eq}$ is an antichain, while the posets $\Delta$ and
$\nabla$ are $\left(p-1\right)$-graded.
Assume that
$\operatorname*{hrefl}\mid_{\operatorname*{Eq}}=\operatorname*{id}$. It is
easy to see that $\operatorname*{hrefl}\left(\Delta\right)=\nabla$.
Let $\mathbb{K}$ be a field such that $N$ is invertible in $\mathbb{K}$.
(b) Define a rational map
$\operatorname*{wing}:\mathbb{K}^{\widehat{\Delta}}\dashrightarrow\mathbb{K}^{\widehat{P}}$
by setting
$\left(\operatorname*{wing}f\right)\left(v\right)=\left\\{\begin{array}[c]{l}f\left(v\right),\
\ \ \ \ \ \ \ \ \ \text{if }v\in\Delta\cup\left\\{1\right\\};\\\ 1,\ \ \ \ \ \
\ \ \ \ \text{if }v\in\operatorname*{Eq};\\\
\dfrac{1}{\left(R_{\Delta}^{p-\deg
v}f\right)\left(\operatorname*{hrefl}v\right)},\ \ \ \ \ \ \ \ \ \ \text{if
}v\in\nabla\cup\left\\{0\right\\}\end{array}\right.$
for all $v\in\widehat{P}$ for all $f\in\mathbb{K}^{\widehat{\Delta}}$. This is
well-defined.
(c) There exists a rational map
$\overline{\operatorname*{wing}}:\overline{\mathbb{K}^{\widehat{\Delta}}}\dashrightarrow\overline{\mathbb{K}^{\widehat{P}}}$
such that the diagram
$\textstyle{\mathbb{K}^{\widehat{\Delta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\operatorname*{wing}}$$\textstyle{\mathbb{K}^{\widehat{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathbb{K}^{\widehat{\Delta}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\operatorname*{wing}}}$$\textstyle{\overline{\mathbb{K}^{\widehat{P}}}}$
commutes.
(d) The rational map $\overline{\operatorname*{wing}}$ defined in Lemma 18.12
(b) satisfies
$\overline{R}_{P}\circ\overline{\operatorname*{wing}}=\overline{\operatorname*{wing}}\circ\overline{R}_{\Delta}.$
(f) Almost every (in the sense of Zariski topology) labelling
$f\in\mathbb{K}^{\widehat{\Delta}}$ satisfying $f\left(0\right)=N$ satisfies
$R_{P}\left(\operatorname*{wing}f\right)=\operatorname*{wing}\left(R_{\Delta}f\right).$
Notice that the hypothesis (87) is actually redundant (it follows from the
other requirements), but we have chosen to state it because it is easily
checked in practice and used in the proof.
###### Example 18.13.
Let $P$ be a positive integer, and let $\mathbb{K}$ be a field of
characteristic $\neq 2$. The hypotheses of Lemma 18.12 are satisfied if we set
$P=\operatorname*{Rect}\left(p,p\right)$,
$\operatorname*{hrefl}=\operatorname*{hrefl}$ (by this, we mean that we define
$\operatorname*{hrefl}$ to be the map $\operatorname*{hrefl}$ defined in Lemma
18.10) and $N=2$. In this case, the posets $\Delta$, $\operatorname*{Eq}$ and
$\nabla$ defined in Lemma 18.12 are precisely the posets
$\Delta\left(p\right)$, $\operatorname*{Eq}\left(p\right)$ and
$\nabla\left(p\right)$ introduced in Definition 18.1. Hence, Lemma 18.12 (when
applied to this setting) yields the parts (b), (c), (d) and (f) of Lemma
18.10.
###### Example 18.14.
Here is another example of a situation in which Lemma 18.12 applies. Namely,
the hypotheses of Lemma 18.12 are satisfied when $p=5$, $N=3$ and $P$ is the
poset with Hasse diagram
|
---|---
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$
(with $\operatorname*{hrefl}:P\rightarrow P$ being the reflection with respect
to the horizontal axis of symmetry of this diagram).
###### Proof of Lemma 18.12 (sketched)..
The proof of Lemma 18.12 is almost completely analogous to the proof of parts
(b), (c), (d) and (f) of Lemma 18.10. Of course, several changes need to be
made to the latter proof to make it apply to Lemma 18.12: for instance,
* •
every appearance of $\operatorname*{Rect}\left(p,p\right)$,
$\Delta\left(p\right)$, $\nabla\left(p\right)$ or
$\operatorname*{Eq}\left(p\right)$ must be replaced by $P$, $\Delta$, $\nabla$
or $\operatorname*{Eq}$, respectively;
* •
many (but not all) appearances of the number $2$ (such as its appearance in
the definition of $a_{i}$) have to be replaced by $N$;
* •
various properties of $P$ now no longer follow from the definition of
$\operatorname*{Rect}\left(p,p\right)$ (because $P$ is no longer
$\operatorname*{Rect}\left(p,p\right)$), but instead have to be derived from
the hypotheses of Lemma 18.12434343Most of the time, this is obvious. For
instance, the fact that $\operatorname*{hrefl}\left(\Delta\right)=\nabla$
follows from (87). The only fact that is not completely trivial is that, for
every $v\in P$ satisfying $\deg v=p+1$, there exist precisely $N$ elements $u$
of $P$ satisfying $u\lessdot v$. Let us prove this fact. We know that
$\operatorname*{hrefl}$ is a poset antiautomorphism of $\widehat{P}$. Hence,
if $u$ and $v$ are two elements of $\widehat{P}$, then we have the following
equivalence of statements: $\left(u\lessdot
v\right)\Longleftrightarrow\left(\operatorname*{hrefl}u\gtrdot\operatorname*{hrefl}v\right).$
(88) We also have assumed that, for every $v\in P$ satisfying $\deg v=p-1$,
there exist precisely $N$ elements $u$ of $P$ satisfying $u\gtrdot v$. In
other words, for every $v\in P$ satisfying $\deg v=p-1$, we have
$\left(\text{the number of elements }u\text{ of }P\text{ satisfying }u\gtrdot
v\right)=N.$ (89) Now, let $v\in P$ be such that $\deg v=p+1$. We need to show
that there exist precisely $N$ elements $u$ of $P$ satisfying $u\lessdot v$.
From (87), we obtain
$\deg\left(\operatorname{hrefl}v\right)=2p-\underbrace{\deg
v}_{=p+1}=2p-\left(p+1\right)=p-1$. Hence, (89) (applied to
$\operatorname*{hrefl}v$ instead of $v$) yields $\left(\text{the number of
elements }u\text{ of }P\text{ satisfying
}u\gtrdot\operatorname*{hrefl}v\right)=N.$ (90) But
$\operatorname*{hrefl}:P\rightarrow P$ is a bijection (since
$\operatorname*{hrefl}$ is an involution). Thus, we can substitute
$\operatorname*{hrefl}u$ for $u$ in “$\left(\text{the number of elements
}u\text{ of }P\text{ satisfying }u\gtrdot\operatorname*{hrefl}v\right)$”. We
thus obtain $\displaystyle\left(\text{the number of elements }u\text{ of
}P\text{ satisfying }u\gtrdot\operatorname*{hrefl}v\right)$
$\displaystyle=\left(\text{the number of elements }u\text{ of }P\text{
satisfying
}\underbrace{\operatorname*{hrefl}u\gtrdot\operatorname*{hrefl}v}_{\begin{subarray}{c}\text{this
is equivalent to }\left(u\lessdot v\right)\\\ \text{(due to
(\ref{pf.Delta.hrefl-general.d.equiv}))}\end{subarray}}\right)$
$\displaystyle=\left(\text{the number of elements }u\text{ of }P\text{
satisfying }u\lessdot v\right).$ Thus, $\displaystyle\left(\text{the number
of elements }u\text{ of }P\text{ satisfying }u\lessdot v\right)$
$\displaystyle=\left(\text{the number of elements }u\text{ of }P\text{
satisfying }u\gtrdot\operatorname*{hrefl}v\right)=N$ (by (90)). In other
words, there exist precisely $N$ elements $u$ of $P$ satisfying $u\lessdot v$.
This completes our proof of the fact that, for every $v\in P$ satisfying $\deg
v=p+1$, there exist precisely $N$ elements $u$ of $P$ satisfying $u\lessdot
v$.;
* •
checking the case when $p\leqslant 2$ is no longer trivial, but needs a bit
more work444444The case when $p=1$ is still obvious (since $\Delta$ and
$\nabla$ are empty sets in this case). The case when $p=2$ can be handled by
the same arguments that were used to deal with the case when $p>2$ (in
particular, the same definition of the $\left(2p+1\right)$-tuple
$\left(a_{0},a_{1},...,a_{2p}\right)$ applies), but the details are slightly
different (instead of the seven cases, there are now only three cases: $\deg
v=3$, $\deg v=2$ and $\deg v=1$)..
∎
## 19 The quarter-triangles
We have now studied the order of birational rowmotion on all four triangles
(two of which are isomorphic as posets) which are obtained by cutting the
rectangle $\operatorname*{Rect}\left(p,p\right)$ along one of its diagonals.
But we can also cut $\operatorname*{Rect}\left(p,p\right)$ along both
diagonals into four smaller triangles. These are isomorphic in pairs, and we
will analyze them now. The following definition is an analogue of Definition
18.1 but using $\operatorname*{Tria}\left(p\right)$ instead of
$\operatorname*{Rect}\left(p,p\right)$:
###### Definition 19.1.
Let $p$ be a positive integer. Define three subsets
$\operatorname*{NEtri}\left(p\right)$, $\operatorname*{Eqtri}\left(p\right)$
and $\operatorname*{SEtri}\left(p\right)$ of
$\operatorname*{Tria}\left(p\right)$ by
$\displaystyle\operatorname*{NEtri}\left(p\right)$
$\displaystyle=\left\\{\left(i,k\right)\in\operatorname*{Tria}\left(p\right)\
\mid\ i+k>p+1\right\\};$ $\displaystyle\operatorname*{Eqtri}\left(p\right)$
$\displaystyle=\left\\{\left(i,k\right)\in\operatorname*{Tria}\left(p\right)\
\mid\ i+k=p+1\right\\};$ $\displaystyle\operatorname*{SEtri}\left(p\right)$
$\displaystyle=\left\\{\left(i,k\right)\in\operatorname*{Tria}\left(p\right)\
\mid\ i+k<p+1\right\\}.$
These subsets $\operatorname*{NEtri}\left(p\right)$,
$\operatorname*{Eqtri}\left(p\right)$ and
$\operatorname*{SEtri}\left(p\right)$ inherit a poset structure from
$\operatorname*{Tria}\left(p\right)$. In the following, we will consider
$\operatorname*{NEtri}\left(p\right)$, $\operatorname*{Eqtri}\left(p\right)$
and $\operatorname*{SEtri}\left(p\right)$ as posets using this structure.
Clearly, $\operatorname*{Eqtri}\left(p\right)$ is an antichain. The posets
$\operatorname*{NEtri}\left(p\right)$ and
$\operatorname*{SEtri}\left(p\right)$ are $\left(p-1\right)$-graded posets
having the form of right-angled triangles.
###### Example 19.2.
Here is the Hasse diagram of the poset $\operatorname*{Tria}\left(4\right)$,
where the elements belonging to $\operatorname*{NEtri}\left(4\right)$ have
been underlined and the elements belonging to
$\operatorname*{Eqtri}\left(4\right)$ have been boxed:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&&\\\&&&&&&\\\&&&&&&\\\&&&&&&\\\&&&&&&\\\&&&&&&\\\&&&&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 5.39996pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 13.79993pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 30.811pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\underline{\left(4,4\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
63.9332pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 94.55539pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 125.17758pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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And here is the Hasse diagram of the poset
$\operatorname*{NEtri}\left(4\right)$ itself:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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Here, on the other hand, is the Hasse diagram of the poset
$\operatorname*{SEtri}\left(4\right)$:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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114.06647pt\raise-26.79993pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-53.59985pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
5.39996pt\raise-53.59985pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
13.79993pt\raise-53.59985pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
22.19989pt\raise-53.59985pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\left(1,1\right)}$}}}}}}}{\hbox{\kern
63.9332pt\raise-53.59985pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
94.55539pt\raise-53.59985pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
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###### Remark 19.3.
Let $p$ be an even positive integer. The poset
$\operatorname*{NEtri}\left(p\right)$ is isomorphic to the poset
$\Phi^{+}\left(B_{p\diagup 2}\right)$ of [StWi11, §3.2]. (For odd $p$, the
poset $\operatorname*{NEtri}\left(p\right)$ does not seem to appear in
[StWi11, §3.2].)
The following conjectures have been verified using Sage for small values of
$p$:
###### Conjecture 19.4.
Let $p$ be an integer $>1$. Then,
$\operatorname*{ord}\left(R_{\operatorname*{SEtri}\left(p\right)}\right)=p$.
###### Conjecture 19.5.
Let $p$ be an integer $>1$. Then,
$\operatorname*{ord}\left(R_{\operatorname*{NEtri}\left(p\right)}\right)=p$.
The approach used to prove Theorem 17.4 allows proving these two conjectures
in the case of odd $p$, but in the even-$p$ case it fails (although the order
of classical rowmotion is again known to be $p$ in the even-$p$ case – see
[StWi11, Conjecture 3.6]). Here is how the proof proceeds in the case of odd
$p$:
###### Proposition 19.6.
Let $p$ be an odd integer $>1$. Let $\mathbb{K}$ be a field. Then,
$\operatorname*{ord}\left(R_{\operatorname*{SEtri}\left(p\right)}\right)=p$.
###### Proposition 19.7.
Let $p$ be an odd integer $>1$. Let $\mathbb{K}$ be a field. Then,
$\operatorname*{ord}\left(R_{\operatorname*{NEtri}\left(p\right)}\right)=p$.
Our proof of Proposition 19.7 rests upon the following fact:
###### Lemma 19.8.
Let $\mathbb{K}$ be a field of characteristic $\neq 2$.
Let $p$ be a positive integer.
(a) Let
$\operatorname*{vrefl}:\Delta\left(p\right)\rightarrow\Delta\left(p\right)$ be
the map sending every $\left(i,k\right)\in\Delta\left(p\right)$ to
$\left(k,i\right)$. This map $\operatorname*{vrefl}$ is an involutive poset
automorphism of $\Delta\left(p\right)$. (In intuitive terms,
$\operatorname*{vrefl}$ is simply reflection across the vertical axis.) We
have
$\operatorname*{vrefl}\left(v\right)\in\operatorname*{NEtri}\left(p\right)$
for every
$v\in\Delta\left(p\right)\setminus\operatorname*{NEtri}\left(p\right)$.
We extend $\operatorname*{vrefl}$ to an involutive poset automorphism of
$\widehat{\Delta\left(p\right)}$ by setting
$\operatorname*{vrefl}\left(0\right)=0$ and
$\operatorname*{vrefl}\left(1\right)=1$.
(b) Define a map
$\operatorname*{dble}:\mathbb{K}^{\widehat{\operatorname*{NEtri}\left(p\right)}}\rightarrow\mathbb{K}^{\widehat{\Delta\left(p\right)}}$
by setting
$\left(\operatorname*{dble}f\right)\left(v\right)=\left\\{\begin{array}[c]{l}\dfrac{1}{2}f\left(1\right),\
\ \ \ \ \ \ \ \ \ \text{if }v=1;\\\ f\left(0\right),\ \ \ \ \ \ \ \ \ \
\text{if }v=0;\\\ f\left(v\right),\ \ \ \ \ \ \ \ \ \ \text{if
}v\in\operatorname*{NEtri}\left(p\right);\\\
f\left(\operatorname*{vrefl}\left(v\right)\right),\ \ \ \ \ \ \ \ \ \
\text{otherwise}\end{array}\right.$
for all $v\in\widehat{\Delta\left(p\right)}$ for all
$f\in\mathbb{K}^{\widehat{\operatorname*{NEtri}\left(p\right)}}$. This is
well-defined. We have
$\left(\operatorname*{dble}f\right)\left(v\right)=f\left(v\right)\ \ \ \ \ \ \
\ \ \ \text{for every }v\in\operatorname*{NEtri}\left(p\right).$ (91)
Also,
$\left(\operatorname*{dble}f\right)\left(\operatorname*{vrefl}\left(v\right)\right)=f\left(v\right)\
\ \ \ \ \ \ \ \ \ \text{for every }v\in\operatorname*{NEtri}\left(p\right).$
(92)
(c) Assume that $p$ is odd. Then,
$R_{\Delta\left(p\right)}\circ\operatorname*{dble}=\operatorname*{dble}\circ
R_{\operatorname*{NEtri}\left(p\right)}.$
###### .
We omit the proofs of Lemma 19.8, Proposition 19.7 and Proposition 19.6 since
neither of them involves any new ideas. The first is analogous to that of
Lemma 17.5 (with $\Delta\left(p\right)$ and
$\operatorname*{NEtri}\left(p\right)$ taking the roles of
$\operatorname*{Rect}\left(p,p\right)$ and
$\operatorname*{Tria}\left(p\right)$, respectively)454545The only non-
straightforward change that must be made to the proof is the following: In
Case 2 of the proof of Lemma 17.5, we used the (obvious) observation that
$\left(i-1,i\right)$ and $\left(i,i-1\right)$ are elements of
$\operatorname*{Rect}\left(p,p\right)$ for every
$\left(i,i\right)\in\operatorname*{Rect}\left(p,p\right)$ satisfying $i\neq
1$. The analogous observation that we need for proving Lemma 19.8 is still
true in the case of odd $p$, but a bit less obvious. In fact, it is the
observation that $\left(i-1,i\right)$ and $\left(i,i-1\right)$ are elements of
$\Delta\left(p\right)$ for every $\left(i,i\right)\in\Delta\left(p\right)$.
This uses the oddness of $p$.. The proof of Proposition 19.7 combines Lemma
19.8 with Theorem 18.7. Proposition 19.6 is derived from Proposition 19.7
using Proposition 8.4. ∎
Nathan Williams suggested that the following generalization of Conjecture 19.5
might hold:
###### Conjecture 19.9.
Let $p$ be an integer $>1$. Let $s\in\mathbb{N}$. Let
$\operatorname*{NEtri}\nolimits^{\prime}\left(p\right)$ be the subposet
$\left\\{\left(i,k\right)\in\operatorname*{NEtri}\left(p\right)\ \mid\
k\geqslant s\right\\}$ of $\operatorname*{NEtri}\left(p\right)$. Then,
$\operatorname*{ord}\left(R_{\operatorname*{NEtri}\nolimits^{\prime}\left(p\right)}\right)\mid
p$.
This conjecture has been verified using Sage for all $p\leqslant 7$. Williams
(based on a philosophy from his thesis [Will13]) suspects there could be a
birational map between
$\mathbb{K}^{\widehat{\operatorname*{NEtri}\nolimits^{\prime}\left(p\right)}}$
and $\mathbb{K}^{\widehat{\operatorname*{Rect}\left(s-1,p-s+1\right)}}$ which
commutes with the respective birational rowmotion operators for all
$s>\dfrac{p}{2}$; this, if shown, would obviously yield a proof of Conjecture
19.9. This already is an interesting question for classical rowmotion; a
bijection between the antichains (and thus between the order ideals) of
$\operatorname*{NEtri}\nolimits^{\prime}\left(p\right)$ and those of
$\operatorname*{Rect}\left(s-1,p-s+1\right)$ was found by Stembridge [Stem86,
Theorem 5.4], but does not commute with classical rowmotion.
## 20 Negative results
Generally, it is not true that if $P$ is an $n$-graded poset, then
$\operatorname*{ord}\left(R_{P}\right)$ is necessarily finite. When
$\operatorname*{char}\mathbb{K}=0$, the authors have proven the
following464646See the ancillary files of the present arXiv preprint
(arXiv:1402.6178) for an outline of the (rather technical) proofs.:
* •
If $P$ is the poset $\left\\{x_{1},x_{2},x_{3},x_{4},x_{5}\right\\}$ with
relations $x_{1}<x_{3}$, $x_{1}<x_{4}$, $x_{1}<x_{5}$, $x_{2}<x_{4}$ and
$x_{2}<x_{5}$ (this is a $5$-element $2$-graded poset), then
$\operatorname*{ord}\left(R_{P}\right)=\infty$.
* •
If $P$ is the “chain-link fence” poset $/\backslash/\backslash/\backslash$
(that is, the subposet
$\left\\{\left(i,k\right)\in\operatorname*{Rect}\left(4,4\right)\ \mid\
5\leqslant i+k\leqslant 6\right\\}$ of
$\operatorname*{Rect}\left(4,4\right)$), then
$\operatorname*{ord}\left(R_{P}\right)=\infty$.
* •
If $P$ is the Boolean lattice
$\left[2\right]\times\left[2\right]\times\left[2\right]$, then
$\operatorname*{ord}\left(R_{P}\right)=\infty$.
The situation seems even more hopeless for non-graded posets.
## 21 The root system connection
A question naturally suggesting itself is: What is it that makes certain
posets $P$ have finite $\operatorname*{ord}\left(R_{P}\right)$, while others
have not? Can we characterize the former posets? It might be too optimistic to
expect a full classification, given that our examples are already rather
diverse (skeletal posets, rectangles, triangles, posets like that in Remark
11.8). As a first step (and inspired by the general forms of the Zamolodchikov
conjecture), we were tempted to study posets arising from Dynkin diagrams. It
appears that, unlike in the Zamolodchikov conjecture, the interesting cases
are not those having $P$ be a product of Dynkin diagrams, but those having $P$
be a positive root poset of a root system, or a parabolic quotient thereof.
The idea is not new, as it was already conjectured by Panyushev [Pan08,
Conjecture 2.1] and proven by Armstrong, Stump and Thomas [AST11, Theorem 1.2]
that if $W$ is a finite Weyl group with Coxeter number $h$, then classical
rowmotion on the set $J\left(\Phi^{+}\left(W\right)\right)$ (where
$\Phi^{+}\left(W\right)$ is the poset of positive roots of $W$) has order $h$
or $2h$ (along with a few more properties, akin to our “reciprocity”
statements)474747Neither [Pan08] nor [AST11] work directly with order ideals
and rowmotion, but instead they study antichains of the poset
$\Phi^{+}\left(W\right)$ (which are called “nonnesting partitions” in [AST11])
and an operation on these antichains called Panyushev complementation. There
is, however, a simple bijection between the set of antichains of a poset $P$
and the set $J\left(P\right)$, and the conjugate of Panyushev complementation
with respect to this bijection is precisely classical rowmotion..
In the case of birational rowmotion, the situation is less simple.
Specifically, the following can be said about positive root posets of
crystallographic root systems (as considered in [StWi11, §3.2])484848We refer
to [StWi11, Definition 3.4] for notations.:
* •
If $P=\Phi^{+}\left(A_{n}\right)$ for $n\geqslant 2$, then
$\operatorname*{ord}\left(R_{P}\right)=2\left(n+1\right)$. This is just the
assertion of Corollary 18.9. Note that for $n=1$, the order
$\operatorname*{ord}\left(R_{P}\right)$ is $2$ instead of
$2\left(1+1\right)=4$.
* •
If $P=\Phi^{+}\left(B_{n}\right)$ for $n\geqslant 1$, then Conjecture 19.4
claims that $\operatorname*{ord}\left(R_{P}\right)=2n$. Note that
$\Phi^{+}\left(B_{n}\right)\cong\Phi^{+}\left(C_{n}\right)$.
* •
We have $\operatorname*{ord}\left(R_{P}\right)=2$ for
$P=\Phi^{+}\left(D_{2}\right)$, and we have
$\operatorname*{ord}\left(R_{P}\right)=8$ for $P=\Phi^{+}\left(D_{3}\right)$.
However, $\operatorname*{ord}\left(R_{P}\right)=\infty$ in the case when
$P=\Phi^{+}\left(D_{4}\right)$. This should not come as a surprise, since
$\Phi^{+}\left(D_{4}\right)$ has a property that none of the
$\Phi^{+}\left(A_{n}\right)$ or
$\Phi^{+}\left(B_{n}\right)\cong\Phi^{+}\left(C_{n}\right)$ have, namely an
element covered by three other elements. On the other hand, the finite orders
in the $\Phi^{+}\left(D_{2}\right)$ and $\Phi^{+}\left(D_{3}\right)$ cases can
be explained by $\Phi^{+}\left(D_{2}\right)\cong\Phi^{+}\left(A_{1}\times
A_{1}\right)\cong\left(\text{two-element antichain}\right)$ and
$\Phi^{+}\left(D_{3}\right)\cong\Phi^{+}\left(A_{3}\right)$.
Nathan Williams has suggested that the behavior of
$\Phi^{+}\left(A_{n}\right)$ and
$\Phi^{+}\left(B_{n}\right)\cong\Phi^{+}\left(C_{n}\right)$ to have finite
orders of $R_{P}$ could generalize to the “positive root posets” of the other
“coincidental types” $H_{3}$ and $I_{2}\left(m\right)$ (see, for example,
Table 2.2 in [Will13]). And indeed, computations in Sage have established that
$\operatorname*{ord}\left(R_{P}\right)=10$ for $P=\Phi^{+}\left(H_{3}\right)$,
and we also have
$\operatorname*{ord}\left(R_{P}\right)=\operatorname{lcm}\left(2,m\right)$ for
$P=\Phi^{+}\left(I_{2}\left(m\right)\right)$ (this is a very easy consequence
of Proposition 7.3).
It seems that minuscule heaps, as considered e.g. in [RuSh12, §6], also lead
to small $\operatorname*{ord}\left(R_{P}\right)$ values. Namely:
* •
The heap $P_{w_{0}^{J}}$ in [RuSh12, Figure 8 (b)] satisfies
$\operatorname*{ord}\left(R_{P}\right)=12$.
* •
The heap $P_{w_{0}^{J}}$ in [RuSh12, Figure 9 (b)] seems to satisfy
$\operatorname*{ord}\left(R_{P}\right)=18$ (this was verified on numerical
examples, as the poset is too large for efficient general computations).
(These two posets also appear as posets corresponding to the “Cayley plane”
and the “Freudenthal variety” in [ThoYo07, p. 2].)
Various other families of posets related to root systems (minuscule posets,
d-complete posets, rc-posets, alternating sign matrix posets) remain to be
studied.
## References
* [AKSch12] Arvind Ayyer, Steven Klee, Anne Schilling, Combinatorial Markov chains on linear extensions, Journal of Algebraic Combinatorics, September 2013, DOI 10.1007/s10801-013-0470-9. Also available as arXiv:1205.7074v3.
http://arxiv.org/abs/1205.7074v3
* [AST11] Drew Armstrong, Christian Stump and Hugh Thomas, A uniform bijection between nonnesting and noncrossing partitions, Trans. Amer. Math. Soc. 365 (2013), pp. 4121–4151, DOI 10.1090/S0002-9947-2013-05729-7. A preprint is available at arXiv:1101.1277v2.
* [BrSchr74] Andries E. Brouwer and A. Schrijver, On the period of an operator, defined on antichains, Math. Centr. report ZW24, Amsterdam (Jun. 1974).
http://www.win.tue.nl/~aeb/preprints/zw24.pdf
* [CaFl95] Peter J. Cameron, Dmitry G. Fon-der-Flaass, Orbits of Antichains Revisited, European Journal of Combinatorics, vol. 16, Issue 6, November 1995, pp. 545–554.
http://www.sciencedirect.com/science/article/pii/0195669895900365
* [EiPr13] David Einstein, James Propp, Combinatorial, piecewise-linear, and birational homomesy for products of two chains, arXiv:1310.5294v1 (preliminary version), October 20, 2013\.
http://arxiv.org/abs/1310.5294v1
* [EiPr14] David Einstein, James Propp, Piecewise-linear and birational toggling, (extended abstract) DMTCS proc. FPSAC 2014. A preprint appears as arXiv:1404.3455v1:
http://arxiv.org/abs/1404.3455v1
* [Flaa93] Dmitry G. Fon-der-Flaass, Orbits of Antichains in Ranked Posets, European Journal of Combinatorics, vol. 14, Issue 1, January 1993, pp. 17–22.
http://www.sciencedirect.com/science/article/pii/S0195669883710036
* [GrRo13] Darij Grinberg, Tom Roby, The order of birational rowmotion, (extended abstract) DMTCS proc. FPSAC 2014. (This is an extended abstract, presented at the FPSAC 2014 conference, of the paper you are reading.)
http://web.mit.edu/~darij/www/algebra/ipbrFPSAC6.pdf
* [Haiman92] Mark D. Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Mathematics, Volume 99, Issues 1–3, 2 April 1992, pp. 79–113.
http://www.sciencedirect.com/science/article/pii/0012365X9290368P
* [Kiri00] Anatol N. Kirillov, Introduction to tropical combinatorics, Physics and combinatorics: Proceedings of the Nagoya 2000 International Workshop, held 21 - 26 August 2000 in Nagoya University. Edited by Anatol N Kirillov (Nagoya University) & Nadejda Liskova. Published by World Scientific Publishing Co. Pte. Ltd., 2001. ISBN #9789812810007, pp. 82–150.
* [KiBe95] A. N. Kirillov, A. D. Berenstein, Groups generated by involutions, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux, Algebra i Analiz, volume 7 (1995), issue 1, pp. 92–152. A preprint is available at:
http://math.uoregon.edu/~arkadiy/bk1.pdf
* [KlLa72] S. L. Kleiman, Dan Laksov, Schubert calculus, The American Mathematical Monthly, vol. 79, no. 10 (December 1972), pp. 1061–1082.
* [Leeu01] Marc van Leeuwen, The Littlewood-Richardson Rule, and Related Combinatorics, Mathematical Society of Japan Memoirs, Volume 11, 2001, pp. 95–145. Possibly newer version at:
http://www-math.univ-poitiers.fr/~maavl/pdf/lrr.pdf
* [OSZ13] Neil O’Connell, Timo Seppäläinen, Nikos Zygouras, Geometric RSK correspondence, Whittaker functions and symmetrized random polymers, Inventiones Mathematicae, October 2013, DOI 10.1007/s00222-013-0485-9.
http://dx.doi.org/10.1007/s00222-013-0485-9
An older preprint version is also available as arXiv:1210.5126v2, June 6,
2013.
http://arxiv.org/abs/1210.5126v2
* [Pan08] Dmitri I. Panyushev, On orbits of antichains of positive roots, Europ. J. Combin. 30 (2009), no. 2, pp. 586–594. Also available at arXiv:0711.3353v2.
* [Post06] Alexander Postnikov, Total positivity, Grassmannians, and networks, October 17, 2007 version.
http://math.mit.edu/~apost/papers/tpgrass.pdf
* [PrRo13] James Propp and Tom Roby, Homomesy in products of two chains, DMTCS proc. FPSAC 2013,
http://www.math.uconn.edu/~troby/ceFPSAC.pdf
* [PrRo14] James Propp and Tom Roby, Homomesy in products of two chains, The Electronic Journal of Combinatorics, Volume 22, Issue 3 (2015), Paper #P3.4. A preprint appeared as arXiv:1310.5201v6.
* [Rhoa10] Brendon Rhoades, Cyclic sieving, promotion, and representation theory, J. Combin. Theory Ser. A, vol. 117, no. 1, (2010), pp. 38–76.
http://www.sciencedirect.com/science/article/pii/S0097316509000703
Also available as arXiv:1005.2568v1.
http://arxiv.org/abs/1005.2568v1
* [RuSh12] David B Rush, XiaoLin Shi, On Orbits of Order Ideals of Minuscule Posets, Journal of Algebraic Combinatorics, May 2013, Volume 37, Issue 3, pp. 545–569. Also available as arXiv:1108.5245v2.
http://arxiv.org/abs/1108.5245v2
* [Rusk92] Frank Ruskey, Generating Linear Extensions of Posets by Transpositions, Journal of Combinatorial Theory, vol. 54, Issue 1, January 1992, pp. 77–101.
http://www.sciencedirect.com/science/article/pii/0095895692900678
* [Russ13] Heather M. Russell, An explicit bijection between semistandard tableaux and non-elliptic $sl_{3}$ webs, Journal of Algebraic Combinatorics 38.4 (2013), pp. 851–862. Also appears as arXiv:1204.1037v1.
http://arxiv.org/abs/1204.1037v1
* [S+09] W. A. Stein et al., _Sage Mathematics Software (Version 6.2.beta2)_ , The Sage Development Team, 2014, http://www.sagemath.org.
* [Sage08] The Sage-Combinat community, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2008.
http://combinat.sagemath.org
* [Stan11] Richard Stanley, Enumerative Combinatorics, volume 1, 2nd edition, Cambridge University Press 2011.
http://math.mit.edu/~rstan/ec/ec1/
* [Stan86] Richard Stanley, Two Poset Polytopes, Discrete & Computational Geometry, 1986, Volume 1, Issue 1, pp. 9–23.
* [Stem86] John R. Stembridge, Trapezoidal Chains and Antichains, European Journal of Combinatorics, Volume 7, Issue 4, October 1986, pp. 377–387.
http://www.sciencedirect.com/science/article/pii/S0195669886800099
* [StWi11] Jessica Striker, Nathan Williams, Promotion and Rowmotion, European Journal of Combinatorics 33 (2012), pp. 1919–1942, DOI 10.1016/j.ejc.2012.05.003.
http://www.sciencedirect.com/science/article/pii/S0195669812000972
Also available as arXiv:1108.1172v3.
http://arxiv.org/abs/1108.1172v3
* [ThoYo07] Hugh Thomas, Alexander Yong, Cominuscule tableau combinatorics, preprint 2013.
http://www.math.uiuc.edu/~ayong/DE.Japan.0730.ps
* [Volk06] Alexandre Yu. Volkov, On the Periodicity Conjecture for Y-systems, Commun. Math. Phys. 276 (2007), pp. 509–517, DOI 10.1007/s00220-007-0343-y.
A preprint of this paper is also available under the name On Zamolodchikov’s
Periodicity Conjecture as arXiv:hep-th/0606094v1:
http://arxiv.org/abs/hep-th/0606094v1
* [Will13] Nathan Williams, Cataland, dissertation at University of Minnesota, August 2013.
https://conservancy.umn.edu/bitstream/159973/1/Williams_umn_0130E_14358.pdf
* [Zeil98] Doron Zeilberger, Dodgson’s Determinant-Evaluation Rule proved by Two-Timing Men and Women, The Electronic Journal of Combinatorics, vol. 4, issue 2 (1997) (The Wilf Festschrift volume), R22.
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r22
Also available as arXiv:math/9808079v1.
http://arxiv.org/abs/math/9808079v1
|
arxiv-papers
| 2014-02-25T14:25:42 |
2024-09-04T02:49:58.800043
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Darij Grinberg, Tom Roby",
"submitter": "Darij Grinberg",
"url": "https://arxiv.org/abs/1402.6178"
}
|
1402.6242
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-027 LHCb-PAPER-2014-003
Precision measurement of the ratio of the $\mathchar 28931\relax^{0}_{b}$ to
$\kern 3.73305pt\overline{\kern-3.73305ptB}{}^{0}$ lifetimes
The LHCb collaboration111Authors are listed on the following pages.
The LHCb measurement of the lifetime ratio of the $\mathchar
28931\relax^{0}_{b}$ baryon to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ meson is updated using data
corresponding to an integrated luminosity of 3.0 fb-1 collected using 7 and 8
TeV centre-of-mass energy $pp$ collisions at the LHC. The decay modes used are
$\mathchar 28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}K^{-}$, where the $\pi^{+}K^{-}$ mass is consistent with that of
the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ meson. The
lifetime ratio is determined with unprecedented precision to be $0.974\pm
0.006\pm 0.004$, where the first uncertainty is statistical and the second
systematic. This result is in agreement with original theoretical predictions
based on the heavy quark expansion. Using the current world average of the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime, the $\mathchar
28931\relax^{0}_{b}$ lifetime is found to be $1.479\pm 0.009\pm 0.010$ ps.
Submitted to Physics Letters B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25, S. Amato2, S. Amerio22, Y. Amhis7, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,m, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel39, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson50, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30, W.
Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,q, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez36,o, A. Camboni36, P. Campana18,38, D. Campora
Perez38, F. Caponio21, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,j, A.
Cardini15, H. Carranza-Mejia50, L. Carson50, K. Carvalho Akiba2, G. Casse52,
L. Cassina20, L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci58, M.
Charles8, Ph. Charpentier38, S.-F. Cheung55, N. Chiapolini40, M.
Chrzaszcz40,26, K. Ciba38, X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M.
Clemencic38, H.V. Cliff47, J. Closier38, C. Coca29, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins38, A. Comerma-Montells36, A. Contu15,38, A. Cook46, M.
Coombes46, S. Coquereau8, G. Corti38, I. Counts56, B. Couturier38, G.A.
Cowan50, D.C. Craik48, M. Cruz Torres60, S. Cunliffe53, R. Currie50, C.
D’Ambrosio38, J. Dalseno46, P. David8, P.N.Y. David41, A. Davis57, I. De
Bonis4, K. De Bruyn41, S. De Capua54, M. De Cian11, J.M. De Miranda1, L. De
Paula2, W. De Silva57, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, D. Derkach55, O. Deschamps5, F. Dettori42, A. Di Canto11,
H. Dijkstra38, S. Donleavy52, F. Dordei11, M. Dorigo39, P. Dorosz26,n, A.
Dosil Suárez37, D. Dossett48, A. Dovbnya43, F. Dupertuis39, P. Durante38, R.
Dzhelyadin35, A. Dziurda26, A. Dzyuba30, S. Easo49, U. Egede53, V.
Egorychev31, S. Eidelman34, S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L.
Eklund51,38, I. El Rifai5, Ch. Elsasser40, S. Esen11, A. Falabella16,f, C.
Färber11, C. Farinelli41, S. Farry52, D. Ferguson50, V. Fernandez Albor37, F.
Ferreira Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M.
Fiorini16,f, C. Fitzpatrick38, M. Fontana10, F. Fontanelli19,j, R. Forty38, O.
Francisco2, M. Frank38, C. Frei38, M. Frosini17,38,g, J. Fu21, E. Furfaro24,l,
A. Gallas Torreira37, D. Galli14,d, S. Gambetta19,j, M. Gandelman2, P.
Gandini59, Y. Gao3, J. Garofoli59, J. Garra Tico47, L. Garrido36, C. Gaspar38,
R. Gauld55, L. Gavardi9, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph.
Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47, L. Giubega29, V.V. Gligorov38,
C. Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, H. Gordon38, M.
Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E. Graugés36,
G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P. Griffith45, L.
Grillo11, O. Grünberg61, B. Gui59, E. Gushchin33, Yu. Guz35,38, T. Gys38, C.
Hadjivasiliou59, G. Haefeli39, C. Haen38, T.W. Hafkenscheid64, S.C. Haines47,
S. Hall53, B. Hamilton58, T. Hampson46, S. Hansmann-Menzemer11, N. Harnew55,
S.T. Harnew46, J. Harrison54, T. Hartmann61, J. He38, T. Head38, V. Heijne41,
K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E. van
Herwijnen38, M. Heß61, A. Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54, W.
Hulsbergen41, P. Hunt55, N. Hussain55, D. Hutchcroft52, D. Hynds51, M.
Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, E. Jans41, P. Jaton39, A.
Jawahery58, F. Jing3, M. John55, D. Johnson55, C.R. Jones47, C. Joram38, B.
Jost38, N. Jurik59, M. Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M.
Karbach38, M. Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C.
Khurewathanakul39, S. Klaver54, O. Kochebina7, I. Komarov39, R.F. Koopman42,
P. Koppenburg41, M. Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11,
M. Kreps48, G. Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k,
V. Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31,38, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, B. Langhans38, T. Latham48, C.
Lazzeroni45, R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A.
Leflat32, J. Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11,
Y. Li3, M. Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G.
Liu38, S. Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40,
H. Lu3, D. Lucchesi22,q, H. Luo50, E. Luppi16,f, O. Lupton55, F. Machefert7,
I.V. Machikhiliyan31, F. Maciuc29, O. Maev30,38, S. Malde55, G. Manca15,e, G.
Mancinelli6, M. Manzali16,f, J. Maratas5, U. Marconi14, C. Marin Benito36, P.
Marino23,s, R. Märki39, J. Marks11, G. Martellotti25, A. Martens8, A. Martín
Sánchez7, M. Martinelli41, D. Martinez Santos42, F. Martinez Vidal63, D.
Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A.
Mazurov16,38,f, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
McSkelly52, B. Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,s, R. Mountain59, F.
Muheim50, K. Müller40, R. Muresan29, B. Muryn27, B. Muster39, P. Naik46, T.
Nakada39, R. Nandakumar49, I. Nasteva1, M. Needham50, N. Neri21, S. Neubert38,
N. Neufeld38, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-
Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R.
Oldeman15,e, G. Onderwater64, M. Orlandea29, J.M. Otalora Goicochea2, P.
Owen53, A. Oyanguren36, B.K. Pal59, A. Palano13,c, F. Palombo21,t, M.
Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51, L. Pappalardo16,
C. Parkes54, C.J. Parkinson9, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, C. Pavel-Nicorescu29, A. Pazos Alvarez37, A. Pearce54, A.
Pellegrino41, M. Pepe Altarelli38, S. Perazzini14,d, E. Perez Trigo37, P.
Perret5, M. Perrin-Terrin6, L. Pescatore45, E. Pesen65, G. Pessina20, K.
Petridis53, A. Petrolini19,j, E. Picatoste Olloqui36, B. Pietrzyk4, T.
Pilař48, D. Pinci25, A. Pistone19, S. Playfer50, M. Plo Casasus37, F. Polci8,
A. Poluektov48,34, E. Polycarpo2, A. Popov35, D. Popov10, B. Popovici29, C.
Potterat36, A. Powell55, J. Prisciandaro39, A. Pritchard52, C. Prouve46, V.
Pugatch44, A. Puig Navarro39, G. Punzi23,r, W. Qian4, B. Rachwal26, J.H.
Rademacker46, B. Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N.
Rauschmayr38, G. Raven42, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S.
Ricciardi49, A. Richards53, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts58, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez
Perez37, S. Roiser38, V. Romanovsky35, A. Romero Vidal37, M. Rotondo22, J.
Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P. Ruiz Valls36, G.
Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B. Saitta15,e,
V. Salustino Guimaraes2, B. Sanmartin Sedes37, R. Santacesaria25, C.
Santamarina Rios37, E. Santovetti24,l, M. Sapunov6, A. Sarti18, C.
Satriano25,m, A. Satta24, M. Savrie16,f, D. Savrina31,32, M. Schiller42, H.
Schindler38, M. Schlupp9, M. Schmelling10, B. Schmidt38, O. Schneider39, A.
Schopper38, M.-H. Schune7, R. Schwemmer38, B. Sciascia18, A. Sciubba25, M.
Seco37, A. Semennikov31, K. Senderowska27, I. Sepp53, N. Serra40, J. Serrano6,
P. Seyfert11, M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52,
L. Shekhtman34, O. Shevchenko43, V. Shevchenko62, A. Shires9, R. Silva
Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, T. Skwarnicki59, N.A.
Smith52, E. Smith55,49, E. Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D.
Sokoloff57, F.J.P. Soler51, F. Soomro39, D. Souza46, B. Souza De Paula2, B.
Spaan9, A. Sparkes50, F. Spinella23, P. Spradlin51, F. Stagni38, S. Stahl11,
O. Steinkamp40, S. Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S.
Stracka23,38, M. Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38,
L. Sun57, W. Sutcliffe53, S. Swientek9, V. Syropoulos42, M. Szczekowski28, P.
Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, E. Teodorescu29, F. Teubert38, C. Thomas55, E. Thomas38, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D.
Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4,53, S. Tourneur39,
M.T. Tran39, M. Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M.
Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin62, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez
Sierra37, S. Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M.
Vesterinen11, B. Viaud7, D. Vieira2, X. Vilasis-Cardona36,o, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß61, H.
Voss10, J.A. de Vries41, R. Waldi61, C. Wallace48, R. Wallace12, S.
Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, A.D. Webber54, D.
Websdale53, M. Whitehead48, J. Wicht38, J. Wiechczynski26, D. Wiedner11, G.
Wilkinson55, M.P. Williams48,49, M. Williams56, F.F. Wilson49, J. Wimberley58,
J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A. Wotton47, S.
Wright47, S. Wu3, K. Wyllie38, Y. Xie50,38, Z. Xing59, Z. Yang3, X. Yuan3, O.
Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, F. Zhang3, L. Zhang59, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3, A. Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
62National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
63Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
64KVI - University of Groningen, Groningen, The Netherlands, associated to 41
65Celal Bayar University, Manisa, Turkey, associated to 38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
## 1 Introduction
The heavy quark expansion (HQE) is a powerful theoretical technique in the
description of decays of hadrons containing heavy quarks. This model describes
inclusive decays and has been used extensively in the analysis of beauty and
charm hadron decays, for example in the extraction of Cabibbo-Kobayashi-
Maskawa matrix elements, such as $|V_{cb}|$ and $|V_{ub}|$ [1]. The basics of
the theory were derived in the late 1980’s [2, *Shifman:1984wx,
*Shifman:1986sm, *Guberina:1986gd]. For $b$-flavoured hadrons, the expansion
of the total decay width in terms of powers of $1/m_{b}$, where $m_{b}$ is the
$b$ quark mass, was derived a few years later [6, *Blok:1992he, *Bigi:1991ir,
*Bigi:1992su]. These developments are summarized in Ref. [10]. It was found
that there were no terms of $\mathcal{O}(1/m_{b})$, that the
$\mathcal{O}(1/m_{b}^{2})$ terms were tiny, and initial estimates of
${\cal{O}}(1/m^{3}_{b})$ [11, 12, *DiPierro:1999tb] effects were small. Thus
differences of only a few percent were expected between the $\mathchar
28931\relax^{0}_{b}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
total decay widths, and hence their lifetimes [11, 14, 15].
In the early part of the past decade, measurements of the ratio of $\mathchar
28931\relax^{0}_{b}$ to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
lifetimes, $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}$, gave results considerably
smaller than this expectation. In 2003 one experimental average gave $0.798\pm
0.052$ [16], while another was $0.786\pm 0.034$ [17, *Franco:2002fc]. Some
authors sought to explain the small value of the ratio by including additional
operators or other modifications [19, *Gabbiani:2003pq, *Gabbiani:2004tp,
*Altarelli:1996gt], while others thought that the HQE could be pushed to
provide a ratio of about 0.9 [23], but not so low as the measured value.
Recent measurements have obtained higher values [24, *Chatrchyan:2013sxa,
*Aaltonen:2010pj]. In fact, the most precise previous measurement from LHCb,
$0.976\pm 0.012\pm 0.006$ [27], based on 1.0 $\mbox{\,fb}^{-1}$ of data,
agreed with the early HQE expectations.
In this paper we present an updated result for $\tau_{\mathchar
28931\relax^{0}_{b}}/\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}}$
using data from 3.0 $\mbox{\,fb}^{-1}$ of integrated luminosity collected with
the LHCb detector from $pp$ collisions at the LHC. Here we add the 2.0
$\mbox{\,fb}^{-1}$ data sample from the 8 TeV data to our previous 1.0
$\mbox{\,fb}^{-1}$ 7 TeV sample [27]. The data are combined and analyzed
together. Larger simulation samples are used than in our previous publication,
and uncertainties are significantly reduced.
The $\mathchar 28931\relax^{0}_{b}$ baryon is detected in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}pK^{-}$ decay mode, discovered by
LHCb [27], while the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ meson
is reconstructed in ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ decays, with $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)\rightarrow\pi^{+}K^{-}$.222Charge-
conjugate modes are implicitly included throughout this Letter. These modes
have the same topology into four charged tracks, thus facilitating
cancellation of systematic uncertainties in the lifetime ratio.
The LHCb detector [28] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes [29] placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6%
at 100$\mathrm{\,Ge\kern-1.00006ptV}$, and impact parameter resolution of
20${\,\upmu\rm m}$ for tracks with large transverse momentum, $p_{\rm
T}$.333We use natural units with $\hbar=c=1$. Different types of charged
hadrons are distinguished using information from two ring-imaging Cherenkov
(RICH) detectors [30]. Photon, electron and hadron candidates are identified
by a calorimeter system consisting of scintillating-pad and preshower
detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons
are identified by a system composed of alternating layers of iron and
multiwire proportional chambers [31]. The trigger [32] consists of a hardware
stage, based on information from the calorimeter and muon systems, followed by
a software stage, which applies a full event reconstruction.
## 2 Event selection and $b$ hadron reconstruction
Events selected for this analysis are triggered by a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decay,
where the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is required at the
software level to be consistent with coming from the decay of a $b$ hadron by
use of either impact parameter (IP) requirements or detachment of the
reconstructed ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay position
from the associated primary vertex.
Events are required to pass a cut-based preselection and then are further
filtered using a multivariate discriminator based on the boosted decision tree
(BDT) technique [33, *AdaBoost]. To satisfy the preselection requirements the
muon candidates must have $p_{\rm T}$ larger than 550 MeV, while the hadron
candidates are required to have $p_{\rm T}$ larger than 250 MeV. Each muon is
required to have $\chi^{2}_{\rm IP}>4$, where $\chi^{2}_{\rm IP}$ is defined
as the difference in $\chi^{2}$ of the primary vertex reconstructed with and
without the considered track. Events must have a $\mu^{+}\mu^{-}$ pair that
forms a common vertex with $\chi^{2}<16$ and that has an invariant mass
between $-48$ and +43 MeV of the known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ mass [1]. Candidate $\mu^{+}\mu^{-}$ pairs are then constrained to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass to improve the
determination of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum.
The two charged final state hadrons must have a vector summed $p_{\rm T}$ of
more than 1 GeV, and form a vertex with $\chi^{2}/{\rm ndf}<10$, where ndf is
the number of degrees of freedom, and a common vertex with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate with $\chi^{2}/{\rm
ndf}<16$. Particle identification requirements are different for the two
modes. Using information from the RICH detectors, a likelihood is formed for
each hadron hypothesis. The difference in the logarithms of the likelihoods,
DLL$(h_{1}-h_{2})$, is used to distinguish between the two hypotheses, $h_{1}$
and $h_{2}$ [30]. In the $\mathchar 28931\relax^{0}_{b}$ decay the kaon
candidate must have DLL$(K-\pi)>4$ and DLL$(K-p)>-3$, while the proton
candidate must have DLL$(p-\pi)>10$ and DLL$(p-K)>-3$. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay, the requirements on the
pion candidate are DLL$(\pi-\mu)>-10$ and DLL$(\pi-K)>-10$, while
DLL$(K-\pi)>0$ is required for the kaon.
The BDT selection uses the smaller value of the DLL($\mu-\pi$) of the
$\mu^{+}$ and $\mu^{-}$ candidates, the $p_{\rm T}$ of each of the two charged
hadrons, and their sum, the $\mathchar 28931\relax^{0}_{b}$ $p_{\rm T}$, the
$\mathchar 28931\relax^{0}_{b}$ vertex $\chi^{2}$, and the $\chi^{2}_{\rm IP}$
of the $\mathchar 28931\relax^{0}_{b}$ candidate with respect to the primary
vertex. The choice of these variables is motivated by minimizing the
dependence of the selection efficiency on decay time; for example, we do not
use the $\chi^{2}_{\rm IP}$ of the proton, the kaon, the flight distance, or
the pointing angle of $\mathchar 28931\relax^{0}_{b}$ to the primary vertex.
To train and test the BDT we use a simulated sample of $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$events for signal and a background data sample from the mass
sidebands in the region $100-200$ MeV below the $\mathchar
28931\relax^{0}_{b}$ signal peak. Half of these events are used for training,
while the other half are used for testing. The BDT selection is chosen to
maximize $S^{2}/(S+B)$, where $S$ and $B$ are the signal and background
yields, respectively. This optimization includes the requirement that the
$\mathchar 28931\relax^{0}_{b}$ candidate decay time be greater than 0.4 ps.
The same BDT selection is used for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{-}K^{+}$ decays. The distributions of the BDT classifier output for
signal and background are shown in Fig. 1. The final selection requires that
the BDT output variable be greater than 0.04.
Figure 1: BDT classifier output for the signal and background. Both training
and test samples are shown; their definitions are given in the text.
The resulting $\mathchar 28931\relax^{0}_{b}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidate invariant mass
distributions are shown in Fig. 2. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidates we also require that
the invariant $\pi^{+}K^{-}$ mass be within $\pm 100$ MeV of the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ mass.
Figure 2: Fits to the invariant mass spectrum of (a)
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}pK^{-}$ and (b)
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}K^{-}$ combinations. The
$\mathchar 28931\relax^{0}_{b}$ and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}$ signals are shown by the (magenta)
solid curves. The (black) dotted lines are the combinatorial backgrounds, and
the (blue) solid curves show the totals. In (a) the $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}K^{-}$ reflections, caused by particle misidentification, are
shown with the (brown) dot-dot-dashed and (red) dot-dashed shapes,
respectively, and the (green) dashed shape represents the doubly misidentified
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\overline{p}$ final state,
where the kaon and proton masses are swapped. In (b) the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}K^{-}$
mode is shown by the (red) dashed curve and the (green) dot-dashed shape
represents the $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ reflection.
In order to measure the number of signal events we need to ascertain the
backgrounds. The background is dominated by random track combinations at
masses around the signal peaks, and their shape is assumed to be exponential
in invariant mass. Specific backgrounds arising from incorrect particle
identification, called “reflections,” are also considered. In the case of the
$\mathchar 28931\relax^{0}_{b}$ decay, these are $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ decays where a kaon is misidentified as a proton and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ decays with
$\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)\rightarrow\pi^{+}K^{-}$ where
the pion is misidentified as a proton. There is also a double
misidentification background caused by swapping the kaon and proton
identifications.
To study these backgrounds, we examine the mass combinations in the sideband
regions from $60-200$ MeV on either side of the $\mathchar
28931\relax^{0}_{b}$ mass peak. Specifically for each candidate in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}pK^{-}$ sideband regions we
reassign to the proton track the kaon or pion mass hypothesis respectively,
and plot them separately. The resulting distributions are shown in Fig. 3.
Figure 3: Invariant mass distributions of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}pK^{-}$ data candidates in the
sideband regions $60-200$ MeV on either side of the $\mathchar
28931\relax^{0}_{b}$ mass peak, reinterpreted as misidentified (a) $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ and (b) $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}K^{-}$ combinations through appropriate mass reassignments. The
(red) dashed curves show the $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}$ contributions and the (green) dot-
dashed curves show $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$
contributions. The (black) dotted curves represent the polynomial background
and the (blue) solid curves the total.
The $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-})$ invariant mass
distribution shows a large peak at the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass. There is also a small
contribution from the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ final
state where the $\pi^{+}$ is misidentified as a $p$. The
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}K^{-})$ distribution, on
the other hand, shows a peak at the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass with a large contribution
from $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays where the
$K^{+}$ is misidentified as a $p$. For both distributions the shapes of the
different contributions are determined using simulation. Fitting both
distributions we find 19 327$\pm$309 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$, and 5613$\pm$285 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ events in the $\mathchar
28931\relax^{0}_{b}$ sideband.
Samples of simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ events are used to find the shapes of these reflected
backgrounds in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}pK^{-}$ mass
spectrum. Using the event yields found in data and the simulation shapes, we
estimate $5603\pm 90$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ and $1150\pm 59$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}K^{-}$ reflection candidates within $\pm 20$ MeV of the
$\mathchar 28931\relax^{0}_{b}$ peak. These numbers are used as Gaussian
constraints in the mass fit described below with the central values as the
Gaussian means and the uncertainties as the widths. Following a similar
procedure we find $1138\pm 48$ doubly-misidentified $\mathchar
28931\relax^{0}_{b}$ decays under the $\mathchar 28931\relax^{0}_{b}$ peak.
This number is also used as a Gaussian constraint in the mass fit.
To determine the number of $\mathchar 28931\relax^{0}_{b}$ signal candidates
we perform an unbinned maximum likelihood fit to the candidate
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}pK^{-}$ invariant mass spectrum
shown in Fig. 2(a). The fit function is the sum of the $\mathchar
28931\relax^{0}_{b}$ signal component, combinatorial background, the
contributions from the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}K^{-}$ reflections and the doubly-misidentified
$\overline{\mathchar
28931\relax^{0}_{b}}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\overline{p}$ decays. The signal is modeled by a triple-Gaussian
function with common means. The fraction and the width ratio for the second
and third Gaussians are fixed to the values obtained in the fit to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ decays, shown in
Fig. 2(b). The effective r.m.s. width is 4.7 MeV. The combinatorial background
is described by an exponential function. The shapes of reflections and doubly-
misidentified contributions are described by histograms imported from the
simulations. The mass fit gives $50\,233\pm 331$ signal and $15\,842\pm 104$
combinatorial background candidates, $5642\pm 88$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ and $1167\pm 58$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}K^{-}$ reflection candidates, and $1140\pm 48$ doubly-
misidentified $\mathchar 28931\relax^{0}_{b}$ candidates within $\pm 20$ MeV
of the $\mathchar 28931\relax^{0}_{b}$ mass peak. The $pK^{-}$ mass spectrum
is consistent with that found previously [27], with a distinct peak near 1520
MeV, together with the other broad resonant and non-resonant structures that
cover the entire kinematic region.
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidate mass
distribution can be polluted by the reflection from $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ decays. Following a similar procedure as for the analysis of
the $\mathchar 28931\relax^{0}_{b}$ mass spectra, we take into account the
reflection under the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ peak.
Figure 2(b) shows the fit to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}K^{-}$ mass distribution. There are signal peaks at both $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ masses on top of the
background. A triple-Gaussian function with common means is used to fit each
signal. The shape of the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}K^{-}$
mass distribution is taken to be the same as that of the signal $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay. The effective r.m.s. width
is 6.5 MeV. An exponential function is used to fit the combinatorial
background. The shape of the $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ reflection is taken from simulation, the yield being Gaussian
constrained in the global fit to the expected value. The mass fit gives
$340\,256\pm 893$ signal and $11\,978\pm 153$ background candidates along with
a negligible $573\pm 27$ contribution of $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ reflection candidates within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peak. All other reflection
contributions are found to be negligible.
## 3 Measurement of the $\mathchar 28931\relax^{0}_{b}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime ratio
The decay time, $t$, is calculated as
$t=m\frac{\vec{d}\cdot\vec{p}}{|\vec{p}|^{2}},$ (1)
where $m$ is the reconstructed invariant mass, $\vec{p}$ the momentum and
$\vec{d}$ the flight distance vector of the particle between the production
and decay vertices. The $b$ hadron is constrained to come from the primary
vertex. To avoid systematic biases due to shifts in the measured decay time,
we do not constrain the two muons to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass.
The decay time distribution of the $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ signal can be described by an exponential function convolved
with a resolution function, $G(t-t^{\prime},\sigma_{\mathchar
28931\relax^{0}_{b}})$, where $t^{\prime}$ is the true decay time, multiplied
by an acceptance function, $A_{\mathchar 28931\relax^{0}_{b}}(t)$:
$F_{\mathchar 28931\relax^{0}_{b}}(t)=A_{\mathchar
28931\relax^{0}_{b}}(t)\times[e^{-t^{\prime}/\tau_{\mathchar
28931\relax^{0}_{b}}}\otimes G(t-t^{\prime},{\sigma_{\mathchar
28931\relax^{0}_{b}}})].$ (2)
The ratio of the decay time distributions of $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ is given by
$R(t)=\frac{A_{\mathchar
28931\relax^{0}_{b}}(t)\times[e^{-t^{\prime}/\tau_{\mathchar
28931\relax^{0}_{b}}}\otimes G(t-t^{\prime},{\sigma_{\mathchar
28931\relax^{0}_{b}}})]}{A_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}(t)\times[e^{-t^{\prime}/\tau_{\kern
0.89996pt\overline{\kern-0.89996ptB}{}^{0}}}\otimes
G(t-t^{\prime},{\sigma_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}})]}.$ (3)
The advantage of measuring the lifetime through the ratio is that the decay
time acceptances introduced by the trigger requirements, selection and
reconstruction almost cancel in the ratio of the decay time distributions. The
decay time resolutions are 40 fs for the $\mathchar 28931\relax^{0}_{b}$ decay
and 37 fs for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay
[27]. They are both small enough in absolute scale, and similar enough for
differences in resolutions between the two modes not to affect the final
result. Thus,
$R(t)=R(0)e^{-t(1/\tau_{\mathchar 28931\relax^{0}_{b}}-1/\tau_{\kern
0.89996pt\overline{\kern-0.89996ptB}{}^{0}})}=R(0)e^{-t\Delta_{\mathchar
28931\relax B}},$ (4)
where $\Delta_{\mathchar 28931\relax B}\equiv 1/\tau_{\mathchar
28931\relax^{0}_{b}}-1/\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}$ is the width difference and
$R(0)$ is the normalization. Since the acceptances are not quite equal, a
correction is implemented to first order by modifying Eq. (4) with a linear
function
$R(t)=R(0)[1+at]e^{-t\Delta_{\mathchar 28931\relax B}},$ (5)
where $a$ represents the slope of the acceptance ratio as a function of decay
time.
The decay time acceptance is the ratio between the reconstructed decay time
distribution for selected events and the generated decay time distribution
convolved with the triple-Gaussian decay time resolutions obtained from the
simulations. In order to ensure that the $p$ and $p_{\rm T}$ distributions of
the generated $b$ hadrons are correct, we weight the simulated samples to
match the data distributions. The simulations do not model the hadron
identification efficiencies with sufficient accuracy for our purposes.
Therefore we further weight the samples according to the hadron identification
efficiencies obtained from $D^{*+}\rightarrow\pi^{+}D^{0},$ $D^{0}\rightarrow
K^{-}\pi^{+}$ events for pions and kaons, and $\mathchar
28931\relax\rightarrow p\pi^{-}$ for protons.
Figure 4: (a) Decay time acceptances (arbitrary scale) from simulation for
(green) circles $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$, and (red) open-boxes $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}(892)$ decays. (b)
Ratio of the decay time acceptances between $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}(892)$ decays obtained
from simulation. The (blue) line shows the result of the linear fit.
The $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ sample is also weighted using signal yields in bins of
$m\left(pK^{-}\right)$.
The decay time acceptances obtained from the weighted simulations are shown in
Fig. 4(a). The individual acceptances in both cases exhibit the same
behaviour. The ratio of the decay time acceptances is shown in Fig. 4(b). For
decay times greater than 7 ps, the acceptance is poorly determined, while
below 0.4 ps the individual acceptances decrease quickly. Thus, we consider
decay times in the range $0.4-7.0$ ps. A $\chi^{2}$ fit to the acceptance
ratio with a function of the form $C(1+at)$ between 0.4 and 7 ps, gives a
slope $a=0.0066\pm 0.0023~{}\rm ps^{-1}$ and an intercept of $C=0.996\pm
0.005$. The $\chi^{2}/\rm ndf$ of the fit is $65/64$.
In order to determine the ratio of $\mathchar 28931\relax^{0}_{b}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetimes, we determine the yield
of $b$ hadrons for both decay modes using unbinned maximum likelihood fits
described in Sec. 2 to the $b$ hadron mass distributions in 22 bins of decay
time of equal width between 0.4 and 7 ps. We use the parameters found from the
time integrated fits fixed in each time bin, with the signal and background
yields allowed to vary, except for the double misidentification background
fraction that is fixed.
The resulting signal yields as a function of decay time are shown in Fig. 5.
Figure 5: Decay time distributions for $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ shown as (blue) circles, and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}(892)$ shown as (green)
squares. For most entries the error bars are smaller than the points.
The subsequent decay time ratio distribution fitted with the function given in
Eq. 5 is shown in Fig. 6.
Figure 6: Decay time ratio between $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}(892)$ decays, and the
fit for $\Delta_{\mathchar 28931\relax B}$ used to measure the $\mathchar
28931\relax^{0}_{b}$ lifetime.
A $\chi^{2}$ fit is used with the slope $a=0.0066~{}\rm ps^{-1}$ fixed, and
both the normalization parameter $R(0)$, and $\Delta_{\mathchar 28931\relax
B}$ allowed to vary. The fitted value of the reciprocal lifetime difference is
$\Delta_{\mathchar 28931\relax B}=17.9\pm 4.3\pm 3.1~{}\rm ns^{-1}.$
Whenever two uncertainties are quoted, the first is statistical and second
systematic. The latter will be discussed in Sec. 4. The $\chi^{2}/\rm ndf$ of
the fit is $20.3/20$. The resulting ratio of lifetimes is
$\frac{\tau_{\mathchar 28931\relax^{0}_{b}}}{\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}=\frac{1}{1+\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}\Delta_{\mathchar 28931\relax
B}}=0.974\pm 0.006\pm 0.004,$
where we use the world average value $1.519\pm 0.007$ ps for $\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}$ [1]. This result is consistent
with and more precise than our previously measured value of $0.976\pm 0.012\pm
0.006$ [27]. Multiplying the lifetime ratio by $\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}$, the $\mathchar
28931\relax^{0}_{b}$ baryon lifetime is
$\tau_{\mathchar 28931\relax^{0}_{b}}=1.479\pm 0.009\pm 0.010~{}\rm ps.$
## 4 Systematic uncertainties
Sources of the systematic uncertainties on $\Delta_{\mathchar 28931\relax B}$,
$\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}$ and the $\mathchar
28931\relax^{0}_{b}$ lifetime are summarized in Table 1.
Table 1: Systematic uncertainties on the $\Delta_{\mathchar 28931\relax B}$, the lifetimes ratio $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern 1.13394pt\overline{\kern-1.13394ptB}{}^{0}}$ and the $\mathchar 28931\relax^{0}_{b}$ lifetime. The systematic uncertainty associated with $\Delta_{\mathchar 28931\relax B}$ is independent of the $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}$ lifetime. Source | $\Delta_{\mathchar 28931\relax B}$ $(\rm ns^{-1})$ | $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}}$ | $\tau_{\mathchar 28931\relax^{0}_{b}}$ $(\rm ps)$
---|---|---|---
Signal shape | 1.5 | 0.0021 | 0.0032
Background model | 0.7 | 0.0010 | 0.0015
Double misidentification | 1.3 | 0.0019 | 0.0029
Acceptance slope | 2.2 | 0.0032 | 0.0049
Acceptance function | 0.2 | 0.0003 | 0.0004
Decay time fit range | 0.3 | 0.0004 | 0.0006
$pK$ helicity | 0.3 | 0.0004 | 0.0006
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime | - | 0.0001 | 0.0068
Total | 3.1 | 0.0044 | 0.0096
The systematic uncertainty due to the signal model is estimated by comparing
the results between the default fit with a triple-Gaussian function and a fit
with a double-Gaussian function. We find a change of $\Delta_{\mathchar
28931\relax B}=1.5~{}\rm ns^{-1}$, which we assign as the uncertainty. Letting
the signal shape parameters free in every time bin results in a change of
$0.4~{}\rm ns^{-1}$. The larger of these two variations is taken as the
systematic uncertainty on the signal shape.
The uncertainties due to the background are estimated by comparing the default
result to that obtained when we allow the exponential background parameter to
float in each time bin. We also replace the exponential background function
with a linear function; the resulting difference is smaller than the assigned
uncertainty due to floating the background shape. The systematic uncertainty
due to the normalization of the double misidentification background is
evaluated by allowing the fraction to change in each time bin.
The systematic uncertainties due to the acceptance slope are estimated by
varying the slope, $a$, according to its statistical uncertainty from the
simulation. An alternative choice of the acceptance function, where a second-
order polynomial is used to parametrize the acceptance ratio between
$\mathchar 28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$, results in a
smaller uncertainty. There is also an uncertainty due to the decay time range
used because of the possible change of the acceptance ratio at short decay
times. This uncertainty is ascertained by changing the fit range to be
$0.7-7.0$ ps and using the difference with the baseline fit. This uncertainty
is greatly reduced with respect to our previous publication [27] due to the
larger fit range, finer decay time bins, and larger signal sample.
In order to correctly model the acceptance, which can depend on the kinematics
of the decay, the $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ simulation is weighted according to the $m(pK^{-})$ distribution
observed in data. As a cross-check, we weight the simulation according to the
two-dimensional distribution of $m(pK^{-})$ and $pK^{-}$ helicity angle and
assign the difference as a systematic uncertainty. In addition, the PDG value
for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime,
$\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}}=1.519\pm 0.007~{}\rm
ps$ [1], is used to calculate the $\mathchar 28931\relax^{0}_{b}$ lifetime;
the errors contribute to the systematic uncertainty. The total systematic
uncertainty is obtained by adding all of the contributions in quadrature.
## 5 Conclusions
We determine the ratio of lifetimes of the $\mathchar 28931\relax^{0}_{b}$
baryon and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ meson to be
$\frac{\tau_{\mathchar 28931\relax^{0}_{b}}}{\tau_{B^{0}}}=0.974\pm 0.006\pm
0.004.$
This is the most precise measurement to date and supersedes our previously
published result [27]. It demonstrates that the $\mathchar
28931\relax^{0}_{b}$ lifetime is shorter than the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime by $-(2.6\pm 0.7)$%,
consistent with the original predictions of the HQE [2, 11, 35, 36, 10], thus
providing validation for the theory. Using the world average measured value
for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime [1], we
determine
$\tau_{\mathchar 28931\relax^{0}_{b}}=1.479\pm 0.009\pm 0.010~{}\rm ps,$
which is the most precise measurement to date.
LHCb has also made a measurement of $\tau_{\mathchar 28931\relax^{0}_{b}}$
using the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax$
final state obtaining $1.415\pm 0.027\pm 0.006$ ps [37]. The two LHCb
measurements have systematic uncertainties that are only weakly correlated,
and we quote an average of the two measurements of $1.468\pm 0.009\pm 0.008$
ps.
## Acknowledgements
We are thankful for many useful and interesting conversations with Prof.
Nikolai Uraltsev who contributed greatly to theories describing heavy hadron
lifetimes; unfortunately he passed away before these results were available.
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages we depend on. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia).
## References
* [1] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001, and 2013 update for 2014 edition
* [2] M. A. Shifman and M. B. Voloshin, Hierarchy of lifetimes of charmed and beautiful hadrons, Sov. Phys. JETP 64 (1986) 698
* [3] M. A. Shifman and M. B. Voloshin, Preasymptotic effects in inclusive weak decays of charmed particles, Sov. J. Nucl. Phys. 41 (1985) 120
* [4] M. A. Shifman and M. B. Voloshin, On annihilation of mesons built from heavy and light quark and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\leftrightarrow B^{0}$ oscillations, Sov. J. Nucl. Phys. 45 (1987) 292
* [5] B. Guberina, R. Rückl, and J. Trampetić, Charmed baryon lifetime differences, Z. Phys. C33 (1986) 297
* [6] B. Blok and M. A. Shifman, The rule of discarding $1/N_{c}$ in inclusive weak decays (I), Nucl. Phys. B399 (1993) 441, arXiv:hep-ph/9207236
* [7] B. Blok and M. A. Shifman, The rule of discarding $1/N_{c}$ in inclusive weak decays (II), Nucl. Phys. B399 (1993) 459, arXiv:hep-ph/9209289
* [8] I. I. Bigi and N. G. Uraltsev, Gluonic enhancements in non-spectator beauty decays: an inclusive mirage though an exclusive possibility, Phys. Lett. B280 (1992) 271
* [9] I. I. Bigi, N. G. Uraltsev, and A. I. Vainshtein, Nonperturbative corrections to inclusive beauty and charm decays: QCD versus phenomenological models, Phys. Lett. B293 (1992) 430, arXiv:hep-ph/9207214
* [10] I. I. Bigi et al., Nonleptonic decays of beauty hadrons: From phenomenology to theory, arXiv:hep-ph/9401298, in $B$ Decays revised 2nd edition, ed. S. Stone, World Scientific, Singapore, 1994 p132-157
* [11] M. Neubert and C. T. Sachrajda, Spectator effects in inclusive decays of beauty hadrons, Nucl. Phys. B483 (1997) 339, arXiv:hep-ph/9603202
* [12] N. G. Uraltsev, On the problem of boosting nonleptonic $b$ baryon decays, Phys. Lett. B376 (1996) 303, arXiv:hep-ph/9602324
* [13] UKQCD collaboration, M. Di Pierro, C. T. Sachrajda, and C. Michael, An exploratory lattice study of spectator effects in inclusive decays of the $\mathchar 28931\relax^{0}_{b}$ baryon, Phys. Lett. B468 (1999) 143, arXiv:hep-lat/9906031
* [14] H.-Y. Cheng, Phenomenological analysis of heavy hadron lifetimes, Phys. Rev. D56 (1997) 2783, arXiv:hep-ph/9704260
* [15] J. L. Rosner, Enhancement of the $\mathchar 28931\relax^{0}_{b}$ decay rate, Phys. Lett. B379 (1996) 267, arXiv:hep-ph/9602265
* [16] M. Battaglia et al., The CKM matrix and the unitarity triangle. Workshop, CERN, Geneva, Switzerland, 13-16 Feb 2002: Proceedings, arXiv:hep-ph/0304132
* [17] C. Tarantino, Beauty hadron lifetimes and B meson CP violation parameters from lattice QCD, Eur. Phys. J. C33 (2004) S895, arXiv:hep-ph/0310241
* [18] E. Franco, V. Lubicz, F. Mescia, and C. Tarantino, Lifetime ratios of beauty hadrons at the next-to-leading order in QCD, Nucl. Phys. B633 (2002) 212, arXiv:hep-ph/0203089
* [19] T. Ito, M. Matsuda, and Y. Matsui, New possibility of solving the problem of lifetime ratio $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}}$, Prog. Theor. Phys. 99 (1998) 271, arXiv:hep-ph/9705402
* [20] F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, $\mathchar 28931\relax^{0}_{b}$ lifetime puzzle in heavy quark expansion, Phys. Rev. D68 (2003) 114006, arXiv:hep-ph/0303235
* [21] F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, Spectator effects and lifetimes of heavy hadrons, Phys. Rev. D70 (2004) 094031, arXiv:hep-ph/0407004
* [22] G. Altarelli, G. Martinelli, S. Petrarca, and F. Rapuano, Failure of local duality in inclusive nonleptonic heavy flavor decays, Phys. Lett. B382 (1996) 409, arXiv:hep-ph/9604202
* [23] N. G. Uraltsev, Topics in the heavy quark expansion, arXiv:hep-ph/0010328
* [24] ATLAS collaboration, G. Aad et al., Measurement of the $\mathchar 28931\relax^{0}_{b}$ lifetime and mass in the ATLAS experiment, Phys. Rev. D87 (2013) 032002, arXiv:1207.2284
* [25] CMS collaboration, S. Chatrchyan et al., Measurement of the $\mathchar 28931\relax^{0}_{b}$ lifetime in pp collisions at $\sqrt{s}=7$ TeV, JHEP 07 (2013) 163, arXiv:1304.7495
* [26] CDF collaboration, T. Aaltonen et al., Measurement of $b$ hadron lifetimes in exclusive decays containing a $J/\psi$ in $p\overline{p}$ collisions at $\sqrt{s}=1.96$TeV, Phys. Rev. Lett. 106 (2011) 121804, arXiv:1012.3138
* [27] LHCb collaboration, R. Aaij et al., Precision measurement of the $\mathchar 28931\relax^{0}_{b}$ baryon lifetime, Phys. Rev. Lett. 111 (2013) 102003, arXiv:1307.2476
* [28] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [29] R. Arink et al., Performance of the LHCb outer tracker, JINST 9 (2014) 01002, arXiv:1311.3893
* [30] M. Adinolfi et al., Performance of the LHCb RICH detector at the LHC, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
* [31] A. A. Alves Jr. et al., Performance of the LHCb muon system, JINST 8 (2013) P02022, arXiv:1211.1346
* [32] R. Aaij et al., The LHCb trigger and its performance in 2011, JINST 8 (2013) P04022, arXiv:1211.3055
* [33] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
* [34] R. E. Schapire and Y. Freund, A decision-theoretic generalization of on-line learning and an application to boosting, Jour. Comp. and Syst. Sc. 55 (1997) 119
* [35] N. Uraltsev, Heavy quark expansion in beauty and its decays, arXiv:hep-ph/9804275
* [36] I. I. Bigi, The QCD perspective on lifetimes of heavy flavor hadrons, arXiv:hep-ph/9508408
* [37] LHCb collaboration, R. Aaij et al., Measurements of the $B^{+}$, $B^{0}$, $B_{s}^{0}$ meson and $\mathchar 28931\relax^{0}_{b}$ baryon lifetimes, arXiv:1402.2554, submitted to JHEP
|
arxiv-papers
| 2014-02-25T17:01:26 |
2024-09-04T02:49:58.834759
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. Anderlini,\n J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, M. Borsato, T.J.V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto,\n J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M. Calvi, M. Calvo Gomez,\n A. Camboni, P. Campana, D. Campora Perez, F. Caponio, A. Carbone, G. Carboni,\n R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G.\n Casse, L. Cassina, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M.\n Charles, Ph. Charpentier, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba,\n X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, I.\n Counts, B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R.\n Currie, C. D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n S. Donleavy, F. Dordei, M. Dorigo, P. Dorosz, A. Dosil Su\\'arez, D. Dossett,\n A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger,\n R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, S. Esen, A. Falabella, C.\n F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, J. Fu, E. Furfaro, A. Gallas Torreira, D. Galli, S.\n Gambetta, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, L. Gavardi, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, A. Gianelle, S. Giani', V. Gibson, L. Giubega, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, L. Grillo, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, T.W. Hafkenscheid, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, L. Henry, J.A.\n Hernando Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, D. Hill, M.\n Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, N. Hussain, D. Hutchcroft, D.\n Hynds, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, N.\n Jurik, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, M.\n Kelsey, I.R. Kenyon, T. Ketel, B. Khanji, C. Khurewathanakul, S. Klaver, O.\n Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, B. Langhans, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, M. Liles,\n R. Lindner, C. Linn, F. Lionetto, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H.\n Lopes, N. Lopez-March, P. Lowdon, H. Lu, D. Lucchesi, H. Luo, E. Luppi, O.\n Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G.\n Manca, G. Mancinelli, M. Manzali, J. Maratas, U. Marconi, C. Marin Benito, P.\n Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, F. Martinez Vidal, D. Martins\n Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov, M.\n McCann, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier,\n M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, M. Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, R.\n Mountain, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik,\n T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neri, S. Neubert, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M.\n Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano,\n F. Palombo, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, L.\n Pappalardo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pearce, A. Pellegrino,\n M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, P. Perret, M. Perrin-Terrin,\n L. Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, A. Pistone, S. Playfer, M. Plo\n Casasus, F. Polci, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve,\n V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H. Rademacker,\n B. Rakotomiaramanana, M. Rama, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, A.B.\n Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A.\n Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz\n Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V.\n Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, G.\n Simi, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E. Smith,\n J. Smith, M. Smith, H. Snoek, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, F. Spinella, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S. Stone, B.\n Storaci, S. Stracka, M. Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah,\n L. Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer,\n V. Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, J.A. de\n Vries, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, G. Wilkinson, M.P. Williams, M. Williams, F.F.\n Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1402.6242"
}
|
1402.6243
|
∎
11institutetext: Ahmed M. Alaa 22institutetext: Electronics and Electrical
Communications Engineering Dept., Cairo University,
Gizah 12316, Egypt
Tel.: +20-0100-2968798
22email: [email protected] 33institutetext: Omar A. Nasr 44institutetext:
Electronics and Electrical Communications Engineering Dept., Cairo University,
Gizah 12316, Egypt
44email: [email protected]
# Globally Optimal Cooperation in Dense Cognitive Radio Networks
Ahmed M. Alaa Omar A. Nasr
(Received: date / Accepted: date)
###### Abstract
The problem of calculating the local and global decision thresholds in hard
decisions based cooperative spectrum sensing is well known for its
mathematical intractability. Previous work relied on simple suboptimal
counting rules for decision fusion in order to avoid the exhaustive numerical
search required for obtaining the optimal thresholds. However, these simple
rules are not globally optimal as they do not maximize the overall global
detection probability by jointly selecting local and global thresholds.
Instead, they maximize the detection probability for a specific global
threshold. In this paper, a globally optimal decision fusion rule for Primary
User signal detection based on the Neyman-Pearson (NP) criterion is derived.
The algorithm is based on a novel representation for the global performance
metrics in terms of the regularized incomplete beta function. Based on this
mathematical representation, it is shown that the globally optimal NP hard
decision fusion test can be put in the form of a conventional one dimensional
convex optimization problem. A binary search for the global threshold can be
applied yielding a complexity of $\mathcal{O}(\log_{2}(N))$, where $N$
represents the number of cooperating users. The logarithmic complexity is
appreciated because we are concerned with dense networks, and thus $N$ is
expected to be large. The proposed optimal scheme outperforms conventional
counting rules, such as the OR, AND, and MAJORITY rules. It is shown via
simulations that, although the optimal rule tends to the simple OR rule when
the number of cooperating secondary users is small, it offers significant SNR
gain in dense cognitive radio networks with large number of cooperating users.
###### Keywords:
Cooperative spectrum sensing Cognitive radio Decision Fusion Optimization
## 1 Introduction
Cognitive radio (CR) is a promising technology offering enhanced spectrum
efficiency via dynamic spectrum access [1], [2]. In a CR network, unlicensed
Secondary Users (SU) can opportunistically occupy the unused spectrum
allocated to a licensed primary user (PU). This is achieved by means of PU
signal detection. Detection of PU signal entails sensing the spectrum occupied
by the licensed user in a continuous manner. Based on the sensing data, the SU
is required to decide whether or not a PU exists. A common problem encountered
in CR systems is the hidden terminal problem [3], where shadowing and
multipath fading affect the strength of the PU signal causing it to be
undetectable. Hence, spatial diversity is applied by utilizing multiple
decisions from several SU terminals using a decision fusion rule. The fusion
rule is applied by a central terminal known as the fusion center. Two basic
approaches for decision combining are discussed in literature: soft decision
(SD) and hard decision (HD) combining. The former relies on adding the sensed
energies, while the latter combines one-bit local decisions to make a final
decision [4].
In this work, we tackle the problem of optimizing the HD combining scheme
based on Neyman-pearson (NP) criterion. While the optimal NP test has been
formulated for the SD combining case [4], it is much more challenging to apply
an optimal NP test for the HD combining scheme. The reason for this is that
every SU employs a local detection threshold, while the fusion center applies
a global threshold to make a final decision on the existence of a PU. Thus,
unlike the simple one-dimensional problem in SD combining, two degrees of
freedom are considered in the HD combining optimization problem. In his
pioneering work, Tsitsiklist [5] showed that the problem is mathematically
intractable and an exhaustive search would be used to obtain local detection
thresholds. In a recent comprehensive survey, Quan et al [6] pointed out that
computing the optimal decision thresholds under the NP criterion is
mathematically intractable. Various suboptimal solutions were presented in
literature. In [7], the problem was solved by simply fixing local thresholds
and obtaining the optimum global threshold or vice versa. Recently, the
problem was revisited in [8], were large deviation analysis was used to
determine a local decision rule to optimize the asymptotic global performance.
However, the intractability of the exact NP optimization problem was again
emphasized. In literature, the adopted HD combining rules are never globally
optimal. Researchers usually employ simple suboptimal AND, OR or MAJORITY
counting rules for global detection [9][10]. Others try to calculate the
optimim local and global thresholds but mainly using exhaustive numerical
methods [11][12]. In [4], the performance of the SD combining scheme with NP
test was compared with an OR-rule based HD combining scheme, which is not
necessarily optimal. The problem of HD and SD performance comparison was
thoroughly studied in [12]. However, the authors used suboptimal counting
rules and stated that the threshold calculations are not trivial as complex
optimization schemes are needed to solve them.
Although simple fusion rules, such as the OR rule, is usually found to be
optimal for cognitive radio networks with small number of cooperating SUs, it
was never verified in literature that the same applies for dense networks with
large number of SUs. In this work, we propose a globally optimal decision
fusion rule for HD combining based on Neyman-pearson criterion. It is shown
that the NP optimal thresholds can be obtained by solving a simple one-
dimensional convex optimization problem. Besides, we obtain a closed form
expression for the local detection threshold as a function of the optimal
global threshold. A simple and efficient algorithm for optimizing global and
local thresholds is proposed. Although the algorithm is general and can be
applied for any number of SUs, it is shown that it offers significant
performance gain compared to the OR rule in networks with large number of
cooperating users.
The rest of the paper is organized as follows. In section 2, we present the
system model. Next, we propose the globally optimal HD combining scheme in
section 3. Simulation results are discussed in section 4. Finally, we draw our
conclusion in section 5.
## 2 System Model
We investigate cooperative spectrum sensing in a CR network with $N$ cognitive
users and a single common receiver (Fusion Center). We assume that the SU
observes $M$ samples for spectrum sensing. Energy detection is adopted as a
spectrum sensing technique. It is assumed that the instantaneous SNR at the
$j^{th}$ node is $\gamma_{j}$ and the primary signalӳ $i^{th}$ sample at the
$j^{th}$ CR is $S_{ji}$, and considered constant with unity power for the
entire sensing period. The additive white noise is $n_{ji}$ $\sim$
$\mathcal{N}(0,1)$. Thus, the $i^{th}$ sample received at the $j^{th}$ CR is a
binary hypothesis give by:
$r_{ji}=\left\\{\begin{array}[]{lr}n_{ji},&\ \mathcal{H}_{o}\\\
\sqrt{\gamma_{j}}\hskip 2.84526ptS_{ji}+n_{ji},&\
\mathcal{H}_{1}\end{array}\right.$ (1)
The conditional distributions on null and alternative hypotheses are:
$r_{ji}\sim\left\\{\begin{array}[]{lr}\mathcal{N}(0,1),&\ \mathcal{H}_{o}\\\
\mathcal{N}(\sqrt{\gamma_{j}},1),&\ \mathcal{H}_{1}\end{array}\right.$ (2)
where $\mathcal{H}_{o}$ denotes the absence of the PU, while $\mathcal{H}_{1}$
denotes the existence of the PU. After applying such signal to an energy
detector and obtaining binary decisions on PU existence, the local false alarm
and detection probabilities at the $j^{th}$ CR are [2]:
$P_{F}(M,\lambda)=P(Y_{j}>\lambda|\mathcal{H}_{o})=\frac{\Gamma(\frac{M}{2},\frac{\lambda}{2})}{\Gamma(\frac{M}{2})},$
and
$P_{D}(M,\lambda,\gamma_{j})=P(Y_{j}>\lambda|\mathcal{H}_{1})=Q_{M/2}(\sqrt{2\gamma_{j}},\sqrt{\lambda})$
(3)
where $\lambda$ is the local threshold, $\Gamma(.,.)$ is the incomplete gamma
function, $\Gamma(.)$ is the gamma function, and $Q_{u}(.)$ is the generalized
Marcum Q-function. We assume Rayleigh fading with an average SNR of
$\overline{\gamma}$. The average SNR is assumed to be the same for all CR
users. The instantaneous SNR is assumed to be constant over the $M$ observable
samples. Different observations perceive different SNR values. The SNR varies
according to the exponential pdf:
$f_{\gamma}(\gamma)=\frac{1}{\overline{\gamma}}e^{-\frac{\gamma}{\overline{\gamma}}},\gamma\geq
0.$ (4)
The reporting channel between the SUs and the fusion center is assumed to be
free of errors.
## 3 Globally Optimal Hard Decision Fusion
In this section, we propose a globally optimal algorithm for HD combining
based on the Neyman-Pearson criterion. The ultimate goal of a Neyman-pearson
test is to maximize the detection probability for a given false alarm
probability. The overall performance of the HD scheme is determined by the
global detection and false alarm probabilities, which are functions of the
local detection and false alarm probabilities given in equation (3). As the
fusion center employs an n-out-of-N rule fusion rule, we let $l$ be the test
statistic denoting the number of votes for the existence of PU from the $N$ SU
votes. Hence, the conditional pdfs follow the binomial distribution as [3]:
$P(l|\mathcal{H}_{o})=\binom{N}{l}\hskip 4.2679ptP_{F}^{l}\hskip
4.2679pt(1-P_{F})^{N-l}$
and
$P(l|\mathcal{H}_{1})=\binom{N}{l}\hskip 4.2679pt\overline{P}_{D}^{l}\hskip
4.2679pt(1-\overline{P_{D}})^{N-l},$ (5)
where $\overline{P}_{D}$ is the local detection probability averaged over the
fading channel pdf as follows:
$\overline{P}_{D}=\int_{0}^{\infty}Q_{M/2}(\sqrt{2\gamma},\sqrt{\lambda})\,\,\frac{1}{\overline{\gamma}}e^{-\frac{\gamma}{\overline{\gamma}}}\,d\gamma.$
(6)
and the global false alarm and detection probabilities $Q_{f}$ and $Q_{d}$ are
[3][12]:
$Q_{f}(n,\lambda)=\sum_{l=n}^{N}\binom{N}{l}\hskip
4.2679ptP_{F}^{l}(\lambda)\hskip 4.2679pt(1-P_{F}(\lambda))^{N-l},$
$Q_{d}(n,\lambda)=\sum_{l=n}^{N}\binom{N}{l}\hskip
4.2679pt\overline{P}_{D}^{l}(\lambda)\hskip
4.2679pt(1-\overline{P}_{D}(\lambda))^{N-l}.$ (7)
The global Neyman-pearson threshold for the discrete observable random
variable $l$ is denoted by $n$. We search for the pair of thresholds
$(n,\lambda)_{opt}$ that maximizes the global detection probability $Q_{d}$
for $Q_{f}$ = $\alpha$. Unlike the conventional Neyman-Pearson detection
schemes, we have two degrees of freedom dictated by the local and global
thresholds.
The cumulative density function (CDF) of the binomial distribution can be
written in the form of the regularized incomplete beta function defined as
[13, eq. 6.6.2]:
$\mathcal{I}(x;a,b)=\frac{\beta(x;a,b)}{\beta(a,b)},$
where $\beta(x;a,b)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt$ is the upper incomplete
beta function and $\beta(a,b)=\int_{0}^{1}t^{a-1}(1-t)^{b-1}dt$ is the beta
function. The CDF of a binomial random variable $x\sim B(N,p)$ is $F(x\leq
X)=\mathcal{I}(1-p;N-X,X+1)$ [13, eq. 6.6.4]. Thus, the cumulative density of
the discrete variable $l$ under $\mathcal{H}_{o}$ hypothesis is given by:
$P(L\leq n|\mathcal{H}_{o})=\mathcal{I}(1-P_{F};N-n,n+1),$ (8)
and the global false alarm probability is given by:
$\displaystyle Q_{f}$ $\displaystyle=$ $\displaystyle
1-P(L<n|\mathcal{H}_{o})=1-P(L\leq n-1|\mathcal{H}_{o})$ (9) $\displaystyle=$
$\displaystyle 1-\mathcal{I}(1-P_{F};N-n+1,n).$
One of the properties of the regularized incomplete beta function is the
symmetry property [13, eq. 6.6.3]:
$1-\mathcal{I}(1-p;a,b)=\mathcal{I}(p,b,a).$
Applying this property to equation (9):
$Q_{f}=\mathcal{I}(P_{F};n,N-n+1),$ (10)
and by using the inverse regularized beta function, we can obtain the local
false alarm probability by setting $Q_{f}$ = $\alpha$:
$P_{F}=\mathcal{I}^{-1}(\alpha;n,N-n+1).$ (11)
The regularized beta function and its inverse are implemented with low
complexity algorithms in mathematical software tools like MATLAB and
MATHEMATICA. The same algorithms can be implemented at the SU recievers.
Similarly, the global detection probability is given by:
$Q_{d}=\mathcal{I}(\overline{P}_{D};n,N-n+1).$ (12)
Before presenting the proposed Neyman-Pearson algorithm, we construct some
auxiliary mathematical tools. We define the functions $\zeta_{M}(x)$ and
$\Phi_{M}(x,a,b)$ as:
$\zeta_{M}(x)=\frac{\Gamma(\frac{M}{2},\frac{x}{2})}{\Gamma(\frac{M}{2})}$
and
$\Phi_{M}(x,a,b)=\mathcal{I}(\zeta_{M}(x);a,b-a+1).$ (13)
With the inverse function given by:
$\Phi_{M}^{-1}(y,a,b)=\zeta_{M}^{-1}(\mathcal{I}^{-1}(y;a,b-a+1)).$ (14)
Where $\zeta^{-1}_{M}(.)$ is the inverse incomplete gamma function. We can
rewrite the global false alarm probability and local threshold in terms of the
$\Phi_{M}(x;a,b)$ function by combining equation (3) with equation (10):
$Q_{f}=\Phi_{M}(\lambda;n,N),$ (15) $\lambda=\Phi_{M}^{-1}(\alpha;n,N).$ (16)
Note that equation (15) is a single equation in two unknowns $n$ and
$\lambda$. Thus, there is an infinte number of $(n,\lambda)$ pairs that solve
(15). We search for the pair that maximizes the expression in (12).
The global detection probability $Q_{d}(n)$ is a log-concave function of the
global threshold $n$. Thus, the global and local threshold pair
$(n,\lambda)_{opt}$ is obtained by solving the convex optimization problem:
$n_{opt}=\underset{n\in\\{1,\cdots,N\\}}{\operatorname{arg\,min}}\,\biggl{(}-\ln\left(\hskip
3.55658pt\mathcal{I}\left(\hskip
2.84526pt\overline{P}_{D}(n);n,N-n+1\right)\right)\biggr{)},$
and
$\lambda_{opt}=\Phi_{M}^{-1}(\alpha;n_{opt},N).$ (17)
Our objective is to prove that the global detection probability in equation
(12) is a log-concave function of $n$. Hence, taking the negative of its
natural logarithm leads to a straight forward convex optimization problem.
Note that the regularized incomplete beta function can be written in terms of
the gauss hypergeometric function ${}_{2}F_{1}\left(.;.;.;.\right)$ as [14,
eq. 8.392]:
$Q_{d}(n)=\frac{{\overline{{P}}_{D}}^{n}}{n\hskip
2.84526pt\beta(n,N-n+1)}\hskip
2.84526pt_{2}F_{1}\left(n;n-N;n+1;\overline{P}_{D}\right).$
Furthermore, the beta function can be obtained in terms of the gamma function
as in [14, eq. 8.384.1] which yields:
$Q_{d}(n)=\frac{{\overline{{P}}_{D}}^{n}\hskip 2.84526pt\Gamma(N+1)}{n\hskip
2.84526pt\Gamma(n)\Gamma(N-n+1)}\hskip
2.84526pt_{2}F_{1}\left(n;n-N;n+1;\overline{P}_{D}\right).$
By replacing the gauss hypergeometric function by the equivalent series
representation [15, eq. (4)]:
$Q_{d}(n)=\frac{{\overline{{P}}_{D}}^{n}\hskip 2.84526pt\Gamma(N+1)}{n\hskip
2.84526pt\Gamma(n)\Gamma(N-n+1)}\hskip
2.84526pt\sum_{k=0}^{\infty}\frac{(n)_{k}(n-N)_{k}}{(n+1)_{k}}\times\frac{{\overline{{P}}_{D}}^{k}}{k!},$
where $(a)_{k}=a(a+1)\cdots(a+k-1)$ is Pochhammer’s symbol, which can be
represented by $(a)_{k}=\frac{\Gamma(a+k)}{\Gamma(a)}$ [15, eq. (1)]. By
simplifying the above expression using the gamma function representation of
the Pochhammer symbols, the function $Q_{d}(n)$ becomes:
$Q_{d}(n)=\sum_{k=0}^{\infty}\Xi(n,k),$
where
$\Xi(n,k)\propto$
$\underbrace{(n-N)_{k}}_{F_{1}(n,k)}\times\underbrace{\frac{1}{n\Gamma(n)}}_{F_{2}(n,k)}\times\underbrace{\frac{1}{(n+k)\Gamma(N-n)}}_{F_{3}(n,k)}\times\underbrace{{\overline{P}_{D}}^{n+k}}_{F_{4}(n,k)}.$
(18)
Thus, the global detection probability is composed of $\Xi(n,k)$ terms that
are summed over $k$. Every $\Xi(n,k)$ term is proportional (within a positive
scale) to the product of the terms $F_{1}(n,k)$, $F_{2}(n,k)$, $F_{3}(n,k)$
and $F_{4}(n,k)$ as depicted by equation (18). We start by studying the
behavior of each $F(n,k)$ term individually.
* •
log-concavity of $F_{1}(n,k)$
In order to prove the log-concavity of Pochhammer’s symbol
$F_{1}(n,k)=(n-N)_{k}$, we take the natural logarithm of the gamma function
representation of $F_{1}(n,k)$ as:
$\ln(F_{1}(n,k))=\ln(\Gamma(n-N+k))-\ln(\Gamma(n-N)).$
Applying the second derevative test, we get:
$\frac{\partial^{2}\ln(F_{1}(n,k))}{\partial
n^{2}}=\psi^{\tiny{(1)}}(n-N+k)-\psi^{\tiny{(1)}}(n-N),$
where $\psi^{\tiny{(1)}}(x)$ is the first order polygamma function [13, eq.
6.4.1]. Based on the property $\psi^{(1)}(x+1)=\psi^{(1)}(x)-\frac{1}{x^{2}}$
[13, eq. 6.4.6], we conclude that $\psi^{(1)}(x+k)<\psi^{(1)}(x),\forall k>0$.
Thus, $\psi^{\tiny{(1)}}(n-N+k)-\psi^{\tiny{(1)}}(n-N)$ is always negative and
the function $F_{1}(n,k)$ is log-concave.
* •
log-concavity of $F_{2}(n,k)$ and $F_{3}(n,k)$
The second derevative test for $F_{2}(n,k)$ is given by:
$\frac{\partial^{2}\ln(F_{2}(n,k))}{\partial
n^{2}}=\psi^{\tiny{(1)}}(n+1)-2\psi^{\tiny{(1)}}(n),$
which is always negative as $\psi^{(1)}(x+1)<\psi^{(1)}(x)$, $\forall x>0$.
Hence, the second derevative test shows the log-concavity of $F_{2}(n,k)$. A
similar analysis can be applied to $F_{3}(n,k)$.
Figure 1: The behavior of local threshold as a function of the global
threshold. Figure 2: The concavity of $Q_{d}(n)$ for various numbers of
cooperating users. Figure 3: Convexity of the objective function.
* •
log-concavity of $F_{4}(n,k)$
Note that $F_{4}(n,k)$ is given by
$F_{4}(n,k)=\int_{0}^{\infty}Q_{M/2}(\sqrt{2\gamma},\sqrt{\lambda})\,\,\frac{1}{\overline{\gamma}}e^{-\frac{\gamma}{\overline{\gamma}}}\,d\gamma.$
The log-concavity of the functions $b\to Q_{M/2}(a,b)$ and $b\to
Q_{M/2}(a,\sqrt{b})$ were shown in [14]. Thus,
$Q_{M/2}(\sqrt{2\gamma},\sqrt{\lambda})$ is a log-concave function of
$\lambda$. By discretization of the integral defining $F_{4}(n,k)$, we obtain
$F_{4}(n,k)=\lim_{\bigtriangleup\gamma\to
0}\sum_{i=0}^{\infty}Q_{M/2}(\sqrt{2i\bigtriangleup\gamma},\sqrt{\lambda})\,\,\frac{1}{\overline{\gamma}}e^{-\frac{i\bigtriangleup\gamma}{\overline{\gamma}}}\,\bigtriangleup\gamma.$
Because the terms
$\frac{1}{\overline{\gamma}}e^{-\frac{i\bigtriangleup\gamma}{\overline{\gamma}}}\,\bigtriangleup\gamma$
in the summation are all positive, and the terms
$Q_{M/2}(\sqrt{2i\bigtriangleup\gamma},\sqrt{\lambda})$ are all log-concave in
$\lambda$, thus $F_{4}(n,k)$ is the sum of positive scaled log-concave
functions, which means that $F_{4}(n,k)$ is also a log-concave function.
Based on the above discussion, we conclude that the function $\Xi(n,k)$ is a
product of log-concave functions. As the product and addition operations
preserve log-concavity [17], $\Xi(n,k)$ and $Q_{d}(n)$ are both log-concave on
all positive values of $n$. Because $Q_{d}(n)$ is a log-concave function of
$n$, we can obtain the global threshold by minimization of the convex function
$-\ln(Q_{d}(n))$.
To sum up, a cognitive radio user needs to perform a simple two step algorithm
in order to obtain the optimal thresholds. Given $\overline{\gamma}$, $M$,
$N$, and assuming that $N$ is odd, the SU applies the following two steps:
Step 1: Obtain the optimal global threshold $n_{opt}$ by applying convex
minimization to the objective function $\biggl{(}-\ln\left(\hskip
3.55658pt\mathcal{I}\left(\hskip
2.84526pt\overline{P}_{D}(n);n,N-n+1\right)\right)\biggr{)}$.
This can be done using a binary search as follows:
1:procedure Global threshold($N,M$)
2: $n_{opt}\leftarrow 0$
3: $i\leftarrow 1$
4: $j\leftarrow\frac{N}{2}$
5: $k\leftarrow 0$
6: $F(n)\leftarrow\biggl{(}-\ln\left(\hskip 3.55658pt\mathcal{I}\left(\hskip
2.84526pt\overline{P}_{D}(n);n,N-n+1\right)\right)\biggr{)}$
7: while $k\not=1$ do
8: if $F(j)\leq F(j+1)\,\ and\,\ F(j)\leq F(j-1)$ then
9: $n_{opt}\leftarrow j$
10: $k\leftarrow 1$
11: else
12: $i\leftarrow i+1$
13: $j\leftarrow j+sign(F(j-1)-F(j+1))\frac{N}{2^{i}}$
14: end if
15: end while
16: return $n_{opt}$
17:end procedure
Step 2: Obtain the optimal local thresholds using the equation
$\lambda_{opt}=\Phi_{M}^{-1}(\alpha;n_{opt},N)$.
The optimization of the objective function is a done using a simple binary
search approach. The feasibility of binary search is due to the convexity of
the set of points representing the discrete objective function $-ln(Q_{d})$.
Thus, the algorithm has a complexity of $\mathcal{O}(\log_{2}(N))$, and it
scales logarithmically with the number of cooperating users. Because we are
mainly concerned with dense networks, the logarithmic complexity is
appreciated. This would be appreciated by CR reciever designers as threshold
optimization has to be done every time the listening or reporting channels
change [12]. Figures 2 depicts the impact of the number of cooperating users
and SNR on $Q_{d}(n)$ for a false alarm probability of 0.01. It is shown that
as more users cooperate, the detection probability improves. It is found that
an OR-rule would be optimal for the case of $N$ = 4 case. However, as $N$
increases, the maximum detection probability becomes interior to the range
$(1,N)$. Figure 3 depicts the convexity of the objective function
$-ln(Q_{d}(n))$ at $N$ = 32. It is shown that increasing SNR will normally
lead to an enhanced detection performance.
## 4 Simulation results
In this section, we aim at characterizing the performance of the proposed
globally optimal algorithm. The optimal fusion rule employs the thresholds
calculated via the optimization problem in (17). We first verify the accuracy
of the analytic model adopted in our work. In figure 4, the simulated
detection probability is plotted versus SNR and compared with the numerical
results obtained from equation (12). It is shown that both results nearly
coincide. In order to verify the optimality of the proposed algorithm, a
comparison is done between the optimal rule and the conventional AND, OR and
MAJORITY rules in figure 5. In all simulations, we set $Q_{F}$ = 0.01. It is
shown that for N = 16, the optimal rule offers 1 dB SNR gain over the OR-rule
and 1.5 dB gain over MAJORITY rule. The optimal scheme significantly
outperforms the AND rule scheme. Moreover, the impact of the number of sensing
samples $M$ (or equivalently, the sensing time) is demonstrated in figure 6.
At an SNR of -2 dB and N = 16, we plot the global detection probability for
$M$ = 6, 12, 18, and 24. It is shown that the maximum detection probability is
significantly boosted from more than 0.6 at $M$ = 6 to more than 0.9 at $M$ =
18. This boost in detection probability comes on the expense of sensing delay.
Figure 7 translates this detection probability boost into an SNR gain for the
same number of cooperating users ($N$ = 16). It is found that increasing the
number of sensing samples from 6 to 24 can offer up to a 4 dB SNR gain. It is
worth mentioning that the proposed scheme offers significant gain only in
networks with large number of cooperating users. As demonstrated by figure 8,
when N = 8, the OR-rule and the optimal fusion rule have nearly equal
performance. The attained SNR gain is only significant when the number of
cooperating users increase to N = 16 and 32. The SNR gain attained in both
cases are 1 dB and 2 dB respectively. Thus, the proposed scheme would be
appreciated in dense cooperative networks.
Figure 4: Simulation results comapred with the proposed analysis. Figure 5:
Comparison between optimal rule and suboptimal counting rules. Figure 6:
Impact of the number of sensing samples on the global detection probability.
Figure 7: SNR gain obtained by increasing the number of sensing samples.
Figure 8: Optimality of the proposed fusion rule in networks with large number
of cooperating users.
## 5 Conclusion
In this paper, we proposed a globally optimal hard decisions fusion scheme for
cooperative spectrum sensing. This problem has been always known for being
complex and mathematically intractable. We have proved that the optimal local
and global Neyman-Pearson thresholds can be obtained by a simple convex
optimization problem. This is achieved by utilizing the mathematical
representation of the global detection and false alarm probabilities in terms
of a regularized incomplete beta function. The log-concavity of global
detection probability as a function of the global threshold paves the way for
constructing a convex objective function. The proposed algorithm has a
complexity of $\mathcal{O}(\log_{2}(N))$. Simulation results verify the
optimality of the proposed scheme. It is shown that the globally optimal
scheme offers significant gain only when the number of cooperating users is
large. Otherwise, one can use a simple OR-rule.
## References
* (1) S. Haykin, ”Cognitive radio: brain-empowered wireless communications,” _IEEE Journal on Selected Areas in Communications_ , vol. 23, pp. 201-220, Feb. 2005.
* (2) A. Ghasemi and E. Sousa, ”Collaborative spectrum sensing for opportunistic access in fading environments,” _First IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks_ , pp. 131 ̵֠136, Nov. 2005.
* (3) Wei Zhang, R. K. Mallik and K. Ben Letaief, ”Cooperative Spectrum Sensing Optimization in Cognitive Radio Networks,” _Proceedings of IEEE International Conference on Communications (ICCҰ8)_ , pp. 411 ̵֠3415, May 2008.
* (4) Jun Ma, Guodong Zhao and Ye Li, ”Soft Combination and Detection for Cooperative Spectrum Sensing in Cognitive Radio Networks,” _IEEE Transactions on Wireless Communications_ , vol. 7, pp. 4502-4507, Nov. 2008.
* (5) J. N. Tsitsiklist, ”Decentralized Detection by a Large Number of Sensors,” _Mathematics of Control, Signals and Systems_ , vol. 1, no. 2, pp. 167ֱ82, 1988.
* (6) Zhi Quan, Shuguang Cui, H. Vincent Poor, and Ali H. Sayed, ”Collaborative Wideband Sensing for Cognitive Radios,” _IEEE Signal Processing Magazine_ , vol. 25, no. 6, pp. 60-73, Nov. 2008.
* (7) Imad Y. Hoballah and Kumar Varshney, ”Neyman-Pearson detection wirh distributed sensors,” _1986 25th IEEE Conference on Decision and Control,_ Syracuse University, Syracuse, New York, Dec. 1986, pp. 237-241.
* (8) Dongliang Duan, Liuqing Yang and Louis L. Scharf, ”Optimal Local Detection for Sensor Fusion by Large Deviatiob Analysis,” _20th European Signal Processing Conference (EUSIPCO 2012)_ , Bucharest, Romania, August 27 - 31, 2012, pp. 744-748.
* (9) Junyang Shen, Tao Jiang, Siyang Liu, and Zhongshan Zhang, ”Maximum Channel Throughput via Cooperative Spectrum Sensing in Cognitive Radio Networks,” _IEEE Transactions on Wireless Communications_ , vol. 8, no. 10, pp. 5166-5175, Oct. 2009.
* (10) Yunfei Chen, ”Analytical Performance of Collaborative Spectrum Sensing Using Censored Energy Detection,” _IEEE Transactions on Wireless Communications_ , vol. 9, no. 12, pp. 3856-3865, Dec. 2010.
* (11) J. Shen, S. Liu, L. Zeng, G. Xie, J. Gao and Y. Liu, ”Optimisation of cooperative spectrum sensing in cognitive radio network,” _IET Communications_ , vol. 3, pp. 1170-1178, March 2008.
* (12) S. Chaudhari, J. Lunden, V. Koivunen, H. V. Poor, ”Cooperative Sensing With Imperfect Reporting Channels: Hard Decisions or Soft Decisions?,” _IEEE transactions on Signal Processing_ , vol. 60, pp. 18-28, Jan. 2012.
* (13) Milton Abramowitz and Irene Stegun, ”Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,” _Dover Publications_ , ISBN 0-486-61272-4, 1964.
* (14) Yin Sun, A. Baricz, and Shidong Zhou, ”On the monotonicity, log-concavity and tight bounds of the generalized Marcum and Nuttall Q−functions” _IEEE Transactions on Information Theory_ , vol. 56, no. 3, pp. 1166 - 1186, March 2010.
* (15) D.Karp and S.M. Sitnik, ”Log-convexity and log-concavity of hypergeometric-like functions,” _Journal of Mathematical Analysis and Applications_ , vol. 364, issue 2, pp. 384–394, Apr. 2010.
* (16) Arpad Baricz, Saminathan Ponnusamy, and Matti Vuoeinen, ”Functional Inequalities for Modified Bessel Functions,” _Journal of Mathematical Analysis and Applications_ , vol. 364, issue 2, pp. 384–394, Mar. 2011.
* (17) Mark Bagnoli and Ted Bergstrom, ”Log-Concave Probability and Its Applications,” _Economic Theory, Springer_ , vol. 26, no. 2, pp. 445-469, Aug. 2005.
* (18) Stephen Boyd and Lieven Vandenberghe, ”Convex Optimization,” _Cambridge University Press_ , 2004.
|
arxiv-papers
| 2014-02-25T17:04:11 |
2024-09-04T02:49:58.846071
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmed M. Alaa and Omar A. Nasr",
"submitter": "Ahmed Alaa",
"url": "https://arxiv.org/abs/1402.6243"
}
|
1402.6248
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-024 LHCb-PAPER-2013-069
Measurement of resonant and $C\\!P$ components in $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
The resonant structure of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ is studied using data corresponding to 3 fb-1 of
integrated luminosity from $pp$ collisions by the LHC and collected by the
LHCb detector. Five interfering $\pi^{+}\pi^{-}$ states are required to
describe the decay: $f_{0}(980),~{}f_{0}(1500),~{}f_{0}(1790),~{}f_{2}(1270)$,
and $f_{2}^{\prime}(1525)$. An alternative model including these states and a
non-resonant ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
component also provides a good description of the data. Based on the different
transversity components measured for the spin-2 intermediate states, the final
state is found to be compatible with being entirely $C\\!P$-odd. The
$C\\!P$-even part is found to be $<2.3$% at 95% confidence level. The
$f_{0}(500)$ state is not observed, allowing a limit to be set on the absolute
value of the mixing angle with the $f_{0}(980)$ of $<7.7^{\circ}$ at 90%
confidence level, consistent with a tetraquark interpretation of the
$f_{0}(980)$ substructure.
Submitted to Phys. Rev. D
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25, S. Amato2, S. Amerio22, Y. Amhis7, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel39, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson50, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30, W.
Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,r, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez36,p, A. Camboni36, P. Campana18,38, D. Campora
Perez38, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,j, A. Cardini15, H.
Carranza-Mejia50, L. Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20,
L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph.
Charpentier38, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38,
X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47,
J. Closier38, C. Coca29, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A.
Comerma-Montells36, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G.
Corti38, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz
Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J. Dalseno46, P.
David8, P.N.Y. David41, A. Davis57, I. De Bonis4, K. De Bruyn41, S. De
Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D.
Derkach55, O. Deschamps5, F. Dettori42, A. Di Canto11, H. Dijkstra38, S.
Donleavy52, F. Dordei11, M. Dorigo39, P. Dorosz26,o, A. Dosil Suárez37, D.
Dossett48, A. Dovbnya43, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49, U. Egede53, V. Egorychev31, S. Eidelman34,
S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5,
Ch. Elsasser40, S. Esen11, A. Falabella16,f, C. Färber11, C. Farinelli41, S.
Farry52, D. Ferguson50, V. Fernandez Albor37, F. Ferreira Rodrigues1, M.
Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M. Fiorini16,f, C. Fitzpatrick38,
M. Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C.
Frei38, M. Frosini17,38,g, J. Fu21, E. Furfaro24,l, A. Gallas Torreira37, D.
Galli14,d, M. Gandelman2, P. Gandini59, Y. Gao3, J. Garofoli59, J. Garra
Tico47, L. Garrido36, C. Gaspar38, R. Gauld55, L. Gavardi9, E. Gersabeck11, M.
Gersabeck54, T. Gershon48, Ph. Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47,
L. Giubega29, V.V. Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38,
A. Gomes1,a, H. Gordon38, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A.
Granado Cardoso38, E. Graugés36, G. Graziani17, A. Grecu29, E. Greening55, S.
Gregson47, P. Griffith45, L. Grillo11, O. Grünberg61, B. Gui59, E. Gushchin33,
Yu. Guz35,38, T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, T.W.
Hafkenscheid64, S.C. Haines47, S. Hall53, B. Hamilton58, T. Hampson46, S.
Hansmann-Menzemer11, N. Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann61,
J. He38, T. Head38, V. Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A.
Hernando Morata37, E. van Herwijnen38, M. Heß61, A. Hicheur1, D. Hill55, M.
Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55, N. Hussain55, D.
Hutchcroft52, D. Hynds51, V. Iakovenko44, M. Idzik27, P. Ilten56, R.
Jacobsson38, A. Jaeger11, E. Jans41, P. Jaton39, A. Jawahery58, F. Jing3, M.
John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N. Jurik59, M.
Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, M.
Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C. Khurewathanakul39, S.
Klaver54, O. Kochebina7, I. Komarov39, R.F. Koopman42, P. Koppenburg41, M.
Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11, M. Kreps48, G.
Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k, V.
Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31,38, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, T. Latham48, C. Lazzeroni45,
R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32, J.
Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11, Y. Li3, M.
Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G. Liu38, S.
Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40, H. Lu3, D.
Lucchesi22,r, J. Luisier39, H. Luo50, E. Luppi16,f, O. Lupton55, F.
Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O. Maev30,38, S. Malde55, G.
Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5, U. Marconi14, P.
Marino23,t, R. Märki39, J. Marks11, G. Martellotti25, A. Martens8, A. Martín
Sánchez7, M. Martinelli41, D. Martinez Santos42, F. Martinez Vidal63, D.
Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A.
Mazurov16,38,f, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
McSkelly52, B. Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,t, R. Mountain59, F.
Muheim50, K. Müller40, R. Muresan29, B. Muryn27, B. Muster39, P. Naik46, T.
Nakada39, R. Nandakumar49, I. Nasteva1, M. Needham50, N. Neri21, S. Neubert38,
N. Neufeld38, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,q, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-
Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R.
Oldeman15,e, G. Onderwater64, M. Orlandea29, J.M. Otalora Goicochea2, P.
Owen53, A. Oyanguren36, B.K. Pal59, A. Palano13,c, F. Palombo21,u, M.
Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51, L. Pappalardo16,
C. Parkes54, C.J. Parkinson9, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, C. Pavel-Nicorescu29, A. Pazos Alvarez37, A. Pearce54, A.
Pellegrino41, G. Penso25,m, M. Pepe Altarelli38, S. Perazzini14,d, E. Perez
Trigo37, P. Perret5, M. Perrin-Terrin6, L. Pescatore45, E. Pesen65, G.
Pessina20, K. Petridis53, A. Petrolini19,j, E. Picatoste Olloqui36, B.
Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50, M. Plo
Casasus37, F. Polci8, G. Polok26, A. Poluektov48,34, E. Polycarpo2, A.
Popov35, D. Popov10, B. Popovici29, C. Potterat36, A. Powell55, J.
Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39,
G. Punzi23,s, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Redford55, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S.
Ricciardi49, A. Richards53, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts58, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez
Perez37, S. Roiser38, V. Romanovsky35, A. Romero Vidal37, M. Rotondo22, J.
Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P. Ruiz Valls36, G.
Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B. Saitta15,e,
V. Salustino Guimaraes2, B. Sanmartin Sedes37, R. Santacesaria25, C.
Santamarina Rios37, E. Santovetti24,l, M. Sapunov6, A. Sarti18, C.
Satriano25,n, A. Satta24, M. Savrie16,f, D. Savrina31,32, M. Schiller42, H.
Schindler38, M. Schlupp9, M. Schmelling10, B. Schmidt38, O. Schneider39, A.
Schopper38, M.-H. Schune7, R. Schwemmer38, B. Sciascia18, A. Sciubba25, M.
Seco37, A. Semennikov31, K. Senderowska27, I. Sepp53, N. Serra40, J. Serrano6,
P. Seyfert11, M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52,
L. Shekhtman34, O. Shevchenko43, V. Shevchenko62, A. Shires9, R. Silva
Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, T. Skwarnicki59, N.A.
Smith52, E. Smith55,49, E. Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D.
Sokoloff57, F.J.P. Soler51, F. Soomro39, D. Souza46, B. Souza De Paula2, B.
Spaan9, A. Sparkes50, F. Spinella23, P. Spradlin51, F. Stagni38, S. Stahl11,
O. Steinkamp40, S. Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S.
Stracka23,38, M. Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38,
L. Sun57, W. Sutcliffe53, S. Swientek9, V. Syropoulos42, M. Szczekowski28, P.
Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, E. Teodorescu29, F. Teubert38, C. Thomas55, E. Thomas38, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D.
Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4,53, S. Tourneur39,
M.T. Tran39, M. Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M.
Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin62, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez
Sierra37, S. Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M.
Vesterinen11, B. Viaud7, D. Vieira2, X. Vilasis-Cardona36,p, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß61, H.
Voss10, J.A. de Vries41, R. Waldi61, C. Wallace48, R. Wallace12, S.
Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, A.D. Webber54, D.
Websdale53, M. Whitehead48, J. Wicht38, J. Wiechczynski26, D. Wiedner11, L.
Wiggers41, G. Wilkinson55, M.P. Williams48,49, M. Williams56, F.F. Wilson49,
J. Wimberley58, J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A.
Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y. Xie50,38, Z. Xing59, Z. Yang3,
X. Yuan3, O. Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, F. Zhang3, L.
Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3, A.
Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
62National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
63Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
64KVI - University of Groningen, Groningen, The Netherlands, associated to 41
65Celal Bayar University, Manisa, Turkey, associated to 38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
## 1 Introduction
$C\\!P$ violation studies in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decay mode complement studies using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ and improve the final accuracy in the $C\\!P$-violating phase,
$\phi_{s}$, measurement [1]. While the $C\\!P$ content was previously shown to
be more than 97.7% $C\\!P$-odd at 95% confidence level (CL), it is important
to determine the size of any $C\\!P$-even components as these could ultimately
affect the uncertainty on the final result for $\phi_{s}$. Since the
$\pi^{+}\pi^{-}$ system can form light scalar mesons, such as the $f_{0}(500)$
and $f_{0}(980)$, we can investigate if these states have a quark-antiquark or
tetraquark structure, and determine the mixing angle between these states [2].
The tree-level Feynman diagram for the process is shown in Fig. 1.
Figure 1: Leading order diagram for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays into
$J/\psi\pi^{+}\pi^{-}$.
We have previously studied the resonance structure in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays using data corresponding to an integrated
luminosity of 1 fb-1 [3].111 Charged conjugated modes are also used when
appropriate. In this paper we use 3 fb-1 of luminosity, and also change the
analysis technique substantially. Here the $\pi^{+}\pi^{-}$ mass, and all
three decay angular distributions are used to determine the resonant and non-
resonant components. Previously the angle between the decay planes of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu+\mu^{-}$ and
$\pi^{+}\pi^{-}$ in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
rest frame, $\chi$, was integrated over. This simplified the analysis, but
sacrificed some precision and also prohibited us from measuring separately the
helicity $+1$ and $-1$ components of any $\pi^{+}\pi^{-}$ resonance, knowledge
of which would permit us to evaluate the $C\\!P$ composition of resonances
with spin greater than or equal to 1. Since one of the particles in the final
state, the $J/\psi$, has spin-1 its three decay amplitudes must be considered,
while the $\pi^{+}\pi^{-}$ system is described as the coherent sum of resonant
and possibly non-resonant amplitudes.
## 2 Amplitude formula for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}h^{+}h^{-}$
The decay of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}h^{+}h^{-}$, where $h$ denotes a pseudoscalar meson, followed by
$J/\psi\rightarrow\mu^{+}\mu^{-}$ can be described by four variables. We take
the invariant mass of $h^{+}h^{-}$ ($m_{hh}$) and three helicity angles
defined as (i) $\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, the
angle between the $\mu^{+}$ direction in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame with respect to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame; (ii)
$\theta_{hh}$, the angle between the $h^{+}$ direction in the $h^{+}h^{-}$
rest frame with respect to the $h^{+}h^{-}$ direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame, and (iii) $\chi$,
the angle between the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$h^{+}h^{-}$ decay planes in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame. Figure 2 shows
these angles pictorially222These definitions are the same for $B_{s}^{0}$ and
$\overline{B}{}_{s}^{0}$, namely, $\mu^{+}$ and $h^{+}$ are used to define the
angles in both cases.. In this paper $hh$ is equivalent to $\pi^{+}\pi^{-}$.
Figure 2: Definition of helicity angles. For details see text.
From the time-dependent decay rate of
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{B}_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}h^{+}h^{-}$ derived in Ref. [4], the time-integrated and flavor-averaged
decay rate is proportional to the function
$\displaystyle
S(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)=$
$\displaystyle|A(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)|^{2}+|\overline{A}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)|^{2}$ $\displaystyle-2{\cal
D}\,\mathcal{R}e\left(\frac{q}{p}A^{*}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)\overline{A}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)\right),$ (1)
where
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{A}$, the
amplitude of
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{B}_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}h^{+}h^{-}$ at proper time $t=0$, is a function of
$m_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\theta_{hh},\chi$, and is summed over all resonant (and possibly non-
resonant) components; $q$ and $p$ are complex parameters that describe the
relation between mass and flavor eigenstates [5]. The interference term arises
because we must sum the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
and $B^{0}_{s}$ amplitudes before squaring. Even when integrating over proper
time, the terms proportional to $\sinh\left(\Delta\Gamma_{s}t/2\right)$ do not
vanish because of the finite $\Delta\Gamma_{s}$ in the $B^{0}_{s}$ system,
where $\Delta\Gamma_{s}$ is the width difference between the light and the
heavy mass eigenstates. The factor $\cal D$ is
${\cal
D}=\frac{\int_{0}^{\infty}{\varepsilon}(t)e^{-\Gamma_{s}t}\sinh\frac{\Delta\Gamma_{s}t}{2}{\rm
d}t}{\int_{0}^{\infty}{\varepsilon}(t)e^{-\Gamma_{s}t}\cosh\frac{\Delta\Gamma_{s}t}{2}{\rm
d}t},$ (2)
where $\Gamma_{s}$ is the average $B^{0}_{s}$ decay width, and
${\varepsilon}(t)$ is the detection efficiency as a function of $t$. For a
uniform efficiency, ${\cal D}=\Delta\Gamma_{s}/(2\Gamma_{s})$ and is $(6.2\pm
0.9)$% [6].
The amplitude, $A_{R}(m_{hh})$, is used to describe the mass line-shape of the
resonance $R$, that in most cases is a Breit-Wigner function. It is combined
with the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ resonance decay
properties to form the expression
${\cal
A}_{R}(m_{hh})=\sqrt{2J_{R}+1}\sqrt{P_{R}P_{B}}F_{B}^{(L_{B})}F_{R}^{(L_{R})}A_{R}(m_{hh})\left(\frac{P_{B}}{m_{B}}\right)^{L_{B}}\left(\frac{P_{R}}{m_{hh}}\right)^{L_{R}}.$
(3)
Here $P_{B}$ is the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum in
the $\overline{B}^{0}_{s}$ rest frame, $P_{R}$ is the momentum of either of
the two hadrons in the dihadron rest frame, $m_{B}$ is the
$\overline{B}^{0}_{s}$ mass, $J_{R}$ is the spin of $R$, $L_{B}$ is the
orbital angular momentum between the $J/\psi$ and $h^{+}h^{-}$ system, and
$L_{R}$ the orbital angular momentum in the $h^{+}h^{-}$ decay, and thus is
the same as the spin of the $h^{+}h^{-}$ resonance. $F_{B}^{(L_{B})}$ and
$F_{R}^{(L_{R})}$ are the Blatt-Weisskopf barrier factors for the
$\overline{B}^{0}_{s}$ and $R$ resonance, respectively [3]. The factor
$\sqrt{P_{R}P_{B}}$ results from converting the phase space of the natural
Dalitz-plot variables $m^{2}_{hh}$ and
$m^{2}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}}$ to that of
$m_{hh}$ and $\cos\theta_{hh}$ [7]. We must sum over all final states, $R$, so
for each ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicity, denoted by
$\lambda$, equal to $0$, $+1$, and $-1$ we have
${\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{\lambda}(m_{hh},\theta_{hh})=\sum_{R}\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\bf
h}_{\lambda}^{R}{\cal A}_{R}(m_{hh})d_{-\lambda,0}^{J_{R}}(\theta_{hh}),$ (4)
where $\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\bf
h}_{\lambda}^{R}$ are the complex coefficients for each helicity amplitude and
the Wigner $d$-functions are listed in Ref. [6].
The decay rates,
$|\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{A}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)|^{2}$, and the interference term,
$A^{*}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)\overline{A}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)$, can be written as functions of
${\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{\lambda}(m_{hh},\theta_{hh})$,
$\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and $\chi$. These
relationships are given in Ref. [4]. In order to use the $C\\!P$ relations, it
is convenient to replace the helicity complex coefficients
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{\lambda}^{R}$
by the complex transversity coefficients
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\tau}^{R}$
using the relations
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{0}^{R}$
$\displaystyle=$
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{0}^{R},$
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{+}^{R}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\parallel}^{R}+\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\perp}^{R}),$
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{-}^{R}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\parallel}^{R}-\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\perp}^{R}).$
(5)
Here
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{0}^{R}$
corresponds to longitudinal polarization of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, and the other two
coefficients correspond to polarizations of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson and $h^{+}h^{-}$ system
transverse to the decay axis:
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\parallel}^{R}$
for parallel polarization of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $h^{+}h^{-}$, and
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\perp}^{R}$
for perpendicular polarization.
Assuming no direct $C\\!P$ violation, as this has not been observed in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays [1], the relation between the $\overline{B}^{0}_{s}$ and
$B^{0}_{s}$ variables is
$\bar{\textbf{a}}^{R}_{\tau}=\eta^{R}_{\tau}\textbf{a}^{R}_{\tau}$, where
$\eta^{R}_{\tau}$ is $C\\!P$ eigenvalue of the $\tau$ transversity component
for the intermediate state $R$, where $\tau$ denotes $0$, $\parallel$, or
$\perp$ component. The final state $C\\!P$ parities for S, P, and D-waves are
given in Table 1.
Table 1: $C\\!P$ parity for different spin resonances. Note that spin-0 only has the transversity component $0$. Spin | $\eta_{0}$ | $\eta_{\parallel}$ | $\eta_{\perp}$
---|---|---|---
0 | $-1$ | – | –
1 | 1 | 1 | $-1$
2 | $-1$ | $-1$ | 1
In this analysis a fit determines the amplitude strength $a_{\tau}^{R}$ and
the phase $\phi_{\tau}^{R}$ of the amplitude
$\textbf{a}^{R}_{\tau}=a_{\tau}^{R}e^{i\phi_{\tau}^{R}}$ (6)
for each resonance $R$ and each transversity $\tau$. For the $\tau=\perp$
amplitude, the $L_{B}$ value of a spin-1 (or -2) resonance is 1 (or 2); the
other transversity components have two possible $L_{B}$ values of 0 and 2 (or
1 and 3) for spin-1 (or -2) resonances. In this analysis the lower one is
used. It is verified that our results are insensitive to the $L_{B}$ choices.
## 3 Data sample and detector
The data sample corresponds to an integrated luminosity of $3\,{\rm fb}^{-1}$
collected with the LHCb detector [8] using $pp$ collisions. One-third of the
data was acquired at a center-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$, and the remainder at
8$\mathrm{\,Te\kern-1.00006ptV}$. The detector is a single-arm forward
spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the
study of particles containing $b$ or $c$ quarks. The detector includes a high-
precision tracking system consisting of a silicon-strip vertex detector
surrounding the $pp$ interaction region, a large-area silicon-strip detector
located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift
tubes [9] placed downstream. The combined tracking system provides a
momentum333We work in units where $c=1$. measurement with relative uncertainty
that varies from 0.4% at 5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$, and impact parameter (IP) resolution of
20$\,\upmu\rm m$ for tracks with large transverse momentum ($p_{\rm T}$).
Different types of charged hadrons are distinguished by information from two
ring-imaging Cherenkov detectors (RICH) [10]. Photon, electron and hadron
candidates are identified by a calorimeter system consisting of scintillating-
pad and preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers [11].
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage that applies a full
event reconstruction [12]. Events selected for this analysis are triggered by
a $J/\psi\rightarrow\mu^{+}\mu^{-}$ decay, where the $J/\psi$ is required at
the software level to be consistent with coming from the decay of a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson by use either of IP
requirements or detachment of the $J/\psi$ from the primary vertex (PV). In
the simulation, $pp$ collisions are generated using Pythia [13,
*Sjostrand:2007gs] with a specific LHCb configuration [15]. Decays of hadronic
particles are described by EvtGen [16], in which final state radiation is
generated using Photos [17]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [18,
*Agostinelli:2002hh] as described in Ref. [20].
## 4 Event selection
Preselection criteria are implemented to preserve a large fraction of the
signal events, and are identical to those used in Ref. [21]. A $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ candidate is reconstructed by combining a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
candidate with two pions of opposite charge. To ensure good track
reconstruction, each of the four particles in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate is required to have
the track fit $\chi^{2}$/ndf to be less than 4, where ndf is the number of
degrees of freedom of the fit. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ candidate is formed by two identified muons
of opposite charge, having $p_{\rm T}$ greater than 500
$\mathrm{\,Me\kern-1.00006ptV}$, and with a geometrical fit vertex $\chi^{2}$
less than 16. Only candidates with dimuon invariant mass between $-48$
$\mathrm{\,Me\kern-1.00006ptV}$ and $+43$ $\mathrm{\,Me\kern-1.00006ptV}$ from
the observed ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass peak are
selected, and are then constrained to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [6] for subsequent use.
Pion candidates are required to each have $p_{\rm T}$ greater than 250
$\mathrm{\,Me\kern-1.00006ptV}$, and the sum, $\mbox{$p_{\rm
T}$}(\pi^{+})+\mbox{$p_{\rm T}$}(\pi^{-})$ larger than 900
$\mathrm{\,Me\kern-1.00006ptV}$. Both pions must have $\chi^{2}_{\rm IP}$
greater than 9 to reject particles produced from the PV. The $\chi^{2}_{\rm
IP}$ is computed as the difference between the $\chi^{2}$ of the PV
reconstructed with and without the considered track. Both pions must also come
from a common vertex with $\chi^{2}{\rm/ndf}<16$, and form a vertex with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ with a $\chi^{2}$/ndf less than
10 (here ndf equals five). Pion candidates are identified using the RICH and
muon systems. The particle identification makes use of the logarithm of the
likelihood ratio comparing two particle hypotheses (DLL). For pion selection
we require DLL$(\pi-K)>-10$ and DLL$(\pi-\mu)>-10$.
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate must have
a flight distance of more than 1.5 $\rm\,mm$. The angle between the combined
momentum vector of the decay products and the vector formed from the positions
of the PV and the decay vertex (pointing angle) is required to be less than
$2.5^{\circ}$.
Events satisfying this preselection are then further filtered using a
multivariate analyzer based on a boosted decision tree (BDT) technique [22].
The BDT uses eight variables that are chosen to provide separation between
signal and background. These are the minimum of DLL($\mu-\pi$) of the
$\mu^{+}$ and $\mu^{-}$, $\mbox{$p_{\rm T}$}(\pi^{+})+\mbox{$p_{\rm
T}$}(\pi^{-})$, the minimum of $\chi^{2}_{\rm IP}$ of the $\pi^{+}$ and
$\pi^{-}$, and the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
properties of vertex $\chi^{2}$, pointing angle, flight distance, $p_{\rm T}$
and $\chi^{2}_{\rm IP}$. The BDT is trained on a simulated sample of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ signal events and a background data sample from the
sideband $5566<m(J/\psi\pi^{+}\pi^{-})<5616$ $\mathrm{\,Me\kern-1.00006ptV}$.
Then the BDT is tested on independent samples. The distributions of BDT
classifier for signal and background samples are shown in Fig. 3. By
maximizing the signal significance we set the requirement that the classifier
is greater than zero, which has a signal efficiency of 95% and rejects 90% of
the background.
Figure 3: Distributions of the BDT classifier for both training and test
samples of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ signal
and background events. The signal samples are from simulation and the
background samples are from data.
The invariant mass of the selected $J/\psi\pi^{+}\pi^{-}$ combinations is
shown in Fig. 4. There is a large peak at the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass and a smaller one at the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass on top of a
background. A double Crystal Ball function with common means models the
radiative tails and is used to fit each of the signals. The known $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass difference [6] is used to
constrain the difference in mean values. Other components in the fit model
take into account contributions from $B^{-}\rightarrow J/\psi K^{-}(\pi^{-})$,
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\eta^{\prime}$ with $\eta^{\prime}\rightarrow\rho^{0}\gamma$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$ with
$\phi\rightarrow\pi^{+}\pi^{-}\pi^{0}$ backgrounds and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi K^{-}\pi^{+}$ and
$\mathchar 28931\relax_{b}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}p$ reflections, where the $K^{-}$ in the former, and both $K^{-}$
and $p$ in the latter, are misidentified as pions. The shape of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$
signal is taken to be the same as that of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$. The combinatorial background
shape is taken from like-sign combinations that are the sum of
$\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$ candidates, and was found to be well
described by an exponential function in previous studies [3, 23]. The shapes
of the other components are taken from simulation with their yields allowed to
vary. The $\mathchar
28931\relax_{b}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}p$ reflection yield in the fit region is constrained to the
expected number $2145\pm 201$, which is obtained from study of the events in
the control region of $5066<m(J/\psi\pi^{+}\pi^{-})<5141$
$\mathrm{\,Me\kern-1.00006ptV}$. The mass fit gives $27396\pm 207$ signal and
$7075\pm 101$ background candidates, leading to the signal fraction $f_{\rm
sig}=(79.5\pm 0.2)\%$, within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak. The effective
r.m.s. mass resolution is 9.9 MeV.
Figure 4: Invariant mass of $J/\psi\pi^{+}\pi^{-}$ combinations. The data have
been fitted with double Crystal Ball signal and several background functions.
The (red) solid curve shows the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal, the (brown) dotted
line shows the combinatorial background, the (green) short-dashed line shows
the $B^{-}$ background, the (purple) dot-dashed curve is $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$,
the (light blue) long-dashed line is the sum of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\eta^{\prime}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$ with
$\phi\rightarrow\pi^{+}\pi^{-}\pi^{0}$ backgrounds and the $\mathchar
28931\relax_{b}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}p$ reflection, the (black) dot-long dashed curve is the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi K^{-}\pi^{+}$
reflection and the (blue) solid curve is the total.
## 5 Probability density function construction
The correlated distributions of four variables $m_{hh}$, $\cos\theta_{hh}$,
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, and $\chi$ are
fitted using the candidates within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak. To improve the
resolution of these variables we perform a kinematic fit constraining the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $J/\psi$ masses to
their world average mass values [6], and recompute the final state momenta.
The overall PDF given by the sum of signal, $S$, and background functions is
$\displaystyle
F(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)$ $\displaystyle=$ $\displaystyle\frac{f_{\rm
sig}}{{\cal{N}}_{\rm
sig}}\varepsilon(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)S(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)$ (7) $\displaystyle+$ $\displaystyle(1-f_{\rm
sig})B(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi),$
where $\varepsilon$ is the detection efficiency, and $B$ is the background PDF
discussed later in Sec. 5.3. The normalization factor for signal is given by
$\displaystyle{\cal{N}}_{\rm sig}$ $\displaystyle=$
$\displaystyle\int\\!\varepsilon(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)S(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)\,{\rm d}\,m_{hh}\,{\rm d}\cos\theta_{hh}\,{\rm
d}\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{\rm d}\chi.$ (8)
The signal function $S$ is defined in Eq. (2), where ${\cal D}=(8.7\pm
1.5)\%$, taking into account the acceptance [24], and choosing a phase
convention $q/p=e^{-i\phi_{s}}$. The phase $\phi_{s}$ is fixed to the standard
model value of $-0.04$ radians [25]. Our results are found to be insensitive
to the value of $\phi_{s}$ used within the 95% CL limits set by the LHCb
measurement [1].
### 5.1 Data distributions of the Dalitz-plot
The event distribution for $m^{2}(\pi^{+}\pi^{-})$ versus
$m^{2}(J/\psi\pi^{+})$ in Fig. 5 shows clear structures in
$m^{2}(\pi^{+}\pi^{-})$. The presence of possible exotic structures in the
$J/\psi\pi^{+}$ system, as claimed in similar decays [26, 27], is investigated
by examining the $J/\psi\pi^{+}$ mass distribution shown in Fig. 6 (a). No
resonant effects are evident. Figure 6 (b) shows the $\pi^{+}\pi^{-}$ mass
distribution. Apart from a large signal peak due to the $f_{0}(980)$, there
are visible structures at about 1450 $\mathrm{\,Me\kern-1.00006ptV}$ and 1800
$\mathrm{\,Me\kern-1.00006ptV}$.
Figure 5: Distribution of $m^{2}(\pi^{+}\pi^{-})$ versus
$m^{2}(J/\psi\pi^{+})$ for all events within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak.
Figure 6: Distributions of (a) $m(J/\psi\pi^{+})$ and (b) $m(\pi^{+}\pi^{-})$
for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ candidate decays within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass. The (red) points with
error bars show the background contribution determined from
$m(J/\psi\pi^{+}\pi^{-})$ fits performed in each bin of the plotted variables.
### 5.2 Detection efficiency
The detection efficiency is determined from a phase space simulation sample
containing $4\times 10^{6}$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ events with $J/\psi\rightarrow\mu^{+}\mu^{-}$. The
efficiency can be parameterized in terms of analysis variables as
$\varepsilon(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)=\varepsilon_{1}(s_{12},s_{13})\times\varepsilon_{2}(m_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\times\varepsilon_{3}(m_{hh},\chi),$ (9)
where $s_{12}\equiv m^{2}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}})$ and $s_{13}\equiv
m^{2}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{-}})$ are functions of
$(m_{hh},\theta_{hh})$; such parameter transformations in $\varepsilon_{1}$
are implemented in order to use the Dalitz-plot based efficiency model
developed in previous publications [3, 21]. The efficiency functions take into
account correlations between $m_{hh}$ and each of the three angles as
determined by the simulation.
The efficiency as a function of the angle $\chi$ is shown in Fig. 7. To
simplify the normalization of the PDF, the efficiency as a function of $\chi$
is parameterized in 26 bins of $m^{2}_{hh}$ as
$\varepsilon_{3}(m_{hh},\chi)=\frac{1}{2\pi}(1+p_{1}\cos\chi+p_{2}\cos
2\chi),$ (10)
where $p_{1}=p_{1}^{0}+p_{1}^{1}m_{hh}^{2}$ and
$p_{2}=p_{2}^{0}+p_{2}^{1}m_{hh}^{2}+p_{2}^{2}m_{hh}^{4}$. A fit to the
simulation determines $p_{1}^{0}=0.0087\pm 0.0051$, $p_{1}^{1}=(-0.0062\pm
0.0019)$ GeV-2, $p_{2}^{0}=0.0030\pm 0.0077$, $p_{2}^{1}=(0.053\pm 0.007)$
GeV-2, and $p_{2}^{2}=(-0.0077\pm 0.0015)$ GeV-4.
Figure 7: Distribution of the angle $\chi$ for the $J/\psi\pi^{+}\pi^{-}$
simulation sample fitted with Eq. (10), used to determine the efficiency
parameters.
The efficiency in $\cos\theta_{J/\psi}$ also depends on $m_{hh}$; we fit the
$\cos\theta_{J/\psi}$ distributions of $J/\psi\pi^{+}\pi^{-}$ simulation
sample with the function
$\varepsilon_{2}(m_{hh},\theta_{J/\psi})=\frac{1+a(m^{2}_{hh})\cos^{2}\theta_{J/\psi}}{2+2a(m^{2}_{hh})/3},$
(11)
giving 26 values of $a$ as a function of $m^{2}_{hh}$. The resulting
distribution in $a$ is shown in Fig. 8 and is best described by a 2nd order
polynomial function
$a(m^{2}_{hh})=a_{0}+a_{1}m^{2}_{hh}+a_{2}m^{4}_{hh},$ (12)
with $a_{0}=0.156\pm 0.020$, $a_{1}=(-0.091\pm 0.018)$ GeV-2 and
$a_{2}=(0.013\pm 0.004)$ GeV-4.
Figure 8: Second order polynomial fit to the acceptance parameter
$a(m^{2}_{hh})$ used in Eq. 11.
The function $\varepsilon_{1}(s_{12},s_{13})$ can be determined from the
simulation after integrating over $\cos\theta_{J/\psi}$ and $\chi$, because
the functions $\varepsilon_{2}$ and $\varepsilon_{3}$ are normalized in
$\cos\theta_{J/\psi}$ and $\chi$, respectively. It is parameterized as a
symmetric 5th order polynomial function given by
$\displaystyle\varepsilon_{1}(s_{12},s_{13})$ $\displaystyle=$ $\displaystyle
1+\epsilon_{1}(x+y)+\epsilon_{2}(x+y)^{2}+\epsilon_{3}xy+\epsilon_{4}(x+y)^{3}+\epsilon_{5}xy(x+y)$
(13)
$\displaystyle+\epsilon_{6}(x+y)^{4}+\epsilon_{7}xy(x+y)^{2}+\epsilon_{8}x^{2}y^{2}$
$\displaystyle+\epsilon_{9}(x+y)^{5}+\epsilon_{10}xy(x+y)^{3}+\epsilon_{11}x^{2}y^{2}(x+y),$
where $x=s_{12}/{\rm GeV}^{2}-18.9$, and $y=s_{13}/{\rm GeV}^{2}-18.9$. The
phase space simulation is generated uniformly in the two-dimensional
distribution of ($s_{12},s_{13})$, therefore the distribution of selected
events reflects the efficiency and is fit to determine the efficiency
parameters $\varepsilon_{i}$. The projections of the fit are shown in Fig. 9,
giving the efficiency as a function of $\cos\theta_{\pi^{+}\pi^{-}}$ versus
$m(\pi^{+}\pi^{-})$ in Fig. 10.
Figure 9: Projections of invariant mass squared of (a) $m^{2}(J/\psi\pi^{+})$
and (b) $m^{2}(J/\psi\pi^{-})$ of the simulated Dalitz plot used to measure
the efficiency parameters. The points represent the simulated event
distributions and the curves the polynomial fit. Figure 10: Parameterization
of the detection efficiency as a function of $\cos\theta_{\pi^{+}\pi^{-}}$ and
$m(\pi^{+}\pi^{-})$. The scale is arbitrary.
### 5.3 Background composition
The main background source is combinatorial and is taken from the like-sign
combinations within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak. The like-sign
combinations also contain the $B^{-}$ background which is peaked at
$\cos\theta_{hh}=\pm 1$. The like-sign combinations cannot contain any
$\rho^{0}$, which is measured to be 3.5% of the total background. To obtain
the $\rho^{0}$ contribution, the background $m(\pi^{+}\pi^{-})$ distribution
shown in Fig. 6 (b), found by fitting the
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-})$ distribution
in bins of $m(\pi^{+}\pi^{-})$, is compared to $m(\pi^{\pm}\pi^{\pm})$
distribution from the like-sign combinations. In this way simulated $\rho^{0}$
background is added into the like-sign candidates. The background PDF $B$ is
the sum of functions for $B^{-}$ ($B_{B^{-}}$) and for the other ($B_{\rm
other}$), given by
$B(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)=\frac{1-f_{B^{-}}}{{\cal N_{\rm other}}}B_{\rm
other}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)+\frac{f_{B^{-}}}{{\cal
N}_{B^{-}}}B_{B^{-}}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi),$ (14)
where ${\cal N}_{\rm other}$ and ${\cal N}_{B^{-}}$ are normalization factors,
and $f_{B^{-}}$ is the fraction of the $B^{-}$ background in the total
background. The $J/\psi\pi^{+}\pi^{-}$ mass fit gives $f_{B^{-}}=(1.7\pm
0.2)\%$.
The $B^{-}$ background is separated because its invariant mass is very close
to the highest allowed limit, resulting in its $\cos\theta_{hh}$ distribution
peaking at $\pm 1$. The function for the $B^{-}$ background is defined as
$\displaystyle
B_{B^{-}}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)=$ $\displaystyle G(m_{hh};m_{0},\sigma_{m})\times
G(|\cos\theta_{hh}|;1,\sigma_{\theta})$ $\displaystyle\times$
$\displaystyle\left(1-\cos^{2}\theta_{J/\psi}\right)\times(1+p_{b1}\cos\chi+p_{b2}\cos
2\chi),$ (15)
where $G$ is the Gaussian function, and the parameters $m_{0}$, $\sigma_{m}$,
$\sigma_{\theta}$, $p_{b1}$, and $p_{b2}$ are determined by the fit. The last
term is the same function for $\chi$.
The function for the other background is
$B_{\rm
other}(m_{hh},\theta_{hh},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\chi)=m_{hh}B_{1}(m_{hh}^{2},\cos\theta_{hh})\times\left(1+\alpha\cos^{2}\theta_{J/\psi}\right)\times(1+p_{b1}\cos\chi+p_{b2}\cos
2\chi),$ (16)
where the function
$B_{1}(m_{hh}^{2},\cos\theta_{hh})=B_{2}(\zeta)\frac{p_{B}}{m_{B}}\times\frac{1+c_{1}q(\zeta)|\cos\theta_{hh}|+c_{2}p(\zeta)\cos^{2}\theta_{hh}}{2[1+c_{1}q(\zeta)/2+c_{2}p(\zeta)/3]}.$
(17)
Here $\zeta\equiv 2(m_{hh}^{2}-m^{2}_{\rm min})/(m^{2}_{\rm max}-m^{2}_{\rm
min})-1$, where $m_{\rm min}$ and $m_{\rm max}$ give the fit boundaries of
$m_{hh}$, $B_{2}(\zeta)$ is a fifth-order Chebychev polynomial; $q(\zeta)$ and
$p(\zeta)$ are both second-order Chebychev polynomials with the coefficients
$c_{1}$ and $c_{2}$ being free parameters. In order to better approximate the
real background in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
signal region, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{\pm}\pi^{\pm}$ candidates are kinematically constrained to the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass, and
$\mu^{+}\mu^{-}$ to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass.
Figure 11: Distribution of $\cos\theta_{J/\psi}$ of the other background and
the fitted function $1+\alpha\cos^{2}\theta_{J/\psi}$. The points with error
bars show the background obtained from candidate mass fits in bins of
$\cos\theta_{J/\psi}$.
The second part $\left(1+\alpha\cos^{2}\theta_{J/\psi}\right)$ is a function
of the $J/\psi$ helicity angle. The $\cos\theta_{J/\psi}$ distribution of
background is shown in Fig. 11; fitting with the function determines the
parameter $\alpha=-0.34\pm 0.03$. A fit to the like-sign combinations added
with additional $\rho^{0}$ background determines the parameters describing the
$m_{hh}$, $\theta_{hh}$, and $\chi$ distributions. Figures 12 and 13 show the
projections of $\cos\theta_{hh}$ and $m_{hh}$, and of $\chi$ of the total
background, respectively.
Figure 12: Projections of (a) $\cos\theta_{\pi\pi}$ and (b)
$m(\pi^{+}\pi^{-})$ of the total background. The (blue) histogram or curve is
projection of the fit, and the points with error bars show the like-sign
combinations added with additional $\rho^{0}$ background. Figure 13:
Distribution of $\chi$ of the total background and the fitted function. The
points with error bars show the like-sign combinations added with additional
$\rho^{0}$ background.
## 6 Final state composition
### 6.1 Resonance models
To study the resonant structures of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ we use the 34 471 candidates with invariant mass lying
within $\pm 20$ $\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak which include
7075$\pm$101 background events. The $\pi^{+}\pi^{-}$ resonance candidates that
could contribute to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ decay are listed in Table 2. The resonances that decay
into a $\pi^{+}\pi^{-}$ pair must be isoscalar ($I=0$), because the $s\bar{s}$
system forming the resonances in Fig. 1 has $I=0$. To test the isoscalar
argument, the isospin-1 $\rho(770)$ meson is also added to the baseline fit.
The non-resonance (NR) is assumed to be S-wave, its shape is defined by Eq.
(3) where the amplitude function $A_{R}(m_{hh})$ is set to be equal to one,
and the Blatt-Weisskopf barrier factors $F_{B}^{(1)}$ and $F_{R}^{(0)}$ are
both set to one.
In the previous analysis [24], we observed a resonant state at $(1475\pm
6)$$\mathrm{\,Me\kern-1.00006ptV}$ with a width of $(113\pm
11)$$\mathrm{\,Me\kern-1.00006ptV}$. We identified it with the $f_{0}(1370)$
though its mass and width values agreed neither with the $f_{0}(1500)$ or the
$f_{0}(1370)$. W. Ochs [28, *Ochs:2013vxa] argues that the better assignment
is $f_{0}(1500)$; we follow his suggestion. In addition, a structure is
clearly visible in the $1800$ MeV region (see Fig. 6 (b)), which was not the
case in our previous analysis [3]. This could be the $f_{0}(1790)$ resonance
observed by BES [30] in ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\phi\pi^{+}\pi^{-}$ decays.
From the measured ratios ${\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{2}^{\prime}(1525)\right)/{\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi\right)$ [31] and ${\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\right)/{\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi\right)$ [3], using the measured $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$
branching fractions [6], the expected $f_{2}^{\prime}(1525)$ fit fraction for
the transversity $0$ component is $(0.45\pm 0.13)\%$, and the ratio of
helicity $\lambda=0$ to $|\lambda|=1$ components, which is equal to the ratio
of transversity $0$ to the sum of $\perp$ and $\parallel$ components, is
$1.9\pm 0.8$, where the uncertainties are dominated by that on
$f_{2}^{\prime}(1525)$ fit fractions in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ decays. This information is used as constraints in the fit.
Table 2: Possible resonance candidates in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\pi^{+}\pi^{-}$ decay mode and their parameters used in the fit. Resonance | Spin | Helicity | Resonance | Mass ($\mathrm{\,Me\kern-1.00006ptV}$) | Width ($\mathrm{\,Me\kern-1.00006ptV}$) | Source
---|---|---|---|---|---|---
| | | formalism | | |
$f_{0}(500)$ | 0 | 0 | BW | $471\pm 21$ | $534\pm 53$ | LHCb [21]
$f_{0}(980)$ | 0 | 0 | Flatté | see text
$f_{2}(1270)$ | 2 | $0,\pm 1$ | BW | $1275.1\pm 1.2$ | $185.1^{+2.9}_{-2.4}$ | PDG [6]
$f_{0}(1500)$ | 0 | 0 | BW | see text
$f_{2}^{\prime}(1525)$ | 2 | $0,\pm 1$ | BW | $1522_{-3}^{+6}$ | $84_{-8}^{+12}$ | LHCb [31]
$f_{0}(1710)$ | 0 | 0 | BW | $1720\pm 6$ | $135\pm 8$ | PDG [6]
$f_{0}(1790)$ | 0 | 0 | BW | $1790_{-30}^{+40}$ | $270_{-30}^{+60}$ | BES [30]
$\rho(770)$ | 1 | $0,\pm 1$ | BW | $775.49\pm 0.34$ | $149.1\pm 0.8$ | PDG [6]
The masses and widths of the resonances are also listed in Table 2. When used
in the fit they are fixed to these central values, except for the parameters
of $f_{0}(980)$ and $f_{0}(1500)$ that are determined by the fit. In addition,
the parameters of $f_{0}(1790)$ are constrained to those determined by the BES
measurement [30].
As suggested by D. V. Bugg [32], the Flatté model [33] for $f_{0}(980)$ is
slightly modified, and is parameterized as
$A_{R}(m_{\pi^{+}\pi^{-}})=\frac{1}{m_{R}^{2}-m^{2}_{\pi^{+}\pi^{-}}-im_{R}(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}F_{KK}^{2}\rho_{KK})},$
(18)
where $m_{R}$ is the $f_{0}(980)$ pole mass, the parameters $g_{\pi\pi}$ and
$g_{KK}$ are the $f_{0}(980)$ coupling constants to $\pi^{+}\pi^{-}$ and
$K^{+}K^{-}$ final states, respectively, and the phase space $\rho$ factors
are given by Lorentz-invariant phase spaces as
$\displaystyle\rho_{\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\sqrt{1-\frac{4m^{2}_{\pi^{\pm}}}{m^{2}_{\pi^{+}\pi^{-}}}}+\frac{1}{3}\sqrt{1-\frac{4m^{2}_{\pi^{0}}}{m^{2}_{\pi^{+}\pi^{-}}}},$
(19) $\displaystyle\rho_{KK}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{\pm}}}{m^{2}_{\pi^{+}\pi^{-}}}}+\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{0}}}{m^{2}_{\pi^{+}\pi^{-}}}}.$
(20)
Compared to the normal Flatté function, a form factor $F_{KK}=\exp(-\alpha
k^{2})$ is introduced above the $KK$ threshold and serves to reduce the
$\rho_{KK}$ factor as $m^{2}_{\pi^{+}\pi^{-}}$ increases, where $k$ is
momentum of each kaon in the $KK$ rest frame, and $\alpha=(2.0\pm 0.25)$ GeV-2
[32]. This parameterization slightly decreases the $f_{0}(980)$ width above
the $KK$ threshold. The parameter $\alpha$ is fixed to $2.0$ GeV-2 as it is
not very sensitive to the fit.
To determine the complex amplitudes in a specific model, the data are fitted
maximizing the unbinned likelihood given as
$\mathcal{L}=\prod_{i=1}^{N}F(m_{hh}^{i},\theta_{hh}^{i},\theta^{i}_{J/\psi},\chi^{i}),$
(21)
where $N$ is the total number of candidates, and $F$ is the total PDF defined
in Eq. (7). In order to converge properly in a maximum likelihood method, the
PDFs of the signal and background need to be normalized. This is accomplished
by first normalizing the $\chi$ and
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ dependent parts
analytically, and then normalizing the $m_{hh}$ and $\cos\theta_{hh}$
dependent parts using a numerical integration over 1000$\times$200 bins.
The fit determines amplitude magnitudes $a_{i}^{R_{i}}$ and phases
$\phi_{i}^{R_{i}}$ defined in Eq. (6). The $a^{f_{0}(980)}_{0}$ amplitude is
fixed to 1, since the overall normalization is related to the signal yield. As
only relative phases are physically meaningful, $\phi_{0}^{f_{0}(980)}$ is
fixed to 0. In addition, due to the averaging of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$, the interference terms
between opposite $C\\!P$ states are cancelled out, making it not possible to
measure the relative phase between $C\\!P$-even and odd states here, so one
$C\\!P$-even phase, $\phi_{\perp}^{f_{2}(1270)}$, is also fixed to 0.
### 6.2 Fit fraction
Knowledge of the contribution of each component can be expressed by defining a
fit fraction for each transversity $\tau$, ${\cal{F}}_{\tau}^{R}$, which is
the squared amplitude of $R$ integrated over the phase space divided by the
entire amplitude squared over the same area. To determine
${\cal{F}}_{\tau}^{R}$ one needs to integrate over all the four fitted
observables in the analysis. The interference terms between different helicity
components vanish, after integrating over the two variables of
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and $\chi$. Thus
we define the transversity fit fraction as
${\cal{F}}^{R}_{\tau}=\frac{\int\left|a^{R}_{\tau}e^{i\phi^{R}_{\tau}}{\cal
A}_{R}(m_{hh})d_{\lambda,0}^{J_{R}}(\theta_{hh})\right|^{2}{\rm d}m_{hh}\;{\rm
d}\cos\theta_{hh}}{\int\left(|{\cal{H}}_{0}(m_{hh},\theta_{hh})|^{2}+|{\cal{H}}_{+}(m_{hh},\theta_{hh})|^{2}+|{\cal{H}}_{-}(m_{hh},\theta_{hh})|^{2}\right){\rm
d}m_{hh}\;{\rm d}\cos\theta_{hh}},$ (22)
where $\lambda=0$ in the $d$-function for $\tau=0$, and $\lambda=1$ for
$\tau=\perp$ or $\parallel$.
Note that the sum of the fit fractions is not necessarily unity due to the
potential presence of interference between two resonances. Interference term
fractions are given by
${\cal{F}}_{\tau}^{RR^{\prime}}=2\mathcal{R}e\left(\frac{\int
a^{R}_{\tau}\;a^{R^{\prime}}_{\tau}e^{i(\phi^{R}_{\tau}-\phi^{R^{\prime}}_{\tau})}{\cal
A}_{R}(m_{hh}){\cal
A}^{*}_{R^{\prime}}(m_{hh})d_{\lambda,0}^{J_{R}}(\theta_{hh})d_{\lambda,0}^{J_{R^{\prime}}}(\theta_{hh}){\rm
d}m_{hh}\;{\rm
d}\cos\theta_{hh}}{\int\left(|{\cal{H}}_{0}(m_{hh},\theta_{hh})|^{2}+|{\cal{H}}_{+}(m_{hh},\theta_{hh})|^{2}+|{\cal{H}}_{-}(m_{hh},\theta_{hh})|^{2}\right){\rm
d}m_{hh}\;{\rm d}\cos\theta_{hh}}\right),$ (23)
and
$\sum_{R,\tau}{\cal{F}}_{\tau}^{R}+\sum^{R>R^{\prime}}_{RR^{\prime},\tau}{\cal{F}}_{\tau}^{RR^{\prime}}=1.$
(24)
Interference between different spin-$J$ states vanishes, when integrated over
angle, because the $d^{J}_{\lambda 0}$ angular functions are orthogonal.
### 6.3 Fit results
In order to compare the different models quantitatively, an estimate of the
goodness of fit is calculated from four-dimensional (4D) partitions of the
four variables, $m(\pi^{+}\pi^{-})$, $\cos\theta_{hh}$, $\cos\theta_{J/\psi}$
and $\chi$. We use the Poisson likelihood $\chi^{2}$ [34] defined as
$\chi^{2}=2\sum_{i=1}^{N_{\rm
bin}}\left[x_{i}-n_{i}+n_{i}\text{ln}\left(\frac{n_{i}}{x_{i}}\right)\right],$
(25)
where $n_{i}$ is the number of events in the four-dimensional bin $i$ and
$x_{i}$ is the expected number of events in that bin according to the fitted
likelihood function. A total of 1845 bins are used to calculate the
$\chi^{2}$, where $41(m_{hh})\times 5(\cos\theta_{hh})\times
3(\cos\theta_{J/\psi})\times 3(\chi)$ equal size bins are used, and $m_{hh}$
is required to be between 0.25 and 2.30 GeV. The $\chi^{2}/\text{ndf}$, and
the negative of the logarithm of the likelihood, $\rm-ln\mathcal{L}$, of the
fits are given in Table 3, where ndf is the number of degree of freedom given
as 1845 subtracted by number of fitting parameters and 1. The nomenclature
describing the models gives the base model first and then “+” for any
additions. The 5R model contains the resonances $f_{0}(980)$, $f_{2}(1270)$,
$f_{2}^{\prime}(1525)$, $f_{0}(1500)$, and $f_{0}(1790)$. If adding NR to 5R
model, two minima with similar likelihoods are found. One minimum is
consistent with the 5R results and has NR fit fraction of $(0.3\pm 0.3)\%$; we
group any fit models that are consistent with this 5R fit into the “Solution
I” category. Another minimum has significant NR fit fraction of $(5.9\pm
1.4)\%$, this model and other consistent models are classified in the
“Solution II” category.
Table 3: Fit $\rm-ln\mathcal{L}$ and $\chi^{2}/\text{ndf}$ of different resonance models. Resonance model | $\rm-ln\mathcal{L}$ | $\chi^{2}/\text{ndf}$
---|---|---
5R (Solution I) | $-93738$ | 2005/1822 = 1.100
5R+NR (Solution I) | $-93741$ | 2003/1820 = 1.101
5R+$f_{0}(500)$ (Solution I) | $-93741$ | 2004/1820 = 1.101
5R+$f_{0}(1710)$ (Solution I) | $-93744$ | 1998/1820 = 1.098
5R+$\rho(770)$ (Solution I) | $-93742$ | 2004/1816 = 1.104
5R+NR (Solution II) | $-93739$ | 2008/1820 = 1.103
5R+NR+$f_{0}(500)$ (Solution II) | $-93741$ | 2004/1818 = 1.102
5R+NR+$f_{0}(1710)$ (Solution II) | $-93745$ | 2004/1818 = 1.102
5R+NR+$\rho(770)$ (Solution II) | $-93746$ | 1998/1814 = 1.101
Among these resonance models, we select the baseline model by requiring each
resonance in the model to have more than 3 standard deviation ($\sigma$)
significance evaluated by the fit fraction divided by its uncertainty. The
baseline fits are 5R in Solution I and 5R+NR in Solution II. No additional
components are significant when added to these baseline fits. Unfortunately,
we cannot distinguish between these two solutions and will quote results for
both of them. In both cases the dominant contribution is S-wave including
$f_{0}(980)$, $f_{0}(1500)$ and $f_{0}(1790)$. The D-wave, $f_{2}(1270)$ and
$f_{2}^{\prime}(1525)$, is only 2.3% for both solutions.
Table 4: Fit fractions (%) of contributing components for both solutions. Component | Solution I | Solution II
---|---|---
$f_{0}(980)$ | $70.3\pm 1.5_{-5.1}^{+0.4}$ | $92.4\pm 2.0_{-16.0}^{+~{}0.8}$
$f_{0}(1500)$ | $10.1\pm 0.8_{-0.3}^{+1.1}$ | $9.1\pm 0.9\pm 0.3$
$f_{0}(1790)$ | $2.4\pm 0.4_{-0.2}^{+5.0}$ | $0.9\pm 0.3_{-0.1}^{+2.5}$
$f_{2}(1270)_{0}$ | $0.36\pm 0.07\pm 0.03$ | $0.42\pm 0.07\pm 0.04$
$f_{2}(1270)_{\|}$ | $0.52\pm 0.15_{-0.02}^{+0.05}$ | $0.42\pm 0.13_{-0.02}^{+0.11}$
$f_{2}(1270)_{\perp}$ | $0.63\pm 0.34_{-0.08}^{+0.16}$ | $0.60\pm 0.36_{-0.09}^{+0.12}$
$f_{2}^{\prime}(1525)_{0}$ | $0.51\pm 0.09_{-0.04}^{+0.05}$ | $0.52\pm 0.09_{-0.04}^{+0.05}$
$f_{2}^{\prime}(1525)_{\|}$ | $0.06_{-0.04}^{+0.13}\pm 0.01$ | $0.11_{-0.07-0.04}^{+0.16+0.03}$
$f_{2}^{\prime}(1525)_{\perp}$ | $0.26\pm 0.18_{-0.04}^{+0.06}$ | $0.26\pm 0.22_{-0.05}^{+0.06}$
NR | - | $5.9\pm 1.4_{-4.6}^{+0.7}$
Sum | 85.2 | 110.6
$\rm-ln\mathcal{L}$ | $-93738$ | $-93739$
$\chi^{2}/\text{ndf}$ | 2005/1822 | $2008/1820$
Table 4 shows the fit fractions from the baseline fits of two solutions, where
systematic uncertainties are included; they will be discussed in Sec. 7.
Figures 14 and 15 show the fit projections of $m(\pi^{+}\pi^{-})$,
$\cos\theta_{\pi\pi}$, $\cos\theta_{J/\psi}$ and $\chi$ from 5R Solution I and
5R+NR Solution II, respectively. Also shown in Figs. 16 and 17 are the
contributions of each resonance as a function of $m(\pi^{+}\pi^{-})$ from the
baseline Solution I and II fits, respectively. Table 5 shows the fit fractions
of the interference terms defined in Eq. (23). In addition, the phases are
listed in Table 6. The other fit results are listed in Table 7 including the
$f_{0}(980)$ mass, the Flatté function parameters $g_{\pi\pi}$,
$g_{KK}/g_{\pi\pi}$, and masses and widths of $f_{0}(1500)$ and $f_{0}(1790)$
resonances.
Figure 14: Projections of (a) $m(\pi^{+}\pi^{-})$, (b) $\cos\theta_{\pi\pi}$, (c) $\cos\theta_{J/\psi}$ and (d) $\chi$ for 5R Solution I. The points with error bars are data, the signal fit is shown with a (red) dashed line, the background with a (black) dotted line, and the (blue) solid line represents the total. Figure 15: Projections of (a) $m(\pi^{+}\pi^{-})$, (b) $\cos\theta_{\pi\pi}$, (c) $\cos\theta_{J/\psi}$ and (d) $\chi$ for 5R+NR Solution II. The points with error bars are data, the signal fit is shown with a (red) dashed line, the background with a (black) dotted line, and the (blue) solid line represents the total. Figure 16: Distribution of $m(\pi^{+}\pi^{-})$ with contributing components labeled from 5R Solution I. Figure 17: Distribution of $m(\pi^{+}\pi^{-})$ with contributing components labeled from 5R+NR Solution II. Table 5: Non-zero interference fraction (%) for both solutions. Components | Solution I | Solution II
---|---|---
$f_{0}(980)$+$f_{0}(1500)$ | 9.50 | $-1.57$
$f_{0}(980)$+$f_{0}(1790)$ | 7.93 | 5.30
$f_{0}(1500)$+$f_{0}(1790)$ | $-2.69$ | $-2.26$
$f_{2}(1270)_{0}$+$f_{2}^{\prime}(1525)_{0}$ | 0.14 | 0.09
$f_{2}(1270)_{\|}$+$f_{2}^{\prime}(1525)_{\|}$ | $-0.09$ | $-0.16$
$f_{2}(1270)_{\perp}$+$f_{2}^{\prime}(1525)_{\perp}$ | 0.03 | 0.05
$f_{0}(980)$+NR | - | $-16.41$
$f_{0}(1500)$+NR | - | 5.26
$f_{0}(1790)$+NR | - | $-0.95$
Table 6: Fitted resonance phase differences (∘). Resonance | Solution I | Solution II
---|---|---
$f_{0}(1500)-f_{0}(980)$ | $138\pm 4$ | $177\pm 6$
$f_{0}(1790)-f_{0}(980)$ | $78\pm 9$ | $95\pm 16$
$f_{2}(1270)_{0}-f_{0}(980)$ | $96\pm 7$ | $123\pm 8$
$f_{2}(1270)_{\|}-f_{0}(980)$ | $-90\pm 11$ | $-84\pm 13$
$f_{2}^{\prime}(1525)_{0}-f_{0}(980)$ | $-132\pm 6$ | $-97\pm 7$
$f_{2}^{\prime}(1525)_{\|}-f_{0}(980)$ | $103\pm 29$ | $130\pm 20$
NR $-f_{0}(980)$ | - | $-104\pm 5$
$f_{2}^{\prime}(1525)_{\perp}-f_{2}(1270)_{\perp}$ | $149\pm 46$ | $145\pm 51$
Table 7: Other fit parameters. The uncertainties are only statistical. Parameter | Solution I | Solution II
---|---|---
$m_{f_{0}(980)}$ ($\mathrm{\,Me\kern-1.00006ptV}$) | $945.4\pm 2.2$ | $949.9\pm 2.1$
$g_{\pi\pi}$ ($\mathrm{\,Me\kern-1.00006ptV}$) | $167\pm 7$ | $167\pm 8$
$g_{KK}/g_{\pi\pi}$ | $3.47\pm 0.12$ | $3.05\pm 0.13$
$m_{f_{0}(1500)}$ ($\mathrm{\,Me\kern-1.00006ptV}$) | $1460.9\pm 2.9$ | $1465.9\pm 3.1$
$\Gamma_{f_{0}(1500)}$ ($\mathrm{\,Me\kern-1.00006ptV}$) | $124\pm 7$ | $115\pm 7$
$m_{f_{0}(1790)}$ ($\mathrm{\,Me\kern-1.00006ptV}$) | $1814\pm 18$ | $1809\pm 22$
$\Gamma_{f_{0}(1790)}$ ($\mathrm{\,Me\kern-1.00006ptV}$) | $328\pm 34$ | $263\pm 30$
In both solutions the $f_{0}(500)$ state does not have a significant fit
fraction. We set an upper limit for the fit fraction ratio between
$f_{0}(500)$ and $f_{0}(980)$ of 0.3% from Solution I and 3.4% from Solution
II, both at 90% CL. A similar situation is found for the $\rho(770)$ state.
When including it in the fit, the fit fraction of $\rho(770)$ is measured to
be $(0.60\pm 0.30^{+0.08}_{-0.14})\%$ in Solution I and $(1.02\pm
0.36^{+0.09}_{-0.15})\%$ from Solution II. The largest upper limit is obtained
by Solution II, where the $\rho(770)$ fit fraction is less than 1.7% at 90%
CL.
Our previous study [3] did not consider the $f_{0}(1790)$ resonance, instead
the NR component filled in the higher mass region near $1800$ MeV. It is found
that including $f_{0}(1790)$ improves the fit significantly in both solutions.
Inclusion of this state reduces $\rm-2ln\mathcal{L}$ by 276 (97) units and
$\chi^{2}$ by 213 (91) units with 4 additional ndf, corresponding to 14 (9)
$\sigma$ Gaussian significance, in Solution I(II), where the numbers are
statistical only. When floating the parameters of $f_{0}(1790)$ resonance in
the fits, we find its mass $m_{f_{0}(1790)}=1815\pm
23$$\mathrm{\,Me\kern-1.00006ptV}$ and width $\Gamma_{f_{0}(1790)}=353\pm
48$$\mathrm{\,Me\kern-1.00006ptV}$ in Solution I, and $m_{f_{0}(1790)}=1793\pm
26$$\mathrm{\,Me\kern-1.00006ptV}$ and $\Gamma_{f_{0}(1790)}=180\pm
83$$\mathrm{\,Me\kern-1.00006ptV}$ in Solution II, where the uncertainties are
statistical only. The values in both solutions are consistent with the BES
results $m_{f_{0}(1790)}=1790_{-30}^{+40}$$\mathrm{\,Me\kern-1.00006ptV}$ and
$\Gamma_{f_{0}(1790)}=270_{-30}^{+60}$$\mathrm{\,Me\kern-1.00006ptV}$ [30] at
the level of $1\sigma$.
Figure 18 compares the total S-wave amplitude strength and phase as a function
of $m(\pi^{+}\pi^{-})$ between the two solutions, showing consistent amplitude
strength but distinct phase. The total S-wave amplitude is calculated as Eq.
(4) summing over all spin-0 component $R$ with $\lambda=0$, where the
$d$-function is equal to 1. The amplitude strength can be well measured from
the $m(\pi^{+}\pi^{-})$ distribution, but this is not the case for the phase,
which is determined from the interference with the small fraction of higher
spin resonances.
Figure 18: S-wave (a) amplitude strength and (b) phase as a function of
$m(\pi^{+}\pi^{-})$ from the 5R Solution I (open) and 5R+NR Solution II
(solid), where the widths of the curves reflect $\pm 1\sigma$ statistical
uncertainties. The reference point is chosen at 980 MeV with amplitude
strength equal to 1 and phase equal to 0.
### 6.4 Angular moments
We define the moments of the cosine of the helicity angle $\theta_{\pi\pi}$,
$\langle Y^{0}_{l}(\cos\theta_{\pi\pi})\rangle$ as the efficiency corrected
and background subtracted $\pi^{+}\pi^{-}$ invariant mass distributions,
weighted by spherical harmonic functions. The moment distributions provide an
additional way of visualizing the presence of different resonances and their
interferences, similar to a partial wave analysis. Figures 19 and 20 show the
distributions of the angular moments for 5R Solution I and 5R+NR Solution II,
respectively. In general the interpretation of these moments [3] is that
$\langle Y^{0}_{0}\rangle$ is the efficiency corrected and background
subtracted event distribution, $\langle Y^{0}_{1}\rangle$ the interference of
the sum of S-wave and P-wave and P-wave and D-wave amplitudes, $\langle
Y^{0}_{2}\rangle$ the sum of the P-wave, D-wave and the interference of S-wave
and D-wave amplitudes, $\langle Y^{0}_{3}\rangle$ the interference between
P-wave and D-wave, $\langle Y^{0}_{4}\rangle$ the D-wave, and $\langle
Y^{0}_{5}\rangle$ the F-wave. The values of $\langle Y^{0}_{1}\rangle$ and
$\langle Y^{0}_{3}\rangle$ are almost zero because the opposite contributions
from $B^{0}_{s}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
decays are summed. Note, in this analysis the P-wave contributions are zero so
the above description simplifies somewhat. The $f_{2}(1270)$ and
$f_{2}^{\prime}(1525)$ interference with S-waves are clearly shown in the
$\langle Y^{0}_{2}\rangle$ plot (see Figs. 19 (c) and 20 (c)).
Figure 19: The $\pi^{+}\pi^{-}$ mass dependence of the spherical harmonic
moments of $\cos\theta_{\pi\pi}$ after efficiency corrections and background
subtraction: (a) $\langle Y^{0}_{0}\rangle$ ($\chi^{2}$/ndf =78/70), (b)
$\langle Y^{0}_{1}\rangle$ ($\chi^{2}$/ndf =37/70), (c) $\langle
Y^{0}_{2}\rangle$ ($\chi^{2}$/ndf =79/70), (d) $\langle Y^{0}_{3}\rangle$
($\chi^{2}$/ndf =42/70), (e) $\langle Y^{0}_{4}\rangle$ ($\chi^{2}$/ndf
=43/70), (f) $\langle Y^{0}_{5}\rangle$ ($\chi^{2}$/ndf =35/70). The points
with error bars are the data points and the solid curves are derived from the
model 5R Solution I.
Figure 20: The $\pi^{+}\pi^{-}$ mass dependence of the spherical harmonic
moments of $\cos\theta_{\pi\pi}$ after efficiency corrections and background
subtraction: (a) $\langle Y^{0}_{0}\rangle$ ($\chi^{2}$/ndf =73/70), (b)
$\langle Y^{0}_{1}\rangle$ ($\chi^{2}$/ndf =36/70), (c) $\langle
Y^{0}_{2}\rangle$ ($\chi^{2}$/ndf =72/70), (d) $\langle Y^{0}_{3}\rangle$
($\chi^{2}$/ndf =43/70), (e) $\langle Y^{0}_{4}\rangle$ ($\chi^{2}$/ndf
=41/70), (f) $\langle Y^{0}_{5}\rangle$ ($\chi^{2}$/ndf =34/70). The points
with error bars are the data points and the solid curves are derived from the
model 5R+NR Solution II.
## 7 Systematic uncertainties
The sources of the systematic uncertainties on the results of the amplitude
analysis are summarized in Table 8 for Solution I and Table 9 for Solution II.
The contributions to the systematic error due to $\phi_{s}$, the function
$\varepsilon(t)$, $\Gamma_{s}$ and $\Delta\Gamma_{s}$[6] uncertainties, and
$L_{B}$ choices for transversity 0 and $\|$ of spin $\geq 1$ resonances, are
negligible. The systematic errors associated to the acceptance or background
modeling are estimated by repeating the fit to the data 100 times. In each fit
the parameters in the acceptance or background function are randomly generated
according to the corresponding error matrix. The uncertainties due to the fit
model include possible contributions from each resonance listed in Table 2 but
not used in the baseline fit models, varying the hadron scale $r$ parameters
in the Blatt-Weisskopf barrier factors for the $B$ meson and $R$ resonance
from 5.0 GeV-1 and 1.5 GeV-1, respectively, to both 3.0 GeV-1, and using
$F_{KK}=1$ in the Flatté function. Compared to the nominal Flatté function,
the new one improves the likelihood fit $\rm-2ln\mathcal{L}$ by 6.8 and 14.0
units for Solution I and Solution II, respectively. The largest variation
among those changes is assigned as the systematic uncertainties for modeling.
Finally, we repeat the data fit by varying the mass and width of resonances
within their errors one at a time, and add the changes in quadrature. To
assign a systematic uncertainty from the possible presence of the $f_{0}(500)$
or $\rho(770)$, we repeat the above procedures using the model that has the
baseline resonances plus $f_{0}(500)$ or $\rho(770)$.
Table 8: Absolute systematic uncertainties for Solution I. Item | Acceptance | Background | Fit model | Resonance parameters | Total
---|---|---|---|---|---
Fit fractions (%)
$f_{0}(980)$ | $\pm 0.17$ | $\pm 0.36$ | ${}_{-5.04}^{+0.00}$ | $\pm 0.03$ | ${}_{-5.1}^{+0.4}$
$f_{0}(1500)$ | $\pm 0.06$ | $\pm 0.14$ | ${}_{-0.29}^{+1.11}$ | $\pm 0.02$ | ${}_{-0.3}^{+1.1}$
$f_{0}(1790)$ | $\pm 0.02$ | $\pm 0.11$ | ${}_{-0.11}^{+4.98}$ | $\pm 0.01$ | ${}_{-0.2}^{+5.0}$
$f_{2}(1270)_{0}$ | $\pm 0.03$ | $\pm 0.01$ | $\pm 0.01$ | $\pm 0.01$ | $\pm 0.03$
$f_{2}(1270)_{\|}$ | $\pm 0.007$ | $\pm 0.009$ | ${}_{-0.020}^{+0.050}$ | $\pm 0.004$ | ${}_{-0.02}^{+0.05}$
$f_{2}(1270)_{\perp}$ | $\pm 0.04$ | $\pm 0.05$ | ${}_{-0.04}^{+0.14}$ | $\pm 0.03$ | ${}_{-0.08}^{+0.16}$
$f_{2}^{\prime}(1525)_{0}$ | $\pm 0.007$ | $\pm 0.012$ | ${}_{-0.000}^{+0.030}$ | $\pm 0.03$ | ${}_{-0.04}^{+0.05}$
$f_{2}^{\prime}(1525)_{\|}$ | $\pm 0.003$ | $\pm 0.004$ | ${}_{-0.020}^{+0.000}$ | $\pm 0.004$ | ${}_{-0.02}^{+0.05}$
$f_{2}^{\prime}(1525)_{\perp}$ | $\pm 0.007$ | $\pm 0.016$ | ${}_{-0.01}^{+0.04}$ | $\pm 0.04$ | ${}^{+0.06}_{-0.04}$
Other fraction (%)
$f_{0}(500)/f_{0}(980)$ | $\pm 0.005$ | $\pm$0.051 | ${}^{+0.150}_{-0.020}$ | $\pm$0.017 | ${}_{-0.06}^{+0.16}$
$\rho(770)$ | $\pm 0.013$ | $\pm 0.065$ | ${}^{+0.040}_{-0.120}$ | $\pm 0.013$ | ${}_{-0.14}^{+0.08}$
$C\\!P$-even | $\pm 0.04$ | $\pm 0.06$ | ${}^{+0.59}_{-0.05}$ | $\pm 0.05$ | ${}^{+0.59}_{-0.10}$
Table 9: Absolute systematic uncertainties for Solution II. Item | Acceptance | Background | Fit model | Resonance parameters | Total
---|---|---|---|---|---
Fit fractions (%)
$f_{0}(980)$ | $\pm 0.12$ | $\pm 0.79$ | ${}_{-15.97}^{+~{}0.00}$ | $\pm 0.00$ | ${}_{-16.0}^{+~{}0.8}$
$f_{0}(1500)$ | $\pm 0.05$ | $\pm 0.15$ | $\pm 0.27$ | $\pm 0.07$ | $\pm 0.3$
$f_{0}(1790)$ | $\pm 0.02$ | $\pm 0.09$ | ${}_{-0.10}^{+2.46}$ | $\pm 0.01$ | ${}_{-0.1}^{+2.5}$
$f_{2}(1270)_{0}$ | $\pm 0.02$ | $\pm 0.01$ | ${}_{-0.03}^{+0.02}$ | $\pm 0.02$ | $\pm 0.04$
$f_{2}(1270)_{\|}$ | $\pm 0.005$ | $\pm 0.009$ | ${}_{-0.010}^{+0.110}$ | $\pm 0.020$ | ${}_{-0.02}^{+0.11}$
$f_{2}(1270)_{\perp}$ | $\pm 0.04$ | $\pm 0.05$ | ${}_{-0.05}^{+0.10}$ | $\pm 0.03$ | ${}_{-0.09}^{+0.12}$
$f_{2}^{\prime}(1525)_{0}$ | $\pm 0.006$ | $\pm 0.012$ | ${}_{-0.010}^{+0.03}$ | $\pm 0.031$ | ${}_{-0.04}^{+0.05}$
$f_{2}^{\prime}(1525)_{\|}$ | $\pm 0.004$ | $\pm 0.008$ | ${}_{-0.040}^{+0.030}$ | $\pm 0.008$ | ${}_{-0.04}^{+0.03}$
$f_{2}^{\prime}(1525)_{\perp}$ | $\pm 0.01$ | $\pm 0.02$ | ${}_{-0.00}^{+0.03}$ | $\pm 0.05$ | ${}^{+0.06}_{-0.05}$
NR | $\pm 0.07$ | $\pm 0.63$ | ${}_{-4.52}^{+0.34}$ | $\pm 0.04$ | ${}_{-4.6}^{+0.7}$
Other fraction (%)
$f_{0}(500)/f_{0}(980)$ | $\pm$0.005 | $\pm$0.051 | ${}^{+0.300}_{-0.120}$ | $\pm$0.017 | ${}_{-0.14}^{+0.31}$
$\rho(770)$ | $\pm 0.015$ | $\pm 0.080$ | ${}^{+0.040}_{-0.120}$ | $\pm 0.016$ | ${}_{-0.15}^{+0.09}$
$C\\!P$-even | $\pm 0.04$ | $\pm 0.06$ | ${}^{+0.66}_{-0.03}$ | $\pm 0.06$ | ${}^{+0.66}_{-0.10}$
## 8 Further results
### 8.1 Fit fraction intervals
The fit fractions shown in Table 4 differ considerably for some of the states
between the two solutions. Table 10 lists the $1\sigma$ regions for the fit
fractions taking into account the differences between the solutions and
including systematic uncertainties. The regions covers both $1\sigma$
intervals of the two solutions.
Table 10: Fit fraction ranges taking $1\sigma$ regions for both solutions including systematic uncertainties. Component | Fit fraction (%)
---|---
$f_{0}(980)$ | $65.0-94.5$
$f_{0}(1500)$ | $8.2-11.5$
$f_{0}(1790)$ | $0.6-7.4$
$f_{2}(1270)_{0}$ | $0.28-0.50$
$f_{2}(1270)_{\|}$ | $0.29-0.68$
$f_{2}(1270)_{\perp}$ | $0.23-1.00$
$f_{2}^{\prime}(1525)_{0}$ | $0.41-0.62$
$f_{2}^{\prime}(1525)_{\|}$ | $0.02-0.27$
$f_{2}^{\prime}(1525)_{\perp}$ | $0.03-0.49$
NR | $0-7.5$
### 8.2 $C\\!P$ content
The only $C\\!P$-even content arises from the $\perp$ projections of the
$f_{2}(1270)$ and $f_{2}^{\prime}(1525)$ resonances, in addition to the 0 and
$\|$ of any possible $\rho(770)$ resonance. The $C\\!P$-even measured values
are $(0.89\pm 0.38_{-0.10}^{+0.59})\%$ and $(0.86\pm 0.42_{-0.10}^{+0.66})\%$
for Solutions I and II, respectively (see Table 4), where the systematic
uncertainty is dominated by the forbidden $\rho(770)$ transversity $0$ and
$\|$ components added in quadrature. To obtain the corresponding upper limit,
the covariance matrix and parameter values from the fit are used to generate
2000 sample parameter sets. For each set, the $C\\!P$-even fraction is
calculated and is then smeared by the systematic uncertainty. The integral of
95% of the area of the distribution yields an upper limit on the $C\\!P$-even
component of 2.3% at 95% CL, where the larger value given by Solution II is
used. The upper limit is the same as our previous measurement [3], while the
current measurement also adds in a possible $f_{2}^{\prime}(1525)$
contribution.
### 8.3 Mixing angle and interpretation of light scalars
The $I=0$ resonanances, $f_{0}(500)$ and $f_{0}(980)$, are thought to be
mixtures of underlying states whose mixing angle has been estimated previously
(see references cited in Ref. [35]). The mixing is parameterized by a normal
2$\times$2 rotation matrix characterized by the angle $\varphi_{m}$, giving in
our case
$\displaystyle|f_{0}(980)\rangle$ $\displaystyle=$
$\displaystyle\;\;\;\cos\varphi_{m}|s\overline{s}\rangle+\sin\varphi_{m}|n\overline{n}\rangle$
$\displaystyle|f_{0}(500)\rangle$ $\displaystyle=$
$\displaystyle-\sin\varphi_{m}|s\overline{s}\rangle+\cos\varphi_{m}|n\overline{n}\rangle,$
$\displaystyle{\rm where~{}}|n\overline{n}\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\left(|u\overline{u}\rangle+|d\overline{d}\rangle\right).$
(26)
In this case only the $|s\overline{s}\rangle$ wave function contributes. Thus
we have [2]
$\tan^{2}\varphi_{m}=\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(500)\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)}\frac{\Phi(980)}{\Phi(500)},$ (27)
where the $\Phi$’s are phase space factors. The phase space in this
pseudoscalar to vector-pseudoscalar decay is proportional to the cube of the
$f_{0}$ momenta. Taking the average of the momentum dependent phase space over
the resonant line shapes results in the ratio of phase space factors
$\frac{\Phi(500)}{\Phi(980)}=1.25$.
Our measured upper limit is
$\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(500),~{}f_{0}(500)\rightarrow\pi^{+}\pi^{-}\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980),~{}f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}<3.4\%~{}{\rm
at~{}90\%~{}CL,}$ (28)
where the larger value of the two solutions (II) is used. This value must be
corrected for the individual branching fractions of the $f_{0}$ resonances
into $\pi^{+}\pi^{-}$. BaBar measures the relative branching ratios of
$f_{0}(980)\rightarrow K^{+}K^{-}$ to $\pi^{+}\pi^{-}$ of $0.69\pm 0.32$ using
$B\rightarrow KKK$ and $B\rightarrow K\pi\pi$ decays [36]. BES has extracted
relative branching ratios using $\psi(2S)\rightarrow\gamma\chi_{c0}$ decays
where the $\chi_{c0}\rightarrow f_{0}(980)f_{0}(980)$, and either both
$f_{0}(980)$’s decay into $\pi^{+}\pi^{-}$ or one into $\pi^{+}\pi^{-}$ and
the other into $K^{+}K^{-}$ [37, *Ablikim:2005kp]. Averaging the two
measurements gives
$\frac{{\cal{B}}\left(f_{0}(980)\rightarrow
K^{+}K^{-}\right)}{{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}=0.35_{-0.14}^{+0.15}$
(29)
Assuming that the $\pi\pi$ and $KK$ decays are dominant we can also extract
${\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)=\left(46\pm
6\right)\%$ (30)
where we have assumed that the only other decays are to $\pi^{0}\pi^{0}$,
$\frac{1}{2}$ of the $\pi^{+}\pi^{-}$ rate, and to neutral kaons, equal to
charged kaons. We use
${\cal{B}}\left(f_{0}(500)\rightarrow\pi^{+}\pi^{-}\right)=\frac{2}{3}$, which
results from isopsin Clebsch-Gordon coefficients, and assuming that the only
decays are into two pions. Since we have only an upper limit on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(500)$, we will only find an
upper limit on the mixing angle, so if any other decay modes of the
$f_{0}(500)$ exist, they would make the limit more stringent. Including
uncertainty of ${\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)$,
our limit is
$\tan^{2}\varphi_{m}=\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(500)\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)}\frac{\Phi(980)}{\Phi(500)}<1.8\%~{}{\rm
at~{}90\%~{}CL},$ (31)
which translates into a limit
$|\varphi_{m}|<7.7^{\circ}~{}{\rm at~{}90\%~{}CL}.$ (32)
This limit is the most constraining ever placed on this mixing angle [21]. The
value of $\tan^{2}\varphi_{m}$ is consistent with the tetraquark model, which
predicts zero within a few degrees[2, 35].
## 9 Conclusions
The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decay can be described by the interfering sum of five
resonant components: $f_{0}(980),f_{0}(1500),f_{0}(1790),f_{2}(1270)$ and
$f_{2}^{\prime}(1525)$. In addition we find that a second model including
these states plus non-resonant ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ also provides a good description of the data. In both
models the largest component of the decay is the $f_{0}(980)$ with the
$f_{0}(1500)$ being almost an order of magnitude smaller. We also find
including the $f_{0}(1790)$ resonance improves the data fit significantly. The
$\pi^{+}\pi^{-}$ system is mostly S-wave, with the D-wave components totaling
only 2.3% in either model. No significant $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho(770)$ decay is observed; a 90% CL upper limit on the fit fraction
is set to be 1.7%.
The most important result of this analysis is that the $C\\!P$ content is
consistent with being purely odd, with the $C\\!P$-even component limited to
2.3% at 95% CL. Also of importance is the limit on the absolute value of the
mixing angle between the $f_{0}(500)$ and $f_{0}(980)$ resonances of
$7.7^{\circ}$ at 90% CL, the most stringent limit ever reported. This is also
consistent with these states being tetraquarks.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centers are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages we depend on. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia).
## References
* [1] LHCb collaboration, R. Aaij et al., Measurement of $C\\!P$ violation and the $B_{s}^{0}$ meson decay width difference with $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ and $B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays, Phys. Rev. D87 (2013) 112010, arXiv:1304.2600
* [2] S. Stone and L. Zhang, Use of $B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}$ decays to discern the $q\bar{q}$ or tetraquark nature of scalar mesons, Phys. Rev. Lett. 111 (2013) 062001, arXiv:1305.6554
* [3] LHCb collaboration, R. Aaij et al., Analysis of the resonant components in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\pi^{+}\pi^{-}$, Phys. Rev. D86 (2012) 052006, arXiv:1204.5643
* [4] L. Zhang and S. Stone, Time-dependent Dalitz-plot formalism for $B_{q}^{0}\rightarrow J/\psi h^{+}h^{-}$, Phys. Lett. B719 (2013) 383, arXiv:1212.6434
* [5] I. I. Bigi and A. Sanda, CP violation, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 9 (2000) 1
* [6] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001
* [7] R. Dalitz, On the analysis of $\tau$-meson data and the nature of the $\tau$-meson, Phil. Mag. 44 (1953) 1068
* [8] LHCb collaboration, A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [9] R. Arink et al., Performance of the LHCb Outer Tracker, JINST 9 (2014) P01002, arXiv:1311.3893
* [10] M. Adinolfi et al., Performance of the LHCb RICH detector at the LHC, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
* [11] A. A. Alves Jr. et al., Performance of the LHCb muon system, JINST 8 (2013) P02022, arXiv:1211.1346
* [12] R. Aaij et al., The LHCb Trigger and its Performance in 2011, JINST 8 (2013) P04022, arXiv:1211.3055
* [13] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 Physics and Manual, JHEP 0605 (2006) 026, arXiv:hep-ph/0603175
* [14] T. Sjöstrand, S. Mrenna, and P. Skands, A brief introduction to PYTHIA 8.1, Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
* [15] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [16] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [17] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [18] Geant4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [19] GEANT4 collaboration, S. Agostinelli et al., GEANT4: A Simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [20] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. Phys. Conf. Ser. 331 (2011) 032023
* [21] LHCb collaboration, R. Aaij et al., Analysis of the resonant components in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$, Phys. Rev. D87 (2013) 052001, arXiv:1301.5347
* [22] A. Hocker et al., TMVA $-$ Toolkit for multivariate data analysis, PoS ACAT (2007) 040, arXiv:physics/0703039
* [23] S. Stone and L. Zhang, Measuring the $C\\!P$ violating phase in $B^{0}_{s}$ mixing using $B^{0}_{s}\rightarrow J/\psi f_{0}(980)$, arXiv:0909.5442
* [24] LHCb collaboration, R. Aaij et al., Measurement of the CP-violating phase $\phi_{s}$ in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays, Phys. Lett. B713 (2012) 378, arXiv:1204.5675
* [25] J. Charles et al., Predictions of selected flavour observables within the standard model, Phys. Rev. D84 (2011) 033005, arXiv:1106.4041
* [26] Belle collaboration, R. Mizuk et al., Dalitz analysis of $B\rightarrow K\pi^{+}\psi^{\prime}$ decays and the $Z(4430)^{+}$, Phys. Rev. D80 (2009) 031104, arXiv:0905.2869
* [27] BESIII collaboration, M. Ablikim et al., Observation of a charged charmonium like structure in $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ at $\sqrt{s}=$4.26 GeV, Phys. Rev. Lett. 110 (2013) 252001, arXiv:1303.5949
* [28] W. Ochs, The status of glueballs, J. Phys. G40 (2013) 043001, arXiv:1301.5183
* [29] W. Ochs, Spectroscopy with glueballs and the role of $f_{0}(1370)$, arXiv:1304.7634
* [30] BES collaboration, M. Ablikim et al., Resonances in $J/\psi\rightarrow\phi\pi^{+}\pi^{-}$ and $\phi K^{+}K^{-}$, Phys. Lett. B607 (2005) 243, arXiv:hep-ex/0411001
* [31] LHCb collaboration, R. Aaij et al., Amplitude analysis and the branching fraction measurement of $\bar{B}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$, Phys. Rev. D87 (2013) 072004, arXiv:1302.1213
* [32] D. V. Bugg, Reanalysis of data on $a_{0}$(1450) and $a_{0}$(980), Phys. Rev. D78 (2008) 074023, arXiv:0808.2706
* [33] S. M. Flatté, On the nature of $0^{+}$ mesons, Phys. Lett. B63 (1976) 228
* [34] S. Baker and R. D. Cousins, Clarification of the use of $\chi^{2}$ and likelihood functions in fits to histograms, Nucl. Instrum. Meth. 221 (1984) 437
* [35] R. Fleischer, R. Knegjens, and G. Ricciardi, Anatomy of $B^{0}_{s,d}\rightarrow J/\psi f_{0}(980)$, Eur. Phys. J. C71 (2011) 1832, arXiv:1109.1112
* [36] BABAR collaboration, B. Aubert et al., Dalitz plot analysis of the decay $B^{\pm}\rightarrow K^{\pm}K^{\pm}K^{\mp}$, Phys. Rev. D74 (2006) 032003, arXiv:hep-ex/0605003
* [37] BES collaboration, M. Ablikim et al., Evidence for $f_{0}(980)f_{0}(980)$ production in $\chi_{c0}$ decays, Phys. Rev. D70 (2004) 092002, arXiv:hep-ex/0406079
* [38] BES collaboration, M. Ablikim et al., Partial wave analysis of $\chi_{c0}\rightarrow\pi^{+}\pi^{-}K^{+}K^{-}$, Phys. Rev. D72 (2005) 092002, arXiv:hep-ex/0508050
|
arxiv-papers
| 2014-02-25T17:24:28 |
2024-09-04T02:49:58.855586
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. Anderlini,\n J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, M. Borsato, T.J.V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto,\n J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M. Calvi, M. Calvo Gomez,\n A. Camboni, P. Campana, D. Campora Perez, F. Caponio, A. Carbone, G. Carboni,\n R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G.\n Casse, L. Cassina, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M.\n Charles, Ph. Charpentier, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba,\n X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, I.\n Counts, B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R.\n Currie, C. D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n S. Donleavy, F. Dordei, M. Dorigo, P. Dorosz, A. Dosil Su\\'arez, D. Dossett,\n A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger,\n R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, S. Esen, A. Falabella, C.\n F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, J. Fu, E. Furfaro, A. Gallas Torreira, D. Galli, S.\n Gambetta, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, L. Gavardi, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, A. Gianelle, S. Giani', V. Gibson, L. Giubega, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, L. Grillo, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, T.W. Hafkenscheid, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, L. Henry, J.A.\n Hernando Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, D. Hill, M.\n Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, N. Hussain, D. Hutchcroft, D.\n Hynds, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, N.\n Jurik, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, M.\n Kelsey, I.R. Kenyon, T. Ketel, B. Khanji, C. Khurewathanakul, S. Klaver, O.\n Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, B. Langhans, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, M. Liles,\n R. Lindner, C. Linn, F. Lionetto, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H.\n Lopes, N. Lopez-March, P. Lowdon, H. Lu, D. Lucchesi, H. Luo, E. Luppi, O.\n Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G.\n Manca, G. Mancinelli, M. Manzali, J. Maratas, U. Marconi, C. Marin Benito, P.\n Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, F. Martinez Vidal, D. Martins\n Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov, M.\n McCann, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier,\n M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, M. Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, R.\n Mountain, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik,\n T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neri, S. Neubert, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M.\n Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano,\n F. Palombo, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, L.\n Pappalardo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pearce, A. Pellegrino,\n M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, P. Perret, M. Perrin-Terrin,\n L. Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, A. Pistone, S. Playfer, M. Plo\n Casasus, F. Polci, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve,\n V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H. Rademacker,\n B. Rakotomiaramanana, M. Rama, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, A.B.\n Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A.\n Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz\n Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V.\n Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, G.\n Simi, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E. Smith,\n J. Smith, M. Smith, H. Snoek, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, F. Spinella, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S. Stone, B.\n Storaci, S. Stracka, M. Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah,\n L. Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer,\n V. Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, J.A. de\n Vries, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, G. Wilkinson, M.P. Williams, M. Williams, F.F.\n Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1402.6248"
}
|
1402.6337
|
# A review of type Ia supernova spectra
J. Parrent11affiliationmark: 22affiliationmark: , B.
Friesen33affiliationmark: , and M. Parthasarathy44affiliationmark:
###### Abstract
SN 2011fe was the nearest and best-observed type Ia supernova in a generation,
and brought previous incomplete datasets into sharp contrast with the detailed
new data. In retrospect, documenting spectroscopic behaviors of type Ia
supernovae has been more often limited by sparse and incomplete temporal
sampling than by consequences of signal-to-noise ratios, telluric features, or
small sample sizes. As a result, type Ia supernovae have been primarily
studied insofar as parameters discretized by relative epochs and incomplete
temporal snapshots near maximum light. Here we discuss a necessary next step
toward consistently modeling and directly measuring spectroscopic observables
of type Ia supernova spectra. In addition, we analyze current spectroscopic
data in the parameter space defined by empirical metrics, which will be
relevant even after progenitors are observed and detailed models are refined.
00footnotetext: 6127 Wilder Lab, Department of Physics & Astronomy, Dartmouth
College, Hanover, NH 03755, USA00footnotetext: Las Cumbres Observatory Global
Telescope Network, Goleta, CA 93117, USA00footnotetext: Homer L. Dodge
Department of Physics and Astronomy, University of Oklahoma, 440 W Brooks,
Norman, OK 73019, USA00footnotetext: Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Post Bag 4, Ganeshkhind, Pune 411007, India
Keywords supernovae : type Ia - general - observational - white dwarfs,
techniques: spectroscopic
## 1 Introduction
The transient nature of extragalactic type Ia supernovae (SN Ia) prevent
studies from conclusively singling out unobserved progenitor configurations
(Roelofs et al., 2008; Li et al., 2011b; Kilic et al., 2013). It remains
fairly certain that the progenitor system of SN Ia comprises at least one
compact C$+$O white dwarf (Chandrasekhar, 1957; Nugent et al., 2011; Bloom et
al., 2012). However, _how_ the state of this primary star reaches a critical
point of disruption continues to elude astronomers. This is particularly so
given that less than $\sim$ 15% of locally observed white dwarfs have a mass a
few 0.1M⊙ greater than a solar mass; very few systems near the formal
Chandrasekhar-mass limit555By “formal” we are referring to the mass limit that
omits stellar rotation (see Appendix of Jeffery et al. 2006)., MCh $\approx$
1.38 M⊙ (Vennes, 1999; Liebert et al., 2005; Napiwotzki et al., 2005;
Parthasarathy et al., 2007; Napiwotzki et al., 2007).
Thus far observational constraints of SN Ia have been inconclusive in
distinguishing between the following three separate theoretical considerations
about possible progenitor scenarios. Along side perturbations in the critical
mass limit or masses of the progenitors, e.g., from rotational support
(Mueller & Eriguchi, 1985; Yoon & Langer, 2005; Chen & Li, 2009; Hachisu et
al., 2012; Tornambé & Piersanti, 2013) or variances of white dwarf (WD)
populations (van Kerkwijk et al., 2010; Dan et al., 2013), the primary WD may
reach the critical point by accretion of material from a low-mass, radially-
confined secondary star (Whelan & Iben, 1973; Nomoto & Sugimoto, 1977; Hayden
et al., 2010a; Bianco et al., 2011; Bloom et al., 2012; Hachisu et al., 2012;
Wheeler, 2012; Mazzali et al., 2013; Chen et al., 2013), and/or through one of
several white dwarf merger scenarios with a close binary companion (Webbink,
1984; Iben & Tutukov, 1984; Paczynski, 1985; Thompson, 2011; Wang et al.,
2013a; Pakmor et al., 2013). In addition, the presence (or absence) of
circumstellar material may not solely rule out particular progenitor systems
as now both single- and double-degenerate systems are consistent with having
polluted environments prior to the explosion (Shen et al., 2013; Phillips et
al., 2013).
Meanwhile, and within the context of a well-observed spectroscopically normal
SN 2011fe, recent detailed models and spectrum synthesis along with SN Ia
rates studies, a strong case for merging binaries as the progenitors of normal
SN Ia has surfaced (c.f., van Kerkwijk et al. 2010; Li et al. 2011c; Blondin
et al. 2012; Chomiuk 2013; Dan et al. 2013; Moll et al. 2013; Maoz et al.
2013; Johansson et al. 2014). However, because no progenitor system has _ever_
been connected to any SN Ia, most observational constraints and trends are
difficult to robustly impose on a standard model picture for even a single
progenitor channel; the SN Ia problem is yet to be confined for each SN Ia
subtype.
As for restricting SN Ia subtypes to candidate progenitor systems: (i)
observed “jumps” between mean properties of SN Ia subtypes signify potential
differences of progenitors and/or explosion mechanisms, (ii) the dispersions
of individual subtypes are thought to arise from various abundance, density,
metallicity, and/or temperature enhancements of the original progenitor
system’s post-explosion ejecta tomography, and (iii) “transitional-type” SN Ia
complicate the already similar overlap of observed SN Ia properties (Nugent et
al., 1995; Lentz et al., 2000; Benetti et al., 2005; Branch et al., 2009;
Höflich et al., 2010; Wang et al., 2012, 2013c; Dessart et al., 2013a).
Moreover, our physical understanding of all observed SN Ia subclasses remains
based entirely on interpretations of idealized explosion models that are so
far constrained and evaluated by “goodness of fit” comparisons to incomplete
observations, particularly for SN Ia spectra at all epochs.
By default, spectra have been a limiting factor of supernova studies due to
associated observational consequences, e.g., impromptu transient targets,
variable intrinsic peak luminosities, a sparsity of complete datasets in
wavelength and time, insufficient signal-to-noise ratios, and the ever-present
obstacle of spectroscopic line blending (Payne-Gaposchkin & Whipple, 1940).
Subsequently, two frequently relied upon empirical quantifiers of SN Ia
spectroscopic diversity have been the rate at which rest-frame 6100 Å
absorption minima shift redward vis-à-vis projected Doppler velocities of the
absorbing Si-rich material (Benetti et al., 2005; Wang et al., 2009a) and
absorption strength measurements (a.k.a. pseudo equivalent widths; pEWs) of
several lines of interest (see Branch et al. 2006; Hachinger et al. 2006;
Silverman et al. 2012b; Blondin et al. 2012). Together these classification
schemes more-or-less describe the same events by two interconnected parameter
spaces (i.e. flux and expansion velocities, Branch et al. 2009; Foley & Kasen
2011; Blondin et al. 2012) that are dependent on a multi-dimensional array of
physical properties. Naturally, the necessary next step for supernova studies
alike is the development of prescriptions for the physical diagnosis of
spectroscopic behaviors (see §2.2 and Kerzendorf & Sim 2014).
For those supernova events that _have_ revealed the observed patterns of SN Ia
properties, the majority are termed “Branch-normal” (Branch et al., 1993; Li
et al., 2011c), while others further away from the norm are historically said
to be “peculiar” (e.g., SN 1991T, 1991bg; see Filippenko 1997 and references
therein). Although, many non-standard events have since obscured the
boundaries between both normal and peculiar varieties of SN Ia, such as SN
1999aa (Garavini et al., 2004), 2000cx (Chornock et al., 2000; Li et al.,
2001; Rudy et al., 2002), 2001ay (Krisciunas et al., 2011), 2002cx (Li et al.,
2003), 2003fg (Howell et al., 2006; Jeffery et al., 2006), 2003hv (Leloudas et
al., 2009; Mazzali et al., 2011), 2004dt (Wang et al., 2006; Altavilla et al.,
2007), 2004eo (Pastorello et al., 2007a), 2005gj (Prieto et al., 2007), 2006bt
(Foley et al., 2010b), 2007ax (Kasliwal et al., 2008), 2008ha (Foley et al.,
2009, 2010a), 2009ig (Foley et al., 2012c; Marion et al., 2013), PTF10ops
(Maguire et al., 2011), PTF11kx (Dilday et al., 2012; Silverman et al.,
2013b), and 2012fr (Maund et al., 2013; Childress et al., 2013c).
The fact that certain subsets of normal SN Ia constitute a near homogenous
group of intrinsically bright events has led to their use as standardizable
distance indicators (Kowal, 1968; Elias et al., 1985a; Branch & Tammann, 1992;
Riess et al., 1999; Perlmutter et al., 1999; Schmidt, 2004; Mandel et al.,
2011; Maeda et al., 2011; Sullivan et al., 2011a; Hicken et al., 2012).
However, this same attribute of homogeneity remains the greatest challenge in
the individual study of SN Ia given that the time-evolving spectrum of a
supernova is unique unto itself from the earliest to the latest epochs.
Because SN Ia are invaluable tools for both cosmology and understanding
progenitor populations, a multitude of large scale surveys, searches, and
observing campaigns666e.g., The Automated Survey for SuperNovae (Assassin),
The Backyard Observatory Supernova Search (BOSS), The Brazilian Supernova
Search (BRASS), The Carnegie Supernova Project (CSP), The Catalina Real-Time
Transient Survey (CRTS), The CHilean Automatic Supernovas sEarch (CHASE), The
Dark Energy Survey (DES), The Equation of State: SupErNovae trace Cosmic
Expansion (ESSENCE) Supernova Survey, The La Silla-QUEST Variability Survey
(LSQ), Las Cumbres Observatory Global Telescope Network (LCOGT), The Lick
Observatory Supernova Search (LOSS), The Mobile Astronomical System of the
Telescope-Robots Supernova Search (MASTER), The Nearby Supernova Factory
(SNfactory), The Optical Gravitational Lensing Experiment (OGLE-IV), The
Palomar Transient Factory (PTF), The Panoramic Survey Telescope and Rapid
Response System (Pan-STARRS), The Plaskett Spectroscopic Supernova Survey
(PSSS), Public ESO Spectroscopic Survey of Transient Objects (PESSTO), The
Puckett Observatory World Supernova Search, The ROTSE Supernova Verification
Project (RSVP), The SDSS Supernova Survey, The Canada-France-Hawaii Telescope
Legacy Survey Supernova Program (SNLS), The Southern inTermediate Redshift ESO
Supernova Search (STRESS), The Texas Supernova Search (TSS); for more, see
http://www.rochesterastronomy.org/snimages/snlinks.html. are continually being
carried out with regularly improved precision. Subsequently, this build-up of
competing resources has also resulted in an ever growing number of new and
important discoveries, with less than complete information for each. In fact,
with so many papers published each year on various aspects of SN Ia, it can be
difficult to keep track of new results and important developments, including
the validity of past and present theoretical explosion simulations and their
related observational interpretations (see Maoz et al. 2013 for the latest).
Here we compile some of the discussions on spectroscopic properties of SN Ia
from the past decade of published works. In §2 we overview the most common
means for studying SN Ia: light curves (§2.1), spectra (§2.2), and detailed
explosion models (§2.3). In particular, we overview how far the well-observed
SN 2011fe has progressed the degree of confidence associated with reading
highly blended SN Ia spectra. Issues of SN Ia diversity are discussed in §3.
Next, in §4 we recall several SN Ia that have made up the bulk of _recent_
advances in uncovering the extent of their properties and peculiarities (see
also the Appendix for a guide of some recent events). Finally, in §5 we
summarize and conclude with some observational lessons of SN 2011fe.
## 2 Common Subfields of Utility
### 2.1 Light curves
The interaction between the radiation field and the ejecta can be interpreted
to zeroth order with the bolometric light curve. For SN Ia, the rise and fall
of the light curve is said to be “powered” by 56Ni produced in the explosion
(Colgate & McKee, 1969; Arnett, 1982; Khokhlov et al., 1993; Mazzali et al.,
1998; Pinto & Eastman, 2000a; Stritzinger & Leibundgut, 2005). Additional
sources _are_ expected to contribute to the overall luminosity behavior at
various epochs777Just a few examples include: C$+$O layer metallicity (Lentz
et al., 2000; Timmes et al., 2003; Meng et al., 2011), interaction with
circumstellar material (CSM, see Quimby et al. 2006b; Patat et al. 2007; Simon
et al. 2007; Kasen 2010; Hayden et al. 2010a; Sternberg et al. 2011; Foley et
al. 2012a; Förster et al. 2012; Shen et al. 2013; Silverman et al. 2013d;
Raskin & Kasen 2013) or an enshrouding C$+$O envelope (Scalzo et al., 2012;
Taubenberger et al., 2013), differences in total progenitor system masses
(Hachisu et al., 2012; Pakmor et al., 2013; Chen et al., 2013), and
directional dependent aspects of binary configurations (e.g., Blondin et al.
2011; Moll et al. 2013)..
For example, Nomoto et al. (2003) has suggested that the variation of the
carbon mass fraction in the C+O WD (C/O), or the variation of the initial WD
mass, causes the diversity of SN Ia brightnesses (see Höflich et al. 2010).
Similarly, Meng et al. (2011) argue that C/O and progenitor metallicity, Z,
are intimately related for a fixed WD mass, and particularly for high
metallicities given that it results in lower 3$\alpha$ burning rates plus an
increased reduction of carbon via 12C($\alpha$,$\gamma$)16O. For Z $>$ Z⊙
($\sim$0.02), Meng et al. (2011) find that both C/O and Z have an
approximately equal influence on 56Ni production since, for a given WD mass,
high progenitor metallicities (a greater abundance of species heavier than
oxygen) and low C/O abundances (low carbon-rich fuel assuming a single-
degenerate scenario) result in a low 56Ni yield and subsequently dimmer SN Ia.
For near solar metallicities or less, the carbon mass fraction plays a
dominant role in 56Ni production (Timmes et al., 2003). This then suggests
that the average C/O ratio in the final state of the progenitor is an
important _physical_ cause, in addition to metallicity, for the observed
width-luminosity relationship (WLR888A WLR is followed when a SN Ia has a
proportionately broader light curve for its intrinsic brightness at maximum
light (Phillips, 1993). Phillips et al. (1999) later extended this correlation
by incorporating measurement of the extinction via late time _B_ $-$ _V_ color
measurements and _B_ $-$ _V_ and _V_ $-$ _I_ measurements at maximum light
(see also Germany et al. 2004; Prieto et al. 2006b). Because lights curves of
faint SN Ia evolve promptly before 15 days post-maximum light, light curve
shape measurements are better suited for evaluating the light curve “stretch”
(Conley et al., 2008).) of normal SN Ia light curves (Umeda et al., 1999a;
Timmes et al., 2003; Nomoto et al., 2003; Bravo et al., 2010; Meng et al.,
2011).
At the same time, the observed characteristics of SN Ia light curves and
spectra can be fairly matched by adopting radial and/or axial shifts in the
distribution of 56Ni, possibly due to a delayed- and/or pulsational-
detonation-like explosion mechanism (see Khokhlov 1991b; Hoflich et al. 1995;
Baron et al. 2008; Bravo et al. 2009; Maeda et al. 2010b; Baron et al. 2012;
Dessart et al. 2013a) or a merger scenario (e.g., Dan et al. 2013; Moll et al.
2013). Central ignition densities are also expected to play a secondary role
in the form of the WLR since they are dependent upon the accretion rate of H
and/or He-rich material and cooling time (Röpke et al., 2005; Höflich et al.,
2010; Meng et al., 2010; Krueger et al., 2010; Sim et al., 2013), in addition
to the spin-down timescales for differentially rotating WDs (Hachisu et al.,
2012; Tornambé & Piersanti, 2013). Generally, discerning which of these
factors dominate the spectrophotometric variation from one SN Ia to another
remains a challenging task (Wang et al., 2012). As a result, astronomers are
still mapping a broad range of SN Ia characteristics and trends (§3).
Meanwhile, cosmological parameters determined by SN Ia light curves depend on
an accurate comparison of nearby and distant events999Most SN Ia distance
determination methods rely on correlating a distance dependent parameter and
one or more distance independent parameters. Subsequently, a number of methods
have been developed to calibrate SN Ia by multi-color light curve shapes
(e.g., Hamuy et al. 1996; Nugent et al. 2002; Knop et al. 2003; Nobili et al.
2005; Prieto et al. 2006b; Jha et al. 2007; Conley et al. 2008; Rodney & Tonry
2009; Burns et al. 2011).. For distant and therefore redshifted SN Ia, a
“K-correction” converts an observed magnitude to that which would be observed
in the rest frame in another bandpass filter, allowing for the comparison of
SN Ia brightnesses at various redshifts (Hogg et al., 2002). Consequently,
K-corrections require the spectral energy distribution (SED) of the SN Ia and
depend on SN Ia broad-band colors and the diversity of spectroscopic features
(Nugent et al., 2002). While some light curve fitters take a K-correction-less
approach (e.g., Guy et al. 2005, 2007; Conley et al. 2008), an SED is still
required. A spectral template time series dataset is usually used since there
exists remarkable homogeneity in the observed optical spectra of “normal” SN
Ia (e.g., Hsiao et al. 2007).
Unfortunately there do remain poorly understood differences regarding
spectroscopic feature strengths and inferred expansion velocities for these
and other types of thermonuclear supernovae (see §2.2 and §3). At best, the
spectroscopic diversity of SN Ia has been determined to be multidimensional
(Hatano et al., 2000; Benetti et al., 2005; Branch et al., 2009; Wang et al.,
2009a). Verily, SN Ia diversity studies require numerous large spectroscopic
datasets in order to subvert many complex challenges faced when interpreting
the data and extracting both projected Doppler velocities and “feature
strength” measurements. However, studies that seek to primarily utilize SN Ia
broad band luminosities need only collect a handful of sporadically sampled
spectra in order to type the supernova event as a bona fide SN Ia. We note
that interests in precision cosmology conflict at this point with the study of
SN Ia. This is primarily because obtaining _UBVRI_ photometry for hundreds of
events is cheaper than collecting complete spectroscopy for a lesser number of
SN Ia at various redshifts.
Nevertheless, the brightness decline rate in the _B_ -band during the first 15
rest-frame days post-maximum light, $\Delta$m15(_B_), has proven useful for
all SN Ia surveys. Phillips (1993) noted that $\Delta$m15(_B_) is well
correlated with the intrinsic luminosity, a.k.a. the width-luminosity
relationship. Previously, Khokhlov et al. (1993) did predict the existence of
a WLR given that the light curve shape is sensitive to the time-dependent
state of the ejected material.
Kasen & Woosley (2007) recently utilized multi-dimensional time-dependent
Monte Carlo radiative transfer calculations of Chandrasekhar-mass SN Ia models
to access the physical relationship between the luminosity and light curve
decline rate. They found that the WLR is largely a consequence of the
radiative transfer inherent to SN Ia atmospheres, whereby the ionization
evolution of iron redirects flux red ward and is hastened for dimmer and/or
cooler SN Ia. Woosley et al. (2007) later explored the diversity of SN Ia
light curves using a grid of 130 one-dimensional models. They concluded that a
WLR is satisfied when SN Ia burn $\sim$ 1.1 M⊙ of material, with iron-group
elements extending out to $\sim$ 8000 km s-1.
Broadly speaking, the shape of the WLR is fundamentally influenced by the
ionization evolution of iron group elements (Kasen & Woosley, 2007). However,
since broad band luminosities are the sum of a supernova SED per wavelength
interval, details of SN Ia diversity risk being “blurred out” for large
samples of SN Ia. Therefore, decoding the spectra of all SN Ia subtypes, in
addition to indirectly constraining detailed explosion models by the WLR, is
of vital importance since variable signatures of iron-peak elements (IPEs)
blend themselves within an SED typically populated by relatively strong
features of overlapping signatures of intermediate-mass elements (IMEs).
### 2.2 Spectra
Supernova spectra detail information about the explosion and its local
environment. To isolate and extract physical details (and determine their
order of influence), several groups have invested greatly in advancing the
computation of synthetic spectra for SN Ia, particularly during the early
phases of homologous expansion (e.g., Mazzali & Lucy 1993; Hauschildt & Baron
1999; Kasen et al. 2002; Thomas et al. 2002; Höflich et al. 2002; Branch 2004;
Sauer et al. 2006; Kasen et al. 2006; Jeffery & Mazzali 2007; Sim et al.
2010a; Thomas et al. 2011a; Hillier & Dessart 2012; Hoffmann et al. 2013;
Pauldrach et al. 2013; Kerzendorf & Sim 2014). Although, even the basic facets
of the supernova radiation environment serve as obstacles for timely
computations of physically accurate, statistically representative, and
robustly certain synthetic spectra (e.g., consequences of expansion).
It is the time-dependent interaction of the radiation field with the expanding
material that complicates drawing conclusions about the explosion physics from
the observations101010There is general consensus that the observed
spectroscopic diversity of most SN Ia are influenced by: different
configurations of 56Ni produced in the events (Colgate & McKee, 1969; Arnett,
1982; Khokhlov et al., 1993; Baron et al., 2012), their effective temperatures
(Nugent et al., 1995), density profiles and the amount of IPEs present within
the outermost layers of ejecta (Hatano et al., 1999a; Baron et al., 2006;
Hachinger et al., 2012), global symmetries of Si-rich material (Thomas et al.,
2002), departures from spherical symmetry for Ca and Si-rich material at high
velocities (Wang et al., 2007; Kasen et al., 2009; Maeda et al., 2010a; Maund
et al., 2013; Moll et al., 2013; Dessart et al., 2013a), efficiencies of flux
redistribution (Kasen et al., 2006; Jack et al., 2012), the radial extent of
stratified material resulting from a detonation phase (Woosley et al., 2007),
host galaxy dust (Tripp & Branch, 1999; Childress et al., 2013a), and the
metallicity of the progenitors (Höflich et al., 1998; Lentz et al., 2000;
Timmes et al., 2003; Howell et al., 2009; Bravo et al., 2010; Jackson et al.,
2010; Wang et al., 2013c).. In a sense, there are two stages during which
direct (and accessible) information about the progenitor system is driven away
from being easily discernible within the post-explosion spectra: explosive
nucleosynthesis and radiation transport111111Some relevant obstacles include:
a high radiation energy density in a low matter density environment, radiative
versus local collisional processes (non-LTE conditions) and effects (Baron et
al., 1996), time-dependent effects and the dominance of line over continuum
opacity (Pinto & Eastman, 2000a, b), and relativistic flows as well as GR
effects on line profiles (Chen et al., 2007; Knop et al., 2009). In addition,
the entire light emission is powered by decay-chain $\gamma$-rays,
interactions with CSM, and is influenced by positrons, fast electrons, and
Auger electrons in later phases (Kozma & Fransson, 1992; Seitenzahl et al.,
2009).. That is to say, the ability to reproduce both the observed light curve
and spectra, as well as the range of observed characteristics among SN Ia, is
essential towards validating and/or restricting any explosion model for a
given subtype.
Fig. 1 : Plotted is the SNFactory’s early epoch dataset of SN 2011fe presented
by Pereira et al. (2013). We have normalized and over-plotted each spectrum at
the 6100 Å P Cygni profile in order to show the relative locations of all ill-
defined features as they evolve with the expansion of the ejecta. The quoted
rise-time to maximum light (dashed black) is from Mazzali et al. (2013).
Moreover, this assumes the sources of observed spectroscopic signatures in all
varieties of SN Ia are known a priori, which is not necessarily the case given
the immense volume of actively contributing atomic line transitions and
continuum processes (Baron et al., 1995, 1996; Kasen et al., 2008; Bongard et
al., 2008; Sauer et al., 2008). In fact, several features throughout the
spectra have been either _tentatively_ associated with a particular blend of
atomic lines or identified with a multiple of conflicting suggestions (e.g.,
forbidden versus permitted lines at late or “nebular” transitional phases, see
Bowers et al. 1997; Branch et al. 2005; Friesen et al. 2012; Dessart et al.
2013b). Meanwhile others are simply misidentified or unresolved due to the
inherent high degeneracy of solutions and warrant improvements to the models
for further study (e.g., Na I versus [Co III]; Dessart et al. 2013b).
For example, the debate over whether or not hydrogen and/or helium are
detected in some early Ibc spectra has been difficult to navigate on account
of the wavelength separation of observed weak features and the number of
plausible interpretations (Deng et al., 2000; Branch et al., 2002b; Anupama et
al., 2005a; Elmhamdi et al., 2006; Parrent et al., 2007; Ketchum et al., 2008;
Soderberg et al., 2008; James & Baron, 2010; Benetti et al., 2011; Chornock et
al., 2011; Dessart et al., 2012; Milisavljevic et al., 2013a, b; Takaki et
al., 2013). Historically, the term “conspicuous” has defined whether or not a
supernova belongs to a particular spectroscopic class. By way of illustration,
_photographic spectrograms of type II events reveal conspicuous emission bands
of hydrogen while type I events do not_ (Minkowski, 1941). With the advent of
CCD cameras in modern astronomy, it has been determined that 6300 Å absorption
features (however weak) in the early spectra of some type Ibc supernovae are
often no less conspicuous than 6100 Å Si II $\lambda$6355 absorption features
in SN Ia spectra, where some 6300 Å features produced by SN Ibc may be due to
Si II and/or higher velocity H$\alpha$ (Filippenko, 1988; Filippenko et al.,
1990; Filippenko, 1992). That is, while SN Ibc are of the type I class, they
do not necessarily lack hydrogen and/or helium within their outer-most layers
of ejecta, hence the conservative definition of type I supernovae as
“hydrogen/helium-poor” events.
This conundrum of which ion signatures construct each observed spectral
feature rests proportionately on the signal-to-noise ratio (S/N) of the data.
However, resolving this spectroscopic dilemma is primarily dependent on the
wavelength and temporal coverage of the observations and traces back to the
pioneering work of McLaughlin (1963) who studied spectra of the type Ib
supernova, SN 1954A, in NGC 4214 (Wellmann, 1955; Branch, 1972; Blaylock et
al., 2000; Casebeer et al., 2000). Contrary to previous interpretations that
supernova spectra were the result of broad, overlapping emission features
(Gaposchkin, 1936; Humason, 1936; Baade, 1936; Walter & Strohmeier, 1937;
Minkowski, 1939; Payne-Gaposchkin & Whipple, 1940; Zwicky, 1942; Baade et al.,
1956), it was D. B. McLaughlin who first began to repeatedly entertain the
idea that “absorption-like” features were present121212Admittedly Minkowski
(1941) had previously mentioned “absorptions and broad emission bands are
developed [in the spectra of supernovae].” Although, this was primarily within
the context of early epoch observations that revealed a featureless, blue
continuum: “Neither absorptions nor emission bands can be definitely seen but
some emission is suspected in the region of H$\alpha$” (Minkowski, 1940). in
regions that “lacked emission” (McLaughlin, 1959, 1960, 1963).
The inherent difficulties in reading supernova spectra and the history of
uncertain line identifications for both conspicuous and _concealed_ absorption
signatures are almost as old as the supernova field itself (Payne-Gaposchkin &
Whipple, 1940; Dessart et al., 2013b). Still, spectroscopic intuitions can
only evolve as far as the data allow. Therefore it is both appropriate and
informative to recall the progression of early discussions on the spectra of
supernovae, during which spectroscopic designations of type I and type II were
first introduced:
> There appears to be a general opinion that the evidence concerning the
> spectrum of the most luminous nova of modern times was so contradictory that
> conclusions as to its spectra nature are impossible. This view is expressed,
> for example, by Miss Cannon: “With the testimony apparently so conflicting,
> it is difficult to form any conception of the class of this spectrum”
> (Gaposchkin, 1936).
> It also seems ill advised to conclude anything regarding the distribution of
> temperature in super-novae from the character of their visible spectra as
> long as a satisfactory explanation of some of the most important features of
> these spectra is completely lacking (Zwicky, 1936).
> The spectrum is not easy to interpret, as true boundaries of the wide
> emission lines are difficult to determine (Humason, 1936).
> Those [emission] bands with distinct maxima and a fairly sharp redward or
> violetward edge, excepting edges due to a drop in plate spectral
> sensitivity, may give an indication of expansion velocity (Popper, 1937).
> Instead of the typical pattern of broad, diffuse emissions dominated by a
> band about 4600 Å, it appeared like a continuum with a few deep and several
> shallow absorption-like minima. Two of the strongest “absorption lines,”
> when provisionally interpreted as $\lambda\lambda$4026, 4472 He I, give
> velocities near $-$5000 km s-1 […] The author is grateful to N. U. Mayall
> and R. Minkowski for the use of spectrograms, and for helpful discussions.
> However, this does not imply agreement with the author’s interpretations
> (McLaughlin, 1959).
> It is hardly necessary to emphasize in detail the difficulties of
> establishing the correct interpretation of a spectrum which may reflect
> unusual chemical composition, whose features may represent emission,
> absorption, or both mixed, and whose details are too ill-defined to admit
> precise measures of wavelengths (Minkowski, 1963).
Given that our general understanding of blended spectral lines remains in a
continual state of improvement, the frequently recurrent part of “the
supernova problem” is pairing observed features with select elements of the
periodic table (Hummer, 1976; Axelrod, 1980; Jeffery & Branch, 1990; Hatano et
al., 1999b; Branch et al., 2000). In fact, it was not until nearly a half-
century after Minkowski (1963), with the discovery and prompt spectroscopic
follow-up of SN 2011fe (Figure 1 and §4.1) that the loose self-similarity of
SN Ia time series spectra from the perceived beginning of the event to near
maximum light was roundly confirmed (Nugent et al. 2011, see also Garavini et
al. 2005; Foley et al. 2012c; Silverman et al. 2012d; Childress et al. 2013c;
Zheng et al. 2013).
While SN 2011fe may not have revealed a direct confirmation on its progenitor
system (Li et al., 2011b), daily spectroscopic records at optical wavelengths
were finally achieved, establishing the most efficient approach for observing
ill-defined features over time (Pereira et al., 2013). This is important given
that UV to NIR line identifications of all observed complexes are highly time-
dependent, are sensitive to most physically relevant effects, continuously
vary between subtypes, and rely on minimal constraint for all observed
events131313See Foley et al. (2012b) for “The First Maximum-light Ultraviolet
through Near-infrared Spectrum of a Type Ia Supernova.”.
Fig. 2 : Top: A schematic representation of how an assumed spherically sharp
and embedded photosphere amounts to a pure line-resonance P Cygni profile
under the conditions of Sobolev line transfer within a geometry of Absorbing,
Emitting, and Occulted regions of material (Jeffery & Branch, 1990; Branch et
al., 2005). The approximate photospheric velocity, $v_{phot}$, is proportional
to the blue ward shift of an unblended absorption minimum. Bottom: Application
of the above P Cygni diagram to SN Ia spectra in terms of which species
dominate and what other species are known to influence the temporal behavior
(Bongard et al., 2008), each of which are constrainable from complete
spectroscopic coverage. For each series of spectra, the black line in bold
represents maximum light.
Even so, this rarely attainable observing strategy does not necessarily
illuminate nor eliminate all degeneracies in spectral feature interpretations.
However the advantage of complimentary high frequency follow-up observations
is that the spectrum solution associated with any proposed explosion scenario
can at least be consistently tested and constrained by the observed rapid
changes over time (“abundance tomography” goals, e.g., Hauschildt & Baron
1999; Stehle et al. 2005; Sauer et al. 2006; Kasen et al. 2006; Hillier &
Dessart 2012; Pauldrach et al. 2013). It then follows that hundreds of well-
observed spectrophotometric datasets serve to carve out the characteristic
information, $f(\lambda;t)$, for each SN Ia between subtypes, in addition to
establishing the perceived boundaries of the SN Ia diversity problem (see Fig.
11 of Blondin et al. 2012 for this concept at maximum light).
For supernovae in general, Figure 1 also serves as a reminder that all
relative strengths evolve continuously over time, where entire features are
always red-shifting across wavelength (line velocity space) during the rise
and fall in brightness. A corollary of this situation is that prescriptions
for taking measurements of spectroscopic behaviors (whereby interpretations
rely on a subjective “goodness of fit”) and robustly associating with any
number of physical causes do not exist. Instead there are two primary means
for interpreting SN Ia spectra and taking measurements of features for the
purposes of extracting physical properties.
_Indirect_ analysis assumes a detailed explosion model and is primarily tasked
with assessing the accuracy and flaws of the model. _Direct_ analysis seeks to
manually measure via spectrum synthesis where one can either assume an initial
post-explosion ejecta composition _or_ give up abundance information
altogether to assess the associated uncertainties and consequences of
supernova line blending via _purposeful_ high parameterizations. For the
latter of these direct inference methods, the conclusions about spectroscopic
interpretations$-$which are supported by remnants of inconsistencies
throughout the literature$-$are summarized as follows.
For the most part, particularly at early epochs and as far as anyone can tell
with current limiting datasets, the features in SN Ia spectra are due to IMEs
and IPEs formed by resonance scattering of continuum and decay-chain photons,
and have P Cygni-type profiles overall (Pskovskii 1969; Mustel 1971; Branch &
Patchett 1973; Kirshner et al. 1973a; see Figure 2). Emission components peak
at or near the rest wavelength and absorption components are blue-shifted
according to the opacity profile of matter at and above the photospheric line
forming region. The combination of these effects can often lead to “trumped”
emission features (Jeffery & Branch, 1990), giving SN Ia spectra their
familiar shapes.
Essentially all _relevant_ atomic species (isotope plus ionization state) are
present somewhere within the ejecta, each with its own 3-dimensional abundance
profile. At optical wavelengths, conditions and abundance tomographies of the
ejecta maintain the dominance of select singly$-$triply ionized subsets of
C$+$O, IMEs, and IPEs (Hatano et al., 1999b). From shortly after the onset of
the explosion to around the time of maximum light, the optical$-$NIR spectrum
of a normal SN Ia consists of a continuum level with superimposed features
that are primarily consistent with strong permitted lines of ions such as O I,
Mg II, Si II, Si III, S II, Ca II, Fe II, Fe III, and trace signatures of C I
and C II (Branch et al., 2006; Thomas et al., 2007; Bongard et al., 2008;
Nugent et al., 2011; Parrent et al., 2012; Hsiao et al., 2013; Mazzali et al.,
2013; Dessart et al., 2013a). After the pre-maximum light phase, blends of Fe
II (in addition to other IPEs) begin to dominate or influence the temporal
behavior of many optical$-$NIR features over timescales from weeks to months
(see Branch et al. 2008 and references therein).
With the above mentioned approximated view of line formation in mind (Figure
2), the real truth is that the time-dependent state of the ejecta and
radiation field _at all locations_ dictates how the material presence within
the line forming regions will be imparted onto the spectral continuum, i.e.
the radiation field and the matter are said to be “coupled.” With the
additional condition of near-relativistic expansion velocities ($\sim$0.1$c$),
line identifications themselves can also be thought of as coupled to the
abundance tomography of ejected material, which includes the projected Doppler
velocities spanned by the recipe of absorbing material. Subsequently, while
spectra can be used for constraining limits of some model parameters, it comes
with a cost of certainty on account of _natural_ uncertainties imparted by the
large expansion velocities and associated expansion opacities.
As an exercise in this point, in Figure 3 we have constructed an early epoch
set of toy model line profiles that are representative of normal SN Ia line
identification procedures (e.g., Branch et al. 2005; Parrent et al. 2011) and
over-plot them with an early optical$-$NIR spectrum (the observed outermost
layers, sans UV) of SN 2011fe. We summarize the take away points of Figure 3
as follows.
Fig. 3 : SYN++ calculation comparisons to the early optical$-$NIR spectrum of
SN 2011fe (Hsiao et al., 2013; Pereira et al., 2013). Calculations are based
on an optical set of photospheric phase spectra (see Parrent et al. 2012) and
are true-to-scale. Bands of color are intended to show overlap between lines
under the simplified however informative assumption of permitted line
scattering under homologous expansion. Some of the weaker lines have not been
highlighted for clarity.
* •
Even without considering weak contributions, at no place along the (UV$-$)
optical$-$NIR spectrum is any observed feature removed from being due to less
than 2 sources (more precisely, see also Bongard et al. 2008). That is, under
the basic assumptions of pure resonance line scattering and homologous
expansion (Figure 2), all features are complex blends of at least 2$+$ ions
and are universally influenced by multiple regions of emitting and/or
absorbing material (e.g., “high[-er] velocity” and “photospheric velocity”
intervals of material, see also Marion et al. 2013).
* •
For supernovae, the components of the spectrum are most easily constrained via
spectrum synthesis, and subsequently measurable (not the converse), when the
bounds of wavelength coverage, $\lambda$a and $\lambda$b, are between
$\sim$2000$-$3500 and 12000 Å, respectively. If $\lambda$b $<$ 7500$-$9500 Å,
then the velocities and relative strengths of several physically relevant ions
(e.g., C I, O I, Mg II, and Ca II) are said to be devoid of useful constraint
and provide a null (or uncertain) measurement for every other overlapping
spectral line signature (i.e. all features). That is, in order to viably
“identify” and measure a single feature, the entire spectrum must be
reproduced. While empirical measurements of certain absorption features are
extremely useful for identifying trends in the observed behavior of SN Ia,
these methods do not suffice to measure the truest underlying atomic recipe
and its time-dependent behavior, much less the “strength” of contributing
lines (e.g., multiple velocity components of Si II in SN 2012fr, §4.2.2).
Specifically, empirical feature strength measurements at least require a
proper modeling of the non-blackbody, IPE-dominated pseudo continuum level
(Bongard et al., 2008) or the use of standardized relative strength parameters
(e.g., Childress et al. 2013b).
* •
Therefore, as in Figure 2, employing stacked Doppler velocity scaled time
series spectra provides useful and timely first-order comparative estimates
for when (epoch) and where (projected Doppler velocity) contributing ions
appear, disappear, and span as the photospheric region recedes inward over
time.
We speak on this only to point out that even simple questions$-$particularly
for homogeneous SN Ia$-$are awash in detection/non-detection ambiguities.
However, it should be noted that a powerful exercise in testing uncertain line
identifications and resolving complex blends can be done, in part, without the
use of additional synthetic spectrum calculations. That is, by comparing a
single observed spectrum to that of other well-observed SN Ia, where the
analysis of the latter offers a greater context for interpretation than the
single spectrum itself, one can deduce whether or not a “mystery” absorption
feature is common to most SN Ia in general. On the other hand, if a matching
absorption feature is not found, then one can infer the presence of either a
newly identified, compositionally consistent ion or the unblended line of an
already accounted for species (resulting from forbidden line emission, non-LTE
effects, and/or when line strengths or expansion velocities differ between
subtypes). Given also the intrinsic dispersion of expansion opacities between
SN Ia, it is likely that an “unidentified” feature is that of a previously
known ion at higher and/or lower velocities. It is this interplay between
expansion opacities and blended absorption features that keep normal and some
peculiar SN Ia within the description of a homogenous set of objects, however
different they may appear.
In fact, when one compares the time series spectra of a broad sample of SN Ia
subtypes, however blended, there is little room for degeneracy among plausible
ion assignments (sans IPEs, e.g., Fe II versus Cr II during post-maximum
phases). In other words, there exists a unique set of ions, common to most SN
Ia atmospheres, that make up the resulting spectrum, where differences in
subtype are associated with differences in temperature and/or the abundance
tomography of the outermost layers (Tanaka et al., 2008). The atomic species
listed in Figure 3 do not so much represent a complete account of the
composition, or the “correct” answer, as they are consistent with the
subsequent time evolution of the spectrum toward maximum light, and therefore
serve to construct characteristic standards for direct comparative diversity
assessments.
Said another way, it is the full time series dataset that enables the best
initial spectrum solution hypothesis, which can be further tested and refined
for the approximate measurement of SN Ia features (Branch et al., 2007a).
Therefore, this idea of a unique set of ions remains open since$-$with current
limiting datasets$-$species with minimal constraint _and_ competing line
transfer processes can be ambiguously present141414See Fig. 9 of Stritzinger
et al. (2013) to see clear detections of permitted Co II lines in the NIR
spectra of the peculiar and faint SN 2010ae., even for data with an infinite
S/N (i.e. sources with few strong lines, or lines predominately found blue
ward of $\sim$6100 Å, e.g., C III, O III, Si IV, Fe I, Co II, Ni II). One can
still circumvent these uncertainties of direct analysis by either using dense
time series observations (e.g., Parrent et al. 2012) or by ruling out spurious
inferred detections by including adjacent wavelength regions into the
spectroscopic analysis (UV$-$optical$-$NIR; see Foley et al. 2012b; Hsiao et
al. 2013; Mazzali et al. 2013).
### 2.3 Models
A detailed account of SN Ia models is beyond the scope of our general review
of SN Ia spectra (for the latest discussions, see Wang & Han 2012; Nomoto et
al. 2013; Hillebrandt et al. 2013; Calder et al. 2013; Maoz et al. 2013).
However, in order to understand the context by which observations are taken
and synthetic comparisons made, here we only mention the surface layer of
matters relating to observed spectra. For some additional recent modeling
work, see Fryer & Diehl (2008), Bravo et al. (2009), Jordan et al. (2009),
Kromer et al. (2010), Blondin et al. (2011), Hachisu et al. (2012), Jordan et
al. (2012), Pakmor et al. (2013), Seitenzahl et al. (2013), Dan et al. (2013),
Kromer et al. (2013b), Moll et al. (2013), and Raskin et al. (2013).
Realistic models are not yet fully ready because of the complicated physical
conditions in the binary stellar evolution that leads up to an expanding SN Ia
atmosphere. For instance, the explosive conditions of the SN Ia problem take
place over a large dynamic range of relevant length-scales (RWD $\sim$ 1R⊕ and
flame-thicknesses of $\sim$ 0.1 cm; Timmes & Woosley 1992; Gamezo et al.
1999), involve turbulent flames that are fundamentally multi-dimensional
(Khokhlov, 1995, 2000; Reinecke et al., 2002a, b; Gamezo et al., 2003, 2005;
Seitenzahl et al., 2013), and consist of uncertainties in both the detonation
velocity (Domínguez & Khokhlov, 2011) and certain nuclear reaction rates,
especially 12C$+$12C (Bravo et al. 2011, however see also Bravo & Martínez-
Pinedo 2012; Chen et al. 2013).
Most synthetic spectra are angle-averaged representations of higher-
dimensional detailed models. Overall, the observed spectra of normal SN Ia
have differed less amongst themselves than that of some detailed models
compared to the data of _normal_ SN Ia. This is not from a lack of efforts,
but is simply telling of the inherent difficulty of the problem and limiting
assumptions and interests of various calculations. Kasen et al. (2008)
reviewed previous work done of N-dimensional SN Ia models and presented the
first high-resolution 3D calculation of a SN Ia spectrum at maximum light.
Their results are still in a state of infancy, however they represent the
first step toward the ultimate goal of SN Ia modeling, i.e. to trace observed
SN Ia properties and infer the details of the progenitor and its subsequent
disruption by comparing 3D model spectra and light curves of 3D explosion
simulations with the best observed temporal datasets.
Still, progress has been made in understanding general observed properties of
SN Ia and their relation to predictions of simulated explosion models. For
example, one-dimensional (1D) numerical models of SN Ia have been used in the
past to test the possible explosion mechanisms such as subsonic flame or
supersonic detonation models, as well as conjoined delayed-detonations (e.g.,
Arnett 1968; Nomoto et al. 1984; Lentz et al. 2001a). The one-dimensional
models disfavor the route of a pure thermonuclear detonation as the mechanism
to explain most SN Ia events (Hansen & Wheeler, 1969; Arnett, 1969; Axelrod,
1980). Such a mechanism produces mostly 56Ni and almost none of the IMEs
observed in the spectra of all SN Ia (e.g., Branch et al. 1982; Filippenko
1997; Gamezo et al. 1999; Pastorello et al. 2007a).
However, one-dimensional models have shown that a detonation _can_ produce
intermediate mass elements if it propagates through a Chandrasekhar-mass WD
that has pre-expanded during an initial deflagration stage (Khokhlov, 1991a;
Yamaoka et al., 1992; Khokhlov et al., 1993; Arnett & Livne, 1994a, b; Wheeler
et al., 1995; Hoflich et al., 1995; Khokhlov et al., 1997). To their
advantage, these deflagration-to-detonation transition (DDT) and pulsating
delayed-detonation (PDD) models _are_ able to reproduce the observed
characteristics of SN Ia, however not without the use of an artificially-set
transition density between stages of burning (Khokhlov, 1991b; Hoflich et al.,
1995; Lentz et al., 2001a, b; Baron et al., 2008; Bravo et al., 2009; Dessart
et al., 2013a). Subsequently, a bulk of the efforts within the modeling
community has been the pursuit of conditions or mechanisms which cause the
burning front to naturally transition from a sub-sonic deflagration to a
super-sonic detonation, e.g., gravitationally confined detonations (Jordan et
al., 2009), prompt detonations of merging WDs, a.k.a. “peri-mergers” (Moll et
al., 2013).
With the additional possibility that the effectively burned portion of the
progenitor is enclosed or obscured by some body of circumstellar or
envelope/disk of material (see Sternberg et al. 2011; Foley et al. 2012a;
Förster et al. 2012; Scalzo et al. 2012; Raskin & Kasen 2013; Silverman et al.
2013d; Dan et al. 2013; Dessart et al. 2013a; Moll et al. 2013), the
intrinsically multi-dimensional nature of the explosion itself is also
expected to manifest signatures of asymmetric plumes of burned material and
pockets of unburned material within a spheroidal debris field of flexible
asymmetries (see Khokhlov 1995; Niemeyer & Hillebrandt 1995; Gamezo et al.
2004; Wang & Wheeler 2008; Patat et al. 2009; Kasen et al. 2009). Add to this
the degeneracy of SN Ia flux behaviors, i.e. colors are sensitive to dust/CSM
extinction and intrinsic dispersions in the same direction (Tripp & Branch,
1999), whether large or small redshift-color dependencies (Saha et al., 1999;
Jha et al., 1999; Parodi et al., 2000; Wang et al., 2008a; Goobar, 2008; Wang
et al., 2009a; Foley & Kasen, 2011; Mohlabeng & Ralston, 2013), and we find
the true difficulty in constraining SN Ia models.
Blondin et al. (2013) recently presented and discussed the photometric and
spectroscopic properties at maximum light of a sequence of 1D DDT explosion
models, with ranges of synthesized 56Ni masses between 0.18 and 0.81 M⊙. In
addition to showing broad consistencies with the diverse array of observed SN
Ia properties, the synthetic spectra of Blondin et al. (2013) predict weaker
absorption features of unburned oxygen (O I $\lambda$7774) at maximum light,
in proportion to the amount of 56Ni produced. This is to be expected (Hoflich
et al., 1995), however constraints on the remaining amount of unburned
material, in addition to its temporal behavior, are more readily seen during
the earliest epochs (within the outermost layers of ejecta) via C II
$\lambda$6580 and O I $\lambda$7774 (Thomas et al., 2007; Parrent et al.,
2011; Nugent et al., 2011). Consequently, temporal spectrum calculations of
detailed explosion models are needed for the purposes of understanding why the
properties of SN Ia are most divergent well before maximum light (Branch et
al., 2006; Dessart et al., 2013a).
Nucleosynthesis in two-dimensional (2D) delayed detonation models of SN Ia
were explored by Maeda et al. (2010a). In particular, they focused on the
distribution of species in an off-center DDT model and found the abundance
tomography to be stratified, with an inner region of 56Ni surrounded by an
off-center shell of electron-capture elements (e.g., Fe54, Ni58). Later, Maeda
et al. (2010b) investigated the late time emission profiles associated with
this off-center inner-shell of material within several observed SN Ia and
found a correlation between _possible_ nebular-line Doppler shifts along the
line-of-sight and the rate-of-decline of Si II velocities at earlier epochs.
Their interpretation is to suggest that some SN Ia subtypes may represent two
different hemispheres of the “same” SN Ia (LVG vs. HVG subtypes; see §3.2).
Moreover, the findings of Maeda et al. (2010b) and Maund et al. (2010b) remain
largely consistent with the additional early and late time observations of the
well-observed SN 2011fe (Smith et al., 2011; McClelland et al., 2013) and
those of larger SN Ia samples (Blondin et al., 2012; Silverman et al., 2013a).
However, even the results of Maeda et al. (2010b) and others that rely on
spectroscopic measurements at all epochs are not without reservation given
that late time emission profiles are subject to more than line-shifts due to
Doppler velocities and ionization balance (Bongard et al., 2008; Friesen et
al., 2012).
Seitenzahl et al. (2013) presented 14 3-dimensional (3D) high resolution
Chandrasekhar-mass delayed-detonations that produce a range of 56Ni (depending
on the location of ignition points) between $\sim$ 0.3 and 1.1 M⊙. For this
set of models, unburned carbon extends down to 4000 km s-1 while oxygen is not
present below 10,000 km s-1. Seitenzahl et al. (2013) conclude that if
delayed-detonations are to viably produce normal SN Ia brightnesses, the
region of ignition cannot be far off-center so as to avoid the over-production
of 56Ni. As noted by Seitenzahl et al. (2013), these models warrant tests via
spectrum synthesis given their 3D nature and possible predictive relations to
the WLR, spectropolarimetry, and C$+$O “footprints” (Howell et al., 2001;
Baron et al., 2003; Thomas et al., 2007; Wang & Wheeler, 2008).
Dessart et al. (2013a) recently compared synthetic light curves and spectra of
a suite of DDT and PDD models. Based on comparisons to SN 2002bo and SN
2011fe, two SN Ia of different spectroscopic subtypes, and based on poor to
moderate agreement between recent DDT models and observed SN Ia diversity
(Blondin et al., 2011), Dessart et al. (2013a) convincingly argue that these
two SN Ia varieties (LVG vs. HVG, as above) are dissimilar enough to be
explained by different explosion scenarios and/or progenitor systems (Wang et
al., 2013c). For SN Ia in general, delineating spectroscopic diversity has
been a difficult issue (Benetti et al., 2005; Branch et al., 2009), and has
only recently been made clear with the belated release of decades-worth of
unpublished data (Blondin et al., 2012; Silverman et al., 2012c).
## 3 Spectroscopic Diversity of SN Ia
Observationally and particularly at optical wavelengths, SN Ia increase in
brightness over $\sim$13 to 23 days before reaching maximum light
($\overline{t}_{rise}$ = 17.38 $\pm$ 0.17; Hayden et al. 2010b). However, it
is not until $\sim$1 year later that the period of observation is said to be
“complete.” From the time of the explosion our perspective as outside
observers begins at the outermost layers if the SN Ia is caught early enough.
In the approximate sense, this is because the line-forming region (the
“photosphere”) recedes as the ejecta expand outward, which in turn means that
the characteristic information for each explosion mechanism and progenitor
channel is specified by the temporal spectrophotometric attributes of the
“inner” and “outer” layers of freshly synthesized and remaining primordial
material. In addition, because the expanding material cools as it expands, the
net flux of photons samples different layers (of different states and
distributions) over time. And since the density profile of the material
roughly declines from the center outward, significant changes within the
spectra for an individual SN Ia take place daily before or near maximum light,
and weekly to monthly thereafter.
Documenting the breadth of temporal spectroscopic properties for each SN Ia is
not only useful for theoretical purposes, but is also necessary for
efficiently typing and estimating the epoch of newly found possible supernova
candidates before they reach maximum light. Several supernova identification
tools have been made that allow for fair estimates of both subtype and epoch
(e.g., SNID; Blondin & Tonry 2007, Gelato; Harutyunyan et al. 2008, Superfit;
Howell et al. 2005). In addition, the spectroscopic goodness-of-fit methods of
Jeffery et al. (2007) allow one to find the “nearest neighbors” of any
particular SN Ia within a sample of objects, enabling the study of so called
“transitional subtype” SN Ia (those attributed with contrasting
characteristics of two or more subtypes).
### 3.1 Data
Fig. 4 : Continual spectroscopic follow-up efficiencies for the most “well-
observed” SN Ia at early phases (not counting multiple spectra per day). Some
of the values reported may be slightly lower for instances of unpublished
data. Dashed lines represent the upper-limit spectroscopic efficiencies and
peak number of pre-maximum light spectra for one and two day follow-up
cadences during the first 25 days post-explosion. See §3.1.
One of the major limitations of spectroscopic studies has been data quality.
For example, the signal-to-noise ratio, S/N, of a spectrum signifies the
quality across wavelength and is usually moderate to high for high-z events.
Similarly, and at least for low-z SN Ia, there should exist a quantity that
specifies the density of spectra within a time series dataset. We suggest
$\mathcal{S}$/$\mathcal{N}$$\bullet$($\mathcal{P}$) $\equiv$ the number of
_continual_ follow-up spectra / the mean number of nights passed between
exposures $\bullet$ (total number of spectra _prior_ to maximum light). In
Figure 4 we apply this quantity to literature data.
An ideal dataset consisting of 25 spectra during the first 25 days post-
explosion would yield $\mathcal{S}$/$\mathcal{N}$$\bullet$($\mathcal{P}$) = 25
(16) (e.g., SN 2011fe), whereas a dataset of spectra at days $-$12, $-$10,
$-$7, $-$4, $+$0, $+$3, $+$8, $+$21, $+$48, $+$119 (a common occurrence) would
be said to have $\mathcal{S}$/$\mathcal{N}$$\bullet$($\mathcal{P}$) = 3.3 (4)
plus follow-up at days $+$21, $+$48, and $+$119\. By including the total
number of spectra prior to maximum light in parentheses, we are anticipating
those cases where $\mathcal{S}$/$\mathcal{N}$ = 1, but with $\mathcal{P}$ = 3,
e.g., a dataset with days $-$12, $-$9, and $-$6 observed. It may serve a
purpose to also add second and third terms to this quantity that take into
account the number of post-maximum light and late time spectra.
Regardless of moniker and definition, a quantity that specifies the density of
spectra observed during the earliest epochs would aid in determining,
quantitatively, which datasets are most valuable for various SN Ia diversity
studies. Clearly such a high follow-up rate for slow-evolving events (e.g., SN
2009dc) or events caught at maximum light are not as imperative. However, when
SN Ia are found and typed early, a high $\mathcal{S}$/$\mathcal{N}$ ensures no
loss of highly time sensitive information, e.g., when high velocity features
and C$+$O signatures dissipate. Since most datasets are less than ideal for
detailed temporal inspections of many events (by default), astronomers have
instead relied upon comparative studies (§3.2); those that maximize sample
sizes by prioritizing the most commonly available spectroscopic observables,
e.g., line velocities of 6100 Å absorption minima near maximum light.
Another limitation of spectroscopic studies has been the localized release of
all published data. The Online Supernova Spectrum Archive
(SuSpect151515http://suspect.nhn.ou.edu/~suspect/; Richardson et al. 2001)
carried the weight of addressing data foraging during the past decade,
collecting a total of 867 SN Ia spectra (1741 SN spectra in all). Many of
these were either at the request of or donation to SuSpect, while some other
spectra were digitized from original publications in addition to original
photographic plates (Casebeer et al., 1998, 2000). Prior to and concurrent
with SuSpect, D. Jeffery managed a collection of SUpernova spectra PENDing
further analysis
(SUSPEND161616http://nhn.nhn.ou.edu/~jeffery/astro/sne/spectra/spectra.html).
With the growing need for a manageable influx of data, the Weizmann
Interactive Supernova Data Repository
(WISeREP171717www.weizmann.ac.il/astrophysics/wiserep/; Yaron & Gal-Yam 2012)
has since served as a replacement and ideal central data hub, and has
increased the number of SN Ia spectra to 7661 (with 7933 publicly available SN
spectra out of 13,334 in all). We encourage all groups to upload published
data to WISeREP, whether or not made available elsewhere.
#### 3.1.1 Samples
By far the largest data releases occurred during the past five years, and are
available on WISeREP and their affiliated archives. Matheson et al. (2008) and
Blondin et al. (2012) presented 2603 optical spectra ($\sim$3700$-$7500 Å on
average) of 462 nearby SN Ia ($\tilde{z}$ = 0.02; $\sim$ 85 Mpc) obtained by
the Center for Astrophysics (CfA) SN group with the F. L. Whipple Observatory
from 1993 to 2008. They note that, of the SN Ia with more than two spectra,
313 SN Ia have eight spectra on average. Silverman et al. (2012a) and the
Berkeley SuperNova Ia Program (BSNIP) presented 1298 optical spectra
($\sim$3300$-$10,400 Å on average) of 582 low-redshift SN Ia (z $<$ 0.2;
$\sim$ 800 Mpc) observed from 1989 to 2008. Their dataset includes spectra of
nearly 90 spectroscopically peculiar SN Ia. Folatelli et al. (2013) released
569 optical spectra of 93 low-redshift SN Ia ($\tilde{z}$ $\sim$ 0.04; $\sim$
170 Mpc) obtained by the Carnegie Supernova Project (CSP) between 2004 and
2009. Notably, 72 CSP SN Ia have spectra earlier than 5 days prior to maximum
light, however only three SN Ia have spectra as early as day $-$12.
These samples provide a substantial improvement and crux by which to explore
particular issues of SN Ia diversity. However, the remaining limitation is
that our routine data collection efforts continue to yield several thousand SN
Ia with few to several spectra by which to dissect and compare SN Ia
atmospheres.
#### 3.1.2 Comparisons of “Well-Observed” SN Ia
Fig. 5 : Early pre-maximum light, rest frame optical spectra of some of the
most well-observed and often referenced SN Ia are plotted, loosely in order of
increasing $\Delta$m15(_B_) (top-down). Subtypes shown include bright SN
2006gz, 2009dc-like super-Chandrasekhar candidate (SCC; purple), high-
ionization, shallow-silicon SN 1991T-like (SS; red), normal SN 1994D, 2005cf,
2011fe-like (CN; black), broad-lined SN 1984A, 2002bo-like (BL; green), and
sub-luminous, low-ionization SN 1991bg, 2004eo-like (CL; blue) SN Ia. The
horizontal dashed lines represent our normalization bounds that were applied
to each spectrum. This ensures a fair comparison of all relevant spectroscopic
features, sans continuum differences. For the SS SN Ia, in Figure 5 and Figure
6 only, we have normalized to the Fe III feature as indicated. For the
purposes of this review, we have only included SN Ia that have received
particular attention within the literature (see §4 and the Appendix). Many
other time series observations can be found in Matheson et al. (2008),
Silverman et al. (2012c), and Blondin et al. (2012). The peculiar PTF09dav is
shown in Figure 8 for comparison, as it is not a prototypical SN Ia, however
appearing similar to SN 1991bg-like events (Sullivan et al., 2010; Kasliwal et
al., 2012). Fig. 6 : 1-week pre-maximum light optical spectroscopic
comparisons. See Figure 5 caption. Fig. 7 : Maximum light optical
spectroscopic comparisons. See Figure 5 caption. Fig. 8 : 1-week post-maximum
light optical spectroscopic comparisons. See Figure 5 caption. Fig. 9 : two
weeks post-maximum light optical spectroscopic comparisons. See Figure 5
caption. Fig. 10 : 1-month post-maximum light optical spectroscopic
comparisons. See Figure 5 caption. Fig. 11 : 2-months post-maximum light
optical spectroscopic comparisons. See Figure 5 caption. Fig. 12 : 100$+$
days post-maximum light optical spectroscopic comparisons. See Figure 5
caption. Fig. 13 : Late-time optical spectroscopic comparisons. See Figure 5
caption.
Given that both quantitative and qualitative spectrum comparisons are at the
heart of SN Ia diversity studies, in Figure 5 \- Figure 13 we plot
spectroscopic temporal snapshots for as many “well-observed” SN Ia as are
currently available on WISeREP (Tables 1 and 2). Because the decline
parameter, $\Delta$m15(_B_), remains a useful parameter for probing
differences of synthesized 56Ni mass, properties of the ejecta, limits of CSM
interaction, etc., we have _loosely_ ordered the spectra with increasing
$\Delta$m15(_B_) (top-down) based on average values found throughout the
literature (Tables 3$-$6) and M${}_{\emph{B}}$(peak) considerations for cases
that are reported as having the same $\Delta$m15(_B_). The spectra have been
normalized with respect to 6100 Å line profiles in order to amplify relative
strengths of the remaining features (see caption of Figure 5). We also denote
the spectroscopic subtype for each object in color in order to show the
overlap of these properties between particular SN Ia subclasses (see §3.2 and
Blondin et al. 2012).
By inspection, the collected spectra show how altogether different and similar
SN Ia (both odd and normal varieties) have come to be since nearly 32 years
ago. With regard to the recent modeling of Blondin et al. (2013) and their
accompanying synthetic spectra, we plot the spectra in Figure 5 \- Figure 11
in the flux-representation of $\lambda^{2}$Fλ for ease of future comparisons.
These juxtapositions should reveal the severity of the SN Ia diversity problem
as well as the future of promising studies and work that lie ahead.
### 3.2 Deciphering 21st Century SN Ia Subtypes
Observationally, the whole of SN Ia are hetero-, homogeneous events (Oke &
Searle, 1974; Filippenko, 1997); some of the observed differences in their
spectra are clear, while other suspected differences are small enough to fall
below associable certainty. Because of this, observational studies have
concentrated on quantitatively organizing a mapping between the most peculiar
and normal events. In this section our aim is to review SN Ia subtypes. In
all, three observational classification schemes will be discussed (Benetti et
al., 2005; Branch et al., 2006; Wang et al., 2009a), as well as the recent
additions of so-called over- and sub-luminous events (see Scalzo et al. 2012,
Foley et al. 2013, Silverman et al. 2013d and references therein). For other
relatively new and truly peculiar subclasses of supernova transients, we refer
the reader to Shen et al. (2010), Kasliwal et al. (2012) and references
therein.
#### 3.2.1 Benetti et al. (2005) Classification
Understanding the origin of the WLR is a key issue for understanding the
diversity of SN Ia light curves and spectra, as well as their use as
cosmological distance indicators. Brighter SN Ia with broader light curves
tend to occur in late-type spiral galaxies, while dimmer, faster declining SN
Ia are preferentially located in an older stellar population and thus the age
and/or metallicity of the progenitor system may be relevant factors affecting
SN Ia properties (Hamuy et al. 1995; Howell 2001; Pan et al. 2013, see also
Hicken et al. (2009a)).
With this in mind, Benetti et al. (2005) studied the observational properties
of 26 well-observed SN Ia (e.g., SN 1984A, 1991T, 1991bg, 1994D) with the
intent of exploring SN Ia diversity. Based on the observed projected Doppler
velocity evolution from the spectra181818The velocity gradient$-$the mean
velocity decline rate $\Delta$v/$\Delta$t$-$of a particular absorption minimum
(e.g., v̇${}_{Si\ II}$) has been redefined to be measured over a fixed phase
range [t0, t1] (Blondin et al., 2012)., in conjunction with characteristics of
the light curve (MB, $\Delta$m15), Benetti et al. (2005) considered three
different groups of SN Ia: (1) “FAINT” SN 1991bg-likes, (2) “low velocity
gradient” (LVG) SN 1991T/1994D-likes, and (3) “high velocity gradient” (HVG)
SN 1984A-like events. The velocity gradient here is based on the time-
evolution of 6100 (“6150”) Å absorption minima as inferred from Si II
$\lambda$6355 line velocities. Overall, HVG SN Ia have higher mean expansion
velocities than FAINT and LVG SN Ia, while LVG SN Ia are brighter than FAINT
and HVG SN Ia on average (Silverman et al., 2012b; Blondin et al., 2012).
Given an apparent separation of SN Ia subgroups from this sample of 26
objects, Benetti et al. (2005) considered it as evidence that LVG, HVG, and
FAINT classifications signify three distinct kinds of SN Ia.
#### 3.2.2 Branch et al. (2006) Classification
Branch et al. (2005, 2006, 2007b, 2008, 2009) published a series of papers
based on systematic, comprehensive, and comparative direct analysis of normal
and peculiar SN Ia spectra at various epochs with the parameterized supernova
synthetic spectrum code, SYNOW191919SYNOW is a simplified spectrum synthesis
code used for the timely determination and measurement of all absorption
features complexes. SYNOW has been updated (SYN++) and can be used as an
automated spectrum fitter (SYNAPPS; see Thomas et al. 2011a and
https://c3.lbl.gov/es/). (Fisher, 2000; Branch et al., 2007a). From the
systematic analysis of 26 spectra of SN 1994D, Branch et al. (2005) infer a
compositional structure that is radially stratified, overall. In addition,
several features are consistent with being due to permitted lines well into
the late post-maximum phases ($\sim$120 days, see Branch et al. 2008; Friesen
et al. 2012). Another highlight of this work is that, barring the usual short
comings of the model, SYNOW is shown to provide a necessary consistency in the
_direct_ quantification of spectroscopic diversity (Branch et al., 2007a).
Consequently, the SYNOW model has been useful for assessing the basic limits
of a spectroscopic “goodness of fit” (Figure 3), with room for clear and
obvious improvements (Friesen et al., 2012).
In their second paper of the series on comparative direct analysis of SN Ia
spectra, Branch et al. (2006) studied the spectra of 24 SN Ia close to maximum
light. Based on empirical pEW measurements of 5750, 6100 Å absorption
features, in addition to spectroscopic modeling with SYNOW, Branch et al.
(2006) organized SN Ia diversity by four spectroscopic patterns: (1) “Core-
Normal” (CN) SN 1994D-likes, (2) “Broad-line” (BL), where one of the most
extreme cases is SN 1984A, (3) “Cool” (CL) SN 1991bg-likes, and (4) “Shallow-
Silicon” (SS) SN 1991T-likes. In this manner, a particular SN Ia is defined by
its spectroscopic similarity to one or more SN Ia prototype via 5750, 6100 Å
features. These spectroscopic subclasses also materialized from analysis of
pre-maximum light spectra (Branch et al., 2007b).
The overlap between both Benetti et al. (2005) and Branch et al. (2006)
classifications schemes comes by comparing Table 1 in Benetti et al. (2005) to
Table 1 of Branch et al. (2006), and it reveals the following SN Ia
descriptors: HVG$-$BL, LVG$-$CN, LVG$-$SS, and FAINT$-$CL. This holds true
throughout the subsequent literature (Branch et al., 2009; Folatelli et al.,
2012; Blondin et al., 2012; Silverman et al., 2012b).
In contrast with Benetti et al. (2005) who interpreted FAINT, LVG, and HVG to
correspond to the “discrete grouping” of _distinctly separate_ SN Ia origins
among these subtypes, Branch et al. (2006) found a continuous distribution of
properties between the four subclasses defined above. We should point out that
this classification scheme of Branch et al. (2006) is primarily tied to the
notion that SN Ia spectroscopic diversity is related to the temperature
sequence found by Nugent et al. 1995. That is, despite the contrast with
Benetti et al. (2005) (continuous versus discrete subgrouping of SN Ia), so
far these classifications say more about the state of the ejecta than the
various number of possible progenitor systems and/or explosion mechanisms (see
also Dessart et al. 2013a). Furthermore, the existence of “transitional”
subtype events support this notion (e.g., SN 2004eo, 2006bt, 2009ig, 2001ay,
and PTF10ops; see appendix).
Branch et al. (2009) later analyzed a larger sample of SN Ia spectra. They
found that SN 1991bg-likes are not a physically distinct subgroup (Doull &
Baron, 2011), and that there are probably many SN 1999aa-like events (A.5)
that similarly may not constitute a physically distinct variety of SN Ia.
With regard to the fainter variety of SN Ia, Doull & Baron (2011) made
detailed comparative analysis of spectra of peculiar SN 1991bg-likes. They
also studied the intermediates, such as SN 2004eo (A.23), and discussed the
spectroscopic subgroup distribution of SN Ia. The CL SN Ia are dim, undergo a
rapid decline in luminosity, and produce significantly less 56Ni than normal
SN Ia. They also have an unusually deep and wide trough in their spectra
around 4200 Å suspected as due to Ti II (Filippenko et al., 1992b), in
addition to a relatively strong 5750 Å absorption (due to more than Si II
$\lambda$5972; see Bongard et al. 2008). Doull & Baron (2011) analyzed the
spectra of SN 1991bg, 1997cn, 1999by, and 2005bl using SYNOW, and found this
group of SN Ia to be fairly homogeneous, with many of the blue spectral
features well fit by Fe II.
#### 3.2.3 Wang et al. (2009a) Classification
Based on the maximum light expansion velocities inferred from Si II
$\lambda$6355 absorption minimum line velocities, Wang et al. (2009a) studied
158 SN Ia, separating them into two groups called “high velocity” (HV) and
“normal velocity” (NV). This classification scheme is similar to those
previous of Benetti et al. (2005) and Branch et al. (2006), where NV and HV SN
Ia are akin to LVG$-$CN and HVG$-$BL SN Ia, respectively. That is, while the
subtype notations differ among authors, memberships between these
classification schemes are roughly equivalent (apart from outliers such as the
HV-CN SN 2009ig, see Blondin et al. 2012).
Explicitly, Benetti et al. (2005) and Wang et al. (2009a) subclassifications
are based on empirically estimated mean expansion velocities near maximum
light ($\pm$4 days; $\pm$500 $-$ 2000 km s-1) of 6100 Å features produced by
an assumed single broad component of Si II. The notion of a single
photospheric layer, much less a single-epoch snapshot, does not realistically
account for the multilayered nature of spectrum formation (Bongard et al.,
2008), its subsequent evolution post-maximum light (Patat et al., 1996; Scalzo
et al., 2012), and potential relations to line-of-sight considerations (Maeda
et al., 2010b; Blondin et al., 2011; Moll et al., 2013). In the strictest
sense of SN Ia sub-classification, “normal” refers to _both_ of these subtypes
since they differ foremost by a continuum of inferred mean expansion
velocities and the extent of expansion opacities, simultaneously.
Furthermore, note from a sample of 13 LVG and 8 HVG SN Ia that Benetti et al.
(2005) found 10 $\lesssim$ v̇${}_{Si\ II}$ (km s-1 day-1) $\lesssim$ 67
($\pm$7) and 75 $\lesssim$ v̇${}_{Si\ II}$ $\lesssim$ 125 ($\pm$20) for each,
respectively. Similarly, and from a sample of 14 LVG and 29 HVG SN Ia,
Silverman et al. (2012b) report that 10 $\lesssim$ v̇${}_{Si\ II}$ $\lesssim$
445 ($\pm$50) and 15 $\lesssim$ v̇${}_{Si\ II}$ $\lesssim$ 290 ($\pm$140) for
LVG and HVG events, respectively. Additionally, the pEW measurements of 5750,
6100 Å absorption features (among others) are seen to share a common
convergence in observed values (Branch et al., 2006; Hachinger et al., 2006;
Blondin et al., 2012; Silverman et al., 2012b). The continually consistent
overlap between the measured properties for these two SN Ia “subtypes” implies
that the notion of a characteristic separation value for v̇${}_{Si\ II}$
$\sim$ 70 km s-1 day-1 (including the inferred maximum light separation
velocity, v0 $\gtrsim$ 12,000 km s-1) is still devoid of any physical
significance beyond overlapping bimodal distributions of LVG$-$CN and HVG$-$BL
SN Ia properties (see §5.3 of Silverman et al. 2012b, §5.2 of Blondin et al.
2012, and Silverman et al. 2012a). Rather, a continuum of _empirically
measured_ properties exists between the extremities of these two particular
_historically-based_ SN Ia classes (e.g., SN 1984A and 1994D). Given also the
natural likelihood for a physical continuum between NV and HV subgroups,
considerable care needs to be taken when concluding on underlying connections
to progenitor systems from under-observed, early epoch snapshots of blended
6100 Å absorption minima.
Hence, the primary obstacle within SN Ia diversity studies has been that it is
not yet clear if the expanse of all observed characteristics of each subtype
has been fully charted. For the observed properties of normal SN Ia, it is at
least true that v̇${}_{Si\ II}$ resides between 10$-$445 km s-1 day-1, with a
median value of $\sim$ 60$-$120 km s-1 day-1 (Benetti et al., 2005; Blondin et
al., 2012; Silverman et al., 2012a), while the rise to peak _B_ -band
brightness ranges from 16.3 to 19 days (Ganeshalingam et al., 2011; Mazzali et
al., 2013).
Recently, Wang et al. (2013c) applied this NV and HV subgrouping to 123
“Branch normal” SN Ia with known positions within their host galaxies and
report that HV SN Ia more often inhabit the central and brighter regions of
their hosts than NV SN Ia. This appears to suggest that a supernova with
“higher velocities at maximum light” is primarily a consequence of a
progenitor with larger than solar metallicities, or that PDD/HVG SN Ia are
primarily found within the galactic distribution of DDT/LVG SN Ia (c.f.
Blondin et al. 2011, 2012; Dessart et al. 2013a). This is seemingly in
contrast to interpretations of Maeda et al. (2010b) who propose, based on both
early epoch and late time considerations, that LVG and HVG SN Ia are possibly
one in the same event where the LVG-to-HVG transition is ascribed to an off-
center ignition.
While it is true that increasing the C$+$O layer metallicity can affect the
blueshift of the 6100 Å absorption feature$-$in addition to lower temperatures
and increased UV line-blocking$-$this is not primarily responsible for the
shift in 6100 Å absorption minima (Lentz et al., 2001a, b), where the
dependence of this effect is not easily decoupled from changes in the
temperature structure (Lentz et al., 2000). However, it is also worthwhile to
point out that, while the early epoch spectra of SN 2011fe (a NV event) are
consistent with a DDT-like composition with a sub-solar C$+$O layer
metallicity (“W7+,” Mazzali et al. 2013) _and_ a PDD-like composition (Dessart
et al., 2013a), the outermost layers of SN 2010jn (a HV event; A.41) are
practically void of unburned material and subsequently already overabundant in
synthesized metals for progenitor metallicity to be well determined (Hachinger
et al., 2013). Therefore, discrepancies between NV and HV SN Ia must still be
largely dependent on more than a single parameter, e.g. differences in
explosion mechanisms (Dessart et al., 2013a; Moll et al., 2013), where
progenitor metallicity is likely to be only one of several factors influencing
the dispersions of each subgroup (Lentz et al., 2000; Höflich et al., 2010;
Wang et al., 2012).
It should be acknowledged again that metallicity-dependent aspects of stellar
evolution are expected to contribute, in part, to the underlying variance of
holistic SN Ia characteristics. However thus far, the seen discrepancies from
metallicities share similarly uncertain degrees of influence as for asymmetry
and line-of-sight considerations of ejecta-CSM interactions for a wide variety
of SN Ia (Lentz et al., 2000; Kasen et al., 2003; Leloudas et al., 2013).
Similar to this route of interpretation for SN Ia subtypes are active galactic
nuclei and the significance of the broad absorption line quasi-stellar objects
(BALQSOs, see de Kool & Begelman 1995; Becker et al. 1997; Elvis 2000; Branch
et al. 2002a; Hamann & Sabra 2004; Casebeer et al. 2008; Leighly et al. 2009;
Elvis 2012).
#### 3.2.4 Additional Peculiar SN Ia Subtypes
Spectroscopically akin to some luminous SS SN Ia are a growing group of events
thought to be “twice as massive,” aka super-Chandrasekhar candidates (SCC,
Howell et al. 2006; Jeffery et al. 2006; Hillebrandt et al. 2007; Hicken et
al. 2007; Maeda et al. 2009; Chen & Li 2009; Yamanaka et al. 2009a; Scalzo et
al. 2010; Tanaka et al. 2010; Yuan et al. 2010; Silverman et al. 2011;
Taubenberger et al. 2011; Kamiya et al. 2012; Scalzo et al. 2012; Hachinger et
al. 2012; Yamanaka et al. 2013). Little is known about this particular class
of over-luminous events, which is partly due to there having been only a
handful of events studied. Thus far, SCC SN Ia are associated with metal-poor
environments (Childress et al., 2011; Khan et al., 2011a). Spectroscopically,
the differences that set these events apart from normal SN Ia are fairly weak
Si II/Ca II signatures and strong C II absorption features relative to the
strength of Si II lines. Most other features are comparable in relative
strengths to those of normal SN Ia, if not muted by either top-lighting or
effects of CSM interaction (Branch et al., 2000; Leloudas et al., 2013), and
are less blended overall due to lower mean expansion velocities. In addition,
there is little evidence to suggest that SCC SN Ia spectra consist of
contributions from physically separate high velocity regions of material
($\gtrsim$ 4000 km s-1 above photospheric). This range of low expansion
velocities ($\sim$5000$-$18,000 km s-1), in conjunction with larger than
normal C II absorption signatures, are difficult to explain with some MCh
explosion models (Scalzo et al. 2012; Kamiya et al. 2012, however see also
Hachisu et al. 2012; Dessart et al. 2013a; Moll et al. 2013 for related
discussions).
Silverman et al. (2013d) recently searched the BSNIP and PTF datasets, in
addition to the literature sample, and compiled a list of 16 strongly CSM
interacting SN Ia (referred to as “Ia-CSM” events). These supernovae obtain
their name from a conspicuous signature of narrow hydrogen emission atop a
weaker hydrogen P Cygni profile that together are superimposed on a loosely
identifiable SS-like SN Ia spectrum (Aldering et al., 2006; Prieto et al.,
2007; Leloudas et al., 2013). Apart from exhibiting similar properties to the
recent PTF11kx (§4.5.1) and SN 2005gj (§4.5.3), Silverman et al. (2013d) find
that SN Ia-CSM have a range of peak absolute magnitudes ($-$21.3 $\leq$
M${}_{\emph{R}}$ $\leq$ $-$19), are a spectroscopically homogenous class, and
all reside in late-type spiral and irregular host-galaxies.
As for peculiar sub-luminous events, Narayan et al. (2011) and Foley et al.
(2013) discussed the heterogeneity of the SN 2002cx-like subclass of SN Ia.
Consisting of around 25 members spectroscopically similar to SN 2002cx (Li et
al., 2003), these new events generally have lower maximum light velocities
spanning from 2000 to 8000 km s-1 and a range of peak luminosities that are
typically lower than those of FAINT SN Ia ($-$14.2 to $-$18.9). In addition,
this class of objects have “hot” temperature structures and$-$in contrast to
SN Ia that follow the WLR$-$have low luminosities for their light curve shape.
This suggests a distinct origin, such as a failed deflagration of a C$+$O
white dwarf (Foley et al., 2009; Jordan et al., 2012; Kromer et al., 2013a) or
double detonations of a sub-Chandrasekhar mass white dwarf with non-degenerate
helium star companion (Fink et al., 2010; Sim et al., 2012; Wang et al.,
2013a). It is estimated that for every 100 SN Ia, there are 31^$+$17_$-$13
peculiar SN 2002cx-like objects in a given volume (Foley et al., 2013).
#### 3.2.5 SN Ia Subtype Summary
Fig. 14 : Top: Peak absolute _B_ -band magnitudes versus $\Delta$m15(_B_) for
most well-observed SN Ia found in the literature. Additional data (grey) taken
from Folatelli et al. (2012), Blondin et al. (2012), and additional points
discussed in Pakmor et al. (2013). Bottom: Expansion velocities at maximum
light ($\pm$ 3 days; via Si II $\lambda$6355 line velocities) versus
$\Delta$m15(_B_). All subtypes have been tagged in accordance with the same
color-scheme as in Figure 5 \- Figure 13. Included for reference are the
brightest, peculiar SN 2002cx-likes (light blue circles). Outliers for each
subtype have been labeled for clarity and reference. We also plot mean values
for the SCC and CL subtypes (larger circles), and include the mean values for
SS, CN, and BL SN Ia (large diamonds) as reported by Blondin et al. (2012).
In Figure 14 we plot average literature values of M${}_{\emph{B}}$(peak),
$\Delta$m15(_B_), and $\mathcal{V}$peak(Si II $\lambda$6355) versus one
another for all known SN Ia subtypes. For M${}_{\emph{B}}$(peak) versus
$\Delta$m15(_B_), the WLR is apparent. We have included the brightest SN
2002cx-likes (Foley et al., 2013) for reference, as these events are suspected
as having separate origins from the bulk of normal SN Ia (Hillebrandt et al.,
2013). We have not included Ia-CSM events given that estimates of expansion
velocities and luminosities, without detailed modeling, are obscured by CSM
interaction. However, it suffices to say for Figure 14 that Ia-CSM are nearest
to SS and SCC SN Ia in both projected Doppler velocities and peak
M${}_{\emph{R}}$ brightness (Silverman et al., 2013d). At a separate end of
these SN Ia diversity plane(s), $\mathcal{V}$peak(Si II $\lambda$6355) versus
$\Delta$m15(_B_) further separates FAINT$-$CL SN Ia and peculiar events away
from the pattern between SCC/SN 1991T-like over-luminous SN Ia and normal
subtypes, where the former tend to be slow-decliners (i.e. typically brighter)
with slower average velocities.
To summarize the full extent of SN Ia subtypes in terms of the qualitative
luminosity and expansion velocity patterns, in Figure 15 we have outlined how
SN Ia relate to one another thus far (for quantitative assessments, see
Blondin et al. 2012; Silverman et al. 2012b; Folatelli et al. 2013). Broadly
speaking, the red ward evolution of SN Ia features span low to high rates of
decline for a large range of luminosities. Shallow Silicon and Super-
Chandrasekhar Candidate SN Ia are by far the brightest, while Ia-CSM SN
exhibit bright H$\alpha$ emission features. These “brightest” SN Ia also show
low to moderate expansion velocities and v̇${}_{Si\ II}$. From BL to CN to
SS/SSC SN Ia, mean peak absolute brightnesses scale up with an overall
decrease in maximum light line velocities. Meanwhile, CL SN Ia fall between
low velocity and high velocity gradients, but lean toward HVG SN Ia in terms
of their photospheric velocity evolution. Comparatively, peculiar SN 2002cx-
like and other sub-luminous events are by far the largest group of
thermonuclear outliers.
Obtaining observations of SN Ia that lie outside the statistical norm is
important for gauging the largest degree by which SN Ia properties diverge in
nature. However, just as imperative for the cause remains filling the gaps of
observed SN Ia properties (e.g., v̇${}_{Si\ II}$, vneb, vC(t), vCa(t),
M${}_{\emph{B}}$(peak), $\Delta$m15(_B_ -band), trise, color evolution) with
well-observed SN Ia. This is especially true for those SN Ia most similar to
one another, aka “nearest neighbors” (Jeffery et al., 2007), and transitional-
type SN Ia.
Fig. 15 : Subtype reference diagram. Dashed lines denote an open transitional
boundary between adjacent spectroscopic subtypes.
### 3.3 Signatures of C$+$O Progenitor Material
If the primary star of most SN Ia is a C$+$O WD, and if the observed range of
SN Ia properties is primarily due to variances in the ejected mass or
abundances of material synthesized in the explosion (e.g., 56Ni), then this
should also be reflected in the remaining amount of carbon and oxygen if MCh
is a constant parameter (see Maeda et al. 2010a; Blondin et al. 2013; Dessart
et al. 2013a). On the other hand, if one assumes that the progenitor system is
the merger of two stars (Webbink, 1984; Iben & Tutukov, 1984; Pakmor et al.,
2013; Moll et al., 2013) or a rapidly rotating WD (Hachisu et al.,
2012)$-$both of which are effectively obscured by an amorphous region and/or
disk of C$+$O material$-$then the properties of C and O absorption features
will be sensitive to the interplay between ejecta and the remaining unburned
envelope (see Livio & Pringle 2011).
Oxygen absorption features (unburned plus burned ejecta) are often present as
O I $\lambda$7774 in the pre-maximum spectra of SN Ia (Figure 5). They may
exhibit similar behavior to those seen in SN 2011fe (§4.1), however current
datasets lack the proper temporal coverage of a large sample of events that
would be necessary to confirm such claims. Still, comparisons of the blue-most
wing in the earliest spectra of many SN Ia to that of SN 2009ig (§4.2.1),
2010jn (A.41), 2011fe (§4.1), 2012cg (A.43), and 2012fr (§4.2.2) may reveal
some indication of HV O I if present and if caught early enough (e.g., SN
1994D; Branch et al. 2005).
Spectroscopic detections of carbon-rich material have been documented since
the discovery of SN 1990N (see Leibundgut et al. 1991; Jeffery et al. 1992;
Branch et al. 2007b; Tanaka et al. 2008) and have been primarily detected as
singly ionized in the optical spectra of LVG$-$CN SN Ia (Parrent et al.,
2011). However, NIR spectra of some SN Ia subtypes have been suspected of
harboring C I absorption features (Höflich et al. 2002; Hsiao et al. 2013, see
also Marion et al. 2006, 2009a), while C III has been tentatively identified
in “hotter” SS/SN 1991T-like SN Ia (Hatano et al., 2002; Garavini et al.,
2004; Chornock et al., 2006).
Observations of the over-luminous SCC SN 2003fg suggested the presence of a
larger than normal C II $\lambda$6580 absorption signature (Howell et al.,
2006). Later in 2006, with the detection of a _conspicuous_ C II $\lambda$6580
absorption “notch” in the early epoch observations of the normal SN Ia, SN
2006D, Thomas et al. (2007) reconsidered the question of whether or not
spectroscopic signatures of carbon were a ubiquitous property of all or at
least some SN Ia subtypes.
As follow-up investigations, Parrent et al. (2011) and Folatelli et al. (2012)
presented studies of carbon features in SN Ia spectra, particularly those of C
II $\lambda\lambda$6580, 7234 (which are easier to confirm than
$\lambda\lambda$4267, 4745). However weak, conspicuous 6300 Å absorption
features were reported in several SN Ia spectra obtained during the pre-
maximum phase. It was shown that most of the objects that exhibit clear
signatures are of the LVG$-$CN SN Ia subtype, while HVG$-$BL SN Ia may either
be void of conspicuous signatures due to severe line blending, or lack carbon
altogether, the latter of which is consistent with DDT models (e.g., Hachinger
et al. 2013) and could also be partially due to increased progenitor
metallicities (Lentz et al., 2000; Meng et al., 2011; Milne et al., 2013).
This requires further study and spectrum synthesis from detailed models.
Thomas et al. (2011b) presented additional evidence of unburned carbon at
photospheric velocities from observations of 5 SN Ia obtained by the Nearby
Supernova Factory. Detections were based on the presence of relatively strong
C II 6300 Å absorption signatures in multiple spectra of each SN, supported by
automated fitting with the SYNAPPS code (Thomas et al., 2011a). They estimated
that at least 22^+10_-6% of SN Ia exhibit spectroscopic C II signatures as
late as day $-$5, i.e. carbon features, whether or not present in all SN Ia,
are not often seen even as early as day $-$5.
Folatelli et al. (2012) later searched through the Carnegie Supernova Project
(CSP) sample and found at least 30% of the objects show an absorption feature
that can be attributed to C II $\lambda$6580\. Silverman & Filippenko (2012)
searched for carbon in the BSNIP sample and found that $\sim$ 11% of the SN Ia
studied show carbon absorption features, while $\sim$ 25% show some indication
of weak 6300 Å absorption. From their sample, they find that if the spectra of
SN Ia are obtained before day $-$5, then the detection percentage is higher
than $\sim$ 30%. Recently it has also been confirmed that “carbon-positive” SN
Ia tend to have bluer near-UV colors than those without conspicuous C II
$\lambda$6580 signatures (Thomas et al., 2011b; Silverman & Filippenko, 2012;
Milne et al., 2013).
Silverman & Filippenko (2012) estimate the range of carbon masses in normal SN
Ia ejecta to be (2 $-$ 30) x 10-3 M⊙. For SN 2006D, Thomas et al. 2007
estimated 0.007 M⊙ of carbon between 10,000 and 14,000 km s-1 as a lower
limit. Thomas et al. 2007 also note that the most vigorous model of Röpke et
al. (2006) left behind 0.085 M⊙ of carbon in the same velocity interval.
However, we are not aware of any subsequent spectrum synthesis for this
particular model that details the state of an associated 6300 Å signature.
In the recent detailed study on SN Ia spectroscopic diversity, Blondin et al.
(2012) searched for signatures of C II $\lambda$6580 in a sample of 2603
spectra of 462 nearby SN Ia and found 23 additional “carbon-positive” SN Ia.
Given that seven of the nine CN SN Ia reported by Blondin et al. (2012) with
spectra prior to day $-$10 clearly exhibit signatures of C II, and that
$\sim$30$-$40% of the SN Ia within their sample are of the CN subtype, it is
likely that at least 30$-$40% of all SN Ia leave behind some amount of carbon-
rich material, spanning velocities between 8000 $-$ 18,000 km s-1 (Parrent et
al., 2011; Pereira et al., 2013; Cartier et al., 2013).
Considering the volume-limited percentage of Branch normal SN Ia estimated by
Li et al. (2011c), roughly 50% or more are expected to contain detectable
carbon-rich material in the outermost layers. If this is true, then it implies
that explosion scenarios that do not naturally leave behind at least a
detectable amount (pEWs $\sim$ 5$-$25 Å) of unprocessed carbon can only
explain half of all SN Ia or less (sans considerations of Ia-CSM progenitors,
subtype-ejecta hemisphere dualities, and effects of varying metallicities; see
below).
Historically, time-evolving signatures of C II $\lambda\lambda$6580, 7234 from
the _computed spectra of some detailed models_ have not revealed themselves to
be consistent with the current interpretations of the observations. This could
be due to an inadequate lower-extent of carbon within the models (Thomas et
al., 2007; Parrent et al., 2011; Blondin et al., 2012) or the limits of the
resolution for the computed spectra (Blondin et al., 2011).
It should be noted that 6300 Å features _are_ present in the non-LTE pre-
maximum light spectra of Lentz et al. (2000) who assessed metallicity effects
on the spectrum for a pure deflagration model (see their Fig. 7). Overall,
Lentz et al. (2000) find that an increase in C$+$O layer metallicities results
in a decreased flux (primarily UV) in addition to a blue ward shift of
absorption minima (primarily the Si II 6100 Å feature). While Lentz et al.
(2000) did not discuss whether or not the weak 6300 Å absorption signatures
are due to C II $\lambda$6580, it is likely the case given that an increase in
C$+$O layer metallicities is responsible for the seen decrease in the strength
of the 6300 Å feature. However, it should be emphasized that the “strength” of
this supposed C II $\lambda$6580 feature appears to be a consequence of how
Lentz et al. (2000) renormalized abundances for metallicity enhancements in
the C$+$O layer. In other words, even though the preponderance of normal SN Ia
with detectable C II $\lambda$6580 notches are of the NV subgroup, the fact
that HV SN Ia are thus far void of 6300 Å notches _does not_ imply robust
consistency with the idea that nearest neighbor HV SN Ia properties are solely
the result of a progenitor with relatively higher metallicities (Lentz et al.,
2001b). Such a claim would need to be verified by exploring a grid of models
with accompanying synthetic spectra.
Additionally, carbon absorption features could signify an origin that is
separate from explosion nucleosynthesis if most SN Ia are the result of a
merger. For example, Moll et al. (2013) recently presented angle-averaged
synthetic spectra for a few “peri-merger” detonation scenarios. In particular,
they find a causal connection between “normal” C II $\lambda$6580 signatures
and the secondary star for both sub- and super-Chandrasekhar mass cases (c.f.
Hicken et al. 2007; Zheng et al. 2013; Dessart et al. 2013a). With constraints
from UV spectra (Milne et al., 2013) and high velocity features (§3.4), peri-
megers can be used to explore the expanse of their spectroscopic influence
within the broader picture of SN Ia diversity.
Coincident with understanding the relevance of remaining carbon-rich material
is the additional goal of grasping the spectroscopic role of species that
arise from carbon-burning below the outermost layers, e.g., magnesium (Wheeler
et al., 1998). While signatures of Mg II $\lambda\lambda$4481, 7890 are
frequently observed at optical wavelengths during the earliest phases prior to
maximum light, these wavelength regions undergo severe line-blending compared
to the NIR signatures of Mg II. Consequently, Mg II $\lambda$10927 has served
as a better investment for measuring the lower regional extent and conditions
during which a DDT is thought to have taken place (e.g., Rudy et al. 2002;
Marion et al. 2003, 2006, 2009a; Hsiao et al. 2013, however see our §4.1).
### 3.4 High Velocity ($>$16,000 km s-1) Features
The spectra of many SN Ia have shown evidence for high-velocity absorption
lines of the Ca II NIR triplet (IR3) in addition to an often concurrent
signature of high-velocity Si II $\lambda$6355 (Mazzali et al., 2005a, b).
Most recently, high velocity features (HVFs) have also been seen in SN 2009ig
(Foley et al., 2012c), SN 2012fr (Childress et al., 2013c), and the SN 2000cx-
like SN 2013bh (Silverman et al., 2013c). Overall, HVFs are more common before
maximum light, display a rich diversity of behaviors (Childress et al.,
2013b), tend to be concurrent with polarization signatures (Leonard et al.,
2005; Tanaka et al., 2010), and may be due to an intrinsically clumpy
distribution of material (Howell et al., 2001; Kasen et al., 2003; Thomas et
al., 2004; Tanaka et al., 2006; Hole et al., 2010).
Maund et al. (2010b) showed that the Si II $\lambda$6355 line velocity decline
rate, v̇${}_{Si\ II}$, is correlated with the polarization of the same line at
day $-$5, p${}_{Si\ II}$, and is consistent with an asymmetric distribution of
IMEs. This interpretation is also complimentary with a previous finding that
v̇${}_{Si\ II}$ is correlated with vneb, the apparent Doppler line shift of
[Fe II] 7155 emitted from the “core” during late times (Maeda et al., 2010b;
Silverman et al., 2013a). For the recent SN 2012fr, high velocity features of
Ca II IR3 and Si II $\lambda$6355 at day $-$11 show concurrent polarization
signatures that decline in strength during post-maximum light phases (Maund et
al., 2013).
As for the _origin_ of HVFs, they may be the result of abundance and/or
density enhancements due to the presence of a circumstellar medium (Gerardy et
al., 2004; Quimby et al., 2006b). If abundance enhancements are responsible,
it could be explained by an overabundant, outer region of Si and Ca
synthesized during a pre-explosion simmering phase (see Piro 2011 and Zingale
et al. 2011). On the other hand, the HVFs in LVG SN Ia spectra could indicate
the presence of an opaque disk. For example, it is plausible that HVFs are due
to magnetically induced merger outflows (Ji et al. 2013, pending abundance
calculations of a successful detonation), or interaction with a tidal tail
and/or secondary star (e.g., Raskin & Kasen 2013; Moll et al. 2013).
Most recently, Childress et al. (2013b) studied 58 low-z SN Ia (z $<$ 0.03)
with well-sampled light curves and spectra near maximum light in order to
access potential relationships between light curve decline rates and empirical
relative strength measurements of Si II and Ca II HVFs. They find a consistent
agreement with Maguire et al. (2012) in that the Ca II velocity profiles
assume a variety of characteristics for a given $\Delta$m15(_B_) solely
because of the overlapping presence of HVFs. In addition, Childress et al.
(2013b) show for their sample that the presence of HVFs is not strongly
related to the overall intrinsic (_B_ $-$ _V_)max colors. It is also seen
that SN Ia with $\Delta$m15(_B_) $>$ 1.4 continue to be void of conspicuous
HVFs, while the strength of HVFs in normal SN Ia is generally larger for
objects with broader light curves. Finally, and most importantly, the strength
of HVFs at maximum light does not uniquely characterize HVF pre-maximum light
behavior.
Notably, Silverman et al. (2013a) find no correlation between nebular velocity
and $\Delta$m15(_B_), and for a given light-curve shape there is a large range
of observed nebular velocities. Similarly Blondin et al. (2012) found no
relation between the FWHM of late time 4700 Å iron emission features and
$\Delta$m15(_B_). This implies the peak brightness of these events _do not_
translate toward uniquely specifying their late time characteristics, however
the data do indicate a correlation between observed (B $-$ V)max and this
particular measure of line-of-sight nebular velocities.
We should also note that while HVG SN Ia do not clearly come with HVFs in the
same sense as for LVG SN Ia, the entire 6100 Å absorption feature for HVG SN
Ia spans across velocity intervals for HVFs detected in LVG SN Ia. This makes
it difficult to regard LVG and HVG subtypes as two separately distinct
explosion scenarios. Instead we can only conclude that HVFs are a natural
component of all normal SN Ia, whether conspicuously separate from a
photospheric region _or_ concealed as an extended region of absorbing material
in the radial direction.
### 3.5 Empirical Diversity Diagnostics
The depth ratio between 5750 and 6100 Å absorption features, $\mathcal{R}$(“Si
II”) (Nugent et al., 1995), has been found to correlate with components of the
WLR. In addition, Benetti et al. (2005) find a rich diversity of
$\mathcal{R}$(Si II) pre-maximum evolution among LVG and HVG SN Ia.
As for some observables _not_ directly related to the decline rate parameter,
Patat et al. (1996) studied a small sample of well observed SN Ia and found no
apparent correlation between the blue-shift of the 6100 Å absorption feature
at the time of maximum and $\Delta$m15(_B_). Similarly, Hatano et al. (2000)
showed that $\mathcal{R}$(Si II) does not correlate well with v10(Si II), the
photospheric velocity derived from the Si II $\lambda$6355 Doppler line
velocities 10 days after maximum light. This could arise from two or more
explosion mechanisms, however Hatano et al. (2000) note that their
interpretation is “rudimentary” on account of model uncertainties and the
limited number of temporal datasets available at that time. In the future, it
would be worthwhile to re-access these trends with the latest detailed
modeling.
Hachinger et al. (2006) made empirical measurements of spectroscopic feature
pEWs, flux ratios, and projected Doppler velocities for 28 well-observed SN
Ia, which include LVG, HVG, and FAINT subtypes. For normal LVG SN Ia they find
similar observed maximum light velocities (via Si II $\lambda$6355; $\sim$
9000$-$10,600 km s-1). Meanwhile the HVG SN Ia in their sample revealed a
large spread of maximum light velocities ($\sim$ 10,300$-$12,500 km s-1),
regardless of the value of $\Delta$m15(_B_). This overlap in maximum light
velocities implies a natural continuum between LVG and HVG SN Ia (enabling
unification through asymmetrical contexts; Maeda et al. 2010b; Maund et al.
2010b). They also note that FAINT SN Ia tend to show slightly smaller
velocities at _B_ -band maximum for larger values of $\Delta$m15(_B_), however
no overreaching trend of maximum light expansion velocities from LVG to HVG to
FAINT SN Ia was apparent from this particular sample of SN Ia.
Hachinger et al. (2006) did find several flux ratios to correlate with
$\Delta$m15(B). In particular, they confirm that the flux ratio,
$\mathcal{R}$(“S II $\lambda$5454, 5640”/“Si II $\lambda$5972”), is a fairly
reliable spectroscopic luminosity indicator in addition to $\mathcal{R}$(Si
II). Hachinger et al. (2006) conclude that these and other flux ratio
comparisons are the result of changes in relative abundances across the three
main SN Ia subtypes. In a follow-up investigation, Hachinger et al. (2009)
argue that the correlation with luminosity is a result of ionization balance,
where dimmer objects tend to have a larger value of $\mathcal{R}$(Si II).
Silverman et al. 2012b later studied correlations between these and other flux
ratios of SN Ia from the BSNIP sample and find evidence to suggest that CSM-
associated events tend to have larger 6100 Å blue-shifts in addition to
broader absorption features at the time of maximum light (see also Arsenijevic
2011; Foley et al. 2012a).
Altavilla et al. (2009) studied the $\mathcal{R}$(Si II) ratio and expansion
velocities of intermediate-redshift supernovae. They find that the comparison
of intermediate-redshift SN Ia spectra with high S/N spectra of nearby SN Ia
_do not_ reveal significant differences in the optical features and the
expansion velocities derived from the Si II and Ca II lines that are within
the range observed for nearby SN Ia. This agreement is also found in the color
and decline of the light curve (see also Mohlabeng & Ralston 2013).
While the use of empirically determined single-parameter descriptions of SN Ia
have proved to be useful in practice, they do not fully account for the
observed diversity of SN Ia (Hatano et al., 2000; Benetti et al., 2004;
Pignata et al., 2004). With regard to SN Ia diversity, it should be
reemphasized that special care needs to be taken with the implementation of
flux ratios and pEWs. Detailed modeling is needed when attempting to draw
connections between solitary characteristics of the observed spectrum and the
underlying radiative environment, where a photon-ray’s route crosses many
radiative contributions that form the spectrum’s various shapes, from UV to IR
wavelengths. For example, the relied upon 5750, 6100 Å features used for
$\mathcal{R}$(Si II) have been shown to be influenced by more than simply Si
II, as well as from more locations (and therefore various temperatures) than a
single region of line formation (Bongard et al., 2008). In fact, it is likely
that a number of effects are at play, e.g., line blending and phase evolution
effects. Furthermore, v10(Si II) is a measure of the 6100 Å absorption minimum
during a phase of intense line blending with no less than Fe II, which imparts
a bewildering array of lines throughout the optical bands (Baron et al., 1995,
1996). Still, parameters such as $\mathcal{R}$(Si II) have served as useful
tools for SN Ia diversity studies in that they often correlate with luminosity
(Bongard et al., 2006) and are relatively accessible empirical measurements
for large samples of under-observed SN Ia. A detailed study on the selection
of global spectral indicators can be found in Bailey et al. 2009.
### 3.6 The Adjacent Counterparts of Optical Wavelengths
#### 3.6.1 Ultraviolet Spectra
SN Ia are known as relatively “weak emitters” at UV wavelengths ($<$ 3500 Å;
Panagia 2007). It has been shown that UV flux deficits are influenced by line-
blanketing effects from IPEs within the outermost layers of ejecta (Sauer et
al., 2008; Hachinger et al., 2013; Mazzali et al., 2013), overall higher
expansion velocities (Foley & Kasen, 2011; Wang et al., 2012), progenitor
metallicity (Höflich et al., 1998; Lentz et al., 2000; Sim et al., 2010b),
viewing angle effects (e.g., Blondin et al. 2011), or a combination of these
(Moll et al., 2013). Although, it is not certain which of these play the
dominant role(s) in controlling UV flux behaviors among all SN Ia.
For SN 1990N and SN 1992A, two extensively studied SN Ia, pre-maximum light UV
observations were made and presented by Leibundgut et al. (1991) and Kirshner
et al. (1993), respectively. These observations revealed their expected
sensitivity to the source temperature and opacity at UV wavelengths.
It was not until recently when a larger UV campaign of high S/N, multi-epoch
spectroscopy of distant SN Ia was presented and compared to that of local SN
Ia (Ellis et al. 2008, see also Milne et al. 2013). Most notably, Ellis et al.
(2008) found a larger intrinsic dispersion of UV properties than could be
accounted for by the span of effects seen in the latest models (e.g.,
metallicity of the progenitor, see Höflich et al. 1998; Lentz et al. 2000).
As a follow-up investigation, Cooke et al. (2011) utilized and presented data
from the STIS spectrograph onboard _Hubble Space Telescope_ (HST) with the
intent of studying near-UV, near-maximum light spectra (day $-$0.32 to day
$+$4) of nearby SN Ia. Between a high-z and low-z sample, they find a
noticeable difference between the mean UV spectrum of each, suggesting that
the cause may be related to different metallicities between the statistical
norm of each sample. Said another way, their UV observations suggest a
plausible measure of two different populations of progenitors (or constituent
scenarios) that could also be dependent on the metallically thereof, including
potentially larger dependencies such as variable 56Ni mass and line blanketing
due to enhanced burning within the outermost layers (Marion et al., 2013). It
should be noted, despite the phase selection criterion invoked by Cooke et al.
(2011), it may not be enough to simply designate a phase range in order to
avoid phase evolution effects (see Fig. 7 of Childress et al. 2013b).
In order to confirm spectroscopic trends at UV wavelengths, a better method of
selection will be necessary as the largest UV difference found by Cooke et al.
(2011) and Maguire et al. (2012) between the samples overlaps with the Si II,
Ca II H&K absorption features (3600$-$3900 Å), i.e. a highly blended feature
that is far too often a poorly understood SN Ia variable, both observationally
(across subtype and phase) and theoretically, within the context of line
formation at UV$-$NIR wavelengths (Mazzali, 2000; Kasen et al., 2003; Thomas
et al., 2004; Foley, 2012; Marion et al., 2013; Childress et al., 2013b).
While it is true that different radiative processes dominate within different
wavelength regions, there are a multitude of explanations for such a
difference between the Si II$-$Ca II blend near 3700 Å. Furthermore, the STIS
UV spectra do not offer a look at either the state of the 6100 Å absorption
feature (is it completely photospheric?$-$the answer requires spectrum
synthesis even for maximum light phases), nor is it clear if the same is true
for Ca II in the NIR where high-velocity components thereof are most easily
discernible (Lentz et al., 2000).
It is important to further reemphasize that the time-dependent behavior of the
sum total of radiative processes that generate a spectrum from a potentially
axially-asymmetric (and as of yet unknown) progenitor system and explosion
mechanism are not well understood, much less easily decipherable with an only
recently obtained continuous dataset for how the spectrum itself evolves over
time at optical wavelengths202020SN 2011fe. See Pereira et al. 2013 and
http://snfactory.lbl.gov/snf/data/. Which is only to say, given the current
lack of certain predictability between particular observational
characteristics of SN Ia (e.g., spectroscopic phase transition times), time
series observations at UV wavelengths would offer a beneficial route for the
essential purposes of hand-selecting the ‘best’ spectrum comparisons in order
to ensure a complete lack of phase evolution effects.
Recently, Wang et al. (2012) presented HST multi-epoch, UV observations of SN
2004dt, 2004ef, 2005M, and 2005cf. Based on comparisons to the results of
Lentz et al. (2000) and Sauer et al. (2008), two studies that show a 0.3
magnitude span of UV flux for a change of two orders of magnitude in
metallicity within the C$+$O layer of a pure deflagration model (W7; Nomoto et
al. 1984), Wang et al. (2012) conclude that the UV excess for a HVG SN Ia, SN
2004dt (A.22), _cannot_ be explained by metallicities or expansion velocities
alone. Rather, the inclusion of asymmetry into a standard model picture of SN
Ia should be a relevant part of their observed diversity (e.g., Kasen et al.
2009; Blondin et al. 2011).
More recently, Mazzali et al. (2013) obtained 10 HST UV$-$NIR spectra of SN
2011fe, spanning $-$13.5 to $+$41 days relative to _B_ -band maximum. They
analyzed the data along side spectrum synthesis results from three explosion
models, namely a ‘fast deflagration’ (W7), a low-energy delayed-detonation
(WS15DD1; Iwamoto et al. 1999), and a third model treated as an intermediary
between the outer-layer density profiles of the other two models (“W7+”). From
the seen discrepancies between W7 and WS15DD1 during the early pre-maximum
phase, in addition to optical flux excess for W7 and a mismatch in observed
velocities for WS15DD1, Mazzali et al. (2013) conclude that their modified W7+
model is able to provide a better fit to the data because of the inclusion of
a high velocity tail of low density material. In addition, and based on a
spectroscopic rise time of $\sim$ 19 days, Mazzali et al. (2013) infer a
$\sim$ 1.4 day period of optical quiescence after the explosion (see Piro &
Nakar 2013; Chomiuk 2013).
#### 3.6.2 Infrared Light Curves and Spectra
By comparing absolute magnitudes at maximum of two dozen SN Ia, Krisciunas
(2005) argue that SN Ia can be best used as standard candles at NIR
wavelengths (which was also suggested by Elias et al. 1985a, b), even without
correction for optical light curve shape. Wood-Vasey et al. (2008) later
confirmed this to be the case from the analysis of 1087 near-IR (JHK)
measurements of 21 SN Ia. Based on their data and data from the literature,
they derive absolute magnitudes of 41 SN Ia in the _H_ -band with rms scatter
of 0.16 magnitudes. Folatelli et al. (2010) find a weak dependence of _J_
-band luminosities on the decline rate from 9 NIR datasets, in addition to _V_
$-$_J_ corrected _J_ -band magnitudes with a dispersion of 0.12 magnitudes.
Mandel et al. (2011) constructed a statistical model for SN Ia light curves
across optical and NIR passbands and find that near-IR luminosities enable the
most ideal use of SN Ia as standard candles, and are less sensitive to dust
extinction as well. Kattner et al. (2012) analyzed the standardizability of SN
Ia in the near-IR by investigating the correlation between observed peak near-
IR absolute magnitude and post-maximum $\Delta$m15(_B_). They confirm that
there is a bimodal distribution in the near-IR absolute magnitudes of fast-
declining SN Ia (Krisciunas et al., 2009) and suggest that applying a
correction to SN Ia peak luminosities for decline rate is likely to be
beneficial in the J and H bands, making SN Ia more precise distance indicators
in the IR than at optical wavelengths (Barone-Nugent et al., 2012).
While optical spectra of SN Ia have received a great deal of attention in the
recent past, infrared datasets (e.g., Kirshner et al. 1973b; Meikle et al.
1996; Bowers et al. 1997; Rudy et al. 2002; Höflich et al. 2002) are either
not obtained, or are not observed at the same epochs or rate as their optical
counterparts. This has only recently begun to change. Thus far, the largest
NIR datasets can be found in Marion et al. (2003) and Marion et al. (2009a).
Marion et al. (2003) obtained NIR spectra (0.8$-$2.5 $\mu$m) of 12 normal SN
Ia, with fairly early coverage. Later, Marion et al. (2009a) presented and
studied a catalogue of NIR spectra (0.7 $-$ 2.5 $\mu$m) of 41 additional SN
Ia. In all, they report an absence of _conspicuous_ signatures of hydrogen and
helium in the spectra, and no indications of carbon via C I $\lambda$10693
(however, see our §4.1). For an extensive review on IR observations, we refer
the reader to Phillips (2012).
### 3.7 Drawing Conclusions about SN Ia Diversity from SN Ia Rates Studies
It has long been perceived that a supernova’s local environment, rate of
occurrence, and host galaxy properties (e.g., WD population) serve as powerful
tools for uncovering solutions to SN Ia origins (Zwicky, 1961; Hamuy et al.,
1995; van den Bergh et al., 2005; Mannucci, 2005; Leaman et al., 2011; Li et
al., 2011a, d). After all, a variety of systems, both standard and exotic
scenarios$-$all unconfirmed$-$offer potential for explaining “oddball” SN Ia,
as well as more normal events, at various distances (z; redshift) and
associations with a particular host galaxy or WD population (Yungelson &
Livio, 2000; Parthasarathy et al., 2007; Hicken et al., 2009a; Hachisu et al.,
2012; Pakmor et al., 2013; Wang et al., 2013c; Pan et al., 2013; Kim et al.,
2014).
Despite this broad extent of the progenitor problem, measurements of the total
cosmic SN Ia rate, RSNIa(z), can be made to gauge the general underlying
behavior of actively contributing systems (Maoz et al., 2012). Further insight
into how various progenitor populations impart their signature onto RSNIa(z)
comes about by considering which scenarios lead to a “prompt” (or a “tardy”)
stellar demise (Scannapieco & Bildsten, 2005; Mannucci, 2005). Whether or not
mergers involve both a (“prompt”) helium-burning or (“tardy”) degenerate
secondary star remains to be seen (Woods et al. 2011; Hillebrandt et al. 2013;
Dan et al. 2013 and references therein). Because brighter SN Ia prefer
younger, metal-poor galaxies, and a linear relation exists between the SN Ia
light curve shape and gas-phase metallicity, the principle finding has been
that the rate of the universally prompt component is proportional to the star
formation rate of the host galaxy, whereas the second delayed component’s rate
is proportional to the stellar mass of the galaxy (Sullivan et al., 2006;
Howell et al., 2007; Sullivan et al., 2010; Zhang, 2011; Pan et al., 2013).
The SN Ia galaxy morphology study of Hicken et al. (2009a) has since
progressed this discussion of linking certain observed SN Ia properties with
their individual environments. Overall, the trend of brighter/dimmer SN Ia
found in younger/older hosts remains, however now with indications that a
continuous distribution of select SN Ia subtypes exist in multiple host galaxy
morphologies and projected distances within each host.
To understand the full form of RSNIa(z), taking into account the delay time
distribution (DTD) for every candidate SN Ia system is necessary (see
Bonaparte et al. 2013; Claeys et al. 2014). Maoz et al. (2010) find that the
DTD peaks prior to 2.2 Gyr and has a long tail out to $\sim$ 10 Gyr. They
conclude that a DTD with a power-law t-1.2 starting at time t = 400 Myr to a
Hubble time can satisfy both constraints of observed cluster SN rates and
iron-to-stellar mass ratios, implying that that half to a majority of all SN
Ia events occur within one Gyr of star formation (see also Strolger et al.
2010; Meng et al. 2011).
In general, the DTD may be the result of binary mergers (Ruiter et al., 2009;
Toonen et al., 2012; Nelemans et al., 2013) and/or a single-degenerate
scenario (Hachisu et al., 2008, 2012; Chen et al., 2013), but with the
consideration that evidence for delay times as short as 100 Myr have been
inferred from SN remnants in the Magellanic Clouds (Badenes et al., 2009; Maoz
& Badenes, 2010). From a recent comparison of low/high-z SN Ia rate
measurements and DTDs of various binary population synthesis models, Graur et
al. (2013) argue that single-degenerate systems are ruled out between 1.8 $<$
z $<$ 2.4. Overall, their results support the existence of a double-degenerate
progenitor channel for SN Ia if the the number of double-degenerate systems
predicted by binary population synthesis models can be “aptly” increased (Maoz
et al., 2010).
However, initial studies have primarily focussed upon deriving the DTD without
taking into account the possible effects of stellar metallicity on the SN Ia
rate in a given galaxy. Given that lower metallicity stars leave behind higher
mass WD stars (Umeda et al., 1999b; Timmes et al., 2003), Kistler et al.
(2013) and Meng et al. (2011) argue that the effects of metallicity may serve
to significantly alter the SN Ia rate (see also Pan et al. 2013). In fact,
models that include the effects of metallicity (e.g., Kistler et al. 2013)
find similar consistencies with the observed RSNIa(z). Notably, recent
spectroscopic studies _do_ indicate a stronger preference of low-metallicity
hosts for super-Chandrasekhar candidate SN Ia (Taubenberger et al., 2011;
Childress et al., 2011), which may just as well be explained by low
metallicity single-degenerate systems (Hachisu et al., 2012). While there are
not enough close binary WD systems in our own galaxy that would result in SCC
DD scenarios (Parthasarathy et al., 2007), sub-Chandrasekhar merging binaries
may be able to account for discrepancies in the observed rate of SN Ia
(Badenes & Maoz, 2012; Kromer et al., 2013b).
Although, we wish to remind the reader that since spectrophotometry of SN Ia
so far offer the best visual insight into these distant extragalactic events,
and because there is no clear consensus on the origin of their observed
spectrophotometric diversity, there is no clear certainty as to what
distribution of progenitor scenarios connect with any kind of SN Ia since none
have been observed prior to the explosion. Furthermore, whether or not
brighter or dimmer SN Ia “tend to” correlate with any property of their hosts
does not alleviate the discussion down to one or two progenitor systems (e.g.,
single- versus double-degenerate systems) since the most often used tool for
probing SN Ia diversity over all distance scales, i.e. the “stretch” of a
light curve, does not necessarily uniquely determine the spectroscopic
subtype. Rather, such correlations reveal the degree of an underlying effect
from samples of uncertain and unknown SN Ia subtype biases, i.e. dust
extinction in star formation galaxies and progenitor ages also evolve along
galaxy mass sequences (Childress et al., 2013a) and the redshift-color
evolution of SN Ia remains an open issue (Mohlabeng & Ralston, 2013; Pan et
al., 2013; Wang et al., 2013b).
While it is important to consider the full redshift range over which various
hierarchies of progenitor and subtype sequences may dominate over others, such
studies are rarely able to incorporate spectroscopic diversity as input (a
“serendipitous” counter-example being Krughoff et al. 2011). This is relevant
given that the landscape of SN Ia spectroscopic diversity has not yet been
seen to be void of line-of-sight discrepancies for all progenitor scenarios
(particularly so for double degenerate detonations/mergers, e.g., Shen et al.
2013; Pakmor et al. 2013; Raskin & Kasen 2013; Moll et al. 2013; Raskin et al.
2013). Ultimately, robust theories should be able to connect spectroscopic
subtypes with individual or dual instances of particular progenitor systems,
which requires detailed spectroscopic modeling.
Thus, the consensus as to how many progenitor channels contribute to SN Ia
populations is still unclear. Broadly speaking, there are likely to be no less
than two to three progenitor scenarios for normal SN Ia so long as single-
degenerate systems remain viable (Hachisu et al., 2012), if not restricted to
explaining Ia-CSM SN alone (see Han & Podsiadlowski 2006; Silverman et al.
2013d; Leloudas et al. 2013). Given also a low observed frequency of massive
white dwarfs and massive double-degenerate binaries near the critical mass
limit with orbital periods short enough to merge within a Hubble time, some
normal SN Ia are still perceived as originating from single-degenerate systems
(Parthasarathy et al., 2007). Meanwhile, some portion of events may also be
the result of a core-degenerate merger (Soker et al., 2013), while some merger
phenomena are possibly accelerated within triple systems (Thompson, 2011;
Kushnir et al., 2013; Dong et al., 2014). It likewise remains unclear whether
or not some double-degenerate mergers predominately result in the production
of a neutron star instead of a SN Ia (Saio & Nomoto, 1985; Nomoto & Kondo,
1991; Piersanti et al., 2003; Saio & Nomoto, 2004; Dan et al., 2013; Tauris et
al., 2013). At present, separately distinct origins for spectroscopically
similar SN Ia cannot be ruled out by even one discovery of a progenitor
system; the spectroscopic diversity is currently too great and too poorly
understood to confirm without greater unanimity among explosion models and
uniformity in data collection efforts.
## 4 Some Recent SN Ia
During the past decade, several normal, interesting, and peculiar SN Ia have
been discovered. For example, the recent SN 2009ig, 2011fe, and 2012fr are
nearby SN Ia that were discovered extremely young with respect to the onset of
the explosion (Nugent et al., 2011; Foley et al., 2012c; Childress et al.,
2013c) and have been extensively studied at all wavelengths, yielding a
clearer understanding of the time-dependent behavior of SN spectroscopic
observations, in addition to a better context by which to compare. Below we
briefly summarize some of the highlighted discoveries during the most recent
decade, during which it has revealed a greater diversity of SN Ia than was
previously known. In the appendix we provide a guide to the recent literature
of other noteworthy SN Ia discoveries. We emphasize that these sections are
not meant to replace reading the original publications, and are only
summarized here as a navigation tool for the reader to investigate further.
### 4.1 SN 2011fe in M101
Thus far, the closest spectroscopically normal SN Ia in the past 25 years, SN
2011fe (PTF11kly), has provided a great amount of advances, including testing
SN Ia distance measurement methods (Matheson et al., 2012; Vinkó et al., 2012;
Lee & Jang, 2012). For example, the early spectroscopy of SN 2011fe showed a
clear and certain time-evolving signature of high-velocity oxygen that varied
on time scales of hours, indicating sizable overlap between C$+$O, Si, and Ca-
rich material and newly synthesized IMEs within the outermost layers (Nugent
et al., 2011).
Parrent et al. (2012) carried out analysis of 18 spectra of SN 2011fe during
its first month. Consequently, they were able to follow the evolution of C II
$\lambda$6580 absorption features from near the onset of the explosion until
they diminished after maximum light, providing strong evidence for overlapping
regions of burned and unburned material between ejection velocities of at
least 10,000 and 16,000 km s-1. At the same time, the evolution of a 7400 Å
absorption feature experienced a declining Doppler-shift until 5 days post-
maximum light, with O I $\lambda$7774 line velocities ranging 11,500 to 21,000
km s-1 (Nugent et al., 2011). Parrent et al. (2012) concluded that incomplete
burning (in addition to progenitor scenarios) is a relevant source of
spectroscopic diversity among SN Ia (Tanaka et al., 2008; Maeda et al.,
2010a).
Pereira et al. (2013) presented high quality spectrophotometric observations
of SN 2011fe, which span from day $-$15 to day $+$97, and discussed
comparisons to other observations made by Brown et al. (2012), Richmond &
Smith (2012), Vinkó et al. (2012), and Munari et al. (2013). From an observed
peak bolometric luminosity of 1.17 $\pm$ 0.04 x 1043 erg s-1, they estimate SN
2011fe to have produced between $\sim$ 0.44 $\pm$ 0.08 $-$ 0.53 $\pm$ 0.11 M⊙
of 56Ni.
By contrast, Pastorello et al. (2007a) and Wang et al. (2009b) estimate 56Ni
production for the normal SN 2005cf (A.26) to be $\sim$ 0.7 M⊙. It is also
interesting to note that SN 2011fe and the fast-declining SN 2004eo produced
similar amount of radioactive nickel, however lower for SN 2004eo ($\sim$ 0.4
M⊙; Mazzali et al. 2008). Pereira et al. (2013) also made comparisons between
SN 2011fe, a SNFactory normal SN Ia (SNF20080514-002) and the broad-lined HV-
CN SN 2009ig (Foley et al., 2012c). Pereira et al. (2013) note similarities
(sans the UV) and notable contrast with respect to high-velocity features,
respectively.
Pereira et al. (2013) calculated v̇${}_{Si\ II}$ for SN 2011fe to be $\sim$ 60
($\pm$ 3) km s-1 day-1, near the high end of low-velocity gradient SN Ia
events (see Benetti et al. 2005; Blondin et al. 2012). Given their high S/N,
time series dataset, Pereira et al. (2013) were also able to place tighter
constraints on the velocities over which C II $\lambda$6580 is observed to be
present in SN 2011fe. They conclude that C II is present down to at least as
low as 8000 km s-1, which is 2000 km s-1 lower than that estimated by Parrent
et al. (2012), and is also $\sim$ 4000$-$6000 km s-1 (or more) lower than what
is predicted by some past and presently favored SN Ia abundance models (e.g.,
W7; Nomoto et al. 1984, and the delayed detonations of Höflich 2006 and Röpke
et al. 2012).
Hsiao et al. (2013) presented and discussed NIR time series spectra of SN
2011fe that span between day $-$15 and day $+$17\. In particular, they report
a detection of C I $\lambda$10693 on the blue-most side of a blended Mg II
$\lambda$10927 absorption feature at roughly the same velocities and epochs as
C II $\lambda$6580 found by Parrent et al. (2011) and Pereira et al. (2013),
which itself is blended on its _blue-most_ side with Si II $\lambda$6355\.
While searches and studies of C I $\lambda$10693 are extremely useful for
understanding the significance of C-rich material from normal to cooler sub-
luminous SN Ia within the greater context of all C I, C II, C III, O I
absorption features (C III for “hotter” SN 1991T-likes), blended C I
$\lambda$10693 absorption shoulders are certainly no more (nor no less) useful
for probing lower velocity boundaries than C II $\lambda$6580 absorption
notches. This is especially true given that C I $\lambda$10693 absorption
features are blended from the _red-most_ side (lower velocities) by the
neighboring Mg II line, which will only serve to obscure the lower velocity
information of the C I profile for the non-extreme cases (e.g., SN 1999by,
Höflich et al. 2002).
Hsiao et al. (2013) used the observed temporal behavior, and later velocity-
plateau, of Mg II $\lambda$10927 to estimate a lower extent of $\sim$ 11,200
km s-1 for carbon-burning products within SN 2011fe. Given that this in
contrast to the refined lower extent of C II at $\sim$ 8000 km s-1 by Pereira
et al. (2013), this _could_ imply (i.e. assuming negligible temperature
differences and/or non-LTE effects) that either some unburned material has
been churned below the boundary of carbon-burning products via turbulent
instabilities (Gamezo et al., 1999, 1999, 2004) and/or the distribution of
emitting and absorbing carbon-rich material is truly globally lopsided (Kasen
et al., 2009; Maeda et al., 2010b; Blondin et al., 2011), and may indicate the
remains of a degenerate secondary star (Moll et al., 2013).
Detailed studies of this nearby, normal, and unreddenned SN 2011fe have given
strong _support_ for double-degenerate scenarios (assuming low environmental
abundances of hydrogen) and have placed strong _constraint_ on single-
degenerate scenarios, i.e. MS and RG companion stars have been strongly
constrained for SN 2011fe (see Shappee et al. 2013 and references therein, and
also Hayden et al. 2010a; Bianco et al. 2011). Nugent et al. (2011), Li et al.
(2011b) and Bloom et al. (2012) confirm that the primary star was a compact
star (R∗ $\lesssim$ 0.1 R⊙, c.f., Bloom et al. 2012; Piro & Nakar 2013;
Chomiuk 2013). From the lack of evidence for an early shock outbreak (Kasen,
2010; Nakar & Sari, 2012), non-detections of radio and X-ray emissions (Horesh
et al., 2012; Chomiuk et al., 2012; Margutti et al., 2012), non-detections of
narrow Ca II H&K or Na D lines or pre-existing dust that could be associated
with the event (Patat et al., 2013; Johansson et al., 2013), and low upper-
limits on hydrogen-rich gas (Lundqvist et al., 2013), the paucity of evidence
for an environment dusted in CSM from a non-degenerate secondary strongly
supports the double degenerate scenario for SN 2011fe. Plus, this inferred
ambient environment is consistent with that of recent merger simulations (Dan
et al., 2012), and could signal an avenue of interpretation for signatures of
carbon-rich material as well (Branch et al., 2005; Dan et al., 2013; Moll et
al., 2013). Specifically, the remaining amount of carbon-rich material
predicted by some explosion models may already be accounted for, and more so
than would be required by the existence of low velocity detections of C I and
C II. If this turns out to be the case, spectroscopic signatures of both C and
O could tap into understanding (i) the sizes of merger C$+$O common envelopes,
(ii) potential downward mixing effects between the envelope and the underlying
ejecta, and/or (iii) test theories on possible asymmetries of C$+$O material
within the post-explosion ejecta of the primary and secondary stars (Livio &
Pringle, 2011), which is expected to depend on the degree of coalescence (Moll
et al., 2013; Raskin & Kasen, 2013).
Of course, this all rests on the assumptions that (i) the surrounding
environment of a single degenerate scenario just prior to the explosion ought
to be contaminated with some amount of CSM, above which it would be detected
(Justham, 2011; Brown et al., 2012), and (ii) the surrounding environment of a
merger remains relatively “clean” (Shen et al., 2013; Raskin & Kasen, 2013).
In this instance, and assuming similarly above that current DDT-like models
roughly fit the outcome of the explosion, absorption signatures of C ($+$ HV O
I) may point to super-massive single-degenerate progenitors with variable
enclosed envelopes and/or disks of material (e.g., Yoon & Langer 2004, 2005;
Hachisu et al. 2012; Scalzo et al. 2012; Tornambé & Piersanti 2013; Dan et al.
2013) or sub-Chandrasekhar mass “peri-mergers” for resolve (see Moll et al.
2013 and references therein).
### 4.2 Other Early Discoveries
#### 4.2.1 SN 2009ig in NGC 1015
Foley et al. (2012c) obtained well-sampled, early UV and optical spectra of SN
2009ig as it was discovered 17 hr after the event (Kleiser et al., 2009;
Navasardyan et al., 2009). SN 2009ig is found to be a normal SN Ia, rising to
_B_ -band maximum in $\sim$ 17.3 days. From the earliest spectra, Foley et al.
(2012c) find Si II $\lambda$6355 line velocities around 23,000 km s-1, which
is exceptionally high for such a spectroscopically normal SN Ia (see also
Blondin et al. 2012). SN 2009ig possess either an overall shallower density
profile than other CN SN Ia, or a buildup of IMEs is present at high
velocities.
Marion et al. (2013) recently analyzed the photospheric to post-maximum light
phase spectra of SN 2009ig, arguing for the presence of additional high-
velocity absorption signatures from not only Si II, Ca II, but also Si III, S
II and Fe II. Whether or not two separate but compositionally equal regions of
line formation is a ubiquitous property of similar SN Ia remains to be seen.
However, it should not be unlikely for primordial amounts of said atomic
species to be present (in addition to singly-ionized silicon and calcium) on
account of possible density and/or abundance enhancements within the outermost
layers (Thomas et al., 2004; Mazzali et al., 2005b, a). For example, simmering
effects during convective phases prior to the explosion may be responsible for
dredging up IMEs later seen as HVFs, which would give favorability to single-
degenerate progenitor scenarios (see Piro 2011; Zingale et al. 2011).
Similarly, it is worthwhile to access the versatility of mergers in producing
high-velocity features.
#### 4.2.2 SN 2012fr in NGC 1365
Childress et al. (2013c) report on their time series spectroscopic
observations of SN 2012fr (Klotz et al., 2012; Childress et al., 2012; Buil,
2012), complete with 65 spectra that cover between $\sim$ 15 days before and
40 days after it reached a peak _B_ -band brightness of $-$19.3. In addition
to the simultaneous spectropolarimetric observations of Maund et al. (2013),
the early to maximum light phase spectra of SN 2012fr reveal one of the
clearest indications that SN Ia of similar type (e.g., SN 1994D, 2001el,
2009ig, 2011fe, and many others; Mazzali05a) tend to have two distinctly
separate regions of Si-, Ca-based material that differ by a range of
separation velocities (Childress et al., 2013b).
Childress et al. (2013c) and Maund et al. (2013) discussed the various
interpretations that have been presented in the past, however no firm
conclusions on the origin of HVFs could be realized given the uncertainties of
current explosion models. Despite this, the most recent advance toward
understanding HVFs is the continual detection of polarization signatures due
to the high-velocity Si II and Ca II absorption features, indicating a
departure from a radially stratified, spherically symmetric geometry at some
layer near or above the “photospheric region” of IMEs.
### 4.3 Super-Chandrasekhar Candidate SN Ia
#### 4.3.1 Over-luminous SN 2003fg (SNLS-03D3bb)
SN 2003fg was discovered as part of the Supernova Legacy Survey (SNLS); z =
0.2440 (Howell et al., 2006). Its peak absolute magnitude was estimated to be
$-$19.94 in _V_ -band, placing SN 2003fg completely outside the
M${}_{\emph{V}}$-distribution of normal low-z SN Ia (2.2 times brighter).
Assuming Arnett’s rule, such a high luminosity corresponds to $\sim$ 1.3 M⊙ of
56Ni, which would be in conflict with SN 2003fg’s spectra since only $\sim$
60% of a Chandrasekhar pure detonation ends up as radioactive nickel
(Steinmetz et al. 1992, however see also Pfannes et al. 2010). Given also the
lower mean expansion velocities, this builds upon the picture of a super-
Chandrasekhar mass progenitor for SN 2003fg and others like it (Howell et al.,
2006; Jeffery et al., 2006).
Yoon & Langer (2005) proposed the formation of super-Chandrasekhar mass WD
stars as a result of rapid rotation. Pfannes et al. (2010) later reworked
these models and found that the “prompt” detonation of a super-Chandrasekhar
mass WD produces enough nickel, as well as a remainder of IMEs in the outer
layers (in contrast to Steinmetz et al. 1992), to explain over-luminous SN Ia.
Hachisu et al. (2012) added to this model by taking into account processes of
binary evolution. Namely, with the inclusion of mass-striping, optically thick
winds of a differentially rotating primary star, Hachisu et al. (2012) find
three critical mass ranges that are each separated according to the spin-down
time of the accreting WD. All three of these single-degenerate scenarios may
explain a majority of events from sub-luminous to over-luminous SN Ia. So far
no super-Chandrasekhar mass WD stars that would result in a SN Ia have been
found in the sample of known WD stars in our Galaxy (Saffer et al. 1998, see
also Kilic et al. 2012). However, this does not so much rule out super-
Chandrasekhar mass models as it suggests that these systems are rare in the
immediate vicinity within our own galaxy.
Hillebrandt et al. (2007) proposed an alternative scenario involving only a
Chandrasekhar-mass WD progenitor to explain the SN 2003fg event. They
demonstrated that an off-center explosion of a Chandrasekhar-mass WD could
explain the super-bright SN Ia. However, in this off-center explosion model it
is not easy to account for the high Ni mass in the outer layers, in addition
to the special viewing direction.
#### 4.3.2 Over-luminous SN 2009dc in UGC 10064
Yamanaka et al. (2009a) presented early phase optical and NIR observations for
SN 2009dc (Puckett et al., 2009; Harutyunyan et al., 2009; Marion et al.,
2009b; Nicolas & Prosperi, 2009). From the peak _V_ -band absolute magnitude
they conclude that SN 2009dc belongs to the most luminous class of SN Ia
($\Delta$m15(_B_) = 0.65), and estimate the 56Ni mass to be 1.2 to 1.6 M⊙.
Based on the JHK photometry Yamanaka et al. (2009a) also find SN 2009dc had an
unusually high NIR luminosity with enhanced fading after $\sim$ day $+$200
(Maeda et al., 2009; Silverman et al., 2011; Taubenberger et al., 2011). The
spectra of SN 2009dc also show strong, long lasting 6300 Å absorption features
(until $\sim$ two weeks post-maximum light) Based on the observed
spectropolarimetric indicators, in combination with photometric and
spectroscopic properties, Tanaka et al. (2010) similarly conclude that the
progenitor mass of SN 2009dc was of super-Chandrasekhar origin and that the
explosion geometry was globally spherically symmetric, with a clumpy
distribution of IMEs.
Silverman et al. (2011) presented an analysis of 14 months of observations of
SN 2009dc and estimate a rise-time of $\sim$ 23 days and $\Delta$m15(_B_) =
0.72. They find a lower limit of the peak bolometric luminosity $\sim$ 2.4 x
1043 erg s-1 and caution that the actual value is likely almost 40% larger.
Based on the high luminosity and low mean expansion velocities of SN 2009dc,
Silverman et al. (2011) derive a mass of more than 2M⊙ for the white dwarf
progenitor and a 56Ni mass of $\sim$ 1.4 to 1.7 M⊙. Taubenberger et al. (2011)
find the minimum 56Ni mass to be 1.8 M⊙, assuming the smallest possible rise-
time of 22 days, and the ejecta mass to be 2.8 M⊙.
Taubenberger et al. (2013) compared photometric and spectroscopic observations
of normal and SCC SN Ia at late epochs, including SN 2009dc, and find a large
diversity of properties spanning through normal, SS, and SCC SN Ia. In
particular the decline in the light curve “radioactive tail” for SCC SN Ia is
larger than normal, along with weaker than normal [Fe III] emission in the
nebular phase spectra. Taubenberger et al. (2013) argue that the weak [Fe III]
emission is indicative of an ejecta environment with higher than normal
densities. Previously, Hachinger et al. (2012) carried out spectroscopic
modeling for SN 2009dc and discussed the model alternatives, such as a 2 M⊙
rotating WD, a core-collapse SN, and a CSM interaction scenario. Overall,
Hachinger et al. (2012) found the interaction scenario to be the most
promising in that it does not require the progenitor to be super-massive.
Taubenberger et al. (2013) furthered this discussion in conjunction with their
late time comparisons and conclude that the models of Hachinger et al. (2012)
do not simultaneously match the peak brightness and decline of SN 2009dc (see
also Yamanaka et al. 2013). Following the interaction scenario of Hachinger et
al. (2012), Taubenberger et al. (2013) propose a non-violent merger model that
produces $\sim$1M⊙ of 56Ni and is enshrouded by $\sim$0.6$-$0.7M⊙ of
C$+$O-rich material. In order to reconcile the low 56Ni production,
Taubenberger et al. (2013) note that additional luminosity from interaction
with CSM is required during the first two months post-explosion. Further
support for CSM interaction comes from the observed suppression of the double
peak in the _I_ -band, which is thought to arise from a breaking of ejecta
stratification in the outermost layers (Kasen, 2006; Kamiya et al., 2012).
It is not yet clear if SN 2003fg, 2006gz (A.32), 2007if (A.34), and SN 2009dc
are the result of a single super-Chandrasekhar mass WD star, given that even
in our galaxy there is no observational evidence for the existence of such a
system. Likewise, there is no direct observational evidence for the presence
of very rapidly rotating massive WD stars, either single WDs or in binary
systems as well. In fact, no double-degenerate close binary systems with a
total mass amounting to super-Chandrasekhar mass configurations that can merge
in Hubble-time have been found (Parthasarathy et al., 2007). Therefore, our
current understanding of the origin of over-luminous SN Ia is limited, and
more observations are needed. For example, progress has been made with the
recent discovery of 24 merging WD systems via the extremely low mass Survey
(see Kilic et al. 2012 and references therein), however it is unclear if any
are systems that would produce a normal SN Ia.
### 4.4 The Peculiar SN 2002cx-like Class of SN
#### 4.4.1 SN 2002cx in CGCG-044-035
Li et al. (2003) considered SN 2002cx as “the most peculiar known SN Ia”
(Wood-Vasey et al., 2002b). They obtained photometric and spectroscopic
observations which revealed it to be unique among all observed SN Ia. Li et
al. (2003) described SN 2002cx as having SN 1991T-like pre-maximum spectrum, a
SN 1991bg-like luminosity, and expansion velocities roughly half those of
normal SN Ia.
Photometrically, SN 2002cx has a broad peak in the _R_ -band, a plateau phase
in the _I_ -band, and a slow late time decline. The _B_ $-$ _V_ color
evolution are described as nearly normal, while the _V_ $-$ _R_ and _V_ $-$
_I_ colors are redder than normal. Spectra of SN 2002cx during early phases
evolve rapidly and are dominated by lines from IMEs and IPEs, but the features
are weak overall. In addition, emission lines are present around 7000 Å during
post-maximum light phases, while the late time nebular spectrum shows narrow
lines of iron and cobalt.
Jha et al. (2006a) presented late time spectroscopy of SN 2002cx, which
includes spectra at 227 and 277 days post-maximum light. They considered it as
a prototype of a new subclass of SN Ia. The spectra do _not_ appear to be
dominated by the forbidden emission lines of iron, which is not expected
during the “nebular phase,” where instead they find a number of P Cygni
profiles of Fe II at exceptionally low expansion velocities of $\sim$ 700 km
s-1 (Branch et al., 2004a). A tentative identification of O I $\lambda$7774 is
also reported for SN 2002cx, suggesting the presence of oxygen-rich material.
Currently, it is difficult to explain all the observed photometric and
spectroscopic properties of SN 2002cx using the standard SN Ia models (see
Foley et al. 2013). However, the spectral characteristics of SN 2002cx support
pure deflagration or failed-detonation models that leave behind a bound
remnant instead of delayed detonations (Jordan et al., 2012; Kromer et al.,
2013a; Hillebrandt et al., 2013).
#### 4.4.2 SN 2005hk in UGC 00272
Phillips et al. (2007) presented extensive multi-color photometry and optical
spectroscopy of SN 2005hk (Quimby et al., 2005). Sahu et al. (2008) also
studied the spectrophotometric evolution SN 2005hk, covering pre-maximum phase
to around 400 days after the event. These datasets reveal that SN 2005hk is
_nearly_ identical in its observed properties to SN 2002cx. Both supernovae
exhibited high ionization SN 1991T-like pre-maximum light spectra but with low
peak luminosities like that of SN 1991bg. The spectra reveal that SN 2005hk,
like SN 2002cx, has expansion velocities that are roughly half those of
typical SN Ia.
The _R_ and _I_ -band light curves of both supernovae are also peculiar for
not displaying the secondary maximum observed for normal SN Ia. Phillips et
al. (2007) constructed a bolometric light curve from 15 days before to 60 days
after _B_ -band maximum. They conclude that the shape and exceptionally low
peak luminosity of the bolometric light curve, low expansion velocities, and
absence of a secondary maximum in the NIR light curves are in reasonable
agreement with model calculations of a three-dimensional deflagration that
produces 0.25 M⊙ of 56Ni. Note however that the low amount of continuum
polarization observed for SN 2005hk ($\sim$ 0.2%$-$0.4%) is far too similar to
that of more normal SN Ia to serve as an explanation for the spectroscopic
peculiarity of SN 2005hk, and possibly other SN 2002cx-like events (Chornock
et al., 2006; Maund et al., 2010a).
#### 4.4.3 Sub-luminous SN 2007qd
McClelland et al. (2010) obtained multi-band photometry and multi-epoch
spectroscopy of SN 2007qd (Bassett et al., 2007). Its observed properties
place it broadly between those of the peculiar SN 2002cx and SN 2008ha (A.37).
Optical photometry indicate a fast rise-time and a peak absolute _B_ -band
magnitude of $-$15.4. McClelland et al. (2010) carried out spectroscopy of SN
2007qd near maximum brightness and detect signatures of IMEs. They find the
photospheric velocity to be 2800 km s-1 near maximum light, and note that this
is $\sim$ 4000 and 7000 km s-1 less than that inferred for SN 2002cx and
normal SN Ia, respectively. McClelland et al. (2010) find that the peak
luminosities of SN 2002cx-like objects are well correlated with their light
curve stretch and photospheric velocities.
#### 4.4.4 SN 2009ku
SN 2009ku was discovered by Pan-STARS-1 as a SN Ia belonging to the peculiar
SN 2002cx class. Narayan et al. (2011) studied SN 2009ku and find that while
its multi-band light curves are similar to that of SN 2002cx, they are
slightly broader and have a later rise to _g_ -band maximum. Its peak
brightness was found to be M${}_{\emph{V}}$ = $-$18.4 and the ejecta velocity
at 18 days after maximum brightness was found to be $\sim$ 2000 km s-1.
Spectroscopically, SN 2009ku is similar to SN 2008ha (A.37). Narayan et al.
(2011) note that the high luminosity and low ejecta velocity for SN 2009ku is
not in agreement with the trend seen for SN 2002cx class of SN Ia. The
spectroscopic and photometric characteristics of SN 2009ku indicate that the
SN 2002cx class of SN Ia are not homogeneous, and that the SN 2002cx class of
events may have a significant dispersion in their progenitor population and/or
explosion physics (see also Kasliwal et al. 2012 for differences between this
class and sub-luminous “calcium-rich” transients).
### 4.5 PTF11kx and the “Ia-CSM” Class of SN Ia
#### 4.5.1 PTF11kx: A case for single-degenerate scenarios?
Dilday et al. (2012) studied the photometric and spectroscopic properties of
another unique SN Ia event, PTF11kx. Using time series, high-resolution
optical spectra, they find direct evidence supporting a single-degenerate
progenitor system based on several narrow, temporal ($\sim$ 65 km s-1)
spectroscopic features of the hydrogen Balmer series, He I, Na I, Ti II, and
Fe II. In addition, and for the first time, PTF11kx observations reveal
strong, narrow, highly time-dependent Ca II absorption features that change
from saturated absorption signatures to emission lines within $\sim$ 40 days.
Dilday et al. (2012) considered the details of these observations and
concluded that the complex CSM environment that enshrouds PTF11kx is strongly
indicative of mass loss or “outflows,” prior to the onset of the explosion of
the progenitor system. Other SN Ia have exhibited narrow, temporal Na D lines
before (e.g., SN 2006X, 2007le; see Simon et al. 2009; Patat et al. 2009,
2010, 2011; Sternberg et al. 2011), but none have been reported as having
signatures of these particular ions, which are clearly present in the high-
resolution spectra of PTF11kx. On the whole, and during the earliest epochs,
Dilday et al. (2012) find that the underlying SN Ia spectroscopic component of
PTF11kx most closely resembles that of SN 1991T (Filippenko et al., 1992a;
Gómez & López, 1998) and 1999aa (Garavini et al., 2004).
As for the late time phases, Silverman et al. (2013b) studied spectroscopic
observations of PTF11kx from 124 to 680 days post-maximum light and find that
its nebular phase spectra are markedly different from those of normal SN Ia.
Specifically, the late time spectra of PTF11kx are void of the strong cobalt
and iron emission features typically seen in other SN 1991T/1999aa-like and
normal SN Ia events (e.g., Ruiz-Lapuente & Lucy 1992; Salvo et al. 2001;
Branch et al. 2003; Stehle et al. 2005; Kotak et al. 2005; McClelland et al.
2013; Silverman et al. 2013a). For the most part, the late time spectra of
PTF11kx are seen to be dominated by broad (FWHM $\sim$ 2000 km s-1) H$\alpha$
emission and strong Ca II emission features that are superimposed onto a
relatively blue, overly luminous continuum level that may be serving to wash
out the underlying SN Ia spectroscopic information. Silverman et al. (2013b)
note that the H$\alpha$ emission increases in strength for $\sim$1 yr before
decreasing. In addition, from the absence of strong H$\beta$, He I, and O I
emission, as well as a larger than normal late time luminosity, Silverman et
al. (2013b) conclude that PTF11kx indeed interacted with some form of CSM
material; possibly of multiply thin shells, shocked into radiative modes of
collisional excitation as the SN ejecta overtakes the slower-moving CSM.
However, it should be noted that it is not yet clear if the CSM originates
from a single-degenerate scenario or a H-rich layer of material that is
ejected prior to a double-degenerate merger event (Shen et al. 2013, see also
Soker et al. 2013).
#### 4.5.2 SN 2002ic
Hamuy et al. (2003) detected a large H$\alpha$ emission in the spectra of SN
2002ic (Wood-Vasey et al., 2002a). Seven days before to 48 days after maximum
light, the optical spectra of SN 2002ic exhibit normal SN Ia spectral features
in addition to the strong H$\alpha$ emission. The H$\alpha$ emission line in
the spectrum of SN 2002ic consists of a narrow component atop a broad
component (FWHM of about 1800 km s-1). Hamuy et al. (2003) argue that the
broad component arose from ejecta$-$CSM interaction. By day $+$48, they find
that the spectrum is similar to that of SN 1990N. Hamuy et al. (2003) argue
that the progenitor system contained a massive AGB star, associated with a few
solar masses of hydrogen-rich CSM.
Kotak et al. (2004) obtained the first high resolution, high S/N spectrum of
SN 2002ic. The resolved H$\alpha$ line has a P Cygni-type profile, indicating
the presence of a dense, slow-moving outflow (about 100 km s-1). They also
find a relatively large and unusual NIR excess and argue that this is the
result of an infrared light-echo originating from the presence of CSM. They
estimate the mass of CSM to be more than 0.3 M⊙, produced by a progenitor mass
loss rate greater than 10-4 M⊙ yr-1. For the progenitor, Kotak et al. (2004)
favor a single-degenerate system with a post-AGB companion star.
Wood-Vasey et al. (2004) obtained pre-maximum and late time photometry of SN
2002ic and find that a non-SN Ia component of the light curve becomes
pronounced about 20 days post-explosion. They suggest the non-SN Ia component
to be due to heating from a shock interaction between SN ejecta and CSM. Wood-
Vasey et al. (2004) also suggest that the progenitor system consisted of a WD
and an AGB star in the protoplanetary nebula phase. Wood-Vasey et al. (2004)
and Sokoloski et al. (2006) proposed that a nova shell ejected from a
recurrent nova progenitor system, creating the evacuated region around the
explosion center of SN 2002ic. They suggest that the periodic shell ejections
due to nova explosions on a WD sweep up the slow wind from the binary
companion, creating density variations and instabilities that lead to
structure in the circumstellar medium. This type of phenomenon may occur in SN
Ia with recurrent nova progenitors, however Schaefer (2011) recently reported
on an ongoing observational campaign of recurrent novae (RN) orbital period
changes between eruptions. For at least two objects (CI Aquilae and U
Scorpii), he finds that the RN _lose_ mass, thus making RN unlikely
progenitors for SN Ia.
Nearly one year after the explosion, Wang et al. (2004) found that the
supernova had become fainter overall, but H$\alpha$ emission had brightened
and broadened compared to earlier observations. From their spectropolarimetry
observations, Wang et al. (2004) find that hydrogen-rich matter is
asymmetrically distributed. Likewise, Deng et al. (2004) also found evidence
of a hydrogen-rich asymmetric circumstellar medium. From their observations of
SN 2002ic, Wang et al. (2004) conclude that the event took place within a
“dense, clumpy, disk-like” circumstellar medium. They suggest that the star
responsible for SN 2002ic could either be a post-AGB star or WD companion (see
also Hachisu et al. 1999; Han & Podsiadlowski 2006).
#### 4.5.3 SN 2005gj
Similar to SN 2002ic, Aldering et al. (2006) argue that SN 2005gj is a SN Ia
in a massive circumstellar envelope (see also Prieto et al. 2007), which is
located in a low metallicity host galaxy with a significant amount of star
formation. Their first spectrum of SN 2005gj shows a blue continuum level with
broad and narrow H$\alpha$ emission. Subsequent spectra reveal muted SN Ia
features combined with broad and narrow H$\gamma$, H$\beta$, H$\alpha$ and He
I $\lambda\lambda$5876, 7065 in emission, where high resolution spectra reveal
narrow P Cygni profiles. An inverted P Cygni profile for [O III] $\lambda$5007
was also detected, indicating top-lighting effects from CSM interaction
(Branch et al., 2000). From their early photometry of SN 2005gj, Aldering et
al. (2006) find that the interaction between the supernova ejecta and CSM was
much weaker for SN 2002ic. Notably, both Aldering et al. (2006) and Prieto et
al. (2007) agree that a SN 1991T-like spectrum can account for many of the
observed profiles with an assumed increase in continuum radiation from
interaction with the hydrogen-rich material.
Aldering et al. (2006) also find that the light curve and measured velocity of
the unshocked CSM imply mass loss as recent as 1998. This is in contrast to SN
2002ic, for which an inner cavity in the circumstellar matter was inferred
(Wood-Vasey et al., 2004). Furthermore, SN 1997cy, SN 2002ic, and SN 2005gj
all exhibit large CSM interactions and are from low-luminosity hosts.
Consistent with this interpretation for CSM interactions is the recent report
by Fox & Filippenko (2013) that a NIR re-brightening, possibly due to emission
from “warm” dust, took place at late times for both SN 2002ic and 2005gj.
Notably, and in contrast to SN 2002ic, Fox & Filippenko (2013) find that the
mid-IR luminosity of SN 2005gj increased to $\sim$ twice its early epoch
brightness.
## 5 Summary and Concluding Remarks
Observations of a significant number of SN Ia during the last two decades have
enabled us to document a larger expanse of their physical properties which is
manifested through spectrophotometric diversity. While in general SN Ia have
long been considered a homogeneous class, they do exhibit up to 3.5 mag
variations in the peak luminosity, whereas “normal” SN Ia dispersions are
$\sim$1 mag, and constitute several marginally distinct subtypes (Blondin et
al., 2012; Scalzo et al., 2012; Silverman et al., 2013d; Foley et al., 2013;
Dessart et al., 2013a). Consequently, the use of normal SN Ia for cosmological
purposes depends on empirical calibration methods (e.g., Bailey et al. 2009),
where one of the most physically relevant methods is the use of the width-
luminosity relation (Phillips, 1993; Phillips et al., 1999).
Understanding the physics and origin of the width-luminosity-relationship of
SN Ia light curves is an important aspect in the modeling of SN Ia (Khokhlov
et al., 1993; Lentz et al., 2000; Timmes et al., 2003; Nomoto et al., 2003;
Kasen & Woosley, 2007; Kasen et al., 2009; Meng et al., 2011; Blondin et al.,
2011). Brighter SN Ia often have broad light curves that decline slowly after
peak brightness. Slightly less bright or dimmer SN Ia have narrower and
relatively rapidly declining light curves. In addition, several SN Ia do not
follow the width-luminosity-relationship (e.g., SN 2001ay, 2004dt, 2010jn,
SCC, CL and SN 2002cx-like SN Ia), which reinforces the notion that a
significant number of physically relevant factors influence the diversity of
SN Ia overall (see Wang et al. 2012; Baron et al. 2012).
Despite the ever increasing number of caught-early supernovae, our perspective
on their general properties and individual peculiarities undergoes a continual
convergence toward a set of predictive standards with which models must be
seen to comply. The most recent observational example is that of SN 2012fr
(Maund et al., 2013; Childress et al., 2013c), a normal/low-velocity-gradient
SN Ia that has been added to the growing list of similar SN Ia that exhibit
stark evidence for a distinctly separate region of “high-velocity” material
($>$16,000 km s-1). While the origin of high velocity features in the spectra
of SN Ia is not well understood, it is concurrent with polarization signatures
in most cases which implies some amount of ejecta density asymmetries (e.g.,
Kasen et al. 2003; Wang & Wheeler 2008; Smith et al. 2011; Maund et al. 2013).
Furthermore, since understanding the temporal behavior of high velocity Si
II/Ca II depends on knowing the same for the photospheric component, studies
that focus on velocity gradients and potential velocity-plateaus of the
photospheric component could make clearer the significance of the physical
separation between these two regions of material (see Patat et al. 1996; Kasen
et al. 2003; Tanaka et al. 2008; Foley et al. 2012c; Parrent et al. 2012;
Scalzo et al. 2012; Childress et al. 2013c; Marion et al. 2013). However, it
is at least certain that all viable models that encompass “normal” SN Ia
conditions must account for the range of properties related to velocity
evolution (see Blondin et al. 2012 and references therein), the occasionally
observed however potentially under-detected signatures of C$+$O material at
both low and high velocities (Thomas et al., 2007; Parrent et al., 2011;
Thomas et al., 2011b; Folatelli et al., 2012; Silverman & Filippenko, 2012;
Piro & Nakar, 2013; Mazzali et al., 2013), a high-velocity region of either
clumps or an amorphous plumage of opaque Si-, Ca-based material (Gamezo et
al., 2004; Leonard et al., 2005; Wang et al., 2007; Maund et al., 2010b; Piro,
2011), and the supposed blue/red-shift of nebular lines emitted from the inner
IPE-rich material (Maeda et al., 2010b; McClelland et al., 2013; Silverman et
al., 2013a).
For at least normal SN Ia, there remain two viable explosion channels (with a
few sub- and super-MCh sub-channels) regardless of the hierarchical dominance
of each at various redshifts and/or ages of galactic constituents (c.f., Röpke
et al. 2012; Hachisu et al. 2012; Seitenzahl et al. 2013; Pakmor et al. 2013;
Moll et al. 2013; Claeys et al. 2014). Also, it may or may not be the case
that some SN Ia are 2$+$ subtypes viewed upon from various lines of sight
amidst variable CSM interaction (Maeda et al., 2010b; Foley et al., 2012a;
Scalzo et al., 2012; Leloudas et al., 2013; Dan et al., 2013; Moll et al.,
2013; Dessart et al., 2013a). However, with the current lack of complete
observational coverage in wavelength, time, and mode (i.e. spectrophotometric
and spectropolarimetric observations) for all SN Ia subtypes and “well-
observed” events, there is a limit for how much constraint can be placed on
many of the proposed explosion models and progenitor scenarios. That is,
despite observational indications for and theoretical consistencies with the
supposition of multiple progenitor channels, the observed diversity of SN Ia
does not yet necessitate that each spectrophotometric subtype be from a
distinctly separate explosive binary scenario than that of others within the
SN Ia family of observed events; particularly so for normal SN Ia.
For the purposes of testing the multifaceted predictions of theoretical
explosion models, time series spectroscopic observations of SN Ia serve to
visualize the post-explosion material of an unknown progenitor system. For
example during the summer of 2011 astronomers bore witness to SN 2011fe, the
best observed normal type Ia supernova of the modern era. The prompt discovery
and follow-up of this nearby event uniquely allowed for a more complete record
of observed properties than all previous well-observed events. More
specifically, the full range (in wavelength and time) of rapid spectroscopic
changes was documented with continual day-to-day follow-up into the object’s
post-maximum light phases and well beyond. However, the observational side of
visualizing other SN Ia remains inefficient without the logistical
coordination of many telescope networks (e.g., LCOGT; Brown et al. 2013),
telescopes large enough to make nearly all SN “nearby” in terms of improved
signal-to-noise ratios (e.g., The Thirty Meter Telescope, The Giant Magellan
Telescope), or a space-based facility dedicated to the study of such time
sensitive UV$-$optical$-$NIR transients.
Existing SN Ia surveys are currently acting toward optimizing a steady flow of
discoveries, while other programs have produced a significant number of
publicly available spectra (Richardson et al., 2001; Matheson et al., 2008;
Blondin et al., 2012; Yaron & Gal-Yam, 2012; Silverman et al., 2012c;
Folatelli et al., 2013). However, for the longterm future we believe it is
imperative to begin a discussion of a larger (digital) network of
international collaboration by way of (data-) cooperative competition like
that done for both The Large Hadron Collider Experiment and Fermi Lab’s
Tevatron, with multiple competing experiments centered about mutual goals and
mutual resources. Otherwise we feel the simultaneous collection of even very
high quality temporal datasets by multiple groups will continue to create an
inefficient pursuit of over-observing the most high profile event(s) of the
year with a less than complete dataset.
Such observational pursuits require an increasingly focused effort toward
observing bright and nearby events. For example, 206 supernovae were reported
in 1999 and 67 were brighter than 18th magnitude while only three reached
$\sim$ 13 magnitude212121Quoted from the archives page of
http://www.rochesterastronomy.org/snimages/.. By 2012 the number of found
supernovae increased to 1045 while 78 were brighter than 16th magnitude and
five brighter than 13th magnitude. This clearly indicates that supernovae
caught early are more prevalent than $\sim$ 15 years ago and it is worthwhile
for multiple groups to continually increase collaborative efforts for the
brightest events. Essentially this could be accomplished without interfering
with spectrum-limited high[er]-z surveys by considering a distance threshold
($\lesssim$ 10$-$30 Mpc) as part of the public domain. Additionally, surveys
that corroborate the immediate release of discoveries would further increase
the number of well-observed events and could be supplemented and sustained
with staggered observations given that there are two celestial hemispheres,
unpredictable weather patterns, and caught-early opportunities nearly every
week during active surveying.
In conclusion, to extract details of the spectroscopic behavior for all SN Ia
subtypes, during all phases, larger samples of well-observed events are
essential, beginning from as close to the onset of the explosion as possible
(e.g., SN 1999ac, 2009ig, 2011fe, 2012cg, 2012fr), where SN Ia homogeneity
diverges the most (see Zheng et al. 2013 for the most recent instance in SN
2013dy). Near-continuous temporal observations are most important for at least
the first 1$-$2 months post-explosion and biweekly to monthly follow-up
thereafter for $\sim$ 1 year. SN Ia spectra are far too complicated to do so
otherwise. Even normal SN Ia deserve UV$-$optical$-$IR spectroscopic follow-up
at a 1:1 to 2:1 ratio between days passed and spectrum taken, whenever
possible, given that fine differences between normal SN Ia detail the variance
in explosion mechanism parameters and initial conditions of their unobserved
progenitor systems. It is through such observing campaigns that the true
diversity to the underlying nature of SN Ia events will be better understood.
Acknowledgements
This work was supported in part by NSF grant AST-0707704, and US DOE Grant DE-
FG02- 07ER41517, and by SFB 676, GRK 1354 from the DFG. Support for Program
number HST-GO- 12298.05-A was provided by NASA through a grant from the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555. We wish to acknowledge the use of the Kurucz & Bell (1995) line
list and colorbrewer2.org for the construction of Figure 3.
This review was made possible by collaborative discussions at the 2011 UC-UC-
HIPACC International AstroComputing Summer School on Computational Explosive
Astrophysics. We would like to thank Dan Kasen and Peter Nugent for organizing
the program and providing the environment for a productive summit, and we hope
that such programs for summer learning opportunities will continue in the
future. We worked on this review during the visits of MP to the Homer L. Dodge
Department of Physics and Astronomy, University of Oklahoma, Norman, OK, USA.,
McDonnell Center for the Space Science, Department of Physics, Washington
University in St. Louis, USA., National Astronomical Observatory of Japan
(NAOJ), Mitaka, Tokyo, Japan., and Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Pune, India.
MP is thankful to Prof. David Branch, Prof. Eddie Baron, Prof. Ramanath
Cowsik, Prof. Shoken Miyama, Prof. Masahiko Hayashi, Prof. Yoichi Takeda,
Prof. Wako Aoki, Prof. Ajit Kembhavi, Prof. Kandaswamy Subramanian, and Prof.
T. Padamanabhan for their kind support, encouragement, and hospitality. JTP
would like to thank the University of Oklahoma Supernova Group, Rollin Thomas,
and Alicia Soderberg for several years of support and many enlightening
discussions on reading supernova spectra. JTP wishes to acknowledge helpful
discussions with B. Dilday, R. A. Fesen, R. Foley, M. L. Graham, D. A. Howell,
G. H. Marion, D. Milisavljevic, P. Milne, D. Sand, and S. Valenti, as well as
S. Perlmutter for an intriguing conversation on “characteristic information of
SN Ia” at the 221st American Astronomical Society Meeting in Long Beach, CA.
JTP is also indebted to Natalie Buckley-Medrano for influential comments on
the text and figures presented here.
Finally, we would like to pay special tribute to our referee, Michael
Childress, whose critical comments and suggestions were substantially helpful
for the presentation of this review.
Table 1 : References for Spectra in Figures 5$-$13 SN Name | References
---|---
SN 1981B | Branch et al. 1983
SN 1986G | Cristiani et al. 1992
SN 1989B | Barbon et al. 1990;
| Wells et al. 1994
SN 1990N | Mazzali et al. 1993;
| Gómez & López 1998
SN 1991T | Filippenko et al. 1992a;
| Gómez & López 1998
SN 1991bg | Filippenko et al. 1992b;
| Turatto et al. 1996
SN 1994D | Patat et al. 1996;
| Ruiz-Lapuente 1997;
| Gómez & López 1998;
| Blondin et al. 2012
SN 1994ae | Blondin et al. 2012
SN 1995D | Sadakane et al. 1996;
| Blondin et al. 2012
SN 1996X | Salvo et al. 2001;
| Blondin et al. 2012
SN 1997br | Li et al. 1999;
| Blondin et al. 2012
SN 1997cn | Turatto et al. 1998
SN 1998aq | Branch et al. 2003
SN 1998bu | Jha et al. 1999;
| Matheson et al. 2008
SN 1998de | Modjaz et al. 2001;
| Matheson et al. 2008
SN 1998es | Matheson et al. 2008
SN 1999aa | Garavini et al. 2004
SN 1999ac | Garavini et al. 2005;
| Phillips et al. 2006;
| Matheson et al. 2008
SN 1999by | Garnavich et al. 2004;
| Matheson et al. 2008
SN 1999ee | Hamuy et al. 2002
SN 2000E | Valentini et al. 2003
SN 2000cx | Li et al. 2001
SN 2001V | Matheson et al. 2008
SN 2001ay | Krisciunas et al. 2011
SN 2001el | Wang et al. 2003
SN 2002bo | Benetti et al. 2004;
| Blondin et al. 2012
SN 2002cx | Li et al. 2003
Table 2 : References for Spectra in Figures 5$-$13 SN Name | References
---|---
SN 2002dj | Pignata et al. 2008;
| Blondin et al. 2012
SN 2002er | Kotak et al. 2005
SN 2003cg | Elias-Rosa et al. 2006;
| Blondin et al. 2012
SN 2003du | Gerardy et al. 2004;
| Anupama et al. 2005b;
| Leonard et al. 2005;
| Stanishev et al. 2007;
| Blondin et al. 2012
SN 2003hv | Leloudas et al. 2009;
| Blondin et al. 2012
SN 2004S | Krisciunas et al. 2007
SN 2004dt | Leonard et al. 2005;
| Altavilla et al. 2007;
| Blondin et al. 2012
SN 2004eo | Pastorello et al. 2007a
SN 2005am | Blondin et al. 2012
SN 2005cf | Garavini et al. 2007;
| Wang et al. 2009b;
| Bufano et al. 2009
SN 2005cg | Quimby et al. 2006b
SN 2005hk | Chornock et al. 2006;
| Phillips et al. 2007;
| Blondin et al. 2012
SN 2005hj | Quimby et al. 2007
SN 2006D | Blondin et al. 2012
SN 2006X | Wang et al. 2008a;
| Yamanaka et al. 2009b;
| Blondin et al. 2012
SN 2006bt | Foley et al. 2010b
SN 2006gz | Hicken et al. 2007
SN 2007ax | Blondin et al. 2012
SN 2007if | Silverman et al. 2011;
| Blondin et al. 2012
SN 2008J | Taddia et al. 2012
SN 2008ha | Foley et al. 2009
SN 2009dc | Silverman et al. 2011;
| Taubenberger et al. 2011
PTF09dav | Sullivan et al. 2011b
SN 2011fe | Parrent et al. 2012
SN 2011iv | Foley et al. 2012b
Table 3 : References for M${}_{\emph{B}}$(Peak) and $\Delta$m15(_B_) plotted in Figure 14: 1981$-$1992 SN Name | References
---|---
SN 1981B | Leibundgut et al. 1993;
| Saha et al. 1996;
| Hamuy et al. 1996;
| Saha et al. 2001b
SN 1984A | Barbon et al. 1989
SN 1986G | Filippenko et al. 1992b;
| Ruiz-Lapuente & Lucy 1992;
| Leibundgut et al. 1993
SN 1989B | Barbon et al. 1990;
| Wells et al. 1994;
| Richmond et al. 1995;
| Saha et al. 1999;
| Contardo et al. 2000;
| Saha et al. 2001a
SN 1990N | Saha et al. 1997;
| Lira et al. 1998;
| Saha et al. 2001a
SN 1991T | Leibundgut et al. 1993;
| Lira et al. 1998;
| Phillips et al. 1999;
| Krisciunas et al. 2004;
| Contardo et al. 2000;
| Saha et al. 2001b;
| Tsvetkov et al. 2011
SN 1991bg | Leibundgut et al. 1993;
| Turatto et al. 1996;
| Mazzali et al. 1997;
| Contardo et al. 2000
SN 1992A | Leibundgut et al. 1993;
| Hamuy et al. 1996;
| Drenkhahn & Richtler 1999;
| Contardo et al. 2000
SN 1992K | Hamuy et al. 1994
SN 1992al | Misra et al. 2005
SN 1992bc | Maza et al. 1994;
| Contardo et al. 2000
SN 1992bo | Maza et al. 1994;
| Contardo et al. 2000
Table 4 : References for M${}_{\emph{B}}$(Peak) and $\Delta$m15(_B_) plotted in Figure 14: 1994$-$1999 SN Name | References
---|---
SN 1994D | Hoflich et al. 1995;
| Richmond et al. 1995;
| Patat et al. 1996;
| Vacca & Leibundgut 1996;
| Drenkhahn & Richtler 1999;
| Contardo et al. 2000
SN 1994ae | Contardo et al. 2000
SN 1995D | Sadakane et al. 1996;
| Contardo et al. 2000
SN 1996X | Phillips et al. 1999;
| Salvo et al. 2001
SN 1997br | Li et al. 1999
SN 1997cn | Turatto et al. 1998
SN 1998aq | Riess et al. 1999;
| Saha et al. 2001a
SN 1998bu | Jha et al. 1999;
| Hernandez et al. 2000;
| Saha et al. 2001a
SN 1998de | Modjaz et al. 2001
SN 1998es | Jha et al. 2006b;
| Tsvetkov et al. 2011
SN 1999aa | Krisciunas et al. 2000;
| Li et al. 2003;
| Tsvetkov et al. 2011
SN 1999ac | Jha et al. 2006b;
| Phillips et al. 2006
SN 1999aw | Strolger et al. 2002
SN 1999by | Vinkó et al. 2001;
| Howell et al. 2001;
| Garnavich et al. 2004;
| Sullivan et al. 2011b
SN 1999ee | Stritzinger et al. 2002;
| Krisciunas et al. 2004
Table 5 : References for M${}_{\emph{B}}$(Peak) and $\Delta$m15(_B_) plotted in Figure 14: 2000$-$2005 SN Name | References
---|---
SN 2000E | Valentini et al. 2003
SN 2000cx | Li et al. 2001;
| Candia et al. 2003;
| Sollerman et al. 2004
SN 2001V | Vinkó et al. 2003
SN 2001ay | Krisciunas et al. 2011
SN 2001el | Krisciunas et al. 2003
SN 2002bo | Benetti et al. 2004;
| Stehle et al. 2005
SN 2002cv | Elias-Rosa et al. 2008
SN 2002cx | Li et al. 2003
SN 2002dj | Pignata et al. 2008
SN 2002er | Pignata et al. 2004
SN 2003cg | Elias-Rosa et al. 2006
SN 2003du | Anupama et al. 2005b;
| Stanishev et al. 2007;
| Tsvetkov et al. 2011
SN 2003fg | Howell et al. 2006;
| Yamanaka et al. 2009b
| Scalzo et al. 2010
SN 2003hv | Leloudas et al. 2009
SN 2004S | Misra et al. 2005;
| Krisciunas et al. 2007
SN 2004dt | Altavilla et al. 2007
SN 2004eo | Pastorello et al. 2007a
SN 2005am | Brown et al. 2005
SN 2005bl | Taubenberger et al. 2008;
| Hachinger et al. 2009
SN 2005cf | Pastorello et al. 2007b;
| Wang et al. 2009b
SN 2005hk | Phillips et al. 2007
Table 6 : References for M${}_{\emph{B}}$(Peak) and $\Delta$m15(_B_) plotted in Figure 14: 2006$-$2012 SN Name | References
---|---
SN 2006X | Wang et al. 2008b
SN 2006bt | Hicken et al. 2009b;
| Foley et al. 2010b
SN 2006gz | Hicken et al. 2007;
| Scalzo et al. 2010
SN 2007ax | Kasliwal et al. 2008
SN 2007if | Scalzo et al. 2010
SN 2007qd | McClelland et al. 2010
SN 2008J | Taddia et al. 2012
SN 2008ha | Foley et al. 2009
SN 2009dc | Yamanaka et al. 2009b;
| Scalzo et al. 2010;
| Silverman et al. 2011;
| Taubenberger et al. 2011
SN 2009ig | Foley et al. 2012c
SN 2009ku | Narayan et al. 2011
SN 2009nr | Khan et al. 2011b;
| Tsvetkov et al. 2011
PTF09dav | Sullivan et al. 2011b
SN 2010jn | Hachinger et al. 2013
SN 2011fe | Richmond & Smith 2012;
| Munari et al. 2013;
| Pereira et al. 2013
SN 20011iv | Foley et al. 2012b
SN 2012cg | Silverman & Filippenko 2012;
| Munari et al. 2013
SN 2012fr | Childress et al. 2013c
## Appendix A Some recent SN Ia, Continued
### A.1 Peculiar SN 1997br in ESO 576-G40
Li et al. (1999) presented observations of the peculiar SN 1991T-like, SN
1997br (Bao Supernova Survey et al., 1997). Hatano et al. (2002) analyzed the
spectra of SN 1997br and raised the question of whether or not Fe III and Ni
III features in the early spectra are produced by 54Fe and 58Ni rather than by
56Fe and 56Ni. In addition, Hatano et al. (2002) discussed the issue of SN
1991T-like events as more powerful versions of normal SN Ia, rather than a
physically distinct subgroup of events.
### A.2 SN 1997cn in NGC 5490
Turatto et al. (1998) studied the faint SN 1997cn, which is located in an
elliptical host galaxy (Li et al., 1997; Turatto et al., 1997). Like SN
1991bg, spectra of SN 1997cn show a deep Ti II trough between 4000 and 5000 Å,
strong Ca II IR3 absorption features, a large $\mathcal{R}$(“Si II”), and slow
mean expansion velocities.
### A.3 SN 1997ff and other “farthest known” SN Ia
With a redshift of z = 1.7, SN 1997ff was the most distant SN Ia discovered at
that time (Riess et al., 2001; Benítez et al., 2002). There have been
$\sim$110 high-z (1 $<$ z $<$ 2) SN Ia discoveries since SN 1997ff (Riess et
al., 2004, 2007; Suzuki et al., 2012), with the most recent and and one of the
most distant SN Ia known being “SN UDS10Wil” at z = 1.914 (Jones et al.,
2013). With these and future observations of “highest-z” SN Ia, constraints on
DTD timescales (Strolger et al., 2010; Graur et al., 2013) and dark energy
(Rubin et al., 2013) will certainly improve.
### A.4 SN 1998aq in NGC 3982
Branch et al. (2003) used SYNOW to study 29 optical spectra of the normal SN
1998aq (Hurst et al., 1998), covering 9 days before to 241 days after maximum
light (days $-$9 and $+$241, respectively). Notably, they find evidence for C
II down to 11,000 km s-1, $\sim$ 3000 km s-1 below the cutoff of carbon in the
pure deflagration model, W7 (Nomoto et al., 1984).
### A.5 SN 1999aa in NGC 2595
From day $-$11 to day $+$58, Garavini et al. (2004) obtained 25 optical
spectra of SN 1999aa (Armstrong & Schwartz, 1999). While SN 1999aa appears SN
1991T-like, Garavini et al. (2004) note that the Ca II absorption feature
strengths are between those of SN 1991T (SS) and the SN 1990N (CN), along with
a phase transition to normal SN Ia characteristics that sets in earlier than
SN 1991T. Subsequently, they suggest SN 1999aa to be a link between SN
1991T-likes and spectroscopically normal SN Ia. Evidence of carbon-rich
material is also found in SN 1999aa; decisively as C II $\lambda$6580,
tentatively as C III $\lambda$4649 (see Parrent et al. 2011). A schematic
representation of their SYNOW fitting results is also presented, showing the
inference of Co II, Ni II, and Ni III during the pre-maximum phases. These
results deserve further study from more detailed models.
### A.6 SN 1999ac in NGC 2841
Between day $-$15 and day $+$42, Garavini et al. (2005) obtained spectroscopic
observations of the unusual SN 1999ac (Modjaz et al., 1999). The pre-maximum
light spectra are similar to that of SN 1999aa-like, while appearing
spectroscopically normal during later epochs. Garavini et al. (2005) find
evidence of a fairly conspicuous, heavily blended C II $\lambda$6580 feature
in the day $-$15 spectrum with approximate ejection velocities $>$16,000 km
s-1. By day $-$9, the C II absorption feature is weak or absent amidst
blending with the neighboring 6100 Å feature. This alone indicates that
studies cannot fully constrain SN Ia models without spectra prior to day
$-$10.
### A.7 SN 1999aw in a low luminosity host galaxy
Strolger et al. (2002) find SN 1999aw to be a luminous, slow-declining SN Ia,
similar to 1999aa-like events. Strolger et al. (2002) derive a peak luminosity
of 1.51 x 1043 and a 56Ni mass of 0.76 M⊙.
### A.8 SN 1999by in NGC 2841
Vinkó et al. (2001) presented and discussed the first three pre-maximum light
spectra of SN 1999by (Arbour et al., 1999), where they find it to be a sub-
luminous SN Ia similar to SN 1991bg (Filippenko et al., 1992b; Leibundgut et
al., 1993; Turatto et al., 1996), SN 1992K (Hamuy et al., 1994), SN 1997cn
(Turatto et al., 1998), and SN 1998de (Modjaz et al., 2001); in addition, the
list of sub-luminous SN Ia include SN 1957A, 1960H, 1971I, 1980I, 1986G (see
Branch et al. 1993; Doull & Baron 2011 and references therein) and several
other recently discovered under-luminous SN Ia (Howell, 2001; McClelland et
al., 2010; Hachinger et al., 2009). Pre-maximum spectra of SN 1999by show
relatively strong features due to O, Mg, and Si, which are due to explosive
carbon burning. In addition, blue wavelength regions reveal spectra dominated
by Ti II and some other IPEs.
Meanwhile, Höflich et al. (2002) studied the infrared spectra of SN 1999by,
covering from day $-$4 to day $+$14\. Post-maximum spectra show features which
can be attributed to incomplete Si burning, while further support for
incomplete burning comes from the detection of a pre-maximum C II absorption
feature (Garnavich et al., 2004). Höflich et al. (2002) analyzed the spectra
through the construction of an extended set of delayed detonation models
covering the entire range of normal to sub-luminous SN Ia. They estimate the
56Ni mass for SN 1999by to be on the order of 0.1 M⊙. Garnavich et al. (2004)
obtained _UBVRIJHK_ light curves of SN 1999by. From the photometry of SN
1999by, the recent Cepheid distance to NGC 2841 (Macri et al., 2001), and
minimal dust extinction along the line-of-sight, Garnavich et al. (2004)
derive a peak absolute magnitude of M${}_{\emph{B}}$ = $-$17.15.
In order to assess the role of material asymmetries as being responsible for
the observed peculiarity of sub-luminous SN Ia, Howell et al. (2001) obtained
polarization spectra of SN 1999by near maximum light. They find relatively low
levels of polarization (0.3%$-$0.8%), however significant enough to be
consistent with a 20% departure from spherical symmetry (Maund et al., 2010b).
### A.9 SN 1999ee in IC 5179
From day $-$10 to day $+$53, Stritzinger et al. (2002) obtained well-sampled
_UBVRIz_ light curves of SN 1999ee (Maza et al., 1999). They find the _B_
-band light curve is broader than normal SN Ia, however sitting toward the
over-luminous end of SN Ia peak brightnesses, with M${}_{\emph{B}}$ = $-$19.85
$\pm$ 0.28 and $\Delta$m15 = 0.94.
Hamuy et al. (2002) obtained optical and infrared spectroscopy of SN 1999ee
between day $-$9 and day $+$42\. Before maximum light, the spectra of SN
1999ee are normal, with relatively strong Si II 6100 Å absorption, however
within the SS subtype (Branch et al., 2009). Hamuy et al. (2002) compared the
infrared spectra of SN 1999ee to that of other SN Ia out to 60 days post-
explosion, and find similar characteristics for SN 1999ee and 1994D (Meikle et
al., 1996).
### A.10 SN 2000E in NGC 6951
Valentini et al. (2003) obtained _UBVRIJHK_ photometry and optical spectra of
SN 2000E, which is located in a spiral galaxy. Optical spectra were obtained
from 6 days before _B_ -band maximum to 122 days after _B_ -band maximum. The
photometric observations span 230$+$ days, starting at day $-$16\. The
photometric light curves of SN 2000E are similar to other SS SN Ia, however SN
2000E is classified as a slowly declining, spectroscopically “normal” SN Ia
similar to SN 1990N. Valentini et al. (2003) estimate the 56Ni mass to be 0.9
M⊙ from the bolometric light curve.
### A.11 SN 2000cx in NGC 524
One of the brightest supernovae observed in the year 2000 was the peculiar SN
2000cx, located in an S0 galaxy (Yu et al., 2000; Li et al., 2001; Candia et
al., 2003). It was classified as a SN Ia with a spectrum resembling that of
the peculiar SN 1991T (Chornock et al., 2000). Sollerman et al. (2004)
obtained late time _BVRIJH_ light curves of SN 2000cx covering 360 to 480 days
after maximum. During these epochs, they find relatively constant NIR
magnitudes, indicating the increasing importance (with time) of the NIR
contribution to the bolometric light curve.
Branch et al. (2004b) decomposed the photospheric-phase spectra of SN 2000cx
with SYNOW. Apart from confirming HVFs of Ca II IR3 (which are consistent with
primordial abundances; Thomas et al. 2004), Branch et al. (2004b) also find
HVFs of Ti II. They attribute the odd behavior of SN 2000cx’s _B_ -band light
curve to the time-dependent behavior of these highly line blanketing Ti II
absorption signatures. Branch et al. (2004b) find an absorption feature near
4530 Å in the spectra of SN 2000cx that can be tentatively associated with
H$\beta$ at high velocities, however this feature is more likely due to C III
$\lambda$4649 or S II/Fe II instead (see Parrent et al. 2011 and references
therein).
Rudy et al. (2002) obtained 0.8$-$2.5 $\mu$m spectra of SN 2000cx at day $-$7
and day $-$8 before maximum light. From the $\lambda$10926 line of Mg II, they
find that carbon-burning has taken place up to $\sim$25,000 km s-1. Given the
SS subtype nature of SN 2000cx, the early epoch IR spectra of Rudy et al.
(2002) are valuable for comparison with other SN Ia IR datasets.
### A.12 SN 2001V in NGC 3987
Vinkó et al. (2003) presented photometry of SN 2001V (Jha et al., 2001). They
find that SN 2001V is over-luminous, relative to the majority of SN Ia.
Spectroscopic observations, spanning from day $-$14 to day $+$106, can be
found in Matheson et al. (2008) and reveal it to be a SS SN Ia, consistent
with its observed brightness.
### A.13 Slowly declining SN 2001ay in IC 4423
Krisciunas et al. (2011) obtained optical and near infrared photometry, and
optical and UV spectra of SN 2001ay (Swift et al., 2001). They find maximum
light Si II and Mg II line velocities of $\sim$ 14,000 km s-1, with Si III and
S II near 9,000 km s-1. SN 2001ay is one of the most slowly declining SN Ia.
However, a $\Delta$m15(_B_) = 0.68 is odd given it is not over-luminous like
SCC SN Ia slow decliners. In fact, the 56Ni yield of 0.58 is comparable to
that of many normal SN Ia. Baron et al. (2012) note this apparent WLR
violation is related to a decrease in $\gamma$-ray trapping deeper within the
ejecta due to an overall outward shift of 56Ni, thus creating a fast rise in
brightness followed by a slow decline caused by enhanced heating of the outer
regions of material, which is a consequence of the larger expansion opacities.
### A.14 SN 2001el in NGC 1448
Krisciunas et al. (2003) obtained well-sampled _UBVRIJHK_ light curves of the
nearby (about 18 Mpc) and normal SN 2001el (Monard et al., 2001), from day
$-$11 to day $+$142\. Because Krisciunas et al. (2003) obtained _UBVRI_ and
_JHK_ light curves, they were able to measure a true optical$-$NIR reddening
value (A${}_{\emph{V}}$ = 0.57 mag along the line-of-sight) for the first
time.
Mattila et al. (2005) obtained early time high resolution and low resolution
optical spectra of SN 2001el. They estimate the mass loss rate (assuming
10$-$50 km s-1 wind velocities) from the progenitor system of SN 2001el to be
no greater than 9 x 10-6 M⊙ yr-1 and 5 x 10-5 M⊙ yr-1, respectively. The low
resolution spectrum was obtained 400 days after maximum light with no apparent
signatures of hydrogen Balmer lines. High velocity Ca II was detected out to
34,000 km s-1, while the 6100 Å absorption feature is suspected of harboring
high velocity Si II (see also Kasen et al. 2003).
### A.15 SN 2002bo in NGC 3190
Between day $-$13 and day $+$102, Benetti et al. (2004) collected optical and
NIR spectra and photometry of the BL SN 2002bo (Cacella et al., 2002;
Krisciunas et al., 2004). Estimates on host galaxy extinction from Na D
equivalent width measurements are consistent with the inferred color excess
determined by comparison to the Lira relation (Lira, 1995; Riess et al., 1996;
Phillips et al., 1999). From the time-evolution of the 6100 Å absorption
feature, Benetti et al. (2004) find that SN 2002bo is an intermediary between
the BL SN 1984A and the CN SN 1994D. Benetti et al. (2004) also discuss SN
2002bo argue that some of the IME high velocity material may be primordial,
while most is produced during the explosion and possibly by prolonged burning
in a delayed detonation. This interpretation is also consistent with a lack of
any clear signatures of unburned carbon. Stehle et al. (2005) studied the
abundance stratification by fitting a series of spectra with a Monte Carlo
code and found that the elements synthesized in different stages of burning
are not completely mixed within the ejecta. In the case of SN 2002bo, they
derived the total mass of 56Ni to be 0.52 M⊙. Similar to SN 2001ay’s fast rise
to maximum light (Baron et al., 2012), Stehle et al. (2005) attribute SN
2002bo’s fast rise to outward mixing of 56Ni.
### A.16 SN 2002cv in NGC 3190
The NIR photometry of SN 2002cv reveal an obscured SN Ia (Di Paola et al.,
2002); more than 8 magnitudes of visual extinction. Both optical and NIR
spectroscopy indicate SN 2002cv is most similar to SN 1991T (Meikle et al.,
2002; Filippenko et al., 2002). It should also be noted that the SS SN 2002cv
and the BL SN 2002bo share the same host galaxy. Elias-Rosa et al. (2008)
obtained and analyzed VRIJHK photometry, in addition to a sampling of optical
and NIR spectroscopy near and after maximum light, and find a best fit value
for the ratio between inferred extinction and reddening, RV = 1.59 $\pm$ 0.07
whereas 3.1 is often assumed for normal SN Ia (however see Tripp 1998; Astier
et al. 2006; Krisciunas et al. 2006; Folatelli et al. 2010). They suggest this
to indicate varying mean grain sizes for the dust along the line of sight
toward SN 2002bo and 2002cv.
### A.17 SN 2002dj in NGC 5018
For two years, and starting from day $-$11, Pignata et al. (2008) monitored
the optical and IR behaviors of the SN 2002bo-like, high-velocity gradient SN
2002dj (Hutchings & Li, 2002). The dataset presented make it one of the most
well-observed SN 1984A-like SN Ia and is a valuable tool for the discussion of
SN Ia diversity.
### A.18 SN 2002er in UGC 10743
From day $-$11 to day $+$215, Kotak et al. (2005) carried out spectroscopic
follow-up for the reddened, CN SN 2002er (Wood-Vasey et al., 2002c). By
contrast with the photometric behavior seen for SN 1992A, 1994D, and 1996X, SN
2002er stands out for its slightly delayed second peak in the _I_ -band and
similarly for _V_ and _R_ -bands. Pignata et al. (2004); Kotak et al. (2005)
estimated the mass of 56Ni to be on the order of 0.6 to 0.7 M⊙, where the
uncertainty in the exact distance to SN 2002er was the primary limitation.
### A.19 SN 2003du in UGC 09391
For 480 days, and starting from day $-$13, Stanishev et al. (2007) monitored
the CN SN 2003du. From modeling of the bolometric light curve, Stanishev et
al. (2007) estimate the mass of 56Ni to between 0.6 and 0.8 M⊙. Like other
normal SN Ia, the early spectra of SN 2003du contain HVFs of Ca II and a 6100
Å feature that departs from being only due to photospheric Si II, suggesting
either a distinctly separate region of HV Si II or the radial extension of
opacities from below.
Tanaka et al. (2011) studied the chemical composition distribution in the
ejecta of SN 2003du by modeling a one year extended time series of optical
spectra. Tanaka et al. (2011) do not find SN 2003du to be as fully mixed as a
some 3D deflagration models. Specifically, from their modeling Tanaka et al.
(2011) that the a core of stable IPEs supersedes 56Ni out to $\sim$ 3000 km
s-1 ($\lesssim$ 0.2 in mass coordinate). Atop this 0.65 M⊙ of 56Ni are layers
of IMEs, while the outermost layers consist of oxygen, some silicon, and no
more than 0.016 M⊙ of carbon above 10,500 km s-1.
### A.20 SN 2003gs in NGC 936
Krisciunas et al. (2009) obtained near-maximum to late time optical and NIR
observations of SN 2003gs, offering a chance to study the post-maximum light
bolometric behavior of a fast declining SN Ia that was sub-luminous at optical
wavelengths, but of standard luminosity in NIR bands at maximum light.
Krisciunas et al. (2009) find $\Delta$m15(_B_) = 1.83 $\pm$ 0.02 and discuss
comparisons to other fast decliners; namely, SN 2003hv (Leloudas et al.,
2009), SN 2004gs (Folatelli et al., 2010; Contreras et al., 2010), SN 2005bl
(Taubenberger et al., 2008; Folatelli et al., 2010; Wood-Vasey et al., 2008),
and SN 2005ke, 2006gt, and 2006mr (Folatelli et al., 2010; Contreras et al.,
2010). In particular, Krisciunas et al. (2009) note that, in contrast to
normal and over-luminous SN Ia, the delay in the time of _J_ -band maximum
from that of the _B_ -band, for fast decliners, is inversely proportional to
the peak NIR magnitude. Furthermore, they discussed the possibility for two
subsets of FAINT$-$CL fast decliners; those that do and do not show a _J_
-band peak before the _B_ -band (see also Kattner et al. 2012); respectively,
SN 1986G, 2003gs, 2003hv, and 2006gt, and SN 1991bg, 1999by, 2005bl, 2005ke,
and 2006mr.
Krisciunas et al. (2009) conclude that differences in NIR opacity within the
outer layers are responsible for dissimilar $\gamma$-ray trapping, and
therefore longer _J_ -band than _B_ -band diffusion times for FAINT$-$CL SN Ia
that are fainter in the NIR. However, the origin (differences of explosion
mechanism and/or progenitor systems) for this apparent ‘bimodal’ difference in
NIR opacity is not clear.
For SN 2003gs, Krisciunas et al. (2009) used _UBVRIJHK_ photometry and
Arnett’s rule (Arnett 1982, but also see Stritzinger & Leibundgut 2005) to
estimate 0.25 M⊙ of 56Ni was produced during the explosion. As for the optical
spectra, SN 2003gs is similar to SN 2004eo (A.23) in that it is found to have
absorption signatures that are consistent with its photometric
characteristics; a FAINT$-$CL SN Ia with a larger than normal $\mathcal{R}$(Si
II) and the presence of Ti II features near 4000$-$4500 Å.
### A.21 SN 2003hv in NGC 1201
Leloudas et al. (2009) studied SN 2003hv out to very late phases (day $+$786).
Notably, this seemingly spectroscopically normal SN Ia has $\Delta$m15(_B_) =
1.61, while the late time light curves show a deficit in flux that follow the
decay of radioactive 56Co, assuming full and instantaneous positron trapping.
Leloudas et al. (2009) consider this as possibly due to a redistribution of
flux for SN 2003hv (a.k.a. an infrared catastrophe, see Axelrod 1980) from a
dense clumping of inner material, and would also explain the flat-topped
nebular emission lines (Motohara et al., 2006; Gerardy et al., 2007).
Mazzali et al. (2011) also studied the nebular spectrum of 2003hv and consider
it as a non-standard event. They note that its late time flux deficit,
compared to normal SN Ia, could be due to SN 2003hv having a lower mean
density structure, possibly consistent with a sub-Chandrasekhar mass origin.
Motohara et al. (2006) presented NIR Subaru Telescope spectra of SN 2003du,
2003hv, and 2005W during their late phase evolution ($+$200 days post-maximum
light). For both SN 2003du and 2003hv, they find a flat-topped [Fe II]
$\lambda$16440 emission feature that is blue-shifted by $\sim$ 2000 km s-1
from the SN rest frame (FWHM $\sim$ 4000 km s-1). Motohara et al. (2006)
further argue that the [Fe II] emission would be rounded on top if the
neutron-rich Fe-peak isotopes produced in the explosion were thoroughly mixed
with the surrounding distribution of 56Ni; for SN 2003du and 2003hv they
suggest that this is not the case. In fact, they find that SN 1991T and 2005W
(see their Fig. 1), at least, may represent instances where the inner most
regions have been thoroughly mixed.
Similarly, Gerardy et al. (2007) looked to address the nature of the
thermonuclear burning front by utilizing late time ($+$135 days) mid-IR
(5.2$-$15.2 $\mu$m) _Spitzer Space Telescope_ spectra of SN 2003hv and 2005df.
In particular, Gerardy et al. (2007) find direct evidence in SN 2005df for a
small inner zone of nickel that is surrounded by 56Co and an asymmetric shell-
like structure of Ar. While it is not clear _why_ a supposed initial
deflagration phase of a DDT explosion mechanism produces little to no mixing
for these two SN Ia, the observations of Gerardy et al. (2007) give strong
support for a stratified abundance tomography like those seen in DDT-like
models; the various species of material are restricted to radially confined
zones, which is inconsistent with the large-scale mixing that is expected to
occur in 3D deflagration models. This is also in agreement with X-ray
observations of the Tycho supernova remnant (Badenes et al., 2006) in addition
to optical and UV line resonance absorption imaging of SNR 1885 in M31 (Fesen
et al., 2007).
### A.22 SN 2004dt in NGC 0799
Wang et al. (2006) and Altavilla et al. (2007) studied the early spectral
evolution of SN 2004dt from more than a week before optical maximum, when line
profiles show matter moving at velocities as high as 25,000 km s-1. The
variation of the polarization across some Si II lines approaches 2%, making SN
2004dt one of the most highly polarized SN Ia observed and an outlier in the
polarization-nebular velocity plane (Maund et al., 2010b). In contrast with
the polarization associated with Si II, Wang et al. (2006) find that the
strong 7400 Å O I$-$Mg II absorption complex shows little or no polarization
signature. Wang et al. (2006) conclude this is due to a spherical geometry of
oxygen-rich material encompassing a lumpy distribution of IMEs.
### A.23 SN 2004eo in NGC 6928
Pastorello et al. (2007a) presented optical and infrared observations of the
transitional normal, CL SN 2004eo (Nakano et al., 2004). The light curves and
spectra appear normal (M${}_{\emph{B}}$ = $-$19.08) while exhibiting low mean
expansion velocities and a fast declining _B_ -band light curve
($\Delta$m15(_B_) = 1.46). The observed properties of SN 2004eo signify it is
intermediate between FAINT, LVG, and HVG SN Ia. Mazzali et al. (2008) also
consider SN 2004eo as a spectroscopically normal SN Ia that produced 0.43
$\pm$ 0.05 M⊙ of 56Ni.
### A.24 SN 2005am in NGC 2811
Between day $-$4 and day $+$69, Brown et al. (2005) obtained UV, optical, and
X-ray observations with the _Swift_ satellite of the SN 1992A-like SN 2005am
(Kirshner et al., 1993; Modjaz et al., 2005). They place an upper limit on SN
2005am’s X-ray luminosity (0.3$-$10 keV) of 6 x 1039 erg s-1.
### A.25 Under-luminous SN 2005bl in NGC 4070
Both Taubenberger et al. (2008) and Hachinger et al. (2009) studied the sub-
luminous SN 2005bl with observations made between day $-6$ and day $+$66, and
carried out spectral analysis (“abundance tomography”) of SN 2005bl (Morrell
et al., 2005). They find it to be one of incomplete burning similar to SN
1991bg and 1999by. Compared to SN 1991bg, a noteworthy difference of SN 1999by
is the likely presence of carbon C II in pre-maximum spectra (Taubenberger et
al., 2008), whereas C I $\lambda$10691 is also clearly detected in NIR spectra
(Höflich et al., 2002). However, this is likely a biased comparison to SN
1991bg given that the earliest spectrum obtained was on day $-$1 (potentially
too late to detect unburned material via C II $\lambda$6580). To our knowledge
no conspicuous C I $\lambda$10691 absorption features have been documented for
other SN Ia. For example, C I $\lambda$10691 is present in the pre-maximum
spectra of SN 2011fe but it is not a conspicuous signature. Similarly, pre-
maximum spectra of SN 2005bl show less conspicuous detections of C I and C II
but still indicate low burning efficiency with a significant amount of
leftover unburned material (Taubenberger et al., 2008). Hachinger et al.
(2009) suggest that a detonation at low pre-expanded densities is responsible
for the abundance stratification of IMEs seen in the spectra of SN 2005bl.
This would also explain the remaining carbon-rich material seen for some CL SN
Ia when caught early enough.
### A.26 SN 2005cf in MCG-01-39-003
Wang et al. (2009b) studied UV$-$optical$-$NIR observations of the normal SN
2005cf (see also Pastorello et al. 2007b). During the early evolution of the
spectrum, HVFs of Ca II and Si II are found to be present above 18,000 km s-1
(confirming observations of Garavini et al. 2007). Gall et al. (2012) studied
the NIR spectra of SN 2005cf at epochs from day $-$10 to day $+$42, which show
clear signatures of Co II during post-maximum phases. In addition, they
attribute fluorescence emission in making the underlying shape of the SED.
### A.27 SN 2005cg in a low-luminosity, star forming host
Quimby et al. (2006b) presented and discussed the spectroscopic evolution and
light curve of the SS SN 2005cg, which was discovered by ROTSE-IIIc. Pre-
maximum spectra reveal HVFs of Ca II and Si II out to $\sim$ 24,000 km s-1 and
Quimby et al. (2006b) find good consistency between observed and modeled Si II
profiles. They interpret the steep rise in the blue wing of the Si II to be an
indication of circumstellar interaction given that abundance estimates for
HVFs suggest modest amounts of swept up material ($\sim$ 10-4 $-$ 10-3 M⊙; see
Quimby et al. 2006b; Branch et al. 2006).
### A.28 SN 2005hj
Quimby et al. (2007) obtained optical spectra of the SS SN 2005hj during pre-
maximum and post-maximum light phases. From a ROTSE-IIIb unfiltered light
curve, SN 2005hj reached an over-luminous peak absolute magnitude of $-$19.6
(assuming z = 0.0574). Interestingly, the sharp and shallow 6100 Å feature
remains fairly stagnant at $\sim$ 10,600 km s-1 near and after maximum light,
with a sudden decrease at later epochs. Similar to Quimby et al. (2006b),
Quimby et al. (2007) find this also consistent with the interpretation that
CSM is influencing spectral profiles of SN 1999aa-like SN Ia (see also Scalzo
et al. 2010, 2012).
### A.29 SN 2006D in MCG-01-33-034
Thomas et al. (2007) obtained the spectra of the spectroscopically normal SN
2006D from day $-$7 to day $+$13\. The spectra show one of the clearest
signatures of carbon-rich material at photospheric velocities observed in a
_normal_ SN Ia (below 10,000 km s-1). The 6300 Å carbon feature becomes weaker
with time and disappears as the photosphere recedes and the SN reaches maximum
brightness. These observations$-$like all SN Ia diversity studies$-$underscore
the importance of obtaining spectra of SN Ia during all phases. If [O I] and
[C I] lines are present in the spectra during post-maximum light phases at
velocities below 10,000 km s-1, this would indicate the presence of unburned
matter. These particular lines have not been detected, however the absence of
said signatures does not imply a complete lack of C+O material at low
velocities (Baron et al., 2003; Kozma et al., 2005).
### A.30 SN 2006X in M100
Wang et al. (2008b) presented _UBVRI_ and _JK_ light curves and optical
spectroscopy of the reddened BL SN 2006X (Stockdale et al., 2006; Immler,
2006; Quimby et al., 2006a) and find high mean expansion velocities during
pre-maximum light phases ($\gtrsim$ 20,000 km s-1). Wang et al. (2008b)
suggest the observed properties of SN 2006X may be due to interaction with
CSM. Yamanaka et al. (2009b) also presented and discussed the early spectral
evolution. They note that the $\mathcal{R}$(Si II) ratio is unusually low for
such a high-velocity gradient SN Ia. However, rather than this being an
indication of low effective temperature, they suggest that the low
$\mathcal{R}$(Si II) value is due to line-blending, likely from a higher
velocity component of Si II. Both Wang et al. (2008b) and Yamanaka et al.
(2009b) find the observed properties of SN 2006X to be consistent with
characteristics of delayed detonation models.
Equipped with high resolution spectra of narrow Na D signatures spanning
$\sim$100 days post-maximum light, Patat et al. (2007) infer the presence of
intervening CSM and argue a mass loss history associated with SN 2006X in the
decades prior to explosion. In fact, at least half of all SN Ia with narrow
rest frame, blue-shifted Na D absorption profiles are associated with high
ejecta velocities (Sternberg et al., 2011; Foley et al., 2012a). This
indicates that CSM outflows are either present in some explosion scenarios
_or_ associated with all progenitors scenarios at some point during the lead
up to the explosion.
Patat et al. (2009) later discussed the VLT spectropolarimetry of SN 2006X. In
particular, they find that the presence of the high-velocity Ca II is
coincident with a relatively high polarization signature ($\sim$1.4%) at day
$-$10, that diminishes by only $\sim$15% near maximum light, and is still
present 41 days later. Patat et al. (2009) note that this day $+$40 detection
is not seen for SN 2001el (Wang et al., 2003) or SN 2004du (Leonard et al.,
2005). As for the high-velocity Si II, its polarization signature is seen to
peak ($\sim$1.1%) at day $-$6, drop by $\sim$30% near maximum, and is
undetected well into the post-maximum phase. While the findings of
spectropolarimetry studies of SN Ia are thought to be associated with, for
example, “deflagration phase plumes” with time-dependent photospheric covering
fractions, Patat et al. (2009) are unable to conclude why SN 2006X exhibits a
sizable post-maximum light re-polarization signature by day $+$39.
### A.31 SN 2006bt in CGCG 108-013
Foley et al. (2010b) obtained optical light curves and spectra of transitional
CN/CL SN 2006bt (Lee & Li, 2006). The _B_ -band decline rate, $\Delta$m15(B) =
1.09, is within the range that is observed for normal SN Ia, however SN 2006bt
shows a larger than normal $\mathcal{R}$(Si II), slightly lower mean expansion
velocities, and a lack of a double peak in the _I_ -band; CL SN 1991bg-like
properties. A tentative C II 6300 feature is identified, however with a
minimum at $\sim$ 6450 Å. Foley et al. (2010b) suggest this inferred lower
projected Doppler velocity could be accounted for by a clump of carbon offset
from the line of sight _at_ photospheric velocities. Because of an association
within a halo population of its passive host galaxy, Foley et al. (2010b)
conclude that the progenitor was also likely to be from an old population of
stars.
### A.32 Over-luminous SN 2006gz in IC 1277
Hicken et al. (2007) studied SN 2006gz (Prieto et al., 2006a) and estimated a
peak intrinsic _V_ -band brightness of $-$19.74 and $\Delta$m15(_B_) = 0.69,
implying M(56Ni) $\sim$ 1.0$-$1.2 M⊙ (assuming $R_{V}$ = 2.1$-$3.1; see also
Maeda et al. 2009). The spectroscopic signatures during early phases are
relatively narrow on account of slightly lower mean expansion velocities. At
two weeks before maximum light, Hicken et al. (2007) attributed a relatively
strong 6300 Å feature to C II $\lambda$6580 that diminishes in strength by day
$-$10 (Prieto et al., 2006b). Compared to a 5 Å equivalent width 6300 Å
absorption feature observed in the CN SN 1990N (Leibundgut et al., 1991;
Jeffery et al., 1992), the absorption feature has an observed equivalent width
of 25 Å in the early spectra of SN 2006gz (Hicken et al., 2007). So far,
spectroscopic modeling that incorporate signatures of C II $\lambda$6580
predict carbon mass fractions, X(C), that span an order of magnitude and are
broadly consistent with both single- and double-degenerate scenarios.
Maeda et al. (2009) obtained Subaru and Keck observations of 2006gz at late-
phases. Interestingly, SN 2006gz shows relatively weak pillars of iron
emission that are usually seen in most SN Ia subtypes.
### A.33 Extremely faint, SN 2007ax in NGC 2577
SN 2007ax was a very faint, red, and peculiar SN Ia. Kasliwal et al. (2008)
find that it shares similarities with a sub-luminous SN 2005ke (Immler et al.,
2006; Hughes et al., 2007; Patat et al., 2012) and also shows clear excess UV
emission $\sim$ 20 days post-maximum light. Based on the small amount of
synthesized 56Ni that is inferred (0.05 $-$ 0.09 M⊙), along with SN Ia-like
expansion velocities near maximum light ($\sim$9000 km s-1), Kasliwal et al.
(2008) conclude that SN 2007ax is not compatible with a number of theoretical
models that have been proposed to explain FAINT$-$CL SN Ia.
### A.34 Over-luminous SN 2007if
Scalzo et al. (2010) find that SN 2007if qualifies as a SCC SN Ia, i.e. it is
over-luminous (M${}_{\emph{V}}$ = $-$20.4), has a slow-rise to peak brightness
(trise = 24 days), the early spectra contain signatures of stronger than
normal C II, and SN 2007if resides in a low-luminosity host (M${}_{\emph{g}}$
= $-$14.10). Despite having a red _B_ $-$ _V_ color ($+$0.16) at _B_ -band
maximum, signs of host reddening via Na D lines appear negligible. Utilizing
Keck observations of the young metal-poor host galaxy, Childress et al. (2011)
concluded that SN 2007if is likely to have originated from a young, metal-poor
progenitor. From the H$\alpha$ line of the host galaxy, Yuan et al. (2010)
derived a redshift of 0.0736.
Based on the bolometric light curve and the sluggish Si II velocity evolution,
Scalzo et al. (2010) conclude that SN 2007if was the death of a super-
Chandrasekhar mass progenitor. They estimate the total mass of the system to
be 2.4 M⊙, with 1.6 M⊙ of 56Ni, and 0.3 to 0.5 M⊙ in the form of a C$+$O
envelope. Given the possibility that other over-luminous events could
potentially stem from similar super-Chandrasekhar mass origins, Scalzo et al.
(2012) searched the SNFactory sample (based on a criterion of SN 1991T/2007if-
like selections) and found four additional super-Chandrasekhar mass
candidates.
### A.35 SN 2007on in NGC 1404
SN 2007on was found associated with the elliptical galaxy, NGC 1404 (Pollas &
Klotz, 2007). Voss & Nelemans (2008) reported the discovery of the progenitor
of SN 2007on based on a detected X-ray source in pre-supernova archival X-ray
images, located 0.9” $\pm$ 1.3” (later 1.15” $\pm$ 0.27”; Roelofs et al. 2008)
from the position of SN 2007on within its host galaxy. However, Roelofs et al.
(2008) later reevaluated the detection of the progenitor of SN 2007on and
concluded that given the offset discrepancy between the X-ray source and the
SN location, the probability for a connection is of order 1 percent. However,
should SN Ia progenitors reveal themselves to be producers of pre-explosive
X-ray sources, Voss & Nelemans (2008) suggest this would be consistent with a
merger model with an accretion disc, formed from the disrupted companion star
rather than an explosion immediately upon or soon after the merger of the two
stars.
### A.36 SN 2008J $-$ heavily reddened SN 2002ic-like in MGC-02-07-033
Taddia et al. (2012) studied SN 2008J (Thrasher et al., 2008), which provides
additional observational evidence for hydrogen-rich CSM around an otherwise SN
1991T-like SS SN Ia. They obtained a NIR spectrum extending up to 2.2 $\mu$m,
and find that SN 2008J is affected by a visual extinction of 1.9 mag.
### A.37 Sub-luminous SN 2008ha in UGC 12682
Foley et al. (2010a) studied the optical spectrum of SN 2008ha near maximum
brightness (Puckett et al., 2008; Soderberg, 2009). It is found to be a dim
thermonuclear SN Ia with uncommonly slow projected expansion velocities.
Carbon features at maximum light indicate that carbon-rich material is present
to significant depths in the SN ejecta. Consequently, Foley et al. (2010a)
conclude that SN 2008ha was a failed deflagration since late time imaging and
spectroscopy also give support to this idea (Kromer et al., 2013a).
### A.38 SN 2009nr in UGC 8255
Khan et al. (2011b) discuss the photometric and spectroscopic observations of
the over-luminous (M${}_{\emph{V}}$ = $-$19.6, $\Delta$m15(_B_) = 0.95) SS SN
2009nr (Balanutsa & Lipunov, 2010). Similarly, Tsvetkov et al. (2011) made
_UBVRI_ photometric observations of SN 2009nr. They estimate that 0.78 $-$
1.07 M⊙ of 56Ni was synthesized during the explosion. Khan et al. (2011b) also
find SN 2009nr is at a projected distance of 13.0 kpc from the nucleus of its
star-forming host galaxy. In turn, this indicates that the progenitor of SN
2009nr was _not_ associated with a young stellar population, i.e. SN 2009nr
may not have originated from a “prompt” progenitor channel as is often assumed
for SN Ia of its subtype.
### A.39 Peculiar, sub-luminous PTF09dav
Sullivan et al. (2011b) studied the peculiar PTF09dav discovered by the
Palomar Transient Factory. Sullivan et al. (2011b) find it to be faint
(M${}_{\emph{B}}$ = $-$15.5) compared to SN 1991bg, and does not satisfy the
faint end of the WLR. Sullivan et al. (2011b) find estimates for both the 56Ni
mass (0.019 M⊙) and ejecta mass (0.36 M⊙) significantly low for thermonuclear
supernovae. The spectra are also consistent with signatures of Sc II, Mn I, Ti
II, Sr II and low velocities of $\sim$6000 km s-1. The host galaxy of PTF09dav
is not clear, however it appears this transient is not associated with
massive, old stellar populations. Sullivan et al. (2011b) conclude that the
observed properties of PTF09dav cannot be explained by the known models of
sub-luminous SN Ia.
Notably, Kasliwal et al. (2012) recently presented late time spectra of
PTF09dav (and other similar low luminosity transients). They confirm that this
class of objects look nothing like SN Ia at all on account of little to no
late-time iron emission, but instead with prominent emission from calcium in
the NIR (Perets et al., 2010), confirming previous suspicions of Sullivan et
al. (2011b).
### A.40 PTF10ops, another peculiar cross-type SN Ia
Maguire et al. (2011) presented optical photometric and spectroscopic
observations of a somewhat peculiar and sub-luminous SN Ia, PTF10ops ($-$17.77
mag). Spectroscopically, this object has been noted as belonging to the CL
class of SN Ia on account of the presence of conspicuous Ti II absorption
features blue ward of 5000 Å, in addition to a larger than normal
$\mathcal{R}$(Si II) ratio (partially indicative of cooler effective
temperatures). Photometrically, PTF10ops overlaps “normal” SN Ia properties in
$\Delta$m15(_B_) (1.12 $\pm$ 0.06 mag) and its rise-time to maximum light
(19.6 days). Maguire et al. (2011) estimate $\sim$0.17 M⊙ of 56Ni was produced
during the explosion, which is well below what is expected for LVG$-$CN SN Ia.
Maguire et al. (2011) also note that either PTF10ops remains without a visible
host galaxy, or it resides within the outskirts of a massive spiral galaxy
located at least 148 kpc away, which would be consistent with a possible
influence of low metallicities or an old progenitor population. Maguire et al.
(2011) suggest the progenitor could have been the merger of two compact
objects (Pakmor et al., 2010), however time series spectrum synthesis is
needed to confirm.
### A.41 SN 2010jn in NGC 2929
The BL SN 2010jn was discovered by the Palomar Transient Factory (PTF10ygu) 15
days before it reached maximum light. Hachinger et al. (2013) performed
spectroscopic analysis of the photospheric phase observations and find that
the outer layers of SN 2010jn are rich in iron-group elements. At such high
velocities ($>$16,000 km s-1), iron-group elements have been tentatively
identified in the spectra of SN Ia before (Hatano et al., 1999a) and may also
be a ubiquitous property of SN Ia. However, more early epoch, time series
observations are needed in order to test and confirm such claims. For SN
2010jn at least, Hachinger et al. (2013) favor a Chandrasekhar-mass delayed
detonation, where the presence of iron-group elements within the outermost
layers may be a consequence of outward mixing via hydrodynamical instabilities
prior to or during the explosion (see Piro 2011, 2012).
### A.42 SN 2011iv in NGC 1404
Foley et al. (2012b) presented the first maximum-light UV through NIR spectrum
of a SN Ia (SN 2011iv; Drescher et al. 2011). Despite having a normal looking
spectrum, SN 2011iv declined in brightness fairly quickly ($\Delta$m15(_B_) =
1.69). Since the UV region of a SN Ia spectrum is extremely sensitive to the
composition of the outer layers, they offer the potential for strong
constraints as soon as observational UV spectroscopic diversity is better
understood.
### A.43 SN 2012cg in NGC 4424
Silverman et al. (2012d) presented early epoch observations of the nearby
spectroscopically normal SN 2012cg (Kandrashoff et al., 2012; Cenko et al.,
2012; Marion et al., 2012), discovered immediately after the event ($\sim$1.5
days after). Compared to the width of other normal SN Ia _B_ -band light
curves, Silverman et al. (2012d) find that SN 2012cg’s light curve relatively
narrow for its peak absolute brightness, with $t_{rise}$ = 17.3 days
(coincident photometry was also presented by Munari et al. 2013). Mean
expansion velocities within 2.5 days of the event were found to be more than
14,000 km s-1, while the earliest observations show high-velocity components
of both Si II and Ca II. The C II $\lambda\lambda$6580, 7234 absorption
features were also detected very early.
Johansson et al. (2013) obtained upper limits on dust emission via far
infrared _Herschel Space Observatory_ flux measurements in the vicinity of the
recent and nearby SN 2011by, 2011fe, and 2012cg. From non-detections during
post-maximum epochs at 70 $\mu$m and 160 $\mu$m band-passes and archival image
measurements, Johansson et al. (2013) exclude dust masses $\gtrsim$ 7 x 10-3
M⊙ for SN 2011fe, and $\gtrsim$ 10-1 M⊙ for SN 2011by and 2012cg for $\sim$
500 K dust temperatures, $\sim$ 1017 cm dust shell radii, and peak SN
bolometric luminosities of $\sim$ 109 L⊙.
### A.44 SN 2000cx-like, SN 2013bh
Silverman et al. (2013c) discussed recent observations of SN 2013bh and found
it similar to SN 2000cx on all accounts, with slightly higher mean expansion
velocities. Silverman et al. (2013c) note that both of these SN Ia reside on
the fringes of their spiral host galaxies. In addition, both SN 2000cx and
2013bh lack narrow Na D lines that would otherwise indicate an environment of
CSM. Given the extreme similarities between SN 2000cx and 2013bh, Silverman et
al. (2013c) suggest identical explosion scenarios for both events.
## References
* Aldering et al. (2006) Aldering, G., et al. 2006, Astrophys. J., 650, 510
* Altavilla et al. (2007) Altavilla, G., et al. 2007, Astron. Astrophys., 475, 585
* Altavilla et al. (2009) —. 2009, Astrophys. J., 695, 135
* Anupama et al. (2005a) Anupama, G. C., Sahu, D. K., Deng, J., Nomoto, K., Tominaga, N., Tanaka, M., Mazzali, P. A., & Prabhu, T. P. 2005a, Astrophys. J. Lett., 631, L125
* Anupama et al. (2005b) Anupama, G. C., Sahu, D. K., & Jose, J. 2005b, Astron. Astrophys., 429, 667
* Arbour et al. (1999) Arbour, R., Papenkova, M., Li, W. D., Filippenko, A. V., & Armstrong, M. 1999, IAU Circ., 7156, 1
* Armstrong & Schwartz (1999) Armstrong, M., & Schwartz, M. 1999, IAU Circ., 7108, 1
* Arnett & Livne (1994a) Arnett, D., & Livne, E. 1994a, Astrophys. J., 427, 315
* Arnett & Livne (1994b) —. 1994b, Astrophys. J., 427, 330
* Arnett (1968) Arnett, W. D. 1968, Nature, 219, 1344
* Arnett (1969) —. 1969, Astrophys. Space Sci., 5, 180
* Arnett (1982) —. 1982, Astrophys. J., 253, 785
* Arsenijevic (2011) Arsenijevic, V. 2011, Mon. Not. R. Astron. Soc., 414, 1617
* Astier et al. (2006) Astier, P., et al. 2006, Astron. Astrophys., 447, 31
* Axelrod (1980) Axelrod, T. S. 1980, PhD thesis, California Univ., Santa Cruz.
* Baade (1936) Baade, W. 1936, Publ. Astron. Soc. Pac., 48, 226
* Baade et al. (1956) Baade, W., Burbidge, G. R., Hoyle, F., Burbidge, E. M., Christy, R. F., & Fowler, W. A. 1956, Publ. Astron. Soc. Pac., 68, 296
* Badenes et al. (2006) Badenes, C., Borkowski, K. J., Hughes, J. P., Hwang, U., & Bravo, E. 2006, Astrophys. J., 645, 1373
* Badenes et al. (2009) Badenes, C., Harris, J., Zaritsky, D., & Prieto, J. L. 2009, Astrophys. J., 700, 727
* Badenes & Maoz (2012) Badenes, C., & Maoz, D. 2012, Astrophys. J. Lett., 749, L11
* Bailey et al. (2009) Bailey, S., et al. 2009, Astron. Astrophys., 500, L17
* Balanutsa & Lipunov (2010) Balanutsa, P., & Lipunov, V. 2010, Central Bureau Electronic Telegrams, 2111, 1
* Bao Supernova Survey et al. (1997) Bao Supernova Survey, Qiao, Q.-Y., Wu, H., Wei, J.-Y., & Li, W.-D. 1997, IAU Circ., 6623, 1
* Barbon et al. (1990) Barbon, R., Benetti, S., Rosino, L., Cappellaro, E., & Turatto, M. 1990, Astron. Astrophys., 237, 79
* Barbon et al. (1989) Barbon, R., Rosino, L., & Iijima, T. 1989, Astron. Astrophys., 220, 83
* Baron et al. (2006) Baron, E., Bongard, S., Branch, D., & Hauschildt, P. H. 2006, Astrophys. J., 645, 480
* Baron et al. (1995) Baron, E., Hauschildt, P. H., & Mezzacappa, A. 1995, ArXiv Astrophysics e-prints
* Baron et al. (1996) Baron, E., Hauschildt, P. H., Nugent, P., & Branch, D. 1996, Mon. Not. R. Astron. Soc., 283, 297
* Baron et al. (2012) Baron, E., Höflich, P., Krisciunas, K., Dominguez, I., Khokhlov, A. M., Phillips, M. M., Suntzeff, N., & Wang, L. 2012, Astrophys. J., 753, 105
* Baron et al. (2008) Baron, E., Jeffery, D. J., Branch, D., Bravo, E., García-Senz, D., & Hauschildt, P. H. 2008, Astrophys. J., 672, 1038
* Baron et al. (2003) Baron, E., Lentz, E. J., & Hauschildt, P. H. 2003, Astrophys. J. Lett., 588, L29
* Barone-Nugent et al. (2012) Barone-Nugent, R. L., et al. 2012, Mon. Not. R. Astron. Soc., 425, 1007
* Bassett et al. (2007) Bassett, B., et al. 2007, Central Bureau Electronic Telegrams, 1137, 1
* Becker et al. (1997) Becker, R. H., Gregg, M. D., Hook, I. M., McMahon, R. G., White, R. L., & Helfand, D. J. 1997, Astrophys. J. Lett., 479, L93
* Benetti et al. (2004) Benetti, S., et al. 2004, Mon. Not. R. Astron. Soc., 348, 261
* Benetti et al. (2005) —. 2005, Astrophys. J., 623, 1011
* Benetti et al. (2011) —. 2011, Mon. Not. R. Astron. Soc., 411, 2726
* Benítez et al. (2002) Benítez, N., Riess, A., Nugent, P., Dickinson, M., Chornock, R., & Filippenko, A. V. 2002, Astrophys. J. Lett., 577, L1
* Bianco et al. (2011) Bianco, F. B., et al. 2011, Astrophys. J., 741, 20
* Blaylock et al. (2000) Blaylock, M., Branch, D., Casebeer, D., Millard, J., Baron, E., Richardson, D., & Ancheta, C. 2000, Publ. Astron. Soc. Pac., 112, 1439
* Blondin et al. (2013) Blondin, S., Dessart, L., Hillier, D. J., & Khokhlov, A. M. 2013, Mon. Not. R. Astron. Soc., 429, 2127
* Blondin et al. (2011) Blondin, S., Kasen, D., Röpke, F. K., Kirshner, R. P., & Mandel, K. S. 2011, Mon. Not. R. Astron. Soc., 417, 1280
* Blondin & Tonry (2007) Blondin, S., & Tonry, J. L. 2007, Astrophys. J., 666, 1024
* Blondin et al. (2012) Blondin, S., et al. 2012, Astron. J., 143, 126
* Bloom et al. (2012) Bloom, J. S., et al. 2012, Astrophys. J. Lett., 744, L17
* Bonaparte et al. (2013) Bonaparte, I., Matteucci, F., Recchi, S., Spitoni, E., Pipino, A., & Grieco, V. 2013, ArXiv e-prints
* Bongard et al. (2006) Bongard, S., Baron, E., Smadja, G., Branch, D., & Hauschildt, P. H. 2006, Astrophys. J., 647, 513
* Bongard et al. (2008) —. 2008, Astrophys. J., 687, 456
* Bowers et al. (1997) Bowers, E. J. C., Meikle, W. P. S., Geballe, T. R., Walton, N. A., Pinto, P. A., Dhillon, V. S., Howell, S. B., & Harrop-Allin, M. K. 1997, Mon. Not. R. Astron. Soc., 290, 663
* Branch (1972) Branch, D. 1972, Astron. Astrophys., 16, 247
* Branch (2004) Branch, D. 2004, in Cosmic explosions in three dimensions, ed. P. Höflich, P. Kumar, & J. C. Wheeler, 132
* Branch et al. (2005) Branch, D., Baron, E., Hall, N., Melakayil, M., & Parrent, J. 2005, Publ. Astron. Soc. Pac., 117, 545
* Branch et al. (2004a) Branch, D., Baron, E., Thomas, R. C., Kasen, D., Li, W., & Filippenko, A. V. 2004a, Publ. Astron. Soc. Pac., 116, 903
* Branch et al. (1982) Branch, D., Buta, R., Falk, S. W., McCall, M. L., Uomoto, A., Wheeler, J. C., Wills, B. J., & Sutherland, P. G. 1982, Astrophys. J. Lett., 252, L61
* Branch et al. (2009) Branch, D., Dang, L. C., & Baron, E. 2009, Publ. Astron. Soc. Pac., 121, 238
* Branch et al. (1993) Branch, D., Fisher, A., & Nugent, P. 1993, Astron. J., 106, 2383
* Branch et al. (2000) Branch, D., Jeffery, D. J., Blaylock, M., & Hatano, K. 2000, Publ. Astron. Soc. Pac., 112, 217
* Branch et al. (1983) Branch, D., Lacy, C. H., McCall, M. L., Sutherland, P. G., Uomoto, A., Wheeler, J. C., & Wills, B. J. 1983, Astrophys. J., 270, 123
* Branch et al. (2002a) Branch, D., Leighly, K. M., Thomas, R. C., & Baron, E. 2002a, Astrophys. J. Lett., 578, L37
* Branch et al. (2007a) Branch, D., Parrent, J., Troxel, M. A., Casebeer, D., Jeffery, D. J., Baron, E., Ketchum, W., & Hall, N. 2007a, in American Institute of Physics Conference Series, Vol. 924, The Multicolored Landscape of Compact Objects and Their Explosive Origins, ed. T. di Salvo, G. L. Israel, L. Piersant, L. Burderi, G. Matt, A. Tornambe, & M. T. Menna, 342–349
* Branch & Patchett (1973) Branch, D., & Patchett, B. 1973, Mon. Not. R. Astron. Soc., 161, 71
* Branch & Tammann (1992) Branch, D., & Tammann, G. A. 1992, Annu. Rev. Astron. Astrophys., 30, 359
* Branch et al. (2002b) Branch, D., et al. 2002b, Astrophys. J., 566, 1005
* Branch et al. (2003) —. 2003, Astron. J., 126, 1489
* Branch et al. (2004b) —. 2004b, Astrophys. J., 606, 413
* Branch et al. (2006) —. 2006, Publ. Astron. Soc. Pac., 118, 560
* Branch et al. (2007b) —. 2007b, Publ. Astron. Soc. Pac., 119, 709
* Branch et al. (2008) —. 2008, Publ. Astron. Soc. Pac., 120, 135
* Bravo et al. (2010) Bravo, E., Domínguez, I., Badenes, C., Piersanti, L., & Straniero, O. 2010, Astrophys. J. Lett., 711, L66
* Bravo et al. (2009) Bravo, E., García-Senz, D., Cabezón, R. M., & Domínguez, I. 2009, Astrophys. J., 695, 1257
* Bravo & Martínez-Pinedo (2012) Bravo, E., & Martínez-Pinedo, G. 2012, Phys. Rev. C, 85, 055805
* Bravo et al. (2011) Bravo, E., Piersanti, L., Domínguez, I., Straniero, O., Isern, J., & Escartin, J. A. 2011, Astron. Astrophys., 535, A114
* Brown et al. (2005) Brown, P. J., et al. 2005, Astrophys. J., 635, 1192
* Brown et al. (2012) —. 2012, Astrophys. J., 753, 22
* Brown et al. (2013) Brown, T. M., et al. 2013, Publ. Astron. Soc. Pac., 125, 1031
* Bufano et al. (2009) Bufano, F., et al. 2009, Astrophys. J., 700, 1456
* Buil (2012) Buil, C. 2012, Central Bureau Electronic Telegrams, 3277, 3
* Burns et al. (2011) Burns, C. R., et al. 2011, Astron. J., 141, 19
* Cacella et al. (2002) Cacella, P., Hirose, Y., Nakano, S., Kushida, Y., Kushida, R., & Li, W. D. 2002, IAU Circ., 7847, 1
* Calder et al. (2013) Calder, A. C., Krueger, B. K., Jackson, A. P., & Townsley, D. M. 2013, Frontiers of Physics
* Candia et al. (2003) Candia, P., et al. 2003, Publ. Astron. Soc. Pac., 115, 277
* Cartier et al. (2013) Cartier, R., et al. 2013, ArXiv e-prints
* Casebeer et al. (2008) Casebeer, D., Baron, E., Leighly, K., Jevremovic, D., & Branch, D. 2008, Astrophys. J., 676, 857
* Casebeer et al. (1998) Casebeer, D., Blaylock, M., Deaton, J., Branch, D., Baron, E., Richardson, D., & Ancheta, C. 1998, in Bulletin of the American Astronomical Society, Vol. 30, American Astronomical Society Meeting Abstracts, 1324
* Casebeer et al. (2000) Casebeer, D., Branch, D., Blaylock, M., Millard, J., Baron, E., Richardson, D., & Ancheta, C. 2000, Publ. Astron. Soc. Pac., 112, 1433
* Cenko et al. (2012) Cenko, S. B., Filippenko, A. V., Silverman, J. M., Gal-Yam, A., Pei, L., Nguyen, M., Carson, D., & Barth, A. J. 2012, Central Bureau Electronic Telegrams, 3111, 2
* Chandrasekhar (1957) Chandrasekhar, S. 1957, An introduction to the study of stellar structure.
* Chen et al. (2007) Chen, B., Kantowski, R., Baron, E., Knop, S., & Hauschildt, P. H. 2007, Mon. Not. R. Astron. Soc., 380, 104
* Chen et al. (2013) Chen, M. C., Herwig, F., Denissenkov, P. A., & Paxton, B. 2013, ArXiv e-prints
* Chen & Li (2009) Chen, W., & Li, X. 2009, Astrophys. J., 702, 686
* Childress et al. (2012) Childress, M., Zhou, G., Tucker, B., Bayliss, D., Scalzo, R., Yuan, F., & Schmidt, B. 2012, Central Bureau Electronic Telegrams, 3275, 2
* Childress et al. (2011) Childress, M., et al. 2011, Astrophys. J., 733, 3
* Childress et al. (2013a) —. 2013a, Astrophys. J., 770, 107
* Childress et al. (2013b) Childress, M. J., Filippenko, A. V., Ganeshalingam, M., & Schmidt, B. P. 2013b, ArXiv e-prints
* Childress et al. (2013c) Childress, M. J., et al. 2013c, Astrophys. J., 770, 29
* Chomiuk (2013) Chomiuk, L. 2013, Proc. Astron. Soc. Aust., 30, 46
* Chomiuk et al. (2012) Chomiuk, L., et al. 2012, Astrophys. J., 750, 164
* Chornock et al. (2006) Chornock, R., Filippenko, A. V., Branch, D., Foley, R. J., Jha, S., & Li, W. 2006, Publ. Astron. Soc. Pac., 118, 722
* Chornock et al. (2000) Chornock, R., Leonard, D. C., Filippenko, A. V., Li, W. D., Gates, E. L., & Chloros, K. 2000, IAU Circ., 7463, 1
* Chornock et al. (2011) Chornock, R., et al. 2011, Astrophys. J., 739, 41
* Claeys et al. (2014) Claeys, J. S. W., Pols, O. R., Izzard, R. G., Vink, J., & Verbunt, F. W. M. 2014, ArXiv e-prints
* Colgate & McKee (1969) Colgate, S. A., & McKee, C. 1969, Astrophys. J., 157, 623
* Conley et al. (2008) Conley, A., et al. 2008, Astrophys. J., 681, 482
* Contardo et al. (2000) Contardo, G., Leibundgut, B., & Vacca, W. D. 2000, Astron. Astrophys., 359, 876
* Contreras et al. (2010) Contreras, C., et al. 2010, Astron. J., 139, 519
* Cooke et al. (2011) Cooke, J., et al. 2011, Astrophys. J. Lett., 727, L35
* Cristiani et al. (1992) Cristiani, S., et al. 1992, Astron. Astrophys., 259, 63
* Dan et al. (2013) Dan, M., Rosswog, S., Brueggen, M., & Podsiadlowski, P. 2013, ArXiv e-prints
* Dan et al. (2012) Dan, M., Rosswog, S., Guillochon, J., & Ramirez-Ruiz, E. 2012, Mon. Not. R. Astron. Soc., 422, 2417
* de Kool & Begelman (1995) de Kool, M., & Begelman, M. C. 1995, Astrophys. J., 455, 448
* Deng et al. (2004) Deng, J., et al. 2004, Astrophys. J. Lett., 605, L37
* Deng et al. (2000) Deng, J. S., Qiu, Y. L., Hu, J. Y., Hatano, K., & Branch, D. 2000, Astrophys. J., 540, 452
* Dessart et al. (2013a) Dessart, L., Blondin, S., Hillier, D. J., & Khokhlov, A. 2013a, ArXiv e-prints
* Dessart et al. (2013b) Dessart, L., Hillier, D. J., Blondin, S., & Khokhlov, A. 2013b, ArXiv e-prints
* Dessart et al. (2012) Dessart, L., Hillier, D. J., Li, C., & Woosley, S. 2012, Mon. Not. R. Astron. Soc., 424, 2139
* Di Paola et al. (2002) Di Paola, A., Larionov, V., Arkharov, A., Bernardi, F., Caratti o Garatti, A., Dolci, M., Di Carlo, E., & Valentini, G. 2002, Astron. Astrophys., 393, L21
* Dilday et al. (2012) Dilday, B., et al. 2012, Science, 337, 942
* Domínguez & Khokhlov (2011) Domínguez, I., & Khokhlov, A. 2011, Astrophys. J., 730, 87
* Dong et al. (2014) Dong, S., Katz, B., Kushnir, D., & Prieto, J. L. 2014, ArXiv e-prints
* Doull & Baron (2011) Doull, B. A., & Baron, E. 2011, Publ. Astron. Soc. Pac., 123, 765
* Drenkhahn & Richtler (1999) Drenkhahn, G., & Richtler, T. 1999, Astron. Astrophys., 349, 877
* Drescher et al. (2011) Drescher, C., Parker, S., Brimacombe, J., Noguchi, T., & Nakano, S. 2011, Central Bureau Electronic Telegrams, 2940, 1
* Elias et al. (1985a) Elias, J. H., Matthews, K., Neugebauer, G., & Persson, S. E. 1985a, Astrophys. J., 296, 379
* Elias et al. (1985b) —. 1985b, Astrophys. J., 296, 379
* Elias-Rosa et al. (2006) Elias-Rosa, N., et al. 2006, Mon. Not. R. Astron. Soc., 369, 1880
* Elias-Rosa et al. (2008) —. 2008, Mon. Not. R. Astron. Soc., 384, 107
* Ellis et al. (2008) Ellis, R. S., et al. 2008, Astrophys. J., 674, 51
* Elmhamdi et al. (2006) Elmhamdi, A., Danziger, I. J., Branch, D., Leibundgut, B., Baron, E., & Kirshner, R. P. 2006, Astron. Astrophys., 450, 305
* Elvis (2000) Elvis, M. 2000, Astrophys. J., 545, 63
* Elvis (2012) Elvis, M. 2012, in Astronomical Society of the Pacific Conference Series, Vol. 460, AGN Winds in Charleston, ed. G. Chartas, F. Hamann, & K. M. Leighly, 186
* Fesen et al. (2007) Fesen, R. A., Höflich, P. A., Hamilton, A. J. S., Hammell, M. C., Gerardy, C. L., Khokhlov, A. M., & Wheeler, J. C. 2007, Astrophys. J., 658, 396
* Filippenko (1988) Filippenko, A. V. 1988, Astron. J., 96, 1941
* Filippenko (1992) —. 1992, Astrophys. J. Lett., 384, L37
* Filippenko (1997) —. 1997, Annu. Rev. Astron. Astrophys., 35, 309
* Filippenko et al. (2002) Filippenko, A. V., Chornock, R., Foley, R. J., & Li, W. 2002, IAU Circ., 7917, 2
* Filippenko et al. (1990) Filippenko, A. V., Porter, A. C., & Sargent, W. L. W. 1990, Astron. J., 100, 1575
* Filippenko et al. (1992a) Filippenko, A. V., et al. 1992a, Astrophys. J. Lett., 384, L15
* Filippenko et al. (1992b) —. 1992b, Astron. J., 104, 1543
* Fink et al. (2010) Fink, M., Röpke, F. K., Hillebrandt, W., Seitenzahl, I. R., Sim, S. A., & Kromer, M. 2010, Astron. Astrophys., 514, A53
* Fisher (2000) Fisher, A. K. 2000, PhD thesis, THE UNIVERSITY OF OKLAHOMA
* Folatelli et al. (2010) Folatelli, G., et al. 2010, Astron. J., 139, 120
* Folatelli et al. (2012) —. 2012, Astrophys. J., 745, 74
* Folatelli et al. (2013) —. 2013, Astrophys. J., 773, 53
* Foley (2012) Foley, R. J. 2012, ArXiv e-prints
* Foley et al. (2010a) Foley, R. J., Brown, P. J., Rest, A., Challis, P. J., Kirshner, R. P., & Wood-Vasey, W. M. 2010a, Astrophys. J. Lett., 708, L61
* Foley & Kasen (2011) Foley, R. J., & Kasen, D. 2011, Astrophys. J., 729, 55
* Foley et al. (2010b) Foley, R. J., Narayan, G., Challis, P. J., Filippenko, A. V., Kirshner, R. P., Silverman, J. M., & Steele, T. N. 2010b, Astrophys. J., 708, 1748
* Foley et al. (2009) Foley, R. J., et al. 2009, Astron. J., 138, 376
* Foley et al. (2012a) —. 2012a, Astrophys. J., 752, 101
* Foley et al. (2012b) —. 2012b, Astrophys. J. Lett., 753, L5
* Foley et al. (2012c) —. 2012c, Astrophys. J., 744, 38
* Foley et al. (2013) —. 2013, Astrophys. J., 767, 57
* Förster et al. (2012) Förster, F., González-Gaitán, S., Anderson, J., Marchi, S., Gutiérrez, C., Hamuy, M., Pignata, G., & Cartier, R. 2012, Astrophys. J. Lett., 754, L21
* Fox & Filippenko (2013) Fox, O. D., & Filippenko, A. V. 2013, ArXiv e-prints
* Friesen et al. (2012) Friesen, B., Baron, E., Branch, D., Chen, B., Parrent, J. T., & Thomas, R. C. 2012, Astrophys. J. Suppl. Ser., 203, 12
* Fryer & Diehl (2008) Fryer, C. L., & Diehl, S. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 391, Hydrogen-Deficient Stars, ed. A. Werner & T. Rauch, 335
* Gall et al. (2012) Gall, E. E. E., Taubenberger, S., Kromer, M., Sim, S. A., Benetti, S., Blanc, G., Elias-Rosa, N., & Hillebrandt, W. 2012, Mon. Not. R. Astron. Soc., 427, 994
* Gamezo et al. (1999) Gamezo, V. N., Desbordes, D., & Oran, E. S. 1999, Combustion and Flame, 116, 154
* Gamezo et al. (2004) Gamezo, V. N., Khokhlov, A. M., & Oran, E. S. 2004, Physical Review Letters, 92, 211102
* Gamezo et al. (2005) —. 2005, Astrophys. J., 623, 337
* Gamezo et al. (2003) Gamezo, V. N., Khokhlov, A. M., Oran, E. S., Chtchelkanova, A. Y., & Rosenberg, R. O. 2003, Science, 299, 77
* Gamezo et al. (1999) Gamezo, V. N., Wheeler, J. C., Khokhlov, A. M., & Oran, E. S. 1999, Astrophys. J., 512, 827
* Ganeshalingam et al. (2011) Ganeshalingam, M., Li, W., & Filippenko, A. V. 2011, Mon. Not. R. Astron. Soc., 416, 2607
* Gaposchkin (1936) Gaposchkin, C. P. 1936, Astrophys. J., 83, 245
* Garavini et al. (2004) Garavini, G., et al. 2004, Astron. J., 128, 387
* Garavini et al. (2005) —. 2005, Astron. J., 130, 2278
* Garavini et al. (2007) —. 2007, Astron. Astrophys., 471, 527
* Garnavich et al. (2004) Garnavich, P. M., et al. 2004, Astrophys. J., 613, 1120
* Gerardy et al. (2004) Gerardy, C. L., et al. 2004, Astrophys. J., 607, 391
* Gerardy et al. (2007) —. 2007, Astrophys. J., 661, 995
* Germany et al. (2004) Germany, L. M., Reiss, D. J., Schmidt, B. P., Stubbs, C. W., & Suntzeff, N. B. 2004, Astron. Astrophys., 415, 863
* Gómez & López (1998) Gómez, G., & López, R. 1998, Astron. J., 115, 1096
* Goobar (2008) Goobar, A. 2008, Astrophys. J. Lett., 686, L103
* Graur et al. (2013) Graur, O., et al. 2013, ArXiv e-prints
* Guy et al. (2005) Guy, J., Astier, P., Nobili, S., Regnault, N., & Pain, R. 2005, Astron. Astrophys., 443, 781
* Guy et al. (2007) Guy, J., et al. 2007, Astron. Astrophys., 466, 11
* Hachinger et al. (2006) Hachinger, S., Mazzali, P. A., & Benetti, S. 2006, Mon. Not. R. Astron. Soc., 370, 299
* Hachinger et al. (2012) Hachinger, S., Mazzali, P. A., Taubenberger, S., Fink, M., Pakmor, R., Hillebrandt, W., & Seitenzahl, I. R. 2012, Mon. Not. R. Astron. Soc., 427, 2057
* Hachinger et al. (2009) Hachinger, S., Mazzali, P. A., Taubenberger, S., Pakmor, R., & Hillebrandt, W. 2009, Mon. Not. R. Astron. Soc., 399, 1238
* Hachinger et al. (2013) Hachinger, S., et al. 2013, Mon. Not. R. Astron. Soc., 429, 2228
* Hachisu et al. (2008) Hachisu, I., Kato, M., & Nomoto, K. 2008, Astrophys. J. Lett., 683, L127
* Hachisu et al. (1999) Hachisu, I., Kato, M., Nomoto, K., & Umeda, H. 1999, Astrophys. J., 519, 314
* Hachisu et al. (2012) Hachisu, I., Kato, M., Saio, H., & Nomoto, K. 2012, Astrophys. J., 744, 69
* Hamann & Sabra (2004) Hamann, F., & Sabra, B. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 311, AGN Physics with the Sloan Digital Sky Survey, ed. G. T. Richards & P. B. Hall, 203
* Hamuy et al. (1995) Hamuy, M., Phillips, M. M., Maza, J., Suntzeff, N. B., Schommer, R. A., & Aviles, R. 1995, Astron. J., 109, 1
* Hamuy et al. (1996) Hamuy, M., Phillips, M. M., Suntzeff, N. B., Schommer, R. A., Maza, J., & Aviles, R. 1996, Astron. J., 112, 2391
* Hamuy et al. (1994) Hamuy, M., et al. 1994, Astron. J., 108, 2226
* Hamuy et al. (2002) —. 2002, Astron. J., 124, 417
* Hamuy et al. (2003) —. 2003, Nature, 424, 651
* Han & Podsiadlowski (2006) Han, Z., & Podsiadlowski, P. 2006, Mon. Not. R. Astron. Soc., 368, 1095
* Hansen & Wheeler (1969) Hansen, C. J., & Wheeler, J. C. 1969, Astrophys. Space Sci., 3, 464
* Harutyunyan et al. (2009) Harutyunyan, A., Elias-Rosa, N., & Benetti, S. 2009, Central Bureau Electronic Telegrams, 1768, 1
* Harutyunyan et al. (2008) Harutyunyan, A. H., et al. 2008, Astron. Astrophys., 488, 383
* Hatano et al. (1999a) Hatano, K., Branch, D., Fisher, A., Baron, E., & Filippenko, A. V. 1999a, Astrophys. J., 525, 881
* Hatano et al. (1999b) Hatano, K., Branch, D., Fisher, A., Millard, J., & Baron, E. 1999b, Astrophys. J. Suppl. Ser., 121, 233
* Hatano et al. (2000) Hatano, K., Branch, D., Lentz, E. J., Baron, E., Filippenko, A. V., & Garnavich, P. M. 2000, Astrophys. J. Lett., 543, L49
* Hatano et al. (2002) Hatano, K., Branch, D., Qiu, Y. L., Baron, E., Thielemann, F., & Fisher, A. 2002, New Astron., 7, 441
* Hauschildt & Baron (1999) Hauschildt, P. H., & Baron, E. 1999, Journal of Computational and Applied Mathematics, 109, 41
* Hayden et al. (2010a) Hayden, B. T., et al. 2010a, Astrophys. J., 722, 1691
* Hayden et al. (2010b) —. 2010b, Astrophys. J., 712, 350
* Hernandez et al. (2000) Hernandez, M., et al. 2000, Mon. Not. R. Astron. Soc., 319, 223
* Hicken et al. (2007) Hicken, M., Garnavich, P. M., Prieto, J. L., Blondin, S., DePoy, D. L., Kirshner, R. P., & Parrent, J. 2007, Astrophys. J. Lett., 669, L17
* Hicken et al. (2009a) Hicken, M., Wood-Vasey, W. M., Blondin, S., Challis, P., Jha, S., Kelly, P. L., Rest, A., & Kirshner, R. P. 2009a, Astrophys. J., 700, 1097
* Hicken et al. (2009b) Hicken, M., et al. 2009b, Astrophys. J., 700, 331
* Hicken et al. (2012) —. 2012, Astrophys. J. Suppl. Ser., 200, 12
* Hillebrandt et al. (2013) Hillebrandt, W., Kromer, M., Röpke, F. K., & Ruiter, A. J. 2013, Frontiers of Physics, 8, 116
* Hillebrandt et al. (2007) Hillebrandt, W., Sim, S. A., & Röpke, F. K. 2007, Astron. Astrophys., 465, L17
* Hillier & Dessart (2012) Hillier, D. J., & Dessart, L. 2012, Mon. Not. R. Astron. Soc., 424, 252
* Hoffmann et al. (2013) Hoffmann, T. L., Sauer, D. N., Pauldrach, A. W. A., & Hultzsch, P. J. N. 2013, ArXiv e-prints
* Höflich (2006) Höflich, P. 2006, Nuclear Physics A, 777, 579
* Höflich et al. (2002) Höflich, P., Gerardy, C. L., Fesen, R. A., & Sakai, S. 2002, Astrophys. J., 568, 791
* Hoflich et al. (1995) Hoflich, P., Khokhlov, A. M., & Wheeler, J. C. 1995, Astrophys. J., 444, 831
* Höflich et al. (1998) Höflich, P., Wheeler, J. C., & Thielemann, F. K. 1998, Astrophys. J., 495, 617
* Höflich et al. (2010) Höflich, P., et al. 2010, Astrophys. J., 710, 444
* Hogg et al. (2002) Hogg, D. W., Baldry, I. K., Blanton, M. R., & Eisenstein, D. J. 2002, ArXiv Astrophysics e-prints
* Hole et al. (2010) Hole, K. T., Kasen, D., & Nordsieck, K. H. 2010, Astrophys. J., 720, 1500
* Horesh et al. (2012) Horesh, A., et al. 2012, Astrophys. J., 746, 21
* Howell (2001) Howell, D. A. 2001, Astrophys. J. Lett., 554, L193
* Howell et al. (2001) Howell, D. A., Höflich, P., Wang, L., & Wheeler, J. C. 2001, Astrophys. J., 556, 302
* Howell et al. (2007) Howell, D. A., Sullivan, M., Conley, A., & Carlberg, R. 2007, Astrophys. J. Lett., 667, L37
* Howell et al. (2005) Howell, D. A., et al. 2005, Astrophys. J., 634, 1190
* Howell et al. (2006) —. 2006, Nature, 443, 308
* Howell et al. (2009) —. 2009, Astrophys. J., 691, 661
* Hsiao et al. (2007) Hsiao, E. Y., Conley, A., Howell, D. A., Sullivan, M., Pritchet, C. J., Carlberg, R. G., Nugent, P. E., & Phillips, M. M. 2007, Astrophys. J., 663, 1187
* Hsiao et al. (2013) Hsiao, E. Y., et al. 2013, Astrophys. J., 766, 72
* Hughes et al. (2007) Hughes, J. P., Chugai, N., Chevalier, R., Lundqvist, P., & Schlegel, E. 2007, Astrophys. J., 670, 1260
* Humason (1936) Humason, M. L. 1936, Publ. Astron. Soc. Pac., 48, 110
* Hummer (1976) Hummer, D. G. 1976, in IAU Symposium, Vol. 70, Be and Shell Stars, ed. A. Slettebak, 281
* Hurst et al. (1998) Hurst, G. M., Armstrong, M., & Arbour, R. 1998, IAU Circ., 6875, 1
* Hutchings & Li (2002) Hutchings, D., & Li, W. D. 2002, IAU Circ., 7918, 1
* Iben & Tutukov (1984) Iben, Jr., I., & Tutukov, A. V. 1984, Astrophys. J. Suppl. Ser., 54, 335
* Immler (2006) Immler, S. 2006, The Astronomer’s Telegram, 751, 1
* Immler et al. (2006) Immler, S., et al. 2006, Astrophys. J. Lett., 648, L119
* Iwamoto et al. (1999) Iwamoto, K., Brachwitz, F., Nomoto, K., Kishimoto, N., Umeda, H., Hix, W. R., & Thielemann, F.-K. 1999, Astrophys. J. Suppl. Ser., 125, 439
* Jack et al. (2012) Jack, D., Hauschildt, P. H., & Baron, E. 2012, Astron. Astrophys., 538, A132
* Jackson et al. (2010) Jackson, A. P., Calder, A. C., Townsley, D. M., Chamulak, D. A., Brown, E. F., & Timmes, F. X. 2010, Astrophys. J., 720, 99
* James & Baron (2010) James, S., & Baron, E. 2010, Astrophys. J., 718, 957
* Jeffery & Branch (1990) Jeffery, D. J., & Branch, D. 1990, in Supernovae, Jerusalem Winter School for Theoretical Physics, ed. J. C. Wheeler, T. Piran, & S. Weinberg, 149
* Jeffery et al. (2006) Jeffery, D. J., Branch, D., & Baron, E. 2006, ArXiv Astrophysics e-prints
* Jeffery et al. (2007) Jeffery, D. J., Ketchum, W., Branch, D., Baron, E., Elmhamdi, A., & Danziger, I. J. 2007, Astrophys. J. Suppl. Ser., 171, 493
* Jeffery et al. (1992) Jeffery, D. J., Leibundgut, B., Kirshner, R. P., Benetti, S., Branch, D., & Sonneborn, G. 1992, Astrophys. J., 397, 304
* Jeffery & Mazzali (2007) Jeffery, D. J., & Mazzali, P. A. 2007, in American Institute of Physics Conference Series, Vol. 924, The Multicolored Landscape of Compact Objects and Their Explosive Origins, ed. T. di Salvo, G. L. Israel, L. Piersant, L. Burderi, G. Matt, A. Tornambe, & M. T. Menna, 401–406
* Jha et al. (2006a) Jha, S., Branch, D., Chornock, R., Foley, R. J., Li, W., Swift, B. J., Casebeer, D., & Filippenko, A. V. 2006a, Astron. J., 132, 189
* Jha et al. (2001) Jha, S., Matheson, T., Challis, P., Kirshner, R., & Berlind, P. 2001, IAU Circ., 7585, 1
* Jha et al. (2007) Jha, S., Riess, A. G., & Kirshner, R. P. 2007, Astrophys. J., 659, 122
* Jha et al. (1999) Jha, S., et al. 1999, Astrophys. J. Suppl. Ser., 125, 73
* Jha et al. (2006b) —. 2006b, Astron. J., 131, 527
* Ji et al. (2013) Ji, S., et al. 2013, Astrophys. J., 773, 136
* Johansson et al. (2013) Johansson, J., Amanullah, R., & Goobar, A. 2013, Mon. Not. R. Astron. Soc.
* Johansson et al. (2014) Johansson, J., Woods, T. E., Gilfanov, M., Sarzi, M., Chen, Y.-M., & Oh, K. 2014, ArXiv e-prints
* Jones et al. (2013) Jones, D. O., et al. 2013, ArXiv e-prints
* Jordan et al. (2009) Jordan, G. C., Meakin, C. A., Hearn, N., Fisher, R. T., Townsley, D. M., Lamb, D. Q., & Truran, J. W. 2009, in Astronomical Society of the Pacific Conference Series, Vol. 406, Numerical Modeling of Space Plasma Flows: ASTRONUM-2008, ed. N. V. Pogorelov, E. Audit, P. Colella, & G. P. Zank, 92
* Jordan et al. (2012) Jordan, IV, G. C., Perets, H. B., Fisher, R. T., & van Rossum, D. R. 2012, Astrophys. J. Lett., 761, L23
* Justham (2011) Justham, S. 2011, Astrophys. J. Lett., 730, L34
* Kamiya et al. (2012) Kamiya, Y., Tanaka, M., Nomoto, K., Blinnikov, S. I., Sorokina, E. I., & Suzuki, T. 2012, Astrophys. J., 756, 191
* Kandrashoff et al. (2012) Kandrashoff, M., et al. 2012, Central Bureau Electronic Telegrams, 3111, 1
* Kasen (2006) Kasen, D. 2006, Astrophys. J., 649, 939
* Kasen (2010) —. 2010, Astrophys. J., 708, 1025
* Kasen et al. (2002) Kasen, D., Branch, D., Baron, E., & Jeffery, D. 2002, Astrophys. J., 565, 380
* Kasen et al. (2009) Kasen, D., Röpke, F. K., & Woosley, S. E. 2009, Nature, 460, 869
* Kasen et al. (2006) Kasen, D., Thomas, R. C., & Nugent, P. 2006, Astrophys. J., 651, 366
* Kasen et al. (2008) Kasen, D., Thomas, R. C., Röpke, F., & Woosley, S. E. 2008, Journal of Physics Conference Series, 125, 012007
* Kasen & Woosley (2007) Kasen, D., & Woosley, S. E. 2007, Astrophys. J., 656, 661
* Kasen et al. (2003) Kasen, D., et al. 2003, Astrophys. J., 593, 788
* Kasliwal et al. (2008) Kasliwal, M. M., et al. 2008, Astrophys. J. Lett., 683, L29
* Kasliwal et al. (2012) —. 2012, Astrophys. J., 755, 161
* Kattner et al. (2012) Kattner, S., et al. 2012, Publ. Astron. Soc. Pac., 124, 114
* Kerzendorf & Sim (2014) Kerzendorf, W. E., & Sim, S. A. 2014, ArXiv e-prints
* Ketchum et al. (2008) Ketchum, W., Baron, E., & Branch, D. 2008, Astrophys. J., 674, 371
* Khan et al. (2011a) Khan, R., Stanek, K. Z., Stoll, R., & Prieto, J. L. 2011a, Astrophys. J. Lett., 737, L24
* Khan et al. (2011b) Khan, R., et al. 2011b, Astrophys. J., 726, 106
* Khokhlov et al. (1993) Khokhlov, A., Mueller, E., & Höflich, P. 1993, Astron. Astrophys., 270, 223
* Khokhlov (1991a) Khokhlov, A. M. 1991a, Astron. Astrophys., 245, 114
* Khokhlov (1991b) —. 1991b, Astron. Astrophys., 245, L25
* Khokhlov (1995) —. 1995, Astrophys. J., 449, 695
* Khokhlov (2000) —. 2000, ArXiv Astrophysics e-prints
* Khokhlov et al. (1997) Khokhlov, A. M., Oran, E. S., & Wheeler, J. C. 1997, Astrophys. J., 478, 678
* Kilic et al. (2012) Kilic, M., Brown, W. R., Allende Prieto, C., Kenyon, S. J., Heinke, C. O., Agüeros, M. A., & Kleinman, S. J. 2012, Astrophys. J., 751, 141
* Kilic et al. (2013) Kilic, M., et al. 2013, ArXiv e-prints
* Kim et al. (2014) Kim, A. G., et al. 2014, ArXiv e-prints
* Kirshner et al. (1973a) Kirshner, R. P., Oke, J. B., Penston, M. V., & Searle, L. 1973a, Astrophys. J., 185, 303
* Kirshner et al. (1973b) Kirshner, R. P., Willner, S. P., Becklin, E. E., Neugebauer, G., & Oke, J. B. 1973b, Astrophys. J. Lett., 180, L97
* Kirshner et al. (1993) Kirshner, R. P., et al. 1993, Astrophys. J., 415, 589
* Kistler et al. (2013) Kistler, M. D., Stanek, K. Z., Kochanek, C. S., Prieto, J. L., & Thompson, T. A. 2013, Astrophys. J., 770, 88
* Kleiser et al. (2009) Kleiser, I., Cenko, S. B., Li, W., & Filippenko, A. V. 2009, Central Bureau Electronic Telegrams, 1918, 1
* Klotz et al. (2012) Klotz, A., et al. 2012, Central Bureau Electronic Telegrams, 3275, 1
* Knop et al. (2003) Knop, R. A., et al. 2003, Astrophys. J., 598, 102
* Knop et al. (2009) Knop, S., Hauschildt, P. H., & Baron, E. 2009, Astron. Astrophys., 501, 813
* Kotak et al. (2004) Kotak, R., Meikle, W. P. S., Adamson, A., & Leggett, S. K. 2004, Mon. Not. R. Astron. Soc., 354, L13
* Kotak et al. (2005) Kotak, R., et al. 2005, Astron. Astrophys., 436, 1021
* Kowal (1968) Kowal, C. T. 1968, Astron. J., 73, 1021
* Kozma & Fransson (1992) Kozma, C., & Fransson, C. 1992, Astrophys. J., 390, 602
* Kozma et al. (2005) Kozma, C., Fransson, C., Hillebrandt, W., Travaglio, C., Sollerman, J., Reinecke, M., Röpke, F. K., & Spyromilio, J. 2005, Astron. Astrophys., 437, 983
* Krisciunas (2005) Krisciunas, K. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 339, Observing Dark Energy, ed. S. C. Wolff & T. R. Lauer, 75
* Krisciunas et al. (2000) Krisciunas, K., Hastings, N. C., Loomis, K., McMillan, R., Rest, A., Riess, A. G., & Stubbs, C. 2000, Astrophys. J., 539, 658
* Krisciunas et al. (2006) Krisciunas, K., Prieto, J. L., Garnavich, P. M., Riley, J.-L. G., Rest, A., Stubbs, C., & McMillan, R. 2006, Astron. J., 131, 1639
* Krisciunas et al. (2003) Krisciunas, K., et al. 2003, Astron. J., 125, 166
* Krisciunas et al. (2004) —. 2004, Astron. J., 128, 3034
* Krisciunas et al. (2007) —. 2007, Astron. J., 133, 58
* Krisciunas et al. (2009) —. 2009, Astron. J., 138, 1584
* Krisciunas et al. (2011) —. 2011, Astron. J., 142, 74
* Kromer et al. (2010) Kromer, M., Sim, S. A., Fink, M., Röpke, F. K., Seitenzahl, I. R., & Hillebrandt, W. 2010, Astrophys. J., 719, 1067
* Kromer et al. (2013a) Kromer, M., et al. 2013a, Mon. Not. R. Astron. Soc., 429, 2287
* Kromer et al. (2013b) —. 2013b, ArXiv e-prints
* Krueger et al. (2010) Krueger, B. K., Jackson, A. P., Townsley, D. M., Calder, A. C., Brown, E. F., & Timmes, F. X. 2010, Astrophys. J. Lett., 719, L5
* Krughoff et al. (2011) Krughoff, K. S., Connolly, A. J., Frieman, J., SubbaRao, M., Kilper, G., & Schneider, D. P. 2011, Astrophys. J., 731, 42
* Kurucz & Bell (1995) Kurucz, R., & Bell, B. 1995, Atomic Line Data (R.L. Kurucz and B. Bell) Kurucz CD-ROM No. 23. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1995., 23
* Kushnir et al. (2013) Kushnir, D., Katz, B., Dong, S., Livne, E., & Fernández, R. 2013, Astrophys. J. Lett., 778, L37
* Leaman et al. (2011) Leaman, J., Li, W., Chornock, R., & Filippenko, A. V. 2011, Mon. Not. R. Astron. Soc., 412, 1419
* Lee & Li (2006) Lee, E., & Li, W. 2006, Central Bureau Electronic Telegrams, 485, 1
* Lee & Jang (2012) Lee, M. G., & Jang, I. S. 2012, Astrophys. J. Lett., 760, L14
* Leibundgut et al. (1991) Leibundgut, B., Kirshner, R. P., Filippenko, A. V., Shields, J. C., Foltz, C. B., Phillips, M. M., & Sonneborn, G. 1991, Astrophys. J. Lett., 371, L23
* Leibundgut et al. (1993) Leibundgut, B., et al. 1993, Astron. J., 105, 301
* Leighly et al. (2009) Leighly, K. M., Hamann, F., Casebeer, D. A., & Grupe, D. 2009, Astrophys. J., 701, 176
* Leloudas et al. (2009) Leloudas, G., et al. 2009, Astron. Astrophys., 505, 265
* Leloudas et al. (2013) —. 2013, ArXiv e-prints
* Lentz et al. (2001a) Lentz, E. J., Baron, E., Branch, D., & Hauschildt, P. H. 2001a, Astrophys. J., 557, 266
* Lentz et al. (2001b) —. 2001b, Astrophys. J., 547, 402
* Lentz et al. (2000) Lentz, E. J., Baron, E., Branch, D., Hauschildt, P. H., & Nugent, P. E. 2000, Astrophys. J., 530, 966
* Leonard et al. (2005) Leonard, D. C., Li, W., Filippenko, A. V., Foley, R. J., & Chornock, R. 2005, Astrophys. J., 632, 450
* Li et al. (2011a) Li, W., Chornock, R., Leaman, J., Filippenko, A. V., Poznanski, D., Wang, X., Ganeshalingam, M., & Mannucci, F. 2011a, Mon. Not. R. Astron. Soc., 412, 1473
* Li et al. (2001) Li, W., et al. 2001, Publ. Astron. Soc. Pac., 113, 1178
* Li et al. (2003) —. 2003, Publ. Astron. Soc. Pac., 115, 453
* Li et al. (2011b) —. 2011b, Nature, 480, 348
* Li et al. (2011c) —. 2011c, Mon. Not. R. Astron. Soc., 412, 1441
* Li et al. (2011d) —. 2011d, Mon. Not. R. Astron. Soc., 412, 1441
* Li et al. (1997) Li, W.-D., Qiu, Y.-L., Qiao, Q.-Y., Zhang, Y., Zhou, W., & Hu, J.-Y. 1997, IAU Circ., 6661, 1
* Li et al. (1999) Li, W. D., et al. 1999, Astron. J., 117, 2709
* Liebert et al. (2005) Liebert, J., Bergeron, P., & Holberg, J. B. 2005, Astrophys. J. Suppl. Ser., 156, 47
* Lira (1995) Lira, P. 1995, Master’s thesis, University of Chile
* Lira et al. (1998) Lira, P., et al. 1998, Astron. J., 115, 234
* Livio & Pringle (2011) Livio, M., & Pringle, J. E. 2011, Astrophys. J. Lett., 740, L18
* Lundqvist et al. (2013) Lundqvist, P., et al. 2013, Mon. Not. R. Astron. Soc., 435, 329
* Macri et al. (2001) Macri, L. M., Stetson, P. B., Bothun, G. D., Freedman, W. L., Garnavich, P. M., Jha, S., Madore, B. F., & Richmond, M. W. 2001, Astrophys. J., 559, 243
* Maeda et al. (2009) Maeda, K., Kawabata, K., Li, W., Tanaka, M., Mazzali, P. A., Hattori, T., Nomoto, K., & Filippenko, A. V. 2009, Astrophys. J., 690, 1745
* Maeda et al. (2010a) Maeda, K., Röpke, F. K., Fink, M., Hillebrandt, W., Travaglio, C., & Thielemann, F.-K. 2010a, Astrophys. J., 712, 624
* Maeda et al. (2010b) Maeda, K., et al. 2010b, Nature, 466, 82
* Maeda et al. (2011) —. 2011, Mon. Not. R. Astron. Soc., 413, 3075
* Maguire et al. (2011) Maguire, K., et al. 2011, Mon. Not. R. Astron. Soc., 418, 747
* Maguire et al. (2012) —. 2012, Mon. Not. R. Astron. Soc., 426, 2359
* Mandel et al. (2011) Mandel, K. S., Narayan, G., & Kirshner, R. P. 2011, Astrophys. J., 731, 120
* Mannucci (2005) Mannucci, F. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 342, 1604-2004: Supernovae as Cosmological Lighthouses, ed. M. Turatto, S. Benetti, L. Zampieri, & W. Shea, 140
* Maoz & Badenes (2010) Maoz, D., & Badenes, C. 2010, Mon. Not. R. Astron. Soc., 407, 1314
* Maoz et al. (2012) Maoz, D., Mannucci, F., & Brandt, T. D. 2012, Mon. Not. R. Astron. Soc., 426, 3282
* Maoz et al. (2013) Maoz, D., Mannucci, F., & Nelemans, G. 2013, ArXiv e-prints
* Maoz et al. (2010) Maoz, D., Sharon, K., & Gal-Yam, A. 2010, Astrophys. J., 722, 1879
* Margutti et al. (2012) Margutti, R., et al. 2012, Astrophys. J., 751, 134
* Marion et al. (2009a) Marion, G. H., Höflich, P., Gerardy, C. L., Vacca, W. D., Wheeler, J. C., & Robinson, E. L. 2009a, Astron. J., 138, 727
* Marion et al. (2003) Marion, G. H., Höflich, P., Vacca, W. D., & Wheeler, J. C. 2003, Astrophys. J., 591, 316
* Marion et al. (2006) Marion, G. H., Höflich, P., Wheeler, J. C., Robinson, E. L., Gerardy, C. L., & Vacca, W. D. 2006, Astrophys. J., 645, 1392
* Marion et al. (2012) Marion, G. H., Kirshner, R. P., Foley, R. J., Challis, P., & Irwin, J. 2012, Central Bureau Electronic Telegrams, 3111, 3
* Marion et al. (2013) Marion, G. H., et al. 2013, ArXiv e-prints
* Marion et al. (2009b) Marion, H., Garnavich, P., Challis, P., Calkins, M., & Peters, W. 2009b, Central Bureau Electronic Telegrams, 1776, 1
* Matheson et al. (2008) Matheson, T., et al. 2008, Astron. J., 135, 1598
* Matheson et al. (2012) —. 2012, Astrophys. J., 754, 19
* Mattila et al. (2005) Mattila, S., Lundqvist, P., Sollerman, J., Kozma, C., Baron, E., Fransson, C., Leibundgut, B., & Nomoto, K. 2005, Astron. Astrophys., 443, 649
* Maund et al. (2010a) Maund, J. R., et al. 2010a, Astrophys. J., 722, 1162
* Maund et al. (2010b) —. 2010b, Astrophys. J. Lett., 725, L167
* Maund et al. (2013) —. 2013, Mon. Not. R. Astron. Soc., 433, L20
* Maza et al. (1994) Maza, J., Hamuy, M., Phillips, M. M., Suntzeff, N. B., & Aviles, R. 1994, Astrophys. J. Lett., 424, L107
* Maza et al. (1999) Maza, J., Hamuy, M., Wischnjewsky, M., Gonzalez, L., Candia, P., & Lidman, C. 1999, IAU Circ., 7172, 1
* Mazzali et al. (2013) Mazzali, P., et al. 2013, ArXiv e-prints
* Mazzali (2000) Mazzali, P. A. 2000, Astron. Astrophys., 363, 705
* Mazzali et al. (2005a) Mazzali, P. A., Benetti, S., Stehle, M., Branch, D., Deng, J., Maeda, K., Nomoto, K., & Hamuy, M. 2005a, Mon. Not. R. Astron. Soc., 357, 200
* Mazzali et al. (1998) Mazzali, P. A., Cappellaro, E., Danziger, I. J., Turatto, M., & Benetti, S. 1998, Astrophys. J. Lett., 499, L49
* Mazzali et al. (1997) Mazzali, P. A., Chugai, N., Turatto, M., Lucy, L. B., Danziger, I. J., Cappellaro, E., della Valle, M., & Benetti, S. 1997, Mon. Not. R. Astron. Soc., 284, 151
* Mazzali & Lucy (1993) Mazzali, P. A., & Lucy, L. B. 1993, Astron. Astrophys., 279, 447
* Mazzali et al. (1993) Mazzali, P. A., Lucy, L. B., Danziger, I. J., Gouiffes, C., Cappellaro, E., & Turatto, M. 1993, Astron. Astrophys., 269, 423
* Mazzali et al. (2011) Mazzali, P. A., Maurer, I., Stritzinger, M., Taubenberger, S., Benetti, S., & Hachinger, S. 2011, Mon. Not. R. Astron. Soc., 416, 881
* Mazzali et al. (2008) Mazzali, P. A., Sauer, D. N., Pastorello, A., Benetti, S., & Hillebrandt, W. 2008, Mon. Not. R. Astron. Soc., 386, 1897
* Mazzali et al. (2005b) Mazzali, P. A., et al. 2005b, Astrophys. J. Lett., 623, L37
* McClelland et al. (2013) McClelland, C. M., Garnavich, P. M., Milne, P. A., Shappee, B. J., & Pogge, R. W. 2013, ArXiv e-prints
* McClelland et al. (2010) McClelland, C. M., et al. 2010, Astrophys. J., 720, 704
* McLaughlin (1959) McLaughlin, D. B. 1959, Astron. J., 64, 130
* McLaughlin (1960) —. 1960, Astron. J., 65, 54
* McLaughlin (1963) —. 1963, Publ. Astron. Soc. Pac., 75, 133
* Meikle et al. (2002) Meikle, P., Mattila, S., Glasse, A., Buckle, J., & Adamson, A. 2002, IAU Circ., 7911, 2
* Meikle et al. (1996) Meikle, W. P. S., et al. 1996, Mon. Not. R. Astron. Soc., 281, 263
* Meng et al. (2010) Meng, X., Yang, W., & Geng, X. 2010, New Astron., 15, 343
* Meng et al. (2011) Meng, X. C., Li, Z. M., & Yang, W. M. 2011, Publ. Astron. Soc. Jpn., 63, L31
* Milisavljevic et al. (2013a) Milisavljevic, D., et al. 2013a, Astrophys. J., 767, 71
* Milisavljevic et al. (2013b) —. 2013b, Astrophys. J. Lett., 770, L38
* Milne et al. (2013) Milne, P. A., Brown, P. J., Roming, P. W. A., Bufano, F., & Gehrels, N. 2013, ArXiv e-prints
* Minkowski (1939) Minkowski, R. 1939, Astrophys. J., 89, 156
* Minkowski (1940) —. 1940, Publ. Astron. Soc. Pac., 52, 206
* Minkowski (1941) —. 1941, Publ. Astron. Soc. Pac., 53, 224
* Minkowski (1963) —. 1963, Publ. Astron. Soc. Pac., 75, 505
* Misra et al. (2005) Misra, K., Kamble, A. P., Bhattacharya, D., & Sagar, R. 2005, Mon. Not. R. Astron. Soc., 360, 662
* Modjaz et al. (1999) Modjaz, M., King, J. Y., Papenkova, M., Friedman, A., Johnson, R. A., Li, W. D., Treffers, R. R., & Filippenko, A. V. 1999, IAU Circ., 7114, 1
* Modjaz et al. (2005) Modjaz, M., Kirshner, R., Challis, P., & Hao, H. 2005, Central Bureau Electronic Telegrams, 112, 1
* Modjaz et al. (2001) Modjaz, M., Li, W., Filippenko, A. V., King, J. Y., Leonard, D. C., Matheson, T., Treffers, R. R., & Riess, A. G. 2001, Publ. Astron. Soc. Pac., 113, 308
* Mohlabeng & Ralston (2013) Mohlabeng, G. M., & Ralston, J. P. 2013, ArXiv e-prints
* Moll et al. (2013) Moll, R., Raskin, C., Kasen, D., & Woosley, S. 2013, ArXiv e-prints
* Monard et al. (2001) Monard, A. G., Bock, G., Wassilieff, A., & Biggs, J. 2001, IAU Circ., 7720, 1
* Morrell et al. (2005) Morrell, N., Folatelli, G., Phillips, M., Contreras, C., & Hamuy, M. 2005, Central Bureau Electronic Telegrams, 139, 1
* Motohara et al. (2006) Motohara, K., et al. 2006, Astrophys. J. Lett., 652, L101
* Mueller & Eriguchi (1985) Mueller, E., & Eriguchi, Y. 1985, Astron. Astrophys., 152, 325
* Munari et al. (2013) Munari, U., Henden, A., Belligoli, R., Castellani, F., Cherini, G., Righetti, G. L., & Vagnozzi, A. 2013, New Astron., 20, 30
* Mustel (1971) Mustel, É. R. 1971, Soviet Astron., 15, 1
* Nakano et al. (2004) Nakano, S., Itagaki, K., & The. 2004, Central Bureau Electronic Telegrams, 88, 1
* Nakar & Sari (2012) Nakar, E., & Sari, R. 2012, Astrophys. J., 747, 88
* Napiwotzki et al. (2005) Napiwotzki, R., et al. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 334, 14th European Workshop on White Dwarfs, ed. D. Koester & S. Moehler, 375
* Napiwotzki et al. (2007) Napiwotzki, R., et al. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 372, 15th European Workshop on White Dwarfs, ed. R. Napiwotzki & M. R. Burleigh, 387
* Narayan et al. (2011) Narayan, G., et al. 2011, Astrophys. J. Lett., 731, L11
* Navasardyan et al. (2009) Navasardyan, H., Cappellaro, E., & Benetti, S. 2009, Central Bureau Electronic Telegrams, 1918, 2
* Nelemans et al. (2013) Nelemans, G., Toonen, S., & Bours, M. 2013, in IAU Symposium, Vol. 281, IAU Symposium, 225–231
* Nicolas & Prosperi (2009) Nicolas, J., & Prosperi, E. 2009, Central Bureau Electronic Telegrams, 1776, 2
* Niemeyer & Hillebrandt (1995) Niemeyer, J. C., & Hillebrandt, W. 1995, Astrophys. J., 452, 769
* Nobili et al. (2005) Nobili, S., et al. 2005, Astron. Astrophys., 437, 789
* Nomoto et al. (2013) Nomoto, K., Kamiya, Y., & Nakasato, N. 2013, in IAU Symposium, Vol. 281, IAU Symposium, 253–260
* Nomoto & Kondo (1991) Nomoto, K., & Kondo, Y. 1991, Astrophys. J. Lett., 367, L19
* Nomoto & Sugimoto (1977) Nomoto, K., & Sugimoto, D. 1977, Publ. Astron. Soc. Jpn., 29, 765
* Nomoto et al. (1984) Nomoto, K., Thielemann, F., & Yokoi, K. 1984, Astrophys. J., 286, 644
* Nomoto et al. (2003) Nomoto, K., Uenishi, T., Kobayashi, C., Umeda, H., Ohkubo, T., Hachisu, I., & Kato, M. 2003, in From Twilight to Highlight: The Physics of Supernovae, ed. W. Hillebrandt & B. Leibundgut, 115–+
* Nugent et al. (2002) Nugent, P., Kim, A., & Perlmutter, S. 2002, Publ. Astron. Soc. Pac., 114, 803
* Nugent et al. (1995) Nugent, P., Phillips, M., Baron, E., Branch, D., & Hauschildt, P. 1995, Astrophys. J. Lett., 455, L147+
* Nugent et al. (2011) Nugent, P. E., et al. 2011, Nature, 480, 344
* Oke & Searle (1974) Oke, J. B., & Searle, L. 1974, Annu. Rev. Astron. Astrophys., 12, 315
* Paczynski (1985) Paczynski, B. 1985, in Astrophysics and Space Science Library, Vol. 113, Cataclysmic Variables and Low-Mass X-ray Binaries, ed. D. Q. Lamb & J. Patterson, 1–12
* Pakmor et al. (2010) Pakmor, R., Kromer, M., Röpke, F. K., Sim, S. A., Ruiter, A. J., & Hillebrandt, W. 2010, Nature, 463, 61
* Pakmor et al. (2013) Pakmor, R., Kromer, M., Taubenberger, S., & Springel, V. 2013, Astrophys. J. Lett., 770, L8
* Pan et al. (2013) Pan, Y.-C., et al. 2013, ArXiv e-prints
* Panagia (2007) Panagia, N. 2007, in American Institute of Physics Conference Series, Vol. 937, Supernova 1987A: 20 Years After: Supernovae and Gamma-Ray Bursters, ed. S. Immler, K. Weiler, & R. McCray, 236–245
* Parodi et al. (2000) Parodi, B. R., Saha, A., Sandage, A., & Tammann, G. A. 2000, Astrophys. J., 540, 634
* Parrent et al. (2007) Parrent, J., et al. 2007, Publ. Astron. Soc. Pac., 119, 135
* Parrent et al. (2011) Parrent, J. T., et al. 2011, Astrophys. J., 732, 30
* Parrent et al. (2012) —. 2012, Astrophys. J. Lett., 752, L26
* Parthasarathy et al. (2007) Parthasarathy, M., Branch, D., Jeffery, D. J., & Baron, E. 2007, New Astron. Rev., 51, 524
* Pastorello et al. (2007a) Pastorello, A., et al. 2007a, Mon. Not. R. Astron. Soc., 377, 1531
* Pastorello et al. (2007b) —. 2007b, Mon. Not. R. Astron. Soc., 376, 1301
* Patat et al. (2009) Patat, F., Baade, D., Höflich, P., Maund, J. R., Wang, L., & Wheeler, J. C. 2009, Astron. Astrophys., 508, 229
* Patat et al. (1996) Patat, F., Benetti, S., Cappellaro, E., Danziger, I. J., della Valle, M., Mazzali, P. A., & Turatto, M. 1996, Mon. Not. R. Astron. Soc., 278, 111
* Patat et al. (2011) Patat, F., Chugai, N. N., Podsiadlowski, P., Mason, E., Melo, C., & Pasquini, L. 2011, Astron. Astrophys., 530, A63
* Patat et al. (2010) Patat, F., Cox, N. L. J., Parrent, J., & Branch, D. 2010, Astron. Astrophys., 514, A78
* Patat et al. (2012) Patat, F., Höflich, P., Baade, D., Maund, J. R., Wang, L., & Wheeler, J. C. 2012, Astron. Astrophys., 545, A7
* Patat et al. (2007) Patat, F., et al. 2007, Science, 317, 924
* Patat et al. (2013) —. 2013, Astron. Astrophys., 549, A62
* Pauldrach et al. (2013) Pauldrach, A. W. A., Hoffmann, T. L., & Hultzsch, P. J. N. 2013, ArXiv e-prints
* Payne-Gaposchkin & Whipple (1940) Payne-Gaposchkin, C., & Whipple, F. L. 1940, Proceedings of the National Academy of Science, 26, 264
* Pereira et al. (2013) Pereira, R., et al. 2013, ArXiv e-prints
* Perets et al. (2010) Perets, H. B., et al. 2010, Nature, 465, 322
* Perlmutter et al. (1999) Perlmutter, S., et al. 1999, Astrophys. J., 517, 565
* Pfannes et al. (2010) Pfannes, J. M. M., Niemeyer, J. C., & Schmidt, W. 2010, Astron. Astrophys., 509, A75
* Phillips (1993) Phillips, M. M. 1993, Astrophys. J. Lett., 413, L105
* Phillips (2012) —. 2012, Proc. Astron. Soc. Aust., 29, 434
* Phillips et al. (1999) Phillips, M. M., Lira, P., Suntzeff, N. B., Schommer, R. A., Hamuy, M., & Maza, J. 1999, Astron. J., 118, 1766
* Phillips et al. (2006) Phillips, M. M., et al. 2006, Astron. J., 131, 2615
* Phillips et al. (2007) —. 2007, Publ. Astron. Soc. Pac., 119, 360
* Phillips et al. (2013) —. 2013, ArXiv e-prints
* Piersanti et al. (2003) Piersanti, L., Gagliardi, S., Iben, Jr., I., & Tornambé, A. 2003, Astrophys. J., 598, 1229
* Pignata et al. (2004) Pignata, G., et al. 2004, Mon. Not. R. Astron. Soc., 355, 178
* Pignata et al. (2008) —. 2008, Mon. Not. R. Astron. Soc., 388, 971
* Pinto & Eastman (2000a) Pinto, P. A., & Eastman, R. G. 2000a, Astrophys. J., 530, 744
* Pinto & Eastman (2000b) —. 2000b, Astrophys. J., 530, 757
* Piro (2011) Piro, A. L. 2011, Astrophys. J. Lett., 738, L5
* Piro (2012) —. 2012, Astrophys. J., 759, 83
* Piro & Nakar (2013) Piro, A. L., & Nakar, E. 2013, Astrophys. J., 769, 67
* Pollas & Klotz (2007) Pollas, C., & Klotz, A. 2007, Central Bureau Electronic Telegrams, 1121, 1
* Popper (1937) Popper, D. M. 1937, Publ. Astron. Soc. Pac., 49, 283
* Prieto et al. (2006a) Prieto, J. L., Depoy, D., & Garnavich, P. 2006a, Central Bureau Electronic Telegrams, 651, 1
* Prieto et al. (2006b) Prieto, J. L., Rest, A., & Suntzeff, N. B. 2006b, Astrophys. J., 647, 501
* Prieto et al. (2007) Prieto, J. L., et al. 2007, ArXiv e-prints
* Pskovskii (1969) Pskovskii, Y. P. 1969, Soviet Astron., 12, 750
* Puckett et al. (2008) Puckett, T., Moore, C., Newton, J., & Orff, T. 2008, Central Bureau Electronic Telegrams, 1567, 1
* Puckett et al. (2009) Puckett, T., Moore, R., Newton, J., & Orff, T. 2009, Central Bureau Electronic Telegrams, 1762, 1
* Quimby et al. (2006a) Quimby, R., Brown, P., Gerardy, C., Odewahn, S. C., & Rostopchin, S. 2006a, Central Bureau Electronic Telegrams, 393, 1
* Quimby et al. (2005) Quimby, R., Höflich, P., Kannappan, S. J., Burket, J., & Li, W. 2005, IAU Circ., 8625, 2
* Quimby et al. (2006b) Quimby, R., Höflich, P., Kannappan, S. J., Rykoff, E., Rujopakarn, W., Akerlof, C. W., Gerardy, C. L., & Wheeler, J. C. 2006b, Astrophys. J., 636, 400
* Quimby et al. (2007) Quimby, R., Höflich, P., & Wheeler, J. C. 2007, Astrophys. J., 666, 1083
* Raskin & Kasen (2013) Raskin, C., & Kasen, D. 2013, ArXiv e-prints
* Raskin et al. (2013) Raskin, C., Kasen, D., Moll, R., Schwab, J., & Woosley, S. 2013, ArXiv e-prints
* Reinecke et al. (2002a) Reinecke, M., Hillebrandt, W., & Niemeyer, J. C. 2002a, Astron. Astrophys., 386, 936
* Reinecke et al. (2002b) —. 2002b, Astron. Astrophys., 391, 1167
* Richardson et al. (2001) Richardson, D., Thomas, R. C., Casebeer, D., Blankenship, Z., Ratowt, S., Baron, E., & Branch, D. 2001, in Bulletin of the American Astronomical Society, Vol. 33, American Astronomical Society Meeting Abstracts, 1428
* Richmond & Smith (2012) Richmond, M. W., & Smith, H. A. 2012, Journal of the American Association of Variable Star Observers (JAAVSO), 40, 872
* Richmond et al. (1995) Richmond, M. W., et al. 1995, Astron. J., 109, 2121
* Riess et al. (1996) Riess, A. G., Press, W. H., & Kirshner, R. P. 1996, Astrophys. J., 473, 88
* Riess et al. (1999) Riess, A. G., et al. 1999, Astron. J., 118, 2675
* Riess et al. (2001) —. 2001, Astrophys. J., 560, 49
* Riess et al. (2004) —. 2004, Astrophys. J., 607, 665
* Riess et al. (2007) —. 2007, Astrophys. J., 659, 98
* Rodney & Tonry (2009) Rodney, S. A., & Tonry, J. L. 2009, Astrophys. J., 707, 1064
* Roelofs et al. (2008) Roelofs, G., Bassa, C., Voss, R., & Nelemans, G. 2008, Mon. Not. R. Astron. Soc., 391, 290
* Röpke et al. (2005) Röpke, F. K., Gieseler, M., & Hillebrandt, W. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 342, 1604-2004: Supernovae as Cosmological Lighthouses, ed. M. Turatto, S. Benetti, L. Zampieri, & W. Shea, 397
* Röpke et al. (2006) Röpke, F. K., Hillebrandt, W., Niemeyer, J. C., & Woosley, S. E. 2006, Astron. Astrophys., 448, 1
* Röpke et al. (2012) Röpke, F. K., et al. 2012, Astrophys. J. Lett., 750, L19
* Rubin et al. (2013) Rubin, D., et al. 2013, Astrophys. J., 763, 35
* Rudy et al. (2002) Rudy, R. J., Lynch, D. K., Mazuk, S., Venturini, C. C., Puetter, R. C., & Höflich, P. 2002, Astrophys. J., 565, 413
* Ruiter et al. (2009) Ruiter, A. J., Belczynski, K., & Fryer, C. 2009, Astrophys. J., 699, 2026
* Ruiz-Lapuente (1997) Ruiz-Lapuente, P. 1997, The Observatory, 117, 312
* Ruiz-Lapuente & Lucy (1992) Ruiz-Lapuente, P., & Lucy, L. B. 1992, Astrophys. J., 400, 127
* Sadakane et al. (1996) Sadakane, K., et al. 1996, Publ. Astron. Soc. Jpn., 48, 51
* Saffer et al. (1998) Saffer, R. A., Livio, M., & Yungelson, L. R. 1998, Astrophys. J., 502, 394
* Saha et al. (1996) Saha, A., Sandage, A., Labhardt, L., Tammann, G. A., Macchetto, F. D., & Panagia, N. 1996, Astrophys. J., 466, 55
* Saha et al. (1997) —. 1997, Astrophys. J., 486, 1
* Saha et al. (2001a) Saha, A., Sandage, A., Tammann, G. A., Dolphin, A. E., Christensen, J., Panagia, N., & Macchetto, F. D. 2001a, Astrophys. J., 562, 314
* Saha et al. (1999) Saha, A., Sandage, A., Tammann, G. A., Labhardt, L., Macchetto, F. D., & Panagia, N. 1999, Astrophys. J., 522, 802
* Saha et al. (2001b) Saha, A., Sandage, A., Thim, F., Labhardt, L., Tammann, G. A., Christensen, J., Panagia, N., & Macchetto, F. D. 2001b, Astrophys. J., 551, 973
* Sahu et al. (2008) Sahu, D. K., et al. 2008, Astrophys. J., 680, 580
* Saio & Nomoto (1985) Saio, H., & Nomoto, K. 1985, Astron. Astrophys., 150, L21
* Saio & Nomoto (2004) —. 2004, Astrophys. J., 615, 444
* Salvo et al. (2001) Salvo, M. E., Cappellaro, E., Mazzali, P. A., Benetti, S., Danziger, I. J., Patat, F., & Turatto, M. 2001, Mon. Not. R. Astron. Soc., 321, 254
* Sauer et al. (2006) Sauer, D. N., Hoffmann, T. L., & Pauldrach, A. W. A. 2006, Astron. Astrophys., 459, 229
* Sauer et al. (2008) Sauer, D. N., et al. 2008, Mon. Not. R. Astron. Soc., 391, 1605
* Scalzo et al. (2012) Scalzo, R., et al. 2012, Astrophys. J., 757, 12
* Scalzo et al. (2010) Scalzo, R. A., et al. 2010, Astrophys. J., 713, 1073
* Scannapieco & Bildsten (2005) Scannapieco, E., & Bildsten, L. 2005, Astrophys. J. Lett., 629, L85
* Schaefer (2011) Schaefer, B. E. 2011, Astrophys. J., 742, 112
* Schmidt (2004) Schmidt, B. P. 2004, Bulletin of the Astronomical Society of India, 32, 269
* Seitenzahl et al. (2009) Seitenzahl, I. R., Taubenberger, S., & Sim, S. A. 2009, Mon. Not. R. Astron. Soc., 400, 531
* Seitenzahl et al. (2013) Seitenzahl, I. R., et al. 2013, Mon. Not. R. Astron. Soc., 429, 1156
* Shappee et al. (2013) Shappee, B. J., Stanek, K. Z., Pogge, R. W., & Garnavich, P. M. 2013, Astrophys. J. Lett., 762, L5
* Shen et al. (2013) Shen, K. J., Guillochon, J., & Foley, R. J. 2013, ArXiv e-prints
* Shen et al. (2010) Shen, K. J., Kasen, D., Weinberg, N. N., Bildsten, L., & Scannapieco, E. 2010, Astrophys. J., 715, 767
* Silverman & Filippenko (2012) Silverman, J. M., & Filippenko, A. V. 2012, Mon. Not. R. Astron. Soc., 425, 1917
* Silverman et al. (2013a) Silverman, J. M., Ganeshalingam, M., & Filippenko, A. V. 2013a, Mon. Not. R. Astron. Soc., 430, 1030
* Silverman et al. (2012a) Silverman, J. M., Ganeshalingam, M., Li, W., & Filippenko, A. V. 2012a, Mon. Not. R. Astron. Soc., 425, 1889
* Silverman et al. (2011) Silverman, J. M., Ganeshalingam, M., Li, W., Filippenko, A. V., Miller, A. A., & Poznanski, D. 2011, Mon. Not. R. Astron. Soc., 410, 585
* Silverman et al. (2012b) Silverman, J. M., Kong, J. J., & Filippenko, A. V. 2012b, Mon. Not. R. Astron. Soc., 425, 1819
* Silverman et al. (2012c) Silverman, J. M., et al. 2012c, Mon. Not. R. Astron. Soc., 425, 1789
* Silverman et al. (2012d) —. 2012d, Astrophys. J. Lett., 756, L7
* Silverman et al. (2013b) —. 2013b, Astrophys. J., 772, 125
* Silverman et al. (2013c) —. 2013c, Mon. Not. R. Astron. Soc.
* Silverman et al. (2013d) —. 2013d, Astrophys. J. Suppl. Ser., 207, 3
* Sim et al. (2012) Sim, S. A., Fink, M., Kromer, M., Röpke, F. K., Ruiter, A. J., & Hillebrandt, W. 2012, Mon. Not. R. Astron. Soc., 420, 3003
* Sim et al. (2010a) Sim, S. A., Kromer, M., Röpke, F. K., Sorokina, E. I., Blinnikov, S. I., Kasen, D., & Hillebrandt, W. 2010a, in Astronomical Society of the Pacific Conference Series, Vol. 429, Numerical Modeling of Space Plasma Flows, Astronum-2009, ed. N. V. Pogorelov, E. Audit, & G. P. Zank, 148
* Sim et al. (2010b) Sim, S. A., Röpke, F. K., Hillebrandt, W., Kromer, M., Pakmor, R., Fink, M., Ruiter, A. J., & Seitenzahl, I. R. 2010b, Astrophys. J. Lett., 714, L52
* Sim et al. (2013) Sim, S. A., et al. 2013, Mon. Not. R. Astron. Soc., 436, 333
* Simon et al. (2007) Simon, J. D., et al. 2007, Astrophys. J. Lett., 671, L25
* Simon et al. (2009) —. 2009, Astrophys. J., 702, 1157
* Smith et al. (2011) Smith, P. S., Williams, G. G., Smith, N., Milne, P. A., Jannuzi, B. T., & Green, E. M. 2011, ArXiv e-prints
* Soderberg (2009) Soderberg, A. 2009, The Astronomer’s Telegram, 1948, 1
* Soderberg et al. (2008) Soderberg, A. M., et al. 2008, Nature, 453, 469
* Soker et al. (2013) Soker, N., Garcia-Berro, E., & Althaus, L. G. 2013, ArXiv e-prints
* Sokoloski et al. (2006) Sokoloski, J. L., Luna, G. J. M., Mukai, K., & Kenyon, S. J. 2006, Nature, 442, 276
* Sollerman et al. (2004) Sollerman, J., et al. 2004, Astron. Astrophys., 428, 555
* Stanishev et al. (2007) Stanishev, V., et al. 2007, Astron. Astrophys., 469, 645
* Stehle et al. (2005) Stehle, M., Mazzali, P. A., Benetti, S., & Hillebrandt, W. 2005, Mon. Not. R. Astron. Soc., 360, 1231
* Steinmetz et al. (1992) Steinmetz, M., Muller, E., & Hillebrandt, W. 1992, Astron. Astrophys., 254, 177
* Sternberg et al. (2011) Sternberg, A., et al. 2011, Science, 333, 856
* Stockdale et al. (2006) Stockdale, C. J., Kelley, M., Sramek, R. A., van Dyk, S. D., Immler, S., Weiler, K. W., Williams, C. L. M., & Panagia, N. 2006, The Astronomer’s Telegram, 729, 1
* Stritzinger & Leibundgut (2005) Stritzinger, M., & Leibundgut, B. 2005, Astron. Astrophys., 431, 423
* Stritzinger et al. (2002) Stritzinger, M., et al. 2002, Astron. J., 124, 2100
* Stritzinger et al. (2013) Stritzinger, M. D., et al. 2013, ArXiv e-prints
* Strolger et al. (2010) Strolger, L.-G., Dahlen, T., & Riess, A. G. 2010, Astrophys. J., 713, 32
* Strolger et al. (2002) Strolger, L.-G., et al. 2002, Astron. J., 124, 2905
* Sullivan et al. (2006) Sullivan, M., et al. 2006, Astrophys. J., 648, 868
* Sullivan et al. (2010) —. 2010, Mon. Not. R. Astron. Soc., 406, 782
* Sullivan et al. (2011a) —. 2011a, Astrophys. J., 737, 102
* Sullivan et al. (2011b) —. 2011b, Astrophys. J., 732, 118
* Suzuki et al. (2012) Suzuki, N., et al. 2012, Astrophys. J., 746, 85
* Swift et al. (2001) Swift, B., Li, W. D., & Schwartz, M. 2001, IAU Circ., 7611, 1
* Taddia et al. (2012) Taddia, F., et al. 2012, Astron. Astrophys., 545, L7
* Takaki et al. (2013) Takaki, K., et al. 2013, Astrophys. J. Lett., 772, L17
* Tanaka et al. (2006) Tanaka, M., Mazzali, P. A., Maeda, K., & Nomoto, K. 2006, Astrophys. J., 645, 470
* Tanaka et al. (2011) Tanaka, M., Mazzali, P. A., Stanishev, V., Maurer, I., Kerzendorf, W. E., & Nomoto, K. 2011, Mon. Not. R. Astron. Soc., 410, 1725
* Tanaka et al. (2008) Tanaka, M., et al. 2008, Astrophys. J., 677, 448
* Tanaka et al. (2010) —. 2010, Astrophys. J., 714, 1209
* Taubenberger et al. (2008) Taubenberger, S., et al. 2008, Mon. Not. R. Astron. Soc., 385, 75
* Taubenberger et al. (2011) —. 2011, Mon. Not. R. Astron. Soc., 412, 2735
* Taubenberger et al. (2013) —. 2013, Mon. Not. R. Astron. Soc., 432, 3117
* Tauris et al. (2013) Tauris, T. M., Sanyal, D., Yoon, S.-C., & Langer, N. 2013, Astron. Astrophys., 558, A39
* Thomas et al. (2004) Thomas, R. C., Branch, D., Baron, E., Nomoto, K., Li, W., & Filippenko, A. V. 2004, Astrophys. J., 601, 1019
* Thomas et al. (2002) Thomas, R. C., Kasen, D., Branch, D., & Baron, E. 2002, Astrophys. J., 567, 1037
* Thomas et al. (2011a) Thomas, R. C., Nugent, P. E., & Meza, J. C. 2011a, Publ. Astron. Soc. Pac., 123, 237
* Thomas et al. (2007) Thomas, R. C., et al. 2007, Astrophys. J. Lett., 654, L53
* Thomas et al. (2011b) —. 2011b, Astrophys. J., 743, 27
* Thompson (2011) Thompson, T. A. 2011, Astrophys. J., 741, 82
* Thrasher et al. (2008) Thrasher, P., Li, W., & Filippenko, A. V. 2008, Central Bureau Electronic Telegrams, 1211, 1
* Timmes et al. (2003) Timmes, F. X., Brown, E. F., & Truran, J. W. 2003, Astrophys. J. Lett., 590, L83
* Timmes & Woosley (1992) Timmes, F. X., & Woosley, S. E. 1992, Astrophys. J., 396, 649
* Toonen et al. (2012) Toonen, S., Nelemans, G., & Portegies Zwart, S. 2012, Astron. Astrophys., 546, A70
* Tornambé & Piersanti (2013) Tornambé, A., & Piersanti, L. 2013, Mon. Not. R. Astron. Soc., 431, 1812
* Tripp (1998) Tripp, R. 1998, Astron. Astrophys., 331, 815
* Tripp & Branch (1999) Tripp, R., & Branch, D. 1999, Astrophys. J., 525, 209
* Tsvetkov et al. (2011) Tsvetkov, D. Y., et al. 2011, Astronomy Letters, 37, 775
* Turatto et al. (1996) Turatto, M., Benetti, S., Cappellaro, E., Danziger, I. J., Della Valle, M., Gouiffes, C., Mazzali, P. A., & Patat, F. 1996, Mon. Not. R. Astron. Soc., 283, 1
* Turatto et al. (1997) Turatto, M., Benetti, S., Pereira, C., & da Silva, L. 1997, IAU Circ., 6667, 1
* Turatto et al. (1998) Turatto, M., Piemonte, A., Benetti, S., Cappellaro, E., Mazzali, P. A., Danziger, I. J., & Patat, F. 1998, Astron. J., 116, 2431
* Umeda et al. (1999a) Umeda, H., Nomoto, K., Kobayashi, C., Hachisu, I., & Kato, M. 1999a, Astrophys. J. Lett., 522, L43
* Umeda et al. (1999b) Umeda, H., Nomoto, K., Yamaoka, H., & Wanajo, S. 1999b, Astrophys. J., 513, 861
* Vacca & Leibundgut (1996) Vacca, W. D., & Leibundgut, B. 1996, Astrophys. J. Lett., 471, L37
* Valentini et al. (2003) Valentini, G., et al. 2003, Astrophys. J., 595, 779
* van den Bergh et al. (2005) van den Bergh, S., Li, W., & Filippenko, A. V. 2005, Publ. Astron. Soc. Pac., 117, 773
* van Kerkwijk et al. (2010) van Kerkwijk, M. H., Chang, P., & Justham, S. 2010, Astrophys. J. Lett., 722, L157
* Vennes (1999) Vennes, S. 1999, Astrophys. J., 525, 995
* Vinkó et al. (2001) Vinkó, J., Kiss, L. L., Csák, B., Fűrész, G., Szabó, R., Thomson, J. R., & Mochnacki, S. W. 2001, Astron. J., 121, 3127
* Vinkó et al. (2003) Vinkó, J., et al. 2003, Astron. Astrophys., 397, 115
* Vinkó et al. (2012) —. 2012, Astron. Astrophys., 546, A12
* Voss & Nelemans (2008) Voss, R., & Nelemans, G. 2008, Nature, 451, 802
* Walter & Strohmeier (1937) Walter, K., & Strohmeier, W. 1937, Astronomische Nachrichten, 263, 399
* Wang & Han (2012) Wang, B., & Han, Z. 2012, New Astron. Rev., 56, 122
* Wang et al. (2013a) Wang, B., Justham, S., & Han, Z. 2013a, ArXiv e-prints
* Wang et al. (2006) Wang, L., Baade, D., Höflich, P., Wheeler, J. C., Kawabata, K., Khokhlov, A., Nomoto, K., & Patat, F. 2006, Astrophys. J., 653, 490
* Wang et al. (2004) Wang, L., Baade, D., Höflich, P., Wheeler, J. C., Kawabata, K., & Nomoto, K. 2004, Astrophys. J. Lett., 604, L53
* Wang et al. (2007) Wang, L., Baade, D., & Patat, F. 2007, Science, 315, 212
* Wang & Wheeler (2008) Wang, L., & Wheeler, J. C. 2008, Annu. Rev. Astron. Astrophys., 46, 433
* Wang et al. (2003) Wang, L., et al. 2003, Astrophys. J., 591, 1110
* Wang et al. (2013b) Wang, S., Geng, J.-J., Hu, Y.-L., & Zhang, X. 2013b, ArXiv e-prints
* Wang et al. (2008a) Wang, X., Li, W., Filippenko, A. V., Foley, R. J., Smith, N., & Wang, L. 2008a, Astrophys. J., 677, 1060
* Wang et al. (2013c) Wang, X., Wang, L., Filippenko, A. V., Zhang, T., & Zhao, X. 2013c, Science, 340, 170
* Wang et al. (2008b) Wang, X., et al. 2008b, Astrophys. J., 675, 626
* Wang et al. (2009a) —. 2009a, Astrophys. J. Lett., 699, L139
* Wang et al. (2009b) —. 2009b, Astrophys. J., 697, 380
* Wang et al. (2012) —. 2012, Astrophys. J., 749, 126
* Webbink (1984) Webbink, R. F. 1984, Astrophys. J., 277, 355
* Wellmann (1955) Wellmann, P. 1955, Z. Astrophys., 35, 205
* Wells et al. (1994) Wells, L. A., et al. 1994, Astron. J., 108, 2233
* Wheeler (2012) Wheeler, J. C. 2012, Astrophys. J., 758, 123
* Wheeler et al. (1995) Wheeler, J. C., Harkness, R. P., Khokhlov, A. M., & Höflich, P. 1995, Phys. Rep., 256, 211
* Wheeler et al. (1998) Wheeler, J. C., Höflich, P., Harkness, R. P., & Spyromilio, J. 1998, Astrophys. J., 496, 908
* Whelan & Iben (1973) Whelan, J., & Iben, Jr., I. 1973, Astrophys. J., 186, 1007
* Wood-Vasey et al. (2002a) Wood-Vasey, W. M., Aldering, G., & Nugent, P. 2002a, IAU Circ., 8019, 2
* Wood-Vasey et al. (2002b) Wood-Vasey, W. M., Aldering, G., Nugent, P., Helin, E. F., Pravdo, S., Hicks, M., & Lawrence, K. 2002b, IAU Circ., 7902, 3
* Wood-Vasey et al. (2002c) Wood-Vasey, W. M., Li, W. D., Swift, B., & Ganeshalingam, M. 2002c, IAU Circ., 7959, 1
* Wood-Vasey et al. (2004) Wood-Vasey, W. M., Wang, L., & Aldering, G. 2004, Astrophys. J., 616, 339
* Wood-Vasey et al. (2008) Wood-Vasey, W. M., et al. 2008, Astrophys. J., 689, 377
* Woods et al. (2011) Woods, T. E., Ivanova, N., van der Sluys, M., & Chaichenets, S. 2011, in Astronomical Society of the Pacific Conference Series, Vol. 447, Evolution of Compact Binaries, ed. L. Schmidtobreick, M. R. Schreiber, & C. Tappert, 127
* Woosley et al. (2007) Woosley, S. E., Kasen, D., Blinnikov, S., & Sorokina, E. 2007, Astrophys. J., 662, 487
* Yamanaka et al. (2009a) Yamanaka, M., et al. 2009a, Astrophys. J. Lett., 707, L118
* Yamanaka et al. (2009b) —. 2009b, Publ. Astron. Soc. Jpn., 61, 713
* Yamanaka et al. (2013) Yamanaka, M., et al. 2013, in IAU Symposium, Vol. 281, IAU Symposium, 319–321
* Yamaoka et al. (1992) Yamaoka, H., Nomoto, K., Shigeyama, T., & Thielemann, F.-K. 1992, Astrophys. J. Lett., 393, L55
* Yaron & Gal-Yam (2012) Yaron, O., & Gal-Yam, A. 2012, Publ. Astron. Soc. Pac., 124, 668
* Yoon & Langer (2004) Yoon, S.-C., & Langer, N. 2004, Astron. Astrophys., 419, 623
* Yoon & Langer (2005) —. 2005, Astron. Astrophys., 435, 967
* Yu et al. (2000) Yu, C., Modjaz, M., & Li, W. D. 2000, IAU Circ., 7458, 1
* Yuan et al. (2010) Yuan, F., et al. 2010, Astrophys. J., 715, 1338
* Yungelson & Livio (2000) Yungelson, L. R., & Livio, M. 2000, Astrophys. J., 528, 108
* Zhang (2011) Zhang, T.-M. 2011, Publ. Astron. Soc. Pac., 123, 251
* Zheng et al. (2013) Zheng, W., et al. 2013, Astrophys. J. Lett., 778, L15
* Zingale et al. (2011) Zingale, M., Nonaka, A., Almgren, A. S., Bell, J. B., Malone, C. M., & Woosley, S. E. 2011, Astrophys. J., 740, 8
* Zwicky (1936) Zwicky, F. 1936, Proceedings of the National Academy of Science, 22, 557
* Zwicky (1942) —. 1942, Astrophys. J., 96, 28
* Zwicky (1961) —. 1961, Publ. Astron. Soc. Pac., 73, 185
|
arxiv-papers
| 2014-02-25T21:00:31 |
2024-09-04T02:49:58.885112
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Parrent, B. Friesen and M. Parthasarathy",
"submitter": "Jerod Parrent",
"url": "https://arxiv.org/abs/1402.6337"
}
|
1402.6353
|
# Approximations of Random Dispersal Operators/Equations by Nonlocal Dispersal
Operators/Equations ††thanks: Partially supported by NSF grant DMS–0907752
Wenxian Shen
Department of Mathematics and Statistics
Auburn University
Auburn, AL 36849, U.S.A.
Xiaoxia Xie
Department of Applied Mathematics
Illinois Institute of Technology
Chicago, IL 60616, U.S.A
Abstract. This paper is concerned with the approximations of random dispersal
operators/equations by nonlocal dispersal operators/equations. It first proves
that the solutions of properly rescaled nonlocal dispersal initial-boundary
value problems converge to the solutions of the corresponding random dispersal
initial-boundary value problems. Next, it proves that the principal spectrum
points of nonlocal dispersal operators with properly rescaled kernels converge
to the principal eigenvalues of the corresponding random dispersal operators.
Finally, it proves that the unique positive time periodic solutions of
nonlocal dispersal KPP equations with properly rescaled kernels converge to
the unique positive time periodic solutions of the corresponding random
dispersal KPP equations.
Key words. Nonlocal dispersal, random dispersal, KPP equation, principal
eigenvalue, principal spectrum point, positive time periodic solution.
Mathematics subject classification. 35K20, 35K57, 45C05, 45J05, 92D25.
## 1 Introduction
Both random dispersal evolution equations (or reaction diffusion equations)
and nonlocal dispersal evolution equations (or differential integral
equations) are widely used to model diffusive systems in applied sciences.
Random dispersal equations of the form
$\begin{cases}\partial_{t}u(t,x)=\Delta u(t,x)+F(t,x,u),\quad&x\in D,\cr
B_{r,b}u(t,x)=0,&x\in\partial D\,\,(x\in\mathbb{R}^{N}\,\,{\rm
if}\,\,D=\mathbb{R}^{N}),\end{cases}$ (1.1)
are usually used to model diffusive systems which exhibit local internal
interactions (i.e. the movements of organisms in the systems occur randomly
between adjacent spatial locations) and have been extensively studied (see [1,
2, 3, 6, 19, 20, 24, 29, 32, 42, 46], etc.). In (1.1), the domain $D$ is
either a bounded smooth domain in $\mathbb{R}^{N}$ or $D=\mathbb{R}^{N}$. When
$D$ is a bounded domain, either $B_{r,b}u=B_{r,D}u:=u$ (in such case,
$B_{r,D}u=0$ on $\partial D$ represents homogeneous Dirichlet boundary
condition), or $B_{r,b}u=B_{r,N}u:=\frac{\partial u}{\partial{\bf n}}$ (in
such case, $B_{r,N}u=0$ on $\partial D$ represents homogeneous Neumann
boundary condition), and when $D=\mathbb{R}^{N}$, it is assumed that
$F(t,x,u)$ is periodic in $x_{j}$ with period $p_{j}$ and
$B_{r,b}u=B_{r,P}u:=u(t,x+p_{j}{\bf e_{j}})-u(t,x)$ with ${\bf
e_{j}}=(\delta_{1j},\delta_{2j},\cdots,\delta_{Nj})$ ($\delta_{ij}=0$ if
$i\not=j$ and $\delta_{ij}=1$ if $i=j$) (in such case, $B_{r,P}u=0$ in
$\mathbb{R}^{N}$ represents periodic boundary condition).
Many applied systems exhibit nonlocal internal interaction (i.e. the movements
of organisms in the systems occur between non-adjacent spatial locations).
Nonlocal dispersal evolution equations of the form
$\begin{cases}\partial_{t}u(t,x)=\nu\int_{D\cup
D_{b}}k(y-x)[u(t,y)-u(t,x)]dy+F(t,x,u),\quad&x\in\bar{D},\cr
B_{n,b}u(t,x)=0,&x\in D_{b}\,\,{\rm if}\,\,D_{b}\not=\emptyset,\end{cases}$
(1.2)
are often used to model diffusive systems which exhibit nonlocal internal
interactions and have been recently studied by many people (see [4, 7, 8, 9,
12, 13, 14, 18, 21, 26, 28, 30, 31, 44], etc.). In (1.2), $D$ is either a
smooth bounded domain of $\mathbb{R}^{N}$ or $D=\mathbb{R}^{N}$; $\nu$ is the
dispersal rate; the kernel function $k(\cdot)$ is a smooth and nonnegative
function with compact support (the size of the support reflects the dispersal
distance) and $\int_{\mathbb{R}^{N}}k(z)dz=1$. When $D$ is bounded, either
$D_{b}=D_{D}:=\mathbb{R}^{N}\backslash{\bar{D}}$ and $B_{n,b}u=B_{n,D}:=u$ (in
such case, $u=0$ on $\mathbb{R}^{N}\backslash\bar{D}$ represents homogeneous
Dirichlet type boundary condition), or $D_{b}=D_{N}:=\emptyset$ (in such case,
nonlocal diffusion takes place only in $\bar{D}$ and hence $D_{N}=\emptyset$
represents homogeneous Neumann type boundary condition); when
$D=\mathbb{R}^{N}$, it is assumed that $F(t,x+p_{j}{\bf e_{j}},u)=F(t,x,u)$,
$D_{b}=D_{P}:=\mathbb{R}^{N}$, and $B_{n,b}u=B_{n,P}u:=u(t,x+p_{j}{\bf
e_{j}})-u(t,x)$ (hence $B_{n,P}u=0$ on $\mathbb{R}^{N}$ represents periodic
boundary condition).
Observe that (1.2) with $D_{b}=D_{D}$ and $B_{n,b}u=B_{n,D}u$ can be rewritten
as
$\partial_{t}u(t,x)=\nu\left[\int_{D}k(y-x)u(t,y)dy-u(t,x)\right]+F(t,x,u),\quad
x\in\bar{D};$ (1.3)
that (1.2) with $D_{b}=D_{N}$ reduces to
$\partial_{t}u(t,x)=\nu\int_{D}k(y-x)\left[u(t,y)-u(t,x)\right]dy+F(t,x,u),\quad
x\in\bar{D};$ (1.4)
and that (1.2) with $D=D_{P}$, $F(t,x,u)$ being periodic in $x_{j}$ with
period $p_{j}$, and $B_{n,b}u=B_{n,P}u$ can be written as
$\begin{cases}\partial_{t}u(t,x)=\nu\int_{\mathbb{R}^{N}}k(y-x)\left[u(t,y)-u(t,x)\right]dy+F(t,x,u),\quad&x\in\mathbb{R}^{N},\\\
u(t,x)=u(t,x+p_{j}{\bf e_{j}}),\quad&x\in\mathbb{R}^{N}\end{cases}$ (1.5)
$(j=1,2,\cdots N)$.
A huge amount of research has been carried out toward various dynamical
aspects of random dispersal evolution equations of the form (1.1). There are
also many research works toward various dynamical aspects of nonlocal
dispersal evolution equations of the form (1.2). It has been seen that random
dispersal evolution equations with Dirichlet, or Neumann, or period boundary
condition and nonlocal dispersal evolution equations with the corresponding
boundary condition share many similar properties. For example, a comparison
principle holds for both equations. There are also many differences between
these two types of dispersal evolution equations. For example, solutions of
random dispersal evolution equations have smoothness and certain compactness
properties, but solutions of nonlocal dispersal evolution equations do not
have such properties. Nevertheless, it is expected that nonlocal dispersal
evolution equations with Dirichlet, or Neumann, or periodic boundary condition
and small dispersal distance possess similar dynamical behaviors as those of
random dispersal evolution equations with the corresponding boundary condition
and that certain dynamics of random dispersal evolution equations with
Dirichlet, or Neumann, or periodic boundary condition can be approximated by
the dynamics of nonlocal dispersal evolution equations with the corresponding
boundary condition and properly rescaled kernels. It is of great theoretical
and practical importance to investigate whether such naturally expected
properties actually hold or not.
The objective of the current paper is to investigate how the dynamics of
random dispersal operators/equations can be approximated by those of nonlocal
dispersal operators/equations from three different perspectives, that is, from
initial-boundary value problem point of view, from spectral problem point of
view, and from asymptotic behavior point of view. To this end, we assume that
$k(\cdot)$ is of the form,
$k(z)=k_{\delta}(z):=\frac{1}{\delta^{N}}k_{0}\left(\frac{z}{\delta}\right)$
(1.6)
for some $k_{0}(\cdot)$ satisfying that $k_{0}(\cdot)$ is a smooth,
nonnegative, and symmetric (in the sense that $k_{0}(z)=k_{0}(z^{\prime})$
whenever $|z|=|z^{\prime}|$) function supported on the unit ball $B(0,1)$ and
$\int_{\mathbb{R}^{N}}k_{0}(z)dz=1$, where $\delta(>0)$ is called the
dispersal distance. We also assume that
$\nu=\nu_{\delta}:=\frac{C}{\delta^{2}},$ (1.7)
where
$C=\Big{(}\frac{1}{2}\int_{\mathbb{R}^{N}}k_{0}(z)z_{N}^{2}dz\Big{)}^{-1}$.
Throughout the rest of this paper, we will distinguish the three boundary
conditions by $i=1,2,3$. Let
$X_{1}=X_{2}=\\{u(\cdot)\in C(\bar{D},\mathbb{R})\\}$
with $\|u\|_{X_{i}}=\max_{x\in\bar{D}}|u(x)|(i=1,2)$,
$X_{3}=\\{u\in C(\mathbb{R}^{N},\mathbb{R})|u(x+p_{j}{\bf e_{j}})=u(x)\\},$
with $\|u\|_{X_{3}}=\max_{x\in\mathbb{R}^{N}}|u(x)|$. Let
$X_{i}^{+}=\\{u\in X_{i}\,|\,u(x)\geq 0\\}$
($i=1,2,3$). For $u^{1}(x),u^{2}(x)\in X_{i}$, we define
$u^{1}\leq u^{2}(u^{1}\geq u^{2})\text{ if }u^{2}-u^{1}\in
X_{i}^{+}(u^{1}-u^{2}\in X_{i}^{+})$
(i=1, 2, 3). Note that $X_{1}=X_{2}$ and the introduction of $X_{2}$ is for
convenience.
First, we investigate the approximations of solutions to the initial-boundary
value problem associated to (1.1), that is,
$\begin{cases}\partial_{t}u(t,x)=\Delta u+F(t,x,u),\quad&x\in D,\cr
B_{r,b}(t,x)u=0,\quad&x\in\partial D\quad(x\in\mathbb{R}^{N}\text{ if
}D=\mathbb{R}^{N}),\cr u(s,x)=u_{0}(x),\quad&x\in\bar{D}\end{cases}$ (1.8)
by solutions to the initial-boundary value problem associated to (1.2) with
$k(\cdot)=k_{\delta}(\cdot)$ and $\nu=\nu_{\delta}$, that is,
$\begin{cases}\partial_{t}u(t,x)=\nu_{\delta}\int_{D\cup
D_{b}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+F(t,x,u),\,\ &x\in\bar{D},\cr
B_{n,b}u(t,x)=0,\quad&x\in D_{b}\,\,\,{\rm if}\,\,D_{b}\not=\emptyset,\cr
u(s,x)=u_{0}(x),\quad&x\in\bar{D},\end{cases}$ (1.9)
where $B_{r,b}=B_{r,D}$ (resp. $B_{n,b}=B_{n,D}$ and $D_{b}=D_{D}$), or
$B_{r,b}=B_{r,N}$ (resp. $D_{b}=D_{N}(=\emptyset$)), or $B_{r,b}=B_{r,P}$
(resp. $B_{n,b}=B_{n,P}$ and $D_{b}=D_{P}$). In the rest of this paper, we
assume
(H0) $D\subset\mathbb{R}^{N}$ is either a bounded $C^{2+\alpha}$ domain for
some $0<\alpha<1$ or $D=\mathbb{R}^{N}$; $k_{\delta}(\cdot)$ is as in (1.6)
and $\nu_{\delta}$ is as in (1.7); $F(t,x,u)$ is $C^{1}$ in $t\in\mathbb{R}$
and $C^{3}$ in $(x,u)\in\mathbb{R}^{N}\times\mathbb{R}$, and when
$D=\mathbb{R}^{N}$, $F$ is periodic in $x_{j}$ with period $p_{j}$, that is,
$F(t,x+p_{j}{\bf e_{j}},u)=F(t,x,u)$ for $j=1,2,\cdots,N$.
Note that, by general semigroup theory (see [22, 35]), for any
$s\in\mathbb{R}$ and any $u_{0}\in X_{i}\cap C^{1}(\bar{D})$ with
$B_{r,b}u_{0}=0$ on $\partial D$, (1.8) with $b=D$ if $i=1$, $b=N$ if $i=2$,
and $b=P$ if $i=3$ has a unique (local) solution, denoted by
$u_{i}(t,x;s,u_{0})$. Similarly, for any $s\in\mathbb{R}$ and any $u_{0}\in
X_{i}$, (1.9) with $b=D$ if $i=1$, $b=N$ if $i=2$, and $b=P$ if $i=3$ has a
unique (local) solution, denoted by $u_{i}^{\delta}(t,x;s,u_{0})$.
Among others, we prove
Theorem A. Assume that for given $1\leq i\leq 3$, $\delta_{0}>0$,
$s\in\mathbb{R}$, $T>0$, and $u_{0}\in X_{i}\cap C^{3}(\bar{D})$ with
$B_{r,b}u_{0}=0$ if $D$ is bounded ($b=D$ if $i=1$ and $b=N$ if $i=2$),
$u_{i}(t,x;s,u_{0})$ and $u_{i}^{\delta}(t,x;s,u_{0})$ exist on $[s,s+T]$ for
all $0<\delta\leq\delta_{0}$. Assume also that $\sup_{s\leq t\leq
s+T,x\in\bar{D},0<\delta\leq\delta_{0}}|u_{i}(t,x;s,u_{0})|<\infty$. Then,
$\lim_{\delta\to
0}\sup_{t\in[s,s+T]}\|u_{i}^{\delta}(t,\cdot;s,u_{0})-u_{i}(t,\cdot;s,u_{0})\|_{X_{i}}=0.$
It should be pointed out that Theorem A is the basis for the study of
approximations of various dynamics of random dispersal evolution equations by
those of nonlocal dispersal evolution equations. It should also be pointed out
that when $F(t,x,u)\equiv 0$ in (1.8) and (1.9), similar results to Theorem A
have been proved in [10] and [11] for the Dirichlet and Neumann boundary
condition cases, respectively.
Secondly, we investigate the principal eigenvalues of time periodic random
dispersal eigenvalue problems of the form
$\begin{cases}-\partial_{t}u+\Delta u+a(t,x)u=\lambda u,\quad&x\in D,\cr
B_{r,b}u=0,\quad&x\in\partial D\,\ (x\in\mathbb{R}^{N}\text{ if
}D=\mathbb{R}^{N}),\cr u(t+T,x)=u(t,x),\quad&x\in D,\end{cases}$ (1.10)
and their nonlocal counterparts of the form
$\begin{cases}-\partial_{t}u+\nu_{\delta}\int_{D\cup
D_{b}}k_{\delta}(y-x)\left[u(t,y)-u(t,x)\right]dy+a(t,x)u=\lambda
u,\quad&x\in\bar{D},\cr B_{n,b}u=0,\quad&x\in D_{b}\,\,{\rm
if}\,\,D_{b}\not=\emptyset,\cr u(t+T,x)=u(t,x),\quad&x\in\bar{D},\end{cases}$
(1.11)
where $a(t+T,x)=a(t,x)$, and when $D=\mathbb{R}^{N}$, $a(t+T,x+p_{j}{\bf
e_{j}})=a(t,x)$ for $j=1,2,\cdots,N$, and $B_{r,b}=B_{r,D}$ (resp.
$B_{n,b}=B_{n,D}$ and $D_{b}=D_{D}$), or $B_{r,b}=B_{r,N}$ (resp.
$D_{b}=D_{N}(=\emptyset)$) or $B_{r,b}=B_{r,P}$ (resp. $B_{n,b}=B_{n,P}$ and
$D_{b}=D_{P}$). We assume that $a(t,x)$ is a $C^{1}$ function in
$(t,x)\in\mathbb{R}\times\mathbb{R}^{N}$.
The eigenvalue problems of (1.10), in particular, their associated principal
eigenvalue problems, are extensively studied and quite well understood (see
[15, 16, 17, 23, 25, 27, 34, 38], etc.). For example, with any one of the
three boundary conditions, it is known that the largest real part, denoted by
$\lambda^{r}(a)$, of the spectrum set of (1.10) is an isolated algebraically
simple eigenvalue with a positive eigenfunction, and for any other $\lambda$
in the spectrum set of (1.10), $\text{Re}\lambda\leq\lambda^{r}(a)$
($\lambda^{r}(a)$ is called the principal eigenvalue of $\eqref{main-random-
eigenvalue}$ in literature).
The eigenvalue problems (1.11) have also been studied recently by many people
(see [5, 12, 27, 36, 38, 39, 40, 41], etc.). Let $\lambda^{\delta}(a)$ be the
largest real part of the spectrum set of (1.11) with any one of the three
boundary conditions. $\lambda^{\delta}(a)$ is called the principal spectrum
point of (1.11). $\lambda^{\delta}(a)$ is also called the principal eigenvalue
of (1.11), if it is an isolated algebraically simple eigenvalue with a
positive eigenfunction (see Definition 3.1 for detail). Note that
$\lambda^{\delta}(a)$ may not be an eigenvalue of (1.11) (see [12], [39] for
examples). Hence the principal eigenvalue of (1.11) may not exist. In [41],
the authors of the current paper studied the dependence of principal spectrum
points or principal eigenvalues (if exist) of nonlocal dispersal operators on
underlying parameters ($\delta,a(\cdot)$, and $\nu$) in a spatially
heterogeneous but temporally homogeneous case. However, the understanding is
still little to many interesting questions regarding the principal spectrum
points or principal eigenvalues (if exist) of (1.11). In this paper, we show
that the principal eigenvalue of (1.10) can be approximated by the principal
spectrum point of (1.11). In fact, we show
Theorem B. $\lim_{\delta\to 0}\lambda^{\delta}(a)=\lambda^{r}(a)$.
We remark that Theorem B is another basis for the study of approximations of
various dynamics of random dispersal evolution equations by those of nonlocal
dispersal evolution equations. We also remark that some necessary and
sufficient conditions are provided in [36] and [37] for $\lambda_{\delta}(a)$
to be the principal eigenvalue of (1.11). Among other, it is proved in [36,
Theorem A] and [37, Theorem 3.1] that $\lambda^{\delta}(a)$ is the principal
eigenvalue of (1.11) if and only if
$\lambda^{\delta}(a)>\max_{x\in\bar{D}}\left\\{-\frac{C}{\delta^{2}}+\frac{1}{T}\int_{0}^{T}a(t,x)dt\right\\}.$
This together with Theorem B implies the following remark.
###### Remark 1.1.
$\lambda^{\delta}(a)$ is the principal eigenvalue of (1.11), provided
$\delta\ll 1$.
Thirdly, we explore the asymptotic dynamics of the following time periodic
dispersal evolution equations,
$\begin{cases}\partial_{t}u=\Delta u+uf(t,x,u),\quad&x\in D,\cr
B_{r,b}u=0,\quad&x\in\partial D\,\ (x\in\mathbb{R}^{N}\text{ if
}D=\mathbb{R}^{N}),\end{cases}$ (1.12)
and
$\begin{cases}\partial_{t}u=\nu_{\delta}\int_{D\cup
D_{b}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+uf(t,x,u),\quad&x\in\bar{D},\cr
B_{n,b}u=0,&x\in D_{b}\,\,\,{\rm if}\,\,D_{b}\not=\emptyset,\end{cases}$
(1.13)
where $D$ is as in (H0). In the rest of this paper, we assume that
(H1) $f$ is $C^{1}$ in $t\in\mathbb{R}$ and $C^{3}$ in
$(x,u)\in\mathbb{R}^{N}\times\mathbb{R}$; $f(t,x,u)<0$ for $u\gg 1$ and
$\partial_{u}f(t,x,u)<0$ for $u\geq 0$; $f(t+T,x,u)=f(t,x,u)$; and when
$D=\mathbb{R}^{N}$, $f(t+T,x,u)=f(t,x+p_{j}{\bf e_{j}},u)=f(t,x,u)$ for
$j=1,2,\cdots,N$.
(H2) For (1.12), $\lambda^{r}(f(\cdot,\cdot,0))>0$, where
$\lambda^{r}(f(\cdot,\cdot,0))$ is the principle eigenvalue of (1.10) with
$a(t,x)=f(t,x,0)$.
(H2)δ For (1.13), $\lambda^{\delta}(f(\cdot,\cdot,0))>0$, where
$\lambda^{\delta}(f(\cdot,\cdot,0))$ is the principle spectrum point of (1.11)
with $a(t,x)=f(t,x,0)$.
Equations (1.12) and (1.13) are widely used to model population dynamics of
species exhibiting random interactions and nonlocal interactions, respectively
(see [4, 14, 33], etc. for (1.12) and [36] for (1.13)). Thanks to the
pioneering works of Fisher [20] and Kolmogorov et al. [29] on the following
special case of (1.12),
$\partial_{t}u=u_{xx}+u(1-u),\quad x\in\mathbb{R},$
(1.12) and (1.13) are referred to as Fisher type or KPP type equations.
The dynamics of (1.12) and (1.13) have been studied in many papers (see [24,
33, 45] and references therein for (1.12), and [36] and references therein for
(1.13)). With conditions (H1) and (H2), it is proved that (1.12) has exactly
two nonnegative time periodic solutions, one is $u\equiv 0$ which is unstable
and the other one, denoted by $u^{*}(t,x)$, is asymptotically stable and
strictly positive (see [45, Theorem 3.1], see also [33, Theorems 1.1, 1.3]).
Similar results for (1.13) under the assumptions (H1) and (H2)δ are proved in
[36, Theorem E]. We denote the strictly positive time periodic solution of
(1.13) by $u_{\delta}^{*}(t,x)$.
Note that, by Theorem B and Remark 1.1, (H2) implies (H2)δ when $0<\delta\ll
1$. Hence, we only assume (H2) in the following theorem. In this paper, we
show that
Theorem C. If (H1) and (H2) hold, then for any $\epsilon>0$, there exists
$\delta_{0}>0$, such that for all $0<\delta<\delta_{0}$, we have
$\sup_{t\in[0,T]}\|u_{\delta}^{*}(t,\cdot)-u^{*}(t,\cdot)\|_{C(\bar{D},\mathbb{R})}\leq\epsilon.$
Theorems A-C in the above show that many important dynamics of random
dispersal equations can be approximated by the corresponding dynamics of
nonlocal dispersal equations, which is of both great theoretical and practical
importance.
The rest of the paper is organized as follows. In section 2, we explore the
approximation of solutions of random dispersal evolution equations by the
solutions of nonlocal dispersal evolution equations and prove Theorem A. In
section 3, we investigate the approximation of principal eigenvalues of time
periodic random dispersal operators by the principal spectrum points of time
periodic nonlocal dispersal operators and prove Theorem B. We study in section
4 the approximation of the asymptotic dynamics of time periodic KPP equations
with random dispersal by the asymptotic dynamics of time periodic KPP
equations with nonlocal dispersal and prove Theorem C.
## 2 Approximation of Initial-boundary Value Problems of Random Dispersal
Equations by Nonlocal Dispersal Equations
In this section, we explore the approximation of solutions to (1.8) by the
solutions to (1.9). We first present some comparison principle for (1.8) and
(1.9). Then we prove Theorem A. Though the ideas of the proofs of Theorem A
for different types of boundary conditions are the same, different techniques
are needed for different boundary conditions. We hence give proofs of Theorem
A for different boundary conditions in different subsections.
### 2.1 Comparison principle for random and nonlocal dispersal evolution
equations
In this subsection, we present a comparison principle for random and nonlocal
evolution equations, which will be applied in the proof of Theorem A in this
section as well as in the proofs of Theorem B and C in sections 3 and 4.
###### Definition 2.1 (Super- and sub- solutions).
A continuous function $u(t,x)$ on $[s,s+T)\times\mathbb{R}^{N}$ is called a
super-solution (sub-solution) of (1.9) on $(s,s+T)$ if for any $x\in\bar{D}$,
$u(t,x)$ is differentiable on $(s,s+T)$ and satisfies that
$\begin{cases}\partial_{t}u(t,x)\geq(\leq)\nu_{\delta}\int_{D\cup
D_{b}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+F(t,x,u),\quad&x\in\bar{D},\\\
B_{n,b}u(t,x)\geq(\leq)0,&x\in D_{b}\,\,{\rm if}\,\,D_{b}\not=\emptyset,\\\
u(s,x)\geq(\leq)u_{0}(x),&x\in\bar{D},\end{cases}$
when $b=D$ or $N$, or that
$\begin{cases}\partial_{t}u(t,x)\geq(\leq)\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+F(t,x,u),\quad&x\in\mathbb{R}^{N},\\\
B_{n,b}u(t,x)=0,&x\in\mathbb{R}^{N},\\\
u(s,x)\geq(\leq)u_{0}(x),&x\in\mathbb{R}^{N},\end{cases}$
when $b=P$.
Super-solutions and sub-solutions of (1.8) on $(s,s+T)$ are defined in an
analogous way.
###### Proposition 2.1 (Comparison principle).
* (1)
Suppose that $u^{-}(t,x)$ and $u^{+}(t,x)$ are sub-solution and super-solution
of (1.8) on $(s,s+T)$, respectively, then
$u^{-}(t,x)\leq u^{+}(t,x)\quad\forall\,\,t\in[s,s+T),\,\,x\in\bar{D}.$
* (2)
Suppose that $u^{-}(t,x)$ and $u^{+}(t,x)$ are sub-solution and super-solution
of (1.9) on $(s,s+T)$, respectively, then
$u^{-}(t,x)\leq u^{+}(t,x)\quad\forall\,\,t\in[s,s+T),\,\,x\in\bar{D}.$
###### Proof.
(1) It follows from comparison principle for parabolic equations.
(2) It follows from [36, Proposition 3.1]. ∎
### 2.2 Proof of Theorem A in the Dirichlet boundary condition case
In this subsection, we prove Theorem A in the Dirichlet boundary case.
Throughout this subsection, we assume (H0), and $B_{r,b}u=B_{r,D}u$ in (1.8),
and $D_{b}=D_{D}(=\mathbb{R}^{N}\backslash\bar{D})$ and $B_{n,b}u=B_{n,D}u$ in
(1.9). Note that $D\cup D_{b}=\mathbb{R}^{N}$ in this case. Without loss of
generality, we assume $s=0$.
###### Proof of Theorem A in the Dirichlet boundary condition case.
Let $u_{0}\in C^{3}(\bar{D})$ with $u_{0}(x)=0$ for $x\in\partial D$. Let
$u_{1}^{\delta}(t,x)$ be the solution of (1.9) with $s=0$ and $u_{1}(t,x)$ be
the solution of (1.8) with $s=0$. Suppose that $u_{1}(t,x)$ and
$u_{1}^{\delta}(t,x)$ exist on $[0,T]$. By regularity of solutions for
parabolic equations, $u_{1}\in
C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\bar{D})\cap
C^{0,2+\alpha}([0,T]\times\bar{D})$. Let $\tilde{u}_{1}$ be an extension of
$u_{1}$ to $[0,T]\times\mathbb{R}^{N}$ satisfying that $\tilde{u}_{1}\in
C^{0,2+\alpha}([0,T]\times\mathbb{R}^{N})$. Define
$L_{\delta}(z)(t,x)=\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)[z(t,y)-z(t,x)]dy.$
Let $G(t,x)=\tilde{u}_{1}(t,x)$ for
$(t,x)\in[0,T]\times\mathbb{R}^{N}\backslash\bar{D}$. Then $\tilde{u}_{1}$
verifies
$\begin{cases}\partial_{t}\tilde{u}_{1}(t,x)=L_{\delta}(\tilde{u}_{1})(t,x)+F_{\delta}(t,x)+F(t,x,\tilde{u}_{1}(t,x)),\quad&x\in\bar{D},\,\,\,\
\quad\,t\in(0,T],\\\ \tilde{u}_{1}(t,x)=G(t,x),\quad\,\
&x\in\mathbb{R}^{N}\backslash\bar{D},t\in[0,T],\\\
\tilde{u}_{1}(0,x)=u_{0}(x),\quad&x\in\bar{D},\end{cases}$
where
$\displaystyle F_{\delta}(t,x)$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-{L}_{\delta}(\tilde{u}_{1})(t,x)$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)(\tilde{u}_{1}(t,y)-\tilde{u}_{1}(t,x))dy.$
Let $w_{1}^{\delta}=\tilde{u}_{1}-u_{1}^{\delta}$. We then have
$\begin{cases}\partial_{t}w_{1}^{\delta}(t,x)=L_{\delta}(w_{1}^{\delta})(t,x)+F_{\delta}(t,x)+a_{1}^{\delta}(t,x)w_{1}^{\delta}(t,x),\quad&x\in\bar{D},\,\,\,\quad\,\
t\in(0,T],\\\
w_{1}^{\delta}(t,x)=G(t,x),\quad&x\in\mathbb{R}^{N}\backslash\bar{D},\,t\in[0,T],\\\
w_{1}^{\delta}(0,x)=0,\quad&x\in\bar{D},\end{cases}$ (2.1)
where
$a_{1}^{\delta}(t,x)=\int_{0}^{1}F_{u}[t,x,u_{1}^{\delta}(t,x)+\theta(\tilde{u}_{1}(t,x)-u_{1}^{\delta}(t,x))]d{\theta}$.
We claim that
$\begin{cases}\sup_{t\in[0,T]}\|F_{\delta}(t,\cdot)\|_{X_{1}}=O({\delta}^{\alpha}),\cr\sup_{t\in[0,T],x\in\mathbb{R}^{N}\setminus\bar{D},{\rm
dist}(x,\partial D)\leq\delta}|G(t,x)|=O(\delta).\end{cases}$ (2.2)
In fact,
$\displaystyle\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)(\tilde{u}_{1}(t,y)-\tilde{u}_{1}(t,x))dy$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}\frac{1}{\delta^{N}}k_{0}\left(\frac{y-x}{\delta}\right)(\tilde{u}_{1}(t,y)-\tilde{u}_{1}(t,x))dy$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}k_{0}(z)(\tilde{u}_{1}(t,x+\delta
z)-\tilde{u}_{1}(t,x))dz$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}k_{0}(z)\left[\frac{\delta^{2}z_{N}^{2}}{2!}\Delta\tilde{u}_{1}(t,x)+O(\delta^{2+\alpha})\right]dz$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\left[\nu_{\delta}\delta^{2}\int_{\mathbb{R}^{N}}k_{0}(z)\frac{z_{N}^{2}}{2}dz\right]\Delta\tilde{u}_{1}(t,x)+O(\delta^{\alpha})$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\Delta\tilde{u}_{1}(t,x)+O(\delta^{\alpha})$
$\displaystyle=O(\delta^{\alpha})\quad\forall\,\,x\in\bar{D},$
and
$\displaystyle|G(t,x)|$ $\displaystyle=|\tilde{u}_{1}(t,x)|$
$\displaystyle\leq\sup_{t\in[0,T],x\in\mathbb{R}^{N}\setminus D,z\in\partial
D,\text{dist}(x,z)\leq\delta}|\tilde{u}_{1}(t,x)-u_{1}(t,z)|$
$\displaystyle=O(\delta)\quad\forall\,\,x\in\mathbb{R}^{N}\backslash\bar{D},\,\,{\rm
dist}(x,\partial D)\leq\delta.$
Therefore, (2.2) holds.
Next, let $\bar{w}$ be given by
$\bar{w}(t,x)=e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta,$
where
$A=\underset{t\in[0,T],x\in\bar{D},0<\delta\leq\delta_{0}}{\max}a_{1}^{\delta}(t,x)$.
By direct calculation, we have
$\begin{cases}\partial_{t}\bar{w}(t,x)=L_{\delta}(\bar{w})+a_{1}^{\delta}(t,x)\bar{w}+\bar{F}_{\delta}(t,x)\quad&x\in\bar{D},\quad\quad\,\,\
\,\,t\in(0,T],\\\
\bar{w}(t,x)=e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta,\quad&x\in\mathbb{R}^{N}\backslash\bar{D},\quad
t\in[0,T],\\\ \bar{w}(0,x)=K_{2}\delta,\quad&x\in\bar{D},\end{cases}$ (2.3)
where
$\bar{F}_{\delta}(t,x)=e^{At}K_{1}{\delta}^{\alpha}+[A-a_{1}^{\delta}(t,x)]e^{At}K_{1}{\delta}^{\alpha}t-a_{1}^{\delta}(t,x)K_{2}\delta.$
By (2.2), there are $0<\tilde{\delta}_{0}\leq\delta_{0}$ and $K_{1},K_{2}>0$
such that
$\begin{cases}F_{\delta}(t,x)\leq\bar{F}_{\delta}(t,x),\quad&x\in\bar{D},\,\,\,\,t\in[0,T],\cr
G(t,x)\leq
e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta,\quad&x\in\mathbb{R}^{N}\backslash\bar{D},\,\,{\rm
dist}(x,\partial D)\leq\delta,\,t\in[0,T],\end{cases}$ (2.4)
when $0<\delta<\tilde{\delta}_{0}$. By (2.1), (2.3), (2.4), and Proposition
2.1, we obtain
$w^{\delta}(t,x)\leq\bar{w}(t,x)=e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta\quad\forall\,x\in\bar{D},\,\,t\in[0,T]$
(2.5)
for $0<\delta<\tilde{\delta}_{0}$.
Similarly, let
$\underline{w}(t,x)=e^{At}(-K_{1}{\delta}^{\alpha}t)-K_{2}\delta$. We can
prove that for $0<\delta<\tilde{\delta}_{0}$ (by reducing $\tilde{\delta}_{0}$
if necessary),
$w^{\delta}(t,x)\geq\underline{w}(t,x)=-e^{At}(K_{1}{\delta}^{\alpha}t)-K_{2}\delta\quad\forall\,\,x\in\bar{D},\,\,t\in[0,T].$
(2.6)
By (2.5) and (2.6) we have
$|w^{\delta}(t,x)|\leq
e^{At}K_{1}{\delta}^{\alpha}t+K_{2}\delta\quad\forall\,\,x\in\bar{D},\,\,t\in[0,T],$
which implies that there is $C(T)>0$ such that for any
$0<\delta<\tilde{\delta}_{0}$,
$\sup_{t\in[0,T]}\|u_{1}(\cdot,t)-u_{1}^{\delta}(\cdot,t)\|_{X_{1}}\leq
C(T){\delta}^{\alpha}.$
Theorem A in the Dirichlet boundary condition case then follows. ∎
###### Remark 2.1.
If the homogeneous Dirichlet boundary conditions $B_{r,D}u=u=0$ on $\partial
D$ and $B_{n,D}u=u=0$ on $\mathbb{R}^{N}\backslash\bar{D}$ are changed to
nonhomogeneous Dirichlet boundary conditions $B_{r,D}u=u=g(t,x)$ on $\partial
D$ and $B_{n,D}u=u=g(t,x)$ on $\mathbb{R}^{N}\backslash\bar{D}$, Theorem A
also holds, which can be proved by the similar arguments as above.
### 2.3 Proof of Theorem A in the Neumann boundary condition case
In this subsection, we prove Theorem A in the Neumann boundary condition case.
Throughout this subsection, we assume (H0), and $B_{r,b}u=B_{r,N}u$ in (1.8),
and $D_{b}=D_{N}=\emptyset$ in (1.9). Without loss of generality, we assume
$s=0$.
We first introduce two lemmas. To this end, for given $\delta>0$ and
$d_{0}>0$, let $D_{\delta}=\\{z\in D|\mathrm{dist}(z,\partial
D)<d_{0}\delta\\}$.
###### Lemma 2.1.
Let $\theta\in
C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\times\mathbb{R}^{N})\cap
C^{0,2+\alpha}([0,T]\times\mathbb{R}^{N})$ and
$\frac{\partial\theta}{\partial{\bf n}}=h$ on $\partial D$, then for $x\in
D_{\delta}$ and $\delta$ small,
$\displaystyle\frac{1}{\delta^{2}}{\int}_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)(\theta(t,y)-\theta(t,x))dy$
$\displaystyle=\frac{1}{\delta}{\int}_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x){\bf n}(\bar{x})\cdot\frac{y-x}{\delta}h(\bar{x},t)dy$
$\displaystyle+{\int}_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)\sum_{|\beta|=2}\frac{D^{\beta}\theta}{2}(\bar{x},t)\left[\left(\frac{y-\bar{x}}{\delta}\right)^{\beta}-\left(\frac{x-\bar{x}}{\delta}\right)^{\beta}\right]dy+O(\delta^{\alpha}),$
where $\bar{x}$ is the orthogonal projection of $x$ on the boundary of $D$ so
that $\|\bar{x}-y\|\leq 2d_{0}\delta$ and ${\bf n}(\bar{x})$ is the exterior
unit normal vector of $\partial D$ at $\bar{x}$.
###### Proof.
See [10, Lemma 3]. ∎
###### Lemma 2.2.
There exist $K>0$ and $\bar{\delta}>0$ such that for $\delta<\bar{\delta}$,
$\underset{\mathbb{R}^{N}\backslash D}{\int}k_{\delta}(y-x){\bf
n}(\bar{x})\frac{y-x}{\delta}dy\geq K\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy.$
###### Proof.
See [10, Lemma 4]. ∎
###### Proof of Theorem A in the Neumann boundary condition case.
Suppose that $u_{0}\in C^{3}(\bar{D})$. Let $u_{2}^{\delta}(t,x)$ be the
solution to (1.9) with $s=0$ and $u_{2}(t,x)$ be the solution to (1.8) with
$s=0$. Assume that $u_{2}(t,x)$ and $u_{2}^{\delta}(t,x)$ exist on $[0,T]$.
Then $u_{2}\in C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\bar{D})$. Let
$\tilde{u}_{2}$ be an extension of $u_{2}$ to $[0,T]\times\mathbb{R}^{N}$
satisfying that $\tilde{u}_{2}\in
C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\mathbb{R}^{N})\cap
C^{0,2+\alpha}([0,T]\times\mathbb{R}^{N})$. Define
$L_{\delta}(z)(t,x)=\nu_{\delta}\int_{D}k_{\delta}(y-x)(z(t,y)-z(t,x))dy,$
and
$\tilde{L}_{\delta}(z)(t,x)=\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)(z(t,y)-z(t,x))dy.$
Set $w_{2}^{\delta}=u_{2}^{\delta}-\tilde{u}_{2}$. Then
$\displaystyle\partial_{t}w_{2}^{\delta}(t,x)$
$\displaystyle=\partial_{t}u_{2}^{\delta}(t,x)-\partial_{t}\tilde{u}_{2}(t,x)$
$\displaystyle=[L_{\delta}(u_{2}^{\delta})(t,x)+F(t,x,u_{2}^{\delta})]-[\Delta\tilde{u}_{2}(t,x)+F(t,x,\tilde{u}_{2})]$
$\displaystyle=L_{\delta}(w_{2}^{\delta})(t,x)+a_{2}^{\delta}(t,x)w_{2}^{\delta}(t,x)+F_{\delta}(t,x),$
where
$a_{2}^{\delta}(t,x)=\int_{0}^{1}F_{u}(t,x,\tilde{u}_{2}(t,x)+\theta(u^{\delta}_{2}(t,x)-\tilde{u}_{2}(t,x)))d\theta$
and
$F_{\delta}(t,x)=\tilde{L}_{\delta}(\tilde{u}_{2})(t,x)-\Delta\tilde{u}_{2}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)(\tilde{u}_{2}(t,y)-\tilde{u}_{2}(t,x))dy.$
Hence $w_{2}^{\delta}$ verifies
$\begin{cases}\partial_{t}w_{2}^{\delta}(t,x)=L_{\delta}(w_{2}^{\delta})(t,x)+a_{2}^{\delta}(t,x)w_{2}^{\delta}(t,x)+F_{\delta}(t,x),&\quad
x\in\bar{D},\\\ w_{2}^{\delta}(0,x)=0,&\quad x\in\bar{D}.\end{cases}$ (2.7)
To prove the theorem, let us pick an auxiliary function $v$ as a solution to
$\begin{cases}\partial_{t}v(t,x)=\Delta
v(t,x)+a_{2}^{\delta}(t,x)v(t,x)+h(t,x),\quad&x\in D,\,\,\,\,t\in(0,T],\\\
\frac{\partial v}{\partial{\bf n}}(t,x)=g(t,x),\quad&x\in\partial
D,\,t\in[0,T],\\\ v(0,x)=v_{0}(x),\quad&x\in D\end{cases}$
for some smooth functions $h(t,x)\geq 1$, $g(t,x)\geq 1$ and $v_{0}(x)\geq 0$
such that $v(t,x)$ has an extension $\tilde{v}(t,x)\in
C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\mathbb{R}^{N})\cap
C^{0,2+\alpha}([0,T]\times\mathbb{R}^{N})$. Then $v$ is a solution to
$\begin{cases}\partial_{t}v(t,x)=L_{\delta}(v)(t,x)+a_{2}^{\delta}(t,x)v(t,x)+H(t,x,\delta),\quad&x\in\bar{D},t\in(0,T],\\\
v(0,x)=v_{0}(x),\quad&x\in\bar{D},t\in[0,T],\end{cases}$ (2.8)
where
$H(t,x,\delta)=\Delta\tilde{v}(t,x)-\tilde{L}_{\delta}(v)(t,x)+\nu_{\delta}\int_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)(\tilde{v}(t,y)-\tilde{v}(t,x))dy+h(t,x).$
By Lemma 2.1 and the first estimate in (2.2), we have the following estimate
for $H(x,t,\delta)$:
$\displaystyle H(t,x,\delta)$
$\displaystyle=\Delta\tilde{v}(t,x)-\tilde{L}_{\delta}(v)(t,x)+\frac{C}{\delta^{2}}\int_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)(\tilde{v}(t,y)-\tilde{v}(t,x))dy+h(t,x)$
$\displaystyle\geq\frac{C}{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x){\bf n}(\bar{x})\frac{y-x}{\delta}g(\bar{x},t)dy$
$\displaystyle\qquad+C\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)\sum_{|\beta|=2}\frac{D^{\beta}\tilde{v}}{2}(\bar{x},t)\left[\left(\frac{y-\bar{x}}{\delta}\right)^{\beta}-\left(\frac{x-\bar{x}}{\delta}\right)^{\beta}\right]dy+1-C_{1}\delta^{\alpha}$
$\displaystyle\geq\frac{C}{\delta}g(\bar{x},t)\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x){\bf n}(\bar{x})\frac{y-x}{\delta}dy-
D_{1}C\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy+\frac{1}{2}$ (2.9)
for some constants $D_{1}$ and $C_{1}$ and $\delta$ sufficiently small such
that $C_{1}\delta^{\alpha}\leq\frac{1}{2}$. Then Lemma 2.2 implies that there
exist $C^{\prime}>0$ and $\delta^{\prime}$ such that
$\frac{1}{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x){\bf
n}(\bar{x})\frac{y-x}{\delta}dy\geq\frac{C^{\prime}}{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy,$
if $\delta<\delta^{\prime}$. This implies that
$\displaystyle
H(x,t,\delta)\geq\left[\frac{CC^{\prime}g(\bar{x},t)}{\delta}-D_{1}\right]\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy+\frac{1}{2},$ (2.10)
if $\delta<\delta^{\prime}$.
We now estimate $F_{\delta}(t,x)$. By Lemmas 2.1, 2.2, the first estimate in
(2.2), and the fact that $\frac{\partial\tilde{u}_{2}}{\partial{\bf n}}=0$ on
$\partial D$, we have
$\displaystyle F_{\delta}(t,x)$
$\displaystyle=O(\delta^{\alpha})+\nu_{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)(\tilde{u}_{2}(t,y)-\tilde{u}_{2}(t,x))dy$
$\displaystyle=O(\delta^{\alpha})+C\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)\sum_{|\beta|=2}\frac{D^{\beta}\theta}{2}(\bar{x},t)\left[\left(\frac{y-\bar{x}}{\delta}\right)^{\beta}-\left(\frac{x-\bar{x}}{\delta}\right)^{\beta}\right]dy$
$\displaystyle\leq
C_{2}\delta^{\alpha}+D_{1}C\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy$
$\displaystyle=C_{2}\delta^{\alpha}+D_{2}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy$ (2.11)
for some $C_{2}>0$ and $D_{2}>0$. Given $\epsilon>0$, let
$v_{\epsilon}=\epsilon v$. By (2.8), $v_{\epsilon}$ satisfies
$\begin{cases}\partial_{t}v_{\epsilon}(t,x)-L_{\delta}(v_{\epsilon})(t,x)-a_{2}^{\delta}(t,x)v_{\epsilon}(t,x)=\epsilon
H(t,x,\delta),&\quad x\in\bar{D},\\\ v_{\epsilon}(0,x)=\epsilon
v_{0}(x),&\quad x\in\bar{D}.\end{cases}$ (2.12)
By (2.10) and (2.3), there exist $C_{3}>0$ and
$0<\tilde{\delta}_{0}<\delta_{0}$ such that for
$0<\delta\leq\tilde{\delta}_{0}$,
$\displaystyle F_{\delta}(t,x)\leq
C\delta^{\alpha}+D_{2}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy\leq\frac{\epsilon}{2}+\frac{C_{3}\epsilon}{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy=\epsilon
H(x,t,\delta)\quad\forall\,x\in\bar{D},\,\,t\in[0,T].$ (2.13)
Then by (2.7), (2.12), (2.13), and Proposition 2.1, we have
$-M\epsilon\leq-v_{\epsilon}\leq w_{2}^{\delta}\leq v_{\epsilon}\leq
M\epsilon\quad\forall\,\,\delta\leq\tilde{\delta}_{0},$
where $\displaystyle M=\max_{t\in[0,T],x\in\bar{D}}v(t,x)$. This implies
$\sup_{t\in[0,T]}\|u_{2}^{\delta}(t,\cdot)-u_{2}(t,\cdot)\|_{X_{2}}\rightarrow
0,\quad\mathrm{as}\,\ \delta\rightarrow 0.$
Theorem A in the Neumann boundary condition is thus proved. ∎
### 2.4 Proof of Theorem A in the periodic boundary condition case
In this subsection, we prove Theorem A in the periodic boundary condition
case. Throughout this subsection, we assume (H0), $B_{r,b}u=B_{r,P}u$ in
(1.8), and $B_{n,b}u=B_{n,P}u$ in (1.9). Without loss of generality again, we
assume $s=0$.
###### Proof of Theorem A in the periodic boundary case.
Suppose that $u_{0}\in X_{3}\cap C^{3}(\mathbb{R}^{N})$. Let
$u_{3}^{\delta}(t,x)$ be the solution to (1.9) with $s=0$ and $u_{3}(t,x)$ be
the solution to (1.8) with $s=0$. Suppose that $u_{3}(t,x)$ and
$u_{3}^{\delta}(t,x)$ exist on $[0,T]$. Set
$w_{3}^{\delta}=u_{3}^{\delta}-u_{3}$. Then $w_{3}^{\delta}$ satisfies
$\begin{cases}\partial_{t}w_{3}^{\delta}(t,x)=\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)(w_{3}^{\delta}(t,y)-w_{3}^{\delta}(t,x))dy+a_{3}^{\delta}(t,x)w_{3}^{\delta}(t,x)+F_{\delta}(t,x),&x\in\mathbb{R}^{N},\,t\in(0,T],\\\
w_{3}^{\delta}(t,x)=w_{3}^{\delta}(t,x+p_{j}\mathbb{e}_{j}),&x\in\mathbb{R}^{N},\,t\in[0,T],\\\
w_{3}^{\delta}(0,x)=0,&x\in\mathbb{R}^{N},\end{cases}$ (2.14)
where
$a_{3}^{\delta}(t,x)=\int_{0}^{1}F_{u}(t,x,u_{3}(t,x)+\theta(u^{\delta}_{3}(t,x)-u_{3}(t,x)))d\theta$
and
$F_{\delta}(t,x)=\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)[u_{3}(t,y)-u_{3}(t,x)]dy-\Delta
u_{3}$. Let
$\bar{w}(t,x)=e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta,$
where $\displaystyle
A=\max_{t\in[0,T],x\in\mathbb{R}^{N},0<\delta\leq\delta_{0}}a_{3}^{\delta}(t,x)$.
Applying the similar approach as in the Dirichlet boundary condition case, we
can show that there are $K_{1}>0$, $K_{2}>0$, and $\delta_{0}>0$ such that for
$0<\delta<\delta_{0}$,
$-\bar{w}(t,x)\leq
w_{3}^{\delta}(t,x)\leq\bar{w}(t,x)\quad\forall\,\,x\in\mathbb{R}^{N},\,\,t\in[0,T].$
Theorem A in the periodic boundary condition case then follows. ∎
## 3 Approximation of Principal Eigenvalues of Time Periodic Random Dispersal
Operators by Nonlocal Dispersal Operators
In this section, we investigate the approximation of principal eigenvalues of
time periodic random dispersal operators by the principal spectrum points of
time periodic nonlocal dispersal operators. We first recall some basic
properties of principal eigenvalues of time periodic random dispersal or
parabolic operators, and basic properties of principal spectrum points of time
periodic nonlocal dispersal operators. We then prove Theorem B.
### 3.1 Basic properties
In this subsection, we present basic properties of principal eigenvalues of
time periodic parabolic operators and basic properties of principal spectrum
points of time periodic nonlocal dispersal operators.
Let
$\mathcal{X}_{1}=\mathcal{X}_{2}=\\{u\in
C(\mathbb{R}\times\bar{D},\mathbb{R})|u(t+T,x)=u(t,x)\\}$
with norm
$\|u\|_{\mathcal{X}_{i}}=\sup_{t\in[0,T]}\|u(t,\cdot)\|_{X_{i}}(i=1,2)$,
$\mathcal{X}_{3}=\\{u\in
C(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})|u(t+T,x)=u(t,x+p_{j}{\bf
e_{j}})=u(t,x)\\}$
with norm $\|u\|_{\mathcal{X}_{3}}=\sup_{t\in[0,T]}\|u(t,\cdot)\|_{X_{3}}$,
and
$\mathcal{X}_{i}^{+}=\\{u\in\mathcal{X}_{i}|u(t,x)\geq 0\\}$
$(i=1,2,3)$. And for $u^{1},u^{2}\in\mathcal{X}_{i}$, we define
$u^{1}\leq u^{2}(u^{1}\geq u^{2})\text{ if
}u^{2}-u^{1}\in\mathcal{X}_{i}^{+}\,(u_{1}-u_{2}\in\mathcal{X}_{i}^{+})$
$(i=1,2,3)$. For given $a(\cdot,\cdot)\in\mathcal{X}_{i}\cap
C^{1}(\mathbb{R}\times\mathbb{R}^{N})$ , let
$L^{\delta}_{i}(a):\mathcal{D}(L^{\delta}_{i}(a))\subset\mathcal{X}_{i}\to\mathcal{X}_{i}$
be defined as follows,
$(L^{\delta}_{1}(a)u)(t,x)=-\partial_{t}u(t,x)+\nu_{\delta}\left[\int_{D}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+a(t,x)u(t,x),\quad(t,x)\in\mathbb{R}\times\bar{D},$
(3.1)
$(L^{\delta}_{2}(a)u)(t,x)=-\partial_{t}u(t,x)+\nu_{\delta}\int_{D}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+a(t,x)u(t,x),\quad(t,x)\in\mathbb{R}\times\bar{D},$
(3.2)
and
$(L^{\delta}_{3}(a)u)(t,x)=-\partial_{t}u(t,x)+\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+a(t,x)u(t,x),\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{N}.$
(3.3)
We first recall the definition of principal spectrum points/eigenvalues of
time periodic nonlocal dispersal operators.
###### Definition 3.1.
Let
$\lambda^{\delta}_{i}(a)=\sup\\{{\rm
Re}\lambda|\lambda\in\sigma(L^{\delta}_{i}(a))\\}$
for $i=1,2,3$.
* (1)
$\lambda^{\delta}_{i}(a)$ is called the principal spectrum point of
$L^{\delta}_{i}(a)$.
* (2)
If $\lambda^{\delta}_{i}(a)$ is an isolated algebraically simple eigenvalue of
$L^{\delta}_{i}(a)$ with a positive eigenfunction, then
$\lambda^{\delta}_{i}(a)$ is called the principal eigenvalue of
$L^{\delta}_{i}(a)$ or it is said that $L^{\delta}_{i}(a)$ has a principal
eigenvalue.
For the time periodic random dispersal operators, let
$a(\cdot,\cdot)\in\mathcal{X}_{i}\cap C^{1}(\mathbb{R}\times\mathbb{R}^{N})$,
and $L_{i}(a):\mathcal{D}(L_{i}(a))\subset\mathcal{X}_{i}\to\mathcal{X}_{i}$
be defined as follows,
$(L_{i}(a)u)(t,x)=-\partial_{t}u(t,x)+\Delta u(t,x)+a(t,x)u(t,x)$
for $i=1,2,3$. Note that for $u\in\mathcal{D}(L_{1}(a))$, $B_{r,D}u=u=0$ on
$\partial D$ and for $u\in\mathcal{D}(L_{2}(a))$, $B_{r,N}u=\frac{\partial
u}{\partial{\bf n}}=0$ on $\partial D$. Let
$\lambda^{r}_{i}(a)=\sup\\{{\rm Re}\lambda|\lambda\in\sigma(L_{i}(a))\\}.$
It is well known that $\lambda^{r}_{i}(a)$ is an isolated algebraically simple
eigenvalue of $L_{i}(a)$ with a positive eigenfunction (see [23]) and
$\lambda^{r}_{i}(a)$ is called the principal eigenvalue of $L_{i}(a)$.
Next we derive some properties of the principal spectrum points of nonlocal
dispersal operators by using the spectral radius of the solution operators of
the associated evolution equations. To this end, for $i=1,2,3$, define
$\Phi_{i}^{\delta}(t,s;a):X_{i}\to X_{i}$ by
$(\Phi_{i}^{\delta}(t,s;a)u_{0})(\cdot)=u_{i}(t,\cdot;s,u_{0},a),\quad
u_{0}\in X_{i},$
where $u_{1}(t,\cdot;s,u_{0},a)$ is the solution to
$\partial_{t}u(t,x)=\nu_{\delta}\left[\int_{D}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+a(t,x)u(t,x),\quad
x\in\bar{D}$ (3.4)
with $u_{1}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{1}$,
$u_{2}(t,\cdot;s,u_{0},a)$ is the solution to
$\partial_{t}u(t,x)=\nu_{\delta}\int_{D}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+a(t,x)u(t,x),\quad
x\in\bar{D}$ (3.5)
with $u_{2}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{2}$, and
$u_{3}(t,\cdot;s,u_{0},a)$ is the solution to
$\partial_{t}u(t,x)=\nu_{\delta}\left[\int_{\mathbb{R}^{N}}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+a(t,x)u(t,x),\quad
x\in\mathbb{R}^{N}$ (3.6)
with $u_{3}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{3}$. By general semigroup
property, $\Phi_{i}^{\delta}(t,s;a)$ ($i=1,2,3$) is well defined.
Let $A_{1}$ be $-\Delta$ with Dirichlet boundary condition acting on
$X_{1}\cap C_{0}(D)$. Let
$X_{1}^{r}=\mathcal{D}(A_{1}^{\alpha})$ (3.7)
for some $0<\alpha<1$ such that $C^{1}(\bar{D})\subset X_{1}^{r}$ with
$\|u\|_{X_{1}^{r}}=\|A_{1}^{\alpha}u\|_{X_{1}}$. Similarly, let $A_{2}$ be
$-\Delta$ with Neumann boundary condition acting on $X_{2}$. Let
$X_{2}^{r}=X_{2}$ (3.8)
with $\|u\|_{X_{2}^{r}}=\|u\|_{X_{2}}$, and
$X_{3}^{r}=X_{3}$ (3.9)
with $\|u\|_{X_{3}^{r}}=\|u\|_{X_{3}}$. Let
$X_{i}^{r,+}=\\{u\in X_{i}^{r}|u(x)\geq 0\\}$
(i=1, 2, 3). Similarly, for $i=1,2,3$, define $\Phi_{i}(t,s;a):X_{i}^{r}\to
X_{i}^{r}$ by
$(\Phi_{i}(t,s;a)u_{0})(\cdot)=u_{i}(t,\cdot;s,u_{0},a),\quad u_{0}\in
X_{i}^{r},$
where $u_{1}(t,\cdot;s,u_{0},a)$ is the solution to
$\begin{cases}\partial_{t}u(t,x)=\Delta u(t,x)+a(t,x)u(t,x),\quad&x\in D,\\\
u(t,x)=0,&x\in\partial D\end{cases}$ (3.10)
with $u_{1}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{1}^{r}$,
$u_{2}(t,\cdot;s,u_{0},a)$ is the solution to
$\begin{cases}\partial_{t}u(t,x)=\Delta u(t,x)+a(t,x)u(t,x),\quad&x\in D,\\\
\frac{\partial u}{\partial{\bf n}}(t,x)=0,&x\in\partial D\end{cases}$ (3.11)
with $u_{2}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{2}^{r}$, and
$u_{3}(t,\cdot;s,u_{0},a)$ is the solution to
$\begin{cases}\partial_{t}u(t,x)=\Delta
u(t,x)+a(t,x)u(t,x),\quad&x\in\mathbb{R}^{N},\\\
u(t,x+p_{j}\mathbb{e_{j}})=u(t,x),&x\in\mathbb{R}^{N}\end{cases}$ (3.12)
with $u_{3}(s,\cdot;s,u_{3},a)=u_{0}(\cdot)\in X_{3}^{r}$.
Let $r(\Phi_{i}^{\delta}(T,0;a))$ be the spectral radius of
$\Phi_{i}^{\delta}(T,0;a)$ and $\lambda_{i}^{\delta}(a)$ be the principal
spectrum point of $L_{i}^{\delta}(a)$. We have the following propositions.
###### Proposition 3.1.
Let $1\leq i\leq 3$ be given. Then
$r(\Phi_{i}^{\delta}(T,0;a))=e^{\lambda_{i}^{\delta}(a)T}.$
###### Proof.
See [41, Proposition 3.3]. ∎
Similarly, let $r(\Phi_{i}(T,0;a))$ be the spectral radius of
$\Phi_{i}(T,0;a)$ and $\lambda_{i}^{r}(a)$ be the principal eigenvalue of
$L_{i}(a)$. Note that $X_{i}^{r}$ is a strongly ordered Banach space with the
positive cone $C=\\{u\in X_{i}^{r}\,|\,u(x)\geq 0\\}$ and by the regularity, a
priori estimates, and comparison principle for parabolic equations,
$\Phi_{i}(T,0;a):X_{i}^{r}\to X_{i}^{r}$ is strongly positive and compact.
Then by the Kreĭn-Rutman Theorem (see [43]), $r(\Phi_{i}(T,0;a))$ is an
isolated algebraically simple eigenvalue of $\Phi_{i}(T,0;a)$ with a positive
eigenfunction $u_{i}^{r}(\cdot)$ and for any
$\mu\in\sigma(\Phi_{i}(T,0;a))\setminus\\{r(\Phi_{i}(T,0;a))\\}$,
$\text{Re}\mu<r(\Phi_{i}(T,0;a)).$
The following propositions then follow.
###### Proposition 3.2.
Let $1\leq i\leq 3$ be given. Then
$r(\Phi_{i}(T,0;a))=e^{\lambda_{i}^{r}(a)T}.$
Moreover, there is a codimension one subspace $Z_{i}$ of $X_{i}^{r}$ such that
$X_{i}^{r}=Y_{i}\oplus Z_{i},$
where $Y_{i}={\rm span}\\{u_{i}^{r}(\cdot)\\}$, and there are $M>0$ and
$\gamma>0$ such that for any $w_{i}\in Z_{i}$, there holds
$\frac{\|\Phi_{i}(nT,0;a)w_{i}\|_{X_{i}^{r}}}{\|\Phi_{i}(nT,0;a)u_{i}^{r}\|_{X_{i}^{r}}}\leq
Me^{-\gamma nT}.$
###### Proposition 3.3.
For given $1\leq i\leq 3$ and $a_{1},a_{2}\in\mathcal{X}_{i}\cap
C^{1}(\mathbb{R}\times\mathbb{R}^{N})$,
$|\lambda_{i}^{\delta}(a_{1})-\lambda_{i}^{\delta}(a_{2})|\leq\max_{t\in[0,T],x\in\bar{D}}|a_{1}(t,x)-a_{2}(t,x)|,$
(3.13)
and
$|\lambda_{i}^{r}(a_{1})-\lambda_{i}^{r}(a_{2})|\leq\max_{t\in[0,T],x\in\bar{D}}|a_{1}(t,x)-a_{2}(t,x)|.$
(3.14)
###### Proof.
Let $a_{0}=\max_{t\in[0,T],x\in\bar{D}}|a_{1}(t,x)-a_{2}(t,x)|$ and
$a_{1}^{\pm}(t,x)=a_{1}(t,x)\pm a_{0}.$
It is not difficult to see that
$\Phi_{i}^{\delta}(t,s;a_{1}^{\pm})=e^{\pm
a_{0}(t-s)}\Phi_{i}^{\delta}(t,s;a_{1}).$
It then follows that
$r(\Phi_{i}^{\delta}(T,0;a_{1}^{\pm}))=e^{(\lambda_{i}^{\delta}(a_{1})\pm
a_{0})T}.$ (3.15)
Observe that by Proposition 2.1, for any $u_{0}\in X_{i}^{+}$,
$\Phi_{i}^{\delta}(T,0;a_{1}^{-})u_{0}\leq\Phi_{i}^{\delta}(T,0;a_{2})u_{0}\leq\Phi_{i}^{\delta}(T,0;a_{1}^{+})u_{0}.$
This implies that
$r(\Phi_{i}^{\delta}(T,0;a_{1}^{-}))\leq r(\Phi_{i}^{\delta}(T,0;a_{2}))\leq
r(\Phi_{i}^{\delta}(T,0;a_{1}^{+})).$
This together with (3.15) implies that
$\lambda_{i}^{\delta}(a_{1})-a_{0}\leq\lambda_{i}^{\delta}(a_{2})\leq\lambda_{i}^{\delta}(a_{1})+a_{0},$
(3.16)
that is, (3.13) holds.
Similarly, we can prove that (3.14) holds. ∎
### 3.2 Proof of Theorem B in the Dirichlet boundary condition case
In this subsection, we prove Theorem B in the Dirichlet boundary condition
case. Throughout this subsection, we assume $B_{r,b}u=B_{r,D}u$ in (1.10), and
$D_{b}=D_{D}(=\mathbb{R}^{N}\setminus\bar{D})$ and $B_{n,b}u=B_{n,D}u$ in
(1.11). Note that for any $a\in\mathcal{X}_{1}\cap
C^{1}(\mathbb{R}\times\mathbb{R}^{N})$, there are $a_{n}\in\mathcal{X}_{1}\cap
C^{3}(\mathbb{R}\times\mathbb{R}^{N})$ such that
$\sup_{t\in[0,T]}\|a_{n}(t,\cdot)-a(t,\cdot)\|_{X_{1}}\to 0$ as $n\to\infty$.
By Proposition 3.3, without loss of generality, we may assume that
$a\in\mathcal{X}_{1}\cap C^{3}(\mathbb{R}\times\mathbb{R}^{N})$.
###### Proof of Theorem B in the Dirichlet boundary condition case.
First of all, for the simplicity in notation, we put
$\Phi(T,0)=\Phi_{1}(T,0;a),\,\ \quad\lambda_{1}^{r}=\lambda_{1}^{r}(a),$
and
$\Phi^{\delta}(T,0)=\Phi_{1}^{\delta}(T,0;a),\quad\lambda_{1}^{\delta}=\lambda_{1}^{\delta}(a).$
Let $u^{r}(\cdot)$ be a positive eigenfunction of $\Phi(T,0)$ corresponding to
$r(\Phi(T,0))$. Without loss of generality, we assume that
$\|u^{r}\|_{X_{1}^{r}}=1$.
We first show that for any $\epsilon>0$, there is $\delta_{1}>0$ such that for
$0<\delta<\delta_{1}$,
$\lambda_{1}^{\delta}\geq\lambda_{1}^{r}-\epsilon.$ (3.17)
In order to do so, choose $D_{0}\subset\subset D$ and $u_{0}\in X_{1}^{r}\cap
C^{3}(\bar{D})$ such that $u_{0}(x)=0$ for $x\in D\backslash D_{0}$, and
$u_{0}(x)>0$ for $x\in\text{Int}D_{0}$. By Proposition 3.2, there exist
$\alpha>0$, $M>0$, and $u^{\prime}\in Z_{1}$, such that
$u_{0}(x)=\alpha u^{r}(x)+u^{\prime}(x),$ (3.18)
and
$\|\Phi(nT,0)u^{\prime}\|_{X_{1}^{r}}\leq Me^{-\gamma
nT}e^{\lambda_{1}^{r}nT}.$ (3.19)
By Theorem A, there is $\delta_{0}>0$ such that for $0<\delta<\delta_{0}$,
there hold
$\big{(}\Phi^{\delta}(nT,0)u^{r}\big{)}(x)\geq\big{(}\Phi(nT,0)u^{r}\big{)}(x)-C^{1}(nT,\delta),$
(3.20)
and
$\big{(}\Phi^{\delta}(nT,0)u^{\prime}\big{)}(x)\leq\big{(}\Phi(nT,0)u^{\prime}\big{)}(x)+C^{2}(nT,\delta),$
(3.21)
where $C^{i}(nT,\delta)\to 0$ as $\delta\to 0$ ($i=1,2$). Hence for
$0<\delta<\delta_{0}$,
$\displaystyle\big{(}\Phi^{\delta}(nT,0)u_{0}\big{)}(x)=$
$\displaystyle\alpha\big{(}\Phi^{\delta}(nT,0)u^{r}\big{)}(x)+\big{(}\Phi^{\delta}(nT,0)u^{\prime}\big{)}(x)$
$\displaystyle\geq$
$\displaystyle\alpha\big{(}\Phi(nT,0)u^{r}\big{)}(x)-\alpha
C^{1}(nT,\delta)-C^{2}(nT,\delta)-\|\Phi(nT,0)u^{\prime}\|_{X_{1}^{r}}$
$\displaystyle\geq$ $\displaystyle\alpha e^{\lambda_{1}^{r}nT}u^{r}(x)-\alpha
C^{1}(nT,\delta)-C^{2}(nT,\delta)-Me^{-\gamma nT}e^{\lambda_{1}^{r}nT}$
$\displaystyle=$ $\displaystyle e^{(\lambda_{1}^{r}-\epsilon)nT}e^{\epsilon
nT}(\alpha u^{r}(x)-Me^{-\gamma nT})-\alpha
C^{1}(nT,\delta)-C^{2}(nT,\delta).$ (3.22)
Note that there exists $m>0$ such that
$u^{r}(x)\geq m>0\quad\text{ for }x\in\bar{D}_{0}.$
Hence for any $0<\epsilon<\gamma$, there is $n_{1}>0$ such that for $n\geq
n_{1}$,
$e^{\epsilon nT}(\alpha u^{r}(x)-Me^{-\gamma nT})\geq u_{0}(x)+1\quad{\rm
for}\quad x\in\bar{D}_{0},$ (3.23)
and there is $\delta_{1}\leq\delta_{0}$ such that for $0<\delta<\delta_{1}$,
$C^{1}(n_{1}T,\delta)+C^{2}(n_{1}T,\delta)\leq
e^{(\lambda_{1}^{r}-\epsilon)n_{1}T}.$ (3.24)
Note that $u_{0}(x)=0$ for $x\in D\backslash D_{0}$ and
$\big{(}\Phi^{\delta}(n_{1}T,0)u_{0}\big{)}(x)\geq 0$ for all $x\in\bar{D}$.
This together with (3.2)-(3.24) implies that for $\delta<\delta_{1}$,
$\big{(}\Phi^{\delta}(n_{1}T,0)u_{0}\big{)}(x)\geq
e^{(\lambda_{1}^{r}-\epsilon)n_{1}T}u_{0}(x),\quad x\in\bar{D}.$ (3.25)
By (3.25) and Proposition 2.1, for any $0<\delta<\delta_{1}$ and $n\geq 1$,
$(\Phi^{\delta}(nn_{1}T,0)u_{0})(\cdot)\geq
e^{(\lambda_{1}^{r}-\epsilon)nn_{1}T}u_{0}(\cdot).$
This together with Proposition 3.1 implies that for $0<\delta<\delta_{1}$,
$e^{\lambda_{1}^{\delta}T}=r(\Phi^{\delta}(T,0))\geq
e^{(\lambda_{1}^{r}-\epsilon)T}.$
Hence (3.17) holds.
Next, we prove that for any $\epsilon>0$, there is $\delta_{2}>0$ such that
for $0<\delta<\delta_{2}$,
$\lambda_{1}^{\delta}\leq\lambda_{1}^{r}+\epsilon.$ (3.26)
To this end, first, choose a sequence of smooth domains $\\{D_{m}\\}$ such
that $D_{1}\supset D_{2}\supset D_{3}\cdots\supset
D_{m}\supset\cdots\supset\bar{D}$, and $\cap_{m=1}^{\infty}D_{m}=\bar{D}$.
Consider the following evolution equation
$\begin{cases}\partial_{t}u(t,x)=\Delta u(t,x)+a(t,x)u(t,x),\quad&x\in
D_{m},\cr u(t,x)=0,\quad&x\in\partial D_{m}.\end{cases}$ (3.27)
Let
$X_{1,m}=\\{u\in C(\bar{D}_{m},\mathbb{R})\\},$
and
$X_{1,m}^{r}=\mathcal{D}(A_{1,m}^{\alpha}),$
where $A_{1,m}$ is $-\Delta$ with Dirichlet boundary condition acting on
$X_{1,m}\cap C_{0}(D_{m})$ and $0<\alpha<1$. We denote the solution of (3.27)
by $u_{m}(t,\cdot;s,u_{0})=(\Phi_{m}(t,s)u_{0})(\cdot)$ with
$u(s,\cdot;s,u_{0})=u_{0}(\cdot)\in X_{1,m}^{r}$. By Proposition 3.2, we have
$r(\Phi_{m}(T,0))=e^{\lambda_{1,m}^{r}T},$
where $\lambda^{r}_{1,m}$ is the principal eigenvalue of the following
eigenvalue problem,
$\begin{cases}-\partial_{t}u+\Delta u+a(t,x)u=\lambda u,\quad&x\in D_{m},\cr
u(t+T,x)=u(t,x),\quad&x\in D_{m},\cr u(t,x)=0,\quad&x\in\partial
D_{m}.\end{cases}$
By the dependence of the principle eigenvalue on the domain perturbation (see
[15]), for any $\epsilon>0$, there exists $m_{1}$ such that
$\lambda_{1,m_{1}}^{r}\leq\lambda_{1}^{r}+\frac{\epsilon}{2}.$ (3.28)
Secondly, let $u_{m_{1}}^{r}(\cdot)$ be a positive eigenfunction of
$\Phi_{m_{1}}(T,0)$ corresponding to $r(\Phi_{m_{1}}(T,0))$. By regularity for
parabolic equations, $u_{m_{1}}^{r}\in C^{3}(\bar{D}_{m_{1}})$. Let
$(\Phi_{m_{1}}^{\delta}(t,0)u^{r}_{m_{1}})(x)$ be the solution to
$\begin{cases}u_{t}=\nu_{\delta}\left[\int_{D_{m_{1}}}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+a(t,x)u(t,x),\quad&x\in\bar{D}_{m_{1}},\\\
u(0,x)=u_{m_{1}}^{r}(x).\end{cases}$ (3.29)
Then by Theorem A,
$\big{(}\Phi_{m_{1}}^{\delta}(nT,0)u_{m_{1}}^{r}\big{)}(x)\leq\big{(}\Phi_{m_{1}}(nT,0)u_{m_{1}}^{r}\big{)}(x)+C(nT,\delta)\quad\forall\,\,x\in\bar{D}_{m_{1}},$
where $C(nT,\delta)\to 0$ as $\delta\to 0$. By Proposition 2.1,
$\big{(}\Phi^{\delta}(nT,0)u_{m_{1}}^{r}|_{\bar{D}}\big{)}(x)\leq\big{(}\Phi_{m_{1}}^{\delta}(nT,0)u_{m_{1}}^{r}\big{)}(x)\quad\forall\,\,x\in\bar{D}.$
It then follows that for $x\in\bar{D}$,
$\displaystyle\big{(}\Phi^{\delta}(nT,0)u_{m_{1}}^{r}|_{\bar{D}}\big{)}(x)$
$\displaystyle\leq\big{(}\Phi_{m_{1}}(nT,0)u_{m_{1}}^{r}\big{)}(x)+C(nT,\delta)$
$\displaystyle=e^{\lambda_{m_{1}}^{r}nT}u_{m_{1}}^{r}(x)+C(nT,\delta)$
$\displaystyle\leq
e^{(\lambda_{1}^{r}+\frac{\epsilon}{2})nT}u_{m_{1}}^{r}(x)+C(nT,\delta)$
$\displaystyle=e^{(\lambda_{1}^{r}+\epsilon)nT}e^{-\frac{\epsilon}{2}nT}u_{m_{1}}^{r}(x)+C(nT,\delta).$
(3.30)
Note that
$\min_{x\in\bar{D}}u_{m_{1}}^{r}(x)>0.$
Hence for any $\epsilon>0$, there is $n_{2}\geq 1$ such that
$e^{-\frac{\epsilon}{2}n_{2}T}\leq\frac{1}{2},$ (3.31)
and there is $\delta_{2}>0$ such that for $0<\delta<\delta_{2}$,
$C(n_{2}T,\delta)\leq\frac{1}{2}e^{(\lambda_{1}^{r}+\epsilon)n_{2}T}u_{m_{1}}^{r}(x)\quad\forall\,x\in\bar{D}.$
(3.32)
By (3.2)-(3.32),
$\big{(}\Phi^{\delta}(n_{2}T,0)u_{m_{1}}^{r}|_{\bar{D}}\big{)}(x)\leq
e^{(\lambda_{1}^{r}+\epsilon)n_{2}T}u_{m_{1}}^{r}(x)\quad\forall\,\,x\in\bar{D}.$
This together with Proposition 2.1 implies that for $0<\delta<\delta_{2}$,
$\big{(}\Phi^{\delta}(nn_{2}T,0)u_{m_{1}}^{r}|_{\bar{D}}\big{)}(x)\leq
e^{(\lambda_{1}^{r}+\epsilon)nn_{2}T}u_{m_{1}}^{r}(x)\quad\forall\,\,x\in\bar{D}.$
(3.33)
This together with Proposition 3.1 implies that
$\lambda_{1}^{\delta}\leq\lambda_{1}^{r}+\epsilon$
for $0<\delta<\delta_{2}$, that is, (3.26) holds.
Theorem B in the Dirichlet boundary condition case then follows from (3.17)
and (3.26). ∎
### 3.3 Proofs of Theorem B in the Neumann and periodic boundary condition
cases
###### Proof of Theorem B in the Neumann boundary condition case.
We assume $B_{r,b}u=B_{r,N}u$ in (1.10), and $D_{b}=D_{N}(=\emptyset)$ in
(1.11). The proof in the Neumann boundary condition case is similar to the
arguments in the Dirichlet boundary condition case (it is simpler). For the
completeness, we give a proof in the following. Without loss of generality, we
may also assume that $a\in\mathcal{X}_{2}\cap
C^{3}(\mathbb{R}\times\mathbb{R}^{N})$.
For the simplicity in notation, put
$\Phi(nT,0)=\Phi_{2}(nT,0;a),\quad\,\ \lambda_{2}^{r}=\lambda_{2}^{r}(a),$
and
$\Phi^{\delta}(nT,0)=\Phi_{2}^{\delta}(nT,0;a),\quad\lambda_{2}^{\delta}=\lambda_{2}^{\delta}(a).$
By Propositions 3.1 and 3.2,
$r(\Phi(T,0))=e^{\lambda_{2}^{r}T},$ (3.34)
and
$r(\Phi^{\delta}(T,0))=e^{\lambda_{2}^{\delta}T}.$ (3.35)
Let $u^{r}(\cdot)$ be a positive eigenfunction of $\Phi(T,0)$ corresponding to
$r(\Phi(T,0))$. By regularity for parabolic equations, $u^{r}\in
C^{3}(\bar{D})$. By Theorem A, we have
$\|\Phi^{\delta}(nT,0)u^{r}-\Phi(nT,0)u^{r}\|_{X_{2}}\leq C(nT,\delta),$
where $C(nT,\delta)\to 0$ as $\delta\to 0$. This implies that for all
$x\in\bar{D}$,
$\displaystyle\big{(}\Phi^{\delta}(nT,0)u^{r}\big{)}(x)$
$\displaystyle\geq\big{(}\Phi(nT,0)u^{r}\big{)}(x)-C(nT,\delta)$
$\displaystyle=e^{\lambda_{2}^{r}nT}u^{r}(x)-C(nT,\delta)$
$\displaystyle=e^{(\lambda_{2}^{r}-\epsilon)nT}e^{\epsilon
nT}u^{r}(x)-C(nT,\delta),$ (3.36)
and
$\displaystyle\big{(}\Phi^{\delta}(nT,0)u^{r}\big{)}(x)$
$\displaystyle\leq\big{(}\Phi(nT,0)u^{r}\big{)}(x)+C(nT,\delta)$
$\displaystyle=e^{\lambda_{2}^{r}nT}u^{r}(x)+C(nT,\delta)$
$\displaystyle=e^{(\lambda_{2}^{r}+\epsilon)nT}e^{-\epsilon
nT}u^{r}(x)+C(nT,\delta).$ (3.37)
Note that
$\min_{x\in\bar{D}}u^{r}(x)>0.$ (3.38)
Hence for any $\epsilon>0$, there is $n_{1}>1$ such that
$\begin{cases}e^{\epsilon
n_{1}T}u^{r}(x)\geq\frac{3}{2}u^{r}(x)\quad\forall\,x\in\bar{D},\\\ \\\
e^{-\epsilon
n_{1}T}u^{r}(x)\leq\frac{1}{2}u^{r}(x)\quad\forall\,x\in\bar{D},\end{cases}$
(3.39)
and there is $\delta_{0}>0$ such that for any $0<\delta<\delta_{0}$,
$C(n_{1}T)\delta<\frac{1}{2}e^{(\lambda_{2}^{r}-\epsilon)n_{1}T}u^{r}(x)\quad\forall\,x\in\bar{D}.$
(3.40)
By (3.3)-(3.40), we have that for any $0<\delta<\delta_{0}$,
$e^{(\lambda_{2}^{r}-\epsilon)n_{1}T}u^{r}(x)\leq\big{(}\Phi^{\delta}(n_{1}T,0)u^{r}\big{)}(x)\leq
e^{(\lambda_{2}^{r}+\epsilon)n_{1}T}u^{r}(x)\quad\forall\,x\in\bar{D}.$
This together with Proposition 2.1 implies that for all $n\geq 1$,
$e^{(\lambda_{2}^{r}-\epsilon)n_{1}nT}u^{r}(x)\leq\big{(}\Phi^{\delta}(n_{1}nT,0)u^{r}\big{)}(x)\leq
e^{(\lambda_{2}^{r}+\epsilon)n_{1}nT}u^{r}(x)\quad\forall\,x\in\bar{D}.$
It then follows that for any $0<\delta<\delta_{0}$,
$e^{(\lambda_{2}^{r}-\epsilon)T}\leq r(\Phi^{\delta}(T,0))\leq
e^{(\lambda_{2}^{r}+\epsilon)T}.$
By Proposition 3.1, we have
$|\lambda_{2}^{\delta}-\lambda_{2}^{r}|<\epsilon\quad\forall\,0<\delta<\delta_{0}.$
Theorem B in the Neumann boundary condition case is thus proved. ∎
###### Proof of Theorem B in the periodic boundary condition case.
We assume $D=\mathbb{R}^{N}$, and $B_{r,b}u=B_{r,P}u$ in (1.10), and
$B_{n,b}u=B_{n,P}u$ in (1.11). It can be proved by the same arguments as in
the Neumann boundary condition case. ∎
## 4 Approximation of Time Periodic Positive Solutions of Random Dispersal
KPP Equations by Nonlocal Dispersal KPP Equations
In this section, we study the approximation of the asymptotic dynamics of time
periodic KPP equations with random dispersal by those of time periodic KPP
equations with nonlocal dispersal. We first recall the existing results about
time periodic positive solutions of KPP equations with random as well as
nonlocal dispersal. Then we prove Theorem C. Throughout this section, we
assume that $D$ is as in (H0), and (H1), (H2) and (H2)δ hold. Recall that,
(H2) implies (H2)δ for $\delta$ sufficiently small by Theorem B.
### 4.1 Basic properties
In this subsection, we present some basic known results for (1.12) and (1.13).
Let $X_{1}^{r}$, $X_{2}^{r}$, and $X_{3}^{r}$ be defined as in (3.7), (3.8),
and (3.9), respectively. For $u_{0}\in X_{i}^{r}$, let
$(U(t,0)u_{0})(\cdot)=u(t,\cdot;u_{0})$, where $u(t,\cdot;u_{0})$ is the
solution to (1.12) with $u(0,\cdot;u_{0})=u_{0}(\cdot)$ and
$B_{r,b}u=B_{r,D}u$ when $i=1$, $B_{r,b}u=B_{r,N}u$ when $i=2$, and
$B_{r,b}u=B_{r,P}u$ when $i=3$. Similarly, for $u_{0}\in X_{i}$, let
$(U^{\delta}(t,0)u_{0})(\cdot)=u^{\delta}(t,\cdot;u_{0})$, where
$u^{\delta}(t,\cdot;u_{0})$ is the solution to (1.13) with
$u^{\delta}(0,\cdot;u_{0})=u_{0}(\cdot)$ and
$D_{b}=D_{D}(=\mathbb{R}^{N}\backslash\bar{D})$, $B_{n,b}u=B_{n,D}u$ when
$i=1$, $D_{b}=D_{N}(=\emptyset)$ when $i=2$, and $B_{n,b}u=B_{n,P}u$ and
$D_{b}=D_{p}(=\mathbb{R}^{N})$ when $i=3$.
###### Proposition 4.1.
* (1)
If $u_{0}\geq 0$, solution $u(t,\cdot;u_{0})$ to (1.12) with
$u(0,\cdot;u_{0})=u_{0}(\cdot)$ exists for all $t\geq 0$ and
$u(t,\cdot;u_{0})\geq 0$ for all $t\geq 0$.
* (2)
If $u_{0}\geq 0$, solution $u(t,\cdot;u_{0})$ to (1.13) with
$u(0,\cdot;u_{0})=u_{0}(\cdot)$ exists for all $t\geq 0$ and
$u(t,\cdot;u_{0})\geq 0$ for all $t\geq 0$.
###### Proof.
(1) Note that $u(\cdot)\equiv 0$ is a sub-solution of (1.12) and
$u(\cdot)\equiv M$ is a super-solution of (1.12) for $M\gg 1$. Then by
Proposition 2.1, there is $M\gg 1$ such that
$0\leq u(t,x;u_{0})\leq M\quad\forall\,\,x\in\bar{D},\,\,t\in(0,t_{\max}),$
where $(0,t_{\max})$ is the interval of existence of $u(t,\cdot;u_{0})$. This
implies that we must have $t_{\max}=\infty$ and hence (1) holds.
(2) It can be proved by similar arguments as in (1). ∎
###### Proposition 4.2.
* (1)
(1.12) has a unique globally stable positive time periodic solution
$u^{*}(t,x)$.
* (2)
(1.13) has a unique globally stable time periodic positive solution
$u^{*}_{\delta}(t,x)$.
###### Proof.
(1) See [45, Theorem 3.1] (see also [33, Theorems 1.1, 1.3]).
(2) See [36, Theorem E]. ∎
###### Remark 4.1.
By Proposition 4.2(2), if there is $u_{0}\in X_{i}^{+}\setminus\\{0\\}$ such
that $(U^{\delta}(nT,0)u_{0})(\cdot)\geq u_{0}(\cdot)$ for some $n\geq 1$,
then we must have
$\lim_{n\to\infty}(U^{\delta}(nT,0)u_{0})(\cdot)=u^{*}_{\delta}(0,\cdot)$ and
hence
$(U^{\delta}(nT,0)u_{0})(\cdot)\leq u^{*}_{\delta}(0,\cdot).$
Similarly, if there is $u_{0}\in X_{i}^{+}\setminus\\{0\\}$ such that
$(U^{\delta}(nT,0)u_{0})(\cdot)\leq u_{0}(\cdot)$ for some $n\geq 1$, then
$(U^{\delta}(nT,0)u_{0})(\cdot)\geq u^{*}_{\delta}(0,\cdot).$
### 4.2 Proof of Theorem C in the Dirichlet boundary condition case
In this subsection, we prove Theorem C in the Dirichlet boundary condition
case. Throughout this subsection, we assume that $B_{r,b}u=B_{r,D}u$ in
(1.12), and $D_{b}=D_{D}$ and $B_{n,b}u=B_{n,D}u$ in (1.13).
###### Proof of Theorem C in the Dirichlet boundary condition case.
First of all, note that it suffices to prove that for any $\epsilon>0$, there
is $\delta_{0}>0$ such that for $0<\delta<\delta_{0}$,
$u_{\delta}^{*}(t,x)-\epsilon\leq u^{*}(t,x)\leq
u_{\delta}^{*}(t,x)+\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
We first show that for any $\epsilon>0$, there is $\delta_{1}>0$ such that for
$0<\delta<\delta_{1}$,
$u^{*}(t,x)\leq
u_{\delta}^{*}(t,x)+\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.1)
To this end, choose a smooth function $\phi_{0}\in C_{0}^{\infty}(D)$
satisfying that $\phi_{0}(x)\geq 0$ for $x\in D$ and
$\phi_{0}(\cdot)\not\equiv 0$. Let $0<\eta\ll 1$ be such that
$u_{-}(x):=\eta\phi_{0}(x)<u^{*}(0,x)\quad{\rm for}\quad x\in\bar{D}.$
Then there is $\epsilon_{0}>0$ such that
$u^{*}(0,x)\geq u_{-}(x)+\epsilon_{0}\quad{\rm for}\quad x\in{\rm
supp}(\phi_{0}).$ (4.2)
By Proposition 4.2, there is $N\gg 1$ such that
$\big{(}U(NT,0)u_{-}\big{)}(x)\geq
u^{*}(NT,x)-\epsilon_{0}/2=u^{*}(0,x)-\epsilon_{0}/2\quad\forall\,\,x\in\bar{D}.$
By Theorem A, there is $\bar{\delta}_{1}>0$ such that for
$0<\delta<\bar{\delta}_{1}$, we have
$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq\big{(}U(NT,0)u_{-}\big{)}(x)-\epsilon_{0}/2\quad\forall\,\,x\in\bar{D}.$
Hence for $0<\delta<\bar{\delta}_{1}$,
$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq
u^{*}(0,x)-\epsilon_{0}\quad\forall\,\,x\in\bar{D}.$ (4.3)
Note that
$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq 0\quad\forall\,\,x\in\bar{D}.$
It then follows from (4.2) and (4.3) that for $0<\delta<\bar{\delta}_{1}$,
$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq
u_{-}(x)\quad\forall\,\,x\in\bar{D}.$
This together with Proposition 4.2 (2) implies that
$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\leq
u_{\delta}^{*}(0,x)\quad\forall\,\,x\in\bar{D}$ (4.4)
(see Remark 4.1).
By Proposition 4.2 (1) again, for $n\gg 1$,
$u^{*}(t,x)\leq(U(nNT+t,0)u_{-})(x)+\epsilon/2\,\,\,\,\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.5)
Fix an $n\gg 1$ such that (4.5) holds. By Theorem A, there is
$0<\tilde{\delta}_{1}\leq\bar{\delta}_{1}$ such that for
$0<\delta<\tilde{\delta}_{1}$,
$\displaystyle(U(nNT+t,0)u_{-})(x)\leq(U^{\delta}(nNT+t,0)u_{-})(x)+C_{1}(\delta),$
(4.6)
where $C_{1}(\delta)\to 0$ as $\delta\to 0$. By (4.4), Proposition 2.1, and
Proposition 4.2 (2),
$\big{(}U^{\delta}(nNT+t,0)u_{-}\big{)}(x)\leq\big{(}U^{\delta}(t,0)u^{*}_{\delta}(0,\cdot)\big{)}(x)=u_{\delta}^{*}(t,x)$
(4.7)
for $t\in[0,T]$ and $x\in\bar{D}$. Let $0<\delta_{1}\leq\tilde{\delta}_{1}$ be
such that
$C_{1}(\delta)<\epsilon/2\quad\forall\,\,0<\delta<\delta_{1}.$ (4.8)
(4.1) then follows from (4.5)-(4.8).
Next, we need to show for any $\epsilon>0$, there is $\delta_{2}>0$ such that
for $0<\delta<\delta_{2}$,
$u^{*}(t,x)\geq
u_{\delta}^{*}(t,x)-\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.9)
To this end, choose a sequence of open sets $\\{D_{m}\\}$ with smooth
boundaries such that $D_{1}\supset D_{2}\supset D_{3}\cdots\supset
D_{m}\supset\cdots\supset\bar{D}$, and $\bar{D}=\cap_{m=1}^{\infty}D_{m}$.
According to Corollary 5.11 in [17], $D_{m}\to D$ regularly and all assertions
of Theorem 5.5 in [17] hold.
Consider
$\begin{cases}\partial_{t}u=\Delta u+uf(t,x,u),\quad&x\in D_{m},\\\
u(t,x)=0,\quad&x\in\partial D_{m}.\end{cases}$ (4.10)
Let $U_{m}(t,0)u_{0}=u(t,\cdot;u_{0})$, where $u(t,\cdot;u_{0})$ is the
solution to (4.10) with $u(0,\cdot;u_{0})=u_{0}(\cdot)$. By Proposition 4.2,
(4.10) has a unique time periodic positive solution $u_{m}^{*}(t,x)$. We first
claim that
$\lim_{m\to\infty}u_{m}^{*}(t,x)\to u^{*}(t,x)\,\,\,\text{uniformly
in}\,\,t\in[0,T]\,\,\text{and}\,\,x\in\bar{D}.$ (4.11)
In fact, it is clear that $u^{*}\in C(\mathbb{R}\times\bar{D},\mathbb{R})$ and
$u_{m}^{*}\in C(\mathbb{R}\times\bar{D}_{m},\mathbb{R})$. By [15, Theorem
7.1],
$\sup_{t\in\mathbb{R}}\|u_{m}^{*}(t,\cdot)-u^{*}(t,\cdot)\|_{L^{q}(D)}\to
0\quad{\rm as}\quad m\to\infty$
for $1\leq q<\infty$. Let $a(t,x)=f(t,x,u^{*}(t,x))$ and
$a_{m}(t,x)=f(t,x,u_{m}^{*}(t,x))$. Then $u^{*}(t,x)$ and $u_{m}^{*}(t,x)$ are
time periodic solutions to the following linear parabolic equations,
$\begin{cases}u_{t}=\Delta u+a(t,x)u,\quad&x\in D,\cr u(t,x)=0,&x\in\partial
D,\end{cases}$ (4.12)
and
$\begin{cases}u_{t}=\Delta u+a_{m}(t,x)u,\quad&x\in D_{m},\cr
u(t,x)=0,&x\in\partial D_{m},\end{cases}$ (4.13)
respectively. Observe that there is $M>0$, such that
$\|a\|_{L^{\infty}([T,2T]\times D)}<M,\,\,\|a_{m}\|_{L^{\infty}([T,2T]\times
D_{m})}<M,\,\,\|u^{*}(0,\cdot)\|_{L^{\infty}(D)}<M,\text{ and
}\|u_{m}^{*}(0,\cdot)\|_{L^{\infty}(D_{m})}<M.$
By [1, Theorem D(1)], $\\{u_{m}^{*}(t,x)\\}$ is equi-continuous on
$[T,2T]\times\bar{D}$. Without loss of generality, we may then assume that
$u_{m}^{*}(t,x)$ converges uniformly on $[T,2T]\times\bar{D}$. But
$u_{m}^{*}(t,\cdot)\to u^{*}(t,\cdot)$ in $L^{q}(D)$ uniformly in $t$. We then
must have
$u_{m}^{*}(t,x)\to u^{*}(t,x)\quad{\rm as}\quad n\to\infty$
uniformly in $(t,x)\in[T,2T]\times\bar{D}$. This together with the time
periodicity of $u_{m}^{*}$ shows that (4.11) holds.
Next, for any $\epsilon>0$, fix $m\gg 1$ such that
$u^{*}(t,x)\geq
u_{m}^{*}(t,x)-\epsilon/3\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.14)
Choose $M\gg 1$ such that for $0<\delta\leq 1$,
$Mu_{m}^{*}(t,x)\geq
u_{\delta}^{*}(t,x)\,\,\,\,\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.15)
Let
$u_{m}^{+}(x)=Mu_{m}^{*}(0,x),\quad u^{+}(x)=u_{m}^{+}(x)|_{\bar{D}}.$
By Proposition 4.2, for fixed $m$ and $\epsilon$, there exists $N\gg 1$, such
that
$u_{m}^{*}(t,x)\geq\big{(}U_{m}(NT+t,0)u_{m}^{+}\big{)}(x)-\epsilon/3\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.16)
By Theorem A, there is $0<\tilde{\delta}_{2}<1$ such that for
$0<\delta<\tilde{\delta}_{2}$,
$(U_{m}(NT+t,0)u_{m}^{+})(x)\geq(U_{m}^{\delta}(NT+t,0)u_{m}^{+})(x)-C_{2}(\delta)\quad\forall\,t\in[0,T],\,\
x\in D_{m},$ (4.17)
where $C_{2}(\delta)\to 0$ as $\delta\to 0$ and
$(U^{\delta}_{m}(t,0)u_{0})(\cdot)=u(t,\cdot;u_{0})$ is the solution to
$\begin{cases}u_{t}(t,x)=\nu_{\delta}\left[\int_{D_{m}}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+u(t,x)f(t,x,u(t,x)),\quad&x\in\bar{D}_{m}\cr
u(0,x)=u_{0}(x),&x\in\bar{D}_{m}.\end{cases}$
Let $0<\delta_{2}<\tilde{\delta}_{2}$ be such that for $0<\delta<\delta_{2}$,
$C_{2}(\delta)<\epsilon/3.$ (4.18)
By Proposition 2.1, for $x\in\bar{D}$ we have
$(U_{m}^{\delta}(NT+t,0)u_{m}^{+})(x)\geq(U^{\delta}(NT+t,0)u^{+})(x),$
and
$(U^{\delta}(NT+t,0)u^{+})(x)=(U^{\delta}(t,0)U^{\delta}(NT,0)u^{+})(x)\geq(U^{\delta}(t,0)u_{\delta}^{*}(0,\cdot))(x)=u_{\delta}^{*}(t,x).$
This together with (4.14), (4.16), (4.17), and (4.18) implies (4.9).
So, for any $\epsilon>0$, there exists
$\delta_{0}=\min\\{\delta_{1},\delta_{2}\\}$, such that for any
$\delta<\delta_{0}$, we have
$|u^{*}(t,x)-u_{\delta}^{*}(t,x)|\leq\epsilon\,\ \text{ uniform in }t>0\text{
and }x\in\bar{D}.$
∎
### 4.3 Proofs of Theorem C in the Neumann and periodic boundary condition
cases
In this subsection, we prove Theorem C in the Neumann and periodic boundary
condition cases.
###### Proof of Theorem C in the Neumann boundary condition case.
We assume $B_{r,b}u=B_{r,N}u$ in (1.10), and $D_{b}=D_{N}(=\emptyset)$ in
(1.11). The proof in the Neumann boundary condition case is similar to the
arguments in the Dirichlet boundary condition case (it is indeed simpler). For
completeness, we provide a proof.
First, we show that for any $\epsilon>0$, there is $\delta_{1}>0$ such that
$u^{*}(t,x)\leq
u_{\delta}^{*}(t,x)+\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D},$ (4.19)
if $0<\delta<\delta_{1}$. Choose a smooth function $u_{-}\in
C^{\infty}(\bar{D})$ with $u_{-}(\cdot)\geq 0$ and $u_{-}(\cdot)\not\equiv 0$
such that
$u_{-}(x)<u^{*}(0,x)\quad\forall\,\,x\in\bar{D}.$
Then there is $\epsilon_{0}>0$ such that
$u^{*}(0,x)\geq u_{-}(x)+\epsilon_{0}\quad\forall\,\,x\in\bar{D}.$ (4.20)
By Proposition 4.2 (1), there is $N\gg 1$ such that
$\big{(}U(NT,0)u_{-}\big{)}(x)\geq
u^{*}(0,x)-\epsilon_{0}/2\quad\forall\,\,x\in\bar{D}.$ (4.21)
By Theorem A, there is $\bar{\delta}_{1}>0$ such that for
$0<\delta<\bar{\delta}_{1}$,
$(U^{\delta}(NT,0)u_{-})(x)\geq(U(NT,0)u_{-})(x)-\epsilon_{0}/2\quad\forall\,\,x\in\bar{D}.$
(4.22)
By (4.20), (4.21) and (4.22),
$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq
u_{-}(x)\quad\forall\,\,x\in\bar{D},$
and then by Proposition 4.2 (2),
$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\leq
u_{\delta}^{*}(0,x)\quad\forall\,\,x\in\bar{D}.$ (4.23)
By Proposition 4.2 (1) again, for any given $\epsilon>0$, $n\gg 1$, and
$0<\delta<\bar{\delta}_{1}$,
$\displaystyle u^{*}(t,x)$
$\displaystyle\leq(U(nNT+t,0)u_{-})(x)+\epsilon/2\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.24)
By Theorem A, there is $0<\delta_{1}\leq\bar{\delta}_{1}$ such that for
$\delta<\delta_{1}$,
$\displaystyle(U(nNT+t,0)u_{-})(x)\leq(U^{\delta}(nNT+t,0)u_{-})(x)+\frac{\epsilon}{2}\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.25)
By Proposition 2.1 and (4.23), we have
$(U^{\delta}(nNT+t,0)u_{-})(x)=(U^{\delta}(t,0)U^{\delta}(nNT,0)u_{-})(x)\leq(U^{\delta}(t,0)u^{*}_{\delta}(t,\cdot))(x)=u_{\delta}^{*}(t,x)$
(4.26)
for $t\in[0,T]$ and $x\in\bar{D}$. (4.19) then follows from (4.24)-(4.26).
Next, we show that for any $\epsilon>0$, there is $\delta_{2}>0$ such that for
$0<\delta<\delta_{2}$,
$u^{*}(t,x)\geq
u_{\delta}^{*}(t,x)-\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.27)
Choose $M\gg 1$ such that $f(t,x,M)<0$ for $t\in\mathbb{R}$ and $x\in\bar{D}$.
Put
$u^{+}(x)=M\quad\forall\,\,x\in\bar{D}.$
Then for all $\delta>0$,
$u_{\delta}^{*}(0,x)\leq u^{+}(x)\quad\forall\,\,x\in\bar{D}.$ (4.28)
By Proposition 4.2, there is $N\gg 1$ such that
$u^{*}(t,x)\geq(U(NT+t,0)u^{+})(x)-\epsilon/2\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.29)
By Theorem A, there is $\delta_{2}>0$ such that for $0<\delta<\delta_{2}$,
$(U(NT+t,0)u^{+})(x)\geq(U^{\delta}(NT+t,0)u^{+})(x)-\frac{\epsilon}{2}\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.30)
By (4.28),
$(U^{\delta}(NT+t,0)u^{+})(x)=(U^{\delta}(t,0)U^{\delta}(NT,0)u^{+})(x)\geq(U^{\delta}(t,0)u_{\delta}^{*}(t,\cdot))(x)=u_{\delta}^{*}(t,x)$
(4.31)
for $t\in[0,T]$ and $x\in\bar{D}$. (4.27) then follows from (4.29)-(4.31).
So, for any $\epsilon>0$, there exists
$\delta_{0}=\min\\{\delta_{1},\delta_{2}\\}$, such that for any
$0<\delta<\delta_{0}$, we have
$|u^{*}(t,x)-u_{\delta}^{*}(t,x)|\leq\epsilon\,\ \text{ uniform in }t>0\text{
and }x\in\bar{D}.$
∎
###### Proof of Theorem C in the periodic boundary condition case.
We assume $D=\mathbb{R}^{N}$, and $B_{r,b}u=B_{r,P}u$ in (1.10), and
$B_{n,b}u=B_{n,P}u$ in (1.11). It can be proved by the similar arguments as in
the Neumann boundary condition case. ∎
Acknowledgments. The authors would like to thank the referees for the valuable
comments and suggestions which improved the presentation of this paper
considerably.
## References
* [1] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607-694.
* [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), pp. 5-49. Lecture Notes in Math., Vol. 466, Springer, Berlin, 1975\.
* [3] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33-76.
* [4] P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl. 332 (2007), no. 1, 428-440.
* [5] P. Bates and G. Zhao, Spectral convergence and Turing patterns for nonlocal diffusion systems, preprint.
* [6] R. S. Cantrell and C. Cosner, Spatial Ecology via reaction-diffusion Equations. Wiley Series in Mathematical and Computational Biology, John Wiley $\&$ Sons, Ltd., Chichester, 2003\.
* [7] R. S. Cantrell, C. Cosner, Y. Lou, and D. Ryan, Evolutionary stability of ideal free dispersal strategies: a nonlocal dispersal model, Can. Appl. Math. Q. 20 (2012), no. 1, 15-38.
* [8] E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9), 86 (2006), no. 3, 271-291.
* [9] C. Cosner, J. Dávila, and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn. 6 (2012), no. 2, 395-405.
* [10] C. Cortazar, M. Elgueta, and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel J. of Math., 170 (2009), 53-60.
* [11] C. Cortazar, M. Elgueta, J. D. Rossi, and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal. 187 (2008), no. 1, 137-156.
* [12] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations 249 (2010), no. 11, 2921-2953.
* [13] J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal. 60 (2005), no.5, 797 - 819.
* [14] J. Coville, J. Dávila, and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal. 39 (2008), no. 5, 1693-1709.
* [15] D. Daners, Domain perturbation for linear and nonlinear parabolic equations, J. Differential Equations 129 (1996), no. 2, 358-402.
* [16] D. Daners, Existence and perturbation of principal eigenvalues for a periodic-parabolic problem, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999), 51-67, Electron. J. Differ. Equ. Conf., 5, Southwest Texas State Univ., San Marcos, TX, 2000\.
* [17] W. Arendt and D. Daners Uniform convergence for elliptic problems on varying domains, Math. Nachr. 280 (2007), no. 1-2, 28-49.
* [18] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153-191, Springer, Berlin, 2003\.
* [19] P. C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979.
* [20] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 335-369.
* [21] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, and G. T. Vickers, Non-local dispersal, Differential Integral Equations 18 (2005), no. 11, 1299-1320.
* [22] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.
* [23] P. Hess, Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics Series, 247, Longman Scientific $\&$ Technical, Harlow; copublished in the United States with John Wiley $\&$ Sons, Inc., New York, 1991\.
* [24] P. Hess and H. Weinberger, Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitnesses, J. Math. Biol. 28 (1990), no. 1, 83-98.
* [25] G. Hetzer, W. Shen, and A. Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations, Rocky Mountain J. Math. 43 (2013), no. 2, 489-513.
* [26] V. Hutson, S. Martinez, K. Mischaikow, and G.T. Vickers, The evolution of dispersal, J. Math. Biol. 47 (2003), no. 6, 483-517.
* [27] V. Hutson, W. Shen and G.T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math. 38 (2008), no. 4, 1147-1175.
* [28] C.-Y. Kao, Y. Lou, and W. Shen, Random dispersal vs non-Local dispersal, Discrete and Contin. Dyn. Syst. 26 (2010), no. 2, 551-596.
* [29] A. Kolmogorov, I. Petrowsky, and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos, Univ., 1(6): 1-25 (1937).
* [30] F. Li, Y. Lou, and Y. Wang, Global dynamics of a competition model with non-local dispersal I: The shadow system, J. Math. Anal. Appl. 412 (2014), no. 1, 485-497.
* [31] W.-T. Li, Y.-J. Sun, Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2302-2313.
* [32] J. D. Murray, Mathematical Biology, Biomathematics 19, Springer-Verlag, Berlin, 1989\.
* [33] G. Nadin, Existence and uniqueness of the solutions of a space-time periodic reaction-diffusion equation, J. Differential Equation 249 (2010), no. 6, 1288-1304.
* [34] G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl. (4) 188 (2009),no. 2, 269-295.
* [35] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983\.
* [36] N. Rawal and W. Shen, Criteria for the existence of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations 24 (2012), no. 4, 927-954.
* [37] N. Rawal, W. Shen, and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, submitted.
* [38] W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations, 235 (2007), no. 1, 262-297.
* [39] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations 249 (2010), no. 4, 747-795.
* [40] W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal. 19 (2012), no. 3, 73-101.
* [41] W. Shen and X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, submitted.
* [42] J. G. Skellam, Random dispersal in theoretical populations, Biometrika 38, (1951) 196-218.
* [43] P. Takáč, A short elementary proof of the Kreĭn-Rutman theorem, Houston J. Math., 20 (1994), no. 1, 93-98.
* [44] G.-B. Zhang, W.-T. Li, and Z.-C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations 252 (2012), no. 9, 5096-5124.
* [45] X.-Q. Zhao, Global attractivity and stability in some monotone discrete dynamical systems, Bull. Austral. Math. Soc. 53 (1996), no. 2, 305-324.
* [46] X.-Q. Zhao, Dynamical Systems in population biology, CMS Books in Mathematics, 16. Springer-Verlag, New York, 2003\.
|
arxiv-papers
| 2014-02-25T21:25:56 |
2024-09-04T02:49:58.921607
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenxian Shen and Xiaoxia Xie",
"submitter": "Xiaoxia Xie",
"url": "https://arxiv.org/abs/1402.6353"
}
|
1402.6407
|
S. Chambi, D. Lemire, O. Kaser, R. GodinBetter bitmap performance with Roaring
bitmaps
Natural Sciences and Engineering Research Council of Canada261437 Daniel
Lemire, LICEF Research Center, TELUQ, Université du Québec, 5800 Saint-Denis,
Office 1105, Montreal (Quebec), H2S 3L5 Canada. Email: [email protected]
# Better bitmap performance with Roaring bitmaps
S. Chambi 1 D. Lemire 2 O. Kaser 3 R. Godin 1 11affiliationmark:
Département d’informatique, UQAM, Montreal, Qc, Canada22affiliationmark:
LICEF Research Center, TELUQ, Montreal, QC, Canada33affiliationmark: Computer
Science and Applied Statistics, UNB Saint John, Saint John, NB, Canada
###### Abstract
Bitmap indexes are commonly used in databases and search engines. By
exploiting bit-level parallelism, they can significantly accelerate queries.
However, they can use much memory, and thus we might prefer compressed bitmap
indexes. Following Oracle’s lead, bitmaps are often compressed using run-
length encoding (RLE). Building on prior work, we introduce the _Roaring_
compressed bitmap format: it uses packed arrays for compression instead of
RLE. We compare it to two high-performance RLE-based bitmap encoding
techniques: WAH (Word Aligned Hybrid compression scheme) and Concise
(Compressed ‘n’ Composable Integer Set). On synthetic and real data, we find
that Roaring bitmaps (1) often compress significantly better (e.g.,
$2\times$) and (2) are faster than the compressed alternatives (up to
$900\times$ faster for intersections). Our results challenge the view that
RLE-based bitmap compression is best.
###### keywords:
performance; measurement; index compression; bitmap index
## 1 Introduction
A bitmap (or bitset) is a binary array that we can view as an efficient and
compact representation of an integer set, $S$. Given a bitmap of $n$ bits, the
$i^{\mathrm{th}}$ bit is set to one if the $i^{\mathrm{th}}$ integer in the
range $[0,n-1]$ exists in the set. For example, the sets $\\{3,4,7\\}$ and
$\\{4,5,7\\}$ might be stored in binary form as 10011000 and 10110000. We can
compute the union or the intersection between two such corresponding lists
using bitwise operations (OR, AND) on the bitmaps (e.g., 10111000 and 10010000
in our case). Bitmaps are part of the Java platform (java.util.BitSet).
When the cardinality of $S$ is relatively large compared to the universe size,
$n$ (e.g., $|S|>n/64$ on 64-bit processors), bitmaps are often superior to
other comparable data structures such as arrays, hash sets or trees. However,
on moderately low density bitmaps ($n/10000<|S|<n/64$), compressed bitmaps
such as Concise can be preferable [1].
Most of the recently proposed compressed bitmap formats are derived from
Oracle’s BBC [2] and use run-length encoding (RLE) for compression: WAH [3],
Concise [1], EWAH [4], COMPAX [5], VLC [6], VAL-WAH [7], etc. Wu et al.’s WAH
is probably the best known. WAH divides a bitmap of $n$ bits into
$\left\lceil\frac{n}{w-1}\right\rceil$ words of $w-1$ bits, where $w$ is a
convenient word length (e.g., $w=32$). WAH distinguishes between two types of
words: words made of just $w-1$ ones (11$\cdots$1) or just $w-1$ zeros
(00$\cdots$0), are _fill words_ , whereas words containing a mix of zeros and
ones (e.g., 101110$\cdots$1) are _literal words_. Literal words are stored
using $w$ bits: the most significant bit is set to zero and the remaining bits
store the heterogeneous $w-1$ bits. Sequences of homogeneous fill words (all
ones or all zeros) are also stored using $w$ bits: the most significant bit is
set to 1, the second most significant bit indicates the bit value of the
homogeneous word sequence, while the remaining $w-2$ bits store the run length
of the homogeneous word sequence.
When compressing a sparse bitmap, e.g., corresponding to the set
$\\{0,2(w-1),4(w-1),\ldots\\}$, WAH can use $2w$ bits per set bit. Concise
reduces this memory usage by half [1]. It uses a similar format except for
coded fill words. Instead of storing the run length $r$ using $w-2$ bits,
Concise uses only $w-2-\left\lceil\log_{2}(w)\right\rceil$ bits, setting aside
$\left\lceil\log_{2}(w)\right\rceil$ bits as _position_ bits. These
$\left\lceil\log_{2}(w)\right\rceil$ position bits encode a number
$p\in[0,w)$. When $p=0$, we decode $r+1$ fill words. When it is non-zero, we
decode $r$ fill words preceded by a word that has its $(p-1)^{\mathrm{th}}$
bit flipped compared to the following fill words. Consider the case where
$w=32$. Concise can code the set $\\{0,62,124,\ldots\\}$ using only 32
bits/integer, in contrast to WAH which requires 64 bits/integer.
Though they reduce memory usage, these formats derived from BBC have slow
random access compared to an uncompressed bitmap. That is, checking or
changing the $i^{\mathrm{th}}$ bit value is an $O(n)$-time operation. Thus,
though they represent an integer set, we cannot quickly check whether an
integer is in the set. This makes them unsuitable for some applications [8].
Moreover, RLE formats have a limited ability to quickly skip data. For
example, suppose that we are computing the bitwise AND between two compressed
bitmaps. If one bitmap has long runs of zeros, we might wish to skip over the
corresponding words in the other bitmap. Without an auxiliary index, this
might be impossible with formats like WAH and Concise.
Instead of using RLE and sacrificing random access, we propose to partition
the space $[0,n)$ into _chunks_ and to store dense and sparse chunks
differently [9]. On this basis, we introduce a new bitmap compression scheme
called _Roaring_. Roaring bitmaps store 32-bit integers in a compact and
efficient two-level indexing data structure. Dense chunks are stored using
bitmaps; sparse chunks use packed arrays of 16-bit integers. In our example
($\\{0,62,124,\ldots\\}$), it would use only $\approx 16$ bits/integer, half
of Concise’s memory usage. Moreover, on the synthetic-data test proposed by
Colantonio and Di Pietro [1], it is at least four times faster than WAH and
Concise. In some instances, it can be hundreds of times faster.
Our approach is reminiscent of O’Neil and O’Neil’s RIDBit external-memory
system. RIDBit is a B-tree of bitmaps, where a list is used instead when a
chunk’s density is too small. However RIDBit fared poorly compared to
FastBit—a WAH-based system [10]: FastBit was up to $10\times$ faster. In
contrast to the negative results of O’Neil et al., we find that Roaring
bitmaps can be several times faster than WAH bitmaps for in-memory processing.
Thus one of our main contributions is to challenge the belief—expressed by
authors such as by Colantonio and Di Pietro [1]—that WAH bitmap compression is
the most efficient alternative.
A key ingredient in the performance of Roaring bitmaps are the new bit-count
processor instructions (such as popcnt) that became available on desktop
processors more recently (2008). Previously, table lookups were often used
instead in systems like RIDBit [11], but they can be several times slower.
These new instructions allow us to quickly compute the density of new chunks,
and to efficiently extract the location of the set bits from a bitmap.
To surpass RLE-based formats such as WAH and Concise, we also rely on several
algorithmic strategies (see § 4). For example, when intersecting two sparse
chunks, we may use an approach based on binary search instead of a linear-time
merge like RIDBit. Also, when merging two chunks, we predict whether the
result is dense or sparse to minimize wasteful conversions. In contrast,
O’Neil et al. report that RIDBit converts chunks after computing them [11].
## 2 Roaring bitmap
We partition the range of 32-bit indexes ($[0,n)$) into chunks of $2^{16}$
integers sharing the same 16 most significant digits. We use specialized
containers to store their 16 least significant bits.
When a chunk contains no more than 4096 integers, we use a sorted array of
packed 16-bit integers. When there are more than 4096 integers, we use a
$2^{16}$-bit bitmap. Thus, we have two types of containers: an array container
for _sparse_ chunks and a bitmap container for _dense_ chunks. The 4096
threshold insures that at the level of the containers, each integer uses no
more than 16 bits: we either use $2^{16}$ bits for more than 4096 integers,
using less than 16 bits/integer, or else we use exactly 16 bits/integer.
The containers are stored in a dynamic array with the shared 16 most-
significant bits: this serves as a first-level index. The array keeps the
containers sorted by the 16 most-significant bits. We expect this first-level
index to be typically small: when $n=$1\,000\,000$$, it contains at most 16
entries. Thus it should often remain in the CPU cache. The containers
themselves should never use much more than 8 kB.
To illustrate the data structure, consider the list of the first 1000
multiples of 62, all integers $[2^{16},2^{16}+100)$ and all even numbers in
$[2\times 2^{16},3\times 2^{16})$. When encoding this list using the Concise
format, we use one 32-bit fill word for each of the 1000 multiples of 62, we
use two additional fill words to include the list of numbers between $2^{16}$
and $2^{16}+100$, and the even numbers in $[2\times 2^{16},3\times 2^{16})$
are stored as literal words. In the Roaring format, both the multiples of 62
and the integers in $[2^{16},2^{16}+100)$ are stored using an array container
using 16-bit per integer. The even numbers in $[2\times 2^{16},3\times
2^{16})$ are stored in a $2^{16}$-bit bitmap container. See Fig. 1.
Array of containers Most significant bits: 0x0000 Cardinality: 1000
062124186248310$\vdots$61 938array container Most significant bits: 0x0001
Cardinality: 100 012345$\vdots$99array container Most significant bits: 0x0002
Cardinality: $2^{15}$ 101010$\vdots$0bitmap container
Figure 1: Roaring bitmap containing the list of the first 1000 multiples of
62, all integers $[2^{16},2^{16}+100)$ and all even numbers in $[2\times
2^{16},3\times 2^{16})$.
Each Roaring container keeps track of its cardinality (number of integers)
using a counter. Thus computing the cardinality of a Roaring bitmap can be
done quickly: it suffices to sum at most $\left\lceil n/2^{16}\right\rceil$
counters. It also makes it possible to support rank and select queries faster
than with a typical bitmap: rank queries count the number of set bits in a
range $[0,i]$ whereas select queries seek the location of the
$i^{\mathrm{th}}$ set bit.
The overhead due to the containers and the dynamic array means that our memory
usage can exceed 16 bits/integer. However, as long as the number of containers
is small compared to the total number of integers, we should never use much
more than 16 bits/integer. We assume that there are far fewer containers than
integers. More precisely, we assume that the density typically exceeds
$0.1\text{\,}\mathrm{\char 37\relax}$ or that $n/|S|>0.001$. When applications
encounter integer sets with lower density (less than
$0.1\text{\,}\mathrm{\char 37\relax}$), a bitmap is unlikely to be the proper
data structure.
The presented Roaring data layout is intentionally simple. Several variations
are possible. For very dense bitmaps, when there are more than $2^{16}-4096$
integers per container, we could store the locations of the zero bits instead
of a $2^{16}$-bit bitmap. Moreover, we could better compress sequences of
consecutive integers. We leave the investigation of these possibilities as
future work.
## 3 Access operations
To check for the presence of a 32-bit integer $x$, we first seek the container
corresponding to $x/2^{16}$, using binary search. If a bitmap container is
found, we access the $(x\bmod{2^{16}})^{\mathrm{th}}$ bit. If an array
container is found, we use a binary search again.
We insert and remove an integer $x$ similarly. We first seek the corresponding
container. When the container found is a bitmap, we set the value of the
corresponding bit and update the cardinality accordingly. If we find an array
container, we use a binary search followed by a linear-time insertion or
deletion.
When removing an integer, a bitmap container might become an array container
if its cardinality reaches 4096. When adding an integer, an array container
might become a bitmap container when its cardinality exceeds 4096. When this
happens, a new container is created with the updated data while the old
container is discarded. Converting an array container to a bitmap container is
done by creating a new bitmap container initialized with zeros, and setting
the corresponding bits. To convert a bitmap container to an array container,
we extract the location of the set bits using an optimized algorithm (see
Algorithm 2).
## 4 Logical operations
We implemented various operations on Roaring bitmaps, including union (bitwise
OR) and intersection (bitwise AND). A bitwise operation between two Roaring
bitmaps consists of iterating and comparing the 16 high-bits integers (keys)
on the first-level indexes. For better performance, we maintain sorted first-
level arrays. Two keys are compared at each iteration. On equality, a second-
level logical operation between the corresponding containers is performed.
This always generates a new container. If the container is not empty, it is
added to the result along with the common key. Then iterators positioned over
the first-level arrays are incremented by one. When two keys are not equal,
the array containing the smallest one is incremented by one position, and if a
union is performed, the lowest key and a copy of the corresponding container
are added to the answer. When computing unions, we repeat until the two first-
level arrays are exhausted. And when computing intersections, we terminate as
soon as one array is exhausted.
Sorted first-level arrays allows first-level comparisons in $O(n_{1}+n_{2})$
time, where $n_{1}$ and $n_{2}$ are the respective lengths of the two compared
arrays. We also maintain the array containers sorted for the same advantages.
As containers can be represented with two different data structures, bitmaps
and arrays, a logical union or intersection between two containers involves
one of the three following scenarios:
Bitmap vs Bitmap:
We iterate over 1024 64-bit words. For unions, we perform 1024 bitwise ORs and
write the result to a new bitmap container. See Algorithm 1. The resulting
cardinality is computed efficiently in Java using the Long.bitCount method.
Algorithm 1 Routine to compute the union of two bitmap containers.
1: input: two bitmaps $A$ and $B$ indexed as arrays of 1024 64-bit integers
2: output: a bitmap $C$ representing the union of $A$ and $B$, and its
cardinality $c$
3: $c\leftarrow 0$
4: Let $C$ be indexed as an array of 1024 64-bit integers
5: for $i\in\\{1,2,\ldots,1024\\}$ do
6: $C_{i}\leftarrow A_{i}\mathrm{~{}OR~{}}B_{i}$
7: $c\leftarrow c+\mathrm{bitCount}(C_{i})$
8: return $C$ and $c$
It might seem like computing bitwise ORs and computing the cardinality of the
result would be significantly slower than merely computing the bitwise ORs.
However, four factors mitigate this potential problem.
1. 1.
Popular processors (Intel, AMD, ARM) have fast instructions to compute the
number of ones in a word. Intel and AMD’s popcnt instruction has a throughput
as high as one operation per CPU cycle.
2. 2.
Recent Java implementations translate a call to Long.bitCount into such fast
instructions.
3. 3.
Popular processors are superscalar: they can execute several operations at
once. Thus, while we retrieve the next data elements, compute their bitwise OR
and store it in memory, the processor can apply the popcnt instruction on the
last result and increment the cardinality counter accordingly.
4. 4.
For inexpensive data processing operations, the processor may not run at full
capacity due to cache misses.
On the Java platform we used for our experiments, we estimate that we can
compute and write bitwise ORs at 700 million 64-bit words per second. If we
further compute the cardinality of the result as we produce it, our estimated
speed falls to about 500 million words per second. That is, we suffer a speed
penalty of about $30\text{\,}\mathrm{\char 37\relax}$ because we maintain the
cardinality. In contrast, competing methods like WAH and Concise must spend
time to decode the word type before performing a single bitwise operation.
These checks may cause expensive branch mispredictions or impair superscalar
execution.
For computing intersections, we use a less direct route. First, we compute the
cardinality of the result, using 1024 bitwise AND instructions. If the
cardinality is larger than 4096, then we proceed as with the union, writing
the result of bitwise ANDs to a new bitmap container. Otherwise, we create a
new array container. We extract the set bits from the bitwise ANDs on the fly,
using Algorithm 2. See Algorithm 3.
Algorithm 2 Optimized algorithm to convert the set bits in a bitmap into a
list of integers. We assume two-complement’s arithmetic. The function bitCount
returns the Hamming weight of the integer.
1: input: an integer $w$
2: output: an array $S$ containing the indexes where a 1-bit can be found in
$w$
3: Let $S$ be an initially empty list
4: while $w\neq 0$ do
5: $t\leftarrow w\textrm{~{}AND~{}}-w$ (cf. [12, p. 12])
6: append bitCount($t-1$) to $S$
7: $w\leftarrow w\textrm{~{}AND~{}}(w-1)$ (cf. [12, p. 11])
8: return $S$
Algorithm 3 Routine to compute the intersection of two bitmap containers. The
function bitCount returns the Hamming weight of the integer.
1: input: two bitmaps $A$ and $B$ indexed as arrays of 1024 64-bit integers
2: output: a bitmap $C$ representing the intersection of $A$ and $B$, and its
cardinality $c$ if $c>4096$ or an equivalent array of integers otherwise
3: $c\leftarrow 0$
4: for $i\in\\{1,2,\ldots,1024\\}$ do
5: $c\leftarrow c+\mathrm{bitCount}(A_{i}\mathrm{~{}AND~{}}B_{i})$
6: if $c>4096$ then
7: Let $C$ be indexed as an array of 1024 64-bit integers
8: for $i\in\\{1,2,\ldots,1024\\}$ do
9: $C_{i}\leftarrow A_{i}\mathrm{~{}AND~{}}B_{i}$
10: return $C$ and $c$
11: else
12: Let $D$ be an array of integers, initially empty
13: for $i\in\\{1,2,\ldots,1024\\}$ do
14: append the set bits in $A_{i}\mathrm{~{}AND~{}}B_{i}$ to $D$ using
Algorithm 2
15: return $D$
Bitmap vs Array:
When one of the two containers is a bitmap and the other one is a sorted
dynamic array, the intersection can be computed very quickly: we iterate over
the sorted dynamic array, and verify the existence of each 16-bit integer in
the bitmap container. The result is written out to an array container. Unions
are also efficient: we create a copy of the bitmap and simply iterate over the
array, setting the corresponding bits.
Array vs Array:
For unions, if the sum of the cardinalities is no more than 4096, we use a
merge algorithm between the two arrays. Otherwise, we set the bits
corresponding to both arrays in a bitmap container. We then compute the
cardinality using fast instructions. If the cardinality is no more than 4096,
we convert the bitmap container to an array container (see Algorithm 2).
For intersections, we use a simple merge (akin to what is done in merge sort)
when the two arrays have cardinalities that differ by less than a factor of
64. Otherwise, we use galloping intersections [8]. The result is always
written to a new array container. Galloping is superior to a simple merge when
one array ($r$) is much smaller than other one ($f$) because it can skip many
comparisons. Starting from the beginning of both arrays, we pick the next
available integer $r_{i}$ from the small array $r$ and seek an integer at
least as large $f_{j}$ in the large array $f$, looking first at the next
value, then looking at a value twice as far, and so on. Then, we use binary
search to advance in the second list to the first value larger or equal to
$r_{i}$.
We can also execute some of these operations _in place_ :
* •
When computing the union between two bitmap containers, we can modify one of
the two bitmap containers instead of generating a new bitmap container.
Similarly, for the intersection between two bitmap containers, we can modify
one of the two containers if the cardinality of the result exceeds 4096.
* •
When computing the union between an array and a bitmap container, we can write
the result to the bitmap container, by iterating over the values of the array
container and setting the corresponding bits in the bitmap container. We can
update the cardinality each time by checking whether the word value has been
modified.
In-place operations can be faster because they avoid allocating and
initializing new memory areas.
When aggregating many bitmaps, we use other optimizations. For example, when
computing the union of many bitmaps (e.g., hundreds), we first locate all
containers having the same key (using a priority queue). If one such container
is a bitmap container, then we can clone this bitmap container (if needed) and
compute the union of this container with all corresponding containers in-
place. In this instance, the computation of the cardinality can be done once
at the end. See Algorithm 4.
Algorithm 4 Optimized algorithm to compute the union of many roaring bitmaps
1: input: a set $R$ of Roaring bitmaps as collections of containers; each
container has a cardinality and a 16-bit key
2: output: a new Roaring bitmap $T$ representing the union
3: Let $T$ be an initially empty Roaring bitmap.
4: Let $P$ be the min-heap of containers in the bitmaps of $R$, configured to
order the containers by their 16-bit keys.
5: while $P$ is not empty do
6: Let $x$ be the root element of $P$. Remove from the min-heap $P$ all
elements having the same key as $x$, and call the result $Q$.
7: Sort $Q$ by descending cardinality; $Q_{1}$ has maximal cardinality.
8: Clone $Q_{1}$ and call the result $A$. The container $A$ might be an array
or bitmap container.
9: for $i\in\\{2,\ldots,|Q|\\}$ do
10: if $A$ is a bitmap container then
11: Compute the in-place union of $A$ with $Q_{i}$: $A\leftarrow
A\mathrm{~{}OR~{}}Q_{i}$. Do not re-compute the cardinality of $A$: just
compute the bitwise-OR operations.
12: else
13: Compute the union of the array container $A$ with the array container
$Q_{i}$: $A\leftarrow A\mathrm{~{}OR~{}}Q_{i}$. If $A$ exceeds a cardinality
of 4096, then it becomes a bitmap container.
14: If $A$ is a bitmap container, update $A$ by computing its actual
cardinality.
15: Add $A$ to the output of Roaring bitmap $T$.
16: return $T$
(a) Compression: uniform distribution (b) Compression: $\mathrm{Beta(0.5,1)}$
distribution (c) Intersection: discretized $\mathrm{Beta(0.5,1)}$ distribution
(d) Union: discretized $\mathrm{Beta(0.5,1)}$ distribution (e) Append: uniform
distribution (f) Removal: uniform distribution
Figure 2: Times and compression measurements: average of 100 runs
## 5 Experiments
We performed a series of experiments to compare the time-space performance of
Roaring bitmaps with the performance of other well-known bitmap indexing
schemes: Java’s BitSet, WAH and Concise. We use the CONCISE Java library for
WAH and Concise (version 2.2) made available by Colantonio and Di Pietro [1].
Our Roaring-bitmap implementation code and data is freely available at
http://roaringbitmap.org/. Our software has been thoroughly tested: our Java
library has been adopted by major database systems like Apache Spark [13] and
Druid [14].
Benchmarks were performed on an AMD FX™-8150 eight-core processor running at
3.60 GHz and having 16 GB of RAM. We used the Oracle Server JVM version 1.7 on
Linux Ubuntu 12.04.1 LTS. All our experiments run entirely in memory.
To account for the just-in-time compiler in Java, we first run tests without
recording the timings. Then we repeat the tests several times and report an
average.
### 5.1 Synthetic experiments
We began by reproducing Colantonio and Di Pietro’s synthetic experiments [1].
However, while they included alternative data structures such as Java’s
HashSet, we focus solely on bitmap formats for simplicity. Our results are
generally consistent with Colantonio and Di Pietro’s results, given the fact
that we have a better processor.
Data sets of $10^{5}$ integers were generated according to two synthetic data
distributions: uniform and discretized $\mathrm{Beta(0.5,1)}$ distributions.
(Colantonio and Di Pietro described the latter as a _Zipfian_ distribution.)
The four schemes were compared on several densities $d$ varying from $2^{-10}$
to $0.5$. To generate an integer, we first picked a floating-point number $y$
pseudo-randomly in $[0,1)$. When we desired a uniform distribution, we added
$\left\lfloor y\times\textrm{max}\right\rfloor$ to the set. In the
$\beta$-distributed case, we added $\left\lfloor
y^{2}\times\textrm{max}\right\rfloor$. The value max represents the ratio
between the total number of integers to generate and the desired density ($d$)
of the set, i.e.: $\textrm{max}=10^{5}/d$. Because results on uniform and
$\mathrm{Beta(0.5,1)}$ distributions are often similar, we do not
systematically present both.
We stress that our data distributions and benchmark closely follow Colantonio
and Di Pietro’s work [1]. Since they used this benchmark to show the
superiority of Concise over WAH, we feel that it is fair to use their own
benchmark to assess our own proposal against Concise.
Figs. 2a and 2b show the average number of bits used by Java’s BitSet and the
three bitmap compression techniques to store an integer in a set. In these
tests, Roaring bitmaps require $50\text{\,}\mathrm{\char 37\relax}$ of the
space consumed by Concise and $25\text{\,}\mathrm{\char 37\relax}$ of WAH
space on sparse bitmaps.
The BitSet class uses slightly more memory even for dense bitmaps in our
tests. This is due to its memory allocation strategy that doubles the size of
the underlying array whenever more storage is required. We could recover this
wasted memory by cloning the newly constructed bitmaps. Our roaring bitmap
implementation has a trim method that can be used to get the same result. We
did not call these methods in these tests.
We also report on intersection and union times. That is, we take two bitmaps
and generate a new bitmap representing the intersection or union. For the
BitSet, it means that we first need to create a copy (using the clone method),
since bitwise operations are in-place. Figs. 2c and 2d present the average
time in nanoseconds to perform intersections and unions between two sets of
integers. Roaring bitmaps are $\times 4-\times 5$ times faster than Concise
and WAH for intersections on all tested densities. Results for unions are
similar except that for moderate densities ($2^{-5}\leq d\leq 2^{-4}$),
Roaring is only moderately ($30\text{\,}\mathrm{\char 37\relax}$) faster than
Concise and WAH. BitSet outperforms the other schemes on dense data, but it is
$>10\times$ slower on sparse bitmaps.
We measured times required by each scheme to add a single element $a$ to a
sorted set $S$ of integers, i.e.: $\forall i\in S:a>i$. Fig. 2e shows that
Roaring requires less time than WAH and Concise. Moreover, WAH and Concise do
not support the efficient insertion of values in random order, unlike Roaring
bitmaps. Finally, we measured the time needed to remove one randomly selected
element from an integers set (Fig. 2f). We observe that Roaring bitmaps have
much better results than the two other compressed formats.
### 5.2 Real-data experiments
Tables 1–2 present results for the five real data sets used an earlier study
of compressed bitmap indexes [15]. There are only two exceptions:
* •
We only use the September 1985 data for the Weather data set (an approach
others have used before [16]), which was otherwise too large for our test
environment.
* •
We omitted the Census2000 data set because it contains only bitmaps having an
average cardinality of 30 over a large universe ($n=$37\,019\,068$$). It is an
ill-suited scenario for bitmaps. Because of the structure overhead, Roaring
bitmaps used $4\times$ as much memory as Concise bitmaps. Still, Roaring
bitmaps were about $4\times$ faster when computing intersections.
The data sets were taken as-is: we did not sort them prior to indexing.
For each data set, a bitmap index was built. Then we chose 200 bitmaps from
the index, using an approach similar to stratified sampling to control for the
large range of attribute cardinalities. We first sampled 200 attributes, with
replacement. For each sampled attribute, we selected one of its bitmaps
uniformly at random. The 200 bitmaps were used as 100 pairs of inputs for 100
pairwise ANDs and ORs; Tables 2b–2c show the time factor increase if Roaring
is replaced by BitSet, WAH or Concise. (Values below 1.0 indicate cases where
Roaring was slower.) Table 2a shows the storage factor increase when Roaring
is replaced by one of the other approaches.
Table 1: Sampled bitmap characteristics and Roaring size. | Census1881 | CensusIncome | Wikileaks | Weather
---|---|---|---|---
Rows | $4\,277\,807$ | $199\,523$ | $1\,178\,559$ | $1\,015\,367$
Density | $1.2\text{\times}{10}^{-3}$ | $1.7\text{\times}{10}^{-1}$ | $1.3\text{\times}{10}^{-3}$ | $6.4\text{\times}{10}^{-2}$
Bits/Item | $18.7$ | $2.92$ | $22.3$ | $5.83$
Table 2: Results on real data
| Census1881 | CensusIncome | Wikileaks | Weather
---|---|---|---|---
Concise | $2.21$ | $1.38$ | $0.79$ | $1.38$
WAH | $2.43$ | $1.63$ | $0.79$ | $1.51$
BitSet | $41.50$ | $2.89$ | $55.45$ | $3.49$
(a) Size expansion if Roaring is replaced with other schemes.
| Census1881 | CensusIncome | Wikileaks | Weather
---|---|---|---|---
Concise | $921.81$ | $6.58$ | $8.30$ | $6.26$
WAH | $841.08$ | $5.89$ | $8.16$ | $5.40$
BitSet | $733.85$ | $0.42$ | $27.91$ | $0.64$
(b) Time increase, for AND, if Roaring is replaced with other schemes.
| Census1881 | CensusIncome | Wikileaks | Weather
---|---|---|---|---
Concise | $33.80$ | $5.41$ | $2.14$ | $3.87$
WAH | $30.58$ | $4.85$ | $2.06$ | $3.39$
BitSet | $28.73$ | $0.43$ | $6.72$ | $0.48$
(c) Time increases, for OR, if Roaring is replaced with other schemes.
Roaring bitmaps are always faster, on average, than WAH and Concise. On two
data sets (Census1881 and Wikileaks), Roaring bitmaps are faster than BitSet
while using much less memory ($40\times$ less). On the two other data sets,
BitSet is more than twice as fast as Roaring, but it also uses three times as
much memory. When comparing the speed of BitSet and Roaring, consider that
Roaring pre-computes the cardinality at a chunk level. Thus if we need the
cardinality of the aggregated bitmaps, Roaring has the advantage. On the
Wikileaks data set, Concise and WAH offer better compression than Roaring (by
about $30\text{\,}\mathrm{\char 37\relax}$). This is due to the presence of
long runs of ones (11$\cdots$1 fill words), which Roaring does not compress.
Results on the Census1881 data set are striking: Roaring is up to $900\times$
faster than the alternatives. This is due to the large differences in the
cardinalities of the bitmaps. When intersecting a sparse bitmap with a dense
one, Roaring is particularly efficient.
## 6 Conclusion
In this paper, we introduced a new bitmap compression scheme called Roaring.
It stores bitmap set entries as 32-bit integers in a space-efficient two-level
index. In comparison with two competitive bitmap compression schemes, WAH and
Concise, Roaring often uses less memory and is faster.
When the data is ordered such that the bitmaps need to store long runs of
consecutive values (e.g., on the Wikileaks set), alternatives such as Concise
or WAH may offer better compression ratios. However, even in these instances,
Roaring might be faster. In future work, we will consider improving the
performance and compression ratios further. We might add new types of
containers. In particular, we might make use of fast packing techniques to
optimize the storage use of the array containers [17]. We could make use of
SIMD instructions in our algorithms [18, 19, 20]. We should also review other
operations beside intersections and unions, such as threshold queries [21].
We plan to investigate further applications in information retrieval. There
are reasons to be optimistic: Apache Lucene (as of version 5.0) has adopted a
Roaring format [22] to compress document identifiers.
## References
* [1] Colantonio A, Di Pietro R. Concise: Compressed ’n’ Composable Integer Set. _Information Processing Letters_ 2010; 110(16):644–650, 10.1016/j.ipl.2010.05.018.
* [2] Antoshenkov G. Byte-aligned bitmap compression. _DCC’95_ , IEEE Computer Society: Washington, DC, USA, 1995; 476.
* [3] Wu K, Stockinger K, Shoshani A. Breaking the curse of cardinality on bitmap indexes. _SSDBM’08_ , Springer: Berlin, Heidelberg, 2008; 348–365.
* [4] Lemire D, Kaser O, Aouiche K. Sorting improves word-aligned bitmap indexes. _Data & Knowledge Engineering_ 2010; 69(1):3–28, 10.1016/j.datak.2009.08.006.
* [5] Fusco F, Stoecklin MP, Vlachos M. NET-FLi: On-the-fly compression, archiving and indexing of streaming network traffic. _Proceedings of the VLDB Endowment_ 2010; 3(2):1382–1393, 10.14778/1920841.1921011.
* [6] Corrales F, Chiu D, Sawin J. Variable length compression for bitmap indices. _DEXA’11_ , Springer-Verlag: Berlin, Heidelberg, 2011; 381–395.
* [7] Guzun G, Canahuate G, Chiu D, Sawin J. A tunable compression framework for bitmap indices. _ICDE’14_ , IEEE, 2014; 484–495.
* [8] Culpepper JS, Moffat A. Efficient set intersection for inverted indexing. _ACM T. Inform. Syst._ Dec 2010; 29(1):1:1–1:25, 10.1145/1877766.1877767.
* [9] Kaser O, Lemire D. Attribute value reordering for efficient hybrid OLAP. _Inform. Sciences_ 2006; 176(16):2304–2336.
* [10] O’Neil E, O’Neil P, Wu K. Bitmap index design choices and their performance implications. _IDEAS’07_ , IEEE, 2007; 72–84.
* [11] Rinfret D, O’Neil P, O’Neil E. Bit-sliced index arithmetic. _Proceedings of the 2001 ACM SIGMOD International Conference on Management of Data_ , SIGMOD ’01, ACM: New York, NY, USA, 2001; 47–57, 10.1145/375663.375669.
* [12] Warren HS Jr. _Hacker’s Delight_. 2nd ed. edn., Addison-Wesley: Boston, 2013\.
* [13] Zaharia M, Chowdhury M, Franklin MJ, Shenker S, Stoica I. Spark: Cluster computing with working sets. _Proceedings of the 2nd USENIX Conference on Hot Topics in Cloud Computing_ , HotCloud’10, USENIX Association: Berkeley, CA, USA, 2010; 10–10.
* [14] Yang F, Tschetter E, Léauté X, Ray N, Merlino G, Ganguli D. Druid: A real-time analytical data store. _Proceedings of the 2014 ACM SIGMOD International Conference on Management of Data_ , SIGMOD ’14, ACM: New York, NY, USA, 2014; 157–168, 10.1145/2588555.2595631.
* [15] Lemire D, Kaser O, Gutarra E. Reordering rows for better compression: Beyond the lexicographical order. _ACM Transactions on Database Systems_ 2012; 37(3), 10.1145/2338626.2338633. Article 20.
* [16] Beyer K, Ramakrishnan R. Bottom-up computation of sparse and iceberg CUBEs. _SIGMOD Record_ 1999; 28(2):359–370, 10.1145/304181.304214.
* [17] Lemire D, Boytsov L. Decoding billions of integers per second through vectorization. _Software: Practice and Experience_ 2015; 45(1), 10.1002/spe.2203.
* [18] Polychroniou O, Ross KA. Vectorized bloom filters for advanced simd processors. _Proceedings of the Tenth International Workshop on Data Management on New Hardware_ , DaMoN ’14, ACM: New York, NY, USA, 2014; 1–6, 10.1145/2619228.2619234.
* [19] Lemire D, Boytsov L, Kurz N. SIMD compression and the intersection of sorted integers. http://arxiv.org/abs/1401.6399 [last checked March 2015] 2014\.
* [20] Inoue H, Ohara M, Taura K. Faster set intersection with SIMD instructions by reducing branch mispredictions. _Proceedings of the VLDB Endowment_ 2014; 8(3).
* [21] Kaser O, Lemire D. Compressed bitmap indexes: beyond unions and intersections. _Software: Practice and Experience_ 2014; 10.1002/spe.2289. In Press.
* [22] Grand A. LUCENE-5983: RoaringDocIdSet. https://issues.apache.org/jira/browse/LUCENE-5983 [last checked March 2015] 2014.
|
arxiv-papers
| 2014-02-26T04:38:22 |
2024-09-04T02:49:58.934668
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Samy Chambi, Daniel Lemire, Owen Kaser, Robert Godin",
"submitter": "Daniel Lemire",
"url": "https://arxiv.org/abs/1402.6407"
}
|
1402.6557
|
7892014201401XXXYY
11institutetext: Leibniz-Institut für Astrophysik, An der Sternwarte 16,
D-14482 Potsdam, Germany
later
# Mean-field dynamos:
the old concept and some recent developments
K.-H. Rädler
[email protected]
(XX January 2014; YY ZZZ 2014)
###### Abstract
This article reproduces the Karl Schwarzschild lecture 2013. Some of the basic
ideas of electrodynamics and magnetohydrodynamics of mean fields in
turbulently moving conducting fluids are explained. It is stressed that the
connection of the mean electromotive force with the mean magnetic field and
its first spatial derivatives is in general neither local nor instantaneous
and that quite a few claims concerning pretended failures of the mean-field
concept result from ignoring this aspect. In addition to the mean-field dynamo
mechanisms of $\alpha^{2}$ and $\alpha\Omega$ type several others are
considered. Much progress in mean-field electrodynamics and
magnetohydrodynamics results from the test-field method for calculating the
coefficients that determine the connection of the mean electromotive force
with the mean magnetic field. As an important example the memory effect in
homogeneous isotropic turbulence is explained. In magnetohydrodynamic
turbulence there is the possibility of a mean electromotive force that is
primarily independent of the mean magnetic field and labeled as Yoshizawa
effect. Despite of many efforts there is so far no convincing comprehensive
theory of $\alpha$ quenching, that is, the reduction of the $\alpha$ effect
with growing mean magnetic field, and of the saturation of mean-field dynamos.
Steps toward such a theory are explained. Finally, some remarks on laboratory
experiments with dynamos are made.
###### keywords:
Cosmic magnetic fields – mean-field electrodynamics – mean-field
magnetohydrodynamics – mean-field dynamos
## 1 Introduction
At the beginning of the last century mankind knew the magnetic field of the
Earth, but nothing about magnetic fields at other celestial bodies. In 1908
George Ellery Hale proposed to interpret line splittings in the spectrum of
the light coming from sunspots, which were not understood at this time, as a
consequence of strong magnetic fields (of a few kilogauss) within them. Eleven
years later, in 1919, Sir Joseph Larmor came up with the idea that magnetic
fields at the Sun could be generated by self-exciting dynamos just as
introduced in engineering for instance by Ernst Werner von Siemens in 1867
111The idea of the self-exciting dynamo has been stated several years before
by Anyos Jedlik, by Søren Hjorth and by Samuel Alfred Varley, in 1867 also by
Charles Wheatstone. Von Siemens is known for having recognized the practical
importance of the dynamo principle.. Of course, Larmor’s proposal was not
readily accepted and there were many attempts to check it or to rule it out. A
dynamo in a homogeneous fluid is quite different from its technical version
built up with insolated wires. More mathematically spoken, a dynamo working in
a singly-connected conducting region is different from that in a multiply-
connected region. In 1934 Thomas George Cowling proved a theorem which we may
now (after some generalizations) formulate by saying that a dynamo can never
work with an axisymmetric magnetic field. Another important theorem traces
back to investigations by Walter M. Elsasser in 1946 and by Edward C. Bullard
and H. Gellman in 1954 and excludes a dynamo in a sphere due to motions
without radial components. Quite a few modifications of such theorems have
been proven in the course of time showing the impossibility of dynamos with
some simple geometrical structures of the magnetic field or the fluid flow.
In 1947 Horace W. Babcock discovered a star with a strong magnetic field (of
34 kilogauss), and in 1958 he published a catalogue of magnetic stars. Later
we learned about a large number of various objects which exhibit magnetic
fields, including galaxies with rather weak but very extended fields (of the
order of $10^{-5}$ gauss) or neutron stars with extremely strong ones (up to
the order of $10^{15}$ gauss).
A rigorous existence proof for a homogeneous dynamo has been delivered by A.
Herzenberg in 1958. The velocity distribution he assumed was, however, far
away from from that expected in the Earth’s interior, in the Sun or in stars.
Already before Herzenberg’s proof, 1955 and 1957, Eugene N. Parker designed a
model for the Sun in which “cyclonic convection” together with rotational
shear, that is differential rotation, produce a magnetic cycle. In 1964
Stanislaus I. Braginsky published his model of the “nearly symmetric dynamo”
which reflects features of the Earth’s magnetic field.
In the early sixties Max Steenbeck in Jena pushed Fritz Krause and myself to
think about the question of how the Sun or the Earth could generate their
magnetic fields. Many conceivable mechanisms were discussed and investigated
in the course of time. At the end the mean-field electrodynamics of
electrically conducting turbulently moving fluids was established. A central
issue of this theory is the $\alpha$ effect, the occurrence of an
electromotive force with a part parallel (or antiparallel) to the mean
magnetic field as a consequence of induction processes caused by irregular
motions. The $\alpha$ effect allows dynamo action. The first paper on this
topic has been published by Steenbeck, Krause and Rädler in 1966
(unfortunately only in German language). Since then mean-field electrodynamics
and, more general, mean-field magnetohydrodynamics have been elaborated in
great detail and dynamo models have been proposed for the Sun, planets,
several types of stars and for galaxies. In this lecture I would like to
explain a few results of this research field. (For comprehensive presentations
see, e.g., Moffatt 1978, Krause and Rädler 1980 or Brandenburg and Subramanian
2005.)
Before doing so, however, I would like to say: It is a great honor for me to
receive the Karl Schwarzschild Medal. I am very grateful to the Board of the
Astronomische Gesellschaft for this distinction. It is also a great honor to
deliver this Schwarzschild lecture.
Let me start with a few remarks about Max Steenbeck (1904-81) and the place,
Jena, where mean-field electrodynamics was born. Max Steenbeck was no geo- or
astrophysicist. He was one of the great pioneers of plasma physics, worked
until the end of the Second World War in the Siemens Company in Berlin, dealt
there with heavy current technology, for example rectifiers, constructed the
first working betatron etc. At the end of the war he has been interned in the
Soviet Union. He spent there (involuntarily) eleven years, dealing in
particular with the separation of Uranium isotopes in the framework of the
Soviet Atomic program. After his return to the G.D.R. he dealt there with
magnetic materials, with nuclear power stations, and in 1959 he established
the Institute for Magnetohydrodynamics in Jena with the idea to deliver
contributions to nuclear fusion research, which looked at that time very
promising. Sometimes, on his frequent rides between Jena and occasionally on
Saturdays and Sundays, he thought about possibilities of processes in the Sun
or in the Earth’s interior that might produce the observed magnetic fields,
and then attacked Fritz Krause and me with his ideas. Since the Institute for
Magnetohydrodynamics was not a place with astrophysical or geophysical
tradition there was, at least at the beginning, no contact to leading
scientists in these fields.
## 2 Mean-field electrodynamics
### 2.1 The basic idea
In what follows we deal with electromagnetic processes in an electrically
conducting moving fluid. Adopting the magnetohydrodynamic approximation we
assume that the electromagnetic fields obey the pre-Maxwell equations
${\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$E$}}=-\partial_{t}{\mbox{\boldmath$B$}}\,,\quad{\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$B$}}=\mu{\mbox{\boldmath$J$}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$B$}}=0$
(1)
and Ohm’s law for moving matter in the form
${\mbox{\boldmath$J$}}=\sigma({\mbox{\boldmath$E$}}+{\mbox{\boldmath$U$}}\times{\mbox{\boldmath$B$}})\,.$
(2)
As usual, $E$ denotes the electric field, $B$ the magnetic field, $J$ the
electric current density, and $U$ the fluid velocity; further $\mu$ means the
magnetic permeability of free space and $\sigma$ the electric conductivity of
the fluid. From (1) and (2) we may derive the induction equation
$\eta{\mbox{\boldmath$\nabla$}}^{2}{\mbox{\boldmath$B$}}+{\mbox{\boldmath$\nabla$}}\times({\mbox{\boldmath$U$}}\times{\mbox{\boldmath$B$}})-\partial_{t}{\mbox{\boldmath$B$}}={\bf
0}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$B$}}=0\,,$ (3)
with $\eta=1/\mu\sigma$ being the magnetic diffusivity. For simplicity we
ignore here any electromotive force independent of electromagnetic fields,
which would act as a battery.
Until further notice we consider the fluid velocity $U$ as given. If the
induction equation is solved and so the magnetic field $B$ is known, we may
calculate the electric field $E$ and the electric current density $J$ without
further integrations.
Thinking of the situation in many astrophysical objects, we assume further
that the fluid velocity $U$ and so also the electromagnetic fields $B$, $E$
and $J$ show irregular fluctuations in space and time. Considering these
fluctuations we simply speak of turbulence (without having a specific
definition of turbulence in mind). We then focus attention on mean fields
defined as averages of these fields and showing simpler dependencies on space
and time coordinates. Space or time or statistical averages or combinations of
them are admitted. It is merely important that the Reynolds averaging rules,
already known from hydrodynamic turbulence theory, are (exactly or
approximately) satisfied. We denote averages by overbars. Let $F$ and $G$ be
quantities showing irregular behavior, that is fluctuations, and put
$F=\overline{F}+f$ and $G=\overline{G}+g$. Then these rules read
$\displaystyle\overline{F+G}=\overline{F}+\overline{G}$ (4)
$\displaystyle\overline{\overline{F}}=\overline{F}\;\mbox{or, what is
equivalent,}\;\overline{f}=0$ (5)
$\displaystyle\overline{F\,G}=\overline{F}\,\overline{G}+\overline{f\,g}$ (6)
$\displaystyle\overline{\partial F/\partial x}=\partial\overline{F}/\partial
x\,,\quad\overline{\partial F/\partial t}=\partial\overline{F}/\partial t\,.$
(7)
We stress that the average of the product of two fluctuating quantities is not
equal to the product of the corresponding mean quantities, but there is an
additional term determined by the fluctuations.
Returning to electrodynamics we subject equations (1) and (2) to averaging. We
find then their mean-field versions
${\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$E$}}}=-\partial_{t}\overline{{\mbox{\boldmath$B$}}}\,,\quad{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}=\mu\overline{{\mbox{\boldmath$J$}}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot\overline{{\mbox{\boldmath$B$}}}=0$
(8)
and
$\overline{{\mbox{\boldmath$J$}}}=\sigma(\overline{{\mbox{\boldmath$E$}}}+\overline{{\mbox{\boldmath$U$}}}\times\overline{{\mbox{\boldmath$B$}}}+{\mbox{\boldmath$\cal{E}$}})\,.$
(9)
When averaging (3) in the same way we obtain
$\displaystyle\\!\\!\\!\\!\eta{\mbox{\boldmath$\nabla$}}^{2}\overline{{\mbox{\boldmath$B$}}}+{\mbox{\boldmath$\nabla$}}\times(\overline{{\mbox{\boldmath$U$}}}\times\overline{{\mbox{\boldmath$B$}}}+{\mbox{\boldmath$\cal{E}$}})-\partial_{t}\overline{{\mbox{\boldmath$B$}}}$
$\displaystyle={\bf 0}\,,$ (10)
$\displaystyle\qquad\qquad\qquad\qquad\qquad{\mbox{\boldmath$\nabla$}}\cdot\overline{{\mbox{\boldmath$B$}}}$
$\displaystyle=0\,.$
$\cal{E}$ is the mean electromotive force due to the fluctuations of the fluid
velocity and the magnetic field,
${\mbox{\boldmath$u$}}={\mbox{\boldmath$U$}}-\overline{{\mbox{\boldmath$U$}}}$
and
${\mbox{\boldmath$b$}}={\mbox{\boldmath$B$}}-\overline{{\mbox{\boldmath$B$}}}$,
that is,
${\mbox{\boldmath$\cal{E}$}}=\overline{{\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}}\,.$
(11)
The form of the equations (8) - (10) agrees widely with that of the original,
not averaged equations (1) - (3). The only, but decisive deviation consists in
the occurrence of the new electromotive force $\cal{E}$.
The crucial point in the elaboration of mean-field electrodynamics is the
determination of that mean electromotive force $\cal{E}$. We first consider
$u$ as given. As for $b$ we may derive from (3) and (10) that
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\eta{\mbox{\boldmath$\nabla$}}^{2}{\mbox{\boldmath$b$}}+{\mbox{\boldmath$\nabla$}}\times(\overline{{\mbox{\boldmath$U$}}}\times{\mbox{\boldmath$b$}}+{\mbox{\boldmath$\epsilon$}})-\partial_{t}{\mbox{\boldmath$b$}}=-{\mbox{\boldmath$\nabla$}}\times({\mbox{\boldmath$u$}}\times\overline{{\mbox{\boldmath$B$}}})$
$\displaystyle\qquad\qquad{\mbox{\boldmath$\epsilon$}}={\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}-\overline{{\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$b$}}=0\,.$
(12)
Clearly $\epsilon$ is the fluctuating part of
${\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}$. Equation (12) tells us
that $b$ is a functional of $u$, $\overline{{\mbox{\boldmath$U$}}}$ and
$\overline{{\mbox{\boldmath$B$}}}$, which is linear (not necessarily linear
and homogeneous) in $\overline{{\mbox{\boldmath$B$}}}$. Consequently,
$\cal{E}$ depends also on these quantities and may be represented as a sum
${\mbox{\boldmath$\cal{E}$}}={\mbox{\boldmath$\cal{E}$}}^{(0)}+{\mbox{\boldmath$\cal{E}$}}^{(B)}$
(13)
of a part ${\mbox{\boldmath$\cal{E}$}}^{(0)}$ independent of
$\overline{{\mbox{\boldmath$B$}}}$ and another part
${\mbox{\boldmath$\cal{E}$}}^{(B)}$ which is is linear and homogeneous in
$\overline{{\mbox{\boldmath$B$}}}$.
Let us assume here that $b$ decays to zero if
$\overline{{\mbox{\boldmath$B$}}}$ vanishes. This implies also the absence of
small-scale dynamos (see section 2.4). Under this assumption
${\mbox{\boldmath$\cal{E}$}}^{(0)}$ decays to zero, too. As a result,
$\cal{E}$ agrees with ${\mbox{\boldmath$\cal{E}$}}^{(B)}$ and it must allow a
representation in the form of the convolution
$\\!{\cal{E}}_{i}({\mbox{\boldmath$x$}},t)=\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{\infty}\\!\\!\\!\\!{\cal
K}_{ij}({\mbox{\boldmath$x$}},t;{\mbox{\boldmath$\xi$}},\tau)\overline{B}_{j}({\mbox{\boldmath$x$}}+{\mbox{\boldmath$\xi$}},t-\tau)\,\mbox{d}^{3}\xi\,\mbox{d}\tau$
(14)
with some tensorial kernel ${\cal K}_{ij}$, which depends on $u$ and
$\overline{{\mbox{\boldmath$U$}}}$. We know the explicit dependence of ${\cal
K}_{ij}$ on $u$ and $\overline{{\mbox{\boldmath$U$}}}$ only for very special
cases, but conclude from the turbulent nature of the velocity fluctuations
that ${\cal K}_{ij}$ vanishes for sufficiently large
$|{\mbox{\boldmath$\xi$}}|$ and $\tau$. As a consequence, $\cal{E}$ in a given
point in space and time depends only on the behavior of
$\overline{{\mbox{\boldmath$B$}}}$ in a certain surroundings of this point,
the extent of which is determined by the correlation length and the
correlation time of $u$.
It is appropriate to split the kernel ${\cal K}_{ij}$ in (14) into two parts,
one symmetric and the other one antisymmetric in $\xi$, and to express the
last one by a derivative of a tensor symmetric in $\xi$. Doing so and
subjecting then (14) to an integration by parts we arrive easily at the
equivalent representation
$\displaystyle\\!\\!\\!\\!{\cal{E}}_{i}({\mbox{\boldmath$x$}},t)$
$\displaystyle=\int_{0}^{\infty}\\!\\!\\!\int_{\infty}\\!\\!\Big{(}{\cal
A}_{ij}({\mbox{\boldmath$x$}},t;{\mbox{\boldmath$\xi$}},\tau)\overline{B}_{j}({\mbox{\boldmath$x$}}+{\mbox{\boldmath$\xi$}},t-\tau)$
(15) $\displaystyle\;+{\cal
B}_{ijk}({\mbox{\boldmath$x$}},t;{\mbox{\boldmath$\xi$}},\tau)\frac{\partial\overline{B}_{j}({\mbox{\boldmath$x$}}+{\mbox{\boldmath$\xi$}},t-\tau)}{\partial
x_{k}}\Big{)}\mbox{d}^{3}\xi\mbox{d}\tau$
with two tensors ${\cal A}_{ij}$ and ${\cal B}_{ijk}$ which are both symmetric
in $\xi$.
Assume for a moment that $\overline{{\mbox{\boldmath$B$}}}$ varies only weakly
in space and time, that is, there are distinct gaps in the spectra of the
length and time scales of the total magnetic field,
$\overline{{\mbox{\boldmath$B$}}}+{\mbox{\boldmath$b$}}$, separating large and
small scales. We speak then of ideal scale separation. In this case relation
(15) turns into a simpler one,
${\cal{E}}_{i}=a_{ij}\overline{B}_{j}+b_{ijk}\frac{\partial\overline{B}_{j}}{\partial
x_{k}}$ (16)
with
$a_{ij}=\int_{0}^{\infty}\\!\\!\int_{\infty}\\!\\!{\cal
A}_{ij}({\mbox{\boldmath$x$}},t;{\mbox{\boldmath$\xi$}},\tau)\,\mbox{d}^{3}\xi\,\mbox{d}\tau$
(17)
and an analogous connection between $b_{ijk}$ and ${\cal B}_{ijk}$.
While relation (15) connects $\cal{E}$ at a given point in space and time with
$\overline{{\mbox{\boldmath$B$}}}$ in an arbitrary spatial surroundings of
this point and at this time and arbitrary past times, relation (16) describes
a local and instantaneous connection of $\cal{E}$ with
$\overline{{\mbox{\boldmath$B$}}}$ and its first spatial derivatives. The
latter relation, which has to be understood as application of the former one
to idealized cases, explains a large number of phenomena in turbulent fluids,
in particular some types of dynamos. It is, however, unable to capture, e.g.,
memory effects, that is, the dependence of the evolution of a mean field not
only on its current values but also on its history (see section 2.6). In what
follows we will, as long as it is appropriate, refer to (16) but switch to
(15) where necessary. We want to stress that many statements on pretended
failures of mean-field electrodynamics or on allegedly narrow limits of its
applicability result from ignoring (15) and considering instead (16) as the
most general relation for the mean electromotive force $\cal{E}$.
Let us finally mention the technical issue that convolutions like (15) turn
under a proper Fourier transformation (or, concerning the time, also a Laplace
transformation) into simpler algebraic relations. Ignore for simplicity any
dependence of ${\cal A}_{ij}$ and ${\cal B}_{ijl}$ on $x$ and $t$. With a
transformation
$\displaystyle F({\mbox{\boldmath$x$}},t)$
$\displaystyle=(2\pi)^{-4}\int\\!\\!\int\hat{F}({\mbox{\boldmath$k$}},\omega)$
(18)
$\displaystyle\qquad\qquad\qquad\exp\big{(}\mbox{i}({\mbox{\boldmath$k$}}\cdot{\mbox{\boldmath$x$}}-\omega
t)\big{)}\mbox{d}^{3}k\,\mbox{d}\omega$
relation (15) turns then into
$\displaystyle\hat{\cal{E}}_{i}({\mbox{\boldmath$k$}},\omega)$
$\displaystyle=\hat{\cal
A}_{ij}({\mbox{\boldmath$k$}},\omega)\,\hat{\overline{B}}_{j}({\mbox{\boldmath$k$}},\omega)$
(19) $\displaystyle\qquad\qquad\qquad+\mbox{i}\,\hat{\cal
B}_{ijk}({\mbox{\boldmath$k$}},\omega)\,\hat{\overline{B}}_{j}({\mbox{\boldmath$k$}},\omega)k_{k}\,.$
### 2.2 A simple example
We consider first the case in which no mean flow exists,
$\overline{{\mbox{\boldmath$U$}}}={\bf 0}$, and the velocity fluctuations $u$
correspond to a homogeneous isotropic turbulence. As for the mean magnetic
field $\overline{{\mbox{\boldmath$B$}}}$ we assume ideal scale separation in
the sense explained above so that (16) applies.
We define homogeneity of the turbulence by the invariance of all averaged
quantities depending on $u$ under arbitrary translations of the $u$ field, and
isotropy by the invariance of all such quantities under arbitrary rotations of
this field about arbitrary axes. We may also fix the $u$ field and subject the
coordinate system in which we describe it to translations or rotations. Then
homogeneity and isotropy occur as invariance of all averaged quantities under
arbitrary translations and arbitrary rotations of the coordinate system. In
particular the tensors $a_{ij}$ and $b_{ijk}$ that occur in (16) have to show
these properties, that is, their components have to be independent of space
coordinates and independent of rotations of the coordinate system. This
qualifies them as space-independent isotropic tensors, that is, they differ
only by simple factors, say $\alpha$ and $\beta$, from the Kronecker tensor
$\delta_{ij}$ and the Levi-Civita tensor $\epsilon_{ijk}$, that is,
$a_{ij}=\alpha\,\delta_{ij}$ and $b_{ijk}=\beta\epsilon_{ijk}$. In this way we
arrive at
${\mbox{\boldmath$\cal{E}$}}=\alpha\,\overline{{\mbox{\boldmath$B$}}}-\beta\,{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}\,,$
(20)
and the mean-field version (9) of Ohm’s law takes the form
$\overline{{\mbox{\boldmath$J$}}}=\sigma_{\rm
m}(\overline{{\mbox{\boldmath$E$}}}+\alpha\overline{{\mbox{\boldmath$B$}}})$
(21)
with the mean-field conductivity $\sigma_{\rm m}$ given by
$\sigma_{\rm m}=\sigma/(1+\mu\sigma\beta)\,.$ (22)
While $\alpha$ is a pseudoscalar, $\beta$ is a scalar.
Homogeneity and isotropy of turbulence do not exclude reflexional symmetry. We
define it by the invariance of all averaged quantities depending on $u$ under
reflexion of the $u$ field at a point. In the case of homogeneity and isotropy
this is equivalent to reflexions at any plane. A reflexion turns a right-
handed structure in the flow field in a left-handed one and vice versa.
Reflexional symmetry in the above sense implies therefore an equipartition of
right-handed and left-handed structures in a fluid flow, that is, the absence
of any preferred handedness. A simple consequence is, e.g., that the mean
kinetic helicity
$\overline{{\mbox{\boldmath$u$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})}$
vanishes. In the case of homogeneous isotropic ref lexionally symmetric
turbulence we may easily show that the pseudoscalar $\alpha$ in (20) and (21)
has to be zero. There is, however, no restriction to the scalar $\beta$.
As long as there are no deviations of the underlying homogeneous isotropic
turbulence from reflexional symmetry the mean-field version of Ohm’s law reads
simply $\overline{{\mbox{\boldmath$J$}}}=\sigma_{\rm
m}\overline{{\mbox{\boldmath$E$}}}$ with $\sigma_{\rm m}$ as given by (22).
The insight, that for mean fields a conductivity different from that relevant
for the original fields applies, can already be found in papers by Sweet
(1950) and Csada (1951). As we will see later (section 2.5) the ratio
$\sigma_{\rm m}/\sigma$ can be much bigger than unity. In the solar convection
zone, e.g., it may take values of the order of $10^{4}$, what explains in
particular the observed life times of sunspots.
Turbulence in rotating bodies, on which we want to focus our attention later,
is subject to the Coriolis force. It deviates therefore not only from isotropy
but also from reflexional symmetry. (This corresponds to the fact that the
Coriolis force is determined, e.g., by a right-hand rule.) Preparing
investigations of this complex situation, we consider here first the more or
less academic case of homogeneous isotropic but not reflexionally symmetric
turbulence, in which (20) and (21) with $\alpha\not=0$ apply. The occurrence
of the electromotive force $\alpha\overline{{\mbox{\boldmath$B$}}}$ is called
“$\alpha$ effect”. It has been first considered by Steenbeck, Krause and
Rädler (1966) in a slightly different context (see section 2.3).
The $\alpha$ effect allows growing mean magnetic fields, that is, dynamo
action. In order to show this we consider the mean-field induction equation
(10) with $\overline{{\mbox{\boldmath$U$}}}={\bf 0}$ and $\cal{E}$ specified
according to (20), that is,
$\eta_{\rm
m}{\mbox{\boldmath$\nabla$}}^{2}\overline{{\mbox{\boldmath$B$}}}+\alpha{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}-\partial_{t}\overline{{\mbox{\boldmath$B$}}}={\bf
0}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot\overline{{\mbox{\boldmath$B$}}}=0\,,$
(23)
where $\eta_{\rm m}$ is the magnetic mean-field diffusivity,
$\eta_{\rm m}=\eta+\beta\,.$ (24)
Simple particular solutions $\overline{{\mbox{\boldmath$B$}}}$ of (23) read
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\overline{{\mbox{\boldmath$B$}}}$
$\displaystyle=B_{0}(\cos kz,\pm\sin kz,0)\exp(\lambda t)\,,$ (25)
$\displaystyle\qquad\qquad\qquad\quad\lambda=-(\eta+\beta)k^{2}\pm\alpha k\,,$
with a wave number $k$, which we consider as positive, and a growth rate
$\lambda$. Growing solutions, i.e., such with $\lambda>0$, are possible as
soon as $|\alpha|>(\eta+\beta)\,k$. We will see later (section 2.5) that this
condition can well be satisfied.
### 2.3 A more realistic example
In a next, somewhat more realistic case, we consider turbulence on a rotating
body. We assume that for a co-rotating observer no mean flow exists,
$\overline{{\mbox{\boldmath$U$}}}={\bf 0}$, but admit slight deviations of the
turbulence, $u$, from homogeneity and isotropy. We further assume that the
inhomogeneity can be described by a vector $g$, e.g., the intensity gradient
${\mbox{\boldmath$\nabla$}}\overline{u^{2}}$ of the turbulence. The anisotropy
depends, of course, apart from $g$ also on the angular velocity $\Omega$ which
defines the Coriolis force. Again, we restrict ourselves to sufficiently small
variations of the mean magnetic field $\overline{{\mbox{\boldmath$B$}}}$ in
space and time so that (16) applies. Considering the deviations of the
turbulence from homogeneity and isotropy as sufficiently small, we assume that
$a_{ij}$ and $b_{ijk}$ are linear in both $g$ and $\Omega$. Studying then the
possible tensorial structures of $a_{ij}$ and $b_{ijk}$ we find
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\mbox{\boldmath$\cal{E}$}}=\alpha_{1}({\mbox{\boldmath$g$}}\cdot{\mbox{\boldmath$\Omega$}})\overline{{\mbox{\boldmath$B$}}}+\alpha_{2}{\mbox{\boldmath$g$}}({\mbox{\boldmath$\Omega$}}\cdot\overline{{\mbox{\boldmath$B$}}})+\alpha_{3}{\mbox{\boldmath$\Omega$}}({\mbox{\boldmath$g$}}\cdot\overline{{\mbox{\boldmath$B$}}})+\gamma{\mbox{\boldmath$g$}}\times\overline{{\mbox{\boldmath$B$}}}$
$\displaystyle\\!\\!-\beta{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}-\delta{\mbox{\boldmath$\Omega$}}\times({\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}})-\delta_{*}{\mbox{\boldmath$\nabla$}}({\mbox{\boldmath$\Omega$}}\cdot\overline{{\mbox{\boldmath$B$}}})$
(26)
with scalar coefficients $\alpha_{1}$, $\alpha_{2}$, … $\delta_{*}$
independent of $g$ and $\Omega$.
The first line of (26) reproduces the momentous result by Steenbeck, Krause
and Rädler (1966). Due to the Coriolis force there is now (at least locally) a
preference of either right-handed or left-handed helical patterns in the
turbulent flow. As a consequence the three contributions to $\cal{E}$ with the
coefficients $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ occur in (26). The
term
$\alpha_{1}({\mbox{\boldmath$g$}}\cdot{\mbox{\boldmath$\Omega$}})\overline{{\mbox{\boldmath$B$}}}$
corresponds to $\alpha\overline{{\mbox{\boldmath$B$}}}$ in relation (20), that
is, in the case of homogeneous isotropic turbulence lacking reflexional
symmetry. However, the $\alpha_{1}$ term is now accompanied by two others, the
$\alpha_{2}$ and $\alpha_{3}$ terms. We speak here again of an $\alpha$
effect, more precisely, if all three terms are considered, of an anisotropic
$\alpha$ effect. On a spherical body, on which $g$ points in radial direction,
$\alpha_{1}({\mbox{\boldmath$g$}}\cdot{\mbox{\boldmath$\Omega$}})$ changes its
sign when moving from the northern hemisphere to the southern one. The
$\alpha$ effect as considered here is crucial for special types of mean-field
dynamo mechanisms (see section 2.4).
The $\gamma$ term in (26) describes the transport of mean magnetic flux by
inhomogeneous turbulence. This effect has been first, for a two-dimensional
turbulence, considered by Zeldovich (1956), later in a more general context by
Rädler (1968a,b). It has been discussed as “pumping of mean magnetic flux” or
(since the mean magnetic flux is expelled from regions of high turbulence
intensity) as “turbulent diamagnetism”.
The $\beta$ term corresponds to that which occurred already in the case of
homogeneous isotropic turbulence, that is in (20), and gives rise to introduce
the mean-field conductivity in the mean-field version of Ohm’s law or the
mean-field diffusivity in the mean-field induction equation.
The effect described by the $\delta$ term in (26) has been first considered by
Rädler (1969a,b). It is often called
“${\mbox{\boldmath$\Omega$}}\times{\mbox{\boldmath$J$}}$ effect”, in what
follows also simply “$\delta$ effect”. Combined with mean shear it is capable
of dynamo action (see section 2.4). Other than the $\alpha$ effect, the
$\delta$ effect requires spatial variations of the mean magnetic field; it
does not occur with a homogeneous field. Apart from this it is in a sense
simpler than the $\alpha$ effect. It needs no gradient of the turbulence
intensity but occurs already with homogeneous turbulence. The sum of the
$\beta$ and $\delta$ terms can also be described by an anisotropic mean-field
conductivity with a non-symmetric conductivity tensor. The $\delta_{*}$ term
is of minor importance. It does not influence the mean-field induction
equation as long as $\delta_{*}$ is spatially constant.
### 2.4 Mean-field dynamo mechanisms
When discussing dynamo mechanisms due to turbulent motions we focus attention
on mean-field dynamos. They are characterized by the ability to generate
magnetic fields with length scales much larger than the typical length scales
of the turbulent motions. Therefore we call them also “large-scale dynamos”.
In this context we should have in mind the finding by Kazantsev (1968)
according to which a homogenous isotropic turbulence, which needs not to
deviate from reflexional symmetry, may under certain conditions maintain
irregular magnetic fields with length scales smaller than or equal to those of
the turbulent motion, which therefore do not contribute to a mean magnetic
field. We speak then of a “small-scale dynamo”. The influence of large-scale
on small-scale dynamos and vice versa is an interesting subject (see, e.g.,
Brandenburg et al. 2012), which we however do not want to discuss here.
Let us consider dynamos due to turbulence on an axisymmetric rotating fluid
body. We restrict attention first to mean magnetic fields that are symmetric
about the axis of rotation. Each such field can be split into a poloidal part
that lies completely in meridional planes, and a toroidal part perpendicular
to them. Within this frame, dynamos can always be understood as an interplay
between the poloidal and the toroidal part of the mean magnetic field. We
admit here a mean motion in the form of differential rotation, that is, a
dependence of the corresponding angular velocity $\Omega$ on radius or
latitude. In addition to induction effects of turbulent motions described,
e.g., by $\alpha$ or $\delta$ effects, we have then also the effect of
rotational shear, which we call “$\Omega$ effect”. While $\alpha$ and $\delta$
effect generate poloidal magnetic fields from toroidal ones and vice versa,
the $\Omega$ effect generates only a toroidal field from a poloidal one.
The simplest mean-field dynamo mechanism is that of $\alpha^{2}$ type, in
which both the generation of the poloidal field from the toroidal one as well
as that of the toroidal field from the poloidal one is due the $\alpha$
effect. The first spherical models of this type were proposed by Steenbeck and
Krause (1969b), and were on the basis of numerical simulations discussed in
view of the Earth and the planets. If we admit in addition the $\Omega$ effect
and assume that it dominates the generation of the toroidal field, we arrive
at dynamos of the $\alpha\Omega$ type. Models of this type have been
investigated also by Steenbeck and Krause (1969a) and applied to explain
essential features of the magnetic solar cycle. The case in which both the
$\alpha$ and the $\Omega$ effect contribute markedly to the generation of the
toroidal field is often labeled as mechanism of $\alpha^{2}\Omega$ type. In
the course of time, a large number of dynamo models of the mentioned types
have been investigated (see, e.g., Rädler 1986).
Mean-field dynamos require not necessarily the $\alpha$ effect. It is easy to
see that there is no dynamo due to the $\delta$ effect alone. However, the
combination of the $\delta$ and $\Omega$ effects is, as demonstrated by Rädler
(1969b, 1976), capable of dynamo action. Meanwhile there are several results
for dynamo models of that type (see, e.g, Rädler 1974, 1986). Also the
combination of another effect which can be described as an anisotropy of the
mean-field conductivity with the $\Omega$ effect can lead to a dynamo (Rädler
1986).
So far we focussed attention on dynamos with axisymmetric mean fields, which
allow a simple description. The mechanisms mentioned here work, however, also
with non-axisymmetric mean fields, and there are quite a few cases in which
such fields are easier to excite than axisymmetric ones (see, e.g., Rädler
1986).
Rogachevskii and Kleorin (2004) claimed that the induction effects that occur
in a turbulent fluid under the influence of a global shear, which are similar
to those in the $\delta\Omega$ mechanism, are also capable of dynamo action,
and they spoke of a “shear-current dynamo”. They calculated the relevant mean-
field coefficients however in a defeasible approximation. Several
investigations on this ”shear-current dynamo” have been carried out, but no
reliable proof for its existence has been given so far.
It has been shown for a wide range of assumptions that the magnetic mean-field
diffusivity $\eta_{\rm m}=\eta+\beta$ is positive so that a growth of a mean
magnetic field due to negative mean diffusivity can be excluded. There is
however a very recent result which might provoke scruples in this respect.
Some properties of mean-field dynamos are reflected by mean-field models
derived from the dynamos investigated by Roberts (1972), working not with
turbulence but with regular three-dimensional flows periodic in, say, $x$ and
$y$ and independent of $z$. It has been shown very recently by Devlen et al.
(2013) that in one of these mean-field models growing mean magnetic fields are
generated with no other mean-field induction effect than a negative mean-field
diffusivity. It remained open whether this result can be extended to mean-
field dynamos working with real turbulence.
### 2.5 Calculation of mean-field coefficients
#### 2.5.1 Second-order correlation approximation
So far we have have considered connections of the mean electromotive force
$\cal{E}$ with the mean magnetic field $\overline{{\mbox{\boldmath$B$}}}$ and
its spatial derivatives that are defined by coefficients like $\alpha$,
$\beta$, $\alpha_{1}$ or $\delta$, but nothing has been said about their
actual values or their dependence on the magnitude or other properties of the
fluctuating motions. In the early days of mean-field electrodynamics many
calculations of $\cal{E}$ were carried out on the basis of equation (12) for
the magnetic fluctuations $b$ but with the $\epsilon$ term canceled. This is
an approximation that can be justified for sufficiently small velocity
fluctuations $u$ only, often called “second-order correlation approximation”
(SOCA) or “first-order smoothing approximation” (FOSA) or “quasilinear
approximation”.
Consider as an example again the case in which the mean velocity
$\overline{{\mbox{\boldmath$U$}}}$ vanishes and the fluctuating velocity $u$
corresponds to a homogeneous isotropic turbulence. Restrict attention further
to small variations of the mean magnetic field
$\overline{{\mbox{\boldmath$B$}}}$ in space and time, that is, ideal scale
separation, and to the in applications most important high-conductivity limit,
defined by $\eta\tau_{\rm c}/\lambda_{\rm c}^{2}\ll 1$, where $\tau_{\rm c}$
and $\lambda_{\rm c}$ are correlation time and correlation length of the
turbulent motions. Within the frame of SOCA we find then
$\displaystyle\alpha$
$\displaystyle=-{\textstyle\frac{1}{3}}\int_{0}^{\infty}\overline{{\mbox{\boldmath$u$}}(t)\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}}(t-\tau))}\,\mbox{d}\tau$
(27) $\displaystyle\beta$
$\displaystyle={\textstyle\frac{1}{3}}\int_{0}^{\infty}\overline{{\mbox{\boldmath$u$}}(t)\cdot{\mbox{\boldmath$u$}}(t-\tau)}\,\mbox{d}\tau\,.$
We may write this also in the form
$\alpha=-{\textstyle{1\over
3}}\overline{{\mbox{\boldmath$u$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})}\,\tau^{(\alpha)}\,,\quad\beta={\textstyle{1\over
3}}\overline{u^{2}}\,\tau^{(\beta)}$ (28)
with $\tau^{(\alpha)}$ and $\tau^{(\beta)}$ defined by equating the
corresponding expressions in (27) and (28). Under reasonable assumptions both
$\tau^{(\alpha)}$ and $\tau^{(\beta)}$ are approximately equal to $\tau_{\rm
c}$. These results are in many respects instructive. In the high-conductivity
limit considered here, however, the application of SOCA can only readily be
justified if the Strouhal number $St=u_{\rm c}\tau_{\rm c}/\lambda_{\rm c}$,
with $u_{\rm c}$ being a characteristic value of $u$, is small compared with
unity. In realistic cases of turbulence it is close to unity.
It is well possible to proceed from the second-order approximation to a third-
order one with $\epsilon$ expressed by second-order results, then to a fourth-
order one with $\epsilon$ expressed by third-order results etc., and it has
been proven that this procedure converges (Krause 1968). Analytic calculations
of that kind are however very tedious and, apart from a few fourth-order
results, no results of practical interest have been gained in this way.
#### 2.5.2 Test-field method
Several other techniques for obtaining results for mean-field coefficients
have been proposed, using assumptions which look to a certain extent plausible
but cannot be justified in a clean way (for a critical review see, e.g.,
Rädler and Rheinhardt 2007). In recent years, with growing possibilities of
numerical calculations, the “test-field method”, established immediately on
the basic equations, brought much progress in the reliable determination of
mean-field coefficients.
The method was developed by Schrinner et al. (2005, 2007) in the context of
this task: Consider a simple geodynamo model, with the magnetic field
maintained by convection. Define mean fields by averaging over the azimuthal
coordinate; they are then axisymmetric. Extract the mean-field coefficients
from the numerical results for this model. Construct a mean-field model with
these coefficients. Compare then the mean fields obtained from the original
model by direct numerical simulations with those obtained from the mean-field
model. In the ideal case they should agree with each other.
Let us sketch the idea of the test-field method for the case of the simple
connection of the mean electromotive force $\cal{E}$ with
$\overline{{\mbox{\boldmath$B$}}}$ and its first spatial derivatives as given
by (16). We choose a set of test fields $\overline{{\mbox{\boldmath$B$}}}^{\rm
T}$ and replace $\overline{{\mbox{\boldmath$B$}}}$ in (12) consecutively by
each of its elements, calculate the corresponding ${\mbox{\boldmath$b$}}^{\rm
T}$ and finally ${\mbox{\boldmath$\cal{E}$}}^{\rm
T}=\overline{{\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}^{\rm T}}$. These
${\mbox{\boldmath$\cal{E}$}}^{\rm T}$ have to obey
$a_{ij}\overline{B}_{j}^{\rm T}+b_{ijk}\partial\overline{B}_{j}^{\rm
T}/\partial x_{k}={\cal{E}}_{i}^{\rm T}\,.$ (29)
With a sufficient number of independent $\overline{{\mbox{\boldmath$B$}}}^{\rm
T}$ we obtain a system of equations which allows us the determination of the
$a_{ij}$ and $b_{ijk}$ from the ${\cal{E}}_{i}^{\rm T}$ calculated for the
chosen set of $\overline{{\mbox{\boldmath$B$}}}^{\rm T}$. It turned out that
there is a high degree of freedom in the choice of the test fields. They need
not to be solenoidal and have not to satisfy specific boundary conditions.
Let us return once more to the coefficients $\alpha$ and $\beta$ for
homogeneous isotropic turbulence. Referring to numerical simulations of
hydrodynamic turbulence in a weakly compressible fluid, Sur et al. (2008) used
the test-field method for the determination of these coefficients. The
turbulence was specified to have an energy input at a wavenumber $k_{\rm f}$,
and to be maximally helical, that is,
$\overline{({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})^{2}}/\overline{{\mbox{\boldmath$u$}}^{2}}=k_{\rm
f}^{2}$. Calculations with different values of the hydrodynamic Reynolds
number $Re=u_{\rm rms}/\nu k_{\rm f}$, where $\nu$ means the kinematic
viscosity, were carried out. In Figs.1 and 2 some results for
$\alpha/\alpha_{0}$ and $\beta/\beta_{0}$ with $\alpha_{0}=-{\textstyle{1\over
3}}u_{\rm rms}$ and $\beta_{0}={\textstyle{1\over 3}}u_{\rm rms}/k_{\rm f}$
are shown in dependence on the magnetic Reynolds number $Rm=u_{\rm rms}/\eta
k_{\rm f}$. In the turbulence considered here the Strouhal number $St$ turned
out to be of the order of unity. So the reported results confirm that (27) and
(28), which were derived for $St\ll 1$ only, apply also with realistic values
of $St$.
The test-field method for the determination of the mean-field coefficients
brought much progress in mean-field electrodynamics and beyond. It has been
extended to a very broad range of assumptions, is in particular not limited to
cases with scale separation (see, e.g., Brandenburg et al. 2008, Rheinhardt
and Brandenburg 2010, 2012).
Figure 1: Normalized mean-field coefficients $\alpha/\alpha_{0}$ and
$\beta/\beta_{0}$ as functions of $Rm$, obtained in test-field calculations by
Sur et al. (2009) based on turbulence simulations with $Re=2.2$ Figure 2:
Same as Fig.1 but simulations with $Re=10\,Rm$
### 2.6 Imperfect scale separation
#### 2.6.1 Apparent discrepancies
In the examples considered so far we have reduced the general representations
(14) or (15) of the mean electromotive force $\cal{E}$ as a convolution
depending on the mean magnetic field $\overline{{\mbox{\boldmath$B$}}}$ in all
space and at the current time and all past times, to the simple local and
instantaneous connection (16) of $\cal{E}$ with
$\overline{{\mbox{\boldmath$B$}}}$ and its first spatial derivatives. On this
level the theory may deliver incomplete or even wrong statements.
One example for that is the aforementioned incomplete agreement of the mean-
field geodynamo models derived immediately from the basic equations and those
constructed with mean-field coefficients determined by the simple version of
the test-field method, which considers only local and instantaneous
connections of $\cal{E}$ with $\overline{{\mbox{\boldmath$B$}}}$ and its first
derivatives (section 2.5.2).
#### 2.6.2 Memory effect
Another interesting example concerns the growth of a mean magnetic field in a
turbulence showing $\alpha$ effect. As Hubbard and Brandenburg (2009) pointed
out, the growth rates obtained in direct numerical simulations clearly differ
from those derived from a dispersion relation with mean-field coefficients
gained in a static approximation, that is, assuming an instantaneous
connection of $\cal{E}$ and $\overline{{\mbox{\boldmath$B$}}}$ as in (16). The
difference disappears if a proper connection of $\cal{E}$ at a given time with
$\overline{{\mbox{\boldmath$B$}}}$ at former times, that is, some memory of
the turbulent system, is taken into account. We know meanwhile many examples
in which such memory effects play an important role and can even be crucial
for the existence of dynamos (Rheinhardt et al. 2014).
For an illustration of the memory effect, Hubbard and Brandenburg (2009)
considered a Roberts flow instead of a real turbulence. They assumed
${\mbox{\boldmath$u$}}\\!=\\!-{\mbox{\boldmath$e$}}\times{\mbox{\boldmath$\nabla$}}\psi+k_{\rm
f}\psi\,{\mbox{\boldmath$e$}}$ and $\psi=(u_{0}/k_{0})\cos k_{0}x\cos k_{0}y$,
where $e$ means the unit vector in $z$ direction and $u_{0}$, $k_{\rm f}$ and
$k_{0}$ are constants, and they restricted attention on the case of a maximal
modulus of the relative helicity
$\overline{{\mbox{\boldmath$u$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})}/\overline{{\mbox{\boldmath$u$}}^{2}}k_{\rm
f}$, which occurs with $k_{\rm f}=\sqrt{2}k_{0}$. They further defined mean
fields by averaging over all $x$ and $y$. Fig 3 shows the normalized growth
rates $\lambda/\lambda_{0}$, with $\lambda_{0}=u_{\rm rms}k_{\rm f}$, as
functions of $Rm$, obtained (i) in direct numerical simulations and (ii) from
the dispersion relation with mean-field coefficients determined in a static
approximation. Note the substantial deviations of the two results for large
$Rm$.
Figure 3: Normalized growth rates $\lambda/\lambda_{0}$ of a mean magnetic
field in a Roberts flow as functions of $Rm$, (i) obtained in direct numerical
simulations and (ii) calculated from a dispersion relation with mean-field
coefficients determined in a static approximation, according to Hubbard and
Brandenburg (2009)
## 3 Mean-field magnetohydrodynamics
### 3.1 Momentum balance and consequences
So far the fluid velocity has been considered as prescribed. We now relax this
assumption and use in addition to the electromagnetic equations (1) and (2),
or the induction equation (3), also the momentum balance. For the sake of
simplicity we restrict ourselves to an incompressible fluid. Admitting a
rotating frame of reference we have then
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\varrho(\partial_{t}{\mbox{\boldmath$U$}}+({\mbox{\boldmath$U$}}\cdot{\mbox{\boldmath$\nabla$}}){\mbox{\boldmath$U$}})=-{\mbox{\boldmath$\nabla$}}P+\varrho\nu{\mbox{\boldmath$\nabla$}}^{2}{\mbox{\boldmath$U$}}-2\varrho{\mbox{\boldmath$\Omega$}}\times{\mbox{\boldmath$U$}}$
$\displaystyle\quad+(1/\mu)({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$B$}})\times{\mbox{\boldmath$B$}}+{\mbox{\boldmath$F$}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$U$}}=0\,.$
(30)
Here $\varrho$ means the mass density, $\nu$ again the kinematic viscosity of
the fluid, and $P$ the hydrodynamic pressure. The angular velocity $\Omega$
defines the rotation of the frame and so the Coriolis force, and $F$ stands
for any external force. The inertial term in (30) is balanced by the pressure
gradient, the viscous force, the Coriolis force, the Lorentz force and
possibly some external force.
Let us focus attention again on turbulent situations. Taking then the average
not only of the induction equation (3) but also of the momentum balance (30),
we find in addition to the mean-field induction equation (10) with the mean
electromotive force $\cal{E}$ given by (11) the mean-field version of the
momentum balance,
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\varrho(\partial_{t}\overline{{\mbox{\boldmath$U$}}}+(\overline{{\mbox{\boldmath$U$}}}\cdot{\mbox{\boldmath$\nabla$}})\overline{{\mbox{\boldmath$U$}}})=-{\mbox{\boldmath$\nabla$}}\overline{P}+\varrho\nu{\mbox{\boldmath$\nabla$}}^{2}\overline{{\mbox{\boldmath$U$}}}-2\varrho{\mbox{\boldmath$\Omega$}}\times\overline{{\mbox{\boldmath$U$}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!+(1/\mu)({\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}})\times\overline{{\mbox{\boldmath$B$}}}+\overline{{\mbox{\boldmath$F$}}}+{\mbox{\boldmath$\cal{F}$}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot\overline{{\mbox{\boldmath$U$}}}=0\,,$
(31)
with a mean ponderomotive force $\cal{F}$,
${\mbox{\boldmath$\cal{F}$}}=-\varrho\overline{({\mbox{\boldmath$u$}}\cdot{\mbox{\boldmath$\nabla$}}){\mbox{\boldmath$u$}}}+(1/\mu)\overline{({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$b$}})\times{\mbox{\boldmath$b$}}}\,.$
(32)
If we ignore the magnetic field we return to pure hydrodynamics. The mean
ponderomotive force $\cal{F}$ covers then, for example, the contribution of
the turbulence to the mean-field viscosity, often discussed as eddy viscosity,
further the $\Lambda$ effect, which, on a rotating body, may drive
differential rotation (Rüdiger 1989), or the anisotropic kinetic $\alpha$
effect (AKA effect, Frisch et al. 1987). We do not want to discuss these
subjects here but focus attention on cases with magnetic field.
Generalizing the considerations on the mean electromotive force $\cal{E}$
explained above, we see that both the electromotive force $\cal{E}$ and the
ponderomotive force $\cal{F}$ may be considered as functionals of fluctuations
${\mbox{\boldmath$u$}}^{(0)}$ and ${\mbox{\boldmath$b$}}^{(0)}$, for example
relating to the case of vanishing mean motion and mean magnetic field, of the
mean velocity $\overline{{\mbox{\boldmath$U$}}}$, the mean magnetic field
$\overline{{\mbox{\boldmath$B$}}}$ and their first spatial derivatives and
also of the angular velocity $\Omega$ that determines the Coriolis force.
These functionals are not necessarily linear in
$\overline{{\mbox{\boldmath$U$}}}$, $\overline{{\mbox{\boldmath$B$}}}$ or
$\Omega$.
Focussing attention on the mean electromotive force $\cal{E}$ we write once
again
${\mbox{\boldmath$\cal{E}$}}={\mbox{\boldmath$\cal{E}$}}^{(0)}+{\mbox{\boldmath$\cal{E}$}}^{(B)}$,
with a part ${\mbox{\boldmath$\cal{E}$}}^{(0)}$ independent of the mean
magnetic field $\overline{{\mbox{\boldmath$B$}}}$. In our short presentation
of mean-field electrodynamics we argued, considering purely hydrodynamic
background turbulence, that the part ${\mbox{\boldmath$\cal{E}$}}^{(0)}$ will
always decay to zero. Now we may no longer exclude magnetohydrodynamic
turbulence, and then this is no longer necessarily the case. A non-zero part
${\mbox{\boldmath$\cal{E}$}}^{(0)}$ of $\cal{E}$ corresponds to a battery. If
such a part exists, the mean-field induction equation is no longer homogeneous
in the mean magnetic field $\overline{{\mbox{\boldmath$B$}}}$ and has always
non-decaying solutions. In the absence of conditions that allow a mean-field
dynamo the magnitude of the corresponding mean magnetic fields is determined
by ${\mbox{\boldmath$\cal{E}$}}^{(0)}$. If a dynamo is possible, they may act
as seed fields.
### 3.2 Yoshizawa effect
An interesting example of a contribution to the mean electromotive force
independent of the mean magnetic field, ${\mbox{\boldmath$\cal{E}$}}^{(0)}$,
has been given by Yoshizawa in 1990. Let us consider magnetohydrodynamic
turbulence in the presence of a mean flow or on a rotating body, that is,
under the influence of the Coriolis force. Assuming originally homogeneous
isotropic turbulence and ideal scale separation in space and time, we may
expect that
${\mbox{\boldmath$\cal{E}$}}^{(0)}=c_{U}\overline{{\mbox{\boldmath$U$}}}+c_{W}{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$U$}}}+c_{\Omega}{\mbox{\boldmath$\Omega$}}$
(33)
with three coefficients $c_{U}$, $c_{W}$ and $c_{\Omega}$. If the turbulence
shows Galilean invariance, $c_{U}$ has to be zero. The two coefficients
$c_{W}$ and $c_{\Omega}$ must be pseudo scalars, and it turns out that they
are closely connected with the cross-helicity
$\overline{{\mbox{\boldmath$u$}}\cdot{\mbox{\boldmath$b$}}}$. The Yoshizawa
effect, that is, an electromotive force like (33) with nonzero $c_{W}$ or
$c_{\Omega}$, is capable of building up and maintaining a mean magnetic field.
Its strength depends on $c_{W}$ and $c_{\Omega}$. As explained above, it may
act as a seed field if there are conditions which allow a further growth of
mean magnetic fields. Of course, the production of cross-helicity in general
depends also on the mean magnetic field, and in that sense the mean
electromotive force under consideration, too, may depend on this mean magnetic
field.
### 3.3 $\alpha$ effect and $\alpha$ quenching
Let us return to the situation as considered in section 2.2, that is, no mean
motion, no Coriolis force and only small variations of
$\overline{{\mbox{\boldmath$B$}}}$ in space and time. Instead of purely
hydrodynamic turbulence we assume however now homogeneous isotropic
magnetohydrodynamic turbulence, for which $b$, like $u$, remains non-zero if
$\overline{{\mbox{\boldmath$B$}}}\to{\bf 0}$. Then $\cal{E}$ has to satisfy
again (20), that is
${\mbox{\boldmath$\cal{E}$}}=\alpha\,\overline{{\mbox{\boldmath$B$}}}-\beta\,({\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}})$.
Solving the equation governing $b$ under SOCA and that for $u$ under an
analogous approximation, further restricting ourselves to the high-
conductivity limit, $\eta\tau_{\rm c}/\lambda_{\rm c}^{2}\ll 1$, and the
analogous low-viscosity limit, $\nu\tau_{\rm c}/\lambda_{\rm c}^{2}\ll 1$, we
find
$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\alpha_{\rm
K}+\alpha_{\rm M}$ $\displaystyle\alpha_{\rm K}$ $\displaystyle=$
$\displaystyle-{\textstyle{1\over
3}}\overline{{\mbox{\boldmath$u$}}^{(0)}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}}^{(0)})}\,\tau^{(\alpha{\rm
K})}$ (34) $\displaystyle\alpha_{\rm M}$ $\displaystyle=$
$\displaystyle{\textstyle{1\over
3\mu\varrho}}\overline{{\mbox{\boldmath$b$}}^{(0)}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$b$}}^{(0)})}\,\tau^{(\alpha{\rm
M})}$
and
$\beta=\beta_{\rm K}={\textstyle{1\over
3}}\overline{{{\mbox{\boldmath$u$}}^{(0)}}^{2}}\,\tau^{(\beta)}\,.$ (35)
(see, e.g., Rädler and Rheinhardt 2007). Here ${\mbox{\boldmath$u$}}^{(0)}$
and ${\mbox{\boldmath$b$}}^{(0)}$ stand for $u$ and $b$ in the limit
$\overline{{\mbox{\boldmath$B$}}}\to{\bf 0}$, and $\tau^{(\alpha{\rm K})}$,
$\tau^{(\alpha{\rm M})}$ and $\tau^{(\beta)}$ are quantities approximately
equal to the correlation time $\tau_{\rm c}$. Within this framework the
$\alpha$ effect has in addition to the kinetic part, which is connected with
the mean kinetic helicity
$\overline{{\mbox{\boldmath$u$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})}$,
a magnetic part connected with the mean current helicity
$\overline{{\mbox{\boldmath$b$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$b$}})}=\overline{\mu{\mbox{\boldmath$j$}}\cdot{\mbox{\boldmath$b$}}}$.
Such a magnetic part has been first considered by Pouquet et al. in 1976.
Remarkably the coefficient $\beta$, which determines the mean-field
diffusivity, has no such magnetic contribution. If we put
${\mbox{\boldmath$b$}}^{(0)}=0$ we return to our old result for purely
hydrodynamic turbulence.
Let us now admit an arbitrarily strong mean magnetic field. It causes an
anisotropy of the turbulence such that the tensor $a_{ij}$ in (16) has the
structure $\alpha_{1}\delta_{ij}+\alpha_{2}e_{i}e_{j}$, where $\alpha_{1}$ and
$\alpha_{2}$ may depend on $|\overline{{\mbox{\boldmath$B$}}}|$, and $e$
stands for the unit vector in the direction of
$\overline{{\mbox{\boldmath$B$}}}$. Considering then (16) but ignoring, for
simplicity, the terms with derivatives of $\overline{{\mbox{\boldmath$B$}}}$,
we find again
${\mbox{\boldmath$\cal{E}$}}=\alpha\,\overline{{\mbox{\boldmath$B$}}}$ with
$\alpha=\alpha_{1}+\alpha_{2}$ being a function of
$|\overline{{\mbox{\boldmath$B$}}}|$. In general we expect a reduction of the
modulus of $\alpha$ with growing $|\overline{{\mbox{\boldmath$B$}}}|$. In this
case we speak of “$\alpha$ quenching”. It limits the growth of the mean
magnetic field and defines a saturation field strength.
The determination of the dependence of $\alpha$ on
$|\overline{{\mbox{\boldmath$B$}}}|$ is a complex task. A simple ansatz that
has been frequently discussed in the past reads
$\alpha=\frac{\alpha_{0}}{1+c\,\overline{B}^{2}/B^{2}_{eq}}\,,$ (36)
where $\alpha_{0}$ is the value of $\alpha$ in the limit
$\overline{{\mbox{\boldmath$B$}}}\to{\bf 0}$, further $c$ a dimensionless
positive constant and $B_{eq}$ the equipartition field strength defined such
that the energies stored in the fluctuating velocity field and in the mean
magnetic field are equal to each other, that is,
$B^{2}_{eq}=\mu\varrho\,\overline{u^{2}}$.
In 1992 Vainshtein and Cattaneo suggested on the basis of analytical
considerations and numerical calculations with an imposed magnetic field a
relation like (36) with $c\approx Rm$, where $Rm$ means again the magnetic
Reynolds number. In the solar convection zone, for example, $Rm$ takes values
of $10^{6}$ or even $10^{9}$, and $\overline{B}/B_{eq}$ values of the order of
unity. Then $\alpha$ would be very close to zero and we could not expect any
dynamo. Therefore this kind of quenching has been called “catastrophic
quenching”.
This finding has initiated many discussions and investigations. Considerable
progress has been made by investigating the simplest possible dynamo systems
with $\alpha$ effect in the nonlinear regime. A fully satisfactory theory of
this subject is, however, still missing.
One important issue in the recent investigations on $\alpha$ quenching are the
hypotheses that $\alpha$ is always a sum of a kinetic part $\alpha_{\rm K}$
and a magnetic part $\alpha_{\rm M}$ and that the latter is determined by the
part of the mean current helicity due to the electric current and magnetic
field fluctuations,
$\overline{{\mbox{\boldmath$j$}}\cdot{\mbox{\boldmath$b$}}}$. The other
important issue is the role of the magnetic helicity in a dynamo. We recall
here that the magnetic helicity, say $H$, is defined as a volume integral over
the magnetic helicity density
$h={\mbox{\boldmath$A$}}\cdot{\mbox{\boldmath$B$}}$, where $A$ is a vector
potential of the magnetic field $B$, that is
${\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$A$}}={\mbox{\boldmath$B$}}$.
If the electromagnetic fields satisfy specific conditions at the surface of
this volume, in particular the magnetic field does not intersect this surface,
$H$ is independent of the special choice of the vector potential $A$, that is,
under gauge transformations of $A$. Then the basic equations imply further
that, in the limit of infinite conductivity, $H$ is a conserved quantity, that
is, does not change in time. Within the mean-field concept the magnetic
helicity density $h$ is the sum of two parts, one originating from the mean
magnetic field $\overline{{\mbox{\boldmath$B$}}}$ and the other from the
fluctuating part of the magnetic field, $b$. The mean part of the latter,
$\overline{{\mbox{\boldmath$a$}}\cdot{\mbox{\boldmath$b$}}}$ with
${\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$a$}}={\mbox{\boldmath$b$}}$,
is closely related to the magnetic contribution $\alpha_{\rm M}$ to $\alpha$,
which is, as explained above, determined by the part
$\overline{{\mbox{\boldmath$j$}}\cdot{\mbox{\boldmath$b$}}}=(1/\mu)\overline{{\mbox{\boldmath$b$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$b$}})}$
of the mean current helicity. If, for example, in the limit of infinite
conductivity $H$ is initially equal to zero and the mean magnetic field grows,
the modulus of $\alpha_{\rm M}$ must grow, too.
With the hypothesis $\alpha=\alpha_{\rm K}+\alpha_{\rm M}$, further the
evolution equation of the mean magnetic helicity density due to fluctuations,
and a few plausible assumptions an evolution equation for $\alpha_{\rm M}$,
$\partial_{t}\alpha_{\rm M}=-2\eta_{\rm t}k_{\rm
f}^{2}\Big{(}\frac{{\mbox{\boldmath$\cal{E}$}}\cdot\overline{{\mbox{\boldmath$B$}}}}{B_{eq}^{2}}+\frac{\alpha_{\rm
M}}{Rm}\Big{)}-{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$F$}}\,,$ (37)
has been derived (see, e.g., Hubbard and Brandenburg 2011). As usual in this
context, we write here $\eta_{\rm t}$ instead of $\beta$, and $k_{\rm f}$
denotes again the wavenumber of the energy-carrying scale in the turbulence.
$\cal{E}$ should be specified to be equal to $(\alpha_{\rm K}+\alpha_{\rm
M})\overline{{\mbox{\boldmath$B$}}}-\eta_{t}{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}$.
As above, $B_{eq}$ is the equipartition field strength, and $F$ means a mean
magnetic helicity flux. In simple models with periodic boundary conditions the
term ${\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$F$}}$ does not change
the total mean magnetic helicity inside a dynamo volume. In general, however,
the mean magnetic helicity flux plays a crucial role, and expressions for $F$
have been elaborated which depend, for example, on differential rotation.
Models incorporating such results reflect indeed many properties of dynamos in
the non-linear regime including saturation field strengths (see, e.g.,
Brandenburg and Subramanian 2005, Hubbard and Brandenburg 2011,2012, Del Sordo
et al. 2013).
## 4 Laboratory experiments
The development of dynamo theory was accompanied and supported by several
laboratory experiments. As early as in 1967, one year after the first paper
about this subject, the $\alpha$ effect has been demonstrated in a liquid
sodium flow in the Institute of Physics in Riga (Steenbeck et al. 1967). The
measurements were carried out at the so-called “$\alpha$ box”, in which a
proper flow geometry has been organized by baffles.
Already at this time there were many discussions on the realization of a
dynamo in a conducting fluid. It was clear from the very beginning that such
an experiment requires a large fluid volume and high flow rates. Only in the
last days of the last century, in December 1999, after expensive preparations,
two dynamos ran successfully with liquid sodium flows, one in Riga (Gailitis
et al. 2000) and one in Karlsruhe (Müller and Stieglitz 2000, 2002). The first
one (Riga) is clearly different from a mean-field dynamo, but the second one
(Karlsruhe) can be well understood as a mean-field dynamo of $\alpha^{2}$ type
(see Rädler et al. 2002).
I do not want to go into the details of these experiments but add a more
general remark on the sometimes underestimated practical value of basic
research. We have learned in geophysically or astrophysically motivated
studies that the self-excitation of magnetic fields in moving electrically
conducting fluids is possible as soon as the magnetic Reynolds number
$Rm=UL/\eta$, with $U$ and $L$ being typical values of fluid velocity and
linear dimensions of the considered device, exceeds a critical value, which
depends on the flow geometry and lies in all investigated cases above unity.
For a long time situations of that kind did not appear in laboratories or in
industrial devices. In the sixties and seventies of the last century, however,
big fast breeder reactors were built with huge circuits of liquid sodium,
which transport the heat produced in the active zone to the places where it is
transformed into electric power. Such devices imply indeed the possibility of
self-excitation of magnetic fields, what constitutes a big danger. These
fields could quickly grow, heavily hamper the sodium flow and so seriously
disturb the whole reactor regime or even cause a catastrophy. At first the
reactor engineers were not aware of that. It was the people thinking about
cosmic dynamos who pointed out this danger. Max Steenbeck, as a Foreign Member
of the Soviet Academy, presented there in 1971 a corresponding memorandum and
initiated so investigations in this field and measures to avoid this danger.
Some years later, independent of that, two papers of British scientists about
this topic have appeared (Bevir 1973, Pierson 1975). This development fostered
the willingness of authorities to support theoretical and experimental
research on dynamos and related topics.
## References
* [Babcock (1947)] Babcock, H.W.: 1947, ApJ, 105, 105
* [Bevir (1973)] Bevir, M.K.: 1973, Journ. British Nuclear Soc., 121, 455
* [Braginsky(1964)] Braginsky, S.I.: 1964, Sov. Phys. JETP 20, 1462
* [Brandenburg(2005)] Brandenburg, A., Subramanian, K.: 2005, Phys. Rep., 417, 1
* [Brandenburg(2008)] Brandenburg, A., Rädler, K.-H., Schrinner, M.: 2008, A&A, 482, 739
* [Brandenburg(2012)] Brandenburg, A., Sokoloff, D., Subramanian, K.: 2012, Space Sci. Rev., 169, 123
* [Bullard(1954)] Bullard, E.C., Gellman, H.: 1954, Phil. Trans. R. Soc. A 247, 213
* [Cowling(1934)] Cowling, T.G.: 1934, MNRAS 94, 39
* [Del Sordo (2013)] Del Sordo, F., Guerrero, G., Brandenburg, A.: 2013, MNRAS, 429, 1686
* [Devlen(2013)] Devlen, E., Brandenburg, A., Mitra, D.: 2013, MNRAS 432, 1651
* [Elsassser(1946)] Elsasser, W.M.: 1946, Phys. Rev., 69, 106
* [Frisch(1987)] Frisch, U., She, Z.S., Sulem, P.L.: 1987, Physica, 28, 382
* [Gailitis(2000)] Gailitis, A., Lielausis, O., Dementev, S., Platacis, E., Cifersons, A., Gerbeth, G., Gundrum, T., Stefani, F., Christen, M., Hänel, H., Will, G.: 2000, Phys. Rev. L., 84, 4365
* [Herzenberg (1958)] Herzenberg, A.: 1958, Phil. Trans. R. Soc. London A, 250, 543
* [Hubbard(2009)] Hubbard, A., Brandenburg, A.: 2009, ApJ, 706, 712
* [Hubbard(2011)] Hubbard, A., Brandenburg, A.: 2011, ApJ, 727, 11
* [Hubbard(2012)] Hubbard, A., Brandenburg, A.: 2012, ApJ 748, 51
* [Kazantsev(1968)] Kazantsev, A.P.: 1968, Sov. Phys. JETP, 16, 1031
* [Krause(1968)] Krause, F.: 1968, ZAMM, 48, 333
* [Krause(1980)] Krause, F., Rädler, K.-H.: 1980, Mean-Field Magnetohydrodynamics and Dynamo Theory (Pergamon Press Oxford)
* [Larmor(1919)] Larmor, J.: 1919, Rep. Brit. Assoc. Adv. Sc. 1919, 159
* [Moffatt(1919)] Moffat, H.K.: 1978, Magnetic Field Generation in Electrically Conducting Fluids (Cambridge University Press)
* [Müller (2000)] Müller, U., Stieglitz, R.: 2000, Naturwissenschaften, 87, 381
* [Müller (2002)] Müller, U., Stieglitz, R.: 2002, Nonlinear Processes in Geophysics, 9, 165
* [Parker (1955)] Parker, E.N.: 1955, Phil. Trans. R. Soc. London A, 250, 543
* [Parker (1957)] Parker, E.N.: 1957, Proc. N.A.S., 43, 8
* [Pierson (1975)] Pierson, E.S.: 1975, Nuclear Science and Engineering, 57, 155
* [Pouquet (1976)] Pouquet, A., Frisch, U., Leorat, J.: 1976, J. Fluid Mech., 77, 321
* [Rädler (1968a)] Rädler, K.-H.: 1968a, Z. Naturforsch. 23a, 1841
* [Rädler (1968b)] Rädler, K.-H.: 1968b, Z. Naturforsch. 23a, 1851
* [Rädler (1969a)] Rädler, K.-H.: 1969a, Monatsber. Dt. Akad. Wiss, 11, 194
* [Rädler (1969b)] Rädler, K.-H.: 1969b, Monatsber. Dt. Akad. Wiss, 11, 272
* [Rädler(1976)] Rädler K.-H.: 1976, in: Bumba, V., Kleczek, J. (eds.), Basic Mechanisms of Solar Activity, Proceedings IAU Symposium No. 71 (D. Reidel Publishing Company Dordrecht) p.323
* [Rädler (1986)] Rädler, K.-H.: 1986, AN 307, 89
* [Rädler et al. (2000)] Rädler, K.-H.: 2000, in: Page, D., Hirsch, J.G. (eds.), From the Sun to the Great Attractor (1999 Guanajuato Lecture in Astrophysics), p.101
* [Rädler (2002)] Rädler, K.-H., Rheinhardt, R., Apstein, E., Fuchs, H.: 2002, Magnetohydrodynamics, 38, 39
* [Rädler et al. (2007)] Rädler, K.-H., Rheinhardt, M.: 2007, Geophys. Astrophys. Fluid Dyn., 101, 117
* [Rheinhardt et al. (2010)] Rheinhardt, M., Brandenburg, A.: 2010, A&A, 520, A28/1-16
* [Rheinhardt et al. (2012)] Rheinhardt, M., Brandenburg, A.: 2012, Astron.Nachr., 333, 71
* [Rheinhardt et al. (2014)] Rheinhardt, M., Devlen, E., Rädler, K.-H., Brandenburg, A.: 2014, submitted to MNRAS, arXiv:1401.5026
* [Rüdiger (1989)] Rüdiger, G.: 1989, Differential Rotation and Stellar Convection (Akademie-Verlag Berlin)
* [Schrinner et al. (2005)] Schrinner, M., Rädler, K.-H., Schmitt, D., Rheinhardt, M., Christensen, U.R.: 2005, AN 326, 245
* [Schrinner et al. (2007)] Schrinner, M., Rädler, K.-H., Schmitt, D., Rheinhardt, M., Christensen, U.R.: 2007, Geophys. Astrophys. Fluid Dyn., 101, 81
* [Steenbeck et al. (1966)] Steenbeck, M., Krause, F., Rädler, K.-H.: 1966, Z. Naturforsch. 21a, 369
* [Steenbeck et al. (1967)] Steenbeck, M., Kirko, I.M., Gailitis, A., Klawina, A.P., Krause, F., Laumanis, I.J., Lielausis, O.A.: 1967, Mber. Dt. Akad. Wiss., 9, 714
* [Steenbeck et al. (1969a)] Steenbeck, M., Krause, F.: 1969a, AN 291, 271
* [Steenbeck et al. (1969b)] Steenbeck, M., Krause, F.: 1969b, AN, 291, 495
* [Sur et al. (2008)] Sur, S., Brandenburg, A., Subramanian, K.: 2008, MNRAS, 385, l15
* [Sweet (1950)] Sweet, P.: 1950, MNRAS, 110, 69
* [Vainshtein (1992)] Vainsthtein, S.I., Cattaneo, F.: 1992, ApJ, 393, 165
* [Yoshizawa (1990)] Yoshizawa, A.: 1990, PhFl B, 2, 1589
* [Zeldovich (1956)] Zeldovich, Y.B.: 1956, J. Exptl. Theoret. Phys., 31, 160
|
arxiv-papers
| 2014-02-26T14:43:44 |
2024-09-04T02:49:58.945858
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K.-H. R\\\"adler",
"submitter": "Karl-Heinz Raedler",
"url": "https://arxiv.org/abs/1402.6557"
}
|
1402.6601
|
11institutetext: Univ. Grenoble Alpes, France 22institutetext: Inria Rhône-
Alpes, France 33institutetext: Institut universitaire de France
44institutetext: Federal University of Rio Grande do Sul (UFRGS), Porto
Alegre, Brazil
# Scheduling data flow program in XKaapi:
A new affinity based Algorithm for Heterogeneous Architectures
Raphaël Bleuse 11 Thierry Gautier 22 João V. F. Lima 44 Grégory Mounié 11
Denis Trystram 1133
###### Abstract
Efficient implementations of parallel applications on heterogeneous hybrid
architectures require a careful balance between computations and
communications with accelerator devices. Even if most of the communication
time can be overlapped by computations, it is essential to reduce the total
volume of communicated data. The literature therefore abounds with ad hoc
methods to reach that balance, but that are architecture and application
dependent. We propose here a generic mechanism to automatically optimize the
scheduling between CPUs and GPUs, and compare two strategies within this
mechanism: the classical Heterogeneous Earliest Finish Time (HEFT) algorithm
and our new, parametrized, Distributed Affinity Dual Approximation algorithm
(DADA), which consists in grouping the tasks by affinity before running a fast
dual approximation. We ran experiments on a heterogeneous parallel machine
with six CPU cores and eight NVIDIA Fermi GPUs. Three standard dense linear
algebra kernels from the PLASMA library have been ported on top of the XKaapi
runtime. We report their performances. It results that HEFT and DADA perform
well for various experimental conditions, but that DADA performs better for
larger systems and number of GPUs, and, in most cases, generates much lower
data transfers than HEFT to achieve the same performance.
###### Keywords:
heterogeneous architectures, scheduling, cost models, dual approximation
scheme, programming tools, affinity
## 1 Introduction
With the recent evolution of processor design, the future generations of
processors will contain hundreds of cores. To increase the performance per
watt ratio, the cores will be non-symmetric with few highly powerful cores
(CPU) and numerous, but simpler, cores (GPU). The success of such machines
will rely on the ability to schedule the workload at runtime, even for small
problem instances.
One of the main challenges is to define a scheduling strategy that may be able
to exploit all potential parallelisms on a heterogeneous architecture composed
of multiple CPUs and multiple GPUs. Previous works demonstrate the efficiency
of strategies such as static distribution [14, 15], centralized list
scheduling with data locality [6], cost models [1, 2, 3, 4] based on
Heterogeneous-Earliest-Finish-Time scheduling (HEFT) [16], and dynamic for
some specific application domains [5, 10]. Locality-aware work stealing [9],
with a careful implementation to overlap communication by computation [13],
improves significantly the performance of compute-bound linear algebra
problems such as matrix product and Cholesky factorization.
Nevertheless, none of the above cited works considers scheduling strategies
from the viewpoint of a compromise between performance and locality. In this
paper, we propose a scheduling algorithm based on dual approximation [12] that
uses a performance model to predict the execution time of tasks during
scheduling decision. This algorithm, called Distributed Affinity Dual
Approximation (DADA), is able to find a compromise between transfers and
performance. It is parametrized by $\alpha$ for tuning this trade-off. The
main advantage of dual approximation algorithms is their theoretical
performance guarantee as they have a constant approximation ratio. On the
contrary, the worst case of HEFT can be arbitrarily bad [12].
We compare these two different scheduling strategies for data-flow task
programming. These strategies are implemented on top of the XKaapi scheduling
framework with performance models for task execution time and transfer
prediction. The contributions of this paper are first the design and
implementation of dual approximation scheduling algorithms (with and without
affinity) and second its evaluation in comparison to the well-known HEFT
algorithm on three dense linear algebra algorithms in double precision
floating-point operations from PLASMA [7]: namely Cholesky, LU, and QR. To our
knowledge, this paper is the first report of experimental evaluations studying
the impact of data transfer model and contention on a machine with up to $8$
GPUs.
The main lesson of this work is that scheduling algorithms need extra
information in order to take the right decisions. Such extra information could
be obtained in a precise communication model to predict processing time of
each task or in a more flexible information such as the affinity in DADA. Even
if HEFT remains a good candidate for scheduling such linear algebra kernels,
DADA is highly competitive against it for multi-GPU systems: the experimental
results demonstrate that it achieves the same range of performances while
reducing significantly the communication volume.
The remainder of this paper is organized as follows. Section 2 provides an
overview of XKaapi runtime system, describes the XKaapi scheduling framework
and the cost model applied for performance prediction. Section 3 details the
two studied scheduling strategies. Section 4 presents our experimental results
on a heterogeneous architecture composed of $12$ CPUs and $8$ GPUs. In Section
5 we briefly survey related works on runtime systems, scheduling strategies
and performance prediction. Finally, Section 6 concludes the paper and
suggests future directions.
## 2 Scheduling framework in XKaapi
The XKaapi111http://kaapi.gforge.inria.fr data-flow model [8] – as in Cilk,
Intel TBB, OpenMP-3.0, or OmpSs [6] – enables non-blocking task creation: the
caller creates the task and proceeds with the program execution. Parallelism
is explicit while the detection of synchronizations is implicit [8]:
dependencies between tasks and memory transfers are automatically managed by
the runtime.
XKaapi runtime is structured around the notion of _worker_ : it is the
internal representation of kernel thread. It executes the code of the tasks
takes local scheduling decisions. Each _worker_ owns a local queue of ready
tasks. Our interface is mainly inspired by work stealing scheduler and
composed of three operations that act on workers’ queues of tasks: _pop_ ,
_push_ and _steal_. In our previous work, we demonstrated the efficiency of
XKaapi locality-aware work stealing as well as the corresponding multi-GPU
runtime support [9] using specialized implementation of these operations. A
new operation, called _activate_ , has been defined to push ready task to a
worker’s queue.
### 2.1 Execution flow
The sketch of the execution mechanism is the following: at each step, either
the own queue of worker is not empty and the worker uses it; or the worker
emits a steal request to a randomly selected worker in order to get a task to
execute. According to the dependencies between tasks, once a worker performs a
task, it calls the _activate_ operations in order to activate the successors
of the task which become ready for execution.
The XKaapi runtime gets information from each internal events (such as start-
end of task execution, or start-end of communication to GPU) to calibrate the
performance model and corrects erroneous predictions due to unpredictable or
unknown behavior (e.g. operating system state or I/O disturbance). StarPU [4]
uses similar runtime measurements in order to correct the performance
predictions in their HEFT implementation.
All of our scheduling strategies follow this sketch. Every worker terminates
its execution when all the tasks of the application have been executed.
### 2.2 Pop, Push, Steal and Activate Operations
A framework interface for scheduling strategies is not a new concept in
heterogeneous systems. Bueno et al. [6] and Augonnet et al. [4] described a
minimal interface to design scheduling strategies with selection at runtime.
However, there is little information available on the comparison of different
strategies. Most of them reported performance on centralized list scheduling
and performance models. Our framework is composed of three classical
operations in work stealing context, plus an action to activate tasks when
predecessors have completed.
* •
The _push_ operation inserts a task into a queue. A worker can push a task
into any other workers’ queue.
* •
A _pop_ removes a task from the local queue owned by the caller worker.
* •
A _steal_ removes a task from the queue of a remote worker. It is called by an
idle thread – the _thief_ – in order to pick tasks from a randomly selected
worker – the _victim_.
* •
The _activate_ operation is called after the completion of a task. The role of
this operation is to allocate the tasks that are ready to be executed. Hence,
most of the scheduling decision are done during this operation.
### 2.3 Performance Model
Cost models depend on a certain knowledge of both application algorithm and
the underlying architecture to predict performance at runtime. In order to
predict performance, we designed a StarPU [3] like performance model for task
execution time and communication. Our task prediction relies on an history-
based model, and transfer time estimation is based on asymptotic bandwidth.
They are associated with scheduling strategies that are based on task
completion time such as HEFT and DADA with and without affinity.
In order to balance efficiently the load, for each processor XKaapi maintains
a shared time-stamp of the predicted date when it has completed its tasks. The
completion date of the last executed task is also kept. The update and
incrementation of the time-stamps are efficiently implemented with atomic
operators.
## 3 Scheduling Strategies
This section introduces the scheduling strategies designed on top of the
XKaapi scheduling framework. We consider a multi-core parallel architecture
with $m$ homogeneous CPUs and $k$ homogeneous GPUs. First, we describe our
implementation of HEFT [16]. Then, we recall the principle of the dual
approximation scheme [11]. We propose a new algorithm – Distributed Affinity
Dual Approximation (DADA) – based on this paradigm which takes into account
the affinity between tasks.
In the following we denote by $p^{CPU}_{i}$ the processing time of task
$T_{i}$ on a CPU and $p^{GPU}_{i}$ on a GPU. We define the speedup $S_{i}$ of
task $T_{i}$ as the ratio $S_{i}=p^{CPU}_{i}\\!/\,p^{GPU}_{i}$.
### 3.1 HEFT within XKaapi
The Heterogeneous Earliest-Finish-Time algorithm (HEFT), proposed by [16], is
a scheduling algorithm for a bounded number of heterogeneous processors. Its
time complexity is in $O(n^{2}\cdot(m+k))$. It has two major phases: _task
prioritizing_ and a _worker selection_. Our XKaapi version of HEFT implements
both phases during the _activate_ operation. The _task prioritizing_ phase
computes for all ready tasks $T_{i}$ its speedup $S_{i}$ relative to an
execution on GPU. Next, it sorts the list of ready tasks by decreasing
speedups. Whereas the original HEFT rule sorts the tasks by decreasing upward
rank (average path length to the end), our rule gives priority on minimizing
the sum of the execution times. In the _worker selection_ phase, the algorithm
selects tasks in the order of their speedup $S_{i}$ and schedules each task on
the worker which minimizes the completion time. Algorithm 1 describes the
basic steps of HEFT over XKaapi.
1
Input : A list of ready tasks $T_{i}$ LR
Output : Tasks $T_{i}$ pushed to the worker’s queues
2
3foreach _$T_{i}\in\textnormal{{LR}}$ _ do
4 $S_{i}\leftarrow p^{CPU}_{i}\\!/\,p^{GPU}_{i}$
5
6 end foreach
7Sort all ready tasks $T_{i}$ by decreasing speedup $S_{i}$
8 foreach _$T_{i}\in\textnormal{{LR}}$ _ do
9 Schedule $T_{i}$ on the worker $w_{j}$ achieving the earliest finish time
10 push of $T_{i}$ into queue of worker $w_{j}$
11 Update processor load time-stamps on worker $w_{j}$
12
13 end foreach
Algorithm 1 HEFT – _activate_ operation.
### 3.2 Dual Approximation and Affinity
#### 3.2.1 Dual Approximation
Let us recall first that a $\rho$-dual approximation scheduling algorithm
considers a guess $\lambda$ (which is an estimation of the optimal makespan)
and either delivers a schedule of makespan at most $\rho\lambda$ or answers
correctly that there exists no schedule of length at most $\lambda$ [11]. The
process is repeated by a classical binary search on $\lambda$ up to a
precision of $\epsilon$. We target $\rho=2$. The dual approximation part of
Algorithm 2 consists in the following steps:
* •
Choice of the initial guess $\lambda$ (lines 2 and 2);
* •
Extract the tasks which fit only into GPUs ($p^{CPU}_{i}>\lambda$), and
symmetrically those which are dedicated to CPUs (line 2);
* •
Keep this schedule if the tasks fit into $\lambda$ (line 2). Otherwise, reject
it if there is a task larger than $\lambda$ on both CPUs and GPUs (line 2);
* •
Add to the tasks allocated to the GPU those which have the largest speedup
$S_{i}$ up to overreaching the threshold $\lambda$ (line 2) which guarantees
the ratio $\rho=2$;
* •
Put all the remaining tasks in the $m$ CPUs and execute them using an
earliest-finish-time scheduling policy (line 2).
#### 3.2.2 Affinity
DADA builds a compromise taking into account both raw performance and
transfers. The principle consists in two successive phases: a first local
phase targeting the reduction of the communications through the abstraction
described below and a second phase which counter-balances the induced
serialization aiming at a global balance. Any algorithm optimizing the
makespan could be used for the second phase. We use a basic dual-
approximation. In order to gain a finer control, the length of the first phase
is controlled by a parameter (denoted by $\alpha$, $0\leq\alpha\leq 1$). A
value of $0$ for $\alpha$ means that the affinity is not taken into account:
DADA is then a basic dual-approximation. While at the opposite a value close
to $1$ allows a length up to $\lambda$ for the first phase, thus giving a
greater weight to affinity.
Each pair (task, computation resource) is given an affinity score. Maximizing
the score over the whole schedule enables to consider local impacts. The
affinity scores are computed using extra information of the runtime. In our
implementation, they were computed using the amount of data updated by each
task. For instance, a task that _writes_ or _modifies_ a data stored on a
resource $R$ has a high score and is prone to be scheduled on $R$.
Input : A list of ready tasks $T_{i}$ LR
Output : Tasks $T_{i}$ pushed to the worker’s queues
1 $lower\leftarrow 0$
2 $upper\leftarrow\sum_{i}max(p^{CPU}_{i},p^{GPU}_{i})$
3 while _$(upper-lower) >\epsilon$_ do
4 $\lambda\leftarrow(upper+lower)/\,2$
5
6 begin _local affinity phase_
7 Schedule tasks of LR per affinity score on its affinity processor, loading
each processor up to overreaching $\alpha\lambda$
8 end
9
10 begin _global balance phase_
11 Schedule LR to minimize finish time using $\lambda$ as hint
12 if _tasks do fit into $(2+\alpha)\lambda$_ then
13 $upper\leftarrow\lambda$
14 Keep current schedule
15
16 else
17 $lower\leftarrow\lambda$
18 Reject current schedule
19
20 end if
21
22 end
23
24 end while
25Push each task $T_{i}$ of LR on queue of worker $w_{j}$ based on the last
fitting schedule and update processor load time-stamps
Algorithm 2 DADA – _activate_ operation.
## 4 Experiments
### 4.1 Experimental setup: Platform & Benchmarks
#### 4.1.1 Platform
All experiments have been conducted on a heterogeneous, multi-GPU system. It
is composed of two hexa-core Intel Xeon X5650 CPUs running at 2.66 GHz with 72
GB of memory. It is enhanced with eight NVIDIA Tesla C2050 GPUs (Fermi
architecture) of 448 GPU cores (scalar processors) running at 1.15 GHz each
(2688 GPU cores total) with 3 GB GDDR5 per GPU (18 GB total). The machine has
$4$ PCIe switches to support up to $8$ GPUs. When $2$ GPUs share a switch,
their aggregated PCIe bandwidth is bounded by the one of a single PCIe 16x.
Experiments using up to $4$ GPUs avoid this bandwidth constraint by using at
most $1$ GPU per PCIe switch.
#### 4.1.2 Benchmarks
All benchmarks ran on top of a GNU/Linux Debian 6.0.2 squeeze with kernel
2.6.32-5-amd64. We compiled with GCC 4.4 and linked against CUDA 5.0 and the
library ATLAS 3.9.39 (BLAS and LAPACK). All experiments use the tile
algorithms of PLASMA [7] for Cholesky (DPOTRF), LU (DGETRF), and QR (DGEQRF).
The QUARK API [17] has been implemented and extended in XKaapi to support task
multi-specialization: the XKaapi runtime maintains the CPU and GPU versions
for each PLASMA task. At the task execution, our QUARK version runs the
appropriate task implementation in accordance with the worker architecture.
The GPU kernels of QR and LU are based on previous works from [1, 2] and
adapted from PLASMA CPU algorithm and MAGMA from [15]. Each running GPU
monopolizes a CPU to manage its worker. The remaining CPU cores are involved
in the application computations.
#### 4.1.3 Methodology
Each experiment has been executed at least 30 times for each set of parameters
and we report on all the figures (Fig. 1, 2, 3 and 4) the mean and the $95\%$
confidence interval. The factorizations have been done in double precision
floating-point operations with a PLASMA internal block (_IB_) of size $128$
and tiles of size $512$. For each of them, we plot the highest performance
obtained on various matrix sizes with the discussed scheduling strategies.
In the following, DADA($\alpha$) represents DADA parametrized by $\alpha$. We
denote by DADA($\alpha$)+CP the algorithm using Communication Prediction as
supplementary information. HEFT strategy always computes the earliest finish
time of a task taking into account the time to transfer data before executing
the task.
### 4.2 Impact of the affinity control parameter $\alpha$
This section highlights the impact of the affinity control parameter $\alpha$
on the compromise between performance and data transfers. The measures have
been done with the Cholesky decomposition on matrices of size $8192\times
8192$ and $16384\times 16384$. However, we present only results for the
smallest size as we observe similar behaviors for both matrix sizes.
Fig. 1 shows both performance (Fig. 1(a) and 1(b)) and total memory transfers
(Fig. 1(c) and 1(d)) for several values of $\alpha$ with respect to the number
of GPUs. Both metrics are shown without (Fig. 1(a) and 1(c)) and with (Fig.
1(b) and 1(d)) communication prediction taken into account. Once affinity is
considered (_i.e._ $\alpha\neq 0$), the higher the value of $\alpha$, the
better the policy scales. Using as little information as possible (_i.e._
DADA($0$) and no communication prediction), the policy performance does not
scale with more than two GPUs due to a too huge amount of transfers.
(a) Performance of DADA($\alpha$).
(b) Performance of DADA($\alpha$)+CP.
(c) Memory transfer of DADA($\alpha$).
(d) Memory transfer of DADA($\alpha$)+CP.
Figure 1: Impact of parameter $\alpha$ on Cholesky (DPOTRF) with matrix of
size $8192\times 8192$.
### 4.3 Comparison of scheduling strategies
We present in this section the results for the three kernels with matrix size
$8192\times 8192$. Other tested sizes have the same behavior. The idea is to
evaluate the behavior of each strategy with different work loads. Both
performance and data transfers of the policies introduced above: HEFT,
DADA($0$), DADA($\alpha$) and DADA($\alpha$)+CP are studied.
#### 4.3.1 Experimental evaluation
Fig. 2 reports the behavior of the Cholesky decomposition (DPOTRF) with
respect to the number of GPUs used. It studies both performance results (Fig.
2(a)) and total memory transfers (Fig. 2(b)). All scheduling algorithms have
similar performances. DADA($\alpha$)+CP slightly better scales with the number
of GPU. As expected DADA($\alpha$)+CP and DADA($\alpha$) are the policies with
the lowest bandwidth footprint up to 6 GPU. Yet, as the number of GPU grows,
the use of communication prediction allows to reduce the communication volume
with sustained high performances.
Fig. 3 reports the behavior of the LU factorization (DGETRF). It studies both
performance results (Fig. 3(a)) and total memory transfers (Fig. 3(b)). Apart
from the performance of DADA+CP for six CPUs and six GPUs (with a large
confidence interval), all scheduling policies sustain the same performance.
Data transfers seem to have a little impact on performance. However,
DADA($\alpha$)+CP generates less memory movements than other strategies.
DADA($0$) is the most costly policy while DADA($\alpha$) and HEFT have similar
impacts.
The total memory transfers have the same shape than for the Cholesky
factorization. Still, the gap between the curves is widening:
DADA($\alpha$)+CP is $3.5$ less demanding in bandwidth than HEFT for only a
slowdown of about $1.13$ in performance for 8 GPU.
Finally, Fig. 4 reports the behavior of the QR factorization (DGEQRF) with
respect to the number of GPUs used. Both performance results (Fig. 4(a)-) and
total memory transfers (Fig. 4(b)) are studied. All dual approximations
(DADA($0$), DADA($\alpha$), DADA($\alpha$)+CP) behave the same and are
outperformed by HEFT. Even the low transfer footprint of both DADA($\alpha$)
is not able to sustain performance. It seems that the dependencies between
tasks for QR factorization have a strong impact on the schedule computed by
all dual approximation algorithms. We are still investigating this particular
point.
(a) Performance ($8192\times 8192$).
(b) Memory Transfer ($8192\times 8192$).
Figure 2: Benchmarks of Cholesky (DPOTRF).
(a) Performance ($8192\times 8192$).
(b) Memory Transfer ($8192\times 8192$).
Figure 3: Benchmarks of LU (DGETRF).
(a) Performance ($8192\times 8192$).
(b) Memory Transfer ($8192\times 8192$).
Figure 4: Benchmarks of QR (DGEQRF).
#### 4.3.2 Discussion
##### Communication prediction
Affinity is a viable alternative to communication modeling. Indeed, DADA
without communication prediction is comparable to HEFT in terms of
performance. Moreover, affinity based policy combined with communication
prediction allows to reduce further more memory transfers (up to a factor
$3.5$ when compared to HEFT).
##### Comparison with work stealing scheduling algorithm
For the sake of completeness, we also tested the work stealing algorithm.
However we did not plot the results in previous figures for the sake of
readability. We briefly discuss them now. The naive work stealing algorithm is
cache unfriendly, especially with small matrices as its random choices are
heavily penalizing [9]. On the contrary, the affinity policies proposed here
are suitable for this case. When scheduling for medium and large matrix sizes,
the impact of modeling inaccuracies grows. Model oblivious algorithms such as
work-stealing behave well by efficiently overlapping communications and
computations while HEFT is induced in error by the imprecise communication
prediction. Hence, our approach is much more robust than work stealing and
HEFT since it does not need a too precise communication model and adapts well
to various matrix sizes.
## 5 Related Works
StarPU [4], OmpSs [6] and QUARK [17] are programming environments or libraries
that enables to automatically schedule tasks with data flow dependencies.
OmpSs is based on OpenMP-like pragmas while StarPU and QUARK are C libraries
of function. QUARK does not schedule tasks on multi-GPUs architecture and
implements a centralized greedy list scheduling algorithm. OmpSs locality-
aware scheduling, similar to our data-aware heuristic from [9], computes an
affinity score based on where the data is located and its size. Then, the task
is placed on the highest affinity resource or in a global list, otherwise.
StarPU scheduler uses the HEFT [16] algorithm to schedule all ready tasks in
accordance with the cost models for data transfer and task execution time [3].
Our data transfer model is based on StarPU model with minor extension. In the
context of dense linear algebra algorithms, PLASMA [7] provides fine-grained
parallel linear algebra routines with dynamic scheduling through QUARK, which
was conceived specially for numerical algorithms on multi-CPUs architecture.
MAGMA [15] implements static scheduling for linear algebra algorithms on
heterogeneous systems composed of GPUs. Recently it has included some methods
with dynamic scheduling in multi-CPU and multi-GPU on top of StarPU, in
addition to the static multi-GPU version. In [14] the authors based their
Cholesky factorization on 2D block cyclic distribution with an owner compute
rule to map tasks to resources. DAGuE [5] is a parallel framework focused on
multi-core clusters and supports single-GPU nodes. Other papers reported
performance results of task-based algorithms with HEFT cost model scheduling
on heterogeneous architectures for the Cholesky [4], LU [1], and QR [2]
factorizations. All the results report evaluation of single floating point
arithmetics with up to $3$ GPUs. Due to the small number of GPUs, such studies
cannot observe contention and scalability.
## 6 Conclusion
We presented in this paper a new scheduling algorithm on top of the XKaapi
runtime. It is based on a dual approximation scheme with affinity and has been
compared to the classical HEFT algorithm for three tile algorithms from PLASMA
on an heterogeneous architecture composed of $8$ GPUs and $12$ CPUs. Both
algorithms attained significant speed up on the three dense linear algebra
kernel. Moreover, if HEFT achieves the best absolute performance with respect
to DADA on QR, while DADA has similar or better performances than HEFT on
Cholesky and LU for large numbers of GPU. Nevertheless, DADA allows to
significantly reduce the data transfers with respect to HEFT. More
interesting, thanks to its affinity criteria DADA can introduce communication
in the scheduling without too precise communication cost model which are
required in HEFT to predict the completion time of tasks.
We would like to extend the experimental evaluations on robustness of
scheduling with respect to uncertainties in cost models, especially on the
communication cost which is very sensitive to contentions that may appear at
runtime. Another interesting issue would be to study other affinity functions.
## Acknowledgments
This work has been partially supported by the French Ministry of Defense –
DGA, the ANR 09-COSI-011-05 Project Repdyn and CAPES/Brazil.
## References
* [1] Agullo, E., Augonnet, C., Dongarra, J., Faverge, M., Langou, J., Ltaief, H., Tomov, S.: Lu factorization for accelerator-based systems. In: IEEE/ACS AICCSA. pp. 217–224. AICCSA ’11, IEEE Computer Society, Washington, DC, USA (2011)
* [2] Agullo, E., Augonnet, C., Dongarra, J., Faverge, M., Ltaief, H., Thibault, S., Tomov, S.: QR Factorization on a Multicore Node Enhanced with Multiple GPU Accelerators. In: IEEE IPDPS. EUA (2011)
* [3] Augonnet, C., Thibault, S., Namyst, R.: Automatic calibration of performance models on heterogeneous multicore architectures. In: Euro-Par. pp. 56–65. Springer-Verlag (2010)
* [4] Augonnet, C., Thibault, S., Namyst, R., Wacrenier, P.A.: StarPU: a unified platform for task scheduling on heterogeneous multicore architectures. Concurrency and Computation: Practice and Experience 23(2), 187–198 (2011)
* [5] Bosilca, G., Bouteiller, A., Danalis, A., Herault, T., Lemarinier, P., Dongarra, J.: DAGuE: A generic distributed DAG engine for High Performance Computing. Parallel Computing 38(1–2), 37–51 (2012)
* [6] Bueno, J., Planas, J., Duran, A., Badia, R.M., Martorell, X., Ayguadé, E., Labarta, J.: Productive Programming of GPU Clusters with OmpSs. In: IEEE IPDPS (2012)
* [7] Buttari, A., Langou, J., Kurzak, J., Dongarra, J.: A class of parallel tiled linear algebra algorithms for multicore architectures. Parallel Computing 35(1), 38–53 (2009)
* [8] Gautier, T., Besseron, X., Pigeon, L.: KAAPI: A thread scheduling runtime system for data flow computations on cluster of multi-processors. In: PASCO’07. ACM, London, Canada (2007)
* [9] Gautier, T., Lima, J.V., Maillard, N., Raffin, B.: XKaapi: A Runtime System for Data-Flow Task Programming on Heterogeneous Architectures. In: IEEE IPDPS. pp. 1299–1308 (2013)
* [10] Hermann, E., Raffin, B., Faure, F., Gautier, T., Allard, J.: Multi-GPU and Multi-CPU Parallelization for Interactive Physics Simulations. In: Euro-Par. vol. 6272, pp. 235–246. Springer (2010)
* [11] Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM 34(1), 144–162 (Jan 1987)
* [12] Kedad-Sidhoum, S., Monna, F., Mounié, G., Trystram, D.: Scheduling independent tasks on multi-cores with gpu accelerators. In: 11th HeteroPar Workshop (2013)
* [13] Lima, J.V.F., Gautier, T., Maillard, N., Danjean, V.: Exploiting Concurrent GPU Operations for Efficient Work Stealing on Multi-GPUs. In: 24th SBAC-PAD. pp. 75–82. IEEE, New York, USA (2012)
* [14] Song, F., Dongarra, J.: A scalable framework for heterogeneous GPU-based clusters. In: ACM SPAA. pp. 91–100. ACM, New York, NY, USA (2012)
* [15] Tomov, S., Dongarra, J., Baboulin, M.: Towards dense linear algebra for hybrid GPU accelerated manycore systems. Parallel Computing 36(5-6), 232–240 (2010)
* [16] Topcuoglu, H., Hariri, S., Wu, M.Y.: Performance-effective and low-complexity task scheduling for heterogeneous computing. IEEE TPDC 13(3), 260–274 (2002)
* [17] YarKhan, A., Kurzak, J., Dongarra, J.: Quark users’ guide: Queueing and runtime for kernels. Tech. Rep. ICL-UT-11-02, University of Tennessee (2011)
|
arxiv-papers
| 2014-02-26T16:37:01 |
2024-09-04T02:49:58.959455
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rapha\\\"el Bleuse and Thierry Gautier and Jo\\~ao V. F. Lima and\n Gr\\'egory Mouni\\'e and Denis Trystram",
"submitter": "Gr\\'egory Mouni\\'e",
"url": "https://arxiv.org/abs/1402.6601"
}
|
1402.6670
|
# Local characterization of polyhedral spaces
Nina Lebedeva N. Lebedeva
Steklov Institute, 27 Fontanka, St. Petersburg, 191023, Russia.
Math. Dept. St. Petersburg State University, Universitetsky pr., 28, Stary
Peterhof, 198504, Russia. [email protected] and Anton Petrunin A. Petrunin
Math. Dept. PSU, University Park, PA 16802, USA [email protected]
###### Abstract.
We show that a compact length space is polyhedral if a small spherical
neighborhood of any point is conic.
N. Lebedeva was partially supported by RFBR grant 14-01-00062.
A. Petrunin was partially supported by NSF grant DMS 1309340.
## 1\. Introduction
In this note we characterize polyhedral spaces as the spaces where every point
has a conic neighborhood. Namely, we prove the following theorem; see Section
2 for all necessary definitions.
1.1. Theorem. A compact length space $X$ is polyhedral if and only if a
neighborhood of each point $x\in X$ admits an open isometric embedding to
Euclidean cone which sends $x$ to the tip of the cone.
Note that we do not make any assumption on the dimension of the space. If the
dimension is finite then the statement admits a simpler proof by induction;
this proof is indicated in the last section.
A priori, it might be not clear why the space in the theorem is even
homeomorphic to a simplicial complex. This becomes wrong if you remove word
“isometric” from the formulation. For example, there are closed 4-dimensional
topological manifold which does not admit any triangulation, see [2, 1.6].
The Theorem 1 is applied in [5], where it is used to show that an Alexandrov
space with the maximal number of extremal points is a quotient of
$\mathbb{R}^{n}$ by a cocompact properly discontinuous isometric action; see
also [4].
Idea of the proof. Let us cover $X$ by finite number of spherical conic
neighborhood and consider its nerve, say $\mathcal{N}$. Then we map
$\mathcal{N}$ barycentrically back to $X$. If we could show that the image of
this map cover whole $X$ that would nearly finish the proof. Unfortunately we
did not manage to show this statement and have make a walk around; this is the
only subtle point in the proof below.
Acknowledgment. We would like to thank Arseniy Akopyan, Vitali Kapovitch,
Alexander Lytchak and Dmitri Panov for their help.
## 2\. Definitions
In this section we give the definition of polyhedral space of arbitrary
dimension. It seems that these spaces were first considered by Milka in [6];
our definitions are equivalent but shorter.
Metric spaces. The distance between points $x$ and $y$ in a metric space $X$
will be denoted as $|x-y|$ or $|x-y|_{X}$. Open $\varepsilon$-ball centered at
$x$ will be denoted as $B(x,\varepsilon)$; i.e.,
$B(x,\varepsilon)=\left\\{\,\left.{y\in
X}\vphantom{|x-y|<\varepsilon}\,\right|\,{|x-y|<\varepsilon}\,\right\\}.$
If $B=B(x,\varepsilon)$ and $\lambda>0$ we use notation $\lambda{\hskip
0.5pt\cdot\hskip 0.5pt}B$ as a shortcut for $B(x,\lambda{\hskip
0.5pt\cdot\hskip 0.5pt}\varepsilon)$.
A metric space is called _length space_ if the distance between any two points
coincides with the infimum of lengths of curves connecting these points.
A _minimizing geodesic_ between points $x$ and $y$ will be denoted by $[xy]$.
Polyhedral spaces. A length space is called _polyhedral space_ if it admits a
finite triangulation such that each simplex is (globally) isometric to a
simplex in Euclidean space.
Note that according to our definition, the polyhedral space has to be compact.
Cones and homotheties. Let $\Sigma$ be a metric space with diameter at most
$\pi$. Consider the topological cone $K=[0,\infty)\times\Sigma/\sim$ where
$(0,x)\sim(0,y)$ for every $x,y\in\Sigma$. Let us equip $K$ with the metric
defined by the rule of cosines; i.e., for any $a,b\in[0,r)$ and $x,y\in\Sigma$
we have
$|(a,x)-(b,y)|_{K}^{2}=a^{2}+b^{2}-2{\hskip 0.5pt\cdot\hskip 0.5pt}a{\hskip
0.5pt\cdot\hskip 0.5pt}b{\hskip 0.5pt\cdot\hskip 0.5pt}\cos|x-y|_{\Sigma}.$
The obtained space $K$ will be called _Euclidean cone over_ $\Sigma$. All the
pairs of the type $(0,x)$ correspond to one point in $K$ which will be called
the _tip_ of the cone. A metric space which can be obtained in this way is
called _Euclidean cone_.
Equivalently, Euclidean cone can be defined as a metric space $X$ which admits
a one parameter family of homotheties $m^{\lambda}\colon X\to X$ for
$\lambda\geqslant 0$ such that for any fixed $x,y\in X$ there are real numbers
$\zeta$, $\eta$ and $\vartheta$ such that $\zeta,\vartheta\geqslant 0$,
$\eta^{2}\leqslant\zeta{\hskip 0.5pt\cdot\hskip 0.5pt}\vartheta$ and
$|m^{\lambda}(x)-m^{\mu}(y)|_{X}^{2}=\zeta{\hskip 0.5pt\cdot\hskip
0.5pt}\lambda^{2}+2{\hskip 0.5pt\cdot\hskip 0.5pt}\eta{\hskip 0.5pt\cdot\hskip
0.5pt}\lambda{\hskip 0.5pt\cdot\hskip 0.5pt}\mu+\vartheta{\hskip
0.5pt\cdot\hskip 0.5pt}\mu^{2}.$
for any $\lambda,\mu\geqslant 0$. The point $m^{0}(x)$ is the tip of the cone;
it is the same point for any $x\in X$.
Once the family of homotheties is fixed, we can abbreviate $\lambda{\hskip
0.5pt\cdot\hskip 0.5pt}x$ for $m^{\lambda}(x)$.
Conic neighborhoods.
2.1. Definition. Let $X$ be a metric space, $x\in X$ and $U$ a neighborhood of
$x$. We say that $U$ is a _conic_ neighborhood of $x$ if $U$ admits an open
distance preserving embedding $\iota\colon U\to K_{x}$ into Euclidean cone
$K_{x}$ which sends $x$ to the tip of the cone.
If $x$ has a conic neighborhood then the cone $K_{x}$ as in the definition
will be called _the cone at $x$_. Note that in this case $K_{x}$ is unique up
to an isometry which sends the tip to the tip. In particular, any conic
neighborhood $U$ of $x$ admits an open distance preserving embedding
$\iota_{U}\colon U\to K_{x}$ which sends $x$ to the tip of $K_{x}$. Moreover,
it is easy to arrange that these embeddings commute with inclusions; i.e., if
$U$ and $V$ are two conic neighborhoods of $x$ and $U\supset V$ then the
restriction of $\iota_{U}$ to $V$ coincides with $\iota_{V}$. The later
justifies that we omit index $U$ for the embedding $\iota\colon U\to K_{x}$.
Assume $x\in X$ has a conic neighborhood and $K_{x}$ is the cone at $x$. Given
a geodesic $[xy]$ in $X$, choose a point
$\bar{y}\in\left[xy\right]\backslash\\{x\\}$ sufficiently close to $x$ and set
$\log[xy]=\frac{|x-y|_{X}}{|x-\bar{y}|_{X}}{\hskip 0.5pt\cdot\hskip
0.5pt}\iota(\bar{y})\in K_{x}.$
Note that $\log[xy]$ does not depend on the choice of $\bar{y}$.
## 3\. Preliminary statements
3.1. Definition. Let $X$ be a metric space and $[px_{1}]$,
$[px_{2}],\dots,[px_{m}]$ are geodesics in $X$. We say that a neighborhood $U$
of $p$ _splits_ in the direction of the geodesics $[px_{1}]$,
$[px_{2}],\dots,[px_{k}]$ if there is an open distance preserving map $\iota$
from $U$ to the product space $E\times K^{\prime}$, such that $E$ is a
Euclidean space and the inclusion
$\iota(U\cap[px_{i}])\subset E\times\\{o^{\prime}\\}$
holds for a fixed $o^{\prime}\in K^{\prime}$ and any $i$.
Splittings and isometric copies of polyhedra. The following lemmas and the
corollary are the key ingredients in the proof.
3.2. Lemma. Let $X$ be a metric space, $p\in X$ and for each
$i\in\nobreak\\{1,\dots,k\\}$ the ball $B_{i}=B(x_{i},r_{i})$, forms a conic
neighborhoods of $x_{i}$. Assume $p\in B_{i}$ for each $i$. Then any conic
neighborhood of $p$ splits in the direction of $[px_{1}],\dots,[px_{k}]$.
In the proof we will use the following statement; its proof is left to the
reader.
3.3. Proposition. Assume $K$ is a metric space which admits cone structures
with different tips $x_{1},\dots,x_{k}$. Then $K$ is isometric to the product
space $E\times K^{\prime}$, where $E$ is a Euclidean space and $K^{\prime}$ is
a cone with tip $o^{\prime}$ and $x_{i}\in E\times\\{o^{\prime}\\}$ for each
$i$.
Proof of Lemma 3. Fix sufficiently small $\varepsilon>0$. For each point
$x_{i}$, consider point $x_{i}^{\prime}\in[px_{i}]$ such that
$|p-x_{i}^{\prime}|=\varepsilon{\hskip 0.5pt\cdot\hskip 0.5pt}|p-x_{i}|$.
Since $\varepsilon$ is sufficiently small, we can assume that $x_{i}^{\prime}$
lies in the conic neighborhood of $p$.
Note that for the right choice of parameters close to 1, the composition of
homotheties with centers at $x_{i}$ and $p$ produce a homothety with center at
$x_{i}^{\prime}$ and these are defined in a fixed conic neighborhood of $p$.
(In particular it states that composition of homotheties of Euclidean space is
a homothety; the proof is the same. The parameters are assumed to be chosen in
such a way that $x_{i}^{\prime}$ stay fixed by the composition.)
These homotheties can be extended to the cone $K_{p}$ at $p$ and taking their
compositions we get the homotheties for all values of parameters with the
centers at $\hat{x}^{\prime}_{i}=\log[px_{i}^{\prime}]\in K_{p}$. It remains
to apply Proposition 3. ∎
From the Lemma 3, we get the following corollary.
3.4. Corollary. Let $X$ be a compact length space and $x\in X$. Suppose
$B=\nobreak B(x,r)$ is a conic neighborhood of $x$ which splits in the
direction of $[px_{1}],\dots\nobreak,[px_{k}]$ and $\iota\colon
B\hookrightarrow E\times K^{\prime}$ be the corresponding embedding. Then the
image $\iota(B)$ is a ball of radius $r$ centered at $\iota(x)\in
E\times\\{o^{\prime}\\}$.
In particular, for any point $q\in B$ such that $|q-p|_{X}=\rho$ and
$\iota(q)\in E\times\\{o^{\prime}\\}$ the ball $B(q,r-\rho)$ is a conic
neighborhood of $q$.
3.5. Lemma. Let $B_{i}=B(x_{i},r_{i})$, $i\in\\{0,\dots,k\\}$ be balls in the
metric space $X$. Assume each $B_{i}$ forms a conic neighborhood of $x_{i}$
and $x_{i}\in B_{j}$ if $i\leqslant j$. Then $X$ contains a subset $Q$ which
contains all $x_{i}$ and is isometric to a convex polyhedron.
Moreover the geodesics in $Q$ do not bifurcate in $X$; i.e., if geodesic
$\gamma\colon[a,b]\to X$ lies in $Q$ and an other geodesics
$\gamma^{\prime}\colon[a,b]\to X$ coincides with $\gamma$ on some interval
then $\gamma^{\prime}=\gamma$.
To illustrate the second statement let us consider tripod $T$; i.e.,
1-dimensional polyhedral space obtained from three intervals by gluing their
left ends together. Let $Q$ be the union of two segments in $T$. Note that $Q$
forms a subset isometric to a real interval; i.e., $Q$ is isometric to
1-dimensional convex polyhedron. On the other hand, $Q$ does not satisfy the
second condition since a geodesic can turn from $Q$ at the triple point.
Proof. To construct $Q=Q_{k}$ we apply induction on $k$ and use the cone
structures on $B_{i}$ with the tip at $x_{i}$ consequently.
For the base case, $k=0$, we take $Q_{0}=\\{x_{0}\\}$.
By the induction hypothesis, there is a set $Q_{k-1}$ containing all
$x_{0},\dots,x_{k-1}$.
Note that $B_{k}$ is strongly convex; i.e., any minimizing geodesic with ends
in $B_{k}$ lies completely in $B_{k}$. In particular $Q_{k-1}\cap B_{k}$ is
convex. Since $x_{i}\in B_{k}$ for all $i<k$, we may assume that
$Q_{k-1}\subset B_{k}$.
Note that the homothety $m_{k}^{\lambda}$ with center $x_{k}$ and
$\lambda\leqslant 1$ is defined for all points in $B_{k}$. Set
$Q_{k}=\left\\{\,\left.{m^{\lambda}_{k}(x)}\vphantom{x\in Q_{k-1}\ \text{and}\
\lambda\leqslant 1}\,\right|\,{x\in Q_{k-1}\ \text{and}\ \lambda\leqslant
1}\,\right\\}.$
Since $Q_{k-1}$ is isometric to a convex polytope, so is $Q_{k}$.
To show that the geodesic $\gamma\colon[a,b]\to X$ in $Q$ can not bifurcate,
it is sufficient to show that if $a<c<b$ then a neighborhood of $p=\gamma(c)$
splits in the direction of $\gamma$.
The point $p$ can be obtained from from $x_{0}$ by a composition of
homotheties
$p=m_{k}^{\lambda_{k}}\circ\cdots\circ m_{1}^{\lambda_{1}}(x_{0}),$
where $0<\lambda_{i}\leqslant 1$. Set $m=m_{k}^{\lambda_{k}}\circ\cdots\circ
m_{1}^{\lambda_{1}}(x_{0})$. We can assume $r_{0}$ to be sufficiently small so
that $m$ is defined on $B_{0}$.
By Lemma 3, $B_{0}$ splits in the directions of
$[x_{0}x_{1}],\dots,[x_{0}x_{k}]$. Since $m$ rescales the distances by fixed
factor, a neighborhood of $p$ also splits. Clearly the Euclidean factor in the
image $m(B_{0})$ covers small neighborhood of $p$ in $Q$. Since $\gamma$ runs
$Q$, a neighborhood of $p$ splits in the direction of $\gamma$. ∎
## 4\. The proof
The proof of Theorem 1 is based on the following lemma.
4.1. Lemma. Assume a length space $X$ is covered by finite number of sets such
that each finite intersection of these sets is isometric to a convex polytope.
Then $X$ is a polyhedral space.
Proof. It is sufficient to show that if any metric space $X$ (not necessary
length metric space) admits a cover as in the lemma then it admits a
triangulation such that each simplex is isometric to a Euclidean simplex.
Denote by $V_{1},\dots V_{n}$ the polytopes in the covering. Let $m$ be the
maximal dimension of $V_{i}$.
We will apply induction on $m$; the base case $m=0$ is trivial.
Now assume $m>0$. Let $W_{1}\dots W_{k}$ denote all the faces of $V_{1},\dots
V_{n}$ of dimension at most $m-1$. Note that the collection $W_{1}\dots W_{k}$
satisfies the assumption of the Lemma. Therefore by induction hypothesis,
$X^{\prime}=\bigcup_{i}W_{i}$ admits the needed triangulation.
It remains to extend this triangulation to each of the $m$-dimensional
polytopes which $X^{\prime}$ cuts from $X$. The later is generously left to
the reader. ∎
Proof of Theorem 1. We need to show the “if” part; the “only if” part is
trivial.
Fix a finite cover of $X$ by open balls $B_{i}=B(x_{i},r_{i})$,
$i\in\\{0,\dots,n\\}$ such that for each $i$, the ball $5{\hskip
0.5pt\cdot\hskip 0.5pt}B_{i}$ forms a conic neighborhood of $x_{i}$.
Given $i\in\\{0,\dots,n\\}$ and $z\in X$ set
$f_{i}(z)=|x_{i}-z|_{X}^{2}-r_{i}^{2}.$
Clearly $f_{i}(z)<0$ if and only if $z\in B_{i}$.
Set
$f(z)=\min_{i}\\{f_{i}(z)\\}.$
It follows that $f(z)<0$ for any $z\in X$.
Consider _Voronoi domains_ $V_{i}$ for the functions $f_{i}$; i.e.,
$V_{i}=\left\\{\,\left.{z\in X}\vphantom{f_{i}(z)\leqslant f_{j}(z)\ \text{for
all}\ j}\,\right|\,{f_{i}(z)\leqslant f_{j}(z)\ \text{for all}\
j}\,\right\\}.$
From above we get that $V_{i}\subset B_{i}$ for each $i$.111It also follows
that $V_{i}$ forms a _strongly convex subset_ of $X$; i.e., any minimizing
geodesic in $X$ with ends in $V_{i}$ lies completely in $V_{i}$. This property
is not needed in our proof, but it is used in the alternative proof; see the
last section.
Given a subset $\sigma\subset\\{0,\dots,n\\}$ set
$V_{\sigma}=\bigcap_{i\in\sigma}V_{i}.$
Note that $V_{\\{i\\}}=V_{i}$ for any $i\in\\{0,\dots,n\\}$.
Let $\mathcal{N}$ be the _nerve_ of the covering $\\{V_{i}\\}$; i.e.,
$\mathcal{N}$ is the abstract simplicial complex with $\\{0,\dots,n\\}$ as the
set of vertexes and such that a subset $\sigma\subset\nobreak\\{0,\dots,n\\}$
forms a simplex in $\mathcal{N}$ if and only if $V_{\sigma}\neq\varnothing$.
Let us fix a simplex $\sigma$ in $\mathcal{N}$. While $\sigma$ is fixed, we
may assume without loss of generality that $\sigma=\\{0,\dots,k\\}$ for some
$k\leqslant n$ and $r_{0}\leqslant r_{1}\leqslant\dots r_{k}$. In particular
$2{\hskip 0.5pt\cdot\hskip 0.5pt}B_{i}\ni x_{0}$ for each $i\leqslant k$.
From above $V_{\sigma}\subset B_{0}$. Since $5{\hskip 0.5pt\cdot\hskip
0.5pt}B_{i}$ is a conic neighborhood of $x_{i}$ and $2{\hskip 0.5pt\cdot\hskip
0.5pt}B_{i}\ni x_{0}$ for each $i\in\sigma$, we can apply Lemma 3 for the
balls $5{\hskip 0.5pt\cdot\hskip 0.5pt}B_{0},\dots,5{\hskip 0.5pt\cdot\hskip
0.5pt}B_{k}$. Denote by $h\colon 5{\hskip 0.5pt\cdot\hskip
0.5pt}B_{0}\hookrightarrow E\times K$ the distance preserving embedding
provided by this lemma. We can assume that the Euclidean factor $E$ has
minimal possible dimension; i.e., the images $h(B_{0}\cap[x_{0}x_{i}])$ span
whole $E$. In this case the projection of $h(V_{\sigma})$ on $E$ is a one-
point set, say $\\{z\\}$. Denote by $x_{\sigma}\in B_{0}$ the point such that
$h(x_{\sigma})=z$. Set $r_{\sigma}=r_{0}$ and
$B_{\sigma}=B(x_{\sigma},r_{\sigma})$. (The point $x_{\sigma}$ plays the role
of _radical center_ of the collection of balls $\\{B_{i}\\}_{i\in\sigma}$.)
According to Corollary 3 the ball $4{\hskip 0.5pt\cdot\hskip 0.5pt}B_{\sigma}$
forms a conic neighborhood of $x_{\sigma}$. Clearly $B_{\sigma}\supset
V_{\sigma}$.
Let $\varphi$ and $\psi$ be faces of $\sigma$; in other words, $\varphi$ and
$\psi$ are subsets in $\sigma=\\{0,\dots,k\\}$. Set $i=\min\varphi$ and
$j=\min\psi$. Assume $i\geqslant j$, in this case $r_{\varphi}=r_{i}\geqslant
r_{j}=r_{\psi}$. From above we get $x_{\varphi}\in B_{i}$, $x_{\psi}\in B_{j}$
and $x_{j}\in 2{\hskip 0.5pt\cdot\hskip 0.5pt}B_{i}$. Therefore $x_{\psi}\in
4{\hskip 0.5pt\cdot\hskip 0.5pt}B_{\varphi}$.
Therefore Lemma 3 provides a subset, say $Q_{\sigma}$ isometric to a convex
polyhedron and contains all $x_{\varphi}$ for $\varphi\subset\sigma$.
It remains to show
1. (a)
$X=\bigcup_{\sigma}Q_{\sigma}$, where the union is taken for all the simplices
$\sigma$ in $\mathcal{N}$.
2. (b)
The intersection of arbitrary collection of $Q_{\sigma}$ is isometric to a
convex polytope.
Once (a) and (b) are proved, Lemma 4 will finish the proof.
Part (b) follows since the geodesics in $Q_{\sigma}$ do not bifurcate in $X$;
see Lemma 3.
Given $p\in X$, set
$\sigma(p)=\left\\{\,\left.{i\in\\{0,\dots,n\\}}\vphantom{p\in
V_{i}}\,\right|\,{p\in V_{i}}\,\right\\}.$
Note that $\sigma(p)$ forms a simplex in $\mathcal{N}$ and $p\in
V_{\sigma(p)}$. Therefore $p\in B_{\sigma(p)}$.
Recall that $B_{\sigma(p)}$ forms a conic neighborhood of $x_{\sigma(p)}$. If
$p\neq x_{\sigma(p)}$ then moving $p$ away from $x_{\sigma(p)}$ in the radial
direction keeps the point in $V_{\sigma(p)}$ till the moment it hits a new
Voronoi domain, say $V_{j}$ with $j\notin\sigma(p)$. Denote this end point by
$p^{\prime}$. In other words, $p^{\prime}$ is the point such that
1. (i)
$p$ lies on the geodesic $[x_{\sigma(p)}p^{\prime}]$;
2. (ii)
$p^{\prime}\in V_{i}$ for any $i\in\sigma(p)$;
3. (iii)
the distance $|x_{\sigma(p)}-p^{\prime}|_{X}$ takes the maximal possible
value.
$x_{2}$$x_{1}$$x_{3}$$p_{0}$$p_{1}$$p_{2}=x_{\\{1,2,3\\}}$$x_{\\{1,2\\}}$$V_{1}$
Start with arbitrary point $p$ and consider the recursively defined sequence
$p=p_{0},p_{1},\dots$ such that $p_{i+1}=p_{i}^{\prime}$.
Note that $\sigma(p)$ forms a proper subset of $\sigma(p^{\prime})$. It
follows that the sequence $(p_{i})$ terminates after at most $n$ steps; in
other words $p_{k}=x_{\sigma(p_{k})}$ for some $k$.
In particular $p_{k}\in Q_{\sigma(p_{k})}$. By construction it follows that
$p_{i}\in Q_{\sigma(p_{k})}$ for each $i\leqslant k$. Hence $p\in
Q_{\sigma(p_{k})}$; i.e., (a) follows. ∎
## 5\. Final remarks
Finite dimensional case. Let $X$ be a compact length space such that each
point $x\in X$ admits a conic neighborhood.
Note that from Theorem 1, it follows in particular that dimension of $X$ is
finite. If we know a priori the dimension (topological or Hausdorff) of $X$ is
finite then one can build an easier proof using induction on the dimension
which we are about to indicate.
Consider the Voronoi domains $V_{i}$ as in the beginning of proof of Theorem
1. Note that all $V_{i}$ are convex and
$\operatorname{dim}V_{\\{i,j\\}}<\operatorname{dim}X$
if $i\neq j$.
By induction hypothesis we can assume that all $V_{\\{i,j\\}}$ are polyhedral
spaces. Cover each $V_{\\{i,j\\}}$ by isometric copies of convex polyhedra
satisfying Lemma 4. Applying the cone construction with center $x_{i}$ over
these copies in $V_{\\{i,j\\}}$ for all $i\neq j$, we get a covering of $X$ by
a finite number of copies of convex polyhedra such that all their finite
intersections are isometric to convex polyhedra. It remains to apply Lemma 4.
Spherical and hyperbolic polyhedral spaces. Analogous characterization holds
for spherical and hyperbolic polyhedral spaces. One needs to use spherical and
hyperbolic rules of cosine in the definition of cone; after that proof goes
without any changes.
Locally compact case. One may define polyhedral space as a complete length
space which admits a locally finite triangulation such that each simplex is
isometric to a simplex in Euclidean space.
In this case a locally compact length space is polyhedral if every point
admits a conic neighborhood. The proof is the same.
One more curvature free result. Our result is curvature free — we do not make
any assumption on the curvature of $X$. Besides our theorem, we are aware
about only one statement of that type — the polyhedral analog of Nash–Kuiper
theorem. It states that any distance nonexpanding map from $m$-dimensional
polyhedral space to the Euclidean $m$-space can be approximated by a piecewise
distance preserving map to the Euclidean $m$-space. In full generality this
result was proved recently by Akopyan [1], his proof is based on earlier
results obtained by Zalgaller [8] and Krat [3]. Akopyan’s proof is sketched in
the lecture notes of the second author [7].
## References
* [1] Akopyan, A. V., A piecewise linear analogue of Nash–Kuiper theorem, a preliminary version (in Russian) can be found on www.moebiuscontest.ru
* [2] Freedman, M. H,. The topology of four-dimensional manifolds. J. Differential Geom. 17 (1982), no. 3, 357--453.
* [3] Krat, S. Approximation problems in Length Geometry, Thesis, 2005
* [4] Lebedeva, N., Number of extremal subsets in Alexandrov spaces and rigidity. Electron. Res. Announc. Math. Sci. 21 (2014), 120--125.
* [5] Lebedeva, N., Alexandrov spaces with maximal number of extremal points, to appear in Geometry and Topology (2015), arXiv:1111.7253.
* [6] Milka, A. D. Multidimensional spaces with polyhedral metric of nonnegative curvature. I. (Russian) Ukrain. Geometr. Sb. Vyp. 5--6 1968 103--114.
* [7] Petrunin, A.; Yashinski, A. Piecewise distance preserving maps. to appear in St. Petersburg Mathematical Journal, arXiv:1405.6606.
* [8] Zalgaller, V. A. Isometric imbedding of polyhedra. (Russian) Dokl. Akad. Nauk SSSR 123 1958 599--601.
|
arxiv-papers
| 2014-02-26T20:26:31 |
2024-09-04T02:49:58.971337
|
{
"license": "Public Domain",
"authors": "Nina Lebedeva and Anton Petrunin",
"submitter": "Anton Petrunin",
"url": "https://arxiv.org/abs/1402.6670"
}
|
1402.6719
|
# High-Time-Resolution Measurements of the Polarization of the Crab Pulsar at
1.38 GHz
Agnieszka Słowikowska11affiliation: Kepler Institute of Astronomy, University
of Zielona Góra, Lubuska 2, 65-265 Zielona Góra, Poland , Benjamin W.
Stappers22affiliation: Jodrell Bank Centre for Astrophysics, University of
Manchester, Manchester M13 9PL, UK , Alice K. Harding33affiliation:
Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA , Stephen L. O’Dell44affiliation: Astrophysics Office, NASA
Marshall Space Flight Center, ZP12, Huntsville, AL 35812, USA , Ronald F.
Elsner44affiliation: Astrophysics Office, NASA Marshall Space Flight Center,
ZP12, Huntsville, AL 35812, USA , Alexander J. van der Horst55affiliation:
Astronomical Institute, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands , Martin C. Weisskopf44affiliation: Astrophysics
Office, NASA Marshall Space Flight Center, ZP12, Huntsville, AL 35812, USA
###### Abstract
Using the Westerbork Synthesis Radio Telescope (WSRT), we obtained high-time-
resolution measurements of the full (linear and circular) polarization of the
Crab pulsar. Taken at a resolution of 1/8192 of the 34-ms pulse period (i.e.,
$4.1~{}\mu{\rm s}$), the 1.38-GHz linear-polarization measurements are in
general agreement with previous lower-time-resolution 1.4-GHz measurements of
linear polarization in the main pulse (MP), in the interpulse (IP), and in the
low-frequency component (LFC). We find the MP and IP to be linearly polarized
at about $24\%$ and $21\%$, with no discernible difference in polarization
position angle. However, and contrary to theoretical expectations and
measurements in the visible, we find no evidence for significant variation
(sweep) in polarization position angle over the MP, the IP, or the LFC.
Although, the main pulse exhibits a small but statistically significant
quadratic variation in the degree of linear polarization. We discuss the
implications which appear to be in contradiction to theoretical expectations.
In addition, we detect weak circular polarization in the main pulse and
interpulse, and strong ($\approx 20\%$) circular polarization in the low-
frequency component, which also exhibits very strong ($\approx 98\%$) linear
polarization at a position angle about $40^{\circ}$ from that of the MP or IP.
The pulse-mean polarization properties are consistent with the LFC being a
low-altitude component and the MP and IP being high-altitude caustic
components. Nevertheless, current models for the MP and IP emission do not
readily account for the observed absence of pronounced polarization changes
across the pulse. Finally, we measure IP and LFC pulse phases relative to the
MP that are consistent with recent measurements, which have shown that the
phases of these pulse components are evolving with time.
neutron stars: general — pulsars: individual — Crab pulsar (catalog ) (PSR
B0531+21) (catalog ) — polarization
††slugcomment: Version 16: 2014.10.20
## 1 Introduction
The Australia Telescope National Facility Pulsar Catalog (Manchester et al.,
2005) lists over 2300 radio pulsars. Several radio studies (e.g., Gould &
Lyne, 1998; Karastergiou & Johnston, 2006; Weltevrede & Johnston, 2008) have
measured the polarization for many of these pulsars. Radio pulsars typically
show moderate-to-strong linear polarization ($p_{L}$), being stronger for
those of higher spin-down energy-loss rate (Weltevrede & Johnston, 2008,
Figure 8). The linear polarization sometimes exhibits a characteristic swing
or sweep of the position angle in an S-like shape near the pulse center, which
is routinely interpreted in terms of the rotating vector model (RVM,
Radhakrishnan & Cooke, 1969). For this model the point of emission is assumed
to be in the polar cap region of the pulsar where a dipolar magnetic-field
line points with a small angle (beamwidth) towards the observer. The two free
parameters of this simple model are the angle between the axes of rotation and
the orientation of the magnetic dipole, and the view angle between the line of
sight and the rotation axis. The variation of the radio position angle from
some pulsars (e.g., Lyne & Graham-Smith, 2006, and references therein) can be
described by this model.
The Crab pulsar, the compact remnant of SN1054, and its pulsar wind nebula
(PWN) are amongst the most intensively studied objects in the sky. The pulsar
is one of the youngest and most energetic and its pulsed emission has been
detected from 10 MHz (Bridle, 1970) up to 400 GeV by VERITAS (Aliu et al.,
2011) and by MAGIC (Aleksić et al., 2012). The PWN is detected at energies up
to 100 TeV (Aharonian et al., 2004, 2006; Allen & Yodh, 2007; Abdo et al.,
2012). Both the pulsar and nebula are predominantly sources of non-thermal
radiation (synchrotron, curvature, and Compton processes), indicated not only
by the broadband spectral continua but also by strong polarization in many
wavelength bands (Lyne & Graham-Smith, 2006; Bühler & Blandford, 2013).
In the visible band, spatially-resolved polarimetry of the nebula, which began
over a half century ago (Oort & Walraven, 1956; Woltjer, 1957), continues
(e.g., Moran et al., 2013b, and references therein). Owing to its brightness,
phase-resolved optical polarimetry of the pulsar has also been possible (Jones
et al., 1981; Smith et al., 1988; Słowikowska et al., 2009). However, phase-
resolved X- and $\gamma$-ray polarimetry measurements of the Crab pulsar
require space-based instruments, which have had limited sensitivity. OSO-8
observations (Silver et al., 1978) of the Crab established only upper limits
to the X-ray (2.6-keV and 5.2-keV) polarization of the pulsed emission.
INTEGRAL IBIS observations (Forot et al., 2008; Moran et al., 2013a) also
detect no significant pulsed $\gamma$-ray (200–800-keV) polarization, although
the off-pulse emission appears highly linearly polarized and is possibly
associated with structures close to the pulsar rather than with the pulsar
itself.
The Crab pulsar’s light curve exhibits different features at different
wavelengths, but it is currently the only pulsar for which the principal
features persist over all wavelengths, from radio to $\gamma$-ray. There are
two principal components—the main pulse (MP) and the interpulse (IP). This
double-peak structure remains more-or-less phase-aligned over all spectral
bands (Moffett & Hankins, 1996; Kuiper et al., 2001). One of several
additional features in the radio band is the low-frequency component (LFC,
e.g., Moffett & Hankins 1996; 1999), having very low amplitude and occurring
$\approx 0.10$ fractional pulse phase ($36^{\circ}$) before the MP. This
component is most prominent around 1.4 GHz, in contrast with the “precursor”
component (Moffett & Hankins, 1996), which precedes the MP by $\approx 0.05$
fractional pulse phase ($19^{\circ}$) at 0.327 and 0.610 GHz (Table 2 of
Backer, Wong & Valanju, 2000).
The MP and IP appear at roughly the same pulse phase from radio to
$\gamma$-ray wavelengths, suggesting that their emission originates from a
similar location in the magnetosphere at all wavebands. Modeling of
$\gamma$-ray light curves from the many pulsars observed by the Fermi Gamma-
ray Space Telescope (Abdo et al., 2013) strongly indicates that the high-
energy emission originates in the outer magnetosphere, at altitudes comparable
to the light-cylinder radius (Romani & Watters, 2010; Pierbattista et al.,
2014; Bai & Spitkovsky, 2010). Outer magnetosphere emission models, such as
the outer-gap (Romani & Yadigaroglu, 1995), slot-gap (Muslimov & Harding,
2004), and current-sheet (Pétri & Kirk, 2005) had been proposed and studied
prior to the Fermi observations, but their emission geometry seems to account
for the characteristics and variety of observed $\gamma$-ray light curves. In
addition, Fermi has discovered a number of $\gamma$-ray millisecond pulsars
whose radio peaks are nearly aligned with their $\gamma$-ray peaks (e.g.,
Espinoza et al., 2013), like the Crab. Modeling both $\gamma$-ray and radio
light curves of these pulsars with the same outer magnetosphere emission
models used to model young pulsars has suggested that their radio emission may
originate from very high altitudes (Venter et al., 2012). Thus, in this paper
we compare the phase-resolved radio polarization observations (§2) that we
have analyzed (§3) with such models (§4).
Manchester (1971) measured the linear polarization of the Crab pulsar’s MP and
precursor components at two radio frequencies. The 0.410-GHz measurements
found the MP to be $20\%$ linearly polarized at position angle $140^{\circ}$
and the precursor to be $80\%$ linearly polarized at position angle
$140^{\circ}$. The 1.664-GHz measurements found the MP to be $24\%$ linearly
polarized at position angle $60^{\circ}$ and the precursor to be completely
absent. As these measurements had rather large uncertainties and were obtained
with a time resolution $1/256$ of the pulse period, they were quite limited
for detecting variation of the linear polarization degree or position angle
within a feature. However, Manchester noted a suggestion of rotation of the
1.664-GHz polarization position angle by about $30^{\circ}$ through the MP.
More recently, Moffett & Hankins (1999) examined the pulse-profile morphology
and polarization properties at three radio frequencies—1.424 GHz, 4.885 GHz,
and 8.435 GHz—with a time resolution of $256~{}\mu{\rm s}$ (about $1/130$ of
the pulse period). The 1.424-GHz measurements found the MP to be $25\%$
linearly polarized at position angle $120^{\circ}$; the IP, $20\%$ at position
angle $120^{\circ}$; and the LFC, $45\%$ at position angle $155^{\circ}$.
Moffett & Hankins note that the polarization position angle “changes across
the full period, although not significantly within components”.
Here we first report our observations (§2), using the Westerbork Synthesis
Radio Telescope (WSRT) in the Netherlands, of the full (linear and circular)
1.38-GHz polarization of the Crab pulsar, at high time resolution. We then
describe the polarimetry analysis and results (§3 and Appendix A) for the
three pulse components, with a primary objective of determining the sweep of
the position angle across each. Next we discuss the implications (§4) of our
measurements and analysis upon theoretical models for the pulsar emission.
Finally, we summarize our conclusions (§5).
## 2 The Observations
The WSRT observations, on 2011 August 8, used 14 25-m-diameter dishes combined
coherently to form the equivalent of a 94-m dish for pulsar observations.
Owing to the interferometric nature of the WSRT, the observations partially
resolve out the radio-bright Nebula, thus improving sensitivity over typical
single-dish observations. Moreover, as the WSRT is an equatorially mounted
telescope, there is no need to correct for parallactic angle.
To combine coherently the dishes, correlated data from observation of a bright
calibrator source is used to determine phase delays amongst dishes. This is
accomplished using initially an unpolarized calibrator to determine delays
between the two linear polarizations separately, followed by observation of a
polarized calibrator to determine any residual delays between the two
polarizations. These procedures accurately calibrate the relative fluxes in
the four Stokes parameters—hence, the polarization properties—but not the
absolute flux. Consequently, we express the Stokes measurements (e.g., Figure
2) in arbitrary units.
The PuMa-II (Karuppusamy et al., 2008) pulsar back-end was used to record
Nyquist-sampled voltages at 8-bit resolution, across a 160-MHz band centered
on 1380 MHz, for PSRs B0531+21 (Crab) and B0355+54, for a total of 144 and 18
minutes, respectively. The data were subsequently coherently de-dispersed and
folded using the DSPSR (van Straten & Bailes, 2011) software package.
Polarization profiles were formed after correcting for (frequency-dependent)
interstellar Faraday rotation (rotation measure RM = $-42.3\pm
0.5~{}\rm{rad~{}m^{-2}}$) of the position angle, using the PSRCHIVE software
package (van Straten et al., 2012). The polarization calibration was already
carried out when forming the coherent sum of the dishes, nevertheless PSR
B0355+54 was observed to verify that no further polarization calibration was
required. Comparison with the profile observed by Gould & Lyne (1998) showed
that the polarization calibration matched exactly. The Crab-pulsar profile was
folded using the Jodrell Bank
Ephemeris111http://www.jb.man.ac.uk/pulsar/crab.html with 8192 bins (about 4.1
$\mu$s/bin) across the pulse profile, matching the time resolution of the data
after dividing into frequency channels and coherently de-dispersing. This time
resolution was chosen also to match approximately the minimum broadening
caused by scattering of the Crab pulsed emission by free electrons in the Crab
Nebula (e.g., Backer, Wong & Valanju, 2000; Kuzmin et al., 2008).
Four Stokes parameters $I\times 10^{3}$, $Q\times 10^{4}$, $U\times 10^{4}$,
and $V\times 10^{4}$ (arbitrary units) as functions of pulse phase $\varphi$,
where the peak of the main pulse (MP) defines $\varphi=0$. The coordinate
system for the Stokes parameters here sets $U=0$ and $Q<0$ for the MP.
Figure 2 displays our measurement of the four Stokes parameters $I$, $Q$, $U$,
and $V$—which fully characterize the polarization—folded on the pulse period.
Unfortunately, we were unable to determine the absolute polarization position
angle for the Crab pulsar observation. Instead, we selected a coordinate
system for the Stokes parameters such that the MP has $U=0$ and $Q<0$.
Inspection of Figure 2 immediately shows that our 1.38-GHz observations detect
the flux and polarization of three components—MP, IP, and LFC. Like the MP,
the IP has $U\approx 0$ and $Q<0$; but the LFC has $U<0$ and $Q\approx 0$:
Thus, the polarization position angles for the MP and the IP are roughly equal
but differ from that of the LFC by about $40^{\circ}$ (cf. Eq. 2). Similarly,
but less obviously, the circular polarization of the MP and the IP are
comparable, but that of the LFC has opposite polarity.
## 3 Analysis and Results
The Stokes parameters have the virtues that they are statistically
independent, typically exhibit Gaussian errors, and are directly
superposable—i.e., each Stokes component ($I$, $Q$, $U$, or $V$) for multiple
sources is the sum of the respective Stokes component for each source. These
properties follow from the fact that the Stokes parameters describe the
polarization state in Cartesian-like coordinates. This has the added virtue
that there is no coordinate singularity at the origin, as occurs for polar-
like coordinates—such as linear-polarization degree $p_{L}$ and position angle
$\psi$. Consequently, we perform all statistical analyses and model fitting
(Appendix A) on (pre-processed, Appendix A.1) raw Stokes data.
It is, of course, straightforward to transform to more customary
parameters—e.g., linear-polarization degree $p_{L}$ (Eq. 1), position angle
$\psi$ (Eq. 2), and circular-polarization (signed) degree $p_{C}$ (Eq. 3):
$p_{L}=\sqrt{(Q^{2}+U^{2})}/I;$ (1) $\psi=\frac{1}{2}\tan^{-1}(U/Q);$ (2)
$p_{C}=V/I.$ (3)
For the three pulse features (MP, IP, and LFC), we estimate
$p_{L}(\varphi_{n})$, $\psi(\varphi_{n})$, and $p_{C}(\varphi_{n})$ at each
datum $n$ by substituting the measured $I_{n}$, $Q_{n}$, $U_{n}$, and $V_{n}$
into Equations 1, 2, and 3.
Direct estimate of customary polarization parameters of the main pulse versus
the phase-angle offset $\Delta\varphi$ from the MP center. From the top, the
plots display measured intensity $I$ data and then directly calculated
fractional linear polarization $p_{L}$, position angle $\psi$, and fractional
circular polarization $p_{C}$. The smooth solid lines show the best-fit phase-
dependent polarization properties based upon forward modeling of the Stokes
data (Table 1).
Direct estimate of customary polarization parameters of the interpulse versus
the phase-angle offset $\Delta\varphi$ from the IP center. From the top, the
plots display measured intensity $I$ data and then directly calculated
fractional linear polarization $p_{L}$, position angle $\psi$, and fractional
circular polarization $p_{C}$. The smooth solid lines show the best-fit phase-
dependent polarization properties based upon forward modeling of the Stokes
data (Table 1).
Figures 3 and 3 display the direct estimates of $I_{n}$, $p_{L\,n}$,
$\psi_{n}$, and $p_{C\,n}$ over the MP and IP, respectively. As the LFC is
quite weak relative to the MP and the IP, the plots for the LFC are too noisy
to display legibly. Even for the stronger features—MP and IP—the RMS noise in
the directly calculated polarization parameters ($p_{L\,n}$, $\psi_{n}$, and
$p_{C\,n})$, which serves as an estimator of the statistical error,
substantially increases away from the pulse center due to the low signal-to-
noise ratio per sample in the pulse wings. In order to deal effectively with
low-signal-to-noise data in the wings of the MP and IP and throughout the
(weaker) LFC, we adopt a more rigorous forward-modeling approach to fit the
measured Stokes data to the modeled $I(\varphi)$, $Q(\varphi)$, $U(\varphi)$,
and $V(\varphi)$:
$Q(\varphi)=I(\varphi)p_{L}(\varphi)\cos(2\psi(\varphi));$ (4)
$U(\varphi)=I(\varphi)p_{L}(\varphi)\sin(2\psi(\varphi));$ (5)
$V(\varphi)=I(\varphi)p_{C}(\varphi).$ (6)
Appendix A describes in some detail our approach for fitting polarization
models to the Stokes data. As Figures 3 and 3 indicate that neither
$p_{L}(\varphi)$, $\psi(\varphi)$, nor $p_{C}(\varphi)$ varies rapidly across
the pulse profile, the approach simply models $p_{L}(\varphi)$,
$\psi(\varphi)$, and $p_{C}(\varphi)$ as Taylor-series expansions in the
phase-angle offset $\Delta\varphi\equiv(\varphi-\varphi_{0})$ from the center
$\varphi_{0}$ of the respective pulse feature (MP, IP, or LFC). Table 1
tabulates the best-fit Taylor-expansion coefficients for the polarization
dependence upon phase-angle offset:
$p_{L}(\varphi)=p_{L0}+p^{\prime}_{L0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L0}(\varphi-\varphi_{0})^{2};$
(7)
$\psi(\varphi)=\psi_{0}+\psi^{\prime}_{0}(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}_{0}(\varphi-\varphi_{0})^{2};$
(8)
$p_{C}(\varphi)=p_{C0}+p^{\prime}_{C0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{C0}(\varphi-\varphi_{0})^{2}.$
(9)
Table 1: Best-fit polarization coefficients for the MP, IP, and LFC, using a single Gaussian for each pulse profile and up-to-quadratic variations in polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$. Parameter | Units | MP | IP | LFC
---|---|---|---|---
$\varphi_{0}-\varphi_{\rm MP}$ | ∘ | $\equiv 0$ | $145.389\pm 0.027$ | $-37.75\pm 0.19$
$p_{L0}$ | $\%$ | $22.98\pm 0.30$ | $21.3\pm 1.0$ | $98.2\pm 6.7$
$p^{\prime}_{L0}$ | $\%/^{\circ}$ | $-0.31\pm 0.19$ | $1.02\pm 0.62$ | $-0.8\pm 2.2$
$p^{\prime\prime}_{L0}$ | $\%/^{\circ}/^{\circ}$ | $0.88\pm 0.22$ | $-0.02\pm 0.63$ | $0.0\pm 1.3$
$\psi_{0}-\psi_{\rm MP}$ | ${}^{\circ}{\rm PA}$ | $\equiv 0$ | $-0.1\pm 1.3$ | $40.8\pm 1.5$
$\psi^{\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}$ | $-0.16\pm 0.20$ | $0.82\pm 0.78$ | $-0.16\pm 0.49$
$\psi^{\prime\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}/^{\circ}$ | $-0.06\pm 0.21$ | $1.00\pm 0.89$ | $-0.21\pm 0.28$
$p_{C0}$ | $\%$ | $-1.25\pm 0.20$ | $-3.15\pm 0.94$ | $20.5\pm 4.9$
$p^{\prime}_{C0}$ | $\%/^{\circ}$ | $0.01\pm 0.13$ | $0.38\pm 0.56$ | $0.3\pm 1.7$
$p^{\prime\prime}_{C0}$ | $\%/^{\circ}/^{\circ}$ | $-0.20\pm 0.15$ | $0.47\pm 0.57$ | $-0.49\pm 0.97$
An important conclusion of this study is that the Stokes data are
consistent—within statistical uncertainties—with constant polarization
position angle $\psi$ across each of the three pulse features (MP, IP, and
LFC) individually. However, the MP does exhibit a small but statistically
significant quadratic variation in the linear-polarization degree $p_{L}$.
While our 1.380-GHz polarimetry of the Crab pulsar has finer time resolution
and better statistical accuracy than previous 1.424-GHz polarimetry (Moffett &
Hankins, 1999), measured values for the polarization degree and position angle
(relative to MP) are mostly similar for the MP and for the IP. The only
significant difference is for the LFC’s linear polarization degree and
position angle. We measured nearly total ($98\%\pm 7\%$) linear polarization
at a $+40.8^{\circ}\pm 1.5^{\circ}$ position-angle offset from the MP, whereas
Moffett & Hankins (1999) found the LFC to be $\approx 40\%$ linearly polarized
at a $\approx+30^{\circ}$ position-angle offset from the MP. We also detect
circular polarization, which is moderately strong in the LFC ($20.5\pm 4.9\%$)
but weak and opposite polarity in the MP ($-1.3\%\pm 0.2\%$) and in the IP
($-3.2\%\pm 0.9\%$). In contrast with Moffett & Hankins, we find no
significant variation in the circular polarization across any of the three
pulse components MP, IP, and LFC.
Another important conclusion—albeit peripheral to the polarimetry—relates to
substructure in the pulse profile of the MP. The fine time resolution and
better statistical accuracy of our radio observation of the Crab pulsar
resulted in measurement of statistically significant substructure (Appendix
A.3) in the profile of the main pulse (Figure 3). The typical width of the
substructure is roughly $10\ \mu$s—i.e., $\leq 0.1$ the width of the MP
profile. As the current analysis utilizes the sum of all data collected during
the observation at a single epoch (2011 August 8), we have not assessed the
temporal behavior of the profile. However, we presume that this substructure
results from sporadic, very strong giant radio pulses (Bhat et al., 2008;
Karuppusamy et al., 2010; Majid et al., 2011; Hankins et al., 2003) occurring
during the 144-minute observation. Although the substructure is readily
apparent in the $I$ profile of the MP, the discernible subpulses contribute
only about $5\%$ of the fluence in the MP over the observation. However, they
likely result from only the strongest giant radio pulses in a distribution of
pulse amplitudes. Note that our conclusions as to the average pulse-phase
dependences of the polarimetry are effectively independent of the precise
modeling of the intensity profile of the MP. On the other hand, inspection of
the Stokes parameters (Figure 3) or polarization parameters (Figure 3)
indicates that the polarization of some of the subpulses (e.g., at phase
offset $\Delta\varphi\approx-2.3^{\circ}$) differs substantially from the
average polarization of the MP.
We also note that our WSRT-measured pulse-phase offsets of the IP and of the
LFC from the MP are in good agreement with contemporaneous measurements at
Jodrell Bank (Lyne et al., 2013). This tends to support the conclusion of Lyne
et al. (2013) that the phase separations of the IP and of the LFC from the MP
are evolving with time. Furthermore, the evolution of phase separations might
contribute to the difference between our measurement of the LFC’s polarization
and earlier measurements (Moffett & Hankins, 1999).
Stokes data $I$, $Q$, $U$, and $V$ versus pulse phase offset $\Delta\varphi$
from the center of the main pulse (MP). The lines represent the best-fit
(minimum-$\chi^{2}$) Stokes functions for a multi-Gaussian profile and up-to-
second-order variations in linear-polarization degree, position angle, and
circular-polarization degree. The pulse profile comprises 2 broad and 4 narrow
Gaussians.
## 4 Implications for Theoretical Models
Emission at altitudes comparable to the light-cylinder radius produce caustic
peaks, formed by cancellation of phase differences due to aberration and
retardation with that due to field-line curvature of radiation along the
trailing magnetic-field lines (Dyks & Rudak, 2003). In outer-magnetosphere
models, peaks in the light curves form when the observer’s sight line sweeps
across one or more bright caustic. The caustics display distinct linear-
polarization characteristics (Dyks et al., 2004), including fast sweeps of
position angle and dips in polarization degree at the peaks, which are caused
by piling up radiation emitted over a large range of altitudes and magnetic-
field directions into the caustics. These characteristics are in fact seen in
the optical polarization of the Crab pulsar (Słowikowska et al., 2009), which
exhibits rapid swings of position angle across both the MP and IP, as well as
dips in polarization degree to the 5% level on the trailing edge of each peak.
From the results presented in this paper, however, the characteristics of the
radio linear polarization of the MP and IP resemble neither those of caustics
in existing geometric models nor those observed in the optical emission. The
lack of position-angle swing in the radio MP and IP is in stark contrast to
the rapid position-angle swings in the optical. The very low circular
polarization and moderate linear polarization observed here in the radio MP
and IP are consistent with caustics, but the observed linear-polarization
values ($\approx 22\%$) in the radio are significantly higher than those in
the optical, and there is only a small variation with phase in the MP. On the
other hand, the radio pulses are much narrower than the optical pulses,
indicating that the radio MP and IP may originate along a smaller range of
altitudes and/or in a subset of field lines.
We have modeled the caustic emission and corresponding linear-polarization
degree $p_{L}$ and position angle $\psi$ for the Crab pulsar, with a
simulation using geometric renditions of standard slot-gap and outer-gap
emission. These geometric emission models assume constant emissivity in the
corotating frame along a set of field lines within the gaps, defined by a gap
width $w$ across field lines in open-volume coordinates (Dyks et al., 2004),
where the width is a fraction of radius of open magnetic field lines. As in
Dyks et al. (2004), the emission is assumed to occur over a fixed radius
range, from minimum $r_{\rm min}$ to maximum $r_{\rm max}$. For the
simulations of Crab polarization here, we explored gap widths
$w=0.002,0.01,0.02,0.05$, $r_{\rm min}=0.3-0.9\,R_{\rm LC}$ and $r_{\rm
max}=0.5-1.2\,R_{\rm LC}$, where $R_{\rm LC}=c/\Omega$ is the light-cylinder
radius. These are smaller ranges of altitude and smaller gap widths than in
standard slot-gap or outer-gap models used in Dyks et al. (2004), which were
$r_{\rm min}=R_{\rm NS}$, $r_{\rm max}=0.95\,R_{\rm LC}$ for the slot gap and
$r_{\rm min}=R_{\rm NC}$, $r_{\rm max}=0.97\,R_{\rm LC}$ for the outer gap.
Here $R_{\rm NS}$ is the neutron star radius and $R_{\rm NC}$ is the radius of
the null-charge surface, at which the magnetospheric charge density in the
corotating frame $\rho_{0}=\mathbf{\Omega\cdot B}/(2\pi c)$ vanishes.
We simulated emission using both retarded-vacuum-dipole (Deutsch, 1955), as in
Dyks et al. (2004), and force-free (Contopoulos & Kalapotharakos, 2010)
magnetic-field geometries, as in Harding et al. (2011). Then we computed light
curves and Stokes parameters for magnetic inclination angles
$\alpha=45^{\circ}-80^{\circ}$, with $5^{\circ}$ resolution for vacuum and
$15^{\circ}$ resolution for force-free magnetospheres, and observer viewing
angles $\zeta=55^{\circ}-80^{\circ}$ (both with respect to the rotation axis).
These ranges of $\alpha$ and $\zeta$ bracket the viewing angle of
$60^{\circ}-65^{\circ}$ suggested by modeling of the X-ray torus (Ng & Romani,
2008). Following Dyks et al. (2004), Blaskiewicz et al. (1991), and Hibschman
& Arons (2001), we assume that the photon electric-field vector is parallel to
the electron acceleration at each point along the field line to determine the
Stokes parameters.
Although simulated light curves for the smaller gap widths produce narrower
caustic peaks with less position-angle swing and depolarization, it is
difficult to produce both $\psi(\varphi)$ and $p_{L}(\varphi)$ curves with no
variation through the peaks. We compared a range of simulated light curves,
$p_{L}$ and $\psi$ to the ones observed, and found that none of the models
agree with the data. For the vacuum magnetospheres, the slot-gap model can
produce appropriately narrow peaks for $w<0.01$, but there is always some
change in $\psi$ through both the MP and IP. At $\zeta=60^{\circ}$, there are
dips in $p_{L}$ at only the first peak for $\alpha<75^{\circ}$ and dips at
both peaks for $\alpha>75^{\circ}$. The outer-gap model produces a change in
$\psi$ mostly in the IP but dips in $p_{L}$ in both peaks. While the force-
free geometry, whose poloidal field lines are straighter than those in vacuum,
can give a flatter position angle for certain inclination and viewing angles,
the model’s $p_{L}$ shows strong variation through the peaks in contradiction
with the data. For the force-free magnetospheres, the slot-gap model produces
much less change in $\psi$ at the peaks for $\zeta=55^{\circ}-65^{\circ}$ and
$\alpha=45^{\circ}-75^{\circ}$, but still not constant as observed. There is
also a high level of depolarization in both peaks but $p_{L}$ is not constant
through the peaks, as in the data. The outer gap in the force-free
magnetosphere also produces changes in $\psi$ and $p_{L}$ in both peaks for
these same ranges of $\alpha$ and $\zeta$.
For comparison with our measurement of the phase-resolved polarization
properties of the Crab pulsar, we simulated 66 total (48 vacuum and 18 force-
free) cases. Based upon inspection of the results of these numerous simulated
cases, the model light curve and polarization characteristics that seem to
resemble most the Crab pulsar radio data is for the case of the slot-gap model
in the force-free magnetosphere with $\alpha=45^{\circ}$ and
$\zeta=60^{\circ}$. Figure 4 displays the results for this model for the MP.
Note that this model does predict a rapid swing in polarization position angle
and degree which we do not see; however, these swings occur on the preceding
wing of the pulse, when the intensity is very low.
Predicted relative variation through the MP of the intensity $I$ (red), linear
polarization degree $p_{L}$ (green), and position angle $\psi$ (blue) for the
slot-gap model, with a force-free magnetosphere. For this case, the magnetic
inclination angle $\alpha=45^{\circ}$ and observer viewing angle
$\zeta=60^{\circ}$ with respect to the spin axis. The ordinate range 0–1
corresponds to zero to peak intensity for $I$, 0%–100% polarization for
$p_{L}$, and $-90^{\circ}$ to $90^{\circ}$ for $\psi$.
In order to explore the possibility that the linear-polarization degree
$p_{L}$ or position angle $\psi$ changes sharply in the preceding wing of the
MP (as in Figure 4), we fit the Stokes data to a simple model of a step jump
in the values of $p_{L}$ and of $\psi$ at a pulse phase $\varphi_{\rm step}$.
$p_{L}(\varphi)=p_{L0}+\Delta p_{L}\;\Theta(\varphi_{\rm step}-\varphi);$ (10)
$\psi(\varphi)=\psi_{0}+\Delta\psi\;\Theta(\varphi_{\rm step}-\varphi).$ (11)
Here, $p_{L0}$ and $\psi_{0}$ are the best-fit values for constant linear-
polarization degree and position angle; $\Delta p_{L}$ and $\Delta\psi$, the
pre-step differences in the value of each; and $\Theta(\varphi_{\rm
step}-\varphi)$, the unit step distribution ($=1$ for $\varphi<\varphi_{\rm
step}$, 0 otherwise). Figure 4 shows the best-fit differences and their
(1-sigma) uncertainties as functions of pulse phase of the step (relative to
pulse center). From this analysis, we conclude that any position-angle swing
must be small—$|\Delta\psi|<10^{\circ}$ for $\varphi_{\rm step}>-3.5^{\circ}$.
A large position-angle swing—$|\Delta\psi|>45^{\circ}$, say—is consistent with
the data (but not required) only for $\varphi_{\rm step}<-4^{\circ}$. Note
that the analysis requires $\Delta p_{L}>0$ for $\varphi_{\rm
step}\geq-2.5^{\circ}$ (and allows it for earlier $\varphi_{\rm step}$), as
this analysis does not include the small positive second derivative
$p^{\prime\prime}_{L0}$ in the linear-polarization degree, which the Taylor-
expansion fit to the MP Stokes data requires (cf. Table 1).
Constraints on a sharp step in the MP linear-polarization degree
($p_{L}(\varphi)$, left) and position angle ($\psi(\varphi)$, right) versus
the putative step’s pulse phase $\varphi_{\rm step}$ (relative to the MP
center). Large position-angle swings ($|\Delta\psi|>45^{\circ}$, say) are
allowed (but not required) only very early ($\varphi_{\rm step}<-4^{\circ}$)
in the pulse—i.e., where the signal-to-noise ratio is low.
It is possible that the radio linear polarization in the MP and LP is very
sensitive to the magnetic-field structure. Existing models explored only the
two extremes of vacuum (accelerating fields but no plasma) and force-free
(plasma but no accelerating fields), neither of which describe real pulsars.
More realistic, dissipative magnetosphere models with finite conductivity now
exist (Kalapotharakos et al., 2012; Li et al., 2012) and should be used to
model light curves and polarization characteristics. It is also possible that
the radio emission in the MP and IP occurs along sets of field lines that lie
deeper within the open/closed field boundary or the current sheet and have
different polarization properties.
The low-frequency component (LFC) is substantially weaker than the MP and IP
at 1.4 GHz. As its name suggests, the LFC is not detected at radio frequencies
higher than a few GHz and has no corresponding component in the visible band.
The nearly complete radio polarization ($p_{L}\approx 98\%$ and $p_{C}\approx
20\%$) of the LFC support the hypothesis that it is a highly coherent, low-
altitude component. Note that the (lower frequency) precursor is also believed
to be a highly coherent, low-altitude component, due to its high polarization
and steep spectrum (Rankin, 1990).
## 5 Conclusions
Our 1.38-GHz observations of the Crab pulsar measured significant linear and
circular polarization in the three most prominent pulse components—the main
pulse (MP), inter pulse (IP), and low-frequency component (LFC). These results
are mostly in agreement with previous measurements of linear polarization at
similar radio frequencies (cf. Moffett & Hankins, 1999). The MP and IP are
moderately linearly polarized ($p_{L}\approx 23\%$ and $21\%$, respectively)
at the same position angle ($\psi_{\rm IP}-\psi_{\rm MP}\approx 0$); they are
weakly circularly polarized ($p_{C}\approx-1.3\%$ and $-3.2\%$, respectively).
In contrast, the LFC is very strongly linearly polarized ($p_{L}\approx
98\%$), at a position angle $+40^{\circ}$ from that of the MP or IP, and
moderately circularly polarized ($p_{C}\approx 20\%$).
The fine time resolution (Period/8192 = 4.1 $\mu$s) and good sensitivity of
the measurements at the Westerbork Synthesis Radio Telescope (WSRT) enabled a
meaningful search for changes in linear-polarization degree $p_{L}$, in
position angle $\psi$, and in circular-polarization degree $p_{C}$ across each
of the three pulse components. Neither the MP, IP, nor LFC exhibits a
statistically significant change in the polarization position angle or
circular polarization across the pulse. For the MP, the linear term (“sweep”)
is well constrained: $\psi^{\prime}_{0\,\rm MP}=(-0.16\pm 0.20)^{\circ}{\rm
PA}/^{\circ}$. Likewise, neither the IP nor LFC displays a statistically
significant change in the polarization degree. However, the MP does show a
small but statistically significant quadratic variation in linear-polarization
degree—$p^{\prime\prime}_{L0\,\rm MP}=(0.88\pm 0.22)\%/^{\circ}/^{\circ}$
about its central value—$p_{L0\,\rm MP}=(23.0\pm 0.3)\%$—for a pulse-average
linear polarization $\overline{p}_{L\,\rm MP}=(23.7\pm 0.3)\%$.
Our analysis of the radio Stokes data shows no strong sweep of the linear-
polarization position angle. This lack of strong position-angle swings
contrasts with the rapid swings observed in the visible band. Current models
for pulsar emission geometries do not readily account for the absence of
substantial variations in both polarization degree and position angle across a
pulse component (§ 4). Thus, alternative models—e.g., dissipative
magnetopheres—should be considered in modeling the radio polarization of the
Crab pulsar’s MP and IP. The nearly complete polarization of the LFC suggest
that it originates at a different location and via a different mechanism than
do the stronger MP and IP.
Finally, the fine time resolution and high signal-to-noise ratio in the MP
data led to detection of statistically significant substructure in its pulse
profile. We surmise that this substructure results from giant radio pulses
occurring during the 144-minute observation.
Acknowledgments
The Westerbork Synthesis Radio Telescope (WSRT) is operated by ASTRON, the
Netherlands Institute for Radio Astronomy, with support from NWO, the
Netherlands Foundation for Scientific Research. AS acknowledges grant
DEC-2011/03/D/ST9/00656 from the Polish National Science Centre; BWS, a
Consolidated Grant from the UK Science and Technology Facilities Council; AKH,
NASA grants Astrophysics Theory 12-ATP12-0169 and Fermi Guest Investigator
11-FERMI11-0052; AJvdH, Advanced Investigator Grant 247295 (PI: R. A. M. J.
Wijers) from the European Research Council; and SLO, RFE and MCW, support by
NASA’s Chandra Program.
## Appendix A Statistical analysis
### A.1 Procedures
As Figure 2 shows, the main pulse (MP), interpulse (IP), and low-frequency
component (LFC) are well separated in the 1.38-GHz data folded on the Crab
pulsar’s period. Consequently, we choose to analyze each of these three
features individually, using phase ranges $(-7.2^{\circ},7.2^{\circ})$ for the
MP, $(134.6^{\circ},156.2^{\circ})$ for the IP, and
$(-52.1^{\circ},-23.3^{\circ})$ for the LFC, where the center of the MP
defines pulse-phase angle $\varphi=0^{\circ}$. We use data over the remaining
phase ranges to measure the off-pulse mean and the root-of-mean-square (RMS)
noise in $I$, $Q$, $U$, and $V$. Upon measuring the off-pulse mean values for
$I$, we noticed that its off-pulse value near the MP is depressed with respect
to the remaining phase ranges. Specifically, in phase ranges
$(-14.4^{\circ},-7.2^{\circ})$ and $(7.2^{\circ},14.4^{\circ})$, the mean $I$
is 0.0273 ($\times 1000)$ less than in other off-pulse ranges. Taking this
into account lowered $\chi^{2}$ by about 300 in fitting the $I$ pulse profile,
but did not significantly alter the fitted polarization properties.
For convenience, we pre-process the raw data by subtracting the respective
off-pulse mean value, under the assumption that the expectation values for
$I$, $Q$, $U$, and $V$ are zero away from pulse features. Furthermore, we take
the RMS noise levels—0.0324, 0.0310, 0.0311, and 0.0307 (each $\times
1000$)—as estimators of the statistical standard deviations $\sigma_{I}$,
$\sigma_{Q}$, $\sigma_{U}$, and $\sigma_{V}$, respectively.
In order to fit the model to the data for each pulse feature, we minimize the
chi-square statistic of the combined Stokes data
$\displaystyle\chi^{2}(\varpi)=\chi_{I}^{2}(\varpi)+\chi_{Q}^{2}(\varpi)+\chi_{U}^{2}(\varpi)+\chi_{V}^{2}(\varpi)=$
$\displaystyle\sum_{n=1}^{N}\left[\frac{(I_{n}-I(\varphi_{n};\varpi))^{2}}{\sigma_{I}^{2}}+\frac{(Q_{n}-Q(\varphi_{n};\varpi))^{2}}{\sigma_{Q}^{2}}+\frac{(U_{n}-U(\varphi_{n};\varpi))^{2}}{\sigma_{U}^{2}}+\frac{(V_{n}-V(\varphi_{n};\varpi))^{2}}{\sigma_{V}^{2}}\right],$
with respect to a set $\varpi$ of $K$ model parameters, leaving $\nu=N-K$
degrees of freedom. We obtain the statistical uncertainty in each parameter,
based upon $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}$. To perform the
$\chi^{2}$ analysis, we used the MathematicaTM (Wolfram, 2013) function
NonlinearModelFit222http://reference.wolfram.com/mathematica/ref/NonlinearModelFit.html,
which finds best-fit model parameters, their errors, correlation matrix
amongst them, etc.
Modeling the Stokes data requires parameterized functions for the pulse
profile $I(\varphi)$, the linear-polarization fraction $p_{L}(\varphi)$, the
polarization position angle $\psi(\varphi)$, and the circular-polarization
fraction $p_{C}(\varphi)$ (cf. Equations 4, 5, and 6 for $Q(\varphi)$,
$U(\varphi)$, and $V(\varphi)$, respectively). As there is no evidence for
rapid changes in polarization degree or position angle over a pulse feature
(cf. Figures 3 and 3), simple Taylor-series expansions suffice:
$\displaystyle p_{L}(\varphi)$ $\displaystyle=$ $\displaystyle
p_{L}(\varphi_{0})+p^{\prime}_{L}(\varphi_{0})(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L}(\varphi_{0})(\varphi-\varphi_{0})^{2}+\cdots$
(A2) $\displaystyle\equiv$ $\displaystyle
p_{L0}+p^{\prime}_{L0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L0}(\varphi-\varphi_{0})^{2}+\cdots;$
$\displaystyle\psi(\varphi)$ $\displaystyle=$
$\displaystyle\psi(\varphi_{0})+\psi^{\prime}(\varphi_{0})(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}(\varphi_{0})(\varphi-\varphi_{0})^{2}+\cdots$
(A3) $\displaystyle\equiv$
$\displaystyle\psi_{0}+\psi^{\prime}_{0}(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}_{0}(\varphi-\varphi_{0})^{2}+\cdots;$
$\displaystyle p_{C}(\varphi)$ $\displaystyle=$ $\displaystyle
p_{C}(\varphi_{0})+p^{\prime}_{C}(\varphi_{0})(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{C}(\varphi_{0})(\varphi-\varphi_{0})^{2}+\cdots$
(A4) $\displaystyle\equiv$ $\displaystyle
p_{C0}+p^{\prime}_{C0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{C0}(\varphi-\varphi_{0})^{2}+\cdots.$
To parameterize the pulse profile, we use a Gaussian (§A.2) for each pulse
feature (MP, IP, or LFC) or multiple Gaussians (§A.3) for the MP.
### A.2 Single-Gaussian fits to the MP, the IP, and to the LFC
To complete the parameterized model for the four Stokes functions, we assume a
Gaussian profile:
$I(\varphi)=I_{0}\exp\left(-\frac{(\varphi-\varphi_{0})^{2}}{2\sigma^{2}_{\varphi}}\right),$
(A5)
with $I_{0}$ the value of $I(\varphi)$ at pulse center, $\sigma_{\varphi}$ the
Gaussian width, and $\varphi_{0}$ the phase at the pulse center. Combining
this parameterization with Equations 4, 5, 6, A2, A3, A4, the full model for
the other three Stokes functions follows:
$\displaystyle Q(\varphi)$ $\displaystyle=$ $\displaystyle
I_{0}\exp\left(-\frac{(\varphi-\varphi_{0})^{2}}{2\sigma^{2}_{\varphi}}\right)[p_{L0}+p^{\prime}_{L0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L0}(\varphi-\varphi_{0})^{2}]$
(A6)
$\displaystyle\times\cos(2[\psi_{0}+\psi^{\prime}_{0}(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}_{0}(\varphi-\varphi_{0})^{2}]);$
$\displaystyle U(\varphi)$ $\displaystyle=$ $\displaystyle
I_{0}\exp\left(-\frac{(\varphi-\varphi_{0})^{2}}{2\sigma^{2}_{\varphi}}\right)[p_{L0}+p^{\prime}_{L0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L0}(\varphi-\varphi_{0})^{2}]$
(A7)
$\displaystyle\times\sin(2[\psi_{0}+\psi^{\prime}_{0}(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}_{0}(\varphi-\varphi_{0})^{2}]);$
$\displaystyle V(\varphi)$ $\displaystyle=$ $\displaystyle
I_{0}\exp\left(-\frac{(\varphi-\varphi_{0})^{2}}{2\sigma^{2}_{\varphi}}\right)[p_{C0}+p^{\prime}_{C0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{C0}(\varphi-\varphi_{0})^{2}].$
(A8)
Stokes data $I$, $Q$, $U$, and $V$ versus pulse phase offset $\Delta\varphi$
from the center of the main pulse (MP). The lines represent the best-fit
(minimum-$\chi^{2}$) Stokes functions for a single-Gaussian profile and up-to-
second-order variations in polarization degree and in position angle.
Stokes data $I$, $Q$, $U$, and $V$ versus pulse phase offset $\Delta\varphi$
from the center of the inter pulse (IP). The lines represent the best-fit
(minimum-$\chi^{2}$) Stokes functions for a single-Gaussian profile and up-to-
second-order variations in polarization degree and in position angle.
Stokes data $I$, $Q$, $U$, and $V$ versus pulse phase offset $\Delta\varphi$
from the center of the low-frequency component (LFC). The lines represent the
best-fit (minimum-$\chi^{2}$) Stokes functions for a single-Gaussian profile
and up-to-second-order variations in polarization degree and in position
angle.
Figures A.2, A.2, and A.2 display Stokes data for the MP, IP, and LFC,
respectively. The lines represent best-fit (minimum-$\chi^{2}$) Stokes
functions (Equations A5, A6, A7, and A8) for a single-Gaussian profile
$I(\varphi)$ and up-to-quadratic variations in linear-polarization degree
$p_{L}(\varphi)$, in position angle $\psi(\varphi)$, and in circular-
polarization degree $p_{C}(\varphi)$. Tables 2, 3, and 4 tabulate the results
of the $\chi^{2}$ analysis for a Gaussian profile and retaining polarization
terms (Equations A6, A7, and A8) through, zeroth, first, and second order,
respectively. For each pulse feature—MP, IP, and LFC—the tables list the
minimum $\chi^{2}$ and degrees of freedom $\nu$ for $I$, $Q$, $U$, and $V$
data sets combined and separately, followed by best-fit estimators and
(1-sigma) uncertainties for the 3 pulse-profile parameters ($I_{0}$,
$\sigma_{\varphi}$, $\varphi_{0}$) and for the relevant polarization
coefficients ($p_{L0}$, $p^{\prime}_{L0}$, $p^{\prime\prime}_{L0}$;
$\psi_{0}$, $\psi^{\prime}_{0}$, $\psi^{\prime\prime}_{0}$; $p_{C0}$,
$p^{\prime}_{C0}$, $p^{\prime\prime}_{C0}$). Note that these tables reference
the pulse-phase angles ($\varphi_{0}$) and polarization position angles
($\psi_{0}$) to the MP, as we set $\varphi_{\rm MP}\equiv 0$ and were unable
to obtain an absolute measurement of position angle $\psi_{\rm MP}$.
Table 2: Best-fit parameters for the MP, IP, and the LFC, using a simple Gaussian for each profile and no variations in polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$. Parameter | Units | MP | IP | LFC
---|---|---|---|---
$\chi^{2}/\nu$ | | $3081./1302$ | $2022./1962$ | $2518./2618$
$\chi^{2}_{I}/\nu_{I}$ | | $1910./324$ | $561./489$ | $577./653$
$\chi^{2}_{Q}/\nu_{Q}$ | | $460./322$ | $534./487$ | $674./651$
$\chi^{2}_{U}/\nu_{U}$ | | $441./322$ | $463./487$ | $603./651$
$\chi^{2}_{V}/\nu_{V}$ | | $269./323$ | $463./488$ | $664./652$
$I_{0}$ | $\times 1000$ | $1.9894\pm 0.0046$ | $0.4414\pm 0.0044$ | $0.0668\pm 0.0031$
$\sigma_{\varphi}$ | ∘ | $1.7801\pm 0.0047$ | $1.947\pm 0.022$ | $3.40\pm 0.14$
$\varphi_{0}-\varphi_{\rm MP}$ | ∘ | $\equiv 0$ | $145.399\pm 0.023$ | $-37.79\pm 0.14$
$p_{L0}$ | $\%$ | $23.67\pm 0.19$ | $21.24\pm 0.81$ | $98.3\pm 5.7$
$\psi_{0}-\psi_{\rm MP}$ | ${}^{\circ}{\rm PA}$ | $\equiv 0$ | $1.0\pm 1.1$ | $40.3\pm 1.2$
$p_{C0}$ | $\%$ | $-1.40\pm 0.18$ | $-2.70\pm 0.78$ | $19.0\pm 4.0$
Table 3: Best-fit parameters for the MP, the IP, and the LFC, using a simple Gaussian for each profile and up-to-linear variations in polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$. Parameter | Units | MP | IP | LFC
---|---|---|---|---
$\chi^{2}/\nu$ | | $3076./1299$ | $2017./1959$ | $2517./2615$
$\chi^{2}_{I}/\nu_{I}$ | | $1910./324$ | $561./489$ | $577./653$
$\chi^{2}_{Q}/\nu_{Q}$ | | $456./320$ | $532./485$ | $674./649$
$\chi^{2}_{U}/\nu_{U}$ | | $440./320$ | $462./485$ | $602./649$
$\chi^{2}_{V}/\nu_{V}$ | | $269./322$ | $463./487$ | $664./651$
$I_{0}$ | $\times 1000$ | $1.9894\pm 0.0046$ | $0.4415\pm 0.0044$ | $0.0668\pm 0.0031$
$\sigma_{\varphi}$ | ∘ | $1.7801\pm 0.0047$ | $1.946\pm 0.022$ | $3.40\pm 0.14$
$\varphi_{0}-\varphi_{\rm MP}$ | ∘ | $\equiv 0$ | $145.389\pm 0.023$ | $-37.74\pm 0.20$
$p_{L0}$ | $\%$ | $23.67\pm 0.19$ | $21.25\pm 0.81$ | $98.3\pm 5.7$
$p^{\prime}_{L0}$ | $\%/^{\circ}$ | $-0.32\pm 0.15$ | $1.09\pm 0.59$ | $-0.9\pm 2.4$
$\psi_{0}-\psi_{\rm MP}$ | ${}^{\circ}{\rm PA}$ | $\equiv 0$ | $0.9\pm 1.1$ | $40.3\pm 1.2$
$\psi^{\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}$ | $-0.15\pm 0.18$ | $0.91\pm 0.78$ | $-0.18\pm 0.48$
$p_{C0}$ | $\%$ | $-1.40\pm 0.18$ | $-2.70\pm 0.78$ | $19.0\pm 4.0$
$p^{\prime}_{C0}$ | $\%/^{\circ}$ | $-0.01\pm 0.14$ | $0.38\pm 0.57$ | $0.3\pm 1.7$
Table 4: Best-fit parameters for the MP, for the IP, and for the LFC, using a simple Gaussian for each profile and up-to-quadratic variations in polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$. Parameter | Units | MP | IP | LFC
---|---|---|---|---
$\chi^{2}/\nu$ | | $3049./1296$ | $2016./1956$ | $2517./2612$
$\chi^{2}_{I}/\nu_{I}$ | | $1909./324$ | $561./489$ | $577./653$
$\chi^{2}_{Q}/\nu_{Q}$ | | $432./318$ | $531./483$ | $674./647$
$\chi^{2}_{U}/\nu_{U}$ | | $440./318$ | $461./483$ | $603./647$
$\chi^{2}_{V}/\nu_{V}$ | | $268./321$ | $462./486$ | $664./650$
$I_{0}$ | $\times 1000$ | $1.9927\pm 0.0047$ | $0.4414\pm 0.0045$ | $0.0666\pm 0.0034$
$\sigma_{\varphi}$ | ∘ | $1.7742\pm 0.0048$ | $1.947\pm 0.023$ | $3.42\pm 0.20$
$\varphi_{0}-\varphi_{\rm MP}$ | ∘ | $\equiv 0$ | $145.389\pm 0.023$ | $-37.75\pm 0.20$
$p_{L0}$ | $\%$ | $22.99\pm 0.23$ | $21.24\pm 0.99$ | $98.1\pm 7.0$
$p^{\prime}_{L0}$ | $\%/^{\circ}$ | $-0.32\pm 0.15$ | $1.03\pm 0.59$ | $-0.9\pm 2.4$
$p^{\prime\prime}_{L0}$ | $\%/^{\circ}/^{\circ}$ | $0.86\pm 0.17$ | $-0.04\pm 0.61$ | $0.1\pm 1.4$
$\psi_{0}-\psi_{\rm MP}$ | ${}^{\circ}{\rm PA}$ | $\equiv 0$ | $-0.1\pm 1.3$ | $40.8\pm 1.4$
$\psi^{\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}$ | $-0.16\pm 0.17$ | $0.82\pm 0.79$ | $-0.16\pm 0.48$
$\psi^{\prime\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}/^{\circ}$ | $-0.06\pm 0.18$ | $1.07\pm 0.80$ | $-0.21\pm 0.28$
$p_{C0}$ | $\%$ | $-1.25\pm 0.22$ | $-3.15\pm 0.96$ | $20.5\pm 4.9$
$p^{\prime}_{C0}$ | $\%/^{\circ}$ | $0.01\pm 0.15$ | $0.38\pm 0.57$ | $0.3\pm 1.7$
$p^{\prime\prime}_{C0}$ | $\%/^{\circ}/^{\circ}$ | $-0.20\pm 0.16$ | $0.47\pm 0.59$ | $-0.49\pm 0.96$
Table 3 documents that, to within statistical uncertainties,
$p^{\prime}_{L0}=0$, $\psi^{\prime}_{0}=0$, and $p^{\prime}_{C0}=0$ for each
of the three pulse features—MP, IP, or LFC. Equivalently, including the three
linear coefficients $p^{\prime}_{L0}=0$, $\psi^{\prime}_{0}=0$, and
$p^{\prime}_{C0}=0$, does not result in a statistically significant reduction
in the value of $\chi^{2}_{\rm min}$ (cf. Tables 2 and 3). In contrast,
including the quadratic parameter $p^{\prime\prime}_{L0}$ does significantly
reduce the value of $\chi^{2}_{\rm min}$ for the MP (cf. Table 4 with Table 3
or 2), but not for the IP nor for the LFC.
### A.3 Comparison of model fits to MP
Table 2 shows that a single-Gaussian profile and constant polarization degree
and position angle provide a statistically adequate fit to the Stokes data for
the IP and for the LFC. However, the simple model does not provide a
statistically adequate fit to the Stokes data for the MP, at least in part due
to the higher signal-to-noise ratio in the MP Stokes data. Consequently, we
here investigate more complicated models in order to improve the goodness of
the $\chi^{2}$ fits to the MP Stokes data. In particular, we investigate using
a multi-Gaussian function for the MP pulse profile. Table 5 lists the minimum
$\chi^{2}$ and degrees of freedom $\nu$ for $I$, $Q$, $U$, and $V$ data sets
combined and separately, followed by best-fit estimators and (1-sigma)
uncertainties for the 9 polarization coefficients ($p_{L0}$,
$p^{\prime}_{L0}$, $p^{\prime\prime}_{L0}$; $\psi_{0}$, $\psi^{\prime}_{0}$,
$\psi^{\prime\prime}_{0}$; $p_{C0}$, $p^{\prime}_{C0}$,
$p^{\prime\prime}_{C0}$) of the Taylor expansion through second order.
Table 5: Comparison of results of fitting the main pulse (MP) profile with a simple Gaussian, with a multi-Gaussian, and with a simple Gaussian after adjusting weightings. The models retain up-to-quadratic variations in the polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$. Parameter | Units | 1-Gaussian | 6-Gaussian | 1-Gaussian (Adj.)
---|---|---|---|---
$\chi^{2}/\nu$ | | $3049./1296$ | $1823./1281$ | $1281./1296$
$\chi^{2}_{I}/\nu_{I}$ | | $1909./324$ | $688./309$ | $324./324$
$\chi^{2}_{Q}/\nu_{Q}$ | | $432./318$ | $430./303$ | $318./318$
$\chi^{2}_{U}/\nu_{U}$ | | $440./318$ | $438./303$ | $318./318$
$\chi^{2}_{V}/\nu_{V}$ | | $268./321$ | $267./306$ | $321./321$
$p_{L0}$ | $\%$ | $22.99\pm 0.23$ | $22.91\pm 0.24$ | $22.98\pm 0.30$
$p^{\prime}_{L0}$ | $\%/^{\circ}$ | $-0.32\pm 0.15$ | $-0.29\pm 0.15$ | $-0.31\pm 0.19$
$p^{\prime\prime}_{L0}$ | $\%/^{\circ}/^{\circ}$ | $0.86\pm 0.17$ | $0.89\pm 0.19$ | $0.88\pm 0.22$
$\psi_{0}$ | ${}^{\circ}{\rm PA}$ | $-89.34\pm 0.27$ | $-89.38\pm 0.29$ | $-89.34\pm 0.32$
$\psi^{\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}$ | $-0.16\pm 0.17$ | $-0.19\pm 0.17$ | $-0.16\pm 0.20$
$\psi^{\prime\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}/^{\circ}$ | $-0.06\pm 0.18$ | $0.05\pm 0.20$ | $-0.06\pm 0.21$
$p_{C0}$ | $\%$ | $-1.25\pm 0.22$ | $-1.27\pm 0.23$ | $-1.25\pm 0.20$
$p^{\prime}_{C0}$ | $\%/^{\circ}$ | $-0.01\pm 0.15$ | $-0.02\pm 0.14$ | $-0.01\pm 0.13$
$p^{\prime\prime}_{C0}$ | $\%/^{\circ}/^{\circ}$ | $-0.20\pm 0.16$ | $-0.18\pm 0.18$ | $-0.20\pm 0.15$
Comparison of the column “MP” in Table 3 with that in Table 4 (or,
equivalently, with the column “1-Gaussian” in Table 5) finds that inclusion of
the three quadratic polarization coefficients—especially
$p^{\prime\prime}_{L0}$—reduces $\chi^{2}_{Q}$ by 42 (from 473 to 431). While
$\psi^{\prime\prime}_{0}=0$ and $p^{\prime\prime}_{C0}=0$ within statistical
uncertainties, $p^{\prime\prime}_{L0}\approx(0.9\pm 0.2)\%/^{\circ}/^{\circ}$
is statistically significant but small.
The main cause of the poor fit of the 1-Gaussian model to the MP data,
however, has nothing to do with polarization. Figure 3 illustrates that, for
the fine time resolution and the high signal-to-noise ratio of the MP data,
substructure in the pulse profile is quite evident. Using a 6-Gaussian (2
broad and 4 narrow) profile for $I(\varphi)$ substantially improves the fit.
Comparing the column “6-Gaussian” with “1-Gaussian” in Table 5 finds that
inclusion of $15=5\times 3$ additional (Gaussian) parameters reduces
$\chi^{2}_{I}$ by 1221 (from 1909 to 688). Even so, the fit to the Stokes data
is not formally acceptable.
It is important to note that the best-fit expectation values and uncertainties
for the polarization coefficients ($p_{L0}$, $p^{\prime}_{L0}$,
$p^{\prime\prime}_{L0}$; $\psi_{0}$, $\psi^{\prime}_{0}$,
$\psi^{\prime\prime}_{0}$; $p_{C0}$, $p^{\prime}_{C0}$,
$p^{\prime\prime}_{C0}$) are rather insensitive to details of the pulse
profile. Thus, we compensate for fine substructure in the pulse profile by
increasing the estimators for the measurement standard deviations until a
statistically acceptable fit is achieved. That is, we adjust $\sigma_{I}$,
$\sigma_{Q}$, $\sigma_{U}$, and $\sigma_{V}$ until (Eq. A.1)
$\chi_{I}^{2}/\nu_{I}$, $\chi_{Q}^{2}/\nu_{Q}$, $\chi_{U}^{2}/\nu_{U}$, and
$\chi_{V}^{2}/\nu_{V}$, respectively, are close to unity. The column
“1-Gaussian (Adj.)” in Table 5 shows the best-fit polarization parameters for
a single-Gaussian profile, with weightings adjusted as described. The only
noticeable effect of this adjustment upon the best-fit polarization parameters
is a small change—typically an increase—in their uncertainties. The
uncertainties quoted in Table 1 (§3) are the typically more conservative
values obtained using the single-Gaussian profiles and adjusted weightings.
## References
* Abdo et al. (2013) Abdo, A. A., Ajello, M., Allafort, A., et al. 2013, ApJS, 208, 17
* Abdo et al. (2012) Abdo, A. A., Allen, B. T., Atkins, R., et al. 2012, ApJ, 750, 63
* Aharonian et al. (2006) Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al. 2006, A&A, 457, 899
* Aharonian et al. (2004) Aharonian, F., Akhperjanian, A., Beilicke, M., et al. 2004, ApJ, 614, 897
* Aleksić et al. (2012) Aleksić, J., Alvarez, E. A., Antonelli, L. A., et al. 2012, A&A, 540, A69
* Aliu et al. (2011) Aliu, E., Arlen, T., Aune, T., et al. 2011, Science, 334, 69
* Allen & Yodh (2007) Allen, B. T., & Yodh, G. B. 2007, AIP Conf. Ser. 921, 528
* Backer, Wong & Valanju (2000) Backer, D. C., Wong, T. & Valanju, J. 2000 ApJ, 543, 740
* Bai & Spitkovsky (2010) Bai, N. & Spitkovsky, A. 2010, ApJ, 715, 1282
* Bhat et al. (2008) Bhat, N. D. R., Tingay, S. J., & Knight, H. S. 2008, ApJ, 676, 1200
* Blaskiewicz et al. (1991) Blaskiewicz, M., Cordes, J. M., & Wasserman, I. 1991, ApJ, 370, 643
* Bridle (1970) Bridle, A. H. 1970, Nature, 225, 1035
* Bühler & Blandford (2013) Bühler, R., & Blandford, R. 2013, arXiv:1309.7046 in preparation
* Contopoulos & Kalapotharakos (2010) Contopoulos, I., & Kalapotharakos, C. 2010, MNRAS, 404, 767
* Dean et al. (2008) Dean, A. J., Clark, D. J., Stephen, J. B., et al. 2008, Science, 321, 1183
* Deutsch (1955) Deutsch, A. J. 1955, Annales d’Astrophysique, 18, 1
* Dyks et al. (2004) Dyks, J., Harding, A. K., & Rudak, B. 2004, ApJ, 606, 1125
* Dyks & Rudak (2003) Dyks, J., & Rudak, B. 2003, ApJ, 598, 1201
* Espinoza et al. (2013) Espinoza, C. M., Guillemot, L., Çelik, Ö., et al. 2013, MNRAS, 430, 571
* Forot et al. (2008) Forot, M., Laurent, P., Grenier, I. A., Gouiffès, C., & Lebrun, F. 2008, ApJ, 688, L29
* Gould & Lyne (1998) Gould, D. M., & Lyne, A. G. 1998, MNRAS, 301, 235
* Hankins et al. (2003) Hankins, T. H., Kern, J. S., Weatherall, J. C., & Eilek, J. A. 2003, Nature, 422, 141
* Harding et al. (2011) Harding, A. K., DeCesar, M. E., Miller, M. C., Kalapotharakos, C. & Contopoulos, I. 2011, Proc. of 2011 Fermi Symposium, eConf C110509 [arXiv:1111.0828]
* Hibschman & Arons (2001) Hibschman, J. A. & Arons, J. 2001, ApJ, 546, 382
* Jones et al. (1981) Jones, D. H. P., Smith, F. G., & Wallace, P. T. 1981, MNRAS, 196, 943
* Kalapotharakos et al. (2012) Kalapotharakos, C., Kazanas, D., Harding, A., & Contopoulos, I. 2012, ApJ, 749, 2
* Karastergiou & Johnston (2006) Karastergiou, A., & Johnston, S. 2006, MNRAS, 365, 353
* Karuppusamy et al. (2008) Karuppusamy, R., Stappers, B., & van Straten, W. 2008, PASP, 120, 191
* Karuppusamy et al. (2010) Karuppusamy, R., Stappers, B. W., & van Straten, W. 2010, A&A, 515, A36
* Kuiper et al. (2001) Kuiper, L., Hermsen, W., Cusumano, G., et al. 2001, A&A, 378, 918
* Kuzmin et al. (2008) Kuzmin, A., Losovsky, B. Y., Jordan, C. A. & Smith, F. G. 2008, A&A, 483, 13
* Li et al. (2012) Li, J., Spitkovsky, A., & Tchekhovskoy, A. 2012, ApJ, 746, 60
* Lyne & Graham-Smith (2006) Lyne, A. G., & Graham-Smith, F. 2006, Pulsar Astronomy, 3rd ed. (Cambridge, UK: Cambridge University Press), ISBN 0521839548
* Lyne et al. (2013) Lyne, A., Graham-Smith, F., Weltevrede, P., et al. 2013, Science, 342, 598
* Majid et al. (2011) Majid, W. A., Naudet, C. J., Lowe, S. T., & Kuiper, T. B. H. 2011, ApJ, 741, 53
* Manchester (1971) Manchester, R. N. 1971, ApJS, 23, 283
* Manchester et al. (2005) Manchester, R. N., Hobbs, G. B., Teoh, A., & Hobbs, M. 2005, AJ, 129, 1993
* Moffett & Hankins (1996) Moffett, D. A., & Hankins, T. H. 1996, ApJ, 468, 779
* Moffett & Hankins (1999) Moffett, D. A., & Hankins, T. H. 1999, ApJ, 522, 1046
* Moran et al. (2013a) Moran, P., Shearer, A., Gouiffès, C., & Laurent, P. 2013a, arXiv:1302.3622
* Moran et al. (2013b) Moran, P., Shearer, A., Mignani, R. P., et al. 2013b, MNRAS, 433, 2564
* Muslimov & Harding (2004) Muslimov, A. G., & Harding, A. K. 2004, ApJ, 606, 1143
* Ng & Romani (2008) Ng, C.-Y. & Romani, R. W. 2008, ApJ, 673, 411
* Oort & Walraven (1956) Oort, J. H., & Walraven, T. 1956, Bull. Astron. Inst. Netherlands, 12, 285
* Pétri & Kirk (2005) Pétri, J., & Kirk, J. G. 2005, ApJ, 627, L37
* Pierbattista et al. (2014) Pierbattista, M., et al. 2014, submitted to A&A [arXiv:1403.3849]
* Radhakrishnan & Cooke (1969) Radhakrishnan, V., & Cooke, D. J. 1969, Astrophys. Lett., 3, 225
* Rankin (1990) Rankin, J. M. 1990, ApJ, 352, 247
* Romani & Watters (2010) Romani, R. W., & Watters, K. P. 2010, ApJ, 714, 810
* Romani & Yadigaroglu (1995) Romani, R. W., & Yadigaroglu, I.-A. 1995, ApJ, 438, 314
* Silver et al. (1978) Silver, E. H., Kestenbaum, H. L., Long, K. S., et al. 1978, ApJ, 225, 221
* Słowikowska et al. (2009) Słowikowska, A., Kanbach, G., Kramer, M., & Stefanescu, A. 2009, MNRAS, 397, 103
* Smith et al. (1988) Smith, F. G., Jones, D. H. P., Dick, J. S. B., & Pike, C. D. 1988, MNRAS, 233, 305
* van Straten & Bailes (2011) van Straten, W., & Bailes, M. 2011, PASA, 28, 1
* van Straten et al. (2012) van Straten, W., Demorest, P., & Oslowski, S. 2012, Astron. Research & Technology, 9, 237
* Venter et al. (2012) Venter, C., Johnson, T. J., & Harding, A. K. 2012, ApJ, 744, 34
* Weltevrede & Johnston (2008) Weltevrede, P., & Johnston, S. 2008, MNRAS, 391, 1210
* Wolfram (2013) Wolfram Research, Inc. 2013, Mathematica, Version 9.0 (Champaign, IL: Wolfram Research)
* Woltjer (1957) Woltjer, L. 1957, Bull. Astron. Inst. Netherlands, 13, 301
|
arxiv-papers
| 2014-02-26T21:39:49 |
2024-09-04T02:49:58.978911
|
{
"license": "Public Domain",
"authors": "Agnieszka Slowikowska, Benjamin W. Stappers, Alice K. Harding, Stephen\n L. O'Dell, Ronald F. Elsner, Alexander J. van der Horst, Martin C. Weisskopf",
"submitter": "Martin C. Weisskopf",
"url": "https://arxiv.org/abs/1402.6719"
}
|
1402.6808
|
# Parallel Lepton Mass Matrices with Texture/Cofactor Zeros
Weijian Wang Department of Physics, North China Electric Power University,
Baoding 071003, P. R. China [email protected]
###### Abstract
In this paper we investigate the parallel texture structures containing
texture zeros in charged lepton mass matrix $M_{l}$ and cofactor zeros in
neutrino mass matrix $M_{\nu}$. These textures are interesting since they are
related to the $Z_{n}$ flavor symmetries. Using the weak basis permutation
transformation, the 15 parallel textures are grouped as 4 classes (class
I,II,III and IV) with the matrices in each class sharing the same physical
implications. Under the current experimental data, the class I, III with
inverted mass hierarchy and class II with normal mass hierarchy are
phenomenologically acceptable. The correlations between some important
physical variables are presented, which are essential for the model selection
and can be text by future experiments. The model realization is illustrated by
means of $Z_{4}\times Z_{2}$ flavor symmetry.
PACS: 14.60.Pq, 12.15.Ff, 11.30.Hv
## I Introduction
The discovery of neutrino oscillations have provided us with convincing
evidences for massive neutrinos and leptonic flavor mixing with high degree of
accuracyneu1 (1, 2, 3). The understanding of the leptonic flavor structure is
one of the major open questions in particle physics. Several attempts have
been proposed to explain the origin of neutrino mass and the observed pattern
of leptonic mixing by introducing the flavor symmetries within the framework
of seesaw modelsseesaw (4). The flavor symmetry often reduces the number of
free parameters and leads to the specific structures of fermion mass matrices
including texture zeroszero (5, 6, 7, 8, 9), hybrid textureshybrid (10, 11),
zero tracesum (12), zero determinantdet (13), vanishing minorsminor1 (14, 15,
16), two traceless submatricestra (17), equal elements or cofactorsco (18),
inverse hybrid textureshyco (19). Among these models, the matrices with
texture or cofactor zeros are particularly interesting due to their
connections to the flavor symmetries. The phenomenological examination of
texture zeros or cofactor zeros in flavor basis have been widely studies in
Ref.zero (5, 6, 14, 15, 16) where the charged lepton mass matrices $M_{l}$ are
diagonal. However, no universal principle is required that the flavor basis is
necessary and the more general cases should be considered in no diagonal
$M_{l}$ basis. In this scenario, the lepton mass matrices with texture zeros
in both lepton mass $M_{l}$ and neutrino mass matrix $M_{\nu}$ have been
systematically investigated by many authorsGCB (7, 8)(for a review, see z1
(9)).
In this paper, we propose the new possible texture structures where there are
two texture zeros in $M_{l}$ and two cofactor zeros in $M_{\nu}$ (We denote
them the matrices with texture/cofactor zeros). It seems that such mass
matrices are rather unusual because one instinctively expects the type of
texture structures to be the same for both $M_{l}$ and $M_{\nu}$. However, one
reminds the type-I seesaw model as $M_{\nu}=-M_{D}M_{R}^{-1}M_{D}^{T}$. Then
the texture or cofactor zeros of $M_{\nu}$ can be attributed to the texture
zeros in $M_{D}$ and $M_{R}$. Generally, this can be realized by $Z_{n}$
flavor symmetryzn (20, 14). Therefore from the point of flavor symmetry, both
texture zeros and cofactor zeros structures manifest the same flavor symmetry
in different ways. It is our main motivation to carry out this work and a
concrete model will be constructed in the following section. Furthermore, we
take the so-called the parallel $Ans\ddot{a}tze$ that the positions of texture
zeros in $M_{l}$ are chosen to be the same as the cofactor zeros in $M_{\nu}$.
Although there is no priori reason requiring the parallel structures, they are
usually regarded in many literatures as an esthetical appeal and the precursor
of the more general cases. The lepton mass matrices with parallel texture zero
structures have been systematically investigated in Ref.GCB (7). Subsequently,
the idea is generalized to more complicated situations such as parallel hybrid
textureshyp (21), parallel cofactor zero texturesmyco (22). In our case, there
exists $C^{2}_{6}=15$ logically possible patterns for two texture/cofactor
zeros in mass matrices. It is indicated that the 15 textures can be grouped
into 4 classes with the matrices in each class connected by $S_{3}$
permutation transformation and sharing the same physical implications. Among
the 4 classes, one of them is not viable phenomenologically. Therefore we
focus on the other three nontrivial classes.
The paper is organized as follow. In Sec. II, we present the classification of
mass matrices and relate them to the current experimental results. In Sec.
III, we diagonalize the mass matrices, confront the numerical results with the
experimental data and discuss their predictions. In Sec. IV, the model
realization is given under the $Z_{4}\times Z_{2}$ flavor symmetry. We
summarize the results in Sec. V.
## II Formalism
### II.1 Weak basis equivalent classes
As shown in Ref.GCB (7), there exists the general weak basis (WB)
transformations leaving gauge currents invariant i.e
$M_{l}\rightarrow M_{l}^{\prime}=W^{\dagger}M_{l}W_{R}\quad\quad\quad
M_{\nu}\rightarrow M_{\nu}^{\prime}=W^{T}M_{\nu}W$ (1)
where the neutrinos are assumed to be Majorana fermions and $W$, $W_{R}$ are
$3\times 3$ unitary matrices. Two matrices related by WB transformations have
the same physical implications. Therefore the parallel matrices with
texture/cofactor zeros located at different positions can be connected by
$S_{3}$ permutation matrix $P$ as a specific WB transformation
$M_{l}^{\prime}=P^{T}M_{l}P\quad\quad\quad M_{\nu}^{\prime}=P^{T}M_{\nu}P$ (2)
It is noted that $P$ changes the positions of cofactor zero elements but still
preserves the parallel structures for both charged lepton and neutrino mass
textures. Then the texture/cofactor zeros matrices are classified into 4
classes:
Class I:
$\begin{split}\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&0/\bigtriangleup\\\
\times&\times&\times\\\
0/\bigtriangleup&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}0/\bigtriangleup&0/\bigtriangleup&\times\\\
0/\bigtriangleup&\times&\times\\\
\times&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&0/\bigtriangleup&\times\\\
0/\bigtriangleup&0/\bigtriangleup&\times\\\
\times&\times&\times\end{array}\right)\\\
\left(\begin{array}[]{ccc}\times&\times&\times\\\
\times&0/\bigtriangleup&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&0/\bigtriangleup\\\
\times&\times&\times\\\
0/\bigtriangleup&\times&0/\bigtriangleup\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&\times\\\
\times&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&0/\bigtriangleup\end{array}\right)\end{split}$ (3)
Class II:
$\begin{split}\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\\\
0/\bigtriangleup&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&0/\bigtriangleup&\times\\\
0/\bigtriangleup&\times&\times\\\
\times&\times&0/\bigtriangleup\end{array}\right)\end{split}$ (4)
Class III:
$\begin{split}\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&0/\bigtriangleup&\times\\\
\times&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&\times&\times\\\
\times&\times&0/\bigtriangleup\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&\times\\\
\times&0/\bigtriangleup&\times\\\
\times&\times&0/\bigtriangleup\end{array}\right)\end{split}$ (5)
Class IV:
$\begin{split}\left(\begin{array}[]{ccc}\times&0/\bigtriangleup&0/\bigtriangleup\\\
0/\bigtriangleup&\times&\times\\\
0/\bigtriangleup&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&0/\bigtriangleup&\times\\\
0/\bigtriangleup&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&0/\bigtriangleup\\\
\times&\times&0/\bigtriangleup\\\
0/\bigtriangleup&0/\bigtriangleup&\times\end{array}\right)\end{split}$ (6)
where ”$0/\bigtriangleup$” at $(i,j)$ position represents the texture zero
condition $M_{ij}=0$ and the cofactor zero condition $C_{ij}=0$; The
”$\times$” denotes arbitrary element. One can check that the matrices with
cofactor zeros in class I are equivalent to the texture zero ones. Choosing
the first matrix of class I as an example, we have
$\begin{split}M_{\nu}=\left(\begin{array}[]{ccc}\bigtriangleup&\times&\bigtriangleup\\\
\times&\times&\times\\\
\bigtriangleup&\times&\times\end{array}\right)\Rightarrow
M_{\nu}^{-1}=\left(\begin{array}[]{ccc}0&\times&0\\\ \times&\times&\times\\\
0&\times&\times\end{array}\right)\Rightarrow
M_{\nu}=\left(\begin{array}[]{ccc}\times&\times&\times\\\ \times&0&0\\\
\times&0&\times\end{array}\right)\end{split}$ (7)
Thus the parallel texture structures of class I are equivalent to the no-
parallel structures with two texture zeros. Although the parallel texture zero
structures has been explored extensivelyGCB (7, 8, 9), the analysis of the no-
parallel two texture zero structure has not yet been reported. On the other
hand, as having been pointed out in Ref.GCB (7, 22), the class IV leads to the
decoupling of a generation of lepton from mixing and thus not experimentally
viable.
### II.2 Useful notations
As we have mentioned, among the 4 classes only class I, II and III are
nontrivial. We represent them as
$\begin{split}M_{l/\nu}^{I}=\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&0\bigtriangleup\\\
\times&\times&\times\\\
0\bigtriangleup&\times&\times\end{array}\right)\quad\quad
M_{l/\nu}^{II}=\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\end{array}\right)\quad\quad
M_{l/\nu}^{III}=\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&0/\bigtriangleup&\times\\\
\times&\times&\times\end{array}\right)\end{split}$ (8)
In the analysis, we consider $M_{l}$ is to be Hermitian and the Majorana
neutrino mass texture $M_{\nu}$ is complex and symmetric. The $M_{l}$ and
$M_{\nu}$ are diagonalized by unitary matrix $V_{l}$ and $V_{\nu}$
$M_{l}=V_{l}M_{l}^{D}V_{l}^{\dagger}\quad\quad
M_{\nu}=V_{\nu}M_{\nu}^{D}V_{\nu}^{T}$ (9)
where $M_{l}^{D}=Diag(m_{e},m_{\mu},m_{\tau})$,
$M_{\nu}^{D}=Diag(m_{1},m_{2},m_{3})$. The Pontecorvo-Maki-Nakagawa-Sakata
matrixPMNS (23) $U_{PMNS}$ is given by
$U_{PMNS}=V_{l}^{\dagger}V_{\nu}$ (10)
and parameterized as
$U_{PMNS}=UP_{\nu}=\left(\begin{array}[]{ccc}c_{12}c_{13}&c_{13}s_{12}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\\\
s_{23}s_{12}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-c_{23}s_{12}s_{13}e^{i\delta}&c_{13}c_{23}\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&e^{i\alpha}&0\\\ 0&0&e^{i(\beta+\delta)}\end{array}\right)$ (11)
where we use the abbreviation $s_{ij}=\sin\theta_{ij}$ and
$c_{ij}=\cos\theta_{ij}$. The ($\alpha$,$\beta$) in $P_{\nu}$ represents the
two Majorana CP-violating phases and $\delta$ denotes the Dirac CP-violating
phase. In order to facilitate our calculation, we treat the Hermitian matrix
$M_{l}$ factorisable. i.e
$M_{l}=K_{l}M_{l}^{r}K_{l}^{\dagger}$ (12)
where $K_{l}$ is the unitary phase matrix parameterized as
$K_{l}=diag(1,e^{i\phi_{1}},e^{i\phi_{2}})$. The $M_{l}^{r}$ becomes a real
symmetric matrix which can be diagonalized by real orthogonal matrix $O_{l}$.
Then we have
$V_{l}=K_{l}O_{l}$ (13)
and
$U_{PMNS}=O_{l}^{T}K_{l}^{\dagger}V_{\nu}$ (14)
From (9), (10) and (14), the neutrino mass matrix $M_{\nu}$ is given by
$M_{\nu}=K_{l}VP_{\nu}M_{\nu}^{D}P_{\nu}V^{T}K_{l}^{\dagger}$ (15)
where $V\equiv O_{l}U$. From (15) and solving the cofactor zero conditions of
$M_{\nu}$
$M_{\nu(pq)}M_{\nu(rs)}-M_{\nu(tu)}M_{\nu(vw)}=0\quad\quad
M_{\nu(p^{\prime}q^{\prime})}M_{\nu(r^{\prime}s^{\prime})}-M_{\nu(t^{\prime}u^{\prime})}M_{\nu(v^{\prime}w^{\prime})}=0$
(16)
we get
$\frac{m_{1}}{m_{2}}e^{-2i\alpha}=\frac{K_{3}L_{1}-K_{1}L_{3}}{K_{2}L_{3}-K_{3}L_{2}}$
(17)
$\frac{m_{1}}{m_{3}}e^{-2i\beta}=\frac{K_{2}L_{1}-K_{1}L_{2}}{K_{3}L_{2}-K_{2}L_{3}}e^{2i\delta}$
(18)
where
$K_{i}=(V_{pj}V_{qj}V_{rk}V_{sk}-V_{tj}V_{uj}V_{vk}V_{wk})+(j\leftrightarrow
k)$ (19)
$L_{i}=(V_{p^{\prime}j}V_{q^{\prime}j}V_{r^{\prime}k}V_{s^{\prime}k}-V_{t^{\prime}j}V_{u^{\prime}j}V_{v^{\prime}k}V_{w^{\prime}k})+(j\leftrightarrow
k)$ (20)
with $(i,j,k)$ a cyclic permutation of (1,2,3). With the help of Eq.(17) and
(18), the magnitudes of neutrino mass radios are given by
$\rho=\Big{|}\frac{m_{1}}{m_{3}}e^{-2i\beta}\Big{|}$ (21)
$\sigma=\Big{|}\frac{m_{1}}{m_{2}}e^{-2i\alpha}\Big{|}$ (22)
with the two Majorana CP-violating phases
$\alpha=-\frac{1}{2}arg\Big{(}\frac{K_{3}L_{1}-K_{1}L_{3}}{K_{2}L_{3}-K_{3}L_{2}}\Big{)}$
(23)
$\beta=-\frac{1}{2}arg\Big{(}\frac{K_{2}L_{1}-K_{1}L_{2}}{K_{3}L_{3}-K_{2}L_{3}}e^{2i\delta}\Big{)}$
(24)
The results of Eq. (21),(22), (23) and (24) imply that the two mass ratio
($\rho$ and $\sigma$) and two Majorana CP-violating phases ($\alpha$ and
$\beta$) are fully determined in terms of the real orthogonal matrix $O_{l}$,
$U$($\theta_{12},\theta_{23},\theta_{13}$ and $\delta$). The neutrino mass
ratios $\rho$ and $\sigma$ are related to the ratio of two neutrino mass-
squared differences defined as
$R_{\nu}\equiv\frac{\delta m^{2}}{\Delta
m^{2}}=\frac{2\rho^{2}(1-\sigma^{2})}{|2\sigma^{2}-\rho^{2}-\rho^{2}\sigma^{2}|}$
(25)
where $\delta m^{2}\equiv m_{2}^{2}-m_{1}^{2}$ and $\Delta m^{2}\equiv\mid
m_{3}^{2}-\frac{1}{2}(m_{1}^{2}+m_{2}^{2})\mid$. The three neutrino mass
eigenvalues $m_{1},m_{2}$ and $m_{3}$ are given by
$m_{2}=\sqrt{\frac{\delta m^{2}}{1-\sigma^{2}}}\quad\quad m_{1}=\sigma
m_{2}\quad\quad m_{3}=\frac{m_{1}}{\rho}$ (26)
In the following numerical analysis, we utilize the recent 3$\sigma$
confidential level global-fit data from the neutrino oscillation
experimentsdata (25).i.e
$\begin{split}\sin^{2}\theta_{12}/10^{-1}=3.08^{+0.51}_{-0.49}\quad\sin^{2}\theta_{23}/10^{-1}=4.25^{+2.16}_{-0.68}\quad\sin^{2}\theta_{13}/10^{-2}=2.34^{+0.63}_{-0.57}\\\
\delta m^{2}/10^{-5}=7.54^{+0.64}_{-0.55}eV^{2}\quad\quad\bigtriangleup
m^{2}/10^{-3}=2.44^{+0.22}_{-0.22}eV^{2}\end{split}$ (27)
for normal hierarchy (NH) and
$\begin{split}\sin^{2}\theta_{12}/10^{-1}=3.08^{+0.51}_{-0.49}\quad\sin^{2}\theta_{23}/10^{-1}=4.25^{+2.22}_{-0.74}\quad\sin^{2}\theta_{13}/10^{-2}=2.34^{+0.61}_{-0.61}\\\
\delta m^{2}/10^{-5}=7.54^{+0.64}_{-0.55}eV^{2}\quad\quad\bigtriangleup
m^{2}/10^{-3}=2.40^{+0.21}_{-0.23}eV^{2}\end{split}$ (28)
for inverted hierarchy(IH). By this time, no constraint is added on the Dirac
CP-violating phase $\delta$ at $3\sigma$ level, however the recent numerical
analysisdata (25) tends to give the best-fit value $\delta\approx 1.40\pi$. In
neutrino oscillation experiments, the CP violation effect is usually reflected
by the Jarlskog rephasing invariant quantityJas (26) defined as
$J_{CP}=s_{12}s_{23}s_{13}c_{12}c_{23}c_{13}^{2}\sin\delta$ (29)
The Majorana nature of neutrino can be determined if any signal of
neutrinoless double decay($0\nu\beta\beta$) is observed, implying the
violation of leptonic number violation. The decay ratio is related to the
effective Majorana neutrino mass $m_{ee}$, which is written as
$m_{ee}=|m_{1}c_{12}^{2}c_{13}^{2}+m_{2}s_{12}^{2}c_{13}^{2}e^{2i\alpha}+m_{3}s_{13}^{2}e^{2i\beta}|$
(30)
Although a $3\sigma$ result of $m_{ee}=(0.11-0.56)$ eV is reported by the
Heidelberg-Moscow CollaborationHM (27), this result is criticizedNND2 (28) and
shall be checked by the forthcoming experiment. It is believed that that the
next generation $0\nu\beta\beta$ experiments, with the sensitivity of $m_{ee}$
being up to 0.01 eVNDD (29), will open the window to not only the absolute
neutrino mass scale but also the Majorana-type CP violation. Besides the
$0\nu\beta\beta$ experiments, a more severe constraint was set from the recent
cosmology observation. Recently, an upper bound on the sum of neutrino mass
$\sum m_{i}<0.23$ eV is reported by Plank CollaborationPlanck (30) combined
with the WMAP, high-resolution CMB and BAO experiments.
## III Numerical analysis
We have proposed a detailed numerical analysis for class I, II and III. In
this section we presented the main predictions of all the classes.
### III.1 Class I
Let’s start from the factorisable formation of charged lepton matrix
$M_{l}^{r}$
$\begin{split}(M_{l}^{r})^{I}=\left(\begin{array}[]{ccc}0&a&0\\\ a&b&c\\\
0&c&d\end{array}\right)\end{split}$ (31)
As proposed in Ref.GCB (7, 22), the coefficients $a,b$ and $c$ are assumed to
be real and positive without losing generality. The real coefficient $d$ is
treated as a free parameter. Then the matrix (31) can be diagonalized by an
orthogonal matrix $O_{l}$
$O_{l}^{T}(M_{l}^{r})^{I}O_{l}=diag(m_{e},-m_{\mu},m_{\tau})$ (32)
where the minus sign in (32) is introduced to facilitate the analytical
calculation and has no physical meaning since it originates from the phase
transformation of Dirac fermions. Following the same strategy of Ref.GCB (7)
and using the invariant Tr$(M_{l}^{r})$, Det$(M_{l}^{r})$ and
Tr$(M_{l}^{r})^{2}$, the nozero elements of $M_{l}^{r}$ can be expressed in
terms of three mass eigenvalues $m_{e},m_{\mu}$, $m_{\tau}$ and $d$
$a=\sqrt{\frac{m_{e}m_{\mu}m_{\tau}}{d}}$ (33) $b=m_{e}-m_{\mu}+m_{\tau}-d$
(34) $c=\sqrt{-\frac{(d-m_{e})(d+m_{\mu})(d-m_{\tau})}{d}}$ (35)
Using the expression (33), (34) and (35), $O_{l}$ can be constructed. Here we
adopt the result of GCB (7) i.e
$\begin{split}O_{l}=\left(\begin{array}[]{ccc}\sqrt{\frac{m_{\mu}m_{\tau}(d-m_{e})}{d(m_{\mu}+m_{e})(m_{\tau}-m_{e})}}&\sqrt{\frac{m_{e}m_{\tau}(m_{\mu}+d)}{d(m_{\mu}+m_{e})(m_{\tau}+m_{\mu})}}&\sqrt{-\frac{m_{e}m_{\mu}(d-m_{\tau})}{d(m_{\tau}-m_{e})(m_{\tau}+m_{\mu})}}\\\
\sqrt{-\frac{m_{e}(m_{e}-d)}{(m_{\mu}+m_{e})(m_{\tau}-m_{e})}}&-\sqrt{-\frac{m_{\mu}(d+m_{\mu})}{(m_{\mu}+m_{e})(m_{\tau}+m_{\mu})}}&\sqrt{\frac{m_{\tau}(m_{\tau}-d)}{(m_{\tau}-m_{e})(m_{\tau}+m_{\mu})}}\\\
-\sqrt{-\frac{m_{e}(d+m_{\mu})(d-m_{\tau})}{d(m_{\mu}+m_{e})(m_{\tau}-m_{e})}}&\sqrt{\frac{m_{\mu}(d-m_{e})(m_{\tau}-d)}{d(m_{\mu}+m_{e})(m_{\tau}-m_{e})}}&\sqrt{\frac{m_{\tau}(d-m_{e})(d+m_{\mu})}{d(m_{\tau}-m_{e})(m_{\tau}+m_{\mu})}}\end{array}\right)\end{split}$
(36)
Replacing the (21), (22), (23), (24) and (25) with the $O_{l}$ obtained in
(43), we can see that the ratios of mass ($\rho,\sigma$), two Majorana CP-
violating phases $(\alpha,\beta)$ and the ratio of mass squared difference
$R_{\nu}$ can be expressed via eight parameters: three mixing angle
$\theta_{12},\theta_{23},\theta_{13}$, one Dirac CP violating phase $\delta$,
three charged lepton mass $(m_{e},m_{\mu},m_{\tau})$ and the parameter $d$.
Here we choose the three charged lepton mass at the electroweak
scale($\mu\simeq M_{Z}$) i.ezzx2 (31)
$m_{e}=0.486570154MeV\quad\quad m_{\mu}=102.7181377MeV\quad\quad
m_{\tau}=1746.17MeV$ (37)
Figure 1: The correlation plots for class I(IH). The blue horizontal bands
represent the 1$\sigma$ uncertainty in determination of
$\theta_{12},\theta_{23}$ and $\theta_{13}$ while they plus the green
horizontal bands correspond to the 2$\sigma$ uncertainty.
In the numerical analysis, a set of random numbers are generated for the three
mixing angles $(\theta_{12},\theta_{23},\theta_{13})$ and mass square
differences ($\delta m^{2},\Delta m^{2}$) in their $3\sigma$ range. We also
randomly vary the parameter $d$ in its appropriate range. Since at 3 $\sigma$
level the Dirac CP-violating phase $\delta$ is unconstrained in neutrino
oscillation experiments, we vary it randomly in the range of $[0,2\pi)$. With
the random number and using Eq. (21), (22) and (25), neutrino mass ratios
$(\rho,\sigma)$ and the mass-squared difference ratio $R_{\nu}$ are
determined. Then the input parameters is empirically acceptable when the
$R_{\nu}$ falls inside the the $3\sigma$ range of experimental data, otherwise
they are ruled out. Finally, we get the value of neutrino mass and Majorana
CP-violating $\alpha$ and $\beta$ though Eq.(23), (24) and (26). Once the the
absolute neutrino mass $m_{1,2,3}$ are obtained , the further constraint from
cosmology should be considered. In this paper, the upper bound on the sum of
neutrino mass $\Sigma m_{i}$ is set to be less than 0.23 eV. It turns out that
class I are phenomenologically acceptable only for inverted mass hierarchy.
Figure 2: The correlation plots $(\theta_{23},\theta_{12})$ and
$(\theta_{23},\theta_{13})$ for class I(NH). The horizontal and vertical lines
respectively denote the 3$\sigma$ upper and lower bound of $\theta_{12}$ and
$\theta_{23}$
The predictions of class I with inverted mass hierarchy are presented in
Fig.1. From the diagrams, one can see that the three neutrino mixing angles
$\theta_{12}$, $\theta_{23}$ and $\theta_{13}$ fully cover their $3\sigma$
experimental data. Although there is no bound on the Dirac CP-violating phase
$\delta$, a numerical preference appears at around $0^{\circ}\sim
50^{\circ}(360^{\circ}\sim 310^{\circ})$. The unrestricted $\delta$ leads to
the $J_{CP}$ varying in the range of $0\sim 0.04$. There also exists a strong
correlation between $\delta$ and the lightest neutrino mass $m_{3}$.
Especially, the range $0.002$eV$<m_{3}<0.02$eV is derived for $\delta$ lying
around $0^{\circ}(360^{\circ})$, indicating that both strong and mild mass
hierarchy are allowed. On the other hand, the mild mass hierarchy is much more
appealing for $100^{\circ}<\delta<260^{\circ}$. Although both Majorana CP-
violating phase $\alpha$ and $\beta$ is allowed in the range of
$-90^{\circ}\sim 90^{\circ}$, there shows a preferable distribution for
$\alpha$ in $\pm 90^{\circ}\sim\pm 50^{\circ}$ and a strong correlation
between $\delta$ and $\beta$. There exists an upper bound of $0.05$eV on the
effective Majorana neutrino mass $m_{ee}$, leaving the possible space for
detecting in future neutrinoless double beta decay $(0\nu\beta\beta)$
experiments.
The class I with inverted hierarchy is ruled out by $3\sigma$ data. To see
this, we show the correlated plots $(\theta_{23},\theta_{12})$ and
$(\theta_{23},\theta_{13})$ in Fig.2. From the diagrams, one can see that even
though $\theta_{13}$ fully covers its $3\sigma$ range, the common parameter
spaces $(\theta_{23},\theta_{12})$ fails to provide a allowed region to
saturate the experimental constraint. Moveover, one always obtains
$\theta_{23}>40^{\circ}$, which means a rather large correction of
$\theta_{12}$ is needed to reconcile the observed PMNS matrix.
### III.2 Class II
The factorisable formation of charged lepton matrix of class I is given by
expression:
$\begin{split}(M_{l}^{r})^{II}=\left(\begin{array}[]{ccc}0&a&c\\\ a&b&0\\\
c&0&d\end{array}\right)\end{split}$ (38)
It can be diagonalized by an orthogonal matrix $O_{l}$
$O_{l}^{T}(M_{l}^{r})^{II}O_{l}=diag(m_{e},-m_{\mu},m_{\tau})$ (39)
Without losing generality, the coefficients $a,c,d$ are set to be real and
positive. Using the invariant Tr$(M_{l}^{r})$, Det$(M_{l}^{r})$ and
Tr$(M_{l}^{r})^{2}$, the nozero elements of $M_{l}^{r}$ are expressed as
$a=\sqrt{-\frac{(m_{e}-m_{\mu}-d)(m_{e}+m_{\tau}-d)(-m_{\mu}+m_{\tau}-d)}{m_{e}-m_{\mu}+m_{\tau}-2d}}$
(40) $b=m_{e}-m_{\mu}+m_{\tau}-d$ (41)
$c=\sqrt{\frac{(d-m_{e})(d+m_{\mu})(d-m_{\tau})}{m_{e}-m_{\mu}+m_{\tau}-2d}}$
(42)
where the parameter $d$ is allowed in the range of $m_{e}-m_{\mu}<d<m_{e}$ and
$m_{\tau}-m_{\mu}<d<m_{\tau}$. Then the $O_{l}$ can be easily constructed as
$\begin{split}O_{l}=\left(\begin{array}[]{ccc}\frac{(b-m_{e})(d-m_{e})}{N_{1}}&\frac{(b+m_{\mu})(d+m_{\mu})}{N_{2}}&\frac{(b-m_{\tau})(d-m_{\tau})}{N_{3}}\\\
-\frac{a(d-m_{e})}{N_{1}}&-\frac{a(d+m_{\mu})}{N_{2}}&-\frac{a(d-m_{\tau})}{N_{3}}\\\
-\frac{c(b-m_{e})}{N_{3}}&-\frac{c(b+m_{\mu})}{N_{3}}&-\frac{c(b-m_{\tau})}{N_{3}}\end{array}\right)\end{split}$
(43)
where $N_{1}$, $N_{2}$ and $N_{3}$ are the normalized coefficients given by
$N_{1}^{2}=(b-m_{e})^{2}(d-m_{e})^{2}+a^{2}(d-m_{e})^{2}+c^{2}(b-m_{\tau})^{2}$
(44)
$N_{2}^{2}=(b+m_{\mu})^{2}(d+m_{\mu})^{2}+a^{2}(d+m_{\mu})^{2}+c^{2}(b+m_{\mu})^{2}$
(45)
$N_{3}^{2}=(b-m_{\tau})^{2}(d-m_{\tau})^{2}+a^{2}(d-m_{\tau})^{2}+c^{2}(b-m_{\tau})^{2}$
(46)
Figure 3: The correlation plots for class II(NH). The blue horizontal bands
represent the 1$\sigma$ uncertainty in determination of
$\theta_{12},\theta_{23}$ and $\theta_{13}$ while they plus the green
horizontal bands correspond to the 2$\sigma$ uncertainty.
The numerical results of class II for normal hierarchy are presented in Fig.3.
We can see from the figures that the three neutrino mixing angle
$\theta_{12}$, $\theta_{23}$ $\theta_{13}$ and Dirac CP-violating phase
$\delta$ vary arbitrarily in its $3\sigma$ range. There exhibits a strong
correlation between $\delta$ and $\theta_{23}$. Only when $\delta$ is located
in the range of $100^{\circ}\sim 260^{\circ}$, the $\theta_{23}$ has the
possibility to be less then $45^{\circ}$. This is particularly interesting
since the recent global fit trends to give the $\theta_{23}<45^{\circ}$ at
2$\sigma$ level. The strong $\delta-\theta_{23}$ correlation is essential for
the model selection and will be confirmed or ruled out by future long-baseline
neutrino oscillation experiments. The similar correlations also holds for
$\delta$, $m_{ee}$ and the lightest neutrino mass $m_{1}$. Moveover, there
exists a constrained range of $0$eV$<m_{1}<$$0.06$eV, indicating that both
strong and mild neutrino mass hierarchy are possible. There are strong
correlations between $\alpha$, $\beta$ and $\delta$. Especially, the Majorana
CP-violating phase $\alpha$ is restricted in the range of
$-5^{\circ}\sim+5^{\circ}$ and $\pm 90^{\circ}\sim\pm 50^{\circ}$. The
effective Majorana neutrino mass $m_{ee}$ is highly constrained in the two
ranges of $0$eV$\sim 0.008$eV and $0.01$eV$\sim 0.025$eV. The later reaches
the accuracy of the future neutrinoless double beta decay $(0\nu\beta\beta)$
experiments. We also observed that the allowed range of Jarlskog rephasing
invariant $|J_{CP}|$ is $0\sim 0.04$, which is potentially detected by future
long-baseline neutrino oscillation experiments.
The IH case, as we can see from Fig.4, is phenomenologically ruled by
$3\sigma$ experimental data. As class I, the theoretical prediction of
$(\theta_{23},\theta_{12})$ common space fails to be located in its
experimental region. Moreover, the possibility distribution of $\theta_{23}$
shows a strong preference of $\theta_{23}<33^{\circ}$ or
$\theta_{23}>50^{\circ}$, which means a large correction of $\theta_{23}$
angle is needed to produce the $2\sigma$ global-fit value.
Figure 4: The correlation plots $(\theta_{23},\theta_{12})$ and
$(\theta_{23},\theta_{13})$ for class II(IH). The horizontal and vertical
lines respectively denote the 3$\sigma$ upper and lower bound of $\theta_{12}$
and $\theta_{23}$ Figure 5: The correlation plots for class III(IH). The blue
horizontal bands represent the 1$\sigma$ uncertainty in determination of
$\theta_{12},\theta_{23}$ and $\theta_{13}$ while they plus the green
horizontal bands correspond to the 2$\sigma$ uncertainty. Figure 6: The
correlation plots $(\theta_{23},\theta_{12})$ and $(\theta_{23},\theta_{13})$
for class III(NH). The horizontal and vertical lines respectively denote the
3$\sigma$ upper and lower bound of $\theta_{12}$ and $\theta_{23}$
### III.3 Class III
In the case of class III, the factorisable charged lepton matrix is written by
$\begin{split}(M_{l}^{r})^{III}=\left(\begin{array}[]{ccc}0&a&b\\\ a&0&c\\\
b&c&d\end{array}\right)\end{split}$ (47)
where $a,b,c$ and $d$ are real number and $b,c$ are set to be positive. The
matrix $(M_{l}^{r})^{III}$ is diagonalized by the orthogonal matrix $O_{l}$
$O_{l}^{T}(M_{l}^{r})^{III}O_{l}=diag(m_{e},-m_{\mu},m_{\tau})$ (48)
Here we choose $a$ as the free parameter because $d$ has been fixed by
Tr$(M_{l}^{r})$. i.e
$d=m_{e}-m_{\mu}+m_{\tau}$ (49)
With the help of other two invariant quantity Det$(M_{l}^{r})$ and
Tr$(M_{l}^{r})^{2}$, $b,c$ are determined by three charged leptonic mass
eigenvalues$(m_{e},m_{\mu},m_{\tau})$ and $a$
$(b\pm
c)^{2}=-(-m_{e}m_{\mu}+m_{e}m_{\tau}-m_{\mu}m_{\tau})-a^{2}\pm\frac{a^{2}(m_{e}-m_{\mu}+m_{\tau})-m_{e}m_{\mu}m_{\tau}}{a}$
(50)
Then diagonalization matrix can be constructed as
$\begin{split}(M_{l}^{r})^{III}=\left(\begin{array}[]{ccc}\frac{O(11)}{N_{1}}&\frac{O(12)}{N_{2}}&\frac{O(13)}{N_{3}}\\\
\frac{O(21)}{N_{1}}&\frac{O(22)}{N_{2}}&\frac{O(23)}{N_{3}}\\\
\frac{O(31)}{N_{1}}&\frac{O(32)}{N_{2}}&\frac{O(33)}{N_{3}}\end{array}\right)\end{split}$
(51)
The matrix elements are given by
$\begin{split}O(11)=&am_{e}^{-1}(bm_{e}^{-1}+ca^{-1})+bm_{e}^{-1}(m_{e}a^{-1}-m_{e}^{-1}a)\\\
O(12)=&-am_{\mu}^{-1}(-bm_{\mu}^{-1}+ca^{-1})-bm_{\mu}^{-1}(-m_{\mu}a^{-1}+m_{\mu}^{-1}a)\\\
O(13)=&am_{\tau}^{-1}(bm_{\tau}^{-1}+ca^{-1})+bm_{\tau}^{-1}(m_{\tau}a^{-1}-m_{\tau}^{-1}a)\\\
&O(21)=bm_{e}^{-1}+ca^{-1}\\\ &O(22)=-bm_{\mu}^{-1}+ca^{-1}\\\
&O(23)=bm_{\tau}^{-1}+ca^{-1}\\\ &O(31)=m_{e}a^{-1}-m_{e}^{-1}a\\\
&O(32)=-m_{\mu}a^{-1}+m_{\mu}^{-1}a\\\ &O(33)=m_{\tau}a^{-1}-m_{\tau}^{-1}a\\\
\end{split}$ (52)
with the normalized coefficients
$\begin{split}N_{1}^{2}=O(11)^{2}+O(21)^{2}+O(31)^{2}\\\
N_{2}^{2}=O(12)^{2}+O(22)^{2}+O(32)^{2}\\\
N_{3}^{2}=O(13)^{2}+O(23)^{2}+O(33)^{2}\end{split}$ (53)
Repeating the previous analysis, the class III with inverted hierarchy are now
found to be acceptable by current experimental data while the NH case are
excluded. In Fig.5, we show the the main predictions for IH case. One can
observe that no bounds are founded on three mixing angles and Dirac CP-
violating phase $\delta$, leading to the Jarlskog rephasing invariant
$0<|J_{CP}|<0.04$. One the other hand, there is a correlation between $\delta$
and the lightest neutrino mass $m_{3}$. One obtains $0$eV$<m_{3}<$$0.05$eV for
$0^{\circ}<\delta<100^{\circ}(260^{\circ}<\delta<360^{\circ})$ while
$0$eV$<m_{3}<$$0.02$eV for $100^{\circ}<\delta<260^{\circ}$, implying that
both strong and mild mass hierarchy are allowed. Interestingly, although the
correlations of $(\delta,\alpha)$ and $(\delta,\beta)$ are complicated, there
exists a lower bound of $0.01$eV on the effective Majorana neutrino mass
$m_{ee}$ which is achievable in future $0\nu\beta\beta$ experiments.
In Fig.6, we present the common space of $(\theta_{23},\theta_{12})$ and
$(\theta_{23},\theta_{13})$ for NH case. One easily observes that parameter
space of $(\theta_{23},\theta_{12})$ is outside the $3\sigma$ allowed region
and a large corrections of $\theta_{23}$ or $\theta_{12}$ is needed.
## IV The $Z_{4}\times Z_{2}$ flavor symmetry realization
In general, all phenomenologically viable lepton mass matrices with with
parallel texture/cofactor zeros can be realized in seesaw models with Abelian
flavor symmetry. The lepton mass matrices of class I are equivalent to the
ones with no-parallel texture zeros. The symmetry realization of such texture
structures has been performed in Ref.zn (20). Thus we only consider class II
and III. In this section, we take the first matrix pattern of class II as a
illustration. It is shown that the lepton mass matrix can be realized based on
the type-I seesaw models with the $Z_{4}\times Z_{2}$ flavor symmetry. We take
the same strategy of Ref.minor (15, 14, 16). In flavor basis, $M_{\nu}$
belonging to class II is realized under $Z_{8}$ symmetryminor1 (14). Different
from Ref.minor1 (14), we build the model under the basis where $M_{l}$ is
nodiagonal. Under the $Z_{4}\times Z_{2}$ symmetry, the three charged lepton
doublets $D_{iL}=(\nu_{iL},l_{iL})$, three right-handed charged lepton
singlets $l_{iR}$ and three right-handed neutrinos $\nu_{iR}$ (where
$i=e,\mu,\tau$) transform as
$\begin{split}\nu_{eR}\sim(\omega,1),\quad\quad\nu_{\mu
R}\sim(1,1),\quad\quad\nu_{\tau R}\sim(\omega^{2},1)\\\
D_{eL}\sim(\omega,-1),\quad\quad D_{\mu L}\sim(1,-1),\quad\quad D_{\tau
L}\sim(\omega^{2},-1)\\\ l_{eR}\sim(\omega^{3},-1),\quad\quad l_{\mu
R}\sim(1,-1),\quad\quad l_{\tau R}\sim(\omega^{2},-1)\end{split}$ (54)
where $\omega=e^{i\pi/2}$. Then, under $Z_{4}$ symmetry, the bilinears of
$\overline{D}_{iL}l_{jR}$, $\overline{D}_{iL}\nu_{jR}$, and
$\nu_{iR}^{T}\nu_{jR}$, transform respectively as
$\left(\begin{array}[]{ccc}-1&-i&i\\\ i&1&-1\\\
-i&-1&1\end{array}\right)\quad\quad\quad\left(\begin{array}[]{ccc}1&-i&i\\\
i&1&-1\\\
-i&-1&1\end{array}\right)\quad\quad\quad\left(\begin{array}[]{ccc}-1&i&-i\\\
i&1&-1\\\ -i&-1&1\end{array}\right)$ (55)
To generate the fermion mass, we need introduce the three Higgs doublets
$\Phi_{12},\Phi_{23},\Phi$ for charged lepton matrix $M_{l}$, one the Higgs
doublet $\Phi^{\prime}$ for Dirac neutrino mass matrix $M_{D}$ and a scalar
singlet $\chi$ for the Majorana neutrino mass matrix $M_{R}$, which transform
under $Z_{4}\times Z_{2}$ symmetry as
$\begin{split}\Phi_{12}\sim(\omega,1),&\quad\quad\Phi_{13}\sim(\omega^{3},1),\quad\quad\Phi\sim(1,1)\\\
&\Phi^{\prime}\sim(1,-1),\quad\quad\quad\quad\chi\sim(\omega,1)\end{split}$
(56)
To maintain the invariant Yukawa Lagrange under the flavor symmetry , the
$\Phi_{12}$ and $\Phi_{13}$ couple to the bilnears $\overline{D}_{eL}l_{\mu
R}$ and $\overline{D}_{eL}l_{\tau R}$ to produce the (1,2) and (1,3) nozero
matrix elements in $M_{l}$ while $\Phi$ couples to $\overline{D}_{\mu L}l_{\mu
R}$ $\overline{D}_{\tau L}l_{\tau R}$ to produce the (2,2) and (3,3) no zero
matrix elements. The zero matrix elements in $M_{l}$ is obtained because there
are no appropriate scalars to generate them. For the Dirac neutrino mass
sector, there exists only one scalar doublet $\Phi^{\prime}$ transforming
invariantly under $Z_{4}$. Therefore the $\Phi^{\prime}$ will contribute only
to the (1,1), (2,2), (3,3) no zero elements leading to a diagonal $M_{D}$.
Here the $Z_{2}$ symmetry is used to distinguish the set of scalar doublets
$(\Phi_{12},\Phi_{13},\phi)$ from $\Phi^{\prime}$ so that they are
respectively in charge of the mass generation of $M_{l}$ and $M_{D}$ without
any crossing. In order to produce the Majorana neutrino mass term, we
introduce a complex scalar singlet $\chi$. The $\chi$ couples to
$\nu_{eR}^{T}\nu_{\tau R}$ while $\chi^{\ast}$ couples to
$\nu_{eR}^{T}\nu_{\mu R}$, leading to the (1,2) and (1,3) no zero elements in
$M_{R}$. From (55), the $\nu_{\mu R}^{T}\nu_{\mu R}$ and $\nu_{\tau
R}^{T}\nu_{\tau R}$ is invariant under $Z_{4}$, thus we can directly write
them in the Lagrange without needing the singlets. The zero elements in
$M_{R}$ are obtained by not introducing other scalar singlets. Therefore the
mass matrices $M_{l}$, $M_{D}$ and $M_{R}$ is given by
$M_{l}\sim\left(\begin{array}[]{ccc}0&\times&\times\\\ \times&\times&0\\\
\times&0&\times\end{array}\right)\quad\quad\quad
M_{D}\sim\left(\begin{array}[]{ccc}\times&0&0\\\ 0&\times&0\\\
0&0&\times\end{array}\right)\quad\quad\quad
M_{R}\sim\left(\begin{array}[]{ccc}0&\times&\times\\\ \times&\times&0\\\
\times&0&\times\end{array}\right)$ (57)
Using the neutrino mass formula of type-I seesaw mechanism
$M_{\nu}=-M_{D}M_{R}M_{D}^{T}$, we obtain
$M_{\nu}\sim\left(\begin{array}[]{ccc}\Delta&\times&\times\\\
\times&\times&\Delta\\\ \times&\Delta&\times\end{array}\right)$ (58)
Together with the $M_{l}$ in (57), we have realized the leptonic mass matrices
of class II with parallel texture/cofactor zeros under $Z_{4}\times Z_{2}$
flavor symmetry. The symmetry realization of class III can be similarly
performed.
## V Conclusion and discussion
We have investigated the parallel texture structures with two texture zeros in
lepton mass matrix $M_{l}$ and two cofactor zeros in neutrino mass matrix
$M_{\nu}$. The 15 possible textures are grouped into class I, II, III, and IV,
where the matrices in each class are related by means of permutation
transformation and share the same physical implications. We found only class
I, II, III are notrivial. Using the recent results of the neutrino oscillation
and cosmology experiments, a phenomenological analysis are systematically
proposed for each class and mass hierarchy. We demonstrate the correlation
plots between Dirac CP-violating phase $\delta$, three mixing angles
$\theta_{12},\theta_{23}$ and $\theta_{13}$, the effective Majorana neutrino
mass $m_{ee}$, the lightest neutrino mass, Majorana CP-violating phase
$\alpha,\beta$ and the neutrino mass ratio, leading to the predictions to be
confirmed by future experiments. A realization of the model base on
$Z_{4}\times Z_{2}$ flavor symmetry is illustrated.
Finally we would like to mention that in the spirit of Ref. GCB (7, 21), the
parallel texture structures are treated as a natural precursor of more general
cases. A systematic analysis of all possible combinations deserves further
study and will be published in pre (32).
###### Acknowledgements.
This work is supported by the Fundamental Research Funds for the Central
Universities The author would like to thank Shu-Yuan Guo for the helpful
discussion.
## References
* (1) Q.R. Ahmad et al.(SNO Collaboration), Phys. Rev. Lett 89, 011301(2002); K. Eguchi et al. (KamLAND Collaboration), Phys. Rev. Lett 90, 021802(2003); M.H. Ahn et al. (K2K Collaboration), Phys. Rev. Lett 90, 041801(2003).
* (2) F.P. An et al. (DAYA-BAY Xollaboration), Phys. Rev. Lett. 108, 171803(2012).
* (3) J.K. Ahn et al. (RENO Collaboration), Phys. Rev. Lett.108, 191802(2012).
* (4) H. Fritzsch, M. Gell-Mann, and P. Minkowski, Phys. Lett. B59, 256(1975); P. Minkowski, Phys. Lett. B67, 421(1977); T. Yanagida, in Proceedings of Workshop on Unified Theory and the Baryon Number of the Universe, edited by O. Sawada and A. Sugamoto(KEK, Tsukuba, 1979), p. 95; M. Gell-Mann, P. Ramond, and Slansky, in Supergravity, edited by P. van. Nieuwenhuizen and D.Z. Freeman (North-Holland, Amsterdam,1979), p. 315; R.N. Mohapatra and G. Senjanovic, Phys, Rev. Lett. 44, 912(1980); J. Schechter and J. W. F. Valle, Phys. Rev. D22, 2227(1980); J. Schechter and J. W. F. Valle, Phys. Rev. D25, 774(1982).
* (5) P.H. Frampton, S. L. Glashow, and D. Marfatia, Phys. Lett. B536, 79(2002); H. Fritzsch, Z.-z. Xing, and S. Zhou, J. High Energy Phys. 09 (2011)083.
* (6) Z,-z. Xing, Phys. Lett. B530, 159(2002); A. Merle, and W. Rodejohann, Phys. Rev. D73, 073012(2006); S. Dev, S. Kumar, S. Verma, and S. Gupta, Phys. Rev. D76, 013002(2007); S. Dev, S. Kumar, S. Verma, and S. Gupta, Nucl. Phys. B784, 103(2007);S. Dev, S. Kumar, Mod. Phys, Lett. A22, 1401(2007); S. Kumar, Phys. Rev. D84, 077301(2011); P.O. Ludl, S. Morisi, and E. Peinado, Nucl. Phys. B857, 411(2012); W. Grimus. and P.O. Ludl, arXiv:1208.4515; D. Meloni, and G. Blankenburg, Nucl. Phys. B867, 749(2013);
* (7) G. C. Branco, D. Emmannuel-Costa, R. Gonz$\acute{a}$lez Felipe, and H. Ser$\hat{o}$dio, Phys. Lett. B670, 340(2009);
* (8) M. Randhawa, G. Ahuja, and M. Gupta, Phys. Lett. B643, 175(2006); G. Ahuja, S. Kumar, M. Randhawa, M. Gupta, and S. Dev, Phys. Rev. D76, 013006(2007); G. Ahuja, M. Gupta, M. Randhawa, and R. Verma, Phys. Rev. D79, 093006(2009); J. A. Acosta, Alfredo Aranda, M. A. Buen-Abad, A. D. Rojas, Phys. Lett. B18, 1413(2013) S. Sharma, P. Fakay, G. Ahuja, M. Gupta, arXiv: 1402.0628; S. Sharma, P. Fakay, G. Ahuja, M. Gupta, arXiv: 1402.1598.
* (9) M. Gupta, G. Ahuja; Int. Jour. Mod. Phys. A23, 1270033(2012).
* (10) S. Kaneko, H. Sawanaka, and M. Tanimoto, J. High Energy Phys. 08 (2005)073; S. Dev, S. Verma, and S. Gupta, Phys. Lett. B687, 53(2010); S. Goswami, S. Khan, and A. Watanable, Phys. Lett. B687, 53(2010), W. Grimus, and P. O. Ludl, arXiv: 1208.4515.
* (11) J.-Y. Liu and S. Zhou, Phys. Rev. D87, 093010(2013).
* (12) X.-G. He and A. Zee, Phys. Rev. D68, 037302(2003).
* (13) G.C. Branco, R. Gonzalez Felipe, F.R. Joaquim, and T. Yanagida, Phys. Lett. B562, 265(2003); B.C. Chauhan, J. Pulido, and M. Picariello, Phys. Rev. D73, 053003(2006).
* (14) L. Lavoura, Phys. Lett. B609, 317(2005); E.I. Lashin and N. Chamoun, Phys. Rev. D78, 073002(2008); E.I. Lashin and N. Chamoun, Phys. Rev. D80, 093004(2009);
* (15) S. Dev, S. Gupta, and R.R. Gautam, Mod. Phys, Lett. A26, 501(2011); S. Dev, S. Gupta, R.R. Gautam, and L. Singh, Phys. Lett. B706, 168(2011); T. Araki, J. Heeck, and J. Kubo, J. High Energy Phys. 07 (2012)083; S. Verma, Nucl. Phys. B854, 340(2012); S. Dev, R.R. Gautam, and L. Singh, arXiv: 1309.4219;
* (16) S. Dev, S. Verma, S. Gupta, and R.R. Gautam, Phys. Rev. D81, 053010(2010); J. Liao,D. Marfatia, K. Whisnant, arXiv: 1311.2639.
* (17) H.A. Alhendi, E.I. Lashin, and A.A. Mudlej, Phys. Rev. D77, 013009(2008).
* (18) S. Dev, R.R. Gautam, and L. Singh, Phys. Rev. D87, 073011(2013).
* (19) S. Dev, R.R. Gautam, and L. Singh, Phys. Rev. D88, 033008(2013); W. Wang, Eur. Phys. J. C73, 2551(2013).
* (20) W. Grimus, A.S. Joshipura, L. Lavoura, M. Tanimoto, Eur. Phys. J. C36, 227(2004).
* (21) S. Dev, S. Gupta, and R.R. Gautam, Phys. Rev. D82, 073015(2010);
* (22) W. Wang, Phys. Lett. B733, 320(2014);
* (23) B. Pontecorvo, Zh. Eksp. Teor. Fiz. 33, 549(1957); Z. Maki, M. Nakagawa, and N. Sakata, Prog. Theor. Phys. 28, 870(1962).
* (24) Z. Z. Xing. Phys. Lett. B550, 178(2002); Z. Z. Xing, and S. Zhou Phys. Lett. B593, 156(2004); S. Zhou, and Z. Z. Xing, Eur. Phys. J. C38, 495(2005);
* (25) F. Capazzi, G.L. Fogli, E. Lisi, A. Marrone, D. Montanino, and A. Palazzo, Phys. Rev. D89, 093018(2014).
* (26) C. Jarlskog, Phys, Rev. Lett. 55, 1039(1985).
* (27) H.V. Klapdor-Kleingrothaus, A. Dietz, H.L. Harney, and I.V. Krivosheina, Mod. Phys. Lett. A16, 2409(2001).
* (28) C.E. Aalseth et al. Mod. Phys. Lett. A17, 1475(2002); F. Feruglio, A. Strumia, and F. Vissani, Nucl. Phys. B637, 345(2002).
* (29) S.M. Bilenky and C. Giunti, Mod. Phys, Lett. A16, 1230015(2012).
* (30) P.A.R. Ade et al. (Planck Collaboration), arXiv: 1303.5076.
* (31) Z. Z. Xing, H. Zhang, and S. Zhou, Phys. Rev. D86, 013013(2012).
* (32) in prepearation.
|
arxiv-papers
| 2014-02-27T07:14:19 |
2024-09-04T02:49:58.989320
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Weijian Wang",
"submitter": "Weijian Wang",
"url": "https://arxiv.org/abs/1402.6808"
}
|
1402.6852
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-026 LHCb-PAPER-2014-001 27 February 2014
Observation of photon polarization in the ${b}\\!\rightarrow{s}{\gamma}$
transition
The LHCb collaboration†††Authors are listed on the following pages.
This Letter presents a study of the flavor-changing neutral current radiative
${{B}^{\pm}}\\!\rightarrow{{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{\pm}}{\gamma}$
decays performed using data collected in proton-proton collisions with the
LHCb detector at $7$ and $8\,$TeV center-of-mass energies. In this sample,
corresponding to an integrated luminosity of $3\,\mbox{\,fb}^{-1}$, nearly
$14\,000$ signal events are reconstructed and selected, containing all
possible intermediate resonances with a
${{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{\pm}}$ final state in the $[1.1,1.9]$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ mass range. The distribution of the
angle of the photon direction with respect to the plane defined by the final-
state hadrons in their rest frame is studied in intervals of
${{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{\pm}}$ mass and the asymmetry between the
number of signal events found on each side of the plane is obtained. The first
direct observation of the photon polarization in the
${b}\\!\rightarrow{s}{\gamma}$ transition is reported with a significance of
$5.2\,\sigma$.
Published in Phys. Rev. Lett. 112, 161801 (2014)
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25, S. Amato2, S. Amerio22, Y. Amhis7, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,m, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel39, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson50, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30, W.
Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,q, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez36,o, A. Camboni36, P. Campana18,38, D. Campora
Perez38, F. Caponio21, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,j, A.
Cardini15, H. Carranza-Mejia50, L. Carson50, K. Carvalho Akiba2, G. Casse52,
L. Cassina20, L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci58, M.
Charles8, Ph. Charpentier38, S.-F. Cheung55, N. Chiapolini40, M.
Chrzaszcz40,26, K. Ciba38, X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M.
Clemencic38, H.V. Cliff47, J. Closier38, C. Coca29, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins38, A. Comerma-Montells36, A. Contu15,38, A. Cook46, M.
Coombes46, S. Coquereau8, G. Corti38, I. Counts56, B. Couturier38, G.A.
Cowan50, D.C. Craik48, M. Cruz Torres60, S. Cunliffe53, R. Currie50, C.
D’Ambrosio38, J. Dalseno46, P. David8, P.N.Y. David41, A. Davis57, I. De
Bonis4, K. De Bruyn41, S. De Capua54, M. De Cian11, J.M. De Miranda1, L. De
Paula2, W. De Silva57, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, D. Derkach55, O. Deschamps5, F. Dettori42, A. Di Canto11,
H. Dijkstra38, S. Donleavy52, F. Dordei11, M. Dorigo39, P. Dorosz26,n, A.
Dosil Suárez37, D. Dossett48, A. Dovbnya43, F. Dupertuis39, P. Durante38, R.
Dzhelyadin35, A. Dziurda26, A. Dzyuba30, S. Easo49, U. Egede53, V.
Egorychev31, S. Eidelman34, S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L.
Eklund51,38, I. El Rifai5, Ch. Elsasser40, S. Esen11, A. Falabella16,f, C.
Färber11, C. Farinelli41, S. Farry52, D. Ferguson50, V. Fernandez Albor37, F.
Ferreira Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M.
Fiorini16,f, C. Fitzpatrick38, M. Fontana10, F. Fontanelli19,j, R. Forty38, O.
Francisco2, M. Frank38, C. Frei38, M. Frosini17,38,g, J. Fu21, E. Furfaro24,l,
A. Gallas Torreira37, D. Galli14,d, S. Gambetta19,j, M. Gandelman2, P.
Gandini59, Y. Gao3, J. Garofoli59, J. Garra Tico47, L. Garrido36, C. Gaspar38,
R. Gauld55, L. Gavardi9, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph.
Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47, L. Giubega29, V.V. Gligorov38,
C. Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, H. Gordon38, M.
Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E. Graugés36,
G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P. Griffith45, L.
Grillo11, O. Grünberg61, B. Gui59, E. Gushchin33, Yu. Guz35,38, T. Gys38, C.
Hadjivasiliou59, G. Haefeli39, C. Haen38, T.W. Hafkenscheid64, S.C. Haines47,
S. Hall53, B. Hamilton58, T. Hampson46, S. Hansmann-Menzemer11, N. Harnew55,
S.T. Harnew46, J. Harrison54, T. Hartmann61, J. He38, T. Head38, V. Heijne41,
K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E. van
Herwijnen38, M. Heß61, A. Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54, W.
Hulsbergen41, P. Hunt55, N. Hussain55, D. Hutchcroft52, D. Hynds51, M.
Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, E. Jans41, P. Jaton39, A.
Jawahery58, F. Jing3, M. John55, D. Johnson55, C.R. Jones47, C. Joram38, B.
Jost38, N. Jurik59, M. Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M.
Karbach38, M. Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C.
Khurewathanakul39, S. Klaver54, O. Kochebina7, I. Komarov39, R.F. Koopman42,
P. Koppenburg41, M. Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11,
M. Kreps48, G. Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k,
V. Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31,38, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, B. Langhans38, T. Latham48, C.
Lazzeroni45, R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A.
Leflat32, J. Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11,
Y. Li3, M. Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G.
Liu38, S. Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40,
H. Lu3, D. Lucchesi22,q, H. Luo50, E. Luppi16,f, O. Lupton55, F. Machefert7,
I.V. Machikhiliyan31, F. Maciuc29, O. Maev30,38, S. Malde55, G. Manca15,e, G.
Mancinelli6, M. Manzali16,f, J. Maratas5, U. Marconi14, C. Marin Benito36, P.
Marino23,s, R. Märki39, J. Marks11, G. Martellotti25, A. Martens8, A. Martín
Sánchez7, M. Martinelli41, D. Martinez Santos42, F. Martinez Vidal63, D.
Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A.
Mazurov16,38,f, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
McSkelly52, B. Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,s, R. Mountain59, F.
Muheim50, K. Müller40, R. Muresan29, B. Muryn27, B. Muster39, P. Naik46, T.
Nakada39, R. Nandakumar49, I. Nasteva1, M. Needham50, N. Neri21, S. Neubert38,
N. Neufeld38, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-
Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R.
Oldeman15,e, G. Onderwater64, M. Orlandea29, J.M. Otalora Goicochea2, P.
Owen53, A. Oyanguren36, B.K. Pal59, A. Palano13,c, F. Palombo21,t, M.
Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51, L. Pappalardo16,
C. Parkes54, C.J. Parkinson9, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, C. Pavel-Nicorescu29, A. Pazos Alvarez37, A. Pearce54, A.
Pellegrino41, M. Pepe Altarelli38, S. Perazzini14,d, E. Perez Trigo37, P.
Perret5, M. Perrin-Terrin6, L. Pescatore45, E. Pesen65, G. Pessina20, K.
Petridis53, A. Petrolini19,j, E. Picatoste Olloqui36, B. Pietrzyk4, T.
Pilař48, D. Pinci25, A. Pistone19, S. Playfer50, M. Plo Casasus37, F. Polci8,
A. Poluektov48,34, E. Polycarpo2, A. Popov35, D. Popov10, B. Popovici29, C.
Potterat36, A. Powell55, J. Prisciandaro39, A. Pritchard52, C. Prouve46, V.
Pugatch44, A. Puig Navarro39, G. Punzi23,r, W. Qian4, B. Rachwal26, J.H.
Rademacker46, B. Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N.
Rauschmayr38, G. Raven42, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S.
Ricciardi49, A. Richards53, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts58, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez
Perez37, S. Roiser38, V. Romanovsky35, A. Romero Vidal37, M. Rotondo22, J.
Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P. Ruiz Valls36, G.
Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B. Saitta15,e,
V. Salustino Guimaraes2, B. Sanmartin Sedes37, R. Santacesaria25, C.
Santamarina Rios37, E. Santovetti24,l, M. Sapunov6, A. Sarti18, C.
Satriano25,m, A. Satta24, M. Savrie16,f, D. Savrina31,32, M. Schiller42, H.
Schindler38, M. Schlupp9, M. Schmelling10, B. Schmidt38, O. Schneider39, A.
Schopper38, M.-H. Schune7, R. Schwemmer38, B. Sciascia18, A. Sciubba25, M.
Seco37, A. Semennikov31, K. Senderowska27, I. Sepp53, N. Serra40, J. Serrano6,
P. Seyfert11, M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52,
L. Shekhtman34, O. Shevchenko43, V. Shevchenko62, A. Shires9, R. Silva
Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, T. Skwarnicki59, N.A.
Smith52, E. Smith55,49, E. Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D.
Sokoloff57, F.J.P. Soler51, F. Soomro39, D. Souza46, B. Souza De Paula2, B.
Spaan9, A. Sparkes50, F. Spinella23, P. Spradlin51, F. Stagni38, S. Stahl11,
O. Steinkamp40, S. Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S.
Stracka23,38, M. Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38,
L. Sun57, W. Sutcliffe53, S. Swientek9, V. Syropoulos42, M. Szczekowski28, P.
Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, E. Teodorescu29, F. Teubert38, C. Thomas55, E. Thomas38, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D.
Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4,53, S. Tourneur39,
M.T. Tran39, M. Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M.
Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin62, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez
Sierra37, S. Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M.
Vesterinen11, B. Viaud7, D. Vieira2, X. Vilasis-Cardona36,o, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß61, H.
Voss10, J.A. de Vries41, R. Waldi61, C. Wallace48, R. Wallace12, S.
Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, A.D. Webber54, D.
Websdale53, M. Whitehead48, J. Wicht38, J. Wiechczynski26, D. Wiedner11, G.
Wilkinson55, M.P. Williams48,49, M. Williams56, F.F. Wilson49, J. Wimberley58,
J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A. Wotton47, S.
Wright47, S. Wu3, K. Wyllie38, Y. Xie50,38, Z. Xing59, Z. Yang3, X. Yuan3, O.
Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, F. Zhang3, L. Zhang59, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3, A. Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
62National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
63Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
64KVI - University of Groningen, Groningen, The Netherlands, associated to 41
65Celal Bayar University, Manisa, Turkey, associated to 38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
The Standard Model (SM) predicts that the photon emitted from the electroweak
penguin loop in ${b}\\!\rightarrow{s}{\gamma}$ transitions is predominantly
left-handed, since the recoiling $s$ quark that couples to a $W$ boson is
left-handed. This implies maximal parity violation up to small corrections of
the order $m_{{s}}/m_{{b}}$. While the measured inclusive
${b}\\!\rightarrow{s}{\gamma}$ rate [1] agrees with the SM calculations, no
direct evidence exists for a nonzero photon polarization in this type of
decays. Several extensions of the SM [2, *Mohapatra:1974gc, *Mohapatra:1974hk,
*Senjanovic:1975rk, *Senjanovic:1978ev, *Mohapatra:1980yp, *Lim:1981kv,
*Everett:2001yy], compatible with all current measurements, predict that the
photon acquires a significant right-handed component, in particular due to the
exchange of a heavy fermion in the penguin loop [10].
This Letter presents a study of the radiative decay
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$, previously
observed at the $B$ factories with a measured branching fraction of $(27.6\pm
2.2)\times 10^{-6}$ [11, 12, 1]. The inclusion of charge-conjugate processes
is implied throughout. Information about the photon polarization is obtained
from the angular distribution of the photon direction with respect to the
normal to the plane defined by the momenta of the three final-state hadrons in
their center-of-mass frame. The shape of this distribution, including the _up-
down asymmetry_ between the number of events with the photon on either side of
the plane, is determined. This investigation is conceptually similar to the
historical experiment that discovered parity violation by measuring the up-
down asymmetry of the direction of a particle emitted in a weak decay with
respect to an axial vector [13]. In
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$ decays, the
up-down asymmetry is proportional to the photon polarization
$\lambda_{\gamma}$ [14, 15] and therefore measuring a value different from
zero corresponds to demonstrating that the photon is polarized. The currently
limited knowledge of the structure of the ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass
spectrum, which includes interfering kaon resonances, prevents the translation
of a measured asymmetry into an actual value for $\lambda_{\gamma}$.
The differential
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$ decay rate
can be described in terms of $\theta$, defined in the rest frame of the final
state hadrons as the angle between the direction opposite to the photon
momentum $\vec{p}_{\gamma}$ and the normal
$\vec{p}_{\pi,\text{slow}}\times\vec{p}_{\pi,\text{fast}}$ to the
${K}^{+}{\pi}^{-}{\pi}^{+}$ plane, where $\vec{p}_{\pi,\text{slow}}$ and
$\vec{p}_{\pi,\text{fast}}$ correspond to the momenta of the lower and higher
momentum pions, respectively. Following the notation and developments of Ref.
[14], the differential decay rate of
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$ can be
written as a fourth-order polynomial in $\cos\theta$
$\displaystyle\frac{\operatorname{d}\\!\Gamma}{\operatorname{d}\\!s\operatorname{d}\\!s_{13}\operatorname{d}\\!s_{23}\operatorname{d}\\!\cos\theta}\propto\sum_{i=0,2,4}a_{i}(s,s_{13},s_{23})\cos^{i}\theta+\lambda_{\gamma}\sum_{j=1,3}a_{j}(s,s_{13},s_{23})\cos^{j}\theta\,,$
(1)
where $s_{ij}=(p_{i}+p_{j})^{2}$ and $s=(p_{1}+p_{2}+p_{3})^{2}$, and $p_{1}$,
$p_{2}$ and $p_{3}$ are the four-momenta of the ${\pi}^{-}$, ${\pi}^{+}$ and
${K}^{+}$ mesons, respectively. The functions $a_{k}$ depend on the resonances
present in the ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass range of interest and their
interferences. The up-down asymmetry is defined as
$\mathcal{A}_{\text{ud}}\equiv\frac{\int_{0}^{1}\operatorname{d}\\!\cos\theta\frac{\operatorname{d}\\!\Gamma}{\operatorname{d}\\!\cos\theta}-\int_{-1}^{0}\operatorname{d}\\!\cos\theta\frac{\operatorname{d}\\!\Gamma}{\operatorname{d}\\!\cos\theta}}{\int_{-1}^{1}\operatorname{d}\\!\cos\theta\frac{\operatorname{d}\\!\Gamma}{\operatorname{d}\\!\cos\theta}}\,,$
(2)
which is proportional to $\lambda_{\gamma}$.
The LHCb detector [16] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
resolution of 20${\,\upmu\rm m}$ for tracks with a few
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ of transverse momentum ($p_{\rm T}$).
Different types of charged hadrons are distinguished by information from two
ring-imaging Cherenkov detectors. Photon, electron and hadron candidates are
identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. The trigger consists of a hardware stage, based on information
from the calorimeter and muon systems, followed by a software stage, which
applies a full event reconstruction.
Samples of simulated events are used to understand signal and backgrounds. In
the simulation, $pp$ collisions are generated using Pythia [17,
*Sjostrand:2007gs] with a specific LHCb configuration [19]. Decays of hadronic
particles are described by EvtGen [20], in which final state radiation is
generated using Photos [21]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [22,
*Agostinelli:2002hh] as described in Ref. [24].
This analysis is based on the LHCb data sample collected from $pp$ collisions
at $7$ and $8$ TeV center-of-mass energies in 2011 and 2012, respectively,
corresponding to $3~{}\mbox{\,fb}^{-1}$ of integrated luminosity. The
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$ candidates
are built from a photon candidate and a hadronic system reconstructed from a
kaon and two oppositely charged pions satisfying particle identification
requirements. Each of the hadrons is required to have a minimum $p_{\rm T}$ of
$0.5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and at least one of them needs to
have a $p_{\rm T}$ larger than $1.2\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
The isolation of the ${K}^{+}{\pi}^{-}{\pi}^{+}$ vertex from other tracks in
the event is ensured by requiring that the $\chi^{2}$ of the three-track
vertex fit and the $\chi^{2}$ of all possible vertices that can be obtained by
adding an extra track differ by more than $2$. The ${K}^{+}{\pi}^{-}{\pi}^{+}$
mass is required to be in the
$[1.1,1.9]\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ range. High transverse
energy ($>3.0\,\mathrm{\,Ge\kern-1.00006ptV}$) photon candidates are
constructed from energy depositions in the electromagnetic calorimeter. The
absence of tracks pointing to the calorimeter is used to distinguish neutral
from charged electromagnetic particles. A multivariate algorithm based on the
energy cluster shape parameters is used to reject approximately $65\,\%$ of
the ${{\pi}^{0}}\\!\rightarrow{\gamma}{\gamma}$ background in which the two
photons are reconstructed as a single cluster, while keeping about $95\,\%$ of
the signal photons. The ${{B}^{+}}$ candidate mass is required to be in the
$[4.3,6.9]\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ range. Backgrounds
that are expected to peak in this mass range are suppressed by removing all
candidates consistent with a
$\bar{{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ or
$\rho^{+}\\!\rightarrow{{\pi}^{+}}{{\pi}^{0}}$ decay when the photon candidate
is assumed to be a ${\pi}^{0}$.
A boosted decision tree [25, 26] is used to further improve the separation
between signal and background. Its training is based on the following
variables: the impact parameter $\chi^{2}$ of the ${{B}^{+}}$ meson and of
each of the final state hadrons, defined as the difference between the
$\chi^{2}$ of a primary vertex (PV) reconstructed with and without the
considered particle; the cosine of the angle between the reconstructed
${{B}^{+}}$ momentum and the vector pointing from the PV to the ${{B}^{+}}$
decay vertex; the flight distance of the ${{B}^{+}}$ meson; and the
${K}^{+}{\pi}^{-}{\pi}^{+}$ vertex $\chi^{2}$.
The mass distribution of the selected
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$ signal is
modeled with a double-tailed Crystal Ball [27] probability density function
(PDF), with power-law tails above and below the $B$ mass. The four tail
parameters are fixed from simulation; the width of the signal peak is fit
separately for 2011 and 2012 data to account for differences in calorimeter
calibration. Combinatorial and partially reconstructed backgrounds are
considered in the fit, the former modeled with an exponential PDF, the latter
described using an ARGUS PDF [28] convolved with a Gaussian function with the
same width as the signal to account for the photon energy resolution. The
contribution to the partially reconstructed background from events with only
one missing pion is considered separately.
The fit of the mass distribution of the selected
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$ candidates
(Fig. 1) returns a total signal yield of $13\,876\pm 153$ events, the largest
sample recorded for this channel to date.
Figure 1: Mass distribution of the selected
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$ candidates.
The blue solid curve shows the fit results as the sum of the following
components: signal (red solid), combinatorial background (green dotted),
missing pion background (black dashed) and other partially reconstructed
backgrounds (purple dash-dotted).
Figure 2 shows the background-subtracted ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass
spectrum determined using the technique of Ref. [29], after constraining the
$B$ mass to its nominal value.
Figure 2: Background-subtracted ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass distribution
of the ${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$
signal. The four intervals of interest, separated by dashed lines, are shown.
No peak other than that of the ${K}_{1}(1270)^{+}$ resonance can be clearly
identified. Many kaon resonances, with various masses, spins and angular
momenta, are expected to contribute and interfere in the considered mass range
[1].
The contributions from single resonances cannot be isolated because of the
complicated structure of the ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass spectrum. The
up-down asymmetry is thus studied inclusively in four intervals of
${K}^{+}{\pi}^{-}{\pi}^{+}$ mass. The
$[1.4,1.6]\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ interval, studied in
Ref. [14], includes the ${K}_{1}(1400)^{+}$, ${K}^{*}_{2}(1430)^{+}$ and
${K}^{*}(1410)^{+}$ resonances with small contributions from the upper tail of
the ${K}_{1}(1270)^{+}$. At the time of the writing of Ref. [14], the
${K}_{1}(1400)^{+}$ was believed to be the dominant $1^{+}$ resonance, so the
${K}_{1}(1270)^{+}$ was not considered. However, subsequent experimental
results [30] demonstrated that the ${K}_{1}(1270)^{+}$ is more prominent than
the ${K}_{1}(1400)^{+}$, hence the
$[1.1,1.3]\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ interval is also studied
here. The $[1.3,1.4]\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ mass interval,
which contains the overlap region between the two $K_{1}$ resonances, and the
$[1.6,1.9]\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ high mass interval,
which includes spin-2 and spin-3 resonances, are also considered.
In each of the four ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass intervals, a simultaneous
fit to the $B$-candidate mass spectra in bins of the photon angle is performed
in order to determine the background-subtracted angular distribution; the
previously described PDF is used to model the mass spectrum in each bin, with
all of the fit parameters being shared except for the yields. Since the sign
of the photon polarization depends on the sign of the electric charge of the
$B$ candidate, the angular variable
$\cos\hat{{\theta}}\equiv\operatorname{charge}({B})\cos\theta\,$ is used. The
resulting background-subtracted $\cos\hat{{\theta}}$ distribution, corrected
for the selection acceptance and normalized to the inverse of the bin width,
is fit with a fourth-order polynomial function normalized to unit area,
$f(\cos\hat{{\theta}};c_{0}\\!=\\!0.5,c_{1},c_{2},c_{3},c_{4})=\sum_{i=0}^{4}c_{i}L_{i}(\cos\hat{{\theta}})\,,$
(3)
where $L_{i}(x)$ is the Legendre polynomial of order $i$ and $c_{i}$ is the
corresponding coefficient. Using Eqs. 1 and 3 the up-down asymmetry defined in
Eq. 2 can be expressed as
$\mathcal{A}_{\text{ud}}=c_{1}-\frac{c_{3}}{4}\,\,.$ (4)
As a cross-check, the up-down asymmetry in each mass interval is also
determined with a counting method, rather than an angular fit, as well as
considering separately the ${{B}^{+}}$ and ${{B}^{-}}$ candidates. All these
checks yield compatible results.
The results obtained from a $\chi^{2}$ fit of the normalized binned angular
distribution, performed taking into account the full covariance matrix of the
bin contents and all of the systematic uncertainties, are summarized in Table
1. These systematic uncertainties account for the effect of choosing a
different fit model, the impact of the limited size of the simulated samples
on the fixed parameters, and the possibility of some events migrating from a
bin to its neighbor because of the detector resolution, which gives the
dominant contribution. The systematic uncertainty associated with the fit
model is determined by performing the mass fit using several alternative PDFs,
while the other two are estimated with simulated pseudoexperiments. Such
uncertainties, despite being of the same size as the statistical uncertainty,
do not substantially affect the fit results since they are strongly correlated
across all angular bins.
The fitted distributions in the four ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass
intervals of interest are shown in Fig. 3. In order to illustrate the effect
of the up-down asymmetry, the results of another fit imposing $c_{1}=c_{3}=0$,
hence forbidding the terms that carry the $\lambda_{\gamma}$ dependence, are
overlaid for comparison.
Figure 3: Distributions of $\cos\hat{{\theta}}$ for
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$ signal in
four intervals of ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass. The solid blue (dashed
red) curves are the result of fits allowing all (only even) Legendre
components up to the fourth power.
The combined significance of the observed up-down asymmetries is determined
from a $\chi^{2}$ test where the null hypothesis is defined as
$\lambda_{\gamma}=0$, implying that the up-down asymmetry is expected to be
zero in each mass interval. The corresponding $\chi^{2}$ distribution has four
degrees of freedom, and the observed value corresponds to a p-value of
$1.7\times 10^{-7}$. This translates into a $5.2\,\sigma$ significance for
nonzero up-down asymmetry. Up-down asymmetries can be computed also for an
alternative definition of the photon angle, obtained using the normal
$\vec{p}_{{{\pi}^{-}}}\times\vec{p}_{{{\pi}^{+}}}$ instead of
$\vec{p}_{\pi,\text{slow}}\times\vec{p}_{\pi,\text{fast}}$. The obtained
values, along with the relative fit coefficients, are listed in Table 2.
Table 1: Legendre coefficients obtained from fits to the normalized background-subtracted $\cos\hat{{\theta}}$ distribution in the four ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass intervals of interest. The up-down asymmetries are obtained from Eq. 4. The quoted uncertainties contain statistical and systematic contributions. The ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass ranges are indicated in ${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ and all the parameters are expressed in units of $10^{-2}$. The covariance matrices are given in the supplementary material. | $[1.1,1.3]$ | $[1.3,1.4]$ | $[1.4,1.6]$ | $[1.6,1.9]$
---|---|---|---|---
$c_{1}$ | $6.3$ | $\pm$ | $1.7$ | $5.4$ | $\pm$ | $2.0$ | $4.3$ | $\pm$ | $1.9$ | $-4.6$ | $\pm$ | $1.8$
$c_{2}$ | $31.6$ | $\pm$ | $2.2$ | $27.0$ | $\pm$ | $2.6$ | $43.1$ | $\pm$ | $2.3$ | $28.0$ | $\pm$ | $2.3$
$c_{3}$ | $-2.1$ | $\pm$ | $2.6$ | $2.0$ | $\pm$ | $3.1$ | $-5.2$ | $\pm$ | $2.8$ | $-0.6$ | $\pm$ | $2.7$
$c_{4}$ | $3.0$ | $\pm$ | $3.0$ | $6.8$ | $\pm$ | $3.6$ | $8.1$ | $\pm$ | $3.1$ | $-6.2$ | $\pm$ | $3.2$
$\mathcal{A}_{\text{ud}}$ | $6.9$ | $\pm$ | $1.7$ | $4.9$ | $\pm$ | $2.0$ | $5.6$ | $\pm$ | $1.8$ | $-4.5$ | $\pm$ | $1.9$
Table 2: Legendre coefficients obtained from fits to the normalized background-subtracted $\cos\hat{{\theta}}$ distribution, using the alternative normal $\vec{p}_{{{\pi}^{-}}}\times\vec{p}_{{{\pi}^{+}}}$ for defining the direction of the photon, in the four ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass intervals of interest. The up-down asymmetries are obtained from Eq. 4. The quoted uncertainties contain statistical and systematic contributions. The ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass ranges are indicated in ${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ and all the parameters are expressed in units of $10^{-2}$. The covariance matrices are given in the supplementary material. | $[1.1,1.3]$ | $[1.3,1.4]$ | $[1.4,1.6]$ | $[1.6,1.9]$
---|---|---|---|---
$c_{1}^{\prime}$ | $-0.9$ | $\pm$ | $1.7$ | $7.4$ | $\pm$ | $2.0$ | $5.3$ | $\pm$ | $1.9$ | $-3.4$ | $\pm$ | $1.8$
$c_{2}^{\prime}$ | $31.6$ | $\pm$ | $2.2$ | $27.4$ | $\pm$ | $2.6$ | $43.6$ | $\pm$ | $2.3$ | $27.8$ | $\pm$ | $2.3$
$c_{3}^{\prime}$ | $0.8$ | $\pm$ | $2.6$ | $0.8$ | $\pm$ | $3.1$ | $-4.4$ | $\pm$ | $2.8$ | $2.3$ | $\pm$ | $2.7$
$c_{4}^{\prime}$ | $3.4$ | $\pm$ | $3.0$ | $7.0$ | $\pm$ | $3.6$ | $8.0$ | $\pm$ | $3.1$ | $-6.6$ | $\pm$ | $3.2$
$\mathcal{A}^{\prime}_{\text{ud}}$ | $-1.1$ | $\pm$ | $1.7$ | $7.2$ | $\pm$ | $2.0$ | $6.4$ | $\pm$ | $1.8$ | $-3.9$ | $\pm$ | $1.9$
To summarize, a study of the inclusive flavor-changing neutral current
radiative ${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{\gamma}$
decay, with the ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass in the
$[1.1,1.9]\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ range, is performed on a
data sample corresponding to an integrated luminosity of $3\,\mbox{\,fb}^{-1}$
collected in $pp$ collisions at $7$ and $8$ TeV center-of-mass energies by the
LHCb detector. A total of $13\,876\pm 153$ signal events is observed. The
shape of the angular distribution of the photon with respect to the plane
defined by the three final-state hadrons in their rest frame is determined in
four intervals of interest in the ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass spectrum.
The up-down asymmetry, which is proportional to the photon polarization, is
measured for the first time for each of these ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass
intervals. The first observation of a parity-violating photon polarization
different from zero at the $5.2\,\sigma$ significance level in
${b}\\!\rightarrow{s}{\gamma}$ transitions is reported. The shape of the
photon angular distribution in each bin, as well as the values for the up-down
asymmetry, may be used, if theoretical predictions become available, to
determine for the first time a value for the photon polarization, and thus
constrain the effects of physics beyond the SM in the
${b}\\!\rightarrow{s}{\gamma}$ sector.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted towards the communities behind the multiple open
source software packages we depend on. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia).
## References
* [1] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001, and 2013 partial update for the 2014 edition
* [2] J. C. Pati and A. Salam, Lepton number as the fourth color, Phys. Rev. D10 (1974) 275, Erratum-ibid. D11 (1975) 703
* [3] R. Mohapatra and J. C. Pati, A natural left-right symmetry, Phys. Rev. D11 (1975) 2558
* [4] R. N. Mohapatra and J. C. Pati, Left-right gauge symmetry and an isoconjugate model of CP violation, Phys. Rev. D11 (1975) 566
* [5] G. Senjanovic and R. N. Mohapatra, Exact left-right symmetry and spontaneous violation of parity, Phys. Rev. D12 (1975) 1502
* [6] G. Senjanovic, Spontaneous breakdown of parity in a class of gauge theories, Nucl. Phys. B153 (1979) 334
* [7] R. N. Mohapatra and G. Senjanovic, Neutrino masses and mixings in gauge models with spontaneous parity violation, Phys. Rev. D23 (1981) 165
* [8] C. Lim and T. Inami, Lepton flavor nonconservation and the mass generation mechanism for neutrinos, Prog. Theor. Phys. 67 (1982) 1569
* [9] L. L. Everett et al., Alternative approach to $b\rightarrow s\gamma$ in the uMSSM, JHEP 01 (2002) 022, arXiv:hep-ph/0112126
* [10] D. Atwood, M. Gronau, and A. Soni, Mixing-induced CP asymmetries in radiative B decays in and beyond the standard model, Phys. Rev. Lett. 79 (1997) 185, arXiv:hep-ph/9704272
* [11] Belle collaboration, S. Nishida et al., Radiative B meson decays into $K\pi\gamma$ and $K\pi\pi\gamma$ final states, Phys. Rev. Lett. 89 (2002) 231801, arXiv:hep-ex/0205025
* [12] BaBar collaboration, B. Aubert et al., Measurement of branching fractions and mass spectra of $B\rightarrow K\pi\pi\gamma$, Phys. Rev. Lett. 98 (2007) 211804, arXiv:hep-ex/0507031
* [13] C. S. Wu et al., Experimental test of parity conservation in beta decay, Phys. Rev. 105 (1957) 1413
* [14] M. Gronau and D. Pirjol, Photon polarization in radiative B decays, Phys. Rev. D66 (2002) 054008, arXiv:hep-ph/0205065
* [15] E. Kou, A. Le Yaouanc, and A. Tayduganov, Determining the photon polarization of the $b\\!\rightarrow\\!s\gamma$ using the $B\\!\rightarrow\\!K_{1}(1270)\gamma\\!\rightarrow\\!(K\pi\pi)\gamma$ decay, Phys. Rev. D83 (2011) 094007, arXiv:1011.6593
* [16] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [17] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [18] T. Sjöstrand, S. Mrenna, and P. Skands, A brief introduction to PYTHIA 8.1, Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
* [19] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [20] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [21] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [22] Geant4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
* [23] Geant4 collaboration, S. Agostinelli et al., Geant4: a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [24] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. Phys. Conf. Ser. 331 (2011) 032023
* [25] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
* [26] R. E. Schapire and Y. Freund, A decision-theoretic generalization of on-line learning and an application to boosting, Jour. Comp. and Syst. Sc. 55 (1997) 119
* [27] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Krakov, PL, Institute of Nuclear Physics, 1986, DESY F31-86-02, Appendix E
* [28] ARGUS collaboration, H. Albrecht et al., Search for hadronic $b\rightarrow u$ decays, Phys. Lett. B241 (1990) 278
* [29] M. Pivk and F. R. Le Diberder, sPlot: a statistical tool to unfold data distributions, Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
* [30] Belle collaboration, H. Yang et al., Observation of $B^{+}\rightarrow K_{1}(1270)^{+}\gamma$, Phys. Rev. Lett. 94 (2005) 111802, arXiv:hep-ex/0412039
## Supplementary material
The covariance matrices obtained from the fit described in the Letter for both
photon angle definitions are shown in Tables 3d and 4d.
Table 3: Covariance matrices (in units of $10^{-3}$) for the fitted values of
$c_{1}$, $c_{2}$, $c_{3}$ and $c_{4}$ of Table 1, for the four
${K}^{+}{\pi}^{-}{\pi}^{+}$ mass intervals.
$\begin{pmatrix}{\phantom{+}}0.31&\phantom{+0.01}&\phantom{+0.09}&\phantom{-0.01}\\\
0.01&0.47&&\\\ 0.09&0.03&0.68&\\\ -0.01&0.16&0.02&0.92\\\ \end{pmatrix}$
(a)
$\begin{pmatrix}0.41&\phantom{+0.02}&\phantom{+0.12}&\phantom{+0.00}\\\
0.02&0.66&&\\\ 0.12&0.04&0.93&\\\ 0.00&0.20&0.04&1.27\\\ \end{pmatrix}$
(b)
$\begin{pmatrix}0.35&\phantom{-0.01}&&\phantom{-0.03}\\\ -0.01&0.52&&\\\
0.14&0.00&0.76&\\\ -0.03&0.23&-0.01&0.99\\\ \end{pmatrix}$
(c)
$\begin{pmatrix}0.34&\phantom{-0.02}&\phantom{0.08}&\phantom{-0.02}\\\
-0.02&0.51&&\\\ 0.08&-0.04&0.75&\\\ -0.02&0.15&-0.04&1.01\\\ \end{pmatrix}$
(d)
Table 4: Covariance matrices (in units of $10^{-3}$) for the fitted values of
$c^{\prime}_{1}$, $c^{\prime}_{2}$, $c^{\prime}_{3}$ and $c^{\prime}_{4}$ of
Table 2, for the four ${K}^{+}{\pi}^{-}{\pi}^{+}$ mass intervals.
$\begin{pmatrix}0.30&\phantom{+0.00}&\phantom{+0.09}&\phantom{+0.02}\\\
0.00&0.47&&\\\ 0.09&0.02&0.68&\\\ 0.02&0.16&0.02&0.92\\\ \end{pmatrix}$
(a)
$\begin{pmatrix}0.41&\phantom{+0.03}&\phantom{+0.12}&\phantom{+0.01}\\\
0.03&0.66&&\\\ 0.12&0.07&0.93&\\\ 0.01&0.20&0.10&1.27\\\ \end{pmatrix}$
(b)
$\begin{pmatrix}0.35&\phantom{+0.01}&\phantom{+0.14}&\phantom{+0.00}\\\
0.01&0.53&&\\\ 0.14&0.05&0.76&\\\ 0.00&0.24&0.03&0.99\\\ \end{pmatrix}$
(c)
$\begin{pmatrix}0.34&\phantom{+0.00}&\phantom{+0.08}&\phantom{+0.02}\\\
0.00&0.51&&\\\ 0.08&0.00&0.75&\\\ 0.02&0.15&-0.01&1.01\\\ \end{pmatrix}$
(d)
|
arxiv-papers
| 2014-02-27T10:24:51 |
2024-09-04T02:49:58.997192
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. Anderlini,\n J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, M. Borsato, T.J.V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto,\n J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M. Calvi, M. Calvo Gomez,\n A. Camboni, P. Campana, D. Campora Perez, F. Caponio, A. Carbone, G. Carboni,\n R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G.\n Casse, L. Cassina, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M.\n Charles, Ph. Charpentier, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba,\n X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, I.\n Counts, B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R.\n Currie, C. D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n S. Donleavy, F. Dordei, M. Dorigo, P. Dorosz, A. Dosil Su\\'arez, D. Dossett,\n A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger,\n R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, S. Esen, A. Falabella, C.\n F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, J. Fu, E. Furfaro, A. Gallas Torreira, D. Galli, S.\n Gambetta, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, L. Gavardi, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, A. Gianelle, S. Giani', V. Gibson, L. Giubega, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, L. Grillo, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, T.W. Hafkenscheid, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, L. Henry, J.A.\n Hernando Morata, E. van Herwijnen, M. He{\\ss}, A. Hicheur, D. Hill, M.\n Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, N. Hussain, D. Hutchcroft, D.\n Hynds, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, N.\n Jurik, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, M.\n Kelsey, I.R. Kenyon, T. Ketel, B. Khanji, C. Khurewathanakul, S. Klaver, O.\n Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, B. Langhans, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, M. Liles,\n R. Lindner, C. Linn, F. Lionetto, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H.\n Lopes, N. Lopez-March, P. Lowdon, H. Lu, D. Lucchesi, H. Luo, E. Luppi, O.\n Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G.\n Manca, G. Mancinelli, M. Manzali, J. Maratas, U. Marconi, C. Marin Benito, P.\n Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, F. Martinez Vidal, D. Martins\n Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov, M.\n McCann, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier,\n M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, M. Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, R.\n Mountain, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik,\n T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neri, S. Neubert, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M.\n Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano,\n F. Palombo, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, L.\n Pappalardo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pearce, A. Pellegrino,\n M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, P. Perret, M. Perrin-Terrin,\n L. Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, A. Pistone, S. Playfer, M. Plo\n Casasus, F. Polci, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve,\n V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H. Rademacker,\n B. Rakotomiaramanana, M. Rama, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, A.B.\n Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A.\n Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz\n Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V.\n Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, G.\n Simi, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E. Smith,\n J. Smith, M. Smith, H. Snoek, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, F. Spinella, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S. Stone, B.\n Storaci, S. Stracka, M. Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah,\n L. Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer,\n V. Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, J.A. de\n Vries, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, G. Wilkinson, M.P. Williams, M. Williams, F.F.\n Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, G. Wormser, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Giovanni Veneziano",
"url": "https://arxiv.org/abs/1402.6852"
}
|
1402.6865
|
# Applications of Structural Balance
in Signed Social Networks
Jérôme Kunegis
University of Koblenz–Landau, Germany
[email protected]
###### Abstract
We present measures, models and link prediction algorithms based on the
structural balance in signed social networks. Certain social networks contain,
in addition to the usual _friend_ links, _enemy_ links. These networks are
called signed social networks. A classical and major concept for signed social
networks is that of structural balance, i.e., the tendency of triangles to be
_balanced_ towards including an even number of negative edges, such as friend-
friend-friend and friend-enemy-enemy triangles. In this article, we introduce
several new signed network analysis methods that exploit structural balance
for measuring partial balance, for finding communities of people based on
balance, for drawing signed social networks, and for solving the problem of
link prediction. Notably, the introduced methods are based on the signed graph
Laplacian and on the concept of signed resistance distances. We evaluate our
methods on a collection of four signed social network datasets.
## 1 Introduction
Signed social networks are such social networks in which social ties can have
two signs: friendship and enmity. Signed social networks have been studied in
sociology and anthropology111See for instance Figure 1, and are now found on
certain websites such as Slashdot222slashdot.org and
Epinions333www.epinions.com. In addition to the usual social network analyses,
the signed structure of these networks allows a new range of studies to be
performed, related to the behavior of edge sign distributions within the
graph. A major observation in this regard is the now classical result of
_balance theory_ by [17], stipulating that signed social networks tend to be
balanced in the sense that its nodes can be partitioned into two sets, such
that nodes within each set are connected only by friendship ties, and nodes
from different sets are only connected by enmity ties. This observation is not
to be understood in an absolute sense – in a large social network, a single
wrongly signed edge would render a network unbalanced. Instead, this is to be
understood as a tendency, which can be exploited to enhance the analytical and
predictive power of network analysis methods for a wide range of applications.
Figure 1: A small example of a signed social network from anthropology: The
tribal groups of the Eastern Central Highlands of New Guinea from the study of
Read [44]. Individual tribes are the vertices of this network, with friendly
relations shown as green edges and antagonistic relations shown as red edges.
In this article, we present ways to measure and exploit structural balance of
signed social networks for graph drawing, measuring conflict, detecting
communities and predicting links. In particular, we introduce methods based on
_algebraic graph theory_ , i.e., the representation of graphs by matrices. In
ordinary network analysis applications, algebraic graph theory has the
advantage that a large range of powerful algebraic methods become available to
analyse networks. In the case of signed networks, an additional advantage is
that structural balance, which is inherently a multiplicative construct as
illustrated by the rule _the enemy of my enemy is my friend_ , maps in a
natural way onto the algebraic representation of networks as matrices. As we
will see, this makes not only signed network analysis methods seamlessly take
into account structural balance theory, it also simplifies calculation with
matrices and vectors, as the multiplication rule is build right into the
definition of their operations.
In the rest of article, the individual methods are not presented in order of
possible applications, but in order of complexity, building on each other. The
breakdown is as follows:
* •
Section 2 introduces the concept of a signed social network, gives necessary
mathematical definitions and presents a set of four signed social networks
that are used throughout the paper.
* •
Section 3 defines structural balance and introduces a basic but novel measure
for quantifying it: the signed clustering coefficient.
* •
Section 4 reviews the problem of drawing signed graphs, and derives from it
the signed Laplacian matrix which arises naturally in that context.
* •
Section 5 gives a proper mathematical definition of the signed Laplacian
matrix, and proves its basic properties.
* •
Section 6 introduces the notion of _algebraic conflict_ , a second way of
quantifying structural balance, based on a spectral analysis of the signed
Laplacian matrix.
* •
Section 7 describes the signed graph clustering problem, and shows how its
solution leads to another derivation of the signed Laplacian matrix.
* •
Section 8 reviews the problem of link prediction in signed networks, and shows
how it can be solved by the _signed resistance distance_.
Section 9 concludes the article. This article is partially based on material
previously published by the author in conference papers [28, 32, 33, 34, 35].
## 2 Background: Signed Social Networks
Negative edges can be found in many types of social networks, to model enmity
in addition to friendship, distrust in addition to trust, or positive and
negative ratings between users. Early uses of signed social networks can be
found in anthropology, where negative edges have been used to denote
antagonistic relationships between tribes [16]. In this context, the
sociological notion of balance is defined as the absence of negative cycles,
i.e., the absence of cycles with an odd number of negative edges [9, 17].
Other cases of signed social networks include student relationships [21] and
voting processes [36].
Recent studies [18] describe the social network extracted from Essembly, an
ideological discussion site that allows users to mark other users as _friends_
, _allies_ and _nemeses_ , and discuss the semantics of the three relation
types. These works model the different types of edges by means of three
subgraphs. Other recent work considers the task of discovering communities
from social networks with negative edges [51].
In trust networks, nodes represent persons or other entities, and links
represent trust relationships. To model distrust, negative edges are then
used. Work in that field has mostly focused on defining global trust measures
using path lengths or adapting PageRank [15, 23, 25, 41, 48].
In applications where users can rate each other, we can model ratings as
_like_ and _dislike_ , giving rise to positive and negative edges, for
instance on online dating sites [6].
An example of a small signed social network is given by the tribal groups of
the Eastern Central Highlands of New Guinea from the study of Read [44] in
Figure 1. This dataset describes the relations between sixteen tribal groups
of the Eastern Central Highlands of New Guinea [16]. Relations between tribal
groups in the Gahuku–Gama alliance structure can be friendly (_rova_) or
antagonistic (_hina_). In addition, four large signed social networks
extracted from the Web will be used throughout the article. All datasets are
part of the Koblenz Network Collection [29]. The datasets are summarized in
Table 1.
Table 1: The signed social network datasets used in this article. The first
four datasets are large; the last one is small and serves as a running
example.
Network Type Vertices ($|V|$) Edges ($|E|$) Percent Negative Slashdot Zoo [32]
Directed 79,120 515,581 23.9% Epinions [40] Directed 131,828 841,372 14.7%
Wikipedia elections [36] Directed 8,297 107,071 21.6% Wikipedia conflicts [5]
Undirected 118,100 2,985,790 19.5% Highland tribes [44] Undirected 16 58 50.0%
#### Definitions
Mathematically, an undirected signed graph can be defined as $G=(V,E,\sigma)$,
where $V$ is the vertex set, $E$ is the edge set, and
$\sigma:E\rightarrow\\{-1,+1\\}$ is the sign function [53]. The sign function
$\sigma$ assigns a positive or negative sign to each edge. The fact that two
edges $u$ and $v$ are adjacent will be denoted by $u\sim v$. The degree of a
node $u$ is defined as the number of its neighbors, and can be written as
$\displaystyle d(u)=\\{v\mid u\sim v\\}.$
A directed signed network will be noted as $G=(V,E,\sigma)$, in which $E$ is
the set of directed edges (or _arcs_).
#### Algebraic Graph Theory
Algebraic graph theory is the branch of graph theory that represents graphs
using algebraic structures in order to exploit the powerful methods of algebra
in graph theory. The main tool of algebraic graph theory is the representation
of graphs as matrices, in particular the adjacency matrix and the Laplacian
matrix. In the following, all matrices are real.
Given a signed graph $G=(V,E,\sigma)$, its adjacency matrix
$\mathbf{A}\in\mathbb{R}^{|V|\times|V|}$ is defined as
$\displaystyle\mathbf{A}_{uv}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\sigma(\\{u,v\\})&\mathrm{when}\\{u,v\\}\in
E\\\ 0&\mathrm{when}\\{u,v\\}\notin E\end{array}\right.$
The adjacency matrix is square and symmetric.
The diagonal degree matrix $\mathbf{D}$ of a signed graph is defined using
$\mathbf{D}_{uu}=d(u)$. Note that the degrees, and thus the matrix
$\mathbf{D}$, is independent of the sign function $\sigma$.
The assumption of structural balance lends itself to using algebraic methods
based on the adjacency matrix of a signed network. To see why this is true,
consider that the square $\mathbf{A}^{2}$ contains at its entry $(u,v)$ a sum
of paths of length two between $u$ and $v$ weighted positively or negatively
depending on whether a third positive edge between $u$ and $v$ would lead to a
balanced or unbalanced triangle.
Finally, the Laplacian matrix of any graph is defined as
$\mathbf{L}=\mathbf{D}-\mathbf{A}$. It is this matrix $\mathbf{L}$ that will
play a central role for graph drawing, graph clustering and link prediction.
## 3 Measuring Structural Balance:
The Signed Clustering Coefficient
In a signed social network, the relationship between two connected nodes can
be positive or negative. When looking at groups of three persons, four
combinations of positive and negative edges are possible (up to permutations),
some being more likely than others. An observation made in actual social
groups is that triangles of positive and negative edges tend to be balanced.
For instance, a triangle of three positive edges is balanced, as is a triangle
of one positive and two negative edges. On the other hand, a triangle of two
positive and one negative edge is not balanced. The case of three negative
edges can be considered balanced, when considering the three persons as three
different groups, or unbalanced, when allowing only two groups.
This characterization of balance can be generalized to the complete signed
network, resulting in the following definition:
###### Definition 1 (Harary, 1953).
A connected signed graph is balanced when its vertices can be partitioned into
two groups such that all positive edges connect vertices within the same
group, and all negative edges connect vertices of the two different groups.
Figure 3 shows a balanced graph partitioned into two vertex sets. The concept
of structural balance can also be illustrated with the phrase _the enemy of my
enemy is my friend_ and its permutations.
Equivalently, unbalanced graphs can be defined as those graphs containing a
cycle with an odd number of negative edges, as shown in Figure 3. To prove
that the balanced graphs are exactly those that do not contain cycles with an
odd number of edges, consider that any cycle in a balanced graph has to cross
sides an even number of times. On the other hand, any balanced graph can be
partitioned into two vertex sets by depth-first traversal while assigning each
vertex to a partition such that the balance property is fulfilled. Any
inconsistency that arises during such a labeling leads to a cycle with an odd
number of negative edges.
Figure 2: The nodes of a graph without negative cycles can be partitioned
into two sets such that all edges inside of each group are positive and all
edges between the two groups are negative. We call such a graph balanced.
Figure 3: An unbalanced graph contains at least one cycle with an odd number
of negative edges. Such a graph cannot be partitioned into two sets with all
negative edges across the sets and positive edges within the sets.
In large signed social networks such as those given in Table 1, it cannot be
expected that the full network is balanced, since already a single unbalanced
triangle makes the full network unbalanced. Instead, we need a measure of
balance that characterizes to what extent a signed network is balanced. To
that end, we extend a well-establish measure in network analysis, the
clustering coefficient, to signed networks, giving the signed clustering
coefficient. We also introduce the relative signed clustering coefficient and
give the values observed in our example datasets.
The clustering coefficient is a characteristic number of a graph taking values
between zero and one, denoting the tendency of the graph nodes to form small
clusters. The clustering coefficient was introduced in [49] and an extension
for positively weighted edges proposed in [22]. The signed clustering
coefficient we define denotes the tendency of small clusters to be _balanced_
, and takes on values between $-1$ and $+1$. The relative signed clustering
coefficient will be defined as the quotient between the two.
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$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c=ab}$$\scriptstyle{a}$$\textstyle{{\bullet}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c=ab}$$\scriptstyle{a}$$\textstyle{{\bullet}}$$\textstyle{{c)}}$$\textstyle{{d)}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$
Figure 4: The four kinds of clustering coefficients. a) Regular clustering
coefficient. b) Directed clustering coefficient. c) Signed clustering
coefficient. d) Signed directed clustering coefficient. Edge $c$ is counted
when edges $a$ and $b$ are present, and for the signed variants, weighted by
$\mathrm{sgn}(abc)$.
The clustering coefficient is defined as the proportion of all incident edge
pairs that are completed by a third edge to form a triangle. Figure 4 gives an
illustration. Given an undirected, unsigned graph $G=(V,E)$ its clustering
coefficient is given by
$\displaystyle c(G)$ $\displaystyle=$ $\displaystyle\frac{\\{(u,v,w)\in
V^{3}\mid u\sim v\sim w\sim u\\}}{\\{(u,v,w)\in V^{3}\mid u\sim v\sim w\\}}$
(2)
To extend the clustering coefficient to negative edges, we assume structural
balance for two incident signed edges. As shown in Figure 4, an edge with sign
$c$ completing two incident edges with signs $a$ and $b$ to form a triangle
must fulfill the equation $c=ab$.
$\displaystyle c_{\mathrm{s}}(G)$ $\displaystyle=$
$\displaystyle\frac{\sum_{u\sim v\sim w\sim
u}\sigma(\\{u,v\\})\sigma(\\{v,w\\})\sigma(\\{w,u\\})}{\\{u,v,w\in V\mid u\sim
v\sim w\\}}$ (3)
Therefore, the signed clustering coefficient denotes to what extent the graph
exhibits a balanced structure. In actual signed social networks, we expect it
to be positive.
Additionally, we define the relative signed clustering coefficient as the
quotient of the signed and unsigned clustering coefficients.
$\displaystyle S(G)=\frac{c_{s}(G)}{c(G)}=\frac{\sum_{u\sim v\sim w\sim
u}\sigma(\\{u,v\\})\sigma(\\{v,w\\})\sigma(\\{w,u\\})}{\\{u,v,w\in V\mid u\sim
v\sim w\sim u\\}}$ (4)
The relative signed clustering coefficient takes on values between $-1$ and
$+1$. It is $+1$ when all triangles are balanced. In networks with negative
relative signed clustering coefficients, structural balance does not hold. In
fact, the relative signed clustering coefficient is closely related to the
number of balanced and unbalanced triangles in a network. If $\Delta^{+}(G)$
is the number of balanced triangles and $\Delta^{-}(G)$ is the number of
unbalanced triangles in a signed network $G$, then
$\displaystyle S(G)$ $\displaystyle=$
$\displaystyle\frac{\Delta^{+}(G)-\Delta^{-}(G)}{\Delta^{+}(G)+\Delta^{-}(G)}.$
The directed signed clustering coefficient and directed relative signed
clustering coefficient can be defined analogously using Expressions (3) and
(4). The signed clustering coefficient and relative signed clustering
coefficient are zero in random networks, when the sign of edges is distributed
equally. The signed clustering coefficients are by definition smaller than
their unsigned counterparts.
Table 2: The values for all variants of the clustering coefficient for the example datasets. The directed variants are not computed for the two undirected datasets. Network | Undirected | Directed
---|---|---
| $c(G)$ | $c_{\mathrm{s}}(G)$ | $S(G)$ | $c(G)$ | $c_{\mathrm{s}}(G)$ | $S(G)$
Slashdot Zoo | 0.0318 | 0.00607 | 19.1% | 0.0559 | 0.00918 | 16.4%
Epinions | 0.1107 | 0.01488 | 13.4% | 0.1154 | 0.01638 | 14.2%
Wikipedia elections | 0.1391 | 0.01489 | 10.9% | 0.1654 | 0.02427 | 14.7%
Wikipedia conflicts | 0.0580 | 0.03342 | 57.6% | – | – | –
Highland tribes | 0.5271 | 0.30289 | 57.5% | – | – | –
Table 2 gives all four variants of the clustering coefficient measured in the
example datasets, along with the relative signed clustering coefficients. The
high values for the relative clustering coefficients show that our
multiplication rule is valid in the examined datasets, and justifies the
structural balance approach.
## 4 Visualizing Structural Balance:
Signed Graph Drawing
To motivate the use of algebraic graph theory based on structural balance, we
consider the problem of drawing signed graphs and show how it naturally leads
to our definition of the Laplacian matrix for signed graphs. We begin by
showing how the signed Laplacian matrix arises naturally in the task of
drawing graphs with negative edges when one tries to place each node near to
its positive neighbors and opposite to its negative neighbors, extending a
standard method of graph drawing in the presence of only positive edges.
The Laplacian matrix turns up in graph drawing when we try to find an
embedding of a graph into a plane in a way that adjacent nodes are drawn near
to each other [1]. In the literature, signed graphs have been drawn using
eigenvectors of the signed adjacency matrix [4]. Instead, our approach
consists of using the Laplacian to draw signed graphs, in analogy with the
unsigned case. To do this, we will stipulate that negative edges should be
drawn as far from each other as possible.
### 4.1 Unsigned Graphs
We now describe the general method for generating an embedding of the nodes of
an unsigned graph into the plane using the Laplacian matrix. Let $G=(V,E)$ be
a connected unsigned graph with adjacency matrix $\mathbf{A}$. We want to find
a two-dimensional drawing of $G$ in which each vertex is drawn near to its
neighbors. This requirement gives rise to the following vertex equation, which
states that every vertex is placed at the mean of its neighbors’ coordinates,
weighted by the sign of the connecting edges. Let
$\mathbf{X}\in\mathbb{R}^{n\times 2}$ be a matrix whose columns are the
coordinates of all nodes in the drawing, then we have for each node $u$:
$\displaystyle\mathbf{X}_{u\bullet}=\left(\sum_{u\sim
v}\mathbf{A}_{uv}\right)^{-1}\sum_{u\sim
v}\mathbf{A}_{uv}\mathbf{X}_{v\bullet}$ (5)
Rearranging and aggregating the equation for all $u$ we arrive at
$\displaystyle\mathbf{D}\mathbf{X}=\mathbf{A}\mathbf{X}$ (6)
or
$\displaystyle\mathbf{L}\mathbf{X}$ $\displaystyle=$
$\displaystyle\mathbf{0}.$
In other words, the columns of $\mathbf{X}$ should belong to the null space of
$\mathbf{L}$, which leads to the degenerate solution of
$\mathbf{X}_{u\bullet}=\mathbf{1}$ for all $u$, i.e., each
$\mathbf{X}_{u\bullet}$ having all components equal, as the all-ones vector
$\mathbf{1}$ is an eigenvector of $\mathbf{L}$ with eigenvalue zero. To
exclude that solution, we require that the columns $\mathbf{X}$ be orthogonal
to $\mathbf{1}$. Additionally, to avoid the degenerate solution
$\mathbf{X}_{u\bullet}=\mathbf{X}_{v\bullet}$ for $u\neq v$, we require that
all columns of $\mathbf{X}$ be orthogonal. This leads to
$\mathbf{X}_{u\bullet}$ being the eigenvectors associated with the two
smallest eigenvalues of $\mathbf{L}$ different from zero. This solution
results in a well-known satisfactory embedding of unsigned graphs. Such an
embedding is related to the resistance distance (or commute-time distance)
between nodes of the graph [1].
Note that Equation (6) can also be transformed to
$\mathbf{X}=\mathbf{D}^{-1}\mathbf{A}\mathbf{X}$, leading to the eigenvectors
of the asymmetric matrix $\mathbf{D}^{-1}\mathbf{A}$. This alternative
derivation is not investigated here.
### 4.2 Signed Graphs
We now extend the graph drawing method described in the previous section to
graphs with positive and negative edges. To adapt Expression (5) to negative
edges, we interpret a negative edge as an indication that two vertices should
be placed on opposite sides of the drawing. Therefore, we take the opposite
coordinates $-\mathbf{X}_{v\bullet}$ of vertices $v$ adjacent to $u$ through a
negative edge, and then compute the mean, as pictured in Figure 5. We may call
this construction _antipodal proximity_.
(a) Unsigned graph (b) Signed graph
Figure 5: Drawing the vertex $u$ at the mean coordinates of its neighbors
$v_{1},v_{2},v_{3}$ by proximity and antipodal proximity. (a) In unsigned
graphs, a vertex $u$ is placed at the mean of its neighbors
$v_{1},v_{2},v_{3}$. (b) In signed graphs, a vertex $u$ is placed at the mean
of its positive neighbors $v_{1},v_{2}$ and antipodal points $-v_{3}$ of its
negative neighbors.
This leads to the vertex equation
$\displaystyle\mathbf{X}_{u\bullet}$ $\displaystyle=\left(\sum_{\\{u,v\\}\in
E}|\mathbf{A}_{ij}|\right)^{-1}\sum_{\\{u,v\\}\in
E}\mathbf{A}_{uv}\mathbf{X}_{v\bullet}$ (7)
resulting in a signed Laplacian matrix $\mathbf{L}=\mathbf{D}-\mathbf{A}$ in
which indeed the definition of the degree matrix
$\mathbf{D}_{uu}=\sum_{v}|\mathbf{A}_{uv}|$ leads to the same equation
$\mathbf{L}\mathbf{X}=\mathbf{0}$ as in the unsigned case.
As we will see in the next section, $\mathbf{L}$ is always positive-
semidefinite, and is positive-definite for graphs that are unbalanced, i.e.,
graphs that contain cycles with an odd number of negative edges. To obtain a
graph drawing from $\mathbf{L}$, we can thus distinguish three cases, assuming
that $G$ is connected:
* •
If all edges are positive, then $\mathbf{L}$ has one eigenvalue zero, and the
eigenvectors of the two smallest nonzero eigenvalues can be used for graph
drawing.
* •
If the graph is unbalanced, $\mathbf{L}$ is positive-definite and we can use
the eigenvectors of the two smallest eigenvalues as coordinates.
* •
If the graph is balanced, its spectrum is equivalent to that of the
corresponding unsigned Laplacian matrix, up to signs of the eigenvector
components. Using the eigenvectors of the two smallest eigenvalues (including
zero), we arrive at a graph drawing with all points being placed on two
parallel lines, reflecting the perfect 2-clustering present in the graph.
### 4.3 Synthetic Examples
Figure 6 shows four small synthetic signed graphs drawn using the eigenvectors
of three characteristic graph matrices. For each synthetic signed graph, let
$\mathbf{A}$ be its adjacency matrix, $\mathbf{L}$ its Laplacian matrix, and
$\mathbf{\bar{L}}$ the Laplacian matrix of the corresponding unsigned graph
$\bar{G}=|G|$, i.e., the same graph as $G$, only that all edges are positive.
For $\mathbf{A}$, we use the eigenvectors corresponding to the two largest
absolute eigenvalues. For $\mathbf{L}$ and $\mathbf{\bar{L}}$, we use the
eigenvectors of the two smallest nonzero eigenvalues. The small synthetic
examples are chosen to display the basic spectral properties of these three
matrices. All graphs contain cycles with an odd number of negative edges.
Column (a) shows all graphs drawn using the eigenvectors of the two largest
eigenvalues of the adjacency matrix $\mathbf{A}$. Column (b) shows the
unsigned Laplacian embedding of the graphs by setting all edge weights to
$+1$. Column (c) shows the signed Laplacian embedding. The embedding given by
the eigenvectors of $\mathbf{A}$ is clearly not satisfactory for graph
drawing. As expected, the graphs drawn using the ordinary Laplacian matrix
place nodes connected by a negative edge near to each other. The signed
Laplacian matrix produces a graph embedding where negative links span large
distances across the drawing, as required.
(1) | | |
---|---|---|---
(2) | | |
(3) | | |
(4) | | |
| (a) $\mathbf{A}$ | (b) $\mathbf{\bar{L}}$ | (c) $\mathbf{L}$
Figure 6: Four small synthetic signed graphs [(1)–(4)] drawn using the
eigenvectors of three graph matrices. (a) the adjacency matrix $\mathbf{A}$,
(b) the Laplacian $\mathbf{\bar{L}}$ of the underlying unsigned graph
$\bar{G}$, (c) the Laplacian $\mathbf{L}$. All graphs shown contain negative
cycles, and their signed Laplacian matrices are positive-definite. Positive
edges are shown as solid green lines and negative edges as red dashed lines.
#### Drawing a Balanced Graph
We assume, in the derivation above, that the eigenvectors corresponding to the
two smallest eigenvalues of the Laplacian $\mathbf{L}$ should be used for
graph drawing. This is true in that it gives the best possible drawing
according to the proximity and distance criteria of positive and negative
edges. If however the graph is balanced, then, as we will see, the smallest
eigenvalue of $\mathbf{L}$ is zero. Unlike the case in unsigned graphs
however, the corresponding eigenvector is not constant but contains values
$\\{\pm 1\\}$ describing the split into two partitions. If we use that
eigenvector to draw the graph, the resulting drawing will place all vertices
on two lines. Such an embedding may be satisfactory in cases where the perfect
balance of the graph is to be visualized. If however positive edges among each
partition’s vertices are also to be visible, the eigenvector corresponding to
the third smallest eigenvalue can be added with a small weight to the first
eigenvector, resulting in a two-dimensional representation of a 3-dimensional
embedding. The resulting three methods are illustrated in Figure 7.
(a) $(\lambda_{1},\lambda_{2})$ (b) $(\lambda_{2},\lambda_{3})$ (c)
$(\lambda_{1}+0.3\lambda_{3},\lambda_{2})$
Figure 7: Three methods for drawing a balanced signed graph, using a small
artificial example network. (a) Using the eigenvector corresponding to the
smallest eigenvalue $\lambda_{1}=0$, intra-cluster structure is lost. (b)
Ignoring the first eigenvalue misses important information about the
clustering. (c) Using a linear combination of both methods gives a good
compromise.
In practice, large graphs are almost always unbalanced as shown in Figure 9
and Table 3, and the two smallest eigenvalues give a satisfactory embedding.
Figure 8 shows large signed networks drawn using the two eigenvectors of the
smallest eigenvalues of the Laplacian matrix $\mathbf{L}$ for three signed
networks.
(a) Slashdot Zoo (b) Epinions (c) Wikipedia elections (d) Wikipedia conflicts
(e) Highland tribes
Figure 8: Signed spectral embedding of networks. For each network, every node
is represented as a point whose coordinates are the corresponding values in
the eigenvectors of the signed Laplacian $\mathbf{L}$ corresponding to the two
smallest eigenvalues. For the Highland tribes network, positive edges are
shown in green and negative edges in red. The edges in the other networks are
not shown.
## 5 Capturing Structural Balance:
The Signed Laplacian
The spectrum Laplacian matrix $\mathbf{L}=\mathbf{D}-\mathbf{A}$ of signed
networks is studied in [19], where it is established that the signed Laplacian
is positive-definite when each connected component of a graph contains a cycle
with an odd number of negative edges. Other basic properties of the Laplacian
matrix for signed graphs are given in [20].
For an unsigned graph, the Laplacian $\mathbf{L}$ is positive-semidefinite,
i.e., it has only nonnegative eigenvalues. In this section, we prove that the
Laplacian matrix $\mathbf{L}$ of a signed graph is positive-semidefinite too,
characterize the graphs for which it is positive-definite, and give the
relationship between the eigenvalue decomposition of the signed Laplacian
matrix and the eigenvalue decomposition of the corresponding unsigned
Laplacian matrix. Our characterization of the smallest eigenvalue of
$\mathbf{L}$ in terms of graph balance is based on [19].
### 5.1 Positive-semidefiniteness of the Laplacian
A Hermitian matrix is positive-semidefinite when all its eigenvalues are
nonnegative, and positive-definite when all its eigenvalues are positive. The
Laplacian matrix of an unsigned graph is symmetric and thus Hermitian. Its
smallest eigenvalue is zero, and thus the Laplacian of an unsigned graph is
always positive-semidefinite but never positive-definite. In the following, we
prove that that Laplacian of a signed graph is always positive-semidefinite,
and positive-definite when the graph is unbalanced.
###### Theorem 1.
The Laplacian matrix $\mathbf{L}$ of a signed graph $G=(V,E,\sigma)$ is
positive-semidefinite.
###### Proof.
We write the Laplacian matrix as a sum over the edges of $G$:
$\displaystyle\mathbf{L}=\sum_{\\{u,v\\}\in E}\mathbf{L}^{\\{u,v\\}}$
where $\mathbf{L}^{\\{u,v\\}}\in\mathbb{R}^{|V|\times|V|}$ contains the four
following nonzero entries:
$\displaystyle\mathbf{L}^{\\{u,v\\}}_{uu}=\mathbf{L}^{\\{u,v\\}}_{vv}$
$\displaystyle=$ $\displaystyle 1$ (8)
$\displaystyle\mathbf{L}^{\\{u,v\\}}_{uv}=\mathbf{L}^{\\{u,v\\}}_{vu}$
$\displaystyle=$ $\displaystyle-\sigma(\\{u,v\\})$
Let $\mathbf{x}\in\mathbb{R}^{|V|}$ be a vertex-vector. By considering the
bilinear form $\mathbf{x}^{\mathrm{T}}\mathbf{L}^{\\{u,v\\}}\mathbf{x}$, we
see that $\mathbf{L}^{\\{u,v\\}}$ is positive-semidefinite:
$\displaystyle\mathbf{x}^{\mathrm{T}}\mathbf{L}^{\\{u,v\\}}\mathbf{x}$
$\displaystyle=$
$\displaystyle\mathbf{x}_{u}^{2}+\mathbf{x}_{v}^{2}-2\sigma(\\{u,v\\})\mathbf{x}_{u}\mathbf{x}_{v}$
$\displaystyle=$
$\displaystyle(\mathbf{x}_{u}-\sigma(\\{u,v\\})\mathbf{x}_{v})^{2}$
$\displaystyle\geq$ $\displaystyle 0$
We now consider the bilinear form
$\mathbf{x}^{\mathrm{T}}\mathbf{L}\mathbf{x}$:
$\displaystyle\mathbf{x}^{\mathrm{T}}\mathbf{L}\mathbf{x}=\sum_{\\{u,v\\}\in
E}\mathbf{x}^{\mathrm{T}}\mathbf{L}^{\\{u,v\\}}\mathbf{x}\geq 0$
It follows that $\mathbf{L}$ is positive-semidefinite. ∎
Another way to prove that $\mathbf{L}$ is positive-semidefinite consists of
expressing it using the incidence matrix of $G$. Assume that for each edge
$\\{u,v\\}$ an arbitrary orientation is chosen. Then we define the incidence
matrix $\mathbf{H}\in\mathbb{R}^{|V|\times|E|}$ of $G$ as
$\displaystyle\mathbf{H}_{u\\{u,v\\}}$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle\mathbf{H}_{v\\{u,v\\}}$ $\displaystyle=$
$\displaystyle-\sigma(\\{u,v\\}).$
Here, the letter $\mathbf{H}$ is the uppercase greek letter Eta, as used for
instance in [14]. We now consider the product
$\mathbf{H}\mathbf{H}^{\mathrm{T}}\in\mathbb{R}^{|V|\times|V|}$:
$\displaystyle(\mathbf{H}\mathbf{H}^{\mathrm{T}})_{uu}$ $\displaystyle=$
$\displaystyle d(u)$ $\displaystyle(\mathbf{H}\mathbf{H}^{\mathrm{T}})_{uv}$
$\displaystyle=$ $\displaystyle-\sigma(\\{u,v\\})$
for diagonal and off-diagonal entries, respectively. Therefore
$\mathbf{H}\mathbf{H}^{\mathrm{T}}=\mathbf{L}$, and it follows that
$\mathbf{L}$ is positive-semidefinite. This result is independent of the
orientation chosen for $\mathbf{H}$.
### 5.2 Positive-definiteness of $\mathbf{L}$
We now show that, unlike the ordinary Laplacian matrix, the signed Laplacian
matrix is strictly positive-definite for some graphs, including most real-
world networks. The theorem presented here can be found in [20], and also
follows directly from an earlier result in [52].
As with the ordinary Laplacian matrix, the spectrum of the signed Laplacian
matrix of a disconnected graph is the union of the spectra of its connected
components. This can be seen by noting that the Laplacian matrix of an
unconnected graph has block-diagonal form, with each diagonal entry being the
Laplacian matrix of a single component. Therefore, we will restrict the
exposition to connected graphs.
Using Definition 1 of structural balance, we can characterize the graphs for
which the signed Laplacian matrix is positive-definite.
###### Theorem 2.
The signed Laplacian matrix of an unbalanced graph is positive-definite.
###### Proof.
We show that if the bilinear form
$\mathbf{x}^{\mathrm{T}}\mathbf{L}\mathbf{x}$ is zero for some vector
$\mathbf{x}\neq\mathbf{0}$, then a bipartition of the vertices as described
above exists.
Let $\mathbf{x}^{\mathrm{T}}\mathbf{L}\mathbf{x}=0$. We have seen that for
every $\mathbf{L}^{\\{u,v\\}}$ as defined in Equation (8) and any
$\mathbf{x}$, $\mathbf{x}^{\mathrm{T}}\mathbf{L}^{\\{u,v\\}}\mathbf{x}\geq 0$.
Therefore, we have for every edge $\\{u,v\\}$:
$\displaystyle\mathbf{x}^{\mathrm{T}}\mathbf{L}^{\\{u,v\\}}\mathbf{x}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\Leftrightarrow(\mathbf{x}_{u}-\sigma(\\{u,v\\})\mathbf{x}_{v})^{2}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\Leftrightarrow\mathbf{x}_{u}$ $\displaystyle=$
$\displaystyle\sigma(\\{u,v\\})\mathbf{x}_{v}$
In other words, two components of $\mathbf{x}$ are equal if the corresponding
vertices are connected by a positive edge, and opposite to each other if the
corresponding vertices are connected by a negative edge. Because the graph is
connected, it follows that all $|\mathbf{x}_{u}|$ must be equal. We can
exclude the solution $\mathbf{x}_{u}=0$ for all $u$ because $\mathbf{x}$ is
not the zero vector. Without loss of generality, we assume that
$|\mathbf{x}_{u}|=1$ for all $u$.
Therefore, $\mathbf{x}$ gives a bipartition into vertices with
$\mathbf{x}_{u}=+1$ and vertices with $\mathbf{x}_{u}=-1$, with the property
that two vertices with the same value of $\mathbf{x}_{u}$ are in the same
partition and two vertices with opposite sign of $\mathbf{x}_{u}$ are in
different partitions, and therefore $G$ is balanced. Equivalently, the signed
Laplacian matrix $\mathbf{L}$ of a connected unbalanced signed graph is
positive-definite. ∎
### 5.3 Balanced Graphs
We now show how the spectrum and eigenvectors of the signed Laplacian of a
balanced graph arise from the spectrum and the eigenvalues of the
corresponding unsigned graph by multiplication of eigenvector components with
$\pm 1$.
Let $G=(V,E,\sigma)$ be a balanced signed graph and $\bar{G}=(V,E)$ the
corresponding unsigned graph. Since $G$ is balanced, there is a vector
$\mathbf{x}\in\\{-1,+1\\}^{|V|}$ such that the sign of each edge $\\{u,v\\}$
is $\sigma(\\{u,v\\})=\mathbf{x}_{u}\mathbf{x}_{v}$.
###### Theorem 3.
If $\mathbf{L}$ is the signed Laplacian matrix of the balanced graph $G$ with
bipartition $\mathbf{x}$ and eigenvalue decomposition
$\mathbf{L}=\mathbf{U}\Lambda\mathbf{U}^{\mathrm{T}}$, then the eigenvalue
decomposition of the Laplacian matrix $\mathbf{\bar{L}}$ of $\bar{G}$ of the
corresponding unsigned graph $\bar{G}$ of $G$ is given by
$\mathbf{\bar{L}}=\mathbf{\bar{U}}\Lambda\mathbf{\bar{U}}^{\mathrm{T}}$ where
$\displaystyle\mathbf{\bar{U}}_{uk}$ $\displaystyle=$
$\displaystyle\mathbf{x}_{u}\mathbf{U}_{uk}.$
###### Proof.
To see that
$\mathbf{\bar{L}}=\mathbf{\bar{U}}\Lambda\mathbf{\bar{U}}^{\mathrm{T}}$, note
that for diagonal elements, we have
$\mathbf{\bar{U}}_{u\bullet}^{\phantom{\mathrm{I}}}\Lambda\mathbf{\bar{U}}_{u\bullet}^{\mathrm{T}}=\mathbf{x}_{u}^{2}\mathbf{U}_{u\bullet}^{\phantom{\mathrm{I}}}\Lambda\mathbf{U}_{u\bullet}^{\mathrm{T}}=\mathbf{U}_{u\bullet}^{\phantom{\mathrm{I}}}\Lambda\mathbf{U}_{u\bullet}^{\mathrm{T}}=\mathbf{L}_{uu}=\mathbf{\bar{L}}_{uu}$.
For off-diagonal elements, we have
$\mathbf{\bar{U}}_{u\bullet}^{\phantom{\mathrm{I}}}\Lambda\mathbf{\bar{U}}_{v\bullet}^{\mathrm{T}}=\mathbf{x}_{u}\mathbf{x}_{v}\mathbf{U}_{u\bullet}^{\phantom{\mathrm{I}}}\Lambda\mathbf{U}_{v\bullet}^{\mathrm{T}}=\sigma(\\{u,v\\})\mathbf{L}_{uv}=-\sigma(\\{u,v\\})\sigma(\\{u,v\\})=-1=\mathbf{\bar{L}}_{uv}$.
We now show that $\mathbf{\bar{U}}\Lambda\mathbf{\bar{U}}^{\mathrm{T}}$ is an
eigenvalue decomposition of $\mathbf{\bar{L}}$ by showing that
$\mathbf{\bar{U}}$ is orthogonal. To see that the columns of
$\mathbf{\bar{U}}$ are indeed orthogonal, note that for any two column indexes
$k\neq l$, we have $\mathbf{\bar{U}}_{\bullet
k}^{\mathrm{T}}\mathbf{\bar{U}}_{\bullet l}^{\phantom{\mathrm{I}}}=\sum_{u\in
V}\mathbf{\bar{U}}_{uk}\mathbf{\bar{U}}_{ul}=\sum_{u\in
V}\mathbf{x}_{u}^{2}\mathbf{U}_{uk}\mathbf{U}_{ul}=\mathbf{U}_{\bullet
k}^{\mathrm{T}}\mathbf{U}_{\bullet l}^{\phantom{\mathrm{I}}}=0$ because
$\mathbf{U}$ is orthogonal. Changing signs in $\mathbf{U}$ does not change the
norm of each column vector, and thus
$\mathbf{\bar{L}}=\mathbf{\bar{U}}\Lambda\mathbf{\bar{U}}^{\mathrm{T}}$ is the
eigenvalue decomposition of $\mathbf{\bar{L}}$. ∎
As shown in Section 5.2, the Laplacian matrix of an unbalanced graph is
positive-definite and therefore its spectrum is different from that of the
corresponding unsigned graph. Aggregating Theorems 2 and 3, we arrive at the
main result of this section.
###### Theorem 4.
The Laplacian matrix of a connected signed graph is positive-definite if and
only if the graph is unbalanced.
###### Proof.
From Theorem 2 we know that every unbalanced connected graph has a positive-
definite Laplacian matrix. Theorem 3 implies that every balanced graph has the
same Laplacian spectrum as its corresponding unsigned graph. Since the
unsigned Laplacian is always singular, the signed Laplacian of a balanced
graph is also singular. Together, these imply that the Laplacian matrix of a
connected signed graph is positive-definite if and only if the graph is
unbalanced. ∎
For a general signed graph that need not be connected, we can therefore make
the following statement: The multiplicity of the eigenvalue zero equals the
number of balanced connected components in $G$ [14].
(a) Slashdot Zoo (b) Epinions (c) Wikipedia elections (d) Wikipedia conflicts
Figure 9: The Laplacian spectra of three signed networks. These plots show
the eigenvalues $\lambda_{1}\leq\lambda_{2}\leq\cdots$ of the Laplacian matrix
$\mathbf{L}$.
The spectra of several large unipartite signed networks are plotted in Figure
9. We can observe that in all cases, the smallest eigenvalue is larger than
zero, implying, as expected, that these graphs are unbalanced.
## 6 Measuring Structural Balance 2:
Algebraic Conflict
The smallest eigenvalue of the Laplacian $\mathbf{L}$ of a signed graph is
zero when the graph is balanced, and larger otherwise. We derive from this
that the smallest Laplacian eigenvalue characterizes the amount of conflict
present in the graph. We will call this number the _algebraic conflict_ of the
graph and denote it $\xi$.
Let $G=(V,E,\sigma)$ be a connected signed graph with adjacency matrix
$\mathbf{A}$, degree matrix $\mathbf{D}$ and Laplacian
$\mathbf{L}=\mathbf{D}-\mathbf{A}$. Let
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{|V|}$ be the eigenvalues of
$\mathbf{L}$. Because $\mathbf{L}$ is positive-semidefinite (Theorem 1), we
have $\lambda_{1}\geq 0$. According to Theorem 2, $\lambda_{1}$ is zero
exactly when $G$ is balanced. Therefore, the value $\lambda_{1}$ can be used
as an invariant of signed graphs that characterizes the conflict due to
unbalanced cycles, i.e., cycles with an odd number of negative edges. We will
call $\xi=\lambda_{1}$ the _algebraic conflict_ of the network. The number
$\xi$ is discussed in [19] and [35], without being given a specific name.
The algebraic conflict $\xi$ for our signed network datasets is compared in
Table 3. All these large networks are unbalanced, and we can for instance
observe that the social networks of the Slashdot Zoo and Epinions are more
balanced than the election network of Wikipedia.
Table 3: The algebraic conflict $\xi$ for several signed unipartite networks. Smaller values indicate a more balanced network; larger values indicate more conflict. Network | $\xi$
---|---
Slashdot Zoo | 0.008077
Epinions | 0.002657
Wikipedia elections | 0.005437
Wikipedia conflicts | 0.0001050
Highland tribes | 0.7775
Figure 10 plots the algebraic conflict of the signed networks against the
relative signed clustering coefficient The number of signed datasets is small,
and thus we cannot make out a correlation between the two measures, although
the data is consistent with a negative between the two measures, as expected.
Figure 10: Scatter plot of the two measures of balance and conflict for the
four signed social networks: The relative signed clustering coefficient $S$
and the algebraic conflict $\xi$. (SZ: Slashdot Zoo, EP: Epinions, EL:
Wikipedia elections, CO: Wikipedia conflict)
#### Monotonicity
From the definition of the algebraic conflict $\xi$, we can derive a simple
theorem stating that adding an edge of any weight to a signed graph can only
increase the algebraic conflict, not decrease it.
###### Theorem 5.
Let $G=(V,E,\sigma)$ be a signed graph and $u,v\in V$ two vertices such that
$\\{u,v\\}\notin E$, and $\xi$ the algebraic of $G$. Furthermore, let
$G^{\prime}=(V,E\cup\\{u,v\\},\sigma^{\prime})$ with
$\sigma^{\prime}(e)=\sigma(e)$ when $e\in E$ and $\sigma(\\{u,v\\})=\sigma$
otherwise be the graph $G$ to which an edge with sign $\sigma$ has been added.
Then, let $\xi^{\prime}$ be the algebraic conflict of $G^{\prime}$. Then,
$\xi\leq\xi^{\prime}$.
###### Proof.
We make use of a theorem stated for instance in [50, p. 97]. This theorem
states that when adding a positive-semidefinite matrix $\mathbf{E}$ of rank
one to a given symmetric matrix $\mathbf{X}$ with eigenvalues
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}$, the new matrix
$\mathbf{X}^{\prime}=\mathbf{X}+\mathbf{E}$ has eigenvalues
$\lambda^{\prime}_{1}\leq\lambda^{\prime}_{2}\leq\cdots\leq\lambda^{\prime}_{n}$
which interlace the eigenvalues of $\mathbf{X}$:
$\displaystyle\lambda_{1}^{\phantom{{}^{\prime}}}\leq\lambda^{\prime}_{1}\leq\lambda_{2}^{\phantom{{}^{\prime}}}\leq\lambda^{\prime}_{2}\leq\cdots\leq\lambda_{n}^{\phantom{{}^{\prime}}}\leq\lambda^{\prime}_{n}$
The Laplacian $\mathbf{L}^{\prime}$ of $G^{\prime}$ can be written as
$\mathbf{L}^{\prime}=\mathbf{L}+\mathbf{E}$, where
$\mathbf{E}\in\mathbb{R}^{|V|\times|V|}$ is the matrix defined by
$\mathbf{E}_{uu}=\mathbf{E}_{vv}=1$ and
$\mathbf{E}_{uv}=\mathbf{E}_{vu}=-\sigma$, and $\mathbf{E}_{uv}=0$ for all
other entries. Then let $\mathbf{e}\in\mathbb{R}^{|V|}$ be the vector defined
by $\mathbf{e}_{u}=1$, $\mathbf{e}_{v}=-\sigma$ and $\mathbf{e}_{w}=0$ for all
other entries. We have $\mathbf{E}=\mathbf{e}\mathbf{e}^{\mathrm{T}}$, and
therefore $\mathbf{E}$ is positive-semidefinite.
Now, due to the interlacing theorem mentioned above, adding a positive-
semidefinite matrix to a given symmetric matrix can only increase each
eigenvalue, but not decrease it. Therefore,
$\lambda_{1}\leq\lambda^{\prime}_{1}$, and thus $\xi\leq\xi^{\prime}$. ∎
We have thus proved that adding an edge of any sign to a signed network can
only increase the algebraic conflict, not decrease it. It also follows that
removing an edge of any sign from a signed network can decrease the algebraic
conflict or leave it unchanged, but not increase it.
## 7 Maximizing Structural Balance:
Signed Spectral Clustering
One of the main application areas of the graph Laplacian are clustering
problems. In spectral clustering, the eigenvectors of matrices associated with
a graph are used to partition the vertices of the graph into well-connected
groups. In this section, we show that in a signed graph, the spectral
clustering problem corresponds to finding clusters of vertices connected by
positive edges, but not connected by negative edges.
Spectral clustering algorithms are usually derived by formulating a minimum
cut problem which is then relaxed [8, 39, 42, 43, 46]. The choice of the cut
function results in different spectral clustering algorithms. In all cases,
the vertices of a given graph are mapped into the space spanned by the
eigenvectors of a matrix associated with the graph.
In this section we derive a signed extension of the ratio cut, which leads to
clustering with the signed Laplacian $\mathbf{L}$. We restrict our proofs to
the case of clustering vertices into two groups; higher-order clusterings can
be derived analogously.
### 7.1 Unsigned Graphs
We first review the derivation of the ratio cut in unsigned graphs. Let
$G=(V,E)$ be an unsigned graph with adjacency matrix $\mathbf{A}$. A cut of
$G$ is a partition of the vertices $V$ into the nonempty sets $V_{1}$ and
$V_{2}$, whose weight is given by
$\displaystyle\mathrm{Cut}(V_{1},V_{2})=|\\{\\{u,v\\}\in E\mid u\in V_{1},v\in
V_{2}\\}|.$
The cut measures how well two clusters are connected. Since we want to find
two distinct groups of vertices, the cut must be minimized. Minimizing
$\mathrm{Cut}(V_{1},V_{2})$ however leads in most cases to solutions
separating very few vertices from the rest of the graph. Therefore, the cut is
usually divided by the size of the clusters, giving the ratio cut:
$\displaystyle\mathrm{RatioCut}(V_{1},V_{2})=\left(\frac{1}{|V_{1}|}+\frac{1}{|V_{2}|}\right)\mathrm{Cut}(V_{1},V_{2})$
To get a clustering, we then solve the following optimization problem:
$\displaystyle\min_{V_{1}\subset V}\quad\mathrm{RatioCut}(V_{1},V\setminus
V_{1})$
Let $V_{2}=V\setminus V_{1}$. Then this problem can be solved by expressing it
in terms of the characteristic vector $\mathbf{x}\in\mathbb{R}^{|V|}$ of
$V_{1}$ defined by:
$\displaystyle\mathbf{x}_{u}=\left\\{\begin{array}[]{ll}+\sqrt{|V_{2}|/|V_{1}|}&\textnormal{
if }u\in V_{1}\\\ -\sqrt{|V_{1}|/|V_{2}|}&\textnormal{ if }u\in
V_{2}\end{array}\right.$ (12)
We observe that
$\mathbf{x}\mathbf{L}\mathbf{x}^{\mathrm{T}}=2|V|\cdot\mathrm{RatioCut}(V_{1},V_{2})$,
and that $\sum_{u}\mathbf{x}_{u}=0$, i.e., $\mathbf{x}$ is orthogonal to the
constant vector. Denoting by $\mathcal{X}$ the vectors $\mathbf{x}$ of the
form given in Equation (12) we have
$\displaystyle\min_{\mathbf{x}\in\mathbb{R}^{|V|}}$
$\displaystyle\mathbf{x}\mathbf{L}\mathbf{x}^{\mathrm{T}}$ (13)
$\displaystyle\mathrm{s.t.}$
$\displaystyle\mathbf{x}\cdot\mathbf{1}=0,\mathbf{x}\in\mathcal{X}$
This can be relaxed by removing the constraint $\mathbf{x}\in\mathcal{X}$,
giving as solution the eigenvector of $\mathbf{L}$ having the smallest nonzero
eigenvalue [39].
### 7.2 Signed Graphs
We now give a derivation of the ratio cut for signed graphs. Let
$G=(V,E,\sigma)$ be a signed graph with adjacency matrix $\mathbf{A}$. We
write $\mathbf{A}^{\oplus}$ and $\mathbf{A}^{\ominus}$ for the adjacency
matrices containing only the positive and negative edges. In other words,
$\mathbf{A}^{\oplus}_{uv}=\max(0,\mathbf{A}_{uv})$,
$\mathbf{A}^{\ominus}_{uv}=\max(0,-\mathbf{A}_{uv})$ and
$\mathbf{A}=\mathbf{A}^{\oplus}-\mathbf{A}^{\ominus}$.
For convenience we define positive and negative cuts that only count positive
and negative edges respectively:
$\displaystyle\mathrm{Cut}^{\oplus}(V_{1},V_{2})$ $\displaystyle=$
$\displaystyle\sum_{u\in V_{1},v\in V_{2}}\mathbf{A}^{\oplus}_{uv}$
$\displaystyle\mathrm{Cut}^{\ominus}(V_{1},V_{2})$ $\displaystyle=$
$\displaystyle\sum_{u\in V_{1},v\in V_{2}}\mathbf{A}^{\ominus}_{uv}$
In these definitions, we allow $V_{1}$ and $V_{2}$ to be overlapping. For a
vector $\mathbf{x}\in\mathbb{R}^{|V|}$, we consider the bilinear form
$\mathbf{x}^{\mathrm{T}}\mathbf{L}\mathbf{x}$. As shown in Equation (5.1),
this can be written in the following way:
$\displaystyle\mathbf{x}^{\mathrm{T}}\mathbf{L}\mathbf{x}=\sum_{\\{u,v\\}\in
E}(\mathbf{x}_{u}-\sigma(\\{u,v\\})\mathbf{x}_{v})^{2}$
For a given partition $V=V_{1}\cup V_{2}$, let $\mathbf{x}\in\mathbb{R}^{|V|}$
be the following vector:
$\displaystyle\mathbf{x}_{u}=\left\\{\begin{array}[]{ll}+\frac{1}{2}\left(\sqrt{\frac{|V_{1}|}{|V_{2}|}}+\sqrt{\frac{|V_{2}|}{|V_{1}|}}\right)&\textnormal{
if }u\in V_{1}\\\
-\frac{1}{2}\left(\sqrt{\frac{|V_{1}|}{|V_{2}|}}+\sqrt{\frac{|V_{2}|}{|V_{1}|}}\right)&\textnormal{
if }u\in V_{2}\end{array}\right.$ (16)
The corresponding bilinear form then becomes:
$\displaystyle\mathbf{x}^{\mathrm{T}}\mathbf{L}\mathbf{x}$ $\displaystyle=$
$\displaystyle\sum_{\\{u,v\\}\in
E}\left(\mathbf{x}_{u}-\sigma(\\{u,v\\})\mathbf{x}_{v}\right)^{2}$
$\displaystyle=$
$\displaystyle|V|\left(\frac{1}{|V_{1}|}+\frac{1}{|V_{2}|}\right)\left(2\cdot\mathrm{Cut}^{\oplus}(V_{1},V_{2})+\mathrm{Cut}^{\ominus}(V_{1},V_{1})+\mathrm{Cut}^{\ominus}(V_{2},V_{2})\right)$
This leads us to define the following signed cut of $(V_{1},V_{2})$:
$\displaystyle\mathrm{SignedCut}(V_{1},V_{2})$ $\displaystyle=$
$\displaystyle\mathrm{Cut}^{\oplus}(V_{1},V_{2})+\frac{1}{2}\left(\mathrm{Cut}^{\ominus}(V_{1},V_{1})+\mathrm{Cut}^{\ominus}(V_{2},V_{2})\right)$
and to define the signed ratio cut as follows:
$\displaystyle\mathrm{SignedRatioCut}(V_{1},V_{2})=\left(\frac{1}{|V_{1}|}+\frac{1}{|V_{2}|}\right)\mathrm{SignedCut}(V_{1},V_{2})$
Therefore, the following minimization problem solves the signed clustering
problem:
$\displaystyle\min_{V_{1}\subset
V}\quad\mathrm{SignedRatioCut}(V_{1},V\setminus V_{1})$
We can now express this minimization problem using the signed Laplacian, where
$\mathcal{X}$ denotes the set of vectors of the form given in Equation (16):
$\displaystyle\min_{\mathbf{x}\in\mathbb{R}^{|V|}}$
$\displaystyle\quad\mathbf{x}\mathbf{L}\mathbf{x}^{\mathrm{T}}$
$\displaystyle\mathrm{s.t.}$ $\displaystyle\quad\mathbf{x}\in\mathcal{X}$
Note that we lose the orthogonality of $\mathbf{x}$ to the constant vector.
This can be explained by the fact that if $G$ contains negative edges, the
smallest eigenvector can always be used for clustering: If $G$ is balanced,
the smallest eigenvalue is zero and its eigenvector equals $(\pm 1)$ and gives
the two clusters separated by negative edges. If $G$ is unbalanced, then the
smallest eigenvalue of $\mathbf{L}$ is larger than zero by Theorem 2, and the
constant vector is not an eigenvalue.
The signed cut $\mathrm{SignedCut}(V_{1},V_{2})$ counts the number of positive
edges that connect the two groups $V_{1}$ and $V_{2}$, and the number of
negative edges that remain in each of these groups. Thus, minimizing the
signed cut leads to clusterings where two groups are connected by few positive
edges and contain few negative edges inside each group. This signed ratio cut
generalizes the ratio cut of unsigned graphs and justifies the use of the
signed Laplacian $\mathbf{L}$ and its particular definition for spectral
clustering of signed graphs.
### 7.3 Signed Clustering using Other Matrices
When instead of normalizing with the number of vertices $|V_{1}|$ we normalize
with the number of edges $\mathrm{vol}(V_{1})$, the result is a spectral
clustering algorithm based on the eigenvectors of $\mathbf{D}^{-1}\mathbf{A}$
introduced by Shi and Malik [46]. The cuts normalized by $\mathrm{vol}(V_{1})$
are called normalized cuts. In the signed case, the eigenvectors of
$\mathbf{D}^{-1}\mathbf{A}$ lead to the signed normalized cut:
$\displaystyle\mathrm{SignedNormalizedCut}(V_{1},V_{2})$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\mathrm{vol}(V_{1})}+\frac{1}{\mathrm{vol}(V_{2})}\right)\mathrm{SignedCut}(V_{1},V_{2})$
A similar derivation can be made for normalized cuts based on
$\mathbf{N}=\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}$, generalizing the
spectral clustering method of Ng, Jordan and Weiss [43]. The dominant
eigenvector of the signed adjacency matrix $\mathbf{A}$ can also be used for
signed clustering [2]. As in the unsigned case, this method is not suited for
very sparse graphs, and does not have an interpretation in terms of cuts.
#### Example
As an application of signed spectral clustering to real-world data, we cluster
the tribes in the Highland tribes network. The resulting graph contains cycles
with an odd number of negative edges, and therefore its signed Laplacian
matrix is positive-definite. We use the eigenvectors of the two smallest
eigenvalues ($\lambda_{1}=1.04$ and $\lambda_{2}=2.10$) to embed the graph
into the plane. The result is shown in Figure 11. We observe that indeed the
positive (green) edges are short, and the negative (red) edges are long.
Looking at only the positive edges, the drawing makes the two connected
components easy to see. Looking at only the negative edges, we recognize that
the tribal groups can be clustered into three groups, with no negative edges
inside any group. These three groups correspond indeed to a higher-order
grouping in the Gahuku–Gama society [16].
Figure 11: The tribal groups of the Eastern Central Highlands of New Guinea
from the study of Read [44] clustered using eigenvectors of the Laplacian
matrix. The three higher-order groups as described by Hage and Harary [16] are
linearly separable.
## 8 Predicting Structural Balance:
Signed Resistance Distance
In the field of network analysis, one of the major applications consists in
predicting the state of an evolving network in the future. When considering
only the network structure, the corresponding learning problem is the link
prediction problem. In this section, we will show that a certain class of link
prediction algorithms based on algebraic graph theory are particularly suited
to signed social networks, since they fulfill three natural requirements that
a link prediction method should follow. We will state the three conditions,
and then present two algebraic link prediction methods: the exponential of the
adjacency matrix and the signed resistance distance. We then finally evaluate
the methods on the task of link prediction.
First however, let us give the correct terminology and define the link
prediction problem for unsigned and signed social networks. Although we state
both problems in terms of social networks, both problems can be extended to
other networks.
Actual social networks are not static graphs, but dynamic systems in which
nodes and edges are added and removed continuously. The main type of change
being the addition of edges, i.e., the appearance of a new tie. Predicting
such ties is a common task. For instance, social networking sites try to
predict who users are likely to already know in order to give good friend
recommendations. Let $G=(V,E)$ be an unsigned social network. The link
prediction then consists of predicting new edges in that network, a link
prediction algorithm is thus a function mapping a given network to edge
predictions. In this work, we will express link prediction functions
algebraically as a map from the $|V|\times|V|$ adjacency matrix of a network
to another $|V|\times|V|$ matrix containing link prediction scores. The
semantics of these scores is that higher values denote a higher likelihood of
link formation. Apart from that, we do not put any other constraint on link
prediction scores. In particular, link prediction scores do not have to be
nonnegative, or restricted to the range $[0,1]$.
In the case of signed social networks, the link prediction problem is usually
restricted to predicting positive edges. This is easily motivated by the
example of a social recommender system, which should recommend friends and not
enemies. Thus, the link prediction problem for signed networks can be
formalized in the same fashion as for unsigned networks, by a function from
the space of adjacency matrices (containing positive and negative entries) to
the space of score matrices. A link prediction function $f$ for signed
networks will thus be denoted as follows:
$\displaystyle
f:\\{-1,0,+1\\}^{|V|\times|V|}\rightarrow\mathbb{R}^{|V|\times|V|}$
A note is in order about the related problem of _link sign prediction_. In the
problem of link sign prediction, a signed (social) network is given, along
with a set of unweighted edges, and the goal is the predict the sign of the
edges [32, 37]. This problem is different from the link prediction problem in
that for each given edge, it is known that the edge is part of the network,
and only its sign must be predicted. By contrast, the link prediction problem
assumes no knowledge about the network and consists in finding the positive
edges.
#### Requirements of a Link Prediction Function
The structure of the link prediction problem implies two requirements for a
link prediction function, in relation with paths connecting any two nodes. In
addition, the presence of negative edges implies a third requirement, in
relation to the edge signs in paths connecting two nodes.
Let $V$ be a fixed set of vertices, and $G_{1}=(V,E_{1})$ and
$G_{2}=(V,E_{2})$ two unsigned networks with the same vertex sets. Let $u,v\in
V$ be two vertices and $f$ a link prediction function. Then, compare the set
set of paths connecting the vertices $u$ and $v$, both in $G_{1}$ and in
$G_{2}$. Two requirements should be fulfilled by $f$:
* •
Path counts: If more paths between $u$ and $v$ are present in $G_{1}$ than
$G_{2}$, than $f$ should return a higher score for the pair $(u,v)$ in $G_{1}$
than in $G_{2}$.
* •
Path lengths: If paths between $u$ and $v$ are longer in $G_{1}$ than in
$G_{2}$, then $f$ should return a lower score for the pair $(u,v)$ in $G_{1}$
than in $G_{2}$.
In addition, the following requirement can be formulated for signed networks.
In this requirement, we will refer to a path as positive when it contains an
even number of negative edges and as negative when it contains an odd number
of negative edges.
* •
Path signs: If paths between $u$ and $v$ are more often positive in $G_{1}$
than in $G_{2}$, than $f$ should return a higher score for the pair $(u,v)$ in
$G_{1}$ than in $G_{2}$.
These three requirements are fulfilled by several link prediction functions,
of which we review one and introduce another in the following.
### 8.1 Signed Matrix Exponential
Let $G=(V,E,\sigma)$ be a signed network with adjacency matrix $\mathbf{A}$.
Its exponential is then defined as
$\displaystyle e^{\alpha\mathbf{A}}$ $\displaystyle=$
$\displaystyle\mathbf{I}+\mathbf{A}+\frac{1}{2}\mathbf{A}^{2}+\frac{1}{6}\mathbf{A}^{3}+\cdots$
This exponential with the parameter $\alpha>0$ is a suitable link prediction
function for signed networks as it can be expressed as a sum over all paths
between any two nodes. Let $P_{G}(u,v,k)$ be the set of paths of length $k$ in
the graph $G$. In this definition, we allow a path to cross a single vertex
multiple times, and set the length of a path as being the number of edges it
contains. Furthermore, let
$\displaystyle(v_{0},v_{1},\ldots,v_{k})\in P_{G}(u,v,k)$
with $u=v_{0}$ and $v=v_{k}$. Then, any power of $\mathbf{A}$ can be expressed
as
$\displaystyle(\mathbf{A}^{k})_{uv}$ $\displaystyle=$
$\displaystyle\sum_{(v_{0},\ldots,v_{k})\in
P_{G}(u,v,k)}\prod_{i=1}^{k}\sigma(\\{v_{i-1},v_{i}\\}).$
In other words, the $k$th power of the adjacency matrix equals a sum over all
paths of length $k$, weighted by the product of their edge signs. This leads
to the following expression for the matrix exponential:
$\displaystyle(e^{\alpha\mathbf{A}})_{uv}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{\alpha^{k}}{k!}\sum_{(v_{0},\ldots,v_{k})\in
P_{G}(u,v,k)}\prod_{i=1}^{k}\sigma(\\{v_{i-1},v_{i}\\}).$
In other words, the matrix exponential is a sum over all paths between any two
nodes, weighted by the function $\alpha^{k}/k!$ of their path length. This
implies that the matrix exponential is a suitable link sign prediction
function for signed networks, since it fulfills all three requirements:
* •
Path counts: The exponential function is a sum over paths and thus counts
paths.
* •
Path lengths: The function $\alpha^{k}/k!$ is decreasing in $k$, for suitably
small values of $\alpha$.
* •
Path signs: Signs are taken into account by multiplication.
Thus, the exponential of the adjacency matrix is a link prediction function
for signed networks.
Other, similar functions can be constructed, for instance the function
$(\mathbf{I}-\alpha\mathbf{A})^{-1}$ is known as the Neumann kernel, in which
$\alpha$ is chosen such that $\alpha^{-1}>|\lambda_{1}|$, $|\lambda_{1}|$
being $\mathbf{A}$’s largest absolute eigenvalue, or equivalently the graph’s
spectral norm [24].
Both the matrix exponential and the Neumann kernel can be applied to the
normalized adjacency matrix
$\mathbf{N}=\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}$, in which each edge
$\\{u,v\\}$ is weighted by $\sqrt{d(u)d(v)}$, i.e., the geometric mean of the
degrees of $u$ and $v$. The rationale behind this normalization is to count a
connection as less important if it is one of many that attaches to a node.
### 8.2 Signed Resistance Distance
The resistance distance is a metric defined on vertices of a graph inspired
from electrical resistance networks. When an electrical current is applied to
an electrical network of resistors, the whole network acts as a single
resistor whose resistance is a function of the individual resistances. In such
an electrical network, any two nodes of the network can be taken as the
endpoint of the total resistance, giving a function defined between every pair
of nodes. As shown in [26], this function is a metric, usually called the
_resistance distance_.
Intuitively, two nodes further apart are separated by a greater equivalent
resistance, while nodes closer to each other lead to a small resistance
distance. This distance function has been used before to perform collaborative
filtering [11, 12, 13], and it fulfills the first two of our assumptions, when
actual edge weights are interpreted as inverse resistances, i.e.,
conductances:
* •
Path counts: Parallel resistances are inverse-additive, and parallel
conductances are additive.
* •
Path lengths: Resistances in series are additive and conductances in series
inverse-additive.
As the resistance distance by default only applies to nonnegative values,
previous works use it on nonnegative data, such as unsigned social networks or
document view counts. In the presence of signed edges, the resistance distance
can be extended by the following formalism, which fulfills the third
requirement on path signs. A positive electrical resistance indicates that the
potentials of two connected nodes will tend to each other: The smaller the
resistance, the more both potentials approach each other. Therefore, a
positive edge can be represented as a unit resistor. If an edge is negative,
we can interpret the connection as consisting of a unit resistor in series
with an _inverting amplifier_ that guarantees its ends to have opposite
voltage, as depicted in Figure 12. In other words, two nodes connected by a
negative edge will tend to opposite voltages.
Figure 12: Interpretation of positive and negative edges as electrical
components. An edge with a negative weight is interpreted as a positive
resistor in series with an inverting component (shown as $\circleddash$).
Thus, a positive edge can be modeled by a unit resistor and a negative edge
can be modeled by a unit resistor in series with a (hypothetical) electrical
component that assures its ends have opposite electrical potential. Note that
the absence of an edge is modeled by the absence of a resistor, which is
equivalent to a resistor with infinite resistance. Thus, actual edge weights
and scores correspond not to resistances, but to inverse resistance, i.e.,
conductances.
We now establish a closed-form expression giving the resistance distance
between all node pairs based on [26].
#### Definitions
The following notation is used.
* •
$\mathbf{J}_{uv}$ is the current flowing through the oriented edge $(u,v)$.
$\mathbf{J}$ is skew-symmetric: $\mathbf{J}_{uv}=-\mathbf{J}_{vu}$.
* •
$\mathbf{v}_{u}$ is the electric potential at node $u$. Potentials are defined
up to an additive constant.
* •
$\mathbf{R}_{uv}$ is the resistance value of edge $(u,v)$. Note that
$\mathbf{R}_{uv}=\mathbf{R}_{vu}$.
In electrical networks, the current entering a node must be equal to the
current leaving that node. This relation is known as _Kirchhoff’s law_ , and
can be expressed as $\sum_{v\sim u}\mathbf{J}_{uv}=0$ for all $u\in V$. We
assume that a current $j$ will be flowing through the network from vertex $a$
to vertex $b$, and therefore we have
$\displaystyle\sum_{(v,a)}\mathbf{J}_{av}$ $\displaystyle=$ $\displaystyle j,$
$\displaystyle\sum_{(v,b)}\mathbf{J}_{bv}$ $\displaystyle=$ $\displaystyle-j.$
Using the identity matrix $\mathbf{I}$, we express these relations as
$\displaystyle\sum_{(v,u)}\mathbf{J}_{uv}=j(\mathbf{I}_{ua}-\mathbf{I}_{ub})$
(17)
The relation between currents and potentials is given by Ohm’s law:
$\mathbf{v}_{u}-\mathbf{v}_{v}=\mathbf{R}_{uv}\mathbf{J}_{uv}$ for all edges
$(u,v)$.
We will now show that the equivalent resistance $\mathbf{\bar{R}}_{ab}$
between $a$ and $b$ in the network can be expressed in terms of the graph
Laplacian $\mathbf{L}$ as
$\displaystyle\mathbf{\bar{R}}_{ab}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})\mathbf{L}^{+}(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})^{\mathrm{T}},$
$\displaystyle=$
$\displaystyle\mathbf{L}^{+}_{aa}+\mathbf{L}^{+}_{bb}-\mathbf{L}^{+}_{ab}-\mathbf{L}^{+}_{ba},$
where $\mathbf{L}^{+}$ is the Moore–Penrose pseudoinverse of $\mathbf{L}$
[26].
The proof follows from recasting Equation (17) as:
$\displaystyle\sum_{(v,u)}\frac{1}{\mathbf{R}_{uv}}(\mathbf{v}_{u}-\mathbf{v}_{v})$
$\displaystyle=j(\mathbf{I}_{ua}-\mathbf{I}_{ub})$
Combining over all $u\in V$:
$\displaystyle\mathbf{D}\mathbf{v}-\mathbf{A}\mathbf{v}$ $\displaystyle=$
$\displaystyle j(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})$
$\displaystyle\mathbf{L}\mathbf{v}$ $\displaystyle=$ $\displaystyle
j(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})$
Let $\mathbf{L}^{+}$ be the Moore–Penrose pseudoinverse of $\mathbf{L}$, then
because $\mathbf{v}$ is contained in the row space of $\mathbf{L}$ [26], we
have $\mathbf{L}^{+}\mathbf{L}\mathbf{v}=\mathbf{v}$, and we get
$\displaystyle\mathbf{v}$ $\displaystyle=$
$\displaystyle\mathbf{L}^{+}j(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})$
Which finally gives the equivalent resistance between $a$ and $b$ as
$\displaystyle\mathbf{\tilde{R}}_{ab}$ $\displaystyle=$
$\displaystyle(\mathbf{v}_{a}-\mathbf{v}_{b})/j$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})^{\mathrm{T}}\mathbf{v}/j$
$\displaystyle=$
$\displaystyle(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})^{\mathrm{T}}\mathbf{L}^{+}(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})$
A symmetry argument shows that
$\mathbf{\tilde{R}}_{ab}=\mathbf{\tilde{R}}_{ba}$ as expected. As shown in
[26], $\mathbf{\tilde{R}}$ is a metric.
The definition of the resistance distance can be extended to signed networks
in the following way.
| | |
---|---|---|---
$\textstyle{(a)}$$\textstyle{\bullet}$$\scriptstyle{r_{1}=+1}$$\textstyle{\bullet}$$\scriptstyle{r_{2}=-1}$$\textstyle{\bullet}$$\textstyle{r=r_{1}+r_{2}=0}$$\textstyle{(b)}$$\textstyle{\bullet}$$\scriptstyle{r_{1}=+1}$$\scriptstyle{r_{2}=-1}$$\textstyle{\bullet}$$\textstyle{r=\frac{r_{1}r_{2}}{r_{1}+r_{2}}=-1/0}$
Figure 13: Applying the sum rules to negative resistance values leads to
contradictions.
Figure 13 shows two examples in which we allow negative resistance values in
Equation (8.2): two parallel edges, and two serial edges. In these examples,
we will use the sum rules that hold for electrical resistances: resistances in
series add up and conductances in parallel also add up.
Therefore, the constructions of Figure 13. would result in a total resistance
of zero for case (a), and an undefined total resistance in case (b). However,
the graph of Figure 13 (a) could result from two users $a$ and $b$ having a
positive and a negative correlation with a third user $c$. Intuitively, the
resulting distance between $a$ and $b$ should take on a negative value. In the
graph of Figure 13 (b), the intuitive result would be $r=-1/2$. What we would
like is for the sign and magnitude of the equivalent resistance to be handled
separately: The sum rules should hold for the _absolute values_ of the
resistance similarity values, while the sign should obey a product rule. These
requirements are summarized in Figure 14.
| | |
---|---|---|---
$\textstyle{(a)}$$\textstyle{\bullet}$$\scriptstyle{r_{1}=+1}$$\textstyle{\bullet}$$\scriptstyle{r_{2}=-1}$$\textstyle{\bullet}$$\textstyle{r=\mathrm{sgn}(r_{1}r_{2})(|r_{1}|+|r_{2}|)=-2}$$\textstyle{(b)}$$\textstyle{\bullet}$$\scriptstyle{r_{1}=+1}$$\scriptstyle{r_{2}=-1}$$\textstyle{\bullet}$$\textstyle{r=\frac{r_{1}r_{2}}{|r_{1}|+|r_{2}|}=-1/2}$
Figure 14: Applying modified sum rules resolves the contradictions.
To achieve the serial sum equation proposed in Figure 14, we propose the
following interpretation of a negative resistance:
* •
An edge carrying a negative resistance value acts like the corresponding
positive resistance in series with a component that negates potentials.
A component that negates electric potential cannot exist in physical
electrical networks, because it violates an invariant of electrical circuit:
The invariant stating that potentials are only defined up to an additive
constant. However, as we will see below, the potential inversion gets canceled
out in the calculations, yielding results independent of any additive
constant. For this reason, we will talk of negative resistances, but avoid the
term resistor in this context.
Before giving a closed-form expression for the signed resistance distance, we
provide three intuitive examples validating our definition in Figure 15.
$\textstyle{(a)}$$\textstyle{A}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{B}$$\textstyle{r>1}$$\textstyle{(b)}$$\textstyle{A}$$\textstyle{B}$$\textstyle{0<r<1}$$\textstyle{(c)}$$\textstyle{A}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{B}$$\textstyle{r<0}$$\textstyle{(d)}$$\textstyle{A}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{B}$$\textstyle{r>0}$
Figure 15: Example configurations of signed resistance values. The total
resistance is to be calculated between the nodes A and B. All edges have unit
absolute resistance. Edges with negative resistance values are shown as dotted
lines. For each case, we formulate a condition that should hold for any signed
resistance distance.
* •
Example (a) shows that, as a path of resistances in series gets longer, the
resulting resistance increases. This conditions applies to the regular
resistance distance as well as to the signed resistance distance. In this
case, the total resistance should be higher than one.
* •
Example (b) shows that a higher number of parallel resistances decreases the
resulting resistance value. Again, this is true for both types of resistances.
In this example, the total resistance should be less than one.
* •
Examples (c) and (d) show that in a path of signed resistances, the total
resistance has the sign of the product of individual resistances. This
condition is particular to the signed resistance distance.
We will now show how Kirchhoff’s law has to be adapted to support our
definition of negative resistances. We adapt Equation (17) by applying the
absolute value to the resistance weight.
$\displaystyle\sum_{(v,u)}\frac{1}{|\mathbf{R}_{uv}|}(\mathbf{v}_{u}-\mathrm{sgn}({\mathbf{R}_{uv}})\mathbf{v}_{v})=0$
where $\mathrm{sgn}(x)$ denotes the sign function. In terms of the matrices
$\mathbf{D}$ and $\mathbf{L}$ we arrive at
$\displaystyle\mathbf{D}_{uu}$ $\displaystyle=$
$\displaystyle\sum_{(v,u)}|1/\mathbf{R}_{uv}|$ $\displaystyle\mathbf{L}$
$\displaystyle=$ $\displaystyle\mathbf{D}-\mathbf{A}$
$\displaystyle\mathbf{\tilde{r}}_{ab}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})\mathbf{L}^{+}(\mathbf{I}_{a\bullet}-\mathbf{I}_{b\bullet})^{\mathrm{T}}$
$\displaystyle=$
$\displaystyle\mathbf{L}^{+}_{aa}+\mathbf{L}^{+}_{bb}-\mathbf{L}^{+}_{ab}-\mathbf{L}^{+}_{ba}.$
The proof follows analogously to the proof for the regular resistance distance
by noting that $\mathbf{v}$ is again contained in the row space of
$\mathbf{L}$.
$\displaystyle\mathbf{L}^{+}\mathbf{L}\mathbf{v}=\mathbf{v}$
From which the result follows.
As with the regular resistance distance, the signed resistance distance is
symmetric: $\mathbf{\tilde{R}}_{ab}=\mathbf{\tilde{R}}_{ba}$.
Due to a duality between electrical networks and random walks [10], the
resistance distance is also known as the commute-time kernel, and its values
can be interpreted as the average time it takes a random walk to _commute_ ,
i.e., to go from a node $u$ to another node $v$ and back to $u$ again.
The matrix $\mathbf{L}^{+}$ will be called the resistance distance kernel.
Similarly, the matrix $\mathbf{e}^{-\alpha\mathbf{L}}$ is known as the heat
diffusion kernel, because it can be derived from a physical process of heat
diffusion. Both of these kernels can be normalized, i.e., they can be applied
to the normalized adjacency matrix
$\mathbf{N}=\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}$, giving the
normalized resistance distance kernel and the normalized heat diffusion
kernel. We note that the normalized heat diffusion kernel is equivalent to the
normalized exponential kernel [47].
The degree matrix $\mathbf{D}$ of a signed graph is defined in this article
using $\mathbf{D}_{uu}=\sum_{v}|\mathbf{A}_{uv}|$ in the general case. In some
contexts, an alternative degree matrix $\mathbf{D}_{\mathrm{alt}}$ is defined
without the absolute value:
$\displaystyle(\mathbf{D}_{\mathrm{alt}})_{uu}=\sum_{v}\mathbf{A}_{uv}$
This leads to an alternative Laplacian matrix
$\mathbf{L}_{\mathrm{alt}}=\mathbf{D}_{\mathrm{alt}}-\mathbf{A}$ for signed
graphs that is not positive-semidefinite. This Laplacian is used in the
context of knot theory [38], to draw graphs with negative edge weights [27],
and to implement constrained clustering, i.e., clustering with _must-link_ and
_must-not-link_ edges [7]. Since $\mathbf{L}_{\mathrm{alt}}$ is not positive-
semidefinite in the general case, it cannot be used as a kernel.
Expressions of the form
$(\sum_{i}|\mathbf{w}_{i}|)^{-1}\sum_{i}\mathbf{w}_{i}\mathbf{x}_{i}$ appeared
several times in the preceding sections. These types of expressions represent
a weighted mean of the values $\mathbf{x}_{i}$, supporting negative values of
the weights $\mathbf{w}_{i}$. These expressions have been used for some time
in the collaborative filtering literature without being connected to the
signed Laplacian, for instance in [45].
### 8.3 Evaluation
We compare the methods shown in Table 4 at the task of link prediction in
signed social networks.
Evaluation is performed using the following methodology. Let $G=(V,E,\sigma)$
be any of the signed networks, and let
$\displaystyle E=E_{\mathrm{a}}\cup E_{\mathrm{b}}$
be a partition of the edge set $E$ into a training set $E_{\mathrm{a}}$ and a
test set $E_{\mathrm{b}}$. The training set is chosen to comprise 75% of all
edges. For the networks in which edge arrival times are known (Epinions,
Wikipedia elections, Wikipedia conflict), the split is made in such a way that
all edges in the training set $E_{\mathrm{a}}$ are older than the edges in the
test set $E_{\mathrm{b}}$. Each link prediction method is then applied to the
training network
$\displaystyle G_{\mathrm{a}}=(V,E_{\mathrm{a}}).$
Let $E^{+}_{\mathrm{b}}$ denote the test edges with positive sign. Then, a
zero test set $E_{\mathrm{z}}$ of edges not in the network at all is
generated, having the same size as $E^{+}_{\mathrm{b}}$. Then, the scores of
each link prediction algorithm are computed for all node pairs in
$E^{+}_{\mathrm{b}}$ and $E_{\mathrm{z}}$, and the accuracy of each link
prediction algorithm evaluated on $E^{+}_{\mathrm{b}}$ and $E_{\mathrm{z}}$
using the area under the curve (AUC) measure [3]. The area under the curve is
a number in the range $[0,1]$ which is larger for better predictions, and
admits a value of 0.5 for a random predictor. The parameters $\alpha$ of the
various link prediction functions are learned using the method described in
[31]. The results of the experiments are shown in Table 5.
Table 4: The link prediction functions evaluated on the signed social network
datasets. Each method is a function of a specific characteristic graph matrix:
$\mathbf{A}$, the adjacency matrix;
$\mathbf{N}=\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}$, the normalized
adjacency matrix; $\mathbf{L}=\mathbf{D}-\mathbf{A}$, the Laplacian matrix;
and
$\mathbf{Z}=\mathbf{I}-\mathbf{N}=\mathbf{D}^{-1/2}\mathbf{L}\mathbf{D}^{-1/2}$,
the normalized Laplacian matrix.
Name | Expression
---|---
Exponential (Exp) | $e^{\alpha\mathbf{A}},0<\alpha$
Neumann kernel (Neu) | $(\mathbf{I}-\alpha\mathbf{A})^{-1},0<\alpha<\mathopen{\parallel}\mathbf{A}\mathclose{\parallel}_{2}^{-1}$
Normalized exponential (N-Exp) | $e^{\alpha\mathbf{A}},0<\alpha$
Normalized Neumann kernel (N-Neu) | $(\mathbf{I}-\alpha\mathbf{N})^{-1},0<\alpha<1$
Resistance distance (Resi) | $\mathbf{L}^{+}$
Heat diffusion (Heat) | $e^{-\alpha\mathbf{L}},0<\alpha$
Normalized resistance distance (N-Resi) | $\mathbf{Z}^{+}$
Normalized heat diffusion | Equivalent to Normalized exponential
Table 5: The full evaluation results. The numbers are the area under the
curve values (AUC); higher values denote better link prediction accuracy. The
best performing link prediction algorithm for each dataset is highlighted in
bold.
Network | Exp | Neu | N-Exp | N-Neu | Resi | Heat | N-Resi
---|---|---|---|---|---|---|---
Slashdot Zoo | 68.98% | 67.71% | 64.87% | 65.68% | 61.64% | 59.11% | 65.71%
Epinions | 75.04% | 73.12% | 78.38% | 78.65% | 63.26% | 63.28% | 78.82%
Wikipedia elections | 57.08% | 55.60% | 60.30% | 61.16% | 51.44% | 50.60% | 60.98%
Wikipedia conflicts | 85.57% | 85.56% | 85.03% | 85.03% | 87.02% | 85.95% | 85.04%
We observe that the best link prediction method depends on the dataset. Each
of the exponential, the normalized Neumann kernel, the resistance distance
kernel and the normalized resistance distance kernel performs best for one or
more datasets.
## 9 Conclusion
We have reviewed network analysis methods for signed social networks – social
networks that allow positive and negative edges. A main theme we found is that
of structural balance, the statement that triangles in a signed social network
tend to be balanced, and on a larger scale the tendency of a whole network to
have a structure conforming to that assumption. We showed how this can be
measured in two different ways: on the scale of triangles by the signed
clustering coefficient, and on the global scale by the algebraic conflict, the
smallest eigenvalue of the graph Laplacian. We also showed how structural
balance can be exploited for graph drawing, graph clustering, and finally for
implementing social recommenders, using signed link prediction algorithms.
As structural balance can be seen as a form of multiplication rule
(illustrated by the phrase _the enemy of my enemy is my friend_), it is
expected that algebraic methods are well-suited to analysing signed social
networks. Indeed, we identified functions of the adjacency matrix $\mathbf{A}$
and of the Laplacian matrix $\mathbf{L}$, which model negative edges in a
natural way.
In a more general sense, signed social networks can be understood as a
stepping stone to the more general topic of _semantic networks_ , in which
edges are labeled by arbitrary predicates. In such networks, the combination
of labels to give a new label, in analogy with the multiplication rule of the
signed edge weights $\\{\pm 1\\}$, cannot be directly mapped by real numbers,
and a general method for that case is still an open problem in network theory.
Certain subproblems have however already be identified, for instance the usage
of split-complex imaginary numbers to represent the _like_ relationship [30].
## Acknowledgments
We thank Andreas Lommatzsch, Christian Bauckhage, Stephan Schmidt, Jürgen
Lerner and Martin Mehlitz. The research leading to these results has received
funding from the European Community’s Seventh Frame Programme under grant
agreement no 257859, ROBUST.
## References
* [1] M. Belkin and P. Niyogi. Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in Neural Information Processing Systems, pages 585–591, 2002.
* [2] P. Bonacich and P. Lloyd. Calculating status with negative relations. Social Networks, (26):331–338, 2004.
* [3] A. P. Bradley. The use of the area under the ROC curve in the evaluation of machine learning algorithms. Pattern Recognition, 30:1145–1159, 1997.
* [4] U. Brandes, D. Fleischer, and J. Lerner. Summarizing dynamic bipolar conflict structures. Trans. on Visualization and Computer Graphics, 12(6):1486–1499, 2006.
* [5] U. Brandes and J. Lerner. Structural similarity: Spectral methods for relaxed blockmodeling. J. Classification, 27(3):279–306, 2010.
* [6] L. Brožovský and V. Petříček. Recommender system for online dating service. In Proc. Conf. Znalosti, pages 29–40, 2007.
* [7] I. Davidson. Knowledge driven dimension reduction for clustering. In Proc. Int. Conf. on Research and Development in Information Retrieval, pages 1034–1039, 2009.
* [8] I. S. Dhillon, Y. Guan, and B. Kulis. Kernel $k$-means: Spectral clustering and normalized cuts. In Proc. Int. Conf. Knowledge Discovery and Data Mining, pages 551–556, 2004.
* [9] P. Doreian and A. Mrvar. A partitioning approach to structural balance. Social Networks, 18:149–168, 1996.
* [10] P. G. Doyle and J. L. Snell. Random Walks and Electric Networks. Math. Ass. of America, 1984.
* [11] F. Fouss, A. Pirotte, J.-M. Renders, and M. Saerens. Random-walk computation of similarities between nodes of a graph with application to collaborative recommendation. Trans. on Knowledge and Data Engineering, 19(3):355–369, 2007.
* [12] F. Fouss, A. Pirotte, and M. Saerens. The application of new concepts of dissimilarities between nodes of a graph to collaborative filtering. In Proc. Workshop on Statistical Approaches for Web Mining, pages 26–37, 2004.
* [13] F. Fouss, A. Pirotte, and M. Saerens. A novel way of computing similarities between nodes of a graph, with application to collaborative recommendation. In Proc. Int. Conf. on Web Intelligence, pages 550–556, 2005.
* [14] K. A. Germina, S. Hameed K., and T. Zaslavsky. On products and line graphs of signed graphs, their eigenvalues and energy. Linear Algebra and Its Applications, 435(10):2432–2450, 2011.
* [15] R. Guha, R. Kumar, P. Raghavan, and A. Tomkins. Propagation of trust and distrust. In Proc. Int. World Wide Web Conf., pages 403–412, 2004.
* [16] P. Hage and F. Harary. Structural Models in Anthropology. Cambridge University Press, 1983.
* [17] F. Harary. On the notion of balance of a signed graph. Michigan Math. J., 2(2):143–146, 1953.
* [18] T. Hogg, D. M. Wilkinson, G. Szabo, and M. J. Brzozowski. Multiple relationship types in online communities and social networks. In Proc. AAAI Spring Symp. on Social Information Processing, 2008\.
* [19] Y. P. Hou. Bounds for the least Laplacian eigenvalue of a signed graph. Acta Math. Sinica, 21(4):955–960, 2005.
* [20] Y. P. Hou, J. S. Li, and Y. Pan. On the Laplacian eigenvalues of signed graphs. Linear and Multilinear Algebra, 1(51):21–30, 2003.
* [21] B. Hu, X.-Y. Jiang, J.-F. Ding, Y.-B. Xie, and B.-H. Wang. A model of weighted network: the student relationships in a class. CoRR, cond-mat/0408125, 2004.
* [22] G. Kalna and D. J. Higham. A clustering coefficient for weighted networks, with application to gene expression data. AI Commun., 20(4):263–271, 2007.
* [23] S. D. Kamvar, M. T. Schlosser, and H. Garcia-Molina. The EigenTrust algorithm for reputation management in P2P networks. In Proc. Int. World Wide Web Conf., pages 640–651, 2003.
* [24] J. Kandola, J. Shawe-Taylor, and N. Cristianini. Learning semantic similarity. In Advances in Neural Information Processing Systems, pages 657–664, 2002.
* [25] C. D. Kerchove and P. V. Dooren. The PageTrust algorithm: How to rank Web pages when negative links are allowed? In Proc. SIAM Int. Conf. on Data Mining, pages 346–352, 2008.
* [26] D. J. Klein and M. Randić. Resistance distance. J. Math. Chemistry, 12(1):81–95, 1993.
* [27] Y. Koren, L. Carmel, and D. Harel. ACE: A fast multiscale eigenvectors computation for drawing huge graphs. In Symp. on Information Visualization, pages 137–144, 2002.
* [28] J. Kunegis. On the Spectral Evolution of Large Networks. PhD thesis, University of Koblenz–Landau, 2011.
* [29] J. Kunegis. KONECT – The Koblenz Network Collection. In Proc. Int. Web Observatory Workshop, pages 1343–1350, 2013.
* [30] J. Kunegis, G. Gröner, and T. Gottron. Online dating recommender systems: The split-complex number approach. In Proc. Workshop on Recommender Systems and the Social Web, pages 37–44, 2012.
* [31] J. Kunegis and A. Lommatzsch. Learning spectral graph transformations for link prediction. In Proc. Int. Conf. on Machine Learning, pages 561–568, 2009.
* [32] J. Kunegis, A. Lommatzsch, and C. Bauckhage. The Slashdot Zoo: Mining a social network with negative edges. In Proc. Int. World Wide Web Conf., pages 741–750, 2009.
* [33] J. Kunegis and S. Schmidt. Collaborative filtering using electrical resistance network models with negative edges. In Proc. Industrial Conf. on Data Mining, pages 269–282, 2007.
* [34] J. Kunegis, S. Schmidt, C. Bauckhage, M. Mehlitz, and S. Albayrak. Modeling collaborative similarity with the signed resistance distance kernel. In Proc. European Conf. on Artificial Intelligence, pages 261–265, 2008.
* [35] J. Kunegis, S. Schmidt, A. Lommatzsch, and J. Lerner. Spectral analysis of signed graphs for clustering, prediction and visualization. In Proc. SIAM Int. Conf. on Data Mining, pages 559–570, 2010.
* [36] J. Leskovec, D. Huttenlocher, and J. Kleinberg. Governance in social media: A case study of the Wikipedia promotion process. In Proc. Int. Conf. on Weblogs and Social Media, pages 98–105, 2010\.
* [37] J. Leskovec, D. Huttenlocher, and J. Kleinberg. Predicting positive and negative links in online social networks. In Proc. Int. Conf. on World Wide Web, pages 641–650, 2010.
* [38] M. Lien and W. Watkins. Dual graphs and knot invariants. Linear Algebra and its Applications, 306(1):123–130, 2000.
* [39] U. v. Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4):395–416, 2007.
* [40] P. Massa and P. Avesani. Controversial users demand local trust metrics: an experimental study on epinions.com community. In Proc. American Association for Artificial Intelligence Conf., pages 121–126, 2005.
* [41] P. Massa and C. Hayes. Page-reRank: Using trusted links to re-rank authority. In Proc. Int. Conf. on Web Intelligence, pages 614–617, 2005.
* [42] M. Meilă and J. Shi. A random walks view of spectral segmentation. In Proc. Int. Conf. on Artificial Intelligence and Statistics, 2001\.
* [43] A. Y. Ng, M. I. Jordan, and Y. Weiss. On spectral clustering: Analysis and an algorithm. In Advances in Neural Information Processing Systems, pages 849–856, 2001.
* [44] K. E. Read. Cultures of the Central Highlands, New Guinea. Southwestern J. of Anthropology, 10(1):1–43, 1954.
* [45] B. M. Sarwar, G. Karypis, J. A. Konstan, and J. Riedl. Item-based collaborative filtering recommendation algorithms. In Proc. Int. World Wide Web Conf., pages 285–295, 2001.
* [46] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 22(8):888–905, 2000.
* [47] A. Smola and R. Kondor. Kernels and regularization on graphs. In Proc. Conf. on Learning Theory and Kernel Machines, pages 144–158, 2003.
* [48] G. Theodorakopoulos and J. S. Baras. Linear iterations on ordered semirings for trust metric computation and attack resiliency evaluation. In Proc. Int. Symp. on Math. Theory of Networks and Systems, pages 509–514, 2006.
* [49] D. J. Watts and S. H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393(1):440–442, 1998.
* [50] J. H. Wilkinson. The Algebraic Eigenvalue Problem. Oxford University Press, 1965.
* [51] B. Yang, W. Cheung, and J. Liu. Community mining from signed social networks. Trans. on Knowledge and Data Engineering, 19(10):1333–1348, 2007\.
* [52] T. Zaslavsky. Signed graphs. Discrete Applied Math., 4:47–74, 1982.
* [53] T. Zaslavsky. Matrices in the theory of signed simple graphs. In Proc. Int. Conf. Discrete Math., pages 207–229, 2008.
|
arxiv-papers
| 2014-02-27T11:32:50 |
2024-09-04T02:49:59.005704
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J\\'er\\^ome Kunegis",
"submitter": "J\\'er\\^ome Kunegis",
"url": "https://arxiv.org/abs/1402.6865"
}
|
1402.6871
|
00footnotetext: Received 15 Dec 201300footnotetext: *Supported by National
Science Foundation of China(11205183,11005117,11225525,11390384 )
# Temperature dependence of the light yield of the LAB-based and mesitylene-
based liquid scintillators
XIA Dong-Mei1,2 YU Bo-Xiang1 Li Xiao-Bo1 SUN Xi-Lei1
DING Ya-Yun1 ZHOU Li1 CAO Jun1 HU Wei1,2 YE Xing-Cheng3
CHEN Hai-Tao4 DING Xue-Feng3 DU Bing1 [email protected] [email protected] 1
(State Key Laboratory of Particle Detection and Electronics (Institute of High
Energy Physics, CAS), Beijing, 100049, China)
2 (University of Chinese Academy of Sciences, Beijing, 100049, China)
3 (Wuhan University, Hubei, 430072, China)
4 (Nanjing University of Aeronautics and Astronautics, Jiangsu, 210016, China)
###### Abstract
We studied the temperature dependence of the light yield of the linear alkyl
benzene (LAB)-based and mesitylene-based liquid scintillators. The light yield
increases by 23% for both liquid scintillators when the temperature is lowered
from $26\;^{\circ}$C to $-40\;^{\circ}$C, correcting for the temperature
response of the photomultiplier tube. The measurements help to understand the
energy response of the liquid scintillator detectors. Especially, the next
generation reactor neutrino experiments for neutrino mass hierarchy, such as
the Jiangmen Underground Neutrino Observatory (JUNO), require very high energy
resolution. As no apparent degradation on the liquid scintillator transparency
was observed, lowering the operation temperature of the detector to $\sim
4\;^{\circ}$C will increase the photoelectron yield of the detector by 13%,
combining the light yield increase of the liquid scintillator and the quantum
efficiency increase of the photomultiplier tubes.
###### keywords:
liquid scintillator, reactor neutrino, linear alkyl benzene, mesitylene
###### pacs:
2
9.40.Mc
00footnotetext: $\scriptstyle\copyright$2013 Chinese Physical Society and the
Institute of High Energy Physics of the Chinese Academy of Sciences and the
Institute of Modern Physics of the Chinese Academy of Sciences and IOP
Publishing Ltd
## 1 Introduction
Organic liquid scintillator (LS) is widely used to detect reactor neutrinos
[2, 3, 4, 5, 6] due to its high light yield and high hydrogen fraction. Liquid
scintillator is made of a solvent and a small amount of fluor, and often with
an additional tiny amount of wavelength shifter. For example, the Daya Bay
undoped liquid scintillator consists of linear alkyl benzene (LAB) as the
solvent, 3 g/L 2,5-diphenyloxazole (PPO) as the fluor, and 15 mg/L
p-bis-(o-methylstyryl)-benzene (bis-MSB) as the wavelength shifter, while the
gadolinium-doped LS has the same recipe but mixed with a Gd complex with 0.1%
Gd in mass [7, 8].
The energy response of the liquid scintillator detector need be well
understood for precision measurements in a reactor neutrino experiment. The
light yield of the liquid scintillator, to which the visible energy of an
event in the detector is proportional, is however temperature dependent
because of the thermal quenching effects. Excited solvent molecules by
ionization may undergo non-radiation transition when colliding with other
molecules. Normally the light yield will increase at lower temperature, when
the viscosity of the solvent rises thus collisions reduce. If quencher
presents in the solution, the situation will be complex since the collisions
may also impede the energy transfer from the exited solvent molecule to the
quencher.
The temperature dependence of the light yield has been studied for some liquid
scintillators [9] but is still scanty in literature. In this study, we will
study the temperature effects of the Daya Bay LS, which is based on the
relative new solvent LAB, over a range from $-40\;^{\circ}$C to
$26\;^{\circ}$C. As a comparison, LS with the same solute fractions but
another solvent, mesitylene, is also measured.
Such studies are of particular interests for the design of the next generation
reactor neutrino experiments such as the Jiangmen Underground Neutrino
Observatory (JUNO). To determine the neutrino mass hierarchy by precisely
measuring the energy spectrum of the reactor neutrinos, JUNO detector requires
a very high energy resolution of 3%/$\sqrt{E({\rm MeV})}$ [10]. Previous
experiments reached (5-6)%/$\sqrt{E}$ [2, 6]. This unprecedent energy
resolution requirement is a major challenge for JUNO. The increasing light
yield of the LS at lower temperature provides an option to operate the
detector at low temperature, e.g. at $\sim 4\;^{\circ}$C, right above the ice
point of the buffer water shielding the neutrino detector. For this purpose,
we also studied the light transmittance of the LS at low temperature.
In this paper, the experimental setup, light yield measurement, correction for
the temperature effects of the photomultiplier tube (PMT) are described in
section 2. The transmittance is studied in section 3, followed by a conclusion
and discussion.
## 2 Temperature dependence of the light yield
### 2.1 Experimental setup
The light yield of the LS is measured via Compton scattering of $\gamma$ rays
from a radioactive source. To improve the precision of the measurement, we tag
the scattered $\gamma$s at a fixed direction. The coincidence of the recoil
electron in the LS and the scattered $\gamma$ selects the events of known
deposited energy in the LS, thus reduces the uncertainty of the light yield
measurement to sub-percent level from $\sim$5% of the common method by fitting
the Compton edge.
Experimental setup for measuring the temperature dependence of the LS light
yield.
Figure 1 shows the scheme of the experimental setup. The LS sample is
contained in a cylindric quartz glass vessel of 5 cm in diameter and 5 cm in
height. The vessel is wrapped with Enhanced Specular Reflector (ESR) to
increase the photon collection efficiency, and is coupled to a CR135 PMT. The
scattered $\gamma$ is tagged by a coincidence detector, which is a plastic
scintillator (PS) detector located about 10 cm from the LS at an angle of
about $30^{\circ}$. The LS is irradiated by $\gamma$-rays from a 137Cs (15
$\mu$Ci) source. The source, the LS vessel, the CR135 PMT, and the coincidence
detector are put in an enclosed thermostat with temperature adjustable from
$-70\;^{\circ}$C to $155\;^{\circ}$C. The signals from the CR135 PMT and the
coincidence detector are recorded by a CAEN DT5751 ADC unit with a self-
trigger function. The relative light yield of the LS is determined by
comparing the peak values measured by the ADC at different temperatures.
To separate the temperature effects of the PMT from that of the LS, a
temperature-resistant optical fibre is coupled to the photocathode of the
CR135 PMT directly, transmitting light from an LED driven by a pulse
generator. The LED is located in an incubator box operating at 25 ∘C, emitting
light at 430 nm wavelength. The pulse generator is at room temperature, which
is almost a constant during the measurement. The LED flashes at a frequency of
10 kHz during the LS light yield measurement and the data is triggered by the
coincidence with the PS detector (noise). The probability of the LED signal
overlapping with a Compton signal is found to be small enough.
The LS sample is bubbled with nitrogen from the bottom of the vessel before
measurement to remove oxygen and water in the LS. Oxygen is a quencher of the
LS. The presence of oxygen reduces the light yield of the LAB-based Daya Bay
LS by up to 11% [11], shortens the time constant of the scintillation [12],
and may change the temperature effects of the LS. Normally the LAB-based LS
contains tens ppm water. To avoid possible impacts on the light transmittance
of the LS at low temperature, water is also removed by bubbling nitrogen. The
LS sample is covered with nitrogen during the measurement. A temperature
sensor is mounted on the LS vessel for monitoring. Data are taken only after
the temperature has reached stable for more than 30 minutes.
### 2.2 Relative light yield of the LS
Figure 2 shows an example of the ADC distribution of the measurement. The
light yield of the LS and the light intensity of the LED are measured by the
ADC peaks, which are fitted with Gaussian functions. The LED intensity is
stable to 1% at constant temperature. The observed shift at different
temperatures shows that the CR135 PMT response suffers from the temperature
variation.
Figure 3 (a) shows the relative light yield of the LAB-based LS before
correcting for the PMT temperature effects. The measurements have been done
three times by different people and with slightly different hardware. The data
labelled ”Third” corresponds to this measurement and the other two were done
before. The three measurements are in good agreement. The temperature response
of the CR135 PMT monitored by the LED is shown in Figure 3 (b).
ADC channel distribution of the LS events and the LED monitoring signal.
(a) The relative light yield of the LAB-based LS before correcting for the PMT
temperature effects, normalized at $26\;^{\circ}$C. (b) The temperature
response of the CR135 PMT, normalized at $26\;^{\circ}$C.
A similar measurement was done for the mesitylene-based LS. After correcting
for the measured temperature response of the PMT shown in Figure 3 (b), the
temperature dependence of the LS light yield are shown in figure 4. The light
yield increases by 23% as the temperature decreases from $26\;^{\circ}$C to
$-40\;^{\circ}$C for both liquid scintillators.
The relative light yield of the LAB-based LS (LLS) and the mesitylene-based LS
(MLS) after correcting for the PMT temperature effects, normalized at
$26\;^{\circ}$C.
The variation of the PMT response at difference temperature might be a
combination of the variation of the PTM gain and the quantum efficiency
changes of the PMT photocathode. The PMT gain decreases as the temperature
increases because of the negative temperature coefficient of the dynodes [13].
Quantum efficiency of the PMT is typically 25% at 430 nm at room temperature
for the CR135, which uses bialkali (SB-K-Cs) photocathode. The quantum
efficiency of the bialkali has almost a constant temperature coefficient of
-0.2%/∘C for photons of wavelength between 200 nm and 550 nm [14]. It will
increase by 13% when the temperature is lowered from $26\;^{\circ}$C to
$-40\;^{\circ}$C. Therefore, the quantum efficiency increase dominates the PMT
temperature effects we have measured. In this measurement, The temperature
stability of the temperature-resistant optical fibre is estimated to be 0.5%
and the stability of the LED intensity is about 1%.
## 3 Temperature dependence of the light transmittance
### 3.1 Experimental setup
For a large detector of $\sim$ 38 m diameter of JUNO, the light transmittance
of the LS is equally important as the light yield. The temperature dependence
of the light transmittance is measured with 430 nm wavelength light, over a
range from $-40\;^{\circ}$C to $26\;^{\circ}$C.
The scheme of the experimental setup is shown in Figure 5. The light source is
a DH-2000 deuterium tungsten halogen lamp, followed by a monochromatic filter
at 430 nm. The light transmits in a temperature-resistant fibre, passes the LS
sample in a cuvette of dimension of 1 cm $\times$ 1 cm $\times$ 3 cm, and is
received by an Ocean Optics QE65000 spectrophotometer. A temperature probe is
attached to the cuvette to monitor the LS temperature. The spectrum of the
light is shown in the bottom panel of figure 5, measured by the
spectrophotometer. The relative transmittance can be obtained by continuously
measuring the light intense as temperature varies.
The surface of the cuvette may frost and bias the transmittance measurement
when the temperature goes down. We observed such phenomenon that the
transmittance started to drop dramatically at certain temperature (e.g.
$-7\;^{\circ}$C for one of our measurements of LAB-based LS), although it is
not visible on the surface of the cuvette by eye. After excluding the possible
causes such as the crystallization of water content in the LAB, precipitation
of scintillation fluor, and freezing of the solvent itself, we improved the
experimental setup by sealing the LS cuvette in a transparent airtight box to
against the frost. The box is flushed with dry nitrogen before the experiment
to remove the vapor in the air, and maintain a small positive pressure with
nitrogen. The nitrogen is released from the box through a bubbler to monitor
the airtightness of the box.
Top: Schematic view of the experimental setup for the light transmittance
measurement. Bottom: The measured light spectrum after passing the
monochromatic filter.
### 3.2 Relative transmittance of the LS
Liquid scintillator exposed to air normally contains water of tens ppm. Water
crystals could be formed in the LS and degrade the transmittance at low
temperature. Water can be removed from the LS thoroughly by bubbling enough
dry nitrogen. The nitrogen bubbling is also necessary to purge the oxygen in
the LS. The water content in the LS can be measured to below 1 ppm with a
831KF Coulometric Moisture Analyzer. Table 1 shows the relationship between
the remaining water content and the volume of nitrogen flushed. Three litres
of nitrogen is needed to purge all the water in the cuvette.
Relationship between the water content in the LS samples and the volume of
nitrogen flushed. LS 0 L 1 L 3 L LLS00 0 $\sim$ 27 ppm 0 $\sim$ 16 ppm $\sim$
0 ppm MLS00 0 $\sim$ 140 ppm 0 $\sim$ 70 ppm $\sim$ 0 ppm
Figure 6 presents the result of the relative transmittance for the LAB-based
and the mysitylene-based LS. We use the temperature instability of the system
without LS in the cuvette as the systematic error of the measurement. For both
the LAB-based and the mysitylene-based LS, transmittance stays stable when the
temperature decreases from $26\;^{\circ}$C to $-40\;^{\circ}$C. The melting
point of the LAB we used is below $-60\;^{\circ}$C and that of the mesitylene
is $-44\;^{\circ}$C. Lowering the temperature to $-40\;^{\circ}$C will not
cause a phase transition of the LS.
Relative transmittance of the LAB-based LS (LLS) and the mysitylene-based LS
(MLS) for light of 430 nm wavelength.
## 4 Conclusion and discussion
The temperature dependence of the light yield and the transmittance of two
liquid scintillators, LAB-based LS and mysitylene-based LS, have been
measured. For both liquid scintillators, when the temperature is lowered from
$26\;^{\circ}$C to $-40\;^{\circ}$C, the light yield increases by 23%, with
the PMT effects corrected, and the transmittance remains stable. Frosting on
the sample vessel at low temperature is observed to have significant impacts
on the measurements. It is avoid by coupling the PMT to the sample vessel with
silicone oil for the light yield measurement, or by putting the vessel in a
nitrogen purged transparent box for the transmittance measurement.
The light yield increase at low temperature provides an option for the next
generation reactor neutrino experiments for neutrino mass hierarchy such as
JUNO, which requires very high energy resolution. When using water as outer
buffer, the operation temperature of the detector could be lowered to $\sim
4\;^{\circ}$C. As no apparent degradation on the liquid scintillator
transparency was observed, lowering the operation temperature to $\sim
4\;^{\circ}$C from $26\;^{\circ}$C will increase the photoelectron yield of
the detector by 13%, in which 9% is from the light yield of the LAB-based
liquid scintillator and $\sim 4$% is from the quantum efficiency of the PMT
with bialkali photocathode.
Operating at even lower temperature, such $-40\;^{\circ}$C, will increase the
photoelectron yield by 32%. But it is more difficult to realize and requires
an oil buffer instead of a water buffer.
## References
* [1]
* [2] Eguchi K et al. (KamLAND collaboration), Phys. Rev. Lett., 2003, 90: 021802.
* [3] An F P et al. (Daya Bay collaboration), Phys. Rev. Lett., 2012, 108: 171803; Chin. Phys. C, 2013, 37(1): 011001.
* [4] Aberle C et al. Nucl. Phys. B (Proc. Suppl.),2012,229: 448.
* [5] Park J S et al. Nucl. Instr. and Meth. A, 2013,707: 45.
* [6] Alimonti G et al. (Borexino collaboration), Astropart. Phys., 2002, 16(3): 205.
* [7] Ding Y Y et al., Nucl. Instr. and Meth. A, 2008, 584: 238.
* [8] Ding Y Y et al., submitted to Nucl. Instr. and Meth. A.
* [9] Buontempo S et al., Nucl. Instr. and Meth. A, 1999, 425: 492.
* [10] Li Y F, Cao J, Wang Y F, and Zhan L, Phys. Rev. D, 2013, 88: 013008.
* [11] XIAO H L et al., Chin. Phys. C, 2010, 34(05): 571.
* [12] LI X B et al., Chin. Phys. C, 2011, 35(11): 1026.
* [13] Moszynski M et al., Nucl. Instr. and Meth. A, 2006, 568: 739.
* [14] Photomultiplier Tubes: Basics and Applications (Third Edition), Hamamatsu Photonics K.K., Japan (2006).
|
arxiv-papers
| 2014-02-27T11:49:37 |
2024-09-04T02:49:59.016432
|
{
"license": "Public Domain",
"authors": "Xia DongMei and Yu BoXiang and Li XiaoBo and Sun XiLei and Ding YaYun\n and Zhou Li and Cao Jun and Hu Wei and Ye XingCheng and Chen HaiTao and Ding\n XueFeng and Du Bing",
"submitter": "Xia Dongmei",
"url": "https://arxiv.org/abs/1402.6871"
}
|
1402.6903
|
# Three Experiments to Analyze the Nature of the Heat Spreader
Seema Sethia1, Shouri Chatterjee2, Sunil Kale3, Amit Gupta4, Smruti R.
Sarangi5
1,2 Department of Electrical Engineering, IIT Delhi
3,4 Department of Mechanical Engineering, IIT Delhi
5 Department of Computer Science and Engineering, IIT Delhi
[email protected], {shouri@ee, srk@mech, agupta@mech,
srsarangi@cse}.iitd.ac.in
###### Abstract
In this paper, we describe ongoing work to investigate the properties of the
heat spreader, and its implication on architecture research. In specific, we
conduct two experiments to quantify the heat distribution across the surface
of a spreader during normal operation. The first experiment uses T-type
thermocouples, to find the temperature difference across different points on
the spreader. We observe about 6∘C difference on average. In the second
experiment, we try to capture the temperature gradients using an infrared
camera. However, this experiment was inconclusive because of some practical
constraints such as the low emissivity of the spreader. We conclude that to
properly model the spreader, it is necessary to conduct detailed finite
element simulations. We describe a method to accurately measure the thermal
conductivity of the heat spreader such that it can be used to compute the
steady state temperature distribution across the spreader.
## I Introduction
An oft-ignored aspect of architecture level thermal modeling is the heat
spreader. The heat spreader is typically a nickel coated copper plate placed
between the die and the heat sink (see Figure 1). Its main role is to
uniformly dissipate the heat generated by the die, and transmit the heat to
the heat sink. The heat sink is a large fin based heat exchanger that is used
to effectively dissipate the heat to the surrounding air. The heat spreader
effectively “spreads out” the heat and reduces the severity and incidence of
thermal hot spots.
Given the fact that the heat spreader is nothing more than a metal plate, and
does not have a lot of inherent complexity, it has not received a lot of
attention by the architecture community. Some prior work such as [1, 2], have
treated it as an isotherm (equal temperature at all points). We experimentally
disprove this hypothesis in this paper. Skadron et. al. [3] treat the spreader
as a mesh of points, where each point is a heat source and two adjacent points
are connected by a thermal resistance in their widely available thermal
modeling tool, HotSpot. However, there has been some recent criticism of the
equivalent thermal circuit based approach adopted by HotSpot in [4, 5]. These
works have reported a mean error of about 10% in HotSpot.
Figure 1: The chip package
We are currently working on developing a new temperature estimation tool.
During the course of this work, we wish to look at the spreader from an
experimental viewpoint. There are several reasons for our belief that the
spreader warrants a more thorough study. (1) The conductivity of silicon is
roughly 100 W/m-K [5], whereas the conductivity of the spreader is about 400
W/m-K. Consequently, the spreader is a far more efficient lateral conductor of
heat than silicon especially at distances of the order of the dimensions of
the die. This has important implications for floor planning, thermal
management, and task allocation in multicores. For example, it is possible for
a set of active cores to heat up a set of relatively quiescent cores by
passing heat through the spreader. This will hurt the performance of the
relatively inactive cores, as well as long term lifetime reliability. (2) We
can use the spreader temperature data that we collect to calibrate temperature
simulators. (3) We can measure some thermal properties of the spreader and use
it to quantify the degradation of the material over time. We can use this
empirical data to perform more accurate FEM simulations. Lastly, for Intel
based chips that integrate the spreader with the die, it is not possible to
study the temperature profile of the die independently. We need to infer its
thermal profile by analyzing the temperature gradients on the spreader.
We performed a simple thought experiment as follows. We consider a die with a
large number of cores (128), and assumed that there is lateral heat conduction
just through the spreader. We use typical parameters from the HotSpot tool
(version 5.0) [3], and simulated a scenario in which each core dissipates
enough power to increase the die temperature measured at the center by 20∘C .
We now turn off a set of cores, and measure the effect that the active cores
have on the inactive cores. We plot the average temperature rise of the
inactive cores in Figure 3 as a fraction of the number of active cores. Figure
3 shows the normalized decrease in the MTTF (Mean Time to Failure) for three
major failure mechanisms: Electro-migration, Thermal Cycling, and Stress
Migration (see Srinivasan et. al. [6]). We observe that lateral heat
conduction can have a significant impact on inactive cores. It can heat them
up by 10 to 20∘C , and can decrease their MTTF by upto 10X.
Figure 2: Mean temp. of the inactive cores | Figure 3: MTTF of the inactive cores
---|---
In this paper, we describe two approaches to measure the temperature
distribution across the spreader during normal operation. We can use these
numbers to calibrate temperature estimation tools. Lastly, we describe an
approach to compute the thermal conductivity of the spreader such that it can
be used to seed FEM simulations.
## II Experiments
For all our experiments we use a 775 pin, 90 nm, Intel Pentium 4(Prescott)
chip mounted on a Dell 00M075 Dimension 4300 Motherboard. It has a nominal
frequency of 3.06GHz, 1MB L2 cache, and has two voltage steppings – 1.25 and
1.388V. We perform three experiments:
1. 1.
Measure the temperature distribution on the surface of the heat spreader using
thermocouples.
2. 2.
Capture the temperature distribution with an infrared camera.
3. 3.
Measure the thermal conductivity of the heat spreader using thermocouples for
accurate steady state FEM simulation.
### II-A Design of Thermocouples
A thermocouple consists of two wires made of different metals/alloys. At the
point of contact with the target material an EMF(voltage differential) is
generated between the two wires because of their differing thermal properties.
This difference in voltage is typically proportional to the temperature of the
target, and can be detected with a simple electronic circuit. Due to their
simplicity and accuracy, they are commonly used to perform accurate
temperature measurements. In our experiments we use 300 $\mu$m T-type
thermocouples made of copper and constantan wires. They operate best in a
temperature range between $\pm 200^{\circ}$C. To eliminate effects of
corrosion and ageing, we used brand new wires. Secondly, to ensure good
contact between the wires, we heat the tip of both wires such that they fuse
together to make a strong junction.
We connected the thermocouples to an Expert EX9018P data acquisition module.
This is a sophisticated analog to digital converter that converts the
thermocouple voltages to digital signals. It can process upto 8 thermocouple
inputs. It has an internal multiplexer that chooses one of them. The final
output is in the RS 485 serial bus format. We subsequently use an Advantech
ADAM 4520 converter to convert the RS 485 signals to RS 232 signals that can
directly be fed to the serial port of a standard PC. The ADAM 4520 chip also
helps to isolate the PC from ground loops and destructive voltage spikes. We
calibrated the thermocouples with distilled boiling water and ice. New Delhi
is 216 meters above sea level. The boiling point at this altitude is 99.304∘C
for a typical atmospheric pressure of 987.56 millibars.
Figure 4: Experimental setup | Figure 5: Position of thermocouples on the spreader | Figure 6: Two attached thermocouples
---|---|---
### II-B Experiment I - Thermocouple based Measurement
In this experiment we place thermocouples at different ends of the integrated
heat spreader. We mostly follow the reference procedure as described in the
Intel Thermal Design Guidelines Document [7] (Appendix D). However, instead of
applying the Kapton adhesive, we apply Halnziye HY 610 thermal paste to
achieve the dual purpose of making the thermocouples stick to the spreader and
provide good thermal conductivity for accurate measurement. This thermal paste
has mild adhesive properties. Figure 6 shows our setup. Secondly, it was not
necessary to drill holes through the heat sink since we do not connect the
thermocouples at the center. We attach them to the middle of the four sides as
shown in Figure 6. We define four positions on the spreader – TCT, TCL, TCB,
and TCR. We allow the setup to reach steady state by having a gap of at least
10 minutes between different measurements. To further minimize the error, it
is necessary to repeat each experiment by interchanging the thermocouples.
This cancels out all sources of linear error. Each such experiment set is
repeated 10 times. We report the mean values. Because of mechanical
constraints, we were not able to attach more than two thermocouples at the
same time (see Figure 6).
We also report the CPU power as measured by the Windows CPUID utility. This is
the temperature at the center of the die [7]. As a benchmark, we use a simple
script that repeatedly performs calculations using the standard Windows
calculator application.
| Left-Bottom (TCB and TCL) | Right-Top(TCR and TCT) | Top-Bottom(TCT and TCB) | Left-Right(TCL and TCR)
---|---|---|---|---
Operation | CPU | TCB | (TCB | CPU | TCT | (TCT | CPU | TCT | (TCT | CPU | TCR | (TCR
| (∘C) | (∘C) | -TCL)(∘C) | (∘C) | (∘C) | -TCR)(∘C) | (∘C) | (∘C) | -TCB)(∘C) | (∘C) | (∘C) | -TCL)(∘C)
Power Off | | 23.90 | 0.15 | | 20.90 | 0.50 | | 24.75 | 0.25 | | 25.80 | 0.40
10 mins later | 57.5 | 39.85 | 4.05 | 52.0 | 35.95 | 3.10 | 52.5 | 39.65 | 7 | 43.5 | 33.50 | 1.35
10 mins after | 89.0 | 58.50 | 6.80 | 90.0 | 55.80 | 6.35 | 88.0 | 62.55 | 14.35 | 74.5 | 56.05 | -1.15
calculator on | | | | | | | | | | | |
10 mins after | 58.5 | 41.25 | 4.20 | 51.5 | 37.85 | 4.45 | 53.5 | 41.40 | 9.85 | 43.5 | 33.45 | 2.00
calculator off | | | | | | | | | | | |
TABLE I: Temperatures of Points on the Surface of the Heat Spreader
Table I shows the collected data at four time instants – power off, 10 minutes
later, 10 minutes after starting the benchmark, and 10 minutes after shutting
it down. We report four sets of readings.
The die temperature varies from 51∘C to 89∘C . The spreader temperature at the
hottest point (TCT) varies from 21∘C to 62.55∘C . As a sanity check we
correlate the temperature values with the layout of the Pentium 4 processor
[8]. TCT is close to the scheduler and trace-cache. In comparison TCB is the
coolest because it abuts the L2 cache. TCL and TCR are closest to the
fetch/decode logic, and floating point units respectively. They show a
moderate amount of activity for our benchmark.
The main take-away point in this experiment is that a large temperature
variation exists across the surface of the spreader. For example, the
difference between TCT and TCB reaches 14.35∘C . The values at TCT and TCL
differ by about 7∘C , and both TCR and TCL are warmer than TCB by about 6∘C .
This experiment gives us an indication of the degree of the temperature
gradients on the surface of the spreader. However, to get a more exact
picture, we need to do a more intrusive experiment.
### II-C Experiment II - IR Camera based Measurement
To get accurate and extensive temperature profiles, we decided to use an IR
camera that can produce a detailed temperature profile of the surface of the
spreader. A similar approach has been used by Martinez et. al. [9] to capture
the temperature profile of a die. The authors in this paper collect their data
by removing the spreader and heat sink. They use an IR transparent oil based
heat sink instead. Note that it is necessary to use some heat removal
mechanism. Otherwise, the temperature of the die will increase to unacceptable
levels, and the processor will shut itself down. We are planning to create
such kind of a setup in the future. However, we observe that in such a setup
we will not get an accurate picture of the temperature dissipation of a die
and the thermal profile of the spreader because the nature of heat transfer is
different. The latest version of the popular thermal modeling tool HotSpot 5
takes this into cognizance. Additionally, secondary heat transfer paths,
especially through the ball grid arrays, become important in this case.
We unsuccessfully try another approach. Our intuition was to remove the heat
sink during regular operation and quickly take an IR photograph of the die.
There will be an intermittent delay of less than a few seconds. However, we
hoped to possibly compensate for the error by trying to back calculate the
original temperature profile using standard results for radiative and
convective heat transfer. We use a Testo 875, 9 Hz IR camera for this purpose.
We initially underestimate the temperature of the spreader greatly. This is
possibly because of the low emissivity of the heat spreader. Consequently, to
increase the emissivity of the spreader, we coat it with Halnziye HY 610
thermal paste that has an emissivity of 0.95. The temperature values increase
by about 15∘C . The average temperature difference is about 10∘C across the
die. However, our original aim of getting a detailed temperature profile was
still not served because the thermal image was heavily dependent on the
uniformity of the thermal paste. As shown in Figure 7 some of the
hottest(darkest) regions are towards TCB (L2 cache). This is not expected to
be the case.
Figure 7: IR Photograph Figure 8: Measurement of thermal conductance | Figure 9: The full measurement setup
---|---
### II-D Experiment III - Measuring Thermal Conductivity
We observe that Experiment II was inconclusive. It only reaffirmed the fact
that a temperature differential exists across the spreader. To get a better
picture, we are in the process of setting up a detailed FEM simulation
framework that will be seeded by parameters obtained from our experiments. We
describe a method to calculate the thermal conductivity of the spreader
material (nickel coated copper plate). The thermal conductivity of a material
is defined as the power that flows across a temperature gradient of 1∘C in an
object that has unit length and unit cross-sectional area. Conceptually, it is
similar to electrical conductivity, and can be used to find the steady state
distribution of temperature.
We use the comparative method. This method proposes to place the unknown
sample (sliver of the spreader material) between two samples (copper wires)
with known thermal conductivity. Both the ends of this ensemble are set to
constant temperatures by dipping them in ice and boiling water respectively.
The thermal conductivities are related by the following equation.
$\frac{\kappa_{c}\Delta T_{w1}A_{w1}}{L_{w1}}=\frac{\kappa_{sp}\Delta
T_{sp}A_{sp}}{L_{sp}}=\frac{\kappa_{c}\Delta T_{w2}A_{w2}}{L_{w2}}$ (1)
Here, $\kappa_{c}$ is the thermal conductivity of copper (400 W/m-K), and
$\kappa_{sp}$ is the unknown thermal conductivity of the spreader. $w_{1}$
refers to copper wire 1, $w_{2}$ refers to copper wire 2, $sp$ refers to the
spreader sample, $A$ represents the cross-sectional area, and $L$ represents
the length of the wire. The temperature gradient, $\Delta T$, is measured
using thermocouples. The intuition behind this equation is that there is a
constant amount of heat flow in the assembly of wires. Note that Equation 1 is
over-constrained. We create two sets of equations – (1) between the spreader
and wire 1, and (2) between the spreader and wire 2. We solve them separately
and report the mean value of thermal conductivity.
Figure 9 shows the measurement setup with the two copper wires, spreader
sample, and six thermocouples. Each adjacent pair of thermocouples measures
the temperature difference across a homogeneous section of material. Note that
Equation 1 can be used only when there is exclusively conductive heat transfer
through the ensemble. We need to reduce convective and radiative heat transfer
to the maximum extent possible. Consequently, we covered the setup with
thermally insulating glass wool. Lastly, we set the reference temperature at
both ends using boiling water and ice respectively. The entire setup is shown
in Figure 9. We allowed upto 4-5 hours for the readings to stabilize. The
experimental procedure (10 repetitions, thermocouple interchange) is the same
as that mentioned in Section II-B.
We obtain a thermal conductivity value of 369 W/m-K ($\pm 0.5$∘C ). For
reference, the thermal conductivity of copper is 400 W/m-K, and the thermal
conductivity of nickel is 90.9 W/m-K.
## III Conclusion and Future Work
In the course of this work, we get an in-vivo estimate of the nature of
temperature gradients on the surface of the heat spreader. We would like to
extend our work to make more detailed and elaborate measurements. In specific,
we would like to drill very fine holes (diameter less than 100 nm) on the heat
sink and measure the temperatures at the center of the spreader also. Lastly,
using empirical data collected from our studies, we wish to correlate our
measurements with FEM based simulations. The final goal is four fold – (1)
Create a corpus of empirically measured temperature data, (2) Propose accurate
temperature simulation methodologies for semi-conductor packages, and (3)
Design new packaging technologies that are more thermally efficient, and
lastly (4) Come up with new architectures that can leverage these advances in
novel packaging technologies.
## References
* [1] S. R. Sarangi, B. Greskamp, A. Tiwari, and J. Torrellas, “Eval: Utilizing processors with variation-induced timing errors,” in _MICRO_ , 2008, pp. 423–434.
* [2] J. Srinivasan and S. Adve, “The importance of heat-sink modeling for dtm and a correction to predictive dtm for multimedia applications,” in _Proceedings of the Fourth Annual Workshop on Duplicating, Deconstructing, and Debunking (WDDD) at ISCA-05_ , 2005.
* [3] K. Skadron, M. R. Stan, W. Huang, S. Velusamy, K. Sankaranarayanan, and D. Tarjan, “Temperature-aware microarchitecture,” in _ISCA_ , 2003, pp. 2–13.
* [4] V. Heriz, J.-H. Park, T. Kemper, S.-M. Kang, and A. Shakouri, “Method of images for the fast calculation of temperature distributions in packaged vlsi chips,” in _Thermal Investigation of ICs and Systems, 2007. THERMINIC 2007\. 13th International Workshop on_ , sept. 2007, pp. 18 –25.
* [5] A. Ziabari, E. Ardestani, J. Renau, and A. Shakouri, “Fast thermal simulators for architecture level integrated circuit design,” in _SemiTherm_ , 2011\.
* [6] J. Srinivasan, S. V. Adve, P. Bose, and J. A. Rivers, “The case for lifetime reliability-aware microprocessors,” in _ISCA_ , 2004, pp. 276–287.
* [7] “Intel pentium 4 processor on 90nm process in the 775-land lga package: Thermal and mechanical design guidelines,” Intel, Tech. Rep. 302553-004, Nov 2005\.
* [8] H. de Vries, “Looking at Intel Prescott die, Part II,” http://chip-architect.com/news/2003_04_20_Looking_at_Intels_Prescott_part2.%html, accessed on September 17th, 2012.
* [9] F. J. Mesa-Martinez, J. Nayfach-Battilana, and J. Renau, “Power model validation through thermal measurements,” in _Proceedings of the 34th annual international symposium on Computer architecture_ , ser. ISCA ’07. New York, NY, USA: ACM, 2007, pp. 302–311.
|
arxiv-papers
| 2014-02-27T13:45:16 |
2024-09-04T02:49:59.022128
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Seema Sethia, Shouri Chatterjee, Sunil Kale, Amit Gupta, Smruti R.\n Sarangi",
"submitter": "Smruti Ranjan Sarangi",
"url": "https://arxiv.org/abs/1402.6903"
}
|
1402.7024
|
# The VST Photometric H$\alpha$ Survey of the Southern Galactic Plane and
Bulge (VPHAS+)
J. E. Drew1, E. Gonzalez-Solares2, R. Greimel3, M. J. Irwin2, A. Kupcu
Yoldas2, J. Lewis2, G. Barentsen1, J. Eislöffel4, H. J. Farnhill1, W. E.
Martin1, J. R. Walsh5, N. A. Walton2, M. Mohr-Smith1, R. Raddi6, S. E. Sale7,
N. J. Wright1, P. Groot8, M. J. Barlow9, R. L. M. Corradi10, J. J. Drake11, J.
Fabregat12, D. J. Frew13, B. T. Gänsicke6, C. Knigge14, A. Mampaso10, R. A. H.
Morris15, T. Naylor16, Q. A. Parker13, S. Phillipps14, C. Ruhland1, D.
Steeghs6, Y.C. Unruh17, J. S. Vink18, R. Wesson19, A. A. Zijlstra20
1School of Physics, Astronomy & Mathematics, University of Hertfordshire,
College Lane, Hatfield, Hertfordshire, AL10 9AB, U.K.
2Institute of Astronomy, Cambridge University, Madingley Road, Cambridge, CB3
OHA, U.K.
3IGAM, Institute of Physics, University of Graz, Universitätsplatz 5, Graz,
Austria
4Thüringer Landessternwarte, Sternwarte 5, 07778, Tautenburg, Germany
5ESO Headquarters, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany
6Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4
7AL, U.K¿
7Rudolf Peierls Centre for Theoretical Physics, Keble Road, Oxford, OX1 3NP
8Afdeling Sterrenkunde, Radboud Universiteit Nijmegen, Faculteit NWI, Postbus
9010, 6500 GL Nijmegen, The Netherlands
9University College London, Department of Physics & Astronomy, Gower Street,
London WC1E 6BT, U.K.
10 Instituto de Astrofisica de Canarias, 38200 La Laguna, Tenerife, Spain
11Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, U.S.A.
12Observatorio Astrónomico, Universidad de Valencia, Catedrático José Beltrán
2, 46980 Paterna, Spain
13Department of Physics & Astronomy, Macquarie University, NSW 2109, Australia
14School of Physics & Astronomy, University of Southampton, Southampton, SO17
1BJ, U.K.
15School of Physics, Bristol University, Tyndall Avenue, Bristol, BS8 1TL,
U.K.
16School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, U.K.
17Department of Physics, Blackett Laboratory, Imperial College London, Prince
Consort Road, London, SW7 2AZ, U.K.
18Armagh Observatory, College Hill, Armagh, Northern Ireland, BT61 9DG, U.K.
19European Southern Observatory, Alonso de Córdova 3107, Casilla 19001,
Santiago, Chile
20Jodrell Bank Centre for Astrophysics, School of Physics & Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
###### Abstract
The VST Photometric H$\alpha$ Survey of the Southern Galactic Plane and Bulge
(VPHAS$+$) is surveying the southern Milky Way in $u,g,r,i$ and H$\alpha$ at
$\sim$1 arcsec angular resolution. Its footprint spans the Galactic latitude
range $-5^{\rm o}<b<+5^{\rm o}$ at all longitudes south of the celestial
equator. Extensions around the Galactic Centre to Galactic latitudes $\pm
10^{\circ}$ bring in much of the Galactic Bulge. This ESO public survey, begun
on 28th December 2011, reaches down to $\sim$20th magnitude (10$\sigma$) and
will provide single-epoch digital optical photometry for $\sim$300 million
stars. The observing strategy and data pipelining is described, and an
appraisal of the segmented narrowband $H\alpha$ filter in use is presented.
Using model atmospheres and library spectra, we compute main-sequence $(u-g)$,
$(g-r)$, $(r-i)$ and $(r-H\alpha)$ stellar colours in the Vega system. We
report on a preliminary validation of the photometry using test data obtained
from two pointings overlapping the Sloan Digital Sky Survey. An example of the
$(u-g,g-r)$ and $(r-H\alpha,r-i)$ diagrams for a full VPHAS+ survey field is
given. Attention is drawn to the opportunities for studies of compact nebulae
and nebular morphologies that arise from the image quality being achieved. The
value of the $u$ band as the means to identify planetary-nebula central stars
is demonstrated by the discovery of the central star of NGC 2899 in survey
data. Thanks to its excellent imaging performance, the VST/OmegaCam
combination used by this survey is a perfect vehicle for automated searches
for reddened early-type stars, and will allow the discovery and analysis of
compact binaries, white dwarfs and transient sources.
###### keywords:
surveys – stars: emission line – Galaxy: stellar content
## 1 Introduction
The $H\alpha$ emission line is well-known as a tracer of diffuse ionized
nebulae and as a marker of pre- or post-main sequence status among spatially-
unresolved stellar sources. Since these objects – both nebulae and stars –
represent relatively short-lived phases of evolution, they amount to a
minority population in a mature galaxy like our own. Their relative scarcity
has in the past stood in the way of developing and testing models for these
crucial evolutionary stages.
In the southern hemisphere, the search for planetary nebulae (PNe) has been
served well by H$\alpha$ imaging surveys carried out by the UK Schmidt
Telescope (Parker et al 2005, 2006 and other more recent works). Nevertheless,
VPHAS+ will have a decisive impact on studies of complex or smaller nebulae of
all types, ranging from optically-detectable ultra-compact and compact HII
regions, to nebulae around YSOs (including associated jets and HH objects),
through PNe, to extended emission from D-type symbiotic stars and supernova
remnants. The superb spatial resolution, dynamic range, and likely photometric
accuracy of the VPHAS+ images warrant a step forward in our knowledge of the
population and detailed characteristics of these object classes.
For southern point sources with emission the situation is very different:
there has been little updating of the available catalogues since the work of
Stephenson & Sanduleak (1971) that was limited to a depth of 12th magnitude.
The major groups of emission line stars that remain as challenges to our
understanding include all types of massive star (O stars, supergiants,
luminous blue variables, Wolf-Rayet stars, various types of Be star), post-AGB
stars, pre-main-sequence stars at all masses, active stars and compact
interacting binaries. Within the disc of the Milky Way, the available samples
of these objects are typically modest and heterogeneous. Fixing this deficit
via a uniform search of the Galactic Plane for these rare object classes
motivated the photometric H$\alpha$ survey of the southern Galactic Plane,
first proposed for the VLT Survey Telescope (VST) in 2004. This paper
describes the realisation of this ESO public survey, now known as the VST
Photometric H$\alpha$ Survey of the Southern Galactic Plane and Bulge
(VPHAS+).
When first proposed, VPHAS (without the plus sign) was envisaged as the
counterpart to the INT/WFC Photometric H$\alpha$ Survey of the Northern
Galactic Plane (IPHAS, Drew et al 2005), that had begun in August 2003. IPHAS
is a digital imaging survey made up of Sloan $r$, $i$ and narrowband $H\alpha$
exposures, reaching to $\sim$20th magnitude, that takes in all Galactic
longitudes north of the celestial equator in the latitude range
$-5^{\circ}<b<+5^{\circ}$. This is all but complete with the new release of a
catalogue of $\sim$200 million unique objects drawn from 93 percent of the
survey’s footprint (Barentsen et al, 2014). During the initial VST public
survey review process, it was agreed that VPHAS should broaden in scope to
also incorporate the Sloan $u$ and $g$ bands (proposed for a separate survey),
that are particularly useful in picking out OB stars, white dwarfs and other
blue-excess objects. With this upgrade to 5 bands, the renamed VPHAS$+$ became
an all-purpose digital optical survey of the southern Galactic Plane, capable
of delivering data at a spatial resolution of $\sim$1 arcsec or better. As
well as fulfilling the role of southern counterpart of IPHAS, VPHAS$+$ is also
the counterpart to UVEX, the UV-Excess Survey of the Northern Galactic Plane
(Groot et al, 2009) that, at the time of writing, continues on the Isaac
Newton Telescope in La Palma.
The final augmentation of the VPHAS$+$ survey footprint came in 2010 on
expanding its footprint to match that of the similarly high spatial-resolution
near-infrared survey, VISTA Variables in the Via Lactea (VVV, Minniti et al
2011). The survey footprint now includes the Galactic Bulge to a latitude of
$|b|<10^{\circ}$, across the longitude range $-10^{\circ}<\ell<+10^{\circ}$.
The VVV $z$, $Y$, $J$, $H$ and $K_{s}$ survey of much of the Bulge and inner
Galactic disc is already complete.
VPHAS$+$ is poised to become the homogeneous digital optical imaging survey of
the Galactic Plane and Bulge, at $\sim$1 arcsec angular resolution, that will
provide a uniform database of stellar spectral energy distributions, from
which a range of colour-magnitude and colour-colour diagrams of well-
established utility can be derived. Once calibrated to the expected precision
of 2 to 3 percent, like its northern counterparts, VPHAS$+$ will be
quantitatively far superior to the photographic surveys of the last century,
and will offer significant added value in the form of calibrated narrowband
H$\alpha$ data. Some of the science enabled is illustrated by the studies that
the northern surveys, IPHAS and UVEX, have already stimulated. These have
included a number of works reporting the discovery of emission line stars,
ranging from young objects (e.g. Valdivieso et al, 2009, Vink et al 2008,
Barentsen et al 2011, Raddi et al 2013) to evolved object classes such as
symbiotic and cataclysmic binaries and compact planetary nebulae (e.g. Corradi
et al 2010, Witham et al 2006, Viironen et al 2009 and Wesson et al 2008). The
diagnostic power to be expected from the blue $u$ and $g$ bands has been
appraised in a series of papers presenting UVEX and early follow-up
spectroscopy (Groot et al 2009, Verbeek et al 2012a and 2012b).
Figure 1: The VPHAS$+$ survey footprint plotted in Galactic coordinates. All
2269 fields are shown in outline. Different colours, as specified in the key,
are used to identify the observations obtained for each field by 1st January
2014 – essentially 2 years after the start of data-taking.
Any comprehensive survey of the Galactic Plane clearly targets the main mass
component of our own Galaxy, made up of stars, gas and dust. When a spatial
resolution of 1 arcsec is combined with wide area coverage spanning 100s of
square degrees, as in IPHAS, UVEX and VPHAS$+$, it becomes possible to exploit
the stellar photometry adaptively to solve for the distribution of both the
stars and dust making up the optically-accessible Galactic disk - by means
that are similar to those already attempted, based on the near-infrared 2MASS
survey (Drimmel & Spergel 2001, Marshall et al 2006). Methods to achieve
3-dimensional mapping of this kind, now incorporating the direct sensitivity
of the H$\alpha$ narrowband to stellar intrinsic colour, are starting to take
shape (Sale et al 2009; Sale 2012). This development coincides with the
approach of the operations phase of Europe’s next major astrometric mission,
Gaia. Indeed, the rich dynamical picture that Gaia will build of the Milky Way
over the next decade will be very effectively complemented by stellar energy
distributions measured for millions of stars from the current generation of
ground-based optical, and near-infrared, wide-field surveys. VPHAS$+$, with
its haul of photometry in 5 optical bands on $\sim$300 million objects, is set
to take its place as one of them.
This paper presents the main features of VPHAS$+$, including a description of
its execution, the data processing and the nature of the photometric colour
information it provides. We begin in the next section with a presentation of
the observing strategy and the data reduction techniques in use. We then turn
to a description and evaluation of the narrowband H$\alpha$ filter procured
for this survey, in section 3. Following this, in section 4, we present
tailored synthetic photometry of main sequence and giant stars that provides
insights into the photometric diagrams that may be generated from survey data.
An exercise in photometric validation is described in Section 5 in which Sloan
Digital Sky Survey (SDSS) data are compared with VST observations. The scene
is then set for an example of VPHAS+ photometry extracted across the entire
square-degree footprint of a single survey field (Section 6). In Sections 7
and 8, we outline the applications of VPHAS$+$ to spatially-resolved nebular
astrophysics and in the time domain. The paper ends in Section 9 with a
summary, examples of early survey exploitation, and a forward look to the
first major data release.
## 2 Survey observations and data processing
### 2.1 VPHAS$+$ specification
The footprint of the survey is shown in figure 1. The OmegaCAM imager (Kuijken
2011) on the VST provides a field size of a full square degree, captured on a
4 x 8 CCD mosaic. After allowing for some modest overlap between adjacent
fields, we arrived at a set of 2269 field centres that will cover the desired
Galactic latitude band $-5^{\circ}<b<+5^{\circ}$ at all southern-hemisphere
Galactic longitudes, as well as incorporate the Galactic Bulge extensions to
$-10^{\circ}<b<+10^{\circ}$ near the Galactic Centre. The survey footprint
extends across the celestial equator by a degree or two to achieve an overlap
with the northern hemisphere surveys IPHAS and UVEX of $\sim$100 sq.deg.
altogether. This is to create the opportunity for some direct photometric
cross-calibration.
The target depth of the survey is to reach to at least $\sim$20th magnitude,
at 10$\sigma$, in each of the Sloan $u$, $g$, $r$ and $i$ broadband filters
and narrowband H$\alpha$. The bright limit consistent with this goal is
typically 12–13th magnitude. Presently, all VPHAS+ photometric magnitudes are
expressed in the Vega system. The original concept was to collect the data in
all 5 bands contemporaneously, in order to build a uniform library of snapshot
photometric spectral energy distributions for 200 million or more stars.
Practical constraints have modified this to the extent that the blue filters
($u$, $g$) are observed separately from the reddest ($i$ and $H\alpha$), with
the $r$ band serving as a linking reference that is observed both with $u$,$g$
and $i$,$H\alpha$. The aim is also to keep the spatial resolution close to 1
arcsecond. OmegaCAM and the Paranal site are well suited to this in that the
camera pixel size is 0.21 arcsec, projected on sky, and the median seeing
achieved is better than 1 arcsec (on occasion falling to as little as 0.6
arcsec).
As a means to obtaining better quality control, and to ensure that only a
minimal fraction of the survey footprint is missed due to the pattern of gaps
between the CCDs in the camera mosaic, every field is imaged at 2 or 3 offset
pointings. This strategy has been carried over from IPHAS and UVEX in the
northern hemisphere, and has the consequence that the majority of imaged
objects will be detected twice some minutes apart. In the $r$ band, there will
be two arbitrarily-separated epochs of data, with typically two detections at
each of them (i.e. 4 altogether).
### 2.2 VPHAS$+$ observations
The VST is a service-observing facility, with all programmes queued for
execution as and when the ambient conditions meet programme requirements.
VPHAS+ survey field acquisition began on 28th December 2011. Normally the
constraint set includes a seeing upper bound of 1.2 arcsec: this is only set
at a lower, more stringent value for fields expected to present a particularly
high density of sources (e.g. in the southern Bulge). In order that the seeing
achieved in the $u$ band is not greatly different from that in $i$ at the
opposite end of the optical range, it is advantageous to separate acquisition
of blue data from red – hence a split between ’blue’ ($u$,$g$,$r$) and ’red’
($H\alpha$,$r$,$i$) observing blocks has been implemented. This split also
permits the use of different moon distance and phase constraints, such that
blue data are obtained when the moon is less than half full at an angular
separation of not less than 60 degrees, while the limits for red data are set
at 0.7 moon illumination and a minimum angle of 50 degrees. Avoiding bright-
moon conditions is important in order to limit the amount of moonlight mixed
in with diffuse H$\alpha$ emission in the reduced images. No requirement has
been placed on the time elapsing between acquisition of blue and red data.
However, the more forgiving constraints on the acquisition of the latter has
meant that these are typically executed sooner than the former, with the
result that many more fields have red data already than have blue (see fig 1).
In all cases, the final constraint is that the sky is required to be clear, if
not necessarily fully photometric.
An impressive feature of the camera, OmegaCAM, is its potential to deliver
remarkably undistorted point-source images all the way across the 1-degree
field of view. To realise this, it is critical that the VST has an actively-
controlled primary. The operational price for this, at the present time, is
that image analysis and correction has to be carried out at every filter
change or after longer slews. The overhead added by this is about 3 minutes.
To reduce the impact of this, observations of sets of 3 neighbouring fields
are scheduled together, so that image analysis need only take place every
15-30 minutes – not much more often than would be essential, in any case, to
compensate for the telescope’s tracking movement. As a result,
’contemporaneous’ in the context of VPHAS+ data-taking means that all 3 blue,
or red, filters are typically exposed within 40-50 minutes of each other (cf.
IPHAS, where the more compact camera allows much faster operation, bringing
this elapsed time down to under 10 minutes). However, the time difference
between the blue and red observing blocks for a given field, i.e. the
$u$/$g$/$r$ data collection and $H\alpha$/$r$/$i$ data collection, can be
anything from a few hours to more than a year.
Figure 2: The transmission profiles of the Sloan $u$, $g$, $r$ and $i$, and
narrowband $H\alpha$ filters used in all VPHAS$+$ observations. The $r$ and
$H\alpha$ profiles are shown as a dashed line and in red respectively just to
clearly distinguish them from each other and the $i$ band. Each profile has
been multiplied by the CCD response function and a model of atmospheric
throughput (Patat et al 2011). The very-nearly grey losses due to the
telescope optics (a further scaling of approximately 0.6) have not been folded
in. Figure 3: An illustration of the VPHAS$+$ offset pattern as it applies to
the segmented H$\alpha$ filter with extra vignetting due to the T bars
separating the 4 segments. The first pointing is to lower right, as drawn – a
conservative estimate of the exposed unvignetted area is shown in black. The
exposed/unvignetted areas of the second and third pointings are shown in green
and blue respectively. The vertical and horzontal scales are numbered in
pixels (RA increasing to the right, declination increasing upwards).
We provide a reminder of the passbands of the Sloan broadband filters in use,
along with that of the narrowband H$\alpha$, in fig 2. They are shown scaled
by a typical CCD response function and a model of the atmospheric transmission
(Patat et al 2011). The exposure times used for the different filters, the
number of exposures in each and the median seeing achieved, up to December
2013, are set out in Table 1. From early on, during the commissioning phase of
the telescope, it became clear that tracking is usually good enough that even
the 150 sec $u$ exposures, our longest, do not have to be guided in most
circumstances. Indeed, experience is showing that it is safer to rely on the
tracking, rather than the autoguider, to maintain good image quality in the
most dense star fields.
Table 1: Observations obtained per survey field. The median seeing quoted is derived from data in hand by December 2013. Filter | Exposure | No. of | median seeing
---|---|---|---
| time (secs) | offsets | (arcsec)
Blue observation blocks
$u$ | 150 | 2 | 1.01
$g$ | 40111Up to 19th February 2013, $g$ exposure times were 30 sec. | 3 | 0.88
$r$ | 25 | 2 | 0.80
Red observation blocks
$H\alpha$ | 120 | 3 | 0.84
$r$ | 25 | 2 | 0.82
$i$ | 25 | 2 | 0.77
The pattern of offsets used for each field is illustrated in fig 3. The shifts
are relatively large, with the outer pointings differing by $-$588 arcsec in
the RA direction and $+$660 arcsec in declination. The choices made have
largely been driven by the characteristics of the narrowband H$\alpha$ filter
(discussed below in section 3), but they also convey the advantage of greatly
increasing the overlaps between neighbouring fields. Just the two outermost
pointings are used when exposing the $u$, $r$ and $i$ filters. This leaves
0.4% of the survey footprint unexposed. This changes to complete coverage on
including the third intermediate offset, as is the policy for the $H\alpha$
and $g$ filters.
In accordance with ESO’s standard procedures, data are evaluated soon after
collection by Paranal staff and graded before transfer to the archive in
Garching and to the Cambridge Astronomy Survey Unit (CASU) in Cambridge. If
the applied constraints are significantly violated, the observation block is
returned to the queue.
### 2.3 Data pipeline
#### 2.3.1 Initial Processing
Figure 4: Flowchart identifying the main VST data processing steps. At the
present time all steps up to the reduced single-band catalogue are undertaken
at CASU. Band merging is performed at the University of Hertfordshire.
From February 29th 2012 raw VST data have been routinely transferred from
Paranal to Garching over the Internet. For each observation the imaging data
are stored in a Multi-Extension FITS file (MEF) with a primary header
describing the overall characteristics of the observation (pointing, filter,
exposure time, etc.) and thirty two image extensions, corresponding to each of
the CCD detectors, with further detector-level information in the secondary
headers. The 32-bit integer raw data files are Rice-compressed at source using
lossless compression (e.g. Sabbey 1998). The files are then checked and
ingested into the ESO raw data archive in Garching. As soon as the data for
any given night become available they are automatically transferred to the
Cambridge Astronomy Survey Unit (CASU) for further checks and subsequent
processing. The VST web pages at CASU provide an external interface for both
monitoring processing status (http://casu.ast.cam.ac.uk/vst/data-processing/)
and overall survey progress and access (http://casu.ast.cam.ac.uk/vstsp/).
The processing sequence is similar to that used for the IPHAS survey of the
northern Galactic Plane (e.g. Gonzalez-Solares et al 2008), while the higher
level control software is based on that developed for the VISTA Data Flow
System (VDFS, Irwin et al 2004). Here we briefly outline the processing steps
illustrated in figure 4, emphasising the main differences relative to the
current VDFS standard. A more detailed description of the VST processing
pipeline is currently in preparation (Yoldas et al 2014).
Science images are first debiassed. Full two-dimensional bias removal is
necessary due to amplifier glow during readout being present in some
detectors. The master bias frames are updated daily from calibration files
taken as part of the operational cycle. The OmegaCAM detectors are linear to
better than 1% over their usable dynamic range removing the need for a
linearity correction. Hence this stage in the pipeline processing (figure 4),
although part of the pipeline architecture, is currently bypassed.
Flatfield images in each band are constructed by combining a series of
twilight sky flats obtained in bright sky conditions. The timescale to obtain
sequences of these for all deployed filters is typically one to two weeks. So
as to adequately trace the variations in the pattern and level of scattered
light in these flats, the master flats derived from them are updated on a
monthly cycle (how these are corrected for scattered light is described in
Section 2.3.3). Four of the detectors, in extensions 29–32, suffer from inter-
detector cross-talk, whereby saturated bright stars in one detector can cause
noticeable positive or negative low level ($\approx$0.1%) ghost objects in
adjacent detectors. The impact of these is minimised in the pipeline by
applying a pre-tabulated cross-talk correction matrix to each of the affected
images.
The flatfield sequences plus bad pixel masks are used to generate the
confidence (weight) maps (e.g. Irwin et al 2004) used later during catalogue
generation and any subsequent image stacking or large area mosaicing. After
flatfielding, science images generally have well behaved sky backgrounds which
makes subsequent image processing straightforward. Where direct scattered
light is present in them, it is an additive phenomenon that is dealt with
automatically during object catalogue generation.
Figure 5: The left-hand panel shows an example of the deduced scattered light
component present in June 2013 $r$-band data. The right-hand panel shows the
corrected outcome. These maps were constructed using 286318 APASS object
matches over the square-degree field. Before correction scattered light
gradients amounting to a 20 to 25 percent variation from corners to centre are
present. This flattens to around $\pm$2 percent.
The redder passbands used in VST observations, in this case the $i$-band, show
fringing patterns at $\approx$2% of the sky background level. Defringing is
done using a standard CASU procedure (Irwin & Lewis 2001). The fringe frames
used are derived from other VST public survey data taken as close as possible
in time at higher Galactic latitude. This approach works because the fringe
pattern induced by sky emission lines at Paranal is quite stable over long
periods. The fringe frames are automatically scaled and subtracted from each
science image reducing the residual fringing level to well below the sky
noise.
Catalogue generation is based on IMCORE222Software publicly available from
http://casu.ast.cam.ac.uk (Irwin, 1985) and makes direct use of the confidence
maps, derived from the flat fields, to suitably weight down unreliable parts
of the images. This step includes object detection, parameterization and
morphological classification, together with generation of a range of quality
control information. Because of the extensive presence of diffuse emission
throughout the southern Galactic Plane, particularly in H$\alpha$, a version
of each affected image is cleaned of nebulosity using the NEBULISER333Software
publicly available from http://casu.ast.cam.ac.uk, see also Irwin (2010) for
the purpose of catalogue generation only. This achieves a more careful removal
of background and ultimately leads to more complete and, on average, more
faithful object detection than in the absence of this step.
With object catalogues available for every VPHAS+ survey image, it is then
possible to improve the rough World Coordinate System (WCS) based on the
telescope pointing and general system characteristics. The WCS is
progressively refined using matches between detected objects and the 2MASS
catalogue (Skrutskie et al 2006). Despite the large field of view, the VST
focal plane is almost free of distortion, and a standard tangent plane
projection yields residual systematics of $\sim$25mas over the entire field.
#### 2.3.2 Photometric Calibration
Provisional photometric calibration is based on a series of standard star
fields observed each night (e.g. Landolt 1992). For each night a zeropoint and
error estimate using the observations of all the standard fields in each
filter is derived. The flatfielding stage nominally places all detectors on a
common internal gain system implying, in principle, that a single zeropoint
suffices to characterise the whole focal surface. Colour equations are used to
transform between the passbands in use on the VST and the Johnson-Cousins
system of the published standard-star photometry. The calibration is currently
in a VST system that uses the SED of Vega as the zero-colour, almost zero-
magnitude, reference object.
The $u$ band data are the most challenging to calibrate. As this part of
Vega’s spectrum, and also the average standard star plus the detector reponse,
are falling rapidly it would be surprising if there were no offsets in $u$ due
to nonlinearities in the required colour transforms and, perhaps, to
degenerate colour transforms for hotter stars. Early experience of working
with $u$ data do indeed suggest that offsets of up to a few tenths of a
magnitude are sometimes present (see Sections 5 and 6).
The colour transforms currently in use to define the VPHAS internal system are
given below.
$\displaystyle u_{VST}$ $\displaystyle=$ $\displaystyle U+0.035\,(U-B)$
$\displaystyle g_{VST}$ $\displaystyle=$ $\displaystyle B-0.405\,(B-V)$
$\displaystyle r_{VST}$ $\displaystyle=$ $\displaystyle R+0.255\,(V-R)$
$\displaystyle i_{VST}$ $\displaystyle=$ $\displaystyle I+0.215\,(R-I)$
$\displaystyle H\alpha_{\,VST}$ $\displaystyle=$ $\displaystyle
R+0.025\,(V-R)$
The transform for the narrowband H$\alpha$ is an approximate initial solution
needed for the subsequent illumination correction stage. At catalogue
bandmerging this is superceded (see Section 4).
Figure 6: Photometric errors in VPHAS$+$ data as a function of magnitude. All
data are drawn from an area of $\sim$0.2 sq.deg in field 1679, positioned
$\sim$20 arcmin E of Westerlund 2. The left hand panel refers to blue
observing block (OB) data, while the right refers to red OB data. The coloured
histogram in each component plot shows the mean absolute deviation of the
magnitude difference, $(m_{1}-m_{2})$, per 0.25-magnitude bin, while the bin
means of the suitably-corrected pipeline estimate of the random error on this
difference are in black. The 10- and 5-$\sigma$ magnitude limits are specified
in the brackets next to each filter name. The rising observed mean deviation
seen in $g$ at the bright end is due to the onset of saturation.
#### 2.3.3 Illumination Correction
The main difficulty in deriving an accurate photometric calibration over the
one degree field arises from the multiplicative systematics caused by
scattered light in the flatfields. The VST (at least up to the introduction of
baffles early in 2014) has proved particularly susceptible to variable
scattered light. Its impact has varied from month to month depending on
conditions prevalent at the time the flat-field sequences were taken. An
illustration of the amount and character of the master-flat correction
required is provided in figure 5.
The scattered light is made up of multiple components with different
symmetries and scales. These range from $\approx$10 arcsec with x-y
rectangular symmetry, e.g. due to scattering off masking strips above the CCD
readout edges, to large fractions of the field due to radial concentration of
light in the optics and to non-astronomical scattered light entering obliquely
in flatfield frames. After some experimentation, and external verification, we
found that the APASS all-sky photometric g,r,i catalogues
(http://www.aavso.org/apass) provide a reliable working solution to the
illumination correction problem inherent in VST data (see fig 5). These
catalogues also provide an independent overall photometric calibration tied to
the SDSS AB magnitude system and will be used in future updates to define an
alternative finer-grained temporal AB magnitude zeropoint.
All filters used are treated in the same manner with colour equations set up
to define transformations between the APASS g,r,i SDSS-like calibration and
the VPHAS+ u,g,r,i,Hα internal system.
Illumination corrections are re-derived for each filter once a month.
Application of these corrections via the master flats reduces the residual
systematics across the entire field to below the 1% level for the broadband
filters and to within 2% for the segmented narrowband H$\alpha$, except in
vignetted regions (see Section 3).
#### 2.3.4 Quality Control
In addition to the usual VDFS quality-control monitoring of average stellar
seeing and ellipticity, sky surface brightness and noise properties, we have
also initiated a more detailed analysis of the image properties based on
inter-detector comparisons.
The well-aligned coplanar detector array coupled with the curved focal surface
is extremely sensitive to imperfections in focus which are relatively easy to
detect using the detector-level average seeing measurement variation available
for each of the 32 detectors. Likewise the variation in average stellar
ellipticity from each detector over the field is used to monitor rotator angle
tracking problems.
All of this information can be used in addition to the observation block (OB)
grades provided by ESO and is incorporated within all data product files and
also the progress database.
### 2.4 Limiting magnitudes and errors
The present convention for VPHAS+ and this paper is that all magnitudes are
expressed in the Vega system, which imposes zero intrinisc colour for A0
stars. The 5-sigma limiting magnitudes commonly achieved per exposure range
from 20.5–21.0 for H$\alpha$ up to 22.2–22.7 for Sloan $g$. The 10$\sigma$
limits are about 1 magnitude brighter.
Every source flux or magnitude determined via the pipeline has a formal error
associated with it. We provide an example of how these compare with empirical
magnitude differences, by extracting a sample of stars from a 0.25 sq.deg
catalogue, cut out from the survey field including Westerlund 2 (field 1679,
see also sec 6) in order to examine the pattern of errors (fig 6). The sky
area chosen is offset from the cluster to the east by $\sim$20 arcmin and
exhibits moderate diffuse ionised nebulosity. In the southern Plane, the
presence of some nebulosity, particularly affecting $r$ and $H\alpha$
exposures, is more the rule than the exception. Sources classified as probable
stars in both $g$ and $r$ (blue filter set, left panel) and $r$ and $i$ (red
filter set, right hand panel) in two consecutive offset exposures have been
selected. The selection also required that each extracted magnitude was
unaffected by vignetting and bad pixels (confidence level $>95$). This step is
particularly important for H$\alpha$ given the extra vignetting introduced by
the cross bars of the segmented filter (see Section 3).
The faintest stars that might have been included in the plots for $i$ and blue
$r$ (top row in fig 6) are absent because of a requirement that every included
source should also be picked up in, respectively, red $r$ and $g$. Fainter
objects than the apparent limits certainly exist in these bands. This feature
follows directly from the typically red colours of Galactic Plane stars at
magnitudes fainter than $\sim$13 that are the target of this survey. For the
same reason, it is not uncommon for the $u$ band source counts to be one or
more orders of magnitude lower than those of the $i$ band. The role of the $u$
band is to pick out the unusual rather than to characterise the routine.
To bring out the systematic effects present, the specific comparison made in
fig 6 is between the bin means of the absolute magnitude differences,
$|m_{1}-m_{2}-\delta|$, between the two exposures, and the expected random
error on the difference derived from the pipeline rms errors on the individual
magnitude measurements. The quantity, $\delta$, is the median magnitude
difference computed from all bright stars down to 18th magnitude ($r$, $i$ and
H$\alpha$), or 19th magnitude ($u$ and $g$). This was small in all cases – the
largest value being 0.011 for $u$. On the other hand, the correction applied
to the pipeline errors was, first, to multiply the single-measurement
magnitude error by $\sqrt{2}$ to give the rms error on the $(m_{1}-m_{2})$
difference, and then to multiply by $\sqrt{2/\pi}$ in order to convert the
measure of dispersion from rms to a mean deviation.
At magnitudes brighter than 18–19 in fig 6, the scatter in the empirical
results can be seen to be appreciably greater than that ’predicted’ for the
random component by the pipeline. The scale of the difference indicates that a
further error component of $\sim$0.01–0.02 magnitudes is present. The amount
and filter-dependence of these levels of error are entirely consistent with
the uncertainties estimated above for the flatfield and illumination
corrections: as noted in section 2.3.3, the VST is presently prone to quite
high and variable levels of scattered light. Practical remedies for this are
under consideration by ESO – when implemented these should tighten up the
error budget.
The enhanced mean magnitude difference seen for $g<13$ in fig 6 is typical of
what is seen as saturation effects begin to set in. For all the other bands,
in this example, saturation sets in at magnitudes a little brighter than 12th.
A safe working assumption across VPHAS+ would be that saturation is never
troublesome at magnitudes fainter than 13, but always an issue for magnitudes
brighter than 12.
Figure 6 identifies the 5$\sigma$ and 10$\sigma$ magnitude limits for each
filter achieved in this representative example. The seeing at the time of
these observations, as measured in the pipeline, ranged from 0.8 to 1 arcsec
(cf Table 1).
## 3 The narrowband $H\alpha$ filter
### 3.1 Overview
Table 2: Summary of filter segment properties Segment | sky quadrant | CCDs covered | centre, mean, corner CWL | mean integrated
---|---|---|---|---
| | | (Å) | throughput (Å)
A | SW | 1 – 8 | 6580.2 – 6585.4 – 6595.3 | 98.64
B | SE | 17 – 24 | 6596.1 – 6585.4 – 6578.9 | 103.38
C | NE | 25 – 32 | 6582.8 – 6591.9 – 6599.8 | 99.74
D | NW | 9 – 16 | 6581.7 – 6594.3 – 6603.8 | 99.49
Figure 7: The segmented H$\alpha$ filter, photographed in the lab soon after
receipt and just prior to measuring its transmission. The filters to either
side of the 2$\times$2 array of H$\alpha$ segments, transmit as $r$-band and
cover the guide CCDs.
A filter required to select a narrow band across a large 27$\times$27 cm2
image plane is a challenging fabrication problem. At the telescope, the filter
in use for VPHAS$+$ is known as NB-659. At the time it was commissioned in
2006, the purchase of a single-piece narrowband filter was offered only by one
supplier and was well beyond budget. This left the 4-segment option as the
achievable alternative.
The H$\alpha$ filter was constructed based on a specification supplied by the
OmegaCAM consortium, setting as goal a central wavelength of 6588 Å, and a
bandpass of 107 Å. It was delivered in the summer of 2009, and was shortly
thereafter tested at the University of Munich Observatory, using the optical
lab set up by the OmegaCAM consortium for filter testing. A photo of the
filter at that time is shown in figure 7. The transmission of each filter
segment was measured at 21 positions forming a coarse radial pattern (fig 8)
using a monochromator beam adjusted to emulate the f-ratio 5.5 VST/OmegaCAM
optical system. The logic of the chosen measurement pattern is to give a good
sampling of the dominantly radial variation ofthe transmission profile due to
the turntable rotation in the filter coating chamber. The diameter of the
monochromator beam used in the measurements was 4-5 mm. This is a more compact
beam than that of starlight at the telescope, which fills a spotsize of up to
12 mm on passing through the filter out of focus. Consequently, the actual
performance will be a somewhat areally-smoothed version of the performance
revealed by the lab measurements and their subsequent simulation. The filter
was shipped to Paranal and VST in the spring of 2011, after some final
selective remeasuring. These confirmed there had been no discernible bandpass
changes in store since delivery almost 2 years earlier.
Figure 8: A map of the positions within each filter segment at which the
transmission was measured. The colour codings are used again in later figures
to distinguish the central positions (black, crosses, sampling roughly 30% of
the segment area), intermediate radii (blue, encircled crosses, sampling
$\sim$half the area), and corners (red crosses, 15% of the area – of which
almost half is lost to vignetting). The dashed lines define the limits of the
strips 2 arcmin wide that experience any vignetting due to the filter T-bars.
They are drawn here as for segment D.
At the time the monochromator measurements were made, a segment naming scheme
was put in place (segments A, B, C and D) which is re-used here. Presently the
filter is housed in magazine B of OmegaCAM, which means that in terms of the
view of the sky, segment A spans the SW section of the image plane, B the SE,
while C and D span the NE and NW respectively. Table 2 identifies the mosaic
CCDs beneath each segment, and sets down the centre-to-corner range in central
wavelength (CWL) and the typical throughput integral. The laboratory tests
showed us that the CWL of segments A, C and D is shortest in the segment
centre, and drifts longwards according to a centro-symmetric pattern, as the
corners and sides are approached. For segment B, the centre-to-corner drift is
reversed, with the result that the corner CWLs are bluer than in the centre of
the glass. Segment B also has the highest mean FWHM, and highest average peak
transmission: integrated over the bandpass this is a difference in throughput
of 0.045 magnitudes relative to A, C and D. The pipeline-applied illumination
correction aims to eliminate this contrast. Area-weighted transmission
profiles for the 4 segments are shown in fig 9, along with the overall mean
profile. The latter is also given numerically in table 3.
Figure 9: The mean transmission profiles of the individual glass segments, A to D (cyan, blue, green and red respectively), making up the $H\alpha$ filter and the overall mean profile (black). Table 3: Mean transmission for NB-659 Wavelength | Transmission | Wavelength | Transmission
---|---|---|---
(Å) | | (Å) |
6456.3 | 0.000 | 6591.5 | 0.962
6461.3 | 0.001 | 6696.5 | 0.961
6465.8 | 0.001 | 6601.0 | 0.960
6470.8 | 0.002 | 6616.0 | 0.955
6475.9 | 0.002 | 6611.0 | 0.945
6480.9 | 0.003 | 6616.1 | 0.928
6485.9 | 0.005 | 6621.1 | 0.896
6490.9 | 0.008 | 6626.1 | 0.839
6496.0 | 0.012 | 6631.1 | 0.736
6501.0 | 0.020 | 6636.2 | 0.609
6506.0 | 0.033 | 6641.2 | 0.466
6511.1 | 0.053 | 6646.2 | 0.230
6516.1 | 0.086 | 6651.2 | 0.219
6521.1 | 0.136 | 6656.3 | 0.133
6526.1 | 0.208 | 6661.3 | 0.081
6531.2 | 0.307 | 6666.3 | 0.048
6536.2 | 0.429 | 6671.3 | 0.029
6541.2 | 0.575 | 6676.4 | 0.018
6546.3 | 0.700 | 6681.4 | 0.011
6551.3 | 0.799 | 6686.4 | 0.007
6556.3 | 0.868 | 6691.5 | 0.005
6561.4 | 0.915 | 6696.5 | 0.003
6566.4 | 0.936 | 6701.5 | 0.002
6571.4 | 0.947 | 6706.5 | 0.002
6576.4 | 0.954 | 6711.5 | 0.001
6581.5 | 0.959 | 6716.5 | 0.001
6586.5 | 0.961 | 6721.6 | 0.000
Compared to the H$\alpha$ filter used in the IPHAS survey, NB-659 has a CWL
that is redder on average by $\sim 20$ Å, it is around 10 percent wider, and
has a higher overall throughput leading to zeropoints $\sim$0.2 higher. The
known variations of bandpass across the 4 segments has implications for how
best to exploit VPHAS+ data. To anticipate these we have carried out two types
of simulation based on the lab measurements in order to identify them. We
describe these next, and summarise the implications in Section 3.4.
### 3.2 Simulation of the main stellar locus in the $(r-H\alpha,r-i)$ diagram
To gain an impression of the extent of the uniformity of performance with
regard to normal main sequence stars, ($r-H\alpha,r-i$) tracks were (1)
computed for each measured $H\alpha$ transmission profile using exactly the
same method as was followed by Drew et al (2005) for the analysis of IPHAS
data, (2) rescaled to a common integrated throughput, mimicking the effect of
the pipeline illumination correction, (3) compared to the mean pattern by
subtracting off the computed mean track. The result of this is shown as fig
10. The track differences picked out in red are from the segment corners
exhibiting the largest CWL shifts. It can be seen that the tracks follow the
same trend to within $\pm$0.02 up to about $r-i=1.2$ (corresponding to M3
spectral type), after which there is a clear fanning out. This shows that the
obtained $r-H\alpha$ excesses should fall within the target photometric
precision range of the survey for all except mid- to late-M stars.
The sensitivity of the M stars to variations in the narrowband transmission
profile is a point of note, while not actually a surprise. It arises from the
great breadth of the feature in M-star spectra created by the absorbing TiO
bands displaced to either side of the narrow H$\alpha$ bandpass, and the fact
that the resulting inter-band flux maximum falls at wavelengths shortward of
H$\alpha$. As these molecular bands strengthen with increasingly late spectral
type, the $r-H\alpha$ apparent excess grows along with the sensitivity to the
exact placement of the bandpass. Viewed in these terms, the $(r-H\alpha,r-i)$
colours of unreddened mid- to late-M dwarfs provide an empirical gauge of
filter bandpass uniformity and/or typical CWL. To minimise the bandpass
sensitivity and hence spread seen at late M, the CWL would need to be lowered
to around 6530 Å or less. In practice, later M dwarfs are sufficiently faint
that they normally appear in $(r-H\alpha,r-i)$ diagrams as a relatively sparse
distribution of scarcely reddened objects – falling within a thinly-populated,
continuation of the unreddened main sequence, redward of $(r-i)=1.0$, rising
from $r-H\alpha\sim 0.5$ up to $\sim 0.8$, (see figs 17, or 20). Reddened M
dwarfs are usually just too faint to be detected.
Figure 10: The r - H$\alpha$ deviations computed for all measured positions on
the H$\alpha$ filter, NB-659, after correcting the data for bandpass
integrated throughput variations. The data are colour-coded according to
position of measurement as in fig 8. The most discrepant corner positions, all
plotted in red, are located in segments C and D.
Selection of mid-to-late M dwarfs is therefore straightforward, but
quantitative interpretation of $(r-H\alpha)$ should be presumed more uncertain
than at earlier spectral types. Similar effects will be seen in the M-giant
spur located at lower $(r-H\alpha)$ in the $(r-H\alpha,r-i)$ diagram (see fig
16). However, as red giants will be picked up by VPHAS+ at large distances
through significant reddening, a precautionary check on the impact of non-zero
extinction on this fanning in colour has been made: tracks of the type
compared in fig 10 were recalculated for $A_{V}=6$ and no noticeable
additional effect was found (see also fig 17).
### 3.3 Simulation of the impact of source radial velocity on in-band
emission line fluxes
Simulations have also been performed to consider how the filter captures
emitted $H\alpha$ flux, as a function of location within the field of view and
source radial velocity. An ideal filter, centred on the mean rest wavelength
of the imaged $H\alpha$ emission and placed in a high f-ratio optical path,
would be insensitive to radial velocity shifts up to a limit proportional to
the FWHM of the bandpass. The desired capabilities of the VPHAS+ H$\alpha$
filter are separation between H$\alpha$ emission line objects and the main
stellar locus – and, better still, a regular mapping of measured $r-H\alpha$
excess onto emission equivalent width (cf Drew et al 2005, figure 6). The two
representative spectra used to investigate how these capabilities are affected
by changing source radial velocity are shown in fig 11.
Figure 11: Top panel: an example of a very bright, simple H$\alpha$ emission
profile (taken from Corradi et al 2010). The emission equivalent width is 220
Å, and the FWHM of the observed profile is close to 390 km s-1. The mean
radial velocity of the line is $+$35 km s-1. The difference between a pure
continuum magnitude and that including the line is somewhat in excess of 1.2.
Lower panel: the contrast of the line relative to continuum is much less here
(EW $\sim$ 20 Å), and the FWHM is somewhat wider at 570 km s-1 (a classical Be
star, taken from Raddi et al, 2013). The mean radial velocity is $-$50 km s-1.
The continuum-only magnitude is fainter by about 0.2 only, here.
Both spectra were blueshifted to $-$500 km s-1 , and then shifted redward in
steps of 100 km s-1 at a time, up to $+$500 km s-1 (altogether a displacement
of 22 Å) – calculating at each step the integral of the spectrum folded
through the filter transmission profile. The resultant in-band fluxes were
converted to magnitudes, and then shifted by the amount required to match the
integrated transmission to the overall mean for the filter (again mimicking
the function of the pipeline illumination correction). In real use, we would
expect the majority of emission line objects to present with FWHM no greater
than either of these examples (interacting binaries and WR stars do present
with much broader emission, however).
The radial velocity range explored was chosen with the following
considerations in mind:- Emission line stars in the thin disk will commonly
have radial velocities falling within the range -100 to +100 km/s. In the
Bulge larger radial velocities may be encountered: excursions to $\pm$200 km
s-1 are observed in CO (Dame, Hartmann & Thaddeus 2001) within $\sim
20^{\circ}$ of longitude of the Galactic Centre, and for a minority of inner-
Galaxy planetary nebulae, radial velocities have been obtained that extend the
range almost to $\pm$300 km s-1 (see Durand, Acker & Zilstra 1998, and
Beaulieu et al 2000).
Figure 12: Results of simulation of the in-band flux as a function of source
radial velocity for the EW = 220 Å emission spectrum (upper panel in fig 11).
The fluxes are expressed as magnitude offsets relative to the peak simulated
in-band flux. The panels representing the segments are arranged as they are
imposed on the plane of the sky, i.e. A covers the SW quadrant, D covers the
NW when the filter is stored in OmegaCAM’s magazine B. The curves are colour-
coded according to measurement position as in fig 8. Galactic sources will
usually fall well within the marked velocity range, $-$200 to $+$200 km s-1.
The more problematic red curves, representing the response of the segment
corners, account for $\sim$8% of each segment’s unvignetted area only. Figure
13: Results for an emission line net equivalent width of 20 Angstroms
(spectrum shown in lower panel in fig 11). Otherwise as fig 12. Figure 14: The
layout of the H$\alpha$ filter, showing the positions of the PN-test
measurements, colour coded according to flux relative to the corners of
segment B. Those shown as black crossed boxes have a scaled flux $\geq 0.95$.
Those in blue are a little lower with a scaled flux in the range 0.90 – 0.94.
The two positions marked in green have fluxes scaling to 0.88 and 0.89, while
the two in red scale to 0.84 and 0.77 (both in segment D). There is no
measurement for the centre of segment C because of a telescope pointing
imprecision. The other centre measurements were obtained $\sim 2$ arcmins away
from the true centres plotted in order to avoid CCD gaps. The dashed lines
mark the limits of the T-bar vignetting.
The results of this exercise are plotted in figs 12 and 13. Fig 12 shows that
segments A and B come very much closer to independence of radial velocity than
segments C and D, in terms of measured $H\alpha$ excess. In all segments,
reasonable fidelity (a flat, or nearly flat response) is achieved around
segment centre – although in B, uniquely, the corners happen to perform a
little better than the centre. Clearly segment D, where the transmission is
centred on longer wavelengths than in the other segments, could yield
measurements in its corners (perhaps 8% of its unvignetted area) of $H\alpha$
magnitude, or of $r-H\alpha$, that underestimate the flux of similarly high
equivalent-width H$\alpha$ emission by up to $\sim$ 0.3 (out of a true excess,
expressed in magnitudes of $\sim 1.2$). Segment C performs similarly, but the
potential flux drop associated with its corners is less pronounced.
We can compare the expectations created by these simulations with the results
of an on-sky experiment in which the planetary nebula (PN) ESO 178-5 (or PNG
327.1 -02.2) has been exposed in H$\alpha$ at a series of positions in the
image plane, placing it well into every corner and also close to the centre of
each of the four filter segments (see fig 14). The observed integrated counts
variation might be predicted to be somewhat stronger than in fig 12 given that
the in-band continuum flux from this PN will be relatively even weaker. But
this will be offset by the additional flux due, in particular, to [NII]
$\lambda$6584\. The PN chosen for this test was picked both because it is
well-calibrated (Dopita & Hua 1997), and because its LSR radial velocity is
quite large and negative ($v_{LSR}=-88.7$ km s-1: on 19th April 2013 when it
was observed, this will have shifted to $-105$ km s-1 at the telescope). It
also happens to possess [NII] $\lambda$6584 emission that is scarcely less
bright than H$\alpha$ (the former has 98% the flux of the latter: Dopita & Hua
also provide a spectrum of this nebula).
Background-subtracted aperture photometry of the PN and a moderately bright
star nearby, serving as a continuum reference, was carried out on the reduced
images. These measurements reveal a pattern of behaviour that essentially
tracks the results shown in figure 12: the continuum reference itself shows a
total count variation of $\pm 5$% across all pointings, while the PN counts,
after scaling to the reference, range from $+5$%, down to $-23$% relative to
the values for the corners of segment B. In the extreme case that all of the
H$\alpha$ and [NII] $\lambda$6548 emission had been shifted out of the
bandpass, the maximum drop for this PN would be $-57$% (the remaining 43%
being attributable to [NII] $\lambda$6584). The pattern across the filter of
the results emerging from this trial is shown in fig 14.
The radial-velocity dependence of the transmitted flux may accordingly become
an issue for objects with very strong H$\alpha$ emission where the aim is
accurate flux determination, unless attention is paid to where the object
falls in the image plane. Qualitatively the issue is less critical: regardless
of where the object is located, the changes in transmission are not so large
that there will be frequent failures to distinguish strong H$\alpha$ emitters
– i.e. they will still appear above the main stellar locus in the
$(r-H\alpha,r-i)$ diagram. In the example simulated, the outer reaches of
segment D would bring $(r-H\alpha)$ down to 0.9 – 1.0, a level that
nevertheless remains clear of the domain that might be occupied by unreddened,
non-emission very late-type M dwarfs ($(r-H\alpha)\sim 0.8)$, cf fig 16). As
further context, we note that nearly continuum-free emission line objects,
such as PNe and HII regions, present with $r-H\alpha\sim 3$.
Where the line emission itself contributes only a minority of the measured
narrowband $H\alpha$ flux the trends seen are much more subdued (fig 13). For
the example shown, only 20 percent of the total in-band flux is attributable
to the net line emission, rather than most of it as in fig 12. Again, the
corners of segments C and D perform least well, in under-representing the
emission flux by up to $\sim$0.05 magnitudes at the most negative likely
Galactic Plane radial velocities. Otherwise, the performance is predicted to
be within the anticipated 0.02–0.03 error budget of the survey.
### 3.4 Implications of the H$\alpha$ filter properties for VPHAS+ and its
exploitation
In summary, for most purposes the H$\alpha$ filter performs as required, and
has very good throughput. For the great majority of stars making up the main
stellar locus, there will be the desired fidelity of $(r-H\alpha)$ colour, and
the great majority of emission line objects will be detected with the same
facility as they are by IPHAS.
There are two caveats to note. First, in Section 3.2 it was shown that
variations in central wavelength across the filter segments will lead to
thickening of the loci traced by mid-to-late M stars. These same variations,
of what is a relatively red H$\alpha$ passband, also introduce the potential
for under-determination of H$\alpha$ fluxes for objects/nebulosity in parts of
the image plane for sources with significantly negative radial velocities
(Section 3.3). This becomes most serious for emission-line sources falling
near the vignetted corners of segments D and C, where a $20-30$ percent under-
counting in H$\alpha$ may occur for radial velocities approaching $-200$ km
s-1. As is always the case for narrowband H$\alpha$ filters, the common
presence of significant [NII] $\lambda\lambda$6548, 6584 emission bracketing
H$\alpha$ in planetary nebulae or HII regions complicates the expected
signature. However, it can be guaranteed in all but rare, exotic circumstances
that the stronger $\lambda$6584 component of the [NII] doublet falls well
within the bandpass.
The outstanding practical consequence of the filter’s transmission
characteristics for the survey strategy are that, for quantitative
reliability, measures of $(r-H\alpha)$ obtained using segments A and B, and
the central zones of C and D (out to $\sim 13$ arcmin) are to be favoured.
This appreciation is half of the reason for the adoption of offsets of several
arcminutes between the 3 successive pointings made in this filter (fig 3) for
each field – our strategy ensures that objects captured in segment corners in
the first pointing are fall close to segment centres in the third.
The rest of the motive is to mitigate the cross-shaped vignetting due to the
blackened T-bars holding the segments in place (fig 7). Each arm of the cross
casts a shadow entirely contained within a strip 4 arcmin wide. By choosing to
offset at least this much in both RA and Dec between the 3 exposures obtained
per field, we raise the probability of at least one high-confidence
$r-H\alpha$ colour measurement per detected source in the final catalogue to
very nearly 100 percent, and the probability of two to over 95 percent.
Finally, we remark that the combination of large offsets and three pointings
has the consequence that the fraction of sky within the survey footprint
missed altogether, due to dead areas between the CCDs and vignetting, is under
0.3 percent.
## 4 Simulation of VPHAS$+$ stellar colours
The five bandpasses of the survey provide the basis for the construction of a
range of magnitude-colour and colour-colour diagrams. To take full advantage
of them, knowledge is needed of the behaviours that can be expected of the
colours of normal stars.
We have simulated colours for solar-metallicity main sequence and giant stars
using the same method as employed by Sale et al. (2009). We adopt the
definition of these two sequences in $(T_{eff},\log g)$ space given by
Straizys & Kuriliene (1981). Then for each spectral type along these
sequences, solar-metallicity model spectra were drawn from the Munari et al.
(2005) library. At a binning of 1 Å the spectra in this library are well
enough sampled to permit the calculation of narrow-band $H\alpha$ relative
magnitudes with confidence, alongside the analogous broadband quantities. More
detail on the broadband filter transmission profiles, shown in Fig. 2, and on
the CCD response curve is provided on the ESO
website444http://www.eso.org/sci/facilities/paranal/instruments/omegacam/doc.
To ensure compliance with the Vega-based zero magnitude scale, we have defined
the synthetic colour arising from a flux distribution $F_{\lambda}$ as
follows:
$m_{1}-m_{2}=-2.5\log\left[\frac{\int T_{1}\lambda F_{\lambda}d\lambda}{\int
T_{1}\lambda F_{\lambda,V}d\lambda}\right]+2.5\log\left[\frac{\int
T_{2}\lambda F_{\lambda}d\lambda}{\int T_{2}\lambda
F_{\lambda,V}d\lambda}\right]$
where $T_{1}$ and $T_{2}$ are the numerical transmission profiles for filters
$1$ and $2$, after multiplying them through by the atmospheric transmission
(Patat et al 2011) and mean OmegaCAM CCD response curves. The SED adopted for
Vega, $F_{\lambda,V}$, is that due to Kurucz
(http://kurucz.harvard.edu/stars.html). Where needed for comparison, we have
also computed colours based on the Pickles (1998, hereafter P98)
spectrophometric stellar library (the approach adopted by Drew et al 2005 for
IPHAS). To maintain precision, the numerical quadrature resamples the more
smoothly varying transmission data onto the sampling interval of the stellar
SED.
The $(r-H\alpha)$ excess is evaluated in exactly the same way as the broadband
colours. Since Vega is an A0V star, its SED at H$\alpha$ incorporates a strong
absorption line feature that reduces the in-band flux below the pure continuum
value. Unlike the broadbands, the $H\alpha$ narrowband has not yet been
standardised and so there is not a formally recognised flux scale. However, we
can specify here that the integrated in-band energy flux for Vega, on adopting
the mean profile for the VST filter, is $1.84\times 10^{-7}$ ergs cm-2 s-1 (at
the top of the Earth’s atmosphere). To assure zero colour relative to the
optical broad bands, this flux is required to correspond to $m_{H\alpha}\simeq
0.03$. The reduction in zeropoint (zpt) that the computed in-band flux implies
relative to the flux captured by the much broader $r$ band – based on folding
Vega’s SED with lab measurements of the filter throughputs corrected for
atmosphere and detector quantum efficiency – is 3.01. Current practice in
VPHAS+ photometric calibration is accordingly to adopt zpt(NB-659) = zpt(r) -
3.01 magnitudes as the default calibration for the narrowband: in section 5
where a direct comparison is made with SDSS spectroscopy, this offset is found
to be satisfactory. When applied, it assures that data obtained in
photometric, or stable, conditions, yield zero $r-H\alpha$ colour for A0
stars.
Figure 15: The expected positions of main sequence and giant stars in the
$(u-g,g-r)$ plane. For the main sequence, tracks are shown for the
monochromatic reddenings $A_{0}=$ 0, 2, 4, 6 and 8 (working from left to
right). The red leak in the $u$ filter starts to lower $u-g$, noticeably from
$A_{0}=6$ (tracks drawn in red). The giant-star tracks, drawn as dashed lines
for $A_{0}=0$, 2 and 4 only, are very similar to their main-squence
counterparts except at the latest types. Figure 16: The expected position of
main sequence and giant stars in the $(r-H\alpha,r-i)$ plane. Tracks are shown
for the monochromatic reddenings $A_{0}=$ 0, 2, 4, 6, 8 and 10 (from left to
right). Solid lines represent the main sequence tracks, while dashed lines are
used for the giant tracks. The lines in red are giant-star tracks derived from
P98 spectrophotometry.
Both main-sequence and giant-star colours have been calculated for a range of
reddenings and optical-IR extinction laws as formulated by Fitzpatrick & Massa
(2007). The unreddened colours for the mean Galactic law ($R=3.1$) are laid
out here in table 4. The Appendix provides additional tables that specify the
colours of main sequence stars at selected reddenings and for two further
representative reddening laws ($R=2.5$ and 3.8). These can be used to
construct intrinsic-colour-specific reddening lines. For the large range in
extinction sampled along many Galactic Plane sightlines, these reddening
trends are slightly curved (see the examples shown in e.g. Sale et al 2009).
In this paper we use $A_{0}$, the monochromatic reddening at 5500 Å to
parameterise the amount of reddening, rather than the band-averaged measure,
$A_{V}$. In most circumstances these quantities are almost identical.
Table 4: VST/OmegaCAM synthetic colours for unreddened main-sequence dwarfs and giants. Sp.type | main sequence (V) | | giants (III)
---|---|---|---
| | | | | | (model) | (P98 library spectra)
| $u-g$ | $g-r$ | $r-i$ | $r-H\alpha$ | | $u-g$ | $g-r$ | $r-i$ | $r-H\alpha$ | $r-i$ | $r-H\alpha$
O6 | -1.494 | -0.313 | -0.145 | 0.071 | | | | | | |
O8 | -1.463 | -0.299 | -0.152 | 0.055 | | | | | | -0.158 | 0.074
O9 | -1.426 | -0.271 | -0.142 | 0.064 | | -1.426 | -0.271 | -0.142 | 0.064 | |
B0 | -1.404 | -0.267 | -0.143 | 0.058 | | -1.404 | -0.267 | -0.143 | 0.058 | |
B1 | -1.296 | -0.236 | -0.130 | 0.052 | | -1.316 | -0.234 | -0.130 | 0.057 | -0.095 | 0.071
B2 | -1.181 | -0.214 | -0.117 | 0.049 | | -1.209 | -0.211 | -0.116 | 0.056 | |
B3 | -1.025 | -0.182 | -0.098 | 0.048 | | -1.046 | -0.182 | -0.098 | 0.054 | -0.035 | 0.083
B5 | -0.799 | -0.133 | -0.071 | 0.043 | | -0.814 | -0.134 | -0.072 | 0.050 | -0.016 | 0.083
B6 | -0.699 | -0.116 | -0.062 | 0.040 | | -0.714 | -0.116 | -0.062 | 0.046 | |
B7 | -0.550 | -0.094 | -0.051 | 0.033 | | -0.568 | -0.095 | -0.051 | 0.041 | |
B8 | -0.361 | -0.071 | -0.039 | 0.022 | | -0.383 | -0.072 | -0.039 | 0.032 | |
B9 | -0.168 | -0.040 | -0.023 | 0.009 | | -0.186 | -0.044 | -0.024 | 0.021 | -0.018 | 0.035
A0 | -0.024 | 0.000 | -0.003 | -0.002 | | -0.030 | -0.009 | -0.006 | 0.011 | 0.012 | 0.034
A1 | 0.007 | 0.015 | 0.004 | -0.004 | | 0.007 | 0.004 | 0.000 | 0.008 | |
A2 | 0.039 | 0.038 | 0.014 | -0.005 | | 0.051 | 0.022 | 0.009 | 0.008 | |
A3 | 0.064 | 0.062 | 0.025 | -0.005 | | 0.085 | 0.045 | 0.019 | 0.007 | 0.037 | 0.063
A5 | 0.096 | 0.130 | 0.056 | 0.008 | | 0.143 | 0.107 | 0.048 | 0.015 | 0.096 | 0.087
A7 | 0.073 | 0.206 | 0.089 | 0.030 | | 0.145 | 0.179 | 0.078 | 0.032 | 0.115 | 0.094
F0 | 0.003 | 0.336 | 0.153 | 0.086 | | 0.091 | 0.317 | 0.144 | 0.084 | 0.156 | 0.102
F2 | -0.021 | 0.396 | 0.182 | 0.111 | | 0.064 | 0.380 | 0.174 | 0.109 | 0.204 | 0.172
F5 | -0.039 | 0.505 | 0.230 | 0.150 | | 0.046 | 0.491 | 0.224 | 0.148 | 0.238 | 0.164
F8 | -0.013 | 0.587 | 0.263 | 0.174 | | | | | | |
G0 | 0.012 | 0.628 | 0.278 | 0.185 | | | | | | 0.333 | 0.213
G2 | 0.011 | 0.628 | 0.279 | 0.185 | | 0.253 | 0.759 | 0.329 | 0.215 | |
G5 | 0.162 | 0.756 | 0.327 | 0.217 | | 0.405 | 0.870 | 0.368 | 0.235 | 0.396 | 0.250
G8 | 0.355 | 0.845 | 0.358 | 0.233 | | 0.531 | 0.944 | 0.395 | 0.247 | 0.419 | 0.247
K0 | 0.523 | 0.938 | 0.396 | 0.248 | | 0.640 | 1.002 | 0.417 | 0.256 | 0.446 | 0.254
K1 | 0.551 | 0.954 | 0.403 | 0.251 | | 0.803 | 1.085 | 0.451 | 0.269 | 0.468 | 0.269
K2 | 0.629 | 0.993 | 0.419 | 0.258 | | 0.963 | 1.159 | 0.483 | 0.281 | 0.508 | 0.293
K3 | 0.779 | 1.062 | 0.447 | 0.269 | | 1.227 | 1.276 | 0.536 | 0.299 | 0.514 | 0.286
K4 | 0.871 | 1.108 | 0.468 | 0.278 | | 1.374 | 1.342 | 0.585 | 0.320 | 0.592 | 0.313
K5 | 1.083 | 1.210 | 0.522 | 0.300 | | 1.578 | 1.420 | 0.630 | 0.338 | 0.714 | 0.337
K7 | 1.387 | 1.402 | 0.724 | 0.387 | | | | | | |
M0 | 1.372 | 1.411 | 0.789 | 0.411 | | 1.697 | 1.454 | 0.686 | 0.360 | 0.827 | 0.411
M1 | 1.335 | 1.439 | 0.934 | 0.467 | | 1.838 | 1.506 | 0.769 | 0.395 | 0.872 | 0.401
M2 | 1.262 | 1.442 | 1.112 | 0.522 | | 1.938 | 1.546 | 0.825 | 0.420 | 0.920 | 0.443
M3 | 1.236 | 1.447 | 1.179 | 0.545 | | 1.980 | 1.556 | 0.933 | 0.444 | 1.165 | 0.471
M4 | 1.248 | 1.457 | 1.168 | 0.543 | | 1.959 | 1.551 | 1.136 | 0.500 | 1.472 | 0.512
M5 | | | | | | 2.009 | 1.569 | 1.296 | 0.556 | 1.739 | 0.560
M6 | | | | | | 2.199 | 1.612 | 1.267 | 0.554 | |
Based on the data from these tables, the main sequence and giant tracks are as
shown in figures 15 and 16. These identify where the main stellar loci will
fall. It is important to note that the OmegaCAM $u$ filter, like all filters
constructed for this challenging band, exhibits a low-level red leak. In this
instance, lab measurements indicate transmission at levels between $10^{-5}$
and $10^{-4}$ within limited windows around $\sim$9000 Å. This is enough to
begin to noticeably, and erroneously, brighten the $u$ magnitudes of normal
stars reddened to $g-r>3$. Because of this, and because the measurement of
very low level leakage is itself subject to proportionately higher
uncertainty, we do not plot or tabulate $u-g$ data beyond $g-r=3$ limit. Very
few detected sources are so extreme. In practice, VPHAS+ $u-g$ is faithful for
extinctions up to $A_{0}\sim 6$, but gradually thereafter it transforms into a
colour that behaves crudely as $-(g-z)$.
The $(r-H\alpha,r-i)$ colour-colour diagram is not subject to such effects,
and therefore remains sound across a wider spread in visual extinctions.
Synthetic tracks are presented in figure 16 for $A_{0}=0$, 2, 4, 6, 8 and 10.
The main sequence tracks shown are similar to those appropriate to IPHAS (cf
figure 6 of Sale et al 2009). But a problem emerges when it comes to the
simulation of red giant colours. Purely theoretical simulation predicts late-K
and M giant colours closely resembling those of dwarfs, whereas simulation
using P98 library spectrophotometry indicates a distinctive flattening of the
M-giant track, peeling away from the steadily rising main sequence track.
Figure 16 points out this contrast. Inspection of table 4 reveals this is a
problem linked mainly to simulation of the $i$ spectral range, which renders
$(r-i)$ progressively larger from late K into the M giant range when the
library spectra are used in place of model atmospheres.
Figure 17: Equatorial VPHAS+ data (upper panel) and IPHAS data (lower panel)
compared to show the different appearance of M-giant $(r-H\alpha,r-i)$
colours. The photometry is extracted from a $\sim$0.2 sq.deg region of sky,
centred on $\ell=35.95^{\circ}$, $b=-3.13^{\circ}$. The magnitude range is
limited to $13<r<18$ in both cases. Telescope-appropriate giant tracks,
computed from the P98 spectrophotometric library, for $A_{0}=$ 2 and 4 are
superimposed in red in both panels. In the upper panel, giant tracks computed
from model atmospheres for the same reddenings are shown in blue. The black
dashed line in each panel is the synthetic early-A reddening line, from
$A_{0}=0$ to $A_{0}=10$. The stars appearing in the IPHAS catalogue as
candidate emission line stars are enclosed in cyan boxes in both panels.
Evidence that M giants are better reproduced by synthetic photometry based on
flux-calibrated spectra is provided by figure 17. This figure also compares
new VPHAS+ data with their crossmatches in the IPHAS survey within a $\sim$0.2
sq.deg equatorial field ($\ell=35.95^{\circ}$, $b=-3.13^{\circ}$), and shows
selected synthetic tracks superimposed. The photometry from the two surveys of
the most densely populated part of the main stellar locus to $(r-i)\simeq 1.5$
substantially overlap, but not perfectly – the response functions describing
the three bandpasses involved in fig 17 undoubtedly differ in detail between
the two telescopes. On cross-correlating either $(r-i)$ or $(r-H\alpha)$
between the two surveys, it becomes clear that the IPHAS colour has the
somewhat larger dynamic range. This is the reason for the slightly more
stretched appearance of both the main locus and the early-A reddening line in
the IPHAS diagram relative to that for VPHAS+.
At $(r-i)>1.5$ in fig 17, it can be seen that the M-giant spurs look very
different. First, the VPHAS+ M giants fall into a nearly flat distribution
lying at lower $(r-H\alpha)$, compared to the more steeply rising higher IPHAS
M-giant sequence. However, as long as the data are interpreted with reference
to telescope-appropriate synthetic photometry, the two datasets will lead to
the same inference. In the example shown in fig 17, the comparisons with
suitable synthetic giant tracks indicate that the maximal extinction in the
field can be no more than $A_{0}\simeq 4$. The extinction measures due to
Marshall et al (2006), based on 2MASS red-giant photometry, indicate a maximum
Galactic extinction of $A_{K}\sim 0.3$ for this pointing. For a typical
Galactic $R=3.1$ reddening law this scales up to $A_{0}\sim 3.3$ (roughly –
see Fitzpatrick & Massa 2009). If model-atmosphere giant tracks are referred
to instead, the M giants would have to be read as demanding visual extinctions
ranging from $\sim$4 upwards.
Fig 17 also demonstrates the broadening in $(r-H\alpha)$ of the VPHAS+ M-giant
sequence that was foretold in Section 3. The IPHAS counterpart is evidently
much sharper, as it rises to higher $r-H\alpha$ with increasing $r-i$. The
main practical impact of this difference is that IPHAS M-giant photometry is
the better starting point for picking apart chemistry differences (Wright et
al 2008). But it is as true of the VPHAS+ $(r-H\alpha,r-i)$ diagram as it is
of its IPHAS equivalent – that M giants at $r-i>1.5$ sit below and apart from
M dwarfs.
Fortuitously, fig 17 identifies an advantage of the generally good seeing
available at the VST. There are two candidate emission line objects apparent
in the IPHAS selection (enclosed in cyan boxes, in fig 17), that drop back
into the main stellar locus in the VPHAS+ data. Inspection of the images shows
that both stars are in close doubles of similar brightness, of under 2 arcsec
separation. Because they are a little better resolved in VPHAS+ ($\sim$0.8
arcsec seeing), than in IPHAS ($\sim$0.9 arcsec seeing), the pipeline makes a
better job of the assigning magnitudes in the different bands to the blend
components. This example nicely illustrates the most common reason for bogus
candidate emission line stars in either VPHAS+ or IPHAS - improperly
disentangled blends. Candidate emission line stars should always be checked
for this kind of problem before spectroscopic follow-up. Otherwise, experience
with IPHAS gives confidence that the selection of emission line objects via
VPHAS+ will be highly efficient (see e.g. Vink et al 2008, Raddi et al 2013).
Finally, it is worth noting that the bright limit of the survey at 12–13th
magnitude effectively excludes any unreddened stars of earlier spectral type
than $\sim$G0. Before more luminous stars of spectral type F and earlier can
enter the survey sensitivity range, they need to be at distances in excess of
1 kpc, typically, where low extinction becomes increasingly improbable. This
constraint bestows a significant selection benefit in that only unreddened or
lightly-reddened subluminous objects, with intrinsically blue colours are left
standing clear near the blue end of the main stellar locus in commonly-
constructed photometric diagrams. In this domain VPHAS+ has important
selection work to do.
## 5 Photometry validation: a comparison of SDSS and VST data
Before the start of survey field acquisition, we obtained observations in all
survey filters of two pointings that fall within the SDSS photometric and
spectroscopic coverage (Abazajian et al 2009). These were centred on RA
20:47:53.7 Dec -06:04:14.5 (J2000) and RA 21:04:25.94 Dec +00:59:15.8 (J2000)
– fields that happen to include a number of white dwarfs and cataclysmic
variables (not discussed further here). The main aim of the data was to verify
VST photometry both by comparison to SDSS photometry and to synthetic
photometry derived from SDSS spectra. The VST observations were obtained on
21/09/2011, during clear weather at a time of generally sub-arcsecond seeing.
The exposure times differ only a little from those now in general survey use:
the $g$ exposures were 30 sec, rather than 40, and $i$ was exposed for 20 sec
rather than 25 - the other times were as given in Section 2.1.
The photometry on the sources in these fields have been pipeline-extracted and
calibrated in the standard way, and have been cross-matched to their SDSS
counterparts. The number of cross matched stars used in this exercise ranges
from $\sim 2500$ ($u$) up to $\sim 10000$ ($r$). For a star to be included, it
must be: unvignetted; to have a star-like point spread function; to lie within
0.5 arcsec of its SDSS counterpart, and to fall within the magnitude range
$16>r_{VST}>19$. The SDSS selection constraints were set to exclude blended
and saturated sources, and sources close to detector edges. In addition, it
was required of every source that, in both surveys, the formal error on the
magnitude measurement is less than 0.03.
Figure 18: The distributions, by band pass, of the measured magnitude
differences between VST test data and SDSS, after correction of the latter
magnitudes from the AB to the Vega zero-magnitude scale. These were obtained
from two VST pointings, away from the Galactic Plane, that overlap the SDSS
footprint. The biggest differences are found in the $u$ band where the VST
data are fainter than the corrected SDSS values by $\sim 0.12$, in the median.
Figure 19: Measured magnitude shifts between VST and SDSS cross-matched
objects as a function of SDSS colour, for the second of the two fields
observed (only). The data in blue are $\Delta u$ versus $(u-g)_{SDSS}$; the
data in green and red are, respectively, $\Delta g$ and $\Delta r$ versus
$(g-r)_{SDSS}$; the data in black are $\Delta i$ versus $(r-i)_{SDSS}$. The
horizontal lines show where the four loci would be expected to lie in the case
that the SDSS and VST broadband filters were identical, and the calibrations
perfect. The data fall into loci that are not far displaced from these
horizontal lines and are almost as flat: this indicates that the colour-
dependent terms that would be needed in equations to transform between the
SDSS and VST systems are small.
In fig 18 we plot the histograms of the magnitude differences between the two
surveys, according to pass band – pooling the data from both pointings. If the
starting assumption is that the VST broadband filters are identical to the
SDSS set, the predicted magnitude for each star in each filter is the measured
SDSS magnitude, less the offset between the AB and Vega scales (essentially
the numerical difference between the magnitude of Vega in the AB system and
its value of 0.02 to 0.03, according to the alternative Vega-based convention
– see table 8 in Fukugita et al 1996). In the $g$, $r$ and $i$ bands, the
predicted and observed magnitudes are well-enough aligned, and the
interquartile spread is consistent with the way the data were selected for
random errors less than 0.03. However in $u$ there is a discrepancy that
exceeds expected error: the median difference deviates by 0.12 magnitudes and
the width of the distribution is twice that arising in the comparisons of the
other bands.
The fuller picture is presented in figure 19 which shows the broadband
magnitude differences as a function of the relevant SDSS colour for the second
of our two fields (only). In all 4 pass bands, including $u$, the colour
dependence can be seen to be very weak in that the loci traced out by the
plotted stars are – to a first approximation – flat. The discrepancy seen in
$u$ is revealed as mainly a zero-point shift, combined with scatter that
exceeds the formal errors. In highly reddened Galactic Plane fields, the
stellar colour effects may become more pronounced as extinction modifies the
effective sampling of the passbands.
Table 5: Mean magnitude offsets between VST photometry and cross-matching synthetic photometry derived from the SDSS database of spectra. The synthetic magnitude scale adopts magnitudes for Vega itself of 0.026 (see Bohlin & Gilliland 2004) Offset | Field 1 (117 stars) | Field 2 (50 stars)
---|---|---
$u_{VST}-u_{syn}$ | 0.07$\pm$0.39 | 0.11$\pm$0.17
$g_{VST}-g_{syn}$ | 0.06$\pm$0.09 | 0.06$\pm$0.04
$r_{VST}-r_{syn}$ | 0.01$\pm$0.06 | 0.03$\pm$0.03
$i_{VST}-i_{syn}$ | -0.06$\pm$0.08 | -0.05$\pm$0.03
As a separate exercise, we have used SDSS spectra to synthesise magnitudes and
colours for stars with cross-matching VST photometry. The spectral type range
present within this much smaller sample runs from B-type through to early
M-type (M1). At wavelengths below 3800 Å falling within the $u$ band, it was
necessary to extrapolate the spectra using appropriately chosen P98 library
data. The result of this comparison is agreement between the VST and
synthesised magnitudes at the $\sim$5 percent level (table 5), with the $u$
band as the outlier exhibiting much more pronounced scatter as well as
somewhat higher offset. This pattern echoes the behaviour apparent in the VST-
SDSS purely photometric comparison of fig 18, using a much larger sample. The
difficulty is not confined to VST $u$ however, in that SDSS $u$ photometry
fares scarcely any better relative to synthesis from the spectra (for the two
fields, offset and scatter are -0.09 $\pm$ 0.37, and -0.04 $\pm$ 0.16). As
more blue survey data are accumulated, it may become clear that the $u$
zeropoint will benefit from being tied to that of $g$ for those fields
observed in the best conditions, as is presently done for $H\alpha$ with
respect to $r$. This option is not yet enacted. For the time being, it must be
acknowledged that pipeline $u$ calibration is more approximate than those of
the other bands.
We have also used the reduced cross-match sample to look at how the VST
photometric $r-H\alpha$ colour compares with its counterpart synthesised from
spectra – looking, in particular, for any trends as a function of distance
from field centre. No such trend is apparent, thereby meeting the expectation
that the narrowband fluxes of normal stars, to early-M spectral type, would
not be affected by the pattern of bandpass shifts discussed earlier in section
3 (cf. fig 10). However we do find that in order to make this detailed
comparison, systematic offsets had to be removed from the VST photometry
first. These were 0.075 in $(r-i)$, in the sense that the VST colours were too
red by this amount, and a 0.02 reduction in $(r-H\alpha)$. The $(r-i)$ offset
is consistent with the broadband magnitude offsets listed in Table 5 and hence
is as expected. The $(r-H\alpha)$ adjustment is small enough (i.e. within the
fit error) that it supports the zeropoint shift of 3.01 magnitudes between the
$r$ and $H\alpha$ bands that was identified in section 4). Once these colour
offsets are applied to the VST data, the rms scatter of the photometric
$(r-H\alpha)$ colour relative to its synthetic counterpart is 0.04 for objects
brighter than $r=19$.
Figure 20: Left, the $(u-g,g-r)$ and right, the $(r-H\alpha,r-i)$ photometric
diagrams pertaining to VPHAS+ survey field 1679. Both diagrams are plotted as
two-dimensional stellar-density histograms, rainbow colour-coded such that
high source densities (80-90 per bin) are red and the lowest densities (one
per bin) are dark blue. The binning is 0.017$\times$0.025 in the left panel,
0.013$\times$0.008 in the right. The synthetic unreddened main sequence is
drawn in, in black, in both panels. The G0V and A3V reddening lines obtained
for $R=3.1$, drawn as black dashed lines, are included in respectively the
$(u-g,g-r)$ and $(r-H\alpha,r-i)$ diagrams as useful aids to interpretation.
## 6 An example of point-source photometry derived from a VPHAS+ field
The extracted point-source photometry from the VST square-degree field is one
of the two main data products from VPHAS+ – the other being the images
themselves, considered below in section 7. We present an example of the two
essential colour-colour diagrams in fig 20, in which band-merged stellar
photometry for field 1679 is compared with the primary diagnostic synthetic
tracks presented in section 4. This field includes the sky area from which
data were taken to construct fig 6 illustrating typical errors. The massive
open cluster, Westerlund 2, is located in the NE of these pointings, and the
field as a whole includes moderate levels of diffuse, complex Hii emission.
The data presented are drawn from a sky area centred on RA 10 24 49 Dec -57 58
00 (J2000) that spans 1.3 sq.deg – the total footprint occupied by the two
offset positions.
Only objects with stellar point-spread functions in $g$, $r$ and $i$, brighter
than $g=20$ are included in fig 20. Where two sets of magnitudes are
available, the mean values have been computed and used. A further requirement
imposed is that the random error in all bands may not exceed 0.1. The same
$\sim$37000 objects are included in both diagrams. In order to obtain the
diagrams shown, the pipeline photometric calibration was checked and refined
as follows: we
* •
cross-matched brighter stars to APASS $g$, $r$ and $i$ photometry
* •
computed the median magnitude offset (applying no colour corrections – it was
shown in fig 19 these are modest)
* •
corrected all $g$, $r$ and $i$ for these offsets;
* •
corrected $u$ by determining the vertical shift needed in the $(u-g,g-r)$
diagram to align the main stellar locus with the unreddened main sequence and
the G0V reddening line
* •
corrected the $H\alpha$ zeropoint and hence all $H\alpha$ magnitudes according
to the requirement that $zpt(H\alpha)=zpt(r)-3.01$.
This resulted in the following broadband corrections:- $\Delta i=-0.004$,
$\Delta r=-0.032$ (red filter set), $\Delta r=-0.033$ (blue filter set),
$\Delta g=0.069$ and $\Delta u=-0.31$. As expected, the correction that had to
be applied to the $u$ photometry was, by far, the largest.
The main stellar locus can be seen to be tightly concentrated in both the blue
and the red diagrams, and to favour lightly reddened G and K stars. The
superimposed synthetic reddening lines (G0V in the $(u-g,g-r)$ diagram, A3V in
$(r-H\alpha,r-i)$) have been drawn adopting the $R=3.1$ reddening law widely
regarded as the Galactic norm. The blue diagram provides examples of three
distinct typical populations falling outside the main stellar locus. Below it,
at $(u-g)>1.5$ and $(g-r)>1.5$ (roughly) the plotted objects will mainly be M
giants. Above the main stellar locus toward the red end, in the ranges
$0<(u-g)<0.5$ and $1.5<(g-r)<2.0$ lie the OB stars in and around Westerlund 2.
Finally, the modest scatter of blue objects lying above the G0V line roughly
in the $0\leq(g-r)\leq 1$ range will include intrinsically-blue lightly-
reddened subluminous objects.
It is interesting to note in the red diagram that there is some evidence that
early-A stars making up the lower edge of the main stellar locus would better
follow a different law, with $R\sim 3.8$ (see the tables in the Appendix).
Indeed a reddening law of this type has been inferred for the OB stars in
Westerlund 2 by Vargas Alvarez et al (2013). Most of the thin scatter of
points below the main stellar locus, and some of the scatter above, in this
same diagram will be the product of inaccurate background subtraction in
H$\alpha$. But many of the objects lying above the main stellar locus will
indeed be emission line objects, and some of the stars below will be white
dwarfs. As expected, the red spurs of M dwarfs and M giants are broader
features than their IPHAS counterparts (cf fig 17 and associated remarks).
For more discussion of these colour-colour diagrams, the reader is referred to
Groot et al (2009, UVEX) and Drew et al (2005, IPHAS).
Figure 21: Two planetary nebulae, NGC 2438 (top) and NGC 2899 (bottom), as
they appear in the SHS and VPHAS+ surveys. The SHS images are shown in the
left-hand panels, with the VPHAS+ images to the right. The bands used to form
them are: NGC 2438 – SHS R/G/B = $H\alpha$/SR/SSS Bj, VPHAS+ R/G/B =
$H\alpha$/$r$/$i$, NGC 2899 – SHS R/G/B = $H\alpha$/SR/SSS Bj, VPHAS+ R/G/B =
$H\alpha$/$r$/$g$. The cut-out image dimensions are 300$\times$300 arcsec2 for
NGC 2438, and 200$\times$180 arcsec2 for NGC 2899.
## 7 Nebular astrophysics with VPHAS+ images
Just over a decade ago the SuperCOSMOS H$\alpha$ Survey (SHS, Parker et al
2005) had only just completed. This was the last survey using photographic
emulsions that the UK Schmidt Telescope undertook. The 3-hour narrowband
H$\alpha$ filter exposures reach a very similar limiting surface brightness to
the 2 minute exposures VPHAS+ is built around. Hence, the differences in
capability are not about sensitivity, as this is roughly the same in the two
surveys. Instead it is about the great improvement in dynamic range on
switching to digital detectors, the good seeing of the VST’s Paranal site, and
the added broad bands.
SHS, with its enormous 5-degree diameter field, has been comprehensively
trawled for southern planetary nebulae (the MASH catalogue, Parker et al 2006,
Miszalski et al 2008). The remaining discovery space for resolved nebulae is
expected to be at low surface brightnesses in locations of high stellar
density, and in the compact domain around and below the limits of the typical
spatial resolution of SHS ($\sim 0.5$ to 3.0 arcsec). Both these conditions
will most often be met in the Galactic Bulge, at a mean distance of $\sim$8
kpc. Data-taking in the Bulge and its maximally-dense star fields is planned
to begin in mid 2014.
Among planetary nebulae (PNe), small angular size is due either to great
distance or to youth – the study of either compact category provides exciting
possibilities. As well as the Bulge, the less-studied outer parts of the
Galactic Plane should be searched. In this respect, IPHAS, with its direct
view to the Galactic Anticentre is better positioned: the ongoing study of the
Anticentre PN population has revealed dozens of new candidates (Viironen et
al. 2009a), including the PN with the largest galactocentric distance to date
(20.8 $\pm$ 3.8 kpc, Viironen et al 2011). By following up such finds to
measure chemical abundances, crucial beacons are obtained for the study of the
Galactic abundance gradient and its much disputed flattening towards the
largest galactocentric radii. VPHAS+ completed the access to the outer Plane
over the longitude range $215^{\circ}<\ell<270^{\circ}$.
Data from both IPHAS and VPHAS+ can make fundamental contributions to the
study of very young PNe – particularly by helping to solve the two-decades-old
puzzle of how PNe already emerge with the observed wide variety of
morphologies (round, elliptical, bipolar, multipolar, point-symmetric, etc. –
see Sahai et al. 2011). What does this variety say about the properties of
their AGB progenitors? Detailed studies of objects in the phases preceding the
PN phase – AGB and post-AGB stars, proto-PNe, and transition or PN-nascent
objects – are underway (e.g. Sanchez Contreras & Sahai 2012). Superb imaging
capabilities like those of the VST, accessed via VPHAS+, will support this
work.
Indeed there is a serious paucity of very small PNe in the existing optical
catalogues: there are no PNe with angular extent less than 3 arcsec in the
MASH catalogue (out of 903 objects; Parker et al 2006), and only 8 PNe in the
catalogue by Tylenda et al (2003, 312 objects) in the size range 1.4 – 3
arcsec. There is just one with a confidently-measured diameter below 1 arcsec
in the larger Strasbourg Catalogue of PNe (1143 objects; Acker et al. 1994),
that happens to be a Bulge PN. IPHAS has demonstrated that extremely young
compact PNe can be reached (Viironen et al, 2009b), while Sabin et al (in
prep.) have found some 20 new PNe with diameters of 1-3 arcsec in by-eye
searchs of IPHAS image mosaics. Even smaller, but brighter, nebulae around
symbiotic stars of the dusty D subtype are emerging – the record so far being
IPHASJ193943.36+262933.1, a new D symbiotic star with an $H\alpha$ extent of
only 0.12 arcsec that has been confirmed via HST imaging and recently studied
with the 10.4m GTC telescope (Rodriguez Flores et al. 2014, submitted to A&A).
Apart from opening up new discoveries, a further benefit of good seeing is the
clearer view of nebular structure that it offers. This is nicely demonstrated
in fig 21. SHS and VPHAS+ detect the main features of the planetary nebulae
NGC 2438 and NGC 2899 to very similar depth – for example, the fainter outer
halo is just detected in both versions of NGC 2438. But, evidently, the VPHAS+
images better resolve the fine sculpting within both nebulae as a consequence
of the seeing FWHM being under a half that prevailing in SHS data. The
extended dynamic range of VPHAS+ helps in this respect, too, in that early
saturation also obliterates detail. This advantage is especially clear in the
images of NGC 2899, where the structure in the bright nebulous lobes is
preserved in VPHAS+, but is entirely bleached out in SHS. The more the level
of detail that can be picked out, the more certain and subtle morphological
classifications and interpretations can become.
The combination of good seeing and high dynamic range also makes the
separation of fainter stars from background nebulosity much easier. This
capability is critically important to the study of the young massive clusters,
still swathed in diffuse Hii emission, where the analysis of stellar content
is very much a focus of continuing research. For example, Feigelson et al
(2013) have offered a critique of the nuisance created by spatially-complex
nebulosity. The obvious answer to this and the problem of dust obscuration is
to turn to selection using NIR and X-ray data. Nevertheless the availability
of imaging data of the high quality seen in VPHAS+ data will make it possible
to extend SEDs for many more stars into the effective-temperature (and
reddening) sensitive optical domain. In addition, understanding the shaping of
the interstellar medium in star-forming environments remains an important part
of the picture (see e.g. Wright et al 2012 on proplyd-like structures in Cyg
OB2). The detail that the VST is capable of revealing both in obscuration and
ionised hydrogen in star-forming regions can be quite exquisite. Here, in fig
22, we illustrate this with an excerpt from VPHAS+ data on the Lagoon Nebula,
showing the fine tracing of the shapes of dark globules and eroding dusty
structures that is achieved.
Figure 22: A cut-out at full resolution from M8, the Lagoon Nebula. This is an
RGB image centred on RA 18 09 36 Dec -24 01 51 (J2000), and spanning
150$\times$150 arcsec2. The filters are combined such that R/G/B =
$H\alpha$/$i$/$r$. Figure 23: The central star of NGC 2899 reveals itself. The
top panel is a 1$\times$1 arcmin2 thumbnail of the centre of NGC 2899 as
imaged through the $u$ filter, while the bottom is the corresponding $r$
thumbnail. The white right-angled bars pick out the position of an extremely
blue, relatively faint star that is clearly present in all $u$ (and $g$)
exposures obtained, but is too faint for detection in $r$.
In planetary and other evolved-star nebulae it is of course important to
identify the ionising object. The search for missing PN central stars is a
quest that VPHAS+ can aid greatly through the provision of spatially well-
resolved $u$ and $g$ data. Indeed inspection of the data used to contruct fig
21 has revealed the probable central star of NGC 2899 for the first time. As
shown in fig 23, there is very evidently a third very-blue star just SW of the
pair of stars that have, in the past, been scrutinised as possible companions
to what is required to be an extremely hot ($T_{eff}>250,000$ K), but probably
faint central star (López et al 1991). This blue object was detected on the
night of 20th December 2012 at a provisional $u$ magnitude of 18.79 $\pm
0.02$. It fades through $g$ (19.36 $\pm 0.02$) to become undetected by the
pipeline, and scarcely visible to eye inspection, in $r$. Its coordinates are
RA 09 27 02.72 Dec -56 -06 22.9 (J2000), just 1.7 arcsec from the more
southerly of the pair of brighter stars examined before by López et al (1991).
Based on the $g$ magnitude and an inferred $V$ flux, we have determined the
central star’s effective temperature, via the well-established Zanstra method.
Using the reddening and integrated H$\alpha$ flux from López et al and Frew,
Bojičić & Parker (2013) respectively, we estimate $T_{\rm z,H}=215\pm 16$ kK.
This is cooler than the temperature given by López et al, based on the
’crossover’ method, but still extraordinarily hot for a central star well down
the white-dwarf cooling track.
It was one of the major science drivers for the merged VPHAS+ survey that $u$
data, supported by $g$, would result in the detection of a broad range of
intrinsically very blue objects – be they PN central stars, interacting
binaries or massive OB and Wolf-Rayet stars. An extreme example like NGC
2899’s central star provides the useful lesson that selection via the
$u-g,g-r$ colour-colour diagram would have failed to pick it out – because of
the non-detection in $r$. In a case like this, the $u,u-g$ colour-magnitude
diagram has to be examined, in tandem with the appropriate images.
## 8 VPHAS+ photometry as a reference set for variability studies
As the northern survey, IPHAS, has progressed over the decade since 2003,
there have been occasions on which it was possible to use the growing database
as a high-quality reference for checking transient reports – particularly of
novae. The most spectacular IPHAS example of this was the nova and variable,
V458 Vul (Wesson et al 2008; Rodriguez-Gil et al 2010) where the eruption
occurred a few months after obtaining H$\alpha$ images revealing a pre-
existing ionised nebula around the star. Indeed there have been several
instances in which photometry of the progenitor object has been extracted from
the IPHAS database and has been used to gain insight into the prior presence
or absence of line emission or to set constraints on likely extinction
(Steeghs et al 2007, Greimel et al 2012). Such opportunities will certainly
arise with VPHAS+ – and be richer given the five filters offered.
In the southern hemisphere novae will be more frequent, as will other
transient events. Furthermore responses to alerts, or the need to demonstrate
long-term flux variations, can bring into use repeats of observations made
necessary by initial quality-control failures. An example of this is provided
by Vink et al (2008) who used repeat IPHAS observations – taken on account of
poor observing conditions – to discuss the LBV candidacy of G79.29$+$0.46.
With the increased attention being given to the reporting and exploitation of
transient objects (including the forthcoming Gaia alerts programme), this use
of VPHAS+ will become more common.
## 9 Summary and concluding remarks
This paper has introduced and defined VPHAS$+$, the VST Photometric $H\alpha$
Survey of the Southern Galactic Plane and Bulge. The data taking, the
rationale behind it, the data processing and data quality have all been
described. The properties and limitations of the survey’s narrowband H$\alpha$
filter, NB-659, have been laid out and simulated in order to anticipate its
performance. In addition we have provided tables of the expected photometric
colours of normal solar-metallicity stars to aid the interpretation of the
survey’s characteristic photometric diagrams – most are to be found in the
Appendix where the effect of changing the adopted reddening law is
illustrated. The VPHAS+ H$\alpha$ filter transmission is redder, wider and
$\sim 20$ % higher-throughput than its IPHAS counterpart – a difference that
feeds through to noticeably different $(r-H\alpha)$ colours for M stars.
We have validated the photometry that is delivered by VST/OmegaCAM and
subsequently pipelined at CASU, using test data taken of a field for which
SDSS photometry is available. We find the agreement is satisfactory, with the
$g$, $r$ and $i$ band calibrations differing by between 0.01 to 0.05
magnitudes. However, for the time being, the pipeline calibration should be
regarded as provisional – it will undoubtedly improve. Examples of the
excellent imaging performance of the VST/OmegaCAM combination relative to
previous surveys have been provided, and we draw attention to the valuable
archival role this first digital survey can fulfill in supporting discoveries
of transient sources.
Exploitation of the survey is now beginning. The detection of a compact
ionized nebula around W26, the extreme M supergiant in Westerlund 1 has
already been published (Wright et al 2014). Applications have been made for
follow-up spectroscopy that will test the quality of selection of specialised
object types that VPHAS+ photometry makes possible. Progress is also being
made via direct analyses of the photometry. For example, Mohr-Smith et al (in
prep.) are conducting a search for OB stars in the vicinity of the massive
cluster Westerlund 2, and they are finding a close match between the
properties of known cluster O stars as derived from VPHAS+ data and those
inferred by Vargas Alvarez et al (2013, see also Drew et al 2013). This and
other early appraisals of the data indicate that VPHAS+ will be an excellent
vehicle for automated searches for reddened early-type stars. Kalari et al (in
prep.) are employing both narrow-band H$\alpha$ and the broadbands to measure
mass-accretion rates in pre-main-sequence stars: they are finding that
H$\alpha$ mass-accretion rates in T Tauri stars compare favourably to rates
determined from the $u$ band in the case of the Lagoon Nebula, NGC 6530.
As the calibration of the survey data improves, the measurement of accurate
integrated H$\alpha$ fluxes for many faint southern PNe and other extended
objects becomes possible, and will extend the work of Frew et al (2013, 2014).
In due course these fluxes can be compared with existing and also new radio
continuum fluxes coming on stream (see e.g. Norris et al 2011) in order to
determine reliable extinction values for many faint nebulae currently lacking
data. This technique has already been applied to the case of W26 in Westerlund
1 (Wright et al 2014).
When it becomes possible to cross-match VVV and VPHAS$+$ data, it will open up
the power of homogeneous photometric mapping of the central parts of the
Galactic Plane in up to 10 photometric bands spanning both the optical and the
near-infrared. Beyond the VVV sky area, there is a synergy to be exploited in
bringing VPHAS+ data together with those of the all-sky 2MASS survey
(Skrutskie et al 2006) and with the UKIDSS Galactic Plane Survey (Lucas et al
2008), in those parts of the first and third Galactic quadrants the latter has
covered. It is worth noting, however, that 2MASS alone is too shallow to link
effectively with VPHAS+ for sightlines where the integrated visual extinction
is less than $\sim 5$ magnitudes. This does mean that the longitude range
$230^{\circ}<\ell<300^{\circ}$, in particular, is presently lacking
sufficiently deep NIR photometry. In the longer term, many of the sources of
interest that VPHAS+ finds will benefit from accurate parallaxes and other
data from ESA’s Gaia mission – given the similar sensitivity limits reached.
Conversely in the meantime, VPHAS+ has already begun to assist ambitious wide-
field spectroscopy programmes such as the Gaia-ESO Survey (Gilmore et al 2012)
through the provision of the wide-field photometry needed for target selection
and field setup.
By the end of 2013, 25% of all observations making up the survey had been
obtained to the required quality, and in May 2013 a first release of single-
band catalogues was made to the ESO archive that contained roughly 10% of the
eventual total (based on data obtained prior to 15 October 2012). By design,
the characteristics of VPHAS+ are similar to those of the IPHAS and UVEX
Galactic plane survey pair in the north. In particular, the double-pass
strategy is shared, with the result that the majority of detected objects are
picked up and measured twice, with no more than $\sim$0.1 percent of objects
missed altogether. This feature has informed the way in which the IPHAS DR2
catalogue (Barentsen et al, 2014) has been constructed – and it is intended
that a first band-merged VPHAS+ catalogue, for public release, will be built
along analogous lines during the second half of 2014. This will incorporate
data from the first 3 seasons of VST observing, and give a complete
photometric account of the Galactic mid-plane. For ease of use, for every
detected source, the catalogue will provide a single recommended set of
magnitudes in up to 5 optical bands.
## Acknowledgments
This paper makes use of public survey data (programme 177.D-3023) obtained via
queue observing at the European Southern Observatory. In respect of the
H$\alpha$ filter, we would very much like to thank Bernard Muschielok for the
benefit of his expertise and support in connection with its laboratory
testing, and Jean-Louis Lizon for his steady hand in correcting some minor
surface defects. The referee of this paper is thanked for constructive
comments that improved its content.
This research made use of the AAVSO Photometric All-Sky Survey (APASS), funded
by the Robert Martin Ayers Sciences Fund. Many elements of the data analysis
contained in this work have been eased greatly by the TOPCAT package created
and maintained by Mark Taylor (Taylor, 2005). The pipeline reduction also
makes significant use of data from the Two Micron All Sky Survey (2MASS),
which is a joint project of the University of Massachusetts and the Infrared
Processing and Analysis Center/California Institute of Technology, funded by
NASA and the NSF.
JED and GB acknowledge the support of a grant from the Science & Technology
Facilities Council of the UK (STFC, ref ST/J001335/1). The research leading to
these results has also benefitted from funding from the European Research
Council under the European Union’s Seventh Framework Programme (FP/2007-2013)
/ ERC Grant Agreement n. 320964 (WDTracer). BTG was also supported in part by
the UK STFC (ST/I001719/1). RLMC and AMR acknowledge funding from the Spanish
AYA2007-66804 and AYA2012-35330 grants. HJF and MM-S both acknowledge STFC
postgraduate studentships. NJW is in receipt of a Royal Astronomical Society
Fellowship. RW acknowledges funding from the Marie Curie Actions of the
European Commission (FP7-COFUND).
## References
* [Abazajian et al2009] Abazajian K. N., et al., 2009, ApJS, 182, 543
* [Acker et al1994] Acker A., Ochsenbein F., Stenholm B., et al. 1994, VizieR Online Data Catalog, 5084, 0
* [Barentsen et al2011] Barentsen G., et al., 2011, MNRAS, 415, 103
* [Barentsen et al2014] Barentsen G., et al., 2014, in preparation
* [Beaulieu et al2000] Beaulieu S. F., Freeman K. C., Kalnajs A. J., Saha P., Zhao H. 2000, AJ, 120, 855
* [Bohlin & Gilliland2004] Bohlin R. C., Gilliland R. L., 2004, AJ, 127, 3508
* [Corradi et al2010] Corradi R. L. M., et al., 2010, A&A, 509, 41
* [Dopita & Hua1997] Dopita M. A., Hua C. T., 1997, ApJS, 108, 515
* [Drew et al2005] Drew J. E., et al., 2005, MNRAS, 362, 753
* [Drew et al2013] Drew J. E., et al., 2013, The Messenger, 154, 41
* [Drimmel & Spergel2001] Drimmel R., Spergel D., 2001, 556, 181
* [Durand et al1998] Durand S., Acker A., Zijlstra A., 1998, A&AS, 132, 13
* [Feigelson et al2013] Feigelson, E. D., et al., 2013, ApJS, 209, 26
* [Finkbeiner et al2004] Finkbeiner D. P., et al., 2004, AJ, 128, 2577
* [Fitzpatrick & Massa2007] Fitzpatrick E. L., Massa D., 2007, ApJ, 663, 320
* [Fitzpatrick & Massa2009] Fitzpatrick E. L., Massa D., 2009, ApJ, 699, 1209
* [Frew et al2013] Frew D. J., Bojičić, I. S., Parker Q. A., 2013, MNRAS, 431, 2
* [Frew et al2014] Frew D. J., Bojičić, I. S., Parker Q. A., Pierce M. J., Gunawardhana M. L. P., Reid W. A., 2014, MNRAS, in press (arxiv:1303.4555)
* [Fukugita et al1996] Fukugita M., Ichikawa T., Gunn J. E., Doi M., Shimasaku K., Schneider D. P., 1996, AJ, 439, 584
* [Gilmore et al2012] Gilmore G., et al., 2012, The Messenger, 147, 25
* [Gonzalez-Solares et al2008] Gonzalez-Solares E., et al., 2008, MNRAS, 388, 89
* [Greimel et al2012] Greimel R., Drew J. E., Steeghs D., Barlow M. J., 2012, ATel, no.4365
* [Groot et al2009] Groot P. J., et al., 2009, MNRAS, 399, 323
* [Hayes1985] Hayes D. S., 1985, in Calibration of fundamental stellar quantities, Proc. IAU Sym. No.111, Dordrecht, D. Reidel Publishing Co., pp225-249
* [Irwin 19851985] Irwin M. J., 1985, MNRAS, 214, 575
* [Irwin et al2004] Irwin M. J., et al., 2004, SPIE, 5493, 41
* [Irwin & Lewis2001] Irwin M. J., Lewis J., 2001, NewAR, 45, 1051
* [Irwin2010] Irwin M. J., 2010, UKIRT Spring Newsletter, p14
* [Kuijken2011] Kuijken K., 2011, The Messenger, 146, 8
* [Lopez et al1991] López J. A., Falcon L. H., Ruiz M. T., Roth M., 1991, A&A, 241, 526
* [Landolt1992] Landolt A. U., 1992, AJ, 104, 340
* [Lucas et al2008] Lucas P. W., 2008, MNRAS, 391, 136
* [Marshall etal2006] Marshall D. J., Robin A. C., Reylé C., Schultheis, M., Picaud, S., 2006, A&A, 453, 635
* [Minniti et al2011] Minniti D., et al., 2010, NewA, 15, 433
* [Miszalski et al2008] Miszalski B., Parker Q. A., Acker A., Birkby J. L., Frew D. J., Kovacevic A., 2008, MNRAS, 384, 525
* [Norris et al2011] Norris R. P., et al., 2011, PASA, 28, 215
* [Parker et al2005] Parker Q.A., et al, 2005, MNRAS, 362, 689
* [Parker et al2006] Parker Q. A., et al, 2006, MNRAS, 373, 79
* [Patat et al2011] Patat F., et al., 2011, A&A, 527, 91
* [Pickles1998] Pickles A. J., 1998, PASP, 110, 863 (P98)
* [Raddi et al2013] Raddi R., et al, 2013, MNRAS, 430, 2169
* [Rodriguez-Gil et al2010] Rodriguez-Gil P., et al, 2010, MNRAS, 407 L21
* [Sahai et al2011] Sahai R., Morris M. R., Villar G. G., 2011, AJ, 141, 134
* [Sale et al2009] Sale S. E., et al., 2009, MNRAS, 392, 497
* [Sale2012] Sale S. E., 2012, MNRAS, 427, 2119
* [Sanchez-Contreras & Sahai2012] Sanchez-Contreras C., Sahai R., 2012, ApJS, 203, 16
* [Schlegel et al1998] Schlegel D. J., Finkbeiner D. P., Davis M., 1998 ApJ, 500, 525
* [Skrutskie et al2006] Skrutskie, M. F., et al., 2006, AJ, 131,1163
* [Steeghs et al2007] Steeghs D., et al., 2007, Atel, no. 795
* [Stephenson & Sanduleak1971] Stephenson C. B., Sanduleak N., 1971 PW&SO 1, 1 (SS71)
* [Straizys & Kuriliene1981] Straizys V., Kuriliene G., 1981, Ap&SS, 80, 353
* [Taylor2005] Taylor M. B., 2005, ASPC, 347, 29
* [Tylenda et al2003] Tylenda R., Si dmiak N., Górny S. K., Corradi R. L. M., Schwarz H. E., 2003, A&A, 405, 627
* [Valdivieso et al2009] Valdivieso, L., et al., 2009, A&A, 497, 973
* [Verbeek et al2012a] Verbeek K., et al., 2012a, MNRAS, 420, 1115
* [Verbeek et al2012b] Verbeek K., et al., 2012b, MNRAS, 426, 1235
* [Viironen et al2009a] Viironen K., et al., 2009a, A&A, 504, 291
* [Viironen et al2009] Viironen K., et al., 2009b, A&A, 502, 113
* [Viironen et al2011] Viironen K., et al., 2011, A&A, 530, 107
* [Vink et al2008] Vink J. S., Drew J. E., Steeghs D., Wright N. J., Martin E. L., Gänsicke B. T., Greimel R., Drake J., 2008, MNRAS, 387, 308
* [Wesson et al2008] Wesson R., et al., 2008, ApJ, 688, L21
* [Witham et al2006] Witham A. R., et al., 2006, MNRAS, 369, 581
* [Wright et al2008] Wright N. J., et al., 2008, MNRAS, 390, 929
* [Wright et al2012] Wright N. J., Drake J. J., Drew J. E., Guarcello M. G., Gutermuth R. A., Hora J. L., Kraemer K. E., 2012, ApJ, 746, L21
* [Wright et al2014] Wright N. J., et al., 2014, MNRAS, 437, L1
## Appendix A Synthetic colour reddening tables
Synthetic colours for main sequence stars, computed as described in Section 4,
are tabulated in full in an online supplement for three representative
reddening laws ($R_{V}=2.5$, 3.1 and 3.8) and a range of reddenings
($A_{0}=0$, 2, 4, 6, 8, 10). The form of the reddening laws used is due to
Fitzpatrick & Massa (2007). As an example of the tables available we include
excerpts from the second and fifth tables that respectively provide
$R_{V}=3.1$ blue-filter and red-filter colours for B,A main-sequence stars.
Two further tables of synthetic colours are included in the supplement for K-M
giants that have been computed using P98 library spectra. Data are provided
for the $R=3.1$ mean Galactic law only, for the limited purposes of (a) giving
an impression of how these luminous red objects may contaminate $(u-g,g-r)$
diagrams at redder $(g-r)$ through $u$ red leak (b) enabling comparisons with
the M-giant spur commonly seen in $(r-H\alpha,r-i)$ colour-colour diagrams.
Table 6: VST/OmegaCAM synthetic colours for B,A main-sequence stars in the $(u-g),(g-r)$ plane reddened with an $R_{V}=3.1$ extinction law. (Full table online.) Spectral | $A_{0}=0$ | $A_{0}=2$ | $A_{0}=4$ | $A_{0}=6$ | $A_{0}=8$
---|---|---|---|---|---
Type | $(u-g)$ | $(g-r)$ | $(u-g)$ | $(g-r)$ | $(u-g)$ | $(g-r)$ | $(u-g)$ | $(g-r)$ | $(u-g)$ | $(g-r)$
$B0V$ | $-1.433$ | $-0.271$ | $-0.692$ | $0.529$ | $0.087$ | $1.301$ | $0.891$ | $2.050$ | $1.632$ | $2.777$
$B1V$ | $-1.324$ | $-0.240$ | $-0.584$ | $0.558$ | $0.195$ | $1.329$ | $0.995$ | $2.076$ | $1.719$ | $2.802$
$B2V$ | $-1.209$ | $-0.218$ | $-0.470$ | $0.579$ | $0.307$ | $1.350$ | $1.104$ | $2.096$ | $1.808$ | $2.821$
$B3V$ | $-1.053$ | $-0.186$ | $-0.315$ | $0.610$ | $0.460$ | $1.379$ | $1.250$ | $2.125$ | $1.923$ | $2.849$
$B5V$ | $-0.828$ | $-0.139$ | $-0.092$ | $0.655$ | $0.680$ | $1.423$ | $1.460$ | $2.166$ | $2.080$ | $2.890$
$B6V$ | $-0.728$ | $-0.121$ | $0.007$ | $0.672$ | $0.776$ | $1.439$ | $1.550$ | $2.182$ | $2.144$ | $2.905$
$B7V$ | $-0.580$ | $-0.100$ | $0.152$ | $0.692$ | $0.918$ | $1.458$ | $1.682$ | $2.200$ | $2.234$ | $2.922$
$B8V$ | $-0.388$ | $-0.076$ | $0.340$ | $0.714$ | $1.101$ | $1.478$ | $1.850$ | $2.219$ | $2.344$ | $2.940$
$B9V$ | $-0.198$ | $-0.046$ | $0.528$ | $0.742$ | $1.285$ | $1.504$ | $2.019$ | $2.244$ | $2.445$ | $2.964$
$A0V$ | $-0.053$ | $-0.005$ | $0.675$ | $0.780$ | $1.431$ | $1.540$ | $2.153$ | $2.277$ | $2.514$ | $2.995$
$A1V$ | $-0.019$ | $0.005$ | $0.709$ | $0.790$ | $1.464$ | $1.550$ | $2.181$ | $2.287$ | $2.525$ | $3.005$
$A2V$ | $0.021$ | $0.025$ | $0.749$ | $0.809$ | $1.505$ | $1.568$ | $2.217$ | $2.304$ | $2.538$ | $3.021$
$A3V$ | $0.038$ | $0.059$ | $0.771$ | $0.840$ | $1.531$ | $1.597$ | $2.241$ | $2.332$ | $2.541$ | $3.048$
$A5V$ | $0.067$ | $0.125$ | $0.805$ | $0.904$ | $1.567$ | $1.658$ | $2.269$ | $2.390$ | $2.523$ | $3.105$
$A7V$ | $0.044$ | $0.199$ | $0.788$ | $0.975$ | $1.554$ | $1.726$ | $2.252$ | $2.456$ | $2.474$ | $3.169$
Table 7: VST/OmegaCAM synthetic colours for B,A main-sequence stars in the $(r-i),(r-H\alpha)$ plane reddened with an $R_{V}=3.1$ extinction law. (Full table online.) Spectral | $A_{0}=0$ | $A_{0}=2$ | $A_{0}=4$ | $A_{0}=6$ | $A_{0}=8$ | $A_{0}=10$
---|---|---|---|---|---|---
Type | $(r-i)$ | $(r-H\alpha)$ | $(r-i)$ | $(r-H\alpha)$ | $(r-i)$ | $(r-H\alpha)$ | $(r-i)$ | $(r-H\alpha)$ | $(r-i)$ | $(r-H\alpha)$ | $(r-i)$ | $(r-H\alpha)$
$B0V$ | $-0.150$ | $0.054$ | $0.278$ | $0.198$ | $0.694$ | $0.316$ | $1.100$ | $0.409$ | $1.496$ | $0.478$ | $1.884$ | $0.526$
$B1V$ | $-0.136$ | $0.048$ | $0.291$ | $0.192$ | $0.708$ | $0.310$ | $1.114$ | $0.403$ | $1.510$ | $0.472$ | $1.898$ | $0.519$
$B2V$ | $-0.123$ | $0.045$ | $0.304$ | $0.188$ | $0.721$ | $0.306$ | $1.126$ | $0.398$ | $1.523$ | $0.466$ | $1.911$ | $0.513$
$B3V$ | $-0.104$ | $0.044$ | $0.323$ | $0.186$ | $0.740$ | $0.303$ | $1.145$ | $0.394$ | $1.541$ | $0.462$ | $1.929$ | $0.508$
$B5V$ | $-0.077$ | $0.039$ | $0.349$ | $0.180$ | $0.765$ | $0.295$ | $1.170$ | $0.386$ | $1.566$ | $0.452$ | $1.954$ | $0.497$
$B6V$ | $-0.068$ | $0.036$ | $0.358$ | $0.177$ | $0.774$ | $0.291$ | $1.179$ | $0.381$ | $1.575$ | $0.448$ | $1.963$ | $0.492$
$B7V$ | $-0.057$ | $0.029$ | $0.369$ | $0.170$ | $0.785$ | $0.284$ | $1.190$ | $0.374$ | $1.586$ | $0.440$ | $1.973$ | $0.484$
$B8V$ | $-0.045$ | $0.018$ | $0.382$ | $0.158$ | $0.797$ | $0.272$ | $1.202$ | $0.362$ | $1.598$ | $0.427$ | $1.985$ | $0.471$
$B9V$ | $-0.028$ | $0.006$ | $0.398$ | $0.145$ | $0.813$ | $0.259$ | $1.218$ | $0.348$ | $1.614$ | $0.413$ | $2.001$ | $0.456$
$A0V$ | $-0.009$ | $-0.005$ | $0.418$ | $0.133$ | $0.833$ | $0.246$ | $1.238$ | $0.334$ | $1.633$ | $0.399$ | $2.020$ | $0.441$
$A1V$ | $-0.003$ | $-0.003$ | $0.423$ | $0.135$ | $0.838$ | $0.248$ | $1.243$ | $0.335$ | $1.638$ | $0.399$ | $2.025$ | $0.442$
$A2V$ | $0.006$ | $-0.004$ | $0.432$ | $0.134$ | $0.847$ | $0.247$ | $1.251$ | $0.334$ | $1.646$ | $0.397$ | $2.033$ | $0.439$
$A3V$ | $0.021$ | $-0.008$ | $0.446$ | $0.130$ | $0.861$ | $0.241$ | $1.265$ | $0.328$ | $1.660$ | $0.391$ | $2.047$ | $0.432$
$A5V$ | $0.051$ | $0.005$ | $0.476$ | $0.141$ | $0.890$ | $0.250$ | $1.293$ | $0.335$ | $1.687$ | $0.396$ | $2.073$ | $0.436$
$A7V$ | $0.083$ | $0.027$ | $0.507$ | $0.160$ | $0.920$ | $0.268$ | $1.322$ | $0.350$ | $1.716$ | $0.410$ | $2.101$ | $0.448$
|
arxiv-papers
| 2014-02-27T19:11:07 |
2024-09-04T02:49:59.033440
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. E. Drew, E. Gonzalez-Solares, R. Greimel, M. J. Irwin, A. Kupcu\n Yoldas, J. Lewis, G. Barentsen, J. Eisloeffel, H. J. Farnhill, W. E. Martin,\n J. R. Walsh, N. A. Walton, M. Mohr-Smith, R. Raddi, S. E. Sale, N. J. Wright,\n P. Groot, M. J. Barlow, R. L. M. Corradi, J. J. Drake, J. Fabregat, D. J.\n Frew, B. T. Gaensicke, C. Knigge, A. Mampaso, R. A. H. Morris, T. Naylor, Q.\n A. Parker, S. Phillipps, C. Ruhland, D. Steeghs, Y.C. Unruh, J. S. Vink, R.\n Wesson, A. A. Zijlstra",
"submitter": "Janet Drew",
"url": "https://arxiv.org/abs/1402.7024"
}
|
1402.7058
|
also at ]Jawaharlal Nehru Centre For Advanced Scientific Research, Jakkur,
Bangalore, India.
# Statistical Properties of the Intrinsic Geometry of Heavy-particle
Trajectories in Two-dimensional, Homogeneous, Isotropic Turbulence
Anupam Gupta [email protected] Department of Physics, University of “Tor
Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy Centre for
Condensed Matter Theory, Department of Physics, Indian Institute of Science,
Bangalore 560012, India Dhrubaditya Mitra [email protected] NORDITA,
Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23,
SE-10691 Stockholm, Sweden Prasad Perlekar [email protected] TIFR Centre
for Interdisciplinary Sciences, 21 Brundavan Colony, Narsingi, Hyderabad
500075, India Rahul Pandit [email protected] [ Centre for
Condensed Matter Theory, Department of Physics, Indian Institute of Science,
Bangalore 560012, India
###### Abstract
We obtain, by extensive direct numerical simulations, trajectories of heavy
inertial particles in two-dimensional, statistically steady, homogeneous, and
isotropic turbulent flows, with friction. We show that the probability
distribution function $\mathcal{P}(\kappa)$, of the trajectory curvature
$\kappa$, is such that, as $\kappa\to\infty$,
$\mathcal{P}(\kappa)\sim\kappa^{-h_{\rm r}}$, with $h_{\rm r}=2.07\pm 0.09$.
The exponent $h_{\rm r}$ is universal, insofar as it is independent of the
Stokes number ${\rm St}$ and the energy-injection wave number $k_{\rm inj}$.
We show that this exponent lies within error bars of their counterparts for
trajectories of Lagrangian tracers. We demonstrate that the complexity of
heavy-particle trajectories can be characterized by the number $N_{\rm
I}(t,{\rm St})$ of inflection points (up until time $t$) in the trajectory and
$n_{\rm I}({\rm St})\equiv\lim_{t\to\infty}\frac{N_{\rm I}(t,{\rm
St})}{t}\sim{\rm St}^{-\Delta}$, where the exponent $\Delta=0.33\pm 0.02$ is
also universal.
turbulence, inertial particle, statistical mechanics
###### pacs:
47.27.-i,05.40.-a
††preprint: NORDITA-2014-21
The transport of particles by turbulent fluids has attracted considerable
attention since the pioneering work of Taylor tay22 . The study of such
transport has experienced a renaissance because (a) there have been tremendous
advances in measurement techniques and direct numerical simulations (DNSs)
tos+bod09 and (b) it has implications not only for fundamental problems in
the physics of turbulence bec+bif+bof+cen+mus+tos06 but also for a variety of
geophysical, atmospheric, astrophysical, and industrial problems sha03 ;
gra+wan13 ; fal+fou+ste02 ; Arm10 ; Csa73 ; eat+fes94 ; pos+abr02 . It is
natural to use the Lagrangian frame of reference fal+gaw+var01 here; but we
must distinguish between (a) Lagrangian or tracer particles, which are
neutrally buoyant and follow the flow velocity at a point, and (b) inertial
particles, whose density $\rho_{p}$ is different from the density $\rho_{f}$
of the advecting fluid. The motion of heavy inertial particles is determined
by the flow drag, which can be parameterized by a time scale $\tau_{\rm s}$,
whose ratio with the Kolmogorov dissipation time $T_{\eta}$ is the Stokes
number ${\rm St}=\tau_{\rm s}/T_{\eta}$; tracer and heavy inertial particles
show qualitatively different behaviors in flows; e.g., the former are
uniformly dispersed in a turbulent flow, whereas the latter cluster
bec+bif+bof+cen+mus+tos06 , most prominently when ${\rm St}\simeq 1$.
Differences between tracers and inertial particles have been investigated in
several studies tos+bod09 , which have concentrated on three-dimensional (3D)
flows and on the clustering or dispersion of these particles.
We present the first study of the statistical properties of the geometries of
heavy-particle trajectories in two-dimensional (2D), homogeneous, isotropic,
and statistically steady turbulence, which is qualitatively different from its
3D counterpart because, if energy is injected at wave number $k_{\rm inj}$,
two power-law regimes appear in the energy spectrum $E(k)$ kra+mon80 ;
pan+per+ray09 ; bof+eck12 , for wave numbers $k<k_{\rm inj}$ and $k>k_{\rm
inj}$. One regime is associated with an inverse cascade of energy, towards
large length scales, and the other with a forward cascade of enstrophy to
small length scales. It is important to study both forward- and inverse-
cascade regimes, so we use $k_{\rm inj}=4$, which gives a large forward-
cascade regime in $E(k)$, and $k_{\rm inj}=50$, which yields both forward- and
inverse-cascade regimes.
For a heavy inertial particle, we calculate the velocity ${\bm{v}}$, the
acceleration ${\bm{a}}=d{\bm{v}}/dt$, with magnitude $a$ and normal and
tangential components $a_{n}$ and $a_{t}$, respectively. The intrinsic
curvature of a particle trajectory is $\kappa=a_{n}/v^{2}$. We find two
intriguing results that shed new light on the geometries of particle tracks in
2D turbulence: First, the probability distribution function (PDF)
$\mathcal{P}(\kappa)$ is such that, as $\kappa\to\infty$,
$\mathcal{P}(\kappa)\sim\kappa^{-h_{\rm r}}$; in contrast, as $\kappa\to 0$,
$\mathcal{P}(\kappa)$ has slope zero; we find that $h_{\rm r}=2.07\pm 0.09$ is
universal, insofar as they are independent of ${\rm St}$ and $k_{\rm inj}$. We
present high-quality data, with two decades of clean scaling, to obtain the
values of these exponents, for different values of ${\rm St}$ and $k_{\rm
inj}$. We obtain data of similar quality for Lagrangian-tracer trajectories
and thus show that $h_{\rm r}$ lies within error bars of its tracer-particle
counterpart. Second, along every heavy-particle track, we calculate the
number, $N_{\rm I}(t,{\rm St})$, of inflection points (at which
${\bm{a}}\times{\bm{v}}$ changes sign) up until time $t$. We propose that
$n_{\rm I}({\rm St})\equiv\lim_{t\to\infty}\frac{N_{\rm I}(t,{\rm St})}{t}$
(1)
is a natural measure of the complexity of the trajectories of these particles;
and we find that $n_{\rm I}\sim{\rm St}^{-\Delta}$, where the exponent
$\Delta=0.33\pm 0.02$ is also universal.
We obtain several other interesting results: (a) At short times the particles
move ballistically but, at large times, there is a crossover to Brownian
motion, at a crossover time $T_{\rm cross}$ that increases monotonically with
${\rm St}$. (b) The PDFs $\mathcal{P}(a)$, $\mathcal{P}(a_{n})$, and
$\mathcal{P}(a_{t})$ all have exponential tails. (c) By conditioning
$\mathcal{P}(\kappa)$ on the sign of the Okubo-Weiss oku70 ; wei91 ;
per+ray+mit+pan11 parameter $\Lambda$, we show that particles in regions of
elongational flow ($\Lambda>0$) have, on average, trajectories with a lower
curvature than particles in vortical regions ($\Lambda<0$).
We write the 2D incompressible Navier-Stokes (NS) equation in terms of the
stream-function $\psi$ and the vorticity
${\bm{\omega}}={\bm{\nabla}}\times{\bm{u}}({\bm{x}},t)$, where
${\bm{u}}\equiv(-\partial_{y}\psi,\partial_{x}\psi)$ is the fluid velocity at
the point ${\bm{x}}$ and time $t$, as follows:
$\displaystyle D_{t}{\bm{\omega}}$ $\displaystyle=$
$\displaystyle\nu\nabla^{2}{\bm{\omega}}-\mu{\bm{\omega}}+F;$ (2)
$\displaystyle\nabla^{2}{\bf\psi}$ $\displaystyle=$
$\displaystyle{\bm{\omega}}.$ (3)
Here, $D_{t}\equiv\partial_{t}+{\bm{u}}\cdot\nabla$, the uniform fluid density
$\rho_{\rm f}=1$, $\mu$ is the coefficient of friction, and $\nu$ the
kinematic viscosity of the fluid. We use a Kolmogorov-type forcing
$F(x,y)\equiv-F_{0}k_{\rm inj}\cos(k_{\rm inj}y)$, with amplitude $F_{0}$ and
length scale $\ell_{\rm inj}\equiv 2\pi/k_{\rm inj}$. (A) For $k<k_{\rm inj}$,
the inverse cascade of energy yields $E(k)\sim k^{-5/3}$; and (B) for
$k>k_{\rm inj}$, there is a forward cascade of enstrophy and $E(k)\sim
k^{-\delta}$, where the exponent $\delta$ depends on the friction $\mu$ (for
$\mu=0$, $\delta=3$). We use $\mu=0.01$ and obtain $\delta=-3.6$. The equation
of motion for a small, spherical, rigid particle (henceforth, a heavy
particle) in an incompressible flow max+ril83 assumes the following simple
form, if $\rho_{\rm p}\gg\rho_{\rm f}$ :
$\frac{d\bf{x}}{dt}={\bm{v}}(t),\hskip
28.45274pt\frac{d{\bm{v}}}{dt}=-\frac{1}{\tau_{\rm
s}}\left[{\bm{v}}(t)-{\bm{u}}(\bf{x}(t),t)\right],$ (4)
where $\bf{x}$, ${\bm{v}}$, and $\tau_{\rm s}=(2R_{\rm p}^{2})\rho_{\rm
p}/(9\nu\rho_{\rm f})$ are, respectively, the position, velocity, and response
time of the particle, and $R_{\rm p}$ is its radius. We assume that $R_{\rm
p}\ll\eta$, the dissipation scale of the carrier fluid, and that the particle
number density is so low that we can neglect interactions between particles,
the particles do not affect the flow, and particle accelerations are so high
that we can neglect gravity. In our DNSs we solve simultaneously for several
species of particles, each with a different value of ${\rm St}$; there are
$N_{\rm p}$ particles of each species. We also obtain the trajectories for
$N_{\rm p}$ Lagrangian particles, each of which obeys the equation
$d(\bf{x})/dt={\bm{u}}\left[\bf{x}(t),t\right]$. The details of our DNS are
given in the Appendix A and parameters in our DNSs are given in Tables(1) and
(2) for $12$ representative values of ${\rm St}$ (we have studied $20$
different values of ${\rm St}$).
In Fig. (1) we show representative particle trajectories of a Lagrangian
tracer (black line) and three different heavy particles with ${\rm St}=0.1$
(red asterisks), ${\rm St}=0.5$ (blue circles), and ${\rm St}=1$ (black
squares) superimposed on a pseudocolor plot of ${\bm{\omega}}$. We expect that
inertial particles move ballistically in the range $0<t\leq\tau_{\rm s}$; for
$t\gg\tau_{\rm s}$, we anticipate a crossover to Brownian behavior, which we
quantify by defining the mean-square displacement $r^{2}(t)=\langle({\bf
x}(t_{0}+t)-{\bf x}(t_{0}))^{2}\rangle_{t_{0},N_{\rm p}}$, where
$\langle\rangle_{t_{0},N_{\rm p}}$ denotes an average over $t_{0}$ and over
the $N_{\rm p}$ particles with a given value of ${\rm St}$. Figure (2)
contains log-log plots of $r^{2}$ versus $t$, for the representative cases
with ${\rm St}=0.1$ (red asterisks) and ${\rm St}=1$ (black squares); both of
these plots show clear crossovers from ballistic ($r^{2}\sim t^{2}$) to
Brownian ($r^{2}\sim t$) behaviors. We define the crossover time $T_{\rm
cross}$ as the intersection of the ballistic and Brownian asymptotes (bottom
inset of Fig. (2)). The top inset of Fig. (2) shows that, in the parameter
range we consider, $T_{\rm cross}$ increases monotonically with ${\rm St}$.
Figure 1: (Color online) Representative particle trajectories of a Lagrangian
tracer (black line) and three different heavy particles with ${\rm St}=0.1$
(red asterisks), ${\rm St}=0.5$ (blue circles), and ${\rm St}=1$ (black
squares) superimposed on a pseudocolor plot of ${\bm{\omega}}$. For the
spatiotemporal evolution of this plot see the animation available at the
location http://www.youtube.com/watch?v=lk3iSHhfTuU Figure 2: (Color online)
Log-log (base 10) plots of $r^{2}$ versus $t/T_{\rm eddy}$ for ${\rm St}=0.1$
(red triangles), and ${\rm St}=1$ (black squares); top inset: plot of $T_{\rm
cross}/T_{\rm eddy}$ versus ${\rm St}$; bottom inset: log-log (base 10) plot
of $r^{2}/t$ versus $t/T_{\rm eddy}$ for tracers (blue curve) and linear fits
to the small- and large-$t$ asymptotes (dashed lines) with slopes $1$ and $0$
in ballistic and Brownian regimes, respectively; the intersection point of
these dashed lines yields $T_{\rm cross}$.
(a)(b)(c)
Figure 3: (Color online) Plots of PDFs of (a) the modulus of $a$ of the
particle acceleration, (b) its tangential component $a_{t}$, and (c) its
normal component $a_{n}$ for ${\rm St}=0$ (blue curve), $0.5$ (red curve), $1$
(green curve), and $2$ (black curve).
In Fig. (3) we present semilog plots of the PDFs $\mathcal{P}(a)$,
$\mathcal{P}(a_{t})$, and $\mathcal{P}(a_{n})$ for some representative values
of ${\rm St}$. Clearly, all of these PDFs have exponential tails, i.e.,
$\mathcal{P}(a,{\rm St})\sim\exp[-a/\alpha({\rm St})]$,
$\mathcal{P}(a_{t},{\rm St})\sim\exp[-a_{t}/\alpha_{\rm t}({\rm St})]$, and
$\mathcal{P}(a_{n},{\rm St})\sim\exp[-a_{n}/\alpha_{\rm n}({\rm St})]$. As
${\rm St}$ increases, the tails of these PDFs fall more and more rapidly,
because the higher the inertia the more difficult is it to accelerate a
particle. Hence, $\alpha$, $\alpha_{\rm t}$, and $\alpha_{\rm n}$ decrease
with ${\rm St}$ [see Table (2)].
Although these acceleration PDFs have exponential tails, $\mathcal{P}(\kappa)$
shows a power-law behavior as $\kappa\to\infty$, as we have mentioned above.
The exponent $h_{\rm r}$ for the right-tail of $\mathcal{P}(\kappa)$ is
especially interesting because it characterizes the parts of a trajectory that
have large values of $\kappa$. If $\mathcal{P}(\kappa)\sim\kappa^{-h_{\rm
r}}$, then its cumulative PDF $\mathcal{Q}(\kappa)\sim\kappa^{-h_{\rm r}+1}$.
We obtain an accurate estimate of $h_{\rm r}$ from $\mathcal{Q}$, which we
obtain by a rank-order method that does not suffer from binning errors mit05a
. We give representative, log-log plots of $\mathcal{Q}$ in Fig. (4), for
${\rm St}=0.1$ (blue asterisks) and ${\rm St}=1$ (red squares); and we
determine $h_{\rm r}$ by fitting a straight line to $\mathcal{Q}$ over a
scaling range of more than two decades; We plot, in the inset, Fig. (4), the
local slope of this scaling range, whose mean value and standard deviation
yield, respectively, $h_{\rm r}$ and its error bars. From such plots we find
that $h_{\rm r}$ does not depend significantly on ${\rm St}$ [Table (2)].
Furthermore, we find that the Lagrangian analog of $h_{\rm r}$, which we
denote by $h_{\rm lagrangian}$, is $2.03\pm 0.09$, i.e., it lies within error
bars of $h_{\rm r}$. By analyzing the $\kappa\to 0$ limit of
$\mathcal{P}(\kappa)$, we find that $\mathcal{P}(\kappa)\sim
A_{0}\kappa^{h_{\rm l}}$, where $A_{0}>0$ is an amplitude and $h_{\rm
l}=0.0\pm 0.1$ (the latter is independent of ${\rm St}$); this indicates that
there is a nonzero probability that the paths of particles have zero
curvature, i.e., they can move in straight lines. The $\kappa\to 0$ limit of
$\mathcal{P}(\kappa)$ seems, therefore, to be different from its counterpart
for 3D fluid turbulence (see Ref. xu+oue+bod07 for Lagrangian tracers and
Ref. akshaypreprint for heavy particles), where $\mathcal{P}(\kappa)\to 0$ as
$\kappa\to 0$. Very-high-resolution DNSs for 2D turbulence must be undertaken
to probe the $\kappa\to 0$ limit of $\mathcal{P}(\kappa)$ by going to even
smaller values of $\kappa$ than we have been able to obtain reliably in our
DNS.
Figure 4: (Color online) Log-log plots of the cumulative PDFs
$\mathcal{Q}(\kappa)$ for ${\rm St}=0.1$ (blue asterisks) and ${\rm St}=1$
(red squares); the inset shows a plot of the local slope of the tail of this
cumulative PDF and the two dashed horizontal lines indicate the maximum and
minimum values of this local slope in the range we use for fitting the
exponent $h_{\rm r}$.
A point in a 2D flow is vortical or strain-dominated if the Okubo-Weiss
parameter $\Lambda=(1/8)(\omega^{2}-\sigma^{2})$ is, respectively, positive or
negative oku70 ; wei91 ; per+ray+mit+pan11 . We now investigate how the
acceleration statistics of heavy particles depends on the sign of $\Lambda$ by
conditioning the PDFs of $a_{t}$ and $\kappa$ on this sign. In particular, we
obtain the conditional PDFs $\mathcal{P}^{+}$ and $\mathcal{P}^{-}$, where the
superscript stands for the sign of $\Lambda$. We find, on the one hand, that
the tail of $\mathcal{P}^{+}(a_{t})$ falls faster than that of
$\mathcal{P}^{-}(a_{t})$ because regions of the trajectory with high
tangential accelerations are associated with strain-dominated points in the
flow. On the other hand, the right tail of $\mathcal{P}^{+}(\kappa)$ falls
more slowly than that of $\mathcal{P}^{-}(\kappa)$, which implies that high-
curvature parts of a particle trajectory are correlated with vortical regions
of the flow. We give plots of $\mathcal{P}^{+}(a_{t})$,
$\mathcal{P}^{+}(\kappa)$, $\mathcal{P}^{-}(a_{t})$, and
$\mathcal{P}^{-}(\kappa)$ in the Appendix A.
We find that ${\bm{a}}\times{\bm{v}}$ (a pseudoscalar in 2D like the
vorticity) changes sign at several inflection points along a particle
trajectory. We use the number of inflection points on a trajectory, per unit
time, $n_{\rm I}({\rm St})$ (see Eq. (1)) as a measure of its complexity. In
Fig. (5) we demonstrate that the limit in Eq. (1) exists by plotting $N_{\rm
I}(t,{\rm St})/t$ as a function of $t$ for ${\rm St}=0.1$ (red asterisks) and
${\rm St}=2$ (black triangles); the mean value of $N_{\rm I}(t,{\rm St})/t$,
between the two vertical dashed lines in Fig. (5), yields our estimate for
$n_{\rm I}({\rm St})$, which is given in the inset as a function of ${\rm St}$
(on a log-log scale); the standard deviation gives the error bars. From this
inset of Fig. (5) we conclude that $n_{\rm I}({\rm St})\sim{\rm
St}^{-\Delta},$ with $\Delta=0.33\pm 0.05$. This exponent $\Delta$ [Table (1)]
is independent of the Reynolds number and $\mu$, within the range of
parameters we have explored. Furthermore, $\Delta$ is independent of whether
our 2D turbulent flow is dominated by forward or the inverse cascades in
$E(k)$, which are controlled by $k_{\rm inj}$.
Figure 5: (Color online) Plots of $N_{\rm I}/(t/T_{\rm eddy})$ versus
$t/T_{\rm eddy}$ for ${\rm St}=0.1$ (red curve) and ${\rm St}=2$ (black
curve); the inset shows a log-log (base 10) plot of $n_{\rm I}$ versus ${\rm
St}$ (blue open circles); the black dotted line has a slope $=-1/3$.
We have repeated all the above studies with a forcing term that yields an
energy spectrum with a significant inverse-cascade part ($k_{\rm inj}=50$);
the parameters for this run are given in Table (1) in the Appendix A and in
Ref. AGthesis . The dependence of all the tails of the PDFs discussed above
and the exponents $h_{\rm l}$ and $h_{\rm r}$ on ${\rm St}$ are similar to
those we have found above for $k_{\rm inj}=4$.
Earlier studies of the geometrical properties of particle tracks have been
restricted to tracers; and they have inferred these properties from tracer
velocities and accelerations. For example, the PDFs of different components of
the acceleration of Lagrangian particles in 2D turbulent flows has been
studied for both decaying wil+kam+fri08 and forced kad+del+bos+sch11 cases;
they have shown exponential tails in periodic domains, but, in a confined
domain, have obtained PDFs with heavier tails kad+bos+sch08 . The PDF of the
curvature of tracer trajectories has been calculated from the same
simulations, which quote an exponent $h_{\rm lagrangian}\simeq 2.25$ (but no
error bars are given). Our work goes well beyond these earlier studies by (a)
investigating the statistical properties of the geometries of the trajectories
of heavy particles in 2D turbulent flows for a variety of parameter ranges and
Stokes numbers, (b) by introducing and evaluating, with unprecedented accuracy
(and error bars), the exponent $h_{\rm r}$, (c) proposing $n_{\rm I}$ as a
measure of the complexity of heavy-particle trajectories and obtaining the
exponent $\Delta$ accurately, (d) by examining the dependence of all these
exponents on ${\rm St}$ and $k_{\rm inj}$, and (e) showing, thereby, that
these exponents are universal (within our error bars).
Our results imply that $n_{\rm I}({\rm St})$ has a power-law divergence, so
the trajectories become more and more contorted, as ${\rm St}\to 0$. This
divergence is suppressed eventually, in any DNS, which can only achieve a
finite value of $Re_{\lambda}$ because it uses only a finite number of
collocation points. Such a suppression is the analog of the finite-size
rounding off of divergences, in say the susceptibility, at an equilibrium
critical point fssprivman . Note also that the limit ${\rm St}\to 0$ is
singular and it is not clear a priori that this limit should yield the same
results, for the properties we study, as the Lagrangian case ${\rm St}=0$.
We hope that our study will lead to experimental studies and accurate
measurements of the exponents $h_{\rm r}$ and $\Delta$, and applications of
these in developing a detailed understanding of particle-laden flows in the
variety of systems that we have mentioned in the introduction.
For 3D turbulent flows, geometrical properties of Lagrangian-particle
trajectories have been studied numerically bra+lil+eck06 ; sca11 and
experimentally xu+oue+bod07 . However, such geometrical properties have not
been studied for heavy particles. The extension of our heavy-particle study to
the case of 3D fluid turbulence is nontrivial and will be given in a companion
paper akshaypreprint .
$Run$ | $N$ | $F_{0}$ | $k_{\rm inj}$ | $\ell_{\rm d}$ | $\lambda$ | $Re_{\lambda}$ | $T_{\rm eddy}$ | $T_{\eta}$ | $T_{\rm inj}$
---|---|---|---|---|---|---|---|---|---
IA | $1024$ | $0.2$ | $50$ | $1.3\times 10^{-3}$ | $0.06$ | $1219$ | $0.98$ | $0.16$ | $2.94$
FA | $1024$ | $0.005$ | $4$ | $5.4\times 10^{-3}$ | $0.2$ | $1322$ | $7$ | $2.9$ | $30.2$
Table 1: The parameters for our DNS runs: $N^{2}$ is the number of collocation points, $N_{\rm p}=10^{4}$ is the number of Lagrangian or inertial particles, $\delta t$ the time step, $\nu=10^{-5}$ the kinematic viscosity, and $\mu=0.01$ the air-drag-induced friction, $F_{0}$ the forcing amplitude, $k_{\rm inj}$ the forcing wave number, $l_{d}\equiv(\nu^{3}/\varepsilon)^{1/4}$ the dissipation scale, $\lambda\equiv\sqrt{\nu E/\varepsilon}$ the Taylor microscale, $Re_{\lambda}=u_{\rm rms}\lambda/\nu$ the Taylor-microscale Reynolds number, $T_{eddy}=(\frac{\sum_{k}E(k)/k}{\sum_{k}E(k)})/u_{rms}$ the eddy-turn-over time, and $T_{\eta}\equiv\sqrt{(\nu/\varepsilon)}$ the Kolmogorov time scale. $T_{\rm inj}\equiv(\ell_{\rm inj}^{2}/E_{\rm inj})^{1/3}$ is the energy-injection time scale, where $E_{\rm inj}=<{\bf f_{\rm u}}\cdot{\bm{u}}>$, is the energy-injection rate, $\ell_{\rm inj}=2\pi/k_{\rm inj}$ is the energy-injection length scale, and ${\bm{f}}_{\omega}=\nabla\times{\bm{f}}_{\rm u}$. $Run$ | ${\rm St}$ | $\alpha$ | $\alpha_{\rm t}$ | $\alpha_{\rm n}$ | $h_{\rm r}$
---|---|---|---|---|---
F1 | $0.1$ | $0.86\pm 0.07$ | $1.45\pm 0.07$ | $0.86\pm 0.07$ | $2.03\pm 0.08$
F2 | $0.2$ | $0.96\pm 0.06$ | $1.66\pm 0.07$ | $0.97\pm 0.06$ | $2.0\pm 0.1$
F3 | $0.3$ | $1.11\pm 0.07$ | $1.87\pm 0.07$ | $1.12\pm 0.06$ | $2.0\pm 0.1$
F4 | $0.4$ | $1.43\pm 0.07$ | $2.15\pm 0.07$ | $1.36\pm 0.09$ | $2.04\pm 0.09$
F5 | $0.5$ | $1.56\pm 0.08$ | $2.27\pm 0.08$ | $1.45\pm 0.09$ | $2.0\pm 0.1$
F6 | $0.6$ | $1.66\pm 0.08$ | $2.36\pm 0.09$ | $1.6\pm 0.1$ | $2.02\pm 0.09$
F7 | $0.7$ | $1.88\pm 0.09$ | $2.51\pm 0.09$ | $1.61\pm 0.09$ | $2.06\pm 0.09$
F8 | $0.8$ | $2.22\pm 0.08$ | $2.73\pm 0.09$ | $1.90\pm 0.09$ | $2.01\pm 0.08$
F9 | $0.9$ | $2.6\pm 0.1$ | $2.9\pm 0.1$ | $2.0\pm 0.1$ | $2.0\pm 0.1$
F10 | $1.0$ | $2.6\pm 0.1$ | $3.3\pm 0.1$ | $2.17\pm 0.09$ | $2.0\pm 0.1$
F11 | $1.5$ | $3.9\pm 0.1$ | $4.3\pm 0.1$ | $3.3\pm 0.1$ | $2.1\pm 0.1$
F12 | $2.0$ | $4.5\pm 0.1$ | $4.7\pm 0.1$ | $3.8\pm 0.1$ | $2.0\pm 0.1$
Table 2: The values of $\alpha$, $\alpha_{\rm n}$, and $\alpha_{\rm t}$ and
the exponent $h_{\rm r}$ for the case $k_{\rm inj}=4$ and for different values
of ${\rm St}$.
## I Acknowledgments
We thank A. Bhatnagar, A. Brandenburg, B. Mehlig, S.S. Ray, and D. Vincenzi
for discussions, and particularly A. Niemi, whose study of the intrinsic
geometrical properties of polymers poly11 , inspired our work on particle
trajectories. The work has been supported in part by the European Research
Council under the AstroDyn Research Project No. 227952 (DM), Swedish Research
Council under grant 2011-542 (DM), NORDITA visiting PhD students program (AG),
and CSIR, UGC, and DST (India) (AG and RP). We thank SERC (IISc) for providing
computational resources. AG, PP, and RP thank NORDITA for hospitality; DM
thanks the Indian Institute of Science for hospitality.
## References
* (1) G. Taylor, Proc. London. Math. Soc. s2-20, 196 (1922).
* (2) F. Toschi and E. Bodenschatz, Ann. Rev. Fluid Mech. 41, 375 (2009).
* (3) R. A. Shaw, Annual Review of Fluid Mechanics 35, 183 (2003).
* (4) W. W. Grabowski and L.-P. Wang, Ann. Rev. Fluid Mech. 45, 293 (2013).
* (5) G. Falkovich, A. Fouxon, and M. Stepanov, Nature, London 419, 151 (2002).
* (6) P. J. Armitage, Astrophysics of Planet Formation (Cambridge University Press, Cambridge, UK, 2010).
* (7) G. T. Csanady, Turbulent Diffusion in the Environmnet (Springer, ADDRESS, 1973), Vol. 3.
* (8) J. Eaton and J. Fessler, Intl. J. Multiphase Flow 20, 169 (1994).
* (9) S. Post and J. Abraham, Intl. J. Multiphase Flow 28, 997 (2002).
* (10) G. Falkovich, K. Gawȩdzki, and M. Vergassola, Rev. Mod. Phys. 73, 913 (2001).
* (11) J. Bec, et al., Phys. Fluids 18, 091702 (2006).
* (12) R. Kraichnan and D. Montgomery, Rep. Prog. Phys. 43, (1980).
* (13) R. Pandit, P. Perlekar, and S.S. Ray, Pramana 73, 179 (2009).
* (14) G. Boffetta and R. E. Ecke, Ann. Rev. Fluid Mech. 44, 427 (2012).
* (15) D. Mitra, J. Bec, R. Pandit, and U. Frisch, Phys. Rev. Lett 94, 194501 (2005).
* (16) P. Perlekar, S.S. Ray, D. Mitra, and R. Pandit, Phys. Rev. Lett 106, 054501 (2011).
* (17) A. Okubo, Deep-Sea. Res. 17, 445 (1970).
* (18) J. Weiss, Physica (Amsterdam) 48D, 273 (1991).
* (19) M. R. Maxey and J. J. Riley, Physics of Fluids 26, 883 (1983).
* (20) A. Gupta, PhD. Thesis, Indian Institute of Science, unpublished (2014).
* (21) M. Wilczek, O. Kamps, and R. Friedrich, Physica D: Nonlinear Phenomena 237, 2090 (2008).
* (22) B. Kadoch, D. del Castillo-Negrete, W. J. T. Bos, and K. Schneider, Phys. Rev. E 83, 036314 (2011).
* (23) B. Kadoch, W. J. T. Bos, and K. Schneider, Phys. Rev. Lett. 100, 184503 (2008).
* (24) W. Braun, F. De Lillo, and B. Eckhardt, Journal of Turbulence 7, (2006).
* (25) A. Scagliarinia, Journal of Turbulence, 12, N25, (2011); DOI: 10.1080/14685248.2011.571261.
* (26) H. Xu, N.T. Ouellette, and E. Bodenschatz, Physical Review Letters 98, 050201 (2007).
* (27) See, e.g., V. Privman, in Chapter I in ”Finite Size Scaling and Numerical Simulation of Statistical Systems,” ed. V. Privman (World Scientific, Singapore, 1990) pp 1-98. Finite-size scaling is used to evaluate inifinte-size-system exponents systematically at conventional critical points; its analog for our study requires several DNSs, over a large range of $Re_{\lambda}$, which lie beyond the scope of our investigation.
* (28) A. Bhatnagar, D. Mitra, A. Gupta, P. Perlekar, and R. Pandit, to be published.
* (29) S. Hu, M. Lundgren, and A.J. Niemi, Phys. Rev. E 83, 061908 (2011)
* (30) C. Canuto, M. Hussaini, A. Quarteroni, and T. Zang, Spectral methods in Fluid Dynamics (Spinger-Verlag, Berlin, 1988).
* (31) S. Cox and P. Matthews, Journal of Computational Physics 176, 430 (2002).
* (32) W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1992).
* (33) P. Perlekar and R. Pandit, New J. Phys. 11, 073003 (2009).
* (34) P. Perlekar, Ph.D. thesis, Indian Institute of Science, Bangalore, India, 2009.
* (35) S. S. Ray, D. Mitra, P. Perlekar, and R. Pandit, Phys. Rev. Lett. 107, 184503 (2011).
* (36) L. Biferale et al., Phys. Rev. Lett. 93, 064502 (2004).
* (37) J. Bec et al., Journal of Fluid Mechanics 550, 349 (2006).
*
## Appendix A Statistical Properties of the Intrinsic Geometry of Heavy-
particle Trajectories in Two-dimensional, Homogeneous, Isotropic Turbulence :
Supplemental Material
In this Supplemental Material we provide numerical details of our direct
numerical simulation (DNS) of Eq. (2) in the main part of this paper. We also
give results of our DNS for the case of the injection wave vector $k_{\rm
inj}=50$, which yields a significant inverse-cascade part in the energy
spectrum $E(k)$. In Fig. (6) we show the energy spectra $E(k)$ for our runs
$\tt FA$ ($k_{\rm inj}=4$) and $\tt IA$ ($k_{\rm inj}=50$).
(a)
(b)
Figure 6: (Color online) Log-log (base 10) plots of the energy spectra $E(k)$
versus $k$ for (a) runs $\tt FA$ ($k_{\rm inj}=4$) and (b) runs $\tt IA$
($k_{\rm inj}=50$).
We perform a DNS of Eq. (2) by using a pseudo-spectral code Can88 with the
$2/3$ rule for dealiasing; and we use a second-order, exponential time
differencing Runge-Kutta method cox+mat02 for time stepping. We use periodic
boundary conditions in a square simulation domain with side $\mathbb{L}=2\pi$,
with $N^{2}$ collocation points. Together with Eq.(2) we solve for the
trajectories of $N_{\rm p}$ heavy particles, for each of which we solve Eq.
(4) with an Euler scheme. The use of an Euler scheme to evolve particles is
justified because, in time $\delta t$, a particle crosses at most one-tenth of
grid spacing. We obtain the Lagrangian velocity at an off-grid particle
position ${\bm{x}}$, from the Eulerian velocity field by using a bilinear-
interpolation scheme Pre+Fla+Teu+Vet92 ; for numerical details see Refs.
per+pan09 ; Per09 ; per+ray+mit+pan11 ; ray+mit+per+pan11 .
We calculate the fluid energy-spectrum $E(k)\equiv\sum_{k-1/2<k^{\prime}\leq
k+1/2}k^{\prime 2}\langle|\hat{\psi}({\bf k^{\prime}},t)|^{2}\rangle_{t}$,
where $\langle\cdot\rangle_{t}$ indicates a time average over the
statistically steady state. The parameters in our simulations are given in
Table(II) of the main part of this paper and in Table(3). These include the
Taylor-microscale Reynolds number, $Re_{\lambda}\equiv u_{\rm
rms}\lambda/\nu$, where $\lambda\equiv\sqrt{\nu E/\varepsilon}$ is the Taylor
microscale and the Stokes number ${\rm St}=\tau_{\rm s}/T_{\eta}$. We use $20$
different values of ${\rm St}$ to study the dependence on ${\rm St}$ of the
PDFs $\mathcal{P}(a)$, $\mathcal{P}(a_{t})$ and $\mathcal{P}(a_{n})$, the
cumulative PDF $\mathcal{Q}(\kappa)$, the mean square displacement, and the
number of inflection points $N_{\rm I}(t,{\rm St})$ at which
${\bm{a}}\times{\bm{v}}$ changes sign along a particle trajectory.
A point in a 2D flow is vortical or strain-dominated if the Okubo-Weiss
parameter $\Lambda=(1/8)(\omega^{2}-\sigma^{2})$ is, respectively, positive or
negative oku70 ; wei91 ; per+ray+mit+pan11 . We investigate how the
acceleration statistics of heavy particles depends on the sign of $\Lambda$ by
conditioning the PDFs of $a_{t}$ and $\kappa$ on this sign. In particular, we
obtain the conditional PDFs $\mathcal{P}^{+}$ and $\mathcal{P}^{-}$, where the
superscript stands for the sign of $\Lambda$. We find, on the one hand, that
the tail of $\mathcal{P}^{+}(a_{t})$ falls faster than that of
$\mathcal{P}^{-}(a_{t})$ because regions of the trajectory with high
tangential accelerations are associated with strain-dominated points in the
flow. On the other hand, the right tail of $\mathcal{P}^{+}(\kappa)$ falls
more slowly than that of $\mathcal{P}^{-}(\kappa)$, which implies that high-
curvature parts of a particle trajectory are correlated with vortical regions
of the flow. We give plots of $\mathcal{P}^{+}(a_{t})$,
$\mathcal{P}^{+}(\kappa)$, $\mathcal{P}^{-}(a_{t})$, and
$\mathcal{P}^{-}(\kappa)$ in Fig. (7) and Fig. (8). These trends hold for all
values of ${\rm St}$ and $k_{\rm inj}$ that we have studied.
Figure 7: (Color online) Semilog (base 10) plots of the PDFs of the tangential
component of the acceleration for ${\rm St}=0.1$ in vortical regions
$\mathcal{P}(a_{t}^{+})$ (red squares) and in strain-dominated regions
$\mathcal{P}(a_{t}^{-})$ (blue asterisks). Figure 8: (Color online) Semilog
(base 10) plots of PDF of the curvature of trajectories for ${\rm St}=0.1$ in
vortical regions $\mathcal{P}(\kappa^{+}\eta)$ (red squares), in strain-
dominated regions $\mathcal{P}(\kappa^{-}\eta)$ (blue asterisks), and in
general (i.e., without conditioning on the sign of $\Lambda$)
$\mathcal{P}(\kappa\eta)$ (black triangles). Figure 9: (Color online) Log-log
(base 10) plots for $k_{\rm inj}=50$ of $r^{2}$ versus $t/T_{\rm eddy}$ for
${\rm St}=0.1$ (red asterisks) and ${\rm St}=1$ (black squares).
In Fig. (9), we plot the square of the mean-squared displacement $r^{2}$
versus time $t$ for $k_{\rm inj}=50$; here too we see a crossove from
ballistic to Brownian behaviors; however, in contrast to the case $k_{\rm
inj}=4$, the crossover time $T_{\rm cross}$ does not depend significantly on
${\rm St}$.
Figure 10: (Color online) Semilog (base 10) plot of the PDF
$\mathcal{P}(\log_{10}(\kappa\eta))$ versus $\log_{10}(\kappa\eta)$, for ${\rm
St}=0.1$ (blue asterisks), 301 ${\rm St}=1$ (red squares) and ${\rm St}=2$
(black circles).
In Fig. (10), we plot the PDF $\mathcal{P}(\log_{10}(\kappa\eta))$ versus
$\log_{10}(\kappa\eta)$, for ${\rm St}=0.1$ (blue asterisks), ${\rm St}=1$
(red squares) and ${\rm St}=2$ (black circles). Such PDFs provide another
convenient way of displaying the power-law behaviors, as $\kappa\to\infty$ and
$\kappa\to 0$, which we have reported in the main part of this paper, where we
have used the cumulative PDF of $\kappa$ to obtain the power-law exponents.
In Table(3) we report the values of $\alpha$, $\alpha_{\rm n}$, $\alpha_{\rm
t}$, and the exponent $h_{\rm r}$ of the right tail of the PDF of the
trajectory curvature, for the case $k_{\rm inj}=50$ and for different values
of ${\rm St}$.
$Run$ | ${\rm St}$ | $\alpha$ | $\alpha_{\rm t}$ | $\alpha_{\rm n}$ | $h_{\rm r}$
---|---|---|---|---|---
I1 | $0.1$ | $0.39\pm 0.06$ | $0.69\pm 0.02$ | $0.40\pm 0.06$ | $2.16\pm 0.09$
I2 | $0.2$ | $0.47\pm 0.05$ | $0.81\pm 0.03$ | $0.46\pm 0.05$ | $2.14\pm 0.09$
I3 | $0.3$ | $0.55\pm 0.04$ | $0.95\pm 0.02$ | $0.54\pm 0.05$ | $2.1\pm 0.1$
I4 | $0.4$ | $0.63\pm 0.04$ | $1.09\pm 0.03$ | $0.61\pm 0.04$ | $2.10\pm 0.08$
I5 | $0.5$ | $0.71\pm 0.04$ | $1.21\pm 0.02$ | $0.68\pm 0.03$ | $2.09\pm 0.09$
I6 | $0.6$ | $0.80\pm 0.03$ | $1.34\pm 0.03$ | $0.77\pm 0.03$ | $2.08\pm 0.09$
I7 | $0.7$ | $0.88\pm 0.04$ | $1.48\pm 0.04$ | $0.85\pm 0.03$ | $2.07\pm 0.09$
I8 | $0.8$ | $0.97\pm 0.03$ | $1.60\pm 0.03$ | $0.94\pm 0.04$ | $2.07\pm 0.09$
I9 | $0.9$ | $1.05\pm 0.03$ | $1.73\pm 0.03$ | $1.01\pm 0.04$ | $2.1\pm 0.1$
I10 | $1.0$ | $1.16\pm 0.03$ | $1.87\pm 0.03$ | $1.10\pm 0.03$ | $2.1\pm 0.1$
Table 3: The values of $\alpha$, $\alpha_{\rm n}$, $\alpha_{\rm t}$, and the
exponent $h_{\rm r}$, for the case $k_{\rm inj}=50$ for different values of
${\rm St}$.
In Table(4) we report the exponent $h_{\rm l}$, which charcterizes
$\mathcal{P}(\kappa\eta)$, as $\kappa\to 0$, in both the cases $k_{\rm inj}=4$
and $k_{\rm inj}=50$. In both these cases and for all the different values of
${\rm St}$ we have studied, $h_{\rm l}=0.0\pm 0.1$.
${\rm St}$ | $0.1$ | $0.2$ | $0.3$ | $0.4$ | $0.5$ | $1.0$
---|---|---|---|---|---|---
$h_{\rm l}$ ($\tt FA$) | $0.0\pm 0.1$ | $0.0\pm 0.1$ | $0.0\pm 0.1$ | $0.0\pm 0.1$ | $0.0\pm 0.1$ | $0.0\pm 0.1$
$h_{\rm l}$ ($\tt IA$) | $0.0\pm 0.1$ | $0.0\pm 0.1$ | $0.0\pm 0.1$ | $0.0\pm 0.1$ | $0.0\pm 0.1$ | $0.0\pm 0.1$
Table 4: The exponent $h_{\rm l}$ that charcterizes $\mathcal{P}(\kappa\eta)$,
as $\kappa\to 0$, in both the cases $k_{\rm inj}=4$ and $k_{\rm inj}=50$ and
for different values of ${\rm St}$.
|
arxiv-papers
| 2014-02-27T20:30:36 |
2024-09-04T02:49:59.051661
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anupam Gupta (Uni of Rome and IISc), Dhrubaditya Mitra (NORDITA),\n Prasad Perlekar (TCIS), and Rahul Pandit (IISc)",
"submitter": "Dhrubaditya Mitra",
"url": "https://arxiv.org/abs/1402.7058"
}
|
1402.7067
|
# Glauber Dynamics: An Approach in a Simple Physical System
Vilarbo da Silva Junior [email protected] Centro de Ciências Exatas e
Tecnológicas, Universidade do Vale do Rio dos Sinos, Caixa Postal 275,
93022-000 São Leopoldo RS, Brazil Alexsandro M. Carvalho
[email protected] Centro de Ciências Exatas e Tecnológicas,
Universidade do Vale do Rio dos Sinos, Caixa Postal 275, 93022-000 São
Leopoldo RS, Brazil
###### Abstract
In this paper, we investigate a special class of stochastic Markov processes,
known as Glauber dynamics. Markov processes are importance, for example, in
the study of complex systems. For this, we present the basic theory of Glauber
dynamics and its application to a simple physical model. The content of this
work was designed in such a way that the reader unfamiliar with the Glauber
dynamics, finds here an introductory material with details and example.
## I Introduction
Probability theory studies the random phenomena and quantifies their
probabilities of occurrence. In principle, when we observe a sequence of
chance experiments, all of the past outcomes could influence our predictions
for the next experiment. For example, the prediction of grades of a student in
a sequence of exams in a course.
In 1906, Andrei Andreyevich Markov markov1 studied an important type of
random process (Markov process). In this process, the outcome of a given
experiment can affect the outcome of the next experiment. In other words, the
past is conditionally independent of the future given the present state of the
process. When presenting the model, Markov did not bother with the
applications. In fact, his intention was to show that the large numbers law is
valid even if the random variables are dependent. Nowadays, there are numerous
applications of which we mention: Biological phenomena gibson , Social science
stander , Electrical engineering kjersti among others. We emphasize that
physics is one of the areas of knowledge that often uses Markov process. For
example, Ehrenfest model Ehrenfest for the diffusion and Glauber dynamics
Glauber for the Ising model.
In this paper, we present a special class of Markov processes known as Glauber
dynamic Glauber ; Daniel . This topic is extremely important since it is the
theoretical basis for the Metropolis Algorithm (or simulated annealing
metropolis ), successfully used to treat and understand problems in physics of
complex systems Newman ; Barabasi .
The organization this article is as follows: In Section II, we focus in
mathematical background (stochastic process and statistical physics). The
generalities of the Glauber dynamics are presented in section III. As an
application, we present in section IV, the explicit implementation of the
Glauber dynamics on the model of localized magnetic nuclei. A physical
interpretation of the model is made in section V. In the sequence, we dedicate
the section VI to the final considerations.
## II Preliminary Concepts
### II.1 Stochastic Process
A continuous time Markov process on a finite or countable state space
$S=\\{s_{0},s_{1},\ldots\\}$ is a sequence of random variable
$X_{0},X_{t_{1}},\ldots$ taking values in $S$, with the property that, for all
$t\geq 0$ and $s_{0},s_{1},\ldots,s_{n},s\in S$ we have
$\mathbb{P}(X_{0}=s_{0}|X_{t_{n}}=s_{n})=\mathbb{P}(X_{0}=s_{0},\ldots,X_{t_{n}}=s_{n})$
(1)
whenever $t>t_{n}>\cdots>t_{1}>0$ and
$\mathbb{P}(X_{0}=s_{0}|X_{t_{n}}=s_{n})>0$. Here
$\mathbb{P}(A|B)=\mathbb{P}(A,B)/\mathbb{P}(B)$ denotes the conditional
probability of occur $A$ given that occurred $B$ and $\mathbb{P}(A,B)$ is the
probability that occur $A$ and $B$ simultaneously. Thus, given the state of
the process at any set of times prior to time $t$, the distribution of the
process at time $t$ depends only on the process at the most recent time prior
to time $t$. This notion is exactly analogous to the Markov property for a
discrete-time process norris .
If the time is discrete, the probability $\mathbb{P}$ associate to the Markov
process is completely determined by a time dependent stochastic matrix $P(t)$
(transition probability matrix) and a stochastic vector $\mu$ (initial
distribution). When $S$ has $n$ states it follows that $P(t)$ is a $n\times n$
matrix and its elements are denoted as
$p_{ij}(t)=\mathbb{P}(X_{t}=j|X_{0}=i)$, i.e, $p_{ij}(t)$ is the transition
probability from $i$ to $j$ in time $t$. Furthermore, the $n$ coordinates of
$\mu$ are $\mu_{i}=\mathbb{P}(X_{0}=i)$ represents the probability of finding
the system in state $i$ initially. A initial distribution $\mu$ are called
invariant or stationary from $P(t)$ if satisfies $\mu P(t)=\mu$ for all $t\geq
0$. This fact indicates the equilibrium state of the system. If there is a
invariant distribution $\mu$ from $P(t)$, them we have
$\lim_{t\rightarrow\infty}P(t)=M$ where all the lines of $M$ are $\mu$.
Now, if time is continuous, the dynamics of $P(t)$ is find as solution of the
initial value problem (Kolmogorov Equation norris )
$\displaystyle\frac{d}{dt}P(t)$ $\displaystyle=$ $\displaystyle QP(t),$ (2)
$\displaystyle P(0)$ $\displaystyle=$ $\displaystyle I,$
where $I$ is the identity matrix and the $Q$ matrix is called $Q$-matrix. The
elements of Q, $q_{ij}$, must satisfy the following conditions:
$\displaystyle 0\leq-q_{ii}<\infty,\;\;$ $\displaystyle\forall$
$\displaystyle\,i$ (3) $\displaystyle q_{ij}>0\;\;$ $\displaystyle\forall$
$\displaystyle\,i\neq j$ (4) $\displaystyle\sum_{j\in S}q_{ij}=0\;\;$
$\displaystyle\forall$ $\displaystyle\,i$ (5)
The $Q$-matrix is also called matrix row sum zero, which complies with its
last property. Each off-diagonal entry $q_{ij}$ we shall interpret as the rate
going from $i$ to $j$ and the diagonal elements are chosen in general as
$q_{ii}=-\sum_{j\neq i}q_{ij}$. For end, the explicit form of transition
probability matrix is $P(t)=e^{tQ}$ (solution of differential equation (2)),
where the exponential matrix is $e^{tQ}=\sum_{k=0}^{\infty}t^{k}Q^{k}/k!$.
Thus, the stochastic vector $\mu(t)$ that describes the probability of finding
the system in their states on time $t$ is $\mu(t)=\mu e^{tQ}$, where $\mu$ is
the initial distribution. The solution $P(t)=e^{tQ}$ shows how basic the
generator matrix $Q$ is to the properties of a continuous time Markov chain.
For example, if $\nu$ is a probability vector and satisfies $\nu Q=0$ means
$\nu$ is a stationary distribution of Markov processnorris .
Some stochastic processes have the property that, when the direction of time
is reversed the behavior of the process remains the same. This class of
stochastic process is known as reversible stochastic process. We say that a
continuous time Markov process $\\{X_{t}\\}_{t\geq 0}$ is reversible with
respect to initial distribution $\mu$ if, for all $n\geq 1$,
$s_{0},s_{1},\ldots,s_{n}\in S$ and $t_{n}>\cdots>t_{1}>0$ occur
$\mathbb{P}(X_{0}=s_{0},X_{t_{1}}=s_{1}\ldots,X_{t_{n}}=s_{n})=\mathbb{P}(X_{t_{n}}=s_{0},X_{t_{n}-t_{n-1}}=s_{1}\ldots,X_{0}=s_{n})$.
Intuitively, if we take a movie of such a process and, then, to run the movie
backwards the process result will be statistically indistinguishable from the
original process. There is a condition for reversibility that can be easily
checked, called detailed balance condition. In more detail, this condition is
obtained when Markov process is reversible with respect to initial
distribution $\mu$, i.e.,
$\mu_{i}q_{ij}=\mu_{j}q_{ij},$ (6)
for all states $i$ and $j$, where $q_{ij}$ denotes the entry of Q-matrix of
$P(t)$ and $\mu_{i}$ the coordinates of $\mu$. A first consequence is that if
$\mu$ and $Q$ satisfies the detailed balance condition then $\mu$ is a
stationary distribution for $P(t)$ norris . This condition has a very clear
intuitive meaning: in equilibrium, we must have the same number of transitions
in both directions ($i\rightarrow j$ and $j\rightarrow i$).
### II.2 Statistical Physics
Here, we present some concepts of equilibrium statistical mechanics. For a
complete treatment, we suggest the references: Huang huang and Reif reif .
Simple systems are macroscopically homogeneous, isotropic, discharged,
chemically inert and sufficiently large. Often, a simple system is called pure
fluid. A composite system is constituted by a set of simple systems separated
by constraints. Constraints are optimal partitions that can be restrictive to
certain variables. The main types of constraints are: adiabatic, fixed and
impermeable.
In relation the equilibrium thermodynamics is important to know its
postulates, which are:
* •
First Postulate: The microscopic state of a pure fluid is completely
characterized by the internal energy $U$, volume $V$ and number of particles
$N$.
* •
Second Postulate: There is a function of all extensive parameters of a
composite system named entropy
$S(U_{1},V_{1},N_{1},\ldots,U_{n},V_{n},N_{n})$, which is defined for all
equilibrium states. On removal of an inner constraints, the extensive
parameters assume values which maximize the entropy.
* •
Third Postulate: The entropy of a composite system is additive on each of its
components. Entropy is a continuous, differentiable and monotonically
increasing function.
* •
The fundamental postulate of statistical mechanics: In a closed statistical
system with fixed energy, all accessible microscopic states are equally
likely.
Figure 1: Simple system $G$ in contact with a thermal reservoir $R$ with
temperature $T$.
Let us consider a simple system $G$ in contact with a thermal reservoir $R$
with temperature $T$, by means of a diathermic constraint fixed and
impermeable (see Fig. 1), where $R$ is very large compared to $G$. If the
composite system $G+R$ is isolated with total energy $E_{0}$, then the
probability distribution
$\mu(T)=(\mu_{1}(T),\mu_{2}(T),\ldots)$ (7)
is characterized by
$\mu_{i}(T)=\frac{e^{-\frac{E_{i}}{T}}}{Z(T)},$ (8)
where $\mu_{i}(T)$ is the probability of finding the system $G$ in particular
microscopic state $i$ with energy $E_{i}$ and temperature $T$ (we choose the
Boltzmann constant $k_{B}=1$, for convenience). The normalization constant
$Z(T)=\sum_{j}e^{-E_{j}/T}$ is called partition function of the system.
Furthermore, the distribution $\mu(T)$ is known as a Gibbs states. Thus the
canonical ensemble consists in a set microscopic states $i$ accessible to the
system $G$ in contact with a thermal reservoir $R$ and temperature $T$, with
probability distribution given by Eq. (8) (Gibbs distribution). Clearly, there
is a energy fluctuation in the canonical ensemble. Using the Gibbs
distribution, we obtain that the average energy of system $G$ (using
$\beta=1/T$) is
$\left<E\right>=-\frac{\partial}{\partial\beta}\log{(Z(\beta))}=\sum_{j}\mu_{j}(\beta)E_{j}$
(9)
and its variance can be write as
$\sigma^{2}(E)=\left<E^{2}\right>-\left<E\right>^{2}=\frac{\partial^{2}}{\partial\beta^{2}}\log{(Z(\beta))},$
(10)
where $E_{j}$ is the energy in a particular microscopic state $j$.
In summary, based on the Gibbs distribution, equilibrium statistical mechanics
indicates that states with lower energy are more likely than those with higher
energy.
## III Glauber Dynamics
We saw in the previous section, the Gibbs state (7) is the equilibrium state
of the system. Thus, a relevant question is: how the system evolves from the
initial state to the Gibbs state? Note that the equilibrium state, in the
context of the Markov process, corresponds to stationary distribution. Thus,
we forward the answer to the question as the solution of Kolgomorov equation
(2). However, for this purpose, we need to know the behavior of the Q-matrix.
An alternative to the shape of the Q-matrix is the Glauber dynamics Glauber .
In order to build a Glauber dynamics is necessary to know the single particle
energy function $E_{i}$ and their accessible microscopic states (states space)
$S=\\{i\\}$. Thus, we write explicitly the partition function $Z(T)$ and Gibbs
states $\mu_{i}(T)$. To describe the time dependent probabilities matrix
$P(t)=e^{tQ(T)}$ which is reversible with respect to the Gibbs state, we need
to propose a $Q(T)$-matrix that satisfies the detailed balance condition (6)
with Gibbs state $\mu(T)$ (for each fixed temperature $T$). There are many
other possibilities sinai , and the optimal choice is often dictated by
special features of the situation under consideration. However, for our
purposes, the one which will serve us best is the one whose $Q(T)$-matrix is
given by Daniel
$\displaystyle\begin{cases}q_{ij}(T)=e^{-\frac{1}{T}(E_{j}-E_{i})}&\mbox{if}\,\,E_{j}>E_{i}\\\
q_{ij}(T)=1&\mbox{if}\,\,E_{j}\leq E_{i}\\\ q_{ii}(T)=-\sum_{j\neq
i}q_{ij}(T),\end{cases}$ (11)
where $E_{i}$ is the single particle energy at particular microscopic state
$i$.
In the App. VII.1, we show that above matrix satisfies the detailed balance
condition with the Gibbs states. Therefore, the time dependent probability
matrix generated by Eq. (11) is a Glauber dynamics for the Gibbs states.
## IV Example: Localized Magnetic Nuclei
The nuclei of certain solids sessoli have integer spin. According to quantum
theory sakurai , each nuclei can have three quantum spin states (with
$\sigma=+1,0$ or $-1$). This quantum number measures the projection of the
nuclear spin along the axis of the crystalline solid. As the charge
distribution is not spherically symmetric, nuclei energy depends on the spin
orientation relative to the local electric field. In Fig. 2, we show a
possible configuration of this magnetic nucleons.
Figure 2: Picture of a magnetic nuclei chain. The up arrow means $\sigma=+1$,
no arrow means $\sigma=0$ and down arrow $\sigma=-1$.
Thus, nuclei in states $\sigma=\pm 1$ and $\sigma=0$ have energy,
respectively, $D>0$ and zero. Therefore, its energy function is given by
$E_{\sigma}=D\sigma^{2},$ (12)
where $D>0$ is electric field intensity and the microscopic states (quantum
states) are characterized by random variables (spins) $\sigma\in
S=\\{+1,0,-1\\}$. So, for each fixed temperature $T$, the partition function
is written as
$Z(T)=\sum_{\sigma\in\\{+1,0,-1\\}}e^{-\frac{1}{T}D\sigma^{2}}=1+2e^{-\frac{1}{T}D}$
(13)
and the Gibbs states coordinates $\mu_{\sigma}(T)$ (8) are given by
$\mu_{\sigma}(T)=\frac{e^{-\frac{1}{T}D\sigma^{2}}}{1+2e^{-\frac{1}{T}D}}.$
(14)
In order to explicit the $Q(T)$-matrix for this physical system, we apply the
energy function (12) at (11). This results (see App. VII.2)
$Q(T)=\left(\begin{array}[]{ccc}-2&1&1\\\
e^{-\frac{1}{T}D}&-2e^{-\frac{1}{T}D}&e^{-\frac{1}{T}D}\\\ 1&1&-2\\\
\end{array}\right).$ (15)
To find $P(t)=e^{tQ(T)}$, we need to diagonalize the $Q(T)$-matrix lay . For
this, we must present a invertible matrix $B$ such as $Q(T)B=BD_{3}$
(equivalently $Q(T)=BD_{3}B^{-1}$), where $B$´s columns is composite by
eigenvector of $Q(T)$, $B^{-1}$ its inverse and $D_{3}$ is a diagonal matrix
($3\times 3$) formed by eigenvalues of $Q(T)$. In this present case, the
characteristic polynomial is
$\displaystyle p(\lambda)$ $\displaystyle=$ $\displaystyle\det{(Q(T)-\lambda
I)}$ (16) $\displaystyle=$
$\displaystyle-e^{-\frac{1}{T}D}\lambda(3+\lambda)(2+e^{\frac{1}{T}D}(1+\lambda)).$
The eigenvalues $\lambda$’s are solution of characteristic equation
$p(\lambda)=0$, i.e,
$\lambda_{1}=-3,\qquad\lambda_{2}=0,\qquad\lambda_{3}=-Z(T),$ (17)
where $Z(T)$ is the partition function (13). Consequently, its associated
eigenvectors are
$v_{1}=\left(\begin{array}[]{c}-1\\\ 0\\\ 1\\\
\end{array}\right),\>v_{2}=\left(\begin{array}[]{c}1\\\ 1\\\ 1\\\
\end{array}\right),\>v_{3}=\left(\begin{array}[]{c}1\\\ -2e^{\frac{1}{T}D}\\\
1\\\ \end{array}\right).$ (18)
We conclude that $Q(T)$-matrix admits a decomposition $Q(T)=BD_{3}B^{-1}$ (see
App. VII.3). Then, $P(t)=Be^{tD_{3}}B^{-1}$ is responsible for describing the
dynamics of transition probabilities between spin. More explicit, $P(t)$ is
$P(t)=\left(\begin{array}[]{ccc}p_{+1+1}(t)&p_{+10}(t)&p_{+1-1}(t)\\\
p_{0+1}(t)&p_{00}(t)&p_{0-1}(t)\\\ p_{-1+1}(t)&p_{-10}(t)&p_{-1-1}(t)\\\
\end{array}\right)$ (19)
where
$\displaystyle p_{+1+1}(t)$ $\displaystyle=$ $\displaystyle
p_{-1-1}(t)=\frac{e^{-3t}}{2}+\frac{e^{-\frac{1}{T}D}}{Z(T)}+\frac{e^{-Z(T)t}}{2Z(T)},$
$\displaystyle p_{0+1}(t)$ $\displaystyle=$ $\displaystyle
p_{0-1}(t)=\frac{e^{-\frac{1}{T}D}}{Z(T)}-\frac{e^{-\frac{1}{T}D-Z(T)t}}{Z(T)},$
$\displaystyle p_{-1+1}(t)$ $\displaystyle=$ $\displaystyle
p_{+1-1}(t)=-\frac{e^{-3t}}{2}+\frac{e^{-\frac{1}{T}D}}{Z(T)}+\frac{e^{-Z(T)t}}{2Z(T)},$
$\displaystyle p_{+10}(t)$ $\displaystyle=$ $\displaystyle
p_{-10}(t)=\frac{1}{Z(T)}-\frac{e^{-Z(T)t}}{Z(T)},$ $\displaystyle p_{00}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{Z(T)}+\frac{2e^{-\frac{1}{T}D-Z(T)t}}{Z(T)}.$
Here,
$p_{\sigma,\widetilde{\sigma}}(t)=\mathbb{P}(X_{t}=\widetilde{\sigma}|X_{0}=\sigma)$
indicates the transition probability from spin state $\sigma$ to
$\widetilde{\sigma}$ in time $t$. For example, $p_{+10}(t)$ is the probability
from $\sigma=+1$ to $\sigma=0$ after time $t$.
It is relatively easy to prove that $P(t)$ satisfies the detailed balance
condition with Gibbs state $\mu(T)$ (14). Therefore, the stochastic Markov
process $\\{X_{t}\\}_{t\geq 0}$ where the random variable $X_{t}$ denotes the
quantum spin state of each located nucleus at time $t$ (i.e. $X_{t}=\sigma$)
is a Glauber dynamics.
## V Results and Discussion
Given the $P(t)$ elements and an initial quantum state, we can follow the
dynamics of the transition probability between quantum spin states. For
example, consider initially the state $\sigma=+1$. Note that we have
$p_{+1+1}(t)\geq p_{+1-1}(t)$ for all $t$. This means that since the nuclei is
in the quantum spin state $\sigma=+1$ is more likely that it remains in such a
state that it “flip” to the quantum spin state $\sigma=-1$. In the Fig. 3, we
present the dynamics of some transition probabilities. For our choice of
parameters $T=1$ and $D=\log{(2)}$ we have
$\lim_{t\rightarrow\infty}\mu(T)(t)=\mu(T)=(1/4\,\,\,1/2\,\,\,1/4)$.
Therefore, in a sample of $N$ nuclei, on average $N/2$ occupy the quantum spin
state $\sigma=0$, $N/4$ occupy $\sigma=+1$ and $N/4$ occupy $\sigma=-1$. In
this figure, we see the convergence (exponential) this limit as well as the
consistency in their values.
Figure 3: Time dependence for some transition probabilities
$p_{\sigma\widetilde{\sigma}}(t)$ for $T=1$ and $D=\log{(2)}$. The solid line
corresponds to $p_{00}(t)$, short dashed line $p_{+1+1}(t)$, medium dashed
line $p_{+10}(t)$ and long dashed line $p_{+1-1}(t)$.
Still on the limit $t\rightarrow\infty$, we have
$\lim_{t\rightarrow\infty}P(t)=\left(\begin{array}[]{ccc}\mu_{+1}(T)&\mu_{0}(T)&\mu_{-1}(T)\\\
\mu_{+1}(T)&\mu_{0}(T)&\mu_{-1}(T)\\\ \mu_{+1}(T)&\mu_{0}(T)&\mu_{-1}(T)\\\
\end{array}\right)$ (20)
where $\mu_{\sigma}(T)$ are the coordinates of Gibbs state $\mu(T)$. This
result is consistent with that shown in the Sec. II.1, i.e,
$\mu(T)=(\mu_{+1}(T)\,\,\mu_{0}(T)\,\,\mu_{-1}(T))$ is the unique equilibrium
state (invariant distribution) for $P(t)$. Then, whatever the initial spin
states distribution $\nu=(\nu_{+1}\,\,\nu_{0}\,\,\nu_{-1})$, we always have
$\lim_{t\rightarrow\infty}\mu(T)(t)=\lim_{t\rightarrow\infty}\nu P(t)=\mu(T)$.
This result, partly, justifies freedom of choice in the initial distribution
of spin states in Monte Carlo simulations (Metropolis algorithm binder1 ).
Note by Eq. (14) that $\mu_{0}(T)\geq\mu_{+1}(T)=\mu_{-1}(T)$. Thus, we can
conclude that after a long time, most of the nuclei are occupying the quantum
spin state $\sigma=0$. This occupation number is due to the fact that the
quantum spin state $\sigma=0$ is less energetic than the other two states
($E_{0}=0$ and $E_{+1}=E_{-1}=D>0$). This is in accordance with a general
physical principle that any physical system tends to occupy the lower energy
state. Additionally, note that if $T\rightarrow\infty$, we prove that
$\lim_{T\rightarrow\infty}\mu_{0}(T)=\lim_{T\rightarrow\infty}\mu_{+1}(T)=\lim_{T\rightarrow\infty}\mu_{-1}(T)=1/3$.
Therefore, for high temperature the three quantum spin states are equally
likely (same occupation number). In the opposite limit, when $T\rightarrow 0$,
$\lim_{T\rightarrow 0}\mu_{0}(T)=1$ and $\lim_{T\rightarrow
0}\mu_{+1}(T)=\lim_{T\rightarrow 0}\mu_{-1}(T)=0$. This indicates that for low
temperatures, all the nuclei tend to occupy the quantum spin state of lower
energy $\sigma=0$ (ground state). Another important verification is the
average number of spin $\left<\sigma\right>$, given by
$\left<\sigma\right>=\sum_{\sigma\in S}\sigma\mu_{\sigma}(T)=0$ where
$S=\\{+1,0,-1\\}$. If we chose a nuclei randomly, is more likely to be found
in the quantum spin state $\sigma=0$. The quantity $m=\left<\sigma\right>$ is
called the magnetization of system reif .
## VI Conclusions
This paper presents a review about continuous time stochastic Markov processes
which are reversible with respect to Gibbs State, called Glauber dynamics.
The main result of our exposition is contained in Sec. IV. We use the theory
developed in the preceding sections applying them in a very simple physical
model. This example, contains the necessary ingredients to illustrate richness
of the method.
We show that, after a long time interval, distribution of quantum spin states
are given by the state Gibbs. This state is a equilibrium state for the
Glauber dynamics. So any initial distribution of spin state will relax in
$\mu(T)$. This, ,in turn, justifying the free choice of the initial
distribution of spin states in computational simulations via Monte Carlo
method.
For a fixed temperature, we verify that the quantum spin state $\sigma=0$ is
the one with the highest number of occupants (most likely). This is consistent
with what is expected physically since $\sigma=0$ is the lowest energy state.
This fact indicates that average value of the random spin variable was zero
and, consequently, a zero magnetization for a sample of this kind of spins.
Finally, for high temperatures, the nuclei are uniformly distributed in the
quantum spin states ($1/3$ for each). On the other hand, for low temperatures,
all the nuclei tend to occupy a quantum spin state $\sigma=0$.
## VII Appendices
### VII.1 Proof of the Detailed Balance Condition
In order to show the equality (6) for $Q(T)$-matrix (11) and Gibbs state (8)
let us assume, without loss of generality, that $E_{j}>E_{i}$. So,
$\displaystyle\mu_{i}(T)q_{ij}(T)$ $\displaystyle=$
$\displaystyle\frac{1}{Z(T)}e^{-\frac{E_{i}}{T}}e^{\frac{1}{T}(E_{j}-E_{i})}$
$\displaystyle=$
$\displaystyle\frac{1}{Z(T)}e^{-\frac{E_{i}}{T}}e^{-\frac{E_{j}}{T}}e^{\frac{E_{i}}{T}}$
$\displaystyle=$
$\displaystyle\frac{1}{Z(T)}e^{-\frac{E_{j}}{T}}e^{-\frac{E_{i}}{T}+\frac{E_{i}}{T}}$
$\displaystyle=$
$\displaystyle\frac{1}{Z(T)}e^{-\frac{E_{j}}{T}}1=\mu_{j}(T)q_{ji}(T),$
because as we are assuming $E_{j}>E_{i}$ it follows that $E_{i}<E_{j}$ then by
(11) $q_{ji}(T)=1$. Therefore, $Q(T)$ and $\mu(T)$ satisfy the detailed
balance condition and hence $P(t)=e^{tQ(T)}$ generated by $Q(T)$ is a Glauber
dynamics.
### VII.2 $Q(T)$-matrix Calculations
In this appendix, we show in detail the entry of $Q(T)$-matrix. As the state
space $S=\\{+1,0,-1\\}$ has tree states our $Q(T)$-matrix is $3\times 3$ and
its elements $q_{\sigma,\widetilde{\sigma}}(T)$, evaluated as prediction in
Eq. (11) with Eq. (12), are given by:
$\displaystyle q_{+10}(T)$ $\displaystyle=$ $\displaystyle
e^{-\frac{1}{T}(E_{0}-E_{+1})}=1,$ $\displaystyle q_{+1-1}(T)$
$\displaystyle=$ $\displaystyle
e^{-\frac{1}{T}(E_{-1}-E_{+1})}=e^{-\frac{1}{T}(D-D)}=1,$ $\displaystyle
q_{+1+1}(T)$ $\displaystyle=$ $\displaystyle-q_{+10}(T)-q_{+1-1}(T)=-2,$
$\displaystyle q_{0+1}(T)$ $\displaystyle=$ $\displaystyle
e^{-\frac{1}{T}(E_{+1}-E_{0})}=e^{-\frac{1}{T}(D-0)}=e^{-\frac{1}{T}D},$
$\displaystyle q_{0-1}(T)$ $\displaystyle=$ $\displaystyle
e^{-\frac{1}{T}(E_{-1}-E_{0})}=e^{-\frac{1}{T}(D-0)}=e^{-\frac{1}{T}D},$
$\displaystyle q_{00}(T)$ $\displaystyle=$ $\displaystyle-
q_{0+1}(T)-q_{0-1}(T)=-2e^{-\frac{1}{T}D},$ $\displaystyle q_{-1+1}(T)$
$\displaystyle=$ $\displaystyle
e^{-\frac{1}{T}(E_{+1}-E_{+1})}=e^{-\frac{1}{T}(D-D)}=1,$ $\displaystyle
q_{-10}(T)$ $\displaystyle=$ $\displaystyle e^{-\frac{1}{T}(E_{0}-E_{-1})}=1,$
$\displaystyle q_{-1-1}(T)$ $\displaystyle=$ $\displaystyle-
q_{-1+1}(T)-q_{-10}(T)=-2.$
because $E_{\pm 1}>E_{0}$.
### VII.3 Explicit Elements: $B$, $D_{3}$ and $B^{-1}$
In Sec. (IV), we discus a decomposition $Q(T)=BD_{3}B^{-1}$. Explicitly,
$B=\left(\begin{array}[]{ccc}-1&1&1\\\ 0&1&-2e^{-\frac{1}{T}D}\\\ 1&1&1\\\
\end{array}\right),\qquad D_{3}=\left(\begin{array}[]{ccc}-3&0&0\\\ 0&0&0\\\
0&0&-Z(T)\\\ \end{array}\right)$
and
$B^{-1}=\left(\begin{array}[]{ccc}-\frac{1}{2}&0&\frac{1}{2}\\\
\frac{e^{-\frac{1}{T}D}}{Z(T)}&\frac{1}{Z(T)}&\frac{e^{-\frac{1}{T}D}}{Z(T)}\\\
\frac{1}{2Z(T)}&-\frac{1}{Z(T)}&\frac{1}{2Z(T)}\\\ \end{array}\right).$
## References
* (1) A. A. Markov, Rasprostranenie zakona bol shih chisel na velichiny, zavisyaschie drug ot druga (Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete), 2-ya seriya 15 135-156 (1906).
* (2) M. C. Gibson, A. B. Patel, R. Nagpal and N. Perrimon, The emergence of geometric order in proliferating metazoan epithelia Nature 442 05014 (2006).
* (3) J. Stander, D. P. Farrington, G. Hill and P. M. E. Altham, Markov Chain Analysis and Specialization in Criminal Careers Br J Criminol 29 317-335 (1989).
* (4) K. Aas, L. Eikvil and R. B. Huseby, Applications of hidden Markov chains in image analysis Pattern Recognition 32 703-713 (1999).
* (5) P. and T.Ehrenfest, Uber zwei bekannte Einwande gegen das Boltzmannsche H-Theorem Physikalische Zeitschrift 8 311-314 (1907).
* (6) R. J. Glauber, Time Dependent Statistics of the Ising Model J Math Phys 4 294-307 (1963).
* (7) S. W. Daniel, An Introduction to Markov Processes (Springer-Verlag, New York, 2000).
* (8) M. Nicholas, R. W. Arianna, R. N. Marshall, T. H. Augusta, T. Edward, Equation of State Calculations by Fast Computing Machines J. Chem. Phys. 21 1087-1093 (1953).
* (9) E. J. Newman, Complex Systems: A Survey cond-mat.stat-mech 79 800-810 (2011).
* (10) R. Albert, A.L. Barabási, Statistical mechanics of complex networks Rev. Mod. Phys. 74 47-97 (2002).
* (11) J. R. Norris, Markov Chains (Cambridge Series in Statistical and Probability Methematics, Cambridge U. Press, 1997.).
* (12) K. Huang, Statistical Mechanics (John Wyley Inc, New York, 1963.).
* (13) F. Reif, Fundamentals of Statistical and Thermal Physics (McGrawHill, New York, 1965.)
* (14) Y. G. Sinai, Probability Theory : An Introductory Course by Yakov G. Sinai ( Springer Textbook Ser, New York, 1992.)
* (15) Lay, David C., Linear Algebra and Its Applications (Addison Wesley, New York, 2005.)
* (16) R. Sessoli and D. Gatteschi, Quantum Tunneling of Magnetization and Related Phenomena in Molecular Materials Angew. Chem. 42 268 (2003).
* (17) J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, New York, 1994.)
* (18) K, Binder, D. W. Heermann, Monte Carlo Simulation in Statistical Physics : An Introductory (Springer-Verlag, New York, 2002.)
|
arxiv-papers
| 2014-02-27T20:56:24 |
2024-09-04T02:49:59.059894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vilardo da Silva Junior and Alexsandro M. Carvalho",
"submitter": "Alexsandro Carvalho",
"url": "https://arxiv.org/abs/1402.7067"
}
|
1402.7108
|
2-categories admitting bicategories of fractions]
On certain 2-categories admitting localisation by bicategories of fractions
D. M. Roberts]David Michael Roberts
School of Mathematical Sciences, University of Adelaide, SA 5005,
DMR was supported financially by ARC grant number DP120100106, and emotionally by Mrs R.
This document is released under a CC0 license <http://creativecommons.org/publicdomain/zero/1.0/>
Pronk's theorem on bicategories of fractions is applied, in almost all cases in the literature, to 2-categories of geometrically presentable stacks on a 1-site.
We give an proof that subsumes all previous such results and which is purely 2-categorical in nature, ignoring the nature of the objects involved.
The proof holds for 2-categories that are not (2,1)-categories, and we give conditions for local essential smallness.
[2000]primary 18D05; secondary 18F10, 18E35
§ INTRODUCTION
The area of higher geometry deals broadly with generalisations of `spaces', be they topological, differential geometric, algebro-geometric etc.,
that can be represented by groupoids (or higher groupoids) in the original category of spaces.
Usually these go by the label differential, topological, algebraic etc. stacks, but when viewed as stacks there are more morphisms between objects than when viewed simply as internal groupoids; there are non-invertible maps of groupoids that become equivalences of the associated stacks.
Pronk, in [7], formulated what it meant to localise a bicategory at a class of morphisms and introduced a bicategory of fractions that exists under certain conditions in order to construct this localisation.
She then went on to show that 2-categories of differentiable, topological and algebraic stacks (of certain sorts) were indeed localisations of the 2-categories of groupoids internal to the appropriate categories.
Since then, many other cases of 2-categorical localisations have been considered, using Pronk's result applied to other categories (for extensive discussion and examples see <cit.>).
However, almost all of them—only two exceptions are known to the author—deal with internal groupoids and/or stacks in some setting.
In this case, the 2-category in question, and the class of morphisms at which one wants to localise, satisfy some properties making available a much simpler calculus of fractions, namely anafunctors.
These were introduced by Makkai [6] for the category of sets sans Choice and in the general internal setting by Bartels [2].
The author's [9] considered the case of a sub-2-category $C\into \Cat(S)$ of the 2-category of categories internal to a subcanonical site $(S,J)$, satisfying some mild closure conditions.
The main result of [9] is that such 2-categories admit a bicategory of fractions at the so-called weak equivalences (also called Morita equivalences), and that anafunctors also calculate this localisation.
This note serves to show that given a 2-category with the structure of a 2-site of a certain form (all covering maps must be representably fully faithful), the same result holds – namely that the bicategory of fractions of Pronk exists.
One can then approach the theory of presentable stacks (on 1-sites) in a formal way, analogous to Street's formal theory of stacks [12] (cf Shulman's [11]).
This result covers all others in the literature dealing with localising 2-categories of internal categories or groupoids.
It may also replicate the result in [8], although the framework therein is conceptually more pleasing; the theorems of this note are definitely sufficient to imply the applications of the abstract framework of [8]
Both [8], and the recent paper [1] (written in parallel with the present note), deal with constructing localisations via fibrancy/projectivity.
Hom-categories in the constructions of localisations in both papers are in fact hom-categories of the original bicategory, and so one is assured of local smallness, a problem when localising any large (bi-)category, using local smallness of the original bicategory.
The present note does not assume existence of enough fibrant objects or projectives to prove local (essential) smallness (<ref>).
It certainly assumes less than the applications in [8] (prestacks on a subcanonical site) or [1] (internal groupoids in a regular category).
Sometimes when calculating the localisation of a 2-category of internal groupoids, various authors use what are variously known as Hilsum-Skandalis morphisms or right principal bibundles (see <cit.> for discussion and references).
In the more general setting of 2-sites as defined here such a definition is not possible, as one has hom-categories that are not groupoids.
Additionally, composition of 1-arrows in the bicategory of internal groupoids and bibundles requires existence of pullback-stable reflexive coequalisers, an assumption not made here.
Also, the definition of a bibundle between internal categories is not clear and the right notion of a map of bibundles (i.e. 2-arrows in the localisation) does not appear to be as simple as in the groupoid case.
The author thanks the organisers of the Australian Category Seminar for the opportunity to present an early version this work in October 2011.
Comments by the referee lead to a rethink of this paper and subsequent strengthening of the results.
§ PRELIMINARIES
Though this paper touches lightly on the theory of bicategories, a knowledge of 2-categories is sufficient (an accessible reference is [5]).
We consider our 2-categories to have one extra piece of structure, namely an analogue of a Grothendieck pretopology.
A fully faithful singleton coverage on a 2-category $K$ is a class $J$ of 1-arrows satisfying the following properties:
* $J$ contains the identity arrows and is closed under composition;
* for all $q\colon u\to x\in J$ and 1-arrows $f\colon y\to x$, there is a square
\[
\xymatrix{
v \ar[d]_k \ar[r] & u\ar[d]^q_{\ }="s" \\
y \ar[r]_{f}^{\ }="t" & x
\ar@{=>}_{\simeq}"s";"t"
\]
with $k\in J$;
* for any $q\colon u\to x$ in $J$ the functor $q_*\colon K(z,u)\to K(z,x)$ is fully faithful;
Morphisms satisfying (iii) are called ff 1-arrows. A 2-site will here denote a 2-category equipped with a fully faithful singleton coverage.
For brevity this paper will use the terminology 2-site even though this has been used elsewhere for something more general.
Note that that $K$ is not necessarily small, but in what follows may sometimes be locally essentially small.
That is, the hom-categories $K(x,y)$ are equivalent to small categories for all objects $x$ and $y$.
One might think about the 1-arrows in $J$ as being something like acyclic fibrations in a category with fibrant objects, without the requirement for the existence of fibrant objects.
We define the analogue of weak equivalences in this setting.
A 1-arrow $x\to y$ in $(K,J)$ is called $J$-locally split if there is a map $u\to y$ in $J$ and a diagram of the form
\[
\xymatrix{
& x\ar[d] \\
u \ar[r]^(.7){\ }="t" \ar[ur]_(.7){\ }="s" & y
\ar@{=>}"s";"t"
\]
with the 2-arrow an isomorphism.
A weak equivalence in $(K,J)$ is an ff 1-arrow that is $J$-locally split.
The class of weak equivalences will be denoted $W_J$.
As an example, take the 2-category $K$ to be $\Cat(S)$ or $\Gpd(S)$ for $(S,T)$ a finitely complete site with singleton pretopology $T$.[Recall that a singleton pretopology $T$ is a class of arrows containing all identity arrows and closed under composition and pullbacks (which must exist for arrows in $T$).]
One can also take the 2-category of Lie groupoids, which is course is not finitely complete – in this instance, $T$ can be taken as the pretopology of surjective submersions.
In each instance the pretopology $J=J(T)$ is defined to be the class of fully faithful functors such that the object component is a cover in $T$.
Then $K$ is a 2-site, as one can take pullbacks of 1-arrows in $J$, and fully faithful functors are closed under pullback.
It is an easy result <cit.> that the resulting weak equivalences in the sense of definition <ref> are the same as weak equivalences between internal categories in the sense of Bunge-Paré [3].
We shall need a more general definition for use later.
Let $A$ be a class of 1-arrows in a 2-category, and $A'$ a subclass.
We say $A'$ is cofinal in $A$ if for every $f\colon x\to y$ in $A$, there is a $g\colon z\to y$ in $A'$ and an $s\colon z\to x$ such that $f\circ s \simeq g$.
If for every object $y$, the arrows in $A'$ with codomain $y$ comprise a set, we say $A'$ is a locally small cofinal class.
Thus $J$ is cofinal in $W_J$, but we will later use classes $J' \subset J$ that do not give the structure of a 2-site as above.
Given a 2-category (or bicategory) $B$ with a class $W$ of 1-arrows, we say that a 2-functor $Q\colon B \to \widetilde{B}$ is a localisation of $B$ at $W$ if it sends the 1-arrows in $W$ to equivalences in $\widetilde{B}$ and is universal with this property.
This latter means that for any bicategory $A$
precomposition with $Q$,
\[
Q^* \colon \Bicat(\widetilde{B},A) \to \Bicat_W(B,A),
\]
is an equivalence of hom-bicategories, with $\Bicat_W$ meaning the full sub-bicategory on those 2-functors sending arrows in $W$ to equivalences.
The definition of a bicategory of fractions of [7] gives a reasonably convenient way to calculate the localisation at a class of arrows, satisfying properties as follows:
* $W$ contains all equivalences;
* $W$ is closed under composition and isomorphism;
* for all $w\colon a' \to a,\ f\colon c \to a$ with $w\in W$ there
exists a 2-commutative square
\[
\xymatrix{
p \ar[d]_v \ar[r] & a'\ar[d]^w_{\ }="s" \\
c \ar[r]_f^{\ }="t" & a
\ar@{=>}_{\simeq}"s";"t"
\]
with $v\in W$;
if $\alpha\colon w \circ f \Rightarrow w \circ g$ is a 2-arrow and $w\in W$ there is a 1-cell $v \in W$ and a 2-arrow $\beta\colon f\circ v \Rightarrow g \circ v$ such that $\alpha\circ v = w \circ \beta$.
when $\alpha$ is an isomorphism, we require $\beta$ to be an isomorphism too; when $v'$ and $\beta'$ form another such pair, there exist 1-cells $u,\,u'$ such that $v\circ u$ and $v'\circ u'$ are in $W$, and an isomorphism $\epsilon\colon v\circ u \Rightarrow v' \circ u'$ such that the following diagram commutes:
\begin{equation}\label{2cf4.diag}
\xymatrix{
f \circ v \circ u \ar@{=>}[r]^{\beta\circ u}
\ar@{=>}[d]_{f\circ \epsilon}^\simeq &
g\circ v \circ u
\ar@{=>}[d]^{g\circ \epsilon}_\simeq \\
f\circ v' \circ u' \ar@{=>}[r]_{\beta'\circ u'} &
g\circ v' \circ u'
\end{equation}
If BF1–BF4 hold, we say $(B,W)$ admits a bicategory of fractions.
Given such a pair $(B,W)$, Pronk constructed a new bicategory $B[W^{-1}]$ with the same objects as $B$ and a functor $U\colon B \to B[W^{-1}]$ that is a localisation of $B$ at $W$.
We will describe the (underlying graphs of the) hom-categories of $B[W^{-1}]$, since this is the most detail we need for the results below.
Let $x$ and $y$ be objects of $B[W^{-1}]$ (which are just objects of $B$).
The 1-arrows from $x$ to $y$ are spans
\[
x \xleftarrow{w} u \xrightarrow{f} y
\]
where $w\in W$.
The 2-arrows $(w_1,f_1) \Rightarrow (w_2,f_2)$ are represented by diagrams
\[
\xymatrix{
& u_1 \ar[dl]_{w_1}^(.6){\ }="s1" \ar[dr]^{f_1}_(.6){\ }="s2"\\
x & v \ar[u]_{p_1} \ar[d]^{p_2} & y\\
& u_2 \ar[ul]^{w_2}_(.6){\ }="t1" \ar[ur]_{f_2}^(.6){\ }="t2"
\ar@{=>}"s1";"t1"^\alpha
\ar@{=>}"s2";"t2"_\beta
\]
where $w_i\circ p_i$ is in $W$ for $i=1,2$ and $\alpha$ is invertible.
Two such diagrams, with data $(v,p_1,p_2,\alpha,\beta)$ and $(v',p'_1,p'_2,\alpha',\beta')$, are equivalent when there exists a diagram
\[
\xymatrix{
u_1 & \ar[l]_{p'_1} v'\\
v \ar[u]^{p_1} \ar[d]_{p_2} &
t \ar[l]_{q}="t1"^{\ }="s2"
\ar[u]_{q'}^(.6){\ }="s1"
\ar[d]^{q'}_(.6){\ }="t2" \\
u_2 & \ar[l]^{p'_2}_{\ } v'
\ar@{=>}"s1";"t1"_{\gamma_1}
\ar@{=>}"s2";"t2"_{\gamma_2}
\]
where $\gamma_1$ and $\gamma_2$ are invertible, $w_1\circ p_1\circ q$ and $w_1\circ p'_1\circ q'$ are in $W$ and
\[
\noindent\makebox[\textwidth]{
\raisebox{36pt}{
\xymatrix{
& \ar[dl]_{w_1}^{\ }="s3" u_1 & \ar[l]_{p'_1} v'\\
x &v \ar[u]_{p_1} \ar[d]^{p_2} &
t \ar[l]_{q}="t1"^{\ }="s2"
\ar[u]_{q'}^(.6){\ }="s1"
\ar[d]^{q'}_(.6){\ }="t2" \\
& \ar[ul]^{w_2}_{\ }="t3" u_2 & \ar[l]^{p'_2}_{\ } v'
\ar@{=>}"s1";"t1"_{\gamma_1}
\ar@{=>}"s2";"t2"_{\gamma_2}
\ar@{=>}"s3";"t3"_\alpha
\; = \;
\raisebox{36pt}{
\xymatrix{
& \ar[dl]_{w_1}^{\ }="s" u_1 \\
x & v' \ar[u]_{p'_1} \ar[d]^{p'_2} & \ar[l]_{q'} t\;,\\
& u_2 \ar[ul]^{w_2}_{\ }="t"
\ar@{=>}"s";"t"_{\alpha'}
\quad
\raisebox{36pt}{
\xymatrix{
v' \ar[r]^{p'_1} & u_1 \ar[dr]^{f_1}_{\ }="s3" \\
t \ar[u]^{q'}_(.6){\ }="s1" \ar[d]_{q'}^(.6){\ }="t2" \ar[r]^{q}="t1"_{\ }="s2" &
v \ar[u]^{p_1}^(.6){\ }
\ar[d]_{p_2}_(.6){\ }& y\\
v' \ar[r]_{p'_2}^{\ } & u_2 \ar[ur]_{f_2}^{\ }="t3"
\ar@{=>}"s1";"t1"^{\gamma_1}
\ar@{=>}"s2";"t2"^{\gamma_2}
\ar@{=>}"s3";"t3"^\beta
\; = \;
\raisebox{36pt}{
\xymatrix{
& u_1 \ar[dr]^{f_1}_{\ }="s3" \\
t \ar[r]^{q'}="t1"_{\ }="s2" &
v' \ar[u]^{p'_1}^(.6){\ }
\ar[d]_{p'_2}_(.6){\ }& y\; .\\
& u_2 \ar[ur]_{f_2}^{\ }="t3"
\ar@{=>}"s3";"t3"^{\beta'}
We then define a 2-arrow of $B[W^{-1}]$ to be an equivalence class of diagrams.
The following lemma is stated in more generality in [14], but we shall merely state it in terms that we need here.
Let $(K,W)$ be a 2-category admitting a bicategory of fractions.
Let $F_i = (x\xleftarrow{w_i} u_i \xrightarrow{f_i} y)$, $i=1,2$ be 1-arrows in $K[W^{-1}]$, and
\[
\xymatrix{
v \ar[d]_{p_2} \ar[r]^{p_1} & u_1\ar[d]^{w_1}_{\ }="s" \\
u_2 \ar[r]_{w_2}^{\ }="t" & x
\ar@{=>}_{\alpha}"s";"t"
\]
be a chosen filler, using BF3, such that $w_i\circ p_i\in W$, $i=1,2$. Then any 2-arrow $F_1 \Rightarrow F_2$ in $K[W^{-1}]$ is represented by a diagram of the form
\[
\xymatrix{
& u_1 \ar[dl]^{\ }="s" & \ar@{=}[l] u_1 \ar[dr]_{\ }="s2"^{f_1}\\
x & v \ar[u]_{p_1} \ar[d]^{p_2}& v' \ar[l]_{q'} \ar[u] \ar[d]& y\\
&u_2 \ar[ul]_{\ }="t" & \ar@{=}[l] u_2 \ar[ur]^{\ }="t2"_{f_2}
\ar@{=>}"s";"t"_\alpha
\ar@{=>}"s2";"t2"^\beta
\]
where $q'\in W$.
The conclusion of the lemma in [14] does not mention that $q'\in W$, but examination of the proof shows that it is so.
§ RESULTS
The first main result is as follows:
A 2-site $(K,J)$ admits a bicategory of fractions for $W_J$.
We verify the conditions in the definition of a bicategory of fractions.
* An internal equivalence $f\colon x\to y$ is clearly $J$-locally split.
Let $g\colon y\to x$ be a pseudoinverse to $f$, and let $w$ be some object of $K$.
Then $g_*$ is a pseudoinverse to $f_*$, where $f_*\colon K(w,x)\to K(w,y)$ is post-composition with $f$ (and analogously with $g_*$).
But then it is a well-known fact that equivalences of categories are fully faithful, and so $f$ is a ff 1-arrow.
* That the composition of ff 1-arrows is again ff, and that ff 1-arrows are closed under isomorphism follows from the analogous fact for fully faithful functors between categories.
So we only need to show the same for $J$-locally split arrows.
Consider the composition $g\circ f$ of two $J$-locally split arrows,
\[
\xymatrix{
u \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^(.35){q}&v \ar[d]^s
\ar@/^.5pc/[dr]_(.5){\ }="s2"^(.4){p}& \\
x\ar[r]_{f}^(.33){\ }="t1" & y \ar[r]_{g}^(.33){\ }="t2" & z
\ar@{=>}"s1";"t1"
\ar@{=>}"s2";"t2"
\]
The cospan $u\xrightarrow{q}y\xleftarrow{s} v$ completes to a 2-commuting square with top arrow $w \to v$ in $J$.
The composite $w \to z$ is in $J$, all 2-arrows are invertible, hence $g\circ f$ is $J$-locally split.
Let $w,f\colon x\to y$ be 1-arrows, $w$ be $J$-locally split and $a\colon w \Rightarrow f$ invertible.
It is immediate from the diagram
\[
\xymatrix{
u \ar[dd] \ar@/^.7pc/[ddrr]_{\ }="s1"^{u} \\
\\
x\ar@/^1pc/[rr]^{w}="t1"_{\ }="s2" \ar@/_1pc/[rr]_{f}^{\ }="t2"
\ar@{=>}"s1";"t1"
\ar@{=>}"s2";"t2"^{a}
\]
that $f$ is also $J$-locally split.
* Let $w\colon x\to y$ be a weak equivalence, and let $f\colon z\to y$ be any other 1-arrow.
From the definition of $J$-locally split, we have the diagram
\[
\xymatrix{
u \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{q}& \\
x\ar[r]_{w}^(.33){\ }="t1"&y
\ar@{=>}"s1";"t1"
\]
We complete the cospan to get a 2-commuting diagram
\[
\xymatrix{
& w \ar[dr]^{p}_{\ }="s2" \ar[dl] \\
u \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{q}& & z \ar[dl]^f_{\ }="t2"\\
x\ar[r]_{w}^(.33){\ }="t1" & y
\ar@{=>}"s1";"t1"
\ar@{=>}"s2";"t2"_\alpha
\]
with $p\in J$, $\alpha$ invertible, and by the trivial observation $J\subset W_J$, we have $p\in W_J$ and a 2-commuting square as required.
* Since $J$-equivalences are ff, given
\[
\xymatrix{
&y \ar[dr]^w \\
x \ar[ur]^f \ar[dr]_g & \Downarrow \alpha & z\\
& y \ar[ur]_w
\]
where $w\in W_J$, there is a unique $\beta\colon f \Rightarrow g$ such that
\[
\raisebox{36pt}{
\xymatrix{
&y \ar[dr]^w \\
x \ar[ur]^f \ar[dr]_g & \Downarrow \alpha & z\\
& y \ar[ur]_w
\equals
\raisebox{36pt}{
\xymatrix{
x \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g&
\Downarrow \beta & y \ar[r]^w & z \,.
\]
The existence of $\beta$ is the first half of BF4, with $v=\id_x$.
Note that if $\alpha$ is an isomorphism, so is $\beta$, since $w$ is ff.
Given $v'\colon t\to x \in W_J$ such that there is a 2-arrow
\[
\xymatrix{
&x \ar[dr]^f \\
t \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow \beta' & y\\
& x \ar[ur]_g
\]
\begin{align*}
\raisebox{36pt}{
\xymatrix{
& x \ar[dr]^f \\
w \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow \beta' & y \ar[r]^w & z\\
& x \ar[ur]_g
\equals &
\raisebox{36pt}{
\xymatrix{
&& y \ar[dr]^w \\
t \ar[r]^{v'} & x \ar[ur]^f \ar[dr]_g & \Downarrow \alpha & z\\
&& y \ar[ur]_w
} \nonumber \\
\equals &
\raisebox{36pt}{
\xymatrix{
t\ar[r]^{v'}& x \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g &\Downarrow \beta &
y \ar[r]^w & z
}\, ,
\end{align*}
then the fact $w$ is ff gives us
\[
\raisebox{36pt}{
\xymatrix{
& x \ar[dr]^f \\
t \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow \beta' & y \\
& x \ar[ur]_g
\equals
\raisebox{36pt}{
\xymatrix{
t\ar[r]^{v'}&x \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g
&\Downarrow \beta
& y
}\, .
\]
This is precisely the diagram (<ref>) with $v=\id_x$, $u=v'$, $u'=\id_w$ and $\epsilon$ the identity 2-arrow.
Hence BF4 holds.
This theorem should be compared with the theorem in the paper [1] (written in independently and in parallel with the present work).
The authors show there that given a class $\Sigma$ of ff arrows in a bicategory satisfying certain conditions, there is a bicategory of fractions for $\Sigma$.
The class of arrows $W_J$ satisfies the conditions for a faithful calculus of fractions <cit.>, using similar arguments as the preceeding proof.
The characterisation of $W_J$ as arising from a class $J$ as in definition <ref> is a means to arrive at a multitude of examples.
One would like to know if the localisation of $K$ at the weak equivalences is locally essentially small.
Let $K$ be a locally essentially small 2-category with a class of 1-arrows $W$ satisfying BF1–BF4.
If there is a locally small cofinal set $V \subset W$ then $K[W^{-1}]$ is locally essentially small.
First given any fraction $x \xleftarrow{w} u \to y$ (with $w\in W_J$) giving a 1-arrow in Pronk's construction <cit.> of $K[W^{-1}]$, there is an isomorphic 1-arrow $x\xleftarrow{v} t\to y$ where $v\in V$.
Thus there are only set-many choices of backwards-pointing arrows with which to form fractions, and by local essential smallness of $K$, only set-many fractions from $x$ to $y$ up to isomorphism.
To show the hom-categories $K[W^{-1}](x,y)$ are locally small, given a pair of fractions $x \xleftarrow{w_i} u_i \xrightarrow{f_i} y$, choose a filler for the cospan $u_1 \to x \leftarrow u_2$, namely
\[
\xymatrix{
v \ar[r]^{p_2} \ar[d]_{p_1} & u_2 \ar[d]^{w_2}_{\ }="s"\\
u_1 \ar[r]_{w_1}^{\ }="t" & x\;.
\ar@{=>}"s";"t"_\alpha
\]
Then arrows in the hom-category are given, by lemma <ref>, by the data of an arrow $q\colon v'\to v \in W$ and a 2-arrow $\beta\colon f_1\circ p_1 \circ q \Rightarrow f_2 \circ p_2 \circ q$ in $K$.
Fixing $q$, there are only set-many arrows as shown, since $K$ is locally essentially small.
Hence if we show a 2-arrow given by $(q,\beta)$ is also given (using the equivalence relation defining the 2-arrows of $K[W^{-1}]$) by $(r,\gamma)$ where $r\in V$, the hom-category is locally small.
Assume we have an arrow given by the data $(q,\beta)$, and we have a second $r\in W$ and a diagram
\[
\xymatrix{
& v'\ar[d]^q_{\ }="s"\\
v_0 \ar[r]_r^{\ }="t" \ar[ur]^s & v
\ar@{=>}"s";"t"_\phi
\]
with $\phi$ invertible.
Defining $\gamma$ as
\[
\xymatrix{
& v \ar[r] & u_1 \ar[dr]_(.3){\ }="s2" \\
v_0 \ar[ur]^r_(.6){\ }="s1" \ar[dr]_r^(.6){\ }="t3" \ar[r]_(.9){\ }="s3"^(.9){\ }="t1" & v' \ar[u] \ar[d]&& y\\
& v \ar[r] & u_2 \ar[ur]^(.3){\ }="t2"
\ar@{=>}"s1";"t1"_{\phi^{-1}}
\ar@{=>}"s2";"t2"_\beta
\ar@{=>}"s3";"t3"_\phi
\]
then one can use the span $v_0 \xleftarrow{=} v_0 \xrightarrow{s} v'$ and the 2-arrows $\phi$ and $\phi^{-1}$ to show that $(r,\gamma)$ defines the same 2-cell of $K[W^{-1}]$ as $(q,\beta)$.
Given a 2-site $(K,J)$, if $K$ is locally essentially small and $J$ has a locally small cofinal set, then $K[W_J^{-1}]$ is locally essentially small.
Notice that local essential smallness in not automatic.
Take the category $\Sch$ of schemes (over a base scheme, if one likes).
The singleton pretopology of fpqc maps (see e.g. <cit.>) is such that $(\Gpd(\Sch),J(\text{\emph{fpqc}}))$ does not satisfy the hypotheses of proposition <ref>.
This is because the class of fpqc maps has no locally small cofinal set <cit.>.
There are even categories that are a priori even better behaved in which proposition <ref> may fail to hold; for example toposes that are well-pointed and that have a natural number object.
Such toposes are models for set theory without the Axiom of Choice, and categories internal to them are simply small categories.
Examples are given by the categories of sets in models of ZF as given by Gitik (cf [15]) and Karagila [4], or the topos constructed in the author's [10].
There are many examples of 2-sites to which the results of this note apply, for instance [9] spends five pages discussing some of them.
This paper will add one example that is not covered by the results of [9].
Let $S$ be a finitely complete category with a singleton pretopology $T$.
Consider the 2-category $\Cat(S)$ of categories internal to $S$, with the structure of a 2-site given by example <ref>.
Note that 2-sites of this form are the ones considered in the many examples in [9].
Now fix a category $X$ in $S$, and consider the lax slice 2-category $\Cat(S)/_lX$.
This 2-category has as objects functors $Z \to X$, 1-arrows triangles
\[
\xymatrix{
Z_1 \ar[rr]^f_(.6){\ }="s" \ar[dr]^(.7){\ }="t" && Z_2 \ar[dl]\\
& X
\ar@{=>}"s";"t"^a
\]
and 2-arrows $(f_1,a_1) \Rightarrow (f_2,a_2)$ given by $f_1\Rightarrow f_2 \colon Z_1 \to Z_2$, commuting with $a_1$ and $a_2$.
Let $J_X$ be the class of 1-arrows $(w,a)$ in $\Cat(S)/_lX$ such that $w\in J$, for $J$ as given in the previous paragraph, and $a$ is invertible.
Likewise we have the slice 2-category $\Cat(S)/X$, where arrows $(f,a)$ have $a$ invertible, and the slice 2-category $\Gpd(S)/X$ where $X$ is an internal groupoid.
These three examples are locally small 2-categories when $S$ is a locally small category.
The class of arrows $J_X$ makes $(\Cat(S)/_lX,J_X)$ a 2-site.
The same statement holds for $(\Cat(S)/X,J_X)$ and $(\Gpd(S)/X,J_X)$, mutatis mutandis.
That $J_X$ is closed under composition and contains identity arrows is immediate.
We can specify a strict pullback of a map $(w,a)\colon U\to Z$ in $J_X$ along $(f,b)\colon Y\to Z$, given a strict pullback
\begin{equation}\label{eq:str_pullback_for_lifting}
\xymatrix{
Y\times_Z U \ar[r]^{\tilde{f}} \ar[d]_{\tilde{w}} &U \ar[d]^w\\
Y \ar[r]_f & Z
\end{equation}
of $w$ along $f$, as follows: let the map $Y\times_Z U \to X$ be the composite $Y\times_Z U \xrightarrow{\tilde{w}} Y \to X$.
The map $Y\times_Z U \to Y$ in $\Cat(S)/_lX$ is $(\tilde{w},\id)$, and as $\tilde{w} \in J$ and $\id$ is invertible, this is in $J_X$ as required.
The map $Y\times_Z U \to U$ is $(\tilde{f},c)$ where $c$ is
\[
\xymatrix{
& U \ar[dr]^(.8){\ }="t1" \ar@/^1pc/[drr]_{\ }="s1"\\
Y\times_Z U \ar[dr]_{\tilde{w}} \ar[ur]^{\tilde{f}} && Z \ar[r] & X\\
& Y \ar[ur]_(.8){\ }="s2" \ar@/_1pc/[urr]^{\ }="t2"
\ar@{=>}"s1";"t1"^{a^{-1}}
\ar@{=>}"s2";"t2"^{b}
\]
It is then easy to check that (<ref>) lifts to a commuting square in $\Cat(S)/_lX$.
To see that an arrow $(w,a)$ in $J_X$ is ff, we use the fact that given a diagram
\[
\xymatrix{
&Z_2 \ar[dr]^{(w,a)}_(.4){\ }="s" \\
Z_1 \ar[ur]^{(f,b)} \ar[dr]_{(g,c)} && Y\\
& Z_2 \ar[ur]_{(w,a)}^(.4){\ }="t"
\ar@{=>}"s";"t"_{\alpha}
\]
in $\Cat(S)/_lX$, we can find a unique $\beta\colon f\Rightarrow g\colon Z_1 \to Z_2$ in $\Cat(S)$ such that $\id_w\circ \beta = \alpha$.
To see that $\beta$ lifts to a 2-arrow in $\Cat(S)/_lX$, we paste $a^{-1}$ and the 2-arrow
\[
\xymatrix{
Z_1 \ar@/^.8pc/[rr]^f_{\ }="s1" \ar@/_.8pc/[rr]_g^{\ }="t1" \ar@/_.4pc/[drr]^(.6){\ }="t2" && Z_2 \ar[r]^w \ar[d]_(.4){\ }="s2"^{\ }="t3" & Y \ar@/^.4pc/[dl]_(.2){\ }="s3"\\
&& X
\ar@{=>}"s1";"t1"^\beta
\ar@{=>}"s2";"t2"^c
\ar@{=>}"s3";"t3"^a
\]
and get $b$, the required condition to give a 2-arrow in $\Cat(S)/_lX$.
The 2-cateories $\Cat(S)/_lX$, $\Cat(S)/X$ and $\Gpd(S)/X$ admit bicategories of fractions for the classes $W_{J_X}$ of weak equivalences.
If we assume that $J$ satisfies the condition WISC from [9] (namely all slices $J/x$ have a weakly initial set), then $W_{J_X}$ has a locally small cofinal class; the localisations above are then locally essentially small.
These 2-categories are not examples of 2-categories of internal categories or groupoids in some 1-category, so are not covered by the results of [9].
Given a 2-site $(K,J)$, if every arrow $j\colon u\to x\in J$ is such that $j^*\colon K(x,z) \to K(u,z)$ is fully faithful, then one can construct a simpler model for the localisation $K[W_J^{-1}]$, where 2-arrows are no longer equivalence classes of diagrams, but given by individual diagrams.
This condition holds for 2-sites of the form $(\Cat(S),J(T))$ where $T$ is a subcanonical singleton pretopology (as well as various sub-2-categories) [9].
This approach will be taken up in future work.
Finally, note that nothing in the above relies on $K$ being a (2,1)-category, namely one with only invertible 2-arrows.
This is usually assumed for results subsumed by theorem <ref>, but is unnecessary in the framework presented here.
[1]
O. Abbad and E.M. Vitale.
Faithful calculus of fractions.
Cahiers de Topologie et Géométrie Différentielle
Catégoriques 54 (2013), 221–239.
[2]
T. Bartels.
Higher gauge theory I: 2-Bundles.
Ph.D. thesis, University of California Riverside.
[3]
M. Bunge and R. Paré.
Stacks and equivalence of indexed categories.
Cahiers Topologie Géom. Différentielle 20 (4)
(1979), 373–399.
[4]
A. Karagila.
Embedding orders into cardinals with $DC_\kappa$.
Fundamenta Mathematicae 226 (2014), 143–156.
[5]
T. Leinster.
Basic bicategories.
arXiv:math.CT/9810017, 1998.
[6]
M. Makkai.
Avoiding the axiom of choice in general category theory.
J. Pure Appl. Algebra 108 (1996), 109–173.
[7]
D. Pronk.
Etendues and stacks as bicategories of fractions.
Compositio Math. 102 (3) (1996), 243–303.
[8]
D. Pronk and M. Warren.
Bicategorical fibration structures and stacks.
Theory and Applications of Categories 29 (29) (2014),
[9]
D. M. Roberts.
Internal categories, anafunctors and localisation.
Theory Appl. Categ. 26 (29) (2012), 788–829.
[10]
D. M. Roberts.
The weak choice principle WISC may fail in the category of
Studia Logica (2015),
[11]
M. Shulman.
Exact completions and small sheaves.
Theory Appl. Categ. 27 (2012), 97–173.
[12]
R. Street.
Two-dimensional sheaf theory.
J. Pure Appl. Algebra 23 (3) (1982), 251–270.
[13]
The Stacks Project Authors.
Stacks Project.
<http://stacks.math.columbia.edu>, 2015.
[14]
M. Tommasini.
Some insights on bicategories of fractions - I,
arXiv:1410.3990, 2014.
[15]
B. van den Berg and I. Moerdijk.
The axiom of multiple choice and models for constructive set
Journal of Mathematical Logic 14 (1).
[16]
A. Vistoli.
Grothendieck topologies, fibred categories and descent
In Fundamental algebraic geometry, Math. Surveys.
Monogr., Volume 123 (Amer. Math. Soc., Providence, RI, 2005), 1–104.
arXiv:math/0412512. Available from
|
arxiv-papers
| 2014-02-28T00:05:39 |
2024-09-04T02:49:59.067835
|
{
"license": "Public Domain",
"authors": "David Michael Roberts",
"submitter": "David Roberts",
"url": "https://arxiv.org/abs/1402.7108"
}
|
1402.7132
|
# Fractal Signatures in Analogs of
Interplanetary Dust Particles
Nisha Katyal1, Varsha Banerjee2 and Sanjay Puri1
1 School of Physical Sciences, Jawaharlal Nehru University,
New Delhi – 110067, India.
2 Department of Physics, Indian Institute of Technology, Hauz Khas,
New Delhi – 110016, India
###### Abstract
Interplanetary dust particles (IDPs) are an important constituent of the
earth’s stratosphere, interstellar and interplanetary medium, cometary comae
and tails, etc. Their physical and optical characteristics are significantly
influenced by the morphology of silicate aggregates which form the core in
IDPs. In this paper we reinterpret scattering data from laboratory analogs of
cosmic silicate aggregates created by Volten et al. [1], to extract their
morphological features. By evaluating the structure factor, we find that the
aggregates are mass fractals with a mass fractal dimension $d_{m}\simeq 1.75$.
The same fractal dimension also characterizes clusters obtained from diffusion
limited aggregation (DLA). This suggests that the analogs are formed by an
irreversible aggregation of stochastically-transported silicate particles.
###### keywords:
silicate cores, interplanetary dust particles, structure factor, mass
fractals, diffusion limited aggregation.
††journal: Journal of Quantitative Spectroscopy & Radiative Transfer,
## 1 Introduction
Fractal geometries provide a description for many forms in Nature such as
coastlines, trees, blood vessels, fluid flow in porous media, burning
wavefronts, dielectric breakdown, diffusion-limited-aggregation (DLA)
clusters, bacterial colonies, colloidal aggregates, etc. [2, 3, 4]. They
exhibit self-similar and scale-invariant properties at all levels of
magnification and are characterized by a non-integer fractal dimension. These
features arise because the underlying processes have an element of
stochasticity in them. Such processes play an important role in shaping the
final morphology, and their origin is distinctive in each physical setting.
Irregular and rough aggregates have also been observed in the astronomical
context. Naturally found cosmic dust aggregates, known as interplanetary dust
particles (IDPs), are collected in earth’s lower stratosphere. They are formed
when dust grains collide in a turbulent circumstellar dust cloud such as the
solar nebula, and are an important constituent of the interstellar medium,
interplanetary medium, cometary comae and tails, etc. Mass spectroscopy
analysis of IDPs have revealed that their primary constituents are (i)
silicates of Fe, Mg, Al and Ca, (ii) complex organic molecules of C, H, O and
N, (iii) small carbonaceous particles of graphite, coal and amorphous carbon
and (iv) ices of CO2, H2O and NH3 [5, 6, 7, 8, 9]. Amongst these, there is an
exclusive abundance of silicates which aggregate to form particle cores. They
have been described as fluffy, loosely-structured particles with high
porosity. The other constituents contribute to the outer covering or the
mantle and are usually contiguous due to flash heating from solar flares and
atmospheric entry [10]. The core, being deep inside retains its morphology.
The latter is believed to have a fractal organization characterized by a
fractal dimension, but this belief is not on firm grounds as yet [11, 12]. As
the core morphology affects the physical and optical characteristics of IDPs,
its understanding has been the focus of several recent works [13, 14, 15, 16,
17, 18].
Two classes of stochastic fractals are found in nature. The first class is
that of surface fractals whose mass $M$ scales with the radius of gyration $R$
in a Euclidean fashion, i.e., $M\sim R^{d}$, where $d$ is the dimensionality.
However, the surface area $S$ increases with the radius as $S\sim R^{d_{s}}$,
where $d_{s}$ is the surface fractal dimension and $d-1\leq d_{s}<d$ [19].
Interfaces generated in fluid flows, burning wavefronts, dielectric breakdown
and deposition processes are examples of surface fractals. The second class is
that of mass fractals which obey the scaling relationship, $M\sim R^{d_{m}}$,
where $d_{m}$ is the mass fractal dimension and $1\leq d_{m}<d$. Examples of
mass fractals are DLA clusters, bacterial colonies and colloidal agregates.
Further, in many situations, mass fractals are bounded by surface fractals [2,
3, 4]. As a matter of fact, the above mass fractals belong to this class.
There are many unanswered questions in the context of fluffy cores or silicate
aggregates of IDPs. For example, are they mass fractals, bounded by surface
fractals? What is their mass and surface fractal dimension? What kind of
aggregation mechanisms are responsible for this morphology? What are the
consequences of fractal organization on the evolution of clusters? In this
paper, we provide answers to some of these questions using the real-space
correlation function $C\left(r\right)$ and the momentum-space structure factor
$S\left(k\right)$. Smooth morphologies are characterized by the Porod law [20,
21]. The signature of fractal domains and interfaces is a power-law decay with
non-integer exponents in $C\left(r\right)$ and $S\left(k\right)$. As typical
experimental morphologies are smooth on some length scales and fractal on
others, the behaviors of $C\left(r\right)$ vs. $r$ and $S\left(k\right)$ vs.
$k$ are characterized by cross-overs from one form to another. We identify
these features in laboratory analogs of cores of IDPs created by Volten et al.
using magnesio-silica grains, by reinterpreting their light-scattering data
[1]. We find that these aggregates are mass fractals with a fractal dimension
$d_{m}\simeq 1.75$. The same fractal dimension characterizes diffusion limited
aggregation (DLA). We therefore infer that aggregation mechanisms of silicate
cores in IDPs are stochastic and irreversible as in DLA.
This paper is organized as follows. In Section 2, we describe the tools for
morphology characterization and their usage to obtain mass and surface fractal
dimensions. In Section 3, we describe the experimental analogs of silicate
cores in IDPs and obtain the structure factor from their light scattering data
to extract fractal properties. In Section 4, we present a simulation of the
DLA cluster, and the evaluation of its structure factor and the corresponding
mass fractal dimension. Finally, we conclude with a summary and discussion of
our results in Section 5.
## 2 Tools for Morphology Characterization
A standard tool to obtain information about sizes and textures of domains and
interfaces is the two-point spatial correlation function [21]:
$C\left(r\right)=\langle\psi\left(\vec{r_{i}}\right)\psi\left(\vec{r_{j}}\right)\rangle-\langle\psi\left(\vec{r_{i}}\right)\rangle\langle\psi\left(\vec{r_{j}}\right)\rangle,$
(1)
where $\psi\left(\vec{r_{i}}\right)$ is an appropriate order parameter and
$r=|\vec{r_{i}}-\vec{r_{j}}|$. (We assume the system to be translationally
invariant and isotropic.) The angular brackets denote an ensemble average.
The scattering of a plane wave by a rough morphology can yield useful
information about the texture of the domains and interfaces in it. Thus,
small-angle scattering experiments (using X-rays, neutrons, etc.) can be used
to probe their nature. The intensity of the scattered wave in these
experiments yields the momentum-space structure factor, which is the Fourier
transform of the correlation function [20, 21, 22, 23]:
$S(\vec{k})=\int\mbox{d}\vec{r}e^{i\vec{k}\cdot\vec{r}}C\left(\vec{r}\right),$
(2)
where $\vec{k}$ is the wave-vector of the scattered beam. The properties of
$C\left(r\right)$ and $S\left(k\right)$ provide deep insights into the nature
of the scattering morphology.
Consider a domain of size $\xi$ formed by spherical particles of size $a$, as
depicted schematically in Fig. 1(a). The typical interfacial width $w$, is
also indicated. This prototypical morphology could represent a colloidal
aggregate, soot particles, a DLA cluster, etc. The correlation function for
such a morphology can be approximated by
$\displaystyle
1-C\left(r\right)=\bar{C}\left(r\right)\simeq\left\\{\begin{array}[]{ll}Ar^{\alpha},&\quad
w\ll r\ll\xi,\\\ Br^{\beta},&\quad r\ll w\ll a,\\\ Cr^{\gamma}&\quad r\ll
a.\end{array}\right.$ (6)
The first term conveys information about the domain texture probed by length
scales $w\ll r\ll\xi$. If the domain has no internal structure, $\alpha$ = 1
signifying Porod decay [20, 21]. For a fractal domain, on the other hand,
$\alpha$ = $d_{m}-d$ where $d_{m}$ is the mass fractal dimension [22, 23]. The
second term conveys information about the properties of the interface, probed
by lengths $a\ll r\ll w$. For fractal interfaces, $0\leq\beta<1$, and $\beta$
is related to the fractal dimension as $d_{s}=d-\beta$ [24]. The third term is
significant only if the building blocks are particles of diameter $a$. In that
case, $\gamma=1$ for $r\lesssim a$, yielding a Porod regime at a microscopic
length scale.
In Fourier space, Eq. (6) translates into the following power-law behavior of
the structure factor:
$\displaystyle
S\left(k\right)\simeq\left\\{\begin{array}[]{ll}\tilde{A}k^{-(d+\alpha)},&\quad\xi^{-1}\ll
k\ll w^{-1},\\\ \tilde{B}k^{-(d+\beta)},&\quad w^{-1}\ll k\ll a^{-1},\\\
\tilde{C}r^{-(d+\gamma)}&\quad a^{-1}\ll k.\end{array}\right.$ (10)
The Porod decay of the form $k^{-\left(d+1\right)}$ in the scattered intensity
is typical of smooth domains or sharp interfaces [20, 21]. A deviation from
this behavior to $S(k)\sim k^{-(d\pm\theta)}$ is indicative of a fractal
structure in the domains or interfaces. When physical structures have multiple
length-scales, one or more terms in Eqs. (6) and (10) may contribute. Their
presence is characterized by cusps in the correlation function, and
corresponding power-laws in the structure factor [25].
We illustrate the power laws and cross-overs discussed above in the context of
the 2-$d$ morphology depicted in Fig. 1(a). It should be noted that both the
domain and the interfacial boundary in this schematic are rough, self-similar
fractals. The structure factor $S\left(k\right)$ vs. $k$ for this morphology
obtained from the Fourier transform of the spherically-averaged correlation
function $C\left(r\right)$ vs. $r$ is plotted in Fig. 1(b) on a log-log scale.
This function exhibits two distinct regimes over large and small values of $k$
as seen from the best fit lines: power-law decay with $S\left(k\right)\sim
k^{-1.71}$ for $\xi^{-1}\ll k\ll w^{-1}$ and a Porod decay with
$S\left(k\right)\sim k^{-3}$ for $a^{-1}\ll k$. With reference to Eqs. (6) and
(10), the power law decay signifies a fractal domain morphology with a mass
fractal dimension $d_{m}\approx 1.71$ while the Porod decay is due to the
smooth surface of the particles. The structure factor corresponding to wave
vectors in the interval $w^{-1}\ll k\ll a^{-1}$ is due to scattering from the
rough fractal interfaces. However it difficult to identify the corresponding
power law with precision due to cross-overs effects from the adjoining mass
fractal and Porod regimes.
## 3 Analysis of Silicate Cores
We now investigate the morphological characteristics of silicate cores using
the correlation function and the structure factor. As real samples are scarce,
it has been customary to create them in the laboratory using a condensation
flow apparatus followed by flash heating to mimic the environment required for
the formation of cosmic silicates and circumstellar dust. A significant
contribution in this context is the work of Volten et al. [1]. They created a
variety of magnesio-silica samples with (relative) concentrations typical of
silicate cores in IDPs [26]. The mixed grains in these samples formed
interconnected, tangled chains ranging from open structures to dense
structures, thereby yielding samples of varied porosities. We calculate
$S\left(k\right)$ for two such analogs: Sample 1 has an equal proportion of Mg
and Si; and Sample 2 has Mg and Si in the ratio 1.4:1. These samples have a
porosity of $\sim 40\%$ [1, 26].
Fig. 2(a) reproduces a prototypical TEM image of an ultra-thin section sliced
through MgSiO particles prepared by Volten et al. (The image is reproduced
from [1] with permission from the authors.) They are organized in the form of
small fluffy aggregates organized in a contiguous but porous morphology [1]
Volten et al. then obtained light-scattering data for the samples using the
Amsterdam light scattering database. The light-scattering properties were
measured at a wavelength of 632.8 nm with the range of scattering angles from
$5^{\circ}$ to $174^{\circ}$, in steps of $1^{\circ}$. These measurements
yielded the scattering matrix elements as a function of scattering angle
$\theta$. The inset of Fig. 2(b) plots scattering phase function $S_{11}$ vs.
$\theta$. We convert this data in terms of the magnitude of the scattering
wave-vector by the transformation $k=4\pi/\lambda\ \mbox{sin}(\theta/2)$. The
transformed data sets are presented in Fig. 2(b) on a log-log scale. The
intermediate-$k$ region is linear on this plot, implying a power-law
dependence between the scattering intensity $S(k)$ and the scattering wave-
vector $k$. The best-fit line (shown alongside) has a slope of $-1.75$. From
Eqs. (6)-(10) and the accompanying discussion, it is clear that the fluffy
aggregates of Samples 1 and 2 are mass fractals with a mass fractal dimension
$d_{m}\simeq 1.75$.
## 4 Diffusion Limited Aggregation
A relevant question now is: What kind of mass-transport mechanisms lead to
fluffy aggregates with $d_{m}\simeq 1.75$? To answer this question, we create
aggregates of particles using the DLA model. We performed this simulation on a
cubic lattice adopting the algorithm introduced by Meakin [27]: (i) A particle
is placed at the origin or the center of the cube. (ii) A new particle is
released at a distance $R$ from the center and performs a random walk. (iii)
On encountering an occupied neighboring site, it adheres irreversibly to it.
Steps (ii) and (iii) are repeated several times to obtain a DLA cluster.
To mimic the morphology (of several small aggregates) observed in the TEM
micrograph of Fig. 2(a), we simulate a DLA cluster ($d=3$) using multiple
seeds. Each seed initiates a sub-cluster using the above procedure. We allow
the sub-clusters to grow till they form a contiguous, yet delicately branched,
self-similar structure. Fig. 3(a) depicts such a prototypical multi-seed
cluster built from $\sim 10^{4}$ particles. We also show a slice of this
morphology ($d=2$) in Fig. 3(b). It contains two initial seeds, marked in red.
Notice here the similarity of this slice with the open contiguous structure in
the TEM image of Fig. 2(a). Next, we quantify the morphology of Fg. 3(a) by
evaluating the spherically-averaged structure factor $S(k)$, which is shown in
Fig. 3(c) on a log-log scale. The power-law behavior at intermediate values of
$k$ fits best to a line with slope $-1.75$. With reference to Eq. (10) and the
discussion thereafter, DLA clusters are mass fractals with $d_{m}\simeq 1.75$.
In view of these observations, we infer that the aggregates obtained in the
experiments of Volten et al. with $d_{m}\simeq 1.75$ are due to irreversible
aggregation of stochastically transported silicate particles.
## 5 Conclusion
Interstellar dust particles (IDPs) found in the earth’s stratosphere are an
important constituent of cosmic matter. These comprise of loosely structured
silicate cores or aggregates ensconced in a mantle of organic and carbonaceous
compounds [5, 6, 7, 8, 9]. The organization and optical characteristics of the
IDPs are greatly influenced by the morphology of the core. In this paper, we
have re-interpreted scattering data from laboratory analogs of silicate cores
in IDPs created by Volten et al. [1] using the correlation function $C(r)$ and
the structure factor $S(k)$. This analysis has provided us a means to quantify
characteristics such as the size and texture of these aggregates. The presence
of fractal architecture is characterized by power laws with non-integer
exponents in the structure factor. We found that the silicate aggregates are
mass fractals with a fractal dimension $d_{m}\simeq 1.75$. This value of
$d_{m}$ is the same as the fractal dimension of aggregates obtained in a
diffusion limited aggregation model. We therefore conclude that the aggregates
are formed by an irreversible aggregation of stochastically transported
silicate particles. We have also studied the effect of density of particles on
the fractal dimension. Our observation is that $d_{m}$ approaches the
Euclidian dimension ($d=3$) with increasing density. Further, we wish to
emphasize that $C(r)$ and $S(k)$ contain information averaged over all domains
and interfaces in contrast to the conventionally used (local) box-counting
procedures. Our estimates of $d_{m}$ are therefore very accurate.
For confirmation of our results and further insights, we require data from
real cosmic dust and cometary particles. We understand that it is difficult to
obtain light scattering data from stellar objects. Alternatively, information
providing depth profiles of these assemblies could also be used to evaluate
the $C(r)$ and $S(k)$. As discussed above, they are excellent tools for
morphology characterization especially due to their direct experimental
relevance. More generally, micro-scale phenomena are characterized by mass-
dependent diffusion, i.e., the diffusion rate $D(m)\sim m^{-\alpha}$, where
$m$ is the mass or number of particles in the cluster and $\alpha$ is a
system-specific parameter [25]. The fractal characteristics of aggregates are
greatly influenced by $\alpha$. We are presently investigating them to
quantify this influence. We hope that such analyses will enhance our
understanding of diffusion mechanisms in mega-scale systems found in the
cosmic environment.
## Acknowledgements
The authors would like to thank the anonymous refeeres for their constructive
comments that helped to improve the quality of the paper. VB would like to
acknowledge the support of DST Grant No. SR/S2/CMP-002/2010.
## References
* [1] Volten, H., Munoz, O., Hovenier, J.W., Rietmeijer, F.J.M., Nuth, J.A., Waters, L.B.F.M., et al, 2007, Astron Astrophys, 470, 377.
* [2] Mandelbrot, B. B., The Fractal Geometry of Nature (W. H. Freeman, 1982).
* [3] Barabasi, A. L. and Stanley, H. E., Fractal Concepts in Surface Growth (Cambridge University, 1995).
* [4] Vicsek, T., Fractal Growth Phenomena (World Scientific, 1992).
* [5] Greenberg, J. M., 1989, From interstellar dust to comet dust and inteplanetary particles, in Highlights of Astronomy, Vol. 8, 241–250.
* [6] Greenberg, J. M., 1998, Earth, Moon and Planets, Vol. 82-83, Issue 0, pp 313-324.
* [7] Rietmeijer, F. J. M., 2002, Chem. Erde, 62, 1.
* [8] Tsuchiyama, A., Uesugi, K., Nakano, T., Okazaki, T., Nakamura. K, Nakamura. T., Noguchi, T. and Yano, H., 2006, Annual Lunar and Planetary Science Conference XXXVII, Texas, abstract no. 2001.
* [9] Cuppen, H. M., and E. Herbst, 2007, Simulation of the Formation and Morphology of Ice Mantles on Interstellar Grains, ApJ, 668, 294.
* [10] Rietmeijer, F. J. M., 1996, Meteoritics Planet Sci., 31, 237.
* [11] Rietmeijer, F. J. M., 1993, Earth Planet. Sci. Lett., 117, 609.
* [12] Rietmeijer, F. J. M. and Nuth III, J. A., 2004, ASSL Vol. 311: The New Rosetta Targets. Observations, Simulations and Instrument Performances, ed. L. Colangeli, E.M. Epifani, and P. Palumbo (Astrophys. Space Sci. Library, Kluwer Academic Publishers), 97-110.
* [13] Messenger, S., Keller, L. P., Stadermann, F. J., Walker, R. M., Zinner, E., 2003, Science, 300, 105.
* [14] Min, M., Waters, L. B. F. M., Koter, A. de., Hovenier, J. W., Keller, L. P., Markwich-Kemper, F., 2007, A & A, 486, 779.
* [15] Henning, T., 2010, Annu. Rev. Astron. Astrophys., 48, 21-46.
* [16] Vaidya, D. B. and Gupta, R., 2011, A&A, 528, A57.
* [17] Botet, R. and Rakesh R., 2013, Earth Planets Space, Vol. 65 (No. 10), pp. 1133.
* [18] Katyal, N., Gupta, R. and Vaidya, D. B., 2013, PASP, Vol. 125, 1443.
* [19] Hurd, A. J., Schaefer, D. W. and Martinn, J. E., 1987, Phys. Rev. A, 35, 2361.
* [20] Porod, G., in Small-Angle X-Ray Scattering, edited by O. Glatter and O. Kratky (Academic Press, New York, 1982); Oono, Y. and Puri, S., Mod. Phys. Lett. B 2, 861 (1988).
* [21] Kinetics of Phase Transitions, edited by S. Puri and V.K. Wadhawan, Taylor and Francis, Boca Raton (2009).
* [22] Sorensen, C. M., 2001, Aerosol Sci. Tech., 35, 648.
* [23] Oh, C. and Sorensen, C. M., 1997, Phys. Rev. E, 193, 17.
* [24] Mildner, D. R. R. and Hall, P. L., 1986, J. Phys. D: Appl. Phys., 19, 1535.
* [25] Shrivastav, G.P., Banerjee, V. and Puri, S. 2010, Eur. Phys. J. B, 78, 217.
* [26] Rietmeijer, F. J. M. (1998) Interplanetary Dust Particles. In Planetary Materials, Reviews in Mineralogy, vol. 36 (J.J. Papike, ed.), 2-1 – 2-95, Mineralogical Society of America, Chantilly, Virginia.
* [27] Meakin, P., 1983, Phys. Rev. A, 27, 1495.
Figure 1: (a) A typical morphology of a domain of size $\xi$ formed by
spherical particles of size $a$ is depicted. (b) Log-log plot of structure
factor of the morphology as shown in (a). A power-law decay with a slope of
-1.71 signifies a fractal morphology whereas a slope of -4 signifies a Porod
law decay due to smooth morphology of the overall domain at that particular
length scale.
Figure 2: (a) TEM image of a section sliced through a fluffy MgSiO particle
consisting of several small aggregates of magnesio-silica grains forming a
contiguous, yet porous structure [1]. (b) Scattering intensity $S(k)$ as a
function of scattering wave-vector $k$ for Samples 1 and 2 on a log-log scale.
A line of slope $-1.75$ fits well to the intermediate-$k$ data. The original
experimental data from Ref. [1], showing the variation of scattering phase
function $S_{11}$ as a function of scattering angle $\theta$, is provided in
the inset.
Figure 3: (a) Computer generated DLA cluster ($d=3$) with multiple seeds
obtained from $\sim 10^{4}$ particles. (b) A slice ($d=2$ of the DLA cluster
depicting two initial seeds (red) and the delicately branched, self-similar
contiguous growth.
(c) Spherically averaged structure factor, $S(k)$ vs. $k$, on a log-log scale
for the DLA cluster in (a). The power-law behavior in the intermediate-$k$
regime yields the mass fractal dimension $d_{m}\simeq 1.75$.
|
arxiv-papers
| 2014-02-28T05:02:50 |
2024-09-04T02:49:59.076123
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nisha Katyal, Varsha Banerjee and Sanjay Puri",
"submitter": "Nisha Katyal",
"url": "https://arxiv.org/abs/1402.7132"
}
|
1402.7152
|
# Non-maximal Tripartite Entanglement Degradation of Dirac and Scalar fields
in Non-inertial frames
Salman Khan† [email protected] Niaz Ali Khan‡ M. K. Khan‡ †Department of
Physics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad,
Pakistan. ‡Department of Physics, Quaid-i-Azam University, Islamabad,
Pakistan.
(December 27, 2013)
###### Abstract
The $\pi$-tangle is used to study the behavior of entanglement of a nonmaximal
tripartite state of both Dirac and scalar fields in accelerated frame. For
Dirac fields, the degree of degradation with acceleration of both one-tangle
of accelerated observer and $\pi$-tangle, for the same initial entanglement,
is different by just interchanging the values of probability amplitudes. A
fraction of both one-tangles and the $\pi$-tangle always survives for any
choice of acceleration and the degree of initial entanglement. For scalar
field, the one-tangle of accelerated observer depends on the choice of values
of probability amplitudes and it vanishes in the range of infinite
acceleration, whereas for $\pi$-tangle this is not always true. The dependence
of $\pi$-tangle on probability amplitudes varies with acceleration. In the
lower range of acceleration, its behavior changes by switching between the
values of probability amplitudes and for larger values of acceleration this
dependence on probability amplitudes vanishes. Interestingly, unlike bipartite
entanglement, the degradation of $\pi$-tangle against acceleration in the case
of scalar fields is slower than for Dirac fields.
PACS: 03.65.Ud, 03.67.Mn, 04.70.Dy
Keywords: Tripartite entanglement, Noninertial frame
Entanglement, Noninertial frames
###### pacs:
03.65.Ud, 03.67.Mn, 04.70.Dy
††preprint:
## I Introduction
One of the potential resources for all kinds of quantum information tasks is
entanglement. It is among the mostly investigated properties of many particles
systems. Since the beginning of the birth of the fields of quantum information
and quantum computation, it has been the pivot in different perspective to
bloom up these fields to be matured for technological purposes Many1 . The
recent development by mixing up the concepts of relativity theory with quantum
information theory brought to fore the relative behavior of entanglement
Alsing1 ; Alsing2 ; Alsing3 ; Fuentes . These studies show that entanglement
not only depends on acceleration of the observer but also strongly depends on
statistics. For practical application in most general scenario, it is
essential to thoroughly investigate the behavior of entanglement and hence of
different protocols (such as teleportation) of quantum information theory
using different statistics in curved spacetime.
The observer dependent character of entanglement under various setup for
different kinds of fields have been studied by a number of authors. For
example, the entanglement between two modes of a free maximally entangled
bosonic and fermionic pairs is studied in Alsing2 ; Alsing3 , between to modes
of noninteracting massless scalar field is analyzed in Fuentes , between free
modes of a free scalar field is investigated in Adesso . Similarly, the
dynamics of tripartite entanglement under different situation for different
fields has also been studied. For example, in Ref. Hwang the degradation of
tripartite entanglement between the modes of free scalar field due to
acceleration of the observer is investigated. All these studies are carried by
taking single mode approximation. The behavior of entanglement in accelerated
frame beyond the single mode approximation is studied in Ref. Bruchi . The
effect of decoherence on the behavior of entanglement in accelerated frame is
studied in Ref. Salman . All these and many other related works show that
entanglement in the initial state is degraded when observed from the frame of
an accelerated observer.
On the other hand, there are studies which show, counter intuitively, that the
Unruh effect not only degrade entanglement shared between an inertial and an
accelerated observer but also amplify it. Ref. Montero studies such
entanglement amplification for a particular family of states for scalar and
Grassman scalar fields beyond the single mode approximation. A similar
entanglement amplification is reported for fermionic system in Ref. Kown .
There are a number of other good papers on the dynamics of entanglement in
accelerated frames which can be found in the list Pan2 .
It is well known that considering correlations between the modes of stationary
observer with both particle and anti-particle modes in the two causally
disconnected regions in the Rindler spacetime provides a broad view for
quantum communications tasks. Such considerations enable the stationary
observer to setup communication with either of the two disconnected regions or
with both at the same time Martin2 . This is possible by considering the
formalism of quantum communication in the limit of beyond single mode
approximation Bruchi . In the same work it is shown that the single mode
approximation holds for some family of states under appropriate constraints.
On the other hand, it has also been suggested that the single mode
approximation is optimal for quantum communication between the stationary
observer and the accelerated observer Hosler . For the purpose of this paper
we will use the later approach.
In this paper, we investigate the dependence of the behavior of a nonmaximal
tripartite entanglement of both Dirac and scalar fields on the acceleration of
the observer frame and on the entanglement parameter that describes the degree
of entanglement in the initial state. We show that the degradation of
entanglement with acceleration not only depends on the degree of initial
entanglement but also depends on the individual values of the normalizing
probability amplitudes of the initial state. We consider three observers
($i=A,B,C$), Alice Bob and Charlie, in Minkowski space such that each of them
observes only one part of the following nonmaximal initial tripartite
entangled state
$\left|\psi_{\omega_{A},\omega_{B},\omega_{C}}\right\rangle=\alpha\left|0_{\omega_{A}}\right\rangle_{A}\left|0_{\omega_{B}}\right\rangle_{B}\left|0_{\omega_{C}}\right\rangle_{C}+\sqrt{1-\alpha^{2}}\left|1_{\omega_{A}}\right\rangle_{A}\left|1_{\omega_{B}}\right\rangle_{B}\left|1_{\omega_{C}}\right\rangle_{C},$
(1)
where $\left|m_{\omega_{i}}\right\rangle$ for $m\in(0,1)$ are the Minkowski
vacuum and first excited states with modes specified by the subscript
$\omega_{i}$ and $\alpha$ is a parameter that specify the degree of
entanglement in the initial state. Under the single mode approximation Bruchi
$\omega_{A}\sim\omega_{B}\sim\omega_{C}=\omega,$ we can write
$\left|m_{\omega_{i}}\right\rangle=\left|m\right\rangle_{i}$.
Instead of being all the time in an inertial frame, if the frame of one of the
observers, say Charlie, suddenly gets some uniform acceleration $a$, then, the
Minkowski vacuum and excited states change from the perspective of the
accelerated observer. The appropriate coordinates for the viewpoint of an
accelerated observer are Rindler coordinates AsPach ; Martin ; Bruchi ; Brown2
. The Rindler spacetime for an accelerated observer splits into two regions,
$\mathrm{I}$ (right) and $\mathrm{II}$ (left), that are separated by Rindler
horizon and thus are causally disconnected from each other. The Rindler
coordinates $(\tau,\xi)$ in region $\mathrm{I}$ are defined in terms of the
Minkowski coordinates $(t,x)$ as follows
$t=\frac{1}{a}e^{a\xi}\sinh(a\tau),\qquad x=\frac{1}{a}e^{a\xi}\cosh(a\tau).$
(2)
An exact similar transformation holds between the coordinates for the Rindler
region $\mathrm{II}$, however, each coordinate differ by an overall minus
sign. These new coordinates allow us to perform a Bogoliubov transformation
between the Minkowski modes of a field and Rindler modes. The Rindler modes in
the two Rindler regions form a complete basis in terms of which the Minkowski
modes can be expanded. Thus any state in Minkowski space can be represented in
Rindler basis as well. However, an accelerated observer in Rindler region
$\mathrm{I}$ has no access to information in Rindler region $\mathrm{II}$. The
degree of entanglement of modes in each Rindler region with the modes of
inertial observers has its own dynamics. To study the behavior of entanglement
in one region, being inaccessible, the modes in other region becomes
irrelevant and thus need to be trace out.
The Minkowski annihilation operator of an arbitrary frequency, observed by
Alice, is related to the two Rindler regions’ operators of frequency, observed
by Charlie, more directly through an intermediate set of modes called Unruh
modes Bruchi . The Unruh modes analytically extend the Rindler region$I$ modes
to region $II$ and the region$II$ modes to region $I$. Since the Unruh modes
exist over all Minkowski space, they share the same vacuum as the Minkowski
annihilation operators. An arbitrary Unruh mode for a give acceleration is
given by
$C_{\omega}=q_{L}C_{\omega,L}+q_{R}C_{\omega,R},$ (3)
where $q_{L}$ and $q_{R}$ are complex numbers satisfying the relation
$\left|q_{L}\right|^{2}+\left|q_{R}\right|^{2}=1$ and the appropriate
relations for the left and right regions’ operators are given by Bruchi
$\displaystyle C_{\omega,R}$ $\displaystyle=\cosh r_{\omega}a_{\omega,I}-\sinh
r_{\omega}a_{\omega,II}^{{\dagger}},$ $\displaystyle C_{\omega,L}$
$\displaystyle=\cosh r_{\omega}a_{\omega,II}-\sinh
r_{\omega}a_{\omega,I}^{{\dagger}},$ (4)
where $a$, $a^{{\dagger}}$ are Rindler particle operators of scalar field in
the two regions. For Grassman case, the transformation relations are given by
$\displaystyle C_{\omega,R}$ $\displaystyle=\cos r_{\omega}c_{\omega,I}-\sinh
r_{\omega}d_{\omega,II}^{{\dagger}},$ $\displaystyle C_{\omega,L}$
$\displaystyle=\cos r_{\omega}c_{\omega,II}-\sinh
r_{\omega}d_{\omega,I}^{{\dagger}},$ (5)
where $c$, $c^{{\dagger}}$ and $d$, $d^{{\dagger}}$ are respectively Rindler
particle and antiparticle operators. The dimensionless parameter $r_{\omega}$
appears in these equations is discussed below. For the purpose of this paper,
in order to recover single mode approximation we will set $q_{R}=1$ and
$q_{L}=0$.
From the viewpoint of accelerated observer, the Minkowski vacuum and excited
states of the Dirac field are found to be, respectively, given by Alsing3 .
$\left|0\right\rangle_{M}=\cos
r\left|0\right\rangle_{I}\left|0\right\rangle_{II}+\sin
r\left|1\right\rangle_{I}\left|1\right\rangle_{II},$ (6)
$\left|1\right\rangle_{M}=\left|1\right\rangle_{I}\left|0\right\rangle_{II}.$
(7)
Similarly, for scalar field the Minkowski vacuum and excited states are given
by
$\left|0\right\rangle_{M}=\frac{1}{\cosh
r}{\displaystyle\sum\limits_{n=0}^{\infty}}\tanh^{n}r\left|n\right\rangle_{I}\left|n\right\rangle_{II},$
(8)
$\left|1\right\rangle_{M}=\frac{1}{\cosh^{2}r}{\displaystyle\sum\limits_{n=0}^{\infty}}\sqrt{n+1}\tanh^{n}r\left|n+1\right\rangle_{I}\left|n\right\rangle_{II}.$
(9)
In the above equations, $\left|\cdot\right\rangle_{I}$ and
$\left|\cdot\right\rangle_{II}$ are Rindler modes in the two causally
disconnected Rindler regions, $\left|n\right\rangle$ represents number states
and $r$ is a dimensionless parameter that depends on acceleration of the
moving observer and modes frequency. For Dirac field, it is given by $\cos
r=(1+e^{-2\pi\omega c/a})^{-1/2}$ such that $0\leq r\leq\pi/4$ for $0\leq
a\leq\infty$ and for scalar field, it is defined as $\cosh r=(1-e^{-2\pi\omega
c/a})^{-1/2}$ such that $0\leq r\leq\infty$ for $0\leq a\leq\infty$. It is
important to note that almost all the previous studies have been focused on
investigating the influence of parameter $r$, as a function of acceleration of
the moving frame by fixing the Rindler frequency, on the degree of
entanglement present in the initial state. Such analysis lead to the
measurement of entanglement in a family of states, all of which share the same
Rindler frequency as seen by an observer with different acceleration. However,
the effect of parameter $r$ on entanglement can also, alternatively, be
interpreted by considering a family of states with different Rindler
frequencies watched by the same observer traveling with fixed acceleration
Bruschi2 .
## II Quantification of Tripartite entanglement
In literature, a number of different criterion for quantifying tripartite
entanglement exist. However, the most popular among them are the residual
three tangle Coffman and $\pi$-tangle Fan ; Vidal . Other measurements for
tripartite entanglement include realignment criterion Rudolph ; Kia and
linear contraction Horodocki . The realignment and linear contraction
criterion are comparatively easy in calculation and are strong criteria for
entanglement measurement. However, these criterion has some limitations and do
not detect the entanglement of all states.
The three tangle is another good quantifier for the entanglement of tripartite
states. This is polynomial invariant Verstraete ; Leifer and it needs an
optimal decomposition of a mixed density matrix. In general, the optimal
decomposition is a tough enough task except in a few rare cases Lohmayer . On
the other hand, the $\pi$-tangle for a tripartite state $|\psi\rangle_{ABC}$
is given by
$\pi_{ABC}=\frac{1}{3}(\pi_{A}+\pi_{B}+\pi_{C}),$ (10)
where $\pi_{A}$ is called residual entanglement and is given by
$\pi_{A}=\mathcal{N}_{A(BC)}^{2}-\mathcal{N}_{AB}^{2}-\mathcal{N}_{AC}^{2}.$
(11)
The other two residual tangles ($\pi_{B},\pi_{C}$) are defined in a similar
way. In Eq. (11), $\mathcal{N}_{AB}(\mathcal{N}_{AC})$ is a two-tangle and is
given as the negativity of mixed density matrix
$\rho_{AB}=Tr_{C}|\psi\rangle_{ABC}\langle\psi|$
$(\rho_{AC}=Tr_{B}|\psi\rangle_{ABC}\langle\psi|)$. The $\mathcal{N}_{A(BC)}$
is a one-tangle and is defined as
$\mathcal{N}_{A(BC)}=\left\|\rho_{ABC}^{T_{A}}\right\|-1$, where
$\left\|O\right\|=\mathrm{tr}\sqrt{OO^{{\dagger}}}$ stands for the trace norm
of an operator $O$ and $\rho_{ABC}^{T_{A}}$ is the partial transposition of
the density matrix over qubit $A$. The one-tangle and the two-tangles satisfy
the following Coffman-Kundu-Wootters (CKW) monogamously inequality relation
Coffman .
$\mathcal{N}_{A(BC)}^{2}\geq\mathcal{N}_{AB}^{2}+\mathcal{N}_{AC}^{2}.$ (12)
In this paper we use the $\pi$-tangle to observe the behavior of entanglement
of the state given in Eq. (1), as a function of acceleration of the observer
and the entanglement parameter $\alpha$.
## III Nonmaximal tripartite entanglement
### III.1 Fermionic Entanglement
To study the influence of acceleration parameter $r$ and the entanglement
parameter $\alpha$ on the entanglement between modes of Dirac field, we
substitute Eqs.(6) and (7) for Charlie part in Eq.(1) and rewrite it in terms
of Minkowski modes for Alice and Bob and Rindler modes for Charlie as follow
$\left|\psi_{ABCI,II}\right\rangle=\alpha(\cos r\text{
}\left|0000\right\rangle+\sin r\text{
}\left|0011\right\rangle)+\sqrt{1-\alpha^{2}}\left|1110\right\rangle,$ (13)
where
$\left|abcd\right\rangle=\left|a\right\rangle_{A}\left|b\right\rangle_{B}\left|c\right\rangle_{CI}\left|d\right\rangle_{CII}$.
Note that for the purpose of writing ease, we have also dropped the frequency
in the subscript of each ket. Being inaccessible to Charlie in Rindler region
I, the modes in Rindler region II must be trace out for investigating the
behavior of entanglement between the modes of inertial observers and the modes
of Charlie in region I. So, tracing out over the forth qubit, leaves the
following mixed density matrix between the modes of Alice, Bob and Charlie,
$\displaystyle\rho_{ABC}$
$\displaystyle=\alpha^{2}\cos^{2}r\left|000\right\rangle\left\langle
000\right|+\alpha\sqrt{1-\alpha^{2}}\cos r(\left|000\right\rangle\left\langle
111\right|+\left|111\right\rangle\left\langle 000\right|)$
$\displaystyle+\alpha^{2}\sin^{2}r\left|001\right\rangle\left\langle
001\right|+(1-\alpha^{2})\left|111\right\rangle\left\langle 111\right|.$ (14)
Taking partial transpose over each qubit in sequence and using the definition
of one-tangle, the three one-tangles can straightforwardly be calculated,
which are given by
$\mathcal{N}_{A(BC)}=\mathcal{N}_{B(AC)}=2\alpha\sqrt{1-\alpha^{2}}\cos r.$
(15) $\mathcal{N}_{C(AB)}=\alpha\sqrt{1-\alpha^{2}}\cos
r-\alpha^{2}\sin^{2}r+\alpha\sqrt{(1-\alpha^{2})\cos^{2}r+\alpha^{2}\sin^{4}r}.$
(16)
Note that $\mathcal{N}_{A(BC)}=\mathcal{N}_{B(AC)}$ shows that the two
subsystems of inertial frames are symmetrical for any values of the parameters
$\alpha$ and $r$. It can easily be checked that all the one-tangles reduce to
$1$ for a maximally entangled initial state with no acceleration, which is a
verification of the result obtained in the rest frames both for Dirac and
Scalar fields Hwang ; Wang . To have a better understanding of the influence
of the two parameters, we plot the one-tangles for different values of
$\alpha$ against $r$ in Fig. $1$(a, b).
Figure 1: (Color Online) The one-tangles (a) $\mathcal{N}_{A(BC)}$ and (b)
$\mathcal{N}_{C(AB)}$ of fermionic modes as a function of the acceleration
parameter $r$ for different values of the entanglement parameter $\alpha$ and
its normalized partners $\sqrt{1-\alpha^{2}}$. The black solid line
corresponds to maximally entangled initial state. The blue solid lines from
top to bottom correspond to $\left|\alpha\right|=\frac{1}{\sqrt{5}}$,
$\frac{1}{\sqrt{10}}$, $\frac{1}{\sqrt{22}}$ and red dashed lines from top to
bottom correspond to $\left|\alpha\right|=\frac{2}{\sqrt{5}}$,
$\frac{3}{\sqrt{10}}$, $\sqrt{\frac{21}{22}}$.
Figure (1a) shows the behavior of $\mathcal{N}_{A(BC)}=\mathcal{N}_{B(AC)}$
and figure (1b) is the plot of $\mathcal{N}_{C(AB)}$. A comparison of the two
figures shows that for maximal entangled initial state ($\alpha=1/\sqrt{2}$)
and hence for all other values of $\alpha$, the $\mathcal{N}_{C(AB)}$ falls
off rapidly with increasing acceleration as compared to $\mathcal{N}_{A(BC)}$.
However, the most interesting feature of the two figures is the different
response of the one-tangles to the parameter $\alpha$. The behavior of
$\mathcal{N}_{A(BC)}$ ($\mathcal{N}_{B(AC)}$) is unchanged by interchanging
the values of $\alpha$ and its normalizing partner $\sqrt{1-\alpha^{2}}$. On
the other hand, $\mathcal{N}_{C(AB)}$ degrades along different trajectories by
switching the values of $\alpha$ and $\sqrt{1-\alpha^{2}}$. This shows an
inequivalence of the quantization for Dirac field in the Minkowski and Rindler
coordinates. Regardless of the amount of acceleration, there is always some
amount of one-tangle left for each subsystem, which ensures the application of
entanglement based quantum information tasks between relatively accelerated
parties. The values chosen for entanglement parameter $\alpha$ and its
normalizing partner $\sqrt{1-\alpha^{2}}$ in figure (1) are
$\frac{1}{\sqrt{2}},\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}},\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}},\frac{1}{\sqrt{22}},\sqrt{\frac{21}{22}}$.
The next step is to evaluate the two-tangles. According to its definition, we
need to take partial trace over each qubit one by one. So, taking partial
trace of the final density matrix of Eq. (14) over Alice’s qubit or Bob’s
qubit leads to the following mixed density matrix
$\rho_{AC(BC)}=\rho_{ABC}^{T_{B(A)}}=\alpha^{2}\cos^{2}r\left|00\right\rangle\left\langle
00\right|+\alpha^{2}\sin^{2}r\left|01\right\rangle\left\langle
01\right|+(1-\alpha^{2})\left|11\right\rangle\left\langle 11\right|.$ (17)
Note that this matrix is diagonal and the partial transpose over either qubit
leaves it unchanged. Similarly, the reduced density matrix $\rho_{AB}$, which
is obtained by taking partial trace over the Charlie qubit, is diagonal. Using
the definition of negativity, one can easily show that there exists no
entanglement between any of these subsystems of the tripartite state
$\rho_{ABC}$. Since this result is valid for a maximally entangled GHZ state
in inertial frame, it shows that the entanglement behavior of subsystems is
independent from the status of the observer and from the degree of initial
entanglement in the state. Also, the zero value of all the two-tangles verify
the validity of the CKW inequality.
Since we now know all the one-tangles and all the two-tangles of the
tripartite state $\rho_{ABC}$, we can find the $\pi$-tangle. As all the two-
tangles are zero, using Eq. (10), it simply becomes
$\displaystyle\pi_{ABC}$
$\displaystyle=\frac{1}{3}(\mathcal{N}_{A(BC)}^{2}+\mathcal{N}_{B(AC)}^{2}+\mathcal{N}_{C(AB)}^{2})$
$\displaystyle=\frac{\alpha^{2}}{3}[\left(\sqrt{(1-\alpha^{2})}\cos^{2}r-\alpha\sin^{2}r+\sqrt{(1-\alpha^{2})\cos^{2}r+\alpha^{2}\sin^{4}r}\right)^{2}$
$\displaystyle+8(1-\alpha^{2})\cos^{2}r].$ (18)
Figure 2: (Color Online) The $\pi$-tangle of fermionic modes as a function of
acceleration parameter $r$ for different values of entanglement parameter
$\alpha$ and its normalized partner $\sqrt{1-\alpha^{2}}$. The black solid
line corresponds to maximally entangled initial state. The blue solid lines
from top to bottom correspond to $\left|\alpha\right|=\frac{1}{\sqrt{5}}$,
$\frac{1}{\sqrt{10}}$, $\frac{1}{\sqrt{22}}$ and the red dashed lines from top
to bottom correspond to $\left|\alpha\right|=\frac{2}{\sqrt{5}}$,
$\frac{3}{\sqrt{10}}$, $\sqrt{\frac{21}{22}}$.
It is straightforward to verify that for inertial frame and maximally
entangled initial state the result of Eq. (18) is $1$. To have a more close
look on how it is effected by the parameters $\alpha$ and $r$, we plot it
against the parameter $r$ for different values of the entanglement parameter
$\alpha$ in Fig. $2$. Like the one-tangles, the $\pi$-tangle exhibit a similar
behavior in response to $\alpha$. Here the solid black line represents the
behavior of $\pi$-tangle against $r$ when the initial state is maximally
entangled. It can be seen that for the same entanglement in the initial state,
interchanging the values of $\alpha$ and its normalizing partner
$\sqrt{1-\alpha^{2}}$ leads to two different degradation curves for
$\pi$-tangle against the acceleration parameter $r$. This degradation behavior
of $\pi$-tangle along two different curves is similar to the degradation of
logarithmic negativity for bipartite fermionic entangled states Pan . It is
interesting to note that the loss of entanglement against the acceleration
parameter is rapid for states of stronger initial entanglement. Nevertheless,
the rate of degradation of $\pi$-tangle is slower than the logarithmic
negativity for bipartite fermionic states.
### III.2 Bosonic Entanglement
To study the behavior of entanglement of nonmaximal initial state of scalar
field, we follow the same procedure as we used to investigate the dynamics of
entanglement of Dirac Field. For Charlie in noninertial frame, the nonmaximal
entangled initial state of Eq. (1) can be rewritten in terms of Minkowski
modes for Alice and Bob and Rindler modes of Fock space for Charlie by using
Eqs. (8) and (9) as follow
$\left|\varphi_{ABCI,II}\right\rangle=\frac{1}{\cosh
r}{\displaystyle\sum\limits_{n=0}^{\infty}}\tanh^{n}r\left[\alpha\left|00nn\right\rangle+\frac{\sqrt{(n+1)(1-\alpha^{2})}}{\cosh
r}\left|11n+1n\right\rangle\right],$ (19)
where, again, the kets
$\left|abcd\right\rangle=\left|a\right\rangle_{A}\left|b\right\rangle_{B}\left|c\right\rangle_{CI}\left|d\right\rangle_{CII}$.
In response to acceleration, for the behavior of entanglement between the
modes of inertial observers and the modes of Charlie in region I, the
inaccessible modes in region II must be trace out. Tracing out over those
modes, leaves the following mixed density matrix
$\displaystyle\varrho_{ABC}$
$\displaystyle=\alpha^{2}\left|00\right\rangle\left\langle 00\right|\otimes
M_{n,n}+(1-\alpha^{2})\left|11\right\rangle\left\langle 11\right|\otimes
M_{n+1,n+1}+$
$\displaystyle\alpha\sqrt{(1-\alpha^{2})}(\left|11\right\rangle\left\langle
00\right|\otimes M_{n+1,n}+\left|00\right\rangle\left\langle 11\right|\otimes
M_{n,n+1}),$ (20)
where
$\displaystyle M_{n,n}$
$\displaystyle=\frac{1}{\cosh^{2}r}{\displaystyle\sum\limits_{n=0}^{\infty}}\tanh^{2n}r\left|n\right\rangle\left\langle
n\right|,$ $\displaystyle M_{n,n+1}$
$\displaystyle=\frac{1}{\cosh^{3}r}{\displaystyle\sum\limits_{n=0}^{\infty}}\sqrt{(n+1)}\tanh^{2n}r\left|n\right\rangle\left\langle
n+1\right|,$ $\displaystyle M_{n+1,n}$
$\displaystyle=\frac{1}{\cosh^{3}r}{\displaystyle\sum\limits_{n=0}^{\infty}}\sqrt{(n+1)}\tanh^{2n}r\left|n+1\right\rangle\left\langle
n\right|,$ $\displaystyle M_{n+1,n+1}$
$\displaystyle=\frac{1}{\cosh^{4}r}{\displaystyle\sum\limits_{n=0}^{\infty}}(n+1)\tanh^{2n}r\left|n+1\right\rangle\left\langle
n+1\right|.$ (21)
The three one-tangles can be computed, as before, by taking partial transpose
of the density matrix of Eq. (20) with respect to each qubit one by one. It is
easy to prove that the two one-tangles which are obtained from partial
transposed of the qubits of inertial observers are equal and is given by
$\mathcal{N}_{A(BC)}=\mathcal{N}_{B(AC)}=\frac{2\alpha\sqrt{1-\alpha^{2}}}{\cosh^{3}r}\sum_{n=0}^{\infty}\sqrt{(n+1)}\tanh^{2n}r.$
(22)
We can write this relation into another more compact form as follow
$\mathcal{N}_{A(BC)}=\frac{2\alpha\sqrt{1-\alpha^{2}}}{\cosh
r\sinh^{2}r}\mathbf{Li}_{-\frac{1}{2}}(\tanh^{2}r),$ (23)
where we have used the following identities
$\displaystyle\sum_{n=0}^{\infty}(n+1)\tanh^{2n}r$ $\displaystyle=\cosh^{4}r$
$\displaystyle\sum_{n=0}^{\infty}\tanh^{2n}r$ $\displaystyle=\cosh^{2}r.$ (24)
The function $\mathbf{Li}_{n}(x)$ in Eq. (23) is a polylogarithm function and
is given by
$\mathbf{Li}_{n}(x)\equiv\sum_{k=1}^{\infty}\frac{x^{k}}{k^{n}}=\frac{x}{1^{n}}+\frac{x^{2}}{2^{n}}+\frac{x^{3}}{3^{n}}+...$
(25)
To compute the one tangle $\mathcal{N}_{C(AB)}$, first we find
$\varrho_{ABC}^{T_{C}}$ from Eq.(20) and then we construct
$(\varrho_{ABC}^{T_{C}})(\varrho_{ABC}^{T_{C}})^{{\dagger}}$, whose explicit
expression is given by
$\displaystyle(\varrho_{ABC}^{T_{C}})(\varrho_{ABC}^{T_{C}})^{{\dagger}}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{\tanh^{4n}r}{\cosh^{4}r}[(\alpha^{4}+\frac{n\alpha^{2}(1-\alpha^{2})\cosh^{2}r}{\sinh^{4}r})\left|00n\right\rangle\left\langle
00n\right|+\frac{\alpha((n+1)(1-\alpha^{2})x)^{\frac{1}{2}}}{\cosh r}$
$\displaystyle(\alpha^{2}\tanh^{2}r+\frac{n(1-\alpha^{2})}{\sinh^{2}r})\\{\left|00n+1\right\rangle\left\langle
11n\right|+\left|11n\right\rangle\left\langle 00n+1\right|\\}$
$\displaystyle+(\frac{\alpha^{2}(1-\alpha^{2})(n+1)}{\cosh^{2}r}+\frac{n^{2}(1-\alpha^{2})^{2}}{\sinh^{4}r})\left|11n\right\rangle\left\langle
11n\right|].$ (26)
The nonvanishing eigenvalues Eq. (26) are
$\left(\frac{\alpha^{4}}{\cosh^{4}r},\Lambda_{n}^{\pm},\text{ \ \
}(n=0,1,2,3,...)\right),$ (27)
where
$\Lambda_{n}^{\pm}=\frac{1}{2}(\xi\pm\sqrt{\eta+\mu}),$ (28)
and
$\displaystyle\xi$
$\displaystyle=\frac{\tanh^{4n}r}{\cosh^{4}r}\left(\frac{n^{2}(1-\alpha^{2})^{2}}{\sinh^{4}r}+\frac{2\alpha^{2}(1-\alpha^{2})(n+1)}{\cosh^{2}r}+\alpha^{4}\tanh^{4}r\right),$
$\displaystyle\mu$
$\displaystyle=\frac{4\alpha^{2}(1-\alpha^{2})(n+1)}{\cosh^{2}r}\frac{\tanh^{8n}r}{\cosh^{8}r}\left(\frac{n(1-\alpha^{2})}{\sinh^{2}r}+\alpha^{2}\tanh^{2}r\right)^{2},$
$\displaystyle\eta$
$\displaystyle=\frac{\tanh^{8n}r}{\cosh^{8}r}\left(\frac{n^{2}(1-\alpha^{2})^{2}}{\sinh^{4}r}-\alpha^{4}\tanh^{4}r\right)^{2}.$
(29)
Using the definition of one-tangle, one can obtain $\mathcal{N}_{C(AB)}$ whose
explicit expression is by
$\mathcal{N}_{C(AB)}=-1+\frac{\alpha^{2}}{\cosh^{2}r}+\sum_{n=0}^{\infty}\frac{\tanh^{2n}r}{\cosh^{2}r}\sqrt{\frac{n^{2}(1-\alpha^{2})^{2}}{\sinh^{4}r}+\frac{2\alpha^{2}(1-\alpha^{2})(n+2)}{\cosh^{2}r}+\alpha^{4}\tanh^{4}r}$
(30)
It is easy to check that the one-tangles results into $1$ for $r=0$ and
maximally entangled initial state.
Figure 3: (Color Online)The one-tangle (a) $\mathcal{N}_{A(BC)}$ and (b)
$\mathcal{N}_{C(AB)}$ of bosonic field as a function of the acceleration
parameter $r$ for different values of entanglement parameter $\alpha$ and its
normalized partners $\sqrt{1-\alpha^{2}}$. The black solid line corresponds to
maximally entangled initial state. The blue solid lines from top to bottom
correspond to $\left|\alpha\right|=\frac{1}{\sqrt{5}}$, $\frac{1}{\sqrt{10}}$,
$\frac{1}{\sqrt{22}}$ and the red dashed lines from top to bottom correspond
to $\left|\alpha\right|=\frac{2}{\sqrt{5}}$, $\frac{3}{\sqrt{10}}$,
$\sqrt{\frac{21}{22}}$.
The dependence of one-tangles on $r$ and $\alpha$, in this case, is shown in
figure ($3$). As can be seen, the one-tangles are strongly effected by the
parameters $\alpha$ and $r$. However, as before, switching between the values
of $\alpha$ and its normalizing partner $\sqrt{1-\alpha^{2}}$ does not effect
the behavior of one-tangle, corresponds to an inertial observer, against $r$
as shown in figure ($3a$). Unlike the fermionic case, the loss in one-tangle
$\mathcal{N}_{A(BC)}$ with acceleration is not uniform through the whole range
of $r$. In fermionic case, it is monotonic strictly decreasing whereas in
bosonic case, it is only monotonic decreasing, however, it never vanishes
completely. On the other hand, figure ($3b$) shows that, like the fermionic
case, the one-tangle $\mathcal{N}_{C(AB)}$ degrades along different curves
against $r$ by interchanging the values of $\alpha$ and $\sqrt{1-\alpha^{2}}$,
however, it vanishes , regardless of the value of $\alpha$, in the asymptotic
limit. The loss in $\mathcal{N}_{C(AB)}$ against $r$ depends on the degree of
entanglement in the initial state, it is faster when the entanglement is
stronger initially.
Similar to the case of Dirac field, we have verified that all the two tangles
for scalar field are also zero, that is,
$\mathcal{N}_{AB}=\mathcal{N}_{AC}=\mathcal{N}_{BC}=0.$ (31)
This verifies that CKW inequality also holds for scalar field. Again, the zero
values of all the two tangles make it easier to find the $\pi$-tangle. Instead
of writing its explicit relation, which is lengthy enough, we want to show its
behavior by plotting it against $r$ for different values of $\alpha$ in figure
($4$).
Figure 4: (Color Online) The $\pi$-tangle of bosonic field as a function of
acceleration parameter $r$ for different values of entanglement parameter
$\alpha$ and its normalized partners $\sqrt{1-\alpha^{2}}$. The black solid
line corresponds to maximally entangled initial state. The blue solid lines
from top to bottom correspond to $\left|\alpha\right|=\frac{1}{\sqrt{5}}$,
$\frac{1}{\sqrt{10}}$, $\frac{1}{\sqrt{22}}$ and the red dashed lines from top
to bottom correspond to $\left|\alpha\right|=\frac{2}{\sqrt{5}}$,
$\frac{3}{\sqrt{10}}$, $\sqrt{\frac{21}{22}}$.
The figure shows that in the range of larger acceleration, the loss of
$\pi$-tangle depends only on the initial value of the degree of entanglement.
This shows that the response of $\pi$-tangle to $r$ is different from
logarithmic negativity for bipartite state because the latter does depend on
the choice of values of $\alpha$ and $\sqrt{1-\alpha^{2}}$. However, for
smaller values of acceleration, it does degrades, like the logarithmic
negativity for bipartite states, along two different trajectories by
interchanging the values of $\alpha$ and $\sqrt{1-\alpha^{2}}$. For every
value of initial entanglement, it has a nonvanishing value at infinite
acceleration. The notable feature of figure ($4$) is that, unlike bipartite
entanglement, the tripartite entanglement for scalar field degrades slowly
with acceleration than for Dirac field and it always remains finite in the
limit of larger values of $r$.
## IV Summary
In this paper, we have investigated the entanglement behavior of nonmaximal
tripartite quantum states in both fermionic and bosonic systems when one of
the parties is traveling with a uniform acceleration. Rindler coordinates are
used for the accelerating party. The behavior of entanglement against the
acceleration parameter and the initial entanglement parameter is quantified
using $\pi$-tangle.
It is shown that the entanglement in tripartite GHZ states does not only
depend on the acceleration and initial entanglement in the states but also
depends, for the same initial entanglement, on the probability amplitudes of
the bases vectors. The one-tangles corresponding to accelerated observer, in
both bosonic and fermionic cases, strongly depends on the entanglement
parameter $\alpha$. However, in the fermionic case, it never vanishes for any
values of $\alpha$ even in the limit of infinite acceleration. Whereas in
bosonic case, regardless of the value of $\alpha$, it vanishes in the range of
infinite acceleration. The two-tangles, in both cases, are always zero, which
means that the acceleration and the degree of initial entanglement do not
affect the entanglement behavior of any of the sub-bipartite systems.
The response of $\pi$-tangle to $r$ and $\alpha$ in the two cases is
considerably different. In fermionic case, for the same initial entanglement,
it strongly depends on the values of $\alpha$ and $\sqrt{1-\alpha^{2}}$. The
difference in degradation against $r$, by interchanging the values of
probability amplitudes, increases with increasing acceleration. However, some
fraction of $\pi$-tangle always survives for all values of $\alpha$ even in
the limit of infinite acceleration. For bosonic case, in the range of large
values of $r$, the $\pi$-tangle just depends on the of initial entanglement,.
However, for small values of $r$, its degradation is different by
interchanging the values of probability amplitudes. Amazingly unlike bipartite
entanglement, the $\pi$-tangle in fermionic case degrades quickly against the
acceleration as compared to bosonic case. The survival of tripartite
entanglement may be used to perform different quantum information task in
situations where execution of such task through bipartite entanglement fails,
for example, between inside and outside of the black hole.
## References
* (1) D. Bouwmeester, A. Ekert, and A. Zeilinger, “The Physics of Quantum Information” (Springer-Verlag, Berlin, 2000); A. Peres and D. R. Terno, Rev. Mod. Phys.76, 93 (2004); C. H. Bennett, et al, Phys. Rev. Lett. 70, 1895 (1993); S. F. Huegla, M. B. Plenio, and J. A. Vaccaro, Phys. Rev. A 65, 042316 (2002); J. L. Dodd, M. A. Nielsen, M. J. Bremner, and R. T. Thew, Phys. Rev. A 65, 040301 (2002).
* (2) P. M. Alsing, David McMahon and G J Milburn, J. Opt. B: Quantum Semiclass. Opt. 6, 834 (2004).
* (3) P.M. Alsing and G. J. Milburn, Phys. Rev. Lett. 91,180404 (2003)
* (4) P. M. Alsing, I. F. Schuller, R. B. Mann, and T. E. Tessier, Phys. Rev. A 74, 032326 (2006).
* (5) I. Fuentes-Schuller and R. B. Mann, Phys. Rev. Lett. 95,120404 (2005)
* (6) G. Adesso, I. Fuentes-Schuller, and M. Ericsson, Phys. Rev. A 76, 062112 (2007).
* (7) M. R. Hwang, D. Park and E. Jung Phys. Rev. A 83, 012111 (2011)
* (8) D. E. Bruschi, J. Louko, E. Martin-Martinez, A. Dragan, and I. Fuentes1, Phy. Rev. A 82, 042332 (2010)
* (9) S. Khan and M. K. Khan, Open Sys. and Information Dyn. 19, 1250013 (2012), S. Khan, J. Mod. Opt. 59, 250 (2012); W. Zhang, J. Jing, arXiv:quant-ph/1103.4903 (2011).
* (10) M. Montero and E. Martín-Martínez, J. High Energy Phys. 07 006 (2011).
* (11) Y. Kwon and J. Chang, Phys. Rev. A 86, 014302 (2012).
* (12) E. Martin-Martinez, Garay, L.J. Leon, J. Phys. Rev. D 82, 064028 (2010); AsPachs, M. Adesso, G. Fuentes, I. Phys. Rev. Lett. 105, 151301 (2010); J. Wang and J. Jing, Phys. Rev. A 83, 022314 (2011); J. Chang and Y. Kwon, Phys. Rev. A 85, 032302 (2012); D. Hosler, C. van de Bruck, and P. Kok, Phys. Rev. A 85, 042312 (2012); M. Montero and E. Martin-Martinez, Phys. Rev. A 85, 024301 (2012); A. Smith and R. B. Mann, Phys. Rev. A 86, 012306 (2012); M. Ramzan and M. K. Khan, Quant. Info. Proc. 11, 443 (2012); M.-Z. Piao and X. Ji, J. Mod. Opt. 59, 21 (2011); S. Khan and M. K. Khan, J. Phys. A: Math. Theor. 44 045305 (2011); M. Montero and E. Martin-Martinez, Phys. Rev. A 83, 062323 (2011); B. Nasr Esfahani, M. Shamirzaie, and M. Soltani, Phys. Rev. D 84, 025024 (2011); Qiyuan Pan and Jiliang Jing, Phys. Rev. D 78, 065015 (2008).
* (13) E. Martin-Martinez, D. Hosler, and M. Montero, Phy. Rev. A 86, 062307 (2012).
* (14) D. Hosler, C. van de Bruck, and P. Kok, Phys. Rev. A 85, 042312 (2012).
* (15) E. G. Brown, K. Cormier, E. Martın-Martınez, and R. B. Mann, Phys. Rev. A 86, 032108 (2012).
* (16) E. Martin-Martinez, Garay, L.J. Leon, J. Phys. Rev. D 82, 064028 (2010).
* (17) AsPachs, M. Adesso, G. Fuentes, I. Phys. Rev. Lett. 105, 151301 (2010).
* (18) D. E. Bruschi, A. Dragan, I. Fuentes and J. Louko, Phys. Rev. D 86, 025026 (2012).
* (19) V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61, 052306 (2000).
* (20) Y. C. Ou and H. Fan, Phys. Rev. A 75 062308 (2007).
* (21) G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002); M. B. Plenio, Phys. Rev. Lett. 95, 090503 (2005).
* (22) O. Rudolph, J. Phys. A Math. Gen. 33, 3951 (2000); Quant. Inf. Proc. 4, 219 (2005).
* (23) K. Chen and L. A. Wu, Phy. Lett. A 306, 14 (2002)
* (24) M. Horodecki, P.Horodecki and R. Horodecki, Open Sys. and Information Dyn. 13, 103 (2006).
* (25) F. Verstraete, J. Dehaene and B. D. Moor, Phys. Rev. A 68, 012103 (2003)
* (26) M. S. Leifer, N. Linden and A. Winter, Phys. Rev. A 69 052304 (2004)
* (27) R. Lohmayer, A. Osterloh, J. Siewert, and A. Uhlmann, Phys. Rev. Lett. 97, 260502 (2006).
* (28) J. Wang and J. Jing, Phys. Rev. A 83, 022314 (2011).
* (29) Q. Pan and J. Jing, Phys. Rev. A 77, 024302, (2008).
|
arxiv-papers
| 2014-02-28T07:16:56 |
2024-09-04T02:49:59.083929
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Salman Khan, Niaz Ali Khan, M.K. Khan",
"submitter": "Salman Khan",
"url": "https://arxiv.org/abs/1402.7152"
}
|
1402.7165
|
# The “magic” angle in the self-assembly of colloids suspended in a nematic
host phase
Sergej Schlotthauer Stranski-Laboratorium für Physikalische und Theoretische
Chemie, Fakultät für Mathematik und Naturwissenschaften, Technische
Universität Berlin, Straße des 17. Juni 115, 10623 Berlin, GERMANY Tillmann
Stieger Stranski-Laboratorium für Physikalische und Theoretische Chemie,
Fakultät für Mathematik und Naturwissenschaften, Technische Universität
Berlin, Straße des 17. Juni 115, 10623 Berlin, GERMANY Michael Melle
Stranski-Laboratorium für Physikalische und Theoretische Chemie, Fakultät für
Mathematik und Naturwissenschaften, Technische Universität Berlin, Straße des
17. Juni 115, 10623 Berlin, GERMANY Marco G. Mazza Max-Planck-Institut für
Dynamik und Selbstorganisation, Am Faßberg 17, 37077 Göttingen, GERMANY
Martin Schoen Stranski-Laboratorium für Physikalische und Theoretische
Chemie, Fakultät für Mathematik und Naturwissenschaften, Technische
Universität Berlin, Straße des 17. Juni 115, 10623 Berlin, GERMANY Department
of Chemical and Biomolecular Engineering, 911 Partners Way, North Carolina
State University, Raleigh, NC 27695, U.S.A.
###### Abstract
Using extensive Monte Carlo (MC) simulations of colloids immersed in a nematic
liquid crystal we compute an effective interaction potential via the local
nematic director field and its associated order parameter. The effective
potential consists of a local Landau-de Gennes (LdG) and a Frank elastic
contribution. Molecular expressions for the LdG expansion coefficients are
obtained via classical density functional theory (DFT). The DFT result for the
LdG parameter $A$ is improved by locating the phase transition through finite-
size scaling theory. We consider effective interactions between a pair of
homogeneous colloids with Boojum defect topology. In particular, colloids
attract each other if the angle between their center-of-mass distance vector
and the far-field nematic director is about $30^{\circ}$ which settles a long-
standing discrepancy between theory and experiment. Using the effective
potential in two-dimensional MC simulations we show that self-assembled
structures formed by the colloids are in excellent agreement with experimental
data.
###### pacs:
61.30.-v,61.30.Jf,82.70Dd,05.10.Ln
If a liquid crystal is in the nematic phase the overall orientation of its
molecules (i.e., mesogens) can be described quantitatively by the non-local
unit vector (i.e., the far-field nematic director) $\bm{\widehat{n}}_{0}$ [1].
Immersing a colloidal particle in this nematic host phase gives rise to a
director field $\bm{\widehat{n}}\left(\bm{r}\right)$ such that sufficiently
close to the colloid’s surface, $\bm{\widehat{n}}\left(\bm{r}\right)$ and
$\bm{\widehat{n}}_{0}$ may differ. The deviation between
$\bm{\widehat{n}}\left(\bm{r}\right)$ and $\bm{\widehat{n}}_{0}$ is caused by
the specific anchoring of mesogens at the surface of the colloid. Depending on
details of the host phase $\bm{\widehat{n}}\left(\bm{r}\right)$ can be of such
dazzling complexity that experts are just beginning to unravel its structural
details [2].
The mismatch between $\bm{\widehat{n}}\left(\bm{r}\right)$ and
$\bm{\widehat{n}}_{0}$ also gives rise to effective interactions between
several colloids that are mediated by the nematic host [3]. These interactions
may therefore be used to self-assemble the colloids into supramolecular
entities in a controlled (i.e., directed) manner. This way ordered assemblies
of colloids of an enormously complex structure with rich symmetries may be
built that would not exist without the ordered structure of the host phase [4,
5].
The complex self-assembled structures formed by the colloids are also of
practical importance. For instance, taking as a specific example dielectric
colloids it could be demonstrated that the propagation of light through a
self-assembled ordered colloidal arrangement is affected in a way similar to
the propagation of electrons in a semiconductor crystal [6]. Hence, ordered
periodic assemblies of colloids are already discussed within the framework of
novel photonic devices with fascinating properties [7].
Clearly, to use the effective interaction potential for the self-assembly of
colloids in a nematic host phase the molecular origin of the potential itself
must be understood. Our motivation to contribute to such an improved
understanding goes back to an observation made some time ago by Poulin and
Weitz [8]. They found experimentally that in a nematic phase the colloidal
center-to-center distance vector $\bm{r}_{12}$ forms a “magic” angle of
$\theta\approx 30^{\circ}$ with $\bm{\widehat{n}}_{0}$ if the mesogens at the
surfaces of the colloidal pair are anchored in a locally planar fashion.
Hence, near an isolated colloid a so-called Boojum defect would arise under
these conditions [9].
This experimental observation has resisted a quantitative theoretical
explanation to date. In previous theoretical attempts a much larger angle of
about $50^{\circ}$ is usually found [8, 10]. This number is based upon
calculations where one employs the electrostatic analog of the Boojum defect
topology [8]. In fact, as stated explicitly by Poulin and Weitz “This
theoretical value is different from the experimentally observed value for
$\theta$ $\ldots$ since the theory is a long-range description that does not
account for short-range effects” [8]. Another motivation for our work is the
more recent experimental observation that between a pair of colloids in a
nematic host repulsive and attractive forces act depending on $\theta$ [10].
For example, at $\theta\approx 30^{\circ}$ the colloids attract each other
whereas at $\theta=0^{\circ}$ and $90^{\circ}$ repulsion between the colloids
is observed.
To unravel the persisting discrepancy between theory and experiment we employ
a combination of Monte Carlo (MC) simulations in the isothermal-isobaric
ensemble, two-dimensional (2D) MC simulations in the canonical ensemble,
classical density functional theory (DFT), concepts of finite-size scaling
(FSS), and Landau-de Gennes (LdG) theory to investigate the effective
interaction between a pair of spherical, chemically homogeneous colloids
mediated by a nematic host phase.
To model the host phase we adopt the so-called Hess-Su model. In this model
mesogen-mesogen interactions are described by an isotropic core where
$\varepsilon$ and $\sigma$ set energy and length scale, respectively.
Superimposed to the isotropic core are anisotropic attractions of respective
strengths $\varepsilon_{1}\varepsilon$ and $\varepsilon_{2}\varepsilon$ where
the dimensionless anisotropy parameters
$2\varepsilon_{1}=-\varepsilon_{2}=0.08$ throughout this work. Under these
conditions the Hess-Su model exhibits isotropic-nematic (IN) phase transitions
[11].
The colloid-mesogen interaction is modeled via short-range repulsive
interactions and an attractive Yukawa tail where we take its inverse Debye
screening length $\lambda\sigma=0.50$ [12]. Mesogens at the colloids’ surfaces
are anchored in a locally planar fashion. This setup is then placed between
structureless, planar solid substrates separated by a distance
$s_{\mathrm{z}}=24\sigma$. Mesogens at the substrates are anchored such that
their longer axes point along the $x$-axis $\bm{\widehat{e}}_{\mathrm{x}}$.
Under these conditions a Boojum defect topology emerges at a single, isolated
colloid. Colloids are immersed in the host phase such that their center-to-
center distance vector is given by $\bm{r}_{12}=\left(x_{12},y_{12},0\right)$.
We employ dimensionless units, that is length is given in units of $\sigma$,
energy in units of $\varepsilon$, and temperature in units of
$\varepsilon/k_{\mathrm{B}}$ ($k_{\mathrm{B}}$ Boltzmann’s constant). Other
derived units are then expressed as combinations of these basic ones as usual
[12]. In particular, we set temperature $T=0.90$ and pressure $P=1.80$ such
that the host phase is nematic at a mean number density $\rho\approx 0.90$.
The hard-core radius of each colloid is $r_{0}=3.00$. Other conditions of the
MC simulations are exactly the same as in Ref. 12 where additonal details of
the model can also be found.
Figure 1: (Color online) (a)–(c) Defect topologies for a colloidal pair with
locally planar surface anchoring of mesogens separated by $\bm{r}_{12}$;
$\cos\theta=\bm{r}_{12}\cdot\bm{\widehat{n}}_{0}/r_{12}$,
$\bm{\widehat{n}}_{0}\cdot\bm{\widehat{e}}_{x}=1$, and
$r_{12}=\left|\bm{r}_{12}\right|$. (d)–(f) As (a)–(c) but projected onto the
$x$–$y$ plane. Attached color bars give $S\left(x,y\right)$ and dashes
indicate $\bm{\widehat{n}}\left(\bm{r}\right)$.
Results of our MC simulations shown in Fig. 1(a) indicate that parts of the
Boojum defects interact forming a torus. As $\theta$ increases, the torus is
“ripped apart” [Fig. 1(b)]. Eventually, a handle-like defect topology emerges
at $\theta\simeq 90^{\circ}$ [Fig. 1(c)].
Defect regions around the colloids are visualized by shading them if the local
nematic order parameter $S\left(\bm{r}\right)\leq 0.20$. We obtain
$S\left(\bm{r}\right)$ numerically as the largest eigenvalue of the local
alignment tensor [12]. The eigenvector $\bm{\widehat{n}}\left(\bm{r}\right)$
associated with $S\left(\bm{r}\right)$ is the director field.
The latter is illustrated in Figs. 1(d)–1(f). The plots indicate relatively
localized regions of low $S\left(\bm{r}\right)$ in the vicinity of the
colloids and that $\bm{\widehat{n}}\left(\bm{r}\right)$ is bent in ways that
depend on the specific defect topology (i.e., on $\theta$).
Naturally, the reduction of $S\left(\bm{r}\right)$ and the bending of
$\bm{\widehat{n}}\left(\bm{r}\right)$ causes the free-energy density
$f\left(\bm{r}\right)$ of the system to increase locally relative to that of
the pure host phase without the colloids. Consequently, we adopt
$\displaystyle\Delta f\left(\bm{r}\right)\\!$ $\displaystyle=$
$\displaystyle\\!A\left(T,\rho\right)S^{2}\left(\bm{r}\right)\\!+\\!B\left(T,\rho\right)S^{3}\left(\bm{r}\right)\\!+\\!C\left(T,\rho\right)S^{4}\left(\bm{r}\right)$
(1)
$\displaystyle+\frac{K}{2}\left\\{\left[\bm{\nabla}\cdot\bm{\widehat{n}}\left(\bm{r}\right)\right]^{2}+\left[\bm{\nabla}\times\bm{\widehat{n}}\left(\bm{r}\right)\right]^{2}\right\\}-f_{0}$
where the first three terms on the right side correspond to a local LdG free-
energy density $f_{\mathrm{LdG}}\left(\bm{r}\right)$,
$f_{0}=AS^{2}+BS^{3}+CS^{4}$ is the LdG free-energy density obtained under the
same thermodynamic conditions but in the absence of the colloids, and $\Delta
f_{\mathrm{LdG}}\left(\bm{r}\right)\equiv
f_{\mathrm{LdG}}\left(\bm{r}\right)-f_{0}$. Coefficients $A$, $B$, and $C$ are
coefficients in the LdG expansion and $S$ is the global nematic order
parameter.
The two terms on the second line of Eq. (1) correspond to the local Frank
free-energy density $f_{\mathrm{el}}\left(\bm{r}\right)$ that accounts for
elastic distortions of the director field where $K$ is an elastic constant. We
consider here the so-called one-constant approximation in which it is assumed
that splay, twist, and bend deformations of
$\bm{\widehat{n}}\left(\bm{r}\right)$ contribute equally to
$f_{\mathrm{el}}\left(\bm{r}\right)$. It has recently been shown [13] that the
one-constant approximation is an excellent approximation for the present model
system because of the small aspect ratio of the mesogens. Under the present
thermodynamic conditions, $K=1.66$. We then obtain $f_{\mathrm{el}}$ by
numerically differentiating $\bm{\widehat{n}}\left(\bm{r}\right)$ [12].
We assume that both the local LdG contribution in Eq. (1) and $f_{0}$ are
governed by the same set $A$, $B$, and $C$. Moreover, $\Delta
f\left(\bm{r}\right)$ in Eq. (1) does not account for either fluctuations in
$S\left(\bm{r}\right)$ or $\bm{\widehat{n}}\left(\bm{r}\right)$ and therefore
constitutes a mean-field expression. Notice also that using in Eq. (1)
$S\left(\bm{r}\right)$ and $\bm{\widehat{n}}\left(\bm{r}\right)$ from MC is
advantageous because then both quantities correspond to an equilibrium
situation. Conventionally, $S\left(\bm{r}\right)$ and
$\bm{\widehat{n}}\left(\bm{r}\right)$ are treated as variational functions in
the ansatz in Eq. (1) which bears the risk that the numerical minimization of
the functional $\Delta
f\left[S\left(\bm{r}\right),\bm{\widehat{n}}\left(\bm{r}\right)\right]$ may
miss the true equilibrium solution.
To use Eq. (1), $A$, $B$, and $C$ are required. Whereas these quantities are
notoriously difficult to compute for reasons described by Eppenga and Frenkel
a long time ago [14], Gupta and Ilg have devised a new approach that works
reliably for mesogens with a relatively large aspect ratio [15]. In practice,
however, we observed that the method of Gupta and Ilg does not work well for
our model fluid where mesogens have a rather small aspect ratio of only
$1.26$.
Because Eq. (1) constitutes a mean-field expression we resort to mean-field
DFT alternatively where [16]
$\beta\Delta
f_{\mathrm{or}}=\rho\int\limits_{-1}^{1}\mathrm{d}x\,\overline{\alpha}\left(x\right)\ln\left[\overline{\alpha}\left(x\right)\right]+\rho^{2}\sum\limits_{\begin{subarray}{c}l=2\\\
l\text{ even}\end{subarray}}^{\infty}S_{l}^{2}u_{l}$ (2)
is the difference in free-energy density of the nematic relative to the
isotropic phase. In Eq. (2), $x=\cos\vartheta$ where $\vartheta$ is the
azimuthal angle, $\beta=1/k_{\mathrm{B}}T$, $\overline{\alpha}\left(x\right)$
is the orientation distribution function, and members of the set
$\left\\{u_{l}\right\\}$ account for the contribution of anisotropic mesogen-
mesogen interactions to the free-energy density. Because of the uniaxial
symmetry of the nematic phase we expand
$\overline{\alpha}\left(x\right)=\frac{1}{2}+\sum\limits_{\begin{subarray}{c}l=2\\\
l\text{
even}\end{subarray}}^{\infty}\frac{2l+1}{2}S_{l}P_{l}\left(x\right)\equiv\frac{1}{2}+\xi\left(x\right)$
(3)
in terms of Legendre polynomials $\left\\{P_{l}\left(x\right)\right\\}$. We
assume $\bm{\widehat{n}}_{0}\cdot\bm{\widehat{e}}_{\mathrm{z}}=1$ and $0\leq
S_{l}\leq 1$ are order parameters.
We then insert the expression on the far right side of Eq. (3) into Eq. (2)
and expand the integrand in terms of $\xi$ around $\xi=0$ (i.e., at the IN
phase transition). Retaining in this expansion only the leading term of $\xi$
for $l=2$ and neglecting terms proportional to $S_{2}^{n}$ ($n\geq 5$) allows
us to rewrite Eq. (2) as
$\Delta
f_{\mathrm{or}}=a\left(\rho\right)\left(T-T^{\ast}\right)S_{2}^{2}-\frac{8\rho
k_{\mathrm{B}}T}{105}S_{2}^{3}+\frac{4\rho k_{\mathrm{B}}T}{35}S_{2}^{4}$ (4)
where $a\left(\rho\right)=2\rho k_{\mathrm{B}}/5$ and $T^{\ast}=-5\rho
u_{2}/2k_{\mathrm{B}}$ is the temperature at which the nematic phase becomes
thermodynamically stable. Assuming that $S=S_{2}$ we equate terms of equal
power in $S$ in $f_{0}$ and Eq. (4) which yields molecular expressions for the
LdG constants $A$, $B$, and $C$. In particular, $A$ changes sign at
$T=T^{\ast}$, $B<0$, and $C>0$ as they must at a first-order phase transition
[1].
One also notices that the value of $T^{\ast}$ depends on $u_{2}$ where the
precise form of $u_{2}$ is a consequence of the level of sophistication at
which pair correlations are treated within mean-field DFT [11]. For example,
at simple mean-field (SMF) level,
$u_{2}=-32\pi\varepsilon_{1}\varepsilon\sigma^{3}/15$ is a constant. At the
more elaborate modified mean-field (MMF) level, $u_{2}$ becomes a function of
$T$ [see Eqs. (3.7) and (3.8) of Ref. 11]. It turns out that at SMF level,
$T^{\ast}$ is underestimated whereas at MMF level it is overestimated.
To overcome this problem we determine $T^{\ast}$ via FSS. Following Ref. 17 we
first calculate the coexistence temperature $T_{\mathrm{IN}}\simeq 1.02$ at
the IN phase transition. It is given as the intersection of the second-order
Binder cumulants of $S$ for different system sizes [17]. From the expression
$T^{\ast}=T_{\mathrm{IN}}-2B^{2}/9aC$ [18] and using $B$, $C$, and $a$ from
DFT, $T^{\ast}$ can easily be determined. Notice also that
$S_{\mathrm{IN}}=-2B/3C=\frac{4}{9}$ irrespective of $T_{\mathrm{IN}}$ [1]
whereas MMF DFT predicts this value of $S_{\mathrm{IN}}$ only to be a
threshold reached for sufficiently high $T_{\mathrm{IN}}$ (Fig. 2 of Ref. 11)
thus pointing to a certain deficiency of LdG theory.
Figure 2: (Color online)
$\mathcal{F}_{\mathrm{el}}=\int\mathrm{d}\bm{r}\,f_{\mathrm{el}}\left(\bm{r}\right)$
($\bullet$) (left ordinate) and
$\Delta\mathcal{F}_{\mathrm{LdG}}=\int\mathrm{d}\bm{r}\,\Delta
f_{\mathrm{LdG}}\left(\bm{r}\right)$ ($\blacksquare$) (right ordinate) as
functions of $\theta$ for $r_{12}=2r_{0}$ (see Fig. 1).
Plots of $\mathcal{F}_{\mathrm{el}}$ and $\Delta\mathcal{F}_{\mathrm{LdG}}$ in
Fig. 2 illustrate the impact of a colloidal pair on the free energy of the
host phase. Both quantities vary nonmonotonically with the angle $\theta$ and
exhibit minima at $\theta\simeq 30^{\circ}$ in agreement with the experimental
findings of Poulin and Weitz [8]. Because deformations of
$\bm{\widehat{n}}\left(\bm{r}\right)$ cost free energy,
$\mathcal{F}_{\mathrm{el}}>0$. Similarly, the presence of the colloids reduces
$S\left(\bm{r}\right)$ such that in some regions $S>S\left(\bm{r}\right)$ (see
Fig. 1). Because the host phase without the colloids is deep in the nematic
phase, $f_{0}<0$ such that $\Delta\mathcal{F}_{\mathrm{LdG}}>0$ as well.
That both $\mathcal{F}_{\mathrm{el}}$ and $\Delta\mathcal{F}_{\mathrm{LdG}}$
become minimal at about the same $\theta$ indicates that destortions of
$\bm{\widehat{n}}\left(\bm{r}\right)$ and a local reduction of nematic order
are coupled. However, deformations of $\bm{\widehat{n}}\left(\bm{r}\right)$
turn out to be more important than reduction of nematic order because
$\mathcal{F}_{\mathrm{el}}$ exceeds $\Delta\mathcal{F}_{\mathrm{LdG}}$ by
between one and two orders of magnitude over the entire range of $\theta$’s.
This conclusion is drawn on the basis of plots in Fig. 2 and by noticing that
for both curves the ground state is the same, namely
$\bm{\widehat{n}}\left(\bm{r}\right)=\bm{\widehat{n}}_{0}$
($\mathcal{F}_{\mathrm{el}}=0$) and $S\left(\bm{r}\right)=S$
($\Delta\mathcal{F}_{\mathrm{LdG}}=0$).
Figure 3: (Color online)
$\Delta\mathcal{F}_{\mathrm{eff}}/\Delta\mathcal{F}_{\mathrm{B}}$ as a
function of relative positions of the colloids in the $x$–$y$ plane (see
attached color bar). The white semicircle at the center represents a reference
colloid.
Results presented in Fig. 2 have been obtained for two colloids in contact
with each other. However, the general physical picture reflected by Fig. 2 is
preserved if besides $\theta$, $r_{12}$ is varied, too. To that end we realize
from Eq. (1) that $\lim_{r_{12}\to\infty}\Delta f\left(\bm{r}\right)=2\Delta
f_{\mathrm{B}}\left(\bm{r}\right)$ where $\Delta
f_{\mathrm{B}}\left(\bm{r}\right)$ is the local free energy density of two
isolated Boojum defects relative to the same ground state used above. Taking
$\Delta\mathcal{F}_{\mathrm{B}}=\int\mathrm{d}\bm{r}\,\Delta
f_{\mathrm{B}}\left(\bm{r}\right)$ allows us to introduce
$\Delta\mathcal{F}_{\mathrm{eff}}\equiv\mathcal{F}_{\mathrm{el}}+\Delta\mathcal{F}_{\mathrm{LdG}}-2\Delta\mathcal{F}_{\mathrm{B}}$
as the effective potential acting between a pair of colloids and mediated by
the nematic host.
A map of $\Delta\mathcal{F}_{\mathrm{eff}}$ in Fig. 3 shows that for
$\theta=0^{\circ}$, $\Delta\mathcal{F}_{\mathrm{eff}}$ is strongly repulsive
in a relatively localized region. This is a consequence of the merger of parts
of the Boojum defect illustrated by Figs. 1(a) and 1(d). In agreement with
plots in Fig. 2 we see that $\Delta\mathcal{F}_{\mathrm{eff}}$ is attractive
if $r_{12}$ is sufficiently small where the absolute minimum of
$\Delta\mathcal{F}_{\mathrm{eff}}$ is found at $\theta\approx 30^{\circ}$.
One also notices from Fig. 3 a small repulsive barrier in
$\Delta\mathcal{F}_{\mathrm{eff}}$ as $\theta$ approaches $90^{\circ}$ and
$7\lesssim r_{12}\lesssim 10$. Hence, a pair of colloids at $\theta\approx
0^{\circ}$ and at sufficiently large $r_{12}$ and
$75^{\circ}\lesssim\theta\lesssim 90^{\circ}$ would repel each other whereas
those forming an angle of $\theta\approx 30^{\circ}$ would attract each other.
These findings are in excellent agreement with experimental observations [Fig.
2(b) of Ref. 10].
Figure 4: (Color online) 2D MC configurations
($\bm{\widehat{n}}_{0}\cdot\bm{\widehat{e}}_{\mathrm{x}}=1$). (a)
$\phi=N_{\mathrm{coll}}\pi r_{0}^{2}/s_{\mathrm{x}}s_{\mathrm{y}}=0.065$, (b)
$\phi=0.234$ ($s_{\mathrm{x}}=s_{\mathrm{y}}=50$).
Taking $\Delta\mathcal{F}_{\mathrm{eff}}$ as an effective, pairwise additive
potential we perform standard Metropolis 2D MC simulations of
$N_{\mathrm{coll}}$ colloids modeling the nematic host phase implicitly.
Technically, $\Delta\mathcal{F}_{\mathrm{eff}}$ is stored at nodes of a
regularly spaced grid in the $x$–$y$ plane; the actual value of
$\Delta\mathcal{F}_{\mathrm{eff}}$ at $\bm{r}_{12}$ is obtained by bilinear
interpolation between the four nearest nodes. The simulations are carried out
in the canonical ensemble. Results in Fig. 4(a) show that at low packing
fraction $\phi$ the colloids tend to form linear chains of an angle of about
$30^{\circ}$ with $\bm{\widehat{n}}_{0}$. At higher $\phi$ the snapshot in
Fig. 4(b) reveals more extended two-dimensional structures. Plots in both
parts of Fig. 4 are in excellent qualitative agreement with experimental
findings (see Fig. 1 of Ref. 10).
To summarize we used a combination of MC simulations, FSS, and mean-field DFT
to compute the effective interaction potential between a pair of colloids
immersed in a nematic liquid crystal. The colloids are chemically homogeneous
and anchor mesogens in a locally planar fashion at their surface. On accound
of the mismatch between this local alignment and $\bm{\widehat{n}}_{0}$ a
Boojum defect topology emerges at an isolated colloid. If two such colloids
approach each other the Boojum defects interact such that the precise topology
changes with the angle $\theta$ formed between the distance vector connecting
the centers of the colloidal pair and $\bm{\widehat{n}}_{0}$.
As a result of the topological change repulsive and attractive effective
interactions arise. These are dominated by the distortion of
$\bm{\widehat{n}}\left(\bm{r}\right)$ whereas the accompanying reduction of
local nematic order is negligible. Most notably, the distribution of regions
in which the effective interaction potential
$\Delta\mathcal{F}_{\mathrm{eff}}$ is attractive or repulsive matches
experimental results reported by Smalyukh et al. despite their much larger
colloids [10].
It is particularly gratifying that the most favorable angle we find is
$\theta\approx 30^{\circ}$ in agreement with the work by Poulin and Weitz [8]
and Smalyukh et al. [10]. Our work therefore offers the first quantitative
theoretical explanation of earlier experimental observations. Moreover, we
show that it is the relatively short-range effects that are responsible for
the observed attraction and repulsion between nematic colloids thereby
confirming the earlier conjecture by Poulin and Weitz [8].
###### Acknowledgements.
We acknowledge financial support from Deutsche Forschungsgemeinschaft through
the International Graduate Research Training Group 1524. S. S. and M. M. are
grateful for discussions with Prof. C. K. Hall (NCSU).
## References
* [1] P. G. de Gennes and J. Prost, The physics of liquid crystals, (Oxford Science Publications, Oxford, 1995).
* [2] V. S. R. Jampani et al., Phys. Rev. E 84, 031703 (2011).
* [3] K. Izaki and Y. Kimura, Phys. Rev. E 87, 062507 (2013).
* [4] U. Ognysta et al., Langmuir 25, 12092 (2009).
* [5] H. Qi and T. Hegmann, J. Mater. Chem. 16, 4197 (2006).
* [6] M. Humar et al., Nat. Photonics 3, 595 (2009).
* [7] I. Muševic et al., J. Phys.: Condens. Matter 23, 284112 (2011).
* [8] P. Poulin and D. A. Weitz, Phys. Rev. E 57, 626 (1998).
* [9] P. Poulin et al., Science 275, 1770 (1997).
* [10] I. I. Smalyukh et al., Phys. Rev. Lett. 95, 157801 (2005).
* [11] S. Giura and M. Schoen, Phys. Rev. E 90, 022507 (2014).
* [12] M. Melle et al., J. Chem. Phys. 136, 194703 (2012).
* [13] T. Stieger et al., J. Chem. Phys. 140, 054905 (2014).
* [14] R. Eppenga and D. Frenkel, Mol. Phys. 52, 1303 (1984).
* [15] B. Gupta and P. Ilg, Polymers, 5, 328 (2013).
* [16] S. Giura et al., Phys. Rev. E 87, 012313 (2013).
* [17] M. Greschek and M. Schoen, Phys. Rev. E 83, 011704 (2011).
* [18] L. Senbetu and C.-W. Woo, Mol. Cryst. Liq. Cryst. 84, 101 (1982).
|
arxiv-papers
| 2014-02-28T08:46:51 |
2024-09-04T02:49:59.092221
|
{
"license": "Public Domain",
"authors": "Sergej Schlotthauer, Tillmann Stieger, Michael Melle, Marco G. Mazza,\n and Martin Schoen",
"submitter": "Marco G. Mazza",
"url": "https://arxiv.org/abs/1402.7165"
}
|
1402.7187
|
# Entanglement detection in coupled particle plasmons
Javier del Pino Departamento de Física Teórica de la Materia Condensada and
Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid,
Madrid E-28049, Spain Instituto de Física Fundamental, IFF-CSIC, Calle
Serrano 113b, Madrid E-28006, Spain Johannes Feist Departamento de Física
Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC),
Universidad Autónoma de Madrid, Madrid E-28049, Spain F.J. García-Vidal
Departamento de Física Teórica de la Materia Condensada and Condensed Matter
Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid E-28049, Spain
Juan Jose García-Ripoll Instituto de Física Fundamental, IFF-CSIC, Calle
Serrano 113b, Madrid E-28006, Spain
###### Abstract
When in close contact, plasmonic resonances interact and become strongly
correlated. In this work we develop a quantum mechanical model for an array of
coupled particle plasmons. This model predicts that when the coupling strength
between plasmons approaches or surpasses the local dissipation, a sizable
amount of entanglement is stored in the collective modes of the array. We also
prove that entanglement manifests itself in far-field images of the plasmonic
modes, through the statistics of the quadratures of the field, in what
constitutes a novel family of entanglement witnesses. Finally, we estimate the
amount of entanglement, the coupling strength and the correlation properties
for a system that consists of two or more coupled nanospheres of silver,
showing evidence that our predictions could be tested using present-day state-
of-the-art technology.
###### pacs:
42.50.Dv, 73.20.Mf, 03.67.Mn
Surface plasmons are hybrid light-matter excitations confined at the interface
between a metal and a dielectric. Due to their small mode volume and strong
electromagnetic (EM) fields, surface plasmons interact very strongly with
quantum optical emitters Chang _et al._ (2006); Dzsotjan _et al._ (2010);
Andersen _et al._ (2011); Gonzalez-Tudela _et al._ (2011), such as quantum
dots Akimov _et al._ (2007), NV-centers Kolesov _et al._ (2009) or inorganic
Gomez _et al._ (2010) and organic molecules Bellessa _et al._ (2004);
Schwartz _et al._ (2011). This, together with their broadband nature, small
size and their inherent quantum properties make them a promising platform for
future integrated quantum information technologies Tame _et al._ (2013).
However, a very important problem lies in the characterization and control of
those quantum properties. So far, several experiments have demonstrated that
coupling photons in and out of plasmonic resonances preserves quantum features
such as single-photon excitations and anti-bunching Akimov _et al._ (2007),
photon-photon entanglement Altewischer _et al._ (2002), energy-time
entanglement Fasel _et al._ (2005) and squeezing Huck _et al._ (2009). In
this work we focus on the quantum properties of the surface plasmon themselves
and in particular in how many-body entanglement can be engineered using arrays
of coupled plasmonic modes.
In this Letter, we present a plasmonic setup that intrinsically exhibits many-
body entanglement and provide a recipe for characterizing it experimentally.
Our results build on a quantum mechanical model for a 1D or a 2D array of
coupled nanoparticles Maier _et al._ (2002a, b); Weick _et al._ (2013) that
includes the dipole-dipole interaction between particle plasmons, the losses
in each nanoparticle and the possibility of injecting energy via coherent or
incoherent light. Using this model we can not only study the transport of
excitations through the plasmonic band, but we also demonstrate the emergence
of stationary entanglement in the array at room temperature. Moreover, we
argue that this entanglement can be detected by measuring fluctuations in the
far-field from the light that is emitted from the plasmonic array.
We introduce three important theoretical ideas. The first one is a quantum
mechanical model for the nanoparticle array that consists of an array of
coupled oscillating dipoles with nearest-neighbor interaction and a local
dissipation that accounts for the losses. This model results in a master
equation for the density matrix associated with the plasmonic array. The
second important idea is that, under very general circumstances, this density
matrix will be Gaussian Weedbrook _et al._ (2012) and all properties of the
array can be deduced from expectation values or “moments” of a finite set of
operators. In practice this implies a single set of exactly solvable ordinary
differential equations that fully describes the evolution of the quantum
surface plasmons. This technique allows us to make predictions not only on the
dynamics of the dipoles (i.e., absorption and transport of energy) but also
about their correlations and the resulting entanglement.
The final idea in this work is a formal study of the experimental observables
that can detect the presence of entanglement in the plasmonic array, the so-
called entanglement witnesses Gühne and Tóth (2009); Horodecki _et al._
(1996); Lewenstein _et al._ (2000); Terhal (2000); Bruß _et al._ (2002). To
this end, we study the plasmonic band and compute the fluctuations of the EM
field in momentum space. We formally prove that the presence of squeezing in
the light with opposite momenta is a signature of entanglement. From an
experimental point of view, this implies that by refocusing the far-field
light emitted from the structure and studying its quantum fluctuations [cf.
Fig. 1], the amount of entanglement that is present in the plasmonic array can
be quantified. This general result is valid even when the Gaussian assumption
or our underlying quantum model breaks down.
Figure 1: An array of interacting nanoparticles gives rise to a set of coupled
plasmonic modes. The far field emission of these modes is collected by a lens.
By correlating the properties of the light at different points in the focal
plane, we get information about the multipartite entanglement.
We model our coupled particle plasmons as a set of $N$ oscillating dipoles
forming a linear 1D array, which interact through nearest-neighbor dipole
coupling and may be subject to external driving. The Hamiltonian reads
($\hbar=1$)
$H=\sum_{n=1}^{N}\frac{\omega}{2}(p_{n}^{2}+x_{n}^{2})+\sum_{\langle
n,m\rangle}gx_{n}x_{m}+\sum_{n=1}^{N}f_{n}(t)x_{n},$ (1)
where $f_{n}(t)$ is a driving force, $x_{n}$ is the dipole moment of the
particle plasmon and $p_{n}$ its associated canonical momentum. $g$ is the
coupling strength between neighboring sites, $\left\langle{n,m}\right\rangle$,
which are separated by a distance $\Lambda$.
We introduce local dissipation by means of a master equation to describe the
evolution of the quantum state or density matrix, $\rho$. This equation groups
all plasmonic losses in a single parameter, $\gamma$, and reads
$\displaystyle\partial_{t}\rho=-\frac{i}{\hbar}[H,\rho]+\sum_{n=1}^{N}\frac{\gamma}{2}(2a_{n}\rho
a_{n}^{\dagger}-a_{n}^{\dagger}a_{n}\rho-\rho a_{n}^{\dagger}a_{n}),$ (2)
where $a_{n}=\frac{1}{\sqrt{2}}(x_{n}+ip_{n})$ are the Fock operators that
diagonalize each individual harmonic oscillator.
Due to the quadratic nature of the problem, we can assume that the ground
state of the array is Gaussian Weedbrook _et al._ (2012), as is usually done
in linear optics. This implies that the density matrix $\rho$ can be
reconstructed from the expectation values,
$\left\langle{O}\right\rangle:=\mathrm{tr}(O\rho)$, of the operators
$O\in\\{x_{n},p_{n},x_{n}x_{m},p_{n}p_{m},x_{n}p_{m}\\}$. Moreover, the
evolution equations for these “moments” form a closed set of first order
different equations that can be exactly solved, as described in detail in
section I of the Supplemental Material. Let us start with the first moment
equations, which describe the dynamics of the effective dipoles
$d_{n}=\left\langle{x_{n}}\right\rangle$. It is straightforward to find a set
of coupled driven classical harmonic oscillators subject to friction
$\ddot{d}_{n}=-\left(\omega^{2}+\frac{\gamma^{2}}{4}\right)d_{n}-2\omega
g\sum_{l}d_{l}-\gamma\dot{d}_{n}+f_{n},$ (3)
where the sum over $l$ is over nearest neighbors of $n$. This is a classical
model that has already been used to describe a particle plasmon array
Brongersma _et al._ (2000) and shows the compatibility of our master equation
with earlier theoretical studies. In particular, our equations must describe
the transport of excitations and absorption of energy by the plasmonic array.
In fact, we can use the available experimental results to extract quantitative
information about the three parameters $g,\omega$ and $\gamma$, which
characterize our modeling.
Regarding transport, let us assume a coherent driving on the first site,
$f_{1}(t)\sim\sin(\nu t)$, and study the asymptotic state of the dipoles as a
function of the distance. From this calculation we can extract a propagation
length, $\xi$, defined as
$\xi=\frac{\sum_{n=1}^{N}n\Lambda\left|\left\langle{x_{n}}\right\rangle\right|}{\sum_{n=1}^{N}\left|\left\langle{x_{n}}\right\rangle\right|}.$
(4)
For the case of a very long chain, this propagation length would determine the
exponential decay of the plasmon population,
$\left\langle{x_{n}}\right\rangle\sim e^{-n\Lambda/\xi}$. In Fig. 2a we show
the propagation length in units of particle spacing, $\xi/\Lambda$, obtained
numerically for a chain of $N=20$ oscillators, as a function of the coupling
strength $g$ and plasmonic loss $\gamma$, under quasi-resonant driving
($\nu=0.99\omega$). Dissipation leads to a finite propagation length, which
grows with $g$ and diverges at the critical point $g/\omega=1/2,\gamma=0$,
where the current model becomes unphysical.
Figure 2: (a) Average propagation length (in units of $\Lambda$) in the 1D
chain of $N=20$ nanoparticles versus coupling strength, $g$, and local
dissipation, $\gamma$. (b) Entanglement in the chain measured by the
logarithmic negativity. (c) Entanglement witness in momentum space.
While the first order moments reproduce predictions of the classical theory,
the second order moments contain information about the non-classicality of the
many-body particle plasmon state. In particular, the matrix of second order
correlations, or covariance matrix, can also be exactly computed (see section
I of the Supplemental Material) and used to quantify the amount of
entanglement present in the plasmonic array. For this purpose let us eliminate
the driving $f_{n}(t)$, whose role is merely to displace the different
oscillator modes, without adding entanglement. In the absence of this driving,
we focus on the second order moment for the covariance matrix
$\sigma_{i,j}=\frac{1}{2}\left\\{\langle R_{i}R_{j}\rangle-\langle
R_{i}\rangle\langle R_{j}\rangle\right\\},$ (5)
where $\mathbf{R}^{T}=(x_{1},\ldots,x_{L},p_{1},\ldots,p_{L})$ is a vector
that groups all positions and momenta.
Let us consider a bipartition of the plasmonic array into two subarrays, A and
B. It is clear that the covariance matrix can be split into boxes that group
the operators of one or the other array,
$\sigma=\left(\begin{matrix}\sigma_{AA}&\sigma_{AB}\\\
\sigma_{BA}&\sigma_{BB}\end{matrix}\right),$ (6)
together with some off-diagonal terms, $\\{\sigma_{AB},\sigma_{BA}\\}$ that
imply some correlation (quantum or classical) between the two arrays. In order
to quantify purely quantum correlations, we compute the so called negativity
Weedbrook _et al._ (2012), $E_{N}[\sigma;A,B]$. A value of
$E_{N}[\sigma;A,B]$ above zero means that the plasmonic array is entangled at
least with respect to this bipartition. Subsequent application of this
criterion to different partitions of the array can be used to ensure true
multipartite entanglement.
The results of this calculation are shown in Fig. 2b for a 1D array of 20
nanoparticles divided into two blocks of 10 consecutive particles. We plot the
negativity as a function of the coupling strength $g$ and the plasmonic loss
$\gamma$. As expected, entanglement grows with $g$ and becomes maximum at the
critical point $g=\omega/2,\gamma=0$, where the propagation length diverges.
The effect of dissipation is to decrease the entanglement, which remains
sizable for moderate coupling strengths, $g\simeq\gamma$.
Unfortunately, the negativity is not an observable. It may be estimated from
the full covariance matrix if a sufficiently accurate reconstruction of this
matrix is available, but this is an experimentally daunting task. It would
therefore be interesting to have an experimental criterion that allows the
detection of entanglement in the plasmonic chain with the least number of
measurements, while being robust to noise and imperfections.
For this task we suggest what is called an entanglement witness Gühne and Tóth
(2009); Horodecki _et al._ (1996); Lewenstein _et al._ (2000); Terhal
(2000); Bruß _et al._ (2002). A witness is an observable $W$ such that when
its expectation value $\langle W\rangle=\mathrm{Tr}(W\rho)$ becomes negative,
we can positively assure that the state $\rho$ is not separable. There are
several such entanglement criteria in the literature of quantum optics. One of
them is the so-called Duan criterion for detecting two-mode squeezing Duan
_et al._ (2003), which was later extended by Hyllus and Eisert Hyllus and
Eisert (2006) to include multipartite entanglement. In this work we develop a
very general but simpler version of this last protocol.
Theorem: Let us take two vectors $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ which
satisfy the following conditions: (i) they are normalized,
$\|{\mathbf{u}_{i}}\|=1$, (ii) have the same modulus element-wise
($|u_{1,i}|=|u_{2,i}|$) and (iii) define two pairs of canonical variables,
$X_{k}=\sum_{j=1}^{L}u_{k,j}x_{j},\ \mathrm{and}\
P_{k}=\sum_{j=1}^{L}u_{k,j}p_{j}.$ (7)
If the two opposite quadratures are squeezed
$\langle\Delta{X}_{1}^{2}\rangle+\langle\Delta{P}_{2}^{2}\rangle<1,$ (8)
then the state is entangled. The demonstration of this theorem is presented in
the Supplemental Material, section II.
While the conditions (i)-(ii) might seem rather artificial, they can be
satisfied by the normal modes of the plasmonic array. The undriven part of
Hamiltonian (1) can be diagonalized using normal modes $\\{X_{k},P_{k}\\}$
(see details in section III of the Supplemental Material)
$H_{0}=\sum_{k}\frac{\omega}{2}(P_{k}^{2}+\lambda_{k}X_{k}^{2}),$ (9)
where $k$ represents the quantized momentum, $k=\pi j/[(N+1)\Lambda]$ with $j$
running from $1$ to $N$. The magnitude $\lambda_{k}=1+2(g/\omega)\cos
k\Lambda$ determines the plasmonic dispersion band,
$\omega_{k}=\omega\sqrt{\lambda_{k}}$.
Therefore, in the case of a 1D linear chain (corresponding to open boundary
conditions) and for a very large number of nanoparticles, $\mathbf{u}_{1}$ and
$\mathbf{u}_{2}$ of the theorem could be the wavefunctions associated to two
eigenmodes with opposite momenta $(k,k^{\prime})=(k,\pi/\Lambda-k)$, which are
equal in modulus and only differ in the fact that one has alternating signs
and the other does not, $u_{1,j}=(-1)^{j}u_{2,j}$. From a practical point of
view, this means that we can detect entanglement by looking for squeezing
among states with momenta $k$ and $(\pi/\Lambda-k)$. In other words, we can
define our entanglement witness
$W_{k}:=\mathrm{min}\\{0,\langle\Delta{X_{k}}^{2}\rangle+\langle\Delta{P}_{\pi/\Lambda-k}^{2}\rangle-1\\},$
(10)
so that $W_{k}<0$ implies entanglement. For the particular case $k=0$, i.e.,
the extrema of the dispersion band, we can find an analytical expression for
the entanglement witness (see details in section III of the Supplemental
Material)
$W_{0}=1+\frac{\frac{2g}{\omega}(\frac{2g}{\omega}-1)}{\frac{\gamma^{2}}{\omega^{2}}+4(1-\frac{2g}{\omega})}-\frac{\frac{2g}{\omega}}{\frac{\gamma^{2}}{\omega^{2}}+4(1+\frac{2g}{\omega})}.$
(11)
Fig. 2c presents the numerical results corresponding to $W_{0}$. As it shown
in the plot, the growth of the witness follows the same trend as that of the
negativity, hence providing the same amount of information.
Figure 3: Absorption versus frequency for a single silver nanosphere (red
line) and a dimer (blue line). In these calculations the radii of the
nanoparticles is set to $R=25$ nm whereas the separation between nanoparticles
in the dimer case is $2$ nm. The dashed grey line represents a Lorentzian fit
to the absorption spectrum of the single nanosphere that is used to estimate
$\gamma$.
In what follows we describe how this entanglement could be measured using
present-day state-of-the-art technology. Squeezing in the plasmonic band is
related to entanglement, and the same applies to far-field images of the
lattice. The light emitted by the plasmons maps the quadratures in the
collective variables $\\{X_{k},P_{k}\\}$ onto the equivalent variables of the
field propagating along directions $\pm k$. This light can be collected by a
large aperture lens, so that each value of the momentum is mapped to a
different point on the focal plane of the lens, as sketched in Fig. 1.
Selecting the photons with the appropriate momenta, we can perform homodyne
detection Weedbrook _et al._ (2012); Welsch _et al._ (1999) to measure the
quadratures and recover the value of $W_{k}$ mentioned above. Moreover, two
important features make this a very useful protocol. The first one is that our
choice of witness (i.e., momentum pairs) is not relevant, as we get similar
results for other values of the momentum. This is a signature that the state
is indeed many-body entangled. The second one is that while we have estimated
$W_{k}$ using Gaussian states, the entanglement witness is valid for any
physical state. In other words, measuring $W_{k}$ detects entanglement
irrespectively of the underlying physical model.
The proposed measurements could be realized using different types of coupled
plasmonic modes. One interesting possibility is provided by already existing
setups with gold or silver nanoparticles Maier _et al._ (2002a, b). Earlier
experiments with such nanoparticles revealed short propagation lengths,
discouraging the use of such arrays for the transport of quantum information.
However, in Fig. 2 it can be appreciated that, while the plasmon propagation
length is related to the coupling strength and local loss, there can be a non-
zero amount of entanglement even when the surface plasmons do not propagate
efficiently. As an example and to provide a quantitative and realistic
estimation, we have calculated the EM coupling between two silver nanospheres
of radii $R=25~{}$nm and separated by a distance of $2~{}$nm. As shown in Fig.
3, we obtain a coupling strength of around $g/\omega\approx 0.15$. By looking
at the absorption spectrum for a single nanoparticle we can also extract a
value for the loss coefficient, $\gamma/\omega\approx 0.08$. These two values
for $g$ and $\gamma$ are fully compatible with earlier works studying larger
arrays Harris _et al._ (2009). For this coupling and the associated plasmonic
loss, we expect a measurable amount of squeezing, $12\%$ (see Fig. 2c), which
would be a conclusive evidence of many-body entanglement within the plasmonic
array.
Summing up, in this work we have studied a quantum model for an array of
particle plasmons. The model, which can be extended to any system of
interacting plasmonic resonances, not only describes the collective resonances
and the transport of excitations through the system, but it also predicts the
existence of many-body entanglement in the system. Using the formalism of
Gaussian states and entanglement witnesses we have provided an experimental
protocol to detect this entanglement and estimated the strength of the
measurement outcomes for realistic setups. The entanglement witness developed
in this work is quite general, as it detects entanglement in far-field images
even for states that are not Gaussian, including coupled surface plasmons that
do not fall within our model. Moreover, some of these ideas can be exported to
other fields, such as nanophotonics, matter waves and the study of coupled
resonators in superconducting circuits.
This work has been funded by the European Research Council (ERC-2011-AdG
Proposal No. 290981). We also acknowledge financial support from EU FP7
project PROMISCE, CAM Research Consortium QUITEMAD (S2009-ESP-1594) and
Spanish MINECO projects FIS2012-33022 and MAT2011-28581-C02-01.
## References
* Chang _et al._ (2006) D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, Phys. Rev. Lett. 97, 053002 (2006).
* Dzsotjan _et al._ (2010) D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, Phys. Rev. B 82, 075427 (2010).
* Andersen _et al._ (2011) M. L. Andersen, S. Stobbe, A. S. Sørensen, and P. Lodahl, Nat. Phys. 7, 215 (2011).
* Gonzalez-Tudela _et al._ (2011) A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
* Akimov _et al._ (2007) A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, Nature 450, 402 (2007).
* Kolesov _et al._ (2009) R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stöhr, A. A. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, Nat. Phys. 5, 470 (2009).
* Gomez _et al._ (2010) D. Gomez, K. Vernon, P. Mulvaney, and T. Davis, Nano Lett. 10, 274 (2010).
* Bellessa _et al._ (2004) J. Bellessa, C. Bonnand, J. C. Plenet, and J. Mugnier, Phys. Rev. Lett. 93, 036404 (2004).
* Schwartz _et al._ (2011) T. Schwartz, J. A. Hutchison, C. Genet, and T. W. Ebbesen, Phys. Rev. Lett. 106, 196405 (2011).
* Tame _et al._ (2013) M. Tame, K. McEnery, Ş. Özdemir, J. Lee, S. Maier, and M. Kim, Nat. Phys. 9, 329 (2013).
* Altewischer _et al._ (2002) E. Altewischer, M. Van Exter, and J. Woerdman, Nature 418, 304 (2002).
* Fasel _et al._ (2005) S. Fasel, F. Robin, E. Moreno, D. Erni, N. Gisin, and H. Zbinden, Phys. Rev. Lett. 94, 110501 (2005).
* Huck _et al._ (2009) A. Huck, S. Smolka, P. Lodahl, A. S. Sørensen, A. Boltasseva, J. Janousek, and U. L. Andersen, Phys. Rev. Lett. 102, 246802 (2009).
* Maier _et al._ (2002a) S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, Phys. Rev. B 65, 193408 (2002a).
* Maier _et al._ (2002b) S. A. Maier, P. G. Kik, and H. A. Atwater, Appl. Phys. Lett. 81, 1714 (2002b).
* Weick _et al._ (2013) G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, Phys. Rev. Lett. 110, 106801 (2013).
* Weedbrook _et al._ (2012) C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).
* Gühne and Tóth (2009) O. Gühne and G. Tóth, Phys. Rep. 474, 1 (2009).
* Horodecki _et al._ (1996) M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 1 (1996).
* Lewenstein _et al._ (2000) M. Lewenstein, B. Kraus, J. Cirac, and P. Horodecki, Phys. Rev. A 62, 052310 (2000).
* Terhal (2000) B. M. Terhal, Phys. Lett. A 271, 319 (2000).
* Bruß _et al._ (2002) D. Bruß, J. I. Cirac, P. Horodecki, F. Hulpke, B. Kraus, M. Lewenstein, and A. Sanpera, J. Mod. Optics 49, 1399 (2002).
* Brongersma _et al._ (2000) M. L. Brongersma, J. W. Hartman, and H. A. Atwater, Phys. Rev. B 62, R16356 (2000).
* Duan _et al._ (2003) L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, in _Quantum Information with Continuous Variables_ (Springer, 2003) pp. 145–153.
* Hyllus and Eisert (2006) P. Hyllus and J. Eisert, New J. Phys. 8, 51 (2006).
* Welsch _et al._ (1999) D.-G. Welsch, W. Vogel, and T. Opatrný, in _Progress in Optics_, Vol. 39, edited by E. Wolf (Elsevier, 1999) pp. 63 – 211.
* Harris _et al._ (2009) N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, J. Phys. Chem. C 113, 2784 (2009).
Supplemental Material
## I. Moment equations
We develop the general framework for studying the steady state of Hamiltonians
with quadratic bosonic operators. In the first place, it is convenient to
define $\boldsymbol{R}^{T}=\left(x_{1},\ldots,x_{N},p_{1},\ldots,p_{N}\right)$
and write $H$ as a quadratic form
$H=\frac{1}{2}\boldsymbol{R}^{T}\boldsymbol{B}\boldsymbol{R}+\boldsymbol{F}(t)^{T}\boldsymbol{R},$
(12)
where $\boldsymbol{B}$ is a real, symmetric matrix and
$\boldsymbol{F}(t)^{T}=\left(f_{1}(t),\ldots,f_{N}(t),0,\ldots,0\right)$
accounts for possible driving forces.
If the time evolution of a density matrix $\rho$ is given by
$\displaystyle\partial_{t}\rho=-i[H,\rho]+\sum_{n}\frac{\gamma}{2}(2a_{n}\rho
a_{n}^{\dagger}-a_{n}^{\dagger}a_{n}\rho-\rho a_{n}^{\dagger}a_{n}),$ (13)
where $a_{n}=\frac{1}{\sqrt{2}}(x_{n}+ip_{n})$, we can easily show that the
time evolution of the mean value of a time-independent operator $O$ is given
in a compact form as
$\partial_{t}\langle
O\rangle=-i\langle\left[O,H\right]\rangle+\sum_{n}\frac{\gamma}{2}\langle[a_{n}^{\dagger},O]a_{n}+a_{n}^{\dagger}\left[O,a_{n}\right]\rangle.$
(14)
Here, we used the cyclic invariance of the trace and
$\mathrm{Tr}\left\\{\dot{\rho}O\right\\}=\partial_{t}\langle O\rangle$. We
apply this idea to the first and second moments of the quadratures,
$O\in\\{x_{n},p_{n},x_{n}x_{m},p_{n}p_{m},x_{n}p_{m}\\}$. Writing Eq. 14 in
the quadrature basis, we get
$\partial_{t}\langle
O\rangle\\!=\\!-i\langle[O,H]\rangle\\!+\\!\sum_{nm}\frac{\Gamma_{nm}}{2}\langle[R_{m}^{\dagger},O]R_{n}\\!+\\!R_{m}^{\dagger}[O,R_{n}]\rangle,$
(15)
where
$\boldsymbol{\Gamma}=\bigoplus_{n=1}^{N}\frac{\gamma}{2}\left(\begin{array}[]{cc}1&-i\\\
i&1\end{array}\right)$ is a matrix that contains the effective dissipation
rates corresponding to operators $R_{n}$ and $R_{m}$ in this expression. The
$\bigoplus$ symbol denotes the direct sum of matrices, so for a set of
matrices $\\{A_{n}\\}$,
$\bigoplus_{n}A_{n}=\mathrm{diag}(A_{1},A_{2},\ldots,A_{n})$. If we now make
use of the commutation relations for quadrature operators, written in compact
form as
$[R_{n},R_{m}]=i\Omega_{nm}\quad\mathrm{with}\quad\boldsymbol{\Omega}=\left(\begin{array}[]{cc}0&\boldsymbol{1}_{N}\\\
-\boldsymbol{1}_{N}&0\end{array}\right),$ (16)
where $\boldsymbol{1}_{N}$ denotes the $N\mathrm{-dimensional}$ identity
matrix, it is straightforward to arrive to a closed set of $2N$ equations for
the first moments $\langle\boldsymbol{R}\rangle$, which we write in matrix
form as
$\partial_{t}\langle\boldsymbol{R}\rangle=\left(\boldsymbol{W}+\boldsymbol{\Omega
F}(t)\right)\langle\boldsymbol{R}\rangle$ (17)
where $\boldsymbol{W}=\boldsymbol{\Omega
B}+\frac{i}{2}(\boldsymbol{\Omega\Gamma}+(\boldsymbol{\Gamma\Omega})^{T})$.
As discussed in the main text, the second moments give information about the
nonclassical properties of the plasmonic array. In the same spirit, we can
write the following $2N\times 2N$ equations for the second moments
$\langle\boldsymbol{C}\rangle=\langle\boldsymbol{R}\boldsymbol{R}^{T}\rangle$
$\partial_{t}\langle\boldsymbol{C}\rangle=\boldsymbol{W}\langle\boldsymbol{C}\rangle+\langle\boldsymbol{C}\rangle\boldsymbol{W}^{T}-2\left(\boldsymbol{\Omega\Gamma\Omega}\right){}^{T}.$
(18)
Here, for simplicity, we have set $\boldsymbol{F}=0$ since its role is merely
to displace the first moments as we discuss in the main text and has no effect
on correlations.
Now, for instance, let us consider our particular case in which,
$\boldsymbol{B}=\boldsymbol{A}\oplus\boldsymbol{1}_{N}$, i.e., the Hamiltonian
may be written in the form
$H=\frac{\omega}{2}\mathbf{p}^{T}\mathbf{p}+\frac{\omega}{2}\mathbf{x}^{T}A\mathbf{x}+\mathbf{f}(t)^{T}\mathbf{x},$
(19)
where $\mathbf{x}^{T}=(x_{1},\ldots,x_{N})$,
$\mathbf{p}^{T}=(p_{1},\ldots,p_{N})$,
$\mathbf{f}(t)^{T}=(f_{1}(t),\ldots,f_{N}(t))$ and $\boldsymbol{A}$ is a
sparse matrix whose diagonal is unity, and the only other nonzero elements are
those connecting nearest neighbor sites, which are given by $2g/\omega$. In
this case we can write for the first moments
$\displaystyle\partial_{t}\langle x_{n}\rangle$ $\displaystyle=\omega\langle
p_{n}\rangle-\frac{\gamma}{2}\langle x_{n}\rangle,$ (20)
$\displaystyle\partial_{t}\langle p_{n}\rangle$ $\displaystyle=-\omega\langle
x_{n}\rangle-\frac{\gamma}{2}\langle p_{n}\rangle-2g\sum_{l}\langle
x_{l}\rangle+f_{n},$ (21)
where the sum over $l$ is over nearest neighbors of $n$. We then obtain the
equation for the effective dipole operator $d_{n}=\langle x_{n}\rangle$,
$\ddot{d}_{n}=-\left(\omega^{2}+\frac{\gamma^{2}}{4}\right)d_{n}-2\omega
g\sum_{l}d_{l}-\gamma\dot{d}_{n}+f_{n},$ (22)
which describe the dynamics of a set of coupled, driven harmonic oscillators
subject to friction. Additionally, the equations for the second moments read
$\displaystyle\partial_{t}\langle x_{n}x_{m}\rangle$
$\displaystyle=\omega\langle x_{n}p_{m}\\!+\\!p_{n}x_{m}\rangle-\gamma\langle
x_{n}x_{m}\rangle$ (23) $\displaystyle\partial_{t}\langle p_{n}p_{m}\rangle$
$\displaystyle=-\omega\langle x_{n}p_{m}+p_{n}x_{m}\rangle-\gamma\langle
p_{n}p_{m}\rangle-$ $\displaystyle-2g\sum_{l}(\langle
p_{n}x_{l}\rangle+\langle p_{m}x_{l}\rangle)$ (24)
$\displaystyle\partial_{t}\langle x_{n}p_{m}\rangle$
$\displaystyle=-\omega\langle x_{n}x_{m}\rangle+\omega\langle
p_{n}p_{m}\rangle-$ $\displaystyle-\frac{\gamma}{2}\langle
x_{n}p_{m}+p_{n}x_{m}\rangle-2g\sum_{l}\langle x_{n}x_{l}\rangle$ (25)
$\displaystyle\partial_{t}\langle x_{n}p_{m}\rangle$
$\displaystyle=\partial_{t}\langle p_{m}x_{n}\rangle.$ (26)
From their steady state solution in the absence of driving,
$\partial_{t}\langle\boldsymbol{R}\rangle=0=\partial_{t}\langle\boldsymbol{C}\rangle$,
we build the covariance matrix.
$\sigma_{i,j}=\frac{1}{2}\left\\{\langle R_{i}R_{j}\rangle-\langle
R_{i}\rangle\langle R_{j}\rangle\right\\}.$ (27)
Then, we consider a bipartition of the plasmonic array into two subarrays, $A$
and $B$, and compute the so-called logarithmic negativity between 2
bipartitions $E_{N}\left[\sigma;\,A,B\right]$. This quantity can be given in
terms of the absolute value of the eigenvalues of the matrix
$i\boldsymbol{\Omega\sigma}$, after performing a non-physical operation known
as partial transposition, as is discussed in detail in Weedbrook _et al._
(2012). It can be shown that this non-physical operation is equivalent to
changing the sign of the $p_{i}$ components of one of the subsystems.
## II. Entanglement witness: Proof
Let us take two vectors $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ which satisfy
these conditions: (i) they are normalized, $\|{\mathbf{u}_{i}}\|=1$, (ii) have
the same modulus element-wise ($|u_{1,i}|=|u_{2,i}|$) and (iii) define two
pairs of canonical variables, $X_{j}=\sum_{i}u_{ji}x_{i},\ \mathrm{and}\
P_{j}=\sum_{i}u_{ji}p_{i}$ with $j=1,2$. If we now compute the fluctuations of
these operators assuming that the state is simply separable,
$\rho=\bigotimes_{i}\rho_{i}$, we have
$\displaystyle\langle{(\Delta{X_{1}})^{2}}\rangle+\langle{(\Delta{P_{2}})^{2}}\rangle=$
$\displaystyle\langle{X_{1}^{\dagger}X_{1}}\rangle-\langle{X^{\dagger}_{1}}\rangle\langle{X_{1}}\rangle+\langle{P_{2}^{\dagger}P_{2}}\rangle-\langle{P_{2}^{\dagger}}\rangle\langle{P_{2}}\rangle$
(28) $\displaystyle=$ $\displaystyle\sum_{i,j}\left[u_{i,1}^{*}u_{j,1}\langle
x_{i}x_{j}\rangle+u_{i,2}^{*}u_{j,2}\langle
p_{i}p_{j}\rangle\right]-\sum_{i,j}\left[u_{i,1}^{*}u_{j,1}\langle
x_{i}\rangle\langle{x_{j}}\rangle+u_{i,2}^{*}u_{j,2}\langle
p_{i}\rangle\langle{p_{j}}\rangle\right]$
$\displaystyle\stackrel{{\scriptstyle(1)}}{{=}}$
$\displaystyle\sum_{i}\left[|u_{i,1}|^{2}\langle\Delta{x}_{i}^{2}\rangle+|u_{i2}|^{2}\langle\Delta{p}_{i}^{2}\rangle\right]\stackrel{{\scriptstyle(2)}}{{=}}\sum_{i}|u_{i,1}|^{2}\left[\langle\Delta{x}_{i}^{2}\rangle+\langle\Delta{p}_{i}^{2}\rangle\right]\stackrel{{\scriptstyle(3)}}{{>}}=\sum_{i}|u_{i,1}|^{2}=1.$
Here we have used various key ideas: In $(1)$ we use the fact that the state
is separable and thus
$\langle{x_{i}x_{j}}\rangle=\langle{x_{i}}\rangle\langle{x_{j}}\rangle$
whenever $i\neq j$. In $(2)$ we use the fact that both vectors have the same
modulus element-wise , $|u_{i1}|=|u_{i2}|$. Finally, in $(3)$ we use the fact
that $\langle\Delta{A}^{2}\rangle+\langle\Delta{B}^{2}\rangle>=\|[A,B]\|$ and
the normalization of the vectors.
This proof can be extended to treat fully all possible cases of separable
states, $\rho=\sum p_{i}\otimes_{j}\rho_{j}^{i}$, which are convex linear
combinations of the previous situation we have shown. In this case the only
difference is that there appear additional cross-terms due to the linear
combinations, but these terms can be shown to be larger than zero, thus
increasing the fluctuations Duan _et al._ (2003).
## III. Normal modes
First, in this section, we are going to show how to diagonalize the effective
model in the dissipation-free case. The diagonalization of the matrix
$\boldsymbol{A}$ through an orthogonal transformation,
$\boldsymbol{A}=\boldsymbol{U}^{T}\boldsymbol{D}\boldsymbol{U}$, allows us to
define new canonical variables
$X_{k}=\sum_{i}u_{k,i}x_{i},\;P_{k}=\sum_{i}u_{k,i}p_{i}.$ (29)
In these new quadratures, the Hamiltonian becomes
$H=\sum_{k}\frac{1}{2}\omega\left(P_{k}^{2}+\lambda_{k}X_{k}^{2}\right)+\sum_{i}u_{k,i}f_{i}(t),$
(30)
where $\lambda_{k}=D_{k,k}$ are the eigenvalues of the quadratic form and we
introduce the effective drivings in momentum space,
$\tilde{f}_{k}=\sum_{i}u_{k,i}f_{i}$. Note that in absence of driving, the
normal frequencies of the problem will be
$\omega_{k}=\omega\lambda_{k},$ (31)
and the new Fock operators will be related to the original ones by a
complicated squeezing transformation
$\tilde{a}_{k}=\sum_{i}\left(\lambda_{k}^{1/2}u_{k,i}x_{i}+i\lambda_{k}^{-1/2}u_{k,i}p_{i}\right),$
(32)
that is the source of the entanglement of this problem.
As a particular instance of the lattice of coupled plasmons we will consider
the case of a one-dimensional lattice of regularly spaced nanoparticles, with
period $\Lambda$, that corresponds to the 1D open-boundary condition case of
$\boldsymbol{A}$. This tridiagonal matrix is diagonalized with the orthogonal
transformation
$u_{i,k}=u_{k,i}=\sqrt{\frac{2}{N+1}}\sin(k_{j}\Lambda i),\ \ \;i,j=1\ldots N$
(33)
where $N$ is the total lattice size and the quasimomenta $k_{j}=\pi
j/(N+1)\Lambda$ determine the eigenfrequencies
$\lambda_{k}=1+2(g/\omega)\cos(k_{j}\Lambda).$ (34)
Notice how for small momenta, when we reach the critical value $g=\omega/2$,
we recover a linear dispersion relation of photon-like quasiparticles with
diverging correlations. In practice, however, $g$ is below this limit and we
obtain a band of massive excitations with a finite correlation length.
Now, as the form of (15) is preserved under the transformation (33) we apply
the ideas exposed previously to the first and second moments of the canonical
variables, obtaining
$\displaystyle\partial_{t}\langle X_{k}\rangle$ $\displaystyle=\omega\langle
P_{k}\rangle-\frac{\gamma}{2}\langle X_{k}\rangle$ (35)
$\displaystyle\partial_{t}\langle P_{k}\rangle$
$\displaystyle=-\omega\lambda_{k}\langle X_{k}\rangle-\frac{\gamma}{2}\langle
P_{k}\rangle-\tilde{f}_{k}(t).$
For the second moments, taking $f=0$ to simplify the expressions, we get the
following closed set
$\displaystyle\partial_{t}\langle X_{k}^{2}\rangle$
$\displaystyle\\!=\omega\langle X_{k}P_{k}+P_{k}X_{k}\rangle-\gamma\langle
X_{k}^{2}\rangle+\frac{\gamma}{2}$ (36) $\displaystyle\partial_{t}\langle
P_{k}^{2}\rangle$ $\displaystyle\\!=\\!-\omega\lambda_{k}\langle
X_{k}P_{k}\\!+\\!P_{k}X_{k}\rangle-\gamma\langle
P_{k}^{2}\rangle+\frac{\gamma}{2}$ (37) $\displaystyle\partial_{t}\langle
X_{k}P_{k}\rangle$ $\displaystyle\\!=\\!-\omega\lambda_{k}\langle
X_{k}^{2}\rangle+\omega\langle
P_{k}^{2}\rangle\\!-\\!\frac{\gamma}{2}\langle\\!X_{k}P_{k}\\!+\\!P_{k}X_{k}\rangle$
(38) $\displaystyle\partial_{t}\langle P_{k}X_{k}\rangle$
$\displaystyle\\!=\partial_{t}\langle X_{k}P_{k}\rangle.$ (39)
In this case, its steady-state solution allows us to compute the fluctuations
needed when computing the proposed entanglement witness
$W_{k}:=\mathrm{min}\\{0,\langle\Delta{X_{k}}^{2}\rangle+\langle\Delta{P}_{(\pi/\Lambda)-k}^{2}\rangle-1\\}$
(40)
given $(\Delta O)^{2}=\langle O^{\dagger}O\rangle-\langle
O^{\dagger}\rangle\langle O\rangle$. In the case of infinite chain length
$N\gg 1$, if we compute this quantity between the extrema of the band we
arrive to
$W_{0}=1+\frac{\frac{2g}{\omega}(\frac{2g}{\omega}-1)}{\frac{\gamma^{2}}{\omega^{2}}+4(1-\frac{2g}{\omega})}-\frac{\frac{2g}{\omega}}{\frac{\gamma^{2}}{\omega^{2}}+4(1+\frac{2g}{\omega})}.$
(41)
|
arxiv-papers
| 2014-02-28T10:30:30 |
2024-09-04T02:49:59.100042
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Javier del Pino, Johannes Feist, F.J. Garc\\'ia-Vidal and Juan Jose\n Garc\\'ia-Ripoll",
"submitter": "Javier Del Pino",
"url": "https://arxiv.org/abs/1402.7187"
}
|
1402.7269
|
# Tauberian Theorem of Laplace Transformation
And
Application of Prime Number Theorem
Lahoucine Elaissaoui
[email protected]
[email protected]
###### Résumé
Dans cet article je donnerai une nouvelle démonstration courte et directe pour
le Théorème des Nombres Premiers. C’est vrai que ce théorème a été
complétement démontré au début du 20ème siecle mais la démonstration était
basé sur des résultats élémentaires (théorème de Chebyshev) et aussi
analytiques compliqués (théorème de Ikehrara), mais ici j’ai pas utilisé le
théorème de Chebyshev ainsi que j’ai remplacé et j’ai généralisé le théorème
de Ikehara grâce à la notion des fonctions à variation bornée qui est ancienne
mais récent dans la théorie analytique des nombres.
## 1 Préliminaires
### 1.1 Les fonctions à variation bornée
Les fonctions à variation bornée joue un rôle très important dans la théorie
de l’intégration au sens de Stieltjes, ici on va s’interésser à les fonctions
à variation bornée sur $\mathbb{R}^{+}$ à valeurs complexes. Soit $x$ un réel
positif et soit $(x_{k})_{k=0,\cdots,n}$ une suite finie et strictement
croissante des réels de l’intervalle $[0,x]$ tels que
$0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}=x$ est une subdivision de l’intervalle
$[0,x]$, on note $\Sigma$ pour cette subdivion et $\mathcal{S}([0,x])$
l’ensemble de toutes les subdivions possibles de $[0,x]$. La fonction
variation totale d’une fonction complexe définie sur $\mathbb{R}^{+}$, notée
$T_{f}$, est la fonction définie par
$T_{f}(x):=\sup_{\Sigma\in\mathcal{S}([0,x])}\sum_{k=1}^{n}|f(x_{k})-f(x_{k-1})|$
(1)
Il est bien clair que la fonction $T_{f}$ est une fonction croissante sur
$\mathbb{R}^{+}$, par conséquent si $T_{f}$ est majorée sur $\mathbb{R}^{+}$
alors on dira que $f$ est à variation bornée sur $\mathbb{R}^{+}$ et on note
$V(f)=\lim_{x\to+\infty}T_{f}(x)\in\mathbb{R}^{+}$
pour la variation totale de la fonction $f$.
###### Propriétés 1.1.
* •
Toute fonction $g$ de classe $\mathcal{C}^{1}$ sur $\mathbb{R}^{+}$ à valeurs
complexes telle que $g^{\prime}\in L^{1}(\mathbb{R}^{+})$ est à variation
bornée, en effet, pour une subdivision $\Sigma:0=x_{0}<x_{1}<\cdots<x_{n}=x$
et puisque $g$ est continue sur chaque intervalle $[x_{i-1},x_{i}]$ (pour
$i=1,\cdots n$) et dérivable sur leurs interieurs topologique alors d’après le
théorème des accroissements finis il existe des $c_{i}$ dans $]x_{i-1},x_{i}[$
tels que
$|g(x_{i})-g(x_{i-1})|=|g^{\prime}(c_{i})||x_{i}-x_{i-1}|$
D’où
$T_{g}(x)=\sup_{\Sigma\in\mathcal{S}([0,x])}\sum_{k=1}^{n}|g^{\prime}(c_{i})||x_{i}-x_{i-1}|$
Or cette somme est une somme de Darboux ce qu’on peut déduire grâce à
l’intégrale de Riemann que
$T_{g}(x)=\int_{0}^{x}|g^{\prime}(t)|dt$
Donc
$V(g)=\int_{0}^{+\infty}|g^{\prime}(t)|dt$
qui est finie puisque $g^{\prime}\in L^{1}(\mathbb{R}^{+})$, alors $g$ est à
variation bornée sur $\mathbb{R}^{+}$.
* •
toute fonction à variation bornée sur $\mathbb{R}^{+}$ est bornée sur
$\mathbb{R}^{+}$, en effet, soit $f$ une fonction à variation bornée sur
$\mathbb{R}^{+}$ alors pour un $x\geq 0$
$\displaystyle|f(x)-f(0)|$
$\displaystyle=\left|\sum_{k=1}^{n}f(x_{i})-f(x_{i-1})\right|$
$\displaystyle\leq\sum_{k=1}^{n}|f(x_{i})-f(x_{i-1})|$ $\displaystyle\leq
T_{f}(x)$ $\displaystyle\leq V(f)<+\infty$
Alors $f$ est bornée sur $\mathbb{R}^{+}$.
On dit qu’une fonction $f$ définie de $\mathbb{R}^{+}$ à valeurs complexes
admet une limite à gauche en $x\in\mathbb{R}^{+}$, notée $f(x^{-})$ si à tout
$\varepsilon>0$ on peut associer un $0\leq\delta<x$ tel que
$a<t<x\Longrightarrow|f(t)-f(x^{-})|<\varepsilon$
Et en plus si $f(x^{-})=f(x)$ on dit que $f$ est continue à gauche en $x$.
On note pour $\mathcal{V}_{b}\mathcal{C}_{g}$ la classe des fonctions,
définies de $\mathbb{R}^{+}$ à valeurs complexes, à variation bornée,
continues à gauche en tout point de $\mathbb{R}^{+}$ et qui s’annullent en
$0$.
### 1.2 Intégrale de Lebesgue-Stieltjes
Le théorème 8.14 page 156 du livre [Rud] a établi le lien entre la théorie de
la mesure et la théorie des fonctions à variation bornée. Donc d’après le même
théorème, soit $f\in\mathcal{V}_{b}\mathcal{C}_{g}$ alors il existe une unique
mesure complexe de Borel $\mu_{f}$ telle que
$f(x)=\mu_{f}([0,x[),\qquad\forall x\geq 0$ (2)
Et en plus pour tout $x\in\mathbb{R}^{+}$ on a
$T_{f}(x)=|\mu_{f}|([0,x[)$ (3)
Où $|\mu_{f}|$ est une mesure positive de Borel, dite la variation totale de
la mesure complexe $\mu_{f}$, qui est finie d’après le théorème 6.4 page 114
de [2].
###### Remarque 1.1.
* •
On peut facilement montrer que $|\mu_{f}|$ est finie autant que
$f\in\mathcal{V}_{b}\mathcal{C}_{g}$, en effet, soit
$f\in\mathcal{V}_{b}\mathcal{C}_{g}$ alors
$\displaystyle|\mu_{f}|(\mathbb{R}^{+})$
$\displaystyle=\lim_{x\to+\infty}|\mu_{f}|([0,x[)$
$\displaystyle=\lim_{x\to+\infty}T_{f}(x)$ $\displaystyle=V(f)<+\infty$
* •
D’une autre part, si $f$ est à valeurs dans $\mathbb{R}$ alors $\mu_{f}$ est
dite une mesure signée alors de la même manière on démontre que cette mesure
est finie.
* •
Soit $f\in\mathcal{V}_{b}\mathcal{C}_{g}$, si $y>x$ alors
$\displaystyle f(y)-f(x)$ $\displaystyle=\mu_{f}([0,y[)-\mu_{f}([0,x[)$
$\displaystyle=\mu_{f}([x,y[)$
Donc
$\mu_{f}(\\{x\\})=f(x^{+})-f(x)$
D’où $f$ est continue en $x$ si et seulement si
$\mu_{f}(\\{x\\})=0$
Le théorème de Radon-Nikodym, voir le théorème 6.12 page 120 de [2], assure
que pour toute mesure complexe $\mu$ il existe une fonction mesurable complexe
$h$ de module égal à $1$ telle que
$d\mu=hd|\mu|.$
Ainsi, on déduit que pour toute fonction
$g:\mathbb{R}^{+}\longrightarrow\mathbb{C}$ mesurable et bornée sur
$\mathbb{R}^{+}$ on a $g\in L_{\mu_{f}}^{1}(\mathbb{R}^{+})$ où
$f\in\mathcal{V}_{b}\mathcal{C}_{g}$. En effet:
$\displaystyle\left|\int_{\mathbb{R}^{+}}gd\mu_{f}\right|$
$\displaystyle\leq\int_{\mathbb{R}^{+}}|g|d|\mu_{f}|$
$\displaystyle\leq\|g\|_{\infty}|\mu_{f}|(\mathbb{R}^{+})$
$\displaystyle<+\infty$
Où
$\|g\|_{\infty}=\sup_{x\in\mathbb{R}^{+}}|g(x)|.$
Maintenant, d’après le théorème 6.1.4 du livre [1] on constate que pour
$f\in\mathcal{V}_{b}\mathcal{C}_{g}$ on a
$\int_{0}^{x}df(t)=\mu_{f}([0,x[),\qquad x\geq 0$ (4)
Soient donc $f\in\mathcal{V}_{b}\mathcal{C}_{g}$ et
$g:\mathbb{R}^{+}\longrightarrow\mathbb{C}$ une fonction de classe
$\mathcal{C}^{1}$ sur $\mathbb{R}^{+}$ telle que $g^{\prime}\in
L^{1}(\mathbb{R}^{+})$, alors d’après le théorème 6.2.2 (grâce au résultat 4)
du même livre on démontre que
$\int_{0}^{+\infty}g(t)df(t)=\mu_{fg}(\mathbb{R}^{+})-\int_{0}^{+\infty}f(t)g^{\prime}(t)dt$
(5)
la mesure complexe $\mu_{fg}$ a bien un sens, en effet d’après les propriétés
1.1 on démontre que $g$ est à variation bornée or le produit de deux éléments
de $\mathcal{V}_{b}\mathcal{C}_{g}$ est un élément de
$\mathcal{V}_{b}\mathcal{C}_{g}$ alors $fg\in\mathcal{V}_{b}\mathcal{C}_{g}$
(car $fg$ est à variation bornée et continue à gauche à chaque point de
$\mathbb{R}^{+}$ et $(fg)(0)=0$), et en plus
$\mu_{fg}([0,x[)=f(x)g(x),\qquad\forall x\in\mathbb{R}^{+}$
Et
$|\mu_{fg}(\mathbb{R}^{+})|\leq|\mu_{fg}|(\mathbb{R}^{+})=\lim_{x\to+\infty}T_{fg}(x)<+\infty$
## 2 Théorème Tauberien de la transformation de Laplace complexe
Dans tout ce qui suit $s=\sigma+it$ où $\sigma,t\in\mathbb{R}$ est un nombre
complexe et $\rho$ est une fonction de la classe
$\mathcal{V}_{b}\mathcal{C}_{g}^{*}:=\\{f\in\mathcal{V}_{b}\mathcal{C}_{g},\quad\
\Im(f)=0\\}$. Ainsi $\mathbb{C}_{*}^{+}$ est l’ensemble des nombres complexes
de partie réelle strictement positive.
On définit la transformation de Laplace-Stieltjes de la fonction $\rho$ par
$\mathcal{L}_{\rho}^{*}(s)=\int_{0}^{+\infty}e^{-sx}d\rho(x),\qquad\sigma>0$
il est bien clair d’après ce qui précéde, puisque $x\mapsto e^{-sx}$ est
continue et bornée sur $\mathbb{R}^{+}$ pour tout $\sigma>0$, que la fonction
$\mathcal{L}_{\rho}^{*}$ est bien définie.
###### Lemme 2.1.
Soit $\rho\in\mathcal{V}_{b}\mathcal{C}_{g}^{*}$ alors $\lim_{s\to
0^{+}}\mathcal{L}_{\rho}^{*}(s)=\lim_{+\infty}\rho.$
preuve:
On pose pour tout $(x,s)\in\mathbb{R}^{+}\times\mathbb{C}_{+}^{*}$
$\phi(x,s)=e^{-sx}$
Alors on a
* •
Pour tout $x\geq 0$ la fonction $s\mapsto\phi(x,s)$ est continue en $0^{+}$.
* •
Pour tout $\sigma>0$ la fonction $x\mapsto\phi(x,s)$ est continue donc
mésurable sur $\mathbb{R}^{+}$.
* •
Pour tout $\sigma>0$ et pour $d\rho$-presque tout $x\in\mathbb{R}^{+}$ on a
$|\phi(x,s)|\leq 1$
Où $1\in L_{d\rho}^{1}(\mathbb{R}^{+})$ car
$\int_{\mathbb{R}^{+}}d\rho(x)=\mu_{\rho}(\mathbb{R}^{+})<+\infty$
Alors la fonction $x\mapsto\phi(x,s)$ est $d\rho$-intégrable sur
$\mathbb{R}^{+}$ et la fonction $\mathcal{L}_{\rho}^{*}$ est est continue en
$0^{+}$. Donc
$\displaystyle\lim_{s\to 0^{+}}\mathcal{L}_{\rho}^{*}(s)$
$\displaystyle=\mathcal{L}_{\rho}^{*}(0)$
$\displaystyle=\int_{\mathbb{R}^{+}}d\rho(x)$
$\displaystyle=\lim_{x\to+\infty}\mu_{\rho}([0,x[)\qquad\text{d'apr\\`{e}s
\ref{e4}}$
$\displaystyle=\lim_{+\infty}\rho\qquad\qquad\qquad\quad\\!\\!\text{d'apr\\`{e}s
\ref{e2}}$
$\blacksquare$
Maintenant on définit la transformation de Laplace complexe d’une fonction
$\rho\in\mathcal{V}_{b}\mathcal{C}_{g}^{*}$ par
$\mathcal{L}_{\rho}(s)=\int_{0}^{+\infty}\rho(x)e^{-sx}dx,\qquad\sigma>0.$
La fonction $\mathcal{L}_{\rho}$ est bien définie, en effet puisque
$\rho\in\mathcal{V}_{b}\mathcal{C}_{g}^{*}$ alors $\rho$ est bornée sur
$\mathbb{R}^{+}$ en plus
$\left|\int_{0}^{+\infty}\rho(x)e^{-sx}dx\right|\leq\|\rho\|_{\infty}\int_{0}^{+\infty}e^{-\sigma
x}dx=\frac{\|\rho\|_{\infty}}{\sigma}<+\infty$
###### Théorème 2.1.
Soit $\rho\in\mathcal{V}_{b}\mathcal{C}_{g}^{*}$, on suppose que
$\mathcal{L}_{\rho}$ est holomorphe sur $\\{\sigma>0\\}$ et admet un
prolongement analytique sur $\\{\sigma\geq 0\\}$ avec un pôle simple au point
$0$. Alors on a $\lim_{+\infty}\rho=Res(\mathcal{L}_{\rho},0).$
Preuve:
Soit $s\in\mathbb{C}_{+}^{*}$, d’après l’équation 5 on a
$\mathcal{L}_{\rho}^{*}(s)=\mu_{\rho
e^{-s\cdot}}(\mathbb{R}^{+})+s\int_{0}^{+\infty}\rho(x)e^{-sx}dx$
Or
$|\mu_{\rho
e^{-s\cdot}}(\mathbb{R}^{+})|=|\mu_{\rho}(\mathbb{R}^{+})|\lim_{x\to+\infty}e^{-\sigma
x}=0$
Donc
$\mathcal{L}_{\rho}^{*}(s)=s\mathcal{L}_{\rho}(s)$
Passons à la limite $s\to 0^{+}$ on a d’après le Lemme 3.1
$\lim_{x\to+\infty}\rho(x)=Res(\mathcal{L}_{\rho},0)$
$\blacksquare$
D’une manière générale, soit $\alpha$ un réel positif alors il est clair,
d’après ce qui précéde, que pour tout
$\rho\in\mathcal{V}_{b}\mathcal{C}_{g}^{*}$ on a $\varrho(x)=\rho(x)e^{-\alpha
x}$ est un élément de $\mathcal{V}_{b}\mathcal{C}_{g}^{*}$. Ainsi on déduit le
résultat suivant:
###### Corollaire 2.1.
Soient $\alpha\in\mathbb{R}^{+}$ et
$\rho\in\mathcal{V}_{b}\mathcal{C}_{g}^{*}$. On suppose que la fonction
$\mathcal{L}_{\rho}$ est holomorphe sur $\\{\sigma>\alpha\\}$ et admet un
prolongement analytique sur $\\{\sigma\geq\alpha\\}$ avec un seul pôle simple
en $s=\alpha$ alors on a
$\rho(x)\underset{x\to+\infty}{\sim}Res(\mathcal{L}_{\rho},\alpha)e^{\alpha
x}.$
preuve:
Soit $\sigma>\alpha$, alors
$\mathcal{L}_{\rho}(s)=\int_{0}^{+\infty}\rho(x)e^{-sx}dx=\int_{0}^{+\infty}\rho(x)e^{-\alpha
x}e^{-(s-\alpha)x}dx$
On pose
$\varrho(x)=\rho(x)e^{-\alpha x},\qquad\forall x\geq 0.$
Alors
$\mathcal{L}_{\rho}(s)=\int_{0}^{+\infty}\varrho(x)e^{-(s-\alpha)x}dx=\mathcal{L}_{\varrho}(s-\alpha)$
Donc
$(s-\alpha)\mathcal{L}_{\rho}(s)=(s-\alpha)\mathcal{L}_{\varrho}(s-\alpha)=z\mathcal{L}_{\varrho}(z)$
D’où quand $s\to\alpha$ on aura $z\to 0$ et d’après le Théorème 2.1 on a
$\lim_{x\to+\infty}\varrho(x)=Res(\mathcal{L}_{\varrho}(z),z=0)=Res(\mathcal{L}_{\rho},\alpha)$
Alors
$\rho(x)\underset{x\to+\infty}{\sim}Res(\mathcal{L}_{\rho},\alpha)e^{\alpha
x}.$
$\blacksquare$
## 3 Théorème des Nombres Premiers (nouvelle démonstration)
Soit $\chi:\mathbb{N}^{*}\longrightarrow\mathbb{R}^{+}$ une fonction
arithmétique positive, on pose pour tout $x\in(1,+\infty)$
$f(x)=\sum_{1\leq n<x}\chi(n)\qquad\text{et}\qquad f(1)=0.$
Il est clair que la fonction $f$ est croissante sur $[1,+\infty)$ et continue
à gauche en tout point de $[1,+\infty)$. Ainsi, les points de discontinuité de
$f$ sont des éléments de $\mathbb{N}^{*}$. Si $f$ est continue en
$x\in\mathbb{N}^{*}$ alors on aura
$f(x^{+})=f(x)$
Donc
$\displaystyle 0$ $\displaystyle=f(x^{+})-f(x)$ $\displaystyle=\sum_{x\leq
n<x^{+}}\chi(n)$ $\displaystyle=\chi(x)$
Alors
$f\ \text{est \ continue \ en }\
x\in\mathbb{N}^{*}\Longleftrightarrow\chi(x)=0$ (6)
Soit maintenant $(a_{k})_{k\in\mathbb{N}}$ une suite croissante des points de
discontinuité de la fonction $f$ sur $[1,+\infty)$ alors $f$ est constante sur
chaque intervalle $I_{k}=(a_{k-1},a_{k}]$ (où $k\in\mathbb{N}^{*}$). En effet,
soit $k\in\mathbb{N}^{*}$ s’il existe $n\in I_{k}$ tel que $f$ est continue en
$n$ alors d’après 6 $\chi(n)=0$ ainsi et d’une manière générale soit
$(\beta_{i})_{i\in\mathbb{N}^{*}}$ une suite strictement croissante des
entiers de $\overset{\circ}{I_{k}}$ (l’interieur de $I_{k}$), alors $f$ est
continue en chaque $\beta_{i}$ d’où $\chi(\beta_{i})=0$ pour tout
$i=1,2,\cdots$ et en conséquent pour tout $x\in(a_{k-1},a_{k}]$ on a
$f(x)=f(a_{k-1}^{+})$ ($k\in\mathbb{N}^{*}$).
Soient $\alpha>1$ un réel et $\rho$ la fonction définie sur $\mathbb{R}^{+}$
par
$\rho(x)=f\left(e^{x}\right)e^{-\alpha x}.$
Soit $k$ un entier strictement positif on note
$(\lambda_{k})_{k\in\mathbb{N}}$ pour la suite croissante des points de
discontinuité de la fonction $\rho$ sur $\mathbb{R}^{+}$ ($\lambda_{k}=\log
a_{k}\in\log\mathbb{N}^{*}$). Alors la fonction $\rho$ est décroissante sur
chaque intervalle $J_{k}=(\lambda_{k-1},\lambda_{k}]$, en effet: soient
$x,y\in J_{k}$ tels que $x>y$, donc puisque $f$ est constante ($\equiv c_{k}$)
sur $J_{k}$ alors $\rho(x)-\rho(y)=c_{k}\left(e^{-\alpha x}-e^{-\alpha
y}\right)<0$ d’où $\rho$ est strictement décroissante sur $J_{k}$ pour tout
$k\in\mathbb{N}^{*}$. D’une autre part, pour tout $k\in\mathbb{N}^{*}$
$\rho(\lambda_{k}^{+})>\rho(\lambda_{k}).$
En effet, puisque la fonction $x\mapsto e^{-\alpha x}$ est continue sur
$\mathbb{R}^{+}$ alors $e^{-\alpha\lambda_{k}^{+}}=e^{-\alpha\lambda_{k}}$
donc
$\rho(\lambda_{k}^{+})-\rho(\lambda_{k})=(f(a_{k}^{+})-f(a_{k}))e^{-\alpha\lambda_{k}}$
et puisque $f$ est discontinue en $a_{k}$ et croissante sur $[1,+\infty)$
alors $f(a_{k}^{+})>f(a_{k})$. D’où
$\rho(\lambda_{k}^{+})>\rho(\lambda_{k}).$
###### Lemme 3.1.
Soit $\alpha>1$ alors $\sum_{n\geq
1}\frac{\chi(n)}{n^{\alpha}}<+\infty\Longrightarrow\rho\in\mathcal{V}_{b}\mathcal{C}_{g}^{*}$
preuve:
Soit $x\in\mathbb{R}^{+}$, on pose $0=x_{0}<x_{1}<\cdots<x_{n}=x$ une
subdivision de l’intervalle $[0,x]$ et on note pour $m$ le plus grand entier
naturel non nul tel que $\lambda_{m-1}<x\leq\lambda_{m}$ où les
$(\lambda_{k})_{k\in\mathbb{N}}$ sont les points de discontinuité de la
fonction $\rho$ définis précédamment, alors
$\sum_{i=1}^{n}|\rho(x_{i})-\rho(x_{i-1})|=\sum_{k=0}^{m}\sum_{\underset{x_{i}\in
J_{k}}{i=1}}^{n}|\rho(x_{i})-\rho(x_{i-1})|$
où $(J_{k})_{k\in\mathbb{N}^{*}}$ sont les intervalles
$(\lambda_{k-1},\lambda_{k}]$ et $J_{0}=[0,\lambda_{0}]$, et on note bien que
$\displaystyle\cup_{k=0}^{m}J_{k}=[0,\lambda_{m}]$ donc puisque $\rho$ est
strictement décroissante sur chaque $J_{k}$ alors
$\displaystyle\sum_{k=0}^{m}\sum_{\underset{x_{i}\in
J_{k}}{i=1}}^{n}|\rho(x_{i})-\rho(x_{i-1})|$
$\displaystyle\leq\rho(0)-\rho(\lambda_{0})+\sum_{k=1}^{m}\left(\rho(\lambda_{k-1}^{+})-\rho(\lambda_{k})\right)-(\rho(x)-\rho(\lambda_{m}))$
$\displaystyle=-\rho(\lambda_{0})+\rho(\lambda_{0}^{+})-\rho(\lambda_{1})+\rho(\lambda_{1}^{+})+\cdots-\rho(\lambda_{m})-\rho(x)+\rho(\lambda_{m})$
$\displaystyle=-\rho(x)+\sum_{k=0}^{m-1}\left(\rho(\lambda_{k}^{+})-\rho(\lambda_{k})\right)$
$\displaystyle=-\rho(x)+\sum_{k=0}^{m-1}\frac{f(a_{k}^{+})-f(a_{k})}{a_{k}^{\alpha}}$
$\displaystyle=-\rho(x)+\sum_{k=0}^{m-1}\frac{\chi(a_{k})}{a_{k}^{\alpha}}$
Où les $(a_{k})_{k\in\mathbb{N}}$ sont les points de discontinuité de la
fonction $f$ et qui sont des éléments de $\mathbb{N}^{*}$. Donc
$\sum_{k=0}^{m-1}\frac{\chi(a_{k})}{a_{k}^{\alpha}}\leq\sum_{1\leq\ell<e^{x}}\frac{\chi(\ell)}{\ell^{\alpha}}$
D’où
$\sum_{i=1}^{n}|\rho(x_{i})-\rho(x_{i-1})|\leq-\rho(x)+\sum_{1\leq\ell<e^{x}}\frac{\chi(\ell)}{\ell^{\alpha}}.$
Alors
$T_{\rho}(x)\leq-\rho(x)+\sum_{1\leq\ell<e^{x}}\frac{\chi(\ell)}{\ell^{\alpha}}.$
Or puisque $\rho$ est une fonction positive alors
$T_{\rho}(x)\leq\sum_{1\leq\ell<e^{x}}\frac{\chi(\ell)}{\ell^{\alpha}}.$
Donc
$\text{la s\'{e}rie}\ \sum_{n\geq 1}\frac{\chi(n)}{n^{\alpha}}\
\text{converge}\ \Longrightarrow\rho\in\mathcal{V}_{b}\mathcal{C}_{g}^{*}.$
$\blacksquare$
Sans perte de généralité le résultat est vrai pour toute fonction arithmétique
$\chi:\mathbb{N}^{*}\longrightarrow\mathbb{R}$ croissante. Dans ce cas, le
Lemme 3.1 peut être reformulé:
$\text{la s\'{e}rie}\ \sum_{n\geq 1}\frac{\chi(n)}{n^{\alpha}}\ \text{est
absolument
convergente}\Longrightarrow\rho\in\mathcal{V}_{b}\mathcal{C}_{g}^{*}.$
où $\alpha>1$.
Maintenant on arrive au résultat le plus important dans cette section:
###### Théorème 3.1.
Soit $\chi:\mathbb{N}^{*}\longrightarrow\mathbb{R}^{+}$ une fonction
arithmétique et soit $\alpha>1$ le plus petit réel tel que la série
$\displaystyle\sum_{n\geq 1}\frac{\chi(n)}{n^{\alpha}}$ soit convergente. On
pose $\displaystyle f(x)=\sum_{1\leq n<x}\chi(n)$ pour tout $x\geq 1$ où
$f(1)=0$ et on suppose que $\mathcal{L}_{\rho}$ est holomorphe sur le demi-
plan complexe $\\{\sigma\geq\beta\\}$ sauf au seul pôle simple en $s=\beta\geq
0$ alors on a
$f(x)\underset{x\to+\infty}{\sim}Res(\mathcal{L}_{\rho},\beta)x^{\alpha+\beta}.$
Où $\rho(x)=f(e^{x})e^{-\alpha x}$ pour tout $x\in\mathbb{R}^{+}$.
Preuve:
Soit $\alpha>1$ un réel tel que la série du terme générale
$\frac{\chi(n)}{n^{\alpha}}$ est convergente alors, d’après le Lemme 3.1, la
fonction $\rho(x)=f(e^{x})e^{-\alpha x}$ est un élément de
$\mathcal{V}_{b}\mathcal{C}_{g}^{*}$. Or d’après le Corollaire 2.1 on déduit
que
$\rho(x)\underset{x\to+\infty}{\sim}Res(\mathcal{L}_{\rho},\beta)e^{\beta x}.$
Ce qui est
$f(e^{x})\underset{x\to+\infty}{\sim}Res(\mathcal{L}_{\rho},\beta)e^{(\alpha+\beta)x}.$
D’où
$f(x)\underset{x\to+\infty}{\sim}Res(\mathcal{L}_{\rho},\beta)x^{\alpha+\beta}.$
Ce qu’il fallait démontrer.
$\blacksquare$
On rappelle que la fonction $\Lambda$ de Von Mangoldt est une fonction
arithmétique définie sur $\mathbb{N}^{*}$ par
$\Lambda(n):=\begin{cases}\log p\quad\text{si}\ n=p^{k},\quad
k\in\mathbb{N}^{*},p\in\mathcal{P}\\\ \\\ \quad
0\qquad\text{sinon}\end{cases}.$
La fonction définie pour tout $x\in[1,+\infty)$ tel que $x\neq p^{k}$ où
$k\in\mathbb{N}^{*}$ et $p\in\mathcal{P}$ par
$\displaystyle\psi(x)=\sum_{n<x}\Lambda(n)$ est dite la fonction de Chebyshev,
ainsi pour démontrer le théorème des nombres premiers il faut et il suffit de
démontrer que
$\psi(x)\underset{x\to+\infty}{\sim}x.$
Il existe une forte relation entre la fonction $\zeta$ de Riemann et la
fonction $\psi$, en effet
$-\frac{\zeta^{\prime}(s)}{\zeta(s)}=\sum_{n\geq
1}\frac{\Lambda(n)}{n^{s}}=s\int_{0}^{+\infty}\psi(e^{x})e^{-sx}dx,\qquad\forall\sigma>1.$
On rappelle aussi que la fonction $\zeta$ est holomorphe sur $\\{\sigma\geq
1\\}$ sauf au $s=1$ qui est le seul pôle simple de la fonction $\zeta$, ainsi
d’après Hadamard et De La Vallée Poussin la fonction $\zeta$ ne s’annulle en
aucun point du demi-plan $\\{\sigma\geq 1\\}$. Alors on déduit que la fonction
$-\frac{\zeta^{\prime}}{\zeta}$ est holomorphe sur $\\{\sigma\geq 1\\}$ sauf
au point $s=1$ qui est le seul pôle simple de résidu égal à $1$.
Et on a le Théorème des Nombres Premiers:
###### Corollaire 3.1.
$\psi(x)\underset{x\to+\infty}{\sim}x.$
preuve:
Soit $\alpha>1$ un réel donné, on pose
$\rho(x)=\psi(e^{x})e^{-\alpha x},\qquad\forall x\in\mathbb{R}^{+}.$
Alors puisque la fonction $-\frac{\zeta^{\prime}}{\zeta}$ est holomorphe sur
$\\{\sigma>1\\}$ alors la série su terme général
$\frac{\Lambda(n)}{n^{\alpha}}$ est convergente pour tout $\alpha>1$.
D’une autre part, soit $\sigma>1$ alors
$\displaystyle\mathcal{L}_{\rho}(s)$
$\displaystyle=\int_{0}^{+\infty}\rho(x)e^{-sx}dx$
$\displaystyle=\int_{0}^{+\infty}\psi(e^{x})e^{-\alpha x}e^{-sx}dx$
$\displaystyle=\int_{0}^{+\infty}\psi(e^{x})e^{-(s+\alpha)x}dx$
$\displaystyle=-\frac{\zeta^{\prime}(s+\alpha)}{(s+\alpha)\zeta(s+\alpha)}$
Alors puisque la fonction
$s\mapsto-\frac{\zeta^{\prime}(s+\alpha)}{(s+\alpha)\zeta(s+\alpha)}$ est
holomorphe sur $\\{\sigma\geq 1-\alpha\\}$ sauf au point $s=1-\alpha$ qui est
le seul pôle simple de cette fonction où
$Res\left(-\frac{\zeta^{\prime}(s+\alpha)}{(s+\alpha)\zeta(s+\alpha)},1-\alpha\right)=1$
Alors d’après le Théorème 3.1 on a
$\psi(x)\underset{x\to+\infty}{\sim}x^{\alpha+1-\alpha}$
C’est à dire
$\psi(x)\underset{x\to+\infty}{\sim}x.$
Ce qu’il fallait démontrer.
$\blacksquare$
## References
* [1] M.Carter et B. Van Brunt, The Lebesgue-Stieltjes Integral a practical introduction, Springer (2000).
* [2] E.C. Titchmarsh, _The Theory of The Riemann Zeta-Function_ 2nd ed, revised by D. R. Heath-Brown, Oxford University Press (1986).
* [3] Walter Rudin, _Analyse réelle et complexe_ , Troisième tirage MASSON Paris New York Barcelone Milan 1980.
|
arxiv-papers
| 2014-02-28T15:04:04 |
2024-09-04T02:49:59.110269
|
{
"license": "Public Domain",
"authors": "Lahoucine Elaissaoui",
"submitter": "Lahoucine Elaissaoui",
"url": "https://arxiv.org/abs/1402.7269"
}
|
1403.0097
|
# Dynamics and control of fast ion crystal splitting in segmented Paul traps
H. Kaufmann1, T. Ruster1, C. T. Schmiegelow1, F. Schmidt-Kaler1, U. G.
Poschinger1 1QUANTUM, Institut für Physik, Universität Mainz, D-55128 Mainz,
Germany [email protected]
###### Abstract
We theoretically investigate the process of splitting two-ion crystals in
segmented Paul traps, i.e. the structural transition from two ions confined in
a common well to ions confined in separate wells. The precise control of this
process by application of suitable voltage ramps to the trap segments is non-
trivial, as the harmonic confinement transiently vanishes during the process.
This makes the ions strongly susceptible to background electric field noise,
and to static offset fields in the direction of the trap axis. We analyze the
reasons why large energy transfers can occur, which are impulsive
acceleration, the presence of residual background fields and enhanced
anomalous heating. For the impulsive acceleration, we identify the diabatic
and adiabatic regimes, which are characterized by different scaling behavior
of the energy transfer with respect to time. We propose a suitable control
scheme based on experimentally accessible parameters. Simulations are used to
verify both the high sensitivity of the splitting result and the performance
of our control scheme. Finally, we analyze the impact of trap geometry
parameters on the crystal splitting process.
###### Contents
1. 1 Introduction
2. 2 Prerequisites
1. 2.1 Electrostatic trap potentials
2. 2.2 Equilibrium positions
3. 2.3 Critical tilt value
3. 3 Intricacies of crystal splitting
1. 3.1 Impulsive acceleration at the critical point
2. 3.2 Uncompensated potential tilt
3. 3.3 Anomalous heating at the critical point
4. 4 Voltage ramps
1. 4.1 Static voltage sets
2. 4.2 Time domain ramps
5. 5 Simulation results
1. 5.1 Dependence on splitting time
2. 5.2 Sensitivity analysis
3. 5.3 Dependence on the limiting voltage
6. 6 Trap geometry optimization
7. 7 Conclusion
## 1 Introduction
Linear crystals of ions trapped in linear Paul traps have allowed for ground-
breaking experiments in the fields of quantum computation, quantum simulation
and precision measurements [1]. Segmented, micro-structured Paul trap arrays
have been proposed as a future hardware platform for scalable quantum
information experiments [2]. Small groups of ions are trapped separately from
each other, such that precise manipulation of the qubits can be accomplished.
Experimental protocols then require ion shuttling operations, in addition to
laser- or microwave-driven logic gates. Essential shuttling operations are
splitting and merging of linear ion crystals. It is important that they are
fast on the typical timescale for quantum gates of 10-100$\mu$s, and in order
to allow for gate operations or readout after the splitting, a low energy
transfer is required. Shuttling of trapped ions in segmented traps has been
realized within a few oscillation cycles of the harmonic trap by time-
dependent control of the trap voltages [3, 4], at energy transfers below one
motional quantum. Crystal splitting in a segmented trap was first demonstrated
in Ref. [5], at energy transfers of about 140 phonons within a splitting time
of 10 ms. With optimizations, splitting has been included to the set of
methods for quantum computing, e.g. for quantum teleportation [6] and
entanglement purification [7]. Currently, the best reported result is a gain
of about two vibrational quanta per ion at a time duration of 55 $\mu$s [4].
The experimental challenge for the control of this process is given by the
fact that the harmonic part of the electrostatic trap potential has to change
its sign during this process and therefore has to cross zero. This situation
of weak confinement reduces the attainable speed and potentially increases the
final motional excitation. In order to make the process more robust and
faster, it is desirable to achieve a large quartic component of the axial
trapping potential.
Trap geometries tailored to improve splitting performance were investigated in
[8]. Optimized geometry parameters for surface electrodes traps were derived
in Ref. [9]. In Ref. [10], robust splitting operations on slow timescales were
carried out by means of real-time observation of the ion positions and
feedback on the segment voltages.
In this work, we analyze the splitting process with the aim of achieving low
energy transfers in segmented miniaturized Paul traps. We reduce our analysis
to the process of splitting ion crystals, as the process of merging ion
crystals is merely the time reversed process. Furthermore, we restrict
ourselves to the case of two ions. For splitting and merging processes with
several ions, the general procedures and conclusions are still valid.
The manuscript is organized as follows: In Sec. 2, we introduce the formalism
for describing the electrostatic potentials during the splitting operations
and the equilibrium positions of the ions, and we analyze the dependence of
the equilibrium positions on the control parameters. In Sec. 3, we give a
detailed explanation of the possible reasons for high energy transfers. Based
on these considerations, a procedure for the design of suitable voltage ramps
is given in Sec. 4. In Sec. 5, we analyze the performance of these ramps by
numerical simulations. Finally, in Sec. 6, we compare typical examples for
trap geometries and discuss the implication for ion splitting.
## 2 Prerequisites
### 2.1 Electrostatic trap potentials
We desire to split a two-ion crystal residing at center segment $C$ along the
trap axis $x$, to obtain two ions stored in separated potential wells at the
position of the splitting segments $S$ neighboring $C$, see Fig. 1.
Figure 1: The process of ion crystal splitting. It is shown schematically how
two ions are moved from the initial center segment $C$ to different
destination segments $S_{R,L}$ by changing a confining electrostatic potential
from a) a strong harmonic confining potential ($\alpha>0$) via b) a
predominantly quartic potential ($\alpha\approx 0$) to c) a double-well
potential ($\alpha<0$). The external potential is determined by the voltages
applied to the respective electrodes. The equilibrium positions are sketched
as dashed lines. The outer electrodes $O$ facilitate the splitting process by
increasing the transient quartic confinement and offer the possibility to
cancel a possible axial background field by application of a differential
voltage. The color coding of the segments and the corresponding voltages is
used throughout the manuscript.
Note that we consider only the spatial dimension along the trap axis, as we
assume that tight radial confinement persists throughout the process and the
ions are always located on the rf node of the trap. Typical distances between
segments range between 50 and 500 $\mu$m, while the initial ion distance is
2-4 $\mu$m. The total external electrostatic potential along the trap axis can
be written as
$\Phi(x)\approx\beta~{}x^{4}+\alpha~{}x^{2}+\gamma~{}x$ (1)
where the coefficients $\alpha,\beta,\gamma$ are given by the the trap
geometry and the voltages applied to the trap segments. This Taylor
approximation is valid as long as the the ions are located sufficiently close
to $x=0$, which is the center of the $C$ segment. Throughout the splitting
process, the external potential is changing from a single well potential
$\alpha_{i}>0$ to a double well potential $\alpha_{f}<0$, crossing the
critical point (CP) at $\alpha=0$. Note that $\beta>0$ is required to
guarantee confinement at $\alpha\leq 0$. The approximation of Eq. 1 holds for
$\alpha\geq 0$ and for $\alpha\lesssim 0$ as long as the separation of the two
potential wells is small compared to the width of segment $C$. When the
distance of the ions from the center of the $C$ segment becomes comparable to
the width of the segment, anharmonic terms of order $>4$ contribute
significantly to the total potential. These are not taken into account here
since the outcome of the splitting process is determined around the CP, as
will be pointed out in the following sections. Furthermore, beyond the CP, the
distance of the separated wells is still increasing monotonically for
decreasing $\alpha$ as long as the variation $\beta$ is sufficiently small,
and the corresponding trap frequencies in these wells are monotonically
increasing. For studies which require precision beyond the CP, the higher
order terms can be taken into account numerically. A cubic term does not
contribute to the potential if the trap is sufficiently symmetric along the
trap axis.
Including Coulomb repulsion, the total electrostatic potential of a two-ion
crystal at a center-of-mass position $x_{0}$ and distance $d$ is given by
$\Phi_{tot}(x_{0},d)=\Phi(x_{0}+d/2)+\Phi(x_{0}-d/2)+\frac{\kappa}{d},$ (2)
with $\kappa=e/4\pi\epsilon_{0}$. At the CP, the harmonic confinement
vanishes, and a weak residual confinement is maintained by the interplay
between Coulomb repulsion and quartic part of the external potential. It is
therefore desirable to maximize $\beta$ at the CP. For a given trap geometry,
the attainable $\beta$ is limited by the voltage range which can be applied to
the trap electrodes 111The maximum voltage is ultimately limited by the
electric breakdown threshold. In practice, as precisely controlled time-
dependent voltage waveforms are to be applied to the trap segments, the
voltage range will be determined by the electrical design, where one faces a
trade-off between voltage range and output bandwidth [11, 12].. The
coefficients of the potential Eq. 1 are given by the segment bias voltages and
the electrostatic properties of the trap:
$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
U_{C}~{}\alpha_{C}+U_{S}\alpha_{S}+U_{O}\alpha_{O}$ (3) $\displaystyle\beta$
$\displaystyle=$ $\displaystyle
U_{C}~{}\beta_{C}+U_{S}\beta_{S}+U_{O}\beta_{O}$ (4) $\displaystyle\gamma$
$\displaystyle=$ $\displaystyle\Delta U_{S}\gamma_{S}+\Delta
U_{O}\gamma_{O}+\gamma^{\prime}$ (5)
An offset parameter $\gamma^{\prime}$ is introduced for taking trap non-
idealities – leading to a symmetry breaking force along the trap axis – into
account, see Sec. 3.2. In contrast to the symmetric quadratic and quartic
contributions, the asymmetric tilt potential is controlled by the differential
voltages $\Delta U_{S,O}$ between the corresponding left and right electrodes
of the respective pair. The segment coefficients are given by Taylor
expansions of the standard potentials $\phi_{n}(x)$, which are the
dimensionless electrostatic potentials along the trap axis if a +1V bias is
applied to segment $n$ and all other segments are grounded [13, 14]:
$\phi_{n,m}(x)=\phi_{n}|_{x_{0}^{(m)}}+\phi_{n}^{\prime}|_{x_{0}^{(m)}}\delta
x+\frac{1}{2}\phi_{n}^{\prime\prime}|_{x_{0}^{(m)}}\delta
x^{2}+\frac{1}{24}\phi_{n}^{(4)}|_{x_{0}^{(m)}}\delta
x^{4}+\mathcal{O}\left(\delta x^{6}\right).$ (6)
with $\delta x=x-x_{0}^{(m)}$, i.e. the Taylor expansions are carried out at
center of segment $m$, $x_{0}^{(m)}$. The coefficients for Eqs. 3,4,5 are
obtained for $m=C,n=C,S,O$:
$\alpha_{n}=\frac{1}{2}f_{n}\phi_{n,C}^{\prime\prime}(0),\qquad\beta_{n}=\frac{1}{24}f_{n}\phi_{n,C}^{(4)}(0),\qquad\gamma_{n}=f_{n}\phi_{n,C}^{\prime}(0),$
(7)
with $f_{C}=1$ and $f_{S,O}=2$ accounting for two $S,O$ segments acting
symmetrically at $x=0$. Note that $\gamma_{C}=0$ by definition.
In the following, for numerical calculations, we use the specific geometry
parameters of a three dimensional microstructured segmented ion trap A as
detailed in Sec.6. There, other traps and their geometry parameters are listed
and analyzed as well.
### 2.2 Equilibrium positions
Figure 2: Ion equilibrium positions near the critical point. a) shows the
equilibrium positions versus the harmonic parameter $\alpha$. In the case of a
perfectly compensated tilt (blue), the ions separate symmetrically, in the
case of a large tilt (red), both ions move towards one side. Panel b) shows a
close-up around the critical point for the same tilt parameters. Additionally,
the minima of the external potential are shown (dashed). In panel c), we
display equilibrium positions and potential minima for tilt parameters
slightly below (blue) and above (red) the critical tilt parameter. In contrast
to the corresponding curves in b), the equilibrium positions exhibit cusps
which lead to strongly enhanced acceleration.
We consider two ions of mass $m$ and charge $e$, with their equilibrium
positions given by the center-of-mass $x_{0}$ and the equilibrium distance
$d$:
$x_{L,R}=x_{0}\pm d/2,$ (8)
determined by minimizing of the total electrostatic potential Eq. 2. The
confinement is characterized by the local trap frequency, which is given by
the curvature of of the external potential at the ion positions:
$\omega=\sqrt{\frac{e}{m}\Phi^{\prime\prime}(x_{L,R})}.$ (9)
The extremal points of the external potential Eq. 1 are given by
$\displaystyle x_{0}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{3^{1/3}\zeta}-\frac{\zeta}{2\cdot 3^{2/3}\beta}$
(10) $\displaystyle x_{0}^{(\pm)}$ $\displaystyle=$
$\displaystyle\frac{(i\sqrt{3}\pm 1)\alpha}{2\cdot 3^{1/3}\zeta}+\frac{(1\mp
i\sqrt{3})\zeta}{4\cdot 3^{2/3}\beta}$ (11) (12)
where
$\zeta(\alpha,\beta,\gamma)=\left(9\beta^{2}\gamma+\sqrt{3}\sqrt{8\alpha^{3}\beta^{3}+27\beta^{4}\gamma^{2}}\right)^{1/3}.$
(13)
Initially, at $\alpha=\alpha_{i}$, the confining harmonic part of the external
potential and the Coulomb repulsion are dominant, thus we can neglect the
quartic potential. The trap frequency is then given by $\omega^{2}=2\alpha
e/m$ at an ion distance of $d=\left(\kappa/\alpha\right)^{1/3}$. At the CP,
$\alpha=0$, and without tilt, $\gamma=0$, the ion distance is determined by
quartic confinement and Coulomb repulsion:
$d_{CP}=\left(2\kappa/\beta\right)^{1/5}.$ (14)
The Coulomb repulsion pushes the ions away from the trap center (where the
curvature of the external potential vanishes), such that a residual harmonic
confinement persists because of the quartic term. The minimum trap frequency
during the splitting process is thus given by [8]
$\omega_{CP}=\beta^{3/10}\left(3e/m\right)^{1/2}\left(2\kappa\right)^{1/5}.$
(15)
Near the CP, the equilibrium distance can be computed from a perturbative
expression up to second order:
$d(\alpha)\approx
d_{CP}-\frac{1}{5}\left(\frac{16}{\beta^{4}\kappa}\right)^{1/5}\alpha+\frac{2}{25}\left(\frac{4}{\beta^{7}\kappa^{3}}\right)^{1/5}\alpha^{2},$
(16)
for $|\alpha|\ll\beta d_{CP}^{2}$ and $|\alpha|\ll\kappa d_{CP}^{-3}$.
The center-of-mass position of the ion crystal near the critical point to
first order in the tilt parameter $\gamma$ is:
$x_{0}(\alpha,\gamma)\approx\gamma\left(-\frac{1}{3\cdot
2^{2/5}\beta^{3/5}\kappa^{2/5}}-\frac{2^{1/5}}{45\cdot\beta^{6/5}\kappa^{4/5}}\alpha+\frac{26\cdot
2^{4/5}}{675\beta^{9/5}\kappa^{6/5}}\alpha^{2}\right)$ (17)
If the ions are sufficiently separated, $\alpha\ll 0$, the Coulomb repulsion
can be neglected and the equilibrium positions approximately coincide with the
extrema of the external potential:
$d_{f}=\sqrt{-2\alpha_{f}/\beta}$ (18)
and the final trap frequency is given by $\omega_{f}^{2}=-4\alpha_{f}e/m$.
### 2.3 Critical tilt value
A static background force along the trap axis can to keep the ions confined in
one common potential well throughout the splitting process. We make use of the
external potential minima Eqs. 12 to obtain an estimate for the tilt parameter
$\tilde{\gamma}$, beyond which the splitting ceases to work. In the following,
we assume $\gamma>0$.
Figure 3: Critically tilted potential, see text such that the Coulomb
repulsion fails to push the right ion across the saddle point.
In the presence of a nonzero potential tilt, an imperfect bifurcation occurs,
i.e. the second potential well opens up at $\tilde{\alpha}<0$, see Fig. 2 c).
We obtain a scaling law for $\tilde{\gamma}$ by calculating at which tilt
parameter the original potential well is deep enough to keep both mutually
repelling ions confined, see Fig. 3. The saddle point where the second
potential well opens can be found by solving $x_{0,c}^{(0)}=x_{0,c}^{(+)}$ for
$\tilde{\alpha}$, yielding
$\tilde{\alpha}=-{\textstyle\frac{3}{2}}\beta^{1/3}|\gamma|^{2/3}$. From this
we obtain its position 222For $\gamma\geq 0$, $x_{0}^{(0)}$ corresponds to the
left potential minimum which always exists, and for $\alpha<\tilde{\alpha}<0$,
$x_{0}^{(+)}$ corresponds to the right potential minimum and $x_{0}^{(-)}$
corresponds to the maximum of the separation barrier. By contrast, for
$\gamma<0$, $x_{0}^{(0)}$ corresponds to the right potential minimum, and for
$\alpha<0<\tilde{\alpha}$, $x_{0}^{(+)}$ corresponds to the left minimum. to
be $x_{0,c}^{(+,0)}={\textstyle\frac{1}{2}}\left(\gamma/\beta\right)^{1/3}$.
At $\tilde{\alpha}$, the left potential minimum is located at twice the
distance from the origin $x_{0,c}^{(-)}=-\left(\gamma/\beta\right)^{1/3}$. The
potential attains the same value as on the saddle point $V(x_{0,c}^{(+,0)})$
at the position
$\tilde{x}_{c}^{(+)}=-{\textstyle\frac{3}{2}}\left(\gamma/\beta\right)^{1/3}$.
The depth of the potential well defined by the saddle point when the right
well opens is therefore
$\Delta
V_{c}=V(x_{0,c}^{(-)})-V(x_{0,c}^{(+,0)})=\frac{27}{16}\left(\frac{\gamma^{4}}{\beta}\right)^{1/3}.$
(19)
We can now define a criterion which determines whether the ions are actually
separated by comparing the Coulomb potential to the depth of the initial well
at the CP, Eq. 19: If the Coulomb repulsion pushes the right ion beyond the
saddle point $x_{0,c}^{(+,0)}$, it will end up in the right potential well,
otherwise the two ions will stay in the left well. Thus, the Coulomb energy at
an ion distance of $x_{0,c}^{(+,0)}-\tilde{x}_{c}^{(+)}$ has to be larger than
the well depth $\Delta V_{c}$. These considerations lead to a critical tilt
value of
$\tilde{\gamma}<\pm~{}C_{\gamma}\left(\kappa^{3}{\beta^{2}}\right)^{1/5}.$
(20)
Despite the fact that the situation depicted Fig. 3 does not actually occur,
as the external force at the saddle point vanishes and therefore cannot
balance the Coulomb force, the obtained scaling behavior is confirmed by
numerical calculations, revealing a prefactor of $C_{\gamma}=$1.06.
The result Eq. 20 enables us to determine the required degree of precision by
which the background axial field has to be corrected. For this calculation,
only the geometry parameters $\beta_{C,S,O}$ are needed. Furthermore, the
sensitivity decreases as $\beta^{2/5}$, which directly characterizes the gain
in robustness when the accessible voltage range is enhanced. For trap A (Sec.
6), we derive a value of $\tilde{\gamma}\approx 3V/m$, corresponding to the
requirement to set $\Delta U_{O}$ more accurately than about 9 mV.
## 3 Intricacies of crystal splitting
### 3.1 Impulsive acceleration at the critical point
Figure 4: Impulsive acceleration at the critical point. a) shows the
equilibrium distance (black) versus time. The red lines depict the approximate
slopes $\dot{d}_{CP}$ within time $\tau_{CP}$ before and beyond the CP. They
illustrate how the impulsive displacement $\delta d_{CP}$ Eq. 23 is obtained
from slope beyond the CP, and why the difference of the slopes, i.e. the
second derivative $\ddot{d}_{CP}$, determines the onset of adiabaticity (see
text). It is also shown how the trap frequency (gray) varies strongly during
the CP trap period. b) compares the final excitation obtained from the simple
approximation Eqs. 25 (dashed), 28 (solid) to simulation results (dots). The
onset of adiabaticity $\chi=1$, is marked with vertical bars. The calculations
are carried out for a harmonic coefficients $\alpha(t)$ linearly varying
around the CP, and different constant values for the quartic coefficient
$\beta$.
A naïve approach towards crystal splitting is the linear interpolation between
two voltage sets pertaining to a single well and a double well, leading to a
constant variation rate of the harmonic coefficient $\alpha$. As this does not
involve a dedicated control of the ion distance, it is equivalent to a rapid
sweep across a structural transition of the ion crystal. This leads to an
unfavorable power-law scaling of the energy transfer with respect to the sweep
time [15], which prevents attaining adiabaticity.
In the following, we derive an approximation for the energy transfer, assuming
the variation of $\alpha$ around the CP to be uniform. We consider the energy
transfer to be caused by impulsive displacement: At the CP, the equilibrium
distance changes most rapidly, while the confinement - and therefore the
restoring forces - are reduced. Fig. 4 a) shows that the situation corresponds
to a harmonic oscillator which is suddenly dragged at uniform speed, causing
displacement and therefore a gain in potential energy. Within the
characteristic timescale set by half a the trap oscillation cycle
$\tau_{CP}=\pi/\omega_{CP}$, this yields the displacement:
$\displaystyle\delta d_{CP}$ $\displaystyle\approx$
$\displaystyle\dot{d}_{CP}\;\xi\;\tau_{CP}/2$ (21) $\displaystyle\approx$
$\displaystyle\left.\frac{\partial
d}{\partial\alpha}\right|_{CP}\dot{\alpha}_{CP}\;\xi\;\tau_{CP}/2$ (22)
$\displaystyle\approx$
$\displaystyle\left(\beta_{CP}^{4}~{}\kappa\right)^{-1/5}\dot{\alpha}_{CP}\;\xi\;\tau_{CP}/2,$
(23)
where Eq. 16 was used in the last line. The factor $\xi$ accounts for the fact
that the trap frequency increases beyond the CP, such that the restoring
forces set in before $\tau_{CP}$ and the resulting displacement is reduced.
This sudden displacement mechanism is sketched in Fig. 4 a). The potential
energy of an ion is consequently increased by
$\displaystyle\delta E$ $\displaystyle=$
$\displaystyle\frac{1}{2}m\omega_{CP}^{2}\left(\delta d_{CP}/2\right)^{2}$
(24) $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{8}\;\xi^{2}\;m\left(\beta_{CP}^{4}~{}\kappa\right)^{-2/5}\dot{\alpha}^{2}_{CP},$
(25)
which serves as an approximation of the final energy transfer.
For a sufficiently small $|\dot{\alpha}|_{CP}$, adiabaticity sets in and the
energy transfer scales exponentially with the splitting time. The reason for
this is that the Coulomb repulsion serves to push the ions outwards, providing
smooth variation of the equilibrium distance as compared to discontinuous
behavior of the minima of the external potential, see Fig. 2 b). It therefore
leads to rapid, but continuous variation of the equilibrium positions with
$\alpha$. The onset of the adiabatic regime is identified by comparing
displacement $\delta d_{CP}$ to the change of the equilibrium distance within
$\tau_{CP}$ below the CP (see Fig. 4 a)), which means that the ion
acceleration around the CP is sufficiently slow to prevent sudden
displacement. We therefore compare the acceleration $\ddot{d}_{CP}$ to the
reference acceleration $d_{CP}\omega_{CP}^{2}$, yielding the adiabaticity
parameter
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\frac{\ddot{d}_{CP}}{d_{CP}\omega_{CP}^{2}}$ (26)
$\displaystyle=$
$\displaystyle\frac{4}{25}\frac{m}{3e}2^{-1/5}\beta_{CP}^{-9/5}\kappa^{-6/5}\dot{\alpha}_{CP}^{2}$
(27)
In the adiabatic regime, $\chi<1$, the energy transfer is given by:
$\delta E^{\prime}\approx\delta
E\exp\left[c^{2}\left(1-\frac{1}{\chi}\right)\right]$ (28)
Numerical simulations are carried out for different constant values for
$\beta$ and a linear variation of $\alpha$ around the CP. The results are
shown in Fig. 4 b). It can be seen that the approximations Eqs. 25,28 hold
over a wide range of splitting times and quartic coefficients, and that large
energy transfers in the regime of 104-106 phonons are readily obtained. The
simulations yield a value of $\xi^{2}\approx$0.1. We conclude that in this
regime, the energy transfer depends only on the ion mass, the variation rate
of $\alpha$ and the quartic confinement at the CP. As can be seen from the
simulation results, still large energy transfers are obtained at the onset of
adiabaticity, such that splitting at energy transfers on the single phonon
level would require splitting times on the order of several hundreds of
$\mu$s.
As we will show in further sections, this problem can be overcome using ramps
that ensure a small ion acceleration $\ddot{d}_{CP}$ at the CP. Note that
$\ddot{d}_{CP}=\left.\frac{\partial^{2}d}{\partial\alpha^{2}}\right|_{CP}\dot{\alpha}_{CP}^{2}+\left.\frac{\partial
d}{\partial\alpha}\right|_{CP}\ddot{\alpha}_{CP}.$ (29)
For sufficiently uniform variation of $\alpha$, the second term can generally
be neglected, such that by using Eq. 16, we obtain
$\ddot{d}_{CP}=\frac{2}{25}\left(\frac{4}{\beta_{CP}^{7}\kappa^{3}}\right)^{1/5}\dot{\alpha}_{CP}^{2}.$
(30)
Thus, the energy transfer can be reduced by ensuring a small variation rate of
$\alpha$ at the CP.
### 3.2 Uncompensated potential tilt
A residual static force along the trap axis, expressed by the coefficient
$\gamma^{\prime}$ in Eq. 5, can originate from stray charges, laser induced
charging of the trap [16], trap geometry imperfections or residual
ponderomotive forces along the trap axis. The behavior of the equilibrium
positions in the presence of an imperfectly compensated tilt, shown in Fig. 2,
reveals a discontinuity for the critical $\tilde{\gamma}$, leading to
diverging acceleration. The divergence of the acceleration impedes us to
perform the splitting process adiabatically for
$|\gamma|\lesssim\tilde{\gamma}$, i.e. the voltages can not be changed
sufficiently slow to suppress motional excitation. Thus, one might encounter
the situation that the tilt is sufficiently well compensated to allow for
splitting, but sufficiently low excitations cannot be obtained irrespectively
of the splitting time and other control parameters. For small tilt parameters,
$|\gamma|\ll\tilde{\gamma}$, we can employ the perturbative expressions Eqs.
16, 17 of the equilibrium positions to obtain
$\frac{\partial^{2}x_{R,L}}{\partial\alpha^{2}}=\frac{\partial^{2}x_{0}}{\partial\alpha^{2}}\pm\frac{1}{2}\frac{\partial^{2}d}{\partial\alpha^{2}}=\gamma\frac{52\cdot
2^{4/5}}{675\beta^{9/5}\kappa^{6/5}}\pm\frac{2}{25}\left(\frac{4}{\beta^{7}\kappa^{3}}\right)^{1/5}$
(31)
We can estimate the tilt parameter at which the acceleration of one of the
ions is twofold compared to the tilt-free case determined by Eq. 30 to be
about 67% of the critical tilt $\tilde{\gamma}$. Due to the divergence of the
acceleration at $\tilde{\gamma}$, we can expect the actual acceleration at
this tilt value to be substantially larger, we thus conclude that a residual
tilt $|\gamma|\ll\tilde{\gamma}$ is required to realize crystal splitting at
low motional excitation. A possible experimental scheme for this has been
demonstrated in [10]: The separation process is performed on a slow (second)
timescale under continuous Doppler cooling and detection. The ion positions
are extracted from the camera image, and a deviation of the center-of-mass
from the initial value is restored by automatic adjustment of the outer
electrode differential voltage $\Delta U_{O}$.
### 3.3 Anomalous heating at the critical point
Microstructured ion traps exhibit anomalous heating, i.e. the mean phonon
number increases due to thermalization with the electrodes at a timescale much
faster than predicted by the assumption that only Johnson-Nyquist noise is
present [17]. This process can be modeled as $\dot{\bar{n}}=\Gamma_{h}$, with
the heating rate $\Gamma_{h}(\omega)=S_{E}(\omega)e^{2}/4m\hbar\omega$ where
the spectral electric-field-noise density $S_{E}$ depends on the trap
frequency $\omega$. A polynomial decrease $S_{E}\propto\omega^{-a}$ is often
assumed, where experimentally determined values for the exponent $a$ range
from 0.5 to 2.5. Additionally, peaked features might arise in the noise
spectrum which are caused by technical sources. Moreover, the absolute values
of the heating rates strongly depend on the properties of the electrode
surfaces. Typical values at trap frequencies in the 1 MHz regime range from
0.1 to tens of phonons per millisecond. As the trap frequency is strongly
decreased around the CP, we can expect a significant amount of excess energy
after the splitting caused by anomalous heating, increasing for longer
splitting durations. We model this contribution by integrating over a time
dependent heating rate:
$\Delta\bar{n}_{th}=\int_{0}^{T}\Gamma_{h}\left(\omega(t)\right)dt.$ (32)
For the simulations that follow we will employ an experimentally determined
relation for trap A (Sec. 6) which is $\Gamma_{h}(\omega)\approx
6.3\cdot\left(\omega/2\pi\textrm{MHz}\right)^{-1.81}ms^{-1}$. This does not
depend on the geometry of trap A but on the properties of our trap apparatus.
In the case of imperfect control of the ion distance around the CP, Sec. 3.1,
or in the presence of an uncompensated tilt, Sec. 3.2, one will attempt to
reduce the motional excitation by splitting very slowly. This might however be
unsuccessful as anomalous heating will strongly contribute to the energy gain
at large splitting times. Experimental procedures for ensuring a sufficient
degree of control are therefore ultimately required.
## 4 Voltage ramps
In this section we explain our scheme for designing voltage ramps for the
splitting process. Our intention is to provide a scheme which can be applied
any given trap geometry. We do explicitly not rely on the precise knowledge of
the electrostatic trap potentials, but rather on quantities which can be
measured with reasonable effort. Furthermore, we describe how a single voltage
level can be used as a tuning parameter to achieve the optimum result. Our
scheme assumes that the tilt potential is perfectly compensated, $\gamma=0$.
We proceed as follows: We first describe how the segment voltages are supposed
to vary with the harmonicity parameter $\alpha$, where we simply fix voltage
levels on a small set of mesh points. We then show how this is used in
conjunction with a chosen distance-versus-time and available distance-
versus-$\alpha$ information to obtain time-domain voltage ramps which can be
employed in the experiment.
### 4.1 Static voltage sets
The calculation of suitable voltage ramps relies on the signs and on the
magnitude ordering of the geometry parameters. In Table 1 we list values for
several different microstructured traps. We assume that any reasonable
segmented trap geometry will exhibit similar characteristics. From the results
of Sec. 3, it is clear that we desire a large positive value of $\beta_{CP}$.
We assume that the voltages which can be applied to the segments are limited
by hardware constraints to the symmetric maximum/minimum values $\pm U_{lim}$.
To achieve the largest possible $\beta$ at the CP, we begin the splitting
protocol by ramping the $O$ segments to $+U_{lim}$, keep them at constant bias
during around the CP, and ramp them back to zero bias after the splitting.
The CP is defined by the condition $\alpha=0$, which is accomplished by
suitable voltages $U_{C,S}$. This leaves one degree of freedom, which can be
eliminated by maximizing $\beta_{CP}$. We solve Eq. 3 for $U_{C}$ :
$U_{C}=\frac{1}{\alpha_{C}}\left(\alpha-\alpha_{O}U_{O}-\alpha_{S}U_{S}\right).$
(33)
The largest possible $\beta_{CP}$ is then given by inserting this result into
Eq. 4 and setting $U_{O}^{(CP)}=+U_{lim},U_{S}^{(CP)}=-U_{lim}$:
$\max_{U_{C},U_{S}}\beta_{CP}=\left(\beta_{O}+\frac{\beta_{C}}{\alpha_{C}}\alpha_{S}-\beta_{S}-\frac{\beta_{C}}{\alpha_{C}}\alpha_{O}\right)U_{lim}$
(34)
Static splitting voltage sets are obtained by fixing the initial, CP and final
voltage configurations and interpolating between these. The procedure consists
of the following steps:
1. 1.
Determine the initial $\alpha_{i}>0$ from Eq. 3 using the initial voltages
$U_{C}^{(i)}<0$ V, $U_{S}^{(i)}=U_{O}^{(i)}=0$ V.
2. 2.
Choose the voltages at the CP such that the maximum $\beta_{CP}$ is attained,
by setting $U_{O}^{(CP)}=+U_{lim},U_{S}^{(CP)}=-U_{lim}$ and $U_{C}^{(CP)}$
from Eq. 33 for $\alpha=0$. 333If the geometry parameters are such that
$U_{C}^{(CP)}$ exceeds $\pm U_{lim}$, set $U_{C}^{(CP)}=-U_{lim}$ and obtain
$U_{S}^{(CP)}$ solving Eq. 3 for $U_{S}$ rather than $U_{C}$. 444If the
magnitude of $U_{S}^{(CP)}$ is chosen smaller than $U_{lim}$, this leads to
smaller values of $\beta_{CP}$ and a larger ion separation at the CP. This
offers the possibility for well-controlled studies of the dependence of the
splitting process on the quartic confinement at the CP..
3. 3.
Determine the desired final voltages. We choose $U_{C}^{(f)}=0$ V,
$U_{S}^{(f)}=U_{S}^{(CP)}=-U_{lim}$ and $U_{O}^{(f)}=0$ V. This choice is
convenient when $U_{C}^{(i)}\approx-U_{lim}$ and ensures that the ions are
finally kept close to the respective centers of the $S$ segments with a trap
frequency similar to the initial one. Obtain $\alpha_{f}$ from Eq. 3.
4. 4.
For approaching the CP, $\alpha_{i}\geq\alpha>0$, set
$U_{S}(\alpha)=\left(1-\frac{\alpha}{\alpha_{i}}\right)U_{S}^{(CP)}$ (35)
and
$U_{O}(\alpha)=\cases{2\left(1-\frac{\alpha}{\alpha_{i}}\right)U_{lim}\qquad\alpha>\frac{\alpha_{i}}{2}\\\
U_{lim}\qquad\ \ \alpha\leq\frac{\alpha_{i}}{2}\\\ }$ (36)
and obtain $U_{C}(\alpha)$ from Eq. 33.
5. 5.
Beyond the CP, $0\geq\alpha\geq\alpha_{f}$, set
$U_{S}(\alpha)=-U_{lim}$ (37)
and
$U_{O}(\alpha)=\cases{U_{lim}\qquad\alpha>\frac{\alpha_{f}}{2}\\\
2\left(1-\frac{\alpha}{\alpha_{f}}\right)U_{lim}\ \ \ \
\alpha\leq\frac{\alpha_{f}}{2}\\\ }$ (38)
and obtain $U_{C}(\alpha)$ from Eq. 33.
Figure 5: Voltage ramp transfer to the time domain: A predefined time-to-
distance function $d(t)$ shown in panel a) is used in conjunction with
$\alpha$-to-distance information $\alpha(d)$ shown in b) to determine the
time-dependent electrode voltages $U_{n}(t)$ using the static voltage sets
$U_{n}(\alpha)$ from panel c). The resulting ramps $U_{n}(t)$ are shown in d).
The dashed curves are corresponding to the case when the voltage ramps are
calculated according to the presented method, but realistic trap potentials
from simulations are used to determine $d_{f}$ and $d(\alpha)$. The dashed
arrows exemplify how a specific value $U_{C}$ is obtained.
### 4.2 Time domain ramps
We now show how to design suitable time-domain voltage ramps $U_{n}(t)$ that
will assure well-controlled splitting. It has been shown in Sec. 3.1 that a
small value of the acceleration at the CP, $\ddot{d}_{CP}$, is required for
achieving a low energy transfer. This in turn is guaranteed by well-controlled
variation of of the distance $d(t)$ throughout the splitting process. As
$d(\alpha)$ is monotonically decreasing with $\alpha$, it can be inverted to
obtain $\alpha(d)$ which is used to compute the final voltage ramp as
$U_{n}(\alpha(d(t)))$ (see Fig. 5.).
Possible choices for $d(t)$ are a sine-squared ramp
$d(t)=d_{i}+\left(d_{f}-d_{i}\right)\sin^{2}\left(\frac{\pi t}{2T}\right)$
(39)
or a polynomial ramp
$d(t)=d_{i}+\left(d_{f}-d_{i}\right)\left(-10\frac{t^{3}}{T^{3}}+15\frac{t^{4}}{T^{4}}-6\frac{t^{5}}{T^{5}}\right)$
(40)
Both ramps fulfill $d(0)=d_{i},d(T)=d_{f},\dot{d}(0)=\dot{d}(T)=0$. The
polynomial ramp, used in the following, additionally fulfills
$\ddot{d}(0)=\ddot{d}(T)=0$, while the second derivative of the sine-squared
ramp displays discontinuities. However, these features presumably play no role
in experiments, as the voltage ramps are generally subject to discretization
and filtering. Different methods can be employed for the determination of
$d(\alpha)$:
* •
The equilibrium distance can be computed by employing realistic trap
potentials from simulation data, using the voltage configuration pertaining to
a given $\alpha$ as determined by the static voltage sets $U_{n}(\alpha)$.
This method requires the simulated potentials to match the actual trap
potential with great precision.
* •
The equilibrium distance can be computed using values from calibration
measurements for the coefficients $\alpha_{n},\beta_{n}$. This circumvents the
need for simulations and accounts for parameter drifts. It yields only valid
values for distances which are small compared to the electrode width, however
we will show in Sec. 5 that this procedure yields useful voltage ramps.
* •
Ion distances can be measured by imaging the ion crystal on a camera, while
voltages configurations for decreasing $\alpha$ values are applied. This is
the most direct method, and it benefits from the availability of a precise
gauge of imaging magnification from measurements of the trap frequency.
## 5 Simulation results
In order to analyze the sensitivity of the splitting process and the
performance of our ramp design protocol, we numerically solve the classical
equations of motion. For the time- and energy-scales and potential shapes
under consideration, we expect quantum effects to play no significant role.
For the case of single-ion shuttling, the occurrence of quantum effects is
thoroughly discussed in Ref. [18].
We perform the simulations using either the Taylor approximation of the
potentials or the realistic potentials from electrostatic simulations [14] for
trap A, which is similar to that described in Ref. [19]. The voltage ramps
$U_{i}(t)$ are used in conjunction with the potentials to yield the equations
of motion for the ion positions $x_{1}<x_{2}$. Employing the Taylor
approximation potential Eq. 1, these read
$-m\ddot{x}_{1,2}=4\beta(t)x_{1,2}^{3}+2\alpha(t)x_{1,2}+\gamma\pm\frac{\kappa}{(x_{2}-x_{1})^{2}},$
(41)
where the coefficients are given by using the voltage ramps in Eqs. 3, 4,5.
For realistic trap potentials, we obtain
$-m\ddot{x}_{1,2}=\sum_{n=C,S,O}U_{n}(t)\left.\frac{d\phi_{n}}{dx}\right|_{x_{1,2}}\pm\frac{\kappa}{(x_{2}-x_{1})^{2}}\\\
$ (42)
The possibility to perform the simulations with approximate and realistic
potentials serves the purpose of verifying the performance of the voltage
ramps. These are determined purely by trap properties around the CP, which are
conveniently accessible by measurements. More precisely, the time-domain
voltage ramps are based on a $d(\alpha)$ dependency given by the Taylor
approximation potential according to Fig. 5, while the resulting energy
transfer pertaining to these ramps can be obtained from simulations using
realistic potentials.
Note that a nonzero tilt can be present in the simulations based on the
realistic potentials by summing separately over electrodes $O_{L}$ and $O_{R}$
and adding the differential voltage $\pm\Delta U_{O}$ given by
$\gamma/\gamma_{O}$ accordingly. The calculations presented here employ the
mass of 40Ca+ ions which we use in our experiments, and all simulations were
performed for a limiting voltage range $U_{lim}=10$ V.
Eqs. 41 or 42 are solved numerically using the NDSolve package from
Mathematica, with the ions starting at rest. The final oscillation of each ion
around its equilibrium position is analyzed and yields the energy transfer
expressed as the mean phonon number $\bar{n}=\Delta E/\hbar\omega_{f}$. We
distinguish several regimes of laser-ion interaction: i) If the vibrational
excitation becomes so large that the average Doppler shift per oscillation
cycle exceeds the natural linewidth of a cycling transition, ion detection by
counting resonance fluorescence photons will be impaired. ii) Measurement of
the energy transfer i.e. by probing on a stimulated Raman transition [3]
typically requires mean phonon numbers below about 300. iii) The Lamb-Dicke
regime of laser-ion interaction, where coherent dynamics on resolved sidebands
can be driven [20] is typically attained below about 10 phonons. The borders
between these regimes depend on the trap frequency, ion mass and the specific
atomic transitions to be driven, thus the regimes are indicated as broad gray
bands in Fig. 6. Note that if final excitations in the measurable regime are
obtained, an electrical counter kick can be applied for bringing the
oscillation to rest [3].
### 5.1 Dependence on splitting time
We first analyze the dependence of the energy transfer on the duration of the
splitting process $T$, the result is shown in Fig. 6. The calculation is
carried out for the ideal case of perfectly compensated potential tilt. We see
that the final excitation becomes sufficiently low to remain in the Lamb-Dicke
regime for typical laser-ion interaction settings at times larger than about
40 $\mu s$, which clearly outperforms the naïve approach of voltage
interpolation from Sec. 3.1.
We also take into account increased anomalous heating around the CP by
employing the averaged heating rate according to Eq. 32. We see that for our
specific heating rates, the limit of about one phonon per ion can not be
overcome, but as the anomalous heating contribution is scaling as $1/T$, the
splitting result becomes rather insensitive with respect to the precise choice
of the $T$ beyond $T=$ 50 $\mu s$.
The simulation results verify our approach of calculating the voltage ramps
using the Taylor approximated potentials. One recognizes that the resulting
energy transfer in this case is larger by a factor of about two throughout the
entire range of splitting durations. As can be seen from Fig. 5, this is due
to the fact that the Taylor expansion leads to an incorrect voltage set
pertaining to the CP, which in turn leads to uncontrolled acceleration as
explained in Sec. 3.1. The discrepancy becomes irrelevant for splitting times
larger than $T=$ 60 $\mu$s. At around $60$ to $70~{}\mu$s the oscillatory
excitation becomes smaller than $\bar{n}=0.1$, corresponding to the limit we
can currently resolve in our experiment. The slight inaccuracy for low phonon
numbers is due to numerical artifacts. Even lower energy transfers at shorter
$T$ could possibly be achieved by ramp engineering, i.e. by the application of
shortcut-to-adiabaticity approaches [18, 21].
Figure 6: Energy transfer versus splitting time: Oscillatory (red) and thermal
excitation (blue), and the sum of both (black) versus the splitting duration
$T$. The solid lines correspond to the calculation using the Taylor
approximation, the dashed lines correspond to the full potential calculation,
see text. Grey bands seperate different regimes of laser-ion interaction, see
text. The thermal excitation was deduced from experimental heating rate data
according to Sec. 3.3. The inset shows the trap frequency (black) and the
corresponding heating rate (red) as a function of normalized time during the
splitting process.
### 5.2 Sensitivity analysis
Figure 7: Mean coherent excitation as a function of the offset voltage at the
center segment at the CP (a) and the tilt force $\gamma$ (b). The tilt voltage
$+\Delta U_{O}$ is applied to the right outer segment and $-\Delta U_{O}$ is
applied to the left outer segment. The mean phonon number for the right ion is
depicted by dashed lines and by solid lines for the left ion. The curves
correspond to different splitting times: $T=60\mu s$ (green), $T=40\mu s$
(black), $T=20\mu s$ (red). The critical tilt is at $\tilde{\gamma}=3~{}$V/m.
Two crucial parameters for the splitting operation are the offset voltage at
the CP $\Delta U_{C}^{(CP)}$ and the potential tilt $\gamma$. Small variations
of these parameters lead to strong coherent excitations as shown in Fig. 7.
The CP voltage offset $\Delta U_{C}^{(CP)}$ serves both for modeling and
compensation of inaccuracies of the trap potentials, leading to a wrongly
determined CP voltage configuration and therefore to increased acceleration.
It is implemented into the simulations by just adding it to $U_{C}^{(CP)}$ as
determined by Eq. 33 in the calculation of the static voltage sets. We see
that even for sufficiently slow splitting, the Lamb-Dicke regime can only be
attained if this voltage offset, and therefore the CP voltages in general, are
correct within a window of about 20 mV, on the other hand it becomes clear
that this voltage serves as convenient fine tuning parameter. The minimum
excitation does not occur at $\Delta U_{C}^{(CP)}=0$, but is slightly shifted
to positive values.
This can be understood by considering that $|\dot{\alpha}|_{CP}$ is increased
for any $\Delta U_{C}^{(CP)}\neq 0$, but $\ddot{\alpha}_{CP}$ is decreased for
$\Delta U_{C}^{(CP)}>0$. With $\partial d/\partial\alpha$, the second term in
Eq. 29 leads to a reduced total acceleration for small positive $\Delta
U_{C}^{(CP)}$. Larger values again lead to increased acceleration because of a
smaller $\beta_{CP}$ value. All other calculations in this work are done using
$\Delta U_{C}^{(CP)}=0$.
For the case of an uncompensated tilt $\gamma^{\prime}$, we observe an even
stronger dependence of the energy transfer. Fine tuning of the voltage
difference on the outer segments $\Delta U_{O}$ on the sub-mV level is
required to reach the single phonon regime. Moreover, we observe that moderate
uncompensated potential tilts reduce the energy transfer to one of the ions,
as its CP acceleration is reduced by a more smooth $x(\alpha)$ dependence.
This might be of interest for specific applications where only the energy
transfer to one of the ions is of importance.
### 5.3 Dependence on the limiting voltage
Figure 8: Dependence on the voltage limit: Oscillatory excitation as a
function of the maximum voltage on the outer segments with all other limiting
voltages remaining unchanged. The curves correspond to different splitting
times: $T=40\mu s$ (green), $T=30\mu s$ (black), $T=20\mu s$ (red).
Finally we study the dependence of the energy transfer on the limiting voltage
$U_{lim}$. We find that by increasing the voltage limit, beyond $U_{lim}=10$ V
used so far, we can obtain lower coherent excitations as shown in Fig. 8. For
this simulation, only the maximum voltage on the outer segments (max $U_{O}$)
is increased and all other limits remain unchanged. We infer that by
increasing the voltage limit on these electrodes up to about $50$ V, one can
reduce the mean phonon number by a factor of $\approx 8$ for $T=60\mu$s. For
lower splitting durations the enhancing factor becomes slightly smaller.
## 6 Trap geometry optimization
We have been showing in Sec. 3 that the outcome of a crystal splitting
operation is strongly determined by magnitude of the quartic confinement
coefficient at the CP $\beta_{CP}$ from Eq. 34. We thus investigate the effect
of the trap geometry on the coefficients $\alpha_{n},\beta_{n},\gamma_{n}$
from Eqs. 7. We calculate the realistic potentials from electrostatic
simulations [14] to infer the geometry parameters according to Eq. 7. In
particular, six different traps designs were studied, four of which are three-
dimensional and two are surface-electrode traps. The results are shown in Tab.
1. The calculations are carried out for a generic simplified geometry shown in
Fig. 9 d), which is essentially determined by the segment width $w$, the slit
height $h$ and the spacer thickness $d$ for the three-dimensional traps. Trap
A ,B[19] and C[13] are similar segmented micro-structured ion traps . Trap B
is subdivided into a loading region of larger geometry, B (wide), and a narrow
processing region, B (narrow). The data for trap C pertains to a wedge segment
of $w=100\mu$m surrounded by larger segments. Trap D is a segmented planar ion
trap [22], the calculations are performed at a distance of $100~{}\mu$m
between the ion and the surface. Trap D2 is a planar ion trap featuring a
segmented ground plane, otherwise identical to trap D. Trap A was used for all
simulations in section 5.
Parameter | Unit | A | B (wide) | B (narrow) | C | D | D2
---|---|---|---|---|---|---|---
$w$ | $\mu$m | 200 | 250 | 125 | 100 | 200 | 200
$h$ | $\mu$m | 400 | 500 | 250 | 200 | - | -
$d$ | $\mu$m | 250 | 125 | 125 | 250 | - | -
$\alpha_{C}$ | 106 m-2 | -3.0 | -2.5 | -9.1 | -6.4 | -1.4 | -12.0
$\beta_{C}$ | 1013 m-4 | 2.7 | 1.7 | 19.9 | 14.4 | 1.5 | -6.5
$\alpha_{S}$ | 106 m-2 | 1.7 | 1.7 | 6.2 | 4.7 | 0.9 | 10.7
$\beta_{S}$ | 1013 m-4 | -3.0 | -1.9 | -22.1 | -14.7 | -1.7 | 5.6
$\gamma_{S}$ | 102 m-1 | 11.0 | 9.3 | 19.2 | 21.6 | 4.1 | 17.8
$\alpha_{O}$ | 106 m-2 | 1.0 | 0.6 | 2.3 | 1.6 | 0.4 | 0.9
$\beta_{O}$ | 1013 m-4 | 0.2 | 0.2 | 2.0 | 1.2 | 0.1 | 0.8
$\gamma_{O}$ | 102 m-1 | 3.2 | 2.2 | 4.3 | 3.2 | 1.2 | 2.2
$\omega_{CP}/2\pi$ | MHz | 0.18 | 0.14 | 0.29 | 0.26 | 0.14 | 0.11
Table 1: Comparison of trap geometry parameters for different linear segmented
Paul traps. Letters A to D denote different traps which are operated at
various institutes, see text. Note that $\gamma_{C}=0$ by definition. The trap
frequency at the critical point is specified for $U_{lim}$=10V and 40Ca+ ions.
For trap A and B (wide) we calculate similar parameters, however the minimum
trap frequency during the splitting is larger for trap A. Trap B (narrow)
exhibits the highest minimum trap frequency of the six geometries as the total
dimensions of this section of the trap are rather small. The wedge segment in
trap C helps to increase the minimum trap frequency but choosing an overall
smaller size seems to be a more favorable solution. The planar trap D has a
similar minimum trap frequency as trap B (wide) and is also suitable for
splitting ion crystals. The segmentation of the ground plane of this trap (D2)
offers an enhanced $\alpha_{C}$, i.e. a large trap frequency. The calculations
show however that for a segmentation of the center electrode, the potentials
become more anharmonic and the Taylor approximation Eq. 1 breaks down. Thus,
the sign and magnitude ordering of the coefficients might be different from
the other geometries, therefore the geometry parameters and the ion height
above the surface should be carefully chosen to allow for successful splitting
operations.
Figure 9: Calculated geometry parameters $\alpha_{n},\beta_{n},\gamma_{n}$ and
the maximum $\beta_{CP}$ at the critical point for a linear segmented Paul
trap with dimensions $h=400~{}\mu$m, $d=250~{}\mu$m as a function of the
segment width $w$. The color code is as above: blue - C, red - S, green - O.
The limiting voltage for the electrodes is $U_{lim}=10V$.
For trap A we calculated the geometry parameters for varying segment width
$w$, the result is shown in Fig. 9. We analyze the dependance of all potential
coefficients on $w$ with parameters $h$ and $d$ held constant. For splitting
operations the optimum segment width would be at about $w=125\mu$m, while the
actual segment width of the trap is $w=200\mu$m. We could therefore obtain a
roughly twofold increase of $\beta_{CP}$ bought at the expense of a reduced
trap frequency for ion storage due to the reduced $\alpha_{C}$ coefficient.
Finally, we investigate the dependence of $\beta_{CP}$ on the overall trap
geometry size. We therefore pick trap parameters $h$ and $d$ from the range of
typical values and determine the optimum segment width $w$ for these. Defining
the effective trap size $d_{eff}=\left(w^{2}+h^{2}+d^{2}\right)^{1/2}$, we
find a scaling behavior of $\beta_{CP}\approx 2.2\cdot 10^{24}V\cdot
d_{eff}^{-4}$, i.e. the best attainable value for the quartic confinement
coefficient scales as the inverse fourth power with the effective trap size,
which is the similar to the presumed distance scaling law for anomalous
heating [17]. We conclude that for a trap architecture aiming at shuttling-
based scalable quantum information, the considerations presented here should
be incorporated into the design process to facilitate crystal splitting
operations.
## 7 Conclusion
We have pointed out the pitfalls for ion crystal splitting: Uncontrolled
separation and uncompensated background fields lead to enhanced acceleration
of the ions when the single well potential is transformed into a double well,
which would require splitting times in the millisecond range to keep the
motional excitation near the single phonon level. This in turn leads to strong
anomalous heating due to the reduced confinement during the splitting process.
We presented a framework to design voltage ramps which allow for coping with
these problems. The scheme does only rely on measured calibration data which
is obtained for the initial situation, where the ions are tightly confined in
a single potential well. We carried out simulations, which elucidate the
energy transfer mechanisms, and verify the performance of our scheme for the
voltage ramp calculation. We showed that excitations near the single phonon
level can be obtained for the specific trap apparatus we use. Furthermore, we
analyzed the suitability of different trap geometries for ion crystal
splitting by means of electrostatic simulations. We concluded that crystal
splitting becomes easier for smaller trap structures, and that dedicated
optimization of the geometry can be helpful. In future work, we envisage to
analyze how crystal splitting can be performed on faster timescales by using
shortcut-to-adiabaticity approaches, with an emphasis on robustness against
experimental imperfections.
## Acknowledgments
We thank René Gerritsma and Georg Jacob for proofreading the manuscript. This
research was funded by the Office of the Director of National Intelligence
(ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the
Army Research Office grant W911NF-10-1-0284. All statements of fact, opinion
or conclusions contained herein are those of the authors and should not be
construed as representing the official views or policies of IARPA, the ODNI,
or the US Government. CTS acknowledges support from the German Federal
Ministry for Education and Research (BMBF) via the Alexander von Humboldt
Foundation.
## References
* [1] Rainer Blatt and David Wineland. Entangled states of trapped atomic ions. Nature, 453(7198):1008–1015, 2008.
* [2] D. Kielpinski, C. Monroe, and D.J. Wineland. Architecture for a large-scale ion-trap quantum computer. Nature, 417:709, 2002.
* [3] A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger. Controlling fast transport of cold trapped ions. Phys. Rev. Lett., 109:080501, 2012.
* [4] R. Bowler, J. Gaebler, Y. Lin, T. R. Tan, D. Hanneke, J. D. Jost, J. P. Home, D. Leibfried, and D. J. Wineland. Coherent diabatic ion transport and separation in a multi-zone trap array. Phys. Rev. Lett., 109:080502, 2012.
* [5] M.A. Rowe, A. Ben-Kish, B. DeMarco, D. Leibfried, V. Meyer, J. Beall, J. Britton, J. Hughes, W.M. Itano, B. Jelenkovic, C. Langer, T. Rosenband, and D.J. Wineland. Transport of quantum states and seperation of ions in a dual rf ion trap. Quantum Inf. and Comput., 2:257, 2002.
* [6] M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri, and D. J. Wineland. Deterministic quantum teleportation of atomic qubits. Nature, 429:737, 2004.
* [7] R Reichle, D Leibfried, E Knill, J Britton, RB Blakestad, JD Jost, C Langer, R Ozeri, S Seidelin, and DJ Wineland. Experimental purification of two-atom entanglement. Nature, 443(7113):838–841, 2006.
* [8] J. P. Home and A. M. Steane. Electrode configurations for fast separation of trapped ions. Quantum Inf. and Comput., 6:289–325, 2006.
* [9] A. H. Nizamani and W. K. Hensinger. Optimum electrode configurations for fast ion separation in microfabricated surface ion traps. Appl. Phys. B, 106:337–338, 2012.
* [10] J. Eble, S. Ulm, P. Zahariev, F. Schmidt-Kaler, and K. Singer. Feedback-optimized operations with linear ion crystals. Journal of the Optical Society of America B, 27, 2010.
* [11] M. T. Baig, M. Johanning, A. Wiese, S. Heidbrink, M. Ziolkowski, and C. Wunderlich. A scalable, fast and multichannel arbitrary waveform generator. arxiv:, 1307.5672, 2013.
* [12] R. Bowler, U. Warring, J. W. Britton, B. C. Sawyer, and J. Amini. Arbitrary waveform generator for quantum information processing with trapped ions. Rev. Sci. Instrum., 84:033108, 2013.
* [13] R. B. Blakestad, C. Ospelkaus, J. H. VanDevender, M. J. Wesenberg, J. Biercuk, D. Leibfried, and D. Wineland. Near-ground-state transport of trapped-ion qubits through a multidimensional array. Phys. Rev. A, 84:032314, 2011.
* [14] K. Singer, U. Poschinger, M. Murphy, P. Ivanov, F. Ziesel, T. Calarco, and F. Schmidt-Kaler. Colloquium : Trapped ions as quantum bits: Essential numerical tools. Rev. Mod. Phys., 82:2609–2632, Sep 2010.
* [15] S Ulm, J Roßnagel, G Jacob, C Degünther, ST Dawkins, UG Poschinger, R Nigmatullin, A Retzker, MB Plenio, F Schmidt-Kaler, et al. Observation of the kibble–zurek scaling law for defect formation in ion crystals. Nature communications, 4, 2013.
* [16] M. Harlander, M. Brownnutt, W. Hänsel, and R. Blatt. Trapped-ion probing of light-induced charging effects on dielectrics. New J. Phys., 12:093035, 2010.
* [17] M. Brownnutt, M. Kumph, P. Rabl, and R. Blatt. Ion-trap measurements of electric-field noise near surfaces. to be published.
* [18] HA Fürst, MH Goerz, UG Poschinger, M Murphy, S Montangero, T Calarco, F Schmidt-Kaler, K Singer, and CP Koch. Controlling the transport of an ion: Classical and quantum mechanical solutions. arXiv preprint arXiv:1312.4156, 2013.
* [19] Stephan Schulz, Ulrich Poschinger, Frank Ziesel, and Ferdinand Schmidt-Kaler. Sideband cooling and coherent dynamics in a microchip multi-segmented ion trap. New J. Phys., 10:045007, 2008.
* [20] Dietrich Leibfried, Brian DeMarco, Volker Meyer, David Lucas, Murray Barrett, Joe Britton, WM Itano, B Jelenković, Chris Langer, Till Rosenband, et al. Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate. Nature, 422(6930):412–415, 2003.
* [21] M Palmero, E Torrontegui, David Guéry-Odelin, and JG Muga. Fast transport of two ions in an anharmonic trap. Physical Review A, 88(5):053423, 2013.
* [22] S. Narayanan, N. Daniilidis, S. A. Möller, R. Clark, F. Ziesel, K. Singer, F. Schmidt-Kaler, and H. Häffner. Electric field compensation and sensing with a single ion in a planar trap. J. Appl. Phys., 110:114909, 2011.
|
arxiv-papers
| 2014-03-01T15:41:10 |
2024-09-04T02:49:59.134386
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Kaufmann, T. Ruster, C. T. Schmiegelow, F. Schmidt-Kaler, U. G.\n Poschinger",
"submitter": "Henning Kaufmann",
"url": "https://arxiv.org/abs/1403.0097"
}
|
1403.0155
|
# Chemistry and Radiative Transfer of Water in Cold, Dense Clouds
Eric Keto1, Jonathan Rawlings2, and Paola Caselli3
1Harvard-Smithsonian Center for Astrophysics, 160 Garden St, Cambridge, MA
02420, USA
2University College London, London, UK
3School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK
E-mail: [email protected] (EK); [email protected] (JR)
[email protected] (PC)
(February 19, 2014)
###### Abstract
The Herschel Space Observatory’s recent detections of water vapor in the cold,
dense cloud L1544 allow a direct comparison between observations and chemical
models for oxygen species in conditions just before star formation. We explain
a chemical model for gas phase water, simplified for the limited number of
reactions or processes that are active in extreme cold ($<$ 15 K). In this
model, water is removed from the gas phase by freezing onto grains and by
photodissociation. Water is formed as ice on the surface of dust grains from O
and OH and released into the gas phase by photodesorption. The reactions are
fast enough with respect to the slow dynamical evolution of L1544 that the gas
phase water is in equilibrium for the local conditions thoughout the cloud. We
explain the paradoxical radiative transfer of the H2O ($1_{10}-1_{01}$) line.
Despite discouragingly high optical depth caused by the large Einstein A
coefficient, the subcritical excitation in the cold, rarefied H2 causes the
line brightness to scale linearly with column density. Thus the water line can
provide information on the chemical and dynamical processes in the darkest
region in the center of a cold, dense cloud. The inverse P-Cygni profile of
the observed water line generally indicates a contracting cloud. This profile
is reproduced with a dynamical model of slow contraction from unstable quasi-
static hydrodynamic equilibrium (an unstable Bonnor-Ebert sphere).
###### keywords:
Interstellar Medium (ISM), Nebulae: ISM Interstellar Medium (ISM), Nebulae:
abundances; Interstellar Medium (ISM), Nebulae, ISM: individual, L1544;
Interstellar Medium (ISM), Nebulae: molecules; Physical Data and Processes,
astrochemistry; Physical Data and Processes, radiative transfer;
## 1 Introduction
Observations of water vapor in the interstellar medium (ISM) by the Infrared
Space Observatory (van Dishoeck et al., 1999) and the Submillimeter Wave
Astronomy Satellite (SWAS) (Bergin et al., 2000) show general agreement with
chemical models for warm ($>300$ K) conditions in the ISM (Melnick et al.,
2000; Neufeld et al., 2000). However, in cold conditions, most of the water is
frozen onto dust grains (Viti et al., 2001; van Dishoeck, Herbst & Neufeld,
2013), and the production of water occurs mainly on the grain surfaces. In
order to test chemical models that include grain-surface chemistry we used the
Heterodyne Instrument for the Far-Infrared (HIFI) (de Graauw et al., 2010) on
the Herschel Space Observatory to observe the H2O ($1_{10}-1_{01}$) line in
the cold, dense cloud L1544 (Caselli et al., 2010, 2012). The first of these
two Herschel observations was made with the wide-band spectrometer (WBS) and
detected water vapor in absorption against the weak continuum radiation of
dust in the cloud. Follow-up observations with higher spectral resolution and
sensitivity, made with the high resolution spectrometer (HRS), confirmed the
absorption and detected a blue-shifted emission line that was predicted by
theoretical modeling (Caselli et al., 2010), but too narrow to be seen by the
WBS in the first observation.
With the better constraints provided by the second observation, we improved
the chemical and radiative transfer modeling in our previous papers. We
modified the radiative transfer code MOLLIE to calculate the line emission in
the approximation that the molecule is sub-critically excited. This assumes
that the collision rate is so slow that every excitation leads immediately to
a radiative de-excitation and the production of one photon which escapes the
cloud, possibly after many absorptions and re-emissions, before another
excitation. The emission behaves as if the line were optically thin with the
line brightness proportional to the column density. This approximation can be
correct even at very high optical depth as long as the excitation rate is slow
enough, C $<$ A/$\tau$, where C is the collision rate, A is the spontaneous
emission rate and $\tau$ the optical depth (Linke et al., 1977). Caselli et
al. (2012) presented the observations and the results of this modeling.
In this paper, we discuss in detail the theory behind the modeling. A
comparison of the spectral line observation with theory requires three models.
First, we require a hydrodynamical model to describe the density, velocity,
and temperature across the cloud. We use a model of slow contraction in quasi-
static unstable equilibrium that we developed in our previous research (Keto &
Field, 2005; Keto & Caselli, 2010). Second, we require a chemical model to
predict the molecular abundance across the varying conditions in the cloud.
Following the philosophy for simplified chemical networks in Keto & Caselli
(2008) or Bethell & Bergin (2009), we extract from a general chemical model
for photo-dissociation regions (Hollenbach et al., 2009) a subset of reactions
expected to be active in cold conditions, principally grain-surface reactions
as well as freeze-out and photodissociation. Third, we require a radiative
transfer model to generate a simulated molecular line. We modify our non-LTE
radiative transfer code MOLLIE to use the escape probability approximation.
This allows better control of the solution in extreme optical depth.
The three models are described in more detail in three sections below. The
relevant equations are included in the appendices.
## 2 The three models
### 2.1 The cold, dense clouds
Given their importance as the nurseries of star formation, the small ($<0.5$
pc), cold ($<15$ K), dense ($n>10^{3}$ cm-3) clouds in low-mass star ($<2$ M⊙)
forming regions such as Taurus are widely studied (Bergin & Tafalla, 2007; di
Francesco et al., 2007). Observations show a unique simplicity. They contain
no internal sources of heat, stars or protostars. Their internal turbulence is
subsonic, barely broadening their molecular line widths above thermal (Myers &
Benson, 1983). With most of their internal energy in simple thermal energy,
and the weak turbulence just a perturbation (Keto et al., 2006; Broderick &
Keto, 2010), the observed density structure approximates the solution of the
Lane-Emden equation for hydrostatic equilibrium (Lada et al., 2003; Kandori et
al., 2005). Correspondingly, most are nearly spherical with an average aspect
ratio of about 1.5 (Jijina, Myers & Adams, 1999). They are heated from the
outside both by cosmic rays and by the UV background of starlight and are
cooled from the inside by long wavelength molecular line and dust continuum
radiation (Evans et al., 2001). Because of their simplicity, we understand the
structure and dynamics of these small, cold, dense clouds better than any
other molecular clouds in the interstellar medium. They are therefore uniquely
useful as a laboratory for testing hypotheses of more complex phenomena such
as the chemistry of molecular gas.
### 2.2 Structure and dynamics
Our physical model for cold, dense clouds is computed with a spherical
Lagrangian hydrodynamic code with the gas temperature set by radiative
equilibrium between heating by external starlight and cosmic rays and cooling
by molecular line and dust radiation. The theory is discussed in Keto & Field
(2005) and Keto & Caselli (2008).
In our previous research (Keto & Caselli, 2010), we generated a dynamical
model for the particular case of L1544 by comparing observations and snapshots
in time out of a theoretical model for the contraction toward star formation.
We began the hydrodynamic evolution with a 10 M⊙ Bonnor-Ebert (BE) sphere with
a central density of $10^{4}$ cm-3 in unstable dynamical equilibrium and in
radiative equilibrium with an external UV field of one Habing flux. In the
early stages of contraction, the cloud evolves most rapidly in the center. As
long as the velocities remain subsonic, the evolving density profile closely
follows a sequence of spherical equilibria or BE spheres with increasing
central densities. We compared modeled CO and N2H+ spectra during the
contraction against those observed in L1544 and determined that the stage of
contraction that best matches the data has a central density of $1\times
10^{7}$ cm-3and a maximum inward velocity just about the sound speed (Keto &
Caselli, 2010). Figure 1 shows the density and velocity at this time along
with the H2O abundance and temperature.
In the present investigation we modify our numerical hydrodynamic code to
include cooling by atomic oxygen. This improves the accuracy of the calculated
gas temperature in the photodissociation region outside the molecular cloud.
The equations governing the cooling by the fine structure lines of atomic
oxygen are presented in the appendix.
### 2.3 Chemistry of H2O in cold conditions
The cold conditions in L1544 allow us to simplify the chemical model for gas
phase water. We include the four oxygen-bearing species most abundant in cold,
dark clouds, O, OH, H2O gas, and H2O ice. Even though all three gas phase
molecules may freeze onto the grains, we consider only one species of ice
because the formation of water from OH and the formation of OH from O are
rapid enough on the grain surface that most of the ice is in the form of H2O.
To provide a back reaction for the freeze-out of atomic oxygen and preserve
detailed balance, we arbitrarily assign a desorption rate for atomic O equal
to that of H2O even though the production of atomic O from H2O ice is not
indicated. Our simplified model is shown in figure 2. The resulting
abundances, calculated as equilibria between creation and destruction, are
shown in figures 3 and 4. Figure 3 shows the abundances near the
photodissociation region (PDR) boundary as a function of the visual
extinction, $A_{V}$. Figure 4 shows the abundances against the log of the
radius to emphasize the center.
Gas phase water is created by UV photodesorption of water ice which also
creates gas phase OH in a ratio H2O/OH = 2 (Hollenbach et al., 2009). In the
outer part of the cloud, the UV radiation derives from the background field of
external starlight. The inward attenuation of the UV flux is modeled from the
visual extinction as $\exp{(-1.8{\rm A_{V}})}$. In the interior where all the
external UV radiation has been attenuated, the only UV radiation is generated
by cosmic ray strikes on H2. In our previous paper (Caselli et al., 2012), we
set this secondary UV radiation to $1\times 10^{-3}$ times the Habing flux
(G0=1) (Hollenbach et al., 2009). In our current model, we use a lower level,
$1\times 10^{-4}$, that is more consistent with estimated rates (Shen et al.,
2004). The difference in abundance for the two rates is shown in figure 4.
H2O and OH are removed from the gas phase by UV photodissociation and by
freezing onto dust grains. To preserve detailed balance with the
photodissociation we include the back reactions, the gas phase production of
H2O, O + H2 $\rightarrow$ OH and OH + H2 $\rightarrow$ H2O, even though these
are not expected to be important in cold gas. Removal of gas phase water by
freeze-out is important in the interior where the higher gas density increases
the dust-gas collision rate, and hence the freeze-out rate.
We assume that the gas-phase ion-neutral reactions that lead to the production
of water are less important at cold temperatures ($<15$ K) than the reactions
that produce water on the surfaces of ice-coated dust grains. Thus, we do not
include gas-phase ion-neutral reactions in the model. This is valid if the
oxygen is quickly removed from the gas-phase by freeze-out and efficiently
converted into water ice on the grain surface.
By leaving out CO, we avoid coupling in the carbon chemistry. Although we
already have a simple model for the carbon chemistry (Keto & Caselli, 2008,
2010), we prefer to keep our oxygen model as simple as possible. This could
create an error of a factor of a few in the abundance of the oxygen species.
Carbon is one-third as abundant as oxygen, and in certain conditions CO is the
dominant carbon molecule. Therefore as much as one-third of the oxygen could
potentially be bound in CO. Ignoring O2 is less of a problem. Created
primarily by the reaction of OH with atomic oxygen, O2 tends to closely follow
the abundance of OH. Since the amount of oxygen in OH should be 1% or less
(figure 3), the abundance of O2 does not affect the abundances of the other
oxygen species, O, OH, and H2O.
Figures 3 and 4 compare the abundances from our simplified network with those
from the more complex network of Hollenbach et al. (2009) (courtesy of E.
Bergin) that includes gas-phase neutral-neutral and ion-neutral reactions. In
this calculation, we hold the cloud at the same time in its dynamical
evolution and allow the chemistry to evolve for 10 Myr from the assumed
starting conditions in which all species are atomic and neutral. Both models
generally agree. The gas-phase water peaks in a region near the boundary. Here
there is enough external UV to rapidly desorb the water from the ice, but not
so much as to dissociate all the molecules. Further inward, the abundance of
water falls as the gas density and the dust-gas collision rate (freeze-out
rate) both increase while the photodesorption rate decreases with the
attenuation of the UV radiation. At high Av, the water is desorbed only by
cosmic rays and the UV radiation they produce in collisions with H2. The
general agreement between the two models suggests that the simple model
includes the processes that are significant in the cold environment. The rate
equations for the processes selected for the simplified model are listed in
the appendix (§B).
Our simple model calculates equilibrium abundances. We can estimate the
equilibrium time scale from the combined rates for creation and destruction
(Caselli et al., 2002),
$t=\frac{t_{creation}t_{destruction}}{t_{creation}+t_{destruction}}$ (1)
where the time scales are the inverses of the rates. Figure 5 shows the
equilibrium time scales for each species as a function of radius. These may be
compared with the time for the hydrodynamic evolution. A cloud with a mass of
10 M⊙ and a central density of $2\times 10^{6}$ cm-3 has a free-fall time,
$t_{ff}=0.03$ Myr using the central density in the standard equation whereas
the sound crossing time is about 2 Myr (Keto & Caselli, 2010). Because the
chemical time scales are all shorter than the dynamical time scales the
chemistry reaches equilibrium before the conditions, density, temperature, and
UV flux change.
In this estimate of the time scale for chemical evolution, we are asking
whether the oxygen chemistry in the contracting molecular cloud can maintain
equilibrium as the cloud evolves dynamically. This is different from the
question of how long it would take for the chemistry to equilibrate if the gas
were held at molecular conditions but evolving from an atomic state.
$\begin{array}[]{cc}\includegraphics[width=234.87749pt]{StructureLVG}\end{array}$
Figure 1: Model of a slowly contracting cloud in quasi-static unstable
equilibrium. The log of the density profile in cm-3 is shown in blue (dotted
line), the fractional abundance of H2O with respect to H2 is shown in green
(dashed line), the velocity as the black (solid) line, and the gas temperature
as the red (dot-dashed) line. The model spectrum is shown in figure 8. Figure
2: Simplified model of the oxygen chemistry in a cold cloud. The model
includes 3 gas-phase species and H2O ice. The significant reactions at cold
temperatures (T $<300$ K) are the freeze-out of molecules colliding with dust
grains, cosmic ray and photodesorption of the ice, and photodissociation of
the gas phase molecules. Figure 3: Abundances of oxygen species as a function
of $A_{V}$ for the model of L1544 based on a slowly contracting Bonnor-Ebert
sphere. The figure emphasizes the variation of abundances in the PDR at the
edge. The figure compares the abundances for the physical conditions in
(figure 1) from two models: Hollenbach et al. (2009) (dashed lines) (courtesy
E. Bergin); and our simplified model (figure 2). Figure 4 shows abundances
from the same models but plotted against log radius to emphasize the
variations in the center. Figure 4: Abundances of oxygen species, same as
figure 3, except plotted against the log of the radius rather than visual
extinction. This figure emphasizes the variations in abundance in the center.
The figure shows the H2O abundance calculated with our simplified model using
two values for the cosmic ray-induced UV photodesorption (equation 19). The
solid green line shows the abundance calculated with factor $\alpha=10^{-4}$.
The dotted line shows the abundance calculated with factor $\alpha=10^{-3}$.
The abundance calculated with the Hollenbach et al. (2009) model assumes
$\alpha=10^{-3}$ (dashed line). Figure 5: Time scales for chemical
equilibrium. From top to bottom, the three lines show the equilibration time
scales for H2O, OH, and O calculated from equation 1 and the reaction rates in
the appendix.
### 2.4 Radiative Transfer
We use our radiative transfer code MOLLIE (Keto, 1990; Keto & Rybicki, 2010)
to compute model H2O spectra to compare with the Herschel observation. Here we
encounter an interesting question. The large Einstein A coefficient of the H2O
($1_{10}-1_{01}$) line results in optical depths across the cloud of several
hundred to a thousand depending on excitation. High optical depths generally
result in radiative trapping and enhanced excitation of the line. In this
case, the line brightness could have a non-linear relationship to the column
density. For example, the line could be saturated. On the other hand, the
large Einstein A means that the critical density for collisional de-excitation
is quite high ($1\times 10^{8}$ cm-3) at the temperatures $<15$ K, higher than
the maximum density ($1\times 10^{7}$ cm-3) in our dynamical model of L1544.
This suggests that the line emission should be proportional to the column
density.
This question was addressed by Linke et al. (1977) who proposed a solution
using the escape probability approximation (Kalkofen, 1984). They assumed a
two level molecule, equal statistical weights in both levels, and the mean
radiation field, $\bar{J}$, set by the escape probability, $\beta$,
$\bar{J}=J_{0}\beta+(1-\beta)S$ (2)
where $J_{0}$ is the continuum from dust and the cosmic microwave background,
$S$ is the line source function, and
$\beta=(1-\exp{(-\tau)})/\tau.$ (3)
After a satisfying bout with three pages of elementary algebra and some
further minor approximations, they show that as long as $C<A/\tau$, the line
brightness is linearly dependent on the column density, no matter whether the
optical depth is low or high, provided that the line is not too bright.
To determine whether the water emission line brightness in L1544 has a non-
linear or linear dependence, we numerically solve the equations for the two-
level molecule with no approximations other than the escape probability and
plot the result. Figures 6 and 7 show the dependence of the antenna
temperature on the density for low and high densities respectively. Since the
column density, the optical depth, and the ratio C/A are all linearly
dependent on the density, any of these may be used on the abcissa. The latter
two are shown just above the axis. Figure 6 shows that the antenna temperature
of the water line emission is linearly dependent on the column density even at
high density or high optical depth. Figure 7 shows that the linear relation
breaks down when C/A is no longer small. The densities in both figures show
that the water line emission in L1544 is in the linear regime.
$\begin{array}[]{cc}\includegraphics[width=234.87749pt]{growth-
low4}\end{array}$
Figure 6: The dependence of the observed antenna temperature of the H2O
($1_{10}-1_{01}$) line on the H2 number density (cm-3). Because the optical
depth and the ratio of the collision rate to spontaneous emission rate (C/A)
are both linearly dependent on the density, the abscissa can be labeled in
these units as well. Both are shown above the axis. The antenna temperature is
linearly dependent on the density or column density even at very high optical
depth as long as the ratio C/A is small.
$\begin{array}[]{cc}\includegraphics[width=234.87749pt]{growth-
high4}\end{array}$
Figure 7: The dependence of the observed antenna temperature of the H2O line
($1_{10}-1_{01}$) on the number density. Same as figure 6 but at higher
densities where the ratio C/A is no longer small and the dependence of the
antenna temperature on the density is no longer linear.
For an intuitive explanation, suppose that a photon is absorbed on average
once per optical depth of one. A photon may be absorbed and another re-emitted
many times in escaping a cloud of high optical depth. The time scale for each
de-excitation is $A^{-1}$. Therefore, the time that it takes a photon to
escape the cloud is $\tau/A$. As long as this time is shorter than the
collisional excitation time ($1/C$), then on average, an emitted photon will
escape the cloud before another photon is created by the next collisional
excitation event and radiative de-excitation. In this case, the line remains
subcritically excited. The molecules are in the lower state almost all the
time. This is the same condition that would prevail if the cloud were
optically thin ($\bar{J}=0$ or $\beta=1$). On this basis, in our earlier paper
we determined the emissivity and opacity of the H2O line in L1544 by setting
$\bar{J}=0$ (Caselli et al., 2012). This approximation was earlier adopted in
analyzing water emission observed by the SWAS satellite (Snell et al., 2000)
where it is referred to as ”effectively optically thin”.
In this current paper, we seek an improved estimate of $\bar{J}>0$ and
$\beta<1$ by using the escape probability formalism as suggested by Linke et
al. (1977). We determine $\beta$ using the local velocity gradient as given by
our hydrodynamical model along with the local opacity using the Sobolev or
large velocity gradient (LVG) approximation (eqn. 3-40 Kalkofen, 1984). We use
the 6-ray approximation for the angle averaging. We allow for one free scaling
parameter on $\beta$ to match the modeled emission line brightness to the
observation. We scale the LVG opacity by 1/2. Because the opacity, column
density, and line brightness, are all linearly related, the scaling could be
considered to derive from any or any combination of these parameters. Given
all the uncertain parameters, for example the mean grain cross-section which
also affects the line brightness (appendix B), this factor of 2 is not
significant.
An alternative method to calculate the excitation is the accelerated
$\Lambda$-iteration algorithm (ALI). We do not know if this method is reliable
with the extremely high optical depth, several hundred to a thousand.
$\Lambda$-iteration generally converges, but whether it converges to the
correct solution cannot be determined from the algorithm itself (eqn. 6-33
Mihalas, 1978).
The excitation may be uncertain, but analysis with the escape probability
method allows us to understand the effect of the uncertainty. For example,
because we know that the dependence of the line brightness on the opacity or
optical depth is linear, we can say that any uncertainty in excitation results
in the same percentage uncertainty in the abundance of the chemical model, or
the pathlength of the structural model.
Once $\bar{J}$ is determined everywhere in the cloud, the equations of
statistical equilibrium are solved to determine the emissivity and opacity.
These are then used in the radiative transfer equation to produce the
simulated spectral line emission and absorption. This calculation is done in
MOLLIE in the same way as if $\bar{J}$ were determined by any other means, for
example, by $\Lambda$-iteration.
Both the emissivity and opacity depend on frequency through the Doppler
shifted line profile function (eqn. 2.14 Kalkofen, 1984) that varies as a
function of position in the cloud. We use a line profile function that is the
thermal width plus a microturbulent Gaussian broadening of 0.08 km s-1 derived
from our CO modeling (Keto & Caselli, 2010). By the approximation of complete
frequency redistribution (eqn. 10-39 Mihalas, 1978), both have the same
frequency dependence. This also implies that each photon emitted after an
absorption event has no memory of the frequency of the absorbed photon. It is
emitted with the frequency probability distribution described by the line
profile function Doppler shifted by the local velocity along the direction of
emission. We also assume complete redistribution in angle.
Figure 8 shows the modeled line profile against the observed profile. The VLSR
is assumed to be 7.16 kms-1, slightly different than 7.2 kms-1 used in Caselli
et al. (2012). The lower value is chosen here as the best fit to the H2O
observation.
The combination of blue-shifted emission and red-shifted absorption is the
inverse P-Cygni profile characteristic of contraction, with the emission and
absorption split by the inward gas motion in the front and rear of the cloud.
The absorption against the dust continuum is unambiguously from the front side
indicating contraction rather than expansion. This profile has also been seen
in other molecules in other low-mass cold, dense clouds, with the absorption
against the dust continuum (Di Francesco et al., 2001).
In L1544, because the inward velocities are below the sound speed, and the H2O
line width is just larger than thermal, the emission is shifted with respect
to the absorption by less than a line width. In the observations, what appears
to be a blue-shifted emission line is just the blue shoulder and wing of the
complete emission, most of which is brighter, redder and wider than the
observed emission.
Our model also shows weaker emission to the red of the absorption line. This
emission is from inward moving gas in the front side of the contracting zone.
Again most of the emission is absorbed by the envelope and only the blue
shoulder of the line is seen. The asymmetry between the red and blue emission
comes about because the absorbing envelope, which is on the front side of the
cloud, is closer in velocity to inward flowing gas (red) on the front side of
the contraction. This is the same effect that produces the blue asymmetric or
double-peaked line profiles seen in contraction in molecular lines without
such significant envelope absorption (Anglada et al., 1987). The model shows
more red emission than is seen in the observations. This red emission may be
absorbed by foreground gas that is not in the model. Figure 1 of Caselli et
al. (2012) shows additional red shifted absorption in H2O and red shifted
emission in CO, both centered around 9 kms-1. The blue wing of this red
shifted water line may be absorbing the red wing of the emission from the
dense cloud.
If L1544 were static, no inward contraction, the emission from the center
would be at the same frequency as the envelope. Because of the extremely high
optical depth, the absorption line is saturated and would absorb all the
emission. We would see only the absorption line. The depth of the absorption
line is set by brightness of the dust continuum which is weak (0.011 K) and
not by the optical depth of the line which is high (few hundred to a
thousand).
In the current radiative transfer calculation, we also use a slightly
different collisional excitation rate than before. The collisional rates for
ortho-H2O are different with ortho and para-H2. In our previous paper (Caselli
et al., 2012) we modeled the H2 ortho-to-para ratio as a lower limit 1:1 or
higher. Here we assume that almost all the hydrogen, 99.9%, is in the para
state. This is suggested by recent chemical models that require a low ortho-
to-para ratio to produce the high deuterium fraction observed in cold, dense
clouds. (e.g Kong et al., 2013; Sipilä, Caselli & Harju, 2013).
## 3 Interpretation
The shape of the line profile (figure 8) is unaffected by any uncertainty in
the excitation which scales the emission across the spectrum. The absorption
is saturated and does not scale with the excitation. Because of the very high
critical density for collisional de-excitation, we know that the line emission
is generated only in the densest gas ($>10^{6}$ cm-3) within a few thousand AU
of the center. Thus the observation of the inverse P-Cygni profile seen in H2O
confirms the model for quasi-hydrostatic contraction with the highest
velocities near the center (figure 1).
The chemical model requires external UV to create the gas phase water by
photodesorption. This confirms the physical model of L1544 as a molecular
cloud bounded by a photodissociation region. The UV flux necessarily creates a
higher temperature, up to about 100 K at the boundary by photoelectric
heating. This helps maintain the pressure balance at the boundary consistent
with the model of a BE sphere.
## 4 Uncertainties
The comparison of the simulated and observed spectral line involves three
models each with multiple parameters. Unavoidably the choice of parameters in
any one of the three models affects not only the choice of other parameters in
the other two models but also the interpretation. It would be a mistake to
focus on the uncertainties in any one of the models to the exclusion of the
others. For example, because of the linear relationship between the line
brightness, the optical depth, and the opacity, uncertainties in the
excitation, pathlength, and abundance, have equal effect on the spectrum. A
factor of two uncertainty in the excitation can be compensated by a factor of
two in the pathlength or a factor of two in the abundance of H2O. The
pathlength is unknown. On the plane of the sky, L1544 has an axial ratio of
2:1, but we are using a spherical model for the cloud. Our rates in the
chemical model involve estimation of the surface density of sites for
desorption and the covering fraction of water ice on the grains. The latter is
assumed to be one even though we know that CO and methane ice, not included in
the simple model, make up a significant fraction of the ice mantle. The
radiative excitation, parameterized as $\beta$ in the escape probability is
also uncertain because of the competing effects of high optical depth and
subcritical excitation.
On a linear plot, a factor of two difference in the brightness of the
simulated and observed spectral line looks to be a damning discrepancy.
However, there is at least this much uncertainty in each of the three models
and this does not significantly affect the conclusions of the study, namely
that the cloud can be modeled as a slowly contracting BE sphere bounded by a
photodissociation region with the gas phase water abundance set by grain
surface reactions.
In this paper, we concentrate on the observation of H2O, but there are also
other constraints that define the model. These are both observational and
theoretical. In an earlier paper, we showed how observations of CO and N2H+
define the physical model with the two spectral lines giving us information on
the outer and inner regions of the cloud respectively. In this regard, the
water emission gives us information in the central few thousand AU of the
cloud where the density approaches or exceeds the critical density for de-
excitation. This small volume of rapid inflow and high density does not much
affect the N2H+ spectrum which is generated in a much larger volume, and has
no affect at all on the CO spectrum. A successful model for L1544 has to
satisfy the constraints of all the data. On the theoretical side, there is an
infinite space of combinations of abundance, density, velocity, and
temperature that would form models that match the data. Only models that are
physically motivated are of interest. It may be tempting to change the
abundances, velocities, or densities arbitrarily, but this is unlikely to be a
useful exercise giving the infinite possibilities. A successful model for
L1544 has to be relevant to plausible theory.
There is a natural prejudice for more complex models that in principle contain
more details. The goal of our simplified models is to enhance our
understanding of the most significant phenomena. In our research on cold,
dense clouds, spanning a number of papers, we have developed simplified models
for the density and temperature structure, for the dynamics including
oscillations, for the CO chemistry, and in this paper simplified models for
H2O chemistry and radiative transfer. Each of these models isolates one or a
few key physical processes and shows how they generate the observables and
operate to control the evolution toward star formation.
$\begin{array}[]{cc}\includegraphics[width=234.87749pt]{SpectrumLVG}\end{array}$
Figure 8: Observed spectrum of H2O (1${}_{10}-1_{01}$) (black lines with
crosses) compared with modeled spectrum (simple red line) for slow contraction
at the time that the central density reaches $1\times 10^{7}$ cm-3. The model
structure is shown in figure 1.
## 5 Conclusions
A simplified chemical model for cold oxygen chemistry primarily by grain
surface reactions is verified by comparing the simulated spectrum of the H2O
($1_{10}-1_{01}$) line against an observation of water vapor in L1544 made
with HIFI spectrometer on the Herschel Space Observatory.
This model reproduces the observed spectrum of H2O, and also approximates the
abundances calculated by a more complete model that includes gas-phase
neutral-neutral and ion-neutral reactions.
The gas phase water is released from ice grains by ultraviolet (UV)
photodesorption. The UV radiation derives from two sources: external starlight
and collisions of cosmic rays with molecular hydrogen. The latter may be
important deep inside the cloud where the visual extinction is high enough
($>50$ mag) to block out the external UV radiation.
Water is removed from the gas phase by photodissociation and freeze-out onto
grains. The former is important at the boundary where the UV from external
starlight is intense enough to create a photodissociation region. Here, atomic
oxygen replaces water as the most abundant oxygen species. In the center where
the external UV radiation is completely attenuated, freeze-out is the
significant loss mechanism.
Time dependent chemistry is not required to match the observations because the
time scale for the chemical processes is short compared to the dynamical time
scale.
The molecular cloud L1544 is bounded by a photodissociation region.
The water emission derives only from the central few thousand AU where the gas
density approaches the critical density for collisional de-excitation of the
water line. In the model of hydrostatic equilibrium, the gas density in the
center is rising with decreasing radius more steeply than the abundance of
water is decreasing by freeze-out. Thus the water spectrum provides unique
information on the dynamics in the very center.
The large Einstein A coefficient ($3\times 10^{-3}$ s-1) of the 557 GHz H2O
($1_{10}-1_{01}$) line results in extremely high optical depth, several
hundred to a thousand. However, the density ($<10^{7}$ cm-3) and temperature
($<15$ K) are low enough that the line is subcritically excited. The result is
that the line brightness under these conditions is directly proportional to
the column density.
## 6 Acknowledgements
The authors acknowledge Simon Bruderer, Fabien Daniel, Michiel Hogerheijde,
Joe Mottram, Floris van der Tak for interesting discussions on the radiative
transfer of water. PC acknowledges the financial support of the European
Research Council (ERC; project PALs 320620), of successive rolling grants
awarded by the UK Science and Technology Funding Council. JR acknowledges the
financial support of the Submillimeter Array Telescope.
## References
* Anglada et al. (1987) Anglada G., Rodriguez L. F., Canto J., Estalella R., Lopez R., 1987, A&A, 186, 280
* Bergin et al. (2000) Bergin E. A. et al., 2000, ApJ Lett, 539, L129
* Bergin & Tafalla (2007) Bergin E. A., Tafalla M., 2007, ARAA, 45, 339
* Bethell & Bergin (2009) Bethell T., Bergin E., 2009, Science, 326, 1675
* Broderick & Keto (2010) Broderick A. E., Keto E., 2010, ApJ, 721, 493
* Caselli et al. (2012) Caselli P. et al., 2012, ApJ Lett, 759, L37
* Caselli et al. (2010) Caselli P. et al., 2010, A&A, 521, L29
* Caselli et al. (2002) Caselli P., Walmsley C. M., Zucconi A., Tafalla M., Dore L., Myers P. C., 2002, ApJ, 565, 344
* Conrath & Gierasch (1984) Conrath B. J., Gierasch P. J., 1984, Icarus, 57, 184
* de Graauw et al. (2010) de Graauw T. et al., 2010, A&A, 518, L6
* di Francesco et al. (2007) di Francesco J., Evans, II N. J., Caselli P., Myers P. C., Shirley Y., Aikawa Y., Tafalla M., 2007, Protostars and Planets V, 17
* Di Francesco et al. (2001) Di Francesco J., Myers P. C., Wilner D. J., Ohashi N., Mardones D., 2001, ApJ, 562, 770
* Evans et al. (2001) Evans, II N. J., Rawlings J. M. C., Shirley Y. L., Mundy L. G., 2001, ApJ, 557, 193
* Fouchet, Lellouch & Feuchtgruber (2003) Fouchet T., Lellouch E., Feuchtgruber H., 2003, Icarus, 161, 127
* Habing (1968) Habing H. J., 1968, Bulletin of the Astronomical Inst. of the Netherlands, 19, 421
* Hollenbach et al. (2009) Hollenbach D., Kaufman M. J., Bergin E. A., Melnick G. J., 2009, ApJ, 690, 1497
* Jijina, Myers & Adams (1999) Jijina J., Myers P. C., Adams F. C., 1999, ApJ Suppl, 125, 161
* Kalkofen (1984) Kalkofen W., 1984, Methods in radiative transfer
* Kandori et al. (2005) Kandori R. et al., 2005, AJ, 130, 2166
* Keto et al. (2006) Keto E., Broderick A. E., Lada C. J., Narayan R., 2006, ApJ, 652, 1366
* Keto & Caselli (2008) Keto E., Caselli P., 2008, ApJ, 683, 238
* Keto & Caselli (2010) Keto E., Caselli P., 2010, MNRAS, 402, 1625
* Keto & Field (2005) Keto E., Field G., 2005, ApJ, 635, 1151
* Keto & Rybicki (2010) Keto E., Rybicki G., 2010, ApJ, 716, 1315
* Keto (1990) Keto E. R., 1990, ApJ, 355, 190
* Kong et al. (2013) Kong S., Caselli P., Tan J. C., Wakelam V., 2013, arxiv:1312.0971
* Lada et al. (2003) Lada C. J., Bergin E. A., Alves J. F., Huard T. L., 2003, ApJ, 586, 286
* Linke et al. (1977) Linke R. A., Goldsmith P. F., Wannier P. G., Wilson R. W., Penzias A. A., 1977, ApJ, 214, 50
* Mathis, Rumpl & Nordsieck (1977) Mathis J. S., Rumpl W., Nordsieck K. H., 1977, ApJ, 217, 425
* Melnick et al. (2000) Melnick G. J. et al., 2000, ApJ Lett, 539, L87
* Mihalas (1978) Mihalas D., 1978, Stellar atmospheres /2nd edition/
* Myers & Benson (1983) Myers P. C., Benson P. J., 1983, ApJ, 266, 309
* Neufeld et al. (2000) Neufeld D. A. et al., 2000, ApJ Lett, 539, L107
* Rawlings et al. (1992) Rawlings J. M. C., Hartquist T. W., Menten K. M., Williams D. A., 1992, MNRAS, 255, 471
* Shen et al. (2004) Shen C. J., Greenberg J. M., Schutte W. A., van Dishoeck E. F., 2004, A&A, 415, 203
* Sipilä, Caselli & Harju (2013) Sipilä O., Caselli P., Harju J., 2013, A&A, 554, A92
* Snell et al. (2000) Snell R. L. et al., 2000, ApJ Lett, 539, L101
* Tielens (2005) Tielens A. G. G. M., 2005, The Physics and Chemistry of the Interstellar Medium
* Tielens & Hollenbach (1985) Tielens A. G. G. M., Hollenbach D., 1985, ApJ, 291, 722
* van Dishoeck et al. (1999) van Dishoeck E. F. et al., 1999, in ESA Special Publication, Vol. 427, The Universe as Seen by ISO, Cox P., Kessler M., eds., p. 437
* van Dishoeck, Herbst & Neufeld (2013) van Dishoeck E. F., Herbst E., Neufeld D. A., 2013, Chemical Reviews, 113, 9043
* Viti et al. (2001) Viti S., Roueff E., Hartquist T. W., Pineau des Forêts G., Williams D. A., 2001, A&A, 370, 557
## Appendix A Cooling by atomic oxygen fine structure lines
The fine structures lines of C+ and atomic O are the major coolants in the
diffuse ($n<1000$ cm-3), photodissociated gas around the molecular clouds. The
more important coolant at temperatures less than 100 K is C+. At higher
temperatures, oxygen becomes increasingly important in the energy balance. The
reason is that the 63.2 and 145.6 $\mu$m fine structure lines of atomic oxygen
have upper states 3P1 and 3P0 that are at 228 K and 326 K above ground,
considerably higher than the 92 K of the upper state, 2P3/2 of the 157.6
$\mu$m fine structure line of C+.
The cooling by atomic oxygen is simple to model because atomic oxygen is a
product of photodissociation and is therefore abundant only in gas with low Av
implying gas densities below the critical densities for collisional de-
excitation, 6400 and 3400 cm-3 for the 63.2 and 145.6 $\mu$m lines
respectively (table 2.7 of Tielens, 2005). At this density, we assume that the
optically thin approximation applies. In this case, every collisional
excitation to an upper state of the fine structure lines results in
spontaneous emission that escapes the cloud and cools the gas,
$\Lambda_{\rm O}=n({\rm O})n({\rm H_{2}})(E_{21}C_{21}+E_{20}C_{20})\ \ {\rm
ergs}\ {\rm cm}^{-3}\ {\rm s}^{-1}$ (4)
where the upward collision rates are,
$C_{21}=1.4\times 10^{-8}\frac{g_{1}}{g_{2}}C_{12}\exp{(-E_{21}/kT)}\sqrt{T}\
\ {\rm cm}^{3}{\rm s}^{-1}$ (5) $C_{20}=1.4\times
10^{-8}\frac{g_{0}}{g_{2}}C_{02}\exp{(-E_{20}/kT)}\sqrt{T}\ \ {\rm cm}^{3}{\rm
s}^{-1}.$ (6)
and the statistical weights are $g_{2}=5$, $g_{1}=3$, and $g_{0}=1$ and the
transition energies are $E_{12}/k=228$K and $E_{02}/k=326$K.
## Appendix B Chemistry
### B.1 Freeze-out
Molecules freeze onto dust grains, sticking when they collide. This process is
easily modeled. We follow Keto & Caselli (2008) to calculate the collision
timescale. The time scale for depletion onto dust may be estimated as
(Rawlings et al., 1992),
$\tau_{on}=(S_{0}R_{dg}n({\rm H_{2}})\sigma V_{T})^{-1}\ {\rm s}$ (7)
Here $S_{0}$ is the sticking coefficient, with $S_{0}=1$ meaning that the
colliding molecule always sticks to the dust in each collision; $R_{dg}$ is
the ratio of the number density of dust grains relative to molecular hydrogen;
$\sigma$ is the mean cross-section of the dust grains; and $V_{T}$ is the
relative velocity between the grains and the gas. If the grains have a power
law distribution of sizes with the number of grains of each size scaling as
the -3.5 power of their radii (Mathis, Rumpl & Nordsieck, 1977), then we can
estimate their mean cross-section as,
$\langle\sigma\rangle=\bigg{(}\int^{a_{2}}_{a_{1}}n(a)da\bigg{)}^{-1}\int^{a_{2}}_{a_{1}}n(a)\sigma(a)da,$
(8)
where $a_{1}$ and $a_{2}$ are the minimum and maximum grain sizes. If
$a_{1}=0.005$ $\mu$m and $a_{2}=0.3$ $\mu$m, then
$\langle\sigma\rangle=3.4\times 10^{-4}$ $\mu{\rm m}^{2}$. Similarly, the
ratio of the number densities of dust and gas may be estimated by computing
the mean mass of a dust grain and assuming the standard gas-to-dust mass ratio
of 100. If the density of the dust is 2 grams cm-3, then the ratio of number
densities is $R_{dg}=4\times 10^{-10}$.
Consistent with Keto & Caselli (2008), our model has a slightly lower value
for the grain cross-section, $1.4\times 10^{-21}$ cm2, than Hollenbach et al.
(2009), $\sigma_{h}=2\times 10^{-21}$ cm2. Both values are per hydrogen
nucleus (2H2 \+ H). Because the ice forms and desorbs off the grain surfaces,
larger values of the average cross-section result in fewer molecules in the
gas phase. The actual properties of grains in cold clouds are somewhat
uncertain.
The relative velocity due to thermal motion is,
$V_{T}=\bigg{(}{{8kT}\over{\pi\mu}}\bigg{)}^{1/2},$ (9)
where $T$ is the temperature and $\mu$ the molecular weight. The freeze-out
rate for species $i$ is,
$f_{i}=\tau_{on}^{-1}n(H_{2})\ \ {\rm cm}^{-3}\ {\rm s}^{-1}$ (10)
### B.2 Gas-phase reactions
The neutral-neutral molecular and photodissociation reactions are from Tielens
& Hollenbach (1985). The reaction rate $k_{1}$ for ${\rm
O+H_{2}}\rightarrow{\rm OH+H}$ is,
$k_{1}=3.1\times 10^{-13}\ (T/300)^{2.7}\exp{(-3150/T)}\ \ {\rm cm}^{3}\ \
{\rm s}^{-1}$ (11)
The reaction $k_{2}$ for ${\rm OH+H_{2}}\rightarrow{\rm H_{2}O+H}$ is,
$k_{2}=2.0\times 10^{-12}(T/300)^{1.57}\exp{(-1736/T)}\ \ {\rm cm}^{3}\ \ {\rm
s}^{-1}$ (12)
OH + H has a rate,
$5.3\times 10^{-18}(T/300)^{-5.22}\exp{(-90/T)}.$ (13)
All three of these reactions have an activation barrier and are irrelevant at
temperatures below 300 K. The photodissociation rate for the destruction of OH
and the formation of O is,
$P_{1}=3.5\times 10^{-10}G_{0}\exp{(-1.7A_{V})}\ \ {\rm s}^{-1}$ (14)
and the rate for the destruction of H2O and formation of OH is,
$P_{2}=5.9\times 10^{-10}G_{0}\exp{(-1.7A_{V})}\\\ \ {\rm s}^{-1}.$ (15)
The unitless parameter $G_{0}=1$ corresponds to the average local interstellar
radiation field in the FUV band (Habing, 1968). $A_{V}$ is the visual
extinction.
### B.3 Desorption
The desorption rates are from Hollenbach et al. (2009). The total desorption
rate includes thermal desorption, photodesorption, and desorption by cosmic
rays. We use equation 2 from Hollenbach et al. (2009) for the rate for thermal
desorption,
$D_{Th}=1.6\times
10^{11}\bigg{(}\frac{E_{i}}{k}\bigg{)}^{1/2}\bigg{(}\frac{m_{H}}{m_{i}}\bigg{)}^{1/2}\exp{\bigg{(}\frac{-E_{i}}{kT_{gr}}\bigg{)}}\
\ {\rm s}^{-1}\ \ {\rm molecule}^{-1}$ (16)
where $E_{i}/k$, the adsorption energy is 800, 1300, and 5770 K for O, OH, and
H2O respectively, and mi/mH is the weight of the species with respect to H.
The thermal desorption rate for water is negligible at the temperatures ($<15$
K) of cold, dense clouds. For the cosmic-ray desorption rate, we use equation
8 from Hollenbach et al. (2009). We include only the cosmic-ray desorption
rate for H2O,
$D_{CR}=4.4\times 10^{-17}{\rm molecule}^{-1}{\rm s}^{-1}.$ (17)
Both the thermal desorption rate and the cosmic ray desorption rate in units
of molecule-1 s-1 are multiplied by the number of molecules on the surface of
grains per molecule of H2 which is $N_{s}f_{s}A_{gr}R_{dg}$ where
$N_{s}=10^{15}$ cm-2 is the number of desorption sites per cm2 on the grain
surface (Hollenbach et al., 2009), $f_{s}=1$ is the fraction of the grain
surface covered by ice, the average surface area of a grain is 4 times the
grain cross-section, $A_{gr}=4\sigma=4\times 3.4\times 10^{-4}$ $\mu$m 2 (Keto
& Caselli, 2008), and the dust-to-gas ratio $R_{dg}=4\times 10^{-10}$ (Keto &
Caselli, 2008). The photodesorption rates are from equations 6 and 7
(Hollenbach et al., 2009),
$D_{UV}=G_{0}F_{0}Y_{i}f_{i}\ \exp{(-1.8A_{V})}\ {\rm s}^{-1}$ (18)
where $F_{0}=10^{8}$ is the number of UV photons per Habing flux, and
$Y_{i}=10^{-3}$ and $2\times 10^{-3}$ are the photodesorption yields per UV
photon per second for the production of OH and H2O respectively from table 1
of Hollenbach et al. (2009). We assume that all the ice is H2O and follow
Hollenbach et al. (2009) in assuming that the photodesorption of this water
ice results in twice as much OH as H2O in the gas phase.
The desorption of water ice does not result in the production of gas phase
oxygen, and we have no oxygen ice in our model. To provide a back reaction to
the freeze-out of atomic oxygen, we arbitrarily assign a desorption rate equal
to that of water. In regions of high extinction ($A_{V}>4$) this results in a
gas phase abundance of atomic oxygen that is approximately the same as
predicted by Hollenbach et al. (2009). This is $<0.001$ of the total oxygen
and has no effect on the other abundances. In the outer part of the cloud
where the UV flux is higher ($A_{V}<4$) most of the atomic oxygen derives from
photodissociation. Here the UV desorption off grains is insignificant.
Additional desorption is caused by the UV photons emitted by hydrogen
excitation by energetic electrons released in the ionization of hydrogen by
cosmic rays. We follow Shen et al. (2004) and scale this process as $10^{-4}$
of one Habing flux, $G_{0}=1$, so that,
$D_{CR\ UV}=\alpha G_{0}F_{0}Y_{i}f_{i}\ {s^{-1}}$ (19)
with $\alpha=10^{-4}$.
### B.4 Equilibrium
In equilibrium, the rate equations in matrix form are,
$\begin{array}[]{lll}\begin{pmatrix}-(f_{O}+k_{1})&P_{1}&0&0\\\
k_{1}&-(f_{OH}+P_{1})&P_{2}&0\\\ 0&k_{2}&-(f_{H_{2}O}+P_{2})&D_{H_{2}O}\\\
f_{O}&f_{OH}&f_{H_{2}O}&-(D_{OH}+D_{H_{2}O})\\\ 1&1&1&1\\\
\end{pmatrix}\begin{pmatrix}O\\\ OH\\\ H_{2}O\\\ ICE\\\
\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\ 1\\\
\end{pmatrix}\end{array}$
where the last row is the conservation equation for oxygen among all the
species. As written, this system is overdetermined, but can be solved by
dropping any one of the first 4 rows.
### B.5 H2O ortho-para ratio
Since the ortho state of H2O is 24K above the para state, the O/P ratio in
thermal equilibrium is very small at lower temperatures (equation 41
Hollenbach et al., 2009). However, when the water molecule is formed, created
from OH on the grain surface for example, it is formed in the ratio of the
available quantum states, ortho:para 3:1. The ortho and para states of H2O
equilibrate by collisions with H or H2. If the chemical equilibrium time scale
is much shorter than the thermal equilibrium time scale, the O/P ratio will
not deviate much from 3:1. Observations generally show ratios close to 3:1
(van Dishoeck, Herbst & Neufeld, 2013).
We have not found previous research on the equilibration of H2O, but an
appreciation of the time scale can be estimated from previous research on the
equilibration of the ortho and para states of molecular hydrogen. The
dissociation energies of H-H and OH-H are not too different nor the
collisional cross-sections of the molecules. Conrath & Gierasch (1984) and
Fouchet, Lellouch & Feuchtgruber (2003) suggest three processes for the
equilibration of the ortho and para states of H2 are: (1) gas phase H
exchange, (2) gas phase paramagnetic conversion with H2, and (3) H exchange on
a surface. We assume that these same processes are applicable to the water.
The rates for these processes scale with the gas density through the collision
rate and scale as the inverse exponential of the temperature. Scaling the
rates for H2 from the conditions in the atmosphere of Jupiter to rarefied,
cold gas of the interstellar medium (10 K and $10^{6}$ cm-3) the time scales
for these processes are all $>1$ Gyr.
In contrast, the chemical time scale is very much shorter (figure 5)
throughout the cloud. In this model, water is dissociated in the gas phase by
photodissociation and also coming off the grain surfaces by photodesorption in
which gas phase OH is produced twice as often as gas phase H2O. The
equilibrium comparison between ortho-para equilibration and chemistry may not
be needed because the equilibration time scale exceeds the expected life times
of the cold, dense, clouds.
|
arxiv-papers
| 2014-03-02T03:07:56 |
2024-09-04T02:49:59.146348
|
{
"license": "Public Domain",
"authors": "Eric Keto, Jonathan Rawlings, Paola Caselli",
"submitter": "Eric Keto",
"url": "https://arxiv.org/abs/1403.0155"
}
|
1403.0177
|
# Vector-valued Hilbert transforms along curves
Guixiang Hong1 and Honghai Liu2∗ 1School of Mathematics and Statistics, Wuhan
University, Wuhan 430072, China and Instituto de Ciencias Matemáticas, CSIC-
UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás
Cabrera 13-15. 28049, Madrid, Spain. [email protected] 2 School of
Mathematics and Information Science, Henan Polytechnic University, Jiaozuo,
Henan 454003, China. [email protected]
(Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.
∗ Corresponding author)
###### Abstract.
In this paper, we show that Hilbert transforms along some curves are bounded
on $L^{p}({\mathbb{R}}^{n};X)$ for some $1<p<\infty$ and some UMD spaces $X$.
In particular, we prove that Hilbert transforms along some curves are
completely $L^{p}$-bounded in the terminology from operator space theory.
Moreover, we obtain the $L^{p}(\mathbb{R}^{n};X)$-boundedness of anisotropic
singular integrals by using the ”method of rotations” of Calderón-Zygmund. All
these results extend already existing related ones.
###### Key words and phrases:
Hilbert transforms along curves, Weighted Hörmander condition, UMD spaces,
Completely bounded, Analytic interpolation.
###### 2010 Mathematics Subject Classification:
Primary 43A32; Secondary 46B99.
## 1\. Introduction
The question of whether the mapping properties of singular integral operators
could be extended to the Lebesgue-Bôhner spaces $L^{p}(\mathbb{R}^{n};X)$
($1<p<\infty$) of vector-valued functions was taken up by several authors in
the 60’s. In [1], Benedek, Calderón and Panzone observed that the boundedness
on $L^{p_{0}}(\mathbb{R}^{n};X)$ for one $1<p_{0}<\infty$ of a singular
integral operator, together with Hörmander’s condition, implies its
boundedness on $L^{p}(\mathbb{R}^{n};X)$ for all $1<p<\infty$. However, to
actually get the $L^{p_{0}}(\mathbb{R}^{n};X)$-boundedness (something that was
immediate for $p_{0}=2$ in the scalar-valued), turned out to be a
significantly difficult task except in the case $X=L^{p_{0}}(\Omega)$ for some
measure space $\Omega$.
The first progress made in this direction is Burkholder’s extension [3] of
Riesz’s classical theorem on the $L^{p}$-boundedness of the Hilbert transform,
where it was shown that if the underlying Banach space $X$ satisfies the so
called UMD-property, then the Hilbert transform is bounded on
$L^{p}(\mathbb{R};X)$ for any $1<p<\infty$. Moreover, the UMD-property was
shown by Bourgain [2] to be necessary for the boundedness of the Hilbert
transform. It is well-known that the Hilbert transform is a prototype of
singular integral operators and Fourier multipliers, its boundedness motivates
McConnell’s [17] and Zimmermann’s [28] results on vector-valued Marcinkiewicz-
Mihlin multipliers, and Hytönen and Weis’s [12] results on vector-valued
singular convolution integrals.
Particularly, if $X$ equals $S_{p}$–the Schatten class, the
$L^{p}(\mathbb{R}^{n};S_{p})$-boundedness is called complete
$L^{p}$-boundedness in the light of noncommutative harmonic analysis. In this
setting, the complete $L^{2}$-boundedness is immediately available because
$S_{2}$ is a Hilbert space, and the Fourier transform (or almost orthogonality
principle) can be adapted. In order to obtain the complete
$L^{p}$-boundedness, so far as we know in the noncommutative harmonic
analysis, there are only two ways. One way is to establish firstly the weak
type $(1,1)$ estimate, and then to use interpolation and the duality argument.
In this way, the convolution kernel need to satisfy the Lipschitz regularity
in order to conduct the pseudo-localization principle as done in [21] (see
also [10] for related results). The other way is to get $(L^{\infty},BMO)$
(the noncommutative BMO space) estimate, then to use interpolation and the
duality argument. In this case, the kernel is required to satisfy the
Hörmander’s condition as done in [18] and [15]. However, to get the complete
$L^{p}$-boundedness is not a trivial work when the kernel does not satisfy the
Lipschitz regularity and the Hörmander condition, see e.g. [9] for more
information.
The purpose of our project is to extend the vector-valued singular integrals
theory to more general setting. We consider vector-valued singular Radon
transforms, which are given by the following principal-valued integral
$\mathscr{T}f(x)={\rm p.v.}\int_{\mathbb{R}^{k}}f(x-\Gamma(t))K(t)dt,\ \ f\in
C_{0}^{\infty}({\mathbb{R}}^{n})\otimes X,$
where $X$ is a Banach space, $K$ is a Calderón-Zygmund kernel in
$\mathbb{R}^{k}$ and $\Gamma:\mathbb{R}^{k}\rightarrow{\mathbb{R}}^{n}$ is a
surface in ${\mathbb{R}}^{n}$ with $\Gamma(0)=0$, $n\geq 2$. Precisely, we are
interested in the boundedness of $\mathscr{T}$ on $L^{p}(\mathbb{R}^{n};X)$,
where $p\in(1,\infty)$ and $X$ is some Banach space. Obviously, $\mathscr{T}$
are classical vector-valued singular convolution integrals if $k=n$ and
$\Gamma(t)=(t_{1},t_{2},\cdots,t_{n})$, and related results have been
introduced in the previous paragraphs. On the other hand, if $X=\mathbb{R}$,
$\mathscr{T}$ are classical singular integrals associated to surfaces, which
have been well-studied by Stein, Nagel, Wainger, Christ and so on, see [27]
for a survey of results through 1978 and [6] through 1999.
In the present paper, we start with the investigation of Hilbert transforms
along curves in the hope of providing the insight and inspiration for
subsequent development of this subject, as the role played by the classical
Hilbert transform in the classical vector-valued Calderón-Zygmund theory.
Vector-valued Hilbert transforms along curves are defined by
$\mathscr{H}f(x)={\rm
p.v.}\int_{\mathbb{R}}f\big{(}x-\Gamma(t)\big{)}\frac{dt}{t},\ \ f\in
C_{0}^{\infty}({\mathbb{R}}^{n})\otimes X.$
In the scalar-valued case, the $L^{2}$-boundedness goes back the work [7] of
Fabes who proved it with $\Gamma(t)=(t^{\alpha},t^{\beta})$ using complex
integration. Then Stein and Wainger [26] obtained the $L^{2}$-boundedness for
all homogeneous curves by using Van der Corput’s estimates for trigonometric
integrals. The first breakthrough was the proof of the $L^{p}$-boundedness in
the papers of Nagel, Rivière and Wainger [19] as well as the paper of Nagel
and Wainger [20] using Stein’s complex interpolation. Since then, many related
results have been obtained, see Stein and Wainger’s survey paper [27] for the
curves having some curvature at the origin, the paper of Carlsson et al [5]
and the references therein for the flat curves in $\mathbb{R}^{2}$. However,
all results about vector-valued singular integrals mentioned previously can
not be directly applied to Hilbert transforms along curves on
$L^{p}(\mathbb{R}^{n};X)$, because they are no longer Calderón-Zygmund
operators. Therefore this study is a move beyond the vector-valued Calderón-
Zygmund theory.
In the present paper, we extend Nagel, Rivière and Wainger as well as Nagel
and Wainger’s results mentioned above to the vector-valued setting by
combining their original arguments and some idea developed recently by Hytönen
and Weis [14] in the vector-valued Calderón-Zygmund theory. To state our
results, we need to recall and introduce some notations. Denote by
$\epsilon_{j}$, $j\in\mathbb{Z}$, the Rademacher system of independent random
variables on a probability space $(\Omega,\Sigma,\mathbf{P})$ verifying
$\mathbf{P}(\epsilon_{j}=1)=\mathbf{P}(\epsilon_{j}=-1)=1/2$. Let
$\mathbb{E}=\int(\cdot)d\mathbf{P}$ be the corresponding expectation. The main
Banach space geometry property of $X$ we are concerned in this paper is the
UMD property (see e.g. [3]), i.e. the following inequality holds:
$\big{(}\mathbb{E}\big{\|}\sum_{k=1}^{N}\epsilon_{k}d_{k}\big{\|}_{X}^{2}\big{)}^{1/2}\leq
C\big{(}\mathbb{E}\big{\|}\sum_{k=1}^{N}d_{k}\big{\|}_{X}^{2}\big{)}^{1/2}$
for all $N\in\mathbb{N}$, all fixed signs $\epsilon_{k}\in\\{-1,1\\}$, all
$X$-valued martingale differences $(d_{k})_{k\geq 0}$. The following notation
is very useful for formulating the main results in this paper.
###### Definition 1.1.
Let $(a,b)\subseteq(0,1)$. We define $\mathcal{I}_{(a,b)}$ to be the set
consisting of UMD spaces with its element $X$ having the form
$X=[H,Y]_{\theta}$ such that $\theta\in(a,b)$, $H$ is a Hilbert space and $Y$
is another UMD space. $\mathcal{I}_{(0,1)}$ is denoted by $\mathcal{I}$ for
simplicity.
###### Remark 1.2.
(i). It is easy to check that all the noncommutative $L_{p}$ spaces
(containing commutative $L^{p}$ spaces) with $1<p<\infty$ belong to the class
$\mathcal{I}_{(|1-\frac{2}{p}|,1)}$. From the reflexivity of UMD space, in
general we have $X\in\mathcal{I}_{(a,b)}$ if and only if
$X^{\ast}\in\mathcal{I}_{(a,b)}$. Furthermore, if
$(a,b)\subseteq(c,d)\subseteq(0,1)$, then
$\mathcal{I}_{(a,b)}\subseteq\mathcal{I}_{(c,d)}$.
(ii). In [23], Rubio de Francia proved that for any UMD lattice $X$ there
exist $\theta\in(0,1)$, a Hilbert space $H$ and another UMD lattice $Y$ such
that $X=[H,Y]_{\theta}$. That means every UMD lattice $X$ belongs to
$\mathcal{I}$. In the same paper, the author also ask the open question “Is
every $B\in UMD$ intermediate between a ’worse’ $B_{0}$ and a Hilbert spaces
?” which in our language means “If $\mathcal{I}$ contains all UMD spaces?”.
The first result is on the Hilbert transform along the homogeneous curves
$\Gamma(t)=(|t|^{\alpha_{1}}sgnt,|t|^{\alpha_{2}}sgnt,\cdots,|t|^{\alpha_{n}}sgnt)$
with each $\alpha_{i}>0$.
###### Theorem 1.3.
Let $X\in\mathcal{I}$ and $1<p<\infty$. Then there exists an absolute constant
$C_{p}$ such that
$\|\mathscr{H}f\|_{L^{p}(X)}\leq C_{p}\|f\|_{L^{p}(X)},\ \
f\in{L^{p}(\mathbb{R}^{n};X)}.$
This is a vector-valued version of Theorem 1 of Nagel, Rivière and Wainger in
[19]. Following the previous remark, Theorem 1.3 implies the complete
boundedness of Hilbert transforms along this kind of curves which is of
independent interest in the operator space theory. This result also partially
generalize the previous result by Rubio de Francia, Ruiz and Torra [22] where
they obtained Theorem 1.3 in the case $X=\ell^{q}$ with $1<q<\infty$. In [22],
the authors used indirectly Benedek, Calderón and Panzone’s strategy mentioned
previously. While the proof of Theorem 1.3 is motivated by the recent
development in the vector-valued Calderón-Zygmund theory [12], see Section 2
for related details.
Let $\delta_{t}$ be a one parameter group of dilations and
$\mathbf{e},\mathbf{f}$ be vectors in $\mathbb{R}^{n}$. A curve $\Gamma(t)$ is
called two-sided homogeneous if the following two conditions hold:
$\Gamma(t)=\left\\{\begin{array}[]{ccc}\delta_{t}\ \mathbf{e},&t>0,\\\
\delta_{-t}\ \mathbf{f},&t<0,\\\ 0,&t=0;\end{array}\right.$ (1.1)
$\\{\xi|\xi\cdot\Gamma(t)\equiv 0,t>0\\}=\\{\xi|\xi\cdot\Gamma(t)\equiv
0,t<0\\}.$
The curve
$\Gamma(t)=(|t|^{\alpha_{1}}sgnt,|t|^{\alpha_{2}}sgnt,\cdots,|t|^{\alpha_{n}}sgnt)$
is a model with
$\delta_{t}x=(t^{\alpha_{1}}x_{1},t^{\alpha_{2}}x_{2},\cdots,t^{\alpha_{n}}x_{n})$,
$\mathbf{e}=\mathbf{1}$ and $\mathbf{f}=-\mathbf{1}$. We will see that the
same argument for this particular curve works for all the curves with the same
dilation but $\mathbf{e}=-\mathbf{f}$. Generalization of Theorem 1.3 to all
two-sided homogeneous curves in turn motivates us to consider the vector-
valued Calderón-Zygmund theory associated to one parameter group of dilations,
which is a project under progress.
As an application, Theorem 1.3 is used to deal with vector-valued anisotropic
singular integrals with homogeneous kernel by Calderón-Zygmund’s rotation
method. This work improves Hytönen’s Theorem 5.2 in [11] in some sense, see
Section 3 for more details.
In the next result, we deal with certain convex curves in $\mathbb{R}^{2}$
with the form $\Gamma(t)=\big{(}t,\gamma(t)\big{)}$, $\gamma(t)$ is some
convex function for $t\geq 0$.
###### Theorem 1.4.
Let $X$ be an UMD lattice belonging to the class $I_{(0,\frac{1}{5})}$,
$\gamma(t)$ be a continuous odd function, twice continuously differentiable,
increasing and convex for $t\geq 0$. Suppose also that $\gamma^{\prime\prime}$
is monotone for $t>0$ and there exists $C>0$ so that $\gamma^{\prime}(t)\leq
Ct\gamma^{\prime\prime}(t)$ for $t>0$. Then for $\frac{5}{3}<p<\frac{5}{2}$,
there exists an absolute constant $C_{p}$ such that
$\|\mathscr{H}f\|_{L^{p}(X)}\leq C_{p}\|f\|_{L^{p}(X)},\ \
f\in{L^{p}(\mathbb{R}^{n};X)}.$
A large class of functions $\gamma(t)$ satisfy the conditions in Theorem 1.4,
such as
$\gamma(t)=sgn(t)|t|^{\alpha},\ (\alpha\geq
2)\quad\textrm{and}\quad\gamma(t)=te^{-1/|t|}.$
The first one is homogeneous, while another one does not have any homogeneity.
This result is a vector-valued extension of Theorem 3.1 of Nagel and Wainger
in [20]. Theorem 1.4 also generalizes the second author’s result [16] in the
case $X=\ell^{q}$ with $5/3<q<5/2$. The proof of Theorem 1.4 is again
motivated by the recent development of the vector-valued Calderón-Zygmund
theory [14]. In fact, in Section 4, we prove a more general version, i.e.
Theorem 1.4 is also true if $X$ satisfies the following weaker condition:
there exist $\theta\in(0,\frac{1}{5})$, Hilbert space $H$ and UMD space $Y$
with property $(\alpha)$ (recalled in Section 4) such that $X=[H,Y]_{\theta}$.
## 2\. Proof of Theorem 1.3
The main arguments in this section are from [27], we will repeat some results
for completeness. Before the proof, we need some notations. Let matrix
$A=diag(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$, then
$\Gamma^{\prime}(t)=A\Gamma(t)/t$ for $t>0$. We also define a norm function
$\rho(x)$ by the unique positive solution of
$\sum^{n}_{i=1}x^{2}_{i}\rho^{-2\alpha_{i}}=1$
and $\rho(0)=0$. This definition was introduced in the pioneering work on
anisotropic singular integrals of Fabes [7]. Obviously,
$\rho(\delta_{t}x)=t\rho(x)$ for $t>0$, $\rho(x)=1$ if and only if the
Euclidean norm $|x|=1$ which means $x$ is on the unit sphere
${\mathbf{S}}^{n-1}$. See also Proposition 1-9 in [27] for more properties of
$\rho$. By a change of variables, we assume $\alpha_{1}=1$ and $\alpha_{i}\geq
1$ for $2\leq i\leq n$, and set
$\Delta=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}$. Without lost of generality,
we assume that $\alpha_{i}\neq\alpha_{j}$ when $i\neq j$, then $\Gamma(t)$
does not lie in a proper subspace of $\mathbb{R}^{n}$. If not, $\Gamma$ lies
in some proper subspace, then the argument of Stein and Wainger in [27,
pp.1262] implies our desired result.
For $z\in\mathbb{C}$, we define an analytic family of operators
${\mathscr{H}}_{z}$ by
$\widehat{\mathscr{H}_{z}f}(\xi)=\\{\rho(\xi)\\}^{z}m_{z}(\xi)\hat{f}(\xi),$
where $m_{z}$ are given by
$m_{z}(\xi)={\rm p.v.}\int_{\mathbb{R}}e^{-2\pi
i\xi\cdot\Gamma(t)}|t|^{z}\frac{dt}{t}.$
Obviously, ${\mathscr{H}}_{0}$ is our original operator $\mathscr{H}$.
As in [27], the desired result will be concluded by analytic interpolation
once we show the following two estimates: For Hilbert space $H$
$\big{\|}{\mathscr{H}}_{z}f\big{\|}_{L^{2}({\mathbb{R}}^{n};H)}\leq
C(z)\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{n};H)},$ (2.1)
where $-1<Re(z)\leq\sigma$ for some $\sigma>0$ and $C(z)$ grows at most
polynomially in $|z|$, and for UMD space $Y$
$\|\mathscr{H}_{z}f\|_{L^{p}(\mathbb{R}^{n};Y)}\leq
C(z,p)\|f\|_{L^{p}(\mathbb{R}^{n};Y)},\ \ 1<p<\infty,$ (2.2)
where $-\beta\leq Re(z)\leq-\eta$ for arbitrarily positive $\eta$ and some
positive $\beta$ as well as $C(z,p)$ grows at most as fast as a polynomial in
$|z|$ for fixed $\eta$.
Indeed, we obtain Theorem 1.3 by performing twice the analytic interpolation
argument in [25] as follows. Let $T_{z}f(x)=e^{z^{2}}{\mathscr{H}}_{z}f(x)$.
Note that $|e^{z^{2}}|=e^{Re(z)^{2}-Im(z)^{2}}$, then by (2.1) there exists a
constant $M_{0}$ which is independent of $Im(z)$ such that
$\big{\|}T_{z}f\big{\|}_{L^{2}(\mathbb{R}^{n};H)}\leq
C(z)e^{-Im(z)^{2}}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{n};H)}\leq
M_{0}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{n};H)}$ (2.3)
when $-1<{\rm Re}(z)<\sigma$. Also, for any UMD space $Y$ and
$q\in(1,\infty)$, by (2.2) there exists a constant $M_{1}$ which is
independent of $Im(z)$ such that
$\big{\|}T_{z}f\big{\|}_{L^{q}(\mathbb{R}^{n};{Y})}\leq
M_{1}\big{\|}f\big{\|}_{L^{q}(\mathbb{R}^{n};{Y})}\quad when\ \ -\beta<{\rm
Re}(z)<0.$ (2.4)
Obviously, this inequality holds in particular with $Y=H$.
For $1<p<\infty$, we choose $\theta_{1}\in(0,1)$, $\sigma_{1}<0$,
$0<\sigma_{0}<\sigma$ and $q_{1}\in(1,\infty)$ such that
$\sigma_{0}(1-\theta_{1})+\sigma_{1}\theta_{1}=:\sigma_{2}>0,\
\frac{1}{p}=\frac{1-\theta_{1}}{2}+\frac{\theta_{1}}{q_{1}}.$
Interpolating between (2.3) and (2.4) with $Y=H$, we have
$\big{\|}T_{z}f\big{\|}_{L^{p}(\mathbb{R}^{n};H)}\leq
C(p,z)\big{\|}f\big{\|}_{L^{p}(\mathbb{R}^{n};H)}\quad when\ \ {\rm
Re}(z)=\sigma_{2}>0.$ (2.5)
Note that $X=[H,Y]_{\theta}$ for some Hilbert space $H$, UMD space $Y$ and
$\theta\in(0,1)$. For fixed $\theta$, we choose $\sigma_{3}<0$ such that
$0=(1-\theta){\sigma_{2}}+\theta\sigma_{3}.$
In the same way, interpolating between (2.5) and (2.4) with $q=p$, we obtain
$\|\mathscr{H}f\|_{L^{p}(\mathbb{R}^{n};X)}=\|T_{0}f\|_{L^{p}(\mathbb{R}^{n};X)}\leq
C\|f\|_{L^{p}(\mathbb{R}^{n};X)}.$
The estimate (2.1) is trivial since Plancherel’s theorem remains true for
Hilbert space valued functions and the original arguments for Lemma 4.2 in
[27] work here. The novelty of the proof lies in the proof of (2.2). In the
case $Y=\ell^{q}$ with $1<q<\infty$, it has been proved in (2.2) in [22] by
Benedek, Calderón and Panzone’s argument since $L^{q}(\ell^{q})$-boundedness
is trivial. For general UMD space, we shall follow Hytönen and Weis’s idea
[14] established recently to prove the $L^{p}(Y)$ estimates simultaneously for
all $1<p<\infty$. The following subsection is devoted to the proof of estimate
(2.2).
### 2.1. The proof of (2.2)
The following proof is essentially the same as [11], we include it here for
the sake of completeness. From Lemma 4.4 of [27], we can write that
$\mathscr{H}_{z}f(x)=K_{z}\ast f(x),$
where
$K_{z}(x)=\int_{\mathbb{R}}h_{z}(x-\Gamma(t))|t|^{z}\frac{dt}{t}\ \text{and}\
\ \hat{h}_{z}(\xi)=\\{\rho(\xi)\\}^{z}.$
It is known that $h_{z}$ is a locally integrable function, $C^{\infty}$ away
from the origin satisfying
$h_{z}(\delta_{\lambda}x)=\lambda^{-\Delta-z}h_{z}(x),\;\lambda>0,\;x\neq 0.$
Moreover, each derivative of $h_{z}(x)$ is bounded by a polynomial in $|z|$,
if $\rho(x)=1$. In particular, $K_{z}$ has the homogeneity property
$\lambda^{\Delta}K_{z}\big{(}\delta_{\lambda}x\big{)}=K_{z}(x)$.
Let
$\hat{\mathcal{D}}_{0}(\mathbb{R}^{n})=\\{\psi\in\mathscr{S}(\mathbb{R}^{n})|\
\hat{\psi}\in\mathscr{D}(\mathbb{R}^{n}),0\notin supp\ \hat{\psi}\\}$. Let
$\eta\in\mathcal{D}(\mathbb{R}^{n})$ have range $[0,1]$, vanish for
$\rho(\xi)\geq 2$ and equal $1$ for $\rho(\xi)\leq 1$. For $j\in\mathbb{Z}$,
we define $\hat{\phi}_{0}(\xi)=\eta(\xi)-\eta(\delta_{2}\xi)$,
$\hat{\varphi}_{j}(\xi)=\hat{\phi}_{0}(\delta_{2^{-j}}\xi)$ and
$\hat{\chi}_{j}(\xi)=\hat{\phi}_{j-1}(\xi)+\hat{\phi}_{j}(\xi)+\hat{\phi}_{j+1}(\xi)$.
Then $\hat{\varphi}_{j}(\xi)$ is supported in the annulus
$\\{2^{j-1}\leq\rho(\xi)\leq 2^{j+1}\\}$, and
$\displaystyle\sum_{j}\hat{\varphi_{j}}(\xi)=1\ \ \text{for}\ \ \xi\neq 0.$
(2.6)
Moreover, since $\hat{\chi}_{j}$ equals 1 on the support of $\hat{\phi}_{j}$,
we have
$\displaystyle\phi_{j}=\phi_{j}\ast\chi_{j}\ast\chi_{j}.$ (2.7)
The estimate (2.2) will be deduced from the following key estimate which will
be shown in the next subsection.
###### Proposition 2.1.
Let $\phi_{0}$ and $K_{z}$ be defined as above. We have
$\displaystyle\int_{\mathbb{R}^{n}}|\phi_{0}\ast
K_{z}(x)|\log^{n}(e+\rho(x))dx\leq C(z).$
With above preparations at hand, we finish the proof of the estimate (2.2).
###### Proof.
For fixed $z$, we denote $K_{z}$ by $K$ for simplicity. Given
$f\in\hat{\mathcal{D}}_{0}(\mathbb{R}^{n})\otimes Y$,
$g\in\hat{\mathcal{D}}_{0}(\mathbb{R}^{n})\otimes Y^{*}$, by (2.6) and (2.7),
we have
$\displaystyle\langle g,K\ast f\rangle$ $\displaystyle=\langle\tilde{K}\ast
g,f\rangle=\sum_{j}\langle\phi_{j}\ast\tilde{K}\ast(\chi_{j}\ast
g),\chi_{j}\ast f\rangle,$
where the summation is finite and $\tilde{K}(x)=K(-x)$. Changing variable and
using the fact $\lambda^{\Delta}K_{z}(\delta_{\lambda}x)=K_{z}(x)$,
$(\phi_{j}\ast\tilde{K})\ast(\chi_{j}\ast
g)(x)=\int_{\mathbb{R}^{n}}\phi_{0}\ast\tilde{K}(y)(\chi_{j}\ast
g)(x-\delta_{2^{-j}}y)dy.$
Hence, by Hölder’s inequality and the Khintchine-Kahane inequality
$\displaystyle\big{|}\langle g,K\ast
f\rangle\big{|}=\big{|}\int_{\mathbb{R}^{n}}\mathbb{E}\langle\sum_{j}\epsilon_{j}\chi_{j}\ast
g(\cdot-\delta_{2^{-j}}y),\sum_{i}\epsilon_{i}\phi_{0}\ast K(y)\chi_{i}\ast
f\rangle dy\big{|}$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g(\cdot-\delta_{2^{-j}}y)\|_{L^{p^{\prime}}(Y^{*})}\mathbb{E}\|\sum_{i}\epsilon_{i}\chi_{i}\ast
f\|_{L^{p}(Y)}|\phi_{0}\ast K(y)|dy.$
It is easy to check that $m=\sum_{j}\epsilon_{j}\hat{\chi}_{j}$ is an
anisotropic multiplier. Hence, by Theorem 3 in [11], we have
$\displaystyle\|\sum_{j}\epsilon_{j}\chi_{j}\ast
f\|_{L^{p}(\mathbb{R}^{n};Y)}\leq C_{p,X}\|f\|_{L^{p}(\mathbb{R}^{n};Y)}.$
(2.8)
By Proposition 2.1 and (2.8), we shall finish the proof by showing
$\displaystyle\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g(\cdot-\delta_{2^{-j}}y)\|_{L^{p^{\prime}}(Y^{*})}\leq
C\log^{n}(e+\rho(y))\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g\|_{L^{p^{\prime}}(Y^{*})}.$
Let $e_{i}$ be the $i$-th standard unit vector. Above estimate is just a
$n$-fold application of
$\displaystyle\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g(\cdot-\delta_{2^{-j}}y_{i}e_{i})\|_{L^{p^{\prime}}(Y^{*})}\leq
C\log(e+\rho(y))\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g\|_{L^{p^{\prime}}(Y^{*})},$
which follows from Lemma 10 of Bourgain [2]. ∎
### 2.2. The proof of Proposition 2.1
The proof of Proposition 2.1 is based on the following two lemmas. The first
one states that the kernel $K_{z}$ satisfies a weighted Hörmander condition,
which will be verified at the end of this subsection.
###### Lemma 2.2.
If $-\beta\leq Re(z)\leq-\eta$, then for sufficiently large constants $C_{0}$
and $C_{1}(z)$, we have
$\displaystyle\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}(e+\rho(x))dx\leq
C_{1}(z)\log^{n}(e+\rho(y))$ (2.9)
for any $y\in\mathbb{R}^{n}\setminus\\{0\\}$. Moreover, $C_{1}(z)$ grows at
most as fast as a polynomial in $|z|$ for a fixed $\eta$.
The second lemma is a kind of decomposition lemma which has been established
in Lemma 4.10 of [14]. We reformulate it in our anisotropic case.
###### Lemma 2.3.
Let $\varphi\in\mathscr{S}(\mathbb{R}^{n})$ with vanishing integral. Then
there exists a decomposition $\varphi=\sum_{m\geq 0}\psi_{m}$ with the
following properties:
$\displaystyle\psi_{m}\in\mathcal{D}(\mathbb{R}^{n}),\;\mathrm{supp}\psi_{m}\subseteq\\{x|\
\rho(x)\leq C2^{\alpha m}\\},\;\int_{\mathbb{R}^{n}}\psi_{m}(y)dy=0,$
where $C$ and $\alpha$ are two universal constants only depending on the norm
$\rho$ and the dimension $n$, and for every $p\in[1,\infty]$ and every $M>0$,
the sequence of Lebesgue norms $\|\psi_{m}\|_{L^{p}}$, as well as
$\|\hat{\psi}_{m}\|_{L^{p}}$, is $\mathcal{O}(2^{-mM})$ as
$m\rightarrow\infty$.
###### Proof.
Let us give a quick explanation of this lemma. From Lemma 4.10 of [12],
$\psi_{m}$ is supported in $\\{x|\ |x|\leq 2^{m}\\}$. Fix $x\in\\{x|\ |x|\leq
2^{m}\\}$, by Proposition 1-9 of [27], if $\rho(x)\geq 1$, then
$\rho(x)\leq c_{1}|x|^{\alpha_{1}}\leq c_{1}2^{a_{1}m}$
and if $\rho(x)\leq 1$, then
$\rho(x)\leq c_{2}|x|^{a_{2}}\leq c_{2}2^{a_{2}m}$
with $c_{1},c_{2},a_{1},a_{2}$ positive constants. We obtain the desired
result by choosing $C=\max\\{c_{1},c_{2}\\}$ and
$\alpha=\max\\{a_{1},a_{2}\\}$. ∎
###### Proof of Proposition 2.1.
The main idea comes from [12], we include most details here for completeness.
By Lemma 2.3, we write $\phi_{0}=\sum_{m\geq 0}\psi_{m}$ with $\psi_{m}$’s
satisfying the properties stated in that lemma. Then we decompose $K_{z}$ into
pieces
$K_{z,m}(x)=K_{z}\ast\psi_{m}(x)$
and estimate each of them respectively.
We first estimate the integral outside the larger ellipsoid
$\mathcal{B}_{1}=\\{x|\ \rho(x)\leq CC_{1}2^{\alpha m}\\}$ with $C_{1}$ fixed
later depending on $C_{0}$. Recall that $\psi_{m}$ is supported in the
ellipsoid $\mathcal{B}_{0}=\\{x|\ \rho(x)\leq C2^{\alpha m}\\}$ and the
integral of $\psi_{m}$ vanishes, by Fubini’s theorem and Lemma 2.2, we obtain
$\displaystyle\int_{\mathcal{B}_{1}^{c}}|K_{z,m}(x)|\log^{n}(e+\rho(x))dx$
$\displaystyle=\int_{\mathcal{B}_{1}^{c}}|\int_{\mathcal{B}_{0}}K_{z}(x-y)\psi_{m}(y)dy|\log^{n}(e+\rho(x))dx$
$\displaystyle\leq\int_{\mathcal{B}_{0}}\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}(e+\rho(x))dx\psi_{m}(y)dy$
$\displaystyle\leq
C_{1}(z)\int_{\mathcal{B}_{0}}\log^{n}(e+\rho(y))\psi_{m}(y)dy\leq
C_{1}(z)\|\psi_{m}\|_{L^{\infty}}\int_{\mathcal{B}_{0}}\log^{n}(e+\rho(y))dy.$
By Lemma 2.3, the last quantity is of order $\mathcal{O}(2^{-m})$ as
$m\rightarrow\infty$ since $\|\psi_{m}\|_{L^{\infty}}\leq C_{M}2^{-mM}$ for
$M>0$ while
$\int_{\mathcal{B}_{0}}\log^{n}(e+\rho(y))dy\leq C2^{mN}$
for a fixed $N$.
Inside the ellipsoid $\mathcal{B}_{1}$, the computation is easier because of
the fact $\|\hat{K}_{z}\|_{L^{\infty}}\leq C(z)$, then
$\displaystyle\int_{\mathcal{B}_{1}}|K_{z,m}(x)|$
$\displaystyle\log^{n}(e+\rho(x))dx\leq\|K_{z,m}\|_{L^{\infty}}\int_{\mathcal{B}_{1}}\log^{n}(e+\rho(x))dx$
$\displaystyle\leq\int_{\mathcal{B}_{1}}\log^{n}(e+\rho(x))dx\|\hat{K}_{z,m}\|_{L^{1}}$
$\displaystyle=\int_{\mathcal{B}_{1}}\log^{n}(e+\rho(x))dx\int_{\mathbb{R}^{n}}|\hat{K}_{z}(\xi)\hat{\psi}_{m}(\xi)|d\xi$
$\displaystyle\leq\|\hat{K}_{z}\|_{L^{\infty}}\|\hat{\psi}_{m}\|_{L^{1}}\int_{\mathcal{B}_{1}}\log^{n}(e+\rho(x))dx\leq
C(z)2^{-m}.$
The last inequality holds due to the same reason that for the case outside the
ellipsoid. Finally, we obtain Proposition 2.1 by summing over $m$. ∎
To complete the proof of Proposition 2.1, we still need to show Lemma 2.2.
###### Proof of Lemma 2.2.
We follow the main sketch provided in [27], but improve related estimates. To
verify $K_{z}$ satisfying (2.9), we may assume that $\rho(y)=1$, it suffices
to prove that
$\int_{\rho(x)\geq
C_{0}}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx\leq C(z).$ (2.10)
In fact, we set $\lambda=\rho(y)$ and $y^{\prime}=y/\lambda$. Obviously,
$\rho(y^{\prime})=1$. By a linear transformation
$x=\delta_{\lambda}x^{\prime}$ and the homogeneity of $K_{z}$, we have
$\displaystyle\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle=$ $\displaystyle\int_{\rho(x^{\prime})\geq
C_{0}}|K_{z}(x^{\prime}-y^{\prime})-K_{z}(x^{\prime})|\log^{n}\big{(}e+\lambda\rho(x^{\prime})\big{)}dx^{\prime}.$
If $\lambda=\rho(y)\geq 6$, it is trivial that
$\log\big{(}e+\lambda\rho(x^{\prime})\big{)}\leq\log\big{(}e+\lambda\big{)}+\log\big{(}e+\rho(x^{\prime})\big{)}\leq\log\big{(}e+\lambda\big{)}\log\big{(}e+\rho(x^{\prime})\big{)},$
where we use the assumption that $C_{0}\geq 6$. Then,
$\displaystyle\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{\rho(x^{\prime})\geq
C_{0}}|K_{z}(x^{\prime}-y^{\prime})-K_{z}(x^{\prime})|\log^{n}\big{(}e+\rho(x^{\prime})\big{)}dx^{\prime}\log^{n}\big{(}e+\rho(y)\big{)}$
$\displaystyle\leq C(z)\log^{n}\big{(}e+\rho(y)\big{)}.$
When $\lambda=\rho(y)<6$, by (2.10), we get
$\displaystyle\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq 2^{n}\int_{\rho(x^{\prime})\geq
C_{0}}|K_{z}(x^{\prime}-y^{\prime})-K_{z}(x^{\prime})|\log^{n}\big{(}e+\rho(x^{\prime})\big{)}dx^{\prime}$
$\displaystyle\leq C(z)\leq C(z)\log^{n}\big{(}e+\rho(y)\big{)}.$
To prove (2.10), we define $K_{z}^{1}$ and $K_{z}^{2}$ by
$K_{z}^{1}(x)=\int_{|t|\leq 1}h_{z}(x-\Gamma(t))|t|^{z}\frac{dt}{t}\
\text{and}\ K_{z}^{2}(x)=K_{z}(x)-K_{z}^{1}(x),$
respectively. We split the integral as
$\displaystyle\int_{\rho(x)\geq
C_{0}}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{\rho(x)\geq
C_{0}}|K_{z}^{1}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle+\int_{\rho(x)\geq
C_{0}}|K_{z}^{1}(x-y)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle+\int_{\rho(x)\geq
C_{0}}|K_{z}^{2}(x-y)-K_{z}^{2}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx.$
To estimate first two summands, we need a estimate related to $h_{z}$, which
can be found in [27, pp.1273]. The homogeneity and smoothness of $h_{z}$ away
from origin imply that
$|h_{z}(x-y)-h_{z}(x)|\leq C(z)\frac{|y|}{\\{\rho(x)\\}^{\Delta+Re(z)+\mu}}$
(2.11)
for some $\mu>0$, provide $|y|/|x|$ is sufficiently small.
We set $\beta=\min\\{\mu,1\\}$. For the first integral, by using Fubini’s
theorem and (2.11), we have
$\displaystyle\int_{\rho(x)\geq
C_{0}}|K_{z}^{1}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{\rho(x)\geq C_{0}}\int_{|t|\leq
1}|h_{z}(x-\Gamma(t))-h_{z}(x)||t|^{Re(z)-1}dt\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{|t|\leq 1}|t|^{Re(z)-1}\int_{\rho(x)\geq
C_{0}}|h_{z}(x-\Gamma(t))-h_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dxdt$
$\displaystyle\leq\int_{|t|\leq 1}|t|^{Re(z)-1}|\Gamma(t)|\int_{\rho(x)\geq
C_{0}}\rho(x)^{-[\Delta+Re(z)+\mu]}\log^{n}\big{(}e+\rho(x)\big{)}dxdt$
$\displaystyle\leq C(z),$
where we use the fact that $-\beta<Re(z)<0$.
The norm function $\rho(x)$ have the property of $\rho(x+y)\leq
c\big{(}\rho(x)+\rho(y)\big{)}$ for some $c>0$(see Proposition 1-9 in [27]).
Specially, we set $C_{0}\geq\max\\{6,3c\\}$. Note that
$\rho(x-y)\geq\frac{1}{c}\rho(x)-\rho(y)\geq\frac{C_{0}}{c}-1\geq 2$ and
$\rho(x)\leq c[\rho(x-y)+\rho(y)]\leq c\rho(x-y)+c$. Using a linear
transformation, we treat the second summand as the first one,
$\displaystyle\int_{\rho(x)\geq
C_{0}}|K_{z}^{1}(x-y)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{\rho(x)\geq
2}|K_{z}^{1}(x)|\log^{n}\big{(}e+c+c\rho(x)\big{)}dx\leq C(z).$
Finally, using Fubini’s theorem, we have
$\displaystyle\int_{\rho(x)\geq
C_{0}}|K_{z}^{2}(x-y)-K_{z}^{2}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq$ $\displaystyle\int_{|t|\geq 1}\int_{\rho(x)\geq
C_{0}}\big{|}h_{z}\big{(}x-y-\Gamma(t)\big{)}-h_{z}\big{(}x-\Gamma(t)\big{)}\big{|}\log^{n}\big{(}e+\rho(x)\big{)}\frac{dxdt}{|t|^{1-Re(z)}}.$
We divide the inner integral above according to the distance between $x$ and
$\Gamma(t)$. Note that $\rho(y)=1$, if $|y|/|x-\Gamma(t)|$ is sufficient
small, that is $|x-\Gamma(t)|$ is away from the origin, we can get that
$\rho(x-\Gamma(t))\geq C_{2}$, where $C_{2}$ is an appropriate constant. In
this case, by (2.11) and a linear transformation, we obtain the following
estimate
$\displaystyle\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq C_{0}\\\
\rho(x-\Gamma(t))\geq
C_{2}\end{subarray}}\big{|}h_{z}\big{(}x-y-\Gamma(t)\big{)}-h_{z}\big{(}x-\Gamma(t)\big{)}\big{|}\log^{n}\big{(}e+\rho(x)\big{)}\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq C\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq
C_{0}\\\ \rho(x-\Gamma(t))\geq
C_{2}\end{subarray}}\frac{|y|}{\\{\rho\big{(}x-\Gamma(t)\big{)}\\}^{\Delta+\mu+Re(z)}}\log^{n}\big{(}e+\rho(x)\big{)}\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq C\int_{|t|\geq 1}\int_{\rho(x)\geq
C_{2}}\frac{1}{\\{\rho(x)\\}^{\Delta+\mu+Re(z)}}\log^{n}\big{(}e+c\rho(x)+ct\big{)}\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq C\int_{|t|\geq 1}\int_{\rho(x)\geq
C_{2}}\frac{1}{\\{\rho(x)\\}^{\Delta+\mu+Re(z)}}\big{\\{}\log^{n}\big{(}e+\rho(x)\big{)}+\log^{n}\big{(}e+t\big{)}\big{\\}}\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq C,$
where we use the fact that for fixed $|t|\geq 1$, $\rho(x)\leq
c[\rho(x-\Gamma(t))+\rho(\Gamma(t))]=c[\rho(x-\Gamma(t))+t]$.
It is trivial that $\rho\big{(}x+y+\Gamma(t)\big{)}\leq
c^{2}[\rho(x)+\rho(y)+\rho(\Gamma(t))]=c^{2}[1+\rho(x)+t]$. Then, the
remainder can be controlled by
$\displaystyle\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq C_{0}\\\
\rho(x-\Gamma(t))\leq
C_{2}\end{subarray}}[|h_{z}(x-y-\Gamma(t))|+|h_{z}(x-\Gamma(t))|]\log^{n}(e+\rho(x))\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq
C_{0}\\\ \rho(x-\Gamma(t))\leq
C_{2}\end{subarray}}|h_{z}\big{(}x-y-\Gamma(t)\big{)}|\log^{n}\big{(}e+\rho(x)\big{)}dx|t|^{Re(z)-1}dt$
$\displaystyle+\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq C_{0}\\\
\rho(x-\Gamma(t))\leq
C_{2}\end{subarray}}|h_{z}\big{(}x-\Gamma(t)\big{)}|\log^{n}\big{(}e+\rho(x)\big{)}dx|t|^{Re(z)-1}dt$
$\displaystyle\leq C\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\leq
c(C_{2}+1)\end{subarray}}|h_{z}(x)|dx|t|^{Re(z)-1}\log^{n}(e+t)dt$
$\displaystyle\leq C(z),$
where we use the fact that $h_{z}$ is locally integrable. ∎
## 3\. Anisotropic singular integrals
It was shown by Calderón and Zygmund [4] that the $L^{p}$-boundedness of
singular integrals with rough kernels can be deduced from the
$L^{p}$-boundedness of the (directional) Hilbert transform using the method of
rotations. In this section, we show a similar phenomenon happens, that is, the
$L^{p}(X)$-boundedness of Hilbert transforms along curve
$\Gamma(t)=(|t|^{\alpha_{1}}sgnt,|t|^{\alpha_{2}}sgnt,\cdots,|t|^{\alpha_{n}}sgnt)$
considered in the previous section implies the $L^{p}(X)$ boundedness of
singular integrals $T_{\Omega}$ with kernels of the form
$K(x)=\Omega(x)\rho(x)^{-\Delta}$, where $\Omega$ is a function on
$\mathbb{R}^{n}\setminus\\{0\\}$ satisfying the homogeneity
$\Omega(\delta_{t}x)=\Omega(x)\ \text{for all}\ t>0,$ size condition
$\int_{\mathbf{S}^{n-1}}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}|\Omega(\omega)|d\omega<\infty,$
(3.1)
and the cancelation condition
$\int_{\mathbf{S}^{n-1}}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}\Omega(\omega)d\omega=0,$
which can be understood from the following change-of-variable formula
$dx=t^{\Delta-1}\sum^{n}_{i=1}\alpha_{i}{\omega}^{2}_{i}dtd\omega.$
###### Theorem 3.1.
Let $X\in\mathcal{I}$. If $\Omega$ is odd, then the operators $T_{\Omega}$
described previously are bounded on $L^{p}(\mathbb{R}^{n};X)$ for
$1<p<\infty$.
Guliev [8] has obtained the boundedness of anisotropic singular integrals with
scalar valued-kernels on UMD lattices. Recently, Hytönen[11] generalized some
work of Guliev to the anisotropic singular integrals with operator-valued
kernels acting on UMD space. While their arguments require that $\Omega(x)$
should satisfy a kind of $L^{\infty}$-Dini condition, which is a much more
restricted condition than ours. So, Theorem 3.1 is a generalization of Hytönen
and Guliev’s result in this sense.
###### Proof.
Changing the variables, we find
$\displaystyle T_{\Omega}f(x)$ $\displaystyle={\rm
p.v.}\int_{\mathbb{R}^{n}}f\big{(}x-\delta_{\rho(y)}\delta_{\rho(y)}^{-1}y\big{)}\Omega(\delta_{\rho(y)}^{-1}y)\\{\rho(y)\\}^{-\Delta}dy$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbf{S}^{n-1}}f\big{(}x-\delta_{t}\omega\big{)}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}\Omega(\omega)d\omega\frac{dt}{t}.$
(3.2)
Note that $\Omega$ is odd, by a linear transformation, we also have
$T_{\Omega}f(x)=\int^{0}_{-\infty}\int_{\mathbf{S}^{n-1}}f\big{(}x+\delta_{(-t)}\omega\big{)}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}\Omega(\omega)d\omega\frac{dt}{t}.$
(3.3)
Using Fubini theorem, and adding (3.2) and (3.3) together, we get
$T_{\Omega}f(x)=\frac{1}{2}\int_{\mathbf{S}^{n-1}}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}\Omega(\omega)\big{[}\int_{-\infty}^{0}f\big{(}x+\delta_{(-t)}\omega\big{)}\frac{dt}{t}+\int_{0}^{\infty}f\big{(}x-\delta_{t}\omega\big{)}\frac{dt}{t}\big{]}d\omega.$
Then, it suffices to prove that
$\|\int_{-\infty}^{0}f\big{(}x+\delta_{(-t)}\omega\big{)}\frac{dt}{t}+\int_{0}^{\infty}f\big{(}x-\delta_{t}\omega\big{)}\frac{dt}{t}\|_{L^{p}(\mathbb{R}^{n};\mathbf{X})}\leq
C_{p}\|f\|_{L^{p}(\mathbb{R}^{n};X)},$
where the constant $C_{p}$ is independent of $\omega$.
For fixed $\omega\in\mathbf{S}^{n-1}$, define $\Gamma_{\omega}(t)$ as the
curve in the form of (1.1) associated to the dilation $\delta_{t}$ with
$\mathbf{e}=\omega$ and $\mathbf{f}=-\omega$, then the quantity inside the
norm of the previous inequality is the Hilbert transform along the curve
$\Gamma_{\omega}(t)$. The same arguments for the proof of Theorem 1.3 work
also for the curve $\Gamma_{\omega}(t)$, and we obtain the desired result. ∎
In the classical case (dilation given by $\delta_{t}x=tx$), it is known that
the boundedness of $T_{\Omega}$ is also obtained for the even function
$\Omega$ under a stronger size condition $\Omega\in
L\log^{+}L(\mathbf{S}^{n-1})$. The main ingredient is the existence of Riesz
transforms $R_{j},\;j=1,2,\cdots,n$, such that
1. (i)
$-\sum^{n}_{j=1}R_{j}\circ R_{j}=I$,
2. (ii)
the kernel of $T_{\Omega}\circ R_{j}$ is still homogeneous, and the associated
$\Omega_{j}$ is an odd function satisfying size condition (3.1).
In the anisotropic setting, it seems very difficult to find some replacements
for Riesz transforms such that similar properties as (i) and (ii) hold. Hence
we leave it as an open problem that whether Theorem 3.1 is still true for the
even function $\Omega$ under a stronger size condition.
## 4\. The proof of Theorem 1.4
The main argument for the proof is similar to that for Theorem 1.3. We first
introduce a family of analytic operators. For $z\in\mathbb{C}$, we define an
analytic family of operators ${\mathscr{H}}_{z}$ by
$\widehat{\mathscr{H}_{z}f}(\xi,\eta)=m_{z}(\xi,\eta)\hat{f}(\xi,\eta),$
where $m_{z}$ are given by
$m_{z}(\xi,\eta)={\rm p.v.}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\frac{dt}{t}.$
Obviously, ${\mathscr{H}}_{0}$ is our original operator $\mathscr{H}$.
Following the idea in [20], it suffices to prove the following two estimates:
$\big{\|}{\mathscr{H}}_{z}f\big{\|}_{L^{2}({\mathbb{R}}^{2};H)}\leq
C_{\delta}\big{[}1+|Im(z)|\big{]}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{2};H)},$
(4.1)
where $Re(z)=\frac{1}{4}-\delta$ for some $\delta>0$, and
$\big{\|}\mathscr{H}_{z}f\big{\|}_{L^{q}({\mathbb{R}}^{2};{Y})}\leq
C\big{[}1+|Im(z)|\big{]}^{2}\big{\|}f\big{\|}_{L^{q}({\mathbb{R}}^{2};{Y})},$
(4.2)
where $Y$ is an UMD lattice, $Re(z)<-1$, $1<q<\infty$, the constant $C$
depends on $Re(z)$ and is independent of $Im(z)$.
Indeed, we finish the proof by analytic interpolation argument [25]. Let
$T_{z}f(x)=e^{z^{2}}{\mathscr{H}}_{z}f(x)$. Note that
$|e^{z^{2}}|=e^{Re(z)^{2}-Im(z)^{2}}$, by (4.1) there exists a constant
$M_{0}$ which is independent of $Im(z)$ such that
$\big{\|}T_{z}f\big{\|}_{L^{2}(\mathbb{R}^{2};H)}\leq
C_{\delta}e^{-Im(z)^{2}}\big{[}1+|Im(z)|\big{]}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{2};H)}\leq
M_{0}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{2};H)}$
when ${\rm Re}(z)=\frac{1}{4}-\delta$. Also, for UMD lattice $Y$ and
$q\in(1,\infty)$, by (4.2) there exists a constant $M_{1}$ which is
independent of $Im(z)$ such that
$\big{\|}T_{z}f\big{\|}_{L^{q}(\mathbb{R}^{2};{Y})}\leq
M_{1}\big{\|}f\big{\|}_{L^{q}(\mathbb{R}^{2};{Y})}\quad when\ \ {\rm
Re}(z)<-1.$
This inequality also holds in particular with $Y=H$.
For $\frac{5}{3}<p\leq 2$, there exist $1<q<\infty$ and
$\theta_{0}\in(0,\frac{1}{5})$ so that
$\frac{1}{p}=\frac{1-\theta_{0}}{2}+\frac{\theta_{0}}{q}\ \ \text{and}\ \
(\frac{1}{4}-\delta)(1-\theta_{0})+(-1-\varepsilon_{0})\theta_{0}=:\sigma_{1}\in(0,\frac{1}{4})$
for some $\varepsilon_{0}>0$ and $0<\delta<\frac{1}{4}$. By interpolation of
analytic operators, we have
$\big{\|}T_{z}f\big{\|}_{L^{p}(\mathbb{R}^{2};H)}\leq
C(z)\big{\|}f\big{\|}_{L^{p}(\mathbb{R}^{2};H)}\quad for\ \ {\rm
Re}(z)=\sigma_{1}\in(0,1/4).$
Given an UMD lattice $X\in\mathcal{I}_{(0,1/5)}$, there exist a
$\theta\in(0,\frac{1}{5})$, a Hilbert space $H$ and another UMD lattice $Y$,
such that
$L^{p}(\mathbb{R}^{2};X)=[L^{p}(\mathbb{R}^{2};H),L^{p}(\mathbb{R}^{2};{Y})]_{\theta}$.
For such a $\theta$ and appropriate $\sigma_{1}$, we choose
$\varepsilon_{1}>0$ such that
$(1-\theta)\sigma_{1}+\theta(-1-\varepsilon_{1})=0$. Using interpolation of
analytic operators once more, we obtain
$\big{\|}\mathscr{H}f\big{\|}_{L^{p}(\mathbb{R}^{2};X)}\leq
C\big{\|}f\big{\|}_{L^{p}(\mathbb{R}^{2};X)}$
for $\frac{5}{3}<p\leq 2$. The duality argument implies the result for $2\leq
p<\frac{5}{2}$. This completes the proof of Theorem 1.4.
The estimate (4.1) holds since Plancherel’s theorem works also for Hilbert
space valued functions and the original argument in [20] can be repeated in
the present situation. The novelty of the proof lies in the estimate (4.2),
for which we need the vector-valued Fourier multiplier theorem established
recently.
Let us firstly recall some notations. A Banach space $X$ satisfies property
$(\alpha)$ if there is a positive constant $C$ such that
$\mathbb{E}\mathbb{E}^{\prime}\bigg{|}\sum^{N}_{k,l=1}\epsilon_{k}\epsilon_{l}^{\prime}\alpha_{kl}x_{kl}\bigg{|}_{X}\leq
C\mathbb{E}\mathbb{E}^{\prime}\bigg{|}\sum^{N}_{k,l=1}\epsilon_{k}\epsilon_{l}^{\prime}x_{kl}\bigg{|}_{X}$
for all $N\in\mathbb{N}$, all vectors $x_{kl}\in X$ and scalars
$|\alpha_{kl}|\leq 1$ $(1\leq k,l\leq N)$, where $\epsilon_{k}$,
$k\in\mathbb{Z}$ and $\epsilon_{l}^{\prime}$, $l\in\mathbb{Z}$ are two
identical independent sequences.
###### Remark 4.1.
The commutative $L^{p}$ spaces satisfy property $(\alpha)$ for all $1\leq
p<\infty$. Also, this property is inherited from $X$ by $L^{p}(\mu,X)$ for
$p\in[1,\infty)$. Every Banach space with a local unconditional structure and
finite cotype, in particular every Banach lattice, has property $(\alpha)$.
Let $m:{\mathbb{R}}^{n}\rightarrow\mathbb{C}$ be a bounded function, the
associated operator $T_{m}$ is defined on the test functions
$f\in{\mathscr{S}}({\mathbb{R}}^{n})\otimes X$ by
$T_{m}f(x)=(m\hat{f})^{\vee}(x).$
The sufficiency part of the following vector-valued Fourier multiplier theorem
was proved by Štrkalj and Weis [24], while the necessity of those conditions
was obtained by Hytönen and Weis [14].
###### Lemma 4.2.
The Marcinkiewicz-Lizorkin condition $|\xi^{\beta}||D^{\beta}m(\xi)|\leq C$
for all $\beta\in\\{0,1\\}^{n}$ is sufficient for the
$L^{p}({\mathbb{R}}^{n};X)$-boundedness of $T_{m}$, $n>1$, if and only if $X$
is an UMD space with property $(\alpha)$.
In view of Lemma 4.2 and Remark 4.1, to prove the estimate (4.2), it suffices
to show that the following functions
$m_{z}(\xi,\eta),\ \xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta),\
\eta\frac{\partial m_{z}}{\partial\eta}(\xi,\eta),\
\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)$
are uniformly bounded on $\mathbb{R}^{2}$ for $Re(z)<-1$.
The uniform boundedness of $m_{z}(\xi,\eta)$ is trivial, it can be showed by
minor modification of the proof of (4.1). Without repetition, we omit the
proof. The following estimates are essentially proved in [20], we include them
here for the sake of completeness.
The boundedness of $\xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta)$.
Integration by part implies that
$\displaystyle\xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta)$
$\displaystyle=$ $\displaystyle-2\pi i\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\xi\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\frac{d}{dt}(e^{-2\pi i\xi
t})e^{-2\pi i\eta\gamma(t)}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle=$ $\displaystyle e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\bigg{|}^{\infty}_{-\infty}$
$\displaystyle+$ $\displaystyle 2\pi i\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle-$ $\displaystyle 2z\eta^{2}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\gamma(t)\gamma^{\prime}(t)dt.$
Note that $Re(z)<-1$, for $t\in{\mathbb{R}}$, we have
$\big{|}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\big{|}=\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\leq
1$. The boundary terms are bounded by $1$.
For $Re(z)<-1$, making the change of variables $u=|\eta|\gamma(t)$, we obtain
$\displaystyle\bigg{|}\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt\bigg{|}$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}}\gamma^{\prime}(t)|\eta|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}dt$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}}\big{(}1+u^{2}\big{)}^{Re(z)}du\leq\pi.$
In a similar way, the second integrated term can be dominated by
$\displaystyle\bigg{|}z\eta^{2}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\gamma(t)\gamma^{\prime}(t)dt\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2|z|\int_{0}^{\infty}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)-1}\eta^{2}\gamma(t)\gamma^{\prime}(t)dt$
$\displaystyle\leq$ $\displaystyle|z|\int_{0}^{\infty}(1+u)^{Re(z)-1}du\leq
1+|Im(z)|.$
Therefore, for $Re(z)<-1$,
$\big{|}\xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta)\big{|}\leq
C\big{[}1+|Im(z)|\big{]}.$
The boundedness of $\eta\frac{\partial m_{z}}{\partial\eta}(\xi,\eta)$.
Integrating by parts, we obtain
$\displaystyle\eta\frac{\partial m_{z}}{\partial\eta}(\xi,\eta)$
$\displaystyle=$ $\displaystyle-2\pi i\ {\rm p.v.}\int_{\mathbb{R}}e^{-2\pi
i[\xi
t+\eta\gamma(t)]}\eta\gamma(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\frac{dt}{t}$
$\displaystyle+$ $\displaystyle 2z\ {\rm p.v.}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\frac{dt}{t}.$
To estimate above two integrals, we follow the argument used in the proof of
(4.1). For the first integral, for any $\varepsilon>0$, it suffices to bound
the following two parts
$\int_{\varepsilon<|t|<t_{0}}|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\frac{dt}{|t|}\
\ \text{and}\ \ \int_{|t|\geq
t_{0}}|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\frac{dt}{|t|}.$
Recall that $t_{0}>0$ was chosen so that $|\eta|\gamma(t_{0})=1$, and
$\gamma(t)\leq t\gamma^{\prime}(t)$ because of the convexity. Thus,
$\displaystyle\int_{\varepsilon<|t|<t_{0}}|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\frac{dt}{|t|}\leq
2|\eta|\int_{0}^{t_{0}}\frac{\gamma(t)}{t}dt\leq
2|\eta|\int_{0}^{t_{0}}\gamma^{\prime}(t)dt\leq 2.$
For $Re(z)<-1$, an elementary calculation implies that
$\int_{|t|\geq
t_{0}}|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\frac{dt}{|t|}\leq
2|\eta|^{2Re(z)+1}\int_{t_{0}}^{\infty}\gamma^{2Re(z)}(t)\frac{\gamma(t)}{t}dt\leq
2.$
Similarly, the second integral can be controlled by
$\displaystyle\bigg{|}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\frac{dt}{t}\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2|z|\int_{0}^{t_{0}}\eta^{2}\gamma^{2}(t)\frac{dt}{t}+2|z|\int_{t_{0}}^{\infty}\eta^{2}\gamma^{2}(t)\big{[}\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)-1}\frac{dt}{t}$
$\displaystyle\leq$ $\displaystyle
2|z|\eta^{2}\int_{0}^{t_{0}}\gamma(t)\gamma^{\prime}(t)dt+2|z|\eta^{2Re(z)}\int_{t_{0}}^{\infty}\gamma^{2Re(z)-1}(t)\gamma^{\prime}(t)dt$
$\displaystyle\leq$ $\displaystyle|z|+\frac{|z|}{|Re(z)|}\leq
2|Re(z)|\big{[}1+|Im(z)|\big{]}.$
Therefore, for $Re(z)<-1$,
$\big{|}\xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta)\big{|}\leq
C\big{[}1+|Im(z)|\big{]}.$
The boundedness of
$\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)$. To deal
with $\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)$, we
rewrite it as
$\displaystyle\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)$
$\displaystyle=$ $\displaystyle-4\pi^{2}\xi\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\gamma(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle-$ $\displaystyle 4\pi iz\xi\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\eta\gamma^{2}(t)dt.$
For the first term, integrating by parts, we obtain
$\displaystyle 4\pi^{2}\xi\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\gamma(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle=$ $\displaystyle 2\pi
i\int_{\mathbb{R}}\frac{d}{dt}\big{(}e^{-2\pi i\xi t}\big{)}e^{-2\pi
i\eta\gamma(t)}[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle=$ $\displaystyle 2\pi ie^{-2\pi i[\xi
t+\eta\gamma(t)]}[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\bigg{|}_{-\infty}^{\infty}$
$\displaystyle-$ $\displaystyle 4\pi^{2}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle-$ $\displaystyle 2\pi i\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle-$ $\displaystyle 4\pi iz\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\eta^{3}\gamma^{2}(t)\gamma^{\prime}(t)dt.$
Obviously, for $Re(z)<-1$, $t\in\mathbb{R}$, $\big{|}2\pi ie^{-2\pi i[\xi
t+\eta\gamma(t)]}[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\big{|}\leq
2\pi|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\leq 2\pi$.
So, the boundary terms are bounded by $2\pi$.
For the first integrated term, making the change of variables
$u=\eta^{2}\gamma^{2}(t)$, we have
$\displaystyle\bigg{|}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2\int_{0}^{\infty}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\eta^{2}\gamma(t)\gamma^{\prime}(t)dt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}\big{(}1+u\big{)}^{Re(z)}du\leq\frac{1}{|Re(z)+1|}.$
The second integrated terms can be treated in the same way, let
$u=\eta\gamma(t)$,
$\bigg{|}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt\bigg{|}\leq\int_{\mathbb{R}}(1+u^{2})^{Re(z)}du\leq\pi.$
Similarly, a trivial calculation shows that
$\displaystyle\bigg{|}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\eta^{3}\gamma^{2}(t)\gamma^{\prime}(t)dt\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2|z|\int_{0}^{\infty}u^{2}\big{(}1+u^{2}\big{)}^{Re(z)-1}du\leq\pi|z|.$
The second term can be handled similarly. Integrating by parts, we decompose
it as
$\displaystyle 4\pi iz\xi\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\eta\gamma^{2}(t)dt$
$\displaystyle=$ $\displaystyle 2\pi
iz\int_{\mathbb{R}}\frac{d}{dt}\big{(}e^{-2\pi i\xi t}\big{)}e^{-2\pi
i\eta\gamma(t)}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt$
$\displaystyle=$ $\displaystyle 2\pi ize^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\bigg{|}_{-\infty}^{\infty}$
$\displaystyle-$ $\displaystyle 4\pi^{2}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt$
$\displaystyle-$ $\displaystyle 4\pi iz\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma(t)\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt$
$\displaystyle-$ $\displaystyle 4\pi iz(z-1)\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-2}\eta^{2}\gamma(t)\gamma^{\prime}(t)dt.$
Obviously, for $Re(z)<-1$, $t\in\mathbb{R}$, $\big{|}ze^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\big{|}\leq|z|$.
The boundary terms are dominated by $4\pi|z|$.
For the first integrated term, by making the change of variables
$u=\eta\gamma(t)$, we have the estimate
$\displaystyle\bigg{|}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt\bigg{|}$
$\displaystyle\leq|z|\int_{\mathbb{R}}u^{2}(1+u^{2})^{Re(z)-1}dt$
$\displaystyle\leq\pi|z|.$
To estimate the second integrated terms, we make the transformation
$u=\eta^{2}\gamma^{2}(t)$ and get
$\displaystyle\bigg{|}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma(t)\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt\bigg{|}$
$\displaystyle\leq|z|\int_{0}^{\infty}(1+u)^{Re(z)-1}du$
$\displaystyle\leq\frac{|z|}{|Re(z)|}.$
Similarly, the third integrated terms can be treated as
$\displaystyle\bigg{|}z(z-1)\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-2}\eta^{4}\gamma^{3}(t)\gamma^{\prime}(t)dt\bigg{|}$
$\displaystyle\leq$
$\displaystyle|z(z-1)|\int_{0}^{\infty}(1+u)^{Re(z)-1}du\leq\frac{|z(z-1)|}{|Re(z)|}.$
Note that for $Re(z)<-1$, we have the following elementary estimates
$|z|\leq|Re(z)|\big{[}1+|Im(z)|\big{]}\ \ \text{and}\ \
|z-1|\leq|Re(z)-1|\big{[}1+|Im(z)|\big{]}.$
Finally, combining the above eight estimates, we obtain
$\big{|}\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)\big{|}\leq
C\big{[}1+Im(z)\big{]}^{2}.$
This completes the proof of Theorem 1.4.
Acknowledgement. The first author is supported in part by MINECO: ICMAT Severo
Ochoa project SEV-2011-0087 and ERC Grant StG-256997-CZOSQP (EU); The second
author is supported in part by NSFC 11371057 and 11471033. The authors would
like to thank the referee for many valuable and useful comments and
suggestions which have improved this paper.
## References
* [1] A. Benedek, A.P. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Natl. Acad. Sci. USA. 48 (1962), no. 3, 356–365.
* [2] J. Bourgain, Vector-valued singular integrals and the $H^{1}$-BMO duality, Probability theory and harmonic analysis, 1-19, Textbooks Pure Appl. Math. Dekker, New York, 1986.
* [3] D.L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II(Chicago, III., 1981), 270-286, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983.
* [4] A.P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), no. 1, 85–139.
* [5] H. Carlsson, M. Christ, A. Córdoba, J. Duoandikoetxea, J.L. Rubio de Francia, J. Vance, S. Wainger and D. Weinberg, $L^{p}$ estimates for maximal functions and Hilbert transforms along flat convex curves in $\mathbb{R}^{2}$, Bull. Amer. Math. Soc. 14 (1986), no. 2, 263–267.
* [6] M. Christ, A. Nagel, E.M. Stein and S. Wainger, Singular and maximal radon transform: Analysis and geometry, Ann. of Math. 150 (1999), no. 2, 489–577.
* [7] E.B. Fabes, Singular integrals and partial differential equations of parabolic type, Studia Math. 28 (1966), no. 1, 81–131.
* [8] V.S. Guliev, Imbedding theorems for spaces of UMD-valued functions, Dokl. Akad. Nauk. 329 (1993), no. 4, 408–410.
* [9] G. Hong and J. Parcet, Necessity of property $(\alpha)$ for for vector-valued Littlewood-Paley sets associated sumsets, in progress.
* [10] G. Hong, L.D. López-Sánchez, J.M. Martell, and J. Parcet, Calderón-Zygmund Operators Associated to Matrix-Valued Kernels, Int. Math. Res. Not. 2014 (2014), no. 5, 1221–1252.
* [11] T. Hytönen, Anisotropic Fourier multipliers and singular integrals for vector-valued functions, Ann. Mat. Pura Appl. 186 (2007), no. 3, 455–468.
* [12] T. Hytönen and L. Weis, Singular convolution integrals with operator-valued kernel, Math. Z. 255 (2007), no. 2, 393–425.
* [13] T. Hytönen, Littlewood-Paley-Stein theory for semigroups in UMD spaces, Rev. Mat. Iberoam. 23 (2007), no. 3, 973–1009.
* [14] T. Hytönen and L. Weis, On the necessity of property $(\alpha)$ for some vector-valued multiplier theorems, Arch. Math.(Basel) 90 (2008), no. 1, 44–52.
* [15] M. Junge, T. Mei and J. Parcet. Smooth Fourier multipliers on group von Neumann algebras, Geom. Funct. Anal. 24 (2014), no. 6, 1913–1980 .
* [16] H. Liu, Hilbert transforms along convex curves for valued functions, ISRN Math. Anal. 2014 (2014), Article ID 827072, doi:10.1155/2014/827072.
* [17] T.R. McConnell, On Fourier multiplier transformations of Banach-valued functions, Trans. Amer. Math. Soc. 285 (1984), no. 2, 739–757.
* [18] T. Mei, Operator valued Hardy spaces, Mem. Amer. Math. Soc. 188 (2007).
* [19] A. Nagel, N.M. Rivière and S. Wainger, On Hilbert transforms along curves, II, Amer. J. Math. 98 (1976), no. 2, 395–403.
* [20] A. Nagel and S. Wainger, Hilbert transforms associated with plane curves, Trans. Amer. Math. Soc. 223 (1976), 235–252.
* [21] J. Parcet, Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory, J. Funct. Anal. 256 (2009), no. 2, 509–593.
* [22] J.L. Rubio de Francia, F.J. Ruiz and J.L. Torra, Calderón-Zygmund theory for operator-valued kernels, Adv. Math. 62 (1986), no. 1, 7–48.
* [23] J.L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Probability and Banach spaces (Zaragoza, 1985), 195-222, Lecture Notes in Math. 1221, Springer, Berlin, 1986\.
* [24] Ž. Štrkalj and L. Weis , On operator-valued Fourier multiplier theorems, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3529–3547.
* [25] E.M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), no. 2, 482–492.
* [26] E.M. Stein and S. Wainger, The estimation of an integrals arising in multiplier transformations, Studia Math. 35 (1970), no. 1, 101–104.
* [27] E.M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer, Math. Soc. 84 (1978), no. 8, 1239–1295.
* [28] F. Zimmermann, On vector-valued Fourier multiplier theorems, Studia Math. T. XCIII (1989), 201-222.
|
arxiv-papers
| 2014-03-02T09:32:52 |
2024-09-04T02:49:59.157304
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Guixiang Hong, Honghai Liu",
"submitter": "Honghai Liu",
"url": "https://arxiv.org/abs/1403.0177"
}
|
1403.0181
|
# Vertex Operators, $\mathbb{C}^{3}$ Curve, and Topological Vertex
###### Abstract
In this article, we prove the conjecture that Kodaira-Spencer theory for the
topological vertex is a free fermion theory. By dividing the $\mathbb{C}^{3}$
curve into core and asymptotic regions and using Boson-Fermion correspondence,
we construct a generic three-leg correlation function which reformulates the
topological vertex in a vertex operator approach. We propose a conjecture of
the correlation function identity which in a degenerate case becomes Zhou’s
identity for a Hopf link.
Jian-feng Wu 1 and Jie Yang2,3
_1 Institute of Theoretical Physics, Department of applied mathematics_
_and physics, Beijing University of Technology, Beijing, 100124, China_
[email protected]
2 _Beijing Center for Mathematics and Information Interdisciplinary Sciences_
3 _School of Mathematical Sciences, Capital Normal University, Beijing,
100048, China_
[email protected]
## 1 Introduction
It has been proposed that the Chern-Simons theory of a gauge group $U(N)$ in
the large $N$ limit is dual to A-model topological string theory [1]. [2]
provided a brane configuration of the knot of Chern-Simons theory. [3]
discovered a quantum structure of Chern-Simons theory and from the well-known
Wess-Zumino-Witten (WZW) model it even discovered the deeper relation between
knot invariants in Chern-Simons theory with $SU(2)$ gauge group and characters
of WZW model. Later the gauge group has been generalized to $U(N)$ [4]. In the
large $N$ limit some knot invariants such as the loop and the Hopf link are
shown to be directly related to symmetric functions such as Schur and skew
Schur functions. In [5] the authors discovered some interesting vertex
structure for some geometrical and physical invariants such as the Donaldson-
Thomas invariants of ${\mathbb{C}}^{3}$ or in physical language, BPS
invariants of D0-D6 branes on ${\mathbb{C}}^{3}$. From statistical point of
view it is the partition function of a crystal melting model. [6] extended
this structure to a general case where there are asymptotic boundaries of
those crystals and related the partition function to a topological vertex.
They managed to build up this connection because of Zhou’s identity [26].
In [7], the authors achieved a B-model approach to the topological vertex
based on the observation of the mirror curve of $\mathbb{C}^{3}$ and the
related symmetries. However, an explicit correspondence between A-model and
B-model is still an open question.
We try to find a more obvious relation between A- and B-model in this article.
In our approach, the curve of $\mathbb{C}^{3}$ is essential. In A-model
description, $\mathbb{C}^{3}$ could be seen as a cotangent bundle $T^{*}S^{3}$
with a single $S^{3}$ as the base. However, as the CS/WZW correspondence [3]
saying, topological invariants in A-model becomes correlation functions in
B-model, where there is an modular $S$ transformation inserted between bra and
ket states. This correspondence strongly implies the mirror curve of
$\mathbb{C}^{3}$ can not be expressed simply in one coordinate chart. We need
at least two coordinate charts , one being related to another by $S$
transformation. Thus a complete B-model mirror curve of $\mathbb{C}^{3}$ has
an asymptotic region (near infinity where the bra state is inserted in) and a
core region (near origin where the ket state is inserted in) with point 1 the
fix point. This means the A-model theory is a union of CFTs in two regions
with a defect inserted at point 1. Surprisingly, if we introduce an excitation
at point 1, the Hamiltonian blows up the excitation and forms a distribution
corresponding to the representation of the excitation. This is very much like
the so-called projective representation of affine algebra as in [23]. Also in
[12], this structure had been introduced without proof.
The CFT considered at hand is a Kodaira-Spencer theory as explained in [7],
which by definition is a bosonic theory, equipped with a broken
$\mathcal{W}_{1+\infty}$ symmetry. A conjecture also was proposed in [7] that
the corresponding fermionic theory is a free fermion theory. We prove in this
article that for the topological vertex case, where the unbroken $W$ symmetry
is $W^{3}_{0}$, this conjecture is true. For other cases, for example,
$W^{4}$, if one would like to keep the integrable structure, the corresponding
fermionic theory is still a free one. It is compatible with Dijkgraaf’s work
[17] two decades ago.
Free fermion has been used in many research areas of physics. In [13] two-
dimensional Yang-Mills theory of $U(N)$ gauge group the Vandermonde of group
measure implies there is a fermionic structure. In two dimensions, due to
Boson-Fermion correspondence, vertex operator is a very useful tool. In
B-model [7] provided a beautiful explanation of a B-brane insertion as a
fermion field and symmetries of the Riemann surface as the sources of
transition function of sections of fiber bundles of fermionic fields. Because
the duality between A-model and B-model, we would expect a similar structure
in the A-model side. In this paper we aim to discover this structure and
approach this topic via a vertex operator formalism.
The structure of this paper is following. In sec. 2 we clarify some notation
to be used in this paper and make some preparation. In sec. 3 we provide a
generating function for the vertex operator. In sec. 4 we obtain the fermionic
expression for $W^{n}_{0}$ and prove the free fermion conjecture for
$W^{3}_{0}$. We also examine the curve of $\mathbb{C}^{3}$ in different
regions and related symplectic transformations. In sec. 5 we solve Hamiltonian
equations for two different coordinate charts and obtain wave functions. We
construct the correlation function with three generic representations inserted
at three points: 0,1,$\infty$ in a single patch. The cyclic symmetry of the
vertex becomes a conjecture of an identity for the correlation function. In
the limitation situation, this correlation function identity becomes Zhou’s
identity of Hopf link. In sec. 6 we point out the future working direction.
## 2 Notations and Preliminaries
A partition $\lambda$ is any sequence
$\lambda=(\lambda_{1}\,,\lambda_{2}\,,\lambda_{3}\,,\cdots)$ of non-negative
integers in weakly decreasing order:
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\cdots.$
The diagram of a partition $\lambda$ may be formally defined as the set of
points $(i,j)\in{\mathbb{Z}}^{2}$ such that $1\leq j\leq\lambda_{i}$. More
often it is convenient to replace the points by squares. It is also called a
Young diagram. The conjugate of a partition $\lambda$ is denoted by
$\lambda^{t}$ whose diagram is obtained by reflection in the main diagonal of
$\lambda$.
A Schur function $s_{\lambda}(z_{i})$ is a symmetric function of
$z_{i}\,,i=1,2,\cdots,\infty$’s and labeled by partition $\lambda$.
Especially, when
$z_{i}=q^{\rho_{i}}=q^{-i+\frac{1}{2}}$
there is a very useful product formula for $s_{\lambda}$
$s_{\lambda}(q^{-\rho})=\frac{q^{||\lambda^{t}||/2}}{\prod_{(i,j)\in\lambda}1-q^{h(i,j)}}\,,$
(2.1)
where $||\lambda^{t}||=\sum_{i}{(\lambda^{t}_{i})^{2}}$ and $h(i,j)$ is the
hook length of square $(i,j)$. Sometimes it is more useful to represent the
hook length as $h(i,j)=a(i,j)+l(i,j)+1$ where $a(i,j)$ and $l(i,j)$ are arm-
length and leg-length respectively and
$\displaystyle a(i,j)=\lambda_{i}-j\,,\quad l(i,j)=\lambda^{t}_{j}-i.$
### 2.1 Zhou’s Hopf link identity
The Hopf link is defined by
$W_{\lambda\mu}=W_{\lambda}\,s_{\mu}(q^{\lambda+\rho}),$ (2.2)
where
$W_{\lambda}=(-1)^{\lambda}q^{\kappa_{\lambda}/2}\,s_{\lambda}(q^{-\rho})=s_{\lambda}(q^{\rho}),$
$\rho=-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\cdots$, and $q=e^{g_{s}}$.
According to the duality between Chern-Simons and A-model topological string
theory, $g_{s}$ is the string coupling constant. $\kappa_{\mu}$ is the
”energy” of the representation $\mu$ and
$\kappa_{\mu}=\sum_{i}{\mu_{i}^{2}}-\sum_{j}{\mu^{t}_{j}}^{2}=\sum_{i\leq\ell(\mu)}\left((\mu+\rho)_{i}^{2}-\rho_{i}^{2}\right)$
where $\ell(\mu)$ is the length of the partition $\mu$, that is $\mu_{1}^{t}$.
Zhou’s Hopf link identity can be written as
$\displaystyle
q^{\kappa_{\mu^{t}}/2}s_{\lambda}(q^{-\rho})s_{\mu^{t}}(q^{-\lambda^{t}-\rho})=\sum_{\eta}s_{\lambda/\eta}(q^{-\rho})s_{\mu/\eta}(q^{-\rho})\,,$
(2.3)
where $s_{\lambda/\eta}(q^{-\rho})$ is the skew Schur function, see [21] for a
detailed definition. For a special case $\lambda=\phi$ it gives rise to an
interesting identity 111$\phi$ denotes the empty Young diagram:
$s_{\mu}(q^{-\rho})=q^{\kappa_{\mu^{t}}/2}s_{\mu^{t}}(q^{-\rho})\,.$ (2.4)
With the help of the useful identity 222[5] and [27] obtained a different
formula
$s_{\mu/\eta}(q^{-\nu-\rho})=(-)^{|\mu|+|\eta|}s_{\mu^{t}/\eta^{t}}(q^{\nu+\rho})\,,$
which is not correct. In [27] there was only a minor typo in the last line in
the derivation of eq. (28). In [5] there was no derivation. We provide an
independent derivation in App. B. We thank Professor Guo-ce Xin for pointing
out the problem for us.[5, 27]:
$s_{\mu/\eta}(q^{-\nu-\rho})=(-)^{|\mu|+|\eta|}s_{\mu^{t}/\eta^{t}}(q^{\nu^{t}+\rho})$
we obtain
$\displaystyle
W_{\lambda\mu}=(-)^{|\lambda|+|\mu|}q^{\kappa_{\lambda}/2+\kappa_{\mu}/2}\sum_{\eta}s_{\lambda/\eta}(q^{-\rho})s_{\mu/\eta}(q^{-\rho}).$
(2.5)
[26] provided a mathematical proof of this Hopf link identity. Since this
identity has a lot of applications in both mathematics and physics, see, e.g.
[6, 28], we expect to uncover its origin from a physical point of view.
However, a concrete physical proof is still an open problem for us.
## 3 Vertex Operators and Generating Functions
Let us first introduce some basic ingredients of conformal field theory (CFT)
of holomorphic boson and also chiral fermion fields.
### 3.1 Bosonization and Fermionization
For a fermion-antifermion system defined on a complex plane, we have the
following chiral fermion fields (Neuve-Schwarz fermions):
$\displaystyle\psi(z)=\sum_{r\in\mathbb{Z}+\frac{1}{2}}\psi_{r}z^{-r-1/2}\,,$
(3.1)
$\displaystyle\psi^{*}(z)=\sum_{s\in\mathbb{Z}+\frac{1}{2}}\psi^{*}_{s}z^{-s-1/2}.$
They have the following operator product expansions (OPEs):
$\displaystyle\psi(z)\psi^{*}(z^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{z-z^{\prime}}:\psi(z)\psi^{*}(z^{\prime}):+\cdots\,$
(3.2) $\displaystyle\psi^{*}(z)\psi(z^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{z-z^{\prime}}:\psi^{*}(z)\psi(z^{\prime}):+\cdots\,\,,$
and also the anti-commutation relations:
$\\{\psi_{n},\psi^{*}_{m}\\}=\delta_{n+m,0}\,,\quad\text{others}=0\,,\quad\quad(\psi_{n})^{*}=\psi^{*}_{-n}\,.$
(3.3)
In another side, the holomorphic bosonic field $\varphi(z)$ is given as
following
$\displaystyle\varphi(z)$ $\displaystyle=$ $\displaystyle q_{0}+p_{0}\ln
z+\sum_{n\neq 0}\frac{a_{n}}{-n}z^{-n},$ (3.4) $\displaystyle[a_{n},a_{m}]$
$\displaystyle=$ $\displaystyle n\delta_{n+m,0}\,\,,\,[p_{0},q_{0}]=1$
$\displaystyle\varphi(z)\varphi(w)$ $\displaystyle=$
$\displaystyle\ln(z-w):\varphi(z)\varphi(w):\,,$ (3.5)
If $\bar{\varphi}$ denotes the corresponding anti-holomorphic bosonic field
then a free bosonic field $\varphi(z,\bar{z})$ can be constructed as:
$\varphi(z,\bar{z})=\varphi(z)-\bar{\varphi}(\bar{z})\,.$
In this article, we only make the holomorphic part of the bosonic field
dynamic and leave the anti-holomorphic part non-dynamic. While this boson is
called a chiral boson and the corresponding vertex operators are called chiral
vertex operators. A chiral vertex operator may be written as
$V_{\alpha}(z)=e^{\alpha\varphi(z)},$
its conjugation is
$(V_{\alpha}(z))^{*}=e^{-\alpha\varphi(z^{*})}=V_{-\alpha}(z^{*}).$
However, we just consider the case that $\alpha=1$ and denote
$\displaystyle V(z)=e^{\varphi(z)}\,\,,\,\,V^{*}(z)=e^{-\varphi(z)}.$ (3.6)
From modes expansion and Heisenberg algebra (3.4), we can calculate the OPEs
of $V(z)$ and $V^{*}(z^{\prime})$:
$\displaystyle:V(z)::V(z^{\prime}):$ $\displaystyle=$
$\displaystyle:VV(z^{\prime}{}):(z-z^{\prime})+reg.$ (3.7)
$\displaystyle:V^{*}(z)::V^{*}(z^{\prime}):$ $\displaystyle=$
$\displaystyle:V^{*}V^{*}(z^{\prime}{}):(z-z^{\prime})+reg.$
$\displaystyle:V^{*}(z)::V(z^{\prime}):$ $\displaystyle=$
$\displaystyle:V^{*}V(z^{\prime}):\frac{1}{(z-z^{\prime})}-\partial{\varphi(z^{\prime})}+reg.$
(3.8) $\displaystyle:V(z)::V^{*}(z^{\prime}):$ $\displaystyle=$
$\displaystyle:VV^{*}(z^{\prime}):\frac{1}{(z-z^{\prime})}+\partial{\varphi(z^{\prime})}+reg.,$
here $reg.$ means regular terms. The singular parts of these OPEs are the same
as those of chiral fermions in eq.(3.2). However chiral fermions have no self-
contractions they are not completely the same as $V$ and $V^{*}$. Nevertheless
according to Pauli’s exclusive principle, namely the fermionic statistics, the
correlation function of fermions is related to Slater determinant
$\displaystyle\langle
vac|\prod_{i=1}^{N}\psi(z_{i})\prod_{j=1}^{N}\psi^{*}(w_{j})|vac\rangle$
$\displaystyle=$ $\displaystyle\langle
0|\prod_{i=1}^{N}V(z_{i})\prod_{j=1}^{N}V^{*}(w_{j})|0\rangle$
$\displaystyle={\rm Det}(\frac{1}{z_{i}-w_{j}})$ $\displaystyle=$
$\displaystyle\frac{\prod_{i<j}(z_{i}-z_{j})(w_{i}-w_{j})}{\prod_{i,j=1}^{N}(z_{i}-w_{j})}\,.$
(3.9)
It reminds us the miraculous boson/fermion correspondence. Therefore during
the calculation of the correlation function, we can replace all fermionic
fields by bosonic vertex operators such that
$\psi(z)\sim V(z)=e^{\varphi(z)}\,,\quad\psi^{*}(z)\sim
V^{*}(z)=e^{-\varphi(z)}\,.$ (3.10)
However, we need to bear it in mind that chiral fermions are not exactly these
vertex operators because of the different self OPEs. Secondly there could be
different numbers of $V$ and $V^{*}$ in the correlation function by carefully
choosing the charges of bra and ket vacua. But if there are different number
of $\psi$ and $\psi^{*}$, the correlation function is automatically vanishing.
In another way, since both fermionic and bosonic theories have the same $U(1)$
symmetry, the charge is measured by the number of $\partial\varphi(z)$ in the
bosonic theory and $\psi\psi^{*}(z)$ in the fermionic theory. Hence we have
the fermionization as follows:
$\partial\varphi(z)=(\psi\psi^{*})(z)\,,$ (3.11)
in terms of the modes expansion that is
$a_{n}=\sum_{r\in{\mathbb{Z}}+1/2}:\psi_{n-r}\psi^{*}_{r}:\,.$
### 3.2 Generating Functions
In the following we denote $V_{+}$ and $V_{-}$ as the positive and negative
modes part of $e^{\varphi}$ and $V_{+}^{*}$ and $V_{-}^{*}$ as the
corresponding part of $e^{-\varphi}$, that is,
$\displaystyle
V_{+}(z)=\exp\left\\{\sum_{n>0}\frac{a_{n}}{-n}z^{-n}\right\\},\quad
V_{-}(z)=\exp\left\\{\sum_{n>0}\frac{a_{-n}}{n}z^{n}\right\\},$ (3.12)
$\displaystyle
V_{+}^{*}(z)=\exp\left\\{\sum_{n>0}\frac{a_{n}}{n}z^{-n}\right\\},\quad
V_{-}^{*}(z)=\exp\left\\{\sum_{n>0}\frac{a_{-n}}{-n}z^{n}\right\\}\,.$
They form four types of generating functions of Schur functions, namely
$\displaystyle\prod_{i}V_{-}(z_{i})$ $\displaystyle=$
$\displaystyle\sum_{\lambda}s_{\lambda}(z_{i})s_{\lambda}(a_{-}),$ (3.13)
$\displaystyle\prod_{i}V_{+}^{*}(z_{i})$ $\displaystyle=$
$\displaystyle\sum_{\lambda}s_{\lambda}(z_{i}^{-1})s_{\lambda}(a_{+}),$ (3.14)
$\displaystyle\prod_{i}V^{*}_{-}(z_{i})$ $\displaystyle=$
$\displaystyle\sum_{\lambda}(-1)^{|\lambda|}s_{\lambda}(z_{i})s_{\lambda^{t}}(a_{-}),$
(3.15) $\displaystyle\prod_{i}V_{+}(z_{i})$ $\displaystyle=$
$\displaystyle\sum_{\lambda}(-1)^{|\lambda|}s_{\lambda}(z_{i}^{-1})s_{\lambda^{t}}(a_{+}).$
(3.16)
We will use these generating functions frequently in our calculation. They
also can be deduced from four basic generating functions such that:
$\displaystyle V_{-}(z)$ $\displaystyle=$
$\displaystyle\sum_{r\in\mathbb{Z}^{+}}s_{(r)}(z)s_{(r)}(a_{-})$ (3.17)
$\displaystyle V_{-}^{*}(z)$ $\displaystyle=$
$\displaystyle\sum_{r\in\mathbb{Z}^{+}}(-)^{r}s_{(r)}(z)s_{(1^{r})}(a_{-})$
$\displaystyle V_{+}(z)$ $\displaystyle=$
$\displaystyle\sum_{r\in\mathbb{Z}^{+}}(-)^{r}s_{(r)}(1/z)s_{(1^{r})}(a_{+})$
$\displaystyle V_{-}^{*}(z)$ $\displaystyle=$
$\displaystyle\sum_{r\in\mathbb{Z}^{+}}s_{(r)}(1/z)s_{(r)}(a_{+})\,,$
where $(r)$ ($(1^{r})$) denotes a length-$r$ horizontal (vertical) Young
diagram. Actually, the Schur polynomials of the horizontal and vertical Young
diagrams are the same as the complete (homogeneous) and elementary symmetric
polynomials respectively, that is,
$s_{(r)}(z)=h_{r}(z),\quad s_{(1^{r})}=e_{r}(z)\,.$
### 3.3 Fermionic Vacua and Maya/Young Correspondence
The vacuum of free fermion theory corresponds to a filled Dirac sea. Firstly,
for the ket vacuum, we denote $|\Omega\rangle$ as the ’fake’ vacuum of the
theory, which is annihilated by all modes of $\psi$ but not $\psi^{*}$. Since
$\psi^{*}$ is the anti-particle field of $\psi$, the positive modes of
$\psi^{*}$ should be understood as creation operators of anti-particles. The
’real’ (physical) ket vacuum should have a natural Dirac sea structure and is
denoted by $|vac\rangle$ and defined as
$|vac\rangle=\psi_{1/2}^{*}\psi_{3/2}^{*}\psi_{5/2}^{*}\cdots|\Omega\rangle\,.$
(3.18)
The bra vacuum could be defined by conjugation of ket vacuum
$\langle vac|=\langle\Omega|\cdots\psi_{-5/2}\psi_{-3/2}\psi_{-1/2}\,.$
A unitary excitation on the ket vacuum $|vac\rangle$ always contains pairs of
particle-anti-particle. To track the sign of the excited state, we would
better define an excited state as follows
$(-)^{\sum_{i}^{n}s_{i}-1/2}\prod_{i=1}^{n}\psi_{-r_{i}}\psi^{*}_{-s_{i}}|vac\rangle\,,$
(3.19)
where the subscripts $r_{i},s_{i}$ are positive half integers
($\mathbb{Z}_{>0}-1/2$).
Since the choice of particle or anti-particle is arbitrary we could choose the
’fake’ vacuum $|\Omega^{\prime}\rangle$ as the one annihilated by all modes of
$\psi^{*}$. Therefore the definition of bra and ket vacua should be changed as
$\displaystyle|vac^{\prime}\rangle=\psi_{1/2}\psi_{3/2}\psi_{5/2}\cdots|\Omega^{\prime}\rangle$
$\displaystyle\langle
vac^{\prime}|=\langle\Omega^{\prime}|\cdots\psi^{*}_{-5/2}\psi^{*}_{-3/2}\psi^{*}_{-1/2}\,.$
Similarly we can define corresponding excited states.
There is a transformation switching these two choices of vacua which is called
an involution and denoted by $\omega$. It acts on the states and operators as
follows
$\displaystyle\omega:|\Omega\rangle\rightarrow|\Omega^{\prime}\rangle\,,\quad\psi^{*}_{n}\rightarrow(-)^{n}\psi_{n}\,,\quad\psi_{n}\rightarrow(-)^{n}\psi^{*}_{n}\,.$
(3.20)
For states defined by eq. (3.19), we can use Maya diagram to demonstrate the
excitations. For example, the Maya diagram corresponding to the vacuum is
shown in fig. 1a. An excited state can be obtained by exchanging certain white
and black dots of the vacuum Maya diagram. For example Fig. 1b. shows an
excited state denoted by $(r_{1}=3/2,r_{2}=1/2,s_{1}=5/2,s_{2}=1/2)$.
There is an amazing correspondence between Maya diagrams and Young diagrams.
For each white/black dot, we assign a unit leftward/downward line segment
connected end to end. Hence a Maya diagram corresponds to a unique Young
diagram in this setup. Thus the subscripts $r_{i},s_{i}$ are Frobenius
coordinates defining the Young diagram. Fig. 1c. shows the corresponding Young
diagram of Maya diagram in Fig. 1b. which is $\\{2,2,1\\}$. As shown in [14],
the excitation states defined in this way are Schur states in fermionic
operator representation.
Figure 1: a. The Maya diagram for the vacuum, b. the Maya diagram for the
excited state $(r_{1}=3/2,r_{2}=1/2,s_{1}=5/2,s_{2}=1/2)$, c. the
corresponding Young diagram of b.
## 4 Curve driving Patch-Shifting
Previously, we have considered the definition of bosonic and fermionic theory
on the complex plane. However, we are more interested in theory on special
Riemann surface with punctures and also defects [15]. By special we mean the
Riemann surface is obtained by gluing various regions, with defects inserted
at fix points.
To obtain Riemann Surface with more complicated topology, we need to define
theories on tubes and pants. A conformal field theory defined on a tube is a
boundary CFT (BCFT) [11] with two boundary conditions. While on a pants, the
corresponding CFT is a BCFT with three boundary conditions.
However, the problem at hand differs from a BCFT problem because the theory is
not defined on a simple Riemann surface but with defect inserted. Precisely,
we have a core region in the toric structure of $\mathbb{C}^{3}$ and three
asymptotic regions which are local patches and can be transited among
themselves. A defect is inserted at point $z=1$333The defect could be anywhere
on the complex plane, however, by global conformal transformation, it can be
fixed at point 1 without loss of generality.. Therefore a BCFT analysis may
fail in this case.
As shown in [7], the Riemann surface corresponding to toric Calabi-Yau could
be defined patch by patch and there are some symplectic transformations of
coordinates of local patches. A simpler case for two-punctured theory can be
obtained by joining two patches together. That is to say, if we define a
theory on a local patch associated with one of these two punctures, then
another theory on another local patch, these two theories can be related to
each other by patch-shifting transformation, which is a symplectic
transformation. The cut and join operation should preserve the symplectic
transformation from one patch to another.
Symplectic transformation is an area-preserving operation which is compatible
with the measure on Riemann surface.
### 4.1 The $\mathcal{W}_{1+\infty}$ Algebra
Now for an infinite cylinder, the symplectic form is
$\Omega=dx\wedge dp\,,$
hence the symplectic transformation of $x$ is
$x\rightarrow x+\epsilon(x)=x+f(p)=x+\sum_{n=0}^{\infty}f_{n}p^{n}\,.$ (4.1)
The transformation of a quantum chiral scalar field associated with the change
of the local coordinate $\delta x=\epsilon(x)$ is implemented by the operator
$\oint T(x)\epsilon(x)\frac{dx}{2\pi i},$ (4.2)
where the stress tensor for chiral boson theory is
$T(x)=\frac{1}{2}[\partial\varphi\partial\varphi](x)\,.$
The observables of the chiral bosonic theory correspond to variations of the
complex structure at infinity. On each patch this is described by the modes of
a chiral boson $\varphi(x)$, defined by
$\partial\varphi(x)=p(x)\,.$
Now the symplectic transformation has an operator expression, namely
$\displaystyle\oint T(x)\epsilon(x)\frac{dx}{2\pi i}$ $\displaystyle=$
$\displaystyle\lim_{x^{\prime}{}\rightarrow x}\oint\frac{1}{4\pi
i}[\partial\varphi\partial\varphi](x)\sum_{n\geq
0}f_{n}[(\partial\varphi)^{n}](x^{\prime}{})dx$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{n\geq
0}(n+2)!f_{n}W_{0}^{n+2}(x)+terms\,\,involve\,\,(\partial^{3}\varphi)\,.$
$W_{0}^{n}$ is the zero mode of free (non-interacting) $\mathcal{W}^{n}$
transformation which is defined by
$\displaystyle W^{n}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{n!}(z\partial_{z}\varphi(z))^{n}$ (4.4) $\displaystyle
W^{n}_{m}$ $\displaystyle=$ $\displaystyle\oint\frac{1}{2\pi
iz}\frac{1}{n!}z^{-m+n}(\partial_{z}\varphi(z))^{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\sum_{k_{i}=-\infty}^{\infty}\delta\left((\sum_{i=1}^{n}k_{i})-m\right)\left(:a_{-k_{1}}a_{-k_{2}}\cdots
a_{-k_{n}}:\right)\,,$
up to some constant ground energy due to normal ordering. In the derivation,
it is useful to apply the coordinate transformation from the cylindrical
coordinates to the complex plane ones $z=e^{x}$. Thus
$z^{-1}dz=dx,\partial_{x}=z\partial_{z}\,,\partial\varphi(x)=z\partial_{z}\varphi(z)=\sum_{n}a_{n}z^{n},\,.$
The appearance of a term containing $\partial^{3}\varphi$ in eq. (4.1)
reflects the non-associativity (and also non-commutation) of operator
expansion product meaning
$[A[BC]](z)\neq[[AB]C](z)\,.$
This is crucial in the derivation of operator formalism of a given integral
formula. Next we will use $W^{3}_{0}$ and $W^{4}_{0}$ as two examples to
explain it.
Firstly, we have
$\displaystyle[\partial_{x}\varphi\partial_{x}\varphi\partial_{x}\varphi](x)=z^{3}[(\partial_{z}\varphi)^{3}](z)$
$\displaystyle=$ $\displaystyle\frac{z^{3}}{2\pi
i}\oint_{z}\frac{1}{z^{\prime}{}-z}\partial_{z^{\prime}{}}\varphi(z^{\prime}{})[\partial_{z}\varphi\partial_{z}\varphi](z)dz^{\prime}{}$
while
$\displaystyle[(\partial_{x}\varphi)^{3}](x)+[\partial_{x}^{3}\varphi](x)$
$\displaystyle=$ $\displaystyle\frac{z^{3}}{2\pi
i}\oint_{z}\frac{1}{z^{\prime}{}-z}[\partial_{z^{\prime}{}}\varphi\partial_{z^{\prime}{}}\varphi](z^{\prime}{})\partial_{z}\varphi(z)dz^{\prime}{}\,.$
Secondly, by using the modes expansion as we defined before, we have the
bosonic operator formalism of (4.1) such that
$\displaystyle W^{3}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\oint_{x}dx\frac{1}{2\pi
i}[\partial_{x}\varphi\partial_{x}\varphi\partial_{x}\varphi](x)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{n,m>0}(a_{-n-m}a_{m}a_{n}+a_{-n}a_{-m}a_{n+m})+\frac{1}{2}\sum_{n>0}a_{0}a_{-n}a_{n}+\frac{1}{6}a_{0}^{3}\,.$
In another way, the bosonic operator formalism of (4.1) gives rise to
$\displaystyle\widetilde{W}^{3}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\oint_{x}dx\frac{1}{2\pi
i}\left([(\partial_{x}\varphi)^{3}](x)+[\partial_{x}^{3}\varphi](x)\right)=W^{3}_{0}+\frac{1}{3}a_{0}\,.$
(4.9)
Here $a_{0}=p_{0}$ is the momentum of the center of mass. Hence the last term
is not an important correction for the spectrum of $W^{3}_{0}$. But it is
fascinating for us that $\frac{1}{3}a_{0}$ also appears in the fermionic side
which seems to provide further evidence for Boson-Fermion correspondence and
we will proceed to that point later.
The non-associative property leads to a severe problem, that is
$[W^{n},W^{m}]\neq 0,\,\,for\,\,n\neq m\,.$
This is a displeasing result since we would expect a $\mathcal{W}_{1+\infty}$
symmetry to generate the integrability, which means there are infinite many
conserved currents commuting with each other. Another problem is that it is
difficult to construct the exact form of $W^{n}$ in terms of bosonic fields.
However, the difference of $W^{3}_{0}$ and $\widetilde{W}^{3}_{0}$ reveals a
simple fact: the difference is just a total derivative! Actually,
$\widetilde{W}_{0}^{3}$ rather than $W^{3}_{0}$ is the one included in the
$\mathcal{W}_{1+\infty}$ algebra.
If we proceed to the fourth $W$ generator, there are three different choices.
An arduous way to find the correct one is to apply the commutation on those
choices with the lower order $W$ generators to check which one gives rise to
zero simultaneously. However, it turns out to be simpler to approach this
problem from the fermionic picture and we will elaborate it next.
Up to a total derivative, we see that $W^{3}$ is the same as
$\widetilde{W}^{3}$. Moreover we can use a $\partial$-cohomology definition of
$\mathcal{W}_{1+\infty}$ algebra which means the algebra is closed upon modulo
all total derivatives. Then we get a good definition of $W^{n}$ algebra in
terms of the bosonic formalism
$W^{n}(z)=\frac{1}{n!}(\partial\varphi)^{n}\,(mod\,\,\partial)\,.$
This observation was known long ago since Dijkgraaf’s paper [17]. However, the
derivation is not exactly the same. Actually it is quite astonishing for us.
Since we are only considering how free chiral boson CFT goes from one patch to
another patch keeping some symplectic symmetries, then we obtain the chiral
boson theory which turns out to be a Kodaira-Spencer-like one as the previous
work [17] by Dijkgraaf.
### 4.2 Fermionic Representation
We want to check the non-associative property in a fermionic picture. Firstly
we consider the $W^{3}_{0}$ case. Since there is no significant difference
between $W^{3}_{0}$ and $\widetilde{W}^{3}_{0}$ it is sufficient to examine
$W^{3}_{0}$.
From fermionization
$\partial_{z}\varphi(z)=\psi\psi^{*}(z)\,,$
we can write down the fermionic formalism of $W^{3}_{0}$ as
$\displaystyle 6W^{3}(w)$ $\displaystyle=$
$\displaystyle\oint_{z}\frac{dz}{2\pi
i(z-w)}[\psi\psi^{*}](z)[-\psi\partial_{w}\psi^{*}-\psi^{*}\partial_{w}\psi](w)$
$\displaystyle=$
$\displaystyle-\oint_{z}\left(\frac{\psi(z)\partial_{w}\psi^{*}(w)}{(z-w)^{2}}+\frac{\psi^{*}(z)\psi(w)}{(z-w)^{3}}\right)\frac{dz}{2\pi
i}$ $\displaystyle+$
$\displaystyle\oint_{z}\left(\frac{\psi^{*}(z)\partial_{w}\psi(w)}{(z-w)^{2}}+\frac{\psi(z)\psi^{*}(w)}{(z-w)^{3}}\right)\frac{dz}{2\pi
i}$ $\displaystyle=$ $\displaystyle
2[\partial\psi^{*}\partial\psi](w)+\frac{1}{2}[\partial^{2}\psi\psi^{*}](w)-\frac{1}{2}[\partial^{2}\psi^{*}\psi](w)\,.$
To get the first equality, we used the well-known result that
$[\partial\varphi\partial\varphi](w)=-[\psi\partial\psi^{*}+\psi^{*}\partial\psi](w)\,.$
Actually, it can be treated as the first generalization of Boson-Fermion
correspondence. The fermionic modes expansion leads to an operator formalism
of $W_{0}^{3}$, namely
$W_{0}^{3}=\frac{1}{2}\sum_{r\in\mathbb{Z}_{>0}-\frac{1}{2}}\left(r^{2}+\frac{1}{12}\right)(\psi_{-r}\psi^{*}_{r}-\psi^{*}_{-r}\psi_{r})\,.$
(4.11)
Similarly, $\widetilde{W}_{0}^{3}$ has a fermionic expression
$6\widetilde{W}^{3}(w)=\frac{3}{2}[\partial^{2}\psi\psi^{*}-\partial^{2}\psi^{*}\psi](w)\,,$
(4.12)
and the operator formalism
$\widetilde{W}_{0}^{3}=\frac{1}{2}\sum_{r\in\mathbb{Z}^{+}-1/2}\left(r^{2}+\frac{3}{4}\right)(\psi_{-r}\psi^{*}_{r}-\psi^{*}_{-r}\psi_{r})=W^{3}_{0}+\frac{1}{3}a_{0}\,,$
(4.13)
where we substituted $a_{0}=\sum_{r}:\psi_{-r}\psi^{*}_{r}:$.
In the last equality of (4.2), the first term could be rewritten as
$2\partial\psi^{*}\partial\psi=\partial(\psi^{*}\partial\psi-\psi\partial\psi^{*})-\psi^{*}\partial^{2}\psi+\psi\partial^{2}\psi^{*}\,.$
The operator formalism perfectly matches with previous bosonic result. A
byproduct of this result is the generalization of Boson-Fermion correspondence
to higher derivatives. For example we have the second generalization formula:
$\displaystyle[\partial^{2}\psi\psi^{*}-\partial^{2}\psi^{*}\psi](z)$
$\displaystyle=$
$\displaystyle\frac{2}{3}[(\partial\varphi)^{3}+\partial^{3}\varphi](z)\,.$
(4.14)
Secondly the OPE method could be easily generalized to $W^{4}_{0}$ case while
there are three ways of multiplication
$[\partial\varphi][(\partial\varphi)^{3}],\,\,[(\partial\varphi)^{2}][\partial\varphi)^{2}]\,,\,\,[(\partial\varphi)^{3}][\partial\varphi]\,.$
Hence there are three kinds of fourth-level Boson-Fermion correspondence as
follows:
$\displaystyle[(\partial\varphi)^{4}]$ $\displaystyle=$
$\displaystyle\left([\frac{3}{2}\partial\psi\partial^{2}\psi^{*}+\frac{1}{6}\partial^{3}\psi\psi^{*}]+\\{\psi\leftrightarrow\psi^{*}\\}\right)$
$\displaystyle\quad+[2\partial\psi^{*}\partial\psi\psi\psi^{*}]$
$\displaystyle\,[(\partial\varphi)^{2}][\partial\varphi)^{2}]$
$\displaystyle=$
$\displaystyle[2\partial^{3}\varphi\partial\varphi+(\partial\varphi)^{4}]$
$\displaystyle=$
$\displaystyle\left([\frac{4}{3}\partial^{3}\psi\psi^{*}-\partial^{2}\psi\partial\psi^{*}]+\\{\psi\leftrightarrow\psi^{*}\\}\right)$
$\displaystyle\quad-[2\partial\psi^{*}\partial\psi\psi\psi^{*}]$
$\displaystyle\,[(\partial\varphi)^{3}][\partial\varphi]$ $\displaystyle=$
$\displaystyle[3\partial^{3}\varphi\partial\varphi+3(\partial^{2}\varphi)^{2}+(\partial\varphi)^{4}]$
$\displaystyle=$
$\displaystyle\frac{5}{3}\left[\partial^{3}\psi\psi^{*}+\partial^{3}\psi^{*}\psi\right]$
$\displaystyle\quad+[2\partial\psi^{*}\partial\psi\psi\psi^{*}]\,.$
These results have not modulo total derivatives yet. It is quite difficult to
write down the operator formalism. All of them contain four-fermion terms
which in general will spoil the integrable structure. If we use the
$\partial$-cohomology definition, these three are the same up to total
derivatives and constant factors. In fermionic picture, the $W^{4}$444We do
not distinguish $W$ and $\tilde{W}$ explicitly from now on. could be
rewritten as
$\displaystyle W^{4}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\oint\frac{dz}{2\pi i(z-w)}W^{3}(z)\psi\psi^{*}(w)$
$\displaystyle=$ $\displaystyle\frac{1}{24}\oint\frac{dz}{2\pi
i(z-w)}\frac{3}{2}[\partial^{2}\psi\psi^{*}-\partial^{2}\psi^{*}\psi](z)[\psi\psi^{*}](w)$
$\displaystyle=$
$\displaystyle\frac{1}{12}(\partial^{3}\psi\psi^{*}+\partial^{3}\psi^{*}\psi)(w)\,.$
A recursive derivation shows that the generic $W^{n}$ can be expressed as
$\displaystyle
W^{n}(z)=\frac{1/2}{(n-1)!}\left(\partial^{n-1}\psi\psi^{*}+(-)^{n}\partial^{n-1}\psi^{*}\psi\right)\,.$
(4.16)
Therefore $W^{n}_{0}$ have distinct expressions for odd and even $n$’s. For
odd $n$,
$W^{n}_{0}\propto\sum_{r\in\mathbb{Z}^{+}-1/2}(\text{even power polynomial
of}\,\,r)(\psi_{-r}\psi^{*}_{r}-\psi^{*}_{-r}\psi_{r})\,,$ (4.17)
and for even $n$
$W^{n}_{0}\propto\sum_{r\in\mathbb{Z}^{+}-1/2}(\text{odd power polynomial
of}\,\,r)(\psi_{-r}\psi^{*}_{r}+\psi^{*}_{-r}\psi_{r})\,\,.$ (4.18)
Especially, the exact form of $W^{4}_{0}$ gives rise to
$W^{4}_{0}=\left(\frac{r^{3}}{6}+\frac{23r}{24}\right)(\psi_{-r}\psi^{*}_{r}+\psi^{*}_{-r}\psi_{r})\,.$
(4.19)
A bonus of this fermionic operator formalism is that it reveals the integrable
structure explicitly inherited from free fermions. The reason is that an
eigenvector of these $W$ operators is formed by those pair excitations of
fermions (3.19) above Dirac sea as discussed before.
This argument actually provides a proof of a conjecture proposed in [7] where
they pointed out that the Kodaira-Spencer chiral bosonic CFT exactly act like
free chiral fermions. We have a stronger result that for a chiral boson action
involving $(\partial\varphi)^{n}$ ($n$ arbitrary) interactions, the
fundamental theory is a free fermion theory.
Although the bosonic theory is in general quite difficult to deal with, the
corresponding fermionic one is rather simple. Moreover the integrable
structure is explicit.
### 4.3 Quantum Curve and Patch-Shifting
Now we consider the quantum curve for $\mathbb{C}^{3}$, which dominants the
behavior of patch-shifting. Actually, the curve under consideration has
distinct representation as a core and an asymptotic part, which are related by
an $S$ transformation.
In [7, 6, 9, 19] and [25], $\mathbb{C}^{3}$ has a mirror manifold defined by
the algebraic equation
$zw-e^{p}-e^{x}-1=0\,,$ (4.20)
the core curve of $\mathbb{C}^{3}$ is understood as
$e^{p}+e^{x}+1=0\,.$ (4.21)
A quantization of this curve is to require a basic commutation relation
$[p,x]=g_{s}\,,\quad p=g_{s}\partial_{x}\,.$
A toric Calabi-Yau three-fold can be treated as gluing of various local
$\mathbb{C}^{3}$. Therefore it is not sufficient to know the core region
geometry of $\mathbb{C}^{3}$ without knowing the asymptotic one of it. In [2,
22], the Ooguri-Vafa operator actually does the work of gluing core and
asymptotic region. The asymptotic region can be obtained by the $S$
transformation of core region geometry. This $S$ transformation is a generator
of the modular group $PSL(2,\mathbb{Z})$. Another generator of the modular
group is $T$ transformation, which plays a role of framing changing, see [7]
for a detailed analysis. Acting on canonical doublet $(x,p)$, $S$ and $T$ have
the matrix representation:
$\displaystyle S=\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)\,\,\,,T=\left(\begin{array}[]{cc}1&1\\\
0&1\end{array}\right)\,.$ (4.26)
It is easy to check the relation that
$ST=\left(\begin{array}[]{cc}0&1\\\
-1&-1\end{array}\right)\,,\quad(ST)^{3}=1\,.$
It is well known that $ST$ transformation generates a $Z_{3}$ subgroup of
$PSL(2,\mathbb{Z})$.
From $S$ transformation we obtain the curve in asymptotic region
$e^{-x}+e^{p}+1=0\,.$ (4.27)
However, there are actually three different asymptotic regions of
$\mathbb{C}^{3}$, reflecting the fact that the toric diagram of
$\mathbb{C}^{3}$ has three legs. Therefore the curve (4.27) should be triply
degenerate. It is easy to check the invariance of (4.27) under $ST$. If we
denote these three patches as $u,v,w$-patch respectively and define a cyclic
relation
$u=p_{w}=g_{s}\partial_{w}\,,\quad v=p_{u}=g_{s}\partial_{u}\,,\quad
w=p_{v}=g_{s}\partial_{v}\,,$
then the $\mathbb{Z}_{3}$ cyclic symmetry is explicit.
From the asymptotic curve in $u$-patch, when $u$ goes to infinity, $v$ should
become $i\pi$. It bothers a lot in further analysis. A more convenient way is
to throw away the $i\pi$ dependence in all three patches, that is, to
reparameterize
$x\rightarrow x+i\pi,\,\,p\rightarrow p+i\pi\,.$
It changes the core geometry to
$e^{x}+e^{p}-1=0\,,$ (4.28)
and the asymptotic geometry to
$e^{-x}+e^{p}-1=0\,.$ (4.29)
Hence the $\mathbb{Z}_{3}$ symmetry is not generated by $ST$ but by the
following $U$-transformation [7],
$U\left(\begin{array}[]{c}u\\\
v\end{array}\right)=ST\left(\begin{array}[]{c}u\\\
v\end{array}\right)+\left(\begin{array}[]{c}0\\\ i\pi\end{array}\right)\,.$
(4.30)
It is straightforward to check that $U$ transformation satisfies
$U^{3}=1\,.$
We claim core curve (4.28) and asymptotic curve (4.29) play important roles in
patch-shifting.
#### 4.3.1 $W^{3}_{0}$ as the generator of $T$
Previously we studied $W$ algebra. However in this article the related
symmetry is $W^{3}_{0}$, since the local patches are joint by $T$
transformation and $W^{3}_{0}$ is the generator of $T$. $T$ acts as follows
$T:\left(\begin{array}[]{c}u\\\
v\end{array}\right)=\left(\begin{array}[]{c}u+v\\\ v\end{array}\right)\,.$
(4.31)
We first notice that $v=g_{s}\partial_{u}$ is expressed as
$v=g_{s}\partial_{u}\varphi(u)$ in a chiral boson theory. Then all arguments
follow as we have discussed in previous section. The current related to the
transformation is simply $W^{3}$. Excitated modes of $W^{3}$ do not contribute
because we only consider the asymptotic region on the $u$-patch (or $v$-patch)
as $u\rightarrow\infty$ (or $v\rightarrow\infty$). Therefore only vacuum state
contributes otherwise the theory will not be unitary. Thus only $W^{3}_{0}$
survives. In summary we conclude that $T$ transformation is generated by
$W^{3}_{0}$ multiplied by $g_{s}$.
A $T^{n}$ transformation of an eigenfunction $f(u)$ has the standard
expression
$e^{ng_{s}W^{3}_{0}}f(u)e^{-ng_{s}W^{3}_{0}}=f(u+nv)\,.$
#### 4.3.2 $S$ transformation on the base
Now let us consider $S$ transformation on the patches. As we argued before,
the $S$ transformation interchanges the canonical pair $(x,p)$ . $x$ and $p$
form a canonical bundle, with the symplectic form as defined previously. The
$S$ transformation also preserves the symplectic structure. If we treat $x$ as
the base and $p$ the fiber, $S$ transformation actually bends $x$ to its
normal direction. From a physical viewpoint, this can be understood as an
insertion of a loop defect which bends the base and fiber simultaneously.
Therefore the Hamiltonians on both sides of the defect should also be related
to each other by $S$ transformation. This is very important in our further
analysis.
## 5 Zhou’s Identity and the Topological Vertex
In this section, we consider the problem how to obtain partition functions
from curves and the underlining symplectic transformations.
### 5.1 Vacuum Partition Function and the Curve of ${\mathbb{C}}^{3}$
Now we look for an eigenfunction of Hamiltonian in the core region
$H_{c}(p,x)=e^{x}+e^{p}-1\,,$
where the subindex $c$ denotes the core. In the local $u$-patch with $u$ being
a coordinate on a cylinder, we have asymptotic curve
$e^{-u}+e^{v}-1=0\,,$
the Hamiltonian in this region in the complex plane coordinates is
$H_{a}(L_{0},u)=\frac{1}{z_{1}}+e^{g_{s}L_{0}}-1\,,$ (5.1)
where $L_{0}=z_{1}\partial_{z_{1}},\,z_{1}=e^{u},\,z_{2}=e^{v},z_{3}=e^{w}$.
It drives the evolution in the asymptotic region that is $z_{1}>1$ region,
while in $z_{1}<1$ region, the core curve $H_{c}$ drives the evolution.
Now we introduce an anti-B-brane at the infinity of $z_{1}$-plane. The next
step is to move it into the core region. This can be treated as an evolution
of Hamiltonian. Unfortunately, there are infinitely many evolution paths in
the spirit of path integral. Moreover it is quite difficult to approach this
problem from a standard Hamiltonian analysis since the Hamiltonian is highly
nonlinear.
Therefore we need to find a new description of Hamiltonian evolution. Notice
the CFT implied by the curve is a Kodaira-Spencer theory, which is equivalent
to a free fermion theory. Since a brane (anti-brane) could be understood as a
fermion (anti-fermion) insertion in local patch [7], the bosonized fermion
(anti-fermion) field will have a representation (ignore the zero modes)
$e^{\pm\varphi(z)}=\exp\left(\pm\sum_{n\neq 0}\frac{a_{-n}}{n}z^{n}\right)\,.$
It means the evolution of Hamiltonian can be replaced by infinitely many
branes insertions in between $z_{1}=1$ and $z_{1}=\infty$. This is due to the
fact radial ordering on a complex plane is a time ordering on cylinder while
the later is controlled by the Hamiltonian. The OPEs of branes and anti-branes
now can be understood as propagators.
However, the positions where B-branes are inserted are arbitrary according to
path integral. We may expect a classical equation of motion to determine the
orbit completely. However, it is difficult to deal with a quantum system where
there are different Hamiltonians in different regions. To simplify the problem
a bit in this article we divide the space into the asymptotic region and the
core region associate an Hamiltonian with each coordinate charts. We may
choose the two coordinate charts to be
$U_{a}=\\{z_{1}=e^{u}\in(0,\infty]\\}\,,\,\,U_{c}=\\{z_{1}^{\prime}{}=e^{u^{\prime}{}}\in[0,\infty)\\}$
where $U_{a}$ is dominated by asymptotic Hamiltonian and $U_{c}$ is dominated
by core Hamiltonian.
In $U_{a}$ chart, we propose the following ansatz equation
$\langle 0|H_{a}\exp\left(\sum_{n>0}\frac{a_{n}}{n}z^{-n}\right)\prod_{i\geq
1}^{\infty}V_{-}(w_{i})|0\rangle\equiv 0\,.$ (5.2)
Applying (5.1) and moving $q^{L_{0}}$ out of the correlation function, we
obtain
$\displaystyle\langle 0|(q^{L_{0}}+\frac{1}{z}-1)\exp\left(\sum_{n>0,i\geq
1}\frac{(w_{i}/z)^{n}}{n}\right)|0\rangle$ $\displaystyle=$
$\displaystyle(q^{L_{0}}+\frac{1}{z}-1)\prod_{i=1}^{\infty}(1-w_{i}/z)^{-1}.$
The vanishing condition gives rise to
$\prod_{i=1}^{\infty}(1-\frac{w_{i}}{qz})^{-1}=\left(1-\frac{1}{z}\right)\prod_{i=1}^{\infty}(1-w_{i}/z)^{-1}\,.$
Suppose $w_{1}=1$ and a recursion relation
$w_{i+1}=q^{-1}w_{i}\,.$
We can prove that it is the solution of the eigen-equation (5.2). The
resulting wave function turns out be a quantum dilogarithm [16], namely
$\Psi^{*}(u)=\exp\left(\sum_{n>0}\frac{-q^{n}e^{-nu}}{n(n)_{q}}\right)\,,$
(5.3)
where
$(n)_{q}=1-q^{n}\,.$
The orbit of the branes insertions are then a set of discrete points at
$\\{w_{i}=q^{-i+1}\\}$, $(i\geq 1)$. This analysis can be generalized to
including the evolution of many anti-branes as well. The insertions of these
anti-branes can be understood as the generating function for bra Schur states,
namely
$\langle
0|\prod_{i}V_{+}^{*}(z_{i})=\sum_{\lambda}\langle\lambda|s_{\lambda}(z_{i})\,.$
We have chosen anti-brane inserted at infinity. Certainly we can consider
brane inserted at infinity where a similar analysis leads to the following
ansatz equation
$\langle 0|H_{a}V_{+}(z)\prod_{i}V_{-}^{*}(w_{i})|0\rangle=0\,.$
Solve the equation we can locate the positions of branes $V_{-}^{*}$’s on the
orbit
$\\{w_{i}=q^{-i+1},\,i\geq 1\\}\,.$
It is then clear how to determine positions of anti-branes (branes) in
$[1,\infty)$. A similar analysis can be done for $U_{c}$ coordinate chart
where the branes insertions are near origin ($e^{u}=0$). Hence it will locate
the orbit points of anti-branes in the region $(0,1]$ by the ansatz equation
$\displaystyle\langle
0|\prod_{i=1}^{\infty}V_{+}^{*}(w_{i})\exp\left(\sum_{n>0}\frac{a_{-n}}{n}z^{n}\right)H_{c}|0\rangle=0\,.$
(5.4)
The solution of this equation gives rise to a set of $w_{i}$’s
$\\{w_{i}=q^{i-1},\,\,i\geq 1\\}\,.$
The next step is to join these two coordinate charts into a single $u$-patch
as we noted. The anti-brane from infinity and the brane from origin meet at
point 1 and annihilate each other identically. It actually gives rise to a
vacuum partition function in $u$-patch, we get
$\displaystyle\langle
0|\prod_{i=1}^{\infty}V_{-}(q^{-i+1})q^{L_{0}}\prod_{j=1}^{\infty}V_{+}^{*}(q^{j-1})|0\rangle$
$\displaystyle=$ $\displaystyle\langle
0|\prod_{i=1}^{\infty}V_{-}(q^{\rho_{i}})\prod_{j=1}^{\infty}V_{+}^{*}(q^{-\rho_{j}})|0\rangle=1\,.$
It is what we expect because when we glue two cylinders into a torus, the
torus vacuum partition function can be chosen as 1 due to normalization.
However, this result is quite different from the one obtained in [5], where
the vacuum partition function is chosen to be MacMahon function.
There is a subtle feature need to be clarified. For the vertex operators $V$
and $V^{*}$, if there are no zero modes, it is not a faithful correspondence
between fermion and boson. However in this article, zero modes will not play
significant roles in many calculations. Only if two charts are joined into a
single patch with a defect at point 1, must the contribution of zero modes be
retrieved. In that case the contribution will highly depend on the
representation of the defect.
It is worth comparing the vertex operator formalism with the definition
fermionic vacuum. An observation is that suppose we define a correspondence
$\displaystyle\psi_{\rho_{i}}\rightarrow
V_{-}(q^{\rho_{i}})\,,\,\,\psi^{*}_{-\rho_{j}}\rightarrow
V_{+}^{*}(q^{-\rho_{i}})\,,$ (5.6)
the Dirac sea structure corresponding to fermionic vacuum now becomes
$\cdots\psi_{-5/2}\psi_{-3/2}\psi_{-1/2}\psi^{*}_{1/2}\psi^{*}_{3/2}\psi^{*}_{5/2}\cdots\,.$
Thus it corresponds to inner product of the fermionic vacuum
$\langle vac|vac\rangle$
as we obtained in sec. 3 where the fermionic vacuum corresponds to the
insertions of branes at infinity and anti-branes at origin.
The last paragraph is only a rough idea about the projective relation from
vertex operators to fermions. We shall have a more concrete derivation of it
in next subsection.
### 5.2 Excited States and the Profile of a Young Diagram
Now we consider excited states in $u$-patch555for excited states in $v$\- or
$w$-patch, the same argument follows. Firstly, suppose there is an excited
state labeled by a Young diagram $\lambda$ inserted at infinity of $u$-patch
and there are no excitations on the other two patches. The partition function
is
$\displaystyle\langle\lambda|\prod_{i>0}V_{-}(q^{\rho_{i}})\prod_{j>0}V_{+}^{*}(q^{-\rho_{j}})|0\rangle=s_{\lambda}(q^{\rho})\,.$
(5.7)
A slightly more complicated case is that besides $\lambda$ there is also an
excited state labeled by $\mu$ at the origin of $u$-patch. The the partition
function is
$\langle\lambda|\prod_{i\geq 1}V_{-}(q^{\rho_{i}})\prod_{j\geq
1}V_{+}^{*}(q^{-\rho_{j}})|\mu\rangle=\sum_{\eta}s_{\lambda/\eta}(q^{\rho})s_{\mu/\eta}(q^{\rho})\,.$
(5.8)
To obtain the equality we have used
$\displaystyle\sum_{\eta}\langle\lambda|\prod_{j}V_{-}(z_{j})|\eta\rangle$
$\displaystyle=$ $\displaystyle\sum_{\eta,\xi}\langle
0|s_{\lambda}(a_{+})s_{\eta}(a_{-})s_{\xi}(a_{-})|0\rangle s_{\xi}(z_{j})$
$\displaystyle=$
$\displaystyle\sum_{\eta,\theta,\xi}c_{\eta\theta}^{\lambda}\langle\theta|\xi\rangle
s_{\xi}(z_{j})=\sum_{\eta,\xi}c_{\eta\xi}^{\lambda}s_{\xi}(z_{j})=\sum_{\eta}s_{\lambda/\eta}(z_{j})\,,$
where $s_{\lambda/\eta}$ is a skew Schur polynomial and
$c_{\eta\xi}^{\lambda}$ is the Littlewood-Richardson coefficient defined by
$s_{\lambda/\eta}=\sum_{\theta}c_{\eta\theta}^{\lambda}s_{\theta}\,.$
Since we argued in previous section, branes and anti-branes can be inserted
not only at the infinity of a given patch, but also at the point 1 (more
precisely, on the unit circle). Although we have calculated the simple case
for excitations near the origin and infinity on a complex plane it would be
quite interesting to ask the question about excitations in the bulk near point
1. It corresponds to the case of joining two charts into a single patch with
some defect inserted at point 1.
In a local patch, the unit circle does not belong to either the asymptotic
region or the core region. Previously we considered the insertion of vacuum at
point 1 the fermionic vacuum becomes products of $V_{-}$ and $V_{+}^{*}$’s,
with all $V_{-}$’s ($V_{+}^{*}$’s) located to the left (right) side of point
1, namely $V_{-}$’s are in the asymptotic region corresponding to the outgoing
modes and $V_{+}^{*}$’s are in the core region corresponding to the incoming
modes.
Now we consider a fermionic excited state labeled by Young diagram $\nu$. In
the fermionic picture, it is an excited state from Dirac sea. The $\nu$ state
can be written down according to the profile of the Young diagram $\nu$. As in
fig. 2, where
$\nu=\\{5,4,2,1\\}$
the corresponding fermionic excitations are
$\cdots\psi_{-11/2}\psi^{*}_{-9/2}\psi_{-7/2}\psi^{*}_{-5/2}\psi_{-3/2}\psi_{-1/2}\psi^{*}_{1/2}\psi_{3/2}\psi^{*}_{5/2}\psi_{7/2}\psi^{*}_{9/2}\cdots\,\,.$
Figure 2: An example of a fermionic excited state and the corresponding Young
diagram
For a general $\nu$, the modes of $\psi$ (white dots) belong to the set
$\left\\{\cdots,\,\,\nu^{t}_{3}-3+\frac{1}{2}\,,\,\nu^{t}_{2}-2+\frac{1}{2},\,\,\nu^{t}_{1}-1+\frac{1}{2}\right\\}\equiv\\{\nu^{t}+\rho\\}\,.$
(5.9)
Similarly the modes of $\psi^{*}$ (black dots) belong to the set
$\left\\{-\nu_{1}+1-\frac{1}{2},\,\,-\nu_{2}+2-\frac{1}{2},\,\,-\nu_{3}+3-\frac{1}{2},\,\cdots\right\\}\equiv\\{-\nu-\rho\\}\,.$
(5.10)
In this fermionic picture, it is clear that presumably, there is an infinity
height fermionic tower at point $e^{u}=1$. This tower will expand to elsewhere
in $u$-patch due to quantum shift.
For the case $\nu=\phi$, the empty set, we have already seen this quantum
shift changes the vacuum Dirac sea to an infinite products of $V$ and
$V^{*}$’s. Actually, it is very simple to deduce from the curve. At point
$z_{1}=1$, the Hamiltonian just becomes
$H(L_{0},1)=q^{L_{0}}\,.$
The fermionic modes expansion becomes
$\psi(1)=\sum_{r\in\mathbb{Z}-\frac{1}{2}}\psi_{r}\,,\,\,\psi^{*}(1)=\sum_{r\in\mathbb{Z}-\frac{1}{2}}\psi^{*}_{r}\,.$
In a quantum manner, all excitations are including in multi-products of these
fields. For the physical vacuum $vac$, it is a multi-product in sequence as
$\cdots\psi_{-5/2}\psi_{-3/2}\psi_{-1/2}\psi^{*}_{1/2}\psi^{*}_{3/2}\psi^{*}_{5/2}\cdots\,\,.$
Hamiltonian at point 1 is a transport operator moving all $\psi$\- fields to
the left of 1 and $\psi^{*}$-fields to the right of 1. Further according to
the bosonization formula, we reproduce the vacuum partition function as
$\displaystyle\langle
0|\prod(q^{L_{0}}V_{-}(1))q^{L_{0}}\prod(V_{+}^{*}(1)q^{L_{0}})|0\rangle\,.$
(5.11)
It is just another expression of (5.1). Here the left (right) transporting
behavior is transferred to left (right) action on the vertex operators. It
proves the projective relation as we mentioned in eq. (5.6).
If we want to generalize the analysis to a generic $\nu$ state, then we just
need to reshuffle (5.11) according to the profile 666here profile means the
sequence of $V_{-}$ and $V^{*}_{+}$’s is determined according to the profile
by the projective relation of the Young diagram of $\nu$. Hence it gives rise
to
$\displaystyle\langle
0|\prod_{\text{profile}\,\,\nu}V_{-}(q^{\nu^{t}+\rho})V_{+}^{*}(q^{-\nu-\rho})|0\rangle\,.$
(5.12)
Moving all $V_{+}^{*}$’s to the right side of all $V_{-}$’s we get
$\displaystyle\langle 0$
$\displaystyle|\prod_{\text{profile}\,\,\nu}V_{-}(q^{\nu^{t}+\rho})V_{+}^{*}(q^{-\nu-\rho})|0\rangle$
$\displaystyle=$ $\displaystyle\prod_{(i,j)\in\nu}\frac{1}{1-q^{h(i,j)}}\equiv
Z_{\nu}(q)\,,$
where
$\displaystyle Z_{\nu}(q)$ $\displaystyle:=$
$\displaystyle\prod_{i,j\in\nu}\frac{1}{1-q^{h(i,j)}}=Z_{\nu^{t}}(q)$
$\displaystyle=$
$\displaystyle(-)^{|\nu|}\prod_{(i,j)\in\nu}\frac{q^{-h(i,j)}}{1-q^{-h(i,j)}}$
$\displaystyle=$
$\displaystyle(-)^{|\nu|}q^{-||\nu||/2-||\nu^{t}||/2}\prod_{(i,j\in\nu)}\frac{1}{1-q^{-h(i,j)}}\,,$
with $h(i,j)$ being the hook length of square $(i,j)$ in $\nu$. Notice that
$Z_{\nu}$ is neither $s_{\nu}(q^{-\rho})$ nor $s_{\nu^{t}}(q^{-\rho})$. Schur
polynomial in variables $\\{q^{1/2},q^{3/2},q^{5/2},\cdots\\}$ is
$\displaystyle
s_{\nu}(q^{-\rho})=q^{\frac{||\nu^{t}||}{2}}\prod_{i,j\in\nu}\frac{1}{1-q^{h(i,j)}}=(-)^{|\nu|}s_{\nu^{t}}(q^{\rho})\,.$
(5.15)
Now we consider $V_{-}$ and $V^{*}_{+}$’s insertions respectively. For the
$V_{-}$’s insertions, we have
$\displaystyle\langle 0|\prod_{i\geq 1}V_{-}(q^{\nu^{t}+\rho_{i}})\,.$ (5.16)
A Young diagram $\nu$ in terms of Frobenius notation is
$\\{r_{1},r_{2},\cdots,r_{d}|s_{1},s_{2},\cdots,s_{d}\\}$
where
$r_{i}=\nu_{i}-i+\frac{1}{2},\,\,s_{i}=\nu^{t}_{i}-i+\frac{1}{2}\,.$
According to the projective relation the fermionic bra state can be
represented as
$\displaystyle\langle$
$\displaystyle\Omega|\cdots\psi_{\nu^{t}_{i}-i+\frac{1}{2}}\psi_{\nu^{t}_{i-1}-i+\frac{3}{2}}\cdots\psi_{\nu^{t}_{1}-\frac{1}{2}}$
$\displaystyle=$ $\displaystyle\langle
vac|(-)^{\sum_{i=1}^{d}(s_{i}-\frac{1}{2})}\prod_{i}^{d}\psi_{s_{i}}\psi^{*}_{r_{i}}=\langle\nu|\,.$
Similarly, for $V_{-}$’s insertions, the corresponding fermionic ket state is
$(-)^{\sum_{i}^{d}(r_{i}-\frac{1}{2})}\prod_{i}^{d}\psi_{-s_{i}}\psi^{*}_{-r_{i}}|vac\rangle=|\nu^{t}\rangle\,.$
(5.18)
The states are compatible with the geometrical observation from infinity to
the origin on one local patch. The $S$ transformation which exchanges
canonical variables (position and momentum) “bends” the project line to its
normal at point 1. Then near infinity, we see the profile of $\nu$, while near
the origin, we find that it reflects to $\nu^{t}$.
This observation defines the following rules:
1\. From infinity to 1, the representation has not been changed.
2\. From 1 to 0, the representation becomes its transpose.
In summary we can consider the patch-shifting and its impacts on the vertex
operator formalism.
We propose a configuration
$\displaystyle\langle\lambda,\nu,\mu\rangle$ $\displaystyle\equiv$
$\displaystyle(-)^{|\nu|}q^{\frac{||\nu||}{2}}\langle\lambda|\prod_{\text{profile}\,\,\nu}V_{-}(q^{\nu^{t}+\rho})V_{+}^{*}(q^{-\nu-\rho})|\mu\rangle$
$\displaystyle=$ $\displaystyle
s_{\nu}(q^{\rho})\sum_{\eta}s_{\lambda/\eta}(q^{\nu^{t}+\rho})s_{\mu/\eta}(q^{\nu+\rho}).$
The factor $(-)^{|\nu|}q^{||\nu||/2}$ comes from zero modes of $V$ and
$V^{*}$. Actually, if we keep the Boson-Fermion correspondence being exact, we
should include the contribution of zero modes. The result of normal ordering
now becomes:
$\displaystyle\prod_{(i,j)\in\nu}\frac{1}{q^{-\nu_{i}-\rho_{i}}-q^{\nu^{t}_{j}+\rho_{j}}}$
$\displaystyle=$
$\displaystyle(-)^{|\nu|}q^{||\nu||/2-||\nu^{t}||}\prod_{(i,j)\in\nu}\frac{1}{1-q^{-h(i,j)}}$
$\displaystyle=(-)^{|\nu|}$ $\displaystyle
q^{\kappa_{\nu}/2}s_{\nu}(q^{\rho})\,.$
Then up to a framing factor $(-)^{|\nu|}q^{\kappa_{\nu}/2}$, the Schur
function $s_{\nu}(q^{\rho})$ occurs as desired.
The states under the shifting from a $u$-patch to a $v$-patch are compatible
with the corresponding curves of different charts on patches.
For example, the insertion of the bra state $\lambda$ at infinity on $u$-patch
is an insertion at point 1 in $v$-patch. Thus patch-shifting leads to bringing
a $\lambda$ state from infinity of $u$ to 1 of $v$.
Then a $\nu$ insertion at point 1 in the $u$-patch becomes a ket state
$\nu^{t}$ inserted at the core region in $v$-patch.
Similarly a ket state $\mu$ inserted in the core region determined by
$e^{u}+e^{v}-1=0$
in the $u$-patch should be transformed to the asymptotic region in $v$-patch
by $S$-transformation, and the $T$ transformation is required to cancel the
divergence. For example
$e^{-u-v}+e^{-v}-1=0$
as $u$ goes to $-\infty$, $v$ becomes $\infty$, this operation moves $\mu$ ket
state to a $\mu^{t}$ bra state along with a factor $q^{\kappa_{\mu}/2}$ due to
the $T$ transformation.
To join the asymptotic region and the core region together into a
T-transformed $v$-patch, we need $T$-transform the core region (with $\nu^{t}$
inserted on) and also the defect (representation $\lambda$). It results in a
further $q^{\kappa_{\nu}/2}$ factor in the expression in $v$-patch. Notice
that there is no further factor corresponding to a $\lambda$ insertion at
point 1 since
$q^{\kappa_{\lambda}/2}q^{\kappa_{\lambda^{t}}/2}=1\,.$
Now we have the following conjecture
$\displaystyle\langle\lambda,\nu,\mu\rangle$ $\displaystyle=$ $\displaystyle
q^{\frac{\kappa_{\mu}+\kappa_{\nu}}{2}}\langle\mu^{t},\lambda,\nu^{t}\rangle$
$\displaystyle=$ $\displaystyle
q^{\frac{\kappa_{\lambda}+\kappa_{\mu}}{2}}\langle\nu,\mu^{t},\lambda^{t}\rangle\,.$
It is our major observation from the curve of $\mathbb{C}^{3}$.
It is difficult to verify this conjecture directly. However, if we let one of
the representations $\lambda$, $\mu$ and $\nu$ be an empty representation
$\phi\equiv 0$, then the resulting identities are just Zhou’s identities [26].
For example, let $\nu=0$. We have
$\displaystyle\langle\lambda,0,\mu\rangle$ $\displaystyle=$
$\displaystyle\sum_{\eta}s_{\lambda/\eta}(q^{\rho})s_{\mu/\eta}(q^{\rho})$
$\displaystyle=$ $\displaystyle
q^{\kappa_{\mu}/2}\langle\mu^{t},\lambda,0\rangle=q^{\kappa_{\mu}/2}s_{\lambda}(q^{\rho})s_{\mu^{t}}(q^{\lambda^{t}+\rho})$
$\displaystyle=$ $\displaystyle q^{(\kappa_{\lambda}+\kappa_{\mu})/2}\langle
0,\mu^{t},\lambda^{t}\rangle$ $\displaystyle=$ $\displaystyle
q^{(\kappa_{\lambda}+\kappa_{\mu})/2}s_{\mu^{t}}(q^{\rho})s_{\lambda^{t}}(q^{\mu^{t}+\rho})\,.$
It is nothing but Zhou’s identity.
We can verify other degenerate cases of (5.2) in detail. Consequently we get
Zhou’s identities in all cases.
### 5.3 The relation with the Topological Vetex
It would be interesting to compare eq. (5.2) with the famous topological
vertex proposed in [6] and further the topological vertex in terms of
symmetric polynomials in [5] and [26].
The topological vertex in [5, 26] is defined as
$C(\lambda,\,\mu,\,\nu)=q^{\kappa_{\lambda}/2}s_{\nu}(q^{\rho})\sum_{\eta}s_{\mu/\eta}(q^{\nu^{t}+\rho})s_{\lambda^{t}/\eta}(q^{\nu+\rho})$
(5.23)
In our configuration
$C(\lambda,\,\mu,\,\nu)=q^{\kappa_{\lambda}/2}\langle\mu,\,\nu,\,\lambda^{t}\rangle\,.$
(5.24)
It means what we have obtained is a reformulation of the topological vertex.
However, the approach here is quite different from that in [6] and [5]. An
direct observation is that our definition as in eq. (5.2) has a very clear
patch meaning rather than a unified topological vertex. The cyclic symmetry of
the topological vertex now becomes the shifting of patches.
## 6 Conclusions
We find an explicit correspondence between A- and B-model for the case of
topological vertex. In our opinion, the mirror curve of $\mathbb{C}^{3}$ is
not a global ly defined chart but a union of two coordinate charts within
defects inserting at point 1. It is crucial for deriving B-model correlation
function, which becomes A-model topological invariant. A new vertex operator
approach to the topological vertex is proposed. On the way of doing this, we
prove the conjecture proposed in [7]. The vertex operator approach can be
treated as an application of projective representation introduced in [23].
Finally, we propose a conjecture on the topological vertex (or in B-model, a
three-leg correlation function) identity (5.2), which becomes Zhou’s
identities of Hopf links in degenerate cases.
There are many further works in this direction. We just list three of them for
instance. Firstly, the identity (5.2) is new and a mathematical proof is not
known to the authors. Secondly, the vertex operator approach could be
generalized to other curves associated to many toric Calabi-Yau manifolds. Due
to the identity (5.2), it is quite free to glue topological vertices to
formulate complicated toric Calabi-Yau’s. This calculation is working in
progress and a future article will contain some applications. Thirdly, it is
natural to ask for a refined version of this approach. However, this is quite
difficult since there the refined curve 777Eynard and Kozcaz provided a mirror
curve for refine topological vertex in [18], the curve has no simple
expression as the topological vertex. is very complicated and related
symplectic transformations are not well-known. Maybe a simpler case could be
considered first. For example, when a background charge is introduced into the
Kodaira-Spencer theory the resulting theory is hence the Feign-Fuchs bosonic
theory. The underlining integrability is controlled by the Calogero-Sutherland
model [10, 24]. In this case, two refined parameters($t$ and $q$) are related
by $t=q^{\alpha}$ (twisted case) and the eigenfunctions are Jack symmetric
functions in the limit $q\rightarrow 1$. A very similar analysis could be done
for this case. We expect a Jack symmetric function expression for the twisted
topological vertex.
## Acknowledgments
We would like to thank Professor Guoce Xin, Professor Ming Yu and Professor
Jian Zhou for valuable comments. The authors are grateful to Morningside
Center of Chinese Academy of Sciences and Kavli Institute for Theoretical
Physics China at the Chinese Academy of Sciences for providing excellent
research environment. This work is also partially supported by Beijing
Municipal Education Commission Foundation (KZ201210028032, KM201210028006),
Beijing Outstanding Person Training Funding (2013A005016000003).
## Appendix A Some Notations on Symmetric Polynomials
In this appendix we just provide a brief review of some symmetric functions.
For detailed description please look up the book by Macdonald [21].
###### Definition 1.
An elementary symmetric polynomial is defined by
$e_{r}(x_{1},x_{2},\cdots)=\sum_{i_{1}<i_{2}<\cdots<i_{r}}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{r}},$ (A.1)
for $r\geq 1$ and $e_{0}=1$.
The generating function for the $e_{r}$ is
$E(t)=\sum_{r\geq 0}e_{r}t^{r}=\prod_{i\geq 1}(1+x_{i}t).$
###### Definition 2.
A complete (homogenous) symmetric polynomial is defined by
$h_{r}(x_{1},x_{2},\cdots)=\sum_{i_{1}\leq i_{2}\leq\cdots\leq
i_{r}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}},$ (A.2)
for $r\geq 1$ and $h_{0}=1$.
The generating function for the $h_{r}$ is
$H(t)=\sum_{r\geq 0}h_{r}t^{r}=\prod_{i\geq 1}\frac{1}{1-x_{i}t}.$
###### Definition 3.
A Schur polynomial $s_{\lambda}$ as a symmetric polynomial in variables
$x_{1},x_{2},\cdots$ corresponding to a partition $\lambda$ is defined by
$s_{\lambda}(x_{1},\cdots,x_{N}):=\sum_{T}\mathbb{x}^{T}$ (A.3)
where $T$ is a semi-standard tableau of shape $\lambda$ and
$\mathbb{x}^{T}=\prod_{i}x_{i}^{n_{i}}$ with $n_{i}$ the number of $i$ filling
in $T$.
###### Definition 4 (Jacobi-Trudi).
The Schur polynomial can be calculated from the elementary or complete
polynomials by
$s_{\nu}(x_{1},x_{2},\cdots,x_{n})=\det(h_{\nu_{i}-i+j})=\det(e_{\nu^{t}_{i}-i+j}).$
(A.4)
Now suppose the variables ($x_{1},x_{2},\cdots$) appear in a formal power
series $E(t)=\prod_{i}(1+x_{i}t)$. We simply denote the Schur function by
$s_{\nu}(E(t)).$
For example
$E(t)=\prod_{i=0}^{\infty}(1+q^{i}t)=\sum_{r=0}^{\infty}e_{r}t^{r}$ (A.5)
where
$e_{r}=\prod_{i=1}^{r}\frac{q^{i-1}}{1-q^{i}}.$ (A.6)
Hence the corresponding Schur function is written as
$s_{\lambda}(1,q,q^{2},\cdots)$. In the q-number notation
$[x]=q^{x/2}-q^{-x/2}$
$s_{\nu}(q^{-\rho})=(-1)^{|\nu|}q^{-\kappa(\nu)/4}\prod_{x\in\nu}\frac{1}{[h(x)]}$
where $h(x)$ is the hook length of the square $x$ and
$\kappa(\nu)=2(n(\nu^{t})-n(\nu))$ with $n(\nu)=\sum_{i}\nu_{i}(i-1)$.
Now let us generalize the formal power series to a more complicated case
$E_{\mu}(t)=\prod_{i=1}^{\infty}(1+q^{\mu_{i}-i+1/2}t)=\prod_{i=1}^{\ell}\frac{1+q^{\mu_{i}-i+1/2}t}{1+q^{-i+1/2}t}\prod_{i=1}^{\infty}(1+q^{-i+1/2}t).$
(A.7)
Recall a very useful identification between multisets of number
$\\{\mu_{i}-i,(d<i\leq\ell)\\}=\\{-1,\cdots,-\ell\\}-\\{-\mu^{t}_{i}+i-1,(1\leq
i\leq d)\\}$ (A.8)
where $d$ is the diagonal of $\nu$. According to Frobenius notation
$\nu=(\alpha_{1},\cdots,\alpha_{d}|\beta_{1},\cdots,\beta_{d})$, it can be
written as
$\\{\mu_{i}-i,(d<i\leq\ell)\\}=\\{-1,\cdots,-\ell\\}-\\{-\beta_{i}-1,(1\leq
i\leq d)\\}.$ (A.9)
(A.7) becomes
$E_{\mu}(t)=\prod_{i=1}^{d(\mu)}\frac{1+q^{\alpha_{i}+1/2}t}{1+q^{-\beta_{i}-1/2}t}\prod_{i=1}^{\infty}(1+q^{-i+1/2}t).$
(A.10)
Therefore
$s_{\nu}(E_{\mu}(t))=s_{\nu}(q^{\mu_{1}-1+1/2},q^{\mu_{2}-2+1/2},\cdots)\,,$
(A.11)
or it can be put in a simple notation $s_{\nu}(q^{\mu+\rho})$ where
$\rho=-\frac{1}{2},-\frac{3}{2},\cdots$. In the Frobenius notation
$s_{\nu}(E_{\mu}(t))=s_{\nu}(q^{\alpha_{1}+\frac{1}{2}},\cdots,q^{\alpha_{d(\mu)}+\frac{1}{2}},q^{-\frac{1}{2}},\cdots,\widehat{q^{-\beta_{1}-\frac{1}{2}}},\cdots,\widehat{q^{-\beta_{d(\mu)}-\frac{1}{2}}},q^{-\beta_{d(\mu)}-\frac{3}{2}},\cdots).$
(A.12)
###### Definition 5.
A skew Schur polynomial $s_{\lambda/\mu}$ as a symmetric function in variables
$x_{1},x_{2},\cdots$ is defined by
$s_{\lambda/\mu}(x_{1},x_{2},\cdots)=\sum_{T}\mathbb{x}^{T}$ (A.13)
where $T$ is a semi-standard tableau of shape $\lambda-\mu$.
The skew Schur function has a property
$s_{\lambda/\mu}(x,y)=\sum_{\nu}s_{\lambda/\nu}(x)s_{\nu/\mu}(y).$
Therefore it can be generalized to $n$ sets of variables
$x^{(1)},\cdots,x^{(n)}$
$s_{\lambda/\mu}(x^{(1)},\cdots,x^{(n)})=\sum_{(\nu)}\prod_{i=1}s_{\nu^{(i)}/\nu^{(i-1)}}(x^{(i)})$
(A.14)
summed over all sequences $(\nu)=(\nu^{(0)},\cdots,\nu^{(n)})$ of partitions
such that $\nu^{(0)}=\mu$, $\nu^{(n)}=\lambda$, and
$\nu^{(0)}\subset\cdots\subset\nu^{(n)}$.
###### Definition 6 (Jacobi-Trudi).
The skew Schur polynomial also can be calculated from the elementary or
complete polynomials by
$s_{\lambda/\mu}(x_{1},x_{2},\cdots,x_{n})=\det(h_{\lambda_{i}-\mu_{j}-i+j})=\det(e_{\lambda_{i}^{t}-\mu^{t}_{j}-i+j}).$
(A.15)
## Appendix B The identity
In this appendix we provide a combinatoric proof of the identity
$s_{\lambda/\mu}(q^{\nu+\rho})=(-1)^{|\lambda|-|\mu|}s_{\lambda^{t}/\mu^{t}}(q^{-\nu^{t}-\rho}).$
(B.1)
According to the definition of $s_{\lambda/\mu}$ (A.15) we only need to prove
$\displaystyle h_{r}(q^{\nu+\rho})=(-1)^{r}e_{r}(q^{-\nu^{t}-\rho}).$
###### Proof.
Now we use the Frobenius notation of a partition
$\nu=(\alpha_{1},\cdots,\alpha_{d}|\beta_{1},\cdots,\beta_{d})$. Suppose
$\nu_{1}=N$, $\nu_{1}^{t}=k$
$\displaystyle E(t,q^{-(\nu^{t}+\rho)})$ $\displaystyle=$
$\displaystyle(1+q^{-(\nu_{1}^{t}-1+1/2)}t)\cdots(1+q^{-(\nu^{t}_{j}-j+1/2)}t)\cdots(1+q^{-(\nu^{t}_{N}-N+1/2)}t)\times$
(B.2) $\displaystyle\times(1+q^{-(-(N+1)+1/2)}t)\cdots$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{d}\frac{1+q^{-(\beta_{i}+1/2)}t}{1+q^{\alpha_{i}+1/2}t}E_{0}(t)$
where $E_{0}(t)=\prod(1+q^{-\rho}t)$. We have used an identity among multisets
of number
$\\{1,2,\cdots,N\\}=\\{j-\nu^{t}_{j},(N\geq
j>d)\\}\cup\\{\alpha_{i}+1(i=1,\cdots,d)\\}.$
Similarly
$\displaystyle H(t,q^{\nu+\rho})$ $\displaystyle=$
$\displaystyle\frac{1}{1-q^{\nu_{1}-1+1/2}t}\cdots\frac{1}{1-q^{\nu_{i}-i+1/2}t}\cdots\frac{1}{1-q^{\nu_{k}-k+1/2}t}\frac{1}{1-q^{-(k+1)+1/2}t}$
(B.3) $\displaystyle=$
$\displaystyle\prod_{i=1}^{d}\frac{1-q^{-(\beta_{i}+1/2)}t}{1-q^{\alpha_{i}+1/2}t}H_{0}(t)$
where $H_{0}(t)=\prod(1-q^{-\rho}t)^{-1}$ and
$\\{1,2,\cdots,k\\}=\\{i-\nu_{i},(k\geq
i>d)\\}\cup\\{\beta_{i}+1(i=1,\cdots,d)\\}.$
The first factor in (B.2) and (B.3) are almost the same except the $+$ and $-$
sign in front of $q$. In addition $\prod(1-q^{-\rho}t)$ and
$\prod(1-q^{\rho}t)^{-1}$ have the same power expansion of $t$. The difference
can be resolved by
$\displaystyle E(-t,q^{-\nu^{t}-\rho})=H(t,q^{\nu+\rho}).$
Therefore we obtain the result we want
$\displaystyle e_{r}(q^{-\nu^{t}-\rho})=(-1)^{r}h_{r}(q^{\nu+\rho}).$
∎
## References
* [1] Rajesh Gopakumar and Cumrun Vafa. On the gauge theory/geometry correspondence. 1999\.
* [2] Hirosi Ooguri and Cumrun Vafa. Knot invariants and topological strings. Nuclear Physics B, 577(3):419–438, 2000.
* [3] Edward Witten. Quantum field theory and the jones polynomial. Communications in Mathematical Physics, 121(3):351–399, 1989.
* [4] Hugh R Morton and Sascha G Lukac. The homfly polynomial of the decorated hopf link. Journal of Knot Theory and Its Ramifications, 12(03):395–416, 2003\.
* [5] Andrei Okounkov, Nikolai Reshetikhin, and Cumrun Vafa. Quantum Calabi-Yau and classical crystals. Progr.Math., 244:597, 2006.
* [6] Mina Aganagic, Albrecht Klemm, Marcos Marino, and Cumrun Vafa. The topological vertex. Communications in mathematical physics, 254(2):425–478, 2005.
* [7] Mina Aganagic, Robbert Dijkgraaf, Albrecht Klemm, Marcos Marino, and Cumrun Vafa. Topological strings and integrable hierarchies. Communications in mathematical physics, 261(2):451–516, 2006.
* [8] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa. Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun.Math.Phys., 165:311–428, 1994.
* [9] Vincent Bouchard and Piotr Sułkowski. Topological recursion and mirror curves. arXiv preprint arXiv:1105.2052, 2011.
* [10] F Calogero. Solution of a three-body problem in one dimension. Journal of Mathematical Physics, 10(12):2191–2196, 2003.
* [11] John Cardy. Boundary conformal field theory. arXiv preprint hep-th/0411189, 2004.
* [12] Amer Iqbal, Can Kozcaz, and Cumrun Vafa. The refined topological vertex. JHEP, 0910:069, 2009.
* [13] Michael R Douglas. Conformal field theory techniques in large n yang-mills theory. pages 119–135, 1995.
* [14] M. Jimbo and T. Miwa. Solitons and infinite dimensional lie algebra. Publ. RIMS, Kyoto Univ., 19:943–1001, 1983.
* [15] Nadav Drukker, Davide Gaiotto, and Jaume Gomis. The virtue of defects in 4d gauge theories and 2d cfts. Journal of High Energy Physics, 2011(6):1–54, 2011.
* [16] Ludvig Dmitrievich Faddeev and Rinat M Kashaev. Quantum dilogarithm. Modern Physics Letters A, 9(05):427–434, 1994.
* [17] Robbert Dijkgraaf. Chiral deformations of conformal field theories. Nuclear physics B, 493(3):588–612, 1997.
* [18] B. Eynard and C. Kozcaz. Mirror of the refined topological vertex from a matrix model. 2011\.
* [19] Kentaro Hori and Cumrun Vafa. Mirror symmetry. arXiv preprint hep-th/0002222, 2000.
* [20] Amer Iqbal, Can Kozcaz, and Khurram Shabbir. Refined topological vertex, cylindric partitions and the u(1) adjoint theory. Nucl. Phys., B838:422–457, 2010.
* [21] I. G. Macdonald. Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford University Press, 2 edition, 1999\.
* [22] Marcos Marino. Chern-Simons theory and topological strings. Rev.Mod.Phys., 77:675–720, 2005.
* [23] Andrei Okounkov. Infinite wedge and random partitions. Selecta Mathematica, 7(1):57–81, 2001.
* [24] Bill Sutherland. Exact results for a quantum many-body problem in one dimension. Physical Review A, 4(5):2019, 1971.
* [25] J. Zhou. Quantum Mirror Curves for $$\\{$$\backslash$mathbb C$\\}$^3$ and the Resolved Confiold. ArXiv e-prints, July 2012.
* [26] Jian Zhou. A conjecture on hodge integrals. arXiv preprint math/0310282, 2003.
* [27] Jian Zhou. Curve counting and instanton counting. arXiv preprint math/0311237, 2003.
* [28] Jian Zhou. Explicit formula for witten-kontsevich tau-function. arXiv preprint arXiv:1306.5429, 2013.
|
arxiv-papers
| 2014-03-02T10:10:52 |
2024-09-04T02:49:59.168451
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian-Feng Wu and Jie Yang",
"submitter": "Jian-Feng Wu",
"url": "https://arxiv.org/abs/1403.0181"
}
|
1403.0231
|
11institutetext: US Naval Research Laboratory, Washington, DC 20375
# On Flux Rope Stability and Atmospheric Stratification in Models of Coronal
Mass Ejections Triggered by Flux Emergence
E. Lee formerly of US Naval Research Laboratory V.S. Lukin M.G. Linton
###### Abstract
Context. Flux emergence is widely recognized to play an important role in the
initiation of coronal mass ejections. The Chen & Shibata (2000) model, which
addresses the connection between emerging flux and flux rope eruptions, can be
implemented numerically to study how emerging flux through the photosphere can
impact the eruption of a pre-existing coronal flux rope.
Aims. The model’s sensitivity to the initial conditions and reconnection
micro-physics is investigated with a parameter study. In particular, we aim to
understand the stability of the coronal flux rope in the context of X-point
collapse, as well as the effects of boundary driving in both unstratified and
stratified atmospheres.
Methods. A modified version of the Chen & Shibata model is implemented in a
code with high numerical accuracy with different combinations of initial
parameters governing the magnetic equilibrium and gravitational stratification
of the atmosphere. In the absence of driving, we assess the behavior of waves
in the vicinity of the X-point. With boundary driving applied, we study the
effects of reconnection micro-physics and atmospheric stratification on the
eruption.
Results. We find that the Chen & Shibata equilibrium can be unstable to an
X-point collapse even in the absence of driving due to wave accumulation at
the X-point. However, the equilibrium can be stabilized by reducing the
compressibility of the plasma, which allows small-amplitude waves to pass
through the X-point without accumulation. Simulations with the photospheric
boundary driving evaluate the impact of reconnection micro-physics and
atmospheric stratification on the resulting dynamics: we show the evolution of
the system to be determined primarily by the structure of the global magnetic
fields with little sensitivity to the micro-physics of magnetic reconnection;
and in a stratified atmosphere, we identify a novel mechanism for producing
quasi-periodic behavior at the reconnection site behind a rising flux rope as
a possible explanation of similar phenomena observed in solar and stellar
flares.
## 1 Introduction
Coronal mass ejections (CMEs) are a common occurrence in the Sun’s atmosphere
that are known to release giga-tons of plasma into interplanetary space. Some
of the ejected plasma can reach the space environment of the Earth and have a
strong and complex influence on space activity by inducing geospace
disruptions that can severely impact spacecraft, power grids, and
communication (Baker et al., 2013). While CMEs are quite commonly observed
(Evans et al., 2013), especially during the peak of the solar cycle, they are
still poorly understood. Some of the biggest CME mysteries pertain to their
origin, propagation, and relation to flares.
The initiation of CMEs has been widely studied and yet remains largely
unexplained (see reviews by Forbes et al., 2006; Chen, 2011). However, many
observational studies of associated features have led to clues about how they
occur and what factors contribute to their destabilization (see review by
Gopalswamy et al., 2006). Prior to an eruption, large-scale shear motions are
often observed in photospheric images, especially about the magnetic neutral
line (Krall et al., 1982) and in the form of sunspot rotations (Tian &
Alexander, 2006). In addition, patches of magnetic flux are found to emerge,
expand, move, fragment, coalesce, and cancel over a wide range of length and
time scales (Sheeley, 1969; Zwaan, 1985; Centeno et al., 2007; Parnell et al.,
2009). It is believed that shear motions, sunspot rotation, and the emergence
of new flux are all related to the injection of magnetic helicity into coronal
magnetic structures that could be directly involved in the eruption (Chae,
2001; Kusano et al., 2002; Démoulin et al., 2002; Pariat et al., 2006; Magara
& Tsuneta, 2008).
In addition to the growing body of observational studies that have improved
our understanding of CMEs, many new insights have also emerged from
theoretical and numerical efforts. CMEs have been modeled in two and three
dimensions using both simple analytical methods and sophisticated
magnetohydrodynamic simulations (see Jacobs & Poedts, 2011, and references
therein). These models differ widely in physical and numerical details, each
making its own choice of how to address the trade-off between complexity and
computational feasibility.
Early theoretical models explained CMEs as a loss of equilibrium, due to
magnetic buoyant instabilities (e.g., van Tend & Kuperus, 1978; Low, 1981;
Demoulin & Priest, 1988), as well as MHD flows (Low, 1984) and reconnection
(Forbes & Isenberg, 1991). Forbes & Priest (1995) proposed a CME model based
on the movement of magnetic footpoints (sources) below a flux rope and the
subsequent development of a singular current sheet, through which a large
magnetic energy release should take place as the flux rope moves continually
outwards. Lin & Forbes (2000) refined their model and computed exact solutions
for the energy release, flux rope height, current sheet length, and
reconnection rate. The Lin & Forbes (hereafter, “LF”) model, while simplistic,
provides an important step forward in CME modeling because it offers exact
solutions to the time-dependent nonlinear problem of a flux rope eruption and
includes more than a heuristic treatment of magnetic reconnection.
Furthermore, it predicts many features (e.g., morphology, current sheet, post-
flare loops, flows, energetics) confirmed by observations (Ciaravella et al.,
2002; Ko et al., 2003; Lin et al., 2005).
A similar two-dimensional flux rope model was proposed by Chen & Shibata
(2000). Like LF, the Chen & Shibata (“CS”) model consists of a two-dimensional
configuration in which a flux rope sits above the photosphere, surrounded by a
line-tied coronal arcade. In both models, the magnetic equilibrium is
destabilized by photospheric driving, causing a current sheet to form in the
flux rope’s wake as it moves outwards. However, whereas the LF model calls for
a somewhat manufactured mechanism for destabilization via large-scale
convergence of the sources, the CS model improves upon the LF model by
incorporating flux emergence as the driver.
While it does not lend itself to a purely analytical treatment, the CS model
is suitable for numerical simulation. The authors report four very different
outcomes based on the position and direction of the driving, showing that the
location of the emergence per se is not a critical factor for destabilizing
the coronal flux rope but rather that the relative orientation of the emerging
flux determines whether the flux rope moves outwards/upwards (CME-like) or
inwards/downwards (failed eruption).
Several subsequent studies have built upon the CS model. For example, Chen et
al. (2004), Shiota et al. (2003), and Shiota et al. (2004) produced synthetic
emission images from CS simulations to compare morphological features,
reconnection in-flows, and coronal dimmings found in actual CME observations.
Moreover, Shiota et al. (2003) and Shiota et al. (2005) were able to identify
the formation, structure, and location of slow and fast shocks in the CMEs
produced in these simulations. Gravitational density stratification in an
isothermal atmosphere was considered by Chen et al. (2004); Shiota et al.
(2004) and also in a later study by Dubey et al. (2006) in spherical
coordinates with axisymmetry.
In this study, we re-examine the CS model using a more sophisticated numerical
tool, a more realistic atmosphere, and higher spatial resolution than previous
studies. Simulations are performed using a high-order spectral element method
with numerically accurate, self-consistent treatments of diffusive transport
(i.e., resistivity, viscosity, and thermal conduction). In addition, we
reformulate the initial conditions to have magnetic fields that are everywhere
continuous and differentiable, and to include a solar-like temperature profile
with a sharp transition region and density stratification.
Through an exploration of physical parameters, we find that the CS magnetic
equilibrium can be unstable even without a flux emergence driver. Linear
theory has shown that sufficient perturbation of the field lines near an
X-point by waves or motion can disrupt the balance between magnetic pressure
and magnetic tension, causing the X-point to collapse and form a reconnecting
current sheet (Priest & Forbes, 2000, chapter 2). Our simulations demonstrate
that under a wide range of conditions the CS equilibrium is susceptible to
such a collapse via nonlinear accumulation of fast magnetosonic waves at the
X-point (McLaughlin et al., 2009). However, we also show that in a
sufficiently incompressible plasma due, for example, to the presence of a
background ”guide” magnetic field co-aligned with the axis of the flux rope,
the X-point collapse does not take place and the CS magnetic equilibrium can
be stabilized. For both stable and unstable configurations, we investigate the
impact of the resistivity model enabling magnetic reconnection below the flux
rope, as well as the plasma parameters in the low solar atmosphere, on the
flux rope’s response to the flux emergence driver. We show that flux emergence
can produce a rising flux rope both in a stratified and an unstratified
atmosphere, though the resulting ejection speed, as well as the plasma
dynamics around the X-point, can be strongly effected by the magnitude of the
guide field and the atmospheric stratification.
## 2 Model
The CS model has a two-dimensional domain with motion and magnetic field
allowed perpendicular (as well as parallel) to the plane of the domain
($\mathbf{v},\mathbf{B}\in\mathbb{R}^{3}$). Therefore, we can write the
magnetic field, normalized to some value $B_{0}$, in terms of a scalar
potential $\psi$ representing the in-plane flux, and an out-of-plane scalar
field:
$\mathbf{B}(x,y;t)=\nabla(-\psi)\times\hat{\mathbf{e}}_{z}+b_{z}\hat{\mathbf{e}}_{z}$
(1)
All quantities are normalized in terms of the first three constants found in
Table 1: $L_{0}=5$ Mm, which is the unit of length; $B_{0}=10$ G, the unit of
magnetic field strength; and $N_{0}=10^{9}$ cm-3, the unit of number density.
Given the Alfén velocity $v_{A}\equiv B_{0}/\sqrt{\mu_{0}m_{p}N_{0}}$, where
$m_{p}$ is the proton mass, we define the unit time as $\tau\equiv
L_{0}/v_{A}$, unit temperature as $T_{0}\equiv B_{0}^{2}/(\mu_{0}k_{B}N_{0})$,
and unit pressure as $P_{0}\equiv B_{0}^{2}/\mu_{0}$. The solar surface
gravity $g_{S}=274$ m/s2 is similarly normalized as $g\equiv
g_{S}(\tau/v_{A})=2.88\cdot 10^{-3}$.
Due to symmetries intrinsic to the model, only half of the domain in the
horizontal direction has to be resolved ($x>0$). Thus, simulations are
performed in a computational domain $(x,y)\in[0,L_{x}]\times[0,L_{y}]$, with
the solar convection zone assumed to be located below the domain ($y<0$).
Table 1: Normalization constants Constant | Value (MKS) | Equivalent Value
---|---|---
$L_{0}$ | $5\cdot 10^{6}$ m | 5 Mm
$B_{0}$ | $10^{-3}$ T | 10 G
$N_{0}$ | $10^{15}$ m-3 | $10^{9}$ cm-3
$v_{A}$ | $6.90\cdot 10^{5}$ m/s | 690 km/s
$\tau$ | 7.25 s | $2\cdot 10^{-3}$ hr
$T_{0}$ | $5.76\cdot 10^{7}$ K | 4.97 keV
$P_{0}$ | $7.96\cdot 10^{-1}$ Pa | 7.96 dyne/cm2
### 2.1 Initial conditions
#### 2.1.1 Magnetic configuration
The initial magnetic configuration prescribed in the CS model consists of a
coronal flux rope of radius $r_{0}$ surrounded by an arcade of “loops” that
are line-tied in the photosphere.
The flux rope contains a current channel that is mirrored by an image current
far below the photosphere (outside the computational domain), and four line
currents produce a potential quadrupolar field just below the photosphere.
Since the bottom boundary of the numerical domain coincides with the
photosphere, the only visible current initially is that within the flux rope.
In the original CS study, the coronal flux rope is given by a flux function
$\psi_{l}$ that results in a discontinuous current density at the edge of the
flux rope. Therefore, we propose the following alternative:
$\displaystyle\psi_{l}=$
$\displaystyle\,\dfrac{r^{2}}{2r_{0}}-\dfrac{(r^{2}-r_{0}^{2})^{2}}{4r_{0}^{3}}\
,$ $r\leq r_{0}$ (2a) $\displaystyle\psi_{l}=$
$\displaystyle\,\dfrac{r_{0}}{2}-r_{0}\ln r_{0}+r_{0}\ln r\ ,$ $r>r_{0}$ (2b)
where $r^{2}=x^{2}+y^{2}$, and the center of the flux rope lies at ($x=0$,
$y=h$).
Our formulation for $\psi_{l}$ lends itself to a continuous current density:
$j_{l}=-\nabla^{2}\psi_{l}=\left\\{\begin{array}[]{l @{\hspace{6mm}}
c}\dfrac{4r^{2}}{r_{0}^{3}}-\dfrac{4}{r_{0}}\ ,\hfil\hskip 17.07164pt&r\leq
r_{0}\\\\[11.38109pt] 0\ ,\hfil\hskip 17.07164pt&r>r_{0}\end{array}\right.$
(3)
The other flux components of the initial configuration, representing the image
current and line currents, respectively, are kept as originally defined:
$\displaystyle\vphantom{\int}\psi_{i}=-\frac{r_{0}}{2}\ln\left[x^{2}+(y+h)^{2}\right]$
(4)
$\displaystyle\psi_{b}=c\,\ln\frac{\left[(x+0.3)^{2}+(y+0.3)^{2}\right]\left[(x-0.3)^{2}+(y+0.3)^{2}\right]}{\left[(x+1.5)^{2}+(y+0.3)^{2}\right]\left[(x-1.5)^{2}+(y+0.3)^{2}\right]}$
(5)
with $r_{0}=0.5$, $h=2$ and $c=0.25628$. All three flux functions are summed
to produce the initial magnetic equilibrium, shown in Fig. 1:
$\psi=\psi_{l}+\psi_{i}+\psi_{b}\ .$ (6)
Figure 1: Contours of $\psi$ in the initial conditions.
In addition to the line currents, which produce the in-plane magnetic field,
we also allow for a uniform background magnetic field out of the plane
$b_{z0}\,\hat{\mathbf{e}}_{z}$. This “guide” field contributes magnetic
pressure but no current.
Figure 2: Contours of $\psi$ (black) and $b_{z}$ (magenta) at $t=0$ for
$b_{z}$ as a function of $r$ (left) and as a function of $\psi$ (right).
In the original CS study, a density spike is applied to support the flux rope
against radial compression: an outward-acting pressure gradient force offsets
the inward-acting Lorentz force due to the flux rope’s poloidal field.
Equivalently, the flux rope can be supported against the radial Lorentz force
by magnetic pressure, as in Shiota et al. (2005): in addition to the
background guide field $b_{z0}$, we apply an additional axial field in the
flux rope which is highest in the center and diminishes over a radius of
$r_{0}$.
We note that if the axial field $b_{z}$ is specified as a function of the flux
rope radius alone the flux rope will not be force-free, as the contours of
$\psi$ are not perfectly circular due to the small but finite contributions of
$\psi_{b}$ and, to a lesser extent, of $\psi_{i}$ to the total flux in the
coronal flux rope. It can be seen from the left panel of Fig. 2 that, given
such a function $b_{z}(r)$, the contours of $\psi$ and $b_{z}$ would not be
well-aligned.
To avoid the misalignment and minimize unbalanced Lorentz forces in the
initial condition, we instead choose to specify $b_{z}$ as a function of
$\psi$, as follows:
$\displaystyle b_{z}=\left\\{\begin{array}[]{l @{\hspace{6mm}}
c}\sqrt{b_{z0}^{2}+\dfrac{10}{3}-8\left(\dfrac{\zeta}{r_{0}}\right)^{2}+6\left(\dfrac{\zeta}{r_{0}}\right)^{4}-\dfrac{4}{3}\left(\dfrac{\zeta}{r_{0}}\right)^{6}}\hfil\hskip
17.07164pt&\zeta\leq r_{0}\\\\[8.53581pt] b_{z0}\ ,\hfil\hskip
17.07164pt&\zeta>r_{0}\end{array}\right.$ (9)
$\displaystyle\zeta^{2}(\psi)=2r_{0}^{2}-\sqrt{3r_{0}^{4}-4r_{0}^{3}(\psi-\psi_{0})}\
.$ (10)
(The derivation of the above equations can be found in Appendix A.) Note that
$\zeta=0$ when $\psi=\psi_{0}-r_{0}/4$, and $\zeta=r_{0}$ when
$\psi=\psi_{0}+r_{0}/2$, where
$\psi_{0}\equiv\left[\psi_{i}+\psi_{b}\right]|_{(x,y)=(0,h)}$.
#### 2.1.2 Unstratified atmosphere
In the case of an unstratified atmosphere, the number density field is
initialized to a uniform value of $n=n_{0}=1(N_{0})$. The pressure field of an
electron-proton plasma can be determined by the following equation of state:
$p=2nT$ (11)
We choose a uniform initial temperature, so the initial pressure $p=p_{0}$ is
also uniform. The free parameter $p_{0}$ is chosen variably in the simulations
to yield temperatures close to coronal values, as well as low plasma
$\beta\equiv 2p/B^{2}$.
An unstratified atmosphere has the advantage of isolating the flux rope
dynamics from the thermodynamics. By controlling $p_{0}$, one essentially
explores different regimes of the plasma $\beta$.
#### 2.1.3 Stratified atmosphere
We also attempt to simulate a solar-like atmosphere by modeling the average
vertical temperature profile as a hyperbolic tangent function, as in Leake &
Arber (2006); Leake & Linton (2013):
$T(y)=\frac{T_{p}}{T_{0}}+\left(\frac{T_{c}-T_{p}}{2T_{0}}\right)\left[1+\tanh\left(\frac{y-y_{\text{\tiny
TR}}/L_{0}}{\Delta y/L_{0}}\right)\right]$ (12)
with photospheric temperature $T_{p}=5000$ K, coronal temperature
$T_{c}=10^{6}$ K, transition region height $y_{\text{\tiny TR}}=2.5$ Mm, and
transition region width $\Delta y=0.5$ Mm.
Given this temperature profile, we seek compatible density and pressure
profiles such that the plasma is in hydrostatic equilibrium:
$\frac{dp}{dy}+ng=0$ (13)
with the constant gravitational acceleration $g$ pointed in the
$-\hat{\mathbf{e}}_{y}$ direction.
We solve (13) using (11) and (12) (see the derivation in Appendix B). The
resulting pressure profile is:
$\begin{split}p(y)=&p_{0}\exp\left\\{\frac{g\Delta
y/L_{0}}{2T_{c}/T_{0}}\left[-\frac{y-y_{\text{\tiny TR}}/L_{0}}{\Delta
y/L_{0}}\right.\right.\\\
&\left.\left.+\frac{T_{c}-T_{p}}{2T_{p}}\ln\left(\frac{T_{p}}{T_{0}}\exp\left[-\frac{2(y-y_{\text{\tiny
TR}}/L_{0})}{\Delta
y/L_{0}}\right]+\frac{T_{c}}{T_{0}}\right)\right]\right\\}.\end{split}$ (14)
Here the parameter $p_{0}$ corresponds to a constant of integration that
shifts the entire pressure profile of the atmosphere. As in the unstratified
case, this affects the values of $\beta$, which should be high ($\sim$ 10) in
the photosphere and low ($\sim$ $10^{-2}$) in the corona. Therefore, we do not
vary $p_{0}$ for the stratified simulations.
Figure 3: Logarithm (base 10) of normalized number density $n$ (green),
normalized pressure $p$ (blue), and temperature $T$ (red) as a function of
height in the initial conditions for a stratified atmosphere.
Profiles of the initial density, pressure, and temperature in a stratified
atmosphere are plotted in Fig. 3. It is evident that all three quantities vary
smoothly and by multiple orders of magnitude. Furthermore, this transition
occurs well below the height of the flux rope core ($y=2$).
### 2.2 Numerical method
We implement this initial configuration in the high-fidelity numerical
simulation framework, HiFi, which makes use of high-order spectral elements
and implicit time-stepping (Lukin, 2008; Lukin & Linton, 2011). As a strong
condition, HiFi requires all variables to be represented by continuous
functions in the initial and boundary conditions. Therefore, the magnetic
field must be everywhere differentiable, implying $\psi\in\mathcal{C}^{2}$. In
addition, the boundary driving of flux needs to be differentiable in both
space and time, in order for the electric and magnetic fields to be smooth.
These conditions are well satisfied by the initial conditions described above.
In this work, HiFi is used to integrate in time the following equations of
visco-resistive MHD:
$\displaystyle\frac{\partial n}{\partial
t}+\nabla\cdot\left(n\mathbf{v}\right)=0$ (15a)
$\displaystyle\frac{\partial(-\psi)}{\partial
t}=-\mathbf{v}\times\mathbf{B}+\eta j_{z}$ (15b) $\displaystyle\frac{\partial
b_{z}}{\partial
t}+\nabla\cdot\left(b_{z}\mathbf{v}-v_{z}\mathbf{B}\right)=\nabla\cdot\left(\eta\nabla
b_{z}\right)$ (15c) $\displaystyle\frac{\partial n\mathbf{v}}{\partial
t}+\nabla\cdot\left\\{n\mathbf{v}\mathbf{v}+p\mathbf{I}-\mu
n\left[\nabla\mathbf{v}+\left(\nabla\mathbf{v}\right)^{T}\right]\right\\}=\mathbf{j}\times\mathbf{B}$
(15d) $\displaystyle\frac{3}{2}\frac{\partial p}{\partial
t}+\nabla\cdot\left(\frac{5}{2}p\mathbf{v}-\kappa\nabla
T\right)=\mathbf{v}\cdot\nabla p+\eta j^{2}+\mu
n\left[\nabla\mathbf{v}+\left(\nabla\mathbf{v}\right)^{T}\right]:\nabla\mathbf{v}$
(15e) with an auxiliary equation (Ampère’s law):
$\nabla\times\mathbf{B}=\mathbf{j}$ (15f)
The normalized transport coefficients found in Eqs. (15f) – namely $\kappa$,
$\mu$, $\eta$ – control the level of dissipation of the MHD fluid quantities
through molecular diffusion: temperature, velocity, and current, respectively.
Each of these three transport parameters is chosen to be compatible with the
resolution and objective of each simulation. Further, for some of the
simulations (see below), we allow the resistivity $\eta$ to be a function of
local current density, $\eta=\eta_{bg}+\eta_{anom}(\mathbf{j})$, where
$\displaystyle\eta_{anom}(\mathbf{j})=$ $\displaystyle\,0$
$|\mathbf{j}|<j_{c}$ $\displaystyle\eta_{anom}(\mathbf{j})=$
$\displaystyle\,\bar{\eta}_{anom}\frac{\left\\{1-\cos\left[\pi(|\mathbf{j}|/j_{c}-1)\right]\right\\}}{2}$
$j_{c}\leq|\mathbf{j}|\leq 2j_{c}$, $\displaystyle\eta_{anom}(\mathbf{j})=$
$\displaystyle\,\bar{\eta}_{anom}$ $|\mathbf{j}|>2j_{c}$
$\eta_{bg}$ is the uniform and time-independent background resistivity, and
$\eta_{anom}$ is some “anomalously enhanced” effective resistivity,
$\bar{\eta}_{anom}\gg\eta_{bg}$, occurring due to micro-physics not captured
by the MHD model whenever the current density rises above the critical current
density $j_{c}$.
#### 2.2.1 Boundary conditions
Table 2: Boundary conditions Boundary | Unstratified | Stratified
---|---|---
Left (reflection) | only vertical flow | only vertical flow
Top/Right (coronal) | only vertical flow, $\nabla_{\hat{n}}\\{n,b_{z},j_{z},p\\}=0$ | only vertical flow, $\nabla_{\hat{n}}\\{n,b_{z},j_{z},p\\}=0$
Bottom (photosphere) | no flow, $\nabla_{\hat{n}}\\{n,b_{z},j_{z},T\\}=0$ | only out-of-plane flow, $\partial_{t}\left[\nabla_{\hat{n}}\\{\ln(n)\\}\right]=0$, $\nabla_{\hat{n}}\\{nv_{z},b_{z},j_{z},T\\}=0$
The bottom boundary of the simulation domain, representing the photosphere, is
perhaps the most important boundary condition affecting the outcome of a
simulation. Flux emergence is achieved by varying the flux function at this
boundary in time, which is equivalent to applying an electric field. This
electric field determines the evolution of the magnetic field, which can be
advected in or out of the domain or resistively dissipated, as described by
Ohm’s Law:
$\frac{\partial\psi}{\partial
t}=E_{z}=-\hat{\mathbf{e}}_{z}\cdot\mathbf{v}\times\mathbf{B}+\eta j_{z}$ (17)
The resistive component, the second term on the right-hand side of (17), is
determined by the geometry of the magnetic field at any given time. Therefore,
by varying the flux at the boundary ($\partial\psi/\partial t$) in a
prescribed way, we also induce cross-field plasma motions
($\mathbf{v}\times\mathbf{B}$).
Chen & Shibata (2000) prescribe two cases of localized boundary driving,
namely, over a region $|x-x_{0}|\leq 0.3$ centered at $x_{0}=0$ (case A) and
at $x_{0}=3.9$ (case B). We apply the same method only for the case of
$x_{0}=0$, but use a formulation that is smoother in time and in space:
$\begin{array}[]{c
c}\psi(x,0;t)=\psi(x,0;0)+\dfrac{\psi_{e}(x)}{2}\left[\dfrac{t}{t_{e}}-\dfrac{\sin\left(2\pi
t/t_{e}\right)}{2\pi}\right]\ ,&t\leq t_{e}\\\\[14.22636pt]
\psi_{e}(x)=\dfrac{c_{e}}{2}\left[1+\cos\dfrac{\pi(x-x_{0})}{0.3}\right]\
,&|x|\leq 0.3\end{array}$ (18)
where $t_{e}$ is the duration over which the electric field drive is applied
at the boundary. For $t>t_{e}$, the photospheric boundary is treated as a
perfect conductor.
We do not allow any in-plane flow on the bottom photospheric boundary
($v_{x}=0$, $v_{y}=0$) and force the normal gradients of $b_{z},j_{z}$, and
temperature to be zero: i.e., $\nabla_{\hat{n}}\equiv\hat{n}\cdot\nabla=0$.
The unstratified atmosphere also has $\nabla_{\hat{n}}n=0$, while the
stratified case imposes a fixed value of the density scale height
$\partial_{t}\left\\{[\nabla_{\hat{n}}n]/n\right\\}=\partial_{t}\left\\{\nabla_{\hat{n}}[\ln(n)]\right\\}=0$.
The left boundary is a symmetry boundary, with odd symmetry required for the
horizontal and out-of-plane components of flow, $v_{x}$ and $v_{z}$, and even
symmetry imposed on all other dependent variables. At the outer boundaries
(top and right), the gradients of density, $b_{z}$, $j_{z}$, and pressure are
zero, and flow is only allowed in the vertical direction. Table 2 provides a
simple reference for the various boundary conditions applied in the
simulations.
#### 2.2.2 Dissipative Boundary Layers
##### Chromosphere.
The flux emergence represented by (18) changes the flux function just at the
boundary but has no direct effect on $\psi$ anywhere else, including just
above it. While Ohm’s Law (17) does relate flux evolution to fluid transport,
it does not guarantee that the flux function and other quantities will be
well-behaved for all time, particularly in the $y$-direction. Therefore, to
allow flux to slip more easily through the region just above the photospheric
driving (effectively, the “chromosphere”), we apply a resistive boundary layer
by enhancing $\eta$ locally according to the following function:
$\eta=\eta_{bg}+\eta_{anom}+\eta_{ph}\exp\left(-\frac{y^{2}}{y_{0}^{2}}\right)$
(19)
where $\eta_{ph}$ is the photospheric value of resistivity, and $y_{0}=0.2$
corresponds to the height of 1 Mm above the photosphere. Conceptually this
boundary layer emulates the enhanced resistivity of the chromosphere due to
collisional impedance by neutrals (Leake & Arber, 2006).
##### Outer corona.
Separately, to mimic an open domain boundary with no wave reflections in the
corona, a viscous boundary layer is prescribed close to the outer coronal (top
and right) boundaries. Starting at a distance of $d=0.5$ inward from the
boundary of the computational domain, $\mu$ is increased gradually towards the
domain boundary (according to a cosine profile) from a background value of
$\mu_{bg}$ up to the outer boundary value of $\mu_{out}=1$. In the same
fashion, near the outer boundaries with $d=0.5$, resistivity is ramped up from
$\eta_{bg}$ to a boundary value of $\eta_{out}=10^{-2}$.
## 3 Results
In this section, we discuss simulations of the undriven equilibrium, as well
as driven simulations of flux emergence in both the stratified and
unstratified cases. While the CS initial conditions describe an approximate
equilibrium, we find that this equilibrium can be unstable to small
perturbations. We discuss the role of MHD waves in destabilizing the flux rope
via X-point collapse and the role played by plasma compressibility in
stabilizing the X-point and the flux rope. Finally, we discuss the driven
simulations of stable and unstable equilibria with different resistivity
models, as well as details of the resulting eruption process.
Figure 4: (No driving.) Four snapshots of current density $j_{z}$ showing
outward propagating MHD waves. Notice the trapping and interference of waves
at the X-point; compounding of these waves here precipitates an X-point
collapse, leading to the formation of a current sheet and initiation of
magnetic reconnection.
### 3.1 Flux rope stability
In Chen & Shibata (2000), the coefficient $c$ (see Eq. 5 above) was determined
by trial and error to yield a magnetic equilibrium such that the center of the
flux rope did not move “for long enough time.” (It can also be derived by
requiring that the vertical component of the Lorentz force is zero at the
center of the flux rope,
$\hat{y}\cdot[\mathbf{j}\times\mathbf{B}]|_{(x,y)=(0,h)}=0$.) Our numerical
experiments confirm that this value for $c$ is indeed appropriate for
equilibrium, but we find that the equilibrium itself is tentative and unstable
to perturbations.
#### 3.1.1 X-Point collapse
At the beginning of each simulation, the flux rope—which is approximately
force-free—goes through a small adjustment to settle into an actual force-free
equilibrium. In Sec. 2, we explained how formulating $b_{z}$ as a function of
$\psi$ forces the contours of $b_{z}$ and $\psi$ to be aligned. Although this
is an improvement on the original setup, the configuration is still not
perfectly force-free because the contours of
$j_{z}=-\nabla^{2}\psi=-\nabla^{2}\psi_{l}$ are still circular and therefore
not aligned with the contours of $\psi$, producing a small but finite Lorentz
force. The adjustment to correct the misalignment, however small it may be, is
sufficient to generate waves that may destabilize the flux rope.
Fig. 4 illustrates the oscillation of current density induced by the fast
magnetosonic waves emanating from the flux rope as it adjusts to the initial
conditions. The distribution of the waves is not uniform because the initial
adjustment is one where the flux rope is squeezed in one direction
(horizontally) while expanding in the other direction (vertically). Therefore,
the waves propagating horizontally are out of phase with those propagating
vertically. It is interesting to note that the propagation of both types of
waves is initially radial but eventually becomes oblique at the flanks of the
active region due to the inhomogeneity of the magnetic pressure in the corona.
The fast waves themselves do not directly destabilize the flux rope. However,
the entire equilibrium can be destabilized when the waves reach the X-point
below the flux rope and cause it to collapse. The role of fast waves in
X-point collapse for a zero-$\beta$ plasma has been studied by McLaughlin et
al. (2009) and their behavior in the neighborhood of X-points has been
investigated by similar earlier studies (McLaughlin & Hood, 2004, 2005, 2006).
As described in these studies, we find that the fast waves approach the
X-point but tend to get trapped there if their phase speed becomes too low at
the X-point. The trapping occurs because the waves are refracted towards the
null and then wrap around it if they cannot pass through it. As a result, the
waves push current towards the null where it accumulates exponentially in the
linear regime. The buildup of waves at the null, however, quickly leads to
nonlinear behavior, forming shocks and jets, which deform the X-point into a
cusp-like geometry, which flattens and forms a current sheet (McLaughlin et
al., 2009). This fast-wave accumulation and resulting collapse of the X-point
are evident in the sequence of figures in Fig. 4.
The consequence of X-point collapse occurring as a result of fast wave
accumulation at the null is that it forms a current sheet separating anti-
parallel fields. The formation of the current sheet, in turn, kicks off
magnetic reconnection that drives itself for as long as there is free magnetic
energy available in the system. When the collapse forms a horizontal current
sheet, the reconnection process draws in the flux rope from above, and it
pulls itself down towards the photosphere to draw in flux from below,
destroying the original configuration. Formation of a vertical current sheet
similarly leads to a CME eruption.
Other factors that may contribute to a collapse of the X-point in the absence
of driving include boundary flows, likely related to the reflection of waves,
and the asymmetric resistivity model, which intentionally biases $\eta$ in the
$y$-direction in order to allow magnetic flux to slip through the photosphere
(see previous section). However, we found these effects to be sub-dominant to
the fast wave accumulation at the X-point in destabilizing the flux rope.
Figure 5: Dependence of flux rope stability on the free parameters $p_{0}$ and
$b_{z0}$ (left), as well as on the plasma $\beta$ and compressibility measure
$\Gamma$ (right). All simulations are performed without driving. The different
symbols signify that the flux rope was stable (black dots); was wobbly but on
average did not rise or sink (black stars); moved upwards (blue triangles); or
moved downwards (red triangles).
#### 3.1.2 Sensitivity to Compressibility
We find that sufficient magnetic pressure and/or gas pressure at the X-point
can suppress the X-point collapse. In the absence of a guide field
($b_{z0}=0$), the magnetic pressure drops to zero at the X-point. Then, with
low gas pressure, the fast wave speed is reduced drastically at the X-point
causing wave refraction and accumulation. However, a parametric exploration of
$p_{0}$ and $b_{z0}$ in an unstratified atmosphere revealed that increasing
either of these parameters helps to stabilize the flux rope. The left panel of
Fig. 5 is a graphical chart of the many simulations that were performed
scanning the parameter space of $p_{0}$ and $b_{z0}$. Black dots represent
simulations in which the flux rope was stable over many hundreds of Alfvén
times (no X-point collapse); blue triangles represent those in which the flux
rope experienced a slow rise (vertical collapse); red triangles represent
those in which the flux rope descended (horizontal collapse); and black stars
represent those in which the flux rope moved up and down but on average
maintained the same height in the atmosphere (oscillatory X-point collapse).
One could argue heuristically that increasing either $p$ or $b_{z}$
effectively decreases the compressibility of the plasma (or increases the
stiffness of the medium), so any motions at the X-point need to do more work
against the gas pressure or magnetic pressure to force a collapse of the
magnetic topology. Therefore, we propose a generalized measure of two-
dimensional plasma compressibility:
$\Gamma=\frac{b_{\perp}^{2}}{2p+b_{z}^{2}}$ (20)
and relate the free parameters of the simulations, $p_{0}$ and $b_{z0}$, to
the magnitude of the in-plane field $b_{\perp}\approx 1$ ($B_{0}$), as well as
to the initial background plasma $\beta$:
$\beta=\frac{2p_{0}}{b_{\perp}^{2}+b_{z0}^{2}}$ (21)
The right panel of Fig. 5 provides an alternative way to assess the effect of
the initial parameters on the stability of the flux rope and the X-point in
terms of dimensionless quantities. Note that $\beta$ and $\Gamma$ are related
to $b_{z}$ such that some combinations of the two are impossible (denoted by
gray shading in the figure):
$b_{z}^{2}=\frac{(1-\Gamma\beta)b_{\perp}^{2}}{\Gamma(1+\beta)}\ ,$ (22)
which implies $\Gamma\beta\leq 1$.
Within the accessible parameter space we observe that above a certain level of
compressibility (approximately 8, determined empirically), the X-point tends
to collapse horizontally and causes the flux rope to descend. Within the range
$4.5<\Gamma<8$, the X-point collapses vertically, causing the flux rope to
move upwards out of equilibrium (though much more slowly than in a driven
eruption). However, if the plasma is “stiffened” beyond a threshold,
$\Gamma\lesssim 3$, the fast waves are able to pass through the X-point as
their phase speed is no longer close to zero. Since the waves no longer
accumulate at the X-point, they do not cause it to collapse and the
equilibrium is preserved.
While it is possible that different perturbations might produce different
empirical thresholds of stability, it has not been the goal of the present
study to determine particular values but rather to show that the X-point
stability can be fundamentally related to the accumulation of fast waves at
the X-point, which can be moderated by changing the background compressibility
of the plasma. Similarly, while the magnitude of the dissipative transport
coefficients within the visco-resistive MHD simulations can have some impact
on the specific stability thresholds via damping of the fast waves emanating
from the flux rope, such damping does not qualitatively change the conclusion
of this parameter study.
Figure 6: Out-of-plane current density $j_{z}$ (color, saturated high and low
values), with magnetic flux contours (solid black), in a simulation of flux
emergence into an unstratified atmosphere. The four panels show snapshots of
the simulation at $100\tau=12$ min, $240\tau=29$ min, $480\tau=58$ min, and
$1800\tau=217.5$ min.
### 3.2 CME eruptions driven by flux emergence
The premise of the CS model is that a stable pre-existing flux rope can be
driven to eruption by magnetic flux emergence. Flux emergence is achieved
through photospheric boundary driving (see Eqs. 18): a small amount of flux is
effectively emerged through the photospheric boundary by applying a time-
dependent electric field. Emerging flux can cause the flux rope to move in
either direction by forcing a destabilization of the X-point, similarly to the
fast waves but more predictably. Within the underlying arcade, when the
emergent flux is oppositely oriented to the local flux, it causes a vertical
collapse of the X-point, leading to a rising flux rope. Oriented in the same
sense as the local flux rope, it causes a horizontal collapse of the X-point,
which forces the flux rope downwards. For emergence outside the arcade,
likewise, it is possible to choose values for the coefficient $c_{e}$ in Eqs.
18 such that the simulation results in a vertical X-point collapse, and when
the sign of $c_{e}$ is reversed, the X-point collapses horizontally. However,
the sign of $c_{e}$ must be carefully chosen based on topological and
geometric considerations, including the sign of the local overlying flux. In
this study, we restrict ourselves to discussing emergence at $x_{0}=0$ alone,
with $c_{e}=1.1$ as in the original CS model.111We note that the original CS
reference Chen & Shibata (2000) quotes $c_{e}=11$ and $c=2.5628$, but these
should have been quoted as a factor of 10 lower, as per personal communication
with P.F. Chen.
To evaluate the impact of reconnection micro-physics, stability of the initial
condition and atmospheric stratification on the system’s response to flux
emergence in the CS model, the simulation study described below has been
performed by changing one model parameter at a time with otherwise identical
numerical and dissipation parameters. In the reference simulation with an
unstratified atmosphere, the background magnetic “guide” field is set to
$b_{z0}=1$, equivalent to $10$ G and of the same order as $b_{\perp}$, such
that the plasma compressibility measure $\Gamma$ is less than unity and the
initial configuration is stable for any plasma pressure profile. To minimize
the impact of the size of the computational domain or the dissipative boundary
layers on the results of the simulations, the domain boundaries are placed at
$L_{x}=4$ and $L_{y}=10$. The computational grid spanning the
$(x,y)\in[0,L_{x}]\times[0,L_{y}]$ domain has $864$ and $1536$ spatial degrees
of freedom in the $x$ and $y$ directions, respectively, distributed non-
uniformly in such a way that the vertically elongated X-point reconnection
current sheet is well-resolved in the $x$-direction, while both magnetic and
thermodynamic structures associated with flux emergence through the
chromosphere can be well resolved in the $y$-direction.
The background resistivity throughout the domain is set to
$\eta_{bg}=10^{-5}$, the photospheric resistivity is set to
$\eta_{ph}=10^{-2}$, there is no anomalous resistivity $\bar{\eta}_{anom}=0$,
the background kinematic viscosity coefficient is set to $\mu_{bg}=10^{-4}$,
and the heat conduction is set to $\kappa=10^{-5}$. The duration of the flux
emergence is taken to be $t_{e}=300$, equivalent to $36.25$ minutes.
Figure 7: Unstratified atmosphere. Left: Height and speed of flux rope center.
Smoothing is performed using a Hanning window of 12 points. The speed is
computed by finite-differencing the smoothed (blue) curve. Right: Temperature
at the X-point (center of current sheet) below the flux rope during the same
period.
#### 3.2.1 Flux emergence in an unstratified atmosphere
To approximate the coronal conditions in the unstratified simulation, the
initial pressure is set to $p_{0}=10^{-2}$, such that the initial $\beta$ is
$\sim 1\%$ throughout the domain. To produce an eruption, the photospheric
electric field drive is applied within the arcade below the X-point to
generate $B_{x}$ opposite to the magnetic field of the arcade. As a result,
the magnetic pressure above the photospheric boundary is reduced causing a
local downflow towards the photosphere. This in turn reduces the plasma
pressure below the X-point, which forces an in-flow at the sides of the
X-point, bringing about its collapse and formation of a reconnection current
sheet (e.g., see Fig. 6).
As shown in Fig. 6, the X-point collapse in this simulation is observed to
occur at $t\approx 100$, forming a current sheet that reaches its maximum
length and strength near $t=200$. Current density then also increases along
the separatrices and the field lines connected to the current sheet. When the
new flux stops emerging ($t>t_{e}$), the current sheet persists at
approximately half to a third of its peak magnitude, slowly diminishing over
time for the duration of the simulation.
As reconnection ensues, the flux rope is nudged out of equilibrium (in the
$+\hat{\mathbf{e}}_{y}$ direction) by the reconnection outflow and continues
to move outwards as reconnection proceeds. The left panel of Fig. 7 tracks the
height of the flux rope center during the eruption by measuring the position
of the magnetic O-point (black dots). The height measurements are smoothed
(blue curve) using a Hanning window convolution over 12-point windows, and the
speed (red curve) is computed by finite-differencing the smoothed height. The
maximum speed of the flux rope is observed to be only about $0.7$ km/s,
quickly slowing down further as the reconnection loses steam. In the right
panel of Fig. 7, the temperature at the X-point, or the current sheet center,
is plotted in mega-Kelvin showing rapid heating early in the eruption due to
Joule heating at the current sheet.
We note that this reference simulation results in a very slowly rising flux
rope which is inconsistent with the original Chen & Shibata (2000) simulation
where the flux rope rise speed of approximately $70$ km/s was observed. To
study the sensitivity of this result to the magnitude of the background
magnetic guide field and the micro-physics of reconnection at the X-point,
represented here by the anomalous resistivity model similar to that of Chen &
Shibata (2000), a series of further simulations has been performed. Figure 8
shows traces of the height of the flux rope center for a set of five
simulations with three different values of the guide magnetic field
$b_{z0}=\\{1.0,0.5,0.25\\}$ and two resistivity models, one with
$\bar{\eta}_{anom}=0$ and another with $\bar{\eta}_{anom}=10^{-2}$ and
$j_{c}=10$, both using the constant background resistivity value
$\eta_{bg}=10^{-5}$.
Figure 8: Unstratified atmosphere. Height of flux rope center for a set of
five simulations with varying magnitude of initial background magnetic guide
field and resistivity models. Three values of the guide magnetic field
$b_{z0}=\\{1.0,0.5,0.25\\}$ and two resistivity models, one with
$\bar{\eta}_{anom}=0$ labeled as “eta const”, and one with
$\bar{\eta}_{anom}=10^{-2}$ and $j_{c}=10$ labeled as “eta anom”, both using
the constant background resistivity value $\eta_{bg}=10^{-5}$, are considered.
The comparison of the five traces clearly demonstrates that the outcome of the
simulations is much more sensitive to the magnitude of the background guide
field, i.e. the global structure and stability of the magnetic configuration,
than to the resistivity model. The two traces with $b_{z0}=1.0$, the initially
stable magnetic configuration, are virtually indistinguishable from each other
despite very different resistivity models. The two traces with $b_{z0}=0.5$
initialized from a marginally stable configuration (see Fig. 5) do show small
differences during the acceleration phase. Here the simulation with anomalous
resistivity allows for slightly faster rise, but both rise much faster than
the $b_{z0}=1.0$ cases. And the initially unstable $b_{z0}=0.25$ case
demonstrates yet faster rise of the flux rope that is comparable to the rise
speed observed in the Chen & Shibata (2000) simulation. (Only the anomalous
resistivity $b_{z0}=0.25$ simulation trace is shown in Fig. 8 because the
corresponding uniform resistivity simulation produces a very intense X-point
current sheet that breaks up due to secondary instabilities (Loureiro et al.,
2007), leading to formation of further spatial sub-structure which we have
chosen not to attempt to resolve. Detailed investigation of such multi-scale
reconnection cases is left for future work.) We note that the choice of
critical current density $j_{c}=10$ for onset of anomalous resistivity is such
that all five simulations achieve $|{\bf j}|>j_{c}$ at the X-point during the
acceleration phase of the flux rope, yet that does not result in significant
acceleration of the flux rope for the $b_{z0}=1.0$ and $b_{z0}=0.5$ cases.
It is also of interest that the rapid rise of the flux rope in the
$b_{z0}=0.25$ case is followed by stagnation at the height of approximately
$19$Mm. Such stagnation is indicative of the system finding a new stable
magnetic equilibrium where the upward force on the flux rope is balanced by
the magnetic tension distributed throughout the overlying magnetic arcade.
#### 3.2.2 Flux emergence in a stratified atmosphere
Introduction of atmospheric stratification, as described in Sec. 2.1.3, leads
to a more realistic equilibrium plasma configuration that is much denser at
the photosphere than in the unstratified corona-like case.
The impact of the flux emergence at the bottom boundary, with and without the
atmospheric stratification, is reflected in the traces of height and speed of
the respective flux rope eruptions. For the stratified atmosphere, the height
and speed of the flux rope as functions of time are shown in the left panel of
Fig. 9 and can be compared to the equivalent traces for the reference
simulation in the left panel of Fig. 7. (Note the different ranges of the time
axes of the two panels.) The two time histories are qualitatively similar,
both showing rapid acceleration of the flux-rope during flux emergence, with a
reduction of the ejection speed by approximately a factor of two once the
driving is turned off. However, both the peak and the post-driving ejection
speed of the CME in the stratified atmosphere are less than half of that
obtained in the unstratified case.
Figure 9: Stratified atmosphere. Left: Height and speed of flux rope center.
Smoothing is performed using a Hanning window of 12 points. The speed is
computed by finite-differencing the smoothed (blue) curve. Right: Temperature
at the X-point (center of current sheet) below the flux rope during the same
period.
Another significant difference between the two cases of flux emergence is
observed by comparing the time traces of the X-point plasma temperature, shown
in the right panels of Fig. 7 and Fig. 9. While in the unstratified atmosphere
there is a notable temperature increase at the X-point at the time of
eruption, in the stratified simulation the temperature decreases instead.
Furthermore, as the flux rope begins to rise between $35$ and $110$ minutes
into the simulation ($300\lesssim t\lesssim 900$) the stratified simulation
shows an oscillatory X-point temperature as long as the flux rope is within
$\approx\>1$ Mm of its original position. The root cause of the overall
X-point cooling can easily be explained as upflows of cold chromospheric
plasma being advected into the coronal reconnection region. Nevertheless, the
observed self-induced quasi-periodic oscillatory behavior of the X-point
temperature is somewhat unexpected.
Figure 10: Evolution in time of the reconnection site behind the CME flux rope
in a stratified atmosphere. Each panel shows a snapshot of the temperature
structure on the left, the density structure on the right, select contours of
the magnetic flux $\psi$ (the same contour values, denoting the same magnetic
field lines, have been chosen for each panel), and arrows denoting the in-
plane plasma flow. The snapshots are made $348\tau=42$ min, $528\tau=64$ min,
$708\tau=86$ min, and $978\tau=118$ min into the simulation. Note that for
illustration purposes both plasma temperature and number density are plotted
using logarithmic color scales with saturated high and low values. Arrows
showing the plasma flow have been scaled by a factor of 25 with respect to the
linear dimensions of the domain so that an arrow of unit length corresponds to
flow of $1.7\times 10^{4}$ km/s.
Fig. 10 shows the evolution in time of plasma temperature, density, and flows
around the X-point during the period of quasi-periodic temperature
oscillations. Continuous upflows of dense cool plasma convected along the
magnetic field lines and into the reconnection region around the X-point are
apparent throughout the evolution. The lower-right panel of this figure makes
clear that this continuous chromospheric upflow results in quasi-periodic
striations of cool dense material alternating with hotter, lower-density
plasma on the recently reconnected field lines rising towards (and with) the
flux rope located above. These striations are the signatures of the same
oscillatory behavior observed on the X-point temperature trace in Fig. 9.
While the origins and parametric robustness of the observed quasi-periodic
phenomenon require further in-depth study that is outside of the scope of this
article, a heuristic explanation of the basic physical mechanism is
straightforward. It results from the competition between the upward directed
tension force in newly reconnected magnetic field lines and the downward
directed gravity acting on the dense, cold plasma deposited onto these same
field-lines by the chromospheric upflows. As in the formation of water
droplets, whenever sufficient amount of plasma accumulates in a small enough
volume in the V-shaped dip of a set of recently reconnected field lines, the
gravitational pull on that plasma overcomes the field’s tension force and a
droplet of plasma forms and falls vertically through the reconnection site
itself. As a result, those flux-rope destined field lines that produce the
droplets end up with lower density hotter plasma, while the field lines that
pass through in between the droplets contain colder and heavier plasma. The
temperature at the X-point, where the reconnection is regularly disrupted by
the droplets, is similarly modulated when the plasma that has been heated by
the reconnection process is periodically replaced by the cold plasma of the
droplets.
Below the reconnection site, the pattern of chromospheric upflows along the
magnetic separatrices and vertical downflows through the X-point creates a
circulation of plasma between reconnection’s outflow and inflow. How, and
whether or not, this circulation pattern contributes to the formation of the
quasi-periodic temperature and density structure described above is left as a
topic for future study.
## 4 Discussion & Conclusions
Coronal mass ejections are eruptive solar events of enormous proportions that
shed plasma and magnetic flux into interplanetary space. The Chen & Shibata
model is a good starting point for understanding how such an eruption can
originate from the destabilization of a global magnetic configuration by local
flux emergence. It helps us to see a connection between flux emergence, a
phenomenon at the solar surface, and flux rope ejection, a phenomenon in the
corona. Many observational studies have shown spatio-temporal correlations
between flux emergence and eruptive events, but few theoretical models to date
have identified a precise single mechanism or sequence of processes whereby
producing magnetic flux at the photosphere dynamically triggers an eruption.
The CS model may assume an oversimplified solar atmosphere and a somewhat
manufactured magnetic topology, but it does proffer a complete story. To
determine the effects of a more realistic solar atmosphere, we have undertaken
an effort to repeat the study using a different numerical suite and allowing
for a stratified atmosphere with the density variation of over four orders of
magnitude, as well as a sharp temperature transition between the chromosphere
and the corona.
We have found that even in the absence of stratification the initial
equilibrium can be unstable to small perturbations. The initial adjustment of
the magnetic equilibrium to slight force imbalances can generate fast waves
that may not be able to propagate through the X-point below the flux rope. In
these cases, the fast waves accumulate in such a way as to collapse the
X-point and initiate reconnection. Thus, the equilibrium can be destabilized
before any photospheric driving is applied. However, we also found that the
stability of the CS equilibrium can be controlled by varying the
compressibility of the plasma, which in a two-dimensional system is determined
by the combination of thermal pressure and the magnitude of the out-of-plane
component of the magnetic field. To quantify this effect, we defined a
generalized measure of compressibility $\Gamma$ and have empirically
determined the equilibrium’s stability boundaries in terms of $\Gamma$.
When emulating flux emergence by applying an electric field at the
photospheric boundary, in the unstratified atmosphere, the results of our
simulations are qualitatively similar to those of the original study. However,
there are also important differences and new findings. As opposed to the
original study, when initialized in a stable configuration, our simulations
show little evidence of significant flux rope acceleration or Joule heating
associated with the reconnection current sheet. Notably, this result appears
to be insensitive to the micro-physics of the reconnection region. By varying
the magnitude of the background out-of-plane magnetic field component and thus
changing the stability of the global magnetic configuration, we also show that
flux rope rise speeds comparable to the original result are possible but
require an unstable magnetic configuration as the initial condition.
We further show that the micro-physics of reconnection is more likely to slow
down than to accelerate the flux rope by comparing simulations with and
without anomalous resistivity. It is well known that current-dependent
anomalous resistivity allows for “fast” magnetic reconnection with only weak
dependence on the magnitude of resistivity itself (Malyshkin et al., 2005).
Yet, for both initially stable and quasi-stable magnetic configurations,
allowing for anomalous resistivity did not result in a substantial increase of
the flux rope rise speed. That is, merely allowing for faster reconnection did
not lead to faster reconnection and faster flux rope ejection. On the other
hand, in magnetic configurations where fast flux-rope ejection is possible,
the simulations with low guide field indicate that the inability of the
magnetic reconnection process to occur sufficiently fast could limit the rise
speed of the flux rope.
In the flux emergence simulations with stable magnetic configuration and
realistic atmospheric stratification, the weakness of the X-point heating and
the slowness of the ejected flux rope are reproduced, and amplified. In these
simulations, changes in the magnetic field structure due to flux emergence
generate persistent chromospheric upflows of cold, dense material that is
convected into and dramatically cools the reconnection current sheet.
In addition to the steady state upflows and cooling, the stratified
simulations also produce another type of behavior: self-induced quasi-periodic
oscillations in the X-point temperature, density, and other fluid quantities.
The quasi-periodic oscillations observed in the stratified simulation are of
transient nature, appearing after the flux emergence drive has been completed
and lasting for just over an hour while the flux rope is within $\approx 1$ Mm
of its initial location. The robustness of this phenomenon will be a subject
of future research, but our initial investigation indicates that a critical
balance between the upward tension force of the reconnected magnetic field and
the downward gravitational pull on the dense chromospheric plasma convected
into the reconnection region has to be achieved in order for the quasi-
periodic oscillations to appear in a simulation. While that may seem to be a
prohibitive constraint, we speculate that in the three-dimensional parameter
space spanned by (1) the height of the X-point, (2) the strength of the
magnetic fields and (3) the horizontal location of the emerging flux relative
to the separatrices of the pre-existing magnetic configuration, all quantities
that can vary greatly throughout the lower solar atmosphere, there is likely
embedded a two-dimensional parameter space where such balance can, indeed, be
achieved.
We note that there is also extensive observational evidence for what has been
called quasi-periodic pulsations (QPP) in solar and stellar flares (e.g., see
Nakariakov & Melnikov, 2009; Mitra-Kraev et al., 2005, and references therein)
with the QPP periodicity time scale varying from fractions of a second to
several minutes, comparable to the period of the oscillations produced in our
simulation. In fact, Nakariakov & Melnikov (2009) have previously resorted to
the water drop formation analogy in describing what they refer to as a class
of “load/unload” models of long multi-minute period QPPs. The plasma droplet
mechanism described in Sec. 3.2.2 above is a much more direct, and novel,
analogy to the same physical process with the potential to provide a new
alternative explanation for the long-duration QPPs.
Finally, we point out that the limitations of the two-dimensional MHD model
used here for modeling a region of potential flaring activity embedded into a
stratified solar atmosphere are many. It is well known that laminar resistive
reconnection cannot account for the observed rates of magnetic energy release,
particle acceleration, or radiation from solar flares, while three-dimensional
effects can substantially alter both the flux-rope stability properties and
the micro-physics of reconnection. Nevertheless, we believe that the careful
and systematic study described in this article is a prerequisite for
performing more complete, and also substantially more challenging and
complicated, studies of CME initiation by flux emergence in the future.
###### Acknowledgements.
E.L. thanks Neil Sheeley for his valuable insights into solving the hyperbolic
function integrals. This work was supported by the NASA SR&T and LWS programs,
as well as ONR 6.1 program. Simulations were performed under grants of
computer time from the US DOD HPC program and the National Energy Research
Scientific Computing Center, which is supported by the US DOE Office of
Science.
## Appendix A
We derive the uniform pressure magnetic flux rope equilibrium with axial field
from a familiar form of the Grad–Shafranov equation:
$\frac{d}{d\psi}\left(\frac{b_{z}^{2}}{2}\right)=-\nabla^{2}\psi=j_{z}$ (23)
In particular, assuming $\psi(r)=\psi_{l}(r)$ given by Eq. (2a):
$\displaystyle\frac{b_{z}^{2}}{2}$ $\displaystyle=\int j_{z}\;d\psi=\int
j_{z}\frac{d\psi}{dr}dr$
$\displaystyle=\int\left[\frac{4r^{2}}{r_{0}^{3}}-\frac{4}{r_{0}}\right]\left[\frac{r}{r_{0}}-\frac{r(r^{2}-r_{0}^{2})}{r_{0}^{3}}\right]dr$
(24)
$\displaystyle=-\frac{2}{3}\left(\frac{r}{r_{0}}\right)^{6}+3\left(\frac{r}{r_{0}}\right)^{4}-4\left(\frac{r}{r_{0}}\right)^{2}+\text{constant,}\,\text{for}\
r\leq r_{0}.$
Requiring that $b_{z}=b_{z0}$ for $r\geq r_{0}$, we can determine the constant
of integration such that $b_{z}$ is continuous:
$b_{z}(r)=\left\\{\begin{array}[]{l @{\hspace{6mm}}
c}\sqrt{b_{z0}^{2}+\dfrac{10}{3}-8\left(\dfrac{r}{r_{0}}\right)^{2}+6\left(\dfrac{r}{r_{0}}\right)^{4}-\dfrac{4}{3}\left(\dfrac{r}{r_{0}}\right)^{6}}\hfil\hskip
17.07164pt&r\leq r_{0}\\\\[8.53581pt] b_{z0}\ .\hfil\hskip
17.07164pt&r>r_{0}\end{array}\right.$ (25)
Suppose, however, we wish to find $b_{z}$ as a function of $\psi$, rather than
of $r$. We then solve for the inverse function $\zeta\equiv\psi_{l}^{-1}(r)$
by replacing $r$ with $\zeta$ in Eq. (2a) and rearranging terms:
$\zeta^{4}-4r_{0}^{2}\zeta^{2}+r_{0}^{4}+4r_{0}^{3}\psi_{l}=0\ .$ (26)
Solving this quadratic equation for $\zeta^{2}$, we find:
$\zeta^{2}=2r_{0}^{2}\pm\sqrt{3r_{0}^{4}-4r_{0}^{3}\psi_{l}}\ .$ (27)
We recover the form of Eq. (10) by rejecting the positive root (to permit
small values of $\zeta$), replacing $\psi_{l}$ by the full functional form of
$\psi=\psi_{l}+\psi_{i}+\psi_{b}$ to approximate a force-free initial
condition with well-aligned contours of constant $\psi$ and $b_{z}$, and
allowing for gauge freedom.
## Appendix B
Here we present a derivation of the pressure profile used in simulations of a
stratified solar atmosphere, given a particular temperature profile (12). To
be physically relevant, we use here dimensional quantities, rather the
normalized code variables.
We begin with the first-order differential equation governing hydrostatic
equilibrium:
$\frac{dp}{dy}+m_{p}ng_{S}=0\ ,$ (28)
which we divide by $p=2nk_{B}T$:
$\frac{d\ln p}{dy}+\frac{m_{p}g_{S}}{2k_{B}T}=0.$ (29)
Then
$\ln\frac{p}{p_{0}}=-\frac{m_{p}g_{S}}{2k_{B}}\int\frac{dy}{T}.$ (30)
We use the profile for temperature $T$ given by (12), but with the following
variable substitution:
$u\equiv\frac{y-y_{\text{\tiny TR}}}{\Delta y}\ ,$ (31)
leading to
$\begin{split}T(u)&=T_{p}+\frac{T_{c}-T_{p}}{2}\left(1+\tanh u\right)\\\
&=T_{p}+\frac{T_{c}-T_{p}}{2}\left(1+\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\right)\\\
&=\frac{T_{p}\,e^{-u}+T_{c}\,e^{u}}{e^{u}+e^{-u}}.\end{split}$ (32)
With algebraic manipulations, we can rewrite (32) as:
$\frac{1}{T}=\frac{1}{T_{c}}+\frac{e^{-2u}(1-T_{p}/T_{c})}{T_{p}\,e^{-2u}+T_{c}}.$
(33)
Then the integral in (30) can be evaluated:
$\begin{split}\int\frac{dy}{T}&=\Delta y\int\frac{du}{T}=\Delta
y\left[\int\frac{du}{T_{c}}+\left(1-\frac{T_{p}}{T_{c}}\right)\int\frac{e^{-2u}\,du}{T_{p}\,e^{-2u}+T_{c}}\right]\\\
&=\Delta
y\left[\frac{u}{T_{c}}-\frac{1}{2}\left(\frac{1}{T_{p}}-\frac{1}{T_{c}}\right)\ln\left(T_{p}e^{-2u}+T_{c}\right)\right].\end{split}$
(34)
Finally, substituting (34) into (30) yields an expression for $p$, in terms of
$u$:
$p(u)=p_{0}\exp\left\\{\frac{m_{p}g_{S}\Delta
y}{2k_{B}T_{c}}\left[\frac{T_{c}-T_{p}}{2T_{p}}\,\ln\left(T_{p}e^{-2u}+T_{c}\right)-u\right]\right\\}.$
(35)
## References
* Baker et al. (2013) Baker, D. N., Li, X., Pulkkinen, A., et al. 2013, Space Weather, 11, 1
* Centeno et al. (2007) Centeno, R., Socas-Navarro, H., Lites, B., et al. 2007, ApJ, 666, L137
* Chae (2001) Chae, J. 2001, ApJ, 560, L95
* Chen (2011) Chen, P. F. 2011, Living Reviews in Solar Physics, 8, 1
* Chen & Shibata (2000) Chen, P. F., & Shibata, K. 2000, ApJ, 545, 524
* Chen et al. (2004) Chen, P. F., Shibata, K., Brooks, D. H., & Isobe, H. 2004, ApJ, 602, L61
* Ciaravella et al. (2002) Ciaravella, A., Raymond, J. C., Li, J., et al. 2002, ApJ, 575, 1116
* Démoulin et al. (2002) Démoulin, P., Mandrini, C. H., Van Driel-Gesztelyi, L., Lopez Fuentes, M. C., & Aulanier, G. 2002, Sol. Phys., 207, 87
* Demoulin & Priest (1988) Demoulin, P., & Priest, E. R. 1988, A&A, 206, 336
* Dubey et al. (2006) Dubey, G., van der Holst, B., & Poedts, S. 2006, A&A, 459, 927
* Evans et al. (2013) Evans, R. M., Pulkkinen, A. A., Zheng, Y., et al. 2013, Space Weather, 11, 333
* Forbes & Isenberg (1991) Forbes, T. G., & Isenberg, P. A. 1991, ApJ, 373, 294
* Forbes & Priest (1995) Forbes, T. G., & Priest, E. R. 1995, ApJ, 446, 377
* Forbes et al. (2006) Forbes, T. G., Linker, J. A., Chen, J., et al. 2006, Space Sci. Rev., 123, 251
* Gopalswamy et al. (2006) Gopalswamy, N., Mikić, Z., Maia, D., et al. 2006, Space Sci. Rev., 123, 303
* Jacobs & Poedts (2011) Jacobs, C., & Poedts, S. 2011, Journal of Atmospheric and Solar-Terrestrial Physics, 73, 1148
* Ko et al. (2003) Ko, Y.-K., Raymond, J. C., Lin, J., et al. 2003, ApJ, 594, 1068
* Krall et al. (1982) Krall, K. R., Smith, Jr., J. B., Hagyard, M. J., West, E. A., & Cummings, N. P. 1982, Sol. Phys., 79, 59
* Kusano et al. (2002) Kusano, K., Maeshiro, T., Yokoyama, T., & Sakurai, T. 2002, ApJ, 577, 501
* Leake & Arber (2006) Leake, J. E., & Arber, T. D. 2006, A&A, 450, 805
* Leake & Linton (2013) Leake, J. E., & Linton, M. G. 2013, ApJ, 764, 54
* Lin & Forbes (2000) Lin, J., & Forbes, T. G. 2000, J. Geophys. Res., 105, 2375
* Lin et al. (2005) Lin, J., Ko, Y.-K., Sui, L., et al. 2005, ApJ, 622, 1251
* Loureiro et al. (2007) Loureiro, N. F., Schekochihin, A. A., & Cowley, S. C. 2007, Phys. Plasmas, 14, 100703
* Low (1981) Low, B. C. 1981, ApJ, 251, 352
* Low (1984) —. 1984, ApJ, 281, 392
* Lukin (2008) Lukin, V. S. 2008, PhD thesis, Princeton University
* Lukin & Linton (2011) Lukin, V. S., & Linton, M. G. 2011, Nonlinear Processes in Geophysics, 18, 871
* Magara & Tsuneta (2008) Magara, T., & Tsuneta, S. 2008, PASJ, 60, 1181
* Malyshkin et al. (2005) Malyshkin, L. M., Linde, T., & Kulsrud, R. M. 2005, Phys. Plasmas, 12, 102902
* McLaughlin et al. (2009) McLaughlin, J. A., De Moortel, I., Hood, A. W., & Brady, C. S. 2009, A&A, 493, 227
* McLaughlin & Hood (2004) McLaughlin, J. A., & Hood, A. W. 2004, A&A, 420, 1129
* McLaughlin & Hood (2005) —. 2005, A&A, 435, 313
* McLaughlin & Hood (2006) —. 2006, A&A, 452, 603
* Mitra-Kraev et al. (2005) Mitra-Kraev, U., Harra, L. K., Williams, D. R., & Kraev, E. 2005, A&A, 436, 1041
* Nakariakov & Melnikov (2009) Nakariakov, V. M., & Melnikov, V. F. 2009, Space Sci. Rev., 149, 119
* Pariat et al. (2006) Pariat, E., Nindos, A., Démoulin, P., & Berger, M. A. 2006, A&A, 452, 623
* Parnell et al. (2009) Parnell, C. E., DeForest, C. E., Hagenaar, H. J., et al. 2009, ApJ, 698, 75
* Priest & Forbes (2000) Priest, E., & Forbes, T. 2000, Magnetic Reconnection: MHD Theory and Applications (Cambridge University Press)
* Sheeley (1969) Sheeley, Jr., N. R. 1969, Sol. Phys., 9, 347
* Shiota et al. (2004) Shiota, D., Isobe, H., Brooks, D. H., Shibata, K., & Chen, P. F. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 325, The Solar-B Mission and the Forefront of Solar Physics, ed. T. Sakurai & T. Sekii, 373
* Shiota et al. (2005) Shiota, D., Isobe, H., Chen, P. F., et al. 2005, ApJ, 634, 663
* Shiota et al. (2003) Shiota, D., Yamamoto, T. T., Sakajiri, T., et al. 2003, PASJ, 55, L35
* Tian & Alexander (2006) Tian, L., & Alexander, D. 2006, Sol. Phys., 233, 29
* van Tend & Kuperus (1978) van Tend, W., & Kuperus, M. 1978, Sol. Phys., 59, 115
* Zwaan (1985) Zwaan, C. 1985, Sol. Phys., 100, 397
|
arxiv-papers
| 2014-03-02T16:26:28 |
2024-09-04T02:49:59.181232
|
{
"license": "Public Domain",
"authors": "E. Lee, V.S. Lukin, and M.G. Linton",
"submitter": "Vyacheslav Lukin",
"url": "https://arxiv.org/abs/1403.0231"
}
|
1403.0299
|
# On the functional Blaschke-Santaló inequality
Youjiang Lin School of Mathematical Sciences, Peking University, Beijing,
100871, China; Department of Mathematics, Department of Mathematics, Shanghai
University, Shanghai, 200444, China [email protected] and Gangsong Leng
Department of Mathematics, Shanghai University, Shanghai, 200444, China
[email protected]
###### Abstract.
In this paper, using functional Steiner symmetrizations, we show that Meyer
and Pajor’s proof of the Blaschke-Santaló inequality can be extended to the
functional setting.
###### Key words and phrases:
Convex body; Polar body; Parallel sections homothety bodies; Mahler
conjecture; Cylinder
2010 Mathematics Subject Classification. 52A10, 52A40.
The authors would like to acknowledge the support from China Postdoctoral
Science Foundation Grant 2013M540806, National Natural Science Foundation of
China under grant 11271244 and National Natural Science Foundation of China
under grant 11271282 and the 973 Program 2013CB834201.
## 1\. Introduction
For a convex body $K\subset\mathbb{R}^{n}$ and a point $z\in\mathbb{R}^{n}$,
the polar body $K^{z}$ of $K$ with respect to $z$ is the convex set defined by
$K^{z}=\\{y\in\mathbb{R}^{n}:\langle y-z,x-z\rangle\leq 1\;{\rm for}\;{\rm
every}\;x\in K\\}$. The Santaló point $s(K)$ of $K$ is a point for which
$V_{n}(K^{s(K)})=\min_{z\in int(K)}V_{n}(K^{z})$, where $V_{n}(K)$ denotes the
volume of set $K$. The Blaschke-Santaló inequality [4, 18, 19] states that
$V_{n}(K)V_{n}(K^{s(K)})\leq V_{n}(B_{2}^{n})^{2}$, where $B_{2}^{n}$ is the
Euclidean ball.
For a log-concave function $f:\mathbb{R}^{n}\rightarrow[0,\infty)$ and a point
$z\in\mathbb{R}^{n}$, its polar with respect to $z$ is defined by
$f^{z}(y)=\inf_{x\in\mathbb{R}^{n}}\frac{e^{-\langle x-z,y-z\rangle}}{f(x)}$.
The Santaló point $s(f)$ of $f$ is the point $z_{0}$ satisfying $\int
f^{z_{0}}=\inf_{z\in\mathbb{R}^{n}}\int f^{z}$.
The functional Blaschke-Santaló inequality of log-concave functions is the
analogue of Blaschke-Santaló inequality of convex bodies.
###### Theorem 1.1.
(Artstein, Klartag, Milman). Let $f:\mathbb{R}^{n}\rightarrow[0,+\infty)$ be a
log-concave function such that $0<\int f<\infty$. Then,
$\int_{\mathbb{R}^{n}}f\int_{\mathbb{R}^{n}}f^{s(f)}\leq(2\pi)^{n}$ with
equality holds exactly for Gaussians.
When $f$ is even, the functional Blaschke-Santaló inequality follows from an
earlier inequality of Ball [2]; and in [9], Fradelizi and Meyer proved
something more general (see also [11]). Lutwak and Zhang [13] and Lutwak et
al. [14] gave other very different forms of the Blaschke-Santaló inequality.
In this paper, we give a more general result than Theorem 1.1, which becomes
into a special case of $\lambda=1/2$ in Theorem 1.2.
###### Theorem 1.2.
Let $f:\mathbb{R}^{n}\rightarrow[0,+\infty)$ be a log-concave function such
that $0<\int f<\infty$. Let $H$ be an affine hyperplane and let $H_{+}$ and
$H_{-}$ denote two closed half-spaces bounded by $H$. If $\lambda\in(0,1)$
satisfies $\lambda\int_{\mathbb{R}^{n}}f=\int_{H_{+}}f$. Then there exists
$z\in H$ such that
$\displaystyle\int_{\mathbb{R}^{n}}f\int_{\mathbb{R}^{n}}f^{z}\leq\frac{1}{4\lambda(1-\lambda)}(2\pi)^{n}.$
(1.1)
In [12], Lehec proved a very general functional version for non-negative Borel
functions, Theorem 1.2 is a particular case of result of Lehec. Lehec’s proof
is by induction on the dimension, and the proof is by functional Steiner
symmetrizations. In fact, Mayer and Pajor [15] have proved the Blaschke-
Santaló inequality for convex bodies, here we show that Meyer and Pajor’s
proof of the Blaschke-Santaló inequality can be extended to the functional
setting. It has recently come to our attention that in a remark of [9],
Fradelizi and Meyer expressed the same idea to prove the functional Blaschke-
Santaló inequality.
## 2\. Notations and background materials
Let $|\cdot|$ denote the Euclidean norm. Let ${\rm int}A$ denote the interior
of $A\subset\mathbb{R}^{n}$. Let ${\rm cl}A$ denote the closure of $A$. Let
${\rm dim}A$ denote the dimension of $A$. A set $C\subset\mathbb{R}^{n}$ is
called a convex cone if $C$ is convex and nonempty and if $x\in C$,
$\lambda\geq 0$ implies $\lambda x\in C$. We define
$C^{\ast}:=\\{x\in\mathbb{R}^{n}:\langle x,y\rangle\leq 0\;\;{\rm for}\;{\rm
all}\;y\in C\\}$ and call this the dual cone of $C$.
For a non-empty convex set $K\subset\mathbb{R}^{n}$ and an affine hyperplane
$H$ with unit normal vector $u$, the Steiner symmetrization $S_{H}K$ of $K$
with respect to $H$ is defined as
$S_{H}K:=\\{x^{\prime}+\frac{1}{2}(t_{1}-t_{2})u:\;x^{\prime}\in
P_{H}(K),\;t_{i}\in I_{K}(x^{\prime})\;{\rm for}\;i=1,2\\}$, where
$P_{H}(K):=\\{x^{\prime}\in H:\;x^{\prime}+tu\in K\;{\rm for}\;{\rm
some}\;t\in\mathbb{R}\\}$ is the projection of $K$ onto $H$ and
$I_{K}(x^{\prime}):=\\{t\in\mathbb{R}:\;x^{\prime}+tu\in K\\}$.
Let $\bar{\mathbb{R}}=\mathbb{R}\cup\\{-\infty,\infty\\}$. For a given
function $f:\mathbb{R}^{n}\rightarrow\bar{\mathbb{R}}$ and for
$\alpha\in\bar{\mathbb{R}}$ we use the abbreviation
$\\{f=\alpha\\}:=\\{x\in\mathbb{R}^{n}:f(x)=\alpha\\}$, and
$\\{f\leq\alpha\\}$, $\\{f<\alpha\\}$ etc. are defined similarly. A function
$f:\mathbb{R}^{n}\rightarrow\bar{\mathbb{R}}$ is called proper if
$\\{f=-\infty\\}=\emptyset$ and $\\{f=\infty\\}\neq\mathbb{R}^{n}$. A function
$\phi$ is called convex if $\phi$ is proper and $\phi(\alpha
x+(1-\alpha)y)\leq\alpha\phi(x)+(1-\alpha)\phi(y)$ for all
$x,y\in\mathbb{R}^{n}$ and for any $0\leq\lambda\leq 1$. A function $f$ is
called log-concave if $f=e^{-\phi}$, where $\phi$ is a convex function. A
function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is called
coercive if $\lim_{|x|\rightarrow+\infty}f(x)=+\infty$. A function $f$ is
called symmetric about $H$ if for any $x^{\prime}\in H$ and $t\in\mathbb{R}$,
$f(x^{\prime}+tu)=f(x^{\prime}-tu)$. A function
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is called unconditional about $z$ if
$f(x_{1}-z_{1},\dots,x_{n}-z_{n})=f(|x_{1}-z_{1}|,\dots,|x_{n}-z_{n}|)$ for
every $(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$. If $z=0$, then $f$ is called
unconditional.
The effective domain of convex function $\phi$ is the nonempty set ${\rm
dom}\phi:=\\{\phi<\infty\\}$. The support of function $f$ is the set ${\rm
supp}f:=\\{f\neq 0\\}$. For log-concave function $f=e^{-\phi}$, it is clear
that ${\rm supp}f={\rm dom}\phi$. The nonempty set ${\rm
epi}\phi:=\\{(x,r)\in\mathbb{R}^{n}\times\mathbb{R}:\;r\geq\phi(x)\\}$ denote
the epigraph of convex function $\phi$.
For an affine subspace $G$ of $\mathbb{R}^{n}$, let $G^{\perp}$ denote the
orthogonal complement of $G$, we have $G^{\bot}=\\{x\in\mathbb{R}^{n}:\langle
x,y-y^{\prime}\rangle=0\;{\rm for\;every\;}y,y^{\prime}\in G\\}$. The Santaló
point $s_{G}(f)$ of $f$ about $G$ is a point satisfying $\int
f^{s_{G}(f)}=\inf_{z\in G}\int f^{z}$. Let $f$ be a log-concave function such
that $0<\int f<\infty$, and let $H_{+}$ and $H_{-}$ be two half-spaces bounded
by an affine hyperplane $H$; let $0<\lambda<1$; we shall say that $H$ is
$\lambda$-separating for $f$ if
$\int_{H_{+}}f\int_{H_{-}}f=\lambda(1-\lambda)\left(\int_{\mathbb{R}^{n}}f\right)^{2}$
and when $\lambda=1/2$, we shall say that $H$ is medial for $f$. For a
function $\phi:\mathbb{R}^{n}\rightarrow\bar{\mathbb{R}}$, its Legendre
transform about $z$ is defined by
$\mathcal{L}^{z}\phi(y)=\sup_{x\in\mathbb{R}^{n}}[\langle
x-z,y-z\rangle-\phi(x)]$. If $f(x)=e^{-\phi(x)}$, where $\phi(x)$ is a convex
function, then $f^{z}(y)=e^{-\mathcal{L}^{z}\phi(y)}$. Since
$\mathcal{L}^{z}(\mathcal{L}^{z}\phi)=\phi$ for a convex function $\phi$,
$(f^{z})^{z}=f$. If $z=0$, we shall use the simpler notation $\mathcal{L}$ for
$\mathcal{L}^{0}$.
Given two functions $f,g:\mathbb{R}^{n}\rightarrow[0,\infty)$, their Asplund
product is defined by $(f\star g)(x)=\sup_{x_{1}+x_{2}=x}f(x_{1})g(x_{2})$.
The $\lambda$-homothety of a function $f$ is defined as $(\lambda\cdot
f)(x)=f^{\lambda}(\frac{x}{\lambda})$. Then, the classical Prékopa inequality
(see Prékopa [16, 17]) can be stated as follows: Given
$f,g:\mathbb{R}^{n}\rightarrow[0,+\infty)$ and $0<\lambda<1$,
$\int(\lambda\cdot f)\star((1-\lambda)\cdot g)\geq\left(\int
f\right)^{\lambda}\left(\int g\right)^{1-\lambda}$. The following lemma, as a
particular case of a result due to Ball [3], was proved by Meyer and Pajor in
[15].
###### Lemma 2.1.
[15] Let $f_{0}$, $f_{1}$, $f_{2}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be
three functions such that $0<\int^{+\infty}_{0}f_{i}<\infty,\;i=0,1,2$, they
are continuous and suppose that $f_{0}\left(\frac{2xy}{x+y}\right)\geq
f_{1}(x)^{\frac{y}{x+y}}f_{2}(y)^{\frac{x}{x+y}}$ for every $x,y>0$. Then one
has
$\frac{1}{\int^{+\infty}_{0}f_{0}(t)dt}\leq\frac{1}{2}\left(\frac{1}{\int^{+\infty}_{0}f_{1}(t)dt}+\frac{1}{\int^{+\infty}_{0}f_{2}(t)dt}\right).$
## 3\. The functional Steiner symmetrization
The familiar definition of Steiner symmetrization for a nonnegative measurable
function $f$ can be stated as following (see [5, 6, 7, 8]):
###### Definition 1.
For a measurable function $f:\mathbb{R}^{n}\rightarrow[0,+\infty)$ and an
affine hyperplane $H\subset\mathbb{R}^{n}$, let $m$ denote the Lebesgue
measure, if $m(\\{f>t\\})<+\infty$ for all $t>0$, then its Steiner
symmetrization is defined as
$\displaystyle S_{H}f(x)=\int_{0}^{\infty}\mathcal{X}_{S_{H}\\{f>t\\}}(x)dt,$
(3.1)
where $\mathcal{X}_{A}$ denotes the characteristic function of set $A$.
Next, we give a approach of defining Steiner symmetrization for coercive
convex functions by the Steiner symmetrization of epigraphs. A similar
functional steiner symmetrization is defined in a remark of AKM’s paper [1]
and studied in an article by Lehec [10]. The idea of our definition is same as
the given definition in a remark at the end of an article by Fradelizi and
Meyer [9].
###### Definition 2.
For a coercive convex function $\phi$ and an affine hyperplane
$H\subset\mathbb{R}^{n}$, we define the Steiner symmetrization $S_{H}\phi$ of
$\phi$ with respect to $H$ as a function satisfying
$\displaystyle{\rm epi}(S_{H}\phi)=S_{\widetilde{H}}({\rm cl}\;{\rm
epi}\phi),$ (3.2)
where $\widetilde{H}=\\{(x^{\prime},s)\in\mathbb{R}^{n+1}:x^{\prime}\in H\\}$
is an affine hyperplane in $\mathbb{R}^{n+1}$.
###### Remark 1.
(i) By Definition 2, for an integrable log-concave function $f=e^{-\phi}$, the
Steiner symmetrization of $f$ can be defined as $S_{H}f:=e^{-(S_{H}\phi)}$. If
we define $S_{H}f$ by Definition 1, then $S_{H}f$ still satisfies (3.2). Thus,
for integrable log-concave functions, the two definitions are essentially
same.
(ii) By Definition 2, for a given $x^{\prime}\in H$ and any $s\in\mathbb{R}$,
we have
$V_{1}\left(\\{(S_{H}\phi)(x^{\prime}+tu)<s\\}\right)=V_{1}\left(\\{\phi(x^{\prime}+tu)<s\\}\right)$.
By the Fubini’s theorem, we have
$\displaystyle\int_{\mathbb{R}}(S_{H}f)(x^{\prime}+tu)dt=\int_{\mathbb{R}}f(x^{\prime}+tu)dt.$
(3.3)
Similarly, $\int_{\mathbb{R}^{n}}S_{H}f=\int_{\mathbb{R}^{n}}f$ is also
established.
###### Proposition 1.
For a coercive convex function $\phi$ and an affine hyperplane
$H\subset\mathbb{R}^{n}$ with outer unit normal vector $u$, then $S_{H}\phi$
has the following properties.
(i) $S_{H}\phi$ is a closed coercive convex function and symmetric about $H$.
(ii) Let $H_{1}$ and $H_{2}$ be two orthogonal hyperplanes in
$\mathbb{R}^{n}$, then $S_{H_{2}}(S_{H_{1}}\phi)$ is symmetric about both
$H_{1}$ and $H_{2}$.
(iii) For any given $x^{\prime}\in H$ and $t\in\mathbb{R}$, let
$\phi_{1}(t):=\phi(x^{\prime}+tu)$ and
$(S\phi_{1})(t):=(S_{H}\phi)(x^{\prime}+tu)$, then $(S\phi_{1})(t)$ satisfies
one of the following three cases. 1).
$(S\phi_{1})(t)=\phi_{1}(t_{1})=\phi_{1}(t_{1}-2t)$ for some
$t_{1}\in\mathbb{R}$. 2).
$(S\phi_{1})(t)=\phi_{1}(t_{0}-2t)\geq\lim_{t\rightarrow
t_{0},\;t<t_{0}}\phi_{1}(t)$ for some $t_{0}\in\mathbb{R}$. 3).
$(S\phi_{1})(t)=\phi_{1}(t_{0}+2t)\geq\lim_{t\rightarrow
t_{0},\;t>t_{0}}\phi_{1}(t)$ for some $t_{0}\in\mathbb{R}$.
###### Proof.
(i) By the fact that $\phi$ is convex if and only if ${\rm epi}\phi$ is
convex, since $\phi$ is convex, ${\rm epi}\phi$ is a convex subset of
$\mathbb{R}^{n+1}$. Since the closure of a convex set is convex, and the
Steiner symmetrization of a convex set is also convex, by (3.2), ${\rm
epi}(S_{H}\phi)$ is a convex subset of $\mathbb{R}^{n+1}$. Therefore,
$S_{H}\phi$ is a convex function. By Definition 2, it is clear that
$S_{H}\phi$ is closed, coercive and symmetric with respect to $H$.
(ii) Since ${\rm epi}(S_{H_{2}}(S_{H_{1}}\phi))$ is symmetric about both
$\widetilde{H_{1}}$ and $\widetilde{H_{2}}$, where
$\widetilde{H_{i}}=\\{(x^{\prime},s)\in\mathbb{R}^{n+1}:x^{\prime}\in
H_{i}\\}$ ($i=1,2$), $S_{H_{2}}(S_{H_{1}}\phi)$ is symmetric about both
$H_{1}$ and $H_{2}$.
(iii) If ${\rm dom}\phi_{1}=\mathbb{R}$, by (3.2) in Definition 2, we have
$\displaystyle{\rm epi}(S\phi_{1})=S_{\widetilde{H}}({\rm cl}\;{\rm
epi}\phi_{1}).$ (3.4)
Thus there exists some $t_{1}\in\mathbb{R}$ satisfying
$\displaystyle(S\phi_{1})(t)=\phi_{1}(t_{1})=\phi_{1}(t_{1}-2t).$ (3.5)
If ${\rm dom}\phi_{1}\neq\mathbb{R}$, then there exist eight cases for ${\rm
dom}\phi_{1}$: 1) $[\alpha,\beta]$; 2) $(\alpha,\beta)$; 3) $(\alpha,\beta]$;
4) $[\alpha,\beta)$; 5) $(-\infty,\beta]$; 6) $(-\infty,\beta)$; 7)
$[\alpha,+\infty)$; 8) $(\alpha,+\infty)$. Here, we only prove our conclusion
for ${\rm dom}\phi_{1}=(\alpha,\beta)$. By the same method we can prove our
conclusion for other cases. For ${\rm dom}\phi_{1}=(\alpha,\beta)$, by
Definition 2, it is clear that $(S\phi_{1})(t)=+\infty$ for
$|t|\geq\frac{\beta-\alpha}{2}$. If $|t|<\frac{\beta-\alpha}{2}$, let
$\lim_{x\rightarrow\alpha,\;x>\alpha}\phi_{1}(x)=b_{1},\;\;\lim_{x\rightarrow\beta,\;x<\beta}\phi_{1}(x)=b_{2}$,
then we consider the following four cases. (a) If $b_{1}=b_{2}=+\infty$, then
by (3.4), there exists some $t_{1}\in\mathbb{R}$ satisfying (3.5). (b) If
$b_{1}<+\infty,\;\;b_{2}=+\infty$, then there exists $\gamma\in(\alpha,\beta)$
such that $\phi_{1}(\gamma)=b_{1}$. Then by (3.4), for
$|t|<\frac{\gamma-\alpha}{2}$, (3.5) is established, for
$|t|\geq\frac{\gamma-\alpha}{2}$, we have
$(S\phi_{1})(t)=\phi_{1}(\alpha+2t)\geq b_{1}$. (c) If
$b_{1}=+\infty,\;\;b_{2}<+\infty$, then there exists $\gamma\in(\alpha,\beta)$
such that $\phi_{1}(\gamma)=b_{2}$. Then by (3.4), for
$|t|<\frac{\beta-\gamma}{2}$, (3.5) is established, for
$|t|\geq\frac{\gamma-\alpha}{2}$, we have
$(S\phi_{1})(t)=\phi_{1}(\beta-2t)\geq b_{2}$. (d) If
$b_{1}<\infty,\;\;b_{2}<+\infty$, we consider three cases. If $b_{1}=b_{2}$,
then (3.5) is established. If $b_{1}>b_{2}$, the proof is same as in (c). If
$b_{1}<b_{2}$, the proof is same as in (b). This completes the proof. ∎
## 4\. The proofs of theorems
In order to prove theorems stated in the introduction, we have to establish
the following six lemmas:
###### Lemma 4.1.
If $f$ be a log-concave function such that $0<\int f<\infty$, then the
function $F$ defined by $F(z):=\int_{\mathbb{R}^{n}}f^{z}(x)dx$ has the
following properties. (i) $F(z)$ is a coercive convex function on
$\mathbb{R}^{n}$ and is strictly convex on ${\rm int}\;{\rm dom}F$; (ii) If
$f(x)$ is even about $z_{0}$, then $F(z)$ is also even about $z_{0}$.
###### Proof.
(i) Step 1. We shall prove $F$ is coercive. Let $f=e^{-\phi}$, for any given
$z\in\mathbb{R}^{n}$ and $r>0$, we have
$\displaystyle
F(z)=\int_{\mathbb{R}^{n}}f^{z}(x+z)dx\geq\int_{rB_{2}^{n}}f^{z}(x+z)dx=\int_{rB_{2}^{n}}e^{-\mathcal{L}\phi(x)+\langle
x,z\rangle}dx.$ (4.1)
Since $f=e^{-\phi}$ is integrable, there is $\gamma>0$ and $h\in\mathbb{R}$
such that
$\displaystyle\phi(x)\geq\gamma\sum_{i=1}^{n}|x_{i}|+h\;\;{\rm for}\;{\rm
any}\;x\in\mathbb{R}^{n}.$ (4.2)
Thus, for $y\in\gamma B_{\infty}^{n}$, where
$B_{\infty}^{n}=\\{x\in\mathbb{R}^{n}:|x_{i}|\leq 1,i=1,\dots,n\\}$,
$\mathcal{L}\phi(y)\leq\sup_{x\in\mathbb{R}^{n}}[\langle
y,x\rangle-\gamma\sum_{i=1}^{n}|x_{i}|-h]\leq-h$. Let
$rB_{2}^{n}\subset\frac{1}{2}\gamma B_{\infty}^{n}$, we have
$rB_{2}^{n}\subset{\rm int}({\rm dom}\mathcal{L}\phi)$. Since function
$g(x):\;=e^{-\mathcal{L}\phi(x)}$ is continuous on $rB_{2}^{n}$. Thus, there
exists $m>0$ such that $g(x)\geq m$ for any $x\in rB_{2}^{n}$. Therefore,
$\displaystyle\int_{rB_{2}^{n}}e^{-\mathcal{L}\phi(x)+\langle
x,z\rangle}dx\geq m\int_{rB_{2}^{n}}e^{\langle x,z\rangle}dx.$ (4.3)
For any $z\in\mathbb{R}^{n}$ and $|z|\geq 1$, let
$z^{\prime}=\frac{r}{2}\frac{z}{|z|}$, we get a closed half-space
$H^{+}=\\{x\in\mathbb{R}^{n}:\langle x-z^{\prime},z\rangle\geq 0\\}$. For any
$x\in H^{+}$, we have $\langle x,z\rangle\geq\langle
z^{\prime},z\rangle=\frac{r}{2}|z|$. Therefore,
$\displaystyle\int_{rB_{2}^{n}}e^{\langle x,z\rangle}dx$ $\displaystyle\geq$
$\displaystyle\int_{(rB_{2}^{n})\cap
H^{+}}e^{\frac{r|z|}{2}}dx=V_{n}((rB_{2}^{n})\cap H^{+})e^{\frac{r|z|}{2}}.$
(4.4)
Since $V_{n}((rB_{2}^{n})\cap H^{+})$ is a positive constant independent of
$z$, by (4.1), (4.3) and (4.4), $F(z)$ is coercive.
Step 2. We shall prove that $F$ is convex and is strictly convex on ${\rm
int}\;{\rm dom}F$. First, we prove $F(z)$ is proper. It is clear that
$F(z)>-\infty$ for any $z\in\mathbb{R}^{n}$. The following claim shows that
$\\{F=\infty\\}\neq\mathbb{R}^{n}$.
###### Claim 1.
For any $z\in{\rm int}\;{\rm supp}f$, $F(z)<\infty$.
Proof of Claim 1. For any $z\in{\rm int}\;{\rm supp}f$, there is a closed ball
$z+rB_{2}^{n}\subset{\rm supp}f$. Since ${\rm supp}f={\rm dom}\phi$, there is
$M\in\mathbb{R}$ such that $M=\sup\\{\phi(y):y\in z+rB_{2}^{n}\\}$. Thus, we
have
$f^{z}(x)\leq\exp\\{-\sup_{y\in(z+rB_{2}^{n})}[\langle
x-z,y-z\rangle-\phi(y)]\\}\leq e^{M}\cdot e^{-r|x-z|^{2}}.$
Therefore, $\int_{\mathbb{R}^{n}}f^{z}(x)dx\leq
e^{M}\int_{\mathbb{R}^{n}}e^{-r|x-z|^{2}}dx<\infty.$ $\Box$
For any $z_{1},z_{2}\in\mathbb{R}^{n}$ and $\alpha\in(0,1)$. Let
$f=e^{-\phi}$, we have
$F(z)=\int_{\mathbb{R}^{n}}e^{-\mathcal{L}\phi(x)+\langle x,z\rangle}dx$.
Since $g_{x}(z):=e^{-\mathcal{L}\phi(x)+\langle x,z\rangle}$ is a convex
function about $z$, we have
$\displaystyle F(\alpha z_{1}+(1-\alpha)z_{2})\leq\alpha
F(z_{1})+(1-\alpha)F(z_{2}).$ (4.5)
If $z_{1},z_{2}\in{\rm int}\;{\rm dom}F$ and $z_{1}\neq z_{2}$, then
inequality (4.5) is a strict inequality. Thus $F(z)$ is strictly convex on
${\rm int}\;{\rm dom}F$.
(ii) Since $f(x)$ is even about $z_{0}$, $f(z_{0}+x)=f(z_{0}-x)$ for any
$x\in\mathbb{R}^{n}$. For any $z\in\mathbb{R}^{n}$, we have
$F(z_{0}+z)=\int_{\mathbb{R}^{n}}f^{z_{0}+z}(x)dx=\int_{\mathbb{R}^{n}}f^{z_{0}-z}(-x+2z_{0})dx=F(z_{0}-z).$
This completes the proof. ∎
###### Remark 2.
By Lemma 4.1, if $f$ is even about $z_{0}$, then $s(f)=z_{0}$.
###### Lemma 4.2.
Let $f$ be a log-concave function such that $0<\int f<\infty$, and let
$G\subset\mathbb{R}^{n}$ be an affine subspace satisfying $G\cap{\rm
int}\;{\rm supp}f\neq\emptyset$. Then there exists a unique point $z_{0}\in G$
satisfying the following two equivalent claims. (i) $F(z_{0})=\min\\{F(z);z\in
G\\}$, where $F(z):=\int_{\mathbb{R}^{n}}f^{z}(x)dx$. (ii) ${\rm
grad}F(z_{0})=\int_{\mathbb{R}^{n}}xf^{z_{0}}(x+z_{0})dx\in G^{\bot}$.
###### Proof.
By Lemma 4.1, $F$ is coercive and strictly convex on ${\rm int}\;{\rm dom}F$,
thus there is a unique minimal point $z_{0}=s_{G}(f)$ on $G$. Let
$f=e^{-\phi}$, then $F(z)=\int_{\mathbb{R}^{n}}e^{-\mathcal{L}\phi(x)+\langle
x,z\rangle}dx$. By the dominated convergence theorem, we have ${\rm
grad}F(z)=\int_{\mathbb{R}^{n}}xe^{-\mathcal{L}\phi(x)+\langle
x,z\rangle}dx=\int_{\mathbb{R}^{n}}xf^{z}(x+z)dx$.
Next, we prove the equivalence of (i) and (ii). Let
$\eta_{1},\dots,\eta_{m}\;(m<n)$ be an orthonormal basis of $G$ and let
$\eta_{m+1},\dots,\eta_{n}$ be an orthonormal basis of $G^{\perp}$. Let
$z=\sum_{i=1}^{n}z_{i}\eta_{i}$, since $z_{0}=s_{G}(f)\in G$, we have
$\left.\frac{\partial F(z)}{\partial
z_{i}}\right|_{z=z_{0}}=\lim_{t\rightarrow
0}\frac{F(z_{0}+t\eta_{i})-F(z_{0})}{t}=0,\;\;i=1,\dots,m$. Hence, ${\rm
grad}F(z_{0})\in G^{\bot}$. On the other hand, if ${\rm grad}F(z_{0})\in
G^{\bot}$, then $\left.\frac{\partial F(z)}{\partial
z_{i}}\right|_{z=z_{0}}=0,\;i=1,\dots,m$. Since $F(z)$ is strictly convex on
$G\cap{\rm int}\;{\rm dom}F$, $z_{0}$ is the unique minimal point on $G$. ∎
###### Remark 3.
In Lemma 4.2, if $G=\mathbb{R}^{n}$, then the lemma shows that the Santaló
point $s(f)$ of $f$ is the barycenter of the function $f^{s(f)}$.
###### Lemma 4.3.
Let $f$ be a log-concave function such $0<\int f<\infty$. Let
$G\subset\mathbb{R}^{n}$ be an affine subspace satisfying $G\cap{\rm
int}\;{\rm supp}f\neq\emptyset$ and $z=s_{G}(f)$. Let $H$ be an affine
hyperplane such that $G\subset H$ and let $g$ be the function defined by
$g^{z}=S_{H}(f^{z})$. Then we have $s_{G}(g)=z=s_{G}(f)$.
###### Proof.
It may be supposed that $z=s_{G}(f)=0$,
$H=\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}:x_{n}=0\\}$ and
$G=\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}:x_{m+1}=\cdots=x_{n}=0\\}$ for
some $m$, $1\leq m\leq n-1$. By Lemma 4.2, we have
$\int_{\mathbb{R}^{n}}xf^{0}(x)dx\in G^{\bot}$. Let
$f^{0}_{x^{\prime}}(t):=f^{0}(x^{\prime}+tu)$ for any $x^{\prime}\in H$, where
$u$ is the unit normal vector of $H$. Thus,
$\int_{H}x_{i}\left(\int_{\mathbb{R}}f^{0}_{x^{\prime}}(t)dt\right)dx^{\prime}=0\;\;\textrm{for}\;\;1\leq
i\leq m$. By $g^{0}=S_{H}(f^{0})$ and (3.3), for every $x^{\prime}\in H$,
$\int_{\mathbb{R}}f^{0}_{x^{\prime}}(t)=\int_{\mathbb{R}}g^{0}_{x^{\prime}}(t)$.
Thus,
$\int_{H}x_{i}\left(\int_{\mathbb{R}}g^{0}_{x^{\prime}}(t)dt\right)dx^{\prime}=0\;\;\textrm{for}\;\;1\leq
i\leq m$, which conversely gives $\int_{\mathbb{R}^{n}}xg^{0}(x)dx\in
G^{\bot}$. Thus, by Lemma 4.2 again, we obtain $s_{G}(g)=0=s_{G}(f)$. ∎
###### Lemma 4.4.
For a log-concave function $f$ such that $0<\int f<\infty$, if $f$ is
symmetric about some affine hyperplane $H$, then, for any $z\in H$, $f^{z}$ is
also symmetric about $H$.
###### Proof.
Let $u$ be the unit normal vector of $H$. For any $x^{\prime},y^{\prime}\in H$
and $s,t\in\mathbb{R}$, since $f(x^{\prime}+su)=f(x^{\prime}-su)$, we have
$\displaystyle f^{z}(y^{\prime}+tu)$ $\displaystyle=$
$\displaystyle\inf_{x^{\prime}+su\in\mathbb{R}^{n}}\frac{\exp\\{-\langle
y^{\prime}+tu-z,x^{\prime}+su-z\rangle\\}}{f(x^{\prime}+su)}$ $\displaystyle=$
$\displaystyle\inf_{x^{\prime}+su\in\mathbb{R}^{n}}\frac{\exp\\{-\langle
y^{\prime}-z-tu,x^{\prime}-z-su\rangle\\}}{f(x^{\prime}-su)}=f^{z}(y^{\prime}-tu).$
This completes the proof. ∎
###### Lemma 4.5.
Let $f$ be a log-concave function such that $0<\int f<\infty$ and let $H$ be
an affine hyperplane satisfying $H\cap{\rm int}\;{\rm supp}f\neq\emptyset$ and
$z\in H\cap{\rm int}\;{\rm supp}f$; let $\lambda$, $0<\lambda<1$ such that $H$
is $\lambda$-separating for $f^{z}$. Then
$\int_{\mathbb{R}^{n}}(S_{H}f)^{z}\geq
4\lambda(1-\lambda)\int_{\mathbb{R}^{n}}f^{z}.$
###### Proof.
It may be supposed that $z=0$ and $H=\\{(x_{1},\dots,x_{n}):x_{n}=0\\}$. For
$y^{\prime}\in H$ and $s\in\mathbb{R}$, let $(y^{\prime},s)$ denote
$y^{\prime}+su$, where $u$ is a unit normal vector of $H$. For $f^{0}$ and
$s\in\mathbb{R}$, we define a new function
$f^{0}_{(s)}(y^{\prime}):=f^{0}(y^{\prime},s),\;{\rm for\;any}\;y^{\prime}\in
H.$
Next we shall prove that for any $y^{\prime}\in H$ and $s,t>0$
$\displaystyle\left(\frac{t}{s+t}\cdot
f_{(s)}^{0}\right)\star\left(\frac{s}{s+t}\cdot
f_{(-t)}^{0}\right)(y^{\prime})\leq(S_{H}f)^{0}_{(\frac{2st}{s+t})}(y^{\prime}).$
(4.6)
###### Claim 2.
For any $x^{\prime}\in H$ and $w\in\mathbb{R}$, if
$(S_{H}f)(x^{\prime}+wu)>0$, then there is some $w_{1}\in\mathbb{R}$ such that
$(S_{H}f)(x^{\prime}+wu)\leq f(x^{\prime}+w_{1}u)$ and
$(S_{H}f)(x^{\prime}+wu)\leq f(x^{\prime}+(w_{1}-2w)u)$.
Proof of Claim 2. Let $f=e^{-\phi}$, since $(S_{H}f)(x^{\prime}+wu)>0$, then
$(S_{H}\phi)(x^{\prime}+wu)<+\infty$. By Proposition 1(iii), there is
$w_{1}\in\mathbb{R}$ such that
$(S_{H}\phi)(x^{\prime}+wu)\geq\phi(x^{\prime}+w_{1}u)$ and
$(S_{H}\phi)(x^{\prime}+wu)\geq\phi(x^{\prime}+(w_{1}-2w)u)$, here we assume
$\phi(x^{\prime}+w_{1}u)$ or $\phi(x^{\prime}+(w_{1}-2w)u)$ equals the limit
in Proposition 1(iii), which doesn’t affect our proof. Hence the claim
follows. $\Box$
For any $y_{1}^{\prime}$, $y_{2}^{\prime}\in H$ such that
$y^{\prime}=y_{1}^{\prime}+y_{2}^{\prime}$, we have
$\displaystyle(S_{H}f)_{(\frac{2st}{s+t})}^{0}(y^{\prime})$ $\displaystyle=$
$\displaystyle\inf_{(x^{\prime},w)\in
H\times\mathbb{R}}\frac{\exp\\{-\langle(y^{\prime},\frac{2st}{s+t}),(x^{\prime},w)\rangle\\}}{(S_{H}f)(x^{\prime},w)}$
$\displaystyle\geq$ $\displaystyle\inf_{(x^{\prime},w)\in
H\times\mathbb{R}}\frac{\exp\\{-\langle(y^{\prime},\frac{2st}{s+t}),(x^{\prime},w)\rangle\\}}{f(x^{\prime},w_{1})^{\frac{t}{s+t}}f(x^{\prime},w_{1}-2w)^{\frac{s}{s+t}}}$
$\displaystyle\geq$ $\displaystyle\inf_{(x^{\prime},w)\in
H\times\mathbb{R}}\frac{\exp\\{-\frac{t}{s+t}\langle(\frac{s+t}{t}y_{1}^{\prime},s),(x^{\prime},w_{1})\rangle\\}}{f(x^{\prime},w_{1})^{\frac{t}{s+t}}}$
$\displaystyle\times\inf_{(x^{\prime},w)\in
H\times\mathbb{R}}\frac{\exp\\{-\frac{s}{s+t}\langle(\frac{s+t}{s}y_{2}^{\prime},-t),(x^{\prime},w_{1}-2w)\rangle\\}}{f(x^{\prime},w_{1}-2w)^{\frac{s}{s+t}}}$
$\displaystyle\geq$ $\displaystyle
f^{0}\left(\frac{s+t}{t}y_{1}^{\prime},s\right)^{\frac{t}{s+t}}f^{0}\left(\frac{s+t}{s}y_{2}^{\prime},-t\right)^{\frac{s}{s+t}},$
where the first inequality is by Claim 2, and the second inequality is by
$\inf(AB)\geq(\inf A)(\inf B)$, and last inequality is by the definition of
the polar of functions. Since $y^{\prime}_{1}$ and $y^{\prime}_{2}$ are
arbitrary, we get (4.6).
Let $F_{0}(w)=\int_{H}(S_{H}f)^{0}_{(w)}$, $F_{1}(s)=\int_{H}f^{0}_{(s)}$ and
$F_{2}(t)=\int_{H}f^{0}_{(-t)}$. By the Prékopa inequality and (4.6), we have
$F_{0}(\frac{2st}{s+t})\geq
F_{1}(s)^{\frac{t}{s+t}}F_{2}(t)^{\frac{s}{s+t}}\;{\rm for}\;{\rm
every}\;s,t>0.$
Now, by Proposition 1(i) and Lemma 4.4, $(S_{H}f)^{0}$ is symmetric about $H$,
we have $\int_{0}^{+\infty}F_{0}=\frac{1}{2}\int_{\mathbb{R}^{n}}(S_{H}f)^{0}$
and since $H$ is $\lambda$-separating for $f^{0}$, we have
$\left(\int_{0}^{+\infty}F_{1}\right)\left(\int_{0}^{+\infty}F_{2}\right)=\lambda(1-\lambda)\left(\int_{\mathbb{R}^{n}}f^{0}\right)^{2}$.
Since $F_{0}$, $F_{1}$, $F_{2}:[0,+\infty)\rightarrow\mathbb{R}^{+}$ satisfy
the hypothesis of Lemma 2.1, and by definitions of $F_{1}$ and $F_{2}$, one
has
$\int_{0}^{+\infty}F_{1}+\int_{0}^{+\infty}F_{2}=\int_{\mathbb{R}^{n}}f^{0}$,
thus, by Lemma 2.1
$\displaystyle\frac{2}{\int_{\mathbb{R}^{n}}(S_{H}f)^{0}}\leq\frac{1}{2}\left(\frac{1}{\int_{0}^{+\infty}F_{1}}+\frac{1}{\int_{0}^{+\infty}F_{2}}\right)=\frac{1}{2\lambda(1-\lambda)\int_{\mathbb{R}^{n}}f^{0}}.$
This gives the desired inequality. ∎
###### Lemma 4.6.
If $f$ is an integrable, unconditional, log-concave function, then
$\int_{\mathbb{R}^{n}}f\int_{\mathbb{R}^{n}}f^{0}\leq(2\pi)^{n}$.
###### Proof.
Let $f_{1}=f$, $f_{2}=f^{0}$ and $f_{3}=e^{-\frac{|x|^{2}}{2}}$, then $f_{1}$,
$f_{2}$ and $f_{3}$ are unconditional. Thus we have
$\int_{\mathbb{R}^{n}}f_{j}=2^{n}\int_{\mathbb{R}_{+}^{n}}f_{j},\;\;j=1,2,3$.
For $(y_{1},\dots,y_{n})\in\mathbb{R}^{n}$, we define
$g_{i}(y_{1},\dots,y_{n})=f_{i}(e^{y_{1}},\dots,e^{y_{n}})e^{\sum_{i=1}^{n}y_{i}}$.
We get $\int_{\mathbb{R}_{+}^{n}}f_{j}=\int_{\mathbb{R}^{n}}g_{j}$, and for
every $s,t\in\mathbb{R}^{n}$, $g_{1}(s)g_{2}(t)\leq
g_{3}\left(\frac{s+t}{2}\right)^{2}$. Hence
$\int_{\mathbb{R}^{n}}f\int_{\mathbb{R}^{n}}f^{0}\leq(2\pi)^{n}$ follows from
Prékopa inequality. ∎
Proof of Theorem 1.2. We proceed by $n$ successive Steiner symmetrizations
until we get an unconditional log-concave function.
Let $u_{1}\in S^{n-1}$, $u_{1}$ orthogonal to $H=H_{1}$ and let
$(u_{i})_{i=2}^{n}\subset S^{n-1}$ such that $(u_{1},\dots,u_{n})$ form an
orthonormal basis for $\mathbb{R}^{n}$. Let $z_{1}=s_{H_{1}}(f)$ and define a
log-concave function $f_{1}$ by the identity
$f_{1}^{z_{1}}=S_{H_{1}}(f^{z_{1}})$. Then $\int f_{1}^{z_{1}}=\int
f^{z_{1}}$. By Proposition 1(i) and Lemma 4.4, $f_{1}$ is symmetric about
$H_{1}$ and by Lemma 4.5, applied to $f^{z_{1}}$, $z=z_{1}$ and $H=H_{1}$,
$\lambda$-separating for $f=(f^{z_{1}})^{z_{1}}$, we get
$\int_{\mathbb{R}^{n}}f_{1}\geq 4\lambda(1-\lambda)\int_{\mathbb{R}^{n}}f$ and
thus $\int f_{1}\int f_{1}^{z_{1}}\geq 4\lambda(1-\lambda)\int f\int
f^{z_{1}}$. Choose now the hyperplane $H_{2}$, orthogonal to $u_{2}$, and
medial for $f_{1}$ and define $z_{2}=s_{(H_{1}\cap H_{2})}(f_{1})$. By Lemma
4.3 we have $z_{1}=s_{H_{1}}(f)=s_{H_{1}}(f_{1})$, we get $\int
f_{1}^{z_{2}}=\min_{z\in H_{1}\cap H_{2}}\int f_{1}^{z}\geq\min_{z\in
H_{1}}\int f_{1}^{z}=\int f_{1}^{z_{1}}$. We define now a new log-concave
function $f_{2}$ by the identity $f_{2}^{z_{2}}=S_{H_{2}}(f_{1}^{z_{2}})$. By
Proposition 1(ii) and Lemma 4.4, $f_{2}$ is symmetric about both $H_{1}$ and
$H_{2}$. Since $H_{2}$ is medial for $f_{1}$, we get by Lemma 4.5 applied to
$f_{1}^{z_{2}}$, $z=z_{2}$ and $H=H_{2}$ that $\int f_{2}\geq\int f_{1}$.
Moreover, we have $\int f_{2}^{z_{2}}=\int S_{H_{2}}(f_{1}^{z_{2}})=\int
f_{1}^{z_{2}}\geq\int f_{1}^{z_{1}}$. It follows that $\int f_{2}\int
f_{2}^{z_{2}}\geq\int f_{1}\int f_{1}^{z_{1}}$.
We continue this procedure by choosing hyperplanes $H_{2},\dots,H_{n}$, points
$z_{2},\dots,z_{n}$, and defining log-concave functions $f_{2},\dots,f_{n}$
such that for $2\leq i\leq n$, we have (i) $H_{i}$ is medial for $f_{i-1}$ and
orthogonal to $u_{i}$; (ii) $z_{i}=s_{(H_{1}\cap H_{2}\cap\dots\cap
H_{i})}(f_{i-1})$; (iii) $f_{i}^{z_{i}}=S_{H_{i}}(f_{i-1}^{z_{i}})$. From (ii)
(iii) and Lemma 4.3, we have $z_{i}=s_{(H_{1}\cap\dots\cap
H_{i})}(f_{i-1})=s_{(H_{1}\cap\dots\cap H_{i})}(f_{i})$. Choosing $H_{i+1}$,
$z_{i+1}$, $f_{i+1}$ according to (i) (ii) (iii), we get thus $\int
f_{i+1}^{z_{i+1}}=\int S_{H_{i+1}}(f_{i}^{z_{i+1}})=\int
f_{i}^{z_{i+1}}\geq\int f_{i}^{s_{(H_{1}\cap\dots\cap H_{i})}(f_{i})}=\int
f_{i}^{z_{i}}$. Now, Lemma 4.5 applied to $f_{i}^{z_{i+1}}$, $z=z_{i+1}$ and
$H_{i+1}$, medial for $f_{i}=(f_{i}^{z_{i+1}})^{z_{i+1}}$, gives $\int
f_{i+1}\geq\int f_{i}$. Thus, $\int f_{i}\int f_{i}^{z_{i}}$ is an increasing
sequence, for $2\leq i\leq n$. Therefore, we have $4\lambda(1-\lambda)\int
f\int f^{z_{1}}\leq\int f_{1}\int f_{1}^{z_{1}}\leq\dots\leq\int f_{n}\int
f_{n}^{z_{n}}$. From Proposition 1(ii), $f_{n}$ is an unconditional function
about $z_{n}$ and $z_{n}\in H_{1}\cap H_{2}\cap\dots\cap H_{n}$ is a center of
symmetry for $f_{n}$. By Lemma 4.6, we have $\int f\int
f^{z_{1}}\leq\frac{(2\pi)^{n}}{4\lambda(1-\lambda)}$, this concludes the
proof. $\Box$
## References
* [1] S. Artstein, B. Klartag, V. D. Milman, On the Santal$\acute{o}$ point of a function and a functional Santal$\acute{o}$ inequality, Mathematika 54 (2004), 33-48.
* [2] K. Ball, Isometric problems in $l_{p}$ and sections of convex sets, Doctoral thesis, University of Cambridge, 1986.
* [3] K. Ball, Logarithmically concave functions and sections of convex sets in $\mathbb{R}^{n}$, Studia Math. 88 (1988), 69-84.
* [4] W. Blaschke, über affine Geometrie 7: Neue Extremeigenschaften von Ellipse und Ellipsoid. Wilhelm Blaschke Gesammelte Werke 3. Thales Verlag, Essen (1985)
* [5] H. J. Brascamp, E. H. Lieb, J. M. Luttinger, A General Rearrangement Inequality for Multiple Integrals, J. Funct. Anal. 17 (1974), 227-237.
* [6] A. Burchard, Steiner symmetrization is continuous in $W^{1,p}$, Geom. Funct. Anal. 7 (1997), 823-860.
* [7] A. Burchard, A short course on rearrangement inequalities, available at http://www.math.utoronto.ca/almut/rearrange.pdf, 2009.
* [8] M. Fortier, Convergence results for rearrangements: Old and new, M.S. Thesis, University of Toronto, December 2010.
* [9] M. Fradelizi, M. Meyer, Some functional forms of Blaschke-Santal$\acute{o}$ inequality, Math. Z. 256 (2007), 379-395.
* [10] J. Lehec, The symmetric property $\tau$ for the Gaussian measure, Ann. Fac. Sci. Toulouse Math. 17(6) (2008), 357-370.
* [11] J. Lehec, Partitions and functional Santaló inequality, Arch. Math. 92 (2009), 89-94.
* [12] J. Lehec, A direct proof of the functional Santaló inequality, C. R. Math. Acad. Sci. Paris, 347 (2009), 55-58.
* [13] E. Lutwak, G. Zhang, Blaschke-Santaló inequalities, J. Differ. Geom. 47(1) (1997), 1-16.
* [14] E. Lutwak, D. Yang, G. Zhang, Moment-entropy inequalities, Ann. Probab. 32 (2004), 757-774.
* [15] M. Meyer, A. Pajor, On the Blaschke Santaló inequality, Arch. Math. 55 (1990), 82-93.
* [16] A. Pr kopa, Logarithmic concave measures with applications to stochastic programming, Acta Sci. Math. (Szeged) 32 (1971), 301-316.
* [17] A. Pr kopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 339-343.
* [18] L. A. Santaló, An affine invariant for convex bodies of n-dimensional space. Port. Math. 8 (1949), 155-161.
* [19] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl., vol. 44, Cambridge University Press, Cambridge, 1993.
|
arxiv-papers
| 2014-03-03T03:19:28 |
2024-09-04T02:49:59.194323
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Youjiang Lin and Gangsong Leng",
"submitter": "Youjiang Lin",
"url": "https://arxiv.org/abs/1403.0299"
}
|
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